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Linear and Non-linear Inversion for Absorptive/Dispersive Medium Parameters

Linear and Non-linear Inversion for Absorptive/Dispersive Medium Parameters. Kristopher Innanen †, †† and Arthur Weglein †† †† University of Houston, † University of British Columbia. M-OSRP Annual Meeting, University of Houston March 31 – April 1, 2004. Acknowledgments.

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Linear and Non-linear Inversion for Absorptive/Dispersive Medium Parameters

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  1. Linear and Non-linear Inversion for Absorptive/Dispersive Medium Parameters Kristopher Innanen†, †† and Arthur Weglein†† ††University of Houston, † University of British Columbia M-OSRP Annual Meeting, University of Houston March 31 – April 1, 2004

  2. Acknowledgments M-OSRP sponsors and members CDSST (UBC) sponsors and members Tad Ulrych Simon Shaw Bogdan Nita

  3. Motivations • Study of the forward scattering series (MOSRP02) suggests that – properly posed – the ISS will attempt to accomplish Q-estimation and Q-compensation in the absence of prior knowledge of medium parameters. • This promise motivates, for ISS-based Q procedures, • investigation of theoretical underpinnings, • investigation of practical issues and data requirements • And it motivates the (skeptical) acceptance of an unusual way • to detect the presence of Q in our data…

  4. Motivations • Study of the forward scattering series (MOSRP02) suggests that – properly posed – the ISS will attempt to accomplish Q-estimation and Q-compensation in the absence of prior knowledge of medium parameters. • This promise motivates, for ISS-based Q procedures, • investigation of theoretical underpinnings, • investigation of practical issues and data requirements • And it motivates the (skeptical) acceptance of an unusual way • to detect the presence of Q in our data…

  5. Key references Linear inversion for multiple parameters. Clayton & Stolt (1981) Raz (1981) Weglein (1985) Absorption/Dispersion. Aki & Richards (2002) [and refs therein] Kjartansson (1979) Innanen and Weglein (2003) Nonlinear inversion & processing. Zhang and Weglein (2002)

  6. Background and review The quantitative inclusion of friction into the modelling of wave propagation is a mixture of physical theory and empiricism: “Keep the constitutive relations linear, impose causality, and keep the amplitude-change per cycle (Q) independent of frequency.”

  7. Background and review Causality dictates that absorption cannot exist without dispersion…

  8. Background and review Constant Q demands that the frequency-dependence of the absorption parameter  be linear. However linear () leads to a divergent Hilbert transform. The challenge is to depart a small amount from one of these assumptions…

  9. Background and review A fairly well-accepted choice leads to the following form, explicitly in terms of the single parameter Q and a reference frequency r: This form relies on a linearization, i.e. it requires …in other words, Q and  can’t both be small.

  10. Background and review Using a Q model of this kind, the forward scattering series (acoustic reference, absorptive/dispersive non-reference media) construction of a viscoacoustic wave field was investigated (M-OSRP02). Indicated: (1) generalized imaging subseries will be a de-propagation subseries, namely it will attempt to accomplish Q compensation (2) generalized inversion subseries will accomplish Q estimation The first step in considering such methodologies is to pose the linear inverse problem for the absorptive/dispersive case.

  11. A viscous linear inverse What should V1look like? The answer depends on our purposes – and we have several here. How many parameters do we wish to include in the problem? What is the dimensionality of the problem? And most importantly: Are we interested in the linear inverse as a valuable product on its own, or as the input to a higher order nonlinear scheme?

  12. A viscous linear inverse Let us begin with the most general case we’ve tackled and go from there. Consider the linear data equations associated with a shot record-like measurement of the scattered wave field, and a V1 that may vary in depth only:

  13. A viscous linear inverse Let us begin with the most general case we’ve tackled and go from there. Consider the linear data equations associated with a shot record-like measurement of the scattered wave field, and a V1 that may vary in depth only:

  14. A viscous linear inverse Alternately, consider the 1D normal incidence linear data equations:

  15. A viscous linear inverse Alternately, consider the 1D normal incidence linear data equations:

  16. A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  17. set =1/Q A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  18. set =1/Q A viscous linear inverse V1is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  19. A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  20. set F(k)i/2-(1/)ln(/r) A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  21. set F(k)i/2-(1/)ln(/r) A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  22. set 1/c2(z)=1/c02[1-(z)] A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  23. set 1/c2(z)=1/c02[1-(z)] A viscous linear inverse V1 is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and

  24. A viscous linear inverse V1is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and expand and linearize (for now!)

  25. A viscous linear inverse V1is obtained from the difference between appropriate wave operators. In our case we have wavenumbers and expand and linearize (for now!)

  26. A viscous linear inverse And so

  27. A viscous linear inverse And so write as F(k) or F(), since k/kr= /r

  28. A viscous linear inverse And so write as F(k) or F(), since k/kr= /r

  29. A viscous linear inverse And so

  30. A viscous linear inverse And so

  31. A viscous linear inverse Write the linear components as

  32. A viscous linear inverse and transforming to the wavenumber domain:

  33. A viscous linear inverse Switch to `=` for now…

  34. A viscous linear inverse Switch to `=` for now…

  35. A viscous linear inverse The 1D normal incidence case cannot in general be used to solve for two parameters. For our purposes consider a case involving contrasts in Q only:

  36. A viscous linear inverse The 1D normal incidence case cannot in general be used to solve for two parameters. For our purposes consider a case involving contrasts in Q only: ( = 0)

  37. A viscous linear inverse The 1D normal incidence case cannot in general be used to solve for two parameters. For our purposes consider a case involving contrasts in Q only:

  38. A viscous linear inverse The 1D normal incidence case cannot in general be used to solve for two parameters. For our purposes consider a case involving contrasts in Q only:

  39. A viscous linear inverse With the tools gathered below we can investigate a set of slightly differing linear inversions for c(z) and/or Q(z). The most general case is to use data with offset and the two parameter perturbation.

  40. A viscous linear inverse With the tools gathered below we can investigate a set of slightly differing linear inversions for c(z) and/or Q(z). The most general case is to use data with offset and the two parameter perturbation.

  41. A viscous linear inverse

  42. A viscous linear inverse

  43. A viscous linear inverse Assume we can deconvolve the source wavelet over the requisite frequency bandwidth.

  44. A viscous linear inverse Assume we can deconvolve the source wavelet over the requisite frequency bandwidth.

  45. A viscous linear inverse Use the definition of the wavenumber k=/c0 …

  46. A viscous linear inverse Use the definition of the wavenumber k=/c0 …

  47. A viscous linear inverse

  48. A viscous linear inverse These are the data equations with which we solve for 1 and 1 from D. We’ve heard and will continue to hear discussion of which parameter to use in our exploitation of the extra degree of freedom. We’ll choose angle for the offset case, frequency for the 1D case. The former comes about as follows…

  49. Offset geometry The wavenumbers and incidence angle  are related as follows:

  50. Offset geometry The wavenumbers and incidence angle  are related as follows:

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