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

# CoGameReg: Cooperative-Game-Based Collaborative Semi-Supervised Regression for Seismic Reservoir Property Prediction
Source: [https://arxiv.org/html/2610.12884](https://arxiv.org/html/2610.12884)
Jiwei Jia1,2††thanks:Corresponding author: jiajiwei@jlu\.edu\.cnLei Li3,4††thanks:Corresponding author: lilei09@cnpc\.com\.cn

###### Abstract

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\.

1School of Mathematics, Jilin University, Changchun, China 2AI for Science and Engineering Center, Shenzhen Loop Area Institute, Shenzhen, China 3BGP Inc\., China National Petroleum Corporation, Zhuozhou, China 4National Engineering Research Center of Oil & Gas Exploration Computer Software, Zhuozhou, China

Keywords:seismic reservoir property prediction; semi\-supervised learning; multi\-attribute seismic data; cooperative game theory; reservoir characterization; quantitative interpretation\.

## 1Introduction

Seismic reservoir property prediction is an important problem in quantitative geophysics and plays a critical role in reservoir characterization and hydrocarbon exploration\. Traditional model\-driven approaches infer reservoir properties through seismic inversion and rock\-physics\-based relationships, providing valuable physical interpretations of subsurface structures and properties\. However, their performance can be affected by assumptions in forward modeling, inversion non\-uniqueness, and accumulated uncertainties during multiple processing stages\[[4](https://arxiv.org/html/2610.12884#bib.bib1),[42](https://arxiv.org/html/2610.12884#bib.bib2),[52](https://arxiv.org/html/2610.12884#bib.bib3)\]\.

With the increasing availability of three\-dimensional seismic surveys, large volumes of seismic data can be obtained, whereas reliable labels derived from well logs remain extremely limited because of the high cost of drilling and logging\. This imbalance provides a practical motivation for learning strategies that can exploit unlabeled seismic observations under limited supervision\. In particular, accurately predicting reservoir properties between sparsely distributed wells remains a major challenge because seismic responses are affected by complex geological structures, heterogeneous reservoir conditions, and nonlinear relationships between seismic attributes and petrophysical properties\.

Recent advances in artificial intelligence have enabled data\-driven approaches for seismic interpretation and reservoir characterization\. Deep learning methods have been successfully applied to seismic denoising, fault detection, seismic facies classification, lithology prediction, and reservoir parameter inversion\[[38](https://arxiv.org/html/2610.12884#bib.bib4),[26](https://arxiv.org/html/2610.12884#bib.bib5),[23](https://arxiv.org/html/2610.12884#bib.bib6),[41](https://arxiv.org/html/2610.12884#bib.bib7)\]\. For reservoir property prediction, recent studies have explored direct learning from multi\-attribute seismic data rather than relying solely on traditional inversion workflows\. Because different seismic attributes capture complementary information related to lithology, structure, and fluid response, multi\-attribute learning provides an effective way to integrate heterogeneous geophysical information\. However, existing deep learning approaches still face challenges associated with limited well labels and generalization across different geological settings\[[46](https://arxiv.org/html/2610.12884#bib.bib8),[47](https://arxiv.org/html/2610.12884#bib.bib9),[18](https://arxiv.org/html/2610.12884#bib.bib10)\]\.

Semi\-supervised learning is therefore a natural candidate for seismic reservoir prediction because it can combine sparse well\-log labels with abundant unlabeled seismic samples\. More generally, semi\-supervised learning has become an important paradigm for improving predictive performance when labeled data are scarce but unlabeled data are abundant\[[8](https://arxiv.org/html/2610.12884#bib.bib11),[15](https://arxiv.org/html/2610.12884#bib.bib12)\]\. In regression settings, where continuous annotations are often expensive to obtain, semi\-supervised regression \(SSR\) is particularly relevant\[[21](https://arxiv.org/html/2610.12884#bib.bib13)\]\. The broader semi\-supervised learning literature includes self\-training\[[45](https://arxiv.org/html/2610.12884#bib.bib14),[24](https://arxiv.org/html/2610.12884#bib.bib15)\], consistency regularization\[[36](https://arxiv.org/html/2610.12884#bib.bib16),[44](https://arxiv.org/html/2610.12884#bib.bib17)\], graph\-based learning\[[51](https://arxiv.org/html/2610.12884#bib.bib18)\], kernel\-based classification\[[12](https://arxiv.org/html/2610.12884#bib.bib19)\], and co\-training\[[6](https://arxiv.org/html/2610.12884#bib.bib20)\]\.

A representative co\-training framework for regression is COREG\[[50](https://arxiv.org/html/2610.12884#bib.bib21)\], which employs two differently configuredkk\-nearest\-neighbor \(kk\-NN\) regressors\. COREG provides a pseudo\-label exchange mechanism based on local mean squared error reduction and explicitly promotes diversity through different distance metrics and/or neighborhood sizes\. The original study also considers fitting linear regression or support vector regression models to the augmented training sets produced by thekk\-NN\-based labeling procedure\.

Committee\-based regression\[[1](https://arxiv.org/html/2610.12884#bib.bib29)\]and S2RMS\[[30](https://arxiv.org/html/2610.12884#bib.bib28)\]already employ multiple regressors for semi\-supervised learning\. Within such settings, deciding which pseudo\-labeled samples to accept and deciding how to combine the regressors’ predictions are related but distinct design questions\. This motivates investigating contribution\-aware weighting for shared pseudo\-label construction alongside local candidate screening\.

A related challenge lies in the reliable exploitation of unlabeled data\. Prior studies have shown that unlabeled data may degrade performance when the semi\-supervised assumption is violated or when pseudo\-labels are inaccurate\[[5](https://arxiv.org/html/2610.12884#bib.bib22),[48](https://arxiv.org/html/2610.12884#bib.bib23),[28](https://arxiv.org/html/2610.12884#bib.bib24),[29](https://arxiv.org/html/2610.12884#bib.bib25)\]\. In co\-training regression, this issue is particularly acute because pseudo\-label quality depends not only on the accuracy of individual regressors, but also on their interaction structure\. Representative SSR approaches address this risk through local error reduction\[[50](https://arxiv.org/html/2610.12884#bib.bib21)\], self\-paced selection with a safe prediction mechanism\[[35](https://arxiv.org/html/2610.12884#bib.bib27)\], and confidence screening based on embedding similarity and prediction dispersion\[[30](https://arxiv.org/html/2610.12884#bib.bib28)\]\. Alongside these sample\-selection mechanisms, this work investigates how estimated regressor contributions can determine the weights used to construct shared pseudo\-labels\.

Motivated by these considerations, we propose CoGameReg \(Cooperative\-Game\-Based Collaborative Semi\-Supervised Regression\) for seismic reservoir property prediction under sparse well control\. CoGameReg treats heterogeneous regressors as players and estimates their contribution\-aware weights once from the labeled training data\. The Tri\-Game mechanism combines coalition\-level rank agreement and mean squared error with pairwise prediction consistency\. The resulting weights determine the shared candidate pseudo\-labels throughout collaborative training and are reused for final prediction fusion\. Candidate admission is evaluated using a common local error\-reduction criterion on each regressor’s current augmented training set\. The central design choice is to use contribution weighting when constructing the additional learning targets, so that model contribution estimates affect both semi\-supervised training and final prediction\. This choice is relevant to seismic reservoir prediction because limited well samples constrain the relationship between reservoir properties and extensive seismic attribute volumes\. Different regressors can produce different estimates from these same inputs, and their relative influence determines the pseudo\-labels used for subsequent learning\. CoGameReg makes this influence explicit through fixed model weights and complements it with local candidate screening\. The contribution lies in this integration within the collaborative learning procedure; neither model agreement nor local screening alone establishes the geological reliability of a pseudo\-label\.

The main contributions of this paper are summarized as follows:

1. 1\.We introduce CoGameReg, a heterogeneous semi\-supervised regression framework in which fixed Shapley\-derived weights determine each regressor’s influence on both shared pseudo\-label construction and final prediction fusion\.
2. 2\.We formulate a Tri\-Game valuation mechanism combining a mixed rank–MSE performance game with a pairwise consistency game\. The resulting model weights are used together with a common k\-NN\-based local error\-reduction criterion for candidate pseudo\-label admission\.
3. 3\.We evaluate the framework on 10 public regression datasets, a synthetic three\-dimensional P\-wave velocity prediction task, and a field water\-saturation application\. The seismic evaluations distinguish quantitative prediction at held\-out wells from qualitative spatial consistency of the three\-dimensional results\.

The experimental results show that CoGameReg is competitive with classical co\-training, self\-training, ensemble, and recent semi\-supervised baselines under limited supervision\. Its advantage on the public benchmarks is most evident at the lowest labeled\-data ratio\. The synthetic and field applications further demonstrate its applicability to seismic reservoir property prediction with sparse well constraints, while also showing that prediction accuracy can vary among blind\-well locations\.

## 2Related Work

We review the literature most closely related to this study from three perspectives: deep learning for seismic reservoir prediction, semi\-supervised regression and co\-training, and game\-theoretic valuation in machine learning\.

### 2\.1Deep Learning for Seismic Reservoir Prediction

Deep learning has become an effective tool for seismic interpretation and reservoir characterization\. Convolutional neural networks and attention\-based architectures have been widely investigated for seismic denoising, fault detection, seismic facies classification, lithology prediction, and reservoir parameter inversion\[[38](https://arxiv.org/html/2610.12884#bib.bib4),[26](https://arxiv.org/html/2610.12884#bib.bib5),[23](https://arxiv.org/html/2610.12884#bib.bib6),[41](https://arxiv.org/html/2610.12884#bib.bib7)\]\. These studies demonstrate the capability of data\-driven models to extract complex nonlinear relationships from seismic observations\.

For quantitative reservoir characterization, recent studies have further explored direct prediction of reservoir\-related properties from multi\-attribute seismic data\. Multi\-attribute learning integrates complementary information from seismic amplitude, frequency, phase, impedance, and other derived attributes, providing richer representations of subsurface geological conditions\. For example, deep learning approaches have been investigated for lithology prediction and fluid\-related property estimation using multi\-seismic information and attribute fusion\[[46](https://arxiv.org/html/2610.12884#bib.bib8),[47](https://arxiv.org/html/2610.12884#bib.bib9)\]\.

Recent studies further illustrate the range of learning strategies used for seismic reservoir characterization\. A convolutionally embedded transformer has been applied to reservoir elastic\-property prediction from field seismic and well\-log data, with comparison against physics\-based amplitude\-versus\-offset inversion\[[3](https://arxiv.org/html/2610.12884#bib.bib51)\]\. A multiresolution approach combining Haar wavelet compression, convolutional and recurrent neural networks, and transfer learning has also been developed for seismic reservoir monitoring\[[2](https://arxiv.org/html/2610.12884#bib.bib52)\]\. These studies address learned seismic representations and elastic\-property estimation\. CoGameReg examines how heterogeneous regressor predictions are combined to form the pseudo\-labels used for learning from unlabeled seismic attributes under sparse well control\.

