Algorithm comparisons for nonstationary phase shift

Robert James Ferguson, Gary F. Margrave

We explore the practical aspects of wavefield extrapolation by nonstationary phaseshift (NSPS), specifically runtime. We examine each of the three NSPS domains, space, dual and Fourier, to find which is the most computationally efficient. Ouranalysis shows that the dual domain is faster than in either the space or Fourier domains for equal operator lengths.

When operator lengths are reduced, that is when the input velocity field issmoothed, the computational effort in the Fourier domain can be greatly reduced. We demonstrate that, for a typical 2-D experiment, significant runtime improvements(8:1) are obtainable when the velocity field is desampled from an interval of 10 m to 160 m (16:1).

A high level programming language (MATLAB) was used to create the prototype of the NSPS extrapolator. Coding a particularly long running program step in C resulted in no run time advantages, though small improvements have been observed for very small data sets (128 traces, 128 samples/trace).

Approximate square root and exponent functions were examined for accuracy and potential for runtime improvement. A 45-degree solution for the square root and the bilinear transform for the exponent were used. Both of these approximations returned disappointing runtimes and, particularly for the bilinear transform, they are prone to error. Error for the square root approximation results from the need to limit scatteringangles to less than 45-degrees. Error for the bilinear transform results in incorrect shifting of the higher frequencies (>20 Hz).