forked from simpeg/simpeg
-
Notifications
You must be signed in to change notification settings - Fork 1
/
plot_vrm_fwd.py
237 lines (200 loc) · 7.9 KB
/
plot_vrm_fwd.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
"""
Predict Response from a Conductive and Magnetically Viscous Earth
=================================================================
Here, we predict the vertical db/dt response over a conductive and
magnetically viscous Earth for a small coincident loop system. Following
the theory, the total response is approximately equal to the sum of the
inductive and VRM responses modelled separately. The SimPEG.VRM module is
used to model the VRM response while an analytic solution for a conductive
half-space is used to model the inductive response.
"""
#########################################################################
# Import modules
# --------------
#
import SimPEG.VRM as VRM
import numpy as np
from SimPEG import mkvc, Mesh, Maps
import matplotlib.pyplot as plt
import matplotlib as mpl
##########################################################################
# Defining the mesh
# -----------------
#
cs, ncx, ncy, ncz, npad = 2., 35, 35, 20, 5
hx = [(cs, npad, -1.3), (cs, ncx), (cs, npad, 1.3)]
hy = [(cs, npad, -1.3), (cs, ncy), (cs, npad, 1.3)]
hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
mesh = Mesh.TensorMesh([hx, hy, hz], 'CCC')
##########################################################################
# Defining the model
# ------------------
#
# Create xi model (amalgamated magnetic property). Here the model is made by
# summing a set of 3D Gaussian distributions. And only active cells have a
# model value.
#
topoCells = mesh.gridCC[:, 2] < 0. # Define topography
xyzc = mesh.gridCC[topoCells, :]
c = 2*np.pi*8**2
pc = np.r_[4e-4, 4e-4, 4e-4, 6e-4, 8e-4, 6e-4, 8e-4, 8e-4]
x_0 = np.r_[50., -50., -40., -20., -15., 20., -10., 25.]
y_0 = np.r_[0., 0., 40., 10., -20., 15., 0., 0.]
z_0 = np.r_[0., 0., 0., 0., 0., 0., 0., 0.]
var_x = c*np.r_[3., 3., 3., 1., 3., 0.5, 0.1, 0.1]
var_y = c*np.r_[20., 20., 1., 1., 0.4, 0.5, 0.1, 0.4]
var_z = c*np.r_[1., 1., 1., 1., 1., 1., 1., 1.]
xi_true = np.zeros(np.shape(xyzc[:, 0]))
for ii in range(0, 8):
xi_true += (
pc[ii]*np.exp(-(xyzc[:, 0]-x_0[ii])**2/var_x[ii]) *
np.exp(-(xyzc[:, 1]-y_0[ii])**2/var_y[ii]) *
np.exp(-(xyzc[:, 2]-z_0[ii])**2/var_z[ii])
)
xi_true += 1e-5
##########################################################################
# Survey
# ------
#
# Here we must set the transmitter waveform, which defines the off-time decay
# of the VRM response. Next we define the sources, receivers and time channels
# for the survey. Our example is similar to an EM-63 survey.
#
waveform = VRM.WaveformVRM.StepOff()
times = np.logspace(-5, -2, 31) # Observation times
x, y = np.meshgrid(np.linspace(-30, 30, 21), np.linspace(-30, 30, 21))
z = 0.5*np.ones(x.shape)
loc = np.c_[mkvc(x), mkvc(y), mkvc(z)] # Src and Rx Locations
src_list_vrm = []
for pp in range(0, loc.shape[0]):
loc_pp = np.reshape(loc[pp, :], (1, 3))
rx_list_vrm = [VRM.Rx.Point(loc_pp, times=times, fieldType='dbdt', fieldComp='z')]
src_list_vrm.append(
VRM.Src.MagDipole(rx_list_vrm, mkvc(loc[pp, :]), [0., 0., 0.01], waveform)
)
survey_vrm = VRM.Survey(src_list_vrm)
##########################################################################
# Problem
# -------
#
# For the VRM problem, we used a sensitivity refinement strategy for cells
# that are proximal to transmitters. This is controlled through the
# *ref_factor* and *ref_radius* properties.
#
# Defining the problem
problem_vrm = VRM.Problem_Linear(
mesh, indActive=topoCells, ref_factor=3, ref_radius=[1.25, 2.5, 3.75]
)
problem_vrm.pair(survey_vrm)
# Predict VRM response
fields_vrm = problem_vrm.fields(xi_true)
n_times = len(times)
n_loc = loc.shape[0]
fields_vrm = np.reshape(fields_vrm, (n_loc, n_times))
