Books russell 1988

Introduction to
Seismic Inversion Methods
Brian H. Russell
Hampson-Russell
Software
Services,
Ltd.
Calgary,
Alberta
Course Notes Series, No. 2
S. N. Domenico, SeriesEditor
Society
of Exploration
Geophysicists

Thesecourse
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arepublished
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andclarity.Questions
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should
bereferred
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totheauthor.
ISBN 978-0-931830-48-8 (Series)
ISBN 978-0-931830-65-5 (Volume)
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88-62743
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Geophysicists
P.O. Box 702740
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¸ 1988bytheSociety
of Exploration
Geophysicists
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Reprinted
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]:nl;roduct1 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
Part 6
Part 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 ProcessingConsiderati ons
4. ! I ntroduc ti on
4.2 Ampl
i rude recovery
4.3 Improvement
of vertical
4.4 Lateral resolution
4.5 Noise attenuation
resolution
Recursive Inversion - Practice
5.1 The recursive inversion method
5.2 Information in the low frequency component
5.3 Seismically derived porosity
Sparse-spike Inversi on
6.1 I ntroduc ti on
6.2 Maximum-likelihood aleconvolution and inversion
6.3 The LI norm method
6.4 Reef Problem
I nversion applied to Thin-beds
7.1 Thin bed analysis
7.Z Inversion compari
son of thin beds
Model-based Inversion
B. 1 I ntroducti on .
8.2 Generalized linear inversion
8.3 Seismic1ithologic roodell
ing (SLIM)
Appendix
8-1 Matrix applications in geophysics
Part 8
2-2
2-6
2-12
2-18
3-1
3-2
3-4
3-8
4-1
4-2
4-4
4-6
4-12
4-14
5-1
5-2
5-10
5-16
6-1
6-2
6-4
6-22
6-30
7-1
7-2
7-4
8-1
8-2
8-4
8-10
8-14

Introduction to Seismic Inversion Methods Brian Russell
Part 9 Travel-time Inversion
g. 1. I ntroducti on
9.2 Numerical examplesof traveltime inversion
9.3 Seismic Tomography
Part 10 Amplitude versus offset (AVO) Inversion
10.1 AVOtheory
10.2 AVOinversion by GLI
Part 11 Velocity Inversion
I ntroduc ti on
Theory and Examples
Part 12 Summary
9-1
9-2
9-4
9-10
10-1
10-2
10-8
11-1
11-2
11-4
12-1

Introduction to Seismic •nversion Methods Brian Russell
PART I - INTRODUCTION
Part 1 - Introduction Page 1 - 1

I NTRODUCT
ION TO SEI 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 seismicdata. It would therefore seemappropriate to begin
by defining what is meantby seismic inversion. The most general definition
is as fol 1ows'
Geophysical inversion involves mapping the physical structure and
properties of the subsurface of the earth using measurements
madeon
the surface of the earth.
The above definition is so broad that it encompasses
virtually all the
work that is done in seismic analysis and interpretation. Thus, in this
course we shall primarily 'restrict our discussion to those inversion methods
which attempt to recover a broadband pseudo-acoustic impedance log from a
band-1 imi ted sei smic trace.
Another way to look at inversion is to consider it as the technique for
creating a model of the earth using the seismic data as input. As such, it
can be consideredas 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-dimensionalmodel). The
relationship betweenforward and inverse modelling is shownin Figure 1.1.
To understandseismic inversion, we must first understandthe 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 andfrequencydomains,
consideringthe three components
of this model:
reflectivity, seismic wavelet, and noise.
Part I - Introduction
_ m i --.
Page 1 - 2

Introduction to Seismic InverSion Methods Brian Russell
FORWARD
MODELL
I NG
i m ß
INVERSEMODELLING
(INVERSION)
_
, ß ß _
Input'
Process:
Output'
EARTH
MODEL
,
MODELLING
ALGORITHM
SEISMIC RESPONSE
i m mlm ii
INVERSION
ALGORITHM
EARTH
MODEL
i ii
Figure1.1 Fo.•ard
' andsInverse
Model,ling
Part I - Introduction Page I - 3

Introduction. to Seismic Inversion Methods Brian l•ussel 1
Once we have an understandingof these concepts and the problems which
can occur, we are in a position to look at the methods
which are currently
ß
used to invert seismic data. These methodsare summarizedin Figure 1.2. The
primary emphasisof the course will be
the ultimate resul.t, as was previously
on poststack seismic inversion where
o
Oiscussed, is a pseudo-impeaance
section.
Wewill start by looking at the most contanon
methods of poststack
inversion, which are basedon 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 schemeChosen. 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 methodwhichwill be aiscussed is
model-basedinversion, wherea geological moael is iteratively upUatedto 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 moveinto the realm
of pretstack. ThesemethoUs,still fairly new, allow us to extract parameters
other than impedance, such as density and shear-wave velocity.
Finally, we will aiscuss the geological aUvantages anU limitations of
each seismic inversion roethoU,looking at examples of each.
Part 1 - Introduction Page i -

Introduction to SelsmicInversion Methods Brian Russell
SE
ISMI
C INV
ERSI
ON
.MET•OS,,,
POSTSTACK
INVERSION
PRESTACK
INVERSION
MODEL-BASED
I RECURSIVE
INVERSION
• ,INVE
SION
- "NARROW
BAND
TRAVELTIME
INVERSION
!TOMOGRAPHY)
SPARSE-
SPIKE
WAV
EF
IEL
D
NVERSIOU
i
LINEAR
METHODS
,,
i i --
I METHODS
]
Figure 1.2 A summary
of current inversion techniques.
Part 1 - Introuuction Page 1 -

Introduction to Seismic Inversion Methods Brtan Russell
PART
2 - THECONVOLUTIONAL
MODEL
Part 2 - The Convolutional Model Page 2 -

Part 2 - The Convolutional Mooel
2.1 Th'e Sei smic Model
The mostbasic andcommonly
used one-Oimensional
moael for the seismic
trace is referreU to as the convolutional moOel, which states that the seismic
trace is simplythe 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.
Anevensimplerassumption
is to consiUerthe noise component
to be zero,
in which case the seismic tr•½e is simply the convolution of a seismic wavelet
with t•e earth ' s refl ecti vi ty,
s(t) = w{t) * r(t).
In seismic processing
we deal exclusively with digital data, that is,
datasampled
at a constant
time interval. If weconsiUer
the relectivity to
consist of a reflection coefficient at each time sample(som• of which can be
zero), andthe waveletto be a smoothfunction in time, convolutioncanbe
thoughtof as "replacing"eachreflection.coefficient with a scaledversionof
the waveletandsumming
the result. Theresult of this process
is illustrated
in Figures 2.1 and2.Z for botha "sparse"
anda "dense"
set of reflection
coefficients. Notice that convolution with the wavelet tends to "smear" the
reflection coefficients. Thatis, thereis a total lossof resolution,which
is the ability to resolve closely spacedreflectors.
Part 2 - The Convolutional Model Page

Introduction to Seismic Inversion Nethods Brian Russell
WAVELET:
(a) '*• • : -'':'
REFLECTIVITY
Figure 2.1
TRACE:
Convolution
of a wavelet with a
(a) •avelet. (b) Reflectivit.y.
sparse"reflectivity.
(c) Resu
1ting SeismicTrace.
(a)
(b')
!
.
i
: !
! : :
i
i ,
ß
: i
! i i
'?t *
c
o o o o o
Fi õure 2.2 Convolution of a wavelet with a sonic-derived "dense"
reflectivity. (a) Wavelet. (b) Reflectivity. (c) SeismicTrace
, i , ß .... ! , m i i L _ - '
Par• 2 - The Convolutional Model Page 2 - 3

Introduction to Seismic Inver'sion Methods Brian Russell
An alternate, but equivalent, way of looking at the seismic trace is in
the frequency domain. If we take the Fourier transform of the previous
ß
equati on, we may write
S(f) = W(f) x R(f),
where S(f) = Fouriertransform
of s(t),
W(f) = Fourier transform of w(t),
R(f) = Fourier transform of r(t), ana f = frequency.
In the aboveequation we see that convolution becomes
multiplication in
the frequencydomain. 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) maythen be simply expressed
esCf)= e
w
where
(f) + er(f),
I•ndicates
amplitude
spectrum,
and
0 indicates phase spectrum. .
In other words, convolution involves multiplying the amplitude spectra
and adding the phase spectra. Figure 2.3 illustrates the convolutional model
in the frequency domain. Notice that the time Oomainproblem of loss of
resolution becomes
one of loss of frequency content in the frequency domain.
Both the high and low frequencies of the reflectivity have been severely
reOuceo by the effects of the seismic wavelet.
Part 2 - The Convolutional Mooel Page ?. - 4

AMPLITUDE SPECTRA PHASE SPECTRA
w (f)
I I
-t-
R (f)
i i , I !
i. iit |11
loo
s (f)
I i!
I
i i
Figure 2.3 Convolution in the frequency domain for
the time series shown in Figure 2.1.
Part 2 - The Convolutional Model Page 2 -

2.g The Reflection Coefficient Series
l_ _ ,m i _ _ , _ _ m_ _,• , _ _ ß _ el
of as the res
within the ear
compres
si onal
i ropedanceto re
impedances by
coefficient at
fo11 aws:
'The reflection coefficient series (or reflectivity, as it is also called)
described
in theprevious
section
is one
of thefundamental
physical
concepts
in the seismicmethod. Basically, each reflection coefficient maybe thought
ponseof the seismic wavelet to an acoustic impeUance change
th, where acoustic impedanceis defined as the proUuctof
velocity and Uensity. Mathematically, converting from acoustic
flectivity involves dividing the difference in the acoustic
the sumof the acoustic impeaances. This gives t•e reflection
the boundary between the two layers. The equation is as
•i+lVi+l- iVi Zi+l-Z
i
i • i+1
where
and
r = reflection coefficient,
/o__density,
V -- compressional velocity,
Z -- acoustic impeUance,
Layer i overlies Layer i+1.
Wemustalso convert from depth to time by integrating the sonic log
transit times. Figure •.4 showsa schematicsonic log, density log, anU
resulting acoustic impedance
for a simplifieU earth moael. Figure 2.$ shows
the resultofconverting
to thereflection
coefficient
series
and
integrating
to time.
It shouldbe pointed out that this formula is true only for the normal
incidence case, that is, for a seismic wavestriking the reflecting interface
at right angles to the beds. Later in this course, we shall consider the case
of nonnormal inciaence.
Part 2 - The Convolutional Model Page 2 - 6

STRATIGRAPHIC SONICLOG
SECTION •T (•usec./mette)
4OO
SHALE ..... DEPTH
ß ß ß ß ß ß
SANOSTONE . . - .. ,
'I ! !_1 ! ! !
UMESTONE
I I I ! I ! I 1
LIMESTONE
2000111
30O 200
I
3600 m/s
_
v--I
V--3600
J
V= 6QO0
I
loo 2.0 3.0
,
OENSITY LOG.
ß •
Fig. 2.4. Borehole
LogMeasurements.
mm mm rome m .am
,mm mm m ----- mm
SHALE ..... OEPTH
•--------'-[
SANDSTONE . . ... ,
! I !11 I1
UMESTONE
I I 1 I I I II
i ! I 1 i I i 1000m
SHALE •.--._--.---- • •.'•
LIMESTONE
2000 m
ACOUSTIC
IMPED,M•CE (2•
(Y•ocrrv x OEaSn•
REFLECTWrrY
V$ OEPTH
VS TWO.WAY
TIME
20K -.25 O Q.2S -.25 O + .2S
I I v ' I
- 1000 m -- NO
,• , ..
- 20o0 m I SECOND
Fig. 2.5. Creation of Reflectivity Sequence.
Part g - The Convolutional Model Page 2 - 7

