.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "content/tutorials/01-models_mapping/plot_1_tensor_models.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_01-models_mapping_plot_1_tensor_models.py: Tensor Meshes ============= Here we demonstrate various ways that models can be defined and mapped to tensor meshes. Some things we consider are: - Surface topography - Adding structures of various shape to the model - Parameterized models - Models with 2 or more physical properties .. GENERATED FROM PYTHON SOURCE LINES 17-20 Import modules -------------- .. GENERATED FROM PYTHON SOURCE LINES 20-30 .. code-block:: Python from discretize import TensorMesh from discretize.utils import active_from_xyz from simpeg.utils import mkvc, model_builder from simpeg import maps import numpy as np import matplotlib.pyplot as plt # sphinx_gallery_thumbnail_number = 3 .. GENERATED FROM PYTHON SOURCE LINES 31-36 Defining the mesh ----------------- Here, we create the tensor mesh that will be used for all examples. .. GENERATED FROM PYTHON SOURCE LINES 36-48 .. code-block:: Python def make_example_mesh(): dh = 5.0 hx = [(dh, 5, -1.3), (dh, 20), (dh, 5, 1.3)] hy = [(dh, 5, -1.3), (dh, 20), (dh, 5, 1.3)] hz = [(dh, 5, -1.3), (dh, 20), (dh, 5, 1.3)] mesh = TensorMesh([hx, hy, hz], "CCC") return mesh .. GENERATED FROM PYTHON SOURCE LINES 49-58 Halfspace model with topography at z = 0 ---------------------------------------- In this example we generate a half-space model. Since air cells remain constant during geophysical inversion, the number of model values we define should be equal to the number of cells lying below the surface. Here, we define the model (*model* ) as well as the mapping (*model_map* ) that goes from the model-space to the entire mesh. .. GENERATED FROM PYTHON SOURCE LINES 58-79 .. code-block:: Python mesh = make_example_mesh() halfspace_value = 100.0 # Find cells below topography and define mapping air_value = 0.0 ind_active = mesh.gridCC[:, 2] < 0.0 model_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define the model model = halfspace_value * np.ones(ind_active.sum()) # We can plot a slice of the model at Y=-2.5 fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) ind_slice = int(mesh.shape_cells[1] / 2) mesh.plot_slice(model_map * model, normal="Y", ax=ax, ind=ind_slice, grid=True) ax.set_title("Model slice at y = {} m".format(mesh.cell_centers_y[ind_slice])) plt.show() .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_001.png :alt: Model slice at y = 2.5 m :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 80-87 Topography, a block and a vertical dyke --------------------------------------- In this example we create a model containing a block and a vertical dyke that strikes along the y direction. The utility *active_from_xyz* is used to find the cells which lie below a set of xyz points defining a surface. .. GENERATED FROM PYTHON SOURCE LINES 87-128 .. code-block:: Python mesh = make_example_mesh() background_value = 100.0 dyke_value = 40.0 block_value = 70.0 # Define surface topography as an (N, 3) np.array. You could also load a file # containing the xyz points [xx, yy] = np.meshgrid(mesh.nodes_x, mesh.nodes_y) zz = -3 * np.exp((xx**2 + yy**2) / 75**2) + 40.0 topo = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)] # Find cells below topography and define mapping air_value = 0.0 ind_active = active_from_xyz(mesh, topo, "N") model_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define the model on subsurface cells model = background_value * np.ones(ind_active.sum()) ind_dyke = (mesh.gridCC[ind_active, 0] > 20.0) & (mesh.gridCC[ind_active, 0] < 40.0) model[ind_dyke] = dyke_value ind_block = ( (mesh.gridCC[ind_active, 0] > -40.0) & (mesh.gridCC[ind_active, 0] < -10.0) & (mesh.gridCC[ind_active, 1] > -30.0) & (mesh.gridCC[ind_active, 1] < 30.0) & (mesh.gridCC[ind_active, 2] > -40.0) & (mesh.gridCC[ind_active, 2] < 0.0) ) model[ind_block] = block_value # Plot fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) ind_slice = int(mesh.shape_cells[1] / 2) mesh.plot_slice(model_map * model, normal="Y", ax=ax, ind=ind_slice, grid=True) ax.set_title("Model slice at y = {} m".format(mesh.cell_centers_y[ind_slice])) plt.show() .