Indicator Geostatistics– A brief revisit
Sanjay SrinivasanCox Visiting FacultyStanford University
Stanford Center for Reservoir ForecastingStanford Center for Reservoir Forecasting
Annual Meeting 2010
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A few words about me……• Associate professor, Department of Petroleum &
Geosystems Eng., UT Austin• Cox visiting faculty, Stanford University, Jan. 2010- Sept.
2010• Assistant Professor, Univ. of Calgary, AB, Feb. 2000 –
Aug. 2002• Ph.D. (Petroleum Eng.), Stanford Univ., 1999• Senior Petroleum Eng., Bechtel Corp., 1989-2006
Motivation
• Become rigorously re-acquainted with the Extended Normal Equation
• Explore the notions of ergodicity and stationarity in the context of mp simulations
• Is it time for application of Extended Normal Equation simulation not Single Extended Normal Equation Simulation?
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Talk Outline
• The indicator paradigm• Extended normal equation as a projection• Projection theorem for deriving extended normal
system• Two reduced cases:
– Traditional indicator kriging– Single extended normal equation
• Application of full extended normal system
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Indicator Paradigm
• Consider the indicator RV:
• Important property:
or better still, given n data:
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I(u ) =1, if A(u) = a0, otherwise
⎧ ⎨ ⎩
E I(u ){ } =1 × Prob A(u ) = a( ) + 0 × Prob A(u ) ≠ a( ) = Prob A(u) = a( )
E I(u) | (n){ }= Prob A(u) = a | (n)( )
Indicator basis function
• Consider the projection defined on the basis of n indicator random variables :
– Given the n indicator random variables, this expansion is the most complete possible.
– There are a total of terms in the above expansion
– There are 2 outcomes for each RV there are outcomes possible for the function .
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ϕ(n)Iα ,α =1,...,n
ϕ(n) = a + bj1
(1)
j1 =1
n
∑ ⋅ I u j1( )+ bj1, j2(2)
j2 =1
n
∑ ⋅ I u j1( )j1 =1
n
∑ I u j2( )+K + bj1, j2,K , jn(n) I( u i )
i =1, n∏
1 +
n1
⎛
⎝ ⎜
⎞
⎠ ⎟ +
n2
⎛
⎝ ⎜
⎞
⎠ ⎟ +L +
nn
⎛
⎝ ⎜
⎞
⎠ ⎟ = 2n
2n ϕ(n)
Indicator Expansion
• The conditional expectation is precisely the projection of the unknown indicator event on to .
• The coefficients solved by projecting to the space defined by
• If instead projection is on a reduced basis Lk:
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E I(u) | (n){ }
I(u ) ϕ(n)2n a,b j1 , j2
, ..., b j1 , j1 ,..., jnI(u )
Ln : 1, I j1
,I j1I j2
,I j1I j2
I j3,K ,I j1
I j2I j3
K I jn{ }
E I | (n){ } ≅ Ek I | (n){ } = Ik* = a + bj1
(1)
j1 =1
n
∑ ⋅ I u j1( )+ bj1, j2(2)
j2 =1
n
∑ ⋅ I u j1( )j1 =1
n
∑ I u j2( )+
K + K bj1, j2,K , jk(k )
jk =1
n
∑j2 =1
n
∑ ⋅ I u j1( )j1 =1
n
∑ I u j2( )K I u jk( )
Indicator Expansion
• Another way to write the expansion:
basis function
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Ik
* = λll =1
nk
∑ ⋅ Vl nk =1 +
n1
⎛
⎝ ⎜
⎞
⎠ ⎟ +
n2
⎛
⎝ ⎜
⎞
⎠ ⎟ +L +
nk
⎛
⎝ ⎜
⎞
⎠ ⎟
V1 = 1{ } ; Ii{ }, Ii ⋅ I j{ } etc.
