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+++ b/_downloads/4a365c21b18d3e18fa9d64572d2219dd/03_HSI_To_Petrophysics.py @@ -0,0 +1,130 @@ +""" +.. _example_predicting_pwave_velocity: + +Predicting P-Wave Velocity from Hyperspectral Data +================================================== + +This example demonstrates the prediction of P-Wave velocities from hyperspectral data using machine learning techniques. Hyperspectral data provides detailed information on rock mineralogy, which influences its petrophysical properties. This script aims to explore and quantify the relationship between spectral features and various petrophysical properties, specifically focusing on P-Wave velocity. +""" + +# %% +# Importing Necessary Libraries +# ----------------------------- +# +# First, we import the required Python packages. We use `dotenv` to manage environment variables, `numpy` and `sklearn` for numerical operations and machine learning, and custom packages `hklearn` and `hycore` for handling hyperspectral data and machine learning workflows. + +from dotenv import dotenv_values +import numpy as np +import sklearn +from sklearn.linear_model import LinearRegression +from sklearn.neural_network import MLPRegressor +from tqdm import tqdm + +# Custom packages for handling hyperspectral data +# Note: As of January 2024, these packages are not publicly available. Please contact the author for more information. +from vector_geology.Packages import hklearn +from vector_geology.Packages import hycore + +# %% +# Loading Data with `hycore.Shed` +# ------------------------------- +# +# The `hycore` package, which organizes hyperspectral data in a structure called Shed, is used here. A Shed makes it convenient to categorize and store hyperspectral data. We use HyLibraries (a data structure within `hylite` for storing spectral libraries) within a Shed to store the hyperspectral data. The corresponding P-Wave velocity measurements are stored as numpy arrays. + +# Load the train and test sheds +config = dotenv_values() +path_to_train = config.get("PATH_TO_HSI_TRAINING") +path_to_test = config.get("PATH_TO_HSI_TESTING") + +train_shed = hycore.loadShed(path_to_train) +test_shed = hycore.loadShed(path_to_test) + +# Load the train and test spectral libraries and P-Wave velocities +train_fx50 = train_shed.results.FX50Lib +train_fenix = train_shed.results.FENIXLib +train_lwir = train_shed.results.LWIRLib +train_vp = train_shed.results.Vp + +test_fx50 = test_shed.results.FX50Lib +test_fenix = test_shed.results.FENIXLib +test_lwir = test_shed.results.LWIRLib +test_vp = test_shed.results.Vp + +# %% +# Creating a `hklearn.Stack` Object +# --------------------------------- +# +# The `hklearn` package, designed to handle hyperspectral data, is used to organize spectral libraries into a structure called Stack. A Stack can manage multiple spectral libraries and integrate data from multiple sensors. We perform hull corrections on the spectra to enhance features of interest. + +# Create Stack objects and apply hull correction to the spectra +hsi_train = hklearn.Stack( + names=['FENIX', 'FX50', 'LWIR'], + data=[train_fenix, train_fx50, train_lwir]).hc( + ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)}, + hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'} +) + +hsi_test = hklearn.Stack( + names=['FENIX', 'FX50', 'LWIR'], + data=[test_fenix, test_fx50, test_lwir]).hc( + ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)}, + hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'} +) + +# %% +# Scaling the Dependent Variable (Y) +# ---------------------------------- +# +# The P-Wave velocities are scaled to facilitate the convergence of the loss function during model training. We use the `StandardScaler` from scikit-learn for this purpose. + +# Scale Y Variable (P-Wave velocity) +y_scaler = sklearn.preprocessing.StandardScaler() +_ = y_scaler.fit(train_vp[:, None]) + +# %% +# Transforming the Independent Variable (X) +# ----------------------------------------- +# +# Hyperspectral data often contains a large number of bands, many of which might be redundant. We apply Principal Component Analysis (PCA) to extract the most significant components. A two-step PCA is performed to ensure that there isn't any inter-sensor correlation. The `Stack` object stores the PCA, which is also applied to the test set. + +# Apply PCA to the X variable (spectral data) +PCA_X = hsi_train.fit_pca(n_components=10, normalise=True) +hsi_test.set_transform(PCA_X) + +# Set the Y-Variable (P-Wave velocity) +hsi_train.set_y(train_vp[:, None, None]) +hsi_test.set_y(test_vp[:, None, None]) + +# %% +# Initializing and Training Models +# -------------------------------- +# +# Different machine learning models are initialized and trained. In this example, we use a simple linear regression (`LinearRegression`) and a Multilayer Perceptron (`MLPRegressor`) from scikit-learn. A dictionary with the parameters for optimization is also initialized. + +# Initialize models +models = dict( + Linear=LinearRegression(), + MLP=MLPRegressor( + hidden_layer_sizes=(180,), + max_iter=1000, + solver='sgd', + learning_rate='adaptive' + ) +) + +# Define parameter ranges for each model +params = dict(Linear={'fit_intercept': [True, False]}, + MLP={"alpha": np.linspace(1e-4, 1e0)}) + +# Train the models +for name, model in tqdm(models.items(), 'Fitting models', leave=True): + hsi_train.fit_model(name, model, xtransform=True, ytransform=y_scaler, grid_search_cv=params[name], n_jobs=-1) + +# %% +# Scoring and Evaluating Models +# ----------------------------- +# +# Finally, the trained models are scored and evaluated. The `Stack.get_score_table()` method is used to display a table with the training, cross-validation, and test scores of the best-performing model. + +# Evaluate models and display score table +hsi_train.get_score_table(hsi_test, y_test=None, style=True) diff --git a/_downloads/4f42b4fbd1192852ada2bb95c8ae557a/01_gravity_gradiometry_inversion_pgi.py b/_downloads/4f42b4fbd1192852ada2bb95c8ae557a/01_gravity_gradiometry_inversion_pgi.py index 33cdddd..52a6927 100644 --- a/_downloads/4f42b4fbd1192852ada2bb95c8ae557a/01_gravity_gradiometry_inversion_pgi.py +++ b/_downloads/4f42b4fbd1192852ada2bb95c8ae557a/01_gravity_gradiometry_inversion_pgi.py @@ -1,169 +1,159 @@ """ -Inversion of Gravity Gradiometry Data on a Tree Mesh ----------------------------------------------------- +.. _example_inversion_ftg_data: -We use the PGI module within SimPEG to invert gravity gradiometry data on a Tree Mesh. This tutorial focusses on: +Inversion of Full Tensor Gravity Gradiometry Data +================================================= - - How to define all the necessary components - - How to initialize the Gaussian Mixture Model Prior - - How to set the parameters - - How to plot the results +This example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, +measured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include: +- Loading necessary modules and data +- Generating a mesh +- Initializing the Gaussian Mixture Model +- Tuning hyper-parameters +- Visualizing the results +The process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example + notebook from SimPEG's documentation (see `SimPEG docs `_ for more information). """ -# %% -# ## Import Modules +import os +import shutil +from dotenv import dotenv_values -import discretize as ds import SimPEG.potential_fields as pf -from SimPEG import ( - maps, - data, - utils, - simulation, - inverse_problem, - inversion, - optimization, - regularization, - data_misfit, - directives, -) -from SimPEG.utils import io_utils -import numpy as np -import pandas as pd -import scipy.interpolate as interp +# Import the SimPEG functions +import discretize as ds import matplotlib as mpl import matplotlib.pyplot as plt -import math as math - -from dotenv import dotenv_values -from mpl_toolkits.axes_grid1.inset_locator import inset_axes - -from discretize.utils import mkvc, refine_tree_xyz, active_from_xyz -from discretize import TensorMesh, TreeMesh -from SimPEG.utils import plot2Ddata, model_builder, surface2ind_topo -from SimPEG import maps -from SimPEG.potential_fields import magnetics as mag +# Import the general modules +import numpy as np +import pandas as pd +from SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives) from SimPEG.potential_fields import gravity as grav -import os -import shutil -from vector_geology import SimpegHelper as SH +# Import the discretize module +from discretize.utils import active_from_xyz -#from get_cpt import get_cmap +# Import SimpegHelper for plotting and resampling data +from vector_geology import SimpegHelper as SH # Plot beautification - formatter = mpl.ticker.ScalarFormatter(useMathText=True) formatter.set_scientific(True) -formatter.set_powerlimits((0,0)) +formatter.set_powerlimits((0, 0)) mpl.rc("axes", titlesize=14, labelsize=12) mpl.rc("xtick", labelsize=12) mpl.rc("ytick", labelsize=12) -# Reproducible science -np.random.seed(518936) - - # %% -# Create directories for the model iterations and output +# Step 1: Create Directories for Storing Model Iterations and Outputs +# ------------------------------------------------------------------- +# +# PGI is an iterative process, and it's useful to save the model at every iteration. This helps in preventing overfitting +# and in adjusting the stopping criteria based on the model iterations. -# Name of the model -name = "Ireland_FTG_NonNeg" -# Names for the directories for the model iterations and output -path_to_mod_iterations = "./" + str(name) + "/Model Iterations" -path_to_output = "./" + str(name) + "/Output" +name = "Ireland_FTG" +path_to_mod_iterations = f"./{name}/Model Iterations" +path_to_output = f"./{name}/Output" -# Check if the model iterations directory exists and remove it if it does if os.path.exists(path_to_mod_iterations): shutil.rmtree(path_to_mod_iterations) - -# Create the model iterations directory. os.makedirs(path_to_mod_iterations) -# Check if the output directory exists and remove it if it does if os.path.exists(path_to_output): shutil.rmtree(path_to_output) - -# Create the output directory. os.makedirs(path_to_output) # %% -# Load the data -# Read gravity and magnetic data from a csv file. -# Inputs: -# file_path: The path to the csv file. -# header: Whether or not the csv file has a header row. - +# Step 2: Load the FTG-G Data +# ---------------------------- +# +# Full Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion. +# +# .. math:: +# G = \left[\begin{array}{cc} +# G_{xx} & G_{xy} & G_{xz}\\ +# G_{xy} & G_{yy} & G_{yz}\\ +# G_{xz} & G_{yz} & G_{zz} +# \end{array}\right] +# config = dotenv_values() file_path = config.get("PATH_TO_GRADIOMETRY") - -header = None - -# Read the csv file into a pandas dataframe and then convert it to a numpy array. FTG_Data = pd.read_csv(file_path, delimiter=",").to_numpy() -# Add Topography -FTG_Data = np.c_[FTG_Data, np.zeros(np.shape(FTG_Data)[0])] -FTG_Data[:,[2, -1]] = FTG_Data[:,[-1, 2]] - -# Format after adding topography -# X, Y, Z, Tyy, Txz, Tyz, Txy, Txx - - -## % -## Resample the data -# The ``SimpegHelper.pf_rs()`` will take in potential field data and resample it onto a new regular grid - # %% -# New sampling interval (Data Units) -inc = 200 - -# Resample the data -[grav_new, nx_new, ny_new] = SH.pf_rs(FTG_Data, inc) - -# Extract the gravity gradiometry data vectors -grav_vec = grav_new[:,3:] +# Step 3: Resample Data onto a Regular Grid +# ----------------------------------------- +# +# Resampling the data onto a regular grid aids in mesh generation and visualization. The `SimpegHelper.pf_rs()` function +# resamples potential field data onto a new regular grid. + +inc = 150 # New sampling interval (ground units) +grav_new, nx_new, ny_new = SH.pf_rs( + FTG_Data, + inc, + bounds=[156000, 168000, 143000, 148500] +) -# Negate the gravity data to the opposite sign to match the coordinate system. # NOTE: This step is necessary for real data since the convention followed by -# the SimPEG forward operator gravity lows over density highs. -# grav_vec = -grav_vec - -# Extract the topography for the active cells. -inv_topo = grav_new[:,[0,1,2]] - - -## Visualize the data +# the SimPEG forward operator is the opposite of the general convention +grav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :] # Adjusting sign convention +inv_topo = grav_new[:, [0, 1, 2]] # Extracting topography -#The ``SimpegHelper.plot_2D_data()`` function will plot the datasets using ``matplotlib``. -#%% -# Plot Gravity Data -SH.plot_2D_data(np.c_[grav_new[:,[0,1,2]], grav_vec[:,-1]], [np.nanmin(grav_vec[:,-1]), np.nanmax(grav_vec[:,-1])], cmap="jet", path_to_output=path_to_output, name=name) +# %% +# Step 4: Visualize the Data +# -------------------------- +# +# The `SimpegHelper.plot_2D_data()` function is used to visualize the datasets with `matplotlib`. + +SH.plot_2D_data( + data=np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]], + clim=[np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])], + cmap="jet", + which_data="FTG", + comp="xx", + path_to_output=path_to_output, + name=name +) +SH.plot_2D_data( + data=np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]], + clim=[np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])], + cmap="terrain", + which_data="Topo", + comp="xx", + path_to_output=path_to_output, + name=name +) -# Plot Topography Data -# SH.plot_2D_data(np.c_[inv_topo[:,[0,1,2]], inv_topo[:,2]], [np.nanmin(inv_topo[:,2]), np.nanmax(inv_topo[:,2])], which_data="Topo", cmap="magma", path_to_output=path_to_output, name=name) -#%% md -## Create a TreeMesh object to invert the data +# %% +# Step 5: Create a TensorMesh Object for Inversion +# ------------------------------------------------ +# +# A 3D TensorMesh is created using the `discretize.TensorMesh` utility to invert the gravity gradiometry data. The mesh +# is designed to have a constant cell size in the core region and expanding/contracting cells in the padding region. +# +# .. math:: +# z_{n+1} = r z_{n} +# +# +# Where $r$ is a multiplicative factor (<1 for contracting, >1 for expanding cells). x and y +# increments can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire +# region is an area of interest). + +dx, dy, dz = inc, inc, 10 # Cell sizes +nz_core, nz_pad = 10, 18 # Number of cells in core and padding +fact = 1.1 # Factor for expanding/contracting cells -# Using the ``discretize.TreeMesh`` utility, create a 3D TreeMesh to invert the gravity gradiometry data. -#%% -dx = inc -dy = inc -dz = 10 -nz_core = 1 -nz_pad = 20 -fact = 1.1 - -# Set the cell sizes to a constant value in the x and y directions and expanding in the z direction. inv_hx = dx * np.ones(nx_new) inv_hy = dy * np.ones(ny_new) inv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)] -# Create the inverse tensor mesh. -inv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:,0]), np.min(inv_topo[:,1]), "C"]) +# Create the inverse tensor mesh +inv_mesh = ds.TensorMesh( + h=[inv_hx, inv_hy, inv_hz], + x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), "C"]) # Drape the topography over the mesh actv = active_from_xyz(inv_mesh, inv_topo) @@ -171,74 +161,72 @@ actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv) nactv = int(actv.sum()) -# Make an x-y grid with the mesh cell centers -# xygrid = np.meshgrid(inv_mesh.cell_centers_x, inv_mesh.cell_centers_y) -# xygrid = np.reshape(xygrid, (2,-1)).T - -# Region Active Array (bool array with the spatial extent of the data) -# RegActvArr = np.empty((len(xygrid)), dtype=bool) +# Print mesh details +print(inv_mesh) -# for i in range(len(xygrid)): -# Dist = np.sqrt(np.sum((inv_topo[:, [0, 1]] - xygrid[i,:])**2, axis=1)) -# MinDist = np.min(Dist) -# RegActvArr[i] = MinDist < 50 - -# Repeating the regional active array for every z (since we only carried out the calculations for x-y, which can be replicated in z) -# RegActvArr = np.tile(RegActvArr, len(inv_mesh.cell_centers_z)) - -# Intersection between Topo active cells and spatial active cells (nactv and actvMap are also updated) -actv = actv -nactv = int(actv.sum()) -actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv) -#%% -inv_mesh -#%% md -## Visualize the mesh +# %% +# Step 6: Visualize the Mesh +# -------------------------- +# +# A slice of the Tensor Mesh object is visualized using the `discretize.TensorMesh.plot_slice()` utility. -#The ``discretize.TreeMesh.plot_slice()`` utility will plot a slice of thee Tree Mesh object generated above. -#%% -# Plotting a mesh slice +# %% # Create a background model. bg = np.ones(nactv) mod = actvMap * bg -# Define the tensor mesh parameters. -props = dict(boxstyle='round,pad=1', facecolor='white') - # Plot the mesh slice. fig = plt.figure(figsize=(6, 6)) ax = fig.add_axes([0.1, 0.1, 0.8, 0.8]) -cplot = inv_mesh.plot_slice(mod, grid=False, ax=ax, normal="Y", ind=22) +cplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal="Y", ind=22) ax.set_title('Tensor Mesh Slice') ax.set_xlabel('x (m)') -ax.set_ylabel('y (m)') +ax.set_ylabel('z (m)') +ax.set_xlim([158000, 159000]) +ax.set_ylim([-500, 100]) ax.ticklabel_format(axis="both") ax.set_aspect('equal') -plt.savefig(path_to_output + "/" + name + "_TreeMeshSlice.pdf", bbox_inches="tight") +plt.savefig( + path_to_output + "/" + name + "_TreeMeshSlice.pdf", + bbox_inches="tight" +) plt.show() -#%% md -## Set up the gravity inverse problem -#%% + +# %% +# Step 7: Set up the Gravity Inverse Problem +# ------------------------------------------ +# +# The gravity inverse problem is set up using SimPEG's `pf.gravity` module to generate a forward model based on the mesh +# and receiver locations. + + +# %% # Create wires for the density model wires = maps.Wires(("density", nactv)) -# Create a Density Mapping +# Create a density map density_map = actvMap * wires.density -# Create Identity Mapping + +# Create an identity map identity_map = maps.IdentityMap(nP=nactv) # Components of the data used as input -gravity_components = ["gyy", "gxz", "gyz", "gxy", "gxx"] +gravity_components = ["gxx", "gyy", "gxz", "gyz", "gxy"] -# Receiver locations -gravity_receiver_locations = grav_new[:, [0, 1, 2]] +# Receiver locations (25 meters above the topography) +gravity_receiver_locations = inv_topo + 120. # Create a gravity receiver list -gravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components) +gravity_receivers = pf.gravity.receivers.Point( + locations=gravity_receiver_locations, + components=gravity_components +) gravity_receiver_list = [gravity_receivers] # Create a gravity source field -gravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list) +gravity_source_field = pf.gravity.sources.SourceField( + receiver_list=gravity_receiver_list +) # Define the gravity survey gravity_survey = pf.gravity.survey.Survey(gravity_source_field) @@ -251,22 +239,31 @@ ind_active=actv ) -# Define uncertainties -# Maximum value of each component -# max_gravity = np.max(np.abs(grav_vec), axis=0) -# Standard deviation of each component -# std_gravity = 0.05 * max_gravity -# std_gravity = np.ones(np.shape(grav_vec)) * std_gravity - -# Create a gravity data object -gravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1) +# Create a gravity data object, with the relative errors and a standard noise floor +gravity_data = data.Data( + survey=gravity_survey, + dobs=grav_vec.flatten(), + noise_floor=5, + relative_error=0.1 +) # Define the misfits associated with the gravity data -gravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem) -#%% md -## Set up the Gaussian Mixture Model (GMM) Prior -#%% -# Defining the Gaussian Mixture Model (GMM) +gravity_misfit = data_misfit.L2DataMisfit( + data=gravity_data, + simulation=gravity_problem +) + +# %% +# Step 8: Setting up the Gaussian Mixture Model (GMM) Prior +# --------------------------------------------------------- +# +# In PGI, the petrophysical data is used as a constraint in the form of a `Gaussian Mixture Model `_ (GMM). +# A GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units (:math:`n`), +# the petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below. +# If you do have a petrophysical dataset, you can fit the GMM to said dataset using the ``gmmref.fit()`` method. + + +# %% # Number of rock units and number of physical properties num_rock_units = 5 num_physical_props = 1 @@ -282,7 +279,8 @@ # Set the background density background_density = 0.0 -# Initialize the GMM fit with random samples, mesh size, and number of physical properties +# Initialize the GMM fit with random samples, mesh size, +# and number of physical properties gmmref.fit(np.random.randn(nactv, num_physical_props)) # Set the mean values of physical property contrasts for each rock unit @@ -299,6 +297,9 @@ density_variance = 8e-5 # Set the covariances of physical properties for each rock unit +# NOTE: Since we don't have petrophysical information for this example, +# we keep the covariances # same for every unit. The GMM will update +# itself during the inversion. gmmref.covariances_ = np.array( [ [[density_variance]], @@ -313,14 +314,15 @@ gmmref.compute_clusters_precisions() # Set the weights for each rock unit +# NOTE: This determines the size of each peak of the n-modal GMM gmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125] -#%% -# GMM Parameters -means_init = gmmref.means_.flatten() -covariances_init = gmmref.covariances_.flatten() -weights_init = gmmref.weights_[0].flatten() -# Plot the 1D GMM +# %% +# Plot the 1D-GMM +# --------------- +# +# The initial density distribution is visualized. + fig = plt.figure(figsize=(6, 6)) ax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2) ax[0].set_xlabel(r"Density Contrast (g/cc)") @@ -328,27 +330,47 @@ ax[0].get_legend().remove() ax[0].set_title(r"Initial DelRho Distribution") ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0)) -ax[0].set_aspect(1/20) -plt.savefig(path_to_output + "/" + name + "_Init_GMM.pdf", bbox_inches="tight") +ax[0].set_aspect(1 / 20) +plt.savefig( + path_to_output + "/" + name + "_Init_GMM.pdf", + bbox_inches="tight" +) plt.show() -#%% md -## Define the inversion parameters + + #%% -# Initial Model +# Step 9: Setting the Hyper-parameters and the Sensitivity Weights +# ----------------------------------------------------------------- +# +# Every PGI is a set of three Maximum-A-Posteriori (`MAP `_) +# problems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both +# the FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise +# the necessary weights for every voxel (as the contribution of every voxel is dependent on its depth from the surface). +# The regularization smallness (:math:`\alpha_s`) and smoothness (:math:`\alpha_i,\ i = x, y, z`) are initialised here. Please check +# `here `_ +# for the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion. +# +# NOTE: The smoothness parameters (`alpha_i`) are static and hence need to be fine-tuned +# through trial and error. +# + initial_model = np.r_[background_density * np.ones(actvMap.nP)] +# %% # Sensitivity weighting # Compute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G) # This is done by computing the square root of the sum of the squared elements of G for each cell, # and then normalizing by the cell volumes and the maximum weight value. + +# %% sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv]) sensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity) # Regularization multipliers smallness_multiplier = 1e-4 -smoothness_x_multiplier = 1e-1 -smoothness_y_multiplier = 1e-1 -smoothness_z_multiplier = 1e-1 +smoothness_x_multiplier = 4 +smoothness_y_multiplier = 4 +smoothness_z_multiplier = 0.25 # Create joint PGI regularization with smoothness regularization_term = regularization.PGI( @@ -361,15 +383,22 @@ alpha_x=smoothness_x_multiplier, alpha_y=smoothness_y_multiplier, alpha_z=smoothness_z_multiplier, + # The second derivative smoothnesses are kept to be zero, since we + # don't want to include second derivatives in the objective function alpha_xx=0.0, alpha_yy=0.0, alpha_zz=0.0, weights_list=[sensitivity_weights_gravity] ) -# Directives for inversion -# Define the directives for the inversion process, based on Astic et al., (2021) +# %% +# Step 10: Initialize the Directives +# ---------------------------------- +# +# Directives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver. + +# %% # Estimate smoothness multipliers alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True) @@ -386,28 +415,37 @@ target_misfits = directives.MultiTargetMisfits(verbose=True) # Add reference model once stable -mref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True) +mref_in_smooth = directives.PGI_AddMrefInSmooth( + wait_till_stable=True, + verbose=True +) # Update smallness parameters, keeping GMM fixed (L2 Approx of PGI) update_smallness_directive = directives.PGI_UpdateParameters( update_gmm=True, - kappa = 0, - nu = 0.5, - zeta = 0 + kappa=0, + nu=0.5, + zeta=0 ) # Update preconditioner update_preconditioner = directives.UpdatePreconditioner() # Save iteration results -save_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output) +save_iteration_directive = directives.SaveOutputEveryIteration( + name=name, + directory=path_to_output +) # Save model iterations -save_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations) +save_model_directive = directives.SaveModelEveryIteration( + name=name, + directory=path_to_mod_iterations +) # Optimization options for the inversion -lower_bound = np.r_[-1 * np.ones(actvMap.nP)] -upper_bound = np.r_[1 * np.ones(actvMap.nP)] +lower_bound = np.r_[-1.0 * np.ones(actvMap.nP)] +upper_bound = np.r_[1.0 * np.ones(actvMap.nP)] optimizer = optimization.ProjectedGNCG( maxIter=20, lower=lower_bound, @@ -433,19 +471,28 @@ save_model_directive ], ) -#%% md -## Run the Inversion! -#%% -# Run the inversion + +# %% +# Step 11: Run the Inversion! +# --------------------------- +# +# The inversion is executed using the initialized settings. + inverted_model = inversion_algo.run(initial_model) -#%% + +# %% +# Step 12: Visualize the 3D Model +# ------------------------------- +# +# The inverted 3D model is visualized using slices in different orientations. + set = 1 save_plots = True # Indices of the depth sections -ind_plot_x = int(len(inv_mesh.cell_centers_x)/6) + 10 -ind_plot_y = int(len(inv_mesh.cell_centers_y)/2) -ind_plot_z = int(len(inv_mesh.cell_centers_z)/2) - 17 +ind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25 +ind_plot_y = int(len(inv_mesh.cell_centers_y) / 2) +ind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15 ind_plot = [ind_plot_x, ind_plot_y, ind_plot_z] @@ -456,29 +503,63 @@ # Plot Density Contrast Model (Z) normal = "Z" model_to_plot = inverted_density_model -SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) +SH.plot_model_slice( + mesh=inv_mesh, + ind_active=actv, + model=model_to_plot, + normal= normal, + ind_plot_arr=ind_plot, + clim=[-1.0, 1.0], + set=set, sec_loc=True, + gdlines=True, + which_prop="Den", + cmap="Spectral", + save_plt=save_plots, + path_to_output=path_to_output, + name=name +) # Plot Inverted Model Slices (Y) normal = "X" model_to_plot = inverted_density_model -SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) # Plot Inverted Model Slices (X) normal = "Y" model_to_plot = inverted_density_model -SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) -#%% -# Updated GMM Parameters -learned_mu = regularization_term.objfcts[0].gmm.means_.flatten() -learned_sigma = regularization_term.objfcts[0].gmm.covariances_.flatten() -learned_pi = regularization_term.objfcts[0].gmm.weights_[0].flatten() +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# %% +# Step 13: Visualize the Quasi-Geological Model +# --------------------------------------------- +# +# The quasi-geological model (as given by `Li et al., 2019 `_) is visualized, showing different rock units classified by the GMM. + +# %% +# Plot Density Contrast Model (Z) +normal = "Z" +model_to_plot = quasi_geology_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# Plot Inverted Model Slices (Y) +normal = "X" +model_to_plot = quasi_geology_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) -# Regularization Parameters -final_alpha_pgi = regularization_term.alpha_pgi +# Plot Inverted Model Slices (X) +normal = "Y" +model_to_plot = quasi_geology_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# %% +# Step 14: Plot the Updated 1D-GMM +# -------------------------------- +# +# The updated GMM, learned through the inversion, is visualized. # Plot the learned GMM -fig = plt.figure(figsize=(6,6)) -ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.5) +fig = plt.figure(figsize=(6, 6)) +ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4) ax[0].hist(inverted_density_model[actv], density=True, bins=1000) ax[0].set_xlabel(r"Density contrast (g/cc)") ax[0].set_ylabel(r"Probability Density Values") @@ -487,5 +568,3 @@ ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0)) plt.savefig(path_to_output + "/" + name + "_Learned_GMM.pdf", bbox_inches="tight") plt.show() -#%% -# sphinx_gallery_thumbnail_number = -4 \ No newline at end of file diff --git a/_downloads/5913f45a04c724f77adb8d78ce7d086e/01_gravity_gradiometry_inversion_pgi.ipynb b/_downloads/5913f45a04c724f77adb8d78ce7d086e/01_gravity_gradiometry_inversion_pgi.ipynb index 55db08e..fda6f17 100644 --- a/_downloads/5913f45a04c724f77adb8d78ce7d086e/01_gravity_gradiometry_inversion_pgi.ipynb +++ b/_downloads/5913f45a04c724f77adb8d78ce7d086e/01_gravity_gradiometry_inversion_pgi.ipynb @@ -15,14 +15,7 @@ "cell_type": "markdown", "metadata": {}, "source": [ - "\n# Inversion of Gravity Gradiometry Data on a Tree Mesh\n\nWe use the PGI module within SimPEG to invert gravity gradiometry data on a Tree Mesh. This tutorial focusses on:\n\n - How to define all the necessary components\n - How to initialize the Gaussian Mixture Model Prior\n - How to set the parameters\n - How to plot the results\n" - ] - }, - { - "cell_type": "markdown", - "metadata": {}, - "source": [ - "## Import Modules\n\n" + "\n\n# Inversion of Full Tensor Gravity Gradiometry Data\n\nThis example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, \nmeasured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include:\n\n- Loading necessary modules and data\n- Generating a mesh\n- Initializing the Gaussian Mixture Model\n- Tuning hyper-parameters\n- Visualizing the results\n\nThe process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example\n notebook from SimPEG's documentation (see [SimPEG docs](https://docs.simpeg.xyz/content/examples/04-grav/plot_grav_inversion_3d.html) for more information).\n" ] }, { @@ -33,14 +26,14 @@ }, "outputs": [], "source": [ - "import discretize as ds\nimport SimPEG.potential_fields as pf\nfrom SimPEG import (\n maps,\n data,\n utils,\n simulation,\n inverse_problem,\n inversion,\n optimization,\n regularization,\n data_misfit,\n directives,\n)\nfrom SimPEG.utils import io_utils\nimport numpy as np\nimport pandas as pd\nimport scipy.interpolate as interp\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\nimport math as math\n\nfrom dotenv import dotenv_values\nfrom mpl_toolkits.axes_grid1.inset_locator import inset_axes\n\nfrom discretize.utils import mkvc, refine_tree_xyz, active_from_xyz\nfrom discretize import TensorMesh, TreeMesh\nfrom SimPEG.utils import plot2Ddata, model_builder, surface2ind_topo\nfrom SimPEG import maps\nfrom SimPEG.potential_fields import magnetics as mag\nfrom SimPEG.potential_fields import gravity as grav\nimport os\nimport shutil\nfrom vector_geology import SimpegHelper as SH\n\n#from get_cpt import get_cmap\n\n# Plot beautification\n\nformatter = mpl.ticker.ScalarFormatter(useMathText=True)\nformatter.set_scientific(True)\nformatter.set_powerlimits((0,0))\n\nmpl.rc(\"axes\", titlesize=14, labelsize=12)\nmpl.rc(\"xtick\", labelsize=12)\nmpl.rc(\"ytick\", labelsize=12)\n\n# Reproducible science\nnp.random.seed(518936)" + "import os\nimport shutil\nfrom dotenv import dotenv_values\n\nimport SimPEG.potential_fields as pf\n# Import the SimPEG functions\nimport discretize as ds\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n# Import the general modules\nimport numpy as np\nimport pandas as pd\nfrom SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives)\nfrom SimPEG.potential_fields import gravity as grav\n# Import the discretize module\nfrom discretize.utils import active_from_xyz\n\n# Import SimpegHelper for plotting and resampling data\nfrom vector_geology import SimpegHelper as SH\n\n# Plot beautification\nformatter = mpl.ticker.ScalarFormatter(useMathText=True)\nformatter.set_scientific(True)\nformatter.set_powerlimits((0, 0))\n\nmpl.rc(\"axes\", titlesize=14, labelsize=12)\nmpl.rc(\"xtick\", labelsize=12)\nmpl.rc(\"ytick\", labelsize=12)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Create directories for the model iterations and output\n\n" + "## Step 1: Create Directories for Storing Model Iterations and Outputs\n\nPGI is an iterative process, and it's useful to save the model at every iteration. This helps in preventing overfitting \nand in adjusting the stopping criteria based on the model iterations.\n\n" ] }, { @@ -51,14 +44,14 @@ }, "outputs": [], "source": [ - "# Name of the model\nname = \"Ireland_FTG_NonNeg\"\n# Names for the directories for the model iterations and output\npath_to_mod_iterations = \"./\" + str(name) + \"/Model Iterations\"\npath_to_output = \"./\" + str(name) + \"/Output\"\n\n# Check if the model iterations directory exists and remove it if it does\nif os.path.exists(path_to_mod_iterations):\n shutil.rmtree(path_to_mod_iterations)\n\n# Create the model iterations directory.\nos.makedirs(path_to_mod_iterations)\n\n# Check if the output directory exists and remove it if it does\nif os.path.exists(path_to_output):\n shutil.rmtree(path_to_output)\n\n# Create the output directory.\nos.makedirs(path_to_output)" + "name = \"Ireland_FTG\"\npath_to_mod_iterations = f\"./{name}/Model Iterations\"\npath_to_output = f\"./{name}/Output\"\n\nif os.path.exists(path_to_mod_iterations):\n shutil.rmtree(path_to_mod_iterations)\nos.makedirs(path_to_mod_iterations)\n\nif os.path.exists(path_to_output):\n shutil.rmtree(path_to_output)\nos.makedirs(path_to_output)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Load the data\nRead gravity and magnetic data from a csv file.\nInputs:\nfile_path: The path to the csv file.\nheader: Whether or not the csv file has a header row.\n\n" + "## Step 2: Load the FTG-G Data\n\nFull Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion.\n\n\\begin{align}G = \\left[\\begin{array}{cc} \n G_{xx} & G_{xy} & G_{xz}\\\\\n G_{xy} & G_{yy} & G_{yz}\\\\\n G_{xz} & G_{yz} & G_{zz}\n \\end{array}\\right]\\end{align}\n\n\n" ] }, { @@ -69,14 +62,14 @@ }, "outputs": [], "source": [ - "config = dotenv_values()\nfile_path = config.get(\"PATH_TO_GRADIOMETRY\")\n\nheader = None\n\n# Read the csv file into a pandas dataframe and then convert it to a numpy array.\nFTG_Data = pd.read_csv(file_path, delimiter=\",\").to_numpy()\n\n# Add Topography\nFTG_Data = np.c_[FTG_Data, np.zeros(np.shape(FTG_Data)[0])]\nFTG_Data[:,[2, -1]] = FTG_Data[:,[-1, 2]]\n\n# Format after adding topography\n# X, Y, Z, Tyy, Txz, Tyz, Txy, Txx\n\n\n## %\n## Resample the data\n# The ``SimpegHelper.pf_rs()`` will take in potential field data and resample it onto a new regular grid" + "config = dotenv_values()\nfile_path = config.get(\"PATH_TO_GRADIOMETRY\")\nFTG_Data = pd.read_csv(file_path, delimiter=\",\").to_numpy()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "New sampling interval (Data Units)\n\n" + "## Step 3: Resample Data onto a Regular Grid\n\nResampling the data onto a regular grid aids in mesh generation and visualization. The `SimpegHelper.pf_rs()` function \nresamples potential field data onto a new regular grid.\n\n" ] }, { @@ -87,14 +80,14 @@ }, "outputs": [], "source": [ - "inc = 200\n\n# Resample the data\n[grav_new, nx_new, ny_new] = SH.pf_rs(FTG_Data, inc)\n\n# Extract the gravity gradiometry data vectors\ngrav_vec = grav_new[:,3:]\n\n# Negate the gravity data to the opposite sign to match the coordinate system.\n# NOTE: This step is necessary for real data since the convention followed by\n# the SimPEG forward operator gravity lows over density highs.\n# grav_vec = -grav_vec\n\n# Extract the topography for the active cells.\ninv_topo = grav_new[:,[0,1,2]]\n\n\n## Visualize the data\n\n#The ``SimpegHelper.plot_2D_data()`` function will plot the datasets using ``matplotlib``." + "inc = 150 # New sampling interval (ground units)\ngrav_new, nx_new, ny_new = SH.pf_rs(\n FTG_Data,\n inc,\n bounds=[156000, 168000, 143000, 148500]\n)\n\n# NOTE: This step is necessary for real data since the convention followed by\n# the SimPEG forward operator is the opposite of the general convention\ngrav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :] # Adjusting sign convention\ninv_topo = grav_new[:, [0, 1, 2]] # Extracting topography" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Plot Gravity Data\n\n" + "## Step 4: Visualize the Data\n\nThe `SimpegHelper.plot_2D_data()` function is used to visualize the datasets with `matplotlib`.\n\n" ] }, { @@ -105,14 +98,14 @@ }, "outputs": [], "source": [ - "SH.plot_2D_data(np.c_[grav_new[:,[0,1,2]], grav_vec[:,-1]], [np.nanmin(grav_vec[:,-1]), np.nanmax(grav_vec[:,-1])], cmap=\"jet\", path_to_output=path_to_output, name=name)\n\n# Plot Topography Data\n# SH.plot_2D_data(np.c_[inv_topo[:,[0,1,2]], inv_topo[:,2]], [np.nanmin(inv_topo[:,2]), np.nanmax(inv_topo[:,2])], which_data=\"Topo\", cmap=\"magma\", path_to_output=path_to_output, name=name)" + "SH.plot_2D_data(\n data=np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]],\n clim=[np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])],\n cmap=\"jet\",\n which_data=\"FTG\",\n comp=\"xx\",\n path_to_output=path_to_output,\n name=name\n)\nSH.plot_2D_data(\n data=np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]],\n clim=[np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])],\n cmap=\"terrain\",\n which_data=\"Topo\",\n comp=\"xx\",\n path_to_output=path_to_output,\n name=name\n)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Create a TreeMesh object to invert the data\n\n" + "## Step 5: Create a TensorMesh Object for Inversion\n\nA 3D TensorMesh is created using the `discretize.TensorMesh` utility to invert the gravity gradiometry data. The mesh\nis designed to have a constant cell size in the core region and expanding/contracting cells in the padding region.\n\n\\begin{align}z_{n+1} = r z_{n}\\end{align}\n\n\nWhere $r$ is a multiplicative factor (<1 for contracting, >1 for expanding cells). x and y \nincrements can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire \nregion is an area of interest).