.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "content/user-guide/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 <sphx_glr_download_content_user-guide_tutorials_01-models_mapping_plot_2_cyl_models.py>`
to download the full example code.
.. rst-class:: sphx-glr-example-title
.. _sphx_glr_content_user-guide_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:: Python
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:: Python
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:: Python
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/user-guide/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_001.png
:alt: Cylindrically Symmetric Model
:srcset: /content/user-guide/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:: Python
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/user-guide/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_002.png
:alt: Cylindrically Symmetric Model
:srcset: /content/user-guide/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 147-178
.. code-block:: Python
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, active_cells=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/user-guide/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_003.png
:alt: Cylindrically Symmetric Model
:srcset: /content/user-guide/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 179-192
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 192-235
.. code-block:: Python
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/user-guide/tutorials/01-models_mapping/images/sphx_glr_plot_2_cyl_models_004.png
:alt: Cylindrically Symmetric Model
:srcset: /content/user-guide/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 5.769 seconds)
**Estimated memory usage:** 295 MB
.. _sphx_glr_download_content_user-guide_tutorials_01-models_mapping_plot_2_cyl_models.py:
.. only:: html
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:download:`Download Jupyter notebook: plot_2_cyl_models.ipynb <plot_2_cyl_models.ipynb>`
.. container:: sphx-glr-download sphx-glr-download-python
:download:`Download Python source code: plot_2_cyl_models.py <plot_2_cyl_models.py>`
.. container:: sphx-glr-download sphx-glr-download-zip
:download:`Download zipped: plot_2_cyl_models.zip <plot_2_cyl_models.zip>`
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