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:
\[\mathbf{u}(\mathbf{m}) = \sigma_0 + (\sigma_1 - \sigma_0) \bigg [ \frac{1}{2} + \pi^{-1} \arctan \bigg ( a \big [ \sqrt{(\mathbf{x_c}-x_0)^2 + (\mathbf{y_c}-y_0)^2} - R \big ] \bigg ) \bigg ]\]The derivative of the mapping with respect to the model parameters is a
numpy.ndarrayof shape (mesh.nC, 5) given by:\[\frac{\partial \mathbf{u}}{\partial \mathbf{m}} = \Bigg [ \frac{\partial \mathbf{u}}{\partial \sigma_0} \;\; \Bigg [ \frac{\partial \mathbf{u}}{\partial \sigma_1} \;\; \Bigg [ \frac{\partial \mathbf{u}}{\partial x_0} \;\; \Bigg [ \frac{\partial \mathbf{u}}{\partial y_0} \;\; \Bigg [ \frac{\partial \mathbf{u}}{\partial R} \Bigg ]\]- 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
- m(
- Returns
scipy.sparse.csr_matrixDerivative 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.