introduction to seismic inversion methods

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Introduction to

Seismic Inversion Methods

Brian H. Russell

Hampson-RusselloftwareServices, td.

Calgary,Alberta

Course Notes Series, No. 2

S. N. Domenico, Series Editor

Society f Exploration eophysicists



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]:nl;roduc 1 on •o Selsmic I nversion •thods Bri an Russell

Table of Contents

PAGE

Part I Introduction 1-2

Part Z The Convolution Model 2-1

Part 3

Part 4

Part 5

P art 6

P art 7

2.1 Tr•e Sei smic Model

2.2 The Reflection Coefficient Series

2.3 The Seismic Wavelet

2.4 The Noise Component

Recursive Inversion - Theory

3.1 Discrete Inversion

3.2 Problems encountered with real

3.3 Continuous Inversion

data

Seismic Processing Considerati ons

4. I ntroduc ti on

4.2 Ampl rude recovery

4.3 Improvement f vertical

4.4 Lateral resolution

4.5 Noise attenuation

resolution

Recursive Inversion - Practice

5.1 The recursive inversion method

5.2 Information in the low frequency component

5.3 Seismically derived porosity

Sparse-spike Inversi on

6.1 I ntroduc ti on

6.2 Maximum-likelihood aleconvolution and inversion

6.3 The L norm method

6.4 Reef Problem

I nversi on appl ed to Thi n-beds

7.1 Thin bed analysis

7.Z Inversion compari on of thin beds

Model-based Inversion

B. 1 I ntroducti on .

8.2 Generalized linear inversion

8.3 Seismic1 thologic roodellng (SLIM)

Appendix8-1 Matrix applications in geophysics

Part 8

2-2

2-6

2-12

2-18

3-1

3-2

3-4

3-8

4-1

4-2

4-4

4-6

4-12

4-14

5-1

5-2

5-10

5-16

6-1

6-2

6-4

6-22

6-30

7-1

7-2

7-4

8-1

8-2

8-4

8-10

8-14



Introduction to Seismic Inversion Methods Brian Russell

Part 9 Travel-time Inversion

g. 1. I ntroducti on

9.2 Numerical examplesof traveltime inversion

9.3 Seismic Tomography

Part 10 Amplitude versus offset (AVO) Inversion

10.1 AVO theory

10.2 AVO nversion by GLI

Part 11 Velocity Inversion

I ntroduc ti on

Theory and Examples

Part 12 Summary

9-1

9-2

9-4

9-10

10-1

10-2

10-8

11-1

11-2

11-4

12-1



Introduction to Seismic •nversion Methods Brian Russell

PART I - INTRODUCTION

Part 1 - Introduction

Page 1 - 1




I NTRODUCTON TO SE SMIC INVERSION METHODS

, __ _• i i _ , . , , • _, l_ , , i.,. _

Part i - Introduction

. .

This course is intended as an overview of the current techniques used in

the inversion of seismic data. It would therefore seemappropriate to begin

by defining what is meant by seismic inversion. The most general definition

is as fol 1 ows'

Geophysical inversion involves mapping the physical structure and

properties of the subsurface of the earth using measurementsmade on

the surface of the earth.

The above definition is so broad that it encompasses irtually all the

work that is done in seismic analysis and interpretation. Thus, in this

course we shall primarily 'restrict our discussion to those inversion methods

which attempt to recover a broadband pseudo-acoustic impedance log from a

band-1 imi ted sei smic trace.

Another way to look at inversion is to consider it as the technique for

creating a model of the earth using the seismic data as input. As such, it

can be considered as the opposite of the forwar• modelling technique, which

involves creating a synthetic seismic section based on a model of the earth

(or, in the simplest case, using a sonic log as a one-dimensional model). The

relationship between forward and inverse modelling is shown n Figure 1.1.

To understandseismic inversion, we must first understand he physical

processes involved in the creation of seismic data. Initially, we will

therefore look at the basic convolutional model of the seismic trace in the

time and frequencydomains, onsidering the three componentsf this model:

reflectivity, seismic wavelet, and noise.

Part I - Introduction

_ m i --.

Page 1 - 2



Introduction to Seismic InverSion Methods Brian Russell

FORWARDMODELL NG

i m ß

INVERSEMODELLINGINVERSION)

, ß ß _

Input'

Process:

Output'

EARTHODEL

,

MODELLING

ALGORITHM

SEISMIC RESPONSE

i m mlm ii

INVERSION

ALGORITHM

EARTHODEL

ii

Figure1.1 Fo.•ard andsInverseodel,ling

Part I - Introduction

Page I - 3



Introduction. to Seismic Inversion Methods Brian l•ussel 1

Once we have an understanding of these concepts and the problems which

can occur, we are in a position to look at the methodswhich are currently

ß

used to invert seismic data. These methods are summarized n Figure 1.2. The

primary emphasis of the course will be

the ultimate resul.t, as was previously

on poststack seismic inversion where

o

Oiscussed, is a pseudo-impeaance

section.

We will start by looking at the most contanonmethods of poststack

inversion, which are based on single trace recursion. To better unUerstand

these recurslye inversion procedures, it is important to look at the

relationship between aleconvolution anU inversion, and how Uependent each

method is on the deconvolution scheme Chosen. Specifically, we will consider

classical "whitening" aleconvolutionmethods, wavelet extraction methods, and

the newer sparse-spike deconvolution methods such as Maximum-likelihood

deconvolution and the L-1 norm metboa.

Another important type of inversion methodwhich will be aiscussed is

model-based inversion, where a geological moael is iteratively upUated to finU

the best fit with the seismic data. After this, traveltime inversion, or

tomography,will be discussedalong with several illustrative examples.

After the discussion on poststack inversion, we shall move nto the realm

of pretstack. These methoUs,still fairly new, allow us to extract parameters

other than impedance, such as density and shear-wave velocity.

Finally, we will aiscuss the geological aUvantages anU limitations of

each seismic inversion roethoU, ooking at examples of each.

Part 1 - Introduction

Page i -



Introduction to Selsmic nversion Methods Brian Russell

SESMI I NV RSI N

.MET•OS ,,

POSTSTACK

INVERSION

PRESTACK

INVERSION

MODEL-BASEDRECURSIVE

INVERSION,INVE ION

- "NARROW

BAND

TRAVELTIME

INVERSION

TOMOGRAPHY)

SPARSE-

SPIKE

WAVFEL

NVERSIOU

LINEAR

METHODS

,,

i i --

I METHODS

Figure 1.2

A summaryof current inversion techniques.

Part 1 - Introuuction

Page 1 -



Introduction to Seismic Inversion Methods Brtan Russell

PART - THECONVOLUTIONALODEL

Part 2 - The Convolutional Model

Page 2 -




Part 2 - The Convolutional Mooel

2.1 Th'e Sei smic Model

The mostbasic and commonlysed one-Oimensionalmoael for the seismic

trace is referreU to as the convolutional moOel, which states that the seismic

trace is simply he convolutionof the earth's reflectivity with a seismic

source function with the adUltion of a noise component. In equation form,

where * implies convolution,

s(t) : w(t) * r(t) + n(t)s

where

and

s (t) = the sei smic trace,

w(t) : a seismic wavelet,

r (t) : earth refl ecti vi ty,

n(t) : additive noise.

An even simpler assumptions to consiUer he noise componento be zero,

in which case the seismic tr•½e is simply the convolution of a seismic wavelet

with t•e earth ' s refl ecti vi ty,

s(t) = w{t) * r(t).

In seismic processingwe deal exclusively with digital data, that is,

data sampledt a constantime interval. If weconsiUerhe relectivity to

consist of a reflection coefficient at each time sample (som• of which can be

zero), and the wavelet to be a smooth function in time, convolutioncan be

thoughtof as "replacing"eachreflection. coefficient with a scaledversion of

the wavelet and summinghe result. The result of this process s illustrated

in Figures 2.1 and2.Z for botha "sparse" nda "dense" et of reflection

coefficients. Notice that convolution with the wavelet tends to "smear" the

reflection coefficients. That is, there is a total loss of resolution,which

is the ability to resolve closely spacedreflectors.


Page



Introduction to Seismic Inversion Nethods Brian Russell

WAVELET:

(a) ' * • : -' :'

REFLECTIVITY

Figure 2.1

TRACE:

Convolution f a wavelet with a

(a) •avelet. (b) Reflectivit.y.

sparse"reflectivity.

(c) Resu ing Seismic Trace.

(a)

(b')

.

i

:

: :

i

i ,

: i

i i

'?t *

c

o o o o o

Fi õure 2.2

Convolution of a wavelet with a sonic-derived "dense"

reflectivity. (a) Wavelet. (b) Reflectivity. (c) Seismic Trace

, i , ß .... , m i i L _ - '

Par• 2 - The Convolutional Model

Page 2 - 3



Introduction to Seismic Inver'sion Methods Brian Russell

An alternate, but equivalent, way of looking at the seismic trace is in

the frequency domain. If we take the Fourier transform of the previous

ß

equati on, we may write

S(f) = W(f) x R(f),

where

S(f) = Fourier ransformf s(t),

W(f) = Fourier transform of w(t),

R(f) = Fourier transform of r(t),

ana f = frequency.

In the above equation we see that convolution becomesmultiplication in

the frequency domain. However, the Fourier transform is a complex function,

and it is normal to consiUer the amplitude and phase spectra of the individual

components. The spectra of S(f) may then be simply expressed

esCf)= ew

where

(f) + er(f),

I •ndicatesmplitudepectrum,nd

0 indicates phase spectrum. .

In other words, convolution involves multiplying the amplitude spectra

and adding the phase spectra. Figure 2.3 illustrates the convolutional model

in the frequency domain. Notice that the time Oomainproblem of loss of

resolution becomesone of loss of frequency content in the frequency domain.

Both the high and low frequencies of the reflectivity have been severely

reOuceo by the effects of the seismic wavelet.

Part 2 - The Convolutional Mooel

Page ?. - 4




AMPLITUDE SPECTRA

PHASE SPECTRA

w (f)

I I

-t-

R (f)

i i , I

i. iit |11

loo

s (f)

I i

I

i i

Figure 2.3

Convolution in the frequency domain for

the time series shown in Figure 2.1.


Page 2 -




2.g The Reflection Coefficient Series

l_ _ ,m i _ _ , _ _ m_ _,• , _ _ ß _ el

of as the res

within the ear

compres i onal

i ropedance o re

impedances by

coefficient at

fo11 aws:

'The reflection coefficient series (or reflectivity, as it is also called)

describedn thepreviousections oneof the fundamentalhysicaloncepts

in the seismic method. Basically, each reflection coefficient maybe thought

ponse of the seismic wavelet to an acoustic impeUance change

th, where acoustic impedance is defined as the proUuct of

velocity and Uensity. Mathematically, converting from acoustic

flectivity involves dividing the difference in the acoustic

the sum of the acoustic impeaances. This gives t•e reflection

the boundary between the two layers. The equation is as

•i+lVi+l - iVi Zi+l- Z

i • i+1

where

and

r = reflection coefficient,

/o__density,

V -- compressional velocity,

Z -- acoustic impeUance,

Layer i overlies Layer i+1.

Wemust also convert from depth to time by integrating the sonic log

transit times. Figure •.4 showsa schematicsonic log, density log, anU

resulting acoustic impedance or a simplifieU earth moael. Figure 2.$ shows

the resultof convertingo thereflection oefficienteries ndntegrating

to time.

It should be pointed out that this formula is true only for the normal

incidence case, that is, for a seismic wave striking the reflecting interface

at right angles to the beds. Later in this course, we shall consider the case

of nonnormal inciaence.

Part 2 - The Convolutional

Model Page 2 - 6




STRATIGRAPHIC SONICLOG

SECTION •T (•usec./mette)

4OO

SHALE ..... DEPTH

ß ß ß ß ß ß

SANOSTONE . . - .. ,

'

I _1

UMESTONE I I I I 1

LIMESTONE

2000111

30O 200

I

3600 m/s

_

v--

V--3600

V= 6QO0

I

loo 2.0 3.0

,

OENSITY LOG.

ß •

Fig. 2.4. Borehole ogMeasurements.

mm mm rome m .am

,mm mm m ----- mm

SHALE ..... OEPTH

•--------'-

SANDSTONE . . ... ,

I 11 I1

UMESTONE I 1 I I I II

i I 1 i I i 1000m

SHALE •.--._--.---- • •.'•

LIMESTONE

2000 m

ACOUSTIC

IMPED,M•CE (2•

(Y•ocrrv x OEaSn•

REFLECTWrrY

V$ OEPTH

VS TWO.WAY

TIME

20K -.25 O Q.2S -.25 O + .2S

I I v ' I

- 1000 m -- NO

,• , ..

- 20o0 m

I SECOND

Fig. 2.5. Creation of Reflectivity Sequence.

Part g - The Convolutional Model

Page 2 - 7



IntroductJ on 1:o Sei stoic Inversion Herhods Bri an Russell

Our best method of observing seJsm•c impedance and reflectivity is •o

derlye them from well log curves. Thus, we maycreate an impedancecurve by

multiplying together •he sonic and density logs from a well. Wemay hen

computehe reflectivlty by using •he formula shown arlier. Often, we do not

have the density log available• to us and must makedo with only the sonJc. The

approxJmatJonof velocJty to •mpedances a reasonable approxjmation, and

seemso holdwell for clas;cics and carbonates not evaporltes, however).

Figure 2.6 shows he sonic and reflectJv•ty traces from a typJcal Alberta well

after they have been Jntegrated to two-way tlme.

As we shall see later, the type of aleconvolution and inversion used is

dependent on the statistical assumptionswhich are made about the seismic

reflectivity and wavelet. Therefore, howcan we describe the reflectivity seen

in a well? The traditional answer has always been that we consider the

reflectivity to be a perfectly random sequence and, from Figure •.6, this

appears to be a good assumption. A ranUomsequencehas the property that its

autocorrelation is a spike at zero-lag. That is, all the components f the

autocorrelation are zero except the zero-lag value, as shown n the following

equati on-

t(Drt = ( 1 , 0 , 0 , ......... )

t

zero-lag.

Let us test this idea on a theoretical random sequence, shown n Figure

2.7. Notice that the autocorrelation of this sequence has a large spike at

ß

the zeroth lag, but that there is a significant noise component at nonzero

lags. To have a truly random sequence, it must be infinite in extent. Also

on this figure is shown the autocorrelation of a well log •erived

reflectivity. Wesee that it is even less "random" han the randomspike

sequence. Wewill discuss this in more detail on the next page.


Page 2 - 8



IntroductJon to Se•.s=•c Inversion Methods Br•an Russell

RFC

F•g. 2.6. Reflectivity equenceerivedrom onJclog.

RANDOM SPIKE SEQUENCE

WELL LOG DERIVED REFLECT1vrrY

AUTOCORRE•JATIONF RANDOMSEQUENCE

AUTOCORRELATION OF REFLECTIVITY

Fig. 2.7.

Autocorrelat4ons of random and well log

der4ved pike sequences.


Page 2-



Introductlon to Sei smic Inversion Methods Brian Russel 1

Therefore, the true earth reflectivity cannot be consideredas being

truly random. For a typical Alberta well we see a number f large spikes

(co•respondingo major ithol ogic change) ticking up above he crowd.A good

way to describe his statistically is as a Bernoulli-Gaussianequence. The

Bernoulli part of this term implies a sparseness n the positions of the

spikes and the Gaussianmplies a randomnessn their amplitudes. Whenwe

generatesuch a sequence, there is a term, lambda, which controls the

sparseness of the spikes. For a lambdaof 0 there are no spikes, and for a

lambda f 1, the sequences perfectly Gaussian in distribution. Figure 2.8

shows a number of such series for different values of lambda. Notice that a

typical Alberta well log reflectivity wouldhavea lambdavalue in the 0.1 to

0.5 range.


Page 2 - 10



I ntroducti on to Sei smic I nversi on Methods Brian Russell

It

tl I I I

LAMBD^•0.01

i I I

•11 I 511 t •tl I

(VERY SPARSE)

11

311 I

LAMBDA--O. 1

4# I

511 I #1 I

TZIIE (KS

1,1

::."• •'•;'" "";'•'l•'••'r'•

LAMBDAI0.5

- "(11

X#E (HS)

LAMBDA-- 1.0 (GAUSSIAN:]

EXAMPLESOF REFLECTIVITIES

Fig. 2.8. Examplesof reflectivities using lambda

factor to be discussed in Part 6.

, , m i ß i


Page 2 -

11



Introduction to Seismic Inversion ,Methods Brian Russell

2.3 The Seismic Wavelet

-- _ ß • ,

Zero Phase and Constant Phase Wavelets

m _ m _ m ß m u , L m _ J

The assumptionha.t there is a single, well-defined wavelet which is

convolved with the reflectivity to produce he seismic trace is overly

simplistic. Morerealistically, the wavelet is both time-varying and complex

in shape. However, he assumption f a simple wavelet is reasonable, and in

this section we shall consider several types of wavelets and their

characteristics.

First, let us consider the Ricker wavelet, which consists of a peak and

two troughs, or side lobes. The Ricker wavelet is dependentonly on its

dominant frequency, that is, the peak frequencyof its a•litude spectrum or

the inverse of the dominantperiod in the time domain the dominantperiod is

found by measuringhe time from trough o trough). TwoRicker wave'lets are

shown n Figures 2.9 and 2.10 of frequencies 20 and 40 Hz. Notice that as the

anq•litude spectrumof a wavelet .is broadened, he wavelet gets narrower in the

timedomain,ndicating n ncrease f resolution.Ourultimatewaveletwould

be a spike, with a flat amplitude spectrum. Sucha wavelet is an unrealistic

goal in seismic processing, but one that is aimed or.

The Rtcker wavelets of Figures 2.9 and 2.10 are also zero-phase, or

perfectly symmetrical. This is a desirable character.stic of wavelets since

the energy is then concentrated at a positive peak, and the convol'ution of the

wavelet with a reflection coefficient will better resolve that reflection. To

get an idea of non-zero-phase wavelets, consider Figure 2.11, where a Ricker

wavelet has been rotated by 90 degree increments, and Figure 2.12, where the

samewavelet has been shifted by 30 degree increments. Notice that the 90

degree rotation displays perfect antis•nmnetry, whereas a 180 degree shift

simply inverts the wavelet. The 30 degree rotations are asymetric.


Page 2- •2




Fig.

Fig.

2.9. 20 Hz Ricker Wavelet'.

•.10. 40 Hz Ricker wavelet.

Fig.

2.11.

Ricker wavelet rotated

by 90 degree increments

Fig.


2.12.

Ricker wavelet rotated

by 30 degree increment

Page 2 - 13




Of course, a typical seismic wavelet contains a larger range of

frequencies than that shownon the Ricker wavelet. Consider the banapass

fil•er shownn Figure 2.13, where we have passed a banaof frequencies

between 15 and 60 Hz. The filter has also had cosine tapers applied between 5

and 15 Hz, and between60 and 80 Hz. The taper reduces the "ringing" effect

that would be noticeable if the wavelet amplitude spectrum was a simple

box-car. The wavelet of Figure 2.13 is zero-phase, and would be excellent as

a stratigraphic wavelet. It is often referred to as an Ormsby avelet.

