# Analysis of GNSS Displacements in Europe and Their Comparison with Hydrological Loading Models
## 3 Results
### Frequency Content and Interannual Variations  
The frequency content of the detrended and 10-day-resampled loading models, along with the GNSS MG3 solution, is provided in Figure 11. The mean Lomb-Scargle [71; 72] periodograms showing the amount of variance per frequency band were computed with Python using the algorithm described in [73]. Before stacking, the individual periodograms were un-normalized from the length of the time series in order to compare all the periodograms in a consistent manner. Moreover, the frequency range we used was the same for every time series (GNSS and models). The GNSS has a higher noise level than the loading models (the GNSS periodogram shows a larger amount of variance than the loading models), which is quite understandable if we consider the multiple sources of noise for the GNSS techniques that were listed in the Introduction section of this article. The estimation of the spectral index delivered by CATS for _cats_d_ and _cats_ is reported in Table 3. In _cats_d_, the spectral index seems to be overestimated when WH noise is taken into account and estimated. Moreover, since for PL noise only, _cats_d_ and _cats_ provide quite different results, we conclude that the index estimation is also greatly affected by the choice of the estimation model (namely including or not the draconitic cycle). As the spectral indices estimated by CATS impact the uncertainties calculation, it is important to confirm the order of magnitude of these values with the profiles of periodograms of Figure 11. First, the GNSS periodogram is well described by simple PL noise at any frequencies (a slope of constant value). This means that the PL noise contribution dominates the WH noise contribution in the plotted range of frequencies. This can be the reason why CATS fails to estimate a WH noise component in the weekly series while it is robust in estimating only PL noise. The periodograms of loading models seem to be closer to a Gauss-Markov process with a very strong annual signal. They also seem to have the same global behaviour as GNSS (especially the slope) in the interannual band, even if they have a lower variance signal. However, for frequencies greater than 1 cpy, the slopes of periodograms correspond to spectral indexes around \(-2\) for hydrological models, while this remains around \(-0.7\) for GNSS. The origin of this change of slope remains unknown, but we can argue that if the models do not contain WH noise, then we only see the coloured noise dominating at high frequencies. As MERRA2 and GLDAS2.1 had initial samples shorter than a day (1 and 3 hours, respectively), we further investigated to search for WH noise in the model time series. We computed the stacked periodograms of the raw model time series in order to reach frequencies around 300 cpy, where there was still no WH noise, which corroborated our previous assumption.  
\begin{table}
\begin{tabular}{l c c c} \hline \hline  & **East** & **North** & **Up** \\ \hline _cats\_d_ (WH + PL) & \(-0.89\) & \(-1.03\) & \(-0.77\) \\ _cats\_d_ (PL) & \(-0.69\) & \(-0.84\) & \(-0.63\) \\ _cats_ (PL) & \(-0.86\) & \(-0.99\) & \(-0.75\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: PL noise mean spectral index for the each of the two CATS-estimated solutions with or without WH noise estimation along PL noise.