russell - introduction to seismic inversion methods

Introduction to

Seismic Inversion Methods

Brian H. Russell Hampson-Russell Software Services, Ltd. Calgary, Alberta

Course Notes Series, No. 2 S. N. Domenico, Series Editor

Society of Exploration Geophysicists

These course notes are published without the normal SEG peer reviews. They have not been examined for accuracy and clarity. Questions or comments by the reader should be referred directly to the author.

ISBN 978-0-931830-48-8 (Series) ISBN 978-0-931830-65-5 (Volume)

Library of Congress Catalog Card Number 88-62743

Society of Exploration Geophysicists P.O. Box 702740

Tulsa, Oklahoma 74170-2740

¸ 1988 by the Society of Exploration Geophysicists All rights reserved. This book or portions hereof may not be reproduced in any form without permission in writing from the publisher.

Reprinted 1990, 1992, 1999, 2000, 2004, 2006, 2008, 2009 Printed in the United States of America

]: nl;roduc t1 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 Consi derati ons

4. ! I ntroduc ti on

4.2 Ampl i rude recovery 4.3 Improvement of 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 I norm method 6.4 Reef Problem

I nversi on appl ied to Thi n-beds

7.1 Thin bed analysis 7.Z Inversion compari son of thin beds

Model-based Inversion

B. 1 I ntroducti on . 8.2 Generalized linear inversion 8.3 Seismic 1 ithologic roodell ing (SLIM) Appendix 8-1 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 examples of traveltime inversion 9.3 Seismic Tomography

Part 10 Amplitude versus offset (AVO) Inversion

10.1 AVO theory 10.2 AVO inversion 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 NTRODUCT ION TO SE I SMI C 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 seem appropriate 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 measurements made on the surface of the earth.

The above definition is so broad that it encompasses virtually 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 in Figure 1.1.

To understand seismic inversion, we must first understand the 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 frequency domains, considering the thre e components of this model: reflectivity, seismic wavelet, and noise.

Part I - Introduction

_ m i --.

Page 1 - 2

Introduction to Seismic InverSion Methods Brian Russell

FORWARD MODELL I NG i m ß

INVERSE MODELLING (INVERSION) _

, ß ß _

Input'

Process:

Output'

EARTH MODEL

,

MODELLING

ALGORITHM

SEISMIC RESPONSE i m mlm ii

INVERSION

ALGORITHM

EARTH MODEL i ii

Figure 1.1 Fo.•ard ' andsInverse Model,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 methods which are currently ß

used to invert seismic data. These methods are summarized in 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 contanon methods 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" aleconvolution methods, 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 method which 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 discussed along with several illustrative examples.

After the discussion on poststack inversion, we shall move into 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, looking at examples of each.

Part 1 - Introduction Page i -

Introduction to Selsmic Inversion Methods Brian Russell

SE I SMI C I NV ERSI ON

.MET•OS ,,,

POSTSTACK

INVERSION

PRESTACK

INVERSION

MODEL-BASED I RECURSIVE INVERSION • ,INVE SION

- "NARROW BAND

TRAVELTIME

INVERSION

!TOMOGRAPHY)

SPARSE- SPIKE

WAV EF I EL D NVERSIOU i

LINEAR

METHODS ,,

i i --

I METHODS ]

Figure 1.2 A summary of current inversion techniques.

Part 1 - Introuuction Page 1 -

Introduction to Seismic Inversion Methods Brtan Russell

PART 2 - THE CONVOLUTIONAL MODEL

Part 2 - The Convolutional Model Page 2 -


Part 2 - The Convolutional Mooel

2.1 Th'e Sei smi c Model

The most basic and commonly used one-Oimensional moael for the seismic trace is referreU to as the convolutional moOel, which states that the seismic

trace is simply the convolution of 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 assumption is to consiUer the noise component to 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 processing we deal exclusively with digital data, that is,

data sampled at a constant time interval. If we consiUer the 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, convolution can be thought of as "replacing" each reflection. coefficient with a scaled version of the wavelet and summing the result. The result of this process is illustrated in Figures 2.1 and 2.Z for both a "sparse" and a "dense" set 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 spaced reflectors.

Part 2 - The Convolutional Model Page

Introduction to Seismic Inversion Nethods Brian Russell

WAVELET:

(a) ' * • • : -' ':'

REFLECTIVITY

Figure 2.1

TRACE:

Convolution of a wavelet with a (a) •avelet. (b) Reflectivit.y.

sparse" reflectivity. (c) Resu 1 ting Sei smic Trace.

(a)

(b')

!

.

i

: !

! : : i i , ß

: 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 transform of 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 becomes multiplication 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) = e w

where

(f) + er(f),

I •ndicates amplitude spectrum, and 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 Oomain problem of loss of

resolution becomes one 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.

Part 2 - The Convolutional Model Page 2 -


2.g The Reflection Coefficient Series l_ _ ,m i _ _ , _ _ m_ _,• , _ _ ß _ el

of as the res

within the ear

compres si onal

i ropedance to re

impedances by coefficient at

fo11 aws:

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

described in the previous section is one of the fundamental physical concepts in the seismic method. Basically, each reflection coefficient may be 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 • i+1

where

and

r = reflection coefficient, /o__ density, V -- compressional velocity,

Z -- acoustic impeUance,

Layer i overlies Layer i+1.

We must also convert from depth to time by integrating the sonic log transit times. Figure •.4 shows a schematic sonic log, density log, anU

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

the result of converting to the reflection coefficient series and integrating 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 P age 2 - 6


STRATIGRAPHIC SONIC LOG SECTION •T (•usec./mette)

4OO

SHALE ..... DEPTH

ß ß ß ß ß ß SANOSTONE . . - .. ,

' I ! !_1 ! ! !

UMESTONE I I I ! I ! I 1

LIMESTONE 2000111

30O 200

I 3600 m/s

_

v-- I V--3600 J

V= 6QO0

I

loo 2.0 3.0 ,

OENSITY LOG.

ß •

Fig. 2.4. Borehole Log Measurements.

mm mm rome m .am

,mm mm m ----- mm

SHALE ..... OEPTH

•--------'- [ SANDSTONE . . ... ,

! I !11 I1 UMESTONE I 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 Convol utional 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 may create an impedance curve by

multiplying together •he sonic and density logs from a well. We may •hen compute the reflectivlty by using •he formula shown earlier. Often, we do not have the density log available• to us and must make do with only the sonJc. The

approxJmatJon of velocJty to •mpedance 1s a reasonable approxjmation, and seems to hold well for clas;cics and carbonates (not evaporltes, however). Figure 2.6 shows the 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 assumptions which are made about the seismic

reflectivity and wavelet. Therefore, how can 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 ranUom sequence has the property that its

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

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

equati on-

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

zero-lag.

Let us test this idea on a theoretical random sequence, shown in 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. We see that it is even less "random" than the random spike sequence. We will discuss this in more detail on the next page.

Part 2 - The Convolutional Model Page 2 - 8

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

RFC

F•g. 2.6. Reflectivity sequence derived from sonJc .log.

RANDOM SPIKE SEQUENCE WELL LOG DERIVED REFLECT1vrrY

AUTOCORRE•JATION OF RANDOM SEQUENCE AUTOCORRELATION OF REFLECTIVITY

Fig. 2.7. Autocorrelat4ons of random and well log der4ved spike sequences.

Part 2 - The Convolutional Model Page 2-

Introductlon to Sei smic Inversion Methods Brian Russel 1

Therefore, the true earth reflectivity cannot be considered as being

truly random. For a typical Alberta well we see a number of large spikes (co•responding to major lithol ogic change) sticking up above the crowd. A good way to describe this statistically is as a Bernoulli-Gaussian sequence. The Bernoulli part of this term implies a sparseness in the positions of the spikes and the Gaussian implies a randomness in their amplitudes. When we generate such a sequence, there is a term, lambda, which controls the sparseness of the spikes. For a lambda of 0 there are no spikes, and for a lambda of 1, the sequence is 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 would have a lambda value in the 0.1 to 0.5 range.


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 I TX#E (HS)

LAMBDA-- 1.0 (GAUSSIAN:]

EXAMPLES OF REFLECTIVITIES

Fig. 2.8. Examples of reflectivities using lambda factor to be discussed in Part 6.

, , m i ß i


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 assumption tha.t there is a single, well-defined wavelet which is convolved with the reflectivity to produce the seismic trace is overly simplistic. More realistically, the wavelet is both time-varying and complex in shape. However, the assumption of 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 dependent only on its

dominant frequency, that is, the peak frequency of its a•litude spectrum or the inverse of the dominant period in the time domain (the dominant period is

found by measuring the time from trough to trough). Two Ricker wave'lets are shown in Figures 2.9 and 2.10 of frequencies 20 and 40 Hz. Notice that as the

anq•litude spectrum of a wavelet .is broadened, the wavelet gets narrower in the time domain, indicating an increase of resolution. Our ultimate wavelet would be a spike, with a flat amplitude spectrum. Such a wavelet is an unrealistic goal in seismic processing, but one that is aimed for.

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

perfectly symmetrical. This is a desirable character. tstic 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

same wavelet 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.

Part 2 - The Convolutional Model 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.

Part 2 - The Convolutional Model

2.12. Ricker wavelet rotated by 30 degree increments

Page 2 - 13


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

frequencies than that shown on the Ricker wavelet. Consider the banapass

fil•er shown in Figure 2.13, where we have passed a bana of frequencies between 15 and 60 Hz. The filter has also had cosine tapers applied between 5

and 15 Hz, and between 60 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 wavelet.

Minimum Phase Wavelets

The concept of minimum-phase is 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-phase is adapted from Treitel and Robinson (1966):

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

the minimum-phase wavelet is the one which has the sharpest leading 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 in Figure 2.14. As

in the aefinition used, notice that the minimum-phase wavelet 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 is also shown.

Part 2 - The Convolutional Model Pa.qe 2 - 14

I•troduct•on to Sei stoic !nversion Nethods. 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

Reg E wayel Speetnm

'188.88 • Trace 1

0.8

188

Fig. 2.14. Minim•-phase equivalent of zero-phase wavelet shown in Fig. 2.13.

