Discussion

We have developed an algorithm to simultaneously invert VSP and seismic time/depth pairs for interval velocity, in cases when the VSP data contains random errors and the seismic data contains both random and correlated errors. Our algorithm utilizes a nonlinear iteration, where we decouple estimation of the correlated depth error and velocity. This decoupling is in general a risky strategy, and although our results are reasonable, we should explore alternatives.

Extension of the algorithm to 2-D and 3-D is the next important issue. Modern wells are deviated, and many operators expend considerable resources to acquire ``source-over-receiver'' VSP data, in order to ensure vertical raypaths Noponen (1995). In this scenario, the subsurface is sparsely sampled by the VSP at any given spatial location. The sparsity is the main reason why we parameterize the interval velocity as a piecewise linear function. Layer-constant velocities are even less parameter-intensive.

We assumed that the errors in VSP data are random. This is likely untrue in practice, but more research is needed. In the future, we may attempt to estimate correlated error in all data sources with the nonlinear iteration. We would also like to incorporate additional data types into our algorithm, such as well logs.