simpeg.regularization.SparseSmoothness.f_m#

SparseSmoothness.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:

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})={\mathbf{G}}_{\mathbf{x}}[\mathbf{m}-{\mathbf{m}}^{(ref)}]$$

where ${\mathbf{G}}_{\mathbf{x}}$ is the partial cell gradient operator along the x-direction (i.e. x-derivative), $\mathbf{m}$ are the discrete model parameters defined on the mesh and ${\mathbf{m}}^{(ref)}$ is the reference model (optional). Similarly for smoothness along y and z.

Parameters:

mnumpy.ndarray

The model.

Returns:

numpy.ndarray

The regularization kernel function.

Notes

The objective function for first-order smoothness regularization along the x-direction is given by:

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}{\mathbf{G}}_{\mathbf{x}}[\mathbf{m}-{\mathbf{m}}^{(ref)}]{\Vert }^2$$

where $\mathbf{m}$ are the discrete model parameters (model), ${\mathbf{m}}^{(ref)}$ is the reference model, ${\mathbf{G}}_{\mathbf{x}}$ is the partial cell gradient operator along the x-direction (i.e. x-derivative), and $\mathbf{W}$ is the weighting matrix. Similar for smoothness along y and z. See the SmoothnessFirstOrder class documentation for more detail.

We define the regularization kernel function ${\mathbf{f}}_{\mathbf{m}}$ as:

$${\mathbf{f}}_{\mathbf{m}}(\mathbf{m})={\mathbf{G}}_{\mathbf{x}}[\mathbf{m}-{\mathbf{m}}^{(ref)}]$$

such that

$${\varphi }_m(\mathbf{m})=\Vert \mathbf{W}{\mathbf{f}}_{\mathbf{m}}{\Vert }^2$$