Self\-supervised pre\-training on unlabeled seismic data followed by few\-shot finetuning has also been investigated for gas\-bearing prediction\[[18](https://arxiv.org/html/2610.12884#bib.bib10)\]\. Limited well supervision nevertheless remains a central constraint\. The present work investigates collaborative SSR in which unlabeled seismic samples are incorporated through contribution\-weighted pseudo\-label generation and local candidate admission\.

### 2\.2SSR and Co\-Training

SSR aims to improve predictive performance by jointly exploiting a small labeled set and a large amount of unlabeled data\[[21](https://arxiv.org/html/2610.12884#bib.bib13)\]\. Representative approaches include safe prediction\[[29](https://arxiv.org/html/2610.12884#bib.bib25)\], semi\-supervised deep kernel learning\[[19](https://arxiv.org/html/2610.12884#bib.bib48)\], and uncertainty\-aware deep regression\[[10](https://arxiv.org/html/2610.12884#bib.bib49)\]\. Within co\-training regression, COREG\[[50](https://arxiv.org/html/2610.12884#bib.bib21)\]uses local error reduction to select pseudo\-labeled samples\. Related collaborative regression methods include committee\-based regression\[[1](https://arxiv.org/html/2610.12884#bib.bib29)\], random\-forest\-based co\-training regression\[[32](https://arxiv.org/html/2610.12884#bib.bib30)\], self\-paced safe co\-training\[[35](https://arxiv.org/html/2610.12884#bib.bib27)\], and embedding\-space learning with pseudo\-label smearing in S2RMS\[[30](https://arxiv.org/html/2610.12884#bib.bib28)\]\.

In the broader semi\-supervised learning literature, co\-training ideas have been extended to tri\-training\[[49](https://arxiv.org/html/2610.12884#bib.bib31)\], self\-paced co\-training for classification\[[34](https://arxiv.org/html/2610.12884#bib.bib26)\], view\- and label\-consistency\-based co\-training\[[16](https://arxiv.org/html/2610.12884#bib.bib32)\], and deep co\-training\[[37](https://arxiv.org/html/2610.12884#bib.bib33),[9](https://arxiv.org/html/2610.12884#bib.bib34)\]\. Applications also include medical imaging, remote sensing, time series, and learning to rank\[[40](https://arxiv.org/html/2610.12884#bib.bib35),[25](https://arxiv.org/html/2610.12884#bib.bib36),[31](https://arxiv.org/html/2610.12884#bib.bib37),[27](https://arxiv.org/html/2610.12884#bib.bib38)\]\.

The present work focuses on coupling coalition\-based contribution allocation with shared pseudo\-label construction and model\-indexed local admission\. Fixed Shapley\-derived weights determine each regressor’s influence on the shared pseudo\-labels and are reused for final prediction fusion, while a common local error\-reduction functional is evaluated on each regressor’s current augmented training set\.

### 2\.3Game\-Theoretic Valuation in Machine Learning

Game\-theoretic methods provide principled tools for quantifying marginal contributions in multi\-agent systems\. The Shapley value\[[39](https://arxiv.org/html/2610.12884#bib.bib39)\]is a classical rule for contribution allocation and has been widely adopted in machine learning, including feature attribution\[[33](https://arxiv.org/html/2610.12884#bib.bib40)\]and data valuation\[[14](https://arxiv.org/html/2610.12884#bib.bib41)\]\. Related coalition\-based ideas have also appeared in unsupervised learning\. For instance, BiGC\[[13](https://arxiv.org/html/2610.12884#bib.bib42)\]formulates clustering through cooperative\-game objectives and characterizes structural coherence via coalition utilities induced by pairwise similarity\.

These studies illustrate applications of cooperative\-game concepts to feature attribution, data valuation, and clustering\. CoGameReg evaluates coalitions of regressors through a mixed rank–MSE performance game and a pairwise consistency game\. The resulting Shapley\-derived contributions are normalized and combined into weights estimated once from the labeled data\. These weights remain fixed throughout collaborative training, where they determine the composition of shared candidate pseudo\-labels, and are reused for final prediction fusion\.

## 3The Proposed Framework

In this section, we present the CoGameReg framework, including an overview of the framework, the Tri\-Game Shapley valuation mechanism, and the collaborative pseudo\-labeling procedure\. Here, “Tri\-Game” denotes three conceptual valuation games—ordinal, accuracy, and consensus\. By Shapley additivity, the ordinal and accuracy games are combined computationally into a mixed performance characteristic function, whereas consensus is represented by a separate characteristic function\.

### 3\.1Framework Overview

CoGameReg constructs a pool ofNNregressors

𝒮=\{ℳi\}i=1N,ℳi∈ℋi,\\mathcal\{S\}=\\\{\\mathcal\{M\}\_\{i\}\\\}\_\{i=1\}^\{N\},\\qquad\\mathcal\{M\}\_\{i\}\\in\\mathcal\{H\}\_\{i\},\(1\)together with a weighted ensemble predictor whose weight vector lies on the probability simplex

f𝐰​\(𝐱\)=∑i=1Nwi​ℳi​\(𝐱\),𝐰∈ΔN−1,f\_\{\\mathbf\{w\}\}\(\\mathbf\{x\}\)=\\sum\_\{i=1\}^\{N\}w\_\{i\}\\mathcal\{M\}\_\{i\}\(\\mathbf\{x\}\),\\qquad\\mathbf\{w\}\\in\\Delta^\{N\-1\},\(2\)whereℋi\\mathcal\{H\}\_\{i\}is the hypothesis class of theii\-th regressor,ΔN−1=\{𝐰∈ℝ≥0N:∑i=1Nwi=1\}\\Delta^\{N\-1\}=\\\{\\mathbf\{w\}\\in\\mathbb\{R\}^\{N\}\_\{\\geq 0\}:\\sum\_\{i=1\}^\{N\}w\_\{i\}=1\\\}\. After initialization on the labeled set𝒟L\\mathcal\{D\}\_\{L\}, the Tri\-Game Shapley valuation mechanism is applied once to estimate an initial contribution\-aware weight vector𝐰\(0\)∈ΔN−1\\mathbf\{w\}^\{\(0\)\}\\in\\Delta^\{N\-1\}\. The obtained weights are fixed throughout collaborative training and reused for final fusion, i\.e\.,𝐰∗=𝐰\(0\)\\mathbf\{w\}^\{\*\}=\\mathbf\{w\}^\{\(0\)\}\.

At iterationtt, collaborative training uses𝒰t−1′⊆𝒟U\\mathcal\{U\}^\{\\prime\}\_\{t\-1\}\\subseteq\\mathcal\{D\}\_\{U\}as the current unlabeled candidate pool\. For each candidate𝐱u∈𝒰t−1′\\mathbf\{x\}\_\{u\}\\in\\mathcal\{U\}^\{\\prime\}\_\{t\-1\}, a pseudo\-label is generated by the fixed contribution\-aware ensemble

y¯u=∑i=1Nwi\(0\)​ℳi\(t−1\)​\(𝐱u\)\.\\bar\{y\}\_\{u\}=\\sum\_\{i=1\}^\{N\}w\_\{i\}^\{\(0\)\}\\mathcal\{M\}\_\{i\}^\{\(t\-1\)\}\(\\mathbf\{x\}\_\{u\}\)\.\(3\)Accepted pseudo\-labeled samples are inserted into model\-specific training sets, and the regressors are retrained in a regressor\-specific batchwise manner\. Figure[1](https://arxiv.org/html/2610.12884#S3.F1)illustrates the overall workflow of CoGameReg, and the detailed pseudocode is provided in Appendix A of the supplementary material\.

Figure 1:Overall workflow of the proposed CoGameReg framework\.
### 3\.2Tri\-Game Shapley Valuation Mechanism

To estimate model\-level authority, CoGameReg formulates the regressors in𝒮\\mathcal\{S\}as players in a transferable\-utility cooperative game\. A characteristic function

v:2𝒮→ℝ,v:2^\{\\mathcal\{S\}\}\\rightarrow\\mathbb\{R\},\(4\)assigns a utility to each coalition𝒢⊆𝒮\\mathcal\{G\}\\subseteq\\mathcal\{S\}, withv⁡\(∅\)=0v\(\\emptyset\)=0\. For regressorℳi∈𝒮\\mathcal\{M\}\_\{i\}\\in\\mathcal\{S\}, its Shapley value is defined as

ϕi​\(v\)=∑𝒢⊆𝒮∖\{ℳi\}\|𝒢\|\!​\(N−\|𝒢\|−1\)\!N\!​\[v⁡\(𝒢∪\{ℳi\}\)−v⁡\(𝒢\)\]\.\\phi\_\{i\}\(v\)=\\sum\_\{\\mathcal\{G\}\\subseteq\\mathcal\{S\}\\setminus\\\{\\mathcal\{M\}\_\{i\}\\\}\}\\frac\{\|\\mathcal\{G\}\|\!\(N\-\|\\mathcal\{G\}\|\-1\)\!\}\{N\!\}\\left\[v\(\\mathcal\{G\}\\cup\\\{\\mathcal\{M\}\_\{i\}\\\}\)\-v\(\\mathcal\{G\}\)\\right\]\.\(5\)The Shapley value provides a principled basis for contribution allocation\.

CoGameReg evaluates model utility from three complementary perspectives: prediction accuracy, ordinal consistency, and cross\-model consensus\. These perspectives are organized into two Shapley valuation components: a performance component combining Rank and MSE utilities, and a consistency component characterizing cross\-model agreement\. After computing the corresponding Shapley\-based contributions, we normalize and aggregate these components to obtain the fixed initial ensemble weights\.

#### 3\.2\.1Consistency Assessment

The consistency component evaluates cross\-regressor consensus\. Inspired by coalition\-based similarity formulations in cooperative\-game learning, we define the consistency utility of a coalition𝒢⊆𝒮\\mathcal\{G\}\\subseteq\\mathcal\{S\}as

vCon​\(𝒢\)=12​∑ℳi∈𝒢∑ℳj∈𝒢j≠iexp⁡\(−‖𝐲^i−𝐲^j‖222​σCon2\),v\_\{\\mathrm\{Con\}\}\(\\mathcal\{G\}\)=\\frac\{1\}\{2\}\\sum\_\{\\mathcal\{M\}\_\{i\}\\in\\mathcal\{G\}\}\\sum\_\{\\begin\{subarray\}\{c\}\\mathcal\{M\}\_\{j\}\\in\\mathcal\{G\}\\\\ j\\neq i\\end\{subarray\}\}\\exp\\\!\\left\(\-\\frac\{\\\|\\hat\{\\mathbf\{y\}\}\_\{i\}\-\\hat\{\\mathbf\{y\}\}\_\{j\}\\\|\_\{2\}^\{2\}\}\{2\\sigma\_\{\\mathrm\{Con\}\}^\{2\}\}\\right\),\(6\)where𝐲^i\\hat\{\\mathbf\{y\}\}\_\{i\}denotes the prediction vector of regressorℳi\\mathcal\{M\}\_\{i\}on the labeled set𝐗L\\mathbf\{X\}\_\{L\}, andσCon\\sigma\_\{\\mathrm\{Con\}\}is set to the median value of the pairwise Euclidean\-distance matrix of these prediction vectors, plusε=10−6\\varepsilon=10^\{\-6\}\.

