## 2 Materials and Methods  
### Processing Time Series from Various Software  
Topocentric GNSS time series are affected by various sources of errors, which are caused mainly by the spatial constellation of satellites, the GNSS signal propagation environment, and tracking stations. Thus, the accuracy of ephemeris, satellite oscillation corrections, Earth rotation parameters, ionosphere and troposphere effects, station stability, the multipath effect, electromagnetic signal interference, etc., have a decisive impact on the quality of time series obtained from GNSS observations [14]Software for processing GNSS observations is usually based on double difference observations (relative positioning) or on absolute resolution (precise point positioning--PPP solution). The first case is called network positioning, where the baselines between all observed stations are estimated simultaneously. The second strategy, precise point positioning, is presented as a way to individually and efficiently estimate the station coordinates directly relative to a global network of GNSS reference stations. PPP calculates \"absolute\" positions at any location relative to a specific reference frame based on undifferentiated observations determined by the orbits of GNSS satellites. The network and PPP approaches can be considered equivalent in terms of obtaining a time series of GNSS coordinates. In both methods, coordinates are usually evaluated relative to the International Terrestrial Reference Frame, the latest version of which, ITRF2020, was released at the end of 2022. It is not widely used yet in software packages. Therefore, for the analysis in this article, we use the previous implementation, ITRF2014. Examples of well-known software packages for precise GNSS positioning and time series generation that use one of the two strategies described above are Bernese, GAMIT, and GIPSY-OASIS/GipsyX. However, other well-known packages, such as NAPEOS v.0.9 (Navigation Package for Earth Orbiting Satellites), PANDA (Position and Navigation Data Analyst), EPOS (Earth Parameter and Orbit Software), and the more recent PRIDE PPP-AR v2.2 (precise point positioning with ambiguity resolution) [15] can also be used. More details about the analysis strategies and physical models used in time series processing can be found in [2; 16]. It should be noted that a GNSS time series might contain various offsets due to geophysical sources (e.g., atmospheric tidal loading, earthquake responses, etc.) or non-geophysical sources, such as changes in antenna height, changes in dissimilar antenna types, phase center modeling errors, inconsistencies in reference frames between analysis centers, etc. [5]. The time series administrator is responsible for detecting significant offsets and setting new offset parameters [17]. Various analytical centers (global, regional, national, or research) provide daily estimates of the positions of GNSS observation stations \((X,Y,Z)\) in the global terrestrial reference frame IGbDX/ITRFXXXX [18]. These coordinates are then transformed into more intuitive and physically meaningful horizontal and vertical displacements \((N,E,U)\) at epoch \(t_{i}\) relative to some initial station coordinates \((X_{0},Y_{0},Z_{0})\) and initial epoch \(t_{0}\). Then, a separate component of the time series \((N,E\ or\ U)\) at discrete epochs \(t_{i}\) can be modeled as \(y_{i}=a+bt_{i}+f(\mathbf{x})\). The residual time series is formed from the differences between the observed (O, e.g., \(N_{i}\)) and calculated \((C,\ \text{e.g. },y_{i})\) coordinates, which characterize the degree of consistency between the model and the real data. Such differences are traditionally associated with observation-calculation (O-C) differences. Residual time series can be cleaned and filtered, but they are usually de-trended by definition, as the slope is one of the parameters estimated.