.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/tutorials/10-vrm/plot_fwd_2_vrm_topsoil.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_10-vrm_plot_fwd_2_vrm_topsoil.py: Forward Simulation of VRM Response on a Tree Mesh ================================================= Here we use the module *SimPEG.electromagnetics.viscous_remanent_magnetization* to predict the characteristic VRM response over magnetically viscous top soil. We consider a small-loop, ground-based survey which uses a coincident loop geometry. For this tutorial, we focus on the following: - How to define the transmitters and receivers - How to define the survey - How to define a diagnostic physical property - How to define the physics for the linear potential fields formulation - How to include surface topography (if desired) - Modeling on an OcTree mesh Note that for this tutorial, we are only modeling the VRM response. A separate tutorial have been developed for modeling both the inductive and VRM responses. .. GENERATED FROM PYTHON SOURCE LINES 25-28 Import modules -------------- .. GENERATED FROM PYTHON SOURCE LINES 28-44 .. code-block:: Python from SimPEG.electromagnetics import viscous_remanent_magnetization as vrm from SimPEG.utils import plot2Ddata from SimPEG import maps from discretize import TreeMesh from discretize.utils import mkvc, refine_tree_xyz, active_from_xyz import numpy as np from scipy.interpolate import LinearNDInterpolator import matplotlib.pyplot as plt import matplotlib as mpl # sphinx_gallery_thumbnail_number = 2 .. GENERATED FROM PYTHON SOURCE LINES 45-52 Defining Topography ------------------- Surface topography is defined as an (N, 3) numpy array. We create it here but the topography could also be loaded from a file. To keep the example simple, we set flat topography at z = 0 m. .. GENERATED FROM PYTHON SOURCE LINES 52-60 .. code-block:: Python [x_topo, y_topo, z_topo] = np.meshgrid( np.linspace(-100, 100, 41), np.linspace(-100, 100, 41), 0.0 ) x_topo, y_topo, z_topo = mkvc(x_topo), mkvc(y_topo), mkvc(z_topo) xyz_topo = np.c_[x_topo, y_topo, z_topo] .. GENERATED FROM PYTHON SOURCE LINES 61-68 Survey ------ Here we define the sources, the receivers and the survey. For this exercise, a coincident loop-loop system measures the vertical component of the VRM response. .. GENERATED FROM PYTHON SOURCE LINES 68-112 .. code-block:: Python # Define the transmitter waveform. This strongly determines the behaviour of the # characteristic VRM response. Here we use a step-off. The off-time begins at # 0 s. waveform = vrm.waveforms.StepOff(t0=0) # Define the time channels for the receivers. The time channels must ALL be # ALL the off-time defined by the waveform. time_channels = np.logspace(-4, -1, 31) # Define the transmitter and receiver locations. This step will define the # receivers 0.5 m above the Earth in the even you use more general topography. x = np.linspace(-40.0, 40.0, 21) y = np.linspace(-40.0, 40.0, 21) x, y = np.meshgrid(x, y) x, y = mkvc(x.T), mkvc(y.T) fun_interp = LinearNDInterpolator(np.c_[x_topo, y_topo], z_topo) z = fun_interp(np.c_[x, y]) + 0.5 # sensor height 0.5 m above surface. locations = np.c_[mkvc(x), y, z] # Define the source-receiver pairs source_list = [] for pp in range(0, locations.shape[0]): # Define dbz/dt receiver loc_pp = np.reshape(locations[pp, :], (1, 3)) receivers_list = [ vrm.receivers.Point( loc_pp, times=time_channels, field_type="dbdt", orientation="z" ) ] dipole_moment = [0.0, 0.0, 1.0] # Define the source source_list.append( vrm.sources.MagDipole( receivers_list, mkvc(locations[pp, :]), dipole_moment, waveform ) ) # Define the survey survey = vrm.Survey(source_list) .. GENERATED