Non-linear forward scattering series expressions for reflected-primary and transmitted-direct wavefield events Kristopher A. Innanen University of Houston.

Non-linear forward scattering series expressions for reflected-primary and transmitted-direct wavefield events Kristopher A. Innanen University of Houston M-OSRP Annual Meeting 11 May, 2006 University of Houston

Acknowledgments M-OSRP sponsors and personnel
Arthur Weglein, Sam Kaplan M-OSRP sponsors and personnel CDSST sponsors and personnel

Plan 1. Introduction: what might we accomplish? 2. Technical Primer : a 1D forward scattering series framework for approximating primaries 3. Reflected primaries and transmitted direct waves due to 3D perturbations 4. Recursive forms and R/T relations 5. A “non-linear overburden, linear diffractor” hybrid approximation 6. Why the FSS? a. Modelling b. Direct inversion of FSS approximations

Introduction The forward scattering series (FSS) is an infinite series expressing one (“actual”) wavefield in terms of another (“reference”) wavefield and a perturbation operator describing the difference between the two. Anything less than the full series is an approximation. The current job is to consider the meaning, validity, and value of wavefield approximations based on the FSS.

Introduction The forward scattering series (FSS) is an infinite series expressing one (“actual”) wavefield in terms of another (“reference”) wavefield and a perturbation operator describing the difference between the two. Anything less than the full series is an approximation. The current job is to consider the meaning, validity, and value of wavefield approximations based on the FSS. linear non-linear

Introduction The linear term, if used to model the wavefield, must give an account of itself with respect to two kinds of non-linearity that might well be present… source receivers reference medium perturbation

Introduction …First, it assumes that the wavefield amplitude is linear in V, which is OK if V2 is small… source receivers reference medium perturbation

Introduction …and second, it assumes that very little propagation takes place in structure that is not incorporated into the reference medium. source receivers reference medium perturbation

Introduction Confounding the linear approximation is as simple as including large, spatially sustained structure in the perturbation. source receivers perturbation large contrast perturbation

Introduction The FSS is not finished at this point, but the linear approximation is. Most importantly: The prescription, or map, linking a known set of terms of the FSS to a known subset of the events of a seismic reflection data set, is gone. That is, the activity of the FSS in creating a primary has been relegated to an a priori unknown set of mathematical operations. Can we re-provision ourselves with this map? Create a prescription for the approximation of meaningful wavefield events given large and spatially sustained perturbations?

Technical primer: 1D primary approximation
Wavefield events due to unincorporated medium structure (“unincorporated” meaning not in the reference medium) that varies in 3D is the goal. The route taken to model these – one of many possible routes – arises as a natural extension of 1D. Take care of some detail in 1D and then move on to multi-D. The concepts are very similar.

Elements of the 1D theory

Elements of the 1D theory source

Elements of the 1D theory multiply the source by G0 and integrate…

Elements of the 1D theory

Elements of the 1D theory Lippmann-Schwinger Born series

Returning to schematics momentarily, the n’th order term of the Born series/FSS has an interpretation in terms of reference medium and scattering interaction… source receivers

If we compute enough of these terms, we will have created an approximation of the wavefield that – if convergent – is not sensitive to the contrast or spatial extent of the perturbation. But the full wavefield contains primaries, multiples etc. To use this formalism to create an approximation of primaries only, the reference propagation/scattering framework needs a closer look… In particular, the explicit form of the Green’s function and its depth dependence is of interest.

The solution to the equation is a causal Green’s function, stipulating that the wave must propagate away from zs.

Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term…

Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term… The first thing to notice is that if we want to model a reflection experiment, we can lose the |.| bars on the leftmost and rightmost Green’s functions…

Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term… Since this cares about whether or not z’’>z’’’ The stipulation that waves propagate only away from the source has consequences because the scatterers are acting like sources… in other words the scattering locations z’, z’’, z’’’ etc. now have to deal with the causal nature of the G0s…

Because Green’s functions appear between scattering interactions in 2nd and higher order FSS terms, this stipulation makes each term care about the relative geometry of these interactions. Take the third order term… …we now need to care whether one  happened above or below another… The stipulation that waves propagate only away from the source has consequences because the scatterers are acting like sources… in other words the scattering locations z’, z’’, z’’’ etc. now have to deal with the causal nature of the G0s…

We evaluate these integrals by allowing the absolute value bars to generate “cases”, which hold for portions of the range of integration. For instance: Each instance of an “interior” Green’s function results in the generation of two (additive) cases.

