Modeling random isotropic vector fields on the sphere: theory and application to the noise in GNSS station position time series

Abstract

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).