IntroductJ on 1:o Sei stoic Inversion Herhods Bri an Russell
Our best methodof 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
compute
the reflectivlty by using •he formulashown
earlier. Often, we do not
have the density log available• to us andmustmakedo with only the sonJc. The
approxJmatJon
of velocJty to •mpedance
1s a reasonable approxjmation, and
seems
to holdwell for clas;cics and carbonates(not evaporltes,however).
Figure 2.6 showsthe sonic and reflectJv•ty traces from a typJcal Alberta well
after they have been Jntegrated to two-waytlme.
As we shall see later, the type of aleconvolution and inversion used is
dependent on the statistical assumptions
which are made about the seismic
reflectivity and wavelet. Therefore, 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 goodassumption. A ranUomsequencehas the property that its
autocorrelation is a spike at zero-lag. That is, all the components
of the
autocorrelation are zero except the zero-lag value, as shownin the following
equati on-
t(Drt = ( 1 , 0 , 0 , ......... )
t
zero-lag.
Let us test this idea on a theoretical randomsequence, shownin Figure
2.7. Notice that the autocorrelation of this sequencehas 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 mustbe infinite in extent. Also
on this figure is shown the autocorrelation of a well log •erived
reflectivity. Wesee that it is even less "random"
than the randomspike
sequence. Wewill discuss this in moredetail on the next page.

IntroductJon to Se•.s=•c Inversion Methods Br•an Russell
RFC
F•g. 2.6. Reflectivity
sequence
derived
from
sonJc
.log.
RANDOM SPIKE SEQUENCE
WELL LOG DERIVEDREFLECT1vrrY
AUTOCORRE•JATION
OF RANDOM
SEQUENCE AUTOCORRELATION OF REFLECTIVITY
Fig. 2.7. Autocorrelat4ons of randomand well log
der4ved
spike sequences.
Part 2 - The Convolutional Model Page 2-

Introductlon to Sei smic Inversion Methods Brian Russel1
Therefore, the true earth reflectivity cannot be consideredas being
truly random. For a typical Alberta well weseea number
of large spikes
(co•responding
to majorlithol ogic change)
stickingupabove
the crowd.A good
way to describethis statistically is as a Bernoulli-Gaussian
sequence. The
Bernoulli part of this term implies a sparsenessin the positionsof the
spikes and the Gaussian
implies a randomness
in their amplitudes. Whenwe
generatesuch a sequence,there is a term, lambda, which controlsthe
sparsenessof the spikes. For a lambda
of 0 there are no spikes, andfor a
lambda
of 1, the sequence
is perfectly Gaussian in distribution. Figure 2.8
shows a numberof 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.

I ntroducti on to Sei smic I nversi on Methods Brian Russell
It
tl I I I
LAMBD^•0.01
i I I
•11 I 511 t •tl I
(VERY SPARSE)
11
311 I
LAMBDA--O. 1
4# I 511 I #1 I
TZIIE (KS !
1,1
::.
•"• •'•;'"
' "";'•'l•'
"••'r'•
LAMBDAI0.5
-• "(11
I
TX#E (HS)
LAMBDA-- 1.0 (GAUSSIAN:]
EXAMPLES
OF REFLECTIVITIES
Fig. 2.8. Examplesof reflectivities using lambda
factor to be discussed in Part 6.
, , m i ß i

Introduction to Seismic Inversion ,Methods Brian Russell
2.3 The Seismic Wavelet
-- _ ß • ,
Zero Phase and Constant Phase Wavelets
m _ m _ m ß m u , L m _ J
The assumption
tha.t there is a single, well-defined wavelet whichis
convolved with the reflectivity to produce
the seismic trace is overly
simplistic. Morerealistically, the wavelet is both time-varying andcomplex
in shape. However,
the assumption
of a simplewavelet 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 peakfrequencyof its a•litude spectrum or
the inverse of the dominantperiod in the time domain(the dominantperiod is
found by measuring
the time fromtroughto trough). TwoRicker wave'lets are
shown
in Figures 2.9 and 2.10 of frequencies20 and40 Hz. Notice that as the
anq•litudespectrum
of a wavelet .is broadened,the wavelet gets narrowerin the
timedomain,
indicating
anincrease
of resolution.Ourultimate
wavelet
would
be a spike, with a flat amplitude spectrum. Sucha wavelet is an unrealistic
goal in seismicprocessing, but onethat is aimedfor.
The Rtcker wavelets of Figures 2.9 and 2.10 are also zero-phase, or
perfectly symmetrical. This is a desirable character.
tstic of wavelets since
the energy is then concentrated at a positive peak, and the convol'ution of the
wavelet with a reflection coefficient will better resolve that reflection. To
get an idea of non-zero-phase wavelets, consider Figure 2.11, wherea 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,whereasa 180 degree shift
simply inverts the wavelet. The 30 degreerotations are asymetric.
Part 2 - The Convolutional Model Page 2- •2

Introduction to SeismicInversion Methods Brian Russell
Fig.
Fig.
2.9. 20 Hz Ricker Wavelet'.
•.10. 40 Hz Ricker wavelet.
Fig. 2.11. Ricker wavelet rotated
by 90 degree increments
Fig.
Part 2 - The Convolutional Model
2.12. Ricker wavelet rotated
by 30 degree increments
Page 2 - 13

Of course, a typical seismic wavelet contains a larger range of
frequencies than that shownon the Ricker wavelet. Consider the banapass
fil•er shown
in Figure 2.13, where we have passed a banaof frequencies
between15 and 60 Hz. The filter has also had cosine tapers applied between5
and 15 Hz, and between60 and 80 Hz. The taper reduces the "ringing" effect
that would be noticeable if the wavelet amplitude spectrum wasa simple
box-car. The wavelet of Figure 2.13 is zero-phase, and would be excellent as
a stratigraphic wavelet. It is often referred to as an Ormsby
wavelet.
Minimum Phase Wavelets
The concept of minimum-phaseis one that is vital to aleconvolution, but
is also a concept that is poorly understood. The reason for this lack of
understanding is that most discussions of the concept stress the mathematics
at the expense of the physical interpretation. The definition we
use of minimum-phase
is adapted from Treitel and Robinson (1966):
For a given set of wavelets, all with the sameamplitude spectrum,
theminimum-phase
wavelet
is theonewhich
hasthesharpest
leading
edge. That is, only wavelets which have positive time values.
The reason that minimum-phase concept is important to us is that a
typical wavelet in dynamite work is close to minimum-phase. Also, the wavelet
from the seismic instruments is also minimum-phase. The minimum-phase
equivalent of the 5/15-60/80 zero-phase wavelet is shownin Figure 2.14. As
in the aefinition used, notice that the minimum-phase
wavelet has no component
prior to time zero and has its energy concentrated as close to the origin as
possible. The phase spectrumof the minimum-waveletis also shown.
Part 2 - The Convolutional Model Pa.qe 2 - 14

I•troduct•onto Seistoic!nversionNethods. Br•anRussell
ql Re• R Zero PhaseI•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.
Reg1) 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
shownin Fig. 2.13.
_
! m,m, i m
Part 2 -Th 'e Convolutional Model
i
Page 2- 15

Let us nowlook at the effect of different waveletson the reflectivity
function itself. Figure 2.15 a anU b shows a numberof different wavelets
conv6lved with the reflectivity (Trace 1) fromthe simpleblockymodel shown
in Figure Z.5. The following wavelets have been used- high
zero-phase (Trace •), low frequency
zero-phase
(Trace ½), high
minimum
phase (Trace 3), low frequency minimumphase (Trace 5).
figure, we can makethe fol 1owing observations:
frequency
frequency
From the
(1) Lowfreq. zero-phasewavelet: (Trace 4)
- Resolution of reflections is poor.
- Identification of onset of reflection is good.
(Z) High freq. zero-phasewavelet: (Trace Z)
- Resolution of reflections is good.
- Identification of onset of reflection is good.
(3) Lowfreq. min. p•ase wavelet- (Trace 5)
- Resolution of reflections i s poor.
- Identification of onset of reflection is poor.
(4) High freq. min. phasewavelet: (Trace 3)
- Resolution of refl ections is good.
- Identification of onset of reflection is poor.
Based on the aboveobservations,wewouldhaveto consider the high
frequency,
zero-phase
wavelet
the best, andthelow-frequency,
minimum
phase
wavelet the worst.

(a)
!ql RegR Zer• Phase
Ua•elet •,'1G-•1• 14z
F
- •.• [' '
•,3 Recj
B miniilium
phue ' '
17 .•
q2 Reg
C Zero
Phase
14aue16('
' •'le-3•4B Hz
e
q• Reg
1) 'minimum
phase " •,leJ3e/4e
h• '
8
e.e •/••/'•-•"v--,._,,
-r
e.•
' ' "s•e
''
,m ,,
Tr'oce
[b)
Fig.
700
2.15. Convolution of four different wavelets shown
in (a) with trace I of (b). The results are
shown on traces 2 to 5 of (b).

g.4 Th•N.
oise.C
o.
mp.o•ne
nt
-
The situation that has been discussed so far is the ideal case. That is,
.
we haveinterpreted every reflection wavelet on a seismictrace as being an
actual reflection from a lithological boundary. Actually, many of the
"wiggles"on a trace are not true reflections, but are actually the result of
seismic noise. Seismic noise can be grouped under two categories-
(i) Random
Noise - noise which is uncorrelated from trace to trace and is
•ue mainly to environmental factors.
(ii) CoherentNoise - noise which is predictable on the seismic trace but
is unwanted. Anexampleis multiple reflection interference.
Randomnoise can be thought of as the additive component
n(t) which was
seenin the equationonpage 2-g. Correcting for this term is the primary
reason for stackingour •ata. Stackingactually uoesan excellent job of
removing ranUomnoise.
Multiples, oneof the major sources of coherentnoise, are caused
by
multiple "bounces"
of the seismicsignal within the earth, as shown
in Figure
2.16. They may be straightforward, as in multiple seafloor bounces
or
"ringing", or extremely
complex,as typified by interbedmultiples. Multiples
cannotbe thoughtof as additive noise andmustbe modeled
as a convolution
with the reflecti vi ty.
Figure
generatedby the simpleblockymodel
this data, it is important that
Multiples maybe partially removed
powerful elimination technique.
aleconvolution, f-k filter.ing,
wil 1 be consi alered in Part 4.
2.17
shownon Figure •. 5.
the multiples be
by stacking, but
Such techniques
and inverse velocity stacking.
shows the theoretical multiple sequence which would be
If we are to invert
effectively removed.
often require a more
include predictive
These techniques

Fig. 2.16. Several multiple generating mechanisms.
TIME TIME
[sec) [sec)
0.7 0.7
REFLECTION R.C.S.
COEFFICIENT WITH ALL
SERIES MULTIPLES
Fig. 2.17. Reflectivi ty sequence
of Fig.
and without multipl es.
Part 2 - TheConvolutionalModel
2.5. with
.
Page 2 - 19

