HV-LSC-ex$$^2$$: velocity field interpolation using extended least-squares collocation

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)$$^2$$ 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$$^2$$ 2 technique.

Variable vertical land motion for sea level rise projections

“…We develop a GNSS displacement rate model as a calibration plane to accommodate for this spatially variable alignment between GNSS and InSAR. We interpolate GNSS rates on a 25 by 25 km grid, with additional points along San Andreas Fault System, using an extended Least-Square Collocation (Hv-LSC-ex 2 [57]) method that takes into account spatial correlation between horizontal displacement components with plate boundaries constraints. Prior to this, we first omit the sites with apparent localized motion based on 2-sigma criteria applied on trend variability metric (dv N orth−South > 0.30 mm/yr, dv East−W est > 0.28 mm.yr, dv V LM > 1 mm/yr), combined horizontal and vertical trend uncertainties (σ 3D > 2 mm/yr) and remaining vertical rates (VLM > 3 mm/yr).…”

“…We observe through trial and error that the remaining trend signal only affects interpolation uncertainties and not its values. Therefore, to obtain more realistic interpolation uncertainties, we use correlation function, starting covariances with mean rate uncertainties estimated prior with a Hector software [54] and moving variance approach within radius of 850 km to adapt to the non-stationary residual horizontal field [57]. The final stochastic parameters for the collocation process are: horizontal c 0 = 0.2mm 2 /yr 2 , d 0 = 225km, vertical c 0 = 0.4mm 2 /yr 2 , d 0 = 52km.…”

“…5). Here, we use all "stable" GNSS sites preselected for the GNSS model to obtain interpolated horizontal motion on the InSAR grid with Hv-LSC-ex 2 [57] method. In the case of only one viewing geometry, we constrain horizontal motion using East-West and North-South GNSS displacements.…”

Determinação do Tensor de Deformação da Crosta em Regiões de Grandes Barragens Utilizando uma Função Covariância Estimada por meio da Colocação por Mínimos Quadrados

“…A MVC do vetor aleatório z, dada pela equação (3.27), é obtida pela soma da MVC do sinal s com a MVC do ruído n, a qual é igual a MVC das observações Lb e dada por (CAMARGO e DALMOLIN, 1995, p. 7;STEFFEN et al, 2022):…”