586 Joint Inversion Overview

8/3/2019 586 Joint Inversion Overview

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Theory 3-110 August 2005

An overview of

Hampson-Russells new

Joint Inversion Program

Dan HampsonBrian RussellKeith Hirsche


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Objective

Objective of Joint Inversion:

To analyze pre-stack CDP gathers and invert for Zp,

Zs, and (optionally) Density ().


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Our current practice is to invert

separately for Zp, Zs, and .

An example of this procedure is

LMR analysis:

Introduction

Gathers

AVO Analysis

RP

Estimate RS

Estimate

Cross-plot

Invert to ZP Invert to ZS

Transform to and


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The problem with this approach is that it ignores the fact that Zp and

Zs should be related.

Introduction

ARCOs original mudrock derivation(Castagna et al, Geophysics, 1985)

For example, we expectthat from Castagnas

equation, Vp and Vs

should be more or less

linearly related, with

variations preciselywhere there are

hydrocarbons.

Similarly, should be

related to Vp by some

form of generalized

Gardners equation.


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The objective of joint inversion is to include some form of coupling

between the variables.

This should add stability to a problem that is ill-conditioned:

- very sensitive to noise

- very non-unique.

A second objective is to create a joint inversion which is consistent

with Strata for the case of zero-offset.

Introduction


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1 2 3( )PP P S DR c R c R c R = + +

We start with the modification of Aki-Richards equation as per

Fatti et al:

Joint Inversion Theory

2

1

2 2

2

2 2 2

3

1 tan

8 sin

1 tan 2 sin2

S

P

c

c

c

V

V

= +

=

= +

=

1

2

12

.

PP

P

SS

S

D

VR

V

VRV

R

= +

= +

=

where:


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To simplify this theory, it is common practice to use the small

reflectivity approximation.

For example, the exact equation for Rp is:

But, if we define: ln( )P PL Z=

[ ]( ) 1 2 ( 1) ( )P P P R i L i L i +

( 1) ( )( )

( 1) ( )

P PP

P P

Z i Z iR i

Z i Z i

+ =

+ +


(natural logarithm)we can show that:

log( )S SL Z=

log( )DL =

[ ]( ) 1 2 ( 1) ( )S S S R i L i L i +

)()1()( iLiLiR DDD +

Similarly:


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In matrix notation for the P-wave reflectivity this is:

(1 2 )P p R D L=

(1) (1)1 1 0

(2) (2)0 1 1 01 2

0 0 1 1

( ) ( )0 0 0

P P

P P

P P

R L

R L

R N L N

=

L

M M

L


or:


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We add the effect of the wavelet by defining the wavelet matrix:

p

T W R=

1

2 1

3 2 1

3 2

0 0 (1)(1) 1 1 0

0 (2)(2) 0 1 1 01 2

0 0 1 10 ( )( ) 0 0 0

P

P

P

W LT

W W LT

W W WW W L NT N

=

L L

L

L MML L


Finally, Fattis equation looks like:

1 2 3( ) (1 2) ( ) (1 2) ( ) ( )P S DT c W DL c W DL c W DL = + +

Note that the wavelet can be different for each angle.


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Now we want to make use of the fact that the resulting Zs and

should be related to Zp.


We use two relationships which should hold for the backgroundwet trend:

constant

ln( ) ln( ) ln( )

S P

S P

V V

Z Z

= =

= +

ln( )ln( ) ln( )

1 1

b

P

P

aVb a

Zb b

=

= ++ +

and:

Constant

GeneralizedGardner


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ln( ) ln( )

ln( ) ln( )

S P c S

P c D

Z k Z k L

m Z m L

= + +

= + +

More generally, we assume the following relationships for the

background trend:


Ln(Zs)

Ln(Zp)

This assumes that the major

trend is linear and that the

outliers are the hydrocarbons:

SL


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Ln(Zs)

Ln(Zp)

Ln()

Ln(Zp)

SLDL

ln( ) ln( )

ln( ) ln( )

S P c S

P c D

Z k Z k L

m Z m L

= + +

= + +

More generally, we assume the following relationships for the

background trend:


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where:

This changes Fattis equation to:

