Note
Go to the end to download the full example code.
Single frequency synthetic study¶
This example aims to provide a full discussion of a single-frequency complex resistivity simulation study. A simulation study usually consists of the following steps:
Generate a Finite-Element mesh for the forward modeling step. This includes defining the bounding geometry of the subsurface to investigate (the boundary), electrode positions on the boundary and embedded in the subsurface, as well as internal geometry.
Internal geometry reflects the geophysical scenario that is being simulated and is usually driven by the scientific research context of the study.
Create a complex-resistivity subsurface model
Generate measurement configurations
Conduct the forward modeling
Add normally-distributed noise to the synthetic data. Remove all data points that have a resulting transfer impedance below 0 (CRMod automatically outputs measurement configurations with positive geometric factors. Therefore, all measurements \(<= 0 \Omega\) are deemed as physically unacceptable.
Conduct an inversion using the synthetic data contaminated with synthetic noise. Usually error estimates are chosen equal to the noise distributions previously added. Depending on the scope of the synthetic study, additional noise levels can be estimated to account for other error components.
Analyse inversion results.
Note that we use a context manager from reda to place output files in a separate output directory
import reda
# imports required for the study
import crtomo
import numpy as np
import shapely.geometry
import pylab as plt
1. Generate forward mesh¶
Generate a simple Finite-Element mesh for forward and inverse modeling. For more information on mesh creation, please refer to Irregular grids. grid_creation.
grid = crtomo.crt_grid.create_surface_grid(
nr_electrodes=30,
spacing=1,
# these lengths determine the size of mesh cells
char_lengths=[0.3, 1, 1, 1]
)
fig, ax = grid.plot_grid()
with reda.CreateEnterDirectory('output_plot_07'):
fig.savefig('grid.jpg', dpi=300, bbox_inches='tight')
This grid was sorted using CutMcK. The nodes were resorted!
Triangular grid found
Create a tdManager instance used for single-frequency forward and inverse modeling
man = crtomo.tdMan(grid=grid)
2. Create complex-resistivity subsurface model¶
pid_mag, pid_pha = man.add_homogeneous_model(
magnitude=100, phase=-5
)
# import IPython
# IPython.embed()
man.parman.modify_area(
pid_mag,
xmin=1, xmax=5,
zmin=-5, zmax=-1,
value=10
)
man.parman.modify_area(
pid_pha,
xmin=1, xmax=5,
zmin=-3, zmax=-2,
value=-30
)
polygon = shapely.geometry.Polygon((
(2, 0), (4, -1), (2, -1)
))
man.parman.modify_polygon(pid_mag, polygon, 3)
with reda.CreateEnterDirectory('output_plot_07'):
fig, axes = plt.subplots(2, 1, figsize=(8, 8), sharex=True)
fig, ax = man.show_parset(
pid_mag,
ax=axes[0],
plot_colorbar=True,
cblabel=r'$\rho~[\Omega m]$',
cmap_name='viridis',
title='Magnitude forward model',
)
fig.savefig('model_magnitude.jpg', dpi=300)
fig, ax = man.show_parset(
pid_pha,
ax=axes[1],
plot_colorbar=True,
cblabel=r'$\varphi~[mrad]$',
cmap_name='turbo',
title='Phase forward model',
)
fig.savefig('model_phase.jpg', dpi=300)
3. Generate Measurement Configurations¶
man.configs.gen_dipole_dipole(skipc=0)
man.configs.gen_dipole_dipole(skipc=1)
man.configs.gen_dipole_dipole(skipc=2)
4. Generate Measurement Configurations¶
rmag_rpha_mod = man.measurements()
rmag = man.measurements()[:, 0]
rpha = man.measurements()[:, 1]
attempting modeling