_

! m,m, i m

Part 2 -Th 'e Convolutional Model i

Page 2- 15


Let us now look at the effect of different wavelets on the reflectivity function itself. Figure 2.15 a anU b shows a number of 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 zero-phase (Trace ½), high minimum phase (Trace 3), low frequency minimum phase (Trace 5).

figure, we can make the fol 1 owing 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 above observations, we would have to consider the high frequency, zero-phase wavelet the best, and the low-frequency, minimum phase wavelet the worst.


(a)


!ql Reg R Zer• Phase Ua•elet •,'1G-•1• 14z

F

- •.• [' ' •,3 Recj B miniilium phue ' '

17 .•

q2 Reg C Zero Phase 14aue16(' ' •'le-3•4B Hz

e

q• Reg 1) 'minimum phase " •,leJ3e/4e h• '

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).



g.4 Th•N. oi se. C o. mp.o•ne nt -

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 Noise - noise which is uncorrelated from trace to trace and is

•ue mainly to environmental factors.

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

is unwanted. An example is multiple reflection interference.

Random noise can be thought of as the additive component n(t) which was

seen in the equation on page 2-g. Correcting for this term is the primary reason for stacking our •ata. Stacking actually uoes an excellent job of removing ranUom noise.

Multiples, one of the major sources of coherent noise, are caused by multiple "bounces" of the seismic signal within the earth, as shown in Figure 2.16. They may be straightforward, as in multiple seafloor bounces or "ringing", or extremely complex, as typified by interbed multiples. Multiples cannot be thought of as additive noise and must be modeled as a convolution with the reflecti vi ty.

Figure generated by 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



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. Refl ectivi ty sequence of Fig. and without mul tipl es.

Part 2 - The Convolutional Model

2.5. with

.

Page 2 - 19

PART 3 - RECURS IVE 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 impedance changes. The formula was written:

Y•i+lV•+l ' •iV! 2i+ 1' Z i ri-- yoi'+lVi+l+ Y•iVi -- -Zi..+l + Z i

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 by 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 i Zi+ 1- Z t 2 Zi+ 1 I + ri- Zi+l + Zi + Zi+l + 2i Zi+l + Zi

I- ri-- Zi+l+ Z i Zi+ 1- Z i 2 Zf[ Zi+l+ Z i Zi+l+ Z i Zi+l+ Z i

Zi+l Z i

l+r. 1

1

Part 3 - Recursive Inversion- Theory

ill, ß , I

Page

Introduction to Seismic Invers-•on Methods Brian Russell

pv-e-

TIME

(sec]

0.7

REFLECTION COEFFICIENT

SERIES

RECOVERED ACOUSTIC IMPEDANCE

Fig. 3.1, Applying the recursive inversion formula to 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 of 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 to start us

off. Assume we can estimate this value for layer one. Then

l+rl , Z2: Zl i r 1 Z3= Z 2 11 + r 2 - r

and so on ...

To find the nth impedance from 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) Frequency Bandl imi 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 and the high frequency components are lost.

Part 3 - Recursive Inversion - Theory Page 3 - 4


0.2 0 V•) 'V,•

•R

R = +0.2

V o: 1000 m Where: --• V,• = 1000 i-o.t

- 1500 m - •ec'.

(a)

- 0.1 '•0.2

R• R=

{ASSUME j•: l)

R•= -0.1 R =+0.2

R: -0.1

V o= 1000 m

-'+ ¾1 = 818 m ii•.

Figure 3.2 Effect of banUlimiting on reflectivity, where (a) shows single reflection coefficient, anU (b) shows bandlimited 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 the 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 the 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 component of 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 ß

spectrum, we may recover much of 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 impedance has the same

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

with depth. This problem of accumulating error is compoundeU by the amplitude problemns introduced 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.

Part 3 - Recursive Inversion - Theory 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) exp 2y r(t) dt. 0

The preceding approximation is valid if r(t) <10.3• which is usually the case. A paper by Berteussen and Ursin (1983), goes into much more detail on

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

paper show that 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 approximation may be made by dropp'ing the logarithmic relationship:

t

1 d Z(t) •_==• Z(t) --2'Z(O) fr(t) dt r(t) --• -dr VO

Part 3 - Recursive Inversion - Theory Page 3 - 8


Fig. 3.5

m i ,, ,m I I IIIII

I + gt ½xp (26•) Difference

-1.0 0.0 0.14 -0.14 -0.9 0.05 0. I? -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

$000 } m MPEDANCE (O I SCR. ) O

r-niL

${300 -• O I FFERENCE o

SO0 O I FFERENCE ( SCALED UP )

T •'•E t SECONOS

C•pari son between impedance c•putatins based on a discrete and a continuous seismic •del.

(Berteussen and Ursin, 1983)

Part 3 - Recursire .Inversion - Theory Page 3 -

Introduction'to Seismic Inversion Methods Brian Russell

PART 4 - SEISMIC PROCESSING CONSIDERATIONS

Part 4 - Seismic Processing Considerations Page 4 - 1

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

4.1 Introduction

Having looked at a simple model'of the seismic trace, anu at the recursire inversion alogorithm in theory, we will now look at the problem of processing real seismic eata in order to get the best results from seismic inversion. We may group the key processing problems into the following categories:

( i ) Amp 1 i tu de rec o very.

(i i) Vertical resolution improvement.

(i i i ) Horizontal resol uti on improvement.

(iv) Noise elimination.

Amplitude problems are a major consideration at the early processing stages and we will look at both deterministic amplitude recovery and surface 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

improvement obtained in migration, using a 3-D example. Finally, we will consider several approaches to noise elimination, especially the elimination of multi pl es.

Simply stateu, to invert our one-dimensional model given in the

approximation of this model (that band-limited reflectivity function) these considerations in minU. Figure 4.1

be useU to 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

shows a processing flow which could



INPUT RAW DATA

DETERMINISTIC AMPLITUDE

CORRECTIONS

,. _•m

mlm

SURFACE-CONS ISTENT

DECONVOLUTIO, N FOLLOWED BY HI GH RESOIJUTI.ON DECON i

i

SURFACE-CONS I STENT AMPt:ITUDE ANAL'YSIS

SURFACE-CONSI STENT STATI CS ANAIJY SIS

VELOCITY ANAUYS IS

APPbY STATICS AND VEUOCITY

MULTIPLE ATTENUATION

STACK ß •

MI GRATI ON ,

Fig. 4.1. Simpl i fied i nversi on processing flow.

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


Inl;roducl:ion 1:o SeJ smlc Invers1 on Nethods BrJ an Russell

4.2 Am.p'l i tu. de.. P,.ecovery

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

•econst•ucting the amplJtudes of the selsmJc t•aces as they would have been Jf the•e were no dJs[urbJng inf'luences present. We normally make the simplJfication that the distortion of the seJsmic amplJtudes may be put into three main categories' sphe•Jcal divergence, absorptJon, and t•ansmJssion loss. Based on a consideration of these three factors, we may wrJte aown an

approximate functJon for the total earth attenuation-

Thus,

data, the

formula.

At: AO* ( b / t) * exp(-at),

where t = time,

A t = recorded amplitude, A 0 = true ampl i tude,

anU a,b = constants.

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

true amplitudes of the data coulU be recovered by using the inverse The deterministic amplitude correction and trace to trace mean

scaling will account for the overall gross changes in amplitude. However, there may still be subtle (or even not-so-subtle) amplitude problems associated with poor surface conditions or other factors. To compensate for these effects, it is often advisable to compute and apply surface-consistent

gain corrections. This correction involves computing a total gain value for each trace and then decomposing this single value in the four components

Aij= Six Rj x G k x MkX •j, 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.

Part 4 - Seismic Processing Considerations Page 4 -

Introduction to Seismic .Inversion Methods Brian Russell

SURFACE

SUEF'A•

CONS Ib'TEh[O{ AND

T |tV•E :

,Ri L-rE R ß

Fig. 4.2. Surface and sub-surface geometry and surface-consistent decomposition. (Mike Graul).

, ,



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

geometry used for this analysis. Notice that the surface-consistent statics anti aleconvolution problem are similar. For the statics problem, the averaging can be •1one by straight summation. For the amplitude problem we must transform the above equation into additive form using the logarithm:

In Aij= In S i + In Rj + In G k + lnkMijX•. The problem can then be treated exactly the same way as in the statics

case. Figure 4.3, from Taner anti Koehler (1981), shows the effect of doing surface consistent amplitude and statics corrections.

4.3 I•mp. rov. ement_ o.[_Ver. t.i.ca.1..Resoluti on

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 sei smic trace, wt= the seismic wavelet, rt= reflection coefficient series, * = convol ution operation.

S(f) • W(f) x R(f) .

The deconvol ution

procedure and consists reflection coefficients.

fol 1 owl ng 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

where Or-- operator -- inverse of w t .

Part 4 - Seismic Processing Considerations ,

Page 4 - 6


ii 11

ß 1'

i

ii

'..,•' •, ," " " ß d.

Preliminary stack bet'ore surface consistent static and ompli- lude corrections.

ß Stock with surface consistent static and amplitude corrections.

Fig. 4.3. Stacks with and without surface-consi stent

corrections. (Taner anu Koehler, 1981).

Part 4 - Seismic Processing Considerations

ß ,

Page 4 - 7


In the frequency domain, this becomes

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

After this extremely simple introduction, it may appear that the deconvolution problem should be easy to solve. This is not the case, and the continuing research into the problem testifies to this. There are two main problems. Is our convolutional model correct, and, if the model is correct, can we derive the true wavelet from the data? The answer to the first

question is that the convolutional model appears to be the best model we have come up with so far. The main problem is in assuming that the wavelet does not vary with time. In our discussion we will assume that the time varying problem is negligible within the zone of interest.

