1. A note: data requirements for inverse theory
Arthur B. Weglein
M-OSRP 2006 Annual Meeting, June 5 ~ June 7, 2007
M-OSRP report pages:

2. Five Key points 1. What does linear in the data mean?
2. Linear in what data?
3. Conservation of dimension (having enough degrees of freedom in your data to “solve” an equation) is not a sufficient condition to define “what data” and being able to solve an equation (in isolation) is not the same as finding a physically meaningful solution or even a linear estimate;

3. Five Key points
4. Solving an equation without the context and framework within which that equation resides, and ignoring the assumptions that lead to that equation is a dangerous path towards an inverse illusion;
5. Implications for data collection and target identification.

4. The ISS answers these questions
Eq.(3) can be expanded in a forward scattering series

5. The ISS answers these questions
The scattered field can be defined as
D, where

6. The Inverse Scattering Series
Then
and

7. 1D Acoustic case
Let’s consider the following logic. Eq. 22 is an exact equation for the linear estimates and , and choosing two (or more) will allow you to solve Eq. 22 for and
For a single reflector model, The RHS of Eq.22 can be rewritten as:
The amplitude of the step-function in Eq. 23 is

8. 1D Acoustic case
Separately,
Eq. 25 is exact

9. The Puzzle exact exact From ISS From Zoepritz:
How can you reconcile Eq.26 being exact with Eq.25 being exact?
We are forced to conclude that consistency between Eq.25 and 26 requires that must be functions of
exact

10. If we think are functions of
and, if you choose two values of , say then Eq.27 will lead to two equations with four unknowns,
What is the problem here? We have forgotten the basic meaning and starting point in defining
In an ISS expansion for a parameter in orders of the data it is critically important to assure that the data in terms of which you are expanding the parameter is sufficient to determine that parameter. The data needed to determine a parameter is dependent upon what other parameters are (or are not) in your model; i.e., it depends on the context within which that parameter resides.
Not a positive moment.

11. Now consider a two parameter world defined by the expansions of in orders of the data. In this case, if we suppose that are expandable in terms of data at two different plane wave angles assuming that such a relationship between and is sufficient to determine then we can write the series
for as follows:
and in a compact notation
where is the portion of linear in the data set Similarly
If the model only allowed bulk modulus changes (no density changes), then

12. The lesson here is that the inverse problem doesn’t start with
Now in the two parameter inverse problem, the data is and then is equal to
and is linearly related to will depend upon which particular angles were chosen, and that is anticipated and perfectly reasonable, since being a linear approximation in the data could (and should) be a different linear estimate depending on the data subset you are considering.
Eq.30, a matrix equation, is the first term in the ISS and determines , the linear estimate of
The lesson here is that the inverse problem doesn’t start with
but with and the latter equation is
driven by a view of what data can determine the operator V

13. This might seem like a somewhat useless academic exercise, since Eq
This might seem like a somewhat useless academic exercise, since Eq. 30 is the equation you would have solved for if you just ignored their dependence entirely. It is anything but that.
There are at least two problems with that conclusion. The value of the above analysis is:
(1) with independent of , you have difficulty claiming or satisfying the important requirement that the first equation in the inverse series is exact;
(2) more importantly you can get into serious conceptual and practical problems in the elastic case if you don’t have very clear grasp of the underlying inverse issues and relationships in the acoustic case.

14. Elastic case
Linear inversion of a 1D elastic medium

15. =

17. Direct nonlinear inversion of a 1D elastic medium

18. and so on, the four components of the data will be coupled in the nonlinear elastic inversion. We cannot perform the direct nonlinear inversion without knowing all components of the data.
As shown in Zhang and Weglein (2005) and this note, when the work on the two parameter acoustic case is extended to the present three parameter elastic case, it is not just simply adding one more parameter, but there are more issues involved.
Even for the linear case, the linear solutions found in (71) (74) are much more complicated than those of the acoustic case. For instance, four different sets of linear parameter estimates are produced from each component of the data. Also, generally four distinct reflector mis-locations arise from the two reference velocities (P-wave velocity and S-wave velocity).

