.. 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_2_cyl_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_2_cyl_models.py: Cylindrical Meshes ================== Cylindrical meshes are useful when the geological problem demonstrates rotational symmetry. In this case, we need only define how the model changes as a funcion of the radial distance and elevation; thus limiting the number of model parameters. Here we demonstrate various ways that models can be defined and mapped to cylindrical meshes. Some things we consider are: - Adding structures of various shape to the model - Parameterized models - Models with 2 or more physical properties .. GENERATED FROM PYTHON SOURCE LINES 19-22 Import modules -------------- .. GENERATED FROM PYTHON SOURCE LINES 22-29 .. code-block:: default from discretize import CylindricalMesh from SimPEG.utils import mkvc from SimPEG import maps import numpy as np import matplotlib.pyplot as plt .. GENERATED FROM PYTHON SOURCE LINES 30-35 Defining the mesh ----------------- Here, we create the tensor mesh that will be used for all examples. .. GENERATED FROM PYTHON SOURCE LINES 35-51 .. code-block:: default def make_example_mesh(): ncr = 20 # number of mesh cells in r ncz = 20 # number of mesh cells in z dh = 5.0 # cell width hr = [(dh, ncr), (dh, 5, 1.3)] hz = [(dh, 5, -1.3), (dh, ncz), (dh, 5, 1.3)] # Use flag of 1 to denote perfect rotational symmetry mesh = CylindricalMesh([hr, 1, hz], "0CC") return mesh .. GENERATED FROM PYTHON SOURCE LINES 52-61 Vertical Pipe and a 2 Layered Earth ----------------------------------- In this example we create a model containing a vertical pipe and a layered Earth. We will see that we need only define the model as a function of r and z. Models of this type are plotted from the center of the mesh to the total radial distance of the mesh. That is why pipes and rings look like blocks. .. GENERATED FROM PYTHON SOURCE LINES 61-92 .. code-block:: default mesh = make_example_mesh() background_value = 100.0 layer_value = 70.0 pipe_value = 40.0 # 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 = background_value * np.ones(ind_active.sum()) ind_layer = (mesh.gridCC[ind_active, 2] > -20.0) & (mesh.gridCC[ind_active, 2] < -0) model[ind_layer] = layer_value ind_pipe = ( (mesh.gridCC[ind_active, 0] < 10.0) & (mesh.gridCC[ind_active, 2] > -50.0) & (mesh.gridCC[ind_active, 2] < 0.0) ) model[ind_pipe] = pipe_value # Plotting fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) mesh.plot_image(model_map * model, ax=ax, grid=True) ax.set_title("Cylindrically Symmetric Model") .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_001.png :alt: Cylindrically Symmetric Model :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 1.0, 'Cylindrically Symmetric Model') .. GENERATED FROM PYTHON SOURCE LINES 93-102 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 102-138 .. code-block:: default mesh = make_example_mesh() background_value = np.log(1.0 / 100.0) layer_value = np.log(1.0 / 70.0) pipe_value = np.log(1.0 / 40.0) # Find cells below topography and define mapping air_value = 0.0 ind_active = mesh.gridCC[:, 2] < 0.0 active_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define the model model = background_value * np.ones(ind_active.sum()) ind_layer = (mesh.gridCC[ind_active, 2] > -20.0) & (mesh.gridCC[ind_active, 2] < -0) model[ind_layer] = layer_value ind_pipe = ( (mesh.gridCC[ind_active, 0] < 10.0) & (mesh.gridCC[ind_active, 2] > -50.0) & (mesh.gridCC[ind_active, 2] < 0.0) ) model[ind_pipe] = pipe_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 # Plotting fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) mesh.plot_image(model_map * model, ax=ax, grid=True) ax.set_title("Cylindrically Symmetric Model") .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_002.png :alt: Cylindrically Symmetric Model :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 1.0, 'Cylindrically Symmetric Model') .. GENERATED FROM PYTHON SOURCE LINES 139-147 Parameterized pipe 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 147-176 .. code-block:: default mesh = make_example_mesh() background_value = 100.0 # background value pipe_value = 40.0 # pipe value rc, zc = 0.0, -25.0 # center of pipe dr, dz = 20.0, 50.0 # dimensions in r, z # Find cells below topography and define mapping air_value = 0.0 ind_active = mesh.gridCC[:, 2] < 0.0 active_map = maps.InjectActiveCells(mesh, ind_active, air_value) # Define the model on subsurface cells model = np.r_[ background_value, pipe_value, rc, dr, 0.0, 1.0, zc, dz ] # add dummy values for phi parametric_map = maps.ParametricBlock(mesh, indActive=ind_active, epsilon=1e-10, p=8.0) # Define a single mapping from model to mesh model_map = active_map * parametric_map # Plotting fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) mesh.plot_image(model_map * model, ax=ax, grid=True) ax.set_title("Cylindrically Symmetric Model") .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_003.png :alt: Cylindrically Symmetric Model :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 1.0, 'Cylindrically Symmetric Model') .. GENERATED FROM PYTHON SOURCE LINES 177-190 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 190-233 .. code-block:: default mesh = make_example_mesh() background_sigma = np.log(100.0) layer_sigma = np.log(70.0) pipe_sigma = np.log(40.0) background_mu = 1.0 pipe_mu = 5.0 # Find cells below topography and define mapping air_value = 0.0 ind_active = mesh.gridCC[:, 2] < 0.0 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_mu]) # Add a conductive and non-permeable layer ind_layer = (mesh.gridCC[ind_active, 2] > -20.0) & (mesh.gridCC[ind_active, 2] < -0) model[ind_layer, 0] = layer_sigma # Add a conductive and permeable pipe ind_pipe = ( (mesh.gridCC[ind_active, 0] < 10.0) & (mesh.gridCC[ind_active, 2] > -50.0) & (mesh.gridCC[ind_active, 2] < 0.0) ) model[ind_pipe] = np.c_[pipe_sigma, pipe_mu] # 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 # Plotting fig = plt.figure(figsize=(5, 5)) ax = fig.add_subplot(111) mesh.plot_image(sigma_map * model, ax=ax, grid=True) ax.set_title("Cylindrically Symmetric Model") .. image-sg:: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_004.png :alt: Cylindrically Symmetric Model :srcset: /content/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 1.0, 'Cylindrically Symmetric Model') .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 4.748 seconds) **Estimated memory usage:** 8 MB .. _sphx_glr_download_content_tutorials_01-models_mapping_plot_2_cyl_models.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_2_cyl_models.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_2_cyl_models.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_