.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/tutorials/13-joint_inversion/plot_inv_3_cross_gradient_pf.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_13-joint_inversion_plot_inv_3_cross_gradient_pf.py: Cross-gradient Joint Inversion of Gravity and Magnetic Anomaly Data =================================================================== Here we simultaneously invert gravity and magentic data using cross-gradient constraint. The recovered density and susceptibility models are supposed to have structural similarity. 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 via combmaps (2 data misfit terms, 2 regularization terms, a coupling term and optimization) - Specifying directives for the inversion - Plotting the recovered model and data misfit Although we consider gravity and magnetic 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 27-30 Import modules -------------- .. GENERATED FROM PYTHON SOURCE LINES 30-55 .. 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 from SimPEG.potential_fields import gravity, magnetics from SimPEG import ( maps, data, data_misfit, inverse_problem, regularization, optimization, directives, inversion, utils, ) np.random.seed(0) .. GENERATED FROM PYTHON SOURCE LINES 56-64 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 64-86 .. code-block:: Python # # storage bucket where we have the data data_source = ( "https://storage.googleapis.com/simpeg/doc-assets/cross_gradient_data.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 + "topo.txt" model_filename = dir_path + "true_model.txt" .. rst-class:: sphx-glr-script-out .. code-block:: none Downloading https://storage.googleapis.com/simpeg/doc-assets/cross_gradient_data.tar.gz saved to: /home/vsts/work/1/s/tutorials/13-joint_inversion/cross_gradient_data.tar.gz Download completed! .. GENERATED FROM PYTHON SOURCE LINES 87-94 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 94-149 .. code-block:: Python # Load topography xyz_topo = np.loadtxt(topo_filename) # Load field data dobs_grav = np.loadtxt(dir_path + "gravity_data.obs") dobs_mag = np.loadtxt(dir_path + "magnetic_data.obs") # Define receiver locations and observed data receiver_locations = dobs_grav[:, 0:3] dobs_grav = dobs_grav[:, -1] dobs_mag = dobs_mag[:, -1] # Plot mpl.rcParams.update({"font.size": 12}) # gravity data fig = plt.figure(figsize=(7, 5)) ax1 = fig.add_axes([0.1, 0.1, 0.73, 0.85]) plot2Ddata(receiver_locations, dobs_grav, 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_grav)), vmax=np.max(np.abs(dobs_grav)) ) 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) # magnetic data fig = plt.figure(figsize=(7, 5)) ax1 = fig.add_axes([0.1, 0.1, 0.73, 0.85]) plot2Ddata(receiver_locations, dobs_mag, ax=ax1, contourOpts={"cmap": "bwr"}) ax1.set_title("Magnetic 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_mag)), vmax=np.max(np.abs(dobs_mag)) ) cbar = mpl.colorbar.ColorbarBase( ax2, norm=norm, orientation="vertical", cmap=mpl.cm.bwr, format="%.1e" ) cbar.set_label("$nT$", rotation=270, labelpad=15, size=12) plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_001.png :alt: Gravity Anomaly :srcset: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_001.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_002.png :alt: Magnetic Anomaly :srcset: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_002.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 150-159 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. For magnetic inversion, the same strategy is performed. .. GENERATED FROM PYTHON SOURCE LINES 159-167 .. code-block:: Python maximum_anomaly_grav = np.max(np.abs(dobs_grav)) uncertainties_grav = 0.01 * maximum_anomaly_grav * np.ones(np.shape(dobs_grav)) maximum_anomaly_mag = np.max(np.abs(dobs_mag)) uncertainties_mag = 0.01 * maximum_anomaly_mag * np.ones(np.shape(dobs_mag)) .. GENERATED FROM PYTHON SOURCE LINES 168-176 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 176-210 .. 