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8/20/2019 Introduction to seismic inversion methods
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Introduction to
Seismic Inversion Methods
Brian H. Russell
Hampson-RusselloftwareServices, td.
Calgary,Alberta
Course Notes Series, No. 2
S. N. Domenico, Series Editor
Society f Exploration eophysicists
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Thesecoursenotes re publishedwithout he normalSEGpeer reviews.
They havenot beenexamined or accuracy nd clarity.Questions r
comments y the readershouldbe referred irectly o the author.
ISBN 978-0-931830-48-8 (Series)
ISBN 978-0-931830-65-5 (Volume)
Library f Congress atalogCardNumber 8-62743
Society f Exploration eophysicists
P.O. Box 702740
Tulsa, Oklahoma 74170-2740
¸ 1988 by the Society f Exploration eophysicists
All rights eserved. hisbookor portions ereofmaynot be reproducedn any ormwithoutpermission
in writing rom he publisher.
Reprinted 990, 1992, 1999, 2000, 2004, 2006, 2008, 2009
Printed in the United States of America
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]:nl;roduc 1 on •o Selsmic I nversion •thods Bri an Russell
Table of Contents
PAGE
Part I Introduction 1-2
Part Z The Convolution Model 2-1
Part 3
Part 4
Part 5
P art 6
P art 7
2.1 Tr•e Sei smic Model
2.2 The Reflection Coefficient Series
2.3 The Seismic Wavelet
2.4 The Noise Component
Recursive Inversion - Theory
3.1 Discrete Inversion
3.2 Problems encountered with real
3.3 Continuous Inversion
data
Seismic Processing Considerati ons
4. I ntroduc ti on
4.2 Ampl rude recovery
4.3 Improvement f vertical
4.4 Lateral resolution
4.5 Noise attenuation
resolution
Recursive Inversion - Practice
5.1 The recursive inversion method
5.2 Information in the low frequency component
5.3 Seismically derived porosity
Sparse-spike Inversi on
6.1 I ntroduc ti on
6.2 Maximum-likelihood aleconvolution and inversion
6.3 The L norm method
6.4 Reef Problem
I nversi on appl ed to Thi n-beds
7.1 Thin bed analysis
7.Z Inversion compari on of thin beds
Model-based Inversion
B. 1 I ntroducti on .
8.2 Generalized linear inversion
8.3 Seismic1 thologic roodellng (SLIM)
Appendix8-1 Matrix applications in geophysics
Part 8
2-2
2-6
2-12
2-18
3-1
3-2
3-4
3-8
4-1
4-2
4-4
4-6
4-12
4-14
5-1
5-2
5-10
5-16
6-1
6-2
6-4
6-22
6-30
7-1
7-2
7-4
8-1
8-2
8-4
8-10
8-14
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Introduction to Seismic Inversion Methods Brian Russell
Part 9 Travel-time Inversion
g. 1. I ntroducti on
9.2 Numerical examplesof traveltime inversion
9.3 Seismic Tomography
Part 10 Amplitude versus offset (AVO) Inversion
10.1 AVO theory
10.2 AVO nversion by GLI
Part 11 Velocity Inversion
I ntroduc ti on
Theory and Examples
Part 12 Summary
9-1
9-2
9-4
9-10
10-1
10-2
10-8
11-1
11-2
11-4
12-1
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Introduction to Seismic •nversion Methods Brian Russell
PART I - INTRODUCTION
Part 1 - Introduction
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Introduction to Seismic Inversion Methods Brian Russell
I NTRODUCTON TO SE SMIC INVERSION METHODS
, __ _• i i _ , . , , • _, l_ , , i.,. _
Part i - Introduction
. .
This course is intended as an overview of the current techniques used in
the inversion of seismic data. It would therefore seemappropriate to begin
by defining what is meant by seismic inversion. The most general definition
is as fol 1 ows'
Geophysical inversion involves mapping the physical structure and
properties of the subsurface of the earth using measurementsmade on
the surface of the earth.
The above definition is so broad that it encompasses irtually all the
work that is done in seismic analysis and interpretation. Thus, in this
course we shall primarily 'restrict our discussion to those inversion methods
which attempt to recover a broadband pseudo-acoustic impedance log from a
band-1 imi ted sei smic trace.
Another way to look at inversion is to consider it as the technique for
creating a model of the earth using the seismic data as input. As such, it
can be considered as the opposite of the forwar• modelling technique, which
involves creating a synthetic seismic section based on a model of the earth
(or, in the simplest case, using a sonic log as a one-dimensional model). The
relationship between forward and inverse modelling is shown n Figure 1.1.
To understandseismic inversion, we must first understand he physical
processes involved in the creation of seismic data. Initially, we will
therefore look at the basic convolutional model of the seismic trace in the
time and frequencydomains, onsidering the three componentsf this model:
reflectivity, seismic wavelet, and noise.
Part I - Introduction
_ m i --.
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Introduction to Seismic InverSion Methods Brian Russell
FORWARDMODELL NG
i m ß
INVERSEMODELLINGINVERSION)
, ß ß _
Input'
Process:
Output'
EARTHODEL
,
MODELLING
ALGORITHM
SEISMIC RESPONSE
i m mlm ii
INVERSION
ALGORITHM
EARTHODEL
ii
Figure1.1 Fo.•ard andsInverseodel,ling
Part I - Introduction
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Introduction. to Seismic Inversion Methods Brian l•ussel 1
Once we have an understanding of these concepts and the problems which
can occur, we are in a position to look at the methodswhich are currently
ß
used to invert seismic data. These methods are summarized n Figure 1.2. The
primary emphasis of the course will be
the ultimate resul.t, as was previously
on poststack seismic inversion where
o
Oiscussed, is a pseudo-impeaance
section.
We will start by looking at the most contanonmethods of poststack
inversion, which are based on single trace recursion. To better unUerstand
these recurslye inversion procedures, it is important to look at the
relationship between aleconvolution anU inversion, and how Uependent each
method is on the deconvolution scheme Chosen. Specifically, we will consider
classical "whitening" aleconvolutionmethods, wavelet extraction methods, and
the newer sparse-spike deconvolution methods such as Maximum-likelihood
deconvolution and the L-1 norm metboa.
Another important type of inversion methodwhich will be aiscussed is
model-based inversion, where a geological moael is iteratively upUated to finU
the best fit with the seismic data. After this, traveltime inversion, or
tomography,will be discussedalong with several illustrative examples.
After the discussion on poststack inversion, we shall move nto the realm
of pretstack. These methoUs,still fairly new, allow us to extract parameters
other than impedance, such as density and shear-wave velocity.
Finally, we will aiscuss the geological aUvantages anU limitations of
each seismic inversion roethoU, ooking at examples of each.
Part 1 - Introduction
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Introduction to Selsmic nversion Methods Brian Russell
SESMI I NV RSI N
.MET•OS ,,
POSTSTACK
INVERSION
PRESTACK
INVERSION
MODEL-BASEDRECURSIVE
INVERSION,INVE ION
- "NARROW
BAND
TRAVELTIME
INVERSION
TOMOGRAPHY)
SPARSE-
SPIKE
WAVFEL
NVERSIOU
LINEAR
METHODS
,,
i i --
I METHODS
Figure 1.2
A summaryof current inversion techniques.
Part 1 - Introuuction
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Introduction to Seismic Inversion Methods Brtan Russell
PART - THECONVOLUTIONALODEL
Part 2 - The Convolutional Model
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Introduction to Seismic Inversion Methods Brian Russell
Part 2 - The Convolutional Mooel
2.1 Th'e Sei smic Model
The mostbasic and commonlysed one-Oimensionalmoael for the seismic
trace is referreU to as the convolutional moOel, which states that the seismic
trace is simply he convolutionof the earth's reflectivity with a seismic
source function with the adUltion of a noise component. In equation form,
where * implies convolution,
s(t) : w(t) * r(t) + n(t)s
where
and
s (t) = the sei smic trace,
w(t) : a seismic wavelet,
r (t) : earth refl ecti vi ty,
n(t) : additive noise.
An even simpler assumptions to consiUer he noise componento be zero,
in which case the seismic tr•½e is simply the convolution of a seismic wavelet
with t•e earth ' s refl ecti vi ty,
s(t) = w{t) * r(t).
In seismic processingwe deal exclusively with digital data, that is,
data sampledt a constantime interval. If weconsiUerhe relectivity to
consist of a reflection coefficient at each time sample (som• of which can be
zero), and the wavelet to be a smooth function in time, convolutioncan be
thoughtof as "replacing"eachreflection. coefficient with a scaledversion of
the wavelet and summinghe result. The result of this process s illustrated
in Figures 2.1 and2.Z for botha "sparse" nda "dense" et of reflection
coefficients. Notice that convolution with the wavelet tends to "smear" the
reflection coefficients. That is, there is a total loss of resolution,which
is the ability to resolve closely spacedreflectors.
Part 2 - The Convolutional Model
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Introduction to Seismic Inversion Nethods Brian Russell
WAVELET:
(a) ' * • : -' :'
REFLECTIVITY
Figure 2.1
TRACE:
Convolution f a wavelet with a
(a) •avelet. (b) Reflectivit.y.
sparse"reflectivity.
(c) Resu ing Seismic Trace.
(a)
(b')
.
i
:
: :
i
i ,
: i
i i
'?t *
c
o o o o o
Fi õure 2.2
Convolution of a wavelet with a sonic-derived "dense"
reflectivity. (a) Wavelet. (b) Reflectivity. (c) Seismic Trace
, i , ß .... , m i i L _ - '
Par• 2 - The Convolutional Model
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Introduction to Seismic Inver'sion Methods Brian Russell
An alternate, but equivalent, way of looking at the seismic trace is in
the frequency domain. If we take the Fourier transform of the previous
ß
equati on, we may write
S(f) = W(f) x R(f),
where
S(f) = Fourier ransformf s(t),
W(f) = Fourier transform of w(t),
R(f) = Fourier transform of r(t),
ana f = frequency.
In the above equation we see that convolution becomesmultiplication in
the frequency domain. However, the Fourier transform is a complex function,
and it is normal to consiUer the amplitude and phase spectra of the individual
components. The spectra of S(f) may then be simply expressed
esCf)= ew
where
(f) + er(f),
I •ndicatesmplitudepectrum,nd
0 indicates phase spectrum. .
In other words, convolution involves multiplying the amplitude spectra
and adding the phase spectra. Figure 2.3 illustrates the convolutional model
in the frequency domain. Notice that the time Oomainproblem of loss of
resolution becomesone of loss of frequency content in the frequency domain.
Both the high and low frequencies of the reflectivity have been severely
reOuceo by the effects of the seismic wavelet.
Part 2 - The Convolutional Mooel
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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
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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 i onal
i ropedance o re
impedances by
coefficient at
fo11 aws:
'The reflection coefficient series (or reflectivity, as it is also called)
describedn thepreviousections oneof the fundamentalhysicaloncepts
in the seismic method. Basically, each reflection coefficient maybe thought
ponse of the seismic wavelet to an acoustic impeUance change
th, where acoustic impedance is defined as the proUuct of
velocity and Uensity. Mathematically, converting from acoustic
flectivity involves dividing the difference in the acoustic
the sum of the acoustic impeaances. This gives t•e reflection
the boundary between the two layers. The equation is as
•i+lVi+l - iVi Zi+l- Z
i • i+1
where
and
r = reflection coefficient,
/o__density,
V -- compressional velocity,
Z -- acoustic impeUance,
Layer i overlies Layer i+1.
Wemust also convert from depth to time by integrating the sonic log
transit times. Figure •.4 showsa schematicsonic log, density log, anU
resulting acoustic impedance or a simplifieU earth moael. Figure 2.$ shows
the resultof convertingo thereflection oefficienteries ndntegrating
to time.
It should be pointed out that this formula is true only for the normal
incidence case, that is, for a seismic wave striking the reflecting interface
at right angles to the beds. Later in this course, we shall consider the case
of nonnormal inciaence.
Part 2 - The Convolutional
Model Page 2 - 6
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Introduction to Seismic Inversion Methods Brian Russell
STRATIGRAPHIC SONICLOG
SECTION •T (•usec./mette)
4OO
SHALE ..... DEPTH
ß ß ß ß ß ß
SANOSTONE . . - .. ,
'
I _1
UMESTONE I I I I 1
LIMESTONE
2000111
30O 200
I
3600 m/s
_
v--
V--3600
V= 6QO0
I
loo 2.0 3.0
,
OENSITY LOG.
ß •
Fig. 2.4. Borehole ogMeasurements.
mm mm rome m .am
,mm mm m ----- mm
SHALE ..... OEPTH
•--------'-
SANDSTONE . . ... ,
I 11 I1
UMESTONE I 1 I I I II
i I 1 i I i 1000m
SHALE •.--._--.---- • •.'•
LIMESTONE
2000 m
ACOUSTIC
IMPED,M•CE (2•
(Y•ocrrv x OEaSn•
REFLECTWrrY
V$ OEPTH
VS TWO.WAY
TIME
20K -.25 O Q.2S -.25 O + .2S
I I v ' I
- 1000 m -- NO
,• , ..
- 20o0 m
I SECOND
Fig. 2.5. Creation of Reflectivity Sequence.
Part g - The Convolutional Model
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IntroductJ on 1:o Sei stoic Inversion Herhods Bri an Russell
Our best method of observing seJsm•c impedance and reflectivity is •o
derlye them from well log curves. Thus, we maycreate an impedancecurve by
multiplying together •he sonic and density logs from a well. Wemay hen
computehe reflectivlty by using •he formula shown arlier. Often, we do not
have the density log available• to us and must makedo with only the sonJc. The
approxJmatJonof velocJty to •mpedances a reasonable approxjmation, and
seemso holdwell for clas;cics and carbonates not evaporltes, however).
Figure 2.6 shows he sonic and reflectJv•ty traces from a typJcal Alberta well
after they have been Jntegrated to two-way tlme.
As we shall see later, the type of aleconvolution and inversion used is
dependent on the statistical assumptionswhich are made about the seismic
reflectivity and wavelet. Therefore, howcan we describe the reflectivity seen
in a well? The traditional answer has always been that we consider the
reflectivity to be a perfectly random sequence and, from Figure •.6, this
appears to be a good assumption. A ranUomsequencehas the property that its
autocorrelation is a spike at zero-lag. That is, all the components f the
autocorrelation are zero except the zero-lag value, as shown n the following
equati on-
t(Drt = ( 1 , 0 , 0 , ......... )
t
zero-lag.
Let us test this idea on a theoretical random sequence, shown n Figure
2.7. Notice that the autocorrelation of this sequence has a large spike at
ß
the zeroth lag, but that there is a significant noise component at nonzero
lags. To have a truly random sequence, it must be infinite in extent. Also
on this figure is shown the autocorrelation of a well log •erived
reflectivity. Wesee that it is even less "random" han the randomspike
sequence. Wewill discuss this in more detail on the next page.
Part 2 - The Convolutional Model
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IntroductJon to Se•.s=•c Inversion Methods Br•an Russell
RFC
F•g. 2.6. Reflectivity equenceerivedrom onJclog.
RANDOM SPIKE SEQUENCE
WELL LOG DERIVED REFLECT1vrrY
AUTOCORRE•JATIONF RANDOMSEQUENCE
AUTOCORRELATION OF REFLECTIVITY
Fig. 2.7.
Autocorrelat4ons of random and well log
der4ved pike sequences.
Part 2 - The Convolutional Model
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Introductlon to Sei smic Inversion Methods Brian Russel 1
Therefore, the true earth reflectivity cannot be consideredas being
truly random. For a typical Alberta well we see a number f large spikes
(co•respondingo major ithol ogic change) ticking up above he crowd.A good
way to describe his statistically is as a Bernoulli-Gaussianequence. The
Bernoulli part of this term implies a sparseness n the positions of the
spikes and the Gaussianmplies a randomnessn their amplitudes. Whenwe
generatesuch a sequence, there is a term, lambda, which controls the
sparseness of the spikes. For a lambdaof 0 there are no spikes, and for a
lambda f 1, the sequences perfectly Gaussian in distribution. Figure 2.8
shows a number of such series for different values of lambda. Notice that a
typical Alberta well log reflectivity wouldhavea lambdavalue in the 0.1 to
0.5 range.
Part 2 - The Convolutional Model
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I ntroducti on to Sei smic I nversi on Methods Brian Russell
It
tl I I I
LAMBD^•0.01
i I I
•11 I 511 t •tl I
(VERY SPARSE)
11
311 I
LAMBDA--O. 1
4# I
511 I #1 I
TZIIE (KS
1,1
::."• •'•;'" "";'•'l•'••'r'•
LAMBDAI0.5
- "(11
X#E (HS)
LAMBDA-- 1.0 (GAUSSIAN:]
EXAMPLESOF REFLECTIVITIES
Fig. 2.8. Examplesof reflectivities using lambda
factor to be discussed in Part 6.
, , m i ß i
Part 2 - The Convolutional Model
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Introduction to Seismic Inversion ,Methods Brian Russell
2.3 The Seismic Wavelet
-- _ ß • ,
Zero Phase and Constant Phase Wavelets
m _ m _ m ß m u , L m _ J
The assumptionha.t there is a single, well-defined wavelet which is
convolved with the reflectivity to produce he seismic trace is overly
simplistic. Morerealistically, the wavelet is both time-varying and complex
in shape. However, he assumption f a simple wavelet is reasonable, and in
this section we shall consider several types of wavelets and their
characteristics.
First, let us consider the Ricker wavelet, which consists of a peak and
two troughs, or side lobes. The Ricker wavelet is dependentonly on its
dominant frequency, that is, the peak frequencyof its a•litude spectrum or
the inverse of the dominantperiod in the time domain the dominantperiod is
found by measuringhe time from trough o trough). TwoRicker wave'lets are
shown n Figures 2.9 and 2.10 of frequencies 20 and 40 Hz. Notice that as the
anq•litude spectrumof a wavelet .is broadened, he wavelet gets narrower in the
timedomain,ndicating n ncrease f resolution.Ourultimatewaveletwould
be a spike, with a flat amplitude spectrum. Sucha wavelet is an unrealistic
goal in seismic processing, but one that is aimed or.
