.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/user-guide/examples/10-pgi/plot_inv_0_PGI_Linear_1D.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_user-guide_examples_10-pgi_plot_inv_0_PGI_Linear_1D.py: Petrophysically guided inversion (PGI): Linear example ====================================================== We do a comparison between the classic least-squares inversion and our formulation of a petrophysically constrained inversion. We explore it through the UBC linear example. .. GENERATED FROM PYTHON SOURCE LINES 12-14 Tikhonov Inversion# #################### .. GENERATED FROM PYTHON SOURCE LINES 14-90 .. code-block:: Python import discretize as Mesh import matplotlib.pyplot as plt import numpy as np from simpeg import ( data_misfit, directives, inverse_problem, inversion, maps, optimization, regularization, simulation, utils, ) # Random seed for reproductibility np.random.seed(1) # Mesh N = 100 mesh = Mesh.TensorMesh([N]) # Survey design parameters nk = 20 jk = np.linspace(1.0, 60.0, nk) p = -0.25 q = 0.25 # Physics def g(k): return np.exp(p * jk[k] * mesh.cell_centers_x) * np.cos( np.pi * q * jk[k] * mesh.cell_centers_x ) G = np.empty((nk, mesh.nC)) for i in range(nk): G[i, :] = g(i) # True model mtrue = np.zeros(mesh.nC) mtrue[mesh.cell_centers_x > 0.2] = 1.0 mtrue[mesh.cell_centers_x > 0.35] = 0.0 t = (mesh.cell_centers_x - 0.65) / 0.25 indx = np.abs(t) < 1 mtrue[indx] = -(((1 - t**2.0) ** 2.0)[indx]) mtrue = np.zeros(mesh.nC) mtrue[mesh.cell_centers_x > 0.3] = 1.0 mtrue[mesh.cell_centers_x > 0.45] = -0.5 mtrue[mesh.cell_centers_x > 0.6] = 0 # simpeg problem and survey prob = simulation.LinearSimulation(mesh, G=G, model_map=maps.IdentityMap()) std = 0.01 survey = prob.make_synthetic_data(mtrue, relative_error=std, add_noise=True) # Setup the inverse problem reg = regularization.WeightedLeastSquares(mesh, alpha_s=1.0, alpha_x=1.0) dmis = data_misfit.L2DataMisfit(data=survey, simulation=prob) opt = optimization.ProjectedGNCG(maxIter=10, cg_maxiter=50, cg_rtol=1e-3) invProb = inverse_problem.BaseInvProblem(dmis, reg, opt) directiveslist = [ directives.BetaEstimate_ByEig(beta0_ratio=1e-5), directives.BetaSchedule(coolingFactor=10.0, coolingRate=2), directives.TargetMisfit(), ] inv = inversion.BaseInversion(invProb, directiveList=directiveslist) m0 = np.zeros_like(mtrue) mnormal = inv.run(m0) .. rst-class:: sphx-glr-script-out .. code-block:: none Running inversion with SimPEG v0.25.2 ================================================= Projected GNCG ================================================= # beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment ----------------------------------------------------------------------------------------------------------------- 0 1.84e+01 2.00e+05 0.00e+00 2.00e+05 0 inf inf 1 1.84e+01 3.36e+02 4.22e+01 1.11e+03 2.43e+06 0 13 7.34e-04 1.79e+03 2 1.84e+01 3.47e+01 4.56e+01 8.73e+02 1.79e+03 0 23 4.41e-04 7.87e-01 Skip BFGS 3 1.84e+00 6.71e+00 4.86e+01 9.61e+01 5.06e+02 0 32 1.91e-04 9.68e-02 Skip BFGS ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 2.2447e+01 <= tolF*(1+|f0|) = 2.0000e+04 0 : |xc-x_last| = 1.5923e-01 <= tolX*(1+|x0|) = 1.0000e-01 0 : |proj(x-g)-x| = 5.0585e+02 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 5.0585e+02 <= 1e3*eps = 1.0000e-02 0 : maxIter = 10 <= iter = 3 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 91-93 Petrophysically constrained inversion # ######################################## .. GENERATED FROM PYTHON SOURCE LINES 93-168 .. code-block:: Python # fit a Gaussian Mixture Model with n components # on the true model to simulate the laboratory # petrophysical measurements n = 3 clf = utils.WeightedGaussianMixture( mesh=mesh, n_components=n, covariance_type="full", max_iter=100, n_init=3, reg_covar=5e-4, ) clf.fit(mtrue.reshape(-1, 1)) # Petrophyically constrained regularization reg = regularization.PGI( gmmref=clf, mesh=mesh, alpha_pgi=1.0, alpha_x=1.0, ) # Optimization opt = optimization.ProjectedGNCG(maxIter=20, cg_maxiter=50, cg_rtol=1e-3) opt.remember("xc") # Setup new inverse problem invProb = inverse_problem.BaseInvProblem(dmis, reg, opt) # directives Alphas = directives.AlphasSmoothEstimate_ByEig(alpha0_ratio=10.0, verbose=True) beta = directives.BetaEstimate_ByEig(beta0_ratio=1e-8) betaIt = directives.PGI_BetaAlphaSchedule( verbose=True, coolingFactor=2.0, warmingFactor=1.0, tolerance=0.1, update_rate=1, progress=0.2, ) targets = directives.MultiTargetMisfits(verbose=True) petrodir = directives.PGI_UpdateParameters() addmref = directives.PGI_AddMrefInSmooth(verbose=True) # Setup Inversion inv = inversion.BaseInversion( invProb, directiveList=[Alphas, beta, petrodir, targets, addmref, betaIt] ) # Initial model same as for WeightedLeastSquares mcluster = inv.run(m0) # Final Plot fig, axes = plt.subplots(1, 3, figsize=(12 * 1.2, 4 * 