Minimum Phase Wavelets

The concept of minimum-phase s one that is vital to aleconvolution, but

is also a concept that is poorly understood. The reason for this lack of

understanding is that most discussions of the concept stress the mathematics

at the expense of the physical interpretation. The definition we

use of minimum-phases adapted from Treitel and Robinson (1966):

For a given set of wavelets, all with the same amplitude spectrum,

the minimum-phaseavelets the onewhich as he sharpesteading

edge. That is, only wavelets which have positive time values.

The reason that minimum-phase concept is important to us is that a

typical wavelet in dynamite work is close to minimum-phase. Also, the wavelet

from the seismic instruments is also minimum-phase. The minimum-phase

equivalent of the 5/15-60/80 zero-phase wavelet is shown n Figure 2.14. As

in the aefinition used, notice that the minimum-phase avelet has no component

prior to time zero and has its energy concentrated as close to the origin as

possible. The phase spectrum of the minimum-wavelet s also shown.


Pa.qe 2 - 14



I•troduct•on to Seistoic nversionNethods. Br•an Russell

ql Re• R Zero Phase I•auel•t

5/15-68Y88 {•

0.6

f1.38 - Trace 1

iii

- e.3e ...... , • ..... ' 2be

1

Trace I

Fig. 2.13. Zero-phase bandpass

wavelet.

Reg 1) min,l• wavelet •/15-68/88 hz

18.00 p Trace I

RegE wayel Speetnm

'188.88 Trace1

0.8

188

Fig.

2.14. Minim•-phase equivalent

of zero-phase wavelet

shown n Fig. 2.13.

_

m,m, i m

Part 2 -Th 'e Convolutional Model

i

Page 2- 15




Let us now ook at the effect of different wavelets on the reflectivity

function itself. Figure 2.15 a anU b shows a numberof different wavelets

conv6lved with the reflectivity (Trace 1) from the simple blocky model shown

in Figure Z.5. The following wavelets have been used- high

zero-phase (Trace •), low frequency ero-phase Trace ½), high

minimum hase (Trace 3), low frequency minimum phase (Trace 5).

figure, we can make the fol 1owing observations:

frequency

frequency

From the

(1) Low freq. zero-phase wavelet: (Trace 4)

- Resolution of reflections is poor.

- Identification of onset of reflection is good.

(Z) High freq. zero-phase wavelet: (Trace Z)

- Resolution of reflections is good.

- Identification of onset of reflection is good.

(3) Low freq. min. p•ase wavelet- (Trace 5)

- Resolution of reflections i s poor.

- Identification of onset of reflection is poor.

(4) High freq. min. phase wavelet: (Trace 3)

- Resolution of refl ec tions is good.

- Identification of onset of reflection is poor.

Based on the aboveobservations, we wouldhave to consider the high

frequency,ero-phase avelet he best, and he low-frequency, inimumhase

wavelet the worst.


Page 2 - 16



(a)


ql RegR Zer• PhaseUa•elet •,'1G-•1• 14z

F

- •.• [' '

•,3 Recj miniiliumhue ' '

17 .•

q2 RegC ZeroPhase4aue16(' •'le-3•4B Hz

e

q• Reg ) 'minimumhase " •,leJ3e/4eh• '

8

e.e •/••/'•-•"v--,._,,r

e.• ' "s•e'

m ,,

Tr'oce

[b)

Fig.

700

2.15. Convolution of four different wavelets shown

in (a) with trace I of (b). The results are

shown on traces 2 to 5 of (b).


Page 2 - 17




g.4 Th•N.oi se.Co.mp.o•net

The situation that has been discussed so far is the ideal case. That is,

.

we have interpreted every reflection wavelet on a seismic trace as being an

actual reflection from a lithological boundary. Actually, many of the

"wiggles" on a trace are not true reflections, but are actually the result of

seismic noise. Seismic noise can be grouped under two categories-

(i) Random oise - noise which is uncorrelated from trace to trace and is

•ue mainly to environmental factors.

(ii) CoherentNoise - noise which is predictable on the seismic trace but

is unwanted. An example s multiple reflection interference.

Randomnoise can be thought of as the additive component (t) which was

seen in the equationon page 2-g. Correcting for this term is the primary

reason for stackingour •ata. Stacking actually uoesan excellent job of

removing ranUomnoise.

Multiples, one of the major sources of coherent noise, are causedby

multiple "bounces" f the seismic signal within the earth, as shownn Figure

2.16. They may be straightforward, as in multiple seafloor bounces r

"ringing", or extremelycomplex,as typified by interbed multiples. Multiples

cannot be thoughtof as additive noise and mustbe modeled s a convolution

with the reflecti vi ty.

Figure

generatedby the simple blocky model

this data, it is important that

Multiples may be partially removed

powerful elimination technique.

aleconvolution, f-k filter.ing,

wil 1 be consi alered in Part 4.

2.17

shown on Figure •. 5.

the multiples be

by stacking, but

Such techniques

and inverse velocity stacking.

shows the theoretical multiple sequence which would be

If we are to invert

effectively removed.

often require a more

include predictive

These techniques


Page 2 - 18




Fig. 2.16. Several multiple generating mechanisms.

TIME TIME

[sec) [sec)

0.7 0.7

REFLECTION R.C.S.

COEFFICIENT WITH ALL

SERIES MULTIPLES

Fig. 2.17.

Reflectivi ty sequence f Fig.

and without mul ipl es.

Part 2 - The ConvolutionalModel

2.5. with

.

Page 2 -

19



PART 3 - RECURSVE INVERSION - THEORY

m•mmm•---' .• ,- - - ' •- - _ - - _- _

Part 3 - Recurstve Inversion - Theory

Page 3 -



•ntroduct•on to SeJsmic Znversion Methods Brian Russell

PART 3 - RECURSIVE INVERSION - THEORY

3.1 Discrete Inversion

, ß , , •

In section 2.2, we saw that reflectivity was defined in terms of

acoustic impedancechanges. The formula was written:

Y•i+lV•+l•iV 2i+ ' Z

ri--yoi'+lVi+l+•iVi - Zi..+lZ

where r -- refl ecti on coefficient,

/0-- density,

V -- compressional velocity,

Z -- acoustic impedance,

and Layer i overlies Layer i+1.

If we have the true reflectivity available to us, it is possible to

recover the a.coustic impedance y inverting the above formula. Normally, the

inverse' formulation is simply written down,but here we will supply the

missing steps for completness. First, notice that:

Also

Ther'efore

Zi+l+Z Zi+ - Z 2 Zi+

I +ri- Zi+lZi + Zi+l2i Zi+lZi

I- ri--

Zi+l+Z Zi+ - Z 2 Zf[

Zi+l+ Z Zi+l+ Z Zi+l+ Z

Zi+l

Z

l+r.

1

1

Part 3 - Recursive nversion- Theory

ill, ß , I

Page



Introduction to Seismic Invers-•onMethods Brian Russell

pv-e-

TIME

(sec]

0.7

REFLECTION

COEFFICIENT

SERIES

RECOVERED

ACOUSTIC

IMPEDANCE

Fig.

3.1,

Applyinghe recursive nversion ormula o a

simple, and exact, reflectivity.

, ß

Part 3 - Recursive

Inversion - Theory

Page 3 -



ntroductt on to Se1 smJc nversi on Methods Brian Russell

•9r• ;• • •;• • • •-•• 9rgr•t-k'k9r9r• •-;• ;• .................................................

Or, the final •esult-

Zi+[= Z

ß

l+r i .

This is called the discrete recursive inversion formula and is the basis

of many current inversion techniques. The formula tells us that if we know

the acoustic impedance f a particular layer and the reflection coefficient at

the base of that layer, we may recover the acoustic impedance of the next

layer. Of course we need an estimate of the first layer impedance o start us

off. Assumewe can estimate this value for layer one. Then

l+rl ,

Z2: l r1

Z3= 112

r

and so on ...

To find the nth impedance rom the first, we simply write the formula as

Figure 3.1 shows the application of the recursive formula to the "

reflection coefficients derived in section 2.2. As expected, the full

acoustic impedance was recovered.

Problems encountered with real data

• ß , m i i • i m

When the recursive inversion formula is applied to real data, we find

that two serious problems are encountered. These problems are as follows-

(i) FrequencyBandl mi ti ng

ß

Referring back to Figure 2.2 we see that the reflectivity is severely

bandlimited when it is convolved with the seismic wavelet. Both the

low frequency components nd the high frequency components re lost.

Part 3 - Recursive Inversion - Theory

Page 3 - 4



Introduction to Seismic nversion Methods Brian Russell

0.2 0 V•) 'V,•

•R

R = +0.2

V :1000 Where:

--• V,•= 1000 i-o.t

- 1500 m

•ec'.

(a)

- 0.1 '•0.2

R•

R=

{ASSUME•: l)

R•= 0.1

R =+0.2

R: -0.1

Vo=1000m

-'+ ¾1 818m

i•.

Figure 3.2 Effect of banUlimitingon reflectivity, where a) shows

single reflection coefficient, anU (b) showsbandlimited

refl ecti on coefficient.

i i m i m I

I __ ___ i _

Part 3 - Recursire Inversion - Theory Page 3 -




(ii) Noise

The inclusion of coherent or random noise into the seismic 'trace will

make he estimate• reflectivity deviate from the true reflectivity.

To get a feeling for the severity of the above limitations on recursire

inversion, let us first use simple models. To illustrate the effect of

bandlimiting, consider Figure 3.Z. It shows the inversion of a single spike

(Figure 3.2 (a)) anU he inversion of this spike convolved with a Ricker

wavelet (Figure 3.2 (b)). Even with this very high frequency banUwidth

wavelet, we have totally lost our abil.ity to recover the low frequency

componentof the acoustic impedance.

In Figure 3.3 the model derived in section Z.2 has been convolved with a

minimum-phase wavelet. Notice that the inversion of the data again shows a

loss of the low frequency component. The loss of the low frequency component

is the most severe problem facing us in the inversion of seismic data, for it

is extremely Oifficult to directly recover it. At the high end of the

spectrum, we may recover muchof the original frequency content using

deconvolution techniques. In part 5 we will address the problem of recovering

the low frequency component.

Next, consider the problem of noise. This noise may be from many

sources, but will always tend to interfere with our recovery of the true

reflectivity. Figure 3.4 shows the effect of adding the full multiple

reflection train (including transmission losses) to the model reflectivity.

As we can see on the diagram, the recovered acoustic impedancehas the same

basic shape as the true acoustic impedance, but becomes ncreasingly incorrect

with depth. This problemof accumulatingerror is compoundeUy the amplitude

problemns ntroduced by the transmission losses.

Part 3 - Recurslye Inversion - Theory Page 3 - 6



Introduction to Seismic Invers,ion Methods Brian Russell

TIME

Fig.

TIME

(see)

Fig.

0.?

RECOVERED

ACOUSTIC

IMPEDANCE

REFLECTION SYNTHETIC

COEFFICIENT (MWNUM-PHASE

SERIES WAVELET)

pv-•,

INVERSION

OF SYNTHETIC

3.3. The effect of bandlimiting on recurslye inversion.

0.7

TIME

(re.c)

REFLECTION RECOVERED R.C.S. RECOVERED

COEFFICIENT ACOUSTIC WITH ALL ACOUSTIC

SERIES IMPEDANCE MULTIPLES IMPEDANCE

3.4. The effect of noise on recursive inversion.


Page 3 -




3.3 Continuous Inversion

A logarithmic relationship is often used to approximate the above

formulas. This is derived by noting that we can write r(t) as a continuous

function in the following way:

Or

r(t) - Z(t+dt) Z{t) _ 1 d Z(t)

ß - Z(t+dt) + Z(•) - •' z'(t)

d In Z(t)

r(t) = • dt

The inverse formula is thus-

t

Z(t) Z(O)xpy r(t)dt.

0

The precedingapproximations valid if r(t) <10.3• which is usually the

case. A paper by Berteussen and Ursin (1983), goes into muchmore detail on

the continuous versus discrete approximation. Figures 3.5 and 3.6 from their

paper show hat the accuracy of the continuous inversion algorithm is within

4% of the correct value between reflection coefficients of -0.5 and +0.3.

If our reflection coefficients are in the order of + or - 0.1, an even

simpler pproximationay e made y dropp'inghe logarithmicelationship:

t

1dZ(t) _==• (t)-2'Z(O)r(t) dt

r(t) -• dr VO


Page 3 - 8




Fig. 3.5

m i ,, ,m I I IIIII

I + gt ½xp26•) Difference

-1.0 0.0 0.14 -0.14

-0.9 0.05 0. ? -0.12

-0.8 0.11 0.20 -0.09

-0.7 0.18 0.25 -0.07

-0.6 0.25 0.30 -0.05

-0.5 0.33 0.37 -0.04 '

-0.4 0.43 0.45 --0.02

-0.3 0.• 0.•5 --0.01

-0.2 0.667 0.670 -0.003

-0.1 0.8182 0.8187 --0.0005

0.0 1.0 1.0 0.0

0.1 1.222 1.221 0.001

0.2 1.500 1.492 0.008

0.3 1.86 1.82 0.04

0.4 2.33 2.23 o.1

0.5 3.0 2.7 0.3

0.6 4.0 3.3 0.7

0.7 5.7 4.1 1.6

0.8 9.0 5.0 4.0

0.9 19.0 6.0 13.0

1.0 co 7.4 •o

Numerical c•pari son of discrete and continuous

i nversi on.

(Berteussen and Ursin, 1983)

Fig. 3.6

$000m PEDANCEOSCR.

r-niL

${300•OFFERENCE

SO0 O FFERENCE SCALEDUP

T •'•E t SECONOS

C•parisonbetweenmpedance•putatins based n a

discrete and a continuous eismic •del.

(Berteussen and Ursin, 1983)

Part 3 - Recursire .Inversion - Theory

Page 3 -



Introduction'to Seismic Inversion Methods Brian Russell

PART 4 - SEISMIC PROCESSINGCONSIDERATIONS

Part 4 - Seismic Processing Considerations

Page 4 - 1



•ntroduction to Seismic •nvers•on Methods B.r.an Russell

4.1 Introduction

Having ookedat a simple model'of the seismic trace, anu at the

recursire inversion alogorithm n theory, we will now ook at the problem of

processingeal seismiceata in order to get the best results fromseismic

inversion. We may group the key processing roblems nto the following

categories:

( i ) Amp tu de rec overy.

(i i) Vertical resolution improvement.

(i i i ) Horizontal resol uti on improvement.

(iv) Noise elimination.

Amplitudeproblemsare a majorconsideration t the early processing

stagesandwewill look at both deterministicamplitudeecovery ndsurface

consistent residual static time corrections. Vertical resolution improvement

will involve a discussion of aleconvolution and wavelet processing techniques.

In our discussion of horizontal resolution we will look at the resolution

improvementbtained in migration, using a 3-D example.Finally, wewill

consider several approacheso noiseelimination, especially the elimination

of multi pl es.

Simply stateu, to invert our

one-dimensional model given in the

approximationof this model (that

band-limited reflectivity function)

these considerations in minU. Figure 4.1

be useU o do preinversion processing.

seismic data we usually assume the

previous section. And to arrive at an

is, that each trace is a vertical,

we must carefully process our data with

showsa processing flow which could


Page 4 - 2




INPUT RAW DATA

DETERMINISTIC

AMPLITUDE

CORRECTIONS

,. _•m

mlm

SURFACE-CONS STENT

DECONVOLUTIO,FOLLOWED

Y HI GH RESOIJUTI.ON DECON

i

i

SURFACE-CONS STENT

AMPt:ITUDE ANAL'YSIS

SURFACE-CONSISTENT

STATI CS ANAIJY IS

VELOCITY ANAUYS S

APPbY STATICS AND VEUOCITY

MULTIPLE ATTENUATION

STACK

ß •

MI GRATI ON

,

Fig. 4.1.

Simplfied nversinprocessinglow.

ll , ß ' ß I , _ i 11 , m - -- m _ • • ,11


Page 4 - 3



Inl;roducl:ion 1:oSeJ mlc Invers1on Nethods BrJan Russell

4.2 Am.p'ltu.de..P,.ecovery

The most dJffJcult job in the p•ocessing of any seismic line is

ß

•econst•uctinghe amplJtudesf the selsmJc•aces as they would avebeenJf

the•e were no dJs[urbJng nf'luences present. We normally make the

simplJficationhat the distortionof the seJsmicmplJtudesay e put into

three main categories'sphe•Jcal ivergence, absorptJon,and t•ansmJssion

loss. Based on a consideration of these three factors, we maywrJte aownan

approximate unctJon or the total earth attenuation-

Thus,

data, the

formula.

At: AO*

b / t) * exp(-at),

where t = time,

A = recordedmplitude,

A = true ampltude,

anU

a,b = constants.

if we estimate the constants in the above equation from the seismic

true amplitudesof the data coulU be recoveredby using the inverse

The deterministic amplitude correction and trace to trace mean

scaling will account or the overall gross changes n amplitude. However,

there may still be subtle (or even not-so-subtle) amplitude problems

associatedwith poorsurface conditions or other factors. To compensateor

these effects, it is often advisable to compute nd apply surface-consistent

gain corrections. This correction involves computing total gain value for

each trace and then decomposinghis single value in the four components

Aij= ixRjxGxMkXj,

where A = Total amplitude factor,

S = Shot component,

R: Receiver component,

G = CDP component, and

M = Offset component,

X = Offset distance,

i,j = shot,receiver pos.,

k = CDP position.


Page 4 -



Introduction to Seismic .Inversion Methods Brian Russell

SURFACE

SUEF'A•

CONS b'TEh[O{

AND

T |tV•E :

,Ri -rE ß

Fig. 4.2.

Surface and sub-surface geometryand

surface-consistent decomposition. (Mike Graul).

, ,

Part 4 - SeismicProcessing onsiderations

Page 4 -

5




Figure 4.g (from Mike Graul's unpublished course notes) shows the

geometry sed or this analysis. Notice that the surface-consistent tatics

anti aleconvolution roblem re similar. For the statics problem, the averaging

can be •1oneby straight summation.For the amplitude problemwemust

transform the above equation into additive form using the logarithm:

InAij=nS + nRj+ nG + nkMijX•.

The problem can then be treated exactly the same way as in the statics

case. Figure 4.3, fromTaner anti Koehler (1981), shows he effect of doing

surface consistent amplitude and statics corrections.

4.3 I•mp.ov. ment_.[_Ver..i.ca.1..Resolutin

Deconvol ution is a process by which an attempt is made to remove the

seismic wavelet from the seismic trace, leaving an estimate of reflectivity.

Let us first discuss the "convolution"part of "deconvolution" starting with

the equation for the convolutional model

In the

st--wt* r t where

frequency domain

st= the seismic race,

wt= the seismicwavelet,

rt= reflection coefficient series,

* = convolution operation.

S(f) • W(f) x R(f) .

The deconvol ution

procedure and consists

reflection coefficients.

fol 1owlng equati on-

rt: st* o

process is simply the reverse of the convolution

of "removing" the wavelet shape to reveal the

We must design an operator to do this, as in the

whereOr--operator- inverse f w .


,

Page 4 - 6




ii 11

ß 1'

i

ii

'..,•' •, ," " " ß d.

Preliminarytack et'oreurfaceonsistenttaticnd mpli-

lude corrections.

ß Stockwith surface onsistenttatic nd amplitude or-

rections.

Fig. 4.3.

Stacks with and without surface-consi stent

corrections. (TaneranuKoehler,1981).