The second problem is much more severe, since it requires solving the ambiguous problem of separating a wavelet and reflectivity sequence when only the seismic trace is known. To get around this problem, all deconvolution or

wavelet estimation programs make certain restrictive assumptions, either 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 techniques only when these phase assumptions are met, and those which do not make restrictive phase assumptions and can be

considered as true wavelet processing methods. In the first category are

(1) Spiking deconvolution, (2) Predictive deconvolution,

(3) Zero phase deconvoluti on, and

(4) Surface-consi stent deconvoluti on.

Part 4 - Seismic Processing Considerations 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 comparison after stack. (a) Zero-phase aleconvolution. (b) Surface-consistent deconvol ution.

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

Part 4 - .Seismic Processing Consioerations Page 4 -

Introduction to Seismic Invers. ion Methods Brian Russell

In the second category are found

(1) Wavelet estimation using a well

(Hampson and Galbraith 1981)

1 og (Strat Decon).

(2) Maximum-1 ikel 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, non 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 the 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-phase inverse of each waveform, and (•) iterate through this procedure to get an optimum result.

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

consistent definition of the waveform in the surface-consistent approach an• the subsequent improvement of the stratigraphic interpretability of the stack.

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

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

wavelet estimation procedure and the wavelet shaping procedure.



Table 4-1 Comparison of Deconvol ution MethoUs m 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.imum phase assumption Random refl ecti vi ty

assumptions.

No assumptions about wavelet•

Zero phase assumption. Random refl ectt vi ty

assumption.

Minimum or zero phase. Random reflecti vi ty

assumption.

No phase assumption. However, well must match sei smi c.

No phase assumption.

Sparse-spike assumption.

WAVELET SHAPING

Ideally shaped to spike. In practice, shaped to minimum

phase, higher frequency output.

Does not whiten data well.

Removes short and long period multiples. Does not affect

phase of wayel et for long lags. ..1_, m

Phase is not altered.

Amplitude spectrum i$ whi tened.

Can shape to desired output.

Phase character i s improved. Ampl i rude spectrum i s

whitened less than in single trace methods.

Phase of wavelet is zeroed.

Amplitude spectrum not whi tened.

Phase of wavelet is zeroed•

Amp 1 i rude spectrum i s whi tened.

Part 4 - Seismic Processing Considerations Page 4 11'


4.4 Lateral Resol uti on

The complete three-dimensional (3-D) diffraction problem is shown in Figure 4.6 for a model study taken from Herman, et al (1982). We will 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 may distinguish 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 has

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

as a warning that migration {preferably 3-D) must be performed on 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 for some of the lateral resolution

problems, we must remember that 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.



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

(Herman et 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 drops close to one. In this case, a coherency enhancement program can be used, which usually involves some type of trace mixing or FK filtering. However, the interpreter must be aware that any mixing of the data will "smear" trace amplitudes, making the 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 methods used in the elimination of multiples are the FK

filtering method, and the newer Inverse Velocity Stacking method. The Inverse Veiocity Stacking method involves the following steps:

(1) Correct the data using the proper NMO velocity, (2) Model the data as a linear sum of parabolic shapes,

(This involves transforming to the Velocity domain),

(3) Filter out the parabolic components with a moveout greater than some pre-determined limit (in the order of 30 msec), and

(4) Perform the inverse transform.

Figure 4.8, taken from Hampson (1986), shows a comparison between the two methods for a typical multiple problem in northern Alberta. The displays are all' co•on offset stacks. Notice that although both methods have performed

well on the outside traces, the Inverse Velocity Stacking method works best on

the inside traces. Figure 4.9, also from Hampson (1986), shows a comparison of final stacks with and without multiple attenuation. It is obvious 'from this

comparison that the result of inverting the section which has not had multiple attenuation would be to introduce spurious velocities into the solution. The

importance of multiple elimination to the preprocessing flow 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.

!lilt tiiti ll!1111iitt i)tt il tli ii/lit t ttl• ill

(b] LINE ld8 - 2-D MIGRATION

IIIIIIll!!1111111111111it I!1111111 I!11111111111illl ill Ii IIIIIIIIIil!111111tllilil!illlllll!111illllllllllllllllllllllli [1111111111111111111111111 III!!1111 I!111111111111111 II II IIIilllllllll!1111111111111111111111111111111111111111111111111 ?•111[•i•• IIIIIIIII !1111111111111111 III I! IIIiill•illlllillllllllllliillllllllllllh

•., }!l!iilll •lllllilllllll i! iiJ :illllllllllllilitiilillit!illllllilll{l•lllliililitl{•{111 ,o

111lllllllllllllllllllll1111llllll Iilllllll!ll!llll I111 illllllllilllllllllllllllllllllllllii{lillllllllllllll{lllll!l{. "• fillllllllll!1111illi!111 IIIIIIIII IIIIIII1111111111 II II Ilillilllllll!1111!1!111111111111111illlllllil!1111111111•111 '•

Col LINE 108 - 3-D MIGR•ATION

F•g. 4.7. Migration of model data shown in F•g. 4.6. - - -- (Herman et al, 1982).

Part 4 - Seismic Processing Considerations ß

Page 4 - 15


AFTER INVERSE VELOCITY STACK

MULTIPLE ATTENUATION INPUT

AFTER F-K MULTIPLE ATTENUATION

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

" "';;:.m;: .... ,;lliml; • .. .

m#l

Fig, 4.8. Common offset 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

1698 - 4

Second real-data set conventional stack without multiple attenuation.

'•" ,• ...... ;•,•<,:u(•:'J,.•J L,.•.,!- •, •, 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'" t••' ' '" •- ..... ,• Interest ,,..,. ,,..,,,_. •,,., .... •.,..., .. ,...,..,.•..,....,,,.,.,.. g •.. ,, ,.

,' , .l•,• ) ' • .'•' ',•' '• .... '. ......•.•_ •.U.•,.., .. • ••,•,•p}•h•?.• r•.•,•. •.} , •.•, ,•,•m,l,•, r ,nm, ""::•"'•'•""""="'""•" .... ";' ,.•,, ,,,.,.•,,,,,.., ,,{. ........ ,,, ... ,,,, ../•.• ,•.•'•, .'•-•%

Fig. 4.9. Second real data stack after inverse velocity stack multiple attenuation. (Hampson, 1986)


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 to invert seismic data. We must remember that all these techniques are baseU on the assumption of a one-aimensional seismic trace model. T•at is, we assume that 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 technique currently used to invert seismic Uata is referred .

to as recursire inversion and goes under such trade names as SEISLOG ana

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

Zi+ 1 Z i <===__===> Zi+l = Z i , ri-- Zi+l+ Z i LIJ where

r i = ith reflection coefficient,

and Z i --/• Vi = density x vel oci ty.

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

inverted using the inverse relationship. However, as was shown in section 3, one of t•e key problems in the recursire inversion of seismic data is the loss of the low-frequency component. Figure 5.1 shows an example of an input seismic section aria the resulting pseuao-acoustic impeaance without the

incorporation of low frequency information. Notice that it resembles a phase-shifteU version of the seismic •ata. The question of introUuclng the low frequency component involves two separate issues. First, where do we get the low-frequency component from, ana, second, how ao we incorporate it?

Part 5 - Recurslye Inversion - Practice Page 5 - 2.


117 112 1e9 leS 1ol 92 93

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_ __ ..• - ,•, _•. • • f •• .• ............ . :•,• m•,•'. • ....... • .... ,.• .... • . •• .........

ß ß ß • ... • ,• ß •- • •, • ,•,..,• :'•l•,fm; ,•v•,• :•,.•.•l.;•.•.'..•l•l;ql .n .................... : ...; •;....: • .. • ................... ' • ]• • '• '•' ',, • •, •' ,,,' ',',•, ",, ',' ",',' •" :•'•'•"•m• i•q•'t•'•'•a .... •., •'. •,•],' •'•,J'•, ,• .• ' ' - '""W',- • -::-= •, '2 ,,• • ., •,•- • ,,• . ,•,•,I,.•.•..,• ....... •.•,,• . .,%•.• . ,• . '-.. ' .,• •, . •i• ....... •. • , • • •-•,• ,, • , ,,.,,• .., ..... •. •.,.,,,,..•,.., ,,,.•,•,•.•.• .... •.,• .... • • ....... ß '•. . •q• • •,•;.• .,• ,.. • •,l•,,..,,, •..•, J I .,,, • .•,• • .... ..,• ....... : ..•..... •.•.•.. :,.. , .... , ,. , .............

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2•Y•' •] ,,•.-..•.•.,'.;.',-,.. .................. • ............ • ................... •'•:.,• ...... • .... - ......... •" ß 7•' . =". .... 7' • '• • '. ' .---- .... - ......... •m:'•' •"• r'u'" •$• .... , ...... r ... •<• • ß • - ' •'•' - ' .'••'•q• "•. •q• • ..... .•,.,• • .... ,_ /. ,,,_ . ; .... •,.:• .- .............. • ...... •%--=: . .•.. ........... • .... , ........... • .....

•4• 7•* • ';•u . :c• i• ,• •.,,•-.•,, •?'..%•.,

•*•'•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- Seismic Data. Heavy lines indicate major reflectors.

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 inversion of data in (a). ß

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)

Part 5 - Recursive Inversion - Practice Page 5 - 3


The low frequency component can be found in one of three ways'

(1) From a filtered sonic log

The sonic log is the best way of deriving low-frequency information in the vicinity of the well. However, it suffers from two main problems' it is usually stretched with respect to the seismic data and it lacks.a lateral component. These problems, discussed in Galbraith and Millington (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 line using Dix' formula. The resulting function will be quite noisy and it is advisable to do some form of two-dimensional filtering on them. In Figure 5.2(a), a 2-D polynomial fit has been done to smooth out the function. This final set of traces represents the filtered

interval velocity in the 0-10 Hz range for each trace and may be added directly to the inverted seismic traces. Refer to rindseth (1979), for more de ta i 1 s.

(3) From a geol ogi cal 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..308 (PV)* 3460 ,

,

i

VELocrrY SURFACE 2rid ORDER POLYN• Frr Figure 5.2 s •mTZ •eH CUT FtT•

tRussell and Lindseth, 1982).

