Transcript Quantifying uncertainty in AVO analysis

Constrained 3 parameter AVO
estimation and uncertainty analysis
Jonathan Downton
and
Larry Lines
Theme
• Demonstrate a constrained 3 term nonlinear AVO
inversion algorithm with reliability estimates
– incorporates constraints suitable for local geologic
conditions
– Degree to which constraints influence the solution is
dependant on S/N, fold, aperture
– reliability quality controls
– Solve for 3 terms
• Density, P- and S- velocity reflectivity
• Can transform to a variety of other attributes
Outline
• Theory
–
–
–
–
–
AVO problem
Bayes Theorem
Demonstrate need for constraints
A priori constraints
Nonlinear inversion algorithm
• Modeling study
• Data example
• Conclusions
Theory – AVO
Use linear approximation of the Zoeppritz equations (Aki and
Richards, 1980)
Or
 1 2 
2
2
  2 sec 1   4 sin 1
d (1 )   

d ( )    1 2 
2
2
 2     sec  2   4 sin  2


  2



 
d
(

)
 N   1 2 
2
2
 sec  N   4 sin  N

 2

 


 






1
1  4 2 sin 2 1
2
1
1  4 2 sin 2  2
2

1
1  4 2 sin 2  N
2
  RVp 
 
  RVs 
 R 
 d 


d=Gm
Where  = average angle of incidence, d() is offset dependant data
 ratio of S velocity to P velocity
RVp, RVs, Rd are the fractional change over the average p-wave velocity, s-wave
velocity and density respectively
Theory - Bayesian Inversion
Bayes’ theorem
Where
P(d | m, I ) P(m | I )
P(m | d, I ) 
P(d | I )
P(m|d,I) = Posterior Probability Distribution Function (PDF)
probability of the parameter vector given the
data and information I
P(d|m,I) = Likelihood function
P(m|I) = A Priori information
P(d|I) = Normalization factor
Theory - Bayesian Inversion
Assuming - Independent data
Uncorrelated uniform Gaussian noise
Likelihood function
2
 NM
 
    Gki mi  d k  
k 1 i 1
 
N

P(d | m, I )   exp 


2 2




Assuming uniform priors, Bayesian inversion is
equivalent to maximum likelihood inversion
Theory - Bayesian Inversion
To find
1d case
1) best estimate of parameters:
find mo for which PDF is
maximum (stationary and
convex)
Best estimate
2) uncertainty of parameters:
width of distribution
 Ideally want uncertainty to
be much smaller than
estimate
m
x
2 uncertainty
Theory – Bayesian Inversion
2 variables lead to a bivariate Gaussian distribution

Q  k  RVp  RVp
2
d
0
RVs  RVs0

  dRVp

 dRVp dRVs
2
1
 dR dR   RVp  RVp
 
2
 dR   RVs  RVs
Vp
Vs
Vs
RVs
RVp
0
0



Theory – Uncertainty Analysis
• Marginalisation equation
allows us to remove
nuisance parameters

P( X | I )   P( X , Y | I )dY

• If we knew value of RVs as
priori information the
uncertainty in RVp would
be much smaller than the
marginalized value
• Important consideration
for 3 parameter AVO
inversion
2dR
Vp
(RVs=0.0)
2dR
Vp
AVO parameter uncertainty
• Solution space for Aki and Richard’s 3 parameter
equation is an ellipsoid defined by the data
covariance matrix Q  mT Cd 1m
AVO parameter uncertainty
• Constraints can be used to reduce the large
uncertainty introduced by the 3rd variable
• One way to do this is with hard constraints
– fix the 3rd variable in terms of a linear combination of
the other two variables.
– Smith and Gidlow (1987) use the Gardner equation
(Gardner et al.1974) to constrain the AVO inversion
problem
Rden=gRVp
where g is Gardner coefficient
Smith and Gidlow (1987) constraint
Constraints
• Instead of hard constraints can use
probabilistic constraints
• constraints calculated from the well control
– The statistics of RIp, RIs and Rd can be estimated and
parameterized using a Gaussian Distribution
– The degree of correlation between RIp, RIs and Rd
define the cross terms or covariances
• The cross-terms have physical significance
Constraints
model a priori constraints with multivariate
Gaussian distribution
 1 T Cm 1 
Pm | I   exp  m
m
2

 2

where
  2 RVp  R R  R R 
Vp Vs
Vp den


2
C m    RVp RVs  RVs  RVs Rden 
2




Rden
R
R
R
R
Vs den
 Vp den

and  is the global scalar
Constraints
Alternatively, Use empirical rock physical
relationships to build covariance matrix
1st derivative of Mudrock line
RVs  mRVp
with correlation coefficient r1
1st derivative of Gardner relationship
Rden  gRVp
with correlation coefficient r2
Rden  fRVs
  2 RVp

  RVp RVs

 RVp Rd en
R R
 2R
R R
Vp Vs
Vs
Vs
d en

1
 RVp Rd en 


 RVs Rd en    2 RVp  m

2

 Rd en 

g

m
m
2
r1
f

g 

f 

g 

2
r 2
AVO parameter uncertainty
Rden
RVs
RVp
3 term AVO inversion
Combining the likelihood function and the a
priori constraints using Bayes’ theorem
leads to the nonlinear equation
1
 T
ε ε
1 
T
m  G G 
C
G
d
m