The Rtcker wavelets of Figures 2.9 and 2.10 are also zero-phase, or
perfectly symmetrical. This is a desirable character.stic of wavelets since
the energy is then concentrated at a positive peak, and the convol'ution of the
wavelet with a reflection coefficient will better resolve that reflection. To
get an idea of non-zero-phase wavelets, consider Figure 2.11, where a Ricker
wavelet has been rotated by 90 degree increments, and Figure 2.12, where the
samewavelet has been shifted by 30 degree increments. Notice that the 90
degree rotation displays perfect antis•nmnetry, whereas a 180 degree shift
simply inverts the wavelet. The 30 degree rotations are asymetric.
Part 2 - The Convolutional Model
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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 increment
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Introduction to Seismic Inversion Methods Brian Russell
Of course, a typical seismic wavelet contains a larger range of
frequencies than that shownon the Ricker wavelet. Consider the banapass
fil•er shownn Figure 2.13, where we have passed a banaof frequencies
between 15 and 60 Hz. The filter has also had cosine tapers applied between 5
and 15 Hz, and between60 and 80 Hz. The taper reduces the "ringing" effect
that would be noticeable if the wavelet amplitude spectrum was a simple
box-car. The wavelet of Figure 2.13 is zero-phase, and would be excellent as
a stratigraphic wavelet. It is often referred to as an Ormsby avelet.
Minimum Phase Wavelets
The concept of minimum-phase s one that is vital to aleconvolution, but
is also a concept that is poorly understood. The reason for this lack of
understanding is that most discussions of the concept stress the mathematics
at the expense of the physical interpretation. The definition we
use of minimum-phases adapted from Treitel and Robinson (1966):
For a given set of wavelets, all with the same amplitude spectrum,
the minimum-phaseavelets the onewhich as he sharpesteading
edge. That is, only wavelets which have positive time values.
The reason that minimum-phase concept is important to us is that a
typical wavelet in dynamite work is close to minimum-phase. Also, the wavelet
from the seismic instruments is also minimum-phase. The minimum-phase
equivalent of the 5/15-60/80 zero-phase wavelet is shown n Figure 2.14. As
in the aefinition used, notice that the minimum-phase avelet has no component
prior to time zero and has its energy concentrated as close to the origin as
possible. The phase spectrum of the minimum-wavelet s also shown.
Part 2 - The Convolutional Model
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I•troduct•on to Seistoic nversionNethods. Br•an Russell
ql Re• R Zero Phase I•auel•t
5/15-68Y88 {•
0.6
f1.38 - Trace 1
iii
- e.3e ...... , • ..... ' 2be
1
Trace I
Fig. 2.13. Zero-phase bandpass
wavelet.
Reg 1) min,l• wavelet •/15-68/88 hz
18.00 p Trace I
RegE wayel Speetnm
'188.88 Trace1
0.8
188
Fig.
2.14. Minim•-phase equivalent
of zero-phase wavelet
shown n Fig. 2.13.
_
m,m, i m
Part 2 -Th 'e Convolutional Model
i
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Introduction to Seismic Inversion Methods Brian Russell
Let us now ook at the effect of different wavelets on the reflectivity
function itself. Figure 2.15 a anU b shows a numberof different wavelets
conv6lved with the reflectivity (Trace 1) from the simple blocky model shown
in Figure Z.5. The following wavelets have been used- high
zero-phase (Trace •), low frequency ero-phase Trace ½), high
minimum hase (Trace 3), low frequency minimum phase (Trace 5).
figure, we can make the fol 1owing observations:
frequency
frequency
From the
(1) Low freq. zero-phase wavelet: (Trace 4)
- Resolution of reflections is poor.
- Identification of onset of reflection is good.
(Z) High freq. zero-phase wavelet: (Trace Z)
- Resolution of reflections is good.
- Identification of onset of reflection is good.
(3) Low freq. min. p•ase wavelet- (Trace 5)
- Resolution of reflections i s poor.
- Identification of onset of reflection is poor.
(4) High freq. min. phase wavelet: (Trace 3)
- Resolution of refl ec tions is good.
- Identification of onset of reflection is poor.
Based on the aboveobservations, we wouldhave to consider the high
frequency,ero-phase avelet he best, and he low-frequency, inimumhase
wavelet the worst.
Part 2 - The Convolutional Model
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(a)
Introduction to Seismic Inversion Methods Brian Russell
ql RegR Zer• PhaseUa•elet •,'1G-•1• 14z
F
- •.• [' '
•,3 Recj miniiliumhue ' '
17 .•
q2 RegC ZeroPhase4aue16(' •'le-3•4B Hz
e
q• Reg ) 'minimumhase " •,leJ3e/4eh• '
8
e.e •/••/'•-•"v--,._,,r
e.• ' "s•e'
m ,,
Tr'oce
[b)
Fig.
700
2.15. Convolution of four different wavelets shown
in (a) with trace I of (b). The results are
shown on traces 2 to 5 of (b).
Part 2 - The Convolutional Model
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g.4 Th•N.oi se.Co.mp.o•net
The situation that has been discussed so far is the ideal case. That is,
.
we have interpreted every reflection wavelet on a seismic trace as being an
actual reflection from a lithological boundary. Actually, many of the
"wiggles" on a trace are not true reflections, but are actually the result of
seismic noise. Seismic noise can be grouped under two categories-
(i) Random oise - noise which is uncorrelated from trace to trace and is
•ue mainly to environmental factors.
(ii) CoherentNoise - noise which is predictable on the seismic trace but
is unwanted. An example s multiple reflection interference.
Randomnoise can be thought of as the additive component (t) which was
seen in the equationon page 2-g. Correcting for this term is the primary
reason for stackingour •ata. Stacking actually uoesan excellent job of
removing ranUomnoise.
Multiples, one of the major sources of coherent noise, are causedby
multiple "bounces" f the seismic signal within the earth, as shownn Figure
2.16. They may be straightforward, as in multiple seafloor bounces r
"ringing", or extremelycomplex,as typified by interbed multiples. Multiples
cannot be thoughtof as additive noise and mustbe modeled s a convolution
with the reflecti vi ty.
Figure
generatedby the simple blocky model
this data, it is important that
Multiples may be partially removed
powerful elimination technique.
aleconvolution, f-k filter.ing,
wil 1 be consi alered in Part 4.
2.17
shown on Figure •. 5.
the multiples be
by stacking, but
Such techniques
and inverse velocity stacking.
shows the theoretical multiple sequence which would be
If we are to invert
effectively removed.
often require a more
include predictive
These techniques
Part 2 - The Convolutional Model
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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.
Reflectivi ty sequence f Fig.
and without mul ipl es.
Part 2 - The ConvolutionalModel
2.5. with
.
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PART 3 - RECURSVE INVERSION - THEORY
m•mmm•---' .• ,- - - ' •- - _ - - _- _
Part 3 - Recurstve Inversion - Theory
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•ntroduct•on to SeJsmic Znversion Methods Brian Russell
PART 3 - RECURSIVE INVERSION - THEORY
3.1 Discrete Inversion
, ß , , •
In section 2.2, we saw that reflectivity was defined in terms of
acoustic impedancechanges. The formula was written:
Y•i+lV•+l•iV 2i+ ' Z
ri--yoi'+lVi+l+•iVi - Zi..+lZ
where r -- refl ecti on coefficient,
/0-- density,
V -- compressional velocity,
Z -- acoustic impedance,
and Layer i overlies Layer i+1.
If we have the true reflectivity available to us, it is possible to
recover the a.coustic impedance y inverting the above formula. Normally, the
inverse' formulation is simply written down,but here we will supply the
missing steps for completness. First, notice that:
Also
Ther'efore
Zi+l+Z Zi+ - Z 2 Zi+
I +ri- Zi+lZi + Zi+l2i Zi+lZi
I- ri--
Zi+l+Z Zi+ - Z 2 Zf[
Zi+l+ Z Zi+l+ Z Zi+l+ Z
Zi+l
Z
l+r.
1
1
Part 3 - Recursive nversion- Theory
ill, ß , I
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pv-e-
TIME
(sec]
0.7
REFLECTION
COEFFICIENT
SERIES
RECOVERED
ACOUSTIC
IMPEDANCE
Fig.
3.1,
Applyinghe recursive nversion ormula o a
simple, and exact, reflectivity.
, ß
Part 3 - Recursive
Inversion - Theory
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ntroductt on to Se1 smJc nversi on Methods Brian Russell
•9r• ;• • •;• • • •-•• 9rgr•t-k'k9r9r• •-;• ;• .................................................
Or, the final •esult-
Zi+[= Z
ß
l+r i .
This is called the discrete recursive inversion formula and is the basis
of many current inversion techniques. The formula tells us that if we know
the acoustic impedance f a particular layer and the reflection coefficient at
the base of that layer, we may recover the acoustic impedance of the next
layer. Of course we need an estimate of the first layer impedance o start us
off. Assumewe can estimate this value for layer one. Then
l+rl ,
Z2: l r1
Z3= 112
r
and so on ...
To find the nth impedance rom the first, we simply write the formula as
Figure 3.1 shows the application of the recursive formula to the "
reflection coefficients derived in section 2.2. As expected, the full
acoustic impedance was recovered.
Problems encountered with real data
• ß , m i i • i m
When the recursive inversion formula is applied to real data, we find
that two serious problems are encountered. These problems are as follows-
(i) FrequencyBandl mi ti ng
ß
Referring back to Figure 2.2 we see that the reflectivity is severely
bandlimited when it is convolved with the seismic wavelet. Both the
low frequency components nd the high frequency components re lost.
Part 3 - Recursive Inversion - Theory
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Introduction to Seismic nversion Methods Brian Russell
0.2 0 V•) 'V,•
•R
R = +0.2
V :1000 Where:
--• V,•= 1000 i-o.t
- 1500 m
•ec'.
(a)
- 0.1 '•0.2
R•
R=
{ASSUME•: l)
R•= 0.1
R =+0.2
R: -0.1
Vo=1000m
-'+ ¾1 818m
i•.
Figure 3.2 Effect of banUlimitingon reflectivity, where a) shows
single reflection coefficient, anU (b) showsbandlimited
refl ecti on coefficient.
i i m i m I
I __ ___ i _
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(ii) Noise
The inclusion of coherent or random noise into the seismic 'trace will
make he estimate• reflectivity deviate from the true reflectivity.
To get a feeling for the severity of the above limitations on recursire
inversion, let us first use simple models. To illustrate the effect of
bandlimiting, consider Figure 3.Z. It shows the inversion of a single spike
(Figure 3.2 (a)) anU he inversion of this spike convolved with a Ricker
wavelet (Figure 3.2 (b)). Even with this very high frequency banUwidth
wavelet, we have totally lost our abil.ity to recover the low frequency
componentof the acoustic impedance.
In Figure 3.3 the model derived in section Z.2 has been convolved with a
minimum-phase wavelet. Notice that the inversion of the data again shows a
loss of the low frequency component. The loss of the low frequency component
is the most severe problem facing us in the inversion of seismic data, for it
is extremely Oifficult to directly recover it. At the high end of the
spectrum, we may recover muchof the original frequency content using
deconvolution techniques. In part 5 we will address the problem of recovering
the low frequency component.
Next, consider the problem of noise. This noise may be from many
sources, but will always tend to interfere with our recovery of the true
reflectivity. Figure 3.4 shows the effect of adding the full multiple
reflection train (including transmission losses) to the model reflectivity.
As we can see on the diagram, the recovered acoustic impedancehas the same
basic shape as the true acoustic impedance, but becomes ncreasingly incorrect
with depth. This problemof accumulatingerror is compoundeUy the amplitude
problemns ntroduced by the transmission losses.
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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
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3.3 Continuous Inversion
A logarithmic relationship is often used to approximate the above
formulas. This is derived by noting that we can write r(t) as a continuous
function in the following way:
Or
r(t) - Z(t+dt) Z{t) _ 1 d Z(t)
ß - Z(t+dt) + Z(•) - •' z'(t)
d In Z(t)
r(t) = • dt
The inverse formula is thus-
t
Z(t) Z(O)xpy r(t)dt.
0
The precedingapproximations valid if r(t) <10.3• which is usually the
case. A paper by Berteussen and Ursin (1983), goes into muchmore detail on
the continuous versus discrete approximation. Figures 3.5 and 3.6 from their
paper show hat the accuracy of the continuous inversion algorithm is within
4% of the correct value between reflection coefficients of -0.5 and +0.3.
If our reflection coefficients are in the order of + or - 0.1, an even
simpler pproximationay e made y dropp'inghe logarithmicelationship:
t
1dZ(t) _==• (t)-2'Z(O)r(t) dt
r(t) -• dr VO
Part 3 - Recursive Inversion - Theory
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Introduction to Seismic Inversion Methods Brian Russell
Fig. 3.5
m i ,, ,m I I IIIII
I + gt ½xp26•) Difference
-1.0 0.0 0.14 -0.14
-0.9 0.05 0. ? -0.12
-0.8 0.11 0.20 -0.09
-0.7 0.18 0.25 -0.07
-0.6 0.25 0.30 -0.05
-0.5 0.33 0.37 -0.04 '
-0.4 0.43 0.45 --0.02
-0.3 0.• 0.•5 --0.01
-0.2 0.667 0.670 -0.003
-0.1 0.8182 0.8187 --0.0005
0.0 1.0 1.0 0.0
0.1 1.222 1.221 0.001
0.2 1.500 1.492 0.008
0.3 1.86 1.82 0.04
0.4 2.33 2.23 o.1
0.5 3.0 2.7 0.3
0.6 4.0 3.3 0.7
0.7 5.7 4.1 1.6
0.8 9.0 5.0 4.0
0.9 19.0 6.0 13.0
1.0 co 7.4 •o
Numerical c•pari son of discrete and continuous
i nversi on.
(Berteussen and Ursin, 1983)
Fig. 3.6
$000m PEDANCEOSCR.
r-niL
${300•OFFERENCE
SO0 O FFERENCE SCALEDUP
T •'•E t SECONOS
C•parisonbetweenmpedance•putatins based n a
discrete and a continuous eismic •del.
(Berteussen and Ursin, 1983)
Part 3 - Recursire .Inversion - Theory
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PART 4 - SEISMIC PROCESSINGCONSIDERATIONS
Part 4 - Seismic Processing Considerations
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•ntroduction to Seismic •nvers•on Methods B.r.an Russell
4.1 Introduction
Having ookedat a simple model'of the seismic trace, anu at the
recursire inversion alogorithm n theory, we will now ook at the problem of
processingeal seismiceata in order to get the best results fromseismic
inversion. We may group the key processing roblems nto the following
categories:
( i ) Amp tu de rec overy.
(i i) Vertical resolution improvement.
(i i i ) Horizontal resol uti on improvement.
(iv) Noise elimination.
Amplitudeproblemsare a majorconsideration t the early processing
stagesandwewill look at both deterministicamplitudeecovery ndsurface
consistent residual static time corrections. Vertical resolution improvement
will involve a discussion of aleconvolution and wavelet processing techniques.
In our discussion of horizontal resolution we will look at the resolution
improvementbtained in migration, using a 3-D example.Finally, wewill
consider several approacheso noiseelimination, especially the elimination
of multi pl es.
Simply stateu, to invert our
one-dimensional model given in the
approximationof this model (that
band-limited reflectivity function)
these considerations in minU. Figure 4.1
be useU o do preinversion processing.
seismic data we usually assume the
previous section. And to arrive at an
is, that each trace is a vertical,
we must carefully process our data with
showsa processing flow which could
Part 4 - Seismic Processing Considerations
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INPUT RAW DATA
DETERMINISTIC
AMPLITUDE
CORRECTIONS
,. _•m
mlm
SURFACE-CONS STENT
DECONVOLUTIO,FOLLOWED
Y HI GH RESOIJUTI.ON DECON
i
i
SURFACE-CONS STENT
AMPt:ITUDE ANAL'YSIS
SURFACE-CONSISTENT
STATI CS ANAIJY IS
VELOCITY ANAUYS S
APPbY STATICS AND VEUOCITY
MULTIPLE ATTENUATION
STACK
ß •
MI GRATI ON
,
Fig. 4.1.
Simplfied nversinprocessinglow.
ll , ß ' ß I , _ i 11 , m - -- m _ • • ,11
Part 4 - Seismic Processing Considerations
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Inl;roducl:ion 1:oSeJ mlc Invers1on Nethods BrJan Russell
4.2 Am.p'ltu.de..P,.ecovery
The most dJffJcult job in the p•ocessing of any seismic line is
ß
•econst•uctinghe amplJtudesf the selsmJc•aces as they would avebeenJf
the•e were no dJs[urbJng nf'luences present. We normally make the
simplJficationhat the distortionof the seJsmicmplJtudesay e put into
three main categories'sphe•Jcal ivergence, absorptJon,and t•ansmJssion
loss. Based on a consideration of these three factors, we maywrJte aownan
approximate unctJon or the total earth attenuation-
Thus,
data, the
formula.
At: AO*
b / t) * exp(-at),
where t = time,
A = recordedmplitude,
A = true ampltude,
anU
a,b = constants.
if we estimate the constants in the above equation from the seismic
true amplitudesof the data coulU be recoveredby using the inverse
The deterministic amplitude correction and trace to trace mean
scaling will account or the overall gross changes n amplitude. However,
there may still be subtle (or even not-so-subtle) amplitude problems
associatedwith poorsurface conditions or other factors. To compensateor
these effects, it is often advisable to compute nd apply surface-consistent
gain corrections. This correction involves computing total gain value for
each trace and then decomposinghis single value in the four components
Aij= ixRjxGxMkXj,
where A = Total amplitude factor,
S = Shot component,
R: Receiver component,
G = CDP component, and
M = Offset component,
X = Offset distance,
i,j = shot,receiver pos.,
k = CDP position.
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SURFACE
SUEF'A•
CONS b'TEh[O{
AND
T |tV•E :
,Ri -rE ß
Fig. 4.2.
Surface and sub-surface geometryand
surface-consistent decomposition. (Mike Graul).
, ,
Part 4 - SeismicProcessing onsiderations
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Introduction to Seismic Inversion Methods Brian Russell
Figure 4.g (from Mike Graul's unpublished course notes) shows the
geometry sed or this analysis. Notice that the surface-consistent tatics
anti aleconvolution roblem re similar. For the statics problem, the averaging
can be •1oneby straight summation.For the amplitude problemwemust
transform the above equation into additive form using the logarithm:
InAij=nS + nRj+ nG + nkMijX•.
