Stochastic Simulation of Patterns using Distance-Based Pattern Modeling

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# Stochastic Simulation of Patterns using Distance-Based Pattern Modeling - PowerPoint PPT Presentation

Multiple-Point Geostatistics. Search Template 9x9 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100. Proposed Method. Filtersim. Stochastic Simulation of Patterns using Distance-Based Pattern Modeling. Mehrdad Honarkhah. Motivation.

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Presentation Transcript

Multiple-Point Geostatistics

Search Template 9x9

Inner Patch 5x5

Multiple-Grids 3

Number of Clusters 100

Proposed Method

Filtersim

Stochastic Simulation of Patterns using Distance-Based Pattern Modeling

Motivation
• Improving pattern reproduction
• Less parameters – less user interaction
• Reduce simulation time
• Checking pattern/MPS reproduction

Introduction

• better, more realistic models should not require an increase in user-set parameters (Filtersim)
• Why use a complexmethod when a much simpler one works just as well ?
Distance-Based Methods:(what will be demonstrated)

In distance-based modeling, many of the tasks usually performed in multiple-point geostatistical algorithms can be carried out in a surprisingly simple yet powerful way

• Second
• First
• Answer queries through several inference mechanisms
• Store and organize a domain of knowledge about the TI

MPS

Realization

Distance-Based

Outline

Implementation details(1) Pattern Database

Pattern Database

3×3 SearchTemplate

Training Image

Implementation details(3) Kernel Mapping

Cartesian Space

Feature Space Projection

Implementation details(4) Kernel K-means Clustering

Cartesian Space

Feature Space Projection

Implementation detailsSummary Example

δ12

Training Image

Multi Dimensional Scaling

δ13

δ14

δ23

δ24

δ34

Kernel Space Mapping

Kernel K-Means

Implementation details(5) Simulation

Recall: Simulation Algorithm in Filtersim

• Classify training image patterns into clusters using filter scores and partition
• Loop through all nodes of the simulation grid
• Retrieve the data event at that node
• Find the most similar cluster prototype to that data event
• Randomly pick a pattern from that cluster
• Paste it on the simulation grid
• end
Implementation details(5) Simulation

Simulation Algorithm in the Proposed Method

• Classify training image patterns into clusters using Kernel k-means
• Loop through all nodes of the simulation grid
• Retrieve that data event at that node
• Find the most similar cluster prototype to Data Event
• Randomly pick a pattern from that cluster
• Paste it on the simulation grid
• end

Note: The basic Filtersim algorithm is maintained.

only the modeling of patterns changes

MPS Simulation Examples (1/2)

Search Template 9x9

Inner Patch 5x5

Multiple-Grids 3

Number of Clusters 100

MPS Simulation Examples (2/2)

Search Template 11x11

Inner Patch 5x5

Multiple-Grids 3

Number of Clusters 100

Reducing user interaction (1)

Automatic Template Size Selection:

• Calculating mean Entropy at different template Dimensions
• Calculating second derivative of the mean Entropy Curve
• Calculating Profile Log-Likelihood of the resulting Curve
• Template Size = Maximum in the Profile
Reducing user interaction (1)

Automatic Template Size Selection:

5 x 5

Training image

Mean Entropy

Log-Likelihood

Template Size

Template Size

1

2

3

4

5

1

2

3

The actual smallest template size is 5 x 5

4

5

Reducing user interaction (1)

Automatic Template Size Selection:

13 x 13

Training image

Mean Entropy

Log-Likelihood

Template Size

Template Size

Reducing user interaction (2)

Find Number of Clusters:

• Eigenvalue decomposition of K =
• Plot
• Calculate Profile Log-Likelihood

Number of Clusters = Maximum in the profile

Comparison with Filtersim

Find Number of Clusters:

• Eigenvalue decomposition of K =
• Plot
• Calculate Profile Log-Likelihood

Number of Clusters = Maximum in the profile

SIMULATION EXAMPLES

Comparison with Filtersim (1/6)

101 x 101 Training Image

Proposed Methodology Realizations

FilterSim Realizations

Search Template 9x9

Inner Patch 5x5

Multiple-Grids 3

Number of Clusters 100

Comparison with Filtersim (1/6)

101 x 101 Training Image

Proposed Methodology Realizations

FilterSim Realizations

Search Template 13x13

Inner Patch 5x5

Multiple-Grids 3

Number of Clusters 100

Comparison with Filtersim (2/6)

101 x 101 Training Image

Proposed Methodology Realizations

FilterSim Realizations

Search Template 9x9

Inner Patch 5x5

Multiple-Grids 3

Number of Clusters 100

Comparison with Filtersim (2/6)

101 x 101 Training Image

Proposed Methodology Realizations

FilterSim Realizations

K = 100

K = 100

Search Template 13x13

Inner Patch 7x7

Multiple-Grids 3

Number of Clusters -----

K = 400

Comparison with Filtersim (3/6)

111 x 111 Training Image

FilterSim Realizations

Proposed Methodology Realizations

Search Template 15x15

Inner Patch 9x9

Multiple-Grids 3

Number of Clusters 1000

Comparison with Filtersim (4/6)

101 x 101 Training Image

FilterSim Realizations

Proposed Methodology Realizations

Search Template 11x11

Inner Patch 5x5

Multiple-Grids 3

Number of Clusters 100

Comparison with Filtersim (5/6)

169 x 169 Training Image

FilterSim Realizations

Proposed Methodology Realizations

Search Template 15x15

Inner Patch 11x11

Multiple-Grids 3

Number of Clusters 200

Comparison with Filtersim (6/6)

69 x 69x39 Training Image

Proposed Methodology Realizations

FilterSim Realizations

Search Template 15x15x9

Inner Patch 9 x 9 x 5

Multiple-Grids 3

Number of Clusters 200

Studying variability between realizations

How can we check the pattern/multiple point statisticalreproduction of the realizations with respect to the training image ?

Studying variability between realizations

Variability between realizations correspond to the variability obtained while honoring the multiple-point information of the training image.

Variability Quantification

• Generate N realizations
• Find a distance function that characterizes the variability between any two realizations
• Map the N realizations into a Cartesian space using MDS
• The cloud of points in that space represents variability
Studying variability between realizations

Measure of Variability between two realizations : l1and l2

f ( l1 ,l2 ) =JSD{ MPH(l1),MPH ( l2) }

Multiple-Point Histogram (MPH)

Using a multiple-point template, scan the realizations

Store the frequency of a specific configuration of outcomes

Results in the multiple-point histogram

Jensen Shannon Divergence (JSD)

A measure of similarity between two probability distributions: pandq

Studying variability between realizations

Red points : Filtersim realizations Black points : Proposed Method realizations

Training Image Used

Filtersim Method

Proposed Method

Studying variability between realizations

Training Image Used

CONCLUSION

Filtersim has less pattern reproduction than our method, and therefore increased artificial variability.

Great way to diagnose pattern reproduction (i.e. by looking at ALL realizations at the same time).

Conclusion
• Distance method: easy to implement, very few user-set parameters
• Distance methods also allow easy evaluation of MPS reproduction
• Future work: use model expansion techniques to generate additional patterns to further increase geological realism and data conditioning