Cheolkyun Jeong, Céline Scheidt, Jef Caers, and Tapan Mukerji Stanford Center for Reservoir Forecasting Modeling Geological Scenario Uncertainty from Seismic Data using Pattern Similarity Hello everyone, today I’m going to talk about ‘modeling geological scenario uncertainty from seismic data using pattern similarity’ Cheolkyun Jeong, Céline Scheidt, Jef Caers, and Tapan Mukerji

𝒅 : seismic data (2D section) Introduction Modeling Geological Scenario Uncertainty from Seismic Data 𝑷 𝒎 𝒓𝒆𝒔 , 𝑺 𝒅 = 𝒌 𝑷 𝒎 𝒓𝒆𝒔 𝑺 𝒌 , 𝒅) 𝑷( 𝑺 𝒌 |𝒅) 𝒅 : seismic data (2D section) 𝑺 𝟏 𝑺 𝟐 𝑷( 𝑺 𝟏 |𝒅) 𝑷( 𝑺 𝟐 |𝒅) We are now focusing on prior distribution and data. Instead of a fixed geological scenario, first, we assess the consistency of all available scenarios with the given observed data. Final goal of this work is to reject low probability scenarios and suggest more likely scenarios based on the given data. This is very essential in actual field situation. For example, here we have a 2D seismic section data. We have no training image and any information yet. Then we generate two scenarios containing different geological features. Through the forward simulation using convolution model, now we have seismic response of models from each scenario. Then our research question is this. What is more likely setting, here we call this scenario? Based on these seismic response, which one is more promising? Note we don’t care about location and matching the data. We are talking about more promising response of patterns. So, we

Forward simulated response Seismic data (2D section) Introduction Modeling Geological Scenario Uncertainty from Seismic Data Forward simulated response Seismic data (2D section) Distance ? Distance = Pattern Similarity (or Dissimilarity) Pattern Histogram (MPH and CHP) and JS divergence Wavelet Transform So, now we directly compare our priors with given data. To estimate it, we generate a set of models from each scenario, and calculate distances between forward modeled seismic responses and obtained seismic data. Then the most important technique is to measure the distance and we propose to use pattern similarity as a distance. We have three different algorithms and I will present first and second ideas and Celine will talk about the third one.

Data-events Frequency Method I: Pattern similarity using MPH Multiple Point Histogram(MPH) recording a histogram of patterns - Basic algorithm of MPH for seismic data 2D Seismic section data (seismograms) Tresholded seismic reflections (positive, negative, and 0) Data-events Frequency Template 2x2 = 𝟑 𝟒 = 6561 Data-events

Method I: Pattern similarity using MPH Cluster-based Histograms of Patterns (CHP) MPH can be problematic in a large template and 3D case MPH is only applicable to categorical variables Thus, instead of recording all patterns, we proposed to use the cluster-based histogram of patterns (CHP) (described in Honarkhah and Caers, 2010) Patterns from the obtained data Prototype CHP from the obtained data Frequency (number of replicates) We now consider one of the multi-resolution grids of the training image and attempt to summarize its statistics. An appealing concept is the multi-point histogram, which records, for binary realizations, within a given template (for 2D the size of the pattern is fixed to 4 × 4 and for 3D to 3 × 3 × 2), every possible pattern configuration and creates a frequency table of patterns (a similar idea is proposed in Lange et al., 2012). However, this becomes problematic for 3D or continuous-valued realizations. Instead, we cluster the patterns into groups using the methodology described in Honarkhah and Caers (2010) (this methodology works for 3D and continuous values). For each cluster, we record the number of patterns and calculate the prototype, which is a representative of each cluster (it could be a mean of the patterns within that cluster or a medoid pattern). The template size for each grid g is determined automatically using a so-called elbow plot (see Honarkhah and Caers, 2010). We repeat this exercise for all multi-resolution grids in the pyramid. We do the same for all realizations. However, for the realizations, the clustering is slightly different. Since we want to make sure that the clustering results between training image and realization are comparable for each multi-resolution grid, we cluster the realization patterns by calculating distances with the training image prototypes and simply assign the pattern to the closets prototype. In this fashion, we obtain the same number of groups/clusters with the same prototypes for realizations and training image. In the example section, we show that such clustering leads to the same results as using the full pattern or multiple-point histogram for a simple 2D binary case, but evidently, what we propose is feasible for large 3D as well as continuous-valued problems. In terms of mathematical notation this operation can be summarized as follows. Each pyramid is now summarized as a list of what we will term cluster-based histograms of patterns CHP, itself a list of frequencies: Prototype 1

