Merge branch 'dev' of https://github.com/simpeg/simpeg into analytics
@@ -2,7 +2,7 @@
|
||||
#
|
||||
|
||||
# You can set these variables from the command line.
|
||||
SPHINXOPTS =
|
||||
SPHINXOPTS = -n -w warnings.txt
|
||||
SPHINXBUILD = sphinx-build
|
||||
PAPER =
|
||||
BUILDDIR = _build
|
||||
|
||||
@@ -0,0 +1,22 @@
|
||||
{# Import the theme's layout. #}
|
||||
{% extends "!layout.html" %}
|
||||
|
||||
{% block extrahead %}
|
||||
{{ super() }}
|
||||
|
||||
<meta name="description" content="Simulation and Parameter Estimation in Geophysics">
|
||||
<meta name="author" content="SimPEG Developers">
|
||||
<meta name="keywords" content="python, geophysics, inversion, electromagnetics, magnetotellurics, magnetics, gravity, DC, flow inverse problems, open source, finite volume">
|
||||
|
||||
|
||||
<script>
|
||||
(function(i,s,o,g,r,a,m){i['GoogleAnalyticsObject']=r;i[r]=i[r]||function(){
|
||||
(i[r].q=i[r].q||[]).push(arguments)},i[r].l=1*new Date();a=s.createElement(o),
|
||||
m=s.getElementsByTagName(o)[0];a.async=1;a.src=g;m.parentNode.insertBefore(a,m)
|
||||
})(window,document,'script','https://www.google-analytics.com/analytics.js','ga');
|
||||
|
||||
ga('create', 'UA-45185336-1', 'auto');
|
||||
ga('send', 'pageview');
|
||||
|
||||
</script>
|
||||
{% endblock %}
|
||||
@@ -1,19 +0,0 @@
|
||||
.. _api_FiniteVolume:
|
||||
|
||||
Finite Volume
|
||||
*************
|
||||
|
||||
Any numerical implementation requires the discretization of continuous functions into discrete approximations. These approximations are typically organized in a mesh, which defines boundaries, locations, and connectivity. Of specific interest to geophysical simulations, we require that averaging, interpolation and differential operators be defined for any mesh. In SimPEG, we have implemented a staggered mimetic finite volume approach (`Hyman and Shashkov, 1999 <http://math.lanl.gov/~mac/papers/numerics/HS99B.pdf>`_). This approach requires the definitions of variables at either cell-centers, nodes, faces, or edges as seen in the figure below.
|
||||
|
||||
.. image:: images/finitevolrealestate.png
|
||||
:width: 400 px
|
||||
:alt: FiniteVolume
|
||||
:align: center
|
||||
|
||||
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
|
||||
api_Mesh
|
||||
api_DiffOps
|
||||
api_InnerProducts
|
||||
@@ -1,36 +0,0 @@
|
||||
.. _api_MeshCode:
|
||||
|
||||
Tensor Mesh
|
||||
===========
|
||||
|
||||
.. automodule:: SimPEG.Mesh.TensorMesh
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
Cylindrical Mesh
|
||||
================
|
||||
|
||||
.. automodule:: SimPEG.Mesh.CylMesh
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
Tree Mesh
|
||||
=========
|
||||
|
||||
.. autoclass:: SimPEG.Mesh.TreeMesh.TreeMesh
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
Curvilinear Mesh
|
||||
================
|
||||
|
||||
.. automodule:: SimPEG.Mesh.CurvilinearMesh
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -0,0 +1,95 @@
|
||||
# application: simpegdocs
|
||||
# version: 1
|
||||
runtime: python27
|
||||
api_version: 1
|
||||
threadsafe: yes
|
||||
|
||||
handlers:
|
||||
|
||||
# favicon
|
||||
- url: /images/logo-block\.ico
|
||||
static_files: /images/logo-block.ico
|
||||
upload: /images/logo-block\.ico
|
||||
|
||||
# all css
|
||||
- url: /(.*\.css)
|
||||
mime_type: text/css
|
||||
static_files: _build/html/\1
|
||||
upload: _build/html/(.*\.css)
|
||||
|
||||
# webfonts
|
||||
- url: /(.*\.(eot|svg|ttf|woff|woff2|otf))
|
||||
static_files: _build/html/\1
|
||||
upload: _build/html/(.*\.(eot|svg|ttf|woff|woff2|otf))
|
||||
|
||||
# javascript
|
||||
- url: /(.*\.js)
|
||||
mime_type: text/javascript
|
||||
static_files: _build/html/\1
|
||||
upload: _build/html/(.*\.js)
|
||||
|
||||
# plain text source
|
||||
- url: /(.*\.txt)
|
||||
mime_type: text/plain
|
||||
static_files: _build/html/\1
|
||||
upload: _build/html/(.*\.txt)
|
||||
|
||||
# images
|
||||
- url: /_images/(.*\.(gif|png|jpg|ico))
|
||||
static_files: _build/html/_images/\1
|
||||
upload: _build/html/_images/(.*\.(gif|png|jpg|ico))
|
||||
|
||||
# redirect en/latest traffic
|
||||
- url: /en/latest/(.*\.html)
|
||||
script: simpegdocs.app
|
||||
|
||||
# raw html
|
||||
- url: /(.*\.html)
|
||||
mime_type: text/html
|
||||
static_files: _build/html/\1
|
||||
upload: _build/html/(.*\.html)
|
||||
|
||||
# serve index files
|
||||
- url: /(.+)/
|
||||
static_files: _build/html/\1/index.html
|
||||
upload: _build/html/(.+)/index.html
|
||||
|
||||
- url: /(.+)
|
||||
static_files: _build/html/\1/index.html
|
||||
upload: _build/html/(.+)/index.html
|
||||
|
||||
- url: /
|
||||
static_files: _build/html/index.html
|
||||
upload: _build/html/index.html
|
||||
|
||||
- url: .*
|
||||
script: simpegdocs.app
|
||||
|
||||
# Recommended file skipping declaration from the GAE tutorials
|
||||
skip_files:
|
||||
- ^(.*/)?app\.yaml
|
||||
- ^(.*/)?app\.yml
|
||||
- ^(.*/)?#.*#
|
||||
- ^(.*/)?.*~
|
||||
- ^(.*/)?.*\.py[co]
|
||||
- ^(.*/)?.*/RCS/.*
|
||||
- ^(.*/)?\..*
|
||||
- ^(.*/)?tests$
|
||||
- ^(.*/)?test$
|
||||
- ^test/(.*/)?
|
||||
- ^COPYING.LESSER
|
||||
- ^README\..*
|
||||
- \.gitignore
|
||||
- ^\.git/.*
|
||||
- \.*\.lint$
|
||||
- ^(.*/)?.*\.doctree$
|
||||
|
||||
libraries:
|
||||
- name: webapp2
|
||||
version: "2.5.2"
|
||||
- name: PIL
|
||||
version: "1.1.7"
|
||||
- name: numpy
|
||||
version: "latest"
|
||||
- name: jinja2
|
||||
version: "latest"
|
||||
@@ -28,7 +28,7 @@ sys.path.append('../')
|
||||
|
||||
# Add any Sphinx extension module names here, as strings. They can be extensions
|
||||
# coming with Sphinx (named 'sphinx.ext.*') or your custom ones.
|
||||
extensions = ['sphinx.ext.todo', 'sphinx.ext.mathjax', 'sphinx.ext.viewcode', 'sphinx.ext.autodoc', 'matplotlib.sphinxext.plot_directive']
|
||||
extensions = ['sphinx.ext.todo', 'sphinx.ext.mathjax', 'sphinx.ext.viewcode', 'sphinx.ext.autodoc', 'sphinx.ext.intersphinx', 'matplotlib.sphinxext.plot_directive']
|
||||
|
||||
# Add any paths that contain templates here, relative to this directory.
|
||||
templates_path = ['_templates']
|
||||
@@ -44,7 +44,7 @@ master_doc = 'index'