1 2 3( ) ( ) ( ) ( )P S DT c W DL c W D L c W D L = + + % %

1 1 2 3

2 2

(1 2) (1 2)

(1 2)

c c kc mc

c c

= + +

=

%

%

Joint inversion theory

Finally, assume we have a series of traces at various angles. Weconcatenate the traces into a single vector to get the system:

1 1 1 1 2 1 1 3 1 1

1 1 2 2 2 2 2 3 2 2

1 2 3

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

P

S

D

N N N N N N N

T c W D c W D c W DL

T c W D c W D c W D L

LT c W D c W D c W D

=

% %

% %

M M M M

% %


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The algorithm looks like this:

(1) Given the following information:

- A set of N angle traces.- A set of N wavelets, one for each angle.

- Initial model values for Zp, Zs, and .

(2) Calculate optimal values for k and m using the actual input logs.

(3) Set up the initial guess:

(4) Solve the system of equations by conjugate gradients.

(5) Calculate the final values of Zp, Zs, and :

[ ] [ ]log( ) 0 0T T

P S D P L L L Z =

exp( )P PZ L=

exp( )S P c S Z kL k L= + +

exp( )P c DmL m L = + +

Joint inversion theory


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Synthetic and real data tests

We now show 2 tests of the joint inversion algorithm:

(1) A synthetic data set, showing variations in fluid content from

pure gas to pure brine.

(2) A real data set from Western Canada.


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We produced a series

of synthetic gathers

corresponding to

varying fluid effects:

Gas/Wet Synthetic Tests

100%

Gas

100%

Wet

Vp Vs

Target Zone


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Zp

Initial guess:

After 50 iterations:

InputModel Error

The result at the GAS location

Zs


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Zp

Initial guess:

After 50 iterations:

InputModelError

The result at the WET location

Zs


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Zp Zs100%

Gas

10%

Gas

0%

Gas


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The synthetic test on a range of CDP gathers

Original offset

gathers

Transformed

to angle

0o 90o

0 18,000


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The synthetic test on a range of CDP gathers

Zp

Zs


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Zp

Vp/Vs


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The synthetic test on a range of CDP gathersInput gathers

Synthetic gathers

Error


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Real Data Test Colony

This test applies the simultaneous inversion algorithm to the Colony data

set from Western Canada:


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Transform to angle gathers:


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Using the known well, create cross plots to determine the optimum

coefficients:


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Zp

Zs


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Zp

Vp/Vs


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Input gathers:

Synthetic data from inversion:



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Input gathers:

Synthetic error from inversion:



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Comparison between real logs and

inversion result at well location

Zp Vp/Vs


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Cross plotting Vp/Vs against Zp using the log curves:

This zone

should

correspondto gas:


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Zp

Vp/Vs

Gas Zone

from log

cross plot


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Extension to PS data

Similarly to the Fatti equation, we can write down a linearized expression

for the PS reflectivity (Stewart, 1990; Larson, 1999):

( )

4 5

2

4

2

5

1

( , )

tan: 4 sin 4 cos cos ,

tan1 2 sin 2 cos cos ,

2

: sin sin .

PS S D R c R c R

where c

c

and

= +

=

= +

=


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Extension to PS data

Combining the expression for the PS reflectivity with the relationships

given earlier, we get:

( )( )

4 4 5

4 4 5

( ) ( ) 2 ( ) ( ) ,: 2 .

PS P S DT c W DL c W D L c W D Lwhere c k c mc

= + + = +

%

%

1 2 3( ) ( ) ( ) ( )PP P S DT c W DL c W D L c W D L = + +

Note that this is exactly the same form as the original equation for TPP:

This means that we can (theoretically) handle any combination of PP andPS traces, at any number of angles.


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Joint Inversion Assumptions

(1) Aki-Richards 3-term equation as modified by Fatti.

(2) Small reflectivity assumption.

(3) Linear relationship between ln(Zs), ln(Zp) and ln() isreasonable.

(4) A constant value of = Vs/Vp used in Fatti coefficients.

(5) NMO-stretch can be handled by angle-dependent wavelets.

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586 Joint Inversion Overview

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