Practical aspects of reflectivity modeling

This paper is intended to help those not familiar with the “lore” of layered earth modeling to avoid some common problems. In the computation of the reflectivity function, an easily incorporated phase‐integral approximation is used away from turning points when the velocity gradient is smaller than the frequency. Hanning windows, or segments thereof, work well for both the slowness integral and the frequency integral. For the quadrature of the slowness integral the Filon method of Frazer is easily coded and vectorizes well; Levin’s Filon method and the Clenshaw‐Curtis‐Filon method of Xu and Mal are more difficult to vectorize, but more powerful because they require fewer evaluations of the reflectivity function. A modification of Strick’s power law is a convenient way to calculate complex frequency‐dependent seismic velocities. The complex frequency technique for avoiding time aliasing is explained by use of the Poisson sum formula. In writing code for vector computers, such as the CRAY, if frequency‐independent velocities are used, the frequency loop should be deepest, whereas if frequency‐dependent velocities are used, then the p‐loops should be deepest.