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8/11/2019 9 AVO Inversion
1/31
Seism ic Inversion app l ied to
Li tho log ic Predict ion
Part 9
AVO Inversion
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9-2
Introduction
In this section, we will look at a model basedapproach to AVO inversion.
We will first look at a flowchart of the method, andthen discuss the theory.
We will work on a simple problem using the wet
and gas cases that we examined earlier. We will then look at a real data example, involving
the Colony sand that has been discussed in earliersections.
Finally, we will discuss a three parameterinversion scheme developed by Kelly et al. (TLE,March and April, 2001), showing examples fromtheir work.
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Common
Offset Stack
Offset
SyntheticWavelet
Least-
Squares
Difference
Final
Model
Good
Fit?
NO
YES
Edited
Logs
Update
Logs using
Inversion Method
Model-based Inversion Flowchart
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Possible approaches to Inversion
The previous slide is fairly straightforward, except for
one box, which specifies that we use an inversion
method. There are many inversion methods that can
be used, including, from simplest to most complex:
Trial and error
Finding a linear model Generalized Linear Inversion (GLI)
Simulated Annealing
Genetic Algorithms
Post-stack inversion of AVO attributes Although each method has its advantages, we will
consider only the second and third methods in this
section. The last method will be discussed in the next
section.
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A Linear Model for Inversion
In model-based inversion, we first need a model that
relates our observations to our parameters. Initially,we will use the Aki-Richards linearized model, as
modified by Shuey:
cbaR)(R P
,sin1
21)D1(2D1a:where 2
,/V/V
V/VDPP
PP
,)1(
sinb
2
2
.tansin
V2
Vc 22
P
P
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9-6
Setting up the Equations
If we have observations from N traces in a CDP
gather about the AVO response, we can write down
N equations with two unknowns, based on the
previous equation:
NPNNNN
2P2222
1P1111
bRac)(RR
bRac)(RRbRac)(RR
Note that the a,b, and c values are not constant but
also depend on the parameters, but we will initially
assume they are constant.
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Matrix form of the equations
We can re-express the equations from the previous
page in matrix form, to make our solution easier:
P
NN
22
11
N
2
1
R
ba
ba
ba
R
R
R
If we write the above equation in the form R = AP, the
solution is P = A -1R. The problem is that N is usually
greater than 2, and a non-square matrix cannot be
inverted.
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More equations than unknowns
The solution to a problem with more equations than
unknowns is well known (Lines and Treitel, 1984) but willnot be derived here. The first step is to multiply both
sides by the matrix transpose. This creates a covariance
matrix, which is square and can be inverted:
.RA)AA(P:Invert)3(
,P)AA(RA:transposebyMultiply)2(
,APR:equationOriginal)1(
T1T
TT
If the inverse is unstable, we must add prewhitening:
10
01Iwhere,RA)IAA(P T1T
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9-9
The Full Solution
Let us now fill in the details of the computation:
N
1k
2
k
N
1k
kk
N
1k
kk
N
1k
2
k
NN
22
11
N21
N21T
bab
baa
ba
ba
ba
bbb
aaaAA
N
1k
kk
N
1k
kk
N
2
1
N21
N21T
Rb
Ra
R
R
R
bbb
aaaRA:And
N
1k
kk
N
1k
kk
1
N
1k
2
k
N
1k
kk
N
1k
kk
N
1k
2
k
Rb
Ra
bab
baa
P:Thus
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9-10
The Smith/Gidlow Method
This method was also proposed by Smith andGidlow, except that they used the following
equation, modified from Aki-Richards:
S
S
P
P
VVb
VVa)(R
.sinV
V4b
,tan
2
1sin
V
V
2
1
8
5a
2
2
P
S
22
2
P
S
where:
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A more complete solution
However, as said before, the coefficients a, b, and c
depend on the parameters that we are trying tosolve. Therefore, a single iteration through the
previous inversion step will not fully solve the
problem. We need to arrive at the solution
iteratively, as follows (note that Smith and Gidlow
only use a single iteration in their method):
(1) Estimate an initial set of values for , , and VP, and
thus work out initial values for a, b, and c.
