My great amigo

I sincerely thankI sincerely thank

Zoli

for being Zolifor being Zoli

this amigo

I sincerely thankI sincerely thank

this amigo

for the kind invitefor the kind invite

and

I wish to, humbly and I wish to, humbly and most sincerely, thank most sincerely, thank

thethe

The Society that The Society that publishes 2 of the very publishes 2 of the very

bestbest

and holds meetings in and holds meetings in exotic placesexotic places

for honoring me with this for honoring me with this marvelous and unique marvelous and unique

adventureadventure

who organised it ALL !!!

Imagine. Angels do exist in the sky.

This tour would have been a This tour would have been a routrout

without

Judy Wall

Tury Taner, what can I say?, he who has done it all.Tury Taner, what can I say?, he who has done it all.

Enders Robinson, he was and is, numero uno.Enders Robinson, he was and is, numero uno.

Sven Treitel, there are no words, except, Sven.Sven Treitel, there are no words, except, Sven.

Arthur Weglein, my friend, my teacher.Arthur Weglein, my friend, my teacher.

Mauricio Sacchi, without whom Tad would be Tad who?Mauricio Sacchi, without whom Tad would be Tad who?

Those marvelous friends, colleagues, students, who must Those marvelous friends, colleagues, students, who must assume full responsibility for making me who I have assume full responsibility for making me who I have become.become.

Tury Taner, what can I say?, he who has done it all.Tury Taner, what can I say?, he who has done it all.

Enders Robinson, he was and is, numero uno.Enders Robinson, he was and is, numero uno.

Sven Treitel, there are no words, except, Sven.Sven Treitel, there are no words, except, Sven.

Arthur Weglein, my friend, my teacher.Arthur Weglein, my friend, my teacher.

Mauricio Sacchi, without whom Tad would be Tad who?Mauricio Sacchi, without whom Tad would be Tad who?

Those marvelous friends, colleagues, students, who must Those marvelous friends, colleagues, students, who must assume full responsibility for making me who I have assume full responsibility for making me who I have become.become.

With additonal thanks toWith additonal thanks to

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and now

the taaaalk

and now

the taaaalk

The role of The role of

AmplitudeAmplitude andand PhasePhase

in in

ProcessingProcessing and and InversionInversion

Tadeusz UlrychTadeusz Ulrych

The role of The role of

AmplitudeAmplitude andand PhasePhase

in in

ProcessingProcessing and and InversionInversion

Tadeusz UlrychTadeusz Ulrych

I have chosen this title,I have chosen this title,

because I can because I can

talk about talk about

ANYTHING !!ANYTHING !!

I have chosen this title,I have chosen this title,

because I can because I can

talk about talk about

ANYTHING !!ANYTHING !!

This presentation This presentation was prepared was prepared

while partying in while partying in the local bar, the local bar,

illustrated in the illustrated in the next slidenext slide

Consider

spectrum phase the is

spectrum amplitude the is

where

tiontransforma Fourier represent Letting

noise"" called generally is and ,stuff" other all" is

x

x

ix

A

eA=]x[X

n

n+sx

x=

=

FF

Definitions

A brief story

Doug Foster arranges a presentation for Monday

Dr. Doug J. Foster This is Me

Sunday evening is slightly brutal

I cannot remember[1] How many participants?[2] Where is my presentation?

I have a Canadian cell with enough credit forONE question

What question do I ask?

How many participants?

or

Where is the presentation?

The answer to

HOW MANY?

is

AMPLITUDE

(goodbye presentation and future invitation)

The answer to

WHERE?

is

PHASE

(Oblivious to the number, I blindly carried on)

WHERE ?WHERE ?

HOW BIG ?HOW BIG ?

WHERE ?WHERE ?

HOW BIG ?HOW BIG ?

x in encoded nInformatio

A in encoded nInformatio x

Relative “Importance” ofRelative “Importance” of

and xxA

Original

?

