methods of seismic data processing,

8/18/2019 Methods of Seismic Data Processing,

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Methods of Seismic Data ProcessingGeophysics 557/657

Course Lecture Notes

420 Pages

Winter 2005

byG.F. Margrave, Associate Professor, P.Geoph.

The CREWES ProjectDepartment of Geology and Geophysics

The University of CalgaryCalgary, Alberta, T2N-1N4

[email protected]


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Table of Contents

Section Title Page NumberChapter 1: Synthetic Seismograms 30 pages The Big Picture 1-2Elastic Waves 1-7

Well Logs 1-9Gardner's Rule 1-11The Wave Equation 1-14Traveling Waveforms 1-17Normal Incidence Reflection Coefficients 1-19Synthetic Seismogram Algorithms 1-23Synthetic Seismogram Examples 1-28P-S Synthetic Seismogram Construction 1-30

Chapter 2: Signal Processing Concepts 76 pages Convolution 2-2Convolution by Replacement 2-5Convolution as a Weighted Sum 2-6Matrix Multiplication by Rows 2-7Matrix Multiplication by Columns 2-8Convolution as a Matrix Operation 2-9Fourier Transforms and Convolution 2-13Fourier Analysis and Synthesis 2-19Fourier Analysis Example 2-21Fourier Transform Pairs 2-23The Dirac Delta Function 2-25The Convolution Theorem 2-27Sampling 2-29The Discrete Fourier Transform 2-33The Fast Fourier Transform 2-37

Filtering 2-38The Z Transform 2-39Crosscorrelation 2-44Autocorrelations 2-46Spectral Estimation 2-48Wavelength Components 2-53Apparent Velocity (or phase velocity) 2-56The 2-D F-K Transform 2-58F-K Transform Pairs 2-62-p Transforms 2-63

Properties and uses of the -p Transform 2-68Inverse -p Transforms 2-71Least Squares -p and f-k Transforms 2-74

Chapter 3: Amplitude Effects 32 pages Seismic Wave Attenuation 3-2True Amplitude Processing 3-8Automatic Gain Correction (AGC) 3-9Trace Equalization (TE) or Trace Balancing 3-13Constant Q Effects 3-14Minimum Phase Intuitively 3-18Minimum Phase and the Hilbert Transform 3-21


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Minimum Phase and Velocity Dispersion 3-25Array Theory 3-27

Chapter 4: The Convolutional Model and Deconvolution 61 pages Bandlimited Reflectivity 4-2The Convolutional Model 4-4

Frequency Domain Spiking Deconvolution 4-12Finding a Wavelet's Inverse 4-20Wiener Spiking Deconvolution 4-23Prediction and Prediction Error Filters 4-28Gapped Predictive Deconvolution 4-32Burg (Maximum Entropy) Deconvolution 4-36The Minimum Phase Equivalent Wavelet 4-39Vibroseis Deconvolution 4-41Deconvolution Pitfalls 4-47Reflectivity Color 4-55Q Example 4-58

Chapter 5: Surface Consistent Methods 29 pages

Seismic Line Coordinates 5-2A Surface Consistent Convolutional Model 5-5Surface Consistent Methods 5-9Statics and Datums 5-12Statics with Uphole Times 5-17Surface Consistent Residual Statics 5-19Refraction Statics 5-25

Chapter 6: Velocity Definitions and Simple Raytracing 26 pages Velocity in Theory and Practice 6-2Instantaneous Velocity 6-3Vertical Traveltime 6-4Vins as a Function of Vertical Traveltime 6-6Average Velocity 6-8Mean Velocity 6-10RMS Velocity 6-11Interval Velocity 6-13Snell's Law 6-18Raytracing in a v(z) Medium 6-20Measurement of the Ray Parameter 6-24Raypaths when v = vo + cz 6-25

Chapter 7: Normal Moveout and Stack 38 pages Normal Moveout 7-2Stacking Velocity 7-5

Normal Moveout and Reflector Dip 7-6NMO for a V(z) Medium 7-10Dix Equation Moveout 7-13Normal Moveout Removal 7-15Extension of NMO and Dip to V(z) 7-17NMO for Multiple Reflections 7-22CMP Stacking 7-27Post Stack Considerations 7-30ZOS: A Model for the CMP Stack 7-34Fresnel Zones 7-36


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Chapter 8: Migration Concepts 52 pages Raytrace Migration of Normal Incidence Seismograms 8-2Time and Depth Migrations, A First Look 8-5Elementary Constant Velocity Migration 8-6Huygen's Principle and Point Diffractors 8-9

The Exploding Reflector Model 8-14F-K Migration, Geometric Approach 8-20F-K Migration, Mathematics 8-25F-K Wavefield Extrapolation 8-27Recursive F-K Wavefield Extrapolation for v = v(z) 8-31The Extrapolation Operator 8-33Vertical Time-Depth Conversions 8-36Time and Depth Migration in Depth 8-37Kirchhoff Migration 8-40Finite Difference Concepts 8-43Finite Difference Migration 8-46

Chapter 9: The Third Dimension 32 pages

Impulse Responses 9-2Wave Propagation 9-6Fresnel Zones 9-7Wavelength Components 9-10Apparent Velocity (or phase velocity) 9-13The F-K Transform 9-15F-K Transform Pairs 9-19F-L transform Computation 9-203-D Migration by Double 2-D 9-24Exploitable Symmetries 9-27Mapping Strategies 9-29Time migration of traveltime maps 9-31

Chapter 10: Seismic Resolution Limits 35 pages Resolution Concepts 10-2Linear v(z) resolution theoru for zero offset seismic data 10-18

Chapter 11: Study Guide 9 pages Geophysics 557 Final Exam Study Guide 11-2Exam Sampler 11-7


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Methods of Seismic Data Processing 1-1

Methods of Seismic Data Processing

Lecture Notes

Geophysics 557

Chapter 1

Synthetic Seismograms


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1-2 Synthetic Seismograms

The Big Picture

Th e s imp les t mode l o f s ei sm ic d at a i s t h at o f a w ave le t

c onvo lv ed wi t h re f l ect i v it y . The p i c ture i s s imp le a nd

ap pea li ng . A c ompact p uls e of sound i s sen t d ow n in t o

t he e ar t h an d s ca l ed c op i es o f i t are re f l ec ted f r om the

ma jor f ormation b ounda ri es .

T hese echoes ar e recorde d o ve r t he e xten t o f the

seismic ex per iment an d a na l yzed . S i nce each echo i s a

sca led c opy of th e s our ce w aveform , s imp le compar ison

m akes i t i s easy t o deduc e t he re lat ive streng th o f the

dif ferent ref lect ing hor izons . T he est imate d set o f

re fle ct i on coe ff ic ie nts i s c al le d t he r ef le ct i vi ty f unct io n

o f t he earth b enea th the s ur ve y.

It s a n i ce concep t bu t is i t v a li d ? How can i t b e

de fen ded f ro m ba si c phys i c al pr inc ip le s ? What

as sumpt i on s ( th er e ar e al w ays a s sump t io n s i n ph ys i cs )

ar e requ i red ? Whe n ar e t hey just i fi ed a nd w hen are t he y

not?


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The Big Picture

Once we s ta rt t o t hi nk about the i dea, we can immediatel y

come up with a lot o f q ue stions suc h as:

• How can we procede if we don t know the source waveform?

• What if several echos are very closely spaced?

• How can we tell where the echo came from?

• I sn t ther e at tenua ti on o f sei sm ic ene rgy and doesn t t hi s

change the source waveform?

• What is convolution anyway? (And why should I care?)

• What about multiple bounce echos? Don t they confuse things?

• How can I decide how much source energy I need?

• What are the limits of the detail that can be resolved?

• What are the tradeoffs with Vibroseis and dynamite?

• What is reflectivity anyway? (And why should I care?)

• If things are so simple, how come seismic processing is so

complicated? Maybe those processors are just fooling us ...

• Why can t I j ust t ru st the seismic p roces so r t o take c are

of these messy deta ils?

I m su re tha t you ca n th in k o f more quest ions a s we ll . A l l

o f t hese q ues ti on s have the ir re levance and I hope to

addr ess m an y of them in th is course . A t t he end , you

shou ld h av e a good u nderstandi ng o f the strength s a nd

weakness es o f th e c onv olu ti on al m ode l a nd t hi s s ho uld

he lp y ou fo rm a hea lthy , scept ica l v ie w o f f in al se is mi c

images.

• What s this band-limited stuff?


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The Big Picture

Se ism i c d at a p roc ess i ng i s typ ic a ll y d iv ided in to many

steps t hou gh t he r ea li ty i s t hat t he s eism i c re f l ec t i on

p roces s d oe s n ot c le anly separ at e i n to d i s cre te

p ac ka ge s . W e have a sou rc e wh ich sends out a

comp li cat ed , l ar ge l y u nk nown wavef orm whi ch e xpand s,

at ten uat es , r ef lec ts , t ransmit s, c hanges m odes , an d

general l y s c at t er s a bou t wh i le a s et of re c ei ve rs

p la c id ly recor d s whatever comes the ir wa y. And

general l y what h i t s t h e re cor der s is f ar mor e

compl i cat ed t h an t h e s impl e d i re c t e ch os t h at w e wan t :

P wave

reflection

Surface wave

S wave

reflection

Receivers

ll kinds o f waves

sweep across the

receivers

Go d wou ld n ot p r oce ss s ei sm ic d at a th e wa y we do . ( I v e

r ec e iv ed a r eve l at i on o n t h at p oin t . ; - } ) In s tead , H e

w ou l d b ac k t he waves d ow n in to the e ar t h undo in g al l

physi c al e f fec t s a t t h e p oi n t where they occu rred . W e

ar e p r evented f ro m do in g th i s l ar ge l y becau s e of

i gn orance of t he subsurf ac e st ructure . T hat i s, i n o rder

t o u nd o the phys i cal e ffects o f wav e p r opaga t io n , we

r equ i re k now l edge of t h e subsurf a ce p r ope r ti e s that

c on tr ol t ho se e ff ec ts . U n fo rt u nat el y , thos e are t h e v e ry

p r oper ti e s wh ic h we hope t o d i scover w i th t h e s ei sm ic

exper iment in the f i rs t p l ac e . P ro blems of th i s so rt a re

c ommon in g eoph ys i cs an d ar e c al l ed i nv ers e p r ob l ems .