Due to the pairwise\-additive structure ofvConv\_\{\\mathrm\{Con\}\}, the exact Shapley value of regressorℳi\\mathcal\{M\}\_\{i\}is

ϕCon,i=12​∑ℳj∈𝒮j≠iSim⁡\(ℳi,ℳj\),\\phi\_\{\\mathrm\{Con\},i\}=\\frac\{1\}\{2\}\\sum\_\{\\begin\{subarray\}\{c\}\\mathcal\{M\}\_\{j\}\\in\\mathcal\{S\}\\\\ j\\neq i\\end\{subarray\}\}\\mathrm\{Sim\}\(\\mathcal\{M\}\_\{i\},\\mathcal\{M\}\_\{j\}\),\(7\)whereSim⁡\(ℳi,ℳj\)\\mathrm\{Sim\}\(\\mathcal\{M\}\_\{i\},\\mathcal\{M\}\_\{j\}\)denotes the Gaussian similarity induced by their predictions on𝐗L\\mathbf\{X\}\_\{L\}\. Equivalently,vCon​\(𝒢\)v\_\{\\mathrm\{Con\}\}\(\\mathcal\{G\}\)is the total edge weight induced by coalition𝒢\\mathcal\{G\}in the prediction\-similarity graph\. The corresponding Shapley allocation assigns each regressor one\-half of its incident edge weight\. Because this factor is common to all regressors, it cancels during component\-wiseL1L\_\{1\}normalization; hence,ϕ~Con,i\\tilde\{\\phi\}\_\{\\mathrm\{Con\},i\}is the normalized weighted\-degree centrality ofℳi\\mathcal\{M\}\_\{i\}in this graph\. This quantity characterizes cross\-model agreement rather than predictive correctness\.

#### 3\.2\.2Performance Assessment

The performance component evaluates predictive behavior with respect to the labeled targets\. Specifically, the Rank utility measures ordinal correctness, whereas the MSE utility captures absolute prediction accuracy\. These two utilities capture complementary aspects of coalition\-level predictive behavior\.

Instead of computing two separate Shapley values, we exploit the additivity property of the Shapley value and define the following mixed characteristic function for each nonempty coalition:

vPerf​\(𝒢\)=α⋅vRank​\(𝒢\)\+β⋅vMSE​\(𝒢\),𝒢≠∅,v\_\{\\mathrm\{Perf\}\}\(\\mathcal\{G\}\)=\\alpha\\cdot v\_\{\\mathrm\{Rank\}\}\(\\mathcal\{G\}\)\+\\beta\\cdot v\_\{\\mathrm\{MSE\}\}\(\\mathcal\{G\}\),\\qquad\\mathcal\{G\}\\neq\\emptyset,\(8\)whereα,β≥0\\alpha,\\beta\\geq 0andα\+β≤1\\alpha\+\\beta\\leq 1\. Within the performance game, the ratio betweenα\\alphaandβ\\betacontrols the balance between ordinal and error\-based criteria, whereasα\+β\\alpha\+\\betadetermines the overall contribution of the performance component in the final fusion\.

Evaluating the Shapley value on this combined function yields the performance contributionϕPerf,i\\phi\_\{\\mathrm\{Perf\},i\}through a single Shapley\-value computation\. The resulting performance contribution of regressorℳi\\mathcal\{M\}\_\{i\}is

ϕPerf,i=ϕi​\(vPerf\)\.\\phi\_\{\\mathrm\{Perf\},i\}=\\phi\_\{i\}\\\!\\left\(v\_\{\\mathrm\{Perf\}\}\\right\)\.\(9\)
For a nonempty coalition∅≠𝒢⊆𝒮\\emptyset\\neq\\mathcal\{G\}\\subseteq\\mathcal\{S\}, let𝐲¯𝒢\\bar\{\\mathbf\{y\}\}\_\{\\mathcal\{G\}\}denote its aggregated prediction on the labeled set, computed as the mean prediction of all regressors in the coalition:

𝐲¯𝒢=1\|𝒢\|​∑ℳi∈𝒢ℳi​\(𝐗L\)\.\\bar\{\\mathbf\{y\}\}\_\{\\mathcal\{G\}\}=\\frac\{1\}\{\|\\mathcal\{G\}\|\}\\sum\_\{\\mathcal\{M\}\_\{i\}\\in\\mathcal\{G\}\}\\mathcal\{M\}\_\{i\}\(\\mathbf\{X\}\_\{L\}\)\.\(10\)The unweighted coalition average is used only for coalition utility evaluation, thereby avoiding circular dependence between Shapley valuation and the weights to be estimated\. The two component utilities are defined as follows\.

- •Rank utility\. The ordinal agreement between the coalition prediction and the labeled targets is measured using Spearman’s rank correlation coefficient: vRank​\(𝒢\)=1\+ρs​\(𝐲L,𝐲¯𝒢\)2\.v\_\{\\mathrm\{Rank\}\}\(\\mathcal\{G\}\)=\\frac\{1\+\\rho\_\{s\}\\\!\\left\(\\mathbf\{y\}\_\{L\},\\bar\{\\mathbf\{y\}\}\_\{\\mathcal\{G\}\}\\right\)\}\{2\}\.\(11\)Here,ρs\\rho\_\{s\}denotes Spearman’s rank correlation coefficient\. If the standard deviation of either𝐲L\\mathbf\{y\}\_\{L\}or𝐲¯𝒢\\bar\{\\mathbf\{y\}\}\_\{\\mathcal\{G\}\}is smaller than10−910^\{\-9\}, or if the computed correlation coefficient is not finite,ρs\\rho\_\{s\}is set to zero\. Consequently, the corresponding rank utility is set tovRank​\(𝒢\)=0\.5v\_\{\\mathrm\{Rank\}\}\(\\mathcal\{G\}\)=0\.5\.
- •MSE utility\. The prediction error of a coalition is evaluated using an exponential transformation of its mean squared error: vMSE​\(𝒢\)=exp⁡\[−MSE⁡\(𝐲L,𝐲¯𝒢\)2​bMSE2\],v\_\{\\mathrm\{MSE\}\}\(\\mathcal\{G\}\)=\\exp\\left\[\-\\frac\{\\mathrm\{MSE\}\\\!\\left\(\\mathbf\{y\}\_\{L\},\\bar\{\\mathbf\{y\}\}\_\{\\mathcal\{G\}\}\\right\)\}\{2b\_\{\\mathrm\{MSE\}\}^\{\\,2\}\}\\right\],\(12\)where bMSE=0\.5​Var​\(𝐲L\)\+ε,ε=10−6\.b\_\{\\mathrm\{MSE\}\}=0\.5\\,\\mathrm\{Var\}\(\\mathbf\{y\}\_\{L\}\)\+\\varepsilon,\\qquad\\varepsilon=10^\{\-6\}\.\(13\)The value ofbMSEb\_\{\\mathrm\{MSE\}\}is computed once from the labeled targets and is kept fixed for all coalition evaluations\. The utility decreases monotonically with coalition MSE and takes values in\(0,1\]\(0,1\]\.

#### 3\.2\.3Contribution\-Aware Model Fusion

After computing the Shapley\-based performance contribution and the consistency contribution for each regressor, we combine them into a unified ensemble weight\. Since the performance Shapley values may be negative, we first apply a zero\-boundedL1L\_\{1\}normalization to preserve relative proportions while filtering out detrimental marginal contributions\. Specifically, the performance Shapley valuesϕPerf,i\\phi\_\{\\mathrm\{Perf\},i\}and consistency scoresϕCon,i\\phi\_\{\\mathrm\{Con\},i\}are separately normalized as

ϕ~Perf,i\\displaystyle\\tilde\{\\phi\}\_\{\\mathrm\{Perf\},i\}=max⁡\(ϕPerf,i,0\)∑k=1Nmax⁡\(ϕPerf,k,0\),\\displaystyle=\\frac\{\\max\(\\phi\_\{\\mathrm\{Perf\},i\},0\)\}\{\\sum\_\{k=1\}^\{N\}\\max\(\\phi\_\{\\mathrm\{Perf\},k\},0\)\},\(14\)ϕ~Con,i\\displaystyle\\tilde\{\\phi\}\_\{\\mathrm\{Con\},i\}=max⁡\(ϕCon,i,0\)∑k=1Nmax⁡\(ϕCon,k,0\)\.\\displaystyle=\\frac\{\\max\(\\phi\_\{\\mathrm\{Con\},i\},0\)\}\{\\sum\_\{k=1\}^\{N\}\\max\(\\phi\_\{\\mathrm\{Con\},k\},0\)\}\.If the denominator of either component\-wise normalization in equation[14](https://arxiv.org/html/2610.12884#S3.E14)is zero, meaning that all zero\-bounded component scores are zero, the corresponding normalized component is set to the uniform distribution, namely,ϕ~Perf,i=1/N\\tilde\{\\phi\}\_\{\\mathrm\{Perf\},i\}=1/Norϕ~Con,i=1/N\\tilde\{\\phi\}\_\{\\mathrm\{Con\},i\}=1/Nfor allii\.

The fixed initial fusion weight is then obtained by aggregating these normalized components:

wi\(0\)=\(α\+β\)​ϕ~Perf,i\+γ​ϕ~Con,i,w\_\{i\}^\{\(0\)\}=\(\\alpha\+\\beta\)\\tilde\{\\phi\}\_\{\\mathrm\{Perf\},i\}\+\\gamma\\tilde\{\\phi\}\_\{\\mathrm\{Con\},i\},\(15\)whereγ=1−α−β≥0\\gamma=1\-\\alpha\-\\beta\\geq 0, so thatα\+β\+γ=1\\alpha\+\\beta\+\\gamma=1\. Here,α\\alphaandβ\\betadetermine the relative composition of the performance game, whileα\+β\\alpha\+\\betaspecifies the total weight assigned to the performance component relative to the consistency component in the final fusion stage\. Because both normalized contribution vectors lie on the probability simplex and\(α\+β\)\+γ=1\(\\alpha\+\\beta\)\+\\gamma=1, the fixed weight vector also satisfies𝐰\(0\)∈ΔN−1\\mathbf\{w\}^\{\(0\)\}\\in\\Delta^\{N\-1\}\. Consequently, the fused prediction lies within the pointwise range of the base\-regressor predictions; this convex\-hull property does not impose a physical bound unless the constituent predictions are themselves bounded\.