1=xx A only,

INTRODUCTIONINTRODUCTIONMathematics is Mathematics is BeautifulBeautiful. . However, it is tiresome to digest.However, it is tiresome to digest.Therefore, this talk contains Therefore, this talk contains asas little of this beauty as possible. little of this beauty as possible.

Please remember, that the magicPlease remember, that the magic

of mathematics lies in its physicalof mathematics lies in its physical

interpretation. For example ….interpretation. For example ….

QuestionQuestion

Why is it true thatWhy is it true that

(-1)1/2

= x

Because,Because,as is well knownas is well known

(-1)(-1)1/2 1/2 = i= i

and i is an operatorand i is an operator

that rotates by 90that rotates by 90oo

Amplitude & Phase

in blind deconvolution

The Enders example

The ManThe Man

Enders Robinson

The canonical model for the The canonical model for the seismogramseismogram

xt = wt ¤qt +nt

xt is the seismogram

is the source signature

is the Greens function, the reflectivity

is ‘everything else’, the noise

This equation,This equation,

xt = wt ¤qt +nt

is 1 equation with 2 unknowns.is 1 equation with 2 unknowns.

This is akin to 7= a + b and what is a This is akin to 7= a + b and what is a and b and b

uniquely uniquely ??

This, of course, is an impossible problem This, of course, is an impossible problem

unless

a priori constraints are known

or, at least, assumed

unless

a priori constraints are known

or, at least, assumed

Some more thoughts Some more thoughts regardingregarding

Phase Phase

OUTLINE for the next few slidesOUTLINE for the next few slides

POCS and only-phase reconstructionPOCS and only-phase reconstruction

Phase and cepstral processingPhase and cepstral processing

Summary Summary

POCSPOCSProjection onto convex setsProjection onto convex sets

POCS attempts to solve anPOCS attempts to solve an

underdetermined, generally nonlinear,underdetermined, generally nonlinear,

inverse probleminverse problem

GG[x]+n=d[x]+n=dwhere where G G is a nonlinear operatoris a nonlinear operator

POCSPOCSProjection onto convex setsProjection onto convex sets

POCS attempts to solve anPOCS attempts to solve an

underdetermined, generally nonlinear,underdetermined, generally nonlinear,

inverse probleminverse problem

GG[x]+n=d[x]+n=dwhere where G G is a nonlinear operatoris a nonlinear operator

A convex set, A convex set, A,A, is one for which the line is one for which the line

joining any two points, joining any two points, xx and and yy, in the set, is, in the set, is

totally within the set. totally within the set.

In other words, a set In other words, a set AA in a vector space is in a vector space is

convex, convex,

iff iff xx and and yy Є Є AA

λx + (λx + (11 - λy) - λy) ЄЄ A A 0 ≤ 0 ≤ λ ≤ λ ≤ 11

A convex set, A convex set, A,A, is one for which the line is one for which the line

joining any two points, joining any two points, xx and and yy, in the set, is, in the set, is

totally within the set. totally within the set.

In other words, a set In other words, a set AA in a vector space is in a vector space is

convex, convex,

iff iff xx and and yy Є Є AA

λx + (λx + (11 - λy) - λy) ЄЄ A A 0 ≤ 0 ≤ λ ≤ λ ≤ 11

Illustrating convex and non-convexIllustrating convex and non-convex

setssets

A convex setA convex set A non-convex setA non-convex set

Alternating POCSAlternating POCSIterative projection onto convex setsIterative projection onto convex sets

Alternating POCSAlternating POCSIterative projection onto convex setsIterative projection onto convex sets

Possible stagnation point whenPossible stagnation point when

one of the sets is non-convexone of the sets is non-convexPossible stagnation point whenPossible stagnation point when

one of the sets is non-convexone of the sets is non-convex

Application of alternating POCSApplication of alternating POCSto the problem of reconstructionto the problem of reconstructionfrom phase-only to obtain thefrom phase-only to obtain theonly-phase imageonly-phase image

Application of alternating POCSApplication of alternating POCSto the problem of reconstructionto the problem of reconstructionfrom phase-only to obtain thefrom phase-only to obtain theonly-phase imageonly-phase image

The image, of finite support , The image, of finite support , isis

a convex set.a convex set.

