Model parametrization in seismic tomography: a choice of consequence for the solution quality

Abstract

To better assess quality of three-dimensional (3-D) tomographic images and to better define possible improvements to tomographic inversion procedures, one must consider not only data quality and numerical precision of forward and inverse solvers but also appropriateness of model parametrization and display of results. The quality of the forward solution, in particular, strongly depends on parametrization of the velocity field and is of great importance both for calculation of travel times and partial derivatives that characterize the inverse problem. To achieve a quality in model parametrization appropriate to high-precision forward and inverse algorithms and to high-quality data, we propose a three-grid approach encompassing a seismic, a forward, and an inversion grid. The seismic grid is set up in such a way that it may appropriately account for the highest resolution capability (i.e. optimal data) in the data set and that the 3-D velocity structure is adequately represented to the smallest resolvable detail apriori known to exist in real earth structure. Generally, the seismic grid is of uneven grid spacing and it provides the basis for later display and interpretation. The numerical grid allows a numerically stable computation of travel times and partial derivatives. Its specifications are defined by the individual forward solver and it might vary for different numerical techniques. The inversion grid is based on the seismic grid but must be large enough to guarantee uniform and fair resolution in most areas. For optimal data sets the inversion grid may eventually equal the seismic grid but in reality, the spacing of this grid will depend on the illumination qualities of our data set (ray sampling) and on the maximum matrix size we can invert for. The use of the three-grid approach in seismic tomography allows to adequately and evenly account for characteristics of forward and inverse solution algorithms, apriori knowledge of earth’s structure, and resolution capability of available data set. This results in possibly more accurate and certainly in more reliable tomographic images since the inversion process may be well-tuned to the particular application and since the three-grid approach allows better assessment of solution quality.