Kristopher A. Innanen University of Houston

1. Non-linear forward scattering series expressions for reflected-primary and transmitted-direct wavefield events Kristopher A. Innanen University of Houston M-OSRP Annual Meeting 11 May, 2006 University of Houston

2. Acknowledgments Arthur Weglein, Sam Kaplan M-OSRP sponsors and personnel CDSST sponsors and personnel

3. Plan 1. Introduction: what might we accomplish? 2. Technical Primer : a 1D forward scattering series framework for approximating primaries 3. Reflected primaries and transmitted direct waves due to 3D perturbations 4. Recursive forms and R/T relations 5. A “non-linear overburden, linear diffractor” hybrid approximation 6. Why the FSS? a. Modelling b. Direct inversion of FSS approximations

4. Introduction The forward scattering series (FSS) is an infinite series expressing one (“actual”) wavefield in terms of another (“reference”) wavefield and a perturbation operator describing the difference between the two. Anything less than the full series is an approximation. The current job is to consider the meaning, validity, and value of wavefield approximations based on the FSS.

5. Introduction The forward scattering series (FSS) is an infinite series expressing one (“actual”) wavefield in terms of another (“reference”) wavefield and a perturbation operator describing the difference between the two. Anything less than the full series is an approximation. The current job is to consider the meaning, validity, and value of wavefield approximations based on the FSS. linear non-linear

6. Introduction The linear term, if used to model the wavefield, must give an account of itself with respect to two kinds of non-linearity that might well be present… source receivers reference medium perturbation

7. Introduction …First, it assumes that the wavefield amplitude is linear in V, which is OK if V2 is small… source receivers reference medium perturbation

8. Introduction …and second, it assumes that very little propagation takes place in structure that is not incorporated into the reference medium. source receivers reference medium perturbation

9. Introduction Confounding the linear approximation is as simple as including large, spatially sustained structure in the perturbation. source receivers perturbation large contrast perturbation

10. Introduction The FSS is not finished at this point, but the linear approximation is. Most importantly: The prescription, or map, linking a known set of terms of the FSS to a known subset of the events of a seismic reflection data set, is gone. That is, the activity of the FSS in creating a primary has been relegated to an a priori unknown set of mathematical operations. Can we re-provision ourselves with this map? Create a prescription for the approximation of meaningful wavefield events given large and spatially sustained perturbations?

11. Technical primer: 1D primary approximation Wavefield events due to unincorporated medium structure (“unincorporated” meaning not in the reference medium) that varies in 3D is the goal. The route taken to model these – one of many possible routes – arises as a natural extension of 1D. Take care of some detail in 1D and then move on to multi-D. The concepts are very similar.

12. Technical primer: 1D primary approximation Elements of the 1D theory

13. Technical primer: 1D primary approximation Elements of the 1D theory

14. Technical primer: 1D primary approximation Elements of the 1D theory

15. Technical primer: 1D primary approximation Elements of the 1D theory source

16. Technical primer: 1D primary approximation Elements of the 1D theory multiply the source by G0 and integrate…

17. Technical primer: 1D primary approximation Elements of the 1D theory

18. Technical primer: 1D primary approximation Elements of the 1D theory Lippmann-Schwinger Born series

19. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

20. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

21. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

22. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

23. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

24. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

25. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

26. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

27. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

28. Technical primer: 1D primary approximation Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

29. Technical primer: 1D primary approximation If we compute enough of these terms, we will have created an approximation of the wavefield that – if convergent – is not sensitive to the contrast or spatial extent of the perturbation. But the full wavefield contains primaries, multiples etc. To use this formalism to create an approximation of primaries only, the reference propagation/scattering framework needs a closer look… In particular, the explicit form of the Green’s function and its depth dependence is of interest.

30. Technical primer: 1D primary approximation The solution to the equation is a causal Green’s function, stipulating that the wave must propagate away from zs.

