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CoGameReg: Cooperative-Game-Based Collaborative Semi-Supervised Regression for Seismic Reservoir Property Prediction

Zhiyu Guo, Jiwei Jia, Lei Li

arXiv:2610.12884v1cs.LG

TL;DR

Sparse well-log labels and abundant seismic observations make reservoir-property prediction challenging. CoGameReg uses fixed Shapley-derived regressor weights to form shared pseudo-labels and combines this with local candidate screening. Its benchmark advantage is most evident at the lowest labeled-data ratio, while results vary across blind-well locations.

  • Problem

    Reservoir-property prediction is difficult when well-log labels are sparse, seismic observations are abundant, and geological complexity makes between-well relationships challenging.

    Source: §1 · pp. 2–3

  • Method

    CoGameReg estimates fixed Shapley-derived weights from labeled data for shared pseudo-label construction and final prediction fusion, with local error-reduction screening for candidates.

    Source: §1 · p. 3, §3.3.2 · p. 11

  • Results

    CoGameReg is competitive with classical co-training, self-training, ensemble, and recent semi-supervised baselines, with its public-benchmark advantage most evident at the lowest labeled-data ratio.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The results support using CoGameReg to combine sparse well supervision with dense seismic observations, with the reported blind-well accuracy varying among locations.

    Source: §1 · p. 4, §5 · p. 34

    The results do not quantify accuracy across the complete prediction volume because reference water saturation is unavailable away from well trajectories.

    Source: §4.3.5 · p. 33

Abstract

from arXiv · show

Accurate estimation of reservoir properties from seismic data remains challenging when well-log labels are sparse but three-dimensional seismic observations are abundant. We present CoGameReg, a cooperative-game-based semi-supervised regression framework that uses contribution-weighted predictions of heterogeneous regressors to construct shared pseudo-labels. This design makes each regressor's influence explicit in the learning targets generated for unlabeled seismic samples under sparse well control. The Tri-Game Shapley mechanism combines a rank-MSE performance game with a pairwise consistency game to estimate nonnegative regressor weights from labeled training data. These weights remain fixed during collaborative training and are reused for final prediction fusion. Candidate pseudo-labels are screened using a common local k-nearest-neighbor error-reduction criterion before retraining. We evaluate CoGameReg on 10 public regression datasets, a synthetic three-dimensional P-wave velocity prediction task, and a field water-saturation application. The benchmark advantage is most pronounced at the 2.5% labeled-data ratio and decreases as supervision increases. CoGameReg obtains the best blind-well metrics in the synthetic experiment. In the field application, predictions clipped to 0-100% yield a pooled coefficient of determination of 0.5942, a root mean square error of 5.7441 percentage points, and a Pearson correlation of 0.8265 across three blind wells. CoGameReg has the highest Pearson correlation among the four field methods. The results support the use of CoGameReg to integrate sparse well labels with dense seismic attributes, with performance varying across metrics and well locations.

1 Introduction

Sparse well-log labels and complex relationships between seismic attributes and reservoir properties make prediction between wells challenging. CoGameReg combines fixed contribution-weighted pseudo-labeling with local candidate screening, and its reported advantage is strongest under the sparsest supervision.

1 Introduction

  • Sparse, costly well-log labels contrast with abundant three-dimensional seismic observations, while complex geology and nonlinear attribute-property relationships challenge prediction between wells.The introduction also notes that traditional inversion approaches can be affected by forward-model assumptions, non-uniqueness, and accumulated uncertainty.Source: §1 · p. 2
  • The design addresses two distinct choices in collaborative regression: which pseudo-labeled samples to accept and how to combine regressors’ predictions.CoGameReg pairs contribution-aware weighting for shared pseudo-labels with local candidate screening.Source: §1 · p. 3
  • CoGameReg uses fixed Shapley-derived weights for both shared pseudo-label construction and final prediction fusion across heterogeneous regressors.The weights are estimated from labeled training data; candidate admission uses a common local error-reduction criterion.Source: §1 · p. 3
  • The study evaluates CoGameReg on 10 public regression datasets, a synthetic three-dimensional P-wave task, and a field water-saturation application.The seismic evaluations distinguish quantitative held-out-well prediction from qualitative spatial consistency.Source: §1 · p. 4
  • CoGameReg is competitive with classical co-training, self-training, ensemble, and recent semi-supervised baselines, with its public-benchmark advantage most evident at the lowest labeled-data ratio.The synthetic and field applications demonstrate applicability under sparse well constraints, while blind-well accuracy varies by location.Source: §1 · p. 4