FROM PYTHON SOURCE LINES 113-121 Defining an OcTree Mesh ----------------------- Here, we create the OcTree mesh that will be used for the tutorial. Since only the very near surface contributes significantly to the response, the dimensions of the domain in the z-direction can be small. Here, we are assuming the magnetic viscosity is negligible below 8 metres. .. GENERATED FROM PYTHON SOURCE LINES 121-148 .. code-block:: Python dx = 2 # minimum cell width (base mesh cell width) in x dy = 2 # minimum cell width (base mesh cell width) in y dz = 1 # minimum cell width (base mesh cell width) in z x_length = 100.0 # domain width in x y_length = 100.0 # domain width in y z_length = 8.0 # domain width in y # Compute number of base mesh cells required in x and y nbcx = 2 ** int(np.round(np.log(x_length / dx) / np.log(2.0))) nbcy = 2 ** int(np.round(np.log(y_length / dy) / np.log(2.0))) nbcz = 2 ** int(np.round(np.log(z_length / dz) / np.log(2.0))) # Define the base mesh hx = [(dx, nbcx)] hy = [(dy, nbcy)] hz = [(dz, nbcz)] mesh = TreeMesh([hx, hy, hz], x0="CCN") # Refine based on surface topography mesh = refine_tree_xyz( mesh, xyz_topo, octree_levels=[2, 2], method="surface", finalize=False ) mesh.finalize() .. rst-class:: sphx-glr-script-out .. code-block:: none /home/ssoler/simpeg/tutorials/10-vrm/plot_fwd_2_vrm_topsoil.py:142: DeprecationWarning: The surface option is deprecated as of `0.9.0` please update your code to use the `TreeMesh.refine_surface` functionality. It will be removed in a future version of discretize. .. GENERATED FROM PYTHON SOURCE LINES 149-160 Defining the Model ------------------ For the linear potential field formulation, the magnetic viscosity characterizing each cell can be defined by an "amalgamated magnetic property" (see Cowan, 2016). Here we define an amalgamated magnetic property model. The model is made by summing a set of 3D Gaussian distributions. For other formulations of the forward simulation, you may define the parameters assuming a log-uniform or log-normal distribution of time-relaxation constants. .. GENERATED FROM PYTHON SOURCE LINES 160-213 .. code-block:: Python # Find cells active in the forward simulation (cells below surface) ind_active = active_from_xyz(mesh, xyz_topo) # Define 3D Gaussian distribution parameters xyzc = mesh.gridCC[ind_active, :] c = 3 * np.pi * 8**2 pc = np.r_[4e-4, 4e-4, 4e-4, 6e-4, 8e-4, 6e-4, 8e-4, 8e-4] x_0 = np.r_[50.0, -50.0, -40.0, -20.0, -15.0, 20.0, -10.0, 25.0] y_0 = np.r_[0.0, 0.0, 40.0, 10.0, -20.0, 15.0, 0.0, 0.0] z_0 = np.r_[0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] var_x = c * np.r_[3.0, 3.0, 3.0, 1.0, 3.0, 0.5, 0.1, 0.1] var_y = c * np.r_[20.0, 20.0, 1.0, 1.0, 0.4, 0.5, 0.1, 0.4] var_z = c * np.r_[1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0, 1.0] # Define model model = np.zeros(np.shape(xyzc[:, 0])) for ii in range(0, 8): model += ( pc[ii] * np.exp(-((xyzc[:, 0] - x_0[ii]) ** 2) / var_x[ii]) * np.exp(-((xyzc[:, 1] - y_0[ii]) ** 2) / var_y[ii]) * np.exp(-((xyzc[:, 2] - z_0[ii]) ** 2) / var_z[ii]) ) # Plot Model mpl.rcParams.update({"font.size": 12}) fig = plt.figure(figsize=(7.5, 7)) plotting_map = maps.InjectActiveCells(mesh, ind_active, np.nan) ax1 = fig.add_axes([0.09, 0.12, 0.72, 0.77]) mesh.plot_slice( plotting_map * model, normal="Z", ax=ax1, ind=0, grid=True, clim=(np.min(model), np.max(model)), pcolor_opts={"cmap": "magma_r"}, ) ax1.set_title("Model slice at z = 0 m") ax1.set_xlabel("x (m)") ax1.set_ylabel("y (m)") ax2 = fig.add_axes([0.83, 0.12, 0.05, 0.77]) norm = mpl.colors.Normalize(vmin=np.min(model), vmax=np.max(model)) cbar = mpl.colorbar.ColorbarBase( ax2, norm=norm, orientation="vertical", cmap=mpl.cm.magma_r ) cbar.set_label("Amalgamated Magnetic Property (SI)", rotation=270, labelpad=15, size=12) plt.show() .. image-sg:: /content/tutorials/10-vrm/images/sphx_glr_plot_fwd_2_vrm_topsoil_001.png :alt: Model slice at z = 0 m :srcset: /content/tutorials/10-vrm/images/sphx_glr_plot_fwd_2_vrm_topsoil_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 214-223 Define the Simulation --------------------- Here we define the formulation for solving Maxwell's equations. We have chosen to model the off-time VRM response. There are two important keyword arguments, *refinement_factor* and *refinement_distance*. These are used to refine the sensitivities of the cells near the transmitters. This improves the accuracy of the forward simulation without having to refine the mesh near transmitters. .. GENERATED FROM PYTHON SOURCE LINES 223-236 .. code-block:: Python # For this example, cells lying within 2 m of a transmitter will be modeled # as if they are comprised of 4^3 equal smaller cells. Cells within 4 m of a # transmitter will be modeled as if they are comprised of 2^3 equal smaller # cells. simulation = vrm.Simulation3DLinear( mesh, survey=survey, indActive=ind_active, refinement_factor=2, refinement_distance=[2.0, 4.0], ) .. GENERATED FROM PYTHON SOURCE LINES 237-240 Predict Data and Plot --------------------- .. GENERATED FROM PYTHON SOURCE LINES 240-299 .. code-block:: Python # Predict VRM response dpred = simulation.dpred(model) # Reshape for plotting n_times = len(time_channels) n_loc = locations.shape[0] dpred = np.reshape(dpred, (n_loc, n_times)) # Plot fig = plt.figure(figsize=(13, 5)) # Index for what time channel you would like to see the data map. time_index = 10 v_max = np.max(np.abs(dpred[:, time_index])) v_min = np.min(np.abs(dpred[:, time_index])) ax11 = fig.add_axes([0.12, 0.1, 0.33, 0.85]) plot2Ddata( locations[:, 0:2], -dpred[:, time_index], ax=ax11, ncontour=30, clim=(v_min, v_max), contourOpts={"cmap": "magma_r"}, ) ax11.set_xlabel("x (m)") ax11.set_ylabel("y (m)") titlestr = "- dBz/dt at t=" + "{:.1e}".format(time_channels[time_index]) + " s" ax11.set_title(titlestr) ax12 = fig.add_axes([0.46, 0.1, 0.02, 0.85]) norm1 = mpl.colors.Normalize(vmin=v_min, vmax=v_max) cbar1 = mpl.colorbar.ColorbarBase( ax12, norm=norm1, orientation="vertical", cmap=mpl.cm.magma_r ) cbar1.set_label("$T/s$", rotation=270, labelpad=15, size=12) # Indicies for some locations you would like to see the decay location_indicies = [0, 65, 217] color_flags = ["k", "r", "b"] legend_str = [] ax2 = fig.add_axes([0.6, 0.1, 0.35, 0.85]) for ii in range(0, len(location_indicies)): ax2.loglog(time_channels, -dpred[location_indicies[ii], :], color_flags[ii], lw=2) legend_str.append( "(" + "{:.1f}".format(locations[location_indicies[ii], 0]) + " m, " + "{:.1f}".format(locations[location_indicies[ii], 1]) + " m)" ) ax2.set_xlim((np.min(time_channels), np.max(time_channels))) ax2.set_xlabel("time [s]") ax2.set_ylabel("-dBz/dt [T/s]") ax2.set_title("Decay Curve") ax2.legend(legend_str) .. image-sg:: /content/tutorials/10-vrm/images/sphx_glr_plot_fwd_2_vrm_topsoil_002.png :alt: - dBz/dt at t=1.0e-03 s, Decay Curve :srcset: /content/tutorials/10-vrm/images/sphx_glr_plot_fwd_2_vrm_topsoil_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none CREATING T MATRIX CREATING A MATRIX .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 13.707 seconds) **Estimated memory usage:** 9 MB .. _sphx_glr_download_content_tutorials_10-vrm_plot_fwd_2_vrm_topsoil.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_fwd_2_vrm_topsoil.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_fwd_2_vrm_topsoil.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_