plot_view_bounds.py

"""
Mesh: Plotting with defining range
==================================

When using a large Mesh with the cylindrical code, it is advantageous
to define a :code:`range_x` and :code:`range_y` when plotting with
vectors. In this case, only the region inside of the range is
interpolated. In particular, you often want to ignore padding cells.
"""

import numpy as np
import matplotlib.pyplot as plt
from scipy.constants import mu_0

from SimPEG import Mesh, Utils, Maps
from SimPEG.EM import FDEM

# Try importing PardisoSolver from pymatsolver otherwise, use SimPEG.SolverLU
try:
    from pymatsolver import PardisoSolver as Solver
except ImportError:
    from SimPEG import SolverLU as Solver


def run(plotIt=True):

    # ## Model Parameters
    #
    # We define a
    # - resistive halfspace and
    # - conductive sphere
    #    - radius of 30m
    #    - center is 50m below the surface

    # electrical conductivities in S/m
    sig_halfspace = 1e-6
    sig_sphere = 1e0
    sig_air = 1e-8

    # depth to center, radius in m
    sphere_z = -50.
    sphere_radius = 30.

    # ## Survey Parameters
    #
    # - Transmitter and receiver 20m above the surface
    # - Receiver offset from transmitter by 8m horizontally
    # - 25 frequencies, logaritmically between $10$ Hz and $10^5$ Hz


    boom_height = 20.
    rx_offset = 8.
    freqs = np.r_[1e1, 1e5]

    # source and receiver location in 3D space
    src_loc = np.r_[0., 0., boom_height]
    rx_loc = np.atleast_2d(np.r_[rx_offset, 0., boom_height])



    # print the min and max skin depths to make sure mesh is fine enough and
    # extends far enough

    def skin_depth(sigma, f):
        return 500./np.sqrt(sigma * f)

    print(
        'Minimum skin depth (in sphere): {:.2e} m '.format(
            skin_depth(sig_sphere, freqs.max())
        )
    )
    print(
        'Maximum skin depth (in background): {:.2e} m '.format(
            skin_depth(sig_halfspace, freqs.min())
        )
    )


    # ## Mesh
    #
    # Here, we define a cylindrically symmetric tensor mesh.
    #
    # ### Mesh Parameters
    #
    # For the mesh, we will use a cylindrically symmetric tensor mesh. To
    # construct a tensor mesh, all that is needed is a vector of cell widths in
    # the x and z-directions. We will define a core mesh region of uniform cell
    # widths and a padding region where the cell widths expand "to infinity".

    # x-direction
    csx = 2  # core mesh cell width in the x-direction
    ncx = np.ceil(1.2*sphere_radius/csx)  # number of core x-cells (uniform mesh slightly beyond sphere radius)
    npadx = 50  # number of x padding cells

    # z-direction
    csz = 1  # core mesh cell width in the z-direction
    ncz = np.ceil(1.2*(boom_height - (sphere_z-sphere_radius))/csz) # number of core z-cells (uniform cells slightly below bottom of sphere)
    npadz = 52  # number of z padding cells

    # padding factor (expand cells to infinity)
    pf = 1.3

    # cell spacings in the x and z directions
    hx = Utils.meshTensor([(csx, ncx), (csx, npadx, pf)])
    hz = Utils.meshTensor([(csz, npadz, -pf), (csz, ncz), (csz, npadz, pf)])

    # define a SimPEG mesh
    mesh = Mesh.CylMesh([hx, 1, hz], x0 = np.r_[0.,0., -hz.sum()/2.-boom_height])


    # ### Plot the mesh
    #
    # Below, we plot the mesh. The cyl mesh is rotated around x=0. Ensure that
    # each dimension extends beyond the maximum skin depth.
    #
    # Zoom in by changing the xlim and zlim.

    # X and Z limits we want to plot to. Try
    xlim = np.r_[0., 2.5e6]
    zlim = np.r_[-2.5e6, 2.5e6]

    fig, ax = plt.subplots(1, 1)
    mesh.plotGrid(ax=ax)

    ax.set_title('Simulation Mesh')
    ax.set_xlim(xlim)
    ax.set_ylim(zlim)

    print(
        'The maximum skin depth is (in background): {:.2e} m. '
        'Does the mesh go sufficiently past that?'.format(
            skin_depth(sig_halfspace, freqs.min())
        )
    )


    # ## Put Model on Mesh
    #
    # Now that the model parameters and mesh are defined, we can define
    # electrical conductivity on the mesh.
    #
    # The electrical conductivity is defined at cell centers when using the
    # finite volume method. So here, we define a vector that contains an
    # electrical conductivity value for every cell center.

