Note
Click here to download the full example code
Naive complex-valued electrical inversion¶
This example presents a quick and dirty proof-of-concept for a complex-valued inversion, similar to Kemna, 2000. The normal equations are solved using numpy, and no optimization with respect to running time and memory consumptions are applied. As such this example is only a technology demonstration and should not be used for real-world inversion of complex resistivity data!
Kemna, A.: Tomographic inversion of complex resistivity – theory and application, Ph.D. thesis, Ruhr-Universität Bochum, doi:10.1111/1365-2478.12013, 2000.
Note
This is a technology demonstration. Don’t use this code for research. If you require a complex-valued inversion, please contact us at info@pygimli.org
import numpy as np
import matplotlib.pyplot as plt
import pygimli as pg
import pygimli.meshtools as mt
from pygimli.physics import ert
For reference we later plot the true complex resistivity model as reference
def plot_fwd_model(axes):
"""This function plots the forward model used to generate the data
"""
# Mesh generation
world = mt.createWorld(
start=[-55, 0], end=[105, -80], worldMarker=True)
conductive_anomaly = mt.createCircle(
pos=[10, -7], radius=5, marker=2
)
polarizable_anomaly = mt.createCircle(
pos=[40, -7], radius=5, marker=3
)
plc = mt.mergePLC((world, conductive_anomaly, polarizable_anomaly))
# local refinement of mesh near electrodes
for s in scheme.sensors():
plc.createNode(s + [0.0, -0.2])
mesh_coarse = mt.createMesh(plc, quality=33)
mesh = mesh_coarse.createH2()
rhomap = [
[1, pg.utils.complex.toComplex(100, 0 / 1000)],
# Magnitude: 50 ohm m, Phase: -50 mrad
[2, pg.utils.complex.toComplex(50, 0 / 1000)],
[3, pg.utils.complex.toComplex(100, -50 / 1000)],
]
rho = pg.solver.parseArgToArray(rhomap, mesh.cellCount(), mesh)
pg.show(
mesh,
data=np.log(np.abs(rho)),
ax=axes[0],
label=r"$log_{10}(|\rho|~[\Omega m])$"
)
pg.show(mesh, data=np.abs(rho), ax=axes[1], label=r"$|\rho|~[\Omega m]$")
pg.show(
mesh, data=np.arctan2(np.imag(rho), np.real(rho)) * 1000,
ax=axes[2],
label=r"$\phi$ [mrad]",
cMap='jet_r'
)
fig.tight_layout()
fig.show()
Create a measurement scheme for 51 electrodes, spacing 1
scheme = ert.createERTData(
elecs=np.linspace(start=0, stop=50, num=51),
schemeName='dd'
)
# Not strictly required, but we switch potential electrodes to yield positive
# geometric factors. Note that this was also done for the synthetic data
# inverted later.
m = scheme['m']
n = scheme['n']
scheme['m'] = n
scheme['n'] = m
scheme.set('k', [1 for x in range(scheme.size())])
Mesh generation for the inversion
world = mt.createWorld(
start=[-15, 0], end=[65, -30], worldMarker=False, marker=2)
# local refinement of mesh near electrodes
for s in scheme.sensors():
world.createNode(s + [0.0, -0.4])
mesh_coarse = mt.createMesh(world, quality=33)
mesh = mesh_coarse.createH2()
for nr, c in enumerate(mesh.cells()):
c.setMarker(nr)
pg.show(mesh)

Out:
(<matplotlib.axes._subplots.AxesSubplot object at 0x7f4b2dc54c88>, None)
Define start model of the inversion [magnitude, phase]
start_model = np.ones(mesh.cellCount()) * pg.utils.complex.toComplex(
80, -0.01 / 1000)
Initialize the complex forward operator
fop = ert.ERTModelling(
sr=False,
verbose=True,
)
fop.setComplex(True)
fop.setData(scheme)
fop.setMesh(mesh, ignoreRegionManager=True)
fop.mesh()
Out:
Mesh: Nodes: 2248 Cells: 4228 Boundaries: 6475
Compute response for the starting model
Regularization matrix
rm = fop.regionManager()
rm.setVerbose(True)
rm.setConstraintType(2)
Wm = pg.matrix.SparseMapMatrix()
rm.fillConstraints(Wm)
Wm = pg.utils.sparseMatrix2coo(Wm)
read-in data and determine error parameters
filename = pg.getExampleFile(
'CR/synthetic_modeling/data_rre_rim.dat', load=False, verbose=True)
data_rre_rim = np.loadtxt(filename)
N = int(data_rre_rim.size / 2)
d_rcomplex = data_rre_rim[:N] + 1j * data_rre_rim[N:]
dmag = np.abs(d_rcomplex)
dpha = np.arctan2(d_rcomplex.imag, d_rcomplex.real) * 1000
fig, axes = plt.subplots(1, 2, figsize=(20 / 2.54, 10 / 2.54))
k = np.array(ert.createGeometricFactors(scheme))
ert.showERTData(
scheme, vals=dmag * k, ax=axes[0], label=r'$|\rho_a|~[\Omega$m]')
ert.showERTData(scheme, vals=dpha, ax=axes[1], label=r'$\phi_a~[mrad]$')
# real part: log-magnitude
# imaginary part: phase [rad]
d_rlog = np.log(d_rcomplex)
# add some noise
np.random.seed(42)
noise_magnitude = np.random.normal(
loc=0,
scale=np.exp(d_rlog.real) * 0.04
)
# absolute phase error
noise_phase = np.random.normal(
loc=0,
scale=np.ones(N) * (0.5 / 1000)
)
d_rlog = np.log(np.exp(d_rlog.real) + noise_magnitude) + 1j * (
d_rlog.imag + noise_phase)
# crude error estimations
rmag_linear = np.exp(d_rlog.real)
err_mag_linear = rmag_linear * 0.04 + np.min(rmag_linear)
err_mag_log = np.abs(1 / rmag_linear * err_mag_linear)
Wd = np.diag(1.0 / err_mag_log)
WdTwd = Wd.conj().dot(Wd)

