.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/tutorials/03-gravity/plot_inv_1b_gravity_anomaly_irls.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_03-gravity_plot_inv_1b_gravity_anomaly_irls.py: Sparse Norm Inversion of Gravity Anomaly Data ============================================= Here we invert gravity anomaly data to recover a density contrast 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 gravity anomaly data in this tutorial, the same approach can be used to invert gradiometry and other types of geophysical data. .. GENERATED FROM PYTHON SOURCE LINES 24-27 Import modules -------------- .. GENERATED FROM PYTHON SOURCE LINES 27-52 .. 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.utils import plot2Ddata, model_builder from simpeg.potential_fields import gravity from simpeg import ( maps, data, data_misfit, inverse_problem, regularization, optimization, directives, inversion, utils, ) # sphinx_gallery_thumbnail_number = 3 .. GENERATED FROM PYTHON SOURCE LINES 53-62 Define File Names ----------------- 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/gravity.tar.gz" .. GENERATED FROM PYTHON SOURCE LINES 62-83 .. code-block:: Python # storage bucket where we have the data data_source = "https://storage.googleapis.com/simpeg/doc-assets/gravity.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 + "gravity_topo.txt" data_filename = dir_path + "gravity_data.obs" model_filename = dir_path + "true_model.txt" .. rst-class:: sphx-glr-script-out .. code-block:: none overwriting /home/vsts/work/1/s/tutorials/03-gravity/gravity.tar.gz Downloading https://storage.googleapis.com/simpeg/doc-assets/gravity.tar.gz saved to: /home/vsts/work/1/s/tutorials/03-gravity/gravity.tar.gz Download completed! .. GENERATED FROM PYTHON SOURCE LINES 84-91 Load Data and Plot ------------------ Here we load and plot synthetic gravity anomaly data. Topography is generally defined as an (N, 3) array. Gravity data is generally defined with 4 columns: x, y, z and data. .. GENERATED FROM PYTHON SOURCE LINES 91-121 .. code-block:: Python # Load topography xyz_topo = np.loadtxt(str(topo_filename)) # Load field data dobs = np.loadtxt(str(data_filename)) # Define receiver locations and observed data receiver_locations = dobs[:, 0:3] dobs = dobs[:, -1] # Plot mpl.rcParams.update({"font.size": 12}) fig = plt.figure(figsize=(7, 5)) ax1 = fig.add_axes([0.1, 0.1, 0.73, 0.85]) plot2Ddata(receiver_locations, dobs, ax=ax1, contourOpts={"cmap": "bwr"}) ax1.set_title("Gravity Anomaly") ax1.set_xlabel("x (m)") ax1.set_ylabel("y (m)") ax2 = fig.add_axes([0.8, 0.1, 0.03, 0.85]) 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, format="%.1e" ) cbar.set_label("\$mgal\$", rotation=270, labelpad=15, size=12) plt.show() .. image-sg:: /content/tutorials/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_001.png :alt: Gravity Anomaly :srcset: /content/tutorials/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 122-131 Assign Uncertainties -------------------- Inversion with SimPEG requires that we define standard deviation on our data. This represents our estimate of the noise in our data. For gravity inversion, a constant floor value is generally applied to all data. For this tutorial, the standard deviation on each datum will be 1% of the maximum observed gravity anomaly value. .. GENERATED FROM PYTHON SOURCE LINES 131-136 .. code-block:: Python maximum_anomaly = np.max(np.abs(dobs)) uncertainties = 0.01 * maximum_anomaly * np.ones(np.shape(dobs)) .. GENERATED FROM PYTHON SOURCE LINES 137-145 Defining the Survey ------------------- Here, we define survey that will be used for this tutorial. Gravity surveys are simple to create. The user only needs an (N, 3) array to define the xyz locations of the observation locations. From this, the user can define the receivers and the source field. .. GENERATED FROM PYTHON SOURCE LINES 145-158 .. code-block:: Python # Define the receivers. The data consist of vertical gravity anomaly measurements. # The set of receivers must be defined as a list. receiver_list = gravity.receivers.Point(receiver_locations, components="gz") receiver_list = [receiver_list] # Define the source field source_field = gravity.sources.SourceField(receiver_list=receiver_list) # Define the survey survey = gravity.survey.Survey(source_field) .. GENERATED FROM PYTHON SOURCE LINES 159-165 Defining the Data ----------------- Here is where we define the data that are inverted. The data are defined by the survey, the observation values and the standard deviation. .. GENERATED FROM PYTHON SOURCE LINES 165-169 .. code-block:: Python data_object = data.Data(survey, dobs=dobs, standard_deviation=uncertainties) .. GENERATED FROM PYTHON SOURCE LINES 170-176 Defining a Tensor Mesh ---------------------- Here, we create the tensor mesh that will be used to invert gravity anomaly data. If desired, we could define an OcTree mesh. .. GENERATED FROM PYTHON SOURCE LINES 176-183 .. 