Introduction

The southeastern Mediterranean region, particularly Greek territory, is one of the world’s most earthquake-prone areas, with intense and complex geodynamic behavior. The Aegean Sea and its surrounding region are frequently referred to as a "natural laboratory of geophysics," due to a wide range of deformation mechanisms found in a relatively small geographical area. Since the advent of space geodesy, the tectonics of Greece have been extensively researched using both campaign acquired and continuous GPS data. The last decades the GNSS technique was a very important tool for studying the kinematics in plate boundaries, co-seismic displacements, interseismic and post-seismic deformation. The growth of international/regional GNSS tracking networks plays a key role that allows for better estimations about the crustal deformation. The earlier geodetic studies have been carried out since the late 1980s, using GPS campaign measurements to estimate the strain rates of the Aegean Sea on a large spatial scale or over long timescales. The first measurement campaigns are based on Satellite Laser Ranging (SLR) technique at six sites within the Aegean region since 1986 as part of WEGENER/MEDLAS network (Noomen et al. 1996; Plag et al. 1998) and GPS campaign type of observations. Most previous studies focus on tectonically active areas such as the Ionian Sea (Kahle et al. 1995; Rossikopoulos et al. 1998; Hollenstein et al. 2006; Ganas et al. 2013), the central Greece (Clarke et al. 1998; Chousianitis et al. 2013) and at the Gulf of Corinth (Briole et al. 2000; Avallone et al. 2004; Lazos et al. 2020), which is one of the most active rift within the Aegean region.

Methodology

This section begins with the GPS data analysis, where we specify the processing strategy that we follow in order to estimate the loosely constrained position solution. We then describe the methodology of the reference frame definition, providing the post-fit statistics of the reference frame realization. Finally, the main seismic events that have an effect on our GPS time series analysis are represented, giving the estimated co-seismic displacements.

Geodetic data analysis

We analyzed the GPS data of a dense GNSS permanent network of 227 European GNSS stations, among them 186 sites located in Greece, 10 in Bulgaria and 7 in Romania and 24 are part of the International GNSS Service (IGS) network. The geographic distribution of Greek GNSS permanent stations is illustrated in Fig. 1. More specific about the Greek GNSS network, 77 to HxGN SmartNet Greece operated by the Metrica SA (https://www.metrica.gr), 46 to Hellenic Position System (HEPOS, https://www.hepos.gr), 21 to the National Observatory of Athens (NOA) with 1 of them is in the contribution with National and Kapodistrian University of Athens (NKUA) known as NOANET (Ganas et al. 2008; Chousianitis et al. 2021), 18 to Dionysos Satellite Observatory (http://dionysos.survey.ntua.gr) of School of Rural, Surveying and Geoinformatics Engineering in cooperation with British Centre for the Observation and Modelling of Earthquakes, Volcanoes and Tectonics (DSO/COMET), 5 to HermesNet (Fotiou et al. 2009) managed by Department of Geodesy and Surveying (DGS) of the School of Rural and Surveying Engineering of Aristotle University of Thessaloniki (https://www.users.auth.gr/cpik), 5 to Corinth Rift Laboratory (CRL, http://crlab.eu), 1 station belongs to Centre National d'Etudes Spatiales (CNES), 1 to Institut National des Sciences de l'Univers (INSU/ENS), 6 to PPGNet (https://www.pecny.cz/CzechGeo/ppgnet) joint network of Geodetic Observatory Pecny—Research Institute of Geodesy, Topography and Cartography, Patras-Charles University (RIGTC GOP), 1 to Technological Educational Institute of Serres, 1 to Technical University of Crete and 4 to EarthScope (1 are with NOA).

