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<p>Accurate reconstruction of the initial in-situ stress field of rock masses is paramount for the stability analysis of deep underground engineering, particularly in high geo-stress environments. However, the inherent correlation between gravitational and tectonic stresses frequently induces severe multicollinearity. This causes the standard Ordinary Least Squares (OLS) inversion to yield distorted solutions. These solutions are characterized by physically unreasonable magnitudes of regression coefficients for tectonic stress fields, physically inconsistent negative coefficients for compressive stress fields, and statistically insignificant regression coefficients for key tectonic stress fields. To address the distorted solutions, this study proposes a new inversion strategy based on iterative Generalized Ridge Regression (GRR). Unlike Standard Ridge Regression (SRR), which applies uniform isotropic shrinkage, the proposed GRR method employs anisotropic regularization. By assigning individual regularization parameters to distinct canonical coefficients of gravitational and tectonic stresses, the method adaptively suppresses noise-dominated components amplified by multicollinearity while preserving valid geomechanical signals. In addition, an optimized Hoerl-Kennard iterative procedure is established, incorporating a multi-criteria termination protocol. Application to the Banzhulin Tunnel demonstrates that this protocol identifies an optimal iteration step that simultaneously satisfies statistical criteria (e.g., <i>p</i>-values ≤ 0.05), ensures essential physical constraints (e.g., positive coefficients for compressive stress fields), and avoids the over-regularization associated with full numerical convergence. Comparative analysis reveals that the GRR framework reduces the Residual Mean Squared Error (Residual MSE) by 14.4% relative to SRR and mitigates predictive drift in deep extrapolation zones. This research provides a statistically reliable and physically consistent methodology for resolving ill-posed inverse problems arising from multicollinearity among gravitational and tectonic stresses.</p>

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Back-Analysis of the Initial In-Situ Stress Field Using Iterative Generalized Ridge Regression (GRR) with Anisotropic Regularization: A Multi-Criteria Optimization Approach

  • 期刊论文

  • 发表日期：

    2026年8月3日

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  • Wei Meng,
  • Dong Chen,
  • Zihan Zhou,
  • Ziquan Chen,
  • Guowen Xu,
  • Chuan He

摘要

Accurate reconstruction of the initial in-situ stress field of rock masses is paramount for the stability analysis of deep underground engineering, particularly in high geo-stress environments. However, the inherent correlation between gravitational and tectonic stresses frequently induces severe multicollinearity. This causes the standard Ordinary Least Squares (OLS) inversion to yield distorted solutions. These solutions are characterized by physically unreasonable magnitudes of regression coefficients for tectonic stress fields, physically inconsistent negative coefficients for compressive stress fields, and statistically insignificant regression coefficients for key tectonic stress fields. To address the distorted solutions, this study proposes a new inversion strategy based on iterative Generalized Ridge Regression (GRR). Unlike Standard Ridge Regression (SRR), which applies uniform isotropic shrinkage, the proposed GRR method employs anisotropic regularization. By assigning individual regularization parameters to distinct canonical coefficients of gravitational and tectonic stresses, the method adaptively suppresses noise-dominated components amplified by multicollinearity while preserving valid geomechanical signals. In addition, an optimized Hoerl-Kennard iterative procedure is established, incorporating a multi-criteria termination protocol. Application to the Banzhulin Tunnel demonstrates that this protocol identifies an optimal iteration step that simultaneously satisfies statistical criteria (e.g., p-values ≤ 0.05), ensures essential physical constraints (e.g., positive coefficients for compressive stress fields), and avoids the over-regularization associated with full numerical convergence. Comparative analysis reveals that the GRR framework reduces the Residual Mean Squared Error (Residual MSE) by 14.4% relative to SRR and mitigates predictive drift in deep extrapolation zones. This research provides a statistically reliable and physically consistent methodology for resolving ill-posed inverse problems arising from multicollinearity among gravitational and tectonic stresses.