|
||||
|
||||
# General information about the project.
|
||||
project = u'SimPEG'
|
||||
copyright = u'2013, SimPEG Developers'
|
||||
copyright = u'2013 - 2016, SimPEG Developers'
|
||||
|
||||
# The version info for the project you're documenting, acts as replacement for
|
||||
# |version| and |release|, also used in various other places throughout the
|
||||
@@ -124,12 +124,12 @@ except Exception, e:
|
||||
# The name of an image file (within the static path) to use as favicon of the
|
||||
# docs. This file should be a Windows icon file (.ico) being 16x16 or 32x32
|
||||
# pixels large.
|
||||
#html_favicon = None
|
||||
html_favicon = './images/logo-block.ico'
|
||||
|
||||
# Add any paths that contain custom static files (such as style sheets) here,
|
||||
# relative to this directory. They are copied after the builtin static files,
|
||||
# so a file named "default.css" will overwrite the builtin "default.css".
|
||||
html_static_path = ['_static']
|
||||
html_static_path = []
|
||||
|
||||
# If not '', a 'Last updated on:' timestamp is inserted at every page bottom,
|
||||
# using the given strftime format.
|
||||
@@ -229,6 +229,12 @@ man_pages = [
|
||||
# If true, show URL addresses after external links.
|
||||
#man_show_urls = False
|
||||
|
||||
# Intersphinx
|
||||
intersphinx_mapping = {'python': ('http://docs.python.org/2', None),
|
||||
'numpy': ('http://docs.scipy.org/doc/numpy/', None),
|
||||
'scipy': ('http://docs.scipy.org/doc/scipy/reference/', None),
|
||||
'matplotlib': ('http://matplotlib.sourceforge.net/', None)}
|
||||
|
||||
|
||||
# -- Options for Texinfo output ------------------------------------------------
|
||||
|
||||
@@ -251,3 +257,36 @@ texinfo_documents = [
|
||||
#texinfo_show_urls = 'footnote'
|
||||
|
||||
autodoc_member_order = 'bysource'
|
||||
|
||||
def supress_nonlocal_image_warn():
|
||||
import sphinx.environment
|
||||
sphinx.environment.BuildEnvironment.warn_node = _supress_nonlocal_image_warn
|
||||
|
||||
def _supress_nonlocal_image_warn(self, msg, node):
|
||||
from docutils.utils import get_source_line
|
||||
|
||||
if not msg.startswith('nonlocal image URI found:'):
|
||||
self._warnfunc(msg, '%s:%s' % get_source_line(node))
|
||||
|
||||
supress_nonlocal_image_warn()
|
||||
|
||||
|
||||
nitpick_ignore = [
|
||||
('py:class', 'IdentityMap'),
|
||||
('py:class', 'BaseSurvey'),
|
||||
('py:class', 'BaseSrc'),
|
||||
('py:class', 'BaseRx'),
|
||||
('py:class', 'Survey'),
|
||||
('py:class', 'FieldsFDEM'),
|
||||
('py:class', 'Fields3D_e'),
|
||||
('py:class', 'Fields3D_b'),
|
||||
('py:class', 'Fields3D_j'),
|
||||
('py:class', 'Fields3D_h'),
|
||||
('py:class', 'SurveyTDEM'),
|
||||
('py:class', 'SrcTDEM'),
|
||||
('py:class', 'EMPropMap'),
|
||||
('py:class', 'Data'),
|
||||
('py:class', 'SurveyDC'),
|
||||
('py:class', 'BaseMTFields'),
|
||||
('py:class', 'SolverLU'),
|
||||
]
|
||||
|
||||
@@ -7,7 +7,7 @@ Examples
|
||||
:maxdepth: 1
|
||||
:glob:
|
||||
|
||||
examples/*
|
||||
../examples/*
|
||||
|
||||
|
||||
External Notebooks
|
||||
@@ -0,0 +1,27 @@
|
||||
.. _api_FiniteVolume:
|
||||
|
||||
Finite Volume
|
||||
*************
|
||||
|
||||
Any numerical implementation requires the discretization of continuous
|
||||
functions into discrete approximations. These approximations are typically
|
||||
organized in a mesh, which defines boundaries, locations, and connectivity. Of
|
||||
specific interest to geophysical simulations, we require that averaging,
|
||||
interpolation and differential operators be defined for any mesh. In SimPEG,
|
||||
we have implemented a staggered mimetic finite volume approach (`Hyman and
|
||||
Shashkov, 1999 <http://math.lanl.gov/~mac/papers/numerics/HS99B.pdf>`_). This
|
||||
approach requires the definitions of variables at either cell-centers, nodes,
|
||||
faces, or edges as seen in the figure below.
|
||||
|
||||
.. image:: ../../images/finitevolrealestate.png
|
||||
:width: 400 px
|
||||
:alt: FiniteVolume
|
||||
:align: center
|
||||
|
||||
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
|
||||
api_Mesh
|
||||
api_DiffOps
|
||||
api_InnerProducts
|
||||
@@ -52,13 +52,15 @@ We can take the derivative of the PDE:
|
||||
|
||||
\nabla_m c(m, u) \partial m + \nabla_u c(m, u) \partial u = 0
|
||||
|
||||
If the forward problem is invertible, then we can rearrange for \\(\\frac{\\partial u}{\\partial m}\\):
|
||||
If the forward problem is invertible, then we can rearrange for
|
||||
\\(\\frac{\\partial u}{\\partial m}\\):
|
||||
|
||||
.. math::
|
||||
|
||||
J = - P \left( \nabla_u c(m, u) \right)^{-1} \nabla_m c(m, u)
|
||||
|
||||
This can often be computed given a vector (i.e. \\(J(v)\\)) rather than stored, as \\(J\\) is a large dense matrix.
|
||||
This can often be computed given a vector (i.e. \\(J(v)\\)) rather than
|
||||
stored, as \\(J\\) is a large dense matrix.
|
||||
|
||||
|
||||
|
||||
@@ -67,13 +69,45 @@ The API
|
||||
|
||||
Problem
|
||||
-------
|
||||
.. automodule:: SimPEG.Problem
|
||||
|
||||
.. autoclass:: SimPEG.Problem.BaseProblem
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. autoclass:: SimPEG.Problem.BaseTimeProblem
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
Fields
|
||||
------
|
||||
|
||||
.. autoclass:: SimPEG.Fields.Fields
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. autoclass:: SimPEG.Fields.TimeFields
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
Survey
|
||||
------
|
||||
.. automodule:: SimPEG.Survey
|
||||
|
||||
.. autoclass:: SimPEG.Survey.BaseSurvey
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. autoclass:: SimPEG.Survey.BaseSrc
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. autoclass:: SimPEG.Survey.BaseRx
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. autoclass:: SimPEG.Survey.BaseTimeRx
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. autoclass:: SimPEG.Survey.Data
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -4,7 +4,10 @@
|
||||
Inner Products
|
||||
**************
|
||||
|
||||
By using the weak formulation of many of the PDEs in geophysical applications, we can rapidly develop discretizations. Much of this work, however, needs a good understanding of how to approximate inner products on our discretized meshes. We will define the inner product as:
|
||||
By using the weak formulation of many of the PDEs in geophysical applications,
|
||||
we can rapidly develop discretizations. Much of this work, however, needs a
|
||||
good understanding of how to approximate inner products on our discretized
|
||||
meshes. We will define the inner product as:
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -14,12 +17,15 @@ where a and b are either scalars or vectors.
|
||||
|
||||
.. note::
|
||||
|
||||
The InnerProducts class is a base class providing inner product matrices for meshes and cannot run on its own.
|
||||
The InnerProducts class is a base class providing inner product matrices
|
||||
for meshes and cannot run on its own.
|
||||
|
||||
|
||||
Example problem for DC resistivity
|
||||
----------------------------------
|
||||
We will start with the formulation of the Direct Current (DC) resistivity problem in geophysics.
|
||||
|
||||
We will start with the formulation of the Direct Current (DC) resistivity
|
||||
problem in geophysics.
|
||||
|
||||
|
||||
.. math::
|
||||
@@ -28,12 +34,13 @@ We will start with the formulation of the Direct Current (DC) resistivity proble
|
||||
|
||||
\nabla\cdot \vec{j} = q
|
||||
|
||||
In the following discretization, \\\( \\sigma \\\) and \\\( \\phi \\\)
|
||||
will be discretized on the cell-centers and the flux, \\\(\\vec{j}\\\),
|
||||
In the following discretization, :math:`\sigma` and :math:`\phi`
|
||||
will be discretized on the cell-centers and the flux, :math:`\vec{j}`,
|
||||
will be on the faces. We will use the weak formulation to discretize
|
||||
the DC resistivity equation.
