Approximation of P-, S1-, and S2-wave reflection coefficients for orthorhombic media

Abstract

Orthorhombic (ORT) media are commonly used in the descriptions of subsurface strata during the exploration and exploitation of unconventional hydrocarbon reservoirs. In anisotropic media, seismic waves propagating in ORT media usually occur as quasi-P, quasi-S1, and quasi-S2 waves. Approximating the phase velocities and reflection coefficients of these waves is of great significance for predicting the properties of unconventional reservoirs, such as their elastic parameters, fracture azimuths, and fracture densities. At the incidence angles of P waves, we have developed formulas to approximate the reflection coefficients of quasi-P, quasi-S1, and quasi-S2 waves based on their first-order phase velocities and polarization vectors. Under weak anisotropy and weak impedance contrast assumptions, in the [ X, Z] and [ Y, Z] symmetry planes, our approximations basically match Rüger’s approximations at small-to-moderate incidence angles. Numerical analyses verify that our approximations are accurate at all azimuth angles under high impedance contrast and strong anisotropy. Two important innovations are provided in our manuscript. First, the errors in the proposed reflection coefficients only lie in the first-order phase velocities and polarization vectors. Second, we use lower-upper factorization to decompose matrix-dependent reflection coefficients into multiple simple nested formulas, which can be used to calculate the reflection coefficients without matrix operations.