Put together one iteration of a naive inversion in log-log transformation d = log(V) m = log(sigma)
def plot_inv_pars(filename, d, response, Wd, iteration='start'):
"""Plot error-weighted residuals"""
fig, axes = plt.subplots(1, 2, figsize=(20 / 2.54, 10 / 2.54))
psi = Wd.dot(d - response)
ert.showERTData(
scheme, vals=psi.real, ax=axes[0],
label=r"$(d' - f') / \epsilon$"
)
ert.showERTData(
scheme, vals=psi.imag, ax=axes[1],
label=r"$(d'' - f'') / \epsilon$"
)
fig.suptitle(
'Error weighted residuals of iteration {}'.format(iteration), y=1.00)
fig.tight_layout()
m_old = np.log(start_model)
d = np.log(pg.utils.toComplex(data_rre_rim))
response = np.log(pg.utils.toComplex(f_0))
# tranform to log-log sensitivities
J = J0 / np.exp(response[:, np.newaxis]) * np.exp(m_old)[np.newaxis, :]
lam = 100
plot_inv_pars('stats_it0.jpg', d, response, Wd)
# only one iteration is implemented here!
for i in range(1):
print('-' * 80)
print('Iteration {}'.format(i + 1))
term1 = J.conj().T.dot(WdTwd).dot(J) + lam * Wm.T.dot(Wm)
term1_inverse = np.linalg.inv(term1)
term2 = J.conj().T.dot(WdTwd).dot(d - response) - lam * Wm.T.dot(Wm).dot(
m_old)
model_update = term1_inverse.dot(term2)
print('Model Update')
print(model_update)
m1 = np.array(m_old + 1.0 * model_update).squeeze()

Out:
--------------------------------------------------------------------------------
Iteration 1
Model Update
[[2.00176959-0.00569047j 2.20440965-0.00632472j 2.62518771-0.00756116j
... 1.84382455-0.0072964j 1.26061497-0.01062452j
1.61506735-0.00875218j]]
Now plot the residuals for the first iteration
m_old = m1
# Response for Starting model
m_re_im = pg.utils.squeezeComplex(np.exp(m_old))
response_re_im = np.array(fop.response(m_re_im))
response = np.log(pg.utils.toComplex(response_re_im))
plot_inv_pars('stats_it{}.jpg'.format(i + 1), d, response, Wd, iteration=1)

And finally, plot the inversion results
fig, axes = plt.subplots(2, 3, figsize=(26 / 2.54, 15 / 2.54))
plot_fwd_model(axes[0, :])
axes[0, 0].set_title('This row: Forward model')
pg.show(
mesh, data=m1.real, ax=axes[1, 0],
cMin=np.log(50),
cMax=np.log(100),
label=r"$log_{10}(|\rho|~[\Omega m])$"
)
pg.show(
mesh, data=np.exp(m1.real), ax=axes[1, 1],
cMin=50, cMax=100,
label=r"$|\rho|~[\Omega m]$"
)
pg.show(
mesh, data=m1.imag * 1000, ax=axes[1, 2], cMap='jet_r',
label=r"$\phi$ [mrad]",
cMin=-50, cMax=0,
)
axes[1, 0].set_title('This row: Complex inversion')
for ax in axes.flat:
ax.set_xlim(-10, 60)
ax.set_ylim(-20, 0)
for s in scheme.sensors():
ax.scatter(s[0], s[1], color='k', s=5)
fig.tight_layout()

Total running time of the script: ( 1 minutes 54.777 seconds)