|
||||
|
||||
We can define in weak form by integrating with a general face function \\\(\\vec{f}\\\):
|
||||
We can define in weak form by integrating with a general face function
|
||||
:math:`\vec{f}`:
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -61,9 +68,16 @@ We can then discretize for every cell:
|
||||
|
||||
.. note::
|
||||
|
||||
We have discretized the dot product above, but remember that we do not really have a single vector \\\(\\mathbf{J}\\\), but approximations of \\\(\\vec{j}\\\) on each face of our cell. In 2D that means 2 approximations of \\\(\\mathbf{J}_x\\\) and 2 approximations of \\\(\\mathbf{J}_y\\\). In 3D we also have 2 approximations of \\\(\\mathbf{J}_z\\\).
|
||||
We have discretized the dot product above, but remember that we do not
|
||||
really have a single vector :math:`\mathbf{J}`, but approximations of
|
||||
:math:`\vec{j}` on each face of our cell. In 2D that means 2
|
||||
approximations of :math:`\mathbf{J}_x` and 2 approximations of
|
||||
:math:`\mathbf{J}_y`. In 3D we also have 2 approximations of
|
||||
:math:`\mathbf{J}_z`.
|
||||
|
||||
Regardless of how we choose to approximate this dot product, we can represent this in vector form (again this is for every cell), and will generalize for the case of anisotropic (tensor) sigma.
|
||||
Regardless of how we choose to approximate this dot product, we can represent
|
||||
this in vector form (again this is for every cell), and will generalize for
|
||||
the case of anisotropic (tensor) sigma.
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -71,14 +85,17 @@ Regardless of how we choose to approximate this dot product, we can represent th
|
||||
-\phi^{\top} v_{\text{cell}} \mathbf{D}_{\text{cell}} \mathbf{F})
|
||||
+ \text{BC}
|
||||
|
||||
We multiply by square-root of volume on each side of the tensor conductivity to keep symmetry in the system. Here \\\(\\mathbf{J}_c\\\) is the Cartesian \\\(\\mathbf{J}\\\) (on the faces that we choose to use in our approximation) and must be calculated differently depending on the mesh:
|
||||
We multiply by square-root of volume on each side of the tensor conductivity
|
||||
to keep symmetry in the system. Here :math:`\mathbf{J}_c` is the Cartesian
|
||||
:math:`\mathbf{J}` (on the faces that we choose to use in our approximation)
|
||||
and must be calculated differently depending on the mesh:
|
||||
|
||||
.. math::
|
||||
\mathbf{J}_c = \mathbf{Q}_{(i)}\mathbf{J}_\text{TENSOR} \\
|
||||
\mathbf{J}_c = \mathbf{N}_{(i)}^{-1}\mathbf{Q}_{(i)}\mathbf{J}_\text{Curv}
|
||||
|
||||
Here the \\\(i\\\) index refers to where we choose to approximate this integral, as discussed in the note above.
|
||||
We will approximate this integral by taking the fluxes clustered around every node of the cell, there are 8 combinations in 3D, and 4 in 2D. We will use a projection matrix \\\( \\mathbf{Q}_{(i)} \\\) to pick the appropriate fluxes. So, now that we have 8 approximations of this integral, we will just take the average. For the TensorMesh, this looks like:
|
||||
Here the :math:`i` index refers to where we choose to approximate this integral, as discussed in the note above.
|
||||
We will approximate this integral by taking the fluxes clustered around every node of the cell, there are 8 combinations in 3D, and 4 in 2D. We will use a projection matrix :math:`\mathbf{Q}_{(i)}` to pick the appropriate fluxes. So, now that we have 8 approximations of this integral, we will just take the average. For the TensorMesh, this looks like:
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -107,10 +124,12 @@ By defining the faceInnerProduct (8 combinations of fluxes in 3D, 4 in 2D, 2 in
|
||||
\sum_{i=1}^{2^d}
|
||||
\mathbf{P}_{(i)}^{\top} \Sigma^{-1} \mathbf{P}_{(i)}
|
||||
|
||||
Where \\\(d\\\) is the dimension of the mesh.
|
||||
The \\\( \\mathbf{M}^f \\\) is returned when given the input of \\\( \\Sigma^{-1} \\\).
|
||||
Where :math:`d` is the dimension of the mesh.
|
||||
The :math:`\mathbf{M}^f` is returned when given the input of :math:`\Sigma^{-1}`.
|
||||
|
||||
Here each \\( \\mathbf{P} \\in \\mathbb{R}^{(d*nC, nF)} \\\) is a combination of the projection, volume, and any normalization to Cartesian coordinates (where the dot product is well defined):
|
||||
Here each :math:`\mathbf{P} ~ \in ~ \mathbb{R}^{(d*nC, nF)}` is a combination
|
||||
of the projection, volume, and any normalization to Cartesian coordinates
|
||||
(where the dot product is well defined):
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -129,7 +148,10 @@ If ``returnP=True`` is requested in any of these methods the projection matrices
|
||||
# In 1D
|
||||
P = [P0, P1]
|
||||
|
||||
The derivation for ``edgeInnerProducts`` is exactly the same, however, when we approximate the integral using the fields around each node, the projection matrices look a bit different because we have 12 edges in 3D instead of just 6 faces. The interface to the code is exactly the same.
|
||||
The derivation for ``edgeInnerProducts`` is exactly the same, however, when we
|
||||
approximate the integral using the fields around each node, the projection
|
||||
matrices look a bit different because we have 12 edges in 3D instead of just 6
|
||||
faces. The interface to the code is exactly the same.
|
||||
|
||||
|
||||
Defining Tensor Properties
|
||||
@@ -137,7 +159,8 @@ Defining Tensor Properties
|
||||
|
||||
**For 3D:**
|
||||
|
||||
Depending on the number of columns (either 1, 3, or 6) of mu, the material property is interpreted as follows:
|
||||
Depending on the number of columns (either 1, 3, or 6) of mu, the material
|
||||
property is interpreted as follows:
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -188,13 +211,16 @@ Which is nice and easy to invert if necessary, however, in the fully anisotropic
|
||||
Taking Derivatives
|
||||
------------------
|
||||
|
||||
We will take the derivative of the fully anisotropic tensor for a 3D mesh, the other cases are easier and will not be discussed here. Let us start with one part of the sum which makes up \\\(\\mathbf{M}^f_\\Sigma\\\) and take the derivative when this is multiplied by some vector \\\(\\mathbf{v}\\\):
|
||||
We will take the derivative of the fully anisotropic tensor for a 3D mesh, the
|
||||
other cases are easier and will not be discussed here. Let us start with one
|
||||
part of the sum which makes up :math:`\mathbf{M}^f_\Sigma` and take the
|
||||
derivative when this is multiplied by some vector :math:`\mathbf{v}`:
|
||||
|
||||
.. math::
|
||||
|
||||
\mathbf{P}^\top \boldsymbol{\Sigma} \mathbf{Pv}
|
||||
|
||||
Here we will let \\\( \\mathbf{Pv} = \\mathbf{y} \\\) and \\\(\\mathbf{y}\\\) will have the form:
|
||||
Here we will let :math:`\mathbf{Pv} = \mathbf{y}` and :math:`\mathbf{y}` will have the form:
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -233,7 +259,9 @@ Here we will let \\\( \\mathbf{Pv} = \\mathbf{y} \\\) and \\\(\\mathbf{y}\\\) wi
|
||||
\end{matrix}
|
||||
\right]
|
||||
|
||||
Now it is easy to take the derivative with respect to any one of the parameters, for example, \\\(\\frac{\\partial}{\\partial\\boldsymbol{\\sigma}_1}\\\)
|
||||
Now it is easy to take the derivative with respect to any one of the
|
||||
parameters, for example,
|
||||
:math:`\frac{\partial}{\partial\boldsymbol{\sigma}_1}`
|
||||
|
||||
.. math::
|
||||
\frac{\partial}{\partial \boldsymbol{\sigma}_1}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right)
|
||||
@@ -247,7 +275,8 @@ Now it is easy to take the derivative with respect to any one of the parameters,
|
||||
\end{matrix}
|
||||
\right]
|
||||
|
||||
Whereas \\\(\\frac{\\partial}{\\partial\\boldsymbol{\\sigma}_4}\\\), for example, is:
|
||||
Whereas :math:`\frac{\partial}{\partial\boldsymbol{\sigma}_4}`, for
|
||||
example, is:
|
||||
|
||||
.. math::
|
||||
\frac{\partial}{\partial \boldsymbol{\sigma}_4}\left(\mathbf{P}^\top\Sigma\mathbf{y}\right)
|
||||
@@ -261,11 +290,12 @@ Whereas \\\(\\frac{\\partial}{\\partial\\boldsymbol{\\sigma}_4}\\\), for example
|
||||
\end{matrix}
|
||||
\right]
|
||||
|
||||
These are computed for each of the 8 projections, horizontally concatenated, and returned.
