1. Construction of a Q-compensation operator directly from measured seismic data
Kristopher Innanen** and Arthur Weglein*
*University of Houston
**University of Houston & University of British Columbia
M-OSRP Annual Meeting
21 April, 2005
University of Houston

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

3. Overview
Review of recent results, and a conjecture about attenuation in the inverse scattering series
What does a Q-compensation operator look like?
Generalizing the scope of the ISS non-linear imaging mechanisms
The form and behavior of a non-linear data driven Q-compensation operator (with simple examples)

4. Previous results & conjectures
2001— Forward scattering series for
attenuative media
2003— Linear inversion for absorptive-
dispersive medium parameters
“…construction of an appropriate linear
inverse, and use of the imaging mechanisms
of the ISS, will lead to Q-compensation in the
absence of an a priori Q estimate.”

5. What does a Q-compensation operator look like?
Q-compensation (with known Q) belongs to the class of ill-conditioned deconvolution problems.

6. What does a Q-compensation operator look like?
Q-compensation (with known Q) belongs to the class of ill-conditioned deconvolution problems.
…i.e., in the conjugate domain we apply the operator 1/MT(k).

7. What does a Q-compensation operator look like?
Operators of this type:
1. Preferentially magnify large k signal components
2. Are often regularized, or stabilized via some approximation
3. Usually contain the relevant medium parameters…

8. What does a Q-compensation operator look like?
Summary:
1. Any/all operators that purport to correct for absorption effects of Q must have this form or similar.
2. Therefore “be on the look-out” for such a form in our current developments.

9. Review The linear inverse.
1. We have considered 1D normal incidence, pre-stack, and both and 2-parameter linear absorptive- dispersive inversion
2. We have reconstructed (linear) complex impedance contrasts by detecting the phase/amplitude imprint of complex reflection coefficients in measured wave field

10. Review The linear inverse.
3. All schemes produced complex estimates of spatial distributions of the linear component of the A/D medium.
We are particularly interested in the effect on the non-linear terms of the ISS of this new aspect of V1

11. Review The linear inverse.
So, construct the simplest possible of these and investigate the consequent behavior…

12. Review The linear inverse.
So, construct the simplest possible of these and investigate the consequent behavior…
1D causal Green’s fn. for homogeneous non-attenuating reference medium
Incident acoustic plane wave

13. Review The linear inverse.
So, construct the simplest possible of these and investigate the consequent behavior…
Data bears the effects of attenuation-dispersion in wave propagation and in interaction with complex impedance contrasts
The linear component of the medium parameters, expressed as a perturbation away from a very simple acoustic reference.

14. Review The linear inverse.
Key point: linear estimation problem must be posed to interpret phase/amplitude data effects as being due to complex medium contrasts (Innanen and Weglein, 2004).

15. Review The linear inverse.
Key point: linear estimation problem must be posed to interpret phase/amplitude data effects as being due to complex medium contrasts (Innanen and Weglein, 2004).

16. Review The leading order imaging subseries.
Incorporate a large subset of the terms in the ISS that are ONLY concerned with correcting the spatial structure of the medium (misplaced by the linear inverse which contains an inaccurate reference medium).

17. Review The leading order imaging subseries.
In 1D the linear component of the velocity perturbation
is computed directly from the data. The ISS may be manipulated so that the order-by-order reconstruction, cast in terms of 1(z),

18. Review The leading order imaging subseries.
…involves a subset of the form
where (Shaw, 2005)

19. Review The leading order imaging subseries.
…involves a subset of the form
where (Shaw, 2005)
We will be particularly interested in this representation…

20. Construction of a Q-compensation operator
This portion of the 1D ISS can be exactly reproduced in the case of the absorptive linear inverse for media with contrasts in Q only, i.e., with a complex linear input:
Some points:
1. As in the case of LOIS, we are building an operator, to act on 1 at the output point z, that cares about what 1 itself is at all points above (0—z).