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The Big Picture

So , f ac ed w i th t he need t o f i nd a s olut ion i n sp i te of

almost total ig nor ance , w e subd iv ide,

c omp ar tm en ta li ze, assume, an d ap pr ox im at e un ti l w e

r ea ch a r es t atement of t h e p r ob l em wh i ch i s s o v as t l y

s imp l if i ed that w e c an actual ly so l ve it . An example of

such a t remen dou s over simpl i f ica t io n i s the

convol ut ion al m od el o f the s e ismi c t ra ce w hi ch is of

cen tral impor t ance t o deconvo lu t i on theor y.

C ontinu ing w it h sweep i ng g ener al i ti e s , we c an gr ou p

most p hy si cal l y ba se d s eismi c p roc es se s i nt o one of t wo

g roups : imagi ng p r oce sse s and deconvol ut ion p r oce s se s .

Imagi ng p rocesses a ttempt t o d ete rmin e the co rrec t

s p at i al p os i t io n o f t h e echos an d ar e t yp i f i ed b y nmo

r emoval , c mp s tac king , an d m igr at ion . Decon volut ion

p r oc e ss e s attempt t o r emov e t h e i l luminat in g w av ef o rm

an d m ax im ize the r e so luti o n of t h e se i smic image .

E x amples ar e g ai n r ec ov er y, s tat i s t i ca l d ec on vo luti on ,

i n v er s e Q f i lt er i ng, an d wave l et p r oce ss i ng .

Deconvolution

techniques


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The Big Picture

In ord er t o u nder stan d t he impl i cat ion s of ou r si m pli f i ed

theor ie s , it is important t o unders t an d as much a s

poss i bl e a bou t t h e m or e r ea li s t ic p hys i c s t h at w e ar e

ap pro x imat i ng. The ref or e , i n add it i on t o studyi n g

ma themat ic a l s imp li fi c at i ons s uc h as t h e c on v ol ut ion a l

model , w e wi l l n ot h es i t at e t o examine of t h e most

import an t p h ys i cal mec han isms i nv ol v ed i n s ei sm ic wav e

propagat ion.

physics of continuous media

anelastic wave theory

elastic wave theory

primaries multiples etc

one way scalar waves

imaging methods

the convolutional model

deconvolution methods


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Elastic Waves

The simple s t e la st i c mat er i al r equ ir e s 2 fundamenta l

c onstants t o descri be th e re la t io n be twe en st ress an d

s t ra in k n own a s H oo ke s l aw :

σii = λΔ + 2μεii, i=x,y,z Δ = εxx+εyy+εzz

σij = μεij, i=x,y,z, i≠j (Sherrif and Geldart,

Exploration Seismology, 1981)

Here

i j

denot e s t he c omponents of t h e st res s tenso r

an d e

i j

t h e c ompon en ts of t h e s tr ai n t enso r. an d μ ar e

c al l ed t he L ame cons tan ts an d μ is a ls o of t en known a s

t he shea r modul us. μ i s z er o f or a f lu id . Other con stants

ar e of ten al s o re ferenced such as Young s modu lus , E ,

P oi ss on s r at io , , an d t he bu l k m odu lus, k . These

c ons tan t s ar e al l r e la t ed i n v ari ous w ays an d any t wo

su ff i ce t o d e sc r i be t h e el as t ic mat er ia l .

E = μ 3λ+2μλ+μ

σ = λ2 λ+μ

k = 3λ+2μ3

The description of elastic wave in such a medium, requires the

application of Newton s second law (f=ma). This leads to the

incorporation of the density,

ρ,

as a necessary constant in the role

of mass in Newton s second law. Thus, analysis of elastic waves

in the most simple elastic solid (homogeneous and isotropic),

requires three parameters: any two of:

, μ

, E,

, and k, plus the

density, ρ.


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Elastic Waves

It is well established in theory

1,2,3

that a homogeneous, isotropic

elastic solid supports two distinct types of body waves:

compressional and shear. Compressional or P waves are

characterized by particle motion parallel to the direction of wave

propagation. Shear or S waves have particle motion transverse to

the direction of wave propagation. P and S waves have velocities

of propagation given by:

1: Waters, Reflection Seismology, 1987

3: Aki and Richards, Quantitave Seismology Theory and Methods,

1980,

α =λ+2μ

ρ β =

μ

ρ

We may choose to regard α and β as fundamental constants

(together with ρ). Some relationships are:

λ = ρ α2–2β2 μ = ρβ2 σ =

α2–2β2

2 α2–β2α

β =

2 1 – σ

1 – 2σ

0.2 0.25 0.3 0.35 0.4 0.451.5

2

2.5

3

3.5

Poisson's ratio

2: Sherrif and Geldart, Exploration Seismology, 1982


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Well Logs

Well logging is a technology designed to make geophysical

measurements in a bore hole. Well logs are the most common way

to get information about the elastic parameters of rocks which are

needed for making synthetic seismograms. Three very common

logs, which are of interest to us, are

SON ... P-wave interval transit time

SSON ... S-wave interval transit time

RHOB ... density

The interval transit time logs are usually provided in units of

microseconds/lu (lu= meters or feet). Thus, the P and S wave

velocities are found as:

α = 10

6

sonβ =

106

sson

Units for density logs can vary. Be careful to work with consistent

units.

Digital well logs are usually packaged in ascii flat files in either GMA

or LAS format. The LAS format is more modern and flexible and is

to be preferred.


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15

0

20

0

25

0

30

0

35

0

1400

1450

1500

1550

1600

1650

Units of log SON

100/08-08-023-23 W4

mannville

coal_1coal_2

coal_3

glauc_ch_top

glauc_ss_top

glauc_base

miss

base

glauc_1

18

0

20

0

22

0

24

0

26

0

28

0

30

0Units of log RHOB

100/08-08-023-23 W4

Well Logs

Here are some example logs from 8-8, an oil well in the Blackfoot

field

1400

1450

1500

1550

1600

1650

mannville

coal_1coal_2

coal_3

glauc_ch_top

glauc_ss_top

glauc_base

miss

base

glauc_1

Faster

More dense

Why do these logs appear to have a negative correlation?


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Gardner s Rule

We ll l og s a re o f ten i nadequat e, i ncomp let e, o r m i ssi ng .

One c ommon ex amp le o f th i s c omes f ro m the f ac t tha t

s on i c lo gs ( SON) a re ru n much m or e f r equ ent l y t h an

dens it y l og s. Thus we a re of ten f aced w it h t he nee d t o

c re at e a s e ismogr am w ithout dens i ty i n formation .

Ga rd ner et a l. (1) , f ol low ed the r ea so nab le approac h o f

s ee ki n g an em pir i cal r el at ionsh i p b etwee n P -wav e

ve lo c it y a nd d ensi t y. B e lo w i s a c r os splo t o f a and r fo r

B l ac kfo ot 8 -8 w hi ch ind icat es a re as on abl e c or rel at ion

ex ists :

1

Ga rdne r , G .H .F ., Ga rd ne r , L . W ., and G regor y, A .R. , 19 74 , Forma ti on

ve lo cit y and den s it y - t he d iagno sti c ba si s f or s tra tigr ap hic tr a ps ,

G eophys ic s , 39 , 770-780

2000 3000 4000 5000 6000 70001800

2000

2200

2400

2600

2800

3000

P-wave velocity


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Gardner s Rule

Gardner et al. sought and found a relationship of the form:

ρ = a αm

Th e c onstan t s a an d m c an be d et e rmin ed f ro m fi t t ing a

s t ra ig h t l ine t o an p l ot o f lo g(

ρ

) v e rs u s l og(

α

) . Be low

ar e t he r es ul t s o f s ev era l s uc h f it s t o B l ac kf oot 8-8 .

2000 3000 4000 5000 6000 70001800

2000

2200

2400

2600

2800

3000

3200

m=.46

m=.30

m=.25

Ga rd ner et al . det ermin ed an d r ecommended m=.25 a s a

r ea sonab le v alue . H owever , as we c an s ee, t h e d at a

support qu i te a ra nge of a lt ernat i ves . ( Th e v al ue of α i s

l ar ge l y dependent o n t he un it s used an d i s n ot quoted

he re . ) Thus , the c ar ef u l ap pli c at i on o f G ardner s ru l e

r equ ires a bi t o f an al ys is .

P-wave velocity


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2500 30001400

1450

1500

1550

1600

1650

m=.30

20002000 2500 30001400

1450

1500

1550

1600

1650

m=.46

2000 2500 30001400

1450

1500

1550

1600

1650

m=.25

Gardner s Rule

Here are the three pseudo density logs from the three fits on the

previous page.

Actual density log from Blackfoot 8-8

Result from a Gardner type regression against P-wave

velocity

Density Density Density


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The Wave Equation

Th e g re at succes s of physi cs in exp l ai n ing o ur w or ld an d

f ue li n g t h e gr ow t h of t ec hn olo gy i s ba se d f un damen ta ll y

u pon d if fer en t i al e qu at io n s an d mor e spec i f ic a ll y p ar t ia l

d i f fer ent i al equat ions . PDE s ar e t he mathemat i ca l

s t atemen t of t h e app l ic at i on of b as i c physi c al l aws t o

c om ple x s ystems. F or e x amp le , a c ons idera ti o n of a

c ons tan t dens i ty f lu id l e ad s t o the s c al ar w av e

e qu at ion wh ic h i s c entr al t o mos t ge op hys ic a l i mag in g

al gor it hms. Th e SWE i s a d i r ec t conseq uence o f

New ton s s econd l aw and H oo ke s l aw a s app l ie d t o t he

f lu id.

∂2Ψ

∂x2

+ ∂2Ψ

∂y2

+ ∂2Ψ

∂z2

– 1

v2

x,y,z

∂2Ψ

∂t2

= f x,y,z,t

He re Y i s t h e p ressure , v i s t h e v el o ci t y o f w av e

p r opaga t io n , a nd f (x ,y ,z , t) r ep re s en t s an y p os s ib l e

sources.

