Methods of Seismic Data Processing,

Methods of Seismic Data Processing Gary F. Margrave Geophysics 557/657 Course Lecture Notes, Winter 2006

The Department of Geology and Geophysics The University of Calgary

Methods of Seismic Data Processing Geophysics 557/657 Course Lecture Notes 420 Pages Winter 2005

by G.F. Margrave, Associate Professor, P.Geoph. The CREWES Project Department of Geology and Geophysics The University of Calgary Calgary, Alberta, T2N-1N4 403-220-4604 [email protected]

Table of Contents Section Title Chapter 1: Synthetic Seismograms The Big Picture Elastic Waves Well Logs Gardner's Rule The Wave Equation Traveling Waveforms Normal Incidence Reflection Coefficients Synthetic Seismogram Algorithms Synthetic Seismogram Examples P-S Synthetic Seismogram Construction

Page Number 30 pages 1-2 1-7 1-9 1-11 1-14 1-17 1-19 1-23 1-28 1-30

Chapter 2: Signal Processing Concepts Convolution Convolution by Replacement Convolution as a Weighted Sum Matrix Multiplication by Rows Matrix Multiplication by Columns Convolution as a Matrix Operation Fourier Transforms and Convolution Fourier Analysis and Synthesis Fourier Analysis Example Fourier Transform Pairs The Dirac Delta Function The Convolution Theorem Sampling The Discrete Fourier Transform The Fast Fourier Transform Filtering The Z Transform Crosscorrelation Autocorrelations Spectral Estimation Wavelength Components Apparent Velocity (or phase velocity) The 2-D F-K Transform F-K Transform Pairs -p Transforms Properties and uses of the -p Transform Inverse -p Transforms Least Squares -p and f-k Transforms

76 pages 2-2 2-5 2-6 2-7 2-8 2-9 2-13 2-19 2-21 2-23 2-25 2-27 2-29 2-33 2-37 2-38 2-39 2-44 2-46 2-48 2-53 2-56 2-58 2-62 2-63 2-68 2-71 2-74

Chapter 3: Amplitude Effects Seismic Wave Attenuation True Amplitude Processing Automatic Gain Correction (AGC) Trace Equalization (TE) or Trace Balancing Constant Q Effects Minimum Phase Intuitively Minimum Phase and the Hilbert Transform

32 pages 3-2 3-8 3-9 3-13 3-14 3-18 3-21

Minimum Phase and Velocity Dispersion Array Theory

3-25 3-27

Chapter 4: The Convolutional Model and Deconvolution Bandlimited Reflectivity The Convolutional Model Frequency Domain Spiking Deconvolution Finding a Wavelet's Inverse Wiener Spiking Deconvolution Prediction and Prediction Error Filters Gapped Predictive Deconvolution Burg (Maximum Entropy) Deconvolution The Minimum Phase Equivalent Wavelet Vibroseis Deconvolution Deconvolution Pitfalls Reflectivity Color Q Example

61 pages 4-2 4-4 4-12 4-20 4-23 4-28 4-32 4-36 4-39 4-41 4-47 4-55 4-58

Chapter 5: Surface Consistent Methods Seismic Line Coordinates A Surface Consistent Convolutional Model Surface Consistent Methods Statics and Datums Statics with Uphole Times Surface Consistent Residual Statics Refraction Statics

29 pages 5-2 5-5 5-9 5-12 5-17 5-19 5-25

Chapter 6: Velocity Definitions and Simple Raytracing Velocity in Theory and Practice Instantaneous Velocity Vertical Traveltime Vins as a Function of Vertical Traveltime Average Velocity Mean Velocity RMS Velocity Interval Velocity Snell's Law Raytracing in a v(z) Medium Measurement of the Ray Parameter Raypaths when v = vo + cz

26 pages 6-2 6-3 6-4 6-6 6-8 6-10 6-11 6-13 6-18 6-20 6-24 6-25

Chapter 7: Normal Moveout and Stack Normal Moveout Stacking Velocity Normal Moveout and Reflector Dip NMO for a V(z) Medium Dix Equation Moveout Normal Moveout Removal Extension of NMO and Dip to V(z) NMO for Multiple Reflections CMP Stacking Post Stack Considerations ZOS: A Model for the CMP Stack Fresnel Zones

38 pages 7-2 7-5 7-6 7-10 7-13 7-15 7-17 7-22 7-27 7-30 7-34 7-36

Chapter 8: Migration Concepts Raytrace Migration of Normal Incidence Seismograms Time and Depth Migrations, A First Look Elementary Constant Velocity Migration Huygen's Principle and Point Diffractors The Exploding Reflector Model F-K Migration, Geometric Approach F-K Migration, Mathematics F-K Wavefield Extrapolation Recursive F-K Wavefield Extrapolation for v = v(z) The Extrapolation Operator Vertical Time-Depth Conversions Time and Depth Migration in Depth Kirchhoff Migration Finite Difference Concepts Finite Difference Migration

52 pages 8-2 8-5 8-6 8-9 8-14 8-20 8-25 8-27 8-31 8-33 8-36 8-37 8-40 8-43 8-46

Chapter 9: The Third Dimension Impulse Responses Wave Propagation Fresnel Zones Wavelength Components Apparent Velocity (or phase velocity) The F-K Transform F-K Transform Pairs F-L transform Computation 3-D Migration by Double 2-D Exploitable Symmetries Mapping Strategies Time migration of traveltime maps

32 pages 9-2 9-6 9-7 9-10 9-13 9-15 9-19 9-20 9-24 9-27 9-29 9-31

Chapter 10: Seismic Resolution Limits Resolution Concepts Linear v(z) resolution theoru for zero offset seismic data

35 pages 10-2 10-18

Chapter 11: Study Guide Geophysics 557 Final Exam Study Guide Exam Sampler

9 pages 11-2 11-7

Methods of Seismic Data Processing Lecture Notes Geophysics 557

Chapter 1 Synthetic Sei sm ogram s

Methods of Seismic Data Processing

1 -1

The Big Picture T h e s i m p l es t m o d e l o f s ei s m i c d at a i s t h at o f a w av e le t c on v o lv e d wi t h re f l ec t i v it y . T h e p i c tu re i s s i m p l e a n d a p pea li n g . A c om p a c t p uls e o f so u n d i s s e n t d ow n in t o t h e e ar t h a n d s ca l ed c op i e s o f i t a re re f l ec t e d f r om t h e m a j or f or m a ti o n b ou n da ri e s .

T he s e e c ho e s ar e r e c o r de d o ve r t he e xt en t o f t he s e i s m i c ex pe r i m e nt an d a na l yz e d . S i nc e e a c h e c ho i s a s c a l e d c o py of t h e s o ur c e w av e f o rm , s i mp l e c o m pa r i s o n m ak e s i t i s e a s y t o de d uc e t he re l a t i v e s t r e ng th o f t he di f f e r e nt r e f l e c t i n g ho r i z on s . T he e s t i m at e d s e t o f r e f l e c t i o n c oe ff i c i e nt s i s c a l l e d t h e r ef l e c t i vi ty f unc t i o n o f t he e a r t h b e ne a t h t he s ur ve y . I t s a n i c e c o n c ep t bu t is i t v a li d ? H o w c a n i t b e d e f en ded f ro m ba si c p h ys i c al p r i n c i p le s ? W h at a s s u m p t i on s ( t h er e a r e a l w ay s a s su mp t io n s i n p h ys i cs ) a r e re q u i r ed ? W h e n a r e t h ey j u s t i fi e d a n d w h en a re t h e y n ot ?

1-2

Synthetic Seismograms

The Big Picture O n c e w e s ta r t t o t hi n k ab o u t th e i de a , w e c a n i mm e d i at e l y c o m e up w i th a l o t o f q ue sti o ns 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 s n't th e r e at te n ua ti o n o f se i s m i c e ne r g y a nd do e s n ' t t hi s c h an g e th e s o ur c e w a vef o r m? • What is convolution anyway? (And why should I care?) • What about multiple bounce echos? Don't they confuse things? • If things are so simple, how come seismic processing is so complicated? Maybe those processors are just fooling us ... • 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?) • What's this band-limited stuff? • W h y c a n' t I j u st t ru st t h e s eis m ic p ro c es so r t o take c a re o f t he se m es sy d e ta i ls ? I 'm su r e th a t y o u ca n th in k o f m o r e q u e st io n s a s w e ll . A l l o f t he se q ue s ti on s h a ve t h e ir r e l e va n ce a n d I h o p e to a d dr ess m an y of t h e m in th is co u r se . A t t he e n d , y o u sh o u ld h av e a g o o d u nd e r s ta n di ng o f t h e st r e n g th s a nd w e a k n e ss e s o f th e c o nv o lu ti on a l m o de l a n d t hi s s ho u ld h e lp y ou fo r m a h e a lt h y , sce p t ica l v ie w o f f in al s e is mi c im a g e s .


1 -3

The Big Picture S e is m i c d at a p r oc e ss i n g i s t y p ic a ll y d i v i d e d i n t o m a n y st e p s t h ou g h t h e r ea li t y i s t h at t h e s e is m i c re f l ec t i on p ro c es s d oe s n ot c le an ly se p ar at e i n t o d i s cr e t e p ac ka ge s . W e h a v e a so u rc e w h i c h s e n d s o u t a co m p l i c at ed , l ar ge l y u n k n ow n wa v ef or m w h i ch e x p an d s, a t t en ua t es , r ef l ec t s , t ra n s m it s , c h an g e s m od e s , a n d ge n e ra l l y s c at t er s a bo u t w h i l e a s et o f re c ei v e rs p la c id ly r e c or d s w h at e v er co m e s t h e ir wa y. A n d ge n e ra l l y w h at h i t s t h e re c or d er s is f ar mor e co m p l i c at ed t h an t h e s i m p l e d i re c t e c h os t h at w e w an t : Receivers

Surface wave

All kinds of waves sweep across the receivers

P-wave reflection

S-wave reflection

G o d wo u l d n ot p r oc e ss s ei s m i c d at a t h e wa y we d o . ( I' v e r ec e iv e d a r ev e l at i on o n t h at p oi n t . ; - } ) I n s t ea d , H e w ou l d b ac k t h e wa v e s d ow n i n t o t h e e ar t h u n d o in g a l l p h y si c al e f f ec t s a t t h e p oi n t w h e re t h e y o c c u rr e d . W e a r e p r ev e n t ed f ro m d o in g t h i s l ar ge l y b e c au s e o f i gn o ra n c e o f t h e s u b s u rf ac e st r u c t u re . T h at i s, i n o rd er t o u nd o t h e p h ys i c al e ff e c t s o f wa v e p r op a ga t io n , we r eq u i r e k n ow l ed g e o f t h e s u b s u rf a ce p r op e r ti e s th a t c on t r ol t h o se e f f ec t s . U n fo rt u na t el y , th o s e a re t h e v e ry p r op e r ti e s w h ic h we h o p e t o d i sc o v er w i t h t h e s ei s m i c e x p er i m e n t in th e f i rs t p l ac e . P ro bl e m s o f t h i s so rt a re c om mo n in g eo p h ys i c s a n d a r e c al l ed " i n v e rs e p r ob l em s" . 1-4


The Big Picture S o , f ac e d w i t h t h e n e e d t o f i n d a s ol u t i on i n s p i t e o f a l m o st t o t al ig n or an c e , we su b d i v i d e, c om p ar t m en t a li z e, a s s u m e , a n d a p p r ox i m at e u n t i l w e r ea ch a r es t at e m e n t o f t h e p r ob l em wh i c h i s s o v as t l y s i m p l if i e d t h a t w e c an a c t u al l y so l v e it . A n ex a m p l e o f such a t r e m en do u s o v e r si m p l i f i ca t io n is th e " c o n v ol u t i on al m o d el " o f th e s e is m i c t ra c e w h i ch is o f c e n t ra l im p or t an c e t o d e c on v o lu t i on t h e or y. C on ti n u i n g w it h sw e ep i n g g en er al i ti e s , we c an gr ou p m o st p h y si c al l y ba se d s ei s m i c p r oc e s se s i n t o o n e o f t wo g ro u p s : im a gi n g p r oc e ss e s a n d d ec o n v ol u t i on p r oc e s se s . I m a gi n g p ro c es s e s a tt e m p t t o d et e rm in e t h e co rr e c t s p at i al p os i t io n o f t h e e c h os a n d a r e t yp i f i ed b y n m o r em o v al , c m p s t ac ki n g , a n d m i gr at i on . D e c on v ol u t i on p r oc e ss e s a t t e m p t t o r e m ov e t h e i l l u m i n at i n g w av ef o rm a n d m ax i m i ze t h e r e so lu ti o n o f t h e s e i sm ic i m ag e . E x am p l e s a r e g ai n r ec ov e r y, s t at i s t i ca l d e c on v o lu ti o n , i n v er s e Q f i l t er i n g, a n d wa v e l et p r oc e ss i n g .

Deconvolution techniques


1 -5

The Big Picture I n o r d e r t o u n der s t an d t h e i m p l i c at i on s o f o u r si m pli f i ed t h e or ie s , it is i m po rt a n t t o u n d e rs t an d a s m u c h a s p o ss i bl e a bo u t t h e m or e r ea li s t ic p h ys i c s t h at w e a r e a p pro x im a t i n g. T h e r ef or e , i n a d d it i on t o st u dy i n g m a th em a t ic a l s i m p l i fi c at i on s s u c h a s t h e c on v ol u t i on a l m o d el , w e wi l l n ot h es i t at e t o ex a m i n e o f t h e m o s t i m p o rt an t p h ys i c al m e c h an i s m s i n v ol v ed i n s ei s m i c w av e p r op a ga t io n .

deconvolution methods the convolutional model

imaging methods one-way scalar waves

primaries, multiples, etc elastic wave theory anelastic wave theory physics of continuous media

1-6


Elastic Waves T h e si m ple s t e la st i c m a t er i al r eq u ir e s 2 f u n d a m en t a l c on st a n t s t o d e s c ri b e t h e re la t io n be t we e n st r e ss a n d s t ra in k n ow n a s H oo ke ' s l aw : σ ii = λΔ + 2μεii, i=x,y,z

Δ = εxx +εyy+εzz (Sherrif and Geldart, Exploration Seismology, 1981)

σ ij = μεij , i=x,y,z, i≠j

H e re σ i j d e n ot e s t h e c om p o n e n t s o f t h e s t r es s t e n so r a n d e i j t h e c o m p on e n t s o f t h e s tr ai n t en so r. λ a n d μ a r e c al l ed t h e L a m e co n s t an t s a n d μ is a ls o o f t en kn o wn a s t h e s h ea r m o d u l u s. μ i s z er o f or a f lu id . O t h er c o n st a n t s a r e o f t e n a l s o re f er e n c ed s u c h a s Yo u n g' s m o d u l u s , E , P oi s s on ' s r at io , σ , a n d t h e bu l k m od u lu s, k . T h e s e c on s t an t s a r e a l l r e la t ed i n v ari o u s w ay s a n d a n y t wo s u ff i c e t o d e sc r i be t h e el as t i c m a t er ia l . E =

μ 3λ+2μ

σ =

λ+μ

λ 2 λ+μ

k =

3λ+2μ 3

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: λ, μ, E, σ, and k, plus the density, ρ.


1 -7

Elastic Waves It is well established in theory1,2,3 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: 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 α2–2β 2

μ = ρβ

2

σ =

α2–2β2 2 α –β 2

2

α = β

21 –σ 1 – 2σ

3.5

3

2.5

2

1.5 0.2

0.25

0.3 0.35 Poisson's ratio

0.4

0.45

1: Waters, Reflection Seismology, 1987 2: Sherrif and Geldart, Exploration Seismology, 1982 3: Aki and Richards, Quantitave Seismology Theory and Methods, 1980,

1-8


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

son

β =

10

6

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.


1 -9

Well Logs Here are some example logs from 8-8, an oil well in the Blackfoot field 1400

100/08-08023-23 W4

1400

mannville

mannville

1450

1500

1450

1500

coal_1 coal_2

coal_1 coal_2 coal_3

coal_3

1550

100/08-08023-23 W4

glauc_ch_top

1550

glauc_ch_top glauc_1 glauc_ss_top

glauc_1 glauc_ss_top

glauc_base

glauc_base

1600

1600 miss

miss base

base

1650 3 5 0

3 2 2 0 5 0 0 0 0 Units of log SON Faster

1 5 0

1650 3 0 0

2 2 2 2 2 1 8 6 4 2 0 8 0 0 0 0 0 0 Units of log RHOB More dense

Why do these logs appear to have a negative correlation?

1-10


Gardner's Rule W e l l l og s a re o f t en i n ad e q u at e , i n c om p l et e , o r m i s si n g . O n e c om mo n e x am p le o f t h i s c om e s f ro m t h e f ac t t h a t s on i c lo gs ( SO N ) a re ru n m u c h m or e f r eq u en t l y t h an d e n s it y l og s. T h u s we a re o f t e n f ac e d w it h t h e n ee d t o c re at e a s e is m o gr am w it h o u t d e n s i t y i n f or m a ti o n . G a rd ner et a l. ( 1 ) , f ol l ow ed th e r ea so n ab l e a p p r oa c h o f s ee ki n g a n em pir i c al r el at i o n sh i p b et w ee n P - w av e v e lo c it y a n d d en si t y. B e lo w i s a c r os sp lo t o f a a n d r fo r B l ac k fo ot 8 - 8 w h i ch i n d i c at es a re as on a bl e c or r el at i on e x is t s : 3000 2800

2600 2400 2200

2000 1800 2000

3000

4000 5000 P-wave velocity

6000

7000

1 Ga rd ner , G .H. F ., G a rd ne r , L . W . , an d G re gor y, A . R. , 19 74, Fo rm a tio n v e lo cit y an d d en s it y - t h e d iag n o st i c b a si s f or s tra tig r ap hic t r a ps , G eo p hy s ics , 39 , 77 0-7 80


1 -11

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

ρ = a αm T h e c on s t an t s a a n d m c an be d et e rm in e d f ro m fi t t i n g a s t ra ig h t l in e t o a n p l ot o f lo g(ρ ) v e rs u s l og ( α ) . B e l ow a r e t h e r es u l t s o f s ev e ra l s u c h f i t s t o B l ac kf o ot 8- 8 . 3200 m=.46

3000

m=.30 2800 m=.25 2600 2400 2200 2000 1800 2000

3000

4000 5000 P-wave velocity

6000

7000

G a rd ner et a l . d et e rm in e d a n d r e co m m e n d e d m = . 2 5 a s a r ea so n ab le v al u e . H ow e v er , a s we c an s e e, t h e d at a s u p p o rt q u i t e a ra n ge o f a lt e rn a t i v es . ( Th e v al u e o f α i s l ar ge l y d e p e n d e n t o n t h e u n it s u s e d a n d i s n ot q u o t ed h e re . ) T h u s , th e c ar ef u l a p pli c at i on o f G ar d n er ' s r u l e r eq u i r es a bi t o f a n a l ys is . 1-12


Gardner's Rule Here are the three pseudo density logs from the three fits on the previous page. 1650

1650

1650

1600

1600

1600

1550

1550

1550

1500

1500

1500

1450

1450

1450

m=.46 1400

2000

2500 Density

m=.30 3000

1400

2000

2500 Density

m=.25 3000

1400

2000

2500 Density

3000

Actual density log from Blackfoot 8-8 Result from a Gardner type regression against P-wave velocity


1 -13

The Wave Equation T h e g re at s u c c es s o f p h y si c s in e x p l ai n i n g o u r w or l d a n d f u e li n g t h e gr ow t h o f t ec h n o lo gy i s ba se d f u n d am en t a ll y u p o n d i f f er en t i al e q u a t io n s a n d m or e sp ec i f ic a ll y p ar t ia l d i f f er en t i al e q u a t io n s . P D E ' s a r e t h e m a t h em a t i ca l s t at e m en t o f t h e a p p l ic at i o n o f b as i c p h y si c al l a ws t o c om ple x s ys t e m s . F or e x am p l e , a c on s i d e ra ti o n o f a c on s t an t d e n s i t y f lu id l e ad s t o t h e 's c al ar w av e e q u at i on ' wh i c h i s c en tr al t o m os t ge op hy s ic a l i m ag i n g a l g or it h ms. T h e S W E i s a d i r ec t c o n s eq uen c e o f N e w to n ' s s e co n d l aw a n d H oo ke 's l aw a s a p p l ie d t o t h e f l u id . ∂2Ψ ∂2 Ψ ∂2 Ψ ∂2 Ψ 1 + + – 2 = f x,y,z,t 2 2 2 2 ∂x ∂y ∂z v x,y,z ∂t H e re Y i s t h e p r e ss u r e , v p r op a ga t io n , a n d f ( x ,y ,z , t) s ou r c es .

i s t h e v el o ci t y o f w av e r e p re s en t s a n y p os s ib l e

T h o u g h i t is h ar d l y o b v i ou s , t h e s o lu t i o n s t o t h i s e q u at i on a re t ra v el i n g w av e s . A gr e at d e al o f in te r es t i n g p h y si c al ef f e ct s ca n b e s t u d i ed w it h t h e S W E i n c lu din g : • • • • • •

1-14

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


The Wave Equation T h e r e is a p ow e rf u l m e t h od o f s ol u t i on o f P D E 's t h a t i s o f c o n s id e r ab le r el e v an c e e x p lo ra t io n s e is m o lo gy . T h i s i s t h e m e th o d o f s o lu t i o n by G r ee n ' s fu n ct i o n s. W e w il l n o t d e v el o p i t h e re b u t s i m p l y st a t e th e im p or t an t r es u l t s . T h e e ss e n c e o f t h e t h eo ry is t o d e v e lo p a s ol u t i on t o t h e P D E o f i n t e re s t f or a " p o in t s ou rc e " a n d t h e n t o s h ow h ow the re s p on s e to a r b it r ar y s ou r c e c on f i g u ra ti o n s ca n be c on st r u c t ed fr om t h e e l em en t ar y s ol u t i on . T h e S W E , w h en s p e ci al i ze d f o r t h e G r ee n ' s f u n c t i on p ro bl e m l o oks li k e: ∂2 G ∂x

2

+

∂2G ∂y

2

+

∂2G ∂z

2

–

1

∂ 2G

v x,y,z ∂t 2

2

= δ x–xo,y–yo ,z–zo,t–t o

T h e t e rm o n t h e ri g h t o f t h e eq u al s ig n i s a D i ra c d el t a f u n c t i on a n d r ep re s en t s a m at h e m a ti c al i m p u l s e a t a s i n gl e p o i n t i n s p ac e , ( xo ,y o , zo ) , a n d a t a n in st a n t o f t i m e , t o . T h e s o lu t i on to t h e G re e n ' s f u n c t i on p ro bl e m , G ( x, y ,z ,t ) , i s k n ow n e x ac t l y f or c o n st a n t v e lo c it y a n d a p pro x im a t el y fo r a n u m b e r o f m or e c o m p l i ca t ed s i tu a t i on s . G c on t a in s a l l p h y si c al e f fe c t s d u e t o t h e i m p u l s iv e s ou r c e a n d i s p r op e rl y c al le d a n " i m p u l s e r es p o n se " . T o o b t ai n t h e r es p o n s e t o g en e ra l s ou r ce c o n fi g u ra t io n s , w e im a g in e t h e so u r ce to b e c om p os e d o f a s et o f s c al e d i m p u l s es . T h en co n s t ru c t t h e G re en ' s f u n c t i on s f or a ll o f t h e se i m p uls e s a n d si m p l y s u p e ri m p o se t h e s e G r ee n ' s f u n c t i on s . T h i s i s a n e x am p l e o f t h e m a t h em a t i ca l p r oc e ss o f " co n v ol u t i on ". W e w il l l ea rn m o re a b ou t c on v o lu ti o n l at e r i n t h i s c ou r s e. F o r n ow , it i s e n ou g h t o v i su a l iz e i t a s a ge n e ra l s u p e r p os i t io n o f s c al e d " i m p u l s e r es p o n se s " . Methods of Seismic Data Processing

1 -15

The Wave Equation T he r e s ul t w e ha ve j us t o b t a i ne d i s s o i m po r t a nt t h at w e r e s t a t e i t i n d i f f e r e nt t er m s : T h e w a v ef ie ld d u e to a so u r ce ha vi n g e xt en d e d sp a t ia l a n d te m p o ra l f o rm c a n b e co n s id e re d t o be t h e c o n v o lu t io n o f t h e e a rt h 's i m p u ls e res p o n se w it h t h e e xt en d e d so ur ce . T h is re su l t h o l d s fo r a n y l i ne a r w a v e e qu a t i o n a n d e x te n d s t o el a st i c, a n is o t ro p i c a n d a t t en u a t in g m ed ia . T h e t wo c om p o n en ts o f t h i s r e su lt , t h e e ar t h ' s i m p u l s e r es p o n se , I r , a n d t h e s ou r c e wa v ef or m , w s , a re b ot h a b s t ra ct

e n t i t ie s th a t

a r e d i f f ic u l t t o q u a n t if y . I r i s g en e r al ly ve r y c om p l i c at e d a n d c on t ai n s a l l p h ys i ca l e ff e c t s. w s i s a c om p l e t e ch a ra c t er iz at i on o f t h e s ou r c e w av e fi e ld a n d c an be co n s id er ed a s t h e sp ec i f ic a t io n o f t h e wa v ef i e ld a t a ll p o i n t s o n a s u rf a ce s u r ro u n d i n g t h e s ou r c e.

Impulse response Response to 3 sources 1-16


Traveling Waveforms T he s im plest m a thematic al w a ve equat io n is th e scala r w a ve e q ua t io n. I n a coustic m e di a o r s im p le e lastic media , compression a l w a ves a r e g o ve rne d b y it . In 1- D , th e s c al ar w a ve equ a tio n is : ∂2 ψ 2

∂z

=

1 ∂2 ψ 2

2

v ∂t

(1)

Where ψ represents the propagating wave. We now show that ψ = f t±z/v

(f is an arbitrary function)

is a solution to (1). ∂2 f 1 ′′ ∂f 1 ′ = ± f , = 2f 2 ∂z v ∂z v ∂ f ∂f ′ ′′ = f , = f 2 ∂t ∂t 2

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.


1 -17

Traveling Waveforms As an example of a waveform, consider the Ricker wavelet defined by:

2

w τ = 1–2 πfdomτ exp – πfdomτ

2

-0.05

0

0.05

τ->

Shown for fdom=30Hz 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 = Wavef o rm trave li ng i n th e - z d ir ec ti o n

w t–z/v =

z=-vt

Wa vef o r m tr a vel i ng i n t he +z d i r e c ti o n

z=vt

Similarly, cos(ω (t-z/v)), cos(k(z - vt)), and cos(ω t-kz) all represent cosine waves traveling in the +z direction. 1

1.01 sec 0.5

1.0 sec

0 -0.5 -1 400

450

500

550

600

z-> (meters) cos 2π30 t–z/1000 1-18

Plotted versus z for t=1.0 and 1.01 (sec) Synthetic Seismograms

Normal Incidence Reflection Coefficients (Adapted from E.S. Krebes, Course Notes in Theoretical Seismology)

Incident displacement

Reflected displacement

Consider a v e r ti c a l l y g t+z/α1 tr a ve li ng com pr es s i o na l Z f t–z/α1 w a ve incident o n a h or i z o nt a l i nterface. I n α1,ρ 1 o r de r t o d e s c r ib e the r e f l e cti o n a n d α2,ρ2 tr a n s mi ss i o n t ha t o c c u r , i t can b e s ho w n tha t t w o h t–z/α2 c o n di t i o ns m us t b e Transmitted s a ti sf i e d : displacement

continuity of displacement:

continuity of normal pressure:

f + g = h

(1)

???

(2)

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 -19

Normal Incidence Reflection Coefficients Consider an infinitesimal elastic element whose ends undergo displacement u1 and u2:

u1

dz Strain =

Δl l

=

u2–u1 dz

∂u ∂z

≈

u2

Now, invoking Hooke's law:

stress = pressure =

Force area

∂u

= k

∂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

=

l

mass l

2

sec2

=

mass 3

l

l

2

sec

2 So, k looks like: k = ρα Thus the pressure continuity equation is:

ρ 1α21

∂f ∂g ∂h + ρ1α21 = ρ2 α22 ∂z ∂z ∂z

But since

∂f ∂z

=

(evaluated at the interface)

–1 ′ ∂ g 1 ′ ∂h –1 ′ f, = g, = h α1 α1 α2 ∂z ∂z

ρ 1α1f′ – ρ1α1g′ = ρ2α2h′ Which can be immediately integrated to give: ρ 1α1f – ρ1α1g = ρ2α2h 1-20

(2) Synthetic Seismograms

Normal Incidence Reflection Coefficients 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:

Where impedance =

f + g = h

(1)

I 1f – I 1g = I 2h

(2)

Ik = ρkαk , k= 1,2

and where f,g, and h are understood to be evaluated at z=0. Multiplying (1) by I2, and subtracting it from (2) leads to: g =

I1–I 2 I1+I 2

f = –Rf

Similarly, we can obtain: h =

2I 1 I1 +I2

f = Tf

The quantities R and T are known as the normal incidence reflection and transmission coefficients: R =

Note that:

I2 –I1 I1 +I2

R+T =


, T =

I 2–I1+2I1 I1+I 2

2I 1 I1+I2 = 1

1 -21

Normal Incidence Reflection Coefficients

R =

I2–I 1 I1+I 2

, T =

2I 1 I1+I 2

R and T are often written in terms of the contrast and average of impedance across the layer: I =

1 2

I1+I 2 , ΔI = I 2–I 1

I 1 = I–.5ΔI , I 2 = I+.5ΔI Straight forward algebra then gives: 1 d ln I ≈ Δz R = 2 2I dz ΔI

T = 1–R =

I–.5ΔI I

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

Δ ρα 2ρα

=

ρΔα+αΔρ 2ρα

=

1 Δα Δρ + 2 α ρ

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.

1-22


Simple "Primaries Only" Impulse Response. Layered Earth, Normal Incidence, Acoustic Model

Impulse Response

k=0

V1,R 1

*R1 t=Δt k=1

V2,R 2

1 –R * 1

* 1–R1 *R2

*R1 t=2Δt 2

k=2

V3,R 3 1– R * 1– R * 1 2

* 1– R1 1– R2 *R3 k=3

Vj,R j

j–1

k = 1

*1– R1 *R2

t=3Δt

2

2

*1– R1 1– R2 *R3

j–1

1– Rk *

* k = 1

1–Rk *Rj k=j

j–1

t=jΔt

*

2

k=1

1– Rk *Rj

Model layers have a Δ Zj =2 t constant traveltime Δ Vj "thickness":

Vn,R n

k=n-1 n–1

n–1

k = 1

1– Rk *

* k=1

1–Rk *Rn k=n


t=nΔt

n–1

*

2

1– Rk *Rn k=1

1 -23

Computation of a 1-D Synthetic Seismic Impulse Response (Including All Multiples) t=Δt

t=2Δt

t=3Δt

t=4Δt

t=5Δt

t=6Δt

t=7Δt

t=8Δt

R0

t R1

z R2

R3 E ar th model i s bui lt of l ay er s of e qua l tr a ve lti me " thic kness" Δ t

R4

Completed node

R5

Current node R6 Note: All Raypaths are actually vertical. They are shown slanted for illustrative purposes.

R7

R8

R9 At the designated point, 6D4 and 6U5 are known and we wish to compute 6U4 and 6D5: 6U4 = R4*6D4 + (1+R4)*6U5

The c omp le te s e is mogr am i s obta ined by r ecur s ive ca lcula tion be ginning i n the up pe r le f t. All nod es on a ny up war d tr ave ling r a y ar e comp le tel y ca lc ul ate d bef ore pr oc ed ing to the nex t d e pth.

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

Adapted from: Reflection Seismolgy, K.H. Waters, 1981 J.H. Wiley 1-24


From Impulse Response to Source Waveform Response Source Waveform Response

Impulse Response

t=Δt

t=Δt

*R 1 2

t=2Δt

* 1–R1 *R2

t=3Δt

* 1–R1 1–R2 *R3

t=jΔt

*

2

j–1

2

2

1–Rk *Rj

*R 1 2

t=2Δt

* 1–R1 *R2

t=3Δt

* 1–R1 1–R2 *R3

2

j–1

t=jΔt

*

k = 1

n–1

t=nΔt

*

2

2

1–Rk *Rj k = 1

2

1–Rk *Rn k = 1

T h e " p r i m ar ie s o n l y" impulse re s p on s e c on s i s ts of a time s er i es o f s c al ed a n d d e la ye d im pu ls e s


n–1

t=nΔt

*

2

1–Rk *Rn k = 1

T o o b t ai n t h e s o u rc e wa v ef o rm re s p on s e f ro m the i m pu ls e re s p on s e , s im ply re p l ac e ea c h s p i ke o f t h e i m pu ls e re s p on s e by t h e p r od uct o f t h e s p i ke and s o u rc e wa v ef o rm . T h i s i s t h e m at h e m a ti c al p ro c es s o f co n v ol u t i on 1 -25

Impulse Responses and Seismograms F o r a li n e ar ea rt h , i t c an be s h ow n t h at i f w e a re g i v en t h e w av e fo rm s i gn a t u re o f a n on - im p u l si v e so u rc e a n d t h e i m p u l se re s p on s e o f a n ea rt h m od el , t h e n :

s t = Ir t •ws t

where:

Ir t

is the earth impulse response

ws t

is the source waveform

st

is the earth response to the source waveform

T h e g e n er al p ro of o f t h i s r es u l t c o m es f ro m " G re e n 's fu n ct i o n a n a ly s is " a n d is t r u e f or a n y l i n e ar wa v e eq ua t i on ( e l as t ic , s c al ar , et c ) G en er al l y I r c on t a in s a l l p h ys i c al ef f ec t s t h e t h e or y i s c ap ab l e o f p r od u c in g , a n d u su a l ly t h a t i s m o re t h a n w e w an t . T he m o s t c o m m o n u s e o f 1 - D s ei s m o gr a m s i s i n t h e i n te rp re t a t io n o f p ro c es s ed s e i sm i c s e c t io n s. I n t h is ca s e m o st of th e p h y si ca l e f f ec ts ( m u lt ip l es , t r a n s m is s io n l o ss es , a tt en u a ti o n ) h a v e b e e n r em o v ed i n t h e p r o ce s si n g. T h e re f o re , c o m m o n p ra c ti ce r e p l a ce s I r ( t ) w it h r ( t ) w h er e :

rt =

Thus:

n o rm a l i n c i d en ce re f le c t i on co ef f i c ie n t s p o s it i o n ed i n 2 - w ay v e rt i c al t r av e l t im e

s t = r t •ws t

s( t ) g i v en b y t h i s re s u lt i s th e m o st c om m on se i s m og ra m c o m p u t e d i n ex p l o ra ti o n g eo p h y si c s . 1-26

1-D


1 - D S y n th e ti c S ei s mog ram Su m ma r y

• A c o m p l et e s ol u t i on , g en er at i n g a l l m u l t ip le s a n d t ra n sm is s i on ef f e c t s, c an b e c on s t r u c te d . S o m e m et h o d s a ls o i n c l u d e a t t e n u at i on . • A ss u m pti o n s: ra y th eo ry , 1 - D , n o rm a l i n c id en c e • G e o p h ys i c al w el l l og s , p r ov i d i n g P - w av e v e l oc i t ie s a n d d e n si t i es , a re u se d . T h e y a re u su a ll y r es am p l ed t o a v a ri ab l e d e p t h l a ye ri n g w it h e q u al D t s te p s . • M e t h od i s i n h er e n t ly a lg or i t h m i c . N o a n a l yt i c c l os ed fo rm s ol u t i on a v ai l ab l e. • I n p r ac t ic e , m ult i p l es a n d t ra n s m is s i on lo s se s a re n o t u s u al ly in cl u d e d . R e f le c t i on c oe f fi c i en ts i n t im e a r e si m ply c on v ol v ed w it h a so u rc e r e sp o n s e.


1 -27

Example of Synthetic Seismogram Creation by Convolution of Reflectivity and Wavelet.

