I sincerely thank
My great amigo
Zoli
for being Zoli
I sincerely thank
this amigo
and
this amigo
for the kind invite
I wish to, humbly and most
sincerely, thank the
The Society that publishes
2 of the very best
and holds meetings in exotic
places
for honoring me with this
marvelous and unique adventure
This tour would have been a rout
without
Judy Wall
who organised it ALL !!!
Imagine. Angels do exist in the sky.
With additonal thanks to
Tury Taner, what can I say?, he who has done it all.
Enders Robinson, he was and is, numero uno.
Sven Treitel, there are no words, except, Sven.
Arthur Weglein, my friend, my teacher.
Mauricio Sacchi, without whom Tad would be Tad who?
Those marvelous friends, colleagues, students, who must
assume full responsibility for making me who I have
become.
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and now
the taaaalk
The role of
Amplitude
and
Phase
in
Processing
and
Inversion
Tadeusz Ulrych
I have chosen this title,
because I can
talk about
ANYTHING !!
This presentation was
prepared while partying
in the local bar,
illustrated in the next
slide
Definitions
Consider x = s + n
n is " all other stuff" , and is generallycalled" noise"
Letting F represent Fourier transformation
X = F [ x] = A x e
i  x
where
A x is the amplitude spectrum
 x is the phase spectrum
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 for
ONE 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 ?
Information encodedin x
HOW BIG ?
Information encodedin Ax
Relative “Importance” of
Ax and x
?
Original
x only, Ax = 1
INTRODUCTION
Mathematics is
Beautiful.
However, it is tiresome to digest.
Therefore, this talk contains
as little of this beauty as possible.
 Please remember, that the magic
of mathematics lies in its physical
interpretation. For example ….
Question
Why is it true that
(-1)1/2
x
=
Because,
as is well known
(-1)1/2 = i
and i is an operator
that rotates by 90o
Amplitude & Phase
in blind deconvolution
The Enders example
The Man
Enders Robinson
The canonical model for the seismogram
x t = w t ¤ qt + n t
xt
is the seismogram
signature
= w t ¤ qist the
+ nsource
t
w t ¤ q t + nistthe Greens function, the reflectivity
t
qt + n t
is ‘everything else’, the noise
This equation,
x t = w t ¤ qt + n t
is 1 equation with 2 unknowns.
This is akin to
7= a + b
and what is a and b
uniquely ?
This, of course, is an impossible problem
unless
a priori constraints are known
or, at least,
assumed
Some more thoughts
regarding
Phase
OUTLINE for the next few slides
 POCS and only-phase reconstruction
 Phase and cepstral processing
 Summary
POCS
Projection onto convex sets
POCS attempts to solve an
underdetermined, generally nonlinear,
inverse problem
G[x]+n=d
where
G is a nonlinear operator
A convex set, A, is one for which the line
joining any two points, x and y, in the set, is
totally within the set.
In other words, a set A in a vector space is
convex,
iff  x and y Є A
λx + (1 - λy) Є A  0 ≤ λ ≤ 1
Illustrating convex and non-convex
sets
A convex set
A non-convex set
Alternating POCS
Iterative projection onto convex sets
Possible stagnation point when
one of the sets is non-convex
Application of alternating POCS
to the problem of reconstruction
from phase-only to obtain the
only-phase image
The image, of finite support , is
a convex set.
The set of constraints, the
thresholded image, is also
another convex set.
Phase-only
Only-phase
Original
Phase in Cepstral analysis
Phase is fundamental in cepstral
processing
 Phase must be unwrapped
 Phase must be detrended
 A serious problem is additive noise
The cepstrum (complex) is defined as
-1
C(n) = F {ln[A(ω)] + iΦ(ω)}
where
-1
F
is the inverse Fourier transform
Application of cepstral analysis to
thin bed blind deconvolution
 Compute cepstrum for each trace
 Stack the cepstra
 Transform back to the time domain
 Deconvolve with estimated wavelet
The original synthetic section
The original reflectivity
The recovered wavelet
Usual approach to
deconvolution with ‘known’
source wavelet
R(f)=X(f)W(f)H/(W(f)W(f)H+k)
f-domain deconvolution
BUT, we can do better!
By utilizing a concept which we,
and particularly
Jon Claerbout and Mauricio Sacchi,
have championed for over a decade.
The principle of
PARSIMONY
some details to follow
The original reflectivity
Sparse deconvolution
f-domain deconvolution
Summary thus far
Phase contains the vital information
about location
Only-phase reconstruction demonstrates
the flexibility of POCS in inverse problems
Proper phase processing leads to useful
cepstral decompositions
Some details concerning
PARSIMONY
or
SPARSENESS
Thanks to Mauricio Sacchi for help with PARSIMONY
The concepr of Sparseness
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 …
An hour-long recording in the night sky
Processing pre-Burg
Processing pre-Burg
1.0
3.0
Frequency (cycles/hour)
5.0
Why extend with 0’s ?
Why not ?
Is this not the least presumptive ?
Only if the star lived for 1 hour
Processing post-Burg
?
?
Question?
How does one turn a
? into mathematics?
John Burg’s answer:
?=
Processing post-Burg
1.0
3.0
Frequency (cycles/hour)
5.0
and the actual fabricated star …
Key points of this part
q Importance of sparseness in the recovery
of low/high frequencies
Spectral Extrapolation
Sparse Inversion
Blind Deconvolution Methods (MED,
ICA etc.,)
q Assumptions for the recovery of missing
frequency components
A few words about the
problem
Recovery of Green’s function from band limited data
*
s(t) = w(t) r(t) + n(t)
s = Wr+ n
Seismogram = Source
Impulse Response + Noise
The required inversion
is performed by
J   (m norm) (d constr.)
We use:
p(m | d)  p(d | m) p(m)
to obtain J
Priors to model sparse signals
Two well-studied priors for the solution of inverse
problems where sparsity is sought:
Laplace
Cauchy
These priors translate into regularization
constraints for the solution of inverse problems
The latter is done via the celebrated Bayes
Theorem
How does it work?
Define a cost function (derived from Bayes) and
minimize it
If all the hyper-parameters of the problem were
properly chosen, the minimization should lead
to solutions that
– a) honor the data
– b) are simple (Sparse)
A sparse solution is associated with a signal
with high frequency content. This is why sparse
solutions are often used for problems of
bandwidth recovery.
Some Math…..
R(r)  | rk |
ql1 norm
k
R(r)   ln(1
qCauchy Norm