MPH using all available patterns Method I: Pattern similarity using MPH Cluster-based Histograms of Patterns (CHP) MPH using all available patterns CHP using prototypes 10.98 % 11.39 % 76.39% 71.57% We now consider one of the multi-resolution grids of the training image and attempt to summarize its statistics. An appealing concept is the multi-point histogram, which records, for binary realizations, within a given template (for 2D the size of the pattern is fixed to 4 × 4 and for 3D to 3 × 3 × 2), every possible pattern configuration and creates a frequency table of patterns (a similar idea is proposed in Lange et al., 2012). However, this becomes problematic for 3D or continuous-valued realizations. Instead, we cluster the patterns into groups using the methodology described in Honarkhah and Caers (2010) (this methodology works for 3D and continuous values). For each cluster, we record the number of patterns and calculate the prototype, which is a representative of each cluster (it could be a mean of the patterns within that cluster or a medoid pattern). The template size for each grid g is determined automatically using a so-called elbow plot (see Honarkhah and Caers, 2010). We repeat this exercise for all multi-resolution grids in the pyramid. We do the same for all realizations. However, for the realizations, the clustering is slightly different. Since we want to make sure that the clustering results between training image and realization are comparable for each multi-resolution grid, we cluster the realization patterns by calculating distances with the training image prototypes and simply assign the pattern to the closets prototype. In this fashion, we obtain the same number of groups/clusters with the same prototypes for realizations and training image. In the example section, we show that such clustering leads to the same results as using the full pattern or multiple-point histogram for a simple 2D binary case, but evidently, what we propose is feasible for large 3D as well as continuous-valued problems. In terms of mathematical notation this operation can be summarized as follows. Each pyramid is now summarized as a list of what we will term cluster-based histograms of patterns CHP, itself a list of frequencies: MPH CHP Variables Categorical only Continuous Patterns in template size 5x3 14,348,907 patterns 602 prototypes Time for 800 models in 3 layers* 10 hours 15 minutes 30% 20% stacked random red: 20 blue:10 light: 2 dark: 3 proportion position size # facies *Matlab runtime using a machine of 3.3GHz (4 CPUs) and 16GB RAM

Obtained data (2D seismic section) Illustration : test case P(Sk| d) using pattern similarity validation Facies Geometry Proportion (Shale/Sand) Channel size (width:thick= 1.5:1) Rockphysics (B/O) Scenario (Random) (Stacked) Random or Stacked Sand:20% Thickness: 10 2 facies 1 9 3 facies 2 10 Thickness: 20 3 11 4 12 Sand:30% 5 13 6 14 7 15 8 16 Obtained data (2D seismic section) S1 S9 S11 S3 S2 S10 S16 S15 This table shows considerable possibilities in the synthetic case. Geologists agree with a deposition of channel or levee or both, but they cannot reach an agreement as a typical subsurface geometry. Additionally, we want to know the presence of oil sand. As the result, now we have 44 scenarios. If we just generate 30 models as priors from each scenario, we have to build a matrix 1320x1320 size. It’s not a trivial task to calculate this number of image distance; however, it is nearly impossible in rejection sampler.

