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

start_re_im = pg.utils.squeezeComplex(start_model)
f_0 = np.array(fop.response(start_re_im))

# Compute the Jacobian for the starting model
J_block = fop.createJacobian(start_re_im)
J_re = np.array(J_block.mat(0))
J_im = np.array(J_block.mat(1))
J0 = J_re + 1j * J_im

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()