Time Domain View

Wavelet Synthetic Seismogram

Reflection Coeficients

0

0.2

0.4

0.6

0.8

1

1.2

Time (secs)

1-28


Example of Synthetic Seismogram Creation by Convolution of Reflectivity and Wavelet.

Frequency Domain View

0 -10 -20

Reflectivity

-30

Wavelet

-40 -50 -60 -70 -80

Synthetic Seismogram

-90 -100

0

50

100 150 Frequency (Hz)


200

250

1 -29

P-S Synthetic Seismogram Construction The SYNTH Algorithm

Define Layered Model

Vp, Vs, and density logs

R e sa m pl e d l o gs

Loop over layers: k=1 to nlayers Iterative Snell's law raytracing

1 ) Ra yt r ac e I nc i de nc e A n gles

P

P

OR

PP

2) Zoeppritz RCs

P

S Free surface

AND

PS

S

S

P

Primary reflections

3) Map RCs to to, apply wavelet.

Input wavelet Re s pons e of l ay e r k

Next layer

+ Accumulated gather after k-1 layers

1-30

=

Accumulated gather aft er k la yer s



Chapter 2 Sign al P rocessi ng


2 -1

Convolution C on v o lu ti o n i s t h e m a t h e m at i c al p r oc e s s o f " s h i ft i n g , s ca li n g , a n d su mm i n g" a w av e fo rm t o p r od u c e a n o u t pu t by s u p er p o si t i on . G e n e ra ll y, t w o i n p u t s i gn a ls a r e re q u i r ed , sa y r a n d w , w it h w b ei n g t h e wa v ef or m a n d r a s e ri e s o f s c al in g c oe f f ic i e n t s. F or ex a m p l e , l e t r= [1 0 0 - .5 .5 0 - 1 ] a n d le t w = [ - . 5 1 - . 5 ] , t h en t h e c on v ol u t i on o f r a n d w i s: j k

0 r0w0

0

1

-.5

1

0

0

-.5

0

0

0

0

0

0

4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

+

0

0

0

0

0

0

0

0

.25

+

r4w0

r4w1 r4w2 = r4*w

0

-.25

0

r5w0

0

0

0

-.25

0

+

-.5

.5

0

+

.25

+ 6

0

r3w1 r3w2 = r3*w

+ 5

0

r2w1 r2w2 = r2*w

r3w0

3

8

r1w1 r1w2 = r1*w

r2w0

2

7

r0w1 r0w2 = r0*w

r1w0

1

O u t p u t sa m p l e n u m b e r 2 3 4 5 6

0

0

0

0

r5w1 r5w2 = r5*w

0 r6w0

0

0

r6w1 r6w2 = r6*w

0

0

0

0

0

.5

-1

1

-.5

.25

-.75

.75

.25

-1

.5

s = r•w -.5 2-2

.5

Signal Processing Concepts

Convolution I n t h e p r ev i ou s s l id e, we d e s cr i be d a t a bl u l ar m e th o d f or c om p u t i n g t h e c on v o lu t i on o f r a n d w t o y i el d s. T h i s c a n b e w ri t t en m a th em a t ic a ll y a s fo ll o ws :

s = r• w sj = Σ rk wj–k k

T o s ee t h a t t h i s s u m m a t io n e x p r es s i on is eq u iv a le n t t o t h e t ab u la r m e t h od , co n s id er t h e e x am p l e o f j =4 :

s4 = r0 w4–0+r 1w4–1+r2w4–2+r3w4–3+r4 w4–4+r 5w4–5+r6w4–6 s4 = r0 w4+r1w3 +r 2w2+r3 w1+r4w0 +r5w–1+r6w–2

N ot e t h at t h e le n g t h o f s i s t h e c om b i n ed l en g t h s o f r a n d w le s s 1 :

length s = length r +length w –1 T h u s , m a t h em a t i ca ll y , e v er yt i m e a c o n v ol u t i on i s p er f or m ed t h e r es u l t i n c re as e s i n l e n gt h . T h i s c re at e s a b i t o f a h e ad e r ( bo okk e ep i n g ) p r ob l em in se i s m i c d at a p ro c es s in g a n d i s n ot u s u al l y a l lo we d . T h a t i s, i f a s e i sm ic t r ac e is c on v ol v ed wi t h a f i l t er o p e r at or , t h e r e su lt i s t r u n c at e d a t t h e s a m e l e n gt h a s t h e se i sm ic t r ac e .


2 -3

Convolution We have seen that the convolution of discretely sampled vectors is written:

Σr w

sj =

k

k

j–k

The analagous result for continuous functions is: ∞

st =

–∞

r τ w t–τ dτ

We now show that the order of convolution is immaterial. Let:

τ′=t–τ, dτ′=–dτ, τ=t–τ′ –∞

Then:

st = –

∞ ∞

And:

So:

st =

–∞

r t–τ ′ w τ′ dτ ′

r t–τ′ w τ′ dτ′

s = r•w = w•r

We also note that convolution is linear in the sense that:

a+b •c = a•c + b•c

2-4


Convolution by Replacement ( e m p ha s is o n in pu t s am ple s )

Consider the discrete convolution of a three point boxcar, b, with an eleven point time series, r. 0.1

r

0.05

1

• 0.5

0 -0.05 -0.10

0.1 0.05 0 -0.05 -0.1 0

0 0 2

4

6

8

10

2

4

12

E ac h i n p u t sa m p l e i s c on s i d e re d se p a ra te l y. T h e b ox c a r i s m u lt i p l i ed by t h e i n p u t sa m p l e re s u lt i n g i n a s c al e d b ox c ar . T h e s c al e d b ox c a r c o n t ri b u t es t o o u t p u t sa m p l e lo c at i on s b eg i n n i n g a t the p os i t io n o f t h e i n p u t sa m p l e . T h us t h e b ox c ar i s sc a le d b y ea c h sa m p l e o f r a n d re p l ic a t ed a t t h e lo c at i on o f th e r s am ple . E ac h o u t p u t sa m p l e r ec e iv e s m u lt i p l e co n t ri b u t i on s w h i c h a r e s u m m e d . I n p u t s am ple s 1 ,2 a n d 6 a r e sh o wn e x p l ic i t l y c o n t ri b u t in g. 2

4

6

8

10

12

0.1

0.1

0

=0 -0.1

-0.1 0

b

2

4

6

8

10

12


14

0

2

4

6

8

10

12

14

2 -5

Convolution as a Weighted Sum ( e m p h as i s o n o u t p u t s am p l es ) Consider the discrete convolution of a three point boxcar, b, with an eleven point time series, r. 0.1

r

0.05

1

• 0.5

0 -0.05 -0.10

b

0 0 2

4

6

8

10

T o co m p u t e a n o u t pu t 0. 1 s am p l e , p o si t i on t h e b ox c ar o v e r s om e r s am p l e s , m u l t ip ly t h e r 0 s am p l e s b y t h e bo x c ar w ei g h t s, a n d s u m . T h e -0.05 c om pu t at i on o f o u t pu t -0.1 0 s am p l e s 1 a n d 7 i s i ll u s t r at ed . T h i s i s a p r oc e ss o f s m o ot h i n g 0.1 o r a v er ag i n g t h e i n p u t .

2

4

12

2

4

6

8

10

12

4

6

8

10

12

14

0 -0.1 0

2-6

2


Matrix Multiplication by Rows Consider the a 4x4 matrix equation such as:

a 11 a12 a13 a 14

b1

c1

a 21 a22 a23 a 24

b2

c2

a 31 a32 a33 a 34

b3

a 41 a42 a43 a 44

b4

=

eqn 1

c3 c4

This is equivalent to the following system of equations:

c1 = a11b 1 + a12b 2 + a 13b 3 + a14b 4 c2 = a21b 1 + a22b 2 + a 23b 3 + a24b 4

eqns 2a-2d

c3 = a31b 1 + a32b 2 + a 33b 3 + a34b 4 c4 = a41b 1 + a42b 2 + a 43b 3 + a44b 4 T h us t h e el e m en t s o f t h e v ec t o r C a r e c om p u t e d b y t ak in g e ac h ro w o f A , m u l t i p l yi n g i t by t h e v e ct o r B , a n d s u m m i n g t h e r e su l t s . T h is p ro c es s i s f am il i ar t o m o s t s t u d e n t s o f li n e ar a l ge br a a s " m at r i x m u l t i p l i ca t io n b y r ow s" . I t c a n be wr i t t en s ym bo li c al l y a s t w o n e s t ed c om pu t at i on lo op s : c=zeros(1,4); for irow=1:4 for jcol=1:4 c(irow)=c(irow) + a(irow,jcol)*b(jcol); end end Methods of Seismic Data Processing

2 -7

Matrix Multiplication by Columns M at r i x m u l t i p li c at i on " b y c ol u mn s" i s l e s s we l l k n ow n t h an th e c o rr es p o n d i n g p ro c es s " b y r ow s" bu t i t p r ov i d e s a u s ef u l in tu it i v e i n s i gh t t o co n v ol u t i on . E x am i n at i on o f e q u at i on s 2 a - 2 d s h ow s t h at t h e c ol u m n s o f A h a v e b ee n m u l t ip li e d by a si n g l e c or re s p on din g e le m e n t o f B . T h u s w e ca n e x p r es s t h e m a t ri x m u l t i p l ic a t io n a s a su m o f c o lu m n v ec t or s , e ac h o n e b ei n g a s c al ed v e rs i on o f a c o lu m n o f A .

a 11

a 12

a 13

a 14

a 21

a 22

a 23

a 24

a 31 a 41

b1 +

a 32 a 42

b2 +

a 33 a 43

b3 +

a 34

c1 b4 =

a 44

c2 c3 c4

W ri t t en a s c om p u t at i on l o op s , t h i s a m o u n t s t o r e v er s in g t h e o rd er o f t h e l oo p s i n t h e m u l t ip l i c at i on 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

2-8


Convolution as a Matrix Operation C on si d e r t h e c on v o lu t i on o f a re f l ec t i v it y se q u e n c e , r , w it h a w av e le t , w , to y i e l d a se i s m i c t ra c e , s . T h i s i s u s u al l y wr i tt e n a s t h e c o n v ol u t i on in t e g ra l: ∞

s(t) =

w(t – τ)r(τ)dτ

–∞

W h e n we h a v e d i s c re t e, fi n i t e l en gt h a p p r ox i m a t io n s t o t h e se q u a n t i ti e s , th e c o n v ol u t i on i s u s u al l y w ri t t e n a s a s u m m a ti o n . I f r j i s t h e re f le c t i vi t y se r ie s wi t h j = 0 , 1 ,. . . n , an d

wk

is

t h e p o ss i b ly

n o n - c au s al

wa v el e t w it h

k =-

m . . .0 . . . m , t h e n : k–m

s k = Δt

Σ

j = k+m

wk–jr j

U s u al l y, in t h e se e xp re s si o n s, t h e Δ t t e rm i s d ro p p e d o r s et t o u n i t y. I t i s u se f u l t o w r it e o u t a fe w t e rm s o f t h i s s u m m a ti o n :

s0 =

+ w0r0 + w–1r1 + w–2r 2 +

s1 =

+ w1r0 + w0 r1 + w–1r 2 +

T h e s am e o p er at i on c an b e a c h i e v ed b y m at r ix m u l t i p l ic a ti o n wh er e th e w av e l et , w , is l oa d ed i n t o a s p e ci a l m at r i x c al l ed a T o ep l i t z o r c on v ol u t io n m a t ri x .


2 -9

Convolution as a Matrix Operation I t i s a s i m p l e ex e rc i s e o f m at r ix m u l ti p l i c at i on b y r ow s t o c h e c k th a t t h e f o ll ow i n g m at r i x e q u at i on c om pu t es t h e c on v ol u t io n o f w w i th r

w0 w–1 w–2 w–3

r0

s0

w1 w0 w–1 w–2

r1

s1

w2 w1 w0 w–1

r2

=

s2

w3 w2 w1 w0

rm

sn

N ot e t h e s ym m e t ry o f t h e W m a t ri x w h i c h h as t h e w av e le t s am ple s c on s t an t a l on g t h e d i ag on al s . A n o t h er w ay t o v i e w W i s t h at e ac h c o lu mn c on t a in s t h e w av e le t w i th t h e z er o t i m e s am ple a l ig n e d o n t h e m ai n d i ag on a l. N ow , im a gi n e d o i n g th e m a t r ix m u l ti p l i c at i on by c ol u m n s i n s t ea d o f r ow s a n d we ge t t h e m o s t i n t u i t iv e v i e w o f c on v ol u t io n " by r ep la c em e n t " .

w0

w–1

s0

w1

w0

s1

w2 w3

r0 +

w1

r1 +

=

s2

w2 sn

2-10


Convolution as a Matrix Operation A s a n e x am p l e o f c o n v ol u t i on by m at r ix m u l t i p l ic a ti o n , h e re i s a n i l l u st r at i on o f t h e c on v ol u t i on o f a r e fl e c t iv i t y s er i es a n d a m i n i m u m p h as e wa v le t t o y i e ld a 1 -D s ei s m o gr am .

=

A s a s ec o n d ex am p l e, h er e i s th e c on v o lu t i on o f a r ef l ec t i v i t y s er i es a n d a z er o p h as e wa vl e t t o y ie l d a z e r o p h as e s ei s m og r am .

=


2 -11

Convolution as a Matrix Operation T h es e e x am p l es o f co n v ol u t i on b y m ar t i x m ult i p l i c at i on s h ow e x p l ic i t l y w h at i s m e an t w h e n we s ay t h at c on v ol u t io n i s a s t at i on a ry p r oc e ss . I n t u i t i v el y , th is p h r as e m e an s t h at t h e o p e ra t io n d o es n ot ch a n g e wi t h t i m e i n s om e se n s e . P re c i se l y, i t m ea n s t h at t h e w av e f or m s i n t h e co l u m n s o f t h e c o n v ol u t io n m at r i x a re a ll id en t i c al . T h at is , th e wa v el e t w h i ch i s s c al e d a n d u s e d t o re p l ac e e ac h r ef l e ct i v i t y s p i k e d o e s n ot c h a n ge w i th t i m e . A s w e s h al l s e e, m a n y p h y si c al p ro ce s s es v i ol at e th is a s su m p t i on a n d i t i s q u i t e p o s si b l e t o g en er al i ze the co n v ol u t i on o p er at i on to m od el n o n s ta t io n ar y p r oc es s e s. W h en t h e a ss u m p t i o n o f s t at i on a ri t y i s m a d e i n t h e c on te x t o f s t at is t i c al d ec o n v ol u t i on t h e or y, it m e an s p r e ci s e ly t h e s a m e t h i n g. W e a s s u m e t h a t t h e t i m e s e ri e s w e m ea su re d ( t h e s ei s m i c t ra c e ) i s re l at e d t o t h a t wh i c h w e w an t ( th e r ef l e ct i v i t y) b y a s ta t io n ar y c o n v ol u t i on o p er at i on . G i v en t h at , we e x p e c t t h at a st a ti o n ar y i n v e rs e o p er at o r wi l l s u f fi c e t o r ec o ve r t h e r e fl e c t iv i t y.

2-12


F ou rie r T ran s fo r ms a n d C o n v ol u t io n C on si d e r t h e f u n c t i on s :

co n v ol u t i on

i n t eg r al

for

iωu

gu = e

Now, let g be a complex sinusoidal function: ∞

Then:

where

ht =

–∞

fτe

Fω =

∞ –∞

iω t–τ

iωt

dτ = e F ω

fτe

–iωτ

dτ

c on t i n u o u s

(1)

(2)

T h i s r em a rk ab l e r es u l t s h ow s t h at i f we c on v o lv e A N Y f u n c t i on , f , wi t h a co m p l e x s in u s o id , t h e r es u l t i s t h e s am e c om p l e x si n u s o id m u lt i p l i ed by a c o m p l ex c oe f f ic i e n t . T h i s c o m p l ex c o ef f i ci e n t , F ( w ) , i s c om pu t ed f ro m f( t ) a n d i s k n ow n a s t h e F o u ri e r T r an s f or m o f f ( t ) . T h o s e w h o h av e st u die d m a t h em a t i c al p h ys i c s wi l l r ec og n i ze t h at t h i s m e an s t h at t h e c om ple x s in u so i d s a r e e ig e n f u n c t io n s o f t h e c on v o lu ti o n o p er at o r a n d t h e F o u ri e r T r an s f or m p ro v id es t h e ei g en v a lu e s .


2 -13

Fourier Transforms and Convolution H e re we s e e t h e r e su l t o f c o n v ol v i n g 1 0 , 3 0 , a n d 7 0 H z c om ple x s in uso id s w it h a 3 0 Hz R i c ke r w av e le t . I n ea ch c as e , o n l y t h e r ea l p ar t s o f t h e c om ple x s i n u s oi d s a r e p l ot t e d . W e se e t h a tt h e 1 0 H z s i n u s oi d i s d im in i s h e d b y 7 3 % , t h e 7 0 H z by 9 3 % , a n d t h e 30 Hz is u n at t e n u at e d . ( T h e d i st o rt i on s i n t h e s i n u s oi d s a re a rt i f ac t s o f t h e d i s p l ay n ot t h e c on v o lu t i o n a l g or i th m . ) 10 Hz.

1

1

0

0 -1

30 Hz. 1 30Hz

0

0.3 0.4 0.5 0.6 0.7 0.8

70 Hz

1

Ricker

0.3 0.4 0.5 0.6 0.7 0.8

Maximum amplitude = 1.0

0 -1

Convolve

1

0.3 0.4 0.5 0.6 0.7 0.8

Maximum amplitude =.064

0

0

-1

-1

0.3 0.4 0.5 0.6 0.7 0.8

1

-1

Maximum amplitude = .27

-1 0.3 0.4 0.5 0.6 0.7 0.8

IN 2-14

0.3 0.4 0.5 0.6 0.7 0.8

OUT Signal Processing Concepts

Fourier Transforms and Convolution H e r e i s t h e a c t u al F o u ri e r a m p l it u de s p e ct r u m o f t h e 3 0 H z R ic k er w av e l et . 10 0

-10

-20 -30

-40

-50 -60 0

20

40

60 Frequency (Hz)

80

100

Since "decibels down" are computed by dbdown = 20*log10(F(ω)/Fmax) we can use the results from the previous figure to compute: dbdown(10Hz) = 20*log10(.27) = -11.4 decibels dbdown(30Hz) = 20*log10(1.0) = 0 decibels dbdown(70Hz) = 20*log10(.064) = -23.9 decibels S o F ( w ) , t h e F ou r ie r T ra n s fo rm o f a fu n ct i o n f ( t ) , i s a q u ic k wa y o f c om pu t in g t h e re l at i v e a t t e n u at i on o f d if f e re n t s i n u s oi d s w h e n t h e y a r e c on v o lv e d wi t h f ( t ) . Methods of Seismic Data Processing

2 -15

Fourier Transforms and Convolution A co n v ol u t i on ca n a f f e ct n o t o n ly t h e a m p l i t u d e o f a s i n u so i d bu t i t s p h as e a s we l l . T h e R i ck e r w av e le t i s kn o wn a s a z er o p h a se f u n c t i on w h ic h m e an s t h a t i t d o es n ot h av e a p h as e ef f ec t . L e t u s r ep ea t t h e a n al ys i s bu t t h i s t im e w i t h a f u n c t i on w h i c h h as a kn o wn p h as e e ff e c t . F or t h i s p u r p os e , w e c o n s id e r a R i c ke r w av l e t w it h a 9 0o p h as e s h i ft . 0.15

0.15

0.1

0.1 0.05

0.05 0 0 -0.05 -0.05

-0.1

-0.1 -0.1

-0.05

0

0.05

30 Hz. Ricker zero phase

0.1

-0.15 -0.1

-0.05

0

0.05

0.1

30 Hz Ricker 90o phase

N o t e t h at z e ro p h a se wa ve f or m s a re a l wa y s s ym m et r i c w h il e 9 0o p h as e r e su lt s i n a n a n t i s ym m e t ri c w av e f or m . W e m i gh t e x p ec t t h e 9 0o R i ck e r t o h av e t h e s am e ef f e c t o n th e a m p l it u de o f s i n u s oi d s b u t so m e a n d d i t i on a l e ff e c t a s we l l. T o s ee , w e re p e at th e a n al y si s o f p as s i n g c om ple x si n u s oi d s th ro u gh i t .

2-16


Fourier Transforms and Convolution H e r e we re p e at t h e r e su lt o f c o n v ol v i n g 1 0 , 3 0 , a n d 7 0 H z c o m p l ex s in uso id s wi t h a 30 Hz R i c ke r wa v el e t b u t t h i s t i m e t h e R i c ke r h a s 9 0o p h as e . T h e a m p li t u d e a tt e n u a ti o n o f t h e s i n u s oi d s i s t h e s am e a s be f or e b u t (When n o w t h e re i s a n a d d i t i on al 9 0o p h as e l ag . c om p ar i n g t h i s f i gu r e w i t h - 2- o f t h i s s e ri e s , n o te t h at t h e re h as b e en a n x - ax i s s c al e c h an g e o n a l l p lo t s. ) R esult with 90o Ricker Result with 0o Ricker

10 Hz. 1

1

0

0

-1 0.45

0.5

0.55

0.6

-1 0.45

0.65

0.5

0.55

0.6

0.65

0.6

0.65

Maximum amplitude = 1.0

30 Hz.

1

Maximum amplitude = .27

1 30Hz

0

Ricker

0

90 o -1 0.45

0.5

0.55

0.6

0.65

Convolve

-1 0.45

0.5

0.55

Maximum amplitude =.064

70 Hz

1

0

0.45

0.5

0.55

0.6

0.65

IN


-1 0.45

0.5

0.55

0.6

0.65

OUT

2 -17

Fourier Transforms and Convolution H e r e i s a c om p le t e d e s c ri p t i on o f t h e 9 0o , 30 H z . R i c ke r i n t h e t i m e d o m ai n a n d a m p l it u de a n d p ha s e s p ec t r u m i n t h e F o u ri e r d om a in . W e h av e s ee n t h at th e F ou r ie r d o m ai n p r ov i d e s a c on v e n i en t d e s cr i p t i on o f t h e e ff e c t o f c on v o lv i n g t h e wa v e l et wi t h c om p l e x s i n u s oi d s . 0.1 0.05

Time Domain

0 -0.05 -0.1 -0.15

-0.1

-0.05

0 Time

0.05

0.1

0.15

0

Fourier Domain Amplitude Spectrum

-20 -40 -60 0

20

40 60 Frequency

80

100

100

Fourier Domain Phase Spectrum

0 -100

0

2-18

20

40 60 Frequency

80

100


&OURIE R ! NALYS IS AND 3 Y NT HES IS 4 H E G R EA T U T IL I T Y O F T H E & OU R I ER T RA N S FO RM C O M E S F R OM I T S A B I LI T Y T O D E CO M P O SE A N Y F U N C T IO N I N T O A S E T O F C AS E T HE C OM P L EX S I N U S OI D S ) N T H E C O N T IN U O U S F R EQ U E N C I ES O F T H E S I N U S OI D S R A N GE F R OM d T O d A N D H A V E A M P L IT U DE S A N D P H AS ES W H I C H A R E C OM P U T ED F R OM T H E F OR W AR D & OU R IE R T R AN S F OR M d

(W

HT E

nI WT

DT

nd

4 H IS E Q U AT I ON C OM P UT ES T H E C OM P L E X C O EF I C I EN T S ( W OF T HE C OM PLE X S I N U SO I D S WHIC H WH E N S UMMED I N T E GR AT E D W IL L Y IE L D H T 5 S U AL L Y ( W I S D EC O M P OS E D I N T O T WO S E P AR AT E R EA L F U N C T IO N S AMPLIT UDE S PEC TRUM

!W (W

F W TAN

PHAS E S PE CT RUM

4 HE I N V ER S E & O U RI E R C ON S T R U C T IO N O F H T A S S I N U S OI D S

HT

2E ( W )M ( W

)M ( W

n

2E ( W

T RA N S FO RM E X P R ES S ES T HE A S U P E R P OS I T IO N O F C O M P L EX

P

d

(W E

I WT

nd

DW

) F W E W I S H T O U S E C YC L IC A L F RE Q U E N C Y F I N S T E AD O F A N GU L A R F R EQ U E N C Y W W P F T H E & O U RI E R T RA N SF O RM P AI R I S d

(F

HT E

n PIFT

DT

nd d

HT

(F E

PIFT

DF

nd

-ETHODS OF 3EISMIC $ATA 0ROCESSING

Fourier Analysis and Synthesis A s a n e x am p l e c on s i d er t h e G a u s s i an fu nct i on :

ht = e

2 –α 2 t

U s in g s t an d a rd t e c h n i q u es o f i n t e gr al c al c u lu s , t h e F o u ri e r t ra n s fo rm o f t h e G a u ss i an c an be s h o wn t o be :

Hω =

half width = 1/α

π – ω /4α α e 2

half width = 2α

h(t)

H(ω)

N o t e t h at t h e h a l f wi d t h s , a s re p re s e n t ed by t h e i r 1 / e p o in t s a re in v e r se l y p ro p or t i on a l. I n f ac t : –1 ΔtΔω = α 2α = 2

T h i s i s a n e x am p l e o f a g en e r al p r op e rt y wh i c h s ay s t h a t t h e " wi d t h " o f a ti m e d om a i n fu n c ti o n is i n v er s el y p r op o rt i on a l t o it s wi d t h i n f re q u e n c y. I t ca n be s h ow n , g iv e n a s u i t ab l e m e as u re o f w i d t h , t h at : ( w id th i n t i m e ) ( w id th i n f r eq u en c y ) > = a co n s t an t B r ac e we l l ( 1 9 7 8 , T h e F o u ri e r T ra n sf o rm and its A p p l ic a ti o n s) s h ow s t h e c o n s ta n t t o b e 1 /2 a n d t h at t h e e q u al i t y h ol d s f or th e G a u s s i an . 2-20


Fourier Analysis Example 0.08 0.06 0.04

H er e i s a m i n i m u m p h as e w av e l et c on s t ru ct e d wi t h a . 0 0 1 s e c s am ple r at e a n d a 3 0 H z d om in a n t f r e q u en cy .

0.02 0 -0.02 -0.04 -0.06 -0.08

0

0.05

0.1

0.15

0.2

time (sec) 1 0.8 0.6 0.4 0.2 0

0

100

200 300 frequency (Hz)

400

500

T h e i s t h e " a m p l it u de s p e ct r u m o f t h e w av e l et d i s p l ay e d wi t h a l i n ea r v e rt i c al s c al e . N o t e t h at t h e f r e q u en c y a x i s s t op s at 500 Hz wh i c h is 1 /( 2 * . 0 0 1 se c ) .

0 -20 -40 -60 -80

3 2 1 0 -1 -2 -3

0

100


400

500

H e re the amplitude sp ec t r u m i s d i s p l ay e d w i t h a d ec i b el v e r t ic al sc a le : db = 2 0 *l o g1 0 ( A ( f ) /A m ax ) T h i s i s t he ph as e s p e c t r um . N o t e t ha t t h e ve r t i c a l s c a l e i s i n r ad i an s .

0

100


400

500

A t t h i s po i nt , F o ur i e r an a l ys i s m a y l o o k l i k e an e xe r c i s e i n g r a ph m a ki n g ; ho w e ve r , i t s ut il i t y w i l l b e c o m e c l e ar o n t h e n e xt pa g e . Methods of Seismic Data Processing

2 -21

Fourier Analysis Example

A

B

Sum of components

Sum of components

80

80

60

60

40

40

20

20

Individual Fourier components 0 0

0.05

0.1 time (sec)

0.15

0. 2

0 0

Cumulative sum of Fourier components 0.05 0.1 0.15

0.2

time (sec)

H e re we s ee t wo e q u i v al e n t wa y s o f v i e wi n g t h e F o u r ie r t ra n s fo rm in fo rm a t i on o n t h e p r e v io u s p ag e. I n A , t h e i n d i v id u al F ou r ie r c om po n e n t s a r e s h ow n f r om 1 0 t o 7 0 H z , p ro p er l y s c al e d f or t h ei r a m p l i t u d e a n d p h as e . T h e s u m o f a l l 1 3 c om p on e n t s y i e ld s th e w av e l et a t t h e t op w h ic h i s q u i t e si m i l ar t o t h e t r u e w av e l et s h o wn o n t h e p r ev i o u s p a g e. A d d i n g i n t h e r em a i n in g f r eq u e n c y c om po n e n t s ( 0 - > 1 0 H z a n d 7 0 - > 5 0 0 H z ) wi l l re c on s t r u c t t h e wa ve l e t e x ac t l y. T h e f i gu r e o n t h e ri g h t c on t ai n s t h e s am e in fo rm a t i on ex c e p t t h at ea c h tr ac e is t h e s u m o f t h e f re q u e n c y c om p o n en ts be t we e n i t s f re q u e n c y a n d 1 0 H z . T h i s g i v es a g oo d i ll u s t ra t io n o f h o w t h e wa v el e t t ak es fo rm a s it s sp ec t r u m i s s u m med . 2-22


Fourier Transform Pairs The table below is reproduced from: Brigham, E.O., 1974, The Fast Fourier Transform, Prentice Hall

N o t e : I t i s a r em a rk ab le f ac t t h at n o s i gn a l c a n h a v e fi n i t e l en g t h ( i . e . c om p a c t s u p p or t ) i n bo t h t h e t i m e a n d fr e q u e n c y d o m a in s .


2 -23

Fourier Transform Pairs The table below is reproduced from: Brigham, E.O., 1974, The Fast Fourier Transform, Prentice Hall

2-24


The Dirac Delta Function T h e D ir ac d el t a fu nct i on wa s i n v en t e d b y P . A. M . D i ra c t o h a n d l e p ro bl e m s i n t h e d e v el op m en t o f q u a n t u m m e c h an i c s . S i n c e t h e n , i ts u n i q u e a b il i t y t o r e p re s en t a " un it s p ik e " i n t h e c on t i n u ou s f u n c ti o n d om a i n . I t ca n b e d e f i n ed a s t h e l im it i n g fo rm o f a s h ar p ly p e ake d f u n c t i on w h os m a x i m u m p ro c ee d s t o i n f i n it y a s it s wi d t h sh r i n ks t o z e r o i n s u c h a w ay t h at it s a re a re m a in s u n i t y. 8

A se ri e s o f b ox c ar s w i t h u nit a r ea c on v e rg e s i n t h e l im it t o t h e d e lt a f u n c t i on :

b4

7 6 5 4 3

b2

2

It can be thought of as: b1

1 0 -0.5

b∞ = δ t

b3

δt = -0.25

0

0.25

0.5

0, t≠0 ∞, t=0

T h e m os t i m p o rt a n t p r op e rt y o f t h e d e l t a f u n c t i on i s i t s be h av i or u nd er i n t e gr at i on . I f f ( t ) is a n y f u n c t i on , t h e n : b a

f t δ t–t0 dt =

f t0 , if a
This is known as the sifting property of the delta function.


2 -25

The Dirac Delta Function Consider the Fourier transform of the delta function: ∞ –∞

δ t–t 0 e

–i ωt

dt = e

–iωt0

T h us i t h a s a c on s t an t , u n i t a m p l i t u d e s p ec t r u m ( a ls o kn o wn a s a " wh i t e " s p e c t ru m) a n d l i n e ar p h a se . Consider the action of the delta function under convolution: ∞ –∞

δ t–t 0 f τ–t dt = f τ–t0

T h u s t h e d e lt a f u n c t i on sh i f t s f ( t ) to p l a ce i t s o ri g in a t t h e l o ca t io n wh e r e t h e a rg u m e n t o f t h e d el t a f u n c t i on v an i s h e s. T h i s i s c a ll e d a " st a ti c sh if t " i n s e is m i c d a t a p r oc e ss i n g . S i n c e c on v ol u t io n c an b e d o n e in th e F o u ri e r d o m ai n b y m u l t i p l ic a t io n o f t ra n sf o rm s , we c an c on c l u d e t h a t a s t at i c s h i ft c an be d o n e b y: fτ FFT Fω Mult IFFT f τ–t o

–iωt o

T h at i s , a s t at i c sh i f t . F i n a ll y, if eq u a t io n a t t h e d ef i n i t io n o f t h e co m p o n en t s :

e sh i f t i s e q u i v al en t t o a l i n e ar p h as e we i n v e rs e F o u ri e r t ra n sf o rm t h e to p o f t h e p ag e, w e en d u p w i t h a d e lt a f u n c t io n i n t e rm s o f i t s F o u ri e r

1 δ (τ − t 0 ) = 2π

∫

∞ −∞

e

iω (τ − t0 )

=

∫

∞ −∞

e

2 π i f ( τ −t0 )

T h us t h e d e l ta f u n c t i on h as u n i t a m p li t u d e sp e c t ru m a n d a p h as e s p e c t ru m t h at is li n e ar i n f r eq u en c y a n d w it h s lo p e p r op o rt i on a l t o t h e t i m e s h i ft . 2-26


The Convolution Theorem C o n s id er t h e c o n t in u ou s c o n v ol u t i on o f f a n d g: ∞

ht =

–∞

f τ g t–τ dτ

(1)

W e c an r e p re s en t f a n d g i n t er m s o f t h e i r sp ec t r a a s: ∞ ∞

fτ =

1 2π

i ωτ

–∞

F ω e dω

1 2π

g t–τ =

an d

Gϖe

–∞

iϖ t–τ

dϖ

S u bs t i t u t in g t h es e i n t o ( 1 ) : ∞

ht = –∞

Interchanging the order of integration

1 2π

∞

1 F ω e dω –∞ 2π iωτ

∞ –∞

iϖ t– τ

Gϖe

dϖ dτ

∞

1 ht = 2π

The term in [ ] is the Dirac delta function.

The d el ta functi on col lap ses one of the f r eq uency inte gra ls

–∞

1 FωGϖ 2π

1 ht = 2π

∞

–∞

e

i ω– ϖ τ

iωt

dτ e dωdϖ i ωt

–∞

1 ht = 2π

∞

F ω G ϖ δ ω–ϖ e dωdϖ ∞

–∞

iωt

F ω G ω e dω

H er e w e h a ve h ( t ) re p r es e n t ed a s t h e i n v e r se F o u r i er t r an s f or m o f " s om e t h i n g " . B y i n fe r en c e , t h a t so m e th in g m u st be t h e F o u ri e r t ra n sf or m o f h . T h u s :

Hω = FωGω


2 -27

The Convolution Theorem T h e r es u l t we h av e j u s t d e ri v e d is o n e o f t h e m o st f u n d am en t a l a n d i m p or t an t in a l l o f si g n al p ro ce s s in g . I t t e ll s u s t h a t we c an c on v ol v e t w o s i gn a ls b y m u l t i p ly i n g t h e ir s p ec t r a a n d i n v e rs e F o u r i er t ra n sf o rm i n g t h e r es u l t . T h e re as on t h at t h is is i m p or t an t is t h at t h e r e i s a n e x t r em e l y f as t a l g or it h m fo r p er f or m i n g t h e d i g it a l F o u ri e r t r an s f or m c al l ed t h e f a st F ou r ie r t ra n sf o rm ( FF T) . Us i n g t h e F FT a c on v ol u t i on c an be d o n e b y: g(t) f(t)

FFT

FFT

F(ω)

G(ω) Multiply

H(ω) IFFT h(t) N o t e t h at m u l t ip l y in g co m p l e x s p e c tr a i s:

H ω = F ω G ω = AF ω e = AF ω AG ω e

iφF ω

iφG ω

AG ω e

i φ F ω + φG ω

T h a t i s w e c an v i ew i t a s m u l t i p l yi n g t h e a m p l it u de s p e ct r a a n d a d din g t h e p h as e s p e c tr a. 2-28


Sampling T h e a n a ly t ic a n al y si s o f c on t i n u ou s s ig n al s i s m o s t u s ef u l f or g ai n in g a c o n c ep t u a l u nd er s t an d i n g o f si g n al p r oc e s si n g . I n a c t u al p ra ct i c e ; h ow e v er , t h e v as t m a j or i t y o f w or k is d o n e wi t h d i s c re t ly s am p l e d f u n c t i on s . T h e p r oc e ss o f s a m p l in g a co n t i n u ou s f u n c t i on i n t i m e ca n be v i ew e d a s a m ult i p l i c at io n b y a s am p l in g c om b. Fr e q u e nc y D o ma in T im e Do m a in

Continuous Gaussian spectrum

Continuous Gaussian

Convolved with

Times

1/Δt

Sampling Comb Comb spacing = Δt

Equals

Sampled Gaussian


F o ur i er t r a ns fo r m of s am pl i ng c om b

Equals

Gaussian Spectrum and aliases

1/Δt

2 -29

Sampling S o w e h av e s e en th a t s am p l i n g i n t h e t i m e d om a i n c a u se s t h e re p l i ca t io n o f th e c on t i n u ou s s p ec t r u m i n t h e f r eq uen c y d om a i n . T h e s p a ci n g b et w e en t h es e s p e c t ra l a l i as es i s 1 / Δ t a n d i t i s c u s t om a ry t o re s t ri c t o u r a t t en t i on t o t h e p ri m ar y fr e q u en cy ba n d l i ei n g b e t we en - 1 /( 2 Δ t) a n d 1 /( 2 Δ t) . T h e f r eq uen c y F n = 1 / ( 2 Δ t ) i s c al l e d t h e N yq u i s t fr e q u e n c y a n d i s t h e l i m i t in g f re q u e n c y o f t h e s am p l ed d at a . Fnyquist = 1/(2Δt)

F

-2Fn

Spectrum of sampled data showing aliasing.