Exact evaluation of the performance Shapley values requires enumeration of2N2^\{N\}coalitions\. In the reported experiments, the ensemble contains five regressors for the field application and six regressors for the benchmark and synthetic experiments, and the fixed contribution weights are evaluated only once before collaborative training\. The exact performance valuation therefore requires 32 coalition evaluations for the field application and 64 for the benchmark and synthetic experiments\. For larger ensembles, exact enumeration can be replaced by Monte Carlo permutation sampling\.

### 3\.3CoGameReg Collaborative Training Procedure

The proposed CoGameReg framework performs iterative collaborative expansion over a locality\-aware candidate pool\. In each iteration, pseudo\-labeled candidates are evaluated and selected in a regressor\-specific batchwise manner, incorporated into corresponding model\-specific training sets, and then used to retrain the regressors; the candidate pool is refilled when needed\.

#### 3\.3\.1Hybrid Unlabeled Pool Initialization and Progressive Expansion

To improve the reliability of early pseudo\-label selection, CoGameReg adopts a hybrid unlabeled\-pool strategy that combines locality\-aware initialization with progressive exploration\. At the start of training, we construct the initial candidate pool𝒰0′\\mathcal\{U\}^\{\\prime\}\_\{0\}by selecting, for each labeled sample𝐱l∈𝒟L\\mathbf\{x\}\_\{l\}\\in\\mathcal\{D\}\_\{L\}, itskpoolk\_\{\\mathrm\{pool\}\}nearest neighbors from𝒟U\\mathcal\{D\}\_\{U\}:

𝒰0′=⋃𝐱l∈𝒟LNNkpool​\(𝐱l,𝒟U\),\\mathcal\{U\}^\{\\prime\}\_\{0\}=\\bigcup\_\{\\mathbf\{x\}\_\{l\}\\in\\mathcal\{D\}\_\{L\}\}\\mathrm\{NN\}\_\{k\_\{\\mathrm\{pool\}\}\}\(\\mathbf\{x\}\_\{l\},\\mathcal\{D\}\_\{U\}\),\(16\)whereNNkpool​\(⋅\)\\mathrm\{NN\}\_\{k\_\{\\mathrm\{pool\}\}\}\(\\cdot\)denotes thekpoolk\_\{\\mathrm\{pool\}\}\-nearest neighbor operator\. The remaining unlabeled set is denoted by

𝒰rem=𝒟U∖𝒰0′\.\\mathcal\{U\}\_\{\\mathrm\{rem\}\}=\\mathcal\{D\}\_\{U\}\\setminus\\mathcal\{U\}^\{\\prime\}\_\{0\}\.\(17\)As training proceeds, if the size of the updated candidate pool𝒰t′\\mathcal\{U\}^\{\\prime\}\_\{t\}falls below the target scales​\|𝒟L\|s\|\\mathcal\{D\}\_\{L\}\|, new samples are added from𝒰rem\\mathcal\{U\}\_\{\\mathrm\{rem\}\}into𝒰t′\\mathcal\{U\}^\{\\prime\}\_\{t\}and removed from𝒰rem\\mathcal\{U\}\_\{\\mathrm\{rem\}\}\.

#### 3\.3\.2Weighted Pseudo\-Labeling, Model\-Specific Acceptance, and Dynamic Batching

Given the initialized regressor pool𝒮=\{ℳ1,…,ℳN\}\\mathcal\{S\}=\\\{\\mathcal\{M\}\_\{1\},\\dots,\\mathcal\{M\}\_\{N\}\\\}, CoGameReg first estimates the fixed initial ensemble weights\{wi\(0\)\}i=1N\\\{w\_\{i\}^\{\(0\)\}\\\}\_\{i=1\}^\{N\}using the Tri\-Game Shapley mechanism described in Section[3\.2](https://arxiv.org/html/2610.12884#S3.SS2)\. For each candidate sample𝐱u∈𝒰t−1′\\mathbf\{x\}\_\{u\}\\in\\mathcal\{U\}^\{\\prime\}\_\{t\-1\}, the pseudo\-label is then generated by the fixed weighted ensemble:

y¯u=∑i=1Nwi\(0\)​ℳi\(t−1\)​\(𝐱u\)\.\\bar\{y\}\_\{u\}=\\sum\_\{i=1\}^\{N\}w\_\{i\}^\{\(0\)\}\\mathcal\{M\}\_\{i\}^\{\(t\-1\)\}\(\\mathbf\{x\}\_\{u\}\)\.\(18\)
With𝐰\(0\)\\mathbf\{w\}^\{\(0\)\}fixed, perturbing only the prediction of regressorℳi\\mathcal\{M\}\_\{i\}byε\\varepsilonchanges the shared pseudo\-label byΔ​y¯u=wi\(0\)​ε\.\\Delta\\bar\{y\}\_\{u\}=w\_\{i\}^\{\(0\)\}\\varepsilon\.Thus,wi\(0\)w\_\{i\}^\{\(0\)\}is the direct linear influence coefficient ofℳi\\mathcal\{M\}\_\{i\}in both candidate pseudo\-label construction and final prediction fusion\. This identity concerns the aggregation step and is not a statement about the total retraining dynamics or generalization\. Letℒi\(t−1\)=𝒟L∪𝒫i\(t−1\)\\mathcal\{L\}\_\{i\}^\{\(t\-1\)\}=\\mathcal\{D\}\_\{L\}\\cup\\mathcal\{P\}\_\{i\}^\{\(t\-1\)\}denote the augmented state associated with regressorℳi\\mathcal\{M\}\_\{i\}\. Candidate admission is model\-indexed because the same local gain functional is evaluated separately on eachℒi\(t−1\)\\mathcal\{L\}\_\{i\}^\{\(t\-1\)\}\. Following the local error\-reduction principle of COREG, we estimate the local mean squared error reduction resulting from accepting\(𝐱u,y¯u\)\(\\mathbf\{x\}\_\{u\},\\bar\{y\}\_\{u\}\)by

δ𝐱u\(i\)=1\|Ωu\|​∑𝐱j∈Ωu\[\(yj−hitemp​\(𝐱j\)\)2−\(yj−hitemp′​\(𝐱j\)\)2\],\\delta\_\{\\mathbf\{x\}\_\{u\}\}^\{\(i\)\}=\\frac\{1\}\{\|\\Omega\_\{u\}\|\}\\sum\_\{\\mathbf\{x\}\_\{j\}\\in\\Omega\_\{u\}\}\\left\[\\left\(y\_\{j\}\-h\_\{i\}^\{\\mathrm\{temp\}\}\(\\mathbf\{x\}\_\{j\}\)\\right\)^\{2\}\-\\left\(y\_\{j\}\-\{h\_\{i\}^\{\\mathrm\{temp\}\}\}^\{\\prime\}\(\\mathbf\{x\}\_\{j\}\)\\right\)^\{2\}\\right\],\(19\)wherehitemph\_\{i\}^\{\\mathrm\{temp\}\}denotes the temporarykk\-NN proxy, common in form across regressors, instantiated onℒi\(t−1\)\\mathcal\{L\}\_\{i\}^\{\(t\-1\)\},hitemp′h\_\{i\}^\{\\mathrm\{temp\}\\prime\}denotes the same proxy after temporarily adding\(𝐱u,y¯u\)\(\\mathbf\{x\}\_\{u\},\\bar\{y\}\_\{u\}\), andΩu\\Omega\_\{u\}denotes the local neighborhood of𝐱u\\mathbf\{x\}\_\{u\}\. The regressor index refers to the augmented state on which the proxy is evaluated rather than to a different proxy model class\.

Becauseδ𝐱u\(i\)\\delta\_\{\\mathbf\{x\}\_\{u\}\}^\{\(i\)\}is the difference between the pre\- and post\-inclusion neighborhood MSE, any candidate satisfyingδ𝐱u\(i\)\>θ\\delta\_\{\\mathbf\{x\}\_\{u\}\}^\{\(i\)\}\>\\thetalowers the neighborhood MSE of the temporary proxy by more thanθ\\thetawhen evaluated individually\. This local implication does not establish monotone reduction of the global loss or test error after batch updating\.

For each regressorℳi\\mathcal\{M\}\_\{i\}, candidates satisfyingδ𝐱u\(i\)\>θ\\delta\_\{\\mathbf\{x\}\_\{u\}\}^\{\(i\)\}\>\\thetaform the feasible set

𝒞i\(t\)=\{\(𝐱u,y¯u\):𝐱u∈𝒰t−1′,δ𝐱u\(i\)\>θ\}\.\\mathcal\{C\}\_\{i\}^\{\(t\)\}=\\left\\\{\(\\mathbf\{x\}\_\{u\},\\bar\{y\}\_\{u\}\):\\mathbf\{x\}\_\{u\}\\in\\mathcal\{U\}^\{\\prime\}\_\{t\-1\},\\ \\delta\_\{\\mathbf\{x\}\_\{u\}\}^\{\(i\)\}\>\\theta\\right\\\}\.\(20\)CoGameReg then selects the topρ\\rhoproportion of𝒞i\(t\)\\mathcal\{C\}\_\{i\}^\{\(t\)\}, ranked byδ𝐱u\(i\)\\delta\_\{\\mathbf\{x\}\_\{u\}\}^\{\(i\)\}, to form the accepted subset𝒜i\(t\)\\mathcal\{A\}\_\{i\}^\{\(t\)\}, and updates the pseudo\-label pool by

𝒫i\(t\)=𝒫i\(t−1\)∪𝒜i\(t\)\.\\mathcal\{P\}\_\{i\}^\{\(t\)\}=\\mathcal\{P\}\_\{i\}^\{\(t\-1\)\}\\cup\\mathcal\{A\}\_\{i\}^\{\(t\)\}\.\(21\)After all regressors complete their model\-specific selections, the candidate pool is updated as

𝒰t′=𝒰t−1′∖⋃i=1N\{𝐱u:\(𝐱u,y¯u\)∈𝒜i\(t\)\}\.\\mathcal\{U\}^\{\\prime\}\_\{t\}=\\mathcal\{U\}^\{\\prime\}\_\{t\-1\}\\setminus\\bigcup\_\{i=1\}^\{N\}\\left\\\{\\mathbf\{x\}\_\{u\}:\(\\mathbf\{x\}\_\{u\},\\bar\{y\}\_\{u\}\)\\in\\mathcal\{A\}\_\{i\}^\{\(t\)\}\\right\\\}\.\(22\)Each regressor maintains its own augmented training setDL∪Pi\(t\)D\_\{L\}\\cup P\_\{i\}^\{\(t\)\}and is retrained independently\. The heterogeneous regressors learn from the shared supervision through their respective inductive biases\. The ensemble weights remain fixed throughout collaborative training and are reused for final prediction fusion, i\.e\.,w∗=w\(0\)w^\{\*\}=w^\{\(0\)\}\. At each iteration, the shared pseudo\-labels are recomputed using these fixed weights and the current regressor predictions\.