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_grav = gravity.receivers.Point(receiver_locations, components="gz") # Define the source field and survey for gravity data source_field_grav = gravity.sources.SourceField(receiver_list=[receiver_grav]) survey_grav = gravity.survey.Survey(source_field_grav) # Define the component(s) of the field we want to simulate as a list of strings. # Here we simulation 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_mag = magnetics.receivers.Point(receiver_locations, components=components) # Define the inducing field H0 = (intensity [nT], inclination [deg], declination [deg]) inclination = 90 declination = 0 strength = 50000 # Define the source field and survey for gravity data source_field_mag = magnetics.sources.UniformBackgroundField( receiver_list=[receiver_mag], amplitude=strength, inclination=inclination, declination=declination, ) survey_mag = magnetics.survey.Survey(source_field_mag) .. GENERATED FROM PYTHON SOURCE LINES 211-217 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 217-226 .. code-block:: Python data_object_grav = data.Data( survey_grav, dobs=dobs_grav, standard_deviation=uncertainties_grav ) data_object_mag = data.Data( survey_mag, dobs=dobs_mag, standard_deviation=uncertainties_mag ) .. GENERATED FROM PYTHON SOURCE LINES 227-233 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 233-241 .. 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 242-252 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. Here, the backgrounds are 1e-6 g/cc and 1e-6 SI for density and susceptibility models, respectively. Note that the background values could be different for density and susceptibility models. .. GENERATED FROM PYTHON SOURCE LINES 252-271 .. code-block:: Python # Define density contrast values for each unit in g/cc. background_dens, background_susc = 1e-6, 1e-6 # Find the indicies 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 # Create Wires Map that maps from stacked models to individual model components # m1 refers to density model, m2 refers to susceptibility wires = maps.Wires(("density", nC), ("susceptibility", nC)) # Define and plot starting model starting_model = np.r_[background_dens * np.ones(nC), background_susc * np.ones(nC)] .. GENERATED FROM PYTHON SOURCE LINES 272-285 Define the Physics ------------------ Here, we define the physics of the gravity and magnetic problems 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 285-304 .. code-block:: Python simulation_grav = gravity.simulation.Simulation3DIntegral( survey=survey_grav, mesh=mesh, rhoMap=wires.density, ind_active=ind_active, engine="choclo", ) simulation_mag = magnetics.simulation.Simulation3DIntegral( survey=survey_mag, mesh=mesh, model_type="scalar", chiMap=wires.susceptibility, ind_active=ind_active, ) .. GENERATED FROM PYTHON SOURCE LINES 305-315 Define the Inverse Problem -------------------------- The inverse problem is defined by 4 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) Coupling: a connection of two different physical property models 4) Optimization: the numerical approach used to solve the inverse problem .. GENERATED FROM PYTHON SOURCE LINES 315-355 .. 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_grav = data_misfit.L2DataMisfit(data=data_object_grav, simulation=simulation_grav) dmis_mag = data_misfit.L2DataMisfit(data=data_object_mag, simulation=simulation_mag) # Define the regularization (model objective function). reg_grav = regularization.WeightedLeastSquares( mesh, active_cells=ind_active, mapping=wires.density ) reg_mag = regularization.WeightedLeastSquares( mesh, active_cells=ind_active, mapping=wires.susceptibility ) # Define the coupling term to connect two different physical property models lamda = 2e12 # weight for coupling term cross_grad = regularization.CrossGradient(mesh, wires, active_cells=ind_active) # combo dmis = dmis_grav + dmis_mag reg = reg_grav + reg_mag + lamda * cross_grad # 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=10, lower=-2.0, upper=2.0, maxIterLS=20, maxIterCG=100, tolCG=1e-3, tolX=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 356-363 