SimPEG.regularization.CrossGradient#

class SimPEG.regularization.CrossGradient(mesh, wire_map, approx_hessian=True, **kwargs)[source]#

Bases: BaseSimilarityMeasure

Cross-gradient regularization for joint inversion.

CrossGradient regularization is used to ensure the location and orientation of non-zero gradients in the recovered model are consistent across two physical property distributions. For joint inversion involving three or more physical properties, a separate instance of CrossGradient must be created for each physical property pair and added to the total regularization as a weighted sum.

Parameters:

meshSimPEG.regularization.RegularizationMesh, discretize.base.BaseMesh: Mesh on which the regularization is discretized. This is not necessarily the same as the mesh on which the simulation is defined.
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.
wire_mapSimPEG.maps.Wires: Wire map connecting physical properties defined on active cells of the RegularizationMesh` to the entire model.
reference_modelNone, (n_param, ) numpy.ndarray: Reference model. If None, the reference model in the inversion is set to the starting model.
unitsNone, str: Units for the model parameters. Some regularization classes behave differently depending on the units; e.g. ‘radian’.
weightsNone, dict: Weight multipliers to customize the least-squares function. Each key points to a (n_cells, ) numpy.ndarray that is defined on the RegularizationMesh.
approx_hessianbool: Whether to use the semi-positive definate approximation for the Hessian.

Notes

Consider the case where the model is comprised of two physical properties \(m_1\) and \(m_2\). Here, we define the regularization function (objective function) for cross-gradient as (Haber and Gazit, 2013):

\[\phi (m_1, m_2) = \frac{1}{2} \int_\Omega \, w(r) \, \Big | \nabla m_1 \, \times \, \nabla m_2 \, \Big |^2 \, dv\]

where \(w(r)\) is a user-defined weighting function. Using the identity \(| \vec{a} \times \vec{b} |^2 = | \vec{a} |^2 | \vec{b} |^2 - (\vec{a} \cdot \vec{b})^2\), the regularization function can be re-expressed as:

\[\phi (m_1, m_2) = \frac{1}{2} \int_\Omega \, w(r) \, \Big [ \, \big | \nabla m_1 \big |^2 \big | \nabla m_2 \big |^2 - \big ( \nabla m_1 \, \cdot \, \nabla m_2 \, \big )^2 \Big ] \, dv\]

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 given by:

\[\phi (m_1, m_2) \approx \frac{1}{2} \sum_i \tilde{w}_i \, \bigg [ \Big | (\nabla m_1)_i \Big |^2 \Big | (\nabla m_2)_i \Big |^2 - \Big [ (\nabla m_1)_i \, \cdot \, (\nabla m_2)_i \, \Big ]^2 \, \bigg ]\]

where \((\nabla m_1)_i\) are the gradients of property \(m_1\) 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.

In practice, we define the model \(\mathbf{m}\) as a discrete vector of the form:

\[\begin{split}\mathbf{m} = \begin{bmatrix} \mathbf{m_1} \\ \mathbf{m_2} \end{bmatrix}\end{split}\]

where \(\mathbf{m_1}\) and \(\mathbf{m_2}\) are the discrete representations of the respective physical properties on the mesh. The discrete regularization function is therefore equivalent to an objective function of the form:

\[\phi (\mathbf{m}) = \frac{1}{2} \Big [ \mathbf{W A} \big ( \mathbf{G \, m_1} \big )^2 \Big ]^T \Big [ \mathbf{W A} \big ( \mathbf{G \, m_2} \big )^2 \Big ] - \frac{1}{2} \bigg \| \mathbf{W A} \Big [ \big ( \mathbf{G \, m_1} \big ) \odot \big ( \mathbf{G \, m_2} \big ) \Big ] \bigg \|^2\]

where exponents are computed elementwise,

\(\mathbf{G}\) is the cell gradient operator (cell centers to faces),

\(\mathbf{A}\) averages vectors from faces to cell centers, and

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

Custom weights and the weighting matrix:

Let \(\mathbf{w_1, \; w_2, \; w_3, \; ...}\) each represent an optional set of custom cell weights. The weighting applied within the objective function is given by:

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

where \(\mathbf{v}\) are the cell volumes. The weighting matrix used to apply weights within the regularization is given by:

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

Each set of custom cell weights is stored within a dict as an (n_cells, ) numpy.ndarray. The weights can be set all at once during instantiation with the weights keyword argument as follows:

>>> reg = CrossGradient(mesh, wire_map, 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

approx_hessian

Whether to use the semi-positive definate approximation for the Hessian.

Methods

`__call__`(model)	Evaluate the cross-gradient regularization function for the model provided.
`calculate_cross_gradient`(model[, ...])	Calculates the magnitudes of the cross-gradient vectors at cell centers.
`deriv`(model)	Gradient of the regularization function evaluated for the model provided.
`deriv2`(model[, v])	Hessian of the regularization function evaluated for the model provided.

Galleries and Tutorials using `SimPEG.regularization.CrossGradient`#

Cross-gradient Joint Inversion of Gravity and Magnetic Anomaly Data

SimPEG.regularization.CrossGradient#

Galleries and Tutorials using SimPEG.regularization.CrossGradient#

Galleries and Tutorials using `SimPEG.regularization.CrossGradient`#