For full waveform inversion based on the inverse scattering approach, point-to-point travel times are required for a complete 3D volume. When the background medium is not homogeneous, these travel times can not easily be stored in a table, as this will require excessive memory storage. Therefore, a smart interpolation scheme is derived for creating any point-to-point travel time from a basic set of travel time realizations. This is achieved with the aid of a neural network. The problem of calculating travel times allows the use of so-called Radialbasis function (RBF) neural networks. RBF networks are a subset of so-called artificial neural networks, which have become quite effective nowadays.
In seismic migration or in full wave-form inversion, a background model is needed in which point to point travel-times have to be calculated. In migration we are only interested in travel-times from surface location to all locations in the object space. These travel-times are calculated once and stored in tables. Calculation of the traveltimes by ray-tracing in a 3D model defined by millions of grid-cells is a time consuming job. However, the time and the storage requirements are not prohibitive since only travel-times from the surface to all points in the object are needed. In full wave-form inversion, particularly in non-linear inversion, the situation is different. First of all, we do need travel-times from every pair of points (source-receiver) in the object space. Secondly, we need the travel-times repeatedly in an iterative loop. It is clear that in 3D, both the storage requirements and the computation time will be excessive. Therefore, we are looking for an approach in which any travel-time can be calculated quickly when needed, without having to resort to point to point ray-tracing in a model defined by millions of grid-cells. The suitable velocity models for fast travel-time calculations are necessarily smooth. They have to be smooth because only in smooth models are travel-times meaningful quantities for calculation of the Green’s functions needed in back-propagation and full wave-form inversion. In spite of the fact that the models are smooth, for accurate ray-tracing they still need to described by many grid-cells, but in terms of information content the models are over-sampled. The purpose of this research is to establish a more direct link between all possible point-to-point (PP) travel-times in the model and a reduced parameter set that in itself would be sufficient to describe the smooth model. Let us first get a feel for the degree of over-sampling one could expect in a smooth background model. For a typical background model, which is suitable for the WKBJ calculation of Green’s functions, and which is accurate enough to serve as a background model for migration or linear inversion, spatial variability of the velocity is described by wavelengths not shorter than approximately 400m. This means that a spatial sampling of 200m would still allow perfect reconstruction of the smooth model on any denser output grid, with the help of spatialsinc functions.