ß ,

Page 4 - 7




In the frequency domain, his becomes

R(f) = W(f) x 1/W(f) .

After this extremely simple introduction, it may appear that the

deconvolution roblemshouldbe easy to solve. This is not the case, and the

continuing research nto the problem testifies to this. There are two main

problems. Is our convolutionalmodel orrect, and, if the model s correct,

can we derive the true wavelet from the data? The answer to the first

question s that the convolutionalmodel ppears o be the best modelwe have

come p with so far. The main problem is in assuminghat the wavelet does

not vary with time. In our discussionwe will assumehat the time varying

problem s negligible within the zoneof interest.

The secondproblem s much more severe, since it requires solving the

ambiguousproblem f separatinga wavelet and reflectivity sequencewhenonly

the seismic trace is known. To get around this problem, all deconvolution or

wavelet estimation programsmakecertain restrictive assumptions, ither about

the wavelet or the reflectivity. There are two classes of deconvolution

methods: those which make restrictive phase assumptions and can be considered

true wavelet processing echniques only when hese phase assumptions re met,

and those which do not make restrictive phase assumptions and can be

considered as true wavelet processingmethods. In the first category are

(1) Spiking deconvolution,

(2) Predictive deconvolution,

(3) Zero phase deconvolution, and

(4) Surface-consi stent deconvoluti on.


Page 4 -




(a)

Fig. 4.4

A comparison of non surface-consistent and surface-consistent

decon on pre-stack data. {a) Zero-phase deconvolution.

{b) Surface-consistent soikinB d•convolution.

(b),

Fig. 4.5 Surface-consistent decon comparisonafter stack.

(a) Zero-phase aleconvolution. (b) Surface-consistent

deconvol ution.

'--'- , ß , ,• ,t ß ß _ , , _ _ ,, , ,_ , ,

Part 4 - .Seismicrocessingonsioerations Page -



Introduction to Seismic Invers. on Methods Brian Russell

In the second category are found

(1) Wavelet estimation using a well

(Hampsonnd Galbraith 1981)

1og (Strat Decon).

(2) Maximum-1 kel ihood aleconvolution.

(Chi et al, lg84)

Let us

surface-consi stent

surface-consi stent

components. We

di recti ons- common

illustrate the effectiveness of one of. the methods,

aleconvolution. Referring to Figure 4.•, notice that a

scheme involves the convolutional proauct of four

must therefore average over four different geometry

source, common receiver, common depth point (CDP), and

con, on offset (COS). The averaging must be performed iteratively and there

are several different ways to perform it. The example in Figures 4.4 ana 4.5

shows an actual surface-consi stent case study which was aone in the following

way'

(a) Compute he autocorrelations of each trace,

(b) average the autocorrelations in each geometry eirection to get four

average autocorrel ati OhS,

(c) derive and apply the minimum-phasenverse of each waveform, and

(•) iterate through this procedure to get an optimum esult.

Two points to note when you are looking at the case study are the

consistent definition of the waveformn the surface-consistent pproachan•

the subsequent improvementof the stratigraphic interpretability of the stack.

We can compareall of the above techniques using Table 4-1 on the next

page. The two major facets of the techniques which will be comparedare the

wavelet estimation procedure and the wavelet shaping procedure.


Page 4 - 10




Table 4-1

Comparison of Deconvolution MethoUs

m ß ß m

METHOD

Spiking

Deconvol ution

Predi cti ve

Deconvol uti on

Zero Phase

Deconvol utton

Surface-cons.

Deconvolution

Stratigraphic

Deconvol ution

Maximum-

L ik el i hood

deconvol ution

WAVELET ESTIMATION

Min.imumhase assumption

Randomefl ecti vi ty

assumptions.

No assumptions about

wavelet•

Zero phaseassumption.

Randomefl ectt vi ty

assumption.

Minimum r zero phase.

Randomeflecti vi ty

assumption.

No phase assumption.

However, well must match

sei smi c.

No phase assumption.

Sparse-spike assumption.

WAVELETSHAPING

Ideally shaped o spike.

In practice, shaped o minimum

phase,higher requency utput.

Does not whiten data well.

Removeshort and ong period

multiples. Does not affect

phase f wayel t for long lags.

..1_, m

Phase is not altered.

Amplitude spectrum $

whi tened.

Canshape o desired output.

Phase haracter s improved.

Ampl rude spectrum i s

whitenedess than in single

trace methods.

Phase of wavelet is zeroed.

Amplitudepectrumot

whi tened.

Phase of wavelet is zeroed•

Amp rude spectrum s

whi tened.


Page 4

11'




4.4 Lateral Resol uti on

The complete three-dimensional (3-D) diffraction problems shown n

Figure 4.6 for a modelstudy taken fromHerman, t al (1982). Wewill look'at

line 108, which cuts obliquely across a fault and also cuts across a reef-like

structure. Note that it misses the second reef structure.

Figure 4.7 shows the result of processing the line. In the stacked

section we maydistinguish two types of diffractions, or lateral events which

do not represent true geology. The first type are due to point reflectors in

the plane of the section, and include the sides of the fault and the sharp

corners at the base of the reef structure which was crossed by the line. The

second type are out-of-t•e-plane diffractions, often called "side-swipe". This

is most noticeable by the appearance of energy from the second reef booy which

was not crossed. In the two-dimensional (2-D) migration, we have correctly

removed the 2-D diffraction patterns, but are still bothere• by the

out-of-the-plane diffractions. The full 3-D migration corrects for these

problems. The final migrated section has also accounted for incorrectly

positioned evehts such as the obliquely dipping fault. This brief summary as

not been intended as a complete summary f the migration procedure, but rather

as a warning that migration {preferably 3-D) must be performedon complex

structural lines for the fol 1 owing reasons:

(a)

(b)

To correctly position dipping events on the seismic section, and

To remove diffracted events.

Although migration can compensate or someof the lateral resolution

problems, we must remember hat this is analogous to the aleconvolution problem

in that not all of the interfering effects may be removed. Therefore, we must

be aware that the true one-dimensional seismic trace, free of any lateral

interference, is impossible to achieve.

Part 4 - Seismic Processing Considerations Page 4 -

12




lol

I

71

131

(a] 3- D MODEL

131

101

108

LINE

ß

ß ß

ß ß

ß

..................................

.............................

.........................................

....................................

{hi 8•8•0 LAYOU•

Fig. 4.6. 3-D model experiment.

i mm _ ml j mm

Part 4 • Seismic Processing Considerations

(Hermant al, 1982).

Page 4 -

13




4.5 Notse Attenuation

As we' discussed in an earlier section, seismic noise can be classified as

either •andom or coherent. Random noise is reduced by the stacking process

quite well unless the signal-to-noise ratio dropsclose to one. In this case,

a coherency nhancementrogram an be used, which usually involves some ype

of trace mixing or FK filtering. However, he interpreter mustbe aware that

any mixing of the data will "smear" trace amplitudes, making he inversion

result on a particular trace less reliable.

Coherent noise is much more difficult to eliminate. One of the major

sources of coherent noise is multiple interference, explained in section 2.4.

Two of the major methodsused in the elimination of multiples are the FK

filtering method,and the newer nverse Velocity Stacking method. The Inverse

VeiocityStackingmethodnvolves he following teps:

(1) Correct the data using the proper NMO elocity,

(2) Model the data as a linear sumof parabolic shapes,

(This involves transforming to the Velocity domain),

(3) Filter out the parabolic omponentsith a moveoutreater hansome

pre-determinedimit (in the order of 30 msec),and

(4) Perform the inverse transform.

Figure 4.8, taken from Hampson1986), shows comparison etween he two

methodsor a typical multiple problemn northernAlberta. The displays are

all' co•on offset stacks. Notice that although both methods have performed

well on the outside traces, the Inverse Velocity Stacking methodworks best on

the inside traces. Figure4.9, also fromHampson1986), shows comparisonf

final stacks with and without multiple attenuation. It is obvious 'from this

comparisonhat the result of inverting the section whichhas not had multiple

attenuation would be to introduce spurious velocities into the solution. The

importance f multiple elimination to the preprocessinglow cannot therefore

be overemphasized.

m i i m , i . i m _ i i _ L ,=•m__ _ i m ß •

Part 4 - Seismic Processing Consideration• Page ½ - 14



Introduction to Seismic Inversion Methods Brian Russell.

liltiitil 1111iitt)ttl lii/littl•ll

(b] LINE d8 - 2-D MIGRATION

IIIIIIll 1111111111111it111111111111111111illllli IIIIIIIIil 111111tllilil illlllll 111

[1111111111111111111111111II 1111111111111111111I I IIilllllllll 11111111111111111

?•111[•i••IIIIIIII1111111111111111II IIiill•illlllillllllllllliillllllllllllh

•., } l iillllllllillllllliJillllllllllllilitiilillit illo

111lllllllllllllllllllll1111llllllilllllll ll llll111llllllllilllllllllllllllllllllllllii{lillllll

"• illllllllll 1111illi 111IIIIIIIIIIIIII1111111111I I lillilllllll 1111 1 111111•

Col LINE 108 - 3-D MIGR•ATION

F•g. 4.7. Migration f model ata shownn F•g. 4.6.

- - -- (Herman t al, 1982).


Page 4 - 15




AFTER

INVERSE VELOCITY STACK

MULTIPLE ATTENUATION

NPUT

AFTER

F-K MULTIPLE

ATTENUATION

J. ' ' ')'%': • t ' 11 1'1 ';.•m,:' :',./-•-•l- •r'm-- all

" "';;:.m;: .... ,;lliml;

.. .

m#l

Fig, 4.8.

Commonoffset stacks calculated from data before multiple

attenuation, after inverse velocity stack multiple attenuation,

and after F-K multiple attenuation. (Hampson, 1986)

888

Zone d

Interest

16984

Second eal-data et conventionaltackwithoutmultiple ttenuation.

'•" • ...... ;•,•<,:u(•:'J,.•J,.•., -, •, I• ,,,, .... •.. •, •,,,•• '•;••

,,t.•/:,.•t.,. ). I',,', ,'; • , , •, ß '1"' ',''. ;•t(•' )"•,'.m,,•""•.

• ,ii%' .t .% .

, ,, , • ..•'•t,..'•"•'i•' • -

---';•-•' "t" 1•%';J• •t•, ß... - ... ; -' ".' ,•..' . 2•>': ..'•, •;,%"'•1

lee "" • "" • • ' "' "•' ß ' ß ' • ....

'" "' Zone of

,,, .t•iill••)•.•);•l',"P,'•)'•"•'".•r'"mm"•""•P"••)r'"••' ' '" •- ..... ,• Interes

,..,.,..,,,_.,,.,... .,...,. ...,..,.•..,....,,,.,.,..•.. , .

,' .l•,•) ' • .'•',•' '• ....

.....•.•_ •.U.•,.., .. •

•,•,•p}•h•?.• •.•,•..} , •.•, ,•,•m,l,•,,nm,"::•"'•'•""""="'""•"...;'

.•,,,,.,.•,,,,,.., ,,{. ........ ,, .. ,,, ../•.• ,•.•'•, .'•-•%

Fig. 4.9.

Second real data stack after inverse velocity stack

multiple attenuation. (Hampson, 1986)


Page 4 - 16



Introduction to Seismic Inverslon Methods Brian Russell

PART 5 - RECURSIVE INVERSION - PRACTICE

_ _ _ _ _ .. . .• ,• _ _

Part 5 - Recursive Inversion - Practice

Page 5 - i




5.1 The Recurslye Inversion Method

We have now reached a point where we may start aiscussing the various

algorithms currently used o invert seismicdata. Wemust ememberhat all

these techniquesare baseUon the assumption of a one-aimensional eismic

trace model. T•at is, we assume hat all the corrections which were aiscussed

in section 4 have been correctly applied, leaving us with a seismic section in

whic• each trace represents a vertical, band-limiteU reflectivity series. In

this section we will look at some of the problems inherent in this assumption.

The most popular techniquecurrently used to invert seismic Uata is referred

.

to as recursire inversion and goes under such trade names as SEISLOGana

VERILOG. The basic equations used are given in part 2, anU can be written

Zi+Z <===__===>i+lZ ,

ri-- i+l+ LIJ

where

r i

= ith reflection coefficient,

and

Z --/•Vi= density veloci y.

The seismic data are simply assumea o fit the forward model and is

inverted using the inverse relationship. However, s wasshownn section 3,

one of t•e key problemsn the recursire inversion of seismic data is the loss

of the low-frequency component. Figure 5.1 shows an exampleof an input

seismic section aria the resulting pseuao-acoustic impeaance without the

incorporationof low frequency information. Notice that it resembles

phase-shifteU version of the seismic ata. The questionof introUuclng the

low frequency omponentnvolves wo separate ssues. First, wheredo we get

the low-frequency omponentrom, ana, second,how o we incorporate t?

Part 5 - Recurslye Inversion - Practice

Page 5 - 2.




1171121e9 eS1ol 92 93

i• 11•I Ittltl =:::•:::::::-•--lll[l•1111t•'• •1tlllttllltlll•l t1 l IIitti llltfltll l•I 1 t•n•'il, , •l••J• •":• •

•'• •" • --'' '

___ ..• - ,•, _•. • • f •• .• ............

:•,• m•,•'. • ....... • .... ,.• .... • . •• .........

ß ß ß • ... •

,• ß - • •, • ,•,..,• :'•l•,fm;•v•,• •,.•.•l.;•.•.'..•l•l;ql .n ................... ...; •;....: • .. • ...................

' • ]• • • •' ,, •, •' ,,,',',•,,, ,'",','" :•'•'•"•m••q•'t•'•'•a...., '. ,•],''•,J'•,• .• ' - '""W',- -::-=

•, '2 ,,• • ., •,•- • ,,• . ,•,•,I,.•.•..,• ....... •.•,,• . .,%•.• . ,• . '-.. ' .,• •, . •i• .......

. • , • • •-•,• ,, • , ,,.,,• .., ..... •. •.,.,,,,..•,.., ,,,.•,•,•.•.• .... .,• .... • .......

'•. . •q• • •,•;.•,• ,.. • •,l•,,..,,, ..•, J I .,,, • .•,• • .....,•...... : .•......•.•.. :,.. , ... , ,. , ............

, , •.•- -. •- (• ••' •'•:; •, / ................... . .... -(•-•( •.•,••(•'••'•"•:•"•'•7 '• . , .

• •'•,:•'•' • x•{

- ,,

2•Y•' •] ,,•.-..•.•.,'.;.',-,.. .................. • ............ • .................. •'•:.,• ...... • ... ......... •" ß7•' . =". .... 7' • '• • '. ' .----

.... - ......... •m:'•' •"• 'u'" •$• .... , ..... .. •<• • ß • - ' •'•' - ' .'••'•q• "•. •q• • .....

.•,.,• • .... ,_ /. ,,,_ . ; .... •,.:• - ............. ...... •%--=: .•.. .............. , .......... • .....

•4• 7•* • ';•u

. :c• • ,• .,,•-.•,, ?'..%•.,

•*•'•d•ti',i l•l•l'i'/lt' i•"'; •:•;•t•l,•i•21.•.l•'*.'•.'l•,•-•ii•.'•'..•,• :b-''? "•''• .... ; '_ ],;,'• ; '-•-•,••-----m'•l• ••"'•I'i•I• ........

•?•'•'• ;•q•.' '•'"•",•h/•'•'}'•' "' c'(•'•'".........

.... •, --.- -••_ ,,.•_.'.';'". :: :: ......

ß" • ..... "• '1 '• ' ' ' ß , -' ' • ..... • ' - ß

•'.•-•-• '•-<•., •

'. ,,,'• ,, ,. ,, ,

(a) Oriœinal- eismic ata. Heavyines indicatemajorreflectors.

0.7

N N N '" "

0.7

0.8

0.9

10

l

12

.3

1.4

1.5

1.6

1.7

(b) Recursive nversion of data in (a).

Figure 5.1

0.8

'I

1.0

i

I 1

I

I

1.2

.I

.

1.3

i

1 4

1.5

1.7

I

I

18

i

I 19

(Galbraith and Millington, 1979)


Page 5 - 3




The ow frequency omponentan be found n oneof three ways'

(1) From a filtered sonic log

The sonic log is the best wayof deriving ow-frequencynformation in

the vicinity of the well. However,t suffers from womainproblems't is

usually stretched with respect o the seismicdata and t lacks.a lateral

component.hese roblems,iscussedn Galbraith ndMillington 1979), are

solved by using a stretching algorithm which stretches the sonic log

information to fit the seismic data at selected control points.

(2) From seismic velocity analysis

In this case, interval velocities are derived from the stacking velocity

functions along a seismic ine usingDix' formula. The resulting function

will be quite noisyand t is advisable to do someormof two-dimensional

filtering on them. In Figure 5.2(a), a 2-D polynomialit has beendone to

smoothout the function. This final set of traces represents the filtered

interval velocity in the 0-10 Hz range or each race and may be added

directly to the inverted seismic races. Refer to rindseth (1979), for more

de ta i 1 s.

(3) From a geological model

Using all

incorporated.

available sources, a blocky geological model

This is a time-consuming method.

can be built and

Part 5 - Recursire Inversion - Practice

Page 5 - 4.



Introduction to Seismic InversiOn Methods Brian Russell

.

70000

(a)

GOOO0

$0000

(pvl 4oooo

'/sgc

( b ) $oooo

ZOOO0

I0000

/ -- V..308PV)*460

,

i

VELocrrY SURFACE 2rid ORDERPOLYN• Frr

Figure 5.2 s •mTZ eH CUT tT•

tRussell and Lindseth, 1982).

Part 5 - Recursivenversion Practice Page5 - 5

ß




Second, the low-frequency component an be added o the high frequency

componenty either adding reflectivity stage or the impedancetage. In

section 2.3, it was shownhat the continuous pproximationo the forwardand

inverse equations was given by

Forward Equati on

1 d 1n Z(t) <::==> Z(t)

r(t) =•- dt -

Inverse Equation

t

=Z(O)xp•0 (t)dt.

Since he previousransformsre nonlinear(becausef the logarithm),

Galbraith and Millington (1979) suggest hat the addition of the low-frequency

componenthouldbe made t the reflectivity stage. In the SEISLOGechnique

they are addedat the velocity stage. However, ue o other considerations,

this should not affect the result too much.

Of course, we are really interested in the seismic velocity rather than

the acoustic impedance.igure5.2(b), fromLindsethlg79), showshat an

approximateinear relationship exists between velocity and acoustic

impedance, given by

V = 0.308 Z + 3460 ft/sec.

Notice that this relationship is good for carbonates and clastics and

poor for evaporitesand should herefore be usedwith caution. A moreexact

relationship may be found by doing crossplots from a well close to the

prospect. However, singa similar relationshipwemayapproximatelyxtract

velocity information from the recoveredacoustic impedance.

Figure 5.3 showsow frequencynformation erived from filtered sonic

logs. The final pseudo-acousticmpedanceog is shown n Figure5.4

including the low-frequencyomponent.Notice that the geologicalmarkers re

moreclearly visible on the final inverted section.

Part 5 - Recurslye Inversion - Practice

Page 5 - 6




Figure 5.3 LowFrequencyomDonenterived from"st.reched:' onic loœ.

0.7

0.8

0.9

l.O

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

19

Figure 5.4 Final inversioncombinin•Figures 5.1(b) and 5.3.

