defining a random function (rf) from a given set of...
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Céline Scheidt, Kwangwon Park and Jef Caers
Stanford University
Defining a Random Function (RF) From a Given Set of Model Realizations
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Introduction
Uncertainty in reservoir modeling requires the construction of ensembles of models
Many applications are “goal”-oriented and thus require the construction of ensembles having particular features
E.g. data conditioning, model updating, optimization, etc.
Ineffective: realizations are generated without the response function in mind
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Motivation Example
Set of Initial Realizations
One realization matching production data
0 500 1000 15000
2
4
6
8x 10
4
Time
Pro
duc
tion
Da
ta
Data Acquisition
How to generate efficiently a new set of realizations that match data ?
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Goals
To generate new realizations without re-running the algorithm/sampler that generated them (Model expansion)
To identify a small set of variables which describe the uncertainty (Parameterization)(useful for inverse modeling, response
analysis, optimization, etc…)
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General Approach
Use of dissimilarity distances and MDSIntegrate information specific to the applicationOptimization or uncertainty quantification problems more efficient
Use of Karhunen-Loeve expansion (KL-expansion) in Feature space
Preserve the spatial variability and the hard data conditioning
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Use of Application-Specific Distances
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In the motivation example previouslyDistance represents dissimilarity of desired properties of the model (here production data)Wish to construct new realizations with small distances to the “reference”
xClose to reference
Far to reference
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K-L Expansion for Gaussian Realizations
2/1 newnew V yx Λ=N(0,1)~newy
[ ]L21 ,,, xxx K=X
New realization:
From Jef Caers
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[ ]
V
VVXXB
X
new
TT
yx
xxx
2/1
L21
expansion -KL
product -Dot
,,, nsrealizatioGaussian
Λ=⇓
Λ==⇓
= K
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MDSUsing K
featurespace
Model expansion
ector)Gaussian v standard a is (L
1 with )(
:expansionLoeve-Karhunen
y
ybbx KVΦ ==ϕ
ΦX)( ii aa ⇒xφx
Φ
21/KKVΦ Λ=
L)(L Matrix Gramor Kernel
)()(
),(
×⇒
⎟⎟⎠
⎞⎜⎜⎝
⎛ −−−==
K
expkK jiT
j,dijiij σ
xxxxxx
TKKK VVK Λ=
Model Expansion in Feature SpaceFrom Jef Caers
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Feature Space
F
)](,),([ ,1, Ldd xx ϕϕ K=Φ
Metric Space
M
Definition of a RF for Non-Euclidean Distances and non-Gaussian Realizations
ϕ
MDS ],,[ ,1, Lddd xxX K=
newnew Φbx =)( ϕ
ϕ-1
],,[ 1 LxxX K=
CNℜ∈ix
Distance Matrix δ:
Non-EuclideanApplication
tailoredPre-image
Reconstruction of
realizations
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The Pre-image ProblemHow to map back realizations from feature space F to metric space M ?
Feature Space
F
Metric Space
M
ϕ-1
newΦb
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The Pre-image Problem
newK
newnewnewd
newd V-Φ
newd
ybbxxx L
1 with )ˆ(minargˆ2
2== ϕ
Model expansion is defined in high-dimensional space F
Mapping function ϕ is unknown, non-linearand non-uniqueOnly approximate solutions are possible
Optimization problem
newΦb
Feature Space
F
Metric Space
Mϕ ϕ-1
newΦb
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ˆnewdx
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The Pre-image ProblemPre-image problem can be solved using an iterative procedure, called fixed point algorithm (Schoelkopf and Smola, 2002)
: non-linear combination of the realizations in Metric space
∑∑
′
′= =+
),ˆ(),ˆ(
ˆ,
,1 ,,
,1,
idnnew
di
L
i ididnnew
dinnewd kb
kbxxxxx
x
1 with ˆL
1,
L
1
== ∑∑== i
optiid
i
opti
newd ββ xx
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1,ˆ +nnewdx
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? ϕ
Fixed point algorithm:Initial guess is defined as linear combination of ’s, with random coefficientsDefine new point using equation from previous slideIterate until convergence
The Pre-image Problem
0,ˆ newdx id ,x
2,ˆ newdx nnew
d,x̂
1,ˆ newdx
newΦb0,ˆ newdx new
dx̂
Metric Space Feature Space
FM
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idi
opti
newd β ,
L
1
ˆ xx ∑=
=
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The Pre-image Problem
1 with ˆL
1,
L
1== ∑∑
== i
optiid
i
opti
newd ββ xx
Pre-image solution:
newdx̂
Summary
Metric Space
M
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Creating a new modelHow to construct the corresponding new realization in physical space?