Two Different Seismic Quality High-resolution seismic Constant velocity and density Thickness: dz = 2.5m Low-resolution seismic Realistic rock-physics model Thickness: dz = 1m *The obtained seismic data is forward simulated by convolution model with 50 Hz frequency

Multi-variate pdf of scenario j in MDS Illustration : test case P(Sk| d) using pattern similarity validation Multi-variate pdf of scenario j in MDS Scenario j 𝒇 𝒅 𝑺 𝒋 = 0.043 ≈𝑷 𝒅 𝑺 𝒋 50 models = 𝑷 𝒅 𝑺 𝒌 𝑷( 𝑺 𝒌 ) 𝒌 𝑷 𝒅 𝑺 𝒌 𝑷( 𝑺 𝒌 ) 𝑷( 𝑺 𝒌 |𝒅) <Park et al., (2013)> − 𝒇 : pdf of data points in MDS map − Each point represents for forward simulated seismic response

Case 1: High-resolution Seismic data 30% 20% stacked random red: 20 blue:10 light: 2 dark: 3 proportion position size # facies Observed seismic P(Sc|seimicdata): 78,681model evaluations   S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 KS 0.03 0.78 0.11 0.07 RS 0.80 0.10

Case 1: High-resolution seismic data Confusion Matrix Difference in # facies Misclassification: 11% Predicted class   Sc1 Sc2 Sc3 Sc4 Sc5 Sc6 Sc7 Sc8 Sc9 Sc10 Sc11 Sc12 Sc13 Sc14 Sc15 Sc16 84 10 2 4 98 86 12 96 90 88 8 100 14 82 16 78 70 10 16 14 12 Actual class

Case 2: Low-resolution Seismic data 30% 20% stacked random red: 20 blue:10 light: 2 dark: 3 proportion position size # facies Observed seismic P(Sc|seimicdata): 97,406 model evaluations   S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S13 S14 S15 KS 0.27 0.73 0.01 0.21 RS 0.10 0.63 0.13

Case 2: Low-resolution seismic data Difference proportion Confusion Matrix Difference position 24 Misclassification: 55% Predicted class   Sc1 Sc2 Sc3 Sc4 Sc5 Sc6 Sc7 Sc8 Sc9 Sc10 Sc11 Sc12 Sc13 Sc14 Sc15 Sc16 54 10 2 4 14 50 24 6 40 16 8 36 20 72 12 66 18 52 32 38 48 26 28 20 26 24 Actual class

Method II: Wavelet Transform Brief overview of wavelet transform on images AK VK Low-pass (average) and high pass (details) filters are passed through the images in different directions high pass HK DK Example of wavelet high pass high pass

Seismic Responses – Wavelet Transform Use of Haar wavelet, 2 levels A1 V1 H1 D1 A2 V2 H2 D2 Location of the coefficients is not important, the distribution is  The distance is defined as the difference in wavelet histograms

JS Divergence On Wavelet Histograms P = (p1,… pN) r1 Wavelet Analysis JS-divergence r2 Q = (q1,… qN) Wavelet Analysis Note: Distances are computed for all directions (H,V,D,A) and all levels The final distance is the average distance.

High-resolution seismic Low-resolution seismic MDS Representation 30% 20% stacked random red: 20 blue:10 light: 2 dark: 3 proportion position size # facies High-resolution seismic Low-resolution seismic  Good distinction of the scenario is observed for both cases

Case 1: High-resolution seismic data Difference proportion Confusion Matrix Difference position Misclassification: 12% Predicted class   Sc1 Sc2 Sc3 Sc4 Sc5 Sc6 Sc7 Sc8 Sc9 Sc10 Sc11 Sc12 Sc13 Sc14 Sc15 Sc16 94 - 6 98 2 4 88 8 96 86 12 82 92 78 84 16 70 100 22 68 8 Actual class 12 12 16 22

Case 1: High-resolution Seismic data Observed seismic 30% 20% stacked random red: 20 blue:10 light: 2 dark: 3 proportion position size # facies P(Sc|seimicdata): 108,647 model evaluations   Sc1 Sc2 Sc3 Sc4 Sc5 Sc6 Sc7 Sc8 Sc9 Sc10 Sc11 Sc12 Sc13 Sc14 Sc15 Sc16 KS 0.9 0.1 RS 0.8 Similar probabilities obtained with CHP

Case 2: Low-resolution seismic data Difference proportion Confusion Matrix Difference position Difference in # facies Misclassification: 43% Predicted class   Sc1 Sc2 Sc3 Sc4 Sc5 Sc6 Sc7 Sc8 Sc9 Sc10 Sc11 Sc12 Sc13 Sc14 Sc15 Sc16 46 6 2 4 12 - 62 14 58 10 8 70 20 60 18 76 74 34 16 30 26 56 64 52 14 14 20 18 Actual class 14 16 26 16 18

