Inverse scattering series and seismic exploration

Abstract

This paper presents an overview and a detailed description of the key logic steps and mathematical-physics framework behind the development of practical algorithms for seismic exploration derived from the inverse scattering series. There are both significant symmetries and critical subtle differences between the forward scattering series construction and the inverse scattering series processing of seismic events. These similarities and differences help explain the efficiency and effectiveness of different inversion objectives. The inverse series performs all of the tasks associated with inversion using the entire wavefield recorded on the measurement surface as input. However, certain terms in the series act as though only one specific task,and no other task, existed. When isolated, these terms constitute a task-specific subseries. We present both the rationale for seeking and methods of identifying uncoupled task-specific subseries that accomplish: (1) free-surface multiple removal; (2) internal multiple attenuation; (3) imaging primaries at depth; and (4) inverting for earth material properties. A combination of forward series analogues and physical intuition is employed to locate those subseries. We show that the sum of the four taskspecific subseries does not correspond to the original inverse series since terms with coupled tasks are never considered or computed. Isolated tasks are accomplished sequentially and, after each is achieved, the problem is restarted as though that isolated task had never existed. This strategy avoids choosing portions of the series, at any stage, that correspond to a combination of tasks,i.e.,

Target-enclosed seismic imaging

“…The required wavefields may also be computed by evaluating an inverse scattering series (Weglein et al, 2003) or a Bremmer series (Malcolm et al, 2009;Davydenko and Verschuur, 2016). Alternatively, we can estimate the required Green's functions directly from the reflection data that are acquired at the earth's surface by solving a multidimensional Marchenko equation (Broggini et al, 2012;Slob et al, 2014;Wapenaar et al, 2014;da Costa Filho et al, 2015;Singh et al, 2015).…”

Nonlinear inverse scattering and three-dimensional near-field optical imaging

“…[9][10][11] Though the algebraic structure of series solutions to different ISPs is similar, their analytic structure and convergence properties are quite different. This is a reflection of the underlying physical difference between the shortrange propagation of evanescent waves in NFT and the longrange propagation of waves in quantum mechanics or acoustics.…”

Scattering of acoustic waves by inhomogeneous obstacles

“…While integral formulations through a Fredholm equation are known in the literature (see, e.g. [1]), a formulation through a Volterra equation is new for the problem at hand. The new formulation is obtained by means of an appropriate fundamental solution.…”