V(z) f-k modelling and resolution simulation

Gary F. Margrave

This paper combines several previous research results to create displays that illustrate the advantage of using v(z)simulations (instead of constant velocity)for acquisition design.First,the v(z)migration theory,based on nonstationary filters,is summarized.Then,nonstationary inverse filter theory is used to create a v(z) modelling theory by inverting the migration theory.For discrete data,the v(z) migration is accomplished y a matrix multiplication in the Fourier domain.Each wavenumber of the f-k spectrum of the data is multiplied y a matrix M which applies the nonstationary migration filter.The nonstationary modelling filter is a matrix M* that is the conjugate-transpose (adjoint)of the migration filter.The operator product MM* is very nearly unity.Numerical simulations show that v(z)f- k modelling produces high quality results with far fewer Fourier wrap-around artifacts than constant velocity modelling.When a modelled response is truncated in space and time (to simulate finite aperture and record length)and then migrated,the result shows the expected geometric artifacts of limited resolution.Resolution simulations with v(z)produce substantially smaller estimates of the minimum resolvable size than do constant velocity simulations.Acquisition designs to specific resolution requirements are less expensive when done with v(z)theory instead of constant-velocity theory.