-Fn

0

Fn

2Fn

S p e ct r u m o f s am p l ed d a t a wi t h m in i m a l a li as i n g .

F

-2Fn

-Fn

0

Fn

2Fn

Primary frequency band

2-30


Sampling T h e u n a li a se d s am p l i n g o f a n y c on t i n u ou s s i gn a l r eq uir e s t h a t t h e s i gn a l h av e i t s p o we r r es t r ic t e d t o a f re q u e n c y b an d: - f m ax < f < fm a x . S u c h s i g n al s a r e s ai d t o b e b an d li m i t e d . A ba n d l im it e d s i gn a l c a n be d i gi t al l y s am p l ed , wi t h ou t a l i as in g , w i t h a s am p l e s i ze o f Δ t = 1 / ( 2 f m ax ) . I t i s a f u n d a m en ta l th eo re m ( T h e S am p l i n g T h eo re m , P ap o u l i s, S i gn a l A n a ly s is , p 1 4 1 , 1 9 8 4 ) t h at s u c h a b an dli m i t e d , c on t i n u o u s , s i gn a l c an b e e x ac t l y re c ov e r ed f ro m i ts d i gi t al s am p l e s b y a p ro c e s s kn o w n a s " s in c f u n c t io n i n t e rp o l at i on " . Ti me D o ma in S ampled band limited function

F r e q ue n cy Do m a in Spectrum of sampled, unaliased, continuous function

Interpolation site C on volve d w i t h a si n c f un c ti on

Mu l t ip l ie d by a b oxca r

Recovers the original continuous function Re cove rs the s p ectr um of the conti nuous fu nction


2 -31

Sampling I n o r d er t o m i n i m i ze a l i as i n g, r aw a n a l og s ei s m i c d at a i s p as s e d t h r ou g h a n a n al og a n t i al i as f i l t er p r i or t o d i g it i za t io n . A t y p ic a l a n t i al i as fi l t er h as a n a m p l i t u d e s p e ct r u m w h i c h be g i n s t o r ol l o f f a t 5 0 % t o 6 0 % o f f n yq uis t a n d re ac h e s v er y la rg e a t t en u at i on ( > 6 0 d b ) a t f n yq uis t . H e re is t he s p ec tr u m o f a n ant ia l ia s f il te r f o r us e pr io r to sampli n g a t . 0 04 s ec .

0 -20 -40 -60 -80 -100 -120 0

2 0

4 0

6 8 10 0 0 0 Frequency (Hz)

120

140

R u l e o f t h u m b : S a m p le y o u r d at a s u c h t h at t h e e x p e c t ed s ig n a l f re q u e n c ie s a re l e ss t h an h al f f n yqu ist .

C om mo n s a m p l in g ra t es and their N y q u i st f re q u e n c ie s

s am ple r at e

N y q u i st

.008 s

6 2 .5 H z

.004 s

125 H z

.002 s

250 H z

.001 s

500 H z

A l i as i n g i s a l s o a p o s s ib i li t y w h e n re s am p l i n g se i s m i c d at a . I f t h e n e w sa m p l e i n t er v al i s m o re c oa rs e t h an t h e o l d , th en a n a n t i al i as f i lt e r s h ou l d be a p p l i ed . 2-32


The Discrete Fourier Transform T h e gr ea t u t il it y o f th e co nt i nu ous F o u ri er tr a n sf o rm t o de c o m p o se f un c t io n s in t o f u n d a m en t a l co m p l ex s in u s o id s ca n be a p p li ed d ir ec tl y to d i sc re t ely s a m p l ed ti m e do m a in fu n c ti o n s. Co n si d er a fu n c ti o n h( t ) w h ic h i s z e ro ev ery w h er e ex c ep t a t N t im e s d ef in e d b y t= k Δ t , k = 0 ,1 , 2 .. . N - 1 , w h e re i t t a k e s t he va lu es h k . T h is fu n c ti o n ca n be w ri tt e n w i th t h e d ira c d e lt a f u n ct i o n a s : N–1

ht =

Σ

h kδ t–kΔt

k = 0

If we now take the Fourier transform of h(t) we have: ∞ N–1

Σ

Hω = –∞

h kδ t–kΔt e

N–1

dt = Σ h k

–i ωt

k = 0

k = 0 N–1

Σ

Hω =

h ke

∞ –∞

δ t–kΔt e

–i ωt

dt

–i ωkΔt

k = 0

He r e w e ha v e a n a na l y tic e x pr e ssi o n fo r t he F o ur i e r tr a n sfo r m o f th e h k sa m p le s w h ic h is d e f in e d f o r a l l ω . We ha v e a lr e a d y se e n th at t he p h e n o me no n o f a li a sin g lim it s th e us a b le fr e qu e n cy b a n d t o -π /Δ t - > + π / Δ t . Fu r t he rmo r e, l in e a r al g e b r a te lls u s th a t N f r e q ue nc ie s in th is b a n d sh o ul d s uf fic e to d e t e r m in e th e N h k . S o w e a r e le a d t o c o ns id e r sa m pl in g th e fr e q u e nc y d o m a in a t ω ν = 2πν/(NΔ t) , ν = 0 ,1, 2 .. . N - 1. N–1

Σ

Hυ =

h ke

–i2πυk/N

k = 0


2 -33

The Discrete Fourier Transform

D i sc r e te e x p on e n t i al s h av e a w e ll k n ow n o r t h og on a li t y p r op e r ty s u c h t h at :

U s in g t h i s , i t i s n o t d if f i c u l t t o sh o w t h a t t h e h k s am p l e s c an be re c ov e re d fr om t h e H ν by :

hk =

1 N

N–1

Σ

υ = 1

H υe

i2πυk/N

Inverse DFT

T h i s r e su lt t og et h e r w it h : N–1

Hυ =

Σ

k = 0

h ke

–i2πυk/N

Forward DFT

f or m t h e d i sc r et e F o u ri e r t r an s f or m p a ir . T h e y a re t h e d i r ec t a n al og t o th e c o n t in u ou s F o u r ie r t ra n sf o rm r el at i on s . L i ke t h e F T, t h e DF T i s c om p l e t e i n t h at t h e h k a r e e x ac t l y r e c ov e ra b l e f ro m t h e i r s p ec t r u m , t h e H ν .

2-34


The Discrete Fourier Transform H e r e i s a p i ct o ri al r e p re s en t a t io n o f t h e d e v el o p m e n t o f t h e D F T fr om t h e c on ti n u o u s c as e : F r e q ue n cy Do m ai n

T im e Do m ai n Sampled band limited function N samples long

Δt

Spectrum of sampled, unaliased, continuous function.

Spectrum is periodic with period 2πN/Δt

-fnyq Co nv o l ve d wi th th e tr a nsf or m o f t h e samp li ng c o mb

1/Δt Times a sampling comb

1/Δf

Δf

DFT

Principle band

The s am pl ed s pe ct ru m

Principle band

T The sampled time series becomes periodic with period T=NΔt

fnyq

ID FT fnyq = 1/(2Δt)


-fnyq T = 1/Δf

fnyq ΔfΔt = 1/N 2 -35

The Discrete Fourier Transform T h e s am p l i n g o f t h e s p ec t r u m o f a d i s cr e t e t i m e s e ri e s c au s e s t h at s e ri e s to b ec o m e p e ri od i c w i t h p e ri od T = N Δ t. T h is h as s i gn a l p r oc e s si n g c on s e q u e n ce s t h at a r e a p pa r en t wh en w e c on s i d er a p p l y in g a f il t e r wi t h t h e D FT a n d t h e c o rr es p o n d i n g c on v o lu ti o n . Time series showing time domain aliases

Principle Period

Filter operator:

T h e c on v o l u t io n o p e r at i on t h at d up li c at e s m u l t ip li c at i on w it h t h e D FT i s c al l ed c i rc u l ar c o n v ol u t i on . N ot e t h at t h e f i lt e r o p er at or p la c ed o n t h e l as t s am p l e o f t h e p r in ci p l e p e ri o d a p p e ar s t o " wr ap a r ou nd " a n d a f f e c t t h e fi r s t s am p l e . T o a v o id t h i s p ro bl e m , i t i s c om m o n t o p ad t h e t i m e s er i es wi t h a l e n gt h o f z e ro s c h os e n w it h t h e l e n gt h o f t h e f il t e r o p e r at or i n m i n d . Principle Period

Zero Pad

2-36


The Fast Fourier Transform T h e f as t F o u ri e r t r an s f orm ( FF T ) is n ot h i n g m or e t h a n a c l ev e r w ay o f c al c u l at i n g t h e D FT wh i c h ge t s i m p r es s iv e p e r fo rm a n c e r es u l t s . T h e c on v ol u t i on o f a n N l en g t h o p er at o r i n t h e t im e d o m ai n r e q u i re s o n t h e o r d er o f N 2 f l oa ti n g p oi n t o p e r at i on s . T h e s am e c om pu t at i on i n t h e f r eq uen c y d o m ai n wi t h t h e F F T r e q u i re s r ou g h l y N * lo g( N ) o p er at i on s . H o w ev e r, we m ust be c ar e f u l wi t h th is s t at e m e n t b ec au s e , ge n e ra ll y , t h e tw o N ' s a r e n ot t h e s am e. T h i s i s be c au s e t h e F F T a l go ri t h m re q u i r es t h at t h e ti m e s er i es le n g t h b e a " m ag i c n u mbe r" w h i c h is u s u al l y a p ow e r o f 2. ( Al s o t h e t w o ti m e se r ie s b e in g c on v ol v ed m u s t be t h e s am e l e n gt h . ) T h i s is a c h ie v e d b y a tt a c h in g a z e r o p ad t o t h e t im e s er i es . T h u s if N i s t h e l e n gt h o f t h e t im e d o m ai n o p e ra t or a n d i f N 2 i s t h e f i rs t p o we r o f 2 g re at e r t h a n N , t h e n we m u s t co m p a re N 2 t o N 2 lo g( N 2 ) . ( O f t e n e v en t h i s i s n o t e n ou gh be c au s e t h e z er o p ad m u s t b e l o n g e n ou g h t o a v o i d o p e ra t or w r ap a ro u n d . ) T h e b ot t om l in e is t h at s h o rt o p e ra t or s ( l e ss t h a t ~ 6 4 p oi n t s ) a re o f t e n a p p l ie d f a st e r wi t h c on v ol u t io n w h i l e lo n g o p er at o rs a re M U C H f as t e r wi t h F FT ' s . T h e d ia gr am b el ow i s a d ap t e d f r om H at t o n e t a l. a n d s h o ws t h e ba s i c t r ad e of f . Time domain Convolution compute time

FFT Operator Length


2 -37

Filtering W e h av e se e n t h at c on v o lu t i on w it h a wa v e f or m su r p r es s e s a n d p o s s ib l y p ha s e sh i f t s s om e f re q u e n c i es re l at i v e t o o t h e rs . T h is f i l t er i n g a c t io n i s o f t en ex p l o it e d t o e n h an c e s i gn a l a n d su r p r es s n o is e . H e re w e se e a co m p a ri s on o f f i v e d i ff e re n t z er o p h as e f i l te r s i n b ot h t h e t im e a n d f re q u e n c y d om a i n s. T h e i n v e rs e r el at i o n sh i p be t we e n t e m p or al w i d t h a n d f re q u e n c y ba n d wi d t h is re ad i l y a p p a re n t . Wavelet 1

Five Generic Wavelets

Wavelet 2 Wavelet3 Wavelet 4 Wavelet 5 -0.15

-0.1

-0.05

0

0.05

0.1

0.15

0 -20 -40

Their -60 Fourier -80 Am pl itu de -100 Spec tra

Wavelet 2 Wavelet3 Wavelet 4 Wavelet 5 Wavelet 1

-120 0

2-38

50


200


The Z Transform T h e p er i od i c i ty o r c i rc u l ar i ty i n h e re n t i n bo t h t i m e a n d f r eq uen c y is n i c el y c ap t u r ed b y a p o we rf u l m e t h od o l og y k n ow n a s t h e Z tr an s f or m . C on si d e r th e t i m e s er i es , [ 1 . 5 - .3 0 . 1 0 ] , w h er e i t is a s su m ed t o s t ar t a t t = 0 a n d i n c r em e n t by ⎯ t . W e re p r es e n t th is s er ie s in t h e Z d o m ai n by a p o ly n om i a l i n z : 0

1

2

3

H z = 1z –.5z –.3z +0z +.1z 1

2

= 1–.5z –.3z +.1z

4

4

S o w e s e e t hat t h e e x p onent o f z gi v e s t h e s am p l e n um b e r a n d he n c e determines t h e s a m pl e t i me ( nΔt ). N ot e a l s o t he f o l lowing: • Negative times correspond to negative exponents of z • M u l t i pl i c a t i on b y z n d e l ay s t h e t i m e s e r i e s by n s a mp l es i f n i s p os i t i ve a nd a dv a n c es i t b y n s am pl e s f o r n e g at i v e n . T h e g re at u t il i t y o f t h e Z t r an s f or m li e s i n it s a b il i t y t o r ep re s en t d i sc r et e co n v ol u t i on a n d t h e D FT a s o p e r at i on s w i th p ol y n om i al s . I t i s n o t d i f fi c u l t t o sh o w t h at t h e c on v ol u t io n o f t wo t i m e s e ri e s , f a n d g, c an b e re al i ze d b y s i m p l y m u l t i p l yi n g t h e i r Z t r an s f or m s a n d r ea d in g o f f t h e r e su l t . ( S ee W at e rs ( p 1 3 3 ) f or a p r oo f. ) 1

1

2

2

F z = f0 +f1z +f2z + ... G z = g0 +g 1z +g 2z + ... Hz = FzGz = 1

2

f0 g0+ f0 g1+g0f1 z + f0g 2+f1g 1+g0f2 z + ... h = f• g Methods of Seismic Data Processing

2 -39

The Z Transform T he f a ct t ha t c o nv o lu t io n i s d o ne b y m u lt i p li c at io n o f Z t r a n s fo r m s i s r e m in is c e nt o f t he F o u r ie r t r a n s fo r m . I n f a ct , i f w e l e t z = e - iω Δ t t h e n t he Z t r a ns f or m b e c om e s : N–1

N–1

Gz =

Σ

gkz

Gω =

k

Σ

gke

–iωkΔ t

k = 0

k = 0

A s wi t h t h e D FT , i f w e n o w c on s i d e r o n l y d i s c re t e f re q u e n c i es ω ν = 2 πν/( N Δ t ) , ν = 0 , 1, 2 . .. N - 1 , t h en w e s ee n t h at t h e Z t r an s f or m , w i t h z = e - iω Δt , is p r ec i s el y t h e D FT . N–1

Gν =

Σ

–i2πυk/N

g ke

k = 0

T he Z transfo r m is more general t ha n th e DF T s in ce z c an b e a ny c o mp le x number. I n f a ct t he D F T amount s to evalua t ing th e Z tranfo r m a t N d iscrete l o ca tio n s around th e un it cir c le in th e comple x z p lane. imag(z) ω2

Complex z plane

ω1

ωo ων

real(z)

ωN-1 ων+1

2-40


The Z Transform C o n s i d e r t h e el e m e n t al c ou pl et F( z) = 1 -a z. N ow i f w e c o n v ol v e F( z ) wi t h an o t h e r ar b i t ra ry t im e s e ri e s g ( z ) , t h e n w e r ep re s e n t t his as : H ( z) = F( z ) G ( z) . Su pp os e t h at o n l y F( z) an d H ( z) ar e k n o wn t o u s an d w e w i sh t o r e c ov e r G ( z) . In t h e z t r a n s fo r m d om ai n we c an s im p l y:

Hz = FzGz

∴ Gz =

Hz Fz

So we define the inverse of any time series as: –1

F z =

1 Fz

For F(z) = 1 -az, this gives: –1

F z =

1 1–az

2

3

= 1+az+ az + az +

T h i s s er i e s , c a l le d t he g e om et r i c s er i e s , i s k n ow n t o c o n v e r ge a b so l u t e ly p ro v i d e d t hat | az | < 1 . S i n c e w e ar e e s p e c i al l y i n t e re s t e d in t h i s re s u l t ev a l u at e d on t h e u n i t c i r c l e ( | z| = 1 ) t h e n w e n e e d | a| < 1 . It i s c ust o m a ry t o t a l k a b ou t t he lo c at i o n of t he "ze r o" of t h i s c o u p l e t d ef i n e d b y:

1–az 0 = 0 ⇒ z0 =

1 a

I f |a | < 1 , t h e n w e se e t h at z o m u st li e o u t s i d e t h e u n i t c i r c le in o r d e r f or th e i n v er s e t o c on v e rg e . S u ch a n i n v e r se is sa i d t o b e s t ab l e ( p h y ic a ll y r ea li za bl e ) . N ot e a l so t h a t F ( z ) i t s el f i s t r i v ia l ly s t ab l e. Methods of Seismic Data Processing

2 -41

The Z Transform A n y c a u s a l, st a b le t i m e se ri es wi th a ca u sa l , s t a bl e in v e rs e i s s a i d t o b e m i n im um p h ase . T h u s o u r e le m en t a l c o u p le t , 1 - a z , is m i n im u m p h a s e wh en e v er |a |< 1 . A n y m o r e c o m p l ex t im e se ri es c a n a lw a y s be fa ct o r ed i n to a se t o f el e m en t a r y c o u p l et s . N–1

Gz =

Σ

k

g kz = z–z 0 z–z1

z–zN–1

k = 0

W e s a y t h a t G ( z ) i s m i ni m u m p h a s e i f a ll i t s e le m e n ta l c o u pl e ts a r e m i ni m u m p ha s e . T h a t i s e q ui v a le n t t o s a y in g t h a t a l l o f t h e r o o ts o f t he p o ly n o mi al G (z ) m us t l i e o u ts i d e t h e u n it c i r cl e i n t h e c o m pl e x z p la ne . I f a ll r o o ts l i e i n s i d e t he u ni t c i r c le , G (z ) i s s a id t o b e m ax im um p ha s e a nd o th e r w is e i t i s m ix e d p h as e . imag(z)

•

z=z0 Complex z plane

real(z) • • z=zN-1

z=z1

A minimum ph as e t ime series ha s al l i ts zeros outside the unit circle. 2-42


The Z Transform T h e z er o s o f F ( z ) c o r re sp on d t o p o l es f o r F -1 ( z ) . T h u s, fo r t h e c a se o f a t i m e se ri es w h o s e Z tr a n s fo rm h as a d e n om i n a t or, we s ee th at t h e s t a bi li t y c o n d it i o n re q u ir es t h a t a l l p o l es a l s o l i e o u ts i d e t h e u n it ci rc l e. T h e m o s t g en e ra l t im e s er ie s c a n b e w ri t te n a s a Z tr a n s fo rm w it h b o t h n u m e ra t o r a n d d en o m i n a t o r s uc h a s:

Hz =

Az Bz

=

z–α 0 z–α 1 z–β 0 z–β 1

W e s a y t h e c o r r e s p o nd in g t im e s e r ie s i s m i n im um p h as e i f a ll α i a n d a ll β i l i e o ut s i d e t h e u n it c i r c le . T h e f ol lo w in g t he o r e m f o ll ow s i m me d ia te l y: T h e r es ul t an t o f t he s e quent i al c onv ol ut i on o f a n y nu m ber o f m i n i mu m phas e time series i s al s o min im um ph a s e. Conversely:

I f a ny ti m e s e r i e s i n a se qu e nce of co nvo l u ti o n s i s n ot mi n i m um ph a s e , th e n th e r e s ulta nt is no t m i nimu m p ha s e .

T h o ug h th e se s ta te me nt s se e m ir o n c la d , k e e p in m in d t he u n sta t e d a s su mp t io n t ha t al l th e se tim e se r ie s h a ve t he s am e sa m p le r a te . T hu s th e r e s a mp li ng o f a ti me s e r ie s is a n o p e r a ti o n w h ic h l ie s o ut sid e t he sco p e o f t he s e th e o r ems . Methods of Seismic Data Processing

2 -43

Crosscorrelation G i v e n t w o si g n al s , r a n d s , t h e c r os s c or re l at i on p ro v id es a n u m e ri c al ch a ra c t er i za t i on o f th ei r s i m i la ri t y. Z e ro l ag : s0

T h e c a lc u l at i on o f s1

s2

s3

s ⊗r s4

s5

•••

r5

•••

s4

s5

Multiply aligned samples and sum: r0

r1

r2

r3

r4

c0 = s0 r0+s1 r1+s2 r2+ F i r s t p o si t i v e l ag : s0

s1

s2

s3

•••


r1

r2

r3

r4

r5

•••

s5

•••

r4

r5

c1 = s0 r1+s1 r2+s2 r3+ F i r st n e ga ti v e l ag : s0

s1

s2

s3

s4


r1

r2

r3

•••

c–1= s 1r0 +s 2r1+s 3r2+ 2-44


C r o ss c o r r el a ti o n T h e ge n e ra l f or m f o r t h e c ro s sc o rr el at i on o f s a n d r c an b e w ri t t en :

cj =

Σs r k

k+j

k

O r, fo r c on t i n u o u s s ig n al s :

cτ =

∞ –∞

s t r t+τ dt

P r op e rt i e s o f cr os s c or re l at i on s : • I f ei t h e r s o r r is a n i n f i n it e l en g t h ra n d om s i gn a l , t h e n c j = 0 f or a l l j . • T h e m ax i m u m o f c d e fi n e s t h e " l ag " a t wh ic h s a n d r a r e m o st s im il ar wh e n a l ig n e d . • A c r os s c or re l at i on c an be c om p u t e d by re v e rs i n g s a n d c o n v ol v in g. C a n y ou p r ov e t h i s ? • Th e a u t o co rr e la t io n c ro ss c o rr el at i on w h en r =s .


is

a

sp ec i al

c as e

ti m e of

2 -45

Autocorrelations T h e a u to c o r r e la t io n, φ , of a sig n a l, s, is a ch a r a ct e r iz a t io n o f i ts se l f sim i la r it y . It c a n b e c o mp u te d a s f o llo w s: Z e ro l ag : The signal s

s0

s1

s2

s3

s4

s5

•••

s5

•••

Multiply aligned samples and sum: s0

A copy of s

s1

s2

s3

s4

φ0 = so +s 1+s2+s3 + 2

2

2

2

F i r st p os i t i v e la g: s0

s1

s2

s3

s4

s5

•••


s1

s2

s3

s4

s5

•••

φ1 = so s1+s1 s2+ s2s 3+ S e c on d p o si t i v e l ag : s0

s1

s2

s3

s4

s5

•••


s1

s2

s3

s4

s5

•••

φ2 = so s2+s1 s3+ s2s 4+ 2-46


Autocorrelations The general form for the autocorrelation of s can be written: length (s)

φj =

Σ

k = 0

sk sk+j

Properties of the autocorrelation: • φ o >= φ j for all j. The zero lag is always largest. • I f s is a n in f i n it e le n gt h r a n d o m se q u en c e , th e n φ o g iv e s th e s u m o f sq u a r es o f th e s eq u e n ce a n d a l l o t h er φ j a r e z er o . • The Fourier transform of the autocorrelation gives the power spectrum (squared amplitude spectrum) of the signal, s. • The autocorrelation has no phase information. The autocorrelation is often normalized such that φ o=1: length (s)

φj =

Σ

k = 0

sks k+j

length (s)

Σ

k = 0


2

sk

2 -47

Spectral Estimation T h e ge n e ra l p r ob l em o f e st i m a t in g a m p l i t u d e o r p o we r s p e ct r a o f a n u n k n ow n s i gn a l em be d d e d i n n oi s e o r o t h e r u n w an t e d s i gn a ls i s c al l ed sp ec t r al es t i m at i on . I t a r is e s i n m a n y c on te x t s i n s ei s m i c d a t a p ro c es s i n g b u t m o s t n ot a bl y i n d ec on v ol u t io n t h e or y. T w o sa m p l e p r ob le m s : • G iv e n a sm a l l n u m b er o f l ag s o f a p os s i bl y i n f in i t e a u t o c orr e la t io n , e s ti m a t e t h e p o we r s p ec t r u m o f t h e u n d e rl y in g p h ys i c al p r oc e s s. • G i v e n a s m al l p o rt i on o f a p os s ib l y in f i n i t e t i m e s e ri e s , es t i m at e t h e a m p l it u de sp ec t r u m o f t h e u n d er l yi n g p h y s ic al p r oc es s . T h e s e t wo p ro bl e m s a r i se re p e at ed ly , a n d i n a v ar i et y o f c on t e x t s , i n s ei s m i c d at a p ro ce s s in g t h e or y. H ow e v er , t h e y a re e s se n t i al l y s i m i l ar d i f f er in g o n l y in t h e n at u r e o f t h e i n p u t : e. g a n a u t o c or re l at i on o r a g e n er al t i m e s er i es . W e s h al l c on s i d e r t w o a p p r oa c h e s : t h e wi n d o we d D FT , a n d t h e m a x im um e n t ro p y s p ec t r u m ( B u rg s p e c t ru m) . C on si d e r t h e co n s t ru c t i on o f a n el e m e n t ar y s e is m o gr am b y co n v ol u t i on :

Wavelet Seismogram Reflectivity

0 2-48

0.2

0.4

0.6

0.8

1


Spectral Estimation I f we c om p u t e t h e f u n c t i on s , w e o b t ai n :

a u t o c or re la t io n s

of

t h es e

t h r ee

Autocorrelation of wavelet

Autocorrelation of seismogram

Autocorrelation of reflectivity

-0.2

-0.1

0

0.1

0.2

W e s ee fr om t h i s re s u l t t h a t t h e a u t oc o rr el at i o n o f t h e s e is m o gr am i s q u i t e s i m i la r t o t h a t o f t h e w av e l et . T h u s i t i s re as on a bl e t o a sk i f we c a n e s t im a t e t h e w av e le t p o we r s p e ct r u m fr om the c e n t ra l l ag s of the a u t oc or re l at i on o f th e se i s m og ra m . F u rt h e rm o r e, w e w i ll d o t h i s wi t h ou t u s in g a n y d i r ec t kn o wl e d ge o f t h e w av e le t . S o, w e wi l l t ak e th e sa m p l es fr om - . 1 t o . 1 o f t h e s e is m o gr am a u t oc o rr el a ti o n a n d c om p ute t h e i r p ow er s p e c t ru m . I f w e s im p ly t r u n c at e t h e a u t oc or re l at i on we o bt ai n t h e r es u l t s h ow n be l ow : Estimate with boxcar window

Exact result

Frequency (Hz) 0 20 40 Methods of Seismic Data Processing

60

80

100 2 -49

Spectral Estimation T h e p r e c ed i n g s p e c t ra l es t i m at e i s n o t b ad b u t c an b e im p ro v ed by t ap e ri n g t h e s am p l es n e ar t h e e d g e o f t h e ch o s en wi n d o w i n st e ad o f s i m p l y t ru n ca t in g. T h e m e t h od o f t ap e r in g i s re f er r ed t o a s " w i n d ow i n g " a n d a n u m b e r o f s p e c i al w in d ow s h av e b e en d e v is e d . boxcar hanning

mwindow

bartlett

boxcar mwindow Hanning Bartlett Exact result H er e a re t h e r e su l ta n t e st im at e s f ro m a p p ly in g t he v a ri o u s w in d ow p ri or t o e s t i ma t in g t h e p ow er w it h t h e D F T. A l l w in d ow s d o a r ea s on ab l e j ob t ho ug h t h e e d ge s e em s t o b e w i t h B a rt le t t a nd H an n in g. 0 2-50

F requency (Hz) 20 40

60

80

100


Spectral Estimation The DFT is a polynomial in z containing no denominator terms. N–1

Gz =

Σ

g kz

k

k

z = e

–i2πυk/N

k = 0

C on s e q u e n t l y, t h e D FT s p e ct r al es t i m at e c o n t ai n s o n l y z e ro s ( n o p o l es ) in t h e z p l an e a n d is so m e t im es c a ll e d a n a l l- z e ro s es t i m at e . A n a l t er n at i v e es t i m at e w as d e v el o p ed by J .P . Burg ( s ee C l aer b ou t , 1976, F u n da m e n t al s o f G e op hy s ic a l D a t a P r oc es s i n g) wh i c h s ee ks t o p r od u c e a s p ec t r al m od e l u si n g a Z t r an s f or m w it h o n ly d en o m i n at o r t e rm s . T h i s m at h e m at i c al d e v el o p m e n t o f t h e B u r g sp ec t r u m , a l so c a ll e d t h e m ax i m um en t r op y sp ec t r al e s t im a t e or a ll - p o l es e st i m a te , i s be yo n d t h e sc o p e o f t h i s p r es e n t at i on . N e v e rt h e l es s , t h i s i n t u i ti v e c on c e p t o f t h e B u rg t e ch niq ue h e l p s u s u n d er s t an d i t s ba si c b eh a v i or . A s a n a l l - p o le s e s ti m a t e, it i s v er y e ff e c t iv e a t m o d e li n g s p e ct r a wh ic h h av e is o la t ed s p i ke s b u t le s s s o f or s m oo t h s p ec t r a. F u r t h e rm o re , B u r g d ev e l op e d t h e m e t h od u s i n g p r ed ic t i on o p er at or s t o p re d i c t t h e t i m e s er i es o u t s id e o f t h e t r u n c at i on ra n ge s o t h at t h e c on c e p t o f a wi n d o w d o es n o t a p p l y t o t h e B u rg s p e ct r u m . T h e F o ur ie r s p ec t r u m p la c e s z e r os c l o se t o t h e u ni t c i rc l e a n d s o c a n m o de l a p hy si c a l p r oc e s s w i t h a s m oo th s p ec t r u m h a vi n g n ot c h es .

Frequency T h e B u rg s p ec t ru m p la c e s p o le s c l o se t o t he u ni t c i rc l e a nd s o c a n m od el s pi k es i n a n u nd er l yi n g p hy s ic a l p ro ce s s


2 -51

Spectral Estimation A s m ig h t b e ex p e c t ed f ro m t h e p r ec e d i n g d is c u s s io n , t h e B u r g s p e ct r u m d oe s n o t d o a g oo d j o b i n t h i s c as e :

E xact result

Burg (maximum entropy) spectrum (l=30)

0

20

40

60

80

100

120

H o we v er , th is d oe s n o t m ea n th a t t h e B u r g s p e ct r u m i s w it h o u t m e ri t . H a t t on e t a l. ( p ag es 3 6 - 3 8 ) gi v e a n e x ce l l en t a n a l ys is s h ow i n g t h e su per i or it y o f th e B u r g t e ch n iq u e o v e r t h e D F T i n t h e c a se o f t h e re s ol u t io n o f t wo c l os e ly s p a ce sp e c t ra l p e ak s. F u r th er m or e , a s w e s h al l s e e, t h e B u r g t e c h n iq u e l ea d s t o a v e ry e f f ec t i v e d e c on v o lu t i on m e t h od .

2-52


Wavelength Components C o n sid er a se rie s o f p la n a r w a v e fro nt s p r op a g a ti ng a s s ho wn be lo w.

λx θ λz

θ

X

λ Wave propagation direction

Z T he di st an c e between w a ve f r o nt s , m e a s ur e d pe r p e nd i c ul a r t o t h e m, i s de fi ne d as t he w a ve l e n g t h, λ . We c an a l s o s pe a k o f t h e w av e l e ng t h "c om po ne n t s " i n t h e va r i ou s c o o r di n at e di r e c t ion s . F or ex am pl e , t he ho r i z on t a l w a ve l e ng th, λ , i s t he d i s t a nc e b e t w e e n x

w av e f r on t s me a s u re d i n t h e x c oo r d i na t e di r e c t ion . Th us :

λx =

λ sin θ


and

λz =

λ cos θ 2 -53

Wavelength Components W e s e e t h at t h e c o mp o ne n ts o f w a ve l e ng t h a r e n e v e r l e ss t h an t h e w a ve le n g th i t s e lf . I n f a ct , f o r a v e r ti ca ll y t r a v e li n g w a v e , λ x i s i nf i ni t e . T h e c o mp o ne n t s a d d a s i n ve r se s qu a r e s :

1 1 1 = + λ2 λ2x λ2z I t is o f t e n c o nv en i e nt t o d e a l wi t h v e c t o r q ua nt i t i e s s o w e d e f i ne t he wa v e nu mb e r , k , a n d i t s c om po ne nt s a s t h e i nv er s e o f t h e w a v e l en g t h a n d i t s c om po ne nt s . –1 k = λ

–1 k x = λx 2

2

–1 k z = λz 2

k = kx + kz k is th e m a g ni tu de o f a ve c to r , k, w h ic h p o in ts in th e d ir e c tio n o f w a v e p r o pa g a t io n an d w h os e co m p o ne n t s a r e th e i nv e r se w a ve l e ng th s.

I n 3- D , w e ha v e p la na r w a v e fr o n ts i ns te a d o f lin e a r b u t a sim p le e xt e ns io n o f th is r e su lt st ill h o ld s: 2

2

2

kx + ky + kz = k

2

The dispersion relation for scalar waves.

Where: –1

k y = λy

2-54


Wavelength Components This geometric relation between components of the wavenumber vector is fundamental to the study of wave propagation. It can be considered as the Fourier domain equivalent of the scalar wave equation. A fundamental result from theory is that the extrapolation of surface recorded data into the subsurface (z direction) requires knowledge of kz. On the surface, we can measure kx, ky, and f, and since fλ=v, this allows kz to be calculated form the dispersion relation:

Since kz must be a real number (in order to be interpreted as an inverse wavelength) we see from this equation another fundamental result. Not all values of (kx,ky,f) can be considered as wavelike. In fact, we must have

in order for a triplet of (kx,ky,f) to be a propagating wave.


2 -55

Apparent Velocity T h e w a ve le n gt h c o m p o n en t s a n d t h e co r re sp o n d i ng w a ve n u m ber s a re cl o se ly r ela te d t o t he w a v e v el o ci t y a n d it s co m p o n en t s w h ic h a r e ca l le d a p p a r en t v el o c it ie s. R e ca ll in g th e ba s ic r el a ti o n , λ f = v , w e se e t h a t t he a d d i ti o n f o rm u l a f o r w a v el en gt h c o m p o n en t s:

1 1 1 = + λ2x λ2z λ2 leads directly to:

1 v

2

=

1 2

vx

+

vx = fλx

where

1 2

vz

and

vz = fλz

If we use wavenumber components, we have:

v =

f k

vx =

f kx

vz =

f kz

N oth i ng p h y si ca l a c tu a l ly p r o p a ga t es a t a n y o f t he a p p a r en t v e lo c it i es . R a t h er, th e y a re s im p l y rel a t ed t o t h e a rbi t ra r y c h o ic e o f c o o rd i n a t e d i re ct i o n s a n d c a n be v is u a l iz e d a s t h e w a ve le n gt h a lo n g a co o rd i na te d ir ec ti o n d i vi d ed by t h e ti m e b et w e en w ave cr es ts ( i .e. t h e p e ri o d o f t h e w a v es .)

