VES inversion for a blocky model

This tutorial shows how an built-in forward operator is used for inversion. A DC 1D (VES) modelling is used to generate data, noisify and invert them.

We import numpy, matplotlib and the 1D plotting function

import numpy as np
import matplotlib.pyplot as plt

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

some definitions before (model, data and error)

nlay = 4  # number of layers
lam = 200.  # (initial) regularization parameter
errPerc = 3.  # relative error of 3 percent
ab2 = np.logspace(-1, 2, 50)  # AB/2 distance (current electrodes)
mn2 = ab2 / 3.  # MN/2 distance (potential electrodes)

initialize the forward modelling operator

f = pg.core.DC1dModelling(nlay, ab2, mn2)

other ways are by specifying a Data Container or am/an/bm/bn distances

synres = [100., 500., 20., 800.]  # synthetic resistivity
synthk = [0.5, 3.5, 6.]  # synthetic thickness (nlay-th layer is infinite)

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

rhoa = f(synthk+synres)
rhoa = rhoa * (pg.randn(len(rhoa), seed=0) * errPerc / 100. + 1.)

create some transformations used for inversion

transThk = pg.trans.TransLog()  # log-transform ensures thk>0
transRho = pg.trans.TransLogLU(1, 1000)  # lower and upper bound
transRhoa = pg.trans.TransLog()  # log transformation for data

set model transformation for thickness and resistivity

f.region(0).setTransModel(transThk)  # 0=thickness
f.region(1).setTransModel(transRho)  # 1=resistivity

generate start model values from median app. resistivity & spread

paraDepth = max(ab2) / 3.  # rule-of-thumb for Wenner/Schlumberger
f.region(0).setStartValue(paraDepth / nlay / 2)
f.region(1).setStartValue(np.median(rhoa))

set up inversion

inv = pg.core.Inversion(rhoa, f, transRhoa, True)  # data vector, fop, verbose
# could also be set by inv.setTransData(transRhoa)

set error model, regularization strength and Marquardt scheme

inv.setRelativeError(errPerc / 100.0)  # alternative: setAbsoluteError in Ohmm
inv.setLambda(lam)  # (initial) regularization parameter
inv.setMarquardtScheme(0.9)  # decrease lambda by factor 0.9
model = f.createStartVector()  # creates from region start value
model[nlay] *= 1.5  # change default model by changing 2nd layer resistivity
inv.setModel(model)  #

run actual inversion and extract resistivity and thickness

model = inv.run()  # result is a pg.Vector, but compatible to numpy array
res, thk = model[nlay-1:nlay*2-1], model[0:nlay-1]
print('rrms={:.2f}%, chi^2={:.3f}'.format(inv.relrms(), inv.chi2()))

Out:

rrms=3.21%, chi^2=1.136

show estimated&synthetic models and data with model response in 2 subplots

fig, ax = plt.subplots(ncols=2, figsize=(8, 6))  # two-column figure
drawModel1D(ax[0], synthk, synres, plot='semilogx', color='r')
drawModel1D(ax[0], thk, res, color='b')
ax[0].grid(True, which='both')
ax[0].set_ylabel('z (m)')
ax[0].set_xlabel(r'$\rho$ ($\Omega$m)')
ax[1].loglog(rhoa, ab2, 'rx-', label='data')  # sounding curve
ax[1].loglog(inv.response(), ab2, 'b-', label='response')
ax[1].set_ylim((max(ab2), min(ab2)))  # downwards according to penetration
ax[1].grid(True, which='both')
ax[1].set_xlabel(r'$\rho_a$ ($\Omega$m)')
ax[1].set_ylabel('AB/2 (m)')
ax[1].legend(loc='best')
plt.show()