Stationary-phase analysis of time-shift extended imaging in a constant-velocity model

Abstract

To estimate the depth errors in a subsurface model obtained from the inversion of seismic data, the stationary-phase approximation in a two-dimensional constant-velocity model with a dipped reflector is applied to migration with a time-shift extension. This produces two asymptotic solutions: one is a straight line, and the other is a curve. If the velocity differs from the true one, a closed-form expression of the depth error follows from the depth and apparent dip of the reflector as well as the position of the amplitude peak at a non-zero time shift, where the two solutions meet and the extended migration image focuses. The results are compared to finite-frequency results from a finite-difference code. A two-dimensional synthetic example with a salt diapir illustrates how depth errors can be estimated in an inhomogeneous model after inverting the seismic data for the velocity model.