.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/tutorials/08-tdem/plot_inv_1_em1dtm.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_08-tdem_plot_inv_1_em1dtm.py: 1D Inversion of Time-Domain Data for a Single Sounding ====================================================== Here we use the module *simpeg.electromangetics.time_domain_1d* to invert time domain data and recover a 1D electrical conductivity model. In this tutorial, we focus on the following: - How to define sources and receivers from a survey file - How to define the survey - Sparse 1D inversion of with iteratively re-weighted least-squares For this tutorial, we will invert 1D time domain data for a single sounding. The end product is layered Earth model which explains the data. The survey consisted of a horizontal loop with a radius of 6 m, located 20 m above the surface. The receiver measured the vertical component of the magnetic flux at the loop's centre. .. GENERATED FROM PYTHON SOURCE LINES 22-25 Import modules -------------- .. GENERATED FROM PYTHON SOURCE LINES 25-52 .. code-block:: Python import os import tarfile import numpy as np import matplotlib.pyplot as plt from discretize import TensorMesh import simpeg.electromagnetics.time_domain as tdem from simpeg.utils import mkvc, plot_1d_layer_model from simpeg import ( maps, data, data_misfit, inverse_problem, regularization, optimization, directives, inversion, utils, ) plt.rcParams.update({"font.size": 16, "lines.linewidth": 2, "lines.markersize": 8}) # sphinx_gallery_thumbnail_number = 2 .. GENERATED FROM PYTHON SOURCE LINES 53-61 Download Test Data File ----------------------- Here we provide the file path to the data we plan on inverting. The path to the data file is stored as a tar-file on our google cloud bucket: "https://storage.googleapis.com/simpeg/doc-assets/em1dtm.tar.gz" .. GENERATED FROM PYTHON SOURCE LINES 61-79 .. code-block:: Python # storage bucket where we have the data data_source = "https://storage.googleapis.com/simpeg/doc-assets/em1dtm.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 data_filename = dir_path + "em1dtm_data.txt" .. rst-class:: sphx-glr-script-out .. code-block:: none Downloading https://storage.googleapis.com/simpeg/doc-assets/em1dtm.tar.gz saved to: /home/vsts/work/1/s/tutorials/08-tdem/em1dtm.tar.gz Download completed! .. GENERATED FROM PYTHON SOURCE LINES 80-86 Load Data and Plot ------------------ Here we load and plot the 1D sounding data. In this case, we have the B-field response to a step-off waveform. .. GENERATED FROM PYTHON SOURCE LINES 86-101 .. code-block:: Python # Load field data dobs = np.loadtxt(str(data_filename), skiprows=1) times = dobs[:, 0] dobs = mkvc(dobs[:, -1]) fig = plt.figure(figsize=(7, 7)) ax = fig.add_axes([0.15, 0.15, 0.8, 0.75]) ax.loglog(times, np.abs(dobs), "k-o", lw=3) ax.set_xlabel("Times (s)") ax.set_ylabel("|B| (T)") ax.set_title("Observed Data") .. image-sg:: /content/tutorials/08-tdem/images/sphx_glr_plot_inv_1_em1dtm_001.png :alt: Observed Data :srcset: /content/tutorials/08-tdem/images/sphx_glr_plot_inv_1_em1dtm_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 1.0, 'Observed Data') .. GENERATED FROM PYTHON SOURCE LINES 102-109 Defining the Survey ------------------- Here we demonstrate a general way to define the receivers, sources, waveforms and survey. For this tutorial, we define a single horizontal loop source as well a receiver which measures the vertical component of the magnetic flux. .. GENERATED FROM PYTHON SOURCE LINES 109-146 .. code-block:: Python # Source loop geometry source_location = np.array([0.0, 0.0, 20.0]) source_orientation = "z" # "x", "y" or "z" source_current = 1.0 # peak current amplitude source_radius = 6.0 # loop radius # Receiver geometry receiver_location = np.array([0.0, 0.0, 20.0]) receiver_orientation = "z" # "x", "y" or "z" # Receiver list receiver_list = [] receiver_list.append( tdem.receivers.PointMagneticFluxDensity( receiver_location, times, orientation=receiver_orientation ) ) # Define the source waveform. waveform = tdem.sources.StepOffWaveform() # Sources source_list = [ tdem.sources.CircularLoop( receiver_list=receiver_list, location=source_location, waveform=waveform, current=source_current, radius=source_radius, ) ] # Survey survey = tdem.Survey(source_list) .. GENERATED FROM PYTHON SOURCE LINES 147-153 Assign Uncertainties and Define the Data