# DEM Mu InversionΒΆ

nversion of Magnetic Susceptibility from FDEM data assuming a fixed trical conductivity

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 from SimPEG import ( Mesh, Maps, Utils, DataMisfit, Regularization, Optimization, Inversion, InvProblem, Directives ) from SimPEG.EM import FDEM, TDEM, mu_0 import numpy as np import matplotlib.pyplot as plt import matplotlib try: from pymatsolver import PardisoSolver as Solver except ImportError: from SimPEG import SolverLU as Solver def run(plotIt=True): """ 1D FDEM Mu Inversion ==================== 1D inversion of Magnetic Susceptibility from FDEM data assuming a fixed electrical conductivity """ # Set up cylindrically symmeric mesh cs, ncx, ncz, npad = 10., 15, 25, 13 # padded cyl mesh 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') # Geologic Parameters model layerz = np.r_[-100., -50.] layer = (mesh.vectorCCz >= layerz[0]) & (mesh.vectorCCz <= layerz[1]) active = mesh.vectorCCz < 0. # Electrical Conductivity sig_half = 1e-2 # Half-space conductivity sig_air = 1e-8 # Air conductivity sig_layer = 1e-2 # Layer conductivity sigma = np.ones(mesh.nCz)*sig_air sigma[active] = sig_half sigma[layer] = sig_layer # mur - relative magnetic permeability mur_half = 1. mur_air = 1. mur_layer = 2. mur = np.ones(mesh.nCz)*mur_air mur[active] = mur_half mur[layer] = mur_layer mtrue = mur[active] # Maps actMap = Maps.InjectActiveCells(mesh, active, mur_air, nC=mesh.nCz) surj1Dmap = Maps.SurjectVertical1D(mesh) murMap = Maps.MuRelative(mesh) # Mapping muMap = murMap * surj1Dmap * actMap # ----- FDEM problem & survey ----- rxlocs = Utils.ndgrid([np.r_[10.], np.r_[0], np.r_[30.]]) bzr = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'real') # bzi = FDEM.Rx.Point_bSecondary(rxlocs, 'z', 'imag') freqs = np.linspace(2000, 10000, 10) #np.logspace(3, 4, 10) srcLoc = np.array([0., 0., 30.]) print( 'min skin depth = ', 500./np.sqrt(freqs.max() * sig_half), 'max skin depth = ', 500./np.sqrt(freqs.min() * sig_half) ) print( 'max x ', mesh.vectorCCx.max(), 'min z ', mesh.vectorCCz.min(), 'max z ', mesh.vectorCCz.max() ) srcList = [ FDEM.Src.MagDipole([bzr], freq, srcLoc, orientation='Z') for freq in freqs ] surveyFD = FDEM.Survey(srcList) prbFD = FDEM.Problem3D_b( mesh, sigma=surj1Dmap * sigma, muMap=muMap, Solver=Solver ) prbFD.pair(surveyFD) std = 0.03 surveyFD.makeSyntheticData(mtrue, std) surveyFD.eps = np.linalg.norm(surveyFD.dtrue)*1e-6 # FDEM inversion np.random.seed(13472) dmisfit = DataMisfit.l2_DataMisfit(surveyFD) regMesh = Mesh.TensorMesh([mesh.hz[muMap.maps[-1].indActive]]) reg = Regularization.Simple(regMesh) opt = Optimization.InexactGaussNewton(maxIterCG=10) invProb = InvProblem.BaseInvProblem(dmisfit, reg, opt) # Inversion Directives betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.) beta = Directives.BetaSchedule(coolingFactor=4, coolingRate=3) betaest = Directives.BetaEstimate_ByEig(beta0_ratio=2.) target = Directives.TargetMisfit() directiveList = [beta, betaest, target] inv = Inversion.BaseInversion(invProb, directiveList=directiveList) m0 = mur_half * np.ones(mtrue.size) reg.alpha_s = 2e-2 reg.alpha_x = 1. prbFD.counter = opt.counter = Utils.Counter() opt.remember('xc') moptFD = inv.run(m0) dpredFD = surveyFD.dpred(moptFD) if plotIt: fig, ax = plt.subplots(1, 3, figsize=(10, 6)) fs = 13 # fontsize matplotlib.rcParams['font.size'] = fs # Plot the conductivity model ax[0].semilogx(sigma[active], mesh.vectorCCz[active], 'k-', lw=2) ax[0].set_ylim(-500, 0) ax[0].set_xlim(5e-3, 1e-1) ax[0].set_xlabel('Conductivity (S/m)', fontsize=fs) ax[0].set_ylabel('Depth (m)', fontsize=fs) ax[0].grid( which='both', color='k', alpha=0.5, linestyle='-', linewidth=0.2 ) ax[0].legend(['Conductivity Model'], fontsize=fs, loc=4) # Plot the permeability model ax[1].plot(mur[active], mesh.vectorCCz[active], 'k-', lw=2) ax[1].plot(moptFD, mesh.vectorCCz[active], 'b-', lw=2) ax[1].set_ylim(-500, 0) ax[1].set_xlim(0.5, 2.1) ax[1].set_xlabel('Relative Permeability', fontsize=fs) ax[1].set_ylabel('Depth (m)', fontsize=fs) ax[1].grid( which='both', color='k', alpha=0.5, linestyle='-', linewidth=0.2 ) ax[1].legend(['True', 'Predicted'], fontsize=fs, loc=4) # plot the data misfits - negative b/c we choose positive to be in the # direction of primary ax[2].plot(freqs, -surveyFD.dobs, 'k-', lw=2) # ax[2].plot(freqs, -surveyFD.dobs[1::2], 'k--', lw=2) ax[2].loglog(freqs, -dpredFD, 'bo', ms=6) # ax[2].loglog(freqs, -dpredFD[1::2], 'b+', markeredgewidth=2., ms=10) # Labels, gridlines, etc ax[2].grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2) ax[2].grid(which='both', alpha=0.5, linestyle='-', linewidth=0.2) ax[2].set_xlabel('Frequency (Hz)', fontsize=fs) ax[2].set_ylabel('Vertical magnetic field (-T)', fontsize=fs) # ax[2].legend(("Obs", "Pred"), fontsize=fs) ax[2].legend( ("z-Obs (real)", "z-Pred (real)"), fontsize=fs ) ax[2].set_xlim(freqs.max(), freqs.min()) ax[0].set_title("(a) Conductivity Model", fontsize=fs) ax[1].set_title("(b) $\mu_r$ Model", fontsize=fs) ax[2].set_title("(c) FDEM observed vs. predicted", fontsize=fs) # ax[2].set_title("(c) TDEM observed vs. predicted", fontsize=fs) plt.tight_layout(pad=1.5) if __name__ == '__main__': run(plotIt=True) plt.show()