Lines indicate major reflectors.

0.9

1.0

1.1

1.2

1.:)

1.4

I$

1.6

1.7

19

(Galbraith and Millington, 1979)


Page 5 - 7




In sugary, the recursive method of seismic inversion may be given by the

fol 1owing flowchart'

I

i

INTRODUCEOWREQUENCIES)

I•.v• o •DO-•CO••c

'

CORRECTOSEUDOELOCITIES,

CONVERTOEPTH

Recursi ve Inversion Procedure

, . _ ß ., . i

A commonmethod of display used for inverted sections is to convert to

actual interval transit times. These transit times are then contoured and

coloured according to a lithological colour scheme. This is an effective way

of presenting the information• especially to those not totally familiar'with

normal seismic sections.


Page 5 - 8




(a)

Frequency

(e)

1

(b)

Fig.

(a) Frequency response of a theoretical differentiator.

(b) Frequency esponseof a theoretical integrator.

Part 5 -Recursire Inversion - Practice

(Russell and Lindseth,

,m ,i m ml , ,

Page 5 - 9

982 )



Introduction to Seismic Inver.si.on Methods Brian Russell

5.2

I nfor marl .n ?•_Th.e.o..wF.r.equ.e.cycompo..ne.t

The key factor which sets inverted data apart from normal seismic data is

the inclusion of the low frequency component, egardless of how his component

is introduced. In this section we will look at the interpretational

advantages of introducing this component. The information in this section is

taken from a paper by Russell and Lindseth (1982).

We start by assuming the extremely simple moael for the

reflectivity-impedance relationship which was introduced in part 5.1. However,

we will neglect the logarithmic relationship of the more complete theory (this

is justifiea for reflection coefficients less that 0.1), so t•at

t

_1dZ(t) =__==>(t)2Z(O)j•(t) t

(t) - • dt- '

If we consider a single harmonic component, we may derive the

response of this tel ationship, which is

dewt jwt jwt -j eJWt

-dt "-- we <===> . dt= w

where w-- 21Tf,

frequency

In words.,differentiation introduces a -6 riB/octaveslope from the high

end of the spectrum o the low, and a +90 degree phase shift. Integration

introduces a -6 dB/octave slope from the low end to the high end, and a -90

degree phase shift. Simpler still, differentiation removesow frequencies

and integration puts them in. Figure 5.5 illustrates these relationships.

But how aoes all this effect our geology? In Figure 5,6 we have

illustrated three basic geological models'

ß

(1) Abrupt 1 thol ogi c change,

(2) Transitional lithologic change, an•

(3) Cyclical change.


Page 5 - 10




(A)MAJOR ITHOLOGICHANGE

V1

Vl

I

i

I.

I

I

I

I

i

I

(B)TRANSITIONAL LITHOLOGIC

CHANGE

V:V•+KZ

i i

(C)CYCLICALCHANGE

• _

Fig. 5.6.

Three ypesof lithological models' (a) Major change,

(b) Transitional, (c) Cyclical. (Russell and Lindseth, 1982).


Page 5- 11




Wemayllustrate the effect of inversion n these hree casesby ooking

at both seismic anU sonic log Uata. To show he loss of high frequencyon the

sonic log, a simple filter is used, nd he associatedhase hift is not

introUuced.

To start with, considera major1 thologic boundarys exempllieu by the

Paleozoicunconformityf Western Canada, changeroma clastic sequenceo

a carbonateequence. igure5.7 showshat most f the information bout he

largestep n velocity s containeUn theD-10iz componentf the sonicog.

In Figure5.8, the seismic ataand inal Uepthnversion re shown.On the

seismic data, a major boundary howsup as simply a large reflection

coefficient, whereas,on the inversion, the large velocity step is shown.

RAWSONIC FILTEREDONICLOGS

VELOCITY FT/SEC

0 10000 10-90HZ O-IOHZ O-CJOHZ

TIME

0.3-

0.5-

Fig. 5.7. Frequencyomponentsf a sonicog.

(Russell and Lindset•, 1982).

L , , , I I ß [ I L


Page 5 - 12




o'- .

ß

(a)

.%;

DEPTH SEISLOG

ß o

DEPTH

(b)

..... ßOP OF

"' . ß""I:'ALEOZOIC

-425'

Fig. 5.8. Major litholgical'change, Saskatchewan example.

(a) Sesimic s_ection, (b) Inverted section.

..... _........ _(R_q•sell... nd L ,pqse_th,_• 98_2)___


Page 5 - 13




To illustrate transitional and cyclic change, a single examplewill be

used. Figure. 5.9 showsa sonic og from an offshore Tertiary basin,

illustrating the rampswhichshow a transitional velocity increase, and the

rapidly varyingcyclic sequences. otice hat the 0-10 Hz componentontains

all the information about the ramps, but the cyclic sequences contained n

the 10-50 Hz component.Only he Oc components lost from the cyclic

componentpon emoval f the low frequencies. Figure 5.10 illustrates the

same oint using he original seismic ata and he final depth nversion.

In summary,he information ontainedn the low frequency omponent f

the sonic og is .lost in the seismic data. This includessuchgeological

information as the dc velocity component,arge jumps in velocity, and linear

velocity ramps. If this information could be recovered nd ncluUeaduring

the inversion process, it would ntroduce his lost geological information.

Fig. 5.9. Sonicog showingyclic and ransitionalstrata.

Part 5 - Recurslye

Inversion - Practice

(Russell

and LinOseth, 1982)

Page 5 - 14



(b)


(a)

SEISMIC SECTION-CYCUC & TRANSITIONAL STRATA

i 1-3500

ß


Page 5 - 15




5.3

Sei smical ly Derived Poros ty

- ILI , ß I

We have shown hat seismic data may be quite adequately inverted to

pseudo-velocity (and hence pseudo-sonic) nformation i f our corrections and

assumptions are reasonable. Thus, we may try to treat the inverted data as

true sonic log information and extract petrophysical data from it,

specifically porosity values. Angeleri and Carpi (1982) have tried just this,

with mixed results. The flow chart for their procedure is shown in Figure

5.11. In their chart, the Wyllie formula and shale correction are given by:

where

At --transit time for fluid saturated rock,

Zstf= pore luid transit time,

btma: ockmatrix ransittime,

Vsh fractional olumef shale,and

btsh: shale ransit ime.

The derivation of porosity was tried on a line which had good well

control. Figure 5.12 shows the plot of well log porosity versus seismic

porosity for each of three wells. Notice that the fit is reasonable in the

clean sands and very poor in the dirty sands. Thus, we may extract porosity

information from the seismic section only under the most favourable

conditions, notably excellent well control and clean sand content.

Part 5 - Recurslye Inversion - Practice Page 5 - 16




F ']w[ttill

•ILI61CAT&$[IS'MI•AT&'

I-"'• '' m.,,•,,ml

-[ ,gnu mill i' •ill. Utl.. 111l lit

•%lOtOG

I IIITEIPllETATII

Fig.

l WlltK :

t ' .

5.11. Porosity eval uati on flow diagram.

(AngeleriandCarpi, 1982).

Fig.

, ,

WELL 2 WELL 3 WELL

__ ClII PNIIVI o..- OPt poeoItrv ..... CPI

ß , , ß ß ' I ,- --

e e I e . e e . . e ß e e e e I i e e e ß i e i ß ß ß e

.

1.4

1.7

1.8,

1.9

5.12. Porosity profiles from seismic data and borehole data.

Shalepercentages al so displayed. (Angel ri andCarpi, 1982).


i ,

Page 5 - 17



Introduction to Sei stoic Inversion Methods Brian Russel 1

PART 6 - SPARSE-SPIKE INVERSION

• { • ...... • I ] m • m

Part 6 - Sparse-spike Inversion

6- 1



Introduction to Seismic Inversion Me.thods Brian Russell

6.1 Introduction

Thebasic theoryof maximum-1kel hood econvoltion (MLD)wasdeveloped

by Dr. Jerry Mendel nd his associatesat USC nUhas beenwell publicised

(Kormylo ndMendel, 983;Chiet el, 1984). A paperby Hampsonnd Russell

(1985) outlined a modification of maximum-likelihoodeconvolutionmelthod

which allowed the method o be moreeasily applied to real seismic •ata. One

of the conclusions f that paper wasthat the method ould be extenoed o use

the sparse eflectivity as the first step of a broadband eismic inversion

technique.This technique,which will be termedmaximum-likelihoodeismic

inversion, is discussed later in these notes.

You will recall that our basic model of the seismic trace is

s(t) = w(t) * r(t) + n(t),

where

s(t) : the seismic trace,

w(t) : a seismic wayelet,

r(t) : earth reflectivity, and

n(t) = addi tire noise.

Notice that the solution to the above equation is indeterminate, since

there are three unknowns o solve for. However, using certain assumptions,

the aleconvolution roblem can be solved. As we have seen, the recursire

method of seismic inversion is basedon classical aleconvolution techniques,

which assume random eflectivity and a minimum r zero-phase wavelet. They

produce higher requency aveleton output,but never ecover he reflection

coefficient series completely. More recent aleconvolutionechniquesmaybe

groupedunder the category f sparse-spike eth•s. That is, they assume

certain modelof the reflectivity and make a wavelet estimate based on this

assumption.


6- 2




ACTUAL REFLECTIVITY

I,:, I ..

POISSON-GAUSSIAN

SERIES OF LARGE

EVENTS

--F

GAUSSIAN BACKGROUND

OF SMALL EVENTS

SONIC-LOG REFLECTIVITY

EXAMPLE

Figure 6.1 The fundamentalssumptionf the maximum-likelihoodethod.

Part 6- Sparse-spike Inversion

6- 3



Intr6duction to Seismic Tnvetsion Methods Brian Russell

These techniques include-

(1) btaximum-Likel ihood deconvolutton and inversion.

(2) L1 norm deconvolution and inversion.

(3) Minimum ntropy deconvol tion (MEO).

From the point of view of seismic inversion, sparse-spike methods have an

advantage over classical methods f deconvolutionbecause the sparse-spike

estimate, with extra constraints, can be used as a full bandwidth estimate of

the reflectivity. We will focus initially on maximum-likelihood

deconvolution, and will then move on to the L1 normmethod of Dr. Doug

O1denburg. The MEDmethod will not be discussed in these notes.

6.2 Maximum-Likelihood Deconvolution and Inversion

i i m ß m m m m I _ ß

Maximum-Li kel i hood Deconvoluti on

I ß ß ß m _ _ l . . • am .. I _

Figure 6.1 illustrates the fundamental assumption of Maximum-Likelihood

deconvolution, which is that the earth' s reflectivity is composed f a series

of large events superimposedon a Gaussian backgroundof smaller events. This

contrasts with spiking decon, which assumesa perfectly randomdistribution of

reflection coefficients. The real sonic-log reflectivity at the bottom of

Figure 6.1 shows that in fact this type of model is not at all unreasonable.

Geologically, the large events correspond to unconformities and major

ß

1 thol ogic boundaries.

From our assumptions about the model, we can derive an objective function

whichmaybe minimized o yield the "optimum" r most ikely reflectivity. and

wavelet combination consistent with the statistical assumption. Notice that

this method gives us estimates of both the sparse reflectivity and wavelet.

,,


m




INPUT

WAVELET

REFLECTIVITY

NOISE

SPIKE SIZE' 9.19

SPl• ••: 50.00

NOISE' 39.00

OB,.ECTIVE' 98.19

Figure6.2(a) Objectiveunctionor onePoSsibleolution o input race.

INPUT

WAVELET

REFLECTIVITY

SPIKE S 7_F: 6.38

SPIKE DENSIq'•, 70.85

NOISE

NOISE: 81.• 5

OBJECTIVE 158.98

Figure 6.2(b)

Objectiveunctionora secondossibleolutiono nput

trace. This alues higherhan .2(a),.ndicatingless

1 kely solution.

, ,,

Part 6 - Sparse-spike

Inversion 6- 5




The objective function j is given by

- R2 N

k=l k=l

where

- 2m n(X)- 2(L-re)In(i-A)

r(k) = reflection coeff. at kth

sample,

m = numberof refl ecti OhS,

ß

L : total numberof samples,

N : sqare root of noise variance,

n : noise at kth sample, and

• = likelihood that a given

sample has a reflection.

Mathematically, the expected behavior of the objective function is

expressed in terms of the parametersshown bove. No assumptions are made

about he wavelet. The reflectivity sequence s postulated o be "sparse",

meaning that the expected number f spi•es is governed by the parameter

lambda, the ratio of the expected numberof nonzer. spikes to the total number

of trace samples. Normally, lambda is a numbermuchsmaller than one. The

other parametersneeded o describe the expected behavior are R, the RMS•size

of the large spi•es, andN, the RMS ize of t•e noise. With these parameters

specified, any glven deconvolution sol ution can be examined to see.whether it

is likely to be the result of a statistical processwith thoseparameters.For

example, f the reflectivity estimate has a number f spikesmucharger than

the expectednumber, hen it is an unlikely result.

In simpler terms, we are looking for the solution with the minimum

number f spikes n its reflectivity and t•e lowestnoisecomponent. igures

6.2(a) and 6.2(b) showwopossiblesolutions or the same nput synthetic

trace. Notice hat the obje6tive function or the onewith the minimumpike

structure is indeed the lowest value.


6- 6



Introduction to Sei smic I nversi.on Methods Bri an Russel1

Original

Model

I terati on I

I terati on 2

Iteration 3

I teration 4

Iteration S

Iteration 6

Iterati on 7

Reflectivity

I, ill.

I ,1.2. -.I

,i.

Synthetic

Figure 6.3.

The Sinl•le Most Likely Addition (SMLA)algorithm illustrated

for a simple reflectivity model.

Part 6 - Sparse-spi ke Inversion

6- 7



Introduction to Seismic Inversion Methods Brian Russel1

Of course, there maybe an infinite number f possible solutions, and it

would take too much omputerime to look at eachone. Therefore, a simpler

method is used to arrive at the answer. Essentially, we start with an initial

wavelet estimate, s'timatehe sparseeflectivity, ' improvehe wavelet and

iterate through this sequence of steps until an acceptably low objective

function is reached. This is shownn block form in Figure 6.4. Thus, there

is a two step procedure-having he wavelet estimate, update he reflectivity,

and then, having the reflectivity estimate, update the wavelet.

These procedures are illustrated on model data in Figures 6.3 an• 6.5.

In Figure 6.3, the proceUure for upUating the reflectivity is shown. It

consists of adding reflection coefficients one by one until an optimum et of

"sparse" coefficients hasbeen ound. Thealgorithm sed or updating the

reflectivity is callee the single-most-likely-addition algorithm (SMLA)since

after each step it tries to find the optimum pike to add. Figure 6.5 shows

the procedure for updating the wavelet phase. The input model is shownat the

top of the figure, and the up•ated reflectivity and phase s shown fter one,

two, five, and ten iterations. Notice that the final result compares

favourably with the model wavelet.

WAVELET

ESTIMATE

ES•TE

REFLECTIVITY

IMPROVE

WAVELET

ESTIMATE

Fiõure 6.4.

Theblock omponentethodf solvingor both

reflectivity andwavelet. Iterate aroundhe

loop unti 1 converRence.


6- 8



Introduction to Seismic Invers.ion Methods Brian Russell

Wayelt Refl ctiVity ' Synthetic

Ill ,I ,

INPUO

INITIAL CUESS

TEN ITERATIONS

Fi õure 6.5.

The procedure for updatinõ the wavelet

in the maximum-likelihood method.

Between each iteration above, a separate

iter. ation on reflectivity (see Fiõure 6.3)

has been done.


6- 9



Introduction o Seismic nversionMethods Brian Russell

Figure 6.6 is an exampleof the algorithm applied to a synthetic

seismogram. Notice that the major reflectors have been recovered fairly well

and that the resultant trace matches he original trace quite accurately. Of

course, the smaller reflection coefficients are missing in the recovered

reflection coefficient series.

Let us now ook at some real data. The first example is a' basal

Cretaceous gas play in Southern Alberta. Figure 6.7(a) and (b) shows the

comparison between the input anU output stack from the aleconvolution

procedure. Also shown re the extracted and final wavelet shapes. The main

things to note are the major increase in detail (frequency content) seen in

the final stack, and the improvement n stratigraphic content.

Figure 6.8 is a comparisonof input and output stacks for a typical

Western Canada basin seismic line. The area is an event of interest between

0.7 anU 0.8 seconds, representing a channel scour within the lower Cretaceous.

Although the scour is visible on both sections, a dramatic improvements seen

in the resolution of the infill of this channel on the deconvolved section.

Within the central portion of the channel, a .positive reflection with a

lateral extent of five traces is clearly visible and is superimposedn the

Uominant negative trough.

INPUT:

V. ,.: --

ESTIMATED:

ttl J':ll'j "'" "

Figure 6.6 Synthetic seismogram test.


6- 10




0.5

0.6

0.7

0.8

'SONIC

SYNTHETIC LOG

iZ.

EXTRACTED WAVELET

0.5

0.6

.

0.8

(b)

(a) Initial seismicwith extracted wavelet.

Final deconvolved seismic with zero-please wavelet.

Figure 6.7

.... - -_ __ ._


11




This is quite possibly a clean channel sand and may or may not be

prospective. However, this feature is entirely absent on the input stack.

Overlying the channel is a linear anomaly which could represent the 'base of a

gas sand, and is muchmore sharply defined on the output section, both in a

lateral and vertical sense.

Finally we have taken the deconvolved output and estimated the

reflectivity. This is shown in Figure 6.9. Although some of the subtle

reflections are missing from this estimated reflectivity, there is no doubt

that all the main reflectors are present. It is interesting to note how

clearly the base of the channel (at 0.7;- seconds)and the base of the

postulated gas sand on top of the channel have been delineated.


6- 12




INPUT

STACK

DECONVOLVED

STACK

0.6

0.7

0.8

0.9

Figure 6.8

An input stack over a channelscour and

the resul ting deconvoled sei smic.

DECONVOLVED

STACK

ESTIMATED

REFLECTIVITY

0.6

0.7

0.8

0.9

Figure 6.9

The deconvolved result from Figure 6.8

and its estimated reflectivity.


m

13




Maximum-Likel ihood Inversion

An obvious extension of the theory is to invert

reflectivity to Uevise a broad-band or "blocky" impedance

data (Hampsonnd Russell, 1985). Given the reflectivity, r(i),

impedance (i) maybe written

Z(i) Z(i_l[1 r(i)]

- r(i) '

the es ti ma ted

from the seismic

the resul ting

Unfortunately, application of thi

from MLD produces unsatisfactory res

additive noise. Although the MLD algor

of the wavelet to produce a broad-band

of this estimate is degraaed by noi

spectrum. The result is that while

s formula to the reflectivity estimates

ults, especially in the presence of

it•m'extrapol ares outsi de the bandwidth

reflectivity estimate, the reliability

se at the low frequency end of the

the short wavelength features of the

impedancemay be properly reconstructed, the overall trenu is poorly resolvea.

This is equivalent to saying that the times of the spires on the reflectivity

estimate are better resolved than their amplituaes.

In order to stabilize the reflectivity estimate, independent knowleUge

of the impedance renU may be input as a constraint. Since r(i) < l, we can

derive a convolutional type equation between acoustic impeUance anU

reflectivity, written

In Z(i) = 2H(i) * r(i) + n(i),

where Z(i) = the known mpedance rend,

• i <0

H(i) :

• i >0

and

n(i) : "errors" in the input trend.