## 4Experimental Results and Analysis

We evaluate CoGameReg in three complementary settings to examine its use for reservoir property prediction under sparse well control\. Experiments on 10 public regression datasets compare predictive performance across datasets and labeled\-data ratios, while ablations examine ensemble fusion, contribution\-based weighting, and collaborative training\. These benchmarks provide methodological evidence for the semi\-supervised regression framework\. The two seismic experiments examine different prediction targets and input attributes\. The synthetic experiment predicts P\-wave velocity, an elastic property, from five attributes derived from post\-stack seismic data\. The field experiment predicts water saturation, a petrophysical property, from the post\-stack seismic volume and inversion\-derived impedance and velocity\-ratio volumes\. Both tasks combine sparse well labels with unlabeled attribute samples using the same learning framework; models are trained separately with their respective inputs, targets, and configurations\. The synthetic experiment uses quantitative blind\-well evaluation, qualitative slice comparisons against the known reference volume, and sensitivity analysis of neighborhood and weighting parameters\. The field experiment compares four methods at common blind wells and on the same seismic section and reservoir horizon\. The field volume comparisons provide spatial context, with saturation reference values available only along well trajectories\.

### 4\.1Benchmark Experiments on Public Datasets

#### 4\.1\.1Experimental Protocol

Table[1](https://arxiv.org/html/2610.12884#S4.T1)summarizes the 10 public benchmark regression datasets used in this study\. The exact benchmark files were selected from the data package distributed with the public implementation of S2RMS\[[30](https://arxiv.org/html/2610.12884#bib.bib28)\]\. The underlying datasets originate from the UCI Machine Learning Repository\[[20](https://arxiv.org/html/2610.12884#bib.bib43)\], Delve\[[11](https://arxiv.org/html/2610.12884#bib.bib44)\], and StatLib\[[7](https://arxiv.org/html/2610.12884#bib.bib45)\]\. Following a unified protocol, we randomly select 2,000 samples from each dataset to form the training pool and use the remaining samples as the test set\. The training pool is further partitioned into labeled and unlabeled subsets according to a labeled\-data ratior∈\{2\.5%,5%,10%\}r\\in\\\{2\.5\\%,5\\%,10\\%\\\}, defined with respect to the training pool, yielding 50, 100, and 200 labeled samples per run, respectively\. Twenty percent of the labeled subset is reserved as an internal validation set for methods that require early stopping or hyperparameter selection\. For most methods, the remaining 80% constitutes the labeled training set\. CoGameReg uses only this training portion both to fit the base regressors and to estimate the initial Tri\-Game Shapley weights; the validation split is not used for weight estimation or collaborative training\. For TabM, early stopping on the validation subset determines the training duration\. The model is then reinitialized and refit from scratch on the combined training and validation subsets for the selected number of epochs\. The CoGameReg weights are kept fixed throughout collaborative training and final prediction\. All methods are evaluated over 10 independent runs using the same random seeds and data splits\.

Table 1:Summary of the 10 public benchmark regression datasets\.We compare CoGameReg against baselines spanning supervised, ensemble, self\-training, classical semi\-supervised, and recent neural and tabular paradigms\. Specifically, the compared methods include XGBoost, KNN, TabM\[[17](https://arxiv.org/html/2610.12884#bib.bib46)\], Avg\. Ensemble, Self\-KNN, COREG\[[50](https://arxiv.org/html/2610.12884#bib.bib21)\], Mean Teacher\[[43](https://arxiv.org/html/2610.12884#bib.bib47)\], SSDKL\[[19](https://arxiv.org/html/2610.12884#bib.bib48)\], and UCVME\[[10](https://arxiv.org/html/2610.12884#bib.bib49)\]\. COREG serves as the most direct methodological reference because it is the classical co\-training baseline for regression, whereas Mean Teacher, SSDKL, and UCVME represent stronger modern semi\-supervised baselines\. XGBoost and TabM provide competitive supervised and tabular\-learning references\.

For the benchmark datasets, performance is evaluated primarily in terms of RMSE\. To complement the per\-dataset comparisons, we conduct Wilcoxon signed\-rank tests across the 10 datasets for each labeled\-data ratio, together with Holm correction and effect\-size analysis\. Detailed definitions of the evaluation metrics and the statistical testing protocol are provided in Appendix B\.1\. The implementation details and fixed hyperparameter settings of all baseline methods and base regressors are reported in Appendix B\.2\.

#### 4\.1\.2Comparison with Other Methods

Table[2](https://arxiv.org/html/2610.12884#S4.T2)reports the RMSE results of all compared methods on the 10 benchmark datasets under three labeled\-data ratios\. Overall, CoGameReg achieves the lowest average RMSE \(0\.10980\.1098\) across the 30 dataset\-ratio settings and yields the lowest or joint\-lowest RMSE in 18 of them\. Specifically, it yields the lowest or joint\-lowest RMSE on 7, 6, and 5 of the 10 datasets at the 2\.5%, 5%, and 10% labeled\-data ratios, respectively\.

Table 2:Comparison of RMSE performance across different methods under varying labeled\-data ratios\.The comparison with classical collaborative and self\-training baselines is particularly informative\. CoGameReg achieves lower RMSE than Self\-KNN, COREG, and Avg\. Ensemble on most datasets and labeled\-data ratios\. These results are consistent with an empirical benefit from extending dual\-regressor co\-training to a multi\-regressor collaborative setting\. The improvement over Avg\. Ensemble also indicates that simple prediction averaging alone does not account for the observed gains; the ablation analysis below further evaluates the contributions of collaboration and contribution\-aware weighting\.

Strong modern baselines remain competitive on several datasets\. Mean Teacher performs strongly on Kin8nm and Space\_ga\. SSDKL remains competitive on Elevators at the 5% and 10% labeled\-data ratios, while TabM becomes increasingly competitive as the labeled\-data ratio increases\. Nevertheless, CoGameReg achieves the lowest or joint\-lowest RMSE on Abalone, Elevators, Folds5x2\_pp, Parkinsons, Wind, and Wine\_quality at multiple labeled\-data ratios\. Overall, CoGameReg achieves competitive predictive performance across the evaluated semi\-supervised regression tasks\.

#### 4\.1\.3Statistical Significance Analysis

To further assess the statistical significance of the observed performance differences, we conduct Wilcoxon signed\-rank tests on the RMSE results across the 10 datasets at each labeled\-data ratio, with Holm correction to account for multiple comparisons\. The results are summarized in Table[3](https://arxiv.org/html/2610.12884#S4.T3)\.

Table 3:Per\-ratio Wilcoxon signed\-rank test results with Holm correction\.At the 2\.5% labeled\-data ratio, CoGameReg has significantly lower RMSE than all nine compared methods after Holm correction\. It achieves a win/tie/loss record of 10/0/0 against XGBoost, KNN, Self\-KNN, COREG, SSDKL, and UCVME; all six comparisons have Holm\-adjustedpp\-values of0\.01760\.0176and large effect sizes \(r=0\.886r=0\.886\)\. The comparisons with TabM and Avg\. Ensemble are also significant, each with a win/tie/loss record of 9/0/1 \(Holm\-p=0\.0176p=0\.0176,r=0\.854r=0\.854\)\. Against Mean Teacher, CoGameReg obtains a win/tie/loss record of 8/1/1 \(Holm\-p=0\.0176p=0\.0176,r=0\.810r=0\.810\)\.

At the 5% labeled\-data ratio, CoGameReg remains statistically superior to XGBoost, KNN, Avg\. Ensemble, Self\-KNN, COREG, SSDKL, and UCVME after Holm correction\. The differences relative to TabM and Mean Teacher, however, are not statistically significant after Holm correction\. This suggests that as the amount of supervision increases, the advantage of CoGameReg over the stronger modern baselines becomes smaller, although it still retains statistically significant advantages over the majority of the compared methods\.

At the 10% labeled\-data ratio, CoGameReg has significantly lower RMSE than KNN, Avg\. Ensemble, Self\-KNN, COREG, SSDKL, and UCVME after Holm correction\. By contrast, the differences relative to XGBoost, TabM, and Mean Teacher are not statistically significant after Holm correction\. Against SSDKL, CoGameReg obtains a win/tie/loss record of 9/0/1 \(Holm\-p=0\.0391p=0\.0391,r=0\.790r=0\.790\)\. Although CoGameReg records more wins than losses against TabM \(6/0/4\), this comparison remains statistically non\-significant \(Holm\-p=0\.846p=0\.846,r=0\.081r=0\.081\)\.

Overall, CoGameReg shows robust and statistically significant improvements over classical co\-training, self\-training, and simple ensemble baselines\. Against recent supervised and semi\-supervised alternatives, its advantage is most pronounced under scarce supervision and becomes increasingly dataset\-dependent as the labeled\-data ratio increases\. This pattern is consistent with the design motivation of CoGameReg, namely, to exploit model complementarity across multiple regressors under scarce supervision\.

#### 4\.1\.4Ablation Study

To isolate the contributions of the main design components of CoGameReg, we conduct an ablation study on the same 10 benchmark datasets under three labeled\-data ratios\. The ablation variants are defined as follows:

- •Split\-Label CoGameReg \(SL\-CoGameReg\): a variant in which 80% of the labeled training set is used to train the base regressors, while the remaining 20% is held out to compute the Tri\-Game Shapley weights\.
- •Average Ensemble \(Avg\. Ensemble\): a purely supervised heterogeneous ensemble with uniform averaging and no semi\-supervised updating\.
- •Shapley Weighted Ensemble \(Shap\. Ensemble\): a purely supervised heterogeneous ensemble that uses Tri\-Game Shapley weights but does not involve semi\-supervised collaboration\.
- •α\\alpha\-Only Shapley Weighted Ensemble \(α\\alpha\-Shap\. Ensemble\): a supervised Tri\-Game weighted ensemble that uses only the Rank component, i\.e\.,α=1\\alpha=1,β=0\\beta=0, andγ=0\\gamma=0\.
- •β\\beta\-Only Shapley Weighted Ensemble \(β\\beta\-Shap\. Ensemble\): a supervised Tri\-Game weighted ensemble that uses only the MSE component, i\.e\.,α=0\\alpha=0,β=1\\beta=1, andγ=0\\gamma=0\.
- •γ\\gamma\-Only Shapley Weighted Ensemble \(γ\\gamma\-Shap\. Ensemble\): a supervised Tri\-Game weighted ensemble that uses only the Consistency component, i\.e\.,α=0\\alpha=0,β=0\\beta=0, andγ=1\\gamma=1\.
- •SSR\-Average Ensemble \(SSR\-Avg\. Ensemble\): semi\-supervised collaborative training with uniform ensemble weights throughout both training and prediction\.
- •SSR\-Decoupled Shapley Weighting \(Decoupled CoGameReg\): semi\-supervised collaborative training with uniform weights during pseudo\-label generation, followed by Shapley\-based weighting only at the final prediction stage\.
- •SSR\-Undecoupled Shapley Weighting \(CoGameReg\): the final CoGameReg variant, in which the initial Tri\-Game Shapley weights are computed from the labeled data and then kept fixed during both semi\-supervised pseudo\-label generation and final prediction\.

