Abstract
Least-squares collocation (LSC) is a widely used method applied in physical geodesy to separate observations into a signal and noise part but has received only little attention when interpolating velocity fields. The advantage of the LSC is the possibility to filter and interpolate as well as extrapolate the observations. Here, we will present several extensions to the traditional LSC technique, which allows the combined interpolation of both horizontal velocity components (horizontal velocity (HV)-LSC), the separation of velocity observations on different tectonic plates, and the removal of stationarity by moving variance (the latter as HV-LSC-ex(tended)^2). Furthermore, the covariance analysis, which is required to find necessary input parameters for the LSC, is extended by finding a suitable variance and correlation length using both horizontal velocity components at the same time. The traditional LSC and all extensions are tested on a synthetic dataset to find the signal at known as well as newly defined points, with stations separated on four different plates with distinct plate velocities. The methodologies are evaluated by calculation of a misfit to the input data, and implementation of a leave-one-out cross-validation and a Jackknife resampling. The largest improvement in terms of reduced misfit and stability of the interpolation can be obtained when plate boundaries are considered. In addition, any small-scale changes can be filtered out using the moving-variance approach and a smoother velocity field is obtained. In comparison with interpolation using the Kriging method, the fit is better using the new HV-LSC-ex^2 technique.
Highlights
Least-squares collocation (LSC) has become a wellknown and widely used stochastic interpolation technique, which is similar to Kriging (e.g. Dermanis 1984; Reguzzoni et al 2005; Fuhrmann 2016)
The method of the standard LSC is a stochastic interpolation technique, and their general framework was presented by Moritz (1980), which is used in the following to summarize the technique
We presented an extension of the standard LSC technique by allowing the combined interpolation of the horizontal velocity field where the correlation between the components is considered
Summary
Collection of ground data in an area of interest is limited by the accessibility of the area Several interpolation techniques are available (e.g. bilinear interpolation, nearest-neighbourhood, spline interpolation, inverse-distance weighted, radial basis function, Kriging) to estimate a continuous surface from a set of observations. Most of these techniques have in common that the interpolated value is only directly based on the surround-. The aim of this paper is the presentation of the new LSC methodology modified for the special purpose to filter and interpolate horizontal velocities including correlation between the velocity field components (horizontal velocity (HV)-LSC). This correlation has not been considered in velocity field interpolations before.
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