Multidimensional least squares collocation with separable covariance functions

Abstract In geodetic applications, stochastic processes are often used in the context of least squares collocation for prediction and filtering or in the context of least squares estimation for efficient decorrelation of signals. The biggest advantage of collocation method is that the covariance function, which serves as basis function, can be derived directly from the characteristics of the signals, meaning it is learned directly from the data. However, the major disadvantage is that memory requirements increase quadratically and the numerical complexity increases with sixth order of the number of data points. Especially with multidimensional data sets, large amounts of data are usually expected. To reduce numerical complexity, various strategies are employed, e.g., utilizing finite covariance functions and filter techniques for gridded data. But when processing spatio-temporal models, data points are often irregularly distributed in space, whereas repeated measurements are taken at fixed epochs. Specifically for such kind of data sets, an efficient approach is proposed. Key idea is the utilization of separable covariance functions, where the covariance matrices are represented as a Kronecker product. This enables a successive solution in space and time and reduces the numerical complexity drastically. This strategy is used to evaluate a spatio-temporal point stack of surface displacements for the period 1992 to 2000, derived from a DInSAR-SBAS analysis of the ERS1 and ERS2 missions in the Lower-Rhine Bay in North Rhine-Westphalia by least squares collocation.