1. Issues in inverse scattering series primary processing: coupled tasks, lateral shifts, order, and band-limitation reconsidered
Kristopher A. Innanen
University of Houston
M-OSRP Annual Meeting
11 May, 2006
University of Houston

2. Acknowledgments M-OSRP sponsors and personnel
CDSST sponsors and personnel

3. Plan questions 2. Review: 1D coupled primary tasks, low
1. Introduction: some ongoing research
questions
2. Review: 1D coupled primary tasks, low
order and high order
3. Error analysis for a single layer
4. Taylor’s series and lateral/vertical shifts
5. Bandlimitation in direct nonlinear reflector
location v. non-linear AVO

4. Introduction and ongoing research
Task-separation is the lynchpin for deriving practical tools for processing seismic data from the ISS at M-OSRP.
The route to achieving an appropriate level of task-separation in a given problem may be circuitous. The coupling of imaging-inversion, for instance, has aided in:
1. Understanding of the activity of the ISS
2. Framework for linking task-separated
(intuitive) algorithm forms with the ISS
formalism

5. Introduction and ongoing research
Meanwhile, we have considered synthetic models whose contrasts warrant the capture of higher order terms in ISS imaging.
Use the coupled framework to evaluate this “capture” with analytic example + error analysis.
Higher order activity has been incorporated into the 2D imaging (Liu et al.). Review potential 2D shift forms for imaging in the kh = 0 setting.

6. Introduction and ongoing research
Task-separation and bandlimitation are closely
connected. If one task has less restrictive bandwidth requirements (e.g. FSME, IMA) than another (e.g., non-linear AVO), invoke a strategy in which the needs of one are not imposed on the needs of the other.

7. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.

8. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.
To be corrected

9. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.
To be corrected
Purely imaging
Purely inversion

10. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.
To be corrected
Purely imaging
Purely inversion
Mixed

11. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.
…produces a coupled algorithm whose reflector location activity is at leading order:

12. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.
To be corrected
Purely imaging
Purely inversion
Mixed

13. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.
To be corrected
Purely imaging
Purely inversion
Higher order
Mixed

14. Review: ISS direct imaging-inversion
The coupled imaging-inversion subseries’, at leading and higher order, has been due to an attempt to capture and study the ISS primary processing capability.
…produces a coupled algorithm whose reflector location activity is at a higher order:

15. Review: ISS direct imaging-inversion
Addressing some coupled v. uncoupled issues:
1. Analytic example and error analysis of
non-linear direct reflector location in the
coupled framework. .
2. A potential lateral/vertical closed form in the kh=0 setting.
3. Bandlimitation and task 3 v. task 4.

16. 1D analytic example and error analysis
Consider a single layer, i.e., two interfaces, one of which will be accurately located linearly and the other of which will not.
Pert. amplitude
Depth z

17. 1D analytic example and error analysis
Exercising the two coupled expressions on this linear input, we have:
Pert. amplitude
Depth z
Linear
Leading order
Higher order

18. 1D analytic example and error analysis
Exercising the two coupled expressions on this linear input, we have:
Linear
Leading order
Higher order

19. 1D analytic example and error analysis
Exercising the two coupled expressions on this linear input, we have:
Anti-derivative operation drops data information into the denominator, adjusting
amplitudes
Linear
Leading order
Higher order

20. 1D analytic example and error analysis
Exercising the two coupled expressions on this linear input, we have:
Meanwhile various degrees of non-linear (data)(data) interaction acts to alter the depth of the second reflector
Linear
Leading order
Higher order

21. 1D analytic example and error analysis
Consider the error in the corrected depth of the second
reflector, for both the leading and higher order subseries:
Linear
Leading order
Higher order

22. 1D analytic example and error analysis
Consider the error in the corrected depth of the second
reflector, for both the leading and higher order subseries:
Linear
Leading order
Higher order

23. 1D analytic example and error analysis
We can plot the error of the leading order location w.r.t. a series of contrasts (i.e., values of c1):

24. 1D analytic example and error analysis
And we can take a closer look at the error for the higher order location:
…hence, at this particular degree of “capture”, the second interface is perfectly corrected. Not true of Nth layer (c.f. early work on Taylor’s series for a box).

