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Stochastic Simulation of Patterns using Distance-Based Pattern ModelingMehrdad Honarkhah

Stochastic Simulation of Patterns using Distance-Based Pattern ModelingMehrdad Honarkhah

Multiple-Point Geostatistics

Search Template 9x9Inner Patch 5x5

Multiple-Grids 3Number of Clusters 100FiltersimProposed Method

Clear 3-DMultiple-Point GeostatisticsSearch Template 15x15Inner Patch 9x9

Multiple-Grids 3Number of Clusters 1000FiltersimProposed Method

Previous MPS MethodsMPSMotivationImproving pattern reproduction

Less parameters less user interaction

Reduce simulation time

Checking pattern/MPS reproduction

Why use a complex method when a much simpler one works just as well ?Introductionbetter, more realistic models should not require an increase in user-set parameters (Filtersim)

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 wayAnswer queries through several inference mechanismsSecondStore and organize a domain of knowledge about the TIFirst Distance-BasedOutlineMPSRealizationBroad Outlook on the WorkflowImplementation details(1) Pattern DatabasePattern DatabaseTraining Image

33 Search TemplateImplementation details(2) MDS mapping

11Implementation details(3) Kernel MappingCartesian SpaceFeature Space Projection

Implementation details(3) Kernel Mapping

Cartesian SpaceFeature Space ProjectionImplementation details(4) Kernel K-means ClusteringFeature Space Projection

Cartesian Space

Implementation details(4) Kernel K-means ClusteringCartesian SpaceFeature Space ProjectionImplementation detailsSummary Example

122423141334Multi Dimensional Scaling

Kernel K-Means

Kernel Space Mapping

Training ImageWorkFlow Summary?Implementation details(5) SimulationRecall: Simulation Algorithm in FiltersimClassify training image patterns into clusters using filter scores and partitionLoop through all nodes of the simulation gridRetrieve the data event at that nodeFind the most similar cluster prototype to that data eventRandomly pick a pattern from that clusterPaste it on the simulation gridendImplementation details(5) SimulationSimulation Algorithm in the Proposed MethodClassify training image patterns into clusters using Kernel k-meansLoop through all nodes of the simulation gridRetrieve that data event at that nodeFind the most similar cluster prototype to Data EventRandomly pick a pattern from that clusterPaste it on the simulation gridendNote: The basic Filtersim algorithm is maintained. only the modeling of patterns changes19How to handle incomplete Data Events ? Using K-Medoid Algorithm select a representative set of patterns: Sr Store their dissimilarity distances Map Incomplete data event with SEQ-MDS using Sr (see further)

Continue with Simulation in reduced dimensions

Implemention details(5) MPS Simulation20MPS Simulation Examples (1/2)

Search Template 9x9Inner Patch 5x5

Multiple-Grids 3Number of Clusters 100MPS Simulation Examples (2/2)Search Template 11x11Inner Patch 5x5

Multiple-Grids 3Number of Clusters 100

WorkFlow SummaryReducing 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:

The actual smallest template size is 5 x 51354212345Mean EntropyLog-Likelihood5 x 5Training imageTemplate SizeTemplate Size

Reducing user interaction (1)Automatic Template Size Selection:Training image13 x 13

Mean EntropyLog-LikelihoodTemplate SizeTemplate SizeAutomatic 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

Issues (1)Issues (2)New SEQ-MDS mapping developed:

Handle Large amount of information

Less memory demanding

Very Fast and efficient

Computational Complexity: O(N3) to O(N)

New SEQ-MDS mapping developed:

Handle Large amount of information

Less memory demanding

Very Fast and efficient

Computational Complexity: O(N3) to O(N)

Issues (2)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 FiltersimFind Number of Clusters:

Eigenvalue decomposition of K =

Plot

Calculate Profile Log-Likelihood

Number of Clusters = Maximum in the profile

SIMULATION EXAMPLESComparison with Filtersim (1/6)

101 x 101 Training ImageSearch Template 9x9Inner Patch 5x5

Multiple-Grids 3Number of Clusters 100

FilterSim RealizationsProposed Methodology RealizationsComparison with Filtersim (1/6)

101 x 101 Training ImageSearch Template 13x13Inner Patch 5x5

Multiple-Grids 3Number of Clusters 100

FilterSim RealizationsProposed Methodology Realizations

Comparison with Filtersim (2/6)101 x 101 Training ImageSearch Template 9x9Inner Patch 5x5

Multiple-Grids 3Number of Clusters 100FilterSim RealizationsProposed Methodology Realizations

Comparison with Filtersim (2/6)101 x 101 Training ImageSearch Template 13x13Inner Patch 7x7

Multiple-Grids 3Number of Clusters -----FilterSim RealizationsProposed Methodology Realizations

K = 100

K = 100K = 400Comparison with Filtersim (3/6)111 x 111 Training ImageSearch Template 15x15Inner Patch 9x9

Multiple-Grids 3Number of Clusters 1000FilterSim RealizationsProposed Methodology Realizations

Comparison with Filtersim (4/6)101 x 101 Training ImageSearch Template 11x11Inner Patch 5x5

Multiple-Grids 3Number of Clusters 100FilterSim RealizationsProposed Methodology Realizations

Comparison with Filtersim (5/6)169 x 169 Training ImageSearch Template 15x15Inner Patch 11x11

Multiple-Grids 3Number of Clusters 200FilterSim RealizationsProposed Methodology Realizations

Comparison with Filtersim (6/6)69 x 69x39 Training ImageSearch Template 15x15x9Inner Patch 9 x 9 x 5

Multiple-Grids 3Number of Clusters 200

FilterSim RealizationsProposed Methodology Realizations

Studying variability between realizationsHow can we check the pattern/multiple point statistical reproduction of the realizations with respect to the training image ?

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

Generate N realizationsFind a distance function that characterizes the variability between any two realizationsMap the N realizations into a Cartesian space using MDSThe cloud of points in that space represents variability

Variability Quantification

Studying variability between realizationsMeasure of Variability between two realizations : l1 and l2f ( l1 , l2 ) = JSD { MPH( l1 ) , MPH ( l2 ) }

Jensen Shannon Divergence (JSD)

A measure of similarity between two probability distributions: p and qMultiple-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

Studying variability between realizationsRed points : Filtersim realizations Black points : Proposed Method realizationsTraining Image Used

Filtersim MethodProposed MethodStudying variability between realizationsCONCLUSION

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).

Training Image UsedStudying variability between realizations

Hard Data ConditioningSame procedure as Filtersim has been implemented:

Weighted similarity measure between patterns and the Data Event

Weight factor:Hard Data = 0.5Frozen nodes = 0.3Non-frozen nodes = 0.2

Find the most similar cluster prototype accordinglyHard Data Conditioning (1/2)

SandShaleHard data conditioning locations

The proposed Method E-typeFiltersim Method E-type

Reference CaseHard Data Conditioning (2/2)

Two neighboring hard data conditioning locationsSandShale

The proposed Method E-typeFiltersim Method E-typeSnesim Method E-type

Hard Data Conditioning (2/2)

The proposed Method E-typeFiltersim Method E-typeSnesim Method E-type3535Zoomed-in E-Type

ConclusionDistance 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 conditioningConclusion

ConclusionDistance 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