The set of constraints, theThe set of constraints, the

thresholded image, is alsothresholded image, is also

another convex set. another convex set.

The image, of finite support , The image, of finite support , isis

a convex set.a convex set.

The set of constraints, theThe set of constraints, the

thresholded image, is alsothresholded image, is also

another convex set. another convex set.

Phase-only

Only-phase

Original

Phase in Cepstral analysisPhase in Cepstral analysis

Phase is fundamental in cepstralPhase is fundamental in cepstral

processingprocessing

Phase must be unwrappedPhase must be unwrapped

Phase must be detrendedPhase must be detrended

A serious problem is additive noiseA serious problem is additive noise

Phase in Cepstral analysisPhase in Cepstral analysis

Phase is fundamental in cepstralPhase is fundamental in cepstral

processingprocessing

Phase must be unwrappedPhase must be unwrapped

Phase must be detrendedPhase must be detrended

A serious problem is additive noiseA serious problem is additive noise

The cepstrum (complex) is The cepstrum (complex) is defined asdefined asThe cepstrum (complex) is The cepstrum (complex) is defined asdefined as

C(n)C(n) = {ln[ = {ln[AA((ωω)] + )] + iiΦ(ω)Φ(ω)}}

-1F

-1Fwhere is the inverse Fourier transform

Application of cepstral analysis toApplication of cepstral analysis tothin bed blind deconvolutionthin bed blind deconvolution

Compute cepstrum for each Compute cepstrum for each tracetrace

Stack the cepstraStack the cepstra

Transform back to the time Transform back to the time domaindomain

Deconvolve with estimated Deconvolve with estimated waveletwavelet

Application of cepstral analysis toApplication of cepstral analysis tothin bed blind deconvolutionthin bed blind deconvolution

Compute cepstrum for each Compute cepstrum for each tracetrace

Stack the cepstraStack the cepstra

Transform back to the time Transform back to the time domaindomain

Deconvolve with estimated Deconvolve with estimated waveletwavelet

The original synthetic sectionThe original synthetic section

The original reflectivityThe original reflectivity

The recovered waveletThe recovered wavelet

Usual approach toUsual approach todeconvolution with ‘known’ deconvolution with ‘known’

source waveletsource wavelet

R(f)=X(f)W(f)R(f)=X(f)W(f)HH/(W(f)W(f)/(W(f)W(f)HH+k)+k)

f-domain deconvolutionf-domain deconvolution

BUT, we can do better!BUT, we can do better!

By utilizing a concept which we,By utilizing a concept which we,

and particularly and particularly

Jon Claerbout and Mauricio Sacchi,Jon Claerbout and Mauricio Sacchi,

have championed for over a decade.have championed for over a decade.

The principle ofThe principle of

PARSIMONYPARSIMONYPARSIMONYPARSIMONY

some details to followsome details to follow

The original reflectivityThe original reflectivity

Sparse deconvolutionSparse deconvolution

f-domain deconvolutionf-domain deconvolution

Summary thus far

Phase contains the vital information about location

Only-phase reconstruction demonstratesthe flexibility of POCS in inverse problems Proper phase processing leads to usefulcepstral decompositions

Summary thus far

Phase contains the vital information about location

Only-phase reconstruction demonstratesthe flexibility of POCS in inverse problems Proper phase processing leads to usefulcepstral decompositions

PARSIMONY

oror

SPARSENESS

PARSIMONY

oror

SPARSENESS

Some details concerningSome details concerning

Thanks to Mauricio Sacchi for help with PARSIMONY

The concepr of Sparseness

I honour the sparse ones ..I honour the sparse ones ..

Nicholas Copernicus Pierre de Laplace Thomas Bayes Sir Harold Jeffreys Edwin Jaynes John Burg

and, of course, the sparsest of them all and, of course, the sparsest of them all … …

An hour-long recording in the night sky

Processing pre-Burg

Processing pre-Burg

Frequency (cycles/hour)

5.03.0

1.0

Why extend with 0’s ?

