High and low order, contrast, and dimensionality in direct non-linear imaging

1. High and low order, contrast, and dimensionality in direct non-linear imaging Kristopher Innanen University of Houston & University of British Columbia M-OSRP Annual Meeting 21 April, 2005 University of Houston

2. Acknowledgments Arthur Weglein, Fang Liu, Simon Shaw, Tad Ulrych M-OSRP sponsors and personnel CDSST sponsors and personnel

3. In this talk Aim 1. The non-linear algorithms that we develop for primaries are based on cascaded, or nested infinite series The name “leading order imaging subseries” is a statement about the algorithm (Shaw, 2005). Review this in preparation for similar issues in, e.g., multidimensional case.

4. In this talk Aim 2. There is a relationship between the number of levels of cascaded series we encounter in non-linear imaging/inversion and the dimensionality of the medium, e.g., 1D-2D To what extent can we duplicate, mimic, predict behavior in 2D of ISS primary processing terms with Taylor’s series analogues?

5. In this talk Aim 3. There is a relationship between “low” and “high” order in our algorithms and accommodation of “low” and “high” contrasts in the medium under investigation Return to the inverse series to find and incorporate terms that will perform the task of depth imaging (reflector location) for a wider range of medium contrasts.

6. In this talk Aim 4. The series terms – the imaging mechanisms! – become far more complicated as we extend from 1D to 2D. And yet we need to determine patterns within… Continue to look for ways to determine fast, comprehensive algorithms from infinite (and increasingly complex) series.

7. Review of leading order imaging Shaw (2003-4) considers the problem Reference medium: c0 Actual medium: c(z)

8. Review of leading order imaging Shaw (2003-4) considers the problem Reference medium: c0 Actual medium: c(z)

9. Review of leading order imaging Shaw (2003-4) considers the problem Characterizes c(z) as a perturbation and develops the extent to which the ISS: can separately accomplish a key task of the processing of primaries: locating reflectors in depth (z)

10. Review of leading order imaging Shaw (2003-4) considers the problem Characterizes c(z) as a perturbation and develops the extent to which the ISS: can separately accomplish a key task of the processing of primaries: locating reflectors in depth 1(z) (z)

11. Review of leading order imaging Finds the following: …and proceeds. Here of particular interest Shaw makes interpretive statements using a Taylor’s series analogy. Namely: LOIS is the construction of a corrective function via a cascaded Taylor’s series

12. Cascaded (or nested) series In other words, what is occurring, in the toy problem under study, is akin to a Taylor’s series for a Heaviside function: …but where in each case of z0 occurs as a further infinite series—a geometric series in the data:

13. Cascaded (or nested) series In other words, what is occurring, in the toy problem under study, is akin to a Taylor’s series for a Heaviside function: …but where in each case of z0 occurs as a further infinite series—a geometric series in the data:

14. Leading, low, and high order: 1D An issue of concept and terminology arises, that will become more important as we move to increased realism and complexity, e.g., 1D-2D.

15. Leading, low, and high order: 1D The “leading order” (or low order) part of Shaw’s imaging algorithm is in the geometric series. It is of infinite order in the overall construction of the corrective function…!

16. y z Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D:

17. y y x z z Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D:

18. y x z Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D:

19. y x z Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D:

20. y x z Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D:

21. Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D: y x z

22. Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D: y x z

23. x y z x z Leading, low, and high order: 2D Extend the Taylor’s series “model” for the imaging mechanism to 2D:

24. y x z Leading, low, and high order: 2D Imagining that the task of 2D imaging is to construct another corrective function, this time an “undulating ridge”, create another Taylor’s series:

25. y x z Leading, low, and high order: 2D Imagining that the task of 2D imaging is to construct another corrective function, this time an “undulating ridge”, create another Taylor’s series:

26. Leading, low, and high order: 2D Imagining that the task of 2D imaging is to construct another corrective function, this time an “undulating ridge”, create another Taylor’s series:

27. Leading, low, and high order: 2D Imagining that the task of 2D imaging is to construct another corrective function, this time an “undulating ridge”, create another Taylor’s series: …a hierarchy of cascaded series; 3 levels.

28. y x z Leading, low, and high order: 2D We have: 1. A series to correct in depth 2. A series to correct laterally for each term in (1.) 3. A non-linear series in the data for each term in (2.) So, different combinations of high/low order could arise in one algorithm. Next talk, analysis of Liu et al.: -Infinite order in depth corrector. -low order in lateral corrector. -E.g.’s of low and high order in D.