|
||||
These are computed for each of the 8 projections, horizontally concatenated,
|
||||
and returned.
|
||||
|
||||
The API
|
||||
-------
|
||||
|
||||
.. automodule:: SimPEG.Mesh.InnerProducts
|
||||
.. autoclass:: SimPEG.Mesh.InnerProducts.InnerProducts
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -3,7 +3,7 @@
|
||||
InvProblem
|
||||
**********
|
||||
|
||||
.. automodule:: SimPEG.InvProblem
|
||||
.. autoclass:: SimPEG.InvProblem.BaseInvProblem
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -12,7 +12,7 @@ InvProblem
|
||||
Inversion
|
||||
*********
|
||||
|
||||
.. automodule:: SimPEG.Inversion
|
||||
.. autoclass:: SimPEG.Inversion.BaseInversion
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -27,7 +27,8 @@ back to conductivity. This is a relatively trivial example (we are just taking
|
||||
the exponential!) but by defining maps we can start to combine and manipulate
|
||||
exactly what we think about as our model, \\\(m\\\). In code, this looks like
|
||||
|
||||
::
|
||||
.. code-block:: python
|
||||
:linenos:
|
||||
|
||||
M = Mesh.TensorMesh([100]) # Create a mesh
|
||||
expMap = Maps.ExpMap(M) # Create a mapping
|
||||
@@ -46,14 +47,15 @@ We will use an example where we want a 1D layered earth as
|
||||
our model, but we want to map this to a 2D discretization to do our forward
|
||||
modeling. We will also assume that we are working in log conductivity still,
|
||||
so after the transformation we want to map to conductivity space.
|
||||
To do this we will introduce the vertical 1D map (:class:`SimPEG.Maps.Vertical1DMap`),
|
||||
To do this we will introduce the vertical 1D map (:class:`SimPEG.Maps.SurjectVertical1D`),
|
||||
which does the first part of what we just described. The second part will be
|
||||
done by the :class:`SimPEG.Maps.ExpMap` described above.
|
||||
|
||||
::
|
||||
.. code-block:: python
|
||||
:linenos:
|
||||
|
||||
M = Mesh.TensorMesh([7,5])
|
||||
v1dMap = Maps.Vertical1DMap(M)
|
||||
v1dMap = Maps.SurjectVertical1D(M)
|
||||
expMap = Maps.ExpMap(M)
|
||||
myMap = expMap * v1dMap
|
||||
m = np.r_[0.2,1,0.1,2,2.9] # only 5 model parameters!
|
||||
@@ -64,7 +66,7 @@ done by the :class:`SimPEG.Maps.ExpMap` described above.
|
||||
from SimPEG import *
|
||||
import matplotlib.pyplot as plt
|
||||
M = Mesh.TensorMesh([7,5])
|
||||
v1dMap = Maps.Vertical1DMap(M)
|
||||
v1dMap = Maps.SurjectVertical1D(M)
|
||||
expMap = Maps.ExpMap(M)
|
||||
myMap = expMap * v1dMap
|
||||
m = np.r_[0.2,1,0.1,2,2.9] # only 5 model parameters!
|
||||
@@ -122,6 +124,8 @@ When these are used in the inverse problem, this is extremely important!!
|
||||
The API
|
||||
=======
|
||||
|
||||
The :code:`IdentityMap` is the base class for all mappings, and it does absolutely nothing.
|
||||
|
||||
.. autoclass:: SimPEG.Maps.IdentityMap
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -130,7 +134,6 @@ The API
|
||||
Common Maps
|
||||
===========
|
||||
|
||||
|
||||
Exponential Map
|
||||
---------------
|
||||
|
||||
@@ -148,7 +151,7 @@ lives (i.e. it varies logarithmically).
|
||||
Vertical 1D Map
|
||||
---------------
|
||||
|
||||
.. autoclass:: SimPEG.Maps.Vertical1DMap
|
||||
.. autoclass:: SimPEG.Maps.SurjectVertical1D
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
@@ -196,8 +199,8 @@ Mesh to Mesh Map
|
||||
:undoc-members:
|
||||
|
||||
|
||||
Some Extras
|
||||
===========
|
||||
Under the Hood
|
||||
==============
|
||||
|
||||
Combo Map
|
||||
---------
|
||||
@@ -188,6 +188,6 @@ other types of meshes in this SimPEG framework.
|
||||
The API
|
||||
=======
|
||||
|
||||
.. automodule:: SimPEG.Mesh.BaseMesh
|
||||
.. autoclass:: SimPEG.Mesh.BaseMesh.BaseMesh
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -0,0 +1,68 @@
|
||||
.. _api_MeshCode:
|
||||
|
||||
Tensor Mesh
|
||||
===========
|
||||
|
||||
.. autoclass:: SimPEG.Mesh.TensorMesh
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
Cylindrical Mesh
|
||||
================
|
||||
|
||||
.. autoclass:: SimPEG.Mesh.CylMesh
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
Tree Mesh
|
||||
=========
|
||||
|
||||
.. autoclass:: SimPEG.Mesh.TreeMesh
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
Curvilinear Mesh
|
||||
================
|
||||
|
||||
.. autoclass:: SimPEG.Mesh.CurvilinearMesh
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
|
||||
Base Rectangular Mesh
|
||||
=====================
|
||||
|
||||
.. autoclass:: SimPEG.Mesh.BaseMesh.BaseRectangularMesh
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
Base Tensor Mesh
|
||||
================
|
||||
|
||||
.. autoclass:: SimPEG.Mesh.TensorMesh.BaseTensorMesh
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
|
||||
Mesh IO
|
||||
=======
|
||||
|
||||
.. automodule:: SimPEG.Mesh.MeshIO
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
|
||||
|
||||
Mesh Viewing
|
||||
============
|
||||
|
||||
.. automodule:: SimPEG.Mesh.View
|
||||
:members:
|
||||
:undoc-members:
|
||||
:show-inheritance:
|
||||
@@ -0,0 +1,29 @@
|
||||
SimPEG PropMaps
|
||||
***************
|
||||
|
||||
The API
|
||||
=======
|
||||
|
||||
Property
|
||||
--------
|
||||
|
||||
.. autoclass:: SimPEG.PropMaps.Property
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
PropMap
|
||||
-------
|
||||
|
||||
.. autoclass:: SimPEG.PropMaps.PropMap
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
PropModel
|
||||
---------
|
||||
|
||||
.. autoclass:: SimPEG.PropMaps.PropModel
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
@@ -91,10 +91,21 @@ The API
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. autoclass:: SimPEG.Regularization.Simple
|
||||
:show-inheritance:
|
||||
:members:
|
||||
|
||||
.. autoclass:: SimPEG.Regularization.Tikhonov
|
||||
:show-inheritance:
|
||||
:members:
|
||||
|
||||
.. autoclass:: SimPEG.Regularization.Sparse
|
||||
:show-inheritance:
|
||||
:members:
|
||||
|
||||
.. autoclass:: SimPEG.Regularization.RegularizationMesh
|
||||
:show-inheritance:
|
||||
:members:
|
||||
|
||||
|
||||
|
||||
@@ -46,6 +46,8 @@ The API
|
||||
=======
|
||||
|
||||
.. autofunction:: SimPEG.Utils.SolverUtils.SolverWrapD
|
||||
:noindex:
|
||||
|
||||
.. autofunction:: SimPEG.Utils.SolverUtils.SolverWrapI
|
||||
:noindex:
|
||||
|
||||
@@ -6,5 +6,6 @@ Utilities
|
||||
|
||||
api_Solver
|
||||
api_Maps
|
||||
api_PropMaps
|
||||
api_Utils
|
||||
api_Tests
|
||||
@@ -21,7 +21,7 @@ Solver Utilities
|
||||
:undoc-members:
|
||||
|
||||
Curv Utilities
|
||||
=============
|
||||
==============
|
||||
|
||||
.. automodule:: SimPEG.Utils.curvutils
|
||||
:members:
|
||||
@@ -51,7 +51,9 @@ Interpolation Utilities
|
||||
Counter Utilities
|
||||
=================
|
||||
|
||||
::
|
||||
.. code-block:: python
|
||||
:linenos:
|
||||
|
||||
class MyClass(object):
|
||||
def __init__(self, url):
|
||||
self.counter = Counter()
|
||||
@@ -69,7 +71,9 @@ Counter Utilities
|
||||
for i in range(300): c.MySecondMethod()
|
||||
c.counter.summary()
|
||||
|
||||
::
|
||||
|
||||
.. code-block:: text
|
||||
:linenos:
|
||||
|
||||
Counters:
|
||||
MyClass.MyMethod : 100
|
||||
@@ -77,6 +81,8 @@ Counter Utilities
|
||||
Times: mean sum
|
||||
MyClass.MySecondMethod : 1.70e-06, 5.10e-04, 300x
|
||||
|
||||
|
||||
|
||||
The API
|
||||
-------
|
||||
|
||||
@@ -35,7 +35,7 @@ The Big Picture
|
||||
Defining a well-posed inverse problem and solving it is a complex task that requires many components that must interact. It is helpful
|
||||
to view this task as a workflow in which various elements are explicitly identified and integrated. The figure below outlines the inversion components that consists of inputs, implementation, and evaluation. The inputs are composed of the geophysical data, the equations which are a mathematical description of the governing physics, and prior knowledge or assumptions about the setting. The implementation consists of two broad categories: the forward simulation and the inversion. The **forward simulation** is the means by which we solve the governing equations given a model and the **inversion components** evaluate and update this model. We are considering a gradient based approach, which updates the model through an optimization routine. The output of this implementation is a model, which, prior to interpretation, must be evaluated. This requires considering, and often re-assessing, the choices and assumptions made in both the input and implementation stages.
