simpeg.maps.ParametricCircleMap#

class simpeg.maps.ParametricCircleMap(mesh, logSigma=True, slope=0.1)[source]#

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:

$$\mathbf{u}(\mathbf{m})={\sigma }_0+({\sigma }_1-{\sigma }_0)[\frac{1}{2}+{\pi }^{-1}\arctan (a[\sqrt{({\mathbf{x}}_{\mathbf{c}}-x_0{)}^2+({\mathbf{y}}_{\mathbf{c}}-y_0{)}^2}-R])]$$

where ${\mathbf{x}}_{\mathbf{c}}$ and ${\mathbf{y}}_{\mathbf{c}}$ 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.

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.

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)