1. Theoretical framework for task-separated processing of primaries in absorptive/dispersive media
Kristopher Innanen and Arthur Weglein
University of Houston
M-OSRP Annual Meeting
12 May, 2006
University of Houston

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

3. Plan 2. Linear inversion: construction of the input
1. Introduction and multi-parameter strategy
2. Linear inversion: construction of the input
3. Non-linear equations for imaging-inverting
primaries over an A-D medium
4. A numerical prototype for task-separated
Q-compensation using the ISS
5. A theoretical framework for Q compensation
without a Q estimate in 1D c/Q media
6. Lessons of the forward scattering series:
diffractions beneath an A-D overburden

4. The resolution problem
Introduction
The resolution problem
The frequency content and amplitude of reflected events has the tendency to change with propagation distance… one well-explored mechanism for this loss is that the waves obey anelastic wave equation.
What happens to the ISS if waves propagate this way? What are the opportunities? What are the obstacles?

5. The resolution problem
Introduction
The resolution problem

6. The resolution problem
Introduction
The resolution problem

7. The resolution problem
Introduction
The resolution problem

8. The resolution problem
Introduction
The resolution problem
What would we like to be able to do?
- correct for attenuation, i.e., enhance amplitudes and resolution
- correct the phase for the distortion produced by dispersion
What are the obstacles? Address with ISS?
- ill-conditioned
- need to know Q

9. The resolution problem
Introduction
The resolution problem
What would we like to be able to do?
- correct for attenuation, i.e., enhance amplitudes and resolution
- correct the phase for the distortion produced by dispersion
What are the obstacles? Address with ISS?
- ill-conditioned
- need to know Q
ISS won’t change that.
ISS will change that.

10. Introduction: a multi-parameter strategy
Study of the forward scattering series for absorptive-dispersive (A-D) media suggests that Q compensation via the ISS should in principle be possible in the absence of an accurate prior knowledge of Q.
The A-D linear inverse and its characteristics:
1. Provides a clue about how this will take place.
2. Tell us where in the data Q information must reside.
3. Is the input to the non-linear machinery.

11. Introduction: a multi-parameter strategy
Meanwhile, this non-linear machinery follows (largely) the same math patterns that underlie the single-parameter acoustic problem.
Hence, similar equations for non-linear direct imaging-inversion may be written down.
The main difference is the activity due to multiple parameters… the appearance of the data in both operator and operand.

12. Introduction: a multi-parameter strategy
The data appears in various forms in the non-linear equations: a new “form” for each parameter.
The data in each form appears as both operator and operand; we have the freedom to be selective about which form is allowed to act.
Processing the data non-linearly in one component of itself, linearly in another, provides a means for task-separated Q-compensation.

13. Linear inversion: a model
Using a simple reflection model in which sources and receivers are on the same side of a 1D medium in which 2 parameters may vary; c, Q.
source
receivers c0
Q0 =  c1 Q1 c2 Q2

14. Linear inversion: a modelc0
Q0 =  c1 Q1 c2 Q2

15. Linear inversion equations
Next, consider line sources and receivers over this medium, transformed over sources to ks (just a choice). Then form a scattering potential assuming a homogeneous acoustic reference medium and a 1D c/Q actual medium:

16. Linear inversion equations
So the ISS, in which the prescription for the construction of V = V1 + V2 + V3 + … begins with the solution for V1:
through
…which, after substitution of Green’s fns etc., results in…

17. Linear inversion equations
sources
receivers c0
Q0 = 
c(z)
Q(z)

18. Linear A-D inversion
Linear A-D inversion: an over-determined solution for the linear components of the two parameters.
The Q information comes from the imprint of the dispersion in the reflection coefficient.
The aspect of the ISS referred to as purposeful perturbation will be in play here. If no unexpected variation of the data w.r.t. angle is detected, 1=0.

19. Direct non-linear inversion eqns.
sources
receivers c0
Q0 = 
c(z)
Q(z)

20. Direct non-linear inversion eqns.
This is data talking to data.
This is data with a velocity aspect (1) talking to
data with a velocity (1) and a Q aspect (1).
This is data with a Q aspect (1) talking to data with a velocity (1) and a Q aspect (1).

21. A numerical prototype for task-separated Q-compensation using the ISS
1’(z)
Pseudo-depth z (m)

22. A numerical prototype for task-separated Q-compensation using the ISS
P’(z)
Pseudo-depth z (m)

23. Set the velocity processing term Z = 0…
A framework for task-separated Q processing
Set the velocity processing term Z = 0…

24. Set the velocity processing term Z = 0…
A framework for task-separated Q processing
Set the velocity processing term Z = 0…

25. A framework for task-separated Q processing
…and use this altered -processing operator in the velocity inversion expression…

26. A framework for task-separated Q processing
The resulting velocity quantity Q has been non-linearly processed for Q tasks, but it remains linear in the data in the sense of velocity tasks such as imaging. This suggests a framework for processing…

27. A framework for task-separated Q processing
where

28. A framework for task-separated Q processing
Development of codes for numerical validation of this algorithm framework is underway….

29. 2D forward scattering with an absorbing overburden
Study the construction of attenuated primaries having propagated through perturbation structure of varying complexity.
For instance, the approximation of diffractions from 2D structure propagating through a 1D overburden. Generalize the overburden perturbation to incorporate a Q model…

30. 2D forward scattering with an absorbing overburden

31. 2D forward scattering with an absorbing overburden
Viscous model

32. 2D forward scattering with an absorbing overburden

33. 2D forward scattering with an absorbing overburden
Numerics: diffractions with attenuation created entirely through non-linear influence of perturbations.
4 examples with Q decreasing.

34. 2D forward scattering with an absorbing overburden
Q1 = 50
Q2 = 50

35. 2D forward scattering with an absorbing overburden
Q1 = 30
Q2 = 30

36. 2D forward scattering with an absorbing overburden
Q1 = 15
Q2 = 15

37. 2D forward scattering with an absorbing overburden
Q1 = 5
Q2 = 5

38. Conclusions
Basic equations for non-linear direct imaging-inversion for the 2-parameter c/Q model are determined.
Linear inversion (creation of input) shows us that these non-linear activities will only occur when angle-dependence of the data is beyond that predicted by other in-model effects.
The basic equations are altered such that we may process the data linearly in part of the data and non-linearly in another.

39. Conclusions
This strategy can, in particular, lead to a framework for task-separated Q processing, and the estimation of an un-attenuated data set.
Numerical validation of the prototype algorithm is in progress.
The FSS is being examined for more spatial complex viscoacoustic models.