19. The three parameters we are seeking to determine
What data? The answer is once again the data needed to determine those three quantities. What H. Zhang’s thesis has demonstrated for the first time is not only an explicit and direct set of equations for improving upon linear estimates of the changes in those elastic properties, but perhaps equally and maybe even more important, is for the first time the absolutely clear data requirements for determining
The data requirements are
for a 2D earth and generalize to a 3 × 3 matrix for a 3D earth with SH and SV shear waves. =

20. The 2D message is delivered in Eq. 77 , Eq
The 2D message is delivered in Eq. 77 , Eq that the first non-linear contribution to requires that data; and, hence the complete determination of those elastic quantities also require that data set.
The logic is as follows:
i.e., linear in the data needed to determine

21. Inverting alone for while mathematically achievable is a challenged and incorrect linear relationship since what you determine from that procedure doesn’t represent the linear estimate of those quantities in terms of a data that can actually determine those quantities.
Solving for from alone is an injured or challenged linear estimate. The inverse scattering series and task specific subseries need to : (1) treat the linear term with respect and then (2) the higher order terms can carry out their purpose.

22. If you injure the linear estimate, the inverse scattering series cannot recover or compensate–it wants the linear estimate to be the linear estimate, and never expects it to be exact or close to exact, but it never expects it to be less than linear, as well. Let linear be linear.
The power and promise of the inverse scattering series derives from its deliberate and physically consistent and explicit nature. It recognizes that when you perturb anything in a medium the associated perturbation in the wavefield is always non-linearly related to that change.

23. The inverse implies that the medium perturbation is itself non-linearly related to the perturbation in the wavefield; including the change in the wavefield on the measurement surface.
That’s it, Eq. 84 and 85. That is all you assume, and that is hard to argue against.
Beyond that point the process and procedure for determining
is out of your hands and away from your control. How you find V1 from D is prescribed and what you do with V1 to determine V2 is also prescribed. That non-linear explicit and direct nature, and the steps to determine those terms , . . are not decision making opportunities for you.

24. If you decide what to do with V1 rather than have the non-linear relationship between data and V decide, then you step away from a single and defined physics into, e.g., the math world of iterative linear inversion or model matching. How do you formulate a multiple removal algorithm concept in iterative linear inverse or model matching scheme? The latter immediately aim to either improve or match the models properties with the subsurface.
From the inverse scattering series perspective, the latter all or nothing strategy is: (1) missing the opportunity to achieve other useful but less daunting tasks, i.e., multiple removal and depth imaging; and (2) moving at the first step straight into the most challenging task: parameter estimation, with all of the pitfalls of insufficient model types and bandwidth sensitivities.

25. For the inverse scattering series the decisions are not under your control or influence. It is away from you and it is carrying out its single-minded purpose. It has one physical model and a single unchanged separation of the earth into a reference medium and the perturbation and an all at once set of direct equations to solve.
Only that 100% physics consistent inverse formalism predicted that you required
to even linearly estimate elastic properties.

26. Iterative linear tries to substitute a set of constantly changed problems with linear updates for a single entirely prescriptive, consistent and explicit nonlinear physics. The latter is the inverse scattering series, the former has an attraction to linear inverses (and generalized inverses) which has no single physical theory and consistency.
Linear inversion and generalized inverse theory are part of standard graduate training in geophysics; and, hence it’s easy to understand trying to recast the actual non-linear problem into a set of iterative linear problems where the tools are familiar. The model matching schemes and iteratively linear inversion are reasonable and sometimes useful but they are more math than physics and have no way to provide the framework for inversion that staying consistent with the physics will provide.

27. The table that follows represents an informal report card of different inverse methods:
where:
V Violates ( or can violate);
H Honors ( respects);
A= good;
F= bad;
C=Forget about it.
The table shows a report card comparing model matching, iterative linear, and the inverse scattering series. There are times that model matching can be an effective tool even though on a scale of F to A it is F in terms of physics driven.
Modeling matching for the subtraction of 3D multiples earns it a F for compute requirements compared at A for ISS.

28. There is a unique and unambiguous data requirement message sent out from the inverse scattering series. Other methods and approaches that look at the inverse problem either linear or beyond linear, e.g., iterative linear or model matching have never and will never provide that clarity and definition. We can model-match Dpp or iteratively invert Dpp until the cows come home and you will find ambiguities and resolution challenges, and when those methods use more data they sometimes produce less ambiguity and better resolution, but we don’t know why.

29. As a final remark it is interesting to note that the first and linear term of the elastic inverse problem was not only influenced by the non-linear term, it was in fact defined by that term. That data requirement message, along with the entire inverse series apparatus, results from the observation that the perturbed wavefield and the perturbation are non-linearly related. Honor and respect that fundamental non-linear relationship and a physics driven set of consistent, deliberate and purposeful algorithms and a clear platform and unambiguous framework (rather than anecdotal experiences) are the dividend and reward.