We evaluate these integrals by allowing the absolute value bars to generate “cases”, which hold for portions of the range of integration. For instance: Each instance of an “interior” Green’s function results in the generation of two (additive) cases. …which means the n’th order term breaks into 2n-1 raw subterms.

So the one third order term becomes 4 subterms: s r z’’’ z’’ z’ s r z’’ z’’’ z’ s r z’’’ z’ s z’’ r z’ z’’ z’’’ Each term representing a permutation of all four possible “scattering geometries”, i.e., all four ways the scatterers can orient themselves in depth. These are representable with diagrams.

So the one third order term becomes 4 subterms: s r z’’’ z’’ z’ s r z’ z’’ z’’’ s r z’’’ z’’ z’ s r z’’’ z’’ z’ Each term representing a permutation of all four possible “scattering geometries”, i.e., all four ways the scatterers can orient themselves in depth. These are representable with diagrams.

The significance of these terms is … what? They themselves are very poor approximations of any part of the wavefield. s r z’’’ z’’ z’ s r z’ z’’ z’’’ s r z’’’ z’’ z’ s r z’’’ z’’ z’

The significance of these terms is … what? They themselves are very poor approximations of any part of the wavefield. s r z’’’ z’’ z’ s r z’ z’’ z’’’ s r z’’’ z’’ z’ s r z’’’ z’’ z’ This is not a multiple…

But the diagram is still meaningful: to some level of approximation, the FSS tends to construct N’th order events as infinite series expansions about “seeding points”, which appear at N’th order. s r z’’’ z’’ z’ s r z’ z’’ z’’’ s r z’’’ z’’ z’ s r z’’’ z’’ z’ So, for instance, a multiple with a single downward reflection (a 3rd order event) is an expansion around the indicated diagram. Weglein, Araujo, early 1990’s

Following this logic, a framework for the approximation of primaries would involve capturing and summing all terms that are high order extensions of a primary “seeding point”. s r z’’’ z’’ z’ s r z’ z’’ z’’’ s r z’’’ z’’ z’ s r z’’’ z’’ z’ If we choose the linear term (with its single direction change) as the seeding point, all subsequent V shaped diagrams are part of the approximation. Weglein, Foster, et al., late 1990’s.

Fortunately the math makes this a practical choice also. s r z’’’ z’’ z’ s r z’ z’’ z’’’ s r z’’’ z’’ z’ s r z’’’ z’’ z’ We can see this by rejecting all reverberating diagrams (as seeding points for multiple generation), and looking at the remaining expressions.

Fortunately the math makes this a practical choice also. s r z’’’ z’’ z’ s r z’’’ z’’ z’ s r z’’’ z’’ z’ We can see this by rejecting all reverberating diagrams (as seeding points for multiple generation), and looking at the remaining expressions.

Fortunately the math makes this a practical choice also. By using a straightforward property of nested integrals, these three expressions can be seen to be basically equal…

This 1D case simplifies drastically, but let’s hold off on that. transmission s r z’’’ z’’ z’ transmission First let’s flag a couple of things to aid in interpretation.

This 1D case simplifies drastically, but let’s hold off on that. scattering interaction w direction change s r z’’’ z’’ z’ scattering interaction w direction change First let’s flag a couple of things to aid in interpretation.

OK now let’s simplify it. The other fortuitous fact of the 1D case is that the n’th nested integral can be written as a single integral to the n’th power.

OK now let’s simplify it. …which means that the diagrams we have been trying to concatenate can be written in a single term at each order!

The answer we’re looking for is going to be the sum of all such terms: …and our example is therefore amenable to a summation strategy similar to that developed by Keys and Shaw…

Recalling, from MOSRP04, that the incorporation of a geometric series in the perturbation under the integral represents a capture of high order terms, suitable for media of large contrast, we can in fact postulate two primary approximations:

Presently we will return to the 1D case for additional analysis, and to try to take greater advantage of these closed forms. For now, let us at least derive some sense that this capture of FSS terms does indeed approximate primaries.

A 1D model: sources receivers c0=1500m/s c1=3000m/s 200m c2=1700m/s c3=1420m/s

FSS-based approximation of primaries: ● Amplitudes good ● Phase excellent ● No non primaries or artifacts

How much of this carries over to 3D media?
Reflected primaries and transmitted direct waves due to 3D perturbations We came up with an ansatz in the 1D case: 1. Looked at at Green’s functions’ role in the FSS 2. Noted the “seeding point” behaviour of the FSS 3. Collected terms connected to the right seeding points, developed series and closed form RP expressions. How much of this carries over to 3D media?