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 363-397 .. code-block:: Python # Defining a starting value for the trade-off parameter (beta) between the data # misfit and the regularization. starting_beta = directives.PairedBetaEstimate_ByEig(beta0_ratio=1e0) # starting_beta.n_pw_iter = 10 # Defining the fractional decrease in beta and the number of Gauss-Newton solves # for each beta value. beta_schedule = directives.PairedBetaSchedule(cooling_factor=5, cooling_rate=1) # Options for outputting recovered models and predicted data for each beta. save_iteration = directives.SimilarityMeasureSaveOutputEveryIteration(save_txt=False) joint_inv_dir = directives.SimilarityMeasureInversionDirective() stopping = directives.MovingAndMultiTargetStopping(tol=1e-6) sensitivity_weights = directives.UpdateSensitivityWeights(every_iteration=False) # Updating the preconditionner if it is model dependent. update_jacobi = directives.UpdatePreconditioner() # The directives are defined as a list. directives_list = [ joint_inv_dir, sensitivity_weights, stopping, starting_beta, beta_schedule, save_iteration, update_jacobi, ] .. GENERATED FROM PYTHON SOURCE LINES 398-404 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 404-412 .. 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 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 ============================================================ # betas lambda f phi_d phi_m phi_sim iterCG Comment --------------------------------------------------------------------------------------------------------------------------------------- x0 has any nan: 0 0 ['8.50e+04', '3.29e+04'] 2.00e+12 4.55e+05 [] [] 0.00e+00 0 1 ['1.70e+04', '6.58e+03'] 2.00e+12 2.26e+04 ['1.31e+04', '3.39e+02'] ['3.73e-01', '1.63e-01'] 8.75e-10 100 2 ['3.40e+03', '8.77e+03'] 2.00e+12 8.48e+03 ['3.42e+03', '9.28e+01'] ['5.83e-01', '1.74e-01'] 7.32e-10 100 Skip BFGS 3 ['6.80e+02', '1.17e+04'] 2.00e+12 5.31e+03 ['1.54e+03', '5.70e+01'] ['7.34e-01', '1.71e-01'] 6.05e-10 100 Skip BFGS 4 ['1.36e+02', '1.56e+04'] 2.00e+12 4.80e+03 ['1.01e+03', '6.92e+01'] ['8.02e-01', '1.68e-01'] 4.98e-10 100 Skip BFGS 5 ['2.72e+01', '2.08e+04'] 2.00e+12 5.11e+03 ['7.77e+02', '1.01e+02'] ['8.70e-01', '1.63e-01'] 4.09e-10 100 6 ['5.44e+00', '2.77e+04'] 2.00e+12 5.88e+03 ['6.26e+02', '1.27e+02'] ['8.92e-01', '1.60e-01'] 3.39e-10 100 7 ['1.09e+00', '3.63e+04'] 2.00e+12 6.98e+03 ['5.28e+02', '2.21e+02'] ['9.30e-01', '1.55e-01'] 3.03e-10 100 8 ['2.17e-01', '7.26e+03'] 2.00e+12 2.39e+03 ['4.60e+02', '3.32e+02'] ['9.39e-01', '1.51e-01'] 2.50e-10 100 9 ['4.35e-02', '9.68e+03'] 2.00e+12 2.49e+03 ['3.61e+02', '3.67e+01'] ['9.70e-01', '1.66e-01'] 2.45e-10 100 10 ['8.70e-03', '1.29e+04'] 2.00e+12 2.92e+03 ['3.44e+02', '5.61e+01'] ['9.81e-01', '1.64e-01'] 2.04e-10 100 ------------------------- STOP! ------------------------- 1 : maxIter = 10 <= iter = 10 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 413-416 Plotting True Model and Recovered Model --------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 416-474 .. code-block:: Python # Load the true model (was defined on the whole mesh) and extract only the # values on active cells. true_model_dens = np.loadtxt(dir_path + "true_model_dens.txt") true_model_dens[~ind_active] = np.NaN true_model_susc = np.loadtxt(dir_path + "true_model_susc.txt") true_model_susc[~ind_active] = np.NaN # Plot True Model fig = plt.figure(figsize=(9, 8)) ax1 = plt.subplot(211) (im,) = mesh.plot_slice(true_model_dens, normal="Y", ax=ax1, grid=True) ax1.set_title("True density model slice at y = 0 m") cbar = plt.colorbar(im, format="%.1e") cbar.set_label("g/cc", rotation=270, labelpad=15, size=12) ax2 = plt.subplot(212) (im,) = mesh.plot_slice( true_model_susc, normal="Y", ax=ax2, grid=True, pcolor_opts={"cmap": "inferno"} ) ax2.set_title("True susceptibility model slice at y = 0 m") cbar = plt.colorbar(im, format="%.1e") cbar.set_label("SI", rotation=270, labelpad=15, size=12) plt.tight_layout() plt.show() # Plot Recovered Model m_dens_joint, m_susc_joint = wires * recovered_model plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan) fig = plt.figure(figsize=(9, 8)) ax1 = plt.subplot(211) (im,) = mesh.plot_slice( plotting_map * m_dens_joint, normal="Y", ax=ax1, clim=(-0.04, 0.03), ) ax1.set_title("Density model slice at y = 0 m") cbar = plt.colorbar(im) cbar.set_label("g/cc", rotation=270, labelpad=15, size=12) ax2 = plt.subplot(212) (im,) = mesh.plot_slice( plotting_map * m_susc_joint, normal="Y", ax=ax2, pcolor_opts={"cmap": "inferno"} ) ax2.set_title("Susceptibility model slice at y = 0 m") cbar = plt.colorbar(im) cbar.set_label("SI", rotation=270, labelpad=15, size=12) plt.tight_layout() plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_003.png :alt: True density model slice at y = 0 m, True susceptibility model slice at y = 0 m :srcset: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_003.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_004.png :alt: Density model slice at y = 0 m, Susceptibility model slice at y = 0 m :srcset: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_004.png :class: sphx-glr-multi-img .. GENERATED FROM PYTHON SOURCE LINES 475-478 Comparing jointly and separatly recovered models ------------------------------------------------ .. GENERATED FROM PYTHON SOURCE LINES 478-538 .. code-block:: Python # Normalized Cross Gradient of Jointly Recovered Susceptibility and Density Models ncg = cross_grad.calculate_cross_gradient(recovered_model, normalized=True) fig = plt.figure(figsize=(9, 4)) ax = plt.subplot(111) (im,) = mesh.plot_slice( plotting_map * ncg, normal="Y", ax=ax, grid=True, ) ax.set_title("Normalized cross gradient for joint inversion slice at y = 0 m") cbar = plt.colorbar(im, format="%.1e") cbar.set_label("|cross grad|", rotation=270, labelpad=15, size=12) plt.show() # Normalized Cross Gradient of Separately Recovered Susceptibility and Density Models m_dens_single = np.loadtxt(dir_path + "single_model_dens.txt") m_susc_single = np.loadtxt(dir_path + "single_model_susc.txt") m_separate = np.r_[m_dens_single[ind_active], m_susc_single[ind_active]] ncg_single = cross_grad.calculate_cross_gradient(m_separate, normalized=True) fig = plt.figure(figsize=(9, 4)) ax = plt.subplot(111) (im,) = mesh.plot_slice( plotting_map * ncg_single, normal="Y", ax=ax, grid=True, ) ax.set_title("Normalized cross gradient for separate inversion slice at y = 0 m") cbar = plt.colorbar(im, format="%.1e") cbar.set_label("|cross grad|", rotation=270, labelpad=15, size=12) plt.show() # Cross Plots Recovered Susceptibility and Density Models fig = plt.figure(figsize=(14, 5)) ax0 = plt.subplot(121) ax0.scatter( plotting_map * m_dens_joint, plotting_map * m_susc_joint, s=4, c="black", alpha=0.1 ) ax0.set_xlabel("Density", size=12) ax0.set_ylabel("Susceptibility", size=12) ax0.tick_params(labelsize=12) ax0.set_title("Joint inversion") ax1 = plt.subplot(122) ax1.scatter(m_dens_single, m_susc_single, s=4, c="black", alpha=0.1) ax1.set_xlabel("Density", size=12) ax1.set_ylabel("Susceptibility", size=12) ax1.tick_params(labelsize=12) ax1.set_title("Separate inversion") plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_005.png :alt: Normalized cross gradient for joint inversion slice at y = 0 m :srcset: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_005.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_006.png :alt: Normalized cross gradient for separate inversion slice at y = 0 m :srcset: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_006.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_007.png :alt: Joint inversion, Separate inversion :srcset: /content/tutorials/13-joint_inversion/images/sphx_glr_plot_inv_3_cross_gradient_pf_007.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (1 minutes 40.218 seconds) **Estimated memory usage:** 8 MB .. _sphx_glr_download_content_tutorials_13-joint_inversion_plot_inv_3_cross_gradient_pf.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_3_cross_gradient_pf.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_3_cross_gradient_pf.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_