2-56


Apparent Velocity x

Receivers on surface

θ

z

λx

A se ri es o f p la n e w a v ec res t s a p p r o a c h a h o r iz ont a l a n d a ve rt ic a l r ec o rd i n g a r ra y . E a ch a rra y s ee s th e a p p a re nt w a v ele n gt h a l o n g t h e s u rf a ce o n w h ic h it i s d e p lo yed a n d c a n m e a su re t h e a p p a ren t v el o c it y o f t h e w a v ef ro n t s a lo n g t h a t s u rf a ce . T h e a n gle θ i s c a l le d th e “ em er gen c e a n g le ”. T h e w a v el en gt h c o m p o n e nt s a re :

λx =

λ sin θ

and

λz

λz =

λ cos θ

And the apparent velocities are:

similarly Methods of Seismic Data Processing

2 -57

Th e 2 - D F - K Tra nsf o rm The f-k transform is a fundamental tool which essentially allows the direct computation of wavenumber components and frequency for a multidimensional wavefield. In 2-D, it can be written:

and in 3-D

The inverse transforms are mathematically similar:

2-D

3-D

Th e s e i n v e rs e t r an s f or m s have the p h y si c al i n t er p r et a t io n of p re s en t i n g a wa v e f i el d as a s u p e rp o s it i on o f in div i d u a l F ou r ie r co m p o n en t s o r " p l an e w av e s" .

2-58


Th e 2 - D F - K T ransform A f ew f - k t r an s f or m s a re k n ow n a n al yt i c al l y. P er h ap s t h e m o s t i n p o rt a n t i s t h e t r an s f or m o f a si n g l e l in ea r e v en t . U s i n g th e m a t h e m at i c s o f D i ra c d el t a fu nct i o n s, a se i sm ic wa v e f i el d c o n s is t i n g o f a n i so l at e d l in ea r e v en t c an be w ri t t e n : x

t

kx

f

• Horizontal events in (x,t) are vertical in (kx,f) and vice-versa. • All events in (x,t) with then same apparent velocity, vx, are collected into a single linear event in (kx,f). The different events are distinguished by their phase spectra but have differing phase spectra. Methods of Seismic Data Processing

2 -59

T h e 2 -D F - K T ransfor m If w e conside r al l possibl e linear events characterized b y vx=v/sin(θ) , then we have: x

kx

θ =0o θ =15o

vt

θ =30o

f/v

θ =50o θ =90o

θ =0o

θ =90o θ =50o θ =30o θ =15o

The previously encountered fact that f/v > kx is reexpressed in this analysis though the fact that a portion of (kx,f) space is not populated. As a general rule of thumb, we see that large kx values are only wavelike at high frequencies. This fact will turn out to be fundamental in describing the ability of seismic images to resolve small features. Small features require large kx values which, in turn, require a large temporal frequency bandwidth. In proceding from analytic to discrete f-k transforms, it turns out that the implementations of the Fourier transform integrals are approximate but the forward and backward DFKT are exact inverses of each other. This fact is a great convenience in data processing and is not generally true of other transforms such as the Radon transform.

2-60


Th e 2- D F - K Tr ansf o rm Wh e n w e pr o c e d e fr om the c o n ti n uo us F - K t r an s f o r m t o th e di s cr e te , a s i t ua ti o n di r e ctl y a na l o g o us t o t he 1 -D c a s e occ ur s. T ha t i s , the a c t of s p at i a l s a m pl ing ind uc es a periodicity i n th e (ω , k ) do ma i n . Un l i k e te m po r a l aliasing, s pa tial aliasing i s a l w a ys pr e s e n t. knyq

-knyq

ω = ωmax ω

ω

ω

Principle Band He r e w e s ee o n e ev e n t s h o wi n g s p a t i al al i as i n g an d an o t h e r wh ic h d o es n ot . G i v e n a s p at i a l s am pl e ra t e of Δx an d an ap p ar e n t v e l oc i t y v a t h e n a l l t e m p o ra l f re q uen c i e s h i g h er t h an :

f

crit

= 2 π ωcrit = va k nyquist =

va 2 Δx

wi l l b e s p a t i al l y al i as e d . Fo r e x c el l e n t i l l u s t r at i on s o f sp at i al a l ia s i n g se e Ha t t on et al . p p 4 3 - 4 5 an d Y i lm az p p 6 2 -6 9 Methods of Seismic Data Processing

2 -61

FK Transform Pairs Space-time domain

FK amplitude spectra

A si n gl e fl at ev ent. Wa vel et i s 30 Hz ( domi nan t) and m in i mum phas e S ix even ts with e me rgen ce a ngle s: 0, 10, 30, 50, 70, & 90 degr ees . V elo city i s 20 00 m /se c. S ix e v en ts w it h e m e r g en c e a n gl es : 0 , - 10 , - 30 , - 50 , - 7 0 , & - 90 d e g re e s . V el oc i ty i s 2 0 0 0 m / se c .

A single diffraction hyperbola. Veolocity is 2000 m/sec

Many diffraction hyperbolae. Veolocity is 2000 m/sec

2-62


τ-p Transforms T h e τ - p t r an s f or m, a l s o kn ow n a s t he R ad on t r a ns f or m o r s l an t s t a c k , i s a v e r y u s ef u l d at a p r oc es s i n g t o o l d ue t o i t s a bi l i ty t o d e c om po s e a s e is m i c m a t r i x i nt o e v e n t s o f c o ns t a nt h o r i zo nt a l s l o w ne s s , p . I t ' s c l o se r e l at i o n t o t he f -k t ra ns f o r m i s c a pt u r e d i n t he " pr o je c t i o n s l i c e t he o re m " w hi c h s ho w s t ha t t h e τ - p t r a ns f o r m m a y b e c om pu t e d f ro m a n f - k t r a ns f o rm w hi ch h as b e en i nt e r po l at ed t o " p ol a r " c o o rd i na t e s . ( S ee D e an s , S . R . , 198 3, Th e R ad on Tr an s fo r m a nd S om e o f I ts A p pl ica ti on s , J o hn W ile y an d S o n s ) . C o ns i de r t he e x pr e s s i on f or a f o rw a r d f -k t r an s f o r m:

φ(k x,f ) =

∞ –∞

Ψ(x,t)e

2π i (k xx – f t)

(1)

dx dt

W e h a v e se en h ow t h is e xp r es si o n t ra n s fo r m s li ne a r ev en t s in ( x ,t ) i n to li n ea r ev en t s in ( kx ,f ) :

sin θ x t = v

kx θ =0o

x

=

sin θ ω v kx

θ =15o θ =30o

vt

θ =50o θ =90o

(x,t) space Methods of Seismic Data Processing

ω /v θ =90o θ =50o o o θ =30 θ =15 o θ =0

(ω,kx) space 2 -63

τ-p Transforms N ote t h a t s in(θ) /v ( ho ri z o n t a l s lo wn es s ) c a n a l s o b e wr it t e n a s d t /d x o r t h e ra y p a ra m et er p . T h u s , ra d i a l l in e s i n t h e f- k t ra n s f o rm a re l i n es o f c o n s t a n t p . T h i s c a n b e ex am in e d f u rt h er b y a s u bs t it u t io n o f va ri a bl es in th e f - k i n te gr a l ( 1 ) :

φ(p,f ) =

∞ –∞

Ψ(x,t)e

2π i f (p x – t)

dx dt

where

p=

kx f

(2)

H e re p h a s b e en ex pli c i t ly i n t r od u c e d a s t h e ra ti o o f k x a n d f a n d h e n c e i s c on st a n t a l on g ra d i al l i n es in ( kx , f) s p ac e . S o φ ( p , f) c an b e re g ard ed a s a " p ol ar c oo rd in a t e" r ep r e se n t at i on o f φ ( kx , f ) . N o w, co n s id er t h e m e an i n g o f e q u at i on 2 f o r c on s t an t p b y p e rf o rm i n g t h e t i n t eg r at i on fi r st :

φ(p,f ) =

∞ –∞

ψ(x,f)e

2π i f p x

dx

(3)

dt

(4)

where

ψ(x,f) =

∞

Ψ(x,t) e

–2π i f t

–∞

Then, compute the inverse Fourier transform (f->t) of (3)

ϕ(p,τ) =

∞

2π i f τ

–∞

φ(p, f ) e

df

(5)

Now, substitute (3) into (5): ∞

ϕ(p,τ) = –∞

2-64

∞ –∞

ψ(x,f )e

2π i f p x

dx e

2π i f τ

df


τ-p Transforms Interchange the order of integration: ∞ ∞

ϕ(p,τ) =

–∞

2π i f (p x + τ)

ψ(x, f )e

df dx

–∞

The inner integral gives Ψ(x, px+τ ), so:

ϕ(p,τ) =

∞ –∞

Ψ(x, px + τ) dx

(6)

Equat io n (6 ) i s th e c o nv e nt io na l e qu a tio n fo r t he τ- p transform (c o mpare w i th Y il ma z (Seis m ic Da t a Processing , 1 987 ) equat io n 7 .5) . S e veral th in g s ca n b e learned from th is d e velo pment: • T h e τ- p t ra ns f o rm c a n b e c o m p u t e d f ro m t h e f - k ( F o u ri e r) t ra ns f o rm by a c o o r d in a t e ch an ge fr o m ( f ,k x ) to ( f, p ) f o l lo w ed b y a n i n v er se F o u r ie r t r a n sf o r m f ro m f- >τ . T h i s a m o u n t s to c h a n gi n g t o p o la r c o ord i n a te s in th e F ou ri er d o m ain . • T h e τ - p t r an s f or m m ay e q u i v al e n t ly b e c o m p u t e d b y e q u at i on ( 6 ) wh i c h is a p ro c es s k n ow n a s " sl an t x s t ac ki n g " F o r f ix e d ( p ,τ ) , eq u a t io n 6 r ep r ese n ts a su m m a t io n t h ro ug h the f un c t io n t Ψ ( x ,t ) along a l in e a r t ra j ec to r y . H en c e it i s c a l le d sl a n t st a c k i ng .

Ψ(x,t) Methods of Seismic Data Processing

2 -65

τ-p Transforms • S i n c e t h e a n al yt i c τ- p t r an s f or m i s c o m p u t ab l e f r om the 2 - D F ou r ie r t r an s f or m ( a n d v ic e - v e rs a) t h e i n f om a ti o n c o n t en t i s t h e s am e in e it h e r d o m ai n . T h e f ac t t h a t t h e 2 - D F o u r i er t r an s f or m i s c o m p l et e m ea n s t h at t h e a n a ly t i c τ - p t ra n s fo rm is a ls o. W e w i ll s e e t h a t t h i s i s n o t t ru e fo r t h e d i g it a l τ- p t ra n sf o rm . -kxnyquist

kx

kxnyquist

F

H e re w e s e e an illustration o f th e r ep r e sentation of ( f , kx ) space i n b ot h re ct an gu l a r an d p o lar co ordinates . T h e ra dial l in e s a r e li n e s o f c onstant p an d a r e a ll s h ow n t o t er m ina t e ( w h er e p o ss i bl e ) a t t h e s a m e c onstant f . T o c ompute t h e d i s cr et e τ- p t ransfor m transfo rm , spectra l v alu e s a r e i n t er p olated f r om t h e r ectangular ( f ,k x ) g r id t o re g u larl y sampled f l ocation s on e a c h ra dia l li n e :

2-66


τ-p Transforms 0

p

F A f t e r i nt er p ola t i on o n t o ra di a l li nes , ( f ,k x ) s p ac e b ec om es ( f, p ) s p ac e . A n inverse Fourier t ransform f r om f t o τ complet e s t h e journey t o (τ, p) s pa c e . C l o s e i ns pe c ti on o f t he f i g ur e o n t h e p r e v i ou s p a ge s h ow s w hy t he d i s c r et e τ - p t ra ns f o r m h a s d i f f i c u l t y e v e n t h ou gh t h e a na l yt i c τ - p t r a ns f o rm i s c o mp l et e. I t i s i m po s s i bl e t o p i c k a s e t o f d i s c re t e p v a l ue s w hi c h c ov er t h e ( f , kx ) g r i d u n i f o r ml y . E i t he r t h e y a r e t o o f a r a p ar t a t t h e g r i d e d ge s o r t h e y a r e t o o c r o w de d n e ar t he c e nt er . I n e i t he r c a s e, it c an b e s h ow n t ha t t he r e i s a l w a ys " i nf o rm at i on l os s " i n g o i ng t o t h e d i s c r e t e ( τ , p ) s p ac e a nd b a c k a g ai n . P u t a no t he r w ay , m e r e l y t ra ns f o r mi ng d at a t o ( τ , p) s p ac e a n d b ac k ( w i t ho ut a ny τ - p p ro c e s s i ng ) w i l l a l w a ys a l t e r t h e d at a i n s om e w a y. T hu s t h e d i s c r e t e τ - p t ra ns f o r m i s n o t c o mp l e t e i n t h e s a me s e ns e t ha t t he d i s c r e t e ( f , kx ) t ra ns f o r m i s .


2 -67

Properties and uses of the τ-p Transform T h e m o st o b v io u s p r o p e r t y o f a τ-p t r a ns fo r m i s t h at i t m a ps a lin e a r e v e n t in ( x, t) to a p o in t in (τ- p ).

p

x

το

το τ

Δx

t

po

Δt

Δt –1 = vapp = p o Δx Less obviously, hyperbolae map to ellipses: x

το

p

το τ

t

2-68


Properties and uses of the τ-p Transform T hu s w e c a n e xp e c t t ha t b an dl i m i t e d p r op a g at i n g b o d y w a ve s i n a c on s t a nt v e l oc i t y e ar t h w i l l ma p t o a c om pa c t r e g i o n of (f , p) s p ac e de f ine d b y p max = 1 / v. In t h e f - k t r a ns for m , t hi s c o r r e s po n ds t o a t r i a ng u l ar r e g i o n. kx

p

f=fmax f

f

T hu s w e e x pe c t t ha t a p pa r e nt v e l o c i t y f i l t e r i ng c a n be d on e in e i t h er d o ma i n by e s s e nt i a l l y m ut i ng ( s ur pr e s s i ng ) t h a t p o r t i on o f t he d o ma i n c o r r es p on di ng t o t he u n d e s i ra b l e v el o c i t i e s . A l ia si n g a f fe c t s t h e (τ,p ) t r a n sf o r m m u ch a s it d o e s t he ( f, k ) t r a n sfo rm . I f t h e (τ,p ) t r a n sf o r m i s co n st r u ct e d by s la n t st a ck i ng i n ( x, t) o r ( x ,f ) th e n it i s no t di r e c tl y a f fe c ted b y ho r iz o n ta l a l ia si ng. Bu t t he c ho i ce o f Δp a n d t he n um be r o f p v a lu e s is a d iff ic ul t o ne a nd l e a d s di r e c tl y t o p a li a sin g . A rule of thumb for Δp is

Δp =

1 fmax (xmax – xmin)

≈

Δk x fmax

T uner, G., 1 99 0, Al i asin g i n t he tau- p t ransform and the remov al of spati al al iased c oherent noi se: Geophy sic s, 5 5 , 1 49 6- 1 50 3


2 -69

Properties and uses of the τ-p Transform Other uses of the τ-p Transform: • S in c e a s la n t s ta ck is l es s a ff ec t ed by s p a t ia l a l ia si ng t h a n a n f - k t ra n s fo r m , i t ca n b e u s ed t o i n te rp o l a te t o f i ne r t ra c e sp a c in g s a n d " u n a li a s " d a ta . U se d in t h is f a s h io n it i s o f te n c a ll ed a " s m a rt i n te rp o l a t o r" . (Y ilmaz , O. , 1987, S eism ic Dat a P roces sing , p435. )

• I t c an b e s ho w n th a t m u lt ip le s a r e n o t p e r i o di c o n a n of fse t tr a c e i n th e (x ,t ) d o ma i n b u t a r e in t he (τ, p) . (T rei tel et al. , 1982, Pl ane- wav e dec ompo si ti on of sei sm ogram s, Geophysi c s,

T h is m e a ns th a t p r e d ict iv e de c o nv o lu ti o n fo r mu lt ip le su r p r e ss io n o f te n w o r k s b e t te r o n (τ ,p ) g a th e r s. 47, 1375- 1401)

• M i g r at i on c an a ls o b e d on e in t h e (τ, p ) d o m ai n .

(Die bol d, J.B. , a nd Stoffa , P.L., 198 1, The tr av el time e qua ti ons, ta u-p ma ppi ng, a nd inve r sion of common midpoi nt da ta , Geophysi cs, 46 , 23 8-2 54 )

2-70


Inverse τ-p Transforms T h e p r o c es s o f re co n st r uc t i o n o f t h e s ei sm i c d a t a i n ( x , t ) s p a c e gi v en it s τ- p t ra n s fo rm is ca l l ed a n in v er se τ- p t ra n s f o rm . T h er e a r e a n u m be r o f wa ys t o d o th i s p r o c es s t h o u gh we sh a l l d i s cu s s o n l y t wo : F o u ri er m et h o d s a nd f il t er ed ba c k p ro j e c ti o n . T he F o u ri er m et h o d i s o bv io u s f ro m t h e d is c u ss io n o f t h e f o rw a rd τ - p t r a n s fo rm . T he m a jo r s t ep i s t h e r e c o n st ru ct io n o f t h e 2 - D ( f, kx ) t ra n sf o rm w h i ch r e q u ir e s a n i nt er po l a t io n o nt o a r ec t a n gu la r g ri d f r o m a p o l a r o n e . T h is w i ll o bv io u s ly h a v e n u m er i ca l d if fi cu lt ie s t h o u gh t he y a re c o n t ro ll a bl e. F o ll o w in g the i n te rp o la t io n , an i n v er s e 2- D F o ur ie r t ra n sf o rm c o m p le te s t h e p ro c es s. F il t e r e d b a ck p r oj e ct io n a vo id s t h e ( f, k x ) d o ma in a n d r e c o ns t r uc ts t h e i m a g e d ir e ct ly w it h a c o nv o lu ti o na l f i lt e r f o ll o w e d b y a n i n ve r s e s la n t s ta ck . C o n s id e r t he e x pr es s io n f o r t h e i n v e r s e 2 -D F o u r ie r t r a n s fo r m :

Ψ(x,t) =

∞ –∞

φ(k x,f )e

–2π i (k xx – f t)

dk x df

(1)

N ow , letting k x = f p and converting t h e wavenumbe r integral i n to a p integral g ives :

Ψ(x,t) =

∞ –∞

f φ(p,f )e


–2π i f p x

e

2π i f t

dp df

(2)

2 -71

Inverse τ-p Transforms T h e t er m i n b ra c ke ts c a n be c o n si d e re d t o b e th e p r o d u c t o f t w o f u n ct i o n s o f f . H e n ce , it m u st b e a c o n v o l u t io n in t im e : ∞

Ψ(x,t) =

–∞

α(t)•β(p,x,t) dp

(3)

where • denotes a convolution over time and

α(t) = β(p,x,t) =

∞ –∞

∞

2π i f t

fe

df

(4)

–∞

φ(p,f )e

2 π i f (t –p x)

df = ϕ(p,t –px)

(5)

N o te t ha t ϕ (p ,τ) i s th e fo r w a r d s la n t st ac k . S ub sti tu tio n o f ( 5) in t o (3 ) r e su lts in :

Ψ(x,t) = α(t)•

∞ –∞

ϕ(p,t – px) dp

(6)

E q u at io n ( 6) e x pr e sse s fi lte r e d b a ck pr o j e c tio n f r o m th e (τ, p) d o ma in t o th e (x ,t ) d o m a in . E a ch p o int in (x ,t ) i s co n st r u cte d b y i nt e g r a ti ng a lo n g a lin e a r t r a j e ct o r y in (τ, p) , j u st lik e t h e fo r w a r d s la nt st a ck . U nl ik e t he fo r w a r d o p e r a ti o n, th e in te g r a ti on i s fo l lo w e d b y a c o nv o lu tio n w h ich is a fo r m o f a f ilt e r .

2-72


Inverse τ-p Transforms A n ot her w ay t o d o t h e i n v e r s e t r an s fo r m i s s u g ge s t e d b y e q u a t i on 2 . Ra t h e r t h an c on v o l v e i n t h e t i me d om ai n w i t h a fi l t e r o p e r at or w e c a n d o t h e r e c on s t r u c t i on i n t he f d o m a i n . Ta ki n g a fo r wa rd Fo u ri e r t r an s f or m ( t -> f ) o f ( 2 ) g i v e s:

F Ψ(x,t) = ψ(x,f) = f

∞ –∞

–2π i f p x

φ(p,f )e

dp

(7)

T o us e (7 ) f o r t he i n ve r s i o n, w e f i r s t t r an s f o rm ϕ (p,τ) t o φ( p, f ) . T he n, f o r e a c h x, w e m ul t i p l y φ(p , f ) b y a p d e pe n de nt p ha s e s hi f t an d i nt e gr a t e o ve r p a nd t he r e s ul t i s s c a l e d b y f . A f t e r c o ns t r u c t i n g ψ (x, f ), an i n ve r s e F o ur i e r t r an s f o r m f r o m f - > t c o mp l e t e s t h e p r oc e s s .


2 -73

Least Squares τ-p and f-k Transforms W e h av e s e e n t hat a c o n v e n i e n t m et hod of i m p l e m e n t i n g f or w ar d a n d i n v e r s e τ-p t ra n s f or m s i s i n t he f r e q u e n c y d o m ai n : ∞

φ(p,f ) =

ψ(x,f) = f

ψ(x,f)e

dx

φ(p,f )e

dp

2π i f p x

–∞ ∞ –∞

–2π i f p x

(1)

(2)

H e r e (1 ) is th e fo r w a r d t r a ns fo r m f r o m ( x,f ) to (p ,f ) a nd ( 2) is t he in ve rse t r a ns fo r m . V ir tu a ll y a ny in te g rat io n c an b e i mp le me n t e d a s a n e q ui va le nt ma t r ix op e r a t io n fo r d isc r e te da t a . C or r e sp o n di ng t o (1 ) a nd ( 2) w e h a ve :

φj (f) = Σ Rj kψ k(f)

Rj k = exp(2π i f p j x k)

ψ k (f) = fΣ R k j φj (f)

Rk j = exp(– 2π i f p j x k)

k

*

j

*

(3)

(4)

These can be written:

φ= Rψ

(5)

ψ = fR φ

(6)

*T

φ1 (f) φ2 (f) φ= φ3 (f)

2-74

ψ 1(f) ψ 2(f) ψ= ψ 3(f)

R1 1 R1 2 R1 3 R2 1 R2 2 R23 R= R3 1 R3 2 R3 3

etc


Least Squares τ-p and f-k Transforms R at h e r t ha n c o mp ut e t h e f or w a rd t r an s f or m d i r e c t l y , t he l e a s t s qu ar e s t e c hn i qu e u s e s e q ua t i on ( 6 ) t o p os e a n i n v e rs e p r o b l e m f or t he τ - p s pe c t r um .

ψ = fR φ *T

(6)

R f ψ = RR –1

φ = RR

*T

*T

–1

φ

Rf ψ –1

(7)

E qu a ti o n (7 ) is t he st a nd a r d le a st sq u a r e s e sti m at e o f th e τ- p s pe ctr um . I t is u su a lly su p e r io r in th e se n se t ha t th e (x ,t ) d o ma in d at a ca n b e r eco n str u c te d fr o m it w it h fe w e r a r t ifa c ts . T hi s fo r m ul a tio n a ssu m e s th a t t he nu m b e r o f p t r a ce s e x ti ma t e d w il l b e n o la r ge r t ha n t he nu m b e r o f x t r a ce s. E ve n w he n th e d a ta is p e r fe c tl y r e g u la r in x a n d th e nu m b e r o f p a nd x tr a c e s ar e t he sa me , th e le as t sq u a r e s m e t ho d is u su a lly s up e r io r b e ca u se th e τ-p t r a n sfo r m i s i nc o mp le t e . T hi s m e a n s t ha t th e fo r w a r d a n d r e v e r se τ- p pr o c e ss e s le a ve a r t ifa c ts in th e d a ta . T he le a st sq u ar e s a p p r o a ch m in im iz e s su ch ar tif a cts .


2 -75

Least Squares τ-p and f-k Transforms T h e m o re i n co m p le te a n d i n co n si st e n t a tr a n s fo rm p a i r a r e, t h e m o r e t h e le a s t s q u a re a pp ro a c h be co m es u s ef u l . T h i s m ea n s i t is e sp e ci a l ly p re f err ed fo r s la nt s t a c ks a l o n g p a r a bo li c a n d h y p er bo li c t ra j ec t o ri es w h ic h a re i n c o m p l et e ev e n i n t h e a n a l y t ic s en s e . A n o t h e r e x a m pl e o f a n i n c om p l e t e t ra n s f or m i s t h e d i s c r e t e F ou r i e r t r an s f or m f or i rr e g u l ar l y s am p l e d d a t a . I t c a n a l s o b e p o s e d a s a n i n v e r s e p r o bl e m :

ψ=F

*T

ϕ = FF

ϕ *T

–1

Fψ

where

F1 1 F1 2 F1 3 F F F F = 2 1 2 2 23 F3 1 F3 2 F3 3

S e e M a r f ur t,

e t a l. ,

( 199 6,

Fm n = exp(2π i k x mxn )

P i tf al ls

of

u si ng

co n v e nt io n a l R a do n

t r a nsf o r ms on p o o rl y s a mpl ed d at a: G e oph y sic s, 6 1, 1 46 7- 148 2)

f or a

m o r e c o mp le t e d i s cu s s io n .

2-76


Methods of Seismic Data Processing Lecture Notes Geophysics 557 Chapter 3 Amplitude Effects


3 -1

Seismic Wave Attenuation As a seismic wave propagates through the earth, it suffers attenuation (amplitude decay) for a number of reasons: Attenuation Mechanism #1: Geometric Spreading (or Spherical Divergence) A s s ei s m ic e n er gy p ro p a ga t e s a w a y f ro m a s o u r ce ( o r f o c a l p o i n t ) th e co n se rv a t i o n o f to t a l en e rg y re q u ir es t ha t t h e e n er gy Surface A2 f o u n d o n t h e wa v e fr o n t s u rf a c e A 1 a t s o m e t im e t 1 e q u a l th at o n su r fa c e A 2 a t so m e ti m e t 2 .

E t 2 = ε t 2 A 2 = E t 1 = ε t1 A 1

Surface A1

w h e re ε i s t h e e n e rg y p e r u n i t ar e a. Si n c e t h e d i s p l ac e m e n t w av e am p l i t u de, u, is p r op or t i on a l t o t h e s q u a r e r oo t o f ε , w e d e d uce :

u2 = u1

A1 R1 t1 = = A2 R 2 t 2

or

ut =

u0 Rt

T h e p r o p er i n t erp r et a ti o n o f th i s r esu l t is t h a t th e w a v e a m p l it u d e d ec a y s a s 1 / R w h er e R i s t h e r a d iu s o f c u rv a tu r e o f t h e w a v e fr o n t. I n t h e c a se o f a co ns t a n t v el o ci t y m e d iu m , R i s si m p ly th e d is t a n ce t ra v el le d ; h o w e v er, i n a l a y er ed m e d iu m , R c a n b e s h o w n t o b e p ro p ort io n a l to ( V2 r ms / V 1 ) t w h e re V 1 i s t h e v elo ci t y o f t h e f irs t la yer. ( N ewm an, Geo phys ics, 1971, p 481-488, Hu br al, P ., and Krey, T., In te rv al V elocit ies f rom Seism ic Refle ct ion Time Meas ure men ts , 1980, So ciety o f Ex plor atio n G eoph ysicist s)

3-2

Amplitude Effects

Seismic Wave Attenuation Th u s we deduce t h a t t h e e f fe c t s of spheri c a l diverg en c e c a n b e a p prox i m ate l y c ompensated f or by applyi n g a " gain c or re c ti o n " g iv e n b y:

2

G t spreading = G0Vrms t t

(Compare with Hatton et al., page 56)

Attenuation Mechanism #2: Absorption (or inelastic attenuation) I n a p e rf e ct l y el as t i c m ed iu m, t h e t ot a l en e rg y o f t h e p r op a ga t in g w av e fi e l d re m a in s a c on s t an t . H o we v er , t h e e ar th is n o t p er f ec t l y e l as t ic a n d p r op ag at i n g s ei s m i c w av e s g ra d u al l y d i e o u t o v e r t i m e . T h e p ri m ar y m e c h an i s m f or t h i s i s th e c o n t in uo u s c on v e r si on o f a s m al l p o r ti o n o f t h e s e is m i c en e r gy t o h e at d u e t o i rr e v er s ib l e a n el as t i c b e h av i or o f ro c ks . I t is c u s t om ar y t o t al k a b ou t t h e p a ra m et e r Q wh i c h c h ar ac t er i ze s t h i s e n er g y l o ss :

Q =

energy energy loss

per frequency cycle

V ar i o u s at t e nu at i on t h eo r i e s e x i s t w i t h t h e s i mp l e s t b e i n g t h e "c o n s t an t Q " t h e or y o f K j ar t a ns son 1 a nd o t he r s . M os t e mp i r i c a l e v i de n c e i s c o ns i s te nt w i t h t h e a s s um pt i on t ha t Q i s i n de p e nd e nt of f r e qu e nc y at l e a s t o ve r t he s e i s m i c b a ndw i d t h . T he c on s t a nt Q t h e o ri es a l l p re d i c t an a mp l i t ud e l os s g i v e n b y :

1 Kjar tan ss o n, E, 1 979, C o ns tan t Q-Wave P ro pa ga tion an d A tten u atio n, JGR , V 84, p4 737-474 8


3 -3

Seismic Wave Attenuation Thus, the constant Q theory refers to a Q which is independent of frequency but predicts an attenuation which is a first order exponential in both time and frequency. f

t

Constant Q Exponential Decay Surface N o t e t h a t Q = ∞ i s a p e rf e c t ly e la st i c m a te r ia l wh i l e Q = 0 i s p e rf e c t ly a b s or p t iv e . A h i gh l y a b so rp t iv e r oc k h as a Q o f 2 0 - 5 0 w h i l e v e ry co m p e t en t l i m e st o n es a n d d o lo m i t es c an h av e Q o f 2 0 0 o r m o re . T h e c om m on fi r s t o r d e r c or re c t io n f o r Q e ff e c t s i s t o a p p l y a s i m p l e , f re q u e n c y i n d e p en d en t , e xp o n e n t i al g ai n c or re c t i on . I f w e wr i t e:

e

– πft/Q

≈ e

–α t

; α=πfdom/Q

= the attenuation constant

Typica l ly , th e a t te n uation constant i s ex p res se d i n db/se c which w oul d be

Assuming f do m of 2 0 a n d a v a l u e o f 1 2 d b /s e c . 3-4

Q o f 1 0 0 l e ad s to a "t y pi c al "

Amplitude Effects

Seismic Wave Attenuation Attenuation Mechanism #3: Transmission losses I n o u r e x a m in a ti o n o f t he t h eo r y o f t h e 1 - D s y n th et ic s ei sm o gr a m w e s a w t h a t t h er e i s a c o n t in u o us a m p l it ud e d ec a y d u e t o t ra n sm i ss io n l o ss es . I f f a c t, w e f o u nd t h a t t h e e a rt h ' s i m p ul se r e sp o n se r e su lt ed i n t h e r ec o r d in g o f t h e n 't h r e f le c t io n c o e ff ic ie n t a t t h e s u rf a c e m u lt i pl ie d b y a t ra n sm i ss io n l o ss t e rm : nth reflection coefficient recorded at surface

=

Rn (Transmission losses) n–1

where transmission losses

=

2

k = 1

1–Rk

T hi s e f f e c t i s h i g h l y d ep e nd e n t u p o n l o c a l g eo l o gy a nd i s d i f f i c ul t t o e s t i m at e w i t h a n y p r e c i s i on . I t i s c u s t om ar y t o i g no re it a n d " ho pe " t ha t i t i s e i t h er s m al l o r i nc l ud e d i n t he " db / s ec " c o r re c tio ns a l r e ad y d i s c us s e d. Attenuation Mechanism #4: Mode Conversion A s wa v es p ro p ag at e i n a n e la st i c m ed i u m, t h ey a re c on s t an t l y be i n g c o n v er t e d f ro m P t o S a n d th e r ev e r se a t ev e r y i m ped a n c e c on t r as t . T h e s e m od e c on v e rs i on s o c c u r bo th u po n re f le c t i on a n d t r an s m i s si on a n d a re d e s cr i be d b y t h e f am o u s Zo ep pri t z e q u at i on s ( s e e A ki a n d R i c h ar d s , 1 98 0 , o r t h e C RE W E S Zo ep pri t z e x p l or er a t w ww . cr e we s .o rg ) . I f , a s i n c on v e n t i on al s ei s m i c , o n l y t h e v e r ti c al c om p o n e n t o f g ro u n d m ot i on i s r ec o rd e d , t h e n i t i s r ar el y p o ss i b le t o a d d re s s t h i s e ff e c t . T h e s ol u t i on i s t o r e co rd a l l t h r ee c om p o n en ts o f g ro u n d m o t io n a n d p r oc e ss t h e d at a a s e la st i c wa v es . T h i s i s t h e s u b j ec t o f l e ad i n g e d g e re s e arc h a r ou n d t h e w or ld . Methods of Seismic Data Processing

3 -5

Seismic Wave Attenuation Attenuation Mechanism #5: Scattering R a n do m s c a tt e r in g o ff s m a ll i r r e g ul ar i ti e s c au s e s t he d is p e r s a l o f s e is m ic w a v e fi e l ds a n d a n a pp a r e nt l o s s o f e n e r g y . I f a f u ll 3 -D w a ve fi e ld h a s b e e n r e co r d e d t h e n s u c h s ca t te r e r s c an b e i m a g e d b y m ig r a ti o n b ut t h e l o s t w a v e fi e ld e n e r g y i s n ot r e s t or ed .

Attenuation Mechanism #6: Refractions and critical angles S n el l' s l a w go v er n 's t h e a n gl es o f ref le ct i o n a n d re fr a ct i o n w h en a w a v e in t era c t s w it h a n i m p ed a n c e c o n tr a st .

θ

θ

sin θ

φ

v1

=

sin φ v2

I n t h e n o r m al c as e wh er e v 2 > v 1 , t h e re ex is t s a "c r i t i c al a n g le " o f i n c i d e n c e b e yo n d wh ic h n o t r a n s m i s si o n o c c u r s .

sin θ crit =

3-6

v1 v2

Amplitude Effects

Seismic Wave Attenuation En e rgy in c id en t a t o r be y o n d t h e cr it i ca l a n gl e i s t h ro w n ba c k to th e su rf a c e a s p o s t- c ri t ic a l r ef le ct i o n s a n d re fr a ct i o n s. I t is n o t a v a i la b le t o i ll um i n a t e d ee pe r re fl ec t o rs . T h i s i s e sp e ci a ll y n o t ic a b le in t h e n ea r su r fa c e w h e re th e v el o c it y c o n t ra s t a t t h e b a se o f t h e w e a t he ri n g ca n a p p ro a ch 1 / 2 o r l es s. S in c e t h e a rc si n o f 1 / 2 i s 3 0 d eg ree s, t h is m ea n s o n l y a n a rr o w co ne o f en e rgy p en et r a te s to th e su b su rf a c e.

