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Transcript of 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
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 2 - The Convolutional Model Page 2 - 10
I ntroducti on to Sei smic I nversi on Methods Brian Russell
It
tl I I I
LAMBD^•0.01
i I I
•11 I 511 t •tl I
(VERY SPARSE)
11
311 I
LAMBDA--O. 1
4# I 511 I #1 I
TZIIE (KS !
1,1
::. •"• •'•;'" ' "";'•'l•' "••'r'• LAMBDAI0.5
- • "(11 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
Part 2 - The Convolutional Model Page 2 - 11
Introduction to Seismic Inversion ,Methods Brian Russell
2.3 The Seismic Wavelet -- _ ß • ,
Zero Phase and Constant Phase Wavelets m _ m _ m ß m u , L m _ J
The 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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 2 - The Convolutional Model Page 2 - 16
(a)
Introduction to Seismic Inversion Methods Brian Russell
!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).
Part 2 - The Convolutional Model Page 2 - 17
Introduction to Seismic Inversion Methods Brian Russell
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
Part 2 - The Convolutional Model Page 2 - 18
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
(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 -
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Part 4 - Seismic Processing Considerations Page 4 - 2
Introduction to Seismic Inversion Methods Brian Russell
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
Part 4 - Seismic Processing Considerations Page 4 - 3
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).
, ,
Part 4 - Seismic Processing Considerations Page 4 - 5
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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 cor- rections.
Fig. 4.3. Stacks with and without surface-consi stent
corrections. (Taner anu Koehler, 1981).
Part 4 - Seismic Processing Considerations
ß ,
Page 4 - 7
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
(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.
Part 4 - Seismic Processing Considerations Page 4 - 10
Introduction to Seismic Inversion Methods Brian Russell
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'
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 4 - Seismic Processing Considerations Page 4 - 12
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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)
Part 4 - Seismic Processing Considerations Page 4 - 16
Introduction to Seismic Inverslon Methods Brian Russell
PART 5 - RECURSIVE INVERSION - PRACTICE _ _ _ _ _ .. . .• ,• _ _
Part 5 - Recursive Inversion - Practice Page 5 - i
Introduction to Seismic Inversion Methods Brian Russell
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.
Introduction to Seismic Inversion Methods Brian Russell
117 112 1e9 leS 1ol 92 93
i• •11• I I Ittltl =:::•:::::::-•--lll[l•1111t• •'• •1tlllttllltl Ill•l 1t 1 l !IIit! 'ti ! llltfltll!!l• I I !1!t•n•'i •l, , •l••J• •":•!• •'• •" • --'' '
_ __ ..• - ,•, _•. • • 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 .,,, • .•,• • .... ..,• ....... : ..•..... •.•.•.. :,.. , .... , ,. , .............
, , •.•- -. •- (• ••' •'•:; •, / .................... . .... -(•-•( •.•,••(•'••'•"•:•"•'•7 '• . , . • •'•,:•'•' • x•{ , - ,,
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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)
Part 5 - Recursive Inversion - Practice Page 5 - 7
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 5 - Recursive Inversion - Practice Page 5 - 8
Introduction to Seismic Inversion Methods Brian Russell
(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
Introduction to Seismic Inversion Methods Brian Russell
(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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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)___
Part 5 - Recursive Inversion - Practice Page 5 - 13
Introduction to Seismic Inversion Methods Brian Russell
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)
Introduction to Seismic Inversion Methods Brian Russell
(a)
SEISMIC SECTION-CYCUC & TRANSITIONAL STRATA
i 1-3500 ß
Part 5 - Recursive Inversion - Practice Page 5 - 15
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6 - Sparse-spike Inversion 6- 2
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
! , ,,
Part 6 - Sparse-spike Inversion 6- 5
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6 - Sparse-spike Inversion 6- 6
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.
Part 6 - Sparse-spike Inversion 6- 8
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.
Part 6 - Sparse-spike Inversion 6- 9
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6 - Sparse-spi ke Inversion 6- 10
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6 - Sparse-spike Inversion 6- 12
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6 - Sparse-spike Inversion 6- 16
Introduction to Seismic Inversion Methods Bri an Russel 1
Figure 6.12 Creation of a seismic model from a sonic-log.
Figure 6.13 Inversion result from Figure 6.12. •- _ ! ...... ii__ - - i - •_! mm i i i ß i i ! It_l I
Part 6 - Sparse-spi•e Inversion 17
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6 - Sparse-spi ke Inversion 6- 18
Introduction to Seismic Inversion Methods Brian Russell
* !
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.
Part 6 - Sparse-spike Inversion 6- 19
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.
Part 6 - Sparse-spike Inversion 6- 20
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
Introduction to Seismic Inversion Methods Brian Russell
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:• .
