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)
def setupProblem( mesh, muMod, sigmaMod, prbtype='e', invertMui=False, sigmaInInversion=False, freq=1. ): rxcomp = ['real', 'imag'] loc = Utils.ndgrid( [mesh.vectorCCx, np.r_[0.], mesh.vectorCCz] ) if prbtype in ['e', 'b']: rxfields_y = ['e', 'j'] rxfields_xz = ['b', 'h'] elif prbtype in ['h', 'j']: rxfields_y = ['b', 'h'] rxfields_xz = ['e', 'j'] rxList_edge = [ getattr(FDEM.Rx, 'Point_{f}'.format(f=f))( loc, component=comp, orientation=orient ) for f in rxfields_y for comp in rxcomp for orient in ['y'] ] rxList_face = [ getattr(FDEM.Rx, 'Point_{f}'.format(f=f))( loc, component=comp, orientation=orient ) for f in rxfields_xz for comp in rxcomp for orient in ['x', 'z'] ] rxList = rxList_edge + rxList_face src_loc = np.r_[0., 0., 0.] if prbtype in ['e', 'b']: src = FDEM.Src.MagDipole( rxList=rxList, loc=src_loc, freq=freq ) elif prbtype in ['h', 'j']: ind = Utils.closestPoints(mesh, src_loc, 'Fz') + mesh.vnF[0] vec = np.zeros(mesh.nF) vec[ind] = 1. src = FDEM.Src.RawVec_e(rxList=rxList, freq=freq, s_e=vec) survey = FDEM.Survey([src]) if sigmaInInversion: wires = Maps.Wires( ('mu', mesh.nC), ('sigma', mesh.nC) ) muMap = Maps.MuRelative(mesh) * wires.mu sigmaMap = Maps.ExpMap(mesh) * wires.sigma if invertMui: muiMap = Maps.ReciprocalMap(mesh)*muMap prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, muiMap=muiMap, sigmaMap=sigmaMap ) # m0 = np.hstack([1./muMod, sigmaMod]) else: prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, muMap=muMap, sigmaMap=sigmaMap ) m0 = np.hstack([muMod, sigmaMod]) else: muMap = Maps.MuRelative(mesh) if invertMui: muiMap = Maps.ReciprocalMap(mesh) * muMap prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, sigma=sigmaMod, muiMap=muiMap ) # m0 = 1./muMod else: prob = getattr(FDEM, 'Problem3D_{}'.format(prbtype))( mesh, sigma=sigmaMod, muMap=muMap ) m0 = muMod prob.pair(survey) return m0, prob, survey