2
N  1


T
e Gm-d
This equation is weakly nonlinear and may be solved
using Newton-Raphson
3 term inversion
• Uncertainty analysis
– Width of probability distribution
• Parameterize using variance or standard deviation (Downton et
al. 2000)
• Want to know if parameter estimate is coming
from the data or the constraints
• examine ratio of unconstrained to constrained variances
• Transform matrix
– Shuey (1985), Fatti et al. (1994), Gray (1999) are
rearrangements of Aki and Richards equation
– Can trivially transform parameter estimates and
uncertainty to different parameterizations common in
the literature
Outline
• Theory
–
–
–
–
–
AVO problem
Bayes Theorem
Demonstrate need for constraints
A priori constraints
Nonlinear inversion algorithm
• Modeling study
• Data example
• Conclusions
Synthetic example
– Vp, Vs and density based on Glauconite well
log from Western Canada
– synthetic gathers generated
• Design model to test the effect of
– Fold, offset, S/N separately
– Constraints generated based on well control
– Compare parameter estimates to reference to
zero offset synthetics
Model Study
Vp
Vs Vp/Vs Density
Synthetic gather with S/N=2
offset
Blackfoot: Constraints
P-velocity
vs. S-velocity
P-velocity
vs. density
Rp Vs.Rd
Rs Vs.Rd
0.25
0.25
0.2
0.2
0.2
0.15
0.15
0.15
0.1
0.1
0.1
0.05
0.05
0.05
0
Rd
0.25
Rd
Rs
Rp Vs.Rs
S-velocity
vs. density
0
0
-0.05
-0.05
-0.05
-0.1
-0.1
-0.1
-0.15
-0.15
1
1
16
-0.2
-0.25
-0.15
1
1
16
-0.2
-0.25
-0.2
-0.1
0
Rp
0.1
0.2
Rs=1.06Rp, r1=.91
1
1
16
-0.2
-0.25
-0.2
-0.1
0
Rp
0.1
0.2
Rd=0.71Rp, r2 =.93
-0.2
-0.1
0
Rs
0.1
0.2
Rd=0.82Rs, r3 =.90
  2 RVp  R R  R R 
 4.176
Vp Vs
Vp den


2
C m    RVp RVs  RVs  RVs Rden   10 4   3.979
2


 2.450


R
den
RVs Rden
 RVp Rden

3.979 2.450
4.790 2.326
2.326 1.640 
3 term AVO inversion
0-45 degrees, S/N=8
Estimate
Actual
3 term AVO inversion
0-45 degrees, S/N=1/4
Estimate
Actual
3 term AVO inversion
0-28 degrees, S/N=8
Estimate
Actual
Outline
• Theory
–
–
–
–
–
AVO problem
Bayes Theorem
Demonstrate need for constraints
A priori constraints
Nonlinear inversion algorithm
• Modeling study
• Data example
• Conclusions
Data Example
• Line part of project shot to explore for Halfway
sand potential (Downton and Tonn, 1998)
– 2 Bright spots
• Producing field
– Expect both velocity and density anomaly
• Uneconomic field (porous sand with low gas saturations)
– Expect velocity anomaly to be bigger than density anomaly
• 3 term constrained AVO inversion performed
– Data has good S/N
– Angles to 45 degrees
• Does density reflectivity differentiate two
anomalies?
A
C
E
F
A
C
E
F
A
C
E
F
A
C
E
F
Conclusions I
• Demonstrated a nonlinear 3 parameter AVO
inversion on synthetic and real data
– incorporates probabilistic a priori constraints
that are calibrated with local well control or
rock physical relationships
• These probabilistic constraints introduce less
bias than the hard constraints implicit in
many two term AVO inversion schemes
• Amount constraints influence solution is
dependent on S/N, fold and offset
Conclusions II
• If constraints dominate the solution
– Still get useful P- and S-impedance reflectivity
estimates
– Density reflectivity is largely a linear combination of Pand S-impedance reflectivity and does not provide
independent information
• Need a combination of large S/N ratio, fold and
offsets to estimate the density reflectivity reliably
– Most datasets probably will not meet this requirement
– Need to view quality controls to understand the
influence of the constraints and the reliability of the
estimates
• The results of the inversion and uncertainty can be
transformed to other attributes common in the
literature
Conclusion
• Demonstrated a nonlinear 3 parameter AVO inversion on synthetic and
real data
– incorporates probabilistic a priori constraints that are calibrated
with local well control or rock physical relationships
• These probabilistic constraints introduce less bias than the hard
constraints implicit in many two term AVO inversion schemes
• Amount constraints influence solution is dependent on S/N,
fold and offset
– If constraints dominate the solution
» Still get useful P- and S-impedance reflectivity estimates
» Density reflectivity is largely a linear combination of P- and Simpedance reflectivity estimates and does not provide independent
information
– Need a combination of large S/N ratio, fold and offsets to
estimate the density reflectivity reliably
» Most datasets probably will not meet this requirement
– Need to view quality controls to understand the influence of the
constraints and the reliability of the estimates
• The results of the inversion and uncertainty can be transformed to
other attributes common in the literature
AVO parameter uncertainty
Constraints
Rd=0
RVp=0
Rden=gRVp
Rden
covm
RVs
RVp
AVO parameter uncertainty
Constraints
Rd=0
RVp=0
Rden=gRVp
Rden
covm
RVs
RVp