29. High order terms for processing primaries Summarizing: when we encounter analysis and examples in 2D etc., there is the administrative issue of clarifying what portions of the algorithm are low order, what are high order. In using the ambiguous term “high-order imaging” we refer to using a larger number of the terms in the innermost series. Leading order implies …and results in effective algorithms that are, yet, sensitive to large contrasts, i.e., data amplitudes for whom D2 is non-negligible.

30. High order terms for processing primaries To capture terms that will involve the data non-linearly at this lowest level within the series mechanisms, re-consider the 3rd order 1D normal incidence, 1 parameter acoustic situation. (In other words, return to the simplest possible framework that will reproduce the math and/or physics that is of interest.)

31. High order terms for processing primaries To capture terms that will involve the data non-linearly at this lowest level within the series mechanisms, re-consider the 3rd order 1D normal incidence, 1 parameter acoustic situation. (In other words, return to the simplest possible framework that will reproduce the math and/or physics that is of interest.)

32. High order terms for processing primaries Last year Innanen et al. (2004) described an attempt to capture large numbers of all primary processing terms; 12 was not among them. In effect, the search is for a generating function that reproduces the terms we are interested in. Try:

33. High order terms for processing primaries Expand it at n=3 to see where we are with this:

34. High order terms for processing primaries Expand it at n=3 to see where we are with this: Good. Grabbed a higher-order imaging term.

35. High order terms for processing primaries Expand it at n=3 to see where we are with this: Good. Got the imaging and mixed-task terms same as Innanen et al. (2004).

36. High order terms for processing primaries Expand it at n=3 to see where we are with this: Oh well. Still an approximation to amplitude term.

37. High order terms for processing primaries: closed-form We have Note that So, using the forward/inverse FT summing of M-OSRP03, we have finally

38. High order terms for processing primaries: numerical example So as ever: 1. Compute the linear inverse directly from the data and the reference Green’s function. 2. Substitute this linear input into the closed form HOI. 3. Compare against the true  and the linear 1.

39. High order terms for processing primaries: numerical example True (z)

40. High order terms for processing primaries: numerical examples Linear 1(z) Low order SII(z) High order HOI(z)

41. High order terms for processing primaries: numerical examples Linear 1(z) Low order SII(z) High order HOI(z)

42. High order terms for processing primaries: summary 1. 2002—2005 analysis and development of non- linear primary processing algorithms that are first order in the interior coefficient- generating series. 2. Effort to extend this to high-order incorporation of coefficient-generating terms in a simply-computable form. 3. Form for high-order imaging is found with little additional algorithmic complexity.

43. Strategies for deriving multi-D non-linear imaging algorithms What is the simplest way of producing closed-form imaging algorithms? Consider the 1D leading order imaging subseries as a demonstration: 1. Compute ~3 orders of all ISS terms 2. Postulate a generating function that reproduces terms of interest from (1.) 3. Identify in an appropriate domain recognizable Taylor’s series and sum

44. Strategies for deriving multi-D non-linear imaging algorithms Can we bring in the Taylor’s series analogues at an earlier stage? Assume that the imaging part of the ISS will be working to construct some …i.e. an imaged output point will be the linear input point plus something to be determined by the ISS.

45. Strategies for deriving multi-D non-linear imaging algorithms Can we bring in the Taylor’s series analogues at an earlier stage? Assume that the “plus something” ISS part will be constructed by a process analogous to a Taylor’s series: …and then use the ISS to try to figure out what this Z is.

46. Strategies for deriving multi-D non-linear imaging algorithms Can we bring in the Taylor’s series analogues at an earlier stage? Knowing that getting at Z is the goal shortens the analysis: we can go far with 1 ISS term only.

47. Strategies for deriving multi-D non-linear imaging algorithms Can we bring in the Taylor’s series analogues at an earlier stage? If we’re right about the ISS creating a correction of the form described, then this should be the Z we’re after: …which is the closed-form LOIS of Shaw (2005).

48. Strategies for deriving multi-D non-linear imaging algorithms Try this with the 2D case, making use of the analysis of Liu et al., 2004. A 2D Taylor’s series:

49. Strategies for deriving multi-D non-linear imaging algorithms Again, assume that an important imaging component of the ISS will construct an output along these lines: via …if this is the case to the same extent that it is in 1D, then our analysis of 2(x,z) should be sufficient to give us the closed-form.

50. Strategies for deriving multi-D non-linear imaging algorithms The second order coefficients of 1(x)(x,z) and 1(y)(x,z)found in the ISS by Liu et al. (2004) are given the restriction kh=0.