|
||||
|
||||
.. image:: InversionWorkflow-PreSimPEG.png
|
||||
.. image:: ../../images/InversionWorkflow-PreSimPEG.png
|
||||
:width: 400 px
|
||||
:alt: Components
|
||||
:align: center
|
||||
@@ -46,24 +46,24 @@ A Comprehensive Framework
|
||||
|
||||
There are an overwhelming amount of choices to be made as one works through the forward modeling and inversion process (see figure above). As a result, software implementations of this workflow often become complex and highly interdependent, making it difficult to interact with and to ask other scientists to pick up and change. Our approach to handling this complexity is to propose a framework, (see below), that compartmentalizes the implementation of inversions into various units. We present it in this specific modular style, as each unit contains a targeted subset of choices crucial to the inversion process.
|
||||
|
||||
.. image:: InversionWorkflow.png
|
||||
.. image:: ../../images/InversionWorkflow.png
|
||||
:width: 400 px
|
||||
:alt: Framework
|
||||
:align: center
|
||||
|
||||
The process of obtaining an acceptable model from an inversion generally requires the geophysicist to perform several iterations of the inversion workflow, rethinking and redesigning each piece of the framework to ensure it is appropriate in the current context. Inversions are experimental and empirical by nature and our software package is designed to facilitate this iterative process. To accomplish this, we have divided the inversion methodology into eight major components (See figure above). The (:class:`SimPEG.Mesh.BaseMesh`) class handles the discretization of the earth and also provides numerical operators. The forward simulation is split into two classes, the (:class:`SimPEG.Survey.BaseSurvey`) and the (:class:`SimPEG.Problem.BaseProblem`). The (:class:`SimPEG.Survey.BaseSurvey`) class handles the geometry of a geophysical problem as well as sources. The (:class:`SimPEG.Problem.BaseProblem`) class handles the simulation of the physics for the geophysical problem of interest. Although created independently, these two classes must be paired to form all of the components necessary for a geophysical forward simulation and calculation of the sensitivity. The (:class:`SimPEG.Problem.BaseProblem`) creates geophysical fields given a source from the (:class:`SimPEG.Survey.BaseSurvey`). The (:class:`SimPEG.Survey.BaseSurvey`) interpolates these fields to the receiver locations and converts them to the appropriate data type, for example, by selecting only the measured components of the field. Each of these operations may have associated derivatives with respect to the model and the computed field; these are included in the calculation of the sensitivity. For the inversion, a (:class:`SimPEG.DataMisfit.BaseDataMisfit`) is chosen to capture the goodness of fit of the predicted data and a (:class:`SimPEG.Regularization.BaseRegularization`) is chosen to handle the non-uniqueness. These inversion elements and an Optimization routine are combined into an inverse problem class (:class:`SimPEG.InvProblem.BaseInvProblem`). (:class:`SimPEG.InvProblem.BaseInvProblem`) is the mathematical statement that will be numerically solved by running an Inversion. The (:class:`SimPEG.Inversion.BaseInversion`) class handles organization and dispatch of directives between all of the various pieces of the framework.
|
||||
The process of obtaining an acceptable model from an inversion generally requires the geophysicist to perform several iterations of the inversion workflow, rethinking and redesigning each piece of the framework to ensure it is appropriate in the current context. Inversions are experimental and empirical by nature and our software package is designed to facilitate this iterative process. To accomplish this, we have divided the inversion methodology into eight major components (See figure above). The :class:`SimPEG.Mesh.BaseMesh.BaseMesh` class handles the discretization of the earth and also provides numerical operators. The forward simulation is split into two classes, the :class:`SimPEG.Survey.BaseSurvey` and the :class:`SimPEG.Problem.BaseProblem`. The :class:`SimPEG.Survey.BaseSurvey` class handles the geometry of a geophysical problem as well as sources. The :class:`SimPEG.Problem.BaseProblem` class handles the simulation of the physics for the geophysical problem of interest. Although created independently, these two classes must be paired to form all of the components necessary for a geophysical forward simulation and calculation of the sensitivity. The :class:`SimPEG.Problem.BaseProblem` creates geophysical fields given a source from the :class:`SimPEG.Survey.BaseSurvey`. The :class:`SimPEG.Survey.BaseSurvey` interpolates these fields to the receiver locations and converts them to the appropriate data type, for example, by selecting only the measured components of the field. Each of these operations may have associated derivatives with respect to the model and the computed field; these are included in the calculation of the sensitivity. For the inversion, a :class:`SimPEG.DataMisfit.BaseDataMisfit` is chosen to capture the goodness of fit of the predicted data and a :class:`SimPEG.Regularization.BaseRegularization` is chosen to handle the non-uniqueness. These inversion elements and an Optimization routine are combined into an inverse problem class :class:`SimPEG.InvProblem.BaseInvProblem`. :class:`SimPEG.InvProblem.BaseInvProblem` is the mathematical statement that will be numerically solved by running an Inversion. The :class:`SimPEG.Inversion.BaseInversion` class handles organization and dispatch of directives between all of the various pieces of the framework.
|
||||
|
||||
The arrows in the figure above indicate what each class takes as a primary argument. For example, both the (:class:`SimPEG.Problem.BaseProblem`) and (:class:`SimPEG.Regularization.BaseRegularization`) classes take a (:class:`SimPEG.Mesh.BaseMesh`) class as an argument. The diagram does not show class inheritance, as each of the base classes outlined have many subtypes that can be interchanged. The (:class:`SimPEG.Mesh.BaseMesh`) class, for example, could be a regular Cartesian mesh (:class:`SimPEG.Mesh.TensorMesh`) or a cylindrical coordinate mesh (:class:`SimPEG.Mesh.CylMesh`), which have many properties in common. These common features, such as both meshes being created from tensor products, can be exploited through inheritance of base classes, and differences can be expressed through subtype polymorphism. Please look at the documentation here for more in-depth information.
