Currently, we offer a few methods of ensuring the inversion adheres to a-priori elevation information for the surface of interest, referred to as smallness regularization:
- weighting grid:
- the Jacobian and surface correction grid is computed everywhere, but scaled by the weighting grid values before being added to the past iteration's topography. Weighting grid values of 0 result in no change to the model element. This is inefficient since the Jacobian elements are calculated, but unused.
- upper/lower confining layers:
- the Jacobian and surface correction grid is computed everywhere, and the surface corrections are added to the model, but if the updated model crosses one of the confining layers, it is instead clipped to the layer. As with the weighting grid, this is inefficient if a large region is confined both above and below, as the Jacobian in these regions is calculated but unused.
- masking the model:
- a mask variable can be added to the model dataset. This excludes model elements from the computation of the Jacobian and correction grid, but the entire model is still used for calculating the misfit. This means the masked portion of the model won't alter during the inversion.
All of the above methods have an issue where the gravity residual will likely be non-zero in the portions of the model that are not allowed to alter. This means the inversion will attempt to minimize this non-zero misfit at the nearest location that is unconstrained. For example, if there is a negative residual anomaly just outside of the unconstrained region, the unconstrained region will become unrealistically overdeepened to account for the negative residual nearby. First, we should check that these effects are similar with all three methods.
The alternative technique is to change the input gravity residuals before the Jacobian / topo-corrections are calculated. Currently this can be done with the regional field estimation, especially with constraint point minimization. This minimize the residual gravity misfit at a-priori measurements. With this, there shouldn't be a jump at the constrained-unconstrained boundary, since the residual misfit should smoothly taper to zero at the constrained boundary. Alternatives would be to apply the weighting grid to the residual prior to each inversion, instead of to the topo-corrections. Same thing with the upper / lower confining layers. If the topo is already equal to the upper confining layer, than don't allow positive residual misfits and vice versa.
Currently, we offer a few methods of ensuring the inversion adheres to a-priori elevation information for the surface of interest, referred to as smallness regularization:
All of the above methods have an issue where the gravity residual will likely be non-zero in the portions of the model that are not allowed to alter. This means the inversion will attempt to minimize this non-zero misfit at the nearest location that is unconstrained. For example, if there is a negative residual anomaly just outside of the unconstrained region, the unconstrained region will become unrealistically overdeepened to account for the negative residual nearby. First, we should check that these effects are similar with all three methods.
The alternative technique is to change the input gravity residuals before the Jacobian / topo-corrections are calculated. Currently this can be done with the regional field estimation, especially with constraint point minimization. This minimize the residual gravity misfit at a-priori measurements. With this, there shouldn't be a jump at the constrained-unconstrained boundary, since the residual misfit should smoothly taper to zero at the constrained boundary. Alternatives would be to apply the weighting grid to the residual prior to each inversion, instead of to the topo-corrections. Same thing with the upper / lower confining layers. If the topo is already equal to the upper confining layer, than don't allow positive residual misfits and vice versa.