_


6• 14




Figure 6.10 Input Model parameters.

Figure 6.11

ß

Maximu•m-Lkel i hood i nversi on result from Figure 6.10.

.m __


6- lb



Introduction to Seismic Inversion 'Methods Brian Russell

The error series n(i) reflects the fact that the trend information is

approximate. We now have two measured time-series: the seismic trace, T(i),

and the log of impedance n Z(i), each with its own wavelet and noise

parameters. The objective function is modified to contain two terms weighted

by their relative noise variances. Minimizing this function gives a solution

for r(i) which attempts a compromiseby simultaneouslymoUelling he seismic

trace while conforming o the known mpedance rend. If both the seismic

noise and the impedancerend noise are modelledas Gaussiansequences, heir

respective variances become tuning" parameterswhich the user can modify to

shift the point at which the compromise occurs. That is, at one extreme only

the seismic nformation s usedand at the ot•er extremeonly the impedance

trend.

In our first example, he methods tested on a simple synthetic. Figure

6.10 shows he sonic og, the derived reflectivity, the zero-phase wavelet

used to generate the synthetic, and finally the synthetic itself. This

example was used nitially because t truly represents a "blocky" impedance

(and therefor.e a "sparse" reflectivity) and therefore satisfies the basic

assumptions of the method.

In Figure 6.11 the maximum-likelihood inversion result is shown. In

this casewe haveuseda smoothedversion of the sonic velocities to provide

the constraint. A visual comparisonwoulU indicate that the extracteU

velocity profile corresponds very well to the input. A moredetailed

comparisonof the two figures shows hat the original and extracted logs do

not matchperfectly. T•ese small. shifts are due to slight amplitude problems

on the extracted reflectivity. It is doubtful that a perfect match could ever

be obtai neU.


6- 16



Introduction to Seismic Inversion Methods

Bri an Russel 1

Figure 6.12 Creation of a seismic model from a sonic-log.

Figure 6.13 Inversion result from Figure 6.12.

•- _ ...... ii__ - - i - •_ mm i i i ß i i It_l I

Part 6 - Sparse-spi•e Inversion

17




Let us now urn our attention to a slightly more realistic synthetic

example. Figure 6.12 showshe applicationof this algorithm o a sonic-log

derivedsynthetic. At the' top of the figure we see a sonic log with 'its

reflectivity sequence below. (In this example,we have assumed hat the

density is constant, but this is not a necessary restriction.) The

reflectivity wascbnvolved ith a zero-phase avelet,bandlimited rom10 to

60 Hz, andthe final synthetic s shownt the bottom f the figure.

The results of the maximum-likelihood inversion method are sbown in

Figure 6.13. The initial log is shownat the top, the constraint is shown n

the middle panel, and the extracted resull• is shownat the bottom of the

diagram. In this calculation, the waveletwasassumednown. Note the blocky

nature of the estimated elocity profile comparedith the actual sonic log

profile. Again, the input and output logs do not matchperfectly.

The fact that the two do not perfectly match s due to slight errors in

the reflectivity sizes whichare amplified by the integration process,and s

partially the effect of the constaint used. Theconstraintshown n Figure

6.13 wascalculatedby applying a 200 ms smoothero the actual log. In

practice, this information could be derived from stacking velocities or from

nearby well control.


6- 18




*

Figure 6.14

An input seismic 1 ne to be inverted.

:

ß

'.

eel'?

e4dl

Figure 6.15

Maximum-Liklihood reflectivity estimate from

seismic in Figure 6.14.

Part 6 - Sparse-spike Inversion 6- 19



Introauction to Seismic Inversion Methods Brian Russell

Finally, we show he results of the algorithm applied to real seismic

data. Figure 6.14 shows portion of t•e input stack. Figure 6.15 shows he

•D extracted reflectivity. Figure 6.16 shows the recoveredacoustic

impedance,wherea linear ramphas beenusedas the constraint. Notice that

the inverted section •isplays a "blocky" character, indicating that the major

features of the impedanceog have been successfullyrecovered. This blocky

impedanceanbe contrastedwith the more traditional narrow-bandinversion

procedures, whichestimate a "smoothed"r frequencyimited version of the

impedance.Finally, Figure 6.17 shows a comparisonetweenhe well itself

and the inverted section.

In summary,maximum-likelihoodnversion is a procedurewhich extracts a

broad-band estimate of the seismic reflectivity and, by the introduction of

1 near constraints, al lows us to invert to an acoustic impedance ection which

retains the major geological features of borehole og data.


6- 20



Introduction to Seismic Inver.sion Methods Brian Russell

Figure 6.16

Inversion of reflectivity shownn Figure 6.15.

SEISMIC NVERSION

WELL

+

SONIC

LOG

Figure 6.17

A comparison of the inverted seismic data and

the sonic log at well location.


..

21




6.3 The L 1 Norm Method

-- __LI _ _ _ i .

Another method of- recursive, single trace inversion which uses a

"sparse-spike"ssumptions the L1 normmethod,evelopedrimarilyby Dr.

DougOldenburg f UBC. nd Inverse Theory andApplications (ITA). This method

is also often referred to as the linear programmingethod, nd this can lead

to confusion. Actually, the two names efer to separateaspects f the

method. Themathematical odelused n the construction f the algorithm is

the minimization f the L1 norm. However, he methodused o solve the

problem is linear programming. The basic theory of this method s found in a

paper by Oldenburg, et el (1983). The first part of the paper discusses he

noi se-free convolutional model,

x(t) --w(t) * r(t), where x(t) = the seismic race,

w(t) --the wavelet, an•

r(t) -- the reflectivity.

The authors point out that if a high-resolution aleconvolution is

performed n the seismic race, the resulting estimateof the reflectivity can

be thought of as an averagedversion of the original reflectivity, as shown t

the top of Figure6.18. This averagedeflectivity is missing oth •e high

and ow frequencyange,and s accurateonly in a band-limitea entral range

of frequencies. Although there are an infinite number f ways in which the

missing frequency components an be supplied, Oldenburg, et al (1983) show

that we can reduce this nonuniqueness by supplying more information to the

problem, such as the layered geological model

r(t) -•,rj6(t l•),

j--

where= 0 if t •l• , an•

=1 ift:• .


6- 22




b

ß ß ß • 1

m m m

0.0

T.IJdE• •J

e f

o .50 joo j25

FR F.,O [HZJ

I

I

Figure 6.18

Synthetic test of L1 Norm nversion, moUified fro•.q

Oldenburg t al (1983). (a) Input impedance,

(b) Input reflectivity, (c) Spectrum f (b),

(d) Low frequencymodel trace, (e) Deconvolutionof (•),

(f) Spectrum f (U), (g) Estimated mpedancerom L1 Norm

method, (•) Estimated reflectivity, (i) Spectrumof (•).


6- 23



Introduction to Seismic Inversion •.le.thods Brian Russell

Mathematically, the previous equation is considered as the constraint to

the inversion problem. Now, the layered earth model equates to a "blocky"

impedance unction, which in turn equates to a "sparse-spiKe" reflectivity

function. The above constraint will thus restrict our inverted result to a

"sparse" structure so that extremely fine structure, such as very small

reflection coefficients, will not be fully inverted.

The other key difference in the linear programmingmethod is that the L1

norm is minimized rather than the L2 norm. The L1 norm is defined as the sum

of the absolute values of the seismic trace. TrueL2 norm, on the other hand,

is defined as the square root of the sumof t•e squares of the seismic trace

values. The two norms are shown below, applied to the trace x:

x1 : x and x2: x

i--1 i:1

The fact that the L1 norm favours a "sparse" structure is shown in the

following simple example. (Taken from the notes to Dr. Oldenburg's 1085 CSEG

convention course' "Inverse theory with application to aleconvolution and

seismogram nversion"). Let f and g be two portions of seismic traces, where'

f: (1,-1,0) and g : (0,%• 0) .

The L2 norms are therefore'

The L1 norms are given by'

-

l - 1 + 1 : 2 and gl = '

Notice that the L1 norm of wavelet g is smaller than the L1 norm of f,

whereas the L2 norms are both the same. Hence, minimizing the L1 norm would

reveal that g is a "preferred" seismic trace based on it's sparseness.


6- 24




(a)

Input sei smic data

(b)

Estimated refl ec ti vi ty

(c)

Final impedance

Figure 6.19

L1 14ormmetboOapplied to real seismic data,


(Walker and Ulrych, 1983

6- 25



Introduction to Seismic Inversion MethoUs Brian Russell

Several other authors had previously considered he L1 normsolution in

deconvolutionClaerboutand Muir, 1973, andTaylor etal., 1979), however,

they consideredhe problemn the time domain.Oldenburgt al.w suggested

solving the problem sing frequency omain onstraints. That is, the reliable

frequency and is honoredhileat the sameimea sparsereflectivity is

created. The results of their. algorithm on synthetic data are shownat the

bottom of Figure 6.18. Theactual implementationf the L1 algorithm o real

seismicdata has been done by Inverse Theory andApplications ITA). The

processing flow •or the linear programmingnversion methods shown elow.

InterPreter'=MP tacl<edection

<r(t)>= r(t)©w(t)

t ß ,i

i

I,,i o,ect,',o,' esidum'm,e,wt) I

ß i i i i i I i i

I Fourierrans••f •rt)> I

i

Scaleata

Const.ints.romtackins•_V'elocitles

ii &

Con,straintsromWellogs

Unear Programing Invemion

Assume( ß n ) t- q, s spame,eflectioneries.

Minimize the sum of absolute reflection strengU•.

FulFBand Reflectivity Series r (t)

Signal to Noise Enhancement and Display Preparation

Integration to Obtain Impedance Sections

Figure 6.19(b) The L1 Norm Linear Programming.)ethod. (Oldenburg,1985).


6- 26



Introduction to Seismic Inver. s,ion Methods Brian Russell

TSN

1,2

tO0 90 80 70 60 50 40 30 20 tO

1,3

1,4

1,5

1,6

1,7

1,:8

.2,0

2ø2

Figure 6.20

Inputseismic atasectiono L1 Normnversion.

(O1 enburg, 1985


6- 27




Figure 6.19 shows he application of the above technique to an actual

seismic line from Alberta. The data consist of 49 traces with a sample rate

of 4 msecand a 10-50 Hz bandwidth. The figure showshe linear programming

reflectivity and impedanceestimates below the input seismic section. It

should be pointed out that a three trace spatial smootherhas been applied to

the final results in both cases.

Finally, let us consider a dataset fromAlberta which has been processeU

through the LP inversion method. The input seismic is shown n Figure 6.2D

and the final inversion in Figure 6.21. The constraints useU here were from

well log data. In the final inversion notice that the impedance has been

superimposed on the final reflectivity estimate using a grey level scale.


6- 28




1.6

1.7

1.8

1.9

2.0

2.1

2.2

Figure 6.21

Reflectivity and grey-level plot of impedance

the L1 Norm nversion of data in Figure 6.20.

Part 6 -Sparse-spike Inversion

for

(O1denburg, 19

6- 2-9



Introduction to Seismic Inversion Methods Br•an Russell

6.4 Reef P roblee

_

On he next few pages is a comparison etweena recursive nversion

procedure (Verilog) anda sparse-spike nversionmethodMLD). The sequence

of pages includes the following:

- a sonic log and its derived reflecti vt ty,

- a synthetic seismogram t both polarities,

- the original seismic line, showing he well location,

- the Verilog inversion, and

- the MLD inversi on.

BaseUon the these data handouts, do the following interpretation

exerc i se:

([) Tie the synthetic to the seismic ine at SP 76. (Hint- use reverse

pol ari ty syntheti c).

(g) Identify and color the following events in the reef zone-

- the Calmar shale (which overlies the Nisku shaly carbonate),

- the 1retort shale, and

- .the porous Leduc reef.

(3) Comparehe reefal events on the seismic and the two inversions. Use

a blocked off version of the sonic log.

(4) Determine for parallelism which section tells you the most about the

reef zone?


6- 30




Rickel, g Phas•

3g Ns, 26 Hz

REFL. DEPTH VELOCI •¾

COEF. lib Eft,/sec.

...,--

...,--

...m

$11qPLE I HTI3tViIL- 2 Ns.

AliPLI •IIi)E I

tiC. Ilql•. - Sonic

Pei.•ri es onlg

Figure 6.22

Sonic Log and synthetic at the reef well.


6- 31




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5

55

57

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;'5

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$5

99

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6- 33



Introduc%ion [o Seismic Inversion Meltotis Brian Russell

0.? IB.G O.G 1.0 1.1 1.2 1.• 1.4

ß

1

25

27

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$1

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113

119

123

127

131

137

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Part 6 - Sparse-spikenversio•

i i i i I • e e t I i e I i e I e i ß l

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34




PART 7 - INVERSION APPLIED TO THIN BEDS

Part 7 - Inversion applied to Thin Beds

Page 7- I



Intro4uction to Seismic Inversion Methods Brian Russell

7.1 Thin Bed Analysis

One of the problems hat we have identified in the inversion of seismic

traces is the loss of resolution caused by the convolution of the seismic

wavelet with the earth's reflectivity. As the time separation between

reflection coefficients becomessmaller, the interference between overlapping

wavelets becomesmore severe. Indeed, in Figure 6.19 it was shown hat the

effect of reflection coefficients one sampleapart and of opposite sign is to

simply apply a phaseshift of 90 degrees to the wavelet. In fact, the effect

is more of a differentiation of the wavelet, which alters the amplitude

spectrum s wel1 as the phase spectrum. In this section we will look closer

at the effect of wavelets on thin beds and how .effectively we can invert these

thin bed s.

The first comprehensive 'ook at thin bed effects was done by Widess

(1973). In this paper he used a model which has become he standard for

discussing thin beds, the wedgemodel. That is, consider a high velocity

laye6 encasedn a low velocity layer (or vice versa) and allow the thickness

of the layer to pinch out to zero. Next create the reflectivity response rom

the impedance, nd convolvewith a wavelet. The thickness of the layer is

given in terms of two-way ime through the layer and is then related to the

dominantperiod of the wavelet. The usual wavelet used s a Ricker becauseof

the simpl city of its shape.

Figure 7.1 is taken from Widess' paper and shows he synthetic section as

the thickness of the layer decreases from twice the dominant period of the

wavelet to 1/ZOth of the dominant period. (Note that what is refertea to as a

wavelength n his plot i s actual y twice the dominant eriod). A few important

points can be noted from Figure 7.1. First, the wavelets start interfering

with eackotherat a thickness ust below two dominanteriods,but remain

Clistinguishable down to about one period.


Page 7- g




PI•OPAGA ION I NdC

ACnOSS TK arO) .

•'------).z _1

--t

Figure 7.1

Effect of bed thickness on

reflection waveshape,where

(a) Thin-bed model,

(b) Wavelet shapesat top

and bottom re fl ectors,

(c) Synthetic seismic

model, anU (d) Tuning

parameters as measured from

resul ting waveshape.

(C) (D)

5O

, ,.

THINBEDREGIME

J PEAK-TO-TROUGH/

AMPLITUDE

2.0

1.0 <

0.8

0.4

/ \

-0.4 ,• . . . . .

-40 0 20 40

MS

TWO-WAY TRUE THICKNESS

(MILLISECONDS)

Figure 7.2 A typical detection and resolution cha•t used

to interpret bed thickness from zero phase seismic data.

('Hardage, 1986

. .. _ i i ,, , i _ - - - -_- - _ - _ ..... l. _


Page 7- 3




Below a thickness alue of oneperiod he waveletsStart merging nto a

single wavelet, and an amplitude increase is observe•. This amplitude

increase is a maximum t 1/4 period, and decreases from this point down... The

amplitude is appraoching ero at 1/•0 period, but note that the resulting

waveform is a gO degree phase shifted version of the original wavelet.

A more quantitative way to measure his information is to plot the peak

to trough amplitude difference and i sochron across the thin bed. This is done

in Figure 7.•, taken from Hardage (1986). This diagram quantifies what has

already been seen qualitatively the seimsic section. That is that the

amplitude is a maximum t a thickness of 1/4 the wavelet dominant period, and

also that this is the lower isochron limit. Thus, 1/4 the dominant period is

considered to be the thin bed threshhold, below which it is difficult to

obtain fully resolved reflection coefficients.

7.2 In. ersion Camparisonf T.hinBees

ß

To test out this theory, a thin bed model was set up and was inverted

using both recursire inversion and maximum-likelihood aleconvolution. The

impedance model is shown n Figure 7.3, and displays a velocity decrease in

the thin bed rather than an increase. This simply inverts the polarity of

Widess' diagram. Notice that the wedge starts at trace 1 with a time

thickness of 100 msec and thins down to a thickness of 2 msec,.or .one time

sample. The resulting synthetic seismogram is shown n Figure 7.4. A 20 Hz

'Ricker wavelet was used to create the synthetic. Since the dominant period

(T) of a 20 Hz Ricker is 50 msec, he wedge has a thicknessof 2T at trace 1,

T at trace 25, T/2 at trace 37, etc.

Parl• 7 - 'inverslYnap'pled 1•oThin'- eds....

Page 7 --'4 '•-




lOO

200

3OO

400

500

4

8 12 16

20 24 28 32 36 40 44 48

ß

Figure7.3 True impedanceromwedgemodel.

o

lOO

200

.

300

ß

400

500

Figure 7.4

Wedgemodel reflectivity convol ved with

20 HZ Ricker wavelet.

Part 7 - Inversion applied to Thin BeUs Page 7- 5




First, let us consider the effect of performing a recursire inversion on

the wedgemodel. The inversion result is shown n Figure 7.5. Note that the

low frequency component as not added into the solution of Figure 7.5, to

better show the effects of the initial recursire phase of the inversion. It

was also felt that the addition of the low frequency componentwould ado

little information to this test. Notice that there'are two major problems

with recursire inversion.. First, the thickness of the beU has only been

resolved down o about 25 msec, which is 1/2 of the dominantperiod. Remember,

that this is a two-way time, therefore we say that the bed thickness itself

has been resolved down to 12.5 msec, or 1/4 period. This theoretical

resolution limit is the sameas that of Widess. Also, the top of the weUge

appears "pulled-up" at the right side of the plot as the inversion has trouble

with the interfering wavelets. A second problem is that, although we know

that there are actually only three distinct velocity units in the section, the

recursire inversion has estimate• at least seven in the vertical =irection.

ß

This result is Uue to the banu-limited nature of the Ricker wavelet. More

Uescriptively, every wiggle on the section has been interpreted as a velocity.

ß

Next, consider a maximum-likelihood inversion of the weOge. The

constraint used was simply a linear ramp. In this case, the shape of the

ß

wedge has been much better defined, due to the broad-band nature of the

inversion. However, notice that the resolution limit has still been observeU.

That is, the maximum-likelihood inversion method also failed to resolve the

bed thickness below 1/4 dominant period. The "pUll-up" observedon the

recursively inverted section is also in evidence here.

In summary,even though sparse-spike methods give an output section that

is visually more appealing than recursively inverted sections, there does not

appear to be a way to break the low resolution limit of 1/4 of the dominant

se smic peri od.


_ i _ i mk

Page 7- 6



Introduction to Seismic Inversi.on Methods Brian Russell

4 8 12 16 20 24 28 32 36 40 44 48

o

300-

400.

