Wide‐angle elastic wave one‐way propagation in heterogeneous media and an elastic wave complex‐screen method

Abstract

In this paper a system of equations for wide‐angle one‐way elastic wave propagation in arbitrarily heterogeneous media is formulated in both the space and wavenumber domains using elastic Rayleigh integrals and local elastic Born scattering theory. The wavenumber domain formulation leads to compact solutions to one‐way propagation and scattering problems. It is shown that wide‐angle scattering in heterogeneous elastic media cannot be formulated as passage through regular phase‐screens, since the interaction between the incident wavefield and the heterogeneities is not local in both the space domain and the wavenumber domain. Our more generally valid formulation is called the “thin‐slab” formulation. After applying the small‐angle approximation, the thin‐slab effect degenerates to that of an elastic complex‐screen (or “generalized phase‐screen”). Compared with scalar phase‐screen, the elastic complex‐screen has the following features. (1) For P‐P scattering and S‐S in‐plane scattering, the elastic complex‐screen acts as two separate scalar phase‐screens for P and S waves respectively. The phase distortions are determined by the P and S wave velocity perturbations respectively. (2) For P‐S and S‐P conversions, the screen is no longer a pure phase‐screen and becomes complex (with both phase and amplitude terms); both conversions are determined by the shear wave velocity perturbation and the shear modulus perturbation. For Poisson solids the S wave velocity perturbation plays a major role. In the special case of α0 = 2β0, S wave velocity perturbation becomes the only factor for both conversions. (3) For the cross‐coupling between in‐plane S waves and off‐plane S waves, only the shear modulus perturbation δμ has influence in the thin‐slab formulation. For the complex‐screen method the cross‐coupling term is neglected because it is a higher order small quantity for small‐angle scattering. Relative to prior derivations of vector phase‐screen method, our method can correctly treat the conversion between P and S waves and the cross‐coupling between differently polarized S waves. A comparison with solutions from three‐dimensional finite difference and exact solutions using eigenfunction expansion is made for two special cases. One is for a solid sphere with only P velocity perturbation; the other is with only S velocity perturbation. The Elastic complex‐screen method generally agrees well with the three‐dimensional finite difference method and the exact solutions. In the limiting case of scalar waves, the derivation in this paper leads to a more generally valid new method, namely, a scalar thin‐slab method. When making the small‐angle approximation to the interaction term while keeping the propagation term unchanged, the thin‐slab method approaches the currently available scalar wide‐angle phase‐screen method.