2 Related Work

Related work spans learned seismic-property prediction, semi-supervised regression and co-training, and game-theoretic valuation. CoGameReg focuses on contribution-weighted shared pseudo-labels alongside local candidate admission.

2.1 Deep Learning for Seismic Reservoir Prediction

  • Prior seismic studies use deep and multi-attribute learning for reservoir-property prediction, while limited well labels and generalization across geological settings remain challenges.The literature also includes learned seismic representations and elastic-property estimation, and seismic reservoir monitoring.Source: §2.1 · pp. 4–5

2.2 SSR and Co-Training

  • Semi-supervised regression and co-training methods include local-error-reduction selection, safe or self-paced strategies, committee methods, and embedding-space pseudo-labeling.The broader co-training literature also covers tri-training, view- and label-consistency approaches, and deep co-training.Source: §2.2 · p. 5
  • CoGameReg couples coalition-based contribution allocation to shared pseudo-label construction, reusing fixed Shapley-derived weights for final prediction fusion and applying local error reduction to each regressor’s current augmented set.The related-work overview distinguishes prediction combination from candidate acceptance as connected but separate design questions.Source: §2.2 · p. 5

2.3 Game-Theoretic Valuation in Machine Learning

  • CoGameReg evaluates regressor coalitions with a mixed rank–MSE performance game and a pairwise consistency game, then combines their Shapley-derived contributions into fixed weights.The Shapley value is presented as a contribution-allocation rule, with prior machine-learning uses including feature attribution [33] and data valuation [14].Source: §2.3 · pp. 5–6

3 The Proposed Framework

The proposed framework combines a Tri-Game Shapley valuation mechanism with collaborative pseudo-labeling. Its valuation comprises three conceptual games, with the first two combined computationally and consensus kept separate.

3 The Proposed Framework

  • The framework comprises an overview, Tri-Game Shapley valuation, and a collaborative pseudo-labeling procedure.Source: §3 · p. 6
  • “Tri-Game” denotes ordinal, accuracy, and consensus valuation games.Source: §3 · p. 6
  • By Shapley additivity, the ordinal and accuracy games are combined computationally into a mixed performance characteristic function, while consensus has a separate characteristic function.Source: §3 · p. 6

3.1 Framework Overview

The framework initializes heterogeneous regressors and estimates contribution-aware weights from labeled data. Collaborative training then generates pseudo-labels from unlabeled candidates and retrains regressors on accepted samples.

3.1 Framework Overview

  • CoGameReg uses a pool of N regressors and a weighted ensemble predictor whose weight vector lies on the probability simplex.Source: §3.1 · p. 6
  • After initialization on labeled data, Tri-Game Shapley valuation estimates the weights once; they remain fixed during collaborative training and are reused for final fusion.Source: §3.1 · p. 6
  • At each iteration, the fixed contribution-aware ensemble generates pseudo-labels for candidates, and accepted samples enter model-specific training sets for regressor-specific batchwise retraining.The candidate pool is refilled when needed.Source: §3.1 · p. 6

3.2 Tri-Game Shapley Valuation Mechanism

CoGameReg values regressors through cooperative-game utilities and combines their contributions into fixed ensemble weights. The Tri-Game mechanism assesses coalition performance and cross-model consistency before using the weights for fusion.