Though i t is h ard l y obv iou s , t h e so lut ions t o th i s

e qu at i on a re t ra v el in g w av es . A gr e at d e al of inte r es ti ng

p hy si c al ef fe ct s ca n b e s tu di ed w it h t h e SWE i nc lu din g :

• propagation of primaries and multiples

• reflection and transmission at interfaces

• head waves and surface waves

• ray theory, Snell s law

• characterization of sources

• arrays of sources and receivers


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The Wave Equation

The re is a p owe rf u l method of s olut ion of PDE s that i s

of cons id e rab le r el e v an ce exp lo ra t io n se ismo lo gy . Th i s i s

t he m ethod o f s o lut ion by Gr een s fun ct ions. W e w il l n ot

devel op i t h ere b ut s imply sta te the imp or t an t r esu l t s .

Th e e ssence o f t h e t heo ry is t o deve lo p a s olut ion t o

t he PDE o f in te re s t f or a po in t s ourc e an d then t o

s how h ow the respon s e t o ar b it r ar y s ou rce

c onf igura ti on s ca n be c onstructed fr om t he e lemen tar y

s olut ion . T he SWE , when speci al i ze d f o r t h e Gr een s

f un ct ion p ro bl em l ooks l i k e:

∂2G

∂x2

+ ∂2G

∂y2

+ ∂2G

∂z2

– 1

v2

x,y,z

∂2G

∂t2

= δ x–xo,y–y

o,z–z

o,t–t

o

Th e t e rm on t he righ t of t h e equ al s ig n i s a D i ra c d el t a

funct ion an d r epre s en t s a m athemati c al impu lse at a

s i ngl e poi n t i n sp ace , ( x

o

,y

o

, z

o

) , a nd at an instant o f

t ime , t o. T he s o lu t i on to t he Green s funct ion pro bl em,

G(x, y ,z ,t ) , i s k now n exact l y f or constant ve lo c it y an d

ap pro x imat el y fo r a number o f m or e compl ica t ed

s i tuat i on s . G c onta in s al l physi c al e f fec ts due t o t he

impu ls iv e s ou rce an d i s p rope rl y c al le d a n impu lse

r espo nse .

To ob t ai n t h e r es pons e t o g en e ra l s ou r ce c on fi gu ra t io n s,

w e imag in e the sou rce to b e c omp ose d o f a s et of s cal ed

impu lses . Then con st ru ct t he G re en s funct ion s f or a ll o f

these impuls e s an d simp ly superimpose these Gr een s

funct ion s . Th i s i s an e x amp l e of t h e mathemat i ca l

p r oc e ss o f convolu t i on . We w il l l ea rn more ab out

c onvo luti on l at er i n th i s c ou r se. Fo r n ow , it i s e nou gh t o

v i sual iz e i t a s a ge ne ra l s upe rpos it io n of s c al e d impu lse

responses .


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The Wave Equation

T he resul t w e ha ve j us t obta ine d i s s o i mportant t hat w e

r es ta te i t i n d if fe re nt terms :

T he wavef ie ld d ue to a sou rce havi ng e xt ended spat ia l

a nd tempora l f o rm can b e cons id e re d t o be the

convo lu t ion o f t he ea rt h s impu ls e response wit h t he

e xt ended sour ce . T his re su lt ho lds fo r a ny l ine ar wa ve

e quat ion and ex tends t o el a st i c, a n isot rop i c a nd

attenuat in g m ed ia .

Th e t wo c omponents of th i s r esu lt , t he e ar th s impu lse

r es po nse , I

r

, a nd t he s ou rce wavef orm , w

s

, are b ot h

abs t ra ct ent i t ie s that ar e d i f f icu l t t o quant if y . I

r

i s

g ene ral ly ve r y c omp l icat ed an d c on t ai n s al l ph ys i ca l

e ff ec ts. w

s

i s a c omplete chara ct er iz at ion of t he s ou rc e

w av efi e ld an d c an be cons ider ed as t he spec i f ica tio n o f

t he wa vef i e ld at a ll poin t s on a s urf ace su r round ing t he

source.

Impulse response

Response to 3 sources


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Traveling Waveforms

The s implest mathematical wave equat ion is the sca lar wave

equation. I n acoustic media or s imple e lastic media ,

compressional waves are governed by it . In 1-D, the scalar

wave equation is :

∂2 ψ

∂z 2

= 1

v2

∂2 ψ

∂ t 2

Where

ψ

represents the propagating wave. We now show that

ψ = f t±z/v

is a solution to (1).

(f is an arbitrary function)

(1)

∂ f

∂ z= ±

1

vf

′

, ∂2 f

∂z 2

= 1

v2

f′′

∂ f

∂t

= f′

, ∂

2f

∂ t2

= f′′

Substitution of the second partials of f into (1) results in an

immediate identity. Thus f is a solution to (1) with the form of f

being arbitrary except that it must be twice differentiable.


23/409


Traveling Waveforms

w τ = 1–2 πfdom

τ2

exp – πfdom

τ2

As an example of a waveform, consider the Ricker wavelet defined

by:

-0.05 0 0.05

Shown for f

dom

=30Hz

τ

->

Note that the Ricker wavelet is centered where its argument equals

zero. Thus w(t+z/v) represents a wavelet centered at t+z/v = 0 or

z = -vt. So we conclude:

w t+z/v = Waveform traveli ng

i n the - z d ir ec ti on

w t–z/v = Waveform travel ing

i n t he +z directi on

Similarly, cos(ω (t-z/v)), cos(k(z - vt)), and cos(ω t-kz) all

represent cosine waves traveling in the +z direction.

400 450 500 550 600-1

-0.5

0

0.5

1

z-> (meters)

cos 2π30 t–z/1000 Plotted versus z for t=1.0 and 1.01 (sec)

1.01 se

1.0 sec

z=-vt

z=vt


24/409


Normal Incidence Reflection Coefficients

(Adapted from E.S. Krebes, Course Notes in Theoretical Seismology)

f t–z/α1 g t+z/α1

h t–z/α2

Z

α1,ρ

1

α2,ρ

2

Consider a vertical ly

trave li ng compr essiona l

wave incident on a

hor izonta l inter face. In

o rde r to descr ibe the

ref le

ction and

transmiss ion that occur,

i t can b e s hown tha t two

condi tions must be

satisfied:

f + g = hontinuity of displacement:

continuity of normal pressure:

???

To develop a form for the second equation, we use Hookes

law which says stress is proportional to strain.

stress = (applied force)/area

strain = (change in length)/length

(1)

(2)

Incident

displacement

Reflected

displacement

Transmitted

displacement


25/409



Consider an infinitesimal elastic element

whose ends undergo displacement u1 and u2:

dzu1

u2

Strain =

Δl

l=

u2–u

1

dz≈

∂ u

∂ z

Now, invoking Hooke s law:

stress = pressure =

Forcearea

= k ∂ u∂ z

Where k is a constant formed from the material constants. To

determine k, we can use dimensional analysis:

pressure =

force units

(length units)2=

mass l sec2l

sec2

l2

= mass

l3

l

sec

2

So k looks like:

k = ρα2

Thus the pressure continuity equation is:

ρ1α

1

2∂ f

∂ z+ ρ

1α

1

2∂ g

∂ z= ρ

2α

2

2 ∂ h

∂ z

Which can be immediately integrated to give:

∂ f

∂ z=

–1α1

f′, ∂ g

∂ z=

1α1

g , ∂ h

∂ z=

–1α2

h′But since

ρ1α

1f – ρ

1α

1g = ρ

2α

2h

ρ1α

1f – ρ

1α

1g = ρ

2α

2h (2)

(evaluated at the interface)


26/409



Assume that an interface occurs at z=0, then if the boundary

conditions are applied there, the two equations determining normal

incidence reflection and transmission are:

f + g = h

I1f – I1g = I2h

Ik

= ρkα

k , k= 1,2

here impedance =

(1)

(2)

Multiplying (1) by

, and subtracting it from (2) leads to:

g = I

1–I

2

I1+I2f = –Rf

h =

2I1

I +I f = T f1 2

R = I2–I1

I1+I2

, T = 2I1

I1+I2

Similarly, we can obtain:

The quantities R and T are known as the normal incidence reflection

and transmission coefficients:

R+T = I

2–I

1+2I

1

I1+I2= 1ote that:

and where f,g, and h are understood to be evaluated at z=0.


27/409



R = I

2–I

1

I1+I2, T =

2I1

I1+I2

R and T are often written in terms of the contrast and average of

impedance across the layer:

I = 1

2I

1+I

2 , ΔI = I

2–I

1

I1 = I–.5ΔI , I2 = I+.5ΔI

Straight forward algebra then gives:

R = ΔI

2I≈

1

2

d ln I

dzΔz

T = 1–R =

I–.5ΔI

I

R =Δ ρα

2ρα=

ρΔα+αΔρ

2ρα=

1

2

Δα

α +

Δρ

ρ

R can be written in terms of ρ and α as:

Note that the definition of R is such that an impedance increase

gives a positive RC but that the reflected pulse is flipped in

polarity.


28/409


t=2Δt

t=3Δt

t=jΔt

t=nΔt

t=Δt

* R1

2

*R2

* R12

R2

2

*R

* Rk2

k = 1

j–1

*Rj

* Rk2

k = 1

n–1

*Rn

*R1

1–

1–1–

1–

1–

Impulse

Response

3

Simple Primaries Only Impulse Response.

Layered Earth, Normal Incidence, Acoustic

V1,R

1

V2,R

2

V3,R3

Vj,R

j

Vn,R

n

k=0

k=1

k=2

k=3

k=j

k=n

k=n-1

Model layers have a

constant traveltime

"thickness":

* R1 *R

2

* R1 R2 *R3

Rkk = 1

n–1

*

Δt=2ΔZ

Vj

j

Rk

k = 1

j–1

* * Rkk = 1

j–1

*Rj

* Rkk = 1

n–1

*Rn

R1 * R2 *

R1 *

*R1

1–

1–

1–

1–

1– 1–

1–1–

1– 1–

Model


29/409


At the designated point, 6D4 and 6U5

are known and we wish to compute

6U4 and 6D5:

6U4 = R4*6D4 + (1+R4)*6U5

6D5 = (1-R4)*6D4 -R4*6U5

The complete se ismogram is

obta ined by recurs ive ca lculation

beginn ing i n the uppe r le f t. A ll

nodes on any upward tr ave ling r ay

are complete ly ca lculated bef ore

proced ing to the nex t depth .

