A SoftMax-Driven Ensemble Framework for Compressional Wave Velocity Prediction

Compressional wave velocity (Vp) is a crucial parameter for reservoir characterization due to its extensive application in lithology distinguishing, porosity assessment, fluid identification, and fracture analysis. However, obtaining Vp directly from sonic logs is often not feasible because sonic tools are frequently unavailable, especially in older wells. This study proposes an ensemble method that combines Random Forest (RF) and Gradient Boosting (GB) using SoftMax-weighted linear combiner (LC). Unlike traditional ensemble techniques, which usually employ fixed weights for building ensemble models, the proposed approach combines the SoftMax function with the proposed model to determine the optimal weight combination, thereby enhancing the accuracy of the ensemble model. A comprehensive set of wireline logs was collected from the Nubian sandstone reservoir in the Sirt Basin, Libya. The data underwent a series of preprocessing steps designed to enhance its quality. The outcomes of the proposed ensemble model demonstrate remarkable effectiveness, characterized by a high coefficient of determination (R² = 0.97) and a relatively low Mean Squared Error (MSE = 18.5). Furthermore, the trained ensemble is further analyzed in the model interpretation and reliability stage using Uncertainty Quantification (UQ) methods, Shapley Additive Explanations (SHAP), Partial Dependence Plots (PDP), and Individual Conditional Expectation (ICE) methods. The SHAP results indicate that Neutron Porosity (NPHI) and Gamma Ray (GR) are the most significant influences on Vp, while PDP and ICE reveal a direct relationship between NPHI and Sonic log (DT), consistent with lithological heterogeneity and variations in pore structure within the Nubian reservoir. Additionally, UQ and k-fold cross-validation demonstrate the model’s reliability and generalization, with low prediction variance and high generalization.

The graphical abstract highlights the comprehensive workflow established in the present study for predicting compressional wave velocity (Vp) within the Nubian Sandstone Formation of the Sirt Basin, Libya. The first step involves acquiring well log data, which comprises gamma ray (GR), neutron porosity (NPHI), induction log (ILM), and micro-spherical focused log (MSFL). The data undergo extensive preprocessing—comprising cleaning, imputation, outlier removal, normalization, and principal component analysis (PCA)—to ensure quality and consistency. Subsequently, two ensemble-based learners, Random Forest (RF) and Gradient Boosting (GB), are trained and integrated through a SoftMax-weighted linear combiner to optimize the contribution of each model. The trained ensemble is further analyzed in the “Model Interpretation & Reliability Analysis” stage, where Shapley Additive explanations (SHAP), Partial Dependence Plots (PDP), and Individual Conditional Expectation (ICE) analyses are employed to interpret model behavior and feature influence. Ensemble Uncertainty Quantification (UQ) and 10-fold cross-validation are applied to assess the model’s reliability and generalization. The final evaluation demonstrates superior predictive performance (R² = 0.97, MSE = 18.5) and reduced overfitting, confirming the reliability of the proposed SoftMax-weighted ensemble framework for accurate Vp estimation in heterogeneous sandstone reservoirs