Variance reduction function for a spatially variable soil property is an important factor that affects the spatial average-based characteristic value, which in turn influences the design of a geotechnical structure in the context of Eurocode 7. This study derives the theoretical and approximate variance reduction functions (VRFs) for a potential inclined slip line in a spatially variable soil. Only stationary random fields are studied but more general spatial variability characteristics including smoothness and hole effect are considered. First, the closed-form one-dimensional (1D) VRFs are investigated, including the VRFs for classical one-parameter autocorrelation models, the non-classical two-parameter Whittle-Matérn (WM) model, and the most general three-parameter cosine Whittle-Matérn (CosWM) model proposed to date. It is found that closed-form solutions are not available or not practical to compute in general and the simple approximate VRF (equal to scale of fluctuation/averaging length) is not adequate for the non-classical autocorrelation models (WM and CosWM). Two approximate VRFs are developed in this study for the 1D WM and 1D CosWM models. For a spatial average over an inclined line, this paper derives the theoretical scales of fluctuation (SOFs) and VRFs for five commonly used 2D autocorrelation models. The theoretical solutions show that the equivalent SOFs proposed in the literature are only applicable under special conditions which are clarified in this paper. More general approximations for the VRF over an inclined line are proposed. The range of applicability for each approximation is stated. The proposed approximate VRFs are shown to be reasonably accurate when they are applied to the spatial average-based characteristic value and for the design of a vertical pile and an inclined soil nail.

空间可变土壤属性的方差减少函数是影响基于空间平均值的特征值的重要因素，进而影响欧洲规范 7 背景下的岩土结构设计。本研究推导了理论和近似方差减少函数（VRF）用于空间变化土壤中潜在的倾斜滑移线。仅研究平稳随机场，但考虑更一般的空间变异特征，包括平滑度和孔效应。首先，研究了闭式一维（1D）VRF，包括经典一参数自相关模型的VRF、非经典两参数Whittle-Matérn（WM）模型和最通用的三参数余弦模型迄今为止提出的 Whittle-Matérn (CosWM) 模型。研究发现，封闭式解一般不可用或不实用，并且简单的近似 VRF（等于波动范围/平均长度）对于非经典自相关模型（WM 和 CosWM）来说是不够的。本研究为 1D WM 和 1D CosWM 模型开发了两个近似 VRF。对于斜线上的空间平均值，本文推导了五种常用二维自相关模型的理论波动尺度 (SOF) 和 VRF。理论解表明，文献中提出的等效SOF仅适用于本文阐明的特殊条件。提出了斜线上 VRF 的更一般的近似值。规定了每个近似值的适用范围。当将所提出的近似 VRF 应用于基于空间平均的特征值以及垂直桩和倾斜土钉的设计时，它们被证明是相当准确的。