Fig. 1
figure 1

Geographic distribution of Greek GNSS permanent stations, fault lines (purple) are from the NOAfaults database (Ganas et al. 2018)

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The GPS data was processed for the time period 2001–2016, using GAMIT/GLOBK software release 10.7 (Herring et al. 2018aa; Herring et al. 2018b). We analyze daily RINEX data in a three-step approach to estimate station position, using double-differencing techniques to eliminate satellite and receiver clock errors. To enhance computation efficiency and consider the GNSS network geometry, the station distribution scheme was optimized into five separate subnetworks. The GNSS stations in each subnetwork are determined every day, based on station availability, and are chosen by geographic location to minimize station baseline lengths, which improves integer phase ambiguity resolution. Particularly, we select to divide the GNSS network following two criteria: (a) to minimize the baseline length between the stations and (b) to availability of the GNSS stations. Some of the most common issues that occurs on geodetic network design concern on:

  • Reference frame definition through the estimation of position coordinates on fiducial sites of the geodetic network.

  • The network geometry, ensuring the optimal observation selection.

  • The definition of weights in observations, as formed in the weight matrix.

  • The densification of local GNSS network, starting with fundamental control networks, i.e., IGS Permanent network.

The location of the 24 IGS and 5 Greek European Permanent Network (EPN) stations is shown in Fig. 2, where used for reference frame definition. These stations coordinates and geodetic velocities are precisely determined by international services such as IERS and IGS and used to connect the five local individual subnetworks (see Fig. 3).

Fig. 2
figure 2

Geographic distribution of IGS (red triangle) and EPN (green circle) GNSS permanent stations used for reference frame definition

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Fig. 3
figure 3

Geographical distribution of Greek GNSS stations per subnetwork a HEPOS network, b DSO/COMET network, c Northern and d Southern subnetwork mainly of operated by HxGN SmartNet Greece and e NOANET including f CRL stations and foreign stations

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As a first step of GPS data processing, a weighted least squares algorithm was used to estimate the station position and satellite orbit parameters, imposing proper constraints to a priori station position coordinates at the level of \(0.05\mathrm{ m}\) to aid phase ambiguity resolution. Integer phase ambiguities are generally well resolved in all the subnetwork solutions, with greater than 94% in wide lane and greater than \(90\%\) in narrow lane of ambiguities are resolved, based on Dong and Bock (1989) methodology.

Figure 4 shows the time series of the percentage of resolved phase ambiguities, in narrow lane for each subnetwork. Notably, the ambiguity resolution for the NOANET (see Fig. 4e) exhibits a lower level in the early years of its processing, primarily attributed to the limited number of GNSS stations available. It is important to emphasize that the solution generated using the IGS precise orbits which become available, with a latency of 12–18 days, where the orbital parameters are adjusted in 24 h arcs. Then loosely constrained solutions from each individual subnetwork were combined into daily, providing a single full-network solution using GLOBK. The reference frame realization was implemented on ITRF2014 (Altamimi et al. 2016) using as fiducial points 24 IGS stations (see Fig. 2), which are located in the surrounding area, in European plate. These sites selected consider the optimal geographic distribution of the GNSS network, the completeness of the observations, and the contribution of these GNSS stations to IGS network. Most of these sites are co-location with other satellite techniques (VLBI, SLR, DORIS).

Fig. 4
figure 4

Phase ambiguity resolution per subnetwork a HEPOS network, b DSO/COMET network, c Northern and d Southern subnetwork mainly of operated by HxGN SmartNet Greece and e NOANET including CRL stations and foreign stations

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The GAMIT software uses the LC or L3 ionosphere-free linear combination of GPS phase observables by applying a double-differencing technique to eliminate phase biases related to drifts in the satellite and receiver clock oscillators (Fotiou and Pikridas 2012). Tropospheric refraction significantly affects the high performance of GPS position estimations with impact mostly on height component, to reduce it we use the GPT2 model for a priori atmospheric parameters (Lagler et al. 2013) and Vienna Mapping Functions 1 (VMF1) grids (Boehm et al. 2006). VMF1 includes the meteorological coefficients as derived from gridded reanalysis of European Centre for Medium-Range Weather, improving the Zenith Wet Delay (ZWD) estimations. The residual zenith wet delays were adjusted at 2 h intervals using piecewise linear function with process noise uncertainty constraint of \(20 \mathrm{mm}/\sqrt{\mathrm{hr}}\), gradients also estimated in the same interval with a priori constraint of \(10 \mathrm{mm}\) at \(10^\circ\) elevation angle. Tidal effects such as solid Earth tides, ocean loading, pole tide, atmospheric loading, and geocenter motion can significantly bias the position estimations and consequently velocity estimation in GNSS stations. To minimize these influences of temporal changes that can be driven to un-realistic, we comply the international recommendations following the IERS Conventions 2003 (McCarthy and Petit 2004). Most predominant tidal forces in semi-annual and annual periods are modeled in the GPS data analysis. The Ocean tide effect can be well calculated using the ocean tide model that can provide by Chalmers University of Technology due to the web service (http://holt.oso.chalmers.se/loading). In our analysis, for ocean tide loading effects we apply the FES2004 model (Lyard et al. 2006), including corrections for center of mass motion. Regarding non-tidal atmospheric loading component, we applied the methodology that is described by Tregoning and van Dam (2005).

