SimPEG.regularization.AmplitudeSmoothnessFirstOrder.f_m

AmplitudeSmoothnessFirstOrder.f_m(m)

Evaluate the regularization kernel function.

For first-order smoothness regularization in the x-direction, the regularization kernel function is given by:

\[\mathbf{f_m}(\mathbf{m}) = \mathbf{G_x \, \bar{m}}\]

where \(\mathbf{G_x}\) is the partial cell gradient operator along the x-direction (i.e. x-derivative), and

\[\mathbf{f_m}(\mathbf{m}) = \mathbf{\bar{m}} = \bigg ( \Big [ \mathbf{m}_p - \mathbf{m}_p^{(ref)} \Big ]^2 + \Big [ \mathbf{m}_s - \mathbf{m}_s^{(ref)} \Big ]^2 + \Big [ \mathbf{m}_t - \mathbf{m}_t^{(ref)} \Big ]^2 \bigg )^{1/2}\]

The global set of model parameters \(\mathbf{m}\) defined at cell centers is ordered according to its primary (\(p\)), secondary (\(s\)) and tertiary (\(t\)) directions as follows:

\[\begin{split}\mathbf{m} = \begin{bmatrix} \mathbf{m}_p \\ \mathbf{m}_s \\ \mathbf{m}_t \end{bmatrix}\end{split}\]

Likewise for the reference model vector. The expression has the same form for smoothness along y and z.

Parameters:

mnumpy.ndarray

The model.

Returns:

numpy.ndarray

The regularization kernel function evaluated for the model provided.