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 184-192 Starting/Reference Model and Mapping on Tensor Mesh --------------------------------------------------- Here, we 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. .. GENERATED FROM PYTHON SOURCE LINES 192-204 .. code-block:: Python # Find the indices of the active cells in forward model (ones below surface) ind_active = active_from_xyz(mesh, xyz_topo) # Define mapping from model to active cells nC = int(ind_active.sum()) model_map = maps.IdentityMap(nP=nC) # model consists of a value for each active cell # Define and plot starting model starting_model = np.zeros(nC) .. GENERATED FROM PYTHON SOURCE LINES 205-218 Define the Physics ------------------ Here, we define the physics of the gravity problem by using the simulation class. .. tip:: Since SimPEG v0.21.0 we can use `Choclo `_ as the engine for running the gravity simulations, which results in faster and more memory efficient runs. Just pass ``engine="choclo"`` when constructing the simulation. .. GENERATED FROM PYTHON SOURCE LINES 218-228 .. code-block:: Python simulation = gravity.simulation.Simulation3DIntegral( survey=survey, mesh=mesh, rhoMap=model_map, ind_active=ind_active, engine="choclo", ) .. GENERATED FROM PYTHON SOURCE LINES 229-238 Define the 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 238-259 .. 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) dmis.W = utils.sdiag(1 / uncertainties) # Define the regularization (model objective function). reg = regularization.Sparse(mesh, active_cells=ind_active, mapping=model_map) reg.norms = [0, 2, 2, 2] # 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=100, lower=-1.0, 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 260-267 Define Inversion Directives --------------------------- Here we define any directiveas 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 267-299 .. 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=1e0) # 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, ) # Options for outputting recovered models and predicted data for each beta. save_iteration = directives.SaveOutputEveryIteration(save_txt=False) # Updating the preconditionner if it is model dependent. update_jacobi = directives.UpdatePreconditioner() # Add sensitivity weights sensitivity_weights = directives.UpdateSensitivityWeights(every_iteration=False) # The directives are defined as a list. directives_list = [ update_IRLS, sensitivity_weights, starting_beta, save_iteration, update_jacobi, ] .. GENERATED FROM PYTHON SOURCE LINES 300-306 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 306-314 .. code-block:: Python # Here we combine the inverse problem and the set of directives inv = inversion.BaseInversion(inv_prob, directives_list) # Run inversion recovered_model = inv.run(starting_model) .. rst-class:: sphx-glr-script-out .. code-block:: none Running inversion with SimPEG v0.22.1 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 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 3.15e+03 2.66e+05 0.00e+00 2.66e+05 2.19e+02 0 1 1.58e+03 1.85e+04 1.16e+01 3.67e+04 2.16e+02 0 2 7.88e+02 8.03e+03 1.62e+01 2.08e+04 2.13e+02 0 Skip BFGS 3 3.94e+02 3.19e+03 2.04e+01 1.12e+04 2.09e+02 0 Skip BFGS 4 1.97e+02 1.21e+03 2.38e+01 5.91e+03 2.00e+02 0 Skip BFGS 5 9.85e+01 4.61e+02 2.64e+01 3.06e+03 1.91e+02 0 Skip BFGS Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 0.0549240340477234 6 4.92e+01 1.86e+02 5.44e+01 2.87e+03 2.04e+02 0 Skip BFGS 7 1.21e+02 9.90e+01 7.46e+01 9.13e+03 2.14e+02 0 8 8.13e+01 4.88e+02 7.44e+01 6.53e+03 2.10e+02 0 9 6.76e+01 3.17e+02 7.54e+01 5.41e+03 2.10e+02 0 10 1.03e+02 2.78e+02 6.78e+01 7.24e+03 2.14e+02 0 11 7.14e+01 4.52e+02 5.83e+01 4.62e+03 2.02e+02 0 12 1.08e+02 2.80e+02 5.20e+01 5.91e+03 2.16e+02 0 Skip BFGS 13 8.02e+01 3.96e+02 4.27e+01 3.82e+03 2.09e+02 0 14 1.24e+02 2.67e+02 3.74e+01 4.88e+03 2.17e+02 0 Skip BFGS 15 9.90e+01 3.39e+02 3.16e+01 3.47e+03 2.12e+02 0 16 1.55e+02 2.58e+02 2.85e+01 4.67e+03 2.18e+02 0 Skip BFGS 17 1.20e+02 3.58e+02 2.50e+01 3.36e+03 2.13e+02 0 18 1.20e+02 2.88e+02 2.36e+01 3.13e+03 2.17e+02 0 Skip BFGS 19 1.83e+02 2.79e+02 2.21e+01 4.32e+03 2.18e+02 0 Skip BFGS 20 1.38e+02 3.82e+02 2.03e+01 3.18e+03 2.13e+02 0 21 1.18e+02 3.01e+02 2.00e+01 2.65e+03 2.17e+02 0 Skip BFGS 22 1.84e+02 2.55e+02 1.94e+01 3.83e+03 2.18e+02 