Part 5 - Recursive Inversion - Practice Page 5 - 5 .

ß


Second, the low-frequency component can be added to the high frequency

component by either adding reflectivity stage or the impedance stage. In section 2.3, it was shown that the continuous approximation to the forward and inverse equations was given by

Forward Equati on

1 d 1 n Z(t) <::==> Z(t) r(t) =•- dt -

Inverse Equation t

= Z(O) exp 2•0 r(t) dt. Since the previous transforms are nonlinear (because of the logarithm),

Galbraith and Millington (1979) suggest that the addition of the low-frequency component should be made at the reflectivity stage. In the SEISLOG technique they are added at the velocity stage. However, due to 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. Figure 5.2(b), from Lindseth (lg79), shows that an approximate linear 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 evaporites and should therefore be used with caution. A more exact relationship may be found by doing crossplots from a well close to the prospect. However, using a similar relationship we may approximately extract velocity information from the recovered acoustic impedance.

Figure 5.3 shows low frequency information derived from filtered sonic logs. The final pseudo-acoustic impedance log is shown in Figure 5.4 including the low-frequency component. Notice that the geological markers are more clearly visible on the final inverted section.

Part 5 - Recurslye Inversion - Practice Page 5 - 6


Figure 5.3 Low Frequency comDonent derived from "st.reched:' sonic 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 inversion combinin• 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)



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

fol 1 owing flowchart'

I i

i

INTRODUCE LOW FREQUENCIES •)

I•.v• •o ••DO-•CO••c • ,

' I CORRECT TO PSEUDO VELOCITIES ß ,

CONVERT TO DEPTH I

Recursi ve Inversion Procedure , . _ ß ., . i

A common method 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.



(a) Frequency

(e)

1

(b)

Fig. (a) Frequency response of a theoretical differentiator.

(b) Frequency response of 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 o.n I ?•_Th.e. L o..w .F.r.equ.e. ncy compo..ne. nt

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

the inclusion of the low frequency component, regardless of how this 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

_ 1 d Z(t) <=__==> Z(t) = 2 Z(O)j• 0 r(t) at r(t) - • dt- '

If we consider a single harmonic component, we may derive the response of this tel ationship, which is

d e jwt jwt jwt -j eJWt -dt "-- jwe <===> . dt= w

where w-- 21Tf,

frequency

In words., differentiation introduces a -6 riB/octave slope from .the high end of the spectrum to 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 removes low 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 i thol ogi c change,

(2) Transitional lithologic change, an•

(3) Cyclical change.

Part 5 - Recursire Inversion - Practice Page 5 - 10


(A) MAJOR LITHOLOGIC CHANGE

V 1

Vl I i I. I I I I i I

(B) TRANSITIONAL LITHOLOGIC CHANGE

V:V•+KZ

i i

(C) CYCLICAL CHANGE

! v• _

Fig. 5.6. Three types of lithological models' (a) Major change, (b) Transitional, (c) Cyclical. (Russell and Lindseth, 1982).

Part 5 - Recursire Inversion - Practice Page 5- 11


We may illustrate the effect of inversion on these three cases by looking at both seismic anU sonic log Uata. To show the loss of high frequency on the

sonic log, a simple filter is used, and the associated phase shift is not introUuced.

To start with, consider a major 1 ithologic boundary as exempl i lieu by the Paleozoic unconformity of Western Canada, a change from a clastic sequence to a carbonate sequence. Figure 5.7 shows that most of the information about the large step in velocity is containeU in the D-10 liz component of the sonic log. In Figure 5.8, the seismic data and final Uepth inversion are shown. On the seismic data, a major boundary shows up as simply a large reflection coefficient, whereas, on the inversion, the large velocity step is shown.

RAW SONIC FILTERED SONIC LOGS VELOCITY FT/SEC 0 10000 10-90HZ O-IOHZ O-CJOHZ

TIME

0.3-

0.5-

Fig. 5.7. Frequency components of a sonic log. (Russell and Lindset•, 1982).

! L , , , I I ß [ I L

Part 5 - Recursire Inversion - Practice Page 5 - 12


o'- .

ß

(a)

.%;

DEPTH SEISLOG

ß o

DEPTH

(b)

..... ß lOP OF "' . ß ""I:'ALEOZOIC

-425'

Fig. 5.8. Major litholgical'change, Saskatchewan example. (a) Sesimic s_ection, (b) Inverted section.

..... _ ......... _(R_q•sell .... and L i,pqse_th,_•!98_2)___



To illustrate transitional and cyclic change, a single example will be

used. Figure. 5.9 shows a sonic log from an offshore Tertiary basin, illustrating the ramps which show a transitional velocity increase, and the rapidly varying cyclic sequences. Notice that the 0-10 Hz component contains all the information about the ramps, but the cyclic sequence is contained in

the 10-50 Hz component. Only the Oc component is lost from the cyclic component upon removal of the low frequencies. Figure 5.10 illustrates the same point using the original seismic data and the final depth inversion.

In summary, the information contained in the low frequency component of the sonic log is .lost in the seismic data. This includes such geological information as the dc velocity component, large jumps in velocity, and linear

velocity ramps. If this information could be recovered and incluUea during the inversion process, it would introduce this lost geological information.

Fig. 5.9. Sonic log showing cyclic and transitional strata.

Part 5 - Recurslye Inversion - Practice

(Russell and LinOseth, 1982)

Page 5 - 14

(b)


(a)

SEISMIC SECTION-CYCUC & TRANSITIONAL STRATA

i 1-3500 ß



5.3 Sei smi cal ly Derived Poros i ty -- ILI , ß I

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

pseudo-velocity (and hence pseudo-sonic) information 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 fluid transit time,

btma: rock matrix transit time,

Vsh = fractional volume of shale, and

btsh: shale transit time.

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[tt 'ill ] !•ILI61C .AT& '$[IS'MI• .AT&' I-"'• ''' m.,,•, _,ml . -[ ,gnu mill i' •ill. Utl.. I 111 ,l lit

•%lOtOG

I IIITEIPllETATII i

Fig.

l! WlltK :

t ' .

5.11. Porosity eval uati on flow diagram. (Angeleri and Carpi, 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. Shale percentage is al so displayed. (Angel eri and Carpi, 1982).

Part 5 - Recursire Inversion - Practice

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

The basic theory of maximum-1 ikel i hood deconvol ution (MLD) was developed by Dr. Jerry Mendel and his associates at USC anU has been well publicised

,

(Kormylo and Mendel, 1983; Chiet el, 1984). A paper by Hampson and Russell (1985) outlined a modification of maximum-likelihood Ueconvolution melthod which allowed the method to be more easily applied to real seismic •ata. One of the conclusions of that paper was that the method could be extenoed to use

the sparse reflectivity as the first step of a broadband seismic inversion technique. This technique, which will be termed maximum-likelihood seismic 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 wayel et,

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 to solve for. However, using certain assumptions,

the aleconvolution problem can be solved. As we have seen, the recursire method of seismic inversion is based on classical aleconvolution techniques, which assume a random reflectivity and a minimum or zero-phase wavelet. They

produce a higher frequency wavelet on output, but never recover the reflection coefficient series completely. More recent aleconvolution techniques may be grouped under the category of sparse-spike meth•s. That is, they assume a certain model of the reflectivity and make a wavelet estimate based on this

assumption.



ACTUAL REFLECTIVITY

I,:, I ..

POISSON-GAUSSIAN SERIES OF LARGE

EVENTS

--F

GAUSSIAN BACKGROUND

OF SMALL EVENTS

SONIC-LOG REFLECTIVITY EXAMPLE

Figure 6.1 The fundamental assumption of the maximum-likelihood method.

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 entropy deconvol ution (MEO).

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

advantage over classical methods of deconvolution because 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 norm method of Dr. Doug O1 denburg. The MED method 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 of a series

of large events superimposed on a Gaussian background of smaller events. This

contrasts with spiking decon, which assumes a perfectly random distribution 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 i thol ogic boundaries.

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

which may be minimized to yield the "optimum" or most likely reflectivity. and wavelet combination consistent with the statistical assumption. Notice that

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

Part 6 - Sparse-spike Inversion m


INPUT

WAVELET

REFLECTIVITY

NOISE

SPIKE SIZE' 9.19

SPl• ••: 50.00

NOISE' 39.00

OB,.ECTIVE' 98.19

Figure 6.2(a) Objective function for one PoSsible solution to input trace.

INPUT

WAVELET

REFLECTIVITY

SPIKE S!7_F: 6.38

SPIKE DENSIq'•, 70.85

NOISE NOISE: 81.• 5

OBJECTIVE :158.98

Figure 6.2(b) Objective function for a second possible solution to input trace. This value is higher than 6.2(a),. indicating a less 1 ikely solution.

! , ,,



The objective function j is given by

- R2 N 2 k=l k=l

ß

where

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

r(k) = reflection coeff. at kth

sample,

m = number of refl ecti OhS, ß

L : total number of 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 parameters shown above. No assumptions are made about the wavelet. The reflectivity sequence is postulated to be "sparse", meaning that the expected number of spi•es is governed by the parameter lambda, the ratio of the expected number of nonzer. o spikes to the total number of trace samples. Normally, lambda is a number much smaller than one. The

other parameters needed to describe the expected behavior are R, the RMS•size

of the large spi•es, and N, the RMS size of t•e noise. With these parameters specified, any glven deconvol ution sol ution can be examined to see.whether it

is likely to be the result of a statistical process with those parameters. For example, if the reflectivity estimate has a number of spikes much larger than the expected number, then it is an unlikely result.

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

number of spikes in its reflectivity and t•e lowest noise component. Figures 6.2(a) and 6.2(b) show two possible solutions for the same input synthetic trace. Notice that the obje6tive function for the one with the minimum spike structure is indeed the lowest value.