# Add an artificial TEM response. An analytic solution for the response near
# the surface of a conductive half-space (Nabighian, 1979) is scaled at each
# location to provide lateral variability in the TEM response.
sig = 1e-1
mu0 = 4*np.pi*1e-7
fields_tem = -sig**1.5*mu0**2.5*times**-2.5/(20*np.pi**1.5)
fields_tem = np.kron(np.ones((n_loc, 1)), np.reshape(fields_tem, (1, n_times)))
c = (
np.exp(-(loc[:, 0]-10)**2/(25**2))*np.exp(-(loc[:, 1]-20)**2/(35**2)) +
np.exp(-(loc[:, 0]+20)**2/(20**2))*np.exp(-(loc[:, 1]+20)**2/(40**2)) +
1.5*np.exp(-(loc[:, 0]-25)**2/(10**2))*np.exp(-(loc[:, 1]+25)**2/(10**2)) +
0.25
)
c = np.kron(np.reshape(c, (len(c), 1)), np.ones((1, n_times)))
fields_tem = c*fields_tem
##########################################################################
# Plotting
# --------
#
# Plotting the model
Fig = plt.figure(figsize=(10, 10))
font_size = 12
plotMap = Maps.InjectActiveCells(mesh, topoCells, 0.) # Maps to mesh
ax1 = 4*[None]
cplot1 = 3*[None]
view_str = ['X', 'Y', 'Z']
param_1 = [ncx, ncy, ncz]
param_2 = [6, 0, 1]
param_3 = [-12, 0, 0]
for qq in range(0, 3):
ax1[qq] = Fig.add_axes([0.07+qq*0.29, 0.7, 0.23, 0.23])
cplot1[qq] = mesh.plotSlice(
plotMap*xi_true, normal=view_str[qq],
ind=int((param_1[qq]+2*npad)/2-param_2[qq]),
ax=ax1[qq], grid=True, pcolorOpts={'cmap': 'gist_heat_r'})
cplot1[qq][0].set_clim((0., np.max(xi_true)))
ax1[qq].set_xlabel('Y [m]', fontsize=font_size)
ax1[qq].set_ylabel('Z [m]', fontsize=font_size, labelpad=-10)
ax1[qq].tick_params(labelsize=font_size-2)
ax1[qq].set_title('True Model (x = {} m)'.format(
param_3[qq]), fontsize=font_size+2
)
ax1[3] = Fig.add_axes([0.89, 0.7, 0.01, 0.24])
norm = mpl.colors.Normalize(vmin=0., vmax=np.max(xi_true))
cbar14 = mpl.colorbar.ColorbarBase(
ax1[3], cmap='gist_heat_r', norm=norm, orientation='vertical'
)
cbar14.set_label(
'$\Delta \chi /$ln$(\lambda_2 / \lambda_1 )$ [SI]',
rotation=270, labelpad=15, size=font_size
)
# Plotting the decay
ax2 = 2*[None]
n = x.shape[0]
for qq in range(0, 2):
ax2[qq] = Fig.add_axes([0.1+0.47*qq, 0.335, 0.38, 0.29])
k = int((n**2-1)/2 - 3*n*(-1)**qq)
di_vrm = mkvc(np.abs(fields_vrm[k, :]))
di_tem = mkvc(np.abs(fields_tem[k, :]))
ax2[qq].loglog(times, di_tem, 'r.-')
ax2[qq].loglog(times, di_vrm, 'b.-')
ax2[qq].loglog(times, di_tem+di_vrm, 'k.-')
ax2[qq].set_xlabel('t [s]', fontsize=font_size)
if qq == 0:
ax2[qq].set_ylabel('|dBz/dt| [T/s]', fontsize=font_size)
else:
ax2[qq].axes.get_yaxis().set_visible(False)
ax2[qq].tick_params(labelsize=font_size-2)
ax2[qq].set_xbound(np.min(times), np.max(times))
ax2[qq].set_ybound(1.2*np.max(di_tem+di_vrm), 1e-5*np.max(di_tem+di_vrm))
titlestr2 = (
"Decay at X = " + '{:.2f}'.format(loc[k, 0]) +
" m and Y = " + '{:.2f}'.format(loc[k, 1]) + " m"
)
ax2[qq].set_title(titlestr2, fontsize=font_size+2)
if qq == 0:
ax2[qq].text(
1.2e-5, 18*np.max(di_tem)/1e5, "TEM", fontsize=font_size, color='r'
)
ax2[qq].text(
1.2e-5, 6*np.max(di_tem)/1e5, "VRM", fontsize=font_size, color='b'
)
ax2[qq].text(
1.2e-5, 2*np.max(di_tem)/1e5, "TEM + VRM", fontsize=font_size, color='k'
)
# Plotting the TEM anomalies
ax3 = 3*[None]
cplot3 = 3*[None]
cbar3 = 3*[None]
for qq in range(0, 3):
ax3[qq] = Fig.add_axes([0.07+0.31*qq, 0.05, 0.24, 0.21])
d = np.reshape(np.abs(fields_tem[:, 10*qq]+fields_vrm[:, 10*qq]), (n, n))
cplot3[qq] = ax3[qq].contourf(x, y, d.T, 40, cmap='magma_r')
cbar3[qq] = plt.colorbar(cplot3[qq], ax=ax3[qq], pad=0.02, format='%.2e')
cbar3[qq].set_label('[T/s]', rotation=270, labelpad=12, size=font_size)
cbar3[qq].ax.tick_params(labelsize=font_size-2)
ax3[qq].set_xlabel('X [m]', fontsize=font_size)
if qq == 0:
ax3[qq].scatter(x, y, color=(0, 0, 0), s=4)
ax3[qq].set_ylabel('Y [m]', fontsize=font_size, labelpad=-8)
else:
ax3[qq].axes.get_yaxis().set_visible(False)
ax3[qq].tick_params(labelsize=font_size-2)
ax3[qq].set_xbound(np.min(x), np.max(x))
ax3[qq].set_ybound(np.min(y), np.max(y))
titlestr3 = "dBz/dt at t=" + '{:.1e}'.format(times[10*qq]) + " s"
ax3[qq].set_title(titlestr3, fontsize=font_size+2)