586 Joint Inversion Overview

An overview of  Hampson-Russell’s new  Joint Inversion Program  Dan Hampson  Brian Russell  Keith Hirsche  10 August 2005

Theory 3-1

Objective

Objective of Joint Inversion: To analyze pre-stack CDP gathers and invert for Zp, Zs, and (optionally) Density (ρ).

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Theory 3-2

Objective

Objective of Joint Inversion: To analyze pre-stack CDP gathers and invert for Zp, Zs, and (optionally) Density (ρ).

10 August 2005

Theory 3-2

Introduction

Our current practice is to invert separately for Zp, Zs, and ρ. An example of this procedure is LMR analysis:

Gathers AVO Analysis

RP Estimate

RS Estimate

Invert to ZP

Invert to ZS

Transform to λρ and µρ Cross-plot 10 August 2005

Theory 3-3

Introduction The problem with this approach is that it ignores the fact that Zp and Zs should be related. For example, we expect that from Castagna’s equation, Vp and Vs should be more or less linearly related, with variations precisely where there are hydrocarbons. Similarly, ρ should be related to Vp by some form of generalized Gardner’s equation. ARCO’s original mudrock derivation  (Castagna et al, Geophysics, 1985) 10 August 2005

Theory 3-4

Introduction

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.

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Theory 3-5

Joint Inversion Theory We start with the modification of Aki-Richards’ equation as per  Fatti et al:

RPP (θ ) = c1 RP + c2 RS + c3 RD where: 2 c1 = 1 + tan θ  2

 RP =

2

c2 = −8γ sin θ  c3 = −

γ   =

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V S V P

1 2

2

2

2

tan θ + 2γ sin θ 

 RS =  R D =

1  ∆ V P



2  V P 1  ∆ V S



2  V S

∆ ρ   ρ 

+ +

∆ ρ  



 ρ  

∆ ρ  



 ρ  

. Theory 3-6

Joint Inversion Theory To simplify this theory, it is common practice to use the “small reflectivity” approximation. For example, the exact equation for Rp is:

 RP (i ) = But, if we define:  LP we can show that:

ZP ( i + 1) − ZP ( i) ZP ( i + 1) + ZP ( i)

= ln( Z P ) (natural logarithm)

RP ( i) ≅ 1 2 [ LP ( i + 1) − LP ( i)] Similarly:

 LS = log( Z S )  L D = log( ρ )

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RS ( i) ≅ 1 2 [ LS ( i + 1) − LS ( i)]  R D (i ) ≅  L D (i + 1) − L D (i ) Theory 3-7

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

RP = (1 2 ) D Lp or:

  RP (1)   −1 1 0   R (2)   0 −1 1  P  =1 2  M   0 0 −1    ( ) R N 0 0 0  P 

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L  LP (1)

 0   LP (2)    1 M    ( ) N L  L P 

 

Theory 3-8

Joint Inversion Theory We add the effect of the wavelet by defining the wavelet matrix:

T = W R p  T (1)   T (2)    =1  M    ( ) T N   

0 W1 0 W W  0 2 1 2 W3 W2 W 1   0 W3 W 2

L  −1

 0  L  0  L  0 L

 −1 1 0   LP (2)    0 −1 1   M    0 0 L  LP ( N )  1

0

L  LP (1)

Finally, Fatti’s equation looks like:

T (θ ) = (1 2) c1W (θ ) DLP + (1 2)c2W (θ ) DLS + c3W (θ ) DLD   Note that the wavelet can be different for each angle. 10 August 2005

Theory 3-9

Joint Inversion Theory Now we want to make use of the fact that the resulting Zs and ρ should be related to Zp. We use two relationships which should hold for the background “wet” trend:

VS V P = γ   = constant

Constant γ

→ ln( Z S ) = ln( Z P ) + ln(γ  ) and:

 ρ  = aV Pb

→ ln( ρ ) = 10 August 2005

b

1+ b

ln( Z P ) +

ln( a )

Generalized Gardner 

1+ b Theory 3-10

Joint Inversion Theory More generally, we assume the following relationships for the background trend:

ln( ZS ) = kln( ZP ) + kc + ∆ LS ln( ρ ) = m ln( Z P ) + mc + ∆LD

This assumes that the major  trend is linear and that the outliers are the hydrocarbons:

Ln(Zs)

∆ LS

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Ln(Zp)

Theory 3-11

Joint Inversion Theory More generally, we assume the following relationships for the background trend:

ln( ZS ) = kln( ZP ) + kc + ∆ LS ln( ρ ) = m ln( Z P ) + mc + ∆LD

Ln(ρ)

