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

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.