|
||||
The arrows in the figure above indicate what each class takes as a primary argument. For example, both the :class:`SimPEG.Problem.BaseProblem` and :class:`SimPEG.Regularization.BaseRegularization` classes take a :class:`SimPEG.Mesh.BaseMesh.BaseMesh` class as an argument. The diagram does not show class inheritance, as each of the base classes outlined have many subtypes that can be interchanged. The :class:`SimPEG.Mesh.BaseMesh.BaseMesh` class, for example, could be a regular Cartesian mesh :class:`SimPEG.Mesh.TensorMesh` or a cylindrical coordinate mesh :class:`SimPEG.Mesh.CylMesh`, which have many properties in common. These common features, such as both meshes being created from tensor products, can be exploited through inheritance of base classes, and differences can be expressed through subtype polymorphism. Please look at the documentation here for more in-depth information.
|
||||
|
||||
|
||||
.. include:: ../CITATION.rst
|
||||
.. include:: ../../../CITATION.rst
|
||||
|
||||
Authors
|
||||
-------
|
||||
|
||||
.. include:: ../AUTHORS.rst
|
||||
.. include:: ../../../AUTHORS.rst
|
||||
|
||||
License
|
||||
-------
|
||||
|
||||
.. include:: ../LICENSE
|
||||
.. include:: ../../../LICENSE
|
||||
@@ -1,5 +1,3 @@
|
||||
.. _api_DC:
|
||||
|
||||
.. math::
|
||||
|
||||
\renewcommand{\div}{\nabla\cdot\,}
|
||||
@@ -38,8 +36,16 @@
|
||||
\renewcommand {\u} { {\vec u} }
|
||||
\newcommand{\I}{\vec{I}}
|
||||
|
||||
|
||||
Direct Current Resistivity
|
||||
**************************
|
||||
|
||||
`SimPEG.DCIP` uses SimPEG as the framework for the forward and inverse
|
||||
direct current (DC) resistivity and induced polarization (IP) geophysical problems.
|
||||
|
||||
|
||||
DC resistivity survey
|
||||
*********************
|
||||
=====================
|
||||
|
||||
Electrical resistivity of subsurface materials is measured by causing an electrical current to flow in the earth between one pair of electrodes while the voltage across a second pair of electrodes is measured. The result is an "apparent" resistivity which is a value representing the weighted average resistivity over a volume of the earth. Variations in this measurement are caused by variations in the soil, rock, and pore fluid electrical resistivity. Surveys require contact with the ground, so they can be labour intensive. Results are sometimes interpreted directly, but more commonly, 1D, 2D or 3D models are estimated using inversion procedures (`GPG <http://www.eos.ubc.ca/courses/eosc350/content/>`_).
|
||||
|
||||
@@ -55,7 +61,7 @@ As direct current (DC) implies, in DC resistivity survey, we assume steady-state
|
||||
|
||||
\curl \e = 0
|
||||
|
||||
Then by taking \\(\\curl\\) for the first equation, we have
|
||||
Then by taking \\(\\div\\) of the first equation, we have
|
||||
|
||||
.. math::
|
||||
|
||||
@@ -137,13 +143,14 @@ Comparing to the analytic function:
|
||||
|
||||
.. plot::
|
||||
|
||||
import simpegDC as DC
|
||||
DC.Examples.Verification.run(plotIt=True)
|
||||
from SimPEG import Examples
|
||||
Examples.DC_Analytic_Dipole.run(plotIt=True)
|
||||
|
||||
API
|
||||
===
|
||||
|
||||
.. automodule:: simpegDC.BaseDC
|
||||
API for DC codes
|
||||
================
|
||||
|
||||
.. automodule:: SimPEG.DCIP.BaseDC
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -9,17 +9,28 @@
|
||||
Frequency Domain Electromagnetics
|
||||
*********************************
|
||||
|
||||
Electromagnetic (EM) geophysical methods are used in a variety of applications from resource exploration, including for hydrocarbons and minerals, to environmental applications, such as groundwater monitoring. The primary physical property of interest in EM is electrical conductivity, which describes the ease with which electric current flows through a material.
|
||||
Electromagnetic (EM) geophysical methods are used in a variety of applications
|
||||
from resource exploration, including for hydrocarbons and minerals, to
|
||||
environmental applications, such as groundwater monitoring. The primary
|
||||
physical property of interest in EM is electrical conductivity, which
|
||||
describes the ease with which electric current flows through a material.
|
||||
|
||||
|
||||
Background
|
||||
==========
|
||||
|
||||
Electromagnetic phenomena are governed by Maxwell's equations. They describe the behavior of EM fields and fluxes. Electromagnetic theory for geophysical applications by Ward and Hohmann (1988) is a highly recommended resource on this topic.
|
||||
Electromagnetic phenomena are governed by Maxwell's equations. They describe
|
||||
the behavior of EM fields and fluxes. Electromagnetic theory for geophysical
|
||||
applications by Ward and Hohmann (1988) is a highly recommended resource on
|
||||
this topic.
|
||||
|
||||
Fourier Transform Convention
|
||||
----------------------------
|
||||
In order to examine Maxwell's equations in the frequency domain, we must first define our choice of harmonic time-dependence by choosing a Fourier transform convention. We use the :math:`e^{i \omega t}` convention, so we define our Fourier Transform pair as
|
||||
|
||||
In order to examine Maxwell's equations in the frequency domain, we must first
|
||||
define our choice of harmonic time-dependence by choosing a Fourier transform
|
||||
convention. We use the :math:`e^{i \omega t}` convention, so we define our
|
||||
Fourier Transform pair as
|
||||
|
||||
.. math ::
|
||||
F(\omega) = \int_{-\infty}^{\infty} f(t) e^{- i \omega t} dt \\
|
||||
@@ -31,6 +42,7 @@ where :math:`\omega` is angular frequency, :math:`t` is time, :math:`F(\omega)`
|
||||
|
||||
Maxwell's Equations
|
||||
===================
|
||||
|
||||
In the frequency domain, Maxwell's equations are given by
|
||||
|
||||
.. math ::
|
||||
@@ -104,19 +116,20 @@ The H-J formulation is in terms of the current density and the magnetic field:
|
||||
|
||||
Discretizing
|
||||
------------
|
||||
|
||||
For both formulations, we use a finite volume discretization
|
||||
and discretize fields on cell edges, fluxes on cell faces and
|
||||
physical properties in cell centers. This is particularly
|
||||
important when using symmetry to reduce the dimensionality of a problem
|
||||
(for instance on a 2D CylMesh, there are :math:`r`, :math:`z` faces and :math:`\theta` edges)
|
||||
|
||||
.. figure:: ../images/finitevolrealestate.png
|
||||
.. figure:: ../../images/finitevolrealestate.png
|
||||
:align: center
|
||||
:scale: 60 %
|
||||
|
||||
For the two formulations, the discretization of the physical properties, fields and fluxes are summarized below.
|
||||
|
||||
.. figure:: ../images/ebjhdiscretizations.png
|
||||
.. figure:: ../../images/ebjhdiscretizations.png
|
||||
:align: center
|
||||
:scale: 60 %
|
||||
|
||||
@@ -150,7 +163,7 @@ API
|
||||
FDEM Problem
|
||||
------------
|
||||
|
||||
.. automodule:: SimPEG.EM.FDEM.FDEM
|
||||
.. automodule:: SimPEG.EM.FDEM.ProblemFDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -169,6 +182,11 @@ FDEM Survey
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
.. automodule:: SimPEG.EM.FDEM.RxFDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
FDEM Fields
|
||||
-----------
|
||||
|
||||
@@ -359,7 +359,7 @@ TDEM - B formulation
|
||||
Field Storage
|
||||
=============
|
||||
|
||||
.. autoclass:: SimPEG.EM.TDEM.SurveyTDEM.FieldsTDEM
|
||||
.. autoclass:: SimPEG.EM.TDEM.BaseTDEM.FieldsTDEM
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
@@ -0,0 +1,33 @@
|
||||
Overview of Electromagnetics in SimPEG
|
||||
**************************************
|
||||
|
||||
|
||||
The API
|
||||
=======
|
||||
|
||||
Physical Properties
|
||||
-------------------
|
||||
|
||||
.. autoclass:: SimPEG.EM.Base.EMPropMap
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
Problem
|
||||
-------
|
||||
|
||||
.. autoclass:: SimPEG.EM.Base.BaseEMProblem
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
Survey
|
||||
------
|
||||
|
||||
.. autoclass:: SimPEG.EM.Base.BaseEMSurvey
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
|
||||
|
||||
@@ -3,22 +3,23 @@ Electromagnetics
|
||||
================
|
||||
|
||||
`SimPEG.EM` uses SimPEG as the framework for the forward and inverse
|
||||
electromagnetics geophysical problems.