Surface source

Base of weathering

Transmitted cone


3 -7

True Amplitude Processing W e h a v e s ee n t h a t a s im p l e m o d el f o r a 1 - D s e is m o g ra m p re d i ct s th at t h e se is m i c d a t a c o n s i st s o f b a n d - l im i t ed re f le ct i o n c o e f ic i en t s:

s t = w t •r t W he r e w is a se is mi c w av e le t, r is th e e ar th 's r e fle cti vi ty e x pr e sse d a s a fu n ct io n o f 2- w a y ve rtic a l t r a ve lti me , s i s th e se i sm o g r a m , an d • d e n ot e s c on v o lut io n . S in ce w g e n e r a ll y co n ta in s sig n if ica n t e ne rg y o nl y o ve r so m e ch a r a ct e r is tic fr equ e n cy b a n d w id th , i f w e vie w th e co n v ol ut io n a s a m ul tip li ca ti o n in th e fr e q u e nc y do m a in , w e se e th a t s i s i nd e e d a b a n dl im ite d ve rsio n o f r . (I f w i s n o t z e r o p h as e th e n th e r e is a p ha s e sh ift a s w e l l.) T h er e a r e m a n y r ea l ea r th w a v e p ro pa ga t io n e ff ec t s w h i ch ca us e t he r a w se is m ic d a t a t o d e vi a t e co n s id e ra b ly f ro m t h is m o d el . T r u e a m p l it u d e p ro c e ss in g i s a " ho ly gr a il " o f t he s ei sm i c d a t a p ro c es si n g w o r ld a n d re fe rs t o a p ro ce ss in g seq u e n ce w h ic h , w h en c o m p le te , y i el d s d a t a w h i ch is a cc u ra t el y re p res en t a bl e a s b a n d li m it ed re fl ec t io n c o e ff ic ie n ts . W h il e n o t y et s tr ic t l y p o ss ib l e, m a n y d a ta p ro ce ss in g f l o w s c o m e q u it e c l o se t o b e i ng t ru e a m p li tu d e p ro c es si n g. G e n er a l ly , t ho u g h n o t e x cl u si v el y , t hi s m ea n s the a vo i d a n ce of s t a ti st ic a l a m p li tu d e c o rr e c t io n s like AGC in fa v o r of d et er mi n is ti c c o rr e c t io n s l i k e s p h er i ca l d i ve rg e nc e a nd e x p o n en ti a l ga i n . I t i s n ot u n c om m on t o f i n d m od e rn p r oc e s se d s ei s m i c d at a wh i c h i s r ou g h ly p r op o rt i on a l t o w el l l og d e ri v ed r ef l ec t i on c oe f fi c i en ts o v e r l im it e d ti m e z o n e s . 3-8

Amplitude Effects

Automatic Gain Correction (AGC) A u t om a t ic g ai n c or re c t io n m e t h od s a t t e m p t t o p e rf o rm n e c e ss ar y a m pli t u d e a d j u s t m e n t s t o se i sm ic d at a b as e d p u r el y o n s t at i st i c s o f t h e o b se r v ed a m p l i t u d e d e c ay . T h e y s h ou l d be c o n t ra st e d wi t h d et e rm in i s t i c m et h o d s w h ic h u s e a p h ys i c al m o d e l o f o n e o r m or e d e c ay p ro c es s es t o d et e rm in e c or re c t i on f ac t or s . G en e r al ly , A G C m e t h od s a r e si m ple a n d e f fi c i en t bu t t e n d t o p ro d u c e u n p h y si c al a m p l i t u d e d i s t or t io n s . T h e y a re u s ef u l i n s i t u at i on s w h er e p h ys i c al l y m e an i n g f u l a m p l i tu d es a re l es s i m p o rt a n t t h an " we l l ba la n c ed t r ac e s. T h er e a r e m an y A G C a l g or it h ms in c om m o n u s e . A s i m p l e , ef f e c ti v e m et h o d i n v ol v e s t h e d e f in it i on o f a t e m p o ra l w in do w s i ze a n d t h e m e as u re m e n t o f t h e t r ac e rm s a m p l it u de o v er t h at w i n d ow . T h e wi n d o w i s t h e n i n c re m e n t e d a n d th e m ea su r m e n t re p e at e d . T h e r e su l t i s a se t o f rm s a m p l i t u d e m ea s u re m e n t s a t d i s c re t e t i m es wh i c h d ef i n es a n ' am p l it u d e m o d e l ' o f t h e t ra c e. T h i s m od el i s t h e n l i n ea rl y i n t e rp o l at e d t o t h e t r ac e sa m p l e r at e a n d t h e A GC ' d t r ac e i s c o m p u t e d b y d i v i d in g t h e o r i gi n al tr ac e b y t h e a m p l it u d e m od e l .

r c s w it h t h eo r et i c a l a mp l it u d e d e c ay

r e f l ec t i o n c o ef fi c i e nt s


3 -9

Automatic Gain Correction (AGC)

i n t er p o l a t ed r m s a m p l i tu d e m o d e l

d i s c re t e r m s m e a s u r e s

A b ov e i s t h e co n s t ru c t i on o f a n r m s a m p l i t u d e m o d e l f ro m m ea su r e s ev e r y . 1 s e c on d s a n d t h e n i n t e rp o l at e d . B e l ow i s t h e a p p l ic a ti o n o f t h at m od e l t o t h e t r ac e .

A G C ' d r e s u l t. T r a c e d i vi de d a m p l i tu de m o de l .

r m s a m p l it ud e m o d e l

T r a c e s h ow i ng a m p l it ud e d e c ay

3-10

Amplitude Effects

Automatic Gain Correction (AGC) B e l ow i s a c om p ar i so n o f a d e t er m i n i s t ic a m p l i t u d e r es t or at i on a n d s ev e ra l d if f e re n t A G C p ro d u c t s. N ot e t h at t h e re l at i v e e v e n t 's t an do u t ' ( t h e a m p l i t u d e r at i o b et w ee n a n y t wo e v e n t s) i s be s t p re s er v ed b y d e t er m i n i st i c m e t h od s a n d s ec o n d ar il y by lo n g A G C o p er at or s .

AGC .4 operator.

sec

AGC .1 operator.

sec

AGC .0 2 5 operator.

sec

D e t er m in i st ic g a i n S y nt he t i c w it h t - 1 a m p l i tu d e d e c a y S y nt h e t ic w i th n o a m p l i t ud e l os s e s

T h e t wo m os t c om m o n m i st a ke s w i th A GC a r e t o u se i t e x cl u s i v l y fo r a l l g ai n a d j u s t m e n t s o r t o a v oi d i t e n t ir e ly . I n t h e f i r st c as e , A GC s h ou l d be u s ed w i th c au t io n if t h e i n t en ded i n t e rp r e t at i on m e t h od p l ac e s e m p ha s is o n r el i ab le a m p l it u d e i n f or m at i on . I n th e s e c on d c as e , A G C o f t e n l e ad s t o su per i or r es i d u al st a t ic s a n d v el o ci t y a n a ly s es si d e f l ow s e v en w h en n e v er u s e d i n a m ai n fl ow .


3 -11

Automatic Gain Correction (AGC) A co m p a ri s on o f A G C o p e ra t or l e n gt h s o n a r ea l , r aw , s e is m i c tr ac e s h o ws h o w t h e c h o ic e o f o p er at or l e n g t h c a n d r as t ic a ll y a f f ec t t h e e v en t c h ar ac t er . A l s o a p p ar e n t is t h e e f fe c t kn o wn a s a n A G C sh a d ow z o n e . T h i s o c c u r s wh en a p ac ka ge o f e n e rg y ( i n t h e c as e t h e f r is t br ea ks ) h as m u ch h i g h er a m p l i tu d e t h an a d j a c en t e v e n t s. T h e a d j a ce n t e n e rg y t e n d s t o h av e a s u r p re s s ed a m p l i t u d e o v er r ou g h ly t h e l e n g th o f t h e A GC o p e r at or . A G C " s h a d ow " z on e s

2. 0 se c A GC 1 .0 se c A GC .5 se c A GC . 2 5 se c A GC R a w t ra ce

A m aj o r c on ce r n wh e n u s in g a n A GC i s th a t se r io u s d i s t or t i on s i n t h e e m be d d e d wa v e l et ca n o cc u r i f t h e A GC o p e ra t or l e n gt h i s s h or t er t h a n t h e s ou rc e w av e f or m . T h i s c a n re s u lt i n a s t ro n g d eg r ad at i on o f t h e p er f or m an c e o f d e co n v ol u t i on a l go ri t h m s . T h i s wi l l b e co m e m o r e c le ar a f t er t h e re ad e r h a s s t u d i ed d e c on v o l u t io n i n t h e n e x t c h ap t e r . 3-12

Amplitude Effects

Trace Equalization (TE) or Trace Balancing T r ac e s fr om r aw f i el d r ec o rd s c an o ft e n h av e w il d l y v ar yi n g t o ta l ( r m s ) p o we r l ev e l s. T h e re a re m a n y p o ss i bl e c au s es i n c l u d i n g: s h ot s t re n g t h v ari a ti o n , g eo p h on e c ou p li n g v ar i t io n , n ea r s u r f ac e ge ol og y c h an g e s, so u rc e - r ec e i v er o ff s e t , a n d m o re . .. E v en i n c a se s wh e r e d et e rm i n i s t i c g ai n i s p r ef e rr e d , s om e s or t o f t r ac e b al an c i n g sh o u l d s t il l b e p er f or m e d . O t h e rw is e , h ig h r m s p ow er t ra c es ( w h i ch a re o f t e n t h e n oi s i es t tr ac e s ) , w il l d o m i n at e i n st a c ki n g a n d c r os sc or re l at i on s . A s im p l e m e th o d is ca l le d t ra c e e q u a li za t io n , o r T E , a n d i s u s u a ll y s y n on o m ou s wi t h t r ac e b al an c i n g . T E i s a v er y s i m p l e p r oc e s s i n w h i ch a l l t r ac e s a re a d ju st e d t o h a v e t h e s am e rm s p ow er l e v el a c c or d i n g t o: o u t pu t t r ac e = i n p u t t ra c e/ ( rm s p ow e r o f i n p u t t ra c e) A c om mo n v ar ia n t o f T E i s t o c om p u t e t h e r m s p o we r o v e r a p ar t i c u la r t im e z o n e i n st e ad o f t h e en t i r e t ra c e. I f t h e t i m e z o n e v ar i es in w id t h , t h en c ar e m u s t b e t ak en t o n or m al i ze t h e rm s p ow er m ea su r e s f or t h i s e f fe c t . C au t i on sh o u l d a l w ay s b e ex e r ci s e d wh en i n t e rp re t in g se i s m i c p l ot s wh e r e a t ra ce e q u al i za t io n o r A G C h as be e n a p p l ie d a s a n o p t i on i n p l ot t i n g . W h i l e t h is m ay b e a c on v en i e n c e i t m e an s t h a t t h e d a t a d i s p l ay m ay n ot t ru l y re p r es e n t t h e d a ta a s s t o re d o n d i s k o r t ap e . F o r ex a m p l e, d at a th a t i s w il d l y u n b al an c e d f ro m t r ac e - t ot ra ce m a y a p p e ar t o h a v e g oo d a m p l it u de v a ri at i on , le ad in g to e r ro n eo u s p r oc e ss i n g d e c i si o n s. Methods of Seismic Data Processing

3 -13

Constant Q Effects S t r i c tl y s p e ak i n g, c on s t an t Q t h e or y r ef e rs t o a Q w h i ch i s i n d e p en den t o f f r eq uen c y b u t m a y s t i l l d e p en d o n t i m e . F o r s i m p l i ci t l y, we wi l l a ls o a ss u me time i n d e p en d en c e . N ot e t h at t h e a t t en u a t io n c an be w ri t t en as :

–πft/Q

exp

= exp

= exp

–πfx/(vQ)

= exp

–πx/(λQ)

–πn λ/Q

w h e r e w e h a ve us e d λ f= v a n d n λ = x/λ i s t he n um b e r o f w a v e le n g th s th a t fit i n th e di sta n ce tr av e le d. T h us , a s a w a v e fo r m p r o p ag a te s, it is co n ta n tl y b e in g at te nu a te d w it h th e hig he r fr equ e n cie s b e in g a tt e n ua t e d fa s te r . I f W (f ) is t he sp e c tr u m o f o u r s o ur c e w av e f or m, w ( τ), t he n a ft e r p r o pa g a t in g a tim e t , t he a m p lit ud e sp e c tr u m o f th e p r o p a g a tin g w a v e fo r m h a s b e c o m e :

Wp f

= W f exp –πft/Q

If w e a ss um e Q=50, a n d a s pecific s h a pe f o r |W (f )| , t h e n w e c a n c ompu te th e ampl it u de s p ec trum o f t h e pr o pagating w aveform at any t im e: 0

Wf

-50 .5 sec -100 1.0 sec -150 1.5 sec -200

-2500 3-14

2.0 sec 50


200

250

Amplitude Effects

Constant Q Effects T hu s w e se e t ha t s e i s mi c d at a m u s t a c t ua l l y c o nt a i n a w av el e t w i t h c o nt i nu ou s l y d e c r e a s i ng b a nd wi d t h. T h i s m e a ns t h e d a t a s i gn al s p e c t r um i s a c t ua ll y a f un c t i o n o f t i m e a n d i s s a i d t o b e n on s t at i o na r y ( o r, e q ui v a l e nt l y , t i m e- v a ri a nt ) . D e pe nd i ng u p on t he v a l ue t a k en t o c ha r ac ter i z e t h e b ac k g r ou n d n o i s e , we o b t ai n t he s e s pe c i f i c m ax i m um s i g n al f r e qu e n c y e s t i m at e s ( b a s e d o n t he p r e c e di n g g r a ph ) : time

.5 sec

1.0 sec

1.5 sec

2.0 sec

100 db down

180 Hz

120 Hz

80 Hz

70 Hz

75 db down

130 Hz

80 Hz

70 Hz

55 Hz

50 db down

80Hz

60 Hz

45 Hz

40 Hz

25 db down

45 Hz

35 Hz

30 Hz

25 Hz

noise

Table showing predicted signal band for Q=50

Here is a repeat of the Q analysis for Q=100. 0

Wf

-50

.5 sec 1.0 sec

-100

1.5 sec -150

2.0 sec

-200

50 time

100

150 frequency (Hz)

200

.5 sec

1.0 sec

1.5 sec

2.0 sec

100 db down

+200 Hz

185 Hz

140 Hz

120 Hz

75 db down

180 Hz

130 Hz

105 Hz

80 Hz

50 db down

110 Hz

80 Hz

70 Hz

60 Hz

25 db down

60 Hz

45 Hz

40 Hz

35 Hz

noise


250

Ta ble sh o wi ng p re d ic t ed sign al ban d fo r Q= 1 0 0

3 -15

Constant Q Effects Thus fa r we have di scu ssed th e e ffect s o f a ttenuatio n o n th e ampl it ud e spectrum o f t he propagati ng waveform but th e p hase effects are a ls o dramatic. Consider a 1-d earth with constant Q properties: x in

A 1-d attenuating earth: {output spectrum} = {input spectrum}*exp(-pi*f*x/(v*Q))

out

• An input impulse suffers attenuation at all non-zero frequencies • The amount of attenuation is proportional to x/v = t • T h e a t te nuation is n e ce ssaril y couple d w i th m in i mu m phase d i sp er si o n ( Futte rman, W.I ., 19 62, Dispe rsi ve Body W av es, JGR, 67 , 52 79 -5 29 1) I f th e e a r t h b eha v e s lin e a r l y, th e n w e ca n s til l a r g u e th a t su p e r p o sit io n h o ld s. T h us t h e i mp u lse r esp o ns e o f an e a r t h w it h r e f le c tiv it y { r } is t he s up e r p o sit io n of a se t o f d e la y e d a nd p r o g r e s siv ly m o r e a t te n ua t e d w a ve f o r m s:

in A three reflector earth

out

T hr e e s u pe r i mp o s e d w av e f or ms a t t e nu a t i o n wi th i nc r e as i n g t i m e . 3-16

s h ow i ng

i nc r e a s i ng

Amplitude Effects

Constant Q Effects Matrix Model of the Linear Attenuating Earth

N on-stationary Q impulse response matrix

Stationary Earth Response

Impulse response of a constant Q Earth

T he c on s t r uc ti on o f a n on s t a t i o na r y mu l t i p l e f r e e s y nt h e t i c s e i smo g r am i s s h ow n f o r a c on s t a nt Q e ar t h ha vi n g 3 r e f l e c t o r s . Th e ma t r i x m ul t i p l i c a t i o n s h ow n he r e i s p e rf or m i ng a c o nvo l ut i on as d e s c r i b e d o n pa g e 2 -1 1. T he c o nvo l ut i on ha s b e e n m ad e n on s t a t i o na r y b y c h an g i ng t h e w a ve l e t i n e ac h c o l um n o f t h e c on vo l ut i on m at r i x . Methods of Seismic Data Processing

3 -17

Minimum Phase, Intuitively I nfinitely m a n y w avelets c a n be c ons t ruc t e d whic h hav e th e s ame a m pl i t ude s pectrum b y making different assumptions a bo u t phas e .

0

-40

-80 0


-0.1

0

0.1

s ec o n d s

0.2

H o we v er , o n l y a f e w o f th es e h av e a n y p r ac t i ca l u s e. T h e m i n i m u m p ha s e w av e l et i s d i st i n g u i sh e d f r om a ll o t h e r s b y be i n g t h e m o st f ro n t- lo ad e d o f t h e " c au s al " wa v e l et s .

3-18

Amplitude Effects

Minimum Phase, Intuitively

in

Linear, causal, attenuating earth

out

t=0

M in i m u m p h a se w a v el et s a r is e n a t u ra l l y i n t h e ea rt h . O nl y t h e a ss u m p t io n s o f ca u s a l it y a n d l in e a ri t y a r e n ee d ed t o s h o w th a t a tt e n u a t io n in th e e a rt h i n a m in i m u m p h a s e p ro ce s s. ( F utte r ma n, 1 962, JG R vol 73 , p 3 917- 393 5) T h e a m p l i t u d e s p e ct r u m a l on e i s s u f f ic i e n t t o d e t er m i n e u n i q u e l y t h e m i n i m u m p h as e w av e le t . T h e p h as e s p e ct r u m , φ ( f ) , m ay b e c om pu t ed a s : φ f = H ln A f where H denotes the Hilbert Transform.


3 -19

Minimum Phase, Intuitively I t is a co m m o n m ist a ke t o th in k t ha t " m in im um ph a se " r e f e r s to a p ar tic ul ar p h a se sp e ct r u m w h ic h, i f pr e se r v e d , m a in ta in s a d a ta se t 's "m in im u m p h a se n e ss ". W e ha v e j us t se en t h at t his is no t th e c a se . In st e a d , m in im um p h a se r e f e r s t o a p a r ti cu la r m at h e ma t ica l r e la ti o ns hi p e x ist in g b e t w e e n th e a mp li tu de a n d p h a se sp e ct r a so th a t k n o w le d g e of e i th e r o n e i s s uf fic ie n t to c o mp u te th e o th e r . W h e n a s ei sm ic d a t a s et i s sa id t o be m in i m u m p h a s e , w e ge n er a l ly m ea n th at th e e m be d d ed w a v el et h as t h i s p ro p er ty a n d n o t t h e tr a c es th e m s el ve s . C er t a in l y t h e ea rt h 's r ef l ec t iv i ty f u n ct i o n i s n o t m i n i m u m p h a s e. T r u e o r F a l s e : If a da t as e t i s min i m um ph a s e a l re a dy , t hen a z er o ph a s e filter w i l l preserve m in i mu m p h as e b e c aus e i t do e s not change t he p h as e i n an y wa y. T rue o r F a l se : I f t he a m p li tu d e s p e ct r u m o f a m i ni m um p h as e d a ta set is c h a nge d , th e n th e p ha s e sp e c tr u m mu s t a ls o ch a n g e t o p r e s e r v e t h e m in im u m p ha s e r ela t io n sh ip . T ru e o r F a l s e : I t h a s b ee n p ro v en be yo n d d ou b t t h a t s ei s m i c d a t a f r om i m p u l s iv e so u rc e s i s m i n i m u m p h a se . T r u e o r F a l s e : Al l phys ic a l processes ar e m i n i mu m phas e . True or False: A minimum phase process can have zero amplitude over part of its spectrum. T r u e o r F als e : A ba n d l i m it e d p r o c es s c a n n e ve r t ru l y b e m i n im u m p h a se .

3-20

Amplitude Effects

Minimum Phase and the Hilbert Transform T h e c o n c ep t o f m i n im u m i s i nt i m a te ly l in ked w i th t h a t o f c a u sa l it y . F or o u r p u rp o s es , a ca u s a l t im e ser ie s i s o n e w h i ch v a n i sh e s f o r t< 0 . T h e i n ve st ig a t io n o f c a u s a l f un c t io n s is f a c il it a t ed b y t h e f o l lo win g F o u ri er tr a n sf o rm p a i r:

1 1 ht = + sgn t 2 2

1.0

h(t)

Graph of the step function h(t)

Thus h(t) is the unit causal function also called the step function. T he F o ur i e r tr a nsf o r m o f the s t e p f u nc t i o n i s : ( See 1984, Sig nal A na lysis, McG raw-Hill for a pr oo f. )

H ω = πδ ω – If f(t) is any causal function, then:

P apo ulis , A. ,

i ω

ft = ftht Taking the Fourier transform of both sides of this result gives:

Fr ω + iFi ω =

1 i Fr ω + iFi ω • πδ ω – ω 2π

Equating real and imaginary parts gives:

Fr ω =

1

Fi ω =

1

2 2

Fr ω +

1 1 Fi ω • ω 2π

Fi ω –

1 1 Fr ω • ω 2π


3 -21

Minimum Phase and the Hilbert Transform T hus , t he s pectrum of a c ausal f unction h as its real a nd i maginary p arts linked b y t he relations :

1 1 ω ω • Fr = π Fi ω

and

1 1 ω ω • Fi = – π Fr ω

If we write out the convolution integrals, we obtain:

1 Fr ω = π

∞

–∞

Fi ϖ ω–ϖ

dϖ

–1 Fi ω = π

∞

–∞

Fr ϖ ω–ϖ

dϖ

T h e s e i n t e gr al s a r e c al l e d H i lb e rt t r an s f or m s a n d we s ay t h at t h e re al a n d im a g in a ry p a rt s o f a c au s al si g n al f o rm a H i l b er t t ra n sf o rm p a ir . I n o u r c as e, w e a c t u a ll y w an t t o r el at e t h e a m p l it u d e a n d p ha s e o f a c au s al s i gn a l t o o n e a n o t h er , n o t t h e r ea l a n d i m ag i n ar y p ar t s. H ow e v er , r ec al l i n g t h at : iφ ω

Fω = Aωe

⇒ ln F ω

= ln A ω

+ iφ ω

The answe r s eems immediate that the p has e an d lo g amplitude spectrum ar e Hilber t t ransfor m p airs. How ever; w e must as k: - Under what circumstances c an w e t ak e the log spectru m? - Does the l og spectrum still correspon d t o a c ausa l time domai n function? T h e a n sw e r to th e f ir st q u e st io n is th a t w e c a n ta k e th e lo g so l o ng a s A (ω ) ≠ 0 . T his is e q u iva l e nt t o sa y in g th a t th e t im e se r ie s f (t ) mu st ha v e a st a b le i nv e r se .

3-22

Amplitude Effects

Minimum Phase and the Hilbert Transform For the second question, consider the z transform of f(t): ∞

Fz =

Σ

fkz

k

k = 0

W e k n ow th a t, s i nce f i s causal, t he n F ( z) c o nta i n s no neg at i ve po w e r s of z . N o w con s i d e r t he w e l l k n ow n s er i e s e xp a ns ion f o r the log a r i t hm :

ln u = 1–u –

1–u 2

2

+

1–u

3

3

+

w h ich is v al id i n th e r e g i o n 0 < | u | < 2 . S in ce F (z ) co n ta i ns o nly p o si tiv e p o w e r s o f z a nd ln ( u) c on t ai ns o nl y p o sit iv e p o w e r s o f u , th e n ln (F ( z )) c o nt a in s o n ly p o si tiv e p o w e r s o f z a n d th e r efo r e i s t he tr an sf o r m o f a ca u sa l ti m e se r i e s. T h e r efo r e w e c o nc lu d e : ∞

–1 φω = π

ln A ϖ ω–ϖ

dϖ

–∞

ln A ω

1 = π

∞

–∞

φϖ ω–ϖ

dϖ

For our purposes, we state the following important result: F or a c au s a l , s t a b l e f unc t i o n w i th a c au s a l , s t a b l e i nv e r s e , t h e p ha s e a nd l og am pl i t ud e s p e c t r um f or m a H i l b e rt t ra ns f or m p a i r . I n pa r t i c u l ar , t h e p ha s e ma y b e c o mp ut e d a s t he Hi lb er t t r a ns for m o f t he l og o f t he a mp l i t ud e s p e c t r um . S uc h a f u nc t i o n i s s a i d t o b e mi ni mu m p ha s e .


3 -23

Minimum Phase and the Hilbert Transform W e h a ve d emonstrated th a t a ca u sal, st a bl e time s eri es w i th a c a us a l, st a bl e in v er se i s comple te ly d et er mi n ed by e it h er it s amp l itude o r phase sp ec t ru m . G i ve n one, t he o th e r ca n be computed . I n p a rt ic u la r, t he p ha s e s p ect ru m i s compu t ed as:

φ ω = H ln A ω w h e r e H d e n o te s t he H il b e rt t r a n s fo r m . W e c a ll e d s uc h a w a ve f or m m i ni m u m p h a s e . T he r e a s o n f o r t h is n a m e c o m e s f r o m a t h e o r e m ( R o b in s o n , E. A . a nd Tr ei te l, S. , 1 98 0, G eo p h ys i ca l S ig n a l A na l ys is , P r en t i ce -H al l) w hi ch s h o w s t ha t, f o r a ll c a u s a l w a ve le t s w it h t h e s a m e a mp li t ud e s p e ct r um , t h e m in i mu m p ha s e w av e le t a r r iv e s t he s o o ne st w i th t h e m o s t e n e r g y . M a th e m at ic a ll y, t hi s i s s t at e d b y p r o vi n g t h a t t h e p ar t ia l e n e r g ie s : p

Ep =

Σ

2

fk

k = 0

a r e l a r g er f o r t h e m i ni m um p ha s e w a ve l e t t ha n f o r a ny o t he r w a ve l e t f or al l p . T hi s pr o of i s e q ui va l e nt t o s a yi n g t h at t he p ha s e de l a y o f t h e mi n i mu m p ha s e w av e l e t i s t h e s m al l e s t po s s i b l e de l a y al l o w e d b y c a us a l i t y , f o r e ac h f r e qu e nc y . R e ca l li n g t h a t t h e H i l b er t t r an s f or m i s j u st a c on v ol u t i on wi t h 1 /ω , i t f ol l ow s th a t t h e m i n i m u m p h as e sp e c t u m fo r a n y p ar t ic u l ar f re q u e n c y is i n f l u en c e d b y th e a m p l i t u d e sp ec t r u m a t a ll f re q u e n c i es . P u t a n o t h e r w ay , a ch a n g e t o t h e a m p l it u de s p e ct r u m a t a p ar t i cu la r fr e q u en cy wi l l c h an g e th e m i n i m u m p h as e sp ec t r u m a t a l l f re q u e n c i es . 3-24

Amplitude Effects

Minimum Phase and Velocity Dispersion W e ha v e sh o w n h o w to c al cu la te t he ph a se o f a m in im um p ha s e w a v e le t g ive n it s a m p lit ud e sp e c tr u m a n d ha v e a lso in di ca te d th a t t he co n st an t Q at te n u a tio n m o d e l i s m in im um p h a se . T hu s, if w e r e p r e se n t a si ng le p r o p ag at in g co m p le x sin u so id a s:

w f,t,z = A f e

–i2πf(t–z/v)

W e c an i n fe r t h e v e lo c it y o f t h i s wa v e b y f ol l ow in g t h e m o t io n o f a p oi n t o f c on st a n t p h as e. W i t h c om ple t e g en e r al it y , w e c an f ol l ow t h e p o in t o f z e r o p h as e b y e q u at i n g t h e p h as e t o z er o a n d s o lv i n g fo r z / t . T h u s w e d e d u c e i t s v el o ci t y t o be z /t = v . I f th e sa m e w av e p r op a ga t es t h r ou g h a co n s t an t Q m e d i u m , t h e n w e h av e :

wQ f,t,z = w f,t,z e or

wQ f,t,z = A f e

where

φQ f

≈

t f ln πQ f0

–πft/Q

e

–πft/Q + iH(–πft/Q)

–i2πf(t–z/v+φQ (f))

(Kjartansson, 1979)

Then solving for the velocity: t–z/v+

t f ln πQ f0

= 0 ⇒


z

t

= v f

≈ v 1+

f 1 ln πQ f0

3 -25

Minimum Phase and Velocity Dispersion Thu s w e s e e t h a t , i n an at tenu at i ng m e di um , velocity becomes f requency de p ende n t , a phenomenon know n as di s per s i on. T h e v e l oc i t y d is p e r si o n p r ed ic t e d b y t h is t h eo ry i s st r on g e st fo r lo w Q v al u es . T h e f i g u re b el o w p l ot s v el oc i t y v e rs u s f re q u e n c y f or d i f fe r en t t h r e e d i ff e re n t Q ' s. N ot e t h e n e ar ly c on s t an t b eh a v i or f o r Q o f 2 0 0 a n d t h e s t ro n g v ar ia t io n f o r Q o f 1 0 .

T h e w or d "d i s pe r s io n " a ri s e s because a p u ls e w i l l t e n d t o s pr e ad out (d i s pe rs e ) as i t s var i ous f requencies pr o pag a t e a t different v e l ocities. It i s t h i s di s p er s ion w h ic h l e a ds t o t h e c h a r acteristic p u l s e s hap e o f a m in imu m phas e wavelet. T h e pu l s e s h a pe i s s t r on g l y influenced b y t h e n e a r s urf a c e b e c au s e i t h as dr a m atic a lly l o we r Q valu es . 0.1 0.05 0 -0.05 -0.1

3-26

0

0.05

0.1

0.15

0.2

Amplitude Effects

Array Theory T h e u se o f a rr a y s o f so ur c es a n d re ce i ve rs i s c o m m o n p l a c e i n ex p l o r a t io n se is m o l o g y . T h e es s en t ia l d e t a i ls o f t h ei r use are s tr a i gh t fo rw a rd c o n s eq u e n ce s of l in e a r s u p er p o s it i o n a n d si gn al p ro c es s in g . C o n si d e r: Thr ee single f re q ue ncy sour ce s a t hal f wa vle ngth spa ci ngs v = 2000 m/sec f = 30 Hz l = 2000/30 = 67 m

+ The sum ma tio n o f t he th ree sou rce wav efields . No te t he d ram atic at te nu atio n o f sp ecific rayp at hs. Tho ug h n ot v ery a ppar ent , t he cent ra l por tio n o f t he wa vef ield also ha s les s cu rv at ure.

λ/2

Tw o m o r e so u rc es at in t er med iat e lo ca tio ns . T he s ou r ce s pa cin g d ecr ea se s to a q ua rt e r- wav e len g t h

+

λ/2


Th e su mm at ion of all wav ef ields in creas es r eje ctio n o f s te eper r aypat h s.

f ive t he

3 -27

Array Theory

He r e is the summati on of fiv e wav efi e lds fr om the pr ev ious page .

λ/2

We increase the array length with two more sources.

+ λ/2

No w t h e arr ay rej ect s mor e ra ypat hs . Lo ng er arra ys rej ect mo re, unt il an in fin ite len g th pas ses only v er tical ra ypat hs .

+ λ/2

3-28

We in creas e t he arra y len gt h ag ain with t wo m ore s ou rces. Thi s is a ve r y l ong ar r a y wi th 9 e le me nts. The e ffe c ts ar e quite dr ama tic . We note the occ ura nc e of seve r a l str ong r e je cti on notche s.

Amplitude Effects

Array Theory T he respo ns e of a n array c an b e a nalyze d b y considering the i dealized r espons e of a seque nce o f uni t spikes. 1/dx

9 8 7

dx

notches a t n/L wher e n=1, 2,.. .

6 5 4 3

L

2 1

L= 9*dx = 9*16.7 = 150.3

Array in space domain

0

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

-1

wavenumber (m )

Fourier transform o f th e array. S in ce th e a r r a y i s p u r e ly a f un ct io n o f x , its r e sp o n se i s p ur ely a fu n cti o n o f k x . T ha t is, it w ill b e i nd e p e n de nt o f k z o r f . H o w e ve r ; in o r d e r t o u se t he ar r a y r e sp o n se ch a r t, w e n e e d a w a y to e st im a te k x fo r a n e ve nt o f in te r e st . We c a n d o t hi s b y pi ck in g a h o r iz o n ta l su r f ac e o f in t e r e st an d me as ur ing t h e ho r i zo n ta l a p p a r e n t w a v e le n g th a lo n g it : S i n c e w e u s u a l l y d on ' t h a v e a λ m on o c h ro m at i c w a v e f ie ld , t h en w e u s u a l ly m e a s u re a p p ar e n t λx h or iz o n t a l v e l oc it y and c om p u t e k x f r om : kx 1 sin θ = = f v va Thu s, we must pick a f requency of interest t o perform the analysis Methods of Seismic Data Processing

3 -29

Array Theory He r e w e se e a n a rra y s i mul a ti on fo r a br oa dba nd w a ve fro nt w i th the wav el e t sh ow n at the ri g ht. T he sa me a rra ys as s i m u l a t e d pre vi ou sl y f or a 30 Hz si ng le f r e qu e nc y so urce a re sh ow n.

Wavelet: 30 Hz, Minimum phase

λ/2

λ/2

λ/2

λ/2

3-30

Amplitude Effects

Array Theory

H e re a re b r oa d -b a nd s n a ps h ot s o f t h e s im ul at io n o f a n i mp ul s i ve s o ur c e a n d f o u r d i f fe re nt a r r ay s. T he s ma l l b o x a t t he t o p o f e ac h c o l um n g i ve s t h e p h y si ca l s i ze o f t h e a rr ay . I m ag es a r e p l ot t e d w i th a s li gh t v e r ti ca l e xa gg e ra t io n a nd e ac h w av e f ro n t i s a ct u a ll y c i rc u l a r. E ac h s o u r ce s o n f i gu r at i on i s s ho w n f ul l- b an d a nd b ro k e n i n to f i ve d i ff e re nt s ub - b an ds . T he a rr a y s a l w ay s a ff ec t h ig h f r e qu e nc ie s m or e s t ro n g ly a n d t he l on g e r a r ra y s p r o du c e a n u n d i s to r t e d w a ve f ro m o n l y f or n e a rl y v e rt i ca l t r a ve l pa t hs . T he f u ll - ba nd i ma g es a r e t h e s am e a s t ho s e o n t he p r e vi ou s p ag e. Methods of Seismic Data Processing 3 -31

Array Theory

A m a j or ef f ec t o f a q u i si ti o n a rr a y s is th a t t h ey r es u lt in a v a r ia b le ( n o n st a t io n a r y ) em b ed d ed w a v el et . F o r a gi v en re fl ec t o r, t h e w a v el et w i ll v a r y w i t h o f fs et . F o r a g iv en t ra c e, th e w a v el et w il l v a ry w it h ti m e. T h is h a s s ign i fi c a n t im p li c a ti o n s fo r d e co n v o l u ti o n t h eo r y w h i ch a s su m es a st a t io n a r y w a v e le t.