qBayesian Cost to minimize:
k
rk2

2
)
J || Wr d ||22 2 R(r)


2

J = Misfit +
(Regularization term derived from prior)
Solution
e.g. for regularization using the Cauchy norm
∇J = ∇{| | Wr - d | |22 + 2 R (r )} = 0
r = [ W W +  Q (r)] W
T
2
-1
T
1
Q ii = 2
 + ri2
The last equation is solved using an iterative
algorithm to cope with the nonlinearity
Damped LS: all the unknown samples are damped by the
same amount
Qii  1
Cauchy: adaptive damping
1
1
Qii  2
r
 2
2
i 0
  ri

1
Qii  2
r
0
2
i 
  ri
Adaptive damping is what leads to sparse solutions

Example: Non-Gaussian Impulse Response
model via a Gaussian Mixture
More area under
green curve
SPARSENESS
Sparsity is controlled by the mixing parameter
Mixing Parameter p=0.8
Data
True impulse
response
Predictd
data
Estimated
impulse response
Mixing Parameter p=0.2
Data
True impulse
response
Predicte
d data
Estimate
impulse response
Key features for proper recovery
of the impulse response
# Sparseness
# Bandwidth
Source BW (p=0.5)
Error = difference between true and estimated impulse response
Source functions used in the
simulation
AR Prediction in the f-domain
?
?
?
AR Gap-filling algorithm
AR Gap-filling algorithm (contd.)
AR Predictive Extension
Time
True
Recovered
Input BL signal
Frequency
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
algorithm
Data without a free surface
R()
1
Data with a free surface
R f ()
1
contains
free-surface multiples.
Free surface demultiple algorithm
R( )
R()
 R()
 R2 ( )
= primaries and
internal multiples
R f ( ) = primaries, free
surface multiples
and internal
1
multiples
R( )
Total upfield R f ( ) 
1  R( )
R f ( )
and, R ( ) 
1  R f ( )
R( )  R f ( )  R f ( )  R f ( ) 
2
3
Free surface demultiple example
t1
2t2
2t1
t1 + t2
t2
R f ( t )  R1 ( t  t1 )  R2'  ( t  t 2 )  R12 ( t  2t1 )  R2'2 ( t  2t 2 ) 
R f ( )  R1e it1  R2' e it 2  R12 e 2 it1  R2'2 e 2 it 2  2 R1 R2' e i ( t1  t 2 ) 
2
2 2 i t 1
f
1
R ( )  R e
' 2 2 i t 2
R e
2
 2R R e
'
1
2
2 i ( t 1

t2 )
 ...
t1 + t2
2t1
2t2
So R f ()  R2f () precisely eliminates all free
surface multiples that have experienced one
downward reflection at the free surface.
The absence of low frequencies (and in fact any
other frequencies) plays absolutely no role in this
prediction.
Please note that this Inverse Scattering approach to the
attenuation of both surface and internal multiples, does
not require knowledge of the velocity structure of the
subsurface
Measurement surface
subsurface
Mississippi Canyon
Water Bottom Top Salt
Base Salt Internal multiple
Water Bottom
Top Salt
Base Salt
Internal multiple algorithm
b3 ( k g , k s , q g  q s ) 
 1 


 2 
2  

 dk1dk2e
iq1 ( e g  e s ) iq2 ( e g  e s )
e
  

 dz1e
i ( q g  q1 ) z1
b1 ( k g , k1 , z1 )

z1


z2
i ( q2  q s ) z3
i (  q1  q2 ) z 2
dz
e
b
(
k
,

k
,
z
)
dz
e
b1 ( k 2 , k s , z3 )
1
1
2
2 
3
 2
where
b1 ( k g , k s , q g  q s )  2iqs D( k g , k s ,  )  2iqsG s ( k g , k s ,  )
and
z3  z 2 ,
z1  z 2
Araújo and Weglein (1994)
The role of phase is clear and is of
central importance.
q Surface multiple attenuation involves
convolution.
q Internal multiple attenuation involves
both convolution and correlation.
Amplitude is, of course, also important.
However, it is much less crucial than Phase.
The reason is that if the Location is wrong,
multiple attenuation will give birth to more
multiples.
(perhaps with the correct amplitude)
Mississippi Canyon
Input
Predicted
Output
multiples (2D)
Input Predicted
Output
multiples (2D)
1.7
Water
bottom
Seconds
Top salt
Base salt
Internal
multiples
3.4
Common Offset Panel (1450 ft)
Common Offset Panel (2350 ft)
It is time to
Fly away
But,
One last slide
Thank you
for your
Patience