simpeg.regularization.AmplitudeSmoothnessFirstOrder.f_m#

AmplitudeSmoothnessFirstOrder.f_m(m)[source]#

Evaluate the regularization kernel function.

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

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})={\mathbf{G}}_{\mathbf{x}}\overline{\mathbf{m}}$$

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

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})=\overline{\mathbf{m}}=([{\mathbf{m}}_p-{\mathbf{m}}_p^{(ref)}{]}^2+[{\mathbf{m}}_s-{\mathbf{m}}_s^{(ref)}{]}^2+[{\mathbf{m}}_t-{\mathbf{m}}_t^{(ref)}{]}^2{)}^{1/2}$$

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:

$$\begin{array}{r}\mathbf{m}=\left[\begin{array}{c}{\mathbf{m}}_p\\ {\mathbf{m}}_s\\ {\mathbf{m}}_t\end{array}\right]\end{array}$$

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.