Figure 7.5 Recursive inversion of wedgemodelshownn Figure 7.4.

4 8 12 16 20 24 28 32 36 40 44 48

' ' • i ' ' I i

100 .................

300

400

500 ,, .

Figure 7.6 Maximum-likelihoodderived impedance f wedgemoUel

shown i n Figure 7.4.


Page 7- 7



[ntroductJon to Seismic Inversion Methods Br•an Russel•

PART 8 - MODEL-BASED NVERSION

_ - _ - m m L ß .... •

Part 8 - Model-based Inversion

Page 8 -




8.1 Introducti on

In the past sections of the course, we have derived reflectivi-ty

information directly from the seismic section and used recursire inversion to

produce a final velocity versus depth model. We have also seen that these

methods can be severely affected by noise, poor amplitude recovery, and the

band-limited nature of seismic data. That is, any problems in the data itsel f

will be included in the final inversion result.

In this chapter, we shall consider the case of builaing a geologic moUel

first and comparing the model to our seismic data. We shall then use the

results of t•is comparison between real and modeled data to iteratively update

the model in such a way as to better match the seismic data. The basic idea

of this approac• is shown in Figure 8.1. Notice that this method is

intuitively very appealing since it avoids the airect inversion of the seismic

data itself. On the other hand, it may be possible to come up with a model

that matches he data'very well, but is incorrect. (This can be seen easily

by noting that there are infinitely manyvelocity/depth pairs that will result

in the same ime value.) This is referred to as the problem of nonuniqueness.

To implement the approach shown in Figure 8.1, we need to answer two

fundamental questions. First, what is the mathematical relationship between

the model data and the seismic data? Second, how do'we update the' model? We

shall consider two approacheso theseproblems, he generalized inear

inversion (GLI) approach outlined in CooRe and Schneider (1983}, and the

Seismic Lithologic (SLIM) method which was developed in Gelland and Larner

(1983).


Page 8 - 2



Introduction to Seismic Inversion Methods' Brian Russell

CALCULATE

ERROR

UPDATE

IMPEDANCE

ERROR

SMALL

ENOUGH

NO

YES

SOLUTION

= ESTIMATE

Model Based Invemion

Figure 8.1

Flowchart for the model based inversion technique.


Page 8 -



Introduction %o Seismic Inversion Methods Brian Russell

8.2 Generalized Linear Inversion

The generalized linear inversion(GLI) method is a methodw•ich can be.

applied to virtually any set of geophysicalmeasurementso determine the

geological situation whichproduced these results. That is, given a set of

geophysicalobservations, the GLI method ill derive the geological model

which best fits t•ese observations in a least squares sense. Mathematically,

if we express the model and observations as vectors

M: (m,m, ..... , mk)=vectorfkmodelarameters,nd

T: (t1, t2, ..... , tnT

vector of n observations.

Then the relationship between the model

in the functional form

and observations can be expressed

t i = F(ml,m , ...... , m )

ß i : 1, ... , n.

functional relationship has been derived between the

nce the

observations and the model, any set of model parameters will produce an

ß

output. But what model?GLI eliminates he need or trial and error by

analyzing the error between he model output and the observations, and then

in such a way as to produce an output which

way, we may iterate towards a solution.

perturbing the model parameters

will produce ess error. In this

Mathematically'

)F(MO)

= F(Mo) aT •M,

MO--nitial odel,

M: true earth model,

AM: change n model parameters,

F(M) : observations,

F(Mo): alculatedaluesrom nitial

•)F(M )

.2 • = changen calculatedalues.

model, and

F(M)

where


Page 8 - 4



Introduction to Seismic Inversion Flethods Brian Russell

IMPEDANCE

4.6 41.5AMPLITUDE

ml

I

ß

,

- ii

,i i,

i

i

ii

,

ß

ß

,

, i

:.

__

IMPEDANCE

(GM/CM3) FT/SEC) 1000

41.5 4.6 41.5 4.6

i i

41.5

b c d e

Figure 8.2

A synthetic test of the GLI approach to model based

inversion.

(a) Input impedance. (b) Reflectivity derived from (a)

with added multiples. (c) Recurslye inversion of (b).

(d) Recurslye inversion of (b)convolved with wavelet.

(e) GLI inversion of (b). (Cooke and Schneider, 1983)

Part 8 - Model-based I nversi on

Page 8 -




But note that the error between the observations and the computed values

i s simply

•F = F(M) F(MO).

Therefore, the above equation can be re expressed as a matrix equation

•F = A AM,

where A: matrix of deri vatives

with n rows anU k columns.

The soluti on to the above equation would appear to be

-1

•M = A •F,

where A l: matrix nverse f A.

However, since there are usually more observations than parameters (that

is, n is usually greater than k) the matrix A is usually not square and

therefore does not have a true inverse. This is referred to as an

overdeterminedcase. To solve the equation in that case, we use a least

squares solution often referred to as the Marquart-Levenburgmethod see Lines

and Treitel (1984)). The solution is given by

•M: (AT'A)-IAZ•F.

Figure 8.1 can be thought of as a flowchart of the GLI method f we make

the impedanceupdate using the method ust described. However, we still must

derive the functional relationship necessary to relate the model to the

observations. The simplest solution which presents itself is the standarO

convol utional model

s(t) = w(t) * r(t), where r(t) = primaries only.

Cooke and Schneider (1983) use a modi ied version of the previous formula

in which multiples and transmission losses are modelled. Figure 8.2 is a

composite from their paper showing he results of an inversion applied to a

single synthetic impedance trace.


Page 8 - 6




• ' • IMP.EDANCE1OOO

(GM/CM3)(FT/$EC)

ß ._ . .:. . . .........•::...., .. ... .. :... O

.o . ß ß ,, ,, ? "e'. ,,

. .:-: . .• ..... : :........:..:.-.-_- ........ , ß ....-. -.

4': ::.•/-.:. i i..::..':...:......:.':i•i.'-'-:....'...'......-...•.•.::

..'." .

• ' 300M$

.

,

Figure 8.3

2-D model to test GLI algorithm. The well on the right

encounters gas sand while the well on the left does not.

(Cooke and Schneider, 1983)

Figure 8.4

AMPLITUDE

Model traces derived from

m)del in Figure 8.3.

{Cooke and Sc)•neider, 1983)

Part 8 - Model-based •nversion

Figure 8.5

IMPEDANCE

(GM/CM3 (FT/SEC)X1000

10 38 10 38

,,,.l A B

GLI inversion of model traces. Compa

with sonic log on right side of Fi•iure

(Cooke and Schneider, 1983

Page 8 - 7




I

I

YES ___•__•J

' ' FINALMObE

- _ ._ x•, • .... r -• •;•,• -.-'%•..

-cx-r. . . . .-. .,'•_;'•.:.

,• . . t .•..

Figure 8.6

I11 ustrated flow chart for the SLIM method.

(Western GeophysicalBrochure)


Page 8 - 9



ntroducti on to Sei stoic nver si on Methods Brian Russel 1

8.3 Sei_smic_ithologicModelling ,SLIM)

Although the n•thod outlined in Cookeand Schneider (1983) showed much

promise, it has not, as far as this author is aware, been implemented

commercially. However, one method that appears very similar and is

commercially available is the Seismic Lithologic Modeling (SLIM) method of

Western Geophysical. Although the details of the algorithm have not been

fully released, the method does involve the perturbation of a model rather

than the direct inversion of a seismic section.

Figure 8.6 shows a flowchart of the SLIM method taken from a Western

brochure. Notice that, as in the GLI method, an initial geological model is

created and comparedwith a seismic section. The model is defined as a series

of layers of variable velocity, density, and thickness at various control

points along the line. Also, the seismic wavelet is either supplied (from a

previous wavelet extraction procedure) or is estimated from the data. The

synthetic model is then comparedwith the seismic data and the least-squared

error sum is computed. The model is perturbed in such a way as to reduce the

error, and the process is repeated until convergence.

The user has total control over the constraints and may incorporate

geological information from any source. The major advantage of this method

over classical recurslye methods is that noise in the seismic section is not

incorporated. However, s in the GLI method, •hesolution is nonunique.

The best examples of applying this method to real data are given in

Gelland and Larner (1983). Figure 8.7 is taken from their paper and shows an

initial Denver basin model which has 73 flat layers derived from the major

boundaries of a sonic log. Beside this is the actual stacked data to be

inverted.

Part 8 - Model-based I nversi on

Page 8 - 10




1.4

1.6

1kit

,1.4

2.0

Initial

Figure 8.7

lkft

Stack

Left' Init)al Denver asinmodel eismic.

Right: Stacked section from DenverBasin.

(Gel and and Larner, 1983).

2.0

.4

1.6

1.8

1.8

2.0 •

Fieldata Synthetic Reflectivity 2.0

Figure 8.8 Left: F•na• SLIM JnversJon of data shown 1n

Figure 8.7 spl iceU into field data.

Right- Final reflectivity from inversion.

' -- _• -- __-__i m - ' -' (Gelfand and Larner, 1983). • .......... .m:

Part 8 - Model based Inversion Page 8 - 11




In Figure 8.8 the stack is again shown in its most complexregion, with

the final synthetic data is shown fter 7 iterations through the program.

Notice the excellent agreement. On the right hand si•e of Figure 8-.9 is the

final reflectivity section from which the pseudo mpedance s derived. Since

this reflectivity is "spi•y", or broad band, it already contains the low

frequencycomponentecessaryor full inversion. Finally, Figure 8.10 shows

the final inversion compared ith a traditional recursire inversion. Note the

'blocky' nature of the parameter ased nversion when comparedwith the

recurs i ve i nvers i on.

I n summary, parameter

which can be thought of

reflectivity is extracted.

propagated hrough the final

based inversion i s an iterative model1 ng scheme

as a geology-baseddeconvolution since the full

I• has the advantage that errors are not

result as in recursire inversion.

Part 8 - Model-based

Inversion

Page 8 - 12




w

1

500-ft 114mile 114mile E

lkft •

-

.5

l m

ß

.7

1.9

Figure 8.9 Impedance section derived from SLIM inversion of

Denver Basln 1 ine shown n Figure 8.7.

{GelfanU and Larner, 1983)

W

1.7

50011 114mileS 114mile

lkft ß ß .• E

19

F gu e 8.10

Traditional recursire inversion of Denver Basin line

from F gur. 8.7.

(Gelfana anU Larner, 1983)


Page 8 - 13




Appendix- Mat_r_ixappljc.•at.ons_inGe•ophy.s.ics

Matrix theory showsup in every aspect of geophysicalproocessing.Before

looking at generalized matrix theory, let us consider he application of

matrices to the solution of a linear equation, probably the most important

application. For example, let

3x1+2x : 1, and

x1- x2 = 2.

By inspection, we see that the solution is

However,we Could .haveexpressed the equations in the matrix form

or

A X = y,

3 2 x1

1 -1 x2 ß

The sol ution is, therefore

or

-1

x = A y,

x1 1 . -2 1

-1/5

1 3 1

x2


Page 8 - 14



Introduction to Seismic Inversion Me.thods Brian Russell

is of little

overde termi neU

problems:

In the above equations we had the same numberof equations as unknowns

and the problem t•erefore had a unique answer. In matrix terms, this means

that the problem can be set up as a square matrix of dimension N x N times a

vector of dimension N. However, in geophysical problemswe are Uealing with

the real earth anU the equations are never as nice. Generally, we either have

fewer equations than unknowns (in which case the situation is called

underdetermined) or more equations than unknownsin which case the situation

is calleU overdetermined). In geophysicalproblems, he underUeterminedase

interest to us since there is no unique solution. The

case is of much nterest since it occurs in the following

( ) Surface consi stent resi dual

(2) Lithological modelling,and

(3) Refracti on model ng.

statics,

The overdetermined system of equations • can

categories- consistent an• inconsistent. These

extending our earlier example.

be split into two separate

are best described by

(a) Con•s.i••t Overd..eterminedn.earEqua.t.on.s

In this case we

equations are simply

reUunUant equations may

square matrix case.

earl ier example,

have more equations than unknowns, but the extra

scaled versions of t•e others. In this case, the

simply be eliminated, reducing the prø•lem o the

For examp.le, consider adding a third equation to our

so that

anU

3x1+ x : 1,

x1- x2 : 2,

5x - 5x : 10.


P age 8 -

15




This may be written in matrix form as

2 x1

x

o

But notice that the third equation is simply five times the second, and

therefore conveys no new information. We may thus reduce the system of

equations back to the original form.

(b) Inco,s,s, en•tO•verd.•ermine.L.i.near qua•i.on?

In this case the extra equations are not scalea versions of other

equations-in the set, but conveyconflicting information. In this case, there

is no solution to the problem which will solve all the equations. This is

usually the case in our seismic wor• and indicates the presence of measurement

noise and errors. As an example, consider a modification to the preceding

equations, so that

3x1+2x -- 1,

x1- x2 -- Z,

ana 5x - $x = 8.

This may be written in matrix form as

3 2 x 1

I 2

- x2

-5 8


Page 8 - 16



Introduction to Seismic Inversion He.thods Brian Russell

'Now the third equation s not reducible to either of the other two, ana

an alternate solution must be found. The most popular aproach is the method

of least squares, which minimizes the sumof the squared error between the

solution and the observed results. That is, if we set the error to

e=Ax-y,

then we si reply mini mize

eTe--e , ez .......

n

, en = e ß

2

Le.

Re expressing the 'preceding equation in terms of the values x, y, and A,

we have

ß E = eTe (y - Ax)T(y Ax)

= yTy xTATyyTAx xTATAx.

We then solve the equation

bE_

bx

The final solution to the least-squares problem is given by the normal

equa i OhS

AT x = A y

or x = (ATA)-lATy


Page 8 - 17




Sei smi c Travel time

Inversion

9,1 Introduction

_ _ • L _ , _ . _

In this section we will look at a type of inversion that goes under

several names, incluUing traveltime inversion, raypath inversion, ana seismic

tomography. The last term tenUs to be overuseU at the moment, so it is

important to use the term correctly. In section 9.3 we shall showan example

whic• may be considerea as seismic tomography. As all of t•e other names

suggest, however, seismic traveltime inversion uses a set of traveltime

measurements to infer the structure of the earth. The parameters which are

extracteU are velocities and depths, aria [herefore a gross model of earth

structure can be derived. Initial)y, this was considered the ultimate goal,

but Jr'has becomeobvious that this accurate set of velocity versus depth

measurements can be used effectively to constrain other types of inversion.

For example, the'velocities could be used as the low frequency component n

recursire inversion, or as the velocity control for a depth migration.

The way in which traveltime inversion is carried out is to first pick a

set of times from a dataset. These picks m•y come from any of three basic

types of seismic datasets-

Surface seismic measurements

- shots and geophones on the surface,

VSP measurements

- shots. on surface, geop•ones in well,

Cross-hol e measurements

- s•ots anU geophonesboth in well.

and

Once the times have been picked, they must be made to fit a model of the

subsurface. In the next section, we will look at some straighforwara examples

of using traveltime picks in order to resolve the earth's velocity and depth

structure.

Part 9 - Travel time I nversi on

Page 9 - 2



Introductfono Sef mJ Inversf n Methods Brjan Russell

9.• Numerical_Exa•mplesf .Travelti In•v.rsion

Considerhe simplest ossiblecase, a constantelocityearth. Figure

9.1 showshe travel paths that would esult from the three geometry

configurationsgiven a square area of dimension L by L. Note that the

traveltimes in Figure 9.1 would simply be:

(1) Surface sei smic'

(z) vsP-

(3) Cross-hol e'

t--Z L p or p-- t/Z L,

t --•L p or

p -- t /i•L, ana

t=Lp

or p=t/L,

where p -- / V.

Obviously, all three sets of measurements ontain the same nformation.

However,f the velocity or slowness) and he depth are both unknown,

neitheronecanbe determinedroma single imemeasurement.nevengreater

ambiguity comes into play if we have a single measurementut more than one

box. In Figureg.g this situation is shown. Notice that the equations ow

would involve three unknowns nd only one measurement.

A moregeneralmodel s proposedn Bishop t al (lg85) an• Bor•inget al

(1986). The earth is represented s a number f boxesof constantsize and

velocity. lthoughhevelocityf each oxs a constant,hevelocity ay

vary from box to box. This is shownn Figure 9.3. The objective is thus to

computehe seismic travel path through each box using the traveltime

measurements.key problem ere s howo allowthe rays to travel through

the boxes. The first order approximationwould be straight rays with no

bending. However, f Snell's law is use4, the problembecomes oredifficult

to sol ve.

Part-g Travel i'i me'-'in'ver's on .....

Pag• g' - 4'




Source Receiver

Figure 9.3

Separation of the earth into small

for sei stoic travel time inversion.

constant vel oci ty blocks

(Bording et al, 1986)

Page 9 -

art g - Traveltime Inversion 5




Let us apply the straight ray approximation n the simplecase of having

simply two blocks of different velocity. In this case, we have coupled

together both surface and VSP measurements. Two possible recorUing

arrangementsare shown n Figures 9.4 and 9.5. The situations illustrated are

obviously oversimplified since we have assumed straight ray approximation in

both boxes. That is, there is no refraction at the velocity discontinuity,

and the reflection point is directly at the center of the two boxes. However,

if we assumehat the velocities vary only slightly, this approximation is

reasonable.

Let us start with the situation illustrated by Figure 9.4. In this case,

t•ere is a single shot with geophonesboth on the surface and in a borehole at

the base of the layer. If we assume that the sides of the boxes are unity in

length (1 cm or m or km. , the travel time equations are

(1) For the. raypath from S to R

where Pl: 1/velocity n box1

P2: 1/velocity n box2

(Z) For he raypathrom to R2:

t2=•Pl P2.

2

Thus, the total problem can be expressed in matrix form as:

• • Pl tl

•r• •]• : or Ap: t .

• 2 P2 t2

The solution to the previous equation is then

p = A-lt.

Unfortunately, a quick try at solving the above equation will show that

the Ueterminant of A is O, which means that the inverse is nonrealizable.

Physically, this is telling us that the two travel paths spene equal

proportions of their paths in eac• box.

Part 9 - Travel time Inversion

Page 9 - 6




P,, S

x P•"% P,'v,

Figure 9.4

Surface

and VSP raypaths

for a single shot.

R

$• St

Figure 9.5

Surface and VSP raypaths

for two separate shots.

Page 9 -

art 9 - Travel time Inversion 7




A simple way to remedy this situation is to move the shot for the second.

raypath. This is shown n Figure 9.5. In this situation, we have moveU he

shot one-half a box length to t•e left for the recorUing in t•e hole. In this

case, the traveltime equations are

(1) For the raypathromS1 to R :

tl: 1•Pl+ P2

(2) Forthe raypathromS to R -

In this case notice from the diagram hat

tan 0 : 1/1 $ : 2/3 = 0 6667, or B : 33 69o

Thus cos 0 = 0.8320

and (see figure) x = 1/(2 x 0.832) = 0.6

y = 3/(• x 0.832) = 1.8

y-x=l.2

Therefore

t2:1.2 Pl + 0.6 P2

Thus, the total problem can be expressed in matrix form as'

1.2

•[• Pl tl

0.6 P2 t2

with sol ution

Pl

P2

1

o.85

0.6 - 2 t 1

-1.2 2 t 2

Problem' Try to solve the above equation when the two velocities are 1.0

and 1.1 kin/sec. T•at is, work out the traveltimes and plug them into the last

matrix equati on.