3.2 Tri-Game Shapley Valuation Mechanism

  • Tri-Game treats regressors as game players and combines a rank–MSE performance contribution with a consistency contribution to set fixed ensemble weights.The coalition predictions used for utility evaluation are unweighted means, avoiding circular dependence on the weights being estimated.Source: §3.2 · p. 8, §3.2.2 · p. 9

3.2.1 Consistency Assessment

  • Consistency utility measures cross-regressor agreement on labeled-set predictions, with Shapley contributions corresponding to normalized weighted degree in a prediction-similarity graph.The graph uses Gaussian similarity between regressors' predictions; this agreement score does not measure predictive correctness.Source: §3.2.1 · p. 8

3.2.2 Performance Assessment

  • Performance assessment combines Spearman rank agreement and an MSE-based utility, with α/β setting their balance and α+β setting performance's overall fusion share.The MSE utility decreases with coalition MSE; the rank utility is set to 0.5 when the labeled targets or coalition predictions have near-zero standard deviation or correlation is non-finite.Source: §3.2.2 · p. 9

3.2.3 Contribution-Aware Model Fusion

  • Zero-bounded L1 normalization removes negative performance contributions before the normalized performance and consistency components are aggregated into weights.The resulting fusion is a convex combination of base predictions and therefore stays within their pointwise range, but this does not itself impose physical bounds.Source: §3.2.3 · p. 10
  • Exact performance valuation requires 32 coalition evaluations for the five-regressor field ensemble and 64 for the six-regressor benchmark and synthetic ensembles.Monte Carlo permutation sampling can replace exact enumeration for larger ensembles.Source: §3.2.3 · p. 10

3.3 CoGameReg Collaborative Training Procedure

CoGameReg iteratively expands a candidate pool and retrains heterogeneous regressors using shared pseudo-labels. Candidates are initialized near labeled samples, while a local error-reduction criterion governs model-specific admission.

3.3 CoGameReg Collaborative Training Procedure

  • Each training iteration evaluates and selects pseudo-labeled candidates in regressor-specific batches, retrains the models, and refills the candidate pool when needed.Each regressor maintains its own augmented training set while learning from shared supervision.Source: §3.3 · p. 11

3.3.1 Hybrid Unlabeled Pool Initialization and Progressive Expansion

  • The initial unlabeled candidate pool contains the k_pool nearest neighbors of each labeled sample, and is replenished from the remaining unlabeled set when it falls below its target size.The replenishment threshold is defined relative to the size of the labeled set.Source: §3.3.1 · p. 11

3.3.2 Weighted Pseudo-Labeling, Model-Specific Acceptance, and Dynamic Batching

  • Fixed contribution weights form shared pseudo-labels, while a common local k-NN proxy evaluates candidate admission separately for each regressor's augmented state.The weights encode each regressor's direct linear influence in pseudo-label construction and final fusion; an individual local error reduction does not establish monotone global-loss or test-error reduction after batch updating.Source: §3.3.2 · pp. 11–12

4 Experimental Results and Analysis

The evaluation covers public regression benchmarks and two seismic reservoir-property tasks with different prediction targets and input attributes. The experimental designs include quantitative blind-well evaluation and field comparisons at common wells.

4 Experimental Results and Analysis

  • Ten public regression datasets assess performance across labeled-data ratios, with ablations of ensemble fusion, contribution-based weighting, and collaborative training.The benchmark experiments provide methodological evidence for the semi-supervised regression framework.Source: §4 · p. 12
  • The synthetic task predicts P-wave velocity from five post-stack seismic attributes, while the field task predicts water saturation from post-stack seismic and inversion-derived attributes.The two tasks use the same framework but are trained separately with their respective inputs, targets, and configurations.Source: §4 · p. 13
  • Synthetic evaluation includes blind-well metrics and slice comparisons against a known reference volume; field methods are compared at common blind wells and on the same seismic section and horizon.Field reference values are available only along well trajectories.Source: §4 · p. 13

4.1 Benchmark Experiments on Public Datasets

Across 10 public datasets, CoGameReg achieved the lowest average RMSE, with its statistical advantage clearest under the scarcest supervision. Ablations favored the complete fixed-weight collaborative design over the tested alternatives.