(2) Use the inversion equation to solve for and RP.
(3) Derive new values for a, b, and c, using Gardnersequation to break the acoustic impedance into and VP.
(4) Invert again using the new a, b, and c values.
(5) Repeat the above procedure until convergence.
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9-13
Inversion Exercise
Let us assume that we have encountered a gas sand on our
seismic data identical to the one that was modelled earlier.Starting with an initial guess that is correct for VPand , but
incorrect for , use the previous equation to iterate towards a
correct solution.
Assume that you have made one measurement at 30o. Note the
following parameters: (remember, the observed reflectivity is thevalue calculated using Shueys full equation, not the Aki-Richards
form of the equation):
3/1
005.0c
071.0R
133.0)30(R
initial2
P
o
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9-14
Inversion Exercise
Fill in the table on the next page for all 10 iterations
(or until convergence) by using the lookup table on
the following page to derive a and b values.
Hints:
First look up a and b for
2
Then work out
Next, compute 2
Look up new values for a and b
Continue through iterations.
The next few slides take you through the firstiteration.
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Computations Starting point
Iteration
2
a b
0 0.333 0 0.333 0.750 0.5631 0.333
2 0.333
3 0.333
4 0.333
5 0.333
6 0.333
7 0.333
8 0.333
9 0.333
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Computations First Iteration
Iteration
2
a b
0 0.333 0 0.333 0.750 0.5631 0.333 -0.133 0.200 0.626 0.465
2 0.333
3 0.333
4 0.333
5 0.333
6 0.333
7 0.333
8 0.333
9 0.333
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Lookup table for a and b values2 a b
0.333 0.750 0.563
0.330 0.747 0.560
0.325 0.742 0.556
0.320 0.736 0.551
0.315 0.731 0.547
0.310 0.726 0.543
0.305 0.722 0.539
0.300 0.717 0.535
0.295 0.712 0.531
0.290 0.707 0.528
0.285 0.702 0.524
0.280 0.697 0.520
0.275 0.693 0.516
0.270 0.688 0.513
0.265 0.683 0.509
0.260 0.679 0.505
0.255 0.674 0.5020.250 0.670 0.498
0.245 0.665 0.495
0.240 0.661 0.491
0.235 0.656 0.488
0.230 0.652 0.484
0.225 0.647 0.481
0.220 0.643 0.478
2 a b
0.215 0.639 0.475
0.210 0.634 0.471
0.205 0.630 0.468
0.200 0.626 0.465
0.195 0.621 0.462
0.190 0.617 0.459
0.185 0.613 0.456
0.180 0.609 0.452
0.175 0.605 0.449
0.170 0.601 0.446
0.165 0.597 0.443
0.160 0.593 0.441
0.155 0.589 0.438
0.150 0.585 0.435
0.145 0.581 0.432
0.140 0.577 0.429
0.135 0.573 0.4260.130 0.569 0.423
0.125 0.565 0.421
0.120 0.561 0.418
0.115 0.558 0.415
0.110 0.554 0.413
0.105 0.550 0.410
0.100 0.546 0.407
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9-18
Answers to Computations
0.4200.565-0.2090.1240.33310
0.4200.565-0.2090.1240.3339
0.4200.565-0.2090.1240.3338
0.4210.565-0.2080.1250.3337
0.4210.566-0.2070.1260.3336
0.4230.568-0.2040.1290.3335
0.4270.574-0.1970.1360.3334
0.4370.588-0.1790.1540.3333
0.4650.626-0.1320.2010.3332
0.5630.75000.3330.3331
ba2
Iteration
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9-19
Graph your Results
0.4
0.3
0.2
0.1
109876543210
Iteration #
2
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9-20
Results of the Inversion
Inversion with Shuey's Equation
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2 4 6 8 10
Iteration Number
Poisson'sRatio
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Generalized Linear Inversion
If a linear model cannot be found, we use the technique
of Generalized Linear Inversion, or GLI. In this method,we linearize the problem in the following way:
p
1k
k
k
mm
ff
.mmparametersmodelinchangem
p,1,...,k,parametersmodelguessinitialm
p,1,...,k,parametersmodeltruem
),(mf)m(ff
N,1,...,jvalues,alculatedc)(mf
1,...N,jns,observatio)m(f
k0k
k0
k
0jj
0j
j
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9-22