There are two types of antenna phase center calibration models, the absolute and the relative model provided by NOAA’s National Geodetic Survey (NGS). IGS absolute phase center variations models (IGS08_1930.atx) for satellite and ground-based GNSS antennas are adopted, improving the position estimations and reducing the systematic errors on long intercontinental baselines that they have found in the relative phase center variation (PCV) model. It must be noted that in short baselines, the relative PCV model do not have negative impact on position estimations. Below, we provide the most important processing models that we adopt in our strategy (listed in Table 1), including the atmospheric delay and mapping functions, tide, and tidal loading models. Aside from the applied models, we document the principal processing parameters that we use following the IGS/EPN recommendations.

Table 1 Processing models and parameters
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Reference frame definition

The geodetic network adjustment was carried out following the sequential least squares approach, applying the stacking of normal equations (Dermanis and Fotiou 1992). The estimated parameters are independent of each other and have been derived by imposing loosely constraints to stations coordinates, orbital solutions and earth orientation parameters (EOP). The observation equations for the \(k=\mathrm{1,2},\dots\) GNSS subnetworks are written as:

$$\begin{array}{*{20}c} {\mathbf{b}_{1} = \mathbf{A}_{1} \mathbf{x} + \mathbf{v}_{1} ,} & {{\text{with}} {\text{ covariance}} {\text{ matrix}} } & {\mathbf{C}_{1} } \\ {\mathbf{b}_{2} = \mathbf{A}_{2} \mathbf{x} + \mathbf{v}_{2} ,} &{} &{\mathbf{C}_{2} } \\ \vdots & {} & \vdots \\ \end{array}$$
(1)

or, in more general notation:

$$\begin{array}{*{20}c} {\mathbf{b} = \mathbf{Ax} + \mathbf{v}} & {{\text{with}} {\text{ covariance}} {\text{ matrix}}} & \mathbf{C} \\ \end{array}$$
(2)
$$\left[ {\begin{array}{*{20}c} {\mathbf{b}_{1} } \\ {\mathbf{b}_{2} } \\ \vdots \\ {\mathbf{b}_{k} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\mathbf{A}_{1} } \\ {\mathbf{A}_{2} } \\ \vdots \\ {\mathbf{A}_{{_{k} }} } \\ \end{array} } \right]\mathbf{x} + \left[ {\begin{array}{*{20}c} {\mathbf{v}_{1} } \\ {\mathbf{v}_{2} } \\ \vdots \\ {\mathbf{v}_{k} } \\ \end{array} } \right]$$
(3)

where \({\mathbf{x}}\) is the vector of unknown parameter improvements, not only the station coordinates, and \({\mathbf{A}}\) is the matrix of given coefficients with full rank, as called design matrix and with \(mathbf{v}\) the vector of observational errors. Here, \(\mathrm{C}\) is the covariance matrix for each \(k\) subnetwork, where the positive-definite weight matrix is expressed as \(\mathbf{P}={\mathbf{C}}^{-1}\) for the \(\mathbf{b}_{k}\) observation vector.

$$\begin{gathered} {\mathbf{P}} = \left[ {\begin{array}{*{20}c} {{\text{P}}_{1} } & 0 & \ldots & 0 \\ 0 & {{\text{P}}_{2} } & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & {{\text{P}}_{k} } \\ \end{array} } \right] \hfill \\ \hfill \\ \end{gathered}$$
(4)

We assume that each observation series was independent between them, and the observation errors \(\mathbf{v}\) were characterized as normal random errors following the Gaussian or normal distribution.

