Membrane Surface Wave Synthetics: Modeling Mode-Mode Scattering

Structural heterogeneity near the earth's surface has considerable effects on the surface waves with period around 20s: they show complicated propagation effects such as curved paths, multiple rays and scattering. We formulate a method in order to reproduce these effects in synthetic surface waves, which is closely connected to an algorithm to invert the observed seismograms proposed by Tanimoto (GJI, 1990). We assume smooth lateral variations in the material parameters so that the surface wave may be expressed in terms of potential functions and local eigenfunctions (e.g., Love wave displacement is given by u_{x=W({ {x}})} Ψ _L,y(x,y), u_{y=-W({ {x}})Ψ L,x(x,y),} u_z=0). We apply the Born approximation to the equations of motion to obtain equations -ρ ω ² u_i⁰ = L_{0(Ψ ⁰⁾} + f_i and -ρ ω ² u_i¹ = L_{0(Ψ ¹⁾} + L_{1(Ψ ⁰⁾} for primary and scattered field Ψ ⁰ and Ψ ¹, where L₀ and L₁ denote the zero-th and first order terms with respect to the lateral derivatives of the elastic constants and the local eigenfunctions in τ _ij,j, respectively. The primary potential function Ψ ⁰ is the solution of 2D scalar equation ∇ ² Ψ ⁰ + k(x,y)² Ψ ⁰⁼⁰ with appropriate source terms and k(x,y) being the local wavenumber (Tanimoto 1990). The effect of scattered potential Ψ ¹ is evaluated reciprocally by u_{i^1(x_r)=∫} G_ji(ξ ; x_r) g_{j(ξ )} dV(ξ ), where G_ji(ξ ; x_r) is reciprocal Green's function for a force applied at the receiver (x_r) and g_{j(ξ )} denotes L_{1(Ψ ⁰⁾}. The reciprocal potential (Ψ _r¹) for G_ji is also the solution of the above scalar equation with proper source terms. Unlike the ray-theoretical approach, the effect of finite wavelength is naturally incorporated in our formulation. The effect of curved paths and multiple rays in a single mode is included in the primary potential Ψ ⁰, and scattering between different modes is considered by Ψ _r¹. Further, Ψ _r¹ can be used for the inversion algorithm of Tanimoto (1990) so that, after correcting for the scattering effect, we may invert the data without additionally solving the scalar equations.