The Matérn family of isotropic covariance functions has been central to the theoretical development and application of statistical models for geospatial data. For global data defined over the whole sphere representing planet Earth, the natural distance between any two locations is the great circle distance. In this setting, the Matérn family of covariance functions has a restriction on the smoothness parameter, making it an unappealing choice to model smooth data. Finding a suitable analogue for modelling data on the sphere is still an open problem. This paper proposes a new family of isotropic covariance functions for random fields defined over the sphere. The proposed family has a parameter that indexes the mean square differentiability of the corresponding Gaussian field, and allows for any admissible range of fractal dimension. Our simulation study mimics the fixed domain asymptotic setting, which is the most natural regime for sampling on a closed and bounded set. As expected, our results support the analogous results (under the same asymptotic scheme) for planar processes that not all parameters can be estimated consistently. We apply the proposed model to a dataset of precipitable water content over a large portion of the Earth, and show that the model gives more precise predictions of the underlying process at unsampled locations than does the Matérn model using chordal distances.

Matérn各向同性协方差函数家族一直是地理空间数据统计模型的理论发展和应用的中心。对于在代表地球的整个球体上定义的全局数据，任意两个位置之间的自然距离就是大圆距。在这种设置下，Matérn协方差函数系列对平滑度参数有所限制，这使其成为对平滑数据建模的诱人选择。寻找合适的模拟物来模拟球体上的数据仍然是一个悬而未决的问题。本文针对球体上定义的随机场提出了新的各向同性协方差函数族。所提出的族具有一个参数，该参数索引对应的高斯场的均方差，并允许分形维数的任何可接受范围。我们的模拟研究模仿了固定域渐近设置，这是在封闭和有界集合上进行采样的最自然的方式。不出所料，我们的结果支持了并非所有参数都能一致估计的平面过程的相似结果（在同一渐近方案下）。我们将提出的模型应用于地球大部分区域的可降水量的数据集，并表明与使用弦距离的Matérn模型相比，该模型对未采样位置的潜在过程提供了更精确的预测。