The normal equation matrix for each \(k\) independent series \(\left(i=\mathrm{1,2},\dots k\right)\) becomes:

$$\mathbf{N}_{k} \hat{\mathbf{x}} = \mathbf{u}_{k} ,\;\mathbf{N}_{k} = \mathbf{A}_{k}^{T} \mathbf{P}_{k} \mathbf{A}_{k} ,\;\mathbf{u}_{k} = \mathbf{A}_{k}^{T} \mathbf{P}_{k} \mathbf{b}_{k}$$
(5)

which can also be written in a more analytic form for each \(k\) independent series:

$$\begin{array}{*{20}c} {\left( {\mathbf{A}_{1}^{T} \mathbf{P}_{1} \mathbf{A}_{1} } \right)\hat{\mathbf{x}} = \mathbf{A}_{1}^{T} \mathbf{P}_{1} \mathbf{b}_{1} } & {} & {\mathbf{N}_{1} \hat{x} = \mathbf{u}_{1} } \\ {\left( {\mathbf{A}_{2}^{T} \mathbf{P}_{2} \mathbf{A}_{2} } \right)\hat{\mathbf{x}} = \mathbf{A}_{2}^{T} \mathbf{P}_{2} \mathbf{b}_{2} } & {} & {\mathbf{N}_{2} \hat{\mathbf{x}} = \mathbf{u}_{2} } \\ \vdots & {{\text{or}}} & \vdots \\ {\left( {\mathbf{A}_{k}^{T} \mathbf{P}_{k} A_{k} } \right)\hat{\mathbf{x}} = \mathbf{A}_{k}^{T} \mathbf{P}_{k} \mathbf{b}_{k} } & {} &{\mathbf {N}_{k} \hat{\mathbf{x}} = \mathbf{u}_{k} } \\ \end{array}$$
(6)

where the solution of the normal equations is:

$$\mathbf{N} \hat{\mathbf{x}} = \mathbf{u}$$
(7)

Equivalent can be expressed as:

$$\begin{array}{*{20}c} {{\mathbf{N}} = {\mathbf{A}}^{T} {\mathbf{PA}} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{1}^{T} } & {{\mathbf{A}}_{2}^{T} } & \ldots & {{\mathbf{A}}_{k}^{T} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\text{P}}_{1} } & 0 & \ldots & 0 \\ 0 & {{\text{P}}_{2} } & {} & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & {{\text{P}}_{k} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{1} } \\ {{\mathbf{A}}_{2} } \\ \vdots \\ {A_{k} } \\ \end{array} } \right] = } \\ {{\text{A}}_{1}^{T} {\text{P}}_{1} {\text{A}}_{1} + {\text{A}}_{2}^{T} {\text{P}}_{2} {\text{A}}_{2} + \ldots + {\text{A}}_{k}^{T} {\text{P}}_{k} {\text{A}}_{k} = } \\ {{\mathbf{N}}_{1} + {\mathbf{N}}_{1} + \ldots + {\mathbf{N}}_{k} } \\ \end{array} ,$$
(8)

respectively, as:

$$\begin{array}{*{20}c} {{\mathbf{u}} = {\mathbf{A}}^{T} {\mathbf{Pb}} = \left[ {\begin{array}{*{20}c} {{\mathbf{A}}_{1}^{T} } & {{\mathbf{A}}_{2}^{T} } & \ldots & {{\mathbf{A}}_{k}^{T} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\text{P}}_{1} } & 0 & \ldots & 0 \\ 0 & {{\text{P}}_{2} } & \ldots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \ldots & {P_{k} } \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\mathbf{b}}_{1} } \\ {{\mathbf{b}}_{2} } \\ \vdots \\ {{\mathbf{b}}_{k} } \\ \end{array} } \right] = } \\ {{\mathbf{A}}_{1}^{T} {\mathbf{P}}_{1} {\mathbf{b}}_{1} + {\mathbf{A}}_{2}^{T} {\mathbf{P}}_{2} {\mathbf{b}}_{2} + \ldots + {\mathbf{A}}_{k}^{T} {\mathbf{P}}_{k} {\mathbf{b}}_{k} = } \\ {{\mathbf{u}}_{1} + {\mathbf{u}}_{2} + \ldots + {\mathbf{u}}_{k} } \\ \end{array}$$
(9)