Introduction to Sei smic I nversi.on Methods Bri an Russel 1

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 Russel 1

Of course, there may be an infinite number of possible solutions, and it

would take too much computer time to look at each one. m Therefore, a simpler method is used to arrive at the answer. Essentially, we start with an initial

wavelet estimate, es'timate the sparse reflectivity, ' improve the wavelet and iterate through this sequence of steps until an acceptably low objective

function is reached. This is shown in block form in Figure 6.4. Thus, there is a two step procedure- having the wavelet estimate, update the 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 set of "sparse" coefficients has been found. The algorithm used for updating the reflectivity is callee the single-most-likely-addition algorithm (SMLA) since

after each step it tries to find the optimum spike to add. Figure 6.5 shows the procedure for updating the wavelet phase. The input model is shown at the

top of the figure, and the up•ated reflectivity and phase is shown after 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. The block component method of solving for both reflectivity and wavelet. Iterate around the loop unti 1 converRence.


Introduction to Seismic Invers.ion Methods Brian Russell

Wayel et Refl ecti Vity ' Synthetic

Ill ,I ,

INPUT MODEL

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.



Figure 6.6 is an example of the algorithm applied to a synthetic

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

and that the resultant trace matches the original trace quite accurately. Of course, the smaller reflection coefficients are missing in the recovered reflection coefficient series.

Let us now look 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 are 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 in stratigraphic content.

Figure 6.8 is a comparison of 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 improvement is 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 superimposed on the Uominant negative trough.

INPUT:

V. ,.: --

ESTIMATED:

ttl J':ll'j ' "'" " ß

Figure 6.6 Synthetic seismogram test.



0.5

0.6

0.7

0.8

'SONIC SYNTHETIC LOG

iZ. i

EXTRACTED WAVELET

0.5

0.6

.

0.8

(b)

(a) Initial seismic with extracted wavelet.

Final deconvolved seismic with zero-please wavelet.

Figure 6.7 .... - -_ __ ._

Part 6- Sparse-spike Inversion 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 much more 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.



INPUT

STACK

DECONVOLVED

STACK

0.6

0.7

0.8

0.9

Figure 6.8 An input stack over a channel scour and the resul ting deconvol ved 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.

Part 6 - Sparse-spike Inversion 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 (Hampson and Russell, 1985). Given the reflectivity, r(i),

impedance Z(i) may be written

Z(i) =Z(i_l )[1 +r(i)] 1 - 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

impedance may 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 trenU 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 impedance trend,

• i <0 H(i) :

• i >0

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

_

Part 6 - Sparse-spike Inversion 6• 14


Figure 6.10 Input Model parameters.

Figure 6.11 ß

Maximu•m-L i kel i hood i nversi on result from Figure 6.10. .m __

Part 6 - Sparse-spike Inversion 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 In 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 compromise by simultaneously moUelling the seismic trace while conforming to the known impedance trend. If both the seismic

noise and the impedance trend noise are modelled as Gaussian sequences, their respective variances become "tuning" parameters which the user can modify to shift the point at which the compromise occurs. That is, at one extreme only

the seismic information is used and at the ot•er extreme only the impedance trend.

In our first example, the method is tested on a simple synthetic. Figure 6.10 shows the sonic log, the derived reflectivity, the zero-phase wavelet used to generate the synthetic, and finally the synthetic itself. This

example was used initially because it 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 case we have used a smoothed version of the sonic velocities to provide the constraint. A visual comparison woulU indicate that the extracteU

velocity profile corresponds very well to the input. A more detailed comparison of the two figures shows that the original and extracted logs do not match perfectly. 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.


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 turn our attention to a slightly more realistic synthetic

example. Figure 6.12 shows the application of this algorithm to a sonic-log derived synthetic. At the' top of the figure we see a sonic log with 'its reflectivity sequence below. (In this example, we have assumed that the density is constant, but this is not a necessary restriction.) The

reflectivity was cbnvolved with a zero-phase wavelet, bandlimited from 10 to 60 Hz, and the final synthetic is shown at the bottom of the figure.

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

Figure 6.13. The initial log is shown at the top, the constraint is shown in

the middle panel, and the extracted resull• is shown at the bottom of the

diagram. In this calculation, the wavelet was assumed known. Note the blocky nature of the estimated velocity profile compared with the actual sonic log profile. Again, the input and output logs do not match perfectly.

The fact that the two do not perfectly match is due to slight errors in the reflectivity sizes which are amplified by the integration process, and is partially the effect of the constaint used. The constraint shown in Figure 6.13 was calculated by applying a 200 ms smoother to the actual log. In practice, this information could be derived from stacking velocities or from nearby well control.



* !

Figure 6.14 An input seismic 1 ine to be inverted.

:

ß

'.

eel'?

e4dl

Figure 6.15 Maximum-Liklihood reflectivity estimate from seismic in Figure 6.14.


Introauction to Seismic Inversion Methods Brian Russell

Finally, we show the results of the algorithm applied to real seismic data. Figure 6.14 shows a portion of t•e input stack. Figure 6.15 shows the •D extracted reflectivity. Figure 6.16 shows the recovered acoustic

impedance, where a linear ramp has been used as the constraint. Notice that the inverted section •isplays a "blocky" character, indicating that the major features of the impedance log have been successfully recovered. This blocky

impedance can be contrasted with the more traditional narrow-band .inversion procedures, which estimate a "smoothed" or frequency limited version of the impedance. Finally, Figure 6.17 shows a comparison between the well itself and the inverted section.

In summary, maximum-likelihood inversion is a procedure which extracts a broad-band estimate of the seismic reflectivity and, by the introduction of

1 inear constraints, al lows us to invert to an acoustic impedance section which retains the major geological features of borehole log data.


Introduction to Seismic Inver.sion Methods Brian Russell

Figure 6.16 Inversion of reflectivity shown in Figure 6.15.

SEISMIC INVERSION

WELL

+ SONIC

LOG

Figure 6.17 A comparison of the inverted seismic data and the sonic log at well location.

Part 6 - Sparse-spike Inversion .. 21


6.3 The L 1 Norm Method -- __LI _ _ _ i .

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

"sparse-spike" assumption is the L1 norm method, developed primarily by Dr. Doug Oldenburg of UBC. and Inverse Theory and Applications (ITA). This method is also often referred to as the linear programming method, and this can lead to confusion. Actually, the two names refer to separate aspects of the method. The mathematical model used in the construction of the algorithm is the minimization of the L1 norm. However, the method used to solve the problem is linear programming. The basic theory of this method is found in a

paper by Oldenburg, et el (1983). The first part of the paper discusses the noi se-free convol utional model,

x(t) --w(t) * r(t), where x(t) = the seismic trace, w(t) --the wavelet, an•

r(t) -- the reflectivity.

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

performed on the seismic trace, the resulting estimate of the reflectivity can be thought of as an averaged version of the original reflectivity, as shown at

the top of Figure 6.18. This averaged reflectivity is missing both t•e high and low frequency range, and is accurate only in a band-limitea central range of frequencies. Although there are an infinite number of ways in which the missing frequency components can 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) --•, rj 6(t -l•), j--!

where •= 0 if t •l• , an• =1 ift:• .



b

ß ß ß • 1 I 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 Inversion, moUified fro•.q Oldenburg et al (1983). (a) Input impedance, (b) Input reflectivity, (c) Spectrum of (b), (d) Low frequency model trace, (e) Deconvolution of (•), (f) Spectrum of (U), (g) Estimated impedance from L1 Norm method, (•) Estimated reflectivity, (i) Spectrum of (•).


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 function, 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 programming method 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. True L2 norm, on the other hand,

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

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

x 1 : x i and x 2: x i 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 inversion"). 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'

- fl - 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.



(a) Input sei smi c data

(b) Estimated refl ec ti vi ty

(c) Final impedance

Figure 6.19 L1 14orm metboO applied to real seismic data,

Part 6 - Sparse-spike Inversion

(Walker and Ulrych, 1983)

6- 25

Introduction to Seismic Inversion MethoUs Brian Russell

Several other authors had previously considered the L1 norm solution in deconvolution (Claerbout and Muir, 1973, and Taylor etal., 1979), however, they considered the problem in the time domain. Oldenburg et al.w suggested solving the problem using frequency domain constraints. That is, the reliable frequency band is honored while at the same time a sparse reflectivity is created. The results of their. algorithm on synthetic data are shown at the

bottom of Figure 6.18. The actual implementation of the L1 algorithm to real seismic data has been done by Inverse Theory and Applications (ITA). The processing flow •or the linear programming inversion method is shown below.

InterPreter'= CMP Stacl<ed section <r(t)>= r(t)©w(t) t ß ,i

i

I,,i co,ect,', ,o,' Residu Pm'm,e o,w (t) I ß i i i i i I i i

I Fourier Trans•• of <•r (t)> I i

Scale Data Const. mints. From $tackins•_V'elocitles I

ii &

Con,straints From 'Well Logs I i

Unear Programing Invemion

Assume r( t ) ß • n ;) (t- •q ), is a spame, reflection series. 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.) Method. (Oldenburg, 1985).


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 Input seismic data section to L1 Norm inversion. (O1 denburg, 1985'



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

seismic line from Alberta. The data consist of 49 traces with a sample rate of 4 msec and a 10-50 Hz bandwidth. The figure shows the linear programming reflectivity and impedance estimates below the input seismic section. It

should be pointed out that a three trace spatial smoother has been applied to the final results in both cases.

Finally, let us consider a dataset from Alberta which has been processeU through the LP inversion method. The input seismic is shown in 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.



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 inversion of data in Figure 6.20.

Part 6 -Sparse-spike Inversion

for

(O1 denburg, 1985

6- 2-9

Introduction to Seismic Inversion Methods Br•an Russell

6.4 Reef P roblee ß _

On the next few pages 'is a comparison between a recursive inversion procedure (Verilog) and a sparse-spike inversion method (MLD). The sequence

!

of pages includes the following:

- a sonic log and its derived reflecti vt ty, - a synthetic seismogram at both polarities,

- the original seismic line, showing the well location, - the Verilog inversion, and - the MLD inversi on.

BaseU on the these data handouts, do the following interpretation exerc i se:

([) Tie the synthetic to the seismic line 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) Compare the 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?



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.