The problem can then be treated exactly the same way as in the statics
case. Figure 4.3, fromTaner anti Koehler (1981), shows he effect of doing
surface consistent amplitude and statics corrections.
4.3 I•mp.ov. ment_.[_Ver..i.ca.1..Resolutin
Deconvol ution is a process by which an attempt is made to remove the
seismic wavelet from the seismic trace, leaving an estimate of reflectivity.
Let us first discuss the "convolution"part of "deconvolution" starting with
the equation for the convolutional model
In the
st--wt* r t where
frequency domain
st= the seismic race,
wt= the seismicwavelet,
rt= reflection coefficient series,
* = convolution operation.
S(f) • W(f) x R(f) .
The deconvol ution
procedure and consists
reflection coefficients.
fol 1owlng equati on-
rt: st* o
process is simply the reverse of the convolution
of "removing" the wavelet shape to reveal the
We must design an operator to do this, as in the
whereOr--operator- inverse f w .
Part 4 - Seismic Processing Considerations
,
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Introduction to Seismic Inversion Methods Brian Russell
ii 11
ß 1'
i
ii
'..,•' •, ," " " ß d.
Preliminarytack et'oreurfaceonsistenttaticnd mpli-
lude corrections.
ß Stockwith surface onsistenttatic nd amplitude or-
rections.
Fig. 4.3.
Stacks with and without surface-consi stent
corrections. (TaneranuKoehler,1981).
Part 4 - Seismic Processing Considerations
ß ,
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Introduction to Seismic Inversion Methods Brian Russell
In the frequency domain, his becomes
R(f) = W(f) x 1/W(f) .
After this extremely simple introduction, it may appear that the
deconvolution roblemshouldbe easy to solve. This is not the case, and the
continuing research nto the problem testifies to this. There are two main
problems. Is our convolutionalmodel orrect, and, if the model s correct,
can we derive the true wavelet from the data? The answer to the first
question s that the convolutionalmodel ppears o be the best modelwe have
come p with so far. The main problem is in assuminghat the wavelet does
not vary with time. In our discussionwe will assumehat the time varying
problem s negligible within the zoneof interest.
The secondproblem s much more severe, since it requires solving the
ambiguousproblem f separatinga wavelet and reflectivity sequencewhenonly
the seismic trace is known. To get around this problem, all deconvolution or
wavelet estimation programsmakecertain restrictive assumptions, ither about
the wavelet or the reflectivity. There are two classes of deconvolution
methods: those which make restrictive phase assumptions and can be considered
true wavelet processing echniques only when hese phase assumptions re met,
and those which do not make restrictive phase assumptions and can be
considered as true wavelet processingmethods. In the first category are
(1) Spiking deconvolution,
(2) Predictive deconvolution,
(3) Zero phase deconvolution, and
(4) Surface-consi stent deconvoluti on.
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(a)
Fig. 4.4
A comparison of non surface-consistent and surface-consistent
decon on pre-stack data. {a) Zero-phase deconvolution.
{b) Surface-consistent soikinB d•convolution.
(b),
Fig. 4.5 Surface-consistent decon comparisonafter stack.
(a) Zero-phase aleconvolution. (b) Surface-consistent
deconvol ution.
'--'- , ß , ,• ,t ß ß _ , , _ _ ,, , ,_ , ,
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In the second category are found
(1) Wavelet estimation using a well
(Hampsonnd Galbraith 1981)
1og (Strat Decon).
(2) Maximum-1 kel ihood aleconvolution.
(Chi et al, lg84)
Let us
surface-consi stent
surface-consi stent
components. We
di recti ons- common
illustrate the effectiveness of one of. the methods,
aleconvolution. Referring to Figure 4.•, notice that a
scheme involves the convolutional proauct of four
must therefore average over four different geometry
source, common receiver, common depth point (CDP), and
con, on offset (COS). The averaging must be performed iteratively and there
are several different ways to perform it. The example in Figures 4.4 ana 4.5
shows an actual surface-consi stent case study which was aone in the following
way'
(a) Compute he autocorrelations of each trace,
(b) average the autocorrelations in each geometry eirection to get four
average autocorrel ati OhS,
(c) derive and apply the minimum-phasenverse of each waveform, and
(•) iterate through this procedure to get an optimum esult.
Two points to note when you are looking at the case study are the
consistent definition of the waveformn the surface-consistent pproachan•
the subsequent improvementof the stratigraphic interpretability of the stack.
We can compareall of the above techniques using Table 4-1 on the next
page. The two major facets of the techniques which will be comparedare the
wavelet estimation procedure and the wavelet shaping procedure.
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Table 4-1
Comparison of Deconvolution MethoUs
m ß ß m
METHOD
Spiking
Deconvol ution
Predi cti ve
Deconvol uti on
Zero Phase
Deconvol utton
Surface-cons.
Deconvolution
Stratigraphic
Deconvol ution
Maximum-
L ik el i hood
deconvol ution
WAVELET ESTIMATION
Min.imumhase assumption
Randomefl ecti vi ty
assumptions.
No assumptions about
wavelet•
Zero phaseassumption.
Randomefl ectt vi ty
assumption.
Minimum r zero phase.
Randomeflecti vi ty
assumption.
No phase assumption.
However, well must match
sei smi c.
No phase assumption.
Sparse-spike assumption.
WAVELETSHAPING
Ideally shaped o spike.
In practice, shaped o minimum
phase,higher requency utput.
Does not whiten data well.
Removeshort and ong period
multiples. Does not affect
phase f wayel t for long lags.
..1_, m
Phase is not altered.
Amplitude spectrum $
whi tened.
Canshape o desired output.
Phase haracter s improved.
Ampl rude spectrum i s
whitenedess than in single
trace methods.
Phase of wavelet is zeroed.
Amplitudepectrumot
whi tened.
Phase of wavelet is zeroed•
Amp rude spectrum s
whi tened.
Part 4 - Seismic Processing Considerations
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4.4 Lateral Resol uti on
The complete three-dimensional (3-D) diffraction problems shown n
Figure 4.6 for a modelstudy taken fromHerman, t al (1982). Wewill look'at
line 108, which cuts obliquely across a fault and also cuts across a reef-like
structure. Note that it misses the second reef structure.
Figure 4.7 shows the result of processing the line. In the stacked
section we maydistinguish two types of diffractions, or lateral events which
do not represent true geology. The first type are due to point reflectors in
the plane of the section, and include the sides of the fault and the sharp
corners at the base of the reef structure which was crossed by the line. The
second type are out-of-t•e-plane diffractions, often called "side-swipe". This
is most noticeable by the appearance of energy from the second reef booy which
was not crossed. In the two-dimensional (2-D) migration, we have correctly
removed the 2-D diffraction patterns, but are still bothere• by the
out-of-the-plane diffractions. The full 3-D migration corrects for these
problems. The final migrated section has also accounted for incorrectly
positioned evehts such as the obliquely dipping fault. This brief summary as
not been intended as a complete summary f the migration procedure, but rather
as a warning that migration {preferably 3-D) must be performedon complex
structural lines for the fol 1 owing reasons:
(a)
(b)
To correctly position dipping events on the seismic section, and
To remove diffracted events.
Although migration can compensate or someof the lateral resolution
problems, we must remember hat this is analogous to the aleconvolution problem
in that not all of the interfering effects may be removed. Therefore, we must
be aware that the true one-dimensional seismic trace, free of any lateral
interference, is impossible to achieve.
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lol
I
71
131
(a] 3- D MODEL
131
101
108
LINE
ß
ß ß
ß ß
ß
..................................
.............................
.........................................
....................................
{hi 8•8•0 LAYOU•
Fig. 4.6. 3-D model experiment.
i mm _ ml j mm
Part 4 • Seismic Processing Considerations
(Hermant al, 1982).
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4.5 Notse Attenuation
As we' discussed in an earlier section, seismic noise can be classified as
either •andom or coherent. Random noise is reduced by the stacking process
quite well unless the signal-to-noise ratio dropsclose to one. In this case,
a coherency nhancementrogram an be used, which usually involves some ype
of trace mixing or FK filtering. However, he interpreter mustbe aware that
any mixing of the data will "smear" trace amplitudes, making he inversion
result on a particular trace less reliable.
Coherent noise is much more difficult to eliminate. One of the major
sources of coherent noise is multiple interference, explained in section 2.4.
Two of the major methodsused in the elimination of multiples are the FK
filtering method,and the newer nverse Velocity Stacking method. The Inverse
VeiocityStackingmethodnvolves he following teps:
(1) Correct the data using the proper NMO elocity,
(2) Model the data as a linear sumof parabolic shapes,
(This involves transforming to the Velocity domain),
(3) Filter out the parabolic omponentsith a moveoutreater hansome
pre-determinedimit (in the order of 30 msec),and
(4) Perform the inverse transform.
Figure 4.8, taken from Hampson1986), shows comparison etween he two
methodsor a typical multiple problemn northernAlberta. The displays are
all' co•on offset stacks. Notice that although both methods have performed
well on the outside traces, the Inverse Velocity Stacking methodworks best on
the inside traces. Figure4.9, also fromHampson1986), shows comparisonf
final stacks with and without multiple attenuation. It is obvious 'from this
comparisonhat the result of inverting the section whichhas not had multiple
attenuation would be to introduce spurious velocities into the solution. The
importance f multiple elimination to the preprocessinglow cannot therefore
be overemphasized.
m i i m , i . i m _ i i _ L ,=•m__ _ i m ß •
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liltiitil 1111iitt)ttl lii/littl•ll
(b] LINE d8 - 2-D MIGRATION
IIIIIIll 1111111111111it111111111111111111illllli IIIIIIIIil 111111tllilil illlllll 111
[1111111111111111111111111II 1111111111111111111I I IIilllllllll 11111111111111111
?•111[•i••IIIIIIII1111111111111111II IIiill•illlllillllllllllliillllllllllllh
•., } l iillllllllillllllliJillllllllllllilitiilillit illo
111lllllllllllllllllllll1111llllllilllllll ll llll111llllllllilllllllllllllllllllllllllii{lillllll
"• illllllllll 1111illi 111IIIIIIIIIIIIII1111111111I I lillilllllll 1111 1 111111•
Col LINE 108 - 3-D MIGR•ATION
F•g. 4.7. Migration f model ata shownn F•g. 4.6.
- - -- (Herman t al, 1982).
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AFTER
INVERSE VELOCITY STACK
MULTIPLE ATTENUATION
NPUT
AFTER
F-K MULTIPLE
ATTENUATION
J. ' ' ')'%': • t ' 11 1'1 ';.•m,:' :',./-•-•l- •r'm-- all
" "';;:.m;: .... ,;lliml;
.. .
m#l
Fig, 4.8.
Commonoffset stacks calculated from data before multiple
attenuation, after inverse velocity stack multiple attenuation,
and after F-K multiple attenuation. (Hampson, 1986)
888
Zone d
Interest
16984
Second eal-data et conventionaltackwithoutmultiple ttenuation.
'•" • ...... ;•,•<,:u(•:'J,.•J,.•., -, •, I• ,,,, .... •.. •, •,,,•• '•;••
,,t.•/:,.•t.,. ). I',,', ,'; • , , •, ß '1"' ',''. ;•t(•' )"•,'.m,,•""•.
• ,ii%' .t .% .
, ,, , • ..•'•t,..'•"•'i•' • -
---';•-•' "t" 1•%';J• •t•, ß... - ... ; -' ".' ,•..' . 2•>': ..'•, •;,%"'•1
lee "" • "" • • ' "' "•' ß ' ß ' • ....
'" "' Zone of
,,, .t•iill••)•.•);•l',"P,'•)'•"•'".•r'"mm"•""•P"••)r'"••' ' '" •- ..... ,• Interes
,..,.,..,,,_.,,.,... .,...,. ...,..,.•..,....,,,.,.,..•.. , .
,' .l•,•) ' • .'•',•' '• ....
.....•.•_ •.U.•,.., .. •
•,•,•p}•h•?.• •.•,•..} , •.•, ,•,•m,l,•,,nm,"::•"'•'•""""="'""•"...;'
.•,,,,.,.•,,,,,.., ,,{. ........ ,, .. ,,, ../•.• ,•.•'•, .'•-•%
Fig. 4.9.
Second real data stack after inverse velocity stack
multiple attenuation. (Hampson, 1986)
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PART 5 - RECURSIVE INVERSION - PRACTICE
_ _ _ _ _ .. . .• ,• _ _
Part 5 - Recursive Inversion - Practice
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5.1 The Recurslye Inversion Method
We have now reached a point where we may start aiscussing the various
algorithms currently used o invert seismicdata. Wemust ememberhat all
these techniquesare baseUon the assumption of a one-aimensional eismic
trace model. T•at is, we assume hat all the corrections which were aiscussed
in section 4 have been correctly applied, leaving us with a seismic section in
whic• each trace represents a vertical, band-limiteU reflectivity series. In
this section we will look at some of the problems inherent in this assumption.
The most popular techniquecurrently used to invert seismic Uata is referred
.
to as recursire inversion and goes under such trade names as SEISLOGana
VERILOG. The basic equations used are given in part 2, anU can be written
Zi+Z <===__===>i+lZ ,
ri-- i+l+ LIJ
where
r i
= ith reflection coefficient,
and
Z --/•Vi= density veloci y.
The seismic data are simply assumea o fit the forward model and is
inverted using the inverse relationship. However, s wasshownn section 3,
one of t•e key problemsn the recursire inversion of seismic data is the loss
of the low-frequency component. Figure 5.1 shows an exampleof an input
seismic section aria the resulting pseuao-acoustic impeaance without the
incorporationof low frequency information. Notice that it resembles
phase-shifteU version of the seismic ata. The questionof introUuclng the
low frequency omponentnvolves wo separate ssues. First, wheredo we get
the low-frequency omponentrom, ana, second,how o we incorporate t?
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1171121e9 eS1ol 92 93
i• 11•I Ittltl =:::•:::::::-•--lll[l•1111t•'• •1tlllttllltlll•l t1 l IIitti llltfltll l•I 1 t•n•'il, , •l••J• •":• •
•'• •" • --'' '
___ ..• - ,•, _•. • • f •• .• ............
:•,• m•,•'. • ....... • .... ,.• .... • . •• .........
ß ß ß • ... •
,• ß - • •, • ,•,..,• :'•l•,fm;•v•,• •,.•.•l.;•.•.'..•l•l;ql .n ................... ...; •;....: • .. • ...................
' • ]• • • •' ,, •, •' ,,,',',•,,, ,'",','" :•'•'•"•m••q•'t•'•'•a...., '. ,•],''•,J'•,• .• ' - '""W',- -::-=
•, '2 ,,• • ., •,•- • ,,• . ,•,•,I,.•.•..,• ....... •.•,,• . .,%•.• . ,• . '-.. ' .,• •, . •i• .......
. • , • • •-•,• ,, • , ,,.,,• .., ..... •. •.,.,,,,..•,.., ,,,.•,•,•.•.• .... .,• .... • .......
'•. . •q• • •,•;.•,• ,.. • •,l•,,..,,, ..•, J I .,,, • .•,• • .....,•...... : .•......•.•.. :,.. , ... , ,. , ............
, , •.•- -. •- (• ••' •'•:; •, / ................... . .... -(•-•( •.•,••(•'••'•"•:•"•'•7 '• . , .
• •'•,:•'•' • x•{
- ,,
2•Y•' •] ,,•.-..•.•.,'.;.',-,.. .................. • ............ • .................. •'•:.,• ...... • ... ......... •" ß7•' . =". .... 7' • '• • '. ' .----
.... - ......... •m:'•' •"• 'u'" •$• .... , ..... .. •<• • ß • - ' •'•' - ' .'••'•q• "•. •q• • .....
.•,.,• • .... ,_ /. ,,,_ . ; .... •,.:• - ............. ...... •%--=: .•.. .............. , .......... • .....
•4• 7•* • ';•u
. :c• • ,• .,,•-.•,, ?'..%•.,
•*•'•d•ti',i l•l•l'i'/lt' i•"'; •:•;•t•l,•i•21.•.l•'*.'•.'l•,•-•ii•.'•'..•,• :b-''? "•''• .... ; '_ ],;,'• ; '-•-•,••-----m'•l• ••"'•I'i•I• ........
•?•'•'• ;•q•.' '•'"•",•h/•'•'}'•' "' c'(•'•'".........
.... •, --.- -••_ ,,.•_.'.';'". :: :: ......
ß" • ..... "• '1 '• ' ' ' ß , -' ' • ..... • ' - ß
•'.•-•-• '•-<•., •
'. ,,,'• ,, ,. ,, ,
(a) Oriœinal- eismic ata. Heavyines indicatemajorreflectors.
0.7
N N N '" "
0.7
0.8
0.9
10
l
12
.3
1.4
1.5
1.6
1.7
(b) Recursive nversion of data in (a).
Figure 5.1
0.8
'I
1.0
i
I 1
I
I
1.2
.I
.
1.3
i
1 4
1.5
1.7
I
I
18
i
I 19
(Galbraith and Millington, 1979)
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The ow frequency omponentan be found n oneof three ways'
(1) From a filtered sonic log
The sonic log is the best wayof deriving ow-frequencynformation in
the vicinity of the well. However,t suffers from womainproblems't is
usually stretched with respect o the seismicdata and t lacks.a lateral
component.hese roblems,iscussedn Galbraith ndMillington 1979), are
solved by using a stretching algorithm which stretches the sonic log
information to fit the seismic data at selected control points.
(2) From seismic velocity analysis
In this case, interval velocities are derived from the stacking velocity
functions along a seismic ine usingDix' formula. The resulting function
will be quite noisyand t is advisable to do someormof two-dimensional
filtering on them. In Figure 5.2(a), a 2-D polynomialit has beendone to
smoothout the function. This final set of traces represents the filtered
interval velocity in the 0-10 Hz range or each race and may be added
directly to the inverted seismic races. Refer to rindseth (1979), for more
de ta i 1 s.
(3) From a geological model
Using all
incorporated.
available sources, a blocky geological model
This is a time-consuming method.
can be built and
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.
70000
(a)
GOOO0
$0000
(pvl 4oooo
'/sgc
( b ) $oooo
ZOOO0
I0000
/ -- V..308PV)*460
,
i
VELocrrY SURFACE 2rid ORDERPOLYN• Frr
Figure 5.2 s •mTZ eH CUT tT•
tRussell and Lindseth, 1982).