Adapted from: Reflection Seismolgy, K.H. Waters, 1981

J.H. Wiley

Computation of a 1-D Synthetic Seismic Impulse Response

(Including All Multiples)

E ar th model i s bui lt of

l ayers of equa l t rave lt ime

thickness

Δ

t

t

z

R0

R1

R2

R3

R4

R5

R6

R7

R8

R9

t=

Δ

t t=2

Δ

t t=3

Δ

t t=4

Δ

t t=5

Δ

t t=6

Δ

t t=7

Δ

t t=8

Δ

t

Note: All Raypaths are

actually vertical. They are

shown slanted for illustrative

purposes.

Completed node

Current node


30/409


t=2Δt

t=3Δt

t=jΔt

t=nΔt

t=Δt

* 1–R12

*R2

* 1–R12

1–R22

*R3

* 1–Rk2

k = 1

j–1

*Rj

* 1–Rk2

k = 1

n–1

*Rn

*R1

Impulse Response

t=2Δt

t=3Δt

t=jΔt

t=nΔt

t=Δt

* 1–R12

*R2

* 1–R12

1–R22

*R3

* 1–Rk2

k = 1

j–1

*Rj

* 1–Rk2

k = 1

n–1

*Rn

*R1

Source Waveform Response

Th e pr imar ie s on l y

impu lse respon s e

c ons i sts o f a t ime

s er i es o f s c al ed an d

de la ye d impu lses

To ob t ai n t h e sou rc e

wavef o rm respon s e

f ro m the impu ls e

response , s im ply

rep l ac e ea ch sp i ke of

t he i mpu ls e respon se

by the p r od uct o f t h e

sp i ke and sou rc e

wavef o rm . Th i s i s t h e

m at hema ti c al p ro ces s

of convolu t i on

From Impulse Response to Source Waveform

Response


31/409


Impulse Responses and Seismograms

For a l i near ea rt h , i t c an be s how n t hat i f w e a re g iven

t he w ave fo rm s i gnat u re of a n on- impu l si v e sou rc e an d

t he impu lse re sp on se o f an ea rt h model , the n:

s t = Ir t •w

s t

ws t is the source waveform

Ir t is the earth impulse response

s t is the earth response to the source waveform

where:

Th e gener al pro of o f th i s r esu l t comes f rom G reen s

fun ct i on ana ly s is an d is t rue f or any l i near wave

eq uat i on (e l as t ic , s cal ar , et c ) G ener al ly Ir c onta ins a l l

p hys i c al ef fec ts t he the or y i s c ap ab le o f p rodu c in g, an d

u sual ly th at i s m ore than w e w an t.

T he mos t common us e o f 1- D s eismogram s i s in t h e

in te rp ret a t io n o f p ro cess ed sei sm ic sec t io ns. I n th is

ca s e mo st of th e phy si ca l eff ec ts (mu lt ip l es ,

tra nsmiss io n l o ss es, a tt en u ati o n) h a ve bee n r emoved in

th e p ro ces si ng. Th eref o re, common p ra c ti ce rep la ces

I

r

( t ) w it h r( t ) w here :

r t = no rmal i nc idence re f lec t i on co ef f i c ien t s

pos it i oned i n 2 -way ve rt i c al t r ave l time

s t = r t •ws thus:

s( t ) g i ven b y th i s resu lt i s the most c omm on 1 -D

se i smog ra m computed i n exp l ora ti on g eophysi c s .


32/409


1-D Syntheti c Seismogram Summary

• A complet e s olut ion , g ener at i ng al l mu l t ip le s an d

t ransmiss ion ef fects, c an b e c onstructed . Some

m ethods a ls o i n cl ude a tt enuat i on .

• A ss umpti on s: ra y theo ry , 1 -D , n o rma l i nc iden c e

• Geophys i c al w el l l og s , p r ov id in g P -wave ve l oc i t ie s

an d d en si t i es , a re use d . T he y a re usu a ll y r es amp l ed t o

a v a ri ab l e d ep th l a ye ri n g w it h e qu al D t s te ps .

• Met hod i s i n her en t ly a lg or it hm ic . No an al yt i c c l os ed

fo rm s olu t i on avai l ab l e.

• In p r ac t ic e , m ult ip les an d t ra nsmis s i on lo s se s a re n o t

usual ly included . Re f lect ion c oe ffi c i ents i n t ime ar e

simply c onvol v ed w it h a sou rc e r e spon s e.


33/409


0 0.2 0.4 0.6 0.8 1 1.2

Wavelet

Synthetic Seismogram

Reflection Coeficients

Time (secs)

Example of Synthetic Seismogram Creation by

Convolution of Reflectivity and Wavelet.

Time Domain View


34/409


0 50 100 150 200 250

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

Frequency (Hz)

Reflectivity

Wavelet

Synthetic Seismogram

Example of Synthetic Seismogram Creation by

Convolution of Reflectivity and Wavelet.

Frequency Domain View


35/409


S

P

S

P SP ORR

Iterative Snell s law raytracing

1) Rayt race

Incidence

Angles

Loop over layers: k=1 to nlayers

Next layer

2) Zoeppritz

RCs

PP

PS

AN DND

Response of

l aye r k

Input

wavelet

3) Map RCs to t

o

,

apply wavelet.

Vp, Vs, and

density logs

Define Layered Model

Resamp led

lo gs

+

=

ccumulated

gather after k-1

layers

Accumulated

gather after k

layers

The SYNTH Algorithm

P-S Synthetic Seismogram Construction

Free

surface

Primary reflections


36/409


Chapter 2

Signal Processing

Methods of Seismic Data Processing

Lecture Notes

Geophysics 557


37/409

2-2 Signal Processing Concepts

Convolution

C onvo luti on i s t h e mathemat i cal p rocess of sh i ft i ng ,

s ca li ng , a n d summ in g a w av e fo rm t o p r odu c e an ou t pu t

by superposi t i on . Genera ll y, two input s i gn a ls ar e

requ i red , sa y r an d w , w it h w b ei ng t he wavef or m an d r

a s eri es of s cal in g c oef f ic ients. F or example , l e t r= [1 0

0 - .5 .5 0 -1 ] an d le t w = [ - .5 1 - .5] , t h en t he

c on volut i on of r an d w i s:

-.5 1

-.5 0 0 0 0 0 0

0

0 0 0 0 0

r0w0

0

= r1*w

0

0 0 0 0 0

.25

-.5 0 0 0

.25

-.25

.5

0 0-.25

0

0 0 0 0 0

0

0 0

.5 -1

.5

+

+

+

+

+

+

-.5 1

-.5 .25 -.75 .75

.25 -1 .5

s = r•w

j

1 2 3 4 5 6 7 8

k

1

2

3

4

5

6

0

r0w1 r0w2

r1w0 r1w1 r1w2

r2w0 r2w1 r2w2

r3w0 r3w1 r3w2

r4w0 r4w1 r4w2

r5w0 r5w1 r5w2

r6w0 r6w1 r6w2

= r0*w

= r2*w

= r3*w

= r4*w

= r5*w

= r6*w

Outpu t sampl e numbe r


38/409


Convolution

In t h e p r ev i ou s s l ide, we descr i be d a t ablu l ar me thod

f or c omput ing t he c onvo lu t i on of r an d w t o y i el d s.

Th i s c a n b e w ri tt en ma themat ic a ll y as fo ll ows :

s = r•ws

j = r

kw

j–kΣk

To s ee t ha t t hi s s ummat io n e xp res s i on is equ iv a le n t t o

t h e t ab u la r met hod , co ns ider t h e e x amp l e of j=4 :

s4

= r0w

4–0+r

1w

4–1+r

2w

4–2+r

3w

4–3+r

4w

4–4+r

5w

4–5+r

6w

4–6

s4

= r0w

4+r

1w

3+r

2w

2+r

3w

1+r

4w

0+r

5w

–1+r

6w

–2

N ot e t h at t h e length of s i s t h e c omb ined l engths of

r a nd w le s s 1 :

length s = length r +length w –1

Thus , mathemat i ca ll y , e v er yt ime a convolu t i on i s

per formed t he r es ul t i nc re as e s i n l en gt h. T hi s c re at es a

b i t of a heade r (bo okk eep ing) p r ob l em in se i smic d at a

proces s in g and i s n ot usual l y al lo wed . Tha t i s, i f a

se i smic t r ac e is c onv ol v ed wi t h a f i l ter ope rat or , t h e

r e sult i s t runcat ed at t h e same l engt h as t he se i smic

t race.


39/409


40/409


Convolution by Replacement

Consider the discrete convolution of a three point boxcar, b, with

an eleven point time series, r.

0 2 4 6 8 10 12-0.1

-0.05

0

0.05

0.1

0 2 40

0.5

1

•

r

b

-0.1

-0.05

0

0.05

0.1

0 2 4 6 8 10 12

0 2 4 6 8 10 12 14

-0.1

0

0.1

0 2 4 6 8 10 12 14

-0.1

0

0.1

=

E ac h input samp le i s c ons idere d sepa ra te l y. Th e

b ox ca r i s mu lt ip l i ed by t he input samp le res ult i ng i n a

s c al e d b ox car . The sc al e d b oxca r cont ributes t o

output samp le lo c at i on s b eginn ing at t h e

p os i t io n of t h e input sample . Th us t he

b ox car i s sc a le d b y eac h samp le of r

an d rep l ica ted at t he lo cat ion o f

the r s am ple . E ac h output

samp le r ece iv e s mu lt i p le

con t ri but i on s wh i ch ar e summed .

Input s am ple s 1 ,2 and 6 ar e

shown exp l ic i t l y cont ribut ing.

( emphas is on in pu t s ample s )


41/409


Convolution as a Weighted Sum

Consider the discrete convolution of a three point boxcar, b, with

an eleven point time series, r.

