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3 Sampling efficiency Key issues SCRF 2012 Rejection Sampler Markov Chain MC Reference d obs d predict Proposed model Forward modeling d predict
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Adaptive Spatial Resampling as a McMC Method for Uncertainty Quantification
in Seismic Reservoir Modeling
Cheolkyun Jeong*, Tapan Mukerji, and Gregoire Mariethoz
Stanford Center for Reservoir Forecasting
How to quantify uncertainty of models?
Why quantify uncertainty?
2
Key issues
SCRF 2012
1. We make decisions under uncertainty2. Modeling subsurface reservoir is a uncertain process
1. In a Bayesian framework, sampling posterior distribution can quantify the uncertainty
2. Rejection sampler is a theoretically perfect method but inefficient
A critical issue is to sample posteriors efficiently: A Markov chain Monte Carlo method as an equivalent posterior sampler
3
Sampling efficiencyKey issues
SCRF 2012
Rejection Sampler Markov Chain MC Reference
dobs dpredict
Proposed model
Forward modeling
dpredict
4SCRF 2012
d
a. Generate and evaluate its likelihood L().
b. Select a subset points .c. Generate a proposal
model d. Evaluate L() and accept or
reject by acceptance criterion.
e. Iterate b. ~ d.
ππ
a
ππ
bc d ππ
d
e
ππ
d *e
Creating a Markov chain: Iterative Spatial Resampling (ISR)Methodology
5
Creating a Markov chain: ISR
SCRF 2012
Methodology
a. Generate and evaluate its likelihood L().
b. Select a subset .c. Generate a proposal
model d. Evaluate L() and accept or
reject by acceptance criterion.
If L() L() orL() L() with P(L()/ L())
6
ASR algorithm in acoustic impedance
Randomly sampled subset points
Adaptively sampled subset points
Randomly sampled subset points
Adaptively sampled subset points
SCRF 2012
(π ΒΏΒΏπππβπ (πΒΏΒΏππππ))ΒΏΒΏ
Methodology β Adaptive Spatial Resampling
spatial error map
7
ASR algorithm in seismic section
SCRF 2012
Methodology β Adaptive Spatial Resampling
Seismogram: obtained data
Seismogram: predicted model
Cross correlation coefficient in each trace
time
time
corr
elat
ion
coef
ficie
nt
CDP
Higher correlationHigher chance
Lower correlationMore perturbation
subset
Reference: faciesReference
Iterative Spatial Resampling Adaptive Spatial Resampling
SCRF 2012
ASR algorithm in acoustic impedance
Methodology β Adaptive Spatial Resampling
Log1
0 RM
SE
Log1
0 RM
SE
Iteration Iteration
9
1. Fraction rate in ASR
SCRF 2012
Methodology β Parameter sensitivity
Log1
0 RM
SE
Iterations
10
2. Number of traces in seismic section
SCRF 2012
Methodology β Parameter sensitivity
Log1
0 SSE
Iterations
11
1. Acoustic impedance for lithofacies characterization
Reference: facies Well dataWells
Predicted seismic data
SCRF 2012
Illustration
Seismic dataacoustic impedance
CDP 25 125
MRayls
MRayls
Vp
π
Bivariate pdf Rockphysics
Reference: facies
Etype of priors
Etype of sampled posteriors (RS)
Variance of sampled posteriors (RS)
100,000 priors
125 posteriors
12SCRF 2012
1. Acoustic impedance: Rejection SamplerIllustration
Reference: facies
1. Acoustic impedance: Results
13
Etyp
eVa
rianc
e
125 posteriors (100,000 eval.)
21 posteriors (500 eval.)
94 posteriors(500 eval.)
Rejection sampling Iterative Spatial Resampling Adaptive Spatial Resampling
SCRF 2012
Illustration
14SCRF 2012
1. Acoustic impedance: ASRIllustration
15
2. Seismograms for facies characterization
Reference: facies Well dataWells
Predicted seismic data
SCRF 2012
Seismic dataseismograms
CDP 25 125
Vp
π
Bivariate pdf Rockphysics
Illustration
Reference: facies
2. Seismogram: Results
16
Etyp
eVa
rianc
e
140 posteriors (100,000 eval.)
29 posteriors (500 eval.)
51 posteriors(500 eval.)
