Indicator Geostatistics– A brief revisit

Sanjay SrinivasanCox Visiting FacultyStanford University

Stanford Center for Reservoir ForecastingStanford Center for Reservoir Forecasting

Annual Meeting 2010

SCRF 2010 2

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