simpeg.maps.ParametricLayer.deriv#

ParametricLayer.deriv(m)[source]#

Derivative of the mapping with respect to the input parameters.

Let $\mathbf{m}=[{\sigma }_0,{\sigma }_1,z_L,h]$ be the set of model parameters the defines a layer within a wholespace. The mapping :math:`mathbf{u}(mathbf{m})`from the parameterized model to all active cells is given by:

$$\mathbf{u}(\mathbf{m})={\sigma }_0+\frac{({\sigma }_1-{\sigma }_0)}{\pi }[\arctan (a({\mathbf{z}}_{\mathbf{c}}-z_L+\frac{h}{2}))-\arctan (a({\mathbf{z}}_{\mathbf{c}}-z_L-\frac{h}{2}))]$$

where ${\mathbf{z}}_{\mathbf{c}}$ is a vectors containing the vertical cell center locations for all active cells in the mesh. The derivative of the mapping with respect to the model parameters is a numpy.ndarray of shape (nAct, 4) given by:

$$\frac{\partial \mathbf{u}}{\partial \mathbf{m}}=[\frac{\partial \mathbf{u}}{\partial {\sigma }_0}\frac{\partial \mathbf{u}}{\partial {\sigma }_1}\frac{\partial \mathbf{u}}{\partial z_L}\frac{\partial \mathbf{u}}{\partial h}]$$

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.