Inverse scattering sub-series direct removal of multiples and depth imaging and inversion of primaries without subsurface information: strategy and recent advances A. B. Weglein 1 , L. Amundsen 2 , F. Liu 1 , K. Innanen 1 , B. Nita 3 , J. Zhang 1 , A. Ramirez 1 and E. Otnes 2 1 Dept. of Physics, U. Houston, M-OSRP; 2 Statoil Research Center; 3 Dept. of Math, Montclair State University SUMMARY This paper provides: (1) a review of the logic and promise behind the isolated task inverse-scattering sub-series concept for achieving all processing objectives directly in terms of data only, without knowing or determining or estimating the properties that govern wave propaga- tion in the actual earth; (2) the recognition that an effective response to pressing seismic challenges requires understanding that those chal- lenges arise when assumptions, and prerequisites behind current lead- ing edge seismic processing are not satisfied and that those failures can be attributed to: (A) insufficient acquisition, and/or (B) compute power and (C) bumping into algorithmic limitations and assumptions, and (3) the status and plans of the inverse series campaign to address the fundamental algorithmic limitations of processing methods, that are not addressed by more complete acquisition and faster computers. BACKGROUND Scattering theory is a form of perturbation theory. It relates a perturba- tion in the properties of a medium to the concomitant perturbation in the wave field. L 0 G 0 = δ and LG = δ represent the equations govern- ing wave propagation in the unperturbed and perturbed media where L 0 and G 0 , and L and G are the unperturbed and perturbed differential operators and Green’s functions, respectively. V = L 0 − L is the per- turbation operator and ψ = G − G 0 , is the scattered field, and is the difference between the unperturbed and perturbed medium’s Green’s functions. In our seismic application the unperturbed medium is called the reference medium, and will be chosen (in this paper) for the ma- rine case to be water. The perturbed medium in our marine context is the actual earth and the domain of the perturbation, V , the difference between earth properties and water, begins at the water bottom. Scattering equation The relationship between G, G 0 , and V is given by an operator iden- tity called the scattering equation or the Lippmann-Schwinger (LS) equation (Goldberger and Watson, 2004; Taylor, 1972; Joachain, 1975, e.g.), G = G 0 + G 0 VG. (1) In general, the forward problem predicts the wave field from the prop- erties of the medium, and the forward problem in scattering theory predicts the wave field from the medium in terms of L, not directly but in a perturbative sense, from L in terms of L 0 and V through G 0 and V . Forward series From (1) a forward scattering series can be written formally as G =(1/(1 − G 0 V ))G 0 = G 0 + G 0 VG 0 + ..., (2) and using ψ = G − G 0 , equation (2) becomes: ψ = G 0 VG 0 + G 0 VG 0 VG 0 + ... = ψ 1 + ψ 2 + ψ 3 + ..., (3) where ψ n is the portion of ψ n’th order in V . In general, the inverse problem is to determine actual medium properties, contained in L, from measurements outside the support of the medium to be identi- fied. The inverse problem in scattering theory assumes that the refer- ence medium L 0 and Green’s function G 0 are chosen and known; and, hence, the inverse problem is to determine L through determining V , the difference between L and L 0 , from measurements of ψ = G − G 0 on a measurement surface outside the support of V . The measurements of ψ = G − G 0 constitute the data, D. Inverse series The inverse scattering series produces V in terms of D, through a series V = V 1 +V 2 +V 3 + ..., (4) where V n is the portion of Vn’th order in the measured values of ψ = D. The equations for V 1 , V 2 , V 3 , ... are derived in (Weglein et al., 1997, 2003,e.g., ): G 0 V 1 G 0 = D G 0 V 2 G 0 = −G 0 V 1 G 0 V 1 G 0 G 0 V 3 G 0 = −G 0 V 1 G 0 V 1 G 0 V 1 G 0 − G 0 V 1 G 0 V 2 G 0 − G 0 V 2 G 0 V 1 G 0 , (5) etc. It is worth noting that: (1) the inverse scattering series (equations (5)) provide a general direct formalism to solve the inverse problem explicitly in terms of data, D and the water speed Green’s function; each V n is explicitly and directly calculated from G 0 and D, (2) there is no updating of the reference nor claim that the reference is proximal to the actual nor any attempt nor need nor interest in moving it towards the actual, and hence the methodology is not e.g., in any way related to nor shares the properties of iterative linear inverse, and (3) the inverse series equations (5) do not in any sense represent the Born approxi- mation, and e.g., the first equation in (5) is the exact equation for V 1 , the second equation in (5) is the exact equation for V 2 , and V 1 is never assumed to be an approximate to V , but rather the linear estimate to V , and the first equation in (5) is the exact relationship for the linear estimate and the data. Hence, equations (5) do not depend upon or launch from an assumption on the linear estimate, they in fact launch from the exact equation for the linear estimate; and hence the inverse series doesn’t begin with an approximation; (4) at every term in equa- tions (5) there is only a single and repeated inverse step of inverting G 0 , on the left hand member, and in every step in the series, which for water speed and Fourier transforms becomes a simple multiplica- tive algebraic and stable operation, essentially a single Stolt FK pre- stack migration at water speed is the only inverse step; the complexity comes from multiplying factors involving the data, D, and the water speed Green’s function on the right hand side of equation (5), and multiplying water speed Green’s functions and data, is not comparable in terms of treacherous numerical and computational challenge, and stability issues to inverting an updated variable background Green’s function, and (5) there is no optimization, no searching algorithm, no invariance such as flat common image gathers, no proxy or surrogate for the actual velocity, nor any other subsurface property, no optimal stacking nor searching for optimal weighted move-out patterns, but in- stead an explicit set of equations for V . It is multi-dimensional, fully non-linear and direct inversion. Equations (5) use the information in the data, D, the amplitude and phase of events in specific distinct task determined linear and non-linear combination, to achieve processing objectives where traditional linear processing methods required sub- surface information, to then allow those goals to be realized without the traditional need for subsurface information. In addition, the right hand sides of equation (5) provides a transparency and a unique window to look into the inner workings of the inverse 2456 SEG/San Antonio 2007 Annual Meeting