b'\nlibgomp: Invalid value for environment variable OMP_NUM_THREADS\n ######### CMod ############\nLicence:\nCopyright \xc2\xa9 1990-2020 Andreas Kemna <kemna@geo.uni-bonn.de>\nCopyright \xc2\xa9 2008-2020 CRTomo development team (see AUTHORS file)\nPermission is hereby granted, free of charge, to any person obtaining a copy of\nthis software and associated documentation files (the \xe2\x80\x9cSoftware\xe2\x80\x9d), to deal in\nthe Software without restriction, including without limitation the rights to\nuse, copy, modify, merge, publish, distribute, sublicense, and/or sell copies\nof the Software, and to permit persons to whom the Software is furnished to do\nso, subject to the following conditions:\n\nThe above copyright notice and this permission notice shall be included in all\ncopies or substantial portions of the Software.\n\nTHE SOFTWARE IS PROVIDED \xe2\x80\x9cAS IS\xe2\x80\x9d, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\nIMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\nFITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\nAUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\nLIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\nOUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE\nSOFTWARE.\n \n \n CRMod Process_ID :: 5275\n OpenMP max threads: 4\n Git-Branch master\n Commit-ID cb89ccd2c4341b9b08bbe7bb08f5aea0d60916ee\n Created Mon-Mar-25-14:34:56-2024\n Compiler \n OS GNU/Linux\n\n Reading Input-Files\n++ check 1\r++ check 2 done!\n\rGetting voltage 1\rGetting voltage 2\rGetting voltage 3\rGetting voltage 4\rGetting voltage 5\rGetting voltage 6\rGetting voltage 7\rGetting voltage 8\rGetting voltage 9\rGetting voltage 10\rGetting voltage 11\rGetting voltage 12\rGetting voltage 13\rGetting voltage 14\rGetting voltage 15\rGetting voltage 16\rGetting voltage 17\rGetting voltage 18\rGetting voltage 19\rGetting voltage 20\rGetting voltage 21\rGetting voltage 22\rGetting voltage 23\rGetting voltage 24\rGetting voltage 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voltage 504\rGetting voltage 505\rGetting voltage 506\rGetting voltage 507\rGetting voltage 508\rGetting voltage 509\rGetting voltage 510\rGetting voltage 511\rGetting voltage 512\rGetting voltage 513\rGetting voltage 514\rGetting voltage 515\rGetting voltage 516\rGetting voltage 517\rGetting voltage 518\rGetting voltage 519\rGetting voltage 520\rGetting voltage 521\rGetting voltage 522\rGetting voltage 523\rGetting voltage 524\rGetting voltage 525\rGetting voltage 526\rGetting voltage 527\rGetting voltage 528\rGetting voltage 529\rGetting voltage 530\rGetting voltage 531\rGetting voltage 532\rGetting voltage 533\rGetting voltage 534\rGetting voltage 535\rGetting voltage 536\rGetting voltage 537\rGetting voltage 538\rGetting voltage 539\rGetting voltage 540\rGetting voltage 541\rGetting voltage 542\rGetting voltage 543\rGetting voltage 544\rGetting voltage 545\rGetting voltage 546\rGetting voltage 547\rGetting voltage 548\rGetting voltage 549\rGetting voltage 550\rGetting voltage 551\rGetting voltage 552\rGetting voltage 553\rGetting voltage 554\rGetting voltage 555\rGetting voltage 556\rGetting voltage 557\rGetting voltage 558\rGetting voltage 559\rGetting voltage 560\rGetting voltage 561\rGetting voltage 562\rGetting voltage 563\rGetting voltage 564\rGetting voltage 565\rGetting voltage 566\rGetting voltage 567\rGetting voltage 568\rGetting voltage 569\rGetting voltage 570\rGetting voltage 571\rGetting voltage 572\rGetting voltage 573\rGetting voltage 574\rGetting voltage 575\rGetting voltage 