The observation equation can be expressed using the linearized form of the pseudo-observations, for all the parameters at the epoch \(\mathrm{t}\) as:

$$\mathbf {b}_{t} = \mathbf {A}_{t} \mathbf {x}_{t} + \mathbf {v}_{t}$$
(10)

where \({\mathbf {b}}_{t}\) is the vector of the differences between the observations and the a priori values, \({\mathbf{A}}_{t}\) is the design matrix, which contains the partial derivatives of each observation with respect to each parameter, and \({\mathbf {x}}_{t}\) is the estimated unknown parameters with \(E\left\{ {\mathbf {v}_{t} } \right\} = 0\), \(E\left\{ {\mathbf {v}_{t}\mathbf {v}_{t}^{T} } \right\} =\mathbf {V}_{t}\), \(E\left\{ {\mathbf {v}_{t} \mathbf {v}_{t + 1}^{T} } \right\} = 0\).

For the reference frame definition, we use a 7-parameter Helmert transformation (HT), where the resulting geocentric coordinates of the GNSS sites are transformed to an a priori reference frame (i.e., ITRFyy), using fiducial control points in the central European plate. Minimum constraint conditions are based on the assumption that there are two reference frames, where the geocentric coordinates of each \(i\) station are known. Helmert transformation (HT) is a key tool that relates to reference frame definition and its parameter estimates may be affected by the network distribution, with the magnitude of these being an indication of the consistency between the solutions. Figure 5 shows the HT residuals in topocentric coordinate system and the post-fit RMS.

Fig. 5
figure 5

Normalized and weighted fitting residuals of Helmert transformation approach in topocentric coordinate system and post-fit RMS error

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The a posteriori RMS error of the 7-parameteres Helmert transformation is \(1.8 \pm 0.7 \mathrm{mm}\), and these statistical values of the reference frame definitions maintain the consistency with the ITRF2014 solutions.

Pre-analysis of the time series

It is well known that the appearance of discontinuities in the position time series has a significant impact on the estimation of geophysical signals. They typically result from changes in the equipment of permanent reference stations or from geophysical phenomena (e.g., seismic phenomena), which cause significant changes in position that are spatially correlated with nearby stations. To mitigate this effect, GPS data are often filtered and processed to remove or correct for discontinuities. This can involve removing or adjusting data that are known to be affected by instrument errors or other artificial sources of discontinuity or using mathematical models to correct for discontinuities caused by natural events. We pre-analyzed the GNSS observations using an automated algorithm for the quality check of RINEX files, to ensure the performance of the tracking of each constellation for the GNSS stations. Also, we search for antenna offsets due to equipment changes that have an impact to height estimation. Various previous studies have shown that two-thirds of the discontinuities are caused by known factors such as geophysical sources (e.g., earthquake displacements), non-geophysical sources (e.g., equipment changes), and unknown factors (e.g., upgrading the receiver firmware or changing the antenna position). The remaining one-third of the sources that cause discontinuities in the position time series are unknown, potentially human-induced, and related to changes in the GNSS data processing strategy or processing parameters such as the cutoff angle (Gazeaux et al. 2013).

To eliminate the effects in time series discontinuities due to (i) earthquake events and (ii) antenna/equipment changes, we use a Heaviside step function, where we separate the time series. In GPS data analysis, we have the choice to include these discontinuities initially in data processing step, as break offsets from each site in specific time or to include it on the next step of GPS time series analysis. Many previous studies focus on discontinuities detection methodology, providing an easy-automated algorithmic processes. In our analysis we use the findoffset program of the Hector software (Bos et al. 2013) to search for potential offset positions and estimate their sizes. In Fig. 6, we depict the peak ground velocity (PGV) as retrieved from United States Geological Survey (USGS) ShakeMap System (Worden and Wald 2016) and the GNSS sites that were available during the seismic event. Peak ground velocity (PGV) is a measure of the maximum velocity of ground motion (cm per second) caused by an earthquake event (EQ). It is used to evaluate the intensity of ground shaking and potential damage to structures. We should note that the most discontinuities are detected on the vertical component related to equipment changes and more especially with antenna changes. On the other hand, the estimated offsets due to co-seismic displacements have in the most cases effect on horizontal component and depend on the nature of the fault which gives the EQ event. Herring et al. (2016) proposed the empirical formula that approximates the radius of influence of an earthquake, where \(d\) is the influenced radius from the EQ event in \(Km\).

