9 AVO Inversion
Short Description
9 AVO Inversion...
Description
S e i s m i c In I n v e r s i o n a p p l ie ied t o L i t h o l o g i c Pr P r ed ed i c t i o n
Part 9 – AVO Inversion
Introduction • In this section, we will look at a model based approach to AVO inversion. • We will first look at a flowchart of the method, and then 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 earlier sections. • Finally, we will discuss a three parameter inversion scheme developed by Kelly et al. (TLE, March and April, 2001), showing examples from their work. 9-2
Introduction • In this section, we will look at a model based approach to AVO inversion. • We will first look at a flowchart of the method, and then 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 earlier sections. • Finally, we will discuss a three parameter inversion scheme developed by Kelly et al. (TLE, March and April, 2001), showing examples from their work. 9-2
Model-based Inversion Flowchart Common
Offset
Wavelet
Synthetic
Offset Stack
Edited Logs
LeastSquares Difference Update Logs using Inversion Method Good
NO
Fit?
YES Final Model
9-3
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. 9-4
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:
R ( ) aR P b c 1 2 2 where : a 1 D 2 ( 1 D ) sin , 1 D
b
sin 2
, 2 ( 1 )
c
V P / V P V P / V P /
V P
2 V P
,
sin2 tan 2 . 9-5
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:
R 1 R ( 1 ) c 1 a1R P b1 R 2 R ( 2 ) c 2 a2 R P b2
R N R ( N ) c N aN R P bN • 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. 9-6
Matrix form of the equations • We can re-express the equations from the previous page in matrix form, to make our solution easier:
R 1 a1 b1 R a b R 2 2 2 P R N aN bN • If we write the above equation in the form R = A P , the solution is P = A -1 R. The problem is that N is usually greater than 2, and a non-square matrix cannot be inverted.
9-7
More equations than unknowns • The solution to a problem with more equations than unknowns is well known (Lines and Treitel, 1984) but will not 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:
( 1 ) Original equation : R AP , ( 2 ) Multiply by transpose : AT R ( AT A )P , ( 3 ) Invert : P ( AT A )1 AT R . • If the inverse is unstable, we must add prewhitening:
1 0 P ( A A I ) A R , where I 0 1 T
1
T
9-8
The Full Solution • Let us now fill in the details of the computation: a1 a2 A A b1 b2 T
a1 b1 N 2 ak aN a2 b2 N k 1 bN bk ak aN bN k 1
a1 a2 And : A R b1 b2 T
Thus :
N 2 ak P N k 1 ba k k k 1
ak bk
2 b k k 1
k 1 N
R 1 N a R k k aN R 2 k 1 N bN b R k k R N k 1
a b k k k 1 N bk 2 k 1 N
N
1
N a R k k k N 1 b R k k k 1
9-9
The Smith/Gidlow Method • This method was also proposed by Smith and Gidlow, except that they used the following equation, modified from Aki-Richards:
R ( ) a
V P
V P
b
V S
V S 2
where:
5 1 V S 1 2 a sin tan 2 , 8 2 V P 2 2
V S b 4 sin 2 . V P 9-10
A more complete solution • However, as said before, the coefficients a, b, and c depend on the parameters that we are trying to solve. 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 Gardner’s equation 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. 9-11
The Simplest Case • But what if we assume that we know R P? The solution can now be written without matrices:
( R ( ) c aR P ) / b
• In the next exercise, we will use the above equation iteratively to invert for a gas sand.
9-12
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 V P and , but incorrect for , use the previous equation to iterate towards a correct solution. o
• Assume that you have made one measurement at 30 . Note the following parameters: (remember, the observed reflectivity is the value calculated using Shuey’s full equation, not the Aki-Richards form of the equation):
R ( 30 o ) 0 .133 R P 0 .071 c 0 .005 2 initial 1 / 3 9-13
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 – Then work out – Next, compute – Look up new values for a and b – Continue through iterations.
• The next few slides take you through the first iteration.
