simpeg.electromagnetics.time_domain.Simulation3DCurrentDensity.getRHSDeriv#

Simulation3DCurrentDensity.getRHSDeriv(tInd, src, v, adjoint=False)[source]#

Derivative of the right-hand side times a vector for a given source and time index.

The right-hand side for a given source at time index k is constructed according to:

$${\mathbf{q}}_k=-\frac{1}{\mathrm{\Delta }{t}_k}[{\mathbf{s}}_{\mathbf{e},k}+{\mathbf{s}}_{\mathbf{e},k-1}]-\frac{1}{\mathrm{\Delta }{t}_k}\mathbf{C}{\mathbf{M}}_{\mathbf{e}\mu }^{\mathbf{-}\mathbf{1}}[{\mathbf{s}}_{\mathbf{m},k}+{\mathbf{s}}_{\mathbf{m},k-1}]$$

where

• $\mathrm{\Delta }{t}_k$ is the step length

• $\mathbf{C}$ is the discrete curl operator

• ${\mathbf{s}}_{\mathbf{m}}$ and ${\mathbf{s}}_{\mathbf{e}}$ are the integrated magnetic and electric source terms, respectively

• ${\mathbf{M}}_{\mathbf{e}\mu }$ is the inner-product matrix for permeabilities projected to edges

See the Notes section of the doc strings for Simulation3DCurrentDensity for a full description of the formulation.

Where $\mathbf{m}$ are the set of model parameters and $\mathbf{v}$ is a vector, this method returns

$$\frac{\partial {\mathbf{q}}_{\mathbf{k}}}{\partial \mathbf{m}}\mathbf{v}$$

Or the adjoint operation

$${\frac{\partial {\mathbf{q}}_{\mathbf{k}}}{\partial \mathbf{m}}}^T\mathbf{v}$$

Parameters:

tIndint

The time index; between [0, n_steps].

srctime_domain.sources.BaseTDEMSrc

The TDEM source object.

vnumpy.ndarray

The vector. (n_param,) for the standard operation. (n_faces,) for the adjoint operation.

adjointbool

Whether to perform the adjoint operation.

Returns:

numpy.ndarray

Derivative of the right-hand sides times a vector. (n_faces,) for the standard operation. (n_param,) for the adjoint operation.