…everything except the closed-form part, it turns out.
Reflected primaries and transmitted direct waves due to 3D perturbations We came up with an ansatz in the 1D case: 1. Looked at at Green’s functions’ role in the FSS 2. Noted the “seeding point” behaviour of the FSS 3. Collected terms connected to the right seeding points, developed series and closed form RP expressions. …everything except the closed-form part, it turns out.

Reflected primaries and transmitted direct waves due to 3D perturbations
Let’s start again, this time without restricting the perturbation to z-dependency. We’ll make 3 types of event approximation. TYPE 1: Downward transmitted direct zs zg

Let’s start again, this time without restricting the perturbation to z-dependency. We’ll make 3 types of event approximation. TYPE 2: Upward transmitted direct zg zs

Let’s start again, this time without restricting the perturbation to z-dependency. We’ll make 3 types of event approximation. TYPE 3: Reflected primary zs ,zg

The FSS equations are similar in form, but the newness of the lateral variability will require that we make some decisions. For instance at 2nd order: …the term still involves 2 interactions with  separated in space, but now separated in 3 dimensions. Before it was just depth. Geometrical considerations could become quite complicated here.

For instance, do we want to partition and then distinguish between these?

For instance, do we want to partition and then distinguish between these? …not specifically.

For instance, do we want to partition and then distinguish between these? …yes definitely!

In terms of primaries v. multiples, ordering in DEPTH is still a powerful discriminator (e.g., Weglein and Dragoset, 2005). So can we choose a 3D framework in which scattering geometry in depth can be preferentially considered? Yes, if the domain in which the Green’s function is represented is correctly chosen…

…as follows:

In terms of primaries v. multiples, ordering in DEPTH is still a powerful discriminator (e.g., Weglein and Dragoset, 2005). So can we choose a 3D framework in which scattering geometry in depth can be preferentially considered? And if we recognize Fourier transforms over the lateral coordinates…

TYPE 1: Downward transmitted direct 1. Place the depth support of  between zs and zg , with zs<zg . 2. Substitute in the above Green’s functions. 3. Take lateral Fourier transforms zs zg

zs zg

So exactly the same choices are available in 3D that were available in 1D with respect to scattering geometry. The differences are: 1. There are now integrals over wavenumber, more and more at higher order 2. Exponential functions under the nested integrals, which vanish as exp(0) = 1 in the 1D case, now remain. 1. and 2. hamper the attempt to create closed-forms. Nevertheless recognizable patterns appear…

Altogether, downward transmitted direct:

TYPE 2: Upward transmitted direct 1. Place the depth support of  between zs and zg , with zs>zg . 2. Substitute in the above Green’s functions. 3. Take lateral Fourier transforms zg zs

Altogether, upward transmitted direct:

TYPE 3: Reflected primary 1. Place the depth support of  below zs and zg . 2. Substitute in the above Green’s functions. 3. Take lateral Fourier transforms ; apply 1D approx. strategy. zg,zs

Altogether, reflected primary:

Fortunately the math makes this a practical choice also. s r z’’’ z’’ z’ s r z’ z’’ z’’’ s r z’’’ z’’ z’ s r z’’’ z’’ z’ We can see this by rejecting all reverberating diagrams (as seeding points for multiple generation), and looking at the remaining expressions.

The math becomes less penetrable, but we can produce expressions, based on 1D thinking, that approximate portions of the 3D wavefield identifiable as transmitted direct events and reflected primary events. We end this section with the notion that we therefore have the formal capacity to do this in 3D, which is good news. However, in the absence of closed-forms, we are in a position where a large number of independent terms, each involving a greater number of integrations than the last, must be computed.

Recursive forms and R/T relations
A continuum of computational burden SERIES FORMS COMPUTABLE RECURSIVELY small burden large burden SERIES FORMS CLOSED FORMS

The transmitted direct event approximations may be written such that the Nth term is a reasonably simple operation on the N-1th term.

Furthermore, the reflected primary approximation can be made via quasi-linear constructions involving the transmitted direct approximations.

RP31 = TDU0 k2 TDD2 RP32 = TDU1 k2 TDD1 RP33 = TDU2 k2 TDD0

Furthermore, the reflected primary approximation can be made from quasi-linear constructions involving the transmitted direct approximations. …this means that since the transmitted direct approximations are expressible recursively, so are the reflected primary approximations.

A “non-linear overburden, linear diffractor” hybrid approximation
Without further intervention or approximation, we at present have that: 1. 1D Earth models permit closed-form FSS approximations of primaries (and ISS processing of primaries) 2. 2D/3D Earth models require series forms Can we leverage the analytic/computational efficiency of the closed-form to see the FSS construct 2D wavefield behaviour, e.g., diffractions…?