newdx̂
Metric Space
M
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Creation of new models in physical space
Unconstrained optimizationApply the same weights as in the previous equation ( )
Ensures hard data conditioning:
Feature constrained optimizationConstrain explicitly the solutions of the inverse problem
Geological constrained optimizationUse of probability mapsUse of Gradual Deformation Method (Hu, 2001) or Probability Perturbation Method (Caers, 2002)
3 different options explored:
1L
1=∑
=i
optiβ
optiβ
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Examples
Present 3 examples of different types of realizations
Continuous, Boolean and Multi-Point
RF generated as follow:Initial ensemble contains 300 realizations
Definition of a connectivity distance between realizations (Park and Caers, 2008)
Gaussian rbf Kernel for the definition of the feature space F
Pre-imageUnconstrained, feature constrained and geologically constrained
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Unconstrained Pre-Image - ExampleDirect sequential simulation (dssim)
Uniform conditional distribution, exp. variogram
Initial Real.:
Connectivity distance
4
3
2
0
2
3
MDS
Kernel techniques
K-L expansion
Pre-image:
Physical realizations
2D projection of modelsFrom metric space
2D projection of modelsFrom Feature space
2D projection of modelsFrom metric space
)( newnewd Φbx =ϕ
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Initial Real.New Real.
∑=
=L
idi
opti
newd β
1,xx
∑=
=L
ii
opti
new β1
xx
ix
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Unconstrained Pre-Image - Example
Analysis of the new generated realizations
Same variability between realizations19 SCRF Affiliate Meeting - April 30, 2009
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300 Boolean realizations
Feature Constrained Optimization
3 Initial real.Connectivity
distance
MDS
Model expansion
Pre-image
Solve pre-image problem
optiβ
0
5
0
5
5
2
5
Initial Real.New Real.
2D projection of modelsFrom metric space
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Can we apply weights directly to binary images?
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Reconstruction of realizations from the pre-image weights
Feature Constrained Optimization
Difficult to apply non-linear weights on binary images
optiβ
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How can we preserve the shape of the objects?
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Boolean realizations
Feature Constrained Optimization
Channel objects are clearly identifiable
3 New real.
Weights obtained
previously (Unconstrained
pre-image)
InverseProximity Transform
Solution: Use a proximity distance
Proximity Transform
optiβ
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5
0
5
5
2
5
Initial Real.New Real.
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Geologically Constrained OptimizationExample 1
optiβ
Unconstrained pre-image
Weights :3 Initial real.
Multiple-Point realizations (snesim)
0
0
0
0
0
0
2D projection of modelsFrom metric space
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Probability maps -> input for snesim as soft data
Geologically Constrained Opt.
Smoothing algorithm
Generation of real. using snesim
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Geologically Constrained Optimization
Multiple-Point realizations (snesim)Reconstruction of 300 new realizations
3 Initial real. 3 New real.Metric space based on connectivity distance
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Example 2
Optimization of the pre-image problem using
PPM
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Geologically Constrained Optimization
Multiple-Point realizationsReconstruction of 1 realization close to one having desired properties
x
oooo
oxoo
Metric space
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Conclusions
We presented a method to generate new realizations by solving the pre-image problem using an ensemble of existing realizations
By construction, new realizations share the same conditioning data and spatial continuityproperties as the existing ones.
Method applies to all types of input geostatistical model (Gaussian-type, Boolean and multi-point techniques).
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Application of this technique is shown in other presentations (Scheidt a