simpeg.maps.ParametricCircleMap.deriv#

ParametricCircleMap.deriv(m, v=None)[source]#

Derivative of the mapping with respect to the input parameters.

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 :math:`mathbf{u}(mathbf{m})`from the parameterized model to all cells within a 2D mesh is given by:

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

The derivative of the mapping with respect to the model parameters is a numpy.ndarray of shape (mesh.nC, 5) given by:

$$\frac{\partial \mathbf{u}}{\partial \mathbf{m}}=[\frac{\partial \mathbf{u}}{\partial {\sigma }_0}[\frac{\partial \mathbf{u}}{\partial {\sigma }_1}[\frac{\partial \mathbf{u}}{\partial x_0}[\frac{\partial \mathbf{u}}{\partial y_0}[\frac{\partial \mathbf{u}}{\partial R}]$$

Parameters:

m(nP) numpy.ndarray

A vector representing a set of model parameters

v(nP) numpy.ndarray

If not None, the method returns the derivative times the vector v

Returns:

scipy.sparse.csr_matrix

Derivative of the mapping with respect to the model parameters. If the input argument v is not None, the method returns the derivative times the vector v.