Numerical verification and extension of an analytic generalized inverse for common‐depth‐point and vertical‐seismic‐profile traveltime equations

Abstract

This paper empirically establishes the range of validity for an analytic generalized inverse (assuming negligible ray bending) associated with one‐dimensional vertical seismic profiles (VSP) or common‐depth‐point (CDP) traveltime equations. Computer tests show that the analytic inverse closely predicts the condition number of [Formula: see text] and roughly predicts some features of the unit covariance matrix for a source offset‐to‐well depth ratio less than 1. The analytic inverse is invalid for offset‐to‐depth ratios greater than 1; i.e., when ray bending becomes severe enough to violate the assumption of negligible ray bending. To overcome the restriction of negligible ray bending, we extend the analytic inverse to traveltime equations which honor Snell’s law. Computer tests show this extended inverse is a good approximation to the actual generalized inverse. The extended inverse appears to be valid for any practical source‐receiver offset and any layered velocity structure. The important implication is that, prior to their execution, VSP or CDP experiments (over approximately one‐dimensional structures) can now be designed for optimal velocity resolution. No numerical inverses need to be computed and the condition number, covariance matrix, resolution matrix, and inverse matrix are closely approximated by analytic formulas. This development promises to allow accurate and efficient velocity analysis of VSP data or CDP gathers by least‐squares traveltime inversion.