SimPEG.regularization.SmoothnessFirstOrder#

class SimPEG.regularization.SmoothnessFirstOrder(mesh, orientation='x', reference_model_in_smooth=False, **kwargs)[source]#

Bases: BaseRegularization

First-order smoothness least-squares regularization.

SmoothnessFirstOrder regularization is used to ensure that values in the recovered model are smooth along a specified direction. When a reference model is included, the regularization preserves gradients/interfaces within the reference model along the direction specified (x, y or z). Optionally, custom cell weights can be used to control the degree of smoothness being enforced throughout different regions the model.

See the Notes section below for a comprehensive description.

Parameters:

meshdiscretize.base.BaseMesh mesh: The mesh on which the regularization is discretized.
orientation{‘x’, ‘y’, ‘z’}: The direction along which smoothness is enforced.
active_cellsNone, (n_cells, ) numpy.ndarray of bool: Boolean array defining the set of RegularizationMesh cells that are active in the inversion. If None, all cells are active.
mappingNone, SimPEG.maps.BaseMap: The mapping from the model parameters to the active cells in the inversion. If None, the mapping is the identity map.
reference_modelNone, (n_param, ) numpy.ndarray: Reference model. If None, the reference model in the inversion is set to the starting model. To include the reference model in the regularization, the reference_model_in_smooth property must be set to True.
reference_model_in_smoothbool, optional: Whether to include the reference model in the smoothness regularization.
unitsNone, str: Units for the model parameters. Some regularization classes behave differently depending on the units; e.g. ‘radian’.
weightsNone, dict: Custom weights for the least-squares function. Each key points to a numpy.ndarray that is defined on the regularization.RegularizationMesh. A (n_cells, ) numpy.ndarray is used to define weights at cell centers, which are averaged to the appropriate faces internally when weighting is applied. A (n_faces, ) numpy.ndarray is used to define weights directly on the faces specified by the orientation input argument.

Notes

We define the regularization function (objective function) for first-order smoothness along the x-direction as:

\[\phi (m) = \frac{1}{2} \int_\Omega \, w(r) \, \bigg [ \frac{\partial m}{\partial x} \bigg ]^2 \, dv\]

where \(m(r)\) is the model and \(w(r)\) is a user-defined weighting function.

For implementation within SimPEG, the regularization function and its variables must be discretized onto a mesh. The discretized approximation for the regularization function (objective function) is expressed in linear form as:

\[\phi (\mathbf{m}) = \frac{1}{2} \sum_i \tilde{w}_i \, \bigg | \, \frac{\partial m_i}{\partial x} \, \bigg |^2\]

where \(m_i \in \mathbf{m}\) are the discrete model parameter values defined on the mesh and \(\tilde{w}_i \in \mathbf{\tilde{w}}\) are amalgamated weighting constants that 1) account for cell dimensions in the discretization and 2) apply any user-defined weighting. This is equivalent to an objective function of the form:

\[\phi (\mathbf{m}) = \frac{1}{2} \Big \| \mathbf{W \, G_x m } \, \Big \|^2\]

where

\(\mathbf{G_x}\) is the partial cell gradient operator along the x-direction, and

\(\mathbf{W}\) is the weighting matrix.

Note that since \(\mathbf{G_x}\) maps from cell centers to x-faces, \(\mathbf{W}\) is an operator that acts on variables living on x-faces.

Reference model in smoothness:

Gradients/interfaces within a discrete reference model \(\mathbf{m}^{(ref)}\) can be preserved by including the reference model the regularization. In this case, the objective function becomes:

\[\phi (\mathbf{m}) = \frac{1}{2} \Big \| \mathbf{W G_x} \big [ \mathbf{m} - \mathbf{m}^{(ref)} \big ] \Big \|^2\]

This functionality is used by setting a reference model with the reference_model property, and by setting the reference_model_in_smooth parameter to True.

Custom weights and the weighting matrix:

Let \(\mathbf{w_1, \; w_2, \; w_3, \; ...}\) each represent an optional set of custom weights defined on the faces specified by the orientation property; i.e. x-faces for smoothness along the x-direction. Each set of weights were either defined directly on the faces or have been averaged from cell centers.

The weighting applied within the objective function is given by:

\[\mathbf{\tilde{w}} = \mathbf{v_x} \odot \prod_j \mathbf{w_j}\]

where \(\mathbf{v_x}\) are cell volumes projected to x-faces. The weighting matrix is given by:

\[\mathbf{W} = \textrm{diag} \Big ( \, \mathbf{\tilde{w}}^{1/2} \Big )\]

Each set of custom weights is stored within a dict as an numpy.ndarray. A (n_cells, ) numpy.ndarray is used to define weights at cell centers, which are averaged to the appropriate faces internally when weighting is applied. A (n_faces, ) numpy.ndarray is used to define weights directly on the faces specified by the orientation input argument. The weights can be set all at once during instantiation with the weights keyword argument as follows:

>>> array_1 = np.ones(mesh.n_cells)  # weights at cell centers
>>> array_2 = np.ones(mesh.n_faces_x)  # weights directly on x-faces
>>> reg = SmoothnessFirstOrder(
>>>     mesh, orientation='x', weights={'weights_1': array_1, 'weights_2': array_2}
>>> )

or set after instantiation using the set_weights method:

>>> reg.set_weights(weights_1=array_1, weights_2=array_2)

The default weights that account for cell dimensions in the regularization are accessed via:

>>> reg.get_weights('volume')

Attributes

`W`	Weighting matrix.
`cell_gradient`	Partial cell gradient operator.
`orientation`	Direction along which smoothness is enforced.

reference_model_in_smooth

Methods

`f_m`(m)	Evaluate the regularization kernel function.
`f_m_deriv`(m)	Derivative of the regularization kernel function.