Ray trace tomographic velocity analysis of surface seismic reflection data (1988)

Recent development of two technologies allows application of a generalized formulation of travel time inversion to very large data sets, such as the surface reflection surveys collected for oil exploration. This generalized formulation uses very small cell sizes, effectively eliminating discretization effects. Inversion of an effective continuum that has no built-in a priori constraints is what places this technique in the category of tomography. In reflection surveys, the generalized formulation investigated here treats the continuous velocity field independently from the reflector locations. The a priori assumption, common with travel time inversions in seismic exploration data, is thus not made: that the velocity field is defined as a series of layers with constant or smoothly varying velocity. This assumption restricts significant velocity variations to occur only at reflector locations. Velocity parameterized as layers is merely one of many geologic constraints that can be added optionally in tomographic inversion. The technologies that enable this generalized approach to travel time inversion are: 1) a computer program capable of tracing rays through a 2-dimensional grid of points and off reflectors with structure, and 2) iterative schemes that efficiently perform damped, constrained generalized matrix inversions over a user-specified wide eigenvalue range for very large model and data sizes. An argument is presented that a variation of Richardson's iteration is preferred to the Conjugate Gradient Iterative Method for performing the matrix inversion. With this generalized formulation, Ray Trace Tomography is a first approach to tomographic transmission analysis. Travel times and ray paths are a valid approximation to the wave equation for broad velocity variations. The method efficiently addresses the characteristics of more general but much more expensive transmission techniques. For example, Ray Trace Tomography demonstrates that an iterative application of a transmission velocity analysis technique, tomography, and a scattering reflector location technique, migration, do not necessarily converge to the optimal solution. To resolve the ambiguity between velocity-reflector depth, velocity and reflector locations must be coupled in one inversion technique. Ray Trace Tomography is able to couple the two. Using it to indicate the absolute resolution between velocity and reflector depth, we find that for certain geometries, reflector depths cannot be resolved where most recorded energy travels within 45° of vertical. Poor resolution of the velocity-reflector depth ambiguity and other problems are inherent to reflection surveys. These problems also exist for other transmission techniques and can be solved only through use of inversion constraints. Ray Trace Tomography can test constraints for possible use in other transmission techniques efficiently. Ray Trace Tomography has difficulty with non-linearities caused by some types of starting model errors, such as small-scale reflector structure. Improved performance with non-linearities is an objective we should seek in other transmission techniques. Not only is Ray Trace Tomography a useful intellectual exercise as a preliminary analysis of transmission inversion, but in many cases it is a viable technique for addressing serious problems with surface seismic reflection data. It can determine an accurate two-dimensional velocity field for migration, such as in the case of gas pockets or fault blocks. In addition, it can resolve between certain velocity and reflector ambiguities such as those occuring in the permafrost region of Alaska. As a comparatively efficient technique, Ray Trace Tomography can serve as a tool for interactive interpretation. The geologist can use the ray tracing to compare various geologic models with the data and then use the inversion to fine-tune the models. The inversion enables the geologist to formulate his geologic knowledge as constraints in the inversion. By analyzing the inversion results, the interpreter will develop an understanding of the validity of the various models and the resolution amoung them.