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_002.png :alt: Model slice at y = 2.5 m :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 129-138 Combo Maps ---------- Here we demonstrate how combo maps can be used to create a single mapping from the model to the mesh. In this case, our model consists of log-conductivity values but we want to plot the resistivity. To accomplish this we must take the exponent of our model values, then take the reciprocal, then map from below surface cell to the mesh. .. GENERATED FROM PYTHON SOURCE LINES 138-183 .. code-block:: Python mesh = make_example_mesh() background_value = np.log(1.0 / 100.0) dyke_value = np.log(1.0 / 40.0) block_value = np.log(1.0 / 70.0) # Define surface topography [xx, yy] = np.meshgrid(mesh.nodes_x, mesh.nodes_y) zz = -3 * np.exp((xx**2 + yy**2) / 75**2) + 40.0 topo = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)] # Find cells below topography air_value = 0.0 ind_active = active_from_xyz(mesh, topo, "N") active_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define the model on subsurface cells model = background_value * np.ones(ind_active.sum()) ind_dyke = (mesh.gridCC[ind_active, 0] > 20.0) & (mesh.gridCC[ind_active, 0] < 40.0) model[ind_dyke] = dyke_value ind_block = ( (mesh.gridCC[ind_active, 0] > -40.0) & (mesh.gridCC[ind_active, 0] < -10.0) & (mesh.gridCC[ind_active, 1] > -30.0) & (mesh.gridCC[ind_active, 1] < 30.0) & (mesh.gridCC[ind_active, 2] > -40.0) & (mesh.gridCC[ind_active, 2] < 0.0) ) model[ind_block] = block_value # Define a single mapping from model to mesh exponential_map = maps.ExpMap() reciprocal_map = maps.ReciprocalMap() model_map = active_map * reciprocal_map * exponential_map # Plot fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) ind_slice = int(mesh.shape_cells[1] / 2) mesh.plot_slice(model_map * model, normal="Y", ax=ax, ind=ind_slice, grid=True) ax.set_title("Model slice at y = {} m".format(mesh.cell_centers_y[ind_slice])) plt.show() .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_003.png :alt: Model slice at y = 2.5 m :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 184-192 Models with arbitrary shapes ---------------------------- Here we show how model building utilities are used to make more complicated structural models. The process of adding a new unit is twofold: 1) we must find the indicies for mesh cells that lie within the new unit, 2) we replace the prexisting physical property value for those cells. .. GENERATED FROM PYTHON SOURCE LINES 192-237 .. code-block:: Python mesh = make_example_mesh() background_value = 100.0 dyke_value = 40.0 sphere_value = 70.0 # Define surface topography [xx, yy] = np.meshgrid(mesh.nodes_x, mesh.nodes_y) zz = -3 * np.exp((xx**2 + yy**2) / 75**2) + 40.0 topo = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)] # Set active cells and define unit values air_value = 0.0 ind_active = active_from_xyz(mesh, topo, "N") model_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define model for cells under the surface topography model = background_value * np.ones(ind_active.sum()) # Add a sphere ind_sphere = model_builder.get_indices_sphere( np.r_[-25.0, 0.0, -15.0], 20.0, mesh.gridCC ) ind_sphere = ind_sphere[ind_active] # So it's same size and order as model model[ind_sphere] = sphere_value # Add dyke defined by a set of points xp = np.kron(np.ones((2)), [-10.0, 10.0, 45.0, 25.0]) yp = np.kron([-1000.0, 1000.0], np.ones((4))) zp = np.kron(np.ones((2)), [-120.0, -120.0, 35.0, 35.0]) xyz_pts = np.c_[mkvc(xp), mkvc(yp), mkvc(zp)] ind_polygon = model_builder.get_indices_polygon(mesh, xyz_pts) ind_polygon = ind_polygon[ind_active] # So same size and order as model model[ind_polygon] = dyke_value # Plot fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) ind_slice = int(mesh.shape_cells[1] / 2) mesh.plot_slice(model_map * model, normal="Y", ax=ax, ind=ind_slice, grid=True) ax.set_title("Model slice at y = {} m".format(mesh.cell_centers_y[ind_slice])) plt.show() .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_004.png :alt: Model slice at y = 2.5 m :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_004.