0 2 4 6 8 10 12-0.1

-0.05

0

0.1

0 2 4 6 8 10 12 14

-0.1

0

0.1

To compute an ou tpu t

s amp le , po si t i on t he

b ox car ove r s om e r

s ampl es , m ul tip ly t h e r

s amp les b y t he bo xc ar

w eights, a nd sum. T he

compu t at i on of ou t pu t

s amp les 1 an d 7 i s

i ll ust rat ed. Th i s i s a

p r oce ss o f smooth ing

or a v er ag in g t h e i np ut .

0 2 4 6 8 10 12-0.1

-0.05

0

0.05

0.1

0 2 40

0.5

1

•

r

b

( emphas i s o n out pu t s amp l es )


42/409


a11 a12 a13 a14

a21 a22 a23 a24

a31 a32 a33 a34

a41

a42

a43

a44

b 1

b 2

b 3

b4

=

c1

c2

c3

c4

Matrix Multiplication

by Rows

Consider the a 4x4 matrix equation such as:

This is equivalent to the following system of equations:

c1

= a11

b1

+ a12

b2

+ a13

b3

+ a14

b4

c2

= a21

b1

+ a22

b2

+ a23

b3

+ a24

b4

c3

= a31

b1

+ a32

b2

+ a33

b3

+ a34

b4

c4 = a41b 1 + a42b 2 + a43b 3 + a44b4Th us t he elemen t s o f t h e v ecto r C ar e c omputed b y

t ak ing e ac h ro w of A, mult ip lyi n g i t by t he ve ct o r B , an d

summing t he r e su l t s . T h is p ro ces s i s f amil i ar t o most

students of l i near a lge br a as m at r i x mult ip l ica t io n b y

r ow s . It c a n be wr i t ten s ymbo li c al l y a s two nested

c om putat ion loops :

c=zeros(1,4);

for irow=1:4

for jcol=1:4

c(irow)=c(irow) + a(irow,jcol)*b(jcol);

end

end

eqn 1

eqns 2a-2d


43/409


M at r i x mu lt ip li cat ion by c ol umn s i s l ess we l l k now n

t han the c o rr espond ing pro ces s by r ow s bu t i t

p r ov ides a u s ef u l intu it i ve ins ight t o convolut i on .

E x am in at i on of e qu at i on s 2 a- 2d s h ow s t h at t h e c ol umns

of A have b ee n mu l t ip l i e d by a sing le c or re spon din g

e lement o f B . Thus w e ca n expres s t he mat ri x

mult ip l ic a t io n as a su m o f c o lum n v ec tor s , e ac h on e

b ei ng a s cal ed ve rs ion of a co lum n o f A.

a11

a21

a31

a41

b1

+

a12

a22

a32

a42

b2

+

a13

a23

a33

a43

b3

+

a14

a24

a34

a44

b4

=

c1

c2

c3

c4

Wri tt en as c omput at i on l o op s , t hi s amoun ts t o r ev er s in g

t he o rder o f t he l oo ps i n t he mu l tip l i cat ion s by r ow s

c=zeros(1,4);

for jcol=1:4

for irow=1:4

c(irow)=c(irow) + a(irow,jcol)*b(jcol);

end

end

Matrix Multiplication

by Columns


44/409


Convolution as a Matrix Operation

C onside r t he c onvo lu t i on o f a re f l ect iv it y sequence , r ,

w it h a w ave le t , w , to yie l d a se i smic t rac e , s . Th i s i s

u su al l y wr i tt e n as t he convol u ti on in tegra l:

s(t) = w(t – τ)r(τ)dτ– ∞

∞

Wh en we have d i sc re te, fi n i te l engt h app rox imat io ns t o

t he se quant i ti es , the co nvolut i on i s u su al l y w ri tten as a

summati on . I f r

j

i s t h e re f le ct ivi t y se r ie s wi t h j =0 ,1 ,. . . n,

an d w

k

is t h e poss i b ly non-cau sal wavel e t w it h k =-

m. . .0 . . .m, then :

sk

= Δt wk–j

rjΣ

j = k+m

k–m

U su al l y, in t he se e xpre s si on s, t h e Δt t e rm i s dropped o r

s et t o un i ty. It i s u se fu l t o w r it e ou t a fe w t e rms of th i s

summat ion:

Th e s am e oper at i on c an b e ach ieved b y m at r ix

mult ip l ic a ti o n wher e the w ave let , w , is l oa ded i n to a

s p eci a l mat r i x c al l ed a Toep l i tz or c on vol u tio n mat ri x .

s0

= + w0r

0 + w

–1r

1 + w

–2r

2 +

s1 = + w1r0 + w0r1 + w–1r +2


45/409


It i s a s imple ex erc i se o f m at r ix mu l tip l i cat ion b y r ow s t o

check that t h e f oll ow i ng m at r i x equat ion c om pu tes t he

c onv olu t io n o f w w ith r

w0 w–1 w–2 w–3

w1 w0 w–1 w–2

w2 w1 w0 w–1

w3

w2

w1

w0

r0

r1

r2

rm

=

s0

s1

s2

sn

N ot e t he s ymmetry o f t h e W mat ri x wh i ch h as t he

w av e le t s ample s c on st an t al on g t he d iag on al s . Ano th er

w ay t o v i ew W i s t h at e ac h c o lu mn c onta in s t h e w ave le t

w ith t he z er o t ime s am ple al igned on t he m ai n d iag on al.

N ow , imagi n e d oi ng the mat r ix mul ti pl ic at i on by c ol umns

instea d o f r ow s an d we ge t t he mos t intu i t iv e v i ew of

c onvolu t io n by r ep lacement .

w0

w1w2

w3

r0 +

w–1

w0w1

w2

r1 + =

s0

s1s2

sn



46/409



=

As an e xamp le o f convolu t i on by m at r ix mul t ip l ic ati on ,

h ere i s a n i l lust rat ion of t he c on v olut ion of a r efl ect iv it y

s er i es a nd a min imum ph as e wav le t t o yi e ld a 1 -D

seismogram.

=

As a s econd ex amp l e, h er e i s the c onvo lu t i on o f a

r ef l ect iv i ty s er i es a nd a z er o phas e wa vl e t t o y ie l d a

ze ro phas e s ei smog r am .


47/409



T hes e e x amp les of co nv ol ut ion b y m ar ti x m ult ip l icat ion

show exp l ic i t l y what i s mean t when we s ay t h at

c onv olu t io n i s a s t at i onary p roce ss . Intu i t ivel y , th is

ph ras e means th at t h e ope ra t io n does n ot change wi t h

t ime i n s ome sense . P rec i se l y, i t mean s t h at t h e

w ave for ms i n t he co lumns of t he convol ut io n m at r i x a re

a ll ident ical . Th at is , the wavel e t wh i ch i s s c al e d and

us ed t o re pl ac e e ac h r ef l e ct iv i ty sp i ke doe s n ot change

w i th t ime. As w e shal l s e e, man y physi c al pro cesses

v i ol at e th is as sumpt ion an d i t i s qu i te possi b l e t o

g ener al i ze t h e convolu t i on oper at i on t o m odel

nonstat ion ar y pr oc ess es.

Wh en t he a ssumpt ion of s t at i on a ri t y i s made i n t h e

c onte x t of s t at is t i cal d econvolut i on theor y, it mean s

p re ci se ly t he s am e th ing. W e as sume t ha t t he t im e s eri es

w e m ea sure d ( the s ei sm ic t rac e) i s re lat ed t o that wh ich

w e w an t (the r ef l e ct iv i ty) b y a s ta t io nar y convolut ion

oper at i on . Gi v en t h at , we expect t h at a st a ti onar y

i n ve rs e oper at o r wi l l s u f fi c e t o r ecove r t h e r e fl e ct iv i ty.


48/409


Fou rie r T ransforms and Convolut ion

C onside r t h e convolut i on in teg r al f o r c ont inuous

funct ions:

Now, let g be a complex sinusoidal function: g u = eiωu

h t = f τ eiω t–τ

dτ–∞

∞

= eiωt

F ω

F ω = f τ e–iωτ

dτ–∞

∞

Then:

where

(1)

(2)

Th i s r ema rk ab l e r esu l t s how s th at i f we c onvo lv e AN Y

funct ion , f , wi t h a comp lex s inuso id, t h e r esu l t i s t h e

s am e c omp lex sinuso id mu lt ip l i ed b y a complex

c oe f fic ie nt . T h is c ompl ex c o ef f ici en t, F(w) , i s c ompu t ed

f rom f( t) an d i s k now n as t he Fo uri e r Tr ans for m o f f ( t ) .

Those who h av e st udie d mat hemat i c al p hys i cs wi l l

r ec ogn i ze t h at th i s mean s t h at t h e c om ple x s inu so id s

ar e e igenfunct io ns of t he c on vo luti on o per at or an d t he

Fou ri e r T r ans for m p rov ides t he ei genva lues .


49/409


30Hz Ricker

-1

0

1

0.3 0.4 0.5 0.6 0.7 0.8-1

0

1

-1

0

1

-1

0

1

-1

0

1

-1

0

1

IN

OUT

Maximum amplitude =.064

Maximum amplitude = .27

Maximum amplitude = 1.0

Convolve

10 Hz.

30 Hz.

70 Hz

Fourier Transforms and Convolution

He re we s ee t he r e su l t of convolv i ng 10 , 30 , a nd 7 0 H z

c om ple x s in uso ids w it h a 30Hz Ri c ke r w av ele t . I n ea ch

c ase , on l y t h e r ea l p ar t s of t h e c om ple x s inusoi d s ar e

p lot ted . We se e thatt he 1 0 H z s inusoi d i s d imin i shed b y

73 , t h e 7 0 H z by 93 , a nd t he 30 Hz is un attenuated .

(The d i st o rt i on s i n t h e s inusoi d s a re a rt i f ac t s o f t h e

d is pl ay not t h e c onvolu t i o n al g or i thm . )

0.3 0.4 0.5 0.6 0.7 0.8

0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 0.8

0.3 0.4 0.5 0.6 0.7 0.8 0.3 0.4 0.5 0.6 0.7 0.8


50/409


51/409



A convolut i on ca n af fect not only t h e ampl i tude of a

s inuso i d bu t i t s phas e as we l l . T he R i ck e r w av e le t i s

known as a zer o phase funct ion w hic h m ean s that i t d oes

n ot h av e a ph as e ef f ec t . L e t u s r epeat t h e anal ys i s but

th i s t ime w i th a funct ion wh ich h as a known ph as e

e ffect . F or th i s purpose , w e cons id e r a Ri c ke r w av le t

w it h a 9 0

o

phas e sh i ft .