$$d = 2.5 \times 10^{ - 3} \times 5^{{M_{w} }}$$
(11)
Fig. 6
figure 6

Horizontal co-seismic displacements (vectors) as calculated by GPS time series analysis and PGV values (contour line) for the strongest EQ events (\({M}_{w}>5.3\)) of the period 2001–2016 were detected in GPS time series. a Methoni, Movri, Efpalion, b Cephalonia, c Samothraki, d Lefkada

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Blewitt et al. (2018) propose an empirical formula that helps in identify the potential time series discontinuities caused by earthquake events, using a distance threshold which is calculated by the simple formula:

$$r = 10^{{\left( {M_{w} /2 - 0.79} \right)}}$$
(12)

where \({M}_{W}\) is the magnitude of the earthquake and the distance \(r\) between the epicenter and the GNSS site expressed in km. It is essential to emphasize that (11) and (12) are exclusively utilized during the initial phase of earthquake offset detection in the GPS data pre-processing level, wherein they are properly integrated into the baseline solution processing step.

During the study period, 8 earthquake events had an impact and were detected on GPS position time series (see supplementary material), are those of Methoni 2008 February 14, Movri 2008 June 8, Efpalion 2010 January, Cephalonia 2014 January and February, Samothraki 2014 May 24, and Lefkada 2015 November 17. In Table 2, we describe in detail the most earthquake events that they have an impact on GNSS stations as recognized by daily position monitoring, with moment magnitude greater than \(5.3{ M}_{w}\). Also, Table 3 describes the co-seismic displacements in topocentric coordinate system (east, north, up), as calculated for the mean position for approximately 10 days before and after the main seismic event, in nearly to the epicenter GNSS stations. The uncertainties of co-seismic displacements are estimated for each component before and after the main shock by applying the law of error propagation to the uncertainties of the coordinates.

Table 2 Catalogue with the strongest EQ events (\({M}_{w}>5.3\)) of the period 2001–2016 was detected in GPS time series. Earthquake parameters are given as reported at the EMSC (European-Mediterranean Seismological Centre) website
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Table 3 Observed co-seismic displacements and their uncertainties at the nearest GNSS stations for of the period 2001–2016
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It is essential to specifically mention that Fig. 6a provides an encompassing view, incorporating four distinct earthquake events in Methoni, Movri, and Efpalion (2010/01/18, 2010/01/22). The Cephalonia doublet, comprising two earthquake events with magnitudes.

\({M}_{w}=6.1\) and \({M}_{w}=6.0\), is illustrated in Fig. 6b as a unified depiction of the co-seismic displacements and the PGV values.

The post-seismic effects, particularly for significant events, can persist over several years due to the extended relaxation time. As a result, these effects tend to smoothen over time and can become challenging to differentiate from the long-term velocity. In the Greek area, many studies focus on post-seismic deformation using GNSS stations and InSAR techniques, which are occurred after strong earthquake events as in the case of 2015 Lefkada \(({M}_{w}=6.5)\) earthquake (Vallianatos and Sakkas 2021). In this study, we do not examine the post-seismic deformation in GNSS site due to the larger events.

