Fu Yang Jon Antoine Elitasep€¦ · r t is reﬂectivity (and residual) u τ are the free...

Fu Yang JonAntoine

ElitaYi

Thursday, May 24, 2012

Sparsity decon in the log domainwith variable gain

Jon Claerbout, Antoine Guitton, and Qiang Fu

SEP 2012 spring meetingMonterey, California

SEP report page 147, page 313

OLD NEWS: We seek sparse deconvolutionsby imposing a hyperbolic penalty function.

NEW: Although FT based, we find theoryfor arbitrary gain(t) and mute(t,x)

AFTER decon.

NEW: We have identified a long-needed regularization.

NEW: Results confirm benefit of “gain after decon”

Sparseness goals

The �2-norm decon forces a whiteness assumptionand forces a “minimum phase” assumption. Both bad.

The sparseness goal should yield a “best” spectrumand (hopefully) the most appropriate phase.

Enhance low frequency only when it aids sparsity.

Seek to integrate reflectivity to obtain log impedance.

Logarithmic parameterization

Lag-log space.

Strange!

D(ω) is the FT of the data.rt is reflectivity (and residual)uτ are the free parameters.. u0 = 0 is mean log spectrum.

rt = FT−1 D(ω) exp

τ �=0

uτZτ

Gain and sparsity

qt = gt rt

where:rt is the physical output of the filtergt is the given gain function, often t2

qt is the gained output, also calledthe “statistical signal” to be sparsified.

qt = gt rt

H(qt) =�

q2t + 1− 1

�(q) =q�

q2 + 1= softclip(q)

rt is the physical output of the filtergt is the given gain functionqt is the gained output,H(q) is the hyperbolic penalty function.Choose gt so that qt ≈ 1.“Sparsity” is 1 /

�t H(qt)

softly clipped residual

What percentile?

rt = FT−1 D(Z) e···+u2Z2+u3Z3+u4Z4+···

duτ= FT−1 D(Z) Zτe···+u2Z2+u3Z3+u4Z4+···

duτ= rt+τ

rt = FT−1 D(Z) e···+u2Z2+u3Z3+u4Z4+···

duτ= rt+τ You think you have seen this before....?

rt = FT−1 D(Z) e···+u2Z2+u3Z3+u4Z4+···

duτ= rt+τ

Residual orthogonal to fitting functionbecomes

Residual orthogonal to itself

No, you likely saw dt+τ .

Physical output gradientw.r.t. lag-log variable

Statistical gradient

amazing result coming11

rt = FT−1 D(Z) e···+u2Z2+u3Z3+u4Z4+···

duτ= rt+τ

qt = rt gt

duτgt = rt+τ gt

A crosscorrelation: Compute it in the Fourier domain.

∆uτ =�

dH(qt)duτ

τ �= 0

dH(qt)dqt

∆uτ =�

(rt+τ ) (gtH�(qt)) τ �= 0

Jon’s favorite theory slide.

the step

∆uτ =�

dH(qt)duτ

τ �= 0

dH(qt)dqt

∆uτ =�

(rt+τ ) (gtH�(qt)) τ �= 0

the step

A crosscorrelation: Compute it in the Fourier domain.

Jon’s favorite theory slide.

the softly clipped residual

∆uτ =�

dH(qt)duτ

τ �= 0

dH(qt)dqt

∆uτ =�

(rt+τ ) (gtH�(qt)) τ �= 0

the step

A crosscorrelation: Compute it in the Fourier domain.At convergence this is a delta function.

Special case: stationary L2 then r(t) is white.Amazing generalization to

(1) non-causal, (2) gain, and (3) sparsity!Jon’s favorite theory slide.

the softly clipped residual

From ∆u to ∆r

Skipping lots of algebra(including a linearization)given the gradient step ∆u = (∆uτ )and the residual r = (rt),the residual perturbation is ∆r = r ∗ ∆u.(“∗” is convolution)and the sparsity perturbation is∆qt = gt ∆rt.

Minimizing H(q + α∆q)

At each qt fit hyperbola to parabola (Taylor series).A sum of parabolas is a parabola. Easy getting α.

α = −�

t ∆qtH�t�

t(∆qt)2H ��t

Update the residual q and unknowns u.Form new Taylor series and iterate.

Recall stationary �2: α = − (∆r · r)/(∆r · ∆r)

Newton’s method.

Quick peek at the algorithm:math to code key

Lower case letters for variables in time and spacelike d = d(t, x), dq = ∆q(t, x), u = uτ .