Rejection sampling Iterative Spatial Resampling Adaptive Spatial Resampling
SCRF 2012
2. Seismogram
Illustration
17SCRF 2012
2. Seismogram results using MDS projectionIllustration
18
1st principal coordinate
2nd p
rinci
pal c
oord
inat
e
SCRF 2012
3. Verification using MDS projectionIllustration
19
3. Finding facies not seen in well data
Reference: facies Well dataWells
Predicted seismic data
SCRF 2012
Seismic dataseismograms
CDP 25 125
Vp
π
Bivariate pdf Rockphysics
Oilsand
Brinesand
Shale
*Not detected oilsand distribution is generated by Gassmannβs equation
Facies Actual Logs
Vp π One model in priorsOne model in posteriors
Illustration
20SCRF 2012
24 posteriors (50,000 eval.) 43 posteriors (1000 eval.)
3. Finding facies not seen in well data
Probability of Oil Sand Probability of Oil Sand
CDP CDP
Probability
Oilsand
Brinesand
Shale
Rejection sampling Adaptive Spatial Resampling
Reference: facies
Illustration
4. ASR as an optimizer
21
Log1
0 RM
SE
Iterations
Reference: facies
SCRF 2012
Keep only better models in a Markov chain: L() L()
Illustration
4. ASR as an optimizer
22SCRF 2012
Illustration
2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and itβs more efficient.
1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making.
23
3. Depending on the acceptation/rejection criterion, it is possible to obtain a chain for sampling posterior or calibrating the most likely earth model.
SCRF 2012
Summary
1. Application in actual dataset: West Africa dataset
24SCRF 2012
Ongoing and Future work
3 wells, Near and Far offset seismic data
<Courtesy to Hess>
Geological Observation
Rockphysics model (Dutta, 2009)Facies 1: Channel DepositionFacies 2: Near channel leveesFacies 3: Medial-distal levees
What we have
1. Application in actual dataset: West Africa dataset
25SCRF 2012
Ongoing and Future work
2D slice : Acoustic Impedance
Geological ObservationBuild Training images
3D study
What we need
2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and itβs more efficient.
1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making.
26
3. Depending on the acceptation/rejection criterion, it is possible to obtain a chain for sampling posterior or calibrating the most likely earth model.
SCRF 2012
Summary
1. Application in actual dataset: West Africa dataset
27SCRF 2012
Ongoing and Future work
Multiple subsurface scenarios
1. P(Tis | Seismogram) using pattern validation
Geologist (1)
Geologist (2)
Geologist (3)
Pattern Validation for finding distances between seismogram images
Generate priors, m Forward model, g(m)
Multiple subsurface scenarios
2. P(RPs | Seismic data) using pattern validation
Rockphysics (1)
Pattern Validation for finding distances between seismogram
Forward model, g(m)
3. Multiple subsurface scenarios
Rockphysics (2)
Generate priors, m
2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and itβs more efficient.
1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making.
30
3. Multiple subsurface scenarios help to choose the most applicable setting for unknown reservoir modeling.
SCRF 2012
Summary
31SCRF 2012
3. Multiple subsurface scenarios1. P(Ti | Seismic data) using pattern validation
32SCRF 2012
3. Multiple subsurface scenarios
[301x301]
1. P(Ti | Seismic data) using pattern validation
33SCRF 2012
3. Multiple subsurface scenarios
Ti1 = 0.0014 at the data location, Ti2 was 2.4498, and Ti3 was 5.6447
According to Bayesian theorem and Park(2011), P(Ti2|data) = 30% and P(Ti3|data) = 70%
Ti2 Ti3
1. P(Ti | Seismic data) using pattern validation
2. P(RPs | Seismic data) using pattern validation
Rockphysics (1)
Pattern Validation for finding distances between seismogram
Forward model, g(m)
3. Multiple subsurface scenarios
Rockphysics (2)
Generate priors, m
35SCRF 2012
3. Multiple subsurface scenarios2. P(RPs | Seismic data) using pattern validation
36SCRF 2012
24 posteriors (50,000 eval.) 43 posteriors (1000 eval.)
3. Finding facies not seen in well data
Probability of Shale Probability of Shale
Probability of Brine Sand Probability of Brine Sand
Probability of Oil Sand Probability of Oil Sand
CDP CDP
Probability
Oilsand
Brinesand
Shale
Rejection sampling Adaptive Spatial Resampling
Illustration
SEG 2011 37
Appendix III : Ti
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