Geodetic GPS time series analysis

It was already known from previous studies that GNSS time series contain tectonic and non-tectonic signals that we need to minimize due to the purposes of our research. Non-tectonic signals refer to a variety of sources of movement or deformation of the Earth’s surface that are not directly related to plate tectonics or other geodynamic processes. These signals can include a range of natural and anthropogenic effects, such as atmospheric effects, ocean tides, and human-induced subsidence. Besides, it is crucial to consider the presence of spatially correlated signals arising from two primary sources: common-mode signals (CMS) and common-mode errors (CME). Common-mode signals are associated with large-scale processes, encompassing phenomena like solid Earth displacements, ocean and atmosphere effects, and satellite orbit mismodeling. These processes contribute to the emergence of correlated signals in the GNSS data. Conversely, common-mode errors stem from inaccuracies in processing, including mismodeling of displacements induced by various factors like solid Earth, ocean, and atmosphere, along with the occurrence of draconitic signals. By understanding and addressing both non-tectonic signals and spatially correlated signals, we can refine our analysis and ensure the accuracy of our research finding.

To estimate the geodetic velocities and their realistic uncertainties from the daily position time series, we applied the robust Median Interannual Difference Adjusted for Skewness (MIDAS) trend estimator (Blewitt et al. 2016), which avoids detecting step discontinuities. MIDAS algorithm provides the most accurate estimations about the geodetic velocities and their uncertainties, as verified to blind tests on synthetic data (Gazeaux et al. 2013). MIDAS is a variant of the Theil‐Sen median trend estimator, which a nonparametric method for estimating the slope of a linear trend in a dataset. It was based on the median of the slopes of all possible pairs of data points in the dataset, rather than the mean of the slopes as in traditional linear regression. The median of slopes between pairs of data of the ordinary Theil-Sen estimator can be expressed as: \(\widehat{v}={\mathrm{median}}_{j>1}\left(\frac{{x}_{j}-{x}_{i}}{{t}_{j}-{t}_{i}}\right)\), where coordinate \({x}_{i}\) is sampled at time \({t}_{i}\). In particular, this process involves ordering the computation of the median trend using slopes from all selected data pairs and  selecting the median as the estimated rate for the entire dataset. Additionally, the median absolute deviation (MAD) is calculated and multiplied by a constant 1.4826 to determine the standard deviation of the rates \(\sigma =1.4826\times MAD\), assuming that the differences between the rates and the median follow a Gaussian distribution. The velocity uncertainty is rescaled by \(1.2533/\sqrt{N}\), where \({\rm N}\) is the number of individual rates calculated within the program. The differences between the MIDAS and a simple linear regression model are very close in the level of \(1 \pm 1\mathrm{ mm}/\mathrm{yr}\) on GNSS stations without discontinuities on their position time series. We do, however, note the greater differences found on GNSS sites which are located in Santorini Island (NOMI, KERA), with differences greater to \(18\mathrm{ mm}/\mathrm{yr}\) on horizontal and \(8\mathrm{ mm}/\mathrm{yr}\) on vertical component, due to the inflation observed during the Santorini unrest of 2011–2012 that caused nonlinear ground motion as also observed in previous studies using GPS and InSAR (Papoutsis et al. 2013).

The geodetic velocities are expressed to the European Terrestrial Reference Frame 2000 (ETRF2000), which is an implementation of the European Terrestrial Reference System of 1989 (ETRS89), following the transformation formula (Altamimi 2018) to link ETRF89 to the ITRS for station velocities:

$$\dot{{\mathbf{X}}}_{{{\text{ETRF}}_{{{\text{yy}}}} }}^{i} = \dot{{\mathbf{X}}}_{{{\text{ITRFyy}}}}^{i} + \left( {\begin{array}{*{20}c} 0 & { - \dot{R}z_{{{\text{yy}}}} } & {\dot{R}y_{{{\text{yy}}}} } \\ {\dot{R}z_{{{\text{yy}}}} } & 0 & { - \dot{R}x_{{{\text{yy}}}} } \\ { - \dot{R}y_{{{\text{yy}}}} } & {\dot{R}x_{{{\text{yy}}}} } & 0 \\ \end{array} } \right){\mathbf{X}}_{{{\text{ITRFyy}}}}^{i}$$
(13)

where \(\dot{X}_{{{\text{ETRF}}_{{{\text{yy}}}} }}^{i} , \dot{X}_{{{\text{ITRFyy}}}}^{i}\) is the 3D vectors of geodetic velocities in \(yy\) reference frame realizations, on ITRS and ETRS89 respectively. \(X_{{{\text{ITRFyy}}}}^{i}\) denotes the 3D position vector with respect to \({\text{ITRFyy}}\). The rotation rate parameters, \(\dot{R}x_{{{\text{yy}}}} , \dot{R}y_{{{\text{yy}}}} , \dot{R}z_{{{\text{yy}}}}\) about each axis of a (clockwise) Cartesian system, are the three components of the Eurasia Euler vector (or angular velocity) expressed in the \({\text{ITRFyy}}\).