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 238-246 Parameterized block model ------------------------- Instead of defining a model value for each sub-surface cell, we can define the model in terms of a small number of parameters. Here we parameterize the model as a block in a half-space. We then create a mapping which projects this model onto the mesh. .. GENERATED FROM PYTHON SOURCE LINES 246-280 .. code-block:: Python mesh = make_example_mesh() background_value = 100.0 # background value block_value = 40.0 # block value xc, yc, zc = -25.0, 0.0, -20.0 # center of block dx, dy, dz = 30.0, 40.0, 30.0 # dimensions in x,y,z # Define surface topography [xx, yy] = np.meshgrid(mesh.nodes_x, mesh.nodes_y) zz = -3 * np.exp((xx**2 + yy**2) / 75**2) + 40.0 topo = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)] # Set active cells and define unit values air_value = 0.0 ind_active = active_from_xyz(mesh, topo, "N") active_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define the model on subsurface cells model = np.r_[background_value, block_value, xc, dx, yc, dy, zc, dz] parametric_map = maps.ParametricBlock(mesh, indActive=ind_active, epsilon=1e-10, p=5.0) # Define a single mapping from model to mesh model_map = active_map * parametric_map # Plot fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) ind_slice = int(mesh.shape_cells[1] / 2) mesh.plot_slice(model_map * model, normal="Y", ax=ax, ind=ind_slice, grid=True) ax.set_title("Model slice at y = {} m".format(mesh.cell_centers_y[ind_slice])) plt.show() .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_005.png :alt: Model slice at y = 2.5 m :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_005.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 281-294 Using Wire Maps --------------- Wire maps are needed when the model is comprised of two or more parameter types (e.g. conductivity and magnetic permeability). Because the model vector contains all values for all parameter types, we need to use "wires" to extract the values for a particular parameter type. Here we will define a model consisting of log-conductivity values and magnetic permeability values. We wish to plot the conductivity and permeability on the mesh. Wires are used to keep track of the mapping between the model vector and a particular physical property type. .. GENERATED FROM PYTHON SOURCE LINES 294-348 .. code-block:: Python mesh = make_example_mesh() background_sigma = np.log(100.0) sphere_sigma = np.log(70.0) dyke_sigma = np.log(40.0) background_myu = 1.0 sphere_mu = 1.25 # Define surface topography [xx, yy] = np.meshgrid(mesh.nodes_x, mesh.nodes_y) zz = -3 * np.exp((xx**2 + yy**2) / 75**2) + 40.0 topo = np.c_[mkvc(xx), mkvc(yy), mkvc(zz)] # Set active cells air_value = 0.0 ind_active = active_from_xyz(mesh, topo, "N") active_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define model for cells under the surface topography N = int(ind_active.sum()) model = np.kron(np.ones((N, 1)), np.c_[background_sigma, background_myu]) # Add a conductive and permeable sphere ind_sphere = model_builder.get_indices_sphere( np.r_[-25.0, 0.0, -15.0], 20.0, mesh.gridCC ) ind_sphere = ind_sphere[ind_active] # So same size and order as model model[ind_sphere, :] = np.c_[sphere_sigma, sphere_mu] # Add a conductive and non-permeable dyke xp = np.kron(np.ones((2)), [-10.0, 10.0, 45.0, 25.0]) yp = np.kron([-1000.0, 1000.0], np.ones((4))) zp = np.kron(np.ones((2)), [-120.0, -120.0, 35.0, 35.0]) xyz_pts = np.c_[mkvc(xp), mkvc(yp), mkvc(zp)] ind_polygon = model_builder.get_indices_polygon(mesh, xyz_pts) ind_polygon = ind_polygon[ind_active] # So same size and order as model model[ind_polygon, 0] = dyke_sigma # Create model vector and wires model = mkvc(model) wire_map = maps.Wires(("log_sigma", N), ("mu", N)) # Use combo maps to map from model to mesh sigma_map = active_map * maps.ExpMap() * wire_map.log_sigma mu_map = active_map * wire_map.mu # Plot fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) ind_slice = int(mesh.shape_cells[1] / 2) mesh.plot_slice(sigma_map * model, normal="Y", ax=ax, ind=ind_slice, grid=True) ax.set_title("Model slice at y = {} m".format(mesh.cell_centers_y[ind_slice])) plt.show() .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_006.png :alt: Model slice at y = 2.5 m :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_1_tensor_models_006.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 6.340 seconds) **Estimated memory usage:** 9 MB .. _sphx_glr_download_content_tutorials_01-models_mapping_plot_1_tensor_models.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_1_tensor_models.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_1_tensor_models.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_