-0.1 -0.05 0 0.05 0.1-0.1

-0.05

0

0.05

0.1

0.15

-0.1 -0.05 0 0.05 0.1-0.15

-0.1

-0.05

0

0.05

0.1

0.15

30 Hz. Ricker zero phase

30 Hz Ricker 90

o

phase

No te t hat ze ro phase wa ve for ms a re a lway s s ymm et r i c

wh il e 9 0

o

ph as e r esu lt s i n an ant i symmetri c w av ef orm.

W e m i gh t expec t t h e 9 0

o

R ick er t o h av e t he s am e ef fect

on the amp l it u de o f s inusoi d s bu t some andd i t ion a l

e ff ec t as we l l. To s ee , w e repeat the an al ysi s o f p ass ing

c om ple x si n usoi d s th rough i t .


52/409


3 H z Ricker

IN O UT

Max imum ampli tude = 64

M a xi m um a m p li tu d e = 2 7

M a x im um a m p lit u de = 1

Convolve

1 Hz

3 Hz

7 Hz

0.45 0.5 0.55 0.6 0.65-1

0

1

0.45 0.5 0.55 0.6 0.65

-1

0

1

0.45 0.5 0.55 0.6 0.65

-1

0

1

0.45 0.5 0.55 0.6 0.65-1

0

1

0.45 0.5 0.55 0.6 0.65 0.45 0.5 0.55 0.6 0.65

-1

0

1

9 o


Here we repeat t h e r e su lt of convolv i ng 10 , 30 , an d 7 0

H z complex s in uso ids wi th a 30Hz R ic ke r wa vel e t b ut t hi s

t ime t he Ri c ke r h a s 9 0

o

phas e . The amp li tude

a ttenuati o n of t h e s inusoi d s i s t h e s am e as be f or e bu t

now the re i s a n add i t ion al 9 0

o

phas e l ag . (When

c omp ar i ng th i s f i gu r e w i th -2- o f th i s s e ri es , n o te t h at

the re h as b een an x -ax i s s cal e c han ge o n al l p lo ts. )

Result with 90o Ricker

Result with 0

o

Ricker


53/409


-0.15 -0.1 -0.05 0 0.05 0.1 0.15-0.1

-0.05

0

0.05

0.1

Time

0 20 40 60 80 100

-60

-40

-20

0

Frequency

0 20 40 60 80 100

-100

0

100

Frequency


Here i s a c omp le te descri pt i on of t h e 9 0

o

, 30 Hz . Ri c ke r

i n t he t ime domai n an d amp l it u de an d p has e s pect rum i n

t he Fou ri e r d oma in. We h av e s ee n t h at the F ou r ie r

dom ai n p rov ides a c onven ien t de scr ipt ion of t he e ffec t

o f c on volv i ng t h e wav e let wi t h c ompl ex s in us oi ds .

Time Domain

Fourier Domain

Amplitude

Spectrum

Fourier Domain

Phase Spectrum


54/409


55/409


Fourier Analysis and Synthesis

As an e x amp l e c on si der t h e Gau ss ian fu nct i on :

h t = e–α2t

2

Us ing s t anda rd techniques o f integr al c al cu lu s , t h e

Fo uri er t rans fo rm of t he Ga uss ian c an be shown t o be :

H ω = π

α e–ω

2/4α

h t

H

half w idth = 1/

half width = 2

No te th at t h e ha l f widths , as rep resented by the i r 1 /e

po in t s are in ve rse l y p ropor t ion a l. In f ac t :

ΔtΔω = α–1

2α = 2

Th is i s a n e xamp le of a g eneral p rop ert y wh ich s ays that

t h e width o f a time d omai n fun c ti o n is i nver s el y

p r opo rt i on a l t o it s wid th i n f requency. It ca n be s how n,

g iv e n a s ui tab l e me as u re of w id th , t h at :

( w idth i n t ime)(width i n f r equ en cy ) > = a co ns tan t

B r acewe l l (1978, Th e Fou ri e r T ra n sf o rm and i t s

App l ic ati o ns) s how s t he consta nt t o b e 1 /2 an d t hat t he

eq ual i ty h ol d s f or the Gau ss ian .


56/409


Fourier Analysis Example

0 0.05 0.1 0.15 0.2-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

time (sec)

0 100 200 300 400 5000

0.2

0.4

0.6

0.81

frequency (Hz)

0 100 200 300 400 500-80

-60

-40

-20

0

frequency (Hz)

0 100 200 300 400 500-3

-2

-10

1

23

frequency (Hz)

Her e i s a min imum p has e

w ave l et c ons t ruct ed w i t h

a .001 s ec s am ple r at e

and a 3 0 H z d ominant

frequency.

Th e i s t h e amp l it u de

spect rum of t he w ave let

d i sp laye d wi t h a l i nea r

v e rt i cal s c al e . No te th at

t h e f requen cy ax i s s t ops

at 500 Hz wh ich i s

1/(2*.001sec).

He re t he ampl i tude

spec tr um i s d isp laye d w it h

a d ec i bel v er tic al sc a le :

d b =

20*log10(A(f)/Amax)

T hi s i s the ph as e spectrum .

Not e t ha t t he vert ical s ca le

i s i n r ad ian s.

A t th is po int, F our ie r ana lys is m ay l oo k l ik e an e xerc ise i n

g ra ph maki ng ; howeve r , i ts ut il it y w il l become cl ear o n t he

next page .


57/409


Fourier Analysis Example

0 0.05 0.1 0.15 0.

2

0

20

40

60

80

time (sec)

0 0.05 0.1 0.15 0.20

20

40

60

80

time (sec)

Sum of components

Sum of components

A

B

Individual Fourier components

Cumulat ive sum of Fourier

components

He re we s ee t wo equ ival en t way s o f v i ewi ng t he Fou r ie r

t rans fo rm info rmat i on on t he prev io u s p ag e. In A, t h e

ind iv idu al F ou r ie r c om ponents ar e s how n f rom 1 0 t o 7 0

Hz , proper l y s c al e d f or t h ei r ampl i tude an d phase . The

s um o f al l 1 3 c omp onents yi e lds the w ave let at t h e t op

wh ic h i s qu i te sim i l ar t o t he t rue w ave let shown o n t he

p r ev ious pag e. Add i ng i n t h e r ema in in g f r equency

c omponen ts ( 0- >10 Hz a nd 70 ->500 Hz) wi l l re c on str uc t

t he wa ve le t e x ac tl y. The f igu re on t he ri gh t c on tai ns t he

s am e info rmat i on except t h at ea ch tr ac e is t h e s um o f

t h e f re qu en cy c omponen ts be tween i t s f re qu en cy and 10

Hz . Th i s g i ves a g oo d i ll ust ra t io n of how the wavel e t

t ak es fo rm as it s spect rum i s s ummed .


58/409


Fourier Transform PairsThe table below is reproduced from:

Brigham, E.O., 1974, The Fast Fourier Transform, Prentice Hall

Not e : I t i s a r ema rk ab le f ac t t h at n o s i gn a l c a n hav e

fi n i te l ength ( i . e . c ompact suppor t ) i n bo t h t he t ime

an d fr equency doma in s .


59/409


Fourier Transform Pairs

The table below is reproduced from:

Brigham, E.O., 1974, The Fast Fourier Transform, Prentice Hall


60/409


The Dirac Delta Function

T he D ir ac del ta fu nct ion wa s i nven te d b y P .A. M. D i ra c t o

hand le p ro blems i n t h e devel opm ent of quantum

mechan ics . Si nce then , i ts un ique ab il i t y t o rep re sen t a

un it s p ik e i n t he c ont inuou s f uncti o n d omai n . It ca n b e

def ined a s t he l imit i ng fo rm o f a s h ar p ly p eake d funct ion

w hos max imum p ro cee ds t o inf in it y a s it s wi d th shr i nks

t o ze ro i n such a w ay t hat it s a re a rema in s un i ty.

b1

b2

b3

b4

-0.5 -0.25 0 0.25 0.50

1

2

3

4

5

6

7

8

b∞

= δ t

A se ri es o f b oxc ar s w i th

u nit ar ea c onve rge s i n

t h e l im it t o t he de lt a

funct ion :

It can be thought of as:

δ t = 0, t≠0∞, t=0

Th e m os t important p rop ert y of t he de l ta funct ion i s i ts

be h av i or u nd er i nt egr at i on . If f ( t) is an y f un ct ion , t he n:

f t δ t–t0 dta

b

=

f t0 , if a


61/409


T

T

f τ

e–iωto

f τ–to

F ω Mult

The Dirac Delta Function

Consider the Fourier transform of the delta function:

Th us i t h a s a c onstan t , un i t ampl i tude s pect rum (a ls o

known as a wh it e s pe ct rum) a n d l in ear p ha se .

Consider the action of the delta function under convolution:

δ t–t0 f τ–t dt

–∞

∞

= f τ–t0

Thu s t he d elt a funct ion sh i f t s f ( t ) to p l ace i ts o ri g in at

t he l oca t io n where t he a rgument o f t he d el ta funct ion

v an ishes. Th is i s c a ll ed a st ati c sh if t i n s eism i c dat a

p r oce ss ing . S ince c onv olu t io n c an b e don e in the

Fo u ri e r dom ai n b y mult ip l ic at io n of t ra nsf o rm s , we c an

c onc lude that a s tat ic s hi ft c an be don e b y:

Th at i s , a s t at i c sh i f t i s equ ival ent t o a l i near ph as e

sh i f t . Fi na ll y, if we invers e Fou ri e r t ra n sf o rm the

equat io n a t t h e to p o f t h e p ag e, w e end u p w i th a

d ef in i t io n o f t h e de lt a funct io n i n t e rms of i t s Fou ri e r

components:

δ t–t0 e–iωt

dt–∞

∞

= e–iωt0

Th us t he d el ta funct ion h as u ni t amp li tude spect rum

an d a phas e spectru m that is l i near i n f r equ enc y and

w it h s lo p e p r op o rt i on a l t o t h e t im e s hi ft .