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Second, the low-frequency component an be added o the high frequency
componenty either adding reflectivity stage or the impedancetage. In
section 2.3, it was shownhat the continuous pproximationo the forwardand
inverse equations was given by
Forward Equati on
1 d 1n Z(t) <::==> Z(t)
r(t) =•- dt -
Inverse Equation
t
=Z(O)xp•0 (t)dt.
Since he previousransformsre nonlinear(becausef the logarithm),
Galbraith and Millington (1979) suggest hat the addition of the low-frequency
componenthouldbe made t the reflectivity stage. In the SEISLOGechnique
they are addedat the velocity stage. However, ue o other considerations,
this should not affect the result too much.
Of course, we are really interested in the seismic velocity rather than
the acoustic impedance.igure5.2(b), fromLindsethlg79), showshat an
approximateinear relationship exists between velocity and acoustic
impedance, given by
V = 0.308 Z + 3460 ft/sec.
Notice that this relationship is good for carbonates and clastics and
poor for evaporitesand should herefore be usedwith caution. A moreexact
relationship may be found by doing crossplots from a well close to the
prospect. However, singa similar relationshipwemayapproximatelyxtract
velocity information from the recoveredacoustic impedance.
Figure 5.3 showsow frequencynformation erived from filtered sonic
logs. The final pseudo-acousticmpedanceog is shown n Figure5.4
including the low-frequencyomponent.Notice that the geologicalmarkers re
moreclearly visible on the final inverted section.
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Figure 5.3 LowFrequencyomDonenterived from"st.reched:' onic loœ.
0.7
0.8
0.9
l.O
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
19
Figure 5.4 Final inversioncombinin•Figures 5.1(b) and 5.3.
Lines indicate major reflectors.
0.9
1.0
1.1
1.2
1.:)
1.4
I$
1.6
1.7
19
(Galbraith and Millington, 1979)
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In sugary, the recursive method of seismic inversion may be given by the
fol 1owing flowchart'
I
i
INTRODUCEOWREQUENCIES)
I•.v• o •DO-•CO••c
'
CORRECTOSEUDOELOCITIES,
CONVERTOEPTH
Recursi ve Inversion Procedure
, . _ ß ., . i
A commonmethod of display used for inverted sections is to convert to
actual interval transit times. These transit times are then contoured and
coloured according to a lithological colour scheme. This is an effective way
of presenting the information• especially to those not totally familiar'with
normal seismic sections.
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(a)
Frequency
(e)
1
(b)
Fig.
(a) Frequency response of a theoretical differentiator.
(b) Frequency esponseof a theoretical integrator.
Part 5 -Recursire Inversion - Practice
(Russell and Lindseth,
,m ,i m ml , ,
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5.2
I nfor marl .n ?•_Th.e.o..wF.r.equ.e.cycompo..ne.t
The key factor which sets inverted data apart from normal seismic data is
the inclusion of the low frequency component, egardless of how his component
is introduced. In this section we will look at the interpretational
advantages of introducing this component. The information in this section is
taken from a paper by Russell and Lindseth (1982).
We start by assuming the extremely simple moael for the
reflectivity-impedance relationship which was introduced in part 5.1. However,
we will neglect the logarithmic relationship of the more complete theory (this
is justifiea for reflection coefficients less that 0.1), so t•at
t
_1dZ(t) =__==>(t)2Z(O)j•(t) t
(t) - • dt- '
If we consider a single harmonic component, we may derive the
response of this tel ationship, which is
dewt jwt jwt -j eJWt
-dt "-- we <===> . dt= w
where w-- 21Tf,
frequency
In words.,differentiation introduces a -6 riB/octaveslope from the high
end of the spectrum o the low, and a +90 degree phase shift. Integration
introduces a -6 dB/octave slope from the low end to the high end, and a -90
degree phase shift. Simpler still, differentiation removesow frequencies
and integration puts them in. Figure 5.5 illustrates these relationships.
But how aoes all this effect our geology? In Figure 5,6 we have
illustrated three basic geological models'
ß
(1) Abrupt 1 thol ogi c change,
(2) Transitional lithologic change, an•
(3) Cyclical change.
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(A)MAJOR ITHOLOGICHANGE
V1
Vl
I
i
I.
I
I
I
I
i
I
(B)TRANSITIONAL LITHOLOGIC
CHANGE
V:V•+KZ
i i
(C)CYCLICALCHANGE
• _
Fig. 5.6.
Three ypesof lithological models' (a) Major change,
(b) Transitional, (c) Cyclical. (Russell and Lindseth, 1982).
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Wemayllustrate the effect of inversion n these hree casesby ooking
at both seismic anU sonic log Uata. To show he loss of high frequencyon the
sonic log, a simple filter is used, nd he associatedhase hift is not
introUuced.
To start with, considera major1 thologic boundarys exempllieu by the
Paleozoicunconformityf Western Canada, changeroma clastic sequenceo
a carbonateequence. igure5.7 showshat most f the information bout he
largestep n velocity s containeUn theD-10iz componentf the sonicog.
In Figure5.8, the seismic ataand inal Uepthnversion re shown.On the
seismic data, a major boundary howsup as simply a large reflection
coefficient, whereas,on the inversion, the large velocity step is shown.
RAWSONIC FILTEREDONICLOGS
VELOCITY FT/SEC
0 10000 10-90HZ O-IOHZ O-CJOHZ
TIME
0.3-
0.5-
Fig. 5.7. Frequencyomponentsf a sonicog.
(Russell and Lindset•, 1982).
L , , , I I ß [ I L
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Introduction to Seismic Inversion Methods Brian Russell
o'- .
ß
(a)
.%;
DEPTH SEISLOG
ß o
DEPTH
(b)
..... ßOP OF
"' . ß""I:'ALEOZOIC
-425'
Fig. 5.8. Major litholgical'change, Saskatchewan example.
(a) Sesimic s_ection, (b) Inverted section.
..... _........ _(R_q•sell... nd L ,pqse_th,_• 98_2)___
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To illustrate transitional and cyclic change, a single examplewill be
used. Figure. 5.9 showsa sonic og from an offshore Tertiary basin,
illustrating the rampswhichshow a transitional velocity increase, and the
rapidly varyingcyclic sequences. otice hat the 0-10 Hz componentontains
all the information about the ramps, but the cyclic sequences contained n
the 10-50 Hz component.Only he Oc components lost from the cyclic
componentpon emoval f the low frequencies. Figure 5.10 illustrates the
same oint using he original seismic ata and he final depth nversion.
In summary,he information ontainedn the low frequency omponent f
the sonic og is .lost in the seismic data. This includessuchgeological
information as the dc velocity component,arge jumps in velocity, and linear
velocity ramps. If this information could be recovered nd ncluUeaduring
the inversion process, it would ntroduce his lost geological information.
Fig. 5.9. Sonicog showingyclic and ransitionalstrata.
Part 5 - Recurslye
Inversion - Practice
(Russell
and LinOseth, 1982)
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(b)
Introduction to Seismic Inversion Methods Brian Russell
(a)
SEISMIC SECTION-CYCUC & TRANSITIONAL STRATA
i 1-3500
ß
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5.3
Sei smical ly Derived Poros ty
- ILI , ß I
We have shown hat seismic data may be quite adequately inverted to
pseudo-velocity (and hence pseudo-sonic) nformation i f our corrections and
assumptions are reasonable. Thus, we may try to treat the inverted data as
true sonic log information and extract petrophysical data from it,
specifically porosity values. Angeleri and Carpi (1982) have tried just this,
with mixed results. The flow chart for their procedure is shown in Figure
5.11. In their chart, the Wyllie formula and shale correction are given by:
where
At --transit time for fluid saturated rock,
Zstf= pore luid transit time,
btma: ockmatrix ransittime,
Vsh fractional olumef shale,and
btsh: shale ransit ime.
The derivation of porosity was tried on a line which had good well
control. Figure 5.12 shows the plot of well log porosity versus seismic
porosity for each of three wells. Notice that the fit is reasonable in the
clean sands and very poor in the dirty sands. Thus, we may extract porosity
information from the seismic section only under the most favourable
conditions, notably excellent well control and clean sand content.
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F ']w[ttill
•ILI61CAT&$[IS'MI•AT&'
I-"'• '' m.,,•,,ml
-[ ,gnu mill i' •ill. Utl.. 111l lit
•%lOtOG
I IIITEIPllETATII
Fig.
l WlltK :
t ' .
5.11. Porosity eval uati on flow diagram.
(AngeleriandCarpi, 1982).
Fig.
, ,
WELL 2 WELL 3 WELL
__ ClII PNIIVI o..- OPt poeoItrv ..... CPI
ß , , ß ß ' I ,- --
e e I e . e e . . e ß e e e e I i e e e ß i e i ß ß ß e
.
1.4
1.7
1.8,
1.9
5.12. Porosity profiles from seismic data and borehole data.
Shalepercentages al so displayed. (Angel ri andCarpi, 1982).
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Introduction to Sei stoic Inversion Methods Brian Russel 1
PART 6 - SPARSE-SPIKE INVERSION
• { • ...... • I ] m • m
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Introduction to Seismic Inversion Me.thods Brian Russell
6.1 Introduction
Thebasic theoryof maximum-1kel hood econvoltion (MLD)wasdeveloped
by Dr. Jerry Mendel nd his associatesat USC nUhas beenwell publicised
(Kormylo ndMendel, 983;Chiet el, 1984). A paperby Hampsonnd Russell
(1985) outlined a modification of maximum-likelihoodeconvolutionmelthod
which allowed the method o be moreeasily applied to real seismic •ata. One
of the conclusions f that paper wasthat the method ould be extenoed o use
the sparse eflectivity as the first step of a broadband eismic inversion
technique.This technique,which will be termedmaximum-likelihoodeismic
inversion, is discussed later in these notes.
You will recall that our basic model of the seismic trace is
s(t) = w(t) * r(t) + n(t),
where
s(t) : the seismic trace,
w(t) : a seismic wayelet,
r(t) : earth reflectivity, and
n(t) = addi tire noise.
Notice that the solution to the above equation is indeterminate, since
there are three unknowns o solve for. However, using certain assumptions,
the aleconvolution roblem can be solved. As we have seen, the recursire
method of seismic inversion is basedon classical aleconvolution techniques,
which assume random eflectivity and a minimum r zero-phase wavelet. They
produce higher requency aveleton output,but never ecover he reflection
coefficient series completely. More recent aleconvolutionechniquesmaybe
groupedunder the category f sparse-spike eth•s. That is, they assume
certain modelof the reflectivity and make a wavelet estimate based on this
assumption.
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ACTUAL REFLECTIVITY
I,:, I ..
POISSON-GAUSSIAN
SERIES OF LARGE
EVENTS
--F
GAUSSIAN BACKGROUND
OF SMALL EVENTS
SONIC-LOG REFLECTIVITY
EXAMPLE
Figure 6.1 The fundamentalssumptionf the maximum-likelihoodethod.
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These techniques include-
(1) btaximum-Likel ihood deconvolutton and inversion.
(2) L1 norm deconvolution and inversion.
(3) Minimum ntropy deconvol tion (MEO).
From the point of view of seismic inversion, sparse-spike methods have an
advantage over classical methods f deconvolutionbecause the sparse-spike
estimate, with extra constraints, can be used as a full bandwidth estimate of
the reflectivity. We will focus initially on maximum-likelihood
deconvolution, and will then move on to the L1 normmethod of Dr. Doug
O1denburg. The MEDmethod will not be discussed in these notes.
6.2 Maximum-Likelihood Deconvolution and Inversion
i i m ß m m m m I _ ß
Maximum-Li kel i hood Deconvoluti on
I ß ß ß m _ _ l . . • am .. I _
Figure 6.1 illustrates the fundamental assumption of Maximum-Likelihood
deconvolution, which is that the earth' s reflectivity is composed f a series
of large events superimposedon a Gaussian backgroundof smaller events. This
contrasts with spiking decon, which assumesa perfectly randomdistribution of
reflection coefficients. The real sonic-log reflectivity at the bottom of
Figure 6.1 shows that in fact this type of model is not at all unreasonable.
Geologically, the large events correspond to unconformities and major
ß
1 thol ogic boundaries.
From our assumptions about the model, we can derive an objective function
whichmaybe minimized o yield the "optimum" r most ikely reflectivity. and
wavelet combination consistent with the statistical assumption. Notice that
this method gives us estimates of both the sparse reflectivity and wavelet.
,,
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INPUT
WAVELET
REFLECTIVITY
NOISE
SPIKE SIZE' 9.19
SPl• ••: 50.00
NOISE' 39.00
OB,.ECTIVE' 98.19
Figure6.2(a) Objectiveunctionor onePoSsibleolution o input race.
INPUT
WAVELET
REFLECTIVITY
SPIKE S 7_F: 6.38
SPIKE DENSIq'•, 70.85
NOISE
NOISE: 81.• 5
OBJECTIVE 158.98
Figure 6.2(b)
Objectiveunctionora secondossibleolutiono nput
trace. This alues higherhan .2(a),.ndicatingless
1 kely solution.
, ,,
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The objective function j is given by
- R2 N
k=l k=l
where
- 2m n(X)- 2(L-re)In(i-A)
r(k) = reflection coeff. at kth
sample,
m = numberof refl ecti OhS,
ß
L : total numberof samples,
N : sqare root of noise variance,
n : noise at kth sample, and
• = likelihood that a given
sample has a reflection.
Mathematically, the expected behavior of the objective function is
expressed in terms of the parametersshown bove. No assumptions are made
about he wavelet. The reflectivity sequence s postulated o be "sparse",
meaning that the expected number f spi•es is governed by the parameter
lambda, the ratio of the expected numberof nonzer. spikes to the total number
of trace samples. Normally, lambda is a numbermuchsmaller than one. The
other parametersneeded o describe the expected behavior are R, the RMS•size
of the large spi•es, andN, the RMS ize of t•e noise. With these parameters
specified, any glven deconvolution sol ution can be examined to see.whether it
is likely to be the result of a statistical processwith thoseparameters.For
example, f the reflectivity estimate has a number f spikesmucharger than
the expectednumber, hen it is an unlikely result.
In simpler terms, we are looking for the solution with the minimum
number f spikes n its reflectivity and t•e lowestnoisecomponent. igures
6.2(a) and 6.2(b) showwopossiblesolutions or the same nput synthetic
trace. Notice hat the obje6tive function or the onewith the minimumpike
structure is indeed the lowest value.
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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.
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Of course, there maybe an infinite number f possible solutions, and it
would take too much omputerime to look at eachone. Therefore, a simpler
method is used to arrive at the answer. Essentially, we start with an initial
wavelet estimate, s'timatehe sparseeflectivity, ' improvehe wavelet and
iterate through this sequence of steps until an acceptably low objective
function is reached. This is shownn block form in Figure 6.4. Thus, there
is a two step procedure-having he wavelet estimate, update he reflectivity,
and then, having the reflectivity estimate, update the wavelet.
These procedures are illustrated on model data in Figures 6.3 an• 6.5.
In Figure 6.3, the proceUure for upUating the reflectivity is shown. It
consists of adding reflection coefficients one by one until an optimum et of
"sparse" coefficients hasbeen ound. Thealgorithm sed or updating the
reflectivity is callee the single-most-likely-addition algorithm (SMLA)since
after each step it tries to find the optimum pike to add. Figure 6.5 shows
the procedure for updating the wavelet phase. The input model is shownat the
top of the figure, and the up•ated reflectivity and phase s shown fter one,
two, five, and ten iterations. Notice that the final result compares
favourably with the model wavelet.
WAVELET
ESTIMATE
ES•TE
REFLECTIVITY
IMPROVE
WAVELET
ESTIMATE
Fiõure 6.4.
Theblock omponentethodf solvingor both
reflectivity andwavelet. Iterate aroundhe
loop unti 1 converRence.
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Wayelt Refl ctiVity ' Synthetic
Ill ,I ,
INPUO
INITIAL CUESS
TEN ITERATIONS
Fi õure 6.5.
The procedure for updatinõ the wavelet
in the maximum-likelihood method.
Between each iteration above, a separate
iter. ation on reflectivity (see Fiõure 6.3)
has been done.
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Figure 6.6 is an exampleof the algorithm applied to a synthetic
seismogram. Notice that the major reflectors have been recovered fairly well
and that the resultant trace matches he original trace quite accurately. Of
course, the smaller reflection coefficients are missing in the recovered
reflection coefficient series.
Let us now ook at some real data. The first example is a' basal
Cretaceous gas play in Southern Alberta. Figure 6.7(a) and (b) shows the
comparison between the input anU output stack from the aleconvolution
procedure. Also shown re the extracted and final wavelet shapes. The main
things to note are the major increase in detail (frequency content) seen in
the final stack, and the improvement n stratigraphic content.
Figure 6.8 is a comparisonof input and output stacks for a typical
Western Canada basin seismic line. The area is an event of interest between
0.7 anU 0.8 seconds, representing a channel scour within the lower Cretaceous.
Although the scour is visible on both sections, a dramatic improvements seen
in the resolution of the infill of this channel on the deconvolved section.
Within the central portion of the channel, a .positive reflection with a
lateral extent of five traces is clearly visible and is superimposedn the
Uominant negative trough.
INPUT:
V. ,.: --
ESTIMATED:
ttl J':ll'j "'" "
Figure 6.6 Synthetic seismogram test.
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0.5
0.6
0.7
0.8
'SONIC
SYNTHETIC LOG
iZ.
EXTRACTED WAVELET
0.5
0.6
.
0.8
(b)
(a) Initial seismicwith extracted wavelet.
Final deconvolved seismic with zero-please wavelet.
Figure 6.7
.... - -_ __ ._
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This is quite possibly a clean channel sand and may or may not be
prospective. However, this feature is entirely absent on the input stack.
Overlying the channel is a linear anomaly which could represent the 'base of a
gas sand, and is muchmore sharply defined on the output section, both in a
lateral and vertical sense.
Finally we have taken the deconvolved output and estimated the
reflectivity. This is shown in Figure 6.9. Although some of the subtle
reflections are missing from this estimated reflectivity, there is no doubt
that all the main reflectors are present. It is interesting to note how
clearly the base of the channel (at 0.7;- seconds)and the base of the
postulated gas sand on top of the channel have been delineated.