\n\n" ] }, { @@ -123,18 +116,21 @@ }, "outputs": [], "source": [ - "# Using the ``discretize.TreeMesh`` utility, create a 3D TreeMesh to invert the gravity gradiometry data." + "dx, dy, dz = inc, inc, 10 # Cell sizes\nnz_core, nz_pad = 10, 18 # Number of cells in core and padding\nfact = 1.1 # Factor for expanding/contracting cells\n\ninv_hx = dx * np.ones(nx_new)\ninv_hy = dy * np.ones(ny_new)\ninv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)]\n\n# Create the inverse tensor mesh\ninv_mesh = ds.TensorMesh(\n h=[inv_hx, inv_hy, inv_hz],\n x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), \"C\"])\n\n# Drape the topography over the mesh\nactv = active_from_xyz(inv_mesh, inv_topo)\nndv = np.nan\nactvMap = maps.InjectActiveCells(inv_mesh, actv, ndv)\nnactv = int(actv.sum())\n\n# Print mesh details\nprint(inv_mesh)" ] }, { - "cell_type": "code", - "execution_count": null, - "metadata": { - "collapsed": false - }, - "outputs": [], + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 6: Visualize the Mesh\n\nA slice of the Tensor Mesh object is visualized using the `discretize.TensorMesh.plot_slice()` utility.\n\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, "source": [ - "dx = inc\ndy = inc\ndz = 10\nnz_core = 1\nnz_pad = 20\nfact = 1.1\n\n# Set the cell sizes to a constant value in the x and y directions and expanding in the z direction.\ninv_hx = dx * np.ones(nx_new)\ninv_hy = dy * np.ones(ny_new)\ninv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)]\n\n# Create the inverse tensor mesh.\ninv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:,0]), np.min(inv_topo[:,1]), \"C\"])\n\n# Drape the topography over the mesh\nactv = active_from_xyz(inv_mesh, inv_topo)\nndv = np.nan\nactvMap = maps.InjectActiveCells(inv_mesh, actv, ndv)\nnactv = int(actv.sum())\n\n# Make an x-y grid with the mesh cell centers\n# xygrid = np.meshgrid(inv_mesh.cell_centers_x, inv_mesh.cell_centers_y)\n# xygrid = np.reshape(xygrid, (2,-1)).T\n\n# Region Active Array (bool array with the spatial extent of the data)\n# RegActvArr = np.empty((len(xygrid)), dtype=bool)\n\n# for i in range(len(xygrid)):\n# Dist = np.sqrt(np.sum((inv_topo[:, [0, 1]] - xygrid[i,:])**2, axis=1))\n# MinDist = np.min(Dist)\n# RegActvArr[i] = MinDist < 50\n\n# Repeating the regional active array for every z (since we only carried out the calculations for x-y, which can be replicated in z)\n# RegActvArr = np.tile(RegActvArr, len(inv_mesh.cell_centers_z))\n\n# Intersection between Topo active cells and spatial active cells (nactv and actvMap are also updated)\nactv = actv\nnactv = int(actv.sum())\nactvMap = maps.InjectActiveCells(inv_mesh, actv, ndv)" + "Create a background model.\n\n" ] }, { @@ -145,14 +141,21 @@ }, "outputs": [], "source": [ - "inv_mesh" + "bg = np.ones(nactv)\nmod = actvMap * bg\n\n# Plot the mesh slice.\nfig = plt.figure(figsize=(6, 6))\nax = fig.add_axes([0.1, 0.1, 0.8, 0.8])\ncplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal=\"Y\", ind=22)\nax.set_title('Tensor Mesh Slice')\nax.set_xlabel('x (m)')\nax.set_ylabel('z (m)')\nax.set_xlim([158000, 159000])\nax.set_ylim([-500, 100])\nax.ticklabel_format(axis=\"both\")\nax.set_aspect('equal')\nplt.savefig(\n path_to_output + \"/\" + name + \"_TreeMeshSlice.pdf\",\n bbox_inches=\"tight\"\n)\nplt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Visualize the mesh\n\n" + "## Step 7: Set up the Gravity Inverse Problem\n\nThe gravity inverse problem is set up using SimPEG's `pf.gravity` module to generate a forward model based on the mesh\nand receiver locations.\n\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Create wires for the density model\n\n" ] }, { @@ -163,14 +166,21 @@ }, "outputs": [], "source": [ - "#The ``discretize.TreeMesh.plot_slice()`` utility will plot a slice of thee Tree Mesh object generated above." + "wires = maps.Wires((\"density\", nactv))\n\n# Create a density map\ndensity_map = actvMap * wires.density\n\n# Create an identity map\nidentity_map = maps.IdentityMap(nP=nactv)\n\n# Components of the data used as input\ngravity_components = [\"gxx\", \"gyy\", \"gxz\", \"gyz\", \"gxy\"]\n\n# Receiver locations (25 meters above the topography)\ngravity_receiver_locations = inv_topo + 120.\n\n# Create a gravity receiver list\ngravity_receivers = pf.gravity.receivers.Point(\n locations=gravity_receiver_locations,\n components=gravity_components\n)\ngravity_receiver_list = [gravity_receivers]\n\n# Create a gravity source field\ngravity_source_field = pf.gravity.sources.SourceField(\n receiver_list=gravity_receiver_list\n)\n\n# Define the gravity survey\ngravity_survey = pf.gravity.survey.Survey(gravity_source_field)\n\n# Set up the gravity simulation problem\ngravity_problem = grav.simulation.Simulation3DIntegral(\n inv_mesh,\n survey=gravity_survey,\n rhoMap=wires.density,\n ind_active=actv\n)\n\n# Create a gravity data object, with the relative errors and a standard noise floor\ngravity_data = data.Data(\n survey=gravity_survey,\n dobs=grav_vec.flatten(),\n noise_floor=5,\n relative_error=0.1\n)\n\n# Define the misfits associated with the gravity data\ngravity_misfit = data_misfit.L2DataMisfit(\n data=gravity_data,\n simulation=gravity_problem\n)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 8: Setting up the Gaussian Mixture Model (GMM) Prior\n\nIn PGI, the petrophysical data is used as a constraint in the form of a [Gaussian Mixture Model](https://scikit-learn.org/stable/modules/mixture.html) (GMM).\nA GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units ($n$),\nthe petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below. \nIf you do have a petrophysical dataset, you can fit the GMM to said dataset using the ``gmmref.fit()`` method.\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Plotting a mesh slice\nCreate a background model.\n\n" + "Number of rock units and number of physical properties\n\n" ] }, { @@ -181,14 +191,14 @@ }, "outputs": [], "source": [ - "bg = np.ones(nactv)\nmod = actvMap * bg\n\n# Define the tensor mesh parameters.\nprops = dict(boxstyle='round,pad=1', facecolor='white')\n\n# Plot the mesh slice.\nfig = plt.figure(figsize=(6, 6))\nax = fig.add_axes([0.1, 0.1, 0.8, 0.8])\ncplot = inv_mesh.plot_slice(mod, grid=False, ax=ax, normal=\"Y\", ind=22)\nax.set_title('Tensor Mesh Slice')\nax.set_xlabel('x (m)')\nax.set_ylabel('y (m)')\nax.ticklabel_format(axis=\"both\")\nax.set_aspect('equal')\nplt.savefig(path_to_output + \"/\" + name + \"_TreeMeshSlice.pdf\", bbox_inches=\"tight\")\nplt.show()" + "num_rock_units = 5\nnum_physical_props = 1\n\n# Create a weighted Gaussian mixture model with specified parameters\ngmmref = utils.WeightedGaussianMixture(\n n_components=num_rock_units,\n mesh=inv_mesh,\n actv=actv,\n covariance_type=\"full\",\n)\n\n# Set the background density\nbackground_density = 0.0\n\n# Initialize the GMM fit with random samples, mesh size, \n# and number of physical properties\ngmmref.fit(np.random.randn(nactv, num_physical_props))\n\n# Set the mean values of physical property contrasts for each rock unit\n# One value (density) for each rock unit\ngmmref.means_ = np.c_[\n [-0.4],\n [-0.2],\n [0.0],\n [0.2],\n [0.4],\n].T\n\n# Set the original variance for density\ndensity_variance = 8e-5\n\n# Set the covariances of physical properties for each rock unit\n# NOTE: Since we don't have petrophysical information for this example, \n# we keep the covariances # same for every unit. The GMM will update \n# itself during the inversion.\ngmmref.covariances_ = np.array(\n [\n [[density_variance]],\n [[density_variance]],\n [[density_variance]],\n [[density_variance]],\n [[density_variance]]\n ]\n)\n\n# Compute the precision (inverse covariance) of each cluster\ngmmref.compute_clusters_precisions()\n\n# Set the weights for each rock unit\n# NOTE: This determines the size of each peak of the n-modal GMM\ngmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Set up the gravity inverse problem\n%%\n Create wires for the density model\n\n" + "## Plot the 1D-GMM\n\nThe initial density distribution is visualized.\n\n" ] }, { @@ -199,14 +209,14 @@ }, "outputs": [], "source": [ - "wires = maps.Wires((\"density\", nactv))\n\n# Create a Density Mapping\ndensity_map = actvMap * wires.density\n# Create Identity Mapping\nidentity_map = maps.IdentityMap(nP=nactv)\n\n# Components of the data used as input\ngravity_components = [\"gyy\", \"gxz\", \"gyz\", \"gxy\", \"gxx\"]\n\n# Receiver locations\ngravity_receiver_locations = grav_new[:, [0, 1, 2]]\n\n# Create a gravity receiver list\ngravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components)\ngravity_receiver_list = [gravity_receivers]\n\n# Create a gravity source field\ngravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list)\n\n# Define the gravity survey\ngravity_survey = pf.gravity.survey.Survey(gravity_source_field)\n\n# Set up the gravity simulation problem\ngravity_problem = grav.simulation.Simulation3DIntegral(\n inv_mesh,\n survey=gravity_survey,\n rhoMap=wires.density,\n ind_active=actv\n)\n\n# Define uncertainties\n# Maximum value of each component\n# max_gravity = np.max(np.abs(grav_vec), axis=0)\n# Standard deviation of each component\n# std_gravity = 0.05 * max_gravity\n# std_gravity = np.ones(np.shape(grav_vec)) * std_gravity\n\n# Create a gravity data object\ngravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1)\n\n# Define the misfits associated with the gravity data\ngravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem)" + "fig = plt.figure(figsize=(6, 6))\nax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2)\nax[0].set_xlabel(r\"Density Contrast (g/cc)\")\nax[0].set_ylabel(r\"Probability Density Values\")\nax[0].get_legend().remove()\nax[0].set_title(r\"Initial DelRho Distribution\")\nax[0].ticklabel_format(axis=\"both\", style=\"scientific\", scilimits=(0, 0))\nax[0].set_aspect(1 / 20)\nplt.savefig(\n path_to_output + \"/\" + name + \"_Init_GMM.pdf\",\n bbox_inches=\"tight\"\n)\nplt.show()" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Set up the Gaussian Mixture Model (GMM) Prior\n%%\n Defining the Gaussian Mixture Model (GMM)\n Number of rock units and number of physical properties\n\n" + "## Step 9: Setting the Hyper-parameters and the Sensitivity Weights\n\nEvery PGI is a set of three Maximum-A-Posteriori ([MAP](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation))\nproblems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both\nthe FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise \nthe necessary weights for every voxel (as the contribution of every voxel is dependent on its depth from the surface). \nThe regularization smallness ($\\alpha_s$) and smoothness ($\\alpha_i,\\ i = x, y, z$) are initialised here. Please check \n[here](https://giftoolscookbook.readthedocs.io/en/latest/content/fundamentals/index.html)\nfor the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion.\n\nNOTE: The smoothness parameters (`alpha_i`) are static and hence need to be fine-tuned \nthrough trial and error.\n\n\n" ] }, { @@ -217,14 +227,14 @@ }, "outputs": [], "source": [ - "num_rock_units = 5\nnum_physical_props = 1\n\n# Create a weighted Gaussian mixture model with specified parameters\ngmmref = utils.WeightedGaussianMixture(\n n_components=num_rock_units,\n mesh=inv_mesh,\n actv=actv,\n covariance_type=\"full\",\n)\n\n# Set the background density\nbackground_density = 0.0\n\n# Initialize the GMM fit with random samples, mesh size, and number of physical properties\ngmmref.fit(np.random.randn(nactv, num_physical_props))\n\n# Set the mean values of physical property contrasts for each rock unit\n# One value (density) for each rock unit\ngmmref.means_ = np.c_[\n [-0.4],\n [-0.2],\n [0.0],\n [0.2],\n [0.4],\n].T\n\n# Set the original variance for density\ndensity_variance = 8e-5\n\n# Set the covariances of physical properties for each rock unit\ngmmref.covariances_ = np.array(\n [\n [[density_variance]],\n [[density_variance]],\n [[density_variance]],\n [[density_variance]],\n [[density_variance]]\n ]\n)\n\n# Compute the precision (inverse covariance) of each cluster\ngmmref.compute_clusters_precisions()\n\n# Set the weights for each rock unit\ngmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125]" + "initial_model = np.r_[background_density * np.ones(actvMap.nP)]" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "GMM Parameters\n\n" + "Sensitivity weighting\nCompute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G)\nThis is done by computing the square root of the sum of the squared elements of G for each cell,\nand then normalizing by the cell volumes and the maximum weight value.\n\n" ] }, { @@ -235,14 +245,21 @@ }, "outputs": [], "source": [ - "means_init = gmmref.means_.flatten()\ncovariances_init = gmmref.covariances_.flatten()\nweights_init = gmmref.weights_[0].flatten()\n\n# Plot the 1D GMM\nfig = plt.figure(figsize=(6, 6))\nax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2)\nax[0].set_xlabel(r\"Density Contrast (g/cc)\")\nax[0].set_ylabel(r\"Probability Density Values\")\nax[0].get_legend().remove()\nax[0].set_title(r\"Initial DelRho Distribution\")\nax[0].ticklabel_format(axis=\"both\", style=\"scientific\", scilimits=(0, 0))\nax[0].set_aspect(1/20)\nplt.savefig(path_to_output + \"/\" + name + \"_Init_GMM.pdf\", bbox_inches=\"tight\")\nplt.show()" + "sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv])\nsensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity)\n\n# Regularization multipliers\nsmallness_multiplier = 1e-4\nsmoothness_x_multiplier = 4\nsmoothness_y_multiplier = 4\nsmoothness_z_multiplier = 0.25\n\n# Create joint PGI regularization with smoothness\nregularization_term = regularization.PGI(\n gmmref=gmmref,\n mesh=inv_mesh,\n wiresmap=wires,\n maplist=[identity_map],\n active_cells=actv,\n alpha_pgi=smallness_multiplier,\n alpha_x=smoothness_x_multiplier,\n alpha_y=smoothness_y_multiplier,\n alpha_z=smoothness_z_multiplier,\n # The second derivative smoothnesses are kept to be zero, since we\n # don't want to include second derivatives in the objective function\n alpha_xx=0.0,\n alpha_yy=0.0,\n alpha_zz=0.0,\n weights_list=[sensitivity_weights_gravity]\n)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 10: Initialize the Directives\n\nDirectives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver.\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Define the inversion parameters\n%%\n Initial Model\n\n" + "Estimate smoothness multipliers\n\n" ] }, { @@ -253,14 +270,14 @@ }, "outputs": [], "source": [ - "initial_model = np.r_[background_density * np.ones(actvMap.nP)]\n\n# Sensitivity weighting\n# Compute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G)\n# This is done by computing the square root of the sum of the squared elements of G for each cell,\n# and then normalizing by the cell volumes and the maximum weight value.\nsensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv])\nsensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity)\n\n# Regularization multipliers\nsmallness_multiplier = 1e-4\nsmoothness_x_multiplier = 1e-1\nsmoothness_y_multiplier = 1e-1\nsmoothness_z_multiplier = 1e-1\n\n# Create joint PGI regularization with smoothness\nregularization_term = regularization.PGI(\n gmmref=gmmref,\n mesh=inv_mesh,\n wiresmap=wires,\n maplist=[identity_map],\n active_cells=actv,\n alpha_pgi=smallness_multiplier,\n alpha_x=smoothness_x_multiplier,\n alpha_y=smoothness_y_multiplier,\n alpha_z=smoothness_z_multiplier,\n alpha_xx=0.0,\n alpha_yy=0.0,\n alpha_zz=0.0,\n weights_list=[sensitivity_weights_gravity]\n)\n\n# Directives for inversion\n# Define the directives for the inversion process, based on Astic et al., (2021)\n\n# Estimate smoothness multipliers\nalphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True)\n\n# Initialize beta and beta/alpha_s schedule\nbeta_directive = directives.BetaEstimate_ByEig(beta0_ratio=1e-2)\nbeta_schedule = directives.PGI_BetaAlphaSchedule(\n coolingFactor=16.0,\n tolerance=0.2,\n progress=0.2,\n verbose=True,\n)\n\n# Define target misfits for geophysical and petrophysical data\ntarget_misfits = directives.MultiTargetMisfits(verbose=True)\n\n# Add reference model once stable\nmref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True)\n\n# Update smallness parameters, keeping GMM fixed (L2 Approx of PGI)\nupdate_smallness_directive = directives.PGI_UpdateParameters(\n update_gmm=True,\n kappa = 0,\n nu = 0.5,\n zeta = 0\n)\n\n# Update preconditioner\nupdate_preconditioner = directives.UpdatePreconditioner()\n\n# Save iteration results\nsave_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output)\n\n# Save model iterations\nsave_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations)\n\n# Optimization options for the inversion\nlower_bound = np.r_[-1 * np.ones(actvMap.nP)]\nupper_bound = np.r_[1 * np.ones(actvMap.nP)]\noptimizer = optimization.ProjectedGNCG(\n maxIter=20,\n lower=lower_bound,\n upper=upper_bound,\n maxIterLS=20,\n maxIterCG=100,\n tolCG=1e-4,\n)\n\n# Inverse problem setup\ninverse_prob = inverse_problem.BaseInvProblem(gravity_misfit, regularization_term, optimizer)\ninversion_algo = inversion.BaseInversion(\n inverse_prob,\n directiveList=[\n alphas_directive,\n beta_directive,\n update_smallness_directive,\n target_misfits,\n beta_schedule,\n mref_in_smooth,\n update_preconditioner,\n save_iteration_directive,\n save_model_directive\n ],\n)" + "alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True)\n\n# Initialize beta and beta/alpha_s schedule\nbeta_directive = directives.BetaEstimate_ByEig(beta0_ratio=1e-2)\nbeta_schedule = directives.PGI_BetaAlphaSchedule(\n coolingFactor=16.0,\n tolerance=0.2,\n progress=0.2,\n verbose=True,\n)\n\n# Define target misfits for geophysical and petrophysical data\ntarget_misfits = directives.MultiTargetMisfits(verbose=True)\n\n# Add reference model once stable\nmref_in_smooth = directives.PGI_AddMrefInSmooth(\n wait_till_stable=True,\n verbose=True\n)\n\n# Update smallness parameters, keeping GMM fixed (L2 Approx of PGI)\nupdate_smallness_directive = directives.PGI_UpdateParameters(\n update_gmm=True,\n kappa=0,\n nu=0.5,\n zeta=0\n)\n\n# Update preconditioner\nupdate_preconditioner = directives.UpdatePreconditioner()\n\n# Save iteration results\nsave_iteration_directive = directives.SaveOutputEveryIteration(\n name=name,\n directory=path_to_output\n)\n\n# Save model iterations\nsave_model_directive = directives.SaveModelEveryIteration(\n name=name,\n directory=path_to_mod_iterations\n)\n\n# Optimization options for the inversion\nlower_bound = np.r_[-1.0 * np.ones(actvMap.nP)]\nupper_bound = np.r_[1.0 * np.ones(actvMap.nP)]\noptimizer = optimization.ProjectedGNCG(\n maxIter=20,\n lower=lower_bound,\n upper=upper_bound,\n maxIterLS=20,\n maxIterCG=100,\n tolCG=1e-4,\n)\n\n# Inverse problem setup\ninverse_prob = inverse_problem.BaseInvProblem(gravity_misfit, regularization_term, optimizer)\ninversion_algo = inversion.BaseInversion(\n inverse_prob,\n directiveList=[\n alphas_directive,\n beta_directive,\n update_smallness_directive,\n target_misfits,\n beta_schedule,\n mref_in_smooth,\n update_preconditioner,\n save_iteration_directive,\n save_model_directive\n ],\n)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "# Run the Inversion!\n%%\n Run the inversion\n\n" + "## Step 11: Run the Inversion! \n\nThe inversion is executed using the initialized settings.\n\n" ] }, { @@ -274,6 +291,13 @@ "inverted_model = inversion_algo.run(initial_model)" ] }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 12: Visualize the 3D Model\n\nThe inverted 3D model is visualized using slices in different orientations.\n\n" + ] + }, { "cell_type": "code", "execution_count": null, @@ -282,14 +306,21 @@ }, "outputs": [], "source": [ - "set = 1\nsave_plots = True\n\n# Indices of the depth sections\nind_plot_x = int(len(inv_mesh.cell_centers_x)/6) + 10\nind_plot_y = int(len(inv_mesh.cell_centers_y)/2)\nind_plot_z = int(len(inv_mesh.cell_centers_z)/2) - 17\n\nind_plot = [ind_plot_x, ind_plot_y, ind_plot_z]\n\n# Extract the results\ninverted_density_model = density_map * inverted_model\nquasi_geology_model = actvMap * regularization_term.objfcts[0].compute_quasi_geology_model()\n\n# Plot Density Contrast Model (Z)\nnormal = \"Z\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (Y)\nnormal = \"X\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (X)\nnormal = \"Y\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)" + "set = 1\nsave_plots = True\n\n# Indices of the depth sections\nind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25\nind_plot_y = int(len(inv_mesh.cell_centers_y) / 2)\nind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15\n\nind_plot = [ind_plot_x, ind_plot_y, ind_plot_z]\n\n# Extract the results\ninverted_density_model = density_map * inverted_model\nquasi_geology_model = actvMap * regularization_term.objfcts[0].compute_quasi_geology_model()\n\n# Plot Density Contrast Model (Z)\nnormal = \"Z\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(\n mesh=inv_mesh, \n ind_active=actv,\n model=model_to_plot,\n normal= normal,\n ind_plot_arr=ind_plot, \n clim=[-1.0, 1.0],\n set=set, sec_loc=True,\n gdlines=True,\n which_prop=\"Den\",\n cmap=\"Spectral\",\n save_plt=save_plots,\n path_to_output=path_to_output,\n name=name\n)\n\n# Plot Inverted Model Slices (Y)\nnormal = \"X\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (X)\nnormal = \"Y\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 13: Visualize the Quasi-Geological Model\n\nThe quasi-geological model (as given by [Li et al., 2019](https://doi.org/10.1190/tle38010060.1)) is visualized, showing different rock units classified by the GMM.\n\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "Updated GMM Parameters\n\n" + "Plot Density Contrast Model (Z)\n\n" ] }, { @@ -300,14 +331,25 @@ }, "outputs": [], "source": [ - "learned_mu = regularization_term.objfcts[0].gmm.means_.flatten()\nlearned_sigma = regularization_term.objfcts[0].gmm.covariances_.flatten()\nlearned_pi = regularization_term.objfcts[0].gmm.weights_[0].flatten()\n\n# Regularization Parameters\nfinal_alpha_pgi = regularization_term.alpha_pgi\n\n# Plot the learned GMM\nfig = plt.figure(figsize=(6,6))\nax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.5)\nax[0].hist(inverted_density_model[actv], density=True, bins=1000)\nax[0].set_xlabel(r\"Density contrast (g/cc)\")\nax[0].set_ylabel(r\"Probability Density Values\")\nax[0].get_legend().remove()\nax[0].set_title(r\"Learned DelRho Distribution\")\nax[0].ticklabel_format(axis=\"both\", style=\"scientific\", scilimits=(0, 0))\nplt.savefig(path_to_output + \"/\" + name + \"_Learned_GMM.pdf\", bbox_inches=\"tight\")\nplt.show()" + "normal = \"Z\"\nmodel_to_plot = quasi_geology_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop=\"QGM\", cmap=\"jet\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (Y)\nnormal = \"X\"\nmodel_to_plot = quasi_geology_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop=\"QGM\", cmap=\"jet\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (X)\nnormal = \"Y\"\nmodel_to_plot = quasi_geology_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop=\"QGM\", cmap=\"jet\", save_plt=save_plots, path_to_output=path_to_output, name=name)" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ - "sphinx_gallery_thumbnail_number = -4\n" + "## Step 14: Plot the Updated 1D-GMM\n\nThe updated GMM, learned through the inversion, is visualized.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Plot the learned GMM\nfig = plt.figure(figsize=(6, 6))\nax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4)\nax[0].hist(inverted_density_model[actv], density=True, bins=1000)\nax[0].set_xlabel(r\"Density contrast (g/cc)\")\nax[0].set_ylabel(r\"Probability Density Values\")\nax[0].get_legend().remove()\nax[0].set_title(r\"Learned DelRho Distribution\")\nax[0].ticklabel_format(axis=\"both\", style=\"scientific\", scilimits=(0, 0))\nplt.savefig(path_to_output + \"/\" + name + \"_Learned_GMM.pdf\", bbox_inches=\"tight\")\nplt.show()" ] } ], diff --git a/_downloads/7c124765436a9efece3eea37d06bd896/03_HSI_To_Petrophysics.ipynb b/_downloads/7c124765436a9efece3eea37d06bd896/03_HSI_To_Petrophysics.ipynb new file mode 100644 index 0000000..d9f67b3 --- /dev/null +++ b/_downloads/7c124765436a9efece3eea37d06bd896/03_HSI_To_Petrophysics.ipynb @@ -0,0 +1,169 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\nfrom pyvista import set_plot_theme\nset_plot_theme('document')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n\n# Predicting P-Wave Velocity from Hyperspectral Data\n\nThis example demonstrates the prediction of P-Wave velocities from hyperspectral data using machine learning techniques. Hyperspectral data provides detailed information on rock mineralogy, which influences its petrophysical properties. This script aims to explore and quantify the relationship between spectral features and various petrophysical properties, specifically focusing on P-Wave velocity.\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Importing Necessary Libraries\n\nFirst, we import the required Python packages. We use `dotenv` to manage environment variables, `numpy` and `sklearn` for numerical operations and machine learning, and custom packages `hklearn` and `hycore` for handling hyperspectral data and machine learning workflows.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "from dotenv import dotenv_values\nimport numpy as np\nimport sklearn\nfrom sklearn.linear_model import LinearRegression\nfrom sklearn.neural_network import MLPRegressor\nfrom tqdm import tqdm\n\n# Custom packages for handling hyperspectral data\n# Note: As of January 2024, these packages are not publicly available. Please contact the author for more information.\nfrom vector_geology.Packages import hklearn\nfrom vector_geology.Packages import hycore" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Loading Data with `hycore.Shed`\n\nThe `hycore` package, which organizes hyperspectral data in a structure called Shed, is used here. A Shed makes it convenient to categorize and store hyperspectral data. We use HyLibraries (a data structure within `hylite` for storing spectral libraries) within a Shed to store the hyperspectral data. The corresponding P-Wave velocity measurements are stored as numpy arrays.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Load the train and test sheds\nconfig = dotenv_values()\npath_to_train = config.get(\"PATH_TO_HSI_TRAINING\")\npath_to_test = config.get(\"PATH_TO_HSI_TESTING\")\n\ntrain_shed = hycore.loadShed(path_to_train)\ntest_shed = hycore.loadShed(path_to_test)\n\n# Load the train and test spectral libraries and P-Wave velocities\ntrain_fx50 = train_shed.results.FX50Lib\ntrain_fenix = train_shed.results.FENIXLib\ntrain_lwir = train_shed.results.LWIRLib\ntrain_vp = train_shed.results.Vp\n\ntest_fx50 = test_shed.results.FX50Lib\ntest_fenix = test_shed.results.FENIXLib\ntest_lwir = test_shed.results.LWIRLib\ntest_vp = test_shed.results.Vp" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Creating a `hklearn.Stack` Object\n\nThe `hklearn` package, designed to handle hyperspectral data, is used to organize spectral libraries into a structure called Stack. A Stack can manage multiple spectral libraries and integrate data from multiple sensors. We perform hull corrections on the spectra to enhance features of interest.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Create Stack objects and apply hull correction to the spectra\nhsi_train = hklearn.Stack(\n names=['FENIX', 'FX50', 'LWIR'],\n data=[train_fenix, train_fx50, train_lwir]).hc(\n ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)},\n hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'}\n)\n\nhsi_test = hklearn.Stack(\n names=['FENIX', 'FX50', 'LWIR'],\n data=[test_fenix, test_fx50, test_lwir]).hc(\n ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)},\n hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'}\n)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Scaling the Dependent Variable (Y)\n\nThe P-Wave velocities are scaled to facilitate the convergence of the loss function during model training. We use the `StandardScaler` from scikit-learn for this purpose.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Scale Y Variable (P-Wave velocity)\ny_scaler = sklearn.preprocessing.StandardScaler()\n_ = y_scaler.fit(train_vp[:, None])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Transforming the Independent Variable (X)\n\nHyperspectral data often contains a large number of bands, many of which might be redundant. We apply Principal Component Analysis (PCA) to extract the most significant components. A two-step PCA is performed to ensure that there isn't any inter-sensor correlation. The `Stack` object stores the PCA, which is also applied to the test set.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Apply PCA to the X variable (spectral data)\nPCA_X = hsi_train.fit_pca(n_components=10, normalise=True)\nhsi_test.set_transform(PCA_X)\n\n# Set the Y-Variable (P-Wave velocity)\nhsi_train.set_y(train_vp[:, None, None])\nhsi_test.set_y(test_vp[:, None, None])" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Initializing and Training Models\n\nDifferent machine learning models are initialized and trained. In this example, we use a simple linear regression (`LinearRegression`) and a Multilayer Perceptron (`MLPRegressor`) from scikit-learn. A dictionary with the parameters for optimization is also initialized.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Initialize models\nmodels = dict(\n Linear=LinearRegression(),\n MLP=MLPRegressor(\n hidden_layer_sizes=(180,),\n max_iter=1000,\n solver='sgd',\n learning_rate='adaptive'\n )\n)\n\n# Define parameter ranges for each model\nparams = dict(Linear={'fit_intercept': [True, False]},\n MLP={\"alpha\": np.linspace(1e-4, 1e0)})\n\n# Train the models\nfor name, model in tqdm(models.items(), 'Fitting models', leave=True):\n hsi_train.fit_model(name, model, xtransform=True, ytransform=y_scaler, grid_search_cv=params[name], n_jobs=-1)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Scoring and Evaluating Models\n\nFinally, the trained models are scored and evaluated. The `Stack.get_score_table()` method is used to display a table with the training, cross-validation, and test scores of the best-performing model.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Evaluate models and display score table\nhsi_train.get_score_table(hsi_test, y_test=None, style=True)" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.10.12" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} \ No newline at end of file diff --git a/_downloads/9c357fcc2a0ec18ad79fc30d38b7a6f0/04_FTG_PGI.py b/_downloads/9c357fcc2a0ec18ad79fc30d38b7a6f0/04_FTG_PGI.py new file mode 100644 index 0000000..cd5ee5a --- /dev/null +++ b/_downloads/9c357fcc2a0ec18ad79fc30d38b7a6f0/04_FTG_PGI.py @@ -0,0 +1,486 @@ + +""" +.. _example_inversion_ftg_data: + +Inversion of Full Tensor Gravity Gradiometry Data +================================================= + +This example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, measured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include: + +- Loading necessary modules and data +- Generating a mesh +- Initializing the Gaussian Mixture Model +- Tuning hyper-parameters +- Visualizing the results + +The process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example notebook from SimPEG's documentation (see [SimPEG docs](https://docs.simpeg.xyz/index.html) for more information). +""" + +import os +import shutil +from dotenv import dotenv_values + +import SimPEG.potential_fields as pf +# Import the SimPEG functions +import discretize as ds +import matplotlib as mpl +import matplotlib.pyplot as plt +# Import the general modules +import numpy as np +import pandas as pd +from SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives) +from SimPEG.potential_fields import gravity as grav +# Import the discretize module +from discretize.utils import active_from_xyz + +# Import SimpegHelper for plotting and resampling data +from vector_geology import SimpegHelper as SH + +# Plot beautification +formatter = mpl.ticker.ScalarFormatter(useMathText=True) +formatter.set_scientific(True) +formatter.set_powerlimits((0, 0)) + +mpl.rc("axes", titlesize=14, labelsize=12) +mpl.rc("xtick", labelsize=12) +mpl.rc("ytick", labelsize=12) + +# %% +# Step 1: Create Directories for Storing Model Iterations and Outputs +# ------------------------------------------------------------------- +# +# PGI is an iterative process, and it's useful to save the model at every iteration. This helps in preventing overfitting and in adjusting the stopping criteria based on the model iterations. + +name = "Ireland_FTG" +path_to_mod_iterations = f"./{name}/Model Iterations" +path_to_output = f"./{name}/Output" + +if os.path.exists(path_to_mod_iterations): + shutil.rmtree(path_to_mod_iterations) +os.makedirs(path_to_mod_iterations) + +if os.path.exists(path_to_output): + shutil.rmtree(path_to_output) +os.makedirs(path_to_output) + +# %% +# Step 2: Load the FTG-G Data +# ---------------------------- +# +# Full Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion. +# $$ +# G = \left[\begin{array}{cc} +# G_{xx} & G_{xy} & G_{xz}\\ +# G_{xy} & G_{yy} & G_{yz}\\ +# G_{xz} & G_{yz} & G_{zz} +# \end{array}\right] +# $$ + +config = dotenv_values() +file_path = config.get("PATH_TO_GRADIOMETRY") +FTG_Data = pd.read_csv(file_path, delimiter=",").to_numpy() + +# %% +# Step 3: Resample Data onto a Regular Grid +# ----------------------------------------- +# +# Resampling the data onto a regular grid aids in mesh generation and visualization. The `SimpegHelper.pf_rs()` function resamples potential field data onto a new regular grid. + +inc = 150 # New sampling interval (ground units) +grav_new, nx_new, ny_new = SH.pf_rs(FTG_Data, inc, bounds=[156000, 168000, 143000, 148500]) + +# NOTE: This step is necessary for real data since the convention followed by +# the SimPEG forward operator is the opposite of the general convention +grav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :] # Adjusting sign convention +inv_topo = grav_new[:, [0, 1, 2]] # Extracting topography + +# %% +# Step 4: Visualize the Data +# -------------------------- +# +# The `SimpegHelper.plot_2D_data()` function is used to visualize the datasets with `matplotlib`. + +SH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]], [np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])], cmap="jet", which_data="FTG", comp="xx", path_to_output=path_to_output, name=name) +SH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]], [np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])], cmap="terrain", which_data="Topo", comp="xx", path_to_output=path_to_output, name=name) + +# %% +# Step 5: Create a TensorMesh Object for Inversion +# ----------------------------------------------- +# +# A 3D TensorMesh is created using the `discretize.TensorMesh` utility to invert the gravity gradiometry data. The mesh is designed to have a constant cell size in the core region and expanding/contracting cells in the padding region. +# +# $$ +# z_{n+1} = r z_{n} +# $$ +# +# Where $r$ is a multiplicative factor ($<1$ for contracting, $>1$ for expanding cells). $x$ and $y$ increments can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire region is an area of interest). + +dx, dy, dz = inc, inc, 10 # Cell sizes +nz_core, nz_pad = 10, 18 # Number of cells in core and padding +fact = 1.1 # Factor for expanding/contracting cells + +inv_hx = dx * np.ones(nx_new) +inv_hy = dy * np.ones(ny_new) +inv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)] + +# Create the inverse tensor mesh +inv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), "C"]) + +# Drape the topography over the mesh +actv = active_from_xyz(inv_mesh, inv_topo) +ndv = np.nan +actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv) +nactv = int(actv.sum()) + +# Print mesh details +print(inv_mesh) + + +# %% +# Step 6: Visualize the Mesh +# -------------------------- +# +# A slice of the Tensor Mesh object is visualized using the `discretize.TensorMesh.plot_slice()` utility. + +# %% +# Create a background model. +bg = np.ones(nactv) +mod = actvMap * bg + +# Plot the mesh slice. +fig = plt.figure(figsize=(6, 6)) +ax = fig.add_axes([0.1, 0.1, 0.8, 0.8]) +cplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal="Y", ind=22) +ax.set_title('Tensor Mesh Slice') +ax.set_xlabel('x (m)') +ax.set_ylabel('z (m)') +ax.set_xlim([158000, 159000]) +ax.set_ylim([-500, 100]) +ax.ticklabel_format(axis="both") +ax.set_aspect('equal') +plt.savefig(path_to_output + "/" + name + "_TreeMeshSlice.pdf", bbox_inches="tight") +plt.show() + +# %% +# Step 7: Set up the Gravity Inverse Problem +# ------------------------------------------ +# +# The gravity inverse problem is set up using SimPEG's `pf.gravity` module to generate a forward model based on the mesh and receiver locations. + + +# %% +# Create wires for the density model +wires = maps.Wires(("density", nactv)) + +# Create a density map +density_map = actvMap * wires.density + +# Create an identity map +identity_map = maps.IdentityMap(nP=nactv) + +# Components of the data used as input +gravity_components = ["gxx", "gyy", "gxz", "gyz", "gxy"] + +# Receiver locations (25 meters above the topography) +gravity_receiver_locations = inv_topo + 120. + +# Create a gravity receiver list +gravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components) +gravity_receiver_list = [gravity_receivers] + +# Create a gravity source field +gravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list) + +# Define the gravity survey +gravity_survey = pf.gravity.survey.Survey(gravity_source_field) + +# Set up the gravity simulation problem +gravity_problem = grav.simulation.Simulation3DIntegral( + inv_mesh, + survey=gravity_survey, + rhoMap=wires.density, + ind_active=actv +) + +# Create a gravity data object, with the relative errors and a standard noise floor +gravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1) + +# Define the misfits associated with the gravity data +gravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem) + +# %% +# Step 8: Setting up the Gaussian Mixture Model (GMM) Prior +# -------------------------------------------------------- +# +# In PGI, the petrophysical data is used as a constraint in the form of a [Gaussian Mixture Model](https://scikit-learn.org/stable/modules/mixture.html). A GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units ($n$), the petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below. If you do have a petrophysical dataset, you can fit the GMM to said dataset using the ``gmmref.fit()`` method. + + + +# %% +# Number of rock units and number of physical properties +num_rock_units = 5 +num_physical_props = 1 + +# Create a weighted Gaussian mixture model with specified parameters +gmmref = utils.WeightedGaussianMixture( + n_components=num_rock_units, + mesh=inv_mesh, + actv=actv, + covariance_type="full", +) + +# Set the background density +background_density = 0.0 + +# Initialize the GMM fit with random samples, mesh size, and number of physical properties +gmmref.fit(np.random.randn(nactv, num_physical_props)) + +# Set the mean values of physical property contrasts for each rock unit +# One value (density) for each rock unit +gmmref.means_ = np.c_[ + [-0.4], + [-0.2], + [0.0], + [0.2], + [0.4], +].T + +# Set the original variance for density +density_variance = 8e-5 + +# Set the covariances of physical properties for each rock unit +# NOTE: Since we don't have petrophysical information for this example, we keep the covariances +# same for every unit. The GMM will update itself during the inversion. +gmmref.covariances_ = np.array( + [ + [[density_variance]], + [[density_variance]], + [[density_variance]], + [[density_variance]], + [[density_variance]] + ] +) + +# Compute the precision (inverse covariance) of each cluster +gmmref.compute_clusters_precisions() + +# Set the weights for each rock unit +# NOTE: This determines the size of each peak of the n-modal GMM +gmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125] + +# %% +# Plot the 1D-GMM +# --------------- +# +# The initial density distribution is visualized. + +fig = plt.figure(figsize=(6, 6)) +ax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2) +ax[0].set_xlabel(r"Density Contrast (g/cc)") +ax[0].set_ylabel(r"Probability Density Values") +ax[0].get_legend().remove() +ax[0].set_title(r"Initial DelRho Distribution") +ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0)) +ax[0].set_aspect(1 / 20) +plt.savefig(path_to_output + "/" + name + "_Init_GMM.pdf", bbox_inches="tight") +plt.show() + +# %% +# Step 9: Setting the Hyper-parameters and the Sensitivity Weights +# ----------------------------------------------------------------- +# +# Every PGI is a set of three Maximum-A-Posteriori ([MAP](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation)) problems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both the FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise the necessary weights for every voxel (as the contribution of every voxel is dependent on it's depth from the surface). The regularization smallness ($\alpha_s$) and smoothness ($\alpha_i,\ i = x, y, z$) are initialised here. Please check [here](https://giftoolscookbook.readthedocs.io/en/latest/content/fundamentals/index.html) for the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion. +# +# NOTE : The smoothness parameters ($\alpha_i$) are static and hence need to be fine-tuned through trial and error. +# + +initial_model = np.r_[background_density * np.ones(actvMap.nP)] + +# %% +# Sensitivity weighting +# Compute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G) +# This is done by computing the square root of the sum of the squared elements of G for each cell, +# and then normalizing by the cell volumes and the maximum weight value. + +# %% +sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv]) +sensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity) + +# Regularization multipliers +smallness_multiplier = 1e-4 +smoothness_x_multiplier = 4 +smoothness_y_multiplier = 4 +smoothness_z_multiplier = 0.25 + +# Create joint PGI regularization with smoothness +regularization_term = regularization.PGI( + gmmref=gmmref, + mesh=inv_mesh, + wiresmap=wires, + maplist=[identity_map], + active_cells=actv, + alpha_pgi=smallness_multiplier, + alpha_x=smoothness_x_multiplier, + alpha_y=smoothness_y_multiplier, + alpha_z=smoothness_z_multiplier, + # The second derivative smoothnesses are kept to be zero, since we + # don't want to include second derivatives in the objective function + alpha_xx=0.0, + alpha_yy=0.0, + alpha_zz=0.0, + weights_list=[sensitivity_weights_gravity] +) + +# %% +# Step 10: Initialize the Directives +# ---------------------------------- +# +# Directives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver. + + +# %% +# Estimate smoothness multipliers +alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True) + +# Initialize beta and beta/alpha_s schedule +beta_directive = directives.BetaEstimate_ByEig(beta0_ratio=1e-2) +beta_schedule = directives.PGI_BetaAlphaSchedule( + coolingFactor=16.0, + tolerance=0.2, + progress=0.2, + verbose=True, +) + +# Define target misfits for geophysical and petrophysical data +target_misfits = directives.MultiTargetMisfits(verbose=True) + +# Add reference model once stable +mref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True) + +# Update smallness parameters, keeping GMM fixed (L2 Approx of PGI) +update_smallness_directive = directives.PGI_UpdateParameters( + update_gmm=True, + kappa=0, + nu=0.5, + zeta=0 +) + +# Update preconditioner +update_preconditioner = directives.UpdatePreconditioner() + +# Save iteration results +save_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output) + +# Save model iterations +save_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations) + +# Optimization options for the inversion +lower_bound = np.r_[-1.0 * np.ones(actvMap.nP)] +upper_bound = np.r_[1.0 * np.ones(actvMap.nP)] +optimizer = optimization.ProjectedGNCG( + maxIter=20, + lower=lower_bound, + upper=upper_bound, + maxIterLS=20, + maxIterCG=100, + tolCG=1e-4, +) + +# Inverse problem setup +inverse_prob = inverse_problem.BaseInvProblem(gravity_misfit, regularization_term, optimizer) +inversion_algo = inversion.BaseInversion( + inverse_prob, + directiveList=[ + alphas_directive, + beta_directive, + update_smallness_directive, + target_misfits, + beta_schedule, + mref_in_smooth, + update_preconditioner, + save_iteration_directive, + save_model_directive + ], +) + +# Step 11: Run the Inversion! +# --------------------------- +# +# The inversion is executed using the initialized settings. + +inverted_model = inversion_algo.run(initial_model) + +# %% +# Step 12: Visualize the 3D Model +# ------------------------------- +# +# The inverted 3D model is visualized using slices in different orientations. + +set = 1 +save_plots = True + +# Indices of the depth sections +ind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25 +ind_plot_y = int(len(inv_mesh.cell_centers_y) / 2) +ind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15 + +ind_plot = [ind_plot_x, ind_plot_y, ind_plot_z] + +# Extract the results +inverted_density_model = density_map * inverted_model +quasi_geology_model = actvMap * regularization_term.objfcts[0].compute_quasi_geology_model() + +# Plot Density Contrast Model (Z) +normal = "Z" +model_to_plot = inverted_density_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# Plot Inverted Model Slices (Y) +normal = "X" +model_to_plot = inverted_density_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# Plot Inverted Model Slices (X) +normal = "Y" +model_to_plot = inverted_density_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# %% +# Step 13: Visualize the Quasi-Geological Model +# --------------------------------------------- +# +# The quasi-geological model (as given by [Li et al., 2019](https://doi.org/10.1190/tle38010060.1))is visualized, showing different rock units classified by the GMM. + +# %% +# Plot Density Contrast Model (Z) +normal = "Z" +model_to_plot = quasi_geology_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# Plot Inverted Model Slices (Y) +normal = "X" +model_to_plot = quasi_geology_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# Plot Inverted Model Slices (X) +normal = "Y" +model_to_plot = quasi_geology_model +SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + +# %% +# Step 14: Plot the Updated 1D-GMM +# -------------------------------- +# +# The updated GMM, learned through the inversion, is visualized. + +# Plot the learned GMM +fig = plt.figure(figsize=(6, 6)) +ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4) +ax[0].hist(inverted_density_model[actv], density=True, bins=1000) +ax[0].set_xlabel(r"Density contrast (g/cc)") +ax[0].set_ylabel(r"Probability Density Values") +ax[0].get_legend().remove() +ax[0].set_title(r"Learned DelRho Distribution") +ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0)) +plt.savefig(path_to_output + "/" + name + "_Learned_GMM.pdf", bbox_inches="tight") +plt.show() diff --git a/_downloads/e253201752a10d57e0e78db6c944841a/04_FTG_PGI.ipynb b/_downloads/e253201752a10d57e0e78db6c944841a/04_FTG_PGI.ipynb new file mode 100644 index 0000000..8c38664 --- /dev/null +++ b/_downloads/e253201752a10d57e0e78db6c944841a/04_FTG_PGI.ipynb @@ -0,0 +1,359 @@ +{ + "cells": [ + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "%matplotlib inline\nfrom pyvista import set_plot_theme\nset_plot_theme('document')" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "\n\n# Inversion of Full Tensor Gravity Gradiometry Data\n\nThis example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, measured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include:\n\n- Loading necessary modules and data\n- Generating a mesh\n- Initializing the Gaussian Mixture Model\n- Tuning hyper-parameters\n- Visualizing the results\n\nThe process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example notebook from SimPEG's documentation (see [SimPEG docs](https://docs.simpeg.xyz/index.html) for more information).\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "import os\nimport shutil\nfrom dotenv import dotenv_values\n\nimport SimPEG.potential_fields as pf\n# Import the SimPEG functions\nimport discretize as ds\nimport matplotlib as mpl\nimport matplotlib.pyplot as plt\n# Import the general modules\nimport numpy as np\nimport pandas as pd\nfrom SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives)\nfrom SimPEG.potential_fields import gravity as grav\n# Import the discretize module\nfrom discretize.utils import active_from_xyz\n\n# Import SimpegHelper for plotting and resampling data\nfrom vector_geology import SimpegHelper as SH\n\n# Plot beautification\nformatter = mpl.ticker.ScalarFormatter(useMathText=True)\nformatter.set_scientific(True)\nformatter.set_powerlimits((0, 0))\n\nmpl.rc(\"axes\", titlesize=14, labelsize=12)\nmpl.rc(\"xtick\", labelsize=12)\nmpl.rc(\"ytick\", labelsize=12)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 1: Create Directories for Storing Model Iterations and Outputs\n\nPGI is an iterative process, and it's useful to save the model at every iteration. This helps in preventing overfitting and in adjusting the stopping criteria based on the model iterations.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "name = \"Ireland_FTG\"\npath_to_mod_iterations = f\"./{name}/Model Iterations\"\npath_to_output = f\"./{name}/Output\"\n\nif os.path.exists(path_to_mod_iterations):\n shutil.rmtree(path_to_mod_iterations)\nos.makedirs(path_to_mod_iterations)\n\nif os.path.exists(path_to_output):\n shutil.rmtree(path_to_output)\nos.makedirs(path_to_output)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 2: Load the FTG-G Data\n\nFull Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion.\n$$\nG = \\left[\\begin{array}{cc} \nG_{xx} & G_{xy} & G_{xz}\\\\\nG_{xy} & G_{yy} & G_{yz}\\\\\nG_{xz} & G_{yz} & G_{zz}\n\\end{array}\\right]\n$$\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "config = dotenv_values()\nfile_path = config.get(\"PATH_TO_GRADIOMETRY\")\nFTG_Data = pd.read_csv(file_path, delimiter=\",\").to_numpy()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 3: Resample Data onto a Regular Grid\n\nResampling the data onto a regular grid aids in mesh generation and visualization. The `SimpegHelper.pf_rs()` function resamples potential field data onto a new regular grid.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "inc = 150 # New sampling interval (ground units)\ngrav_new, nx_new, ny_new = SH.pf_rs(FTG_Data, inc, bounds=[156000, 168000, 143000, 148500])\n\n# NOTE: This step is necessary for real data since the convention followed by\n# the SimPEG forward operator is the opposite of the general convention\ngrav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :] # Adjusting sign convention\ninv_topo = grav_new[:, [0, 1, 2]] # Extracting topography" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 4: Visualize the Data\n\nThe `SimpegHelper.plot_2D_data()` function is used to visualize the datasets with `matplotlib`.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "SH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]], [np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])], cmap=\"jet\", which_data=\"FTG\", comp=\"xx\", path_to_output=path_to_output, name=name)\nSH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]], [np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])], cmap=\"terrain\", which_data=\"Topo\", comp=\"xx\", path_to_output=path_to_output, name=name)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 5: Create a TensorMesh Object for Inversion\n\nA 3D TensorMesh is created using the `discretize.TensorMesh` utility to invert the gravity gradiometry data. The mesh is designed to have a constant cell size in the core region and expanding/contracting cells in the padding region.\n\n$$\nz_{n+1} = r z_{n}\n$$\n\nWhere $r$ is a multiplicative factor ($<1$ for contracting, $>1$ for expanding cells). $x$ and $y$ increments can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire region is an area of interest).\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "dx, dy, dz = inc, inc, 10 # Cell sizes\nnz_core, nz_pad = 10, 18 # Number of cells in core and padding\nfact = 1.1 # Factor for expanding/contracting cells\n\ninv_hx = dx * np.ones(nx_new)\ninv_hy = dy * np.ones(ny_new)\ninv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)]\n\n# Create the inverse tensor mesh\ninv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), \"C\"])\n\n# Drape the topography over the mesh\nactv = active_from_xyz(inv_mesh, inv_topo)\nndv = np.nan\nactvMap = maps.InjectActiveCells(inv_mesh, actv, ndv)\nnactv = int(actv.sum())\n\n# Print mesh details\nprint(inv_mesh)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 6: Visualize the Mesh\n\nA slice of the Tensor Mesh object is visualized using the `discretize.TensorMesh.plot_slice()` utility.\n\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Create a background model.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "bg = np.ones(nactv)\nmod = actvMap * bg\n\n# Plot the mesh slice.\nfig = plt.figure(figsize=(6, 6))\nax = fig.add_axes([0.1, 0.1, 0.8, 0.8])\ncplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal=\"Y\", ind=22)\nax.set_title('Tensor Mesh Slice')\nax.set_xlabel('x (m)')\nax.set_ylabel('z (m)')\nax.set_xlim([158000, 159000])\nax.set_ylim([-500, 100])\nax.ticklabel_format(axis=\"both\")\nax.set_aspect('equal')\nplt.savefig(path_to_output + \"/\" + name + \"_TreeMeshSlice.pdf\", bbox_inches=\"tight\")\nplt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 7: Set up the Gravity Inverse Problem\n\nThe gravity inverse problem is set up using SimPEG's `pf.gravity` module to generate a forward model based on the mesh and receiver locations.\n\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Create wires for the density model\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "wires = maps.Wires((\"density\", nactv))\n\n# Create a density map\ndensity_map = actvMap * wires.density\n\n# Create an identity map\nidentity_map = maps.IdentityMap(nP=nactv)\n\n# Components of the data used as input\ngravity_components = [\"gxx\", \"gyy\", \"gxz\", \"gyz\", \"gxy\"]\n\n# Receiver locations (25 meters above the topography)\ngravity_receiver_locations = inv_topo + 120.\n\n# Create a gravity receiver list\ngravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components)\ngravity_receiver_list = [gravity_receivers]\n\n# Create a gravity source field\ngravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list)\n\n# Define the gravity survey\ngravity_survey = pf.gravity.survey.Survey(gravity_source_field)\n\n# Set up the gravity simulation problem\ngravity_problem = grav.simulation.Simulation3DIntegral(\n inv_mesh,\n survey=gravity_survey,\n rhoMap=wires.density,\n ind_active=actv\n)\n\n# Create a gravity data object, with the relative errors and a standard noise floor\ngravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1)\n\n# Define the misfits associated with the gravity data\ngravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 8: Setting up the Gaussian Mixture Model (GMM) Prior\n\nIn PGI, the petrophysical data is used as a constraint in the form of a [Gaussian Mixture Model](https://scikit-learn.org/stable/modules/mixture.html). A GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units ($n$), the petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below. If you do have a petrophysical dataset, you can fit the GMM to said dataset using the ``gmmref.fit()`` method.\n\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Number of rock units and number of physical properties\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "num_rock_units = 5\nnum_physical_props = 1\n\n# Create a weighted Gaussian mixture model with specified parameters\ngmmref = utils.WeightedGaussianMixture(\n n_components=num_rock_units,\n mesh=inv_mesh,\n actv=actv,\n covariance_type=\"full\",\n)\n\n# Set the background density\nbackground_density = 0.0\n\n# Initialize the GMM fit with random samples, mesh size, and number of physical properties\ngmmref.fit(np.random.randn(nactv, num_physical_props))\n\n# Set the mean values of physical property contrasts for each rock unit\n# One value (density) for each rock unit\ngmmref.means_ = np.c_[\n [-0.4],\n [-0.2],\n [0.0],\n [0.2],\n [0.4],\n].T\n\n# Set the original variance for density\ndensity_variance = 8e-5\n\n# Set the covariances of physical properties for each rock unit\n# NOTE: Since we don't have petrophysical information for this example, we keep the covariances\n# same for every unit. The GMM will update itself during the inversion.\ngmmref.covariances_ = np.array(\n [\n [[density_variance]],\n [[density_variance]],\n [[density_variance]],\n [[density_variance]],\n [[density_variance]]\n ]\n)\n\n# Compute the precision (inverse covariance) of each cluster\ngmmref.compute_clusters_precisions()\n\n# Set the weights for each rock unit\n# NOTE: This determines the size of each peak of the n-modal GMM\ngmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Plot the 1D-GMM\n\nThe initial density distribution is visualized.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "fig = plt.figure(figsize=(6, 6))\nax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2)\nax[0].set_xlabel(r\"Density Contrast (g/cc)\")\nax[0].set_ylabel(r\"Probability Density Values\")\nax[0].get_legend().remove()\nax[0].set_title(r\"Initial DelRho Distribution\")\nax[0].ticklabel_format(axis=\"both\", style=\"scientific\", scilimits=(0, 0))\nax[0].set_aspect(1 / 20)\nplt.savefig(path_to_output + \"/\" + name + \"_Init_GMM.pdf\", bbox_inches=\"tight\")\nplt.show()" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 9: Setting the Hyper-parameters and the Sensitivity Weights\n\nEvery PGI is a set of three Maximum-A-Posteriori ([MAP](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation)) problems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both the FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise the necessary weights for every voxel (as the contribution of every voxel is dependent on it's depth from the surface). The regularization smallness ($\\alpha_s$) and smoothness ($\\alpha_i,\\ i = x, y, z$) are initialised here. Please check [here](https://giftoolscookbook.readthedocs.io/en/latest/content/fundamentals/index.html) for the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion.\n\n NOTE : The smoothness parameters ($\\alpha_i$) are static and hence need to be fine-tuned through trial and error.\n\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "initial_model = np.r_[background_density * np.ones(actvMap.nP)]" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Sensitivity weighting\nCompute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G)\nThis is done by computing the square root of the sum of the squared elements of G for each cell,\nand then normalizing by the cell volumes and the maximum weight value.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv])\nsensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity)\n\n# Regularization multipliers\nsmallness_multiplier = 1e-4\nsmoothness_x_multiplier = 4\nsmoothness_y_multiplier = 4\nsmoothness_z_multiplier = 0.25\n\n# Create joint PGI regularization with smoothness\nregularization_term = regularization.PGI(\n gmmref=gmmref,\n mesh=inv_mesh,\n wiresmap=wires,\n maplist=[identity_map],\n active_cells=actv,\n alpha_pgi=smallness_multiplier,\n alpha_x=smoothness_x_multiplier,\n alpha_y=smoothness_y_multiplier,\n alpha_z=smoothness_z_multiplier,\n # The second derivative smoothnesses are kept to be zero, since we\n # don't want to include second derivatives in the objective function\n alpha_xx=0.0,\n alpha_yy=0.0,\n alpha_zz=0.0,\n weights_list=[sensitivity_weights_gravity]\n)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 10: Initialize the Directives\n\nDirectives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver.\n\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Estimate smoothness multipliers\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True)\n\n# Initialize beta and beta/alpha_s schedule\nbeta_directive = directives.BetaEstimate_ByEig(beta0_ratio=1e-2)\nbeta_schedule = directives.PGI_BetaAlphaSchedule(\n coolingFactor=16.0,\n tolerance=0.2,\n progress=0.2,\n verbose=True,\n)\n\n# Define target misfits for geophysical and petrophysical data\ntarget_misfits = directives.MultiTargetMisfits(verbose=True)\n\n# Add reference model once stable\nmref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True)\n\n# Update smallness parameters, keeping GMM fixed (L2 Approx of PGI)\nupdate_smallness_directive = directives.PGI_UpdateParameters(\n update_gmm=True,\n kappa=0,\n nu=0.5,\n zeta=0\n)\n\n# Update preconditioner\nupdate_preconditioner = directives.UpdatePreconditioner()\n\n# Save iteration results\nsave_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output)\n\n# Save model iterations\nsave_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations)\n\n# Optimization options for the inversion\nlower_bound = np.r_[-1.0 * np.ones(actvMap.nP)]\nupper_bound = np.r_[1.0 * np.ones(actvMap.nP)]\noptimizer = optimization.ProjectedGNCG(\n maxIter=20,\n lower=lower_bound,\n upper=upper_bound,\n maxIterLS=20,\n maxIterCG=100,\n tolCG=1e-4,\n)\n\n# Inverse problem setup\ninverse_prob = inverse_problem.BaseInvProblem(gravity_misfit, regularization_term, optimizer)\ninversion_algo = inversion.BaseInversion(\n inverse_prob,\n directiveList=[\n alphas_directive,\n beta_directive,\n update_smallness_directive,\n target_misfits,\n beta_schedule,\n mref_in_smooth,\n update_preconditioner,\n save_iteration_directive,\n save_model_directive\n ],\n)\n\n# Step 11: Run the Inversion! \n# ---------------------------\n# \n# The inversion is executed using the initialized settings.\n\ninverted_model = inversion_algo.run(initial_model)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 12: Visualize the 3D Model\n\nThe inverted 3D model is visualized using slices in different orientations.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "set = 1\nsave_plots = True\n\n# Indices of the depth sections\nind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25\nind_plot_y = int(len(inv_mesh.cell_centers_y) / 2)\nind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15\n\nind_plot = [ind_plot_x, ind_plot_y, ind_plot_z]\n\n# Extract the results\ninverted_density_model = density_map * inverted_model\nquasi_geology_model = actvMap * regularization_term.objfcts[0].compute_quasi_geology_model()\n\n# Plot Density Contrast Model (Z)\nnormal = \"Z\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (Y)\nnormal = \"X\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (X)\nnormal = \"Y\"\nmodel_to_plot = inverted_density_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop=\"Den\", cmap=\"Spectral\", save_plt=save_plots, path_to_output=path_to_output, name=name)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 13: Visualize the Quasi-Geological Model\n\nThe quasi-geological model (as given by [Li et al., 2019](https://doi.org/10.1190/tle38010060.1))is visualized, showing different rock units classified by the GMM.\n\n" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "Plot Density Contrast Model (Z)\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "normal = \"Z\"\nmodel_to_plot = quasi_geology_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop=\"QGM\", cmap=\"jet\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (Y)\nnormal = \"X\"\nmodel_to_plot = quasi_geology_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop=\"QGM\", cmap=\"jet\", save_plt=save_plots, path_to_output=path_to_output, name=name)\n\n# Plot Inverted Model Slices (X)\nnormal = \"Y\"\nmodel_to_plot = quasi_geology_model\nSH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop=\"QGM\", cmap=\"jet\", save_plt=save_plots, path_to_output=path_to_output, name=name)" + ] + }, + { + "cell_type": "markdown", + "metadata": {}, + "source": [ + "## Step 14: Plot the Updated 1D-GMM\n\nThe updated GMM, learned through the inversion, is visualized.\n\n" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": { + "collapsed": false + }, + "outputs": [], + "source": [ + "# Plot the learned GMM\nfig = plt.figure(figsize=(6, 6))\nax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4)\nax[0].hist(inverted_density_model[actv], density=True, bins=1000)\nax[0].set_xlabel(r\"Density contrast (g/cc)\")\nax[0].set_ylabel(r\"Probability Density Values\")\nax[0].get_legend().remove()\nax[0].set_title(r\"Learned DelRho Distribution\")\nax[0].ticklabel_format(axis=\"both\", style=\"scientific\", scilimits=(0, 0))\nplt.savefig(path_to_output + \"/\" + name + \"_Learned_GMM.pdf\", bbox_inches=\"tight\")\nplt.show()" + ] + } + ], + "metadata": { + "kernelspec": { + "display_name": "Python 3", + "language": "python", + "name": "python3" + }, + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.10.12" + } + }, + "nbformat": 4, + "nbformat_minor": 0 +} \ No 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a/_sources/examples/03_forward_engines/01_gravity_gradiometry_inversion_pgi.rst.txt +++ b/_sources/examples/03_forward_engines/01_gravity_gradiometry_inversion_pgi.rst.txt @@ -18,75 +18,57 @@ .. _sphx_glr_examples_03_forward_engines_01_gravity_gradiometry_inversion_pgi.py: -Inversion of Gravity Gradiometry Data on a Tree Mesh ----------------------------------------------------- +.. _example_inversion_ftg_data: -We use the PGI module within SimPEG to invert gravity gradiometry data on a Tree Mesh. This tutorial focusses on: +Inversion of Full Tensor Gravity Gradiometry Data +================================================= - - How to define all the necessary components - - How to initialize the Gaussian Mixture Model Prior - - How to set the parameters - - How to plot the results +This example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, +measured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include: -.. GENERATED FROM PYTHON SOURCE LINES 16-17 +- Loading necessary modules and data +- Generating a mesh +- Initializing the Gaussian Mixture Model +- Tuning hyper-parameters +- Visualizing the results -## Import Modules +The process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example + notebook from SimPEG's documentation (see `SimPEG docs `_ for more information). -.. GENERATED FROM PYTHON SOURCE LINES 17-69 +.. GENERATED FROM PYTHON SOURCE LINES 19-49 .. code-block:: default - import discretize as ds + import os + import shutil + from dotenv import dotenv_values + import SimPEG.potential_fields as pf - from SimPEG import ( - maps, - data, - utils, - simulation, - inverse_problem, - inversion, - optimization, - regularization, - data_misfit, - directives, - ) - from SimPEG.utils import io_utils - import numpy as np - import pandas as pd - import scipy.interpolate as interp + # Import the SimPEG functions + import discretize as ds import matplotlib as mpl import matplotlib.pyplot as plt - import math as math - - from dotenv import dotenv_values - from mpl_toolkits.axes_grid1.inset_locator import inset_axes - - from discretize.utils import mkvc, refine_tree_xyz, active_from_xyz - from discretize import TensorMesh, TreeMesh - from SimPEG.utils import plot2Ddata, model_builder, surface2ind_topo - from SimPEG import maps - from SimPEG.potential_fields import magnetics as mag + # Import the general modules + import numpy as np + import pandas as pd + from SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives) from SimPEG.potential_fields import gravity as grav - import os - import shutil - from vector_geology import SimpegHelper as SH + # Import the discretize module + from discretize.utils import active_from_xyz - #from get_cpt import get_cmap + # Import SimpegHelper for plotting and resampling data + from vector_geology import SimpegHelper as SH # Plot beautification - formatter = mpl.ticker.ScalarFormatter(useMathText=True) formatter.set_scientific(True) - formatter.set_powerlimits((0,0)) + formatter.set_powerlimits((0, 0)) mpl.rc("axes", titlesize=14, labelsize=12) mpl.rc("xtick", labelsize=12) mpl.rc("ytick", labelsize=12) - # Reproducible science - np.random.seed(518936) - @@ -94,34 +76,29 @@ We use the PGI module within SimPEG to invert gravity gradiometry data on a Tree +.. GENERATED FROM PYTHON SOURCE LINES 50-55 -.. GENERATED FROM PYTHON SOURCE LINES 70-71 +Step 1: Create Directories for Storing Model Iterations and Outputs +------------------------------------------------------------------- -Create directories for the model iterations and output +PGI is an iterative process, and it's useful to save the model at every iteration. This helps in preventing overfitting +and in adjusting the stopping criteria based on the model iterations. -.. GENERATED FROM PYTHON SOURCE LINES 71-92 +.. GENERATED FROM PYTHON SOURCE LINES 55-68 .. code-block:: default - # Name of the model - name = "Ireland_FTG_NonNeg" - # Names for the directories for the model iterations and output - path_to_mod_iterations = "./" + str(name) + "/Model Iterations" - path_to_output = "./" + str(name) + "/Output" + name = "Ireland_FTG" + path_to_mod_iterations = f"./{name}/Model Iterations" + path_to_output = f"./{name}/Output" - # Check if the model iterations directory exists and remove it if it does if os.path.exists(path_to_mod_iterations): shutil.rmtree(path_to_mod_iterations) - - # Create the model iterations directory. os.makedirs(path_to_mod_iterations) - # Check if the output directory exists and remove it if it does if os.path.exists(path_to_output): shutil.rmtree(path_to_output) - - # Create the output directory. os.makedirs(path_to_output) @@ -131,141 +108,157 @@ Create directories for the model iterations and output -.. GENERATED FROM PYTHON SOURCE LINES 93-98 +.. GENERATED FROM PYTHON SOURCE LINES 69-81 -Load the data -Read gravity and magnetic data from a csv file. -Inputs: -file_path: The path to the csv file. -header: Whether or not the csv file has a header row. +Step 2: Load the FTG-G Data +---------------------------- -.. GENERATED FROM PYTHON SOURCE LINES 98-120 +Full Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion. -.. code-block:: default +.. math:: + G = \left[\begin{array}{cc} + G_{xx} & G_{xy} & G_{xz}\\ + G_{xy} & G_{yy} & G_{yz}\\ + G_{xz} & G_{yz} & G_{zz} + \end{array}\right] +.. GENERATED FROM PYTHON SOURCE LINES 81-86 - config = dotenv_values() - file_path = config.get("PATH_TO_GRADIOMETRY") +.. code-block:: default - header = None - # Read the csv file into a pandas dataframe and then convert it to a numpy array. + config = dotenv_values() + file_path = config.get("PATH_TO_GRADIOMETRY") FTG_Data = pd.read_csv(file_path, delimiter=",").to_numpy() - # Add Topography - FTG_Data = np.c_[FTG_Data, np.zeros(np.shape(FTG_Data)[0])] - FTG_Data[:,[2, -1]] = FTG_Data[:,[-1, 2]] - - # Format after adding topography - # X, Y, Z, Tyy, Txz, Tyz, Txy, Txx - - ## % - ## Resample the data - # The ``SimpegHelper.pf_rs()`` will take in potential field data and resample it onto a new regular grid +.. GENERATED FROM PYTHON SOURCE LINES 87-92 +Step 3: Resample Data onto a Regular Grid +----------------------------------------- -.. GENERATED FROM PYTHON SOURCE LINES 121-122 +Resampling the data onto a regular grid aids in mesh generation and visualization. The `SimpegHelper.pf_rs()` function +resamples potential field data onto a new regular grid. -New sampling interval (Data Units) - -.. GENERATED FROM PYTHON SOURCE LINES 122-142 +.. GENERATED FROM PYTHON SOURCE LINES 92-105 .. code-block:: default - inc = 200 - - # Resample the data - [grav_new, nx_new, ny_new] = SH.pf_rs(FTG_Data, inc) - # Extract the gravity gradiometry data vectors - grav_vec = grav_new[:,3:] + inc = 150 # New sampling interval (ground units) + grav_new, nx_new, ny_new = SH.pf_rs( + FTG_Data, + inc, + bounds=[156000, 168000, 143000, 148500] + ) - # Negate the gravity data to the opposite sign to match the coordinate system. # NOTE: This step is necessary for real data since the convention followed by - # the SimPEG forward operator gravity lows over density highs. - # grav_vec = -grav_vec + # the SimPEG forward operator is the opposite of the general convention + grav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :] # Adjusting sign convention + inv_topo = grav_new[:, [0, 1, 2]] # Extracting topography - # Extract the topography for the active cells. - inv_topo = grav_new[:,[0,1,2]] - ## Visualize the data - #The ``SimpegHelper.plot_2D_data()`` function will plot the datasets using ``matplotlib``. +.. GENERATED FROM PYTHON SOURCE LINES 106-110 +Step 4: Visualize the Data +-------------------------- +The `SimpegHelper.plot_2D_data()` function is used to visualize the datasets with `matplotlib`. -.. GENERATED FROM PYTHON SOURCE LINES 143-144 - -Plot Gravity Data - -.. GENERATED FROM PYTHON SOURCE LINES 144-148 +.. GENERATED FROM PYTHON SOURCE LINES 110-130 .. code-block:: default - SH.plot_2D_data(np.c_[grav_new[:,[0,1,2]], grav_vec[:,-1]], [np.nanmin(grav_vec[:,-1]), np.nanmax(grav_vec[:,-1])], cmap="jet", path_to_output=path_to_output, name=name) - # Plot Topography Data - # SH.plot_2D_data(np.c_[inv_topo[:,[0,1,2]], inv_topo[:,2]], [np.nanmin(inv_topo[:,2]), np.nanmax(inv_topo[:,2])], which_data="Topo", cmap="magma", path_to_output=path_to_output, name=name) + SH.plot_2D_data( + data=np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]], + clim=[np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])], + cmap="jet", + which_data="FTG", + comp="xx", + path_to_output=path_to_output, + name=name + ) + SH.plot_2D_data( + data=np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]], + clim=[np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])], + cmap="terrain", + which_data="Topo", + comp="xx", + path_to_output=path_to_output, + name=name + ) -.. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_001.png - :alt: Gravity Anomaly - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_001.png - :class: sphx-glr-single-img +.. rst-class:: sphx-glr-horizontal + * + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_001.png + :alt: FTG ($G_{xx}$) + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_001.png + :class: sphx-glr-multi-img -.. GENERATED FROM PYTHON SOURCE LINES 149-150 + * -# Create a TreeMesh object to invert the data + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_002.png + :alt: Topography + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_002.png + :class: sphx-glr-multi-img -.. GENERATED FROM PYTHON SOURCE LINES 150-152 -.. code-block:: default - # Using the ``discretize.TreeMesh`` utility, create a 3D TreeMesh to invert the gravity gradiometry data. +.. GENERATED FROM PYTHON SOURCE LINES 131-144 +Step 5: Create a TensorMesh Object for Inversion +------------------------------------------------ +A 3D TensorMesh is created using the `discretize.TensorMesh` utility to invert the gravity gradiometry data. The mesh +is designed to have a constant cell size in the core region and expanding/contracting cells in the padding region. +.. math:: + z_{n+1} = r z_{n} +Where $r$ is a multiplicative factor (<1 for contracting, >1 for expanding cells). x and y +increments can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire +region is an area of interest). -.. GENERATED FROM PYTHON SOURCE LINES 153-193 +.. GENERATED FROM PYTHON SOURCE LINES 144-167 .. code-block:: default - dx = inc - dy = inc - dz = 10 - nz_core = 1 - nz_pad = 20 - fact = 1.1 - # Set the cell sizes to a constant value in the x and y directions and expanding in the z direction. + dx, dy, dz = inc, inc, 10 # Cell sizes + nz_core, nz_pad = 10, 18 # Number of cells in core and padding + fact = 1.1 # Factor for expanding/contracting cells + inv_hx = dx * np.ones(nx_new) inv_hy = dy * np.ones(ny_new) inv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)] - # Create the inverse tensor mesh. - inv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:,0]), np.min(inv_topo[:,1]), "C"]) + # Create the inverse tensor mesh + inv_mesh = ds.TensorMesh( + h=[inv_hx, inv_hy, inv_hz], + x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), "C"]) # Drape the topography over the mesh actv = active_from_xyz(inv_mesh, inv_topo) @@ -273,183 +266,120 @@ Plot Gravity Data actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv) nactv = int(actv.sum()) - # Make an x-y grid with the mesh cell centers - # xygrid = np.meshgrid(inv_mesh.cell_centers_x, inv_mesh.cell_centers_y) - # xygrid = np.reshape(xygrid, (2,-1)).T - - # Region Active Array (bool array with the spatial extent of the data) - # RegActvArr = np.empty((len(xygrid)), dtype=bool) - - # for i in range(len(xygrid)): - # Dist = np.sqrt(np.sum((inv_topo[:, [0, 1]] - xygrid[i,:])**2, axis=1)) - # MinDist = np.min(Dist) - # RegActvArr[i] = MinDist < 50 - - # Repeating the regional active array for every z (since we only carried out the calculations for x-y, which can be replicated in z) - # RegActvArr = np.tile(RegActvArr, len(inv_mesh.cell_centers_z)) - - # Intersection between Topo active cells and spatial active cells (nactv and actvMap are also updated) - actv = actv - nactv = int(actv.sum()) - actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv) - + # Print mesh details + print(inv_mesh) +.. rst-class:: sphx-glr-script-out -.. GENERATED FROM PYTHON SOURCE LINES 194-195 + .. code-block:: none -.. code-block:: default - inv_mesh - - - - - -.. raw:: html - -

- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
TensorMesh111,930 cells
MESH EXTENTCELL WIDTHFACTOR
dirnCminmaxminmaxmax
x78155,500.00171,100.00200.00200.001.00
y35142,500.00149,500.00200.00200.001.00
z41-635.02635.0210.0067.271.10
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- -.. GENERATED FROM PYTHON SOURCE LINES 196-197 - -# Visualize the mesh - -.. GENERATED FROM PYTHON SOURCE LINES 197-199 + TensorMesh: 137,862 cells -.. code-block:: default + MESH EXTENT CELL WIDTH FACTOR + dir nC min max min max max + --- --- --------------------------- ------------------ ------ + x 81 156,000.00 168,150.00 150.00 150.00 1.00 + y 37 143,000.00 148,550.00 150.00 150.00 1.00 + z 46 -551.59 551.59 10.00 55.60 1.10 - #The ``discretize.TreeMesh.plot_slice()`` utility will plot a slice of thee Tree Mesh object generated above. +.. GENERATED FROM PYTHON SOURCE LINES 168-172 +Step 6: Visualize the Mesh +-------------------------- +A slice of the Tensor Mesh object is visualized using the `discretize.TensorMesh.plot_slice()` utility. -.. GENERATED FROM PYTHON SOURCE LINES 200-202 +.. GENERATED FROM PYTHON SOURCE LINES 174-175 -Plotting a mesh slice Create a background model. -.. GENERATED FROM PYTHON SOURCE LINES 202-219 +.. GENERATED FROM PYTHON SOURCE LINES 175-195 .. code-block:: default bg = np.ones(nactv) mod = actvMap * bg - # Define the tensor mesh parameters. - props = dict(boxstyle='round,pad=1', facecolor='white') - # Plot the mesh slice. fig = plt.figure(figsize=(6, 6)) ax = fig.add_axes([0.1, 0.1, 0.8, 0.8]) - cplot = inv_mesh.plot_slice(mod, grid=False, ax=ax, normal="Y", ind=22) + cplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal="Y", ind=22) ax.set_title('Tensor Mesh Slice') ax.set_xlabel('x (m)') - ax.set_ylabel('y (m)') + ax.set_ylabel('z (m)') + ax.set_xlim([158000, 159000]) + ax.set_ylim([-500, 100]) ax.ticklabel_format(axis="both") ax.set_aspect('equal') - plt.savefig(path_to_output + "/" + name + "_TreeMeshSlice.pdf", bbox_inches="tight") + plt.savefig( + path_to_output + "/" + name + "_TreeMeshSlice.pdf", + bbox_inches="tight" + ) plt.show() -.. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_002.png + +.. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_003.png :alt: Tensor Mesh Slice - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_002.png + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_003.png :class: sphx-glr-single-img -.. GENERATED FROM PYTHON SOURCE LINES 220-223 +.. GENERATED FROM PYTHON SOURCE LINES 196-201 -# Set up the gravity inverse problem -%% - Create wires for the density model +Step 7: Set up the Gravity Inverse Problem +------------------------------------------ -.. GENERATED FROM PYTHON SOURCE LINES 223-266 +The gravity inverse problem is set up using SimPEG's `pf.gravity` module to generate a forward model based on the mesh +and receiver locations. + +.. GENERATED FROM PYTHON SOURCE LINES 204-205 + +Create wires for the density model + +.. GENERATED FROM PYTHON SOURCE LINES 205-256 .. code-block:: default wires = maps.Wires(("density", nactv)) - # Create a Density Mapping + # Create a density map density_map = actvMap * wires.density - # Create Identity Mapping + + # Create an identity map identity_map = maps.IdentityMap(nP=nactv) # Components of the data used as input - gravity_components = ["gyy", "gxz", "gyz", "gxy", "gxx"] + gravity_components = ["gxx", "gyy", "gxz", "gyz", "gxy"] - # Receiver locations - gravity_receiver_locations = grav_new[:, [0, 1, 2]] + # Receiver locations (25 meters above the topography) + gravity_receiver_locations = inv_topo + 120. # Create a gravity receiver list - gravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components) + gravity_receivers = pf.gravity.receivers.Point( + locations=gravity_receiver_locations, + components=gravity_components + ) gravity_receiver_list = [gravity_receivers] # Create a gravity source field - gravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list) + gravity_source_field = pf.gravity.sources.SourceField( + receiver_list=gravity_receiver_list + ) # Define the gravity survey gravity_survey = pf.gravity.survey.Survey(gravity_source_field) @@ -462,18 +392,20 @@ Create a background model. ind_active=actv ) - # Define uncertainties - # Maximum value of each component - # max_gravity = np.max(np.abs(grav_vec), axis=0) - # Standard deviation of each component - # std_gravity = 0.05 * max_gravity - # std_gravity = np.ones(np.shape(grav_vec)) * std_gravity - - # Create a gravity data object - gravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1) + # Create a gravity data object, with the relative errors and a standard noise floor + gravity_data = data.Data( + survey=gravity_survey, + dobs=grav_vec.flatten(), + noise_floor=5, + relative_error=0.1 + ) # Define the misfits associated with the gravity data - gravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem) + gravity_misfit = data_misfit.L2DataMisfit( + data=gravity_data, + simulation=gravity_problem + ) + @@ -481,14 +413,21 @@ Create a background model. -.. GENERATED FROM PYTHON SOURCE LINES 267-271 +.. GENERATED FROM PYTHON SOURCE LINES 257-264 -# Set up the Gaussian Mixture Model (GMM) Prior -%% - Defining the Gaussian Mixture Model (GMM) - Number of rock units and number of physical properties +Step 8: Setting up the Gaussian Mixture Model (GMM) Prior +--------------------------------------------------------- -.. GENERATED FROM PYTHON SOURCE LINES 271-317 +In PGI, the petrophysical data is used as a constraint in the form of a `Gaussian Mixture Model `_ (GMM). +A GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units (:math:`n`), +the petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below. +If you do have a petrophysical dataset, you can fit the GMM to said dataset using the ``gmmref.fit()`` method. + +.. GENERATED FROM PYTHON SOURCE LINES 267-268 + +Number of rock units and number of physical properties + +.. GENERATED FROM PYTHON SOURCE LINES 268-320 .. code-block:: default @@ -506,7 +445,8 @@ Create a background model. # Set the background density background_density = 0.0 - # Initialize the GMM fit with random samples, mesh size, and number of physical properties + # Initialize the GMM fit with random samples, mesh size, + # and number of physical properties gmmref.fit(np.random.randn(nactv, num_physical_props)) # Set the mean values of physical property contrasts for each rock unit @@ -523,6 +463,9 @@ Create a background model. density_variance = 8e-5 # Set the covariances of physical properties for each rock unit + # NOTE: Since we don't have petrophysical information for this example, + # we keep the covariances # same for every unit. The GMM will update + # itself during the inversion. gmmref.covariances_ = np.array( [ [[density_variance]], @@ -537,6 +480,7 @@ Create a background model. gmmref.compute_clusters_precisions() # Set the weights for each rock unit + # NOTE: This determines the size of each peak of the n-modal GMM gmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125] @@ -545,19 +489,19 @@ Create a background model. -.. GENERATED FROM PYTHON SOURCE LINES 318-319 -GMM Parameters +.. GENERATED FROM PYTHON SOURCE LINES 321-325 -.. GENERATED FROM PYTHON SOURCE LINES 319-334 +Plot the 1D-GMM +--------------- + +The initial density distribution is visualized. + +.. GENERATED FROM PYTHON SOURCE LINES 325-341 .. code-block:: default - means_init = gmmref.means_.flatten() - covariances_init = gmmref.covariances_.flatten() - weights_init = gmmref.weights_[0].flatten() - # Plot the 1D GMM fig = plt.figure(figsize=(6, 6)) ax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2) ax[0].set_xlabel(r"Density Contrast (g/cc)") @@ -565,57 +509,88 @@ GMM Parameters ax[0].get_legend().remove() ax[0].set_title(r"Initial DelRho Distribution") ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0)) - ax[0].set_aspect(1/20) - plt.savefig(path_to_output + "/" + name + "_Init_GMM.pdf", bbox_inches="tight") + ax[0].set_aspect(1 / 20) + plt.savefig( + path_to_output + "/" + name + "_Init_GMM.pdf", + bbox_inches="tight" + ) plt.show() + + .. rst-class:: sphx-glr-horizontal * - .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_003.png + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_004.png :alt: 01 gravity gradiometry inversion pgi - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_003.png + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_004.png :class: sphx-glr-multi-img * - .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_004.png + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_005.png :alt: Initial DelRho Distribution - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_004.png + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_005.png :class: sphx-glr-multi-img -.. GENERATED FROM PYTHON SOURCE LINES 335-338 +.. GENERATED FROM PYTHON SOURCE LINES 342-356 + +Step 9: Setting the Hyper-parameters and the Sensitivity Weights +----------------------------------------------------------------- -# Define the inversion parameters -%% - Initial Model +Every PGI is a set of three Maximum-A-Posteriori (`MAP `_) +problems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both +the FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise +the necessary weights for every voxel (as the contribution of every voxel is dependent on its depth from the surface). +The regularization smallness (:math:`\alpha_s`) and smoothness (:math:`\alpha_i,\ i = x, y, z`) are initialised here. Please check +`here `_ +for the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion. -.. GENERATED FROM PYTHON SOURCE LINES 338-436 +NOTE: The smoothness parameters (`alpha_i`) are static and hence need to be fine-tuned +through trial and error. + + +.. GENERATED FROM PYTHON SOURCE LINES 356-359 .. code-block:: default + initial_model = np.r_[background_density * np.ones(actvMap.nP)] - # Sensitivity weighting - # Compute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G) - # This is done by computing the square root of the sum of the squared elements of G for each cell, - # and then normalizing by the cell volumes and the maximum weight value. + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 360-364 + +Sensitivity weighting +Compute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G) +This is done by computing the square root of the sum of the squared elements of G for each cell, +and then normalizing by the cell volumes and the maximum weight value. + +.. GENERATED FROM PYTHON SOURCE LINES 366-394 + +.. code-block:: default + sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv]) sensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity) # Regularization multipliers smallness_multiplier = 1e-4 - smoothness_x_multiplier = 1e-1 - smoothness_y_multiplier = 1e-1 - smoothness_z_multiplier = 1e-1 + smoothness_x_multiplier = 4 + smoothness_y_multiplier = 4 + smoothness_z_multiplier = 0.25 # Create joint PGI regularization with smoothness regularization_term = regularization.PGI( @@ -628,16 +603,36 @@ GMM Parameters alpha_x=smoothness_x_multiplier, alpha_y=smoothness_y_multiplier, alpha_z=smoothness_z_multiplier, + # The second derivative smoothnesses are kept to be zero, since we + # don't want to include second derivatives in the objective function alpha_xx=0.0, alpha_yy=0.0, alpha_zz=0.0, weights_list=[sensitivity_weights_gravity] ) - # Directives for inversion - # Define the directives for the inversion process, based on Astic et al., (2021) - # Estimate smoothness multipliers + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 395-399 + +Step 10: Initialize the Directives +---------------------------------- + +Directives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver. + +.. GENERATED FROM PYTHON SOURCE LINES 402-403 + +Estimate smoothness multipliers + +.. GENERATED FROM PYTHON SOURCE LINES 403-475 + +.. code-block:: default + alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True) # Initialize beta and beta/alpha_s schedule @@ -653,28 +648,37 @@ GMM Parameters target_misfits = directives.MultiTargetMisfits(verbose=True) # Add reference model once stable - mref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True) + mref_in_smooth = directives.PGI_AddMrefInSmooth( + wait_till_stable=True, + verbose=True + ) # Update smallness parameters, keeping GMM fixed (L2 Approx of PGI) update_smallness_directive = directives.PGI_UpdateParameters( update_gmm=True, - kappa = 0, - nu = 0.5, - zeta = 0 + kappa=0, + nu=0.5, + zeta=0 ) # Update preconditioner update_preconditioner = directives.UpdatePreconditioner() # Save iteration results - save_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output) + save_iteration_directive = directives.SaveOutputEveryIteration( + name=name, + directory=path_to_output + ) # Save model iterations - save_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations) + save_model_directive = directives.SaveModelEveryIteration( + name=name, + directory=path_to_mod_iterations + ) # Optimization options for the inversion - lower_bound = np.r_[-1 * np.ones(actvMap.nP)] - upper_bound = np.r_[1 * np.ones(actvMap.nP)] + lower_bound = np.r_[-1.0 * np.ones(actvMap.nP)] + upper_bound = np.r_[1.0 * np.ones(actvMap.nP)] optimizer = optimization.ProjectedGNCG( maxIter=20, lower=lower_bound, @@ -707,21 +711,25 @@ GMM Parameters -.. GENERATED FROM PYTHON SOURCE LINES 437-440 -# Run the Inversion! -%% - Run the inversion +.. GENERATED FROM PYTHON SOURCE LINES 476-480 -.. GENERATED FROM PYTHON SOURCE LINES 440-441 +Step 11: Run the Inversion! +--------------------------- + +The inversion is executed using the initialized settings. + +.. GENERATED FROM PYTHON SOURCE LINES 480-483 .. code-block:: default + inverted_model = inversion_algo.run(initial_model) + .. rst-class:: sphx-glr-script-out .. code-block:: none @@ -732,138 +740,111 @@ GMM Parameters SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv. ***Done using the default solver SolverLU and no solver_opts.*** - Alpha scales: [13303597.345293514, 0.0, 13291291.128584256, 0.0, 96004.25899069471, 0.0] + Alpha scales: [410030855.8629278, 0.0, 423924847.8682445, 0.0, 221301.76852332294, 0.0] - SimPEG.SaveOutputEveryIteration will save your inversion progress as: '###-Ireland_FTG_NonNeg-2023-12-14-16-07.txt' - SimPEG.SaveModelEveryIteration will save your models as: './Ireland_FTG_NonNeg/Model Iterations/###-Ireland_FTG_NonNeg-2023-12-14-16-07.npy' + SimPEG.SaveOutputEveryIteration will save your inversion progress as: '###-Ireland_FTG-2024-01-26-10-56.txt' + SimPEG.SaveModelEveryIteration will save your models as: './Ireland_FTG/Model Iterations/###-Ireland_FTG-2024-01-26-10-56.npy' model has any nan: 0 =============================== Projected GNCG =============================== # beta phi_d phi_m f |proj(x-g)-x| LS Comment ----------------------------------------------------------------------------- x0 has any nan: 0 - 0 3.79e-08 1.59e+04 0.00e+00 1.59e+04 2.06e+02 0 - geophys. misfits: 6396.9 (target 6825.0 [True]) | smallness misfit: 31833.0 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [6396.9]; minimum progress targets: [12694.7] - Warming alpha_pgi to favor clustering: 0.00010669177094483827 - mref changed in 29837 places - 1 3.79e-08 6.40e+03 1.12e+10 6.82e+03 1.64e+02 0 - geophys. misfits: 3443.7 (target 6825.0 [True]) | smallness misfit: 32765.9 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3443.7]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.00021145111553427776 - mref changed in 14173 places - 2 3.79e-08 3.44e+03 1.09e+10 3.86e+03 6.78e+01 0 - geophys. misfits: 3443.1 (target 6825.0 [True]) | smallness misfit: 32692.5 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3443.1]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.0004191377804309684 - mref changed in 2336 places - 3 3.79e-08 3.44e+03 1.08e+10 3.85e+03 5.91e+01 2 Skip BFGS - geophys. misfits: 3434.6 (target 6825.0 [True]) | smallness misfit: 32301.7 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3434.6]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.0008328763975470134 - mref changed in 9637 places - 4 3.79e-08 3.43e+03 1.09e+10 3.85e+03 7.63e+01 0 - geophys. misfits: 3410.0 (target 6825.0 [True]) | smallness misfit: 32353.0 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3410.]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.001666997423849284 - mref changed in 2304 places - 5 3.79e-08 3.41e+03 1.08e+10 3.82e+03 6.27e+01 1 - geophys. misfits: 3405.8 (target 6825.0 [True]) | smallness misfit: 32305.6 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3405.8]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.003340537806642481 - mref changed in 548 places - 6 3.79e-08 3.41e+03 1.08e+10 3.82e+03 6.47e+01 3 Skip BFGS - geophys. misfits: 3394.3 (target 6825.0 [True]) | smallness misfit: 32227.0 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3394.3]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.006716901289141315 - mref changed in 853 places - 7 3.79e-08 3.39e+03 1.08e+10 3.80e+03 5.97e+01 2 Skip BFGS - geophys. misfits: 3379.7 (target 6825.0 [True]) | smallness misfit: 32130.3 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3379.7]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.013563981510234007 - mref changed in 2484 places - 8 3.79e-08 3.38e+03 1.08e+10 3.79e+03 4.83e+01 1 Skip BFGS - geophys. misfits: 3375.7 (target 6825.0 [True]) | smallness misfit: 32093.2 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3375.7]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.02742395715929332 - mref changed in 1765 places - 9 3.79e-08 3.38e+03 1.08e+10 3.79e+03 3.75e+01 1 Skip BFGS - geophys. misfits: 3375.5 (target 6825.0 [True]) | smallness misfit: 32073.9 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3375.5]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.05544992823765521 - mref changed in 451 places - 10 3.79e-08 3.38e+03 1.08e+10 3.79e+03 3.79e+01 2 - geophys. misfits: 3366.7 (target 6825.0 [True]) | smallness misfit: 32451.3 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3366.7]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.11240918288836876 - mref changed in 3181 places - 11 3.79e-08 3.37e+03 1.10e+10 3.78e+03 3.36e+01 0 Skip BFGS - geophys. misfits: 3365.8 (target 6825.0 [True]) | smallness misfit: 32609.1 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3365.8]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.22793964709919495 - mref changed in 332 places - 12 3.79e-08 3.37e+03 1.12e+10 3.79e+03 3.18e+01 2 - geophys. misfits: 3366.4 (target 6825.0 [True]) | smallness misfit: 33336.6 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3366.4]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.4621242063572519 - mref changed in 563 places - 13 3.79e-08 3.37e+03 1.16e+10 3.81e+03 2.79e+01 1 - geophys. misfits: 3366.1 (target 6825.0 [True]) | smallness misfit: 34792.5 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3366.1]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 0.9369936514939342 - mref changed in 471 places - 14 3.79e-08 3.37e+03 1.27e+10 3.85e+03 3.56e+01 1 - geophys. misfits: 3366.7 (target 6825.0 [True]) | smallness misfit: 34950.6 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3366.7]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 1.8994705672039287 - mref changed in 56 places - 15 3.79e-08 3.37e+03 1.49e+10 3.93e+03 5.04e+01 5 Skip BFGS - geophys. misfits: 3384.5 (target 6825.0 [True]) | smallness misfit: 37831.4 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3384.5]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 3.830382755121003 - mref changed in 122 places - 16 3.79e-08 3.38e+03 2.08e+10 4.17e+03 7.88e+01 1 - geophys. misfits: 3396.2 (target 6825.0 [True]) | smallness misfit: 35625.0 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3396.2]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 7.697450612448777 - mref changed in 431 places - 17 3.79e-08 3.40e+03 3.60e+10 4.76e+03 1.17e+02 0 - geophys. misfits: 3508.6 (target 6825.0 [True]) | smallness misfit: 30940.9 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3508.6]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 14.973218112151427 - mref changed in 299 places - 18 3.79e-08 3.51e+03 5.79e+10 5.70e+03 1.35e+02 0 - geophys. misfits: 3692.8 (target 6825.0 [True]) | smallness misfit: 29180.6 (target: 27300.0 [False]) - Beta cooling evaluation: progress: [3692.8]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 27.673539996945806 - mref changed in 231 places - 19 3.79e-08 3.69e+03 9.25e+10 7.20e+03 1.44e+02 0 - geophys. misfits: 4215.7 (target 6825.0 [True]) | smallness misfit: 26238.4 (target: 27300.0 [True]) + 0 5.50e-11 1.76e+04 0.00e+00 1.76e+04 2.82e+02 0 + geophys. misfits: 7363.5 (target 7492.5 [True]) | smallness misfit: 53932.6 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [7363.5]; minimum progress targets: [14114.6] + Warming alpha_pgi to favor clustering: 0.0001017520433220537 + mref changed in 46923 places + 1 5.50e-11 7.36e+03 2.93e+12 7.52e+03 2.40e+02 0 + geophys. misfits: 2878.4 (target 7492.5 [True]) | smallness misfit: 56886.0 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [2878.4]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.00026486260428953257 + mref changed in 24033 places + 2 5.50e-11 2.88e+03 2.82e+12 3.03e+03 1.76e+02 0 + geophys. misfits: 1902.3 (target 7492.5 [True]) | smallness misfit: 59150.1 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1902.3]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.0010432238094228118 + mref changed in 13728 places + 3 5.50e-11 1.90e+03 2.90e+12 2.06e+03 1.25e+02 0 Skip BFGS + geophys. misfits: 1377.6 (target 7492.5 [True]) | smallness misfit: 60406.7 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1377.6]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.005673861421188311 + mref changed in 9162 places + 4 5.50e-11 1.38e+03 2.90e+12 1.54e+03 8.08e+01 0 + geophys. misfits: 1370.3 (target 7492.5 [True]) | smallness misfit: 60168.8 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1370.3]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.031023047077501028 + mref changed in 1935 places + 5 5.50e-11 1.37e+03 2.88e+12 1.53e+03 1.12e+02 2 Skip BFGS + geophys. misfits: 1367.2 (target 7492.5 [True]) | smallness misfit: 60140.0 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1367.2]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.1700062887019205 + mref changed in 493 places + 6 5.50e-11 1.37e+03 2.88e+12 1.53e+03 1.15e+02 4 Skip BFGS + geophys. misfits: 1323.8 (target 7492.5 [True]) | smallness misfit: 60296.1 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1323.8]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.9622012648565086 + mref changed in 1397 places + 7 5.50e-11 1.32e+03 2.86e+12 1.48e+03 1.00e+02 2 Skip BFGS + geophys. misfits: 1245.7 (target 7492.5 [True]) | smallness misfit: 61068.6 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1245.7]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 5.787288226899011 + mref changed in 5580 places + 8 5.50e-11 1.25e+03 2.92e+12 1.41e+03 6.00e+01 0 + geophys. misfits: 1241.2 (target 7492.5 [True]) | smallness misfit: 61073.9 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1241.2]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 34.93527456181403 + mref changed in 510 places + 9 5.50e-11 1.24e+03 3.02e+12 1.41e+03 6.36e+01 3 Skip BFGS + geophys. misfits: 1234.0 (target 7492.5 [True]) | smallness misfit: 61269.3 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1234.]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 212.1246047516577 + mref changed in 306 places + 10 5.50e-11 1.23e+03 3.66e+12 1.44e+03 6.63e+01 3 + geophys. misfits: 1215.3 (target 7492.5 [True]) | smallness misfit: 68474.0 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1215.3]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 1307.8129600116547 + mref changed in 153 places + 11 5.50e-11 1.22e+03 8.38e+12 1.68e+03 5.77e+01 1 Skip BFGS + geophys. misfits: 1206.7 (target 7492.5 [True]) | smallness misfit: 67958.3 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1206.7]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 8120.550411338 + mref changed in 11 places + 12 5.50e-11 1.21e+03 4.07e+13 3.44e+03 2.13e+02 0 + geophys. misfits: 1463.9 (target 7492.5 [True]) | smallness misfit: 43152.9 (target: 44885.0 [True]) All targets have been reached - Beta cooling evaluation: progress: [4215.7]; minimum progress targets: [8190.] - Warming alpha_pgi to favor clustering: 44.80242420960896 - mref changed in 308 places + Beta cooling evaluation: progress: [1463.9]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 41561.559728045744 + mref changed in 31 places ------------------------- STOP! ------------------------- - 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5869e+03 - 0 : |xc-x_last| = 6.4899e+00 <= tolX*(1+|x0|) = 1.0000e-01 - 0 : |proj(x-g)-x| = 1.4333e+02 <= tolG = 1.0000e-01 - 0 : |proj(x-g)-x| = 1.4333e+02 <= 1e3*eps = 1.0000e-02 - 1 : maxIter = 20 <= iter = 20 + 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.7644e+03 + 0 : |xc-x_last| = 1.1471e+01 <= tolX*(1+|x0|) = 1.0000e-01 + 0 : |proj(x-g)-x| = 2.0926e+02 <= tolG = 1.0000e-01 + 0 : |proj(x-g)-x| = 2.0926e+02 <= 1e3*eps = 1.0000e-02 + 0 : maxIter = 20 <= iter = 13 ------------------------- DONE! ------------------------- -.. GENERATED FROM PYTHON SOURCE LINES 442-470 +.. GENERATED FROM PYTHON SOURCE LINES 484-488 + +Step 12: Visualize the 3D Model +------------------------------- + +The inverted 3D model is visualized using slices in different orientations. + +.. GENERATED FROM PYTHON SOURCE LINES 488-532 .. code-block:: default + set = 1 save_plots = True # Indices of the depth sections - ind_plot_x = int(len(inv_mesh.cell_centers_x)/6) + 10 - ind_plot_y = int(len(inv_mesh.cell_centers_y)/2) - ind_plot_z = int(len(inv_mesh.cell_centers_z)/2) - 17 + ind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25 + ind_plot_y = int(len(inv_mesh.cell_centers_y) / 2) + ind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15 ind_plot = [ind_plot_x, ind_plot_y, ind_plot_z] @@ -874,17 +855,32 @@ GMM Parameters # Plot Density Contrast Model (Z) normal = "Z" model_to_plot = inverted_density_model - SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + SH.plot_model_slice( + mesh=inv_mesh, + ind_active=actv, + model=model_to_plot, + normal= normal, + ind_plot_arr=ind_plot, + clim=[-1.0, 1.0], + set=set, sec_loc=True, + gdlines=True, + which_prop="Den", + cmap="Spectral", + save_plt=save_plots, + path_to_output=path_to_output, + name=name + ) # Plot Inverted Model Slices (Y) normal = "X" model_to_plot = inverted_density_model - SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) # Plot Inverted Model Slices (X) normal = "Y" model_to_plot = inverted_density_model - SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + @@ -893,47 +889,104 @@ GMM Parameters * - .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_005.png - :alt: Inverted Den Model Slice at Z = -4.26E+02 m - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_005.png + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_006.png + :alt: Inverted Den Model Slice at Z = -2.12E+02 m + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_006.png :class: sphx-glr-multi-img * - .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_006.png - :alt: Inverted Den Model Slice at X = 1.60E+05 m - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_006.png + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_007.png + :alt: Inverted Den Model Slice at X = 1.62E+05 m + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_007.png :class: sphx-glr-multi-img * - .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_007.png + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_008.png :alt: Inverted Den Model Slice at Y = 1.46E+05 m - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_007.png + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_008.png :class: sphx-glr-multi-img -.. GENERATED FROM PYTHON SOURCE LINES 471-472 +.. GENERATED FROM PYTHON SOURCE LINES 533-537 + +Step 13: Visualize the Quasi-Geological Model +--------------------------------------------- -Updated GMM Parameters +The quasi-geological model (as given by `Li et al., 2019 `_) is visualized, showing different rock units classified by the GMM. -.. GENERATED FROM PYTHON SOURCE LINES 472-490 +.. GENERATED FROM PYTHON SOURCE LINES 539-540 + +Plot Density Contrast Model (Z) + +.. GENERATED FROM PYTHON SOURCE LINES 540-554 .. code-block:: default - learned_mu = regularization_term.objfcts[0].gmm.means_.flatten() - learned_sigma = regularization_term.objfcts[0].gmm.covariances_.flatten() - learned_pi = regularization_term.objfcts[0].gmm.weights_[0].flatten() + normal = "Z" + model_to_plot = quasi_geology_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + + # Plot Inverted Model Slices (Y) + normal = "X" + model_to_plot = quasi_geology_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + + # Plot Inverted Model Slices (X) + normal = "Y" + model_to_plot = quasi_geology_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_009.png + :alt: Quasi-Geological Model Slice at Z = -2.12E+02 m + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_009.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_010.png + :alt: Quasi-Geological Model Slice at X = 1.62E+05 m + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_010.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_011.png + :alt: Quasi-Geological Model Slice at Y = 1.46E+05 m + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_011.png + :class: sphx-glr-multi-img + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 555-559 + +Step 14: Plot the Updated 1D-GMM +-------------------------------- + +The updated GMM, learned through the inversion, is visualized. + +.. GENERATED FROM PYTHON SOURCE LINES 559-571 + +.. code-block:: default - # Regularization Parameters - final_alpha_pgi = regularization_term.alpha_pgi # Plot the learned GMM - fig = plt.figure(figsize=(6,6)) - ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.5) + fig = plt.figure(figsize=(6, 6)) + ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4) ax[0].hist(inverted_density_model[actv], density=True, bins=1000) ax[0].set_xlabel(r"Density contrast (g/cc)") ax[0].set_ylabel(r"Probability Density Values") @@ -950,30 +1003,26 @@ Updated GMM Parameters * - .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_008.png + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_012.png :alt: 01 gravity gradiometry inversion pgi - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_008.png + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_012.png :class: sphx-glr-multi-img * - .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_009.png + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_013.png :alt: Learned DelRho Distribution - :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_009.png + :srcset: /examples/03_forward_engines/images/sphx_glr_01_gravity_gradiometry_inversion_pgi_013.png :class: sphx-glr-multi-img -.. GENERATED FROM PYTHON SOURCE LINES 491-491 - -sphinx_gallery_thumbnail_number = -4 - .. rst-class:: sphx-glr-timing - **Total running time of the script:** ( 9 minutes 3.455 seconds) + **Total running time of the script:** ( 10 minutes 14.392 seconds) .. _sphx_glr_download_examples_03_forward_engines_01_gravity_gradiometry_inversion_pgi.py: diff --git a/_sources/examples/03_forward_engines/03_HSI_To_Petrophysics.rst.txt b/_sources/examples/03_forward_engines/03_HSI_To_Petrophysics.rst.txt new file mode 100644 index 0000000..d407cb0 --- /dev/null +++ b/_sources/examples/03_forward_engines/03_HSI_To_Petrophysics.rst.txt @@ -0,0 +1,321 @@ + +.. DO NOT EDIT. +.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. +.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: +.. "examples/03_forward_engines/03_HSI_To_Petrophysics.py" +.. LINE NUMBERS ARE GIVEN BELOW. + +.. only:: html + + .. note:: + :class: sphx-glr-download-link-note + + :ref:`Go to the end ` + to download the full example code + +.. rst-class:: sphx-glr-example-title + +.. _sphx_glr_examples_03_forward_engines_03_HSI_To_Petrophysics.py: + + +.. _example_predicting_pwave_velocity: + +Predicting P-Wave Velocity from Hyperspectral Data +================================================== + +This example demonstrates the prediction of P-Wave velocities from hyperspectral data using machine learning techniques. Hyperspectral data provides detailed information on rock mineralogy, which influences its petrophysical properties. This script aims to explore and quantify the relationship between spectral features and various petrophysical properties, specifically focusing on P-Wave velocity. + +.. GENERATED FROM PYTHON SOURCE LINES 11-15 + +Importing Necessary Libraries +----------------------------- + +First, we import the required Python packages. We use `dotenv` to manage environment variables, `numpy` and `sklearn` for numerical operations and machine learning, and custom packages `hklearn` and `hycore` for handling hyperspectral data and machine learning workflows. + +.. GENERATED FROM PYTHON SOURCE LINES 15-28 + +.. code-block:: default + + + from dotenv import dotenv_values + import numpy as np + import sklearn + from sklearn.linear_model import LinearRegression + from sklearn.neural_network import MLPRegressor + from tqdm import tqdm + + # Custom packages for handling hyperspectral data + # Note: As of January 2024, these packages are not publicly available. Please contact the author for more information. + from vector_geology.Packages import hklearn + from vector_geology.Packages import hycore + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 29-33 + +Loading Data with `hycore.Shed` +------------------------------- + +The `hycore` package, which organizes hyperspectral data in a structure called Shed, is used here. A Shed makes it convenient to categorize and store hyperspectral data. We use HyLibraries (a data structure within `hylite` for storing spectral libraries) within a Shed to store the hyperspectral data. The corresponding P-Wave velocity measurements are stored as numpy arrays. + +.. GENERATED FROM PYTHON SOURCE LINES 33-53 + +.. code-block:: default + + + # Load the train and test sheds + config = dotenv_values() + path_to_train = config.get("PATH_TO_HSI_TRAINING") + path_to_test = config.get("PATH_TO_HSI_TESTING") + + train_shed = hycore.loadShed(path_to_train) + test_shed = hycore.loadShed(path_to_test) + + # Load the train and test spectral libraries and P-Wave velocities + train_fx50 = train_shed.results.FX50Lib + train_fenix = train_shed.results.FENIXLib + train_lwir = train_shed.results.LWIRLib + train_vp = train_shed.results.Vp + + test_fx50 = test_shed.results.FX50Lib + test_fenix = test_shed.results.FENIXLib + test_lwir = test_shed.results.LWIRLib + test_vp = test_shed.results.Vp + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 54-58 + +Creating a `hklearn.Stack` Object +--------------------------------- + +The `hklearn` package, designed to handle hyperspectral data, is used to organize spectral libraries into a structure called Stack. A Stack can manage multiple spectral libraries and integrate data from multiple sensors. We perform hull corrections on the spectra to enhance features of interest. + +.. GENERATED FROM PYTHON SOURCE LINES 58-74 + +.. code-block:: default + + + # Create Stack objects and apply hull correction to the spectra + hsi_train = hklearn.Stack( + names=['FENIX', 'FX50', 'LWIR'], + data=[train_fenix, train_fx50, train_lwir]).hc( + ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)}, + hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'} + ) + + hsi_test = hklearn.Stack( + names=['FENIX', 'FX50', 'LWIR'], + data=[test_fenix, test_fx50, test_lwir]).hc( + ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)}, + hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'} + ) + + + + + +.. rst-class:: sphx-glr-script-out + + .. code-block:: none + + Removing hull: 0%| | 0/3771 [00:00 + + + + + + + + + + + + + + + + + + + + + + + + +