δ τ

− t 0( ) =

1

2π e

iω τ − t 0

( )

−∞

∞

∫ = e2π i f τ −t

0( )

−∞

∞

∫


62/409


H ω = F ω G ω

h t = 1

2πF ω G ω e

iωt

dω–∞

∞

The Convolution Theorem

Con sider t h e c on tinu ou s c on vol ut ion of f an d g:

h t = f τ g t–τ dτ–∞

∞

We c an rep re s ent f a nd g i n t erms o f the i r spect r a

a s:

f τ = 1

2πF ω e

iωτ

dω–∞

∞

Subs ti tu tin g t hes e i n to (1 ):

g t–τ = 1

2πG ϖ e

iϖ t–τ

dϖ–∞

∞

h t = 1

2π F ω e

iωτ

dω–∞

∞ 1

2π G ϖ e

iϖ t–τ

dϖ–∞

∞

dτ

–∞

∞

Interchanging

the order of

integration

The term in [ ] is the

Dirac delta function.

The delta function

col lapses one of the

frequency integrals

Her e w e have h( t ) rep resented a s the inverse Fou r i er

t r an s form of s ometh in g . By i n fe r en c e, t ha t somethin g

mu st be t he Fouri er t ransf or m of h. Thus :

h t = 1

2π F ω G ϖ

1

2π e

i ω–ϖ τ

dτ–∞

∞

eiωt

dωdϖ

–∞

∞

h t =

1

2π F ω G ϖ δ ω–ϖ eiωt

dωdϖ–∞

∞

(1)

an d


63/409


Multiply

f(t)

g(t)

FFT FFT

IFFT

F(

ω

) G(

ω

)

H(ω)

h(t)

The Convolution Theorem

Th e r esu l t we h av e j us t de ri v ed is one o f t h e most

fu ndamen ta l an d imp or tan t in al l of si gn al pro ce ss in g . It

t e ll s u s tha t we c an c on vol ve tw o s ign a ls b y mult ip ly i ng

the ir s p ect r a an d invers e Fou r i er t ra n sf o rm ing t he

r esu l t . Th e re as on th at t h is is impor tan t is t h at there i s

an ext reme ly f as t al gor it h m fo r per form ing t he d ig it a l

Fo u ri e r t r ans for m c al l ed t he f a st F ou r ie r t ransf o rm

( FF T) . Us i ng t he F FT a c on volut i on c an be don e b y:

H ω = F ω G ω = A F ω eiφ

F ω

A G ω eiφ

G ω

= A F ω A G ω ei φ F ω +φG ω

Not e t h at mul t ip l y in g comp le x spec tr a i s:

Th a t i s w e c an v i ew i t a s mult ip lyi n g t he amp l it u de

s p ect r a an d addin g t h e p h as e s pe ctr a.


64/409


Sampling

T h e an a ly t ic an al y si s o f c on ti nu ous s ig n al s i s mos t u s ef u l

f or g ai n ing a conceptua l u nd er s tand ing of signal

p r ocessi ng . In ac tual p ra ct ice ; h owever , t h e v as t

ma jor i t y of w or k is done wi t h d i sc re t ly s amp led

f un ct ion s . Th e p r oc e ss o f s ampl in g a cont in uou s f un ct ion

i n t ime ca n be v i ewed as a m ult ip l i cat io n b y a s amp l in g

comb.

Continuous

Gaussian

Sampling

Comb

Sampled

Gaussian

Times

Equals

Comb spacing = Δt

Timme Doomainn Frrequenccy Domaa inn

Convolved with

Continuous

Gaussian

spectrum

1/Δt

Fo ur i er

t ransform

of

s am pl i ng

comb

Gaussian

Spectrum

and aliases

1/Δt

Equals


65/409


F

-2Fn

-Fn

2Fnn

F

-2Fn

-Fn

2Fnn

Primary

frequency band

Sampling

S o w e h av e s een that s amp l ing i n t h e t ime d oma in

cause s the rep l i ca t io n of the c ont inuou s spect rum i n

t he f r eq uency d omai n . Th e spaci n g b etween thes e

spect ra l al i as es i s 1/

Δ

t and i t i s customa ry t o res t ri c t

o u r a t tent i on t o the primar y frequency band l i ei n g

b etwe en - 1 /(2

Δ

t) an d 1 /(2

Δ

t) . Th e f r eq uency Fn =

1/(2Δt ) i s c al l ed t he N yqu i st frequency an d i s t h e

l im it in g f re qu en cy o f t h e s amp l ed d at a .

Fnyquist = 1/(2Δt)

Spectrum of

sampled

data

showing

aliasing.

Spect r um of

sampled

dat a wi t h

min ima l

a l ias ing .


66/409


Sampling

The u na li a se d s ampl in g o f a n y c on ti nu ou s s i gn a l r eq uir e s

tha t t he s ign a l h av e i ts powe r r es tr ic ted t o a f requency

b an d: - fmax < f < fmax . Such s ignal s ar e s ai d t o b e

b an d lim ited . A band l imit e d s i gn a l c a n be d i gi t al l y

s amp l ed , wi thou t al i as in g , w i th a s amp l e s i ze of

Δ

t=1/(2fmax ) . It i s a fundamenta l theo re m (The

S amp l ing Theo rem , P apou l i s, Si gn a l Ana ly s is , p 141,

1984) t hat such a b an dlim ited, c ont inuous, s ign a l c an

b e ex ac t l y re cove red f ro m i ts d i gi t al s amp les b y a

p roce s s know n as s in c f un ct io n i n te rpol at i on .

Spectrum

of sampled,

unaliased,

continuous

function

Multiplied

by a

boxcar

Recovers the

spectrum of

the continuous

function

Sampled band limited

function

Convolved

w i th a si n c

function

Recovers the original

continuous function

Interpolation

site

Tiimee Domaa inn Frequeencyy Doomainn


67/409


Sampling

In or der t o min imize al ias i ng, r aw ana l og s ei sm ic d at a i s

p assed th rough a n anal og an t i al i as f i l ter p r i or t o

d i g it iza t io n . A typ ic a l an t i al ias fi l t er h as a n ampl i tude

spect rum wh ich beg ins t o r ol l of f at 50 t o 60 o f

f nyq uis t and re ac he s v er y la rg e at tenu at ion (>60db) a t

fnyquist .

0 20

40

60

80

100

120 140-120

-100

-80

-60

-40

-20

0

Frequency (Hz)

Here is t he

spectrum of an

antial ias filter

for use pr io r to

sampling at

.004 sec .

R u l ee o f t h u m b : Samp le your d at a such th at t he

e xp ected s ig na l f reque nc ie s a re l e ss t h an h al f f

n yqu ist.

Al i as i ng i s al s o a pos s ib i li t y when re s amp l ing se ismic

d at a . If t h e new samp le i n ter v al i s mo re c oa rs e th an t he

ol d , then an an t i al i as f i lt e r s h ou l d be ap pl ied .

. 008 s 62 .5 H z

.004 s

.002 s

.001 s

125 H z

250 H z

500 H z

sample

ra te

Nyquist

C ommon sampl in g

ra t es and the i r

Nyqu ist f requenc ie s


68/409


The Discrete Fourier Transform

T he gr ea t u t il it y o f th e cont inu ous F ouri er tr an sf orm t o

decompose f unct ions in to fundamenta l comp lex s inuso id s

ca n be app li ed d ir ec tl y to d i sc re t ely sampled time

domain fu nc ti on s. Con si d er a fu nc ti o n h( t ) w h ic h i s z e ro

ev eryw her e ex cep t a t N t im es d ef in ed b y t=kΔt , k=0,1 ,2

.. . N -1 , where i t takes t he valu es h

k

. T h is functi o n ca n

be w ri tt e n w i th t he d irac d e lt a f un ct io n a s :

If we now take the Fourier transform of h(t) we have:

H ω =

H ω = hke

–iωkΔt

Σk = 0

N–1

Her e w e have a n a na lytic expre ssi on fo r t he Four ie r

transfo rm o f th e h

k

samp le s wh ic h is def ined fo r a l l

ω

.

We have a lready see n th at t he phenomeno n o f a li a sin g

lim it s th e usab le fre quency band t o -

π

/

Δ

t - > +

π

/

Δ

t .

Fu rt he rmo r e, l in ea r al ge br a te l ls u s th a t N f re quenc ie s in

th is band sh oul d s uf fic e to determin e th e N h

k

. S o w e a re

le ad t o c ons id er sampl in g th e frequenc y domain a t

ω

ν

=

2

πν

/(N

Δ

t) ,

ν

= 0 ,1, 2 .. . N -1.

h t = hkδ t–kΔtΣ

k = 0

N–1

hkδ t–kΔtΣ

k = 0

N–1

e–iωt

dt

–∞

∞

= hk

δ t–kΔt e–iωt

dt–∞

∞

Σk = 0

N–1

Hυ

= hke

–i2πυk/NΣk = 0

N–1


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D isc re te e xp onent ial s h av e a w e ll k now n orthog on a li ty

p r ope r ty su ch t h at :

U s ing th i s , i t i s n o t d if f icu l t t o sho w that t h e h

k

samples

c an be re cov ere d fr om t he H

ν

by :

h k = 1

NHυe

i2πυk/NΣυ = 1

N–1

Hυ = h ke–i2πυk/NΣ

k = 0

N–1

Th i s r e su lt t og et he r w it h :

Inverse DFT

Forward DFT

f or m th e d i sc r et e Fou ri e r t r ans for m pa ir . They a re t he

d i rec t a n al og t o the cont inu ou s Fou r ie r t ra n sf o rm

r el at ion s. L ike t h e F T, t he DF T i s c omp lete i n t hat t he h

k

ar e e x ac t l y r e cov e rab l e f rom t he ir s p ec tr um , t h e H

ν

.


70/409


Principleband

Spectrum

of sampled,

unaliased,

continuous

function.