4.1.1 Experimental Protocol

  • The protocol used 2,000 training-pool samples per dataset, three label ratios, and nine baselines, with evaluation over 10 runs using shared seeds and splits.The ratios were 2.5%, 5%, and 10%; remaining samples formed each test set.Source: §4.1.1 · pp. 13–14

4.1.2 Comparison with Other Methods

  • Mean Teacher, SSDKL, and TabM remained competitive on some datasets, while CoGameReg achieved lowest or joint-lowest RMSE on several datasets at multiple label ratios.The named datasets include Abalone, Elevators, Folds5x2 pp, Parkinsons, Wind, and Wine quality.Source: §4.1.2 · pp. 14–15
  • Across 30 dataset-ratio settings, CoGameReg averaged RMSE 0.1098 and was lowest or joint-lowest in 18; it beat COREG, Self-KNN, and averaging on most settings.The comparisons cover 10 public regression datasets at three labeled-data ratios.Source: §4.1.2 · p. 14, Table 2 · p. 15

4.1.3 Statistical Significance Analysis

  • At 2.5% labels, CoGameReg significantly beat all nine methods after Holm correction; its advantage over stronger baselines narrowed at 5% and became dataset-dependent at 10%.At 5%, differences from TabM and Mean Teacher were not significant; at 10%, differences from XGBoost, TabM, and Mean Teacher were not significant.Source: §4.1.3 · pp. 15, 17, Table 3 · p. 16

4.1.4 Ablation Study

  • The complete CoGameReg variant had the lowest ablation average RMSE, 0.1098, and was best or tied-best in 12 of 30 settings.It outperformed five alternative variants in 24 to 30 settings.Source: Table 4 · p. 18, §4.1.4 · p. 18
  • Shapley weighting reduced average RMSE to 0.1106 from 0.1156 for averaging; γ-only and α-only variants scored 0.1112 and 0.1120, versus 0.1234 for β-only.The results support an average advantage for combining the Tri-Game components, but not a consistent advantage in every individual setting.Source: §4.1.4 · pp. 18–19
  • Decoupled Shapley weighting averaged RMSE 0.1111 versus 0.1106 for supervised Shapley fusion, while full CoGameReg reached 0.1098; the split-label variant averaged 0.1175.The authors associate the full variant's benefit with using fixed contribution-aware weights for pseudo-label generation and final prediction, and initializing weights and regressors with all available labeled data.Source: §4.1.4 · p. 19

4.2 Synthetic 3D Seismic Reservoir Property Prediction

In the synthetic 3D experiment, CoGameReg achieves the best reported blind-well metrics and shows improved spatial coherence, while fine-scale heterogeneity remains incompletely recovered.

4.2.1 3D Seismic Data Configuration and Evaluation Metrics

  • The experiment predicts P-wave velocity from five seismic attributes, using spatially separated training, validation, and blind-test wells and evaluating blind wells with RMSE, R2, and SRC.Full-field ground truth also supports qualitative structural comparisons on representative slices.Source: §4.2.1 · pp. 19, 21, Figure 2 · p. 20, Figure 3 · p. 21