 train set5-fold CVtest set
Linear0.6860.6620.614
MLP0.8760.8220.695
+ + +
+
+ + +.. rst-class:: sphx-glr-timing + + **Total running time of the script:** ( 0 minutes 34.561 seconds) + + +.. _sphx_glr_download_examples_03_forward_engines_03_HSI_To_Petrophysics.py: + +.. only:: html + + .. container:: sphx-glr-footer sphx-glr-footer-example + + + + + .. container:: sphx-glr-download sphx-glr-download-python + + :download:`Download Python source code: 03_HSI_To_Petrophysics.py <03_HSI_To_Petrophysics.py>` + + .. container:: sphx-glr-download sphx-glr-download-jupyter + + :download:`Download Jupyter notebook: 03_HSI_To_Petrophysics.ipynb <03_HSI_To_Petrophysics.ipynb>` + + +.. only:: html + + .. rst-class:: sphx-glr-signature + + `Gallery generated by Sphinx-Gallery `_ diff --git a/_sources/examples/03_forward_engines/04_FTG_PGI.rst.txt b/_sources/examples/03_forward_engines/04_FTG_PGI.rst.txt new file mode 100644 index 0000000..8c9f784 --- /dev/null +++ b/_sources/examples/03_forward_engines/04_FTG_PGI.rst.txt @@ -0,0 +1,951 @@ + +.. DO NOT EDIT. +.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. +.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: +.. "examples/03_forward_engines/04_FTG_PGI.py" +.. LINE NUMBERS ARE GIVEN BELOW. + +.. only:: html + + .. note:: + :class: sphx-glr-download-link-note + + :ref:`Go to the end ` + to download the full example code + +.. rst-class:: sphx-glr-example-title + +.. _sphx_glr_examples_03_forward_engines_04_FTG_PGI.py: + + +.. _example_inversion_ftg_data: + +Inversion of Full Tensor Gravity Gradiometry Data +================================================= + +This example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, measured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include: + +- Loading necessary modules and data +- Generating a mesh +- Initializing the Gaussian Mixture Model +- Tuning hyper-parameters +- Visualizing the results + +The process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example notebook from SimPEG's documentation (see [SimPEG docs](https://docs.simpeg.xyz/index.html) for more information). + +.. GENERATED FROM PYTHON SOURCE LINES 18-48 + +.. code-block:: default + + + import os + import shutil + from dotenv import dotenv_values + + import SimPEG.potential_fields as pf + # Import the SimPEG functions + import discretize as ds + import matplotlib as mpl + import matplotlib.pyplot as plt + # Import the general modules + import numpy as np + import pandas as pd + from SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives) + from SimPEG.potential_fields import gravity as grav + # Import the discretize module + from discretize.utils import active_from_xyz + + # Import SimpegHelper for plotting and resampling data + from vector_geology import SimpegHelper as SH + + # Plot beautification + formatter = mpl.ticker.ScalarFormatter(useMathText=True) + formatter.set_scientific(True) + formatter.set_powerlimits((0, 0)) + + mpl.rc("axes", titlesize=14, labelsize=12) + mpl.rc("xtick", labelsize=12) + mpl.rc("ytick", labelsize=12) + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 49-53 + +Step 1: Create Directories for Storing Model Iterations and Outputs +------------------------------------------------------------------- + +PGI is an iterative process, and it's useful to save the model at every iteration. This helps in preventing overfitting and in adjusting the stopping criteria based on the model iterations. + +.. GENERATED FROM PYTHON SOURCE LINES 53-66 + +.. code-block:: default + + + name = "Ireland_FTG" + path_to_mod_iterations = f"./{name}/Model Iterations" + path_to_output = f"./{name}/Output" + + if os.path.exists(path_to_mod_iterations): + shutil.rmtree(path_to_mod_iterations) + os.makedirs(path_to_mod_iterations) + + if os.path.exists(path_to_output): + shutil.rmtree(path_to_output) + os.makedirs(path_to_output) + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 67-78 + +Step 2: Load the FTG-G Data +---------------------------- + +Full Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion. +$$ +G = \left[\begin{array}{cc} +G_{xx} & G_{xy} & G_{xz}\\ +G_{xy} & G_{yy} & G_{yz}\\ +G_{xz} & G_{yz} & G_{zz} +\end{array}\right] +$$ + +.. GENERATED FROM PYTHON SOURCE LINES 78-83 + +.. code-block:: default + + + config = dotenv_values() + file_path = config.get("PATH_TO_GRADIOMETRY") + FTG_Data = pd.read_csv(file_path, delimiter=",").to_numpy() + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 84-88 + +Step 3: Resample Data onto a Regular Grid +----------------------------------------- + +Resampling the data onto a regular grid aids in mesh generation and visualization. The `SimpegHelper.pf_rs()` function resamples potential field data onto a new regular grid. + +.. GENERATED FROM PYTHON SOURCE LINES 88-97 + +.. code-block:: default + + + inc = 150 # New sampling interval (ground units) + grav_new, nx_new, ny_new = SH.pf_rs(FTG_Data, inc, bounds=[156000, 168000, 143000, 148500]) + + # NOTE: This step is necessary for real data since the convention followed by + # the SimPEG forward operator is the opposite of the general convention + grav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :] # Adjusting sign convention + inv_topo = grav_new[:, [0, 1, 2]] # Extracting topography + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 98-102 + +Step 4: Visualize the Data +-------------------------- + +The `SimpegHelper.plot_2D_data()` function is used to visualize the datasets with `matplotlib`. + +.. GENERATED FROM PYTHON SOURCE LINES 102-106 + +.. code-block:: default + + + SH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]], [np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])], cmap="jet", which_data="FTG", comp="xx", path_to_output=path_to_output, name=name) + SH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]], [np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])], cmap="terrain", which_data="Topo", comp="xx", path_to_output=path_to_output, name=name) + + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_001.png + :alt: FTG ($G_{xx}$) + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_001.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_002.png + :alt: Topography + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_002.png + :class: sphx-glr-multi-img + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 107-117 + +Step 5: Create a TensorMesh Object for Inversion +----------------------------------------------- + +A 3D TensorMesh is created using the `discretize.TensorMesh` utility to invert the gravity gradiometry data. The mesh is designed to have a constant cell size in the core region and expanding/contracting cells in the padding region. + +$$ +z_{n+1} = r z_{n} +$$ + +Where $r$ is a multiplicative factor ($<1$ for contracting, $>1$ for expanding cells). $x$ and $y$ increments can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire region is an area of interest). + +.. GENERATED FROM PYTHON SOURCE LINES 117-139 + +.. code-block:: default + + + dx, dy, dz = inc, inc, 10 # Cell sizes + nz_core, nz_pad = 10, 18 # Number of cells in core and padding + fact = 1.1 # Factor for expanding/contracting cells + + inv_hx = dx * np.ones(nx_new) + inv_hy = dy * np.ones(ny_new) + inv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)] + + # Create the inverse tensor mesh + inv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), "C"]) + + # Drape the topography over the mesh + actv = active_from_xyz(inv_mesh, inv_topo) + ndv = np.nan + actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv) + nactv = int(actv.sum()) + + # Print mesh details + print(inv_mesh) + + + + + + +.. rst-class:: sphx-glr-script-out + + .. code-block:: none + + + TensorMesh: 137,862 cells + + MESH EXTENT CELL WIDTH FACTOR + dir nC min max min max max + --- --- --------------------------- ------------------ ------ + x 81 156,000.00 168,150.00 150.00 150.00 1.00 + y 37 143,000.00 148,550.00 150.00 150.00 1.00 + z 46 -551.59 551.59 10.00 55.60 1.10 + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 140-144 + +Step 6: Visualize the Mesh +-------------------------- + +A slice of the Tensor Mesh object is visualized using the `discretize.TensorMesh.plot_slice()` utility. + +.. GENERATED FROM PYTHON SOURCE LINES 146-147 + +Create a background model. + +.. GENERATED FROM PYTHON SOURCE LINES 147-164 + +.. code-block:: default + + bg = np.ones(nactv) + mod = actvMap * bg + + # Plot the mesh slice. + fig = plt.figure(figsize=(6, 6)) + ax = fig.add_axes([0.1, 0.1, 0.8, 0.8]) + cplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal="Y", ind=22) + ax.set_title('Tensor Mesh Slice') + ax.set_xlabel('x (m)') + ax.set_ylabel('z (m)') + ax.set_xlim([158000, 159000]) + ax.set_ylim([-500, 100]) + ax.ticklabel_format(axis="both") + ax.set_aspect('equal') + plt.savefig(path_to_output + "/" + name + "_TreeMeshSlice.pdf", bbox_inches="tight") + plt.show() + + + + +.. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_003.png + :alt: Tensor Mesh Slice + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_003.png + :class: sphx-glr-single-img + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 165-169 + +Step 7: Set up the Gravity Inverse Problem +------------------------------------------ + +The gravity inverse problem is set up using SimPEG's `pf.gravity` module to generate a forward model based on the mesh and receiver locations. + +.. GENERATED FROM PYTHON SOURCE LINES 172-173 + +Create wires for the density model + +.. GENERATED FROM PYTHON SOURCE LINES 173-211 + +.. code-block:: default + + wires = maps.Wires(("density", nactv)) + + # Create a density map + density_map = actvMap * wires.density + + # Create an identity map + identity_map = maps.IdentityMap(nP=nactv) + + # Components of the data used as input + gravity_components = ["gxx", "gyy", "gxz", "gyz", "gxy"] + + # Receiver locations (25 meters above the topography) + gravity_receiver_locations = inv_topo + 120. + + # Create a gravity receiver list + gravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components) + gravity_receiver_list = [gravity_receivers] + + # Create a gravity source field + gravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list) + + # Define the gravity survey + gravity_survey = pf.gravity.survey.Survey(gravity_source_field) + + # Set up the gravity simulation problem + gravity_problem = grav.simulation.Simulation3DIntegral( + inv_mesh, + survey=gravity_survey, + rhoMap=wires.density, + ind_active=actv + ) + + # Create a gravity data object, with the relative errors and a standard noise floor + gravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1) + + # Define the misfits associated with the gravity data + gravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem) + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 212-216 + +Step 8: Setting up the Gaussian Mixture Model (GMM) Prior +-------------------------------------------------------- + +In PGI, the petrophysical data is used as a constraint in the form of a [Gaussian Mixture Model](https://scikit-learn.org/stable/modules/mixture.html). A GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units ($n$), the petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below. If you do have a petrophysical dataset, you can fit the GMM to said dataset using the ``gmmref.fit()`` method. + +.. GENERATED FROM PYTHON SOURCE LINES 220-221 + +Number of rock units and number of physical properties + +.. GENERATED FROM PYTHON SOURCE LINES 221-271 + +.. code-block:: default + + num_rock_units = 5 + num_physical_props = 1 + + # Create a weighted Gaussian mixture model with specified parameters + gmmref = utils.WeightedGaussianMixture( + n_components=num_rock_units, + mesh=inv_mesh, + actv=actv, + covariance_type="full", + ) + + # Set the background density + background_density = 0.0 + + # Initialize the GMM fit with random samples, mesh size, and number of physical properties + gmmref.fit(np.random.randn(nactv, num_physical_props)) + + # Set the mean values of physical property contrasts for each rock unit + # One value (density) for each rock unit + gmmref.means_ = np.c_[ + [-0.4], + [-0.2], + [0.0], + [0.2], + [0.4], + ].T + + # Set the original variance for density + density_variance = 8e-5 + + # Set the covariances of physical properties for each rock unit + # NOTE: Since we don't have petrophysical information for this example, we keep the covariances + # same for every unit. The GMM will update itself during the inversion. + gmmref.covariances_ = np.array( + [ + [[density_variance]], + [[density_variance]], + [[density_variance]], + [[density_variance]], + [[density_variance]] + ] + ) + + # Compute the precision (inverse covariance) of each cluster + gmmref.compute_clusters_precisions() + + # Set the weights for each rock unit + # NOTE: This determines the size of each peak of the n-modal GMM + gmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125] + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 272-276 + +Plot the 1D-GMM +--------------- + +The initial density distribution is visualized. + +.. GENERATED FROM PYTHON SOURCE LINES 276-288 + +.. code-block:: default + + + fig = plt.figure(figsize=(6, 6)) + ax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2) + ax[0].set_xlabel(r"Density Contrast (g/cc)") + ax[0].set_ylabel(r"Probability Density Values") + ax[0].get_legend().remove() + ax[0].set_title(r"Initial DelRho Distribution") + ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0)) + ax[0].set_aspect(1 / 20) + plt.savefig(path_to_output + "/" + name + "_Init_GMM.pdf", bbox_inches="tight") + plt.show() + + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_004.png + :alt: 04 FTG PGI + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_004.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_005.png + :alt: Initial DelRho Distribution + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_005.png + :class: sphx-glr-multi-img + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 289-296 + +Step 9: Setting the Hyper-parameters and the Sensitivity Weights +----------------------------------------------------------------- + +Every PGI is a set of three Maximum-A-Posteriori ([MAP](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation)) problems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both the FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise the necessary weights for every voxel (as the contribution of every voxel is dependent on it's depth from the surface). The regularization smallness ($\alpha_s$) and smoothness ($\alpha_i,\ i = x, y, z$) are initialised here. Please check [here](https://giftoolscookbook.readthedocs.io/en/latest/content/fundamentals/index.html) for the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion. + + NOTE : The smoothness parameters ($\alpha_i$) are static and hence need to be fine-tuned through trial and error. + + +.. GENERATED FROM PYTHON SOURCE LINES 296-299 + +.. code-block:: default + + + initial_model = np.r_[background_density * np.ones(actvMap.nP)] + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 300-304 + +Sensitivity weighting +Compute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G) +This is done by computing the square root of the sum of the squared elements of G for each cell, +and then normalizing by the cell volumes and the maximum weight value. + +.. GENERATED FROM PYTHON SOURCE LINES 306-334 + +.. code-block:: default + + sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv]) + sensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity) + + # Regularization multipliers + smallness_multiplier = 1e-4 + smoothness_x_multiplier = 4 + smoothness_y_multiplier = 4 + smoothness_z_multiplier = 0.25 + + # Create joint PGI regularization with smoothness + regularization_term = regularization.PGI( + gmmref=gmmref, + mesh=inv_mesh, + wiresmap=wires, + maplist=[identity_map], + active_cells=actv, + alpha_pgi=smallness_multiplier, + alpha_x=smoothness_x_multiplier, + alpha_y=smoothness_y_multiplier, + alpha_z=smoothness_z_multiplier, + # The second derivative smoothnesses are kept to be zero, since we + # don't want to include second derivatives in the objective function + alpha_xx=0.0, + alpha_yy=0.0, + alpha_zz=0.0, + weights_list=[sensitivity_weights_gravity] + ) + + + + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 335-339 + +Step 10: Initialize the Directives +---------------------------------- + +Directives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver. + +.. GENERATED FROM PYTHON SOURCE LINES 342-343 + +Estimate smoothness multipliers + +.. GENERATED FROM PYTHON SOURCE LINES 343-413 + +.. code-block:: default + + alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True) + + # Initialize beta and beta/alpha_s schedule + beta_directive = directives.BetaEstimate_ByEig(beta0_ratio=1e-2) + beta_schedule = directives.PGI_BetaAlphaSchedule( + coolingFactor=16.0, + tolerance=0.2, + progress=0.2, + verbose=True, + ) + + # Define target misfits for geophysical and petrophysical data + target_misfits = directives.MultiTargetMisfits(verbose=True) + + # Add reference model once stable + mref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True) + + # Update smallness parameters, keeping GMM fixed (L2 Approx of PGI) + update_smallness_directive = directives.PGI_UpdateParameters( + update_gmm=True, + kappa=0, + nu=0.5, + zeta=0 + ) + + # Update preconditioner + update_preconditioner = directives.UpdatePreconditioner() + + # Save iteration results + save_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output) + + # Save model iterations + save_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations) + + # Optimization options for the inversion + lower_bound = np.r_[-1.0 * np.ones(actvMap.nP)] + upper_bound = np.r_[1.0 * np.ones(actvMap.nP)] + optimizer = optimization.ProjectedGNCG( + maxIter=20, + lower=lower_bound, + upper=upper_bound, + maxIterLS=20, + maxIterCG=100, + tolCG=1e-4, + ) + + # Inverse problem setup + inverse_prob = inverse_problem.BaseInvProblem(gravity_misfit, regularization_term, optimizer) + inversion_algo = inversion.BaseInversion( + inverse_prob, + directiveList=[ + alphas_directive, + beta_directive, + update_smallness_directive, + target_misfits, + beta_schedule, + mref_in_smooth, + update_preconditioner, + save_iteration_directive, + save_model_directive + ], + ) + + # Step 11: Run the Inversion! + # --------------------------- + # + # The inversion is executed using the initialized settings. + + inverted_model = inversion_algo.run(initial_model) + + + + + +.. rst-class:: sphx-glr-script-out + + .. code-block:: none + + SimPEG.InvProblem will set Regularization.reference_model to m0. + SimPEG.InvProblem will set Regularization.reference_model to m0. + + SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv. + ***Done using the default solver SolverLU and no solver_opts.*** + + Alpha scales: [418083224.1567727, 0.0, 430664170.54921407, 0.0, 241249.22858700654, 0.0] + + SimPEG.SaveOutputEveryIteration will save your inversion progress as: '###-Ireland_FTG-2024-01-25-16-16.txt' + SimPEG.SaveModelEveryIteration will save your models as: './Ireland_FTG/Model Iterations/###-Ireland_FTG-2024-01-25-16-16.npy' + model has any nan: 0 + =============================== Projected GNCG =============================== + # beta phi_d phi_m f |proj(x-g)-x| LS Comment + ----------------------------------------------------------------------------- + x0 has any nan: 0 + 0 5.49e-11 1.76e+04 0.00e+00 1.76e+04 2.82e+02 0 + geophys. misfits: 7385.5 (target 7492.5 [True]) | smallness misfit: 53905.8 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [7385.5]; minimum progress targets: [14114.6] + Warming alpha_pgi to favor clustering: 0.00010144902120428576 + mref changed in 46912 places + 1 5.49e-11 7.39e+03 2.95e+12 7.55e+03 2.40e+02 0 + geophys. misfits: 2841.6 (target 7492.5 [True]) | smallness misfit: 56915.4 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [2841.6]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.0002674939266088537 + mref changed in 23999 places + 2 5.49e-11 2.84e+03 2.84e+12 3.00e+03 1.75e+02 0 + geophys. misfits: 1907.3 (target 7492.5 [True]) | smallness misfit: 59165.4 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1907.3]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.0010507772902925534 + mref changed in 13880 places + 3 5.49e-11 1.91e+03 2.93e+12 2.07e+03 1.26e+02 0 Skip BFGS + geophys. misfits: 1373.7 (target 7492.5 [True]) | smallness misfit: 60358.7 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1373.7]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.005731251890475955 + mref changed in 8975 places + 4 5.49e-11 1.37e+03 2.93e+12 1.53e+03 8.01e+01 0 + geophys. misfits: 1371.4 (target 7492.5 [True]) | smallness misfit: 60091.0 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1371.4]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.031312537491235436 + mref changed in 1888 places + 5 5.49e-11 1.37e+03 2.90e+12 1.53e+03 1.11e+02 2 Skip BFGS + geophys. misfits: 1367.5 (target 7492.5 [True]) | smallness misfit: 60077.9 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1367.5]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.17156327747832578 + mref changed in 534 places + 6 5.49e-11 1.37e+03 2.90e+12 1.53e+03 1.14e+02 4 Skip BFGS + geophys. misfits: 1322.5 (target 7492.5 [True]) | smallness misfit: 60252.3 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1322.5]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 0.9719405715672758 + mref changed in 1387 places + 7 5.49e-11 1.32e+03 2.89e+12 1.48e+03 9.98e+01 2 Skip BFGS + geophys. misfits: 1247.2 (target 7492.5 [True]) | smallness misfit: 61007.7 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1247.2]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 5.838731886914972 + mref changed in 5572 places + 8 5.49e-11 1.25e+03 2.94e+12 1.41e+03 5.79e+01 0 + geophys. misfits: 1242.3 (target 7492.5 [True]) | smallness misfit: 61012.7 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1242.3]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 35.21337400407953 + mref changed in 503 places + 9 5.49e-11 1.24e+03 3.04e+12 1.41e+03 6.17e+01 3 Skip BFGS + geophys. misfits: 1241.1 (target 7492.5 [True]) | smallness misfit: 61724.2 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1241.1]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 212.57758849285366 + mref changed in 381 places + 10 5.49e-11 1.24e+03 3.69e+12 1.44e+03 7.45e+01 2 + geophys. misfits: 1222.1 (target 7492.5 [True]) | smallness misfit: 68747.8 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1222.1]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 1303.254486428423 + mref changed in 151 places + 11 5.49e-11 1.22e+03 8.42e+12 1.68e+03 6.22e+01 1 + geophys. misfits: 1210.3 (target 7492.5 [True]) | smallness misfit: 67778.5 (target: 44885.0 [False]) + Beta cooling evaluation: progress: [1210.3]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 8067.926413603839 + mref changed in 11 places + 12 5.49e-11 1.21e+03 4.04e+13 3.43e+03 2.13e+02 0 + geophys. misfits: 1464.6 (target 7492.5 [True]) | smallness misfit: 42906.7 (target: 44885.0 [True]) + All targets have been reached + Beta cooling evaluation: progress: [1464.6]; minimum progress targets: [8991.] + Warming alpha_pgi to favor clustering: 41274.01718048347 + mref changed in 39 places + ------------------------- STOP! ------------------------- + 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.7644e+03 + 0 : |xc-x_last| = 1.1470e+01 <= tolX*(1+|x0|) = 1.0000e-01 + 0 : |proj(x-g)-x| = 2.0919e+02 <= tolG = 1.0000e-01 + 0 : |proj(x-g)-x| = 2.0919e+02 <= 1e3*eps = 1.0000e-02 + 0 : maxIter = 20 <= iter = 13 + ------------------------- DONE! ------------------------- + + + + +.. GENERATED FROM PYTHON SOURCE LINES 414-418 + +Step 12: Visualize the 3D Model +------------------------------- + +The inverted 3D model is visualized using slices in different orientations. + +.. GENERATED FROM PYTHON SOURCE LINES 418-448 + +.. code-block:: default + + + set = 1 + save_plots = True + + # Indices of the depth sections + ind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25 + ind_plot_y = int(len(inv_mesh.cell_centers_y) / 2) + ind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15 + + ind_plot = [ind_plot_x, ind_plot_y, ind_plot_z] + + # Extract the results + inverted_density_model = density_map * inverted_model + quasi_geology_model = actvMap * regularization_term.objfcts[0].compute_quasi_geology_model() + + # Plot Density Contrast Model (Z) + normal = "Z" + model_to_plot = inverted_density_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + + # Plot Inverted Model Slices (Y) + normal = "X" + model_to_plot = inverted_density_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + + # Plot Inverted Model Slices (X) + normal = "Y" + model_to_plot = inverted_density_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name) + + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_006.png + :alt: Inverted Den Model Slice at Z = -2.12E+02 m + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_006.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_007.png + :alt: Inverted Den Model Slice at X = 1.62E+05 m + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_007.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_008.png + :alt: Inverted Den Model Slice at Y = 1.46E+05 m + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_008.png + :class: sphx-glr-multi-img + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 449-453 + +Step 13: Visualize the Quasi-Geological Model +--------------------------------------------- + +The quasi-geological model (as given by [Li et al., 2019](https://doi.org/10.1190/tle38010060.1))is visualized, showing different rock units classified by the GMM. + +.. GENERATED FROM PYTHON SOURCE LINES 455-456 + +Plot Density Contrast Model (Z) + +.. GENERATED FROM PYTHON SOURCE LINES 456-470 + +.. code-block:: default + + normal = "Z" + model_to_plot = quasi_geology_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + + # Plot Inverted Model Slices (Y) + normal = "X" + model_to_plot = quasi_geology_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + + # Plot Inverted Model Slices (X) + normal = "Y" + model_to_plot = quasi_geology_model + SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name) + + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_009.png + :alt: Quasi-Geological Model Slice at Z = -2.12E+02 m + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_009.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_010.png + :alt: Quasi-Geological Model Slice at X = 1.62E+05 m + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_010.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_011.png + :alt: Quasi-Geological Model Slice at Y = 1.46E+05 m + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_011.png + :class: sphx-glr-multi-img + + + + + +.. GENERATED FROM PYTHON SOURCE LINES 471-475 + +Step 14: Plot the Updated 1D-GMM +-------------------------------- + +The updated GMM, learned through the inversion, is visualized. + +.. GENERATED FROM PYTHON SOURCE LINES 475-487 + +.. code-block:: default + + + # Plot the learned GMM + fig = plt.figure(figsize=(6, 6)) + ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4) + ax[0].hist(inverted_density_model[actv], density=True, bins=1000) + ax[0].set_xlabel(r"Density contrast (g/cc)") + ax[0].set_ylabel(r"Probability Density Values") + ax[0].get_legend().remove() + ax[0].set_title(r"Learned DelRho Distribution") + ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0)) + plt.savefig(path_to_output + "/" + name + "_Learned_GMM.pdf", bbox_inches="tight") + plt.show() + + + +.. rst-class:: sphx-glr-horizontal + + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_012.png + :alt: 04 FTG PGI + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_012.png + :class: sphx-glr-multi-img + + * + + .. image-sg:: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_013.png + :alt: Learned DelRho Distribution + :srcset: /examples/03_forward_engines/images/sphx_glr_04_FTG_PGI_013.png + :class: sphx-glr-multi-img + + + + + + +.. rst-class:: sphx-glr-timing + + **Total running time of the script:** ( 10 minutes 53.107 seconds) + + +.. _sphx_glr_download_examples_03_forward_engines_04_FTG_PGI.py: + +.. only:: html + + .. container:: sphx-glr-footer sphx-glr-footer-example + + + + + .. container:: sphx-glr-download sphx-glr-download-python + + :download:`Download Python source code: 04_FTG_PGI.py <04_FTG_PGI.py>` + + .. container:: sphx-glr-download sphx-glr-download-jupyter + + :download:`Download Jupyter notebook: 04_FTG_PGI.ipynb <04_FTG_PGI.ipynb>` + + +.. only:: html + + .. rst-class:: sphx-glr-signature + + `Gallery generated by Sphinx-Gallery `_ diff --git a/_sources/examples/03_forward_engines/index.rst.txt b/_sources/examples/03_forward_engines/index.rst.txt index c1292e2..90549f6 100644 --- a/_sources/examples/03_forward_engines/index.rst.txt +++ b/_sources/examples/03_forward_engines/index.rst.txt @@ -2,8 +2,8 @@ .. _sphx_glr_examples_03_forward_engines: -Forward GeoPhysics Engines and Inversion -======================================== +GeoPhysics and Inversion +======================== Our tutorials are crafted to guide you through the process of applying geophysical forward modeling and inversion techniques, using advanced tools like `gempy` and `simpeg`. Here’s what you can expect: @@ -26,7 +26,7 @@ Our tutorials are crafted to guide you through the process of applying geophysic .. raw:: html -
+
.. only:: html @@ -37,7 +37,7 @@ Our tutorials are crafted to guide you through the process of applying geophysic .. raw:: html -
Inversion of Gravity Gradiometry Data on a Tree Mesh
+
Inversion of Full Tensor Gravity Gradiometry Data
@@ -58,6 +58,23 @@ Our tutorials are crafted to guide you through the process of applying geophysic
+.. raw:: html + +
+ +.. only:: html + + .. image:: /examples/03_forward_engines/images/thumb/sphx_glr_03_HSI_To_Petrophysics_thumb.png + :alt: + + :ref:`sphx_glr_examples_03_forward_engines_03_HSI_To_Petrophysics.py` + +.. raw:: html + +
Predicting P-Wave Velocity from Hyperspectral Data
+
+ + .. raw:: html @@ -68,4 +85,5 @@ Our tutorials are crafted to guide you through the process of applying geophysic /examples/03_forward_engines/01_gravity_gradiometry_inversion_pgi /examples/03_forward_engines/02_model_1_gempy_fw_gravity + /examples/03_forward_engines/03_HSI_To_Petrophysics diff --git a/_sources/examples/03_forward_engines/sg_execution_times.rst.txt b/_sources/examples/03_forward_engines/sg_execution_times.rst.txt index b6bbb7c..d9c948f 100644 --- a/_sources/examples/03_forward_engines/sg_execution_times.rst.txt +++ b/_sources/examples/03_forward_engines/sg_execution_times.rst.txt @@ -6,10 +6,12 @@ Computation times ================= -**09:03.455** total execution time for **examples_03_forward_engines** files: +**10:14.392** total execution time for **examples_03_forward_engines** files: +-----------------------------------------------------------------------------------------------------------------------------------+-----------+--------+ -| :ref:`sphx_glr_examples_03_forward_engines_01_gravity_gradiometry_inversion_pgi.py` (``01_gravity_gradiometry_inversion_pgi.py``) | 09:03.455 | 0.0 MB | +| :ref:`sphx_glr_examples_03_forward_engines_01_gravity_gradiometry_inversion_pgi.py` (``01_gravity_gradiometry_inversion_pgi.py``) | 10:14.392 | 0.0 MB | +-----------------------------------------------------------------------------------------------------------------------------------+-----------+--------+ | :ref:`sphx_glr_examples_03_forward_engines_02_model_1_gempy_fw_gravity.py` (``02_model_1_gempy_fw_gravity.py``) | 00:00.000 | 0.0 MB | +-----------------------------------------------------------------------------------------------------------------------------------+-----------+--------+ +| :ref:`sphx_glr_examples_03_forward_engines_03_HSI_To_Petrophysics.py` (``03_HSI_To_Petrophysics.py``) | 00:00.000 | 0.0 MB | ++-----------------------------------------------------------------------------------------------------------------------------------+-----------+--------+ diff --git a/_sources/examples/index.rst.txt b/_sources/examples/index.rst.txt index bc595e5..1974e91 100644 --- a/_sources/examples/index.rst.txt +++ b/_sources/examples/index.rst.txt @@ -180,8 +180,8 @@ Prepare to enhance your understanding and skills in geological modeling with a f -Forward GeoPhysics Engines and Inversion -======================================== +GeoPhysics and Inversion +======================== Our tutorials are crafted to guide you through the process of applying geophysical forward modeling and inversion techniques, using advanced tools like `gempy` and `simpeg`. Here’s what you can expect: @@ -204,7 +204,7 @@ Our tutorials are crafted to guide you through the process of applying geophysic .. raw:: html -
+
.. only:: html @@ -215,7 +215,7 @@ Our tutorials are crafted to guide you through the process of applying geophysic .. raw:: html -
Inversion of Gravity Gradiometry Data on a Tree Mesh
+
Inversion of Full Tensor Gravity Gradiometry Data
@@ -236,6 +236,23 @@ Our tutorials are crafted to guide you through the process of applying geophysic
+.. raw:: html + +
+ +.. only:: html + + .. image:: /examples/03_forward_engines/images/thumb/sphx_glr_03_HSI_To_Petrophysics_thumb.png + :alt: + + :ref:`sphx_glr_examples_03_forward_engines_03_HSI_To_Petrophysics.py` + +.. raw:: html + +
Predicting P-Wave Velocity from Hyperspectral Data
+
+ + .. raw:: html diff --git a/contributing.html b/contributing.html index a942d53..5d857ad 100644 --- a/contributing.html +++ b/contributing.html @@ -129,7 +129,7 @@