3-32

Amplitude Effects


3 -33


Chapter 4 The C onvolutional Model and Deconvolution


4 -1

Bandlimited Reflectivity T h e u l t im a t e g oa l o f s e is m i c d at a p ro ce s s in g i s t o d e t er m i n e t h e ea rt h ' s re f le c t i v it y a t a f u n c t i on o f p o si t i on b e n ea t h t h e su r v e y. S i n c e s ei s m i c so u rc e s d o n ot ge n e ra te u s e f u l p ow e r a t a l l f re q u e n c i es , i t i s g en e r al ly a c c ep t e d t h at a n y r ef l ec t i v i t y es t i m at e m u s t b e " b an d l i m i te d ". T h i s m e an s th a t t h e b e st p os s ib l e r es u l t f r om f u ll y p r oc e ss e d se i sm ic d at a i s t h at i t r ep r e se n t s b an d l i m i t ed re f l ec t i v i ty . W e c an t h i n k o f t h i s r es u l t a s be i n g t h e t ru e ( br oa d ba n d ) r e fl e c t iv i t y c on v o lv e d wi t h a z er o p h as e w av e l et . E v en t h i s m od e s t g oa l i s ra re l y f u l ly re al i ze d . D e co n v ol u t i on i s o n e o f o u r m a j or t oo ls f o r a c h ie v i n g t h i s e n d . S h o r tc o m i n gs i n o u r t h e or y a n d a l go ri t h m s a n d l ac k o f k n ow le d g e t o g u i d e t h e m u s u al l y m e an s t h a t o u r f i n al e s t im a t e wi l l h a v e so m e u n des i re d p h as e r ot at i on o r a n i n co rr e c t a m pli t u d e s p e c t ru m . T h e f i gu r es be l ow a n d o n t h e n e x t p a ge il l u s t ra t e th es e c on c e p t s . Embedded wavelet

Nonwhite (20 Hz dominant) and minimum phase reflection coefficients Bandlimited (10-70Hz) and 60° phase rotated Reflection coefficients Bandlimited (10-70Hz) Reflection coefficients

Reflection coefficients

4-2

The Convolutional Model and Deconvolution

Bandlimited Reflectivity T h e c on s e q u e n c e o f a li m i t e d f re q u e n c y b an d i s lo s s o f r es ol u t i on . T h a t is we c an n o t d is t i n g u is h c l os e ly sp a c ed r ef l ec t i v i t y s p i ke s. A n u n kn o wn p ha s e r ot at i on m ake s i t d i f f ic u l t t o d e t e rm i n e t h e p re c i se l o ca t io n o f a r ef l ec t i v i t y sp ik e o r it s a m p l i t u d e . N o t e t h at t h e p r es e n c e o f a p h as e r ot a te d w av e le t c an n o t b e d et e c t ed w it h t h e p h a se s p e c t ru m o f t h e t r ac e a l on e . Amplitude Spectra

Reflection coefficients

Bandlimited (10-70Hz) Reflection coefficients Bandlimited (10-70Hz) and 60° phase rotated Reflection coefficients Nonwhite (20 Hz dominant) and minimum phase reflection coefficients

Phase Spectra

Reflection coefficients Bandlimited (10-70Hz) Reflection coefficients Bandlimited (10-70Hz) and 60° phase rotated Reflection coefficients Nonwhite (20 Hz dominant) and minimum phase reflection coefficients


4 -3

The Convolutional Model T h e m a j o ri ty o f th e th e o ry o f th e d ec o n v o lu t io n o f s eis m ic d ata is ba s ed o n a ser ie s o f si m p li fy i n g a s su m p t io n s c o n ce rn in g t he n a t u re o f th a t d a t a . T h es e a s su m p t io n s a r e u su a l ly en c a p su l a te d a n d ref er en c ed a s " T h e C o n v o lu t i o n a l M o d el " . W e h a v e a l rea d y s ee n t h a t , in a l in e a r 1 - D ea rt h, w e c a n w r it e th e t he c o n st ru c t io n o f a s y n th e ti c sei s m o gra m a s a co nv o l u ti o n o f a s o u rc e w a v e fo r m a n d a n i m p u ls e r es p o n se : where:

s t = Ir t •ws t Ir t is the earth impulse response ws t is the source waveform s t is the earth response to the source waveform

T h e a ss u m pti o n o f l i n ea ri t y s i m p l y m ea n s t h at a li n e ar c om b i n at i on o f so lu ti o n s t o t h e g ov e r n in g 1 - D w av e e q u at i on i s a l s o a s ol u t i on . W h i le th is i s a n i m p o rt a n t r es u l t f ro m p h y s ic s , f or t h e p u r p os e o f p r ov i d i n g a b as e f or d ec o n v ol u t i on t h e ory , i t i s p r ac t i c al ly u se l es s . T h e p r ob l em i s t h at A L L o f t h e p h ys i c s a n d g eo lo gy o f t h e p r ob l em is c o n t ai n ed i n t h e i m p u l s e re s p on s e . T h at is , i f w e c o n si d e r a n a tt e n u a ti n g e ar th , w i t h m u l t i p l es a n d t ra n s m i ss i on l o ss e s , t h e n a ll o f t h e s e ef f ec t s a r e c on t a in e d in t h e im p u l se r e sp o n s e. I n f ac t , t h e c on v o lu ti o n al r es u l t a b ov e , is v a li d in 2- D o r 3 - D a n d t h e re f or e t h e i m p u l s e r es p o n se c an a ls o c o n t ai n s u c h e ff e c t s a s e l as t ic m od e co n v e rs i on s a n d sp h er i c al d i v e rg en c e i n a d d i t i on t o th o s e a lr e ad y m en t i on ed . S o , a l t h ou gh t h i s re s u l t c an be p ro v en f ro m a v e ry g en e r al t h e or y, it i s t o o ge n e ra l t o b e o f u s e t o u s. I n s t ea d , w e m u s t m ak e a n u m b e r o f si m pli f yi n g a s su mp t io n s t o f ra m e t h e c on te x t o f d e c on v o l u t io n t h e or y. 4-4


The Convolutional Model S h er if f a nd G eldart (Exploration Seismology, 1995, Cambridge University Press) p re se nt th e convolutional mod el by d ecomposi n g t h e e a rt h' s impuls e r esponse a s :

Ir t = ns t • p t • e t where

ns t

re presents n e a r surf a c e e f f ect s be nea t h b o t h t h e source a n d receiver

pt

r epre se n t s a ll e ff e c ts n o t o th e r wise mo d ele d su c h as m u lti p le s , a b s o r p ti o n , mode c o nv e r s io n s, e t c.

et

i s t h e "i mp u l s e re s p o n s e" ( t h e i r t e rm ) o f t h e t ar g et r ef l e c t or s . " t h i s i s t h e s ig n a l t h a t s ei s m i c re f l e c t io n wo r k is i n t e n d e d t o f i n d ".

T hi s t e r m in o lo g y i l l u s tr a t e s s o m e o f t h e t y p ic a l c o n fu s io n s u r r o un di ng t h e c o nv ol ut io na l m o d e l. C on s id e r t h e ir d e fi n it i o n o f e ( t ) . I f i t i s t r u ly t h e i m p u ls e r e s p o ns e o f t h e t ar g e t r e fl e ct o r s t h e n i t c o n ta in s a ll m ul t ip le s , a b s o r pt io n, m o d e c on ve r s io ns a s w e l l a s p r im ar i e s f r o m t ha t z o ne . T h is m e a ns i t i s N OT t h e s ig n a l w e w i s h t o u n co ve r a n d t h us t he i r d e f in i ti o n i s s e l f- c o nt r a d ic t o r y . A ls o , p ( t) i s s up p os e d t o b e a c o n vo lu ti o n a l o pe r a t or w h ic h m o de l s a d i v e r s e r a n g e o f e ff e ct s w i th o u t a ny j u s t if i c at io n t h a t t h is i s e ve n p os s ib le . I n f a ct , m o s t o f t h e m e nt io ne d e f fe ct s a r e n on s t at io na r y ( s e e 2 - 12 f o r a d e fi n it i on ) a n d t h e r e fo r e c a n no t b e m o d e le d a s a c o nv o lu t io n. T hi s i s t h e p r e s e n ta ti o n i n a n e x ce ll e n t, h ig h ly r eg ar d e d r e fe r e n ce w o r k s o i t i s u nd e r s ta n da b le t h a t t h e r e i s a g r e at d e a l o f c o nf u s io n s u r r o un di ng t he c o nv o lu t io na l m o d e l i n t h e i n du s t r y. Methods of Seismic Data Processing

4 -5

The Convolutional Model W e n o w m o di f y t h e m o de l o f S he r i f f a nd G e l da r t w i t h t he i n t e nt o f p re s e r v i ng it s s pi r i t b ut m a ki n g i t l o gi ca l l y c o ns i s t e nt . F i rs t w e c o m bi n e t h e s o ur c e w a v e f o rm a nd t he n e a r s u rf a c e e f f e c t s i n t o a n e qu i va l e nt w a v e l et :

we t = ws t • ns t N e x t we d is c ar d p ( t ) a s c on t ai n i n g n o n s t at i on ar y e ff e c t s w h ic h a re b ey on d t h e s c op e o f t h e m o d el a n d a l l ow e( t ) t o be a n i m pu ls e r e sp o n s e i n a l i m i te d s en s e o f t h e t ar ge t r e f le c t or s:

s t = we t • e t + noise t He r e we h a v e al s o in t ro d u c e d a d d i t i v e , s t at i on ar y, w h i t e n o i se . T h e ea r t h ' s im p u l s e r es p o n s e i s f u rt her a s su m e d t o b e:

e t = m t •r t where:

rt

= the earth's primary reflection series

mt

= the subset of the earth's multiple reflection response which can be modeled as a stationary process.

Thus we can write:

s t = we t • m t • r t + noise t

4-6


The Convolutional Model N ote t h a t t h e m u lt i p le t e rm c a n be e q ua ll y wel l a s so ci a t e d wi t h t h e wa v e le t i ns t ea d o f t h e re fl e ct i vi t y so t ha t we c a n wr it e :

s t = wm t • r t + noise t wm t = we t • m t

T hi s r e s ul t i s a g o od s ta r t in g p o in t f o r d e c on vo lu ti on t h e o r y s in ce i t p r e s e n ts t he s e is m i c t r a ce a s t h e c o n vo lu ti o n o f a w a ve le t w it h t h e e a r th 's r e fl e c t i v it y . I t i s e m ph a s iz e d t ha t o ur g o a l i s d e d uc in g t h e e a r th 's r e fl e ct i vi t y a nd N OT i t s i m p ul s e r e sp o n s e . T he t w o a r e v e r y d if f e r e n t. W e r e ma r ke d t h a t t h e n o i s e i s m o de l e d a s b e i ng " s ta t i o na r y" a nd " w hi t e " i n n a t ur e . S t a t i o na ry i n t h i s c o nt e x t m e a ns t ha t t h e b a s i c f e a t ur e s o f t he s pe c t r um d o n ot c ha ng e w i t h t i me . T ha t i s , i f we ex t r ac t e d s pe c t r a f r om s ma l l w i nd ow s r a ng i ng u p a n d d ow n n ( t ) w e w o ul d f i nd e s s e nt i a l l y t he s am e s p ec t r al s ha p e . G au s s i a n o r u n i f or m l y d i s tri b ut ed n o i s e c a n b e s ho w n t o h a v e t h i s p r o pe r t y . T h e c on v o l ut i o n o f t w o s t a t i on ar y s i g na l s i s a l s o s t at i o na r y. A n e x a mp l e o f a n o ns t a t i on ar y s i g na l i s t h e i m pu l s e r e s p on s e f r om a c o ns t a nt Q e ar t h . A s we h a v e s e e n, t he s p ec t ra l r e s po ns e c ha ng e s s y s t e ma t i c al l y wi th t i me . A w h i t e s p e c t ru m is o n e t h a t h as c on s t an t p o we r a t a l l f re q u e n c i es ( e . g . " w h i t e n o is e " ) . A n i n fi n i t e l e n gt h s i gn a l o f G a u ss i an o r u n if o rm l y d i st r i bu t e d n o is e c an b e s h ow n t o h av e a w h it e s p e c t ru m . F i n it e l e n gt h n oi s e s eq u en c e s h av e a p p r ox i m at e l y w h i t e sp e c t ra w h en s m oo t h ed w it h a sh o r t o p er at or .


4 -7

The Convolutional Model T ho u g h n o t s t r ic t l y p a r t o f t he c o n vo lu ti o n a l m o d e l, a f u r th e r a s s u mt io n i s o ft e n m a d e ( i n t he c o nt e xt o f d e co nv o lu t io n t h e o r y ) t h at t he r e fl e c t i v it y , r ( t) , i s a w h it e a n d s ta ti o na r y t i me s e r ie s . I t c an b e e as i l y d e mo n s tr a t e d u s in g s o n ic lo g s t ha t real e a r th r e fl e ct i vi t y d oe s n ot h a ve a w hi te s p e c tr u m b ut i n s te a d s h o w s c o n s id e r a b le s p e c tr a l c ol or e vi d e n ce d b y a p r o no un ce d r o ll o ff i n p o w e r a t t h e l o w f r e q u e nc ie s . H e re w e s ee a n e xa m p le o f a re a l r ef l ec t iv i t y ( in t im e , c o m p u t ed fr o m a s o n ic lo g a ss u m in g c o n s t a n t d en s it y) a n d i ts F o u ri er s p ec t ru m : 0.4

0

0.2

-10 -20

0 -30

-0.2

0

0.2

0.4

0.6

0.8 1 Time (sec)

1.2

1.4

1.6

1.8

0

50

100

Re al r cs com pute d fro m a s oni c lo g a t cons ta nt de ns it y

S pe ctr um o f the re a l rcs . Not e the 20db rol l of f f rom 10 0 to 0 Hz .

C o nt r a s t t hi s w i t h a c o mp ut e r g e ne r at e d r ef le c ti v i t y d e s ig n e d wi th a w hi t e s pe c t r um : 0.2

150

Frequency (Hz)

r a nd o m

0

0.1

-10

0 -20

-0.1 -30

-0.2

0

0.2

0.4

0.6

0.8 1 Time (sec)

1.2

C omp u t e r ge n era t ed r an d om rcs.

4-8

1.4

1.6

1.8

0

50

100

150

Frequency (Hz)

p se u d o

S pe c tr u m o f t h e p s eu d o r a nd om r cs . N o te t he e s s en ti a ll y f la t ( wh it e) s pe ct ru m.


The Convolutional Model I n o u r ba s ic c o n v o l ut i o n a l m o d e l, w e a ss u m ed th e e ff ec t s o f m u lt i p les c o u ld be tr ea t ed a s a co nv o l u ti o n o f t h e s o u rc e w a ve fo rm w it h a " m u l ti p le o p er a t o r" :

wm t = w t • m t I n o u r d e v e l op me nt o f t h e 1 - D s e i s mo g r am , w e e x a mi ne d a n a l g o ri t h m w hi c h i s c ap ab l e o f g en e ra t i ng a l l p o s s i b l e m u l t i pl e s . C o u ld t h i s o p e r at i o n h av e be e n p e r f or m e d a s a c o nv o l ut i o n? T he ge n e ra l a ns w e r t o t hi s q u e s t i o n i s " n o" b e c a us e t he m ul t i p l e t r a i n gr o w s i n l e ng t h a s t i m e i nc r e a s es a n d i s t h us f u nd a me n t al l y n on -s t a t i on ar y . H o w e v e r , c e r t a i n c l as s e s o f m u l ti pl e s c a n b e m od el e d b y a c o nv o l ut i on i n c l ud i n g s u rf a c e gh os ts a nd w a t e r b o t t o m m u l t i pl e s . I n g e ne r al , i f w e r es tri c t o u r a t t e nt i on t o t h e p o r t i o n o f a n i m pu l s e r e s po ns e l a t e r i n t i me t ha n a m a jo r m ul t i p l e g e ne r at o r , t he n t he m u l t i pl e c o n t ri b u t i on f r om t h a t g e ne r at i ng i n t e rf a c e c a n b e m od e le d a s t h e c o nv o l ut i o n o f a m u l t i pl e o pe r at o r w it h t he s o ur c e w av ef o r m. H o w ev er , a s a no t he r c a v e a t , e v e n wa t e r b ot t om m u l ti pl e s o n f a r o f f s e t t r a c e s s h ow n o n- pe r i od i c s pa c i ng a n d s o v i o l a t e o ur m o de l . Summary of assumptions: • Ea rt h's i m p u l s e r e s p on se c on s i s t s o f a r e f le c t i v i t y s e ri e s p o s si b l y c o n v ol v e d w it h a m u l t i p l e o p e ra t or . It is al s o s t at i o n ar y. • T h e e ff ec t o f t h e source wavefo rm m a y be modeled a s a simple s ta t ionary convolu ti o n wit h t he e a rt h' s impuls e re sponse . • Any noise is additive, white, and stationary. • Optionally, Earth's reflectivity series is white and stationary. Methods of Seismic Data Processing

4 -9

The Convolutional Model Here we i llust rate the steps involve d i n the const ructio n o f a multiple-fr ee synthetic s ei smic t race u sing a pseudo random re fl ectivity: 0.2 Pseudo random reflectivity

0.1 0 -0.1 -0.2

0

0.2

0.4

0.6

0.1

0

1

1.2

1.4

1.6

1.8

0.1

M in i m u m p h as e w a v el e t t o s ca l e

0 -0.1

0.8

Time (sec)

Minimum phase wavelet enlarged

0 -0.1

0.2

Time (sec)

0.02

0

0.05

0.1

Time (sec)

0.15

0.2

0.01 Noise free seismogram

0 -0.01 -0.02

0

0.2

0.4

0.6 Time 0.8(sec) 1 1.2 Amplitude Spectra

1.4

1.6

1.8

0 -20

Wavelet

-40 -60

Reflectivity

S eismogram

-80 -100 20 4-10

40

60

80 100 120 140 Frequency (Hz) The Convolutional Model and Deconvolution

The Convolutional Model 0.02 Noise free s e i s m o g ra m with noise superimposed

0.01 0 -0.01 -0.02

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time (sec)

0.02 Noisy seismogram

0.01 0 -0.01 -0.02

0

0.2

0.4

0.6

0.8

1

Time (sec)

1.2

1.4

1.6

1.8

Amplitude Spectra -20

Reflectivity

-40 -60

Noisy seismogram Wavelet

-80 20

40

60


120

140

De f i n i ti o n : “ Th e em be d d e d wa v el e t ”. A s y o u n o w k n ow , th er e a re m an y “ wa v el e t s” i n e x p lo ra t io n s ei s m ol o gy . T h e p h ra s e e m b ed ded w av e l et r e fe r s t o a w av e l e t d e r i ve d b y f i tt i ng an y s e i s mi c t r a c e t o t h e c o nv o l ut i o n a l m o de l . T ha t i s , t h e e m b ed de d w a v e l e t i s w h a te v e r s i g n a l m us t b e c on v o l ve d w i t h t h e r e f l e ct i vi t y t o g i ve t h e t r a c e u nd e r c o n s i de r a t i o n. E v e n w he n t h e c o n vol u t ion a l m o d e l i s a p o o r f i t t o t h e d a t a , a n e mb e dd e d w a v el e t c an s t i l l b e e s t i ma t e d i n t he l e a s t -s qu a r e s e ns e . Methods of Seismic Data Processing

4 -11

Frequency Domain Spiking Deconvolution Perhaps the ea siest d e c o nv o lu t io n t e ch n iq u e to c o nc e p tu a li z e is th e fr eq ue n c y d o m a in m eth o d . I t i s su g ge st e d b y t he sp e c tr a we e x a m in e d in o u r d isc u ssi o n o f t he co n v o lu ti o na l m o d e l: Reflectivity

-10

-20 Wavelet

Noisy seismogram

-30

-40

-50 N oise free seismogram -60

-70 20

40

60

80 Frequency (Hz)

100

120

140

H e r e w e s e e t he b as i c i d e a t ha t u n d e r l i es a l l d e c o nv o l ut i on c on c e pt s : T he a mp l i t ud e s pe c t r al s ha p e o f t h e s e i s m i c t ra c e ( s e i s mo g ra m i n t h i s c a s e ) i s e s s e nt i a l l y s i mi l a r t o t ha t o f t h e u nk no wn w a v e l e t . G i v e n t hi s , a l l t ha t r e m ai ns i s t o d e du c e t he w av e l e t ' s p h a s e a nd t h en w e c a n d e s i g n a n i n v e r s e f o r i t. W e o b s e r v e t ha t t h e n o i s y s e i m og r a m i nt ro du c e s a f u r t he r c o mp l i c at i o n i n t ha t w e m us t r e s t r i c t o u r a t t e n t i on t o t he s i g na l f r e q ue nc y ba nd . N o t e t h a t w e a re re l yi n g o n t h e r ef l e c ti v i t y t o h a v e a w h it e s p e c t ru m so t h at w e c an a t t ri b u t e a ll s p e c t ra l " c h ar ac t e r" t o t h e w av e le t . 4-12


Frequency Domain Spiking Deconvolution I f w e c a n c o m p u te t he a m p li t u de s pe ct r um o f t h e w a v e le t b y s m o o t h in g th e a m p li tu d e sp e ct ru m o f t h e s eis m ic tr a c e, t h en w e ca n i nv o ke th e m i ni m u m p h a s e a s su m p t io n to c o m p le te ly s p ec if y t h e u n k n o w n w a v el et . H er e i s t h e h e lp f il e f ro m t h e M a t la b ro u t in e , d ec o n f , w h i ch do es fr eq u en c y d o m ain d e co n v o l u ti o n : % % % % % % % % % % % % % % % % % % % % % %

deconf algorithm • Compute the power spectrum of the design trace. Add in th e stab power. • Con volve the power DECONF performs a frequency domain deconvolution of the input trace spectrum with a boxcar smoother to estimate the trin= input trace to be deconvolved wavelet power spectrum. trdsign= input trace to be used for operator design • Compute the wavelet phase n= number of points in frequency domain boxcar smoother spectrum with the Hilbert stab= stabilization factor expressed as a fraction of the zero lag of the autocorrelation. This is equivalent to being transform. a fraction of the mean power. • Compute the spectrum of ********* default= .0001 ********** the input trace. phase= 0 ... zero phase whitening is performed • Divide the input trace 1 ... minimum phase deconvolution is performed spectrum by the estimated ************** default= 1 *************** wavelet spectru m. trout= output trace which is the deconvolution of trin • Inverse FFT to give specinv= output inverse operator spectrum. The time domain deconvolved trace. [trout,specinv]=deconf(trin,trdsign,n,stab,phase) [trout,specinv]=deconf(trin,trdsign,n,stab) [trout,specinv]=deconf(trin,trdsign,n)

operator can be recovered by real(ifft(fftshift(specinv)))

W e n ot e t h a t t h e d e c on v ol u t i on o p e r at or c an be d es i gn e d o n o n e t r ac e a n d a p p l i ed t o a n o t h e r. T h i s i s t o s i m u l at e t h e p ra c ti c e o f d e s ig n i n g th e o p er at o r o n a s e gm en t o f t h e t r ac e t o a v o id l e t t in g su ch t h i n g s a s s u r f ac e wa v es i n f l u en c e t h e d e s ig n . T h e o t h er s i g n if i c an t p a ra m et e rs a r e : t h e l en g t h o f a b ox c ar s m o ot h e r, a s t ab i li z at io n f ac t or , a n d a f l ag fo r z er o o r m in i m um p h a se . I n o r d e r t o s p e c if y n , w e re c al l t h at t h e f r eq u en c y s am p l e s i ze o f th e D F T s p e c t ru m i s Δ f = 1 / T w h e re T i s t h e t ra ce l en gt h i n s e co n d s . T h u s , a s m o ot h e r o f l e n gt h F s mo ot h ( i n H er t z) wi l l h av e a n u m b er o f p o in ts gi v e n b y: F

n smooth =


smooth

Δf

= TFsmooth

4 -13

Frequency Domain Spiking Deconvolution T he s t a b il i z a ti o n f a ct o r i s d e s ig n e d t o p r e ve n t t h e o pe r at o r d e s ig n f r o m b e in g u n du ly i n fl u e nc e d b y n oi s e a nd t o a vo id t h e p o s si b il i t y t h a t a d iv i s io n b y z e r o m ig ht o cc ur w he n t he s pe c tr u m i s i nv e r te d . I t c a n b e t h o ug h t o f a s w hi t e n oi s e a d de d t o t h e s p e c tr u m w it h a c e r t a in p o w e r l e v e l. T ha t p o w e r l e v e l i s : stab power (db below mean power) = 10*log10(stab) So the default stab o f .0001 is a d b level o f 10*(-4) o r 4 0 dbdown f ro m mea n pow er . If we c hoo s e a f r e quenc y s moot h er o f 1 0 Hz -> 10*1.6 = 16 points, and de f aul t th e s t a b f ac t or , t hen, th e deconvolution of t h e n oi s e f r ee s e i smog r a m giv es : 0.1

0.08

exact rcs

0.06

0.04 deconf estimate 0.02 noise free seismogram

0

-0.02

0

4-14

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8


Frequency Domain Spiking Deconvolution In the frequency domain our result looks like:

Exact rcs

0

-20

-40

deconf estimate

-60 N oise free seismogram -80

-100

0

50


200

250

W e c a n s e e t h at t h e e st i m at e is q u i t e go od . W e c an b e m o re p r e ci s e a b ou t h ow g oo d i t i s b y u si n g t h e M a t la b f u n c t io n m x c or r wh i c h c om p a re s t w o t i m e s er i es a n d re t u r n s t h e m ax i m u m o f t h ei r c ro s s co rr e la t io n a n d t h e l ag a t wh ic h it o c c u r s. T h e r e su l t s i n : m ax c or re l at i on = . 3 9 a t l ag o f . 1 s am ple s If w e n o w r u n th e s am e pr o c e ss w it h t he sa m e pa r a m e t e r s o n t he n o isy se i sm o g r a m w e o b t a in q ui te a di ffe r e n t r e s ul t a s s ho w n o n th e n e xt p ag e .


4 -15

Frequency Domain Spiking Deconvolution

0.18 0.16 0.14 0.12

exact rcs

0.1 deconf stab =.5

0.08 0.06

deconf stab = .01

0.04 deconf, stab=.0001

0.02

noisy seismogram

0 -0.02

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0.12 High cut filtered: 70-80 Hz rolloff 0.1 deconf stab =.5 0.08 deconf stab = .01

0.06 0.04

deconf stab=.0001 0.02 exact rcs

0 -0.02 -0.04 Here are some -0.06 0 0.2 0.4 results from maxcorr:

4-16

Filtered results

Max corr

deconf stab=.0001 0.6deconf 0.8 stab1 = .01 1.2 deconf stab =.5

0.0603 1.4

Lag 1.3000

1.6 1.8 1.6000 0.0728 0.1510

3.0000


Frequency Domain Spiking Deconvolution W e c a n d e v e lo p a s i m p l e m a t h em a t i ca l m o d el fo r d e c on v o lu t i on i n t h e f r eq u en c y d o m ai n . F ir s t , t h e c on v o lu ti o n al m o d e l f or a m u l t i p l e- fr e e se i s m i c t ra c e i s (1) w h er e r i s r ef l e ct i v i t y, w i s t h e wa v e l et , a n d n i s a d d i t iv e n oi s e . I n t h e f re q u e n c y d om a in , t h i s be c om es (2) G e n e ra ll y , t h er e w il l b e a r an g e o f fr e q u en ci e s , c al l ed t h e s i gn a l ba n d , o v er w h i c h t h e R ( f ) W ( f ) t e r m d o m i n at e s o v e r N ( f ) . D en o t in g t h e b ou n d s o f t h is f re q u e n c y b an d b y f m i n a n d f m a x, w e c an h av e t h e a p p r ox i m at i on ( 3) w h er e t h e v er t i c al b ars ( e. g . |S ( f) | ) d e n ot e a b s ol u t e v al u e s o r a m pli t u d e s p e ct r a. N o t e th a t b y u s i n g a m p li t u d e s p ec t r a, we a r e d i s c ard in g t h e p o s si b i li t y o f e st i m a t in g t h e wa ve l e t p h a se d ir e ct l y f ro m t h e d at a. T h e n e x t st e p , s p ec t r al s m oo th in g i s d if f i c u l t t o fu ll y j u s t i fy m at h e m a ti c al l y. D e n ot i n g a sm o o t h e d s p e c t ru m b y a n o v er b ar , t h e " w h i t e re f le c t i v it y " a ss u mp t io n m ea n s t h at

R (f ) ≈ 1

(4)

W e t h en a r g u e t h at s m oo t h i n g |S ( f ) | y i e l d s a n e s t im a t e o f t h e a m p l it u d e s p e c t ru m o f t h e em b e d d e d wa v el e t . T h o u g h we k n ow t h i s i s n ot p r e ci s e ly t ru e , i t i s a p pro x im a t el y s o i n m a n y u s ef u l s i t u at i on s . Methods of Seismic Data Processing

4 -17

Frequency Domain Spiking Deconvolution T h us we h av e t h e e s t im a t e

W (f ) est = S(f ) ≈ W (f )

(5)

T h e a m pli t u d e sp ec t r u m o f th e d e c on v ol u t i on o p er at o r i s j u s t t h e in v e r se o f t h i s (6) G e n e ra ll y , th is s p e c t ra l d i v is i on is p r ob l em a t i c i f t h er e a r e f r eq u e n c i es wh er e t h e e s t im a t ed w av e l et ' s s p e c t ru m i s v er y sm a l l. W h er e i t is s m a ll u su a ll y m e an s t h at t h er e w as n ot m u c h ra d i at e d s ou r c e p o w er a n d s o n o is e i s l i ke ly d o m i n an t . S i n c e t h e s e s m al l v a l u es a r e i n v er t e d , t h e y be c om e v er y i m p o rt an t i n D( f) . G i v e n t h e s e c on s i d e ra ti o n s, i t i s c u s t om a r y t o a d d a s m a ll c on s t an t t o t h e e s t im a t ed wa v el e t' s a m p l it u d e s p e c t ru m p ri or t o i n v er s i on . T h en (7)

w h er e :

( 8)

T h e c o n s ta n t μ i s c al l e d t h e " w h it e n o i se f ac t or " o r " s t ab i li t y fa c t or" a n d is a sm a l l p os i t iv e n u m b e r u s u a ll y b et w ee n .0 1 a n d . 0 00 0 0 1 . L as t l y, w e m ust e s t im a t e th e p h a s e s p e c t ru m o f D ( f ) . U n d e r th e m in i m u m p h as e a s s u m p t i on a n d u s i n g H t o d e n ot e t h e H i lb e rt tr an s f or m , w e h a v e (9)

4-18


Frequency Domain Spiking Deconvolution w h er e w e u s e |D ( f ) | a s gi v e n b y e q u at i on ( 7 ) . N ot e t h a t t h e s t ab i li t y f ac t or a l so g u a rd s a g ai n s t t ak i n g t h e l og ar it h m o f z er o i n e q u at i on ( 9 ) . S o , we n o w h av e t h e a m p li t u d e a n d p h as e s p ec t r u m o f t h e d e c on v ol u t i on o p er at or a n d w e a r e re ad y t o a p ply i t t o t h e s ei s m i c t r ac e. A g ai n i n t h e f re q u e n c y d om a in , t h i s i s (10) I f we s u bs t i t u t e e q u a t io n ( 3 ) in to eq u at i on ( 1 0 ) w e c an o b t ai n a n ex pre s si on fo r th e e m b ed d ed wa v el e t r em a in in g a f t er d e c on v o l u t io n (11) N e g le c t i n g t h e n oi s e t e rm , we es t i m at e t h e e m b ed ded w av e le t a s (12) As s u m in g a b an d p a ss fi l t er i s a p p l ie d fo l lo wi n g d e c on v o lu t i on , w e c an r eg ar d W D ( f ) a s ef f e ct i v e ly z e ro o u ts i d e t h is b an d w i d t h . E q u a t io n ( 1 2 ) c an b e f u r t h er w ri t t en a s (13)

I n t h e l as t s t e p , t h e a p p ro x im a t e u n i t y f ol lo ws o n l y i f t h e a s s u m p t i on s o f s t at i on ar y w av e le t , w h i t e r ef l e ct i v i t y, a n d m in i m um p h as e a re a p p r ox i m a t el y v al i d . I f th e f i rs t t w o f ai l th en w e e x p e c t a n on - wh i t e a m p l i t u d e sp ec t r u m f or W D ( f ) a n d i f t h e l as t f ai ls t h e n we ex p e c t a re s id u al p h a se sp e c t ru m . Methods of Seismic Data Processing

4 -19

Finding a Wavelet's Inverse If w symbolizes a wavelet and x is its unknown inverse, then the two are related by:

w• x = 1

Here, the • d enotes convolution and 1 i s a unit vector. In matri x n ot ation, this i s written:

w0 0 0 0 0

x0

w1 w0 0 0 0

x1

w2 w1 w0 0 0

x2

w3 w2 w1

x3

0

1 =

0 0

w0 Here we have assumed that both w and x are causal. In general such an inverse will require infinitly many terms to produce and exact result so we will look for an approximate finite length inverse. If n is the length of the inverse and m is the length of the wavelet, then the above matrix equation is: n n m

4-20

W

X = m

D

Vector D is the desired output which, in this case is a spike at zero lag.


Finding a Wavelet's Inverse T h u s w e h a v e c h o s en n
T

W WX = W D T

–1

The normal equations T

X = WW WD

The estimated X

w0 w1 w2 w3

w0 0 0 0 0

x0

w0 w1 w2 w3

1

0 w0 w1 w2

w1 w0 0 0 0

x1

0 w0 w1 w2

0

0 0 w0 w1

w2 w1 w0 0 0

x2 =

0 0 w0 w1

0

0 0 0

w3 w2 w1

x3

0 0 0

w0

0 w0

w0

M u l t i p li c at i on by W T d oe s a cr os s c or re l at i on be c au s e i t c an b e e as i l y s ee n t o b e c o n v ol u t i on w it h t h e t i m e r ev e rs e d w av e l et . T h i s c an be se e n t o b e:

φ0 φ1 φ2 φ3

x0

φ1 φ0 φ1 φ2

x1

φ2 φ1 φ0 φ1

x2

φ3 φ2 φ1

x3

w0 =

0 0

φ0 Where

φj

is the jth lag of the autocorrelation of w.


4 -21

Finding a Wavelet's Inverse We have seen that the process of finding the m-length causal inverse, x, of a causal wavelet, w, reduces to solving the m by m linear system:

φ0 φ1 φ2 φ3

x0

φ1 φ0 φ1 φ2

x1

φ2 φ1 φ0 φ1

x2

φ3 φ2 φ1

x3

=

w0 0 0

Where φj is the jth lag of the autocorrelation of w.

φ0 This remarkable result says that we don't need to know the wavelet itself, just m lags of its autocorrelation. And, if we are content to be off by an arbitrary scale factor, then we can replace wo by 1. How this is possible is a consequence of the following facts: • A causal, stable wavelet with a causal, stable inverse IS minimum phase. (Karl, J.H., An Introduction to Digital Signal Processing, Academic Press, 1989, see pages 35-37) • Th e F o ur i e r tr a ns f o r m o f t he a u tocorrelati o n i s t he p o w e r s p e c tr u m of th e w a ve l e t (Wiener-K h i n tc h i ne Theorem) . T hu s th e ph a s e i nfor m a ti o n is no t p r e s e nt i n t he a ut ocor r e lat i o n. T hu s , th e problem o f e s t i ma ting th e inverse t o a mini m um p ha s e wavelet i s r e d uce d to o n e o f e s ti m a ti n g th e a ut ocor r e lat i o n o f th e un k no w n w av e l e t. Mo s t te c h niqu e s d o s o imperfectly.