Normal Equations
• The implication of the projection theorem is that:
or in terms of projections:
which is a system of normal equations
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I − Ik*( ), Vl = 0 ⇔ I ⋅ Vl = Ik
*⋅ Vl
E Ik
* Vl{ }= E I Vl{ } , ∀ l =1 ,.., nk
nk
Examples
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V1 = 1 ⇒ E Ik
* 1{ }= λll =1
nk
∑ ⋅ E Vl{ }= E I{ }
Vl = I j , j =1,..,n ⇒ E Ik
* I j{ }= λll =1
nk
∑ ⋅ E Vl ⋅ I j{ }= E I ⋅ I j{ }
Vl = I j1
⋅ I j2, j1, j2 =1,..,n ⇒ E Ik
* I j1I j 2{ }= λl
l =1
nk
∑ ⋅ E Vl ⋅ I j1I j2{ }= E I ⋅ I j1
I j2{ }
λl
l =1
nk
∑ ⋅ E Vl ⋅ Vl'{ }= E I ⋅ Vl
'{ }, l =1,..,nk
k +1
Two simple cases
Applying
Substituting in:
Restricting to
with the normal system:
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V1 = 1 ⇒ λ1 + λll =2
nk
∑ ⋅ E Vl{ }= E I{ }
⇒λ1 = E I{ }− λll =2
nk
∑ ⋅ ml , ml = E Vl{ } , l >1
Ik
* = λll =1
nk
∑ ⋅ Vl =λ1 + λll =2
nk
∑ ⋅ Vl
Ik
* − E I{ } = λll =2
nk
∑ ⋅ Vl − ml( )
Vl = I j , j =1,..,n ⇒ E I | (n){ }− E I{ }[ ] = λj
j =1
n
∑ ⋅ I j − E Ij{ }[ ]
λl
j =1
n
∑ ⋅ Cov Ij ⋅ I i{ }= Cov I ⋅ I i{ }, i =1,..,n
SIK
Second CaseIn general the extended equations are for all 2n
realizations of the n data.If instead, the estimate corresponding to a specific realization of the indicator RVs, say
, then:
Unbiasedness:
Orthogonality:
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Io*
D = i1,i2 ,...,in{ }=1 Io* = ϕ D( ) =λo +λ1⋅ D
Two bases 1,D – two equations to obtain two unknownsIo
* − po =λ1⋅ D − E D{ }( )Io − Io
*, D = Io − po − λ1⋅ D − E D{ }( ), D = 0
⇒ λ1 =E IoD( )− poE D( )
Var D{ } =E IoD( )− E Io( )E D( )
Var D{ }
Single Normal Equation
Application
Central node
Training Image
Eroded Set
Template
ImplementationIndicator Kriging
•4-data configuration shown in the template is broken down into 2 point configurations between pairs of data and between the dataand the unknown•TI scanned using these 2 point sub-templates in order to get
•The weights λ1, λ2, λ3 and λ4 are calculated once for the particular configuration of data•Data template is place at each location on the eroded grid S. Actual values at the data locations are multiplied by weights inorder to calculate p*
Cov I j ⋅ Ii{ } and Cov I⋅ Ii{ }
p* = E I | (n){ }= E I{ }+ λ jj =1
n
∑ ⋅ I j − E I j{ }[ ]
ImplementationSingle Extended Normal Equation Kriging (SNEK)
•At each node u on the eroded set S, the data event D at the data nodes on the template are recorded.•The TI is scanned for computing the frequency of the data event D. This yields E(D).•Corresponding to each occurrence of D, the frequency of the outcome Io=1 on the central node is also recorded - E(Io,D)• p* is then calculated.
pSNEKo
* =E IoD( )E D( ) = E Io | D( )
Results• The base case TI is 2000 x 2000. • Squared error at each estimation location calculated as:
• Histogram of E plotted. Three measures are retained from the histogram:– Mean squared error– Std. Deviation of squared error– Inter-decile range of squared error
• A smaller size TI (1500 x 1500) is sub-sampled from the original TI
• The calculation of λ’s for IK, E(D) & E(Io,D) for SNEK, p* and E is repeated. The procedure is repeated for other grid sizes.
E(u) = i(u) − p*(u)( )2
Maps of p*
2000 grid
500 grid
SNEK IK
Statistics of (i-p*)2 as a function of TI size
SNEK
IK
Some remarks
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• There is a compromise between the optimality of the estimate and the stability/ergodicity of the computed statistics (covariances, mp proportions etc.)
• SNEK is more prone to instability of the inferred statistics especially when the TI size is small
Map of (i-p*)2 - IK Map of (i-p*)2 - SNEK
New Training Image
Map of p* - IKbefore order relations
correctionsMap of p* - after
correcting order relations Map of p* SNEK
Template
Central node
Error statistics• Squared error E at each estimation location is
calculated.• The average of squared error is calculated within the
window of a particular size (say 5 x 5) centered at uo:
• Histogram of plotted. Three measures are retained from the histogram:– Mean squared error– Std. Deviation of squared error– Inter-decile range of squared error
E
E =1
n window
e(u)u ∈ ( uo ± windowsize )
∑
E
Statistics of within moving windows
SNEK
IK
E
How about the full extended equation?
