.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/tutorials/04-magnetics/plot_inv_2a_magnetics_induced.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_tutorials_04-magnetics_plot_inv_2a_magnetics_induced.py: Sparse Norm Inversion for Total Magnetic Intensity Data on a Tensor Mesh ======================================================================== Here we invert total magnetic intensity (TMI) data to recover a magnetic susceptibility model. We formulate the inverse problem as an iteratively re-weighted least-squares (IRLS) optimization problem. For this tutorial, we focus on the following: - Defining the survey from xyz formatted data - Generating a mesh based on survey geometry - Including surface topography - Defining the inverse problem (data misfit, regularization, optimization) - Specifying directives for the inversion - Setting sparse and blocky norms - Plotting the recovered model and data misfit Although we consider TMI data in this tutorial, the same approach can be used to invert other types of geophysical data. .. GENERATED FROM PYTHON SOURCE LINES 25-28 Import modules -------------- .. GENERATED FROM PYTHON SOURCE LINES 28-53 .. code-block:: Python import os import numpy as np import matplotlib as mpl import matplotlib.pyplot as plt import tarfile from discretize import TensorMesh from discretize.utils import active_from_xyz from SimPEG.potential_fields import magnetics from SimPEG.utils import plot2Ddata, model_builder from SimPEG import ( maps, data, inverse_problem, data_misfit, regularization, optimization, directives, inversion, utils, ) # sphinx_gallery_thumbnail_number = 3 .. GENERATED FROM PYTHON SOURCE LINES 54-63 Load Data and Plot ------------------ File paths for assets we are loading. To set up the inversion, we require topography and field observations. The true model defined on the whole mesh is loaded to compare with the inversion result. These files are stored as a tar-file on our google cloud bucket: "https://storage.googleapis.com/simpeg/doc-assets/magnetics.tar.gz" .. GENERATED FROM PYTHON SOURCE LINES 63-83 .. code-block:: Python # storage bucket where we have the data data_source = "https://storage.googleapis.com/simpeg/doc-assets/magnetics.tar.gz" # download the data downloaded_data = utils.download(data_source, overwrite=True) # unzip the tarfile tar = tarfile.open(downloaded_data, "r") tar.extractall() tar.close() # path to the directory containing our data dir_path = downloaded_data.split(".")[0] + os.path.sep # files to work with topo_filename = dir_path + "magnetics_topo.txt" data_filename = dir_path + "magnetics_data.obs" .. rst-class:: sphx-glr-script-out .. code-block:: none Downloading https://storage.googleapis.com/simpeg/doc-assets/magnetics.tar.gz saved to: /home/vsts/work/1/s/tutorials/04-magnetics/magnetics.tar.gz Download completed! .. GENERATED FROM PYTHON SOURCE LINES 84-91 Load Data and Plot ------------------ Here we load and plot synthetic TMI data. Topography is generally defined as an (N, 3) array. TMI data is generally defined with 4 columns: x, y, z and data. .. GENERATED FROM PYTHON SOURCE LINES 91-124 .. code-block:: Python topo_xyz = np.loadtxt(str(topo_filename)) dobs = np.loadtxt(str(data_filename)) receiver_locations = dobs[:, 0:3] dobs = dobs[:, -1] # Plot fig = plt.figure(figsize=(6, 5)) v_max = np.max(np.abs(dobs)) ax1 = fig.add_axes([0.1, 0.1, 0.75, 0.85]) plot2Ddata( receiver_locations, dobs, ax=ax1, ncontour=30, clim=(-v_max, v_max), contourOpts={"cmap": "bwr"}, ) ax1.set_title("TMI Anomaly") ax1.set_xlabel("x (m)") ax1.set_ylabel("y (m)") ax2 = fig.add_axes([0.85, 0.05, 0.05, 0.9]) norm = mpl.colors.Normalize(vmin=-np.max(np.abs(dobs)), vmax=np.max(np.abs(dobs))) cbar = mpl.colorbar.ColorbarBase( ax2, norm=norm, orientation="vertical", cmap=mpl.cm.bwr ) cbar.set_label("$nT$", rotation=270, labelpad=15, size=12) plt.show() .. image-sg:: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_001.png :alt: TMI Anomaly :srcset: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 125-134 Assign Uncertainty ------------------ Inversion with SimPEG requires that we define standard deviation on our data. This represents our estimate of the noise in our data. For magnetic inversions, a constant floor value is generall applied to all data. For this tutorial, the standard deviation on each datum will be 2% of the maximum observed magnetics anomaly