4-22


Wiener Spiking Deconvolution T he o r i g i na l d e c o nv o l ut i on t e c h ni qu e, a n d s t i l l t h e w or k ho r s e o f t h e m e t ho d ol o g y i s a t i m e d o ma i n m e t h od r e f e rr e d t o a s W i e ne r d e c on v ol u t i o n. I t re s t s o n t h e t i m e d o ma i n c o mp u t at i o n o f t h e i nv e r s e o f a m i n i mu m p h as e f i l t e r g i v e n i t s a ut o c or r e l at i on . B e l o w we s e e t h e a ut o c or r e l at io ns o f t he s yn t he t i c t r ac e w hi c h w e h a v e b e e n ex a m i ni ng : 6

5 autocorrelation of wavelet 4 autocorrelation of noisy seismogram

3 2

a u t oc o r r el at io n o f n o is e f r ee s e i sm o gr am

1

autocorrelation of synthetic reflectivity

0

-1 -2

-1.5

-1

-0.5

0

0.5

1

1.5

2

T hu s we are r e m in d e d of the f ac t th a t th e a ut o co r r e la ti o n o f t he s e is m o g r a m i s v e r y s i mi l ar t o t h e a ut o co r r e la ti o n o f t he w a ve l e t. T hi s i s a c o n s e q ue n ce o f o ur a s su m pt io n t ha t t he r ef l e ct iv i ty i s a r a nd o m , w hi t e s e q ue n c e a n d c an b e d e mo n s tr a t e d m a th e ma ti ca ll y a s f o ll ow s: Methods of Seismic Data Processing

4 -23

Wiener Spiking Deconvolution Recall the expression for the convolutional model:

s t = wm t •r t + n t w hi c h e x pr e s s e s t h e s e i s mi c t r a c e , s , a s a c o n v ol u t i on b e t w e en a wa v e l e t w i t h a p os s i b l e m u l t i pl e t r a i n, w m , a nd a re f l e c t i c i ty , r , p l us a d di t i v e ra nd om n o i s e , n . S i n c e a n a u t oc o r r e l at i o n i s f o rm e d b y t i m e re v e r s i ng t h e t r a c e a nd c o nv o l v i ng i t w i t h i t s e l f , w e h a v e :

A s t = s t • s –t = wm t •r t + n t • wm –t •r –t +n –t = wm t •r t •wm –t •r –t + wm t •r t •n –t + n t •wm –t • r –t + n t •n –t S in ce th e o r d e r o f co n vo lu ti o n i s u ni mp o r t a nt , t he f ir s t te r m in th is e x p r e ss io n c a n b e se e n t o b e th e c o nv o lu tio n o f th e a ut o co r r ela ti o ns o f w m a nd r . T h e se c o nd a n d th ir d te r m s b o th inv o lv e th e cr o ss co r r e l at io n s b e t w e e n t w o r a n do m se q u e n ce s, r an d n, a n d h e n ce a r e z e ro w h il e th e la st te r m is t he au to c o r r a lt io n o f n . T hu s

A s t = A w t •A r t + A n t Since r and n are both random sequences by assumption, their autocorrelations are delta functions and we obtain:

A s t = A w t + pnδ t w h e r e p n is t he m e a n n o ise p o w e r . S o w e se e th a t th e a u to co r re la ti o n o f se i sm o g r a m a n d w a v e le t s ho u ld b e e q u al e xce pt f o r th e p o ss ib i lit y o f a sli g ht i nc r e a se in th e z e r o la g po w e r. 4-24


Wiener Spiking Deconvolution T he p r oo f t ha t t he a ut o c or r e l at i on o f t he w av l e t c a n b e o bt a i ne d f ro m t h at o f t he s e i s mo g r am r e l i e s o n s t a t i s t i c a l p ro pe r t i e s t ha t c an n e v e r b e ex a c t l y s at i s f i e d i n p r a c t i c e . T h e re f o r e t w o p r o bl e m s a r i s e : w e m u s t c ho o s e h o w m an y l a g s o f t h e a u t o c o r r el a t i o n t o a l l o w i nt o t he s o l ut i o n, a n d we m u s t g ua r d a g ai n s t t h e p o s s i b i l i t y t ha t t he s pe c tru m o f t he t r un c at e d a ut o c or r e l at io n m i g ht c o nt a i n z e r o s . T he f i r s t i s " s o l v e d " b y m ak i ng t he n u m be r o f l ag s a u s e r p a r am e t e r, w hi l e t h e s e c o nd r e qu i r e s t h e a dd i ti on o f a " s ta b " f ac t or t o t he z e r o l ag o f t he a ut o c o rr e l a t i on . T hu s t he n o r m al e qu a t i o ns w hi ch m us t b e s o lv e d f o r t h e w av el e t i nv er s e a r e m o di f i e d t o :

φ0+λ

φ1

φ2

φ3

x0

φ1

φ0+λ

φ1

φ2

x1

φ2

φ1

φ0+λ

φ1

x2

φ3

φ2

φ1

x3

1 =

0 0

φ0 +λ W h e r e φ is t h e a u tocorrelati o n o f th e s e is m ic t r a c e , λ i s t he s ta b f a ct o r , a n d x is t h e un k n o w n in ver se o p e ra to r . In c o mp a r i ng t hi s a lg o r i th m with fr e q u e n cy d o m a in d e c o n, i t is n o te d t h a t t h e y a r e nea r ly t h e F o u r ier e q u iv a le n t s o f o n e a n o th e r . W i nd o w i n g t h e a u t o co r re la t io n i n W iener d eco n i s e q u iv a le n t to s m o o th in g t h e p o w e r sp e c tr u m in f r e q u e n cy d e c o n. T he n u m b e r o f la g s i n t h e a u tocorrelati o n a n d th e n um be r o f p o in ts in t h e fr equ e n c y d o m a in sm o o t her a r e in v e r s e ly re la t e d . R ea so n in g v e r y lo o sely , w e h a ve:

nlagsΔt ≈

1 nsmoothΔf

⇒ nlags ≈


nsamps T = n smooth nsmooth Δt 1

4 -25

Wiener Spiking Deconvolution S i nc e o ur s y nt h e t i c ha s a 2 mi l s a mp l e ra t e , i t h as r o ug h l y 8 0 0 s am pl e s , s o w e e x pe c t t he 1 6 po i n t s mo o t he r w e us e d t o b e s i mi lar t o 8 0 0/1 6 = 5 0 l ag s o f t h e a ut o c o r r e l at i on . He r e a r e i s t he h e l p f i l e f r o m t he M a t l a b r o ut i n e d e c o nw : % % % % % % % % % % % % % % % % %

[trout,x]=deconw(trin,trdsign,n,stab) [trout,x]=deconw(trin,trdsign,n) routine performs a Weiner style deconvolution of the input trace trin= input trace to be deconvolved trdsign= input trace to be used for operator design n= number of autocorrelogram lags to use (and length of inverse operator stab= stabilization factor expressed as a fraction of the zero lag of the autocorrelation. ********* default= .0001 **********

Algo rith m: • Co mpu te th e a uto co rre latio n o f th e inp u t seismic trace . • Wind ow th e au to correla tion (bo xca r) to o nly n la gs • Se t up th e n o rm al equ a tio n s fo r th e wie ne r in verse, a dd th e sta b fac tor to th e d ia gon al, a n d solve • Co nvo lve th e in ve rse o pe ra to r w ith th e se ism ic tra ce.

trout= output trace which is the deconvolution of trin x= output inverse operator used to deconvolve trin

U s i ng e s s e nt i a l l y c o m pa r a b l e p ar a me t e r s to the f r e qu e n c y d o ma i n e x a mp l e , w e o b t ai n f o r t he n o i s e f r e e c a s e: 0.1 0.08 0.06

Exact rcs

0.04

deconw estimate 0.02

Noise free seismogram

0 -0.02

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

F r o m m a xc o r r , w e o b ta in a m a xi m um c r o ss c o r r e la ti on c o e f fi c i e nt b e tw e e n t h e e s ti ma te d r cs a n d t h e e xa ct o ne s o f . 3 9 a t a l a g o f . 2 s e c o nd s . V e r y c l os e t o t h e r e s u lt f r o m d e co n f. 4-26


Wiener Spiking Deconvolution In the frequency domain these results look like:

Exact rcs

0

-20

-40 deconw estimate

-60 Noise free seismogram -80

-100

0

50


200

250

H e r e a re so m e sa m p le d e c o ns o f th e n o is y t r a c e w hi ch h a ve a lr ead y be e n f ilt e r ed b a ck t o 7 0H z . High cut filtered: 70-80 Hz rolloff 0.14 0.12

exact rcs

0.1 deconw stab =.5 0.08 0.06

deconw stab = .01

0.04 deconw stab=.0001 0.02 0 -0.02

noisy seismogram 0

0.2

0.4

0.6

0.8

1

Filtered results Here are some results from maxcorr:

1.2

1.4

1.6

Max corr

1.8 Lag

deconw stab=.0001

0.1262

deconw stab = .01

0.1414

1.6000

deconw stab =.5

0.1802

3.0000


1.4000

4 -27

Prediction and Prediction Error Filters A f or wa r d p re d i c t io n f i lt e r i s a l i n ea r c on v ol u t i on a l o p er at or wh i c h i s d e s i gn e d t o p r ed i c t t h e n e x t el e m en t i n a s eq u e n c e g iv e n t h e v a l u es p r e c ed i n g i t . W e c an w ri t e t h is p ro ce s s u s in g t h e m a t ri x f or m o f c on v ol u t i on as :

w0 0 0

0

x0

w1

w1 w0 0

0

x1

w2

w2 w1 w0

0

x2

=

w3

Eqn 1

0 wm

w0

xN

wm+1

M u lt iplyin g b o th s id es o f t h is by t he transpose o f t h e Toepl it z W m a tr ix a n d f o rming t h e normal equations a s w e d id f o r in v ers e f il te ri ng gi v es:

φ0 φ1 φ2

φN

x0

φ1

φ1 φ0 φ1

x1

φ2

φ2 φ1 φ0

x2

φN

φ0

xN

=

φ3

Eqn 2

φN+1

H e r e , i n c on t r as t t o t he n o rm al e qu at i o ns f o r i n v e rs e f i l t e r i ng , we h a v e t he s i g na l a ut o c o rr e l a t i on a pp e ar i ng o n b ot h s i d e s o f t he e qu a t i o n. T h e s o l ut i o n t o t h e s e e qu at i o ns g i v e s a p r e di c tio n f i l t e r x , wh i c h, i n p r a c t i c e , i s u s e d t o p r e di c t v a l ue s " o f f t h e e n ds " o f t he s e qu e nc e o n wh i c h i t w as d e s i g ne d. W e m i g ht s us pe c t t h at s i nc e t he r e i s n o p ha s e i nf o r ma t i on go i ng i nt o t he p r e di c t i o n f i l t e r d e s i g n t ha t t he f i l t e r w i l l be m in i mu m p h a s e a n d t ha t i s i nd e e d t h e c a s e . 4-28


Prediction and Prediction Error Filters W e n ow w is h t o d ra w a p ar al l el b et w ee n p r e d ic t i on f i lt e ri n g a n d t h e d e s ig n o f i n v er s e fi l t e rs . I t t u r n s o u t t h at t h e r e la t io n s h ip i s n ot wi t h p r ed i c t i on f il t e rs bu t w it h a c l os el y re l at ed f il t e r, t h e p r ed i c t i on e r ro r f i l t er . T o d e r iv e t h i s , n ot e t h a t e q u at i on 1 c an be wr it t e n a s t h e f ol l ow i n g ex pre s si on w it h z t ra n sf o rm s : –1

w z x z = z w z – w0 Now, we can reformulate this into: –1

–1

w z x z – z w z = –z w0 N o t e t h a t t h e l e ft h a n d si d e is e s se n ti a l ly t h e d if f ere n c e be t wee n t h e p re d ic t ed va lu e s, w ( z ) x ( z ) , a n d t h ei r a c tu al v a lu e s, z - 1w( z ) . H e nc e it i s t er m ed t h e p re d ic t io n e rr o r . M a n ip u l a t i n g f u rt h e r: –1

multiply by z

–1

w z z –x z

= z w0

w z 1–zx z

= w0

eqn 3

T h e o p e ra t o r , 1 -z x (z ), i s c a l l e d a p r e di c ti on e rr o r f i l t e r o f u n i t l ag b e c a us e i t a s s e rt s t ha t w e c a n o pe r at e o n w (z ) t o y i el d w 0 f o l l ow e d b y a s eq ue nc e o f z e r os . T h at i s , w e c a n' t p os s i b l y p r e di c t t he f i r s t v a lu e i n a s e qu e n c e, s o t he e rr o r i n t ha t p r e di c t i o n m u s t a l w a ys b e 1 0 0 % , h o w e v e r , w e a s s e r t t hr o ug h e qu at i o n 3 , t ha t a l l o t h e r v a l ue s c an b e p r ed i c t e d w i t ho ut e rr o r . O f c o ur s e t hi s wo n' t be p o s s i bl e i n g e ne r al a n d w e w i l l o b t ai n a l e as t s qu a r e s s o l ut i o n w hi ch m i ni mi z e s t h e p r e di c ti on e rr o r .


4 -29

Prediction and Prediction Error Filters F o r m al ly , e q u a tio n 3 is i de n t ica l, to w i th in a sca le fa ct o r , o f th e z t r a ns fo r m e xp r e s sio n fo r th e de s ig n o f an in ve r se f ilt e r fo r w . T ha t is, w - 1 m u st s a tis fy : –1

wzw z = 1 If we write the matrix expression for equation 3, we have:

w0 0 0

0

1

w0

w1 w0 0

0

–x 0

0

w2 w1 w0

0

–x 1 =

0

–x N

0

0 wm

w0

Forming the normal equations as before leads to: 2

1

w0

φ1 φ0 φ1

–x 0

0

φ2 φ1 φ0

–x 1 =

0

–x N

0

φ0 φ1 φ2

φN

φN

φ0

eqn 4

A s ex p e ct e d , eq u ati o n 4 is ne a r ly i d en t i ca l t o t h e n o rm a l eq ua ti o n s f o r a W i en e r in v e rs e fi l te r. T h u s we m ake t wo c o n c lu s io n s: • P re d ic t io n fi lt e rs a n d p re d i ct i o n er ro r f il t ers a r e m i n im u m p h a se . • S p i ki n g ( W i en er ) d e co n v o lu t i o n i s i d en t ic a l t o u n it l a g p re d i ct i o n e rr o r fi l te ri n g. • T h u s d e c o n v o lu t i o n r em ove s t h e p re d ic t a bl e p a rt o f t h e tr a c e. 4-30


Prediction and Prediction Error Filters H a vi ng d e si g ne d a pr edi ct io n fi lt e r to pr e di ct o n e sa m p le a h e a d, it is a s im pl e m a tte r to d e s ig n o ne to p r e d ic t α sa m p le s a h e a d b y m o di fy in g e q u at io n 1 t o g iv e :

w0 0 0

0

x0

wα

w1 w0 0

0

x1

wα+1

w2 w1 w0

0

x2 = wα+2

eqn 5

0 wm

w0

xN

P ro c ed i n g a s be fo r e, w e eq u i va le nt t o e q u a ti o n 2 :

φ0 φ1 φ2

φN

wα+m

fo r m t h e no rm a l e q ua t i o n s

x0

φα

φ1 φ0 φ1

x1

φα+1

φ2 φ1 φ0

x2 = φα+2

φN

φ0

xN

eqn 6

φα+N

T he s ol u t i o n t o e q ua t i o n 6 g i ve s a N + 1 l on g p r e di c ti on o pe r a t o r w hi c h a t t e m pt s t o pr e d i c t α s a mp l e s a he a d. It i s c a l l e d a g a p pe d p r e di c t ion o p e ra t o r an d pl a y s an e s s e n t i a l r o l e i n t he s ur pr e s s ion o f m ul t i p l e s w h i c h f i t t h e c o nv ol u t i o na l mo de l .


4 -31

Gapped Predictive Deconvolution W e h av e s ee n t h a t W i en e r sp ik i n g d ec on vo l u t io n i s e q u i v al en t t o u nit l ag p re d i c t i on er r or f i l t er i n g. A s i m i l ar t e ch n iq u e is t o u s e p r ed i c t i on f i l t er s o f so m e l ag o t h e r t h an z e ro t o c om p u t e t h e p r e d i ct a bl e p ar t o f a s ei s m i c t ra c e a n d s u bt r ac t i t f ro m t h e o r ig i n al t ra c e. T h u s, i f t h e l ag u s e d is 1 , w e s h ou l d ge t t h e s am e r e su l t a s W i en e r s p i ki n g d ec o n v ol u t i on . T h is t e c h n iq u e i s m o st u s e f u l in a t t e n u at i n g m u l t ip le s t h at f i t t h e c on v ol u t i on a l m od e l . S u c h a m u l t i p l e is t h e w at e r b ot t o m m u l t i p l e wh i c h c an b e si m ula t ed (Backus, M.M., Geophysics, vol 24, p233-261, 1959) b y c on v o l v in g o u r se i s m og ra m wi t h t h e i m p u l s e r es p o n se o f a n o c ea n . 2 s e c on d s d e e p ( 2 - wa y t i m e ) a n d a n o c e an bo t t om rc o f . 4 ( h u ge ) . 0.1

Noise free seismogram with water bottom multiple

0.08 0.06 0.04

Water bottom impulse response. Scaled by .1

0.02

Noise free seismogram

0 -0.02

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

I t is d if fic u lt to se e t he e f fe c ts o n t h e s e is m o g r a m b u t if y o u lo o k cl o se l y a t .4 se c o nd s be h in d a m a j o r r e f le c to r, t hen y o u sh o u ld s e e a r eve rse p o l a r it y im a g e o f it su p e r i m po s e d o n th e se i sm o g r a m .

4-32


Gapped Predictive Deconvolution On the autocorrelations, we see a significant new side lobe has developed at a lag of .4 seconds. 4

A ut o c o r r ela ti o n of wa v ele t

3

A u t o c o r re lat i on of n o i se fr e e sy nt h et ic p l us wa te r bo t to m m u lt ip le

2 1 0 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

A u t o c o r re lat i on of n o is e fr e e sy nt h et ic

B a s e d o n t he s e d is p la ys w e a r e l e a d t o s e le ct a p r ed i c ti o n g a p o f 1 80 s a m pl e s ( . 3 6 s e co n ds ) a nd a n o pe r a t or l e n g th o f 5 0 s a m pl e s ( a s i n s p ik in g d e c o n) . H e r e i s t h e h e lp f il e f r o m t h e M a t la b f u n ct io n d e co n pr : % [trout,x]= deconpr(trin,trdsign,nop,nlag,stab) % [trout,x]= deconpr(trin,trdsign,nop,nlag) % % DECONPR performs Wiener predictive deconvolution by calling % PREDICT to design a prediction filter, nop long with lag nlag % and stab factor, using trdsign. The predicted part of trin, trinhat, % then formed by convolving the prediction operator with trin, % and trout is computed by delaying trinhat by nlag samples and % subtracting it from trin. The prediction operator is returned % in x. % % trin= input trace to be deconvolved % trdsign= input trace used to design the prediction operator % nop= number of points in the prediction operator % nlag= prediction lag distance in samples % stab= stabilazation factor expressed as a fraction of the zero % lag of the autocorrelation. % ************ default= .0001 *********** % % trout= deconvolved output trace % x= prediction operator % % See also: Peacock and Treitel, Geophysics vol 34, 1968 % and the description of PREDICT


Alg o rit h m: • De sig n a ga ppe d, min im um ph as e pre dict io n filt er ( w it h st ab f act o r) fr om t he au toco rr ela tio n of tr ds ig n. • Co n vo lv e t h e p re dict io n op era to r w ith t ri n t o f or m th e pre dict ab le pa rt . • Su bt ract t h e p re dica tb le par t of t rin f ro m tr in t o fo rm t he ou tpu t t ra ce.

4 -33

Gapped Predictive Deconvolution S o , ru n n in g d ec o n p r o n th e n o i se f r ee s y n t h et i c w it h m u l t i p l es u s in g t h e g ap a n d o p er at o r l en g t h m en ti o n ed a n d a d e f au l t s t ab f ac t o r gi v e s: 0.1

Estimate of multiples

0.08

True multiple free seismogram

0.06 0.04

Estimate of multiple free seismogram

0.02

Seismogram with multiples

0 -0.02

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

I f w e e x a m in e t he a ut o co r r ela tio n s, w e se e th a t th e p e r io d ic ity in th e a ut o co r r e l a tio n s a t la g o f .4 se co n d s h a s b e e n su r p r e ss e d . 4

True autocorrelation with out multiples

3 2

Es t imat e o f au to c o rr ela tio n w ith o ut mu lt ipl es A utocorrelation with multiples

1 0 -1 -0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8 Max Coeff

Re su lts fr om us ing max corr t o com par e with t he nois e f ree, m ult iple free se ismogram

4-34

With multiples After deconpr

0.9358

lag -0.1000

0.9736 -0.1000


Gapped Predictive Deconvolution H e re w e c om pa r e t h e re s u l t s f r o m f o l l ow i n g ou r pr e v i ou s de con pr b y a de c o nw ( n = 5 0 ) wi t h a s i ng l e de c o nw w i t h n= 1 8 0 + 50 = 2 3 0. 0.16 0.14 exact rcs

0.12 0.1

deconw (n 230)

0.08

de conp r (la g 1 80, n 50) followe d by de conw ( n 50 )

0.06 0.04

Noi se fre e se ismogram w ith mult ipl es

0.02

No ise f ree s eis mo gra m

0 -0.02

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Autocorrelations 7 exact rcs

6 5

deconw (n 230)

4 d econp r (la g 1 80 , n 5 0) fol lowed by d econw (n 5 0) N oise fre e se i smog ram wi th mul ti pl e s

3 2 1

No is e f re e s eis m ogr am

0 -1 -0.6

-0.4

-0.2

0

Re sults from using maxcor r to compare the t wo de cons on thi s page w ith the e xac t r c s


0.2

0.4

0.6

0.8 Max Coeff

deconpr + deconw long deconw

lag

0.3451

-0.3000

0.3892

-0.2000

4 -35

Burg (Maximum Entropy) Deconvolution B ur g d e c on v o lut io n i s clo se ly r e la te d to W ie n e r sp ik in g d e co n vo l ut io n in th a t it i s a c co m pl ish e d u sin g p r e d ic tio n e r r o r f ilt e r s o f un it la g . T he t e ch n iq ue w a s d e si g ne d b y J. P. Bu r g ( C la e rbout, J. F ., 19 7 6, F u ndamentals of G eophysi ca l Da t a P roces si ng, Mc G raw-Hill) in r e sp o ns e to d o ub t s a b o u t t he W i e ne r te c hn iq u e o f w in d o w in g th e a u to co r re la ti o n. He r e a so n e d t h at w in d o w in g th e a u to co r re la ti o n ca u se d th e no rma l e q u a tio n s to d e si g n a p r e d ict io n filt e r a s th o u g h t he d a ta ha d th e pr o p e r t y th a t it s a ut o co r r ela tio n va n ish e d a ft e r n la g s . I t ce rta in ly se ems r e a so n ab le to e x p e ct a b ette r sp e ct r a l e st im a tio n fr o m a n a lg o r it h m th a t e xp e c ts t he a ut o co r r ela tio n to co n ti nu e in s o me r e a s o na b l e w a y . B u r g f o un d a te ch niq u e w h ich d e si g ne d a p r e d ic tio n e r r o r f ilt e r d ir e c tl y fr o m th e d a ta r a th e r t h an f ir st fo r m in g t he au to c o r r e l at io n a nd w in d o w in g it. H is t e c hn iq ue w i ll n o t b e de ve l o pe d h e r e b u t w e w ill qu o te th e fo l lo w in g p r o p e r ti e s: • The Burg pr e diction e r r or f i lter m in imiz e s t he s quar e d er r o r f r om f o r wa r d and ba c k wa r d pr ediction.

sum

• T h e s o- c al l ed B u r g s p e c t r u m i s c o m p u t e d as t h e in v er s e of t h e s p e c t ru m o f t h e p r e d i c t i on e r r or f i l t er . ( T h i s i s n o t e x p l i c i t ly d on e i n B u rg d e c on v o l u t i on . ) • The Burg prediction error filter is minimum phase. • T ho u g h n o t c o m pu te d d ir e c tl y , t h e B ur g t h e o r y c an b e s h o w n t o b e e q u iv a le n t t o a W ie ne r t he o r y w h ic h , i n s te a d o f t r un ca ti n g t h e a u to co r r e la t io n, e x tr a po la t e s i t i n a w a y w h ic h m a x im iz e s t he r a n do m ne ss ( e n tr o py ) o f t he i m pl ie s s i g na l. ( Ka n a s ew i ch , E.R ., 1 9 81 , T i m e S e qu e n ce A n a ly s is in G e o p h y si cs ( 3r d E d it i o n ), U ni v ers it y o f A l be rt a P res s )

4-36


Burg (Maximum Entropy) Deconvolution Here is the help file from the Matlab function deconb: % % % % % % % % % % % % %

A l go ri t hm : • De si g n a un i t l a g pr e di c ti on e rr or f i l te r of routine performs a Burg scheme deconvolution of the l e ng th l on t rds i g n. input trace • Co nv ol ve t he pr e di ct i on trin= input trace to be deconvolved e r ro r f i l te r wi t h tr i n to trdsign= input trace to be used for operator design f or m the o utp ut t ra ce . [trout,pefilt]=deconb(trin,trdsign,l)

l= prediction error filter length (and length of inverse operator

trout= output trace which is the deconvolution of trin pefilt= output inverse operator used to deconvolve trin

N ot e t ha t t h e r e i s no s t a b f a c t o r i nv ol v e d ( t he a l g o r i t hm i s al w a y s s t a b l e ) a nd t h at w e m us t c ho o s e t he l e n g t h o f a pr e d i c t i o n e r r o r f i l t e r i n s t e ad o f t he n umb e r o f l a g s on an a ut o c o r r e l at i on f unc t i o n. Ho w e ve r , as a r ou g h g ue s s , w e m i g ht c o ns i d e r l t o b e s i m i l ar t o t h e n um b e r o f l a g s . He r e i s t h e r es u l t f r om d ec o n v o lv in g o u r n oi s e fr e e s yn t h et i c w i t h l = 5 0 . It ac h i e v e s a m a x i m u m c ro s s c or r e la t i on o f . 4 3 5 5 at a l ag o f ze r o, c o n s id er a bl y be t t e r t h a n t he ot her al go r it hm s . 0.1 0.08

exact rcs

0.06 0.04

estimated rcs from deconb

0.02

noise free synthetic

0 -0.02

0

0.2

0.4

0.6

0.8


1

1.2

1.4

1.6

1.8

4 -37

Burg (Maximum Entropy) Deconvolution B e lo w i s t h e r es u l t fr o m d ec o n v o l v in g t h e n o is y s ei sm ogr a m w it h t h r ee d i ff er en t p red ic t io n f il t er le n gt h s . A l l re su l t s h a v e be en h i gh c ut fi l te re d a t 7 0 H z . 0.16 0.14 exact rcs

0.12 0.1

deconb l=12 0.08 deconb l=25

0.06 0.04

deconb l=50 0.02 noisy seismogram

0 -0.02

0

0.2

0.4

0.6

R esu lt s fr o m u sin g m axc o r r t o c o mp ar e e ac h de c o nv o lu t io n w it h t h e exa ct r c s:

0.8

1

1.2

1.4

1.6

1.8

max corr

lag

deconb l=50

0.2227

-1.4000

deconb l=25

0.2168

-1.4000

deconb l=12

0.2093

-1.4000

S o w e s ee t h at , a t l e a s t o n t h is s y nt he t ic , t he B u r g a lg o r it hm d o e s a n e xc e ll e n t j ob , i s v e r y s t a b le , a n d n o t v e r y s e ns i ti v e t o t h e c h o ic e o f t he p a r am e t e r l .

4-38


The Minimum Phase Equivalent Wavelet A n y wa v el e t, n o m a t te r w h at i t s a m p l it u d e o r p h as e s p e ct r u m , c an be s ai d t o h av e a re l at ed w av e l et c al l ed i t s m i n i m u m p h as e eq u iv a l en t . I f t h e gi v e n w av e le t h a s a n a m p li t u d e sp ec t r u m wh i c h is p os i t iv e d ef i n i t e, t h en i t s m in im u m p h a se e q u i v al en t h a s t h e s a m e a m p l i t u d e s p e ct r u m b u t a p h a s e s p ec t r u m c om p u t e d a s t h e H i l be r t t ra n s fo rm o f t h e lo ga ri t h m o f t h e a m p l i tu d e s p e c t ru m. I f w e l e t w( t ) d en o t e t h e a r b it r ar y w av e l et a n d F a n d H b e t h e f or wa r d F ou r ie r a n d H il b e rt t r an s f or m s re s p ec t i v e ly , then: F [ w ( t ) ] = W ( f) = A ( f ) e x p ( i φ ( f) ) . I n t h i s e x p r es s i on , W ( f ) is t h e c o m p l ex - v a lu ed F o u r ie r s p e ct r u m a n d A ( f ) a n d φ ( f ) a re t h e re al - v al u e d a m p l i t u d e a n d p h as e s p e c t ra . N ow , i f W ( f ) i s p os i t iv e e v er yw h e re , t h e n t h e m in i m u m p h a s e eq uiv a le n t w av e l et h as a F ou r ie r s p e ct r u m gi v e n b y: W m i n( f ) = A ( f ) e x p ( i φ m i n( f ) ) . w h er e φ m in ( f) = H [ l n ( A( f ) ) ] . I n th e m or e g en e r al c as e , w h en A ( f ) m i gh t h av e a z er o s om e wh e r e o r w h e n l ar g e p or t i on s o f i t s d o m ai n a r e d om in a t ed b y n oi s e, it is c u s to m ar y t o c om p ute t h e m i n i m u m p h as e e q u i v al en t b y: W m i n( f ) = A μ ( f ) ex p( iφ μ _ m in ( f) ). φ μ _ m i n( f ) = H [ l n ( Aμ ( f ) ) ] . A μ ( f ) = A ( f) + μ m a x ( A( f ) ) I n t h e f i n al ex p r e ss i on , μ i s a s m a ll n u m b er , t yp i c al l y be t w ee n 1 0- 1 a n d 1 0- 6, w h os e e x ac t v a l u e d ep en ds o n t h e s i gn a l- to - n oi s e r at i o a n d t h e s p e c t ra l s h ap e o f t h e s ig n al sp ec t r u m . Methods of Seismic Data Processing

4 -39

The Minimum Phase Equivalent Wavelet I n i t s m o s t g en er al fo rm , t h e m in i m u m p h as e e q u i v al en t w av e le t i s n o t u n i q u e b ec au s e o f i ts d ep en d e n c e o n t h e " w h i t e n oi s e f ac t or " μ . T h e s im u la t io n be l ow s h o ws a [ 1 0 ,2 0 , 6 0 , 7 0] z er o- ph as e O rm sb y w av e l et a n d t h re e o f i ts m i n i m u m p h as e e q u i v al en t s . N o t e t h at a l l h av e d i st i n c l y d i f f er e n t p h as e a s a r e su l t o f t h e i r d i f fe r in g μ v al u e s . 0 20 40 60 80 -100

Minimum phase equivalent μ=.01 Minimum phase equivalent μ=.0001 Minimum phase equivalent μ=.000001

Original zero phase wavelet

-120 -140 0

5 0

100 15 Frequency0(Hz)

200

250

4 Minimum phase equivalent μ=.01

3 2

Minimum phase equivalent μ=.0001

1

Minimum phase equivalent μ=.000001

0

Original zero phase wavelet

-0.1 4-40

0

0.1 0. 0.3 time (seconds) 2 The Convolutional Model and Deconvolution

Vibroseis Deconvolution E x p l or at i on w it h V i br os e is ® s ou rc e s i s f u n d am en t a ll y d i f f er en t f ro m t h e u s e o f e x p l os i v e s ou r c es a n d n e e d s s p e ci a l c on s i d e ra ti o n i n o u r t h eo re t i ca l d e v e lo p m e n t . I n s t e ad o f a n u nkn ow n i m p u l s i ve s ou r c e w av e f or m , v i br os e is a t te m p t s t o c re at e a k n ow n ex t e n d e d s ou r c e kn o wn a s a sw e ep . A s w ee p i s t y p i c al ly a s ig n al w h i ch m o v es c o n t in u o u s ly t h r ou g h a s p ec i f i ed fr e q u e n c y b an d g en e r at i n g o n l y o n e f r eq u en c y i n s t an t an e ou sl y. T y p ic a ll y s we e p s a r e l i n ea r ( t h e s am e t i m e i s s w ep t a t ea ch f re q u e n c y) b u t n o n - l in e ar sw e ep s , w h i c h e m p h a s iz e t h e h i gh f re q u e n c i es , a re a ls o c om mo n . H er e i s a 1 0 - 7 0 H z , 8 s e c on d , li n e ar s w ee p :

sweep Time (sec)

T h e c o n v ol u t i on al m od e l s t il l f i t s t h i s so u rc e e q u a ll y we l l a s t h e im p u l si v e so u r ce . T h a t i s , gi v e n a r ef l ec t i v i t y r ( t ) , w e c an si m u l at e th e ea rt h ' s r es p o n s e by c o n v ol v i n g t h e s we e p w i t h r ( t ) : reflectivity

sweep c on vo l ve d with r e f l e c ti vi t y

O b v io u s ly , t h i s is a d i f f er en t s or t o f re c or d t h an t h e i m p u l s iv e s ou r c e a n d i s m u c h m o r e d i f f ic u l t t o in te r p re t e b ec au se t h e s ou rc e wa v e f or m i s s o e x t en d ed . W e n e ed a m e t h od o f co l la p s in g t h e s ou r ce t o a c om p a c t p u l se . T h a t t u r n s o u t t o b e t h e c ro ss c or re la t io n m et h o d . Methods of Seismic Data Processing

4 -41

Vibroseis Deconvolution According to the convolutional model, the vibroseis record is:

M o st o f t h e se t e rm s we re d e f i n ed a l r ea d y i n o u r d i s c u s si on o f t h e c on v ol u t i on a l m od e l . W e r e p e at t h e d e f i n it i on s h e re : the uncorrelated vibroseis record the vibroseis sweep near s ur fa c e e ff ec ts a n d v ib rator d i st o rt io n a convolutional approximati o n t o Q ef fec t s the subset of all multiples which are convolutional the desired reflectivity zero mean, white noise Now we cross correlate with the sweep and rewrite the model as:

where is the correlated vibroseis record is the autocorrelation of the sweep (Klauder wavelet) i s t h e ef f e ct i v e “ e art h f i lt e r” R o u gh ly s p ea ki n g , t h i s s ay s t h a t we c an u s e t h e c on v o lu ti o n al m o d el fo r co rr e la t ed v ib r os e is d a t a i f w e r eg ar d t h e s ou r c e w av e f or m a s t h e a u t o c or re l at i on o f t h e s we e p . T h is is o f t en c al l ed t h e K l au d e r wa v el e t .

