Homotopy analysis of the Lippmann–Schwinger equation for seismic wavefield modelling in strongly scattering media

Abstract

SUMMARY We present an application of the homotopy analysis method for solving the integral equations of the Lippmann–Schwinger type, which occurs frequently in acoustic and seismic scattering theory. In this method, a series solution is created which is guaranteed to converge independent of the scattering potential. This series solution differs from the conventional Born series because it contains two auxiliary parameters ϵ and h and an operator H that can be selected freely in order to control the convergence properties of the scattering series. The ϵ-parameter which controls the degree of dissipation in the reference medium (that makes the wavefield updates localized in space) is known from the so-called convergent Born series theory; but its use in conjunction with the homotopy analysis method represents a novel feature of this work. By using H = I (where I is the identity operator) and varying the convergence control parameters h and ϵ, we obtain a family of scattering series which reduces to the conventional Born series when h = −1 and ϵ = 0. By using H = γ where γ is a particular pre-conditioner and varying the convergence control parameters h and ϵ, we obtain another family of scattering series which reduces to the so-called convergent Born series when h = −1 and ϵ ≥ ϵc where ϵc is a critical dissipation parameter depending on the largest value of the scattering potential. This means that we have developed a kind of unified scattering series theory that includes the conventional and convergent Born series as special cases. By performing a series of 12 numerical experiments with a strongly scattering medium, we illustrate the effects of varying the (ϵ, h, H)-parameters on the convergence properties of the new homotopy scattering series. By using (ϵ, h, H) = (0.5, −0.8, I) we obtain a new scattering series that converges significantly faster than the convergent Born series. The use of a non-zero dissipation parameter ϵ seems to improve on the convergence properties of any scattering series, but one can now relax on the requirement ϵ ≥ ϵc from the convergent Born series theory, provided that a suitable value of the convergence control parameter h and operator H is used.