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Part 6- Sparse-spike Inversion

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Introduction to Seismic Inversion Methods Brian Russell ***__********************************************************

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Introduc%ion [o Seismic Inversion Meltotis Brian Russell

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Part 6 - Sparse-spike Inversio•

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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 that 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 becomes smaller, the interference between overlapping

wavelets becomes more severe. Indeed, in Figure 6.19 it was shown that the

effect of reflection coefficients one sample apart and of opposite sign is to

simply apply a phase shift of 90 degrees to the wavelet. In fact, the effect is more of a differentiation of the wavelet, which alters the amplitude

spectrum as wel 1 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 l'ook at thin bed effects was done by Widess (1973). In this paper he used a model which has become the standard for

discussing thin beds, the wedge model. That is, consider a high velocity laye6 encased in a low velocity layer (or vice versa) and allow the thickness of the layer to pinch out to zero. Next create the reflectivity response from the impedance, and convolve with a wavelet. The thickness of the layer is given in terms of two-way time through the layer and is then related to the dominant period of the wavelet. The usual wavelet used is a Ricker because of the simpl i city of its shape.

Figure 7.1 is taken from Widess' paper and shows the 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 in his plot i s actual ly twice the dominant period). A few important points can be noted from Figure 7.1. First, the wavelets start interfering with eack other at a thickness just below two dominant periods, but remain Clistinguishable down to about one period.

Part 7 - Inversion applied to Thin Beds Page 7- g


PI•OPAGA ! ION I NdC ACnOSS TK arO) .

•'------ •).z _1 I

--t

Figure 7.1 Effect of bed thickness on reflection waveshape, where (a) Thin-bed model, (b) Wavelet shapes at top and bottom re fl ectors, (c) Synthetic seismic model, anU (d) Tuning parameters as measured from resul ting waveshape.

(C) (D)

5O , ,.

THIN BED REGIME

J PEAK-TO-TROUGH/ AMPLITUDE

2.0

1.0 <

0.8

0.4

/ \ -0.4 ,• i . . . . .

-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. _

Part 7 - Inversion applied to Thin Beds Page 7- 3


Below a thickness value of one period the wavelets Start merging into a single wavelet, and an amplitude increase is observe•. This amplitude

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

amplitude is appraoching zero 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 this 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 at 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. version Camparison of T.hin Bees

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 in 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 in 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, the wedge has a thickness of 2T at trace 1, T at trace 25, T/2 at trace 37, etc.

Parl• '7 - 'inverslYn 'ap'pl led 1•o Thin'- Beds ..... Page 7 --'4 '•-


lOO

200

3OO

400

500

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

ß

Figure 7.3 True impedance from wedge model.

o

lOO

200

.

300

ß

400

500

Figure 7.4 Wedge model 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 wedge model. The inversion result is shown in Figure 7.5. Note that the

low frequency component was 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 component would 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 to about 25 msec, which is 1/2 of the dominant period. 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 same as 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" observed on 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 i smi c peri od.

Part 7 - Inversion applied to Thin Beds

_ 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 wedge model shown in 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-likelihood derived impedance of wedge moUel shown i n Figure 7.4.

Part 7 - Inversion applied to Thin Beds Page 7- 7

[ntroductJon to Seismic Inversion Methods Br•an Russel•

PART 8 - MODEL-BASED INVERSION _ - _ - 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 the data'very well, but is incorrect. (This can be seen easily

by noting that there are infinitely many velocity/depth pairs that will result

in the same time 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 approaches to these problems, the generalized linear inversion (GLI) approach outlined in CooRe and Schneider (1983}, and the Seismic Lithologic (SLIM) method which was developed in Gelland and Larner

(1983).

Part 8 - Model-based Inversion 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.

Part 8 - Model-based Inversion Page 8 -

Introduction %o Seismic Inversion Methods Brian Russell

8.2 Generalized Linear Inversion

The generalized linear inversion(GLI) method is a method w•ich can be. applied to virtually any set of geophysical measurements to determine the geological situation which produced these results. That is, given a set of geophysical observations, the GLI method will 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 1, m 2, ..... , mk) T= vector of k model parameters, and

T : (t 1, t 2, ..... , tn )T : vector of n observations.

Then the relationship between the model in the functional form

and observations can be expressed

t i = F(ml, m 2, ...... , m k) ß i : 1, ... , n.

functional relationship has been derived between the Once the

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

output. But what model? GLI eliminates the need for trial and error by analyzing the error between the 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 less error. In this Mathematically'

)F(M O) = F(Mo) + aT •M,

MO-- Initial •odel, M: true earth model,

AM: change in model parameters, F(M) : observations,

F(Mo): calculated values from initial

•)F(M O) .2 • = change in calculated values.

model, and

F(M)

where


Introduction to Seismic Inversion Flethods Brian Russell

IMPEDANCE 4.6 41.5 AMPLITUDE ß

ml I

ß

,

- ii

,i i,

i

i

ii

,

ß

ß

,

, i

:.

__

IMPEDANCE

(GM/CM3) (FT/SEC) X1000 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 inverse of 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

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

squares solution often referred to as the Marquart-Levenburg method (see Lines and Treitel (1984)). The solution is given by

•M: (AT'A)-IA T Z•F.

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

the impedance update using the method just 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 lied version of the previous formula

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

composite from their paper showing the results of an inversion applied to a single synthetic impedance trace.



• ' • IMP.EDANCE x1OOO (G M/CM3)(FT/$EC)

ß ._ . .:. . . :.........•:: :...., .. .... .. :... lO ,.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 a gas sand while the well on the left does not.

(Cooke and Schnei der, 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. Compare with sonic log on right side of Fi•iure 8.3.

(Cooke and Schneider, 1983)

Page 8 - 7

Introduction to Seismic Inver. sJon Methods Brian Russell

In Figure 8.2, notice that the advantage of incorporating multiples in the solution is that, although they are modelled in co•uting the seismic

response, they are not included in the model parameters. This is a big advantage over recursire methods, since those methods incorporate the multiples into the solution if they are not removed from the section.

Another important feature of this particular method is the

parameterization used. Instead of assigning a different value of velocity at each time sample, large geological blocks were defined. Each block was

assigned a starting impedance value, impedance gradient, and a thickness in time. This reduceU the number of parameters and therefore simplified the

computation. However, there is enough flexibility in this modelling approach to derive a fairly detailed geological inversion. We will now look at both a

synthetic and real example from Cooke and Schneider (1983).

A 2-0 synthetic example was next considered by Cooke and Schneider

(1983). Figure 8.3 shows the model, which consisted of two gas sands encased in shale. One well encountered the sand and the other missed. The impedance

profile of the discovery well is shown on the right. Figure 8.4 shows synthetic traces over the two wells, in which a noise component has been added. Finally, Figure 8.5 shows the initial guess and the final solution,

for which the gradients have been set to zero. Notice that although the

solution is not perfect, the gas sand has been delineated.



I I

YES ___•__•J

' ' FINAL MObE L

- _ ._ x•, • .... r -• •;•,• -.-'%•..

-cx-r. . . . .-. .,'•_;'•.:. ß -,• . . t .•..

Figure 8.6 I11 ustrated flow chart for the SLIM method.

(Western Geophysical Brochure)


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

8.3 Sei_smic L_ithologic Modelling (,SLIM)

Although the n•thod outlined in Cooke and 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 compared with 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 compared with 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, as in the GLI method, t•he solution is nonunique.

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

Gelland and L arner (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 Basin model seismic. Right: Stacked section from Denver Basin.

(Gel fand and Larner, 1983).

2.0

.4

1.6

1.8 1.8

2.0 • Field data 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.

' -- _• -- __ --__ ii 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 complex region, with the final synthetic data is shown after 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 impedance is derived. Since

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

frequency component necessary for full inversion. Finally, Figure 8.10 shows the final inversion compared with a traditional recursire inversion. Note the

'blocky' nature of the parameter based inversion when compared with the recurs i ve i nvers i on.

I n summary, parameter

which can be thought of

reflectivity is extracted.

propagated through the final

based inversion i s an iterative model 1 ing scheme

as a geology-based deconvolution since the full I• has the advantage that errors are not

result as in recursire inversion.



w 1

500-ft N 114 mile S 114 mile S E lkft • -

.5

l m ß

.7

1.9

Figure 8.9 Impedance section derived from SLIM inversion of Denver Basl n 1 ine shown i n Figure 8.7.

{GelfanU and Larner, 1983)

W

1.7

50011 N 114 mile S 114 mile S lkft ß ß .• E

19

F i gu re 8.10 Traditional recursire inversion of Denver Basin line from F i gur. e 8.7.

(Gelfana anU Larner, 1983)



Appendix 8-! Mat_r_ix .appljc.•at. ions_in Ge•ophy.s.ics

Matrix theory shows up in every aspect of geophysical proocessing. Before looking at generalized matrix theory, let us consider the application of matrices to the solution of a linear equation, probably the most important

application. For example, let

3x1+ 2x 2 : 1, and

x 1- x 2 = 2.

By inspection, we see that the solution is

However, we Could .have expressed the equations in the matrix form

or

A X = y,

3 2 x 1

1 -1 x 2 ß

The sol ution is, therefore

or

-1 x = A y,

x 1 1 . -2 1 -1/5

1 3 1 x 2


Introduction to Seismic Inversion Me.thods Brian Russell

is of little

overde termi neU

problems:

In the above equations we had the same number of 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 problems we 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 unknowns (in which case the situation

is calleU overdetermined). In geophysical problems, the underUetermined case interest to us since there is no unique solution. The

case is of much interest since it occurs in the following

(!) Surface consi stent resi dual

(2) Lithological modelling, and (3) Refracti on model 1 i 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 s••t Overd..etermined L in.ear Equa.t. ion.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 to the

For examp.le, consider adding a third equation to our so that

anU

3x1+ 2x 2 : 1,

x 1- x 2 : 2,

5x 1- 5x 2 : 10.

Part 8 - Model-based Inversion P age 8 - 15


This may be written in matrix form as

2 x 1

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, ns, is, ten•t O•verd. e•ermine. d L.i.near Equa•i.on?