Yes, for certain kinds of Earth models. In particular, for situations in which: 1. The circumstantial form of non-linear predominates, and 2. The overburden is essentially 1D. Schematically…

overburden target

overburden target 2D, small contrast. No circumstantial nonlinearity.

Linear approximation of a single scatterer: sources receivers

overburden target

overburden target Fully 1D… Closed forms available.

overburden target Fully 2D/3D… As far as we know we require a series.

overburden target 2D model, but the portion that drives circumstantial non-linearity is 1D…

Suppose we were to break up the perturbation into two additive components: sources receivers

Non-linear terms would then accumulate as follows (loosely!). Non-linear in A(z) only (which produces closed-forms) is: …one A is “pulled out”: A(1+A+A2+A3+…), and the (.) terms ACT on it.

If the goal is “non-linear overburden (A), linear diffractor (B)”, then can we arrange to segment and capture only the related set of terms: …such that the B is “pulled out”: B(1+A+A2+A3+…), and the (.) terms ACT on it in the same way?

The first few FSS terms of the A only subset are:

The first few FSS terms of the A only subset are: …slightly different, since we have forced kg  ks in preparation for the 2D target…

…and in fact the same patterns are followed when we grab ahold of B: These patterns are slightly more complicated, but noticing the Taylor’s series expansion of the product of 2 exponentials in the sum of these terms…

…and in fact the same patterns are followed when we grab ahold of B: so, again in the case of the single scatterer B = B0 c.f.

A hybrid approximation is possible that showcases the construction of diffractions by the FSS, but yet allows the analytical/computational benefits of the closed-form. The resulting expression tells us about what is going on here. Something akin to a new transmitting Green’s function is being constructed, that is highly non-linear in the perturbation. The results so far are “leading order”. Similar manipulations as have been discussed elsewhere (Innanen, 2005; Liu et al., 2005, etc.) are possible to accommodate larger contrast media.

A numeric example Large contrast 2-layer overburden, small contrast scattering target:

A numeric example Three shot records. Result without the overburden.

A numeric example Three shot records. Result including the overburden.

Why a primary approximation with the FSS?
We have seen, and will see, some results due to a subseries of the ISS that couples the imaging-inversion problem: - framework for task-separated Q processing - way-point for derivation of imaging-only algorithms It is the result of the capture of a subset of the full ISS. It turns out that a direct order-by-order inversion of the primaries subset of the FSS exactly re-creates the coupled imaging-inversion algorithm. There are degrees of symmetry between FSS/ISS methods, and this coupling entity experiences a very high degree.

Why a primary approximation with the FSS?
This is part of the basis for planned research. Numerics. -Recursively compute RP, TDD, TDU for 2D models. -Develop empirical evidence of satisfactory event approximation. 2. Direct inversion. -Carry out order by order inversion of FSS expressions with varying levels of complexity. -Look for mechanisms of direct non-linear processing of diffractions (Liu et al.) 3. Explore capability of FSS for modelling reflections and transmissions from/through complex structure.

Conclusions The forward scattering series is a 3D volume-
scattering wave-theoretic forward modelling procedure. Inverse scattering series methods often derive from first understanding its activity w.r.t. a reflection seismic event. 2. Depth-ordering of scattering geometries leads in 1D to a non-linear framework for approximating reflected primaries: analytic/numerical validation is possible. 3. With the correct choice of domain(s) in which to represent the Green’s function, these ideas are extendable to 2D/3D. Closed forms are not available, but recursive forms are. Order-by-order R-T relations appear.

Conclusions 4. Hybrid approximations that discriminate between portions of the perturbation, treating (e.g.) the overburden non-linearly and the scattering target linearly, are possible through a term-by-term analysis of the FSS. 5. If the portion of the model (e.g. the overburden) to be treated non-linearly is 1D, even if the rest of the model is not, closed-forms are available. 6. Within this framework, the class of FSS primary approximation terms under study appear to be acting to effectively create new, transmission-only Green’s functions that are highly non-linear in the perturbation.

Conclusions 7. These lessons tell us how to use the FSS efficiently to model meaningful events in a transmitted or reflected seismic data set. 8. They also inform, directly and indirectly, the generation of direct non-linear methods for processing primaries, based on the inverse scattering series. 9. This latter fact is the main area of current study w.r.t. diffractions and other 2D effects.

Non-linear forward scattering series expressions for reflected-primary and transmitted-direct wavefield events Kristopher A. Innanen University of Houston.

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Non-linear forward scattering series expressions for reflected-primary and transmitted-direct wavefield events Kristopher A. Innanen University of Houston.

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