25. Closed-forms via Taylor’s series analogy
Much of the activity of the ISS is Taylor’s series – like. Can we use this aspect to predict closed-form partial algorithms from low order behaviour of the ISS?
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.

26. Closed-forms via Taylor’s series analogy
Much of the activity of the ISS is Taylor’s series – like. Can we use this aspect to predict closed-form partial algorithms from low order behaviour of the ISS?
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.

27. Closed-forms via Taylor’s series analogy
Knowing that getting at Z is the goal shortens the analysis. We can make considerable progress using only the 2nd order ISS term …

28. Closed-forms via Taylor’s series analogy
If we’re right about the ISS creating a correction of the form described, then this should be the Z we’re after:
…the closed-form leading order imaging subseries.

29. Closed-forms via Taylor’s series analogy
Try this with the 2D case, making use of the analysis of Liu et al., A 2D Taylor’s series:

30. Closed-forms via Taylor’s series analogy
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.

31. Closed-forms via Taylor’s series analogy
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.

32. Closed-forms via Taylor’s series analogy
Based on the 1D case, this suggests that a “kh=0” version of the 2D/3D leading-order imaging algorithm be given in closed form by
where

33. Closed-forms via Taylor’s series analogy
Comments
1. A Taylor’s series analogy can predict the
cumulative effect of a subseries of the ISS
through analysis of low order terms only.
2. A potential lateral/vertical closed form in the kh=0 setting seems plausible when it is
compared with its vertical only counter-
part.
3. Numerical study of its impact is underway.

34. Bandlimitation: amplitude, location, and differential benefit
Shaw et al. (2004; 2005) have studied the
issue of bandlimited data on the direct non-
linear imaging problem.
Further questions:
1. Can we detect a “differential benefit”
between imaging and inversion given a
certain level of missing low freq.?
2. What interventions are needed or
useful? The “adaptive subtraction”
of the direct non-linear imaging problem?

35. Bandlimitation: amplitude, location, and differential benefit
Approach: set up and study one of the hardest
bandlimited problems we can.
1D normal incidence
Not low contrast
Sustained over large depths
Noise
Cumulatively increasing perturbation
Begin dropping 1/10 Hz from the low end…

36. Bandlimitation: amplitude, location, and differential benefit
Numerical examples:
Large contrast, 400—1200m perturbation.
Large contrast, 100—300m perturbation.
(both with 3% Gaussian noise)
Two interventions:

37. Bandlimitation: amplitude, location, and differential benefit
Model:
sources
receivers
c0=1500m/s
400m
c1=1900m/s
800m
c2=2150m/s
1200m
c3=1420m/s

38. 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, No intervention

39. Direct non-linear output 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Actual perturbation
Non-linear AVO
Linear perturbation
Direct non-linear output
HO coupled II
m model, No intervention

40. Bandlimitation: amplitude, location, and differential benefit
Raw results with noise + loss of low
frequency. 0—1/2Hz.

41. 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, No intervention

42. 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, No intervention

43. 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, No intervention

44. 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, No intervention

45. 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, No intervention

46. 400-1200m model, No intervention
Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, No intervention

47. Bandlimitation: amplitude, location, and differential benefit
Added mean intervention and
differential benefit. ~0.5Hz.

48. Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, added mean intervention

49. Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, no intervention

50. Bandlimitation: amplitude, location, and differential benefit
Relationship between reflector depths and
effect of missing low frequency.
Smaller model, added mean intervention.

51. Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model

52. Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, no intervention

53. Bandlimitation: amplitude, location, and differential benefit
Data (Hz)
Non-linear AVO
HO coupled II
m model, added mean intervention

54. Bandlimitation: amplitude, location, and differential benefit
Comments:
1. The low end of the spectrum plays a key
role in ISS imaging and target ID.
2. There is a continuous decay in reflector
location accuracy as interval of missing low
frequency increases.
3. The fraction of the distance to the correct
depth the reflector goes may be predictable
in terms of data properties
4. Impact depends on the combination of
depth to target and sustained contrast.

55. Bandlimitation: amplitude, location, and differential benefit
Comments:
5. Differential benefit is present between
target ID and reflector location at a fixed
interval of missing low frequency.
6. Various water-bottom intervention
approaches and “frequency extrapolations”
show marked improvement.
7. Techniques of bandlimited impedance
inversion employ low level prior
information, to fill in the spectrum with
values that are “maximally non-committal”.