1.2)) for i in range(prob.G.shape[0]): axes[0].plot(prob.G[i, :]) axes[0].set_title("Columns of matrix G") axes[1].hist(mtrue, bins=20, linewidth=3.0, density=True, color="k") axes[1].set_xlabel("Model value") axes[1].set_xlabel("Occurence") axes[1].hist(mnormal, bins=20, density=True, color="b") axes[1].hist(mcluster, bins=20, density=True, color="r") axes[1].legend(["Mtrue Hist.", "L2 Model Hist.", "PGI Model Hist."]) axes[2].plot(mesh.cell_centers_x, mtrue, color="black", linewidth=3) axes[2].plot(mesh.cell_centers_x, mnormal, color="blue") axes[2].plot(mesh.cell_centers_x, mcluster, "r-") axes[2].plot(mesh.cell_centers_x, invProb.reg.objfcts[0].reference_model, "r--") axes[2].legend(("True Model", "L2 Model", "PGI Model", "Learned Mref")) axes[2].set_ylim([-2, 2]) plt.show() .. image-sg:: /content/user-guide/examples/10-pgi/images/sphx_glr_plot_inv_0_PGI_Linear_1D_001.png :alt: Columns of matrix G :srcset: /content/user-guide/examples/10-pgi/images/sphx_glr_plot_inv_0_PGI_Linear_1D_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Running inversion with SimPEG v0.25.2 Alpha scales: [np.float64(0.5312463186890524), np.float64(0.0)] ================================================= Projected GNCG ================================================= # beta phi_d phi_m f |proj(x-g)-x| LS iter_CG CG |Ax-b|/|b| CG |Ax-b| Comment ----------------------------------------------------------------------------------------------------------------- 0 3.15e-02 2.00e+05 0.00e+00 2.00e+05 0 inf inf 1 3.15e-02 3.23e+01 3.53e+02 4.34e+01 2.43e+06 0 14 1.51e-04 3.67e+02 geophys. misfits: 32.3 (target 20.0 [False]) | smallness misfit: 2973.5 (target: 100.0 [False]) mref changed in 26 places Beta cooling evaluation: progress: [32.3]; minimum progress targets: [160000.] 2 3.15e-02 6.05e+00 5.13e+01 7.67e+00 3.67e+02 0 45 3.51e-04 1.29e-01 geophys. misfits: 6.1 (target 20.0 [True]) | smallness misfit: 1908.1 (target: 100.0 [False]) mref changed in 1 places Beta cooling evaluation: progress: [6.1]; minimum progress targets: [25.8] Warming alpha_pgi to favor clustering: 3.3033002338060204 3 3.15e-02 6.18e+00 7.39e+01 8.51e+00 3.13e+00 0 50 3.00e+01 9.38e+01 geophys. misfits: 6.2 (target 20.0 [True]) | smallness misfit: 916.9 (target: 100.0 [False]) mref changed in 0 places Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells Beta cooling evaluation: progress: [6.2]; minimum progress targets: [22.] Warming alpha_pgi to favor clustering: 10.68301488348927 4 3.15e-02 6.28e+00 1.09e+02 9.73e+00 9.40e+01 0 50 6.75e+00 6.35e+02 geophys. misfits: 6.3 (target 20.0 [True]) | smallness misfit: 383.0 (target: 100.0 [False]) mref changed in 0 places Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells Beta cooling evaluation: progress: [6.3]; minimum progress targets: [22.] Warming alpha_pgi to favor clustering: 34.00052609747121 5 3.15e-02 6.48e+00 1.77e+02 1.21e+01 6.35e+02 0 50 1.49e-01 9.48e+01 geophys. misfits: 6.5 (target 20.0 [True]) | smallness misfit: 290.8 (target: 100.0 [False]) mref changed in 0 places Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells Beta cooling evaluation: progress: [6.5]; minimum progress targets: [22.] Warming alpha_pgi to favor clustering: 104.8909955019337 6 3.15e-02 6.68e+00 2.54e+02 1.47e+01 1.03e+02 0 50 2.78e+01 2.88e+03 geophys. misfits: 6.7 (target 20.0 [True]) | smallness misfit: 139.3 (target: 100.0 [False]) mref changed in 0 places Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells Beta cooling evaluation: progress: [6.7]; minimum progress targets: [22.] Warming alpha_pgi to favor clustering: 313.91090731658386 7 3.15e-02 8.71e+00 3.59e+02 2.00e+01 2.88e+03 0 50 7.01e-01 2.02e+03 geophys. misfits: 8.7 (target 20.0 [True]) | smallness misfit: 79.4 (target: 100.0 [True]) All targets have been reached mref changed in 0 places Add mref to Smoothness. Changes in mref happened in 0.0 % of the cells Beta cooling evaluation: progress: [8.7]; minimum progress targets: [22.] Warming alpha_pgi to favor clustering: 720.7844934364794 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 3.8498e+00 <= tolF*(1+|f0|) = 2.0000e+04 0 : |xc-x_last| = 1.1439e-01 <= tolX*(1+|x0|) = 1.0000e-01 0 : |proj(x-g)-x| = 2.8766e+03 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 2.8766e+03 <= 1e3*eps = 1.0000e-02 0 : maxIter = 20 <= iter = 7 ------------------------- DONE! ------------------------- .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 5.890 seconds) **Estimated memory usage:** 325 MB .. _sphx_glr_download_content_user-guide_examples_10-pgi_plot_inv_0_PGI_Linear_1D.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_0_PGI_Linear_1D.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_0_PGI_Linear_1D.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_inv_0_PGI_Linear_1D.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_