576\rGetting voltage 577\rGetting voltage 578\rGetting voltage 579\rGetting voltage 580\rGetting voltage 581\rGetting voltage 582\rGetting voltage 583\rGetting voltage 584\rGetting voltage 585\rGetting voltage 586\rGetting voltage 587\rGetting voltage 588\rGetting voltage 589\rGetting voltage 590\rGetting voltage 591\rGetting voltage 592\rGetting voltage 593\rGetting voltage 594\rGetting voltage 595\rGetting voltage 596\rGetting voltage 597\rGetting voltage 598\rGetting voltage 599\rGetting voltage 600\rGetting voltage 601\rGetting voltage 602\rGetting voltage 603\rGetting voltage 604\rGetting voltage 605\rGetting voltage 606\rGetting voltage 607\rGetting voltage 608\rGetting voltage 609\rGetting voltage 610\rGetting voltage 611\rGetting voltage 612\rGetting voltage 613\rGetting voltage 614\rGetting voltage 615 No electrode cap file\n \n Rescheduling..\n less nodes than wavenumbers\n OpenMP threads: 3( 4)\n\n\r Calculating Potentials : Wavenumber 3 \r Calculating Potentials : Wavenumber 3 \r Calculating Potentials : Wavenumber 3 \r Calculating Potentials : Wavenumber 4 \r Calculating Potentials : Wavenumber 5 \r Calculating Potentials : Wavenumber 6 \r Calculating Potentials : Wavenumber 7 \r Calculating Potentials : Wavenumber 8 \r Calculating Potentials : Wavenumber 9 \r Calculating Potentials : Wavenumber 10 \r Calculating Potentials : Wavenumber 11 done, now processing\n solution time 0d/ 0h/ 0m/ 0s/ 998ms\n\n Modelling completedSTOP 0\n'
reading voltages
Let’s plot the results
with reda.CreateEnterDirectory('output_plot_07'):
fig, axes = plt.subplots(3, 1)
ax = axes[0]
ax.set_title('Measured transfer resistances', loc='left', fontsize=8)
ax.hist(rmag_rpha_mod[:, 0], 100)
ax.set_xlabel(r'$R~[\Omega]$')
ax.set_ylabel('Count [-]')
ax = axes[1]
ax.set_title(
'Measured transfer resistances (log10)', loc='left', fontsize=8
)
ax.hist(rmag_rpha_mod[:, 0], 100)
ax.set_xlabel(r'$R~[\Omega]$')
ax.set_xscale('log')
ax.set_ylabel('Count [-]')
ax = axes[2]
ax.set_title('Phase measurements', loc='left', fontsize=8)
ax.hist(rmag_rpha_mod[:, 1], 100)
ax.set_xlabel('Phases [mrad]')
ax.set_ylabel('Count [-]')
fig.tight_layout()
fig.savefig('modeled_data.jpg', dpi=300)
5. Add noise to synthetic data¶
Important: ALWAYS initialize the random number generator using a seed!
np.random.seed(2048)
# absolute component in [Ohm ]
noise_level_rmag_absolute = 0.01
# relative component [0, 1]
noise_level_rmag_relative = 0.05
noise_rmag = np.random.normal(
loc=0,
scale=rmag * noise_level_rmag_relative + noise_level_rmag_absolute
)
rmag_with_noise = rmag + noise_rmag
# 0.5 mrad absolute noise level
noise_level_phases = 0.5
noise_rpha = np.random.normal(
loc=0,
scale=noise_level_phases
)
rpha_with_noise = rpha + noise_rpha
# register the noise-added data as new measurements and mark them for use in a
# subsequent inversion
man.register_measurements(rmag_with_noise, rpha_with_noise)
# Remove physically implausible negative magnitude values
indices = np.where(rmag_with_noise <= 0)[0]
man.configs.delete_data_points(indices)
man.save_measurements('volt.dat')
Deleting configurations:
[[ 2 3 11 10]
[ 2 3 13 12]
[ 3 4 12 11]
[ 3 4 15 14]
[ 2 4 16 14]
[ 3 5 17 15]]
prepare the inversion note that we create another tdMan class instance for the inversion. This is not necessarily required, but serves nicely to separate the different steps of the inversion
tdman = crtomo.tdMan(grid=grid)
# inversion settings can be changed here:
print(tdman.crtomo_cfg)
# for example, let's change the relative magnitude error estimate to 7 %
tdman.crtomo_cfg['mag_rel'] = 5