The results in Table[4](https://arxiv.org/html/2610.12884#S4.T4)show that the final CoGameReg variant achieves the lowest average RMSE \(0\.10980\.1098\) among all ablation variants and attains the best or tied\-best result in 12 of the 30 dataset\-ratio settings\. It outperforms Split\-Label CoGameReg, Average Ensemble, Shapley Weighted Ensemble, SSR\-Average Ensemble, and SSR\-Decoupled Shapley Weighting in 29, 28, 24, 30, and 27 settings, respectively\.

Table 4:Comparison of RMSE performance across different ablation variants of CoGameReg under varying labeled\-data ratios\.First, contribution\-aware fusion remains beneficial even without semi\-supervised updating: compared with Average Ensemble \(0\.11560\.1156\), Shapley Weighted Ensemble reduces the average RMSE to0\.11060\.1106\. This suggests that explicitly modeling regressor contributions is more effective than uniform averaging in the purely supervised setting\.

Second, the single\-component variants reveal that the three Tri\-Game components play distinct empirical roles\. Theγ\\gamma\-only variant achieves a competitive average RMSE \(0\.11120\.1112\), indicating that cross\-model consensus provides a useful signal for supervised multi\-model fusion\. Theα\\alpha\-only variant is also competitive \(0\.11200\.1120\), suggesting that ordinal information provides a complementary signal\. By contrast, theβ\\beta\-only variant performs substantially worse \(0\.12340\.1234\), showing that the MSE\-based utility alone is insufficient for achieving strong performance\. Among the purely supervised weighting variants, the full Shapley Weighted Ensemble achieves the lowest average RMSE \(0\.11060\.1106\), slightly outperforming theγ\\gamma\-only andα\\alpha\-only variants on average\. Although theγ\\gamma\-only variant achieves a lower RMSE than the full Shapley Weighted Ensemble in 19 of the 30 individual settings, the full combination has the lower overall average RMSE\. Thus, the results support an average advantage for combining the three utility components, but not a consistent advantage across individual settings\. More importantly, the final CoGameReg variant achieves a lower average RMSE \(0\.10980\.1098\) than all single\-component and purely supervised variants\.

Third, the effect of semi\-supervised collaboration depends on how the contribution\-aware weights are incorporated\. SSR\-Decoupled Shapley Weighting achieves an average RMSE of0\.11110\.1111, which is slightly higher than that of the purely supervised Shapley Weighted Ensemble \(0\.11060\.1106\)\. Therefore, decoupled pseudo\-label training does not provide an additional average gain over purely supervised weighted fusion in this comparison\. In contrast, the final CoGameReg variant reduces the average RMSE to0\.10980\.1098and outperforms both\. This suggests that the benefit of semi\-supervised collaboration is associated with using the fixed contribution\-aware weights during pseudo\-label generation and final prediction, rather than applying Shapley weights only at final fusion\.

Finally, among the collaborative CoGameReg variants, Split\-Label CoGameReg has the highest average RMSE \(0\.11750\.1175\)\. This suggests that, when labeled data are scarce, splitting the labeled set into separate subsets for base\-regressor training and contribution\-weight estimation may be suboptimal\. Such a split reduces the effective amount of labeled data available to train each base regressor, while the Shapley weights estimated from a smaller labeled subset may also become less stable\. This observation supports the final design choice of initializing both the base regressors and the contribution\-aware weights using all available labeled data and then keeping these weights fixed throughout collaborative training\.

Overall, the ablation results suggest that the best average performance is obtained when multi\-model ensemble diversity, multi\-component Tri\-Game Shapley valuation, and fixed\-initial\-weight semi\-supervised collaborative training are combined\. No single\-component or intermediate variant matches the average RMSE of the complete CoGameReg configuration\.

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/synthetic_input/GT.png)![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/synthetic_input/inst_amp.png)![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/synthetic_input/inst_phase.png)\(a\) Ground\-truthVpV\_\{p\}\(b\) Instantaneous
amplitude\(c\) Instantaneous
phase![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/synthetic_input/inst_quard.png)![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/synthetic_input/integral_band.png)![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/synthetic_input/sweetness.png)\(d\) Instantaneous
quadrature\(e\) Integral band\(f\) SweetnessFigure 2:Ground\-truth P\-wave velocity and five seismic attributes used as inputs for the synthetic experiment\. Each panel is displayed with its own color scale\.

### 4\.2Synthetic 3D Seismic Reservoir Property Prediction

#### 4\.2\.13D Seismic Data Configuration and Evaluation Metrics

To investigate the applicability of CoGameReg to seismic reservoir characterization, we further conduct experiments on a simulated 3D seismic reservoir property prediction task in the petroleum exploration domain, for which full\-field ground truth is available\. This 3D setting enables us to assess local prediction quality at blind wells and to visually examine structural consistency on representative slices\.

We use the Stanford VI\-E synthetic geological model\[[22](https://arxiv.org/html/2610.12884#bib.bib50)\]as the simulation benchmark\. The dataset consists of a large\-scale 3D reservoir volume with approximately6×1066\\times 10^\{6\}cells, with a grid size of150×200×200150\\times 200\\times 200\. The horizontal and vertical resolutions are25​m25\\,\\mathrm\{m\}and1​m1\\,\\mathrm\{m\}, respectively\. The prediction target is P\-wave velocity, whose ground\-truth distribution is shown in Figure[2](https://arxiv.org/html/2610.12884#S4.F2)\(a\)\. The available wells are divided into training, validation, and blind\-test sets, with their spatial distribution shown in Figure[3](https://arxiv.org/html/2610.12884#S4.F3)\. This well split is generated by random sampling subject to spatial coverage and separation constraints\. Specifically, the blind test wells are selected to be spatially well distributed across the survey area and sufficiently separated from one another, so that the evaluation better reflects generalization at representative blind locations rather than interpolation among clustered wells\. The training wells are used for model fitting, the validation wells for parameter selection and sensitivity analysis, and the blind test wells are reserved exclusively for final evaluation\. The five seismic attributes used as model inputs are displayed in Figure[2](https://arxiv.org/html/2610.12884#S4.F2)\(b–f\)\. The five seismic attribute volumes were derived from the post\-stack seismic data using GeoEast and supplied as precomputed inputs for P\-wave velocity prediction\. The present experiment evaluates how CoGameReg combines these attributes with sparse well labels to predict the target property\.

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/well_distribution.png)Figure 3:Spatial distribution of the training, validation, and blind\-test wells in the synthetic experiment\.In this setting, quantitative evaluation is performed at blind wells using RMSE,R2R^\{2\}, and Spearman rank correlation \(SRC\)\. Because full\-field ground truth is available, we also provide qualitative structural comparisons on representative slices to examine geological continuity and lateral variation\. For the internal heterogeneous ensemble of CoGameReg, we use the same six representative base regressors as in the benchmark experiments: XGBoost, SVR, MLP, Ridge, Linear Regression, and KNN\. The external baselines include XGBoost, KNN, TabM, Avg\. Ensemble, Self\-KNN, COREG, Mean Teacher, SSDKL, and UCVME\. A task\-specific Tri\-Game coefficient setting was selected using the validation wells\. The complete synthetic\-data configuration and CoGameReg parameter settings are provided in Appendix B\.2 of the supplementary material, and the corresponding sensitivity analysis is reported in Appendix C\.

#### 4\.2\.2Blind\-Well Evaluation and Qualitative Comparison

Table[5](https://arxiv.org/html/2610.12884#S4.T5)reports the blind\-well quantitative results for all compared methods\. Figure[4](https://arxiv.org/html/2610.12884#S4.F4)focuses on representative supervised and classical semi\-supervised regression approaches\. From left to right, the first two columns show XGBoost and KNN as supervised references, followed by Self\-KNN and COREG as self\-training and co\-training baselines\. CoGameReg is placed immediately to the left of the ground truth to facilitate direct comparison\. Additional results for the remaining methods are presented in Figure S3 of Appendix D\.

Table 5:Quantitative comparison of reservoir property prediction on the simulated 3D dataset, evaluated on blind wells\.CoGameReg achieves an RMSE of 0\.4137, anR2R^\{2\}of 0\.5065, and an SRC of 0\.5747, yielding the best values among the methods reported in Table 5\. Relative to XGBoost, KNN, Self\-KNN, and COREG, its RMSE is reduced by approximately 9\.7%, 13\.9%, 12\.2%, and 10\.6%, respectively\. Both Self\-KNN and COREG improve upon the supervised KNN baseline, whereas CoGameReg achieves lower prediction error than all four reference methods\. These comparisons support the effectiveness of the complete collaborative framework for blind\-well prediction in this synthetic experiment\.

The first two columns of Figure[4](https://arxiv.org/html/2610.12884#S4.F4)show that XGBoost and KNN recover the main spatial patterns, including the curved bands at Time = 9, the meandering structures at Time = 110, and the branching patterns at Time = 188\. However, local fluctuations remain visible within and around the principal structures\. The Self\-KNN and COREG predictions in the next two columns preserve similar large\-scale geometries, but background mottling and irregular local variations are still evident\. Their quantitative improvements over KNN therefore do not translate into uniformly clearer structures throughout the displayed slices\.

The CoGameReg predictions in the fifth column exhibit less mottled background variation than the four reference predictions, particularly at Time = 9 and Time = 110\. The background surrounding the bodies in the upper part of the Inline = 121 section is also smoother, making the principal structures more clearly distinguishable\. Comparison with the rightmost ground\-truth column shows that CoGameReg preserves the main channel patterns, although narrow internal bands, smoothed boundaries, and incomplete recovery of fine\-scale heterogeneity remain apparent\. These limitations are particularly visible in the Time = 188 slice and the lower part of the inline section\. The visual results thus complement the blind\-well evaluation by showing improved spatial coherence\.

Taken together, the results demonstrate an advantage over the supervised and classical semi\-supervised references highlighted in the main figure, while maintaining competitive performance relative to the remaining methods\.

