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who organised it ALL !!!Imagine. Angels do exist in the sky.
This tour would have been a routwithout
Judy Wall
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 Radon would not be sparse.
Those marvelous friends, colleagues, students, who must assume full responsibility for making me who I have become.
With wholehearted thanks to
The role of
Amplitude and Phase
in
Processing and Inversion
Tadeusz Ulrych
This presentation was prepared while partying
in the local bar, illustrated in the next
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
nn+sx
x=
=
Φ
ΦωFF
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
INTRODUCTIONMathematics is Beautiful. However, it is tiresome to digest.Therefore, this talk contains
as little of this beauty as possible.
Please remember, that the magicof mathematics lies in its physicalinterpretation. For example ….
The canonical model for the seismogram
xt = wt ¤ qt + nt
x t
wt
is the seismogram
¤ qt
is the source signature
is the Greens function, the reflectivity
nt is ‘everything else’, the noise
This equation,
xt = wt ¤ qt + nt
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
Enders’ solution(obtained 45 years ago)
is called
Spiking Deconvolution
It is used in virtually unaltered form
~ 30 million times/day
POCSProjection onto convex sets
POCS attempts to solve anunderdetermined, generally nonlinear,inverse problem
G[x]+n=dwhere G is a nonlinear operator
Application of alternating POCSto the problem of reconstructionfrom phase-only to obtain theonly-phase image
The image, of finite support , isa convex set.The set of constraints, thethresholded image, is alsoanother convex set.
Phase in Cepstral analysis
Phase is fundamental in cepstralprocessing
Phase must be unwrapped
Phase must be detrended
A serious problem is additive noise
The cepstrum (complex) is defined as
C(n) = {ln[A(ω)] + iΦ(ω)}-1F
-1Fwhere is the inverse Fourier transform
Application of cepstral analysis tothin bed blind deconvolution
Compute cepstrum for each trace
Stack the cepstra
Transform back to the time domain
Deconvolve with estimated wavelet
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
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 …
Importance of sparseness in the recovery of low/high frequencies
Spectral ExtrapolationSparse InversionBlind Deconvolution Methods (MED,
ICA etc.,)
Key points of this part
A few words about the problem
n(t)r(t)w(t)s(t) +=
nWrs +=
*
Recovery of Green’s function from band limited data
The required inversionis performed by
.)()( constrnormJ dm += λ
We use:
)()|()|( mmddm ppp ∝
to obtain J
Priors to model sparse signalsTwo well-studied priors for the solution of inverse problems where sparsity is sought:
LaplaceCauchy
These priors translate into regularizationconstraints for the solution of inverse problemsThe latter is done via the celebrated Bayes Theorem
Some Math…..
l1 norm
Cauchy Norm
Bayesian Cost to minimize:
R(r) = | rk |k
∑
R(r) = ln(1+rk
2
β 2 )k
∑
J =|| Wr − d ||22 +µ2R(r )
J = Misfit + (Regularization term derived from prior)
µ2
Solution
2i
2ii
T12T
222
r+1
=Q
WQ(r)+WW=r
0=rR+dWr=J
β
µ
µ
-][
)}(||{|| -∇∇
e.g. for regularization using the Cauchy norm
The last equation is solved using an iterative algorithm to cope with the nonlinearity
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.?
Water Bottom Top Salt Base Salt Internal multiple
Water Bottom
Top Salt
Base Salt
Mississippi Canyon
Internal multiple algorithm
2123
1
321322112
111121
2
3
22
21
2
32
1
221
1121
zzzz
kkGiqkkDiqqqkkb
zkkbedzzkkbedz
zkkbedzeedkdk
qqkkb
sgsssgssgsg
zs
zqqiz
zqqi
gzqqieeiqeeiq
sgsg
s
gsgsg
>>
−=−=+
−−
−
=+
∫∫
∫∫ ∫∞
+
∞−
−−
∞
∞−
+∞
∞−
∞
∞−
++
,and
),,(),,(),,(where
),,(),,(
),,(
),,(
)()(
)()()(
ωω
π
Araújo and Weglein (1994)