Ln(Zs)

∆ L D

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∆ LS

Ln(Zp)

Ln(Zp)

Theory 3-12

Joint inversion theory This changes Fatti’s equation to:

T (θ ) = c%1W (θ ) DLP + c%2W (θ ) D∆LS + c3W (θ ) D∆L D where:

c%1 = (1 2) c1 + (1 2) kc2 + mc3 c%2 = (1 2) c2

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

c%2 (θ1 )W (θ1 ) D c3 (θ1 )W (θ1 ) D    T (θ1 )   c%1 (θ1 )W (θ1 ) D  T (θ )   c% (θ )W (θ ) D c% (θ )W (θ ) D c (θ )W (θ ) D     LP  1 2 2 2 2 3 2 2  = 1 2   ∆ LS    M    M M M  ∆ L D    %  T (θ N)   c1 (θ N)W (θ N) D c%2 (θ N)W (θ N) D c3 (θ N)W (θ N) D    10 August 2005

Theory 3-13

Joint inversion theory 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 ρ. (2) Calculate optimal values for k and m using the actual input logs. (3) Set up the initial guess:

[

LP

∆ LS

T

T 

∆ LD ] = [log( ZP )   0 0]

(4) Solve the system of equations by conjugate gradients. (5) Calculate the final values of Zp, Zs, and ρ:

 Z P = exp( LP ) ZS = exp( kLP + kc + ∆ L S) 10 August 2005

 ρ  = exp( mLP + mc + ∆LD )

Theory 3-14

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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Theory 3-15

Gas/Wet Synthetic Tests Vp

Vs

ρ

σ

We produced a series of synthetic gathers corresponding to varying fluid effects:

Target Zone

100% Gas

100% Wet

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Theory 3-16

The result at the GAS location Initial guess: Zp

Zs

 ρ 

Model

Input

Error 

σ 

After 50 iterations:

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Theory 3-17

The result at the WET location Initial guess: Zp

Zs

 ρ 

Model

Input

Error 

σ 

After 50 iterations:

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Theory 3-18

Zp

Zs

 ρ 

σ  100% Gas

10% Gas

0% Gas

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Theory 3-19

The synthetic test on a range of CDP gathers

0’

18,000’

Original offset gathers

0o

90o

Transformed to angle

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Theory 3-20

The synthetic test on a range of CDP gathers

Zp

Zs

ρ

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Theory 3-21

Zp

Vp/Vs

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Theory 3-22

Input gathers

The synthetic test on a range of CDP gathers

Synthetic gathers

Error 

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Theory 3-23

Real Data Test – Colony This test applies the simultaneous inversion algorithm to the Colony data set from Western Canada:

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Theory 3-24

Real Data Test – Colony Transform to angle gathers:

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Theory 3-25

Real Data Test – Colony Using the known well, create cross plots to determine the optimum coefficients:

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Theory 3-26

Real Data Test – Colony Zp

Zs

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Theory 3-27

Real Data Test – Colony Zp

Vp/Vs

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Theory 3-28

Real Data Test – Colony Input gathers:

Synthetic data from inversion:

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Theory 3-29

Real Data Test – Colony Input gathers:

Synthetic error from inversion:

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Theory 3-30

Comparison between real logs and inversion result at well location Zp

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ρ

Vp/Vs

Theory 3-31

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

This zone should correspond to gas:

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Theory 3-32

Zp

Vp/Vs

Gas Zone from log cross plot 10 August 2005

Theory 3-33

Extension to PS data Similarly to the Fatti equation, we can write down a linearized expression for the PS reflectivity (Stewart, 1990; Larson, 1999):

RPS (θ , φ ) = c4 RS + c5 RD

tan φ 

 4 sin 2 φ − 4γ cosθ cosφ  ,  γ   − tan φ  1 + 2 sin 2 φ − 2γ cosθ cosφ  ,  c5 = 2γ  

where :

c4 =

and  :

φ = sin −1 (γ sin θ ) .

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Theory 3-34

Extension to PS data Combining the expression for the PS reflectivity with the relationships given earlier, we get:

TPS (θ ) = c%4W (θ ) DLP + ( c4 2 ) W (θ ) D∆LS + c5W (θ ) D∆LD  , where : c%4 = k ( c4 2 ) + mc5. Note that this is exactly the same form as the original equation for TPP:

TPP (θ ) = c1W (θ ) DLP + c2W (θ ) D ∆LS + c3W (θ ) D ∆L D This means that we can (theoretically) handle any combination of PP and PS traces, at any number of angles.

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Theory 3-35