PART 3 - RECURS
IVE INVERSION - THEORY
m•mmm•---' .• ,- - - ' •- - _ - - _- _
Part 3 - Recurstve Inversion - Theory Page 3 -

•ntroduct•on to SeJsmic Znversion Methods Brian Russell
PART 3 - RECURSIVE INVERSION - THEORY
3.1 Discrete Inversion
, ! ß , , •
In section 2.2, we saw that reflectivity was defined in terms of
acoustic impedancechanges. The formula was written:
Y•i+lV•+l
' •iV! 2i+
1'Z
i
ri--yoi'+lVi+l+
Y•iVi---Zi..+l
+Z
i
where r -- refl ecti on coefficient,
/0-- density,
V -- compressionalvelocity,
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.cousticimpedance
by inverting the aboveformula. Normally, the
inverse' formulation is simply written down,but here we will supply the
missing steps for completness. First, notice that:
Also
Ther'efore
Zi+l+Z
i Zi+
1-Z
t 2Zi+
1
I +ri- Zi+l
+Zi + Zi+l
+2i Zi+l
+Zi
I- ri--
Zi+l+
Z
i Zi+
1-Z
i 2Zf[
Zi+l+Zi Zi+l+Zi Zi+l+Zi
Zi+l
Zi
l+r.
1
1
Part 3 - RecursiveInversion- Theory
ill, ß , I
Page

Introduction to SeismicInvers-•onMethods Brian Russell
pv-e-
TIME
(sec]
0.7
REFLECTION
COEFFICIENT
SERIES
RECOVERED
ACOUSTIC
IMPEDANCE
Fig. 3.1, Applying
the recursiveinversion
formula
to a
simple,andexact, reflectivity.
, ! ß
Part 3 - Recursive Inversion - Theory Page 3 -

!ntroductt on to Se1 smJc ! nversi on Methods Brian Russell
•9r• ;• • •;• • • •-•• 9rgr•t-k'k9r9r• •-;• ;• .................................................
Or, the final •esult-
Zi+[=Z
ß
l+r i .
This is called the discrete recursive inversion formula and is the basis
of many current inversion techniques. The formula tells us that if we know
the acoustic impedance
of a particular layer and the reflection coefficient at
the base of that layer, we mayrecover the acoustic impedance of the next
layer. Of course we need an estimate of the first layer impedanceto start us
off. Assumewe can estimate this value for layer one. Then
l+rl ,
Z2:
Zli r1 Z3=
Z
211
+r2
- r
and so on ...
To find the nth impedancefrom 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 impedancewas 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 problemsare encountered. These problemsare as follows-
(i) FrequencyBandl
imiti ng
_ ß
Referring back to Figure 2.2 we see that the reflectivity is severely
bandlimited when it is convolved with the seismic wavelet. Both the
low frequency components
and the high frequency components
are lost.
Part 3 - Recursive Inversion - Theory Page 3 - 4

0.2 0 V•) 'V,•
•R
R = +0.2
V
o:1000
m Where:
--• V,•=1000 i-o.t
- 1500 m
- •ec'.
(a)
- 0.1 '•0.2
R•
R=
{ASSUME
j•: l)
R•=
-0.1
R =+0.2
R: -0.1
V
o=1000m
-'+ ¾1
=818m
ii•.
Figure 3.2 Effect of banUlimitingon reflectivity, where(a) shows
single reflection coefficient, anU (b) shows
bandlimited
refl ecti on coefficient.
i i m i m I
I __ ___ i _
Part 3 - Recursire Inversion - Theory Page3 -

(ii) Noise
The inclusion of coherent or random noise into the seismic 'trace will
makethe estimate• reflectivity deviate from the true reflectivity.
To get a feeling for the severity of the abovelimitations 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)) anUthe 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 ourabil.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
ß
spectrum, we may recover muchof the original frequency content using
deconvolution techniques. In part 5 we will address the problemof 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 showsthe effect of adding the full multiple
reflection train (including transmission losses) to the modelreflectivity.
As we can see on the diagram, the recovered acoustic impedancehas the same
basic shape as the true acoustic impedance, but becomesincreasingly incorrect
with depth. This problemof accumulatingerror is compoundeU
by the amplitude
problemns
introduced by the transmission losses.
Part 3 - Recurslye Inversion - Theory Page 3 - 6

Introduction to Seismic Invers,ion Methods Brian Russell
TIME
Fig.
TIME
(see)
Fig.
0.?
RECOVERED
ACOUSTIC
IMPEDANCE
REFLECTION SYNTHETIC
COEFFICIENT (MWNUM-PHASE
SERIES WAVELET)
pv-•,
INVERSION
OF SYNTHETIC
3.3. The effect of bandlimiting on recurslye inversion.
0.7
TIME
(re.c)
REFLECTION RECOVERED R.C.S. RECOVERED
COEFFICIENT ACOUSTIC WITH ALL ACOUSTIC
SERIES IMPEDANCE MULTIPLES IMPEDANCE
3.4. The effect of noise on recursive inversion.
Part 3 - Recursive Inversion - Theory Page 3 -

3.3 Continuous Inversion
A logarithmic relationship is often used to approximate the above
formulas. This is derived by noting that we can write r(t) as a continuous
function in the following way:
Or
r(t) - Z(t+dt)
- Z{t)_ 1dZ(t)
ß - Z(t+dt) + Z(•) - •' z'(t)
! d In Z(t)
r(t) =• dt
The inverse formula is thus-
t
Z(t)
=Z(O)
exp
2yr(t)dt.
0
Theprecedingapproximation
is valid if r(t) <10.3• whichis usually the
case. A paper by Berteussenand Ursin (1983), goes into muchmore detail on
the continuous versus discrete approximation. Figures 3.5 and 3.6 from their
paper showthat the accuracyof the continuous inversion algorithm is within
4% of the correct value between reflection coefficients of -0.5 and +0.3.
If our reflection coefficients are in the order of + or - 0.1, an even
simpler
approximation
may
bemade
bydropp'ing
thelogarithmic
relationship:
t
1dZ(t)
•_==•
Z(t)
--2'Z(O)
fr(t)dt
r(t)
--•-dr VO
Part 3 - Recursive Inversion - Theory Page 3 - 8

Fig. 3.5
m i ,, ,m I I IIIII
I +gt ½xp
(26•) Difference
-1.0 0.0 0.14 -0.14
-0.9 0.05 0.I? -0.12
-0.8 0.11 0.20 -0.09
-0.7 0.18 0.25 -0.07
-0.6 0.25 0.30 -0.05
-0.5 0.33 0.37 -0.04 '
-0.4 0.43 0.45 --0.02
-0.3 0.• 0.•5 --0.01
-0.2 0.667 0.670 -0.003
-0.1 0.8182 0.8187 --0.0005
0.0 1.0 1.0 0.0
0.1 1.222 1.221 0.001
0.2 1.500 1.492 0.008
0.3 1.86 1.82 0.04
0.4 2.33 2.23 o.1
0.5 3.0 2.7 0.3
0.6 4.0 3.3 0.7
0.7 5.7 4.1 1.6
0.8 9.0 5.0 4.0
0.9 19.0 6.0 13.0
1.0 co 7.4 •o
Numericalc•pari sonof discrete and continuous
i nversi on.
(Berteussen and Ursin, 1983)
Fig. 3.6
$000
} m
MPEDANCE
(O
ISCR.
)
O
r-niL
${300
-•O
IFFERENCE
o
SO0 OIFFERENCE( SCALEDUP)
T •'•E t SECONOS
C•parisonbetween
impedance
c•putatins based
ona
discrete anda continuous
seismic•del.
(BerteussenandUrsin, 1983)
Part 3 - Recursire .Inversion - Theory Page 3 -

Introduction'to Seismic Inversion Methods Brian Russell
PART4 - SEISMIC PROCESSING
CONSIDERATIONS
Part 4 - Seismic Processing Considerations Page 4 - 1

•ntroduction to Seismic •nvers•on Methods B.r.
ian Russell
4.1 Introduction
Havinglookedat a simple model'of the seismic trace, anu at the
recursire inversion alogorithmin theory, wewill nowlook at the problem of
processing
real seismiceata in order to get the bestresults fromseismic
inversion. We may group the keyprocessing
problemsinto the following
categories:
(i ) Amp
1i tude recovery.
(i i) Vertical resolution improvement.
(i i i ) Horizontal resoluti on improvement.
(iv) Noise elimination.
Amplitudeproblemsare a majorconsideration
at the early processing
stages
andwewill lookat both deterministicamplitude
recovery
andsurface
consistent residual static time corrections. Vertical resolution improvement
will involve a discussion of aleconvolution and wavelet processingtechniques.
In our discussion of horizontal resolution wewill look at the resolution
improvement
obtainedin migration,using a 3-Dexample.Finally, wewill
consider severalapproaches
to noiseelimination,especiallythe 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 useUto do preinversion processing.
seismic data we usually assume the
previous section. Andto arrive at an
is, that each trace is a vertical,
we must carefully process our data with
showsa processing flow which could

INPUT RAW DATA
DETERMINISTIC
AMPLITUDE
CORRECTIONS
,. _•m
mlm
SURFACE-CONSISTENT
DECONVOLUTIO,
NFOLLOWED
BY HI GH RESOIJUTI.ON DECON
i
i
SURFACE-CONS
I STENT
AMPt:ITUDE ANAL'YSIS
SURFACE-CONSISTENT
STATI CS ANAIJY
SIS
VELOCITY ANAUYS
IS
APPbY STATICS AND VEUOCITY
MULTIPLE ATTENUATION
STACK
ß •
MI GRATI ON
,
Fig. 4.1. Simpl
ifiedinversi
onprocessing
flow.
ll , ß ' ß I , _ i 11 , m - -- m _ • • ,11

Inl;roducl:ion 1:oSeJ
smlc Invers1on Nethods BrJan Russell
4.2 Am.p'l
i tu.de..
P,.ecovery
The most dJffJcult job in the p•ocessing of any seismic line is
ß
•econst•ucting
the amplJtudes
of the selsmJc
t•acesas theywould
havebeen
Jf
the•e were no dJs[urbJnginf'luences present. We normally make the
simplJfication
that thedistortionof the seJsmic
amplJtudes
may
beputinto
three main categories'sphe•Jcal
divergence,absorptJon,and t•ansmJssion
loss. Basedon a consideration of these three factors, we maywrJte aownan
approximate
functJonfor the total earth attenuation-
Thus,
data, the
formula.
At:AO*
( b / t) * exp(-at),
where t = time,
A
t = recorded
amplitude,
A
0 = true ampl
itude,
anU
a,b = constants.
if we estimate the constants in the above equation from the seismic
true amplitudes
of the data coulUbe recoveredby usingthe inverse
The deterministic amplitude correction and trace to trace mean
scalingwill account
for the overall gross changesin amplitude. However,
there may still be subtle (or even not-so-subtle) amplitudeproblems
associated
with poorsurface conditions or other factors. To compensate
for
these effects, it is often advisable to compute
andapply surface-consistent
gaincorrections. Thiscorrection involves computing
a total gainvaluefor
each trace andthen decomposing
this single value in the four components
Aij=
Six
Rj
xG
kxMkX
•j,
where A = Total amplitude factor,
S = Shot component,
R: Receiver component,
G = CDPcomponent, and
M = Offset component,
X = Offset distance,
i,j = shot,receiver pos.,
k = CDPposition.
Part 4 - Seismic ProcessingConsiderations Page 4 -

