The Design of 2-D Explicit Depth Extrapolators Using the Cauchy Norm

In this paper, we present a novel method for designing explicit wavefield extrapolators to perform both prestack and poststack depth migrations of seismic data. This method is achieved through the design of finite impulse response filters that perform the depth migration in the frequency-space ($F-X$) domain. The design method works on finding the regularized least square solution for the filter impulse response by using the Cauchy norm as the regularization cost function. This cost function offers an adaptive damping on the approximated filter impulse response coefficients yielding a more accurate approximation of the filter coefficients. It also insures the stability of the recursive migration process through which the designed filters will be used and prevent both over shooting and dampening of the wavenumber response of the filters designed. To test the designed filters, we then conducted poststack depth migration to the well-known SEG/EAGE salt model zero-offset data set, where subsalt structures were accurately migrated. The filters were also designed and used to perform prestack migration on the challenging Marmousi model data set successfully. The obtained results indicate that these filters can be used to perform stable and efficient prestack and poststack explicit wavefield extrapolations.