Sampled band limited

function N samples long

Times a

sampling

comb

Convo lved wi th the

tr ansform o f t he

sampli ng comb

The sampled time series

becomes periodic

with period T=NΔt

The

sampled

spectrum

Principle

band

DFT

II DFT

Spectrum is

periodic

with

period

2πN/Δt

Δ

t

1/Δf

T

Δf

f

nyq

f

nyq

f

nyq

f

nyq

1/Δ

t

f

nyq

=

1/(2Δ

t) T =

1/Δ

f

Δ

f

Δ

t = 1/N

Timme Doomaiin Frequeencyy Doomaiin


Here i s a p ict ori al r e pre sentat io n of t he dev elopment of

t he D FT fr om t he c ontinuous c as e :


71/409



Th e s ampl ing o f t h e spect rum of a d i scre te t ime se ri e s

c auses t h at s e ri e s to b ecome pe ri od i c w i th pe ri od T =

N

Δ

t. Th is h as s ign a l p rocessi ng c onsequence s t hat ar e

ap par en t when w e c ons ider app ly in g a f il t e r wi t h t h e

D FT and t he co rr espond ing c onvolu ti o n.

Filter operator:

Time series

showing time

domain aliases

Principle Period

Th e c onvo lu tio n ope rat i on t h at d up li c at e s mul t ip li c at i on

w it h t he D FT i s c al l ed c i rc ul ar c onvolu ti on . N ot e t hat t he

f i lt er oper at or p la ced on t he l as t s amp l e o f t he p r incip l e

pe ri o d appear s t o wr ap ar ou nd an d af fect t h e fi r s t

s amp le . To av o id th i s p ro blem, i t i s c ommon t o p ad t he

t ime s er ies wi th a l e ngt h of ze ro s c hos en w it h t he le ngt h

of t he f il te r op er at or i n mi nd .

Principle Period

Zero

Pad


72/409


The Fast Fourier Transform

T he f as t Fo uri er t rans form ( FFT ) is n oth ing m or e than a

c lev er w ay of c al cu l at i ng t he D FT wh ich ge ts impres s iv e

pe r fo rmanc e r esu l ts . Th e c on volut i on of an N l ength

oper at o r i n t h e t ime domai n requ i re s on the or der o f N

2

f loa ti ng p oi nt op erat ion s. T he s am e c om pu tat ion i n t he

f r eq uen c y domai n wi t h t h e FFT r eq ui re s r ou gh ly N* lo g(N)

oper at i on s . How eve r, we m ust be c arefu l wi t h th is

s t atement b ec ause , gene ra ll y , t h e tw o N' s ar e n ot t h e

s ame. Th i s i s bec au s e t he FFT a l go ri t hm requ i res t h at

t h e time s er i es length b e a mag i c numbe r wh ich is

usual l y a p owe r of 2. ( Al s o the two time se r ie s be in g

c onv ol ved mus t be t he s am e le ngth . ) Th i s is ac hieve d b y

a ttach in g a ze r o p ad t o t he t im e s er i es . Thu s if N i s t h e

lengt h o f t h e t im e domai n ope ra tor an d i f N 2 i s t h e f i rs t

powe r o f 2 g re at e r than N , then we must compare N2 t o

N2 lo g(N2) . (Often e ven th i s i s no t en ough bec au s e t he

z er o p ad must b e long enough t o avo id ope ra t or w r ap

a round. ) The b ot t om l in e is t h at short ope ra t or s ( l ess

tha t ~64 p oints) a re of ten app l ie d f a st e r wi t h

c onv olu t io n wh i le lo ng oper at o rs a re MUCH f aste r wi t h

F FT' s. Th e dia gr am b el ow i s adapted f r om H atton e t a l.

a n d s hows t h e bas i c t r ad e of f .

Convolution

compute time

Operator Length

FFT

Time domain


73/409


-120

-100

-80

-60

-40

-20

0

0 50 100 150 200Frequency (Hz)

Wavelet 5avelet 4avelet3avelet 2

Wavelet 1

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Wavelet 1

Wavelet 2

Wavelet3

Wavelet 4

Wavelet 5

Five

Generic

Wavelets

Their

Fourier

Amplitude

Spectra

Filtering

We h av e se en t h at c onvo lu t i on w it h a wavefor m

surp res se s an d pos s ib ly p has e sh i ft s s om e f requenc ies

re l at iv e t o ot he rs . T h is f i l ter in g a ct io n i s of t en ex pl oit e d

t o enhan ce s i gn a l an d surpres s n o is e . He re w e se e a

compari s on of f i ve d i ff ere n t zer o phas e f i l te r s i n b ot h

t h e t ime and f re qu en cy domai ns. Th e i nv ers e r el at io nsh i p

be tween tempor al w idth and f requency bandwid th is

r ead i l y appa rent .


74/409


The Z Transform

The per i od i c i ty or c i rcu l ar i ty i nhe ren t i n bo t h t ime and

f req uen c y is n ic el y c ap tu red b y a p owe rf u l me thod ol og y

know n as t he Z tr ans form . C onsi de r the t ime s er i es , [ 1 -

. 5 - .3 0 . 1 0 ] , wher e i t is as sum ed t o s t ar t a t t =0 and

inc rement by t . We rep resent th is s er ie s in t h e Z

domai n by a p o ly nom ia l i n z:

= 1–.5z1

–.3z2

+.1z4

H z = 1z0–.5z

1–.3z

2+0z

3+.1z

4

• Negative times correspond to negative exponents of z

• Mu l t i pl i cat ion b y z

n

de l ay s the t ime ser ies by n

s amp l es i f n i s p os i t i ve a nd a dvances i t b y n s am pl e s

f o r n eg at iv e n.

The g re at u t il i t y of t h e Z t r ans for m li e s i n it s ab il i t y t o

r ep re s en t d i sc r et e convol ut ion an d t h e DFT as op e rat i on s

w i th p olynom i al s . It i s n o t d i f fi cu l t t o show that t h e

c onv olu t io n of t wo t ime s e ri e s , f an d g, c an b e re al i ze d

b y s imp ly mult ip lyi ng the i r Z t rans f or ms an d r ea d ing of f

t he r esu l t. ( See W at ers ( p 133) f or a p roo f. )

h = f•g

=H z = F z G zf0g0+ f0g1+g0f1 z

1

+ f0g2+f1g1+g0f2 z2

+ ...

F z = f0+f

1z

1+f

2z

2+ ... G z = g0+g1z

1

+g2z2

+ ...

S o w e see t hat the exponent o f z gives the s ample n umber

and hence determines the sampl e t ime ( n

Δ

t ). Note also

the fol lowing:


75/409


The Z Transform

T he fa ct t ha t co nv o lut io n is d one b y m u lt ip lic at io n of Z

tr ans fo rms is r em in is c ent o f t he Fou r ie r tr ans fo rm . In

fa ct , i f w e l e t z = e

- i

ωΔ

t

th e n t he Z t ra ns f orm b ec omes :

As wi t h t he D FT , i f w e now c ons ider on l y d i sc re t e

frequencies ω

ν

= 2πν/(NΔ t ) , ν = 0 ,1, 2 . .. N-1 , t h en w e

s ee n th at t h e Z t r ans form , w i th z = e

- iω Δt

, is p r ec i sel y

t he D FT .

G z = gkzkΣ

k = 0

N–1

G ω = gke

–iωkΔtΣk = 0

N–1

G ν = gke–i2πυk/NΣ

k = 0

N–1

The Z transform is more general t ha n the DFT s in ce z c an be

any comp lex number. I n fact the DFT amounts to eva luating

the Z tranfo rm at N d iscrete loca tions around the un it cir cle

in the complex z p lane.

ω

o

ω

1

ω

ν

ω

2

ω

N-1

ω

ν+1

Complex z plane

real(z)

imag(z)


76/409


The Z Transform

Consider th e elemental c ou pl et F(z) = 1 -a z. N ow i f w e

convol ve F(z ) wi t h anothe r ar bi tra ry t ime se ri e s g (z ),

t he n we r eprese nt t his as : H (z) = F(z)G(z) . Su ppos e t hat

on ly F(z) an d H(z) ar e known t o u s an d w e w ish t o

r ec ov er G (z) . In t he z transfo rm domai n we c an s imp ly:

H z = F z G z ∴ G z =H z

F z

F–1

z = 1F z

So we define the inverse of any time series as:

For F(z) = 1 -az, this gives:

F–1

z = 1

1–az= 1+az+ az

2

+ az 3

+

Th is s er ies , ca l le d t he geometr ic s er ies , i s known t o

converge absolute ly provided t hat | az | < 1 . S ince w e ar e

especial ly interested in th is resu l t eva luate d on th e un it

c irc le ( | z| = 1 ) then w e need | a| < 1 . It i s c ustomary t o

ta lk abou t t he locat ion of t he ze ro of th is couplet

def ined by:

1–az0

= 0 ⇒ z0

= 1

aI f |a| < 1 , then w e se e t h at z

o

mu st l i e outs ide t he un i t

c i r c le in orde r f or the inver s e t o c onverge . S u ch a n

inverse is sa id t o b e s tab le (phy ic a ll y r ea li za bl e ) . N ot e

a l so t ha t F( z ) i ts el f i s t ri via l ly s t ab l e.


77/409


The Z Transform

Any causal, st ab le t ime se ri es wi th a cau sa l , s ta bl e in ve rs e

i s sa id t o b e min imum phase . Thus our e lementa l couplet ,

1- az , is m in imum phase whenever |a |


78/409


The resul tant o f the sequent ial convolut i on of any

numbe r o f minimum phase time series i s al so

minimum phase.

The Z Transform

The zero s o f F(z) corre sp on d t o poles f or F

-1

( z) . Th us, fo r

the case of a t ime se ri es whose Z transform h as a

denominator, we s ee th at t he st abi li ty condit i on req uir es

that a l l po les a lso l i e ou ts ide the un it ci rc l e. The most

g enera l t im e s er ie s can b e w ri t te n a s a Z transform w it h

both numera to r and denominator such as:

H z =A z

B z=

z–α0 z–α1

z–β 0 z–β 1

We say the cor respond in g t ime s er ie s is min im um p h ase

if a ll α

i

a n d a ll β

i

l i e o uts id e th e un it ci r c le . T h e

f ol low ing t heo r em fo ll ow s immed ia te l y:

Conversely:

I f a ny time series i n a se qu ence of co nvoluti

methods of seismic data processing,

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