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Model Roughness
Spatial Gradient Roughness Operator
The gradient roughness operator output by Mamba2D in the Penalty File (.penalty) approximates the integral of the model spatial gradient over each parameter using the formula
where
Minimum Gradient Support (MGS) Roughness Operator
Minimum gradient support regularization can used to find models that have
relatively sharper gradients between blocky regions of nearly constant
conductivity (e.g., [PoZh99]). This is accomplished by using roughness
penalty relaxation weights that are inversely proportional to gradients in the
previous model iteration
The complete model roughness norm is updated to use this diagonal weighting matrix via
This is applied iteratively with a new MGS weighting matrix computed for each
iteration using the model vector from the previous iteration. Since the
elements of
The
Because the MGS weights can greatly reduce the model regularization constraint, they can greatly destabilize the inversion and their application must be done with care. One strategy is to perform a normal smooth inversion to find an acceptable model, then turn on the MGS regularization for a few final model polishing iterations.
See [BKS12] for more details and a variant of this that uses gradients in seismic models for the term in the denominator.
Anisotropy Roughness Operator
For anisotropic models, the roughness is augmented by splitting the model vector into anisotropic subsets
so that
where the last term on the right is used to penalize anisotropy and can
be arbitrarily dialed up or down with the scalar parameter
For the case of transversely isotropic models, there are only two anisotropic components instead of the three components shown above.