    # create a vector that has one entry for every cell center
    sigma = sig_air*np.ones(mesh.nC)  # start by defining the conductivity of the air everwhere
    sigma[mesh.gridCC[:, 2] < 0.] = sig_halfspace  # assign halfspace cells below the earth

    # indices of the sphere (where (x-x0)**2 + (z-z0)**2 <= R**2)
    sphere_ind =(
        (mesh.gridCC[:,0]**2 + (mesh.gridCC[:,2] - sphere_z)**2) <=
        sphere_radius**2
    )
    sigma[sphere_ind] = sig_sphere  # assign the conductivity of the sphere

    # Plot a cross section of the conductivity model
    fig, ax = plt.subplots(1, 1)
    cb = plt.colorbar(mesh.plotImage(np.log10(sigma), ax=ax, mirror=True)[0])

    # plot formatting and titles
    cb.set_label('$\log_{10}\sigma$', fontsize=13)
    ax.axis('equal')
    ax.set_xlim([-120., 120.])
    ax.set_ylim([-100., 30.])
    ax.set_title('Conductivity Model')


    # ## Set up the Survey
    #
    # Here, we define sources and receivers. For this example, the receivers
    # are magnetic flux recievers, and are only looking at the secondary field
    # (eg. if a bucking coil were used to cancel the primary). The source is a
    # vertical magnetic dipole with unit moment.


    # Define the receivers, we will sample the real secondary magnetic flux
    # density as well as the imaginary magnetic flux density

    bz_r = FDEM.Rx.Point_bSecondary(
        locs=rx_loc, orientation='z', component='real'
    )  # vertical real b-secondary
    bz_i = FDEM.Rx.Point_b(
        locs=rx_loc, orientation='z', component='imag'
    )  # vertical imag b (same as b-secondary)

    rxList = [bz_r, bz_i]  # list of receivers


    # Define the list of sources - one source for each frequency. The source is
    # a point dipole oriented in the z-direction

    srcList = [
        FDEM.Src.MagDipole(rxList, f, src_loc, orientation='z') for f in freqs
    ]

    print(
        'There are {nsrc} sources (same as the number of frequencies - {nfreq}). '
        'Each source has {nrx} receivers sampling the resulting b-fields'.format(
            nsrc = len(srcList),
            nfreq = len(freqs),
            nrx = len(rxList)
        )
    )


    # ## Set up Forward Simulation
    #
    # A forward simulation consists of a paired SimPEG problem and Survey.
    # For this example, we use the E-formulation of Maxwell's equations,
    # solving the second-order system for the electric field, which is defined
    # on the cell edges of the mesh. This is the `prob` variable below. The
    # `survey` takes the source list which is used to construct the RHS for the
    # problem. The source list also contains the receiver information, so the
    # `survey` knows how to sample fields and fluxes that are produced by
    # solving the `prob`.

    # define a problem - the statement of which discrete pde system we want to
    # solve
    prob = FDEM.Problem3D_e(mesh, sigmaMap=Maps.IdentityMap(mesh))
    prob.solver = Solver

    survey = FDEM.Survey(srcList)

    # tell the problem and survey about each other - so the RHS can be
    # constructed for the problem and the
    # resulting fields and fluxes can be sampled by the receiver.
    prob.pair(survey)


    # ### Solve the forward simulation
    #
    # Here, we solve the problem for the fields everywhere on the mesh.
    fields = prob.fields(sigma)


    # ### Plot the fields
    #
    # Lets look at the physics!

    # log-scale the colorbar
    from matplotlib.colors import LogNorm

    # sphinx_gallery_thumbnail_number = 3
    fig, ax = plt.subplots(1, 2, figsize=(12, 6))

    def plotMe(field, ax):
        plt.colorbar(mesh.plotImage(
            field, vType='F', view='vec',
            range_x=[-100., 100.], range_y=[-180., 60.],
            pcolorOpts={
                    'norm': LogNorm(), 'cmap': plt.get_cmap('viridis')
                },
            streamOpts={'color': 'k'},
            ax=ax, mirror=True
        )[0], ax=ax)

    plotMe(fields[srcList[0], 'bSecondary'].real, ax[0])
    ax[0].set_title('Real B-Secondary, {}Hz'.format(freqs[0]))

    plotMe(fields[srcList[1], 'bSecondary'].real, ax[1])
    ax[1].set_title('Real B-Secondary, {}Hz'.format(freqs[1]))

    plt.tight_layout()

    if plotIt:
        plt.show()

if __name__ == '__main__':
    run(plotIt=True)