tdman.crtomo_cfg['mag_abs'] = 0.01
tdman.crtomo_cfg['pha_abs'] = 0.5
tdman.read_voltages('volt.dat')
1 ! mswitch
../grid/elem.dat ! elem
../grid/elec.dat ! elec
../mod/volt.dat ! volt
../inv ! inv_dir
F ! difference inversion? ! diff_inv
! prior_model
iseed variance ! iseed_var
0 ! # cells in x-direction ! cells_x
0 ! # cells in z-direction ! cells_z
1.000 ! smoothing parameter in x-direction ! ani_x
1.000 ! smoothing parameter in z-direction ! ani_z
20 ! max. nr of iterations ! max_it
F ! DC inversion? ! dc_inv
T ! robust inversion? ! robust_inv
F ! final phase improvement? ! fpi_inv
5 ! mag_rel
1e-3 ! mag_abs
0 ! pha_a1
0 ! pha_b
0 ! pha_rel
0 ! pha_abs
F ! hom_bg
10.00 ! hom_mag
0.00 ! hom_pha
F ! another_ds
1 ! d2_5
F ! fic_sink
10000 ! fic_sink_node
F ! boundaries
boundary.dat ! boundaries_file
1 ! mswitch2
lambda ! lambda
6. Conduct the actual inversion¶
this command actually calls CRTomo and conducts the actual inversion
tdman.invert()
Attempting inversion in directory: /tmp/tmpmep4ulw2
Using binary: /usr/bin/CRTomo_dev
Calling CRTomo
Inversion attempt finished
Attempting to import the results
Reading inversion results
is robust True
Info: res_m.diag not found: /tmp/tmpmep4ulw2/inv/res_m.diag
/home/runner/work/crtomo_tools/crtomo_tools/examples/02_simple_inversion
Info: ata.diag not found: /tmp/tmpmep4ulw2/inv/ata.diag
/home/runner/work/crtomo_tools/crtomo_tools/examples/02_simple_inversion
Statistics of last iteration:
iteration 4
main_iteration 4
it_type DC/IP
type main
dataRMS 0.9978
magRMS 1.731
phaRMS 1.289
lambda 117.0
roughness 3.645
cgsteps 116.0
nrdata 609.0
steplength 0.0
stepsize 0.9323
l1ratio 1.0
Name: 28, dtype: object
7. Analyse inversion results¶
# The convergence behavior of the inversion is stored in a pandas.DataFrame:
print(tdman.inv_stats)
# save the RMS values of the final iteration
final_rms_all, final_rms_mag, final_rms_pha = tdman.inv_stats.iloc[-1][
['dataRMS', 'magRMS', 'phaRMS']
].values
print('Final RMS mag+pha:', final_rms_all)
print('Final RMS mag:', final_rms_mag)
print('Final RMS pha:', final_rms_pha)
iteration main_iteration it_type ... steplength stepsize l1ratio
0 0 0 DC/IP ... NaN NaN 1.25
1 1 1 DC/IP ... NaN 1060.0000 NaN
2 2 1 DC/IP ... NaN 1410.0000 NaN
3 3 1 DC/IP ... NaN 1570.0000 NaN
4 4 1 DC/IP ... NaN 1650.0000 NaN
5 5 1 DC/IP ... NaN 1670.0000 NaN
6 6 1 DC/IP ... NaN 1700.0000 NaN
7 7 1 DC/IP ... NaN 1700.0000 NaN
8 8 1 DC/IP ... NaN 1580.0000 NaN
9 9 1 DC/IP ... NaN 1210.0000 NaN
10 10 1 DC/IP ... NaN 791.0000 NaN
11 1 1 DC/IP ... 1.0 1582.0000 1.16
12 0 2 DC/IP ... NaN 30.7000 NaN
13 1 2 DC/IP ... NaN 43.4000 NaN
14 2 2 DC/IP ... NaN 53.0000 NaN
15 3 2 DC/IP ... NaN 66.2000 NaN
16 4 2 DC/IP ... NaN 88.2000 NaN
17 5 2 DC/IP ... NaN 110.0000 NaN
18 6 2 DC/IP ... NaN 125.0000 NaN
19 7 2 DC/IP ... NaN 55.1000 NaN
20 2 2 DC/IP ... 8.0 6.3490 0.05
21 0 3 DC/IP ... NaN 98.4000 NaN
22 1 3 DC/IP ... NaN 113.0000 NaN
23 2 3 DC/IP ... NaN 49.2000 NaN
24 3 3 DC/IP ... 0.0 98.4300 1.00
25 0 4 DC/IP ... NaN 0.9320 NaN
26 1 4 DC/IP ... NaN 3.0600 NaN
27 2 4 DC/IP ... NaN 0.4660 NaN
28 4 4 DC/IP ... 0.0 0.9323 1.00
[29 rows x 14 columns]
Final RMS mag+pha: 0.9978
Final RMS mag: 1.731
Final RMS pha: 1.289
The primary convergence goal of the inversion is to reach an RMS near 1. This would indicate that the data is described by the subsurface model within their data bounds. We do not care about update loops of the inversion. Therefore, filter the convergence information and plot it.
