Abstract
SUMMARYWe 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.
Highlights
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
The -parameter which controls the degree of dissipation in the reference medium 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
Which is convenient for uncertainty estimation (Eikrem et al 2019) and (3) it is compatible with the use of domain decomposition and renormalization methods from modern physics (Jakobsen & Wu 2016, 2018).the integral equation approach can be less efficient than the differential equation approach, depending on how it is implemented (Jakobsen & Wu 2018; Malovichko et al 2018; Jakobsen et al 2019)
Summary
There exist a range of different numerical methods for seismic wavefield modelling (Carcione et al 2002), including differential equation methods (e.g., Robertsson et al 2012) and integral equation methods (Pike & Sabatier 2000; Jakobsen 2012; Oristaglio & Blok 2012; Jakobsen & Wu 2016; Malovichko et al 2018). It will be demonstrated that the HAM can be used to construct a scattering series solution of the Lippmann–Schwinger equation in the context of seismic wavefield modelling Such convergent scattering series have been developed on the basis of renormalization methods in the past (Abubakar & Habashy 2013; Osnabrugge et al 2016; Jakobsen et al 2019a), we think it is interesting to study convergence properties of the direct scattering series solution from different perspectives, since this may give us new ideas and insights that may be useful for future studies of nonlinear inverse scattering as well as direct scattering problems.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
