EM: TDEM: 1D: Inversion

Here we will create and run a TDEM 1D inversion.

Out:

SimPEG.InvProblem will set Regularization.mref to m0.

    SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv.
    ***Done using same Solver and solverOpts as the problem***
model has any nan: 0
============================ Inexact Gauss Newton ============================
  #     beta     phi_d     phi_m       f      |proj(x-g)-x|  LS    Comment
-----------------------------------------------------------------------------
x0 has any nan: 0
   0  2.85e+03  2.66e+03  0.00e+00  2.66e+03    3.19e+03      0
   1  2.85e+03  2.71e+02  7.54e-02  4.86e+02    4.20e+02      0
   2  5.70e+02  1.12e+02  1.09e-01  1.75e+02    2.96e+02      0   Skip BFGS
   3  5.70e+02  2.24e+01  1.70e-01  1.19e+02    1.29e+01      0   Skip BFGS
   4  1.14e+02  2.16e+01  1.71e-01  4.10e+01    7.89e+01      0   Skip BFGS
   5  1.14e+02  1.27e+01  1.97e-01  3.52e+01    4.23e+00      0
------------------------- STOP! -------------------------
1 : |fc-fOld| = 5.7763e+00 <= tolF*(1+|f0|) = 2.6573e+02
1 : |xc-x_last| = 1.8143e-01 <= tolX*(1+|x0|) = 3.0149e+00
0 : |proj(x-g)-x|    = 4.2331e+00 <= tolG          = 1.0000e-01
0 : |proj(x-g)-x|    = 4.2331e+00 <= 1e3*eps       = 1.0000e-02
1 : maxIter   =       5    <= iter          =      5
------------------------- DONE! -------------------------

import numpy as np
from SimPEG import (Mesh, Maps, SolverLU, DataMisfit, Regularization,
                    Optimization, InvProblem, Inversion, Directives, Utils)
import SimPEG.EM as EM
import matplotlib.pyplot as plt


def run(plotIt=True):

    cs, ncx, ncz, npad = 5., 25, 15, 15
    hx = [(cs, ncx),  (cs, npad, 1.3)]
    hz = [(cs, npad, -1.3), (cs, ncz), (cs, npad, 1.3)]
    mesh = Mesh.CylMesh([hx, 1, hz], '00C')

    active = mesh.vectorCCz < 0.
    layer = (mesh.vectorCCz < 0.) & (mesh.vectorCCz >= -100.)
    actMap = Maps.InjectActiveCells(mesh, active, np.log(1e-8), nC=mesh.nCz)
    mapping = Maps.ExpMap(mesh) * Maps.SurjectVertical1D(mesh) * actMap
    sig_half = 2e-3
    sig_air = 1e-8
    sig_layer = 1e-3
    sigma = np.ones(mesh.nCz)*sig_air
    sigma[active] = sig_half
    sigma[layer] = sig_layer
    mtrue = np.log(sigma[active])

    rxOffset = 1e-3
    rx = EM.TDEM.Rx.Point_dbdt(
        np.array([[rxOffset, 0., 30]]),
        np.logspace(-5, -3, 31),
        'z'
    )
    src = EM.TDEM.Src.MagDipole([rx], loc=np.array([0., 0., 80]))
    survey = EM.TDEM.Survey([src])
    prb = EM.TDEM.Problem3D_e(mesh, sigmaMap=mapping)

    prb.Solver = SolverLU
    prb.timeSteps = [(1e-06, 20), (1e-05, 20), (0.0001, 20)]
    prb.pair(survey)

    # create observed data
    std = 0.05

    survey.dobs = survey.makeSyntheticData(mtrue, std)
    survey.std = std
    survey.eps = 1e-5*np.linalg.norm(survey.dobs)

    dmisfit = DataMisfit.l2_DataMisfit(survey)
    regMesh = Mesh.TensorMesh([mesh.hz[mapping.maps[-1].indActive]])
    reg = Regularization.Tikhonov(regMesh, alpha_s=1e-2, alpha_x=1.)
    opt = Optimization.InexactGaussNewton(maxIter=5, LSshorten=0.5)
    invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt)

    # Create an inversion object
    beta = Directives.BetaSchedule(coolingFactor=5, coolingRate=2)
    betaest = Directives.BetaEstimate_ByEig(beta0_ratio=1e0)
    inv = Inversion.BaseInversion(invProb, directiveList=[beta, betaest])
    m0 = np.log(np.ones(mtrue.size)*sig_half)
    prb.counter = opt.counter = Utils.Counter()
    opt.remember('xc')

    mopt = inv.run(m0)

    if plotIt:
        fig, ax = plt.subplots(1, 2, figsize=(10, 6))
        ax[0].loglog(rx.times, survey.dtrue, 'b.-')
        ax[0].loglog(rx.times, survey.dobs, 'r.-')
        ax[0].legend(('Noisefree', '$d^{obs}$'), fontsize=16)
        ax[0].set_xlabel('Time (s)', fontsize=14)
        ax[0].set_ylabel('$B_z$ (T)', fontsize=16)
        ax[0].set_xlabel('Time (s)', fontsize=14)
        ax[0].grid(color='k', alpha=0.5, linestyle='dashed', linewidth=0.5)

        plt.semilogx(sigma[active], mesh.vectorCCz[active])
        plt.semilogx(np.exp(mopt), mesh.vectorCCz[active])
        ax[1].set_ylim(-600, 0)
        ax[1].set_xlim(1e-4, 1e-2)
        ax[1].set_xlabel('Conductivity (S/m)', fontsize=14)
        ax[1].set_ylabel('Depth (m)', fontsize=14)
        ax[1].grid(color='k', alpha=0.5, linestyle='dashed', linewidth=0.5)
        plt.legend(['$\sigma_{true}$', '$\sigma_{pred}$'])


if __name__ == '__main__':
    run()
    plt.show()

Total running time of the script: ( 0 minutes 13.711 seconds)