Introduction to Seismic .Inversion Methods Brian Russell
SURFACE
SUEF'A•
CONS
Ib'TEh[O{
AND
T |tV•E :
,Ri
L-rE
R ß
Fig. 4.2. Surface and sub-surfacegeometryand
surface-consistentdecomposition.(Mike Graul).
, ,
Part 4 - Seismic
Processing
Considerations Page 4 - 5

Figure 4.g (from Mike Graul's unpublished course notes) shows the
geometry
usedfor this analysis. Notice that the surface-consistent
statics
antialeconvolution
problem
are similar. For the statics problem,the averaging
canbe •1oneby straight summation.For the amplitudeproblemwemust
transform the aboveequation into additive form using the logarithm:
InAij=
InS
i +InRj
+InG
k+lnkMijX•.
The problem can then be treated exactly the sameway as in the statics
case. Figure 4.3, fromTaneranti Koehler (1981), shows
the effect of doing
surface consistent amplitude and statics corrections.
4.3 I•mp.
rov.
ement_
o.[_Ver.
t.i.ca.1..Resoluti
on
Deconvolution 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 discussthe "convolution"part of "deconvolution" starting with
the equation for the convolutional model
In the
st--wt*rt where
frequency domain
st= theseismic
trace,
wt=the seismic
wavelet,
rt= reflectioncoefficientseries,
* = 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
of w
t .
Part 4 - Seismic Processing Considerations
,
Page 4 - 6

ii 11
ß 1'
i
ii
'..,•'•, ," " " ß d.
Preliminary
stack
bet'ore
surface
consistent
static
and
ompli-
lude corrections.
ßStockwithsurface
consistent
static
andamplitude
cor-
rections.
Fig. 4.3. Stacks with and without surface-consi stent
corrections. (TaneranuKoehler,1981).
Part 4 - Seismic Processing Considerations
ß ,
Page4 - 7

In the frequencydomain,this becomes
R(f) = W(f) x 1/W(f) .
After this extremelysimple introduction, it may appear that the
deconvolution
problemshouldbe easyto solve. This is not the case, and the
continuingresearchinto the problem testifies to this. Thereare two main
problems. Is our convolutional
model
correct, and, if the modelis correct,
can we derive the true wavelet from the data? The answer to the first
questionis that the convolutional
model
appears
to be the bestmodelwe have
come
upwith so far. The main problemis in assuming
that the wavelet does
not vary with time. In our discussionwewill assume
that the time varying
problem
is negligible within the zoneof interest.
The second
problemis much more severe, since it requires solving the
ambiguous
problem
of separatinga waveletandreflectivity sequence
whenonly
the seismic trace is known. To get around this problem, all deconvolution or
wavelet estimation programs
makecertain restrictive assumptions,
either about
the wavelet or the reflectivity. There are two classes of deconvolution
methods: those which makerestrictive phase assumptions and can be considered
,
true wavelet processingtechniquesonly whenthese phaseassumptions
are met,
and those which do not make restrictive phase assumptions and can be
consideredas true wavelet processingmethods. In the first category are
(1) Spiking deconvolution,
(2) Predictive deconvolution,
(3) Zero phasedeconvolution, and
(4) Surface-consi stent deconvolution.
Part 4 - Seismic Processing Considerations Page 4 -

(a)
Fig. 4.4 A comparison of non surface-consistent and surface-consistent
decon on pre-stack data. {a) Zero-phase deconvolution.
{b) Surface-consistent soikinB d•convolution.
(b),
Fig. 4.5 Surface-consistent deconcomparisonafter stack.
(a) Zero-phase aleconvolution. (b) Surface-consistent
deconvol ution.
'--'- , ß , ,• ,t ß ß _ , , _ _ ,, , ,_ , ,
Part 4 - .Seismic
Processing
Consioerations Page
4 -

Introduction to Seismic Invers.ion Methods Brian Russell
In the secondcategory are found
(1) Wavelet estimation using a well
(Hampson
andGalbraith 1981)
1og (Strat Decon).
(2) Maximum-1
ikel ihood aleconvolution.
(Chi et al, lg84)
Let us
surface-consi stent
surface-consi stent
components. We
di recti ons- common
illustrate the effectiveness of one of. the methods,
aleconvolution. Referring to Figure 4.•, notice that a
scheme involves the convolutional proauct of four
must therefore average over four different geometry
source, commonreceiver, commondepth point (CDP), and
con,
non offset (COS). The averaging must be performed iteratively and there
are several different ways to perform it. The example in Figures 4.4 ana 4.5
shows an actual surface-consi stent case study which was aone in the following
way'
(a) Computethe autocorrelations of each trace,
(b) average the autocorrelations in each geometryeirection to get four
average autocorrel ati OhS,
(c) derive and apply the minimum-phase
inverse of each waveform, and
(•) iterate through this procedure to get an optimumresult.
Twopoints to note when you are looking at the case study are the
consistent definition of the waveform
in the surface-consistent
approachan•
the subsequentimprovementof the stratigraphic interpretability of the stack.
Wecan 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.

Table 4-1 Comparison of Deconvolution MethoUs
m m ß ß m
METHOD
Spiking
Deconvol ution
Predi cti ve
Deconvol uti on
Zero Phase
Deconvol utton
Surface-cons.
Deconvolution
Stratigraphic
Deconvol ution
Maximum-
L ik el i hood
deconvol ution
WAVELET ESTIMATION
Min.imum
phaseassumption
Random
refl ecti vi ty
assumptions.
No assumptionsabout
wavelet•
Zero phaseassumption.
Random
refl ectt vi ty
assumption.
Minimum
or zero phase.
Random
reflecti vi ty
assumption.
No phaseassumption.
However, well must match
sei smic.
No phaseassumption.
Sparse-spikeassumption.
WAVELETSHAPING
Ideally shaped
to spike.
In practice, shaped
to minimum
phase,higherfrequency
output.
Doesnot whiten data well.
Removes
short andlong period
multiples. Doesnot affect
phase
of wayel
et for longlags.
..1_, m
Phase is not altered.
Amplitudespectrumi$
whi tened.
Canshape
to desiredoutput.
Phase
characteri s improved.
Ampl
i rude spectrumi s
whitened
less thanin single
trace methods.
Phase of wavelet is zeroed.
Amplitude
spectrum
not
whi tened.
Phase of wavelet is zeroed•
Amp
1i rude spectrumi s
whi tened.
Part 4 - Seismic Processing Considerations Page 4 11'

4.4 Lateral Resol uti on
The complete three-dimensional(3-D) diffraction problem
is shownin
Figure4.6 for a modelstudytaken fromHerman,
et al (1982). We
will look'at
line 108, whichcuts obliquely acrossa fault andalso cuts acrossa reef-like
structure. Note that it misses the second reef structure.
Figure 4.7 shows the result of processingthe line. In the stacked
section wemaydistinguish 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
secondtype are out-of-t•e-plane diffractions, often called "side-swipe". This
is most noticeable by the appearance of energy from the second reef booy which
was not crossed. In the two-dimensional (2-D) migration, we have correctly
removed the 2-D diffraction patterns, but are still bothere• by the
out-of-the-plane diffractions. The full 3-D migration corrects for these
problems. The final migrated section has also accounted for incorrectly
positioned evehts such as the obliquely dipping fault. This brief summary
has
not been intended as a complete summary
of the migration procedure, but rather
as a warning that migration {preferably 3-D) mustbe performedon complex
structural lines for the fol 1owing reasons:
(a)
(b)
To correctly position dipping events on the seismic section, and
To remove diffracted events.
Although migration can compensatefor someof the lateral resolution
problems, we must rememberthat this is analogousto the aleconvolutionproblem
in that not all of the interfering effects may be removed. Therefore, we must
be aware that the true one-dimensional seismic trace, free of any lateral
interference, is impossible to achieve.

lol
I
71
131
(a] 3- D MODEL
131
101
108
LINE
ß
ß ß
ß ß
ß
..................................
.............................
.........................................
....................................
{hi 8•8•0 LAYOU•
Fig. 4.6. 3-D model experiment.
i mm _ ml j mm
Part 4 • Seismic Processing Considerations
(Herman
et al, 1982).
Page 4 - 13

4.5 Notse Attenuation
As we'discussed in an earlier section, seismic noise can be classified as
either •andom'or coherent. Random noise is reduced by the stacking process
quite well unlessthe signal-to-noiseratio dropsclose to one. In this case,
a coherency
enhancement
program
canbe used, whichusually involves some
type
of trace mixingor FKfiltering. However,
the interpreter mustbe aware that
anymixingof the data will "smear"trace amplitudes, makingthe inversion
result on a particular trace less reliable.
Coherent noise is muchmore difficult to eliminate. One of the major
sources of coherent noise is multiple interference, explained in section 2.4.
Two of the major methodsusedin the elimination of multiples are the FK
filtering method,and the newerInverse Velocity Stackingmethod. The Inverse
VeiocityStacking
method
involves
the following
steps:
(1) Correct the data using the proper NMO
velocity,
(2) Model the data as a linear sumof parabolic shapes,
(This involves transformingto the Velocity domain),
(3) Filter outtheparabolic
components
witha moveout
greaterthansome
pre-determined
limit (in the orderof 30 msec),and
(4) Perform the inverse transform.
Figure 4.8, taken fromHampson
(1986), shows
a comparison
between
the two
methods
for a typical multiple problem
in northernAlberta. Thedisplays 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 insidetraces. Figure4.9, also fromHampson
(1986), shows
a comparison
of
final stacks with andwithout multiple attenuation. It is obvious'from this
comparison
that the result of inverting the sectionwhichhas not hadmultiple
attenuation would be to introduce spurious velocities into the solution. The
importance
of multiple elimination to the preprocessing
flow cannottherefore
be overemphasized.
m i i m , i . i m _ i i _ L ,=•m__ _ i m ß •
Part 4 - Seismic Processing Consideration• Page ½ - 14

Introduction to Seismic Inversion Methods Brian Russell.
!lilt
tiiti
ll!1111iitt
i)tt
il
tli
ii/lit
tttl•
ill
(b] LINEld8 - 2-D MIGRATION
IIIIIIll!!1111111111111it
I!1111111
I!11111111111illl
ill
Ii
IIIIIIIIIil!111111tllilil!illlllll!111illlll
[1111111111111111111111111
III!!1111
I!111111111111111
II
II
IIIilllllllll!111111111111111111111
?•111[•i••
IIIIIIIII
!1111111111111111
III
I!
IIIiill•illlllillllllllllliillllllllllllh
•., }!l!iilll
•lllllilllllll
i!
iiJ
:illllllllllllilitiilillit!illllll
,o
111lllllllllllllllllllll1111llllll
Iilllllll!ll!llll
I111
illllllllilllllllllllllllllllllllllii{lilllllllllllll
"•fillllllllll!1111illi!111
IIIIIIIII
IIIIIII1111111111
II
II
Ilillilllllll!1111!1!111111111
'•
ColLINE 108 - 3-D MIGR•ATION
F•g. 4.7. Migration
of model
datashown
in F•g. 4.6.
- - -- (Herman
et al, 1982).
Part 4 - Seismic ProcessingConsiderations
ß
Page 4 - 15

AFTER
INVERSE VELOCITY STACK
MULTIPLE ATTENUATION
INPUT
AFTER
F-KMULTIPLE
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
1698
-4
Second
real-data
setconventional
stack
without
multiple
attenuation.
'•"
,• ...... ;•,•<,:u(•:'J,.•J
L,.•.,!-
•, •, I• ,,,,..... •.. •,•,,,•• '•;••
•,,t.•/:,.•t.,. ). I',,', ,'; • , , •, ß '1"' ',''. ;•t(•' )"•,'.m,,•""•.
• ,ii%' .t .%'.
, ,,,, • ..•'•t,..'•"•'i•' •-
---';•-•' "t" 1•%';J• •t•, ß
.... -.... ; -'".' ,•..''. 2•>
.':'..'•, •;,%"'•1
lee "" • "" •• ' "' "•' ß ' ß ' • ....
'" "' Zone of
,,,.t•iill••)•.•);•l',"P,'•)'•"•'".•r'"mm"•""•P"•
"•)r'"
t••' ''"•- ..... ,• Interest
,,..,.
,,..,,,_.
•,,.,
....
•.,...,
..,...,..,.•..,....,,,.,.,..
g •..,,,.
,', .l•,•) '• .'•'
',•' '•....
'.
......•.•_ •.U.•,.., .. •
••,•,•p}•h•?.•r•.•,•.
•.} ,•.•,,•,•m,l,•,
r ,nm,
""::•"'•'•""""="'""•"
....
";'
,.•,,
,,,.,.•,,,,,.., ,,{.........,,,
...,,,, ../•.• ,•.•'•, .'•-•%
Fig. 4.9. Second real data stack after inverse velocity stack
multiple attenuation. (Hampson, 1986)

Introduction to Seismic Inverslon Methods Brian Russell
PART 5 - RECURSIVE INVERSION - PRACTICE
_ _ _ _ _ .. . .• ,• _ _
Part 5 - Recursive Inversion - Practice Page 5 - i

5.1 The Recurslye Inversion Method
Wehave nowreached a point where we may start aiscussing the various
algorithms currently usedto invert seismicdata. Wemustremember
that all
these techniquesare baseUon the assumptionof a one-aimensional
seismic
trace model. T•at is, we assume
that all the corrections which were aiscussed
in section 4 have been correctly applied, leaving us with a seismic section in
whic• eachtrace representsa vertical, band-limiteUreflectivity series. In
this section we will look at someof the problems inherent in this assumption.
The mostpopulartechniquecurrently usedto invert seismicUata is referred
.
to as recursire inversion and goes under such trade namesas SEISLOGana
VERILOG.The basic equations usedare given in part 2, anUcan be written
Zi+
1Z
i <===__===>
Zi+l
=Z
i ,
ri--
Zi+l+
Z
i LIJ
where
ri= ith reflection coefficient,
and
Z
i --/•Vi= density
x vel
oci
ty.
The seismic data are simply assumeato fit the forward model and is
inverted usingthe inverserelationship. However,
as wasshown
in section 3,
oneof t•e key problems
in the recursire inversion of seismicdata is the loss
of the low-frequency component. Figure 5.1 shows an example
of an input
seismic section aria the resulting pseuao-acoustic impeaance without the
incorporationof low frequency information. Notice that it resembles
a
phase-shifteU versionof the seismic•ata. Thequestionof introUuclng the
lowfrequency
component
involvestwo separateissues. First, where
doweget
the low-frequency
component
from,ana, second,how
aoweincorporateit?
Part 5 - Recurslye Inversion - Practice Page 5 - 2.

1171121e9leS1ol 92 93
i••11•
I IIttltl=:::•:::::::-•--lll[l•1111t•
•'••1tlllttllltl
Ill•l
1t
1 l!IIit!
'ti
!llltfltll!!l•
I I !1!t•n•'i
•l, , •l••J• •":•!•
•'• •" • --'' '
___ ..• - ,•, _•. • •f •• .• ............
. :•,• m•,•'. • ....... • ....,.• .... • . •• .........
ß ß ß • ...•
,• ß•- • •,• ,•,..,• :'•l•,fm;
,•v•,•:•,.•.•l.;•.•.'..•l•l;ql .n
....................
: ...; •;....: • .. • ...................
'• ]••'•'•'',,••,•',,,'
',',•,
",,','",','
•":•'•'•"•m•
i•q•'t•'•'•a
....
•.,•'.•,•],'
•'•,J'•,
,• .•'' - '""W',-
• -::-=
•, '2 ,,• • ., •,•- • ,,• . ,•,•,I,.•.•..,• ....... •.•,,•. .,%•.• . ,• . '-.. ' .,• •, . •i• .......
•. • , • • •-•,• ,, • , ,,.,,• .., ..... •. •.,.,,,,..•,.., ,,,.•,•,•.•.• ....
•.,• ....
• •.......
ß '•. . •q• • •,•;.•
.,• ,.. ••,l•,,..,,,
•..•, J I .,,,• .•,• •....
..,•
....... :..•.....
•.•.•.. :,.. ,.... , ,. ,.............
, , •.•- -. •-(• ••' •'•:;•, /.................... ..... -(•-•( •.•,••(•'••'•"•:•"•'•7 '• . , .
• •'•,:•'•' • x•{
, - ,,
2•Y•' •] ,,•.-..•.•.,'.;.',-,.. .................. • ............ •................... •'•:.,• ...... •....- ......... •" ß7•' . =". .... 7' • '• • '. ' .----
.... - ......... •m:'•' •"•r'u'" •$• .... ,......
r... •<• • ß • - ' •'•' - ' .'••'•q• "•. •q• • .....
.•,.,• •.... ,_ /. ,,,_ . ; .... •,.:•.-
..............
• ...... •%--=:
. .•.. ...........
•.... ,........... • .....
•4• 7•* • ';•u
.:c•
i• ,••.,,•-.•,,
•?'..%•.,
•*•'•d•ti',i l•l•l'i'/lt' i•"'; •:•;•t•l,•i•21.•.l•'*.'•.'l•,•-•ii•.'•'..•,•:b-''? "•''• .... ; '_],;,'• ; '-•-•,••-----m'•l• ••"'•I'i•I• ........
•?•'•'• ;•
•q
••.
('•'•'"•",•h/•'•'}
••'•'•"' c'
((•'•'"
..........
.... •, --.- -••_ ,,.•_.'.';'". :: :: ......
ß" • ..... "• '1 '• ' ' ' ß , -' ' • ..... • ' - ß
•'.•-•-• '•-<•., •
'. ,,,'•,, ,. ,, ,
(a) Oriœinal-
Seismic
Data. Heavy
lines 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) Recursiveinversionof data in (a).
ß
Figure 5.1
0.8
'I
1.0
i
I 1
I
I
1.2
.I
.!
1.3
i
!
1 4
1.5
1.7
I
I
18
i
I 19
(Galbraith andMillington, 1979)
Part 5 - Recursive Inversion - Practice Page 5 - 3

Thelowfrequency
component
canbe foundin oneof three ways'
(1) From a filtered sonic log
The sonic log is the bestwayof derivinglow-frequency
information in
the vicinity of thewell. However,
it suffers fromtwomainproblems'
it is
usually stretchedwith respectto the seismic
dataandit lacks.a lateral
component.
These
problems,
discussed
in Galbraith
andMillington(1979),are
solved by using a stretching algorithm which stretches the sonic log
information to fit the seismic data at selected control points.
(2) Fromseismic velocity analysis
In this case, interval velocities are derivedfromthe stackingvelocity
functions along a seismic
line usingDix' formula. The resulting function
will bequite noisyandit is advisable to dosome
formof two-dimensional
filtering on them. In Figure5.2(a), a 2-D polynomial
fit hasbeendone to
smooth
out the function. This final set of traces represents the filtered
interval velocity in the 0-10 Hz rangefor eachtrace and may be added
directly to the invertedseismictraces. Referto rindseth(1979), for more
deta i 1s.
(3) Froma geological model
Using all
incorporated.
available sources, a blocky geological model
This is a time-consuming method.
can be built and
Part 5 - Recursire Inversion - Practice Page 5 - 4.

Introduction to Seismic InversiOn Methods Brian Russell
. .
70000
(a)
GOOO0
$0000
(pvl 4oooo
'/sgc
( b) $oooo
ZOOO0
I0000
/ --V..308
(PV)*
3460
,
,
i
VELocrrY SURFACE2ridORDERPOLYN• Frr
Figure 5.2 s•mTZ•eH CUT
FtT•
tRussell and Lindseth, 1982).
Part 5 - Recursive
Inversion
- Practice Page
5 - 5
.
ß

Second, the low-frequencycomponent
can be addedto the high frequency
component
by either adding reflectivity stage or the impedance
stage. In
section2.3, it wasshown
that the continuous
approximation
to the forwardand
inverse equations was given by
Forward Equati on
1 d 1nZ(t) <::==> Z(t)
r(t) =•- dt -
Inverse Equation
t
=Z(O)
exp
2•0
r(t)
dt.
Sincethe previous
transforms
are nonlinear(because
of thelogarithm),
Galbraith andMillington (1979) suggestthat the addition of the low-frequency
component
shouldbemade
at the reflectivity stage. In the SEISLOG
technique
they are added
at the velocity stage. However,
dueto other considerations,
this should not affect the result too much.
Of course, weare really interested in the seismicvelocity rather than
the acousticimpedance.
Figure5.2(b), from
Lindseth
(lg79), shows
that an
approximate
linear relationship exists between velocity and acoustic
impedance, given by
V = 0.308 Z + 3460 ft/sec.
Notice that this relationship is goodfor carbonates and clastics and
poor for evaporitesandshouldthereforebe usedwith caution. A moreexact
relationship may be found by doing crossplots from a well close to the
prospect. However,
usinga similar relationship
wemayapproximately
extract
velocity informationfromthe recoveredacousticimpedance.
Figure 5.3 shows
lowfrequency
information
derivedfrom filtered sonic
logs. The final pseudo-acoustic
impedance
log is shown
in Figure5.4
includingthe low-frequency
component.
Noticethat the geologicalmarkers
are
moreclearly visible on the final invertedsection.
Part 5 - Recurslye Inversion - Practice Page 5 - 6

Figure 5.3 Low
Frequency
comDonent
derivedfrom"st.reched:'
sonicloœ.
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•Figures5.1(b) and5.3.
Lines indicate major reflectors.
0.9
1.0
1.1
1.2
1.:)
1.4
I$
1.6
1.7
19
(Galbraith andMillington, 1979)

In sugary, the recursive methodof seismic inversion maybe given by the
fol 1owing flowchart'
I
i
i
INTRODUCE
LOW
FREQUENCIES
•)
I•.v••o
••DO-•CO••c
•
,
'
ICORRECT
TO
PSEUDO
VELOCITIES
ß ,
CONVERT
TO
DEPTH
I
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 accordingto a lithological colour scheme. This is an effective way
of presentingthe information• especially to those not totally familiar'with
normal seismic sections.

(a)
Frequency
(e)
1
(b)
Fig. (a) Frequencyresponse of a theoretical differentiator.
(b) Frequency
responseof a theoretical integrator.
Part 5 -Recursire Inversion - Practice
(Russell andLindseth,
,m ,i m ml , ,
Page 5 - 9
!982 )

Introduction to Seismic Inver.si.on Methods Brian Russell
5.2
I nfor.marl
o.nI?•_Th.e.
Lo..w
.F.r.equ.e.
ncycompo..ne.
nt
The key factor which sets inverted data apart from normal seismic data is
the inclusion of the low frequency component,regardless of howthis 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 paperby Russell andLindseth (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 morecomplete theory (this
is justifiea for reflection coefficients less that 0.1), so t•at
t
_1dZ(t)
<=__==>
Z(t)
=2Z(O)j•
0r(t)
at
r(t) - • dt- '
If we consider a single harmonic component,we mayderive the
response of this tel ationship, which is
de
jwt jwt jwt -j eJWt
-dt "--jwe <===> . dt= w
where w-- 21Tf,
frequency
In words.,differentiation introducesa -6 riB/octaveslope from.the high
end of the spectrumto the low, and a +90 degree phaseshift. Integration
introduces a -6 dB/octave slope from the low end to the high end, and a -90
degreephaseshift. Simpler still, differentiation removes
low frequencies
and integration puts themin. 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 1i thol ogic change,
(2) Transitional lithologic change, an•
(3) Cyclical change.
Part 5 - Recursire Inversion - Practice Page 5 - 10

(A)MAJOR
LITHOLOGIC
CHANGE
V1
Vl I
i
I.
I
I
I
I
i
I
(B)TRANSITIONAL LITHOLOGIC
CHANGE
V:V•+KZ
i i
(C)CYCLICAL
CHANGE
!
v• _
Fig. 5.6. Threetypesof lithological models' (a) Majorchange,
(b) Transitional, (c) Cyclical. (RussellandLindseth, 1982).
Part 5 - Recursire Inversion - Practice Page 5- 11

Introduction to SeismicInversionMethods Brian Russell
We
may
illustrate the effect of inversion
onthesethreecases
bylooking
at both seismicanUsonic log Uata. To show
the loss of high frequencyon the
sonic log, a simplefilter is used,
andtheassociated
phase
shift is not
introUuced.
To start with, considera major1ithologicboundary
as exempl
i lieu bythe
Paleozoicunconformity
of WesternCanada,
a change
froma clastic sequence
to
a carbonate
sequence.
Figure5.7 shows
thatmost
of theinformation
about
the
largestepin velocityis containeU
in theD-10
lizcomponent
of thesonic
log.
In Figure
5.8, the seismic
dataand
final Uepth
inversion
areshown.
On the
seismicdata, a major boundary
shows
up as simplya largereflection
coefficient, whereas,
on the inversion,the large velocity step is shown.
RAW
SONIC FILTERED
SONIC
LOGS
VELOCITY FT/SEC
0 10000 10-90HZ O-IOHZ O-CJOHZ
TIME
0.3-
0.5-
Fig. 5.7. Frequency
components
ofasonic
log.
(Russell andLindset•, 1982).
! L , , , I I ß [ I L
Part 5 - Recursire Inversion - Practice Page 5 - 12

o'- .
ß
(a)
.%;
DEPTH SEISLOG
ß o
DEPTH
(b)
..... ß
lOP OF
"' . ß""I:'ALEOZOIC
-425'
Fig. 5.8. Major litholgical'change, Saskatchewan example.
(a) Sesimic s_ection, (b) Inverted section.
..... _
......... _(R_q•sell
....
and Li,pqse_th,_•!98_2)___

To illustrate transitional and cyclic change, a single examplewill be
used. Figure. 5.9 showsa soniclog from an offshore Tertiary basin,
illustrating the ramps
whichshowa transitional velocity increase,andthe
rapidly varyingcyclic sequences.
Noticethat the0-10Hzcomponent
contains
all the informationaboutthe ramps, but the cyclic sequence
is containedin
the 10-50 Hz component.
Onlythe Oc component
is lost from the cyclic
component
upon
removal
of thelow frequencies. Figure 5.10 illustrates the
same
pointusingthe original seismic
dataandthe final depthinversion.
In summary,
the information
contained
in the lowfrequency
component
of
the soniclog is .lostin the seismic data. This includessuchgeological
information as the dc velocity component,
large jumpsin velocity, and linear
velocity ramps. If this informationcould be recovered
andincluUea
during
the inversionprocess,it wouldintroducethis lost geologicalinformation.
Fig. 5.9. Sonic
log showing
cyclicandtransitionalstrata.
Part 5 - Recurslye Inversion - Practice
(Russell and LinOseth, 1982)
Page 5 - 14

(b)
(a)
SEISMIC SECTION-CYCUC & TRANSITIONAL STRATA
i 1-3500
ß

5.3 Sei smical ly Derived Porosi ty
-- ILI , ß I
Wehave shownthat seismic data may be quite adequately inverted to
pseudo-velocity (and hencepseudo-sonic)information i f our corrections and
assumptions are reasonable. Thus, we may try to treat the inverted data as
true sonic log information and extract petrophysical data from it,
specifically porosity values. Angeleri andCarpi (1982) havetried just this,
with mixed results. The flow chart for their procedureis 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
fluidtransittime,
btma:rock
matrix
transittime,
Vsh
= fractional
volume
of shale,and
btsh:shale
transittime.
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 sandsand very poor in the dirty sands. Thus, we mayextract porosity
information from the seismic section only under the most favourable
conditions, notably excellent well control and clean sandcontent.
Part 5 - Recurslye Inversion - Practice Page 5 - 16

F ']w[tt
'ill
]
!•ILI61C
.AT&
'$[IS'MI•
.AT&'
I-"'•''' m.,,•,
_,ml
.
-[ ,gnu mill i' •ill. Utl..I 111
,l lit
•%lOtOG
IIIITEIPllETATII
i
Fig.
l! WlltK :
t ' .
5.11. Porosity evaluati 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.
Shalepercentage
is alsodisplayed. (Angel
eri andCarpi, 1982).
Part 5 - Recursire Inversion - Practice
i ,
Page 5 - 17

Introduction to Sei stoic Inversion Methods Brian Russel 1
PART 6 - SPARSE-SPIKE INVERSION
• { • ...... • I ] m • m
Part 6 - Sparse-spike Inversion 6- 1

Introduction to Seismic Inversion Me.thods Brian Russell
6.1 Introduction
Thebasictheoryof maximum-1
ikeli hood
deconvol
ution (MLD)
wasdeveloped
byDr. JerryMendel
and his associatesat USC
anUhasbeen
well publicised
,
(Kormylo
andMendel,
1983;Chiet el, 1984). A paperbyHampson
and Russell
(1985)outlineda modification of maximum-likelihood
Ueconvolution
melthod
which allowedthe method
to be moreeasily applied to real seismic•ata. One
of the conclusions
of that paperwasthat the method
couldbe extenoed
to use
the sparse
reflectivity as the first step of a broadband
seismic inversion
technique.Thistechnique,which will betermed
maximum-likelihood
seismic
inversion, is discussed later in these notes.
Youwill recall that our basic modelof 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
to solve for. However, using certain assumptions,
the aleconvolution
problem can be solved. As we haveseen, the recursire
method of seismic inversion is basedon classical aleconvolutiontechniques,
which assume
a random
reflectivity and a minimum
or zero-phasewavelet. They
produce
a higherfrequency
wavelet
onoutput,butneverrecover
thereflection
coefficient series completely. More recent aleconvolution
techniques
maybe
grouped
under the category
of sparse-spike
meth•s. Thatis, theyassume
a
certain modelof the reflectivity and make a wavelet estimate basedon this
assumption.

ACTUAL REFLECTIVITY
I,:, I ..
POISSON-GAUSSIAN
SERIES OF LARGE
EVENTS
--F
GAUSSIAN BACKGROUND
OF SMALL EVENTS
SONIC-LOG REFLECTIVITY
EXAMPLE
Figure6.1 Thefundamental
assumption
of the maximum-likelihood
method.
Part 6- Sparse-spike Inversion 6- 3

Intr6duction to Seismic Tnvetsion Methods Brian Russell
These techniques include-
(1) btaximum-Likel ihood deconvolutton and inversion.
(2) L1 norm deconvolution and inversion.
(3) Minimum
entropy deconvol
ution (MEO).
From the point of view of seismic inversion, sparse-spike methodshave an
advantage over classical methods
of deconvolution
because the sparse-spike
estimate, with extra constraints, can be used as a full bandwidth estimate of
the reflectivity. We will focus initially on maximum-likelihood
deconvolution, and will then move on to the L1 normmethodof Dr. Doug
O1denburg. The MEDmethodwill not be discussed in these notes.
6.2 Maximum-Likelihood Deconvolution and Inversion
i i m ! ß m m m m I _ ß
Maximum-Li kel i hood Deconvoluti on
I ß ß ß m _ _ l! . . • am .. I _
Figure 6.1 illustrates the fundamental assumption of Maximum-Likelihood
deconvolution, which is that the earth' s reflectivity is composed
of a series
of large events 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 showsthat in fact this type of model is not at all unreasonable.
Geologically, the large events correspond to unconformities and major
ß
1i thol ogic boundaries.
From our assumptions about the model, we can derive an objective function
whichmaybe minimized
to yield the "optimum"
or mostlikely reflectivity. and
wavelet combination consistent with the statistical assumption. Notice that
this method gives us estimates of both the sparse reflectivity and wavelet.
,,
Part 6 - Sparse-spike Inversion m

INPUT
WAVELET
REFLECTIVITY
NOISE
SPIKESIZE' 9.19
SPl• ••: 50.00
NOISE' 39.00
OB,.ECTIVE' 98.19
Figure6.2(a) Objective
function
for onePoSsible
solution
to inputtrace.
INPUT
WAVELET
REFLECTIVITY
SPIKE S!7_F: 6.38
SPIKE DENSIq'•, 70.85
NOISE
NOISE: 81.• 5
OBJECTIVE
:158.98
Figure6.2(b) Objective
function
forasecond
possible
solution
toinput
trace.This
value
is higher
than
6.2(a),.
indicating
a less
1ikely solution.
! , ,,

The objective function j is given by
-R2 N
2
k=l k=l
ß
where
- 2mln(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
• = likelihoodthat a given
sample has a reflection.
Mathematically, the expected behavior of the objective function is
expressed in termsof the parametersshown
above. Noassumptionsare made
aboutthe wavelet. The reflectivity sequenceis postulatedto be "sparse",
meaningthat the expected number
of spi•es is governedby the parameter
lambda, the ratio of the expected numberof nonzer.
o spikes to the total number
of trace samples. Normally, lambda is a numbermuchsmaller than one. The
other parametersneededto describe the expectedbehavior are R, the RMS•size
of the large spi•es, andN, the RMS
size of t•e noise. Withthese parameters
specified, any glven deconvol
ution solution can be examinedto see.whether it
is likely to be the result of a statistical process
with thoseparameters.
For
example,
if the reflectivity estimatehas a number
of spikesmuch
larger than
the expectednumber,
then it is an unlikely result.
In simpler terms, we are looking for the solution with the minimum
number
of spikesin its reflectivity and t•e lowestnoisecomponent.
Figures
6.2(a) and 6.2(b) show
twopossiblesolutionsfor the sameinput synthetic
trace. Noticethat theobje6tive functionfor theone
withtheminimum
spike
structure is indeed the lowest value.

Introduction to Seismic I nversi.on Methods Bri an Russel1
Original
Model
I terati on I
I terati on 2
Iteration 3
I teration 4
Iteration S
Iteration 6
Iterati on 7
Reflectivity
I, ill.
I ,1.2. -.I
,i.
Synthetic
Figure 6.3. The Sinl•le MostLikely Addition (SMLA)algorithm illustrated
for a simple reflectivity model.
Part 6 - Sparse-spi ke Inversion 6- 7

Introduction to Seismic Inversion Methods Brian Russel1
Of course, there maybe an infinite number
of possible solutions, and it
would take too much
computer
time to look at eachone.
mTherefore, a simpler
method is used to arrive at the answer. Essentially, we start with an initial
wavelet estimate,
es'timate
thesparse
reflectivity,'improve
the wavelet and
iterate throughthis sequence of steps until an acceptablylow objective
function is reached. This is shown
in block formin Figure 6.4. Thus, there
is a twostep procedure-
havingthe waveletestimate,updatethe reflectivity,
and then, having the reflectivity estimate, update the wavelet.
Theseproceduresare illustrated on model data in Figures 6.3 an• 6.5.
In Figure 6.3, the proceUurefor upUatingthe reflectivity is shown. It
consists of addingreflection coefficients oneby oneuntil an optimum
set of
"sparse"coefficients hasbeenfound. Thealgorithm
usedfor updatingthe
reflectivity is callee the single-most-likely-addition algorithm (SMLA)since
after each step it tries to find the optimum
spike to add. Figure 6.5 shows
the procedure for updating the wavelet phase. The input model is shownat the
top of the figure, andthe up•atedreflectivity andphaseis shown
after one,
two, five, and ten iterations. Notice that the final result compares
favourably with the model wavelet.
WAVELET
ESTIMATE
ES•TE
REFLECTIVITY
IMPROVE
WAVELET
ESTIMATE
Fiõure 6.4.
The
block
component
method
of solving
forboth
reflectivityand
wavelet.Iteratearound
the
loop unti1 converRence.

Introduction to Seismic Invers.ion Methods Brian Russell
Wayel
et Refl
ecti
Vity' Synthetic
Ill ,I ,
INPUT
MOD
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.

Introduction
to Seismic
InversionMethods BrianRussell
Figure 6.6 is an exampleof the algorithm applied to a synthetic
seismogram. Notice that the major reflectors have beenrecovered fairly well
and that the resultant trace matchesthe original trace quite accurately. Of
course, the smaller reflection coefficients are missing in the recovered
reflection coefficient series.
Let us nowlook at some real data. The first example is a' basal
Cretaceous gas play in Southern Alberta. Figure 6.7(a) and (b) shows the
comparisonbetween the input anU output stack from the aleconvolution
procedure. Also shown
are the extracted and final wavelet shapes. The main
things to note are the major increase in detail (frequency content) seen in
the final stack, and the improvement
in stratigraphic content.
Figure6.8 is a comparison
of input and output stacksfor a typical
Western Canada basin seismic line. The area is an event of interest between
0.7 anU0.8 seconds, representing a channel scour within the lower Cretaceous.
Althoughthe scour is visible on both sections, a dramatic improvement
is seen
in the resolution of the infill of this channel on the deconvolved section.
Within the central portion of the channel, a .positive reflection with a
lateral extent of five traces is clearly visible andis superimposed
on the
Uominantnegative trough.
INPUT:
V. ,.: --
ESTIMATED:
ttl J':ll'j' "'" "
ß
Figure 6.6 Synthetic seismogram test.

0.5
0.6
0.7
0.8
'SONIC
SYNTHETIC LOG
iZ.i
EXTRACTED WAVELET
0.5
0.6
.
0.8
(b)
(a) Initial seismicwith extractedwavelet.
Final deconvolved seismic with zero-please wavelet.
Figure 6.7
.... - -_ __ ._
Part 6- Sparse-spike Inversion 11

This is quite possibly a clean channel sand and may or may not be
prospective. However,this feature is entirely absent on the input stack.
Overlying the channel is a linear anomalywhich 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.

INPUT
STACK
DECONVOLVED
STACK
0.6
0.7
0.8
0.9
Figure 6.8 An input stack over a channelscourand
the resulting deconvol
ved seismic.
DECONVOLVED
STACK
ESTIMATED
REFLECTIVITY
0.6
0.7
0.8
0.9
Figure 6.9 The deconvolved result from Figure 6.8
and its estimated reflectivity.
Part 6 - Sparse-spike Inversion m 13

Maximum-Likel ihood Inversion
An obvious extension of the theory is to invert
reflectivity to Uevise a broad-band or "blocky" impedance
data (Hampson
andRussell, 1985). Given the reflectivity, r(i),
impedance
Z(i) maybe written
Z(i)
=Z(i_l
)[1
+r(i)]
1 - r(i) '
the es ti mated
from the seismic
the resul ting
Unfortunately, application of thi
from MLD produces unsatisfactory res
additive noise. Although the MLDalgor
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
impedancemaybe 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, independentknowleUge
of the impedancetrenU maybe input as a constraint. Since r(i) < l, we can
derive a convolutional type equation between acoustic impeUance anU
reflectivity, written
In Z(i) = 2H(i) * r(i) + n(i),
where Z(i) = the known
impedance
trend,
• i <0
H(i) :
• i >0
and n(i) : "errors" in the input trend.
_
Part 6 - Sparse-spike Inversion 6• 14

Figure 6.10 Input Modelparameters.
Figure 6.11
ß
Maximu•m-L
i keli hoodi nversion result from Figure 6.10.
.m __
Part 6 - Sparse-spike Inversion
6- lb

Introduction to Seismic Inversion 'Methods Brian Russell
The error series n(i) reflects the fact that the trend information is
approximate. Wenowhave two measured time-series: the seismic trace, T(i),
and the log of impedanceIn 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) whichattemptsa compromise
by simultaneously
moUellingthe seismic
trace while conformingto the knownimpedancetrend. If both the seismic
noise andthe impedance
trend noise are modelledas Gaussiansequences,
their
respective variances become
"tuning" parameterswhichthe user can modify to
shift the point at which the compromiseoccurs. That is, at one extremeonly
the seismicinformationis usedandat the ot•er extremeonly the impedance
trend.
In our first example,the method
is tested ona simplesynthetic. Figure
6.10 showsthe soniclog, the derivedreflectivity, the zero-phasewavelet
used to generate the synthetic, and finally the synthetic itself. This
example was usedinitially becauseit truly representsa "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 casewehaveuseda smoothed
version of the sonicvelocities to provide
the constraint. A visual comparisonwoulU indicate that the extracteU
velocity profile correspondsvery well to the input. A moredetailed
comparisonof the two figures shows
that the original andextracted logs do
not matchperfectly. T•ese small. shifts are dueto slight amplitudeproblems
on the extracted reflectivity. It is doubtful that a perfect matchcould ever
be obtai neU.

Introduction to Seismic Inversion Methods Bri an Russel 1
Figure 6.12 Creation of a seismic model from a sonic-log.
Figure 6.13 Inversion result from Figure 6.12.
•- _ ! ...... ii__ - - i - •_! mm i i i ß i i ! It_l I
Part 6 - Sparse-spi•e Inversion 17

Let us nowturn our attention to a slightly more realistic synthetic
example. Figure6.12 shows
the applicationof this algorithmto a sonic-log
derivedsynthetic. At the' top of the figure we seea soniclog with'its
reflectivity sequencebelow. (In this example,
wehave assumedthat the
density is constant, but this is not a necessary restriction.) The
reflectivity wascbnvolved
with a zero-phase
wavelet,bandlimited
from10 to
60Hz, andthe final syntheticis shown
at thebottom
of the figure.
The results of the maximum-likelihoodinversion methodare sbown in
Figure 6.13. The initial log is shownat the top, the constraint is shownin
the middle panel, and the extracted resull• is shownat the bottom of the
diagram. In this calculation, the waveletwasassumed
known. Notethe blocky
nature of the estimated
velocityprofile compared
with the actual sonic log
profile. Again, the input andoutput logs donot matchperfectly.
The fact that the twodo not perfectly match
is dueto slight errors in
the reflectivity sizes whichare amplified bythe integration process,andis
partially the effect of the constaintused. Theconstraintshownin Figure
6.13wascalculated
by applying a 200 ms smoother
to the actuallog. In
practice, this information could be derived from stacking velocities or from
nearby well control.
Part 6 - Sparse-spi ke Inversion 6- 18

* !
Figure 6.14 An input seismic 1ine to be inverted.
:
ß
'.
eel'?
e4dl
Figure 6.15 Maximum-Liklihood reflectivity estimate from
seismic in Figure 6.14.

Introauction to Seismic Inversion Methods Brian Russell
Finally, we show
the results of the algorithm appliedto real seismic
data. Figure6.14 shows
a portionof t•e input stack. Figure6.15 showsthe
•D extracted reflectivity. Figure 6.16 shows the recoveredacoustic
impedance,
wherea linear ramphasbeenusedas the constraint. Notice that
the invertedsection•isplays a "blocky" character, indicating that the major
features of the impedance
log havebeensuccessfullyrecovered. This blocky
impedance
canbecontrasted
with the more traditional narrow-band
.inversion
procedures,whichestimatea "smoothed"
or frequency
limited version of the
impedance.
Finally, Figure 6.17 showsa comparison
between
the well itself
and the inverted section.
In summary,
maximum-likelihood
inversion is a procedurewhichextracts a
broad-bandestimate of the seismic reflectivity and, by the introduction of
1inear constraints, allows us to invert to an acoustic impedance
section which
retains the majorgeological features of boreholelog data.

Introduction to Seismic Inver.sion Methods Brian Russell
Figure 6.16 Inversionof reflectivity shown
in Figure 6.15.
SEISMICINVERSION
WELL
+
SONIC
LOG
Figure 6.17 A comparison of the inverted seismic data and
the sonic log at well location.
Part 6 - Sparse-spike Inversion .. 21

6.3 The L1 Norm Method
-- __LI _ _ _ i .
Another method of- recursive, single trace inversion which usesa
"sparse-spike"
assumption
is theL1norm
method,
developed
primarily
by Dr.
DougOldenburg
of UBC.
andInverse Theory andApplications(ITA). This method
is also often referred to as the linear programming
method,
andthis can lead
to confusion. Actually, the two namesrefer to separate
aspects
of the
method. Themathematical
model
usedin the construction
of the algorithm is
the minimization
of the L1 norm. However,the methodusedto solvethe
problem is linear programming.The basic theory of this methodis foundin a
paper by Oldenburg,et el (1983). The first part of the paper discussesthe
noise-free convolutional model,
x(t) --w(t) * r(t), where x(t) = the seismictrace,
w(t) --the wavelet, an•
r(t) -- the reflectivity.
The authors point out that if a high-resolution aleconvolution is
performed
onthe seismictrace, the resulting estimateof the reflectivity can
be thoughtof as an averagedversion of the original reflectivity, as shown
at
the topof Figure6.18. Thisaveraged
reflectivity is missing
botht•e high
andlowfrequency
range,andis accurate
onlyin a band-limitea
central range
of frequencies. Althoughthere are an infinite number
of waysin which the
missing frequencycomponents
can be supplied, Oldenburg, et al (1983) show
that we can reduce this nonuniqueness by supplying more information to the
problem, such as the layered geological model
r(t)
--•,rj6(t
-l•),
j--!
where
•= 0if t •l• , an•
=1ift:• .

b
ß ß ß • 1
I m m m
0.0
T.IJdE•(•J
e f
o .50 joo j25
FRF.,O [HZJ
I !
I
Figure 6.18 Synthetic test of L1 NormInversion, moUified fro•.q
Oldenburg
et al (1983). (a) Input impedance,
(b) Input reflectivity, (c) Spectrum
of (b),
(d) Lowfrequencymodeltrace, (e) Deconvolutionof (•),
(f) Spectrum
of (U), (g) Estimatedimpedance
fromL1 Norm
method,(•) Estimatedreflectivity, (i) Spectrumof (•).