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INPUT
STACK
DECONVOLVED
STACK
0.6
0.7
0.8
0.9
Figure 6.8
An input stack over a channelscour and
the resul ting deconvoled sei smic.
DECONVOLVED
STACK
ESTIMATED
REFLECTIVITY
0.6
0.7
0.8
0.9
Figure 6.9
The deconvolved result from Figure 6.8
and its estimated reflectivity.
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Maximum-Likel ihood Inversion
An obvious extension of the theory is to invert
reflectivity to Uevise a broad-band or "blocky" impedance
data (Hampsonnd Russell, 1985). Given the reflectivity, r(i),
impedance (i) maybe written
Z(i) Z(i_l[1 r(i)]
- r(i) '
the es ti ma ted
from the seismic
the resul ting
Unfortunately, application of thi
from MLD produces unsatisfactory res
additive noise. Although the MLD algor
of the wavelet to produce a broad-band
of this estimate is degraaed by noi
spectrum. The result is that while
s formula to the reflectivity estimates
ults, especially in the presence of
it•m'extrapol ares outsi de the bandwidth
reflectivity estimate, the reliability
se at the low frequency end of the
the short wavelength features of the
impedancemay be properly reconstructed, the overall trenu is poorly resolvea.
This is equivalent to saying that the times of the spires on the reflectivity
estimate are better resolved than their amplituaes.
In order to stabilize the reflectivity estimate, independent knowleUge
of the impedance renU may be input as a constraint. Since r(i) < l, we can
derive a convolutional type equation between acoustic impeUance anU
reflectivity, written
In Z(i) = 2H(i) * r(i) + n(i),
where Z(i) = the known mpedance rend,
• i <0
H(i) :
• i >0
and
n(i) : "errors" in the input trend.
_
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Figure 6.10 Input Model parameters.
Figure 6.11
ß
Maximu•m-Lkel i hood i nversi on result from Figure 6.10.
.m __
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The error series n(i) reflects the fact that the trend information is
approximate. We now have two measured time-series: the seismic trace, T(i),
and the log of impedance n Z(i), each with its own wavelet and noise
parameters. The objective function is modified to contain two terms weighted
by their relative noise variances. Minimizing this function gives a solution
for r(i) which attempts a compromiseby simultaneouslymoUelling he seismic
trace while conforming o the known mpedance rend. If both the seismic
noise and the impedancerend noise are modelledas Gaussiansequences, heir
respective variances become tuning" parameterswhich the user can modify to
shift the point at which the compromise occurs. That is, at one extreme only
the seismic nformation s usedand at the ot•er extremeonly the impedance
trend.
In our first example, he methods tested on a simple synthetic. Figure
6.10 shows he sonic og, the derived reflectivity, the zero-phase wavelet
used to generate the synthetic, and finally the synthetic itself. This
example was used nitially because t truly represents a "blocky" impedance
(and therefor.e a "sparse" reflectivity) and therefore satisfies the basic
assumptions of the method.
In Figure 6.11 the maximum-likelihood inversion result is shown. In
this casewe haveuseda smoothedversion of the sonic velocities to provide
the constraint. A visual comparisonwoulU indicate that the extracteU
velocity profile corresponds very well to the input. A moredetailed
comparisonof the two figures shows hat the original and extracted logs do
not matchperfectly. T•ese small. shifts are due to slight amplitude problems
on the extracted reflectivity. It is doubtful that a perfect match could ever
be obtai neU.
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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
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Let us now urn our attention to a slightly more realistic synthetic
example. Figure 6.12 showshe applicationof this algorithm o a sonic-log
derivedsynthetic. At the' top of the figure we see a sonic log with 'its
reflectivity sequence below. (In this example,we have assumed hat the
density is constant, but this is not a necessary restriction.) The
reflectivity wascbnvolved ith a zero-phase avelet,bandlimited rom10 to
60 Hz, andthe final synthetic s shownt the bottom f the figure.
The results of the maximum-likelihood inversion method are sbown in
Figure 6.13. The initial log is shownat the top, the constraint is shown n
the middle panel, and the extracted resull• is shownat the bottom of the
diagram. In this calculation, the waveletwasassumednown. Note the blocky
nature of the estimated elocity profile comparedith the actual sonic log
profile. Again, the input and output logs do not matchperfectly.
The fact that the two do not perfectly match s due to slight errors in
the reflectivity sizes whichare amplified by the integration process,and s
partially the effect of the constaint used. Theconstraintshown n Figure
6.13 wascalculatedby applying a 200 ms smoothero the actual log. In
practice, this information could be derived from stacking velocities or from
nearby well control.
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*
Figure 6.14
An input seismic 1 ne to be inverted.
:
ß
'.
eel'?
e4dl
Figure 6.15
Maximum-Liklihood reflectivity estimate from
seismic in Figure 6.14.
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Finally, we show he results of the algorithm applied to real seismic
data. Figure 6.14 shows portion of t•e input stack. Figure 6.15 shows he
•D extracted reflectivity. Figure 6.16 shows the recoveredacoustic
impedance,wherea linear ramphas beenusedas the constraint. Notice that
the inverted section •isplays a "blocky" character, indicating that the major
features of the impedanceog have been successfullyrecovered. This blocky
impedanceanbe contrastedwith the more traditional narrow-bandinversion
procedures, whichestimate a "smoothed"r frequencyimited version of the
impedance.Finally, Figure 6.17 shows a comparisonetweenhe well itself
and the inverted section.
In summary,maximum-likelihoodnversion is a procedurewhich extracts a
broad-band estimate of the seismic reflectivity and, by the introduction of
1 near constraints, al lows us to invert to an acoustic impedance ection which
retains the major geological features of borehole og data.
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Figure 6.16
Inversion of reflectivity shownn Figure 6.15.
SEISMIC NVERSION
WELL
+
SONIC
LOG
Figure 6.17
A comparison of the inverted seismic data and
the sonic log at well location.
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6.3 The L 1 Norm Method
-- __LI _ _ _ i .
Another method of- recursive, single trace inversion which uses a
"sparse-spike"ssumptions the L1 normmethod,evelopedrimarilyby Dr.
DougOldenburg f UBC. nd Inverse Theory andApplications (ITA). This method
is also often referred to as the linear programmingethod, nd this can lead
to confusion. Actually, the two names efer to separateaspects f the
method. Themathematical odelused n the construction f the algorithm is
the minimization f the L1 norm. However, he methodused o solve the
problem is linear programming. The basic theory of this method s found in a
paper by Oldenburg, et el (1983). The first part of the paper discusses he
noi se-free convolutional model,
x(t) --w(t) * r(t), where x(t) = the seismic race,
w(t) --the wavelet, an•
r(t) -- the reflectivity.
The authors point out that if a high-resolution aleconvolution is
performed n the seismic race, the resulting estimateof the reflectivity can
be thought of as an averagedversion of the original reflectivity, as shown t
the top of Figure6.18. This averagedeflectivity is missing oth •e high
and ow frequencyange,and s accurateonly in a band-limitea entral range
of frequencies. Although there are an infinite number f ways in which the
missing frequency components an be supplied, Oldenburg, et al (1983) show
that we can reduce this nonuniqueness by supplying more information to the
problem, such as the layered geological model
r(t) -•,rj6(t l•),
j--
where= 0 if t •l• , an•
=1 ift:• .
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b
ß ß ß • 1
m m m
0.0
T.IJdE• •J
e f
o .50 joo j25
FR F.,O [HZJ
I
I
Figure 6.18
Synthetic test of L1 Norm nversion, moUified fro•.q
Oldenburg t al (1983). (a) Input impedance,
(b) Input reflectivity, (c) Spectrum f (b),
(d) Low frequencymodel trace, (e) Deconvolutionof (•),
(f) Spectrum f (U), (g) Estimated mpedancerom L1 Norm
method, (•) Estimated reflectivity, (i) Spectrumof (•).
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Introduction to Seismic Inversion •.le.thods Brian Russell
Mathematically, the previous equation is considered as the constraint to
the inversion problem. Now, the layered earth model equates to a "blocky"
impedance unction, which in turn equates to a "sparse-spiKe" reflectivity
function. The above constraint will thus restrict our inverted result to a
"sparse" structure so that extremely fine structure, such as very small
reflection coefficients, will not be fully inverted.
The other key difference in the linear programmingmethod is that the L1
norm is minimized rather than the L2 norm. The L1 norm is defined as the sum
of the absolute values of the seismic trace. TrueL2 norm, on the other hand,
is defined as the square root of the sumof t•e squares of the seismic trace
values. The two norms are shown below, applied to the trace x:
x1 : x and x2: x
i--1 i:1
The fact that the L1 norm favours a "sparse" structure is shown in the
following simple example. (Taken from the notes to Dr. Oldenburg's 1085 CSEG
convention course' "Inverse theory with application to aleconvolution and
seismogram nversion"). Let f and g be two portions of seismic traces, where'
f: (1,-1,0) and g : (0,%• 0) .
The L2 norms are therefore'
The L1 norms are given by'
-
l - 1 + 1 : 2 and gl = '
Notice that the L1 norm of wavelet g is smaller than the L1 norm of f,
whereas the L2 norms are both the same. Hence, minimizing the L1 norm would
reveal that g is a "preferred" seismic trace based on it's sparseness.
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(a)
Input sei smic data
(b)
Estimated refl ec ti vi ty
(c)
Final impedance
Figure 6.19
L1 14ormmetboOapplied to real seismic data,
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(Walker and Ulrych, 1983
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Several other authors had previously considered he L1 normsolution in
deconvolutionClaerboutand Muir, 1973, andTaylor etal., 1979), however,
they consideredhe problemn the time domain.Oldenburgt al.w suggested
solving the problem sing frequency omain onstraints. That is, the reliable
frequency and is honoredhileat the sameimea sparsereflectivity is
created. The results of their. algorithm on synthetic data are shownat the
bottom of Figure 6.18. Theactual implementationf the L1 algorithm o real
seismicdata has been done by Inverse Theory andApplications ITA). The
processing flow •or the linear programmingnversion methods shown elow.
InterPreter'=MP tacl<edection
<r(t)>= r(t)©w(t)
t ß ,i
i
I,,i o,ect,',o,' esidum'm,e,wt) I
ß i i i i i I i i
I Fourierrans••f •rt)> I
i
Scaleata
Const.ints.romtackins•_V'elocitles
ii &
Con,straintsromWellogs
Unear Programing Invemion
Assume( ß n ) t- q, s spame,eflectioneries.
Minimize the sum of absolute reflection strengU•.
FulFBand Reflectivity Series r (t)
Signal to Noise Enhancement and Display Preparation
Integration to Obtain Impedance Sections
Figure 6.19(b) The L1 Norm Linear Programming.)ethod. (Oldenburg,1985).
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TSN
1,2
tO0 90 80 70 60 50 40 30 20 tO
1,3
1,4
1,5
1,6
1,7
1,:8
.2,0
2ø2
Figure 6.20
Inputseismic atasectiono L1 Normnversion.
(O1 enburg, 1985
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Figure 6.19 shows he application of the above technique to an actual
seismic line from Alberta. The data consist of 49 traces with a sample rate
of 4 msecand a 10-50 Hz bandwidth. The figure showshe linear programming
reflectivity and impedanceestimates below the input seismic section. It
should be pointed out that a three trace spatial smootherhas been applied to
the final results in both cases.
Finally, let us consider a dataset fromAlberta which has been processeU
through the LP inversion method. The input seismic is shown n Figure 6.2D
and the final inversion in Figure 6.21. The constraints useU here were from
well log data. In the final inversion notice that the impedance has been
superimposed on the final reflectivity estimate using a grey level scale.
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1.6
1.7
1.8
1.9
2.0
2.1
2.2
Figure 6.21
Reflectivity and grey-level plot of impedance
the L1 Norm nversion of data in Figure 6.20.
Part 6 -Sparse-spike Inversion
for
(O1denburg, 19
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6.4 Reef P roblee
_
On he next few pages is a comparison etweena recursive nversion
procedure (Verilog) anda sparse-spike nversionmethodMLD). The sequence
of pages includes the following:
- a sonic log and its derived reflecti vt ty,
- a synthetic seismogram t both polarities,
- the original seismic line, showing he well location,
- the Verilog inversion, and
- the MLD inversi on.
BaseUon the these data handouts, do the following interpretation
exerc i se:
([) Tie the synthetic to the seismic ine at SP 76. (Hint- use reverse
pol ari ty syntheti c).
(g) Identify and color the following events in the reef zone-
- the Calmar shale (which overlies the Nisku shaly carbonate),
- the 1retort shale, and
- .the porous Leduc reef.
(3) Comparehe reefal events on the seismic and the two inversions. Use
a blocked off version of the sonic log.
(4) Determine for parallelism which section tells you the most about the
reef zone?
Part 6- Sparse-spike Inversion
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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
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Part 6- Sparse-spike Inversion
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***__********************************************************
0.7 0.8 .la.El 1 .:0 1.1 1..2 1. E)
Part 6- Sparse-spike Inversion
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0.? IB.G O.G 1.0 1.1 1.2 1.• 1.4
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PART 7 - INVERSION APPLIED TO THIN BEDS
Part 7 - Inversion applied to Thin Beds
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7.1 Thin Bed Analysis
One of the problems hat we have identified in the inversion of seismic
traces is the loss of resolution caused by the convolution of the seismic
wavelet with the earth's reflectivity. As the time separation between
reflection coefficients becomessmaller, the interference between overlapping
wavelets becomesmore severe. Indeed, in Figure 6.19 it was shown hat the
effect of reflection coefficients one sampleapart and of opposite sign is to
simply apply a phaseshift of 90 degrees to the wavelet. In fact, the effect
is more of a differentiation of the wavelet, which alters the amplitude
spectrum s wel1 as the phase spectrum. In this section we will look closer
at the effect of wavelets on thin beds and how .effectively we can invert these
thin bed s.
The first comprehensive 'ook at thin bed effects was done by Widess
(1973). In this paper he used a model which has become he standard for
discussing thin beds, the wedgemodel. That is, consider a high velocity
laye6 encasedn a low velocity layer (or vice versa) and allow the thickness
of the layer to pinch out to zero. Next create the reflectivity response rom
the impedance, nd convolvewith a wavelet. The thickness of the layer is
given in terms of two-way ime through the layer and is then related to the
dominantperiod of the wavelet. The usual wavelet used s a Ricker becauseof
the simpl city of its shape.
Figure 7.1 is taken from Widess' paper and shows he synthetic section as
the thickness of the layer decreases from twice the dominant period of the
wavelet to 1/ZOth of the dominant period. (Note that what is refertea to as a
wavelength n his plot i s actual y twice the dominant eriod). A few important
points can be noted from Figure 7.1. First, the wavelets start interfering
with eackotherat a thickness ust below two dominanteriods,but remain
Clistinguishable down to about one period.
Part 7 - Inversion applied to Thin Beds
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PI•OPAGA ION I NdC
ACnOSS TK arO) .
•'------).z _1
--t
Figure 7.1
Effect of bed thickness on
reflection waveshape,where
(a) Thin-bed model,
(b) Wavelet shapesat top
and bottom re fl ectors,
(c) Synthetic seismic
model, anU (d) Tuning
parameters as measured from
resul ting waveshape.
(C) (D)
5O
, ,.
THINBEDREGIME
J PEAK-TO-TROUGH/
AMPLITUDE
2.0
1.0 <
0.8
0.4
/ \
-0.4 ,• . . . . .
-40 0 20 40
MS
TWO-WAY TRUE THICKNESS
(MILLISECONDS)
Figure 7.2 A typical detection and resolution cha•t used
to interpret bed thickness from zero phase seismic data.
('Hardage, 1986
. .. _ i i ,, , i _ - - - -_- - _ - _ ..... l. _
Part 7 - Inversion applied to Thin Beds
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Below a thickness alue of oneperiod he waveletsStart merging nto a
single wavelet, and an amplitude increase is observe•. This amplitude
increase is a maximum t 1/4 period, and decreases from this point down... The
amplitude is appraoching ero at 1/•0 period, but note that the resulting
waveform is a gO degree phase shifted version of the original wavelet.
A more quantitative way to measure his information is to plot the peak
to trough amplitude difference and i sochron across the thin bed. This is done
in Figure 7.•, taken from Hardage (1986). This diagram quantifies what has
already been seen qualitatively the seimsic section. That is that the
amplitude is a maximum t a thickness of 1/4 the wavelet dominant period, and
also that this is the lower isochron limit. Thus, 1/4 the dominant period is
considered to be the thin bed threshhold, below which it is difficult to
obtain fully resolved reflection coefficients.
7.2 In. ersion Camparisonf T.hinBees
ß
To test out this theory, a thin bed model was set up and was inverted
using both recursire inversion and maximum-likelihood aleconvolution. The
impedance model is shown n Figure 7.3, and displays a velocity decrease in
the thin bed rather than an increase. This simply inverts the polarity of
Widess' diagram. Notice that the wedge starts at trace 1 with a time
thickness of 100 msec and thins down to a thickness of 2 msec,.or .one time
sample. The resulting synthetic seismogram is shown n Figure 7.4. A 20 Hz
'Ricker wavelet was used to create the synthetic. Since the dominant period
(T) of a 20 Hz Ricker is 50 msec, he wedge has a thicknessof 2T at trace 1,
T at trace 25, T/2 at trace 37, etc.
Parl• 7 - 'inverslYnap'pled 1•oThin'- eds....
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lOO
200
3OO
400
500
4
8 12 16
20 24 28 32 36 40 44 48
ß
Figure7.3 True impedanceromwedgemodel.
o
lOO
200
.
300
ß
400
500
Figure 7.4
Wedgemodel reflectivity convol ved with
20 HZ Ricker wavelet.
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First, let us consider the effect of performing a recursire inversion on
the wedgemodel. The inversion result is shown n Figure 7.5. Note that the
low frequency component as not added into the solution of Figure 7.5, to
better show the effects of the initial recursire phase of the inversion. It
was also felt that the addition of the low frequency componentwould ado
little information to this test. Notice that there'are two major problems
with recursire inversion.. First, the thickness of the beU has only been
resolved down o about 25 msec, which is 1/2 of the dominantperiod. Remember,
that this is a two-way time, therefore we say that the bed thickness itself
has been resolved down to 12.5 msec, or 1/4 period. This theoretical
resolution limit is the sameas that of Widess. Also, the top of the weUge
appears "pulled-up" at the right side of the plot as the inversion has trouble
with the interfering wavelets. A second problem is that, although we know
that there are actually only three distinct velocity units in the section, the
recursire inversion has estimate• at least seven in the vertical =irection.
