Coupling the Imaging & Inversion Tasks:

Coupling the Imaging & Inversion Tasks: …some simple insights into the theory and numerics of the inverse scattering series Kristopher Innanen†, Bogdan Nita††, Tad Ulrych† and Arthur Weglein†† ††University of Houston, † University of British Columbia M-OSRP Annual Meeting, University of Houston March 31 – April 1, 2004

Acknowledgments M-OSRP sponsors and members CDSST (UBC) sponsors and members Simon Shaw Ken Matson

Motivations Thus far we have worked on algorithms and interpretations that involve the separation of the tasks of imaging and inversion. However: 1. We remain interested in the workings of the inverse scattering series terms (and others) whether separated or not. 2. Questions of model [in]dependence may involve coupling tasks: e.g. locating R in depth. 3. Patterns in the derivation of the terms of the inverse scattering series imply a connectivity between the mechanisms that accomplish the two tasks. 4. Some scattered questions: e.g. what is the meaning of a truncation of terms prior to numerical convergence?

Motivations Thus far we have worked on algorithms and interpretations that involve the separation of the tasks of imaging and inversion. However: 1.We remain interested in the workings of the inverse scattering series terms (and others) whether separated or not. 2. Questions of model [in]dependence may involve coupling tasks: e.g. locating R in depth. 3. Patterns in the derivation of the terms of the inverse scattering series imply a connectivity between the mechanisms that accomplish the two tasks. 4. Some scattered questions: e.g. what is the meaning of a truncation of terms prior to numerical convergence?

Motivations Thus far we have worked on algorithms and interpretations that involve the separation of the tasks of imaging and inversion. However: 1. We remain interested in the workings of the inverse scattering series terms (and others) whether separated or not. 2. Questions of model [in]dependence may involve coupling tasks: e.g. locating R in depth. 3. Patterns in the derivation of the terms of the inverse scattering series imply a connectivity between the mechanisms that accomplish the two tasks. 4. Some scattered questions: e.g. what is the meaning of a truncation of terms prior to numerical convergence?

Key references Imaging & inversion subseries. Weglein et al. (2002) Shaw et al. (2003) Zhang and Weglein (2003) …and a lot of this year’s report. Useful tools from linear inverse theory. Walker and Ulrych (1983) Oldenburg et al. (1983) Hansen (1999) (singular value decomposition)

Background and review A task-separated form of the inverse scattering series was developed using an “integrate by parts” mentality that diagrammatically appears to distinguish between the geometry of scattering interactions; esp. • separated vs. self-interaction • relative geometry. We will consider a “coupled” version of these terms, but one that maintains the mechanisms of uncoupling very close to its heart.

Background and review The original casting, to third order, looks like this: (MOSRP02 and earlier notes)

Background and review The original casting, to third order, looks like this: 1st order 2nd order 3rd order

Background and review …which has an associated inversion subseries:

Background and review …and a leading order imaging subseries:

Background and review Leading order imaging subseries: Analysis indicates that this corresponds to a stretch of 1 spatially. It corrects reflector locations, and has no interest in changing the linear amplitudes…

Background and review Inversion subseries: Has no ability to alter anything except parameter amplitudes: INVERSION (…framework for the study of parameter correction via INTER-EVENT COMMUNICATION)

Re-coupling the tasks Many of the terms in the ISS seem to follow patterns – similar operations are carried out repeatedly at varying orders:

Re-coupling the tasks So consider the following construction: where

Re-coupling the tasks So consider the following construction: Try expanding it over several orders…

Re-coupling the tasks Expand SII over 3 orders: So the terms associated with the formula: (1) Reproduce the leading order imaging subseries (LOIS) (2) Approximate the inversion subseries

Re-coupling the tasks So consider the following construction: This expression appears to be intimately associated with imaging and inversion. See this by expanding it over several orders…

Re-coupling the tasks Some Basic Questions 1. Does it have a closed form? 2. How does it intend to operate upon the data? 3. Can it be stably implemented numerically? 4. What is the impact of its approximate form?