Generalized Linear Inversion
In matrix form, for N=3 observations and P=2
parameters, we have:
Am.for,m
m
m
f
m
f
m
f
m
fm
f
m
f
ff
f
2
1
2
3
1
3
2
2
1
2
2
1
1
1
3
2
1
Since we usually have more observations thanunknown model parameters, the solution can be found
by the least-squares method discussed earlier:
fA)AA(m T1T
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9-23
Real Data Example - Procedure
Now we will look at a real data example of inversion,
using a method that is similar to the one just described,
except that the wavelet is taken into account. The
inversion involves the following steps:
(1) If S-wave log is not available, estimate using Mudrock line(2) Extract a suitable wavelet
(3) Correlate the data using the zero offset seismic trace and
synthetic
(4) Block the log, while honouring the major boundaries
(5) Compute S-wave value in zone of interest via Biot-Gassmann
(6) Use inversion to modify the thickness, density, P-wave velocity,
and S-wave velocity in each of the blocked zones.
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9-24
Real Data Example - Initial Model
This slide shows the initial setup for the inversion. The blocked logs are shown
on the left along with the zero offset correlation. The mudrock line was used
for the S-wave log, except in the gas zone, where Biot-Gassmann was used.
Finally, the real common offset stack is shown on the right.
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9-25
Real Data Example - with synthetic
Here is the samedisplay as the
previous slide
except that the
synthetic has been
inserted in the
middle. Notice that
there is a
reasonable fit at the
zone of interest, but
not below the zone
of interest.
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9-26
Real Data Example - Inversion
We now perform inversion by changing
the thickness, density, and P- and S-wave
velocities in each of the blocked layers.
The figure above shows the decrease inthe least-squared error between the real
data and the resulting synthetic. Notice
the convergence of the error. The figure
on the left shows the wavelet used in the
modelling and inversion.
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9-27
Real Data Example - Final Logs
Here is a comparison between the final inverted logs (in red) and the initial
logs (in black). The zero offset synthetic has also been recalculated on the
right. Notice the better zero offset fit.
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Real Data Example - Final Display
Here is the final display, showing the inverted logs on the left in red (the original
logs are in black), the updated offset synthetic in the middle, and the original
data on the right. Notice the excellent fit between synthetic and real data.
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9-29
Conclusions
This has been a overview of several methods forinverting prestack amplitudes to derive velocity,
density, and Poissons ratio.
We first considered a method which used a linear
model between the observations and theparameters.
We considered an example of this method, and
showed how it was related to the Smith-Gidlow
method. We then looked at the Generalized Linear Inverse
approach to linearizing problems.
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Answers to Computations
0.4200.565-0.2090.1240.33310
0.4200.565-0.2090.1240.3339
0.4200.565-0.2090.1240.3338
0.4210.565-0.2080.1250.3337
0.4210.566-0.2070.1260.3336
0.4230.568-0.2040.1290.3335
0.4270.574-0.1970.1360.3334
0.4370.588-0.1790.1540.3333
0.4650.626-0.1320.2010.3332
0.5630.75000.3330.3331
ba2
Iteration
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Results of the Inversion
Inversion with Shuey's Equation
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 2 4 6 8 10
Iteration Number
Poisson'sRatio