Applications Of A New, Fast Reflectivity Method To Marine Horizontal And Vertical Seismic Profiling
Authors: Tango G.J. , Schmidt H. 
Published: October 2008

ABSTRACT
Previous work has shown the value of exact-wave theoretic solutions when seeking complete synthetic horizontal and vertical seismic profiles. To date, discrete wave number reflectivity methods have not been widely used for most exploration applications because of numeric instabilities at large-arrival angles and frequencies, prohibitive run times, and difficulties in correctly implementing the algorithm for geomodeling needs. This paper considers marine seismic applications of a new, efficient and unconditionally stable algorithm for exact and complete point and line source reflectivity synthetics, based on an extension by Schmidt (1982) of the original Green's function integral transform solution of Pekeris (Ewing et al., 1957; DiNapoli and Fuchs-Mueller, 1971). Basic theory is reviewed, and several examples are considered, indicating correct guidelines for computational parameters in successful geotechnical, crustal, and exploration seismic simulations.

INTRODUCTION
Over the past 10 years, high-resolution seismic profiling has proven an increasingly valuable tool in marine exploration and production evaluation, as well as in deep crustal marine reconnaissance (Tango, 1981; Balch and Lee, 1984; Stephen, 1977). The technology for theoretically understanding broadband seismo-acoustic propagation to (nb) vertical arrays in fluid or solid media has developed rapidly in recent years. Many of the representative numerical classes of synthetic seismic algorithms have been presented recently in the advanced monograph by Toksoz and Stewart (1984), as well as in several industry discussions. Of all the competing ray: mode, brute force, and wave theoretic methods, the Green's function-integral transform technique-of which the reflectivity/discrete wave number and FFP techniques of Fuchs-Muller (1971), Kennett-Stephen (1977), and Bouchon-Apsel (1979) are examples-is currently experiencing a particular resurgence, due to availability of fast memory hardware and attached processor software. Unfortunately, apparently little discussion has been given to the crucial intermediate stage between mathematical theory and experimental application, i.e., the specific numerical computational requirements inevitably necessary in any synthetic wave propagation program.
Computational, experimental, and environmental sampling requirements must be particularly considered for FFP reflectivity/discrete wave number VSP algorithms. In this case, sampling, truncation/smoothing, and discretization play critical roles in determining mathematical correctness and completeness, as well as environmental and experimental validity of the solution. In numerous published cases (and more frequently in unpublished attempts), computational errors in full-wave methods can lead to uninterpretable, and even unphysical results, including (more dangerously) apparently plausible but in fact a mixture of partly or wholly fictitious artifacts: Repltitl0n of these-errors has often led to curtailment or abandonment of a method otherwise efficient and exact for the appropriate class of problems.
This paper outlines theoretical and applications advantages of a new class of reflectivity-discrete wave number-FFP algorithms. We present an encapsulized review of the new direct-global-matrix (DGM) algorithm in the context of previous theory; selected computational requirements that follow from theory for correct algorithm use in the context of three applications examples from marine crustal, exploration, and geotechnical seismology.