While the theory of random isotropic scalar fields on the sphere is well established, it has not been fully extended to the case of vector fields yet. In this contribution, several theoretical results are thus generalized to random isotropic vector fields on the sphere, including an equivalent of the Wiener–Khinchin theorem, which relates the distance-dependent covariance of the field’s components in a particular rotationally invariant basis to the covariance of the vector spherical harmonic coefficients of the field, i.e., its angular power spectrum. A parametric model, based on a stochastic partial differential equation, is proposed to represent the spatial covariance and angular power spectrum of such fields. Such a model is adjusted, with minor modifications, to empirical spatial correlations of the white noise and flicker noise components of 3D displacement time series of ground global navigation satellite system (GNSS) tracking stations. The obtained spatial correlation model may find several applications such as enhanced detection of offsets in GNSS station position time series, enhanced estimation of long-term ground deformation (i.e., station velocities), enhanced isolation of station-specific displacements (i.e., spatial filtering) and more realistic assessment of uncertainties in all GNSS network-based applications (e.g., estimation of crustal strain rates, of glacial isostatic adjustment models or of tectonic plate motion models).

虽然球体上随机各向同性标量场的理论已经很成熟，但尚未完全扩展到矢量场的情况。在这篇文章中，一些理论结果被推广到球体上的随机各向同性矢量场，包括维纳-辛钦定理的等价物，该定理将特定旋转不变基础上场分量的距离相关协方差与场的矢量球谐系数，即其角功率谱。提出了一种基于随机偏微分方程的参数模型来表示此类场的空间协方差和角功率谱。这种模型经过微小的修改，根据地面全球导航卫星系统 (GNSS) 跟踪站 3D 位移时间序列的白噪声和闪烁噪声分量的经验空间相关性进行调整。所获得的空间相关模型可以找到多种应用，例如增强 GNSS 站位置时间序列中的偏移检测、增强长期地面变形（即站速度）的估计、增强站特定位移的隔离（即空间滤波）对所有基于 GNSS 网络的应用中的不确定性进行更现实的评估（例如地壳应变率、冰川均衡调整模型或构造板块运动模型的估计）。