Upper case for frequency domain likeR = R(ω, x), and dU = ∆U(ω).

Asterisk ∗ means multiply within an implied loopon t or ω.

The algorithm is brief.

D = FT(d)

U = 0.

Remove the mean from U(omega).

Iteration {

dU = 0

For all x

r = iFT( D * exp(U))

q = g * r

dU = dU + conjg(FT(r)) * FT(g*softclip(q))

Remove the mean from dU(omega)

For all x

dR = FT(r) * dU

dq = g * iFT(dR)

Newton iteration for finding alfa {

H’ = softclip( q )

H’’ = 1/(1+q^2)^1.5

alfa= - Sum( dq * H’ ) / Sum( dq^2 * H’’)

q = q + alfa * dq

U = U + alfa * dU

Instability! Yikes!

Sometimes there are time shifts.Sometimes the polarity is wrong.

I’m going to work on velocity instead.

Instability! Yikes!

Try preconditioning.Try regularization.

I tried them.I’d rather do Q tomography.

Instability! Yikes!Masking the gradient fails.Here are the sample historiesyou asked for.

I’m going to Houston.

Instability! Yikes!

Antoine! Help!22

Instability. Yikes!

CAntoine: I changed the gainby 10% and the spike jumpedfrom B to C.

Jon: Awful! I thought I hada great starting solution at B

Jon: Make me a movie as afunction of iteration.

Instability. Yikes!

C10 iterations:good spike at B,A&C small

200 iterations:maybe spikes at Amaybe spikes at Bmaybe spikes at Cothers small

with Antoine and Qiang Fu

“But when it’s good, it’s really good!Let’s look at some of the results.”

We’ll return to the stability problem later.

Bubble

Ricker Gain after filterGain before filter

26 first seen by Yang ZhangThursday, May 24, 2012

Prepare to comparegain before with gain after

data t-squared gain decon

data new decon t-squared gainqtrt

Compare

Gain after decon.

Gain before decon.

Scale up by 10x

Estimated shot

Gain after decon.

Gain before decon.

Scale up by 5x

Gain before filter

Low frequencyprecursor

bubble

Produced by Antoine

Instability. Yikes!

C10 iterations,spikes at B,A&C small

200 iterations,maybe spikes at Amaybe spikes at Bmaybe spikes at Cothers small

Nonlinearity?

Instability. Yikes!

C10 iterations,spikes at B,A&C small

200 iterations,maybe spikes at Amaybe spikes at Bmaybe spikes at Cothers small

Nonlinearity?

Null space!!

Nobody has proven it is a null space problem.

But I think it is,

so I must come up with a regularization.

Basic Regularization

0 ≈ wτ (uτ − uτ )weights

prior model

But how to choose them?

Fancier Regularization

0 ≈�

�k wk,τ (uτ − uτ )

0 ≈ W(u− u)

but what to choose for W and u ?

unknown matrix

Intuitive Regularization

Reduces the phase near t=0, more like Ricker there.

36the happy discovery slide

0 ≈ wτ (uτ − u−τ )

Choose big wτ where |τ ≈ 0|.

rm(1)rm(2)rm(3)rm(4)rm(5)rm(6)

0 0 0 0 0 00 +1 0 0 0 −10 0 +1 0 −1 00 0 0 0 0 00 0 −1 0 +1 00 −1 0 0 0 +1

u(1)u(2)u(3)u(4)u(5)u(6)

Regularization

FFT notation in matrix,Fortran notation in vectors.

Report deadline

Only Antoine has seen the results

(if he hasn’t been too busy at work).

Any student had too much synthetic data?

Theory innovations

• Two-sided filters escape minimum phase.

• Use sparsity goal instead of whiteness.

• Apply gain and mute AFTER filtering.

Conclusions from testing

• Value of gain theory confirmed by two examples.

• Sparsity is not powerful enough to ensure a “best’’ phase. Regularization is needed.

• A long-needed regularization is identified.

Fu YangAntoine Elita

I’d like to thank the team.

We thank Western Geophysical for the Gulf of Mexico data set.

Jon Claerbout and Qiang Fu thank the sponsors of the Stanford Exploration Project.

Antoine Guitton thanks Repsol Sinopec Brasil SA and Geo Imaging Solucoes Tecnologicas em Geociencias Ltda.

We'd like to thank Yang Zhang for continued interest.

ACKNOWLEDGEMENT

The end

Fu Yang Jon Antoine Elitasep€¦ · r t is reﬂectivity (and residual) u τ are the free...

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