Why not ?

Is this not the least presumptive ?

Only if the star lived for 1 hour

Processing post-Burg

? ?

John Burg’s answer:

Question?

How does one turn a ? into mathematics?

? =

1.0 3.0 5.0

Frequency (cycles/hour)

Processing post-Burg

and the actual fabricated star …

Importance of sparseness in the recovery Importance of sparseness in the recovery

of low/high frequenciesof low/high frequenciesSpectral ExtrapolationSpectral Extrapolation Sparse InversionSparse InversionBlind Deconvolution Methods (MED, Blind Deconvolution Methods (MED,

ICA etc.,) ICA etc.,)

Assumptions for the recovery of missing Assumptions for the recovery of missing frequency componentsfrequency components

Key points of this part

A few words about the A few words about the problemproblem

n(t)r(t)w(t)s(t) +=

Seismogram = Source Impulse Response + NoiseSeismogram = Source Impulse Response + Noise

nWrs +=

*

Recovery of Green’s function fromRecovery of Green’s function from b band and limited datalimited data

The required inversionThe required inversionis performed byis performed by

.)()( constrnormJ dm

We use:

)()|()|( mmddm ppp

to obtain J

Priors to model sparse signalsPriors to model sparse signals

Two well-studied priors for the solution of inverse Two well-studied priors for the solution of inverse problems where sparsity is sought:problems where sparsity is sought:

LaplaceLaplace

CauchyCauchy

These priors translate into These priors translate into regularizationregularization constraints constraints for the solution of inverse problemsfor the solution of inverse problems

The latter is done via the celebrated The latter is done via the celebrated Bayes Bayes TheoremTheorem

How does it workHow does it work??Define a cost function (derived from Bayes) and Define a cost function (derived from Bayes) and minimize itminimize it

If all the hyper-parameters of the problem were If all the hyper-parameters of the problem were properly chosen, the minimization should lead to properly chosen, the minimization should lead to solutions thatsolutions that– a) honor the data a) honor the data – b) are simple (Sparse)b) are simple (Sparse)

A sparse solution is associated with a signal with A sparse solution is associated with a signal with high frequency content. This is why sparse solutions high frequency content. This is why sparse solutions are often used for problems of bandwidth recoveryare often used for problems of bandwidth recovery. .

Some Math…..Some Math…..

l1l1 norm norm

Cauchy NormCauchy Norm

Bayesian Cost to minimizeBayesian Cost to minimize::

R(r) | rk |k

R(r) ln(1rk

2

2 )k

J || Wr d ||22 2R(r)

J = Misfit + (Regularization term derived from prior)

2

SolutionSolution

2

i

2ii

T12T

22

2

r+

1=Q

WQ(r)+WW=r

0=rR+dWr=J

-][

)}(||{|| -∇∇

e.g. for regularization using the Cauchy norm

The last equation is solved using an iterative algorithm to cope with the

nonlinearity

Damped LS: all the unknown samples are damped by the Damped LS: all the unknown samples are damped by the same amountsame amount

Cauchy: adaptive dampingCauchy: adaptive damping

AdaptiveAdaptive damping is what leads to sparse solutionsdamping is what leads to sparse solutions

Qii 1

Qii 1

2 ri2 ri 0

1

2

Qii 1

2 ri2 ri 0

Example: Non-Gaussian Impulse ResponseExample: Non-Gaussian Impulse Responsemodel via a Gaussian Mixturemodel via a Gaussian Mixture

More area under green curve

Sparsity is controlled by the mixing parameterSparsity is controlled by the mixing parameter

SPA

RSE

NES

S

Mixing Parameter Mixing Parameter p=0.8p=0.8

Data

True impulse response

Predictd data

Estimated impulse response

Mixing Parameter Mixing Parameter p=0.2p=0.2

Data

True impulse response

Predicted data

Estimate impulse response

Key features for proper recovery Key features for proper recovery of the impulse responseof the impulse response

# Sparseness# Sparseness

# Bandwidth# Bandwidth

Source BW (Source BW (p=0.5)p=0.5)

Error = difference between true and estimated impulse response

Source functions used in Source functions used in the simulationthe simulation

AR Prediction in the f-domain

?

