.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/examples/04-dcip/plot_inv_dcip_dipoledipole_3Dinversion_twospheres.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_content_examples_04-dcip_plot_inv_dcip_dipoledipole_3Dinversion_twospheres.py: 3D DC inversion of Dipole Dipole array ====================================== This is an example for 3D DC inversion. The model consists of 2 spheres, one conductive, the other one resistive compared to the background. We restrain the inversion to the Core Mesh through the use an Active Cells mapping that we combine with an exponetial mapping to invert in log conductivity space. Here mapping, :math:`\mathcal{M}`, indicates transformation of our model to a different space: .. math:: \sigma = \mathcal{M}(\mathbf{m}) Following example will show you how user can implement a 3D DC inversion. .. GENERATED FROM PYTHON SOURCE LINES 18-58 .. code-block:: Python import discretize from SimPEG import ( maps, utils, data_misfit, regularization, optimization, inverse_problem, directives, inversion, ) from SimPEG.electromagnetics.static import resistivity as DC, utils as DCutils import numpy as np import matplotlib.pyplot as plt try: from pymatsolver import Pardiso as Solver except ImportError: from SimPEG import SolverLU as Solver np.random.seed(12345) # 3D Mesh ######### # Cell sizes csx, csy, csz = 1.0, 1.0, 0.5 # Number of core cells in each direction ncx, ncy, ncz = 41, 31, 21 # Number of padding cells to add in each direction npad = 7 # Vectors of cell lengths in each direction with padding hx = [(csx, npad, -1.5), (csx, ncx), (csx, npad, 1.5)] hy = [(csy, npad, -1.5), (csy, ncy), (csy, npad, 1.5)] hz = [(csz, npad, -1.5), (csz, ncz)] # Create mesh and center it mesh = discretize.TensorMesh([hx, hy, hz], x0="CCN") # 2-spheres Model Creation .. GENERATED FROM PYTHON SOURCE LINES 59-165 .. code-block:: Python # Spheres parameters x0, y0, z0, r0 = -6.0, 0.0, -3.5, 3.0 x1, y1, z1, r1 = 6.0, 0.0, -3.5, 3.0 # ln conductivity ln_sigback = -5.0 ln_sigc = -3.0 ln_sigr = -6.0 # Define model # Background mtrue = ln_sigback * np.ones(mesh.nC) # Conductive sphere csph = ( np.sqrt( (mesh.gridCC[:, 0] - x0) ** 2.0 + (mesh.gridCC[:, 1] - y0) ** 2.0 + (mesh.gridCC[:, 2] - z0) ** 2.0 ) ) < r0 mtrue[csph] = ln_sigc * np.ones_like(mtrue[csph]) # Resistive Sphere rsph = ( np.sqrt( (mesh.gridCC[:, 0] - x1) ** 2.0 + (mesh.gridCC[:, 1] - y1) ** 2.0 + (mesh.gridCC[:, 2] - z1) ** 2.0 ) ) < r1 mtrue[rsph] = ln_sigr * np.ones_like(mtrue[rsph]) # Extract Core Mesh xmin, xmax = -20.0, 20.0 ymin, ymax = -15.0, 15.0 zmin, zmax = -10.0, 0.0 xyzlim = np.r_[[[xmin, xmax], [ymin, ymax], [zmin, zmax]]] actind, meshCore = utils.mesh_utils.extract_core_mesh(xyzlim, mesh) # Function to plot cylinder border def getCylinderPoints(xc, zc, r): xLocOrig1 = np.arange(-r, r + r / 10.0, r / 10.0) xLocOrig2 = np.arange(r, -r - r / 10.0, -r / 10.0) # Top half of cylinder zLoc1 = np.sqrt(-(xLocOrig1**2.0) + r**2.0) + zc # Bottom half of cylinder zLoc2 = -np.sqrt(-(xLocOrig2**2.0) + r**2.0) + zc # Shift from x = 0 to xc xLoc1 = xLocOrig1 + xc * np.ones_like(xLocOrig1) xLoc2 = xLocOrig2 + xc * np.ones_like(xLocOrig2) topHalf = np.vstack([xLoc1, zLoc1]).T topHalf = topHalf[0:-1, :] bottomHalf = np.vstack([xLoc2, zLoc2]).T bottomHalf = bottomHalf[0:-1, :] cylinderPoints = np.vstack([topHalf, bottomHalf]) cylinderPoints = np.vstack([cylinderPoints, topHalf[0, :]]) return cylinderPoints # Setup a synthetic Dipole-Dipole Survey # Line 1 xmin, xmax = -15.0, 15.0 ymin, ymax = 0.0, 0.0 zmin, zmax = 0, 0 endl = np.array([[xmin, ymin, zmin], [xmax, ymax, zmax]]) survey1 = DCutils.generate_dcip_survey( endl, "dipole-dipole", dim=mesh.dim, a=3, b=3, n=8 ) # Line 2 xmin, xmax = -15.0, 15.0 ymin, ymax = 5.0, 5.0 zmin, zmax = 0, 0 endl = np.array([[xmin, ymin, zmin], [xmax, ymax, zmax]]) survey2 = DCutils.generate_dcip_survey( endl, "dipole-dipole", dim=mesh.dim, a=3, b=3, n=8 ) # Line 3 xmin, xmax = -15.0, 15.0 ymin, ymax = -5.0, -5.0 zmin, zmax = 0, 0 endl = np.array([[xmin, ymin, zmin], [xmax, ymax, zmax]]) survey3 = DCutils.generate_dcip_survey( endl, "dipole-dipole", dim=mesh.dim, a=3, b=3, n=8 ) # Concatenate lines survey = DC.Survey(survey1.source_list + survey2.source_list + survey3.source_list) # Setup Problem with exponential mapping and Active cells only in the core mesh expmap = maps.ExpMap(mesh) mapactive = maps.InjectActiveCells(mesh=mesh, indActive=actind, valInactive=-5.0) mapping = expmap * mapactive problem = DC.Simulation3DCellCentered( mesh, survey=survey, sigmaMap=mapping, solver=Solver, bc_type="Neumann" ) data = problem.make_synthetic_data(mtrue[actind], relative_error=0.05, add_noise=True) # Least Squares Inversion .. GENERATED FROM PYTHON SOURCE LINES 166-241 .. code-block:: Python # Initial Model m0 = np.median(ln_sigback) * np.ones(mapping.nP) # Data Misfit dmis = data_misfit.L2DataMisfit(simulation=problem, data=data) # Regularization regT = regularization.WeightedLeastSquares( mesh, active_cells=actind, alpha_s=1e-6, alpha_x=1.0, alpha_y=1.0, alpha_z=1.0 ) # Optimization Scheme opt = optimization.InexactGaussNewton(maxIter=10) # Form the problem opt.remember("xc") invProb = inverse_problem.BaseInvProblem(dmis, regT, opt) # Directives for Inversions beta = directives.BetaEstimate_ByEig(beta0_ratio=1.0) Target = directives.TargetMisfit() betaSched = directives.BetaSchedule(coolingFactor=5.0, coolingRate=2) inv = inversion.BaseInversion(invProb, directiveList=[beta, Target, betaSched]) # Run Inversion minv = inv.run(m0) # Final Plot ############ fig, ax = plt.subplots(2, 2, figsize=(12, 6)) ax = utils.mkvc(ax) cyl0v = getCylinderPoints(x0, z0, r0) cyl1v = getCylinderPoints(x1, z1, r1) cyl0h = getCylinderPoints(x0, y0, r0) cyl1h = getCylinderPoints(x1, y1, r1) clim = [(mtrue[actind]).min(), (mtrue[actind]).max()] dat = meshCore.plot_slice( ((mtrue[actind])), ax=ax[0], normal="Y", clim=clim, ind=int(ncy / 2) ) ax[0].set_title("Ground Truth, Vertical") ax[0].set_aspect("equal") meshCore.plot_slice((minv), ax=ax[1], normal="Y", clim=clim, ind=int(ncy / 2)) ax[1].set_aspect("equal") ax[1].set_title("Inverted Model, Vertical") meshCore.plot_slice( ((mtrue[actind])), ax=ax[2], normal="Z", clim=clim, ind=int(ncz / 2) ) ax[2].set_title("Ground Truth, Horizontal") ax[2].set_aspect("equal") meshCore.plot_slice((minv), ax=ax[3], normal="Z", clim=clim, ind=int(ncz / 2)) ax[3].set_title("Inverted Model, Horizontal") ax[3].set_aspect("equal") for i in range(2): ax[i].plot(cyl0v[:, 0], cyl0v[:, 1], "k--") ax[i].plot(cyl1v[:, 0], cyl1v[:, 1], "k--") for i in range(2, 4): ax[i].plot(cyl1h[:, 0], cyl1h[:, 1], "k--") ax[i].plot(cyl0h[:, 0], cyl0h[:, 1], "k--") fig.subplots_adjust(right=0.8) cbar_ax = fig.add_axes([0.85, 0.15, 0.05, 0.7]) cb = plt.colorbar(dat[0], ax=cbar_ax) cb.set_label("ln conductivity") cbar_ax.axis("off") plt.show() .. image-sg:: /content/examples/04-dcip/images/sphx_glr_plot_inv_dcip_dipoledipole_3Dinversion_twospheres_001.png :alt: Ground Truth, Vertical, Ground Truth, Horizontal, Inverted Model, Vertical, Inverted Model, Horizontal :srcset: /content/examples/04-dcip/images/sphx_glr_plot_inv_dcip_dipoledipole_3Dinversion_twospheres_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none SimPEG.InvProblem will set Regularization.reference_model to m0. SimPEG.InvProblem will set Regularization.reference_model to m0. SimPEG.InvProblem will set Regularization.reference_model to m0. SimPEG.InvProblem will set Regularization.reference_model to m0. SimPEG.InvProblem will set Regularization.reference_model to m0. SimPEG.InvProblem will set Regularization.reference_model to m0. SimPEG.InvProblem will set Regularization.reference_model to m0. SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv. ***Done using same Solver, and solver_opts as the Simulation3DCellCentered 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 2.69e+01 5.01e+03 0.00e+00 5.01e+03 1.09e+03 0 1 2.69e+01 7.03e+02 1.34e+01 1.06e+03 1.02e+02 0 2 5.38e+00 5.08e+02 1.84e+01 6.07e+02 1.04e+02 0 Skip BFGS 3 5.38e+00 2.19e+02 4.07e+01 4.38e+02 5.08e+01 0 4 1.08e+00 1.86e+02 4.28e+01 2.32e+02 4.81e+01 0 5 1.08e+00 1.36e+02 5.75e+01 1.97e+02 6.16e+01 0 6 2.15e-01 1.22e+02 6.06e+01 1.35e+02 5.56e+01 0 7 2.15e-01 1.10e+02 6.67e+01 1.25e+02 6.60e+01 0 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 5.0146e+02 1 : |xc-x_last| = 4.8861e+00 <= tolX*(1+|x0|) = 7.5300e+01 0 : |proj(x-g)-x| = 6.6048e+01 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 6.6048e+01 <= 1e3*eps = 1.0000e-02 0 : maxIter = 10 <= iter = 8 ------------------------- DONE! ------------------------- .. rst-class:: sphx-glr-timing **Total running time of the script:** (1 minutes 17.631 seconds) **Estimated memory usage:** 267 MB .. _sphx_glr_download_content_examples_04-dcip_plot_inv_dcip_dipoledipole_3Dinversion_twospheres.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_inv_dcip_dipoledipole_3Dinversion_twospheres.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_dcip_dipoledipole_3Dinversion_twospheres.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_