4-42


Vibroseis Deconvolution T h u s , i n t h e s im p le s t c as e , w e ex pec t a c o rr el at e d v i br os e is r e co rd t o be t h e s we e p a u t o c or re l at i on c on v o lv e d wi t h 0.25 t h e re f le c t iv i t y . 0.2 0.15

autocorrlation of 10-70 sweep (Klauder wavelet)

wv t

30 Hz minimum phase pulse

wm t

0.1

0.25

0.1 0.05 0 -0.05 -0.1 -0.15 -0.1

-0.05

0

0.05

0.15

0.2

0.5

wm t • r t

0.4

0.3

wv t • r t

0.2

0.1

rt

0

-0.1

-0.2

0

0.2

0.4


0.6

0.8

1

1.2

4 -43

Vibroseis Deconvolution D e co n v ol u t i on o f t h e v i br os e i s s yn t h e t ic p r es e n t s a s p e ci a l p r ob le m si n c e w e c an n o t a ss u m e t h e wa v el e t i s m i n i m u m p h as e. M o st a p p ro ac h e s to t h i s p ro bl e m i n v ol v e a t t e m p t i n g t o m o d i f y t h e t h e c or re l at ed v i br os e is r ec o rd s o th a t i t s e m be d d e d wa v el e t i s m or e n e ar ly m i n i m u m p h a se . A n i m med i at e p ro bl e m a ri s es be c au s e a m i n i m u m p h a se wa v ef o rm c an no t be ba n d l im it e d y e t t h e v i b ro se i s w av e le t i s e x p l i ci t l y ba n d l im it e d . T h is m e an s t h at A L L m e t h od s wh i c h a t t em pt t o p r ec on d it i o n t h e e m b ed ded v i br os e is wa v ef o rm m u s t em p lo y a wh it e n o i se o r " st a b" f ac t or t o e x te n d t h e sp ec t r u m . I t i s u s u a ll y go od p r oc t i c e t o e n s u re t h at t h i s f ac t or i s t h e sa m e a s t h a t u s ed l at e r i n t h e d ec on a l g or it h m. G i v e n t h i s, a n d a s su m in g t h at t h e e m be d d e d w av e le t i s t h e K la u d er w av e le t , i t i s a s t ra i g h t f or wa rd ex e rc i s e in s i gn a l p ro ce s s in g t o d es i g n a c o n v er s io n o p er at or w h i ch c on v e r ts th e Kl au der w av e le t t o i t s m i n i m u m p h as e e q u i v al en t : 0.25

spectrum of minimum phase equivalent of Klauder wavelet

0

0.2 -20 0.15

Klauder wavelet

0.1

-40

0.05 -60 0 -0.05

-80

-0.1

minimum phase equivalent of Klauder wavelet

-0.15 -0.2 -0.1

4-44

-0.05

0

0.05

-100

0.1

-120 0

Spectrum of Klauder wavelet 50

100 Frequency (Hz)

150

200

250


Vibroseis Deconvolution A t t h i s p oi n t it i s a p p r op r i at e t o a s k w h y t h e v ib ro s ei s r e co rd sh o u l d be d e c on v o lv e d a t a ll . A f t e r a ll , t h e s i g n al i s ge n e ra t ed w i th a wh i t e sp e c t ru m o v e r t h e sw e p t ba n d a n d i s n o m i n al l y z e r o e ls e wh e r e. T h us t h e z e ro p h as e v i b ro se i s sy n t h et i c w h ic h c o n si s t s o f Kl a u d er w av el e t c o n v ol v ed w it h re f l ec t i v i ty i s a lr e ad y o p ti m a l. T h e a n sw e r, o f c o u rs e , l i es i n t h e o t h e r e ar t h f il t e ri n g e f f ec t s s u c h a s t h e n e ar s u r fa c e ef f ec t , m u l t i p l es , a n d a bs or p t i on ( Q ) . T h u s , a m in i m a l v i b ro se i s m o d e l f or d e c on v o l u t io n t h e or y i s:

sx t = wv t • n s t • r t U n l i ke u n l i ke t h e i m p uls i v e c a se , t h e g oa l o f v ib r os ei s d e c on v o l u t io n i s t o re c ov e r r ( t ) o n ly o v e r t h e s we p t b an d, e v en i n t h e n o is e fr e e c as e . W it h t h i s i n m i n d , we u s e t h e 3 0 H z m i n im u m p h as e wa v el e t t o re p r es e n t t h e n e ar s u rf ac e ef f e ct s a n d t h e K la u d e r w av e le t fo r t h e s ou rc e a n d c r ea t e t h es e s yn t h e t i c s e is m o gr am s : 0.5

30 Hz min ph s an d m in p hs Klau der wav lets con v olv ed wit h r eflect iv ity

0.4

30 Hz min phs and Klauder wavlets convolved with reflectivity

0.3

30 Hz min phs wavlet convolved with reflectivity

0.2

Klauder convolved with reflectivity

0.1

0

-0.1

r eflectivity

0

0.2

0.4

0.6


0.8

1

4 -45

Vibroseis Deconvolution T h e r es u l t s o f W ei n e r d ec o n o n t h e s e sy n t h e ti c s i s b el ow . N o t e t h a t we c om pa r e t o t h e r e fl e c t iv i t y c on v o lv e d w it h t h e K la u d e r w av e le t a n d n o t t o t h e r ef l ec t i v i t y it s e lf . A l l o f t h e s e d e c on v ol u t i on s h av e b e en f i lt e re d ba c k to t h e s w ep t b an d . 0.45

A

0.4 0.35

Deco n o f 30 Hz min phs an d Kla ud er wav let s co nv olv ed wit h ref lectiv it y

B

0.3 0.25

C

0.2 0.15 0.1

-0.05

0

0.2

0.4

0.6

0.8

De con of 30 Hz min p hs wa vle t convolve d wi th r ef le ctivi ty

D

Decon of Klauder convolved with reflectivity

E

Klauder convolved with reflectivity

0.05 0

Decon of 30 Hz min phs and min phs Klauder wavlets convolved with reflectivity

1

Max corr

W e c an s ee t h at t h e m in im u m p h as i n g o f A t h e v i br os e i s re c or d p ro d u c es a b e t te r d e c on b u t it a p p e ars t h e re s u l t h a s a 9 0 B d e gr e e p h a s e r ot at i on . I n f ac t t h i s is t h e C c as e : 0.15 D

0.3795

Lag 0.8000

0.3712

-2.3000

0.4978

-0.1000

-0.4138 -9.1000

-90° rotation of A

0.1 0.05 E

0 -0.05 4-46

0

0.2

0.4

0.6

0.8

1


Deconvolution Pitfalls T h e assumption s behind deconvolution t h e or y h e l p u s t o u nde rs t a n d i t s b as i s an d , sometimes , t o anticipa t e p r ob lem s b e f or e t h e y a ri s e . H e re w e w i l l examine s o m e c o m mon deconvolution "pitfalls". Mixed-wavetypes in the design gate. T h e m o st c o mm o n e x a mp le h e r e is th e o cc ur an ce o f a su r f ac e w av e o r si mi la r co h e r e n t n o ise t r a in i n t he d e si g n g a t e . T h e s im ul a te d su r fa ce w a v e b e g i ns a t . 2 se c o nd s.

Surface waveform

Reflection waveform 0

0.2

0.3 0.25

0.4

0.6

0.8

1

1.2

Contaminated with surface wave

-10

0.2 -20

Contaminated with surface wave

0.15 0.1

-30

-40

0.05

Simple synthetic

0

S imple synthetic

-50

-0.05 -60 -0.1 0

0.2

0.4

0.6

0.8

1

1.2

0

20

40

60 Frequency (Hz)

80

100

Deconvolution Results

Contaminated with surface wave S imple synthetic

RCs

0 0.2 0.4 0.6 Methods of Seismic Data Processing

0.8

1

1.2 4 -47

Deconvolution Pitfalls Here is a closeup of the ends of the traces so that the considerable phase distortion can be appreciated: Deconvolution Results Contaminated with surface wave

Simple synthetic

RCs

0.7

0.75

0.8

0.85

0.9

0.95

1

S o we s e e t h at t he pr e s e n c e o f t h e s ur fac e wa ve ha s c a us e d a g r e a t de a l o f ph as e d i s t o r t i o n e v e n qu i t e f a r f r o m t he on s e t of t h e w a ve . S i n c e t he p ha s e c or r e c ti on s ap pl i e d b y m i ni m um p ha s e d e c o nvo l u t i o n ar e de d uc e d f r o m a s mo o t he d r e p re s e n t a t i o n of t h e a mp l i t ud e s pe ct r um , t he pr e s e n c e of t h e s ur f a c e w av e p e ak i n t h e am p l i t ud e s pe c tru m c a u s e s e rr o ne o us p ha s e s t o b e c o mp ut e d .

4-48


Deconvolution Pitfalls Filtering before deconvolution. T h e i s s u e of f i l t e ri n g b e f or e d e c on v ol u t i o n c an b e a c o m p l e x on e w h i c h t ak e s s om e s u r p r is i n g t wi s t s . It m i g h t s e e m t h at on e c o u l d d og m a t i c al l y i n s i st t h a t a l l f i l t e ri n g b e f or e d e c on v ol u t i on m u s t be m i n i m u m p h a s e. How e v e r; a s w e sh al l s e e, t hat i s o f t en i n c or r ec t . I t g re a t ly h e l p s t h e d ec i s i on p r oc e s s t o c o n s i d e r w h e t h e r t h e u n f i l t er e d d at a i s i n t h e " m i n i m u m p has e s t at e " o r n o t . W e w il l s a y t h a t s ei sm i c d a t a i s in t h e " m i ni m u m p h a se s ta t e " i f t h ere is a s in g le e m be dd e d w a v el et a nd t h a t w ave le t i s m i n im u m p h a se . I f d a t a is in t he m in i m u m p h a s e st a t e, t h en a ny f il te ri n g sh o u l d be m i n im u m p h a se t o t ry t o p r es erv e t h a t st a t e. I f n o t , z er o p h a s e fi lt er in g m i gh t a ct u a l ly be pr ef er red i f it c a n be a rg u ed t o m o v e t h e d a t a t o wa rd s t he m i ni m u m p h a se s ta t e . T he s u r fa ce w a v e s y nt he t ic w h ic h w e p r e se nt e d e a r li e r i s n o t i n t he m i ni m um p ha s e s ta t e b e ca u s e i t c o n ta in s t w o e m b e d de d w a ve le t s : t he m in im u m p h as e r e fl e c ti o n w a ve le t a n d t he n o n -m in im um p h as e s u r f ac e w a v e fo r m . T he r ef o r e , m in im um p ha s e f i l te r in g before d e co nv o lu t io n m ig h t n o t b e a p p r op r ia te . I n f a ct , a z e r o p ha s e f il t e r d e si g n e d t o k no ck d o w n t h e s u r fa ce w a v e p e ak s h o ul d m o ve t h e d at a t o w ar d s t he m in i mu m p h as e s t at e . T h e e xa m pl e o n t he n e x t p ag e s h o w s t h at t hi s i s th e ca s e .


4 -49

Deconvolution Pitfalls 0.6

0.5

Deconv ol ved surfac e w ave syn theti c w it h mi nim um phase f il teri ng to remov e the surfac e w ave.

0.4

Deconvolved surface wave synthetic with zero phase filtering to remove the surface wave.

0.3

0.2

Deconvolved synthetic with surface wave present.

0.1

Deconvolved minimum phase synthetic

RCs

0

-0.1

0

0.2

0.4

0.6

0.8

1

1.2

Surface wave spectral peak

-10 -20 -30 -40

Uncontaminated spectrum

-50 -60

After filtering out the surface wave

-70 -80 -90 0

4-50

20

40

60 Frequency (Hz)

80

100


Deconvolution Pitfalls Design gate considerations. T y p ic a ll y t h e d e c o nv ol ut i o n o p e r a to r i s d e si g ne d o ve r a s u b s et o f t h e t r ac e c ho s e n f o r i t s h i g h s ig n al t o n o is e r a t io . C o n s id e r a ti o n s : - include the zone of interest - include large dominant reflectors - exclude surface waves and below basement - don't design on noise - highly non stationary data should avoid very long gates - o p e r a t o r l e n g th s ho u l d be n o m o r e th a n 1/ 3 to 1/ 2 o f th e ga te le n g t h 0.5

1 80 m il o per ato r d es igne d o ver .2 to .4 s eco n ds

0.4

0.3

60 mil operator designed over .2 to .4 seconds

0.2

60 mil operator designed over .4 to .6 seconds

0.1

60 mil operator designed over entire trace

0

-0.1

RCs

0

0.2

0.4

0.6


0.8

1

1.2

4 -51

Deconvolution Pitfalls Filtering after deconvolution. R e al s e i s mi c d at a a l w ay s c o nt a i ns a g r e at d e a l o f a pp ar e nt l y r a nd o m n o i s e . W e ' v e a l r e ad y m e nt i o ne d t he i n ad v i s ab i l i t y o f d e s i g ni ng t he o p e ra t o r o n n oi s y d a t a. A l s o , i t i s a l m os t a l w a ys n e c e s s ar y t o f i l t e r d a t a b ac k t o s o me s i g na l b an d a f t e r d e c on v ol u t i o n. M o s t d e c o n a l g or i t h ms c a nn ot d i s t i n gu i s h s i g n al f ro m n o i s e a n d s o w hi ten b o t h. T hi s c a n h a v e a d i s a s t r ou s e f f e c t o n s u c h n oi s e s e ns i t i v e p r o g ra ms a s r e s i du al s t a t i c s . 0

noisy data spectrum -20 -40

noiseless data spectrum

-60 -80

0

50


200

250

0.2

0.1

noisy seismogram

0

noiseless seismogram

-0.1

0

4-52

0.2

0.4

0.6

0.8

1

1.2


Deconvolution Pitfalls 0.5

RCs filt er ed b ack to 60 Hz.

0.4

Deconvoled noisey seismogram, filtered back to 60 Hz.

0.3

0.2

Deconvoled noisey seismogram

0.1

Deconvoled noiseless seismogram RCs

0

-0.1

0

0.2

0.4

0.6

0.8

1

1.2

T h o ug h th e r esu lts o f th e d e c o nv o lu tio n o f n o is y da t a a r e a lm o st a lw a y s b ette r w h e n f ilt e r e d b a c k , in g e n e r a l, th e m a tch t o th e R C' s is st ill m u ch w o r se th a n w i th o ut n oi se .


4 -53

Deconvolution Pitfalls Iterative deconvolution. I t i s o f te n a s s u m e d t h at d e c on v o lu t i o n i s s om e t h i n g t h a t n e ed s d oi n g o n c e a n d i s t h e n b e st f or go t te n . T h i s a t t i t u d e u su a l ly le ad s t o u n der wh i t e n e d d at a w it h r es i d u al p h as e ro t at i on s . S i n c e t h e a s s u m p t i on s o f d e c on v o lu t i on a re n e v er p r ec i s el y m e t, i t i s o f t en u s e f u l t o a p p l y s ev e r al d i ff e re n t d e c on s f or d i f fe r en t p u r p o se s . F o r e x am ple , we m ay u se p r ed i c t i v e d e c on t o a t t ac k a m u l t i p l e a n d t h e n s p i k in g d ec on v ol u t io n t o s h ar p en r es ol u t i on . M or e i m p or t an t l y, d ec on v ol u t io n a l g or i th m s c an n o t d i s t in g u i s h b e tw e en s ig n al a n d n oi s e. T h u s w e m u s t t h in k o f t h e m a s wh i t e n in g t h e s p e ct r u m o f si g n al p l u s n o is e . I f d e co n v ol u t i on i s t h e n fo l lo we d b y a n y p r oc e s s w h i ch c an r ej e c t n oi s e w h i le re t ai n i n g s i g n al , t h e re s u l ti n g d at a w il l h a v e a n o n - w h it e , l ow e r r e so l u t io n , s p e ct r u m . T h e m o st c om m o n e x am p le o f t h i s i s C M P s t ac ki n g . T h us i t i s o f t e n n e ce s s ary t o r u n a p o s t - st a c k d e c on v ol u t i on o r w h it e n i n g s t e p t o e n su re m ax i m u m r e so lu ti o n . I f p o s ts t ac k m in i m u m p h as e d ec o n v ol u t i on i s d es i r ed , c ar e s h ou l d b e t ak en t o en s u r e t h at z er o- p ha s e f i lt e r in g wa s n ot d on e a f t er t h e p re - s t ac k d e c on v o lu t i on .

4-54


Reflectivity Color Here i s a r e fle c tiv it y series c o mp ut e d fr o m a n A lb er ta well a nd p lo tt e d versus t im e . 0.06 0.04 0.02 0 -0.02 -0.04 -0.06

0

0.2

0.4

0.6

0.8

1

Time (sec) Since standard deconvolution algorithms reflectivity spectrum, we are motivated spectrum and see if it is white.

assume a white to compute the

White spectrum is flat

0 -10

Anti-alias filter rolloff

-20 -30 Low fr e que ncy de ca y i s ty pi ca l we ll l og beha vi er a nd indic a tes consi de r abl e spe ct ra l col or .

-40 -50 -60

0

50


200

250

So we see that this spectrum is non-white. What color is it?


4 -55

Reflectivity Color Example of a reflectivity estimate via Weiner deconvolution for a non-white reflectivity. The traditional whitened estimate is shown (A) along with a color corrected estimate (B) and the original well log (C).

0.25

A

0.2

0.15

B

0.1

0.05

C

0

-0.05

-0.1 0

0.2

0.4

0.6

0.8

1

Time (sec) 4-56


Reflectivity Color A: Reflectivity estimate from normal Weiner decon B: A convolved with a zero phase color restoration operator C: A convolved with a minimum phase color restoration operator D: Original well log at 2 ms sample rate N u m b ers gi ve ( M a x im u m co r re la t io n c o ef f, l a g a t m a x { s a m p l es} ) f o r t h e c o rr el a ti o n be tw ee n a n e st im a t e a n d t h e a n sw e r g iv e n in D 0.35

0.3

A (.8037,-.2)

0.25

0.2

B (.8470,-.2)

0.15

0.1

C (.8816,-1.8)

0.05

D

0

-0.05

-0.1

0

0.1

0.2

0.3

0.4

0.5

Methods of Seismic Data Processing Time (seconds)

0.6

0.7

0.8

0.9

1 4 -57

Q Example H e r e i s a n e xa mp l e com p ar i n g a s ta tion a r y, mi n i m um ph a s e s y nt he t i c w i th a s e r i e s o f c o nsta nt Q s y n th e ti c s . E a c h c o n s ta nt Q s y nth e ti c ha s th e s am e 30 H z , mini m um ph a s e w a ve l e t c o nv ol v e d w ith i t as t he s t at i o na r y s y nth e ti c do e s .

0.14

wavelet

0.12

Q=25

0.1

Q=50

0.08

Q=100

0.06

Q=150

0.04

Q=200

0.02

stationary

0 -0.02

RC's 0

0.2

0.4

0.6

0.8

1

1.2

T h e e ff e c t o f Q a t t e n u at i on c an b e s e en t o h a v e a t l ea s t t h r ee c h ar ac t e ri s t ic s : a p r og r es s iv e l os s o f f r eq u e n c y c on t e n t w i t h i n c r ea s i n g t i m e , a p ro g re ss i v e l os s o f o v e r al l a m p l i t u d e , a n d a p r og r es s iv e t i m e d e l ay . T h e c on s t r u c ti o n o f o n e o f t h e se s yn t h e t i cs i s d et a il e d o n t h e n e x t p a ge .

4-58


Q Example E a c h o f t h e Q sy n t heti c w a s c r e a t e d b y f ir s t co n st r u ct in g t he " Q m a t r ix " w h ic h a pp l ie s a Q r e s po n s e t o a t im e s e r ies v ia a g e n e ra liz ed co n v o lu ti o n. H e re t h e p r o c e s s i s d e p ict e d g r a p h ic a ll y fo r th e Q =2 5 ca se.

=

input time

Q= 25 RCs in seismogram time E a c h c ol u mn o f t h e Q m a t r i x c o nt a i ns t h e Q = 2 5 i m pu l s e r e s p o ns e f o r t he i np ut t i me o f t he c o l u mn c on vo l ve d w i t h t he 3 0 Hz m i ni m um pha s e s o ur ce wa ve f o r m . If w e F o ur i e r t r an s f o rm e a c h c o l um n, w e c a n s e e d i r e c t l y t he Q a m pl i t u de a nd p ha s e r e s po ns e : Q Matrix

input time

Am pl itude s p ec tr um o f Q m a tr ix Methods of Seismic Data Processing

input time

Phase spectrum of Q matrix 4 -59

Q Example T h e f i rs t s t ep in d ec o n v ol v i n g t he Q s yn th et i c s is t o d e t er m i n e e x p on e n t i al g ai n c o rr ec t i on s a n d a p p l y t h e m . T h e re s u lt i s: 0.14 0.12

Q=25

db/sec = 17

0.1

Q=50

db/sec = 11

0.08

Q=100 db/sec = 5

0.06

Q=150 db/sec = 2

0.04

Q=200 db/sec = 0

0.02

stationary RC's

0 0.02 0

0. 2

0.4

0.6

0. 8

1

1.2

T h e s e g a i n f ac tor s w er e de t e r m i ne d e m pi r i c a l l y a s i s s t an da r d p r ac t i c e . I t a pp e ar s t ha t Q =2 5 m ay b e a b i t u nd e r g a i ne d .

4-60


Q Example N ex t w e r un W e i ne r d e c o nv o l ut i on w i t h t he s am e p a r am e t e rs f o r e ac h t r a c e ( 3 0 l a gs a n d . 0 0 0 1 wh i t e n o i s e ). 0.14

MaxCC

Lag

0.12

Q=25

0.1557

-0.5000

0.1

Q=50

-0.2965

-3.6000

0.08

Q=100

-0.3676

-3.0000

0.06

Q=150

-0.3733

-2.8000

0.04

Q=200

-0.3693

-2.6000

0.02

Q=∞

-0.4811

-2.3000

0

R C's

-0.02

0

0.2

0.4

0.6

0.8

1

1.2

I t 's c lea r f ro m t h is e xa m p l e t h a t t h e d ec o n v o l ut i o n re su l t d eg ra d e s s te a d il y w it h d ec re a si n g Q . Kee p in m i n d t ha t t h is i s a " be st ca s e" sc en a r io : no n o i se , n o m u lt ip l es , m i ni m u m p h a s e so u r ce , a n d w h it e re fl ec ti v it y . A l s o , ev en t h e Q= 2 5 c a se i s n o t a n u n rea s o n a b le a t t en u a ti o n le ve l be ca us e t he m a x i m u m t im e i n t h e sy n t h et ic is o n l y 1 s ec o n d . S i n ce t/ Q d et er m in e s t h e a c tu a l a t te n ua t i o n , 1 / 2 5 is t he s a m e a s 2 / 5 0 o r 3/ 75 .


4 -61

Geophysics 557 Final Exam Study Guide What are the expressions for P and S wave velocities in terms of the Lame constants? How does the Vp/Vs ratio depend on poison's ratio? How is the normal incidence reflection coefficient (for P waves) related to impedance? What is impedance? How are normal incidence reflection and transmission coefficients related? What is meant by the term "impulse response"? What physical effects are modeled in construction of a normal incidence seismogram a discussed in lecture? Under what conditions can the earth response to a real source be modeled as a convolution of a source waveform with an impulse response? What is the major use for 1-D synthetic seismograms? Why are multiples and transmission losses not typically included in such models? What information is typically input to the 1-D synthetic seismogram computation? Describe convolution by replacement. Describe convolution as a weighted sum. Use the integral form of convolution to prove that convolution is linear. That is, the convolution of c with a+b is equal to the convolution of c with a plus the convolution of c with b. What happens when any function is convolved with a complex sinusoid? What is technical meaning of a phrase " a 90 degree wavelet" ? 45 degree? any degree? What is the definition of the 1-D Fourier transform? The inverse transform? How is the Fourier phase spectrum defined? The amplitude spectrum? How are the "time width" and "frequency width" of a function related? What is a Dirac Delta function? What is its Fourier transform? When g and f are convolved in time, what happens to their Fourier spectra? What happens to their amplitude spectra? Their phase spectra? How is the Nyquist frequency related to the temporal sample rate? What is the frequency sample rate of a time series of length T? To avoid aliasing a signal of frequency Fmax, what sample rate must be used? From a practical viewpoint, the signal frequencies in your data should be less than a constant, c, times Fnyquist. What is a good value for c? What sample rate should you use for 80 Hz signal? For 130 Hz? For 40 Hz? What is an anti aliasing filter? When should it be used? What are the two main purposes of a zero pad when doing a discrete Fourier transform (DFT)? What is circular convolution? What is the relationship between the DFT and the fast Fourier transform (FFT)? How is the Z transform related to the DFT? A filter which has a pole close to the unit circle at some frequency does what to that frequency? How about a zero close to the unit circle? Where are the zeros of a minimum phase filter located in the z plane? What is the definition of minimum phase in terms of causality and stability? How is an inverse filter defined in the z plane? What is meant by a stable inverse?

11-2

Study Guide

What is meant by the statement that "This seismic data is minimum phase"? (Note that the statement is technically always false but it has a practical, definite meaning.) In order to assert that seismic data is "minimum phase" at some stage of the processing what conditions must be met? How is cross correlation defined? What does it mean? What does the cross correlation lag mean? What is an autocorrelation? What is the expected autocorrelation of a random sequence? What are the two central problems of spectral estimation? What is the role of the "window" in spectral estimation? What kind of spectrum is well modeled by the Burg spectrum? What is the 2-D Fourier transform of a linear event with apparent velocity v? Draw a sketch showing (x,t) space and (f-k) space for a range of different apparent velocities. What is the most likely apparent velocity and where is it found in (f,k) space? What is spatial aliasing? For a given apparent velocity and spatial sample rate, what is the critical frequency at which spatial aliasing begins? How can convolution be expressed as a matrix operation? Draw a diagram showing the Toeplitz matrix symmetry. Describe the six basic modes of seismic attenuation. Geometric spreading corresponds to what conservation law? Under geometric spreading, amplitude decreases proportional to what? How is Q defined? What is the formula for amplitude loss in a constant Q theory? What is the formula for transmission losses in a layered medium? What phenomenon is responsible for trapping large amounts of seismic energy in the near surface? What is true amplitude processing? What do constant Q models predict about the signal bandwidth of seismic data? The phase effects associated with Q attenuation are known as what? What assumptions are required to derive these phase effects? What is unique about a minimum phase wavelet? What is the most important property of minimum phase wavelets from the viewpoint of deconvolution theory? What is meant by velocity dispersion? What is the convolutional model? Write a mathematical expression for the model as it is applied in deconvolution theory. Define each term and state the assumptions which constrain each. Can all types of multiples be included in the convolutional model? Why or why not? The convolutional model expects the seismic trace to be stationary, What is meant by this? Is it a reasonable expectation? What are the essential steps of frequency domain spiking deconvolution? How is the seismic wavelet estimated in frequency domain spiking deconvolution? If a wavelet is known to be minimum phase, then its inverse can be found by solving a set of matrix equations whose left hand side involves not the wavelet itself but a statistical measure of it. What is this measure? Explain intuitively how this result is equivalent to computing a wavelet's phase spectrum from its amplitude spectrum under the minimum phase assumption.


11- 3

Wiener spiking deconvolution assumes that the autocorrelation of the seismogram is similar to the autocorrelation of the wavelet. Justify this assumption by argument from the convolutional model. What is meant by the 'stab' factor or 'white noise' factor in deconvolution? It is customary in seismic data processing to follow deconvolution by a bandpass filter. Is this a sensible practice? Either justify it or refute it by argument from deconvolution theory. What is the relationship between prediction error filters and spiking deconvolution? Explain why deconvolution keeps the prediction error and rejects the predictable part of a seismic trace. How is gapped predictive deconvolution implemented? What is a typical example of a type of multiple which it is designed to attenuate? How can the prediction gap be chosen? How are midpoint and offset defined in terms of source and receiver coordinate? Illustrate with a diagram. The near surface is generally assumed to cause effects which are a function of what coordinates? The subsurface effects are often assumed to be a function of what coordinates? How is a static delay defined? What physical effect is often used to justify the assumption of static delays in the near surface? What is the definition of source static? Receiver static? What is meant by the term "datum"? How does its choice effect the statics application? What datum is most appropriate for pre stack processing? What are surface consistent methods? Why are they useful? List some examples of common surface consistent applications? What is the definition of vertical traveltime? How can instantaneous velocity be computed as a function of vertical traveltime? What is a time-depth curve and what is it used for? How is average velocity defined? In what sense is it an average? What is the mean velocity and how is it defined? Define Vrms in terms of instantaneous velocity and as is relates to the mean and average velocities. Which is always greater the average or the rms velocity? What is interval velocity? Define at least two type of interval velocities. What is the expression for the addition of two interval velocities? What is the expression for the "Dix" interval velocity calculated from two closely positioned rms velocity measurements? Under what conditions can an interval velocity be said to approximate a local wave propagation velocity? How can imaginary interval velocities result from a Dix interval velocity calculation? Derive the traveltime equation for normal moveout. Use a diagram to show the meaning of all quantities. What is the shape of the traveltime curve? What is the shape of the wavefront in the nmo experiment as it approaches the receivers? How must the nmo equation be modified if the reflector is dipping? What is the stacking velocity for a dipping reflector beneath a constant velocity overburden?

11-4

Study Guide

How must the nmo equation for a dipping reflector be modified to take the azimuth of the seismic line into account? What is the definition of stacking velocity? Explain why stacking velocity is always a function of offset. Using a diagram, derive the geometric relation between wavelength components and wavelength for a periodic planar wavefront. What are wavenumbers? What is apparent velocity? What are the mathematical limits (upper and lower) of apparent velocity? How does apparent velocity relate to wavelength components and wavenumbers? What is Snell's law? Snell's law can be considered as the conservation of what quantity? What is a v(z) medium? When raytracing in a v(z) medium, what quantity is conserved and how is it defined? Derive the distance traveled and traveltime integrals for raytracing in a v(z) medium. How can the ray parameter be measured? When velocity increases linearly with depth, what shape are the raypaths? The wavefronts? (Exact equations not necessary.) For the nmo experiment in a v(z) medium, explain how the result that stacking velocities may be approximated by rms velocities arises. What assumptions are required? In practice, when can we expect it to be roughly valid? The Dix equation moveout can be interpreted as allowing the replacement of the real v(z) medium by a constant one with properly chosen parameters. Explain this. Explain why interpolation of trace sample values is needed in nmo removal. What is moveout stretch? Why does it arise? What are residual statics? How are they computed? What is their purpose? What processes should be run on seismic data prior to attempting a residual statics solution? What is velocity analysis? How is it performed? What processes should be run on seismic data prior to attempting a velocity analysis? Do statics and moveout removal commute? That is, do you get the same result regardless of the order of the processes? If not what is the preferred order? In the extension of nmo and dip to v(z), what quantity must be measured in addition to stacking velocity in order to allow the computation of apparent dip and the "dip correction" of stacking velocities? What can be said about the staking velocities of multiples? Where will they be found on a stacking velocity analysis chart? After stacking, the power of random noise can be expected to be reduced by what factor? Considered as an "f-k" process, stacking can be said to pass what portion of the offset wavenumber spectrum? Are "f-k" filters applied to cmp gathers likely to improve a stack? What if they are applied to shot or receiver gathers? What is a zero offset section? How does is serve as a model for a stack? What is the relation between traveltime gradient measured on a stack and the normal incidence ray parameter? What information is needed for the raytrace migration of a normal incidence seismogram?


11- 5

What are the algorithmic steps in normal incidence raytrace migration? Time migration processes are biased towards what class of rays? How are these rays handled? When is time migration a valid process? When is depth migration a valid process? What is the migrator's formula? How can it be used? Explain post stack migration by replacement of each point with a wavefront. How are the wavefronts defined? What is meant by "migration dip"? Should data from an area where all geologic dips are less than 5 degrees be migrated? Why? Should a steep dip algorithm or a low dip algorithm be used? What is Huygen’s principle? What is the traveltime curve of a point diffractor for post stack migration? For pre stack migration? What is a diffraction chart? Explain how diffraction curves can be used to construct a zos image from a geologic model? What is the exploding reflector model? Why is it useful? In the exploding reflector model, what is the mathematical expression for the migrated section? For the zos image? Using the exploding reflector model to explain wavefield extrapolation. What is the mathematical expression for an extrapolated section? What is the relationship between any extrapolated section and the migrated depth section? What is the dispersion relation? Use the dispersion relation to derive the mapping which defines f-k migration. Use a diagram to illustrate the mapping of the f-kx spectrum to the kx-kz spectrum. What is the meaning of the evanescent boundary? What determines the maximum kx wavenumber after migration? What determines it before migration? Draw a flow chart for f-k migration. What is f-k wavefield extrapolation? Derive the expression for the f-k phase shift required to shift the datum by ∆z. Explain how recursive f-k phase shifting can be used to create a v(z) migration algorithm. Draw a flow chart. What is the geometric shape of the wavefield extrapolation operator? What are the two distinct components of the operator? How can is be applied in the (x,t) domain? What is the major distinction between time and depth migration? Can time migration produce a depth section and depth migration a time section? Explain. What is Kirchoff migration? What is the shape of the Kirchoff migration operator (constant velocity) when applied post stack? Pre stack? Describe a general method to determine the shape of the Kirchoff migration operator. Your method must be valid for any (x,y,z) location, any velocity, and pre or post stack. What is pre stack time migration? When is it a valid process? Should it be inferior, the same, or superior to stacking and post stack time migration? What is DMO? Describe a flow using DMO that should give similar results to pre stack time migration. What are the strengths and weaknesses of the DMO approach? Under what circumstance is DMO->stack->migration exactly the same as pre stack migration? What kind of velocities should be input to the NMO removal step in a flow involving DMO? How can these velocities be obtained? Describe, without equations, the essential steps in CSP migration? How does CSP analysis effect velocity resolution?

11-6

Study Guide

What is the central (most difficult and most important) problem in the application of depth migration to the thrust belt? Describe at least one approach to solving this problem. What is wavelet processing? What are the essential steps in wavelet processing? When should it be done in a processing flow? When is it necessary? What are two common methods of wavelet estimation? What is impedance inversion? When should it be run in a processing sequence? How can the convolutional model (from deconvolution theory) be used to justify impedance inversion? What is the major computation involved in impedance inversion? Describe at least two common problems with impedance inversions that are difficult to solve. What is the expected behavior of the amplitude spectrum of the radiated waves from a dynamite sources as a function of charge size? Explain how Q effects necessarily lead to a time variant (i.e. non-stationary) signal bandwidth. What is the relationship between spectral width and wavelet width? What is a "corner frequency"? When can it be observed? What does it mean? For a constant velocity earth, what are the equations which express the limits of observable scattering angle due to aperture, record length, and spatial aliasing? Make a sketch of their basic form. Staring from the theory of f-k migration, derive an expression for the maximum kx after migration as a function of frequency, velocity, and scattering angle. Explain the relevance of this to the problem of resolving small horizontal features? What steps can be taken in recording or processing to increase horizontal resolution?


11- 7

Exam sampler. There will be between 30-35 multiple choice questions and 48 short answer questions. PLEASE ANSWER ALL OF THE FOLLOWING QUESTIONS. There are a total of 100 marks (points) for the examination. You have about 100 minutes for the exam. Write all work directly on the examination sheets. If you need more room, you may attach a work sheet with your name and the question number on it. PLEASE HAND IN THE EXAMINATION SHEET AND ALL WORK SHEETS WITH YOUR ANSWERS.

Multiple Choice Questions (2 points each) INSTRUCTIONS: For each question, there are two best (most correct) responses. Choosing both correct responses and no incorrect ones is worth two points. One correct and one incorrect is worth one point, and any other result (including more than two selections) is worth zero. Write your answers in the space provided below each question. 1) The 1-D synthetic seismogram, as discussed in lecture: a) can be made to contain all possible multiples. b) is useful for modeling AVO and converted waves. c) applies the source waveform of a band limited source by correlation. d) is an excellent model of a trace on a stacked and migrated section, provided that all possible multiples are included in the solution. e) is based on ray theory and normal incidence reflection and transmission coefficients. answer _________ 13) Average velocity: a) characterizes the shape of diffraction curves on a cmp stack. b) is depth divided by the vertical traveltime to that depth. c) is a mathematical average over depth. d) can be measured on an f-k plot. e) is a mathematical average over traveltime. answer _________ 20) Post-stack F-K migration: a) easily handles variable velocity. b) is useful to explain the transformation of the data spectrum under migration. c) shows that a constant frequency, f, maps to a hyperbola in (kx,kz). d) is a steep dip algorithm.

11-8

Study Guide

e) works by applying a phase shift to the f-k transform. answer _________

31) Minimum phase: a) is the state of all raw seismic data. b) is the desired state of final migrated sections. c) means that the phase is the Hilbert transform of the log of the amplitude spectrum. d) is the smallest possible phase. e) is only possible for time series which have inverses. answer _________


11- 9

Short answer problems (10 points each) Please work directly on the examination sheets. Show all work. 1) Deconvolution Suppose a dynamite dataset has mistakenly had a zero phase bandpass filter applied before deconvolution. Assuming that the original data was noise free, with a minimum phase wavelet and a white reflectivity: a) Write a time domain convolutional expression for a single trace after the bandpass filter was applied but before deconvolution. Then rewrite this expression in the frequency domain.

b) By working through the steps in frequency domain deconvolution, derive a frequency domain expression for the embedded wavelet remaining after standard minimum phase deconvolution. Show each step in the frequency domain deconvolution process and indicate any smoothing but you may assume that stab factors (white noise) are not needed.

c) Derive an expression for a correction filter which can be applied after deconvolution to give the desired result from the deconvolution process.

11-10

Study Guide


11-11

Methods of Seismic Data Processing,

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