21. Construction of a Q-compensation operator
This portion of the 1D ISS can be exactly reproduced in the case of the absorptive linear inverse for media with contrasts in Q only, i.e., with a complex linear input:
Some points:
2. With a complex 1, part of this operator will have the form of the 1/MT(k) that we have discussed.

22. Construction of a Q-compensation operator
This portion of the 1D ISS can be exactly reproduced in the case of the absorptive linear inverse for media with contrasts in Q only, i.e., with a complex linear input:
Some points:
2. With a complex 1, part of this operator will have the form of the 1/MT(k) that we have discussed.

23. Construction of a Q-compensation operator
This portion of the 1D ISS can be exactly reproduced in the case of the absorptive linear inverse for media with contrasts in Q only, i.e., with a complex linear input:
Some points:
3. This is a non-linear operator that is constructed directly from the measured data (via 1).

24. An illustrative example
Consider a gradual progression from data through this “algorithm”: discover its character and capability.

25. An illustrative example
Consider a gradual progression from data through this “algorithm”: discover its character and capability.

26. An illustrative example
Consider a gradual progression from data through this “algorithm”: discover its character and capability. MODEL & DATA:

27. An illustrative example
Consider a gradual progression from data through this “algorithm”: discover its character and capability. MODEL & DATA:
Smoothed, attenuated signal for all portions of the wave field that have propagated within the Q   part of the medium…

28. An illustrative example
Consider a gradual progression from data through this “algorithm”: discover its character and capability. LINEAR INVERSE:

29. An illustrative example
Consider a gradual progression from data through this “algorithm”: discover its character and capability. LINEAR INVERSE:
Non-linear reconstruction of  with sharp boundaries from this smooth 1 involves a flavor of Q-compensation.
And we think we know where this resides…

30. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

31. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

32. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

33. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

34. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

35. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

36. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

37. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

38. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

39. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

40. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

41. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

42. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

43. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

44. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

45. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

46. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

47. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

48. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

49. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

50. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

51. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

52. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

53. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

54. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

55. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

56. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

57. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

58. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

59. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

60. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

61. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

62. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

63. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

64. An illustrative example
Non-linear operator
in k domain…
…derived from
the cumulative
effect of 1(z),
z <|

65. An illustrative example
We construct an operator directly from the attenuated data.
Correction operator “intensity” is related to the cumulative effect of data above.
Ill-conditioned, as expected: can it be implemented?
…continue with our example.

66. Non-linear direct Q-compensation
As with most Q-compensation efforts, raw application of the correction operator is rarely indicated. Try employing a crude regularization:
e.g. output z=500m

67. Non-linear direct Q-compensation
As with most Q-compensation efforts, raw application of the correction operator is rarely indicated. Try employing a crude regularization:
e.g. output z=1000m

68. Non-linear direct Q-compensation
Finally: apply the operator to our linear input 1:

69. Non-linear direct Q-compensation
Finally: apply the operator to our linear input 1:
1’(z)
Pseudo-depth z (m)

70. Non-linear direct Q-compensation
Finally: apply the operator to our linear input 1:
LOQC’(z)
Pseudo-depth z (m)

71. Non-linear direct Q-compensation
Looking closer:
Un-attenuated
Attenuated
Corrected
Pseudo-depth z (m)

72. Non-linear direct Q-compensation
Summary
1. The erstwhile imaging mechanisms of the ISS
may be used to perform absorption compensation on reflected seismic primaries
2. Requires a linear inverse (input) that treats the phase/amplitude of events as being due to complex impedance contrasts
3. Non-linearly construct (and regularize) an operator directly from the data via the linear inverse: no prior knowledge of Q

73. Non-linear direct Q-compensation
Summary
4. Analyse the operator and numerics using examples that purposefully isolate the Q issues: model that involves contrasts in Q only, constant reference wavespeed c0 everywhere
5. Results indicate corrective behavior for all events considered; operator is flexible and grows/shrinks in intensity exactly in line with what the ISS “considers” necessary at any given output point
6. “Leading order” nature of the corrector is visible: compensation is not complete. Tie in “high-order” primary terms… future work