0 Skip BFGS 23 1.49e+02 3.31e+02 1.81e+01 3.04e+03 2.16e+02 0 24 2.31e+02 2.65e+02 1.78e+01 4.38e+03 2.18e+02 0 Skip BFGS 25 1.85e+02 3.41e+02 1.69e+01 3.46e+03 2.16e+02 0 26 2.83e+02 2.72e+02 1.67e+01 5.00e+03 2.18e+02 0 Skip BFGS 27 2.21e+02 3.55e+02 1.60e+01 3.90e+03 2.15e+02 0 28 3.35e+02 2.80e+02 1.60e+01 5.64e+03 2.18e+02 0 29 2.58e+02 3.66e+02 1.54e+01 4.33e+03 2.15e+02 0 30 3.90e+02 2.82e+02 1.54e+01 6.29e+03 2.18e+02 0 Skip BFGS 31 2.94e+02 3.82e+02 1.48e+01 4.73e+03 2.14e+02 0 32 2.94e+02 2.88e+02 1.48e+01 4.65e+03 2.16e+02 0 Skip BFGS 33 4.45e+02 2.82e+02 1.47e+01 6.84e+03 2.18e+02 0 Skip BFGS 34 3.28e+02 3.97e+02 1.43e+01 5.09e+03 2.13e+02 0 35 3.28e+02 2.89e+02 1.43e+01 5.00e+03 2.17e+02 0 Reach maximum number of IRLS cycles: 30 ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 2.6622e+04 1 : |xc-x_last| = 3.4464e-02 <= tolX*(1+|x0|) = 1.0000e-01 0 : |proj(x-g)-x| = 2.1651e+02 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 2.1651e+02 <= 1e3*eps = 1.0000e-02 0 : maxIter = 100 <= iter = 36 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 315-318 Recreate True Model ------------------- .. GENERATED FROM PYTHON SOURCE LINES 318-347 .. code-block:: Python # Define density contrast values for each unit in g/cc background_density = 0.0 block_density = -0.2 sphere_density = 0.2 # Define model. Models in SimPEG are vector arrays. true_model = background_density * np.ones(nC) # You could find the indicies of specific cells within the model and change their # value to add structures. ind_block = ( (mesh.gridCC[ind_active, 0] > -50.0) & (mesh.gridCC[ind_active, 0] < -20.0) & (mesh.gridCC[ind_active, 1] > -15.0) & (mesh.gridCC[ind_active, 1] < 15.0) & (mesh.gridCC[ind_active, 2] > -50.0) & (mesh.gridCC[ind_active, 2] < -30.0) ) true_model[ind_block] = block_density # You can also use SimPEG utilities to add structures to the model more concisely ind_sphere = model_builder.get_indices_sphere( np.r_[35.0, 0.0, -40.0], 15.0, mesh.gridCC ) ind_sphere = ind_sphere[ind_active] true_model[ind_sphere] = sphere_density .. GENERATED FROM PYTHON SOURCE LINES 348-351 Plotting True Model and Recovered Model --------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 351-403 .. code-block:: Python # Plot True Model fig = plt.figure(figsize=(9, 4)) plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan) ax1 = fig.add_axes([0.1, 0.1, 0.73, 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("\$g/cm^3\$", rotation=270, labelpad=15, size=12) plt.show() # Plot Recovered Model fig = plt.figure(figsize=(9, 4)) plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan) ax1 = fig.add_axes([0.1, 0.1, 0.73, 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 ) cbar.set_label("\$g/cm^3\$", rotation=270, labelpad=15, size=12) plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/tutorials/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_002.png :alt: Model slice at y = 0 m :srcset: /content/tutorials/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_002.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_003.png :alt: Model slice at y = 0 m :srcset: /content/tutorials/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_003.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 404-407 Plotting Predicted Data and Normalized Misfit --------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 407-449 .. code-block:: Python # Predicted data with final recovered model # SimPEG uses right handed coordinate where Z is positive upward. # This causes gravity signals look "inconsistent" with density values in visualization. dpred = inv_prob.dpred # Observed data | Predicted data | Normalized data misfit data_array = np.c_[dobs, dpred, (dobs - dpred) / uncertainties] fig = plt.figure(figsize=(17, 4)) plot_title = ["Observed", "Predicted", "Normalized Misfit"] plot_units = ["mgal", "mgal", ""] 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, 0.23, 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 + 0.25, 0.11, 0.01, 0.85]) 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/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_004.png :alt: Observed, Predicted, Normalized Misfit :srcset: /content/tutorials/03-gravity/images/sphx_glr_plot_inv_1b_gravity_anomaly_irls_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 29.976 seconds) **Estimated memory usage:** 9 MB .. _sphx_glr_download_content_tutorials_03-gravity_plot_inv_1b_gravity_anomaly_irls.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_1b_gravity_anomaly_irls.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_1b_gravity_anomaly_irls.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_