Figure 4:Qualitative comparison of P\-wave velocity predictions on representative slices of the synthetic 3D dataset\. From top to bottom, the rows correspond to Time = 9, Time = 110, Time = 188, and Inline = 121\. From left to right, the columns show XGBoost, KNN, Self\-KNN, COREG, CoGameReg, and the ground truth\. CoGameReg is placed next to the ground truth for direct visual comparison\. All panels use the same P\-wave velocity color scale as Figure[2](https://arxiv.org/html/2610.12884#S4.F2)\(a\)\.

### 4\.3Real Field Application in the Western Field Area

#### 4\.3\.1Geological setting and data

The field data set was acquired from a proprietary tight\-gas survey in western China, hereafter referred to as the Western Field Area\. To comply with data\-confidentiality requirements, the specific field name, original well identifiers, and absolute survey coordinates are not disclosed\. The 18 wells used in this study are referred to as Well 1 through Well 18\.

The target interval is a Permian tight\-sandstone reservoir and constitutes one of the principal exploration intervals in the study area\. The regional structure is generally gentle, although locally developed low\-relief structures and small\-displacement faults are present\. The survey can be broadly divided into a western structural low, a central slope, and an eastern uplift\.

The target interval was deposited in a braided\-river delta\-plain environment, with sediment supply predominantly from south to north\. The succession consists mainly of interbedded sandstone and mudstone\. The sandstone bodies are thin and laterally variable, generally less than 15 m thick and averaging approximately 11 m\. The gas\-bearing intervals are typically 2–8 m thick, and the distributary channels are approximately 0\.9–1\.3 km wide\. Moreover, sandstone development does not necessarily correspond directly to gas\-bearing intervals\. These characteristics make the spatial prediction of water saturation particularly challenging\. Water saturation is selected as the target property because it integrates information from pore\-fluid distribution and reservoir heterogeneity, while its estimation from seismic observations remains challenging due to the indirect relationship between seismic responses and fluid properties\.

The available seismic data consist of a three\-dimensional post\-stack time\-domain seismic volume, covering approximately 140 km2, with a bin size of 20 m×\\times20 m\. The dominant frequency of the complete seismic volume is approximately 30 Hz, whereas the dominant frequency near the target interval ranges from approximately 35 to 45 Hz\. The processed seismic data exhibit a relatively high signal\-to\-noise ratio and no obvious acquisition footprint within the target interval\.

A total of 18 wells with valid water\-saturation logs were retained for the field experiment\. The available well\-log data include acoustic transit\-time, measured shear\-wave, and ground truth water\-saturation logs\. Before model construction, the well logs were subjected to quality control, borehole\-effect correction, outlier removal, and interwell standardization\. Synthetic seismograms were generated to establish the well\-to\-seismic relationships before the logs were resampled to the seismic sampling interval\.

#### 4\.3\.2Well\-to\-seismic calibration and experimental protocol

The wells were partitioned at the well level, rather than at the sample level, to evaluate spatial generalization between different well locations\. Well 1 through Well 10 were used for model training, Well 11 through Well 15 were used for hyperparameter validation, and Well 16 through Well 18 were reserved for blind\-well evaluation\. The well partition is summarized in Table[6](https://arxiv.org/html/2610.12884#S4.T6)\.

Table 6:Well partition used in the Western Field Area experiment\. The well names are anonymized for data\-confidentiality reasons\.Only the water\-saturation samples from the training wells were used as labeled data during model construction\. The validation\-well logs were used only for hyperparameter selection and model configuration\. The blind\-test\-well logs were completely withheld from model training, ensemble\-weight estimation, pseudo\-label generation, hyperparameter selection, and model selection\. They were accessed only after the complete modeling workflow had been fixed\.

Five seismic\-derived volumes were used as input features for water\-saturation prediction: the post\-stack seismic volume, P\-wave impedance from post\-stack geostatistical inversion, S\-wave impedance from pre\-stack geostatistical inversion, and twoVp/VsV\_\{p\}/V\_\{s\}volumes obtained from pre\-stack geostatistical and Bayesian inversion, respectively\. The post\-stack seismic volume represents the reflection response, whereas the inversion\-derived volumes provide estimates of impedance and velocity\-ratio properties\. The twoVp/VsV\_\{p\}/V\_\{s\}inputs describe the same elastic\-property ratio through different inversion procedures\. Their joint use allows the regression framework to combine these seismic\-derived descriptions when estimating water saturation\. The input volumes were obtained through seismic processing and inversion using GeoEast\. These precomputed volumes were used as the common inputs to XGBoost, COREG, UCVME, and CoGameReg\. The field experiment therefore evaluates the regression frameworks for water\-saturation prediction given the same seismic\-derived information\. Sections of the five input volumes along a common arbitrary seismic line are shown in Figure[5](https://arxiv.org/html/2610.12884#S4.F5)\. The same line is used for the XGBoost, COREG, UCVME, and CoGameReg prediction sections in Figure[7](https://arxiv.org/html/2610.12884#S4.F7), and its trajectory is marked by the red polyline in Figure[8](https://arxiv.org/html/2610.12884#S4.F8)\. This common profile provides a direct spatial reference for examining the input attributes alongside the predicted water\-saturation distributions\.

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Input_feature/all_stack_pstm.png)

\(a\) Post\-stack seismic

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Input_feature/Gi.png)

\(b\) P\-wave impedance: post\-stack geostatistical inversion

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Input_feature/Is.png)

\(c\) S\-wave impedance: pre\-stack geostatistical inversion

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Input_feature/Vp_Vs.png)

\(d\)Vp/VsV\_\{p\}/V\_\{s\}: pre\-stack geostatistical inversion

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Input_feature/VpVsBYS.png)

\(e\)Vp/VsV\_\{p\}/V\_\{s\}: pre\-stack Bayesian inversion

Figure 5:Vertical sections of the five seismic\-derived volumes used as input features for water\-saturation prediction in the Western Field Area\. Each panel is displayed with its own color scale\.Seismic samples without corresponding water\-saturation labels were treated as unlabeled data\. These samples formedDUD\_\{U\}, from which the bounded locality\-aware candidate poolUt′U^\{\\prime\}\_\{t\}was initialized and progressively refilled during collaborative learning\. The trained model was subsequently applied to the complete seismic volume\. After preprocessing, the field experiment contained 1,248 labeled training samples, 625 validation samples, 393 blind\-test samples, and approximately8\.2×1078\.2\\times 10^\{7\}unlabeled seismic samples, illustrating the pronounced imbalance between the available well\-log labels and the large\-scale seismic observations\. For the field experiment, the base\-model pool comprised XGBoost, SVR, MLP, Ridge regression, and KNN regression\. The complete CoGameReg configuration is provided in Appendix B\.2 of the supplementary material\.

The field comparison includes XGBoost as a supervised baseline, COREG as a classical co\-training regression baseline, and UCVME as a modern semi\-supervised baseline\. UCVME is included because it achieved the lowest RMSE and highestR2R^\{2\}among the competing methods in the synthetic experiment\. All four methods use the same five input volumes and training/validation/blind\-test well partition\. XGBoost uses the labeled training samples for model fitting, whereas the semi\-supervised methods additionally use unlabeled seismic samples\. Their predictions are evaluated using the same valid\-sample and output post\-processing rules\. The COREG parameter settings used in the synthetic and field experiments are provided in Appendix B\.2\. The spatial predictions are compared on the same vertical section and reservoir horizon in Figures[7](https://arxiv.org/html/2610.12884#S4.F7)and[8](https://arxiv.org/html/2610.12884#S4.F8), respectively, to examine differences in the spatial distribution and continuity of the predicted water saturation\.

Because CoGameReg is designed as a general semi\-supervised regression framework rather than a water\-saturation\-specific model, no physical range constraint was imposed during model training\. In the field application, the predicted values were post\-processed according to the physical definition of water saturation\. Specifically, predictions from all four methods were clipped to the valid range of 0–100% before visualization and quantitative evaluation on the validation and blind\-test wells\. This post\-processing step only enforces the valid physical range of the target property and does not affect the learning procedure or pseudo\-label generation strategy of CoGameReg\.

Figure[6](https://arxiv.org/html/2610.12884#S4.F6)shows the anonymized spatial distribution of the 18 wells\. The training, validation, and blind\-test wells are distributed across different parts of the survey, allowing the field experiment to evaluate the spatial generalization of CoGameReg between different well locations\.

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/well_distribution_real.png)Figure 6:Spatial distribution and data partition of the 18 anonymized wells in the Western Field Area\. Well 1–Well 10 were used for training, Well 11–Well 15 for validation, and Well 16–Well 18 for blind\-well evaluation\. Different colors denote the three well subsets\.
#### 4\.3\.3Three\-dimensional prediction

The trained CoGameReg model was applied to the seismic volume to generate spatially distributed water\-saturation estimates\. Predictions from XGBoost, COREG, and UCVME provide references for examining the resulting spatial patterns\. The vertical and horizon\-based comparisons are presented below, whereas blind\-well predictive accuracy is evaluated separately in Section[4\.3\.4](https://arxiv.org/html/2610.12884#S4.SS3.SSS4)\.

Figure[7](https://arxiv.org/html/2610.12884#S4.F7)compares the predictions from XGBoost, COREG, UCVME, and CoGameReg along an arbitrary seismic line crossing nine wells\. All four methods exhibit broadly layered saturation patterns, with differences in the strength and continuity of local anomalies\. COREG contains more fragmented local patches in parts of the displayed interval, whereas UCVME and CoGameReg show more laterally continuous bands in those areas\. The projected well locations and reference water\-saturation tracks provide spatial context for inspecting the predictions near well control\. Because the section includes training, validation, and blind\-test wells, the displayed well tracks have different roles in model development and evaluation\.

Figure[8](https://arxiv.org/html/2610.12884#S4.F8)presents maps obtained by sampling the three\-dimensional water\-saturation volumes predicted by XGBoost, COREG, UCVME, and CoGameReg along the same reservoir horizon\. The overlaid well symbols represent fluid interpretations derived from well logs at the corresponding horizon, categorized as Gas, Poor Gas, and Water\. These interpretations are projected onto the maps at the well locations to provide geological reference information\. For example, the gas interpretation at Well 12 and the poor\-gas interpretation at Well 5 occur within relatively low predicted water\-saturation regions, whereas the water interpretations at Wells 3, 9, and 18 occur within a higher\-saturation background\.

All four methods reproduce the broad spatial contrast between relatively low and high saturation regions, while their local spatial patterns differ\. XGBoost and COREG show more small\-scale variations within and around the main upper anomaly near Wells 16, 7, 8, and 12\. UCVME and CoGameReg show smoother and more connected patterns in this region\. Similar differences are visible along the anomalous zone near Wells 13, 5, and 14\. The well interpretations provide geological context for these predicted distributions, while the comparison between methods highlights differences in their spatial continuity\. The red polyline marks the trajectory of the vertical section shown in Figure[7](https://arxiv.org/html/2610.12884#S4.F7), linking the horizon\-based distributions to the vertical prediction patterns\.

