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

Acknowledgments M-OSRP sponsors and personnel CDSST sponsors and personnel

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

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

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.

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.

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.

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

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

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

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:

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

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

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:

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.

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

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

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

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

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

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

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

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):

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).

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.

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.

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 …

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.

Closed-forms via Taylor’s series analogy Try this with the 2D case, making use of the analysis of Liu et al., 2004. A 2D Taylor’s series:

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.

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.

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

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.

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?

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…

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:

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

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

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 400-1200m model, No intervention

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

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

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

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

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

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

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

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

Bandlimitation: amplitude, location, and differential benefit Data (Hz) Non-linear AVO HO coupled II 400-1200m model, added mean intervention

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

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

Bandlimitation: amplitude, location, and differential benefit Data (Hz) Non-linear AVO HO coupled II 100-300m model

Bandlimitation: amplitude, location, and differential benefit Data (Hz) Non-linear AVO HO coupled II 100-300m model, no intervention

Bandlimitation: amplitude, location, and differential benefit Data (Hz) Non-linear AVO HO coupled II 100-300m model, added mean intervention

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.

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”.