Case 2: Low-resolution Seismic data Observed seismic 30% 20% stacked random red: 20 blue:10 light: 2 dark: 3 proportion position size # facies P(Sc|seimicdata): 113,041 model evaluations   Sc1 Sc2 Sc3 Sc4 Sc5 Sc6 Sc7 Sc8 Sc9 Sc10 Sc11 Sc12 Sc13 Sc14 Sc15 Sc16 KS 0.7 0.1 0.2 RS 0.5 0.3 Similar probabilities obtained with CHP, except for Sc.1 (0.3 for Sc1)

Comparison of both approaches High-quality seismic: Both methods show a good distinction of the different scenarios Misclassification in the confusion matrix is 11% & 12% Misclassification mostly observed for: Number of facies for CHP Proportion and facies positioning (stacking vs. random) for the wavelet transform Low-quality seismic: Misclassification in the confusion matrix increases (55% & 43%) Proportion and facies positioning (stacking vs. random) for both approaches

Conclusions & Future work Successful application of pattern similarity approaches to estimate probabilities of multiple scenarios given observed data Cluster-based histogram of patterns (CHP) Wavelet Analysis Results of distance-based scenario modeling is comparable to rejection sampling for the cases investigated At only a fraction of the cost Future work Application of both pattern similarity approaches to real 3D case

Illustration : Future test case Application in actual case Seismic section at Inline 1301 and Xline 984 tied in well E3 Target Horizon*: Orange Proximal This table shows considerable possibilities in the synthetic case. Geologists agree with a deposition of channel or levee or both, but they cannot reach an agreement as a typical subsurface geometry. Additionally, we want to know the presence of oil sand. As the result, now we have 44 scenarios. If we just generate 30 models as priors from each scenario, we have to build a matrix 1320x1320 size. It’s not a trivial task to calculate this number of image distance; however, it is nearly impossible in rejection sampler. Distal *target horizon is color-coded by inverted impedance <Courtesy to Hess>

3. JS divergence for calculating a distance between seismic responses Appendix 3. JS divergence for calculating a distance between seismic responses - Jensen – Shannon divergence (JS divergence) Measuring the similarity between two frequency distributions. 𝑴𝑷𝑯 𝟏: 𝒉 𝒊 from 𝒎 𝒊 𝑴𝑷𝑯 𝟐: 𝒉 𝒋 from 𝒎 𝒋 Frequency Prototype Prototype 𝒅 𝑱𝑺 𝒉 𝒊 , 𝒉 𝒋 = 𝟏 𝟐 𝒊 𝒉 𝒊 𝒍𝒐𝒈 𝒉 𝒊 𝒉 𝒋 + 𝟏 𝟐 𝒊 𝒉 𝒋 𝒍𝒐𝒈 𝒉 𝒋 𝒉 𝒊 𝑫= {𝒅 𝒊𝒋 } , 𝒅 𝒊𝒋 = 𝒅 𝑱𝑺 𝒉 𝒊 , 𝒉 𝒋 ∀ 𝒊=𝟏,𝟐,…, 𝑳×𝒌+𝟏 , 𝒋=𝟏,𝟐,…,( 𝑳×𝒌+𝟏) where L realizations of each scenario k and the given data

*Not detected oilsand distribution is generated by Gassmann’s equation Appendix Synthetic case setting Well data Rockphysics Reference Facies Obtained Seismic data *The obtained seismic data is forward simulated by convolution model with 50 Hz frequency. Actual Logs Vp 𝝆 Vp 𝝆 Bivariate pdf In this case, we assume one oil sand distribution away from the two well locations (CDP25 and CDP125). We have seismogram data, well logs without oil sand information. In this task, we applied realistic rock physics relationship from actual well logs, and generated oil sand properties from the brine sand properties at the wells using Gassmann’s equation. *Not detected oilsand distribution is generated by Gassmann’s equation