Page 9 - 8




Initial

Model

Layer

Stripping

.Inversion

Estimate elocity t well

using onicogandVSP

Pick seismic

reflection imes,

Estimate(x,z)byusing (xo,z),

the reflection traveltimes and the

theassumptionf verticalays

Start with

top ayer

Computerorward

model raveltimes,,

by normal aytracing

Perturb (x,z)

by east quares

or manually

It- fll'

Add another

layer

Final

Seismic Model

layers een

ii

Models omplete,,

Figure 9.6

A possible lowchart for seismic raveltim inversion.

(Lines et al, 1988)


Page 9 -




9.3 Seism•ic T..omo.raphy

The term tomographyas first used in the medical ield for the imaging

of human issue using Nuclear MagneticResonanceNMR) and other physical

measurements. In the seismic field it has come to mean the reconstruction of

the velocity field of the earth by the analysis of traveltime measurements.

Excellent overviews of tomographyare g.iven in Bording et al (1986), and Lines

et al (1988}. You will find t•at the latter paper introduces the term

"cooperative inversion" since both seismic and gravity measurements re used

in the inversion, but that much of the technique used by the authors can be

cons alered sei smic tomography.

Figure 9.6, taken from the paper'by Lines et al (1988), shows the flow

chart that they propose for performing traveltime inversion. This method can

be considered quite general, even though many variations of it are used in the

industry. Basically, the process starts with an estimate of the model which,

in the flowchart shown n Figure 9.6, is deriveU from the sonic log and VSP

measurements. Next, traveltime picks are made from the seismic data. In this

case, stacked CDPdata is usecl, but the shot profiles (or CDPprofiles) could

,

also be used. As well, travel time picks can be made from VSP data and

refraction arrivals. In the next stage of the process, the model is

raytraced, and an error is computed between the computed and observea

traveltimes. Based on the error computed, a new model is computeU. This is

done using the GLI technique described in Chapter 7 of these notes. In the

procedure shown n Figure 9.6, the inversion is done layer by layer until the

model is complete.

Although any traveltime inversion can be considered tomography, Dr. Rob

Stewart (personal communication) points out that to be analagous with the

medical field, where physical measurements are taken completely around t'he

imaged object, a true seismic tomography experiment would involve aata on more

than one side of the portion of the earth to be imaged, such as surface

seismic and VSP.


Page 9 - 10




•0

WlC

_ m m i roll i

-1Z 0

x

I• vsP SOURCE

..2

• 3-D SOURCE • GEOPHONE

Figure 9.7

Surface geometry for tomographic maging example.

(Chiu and Stewart, 1987)

Une89 89D•B 89 89UneDP 8901 8921 8960 8980 CDP

0.0 ..........

•::•=•'•: •.::"--'::':.-:'::.i•r.:iE)•".Z• ;.".•h..

0.1 ---': ..... -" '•'•":'":

Well C VSP

Depth (m)

185 9O7 205 460 730 895

0

fi'•L .o.• ß • mo•w,• .'•.' •

:•(;:• • ....... • .• --'-..

oJ

0.4

o.5

ß

. ..

(b)

•1o ?6o 895

(a)

Fi gue 9.8

(a) Picked events on 3-D seismic..

(b) Picked events on VSP.

Part 9 - Traveltime Inversion

(Stewart and Chui,

.....

Page 9 - 11

19




An example of using multiple datasets for seismic tomography is found in

Stewart and Chiu (1986), and Chiu and Stewart (1987). The objective was a

Glauconitic channel sand which containeU heavy oil. Since this was a

development survey, a lot of measurementswere available to image the

subsurface, including well log data, VSP, and 3-D seismic. Figure 9.7 shows

the •ensity of information along a portion of one seismic line. Figure 9.8

shows the various datasets used in the tomographic maging. Figure 9.8(a)

shows he stacked seismic data with the key events indicated and Figure 9.8{b)

shows he picked VSP from well C. Finally, Figure 9.9 shows he well l'ogs and

synthetic from a different well, clearly indicating the Glauconitic channel.

The tomographic technique involved picking events from both the VSP first

arrivals and the prestack 3-D seismic data. Traveltime inversion was done by

the technique described in Chiu and Stewart (1987). The method involves

starting with a simple modelof the subsurfaceand perturbing this model using

the errors between the picked traveltimes and the raypath times through the

model. This method differs from the method shown in Figure 9.6 since

raytracing is done a nonzero source to receiver offset, and also the VSP data.

To test the method, Chiu and Stewart created a synthetic model. Figures

g.10(a) and (b) show raytrace plots for the VSP and surface Uata,

respectively, through this model.

ZERO PHASE

BANDPASS

10/15 - 80/110 Hz

NORMAL

Figure g. g Wel1

RFC

DENSITY (kg/m 3 )

VE OCITY m/sec)

030O

till ß

SOIl

IO# ß

log curves and synthetic showing Glauconitic channel.

(Stewart and Chiu 1987)

lime

(sec)

Part 9 - Travel time I nversi on

Page 9 - 12




0.0

offset (krn)

1.o 2.0

# $Oul•CE

A •OPHONE'

i

i

i i i i

(a) {•)

Figure 9.10

(a) Surface raypaths through model used to test inversion.

(b) VSP raypaths through model.

(Chiu and Stewart, 1987)

Offset (km)

0.0 1.0 2.0

, ii i 1 - ---

a

Voity

0.0 2.0 4.O

Figure 9.11

Results of tomographic nversion of model data

using VSPand surface data. (Chiu and Stewart,

1987)


Page 9 - 13



Introduction o Seismic nversionMethods Brian Russell

Figure9.11 showshe results of the inversionprocess singboth he VSP

and surface seismic data. To make he test morerealistic, random oise was

addedo travel'timeicks. Notice that the correct esult hasbeen btained

in four iterations.

Let us now return to the case study described initially. The final

velocity/depth model is shownn Figure 9.12. Notice that the velocities fit

quite well with the averaged onic log velocities. This velocity model was

used o produce oth a depthmigrated seismic section, shownn Figure 9.13,

and a full seismic nversionbasedon the maximum-likelihoodechnique. The

final inversion is not showndue to colour reproduction limitations.

As can be seen in Figure 9.13, the Glauconitic channels have been well

delineated. The depth tie is also excellent. The conclusion that the authors

makes that if severaltypesof geophysical easurementsan be intergrated,

the result is an improvedproduct. Each set of data acts as a constraint on

the others.


Page 9 - 14



Introduction to Seismic Inversi on Methods Brian Russell

Offsetkm) Velocity an/s)

-1.0 0.0 LO 0.0 3.0 6.0

TC)iliO6RAPmCC)

INVERSION

- SONCL06 (1:2)

Figure 9.12

Results of tomographic inversion of G1auconitic channel.

(ChiU ndStewart,

1987)

. : •m,,, .......... J•

ß ß . ... l..,.,.,;,,•. ' 't ''•"','

ß -.--:'

._:_.4sl•l_,' i,?a•..:. .,,t.:,

800 ..

:

900 :'": ""' ""' ....... '

Depth

(m) 1000

11oo

1200

1300

1400

F gure g. 13

Depth igrationf seismicatashownn-Figure.8(a).

Tomographicelocities of Figure9.12 havebeenused.

(Stewart an• Chiu,


Page 9 - 15

1986)




PART 10 - AMPLITUDE VERSUS OFFSET INVERSION

Part 10 - Amplitude versus Offset Inversion

Page 10 - 1




10.1 AV.Oheo.•y.._

Until now, we have discusseO only the inversion of zero-incidence seismic

traces. That is, we have considered each reflection coefficient to be the

result of a seismic ray striking the interface between two layers at zero

degrees. In this case, the 'reflection coefficient is a simple function of the

P-wave velocity and density in each of the layers. The formula, which we have

seen many times, is simply

i+lvi+ - ivi zi+- zi

ri= Yoi+iVi+l+OV •Zi+l Z

where r: reflection coefficient

yo: density,

V = P-wave vel oci ty,

Z: acoustic impedance,

and Layer i overlies Layer i+1.

When we allow the seismic ray to strike the boundary at nonzero incidence

angles, as in a common hot recording, a much .more complicated situation

results. In this case, there is P- to S-wave conversion and the reflection

coefficient becomes a function of the P-wave velocity, S-wave velocity, and

density of each of the layers. Indeed, there are now four curves that can be

derived: reflected P-wave amplitude, transmitteU P-wave amplitude, reflected

S-waveamplitude,and transmittedS-wave amplitude. The variation of

ß

amplitude with offset also involves another physical parameter called

Poisson's ratio, which is related to P-and S-wave velocity by the formula

(Vp VS•- Z .

•' =-

Poisson's ratio can theoretically vary between 0 and 0.5.


Page 10 - 2




$i

Sr

at, •t

BOUNDARY

(X2' 2

t

$•

Figure 10.1

Reflectedandtransmitted ays created when P-wave

strikes the boundary etweenwo layers.

(Waters, 1981).

•o, +,•sin2•,' 'cos2•,-•x,n-•:/D,/•-cos2+,/

Figure 10.2

Zoeppritzequations hichdescribe he amplitudes

of the rays shownn Figure 10.1.

(Waters, 1981 .

Part 10 - Ampl rude versus Offset Inversion

Page 10- 3




The equations from which the ampl'itude variations can be derived are

callea the Zoeppritz equations. They are derived from the continuity of

displacement and stress in both the normal and tangential directions across an

interface between two layers. Figure 10.1 shows he seismic rays across a

boundary, and Figure 10.2 gives the final form of the equations. They are

taken from • textbook by Waters (however, someof the signs were wrong, and

they are fixed in the diagram). Since we have four equations with four

unknowns, hey can be rearranged in the form of a ½ x 4 matrix equation

Ax--y

with soluti on

x = A-ly

Over the years, several authors have discussed amplitude versus offset

effects. However, these authors concluded that the effect would be negligible

on seismic data. In a landmark paper, Ostrander (1984) showed hat for a

significant change n Poisson's ratio, a major change n the P-waveamplitude

coefficient can be seen as a function of offset. This Poisson's ratio change

is most noticeable in a gas sand, where the ratio can change from 0.4 in the

encasing shales to as low as 0.1 in the gas sand itself. Ostrander showed

that, in such extreme cases, the P-wave reflection coefficient can go from

positive to negative for a decrease in Poisson's ratio coupled with an

increase in P-wave velocity, or from negative to positive for an increase in

Poisson's ratio coupled with a decrease in P-wave velocity.

Figure 10.3(a) shows he gas sand model that Ostrander used and Figure

10.3(b) shows the result of amplitude versus offset modelling of the P-wave

reflection coefficients. Figures 10.5(a) and (b), also taken from Ostrander,

shows that this effect can inUeed be observeU on a common offset stack.

Figure 10.5(a) shows stackeO seismic section witl• three apparent "bright

spot" anomalies. Unfortunately, only wells A anU B were productive. The three

common ffset stacks, shown n Figure 10.5(b), indicate that only locations A

and B actually Uisplay an AVOeffect.


Page 10 - 4




GAS ':*"*•*•t Vl• =8.000

/32 -"2.14

:o.,

;..•.• :., ß

SHALE •---'

$=10.000

/4) =2.40

(•'3 =0.4

Figure 10.3 (a) Synthetic gas sand model.

(Ostrander,

1984)

0.41

0.3

IN SAND

0.2

t.., 0.1

0

0

..,

-0.2

I0 o

ANGLE F NCIDENCE •,-'-e'

20 o 30 ø 40 ø

NO GAS

., ,,,,,.ooo.o.o.... ..ooo.,o.,.,o.,-'.ø.,,,o,,o*o .....

-0.3

-0.,4

Figure 10.3 (b)

for reflections from top and bottom interfaces of model

s•own in Figure 10.3 (a).

• (Ostrander, 1984)

, , , IlL _ -- _, 11 i , i m m im , ß

Part 10 - Amplitudeversus Offset Inversion Page 10 -

Computedeflection coefficients as a function of offset




The method useU to identify this effect is only partly qualitative, and

can be diagrammedas shownbelow-

INPUT SEISMIC

SHOT PROFILES

COFFSTACK

BUI L D MODEL

...

VISUAL

COMPARISON -

MODEL MATCHES

REALITY

. .

COMPUTE

SYNTHETIC

. •

NO

im m

MANUALLY

CHANGE PARAMETERS

,m

Figure 10.4

Flow Chart for Manual AVO Inversion

Obviously, this visual meth'odf comparisoneavesmuch o be Oesired.

We will therefore look at several methods for the qualitative inversion of AVO

data, both of which have been looked at previously in the context of

normal-incidence inversion.

ß


Page 10- 6




1•0 170 160 1•50 140 130 120 110 100 90 •0 70

0.0 ' , .... • , -- • , • • • • ' ' -' • •- ' -• " • •

-• • •. : ' 't .". '.'..' t'-' " : "• :. ' ' : .... I; ' ' ' 0.0

'*O..' m * re- ß ß l, ß ' ß I , I i I, mm I.. i ,.

' i i i. ii I I ß IIII II i ß

..,.,,.,..•..,.,,...,",',,-.....-,,,,'.,,.-,',,,,,....,,.,....,,'........-•~.,......,,.'""' ... .'_'•"' . ... •,,., ................

'i:

,:•:;.-':, • ..=•..•:.'':."•i '.•.•.'-'?'?'?'?'?'•L--•,•.•'•.•.•..,'::... ß .._ .o..•:?;i•. -- ,..•.•... _'_-•..•

' --?.-•" ":'.. : ' • ...... "=-" il •;,.•:.?:•-•=;.•'.'•..='.:•:-1: . . •-..c• ,•.•. .-...,_•....:,•.•.:..•,.•;,..-.

• • .*...... . ß .- -...- - . ; ...... ..• .•:..;,-.;:.•.. _•:_- .•...4..... <?--r..-.. . .:.;. ,""-•.•r..•_-•".::

0.5 •'.l_'.-•. .: : •_•_•..._. .... _ -:.:...:..._...•...... . _:... ;._.•.=0.5

•,'*' ':'.:-r-'_.•.•;...... . .

1.0

2.5

I.,...,:,•....,;.•......... . ...-..

...

....,.., t,,,

"_,,,d,.•,•l.leeile*e,,I,'t:lit•ll""'It•'•l';I;,d

.......... 2.0

. ..

,•e.•. Illl•lll.1111el'•-ß .le

- .. • '."-'•1,.'•1•-.

........... ;.;... :.....i:-;

ß.:--;•=....=:;;:.1•... ...,;•.".....

' ........ "•"":":........................5

,..,- .1,•.----?'"1 ß ß - .... ,.....-.,.-.,.-. -,., .......... ... ........

"' ' "1;;::=... ":".... ""':"'"1'.'-'---' "•':,';;;;":',::..."

ß ........::.;:..

.. .... :.-.'.. . '}i:.;.=i.•;-."':;::.'.:•:.'

ß ß

Figure 10.5(a)

S•acked seismic line showing"brigh• spot" anomalies.

Loca%ions A an• B are known gas.

(Osl:rander, 984)

.... titIll,

,e*'11:,l:, ol,, ....

' ' I• ' ........

ß .

. ,

6952' 1012 ß

SP 80

":l:1111il•

ß .

eellie

;;;;;;;;i;il

,,

..111tl•

6•$•' 101

Fiõure 10.5(D)

Commonffset sl;acksover locations A, B, andC from

stacked section in Figure 10.5/a). Notice the AVO

increase on A and B.

(Os ;ranOer, 1984)

Par[ 10 - A,•pli[ude versus Offset Inversion

Page 10 - 7



Introduction to Sei stoic Inversi on Methods Bri an Russell

10.2 AVO Inversion by GLI

Recall that in the theory of generalized linear inversion (.GLI) there

were three important components' geological model of the earth, a physical

relationship between the earth and a set of geophysical measurements,anU a

set of geophysicalobservations. This methodWasdiscussean both chapters8

anU 9, applied to stacked data inversion an• traveltime inversion,

respectively. Now, let us apply the method to unstacked data. The result

wil 1 be cal led AVO inversi on.

In secti on 10.1, the three components needed to perform GLI inversion on

AVOUata we,re escribed. Ourmodel of the earth is a series of layers with

t•e el astic 'parameters of P-and S-wave vel oci ty, density, and Poi sson'S ratio.

Our physical relationship between this model ana seismic CDP profiles was

derived using the Zoeppritz equations. And, finally, the observations are the

picked amplitudes and times of events on a CDP profile or common ffset stack.

By computingderivatives from the Zoeppritz matrix, it is possible to set up a

GLI solution to t•e AVO problem similar to the solution found for zero-offset

data. This solution is

aF Mo•)

FIM) F(M)+ •)M bM

where

Mo: nitial earthmodel,

M: true earth model,

AM: change in model parameters,

F(M) -- AVOobservations,

F(MO): oeppritz alues rom nitial model, nd

•)F(M )

i)--••: changen calculatedalues.

The implementation is simply a variation of the manualmethod, anO is

sinownon the next page.


Page 10 - 8




INPUT SEISMIC

SHOT PROFILES

COFFSTACK

i

PICK

AMPLITUDES

COMPUTE

ERROR

COMPUTE

SYNTHETIC

STORE COMPUTED

AMPL TUDES

i t •

COMPUTE MODEL

PARAMETER CHANGE

US NG GLI

NO

ERROR

YES

MODEL MATCHES

REAbITY

,

Figure 10.6

AVO inversion by the GLI method


Page 10 -




Wewill now ook at an example of GLI inversion of amplitude versus

offset data. First, consider the integrated well logs shownon the left hand

panel of Figure 10.7. Actually., only the sonic log or P-wave og wasrecorded

in the field. The density log was derived from the sonic using Gardner's

equation, the Potsson atio was ixed at 0.25, and the S-wavewasderived from

the P-wave and Potsson ratio logs. On the logs, three layers have been

blocked at depth and a significant Poisson's ratio changehas been ntroduced

in the middle block. On the right hand side of Figure 10.7, notice that the

amplitudeversusoffset curves have been displayed for the third layer. As

predicted earlier, the P-wave reflection coefficient displays a strong

increase of amplitude with offset.

Figure 10.8 showshe same et of blocked ogs on the left, but showshe

seismic response f the amplitude change on the right. This synthetic was

produced y simply' eplacing the zero-incidence mplitudes ith the amplitudes

derived from the Zoeppritz calculations. The events between 00 and 700 msec

display a pronouncedmplitude changewit h offset.

ZOEPPRII'ZIHI:LE NTERFFJCE

TESTLOESTLOESi'DEE•-S t•;$TPO

• 2• 2,5 4;8

,,,

Eq, , t: 3 Ti,•: $7• Depth: 795

589 1998

Of*f's•:c'.........................

. i Reflected P-Wave

..... Transmitted P-Wave -9.8)

...... Ref'l ected S-Wave

........... Transmitted S-Wave

ß , mm i

m

Figure 10.7 Blockedwell logs on left, with computed oeppritz curves

for layer 3 on right.

_ _

Part 10 -^mplJtude versus Offset Inve•sJon Page 10- 10



Introducti on to Sei smi c Inversi on Methods Brian Russel 1

20EPPRITZ •EFLECTIUITY tlODEL

TESTLOG TESTLOG IEH$ITY1 S-I, FIUE POISSOH1

u•Ym u•/m 9/½½ usam

...... ,L -

ß•..o...-.. .............. , ....,....