Re-coupling the tasks Closed form: It seems that we may compute this quantity by taking the inverse Fourier transform of a forward, Fourier-like, transform, of the linear input; the kernel of the forward transform is dependent on the second integral of the data.

? Re-coupling the tasks Look more closely at the detail of SII: how does it image and invert simultaneously?

Re-coupling the tasks Let H(z) be the integral of 1(z) – for a piecewise constant Born approx., H is piecewise linear…

Re-coupling the tasks Start by considering H(z) away from its characteristic discontinuities: i.e. linear… then, what does Kn dn/dzn Hn(z) accomplish? No z-dependence! …away from discontinuities the “engine” Kn dn/dzn Hn(z) attempts no structural change to a piecewise constant quantity. REPRODUCES THE INVERSION SUBSERIES.

Re-coupling the tasks Start by considering H(z) away from its characteristic discontinuities: i.e. linear… then, what does Kn dn/dzn Hn(z) accomplish? No z-dependence! …away from discontinuities the “engine” Kn dn/dzn Hn(z) attempts no structural change to a piecewise constant quantity. REPRODUCESTHE INVERSION SUBSERIES.

Re-coupling the tasks Next consider H(z) near its discontinuities. Focus on a general piecewise linear signal portion: Numerically the effect is of a weighted set of derivative operators – construction of the discontinuous 1 correction. IMAGING…

Re-coupling the tasks So from a signal-processing point of view: 1. The simultaneous imaging and inversion subseries acts as a flexible operator that behaves very differently depending on the input 2. In slowly-varying regions it acts to change the amplitudes: inversion 3. At discontinuities it outputs weighted derivatives of the integral of the data: imaging

Re-coupling the tasks To answer this, construct 1D normal incidence wave field data based on some layered Earth models: Model 1. (basic) Model 2. (structure) Model 3. (contrast) Model 4. (contrast) z 

Re-coupling the tasks …which produce associated full-bandwidth data: Data 1. (basic) Data 2. (structure) Data 3. (contrast) Data 4. (contrast) z 

Re-coupling the tasks …finally from which we construct the linear inverse 1: Linear 1. (basic) Linear 2. (structure) Linear 3. (contrast) Linear 4. (contrast) z 

Linear 1. (basic) Linear 2. (structure) Linear 3. (contrast) Linear 4. (contrast) Re-coupling the tasks Clearly, if SII is going to image and invert, in this situation it must construct a sequence of piecewise constant functions, such that when added to the linear inverse, the true perturbation is recovered.

Re-coupling the tasks Brute implementation… Truncation of SII after third order…

Re-coupling the tasks Brute implementation… Truncation of SII after third order… unstable!

Re-coupling the tasks It seems natural to attribute this to the high-wavenumber portions of the derivative operators. We appeal to TSVD-type operator regularization: d2/dz2 d/dz d3/dz3 d4/dz4

Re-coupling the tasks Then for one of the examples, compute ~70 terms:

Re-coupling the tasks The accuracy depends on the contrast (leading order imaging & partial inversion):

Re-coupling the tasks Basic Questions 1. Does it have a closed form? 2. How does it intend to operate upon the data? 3. Can it be stably implemented numerically? 4. What is the impact of its approximate form? This subseries may be manipulated theoretically and numerically to produce stable output, with accuracy depending on contrast. The imaging and inversion are both carried out with a single flexible operator that knows when to image, when to invert.

The relationship between resolution and truncation order We have seen the “mess” prior to convergence of this subseries. Is there interpretable behavior here? Especially: is z the best domain to observe this?

Resolution and truncation order The numeric behaviour of the series, and how it converges, can be qualitatively predicted/described. Some things we know: (1) we will be constructing a discontinuous function, i.e., in the k-domain it will have elements resembling ~F(k)eikC (2) we will be doing it via a series of derivatives: i.e., ~ … + (ik)n G(k) + …

Resolution and truncation order In other words, this is not far from the construction of an exponential function from an infinite series of polynomials: n=1 x 

Resolution and truncation order In other words, this is not far from the construction of an exponential function from an infinite series of polynomials: n=1 n=6 x  x 

Coupling the Imaging & Inversion Tasks: