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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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Introduction to Seismic Inversion Methods Brian Russell
8.1 Introducti on
In the past sections of the course, we have derived reflectivi-ty
information directly from the seismic section and used recursire inversion to
produce a final velocity versus depth model. We have also seen that these
methods can be severely affected by noise, poor amplitude recovery, and the
band-limited nature of seismic data. That is, any problems in the data itsel f
will be included in the final inversion result.
In this chapter, we shall consider the case of builaing a geologic moUel
first and comparing the model to our seismic data. We shall then use the
results of t•is comparison between real and modeled data to iteratively update
the model in such a way as to better match the seismic data. The basic idea
of this approac• is shown in Figure 8.1. Notice that this method is
intuitively very appealing since it avoids the airect inversion of the seismic
data itself. On the other hand, it may be possible to come up with a model
that matches 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.
Part 8 - Model-based Inversion
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Introduction to Seismic Inversion Methods Brian Russell
• ' • 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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Introduction to Seismic Inver. sJon Methods Brian Russell
In Figure 8.2, notice that the advantage of incorporating multiples in
the solution is that, although they are modelled in co•uting the seismic
response, hey are not included in the model parameters. This is a big
advantage over recursire methods, since those methods incorporate the
multiples into the solution if they are not removed rom the section.
Another important feature of this particular method is the
parameterizationused. Instead of assigninga different value of velocity at
each time sample, large geological blocks were defined. Each block was
assigneda starting impedance alue, impedancegradient, and a thickness in
time. This reduceU the numberof parameters and therefore simplified the
computation.However,here is enough flexibility in this modellingapproach
to derive a fairly detailed geological inversion. We will now look at both a
syntheticandreal exampleromCooke ndSchneider1983).
A 2-0 synthetic example was next considered by Cooke and Schneider
(1983). Figure 8.3 shows he model, which consisted of two gas sands encased
in shale. One well encountered the sand and the other missed. The impedance
profile of the discovery well is shown n the right. Figure 8.4 shows
synthetic traces over the two wells, in which a noise component as been
added. Finally, Figure 8.5 shows he initial guess and the final solution,
for which the gradients have been set to zero. Notice that although the
solution is not perfect, the gas sand has been delineated.
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Introduction to Seismic Inversion Methods Brian Russell
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.
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Introduction to Seismic Inversion Methods Brian Russell
1.4
1.6
1kit
,1.4
2.0
Initial
Figure 8.7
lkft
Stack
Left' Init)al Denver 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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Introduction to Seismic Inversion Methods Brian Russell
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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Introduction to Seismic Inversion Methods Brian Russell
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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Introduction to Seismic Inversion He.thods Brian Russell
'Now the third equation s not reducible to either of the other two, ana
an alternate solution must be found. The most popular aproach is the method
of least squares, which minimizes the sumof the squared error between the
solution and the observed results. That is, if we set the error to
e=Ax-y,
then we si reply mini mize
eTe--e , ez .......
n
, en = e ß
2
Le.
Re expressing the 'preceding equation in terms of the values x, y, and A,
we have
ß E = eTe (y - Ax)T(y Ax)
= yTy xTATyyTAx xTATAx.
We then solve the equation
bE_
bx
The final solution to the least-squares problem is given by the normal
equa i OhS
AT x = A y
or x = (ATA)-lATy
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