In this case the extra equations are not scalea versions of other

equations-in the set, but convey conflicting 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 Z -- 1,

x 1- x 2 -- Z,

ana 5x 1- $x 2 = 8. This may be written in matrix form as

3 2 x I 1

I 2 - x 2

-5 8


Introduction to Seismic Inversion He.thods Brian Russell

'Now the third equation is 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 sum of 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 mi ze

eTe-- (e I , ez , ....... n

, e n ) = e i 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 _ xTATy _ yTAx + xTATAx.

We then solve the equation

bE_

bx i

The final solution to the least-squares problem is given by the normal

eq ua ti OhS

AT A x = A T y

or x = (ATA)-lATy .


Introduction to Seismic Inversion Methods Br•an Russell

PART g - TRAVELTIME INVERSION

Part g - Traveltime Inversion

ml ß i ii

Page 9 - I


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 show an 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 become obvious 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 in

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 geophones both 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


(a) (b)

Figure 9.1 Travel paths through a single, constant velocity block.

(a.) Surface recording, (b) VSP recording, (c)Cross-hole recording

s•

$ R

(b)

(c)

Figure 9.2 Travel paths through two blocks of slightly differing veloc-ity.

(a) Surface recording, (b) VSP recording, (c) Cross-hole recording

Part 9 - Travel time Inversion Page 9 -

Introductfon to Sef smJ c Inversf on Methods Brj an Russell

9.• Numerical_Exa•mples of .Travelti • In•v. ersion

Consider the simplest possible case, a constant velocity earth. Figure 9.1 shows the travel paths that would result from the three geometry configurations given a square area of dimension L by L. Note that the traveltimes in Figure 9.1 would simply be:

(1) Surface sei smi c'

(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 contain the same information.

However, if the velocity (or slowness p) and the depth are both unknown, neither one can be determined from a single time measurement. An even greater ambiguity comes into play if we have a single measurement but more than one box. In Figure g.g this situation is shown. Notice that the equations now would involve three unknowns and only one measurement.

A more general model is proposed in Bishop et al (lg85) an• Bor•ing et al (1986). The earth is represented as a number of boxes of constant size and velocity. Although the velocity of each box is a constant, the velocity may vary from box to box. This is shown in Figure 9.3. The objective is thus to compute the seismic travel path through each box using the traveltime

measurements. A key problem here is how to allow the rays to travel through the boxes. The first order approximation would be straight rays with no bending. However, i f Snell's law is use4, the problem becomes more difficult to sol ve.

Part-g Travel i'i me'-'i n'ver's i 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 - Part g - Traveltime Inversion 5


Let us apply the straight ray approximation in the simple case of having simply two blocks of different velocity. In this case, we have coupled together both surface and VSP measurements. Two possible recorUing arrangements are shown in Figures 9.4 and 9.5. The situations illustrated are

obviously oversimplified since we have assumed a 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 assume that 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 geophones both 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. m ), the travel time equations are

(1) For the. raypath from S to R

where Pl: 1/velocity in box 1 P2: 1/velocity in box 2

(Z) For the raypath from S to R2:

t2= q• Pl + • P2. 2 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 - Part 9 - Travel time Inversion 7


A simple way to remedy this situation is to move the shot for the second.

raypath. This is shown in Figure 9.5. In this situation, we have moveU the

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 raypath from S 1 to R 1:

tl: 1•Pl + •P2 (2) For the raypath from S 2 to R 2-

In this case notice from the diagram that

tan 0 : 1/1 $ : 2/3 = 0 6667, or B : 33 69 o

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.



Initial Model

Layer Stripping .Inversion

Estimate velocity at well using sonic log and VSP

Pick seismic

reflection times, t

Estimate V(x,z) by using V(xo,z), the reflection traveltimes and the

the assumption of vertical rays

Start with top layer

Computer forward model traveltimes, f, by normal ray tracing

Perturb V(x,z) by least squares

or manually

It- fll'

Add another

layer

Final

Seismic Model

layers been

ii

Model is complete,, I

Figure 9.6 A possible flow chart for seismic traveltim inversion.

(Lines et al, 1988)

Part 9 - Travel time Inversion Page 9 -


9.3 Sei sm•i •c .T..omo. gr aphy

The term tomography was first used in the medical field for the imaging of human tissue using Nuclear Magnetic Resonance (NMR) 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 tomography are 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 are used

in the inversion, but that much of the technique used by the authors can be

cons i alered sei smi c 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 in 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 CDP data is usecl, but the shot profiles (or CDP profiles) 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 in 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.



•0 WlC

_ m m ß i roll i

-1Z 0

x I• vsP SOURCE

..2

• 3-D SOURCE • GEOPHONE

Figure 9.7 Surface geometry for tomographic imaging example.

(Chiu and Stewart, 1987)

Une 89 89 D•B 89 89 Une CDP 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

1986)


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 measurements were 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 imaging. Figure 9.8(a)

shows the stacked seismic data with the key events indicated and Figure 9.8{b)

shows the picked VSP from well C. Finally, Figure 9.9 shows the 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 model of the subsurface and 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 Wel 1

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 inversion of model data using VSP and surface data. (Chiu and Stewart, 1987)



Figure 9.11 shows the results of the inversion process using both the VSP and surface seismic data. To make the test more realistic, random noise was added to travel'time picks. Notice that the correct result has been obtained in four iterations.

Let us now return to the case study described initially. The final velocity/depth model is shown in Figure 9.12. Notice that the velocities fit

quite well with the averaged sonic log velocities. This velocity model was used to produce both a depth migrated seismic section, shown in Figure 9.13, and a full seismic inversion based on the maximum-likelihood technique. The final inversion is not shown due 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

make is that if several types of geophysical measurements can be intergrated, the result is an improved product. Each set of data acts as a constraint on the others.


Introduction to Seismic Inversi on Methods Brian Russell

Offset (km) Velocity Oan/s) -1.0 0.0 LO 0.0 3.0 6.0

TC)iliO6RAPmC (C) INVERSION

- SONC L06 (1:2)

Figure 9.12 Results of tomographic inversion of G1 auconitic channel.

(ChiU and Stewart, 1987)

. : •m,,, .......... J• ß ß . ... l..,.,.,;,,•. ' 't ''•"','

ß -.--:' ._:_.4sl•l • ,_, i!' ,i,? ,a•.. ,:. I.,,t.:, ?

800 .. :

900 :'": ""' ""' ....... '

Depth (m) 1000

11oo

1200

1300

1400

F i gure g. 13 Depth migration of seismic aata shown in-Figure 9.8(a). Tomographic velocities of Figure 9.12 have been used.

(Stewart an• Chiu,


1986)


PART 10 - AMPLITUDE VERSUS OFFSET INVERSION

Part 10 - Amplitude versus Offset Inversion Page 10 - 1


10.1 AV.O Theo.•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+ yO iV i •Zi+l + Z i

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 shot 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-wave amplitude, and transmitted S-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• 2 - Z . •' =-

Poisson's ratio can theoretically vary between 0 and 0.5.



$i S r

at, •t

BOUNDARY

(X2' •2

t

$•

Figure 10.1 Reflected and transmitted rays created when a P-wave strikes the boundary between two layers.

(Waters, 1981).

•o, 2+, - •sin2•, - ' 'cos2•,- - •x,n-•:/ •D,/ •-cos2+,/

Figure 10.2 Zoeppritz equations which describe the amplitudes of the rays shown in Figure 10.1.

(Waters, 1981 ).

Part 10 - Ampl i 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 the seismic rays across a

boundary, and Figure 10.2 gives the final form of the equations. They are taken from • textbook by Waters (however, some of the signs were wrong, and

they are fixed in the diagram). Since we have four equations with four

unknowns, they 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 that for a

significant change in Poisson's ratio, a major change in the P-wave amplitude 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 the 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 a stackeO seismic section witl• three apparent "bright spot" anomalies. Unfortunately, only wells A anU B were productive. The three

common offset stacks, shown in Figure 10.5(b), indicate that only locations A ,

and B actually Uisplay an AVO effect.



GAS *':*"*•* t •t Vl• Z =8.000 /32 -"2.14

:o., ;..•.• ::., ß

SHALE •---' $=10.000 /4) 3 =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 OF INCIDENCE - •,-'-e' 20 o 30 ø 40 ø

NO GAS ., ,,, ,,, .,.o o o.o.o ..... .. ooo., ß o.,.,o.,-'. oø ,.,,,o .,, 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 - Amplitude versus Offset Inversion Page 10 -

Computed reflection coefficients as a function of offset


The method useU to identify this effect is only partly qualitative, and

can be diagrammed as shown below-

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'od of comparison leaves much to 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

Part 10 - Amplitude versus Offset Inversion Page 10- 6


1•0 170 160 1•50 140 130 120 110 100 90 •0 70 0.0 ' , .... • , -- • , • • ! • • ' ' -' • •- ' -• " • • -• • •. !: ' 't .". '.'..' t'-' " : "• :. ' ' : .... I; ' ' ' 0.0 I'*O..' ß m * re- ß ß l, ß ' ß I , I i I, m m I.. i ,. ß ' 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.,...,:,•....,;.•. ......... :. ,...-.. ß

...

....,.., t,,, "_,,,d,.•, I•l.leeile*e,,I,'t ! :lit I•ll""' IIt•'•l';I;,d .......... 12.0

. ..

,•e.•. Illl•lll.1111el'•- ß ß ..le - .. • '."-'•1, ø.'•1•-. • ............ ;.; .... :.....i:-;

ß '.:--;•=....=:;;:.1• .... ...,;•.".....

' ......... "•"":": ....................... 2..5 ,..,- .1,• ..----?'" '1 ß ß - ..... ,.....-.,.-.,.-. .-,., ........... ,... ......... "' ' "1;;::= .... :":" ..... ;""':"' ""1'.'-'-- '-' ' "•':,';;;;":',:: .... :"

ß ....... ;.::.;:.. 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•$•' 1012'

Fiõure 10.5(D) Common offset sl;acks over locations A, B, and C 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' a geological model of the earth, a physical

relationship between the earth and a set of geophysical measurements, anU a ß

set of geophysical observations. This method Was discussea in both chapters 8 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

AVO Uata we,re described. Our model 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 offset stack.