These comparisons describe differences in the continuity and morphology of the predicted saturation distributions\. They provide qualitative spatial context for the well\-based evaluation\. Because reference water saturation is unavailable away from the wells, greater visual continuity alone does not establish higher prediction accuracy throughout the seismic volume\.

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Traverse_XGB.png)

\(a\) XGBoost

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Traverse_COREG.png)

\(b\) COREG

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Traverse_UCVME.png)

\(c\) UCVME

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Traverse_ours.png)

\(d\) CoGameReg

Figure 7:Comparison of time\-domain water\-saturation sections predicted by \(a\) XGBoost, \(b\) COREG, \(c\) UCVME, and \(d\) CoGameReg along the same arbitrary seismic line crossing nine wells in the Western Field Area\. Projected well locations and reference water\-saturation logs are displayed along the sections\. The line includes training, validation, and blind\-test wells\.![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Horizon_XGB.png)

\(a\) XGBoost

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Horizon_COREG.png)

\(b\) COREG

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Horizion_UCVME.png)

\(c\) UCVME

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/Horizon_ours.png)

\(d\) CoGameReg

Figure 8:Horizon\-based water\-saturation maps extracted from the three\-dimensional volumes predicted by \(a\) XGBoost, \(b\) COREG, \(c\) UCVME, and \(d\) CoGameReg in the Western Field Area\. All four panels show the same reservoir horizon and use the same spatial extent and water\-saturation color scale\. The background colors represent predicted water saturation\. Colored well symbols denote Gas, Poor Gas, and Water interpretations derived from well logs at the corresponding horizon and projected onto the maps for reference\. The red polyline indicates the trajectory of the vertical section shown in Figure[7](https://arxiv.org/html/2610.12884#S4.F7)\.
#### 4\.3\.4Blind\-well evaluation

The blind\-well performance of XGBoost, COREG, UCVME, and CoGameReg was evaluated at the same three blind\-test wells, which were excluded from model training and hyperparameter selection\. The predicted water\-saturation curves were sampled at the corresponding well locations and compared with the ground truth water\-saturation logs after well\-to\-seismic calibration and temporal resampling\. Quantitative metrics were calculated over 2265–2409 ms, inclusive, with 1 ms sampling, using samples valid for the reference and all four methods\. For visualization, the curve comparisons in Figure[9](https://arxiv.org/html/2610.12884#S4.F9)and Appendix E show the interval of 2265–2375 ms\.

Following the physical post\-processing described in subsection[4\.3\.2](https://arxiv.org/html/2610.12884#S4.SS3.SSS2), the displayed predictions were bounded between 0% and 100% using the same predefined clipping rule for all wells\.

Figure[9](https://arxiv.org/html/2610.12884#S4.F9)compares the ground truth water\-saturation curves with predictions from XGBoost, COREG, UCVME, and CoGameReg at the three blind wells\. Overall, CoGameReg reproduces the principal vertical variations and identifies the major low\-water\-saturation intervals at previously unseen well locations\. The agreement nevertheless varies among wells, reflecting the strong lateral heterogeneity of the thin tight\-sandstone reservoir\. Additional comparisons between the ground truth and predicted water\-saturation curves at the training and validation wells are provided in Appendix E of the supplementary material\.

![Refer to caption](https://arxiv.org/html/2610.12884v1/figures/picture/Real_field/test_well_sw_curve_comparison.png)Figure 9:Comparison of ground\-truth water\-saturation curves with predictions from XGBoost, COREG, UCVME, and CoGameReg at the three blind\-test wells \(Wells 16–18\) in the Western Field Area\.For CoGameReg, Well 16 and Well 17 haveR2R^\{2\}values of 0\.5653 and 0\.6748 and Pearson correlation coefficients of 0\.8326 and 0\.8684, respectively, exceeding the corresponding values at Well 18\. These results indicate that the main vertical variations of water saturation are reproduced at these two independent well locations\. At Well 18, theR2R^\{2\}value is relatively low at 0\.2700, but its RMSE is only 3\.6080 percentage points, which is the lowest among the three blind wells\. This combination is consistent with the comparatively limited variation of the reference curve at Well 18, becauseR2R^\{2\}becomes sensitive to relatively small prediction deviations when the target variance is low\. When all 393 valid blind\-well samples are pooled, CoGameReg obtains anR2R^\{2\}of 0\.5942, an RMSE of 5\.7441 percentage points, an MAE of 2\.1742 percentage points, and a Pearson correlation coefficient of 0\.8265\. The overall bias, defined as the mean difference between the prediction and reference, is 0\.7693 percentage points\.

For CoGameReg, macro\-average statistics were also calculated by assigning equal weight to each of the three blind wells\. The macro\-averageR2R^\{2\}is0\.5034±0\.20940\.5034\\pm 0\.2094, and the macro\-average RMSE is5\.5573±1\.80275\.5573\\pm 1\.8027percentage points\. These values are reported as the mean±\\pmsample standard deviation across wells, describing the variation in prediction performance among the three blind\-well locations\.

Table[7](https://arxiv.org/html/2610.12884#S4.T7)compares the four methods using common valid samples from the validation and blind\-test wells\. On the validation wells, CoGameReg achieves the lowest MAE \(1\.9746 percentage points\) and highest Pearson correlation \(0\.7807\), while UCVME obtains a slightly higherR2R^\{2\}\(0\.5416 versus 0\.5407\) and lower RMSE \(4\.4086 versus 4\.4128 percentage points\)\. On the blind\-test wells, CoGameReg has the highest Pearson correlation \(0\.8265\), whereas COREG achieves the highest pooledR2R^\{2\}\(0\.6148\) and lowest RMSE \(5\.5969 percentage points\), and XGBoost has the lowest MAE \(2\.1533 percentage points\)\. These results show that no single method performs best across all reported metrics on both subsets\.

Taken together, the blind\-well results demonstrate that CoGameReg provides acceptable cross\-well generalization for the real field data and reproduces the major vertical water\-saturation trends\. However, the variable performance among the blind wells indicates that the field results should be interpreted as data\-driven reservoir\-property estimates\.

Table 7:Pooled water\-saturation prediction results for four representative methods in the field experiment\.Notes:NNdenotes the number of common valid samples\. Bias is defined asmean⁡\(S^w−Sw\)\\operatorname\{mean\}\(\\widehat\{S\}\_\{w\}\-S\_\{w\}\)\. Pooled metrics are computed after concatenating the valid samples across wells\. Bold indicates the bestR2R^\{2\}, RMSE, MAE, or Pearsonrramong the four displayed methods within each subset; bias is not ranked\.

#### 4\.3\.5Geological interpretation and limitations

The field application addresses the prediction of water saturation in a thin, laterally variable tight\-sandstone reservoir\. The five input volumes provide seismic and inversion\-derived information with different sensitivities to elastic properties\. In particular, the two Vp/Vs volumes estimate the same physical quantity through different inversion procedures and may contain both complementary information and correlated errors\. The Shapley\-derived weights allocate influence among regressors; they do not by themselves remove attribute redundancy or resolve the nonuniqueness of estimating water saturation from seismic information\. Cross\-model agreement can therefore provide a useful fusion criterion while remaining susceptible to errors shared by the participating models\.

The locality used for candidate selection is defined in seismic\-attribute space\. Similar attribute vectors can occur at spatially separated locations or in different lithologic and fluid settings\. Consequently, a reduction in local proxy error does not by itself establish compatibility across geological facies\. The fixed initial weights summarize regressor behavior at the labeled wells and do not provide location\-dependent reliability estimates\. Increasing distance from well control may coincide with changes in facies, rock properties, or seismic response, but the present blind\-well comparison does not establish a quantitative relationship between prediction error and physical distance\. The reported results should therefore be interpreted within the geological and attribute conditions represented by the available well control\.

Finite seismic bandwidth also limits the interpretation of predicted saturation detail\. The target interval contains sandstone bodies averaging approximately 11 m in thickness and gas\-bearing intervals of 2–8 m, while the dominant frequency near the reservoir is approximately 35–45 Hz\. Thin\-bed interference and the difference in scale between well logs and seismic observations can contribute to smoothed boundaries and underestimated saturation contrasts\. The 1 ms sampling used in the well comparison specifies the discretization interval, not the vertical resolving power of the seismic data\. Fine\-scale variations in the predicted saturation should therefore be interpreted with reference to the bandwidth, well\-to\-seismic calibration, and assumptions underlying the input volumes\.

The maps and sections provide spatial context for the blind\-well results\. Because seismic and inversion volumes are themselves model inputs, agreement with their patterns is a consistency check rather than independent validation\. Moreover, greater continuity can accompany smoothing of genuine geological contrasts\. Reference water saturation is unavailable away from the well trajectories, so the present results do not quantify the accuracy of the complete prediction volume\. CoGameReg provides a conditional reservoir\-property estimate whose quantitative support comes from the blind\-well comparison; its spatial interpretation should be checked against independent geological evidence\.

## 5Conclusion

This study developed CoGameReg for seismic reservoir property prediction under sparse well control\. Its central contribution is the use of Shapley\-derived model weights, estimated once from the labeled training data and then held fixed, in both shared pseudo\-label construction and final prediction fusion\. These weights determine how heterogeneous regressors contribute to the additional learning targets generated from unlabeled seismic attributes\. Local error\-reduction screening complements this model\-level weighting by controlling candidate admission\. This combination provides a practical way to use sparse well supervision together with dense seismic observations\. The public benchmark results show that CoGameReg achieves its largest performance advantage at the lowest labeled\-data ratio, with the gap relative to competing methods narrowing as the proportion of labeled data increases\. The benchmark ablations support placing contribution\-aware weighting within the pseudo\-labeling loop\. CoGameReg achieves the best blind\-well metrics in the synthetic experiment\. In the field application, the clipped predictions reproduce major water\-saturation trends at the three blind wells and yield the highest pooled Pearson correlation among the four compared methods\. The field results therefore support applicability under the evaluated conditions, with performance depending on the metric and well location\.

## 6Acknowledgements

This work was supported by CNPC Science and Technology Major Project \(No\. SG0023Z0010050100111\) and CNPC Youth Science and Technology Special Project ”Research on Rock Physics and Facies\-Driven Intelligent Seismic Reservoir Characterization Methods” \(No\. 2025DQ03007\)\.

## 7Data and Materials Availability

The source code for CoGameReg is available at[https://github\.com/ACE020215/CoGameReg](https://github.com/ACE020215/CoGameReg)\. The public benchmark data sets and the Stanford VI\-E synthetic reservoir data are available from the cited sources\. The proprietary field seismic and well\-log data are confidential and cannot be released\.

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