Real geo-data are three-dimensional (3D) and spatially varied, but measurements are often sparse due to time, resource, and/or technical constraints. In these cases, the quantities of interest at locations where measurements are missing must be interpolated from the available data. Several powerful methods have been developed to address this problem in real-world applications over the past several decades, such as two-point geo-statistical methods (e.g., kriging or Gaussian process regression, GPR) and multiple-point statistics (MPS). However, spatial interpolation remains challenging when the number of measurements is small because a suitable covariance function is difficult to select and the parameters are challenging to estimate from a small number of measurements. Note that a covariance function form and its parameters are key inputs for some methods (e.g., kriging or GPR). MPS is a non-parametric simulation method that combines training images as prior knowledge for sparse measurements. However, the selection of a suitable training image for continuous geo-quantities (e.g., soil or rock properties) faces certain difficulties and may become increasingly complicated when the geo-data to be interpolated are high-dimensional (e.g., 3D) and exhibit non-stationary (e.g., with unknown trends or non-stationary covariance structure) and/or anisotropic characteristics. This paper proposes a non-parametric approach that systematically combines compressive sensing and variational Bayesian inference for statistical interpolation of 3D geo-data. The method uses sparse measurements and their locations as the input and provides interpolated values at unsampled locations with quantified interpolation uncertainty as the output. The proposed method is illustrated using a series of numerical 3D examples, and the results indicate a reasonably good performance.

真实的地理数据是三维（3D）且在空间上变化，但是由于时间，资源和/或技术限制，测量常常稀疏。在这些情况下，必须从可用数据中内插缺少测量值的位置上的关注数量。在过去的几十年中，已经开发出了几种强大的方法来解决实际应用中的此问题，例如两点地统计方法（例如，克里格法或高斯过程回归，GPR）和多点统计（MPS）。但是，当测量次数较少时，空间插值仍然具有挑战性，因为很难选择合适的协方差函数，并且很难从少量测量中估计参数。请注意，协方差函数形式及其参数是某些方法（例如克里金法或GPR）的关键输入。MPS是一种非参数模拟方法，将训练图像作为稀疏测量的先验知识进行组合。但是，为连续的地理量（例如，土壤或岩石性质）选择合适的训练图像面临某些困难，并且当要插值的地理数据是高维（例如3D）且显示不出时，可能会变得越来越复杂。 -平稳（例如，具有未知趋势或非平稳协方差结构）和/或各向异性特征。本文提出了一种非参数方法，该方法将压缩感测和变分贝叶斯推理系统地结合起来，用于3D地理数据的统计插值。该方法使用稀疏测量及其位置作为输入，并提供未采样位置的插值和量化插值不确定性作为输出。使用一系列数值3D实例对提出的方法进行了说明，结果表明性能相当不错。