Axially symmetric processes on spheres, for which the second-order dependency structure may substantially vary with shifts in latitude, are a prominent alternative to model the spatial uncertainty of natural variables located over large portions of the Earth. In this paper, we focus on Karhunen–Loève expansions of axially symmetric Gaussian processes. First, we investigate a parametric family of Karhunen–Loève coefficients that allows for versatile spatial covariance functions. The isotropy as well as the longitudinal independence can be obtained as limit cases of our proposal. Second, we introduce a strategy to render any longitudinally reversible process irreversible, which means that its covariance function could admit certain types of asymmetries along longitudes. Then, finitely truncated Karhunen–Loève expansions are used to approximate axially symmetric processes. For such approximations, bounds for the \(L^2\)-error are provided. Numerical experiments are conducted to illustrate our findings.

球面上的轴对称过程的二阶相依结构可能会随着纬度的变化而显着变化，这是对位于地球大部分区域的自然变量的空间不确定性进行建模的一个重要选择。在本文中，我们将重点放在轴向对称高斯过程的Karhunen-Loève展开上。首先，我们研究Karhunen-Loève系数的参数族，该参数族可实现通用的空间协方差函数。各向同性以及纵向独立性可以作为我们建议的极限情况。其次，我们引入一种使任何纵向可逆过程不可逆的策略，这意味着其协方差函数可以沿经度允许某些类型的不对称性。然后，有限截断的Karhunen-Loève展开用于近似轴向对称过程。对于这样的近似值，\（L ^ 2 \） -错误 进行数值实验以说明我们的发现。