9 avo inversion

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


2/31

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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9-3

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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9-4

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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9-5

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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9-7

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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9-8

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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9-11

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.


12/31


13/31

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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9-15

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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9-16

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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9-17

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


21/31

9-21

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


22/31

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


23/31

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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9-28

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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9-30

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


31/31

9-31

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

9 avo inversion

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