|
||||
electromagnetics geophysical problems.
|
||||
|
||||
To solve for predicted data, we follow the framework shown below. The model is
|
||||
what we invert for. This is mapped to a physical property on the simulation
|
||||
mesh. A source which is used to excite the system is specified. Having a model
|
||||
and a source, we can solve Maxwell's equations for fields. We sample these
|
||||
fields with recievers to give us predicted data.
|
||||
fields with recievers to give us predicted data.
|
||||
|
||||
|
||||
.. image:: ../images/simpegEM_noMath.png
|
||||
.. image:: ../../images/simpegEM_noMath.png
|
||||
:scale: 50%
|
||||
|
||||
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
|
||||
api_basic
|
||||
api_FDEM
|
||||
api_TDEM
|
||||
api_Utils
|
||||
@@ -16,6 +16,6 @@ DC Analytic Dipole
|
||||
from SimPEG import Examples
|
||||
Examples.DC_Analytic_Dipole.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/DC_Analytic_Dipole.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/DC_Analytic_Dipole.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -31,6 +31,6 @@ Created by @fourndo
|
||||
from SimPEG import Examples
|
||||
Examples.DC_Forward_PseudoSection.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/DC_Forward_PseudoSection.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/DC_Forward_PseudoSection.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -21,6 +21,6 @@ Here we will create and run a FDEM 1D inversion.
|
||||
from SimPEG import Examples
|
||||
Examples.EM_FDEM_1D_Inversion.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/EM_FDEM_1D_Inversion.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/EM_FDEM_1D_Inversion.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -21,6 +21,6 @@ Here we plot the magnetic flux density from a harmonic dipole in a wholespace.
|
||||
from SimPEG import Examples
|
||||
Examples.EM_FDEM_Analytic_MagDipoleWholespace.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/EM_FDEM_Analytic_MagDipoleWholespace.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/EM_FDEM_Analytic_MagDipoleWholespace.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -17,10 +17,13 @@ current inside a steel-cased. The model is based on the Schenkel and
|
||||
Morrison Casing Model, and the results are used in a 2016 SEG abstract by
|
||||
Yang et al.
|
||||
|
||||
- Schenkel, C.J., and H.F. Morrison, 1990, Effects of well casing on potential field measurements using downhole current sources: Geophysical prospecting, 38, 663-686.
|
||||
.. code-block:: text
|
||||
|
||||
Schenkel, C.J., and H.F. Morrison, 1990, Effects of well casing on potential field measurements using downhole current sources: Geophysical prospecting, 38, 663-686.
|
||||
|
||||
|
||||
The model consists of:
|
||||
|
||||
- Air: Conductivity 1e-8 S/m, above z = 0
|
||||
- Background: conductivity 1e-2 S/m, below z = 0
|
||||
- Casing: conductivity 1e6 S/m
|
||||
@@ -53,6 +56,6 @@ citation would be much appreciated!
|
||||
from SimPEG import Examples
|
||||
Examples.EM_Schenkel_Morrison_Casing.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/EM_Schenkel_Morrison_Casing.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/EM_Schenkel_Morrison_Casing.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -21,6 +21,6 @@ Here we will create and run a TDEM 1D inversion.
|
||||
from SimPEG import Examples
|
||||
Examples.EM_TDEM_1D_Inversion.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/EM_TDEM_1D_Inversion.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/EM_TDEM_1D_Inversion.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -47,6 +47,6 @@ Here we reproduce the results from Celia1990_ demonstrating the head-based formu
|
||||
from SimPEG import Examples
|
||||
Examples.FLOW_Richards_1D_Celia1990.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/FLOW_Richards_1D_Celia1990.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/FLOW_Richards_1D_Celia1990.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -21,6 +21,6 @@ Here we go over the basics of creating a linear problem and inversion.
|
||||
from SimPEG import Examples
|
||||
Examples.Inversion_Linear.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Inversion_Linear.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Inversion_Linear.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -10,7 +10,7 @@
|
||||
|
||||
|
||||
MT: 1D: Inversion
|
||||
=======================
|
||||
=================
|
||||
|
||||
Forward model 1D MT data.
|
||||
Setup and run a MT 1D inversion.
|
||||
@@ -22,6 +22,6 @@ Setup and run a MT 1D inversion.
|
||||
from SimPEG import Examples
|
||||
Examples.MT_1D_ForwardAndInversion.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/MT_1D_ForwardAndInversion.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/MT_1D_ForwardAndInversion.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -10,7 +10,7 @@
|
||||
|
||||
|
||||
MT: 3D: Forward
|
||||
=======================
|
||||
===============
|
||||
|
||||
Forward model 3D MT data.
|
||||
|
||||
@@ -21,6 +21,6 @@ Forward model 3D MT data.
|
||||
from SimPEG import Examples
|
||||
Examples.MT_3D_Foward.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/MT_3D_Foward.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/MT_3D_Foward.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -20,6 +20,6 @@ Mesh: Basic Forward 2D DC Resistivity
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_Basic_ForwardDC.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_Basic_ForwardDC.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_Basic_ForwardDC.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -22,6 +22,6 @@ You can use M.PlotImage to plot images on all of the Meshes.
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_Basic_PlotImage.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_Basic_PlotImage.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_Basic_PlotImage.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -21,6 +21,6 @@ Here we show SimPEG used to create three different types of meshes.
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_Basic_Types.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_Basic_Types.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_Basic_Types.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -52,6 +52,6 @@ field separating as the time increases.
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_Operators_CahnHilliard.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_Operators_CahnHilliard.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_Operators_CahnHilliard.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -26,6 +26,6 @@ on an 8x8 mesh (2^3).
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_QuadTree_Creation.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_QuadTree_Creation.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_QuadTree_Creation.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -21,6 +21,6 @@ Mesh: QuadTree: FaceDiv
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_QuadTree_FaceDiv.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_QuadTree_FaceDiv.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_QuadTree_FaceDiv.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -26,6 +26,6 @@ on an 8x8 mesh (2^3).
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_QuadTree_HangingNodes.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_QuadTree_HangingNodes.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_QuadTree_HangingNodes.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -38,6 +38,6 @@ notation::
|
||||
from SimPEG import Examples
|
||||
Examples.Mesh_Tensor_Creation.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Mesh_Tensor_Creation.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Mesh_Tensor_Creation.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -9,6 +9,10 @@
|
||||
.. --------------------------------- ..
|
||||
|
||||
|
||||
|
||||
Utils: surface2ind_topo
|
||||
=======================
|
||||
|
||||
Here we show how to use :code:`Utils.surface2ind_topo` to identify cells below
|
||||
a topographic surface.
|
||||
|
||||
@@ -19,6 +23,6 @@ a topographic surface.
|
||||
from SimPEG import Examples
|
||||
Examples.Utils_surface2ind_topo.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Utils_surface2ind_topo.py
|
||||
.. literalinclude:: ../../../SimPEG/Examples/Utils_surface2ind_topo.py
|
||||
:language: python
|
||||
:linenos:
|
||||
@@ -35,8 +35,8 @@ Here we reproduce the results from Celia et al. (1990):
|
||||
|
||||
.. plot::
|
||||
|
||||
from SimPEG.FLOW.Examples import Celia1990
|
||||
Celia1990.run()
|
||||
from SimPEG import Examples
|
||||
Examples.FLOW_Richards_1D_Celia1990.run()
|
||||
|
||||
Richards
|
||||
========
|
||||
@@ -0,0 +1,14 @@
|
||||
Induced Polarization
|
||||
********************
|
||||
|
||||
Todo: docs for IP!
