586 joint inversion overview
TRANSCRIPT
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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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Theory 3-210 August 2005
Objective
Objective of Joint Inversion:
To analyze pre-stack CDP gathers and invert for Zp,
Zs, and (optionally) Density ().
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Theory 3-310 August 2005
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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Theory 3-410 August 2005
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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Theory 3-510 August 2005
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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Theory 3-710 August 2005
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
+ =
+ +
Joint Inversion Theory
(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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Theory 3-810 August 2005
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
Joint Inversion Theory
or:
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Theory 3-910 August 2005
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
Joint Inversion Theory
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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Theory 3-1010 August 2005
Now we want to make use of the fact that the resulting Zs and
should be related to Zp.
Joint Inversion Theory
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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Theory 3-1110 August 2005
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:
Joint Inversion Theory
Ln(Zs)
Ln(Zp)
This assumes that the major
trend is linear and that the
outliers are the hydrocarbons:
SL
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Joint Inversion Theory
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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Theory 3-1310 August 2005
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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Theory 3-1410 August 2005
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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Theory 3-1610 August 2005
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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Theory 3-2010 August 2005
The synthetic test on a range of CDP gathers
Original offset
gathers
Transformed
to angle
0o 90o
0 18,000
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Theory 3-2110 August 2005
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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Theory 3-2410 August 2005
Real Data Test Colony
This test applies the simultaneous inversion algorithm to the Colony data
set from Western Canada:
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Real Data Test Colony
Transform to angle gathers:
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Real Data Test Colony
Using the known well, create cross plots to determine the optimum
coefficients:
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Real Data Test Colony
Zp
Zs
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Real Data Test Colony
Zp
Vp/Vs
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Input gathers:
Synthetic data from inversion:
Real Data Test Colony
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Input gathers:
Synthetic error from inversion:
Real Data Test Colony
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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.