simpeg.maps.ParametricCircleMap

class simpeg.maps.ParametricCircleMap(mesh, logSigma=True, slope=0.1)

Bases: IdentityMap

Mapping for a parameterized circle.

Define the mapping from a parameterized model for a circle in a wholespace to all cells within a 2D mesh. For a circle within a wholespace, the model is defined by 5 parameters: the background physical property value (\(\sigma_0\)), the physical property value for the circle (\(\sigma_c\)), the x location \(x_0\) and y location \(y_0\) for center of the circle, and the circle’s radius (\(R\)).

Let \(\mathbf{m} = [\sigma_0, \sigma_1, x_0, y_0, R]\) be the set of model parameters the defines a circle within a wholespace. The mapping \(\mathbf{u}(\mathbf{m})\) from the parameterized model to all cells within a 2D mesh is given by:

\[\mathbf{u}(\mathbf{m}) = \sigma_0 + (\sigma_1 - \sigma_0) \bigg [ \frac{1}{2} + \pi^{-1} \arctan \bigg ( a \big [ \sqrt{(\mathbf{x_c}-x_0)^2 + (\mathbf{y_c}-y_0)^2} - R \big ] \bigg ) \bigg ]\]

where \(\mathbf{x_c}\) and \(\mathbf{y_c}\) are vectors storing the x and y positions of all cell centers for the 2D mesh and \(a\) is a user-defined constant which defines the sharpness of boundary of the circular structure.

Parameters:

meshdiscretize.BaseMesh

A 2D discretize mesh

logSigmabool

If True, parameters \(\sigma_0\) and \(\sigma_1\) represent the natural log of the physical property values for the background and circle, respectively.

slopefloat

A constant for defining the sharpness of the boundary between the circle and the wholespace. The sharpness increases as slope is increased.

Examples

Here we define the parameterized model for a circle in a wholespace. We then create and use a ParametricCircleMap to map the model to a 2D mesh.

>>> from simpeg.maps import ParametricCircleMap
>>> from discretize import TensorMesh
>>> import numpy as np
>>> import matplotlib.pyplot as plt

>>> h = 0.5*np.ones(20)
>>> mesh = TensorMesh([h, h])

>>> sigma0, sigma1, x0, y0, R = 0., 10., 4., 6., 2.
>>> model = np.r_[sigma0, sigma1, x0, y0, R]
>>> mapping = ParametricCircleMap(mesh, logSigma=False, slope=2)

>>> fig = plt.figure(figsize=(5, 5))
>>> ax = fig.add_subplot(111)
>>> mesh.plot_image(mapping * model, ax=ax)

(Source code, png, pdf)

Attributes

is_linear	Determine whether or not this mapping is a linear operation.
logSigma	Whether the input needs to be transformed by an exponential
mesh	The mesh used for the mapping
nP	Number of parameters the mapping acts on; i.e. 5.
shape	Dimensions of the mapping operator
slope	Sharpness of the boundary.

Methods

deriv(m[, v])	Derivative of the mapping with respect to the input parameters.
dot(map1)	Multiply two mappings to create a simpeg.maps.ComboMap.
inverse(D)	The transform inverse is not implemented.
test([m, num, random_seed])	Derivative test for the mapping.