Navigation

User Guide

@@ -158,7 +158,7 @@

Quick search

- ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/01_readers/01_read_OMF_sites.html b/examples/01_readers/01_read_OMF_sites.html index e263ac7..fddf634 100644 --- a/examples/01_readers/01_read_OMF_sites.html +++ b/examples/01_readers/01_read_OMF_sites.html @@ -71,30 +71,30 @@

Load OMF Project
def load_omf():
     config = dotenv_values()
     path = config.get('PATH_TO_MODEL_2')
-    omf_project = omfvista.load_project(path)
-    return omf_project
+    omf_project = omfvista.load_project(path)
+    return omf_project
 
-omf_project = load_omf()
+omf_project = load_omf()

Visualize OMF with PyVista¶

Utilize PyVista for an interactive visualization of the OMF project.

-
omf_project.plot(multi_colors=True, show_edges=True, notebook=False)
+
omf_project.plot(multi_colors=True, show_edges=True, notebook=False)
 

 

 01 read OMF sites

Convert OMF to Unstructured Single Block¶

Convert the loaded OMF project into an unstructured single block for further processing.

-
block_name = omf_project.get_block_name(4)
-polydata_obj: pyvista.PolyData = omf_project[block_name]
-unstruct_pyvista: pyvista.UnstructuredGrid = polydata_obj.cast_to_unstructured_grid()
-cells_pyvista = unstruct_pyvista.cells.reshape(-1, 4)[:, 1:]
+
block_name = omf_project.get_block_name(4)
+polydata_obj: pyvista.PolyData = omf_project[block_name]
+unstruct_pyvista: pyvista.UnstructuredGrid = polydata_obj.cast_to_unstructured_grid()
+cells_pyvista = unstruct_pyvista.cells.reshape(-1, 4)[:, 1:]
 
 unstruct: subsurface.UnstructuredData = subsurface.UnstructuredData.from_array(
-    vertex=unstruct_pyvista.points,
+    vertex=unstruct_pyvista.points,
     cells=cells_pyvista,
 )
 
@@ -109,8 +109,8 @@ 
Convert OMF to Unstructured Single Block¶

 
Finally, visualize the converted unstructured data using Subsurface and PyVista.

 
ts = TriSurf(mesh=unstruct)
-subsurface_mesh = to_pyvista_mesh(ts)
-pv_plot([subsurface_mesh], image_2d=False)
+subsurface_mesh = to_pyvista_mesh(ts)
+pv_plot([subsurface_mesh], image_2d=False)
 

 

 01 read OMF sites
<pyvista.plotting.plotting.Plotter object at 0x7fd53b397f70>
@@ -170,7 +170,7 @@ 
Navigation

 
 
 
  • Structural Modeling
    • 
-
  • Forward GeoPhysics Engines and Inversion
    • 
+
  • GeoPhysics and Inversion
    • 
 
  • Probabilistic Modeling
    • 
 
 
    User Guide
    
@@ -195,7 +195,7 @@ 
    Quick search
    
       
    
     
    
     
    
-      ©2023-2023, Vector Geology Team.
+      ©2023-2024, Vector Geology Team.
       
       |
       Powered by Sphinx 7.1.1
diff --git a/examples/01_readers/02_model_1_OMF_to_subsurface.html b/examples/01_readers/02_model_1_OMF_to_subsurface.html
index a297378..a70a890 100644
--- a/examples/01_readers/02_model_1_OMF_to_subsurface.html
+++ b/examples/01_readers/02_model_1_OMF_to_subsurface.html
@@ -65,7 +65,7 @@ 
    Importing Required Libraries
 
    Loading the OMF Project¶
    
 
    Load the OMF project file using a custom function.
    
-
    omf = load_omf("PATH_TO_MODEL_1")
+
    omf = load_omf("PATH_TO_MODEL_1")
    @@ -74,7 +74,7 @@

    Visualizing the OMF Project with PyVista
    # Replace `False` with a condition or toggle to enable plotting.
 if False:
-    omf.plot(multi_colors=True, show_edges=True, notebook=False)
+    omf.plot(multi_colors=True, show_edges=True, notebook=False)
    @@ -89,19 +89,19 @@

    Visualizing Unstructured Data with Subsurface and PyVistadataset: xarray.Dataset = None -for e in range(omf.n_blocks): - block_name = omf.get_block_name(e) +for e in range(omf.n_blocks): + block_name = omf.get_block_name(e) print(block_name) - polydata_obj: pyvista.PolyData = omf[block_name] + polydata_obj: pyvista.PolyData = omf[block_name] # Check if the polydata is a mesh and if is not continue - print(polydata_obj.get_cell(0).type) - unstruct_pyvista: pyvista.UnstructuredGrid = polydata_obj.cast_to_unstructured_grid() + print(polydata_obj.get_cell(0).type) + unstruct_pyvista: pyvista.UnstructuredGrid = polydata_obj.cast_to_unstructured_grid() - grid = unstruct_pyvista - cell_data = {name: grid.cell_data[name] for name in grid.cell_data} - match polydata_obj.get_cell(0).type: - case pyvista.CellType.TRIANGLE: - cells_pyvista = unstruct_pyvista.cells.reshape(-1, 4)[:, 1:] + grid = unstruct_pyvista + cell_data = {name: grid.cell_data[name] for name in grid.cell_data} + match polydata_obj.get_cell(0).type: + case pyvista.CellType.TRIANGLE: + cells_pyvista = unstruct_pyvista.cells.reshape(-1, 4)[:, 1:] new_cell_data = { **{ "Formation_Major_": e, @@ -109,39 +109,39 @@

    Visualizing Unstructured Data with Subsurface and PyVista**cell_data } unstruct: subsurface.UnstructuredData = subsurface.UnstructuredData.from_array( - vertex=unstruct_pyvista.points, + vertex=unstruct_pyvista.points, cells=cells_pyvista, cells_attr=pd.DataFrame(new_cell_data) ) - ts = TriSurf(mesh=unstruct) - s = to_pyvista_mesh(ts) + ts = TriSurf(mesh=unstruct) + s = to_pyvista_mesh(ts) if e == 5: - meshes_far.append(s) # * This mesh is far from the rest. I am still unsure what is meant to represent. + meshes_far.append(s) # * This mesh is far from the rest. I am still unsure what is meant to represent. else: if False: to_netcdf( base_data=unstruct, path=f"./{block_name}.nc", ) - meshes.append(s) + meshes.append(s) - case pyvista.CellType.LINE: + case pyvista.CellType.LINE: if e > 11: continue continue # To ignore wells for now - cells_pyvista = unstruct_pyvista.cells.reshape(-1, 3)[:, 1:] + cells_pyvista = unstruct_pyvista.cells.reshape(-1, 3)[:, 1:] unstruct: subsurface.UnstructuredData = subsurface.UnstructuredData.from_array( - vertex=unstruct_pyvista.points, + vertex=unstruct_pyvista.points, cells=cells_pyvista, cells_attr=pd.DataFrame(cell_data) ) line = LineSet(data=unstruct) - s = to_pyvista_line(line, radius=100, as_tube=True, spline=False) + s = to_pyvista_line(line, radius=100, as_tube=True, spline=False) if e <= 10: - lines_1.append(s) + lines_1.append(s) elif e <= 11: - lines_far.append(s) + lines_far.append(s) if False: # Replace with condition for exporting to Liquid Earth base_structs_to_binary_file("leapfrog1", unstruct) @@ -185,22 +185,22 @@

    Visualizing Unstructured Data with Subsurface and PyVista

    Visualize Unstructured Data:¶

    Use Subsurface and PyVista to visualize the unstructured data.

    -
    plotter = init_plotter()
+
    plotter = init_plotter()
 if plot_model_area := True:
-    for mesh in meshes[:1]:
-        plotter.add_mesh(mesh, cmap="magma", opacity=1)
+    for mesh in meshes[:1]:
+        plotter.add_mesh(mesh, cmap="magma", opacity=1)
 
     for line in lines_far:
-        plotter.add_mesh(line, cmap="viridis", opacity=1)
+        plotter.add_mesh(line, cmap="viridis", opacity=1)
 else:
     # * This seems to go together
-    for mesh in meshes_far:
-        plotter.add_mesh(mesh, cmap="magma", opacity=0.7)
+    for mesh in meshes_far:
+        plotter.add_mesh(mesh, cmap="magma", opacity=0.7)
 
     for line in lines_1:
-        plotter.add_mesh(line, cmap="viridis", opacity=1)
+        plotter.add_mesh(line, cmap="viridis", opacity=1)
 
-plotter.show()
+plotter.show()
 
    
 
    
 02 model 1 OMF to subsurface
@@ -265,7 +265,7 @@ 
    Navigation
    
 
 
 
  • Structural Modeling
    • 
-
  • Forward GeoPhysics Engines and Inversion
    • 
+
  • GeoPhysics and Inversion
    • 
 
  • Probabilistic Modeling
    • 
 
 
    User Guide
    
@@ -290,7 +290,7 @@ 
    Quick search
    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/01_readers/03_model_2_OMF_to_subsurface.html b/examples/01_readers/03_model_2_OMF_to_subsurface.html index 88539af..46af976 100644 --- a/examples/01_readers/03_model_2_OMF_to_subsurface.html +++ b/examples/01_readers/03_model_2_OMF_to_subsurface.html @@ -67,10 +67,10 @@

    Load OMF Project
    def load_omf():
     config = dotenv_values()
     path = config.get('PATH_TO_MODEL_2')
-    omf_project = omfvista.load_project(path)
-    return omf_project
+    omf_project = omfvista.load_project(path)
+    return omf_project
 
-omf_project = load_omf()
+omf_project = load_omf()

    @@ -78,7 +78,7 @@

    Load OMF Project¶

    Optionally, visualize the OMF project using PyVista. This step can be skipped or modified as needed.

    if False:  # Change to True to enable visualization
-    omf_project.plot(multi_colors=True, show_edges=True, notebook=False)
+    omf_project.plot(multi_colors=True, show_edges=True, notebook=False)
    @@ -88,43 +88,43 @@

    Convert OMF to Unstructured Single Block
    meshes = []
 lines = []
 
-for block_index in range(omf_project.n_blocks):
-    block_name = omf_project.get_block_name(block_index)
-    polydata_obj: pyvista.PolyData = omf_project[block_name]
+for block_index in range(omf_project.n_blocks):
+    block_name = omf_project.get_block_name(block_index)
+    polydata_obj: pyvista.PolyData = omf_project[block_name]
 
     # Skip if the polydata is not a mesh
-    if not isinstance(polydata_obj, pyvista.PolyData):
+    if not isinstance(polydata_obj, pyvista.PolyData):
         continue
 
-    unstruct_pyvista: pyvista.UnstructuredGrid = polydata_obj.cast_to_unstructured_grid()
-    cell_data = {name: unstruct_pyvista.cell_data[name] for name in unstruct_pyvista.cell_data}
+    unstruct_pyvista: pyvista.UnstructuredGrid = polydata_obj.cast_to_unstructured_grid()
+    cell_data = {name: unstruct_pyvista.cell_data[name] for name in unstruct_pyvista.cell_data}
 
     # Process based on cell type
-    match polydata_obj.cell_type(0):
-        case pyvista.CellType.TRIANGLE:
+    match polydata_obj.cell_type(0):
+        case pyvista.CellType.TRIANGLE:
             # Process triangle mesh
-            cells_pyvista = unstruct_pyvista.cells.reshape(-1, 4)[:, 1:]
+            cells_pyvista = unstruct_pyvista.cells.reshape(-1, 4)[:, 1:]
             new_cell_data = {"Formation_Major_": block_index, **cell_data}
             unstruct = subsurface.UnstructuredData.from_array(
-                vertex=unstruct_pyvista.points,
+                vertex=unstruct_pyvista.points,
                 cells=cells_pyvista,
                 cells_attr=pd.DataFrame(new_cell_data)
             )
-            ts = TriSurf(mesh=unstruct)
+            ts = TriSurf(mesh=unstruct)
             meshes.append(to_pyvista_mesh(ts))
 
-        case pyvista.CellType.LINE:
+        case pyvista.CellType.LINE:
             # Process line data
             if "Formation_Major" not in cell_data.keys():
                 continue
-            cells_pyvista = unstruct_pyvista.cells.reshape(-1, 3)[:, 1:]
+            cells_pyvista = unstruct_pyvista.cells.reshape(-1, 3)[:, 1:]
             unstruct = subsurface.UnstructuredData.from_array(
-                vertex=unstruct_pyvista.points,
+                vertex=unstruct_pyvista.points,
                 cells=cells_pyvista,
                 cells_attr=pd.DataFrame(cell_data)
             )
-            line = LineSet(data=unstruct)
-            lines.append(to_pyvista_line(line, radius=100, as_tube=True, spline=False))
+            line = LineSet(data=unstruct)
+            lines.append(to_pyvista_line(line, radius=100, as_tube=True, spline=False))
 
 # Export to desired format here if necessary
    @@ -149,15 +149,15 @@

    Convert OMF to Unstructured Single Block

    Visualize Unstructured Data¶

    Visualize the unstructured data using Subsurface and PyVista.

    -
    plotter = init_plotter()
+
    plotter = init_plotter()
 
-for mesh in meshes:
-    plotter.add_mesh(mesh, cmap="magma", opacity=0.7)
+for mesh in meshes:
+    plotter.add_mesh(mesh, cmap="magma", opacity=0.7)
 
-for line in lines:
-    plotter.add_mesh(line, cmap="viridis", opacity=1)
+for line in lines:
+    plotter.add_mesh(line, cmap="viridis", opacity=1)
 
-plotter.show()
+plotter.show()
 
    
 
    
 03 model 2 OMF to subsurface
    Total running time of the script: ( 0 minutes  1.332 seconds)
    
@@ -214,7 +214,7 @@ 
    Navigation
    
 
 
 
  • Structural Modeling
    • 
-
  • Forward GeoPhysics Engines and Inversion
    • 
+
  • GeoPhysics and Inversion
    • 
 
  • Probabilistic Modeling
    • 
 
 
    User Guide
    
@@ -239,7 +239,7 @@ 
    Quick search
    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/01_readers/04_model_1_read_gravity.html b/examples/01_readers/04_model_1_read_gravity.html index 1c42442..e428f0d 100644 --- a/examples/01_readers/04_model_1_read_gravity.html +++ b/examples/01_readers/04_model_1_read_gravity.html @@ -137,7 +137,7 @@

    Navigation

  • Structural Modeling
    • -
  • Forward GeoPhysics Engines and Inversion
    • +
  • GeoPhysics and Inversion
  • Probabilistic Modeling

    • User Guide

    @@ -162,7 +162,7 @@

    Quick search

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/01_readers/index.html b/examples/01_readers/index.html index 057835d..76beb1e 100644 --- a/examples/01_readers/index.html +++ b/examples/01_readers/index.html @@ -98,7 +98,7 @@

    Navigation

  • Structural Modeling
    • -
  • Forward GeoPhysics Engines and Inversion
    • +
  • GeoPhysics and Inversion
  • Probabilistic Modeling

    • User Guide

    @@ -123,7 +123,7 @@

    Quick search

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/01_readers/sg_execution_times.html b/examples/01_readers/sg_execution_times.html index 7e16450..32f50fb 100644 --- a/examples/01_readers/sg_execution_times.html +++ b/examples/01_readers/sg_execution_times.html @@ -91,7 +91,7 @@

    Navigation

    User Guide

    @@ -116,7 +116,7 @@

    Quick search

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/02_structural_modeling/01_model_1_gempy_step_by_step.html b/examples/02_structural_modeling/01_model_1_gempy_step_by_step.html index a4d20af..36171c5 100644 --- a/examples/02_structural_modeling/01_model_1_gempy_step_by_step.html +++ b/examples/02_structural_modeling/01_model_1_gempy_step_by_step.html @@ -305,7 +305,7 @@

    Navigation

  • Construct Model 1 with Helper Functions
    • -
  • Forward GeoPhysics Engines and Inversion
    • +
  • GeoPhysics and Inversion
  • Probabilistic Modeling

    • User Guide

    @@ -330,7 +330,7 @@

    Quick search

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/02_structural_modeling/02_model_1_gempy.html b/examples/02_structural_modeling/02_model_1_gempy.html index 8eccddd..95bf216 100644 --- a/examples/02_structural_modeling/02_model_1_gempy.html +++ b/examples/02_structural_modeling/02_model_1_gempy.html @@ -22,7 +22,7 @@ - + @@ -343,7 +343,7 @@

    Navigation

  • Construct Model 1 with Helper Functions
    • -
  • Forward GeoPhysics Engines and Inversion
    • +
  • GeoPhysics and Inversion
  • Probabilistic Modeling

    • User Guide

    @@ -368,7 +368,7 @@

    Quick search

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/02_structural_modeling/index.html b/examples/02_structural_modeling/index.html index b786142..4d9c573 100644 --- a/examples/02_structural_modeling/index.html +++ b/examples/02_structural_modeling/index.html @@ -93,7 +93,7 @@

    Navigation

  • Construct Model 1 with Helper Functions
    • -
  • Forward GeoPhysics Engines and Inversion
    • +
  • GeoPhysics and Inversion
  • Probabilistic Modeling

    • User Guide

    @@ -118,7 +118,7 @@

    Quick search

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/02_structural_modeling/sg_execution_times.html b/examples/02_structural_modeling/sg_execution_times.html index b8cea60..2c1ddc7 100644 --- a/examples/02_structural_modeling/sg_execution_times.html +++ b/examples/02_structural_modeling/sg_execution_times.html @@ -83,7 +83,7 @@

    Navigation

    User Guide

    @@ -108,7 +108,7 @@

    Quick search

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/03_forward_engines/01_gravity_gradiometry_inversion_pgi.html b/examples/03_forward_engines/01_gravity_gradiometry_inversion_pgi.html index 944228e..e5a5ef5 100644 --- a/examples/03_forward_engines/01_gravity_gradiometry_inversion_pgi.html +++ b/examples/03_forward_engines/01_gravity_gradiometry_inversion_pgi.html @@ -5,7 +5,7 @@ - Inversion of Gravity Gradiometry Data on a Tree Mesh — Vector Geology 0.1.dev16+g525c9e2 documentation + Inversion of Full Tensor Gravity Gradiometry Data — Vector Geology 0.1.dev16+g525c9e2 documentation @@ -19,11 +19,12 @@ + - + @@ -45,165 +46,151 @@

    Go to the end to download the full example code

    -
    -

    Inversion of Gravity Gradiometry Data on a Tree Mesh¶

    -

    We use the PGI module within SimPEG to invert gravity gradiometry data on a Tree Mesh. This tutorial focusses on:

    -
    -
      -

    • How to define all the necessary components

      • -

    • How to initialize the Gaussian Mixture Model Prior

      • -

    • How to set the parameters

      • -

    • How to plot the results

      • +
      +

      Inversion of Full Tensor Gravity Gradiometry Data¶

      +

      This example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, +measured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include:

      +
        +

      • Loading necessary modules and data

        • +

      • Generating a mesh

        • +

      • Initializing the Gaussian Mixture Model

        • +

      • Tuning hyper-parameters

        • +

      • Visualizing the results

      -
    -

    ## Import Modules

    -
    import discretize as ds
+
    
+
    The process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example
    notebook from SimPEG’s documentation (see SimPEG docs for more information).
    
+
    
+
    
+
    import os
+import shutil
+from dotenv import dotenv_values
+
 import SimPEG.potential_fields as pf
-from SimPEG import (
-    maps,
-    data,
-    utils,
-    simulation,
-    inverse_problem,
-    inversion,
-    optimization,
-    regularization,
-    data_misfit,
-    directives,
-)
-from SimPEG.utils import io_utils
-import numpy as np
-import pandas as pd
-import scipy.interpolate as interp
+# Import the SimPEG functions
+import discretize as ds
 import matplotlib as mpl
 import matplotlib.pyplot as plt
-import math as math
-
-from dotenv import dotenv_values
-from mpl_toolkits.axes_grid1.inset_locator import inset_axes
-
-from discretize.utils import mkvc, refine_tree_xyz, active_from_xyz
-from discretize import TensorMesh, TreeMesh
-from SimPEG.utils import plot2Ddata, model_builder, surface2ind_topo
-from SimPEG import maps
-from SimPEG.potential_fields import magnetics as mag
+# Import the general modules
+import numpy as np
+import pandas as pd
+from SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives)
 from SimPEG.potential_fields import gravity as grav
-import os
-import shutil
-from vector_geology import SimpegHelper as SH
+# Import the discretize module
+from discretize.utils import active_from_xyz
 
-#from get_cpt import get_cmap
+# Import SimpegHelper for plotting and resampling data
+from vector_geology import SimpegHelper as SH
 
 # Plot beautification
-
 formatter = mpl.ticker.ScalarFormatter(useMathText=True)
 formatter.set_scientific(True)
-formatter.set_powerlimits((0,0))
+formatter.set_powerlimits((0, 0))
 
 mpl.rc("axes", titlesize=14, labelsize=12)
 mpl.rc("xtick", labelsize=12)
 mpl.rc("ytick", labelsize=12)
-
-# Reproducible science
-np.random.seed(518936)
 
    
 
    
-
    Create directories for the model iterations and output
    
-
    # Name of the model
-name = "Ireland_FTG_NonNeg"
-# Names for the directories for the model iterations and output
-path_to_mod_iterations = "./" + str(name) + "/Model Iterations"
-path_to_output = "./" + str(name) + "/Output"
-
-# Check if the model iterations directory exists and remove it if it does
+
    
+
    Step 1: Create Directories for Storing Model Iterations and Outputs¶
    
+
    PGI is an iterative process, and it’s useful to save the model at every iteration. This helps in preventing overfitting
+and in adjusting the stopping criteria based on the model iterations.
    
+
    name = "Ireland_FTG"
+path_to_mod_iterations = f"./{name}/Model Iterations"
+path_to_output = f"./{name}/Output"
+
 if os.path.exists(path_to_mod_iterations):
     shutil.rmtree(path_to_mod_iterations)
-
-# Create the model iterations directory.
 os.makedirs(path_to_mod_iterations)
 
-# Check if the output directory exists and remove it if it does
 if os.path.exists(path_to_output):
     shutil.rmtree(path_to_output)
-
-# Create the output directory.
 os.makedirs(path_to_output)
 
    
 
    
-
    Load the data
-Read gravity and magnetic data from a csv file.
-Inputs:
-file_path: The path to the csv file.
-header: Whether or not the csv file has a header row.
    
+
    
+
    
+
    Step 2: Load the FTG-G Data¶
    
+
    Full Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion.
    
+
    
+\[\begin{split}G = \left[\begin{array}{cc}
+G_{xx} & G_{xy} & G_{xz}\\
+G_{xy} & G_{yy} & G_{yz}\\
+G_{xz} & G_{yz} & G_{zz}
+\end{array}\right]\end{split}\]
    
 
    config = dotenv_values()
 file_path = config.get("PATH_TO_GRADIOMETRY")
-
-header = None
-
-# Read the csv file into a pandas dataframe and then convert it to a numpy array.
 FTG_Data = pd.read_csv(file_path, delimiter=",").to_numpy()
-
-# Add Topography
-FTG_Data = np.c_[FTG_Data, np.zeros(np.shape(FTG_Data)[0])]
-FTG_Data[:,[2, -1]] = FTG_Data[:,[-1, 2]]
-
-# Format after adding topography
-# X, Y, Z, Tyy, Txz, Tyz, Txy, Txx
-
-
-## %
-## Resample the data
-# The ``SimpegHelper.pf_rs()`` will take in potential field data and resample it onto a new regular grid
 
    
 
    
-
    New sampling interval (Data Units)
    
-
    inc = 200
-
-# Resample the data
-[grav_new, nx_new, ny_new] = SH.pf_rs(FTG_Data, inc)
-
-# Extract the gravity gradiometry data vectors
-grav_vec = grav_new[:,3:]
+
    
+
    
+
    Step 3: Resample Data onto a Regular Grid¶
    
+
    Resampling the data onto a regular grid aids in mesh generation and visualization. The SimpegHelper.pf_rs() function
+resamples potential field data onto a new regular grid.
    
+
    inc = 150  # New sampling interval (ground units)
+grav_new, nx_new, ny_new = SH.pf_rs(
+    FTG_Data,
+    inc,
+    bounds=[156000, 168000, 143000, 148500]
+)
 
-# Negate the gravity data to the opposite sign to match the coordinate system.
 # NOTE: This step is necessary for real data since the convention followed by
-# the SimPEG forward operator gravity lows over density highs.
-# grav_vec = -grav_vec
-
-# Extract the topography for the active cells.
-inv_topo = grav_new[:,[0,1,2]]
-
-
-## Visualize the data
-
-#The ``SimpegHelper.plot_2D_data()`` function will plot the datasets using ``matplotlib``.
-
    
-
    
-
    Plot Gravity Data
    
-
    SH.plot_2D_data(np.c_[grav_new[:,[0,1,2]], grav_vec[:,-1]], [np.nanmin(grav_vec[:,-1]), np.nanmax(grav_vec[:,-1])], cmap="jet", path_to_output=path_to_output, name=name)
-
-# Plot Topography Data
-# SH.plot_2D_data(np.c_[inv_topo[:,[0,1,2]], inv_topo[:,2]], [np.nanmin(inv_topo[:,2]), np.nanmax(inv_topo[:,2])], which_data="Topo", cmap="magma", path_to_output=path_to_output, name=name)
+# the SimPEG forward operator is the opposite of the general convention
+grav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :]  # Adjusting sign convention
+inv_topo = grav_new[:, [0, 1, 2]]  # Extracting topography
 
    
 
    
-Gravity Anomaly
    # Create a TreeMesh object to invert the data
    
-
    # Using the ``discretize.TreeMesh`` utility, create a 3D TreeMesh to invert the gravity gradiometry data.
+
    
+
    
+
    Step 4: Visualize the Data¶
    
+
    The SimpegHelper.plot_2D_data() function is used to visualize the datasets with matplotlib.
    
+
    SH.plot_2D_data(
+    data=np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]],
+    clim=[np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])],
+    cmap="jet",
+    which_data="FTG",
+    comp="xx",
+    path_to_output=path_to_output,
+    name=name
+)
+SH.plot_2D_data(
+    data=np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]],
+    clim=[np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])],
+    cmap="terrain",
+    which_data="Topo",
+    comp="xx",
+    path_to_output=path_to_output,
+    name=name
+)
 
    
 
    
-
    dx = inc
-dy = inc
-dz = 10
-nz_core = 1
-nz_pad = 20
-fact = 1.1
-
-# Set the cell sizes to a constant value in the x and y directions and expanding in the z direction.
+
      
+
    • FTG ($G_{xx}$)
      • 
+
    • Topography
      • 
+
    
+
    
+
    
+
    Step 5: Create a TensorMesh Object for Inversion¶
    
+
    A 3D TensorMesh is created using the discretize.TensorMesh utility to invert the gravity gradiometry data. The mesh
+is designed to have a constant cell size in the core region and expanding/contracting cells in the padding region.
    
+
    
+\[z_{n+1} = r z_{n}\]
    
+
    Where $r$ is a multiplicative factor (<1 for contracting, >1 for expanding cells). x and y
+increments can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire
+region is an area of interest).
    
+
    dx, dy, dz = inc, inc, 10  # Cell sizes
+nz_core, nz_pad = 10, 18  # Number of cells in core and padding
+fact = 1.1  # Factor for expanding/contracting cells
+
 inv_hx = dx * np.ones(nx_new)
 inv_hy = dy * np.ones(ny_new)
 inv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)]
 
-# Create the inverse tensor mesh.
-inv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:,0]), np.min(inv_topo[:,1]), "C"])
+# Create the inverse tensor mesh
+inv_mesh = ds.TensorMesh(
+    h=[inv_hx, inv_hy, inv_hz],
+    x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), "C"])
 
 # Drape the topography over the mesh
 actv = active_from_xyz(inv_mesh, inv_topo)
@@ -211,132 +198,77 @@
 actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv)
 nactv = int(actv.sum())
 
-# Make an x-y grid with the mesh cell centers
-# xygrid = np.meshgrid(inv_mesh.cell_centers_x, inv_mesh.cell_centers_y)
-# xygrid = np.reshape(xygrid, (2,-1)).T
-
-# Region Active Array (bool array with the spatial extent of the data)
-# RegActvArr = np.empty((len(xygrid)), dtype=bool)
-
-# for i in range(len(xygrid)):
-#     Dist = np.sqrt(np.sum((inv_topo[:, [0, 1]] - xygrid[i,:])**2, axis=1))
-#     MinDist = np.min(Dist)
-#     RegActvArr[i] = MinDist < 50
-
-# Repeating the regional active array for every z (since we only carried out the calculations for x-y, which can be replicated in z)
-# RegActvArr = np.tile(RegActvArr, len(inv_mesh.cell_centers_z))
-
-# Intersection between Topo active cells and spatial active cells (nactv and actvMap are also updated)
-actv = actv
-nactv = int(actv.sum())
-actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv)
+# Print mesh details
+print(inv_mesh)
 
    
 
    
-
    inv_mesh
+
    TensorMesh: 137,862 cells
+
+                    MESH EXTENT             CELL WIDTH      FACTOR
+dir    nC        min           max         min       max      max
+---   ---  ---------------------------  ------------------  ------
+ x     81    156,000.00    168,150.00    150.00    150.00    1.00
+ y     37    143,000.00    148,550.00    150.00    150.00    1.00
+ z     46       -551.59        551.59     10.00     55.60    1.10
 
    
 
    
-
    
-
-  
-    
-    
-  
-  
-    
-    
-    
-    
-    
-  
-  
-    
-    
-    
-    
-    
-    
-    
-  
-  
-    
-    
-    
-    
-    
-    
-    
-  
-  
-    
-    
-    
-    
-    
-    
-    
-  
-  
-    
-    
-    
-    
-    
-    
-    
-  
-
    TensorMesh111,930 cells
    MESH EXTENTCELL WIDTHFACTOR
    dirnCminmaxminmaxmax
    x78155,500.00171,100.00200.00200.001.00
    y35142,500.00149,500.00200.00200.001.00
    z41-635.02635.0210.0067.271.10
    
-
-
    
-
    
-
    # Visualize the mesh
    
-
    #The ``discretize.TreeMesh.plot_slice()`` utility will plot a slice of thee Tree Mesh object generated above.
-
    
-
    
-
    Plotting a mesh slice
-Create a background model.
    
+
    
+
    
+
    Step 6: Visualize the Mesh¶
    
+
    A slice of the Tensor Mesh object is visualized using the discretize.TensorMesh.plot_slice() utility.
    
+
    Create a background model.
    
 
    bg = np.ones(nactv)
 mod = actvMap * bg
 
-# Define the tensor mesh parameters.
-props = dict(boxstyle='round,pad=1', facecolor='white')
-
 # Plot the mesh slice.
 fig = plt.figure(figsize=(6, 6))
 ax = fig.add_axes([0.1, 0.1, 0.8, 0.8])
-cplot = inv_mesh.plot_slice(mod, grid=False, ax=ax, normal="Y", ind=22)
+cplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal="Y", ind=22)
 ax.set_title('Tensor Mesh Slice')
 ax.set_xlabel('x (m)')
-ax.set_ylabel('y (m)')
+ax.set_ylabel('z (m)')
+ax.set_xlim([158000, 159000])
+ax.set_ylim([-500, 100])
 ax.ticklabel_format(axis="both")
 ax.set_aspect('equal')
-plt.savefig(path_to_output + "/" + name + "_TreeMeshSlice.pdf", bbox_inches="tight")
+plt.savefig(
+    path_to_output + "/" + name + "_TreeMeshSlice.pdf",
+    bbox_inches="tight"
+)
 plt.show()
 
    
 
    
-Tensor Mesh Slice
    # Set up the gravity inverse problem
-%%
    
-
    
-
    Create wires for the density model
    
-
    
+Tensor Mesh Slice
    
+
    
+
    Step 7: Set up the Gravity Inverse Problem¶
    
+
    The gravity inverse problem is set up using SimPEG’s pf.gravity module to generate a forward model based on the mesh
+and receiver locations.
    
