Dec 30, 2015
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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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
WHERE ?WHERE ?
HOW BIG ?HOW BIG ?
WHERE ?WHERE ?
HOW BIG ?HOW BIG ?
x in encoded nInformatio
A in encoded nInformatio x
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 ….
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
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
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 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
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)
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
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
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 … …
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
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 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