Implicit structural inversion for lithology using a gridded model

The aim of this PhD project was to develop a method for implicit structural inversion of geophysical data, using a gridded model. With the term {\em structural inversion} we mean an inversion to obtain an image of the (gridded) sub-surface, which can directly be interpreted in terms of sharp boundaries between different lithologies. {\em Implicit} indicates that we neither assume a fixed number of anomalies, nor enforce a fixed structure explicitly. We model the sub-surface with a grid of homogeneous, rectangular prisms. The values of the physical property of these cells are the parameters inverted for. The structures in the model should be apparent from large contrasts in parameter values between neighboring cells. We developed a method for such an implicit structural inversion, using Linear Programming (LP). All the models investigated comprised of two lithologies, with one or more homogeneous anomalous regions embedded within a homogeneous background. The contrast between rock properties is used a constraint on the parameters. Ideally each cell will ultimately be identifiable as belonging to one of the lithologies. This LP-based method (using the L1-norm data misfit) was tested on synthetic gravity data and was able to reconstruct important features of the test models. The results compared favorably with a traditional inversion using Truncated Singular Values. This method does have the limitation that the minimum depth to the top needs to be known a priori. This method was then extended to simultaneously invert for both a linear trend in the data and for the density contrasts of the cells. The method was used to invert a field gravity data set. The results are in agreement with the ones obtained by the industry using detailed forward modeling. The same LP-based method was applied to invert synthetic seismic cross-well first arrival travel-times for the absolute inter-well slowness distribution. This is a non-linear problem due to ray-bending in a heterogeneous medium. For a simple sub-surface model, the implicit structural inversion using the true rays, or using the iterative scheme, both produced very good models of the sub-surface. However, a regularisation parameter needs to be chosen properly. A more complicated model with poor ray-coverage, was difficult to retrieve using only seismic data, however, a joint inversion of seismic data (using the true rays) together with gravity data gave encouraging results. Another method, this time using the L2-norm of the data misfit, was developed in parallel. In an iterative scheme, a reference model was constructed, using information regarding the density contrasts, which the inversion result should resemble. The method needs 5 tuning constants; when proper (wide) ranges are supplied, a parameter search for the constants, yields their optimal values to be used for the inversion. This search can be steered using the resulting data misfits. One of the tuning parameter is dynamically increased during the iterations to ensure a result with clear structure. Experiments on synthetic gravity data gave good inversion results, even when the top of the anomaly was unknown.