Approximating the P-S reflection coefficient for small incidence angles

Alexandru Vant and R. James Brown


We derive an approximation to the Zoeppritz equations for the converted-wave reflection coefficient, RPS, given by:


where α, β, ρ, μ and i denote P-wave velocity, S-wave velocity, density, shear modulus, and P-wave angle of incidence, respectively; Δρ = ρ 2 - ρ 1 , Δμ = μ 2 - μ 1 , and subscripts 1 (upper) and 2 (lower) denote the two homogeneous isotropic media.

The approximation is made assuming small angles of incidence rather then small changes in the elastic parameters. The initial goal of this research is to derive a simple expression for the P-SV reflection coefficient. By doing that we hope to obtain simple relations that govern the change in polarity in the case of converted P-SV waves.

Using this simplified expression, we have investigated the conditions under which RPP and RPS have the same sign (the unusual situation) because this will mean that features that should be correlated on P-P and P-S sections will have opposite polarity. We obtained the following necessary conditions for this to occur: either Δρ > 0,Δβ < 0 and Δμ < 0, or Δρ < 0,Δβ > 0 and Δμ > 0.

We also reviewed other approximations to the Zoeppritz equations. From them we chose to code the expressions derived by Aki and Richards and by Wang. The accuracy of the new approximation was then tested by comparing its results with the results of these previous approximations for three interface models. For these three models, the new approximation proved to be the most accurate one out to beyond 15 degrees incidence and, in some cases, beyond 30 degrees.

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