Part 6- Sparse-spike Inversion 6- 22
Introduction to Seismic Inversion Methods Brian Russell
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 (•).
Part 6- Sparse-spike Inversion 6- 23
Introduction to Seismic Inversion •.le.thods Brian Russell
Mathematically, the previous equation is considered as the constraint to
the inversion problem. Now, the layered earth model equates to a "blocky" impedance 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.
Part 6 - Sparse-spike Inversion 6- 24
Introduction to Seismic Inversion Methods Brian Russell
(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).
Part 6- Sparse-spike Inversion 6- 26
Introduction to Seismic Inver. s,ion Methods Brian Russell
TSN
1,2
tO0 90 80 70 60 50 40 30 20 tO
1,3
1,4
1,5
1,6
1,7
1,:8
.2,0
2ø2
Figure 6.20 Input seismic data section to L1 Norm inversion. (O1 denburg, 1985'
Part 6 - Sparse-spike Inversion 6- 27
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6 - Sparse-spike Inversion 6- 28
Introduction to Seismic Inversion Methods Brian Russell
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?
Part 6- Sparse-spike Inversion 6- 30
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 6- Sparse-spike Inversion 6- 31
Introduction to Seismic Inversion Methods Brian Russell
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Introduction to Seismic Inversion Methods Brian Russell ***__********************************************************
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Part 6- Sparse-spike Inversion 6- 33
Introduc%ion [o Seismic Inversion Meltotis Brian Russell
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Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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 '•-
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
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
Part 8 - Model-based Inversion Page 8 - 4
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 -
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 8 - Model-based Inversion Page 8 - 6
Introduction to Seismic Inversion Methods Brian Russell
• ' • 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.
Part 8 - Model-based Inversion Page 8 - 8
Introduction to Seismic Inversion Methods Brian Russell
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)
Part 8 - Model-based Inversion Page 8 - 9
! 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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 8 - Model-based Inversion Page 8 - 12
Introduction to Seismic Inversion Methods Brian Russell
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)
Part 8 - Model-based Inversion Page 8 - 13
Introduction to Seismic Inversion Methods Brian Russell
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
Part 8 - Model-based Inversion Page 8 - 14
Introduction to Seismic Inversion Me.thods Brian Russell
is of little
overde termi neU
problems:
In the above equations we had the same 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
Introduction to Seismic Inversion Methods Brian Russell
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
Part 8 - Model-based Inversion Page 8 - 16
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 .
Part 8 - Model-based Inversion Page 8 - 17
Introduction to Seismic Inversion Methods Br•an Russell
PART g - TRAVELTIME INVERSION
Part g - Traveltime Inversion
ml ß i ii
Page 9 - I
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
(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'
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 9 - Travel time Inversion Page 9 - 8
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 9 - Travel time Inversion Page 9 - 10
Introduction to Seismic Inversion Methods Brian Russell
•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)
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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)
Part 9 - Travel time Inversion Page 9 - 13
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 9 - Travel time Inversion Page 9 - 14
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,
Part 9 - Travel time Inversion Page 9 - 15
1986)
Introduction to Seismic Inversion Methods Brian Russell
PART 10 - AMPLITUDE VERSUS OFFSET INVERSION
Part 10 - Amplitude versus Offset Inversion Page 10 - 1
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 10 - Amplitude versus Offset Inversion Page 10 - 2
Introduction to Seismic Inversion Methods Brian Russell
$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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 10 - Amplitude versus Offset Inversion Page 10 - 4
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 10 - Amplitude versus Offset Inversion Page 10 - 8
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 10 - Amplitude versus Offset Inversion Page 10- 11
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 10 - Amplitude versus Offset Inversion Page 10- 14
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 10 - Amplitude versus Offset Inversion Page 10 - 15
[nt•oduct• on t• $e• sm•c [nve•sJ on Methods B• an Russe• ]
PART 11 - VEI:OCITY INVERSION
Part 11 - Velocity Inversion Page 11 -
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 11 - Velocity Inversion Page 11 -
Introduction to Seismic Inversion Methods Brian Russell
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 )
Part 11 - Velocity Inversion Page 11 -
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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
Introduction to Seismic Inversion Methods Brian Russell
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-
Introduction to Seismic Inversion Methods Brian Russell
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 -
Introduction to Seismic Inversion Methods Brian Russell
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.
Part 12 - Summary Page 12 - 9
Introducti on to Sei stoic Inversi on Methods Bri an Russell .
Herman, A.J., Anania, R.M., Chun, J.H., Jacewitz, C.A., and
Pepper, R.E.F., 1982, A fast three-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.
Part 12 - Summary Page 12 - 10
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,
Part 12 - Summary Page 12 - 11
ISBN: 978-0-931830-65-5 90000
306
ISBN 978-0-931830-48-8 (Series) ISBN 978-0-931830-65-5 (Volume)