ß
This result is Uue to the banu-limited nature of the Ricker wavelet. More
Uescriptively, every wiggle on the section has been interpreted as a velocity.
ß
Next, consider a maximum-likelihood inversion of the weOge. The
constraint used was simply a linear ramp. In this case, the shape of the
ß
wedge has been much better defined, due to the broad-band nature of the
inversion. However, notice that the resolution limit has still been observeU.
That is, the maximum-likelihood inversion method also failed to resolve the
bed thickness below 1/4 dominant period. The "pUll-up" observedon the
recursively inverted section is also in evidence here.
In summary,even though sparse-spike methods give an output section that
is visually more appealing than recursively inverted sections, there does not
appear to be a way to break the low resolution limit of 1/4 of the dominant
se smic peri od.
Part 7 - Inversion applied to Thin Beds
_ i _ i mk
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4 8 12 16 20 24 28 32 36 40 44 48
o
300-
400.
Figure 7.5 Recursive inversion of wedgemodelshownn Figure 7.4.
4 8 12 16 20 24 28 32 36 40 44 48
' ' • i ' ' I i
100 .................
300
400
500 ,, .
Figure 7.6 Maximum-likelihoodderived impedance f wedgemoUel
shown i n Figure 7.4.
Part 7 - Inversion applied to Thin Beds
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[ntroductJon to Seismic Inversion Methods Br•an Russel•
PART 8 - MODEL-BASED NVERSION
_ - _ - m m L ß .... •
Part 8 - Model-based Inversion
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8.1 Introducti on
In the past sections of the course, we have derived reflectivi-ty
information directly from the seismic section and used recursire inversion to
produce a final velocity versus depth model. We have also seen that these
methods can be severely affected by noise, poor amplitude recovery, and the
band-limited nature of seismic data. That is, any problems in the data itsel f
will be included in the final inversion result.
In this chapter, we shall consider the case of builaing a geologic moUel
first and comparing the model to our seismic data. We shall then use the
results of t•is comparison between real and modeled data to iteratively update
the model in such a way as to better match the seismic data. The basic idea
of this approac• is shown in Figure 8.1. Notice that this method is
intuitively very appealing since it avoids the airect inversion of the seismic
data itself. On the other hand, it may be possible to come up with a model
that matches he data'very well, but is incorrect. (This can be seen easily
by noting that there are infinitely manyvelocity/depth pairs that will result
in the same ime value.) This is referred to as the problem of nonuniqueness.
To implement the approach shown in Figure 8.1, we need to answer two
fundamental questions. First, what is the mathematical relationship between
the model data and the seismic data? Second, how do'we update the' model? We
shall consider two approacheso theseproblems, he generalized inear
inversion (GLI) approach outlined in CooRe and Schneider (1983}, and the
Seismic Lithologic (SLIM) method which was developed in Gelland and Larner
(1983).
Part 8 - Model-based Inversion
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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
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Introduction %o Seismic Inversion Methods Brian Russell
8.2 Generalized Linear Inversion
The generalized linear inversion(GLI) method is a methodw•ich can be.
applied to virtually any set of geophysicalmeasurementso determine the
geological situation whichproduced these results. That is, given a set of
geophysicalobservations, the GLI method ill derive the geological model
which best fits t•ese observations in a least squares sense. Mathematically,
if we express the model and observations as vectors
M: (m,m, ..... , mk)=vectorfkmodelarameters,nd
T: (t1, t2, ..... , tnT
vector of n observations.
Then the relationship between the model
in the functional form
and observations can be expressed
t i = F(ml,m , ...... , m )
ß i : 1, ... , n.
functional relationship has been derived between the
nce the
observations and the model, any set of model parameters will produce an
ß
output. But what model?GLI eliminates he need or trial and error by
analyzing the error between he model output and the observations, and then
in such a way as to produce an output which
way, we may iterate towards a solution.
perturbing the model parameters
will produce ess error. In this
Mathematically'
)F(MO)
= F(Mo) aT •M,
MO--nitial odel,
M: true earth model,
AM: change n model parameters,
F(M) : observations,
F(Mo): alculatedaluesrom nitial
•)F(M )
.2 • = changen calculatedalues.
model, and
F(M)
where
Part 8 - Model-based Inversion
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Introduction to Seismic Inversion Flethods Brian Russell
IMPEDANCE
4.6 41.5AMPLITUDE
ml
I
ß
,
- ii
,i i,
i
i
ii
,
ß
ß
,
, i
:.
__
IMPEDANCE
(GM/CM3) FT/SEC) 1000
41.5 4.6 41.5 4.6
i i
41.5
b c d e
Figure 8.2
A synthetic test of the GLI approach to model based
inversion.
(a) Input impedance. (b) Reflectivity derived from (a)
with added multiples. (c) Recurslye inversion of (b).
(d) Recurslye inversion of (b)convolved with wavelet.
(e) GLI inversion of (b). (Cooke and Schneider, 1983)
Part 8 - Model-based I nversi on
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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 nverse f A.
However, since there are usually more observations than parameters (that
is, n is usually greater than k) the matrix A is usually not square and
therefore does not have a true inverse. This is referred to as an
overdeterminedcase. To solve the equation in that case, we use a least
squares solution often referred to as the Marquart-Levenburgmethod see Lines
and Treitel (1984)). The solution is given by
•M: (AT'A)-IAZ•F.
Figure 8.1 can be thought of as a flowchart of the GLI method f we make
the impedanceupdate using the method ust described. However, we still must
derive the functional relationship necessary to relate the model to the
observations. The simplest solution which presents itself is the standarO
convol utional model
s(t) = w(t) * r(t), where r(t) = primaries only.
Cooke and Schneider (1983) use a modi ied version of the previous formula
in which multiples and transmission losses are modelled. Figure 8.2 is a
composite from their paper showing he results of an inversion applied to a
single synthetic impedance trace.
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• ' • IMP.EDANCE1OOO
(GM/CM3)(FT/$EC)
ß ._ . .:. . . .........•::...., .. ... .. :... O
.o . ß ß ,, ,, ? "e'. ,,
. .:-: . .• ..... : :........:..:.-.-_- ........ , ß ....-. -.
4': ::.•/-.:. i i..::..':...:......:.':i•i.'-'-:....'...'......-...•.•.::
..'." .
• ' 300M$
.
,
Figure 8.3
2-D model to test GLI algorithm. The well on the right
encounters gas sand while the well on the left does not.
(Cooke and Schneider, 1983)
Figure 8.4
AMPLITUDE
Model traces derived from
m)del in Figure 8.3.
{Cooke and Sc)•neider, 1983)
Part 8 - Model-based •nversion
Figure 8.5
IMPEDANCE
(GM/CM3 (FT/SEC)X1000
10 38 10 38
,,,.l A B
GLI inversion of model traces. Compa
with sonic log on right side of Fi•iure
(Cooke and Schneider, 1983
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I
I
YES ___•__•J
' ' FINALMObE
- _ ._ x•, • .... r -• •;•,• -.-'%•..
-cx-r. . . . .-. .,'•_;'•.:.
,• . . t .•..
Figure 8.6
I11 ustrated flow chart for the SLIM method.
(Western GeophysicalBrochure)
Part 8 - Model-based Inversion
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ntroducti on to Sei stoic nver si on Methods Brian Russel 1
8.3 Sei_smic_ithologicModelling ,SLIM)
Although the n•thod outlined in Cookeand Schneider (1983) showed much
promise, it has not, as far as this author is aware, been implemented
commercially. However, one method that appears very similar and is
commercially available is the Seismic Lithologic Modeling (SLIM) method of
Western Geophysical. Although the details of the algorithm have not been
fully released, the method does involve the perturbation of a model rather
than the direct inversion of a seismic section.
Figure 8.6 shows a flowchart of the SLIM method taken from a Western
brochure. Notice that, as in the GLI method, an initial geological model is
created and comparedwith a seismic section. The model is defined as a series
of layers of variable velocity, density, and thickness at various control
points along the line. Also, the seismic wavelet is either supplied (from a
previous wavelet extraction procedure) or is estimated from the data. The
synthetic model is then comparedwith the seismic data and the least-squared
error sum is computed. The model is perturbed in such a way as to reduce the
error, and the process is repeated until convergence.
The user has total control over the constraints and may incorporate
geological information from any source. The major advantage of this method
over classical recurslye methods is that noise in the seismic section is not
incorporated. However, s in the GLI method, •hesolution is nonunique.
The best examples of applying this method to real data are given in
Gelland and Larner (1983). Figure 8.7 is taken from their paper and shows an
initial Denver basin model which has 73 flat layers derived from the major
boundaries of a sonic log. Beside this is the actual stacked data to be
inverted.
Part 8 - Model-based I nversi on
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1.4
1.6
1kit
,1.4
2.0
Initial
Figure 8.7
lkft
Stack
Left' Init)al Denver asinmodel eismic.
Right: Stacked section from DenverBasin.
(Gel and and Larner, 1983).
2.0
.4
1.6
1.8
1.8
2.0 •
Fieldata Synthetic Reflectivity 2.0
Figure 8.8 Left: F•na• SLIM JnversJon of data shown 1n
Figure 8.7 spl iceU into field data.
Right- Final reflectivity from inversion.
' -- _• -- __-__i m - ' -' (Gelfand and Larner, 1983). • .......... .m:
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In Figure 8.8 the stack is again shown in its most complexregion, with
the final synthetic data is shown fter 7 iterations through the program.
Notice the excellent agreement. On the right hand si•e of Figure 8-.9 is the
final reflectivity section from which the pseudo mpedance s derived. Since
this reflectivity is "spi•y", or broad band, it already contains the low
frequencycomponentecessaryor full inversion. Finally, Figure 8.10 shows
the final inversion compared ith a traditional recursire inversion. Note the
'blocky' nature of the parameter ased nversion when comparedwith the
recurs i ve i nvers i on.
I n summary, parameter
which can be thought of
reflectivity is extracted.
propagated hrough the final
based inversion i s an iterative model1 ng scheme
as a geology-baseddeconvolution since the full
I• has the advantage that errors are not
result as in recursire inversion.
Part 8 - Model-based
Inversion
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Introduction to Seismic Inversion Methods Brian Russell
w
1
500-ft 114mile 114mile E
lkft •
-
.5
l m
ß
.7
1.9
Figure 8.9 Impedance section derived from SLIM inversion of
Denver Basln 1 ine shown n Figure 8.7.
{GelfanU and Larner, 1983)
W
1.7
50011 114mileS 114mile
lkft ß ß .• E
19
F gu e 8.10
Traditional recursire inversion of Denver Basin line
from F gur. 8.7.
(Gelfana anU Larner, 1983)
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Introduction to Seismic Inversion Methods Brian Russell
Appendix- Mat_r_ixappljc.•at.ons_inGe•ophy.s.ics
Matrix theory showsup in every aspect of geophysicalproocessing.Before
looking at generalized matrix theory, let us consider he application of
matrices to the solution of a linear equation, probably the most important
application. For example, let
3x1+2x : 1, and
x1- x2 = 2.
By inspection, we see that the solution is
However,we Could .haveexpressed the equations in the matrix form
or
A X = y,
3 2 x1
1 -1 x2 ß
The sol ution is, therefore
or
-1
x = A y,
x1 1 . -2 1
-1/5
1 3 1
x2
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Introduction to Seismic Inversion Me.thods Brian Russell
is of little
overde termi neU
problems:
In the above equations we had the same numberof equations as unknowns
and the problem t•erefore had a unique answer. In matrix terms, this means
that the problem can be set up as a square matrix of dimension N x N times a
vector of dimension N. However, in geophysical problemswe are Uealing with
the real earth anU the equations are never as nice. Generally, we either have
fewer equations than unknowns (in which case the situation is called
underdetermined) or more equations than unknownsin which case the situation
is calleU overdetermined). In geophysicalproblems, he underUeterminedase
interest to us since there is no unique solution. The
case is of much nterest since it occurs in the following
( ) Surface consi stent resi dual
(2) Lithological modelling,and
(3) Refracti on model ng.
statics,
The overdetermined system of equations • can
categories- consistent an• inconsistent. These
extending our earlier example.
be split into two separate
are best described by
(a) Con•s.i••t Overd..eterminedn.earEqua.t.on.s
In this case we
equations are simply
reUunUant equations may
square matrix case.
earl ier example,
have more equations than unknowns, but the extra
scaled versions of t•e others. In this case, the
simply be eliminated, reducing the prø•lem o the
For examp.le, consider adding a third equation to our
so that
anU
3x1+ x : 1,
x1- x2 : 2,
5x - 5x : 10.
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This may be written in matrix form as
2 x1
x
o
But notice that the third equation is simply five times the second, and
therefore conveys no new information. We may thus reduce the system of
equations back to the original form.
(b) Inco,s,s, en•tO•verd.•ermine.L.i.near qua•i.on?
In this case the extra equations are not scalea versions of other
equations-in the set, but conveyconflicting information. In this case, there
is no solution to the problem which will solve all the equations. This is
usually the case in our seismic wor• and indicates the presence of measurement
noise and errors. As an example, consider a modification to the preceding
equations, so that
3x1+2x -- 1,
x1- x2 -- Z,
ana 5x - $x = 8.
This may be written in matrix form as
3 2 x 1
I 2
- x2
-5 8
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'Now the third equation s not reducible to either of the other two, ana
an alternate solution must be found. The most popular aproach is the method
of least squares, which minimizes the sumof the squared error between the
solution and the observed results. That is, if we set the error to
e=Ax-y,
then we si reply mini mize
eTe--e , ez .......
n
, en = e ß
2
Le.
Re expressing the 'preceding equation in terms of the values x, y, and A,
we have
ß E = eTe (y - Ax)T(y Ax)
= yTy xTATyyTAx xTATAx.
We then solve the equation
bE_
bx
The final solution to the least-squares problem is given by the normal
equa i OhS
AT x = A y
or x = (ATA)-lATy
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Sei smi c Travel time
Inversion
9,1 Introduction
_ _ • L _ , _ . _
In this section we will look at a type of inversion that goes under
several names, incluUing traveltime inversion, raypath inversion, ana seismic
tomography. The last term tenUs to be overuseU at the moment, so it is
important to use the term correctly. In section 9.3 we shall showan example
whic• may be considerea as seismic tomography. As all of t•e other names
suggest, however, seismic traveltime inversion uses a set of traveltime
measurements to infer the structure of the earth. The parameters which are
extracteU are velocities and depths, aria [herefore a gross model of earth
structure can be derived. Initial)y, this was considered the ultimate goal,
but Jr'has becomeobvious that this accurate set of velocity versus depth
measurements can be used effectively to constrain other types of inversion.
For example, the'velocities could be used as the low frequency component n
recursire inversion, or as the velocity control for a depth migration.
The way in which traveltime inversion is carried out is to first pick a
set of times from a dataset. These picks m•y come from any of three basic
types of seismic datasets-
Surface seismic measurements
- shots and geophones on the surface,
VSP measurements
- shots. on surface, geop•ones in well,
Cross-hol e measurements
- s•ots anU geophonesboth in well.
and
Once the times have been picked, they must be made to fit a model of the
subsurface. In the next section, we will look at some straighforwara examples
of using traveltime picks in order to resolve the earth's velocity and depth
structure.
Part 9 - Travel time I nversi on
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9.• Numerical_Exa•mplesf .Travelti In•v.rsion
Considerhe simplest ossiblecase, a constantelocityearth. Figure
9.1 showshe travel paths that would esult from the three geometry
configurationsgiven a square area of dimension L by L. Note that the
traveltimes in Figure 9.1 would simply be:
(1) Surface sei smic'
(z) vsP-
(3) Cross-hol e'
t--Z L p or p-- t/Z L,
t --•L p or
p -- t /i•L, ana
t=Lp
or p=t/L,
where p -- / V.
Obviously, all three sets of measurements ontain the same nformation.
However,f the velocity or slowness) and he depth are both unknown,
neitheronecanbe determinedroma single imemeasurement.nevengreater
ambiguity comes into play if we have a single measurementut more than one
box. In Figureg.g this situation is shown. Notice that the equations ow
would involve three unknowns nd only one measurement.
A moregeneralmodel s proposedn Bishop t al (lg85) an• Bor•inget al
(1986). The earth is represented s a number f boxesof constantsize and
velocity. lthoughhevelocityf each oxs a constant,hevelocity ay
vary from box to box. This is shownn Figure 9.3. The objective is thus to
computehe seismic travel path through each box using the traveltime
measurements.key problem ere s howo allowthe rays to travel through
the boxes. The first order approximationwould be straight rays with no
bending. However, f Snell's law is use4, the problembecomes oredifficult
to sol ve.
Part-g Travel i'i me'-'in'ver's on .....
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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)
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Let us apply the straight ray approximation n the simplecase of having
simply two blocks of different velocity. In this case, we have coupled
together both surface and VSP measurements. Two possible recorUing
arrangementsare shown n Figures 9.4 and 9.5. The situations illustrated are
obviously oversimplified since we have assumed straight ray approximation in
both boxes. That is, there is no refraction at the velocity discontinuity,
and the reflection point is directly at the center of the two boxes. However,
if we assumehat the velocities vary only slightly, this approximation is
reasonable.
Let us start with the situation illustrated by Figure 9.4. In this case,
t•ere is a single shot with geophonesboth on the surface and in a borehole at
the base of the layer. If we assume that the sides of the boxes are unity in
length (1 cm or m or km. , the travel time equations are
(1) For the. raypath from S to R
where Pl: 1/velocity n box1
P2: 1/velocity n box2
(Z) For he raypathrom to R2:
t2=•Pl P2.
2
Thus, the total problem can be expressed in matrix form as:
• • Pl tl
•r• •]• : or Ap: t .
• 2 P2 t2
The solution to the previous equation is then
p = A-lt.
Unfortunately, a quick try at solving the above equation will show that
the Ueterminant of A is O, which means that the inverse is nonrealizable.
Physically, this is telling us that the two travel paths spene equal
proportions of their paths in eac• box.
