A FABLE OF PROCESSING VERSUS INVERSION

Everybody knows about getting travel times correct. Let us define a data space ${\bf d}$, a model space ${\bf m}$,and an operator ${\bf K}$ called ``Kinematics'' that gets the travel times correct. The conjugate-transpose ${\bf K}'$ gives processing operators like ``moveout and stack'', and like the sequence ``normal moveout, dip moveout, stack, zero-offset migration''.

Different people have different ideas about amplitudes. We will have four different amplitudes to deal with here. Think of them as diagonal matrices ${\bf A}$,${\bf B}$,${\bf C}$,and ${\bf D}$that we all know how to invert.

Professor Fussy Logic, a deep thinker from The Continent, is not satisfied with kinematics because he likes to get amplitudes correct. This gives him lots of weighting functions. Some of his weights are in model space and some in data space. Some of his weights are best represented in Fourier space, such as $\sqrt{-i\omega}$.We cannot fully decipher all his thoughts about amplitudes but to some (unknown) level of approximation we can think about his work as  

$${\bf A K B m}\quad = \quad{\bf d}$$ (3)

Professor Fussy Logic's view of seismic data processing is that $({\bf AKB})$ is a linear operator that his students can pass to the conjugate-gradient solver which must provide the correct answer. Since the data space and model space have about a million pixels, the CG solver needs about a million iterative applications of the seismic operator ${\bf AKB}$ and its adjoint. We hope his students can get good results after a dozen CG iterations (which is computationally feasible) but we are fearful that Professor Fussy Logic's modeling might not include all the messy aspects of real data and that any near null space could lead to disaster.

Dr. Mucho Moola, is founder-owner of a successful geophysical contracting company. (It would be impolitic of me to reveal his exact identity but you can think of him as an MIT PhD of the '60s or an SEP PhD of the '70s.) Dr. Mucho Moola likes to get amplitudes correct too, but he could tell you a dozen reasons why that has nothing to do with a correct modeling theory. Starting from his notion that ``all data is important'' and from the kinematic operator ${\bf K}$,by various schemes that are ``company confidential'' he finds weighting functions ${\bf C}$ and ${\bf D}$ such that  

$$({\bf CKD})'({\bf CKD}) \quad\approx\quad {\bf I}$$ (4)

In competitive bidding situations, he offers oil companies the simple process  

$$\hat {\bf m} \quad = \quad({\bf CKD})' {\bf d}$$ (5)

For very important surveys he has a special secret nonlinear process ${\bf Q}$that is really just a combination of AGC and decon which he uses to supplement equation (5). For lots of extra bucks, oil companies can purchase his custom result  

$$\hat{ \hat {\bf m}} \quad = \quad Q( ({\bf CKD})' Q({\bf d}) )$$ (6)

Ms. Sepper, a new student at SEP, wants to reconcile logic with success. She figures that Professor Fussy Logic has the problem formulated right but his solution is impractical whereas Dr. Mucho Moola is successful (despite ignoring the physical theory) because he has a good approximation to the inverse of a matrix with the same kinematics as Professor Fussy Logic's matrix. She sets out to apply Dr. Mucho Moola's inverse method to Professor Fussy Logic's problem. Starting with equation (3), she premultiplies by ${\bf CA}^{-1}$and she also inserts ${\bf I}={\bf DD}^{-1}$ in a strategic location getting  

$$({\bf CKD}) {\bf D}^{-1} {\bf Bm} \quad = \quad{\bf CA}^{-1}{\bf d}$$ (7)

Then she premultiplies by ${\bf B}^{-1}{\bf D}({\bf CKD})'$,and invokes Dr. Mucho Moola's approximation (4) getting  

$${\bf m}_{\rm sep} \quad = \quad {\bf B}^{-1}{\bf DD}' {\bf K}' {\bf C}'{\bf C} {\bf A}^{-1}{\bf d}$$ (8)

Recognizing the importance of her result, Ms. Sepper plans to start a new geophysical contracting company specializing in the high end of the market.

1.

According to Ms. Sepper's formula, whose weights are more important, Professor Fussy Logic's ${\bf A}$ and ${\bf B}$or Dr. Mucho Moola's ${\bf C}$ and ${\bf D}$?

2.

In view of competition from Dr. Mucho Moola's secret Q process, how should Ms. Sepper advertise her product?

3.

How much would you like to invest in the new company? Why?