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 modelequates to a "blocky"
impedance
function, which in turn equates to a "sparse-spiKe" reflectivity
function. The above constraint will thus restrict our inverted result to a
"sparse" structure so that extremely fine structure, such as very small
reflection coefficients, will not be fully inverted.
The other key difference in the linear 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 shownbelow, applied to the trace x:
x1 : xi and x2: xi
i--1 i:1
The fact that the L1 normfavours a "sparse" structure is shown in the
following simple example. (Takenfrom the notes to Dr. Oldenburg's1085CSEG
convention course' "Inverse theory with application to aleconvolution and
seismograminversion"). Let f and g be two portions of seismic traces, where'
f: (1,-1,0) andg: (0,%•,0) .
The L2 norms are therefore'
The L1 normsare given by'
-
fl - 1 + 1 : 2 and gl = '
Notice that the L1 normof wavelet g is smaller than the L1 normof f,
whereas the L2 norms are both the same. Hence, minimizing the L1 norm would
reveal that g is a "preferred" seismic trace basedon it's sparseness.

(a) Input sei smic data
(b) Estimated refl ec ti vi ty
(c) Final impedance
Figure 6.19 L1 14orm
metboOapplied to real seismic data,
Part 6 - Sparse-spike Inversion
(Walker andUlrych, 1983)
6- 25

Introduction to Seismic Inversion MethoUs Brian Russell
Several other authorshadpreviouslyconsidered
the L1 norm
solution in
deconvolution
(Claerboutand Muir, 1973, andTayloretal., 1979), however,
they considered
the problem
in the timedomain.Oldenburg
et al.w suggested
solvingthe problem
usingfrequency
domain
constraints. That is, the reliable
frequency
bandis honored
whileat thesame
timea sparsereflectivity is
created. The results of their. algorithm on synthetic data are shown
at the
bottom of Figure6.18. Theactual implementation
of the L1 algorithmto real
seismicdata hasbeen done by Inverse Theory andApplications(ITA). The
processingflow •or the linear programming
inversion method
is shown
below.
InterPreter'=
CMP
Stacl<ed
section
<r(t)>= r(t)©w(t)
t ß ,i
i
I,,i
co,ect,',
,o,'
Residu
Pm'm,e
o,w
(t) I
ß i i i i i I i i
I Fourier
Trans••
of
<•r
(t)>I
i
Scale
Data
Const.
mints.
From
$tackins•_V'elocitles
I
ii &
Con,straints
From
'Well
Logs
I
i
Unear Programing Invemion
Assume
r(
t)ß
• n;)
(t-•q
),is
aspame,
reflection
series.
Minimizethe sum of absolute reflectionstrengU•.
FulFBandReflectivitySeries r (t)
Signal to Noise Enhancement and Display Preparation
Integrationto Obtain Impedance Sections
Figure 6.19(b) TheL1 Norm(Linear Programming.)
Method. (Oldenburg,1985).

Introduction to Seismic Inver. s,ion Methods Brian Russell
TSN
1,2
tO0 90 80 70 60 50 40 30 20 tO
1,3
1,4
1,5
1,6
1,7
1,:8
.2,0
2ø2
Figure 6.20 Inputseismic
datasection
to L1Norm
inversion.(O1
denburg, 1985'

Figure 6.19 showsthe application of the abovetechnique to an actual
seismic line fromAlberta. The data consist of 49 traces with a sample rate
of 4 msec
anda 10-50 Hz bandwidth. The figure shows
the linear programming
reflectivity and impedanceestimates below the input seismic section. It
should be pointed out that a three trace spatial smootherhas beenapplied to
the final results in both cases.
Finally, let us considera dataset fromAlberta whichhas beenprocesseU
through the LP inversion method. The input seismic is shownin Figure 6.2D
and the final inversion in Figure 6.21. The constraints useU here were from
well log data. In the final inversion notice that the impedance has been
superimposed on the final reflectivity estimate using a grey level scale.

1.6
1.7
1.8
1.9
2.0
2.1
2.2
Figure 6.21 Reflectivity andgrey-level plot of impedance
the L1 Norminversion of data in Figure 6.20.
Part 6 -Sparse-spike Inversion
for
(O1
denburg, 1985
6- 2-9

Introduction to Seismic Inversion Methods Br•an Russell
6.4 Reef Probleeß _
Onthe nextfewpages'is a comparison
betweena recursiveinversion
procedure(Verilog) anda sparse-spike
inversionmethod
(MLD). The sequence
!
of pages includes the following:
- a sonic log and its derived reflecti vtty,
- a synthetic seismogram
at both polarities,
- the original seismic line, showing
the well location,
- the Verilog inversion, and
- the MLD inversi on.
BaseUon the these data handouts, do the following interpretation
exerc i se:
([) Tie the syntheticto the seismicline at SP76. (Hint- use reverse
polari ty syntheti c).
(g) Identify andcolor the following events in the reef zone-
- the Calmar shale (which overlies the Nisku shaly carbonate),
- the 1retort shale, and
- .the porous Leduc reef.
(3) Compare
the reefal events on the seismicandthe two inversions. Use
a blocked off version of the sonic log.
(4) Determinefor parallelism which section tells you the most about the
reef zone?

Rickel, g Phas•
3g Ns, 26 Hz
REFL. DEPTH VELOCI •¾
COEF. lib Eft,/sec.
...,--
...,--
...m
$11qPLE I HTI3tViIL- 2 Ns.
AliPLI •IIi)E I
tiC. Ilql•. - Sonic
Pei.•ri es onlg
Figure 6.22 Sonic Log and synthetic at the reef well.

.47
'49
5!
55
57
?!
'?•
;'5
??
B5
$5
99
•41
Part 6- Sparse-spike Inversion
ß
i
32

***__********************************************************
0.7 0.8 .la.El 1 .:0 1.1 1..2 1. E)

Introduc%ion [o Seismic Inversion Meltotis Brian Russell
0.? IB.G O.G 1.0 1.1 1.2 1.• 1.4
ß
1
25
27
31
•x
$1
$?
67
?.1
?$
'111
113
119
123
127
131
137
i•i
ß .
Part6 - Sparse-spike
Inversio•
i i i i I • e e t I i e I i e I e i ß l
e ß e I
ß
i
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IIIII
i iiiiiii
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.., 34

PART 7 - INVERSION APPLIED TO THIN BEDS
Part 7 - Inversion applied to Thin Beds Page 7- I

Intro4uction to Seismic Inversion Methods Brian Russell
7.1 Thin Bed Analysis
One of the problemsthat we have identified in the inversion of seismic
traces is the loss of resolution causedby the convolution of the seismic
wavelet with the earth's reflectivity. As the time separation between
reflection coefficients becomessmaller, the interference between overlapping
wavelets becomes
more severe. Indeed, in Figure 6.19 it was shownthat the
effect of reflection coefficients one sampleapart andof 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
as wel1 as the phase spectrum. In this section wewill look closer
at the effect of wavelets on thin beds and how.effectively we can invert these
thin bed s.
The first comprehensivel'ookat thin bedeffects was done by Widess
(1973). In this paper he used a model which has becomethe standard for
discussing thin beds, the wedge
model. That is, considera high velocity
laye6 encased
in a lowvelocity layer (or vice versa) andallow the thickness
of the layer to pinchout to zero. Nextcreate the reflectivity response
from
the impedance,
and convolvewith a wavelet. The thickness of the layer is
given in termsof two-waytime throughthe layer andis then related to the
dominantperiod of the wavelet. The usual wavelet usedis a Ricker becauseof
the simpli city of its shape.
Figure 7.1 is taken fromWidess' paper andshows
the synthetic section as
the thickness of the layer decreases from twice the dominant period of the
wavelet to 1/ZOth of the dominant period. (Note that what is refertea to as a
wavelength
in his plot i s actually twice the dominant
period). A few important
points can be noted from Figure 7.1. First, the wavelets start interfering
witheackotherat a thicknessjust belowtwo dominant
periods,butremain
Clistinguishable downto about one period.
Part 7 - Inversion applied to Thin Beds Page 7- g

PI•OPAGA! ION I NdC
ACnOSS TK arO) .
•'------
•).z _1
I
--t
Figure 7.1 Effect of bed thickness on
reflection waveshape,where
(a) Thin-bed model,
(b) Waveletshapesat top
and bottom re fl ectors,
(c) Synthetic seismic
model,anU (d) Tuning
parameters as measured from
resulting waveshape.
(C) (D)
5O
, ,.
THIN
BED
REGIME
J PEAK-TO-TROUGH/
AMPLITUDE
2.0
1.0 <
0.8
0.4
/
-0.4 ,•i . . . . .
-40 0 20 40
MS
TWO-WAY TRUE THICKNESS
(MILLISECONDS)
Figure 7.2 A typical detection and resolution cha•t used
to interpret bed thickness from zero phase seismic data.
('Hardage,1986)
. .. _ i i ,, , i _ - - - -_- - _ - _ ..... l. _
Part 7 - Inversion applied to Thin Beds Page 7- 3

Below a thickness
valueof oneperiodthe wavelets
Start merginginto a
single wavelet, and an amplitude increase is observe•. This amplitude
increase is a maximum
at 1/4 period, and decreases from this point down... The
amplitude is appraoching
zero at 1/•0 period, but note that the resulting
waveform is a gOdegree phase shifted version of the original wavelet.
A morequantitative wayto measurethis 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 diagramquantifies what has
already been seen qualitatively the seimsic section. That is that the
amplitude is a maximum
at a thickness of 1/4 the wavelet dominant period, and
also that this is the lower isochron limit. Thus, 1/4 the dominantperiod is
considered to be the thin bed threshhold, below which it is difficult to
obtain fully resolved reflection coefficients.
7.2 In.
versionCamparison
of 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
impedancemodel is shownin 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 downto a thickness of 2 msec,.or .one time
sample. The resulting synthetic seismogram is shownin Figure 7.4. A 20 Hz
'Ricker wavelet wasusedto create the synthetic. Since the dominant period
(T) of a 20 HzRickeris 50 msec,the wedgehasa thicknessof 2T at trace 1,
T at trace 25, T/2 at trace 37, etc.
Parl•'7 - 'inverslYn
'ap'pl
led 1•o
Thin'-
Beds
..... Page 7 --'4 '•-

lOO
200
3OO
400
500
4 8 12 16 20 24 28 32 36 40 44 48
ß
Figure7.3 Trueimpedance
fromwedge
model.
o
lOO
200
.
300
ß
400
500
Figure 7.4 Wedgemodel reflectivity convolved with
20 HZ Ricker wavelet.
Part 7 - Inversion applied to Thin BeUs Page7- 5

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