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

Acknowledgments M-OSRP sponsors and personnel CDSST sponsors and personnel

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

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?

The resolution problem Introduction The resolution problem

The resolution problem Introduction The resolution problem

The resolution problem Introduction The resolution problem

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

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.

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.

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.

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.

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

Linear inversion: a model c0 Q0 =  c1 Q1 c2 Q2

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:

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…

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

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.

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

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).

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

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

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

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

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

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…

A framework for task-separated Q processing where

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

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…

2D forward scattering with an absorbing overburden

2D forward scattering with an absorbing overburden Viscous model

2D forward scattering with an absorbing overburden

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

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

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

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

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

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.

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.