A simple, exact solution for the P‐SV wave conversion point via prestack migration

Abstract

The conversion point (CP) is the lateral position of the reflection point, relative to the source-receiver midpoint, for a laterally homogeneous earth (horizontal reflectors) and the P-SV converted-wave (C-wave) geometry depicted in Figure 1. The source emits a P-wave, which propagates to the horizontal reflector where it reflects as an SV-wave and then propagates to the receiver. Common conversion point (CCP) binning and stacking are tools that use CP modeling for processing C-wave data. Applications include iterative statics and velocity analysis procedures. Acquisition design methods also use CP modeling (see Thomsen, 1999). Calculation of the CP is complicated, unfortunately, and exact solutions exist only for homogeneous and isotropic media. Thomsen (1999) developed asymptotic expansions that approximate the CP in vertically inhomogeneous isotropic and anisotropic media. Yuan and Li (2001) extended these expressions, improving their accuracy for large source-receiver offsets. Exact solutions for the CP exist only in simple media, and that is the subject of this paper. Tessmer and Behle (1988) used Fermat9s principle to find a quartic equation for the exact CP in homogeneous, isotropic media. I show that the equation for the C-wave prestack migration impulse response yields a cubic equation for the exact CP in homogeneous, isotropic media. Analytically solving the cubic equation is less cumbersome than analytically solving the quartic equation, and I have obtained the exact solution of the new cubic equation in simple trigonometric form. The Tessmer and Behle (1998) quartic CP equation depends upon the reflector depth, whereas the new cubic CP equation depends instead upon the two-way reflection traveltime of the C-wave. Thus, the new result directly specifies the CP for each sample of a prestack seismic trace (assuming reflectors are horizontal).