Wave Theoretic Approaches to Multiple Attenuation: Concepts, Status, Open Issues, and Plans: Part II
Authors: Weglein A.B. , Matson K.H. , Berkhout A.J. 
Published: October 2008

Abstract
The inverse scattering series represents the only direct multidimensional inversion procedure. The directness of the method (towards a single objective) implies a purposefulness and focus. If the objective is viewed as being achieved through an ordered sequence of steps, we can then imagine that these steps themselves reside in the algorithm. The logic behind the resulting free-surface and internal multiple attenuation algorithms is revisited and an informal comparison with the evolution of the feedback method is presented. The inverse scattering multiple attenuation algorithms are illustrated using field-data examples.

Introduction
The inverse scattering method for attenuating free-surface and internal multiples (Ref. 1, Ref. 2, Ref. 3) provides a unique set of algorithms for the removal of all free-surface and internal multiples with absolutely no subsurface information, interpretive intervention, iteration, updating, muting, or velocity or event picking. These algorithms derive from identifying terms (and portions of terms) of the multidimensional inverse series for seismic data (Ref. 4) that carry out specific tasks, within the overall inversion process, in a purposeful and direct manner. This concept of associating certain terms (and subseries) with task-separated inverse processes allows great benefit to derive from reaching one (or more) of these goals under circumstances when all of these objectives are not achievable. Further, the fact that each term has a well-defined specific function, within this four distinct task separated inversion framework, allows the prediction of the effect of different portions of the series - independent of the nature of the target. For example, the individual terms in the free-surface demultiple subseries each eliminate a different specific order of free-surface multiple - completely and totally independent of the nature of the earth. These terms carry out their assigned purpose not only independent of the nature of the earth's structure and lithology, but also independent of whether the earth is acoustic, elastic or anelastic.
A recent set of papers (Ref. 5, Ref. 6) provided synthetic data tests as an empirical comparison of these inverse scattering free-surface and internal multiple methods and the feedback method pioneered by Berkhout (Ref. 7) and developed by Verschuur et al. (Ref. 8). References (5) and (6) are comparison papers and mainly consist of numerical and synthetic data examples. One objective of the current paper is to continue this analysis and synthesis.

Scattering theory
Scattering theory is a form of perturbation theory. It relates the actual impulse response, G, and the reference impulse response, G0, to the difference between the actual and reference media, which is characterized by the operator, V. G0 and G satisfy the differential equations
Mathematical equations (1) and (2) (Available in full paper)
where L0, L are the differential operators describing reference and actual propagation, ? represents an impulsive source, and
Mathematical equation (3) (Available in full paper)
The fundamental relationship between G, G0 and V is
Mathematical equation (4) (Available in full paper)
The forward problem starts with G0 and V and produces G; the inverse problem starts with G0 and measurements of G (on a surface outside of V) to determine V.