inv_stats_main = tdman.inv_stats.query('type=="main"').reset_index()
fig, ax = plt.subplots(figsize=(6, 4))
ax.plot(
inv_stats_main['dataRMS'], '.-', label=r'$\mathbf{RMS}_{\mathrm{all}}$')
ax.plot(
inv_stats_main['magRMS'], '.-', label=r'$\mathbf{RMS}_{\mathrm{mag}}$')
ax.plot(
inv_stats_main['phaRMS'], '.-', label=r'$\mathbf{RMS}_{\mathrm{pha}}$')
ax.axhline(y=1.0, color='black', linestyle='dotted', label='target RMS')
ax.legend()
ax.set_xlabel('Iteration number')
ax.set_ylabel('RMS~[-]')
fig.tight_layout()
with reda.CreateEnterDirectory('output_plot_07'):
fig.savefig(
'inversion_convergence_evolution.jpg', dpi=300, bbox_inches='tight'
)
As a short reminder, this dictionary contains all information on where to find data/results in the tdMan instance
print(tdman.a)
{'forward_model': None, 'measurements': [0, 1], 'measurement_errors': None, 'error_norm_factor_mag': None, 'error_norm_factor_pha': None, 'sensitivities': None, 'potentials': None, 'inversion': {'rmag': [0, 1, 2, 3, 4], 'rpha': [5, 6, 7, 8, 9], 'cre_cim': [[10, 11], [12, 13], [14, 15], [16, 17], [18, 19]], 'cre': [10, 12, 14, 16, 18], 'cim': [11, 13, 15, 17, 19], 'fwd_response_rmag': [2, 5, 8, 11, 14], 'fwd_response_rpha': [3, 6, 9, 12, 15], 'fwd_response_wdfak': [4, 7, 10, 13, 16], 'l1_dw_log10_norm': 20}, 'prior_model': None}
We can now visualize the inversion results. First, plot the L1-Coverage to get an idea on the information distribution in the subsurface. We will later use the coverage to add transparency (‘alpha’) to our plots, or to apply a filter mask to the results. The intention here is to visually remove all pixels that definitively do not hold useful subsurface information. Be careful: High sensitivities do not always imply good recovery of the subsurface!
fig, ax = tdman.show_parset(
tdman.a['inversion']['l1_dw_log10_norm'],
plot_colorbar=True,
cmap_name='magma',
xmin=-2,
xmax=32,
zmin=-8,
title='L1-Coverage (normalized)',
cblabel=r'$log_{10}(Cov_{L1}~[-])$',
)
with reda.CreateEnterDirectory('output_plot_07'):
fig.savefig('inversion_coverage_l1.jpg', dpi=300, bbox_inches='tight')
log10_threshold = -2
mask = (tdman.parman.parsets[
tdman.a['inversion']['l1_dw_log10_norm']
] > log10_threshold).astype(int)
Plot Magnitude results:
fig, axes = plt.subplots(3, 1, figsize=(8, 11), sharex=True)
fig, ax = tdman.show_parset(
tdman.a['inversion']['rmag'][-1],
ax=axes[0],
cmap_name='viridis',
plot_colorbar=True,
cblabel=r'$\rho~[\Omega m]$',
xmin=-2,
xmax=32,
zmin=-8,
title='Magnitude inversion (rms-all: {}, rms-mag: {})'.format(
final_rms_all,
final_rms_mag
),
)
ax.set_title('No mask', loc='left', fontsize=8)
fig, ax = tdman.show_parset(
tdman.a['inversion']['rmag'][-1],
ax=axes[1],
cmap_name='viridis',
plot_colorbar=True,
cblabel=r'$\rho~[\Omega m]$',
xmin=-2,
xmax=32,
zmin=-8,
title='Magnitude inversion (rms-all: {}, rms-mag: {})'.format(
final_rms_all,
final_rms_mag
),
cid_alpha=tdman.a['inversion']['l1_dw_log10_norm'],
# default is: 3
alpha_sens_threshold=2,
)
fig, ax = tdman.show_parset(
tdman.a['inversion']['rmag'][-1],
ax=axes[2],
cmap_name='viridis',
plot_colorbar=True,
cblabel=r'$\rho~[\Omega m]$',
xmin=-2,
xmax=32,
zmin=-8,
title='Magnitude inversion (rms-all: {}, rms-mag: {})'.format(
final_rms_all,
final_rms_mag
),
cid_alpha=mask,
)
with reda.CreateEnterDirectory('output_plot_07'):
fig.savefig('inversion_result_rmag_cov_mask.jpg')
Phase inversion results:
fig, ax = tdman.show_parset(
tdman.a['inversion']['rpha'][-1],
cmap_name='turbo',
plot_colorbar=True,
cblabel=r'$\phi~[mrad]$',
xmin=-2,
xmax=32,
zmin=-8,
title='Phase inversion result (rms-all: {}, rms-pha: {})'.format(
final_rms_all,
final_rms_mag
),
cid_alpha=mask,
)
with reda.CreateEnterDirectory('output_plot_07'):
fig.savefig('inversion_result_rpha.jpg')
# sphinx_gallery_thumbnail_number = -1
Total running time of the script: (1 minutes 56.463 seconds)