value. .. GENERATED FROM PYTHON SOURCE LINES 134-139 .. code-block:: Python maximum_anomaly = np.max(np.abs(dobs)) std = 0.02 * maximum_anomaly * np.ones(len(dobs)) .. GENERATED FROM PYTHON SOURCE LINES 140-148 Defining the Survey ------------------- Here, we define survey that will be used for the simulation. Magnetic surveys are simple to create. The user only needs an (N, 3) array to define the xyz locations of the observation locations, the list of field components which are to be modeled and the properties of the Earth's field. .. GENERATED FROM PYTHON SOURCE LINES 148-174 .. code-block:: Python # Define the component(s) of the field we are inverting as a list. Here we will # invert total magnetic intensity data. components = ["tmi"] # Use the observation locations and components to define the receivers. To # simulate data, the receivers must be defined as a list. receiver_list = magnetics.receivers.Point(receiver_locations, components=components) receiver_list = [receiver_list] # Define the inducing field H0 = (intensity [nT], inclination [deg], declination [deg]) inclination = 90 declination = 0 strength = 50000 source_field = magnetics.sources.UniformBackgroundField( receiver_list=receiver_list, amplitude=strength, inclination=inclination, declination=declination, ) # Define the survey survey = magnetics.survey.Survey(source_field) .. GENERATED FROM PYTHON SOURCE LINES 175-181 Defining the Data ----------------- Here is where we define the data that is inverted. The data is defined by the survey, the observation values and the standard deviations. .. GENERATED FROM PYTHON SOURCE LINES 181-185 .. code-block:: Python data_object = data.Data(survey, dobs=dobs, standard_deviation=std) .. GENERATED FROM PYTHON SOURCE LINES 186-192 Defining a Tensor Mesh ---------------------- Here, we create the tensor mesh that will be used to invert TMI data. If desired, we could define an OcTree mesh. .. GENERATED FROM PYTHON SOURCE LINES 192-199 .. code-block:: Python dh = 5.0 hx = [(dh, 5, -1.3), (dh, 40), (dh, 5, 1.3)] hy = [(dh, 5, -1.3), (dh, 40), (dh, 5, 1.3)] hz = [(dh, 5, -1.3), (dh, 15)] mesh = TensorMesh([hx, hy, hz], "CCN") .. GENERATED FROM PYTHON SOURCE LINES 200-208 Starting/Reference Model and Mapping on Tensor Mesh --------------------------------------------------- Here, we would create starting and/or reference models for the inversion as well as the mapping from the model space to the active cells. Starting and reference models can be a constant background value or contain a-priori structures. Here, the background is 1e-4 SI. .. GENERATED FROM PYTHON SOURCE LINES 208-224 .. code-block:: Python # Define background susceptibility model in SI. Don't make this 0! # Otherwise the gradient for the 1st iteration is zero and the inversion will # not converge. background_susceptibility = 1e-4 # Find the indecies of the active cells in forward model (ones below surface) active_cells = active_from_xyz(mesh, topo_xyz) # Define mapping from model to active cells nC = int(active_cells.sum()) model_map = maps.IdentityMap(nP=nC) # model consists of a value for each cell # Define starting model starting_model = background_susceptibility * np.ones(nC) .. GENERATED FROM PYTHON SOURCE LINES 225-231 Define the Physics ------------------ Here, we define the physics of the magnetics problem by using the simulation class. .. GENERATED FROM PYTHON SOURCE LINES 231-242 .. code-block:: Python # Define the problem. Define the cells below topography and the mapping simulation = magnetics.simulation.Simulation3DIntegral( survey=survey, mesh=mesh, model_type="scalar", chiMap=model_map, ind_active=active_cells, ) .. GENERATED FROM PYTHON SOURCE LINES 243-252 Define Inverse Problem ---------------------- The inverse problem is defined by 3 things: 1) Data Misfit: a measure of how well our recovered model explains the field data 2) Regularization: constraints placed on the recovered model and a priori information 3) Optimization: the numerical approach used to solve the inverse problem .. GENERATED FROM PYTHON SOURCE LINES 252-280 .. code-block:: Python # Define the data misfit. Here the data misfit is the L2 norm of the weighted # residual between the observed data and the data predicted for a given model. # Within the data misfit, the residual between predicted and observed data are # normalized by the data's standard deviation. dmis = data_misfit.L2DataMisfit(data=data_object, simulation=simulation) # Define the regularization (model objective function) reg = regularization.Sparse( mesh, active_cells=active_cells, mapping=model_map, reference_model=starting_model, gradient_type="total", ) # Define sparse and blocky norms p, qx, qy, qz reg.norms = [0, 0, 0, 0] # Define how the optimization problem is solved. Here we will use a projected # Gauss-Newton approach that employs the conjugate gradient solver. opt = optimization.ProjectedGNCG( maxIter=20, lower=0.0, upper=1.0, maxIterLS=20, maxIterCG=10, tolCG=1e-3 ) # Here we define the inverse problem that is to be solved inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt) .. GENERATED FROM PYTHON SOURCE LINES 281-288 Define Inversion Directives --------------------------- Here we define any directives that are carried out during the inversion. This includes the cooling schedule for the trade-off parameter (beta), stopping criteria for the inversion and saving inversion results at each iteration. .. GENERATED FROM PYTHON SOURCE LINES 288-323 .. code-block:: Python # Defining a starting value for the trade-off parameter (beta) between the data # misfit and the regularization. starting_beta = directives.BetaEstimate_ByEig(beta0_ratio=5) # Options for outputting recovered models and predicted data for each beta. save_iteration = directives.SaveOutputEveryIteration(save_txt=False) # Defines the directives for the IRLS regularization. This includes setting # the cooling schedule for the trade-off parameter. update_IRLS = directives.Update_IRLS( f_min_change=1e-4, max_irls_iterations=30, coolEpsFact=1.5, beta_tol=1e-2, ) # Updating the preconditioner if it is model dependent. update_jacobi = directives.UpdatePreconditioner() # Setting a stopping criteria for the inversion. target_misfit = directives.TargetMisfit(chifact=1) # Add sensitivity weights sensitivity_weights = directives.UpdateSensitivityWeights(every_iteration=False) # The directives are defined as a list. directives_list = [ sensitivity_weights, starting_beta, save_iteration, update_IRLS, update_jacobi, ] .. GENERATED FROM PYTHON SOURCE LINES 324-330 Running the Inversion --------------------- To define the inversion object, we need to define the inversion problem and the set of directives. We can then run the inversion. .. GENERATED FROM PYTHON SOURCE LINES 330-340 .. code-block:: Python # Here we combine the inverse problem and the set of directives inv = inversion.BaseInversion(inv_prob, directives_list) # Print target misfit to compare with convergence # print("Target misfit is " + str(target_misfit.target)) # Run the inversion recovered_model = inv.run(starting_model) .. rst-class:: sphx-glr-script-out .. code-block:: none SimPEG.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv. ***Done using the default solver Pardiso and no solver_opts.*** model has any nan: 0 =============================== Projected GNCG =============================== # beta phi_d phi_m f |proj(x-g)-x| LS Comment ----------------------------------------------------------------------------- x0 has any nan: 0 0 2.04e+05 2.10e+04 0.00e+00 2.10e+04 1.39e+02 0 1 1.02e+05 5.02e+02 9.99e-03 1.52e+03 1.24e+02 0 Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 0.0014103872270714315 2 5.09e+04 1.79e+02 2.06e-02 1.23e+03 1.85e+02 0 Skip BFGS 3 1.31e+05 9.21e+01 2.90e-02 3.89e+03 1.87e+02 0 4 8.18e+04 5.77e+02 2.64e-02 2.74e+03 1.84e+02 0 5 6.60e+04 3.34e+02 2.93e-02 2.27e+03 1.85e+02 0 6 1.17e+05 1.86e+02 2.64e-02 3.28e+03 1.76e+02 0 7 1.92e+05 2.26e+02 1.47e-02 3.04e+03 1.55e+02 0 8 3.16e+05 2.24e+02 7.57e-03 2.62e+03 1.35e+02 0 9 5.09e+05 2.37e+02 4.51e-03 2.54e+03 1.57e+02 0 10 7.85e+05 2.66e+02 2.88e-03 2.53e+03 1.39e+02 0 11 6.53e+05 3.16e+02 1.77e-03 1.47e+03 1.81e+02 0 12 1.36e+06 1.34e+02 1.22e-03 1.79e+03 1.17e+02 0 13 2.31e+06 2.06e+02 7.58e-04 1.95e+03 1.67e+02 0 14 3.80e+06 2.24e+02 5.15e-04 2.18e+03 1.23e+02 0 15 3.26e+06 2.98e+02 3.37e-04 1.40e+03 1.69e+02 0 16 6.91e+06 1.29e+02 2.35e-04 1.75e+03 1.20e+02 0 17 1.16e+07 2.15e+02 1.53e-04 1.98e+03 1.59e+02 0 18 1.90e+07 2.26e+02 1.04e-04 2.20e+03 1.36e+02 0 19 1.63e+07 2.99e+02 7.03e-05 1.44e+03 1.64e+02 0 20 3.26e+07 1.44e+02 4.85e-05 1.72e+03 1.24e+02 0 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 2.8263e+02 <= tolF*(1+|f0|) = 2.1023e+03 1 : |xc-x_last| = 1.1382e-02 <= tolX*(1+|x0|) = 1.0219e-01 0 : |proj(x-g)-x| = 