+
    Create wires for the density model
    
 
    wires = maps.Wires(("density", nactv))
 
-# Create a Density Mapping
+# Create a density map
 density_map = actvMap * wires.density
-# Create Identity Mapping
+
+# Create an identity map
 identity_map = maps.IdentityMap(nP=nactv)
 
 # Components of the data used as input
-gravity_components = ["gyy", "gxz", "gyz", "gxy", "gxx"]
+gravity_components = ["gxx", "gyy", "gxz", "gyz", "gxy"]
 
-# Receiver locations
-gravity_receiver_locations = grav_new[:, [0, 1, 2]]
+# Receiver locations (25 meters above the topography)
+gravity_receiver_locations = inv_topo + 120.
 
 # Create a gravity receiver list
-gravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components)
+gravity_receivers = pf.gravity.receivers.Point(
+    locations=gravity_receiver_locations,
+    components=gravity_components
+)
 gravity_receiver_list = [gravity_receivers]
 
 # Create a gravity source field
-gravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list)
+gravity_source_field = pf.gravity.sources.SourceField(
+    receiver_list=gravity_receiver_list
+)
 
 # Define the gravity survey
 gravity_survey = pf.gravity.survey.Survey(gravity_source_field)
@@ -349,26 +281,29 @@
     ind_active=actv
 )
 
-# Define uncertainties
-# Maximum value of each component
-# max_gravity = np.max(np.abs(grav_vec), axis=0)
-# Standard deviation of each component
-# std_gravity = 0.05 * max_gravity
-# std_gravity = np.ones(np.shape(grav_vec)) * std_gravity
-
-# Create a gravity data object
-gravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1)
+# Create a gravity data object, with the relative errors and a standard noise floor
+gravity_data = data.Data(
+    survey=gravity_survey,
+    dobs=grav_vec.flatten(),
+    noise_floor=5,
+    relative_error=0.1
+)
 
 # Define the misfits associated with the gravity data
-gravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem)
+gravity_misfit = data_misfit.L2DataMisfit(
+    data=gravity_data,
+    simulation=gravity_problem
+)
 
    
 
    
-
    # Set up the Gaussian Mixture Model (GMM) Prior
-%%
    
-
    
-
    Defining the Gaussian Mixture Model (GMM)
-Number of rock units and number of physical properties
    
-
    
+
    
+
    
+
    Step 8: Setting up the Gaussian Mixture Model (GMM) Prior¶
    
+
    In PGI, the petrophysical data is used as a constraint in the form of a Gaussian Mixture Model (GMM).
+A GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units (\(n\)),
+the petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below.
+If you do have a petrophysical dataset, you can fit the GMM to said dataset using the gmmref.fit() method.
    
+
    Number of rock units and number of physical properties
    
 
    num_rock_units = 5
 num_physical_props = 1
 
@@ -383,7 +318,8 @@
 # Set the background density
 background_density = 0.0
 
-# Initialize the GMM fit with random samples, mesh size, and number of physical properties
+# Initialize the GMM fit with random samples, mesh size,
+# and number of physical properties
 gmmref.fit(np.random.randn(nactv, num_physical_props))
 
 # Set the mean values of physical property contrasts for each rock unit
@@ -400,6 +336,9 @@
 density_variance = 8e-5
 
 # Set the covariances of physical properties for each rock unit
+# NOTE: Since we don't have petrophysical information for this example,
+# we keep the covariances # same for every unit. The GMM will update
+# itself during the inversion.
 gmmref.covariances_ = np.array(
     [
         [[density_variance]],
@@ -414,50 +353,60 @@
 gmmref.compute_clusters_precisions()
 
 # Set the weights for each rock unit
+# NOTE: This determines the size of each peak of the n-modal GMM
 gmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125]
 
    
 
    
-
    GMM Parameters
    
-
    means_init = gmmref.means_.flatten()
-covariances_init = gmmref.covariances_.flatten()
-weights_init = gmmref.weights_[0].flatten()
-
-# Plot the 1D GMM
-fig = plt.figure(figsize=(6, 6))
+
    
+
    
+
    Plot the 1D-GMM¶
    
+
    The initial density distribution is visualized.
    
+
    fig = plt.figure(figsize=(6, 6))
 ax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2)
 ax[0].set_xlabel(r"Density Contrast (g/cc)")
 ax[0].set_ylabel(r"Probability Density Values")
 ax[0].get_legend().remove()
 ax[0].set_title(r"Initial DelRho Distribution")
 ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0))
-ax[0].set_aspect(1/20)
-plt.savefig(path_to_output + "/" + name + "_Init_GMM.pdf", bbox_inches="tight")
+ax[0].set_aspect(1 / 20)
+plt.savefig(
+    path_to_output + "/" + name + "_Init_GMM.pdf",
+    bbox_inches="tight"
+)
 plt.show()
 
    
 
    
 
      
-
    • 01 gravity gradiometry inversion pgi
      • 
-
    • Initial DelRho Distribution
      • 
+
    • 01 gravity gradiometry inversion pgi
      • 
+
    • Initial DelRho Distribution
      • 
 
    
-
    # Define the inversion parameters
-%%
    
-
    
-
    Initial Model
    
-
    
+
    
+
    
+
    Step 9: Setting the Hyper-parameters and the Sensitivity Weights¶
    
+
    Every PGI is a set of three Maximum-A-Posteriori (MAP)
+problems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both
+the FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise
+the necessary weights for every voxel (as the contribution of every voxel is dependent on its depth from the surface).
+The regularization smallness (\(\alpha_s\)) and smoothness (\(\alpha_i,\ i = x, y, z\)) are initialised here. Please check
+here
+for the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion.
    
+
    NOTE: The smoothness parameters (alpha_i) are static and hence need to be fine-tuned
+through trial and error.
    
 
    initial_model = np.r_[background_density * np.ones(actvMap.nP)]
-
-# Sensitivity weighting
-# Compute the sensitivity weights for each cell based on the gravity problem's sensitivity matrix (G)
-# This is done by computing the square root of the sum of the squared elements of G for each cell,
-# and then normalizing by the cell volumes and the maximum weight value.
-sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv])
+
    
+
    
+
    Sensitivity weighting
+Compute the sensitivity weights for each cell based on the gravity problem’s sensitivity matrix (G)
+This is done by computing the square root of the sum of the squared elements of G for each cell,
+and then normalizing by the cell volumes and the maximum weight value.
    
+
    sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv])
 sensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity)
 
 # Regularization multipliers
 smallness_multiplier = 1e-4
-smoothness_x_multiplier = 1e-1
-smoothness_y_multiplier = 1e-1
-smoothness_z_multiplier = 1e-1
+smoothness_x_multiplier = 4
+smoothness_y_multiplier = 4
+smoothness_z_multiplier = 0.25
 
 # Create joint PGI regularization with smoothness
 regularization_term = regularization.PGI(
@@ -470,17 +419,21 @@
     alpha_x=smoothness_x_multiplier,
     alpha_y=smoothness_y_multiplier,
     alpha_z=smoothness_z_multiplier,
+    # The second derivative smoothnesses are kept to be zero, since we
+    # don't want to include second derivatives in the objective function
     alpha_xx=0.0,
     alpha_yy=0.0,
     alpha_zz=0.0,
     weights_list=[sensitivity_weights_gravity]
 )
-
-# Directives for inversion
-# Define the directives for the inversion process, based on Astic et al., (2021)
-
-# Estimate smoothness multipliers
-alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True)
+
    
+
    
+
    
+
    
+
    Step 10: Initialize the Directives¶
    
+
    Directives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver.
    
+
    Estimate smoothness multipliers
    
+
    alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True)
 
 # Initialize beta and beta/alpha_s schedule
 beta_directive = directives.BetaEstimate_ByEig(beta0_ratio=1e-2)
@@ -495,28 +448,37 @@
 target_misfits = directives.MultiTargetMisfits(verbose=True)
 
 # Add reference model once stable
-mref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True)
+mref_in_smooth = directives.PGI_AddMrefInSmooth(
+    wait_till_stable=True,
+    verbose=True
+)
 
 # Update smallness parameters, keeping GMM fixed (L2 Approx of PGI)
 update_smallness_directive = directives.PGI_UpdateParameters(
     update_gmm=True,
-    kappa = 0,
-    nu = 0.5,
-    zeta = 0
+    kappa=0,
+    nu=0.5,
+    zeta=0
 )
 
 # Update preconditioner
 update_preconditioner = directives.UpdatePreconditioner()
 
 # Save iteration results
-save_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output)
+save_iteration_directive = directives.SaveOutputEveryIteration(
+    name=name,
+    directory=path_to_output
+)
 
 # Save model iterations
-save_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations)
+save_model_directive = directives.SaveModelEveryIteration(
+    name=name,
+    directory=path_to_mod_iterations
+)
 
 # Optimization options for the inversion
-lower_bound = np.r_[-1 * np.ones(actvMap.nP)]
-upper_bound = np.r_[1 * np.ones(actvMap.nP)]
+lower_bound = np.r_[-1.0 * np.ones(actvMap.nP)]
+upper_bound = np.r_[1.0 * np.ones(actvMap.nP)]
 optimizer = optimization.ProjectedGNCG(
     maxIter=20,
     lower=lower_bound,
@@ -544,11 +506,10 @@
 )
 
    
 
    
-
    # Run the Inversion!
-%%
    
-
    
-
    Run the inversion
    
-
    
+
    
+
    
+
    Step 11: Run the Inversion!¶
    
+
    The inversion is executed using the initialized settings.
    
 
    inverted_model = inversion_algo.run(initial_model)
 
    
 
    
@@ -558,132 +519,101 @@
                     SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
                     ***Done using the default solver SolverLU and no solver_opts.***
 
-Alpha scales: [13303597.345293514, 0.0, 13291291.128584256, 0.0, 96004.25899069471, 0.0]
+Alpha scales: [410030855.8629278, 0.0, 423924847.8682445, 0.0, 221301.76852332294, 0.0]
 <class 'SimPEG.regularization.pgi.PGIsmallness'>
-SimPEG.SaveOutputEveryIteration will save your inversion progress as: '###-Ireland_FTG_NonNeg-2023-12-14-16-07.txt'
-SimPEG.SaveModelEveryIteration will save your models as: './Ireland_FTG_NonNeg/Model Iterations/###-Ireland_FTG_NonNeg-2023-12-14-16-07.npy'
+SimPEG.SaveOutputEveryIteration will save your inversion progress as: '###-Ireland_FTG-2024-01-26-10-56.txt'
+SimPEG.SaveModelEveryIteration will save your models as: './Ireland_FTG/Model Iterations/###-Ireland_FTG-2024-01-26-10-56.npy'
 model has any nan: 0
 =============================== Projected GNCG ===============================
   #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
 -----------------------------------------------------------------------------
 x0 has any nan: 0
-   0  3.79e-08  1.59e+04  0.00e+00  1.59e+04    2.06e+02      0
-geophys. misfits: 6396.9 (target 6825.0 [True]) | smallness misfit: 31833.0 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [6396.9]; minimum progress targets: [12694.7]
-Warming alpha_pgi to favor clustering:  0.00010669177094483827
-mref changed in  29837  places
-   1  3.79e-08  6.40e+03  1.12e+10  6.82e+03    1.64e+02      0
-geophys. misfits: 3443.7 (target 6825.0 [True]) | smallness misfit: 32765.9 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3443.7]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.00021145111553427776
-mref changed in  14173  places
-   2  3.79e-08  3.44e+03  1.09e+10  3.86e+03    6.78e+01      0
-geophys. misfits: 3443.1 (target 6825.0 [True]) | smallness misfit: 32692.5 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3443.1]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.0004191377804309684
-mref changed in  2336  places
-   3  3.79e-08  3.44e+03  1.08e+10  3.85e+03    5.91e+01      2   Skip BFGS
-geophys. misfits: 3434.6 (target 6825.0 [True]) | smallness misfit: 32301.7 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3434.6]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.0008328763975470134
-mref changed in  9637  places
-   4  3.79e-08  3.43e+03  1.09e+10  3.85e+03    7.63e+01      0
-geophys. misfits: 3410.0 (target 6825.0 [True]) | smallness misfit: 32353.0 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3410.]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.001666997423849284
-mref changed in  2304  places
-   5  3.79e-08  3.41e+03  1.08e+10  3.82e+03    6.27e+01      1
-geophys. misfits: 3405.8 (target 6825.0 [True]) | smallness misfit: 32305.6 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3405.8]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.003340537806642481
-mref changed in  548  places
-   6  3.79e-08  3.41e+03  1.08e+10  3.82e+03    6.47e+01      3   Skip BFGS
-geophys. misfits: 3394.3 (target 6825.0 [True]) | smallness misfit: 32227.0 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3394.3]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.006716901289141315
-mref changed in  853  places
-   7  3.79e-08  3.39e+03  1.08e+10  3.80e+03    5.97e+01      2   Skip BFGS
-geophys. misfits: 3379.7 (target 6825.0 [True]) | smallness misfit: 32130.3 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3379.7]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.013563981510234007
-mref changed in  2484  places
-   8  3.79e-08  3.38e+03  1.08e+10  3.79e+03    4.83e+01      1   Skip BFGS
-geophys. misfits: 3375.7 (target 6825.0 [True]) | smallness misfit: 32093.2 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3375.7]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.02742395715929332
-mref changed in  1765  places
-   9  3.79e-08  3.38e+03  1.08e+10  3.79e+03    3.75e+01      1   Skip BFGS
-geophys. misfits: 3375.5 (target 6825.0 [True]) | smallness misfit: 32073.9 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3375.5]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.05544992823765521
-mref changed in  451  places
-  10  3.79e-08  3.38e+03  1.08e+10  3.79e+03    3.79e+01      2
-geophys. misfits: 3366.7 (target 6825.0 [True]) | smallness misfit: 32451.3 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3366.7]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.11240918288836876
-mref changed in  3181  places
-  11  3.79e-08  3.37e+03  1.10e+10  3.78e+03    3.36e+01      0   Skip BFGS
-geophys. misfits: 3365.8 (target 6825.0 [True]) | smallness misfit: 32609.1 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3365.8]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.22793964709919495
-mref changed in  332  places
-  12  3.79e-08  3.37e+03  1.12e+10  3.79e+03    3.18e+01      2
-geophys. misfits: 3366.4 (target 6825.0 [True]) | smallness misfit: 33336.6 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3366.4]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.4621242063572519
-mref changed in  563  places
-  13  3.79e-08  3.37e+03  1.16e+10  3.81e+03    2.79e+01      1
-geophys. misfits: 3366.1 (target 6825.0 [True]) | smallness misfit: 34792.5 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3366.1]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  0.9369936514939342
-mref changed in  471  places
-  14  3.79e-08  3.37e+03  1.27e+10  3.85e+03    3.56e+01      1
-geophys. misfits: 3366.7 (target 6825.0 [True]) | smallness misfit: 34950.6 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3366.7]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  1.8994705672039287
-mref changed in  56  places
-  15  3.79e-08  3.37e+03  1.49e+10  3.93e+03    5.04e+01      5   Skip BFGS
-geophys. misfits: 3384.5 (target 6825.0 [True]) | smallness misfit: 37831.4 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3384.5]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  3.830382755121003
-mref changed in  122  places
-  16  3.79e-08  3.38e+03  2.08e+10  4.17e+03    7.88e+01      1
-geophys. misfits: 3396.2 (target 6825.0 [True]) | smallness misfit: 35625.0 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3396.2]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  7.697450612448777
-mref changed in  431  places
-  17  3.79e-08  3.40e+03  3.60e+10  4.76e+03    1.17e+02      0
-geophys. misfits: 3508.6 (target 6825.0 [True]) | smallness misfit: 30940.9 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3508.6]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  14.973218112151427
-mref changed in  299  places
-  18  3.79e-08  3.51e+03  5.79e+10  5.70e+03    1.35e+02      0
-geophys. misfits: 3692.8 (target 6825.0 [True]) | smallness misfit: 29180.6 (target: 27300.0 [False])
-Beta cooling evaluation: progress: [3692.8]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  27.673539996945806
-mref changed in  231  places
-  19  3.79e-08  3.69e+03  9.25e+10  7.20e+03    1.44e+02      0
-geophys. misfits: 4215.7 (target 6825.0 [True]) | smallness misfit: 26238.4 (target: 27300.0 [True])
+   0  5.50e-11  1.76e+04  0.00e+00  1.76e+04    2.82e+02      0
+geophys. misfits: 7363.5 (target 7492.5 [True]) | smallness misfit: 53932.6 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [7363.5]; minimum progress targets: [14114.6]
+Warming alpha_pgi to favor clustering:  0.0001017520433220537
+mref changed in  46923  places
+   1  5.50e-11  7.36e+03  2.93e+12  7.52e+03    2.40e+02      0
+geophys. misfits: 2878.4 (target 7492.5 [True]) | smallness misfit: 56886.0 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [2878.4]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.00026486260428953257
+mref changed in  24033  places
+   2  5.50e-11  2.88e+03  2.82e+12  3.03e+03    1.76e+02      0
+geophys. misfits: 1902.3 (target 7492.5 [True]) | smallness misfit: 59150.1 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1902.3]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.0010432238094228118
+mref changed in  13728  places
+   3  5.50e-11  1.90e+03  2.90e+12  2.06e+03    1.25e+02      0   Skip BFGS
+geophys. misfits: 1377.6 (target 7492.5 [True]) | smallness misfit: 60406.7 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1377.6]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.005673861421188311
+mref changed in  9162  places
+   4  5.50e-11  1.38e+03  2.90e+12  1.54e+03    8.08e+01      0
+geophys. misfits: 1370.3 (target 7492.5 [True]) | smallness misfit: 60168.8 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1370.3]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.031023047077501028
+mref changed in  1935  places
+   5  5.50e-11  1.37e+03  2.88e+12  1.53e+03    1.12e+02      2   Skip BFGS
+geophys. misfits: 1367.2 (target 7492.5 [True]) | smallness misfit: 60140.0 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1367.2]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.1700062887019205
+mref changed in  493  places
+   6  5.50e-11  1.37e+03  2.88e+12  1.53e+03    1.15e+02      4   Skip BFGS
+geophys. misfits: 1323.8 (target 7492.5 [True]) | smallness misfit: 60296.1 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1323.8]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.9622012648565086
+mref changed in  1397  places
+   7  5.50e-11  1.32e+03  2.86e+12  1.48e+03    1.00e+02      2   Skip BFGS
+geophys. misfits: 1245.7 (target 7492.5 [True]) | smallness misfit: 61068.6 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1245.7]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  5.787288226899011
+mref changed in  5580  places
+   8  5.50e-11  1.25e+03  2.92e+12  1.41e+03    6.00e+01      0
+geophys. misfits: 1241.2 (target 7492.5 [True]) | smallness misfit: 61073.9 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1241.2]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  34.93527456181403
+mref changed in  510  places
+   9  5.50e-11  1.24e+03  3.02e+12  1.41e+03    6.36e+01      3   Skip BFGS
+geophys. misfits: 1234.0 (target 7492.5 [True]) | smallness misfit: 61269.3 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1234.]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  212.1246047516577
+mref changed in  306  places
+  10  5.50e-11  1.23e+03  3.66e+12  1.44e+03    6.63e+01      3
+geophys. misfits: 1215.3 (target 7492.5 [True]) | smallness misfit: 68474.0 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1215.3]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  1307.8129600116547
+mref changed in  153  places
+  11  5.50e-11  1.22e+03  8.38e+12  1.68e+03    5.77e+01      1   Skip BFGS
+geophys. misfits: 1206.7 (target 7492.5 [True]) | smallness misfit: 67958.3 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1206.7]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  8120.550411338
+mref changed in  11  places
+  12  5.50e-11  1.21e+03  4.07e+13  3.44e+03    2.13e+02      0
+geophys. misfits: 1463.9 (target 7492.5 [True]) | smallness misfit: 43152.9 (target: 44885.0 [True])
 All targets have been reached
-Beta cooling evaluation: progress: [4215.7]; minimum progress targets: [8190.]
-Warming alpha_pgi to favor clustering:  44.80242420960896
-mref changed in  308  places
+Beta cooling evaluation: progress: [1463.9]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  41561.559728045744
+mref changed in  31  places
 ------------------------- STOP! -------------------------
-1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.5869e+03
-0 : |xc-x_last| = 6.4899e+00 <= tolX*(1+|x0|) = 1.0000e-01
-0 : |proj(x-g)-x|    = 1.4333e+02 <= tolG          = 1.0000e-01
-0 : |proj(x-g)-x|    = 1.4333e+02 <= 1e3*eps       = 1.0000e-02
-1 : maxIter   =      20    <= iter          =     20
+1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.7644e+03
+0 : |xc-x_last| = 1.1471e+01 <= tolX*(1+|x0|) = 1.0000e-01
+0 : |proj(x-g)-x|    = 2.0926e+02 <= tolG          = 1.0000e-01
+0 : |proj(x-g)-x|    = 2.0926e+02 <= 1e3*eps       = 1.0000e-02
+0 : maxIter   =      20    <= iter          =     13
 ------------------------- DONE! -------------------------
 
    
 
    
+
    +
    +

    Step 12: Visualize the 3D Model¶

    +

    The inverted 3D model is visualized using slices in different orientations.

    set = 1
 save_plots = True
 
 # Indices of the depth sections
-ind_plot_x = int(len(inv_mesh.cell_centers_x)/6) + 10
-ind_plot_y = int(len(inv_mesh.cell_centers_y)/2)
-ind_plot_z = int(len(inv_mesh.cell_centers_z)/2) - 17
+ind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25
+ind_plot_y = int(len(inv_mesh.cell_centers_y) / 2)
+ind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15
 
 ind_plot = [ind_plot_x, ind_plot_y, ind_plot_z]
 
@@ -694,35 +624,70 @@
 # Plot Density Contrast Model (Z)
 normal = "Z"
 model_to_plot = inverted_density_model
-SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
+SH.plot_model_slice(
+    mesh=inv_mesh,
+    ind_active=actv,
+    model=model_to_plot,
+    normal= normal,
+    ind_plot_arr=ind_plot,
+    clim=[-1.0, 1.0],
+    set=set, sec_loc=True,
+    gdlines=True,
+    which_prop="Den",
+    cmap="Spectral",
+    save_plt=save_plots,
+    path_to_output=path_to_output,
+    name=name
+)
 
 # Plot Inverted Model Slices (Y)
 normal = "X"
 model_to_plot = inverted_density_model
-SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
 
 # Plot Inverted Model Slices (X)
 normal = "Y"
 model_to_plot = inverted_density_model
-SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1, 1], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
      -
    • Inverted Den Model Slice at Z = -4.26E+02 m
      • -
    • Inverted Den Model Slice at X = 1.60E+05 m
      • -
    • Inverted Den Model Slice at Y = 1.46E+05 m
      • +
    • Inverted Den Model Slice at Z = -2.12E+02 m
      • +
    • Inverted Den Model Slice at X = 1.62E+05 m
      • +
    • Inverted Den Model Slice at Y = 1.46E+05 m
    -

    Updated GMM Parameters

    -
    learned_mu = regularization_term.objfcts[0].gmm.means_.flatten()
-learned_sigma = regularization_term.objfcts[0].gmm.covariances_.flatten()
-learned_pi = regularization_term.objfcts[0].gmm.weights_[0].flatten()
+
    +
    +

    Step 13: Visualize the Quasi-Geological Model¶

    +

    The quasi-geological model (as given by Li et al., 2019) is visualized, showing different rock units classified by the GMM.

    +

    Plot Density Contrast Model (Z)

    +
    normal = "Z"
+model_to_plot = quasi_geology_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name)
 
-# Regularization Parameters
-final_alpha_pgi = regularization_term.alpha_pgi
+# Plot Inverted Model Slices (Y)
+normal = "X"
+model_to_plot = quasi_geology_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name)
 
-# Plot the learned GMM
-fig = plt.figure(figsize=(6,6))
-ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.5)
+# Plot Inverted Model Slices (X)
+normal = "Y"
+model_to_plot = quasi_geology_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name)
+
    +
    +
      +
    • Quasi-Geological Model Slice at Z = -2.12E+02 m
      • +
    • Quasi-Geological Model Slice at X = 1.62E+05 m
      • +
    • Quasi-Geological Model Slice at Y = 1.46E+05 m
      • +
    +
    +
    +

    Step 14: Plot the Updated 1D-GMM¶

    +

    The updated GMM, learned through the inversion, is visualized.

    +
    # Plot the learned GMM
+fig = plt.figure(figsize=(6, 6))
+ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4)
 ax[0].hist(inverted_density_model[actv], density=True, bins=1000)
 ax[0].set_xlabel(r"Density contrast (g/cc)")
 ax[0].set_ylabel(r"Probability Density Values")
@@ -734,11 +699,10 @@
      -
    • 01 gravity gradiometry inversion pgi
      • -
    • Learned DelRho Distribution
      • +
    • 01 gravity gradiometry inversion pgi
      • +
    • Learned DelRho Distribution
    -

    sphinx_gallery_thumbnail_number = -4

    -

    Total running time of the script: ( 9 minutes 3.455 seconds)

    +

    Total running time of the script: ( 10 minutes 14.392 seconds)

    Gallery generated by Sphinx-Gallery

    +
    @@ -778,9 +743,27 @@

    Navigation

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/03_forward_engines/02_model_1_gempy_fw_gravity.html b/examples/03_forward_engines/02_model_1_gempy_fw_gravity.html index 392126d..b01ba84 100644 --- a/examples/03_forward_engines/02_model_1_gempy_fw_gravity.html +++ b/examples/03_forward_engines/02_model_1_gempy_fw_gravity.html @@ -22,8 +22,8 @@ - - + + @@ -445,9 +445,10 @@

    Navigation

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/03_forward_engines/03_HSI_To_Petrophysics.html b/examples/03_forward_engines/03_HSI_To_Petrophysics.html new file mode 100644 index 0000000..d37aa7f --- /dev/null +++ b/examples/03_forward_engines/03_HSI_To_Petrophysics.html @@ -0,0 +1,325 @@ + + + + + + + + Predicting P-Wave Velocity from Hyperspectral Data — Vector Geology 0.1.dev16+g525c9e2 documentation + + + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    +
    + + +
    + +
    +

    Note

    +

    Go to the end +to download the full example code

    +
    +
    +

    Predicting P-Wave Velocity from Hyperspectral Data¶

    +

    This example demonstrates the prediction of P-Wave velocities from hyperspectral data using machine learning techniques. Hyperspectral data provides detailed information on rock mineralogy, which influences its petrophysical properties. This script aims to explore and quantify the relationship between spectral features and various petrophysical properties, specifically focusing on P-Wave velocity.

    +
    +

    Importing Necessary Libraries¶

    +

    First, we import the required Python packages. We use dotenv to manage environment variables, numpy and sklearn for numerical operations and machine learning, and custom packages hklearn and hycore for handling hyperspectral data and machine learning workflows.

    +
    from dotenv import dotenv_values
+import numpy as np
+import sklearn
+from sklearn.linear_model import LinearRegression
+from sklearn.neural_network import MLPRegressor
+from tqdm import tqdm
+
+# Custom packages for handling hyperspectral data
+# Note: As of January 2024, these packages are not publicly available. Please contact the author for more information.
+from vector_geology.Packages import hklearn
+from vector_geology.Packages import hycore
+
    +
    +
    +
    +

    Loading Data with hycore.Shed¶

    +

    The hycore package, which organizes hyperspectral data in a structure called Shed, is used here. A Shed makes it convenient to categorize and store hyperspectral data. We use HyLibraries (a data structure within hylite for storing spectral libraries) within a Shed to store the hyperspectral data. The corresponding P-Wave velocity measurements are stored as numpy arrays.

    +
    # Load the train and test sheds
+config = dotenv_values()
+path_to_train = config.get("PATH_TO_HSI_TRAINING")
+path_to_test = config.get("PATH_TO_HSI_TESTING")
+
+train_shed = hycore.loadShed(path_to_train)
+test_shed = hycore.loadShed(path_to_test)
+
+# Load the train and test spectral libraries and P-Wave velocities
+train_fx50 = train_shed.results.FX50Lib
+train_fenix = train_shed.results.FENIXLib
+train_lwir = train_shed.results.LWIRLib
+train_vp = train_shed.results.Vp
+
+test_fx50 = test_shed.results.FX50Lib
+test_fenix = test_shed.results.FENIXLib
+test_lwir = test_shed.results.LWIRLib
+test_vp = test_shed.results.Vp
+
    +
    +
    +
    +

    Creating a hklearn.Stack Object¶

    +

    The hklearn package, designed to handle hyperspectral data, is used to organize spectral libraries into a structure called Stack. A Stack can manage multiple spectral libraries and integrate data from multiple sensors. We perform hull corrections on the spectra to enhance features of interest.

    +
    # Create Stack objects and apply hull correction to the spectra
+hsi_train = hklearn.Stack(
+    names=['FENIX', 'FX50', 'LWIR'],
+    data=[train_fenix, train_fx50, train_lwir]).hc(
+    ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)},
+    hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'}
+)
+
+hsi_test = hklearn.Stack(
+    names=['FENIX', 'FX50', 'LWIR'],
+    data=[test_fenix, test_fx50, test_lwir]).hc(
+    ranges={'FENIX': (450., 2500.), 'FX50': (10, -10), 'LWIR': (10, -10)},
+    hull={'FENIX': 'upper', 'FX50': 'lower', 'LWIR': 'lower'}
+)
+
    +
    +
    Removing hull:   0%|          | 0/3771 [00:00<?, ?it/s]
+Removing hull:   0%|          | 1/3771 [00:01<1:12:24,  1.15s/it]
+
+
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+
+
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+
+
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+
+
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+
+
+Removing hull:   0%|          | 0/1028 [00:00<?, ?it/s]
+
    +
    +
    +
    +

    Scaling the Dependent Variable (Y)¶

    +

    The P-Wave velocities are scaled to facilitate the convergence of the loss function during model training. We use the StandardScaler from scikit-learn for this purpose.

    +
    # Scale Y Variable (P-Wave velocity)
+y_scaler = sklearn.preprocessing.StandardScaler()
+_ = y_scaler.fit(train_vp[:, None])
+
    +
    +
    +
    +

    Transforming the Independent Variable (X)¶

    +

    Hyperspectral data often contains a large number of bands, many of which might be redundant. We apply Principal Component Analysis (PCA) to extract the most significant components. A two-step PCA is performed to ensure that there isn’t any inter-sensor correlation. The Stack object stores the PCA, which is also applied to the test set.

    +
    # Apply PCA to the X variable (spectral data)
+PCA_X = hsi_train.fit_pca(n_components=10, normalise=True)
+hsi_test.set_transform(PCA_X)
+
+# Set the Y-Variable (P-Wave velocity)
+hsi_train.set_y(train_vp[:, None, None])
+hsi_test.set_y(test_vp[:, None, None])
+
    +
    +
    +
    +

    Initializing and Training Models¶

    +

    Different machine learning models are initialized and trained. In this example, we use a simple linear regression (LinearRegression) and a Multilayer Perceptron (MLPRegressor) from scikit-learn. A dictionary with the parameters for optimization is also initialized.

    +
    # Initialize models
+models = dict(
+    Linear=LinearRegression(),
+    MLP=MLPRegressor(
+        hidden_layer_sizes=(180,),
+        max_iter=1000,
+        solver='sgd',
+        learning_rate='adaptive'
+    )
+)
+
+# Define parameter ranges for each model
+params = dict(Linear={'fit_intercept': [True, False]},
+              MLP={"alpha": np.linspace(1e-4, 1e0)})
+
+# Train the models
+for name, model in tqdm(models.items(), 'Fitting models', leave=True):
+    hsi_train.fit_model(name, model, xtransform=True, ytransform=y_scaler, grid_search_cv=params[name], n_jobs=-1)
+
    +
    +
    Fitting models:   0%|          | 0/2 [00:00<?, ?it/s]
+Fitting models:  50%|#####     | 1/2 [00:00<00:00,  1.06it/s]
+Fitting models: 100%|##########| 2/2 [00:32<00:00, 18.92s/it]
+Fitting models: 100%|##########| 2/2 [00:32<00:00, 16.22s/it]
+
    +
    +
    +
    +

    Scoring and Evaluating Models¶

    +

    Finally, the trained models are scored and evaluated. The Stack.get_score_table() method is used to display a table with the training, cross-validation, and test scores of the best-performing model.

    +
    # Evaluate models and display score table
+hsi_train.get_score_table(hsi_test, y_test=None, style=True)
+
    +
    +
    + + + + + + + + + + + + + + + + + + + + + + + + +
     train set5-fold CVtest set
    Linear0.6860.6620.614
    MLP0.8760.8220.695
    + +
    +
    +

    Total running time of the script: ( 0 minutes 34.561 seconds)

    + +

    Gallery generated by Sphinx-Gallery

    +
    +
    + + +
    + +
    +
    + +
    +
    +
    + ©2023-2024, Vector Geology Team. + + | + Powered by Sphinx 7.1.1 + & Alabaster 0.7.13 + + | + Page source +
    + + + + + + \ No newline at end of file diff --git a/examples/03_forward_engines/04_FTG_PGI.html b/examples/03_forward_engines/04_FTG_PGI.html new file mode 100644 index 0000000..202b993 --- /dev/null +++ b/examples/03_forward_engines/04_FTG_PGI.html @@ -0,0 +1,699 @@ + + + + + + + + Inversion of Full Tensor Gravity Gradiometry Data — Vector Geology 0.1.dev16+g525c9e2 documentation + + + + + + + + + + + + + + + + + + + + + + + + + +
    +
    +
    + + +
    + +
    +

    Note

    +

    Go to the end +to download the full example code

    +
    +
    +

    Inversion of Full Tensor Gravity Gradiometry Data¶

    +

    This example illustrates the inversion of sub-surface density models from Full-Tensor Gravity Gradiometry (FTG-G) data, measured over the Irish Midlands, using Petrophysically and Geologically guided Inversion (PGI). The key steps include:

    +
      +

    • Loading necessary modules and data

      • +

    • Generating a mesh

      • +

    • Initializing the Gaussian Mixture Model

      • +

    • Tuning hyper-parameters

      • +

    • Visualizing the results

      • +
    +

    The process is based on the PGI method introduced by Astic and Oldenburg (2019), and it extends the original example notebook from SimPEG’s documentation (see [SimPEG docs](https://docs.simpeg.xyz/index.html) for more information).

    +
    import os
+import shutil
+from dotenv import dotenv_values
+
+import SimPEG.potential_fields as pf
+# Import the SimPEG functions
+import discretize as ds
+import matplotlib as mpl
+import matplotlib.pyplot as plt
+# Import the general modules
+import numpy as np
+import pandas as pd
+from SimPEG import (maps, data, utils, inverse_problem, inversion, optimization, regularization, data_misfit, directives)
+from SimPEG.potential_fields import gravity as grav
+# Import the discretize module
+from discretize.utils import active_from_xyz
+
+# Import SimpegHelper for plotting and resampling data
+from vector_geology import SimpegHelper as SH
+
+# Plot beautification
+formatter = mpl.ticker.ScalarFormatter(useMathText=True)
+formatter.set_scientific(True)
+formatter.set_powerlimits((0, 0))
+
+mpl.rc("axes", titlesize=14, labelsize=12)
+mpl.rc("xtick", labelsize=12)
+mpl.rc("ytick", labelsize=12)
+
    +
    +
    +

    Step 1: Create Directories for Storing Model Iterations and Outputs¶

    +

    PGI is an iterative process, and it’s useful to save the model at every iteration. This helps in preventing overfitting and in adjusting the stopping criteria based on the model iterations.