Object ----------------------------------------------- Here is where we define the data that are inverted. The data are defined by the survey, the observation values and the uncertainties. .. GENERATED FROM PYTHON SOURCE LINES 153-161 .. code-block:: Python # 5% of the absolute value uncertainties = 0.05 * np.abs(dobs) * np.ones(np.shape(dobs)) # Define the data object data_object = data.Data(survey, dobs=dobs, standard_deviation=uncertainties) .. GENERATED FROM PYTHON SOURCE LINES 162-168 Defining a 1D Layered Earth (1D Tensor Mesh) -------------------------------------------- Here, we define the layer thicknesses for our 1D simulation. To do this, we use the TensorMesh class. .. GENERATED FROM PYTHON SOURCE LINES 168-176 .. code-block:: Python # Layer thicknesses inv_thicknesses = np.logspace(0, 1.5, 25) # Define a mesh for plotting and regularization. mesh = TensorMesh([(np.r_[inv_thicknesses, inv_thicknesses[-1]])], "0") .. GENERATED FROM PYTHON SOURCE LINES 177-189 Define a Starting and Reference Model ------------------------------------- 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 starting model is log(0.1) S/m. Define log-conductivity values for each layer since our model is the log-conductivity. Don't make the values 0! Otherwise the gradient for the 1st iteration is zero and the inversion will not converge. .. GENERATED FROM PYTHON SOURCE LINES 189-197 .. code-block:: Python # Define model. A resistivity (Ohm meters) or conductivity (S/m) for each layer. starting_model = np.log(0.1 * np.ones(mesh.nC)) # Define mapping from model to active cells. model_mapping = maps.ExpMap() .. GENERATED FROM PYTHON SOURCE LINES 198-201 Define the Physics using a Simulation Object -------------------------------------------- .. GENERATED FROM PYTHON SOURCE LINES 201-206 .. code-block:: Python simulation = tdem.Simulation1DLayered( survey=survey, thicknesses=inv_thicknesses, sigmaMap=model_mapping ) .. GENERATED FROM PYTHON SOURCE LINES 207-217 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 217-242 .. 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. # The weighting is defined by the reciprocal of the uncertainties. dmis = data_misfit.L2DataMisfit(simulation=simulation, data=data_object) dmis.W = 1.0 / uncertainties # Define the regularization (model objective function) reg_map = maps.IdentityMap(nP=mesh.nC) reg = regularization.Sparse(mesh, mapping=reg_map, alpha_s=0.01, alpha_x=1.0) # set reference model reg.reference_model = starting_model # Define sparse and blocky norms p, q reg.norms = [1, 0] # Define how the optimization problem is solved. Here we will use an inexact # Gauss-Newton approach that employs the conjugate gradient solver. opt = optimization.ProjectedGNCG(maxIter=100, maxIterLS=20, maxIterCG=30, tolCG=1e-3) # Define the inverse problem inv_prob = inverse_problem.BaseInvProblem(dmis, reg, opt) .. GENERATED FROM PYTHON SOURCE LINES 243-250 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 250-281 .. 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=1e1) # Update the preconditionner update_Jacobi = directives.UpdatePreconditioner() # Options for outputting recovered models and predicted data for each beta. save_iteration = directives.SaveOutputEveryIteration(save_txt=False) # Directives for the IRLS update_IRLS = directives.Update_IRLS( max_irls_iterations=30, minGNiter=1, coolEpsFact=1.5, update_beta=True ) # Updating the preconditionner if it is model dependent. update_jacobi = directives.UpdatePreconditioner() # Add sensitivity weights sensitivity_weights = directives.UpdateSensitivityWeights() # The directives are defined as a list. directives_list = [ sensitivity_weights, starting_beta, save_iteration, update_IRLS, update_jacobi, ] .. GENERATED FROM PYTHON SOURCE LINES 282-288 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 288-296 .. code-block:: Python # Here we combine the inverse problem and the set of directives inv = inversion.BaseInversion(inv_prob, directives_list) # Run the inversion recovered_model = inv.run(starting_model) .. rst-class:: sphx-glr-script-out .. code-block:: none Running inversion with SimPEG v0.22.0 simpeg.InvProblem is setting bfgsH0 to the inverse of the eval2Deriv. ***Done using same Solver, and solver_opts as the Simulation1DLayered problem*** 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 1.14e+03 4.01e+03 0.00e+00 