|
||||
|
||||
|
||||
API for IP codes
|
||||
================
|
||||
|
||||
.. automodule:: SimPEG.DCIP.BaseIP
|
||||
:show-inheritance:
|
||||
:members:
|
||||
:undoc-members:
|
||||
:inherited-members:
|
||||
@@ -1,26 +0,0 @@
|
||||
.. _examples_Inversion_IRLS:
|
||||
|
||||
.. --------------------------------- ..
|
||||
.. ..
|
||||
.. THIS FILE IS AUTO GENEREATED ..
|
||||
.. ..
|
||||
.. SimPEG/Examples/__init__.py ..
|
||||
.. ..
|
||||
.. --------------------------------- ..
|
||||
|
||||
|
||||
Inversion: Linear Problem
|
||||
=========================
|
||||
|
||||
Here we go over the basics of creating a linear problem and inversion.
|
||||
|
||||
|
||||
|
||||
.. plot::
|
||||
|
||||
from SimPEG import Examples
|
||||
Examples.Inversion_IRLS.run()
|
||||
|
||||
.. literalinclude:: ../../SimPEG/Examples/Inversion_IRLS.py
|
||||
:language: python
|
||||
:linenos:
|
||||
|
Before Width: | Height: | Size: 49 KiB After Width: | Height: | Size: 49 KiB |
|
Before Width: | Height: | Size: 58 KiB After Width: | Height: | Size: 58 KiB |
|
Before Width: | Height: | Size: 30 KiB After Width: | Height: | Size: 30 KiB |
|
After Width: | Height: | Size: 19 KiB |
|
Before Width: | Height: | Size: 45 KiB After Width: | Height: | Size: 45 KiB |
|
Before Width: | Height: | Size: 23 KiB After Width: | Height: | Size: 23 KiB |
@@ -41,15 +41,16 @@ About SimPEG
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
|
||||
api_bigPicture
|
||||
api_installing
|
||||
content/api_core/api_bigPicture
|
||||
content/api_core/api_installing
|
||||
|
||||
Examples
|
||||
********
|
||||
|
||||
.. toctree::
|
||||
:maxdepth: 2
|
||||
api_Examples
|
||||
|
||||
content/api_core/api_Examples
|
||||
|
||||
Packages
|
||||
********
|
||||
@@ -57,9 +58,11 @@ Packages
|
||||
.. toctree::
|
||||
:maxdepth: 3
|
||||
|
||||
em/index
|
||||
mt/index
|
||||
flow/index
|
||||
content/em/index
|
||||
content/dc/index
|
||||
content/ip/index
|
||||
content/mt/index
|
||||
content/flow/index
|
||||
|
||||
Finite Volume
|
||||
*************
|
||||
@@ -67,7 +70,7 @@ Finite Volume
|
||||
.. toctree::
|
||||
:maxdepth: 3
|
||||
|
||||
api_FiniteVolume
|
||||
content/api_core/api_FiniteVolume
|
||||
|
||||
Forward Problems
|
||||
****************
|
||||
@@ -75,7 +78,7 @@ Forward Problems
|
||||
.. toctree::
|
||||
:maxdepth: 3
|
||||
|
||||
api_ForwardProblem
|
||||
content/api_core/api_ForwardProblem
|
||||
|
||||
Inversion Components
|
||||
********************
|
||||
@@ -83,7 +86,7 @@ Inversion Components
|
||||
.. toctree::
|
||||
:maxdepth: 3
|
||||
|
||||
api_InversionComponents
|
||||
content/api_core/api_InversionComponents
|
||||
|
||||
Utility Codes
|
||||
*************
|
||||
@@ -91,7 +94,7 @@ Utility Codes
|
||||
.. toctree::
|
||||
:maxdepth: 3
|
||||
|
||||
api_Utilities
|
||||
content/api_core/api_Utilities
|
||||
|
||||
|
||||
Project Index & Search
|
||||
|
||||
@@ -0,0 +1,11 @@
|
||||
indexes:
|
||||
|
||||
# AUTOGENERATED
|
||||
|
||||
# This index.yaml is automatically updated whenever the dev_appserver
|
||||
# detects that a new type of query is run. If you want to manage the
|
||||
# index.yaml file manually, remove the above marker line (the line
|
||||
# saying "# AUTOGENERATED"). If you want to manage some indexes
|
||||
# manually, move them above the marker line. The index.yaml file is
|
||||
# automatically uploaded to the admin console when you next deploy
|
||||
# your application using appcfg.py.
|
||||
@@ -0,0 +1,90 @@
|
||||
#!/usr/bin/env python
|
||||
#
|
||||
# Copyright 2007 Google Inc.
|
||||
#
|
||||
# Licensed under the Apache License, Version 2.0 (the "License");
|
||||
# you may not use this file except in compliance with the License.
|
||||
# You may obtain a copy of the License at
|
||||
#
|
||||
# http://www.apache.org/licenses/LICENSE-2.0
|
||||
#
|
||||
# Unless required by applicable law or agreed to in writing, software
|
||||
# distributed under the License is distributed on an "AS IS" BASIS,
|
||||
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
|
||||
# See the License for the specific language governing permissions and
|
||||
# limitations under the License.
|
||||
#
|
||||
import cgi
|
||||
import datetime
|
||||
import webapp2
|
||||
import logging
|
||||
|
||||
from google.appengine.ext import ndb
|
||||
from google.appengine.api import users
|
||||
from google.appengine.api import mail
|
||||
from google.appengine.api import urlfetch
|
||||
|
||||
import os
|
||||
import jinja2
|
||||
import urllib, hashlib
|
||||
import json
|
||||
|
||||
TEMPLATEFOLDER = '_build/html/'
|
||||
|
||||
JINJA_ENVIRONMENT = jinja2.Environment(
|
||||
loader=jinja2.FileSystemLoader(os.path.join(os.path.dirname(__file__).split('/')[:-1])),
|
||||
extensions=['jinja2.ext.autoescape'],
|
||||
autoescape=False)
|
||||
|
||||
def setTemplate(self, template_values, templateFile, _templateFolder=TEMPLATEFOLDER):
|
||||
# add Defaults
|
||||
template_values['_templateFolder'] = _templateFolder
|
||||
template_values['_year'] = str(datetime.datetime.now().year)
|
||||
path = os.path.normpath(_templateFolder+templateFile)
|
||||
template = JINJA_ENVIRONMENT.get_template(path)
|
||||
resp = self.response.write(template.render(template_values))
|
||||
|
||||
# if resp is None:
|
||||
# self.redirect('/error.html', permanent=True)
|
||||
|
||||
class Images(webapp2.RequestHandler):
|
||||
def get(self):
|
||||
self.redirect('/'+self.request.path)
|
||||
|
||||
class Redirect(webapp2.RequestHandler):
|
||||
def get(self):
|
||||
path = str(self.request.path).split(os.path.sep)[3:]
|
||||
self.redirect(('/%s'%os.path.sep.join(path)), permanent=True)
|
||||
|
||||
|
||||
class MainPage(webapp2.RequestHandler):
|
||||
def get(self):
|
||||
setTemplate(self, {"indexPage":True}, 'index.html')
|
||||
|
||||
# class Error(webapp2.RequestHandler):
|
||||
# def get(self):
|
||||
# setTemplate(self, {}, 'error.html', _templateFolder='_templates/')
|
||||
# # self.redirect('/error.html', permanent=True)
|
||||
|
||||
from webapp2 import Route, RedirectHandler
|
||||
# pointers = [
|
||||
# Route('/en/latest/.*', RedirectHandler, defaults={'_uri': '/.*'}),
|
||||
# # Route('/en/latest', RedirectHandler, defaults={'_uri': '/en/latest/'}),
|
||||
# Route('/en/latest/', RedirectHandler, defaults={'_uri': '/'}),
|
||||
#
|
||||
# ('/.*', MainPage),
|
||||
# ('/', MainPage),
|
||||
# ('', MainPage),
|
||||
# ('/_images/.*', Images),
|
||||
# ]
|
||||
|
||||
app = webapp2.WSGIApplication([
|
||||
('/_images/.*', Images),
|
||||
('/en/latest/.*',Redirect),
|
||||
('/', MainPage),
|
||||
# ('/.*', Error),
|
||||
], debug=True)
|
||||
|
||||
|
||||
# app.error_handlers[404] = Error
|
||||
|
||||