NUMERICAL VERIFICATION AND EXTENSION OF AN ANALYTIC GENERALIZED INVERSE FOR COMMON-DEPTH-POINT AND VERTICAL-SEISMIC-PROFILE TRAVELTIME EQUATIONS.

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 the covariance matrix 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, the authors extend the analytic inverse to traveltime equations which honor Snell's law. Computer tests show that 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.