4.2.2 Blind-Well Evaluation and Qualitative Comparison

  • CoGameReg records the best reported blind-well results: RMSE 0.4137, R2 0.5065, and SRC 0.5747.Its RMSE is lower than XGBoost, KNN, Self-KNN, and COREG by approximately 9.7%, 13.9%, 12.2%, and 10.6%, respectively.Source: Table 5 · p. 22, §4.2.2 · p. 22
  • CoGameReg shows less mottled background variation and clearer principal structures than the reference predictions, but narrow bands, smoothed boundaries, and fine-scale heterogeneity remain.The remaining limitations are particularly visible at Time = 188 and in the lower inline section.Source: §4.2.2 · pp. 22–23, Figure 4 · p. 23
  • XGBoost and KNN recover large-scale slice patterns, while Self-KNN and COREG retain similar geometries but show mottling and irregular local variation.Quantitative improvements over KNN do not translate into uniformly clearer structures across the displayed slices.Source: §4.2.2 · p. 22, Figure 4 · p. 23

4.3 Real Field Application in the Western Field Area

The Western Field Area experiment evaluates water-saturation prediction from seismic-derived inputs using well-level training, validation, and blind-test partitions. CoGameReg reproduces major blind-well trends, but performance varies by metric and location, and the spatial volumes lack off-well reference validation.

4.3.1 Geological setting and data

  • The target is water saturation in a thin, laterally variable Permian tight-sandstone reservoir, where sandstone development does not necessarily match gas-bearing intervals.The study uses 18 wells with valid water-saturation logs in the anonymized Western Field Area.Source: §4.3.1 · pp. 23–24

4.3.2 Well-to-seismic calibration and experimental protocol

  • Five seismic-derived volumes were used with 10 training wells, five validation wells, and three blind-test wells; about 8.2 × 10^7 seismic samples were unlabeled.The blind-well logs were withheld from training, weight estimation, pseudo-label generation, and model selection.Source: §4.3.2 · pp. 24–25, Table 6 · p. 24

4.3.3 Three-dimensional prediction

  • UCVME and CoGameReg show smoother, more connected patterns than XGBoost and COREG around the mapped anomalies, while COREG has fragmented patches in parts of the vertical section.The authors treat these spatial comparisons as qualitative context, not evidence of higher accuracy away from wells.Source: §4.3.3 · p. 28, Figure 7 · p. 29, Figure 8 · p. 30

4.3.4 Blind-well evaluation

  • CoGameReg pooled blind-well scores were R2 0.5942, RMSE 5.7441 percentage points, and Pearson correlation 0.8265, the highest Pearson correlation among four methods.No method leads across all reported metrics and subsets; CoGameReg's blind-well performance varies by location.Source: §4.3.4 · pp. 31–32, Table 7 · p. 32

4.3.5 Geological interpretation and limitations

  • Off-well accuracy is unquantified because reference saturation is unavailable away from well trajectories; fine-scale patterns require bandwidth and calibration context.The authors also caution that greater spatial continuity can reflect smoothing rather than preservation of genuine geological contrasts.Source: §4.3.3 · p. 28, §4.3.5 · p. 33

5 Conclusion

CoGameReg uses fixed Shapley-derived regressor weights in shared pseudo-label construction and final prediction fusion under sparse well control. Its largest benchmark advantage occurs at the lowest labeled-data ratio, while field performance varies across metrics and wells.

5 Conclusion

  • CoGameReg's benchmark advantage is largest at the 2.5% labeled-data ratio and narrows as the proportion of labeled data increases.Benchmark ablations support placing contribution-aware weighting within the pseudo-labeling loop.Source: §5 · p. 34
  • CoGameReg achieves the best blind-well metrics in the synthetic experiment and the highest pooled Pearson correlation among four methods in the field application.The conclusion reports that field performance depends on metric and well location.Source: §5 · p. 34
  • Shapley-derived weights estimated once from labeled training data are held fixed and used for both shared pseudo-label construction and final prediction fusion.Local error-reduction screening complements the model-level weighting by controlling candidate admission.Source: §5 · p. 34
  • The evaluated results support CoGameReg's applicability under the tested sparse-well conditions, with field estimates reproducing major water-saturation trends at three blind wells.The conclusion describes the field predictions as clipped and notes variation by metric and well location.Source: §5 · p. 34
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