By computing derivatives 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

a F (Mo•) FIM) : F(M D) + •)M bM , where Mo: initial earth model,

M: true earth model,

AM: change in model parameters,

F(M) -- AVO observations,

F(MO): Zoeppritz values from initial model, and

•)F(M O) i)--••: change in calculated values.

The implementation is simply a variation of the manual method, anO is sinown on the next page.



INPUT SEISMIC

SHOT PROFILES

COFFSTACK J

i

PICK

AMPLITUDES

COMPUTE

ERROR

COMPUTE

SYNTHETIC

STORE COMPUTED

AMPL I TUDES

i t • COMPUTE MODEL

PARAMETER CHANGE

US I NG GLI

NO

ERROR

YES

MODEL MATCHES

REAbITY ,

Figure 10.6 AVO inversion by the GLI method

Part 10 - Amplitude versus Offset Inversion Page 10 -


We will now look at an example of GLI inversion of amplitude versus

offset data. First, consider the integrated well logs shown on the left hand

panel of Figure 10.7. Actually., only the sonic log or P-wave log was recorded in the field. The density log was derived from the sonic using Gardner's

equation, the Potsson ratio was fixed at 0.25, and the S-wave was derived from the P-wave and Potsson ratio logs. On the logs, three layers have been

blocked at depth and a significant Poisson's ratio change has been introduced in the middle block. On the right hand side of Figure 10.7, notice that the

amplitude versus offset 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 shows the same set of blocked logs on the left, but shows the seismic response of the amplitude change on the right. This synthetic was produced by simply' replacing the zero-incidence amplitudes with the amplitudes derived from the Zoeppritz calculations. The events between 600 and 700 msec display a pronounced amplitude change wit h offset.

ZOEPPRII'Z $IHI:LE INTERFFJCE TESTLO TESTLO TESi'DE lE•-S t•;$TPO

• 2• 2,5 4;8 ,,,

Eq, m, nt: 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 Blocked well logs on left, with computed Zoeppritz 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, IFIUE! POISSOH1 u•Ym u•/m 9/½½ usam

...... ,L -

ß .•..o...-.. ....... ........ , ....,....

•ee-. • ......................... -.•" ......... ......... ...... ... ..... •.'.........,. :: ......................... •...• ........... "

268 268 2.5 468 .$

MODEL 1 (meters.) EU 909 727 545 363 181 .

Figure 10.8 Left-

Right:

A "blowup" of the blocked logs shown in Figure 10.7.

A synthetic common offset stack and the AVO curves shown on the right of Figure 10.7.



However, does the change seen in Figure 10.8 reflect the reality of the

situation? Figure 10.9 shows a set of CDP gathers which correspond to the model. The gathers are a realistic modelled dataset and were generated with no change in Poisson's ratio. Since the gathers are noisy and contain fewer traces than the synthetic CDP profile shown in Figure 10.8, they were used to create a common offset stack. The geometry of this st. ack is described in

Ostrander's paper, and the resulting gathers are often referred to' as Ostrander gathers. Traces within a CDP/offset window were gathered and stacked, resulting in increased signal to noise. Figure 10.10 shows a display of the logs, synthetic model, and common offset stack. The mismatch in amplitudes is now obvious.

ß

ß Next, the amplitudes of the event on the contanon offset stack

corresponding to the event displayed in Figure 10.7 were picked. The event above the anomalous layer was also picked. The picks were then used along

with the computed amplitude 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 seismic picks to the magnitude of the actual 'amplitudes.

Figure 10.9 CDP gathers from a seismic dataset corresponding to synthetic shown in Figure 10.8.

Part 10 -Amplituae versus Offset Inversion Page 10 - 12

I ntroducti on to Sei stoic I nversi on Methods B ri an Russel 1

I NVER$] ON FULL HOIIEL

TESTL TESTL I)EN$I $-WI:IU POI$$ _ us/m u•/• g/cc us/•

I t ß ß I I I

EU rIO]]EL 1 (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,non offset stack created from the CDP gathers of Figure 10.9 (right panel). T•e left panel shows the blocked well logs from which the synthetic was created.

Part 10 - Amplitude versus Offset Inversion Page l(J- 13


The results of this inversion are shown in Figure 10.11. The figure

shows the change in Poisson's ratio before and after inversion (dashed line before, solid line = after) on the left hand side. On the upper right is shown the match between the observed picks in the upper layer (shown as small squares) and the final theoretical curve {s•own as a solid line). The lower right shows the same thing for the lower layer.

Finally, Figure 10.1Z shows the comparison between the coanon offset stack and the synthetic model after the model has been recomputed with the new amplitude changes from the updated Poisson's ratio. Notice the improvement in 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. bove Laver

. . ..

O•'•'r•

Event (3) Belo4a Laver

O O 0

e.• e S5e O('•set ( m )

Figure 10.11 The results of a GLI inversion between the computed, amplitudes of Figure 10.7 and the picked amplitudes from the conmon offset stack of Figure 10.10. The dashed line on the plot on the left is the Poisson's ratio before inversion, and the solid line is after inversion. The plots on the right show the new computed curves with the picks (squares) superimposed.



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 using the new Poisson's ratio value.


[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 - Vel oci..ty I.n.v. ersi on __

11.1 ! ntroduc ti on

The last

inversion. Alth

acutally fit in

been dtscussing

topic to be discussed in 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 techniques have 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.



KII. OFEœT KILOFEET

-2 -1 0 I 2 -2 -I 0 I

,

(a) (b)

2

Figure 11.1 The effect of the velocity inversion method on 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 )



11.2 Theory an d .Examples

The velocity inversion procedure is referred to as an inverse scattering

problem, in which the interior of the earth is mapped by inver. ting 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 downward continuation of the wave equation.

The initial work in this area was done by Norman Bleistein and Jack Cohen

at the University of Denver. In their initial paper, Cohen and 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 method solves for only the reflection strength of

the mapped interfaces.

In their more recent 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 to 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 mathematics between the velocity inversion methods and migration methods, the results look very similar to those of

migration. For example, Figure 11.1, from Bleistein and Cohen (1982), shows the input an• inverted result for a g-D buried focus. Note that, as in

migration, the "bow-tie" has been imaged to 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,• 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 Page 11 - 5


Figure 11.2, also from Bleistein and Cohen (1982), shows tl•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 an example of applying the velocity inversion technique to 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, this technique cannot be classed with the other methods which

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 on Page 11 - 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-dimensional model, 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 impedance profile. Flowcharts for these methoUs are

shown in Figures 12.1, 12.2, and 12.3. Let us initially summarize the

advantages and disadvantages of the three methods of single trace inversion which have been discussed:

(1) Recursire Inversion _ ! ,• _ - •

A dv an 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 component must be derived from a separate source.

(2) Spar. se-SP.i kg_.Invers. ion

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 IMPEDANCE

IMPEDANCE

SCALE TO VELOCITY

AND DEPTH

DISPLAY

Fiõure 12.1 Band-Limited Inversion (Recursive)

Part 12 - Summary

ß ß ,

Page 12- 3

Introduction to Seismic Inversion Methods Brian Russell ,

Dô sadvantages'

(i) Statistical nature of the sparse-spike methods used are subject to

probl eros i n noisy Uata.

Final output lacks much of the fine detail seen on recursively inverted data. Only the "blocky" component is inverted.

(3) Model -Base• I nver si on

Advantages'

(i) A complete solution, including low frequency information, 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 complete solution is arrived at iteratively and may never be reached ( i.e. the sol ution may not converge).

(ii)

The

velocity inversion, and amplitude versus offset inversion. methods, but cannot be compared directly with the three

(comparing apples 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 model correctly 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


INTRODUCE

LINEAR CONSTRAINTS

EXTRACT

SPARSE REFLECTIVITY

INVERT TO IMPEDANCE

I vELøcmTY ! AND DL_•.••_•.

m i i m i

Fiõure 12.2 Broad-Band Inversion (Sparse-Spike)

Part 12 - Sugary Page 12-


The velocity inversion method was shown to 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 methods will 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

Introduction to Seismic Inversion Methods Brian Russell 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'•- •'" .... Summary 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 from 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' SEG Continuing 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, D., 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 Canauian CSEG meeting, Ca.!gary, Alberta.


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-dimensional modeling technique and fundamentals of three-dimensional frequency-Uomain migration:

Geophysics, v. 47, p. 1627-1644.

Jones, I.F., and Levy, S., 1987, Signal=to-noise ratio enhancement in

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 from 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. 163 7-1648.


Introduction %o Seismic Inversion Methods Brian Russell

Russell, B.H., and Lindseth, R.O., 1982, The information content of synthetic sonic logs - A frequency domain approach- presented at the 1982 convention of the EAEG, Cannes, France.

Shuey, R.T., 1985, A simplification of the Zoeppritz equations: Geophysics, v. 50, p. 609-614.

Stewart, R.R., and Chiu, S.K.L., 1986, Tomography-based imaging of a heavy oil reservoir using well-logs, VSP and 3-D Seismic data- 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, J.F., 1979, Deconvolution with the L1 norm: Geophysics, v. 44, p. 39-52.

Trei tel, S., and Robinson, E.A., 1966, The design of hi gh-resol ution digital filters ß IEEE Transactions on Geo•cience Electronics, v. GE-4, No. 1, p. 25-38.

Walker, C., and Ulrych, T.J., 1983, Autoregressive recovery of the

acoustic impedance- Geophysics, v. 48, p. 1338- 1350.

Waters, K.H.,

exploration

1981, Reflecti on seismol ogy, a tool

(second edition)- Wiley, New York.

for energy resource

Western Geophysical Co., Brochure.

1983, Sei smi c L i thol ogi c MoUel i ng: Technical

Widess, M.B., 1973,

p. 1176- 1180.

How thin is a thin bed?- Geophysics, v. 38,


ISBN: 978-0-931830-65-5 90000

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russell - introduction to seismic inversion methods

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