•ee-..........................•"........

.....................'.........,.

::...........................•..........

"

268 268 2.5 468 .$

MODEL (meters.)

EU 909 727 545 363 181

.

Figure 10.8 Left-

Right:

A "blowup" f the blocked ogs shownn Figure 10.7.

A synthetic commonoffset stack and the AVO curves

shown on the right of Figure 10.7.


Page 10- 11




However, does the change seen in Figure 10.8 reflect the reality of the

situation? Figure 10.9 showsa set of CDPgathers which correspond to the

model. The gathers are a realistic modelled dataset and were generatedwith

no change n Poisson's ratio. Since the gathers are noisy and contain fewer

traces han he synthetic DP rofile shownn Figure 0.8, theywereusedo

create a common ffset stack. The geometry of this st.ack is described in

Ostrander's aper, and the resulting gathersare often referred o' as

Ostrander gathers. Traces within a CDP/offset window were gathered and

stacked, resulting in increased signal to noise. Figure 10.10 showsa display

of the logs, synthetic model, and commonffset stack. The mismatch in

amplitudes is now obvious.

ß

ß Next, the amplitudes of the event on the contanon offset stack

corresponding o the event displayed in Figure 10.7 were picked. The event

above the anomalous ayer was also picked. The picks were then used along

with the computedamplitude versus offset curve to invert the data by the GLI

method. In the inversion, two parameters were allowed to vary- the Poisson's

ratio in the layer of interest, and a scalar which relates the magnitude of

the seismicicks o themagnitudef theactualamplitudes.

Figure 10.9 CDPgathers from a seismic dataset corresponding to

synthetic shown n Figure 10.8.

Part 10 -Amplituae versus Offset Inversion

Page 10 - 12



I ntroductin o Sei toic nversinMethods B i an Russel

I NVER$]ON FULL HOIIEL

TESTL TESTL )EN$I $-WI:IU OI$$

_ us/m u•/• g/cc us/•

I ß

EU

rIO]]EL (meters)

909 727 545 353 181 0

COFFSTK1 n,elers )

838 6•4 498 :)32 li•E; 0

50

I

2•0 2•0 2.5 4•0 ,5

Figure 10.10 A comparison of the synthetic coneon offset stack from

Figure 10.8 {middle panel) with a con,nonoffset stack

created from the CDPgathers of Figure 10.9 (right panel).

T•e left panel shows he blocked well logs from which

the synthetic was created.


Page l(J- 13




The results of this inversion are shown in Figure 10.11. The figure

showshe changen Poisson'satio before nd fter inversiondashedine

before, solid line = after) on the left hand side. On he upper ight is

shown he match etweenhe observedicks n the upperayer (showns small

squares)nd he final theoretical urve{s•ownsa solid ine). The ower

right showshe same hing for the lower layer.

Finally, Figure10.1Zshows he comparisonetween he coanonffset

stackand he syntheticmodel fter the model asbeen ecomputedith the new

amplitudehangesrom he updatedoisson'satio. Notice he improvementn

the match.

II•'RSIOH SIN•E LI•ER: I101•ELI

70O

6,8

i i i i

Poi•s•s Ratio

. ß

e .e•

0.042

6.666

0.048

6.624

6.606

Ewnt (2) P. ove Laver

. . ..

O•'•'r•

Event (3) Belo4aLaver

O O

0

e.• e S5e

O('•set ( m

Figure 10.11

The esultsof a GLI nversion etweenhe computed,mplitudes

of Figure 0.7and he picked mplitudesrom heconmon

offset stackof Figure10.10. Thedashedine on he plot

on the left is the Poisson'satio before nversion,and he

solid line is after inversion. Theplots on the right show

the new omputedurves ith the picks squares)uperimposed.


Page 10- 14




IHUER$IOH FULL MODœL

MOIlEL2 me•er$) COFFSTKI ( meters )

EU 909 727 545 363 lB1 B 838 664 498 332 166

Figure 10.12

A replot of Figure 10.10, where the synthetic has been

recomputed sing the newPoisson's ratio value.


Page 10 -

15



[nt•oduct• on t• $e• sm•c [nve•sJ on Methods B• an Russe• ]

PART 11 - VEI:OCITY INVERSION

Part 11 - Velocity Inversion

Page 11 -




Part 11 - Veloci..ty .n.v.rsi on

_

11.1 ntroduc ti on

The last

inversion. Alth

acutally fit in

been dtscussing

topic to be discussed n these notes is the topic of velocity

ough this technique is referred to as inversion, it does not

to the narrower category of inversion techniques that we have

in this course. These techniqueshave all involved inputting

a stacked, or unstacked, seismic dataset and inverting to a velocity versus

depth section. The output of the velocity inversion described here is the

seismic section properly positioneU in depth, but still plotted as seismic

amplitudes, and still band-limiteU. As such this technique is closer to that

of depth migration.

In this section, we will look briefly at the theory of velocity

inversion, and then look at a few examples. An excellent review article on

this subject is given in Bleistein and Cohen 1982). In this article, the

theory of the method is reviewed and there is also an extensive literature

summary. Our discussion here will follow that article.

Part 11

- Velocity Inversion

Page 11 -




KII.OFEœT KILOFEET

-2 -1 0 I 2 -2 -I 0 I

,

(a) (b)

2

Figure 11.1

The effect of the velocity inversion methodon synthetic

data. (a) A "buried focus"effect, (b) The output from

the velocity inversion method.

(Bleistein and Cohen

KILOFEET KILOFEET

o 1 -1 o 1

1982 )

m

uJ

LL

o

....

C) ß

ii'1

(a) (b)

Figure 11.2

A second example of the effect of velocity inversion on

synthetic data. (a) Input section with diffraction,

(b) Output from velocity inversion.

(Bleistein and Cohen

......

1982 )

Part 11 - Velocity Inversion

Page 11 -




11.2 Theory and .Examples

The velocity inversion procedure is referred to as an inverse scattering

problem, in which the interior of the earth is mapped by inver. ing the

observations from multiple acoustic sources. (This is a long way of saying

that the seismic section is inverted ) Thus, the starting point for this

method is the acoustic wave equation. The difference between this technique

and classical migration is that perturbation techniques and integral

transforms are used rather than downwardcontinuation of the wave equation.

The initial work in this area was done by NormanBleistein and Jack Cohen

at the University of Denver. In their initial paper, Cohenand Bleistein

(197g), they employed only a perturbation technique in the inversion of

seismic data. In simple terms, this technique involves using a constant

velocity in the wave equation, perturbing this constant velocity by a small

amount, and then, by observing the backscattered wavefield, solving for the

perturbed velocity. This methodsolves for only the reflection strength of

the mapped nterfaces.

In their morerecent paper, Bleistein and Cohen 1982), a more accurate

solution was proposed which al.so solves for transmission losses and

refraction. Clayton and Stolt (1981) have applied a similar method o the

inversion of seismic data. Their method is referred to as the Born-WKBJ

method, and thus this approach to inversion is often cal led Born inversion.

Despite the differences in the mathematicsbetween he velocity inversion

methodsand migration methods, the results look very similar to those of

migration. For example, igure 11.1, fromBleistein andCohen 1982), shows

the input an• inverted result for a g-D buried focus. Note that, as in

migration, the "bow-tie" has been imaged o a synclinal feature.

Part 11 - Velocity Inversion Page 11 - 4



Introduction •o Seismic Inversion Methods Brian Russell

(a)

q ()f . []

ß

ß

ß

I,;,(11). iJ ll;11]ll. () I I 3l) ]. l, I ;i'llroll. IJ. I I,• O0. II

ß , •

I I11.)[11J. fl

½J

qlOO

.-.,,

c)

6500 8900 I 1 300 1 37.r.,P 16' ,' ..[]

- t ,

18b•G

ß

(b)

Figure 11.3 The effect of velocity inversion on real data.

(a) Input section (Marathon Oil), (•) Output section.

(Bleistein and Cohen 1982)

Part 11 - Velocity Inversion Page11 - 5




Figure 11.2, also from Bleistein and Cohen (1982), showsl•e velocity

inversion of a diffraction tail from a geological discontinuity. Notice that

the diffraction tail has been "collapsea", again as in migration.

Finally, Figure 11.3 shows n example of applying the velocity inversion

technique o a real dataset. Again, note the similarity with classical depth

migration. The fact that this section is plotted as wiggle trace only makes

the plot di fficul t to evaluate.

In summary,his technique cannotbe classed with the other methodswhich

have been discussed in this'course due to its similarity with depth migration.

However, research in'this area is continuing at a steady pace, and the

technique promises much for the future.

Part 11 - Velocity

I nversi n Page11 - 6



Introduction to Seismic Inversion Nethods Brian Russell

PART 12 - SUMMARY

Part 12 - Summary

Page 12 - i




lZ.1 Sgmmary

In these notes, we have reviewed the current methods used in the

inversion of seismic data. The basic model used in most of these methods is

the one-dimensionalmodel, which states that the seismic trace is simply the

convolution of a zero phase wavelet with a reflectivity sequence derived from

the earth's acoustic impedanceprofile. Flowcharts for these methoUs are

shown n Figures 12.1, 12.2, and 12.3. Let us initially summarize he

advantages and disadvantages of the three methodsof single trace inversion

which have been discussed:

(1) Recursire Inversion

_ ,• _ - •

Advan tage s:

(i) Utilizes the complete seismic trace in its calculation.

(l i ) A robust procedure when used on clean seismic data.

(iii) Output is in wiggle trace format similar to seismic data.

Di sadvantages:

(i) Errors are propagated through the recurslye solution if there are

phase, amplitude, or noise problems.

(i i) The low frequency componentmust be derived from a separate source.

(2) Spar.e-SP.ig_.Invers.on

Advantages-

(i) The data itself is used in the calculation, as

i nver si on.

(ii) A geological looking inversion is produced.

(iii) The low frequency information is included mathematically

solution.

in recursi ve

in the

Part 12 - Summary

Page 12




BAND-LIMITED

SEISMIC

TRACE

INTRODUCE

LOW

FREQUENCY

COMPONENT

REFL

COEFF.

I INVERT

I TO MPEDANCE

IMPEDANCE

SCALE TO

VELOCITY

AND DEPTH

DISPLAY

Fiõure 12.1

Band-Limited Inversion (Recursive)

Part 12 - Summary

ß ß ,

Page 12- 3




,

Dôsadvantages'

(i)

Statistical nature of the sparse-spike methods used are subject to

probl eros i n noisy Uata.

Final output lacks muchof the fine detail seen on recursively

inverted data. Only the "blocky" components inverted.

(3) Model Base• I nver si on

Advantages'

(i) A complete solution, including low frequency nformation, is possible

to ob rain.

(ii) Errors are distributed through the sol ution.

(iii) Multiple and attenuation effects can be modelled.

Di sadvantages'

(i) A completesolution is arrived at iteratively andmayneverbe

reached ( i.e. the sol ution maynot converge).

(ii)

The

velocity inversion, and amplitude versus offset inversion.

methods, but cannot be compareddirectly with the three

(comparing pples with oranges?).

,

The traveltime inversion method was

accurate velocity versus depth model.

constraint for either one of the

migration.

It is possible that more than one forward modelcorrectly fits the

data (nonuniqueness).

other methods which were considered were traveltime inversion,

All are important

previous methods

an excellent method for finding an

These velocities make an excellent

classical inversion methods or for a depth

Part 12 - Summary

Page 12 - 4



Introduction o SeismicnversionMethods Brian Russell

INTRODUCE

LINEAR

CONSTRAINTS

EXTRACT

SPARSE

REFLECTIVITY

INVERT

TO IMPEDANCE

I vELøcmTY

ANDL_•.••_•.

m i i m i

Fiõure 12.2

Broad-Band nversion (Sparse-Spike)

Part 12 - Sugary

Page 12-




The velocity inversion methodwas showno be very similar to depth

migration. The output from this method could therefore be used as input to

one of the other three classical methods of inversion.

Finally, amplitude versus offset inversion adds an extra dimension to the

inversion problem since it is truly a lithologic inversion rather than a

velocity inversion method. This method is definitely the method of the

future, but still has a number of hurdles to overcome. This author's humble

opinion is that once the interpreter is able to do a complete lithological

inversion on their seismic datasets, the other methodswill be replaced.

The other conclusion from this course is that the more separate datasets

(surface seismic, VSP, well log, gravity, etc..) the interpreter can use in an

inversion, the better the final product will be.

Part 12 - Sumnary

Page 12- 6




ß

MODEL IMPEDANCE

TRACE ESTIMATE

CALCULATE

ERROR

UPDATE

IMPEDANCE

ERROR

SMALL

ENOUGH

NO

YES

ON

=ESTIMATE

Fiõure 12.3

Mode 1-Based Inversion

Part 1'•- •'" ....

ummary

Page 12-



Introduction to Sei stoic Inversion Herhods Brian Russell

REFERENCES

Angeleri, G.P., and Carpi, R., 1982, Porosity prediction from

seismic data' Geophys.Prosp., v.30, p.$80-607.

Berteussen, K.A., and Ursin, B., 1983, Approximate computation of

the acoustic impedance rom seismic data- Geophysics, v. 48,

p. 1351-1358.

Bishop, T.N., Bube, K.P., Cutler, R.T., Langan, RT., Love, P.L.,

Resnick, J.R., Shuey, R.T., SpinUler, D.A., and Wyld, H.W., 1985,

Tomographic determination of velocity and depth in laterally

varying media- Geophysics, v. 50, p. 903- 923.

Bleistein, N., and Cohen, J.K., 198•, The velocity inversion problem-

Present status, new directions: Geophysics, v.47. p.1497-1511.

Bording, R.P., Lines, L.R., Scales, J.A., ana Treitel, S., 1986,

Principles of travel time tomography' SEGContinuing EUucation notes,

Geophysical inversion and applications.

Chi, C., Mendel, J.M., and Hampson,D., 1984, A computationally fast

approach to maximum-likelihood aleconvolution: Geophysics, v. 49,

p. 550-565.

Chiu, S.K., and Stewart, R.R., 1987, Tomographic determination of three-

dimensional seismic velocity structure using well logs, vertical

seismic profiles, and surface seismic data: Geophysics, v.52,

p. 1085-1098.

Claerbout, J.F., and Muir, F., 1973,

Geophysics, v. 38, p. 8Z6-844.

Robust Modeling with erratic

data:

Part 12 - Summary

Page 12 -




Clayton, R.W., and Stolt, R.H., 1981, A born WKBJ

acoustic reflection data: Geophysics, v. 46,

inversion method for

1559-1568.

Cohen, J.K., and Bleistein, N, 1979, Velocity inversion procedure for

acoustic waves: Geophysics, v. 44, p. 1077-1087.

Cooke, D.A., and Schneider, W.A., 1983, Generalized linear inversion

of reflection seismic data: Geophysics, v. 48, p. 665-676.

Galbraith, J.M., and Millington, G.F., 1979, Low frequency recovery in

the inversion of seismograms: Journal of the CSEG, V. 15, p. 30-39.

Gelland, V., and Larner, K., 1983, Seismic litholic modeling:

presented at the 1983 convention of the CSEG,Las Vegas.

Graul, M., Deconvolution and wavelet processing:

notes.

Unpubished SEG course

Hardage, R., 1986, Seismic Stratigraphy:

London - Amsterdam.

Geophysical Press,

.Hampson, ., and Galbraith, M., 1981, Wavelet extraction by sonic-log

correltation: Journal of the CSEG, v. 17, p. 24- 42.

Hampson, D., 1986, Inverse velocity stacking for multiple elimination:

Journal of the CSEG, V. 22, p. 44-55.

Hampson, D., and Russell, B., 1985, Maximum-Likelihood seismic

inversion (abstract no. SP-16)- National CanauianCSEGmeeting,

Ca. gary, Alberta.

Part 12 - Summary

Page 12 - 9



Introducti on to Sei stoic Inversi on Methods Bri an Russell

.

Herman, A.J., Anania, R.M., Chun, J.H., Jacewitz, C.A., and

Pepper, R.E.F., 1982, A fast three-dimensionalmodeling technique

and fundamentals of three-dimensional frequency-Uomainmigration:

Geophysics, v. 47, p. 1627-1644.

Jones, I.F., and Levy, S., 1987, Signal=to-noise ratio enhancement n

multi channel seismic data via the Karhunen-Loeve transform,

Geophysical Propecting, v. 35, p. 12-32.

Kormyl o, J., anu Mendel., J.M.,

deconvolution- IEEE Trans.

v. IT - 28, p. 482 - 488.

1983, Maximum-likelihood seismic

on Geoscience and Remote Sensing,

Lines, L.R., Schultz, A.K., and Treitel, S., 1988, Cooperative inversion

of geophysical data: Geophysics, v. 53, p. 8- 20.

Lines, L.R., and Tritel, S., 1984, A review of least-squares

anU its application to geophysical problems' Geophysical

Prospecting, v. 32, p. 159-186.

inversion

Lindseth, R.O., 1979, Synthetic sonic logs - a process for stratigraphic

interpretation: Geophysics, v. 44, p. 3- 26.

Oldenburg, D.W., 1985, Inverse theory with applica.tion to aleconvolution

and seismogram inversion. Unpublished course notes.

Oldenburg, D.W., Scheuer, T., and Levy, S., 1983, Recovery of the acoustic

impedance rom reflection seismograms:Geophysics, v. 48, p. 1318-1337.

Ostrander, W.J., 1984, Plane wave reflection coefficients

at non-normal angles of incidence: Geophysics, v. 49,

for gas sands

p. 1637-1648.

Part 12 - Summary

Page 12 - 10



Introduction %oSeismic Inversion Methods Brian Russell

Russell, B.H., and Lindseth, R.O., 1982, The information content of

synthetic sonic logs - A frequencydomain pproach- presentedat

the 1982 convention of the EAEG, Cannes, France.

Shuey,R.T., 1985, A simplification of the Zoeppritz equations:

Geophysics, . 50, p. 609-614.

Stewart, R.R., and Chiu, S.K.L., 1986, Tomography-basedmagingof a

heavyoil reservoir usingwell-logs, VSPand3-D Seismicdata-

Journal of the CSEG, v. 22, p. 73-86.

Taner, M.T., an• Koehler, F., 1981,

Geophysics,v. 46, p. 17-22.

Surface consistent corrections-

Taylor, H.L., Banks,S.C., and McCoy, .F., 1979, Deconvolutionwith

the L1 norm: Geophysics,v. 44, p. 39-52.

Trei tel, S., and Robinson,E.A., 1966, The design of hi gh-resolution

digital filters ß IEEETransactions n Geo•cience lectronics,

v. GE-4, No. 1, p. 25-38.

Walker, C., and Ulrych, T.J., 1983, Autoregressive recovery of the

acoustic impedance- eophysics, . 48, p. 1338- 1350.

Waters, K.H.,

exploration

1981, Reflecti on seismology, a tool

(second edition)- Wiley, NewYork.

for energy resource

Western Geophysical Co.,

Brochure.

1983, Sei smic L thol ogi c MoUel ng:

Technical

Widess, M.B., 1973,

p. 1176- 1180.

How thin is a thin bed?-

Geophysics, v. 38,

Part 12 - Summary

Page 12 - 11

introduction to seismic inversion methods

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