VES inversion for a smooth model

This tutorial shows how an built-in forward operator is used for an Occam type (smoothness-constrained) inversion with fixed regularization (most natural). A direct current (DC) one-dimensional (1D) VES (vertical electric sounding) modelling operator is used to generate data, add noise and inversion.

We import numpy numerics, mpl plotting, pygimli and the 1D plotting function

import numpy as np
np.random.seed(1337)

import matplotlib.pyplot as plt

import pygimli as pg
from pygimli.viewer.mpl import drawModel1D

synres = [100., 500., 20., 800.]  # synthetic resistivity
synthk = [4, 6, 10]  # synthetic thickness (lay layer is infinite)
ab2 = np.logspace(-1, 2, 25)  # 0.1 to 100 in 25 steps (8 points per decade)
fBlock = pg.core.DC1dModelling(len(synres), ab2, ab2/3)
rhoa = fBlock(synthk+synres)
# The data are noisified using a
errPerc = 3.  # relative error of 3 percent
rhoa = rhoa * (np.random.randn(len(rhoa)) * errPerc / 100. + 1.)

The forward operator can be called by f.response(model) or simply f(model)

thk = np.logspace(-0.5, 0.5, 30)
f = pg.core.DC1dRhoModelling(thk, ab2, ab2/3)

Create some transformations used for inversion

transRho = pg.trans.TransLogLU(1, 1000)  # lower and upper bound
transRhoa = pg.trans.TransLog()  # log transformation also for data

Set up inversion

inv = pg.core.Inversion(rhoa, f, transRhoa, transRho, False)  # data vector, f, ...
# The transformations can also be omitted and set individually by
# inv.setTransData(transRhoa)
# inv.setTransModel(transRho)
inv.setRelativeError(errPerc / 100.0)

Create a homogeneous starting model

model = pg.Vector(len(thk)+1, np.median(rhoa))  # uniform values
inv.setModel(model)  #

Set pretty large regularization strength and run inversion

print("inversion with lam=200")
inv.setLambda(100)
res100 = inv.run()  # result is a pg.Vector, but compatible to numpy array
print('rrms={:.2f}%, chi^2={:.3f}'.format(inv.relrms(), inv.chi2()))
# Decrease the regularization (smoothness) and start (from old result)
print("inversion with lam=20")
inv.setLambda(10)
res10 = inv.run()  # result is a pg.Vector, but compatible to numpy array
print('rrms={:.2f}%, chi^2={:.3f}'.format(inv.relrms(), inv.chi2()))
# We now optimize lambda such that data are fitted within noise (chi^2=1)
print("chi^2=1 optimized inversion")
resChi = inv.runChi1()  # ends up in a lambda of about 3
print("optimized lambda value:", inv.getLambda())
print('rrms={:.2f}%, chi^2={:.3f}'.format(inv.relrms(), inv.chi2()))

Out:

inversion with lam=200
rrms=5.49%, chi^2=3.373
inversion with lam=20
rrms=3.64%, chi^2=1.454
chi^2=1 optimized inversion
optimized lambda value: 0.7179364718731468
rrms=3.01%, chi^2=0.994

Show model (inverted and synthetic) as well as model response and data

fig, ax = plt.subplots(ncols=2, figsize=(8, 6))  # two-column figure
drawModel1D(ax[0], synthk, synres, color='b', label='synthetic',
            plot='semilogx')
drawModel1D(ax[0], thk, res100, color='g', label=r'$\lambda$=100')
drawModel1D(ax[0], thk, res10, color='c', label=r'$\lambda$=10')
drawModel1D(ax[0], thk, resChi, color='r',
            label=r'$\chi$={:.2f}'.format(inv.getLambda()))
ax[0].grid(True, which='both')
ax[0].legend(loc='best')
ax[1].loglog(rhoa, ab2, 'rx-', label='measured')
ax[1].loglog(inv.response(), ab2, 'b-', label='fitted')
ax[1].set_ylim((max(ab2), min(ab2)))
ax[1].grid(True, which='both')
ax[1].set_xlabel(r'$\rho_a$ [$\Omega$m]')
ax[1].set_ylabel('AB/2 [m]')
ax[1].yaxis.set_label_position('right')
ax[1].yaxis.set_ticks_position('right')
ax[1].legend(loc='best')
plt.show()