Velocity determination from offset VSP data: Tests of the method

Abstract

A previously reported iterative generalized inversion to determine interval velocities from vertical seismic profiling (VSP) data is tested with respect to the uniqueness of the solution and the selection of the optimal value for a damping parameter. The model used here consists of flat, horizontal layers. For error‐free data the solution for the velocities of any two adjacent layers is definitely nonunique if there is no travel time information for those layers. For these layers, the solution depends strongly on the initial estimates. If the same initial values are used for several adjacent layers without travel times, the final velocities differ from the initial ones by a constant amount and give an approximate rms velocity. The effect of errors is analyzed in connection with the optimal value for the damping parameter λ. During the initial stages of the inversion the effect of λ is to assure the convergence of an iterative solution to a nonlinear problem. As the iterations proceed, the value of λ is decreased, and at a certain stage the problem may be considered linear. At this point the selection of the optimal value of λ becomes critical because it determines the effect of errors in the final velocity structure. When errors are added to synthetic travel times, the effect of λ is seen clearly: a value of λ that is too small introduces large errors in the velocities; a value of λ that is too large reduces the resolution of the velocity structure. In general, there is no objective way to determine the optimal λ that will minimize the effect of errors on velocities. However, the differences between synthetic times and times computed for the inversion velocities remain almost the same for λ below a certain threshold. This gives an indication of the best value of λ consistent with the observations when they are contaminated by errors. When dealing with actual data there are two additional pieces of evidence that can be used to determine the optimal λ: the velocities obtained from the sonic logs, and the comparison between the actual VSP and the synthetic VSPs computed for different values of λ. The optimal λ should be the one that gives the best agreement between actual and synthetic VSPs. Both the location of the reflections and their relative amplitudes should be taken into account in the comparison.