1.2393e+02 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.2393e+02 <= 1e3*eps = 1.0000e-02 1 : maxIter = 20 <= iter = 20 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 341-344 Recreate True Model ------------------- .. GENERATED FROM PYTHON SOURCE LINES 344-357 .. code-block:: Python background_susceptibility = 0.0001 sphere_susceptibility = 0.01 true_model = background_susceptibility * np.ones(nC) ind_sphere = model_builder.get_indices_sphere( np.r_[0.0, 0.0, -45.0], 15.0, mesh.cell_centers ) ind_sphere = ind_sphere[active_cells] true_model[ind_sphere] = sphere_susceptibility .. GENERATED FROM PYTHON SOURCE LINES 358-361 Plotting True Model and Recovered Model --------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 361-412 .. code-block:: Python # Plot True Model fig = plt.figure(figsize=(9, 4)) plotting_map = maps.InjectActiveCells(mesh, active_cells, np.nan) ax1 = fig.add_axes([0.08, 0.1, 0.75, 0.8]) mesh.plot_slice( plotting_map * true_model, normal="Y", ax=ax1, ind=int(mesh.shape_cells[1] / 2), grid=True, clim=(np.min(true_model), np.max(true_model)), pcolor_opts={"cmap": "viridis"}, ) ax1.set_title("Model slice at y = 0 m") ax2 = fig.add_axes([0.85, 0.1, 0.05, 0.8]) norm = mpl.colors.Normalize(vmin=np.min(true_model), vmax=np.max(true_model)) cbar = mpl.colorbar.ColorbarBase( ax2, norm=norm, orientation="vertical", cmap=mpl.cm.viridis, format="%.1e" ) cbar.set_label("SI", rotation=270, labelpad=15, size=12) plt.show() # Plot Recovered Model fig = plt.figure(figsize=(9, 4)) plotting_map = maps.InjectActiveCells(mesh, active_cells, np.nan) ax1 = fig.add_axes([0.08, 0.1, 0.75, 0.8]) mesh.plot_slice( plotting_map * recovered_model, normal="Y", ax=ax1, ind=int(mesh.shape_cells[1] / 2), grid=True, clim=(np.min(recovered_model), np.max(recovered_model)), pcolor_opts={"cmap": "viridis"}, ) ax1.set_title("Model slice at y = 0 m") ax2 = fig.add_axes([0.85, 0.1, 0.05, 0.8]) norm = mpl.colors.Normalize(vmin=np.min(recovered_model), vmax=np.max(recovered_model)) cbar = mpl.colorbar.ColorbarBase( ax2, norm=norm, orientation="vertical", cmap=mpl.cm.viridis, format="%.1e" ) cbar.set_label("SI", rotation=270, labelpad=15, size=12) plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_002.png :alt: Model slice at y = 0 m :srcset: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_002.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_003.png :alt: Model slice at y = 0 m :srcset: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_003.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 413-416 Plotting Predicted Data and Misfit ---------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 416-456 .. code-block:: Python # Predicted data with final recovered model dpred = inv_prob.dpred # Observed data | Predicted data | Normalized data misfit data_array = np.c_[dobs, dpred, (dobs - dpred) / std] fig = plt.figure(figsize=(17, 4)) plot_title = ["Observed", "Predicted", "Normalized Misfit"] plot_units = ["nT", "nT", ""] ax1 = 3 * [None] ax2 = 3 * [None] norm = 3 * [None] cbar = 3 * [None] cplot = 3 * [None] v_lim = [np.max(np.abs(dobs)), np.max(np.abs(dobs)), np.max(np.abs(data_array[:, 2]))] for ii in range(0, 3): ax1[ii] = fig.add_axes([0.33 * ii + 0.03, 0.11, 0.25, 0.84]) cplot[ii] = plot2Ddata( receiver_list[0].locations, data_array[:, ii], ax=ax1[ii], ncontour=30, clim=(-v_lim[ii], v_lim[ii]), contourOpts={"cmap": "bwr"}, ) ax1[ii].set_title(plot_title[ii]) ax1[ii].set_xlabel("x (m)") ax1[ii].set_ylabel("y (m)") ax2[ii] = fig.add_axes([0.33 * ii + 0.27, 0.11, 0.01, 0.84]) norm[ii] = mpl.colors.Normalize(vmin=-v_lim[ii], vmax=v_lim[ii]) cbar[ii] = mpl.colorbar.ColorbarBase( ax2[ii], norm=norm[ii], orientation="vertical", cmap=mpl.cm.bwr ) cbar[ii].set_label(plot_units[ii], rotation=270, labelpad=15, size=12) plt.show() .. image-sg:: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_004.png :alt: Observed, Predicted, Normalized Misfit :srcset: /content/tutorials/04-magnetics/images/sphx_glr_plot_inv_2a_magnetics_induced_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 34.226 seconds) **Estimated memory usage:** 8 MB .. _sphx_glr_download_content_tutorials_04-magnetics_plot_inv_2a_magnetics_induced.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_2a_magnetics_induced.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_2a_magnetics_induced.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_