    +
    name = "Ireland_FTG"
+path_to_mod_iterations = f"./{name}/Model Iterations"
+path_to_output = f"./{name}/Output"
+
+if os.path.exists(path_to_mod_iterations):
+    shutil.rmtree(path_to_mod_iterations)
+os.makedirs(path_to_mod_iterations)
+
+if os.path.exists(path_to_output):
+    shutil.rmtree(path_to_output)
+os.makedirs(path_to_output)
+
    +
    +
    +
    +

    Step 2: Load the FTG-G Data¶

    +

    Full Tensor Gradiometry data, a 2-Tensor, contains five independent components due to its symmetric and traceless nature. These components are used for the inversion. +$$ +G = left[begin{array}{cc} +G_{xx} & G_{xy} & G_{xz}\ +G_{xy} & G_{yy} & G_{yz}\ +G_{xz} & G_{yz} & G_{zz} +end{array}right] +$$

    +
    config = dotenv_values()
+file_path = config.get("PATH_TO_GRADIOMETRY")
+FTG_Data = pd.read_csv(file_path, delimiter=",").to_numpy()
+
    +
    +
    +
    +

    Step 3: Resample Data onto a Regular Grid¶

    +

    Resampling the data onto a regular grid aids in mesh generation and visualization. The SimpegHelper.pf_rs() function resamples potential field data onto a new regular grid.

    +
    inc = 150  # New sampling interval (ground units)
+grav_new, nx_new, ny_new = SH.pf_rs(FTG_Data, inc, bounds=[156000, 168000, 143000, 148500])
+
+# NOTE: This step is necessary for real data since the convention followed by
+# the SimPEG forward operator is the opposite of the general convention
+grav_vec = grav_new[:, 3:] * np.array([-1.0, -1.0, 1.0, 1.0, -1.0])[None, :]  # Adjusting sign convention
+inv_topo = grav_new[:, [0, 1, 2]]  # Extracting topography
+
    +
    +
    +
    +

    Step 4: Visualize the Data¶

    +

    The SimpegHelper.plot_2D_data() function is used to visualize the datasets with matplotlib.

    +
    SH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], grav_vec[:, 0]], [np.nanmin(grav_vec[:, 0]), np.nanmax(grav_vec[:, 0])], cmap="jet", which_data="FTG", comp="xx", path_to_output=path_to_output, name=name)
+SH.plot_2D_data(np.c_[grav_new[:, [0, 1, 2]], inv_topo[:, -1]], [np.nanmin(inv_topo[:, -1]), np.nanmax(inv_topo[:, -1])], cmap="terrain", which_data="Topo", comp="xx", path_to_output=path_to_output, name=name)
+
    +
    +
      +
    • FTG ($G_{xx}$)
      • +
    • Topography
      • +
    +
    +
    +

    Step 5: Create a TensorMesh Object for Inversion¶

    +

    A 3D TensorMesh is created using the discretize.TensorMesh utility to invert the gravity gradiometry data. The mesh is designed to have a constant cell size in the core region and expanding/contracting cells in the padding region.

    +

    $$ +z_{n+1} = r z_{n} +$$

    +

    Where $r$ is a multiplicative factor ($<1$ for contracting, $>1$ for expanding cells). $x$ and $y$ increments can be altered as well, but we choose not to, as the data are not spatially clustered (i.e., the entire region is an area of interest).

    +
    dx, dy, dz = inc, inc, 10  # Cell sizes
+nz_core, nz_pad = 10, 18   # Number of cells in core and padding
+fact = 1.1  # Factor for expanding/contracting cells
+
+inv_hx = dx * np.ones(nx_new)
+inv_hy = dy * np.ones(ny_new)
+inv_hz = [(dz, nz_pad, -fact), (dz, nz_core), (dz, nz_pad, fact)]
+
+# Create the inverse tensor mesh
+inv_mesh = ds.TensorMesh([inv_hx, inv_hy, inv_hz], x0=[np.min(inv_topo[:, 0]), np.min(inv_topo[:, 1]), "C"])
+
+# Drape the topography over the mesh
+actv = active_from_xyz(inv_mesh, inv_topo)
+ndv = np.nan
+actvMap = maps.InjectActiveCells(inv_mesh, actv, ndv)
+nactv = int(actv.sum())
+
+# Print mesh details
+print(inv_mesh)
+
    +
    +
    TensorMesh: 137,862 cells
+
+                    MESH EXTENT             CELL WIDTH      FACTOR
+dir    nC        min           max         min       max      max
+---   ---  ---------------------------  ------------------  ------
+ x     81    156,000.00    168,150.00    150.00    150.00    1.00
+ y     37    143,000.00    148,550.00    150.00    150.00    1.00
+ z     46       -551.59        551.59     10.00     55.60    1.10
+
    +
    +
    +
    +

    Step 6: Visualize the Mesh¶

    +

    A slice of the Tensor Mesh object is visualized using the discretize.TensorMesh.plot_slice() utility.

    +

    Create a background model.

    +
    bg = np.ones(nactv)
+mod = actvMap * bg
+
+# Plot the mesh slice.
+fig = plt.figure(figsize=(6, 6))
+ax = fig.add_axes([0.1, 0.1, 0.8, 0.8])
+cplot = inv_mesh.plot_slice(mod, grid=True, ax=ax, normal="Y", ind=22)
+ax.set_title('Tensor Mesh Slice')
+ax.set_xlabel('x (m)')
+ax.set_ylabel('z (m)')
+ax.set_xlim([158000, 159000])
+ax.set_ylim([-500, 100])
+ax.ticklabel_format(axis="both")
+ax.set_aspect('equal')
+plt.savefig(path_to_output + "/" + name + "_TreeMeshSlice.pdf", bbox_inches="tight")
+plt.show()
+
    +
    +Tensor Mesh Slice
    +
    +

    Step 7: Set up the Gravity Inverse Problem¶

    +

    The gravity inverse problem is set up using SimPEG’s pf.gravity module to generate a forward model based on the mesh and receiver locations.

    +

    Create wires for the density model

    +
    wires = maps.Wires(("density", nactv))
+
+# Create a density map
+density_map = actvMap * wires.density
+
+# Create an identity map
+identity_map = maps.IdentityMap(nP=nactv)
+
+# Components of the data used as input
+gravity_components = ["gxx", "gyy", "gxz", "gyz", "gxy"]
+
+# Receiver locations (25 meters above the topography)
+gravity_receiver_locations = inv_topo + 120.
+
+# Create a gravity receiver list
+gravity_receivers = pf.gravity.receivers.Point(gravity_receiver_locations, components=gravity_components)
+gravity_receiver_list = [gravity_receivers]
+
+# Create a gravity source field
+gravity_source_field = pf.gravity.sources.SourceField(receiver_list=gravity_receiver_list)
+
+# Define the gravity survey
+gravity_survey = pf.gravity.survey.Survey(gravity_source_field)
+
+# Set up the gravity simulation problem
+gravity_problem = grav.simulation.Simulation3DIntegral(
+    inv_mesh,
+    survey=gravity_survey,
+    rhoMap=wires.density,
+    ind_active=actv
+)
+
+# Create a gravity data object, with the relative errors and a standard noise floor
+gravity_data = data.Data(gravity_survey, dobs=grav_vec.flatten(), noise_floor=5, relative_error=0.1)
+
+# Define the misfits associated with the gravity data
+gravity_misfit = data_misfit.L2DataMisfit(data=gravity_data, simulation=gravity_problem)
+
    +
    +
    +
    +

    Step 8: Setting up the Gaussian Mixture Model (GMM) Prior¶

    +

    In PGI, the petrophysical data is used as a constraint in the form of a [Gaussian Mixture Model](https://scikit-learn.org/stable/modules/mixture.html). A GMM is a multimodal probabilistic distribution which is just a weighted sum of multiple gaussian distributions. Given the number of rock units ($n$), the petrophysical distribution can be displayed as an $n$-modal GMM. If you have no petrophysical information available, you can initialize the GMM as below. If you do have a petrophysical dataset, you can fit the GMM to said dataset using the gmmref.fit() method.

    +

    Number of rock units and number of physical properties

    +
    num_rock_units = 5
+num_physical_props = 1
+
+# Create a weighted Gaussian mixture model with specified parameters
+gmmref = utils.WeightedGaussianMixture(
+    n_components=num_rock_units,
+    mesh=inv_mesh,
+    actv=actv,
+    covariance_type="full",
+)
+
+# Set the background density
+background_density = 0.0
+
+# Initialize the GMM fit with random samples, mesh size, and number of physical properties
+gmmref.fit(np.random.randn(nactv, num_physical_props))
+
+# Set the mean values of physical property contrasts for each rock unit
+# One value (density) for each rock unit
+gmmref.means_ = np.c_[
+    [-0.4],
+    [-0.2],
+    [0.0],
+    [0.2],
+    [0.4],
+].T
+
+# Set the original variance for density
+density_variance = 8e-5
+
+# Set the covariances of physical properties for each rock unit
+# NOTE: Since we don't have petrophysical information for this example, we keep the covariances
+# same for every unit. The GMM will update itself during the inversion.
+gmmref.covariances_ = np.array(
+    [
+        [[density_variance]],
+        [[density_variance]],
+        [[density_variance]],
+        [[density_variance]],
+        [[density_variance]]
+    ]
+)
+
+# Compute the precision (inverse covariance) of each cluster
+gmmref.compute_clusters_precisions()
+
+# Set the weights for each rock unit
+# NOTE: This determines the size of each peak of the n-modal GMM
+gmmref.weights_ = np.ones((nactv, 1)) * np.c_[0.125, 0.125, 0.5, 0.125, 0.125]
+
    +
    +
    +
    +

    Plot the 1D-GMM¶

    +

    The initial density distribution is visualized.

    +
    fig = plt.figure(figsize=(6, 6))
+ax = gmmref.plot_pdf(flag2d=False, plotting_precision=1000, padding=0.2)
+ax[0].set_xlabel(r"Density Contrast (g/cc)")
+ax[0].set_ylabel(r"Probability Density Values")
+ax[0].get_legend().remove()
+ax[0].set_title(r"Initial DelRho Distribution")
+ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0))
+ax[0].set_aspect(1 / 20)
+plt.savefig(path_to_output + "/" + name + "_Init_GMM.pdf", bbox_inches="tight")
+plt.show()
+
    +
    +
      +
    • 04 FTG PGI
      • +
    • Initial DelRho Distribution
      • +
    +
    +
    +

    Step 9: Setting the Hyper-parameters and the Sensitivity Weights¶

    +

    Every PGI is a set of three Maximum-A-Posteriori ([MAP](https://en.wikipedia.org/wiki/Maximum_a_posteriori_estimation)) problems, being solved iteratively. The solver tries to minimize the L2 error of an objective function containing both the FTG Data and the petrophysical GMM. In this section we tune the necessary hyper-parameters, as well as initialise the necessary weights for every voxel (as the contribution of every voxel is dependent on it’s depth from the surface). The regularization smallness ($alpha_s$) and smoothness ($alpha_i,i = x, y, z$) are initialised here. Please check [here](https://giftoolscookbook.readthedocs.io/en/latest/content/fundamentals/index.html) for the physical meaning of these parameters and the fundamentals of a Tikhonov regularized inversion.

    +

    <strong> NOTE </strong> : The smoothness parameters ($alpha_i$) are static and hence need to be fine-tuned through trial and error.

    +
    initial_model = np.r_[background_density * np.ones(actvMap.nP)]
+
    +
    +

    Sensitivity weighting +Compute the sensitivity weights for each cell based on the gravity problem’s sensitivity matrix (G) +This is done by computing the square root of the sum of the squared elements of G for each cell, +and then normalizing by the cell volumes and the maximum weight value.

    +
    sensitivity_weights_gravity = np.sum(gravity_problem.G ** 2.0, axis=0) ** 0.5 / (inv_mesh.cell_volumes[actv])
+sensitivity_weights_gravity = sensitivity_weights_gravity / np.nanmax(sensitivity_weights_gravity)
+
+# Regularization multipliers
+smallness_multiplier = 1e-4
+smoothness_x_multiplier = 4
+smoothness_y_multiplier = 4
+smoothness_z_multiplier = 0.25
+
+# Create joint PGI regularization with smoothness
+regularization_term = regularization.PGI(
+    gmmref=gmmref,
+    mesh=inv_mesh,
+    wiresmap=wires,
+    maplist=[identity_map],
+    active_cells=actv,
+    alpha_pgi=smallness_multiplier,
+    alpha_x=smoothness_x_multiplier,
+    alpha_y=smoothness_y_multiplier,
+    alpha_z=smoothness_z_multiplier,
+    # The second derivative smoothnesses are kept to be zero, since we
+    # don't want to include second derivatives in the objective function
+    alpha_xx=0.0,
+    alpha_yy=0.0,
+    alpha_zz=0.0,
+    weights_list=[sensitivity_weights_gravity]
+)
+
    +
    +
    +
    +

    Step 10: Initialize the Directives¶

    +

    Directives are set, including instructions on the bounds, update factors, and other hyperparameters for the inversion solver.

    +

    Estimate smoothness multipliers

    +
    alphas_directive = directives.AlphasSmoothEstimate_ByEig(verbose=True)
+
+# Initialize beta and beta/alpha_s schedule
+beta_directive = directives.BetaEstimate_ByEig(beta0_ratio=1e-2)
+beta_schedule = directives.PGI_BetaAlphaSchedule(
+    coolingFactor=16.0,
+    tolerance=0.2,
+    progress=0.2,
+    verbose=True,
+)
+
+# Define target misfits for geophysical and petrophysical data
+target_misfits = directives.MultiTargetMisfits(verbose=True)
+
+# Add reference model once stable
+mref_in_smooth = directives.PGI_AddMrefInSmooth(wait_till_stable=True, verbose=True)
+
+# Update smallness parameters, keeping GMM fixed (L2 Approx of PGI)
+update_smallness_directive = directives.PGI_UpdateParameters(
+    update_gmm=True,
+    kappa=0,
+    nu=0.5,
+    zeta=0
+)
+
+# Update preconditioner
+update_preconditioner = directives.UpdatePreconditioner()
+
+# Save iteration results
+save_iteration_directive = directives.SaveOutputEveryIteration(name=name, directory=path_to_output)
+
+# Save model iterations
+save_model_directive = directives.SaveModelEveryIteration(name=name, directory=path_to_mod_iterations)
+
+# Optimization options for the inversion
+lower_bound = np.r_[-1.0 * np.ones(actvMap.nP)]
+upper_bound = np.r_[1.0 * np.ones(actvMap.nP)]
+optimizer = optimization.ProjectedGNCG(
+    maxIter=20,
+    lower=lower_bound,
+    upper=upper_bound,
+    maxIterLS=20,
+    maxIterCG=100,
+    tolCG=1e-4,
+)
+
+# Inverse problem setup
+inverse_prob = inverse_problem.BaseInvProblem(gravity_misfit, regularization_term, optimizer)
+inversion_algo = inversion.BaseInversion(
+    inverse_prob,
+    directiveList=[
+        alphas_directive,
+        beta_directive,
+        update_smallness_directive,
+        target_misfits,
+        beta_schedule,
+        mref_in_smooth,
+        update_preconditioner,
+        save_iteration_directive,
+        save_model_directive
+    ],
+)
+
+# Step 11: Run the Inversion!
+# ---------------------------
+#
+# The inversion is executed using the initialized settings.
+
+inverted_model = inversion_algo.run(initial_model)
+
    +
    +
    SimPEG.InvProblem will set Regularization.reference_model to m0.
+SimPEG.InvProblem will set Regularization.reference_model to m0.
+
+                    SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
+                    ***Done using the default solver SolverLU and no solver_opts.***
+
+Alpha scales: [418083224.1567727, 0.0, 430664170.54921407, 0.0, 241249.22858700654, 0.0]
+<class 'SimPEG.regularization.pgi.PGIsmallness'>
+SimPEG.SaveOutputEveryIteration will save your inversion progress as: '###-Ireland_FTG-2024-01-25-16-16.txt'
+SimPEG.SaveModelEveryIteration will save your models as: './Ireland_FTG/Model Iterations/###-Ireland_FTG-2024-01-25-16-16.npy'
+model has any nan: 0
+=============================== Projected GNCG ===============================
+  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
+-----------------------------------------------------------------------------
+x0 has any nan: 0
+   0  5.49e-11  1.76e+04  0.00e+00  1.76e+04    2.82e+02      0
+geophys. misfits: 7385.5 (target 7492.5 [True]) | smallness misfit: 53905.8 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [7385.5]; minimum progress targets: [14114.6]
+Warming alpha_pgi to favor clustering:  0.00010144902120428576
+mref changed in  46912  places
+   1  5.49e-11  7.39e+03  2.95e+12  7.55e+03    2.40e+02      0
+geophys. misfits: 2841.6 (target 7492.5 [True]) | smallness misfit: 56915.4 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [2841.6]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.0002674939266088537
+mref changed in  23999  places
+   2  5.49e-11  2.84e+03  2.84e+12  3.00e+03    1.75e+02      0
+geophys. misfits: 1907.3 (target 7492.5 [True]) | smallness misfit: 59165.4 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1907.3]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.0010507772902925534
+mref changed in  13880  places
+   3  5.49e-11  1.91e+03  2.93e+12  2.07e+03    1.26e+02      0   Skip BFGS
+geophys. misfits: 1373.7 (target 7492.5 [True]) | smallness misfit: 60358.7 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1373.7]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.005731251890475955
+mref changed in  8975  places
+   4  5.49e-11  1.37e+03  2.93e+12  1.53e+03    8.01e+01      0
+geophys. misfits: 1371.4 (target 7492.5 [True]) | smallness misfit: 60091.0 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1371.4]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.031312537491235436
+mref changed in  1888  places
+   5  5.49e-11  1.37e+03  2.90e+12  1.53e+03    1.11e+02      2   Skip BFGS
+geophys. misfits: 1367.5 (target 7492.5 [True]) | smallness misfit: 60077.9 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1367.5]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.17156327747832578
+mref changed in  534  places
+   6  5.49e-11  1.37e+03  2.90e+12  1.53e+03    1.14e+02      4   Skip BFGS
+geophys. misfits: 1322.5 (target 7492.5 [True]) | smallness misfit: 60252.3 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1322.5]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  0.9719405715672758
+mref changed in  1387  places
+   7  5.49e-11  1.32e+03  2.89e+12  1.48e+03    9.98e+01      2   Skip BFGS
+geophys. misfits: 1247.2 (target 7492.5 [True]) | smallness misfit: 61007.7 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1247.2]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  5.838731886914972
+mref changed in  5572  places
+   8  5.49e-11  1.25e+03  2.94e+12  1.41e+03    5.79e+01      0
+geophys. misfits: 1242.3 (target 7492.5 [True]) | smallness misfit: 61012.7 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1242.3]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  35.21337400407953
+mref changed in  503  places
+   9  5.49e-11  1.24e+03  3.04e+12  1.41e+03    6.17e+01      3   Skip BFGS
+geophys. misfits: 1241.1 (target 7492.5 [True]) | smallness misfit: 61724.2 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1241.1]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  212.57758849285366
+mref changed in  381  places
+  10  5.49e-11  1.24e+03  3.69e+12  1.44e+03    7.45e+01      2
+geophys. misfits: 1222.1 (target 7492.5 [True]) | smallness misfit: 68747.8 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1222.1]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  1303.254486428423
+mref changed in  151  places
+  11  5.49e-11  1.22e+03  8.42e+12  1.68e+03    6.22e+01      1
+geophys. misfits: 1210.3 (target 7492.5 [True]) | smallness misfit: 67778.5 (target: 44885.0 [False])
+Beta cooling evaluation: progress: [1210.3]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  8067.926413603839
+mref changed in  11  places
+  12  5.49e-11  1.21e+03  4.04e+13  3.43e+03    2.13e+02      0
+geophys. misfits: 1464.6 (target 7492.5 [True]) | smallness misfit: 42906.7 (target: 44885.0 [True])
+All targets have been reached
+Beta cooling evaluation: progress: [1464.6]; minimum progress targets: [8991.]
+Warming alpha_pgi to favor clustering:  41274.01718048347
+mref changed in  39  places
+------------------------- STOP! -------------------------
+1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 1.7644e+03
+0 : |xc-x_last| = 1.1470e+01 <= tolX*(1+|x0|) = 1.0000e-01
+0 : |proj(x-g)-x|    = 2.0919e+02 <= tolG          = 1.0000e-01
+0 : |proj(x-g)-x|    = 2.0919e+02 <= 1e3*eps       = 1.0000e-02
+0 : maxIter   =      20    <= iter          =     13
+------------------------- DONE! -------------------------
+
    +
    +
    +
    +

    Step 12: Visualize the 3D Model¶

    +

    The inverted 3D model is visualized using slices in different orientations.

    +
    set = 1
+save_plots = True
+
+# Indices of the depth sections
+ind_plot_x = int(len(inv_mesh.cell_centers_x) / 6) + 25
+ind_plot_y = int(len(inv_mesh.cell_centers_y) / 2)
+ind_plot_z = int(len(inv_mesh.cell_centers_z) / 2) - 15
+
+ind_plot = [ind_plot_x, ind_plot_y, ind_plot_z]
+
+# Extract the results
+inverted_density_model = density_map * inverted_model
+quasi_geology_model = actvMap * regularization_term.objfcts[0].compute_quasi_geology_model()
+
+# Plot Density Contrast Model (Z)
+normal = "Z"
+model_to_plot = inverted_density_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
+
+# Plot Inverted Model Slices (Y)
+normal = "X"
+model_to_plot = inverted_density_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
+
+# Plot Inverted Model Slices (X)
+normal = "Y"
+model_to_plot = inverted_density_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [-1.0, 1.0], set, sec_loc=True, gdlines=True, which_prop="Den", cmap="Spectral", save_plt=save_plots, path_to_output=path_to_output, name=name)
+
    +
    +
      +
    • Inverted Den Model Slice at Z = -2.12E+02 m
      • +
    • Inverted Den Model Slice at X = 1.62E+05 m
      • +
    • Inverted Den Model Slice at Y = 1.46E+05 m
      • +
    +
    +
    +

    Step 13: Visualize the Quasi-Geological Model¶

    +

    The quasi-geological model (as given by [Li et al., 2019](https://doi.org/10.1190/tle38010060.1))is visualized, showing different rock units classified by the GMM.

    +

    Plot Density Contrast Model (Z)

    +
    normal = "Z"
+model_to_plot = quasi_geology_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name)
+
+# Plot Inverted Model Slices (Y)
+normal = "X"
+model_to_plot = quasi_geology_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name)
+
+# Plot Inverted Model Slices (X)
+normal = "Y"
+model_to_plot = quasi_geology_model
+SH.plot_model_slice(inv_mesh, actv, model_to_plot, normal, ind_plot, [0, 4], set, sec_loc=True, gdlines=True, which_prop="QGM", cmap="jet", save_plt=save_plots, path_to_output=path_to_output, name=name)
+
    +
    +
      +
    • Quasi-Geological Model Slice at Z = -2.12E+02 m
      • +
    • Quasi-Geological Model Slice at X = 1.62E+05 m
      • +
    • Quasi-Geological Model Slice at Y = 1.46E+05 m
      • +
    +
    +
    +

    Step 14: Plot the Updated 1D-GMM¶

    +

    The updated GMM, learned through the inversion, is visualized.

    +
    # Plot the learned GMM
+fig = plt.figure(figsize=(6, 6))
+ax = regularization_term.objfcts[0].gmm.plot_pdf(flag2d=False, plotting_precision=500, padding=0.4)
+ax[0].hist(inverted_density_model[actv], density=True, bins=1000)
+ax[0].set_xlabel(r"Density contrast (g/cc)")
+ax[0].set_ylabel(r"Probability Density Values")
+ax[0].get_legend().remove()
+ax[0].set_title(r"Learned DelRho Distribution")
+ax[0].ticklabel_format(axis="both", style="scientific", scilimits=(0, 0))
+plt.savefig(path_to_output + "/" + name + "_Learned_GMM.pdf", bbox_inches="tight")
+plt.show()
+
    +
    +
      +
    • 04 FTG PGI
      • +
    • Learned DelRho Distribution
      • +
    +

    Total running time of the script: ( 10 minutes 53.107 seconds)

    + +

    Gallery generated by Sphinx-Gallery

    +
    +
    + + +
    + +
    +
    + +
    +
    +
    + ©2023-2024, Vector Geology Team. + + | + Powered by Sphinx 7.1.1 + & Alabaster 0.7.13 + + | + Page source +
    + + + + + + \ No newline at end of file diff --git a/examples/03_forward_engines/index.html b/examples/03_forward_engines/index.html index 70a7393..a50aa30 100644 --- a/examples/03_forward_engines/index.html +++ b/examples/03_forward_engines/index.html @@ -5,7 +5,7 @@ - Forward GeoPhysics Engines and Inversion — Vector Geology 0.1.dev16+g525c9e2 documentation + GeoPhysics and Inversion — Vector Geology 0.1.dev16+g525c9e2 documentation @@ -22,7 +22,7 @@ - + @@ -40,8 +40,8 @@
    -
    -

    Forward GeoPhysics Engines and Inversion¶

    +
    +

    GeoPhysics and Inversion¶

    Our tutorials are crafted to guide you through the process of applying geophysical forward modeling and inversion techniques, using advanced tools like gempy and simpeg. Here’s what you can expect:

    • Geophysical Forward Modeling: Learn how to transform structural models into geophysical models. These tutorials will show you how to use gempy to simulate geophysical responses based on your structural models. Whether you are a beginner or looking to refine your modeling skills, these guides offer comprehensive insights.

      • @@ -53,12 +53,15 @@
    -
    -

    Inversion of Gravity Gradiometry Data on a Tree Mesh

    -
    Inversion of Gravity Gradiometry Data on a Tree Mesh
    +
    +

    Inversion of Full Tensor Gravity Gradiometry Data

    +
    Inversion of Full Tensor Gravity Gradiometry Data

    Model 1 Forward Gravity

    Model 1 Forward Gravity
    +
    +

    Predicting P-Wave Velocity from Hyperspectral Data

    +
    Predicting P-Wave Velocity from Hyperspectral Data
    @@ -91,9 +94,10 @@

    Navigation

    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/examples/03_forward_engines/sg_execution_times.html b/examples/03_forward_engines/sg_execution_times.html index 3f3e81c..901cf4c 100644 --- a/examples/03_forward_engines/sg_execution_times.html +++ b/examples/03_forward_engines/sg_execution_times.html @@ -40,17 +40,21 @@

    Computation times¶

    -

    09:03.455 total execution time for examples_03_forward_engines files:

    +

    10:14.392 total execution time for examples_03_forward_engines files:

    - - + + + + + +

    Inversion of Gravity Gradiometry Data on a Tree Mesh (01_gravity_gradiometry_inversion_pgi.py)

    09:03.455

    Inversion of Full Tensor Gravity Gradiometry Data (01_gravity_gradiometry_inversion_pgi.py)

    10:14.392

    0.0 MB

    Model 1 Forward Gravity (02_model_1_gempy_fw_gravity.py)

    00:00.000

    0.0 MB

    Predicting P-Wave Velocity from Hyperspectral Data (03_HSI_To_Petrophysics.py)

    00:00.000

    0.0 MB
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    Structural ModelingConstruct Model 1 with Helper Functions

    Construct Model 1 with Helper Functions

    -
    -

    Forward GeoPhysics Engines and Inversion¶

    +
    +

    GeoPhysics and Inversion¶

    Our tutorials are crafted to guide you through the process of applying geophysical forward modeling and inversion techniques, using advanced tools like gempy and simpeg. Here’s what you can expect:

    Probabilistic Modeling¶

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    0.1.dev16+g525c9e2

    Date:
    -

    14 December 2023

    +

    26 January 2024

    Source:

    github.com/vector-raw-materials/vector-geology

    @@ -89,7 +89,7 @@

    Explore Our GuidesVector Geology Examples @@ -157,8 +157,8 @@

    Structural ModelingConstruct Model 1 with Helper Functions

    Construct Model 1 with Helper Functions

    -
    -

    Forward GeoPhysics Engines and Inversion¶

    +
    +

    GeoPhysics and Inversion¶

    Our tutorials are crafted to guide you through the process of applying geophysical forward modeling and inversion techniques, using advanced tools like gempy and simpeg. Here’s what you can expect:

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    - ©2023-2023, Vector Geology Team. + ©2023-2024, Vector Geology Team. | Powered by Sphinx 7.1.1 diff --git a/objects.inv b/objects.inv index 8be482d21f800d2a94e1b81f2be38039f4aac968..0f3962404a9b6a9c344ded373e3ae68d8acfdcaf 100644 GIT binary patch delta 1470 zcmV;v1ws1$3cL)Ekblc^qc{}Z`zutH*{Zw!SYmC%al;?C;a+*e<;Nfk^0sxmF=(T@2@FWI*^ja3=9Vb~# z5KZ4mUQm{zJ&#a`IaW?$yd$x6)ATF8ep{nG%a1q@5lJJO%73OMigNr$c_p7AWeMSV z)pMCeRE552yukJo6w#oH_Nbcx+k?9@j=>8_3zmcX+O>RY*WkObXrX60327kNpzMg> z2znzi3#5rVgEe}1enc+}ZH0f99jbx`#Sn8B2C6Wy4YO^(fGzw-d;+if(>`Yjc&IW# zEh7QXaXi4Ta(|Z-9E$Jy75(*ehYHF`(H}-9gL-`b7HtxXXco}e4-r>nmou=f_<^P7 z*ejH=t*`1%KZcjT3`gxISzF5Se{Ow%d%|KC(LlVa**pz;HH_tj^ThT9 zTDzW(lz$(<1Bt~pBeu>okU5=Br)7#*3~CI$j65};C*!$J9arn=X*BPsGNZcS*qeX0 znd6?!3ba4kcNi}hHDW!p?rK8~bm4!9Izt*if-gH@4WTyZ?&ThB?;oAPwHDVHP|HFb z>D(L%gl(H7%Wb;_Wew)K3SBIxSKvleXKaUeG=HWAr@>$g<_8-5D;yW9PfQZq206n~ zjc)a21)#`H5SQwdzbDDIazfaxksYo`L2>E}?c9T&&%OS=$<$O8x3*JeG@4}=*=6W1B#;b3xNjg#$BdKDXjT$3u9mCmm8!bzUu_^W=*a zDt}7RmjhvWwJ^JFg-D0+M)!nm&vCC2Njui9Gu^89CGomAOHQxHo5~4{nw)%RL79-W zPVyqbq8cYDPKRZ6uemuLx&~pBic)mgUAE|+@_iwO+JHT)p#IRFI;AM8YZkq)6K1up z8Cn+v=fPoUShqC=O#OX}BeE#6!#j%N9DjxEn8r+0-g-DtSI6pC#FBlZwz5%$Ii=BV zWJoANLLIW50r1-rHpcoDijYtlLlF|X;_Y@H33(`doS;wXI#(7esQkdCBElK${CE-S zl-x{0Q&P~C(7GYCx{%O_-gDw&(Tc7#Hto2mS=P9iW?(GPkzG`cbDAsBtSncsbAJl8 zQ%PPkR!*UG26xSKF7H#3=1tY4^5(Exov@oEbXMabCry)8*Z4PRBfN%o^P=Jx!Xdlch zS{wEE+Ix-;)txW=jrxi0G~dQbV1Hy$G2=|;Y9cGa4IDTfY@V%C*f@3FXvTRPB+?wh zBT8!~({8b8ZzmKDr}^}e-K+asScY@i_$$^H`AU_kb3rt(kfcIsEJb3Myj%Uz0<$jF}#D}j-p8+PnP1Y09m2Ri>5Vt{HaFn|53QTr}s ztUQgk0j}tNqHefYE;wHjxhLbwVq%VlaJ_GGCh5M6ysm=WSAtMt;s?uVkRRLcOg!y|tQl1fymsszqW zPV_P~pf*1lDau+;A|=|JUw0TQfco8PESvTJ)?3li^B@!?&F^n?u)crl{FMqalVy)-P=ms6Rc){X;2+Jrl zb=mEz=J!B(L6oQwWzV`)8{{&h?x@ybnHV4K5a7ptBR>o_W>f3GG59VB0Z&Ap!Zgkb z107$&hs+`f6SOU^dt+$h4#pKIDbHl%tM@_p?o>;fO7PXpY7eA7e?jN5^%Y||Ut70G zzbp25Pk(bN6-_6Hi#pNtZw)NlIXNjpTYm+Ri7xq=0H|}5=2iRtdLYHFj*4(t`aQ0R zq?j!Y?GnMnv>VSK)qVHAybGP5UT=Y_3fs71avWD)kl2b^05DrvezGG}UE> zGr|VeJ(wM}N2q?>TqI{S;b~P6W|F)qu-2J`VKxP9GK269|NgoyZT*ng2B7rRq+_pp%hXc+cLPL+TlF(d}6HDl(Ba~c8$kWM0aS3!m zu{3V`b!n@tgP3RFl%6QuF8DdEZ9al56Mr~>LJQiEw~Sj**nq)(_q&$&dHb=Ls;hrl z!!9{vH%sV33!g0Z&UEj$LN3H-g5|Wee^I)hTP1X`t{a<Navigation

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14: Plot the Updated 1D-GMM": [[14, "step-14-plot-the-updated-1d-gmm"], [11, "step-14-plot-the-updated-1d-gmm"]], "Predicting P-Wave Velocity from Hyperspectral Data": [[13, "predicting-p-wave-velocity-from-hyperspectral-data"]], "Importing Necessary Libraries": [[13, "importing-necessary-libraries"]], "Loading Data with hycore.Shed": [[13, "loading-data-with-hycore-shed"]], "Creating a hklearn.Stack Object": [[13, "creating-a-hklearn-stack-object"]], "Scaling the Dependent Variable (Y)": [[13, "scaling-the-dependent-variable-y"]], "Transforming the Independent Variable (X)": [[13, "transforming-the-independent-variable-x"]], "Initializing and Training Models": [[13, "initializing-and-training-models"]], "Scoring and Evaluating Models": [[13, "scoring-and-evaluating-models"]], "Step 11: Run the Inversion!": [[11, "step-11-run-the-inversion"]], "GeoPhysics and Inversion": [[15, "geophysics-and-inversion"], [21, "geophysics-and-inversion"], [23, "geophysics-and-inversion"]], "Vector Geology Examples": [[21, "vector-geology-examples"], [23, "vector-geology-examples"]], "vector-geology Documentation": [[23, "vector-geology-documentation"]], "Vector Geology Repository": [[23, "vector-geology-repository"]], "Overview": [[23, "overview"]], "Key Features": [[23, "key-features"]], "Getting Started": [[23, "getting-started"]], "Prerequisites": [[23, "prerequisites"]], "Note!": [[23, null]], "Explore Our Guides": [[23, "explore-our-guides"]], "Galleries": [[23, null]], "Requirements": [[23, "requirements"]], "Optional requirements": [[23, "optional-requirements"]], "Acknowledgements": [[23, "acknowledgements"]]}, "indexentries": {}}) \ No newline at end of file