About

The ULR20 GNSS solution is the result of the reanalysis of 26 years of GNSS data from 2000 to 2025 that has been undertaken within the framework of the SPOTGINS collaborative project. Its associated vertical velocity field is expressed in IGS2020 reference frame.

The daily PPP position time series have been produced using the GINS software (CNES) and precise GPS & Galileo orbit/clock products of the CNES-CLS IGS Analysis Center.

Position time series expressed in IGS2020 were then analysed using PyTRF python library (Rebischung, IGN/IPGP). After the substraction of non-tidal atmospheric and oceanic loading displacements in the time series (provided by the EOST Loading Service), both a functional and a stochastic model were adjusted including long-term linear trends, position offset discontinuities, and periodic signals, following the equation:

$$ \begin{align} x(t) = & x_{ref}+ v_{x}(t-t_{ref}) & \textit{reference position and velocity}\\ & + \sum_{i=1}^{N_{O}} a_{i}H(t-t_{i}) & \textit{position offsets}\\ & + \sum_{j=1}^3 s_{j}\sin(\frac{2\pi}{\tau_{j}}t)) + c_{j}\cos(\frac{2\pi}{\tau_{j}}t)) & \textit{seasonnal signals} \\ & + \sum_{d=1}^8 s_{d}\sin(\frac{2\pi}{\tau_{d}}t)) + c_{d}\cos(\frac{2\pi}{\tau_{d}}t)) & \textit{draconitic signals}\\ & + \sum_{f=1}^3 s_{f}\sin(\frac{2\pi}{\tau_{f}}t)) + c_{f}\cos(\frac{2\pi}{\tau_{f}}t)) & \textit{fortnightly signals} \\ & + \sum_{k=1}^{N_{PSD}}PSD_{k}(t) & \textit{post-seismic deformation signals} \end{align} $$

$$ \begin{align} x_{ref} & \text{ is the position at the reference epoch} t_{ref} \\ v_{x} & \text{ is the linear velocity} \\ H(t-t_{i}) = & \begin{cases} 0 & \text{if } t \lt t_{i}\\ \\ 1 & \text{if } t \geq t_{i} \end{cases} \\ \tau_{j} = & \frac{1}{j} \text{ years} \\ \tau_{d} = & \frac{P_{D}}{365.25} \text{ years}, P_{D} \text{ being the period in days of the draconitics} \\ \tau_{f} = & \frac{P_{F}}{365.25} \text{ years}, P_{F} \text{ being the period in days of the fortnightly signals}\\ PSD_{k}(t) = & \begin{cases} a_{k} \log(1+ \frac{t-t_{k}}{\tau_{k}}) \text{ if PSD model is log} \\ \\ a_{k}(1- \exp(-\frac{t-t_{k}}{\tau_{k}})) \text{ if PSD model is exp} \\ \\ a_{1k} \log(1+ \frac{t-t_{k}}{\tau_{1k}}) + a_{2k}(1- \exp(-\frac{t-t_{k}}{\tau_{2k}})) \text{ if PSD model is log+exp} \\ \\ a_{1k} \log(1+ \frac{t-t_{k}}{\tau_{1k}}) + a_{2k} \log(1+ \frac{t-t_{k}}{\tau_{2k}}) \text{ if PSD model is log+log} \\ \\ a_{1k}(1- \exp(-\frac{t-t_{k}}{\tau_{1k}})) + a_{2k}(1- \exp(-\frac{t-t_{k}}{\tau_{2k}})) \text{ if PSD model is exp+exp} \end{cases} \end{align} $$