4.01e+03 8.04e+02 0 1 5.69e+02 1.11e+03 1.28e-01 1.18e+03 9.91e+02 1 2 2.84e+02 6.33e+02 1.33e-01 6.71e+02 1.15e+03 1 3 1.42e+02 4.04e+02 1.45e-01 4.25e+02 1.66e+03 0 Skip BFGS Reached starting chifact with l2-norm regularization: Start IRLS steps... irls_threshold 2.654608696171035 4 7.11e+01 1.34e+01 1.22e-01 2.21e+01 2.03e+02 0 Skip BFGS 5 3.96e+02 3.39e+00 1.24e-01 5.24e+01 8.55e+01 0 Skip BFGS 6 1.68e+03 4.80e+00 1.29e-01 2.21e+02 1.74e+02 0 Skip BFGS 7 4.48e+03 9.25e+00 1.32e-01 6.01e+02 3.26e+02 0 8 7.19e+03 2.57e+01 1.34e-01 9.92e+02 3.67e+02 0 9 4.56e+03 5.99e+01 1.46e-01 7.24e+02 6.87e+01 0 10 2.76e+03 6.71e+01 1.56e-01 4.98e+02 8.41e+01 0 Skip BFGS 11 1.71e+03 6.37e+01 1.55e-01 3.29e+02 8.34e+01 0 Skip BFGS 12 1.19e+03 4.80e+01 1.44e-01 2.20e+02 5.57e+01 0 13 8.94e+02 4.13e+01 1.40e-01 1.67e+02 3.05e+02 0 14 6.74e+02 4.09e+01 1.30e-01 1.29e+02 3.93e+02 0 15 5.17e+02 3.95e+01 1.17e-01 1.00e+02 3.84e+02 0 Skip BFGS 16 4.15e+02 3.62e+01 1.04e-01 7.94e+01 2.64e+02 0 Skip BFGS 17 4.15e+02 3.38e+01 9.24e-02 7.22e+01 8.15e+01 0 Skip BFGS 18 3.43e+02 3.44e+01 8.30e-02 6.29e+01 1.29e+01 0 Skip BFGS 19 2.83e+02 3.44e+01 7.78e-02 5.64e+01 1.67e+01 0 Skip BFGS 20 2.83e+02 3.38e+01 7.56e-02 5.52e+01 1.25e+01 0 21 2.83e+02 3.34e+01 7.41e-02 5.43e+01 1.20e+01 0 Skip BFGS 22 2.83e+02 3.32e+01 7.30e-02 5.38e+01 1.24e+01 0 23 2.83e+02 3.30e+01 7.22e-02 5.34e+01 1.20e+01 0 Skip BFGS 24 2.83e+02 3.29e+01 7.17e-02 5.31e+01 1.10e+01 0 Skip BFGS Minimum decrease in regularization.End of IRLS ------------------------- STOP! ------------------------- 1 : |fc-fOld| = 0.0000e+00 <= tolF*(1+|f0|) = 4.0084e+02 1 : |xc-x_last| = 1.9934e-03 <= tolX*(1+|x0|) = 1.2741e+00 0 : |proj(x-g)-x| = 1.0990e+01 <= tolG = 1.0000e-01 0 : |proj(x-g)-x| = 1.0990e+01 <= 1e3*eps = 1.0000e-02 0 : maxIter = 100 <= iter = 25 ------------------------- DONE! ------------------------- .. GENERATED FROM PYTHON SOURCE LINES 297-299 Plotting Results --------------------- .. GENERATED FROM PYTHON SOURCE LINES 299-346 .. code-block:: Python # Load the true model and layer thicknesses true_model = np.array([0.1, 1.0, 0.1]) true_layers = np.r_[40.0, 40.0, 160.0] # Extract Least-Squares model l2_model = inv_prob.l2model print(np.shape(l2_model)) # Plot true model and recovered model fig = plt.figure(figsize=(8, 9)) x_min = np.min( np.r_[model_mapping * recovered_model, model_mapping * l2_model, true_model] ) x_max = np.max( np.r_[model_mapping * recovered_model, model_mapping * l2_model, true_model] ) ax1 = fig.add_axes([0.2, 0.15, 0.7, 0.7]) plot_1d_layer_model(true_layers, true_model, ax=ax1, show_layers=False, color="k") plot_1d_layer_model( mesh.h[0], model_mapping * l2_model, ax=ax1, show_layers=False, color="b" ) plot_1d_layer_model( mesh.h[0], model_mapping * recovered_model, ax=ax1, show_layers=False, color="r" ) ax1.set_xlim(0.01, 10) ax1.grid() ax1.set_title("True and Recovered Models") ax1.legend(["True Model", "L2-Model", "Sparse Model"]) plt.gca().invert_yaxis() # Plot predicted and observed data dpred_l2 = simulation.dpred(l2_model) dpred_final = simulation.dpred(recovered_model) fig = plt.figure(figsize=(7, 7)) ax1 = fig.add_axes([0.15, 0.15, 0.8, 0.75]) ax1.loglog(times, np.abs(dobs), "k-o") ax1.loglog(times, np.abs(dpred_l2), "b-o") ax1.loglog(times, np.abs(dpred_final), "r-o") ax1.grid() ax1.set_xlabel("times (Hz)") ax1.set_ylabel("|Hs/Hp| (ppm)") ax1.set_title("Predicted and Observed Data") ax1.legend(["Observed", "L2-Model", "Sparse"], loc="upper right") plt.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /content/tutorials/08-tdem/images/sphx_glr_plot_inv_1_em1dtm_002.png :alt: True and Recovered Models :srcset: /content/tutorials/08-tdem/images/sphx_glr_plot_inv_1_em1dtm_002.png :class: sphx-glr-multi-img * .. image-sg:: /content/tutorials/08-tdem/images/sphx_glr_plot_inv_1_em1dtm_003.png :alt: Predicted and Observed Data :srcset: /content/tutorials/08-tdem/images/sphx_glr_plot_inv_1_em1dtm_003.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-script-out .. code-block:: none (26,) .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 24.001 seconds) **Estimated memory usage:** 9 MB .. _sphx_glr_download_content_tutorials_08-tdem_plot_inv_1_em1dtm.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_1_em1dtm.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_inv_1_em1dtm.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_