A mathematical model for small-scale spatial variations in gravity above the Earth’s surface is presented. Gravity variations are treated as a Gaussian random process arising from underground density variations which are assumed to be a Gaussian random process. Expressions for two-point spatial statistics are calculated for both the vertical component of gravity and the vertical gradient of the vertical component. Results are given for two models of density variations: a delta-correlated model and a fractal model. The effect of an outer scale in the fractal model is investigated. It is shown how the results can be used to numerically generate realisations of gravity variations with fractal properties. Such numerical modelling could be useful for investigating the feasibility of using gravity surveys to locate and characterise underground structures; this is explored through the simple example of a tunnel detection scenario.

中文翻译：

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模拟随机地下密度变化的引力效应

提出了一种用于地球表面上方重力小范围空间变化的数学模型。重力变化被视为由地下密度变化引起的高斯随机过程，而地下密度变化被认为是高斯随机过程。针对重力的垂直分量和垂直分量的垂直梯度都计算出两点空间统计的表达式。给出了两种密度变化模型的结果：δ相关模型和分形模型。研究了外部尺度对分形模型的影响。它显示了如何将结果用于以数字方式生成具有分形特性的重力变化的实现。这样的数值模型可能有助于研究使用重力测量来定位和表征地下结构的可行性。通过隧道检测方案的简单示例对此进行了探讨。