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Central node
Ik* = λo + λ1⋅ I1 + λ2 ⋅ I2 + λ3 ⋅ I3 + λ4 ⋅ I1I2 + λ5 ⋅ I1I3 + λ6 ⋅ I2I3 + λ7 ⋅ I1I2I3Expansion:
System: λo + λ1⋅ E I1{ }+ λ2 ⋅ E I2{ }+ λ3 ⋅ E I3{ }+ λ4 ⋅ E I1I2{ }+
+ λ5 ⋅ E I1I3{ }+ λ6 ⋅ E I2I3{ }+ λ7 ⋅ E I1I2I3{ }= E Io{ } Vl = 1{ }
λoE I1{ }+ λ1⋅ E I12{ }+ λ2 ⋅ E I1,I2{ }+ λ3 ⋅ E I1,I3{ }+
λ4 ⋅ E I1,I1I2{ }+ λ5 ⋅ E I1,I1I3{ }+ λ6 ⋅ E I1,I2I3{ }+ λ7 ⋅ E I1,I1I2I3{ }= E I1,Io{ } Vl = I1{ }
Similarly for I2, I3
Extended Normal System
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λoE I1I2{ }+ λ1⋅ E I1I2,I1{ }+ λ2 ⋅ E I1I2,I2{ }+ λ3 ⋅ E I1I2,I3{ }+ λ4 ⋅ E I1I2,I1I2{ }+
+ λ5 ⋅ E I1I2,I1I3{ }+ λ6 ⋅ E I1I2,I2I3{ }+ λ7 ⋅ E I1I2,I1I2I3{ } = E I1I2,Io{ }
Similarly for I2I3 , I1I3
Vl = I1I2{ }
λoE I1I2I3{ }+ λ1⋅ E I1I2I3,I1{ }+ λ2 ⋅ E I1I2I3,I2{ }+ λ3 ⋅ E I1I2I3,I3{ }+ λ3 ⋅ E I1I2I3,I4{ }+
λ4 ⋅ E I1I2I3,I1I2{ }+ λ5 ⋅ E I1I2I3,I1I3{ }+ λ6 ⋅ E I1I2I3,I2I3{ }+ λ7 ⋅ E I1I2I3,I1I2I3{ } = E I1I2I3,Io{ } Vl = I1I2I3{ }
A system of 8 equations to be solved for 8 unknown λ’s
This system needs to be solved only once. The weights can be applied to any data event on the template nodes.
Results
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Training Image
0.4059139780.2815860220.2946297650.2814027370.2086693550.281219453
0.208486070.20848607
1 0.413703568 0.40023216 0.413520283 0.210441105 0.412787146 0.210074536 0.210074536
0.413703568 0.40591398 0.210441105 0.412787146 0.210441105 0.412787146 0.210074536 0.210074536
0.40023216 0.210441105 0.40591398 0.210074536 0.210441105 0.210074536 0.210074536 0.210074536
0.413520283 0.412787146 0.210074536 0.40591398 0.210074536 0.412787146 0.210074536 0.210074536
0.210441105 0.210441105 0.210441105 0.210074536 0.21044111 0.210074536 0.210074536 0.210074536
0.412787146 0.412787146 0.210074536 0.412787146 0.210074536 0.41278715 0.210074536 0.210074536
0.210074536 0.210074536 0.210074536 0.210074536 0.210074536 0.210074536 0.21007454 0.210074536
0.210074536 0.210074536 0.210074536 0.210074536 0.210074536 0.210074536 0.210074536 0.210074536
λ0
λ1
λ2
λ3
λ4
λ5
λ6
λ7
1 0.102020729i1 -0.017568207i2 0.340700923i3 -0.015784568
i1i2 0.074846892i1i3 0.290132525i2i3 0
i1i2i3 0.218090259
Weight assigned to
12
3
Partial template
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Suppose one template node is un-informed
The system does not have to be re-solved
Re-sum the weights to obtain the new weights for the altered template
e.g. if node 3 is un-informed, thenλ1 = λ1
old + λ1,3old = −0.017 + 0.290 = 0.2725
λ2 = λ2old + λ2,3
old = 0.341+ 0 = 0.341
λ1,2 = λ1,2old + λ1,2,3
old = 0.075 + 0.218 = 0.293
Concluding Remarks
• Various algorithms for mp simulation have to be understood within the framework of extended indicator bases functions
• Issues of ergodicity and stability of computed statistics are key for the successful implementation of mp algorithms
• With advent of high performance computers, is it time to re-embark on the journey of ENESIM?Pro – No repeated scanning for simulation patternsCon - For a 27 node template – 227 size matrix to be
invertedSCRF 2010 27