Spatio-Temporal Covariance and Cross-Covariance Functions of the Great Circle Distance on a Sphere

In this article, we propose stationary covariance functions for processes that evolve temporally over a sphere, as well as cross-covariance functions for multivariate random fields defined over a sphere. For such processes, the great circle distance is the natural metric that should be used to describe spatial dependence. Given the mathematical difficulties for the construction of covariance functions for processes defined over spheres cross time, approximations of the state of nature have been proposed in the literature by using the Euclidean (based on map projections) and the chordal distances. We present several methods of construction based on the great circle distance and provide closed-form expressions for both spatio-temporal and multivariate cases. A simulation study assesses the discrepancy between the great circle distance, chordal distance, and Euclidean distance based on a map projection both in terms of estimation and prediction in a space-time and a bivariate spatial setting, where the space is in this case the Earth. We revisit the analysis of Total Ozone Mapping Spectrometer (TOMS) data and investigate differences in terms of estimation and prediction between the aforementioned distance-based approaches. Both simulation and real data highlight sensible differences in terms of estimation of the spatial scale parameter. As far as prediction is concerned, the differences can be appreciated only when the interpoint distances are large, as demonstrated by an illustrative example. Supplementary materials for this article are available online.

本文针对球面随时间演化的随机过程，提出了平稳协方差函数；同时针对定义于球面的多元随机场，提出了互协方差函数。对于此类过程，大圆距离是描述空间相关性的天然度量。鉴于构造跨时间球面过程的协方差函数存在数学难点，现有文献已提出两类近似方法：采用基于地图投影的欧氏距离与弦距。

本文给出若干基于大圆距离的构造方案，并推导得到时空与多元场景下的闭式表达式。通过模拟实验，我们评估了基于地图投影的欧氏距离、弦距与大圆距离，在以地球为空间维度的时空与二元空间设定中，于估计与预测任务上的表现差异。随后我们重新分析了总臭氧测绘光谱计（Total Ozone Mapping Spectrometer, TOMS）数据集，探究了上述基于距离的方法在估计与预测环节的差异。

模拟实验与真实数据均显示，空间尺度参数的估计结果存在显著差异。就预测任务而言，仅当点间距离较大时，不同方法的预测差异才会显现，该结论可通过示例得到验证。本文的补充材料可在线获取。