Hello Calgary

I sincerely thank

Bill Nickerson&

Larry Lines

for hosting & feasting

I wish to thank

the

SEG

and holds meetings in exotic places

for honoring me with this marvelous and unique adventure

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

Without these humans,

Tad Ulrych

would be

Tad Who?

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

Presenter

Presentation Notes

hjhjhhjjh

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

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 ?

HOW BIG ?

xΦ in encoded nInformatio

A in encoded nInformatio x

Relative “Importance” of

and xxA Φ

Original

?

1=xx A only, Φ

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 ….

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

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 used 2

1] The reflectivity is white

2] The wavelet is minimum phase

Enders’ solution(obtained 45 years ago)

is called

Spiking Deconvolution

It is used in virtually unaltered form

~ 30 million times/day

Some more thoughts regarding

Phase

OUTLINE for the next few slides

POCS and only-phase reconstruction

Phase and cepstral processing

POCSProjection onto convex sets

POCS attempts to solve anunderdetermined, generally nonlinear,inverse problem

G[x]+n=dwhere G is a nonlinear operator

Illustrating convex and non-convexsets

A convex set A non-convex set

Alternating POCSIterative projection onto convex sets

Possible stagnation point whenone of the sets is non-convex

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-only

Only-phase

Original

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

The original synthetic section

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

The original reflectivity

The recovered wavelet

Usual approach todeconvolution with ‘known’

source wavelet

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

The original synthetic section

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 synthetic section

The original reflectivity

Sparse deconvolution

f-domain deconvolution

PARSIMONY

or

SPARSENESS

Some details concerning

Thanks to Mauricio Sacchi for help with PARSIMONY

The concept 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

Frequency (cycles/hour)

5.03.01.0

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 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

Example: Non-Gaussian Impulse Responsemodel via a Gaussian Mixture

More area under green curve

Sparsity is controlled by the width of he Cauchy pdf

SPAR

SENE

SS

Width β=βData

True impulse response

Predictd data

Estimated impulse response

Width β=βsparse

Data

True impulse response

Predicted data

Estimate impulse response

Width β=βGauss

AR Prediction in the f-domain

? ??

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

Internal Multiples

A vision of Arthur Weglein

Water Bottom Top Salt Base Salt Internal multiple

Water Bottom

Top Salt

Base Salt

Mississippi Canyon

What, pictorially, is the

pattern of the internal?

V is the primary

W is the internal

Hence, Arthur and colleagues,

find the W in the math

and remove it.

If you want to see the math,

here is a glimpse.

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)

Thank you for your

Patience