31. Technical primer: 1D primary approximation Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term…

32. Technical primer: 1D primary approximation Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term…

33. Technical primer: 1D primary approximation Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term… The first thing to notice is that if we want to model a reflection experiment, we can lose the |.| bars on the leftmost and rightmost Green’s functions…

34. Technical primer: 1D primary approximation Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term… Since this cares about whether or not z’’>z’’’ The stipulation that waves propagate only away from the source has consequences because the scatterers are acting like sources… in other words the scattering locations z’, z’’, z’’’ etc. now have to deal with the causal nature of the G0s…

35. Technical primer: 1D primary approximation Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term… …we now need to care whether one  happened above or below another… The stipulation that waves propagate only away from the source has consequences because the scatterers are acting like sources… in other words the scattering locations z’, z’’, z’’’ etc. now have to deal with the causal nature of the G0s…

36. Technical primer: 1D primary approximation We evaluate these integrals by allowing the absolute value bars to generate “cases”, which hold for portions of the range of integration. For instance: Each instance of an “interior” Green’s function results in the generation of two (additive) cases.

37. Technical primer: 1D primary approximation We evaluate these integrals by allowing the absolute value bars to generate “cases”, which hold for portions of the range of integration. For instance: Each instance of an “interior” Green’s function results in the generation of two (additive) cases. …which means the n’th order term breaks into 2n-1 raw subterms.

38. s r z’’’ z’’ z’ Technical primer: 1D primary approximation So the one third order term becomes 4 subterms: s r z’’ z’’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Each term representing a permutation of all four possible “scattering geometries”, i.e., all four ways the scatterers can orient themselves in depth. These are representable with diagrams.

39. s r z’’’ z’’ s r z’ z’’ z’’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Technical primer: 1D primary approximation So the one third order term becomes 4 subterms: Each term representing a permutation of all four possible “scattering geometries”, i.e., all four ways the scatterers can orient themselves in depth. These are representable with diagrams.

40. s r z’’’ z’’ s r z’ z’’ z’’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Technical primer: 1D primary approximation The significance of these terms is … what? They themselves are very poor approximations of any part of the wavefield.

41. s r z’’’ z’’ s r z’ z’’ z’’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Technical primer: 1D primary approximation The significance of these terms is … what? They themselves are very poor approximations of any part of the wavefield. This is not a multiple…

42. s r z’’’ z’’ s r z’ z’’ z’’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Technical primer: 1D primary approximation But the diagram is still meaningful: to some level of approximation, the FSS tends to construct N’th order events as infinite series expansions about “seeding points”, which appear at N’th order. So, for instance, a multiple with a single downward reflection (a 3rd order event) is an expansion around the indicated diagram. Weglein, Araujo, early 1990’s

43. s r z’’’ z’’ s r z’ z’’ z’’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Technical primer: 1D primary approximation Following this logic, a framework for the approximation of primaries would involve capturing and summing all terms that are high order extensions of a primary “seeding point”. If we choose the linear term (with its single direction change) as the seeding point, all subsequent V shaped diagrams are part of the approximation. Weglein, Foster, et al., late 1990’s.

44. s r z’’’ z’’ s r z’ z’’ z’’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Technical primer: 1D primary approximation Fortunately the math makes this a practical choice also. We can see this by rejecting all reverberating diagrams (as seeding points for multiple generation), and looking at the remaining expressions.

45. s r z’’’ z’’ z’ s r z’ z’’’ z’’ s r z’ z’’ z’’’ Technical primer: 1D primary approximation Fortunately the math makes this a practical choice also. We can see this by rejecting all reverberating diagrams (as seeding points for multiple generation), and looking at the remaining expressions.

46. Technical primer: 1D primary approximation Fortunately the math makes this a practical choice also. By using a straightforward property of nested integrals, these three expressions can be seen to be basically equal…

47. Technical primer: 1D primary approximation Fortunately the math makes this a practical choice also. By using a straightforward property of nested integrals, these three expressions can be seen to be basically equal…

48. Technical primer: 1D primary approximation Fortunately the math makes this a practical choice also. By using a straightforward property of nested integrals, these three expressions can be seen to be basically equal…

49. s r z’’’ z’’ z’ Technical primer: 1D primary approximation This 1D case simplifies drastically, but let’s hold off on that. transmission transmission First let’s flag a couple of things to aid in interpretation.

50. s r z’’’ z’’ z’ Technical primer: 1D primary approximation This 1D case simplifies drastically, but let’s hold off on that. scattering interaction w direction change scattering interaction w direction change First let’s flag a couple of things to aid in interpretation.