Part 9 - Travel time Inversion
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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.
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A simple way to remedy this situation is to move the shot for the second.
raypath. This is shown n Figure 9.5. In this situation, we have moveU he
shot one-half a box length to t•e left for the recorUing in t•e hole. In this
case, the traveltime equations are
(1) For the raypathromS1 to R :
tl: 1•Pl+ P2
(2) Forthe raypathromS to R -
In this case notice from the diagram hat
tan 0 : 1/1 $ : 2/3 = 0 6667, or B : 33 69o
Thus cos 0 = 0.8320
and (see figure) x = 1/(2 x 0.832) = 0.6
y = 3/(• x 0.832) = 1.8
y-x=l.2
Therefore
t2:1.2 Pl + 0.6 P2
Thus, the total problem can be expressed in matrix form as'
1.2
•[• Pl tl
0.6 P2 t2
with sol ution
Pl
P2
1
o.85
0.6 - 2 t 1
-1.2 2 t 2
Problem' Try to solve the above equation when the two velocities are 1.0
and 1.1 kin/sec. T•at is, work out the traveltimes and plug them into the last
matrix equati on.
Part 9 - Travel time Inversion
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Initial
Model
Layer
Stripping
.Inversion
Estimate elocity t well
using onicogandVSP
Pick seismic
reflection imes,
Estimate(x,z)byusing (xo,z),
the reflection traveltimes and the
theassumptionf verticalays
Start with
top ayer
Computerorward
model raveltimes,,
by normal aytracing
Perturb (x,z)
by east quares
or manually
It- fll'
Add another
layer
Final
Seismic Model
layers een
ii
Models omplete,,
Figure 9.6
A possible lowchart for seismic raveltim inversion.
(Lines et al, 1988)
Part 9 - Travel time Inversion
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9.3 Seism•ic T..omo.raphy
The term tomographyas first used in the medical ield for the imaging
of human issue using Nuclear MagneticResonanceNMR) and other physical
measurements. In the seismic field it has come to mean the reconstruction of
the velocity field of the earth by the analysis of traveltime measurements.
Excellent overviews of tomographyare g.iven in Bording et al (1986), and Lines
et al (1988}. You will find t•at the latter paper introduces the term
"cooperative inversion" since both seismic and gravity measurements re used
in the inversion, but that much of the technique used by the authors can be
cons alered sei smic tomography.
Figure 9.6, taken from the paper'by Lines et al (1988), shows the flow
chart that they propose for performing traveltime inversion. This method can
be considered quite general, even though many variations of it are used in the
industry. Basically, the process starts with an estimate of the model which,
in the flowchart shown n Figure 9.6, is deriveU from the sonic log and VSP
measurements. Next, traveltime picks are made from the seismic data. In this
case, stacked CDPdata is usecl, but the shot profiles (or CDPprofiles) could
,
also be used. As well, travel time picks can be made from VSP data and
refraction arrivals. In the next stage of the process, the model is
raytraced, and an error is computed between the computed and observea
traveltimes. Based on the error computed, a new model is computeU. This is
done using the GLI technique described in Chapter 7 of these notes. In the
procedure shown n Figure 9.6, the inversion is done layer by layer until the
model is complete.
Although any traveltime inversion can be considered tomography, Dr. Rob
Stewart (personal communication) points out that to be analagous with the
medical field, where physical measurements are taken completely around t'he
imaged object, a true seismic tomography experiment would involve aata on more
than one side of the portion of the earth to be imaged, such as surface
seismic and VSP.
Part 9 - Travel time Inversion
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•0
WlC
_ m m i roll i
-1Z 0
x
I• vsP SOURCE
..2
• 3-D SOURCE • GEOPHONE
Figure 9.7
Surface geometry for tomographic maging example.
(Chiu and Stewart, 1987)
Une89 89D•B 89 89UneDP 8901 8921 8960 8980 CDP
0.0 ..........
•::•=•'•: •.::"--'::':.-:'::.i•r.:iE)•".Z• ;.".•h..
0.1 ---': ..... -" '•'•":'":
Well C VSP
Depth (m)
185 9O7 205 460 730 895
0
fi'•L .o.• ß • mo•w,• .'•.' •
:•(;:• • ....... • .• --'-..
oJ
0.4
o.5
ß
. ..
(b)
•1o ?6o 895
(a)
Fi gue 9.8
(a) Picked events on 3-D seismic..
(b) Picked events on VSP.
Part 9 - Traveltime Inversion
(Stewart and Chui,
.....
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An example of using multiple datasets for seismic tomography is found in
Stewart and Chiu (1986), and Chiu and Stewart (1987). The objective was a
Glauconitic channel sand which containeU heavy oil. Since this was a
development survey, a lot of measurementswere available to image the
subsurface, including well log data, VSP, and 3-D seismic. Figure 9.7 shows
the •ensity of information along a portion of one seismic line. Figure 9.8
shows the various datasets used in the tomographic maging. Figure 9.8(a)
shows he stacked seismic data with the key events indicated and Figure 9.8{b)
shows he picked VSP from well C. Finally, Figure 9.9 shows he well l'ogs and
synthetic from a different well, clearly indicating the Glauconitic channel.
The tomographic technique involved picking events from both the VSP first
arrivals and the prestack 3-D seismic data. Traveltime inversion was done by
the technique described in Chiu and Stewart (1987). The method involves
starting with a simple modelof the subsurfaceand perturbing this model using
the errors between the picked traveltimes and the raypath times through the
model. This method differs from the method shown in Figure 9.6 since
raytracing is done a nonzero source to receiver offset, and also the VSP data.
To test the method, Chiu and Stewart created a synthetic model. Figures
g.10(a) and (b) show raytrace plots for the VSP and surface Uata,
respectively, through this model.
ZERO PHASE
BANDPASS
10/15 - 80/110 Hz
NORMAL
Figure g. g Wel1
RFC
DENSITY (kg/m 3 )
VE OCITY m/sec)
030O
till ß
SOIl
IO# ß
log curves and synthetic showing Glauconitic channel.
(Stewart and Chiu 1987)
lime
(sec)
Part 9 - Travel time I nversi on
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0.0
offset (krn)
1.o 2.0
# $Oul•CE
A •OPHONE'
i
i
i i i i
(a) {•)
Figure 9.10
(a) Surface raypaths through model used to test inversion.
(b) VSP raypaths through model.
(Chiu and Stewart, 1987)
Offset (km)
0.0 1.0 2.0
, ii i 1 - ---
a
Voity
0.0 2.0 4.O
Figure 9.11
Results of tomographic nversion of model data
using VSPand surface data. (Chiu and Stewart,
1987)
Part 9 - Travel time Inversion
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Figure9.11 showshe results of the inversionprocess singboth he VSP
and surface seismic data. To make he test morerealistic, random oise was
addedo travel'timeicks. Notice that the correct esult hasbeen btained
in four iterations.
Let us now return to the case study described initially. The final
velocity/depth model is shownn Figure 9.12. Notice that the velocities fit
quite well with the averaged onic log velocities. This velocity model was
used o produce oth a depthmigrated seismic section, shownn Figure 9.13,
and a full seismic nversionbasedon the maximum-likelihoodechnique. The
final inversion is not showndue to colour reproduction limitations.
As can be seen in Figure 9.13, the Glauconitic channels have been well
delineated. The depth tie is also excellent. The conclusion that the authors
makes that if severaltypesof geophysical easurementsan be intergrated,
the result is an improvedproduct. Each set of data acts as a constraint on
the others.
Part 9 - Travel time Inversion
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Offsetkm) Velocity an/s)
-1.0 0.0 LO 0.0 3.0 6.0
TC)iliO6RAPmCC)
INVERSION
- SONCL06 (1:2)
Figure 9.12
Results of tomographic inversion of G1auconitic channel.
(ChiU ndStewart,
1987)
. : •m,,, .......... J•
ß ß . ... l..,.,.,;,,•. ' 't ''•"','
ß -.--:'
._:_.4sl•l_,' i,?a•..:. .,,t.:,
800 ..
:
900 :'": ""' ""' ....... '
Depth
(m) 1000
11oo
1200
1300
1400
F gure g. 13
Depth igrationf seismicatashownn-Figure.8(a).
Tomographicelocities of Figure9.12 havebeenused.
(Stewart an• Chiu,
Part 9 - Travel time Inversion
Page 9 - 15
1986)
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PART 10 - AMPLITUDE VERSUS OFFSET INVERSION
Part 10 - Amplitude versus Offset Inversion
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10.1 AV.Oheo.•y.._
Until now, we have discusseO only the inversion of zero-incidence seismic
traces. That is, we have considered each reflection coefficient to be the
result of a seismic ray striking the interface between two layers at zero
degrees. In this case, the 'reflection coefficient is a simple function of the
P-wave velocity and density in each of the layers. The formula, which we have
seen many times, is simply
i+lvi+ - ivi zi+- zi
ri= Yoi+iVi+l+OV •Zi+l Z
where r: reflection coefficient
yo: density,
V = P-wave vel oci ty,
Z: acoustic impedance,
and Layer i overlies Layer i+1.
When we allow the seismic ray to strike the boundary at nonzero incidence
angles, as in a common hot recording, a much .more complicated situation
results. In this case, there is P- to S-wave conversion and the reflection
coefficient becomes a function of the P-wave velocity, S-wave velocity, and
density of each of the layers. Indeed, there are now four curves that can be
derived: reflected P-wave amplitude, transmitteU P-wave amplitude, reflected
S-waveamplitude,and transmittedS-wave amplitude. The variation of
ß
amplitude with offset also involves another physical parameter called
Poisson's ratio, which is related to P-and S-wave velocity by the formula
(Vp VS•- Z .
•' =-
Poisson's ratio can theoretically vary between 0 and 0.5.
Part 10 - Amplitude versus Offset Inversion
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$i
Sr
at, •t
BOUNDARY
(X2' 2
t
$•
Figure 10.1
Reflectedandtransmitted ays created when P-wave
strikes the boundary etweenwo layers.
(Waters, 1981).
•o, +,•sin2•,' 'cos2•,-•x,n-•:/D,/•-cos2+,/
Figure 10.2
Zoeppritzequations hichdescribe he amplitudes
of the rays shownn Figure 10.1.
(Waters, 1981 .
Part 10 - Ampl rude versus Offset Inversion
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The equations from which the ampl'itude variations can be derived are
callea the Zoeppritz equations. They are derived from the continuity of
displacement and stress in both the normal and tangential directions across an
interface between two layers. Figure 10.1 shows he seismic rays across a
boundary, and Figure 10.2 gives the final form of the equations. They are
taken from • textbook by Waters (however, someof the signs were wrong, and
they are fixed in the diagram). Since we have four equations with four
unknowns, hey can be rearranged in the form of a ½ x 4 matrix equation
Ax--y
with soluti on
x = A-ly
Over the years, several authors have discussed amplitude versus offset
effects. However, these authors concluded that the effect would be negligible
on seismic data. In a landmark paper, Ostrander (1984) showed hat for a
significant change n Poisson's ratio, a major change n the P-waveamplitude
coefficient can be seen as a function of offset. This Poisson's ratio change
is most noticeable in a gas sand, where the ratio can change from 0.4 in the
encasing shales to as low as 0.1 in the gas sand itself. Ostrander showed
that, in such extreme cases, the P-wave reflection coefficient can go from
positive to negative for a decrease in Poisson's ratio coupled with an
increase in P-wave velocity, or from negative to positive for an increase in
Poisson's ratio coupled with a decrease in P-wave velocity.
Figure 10.3(a) shows he gas sand model that Ostrander used and Figure
10.3(b) shows the result of amplitude versus offset modelling of the P-wave
reflection coefficients. Figures 10.5(a) and (b), also taken from Ostrander,
shows that this effect can inUeed be observeU on a common offset stack.
Figure 10.5(a) shows stackeO seismic section witl• three apparent "bright
spot" anomalies. Unfortunately, only wells A anU B were productive. The three
common ffset stacks, shown n Figure 10.5(b), indicate that only locations A
and B actually Uisplay an AVOeffect.
Part 10 - Amplitude versus Offset Inversion
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GAS ':*"*•*•t Vl• =8.000
/32 -"2.14
:o.,
;..•.• :., ß
SHALE •---'
$=10.000
/4) =2.40
(•'3 =0.4
Figure 10.3 (a) Synthetic gas sand model.
(Ostrander,
1984)
0.41
0.3
IN SAND
0.2
t.., 0.1
0
0
..,
-0.2
I0 o
ANGLE F NCIDENCE •,-'-e'
20 o 30 ø 40 ø
NO GAS
., ,,,,,.ooo.o.o.... ..ooo.,o.,.,o.,-'.ø.,,,o,,o*o .....
-0.3
-0.,4
Figure 10.3 (b)
for reflections from top and bottom interfaces of model
s•own in Figure 10.3 (a).
• (Ostrander, 1984)
, , , IlL _ -- _, 11 i , i m m im , ß
Part 10 - Amplitudeversus Offset Inversion Page 10 -
Computedeflection coefficients as a function of offset
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The method useU to identify this effect is only partly qualitative, and
can be diagrammedas shownbelow-
INPUT SEISMIC
SHOT PROFILES
COFFSTACK
BUI L D MODEL
...
VISUAL
COMPARISON -
MODEL MATCHES
REALITY
. .
COMPUTE
SYNTHETIC
. •
NO
im m
MANUALLY
CHANGE PARAMETERS
,m
Figure 10.4
Flow Chart for Manual AVO Inversion
Obviously, this visual meth'odf comparisoneavesmuch o be Oesired.
We will therefore look at several methods for the qualitative inversion of AVO
data, both of which have been looked at previously in the context of
normal-incidence inversion.
ß
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1•0 170 160 1•50 140 130 120 110 100 90 •0 70
0.0 ' , .... • , -- • , • • • • ' ' -' • •- ' -• " • •
-• • •. : ' 't .". '.'..' t'-' " : "• :. ' ' : .... I; ' ' ' 0.0
'*O..' m * re- ß ß l, ß ' ß I , I i I, mm I.. i ,.
' i i i. ii I I ß IIII II i ß
..,.,,.,..•..,.,,...,",',,-.....-,,,,'.,,.-,',,,,,....,,.,....,,'........-•~.,......,,.'""' ... .'_'•"' . ... •,,., ................
'i:
,:•:;.-':, • ..=•..•:.'':."•i '.•.•.'-'?'?'?'?'?'•L--•,•.•'•.•.•..,'::... ß .._ .o..•:?;i•. -- ,..•.•... _'_-•..•
' --?.-•" ":'.. : ' • ...... "=-" il •;,.•:.?:•-•=;.•'.'•..='.:•:-1: . . •-..c• ,•.•. .-...,_•....:,•.•.:..•,.•;,..-.
• • .*...... . ß .- -...- - . ; ...... ..• .•:..;,-.;:.•.. _•:_- .•...4..... <?--r..-.. . .:.;. ,""-•.•r..•_-•".::
0.5 •'.l_'.-•. .: : •_•_•..._. .... _ -:.:...:..._...•...... . _:... ;._.•.=0.5
•,'*' ':'.:-r-'_.•.•;...... . .
1.0
2.5
I.,...,:,•....,;.•......... . ...-..
...
....,.., t,,,
"_,,,d,.•,•l.leeile*e,,I,'t:lit•ll""'It•'•l';I;,d
.......... 2.0
. ..
,•e.•. Illl•lll.1111el'•-ß .le
- .. • '."-'•1,.'•1•-.
........... ;.;... :.....i:-;
ß.:--;•=....=:;;:.1•... ...,;•.".....
' ........ "•"":":........................5
,..,- .1,•.----?'"1 ß ß - .... ,.....-.,.-.,.-. -,., .......... ... ........
"' ' "1;;::=... ":".... ""':"'"1'.'-'---' "•':,';;;;":',::..."
ß ........::.;:..
.. .... :.-.'.. . '}i:.;.=i.•;-."':;::.'.:•:.'
ß ß
Figure 10.5(a)
S•acked seismic line showing"brigh• spot" anomalies.
Loca%ions A an• B are known gas.
(Osl:rander, 984)
.... titIll,
,e*'11:,l:, ol,, ....
' ' I• ' ........
ß .
. ,
6952' 1012 ß
SP 80
":l:1111il•
ß .
eellie
;;;;;;;;i;il
,,
..111tl•
6•$•' 101
Fiõure 10.5(D)
Commonffset sl;acksover locations A, B, andC from
stacked section in Figure 10.5/a). Notice the AVO
increase on A and B.
(Os ;ranOer, 1984)
Par[ 10 - A,•pli[ude versus Offset Inversion
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10.2 AVO Inversion by GLI
Recall that in the theory of generalized linear inversion (.GLI) there
were three important components' geological model of the earth, a physical
relationship between the earth and a set of geophysical measurements,anU a
set of geophysicalobservations. This methodWasdiscussean both chapters8
anU 9, applied to stacked data inversion an• traveltime inversion,
respectively. Now, let us apply the method to unstacked data. The result
wil 1 be cal led AVO inversi on.
In secti on 10.1, the three components needed to perform GLI inversion on
AVOUata we,re escribed. Ourmodel of the earth is a series of layers with
t•e el astic 'parameters of P-and S-wave vel oci ty, density, and Poi sson'S ratio.
Our physical relationship between this model ana seismic CDP profiles was
derived using the Zoeppritz equations. And, finally, the observations are the
picked amplitudes and times of events on a CDP profile or common ffset stack.
By computingderivatives from the Zoeppritz matrix, it is possible to set up a
GLI solution to t•e AVO problem similar to the solution found for zero-offset
data. This solution is
aF Mo•)
FIM) F(M)+ •)M bM
where
Mo: nitial earthmodel,
M: true earth model,
AM: change in model parameters,
F(M) -- AVOobservations,
F(MO): oeppritz alues rom nitial model, nd
•)F(M )
i)--••: changen calculatedalues.
The implementation is simply a variation of the manualmethod, anO is
sinownon the next page.
Part 10 - Amplitude versus Offset Inversion
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INPUT SEISMIC
SHOT PROFILES
COFFSTACK
i
PICK
AMPLITUDES
COMPUTE
ERROR
COMPUTE
SYNTHETIC
STORE COMPUTED
AMPL TUDES
i t •
COMPUTE MODEL
PARAMETER CHANGE
US NG GLI
NO
ERROR
YES
MODEL MATCHES
REAbITY
,
Figure 10.6
AVO inversion by the GLI method
Part 10 - Amplitude versus Offset Inversion
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Wewill now ook at an example of GLI inversion of amplitude versus
offset data. First, consider the integrated well logs shownon the left hand
panel of Figure 10.7. Actually., only the sonic log or P-wave og wasrecorded
in the field. The density log was derived from the sonic using Gardner's
equation, the Potsson atio was ixed at 0.25, and the S-wavewasderived from
the P-wave and Potsson ratio logs. On the logs, three layers have been
blocked at depth and a significant Poisson's ratio changehas been ntroduced
in the middle block. On the right hand side of Figure 10.7, notice that the
amplitudeversusoffset curves have been displayed for the third layer. As
predicted earlier, the P-wave reflection coefficient displays a strong
increase of amplitude with offset.
Figure 10.8 showshe same et of blocked ogs on the left, but showshe
seismic response f the amplitude change on the right. This synthetic was
produced y simply' eplacing the zero-incidence mplitudes ith the amplitudes
derived from the Zoeppritz calculations. The events between 00 and 700 msec
display a pronouncedmplitude changewit h offset.
ZOEPPRII'ZIHI:LE NTERFFJCE
TESTLOESTLOESi'DEE•-S t•;$TPO
• 2• 2,5 4;8
,,,
Eq, , t: 3 Ti,•: $7• Depth: 795
589 1998
Of*f's•:c'.........................
. i Reflected P-Wave
..... Transmitted P-Wave -9.8)
...... Ref'l ected S-Wave
........... Transmitted S-Wave
ß , mm i
m
Figure 10.7 Blockedwell logs on left, with computed oeppritz curves
for layer 3 on right.
_ _
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Introducti on to Sei smi c Inversi on Methods Brian Russel 1
20EPPRITZ •EFLECTIUITY tlODEL
TESTLOG TESTLOG IEH$ITY1 S-I, FIUE POISSOH1
u•Ym u•/m 9/½½ usam
...... ,L -
ß•..o...-.. .............. , ....,....
•ee-..........................•"........
.....................'.........,.
::...........................•..........
"
268 268 2.5 468 .$
MODEL (meters.)
EU 909 727 545 363 181
.
Figure 10.8 Left-
Right:
A "blowup" f the blocked ogs shownn Figure 10.7.
A synthetic commonoffset stack and the AVO curves
shown on the right of Figure 10.7.
Part 10 - Amplitude versus Offset Inversion
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However, does the change seen in Figure 10.8 reflect the reality of the
situation? Figure 10.9 showsa set of CDPgathers which correspond to the
model. The gathers are a realistic modelled dataset and were generatedwith
no change n Poisson's ratio. Since the gathers are noisy and contain fewer
traces han he synthetic DP rofile shownn Figure 0.8, theywereusedo
create a common ffset stack. The geometry of this st.ack is described in
Ostrander's aper, and the resulting gathersare often referred o' as
Ostrander gathers. Traces within a CDP/offset window were gathered and
stacked, resulting in increased signal to noise. Figure 10.10 showsa display
of the logs, synthetic model, and commonffset stack. The mismatch in
amplitudes is now obvious.
ß
ß Next, the amplitudes of the event on the contanon offset stack
corresponding o the event displayed in Figure 10.7 were picked. The event
above the anomalous ayer was also picked. The picks were then used along
with the computedamplitude versus offset curve to invert the data by the GLI
method. In the inversion, two parameters were allowed to vary- the Poisson's
ratio in the layer of interest, and a scalar which relates the magnitude of
the seismicicks o themagnitudef theactualamplitudes.
Figure 10.9 CDPgathers from a seismic dataset corresponding to
synthetic shown n Figure 10.8.
Part 10 -Amplituae versus Offset Inversion
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I ntroductin o Sei toic nversinMethods B i an Russel
I NVER$]ON FULL HOIIEL
TESTL TESTL )EN$I $-WI:IU OI$$
_ us/m u•/• g/cc us/•
I ß
EU
rIO]]EL (meters)
909 727 545 353 181 0
COFFSTK1 n,elers )
838 6•4 498 :)32 li•E; 0
50
I
2•0 2•0 2.5 4•0 ,5
Figure 10.10 A comparison of the synthetic coneon offset stack from
Figure 10.8 {middle panel) with a con,nonoffset stack
created from the CDPgathers of Figure 10.9 (right panel).
T•e left panel shows he blocked well logs from which
the synthetic was created.
Part 10 - Amplitude versus Offset Inversion
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Introduction to Seismic Inversion Methods Brian Russell
The results of this inversion are shown in Figure 10.11. The figure
showshe changen Poisson'satio before nd fter inversiondashedine
before, solid line = after) on the left hand side. On he upper ight is
shown he match etweenhe observedicks n the upperayer (showns small
squares)nd he final theoretical urve{s•ownsa solid ine). The ower
right showshe same hing for the lower layer.
Finally, Figure10.1Zshows he comparisonetween he coanonffset
stackand he syntheticmodel fter the model asbeen ecomputedith the new
amplitudehangesrom he updatedoisson'satio. Notice he improvementn
the match.
II•'RSIOH SIN•E LI•ER: I101•ELI
70O
6,8
i i i i
Poi•s•s Ratio
. ß
e .e•
0.042
6.666
0.048
6.624
6.606
Ewnt (2) P. ove Laver
. . ..
O•'•'r•
Event (3) Belo4aLaver
O O
0
e.• e S5e
O('•set ( m
Figure 10.11
The esultsof a GLI nversion etweenhe computed,mplitudes
of Figure 0.7and he picked mplitudesrom heconmon
offset stackof Figure10.10. Thedashedine on he plot
on the left is the Poisson'satio before nversion,and he
solid line is after inversion. Theplots on the right show
the new omputedurves ith the picks squares)uperimposed.
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IHUER$IOH FULL MODœL
MOIlEL2 me•er$) COFFSTKI ( meters )
EU 909 727 545 363 lB1 B 838 664 498 332 166
Figure 10.12
A replot of Figure 10.10, where the synthetic has been
recomputed sing the newPoisson's ratio value.
Part 10 - Amplitude versus Offset Inversion
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[nt•oduct• on t• $e• sm•c [nve•sJ on Methods B• an Russe• ]
PART 11 - VEI:OCITY INVERSION
Part 11 - Velocity Inversion
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Part 11 - Veloci..ty .n.v.rsi on
_
11.1 ntroduc ti on
The last
inversion. Alth
acutally fit in
been dtscussing
topic to be discussed n these notes is the topic of velocity
ough this technique is referred to as inversion, it does not
to the narrower category of inversion techniques that we have
in this course. These techniqueshave all involved inputting
a stacked, or unstacked, seismic dataset and inverting to a velocity versus
depth section. The output of the velocity inversion described here is the
seismic section properly positioneU in depth, but still plotted as seismic
amplitudes, and still band-limiteU. As such this technique is closer to that
of depth migration.
In this section, we will look briefly at the theory of velocity
inversion, and then look at a few examples. An excellent review article on
this subject is given in Bleistein and Cohen 1982). In this article, the
theory of the method is reviewed and there is also an extensive literature
summary. Our discussion here will follow that article.
Part 11
- Velocity Inversion
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KII.OFEœT KILOFEET
-2 -1 0 I 2 -2 -I 0 I
,
(a) (b)
2
Figure 11.1
The effect of the velocity inversion methodon synthetic
data. (a) A "buried focus"effect, (b) The output from
the velocity inversion method.
(Bleistein and Cohen
KILOFEET KILOFEET
o 1 -1 o 1
1982 )
m
uJ
LL
o
....
C) ß
ii'1
(a) (b)
Figure 11.2
A second example of the effect of velocity inversion on
synthetic data. (a) Input section with diffraction,
(b) Output from velocity inversion.
(Bleistein and Cohen
......
1982 )
Part 11 - Velocity Inversion
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11.2 Theory and .Examples
The velocity inversion procedure is referred to as an inverse scattering
problem, in which the interior of the earth is mapped by inver. ing the
observations from multiple acoustic sources. (This is a long way of saying
that the seismic section is inverted ) Thus, the starting point for this
method is the acoustic wave equation. The difference between this technique
and classical migration is that perturbation techniques and integral
transforms are used rather than downwardcontinuation of the wave equation.
The initial work in this area was done by NormanBleistein and Jack Cohen
at the University of Denver. In their initial paper, Cohenand Bleistein
(197g), they employed only a perturbation technique in the inversion of
seismic data. In simple terms, this technique involves using a constant
velocity in the wave equation, perturbing this constant velocity by a small
amount, and then, by observing the backscattered wavefield, solving for the
perturbed velocity. This methodsolves for only the reflection strength of
the mapped nterfaces.
In their morerecent paper, Bleistein and Cohen 1982), a more accurate
solution was proposed which al.so solves for transmission losses and
refraction. Clayton and Stolt (1981) have applied a similar method o the
inversion of seismic data. Their method is referred to as the Born-WKBJ
method, and thus this approach to inversion is often cal led Born inversion.
Despite the differences in the mathematicsbetween he velocity inversion
methodsand migration methods, the results look very similar to those of
migration. For example, igure 11.1, fromBleistein andCohen 1982), shows
the input an• inverted result for a g-D buried focus. Note that, as in
migration, the "bow-tie" has been imaged o a synclinal feature.
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(a)
q ()f . []
ß
ß
ß
I,;,(11). iJ ll;11]ll. () I I 3l) ]. l, I ;i'llroll. IJ. I I,• O0. II
ß , •
I I11.)[11J. fl
½J
qlOO
.-.,,
c)
6500 8900 I 1 300 1 37.r.,P 16' ,' ..[]
- t ,
18b•G
ß
(b)
Figure 11.3 The effect of velocity inversion on real data.
(a) Input section (Marathon Oil), (•) Output section.
(Bleistein and Cohen 1982)
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Introduction to Seismic Inversion Methods Brian Russell
Figure 11.2, also from Bleistein and Cohen (1982), showsl•e velocity
inversion of a diffraction tail from a geological discontinuity. Notice that
the diffraction tail has been "collapsea", again as in migration.
Finally, Figure 11.3 shows n example of applying the velocity inversion
technique o a real dataset. Again, note the similarity with classical depth
migration. The fact that this section is plotted as wiggle trace only makes
the plot di fficul t to evaluate.
In summary,his technique cannotbe classed with the other methodswhich
have been discussed in this'course due to its similarity with depth migration.
However, research in'this area is continuing at a steady pace, and the
technique promises much for the future.
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PART 12 - SUMMARY
Part 12 - Summary
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lZ.1 Sgmmary
In these notes, we have reviewed the current methods used in the
inversion of seismic data. The basic model used in most of these methods is
the one-dimensionalmodel, which states that the seismic trace is simply the
convolution of a zero phase wavelet with a reflectivity sequence derived from
the earth's acoustic impedanceprofile. Flowcharts for these methoUs are
shown n Figures 12.1, 12.2, and 12.3. Let us initially summarize he
advantages and disadvantages of the three methodsof single trace inversion
which have been discussed:
(1) Recursire Inversion
_ ,• _ - •
Advan tage s:
(i) Utilizes the complete seismic trace in its calculation.
(l i ) A robust procedure when used on clean seismic data.
(iii) Output is in wiggle trace format similar to seismic data.
Di sadvantages:
(i) Errors are propagated through the recurslye solution if there are
phase, amplitude, or noise problems.
(i i) The low frequency componentmust be derived from a separate source.
(2) Spar.e-SP.ig_.Invers.on
Advantages-
(i) The data itself is used in the calculation, as
i nver si on.
(ii) A geological looking inversion is produced.
(iii) The low frequency information is included mathematically
solution.
in recursi ve
in the
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Introduction to Seismic Inversion Methods Brian Russell
BAND-LIMITED
SEISMIC
TRACE
INTRODUCE
LOW
FREQUENCY
COMPONENT
REFL
COEFF.
I INVERT
I TO MPEDANCE
IMPEDANCE
SCALE TO
VELOCITY
AND DEPTH
DISPLAY
Fiõure 12.1
Band-Limited Inversion (Recursive)
Part 12 - Summary
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Introduction to Seismic Inversion Methods Brian Russell
,
Dôsadvantages'
(i)
Statistical nature of the sparse-spike methods used are subject to
probl eros i n noisy Uata.
Final output lacks muchof the fine detail seen on recursively
inverted data. Only the "blocky" components inverted.
(3) Model Base• I nver si on
Advantages'
(i) A complete solution, including low frequency nformation, is possible
to ob rain.
(ii) Errors are distributed through the sol ution.
(iii) Multiple and attenuation effects can be modelled.
Di sadvantages'
(i) A completesolution is arrived at iteratively andmayneverbe
reached ( i.e. the sol ution maynot converge).
(ii)
The
velocity inversion, and amplitude versus offset inversion.
methods, but cannot be compareddirectly with the three
(comparing pples with oranges?).
,
The traveltime inversion method was
accurate velocity versus depth model.
constraint for either one of the
migration.
It is possible that more than one forward modelcorrectly fits the
data (nonuniqueness).
other methods which were considered were traveltime inversion,
All are important
previous methods
an excellent method for finding an
These velocities make an excellent
classical inversion methods or for a depth
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Introduction o SeismicnversionMethods Brian Russell
INTRODUCE
LINEAR
CONSTRAINTS
EXTRACT
SPARSE
REFLECTIVITY
INVERT
TO IMPEDANCE
I vELøcmTY
ANDL_•.••_•.
m i i m i
Fiõure 12.2
Broad-Band nversion (Sparse-Spike)
Part 12 - Sugary
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Introduction to Seismic Inversion Methods Brian Russell
The velocity inversion methodwas showno be very similar to depth
migration. The output from this method could therefore be used as input to
one of the other three classical methods of inversion.
Finally, amplitude versus offset inversion adds an extra dimension to the
inversion problem since it is truly a lithologic inversion rather than a
velocity inversion method. This method is definitely the method of the
future, but still has a number of hurdles to overcome. This author's humble
opinion is that once the interpreter is able to do a complete lithological
inversion on their seismic datasets, the other methodswill be replaced.
The other conclusion from this course is that the more separate datasets
(surface seismic, VSP, well log, gravity, etc..) the interpreter can use in an
inversion, the better the final product will be.
Part 12 - Sumnary
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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'•- •'" ....
ummary
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Introduction to Sei stoic Inversion Herhods Brian Russell
REFERENCES
Angeleri, G.P., and Carpi, R., 1982, Porosity prediction from
seismic data' Geophys.Prosp., v.30, p.$80-607.
Berteussen, K.A., and Ursin, B., 1983, Approximate computation of
the acoustic impedance rom seismic data- Geophysics, v. 48,
p. 1351-1358.
Bishop, T.N., Bube, K.P., Cutler, R.T., Langan, RT., Love, P.L.,
Resnick, J.R., Shuey, R.T., SpinUler, D.A., and Wyld, H.W., 1985,
Tomographic determination of velocity and depth in laterally
varying media- Geophysics, v. 50, p. 903- 923.
Bleistein, N., and Cohen, J.K., 198•, The velocity inversion problem-
Present status, new directions: Geophysics, v.47. p.1497-1511.
Bording, R.P., Lines, L.R., Scales, J.A., ana Treitel, S., 1986,
Principles of travel time tomography' SEGContinuing EUucation notes,
Geophysical inversion and applications.
Chi, C., Mendel, J.M., and Hampson,D., 1984, A computationally fast
approach to maximum-likelihood aleconvolution: Geophysics, v. 49,
p. 550-565.
Chiu, S.K., and Stewart, R.R., 1987, Tomographic determination of three-
dimensional seismic velocity structure using well logs, vertical
seismic profiles, and surface seismic data: Geophysics, v.52,
p. 1085-1098.
Claerbout, J.F., and Muir, F., 1973,
Geophysics, v. 38, p. 8Z6-844.
Robust Modeling with erratic
data:
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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, ., and Galbraith, M., 1981, Wavelet extraction by sonic-log
correltation: Journal of the CSEG, v. 17, p. 24- 42.
Hampson, D., 1986, Inverse velocity stacking for multiple elimination:
Journal of the CSEG, V. 22, p. 44-55.
Hampson, D., and Russell, B., 1985, Maximum-Likelihood seismic
inversion (abstract no. SP-16)- National CanauianCSEGmeeting,
Ca. gary, Alberta.
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Introducti on to Sei stoic Inversi on Methods Bri an Russell
.
Herman, A.J., Anania, R.M., Chun, J.H., Jacewitz, C.A., and
Pepper, R.E.F., 1982, A fast three-dimensionalmodeling technique
and fundamentals of three-dimensional frequency-Uomainmigration:
Geophysics, v. 47, p. 1627-1644.
Jones, I.F., and Levy, S., 1987, Signal=to-noise ratio enhancement n
multi channel seismic data via the Karhunen-Loeve transform,
Geophysical Propecting, v. 35, p. 12-32.
Kormyl o, J., anu Mendel., J.M.,
deconvolution- IEEE Trans.
v. IT - 28, p. 482 - 488.
1983, Maximum-likelihood seismic
on Geoscience and Remote Sensing,
Lines, L.R., Schultz, A.K., and Treitel, S., 1988, Cooperative inversion
of geophysical data: Geophysics, v. 53, p. 8- 20.
Lines, L.R., and Tritel, S., 1984, A review of least-squares
anU its application to geophysical problems' Geophysical
Prospecting, v. 32, p. 159-186.
inversion
Lindseth, R.O., 1979, Synthetic sonic logs - a process for stratigraphic
interpretation: Geophysics, v. 44, p. 3- 26.
Oldenburg, D.W., 1985, Inverse theory with applica.tion to aleconvolution
and seismogram inversion. Unpublished course notes.
Oldenburg, D.W., Scheuer, T., and Levy, S., 1983, Recovery of the acoustic
impedance rom reflection seismograms:Geophysics, v. 48, p. 1318-1337.
Ostrander, W.J., 1984, Plane wave reflection coefficients
at non-normal angles of incidence: Geophysics, v. 49,
for gas sands
p. 1637-1648.
Part 12 - Summary
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8/20/2019 Introduction to seismic inversion methods
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Introduction %oSeismic Inversion Methods Brian Russell
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Part 12 - Summary
Page 12 - 11
8/20/2019 Introduction to seismic inversion methods
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