?

?

AR Gap-filling algorithm

AR Gap-filling algorithm (contd.)

True

Recovered

Input BL signal

Time Frequency

AR Predictive Extension

Summary

[1] The eye is attracted to the light,

but the mystery lies in the shadows.

[2] Gaussian pdf’s imply Least Squares.

[3] The mystery, the , lies in the heavy tails of nonGaussian pdf’s.

?

The role of Phase

in the attenuation of

Surface and Internal

Multiples

The next few slides have been

supplied by Arthur Weglein

a friend and mentor

The 1D FS multiple removal The 1D FS multiple removal algorithmalgorithm

Data without a free surface

1

1

)(R

)(fR

Data with a free surface

contains free-surface multiples.

Free surface demultiple algorithm

1)(R

Total upfield

and,

)( R )(2R

)()()()(

)(1

)()(

)(1

)()(

32

fff

f

f

f

RRRR

R

RR

R

RR

)(R = primaries and internal multiples

)(fR = primaries, free surface multiples and internal

multiples

Free surface demultiple example

)('

2122'

222

1'21

22'

21212

'211

212121 2)(

)2()2()()()(ttititititi

f

f

eRReReReReRR

ttRttRttRttRtR

t1 t2t1 + t2 2t1 2t2

...2)()(2'

21

22'

2

22

1

2 2121 ttititi

feRReReRR

)()( 2 ff RR So precisely eliminates all free So precisely eliminates all free surface multiples that have experienced one surface multiples that have experienced one downward reflection at the free surface.downward reflection at the free surface.

The absence of low frequencies (and in fact The absence of low frequencies (and in fact any other frequencies) plays absolutely no any other frequencies) plays absolutely no role in this predictionrole in this prediction..

t1 + t2 2t1 2t2

Please note that this Inverse Scattering approach to the Please note that this Inverse Scattering approach to the attenuation of both surface and internal multiples, does attenuation of both surface and internal multiples, does not require knowledge of the velocity structure of thenot require knowledge of the velocity structure of the

subsurfacesubsurface

subsurface

Measurement surface

Water Bottom Top Salt Base Salt Internal multiple

Water Bottom

Top Salt

Base Salt

Mississippi Canyon Mississippi Canyon

Internal multiple algorithm

2123

1

321322112

111121

2

3

22

2

1

2

32

1

221

1121

zzzz

kkGiqkkDiqqqkkb

zkkbedzzkkbedz

zkkbedzeedkdk

qqkkb

sgsssgssgsg

z

szqqi

zzqqi

gzqqieeiqeeiq

sgsg

s

gsgsg

,

and

),,(),,(),,(

where

),,(),,(

),,(

),,(

)()(

)()()(

Araújo and Weglein (1994)

Surface multipleSurface multiple attenuation attenuation involves involves

convolution.convolution.

Internal multipleInternal multiple attenuation attenuation involves involves

both convolution and correlation.both convolution and correlation.

The role of phase is clear and is ofThe role of phase is clear and is ofcentral importance.central importance.

Amplitude is, of course, also important.Amplitude is, of course, also important.

However, it is much less crucial than Phase.However, it is much less crucial than Phase.

The reason is that if the Location is wrong,The reason is that if the Location is wrong,

multiple attenuation will give birth to moremultiple attenuation will give birth to more

multiples.multiples.

(perhaps with the correct amplitude)(perhaps with the correct amplitude)

Mississippi Canyon Mississippi Canyon

1.7

3.4

Sec

ond

s

Common Offset Panel (2350 ft) Common Offset Panel (1450 ft)

Predicted multiples (2D)

Input Output Predictedmultiples (2D)

Input Output

Waterbottom

Top salt

Base salt

Internalmultiples

It is time toIt is time toIt is time toIt is time to

Fly away

But,But,

One last slideOne last slide

Thank you Thank you for yourfor your

PatiencePatience