9-14
Computations – Starting point Iteration
a
0
0.333
1
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
0
0.333
0.750
b 0.563
9-15
Computations – First Iteration Iteration
a
b
0
0.333
0
0.333
0.750
0.563
1
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 9-16
Lookup table for a and b values 2
a
b
0.333
0.750
0.563
0.330
0.747
0.325
2
a
b
0.215
0.639
0.475
0.560
0.210
0.634
0.471
0.742
0.556
0.205
0.630
0.468
0.320
0.736
0.551
0.200
0.626
0.465
0.315
0.731
0.547
0.195
0.621
0.462
0.310
0.726
0.543
0.190
0.617
0.459
0.305
0.722
0.539
0.185
0.613
0.456
0.300
0.717
0.535
0.180
0.609
0.452
0.295
0.712
0.531
0.175
0.605
0.449
0.290
0.707
0.528
0.170
0.601
0.446
0.285
0.702
0.524
0.165
0.597
0.443
0.280
0.697
0.520
0.160
0.593
0.441
0.275
0.693
0.516
0.155
0.589
0.438
0.270
0.688
0.513
0.150
0.585
0.435
0.265
0.683
0.509
0.145
0.581
0.432
0.260
0.679
0.505
0.140
0.577
0.429
0.255
0.674
0.502
0.135
0.573
0.426
0.250
0.670
0.498
0.130
0.569
0.423
0.245
0.665
0.495
0.125
0.565
0.421
0.240
0.661
0.491
0.120
0.561
0.418
0.235
0.656
0.488
0.115
0.558
0.415
0.230
0.652
0.484
0.110
0.554
0.413
0.225
0.647
0.481
0.105
0.550
0.410
0.220
0.643
0.478
0.100
0.546
0.407
9-17
Answers to Computations Iteration
a
b
1
0.333
0.333
0
0.750
0.563
2
0.333
0.201
-0.132
0.626
0.465
3
0.333
0.154
-0.179
0.588
0.437
4
0.333
0.136
-0.197
0.574
0.427
5
0.333
0.129
-0.204
0.568
0.423
6
0.333
0.126
-0.207
0.566
0.421
7
0.333
0.125
-0.208
0.565
0.421
8
0.333
0.124
-0.209
0.565
0.420
9
0.333
0.124
-0.209
0.565
0.420
10
0.333
0.124
-0.209
0.565
0.420 9-18
Graph your Results 0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
Iteration # 9-19
Results of the Inversion Inversion with Shuey's Equation 0.35 0.3 o 0.25 i t a R 0.2 s ' n o s 0.15 s i o P 0.1
0.05 0 0
2
4
6
8
10
Iteration Number 9-20
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:
f f mk k 1 mk p
f j ( m ) observations, j 1,...N, f j (m0 ) c alculated values, j 1,..., N, f f j ( m ) f j (m0 ),
mk true model parameters, k 1,..., p, m0 k initial guess model parameters, k 1,..., p, m change in model parameters mk m0 k .
9-21
Generalized Linear Inversion • In matrix form, for N=3 observations and P=2 parameters, we have:
f 1 f 1 m1 f f 2 2 m f 3 f 1 3 m1
f 1 m2 f 2 m1 , or f Am. m2 m2 f 3 m2
• Since we usually have more observations than unknown model parameters, the solution can be found by the least-squares method discussed earlier: m ( AT A )1 AT f
9-22
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. 9-23
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. 9-24 Finally, the real common offset stack is shown on the right.
Real Data Example - with synthetic Here is the same display 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.
9-25
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 in the 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. 9-26
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. 9-27
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. 9-28
Conclusions • This has been a overview of several methods for inverting prestack amplitudes to derive velocity, density, and Poisson’s ratio. • We first considered a method which used a linear model between the observations and the parameters. • 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. 9-29
Answers to Computations Iteration
a
b
1
0.333
0.333
0
0.750
0.563
2
0.333
0.201
-0.132
0.626
0.465
3
0.333
0.154
-0.179
0.588
0.437
4
0.333
0.136
-0.197
0.574
0.427
5
0.333
0.129
-0.204
0.568
0.423
6
0.333
0.126
-0.207
0.566
0.421
7
0.333
0.125
-0.208
0.565
0.421
8
0.333
0.124
-0.209
0.565
0.420
9
0.333
0.124
-0.209
0.565
0.420
10
0.333
0.124
-0.209
0.565
0.420 9-30
View more...
Comments