Results and discussion

This section starts by describing the geodetic velocity field in Greece, using a long-term GPS time series analysis. Then, we evaluate our results with previously published studies, proving a more dense and accurate velocity field that can be used for a plethora of geodetic and geophysical studies.

Greek geodetic velocity field

The estimated horizontal velocities and their uncertainties with respect to Eurasia are shown in Fig. 7. The final geodetic velocity includes 215 GNSS stations out of a total of 227 initial process locations, with the exclusion of 12 sites due to a small time span of data availability. Velocities with respect to Eurasia plate selected because they can provide a clear view of the geophysical processes being performed on the crustal earth in Greek area. The role of the GPS velocities is very important on a plenty of geodetic applications as the semi-kinematic reference frame realization and for geophysical applications, providing the true deformation field. In semi-kinematic datum, the knowledge of the deformation is crucial from the time-depending transformation formula between the observed/measurement epoch and the reference epoch where the datum is referred (Bitharis et al. 2019). For the scope of geophysics, the velocity field has enabled extensive research into the accumulation of strain, plate coupling, and transient motion in various subduction zone configurations, as well as the examination of the relative significance of the various processes occurring within a subduction zone.

Fig. 7
figure 7

Horizontal velocity field in Greece and 95% confidence error ellipses, expressed in the ETRF2000 reference frame. Histograms of velocities uncertainties in topocentric components in east and north

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As a result, the northern part of Greece is more consistent to the stable European plate, with small horizontal velocities with mean magnitude less than sub-centimeter. The velocities at the southern part of Greece are systematically greater than those that occurred in the northern part. Aegean Sea and more especially Crete Island are moving away from the most stable part of Europe with southwest direction and value of about \(3.2\mathrm{ cm}/\mathrm{yr}.\) We find a simple pattern with the magnitude and orientation of the velocity field, which confirms the role of the tectonic settings in the region, as the Hellenic Trench, the North Anatolian Fault (NAF), and the Cephalonia–Lefkada Transform Fault Zone (CLTFZ) in the central Ionian. The Aegean Sea and the southern part of Greece move westward along the bounding strike-slip faults, with primary tectonic setting the prolongation of the North Anatolian Fault (NAF), which controls the geodynamic evolution of region.

The vertical velocities and their uncertainties over the Greek territory are illustrated in Fig. 8. In contrast with the horizontal velocities, the vertical component does not show any predominant pattern. The magnitude is considerably smaller than the velocities in the horizontal component due to the geodynamic behavior in Greece.

Fig. 8
figure 8

Vertical velocity field in Greece. Histogram of vertical velocities uncertainties

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The GPS vertical velocities are a very interesting product in sea-level applications and can be used as a correction term in vertical land motion (VLM) providing the absolute sea-level rise. The sea-level rise can be estimated using tide gauge records, which measure the change in sea level over time at a specific location in coast. However, tide gauge records are often limited in their spatial coverage and can be affected by local factors such as land subsidence or uplift. Satellite altimetry is another method for measuring sea-level change, which uses radar or laser instruments to measure the height of the sea surface from space. However, satellite altimetry data are affected by errors such as signal penetration through the atmosphere and ocean waves, and it is limited in the temporal resolution. By using GPS vertical velocities as a correction term, it is possible to correct for the effects of VLM on the measurements made by tide gauges or satellite altimetry, which can improve the accuracy of the sea-level rise estimates. Measuring and analyzing GPS vertical velocities is a challenging task that poses several limitations. These limitations include signal errors, spatial and temporal resolution, network density, processing techniques, and noise and outliers. To overcome these limitations, we need a long-lasting time series (beyond 8 years), to minimize the impact of transient and seasonal signals on vertical geodetic velocities estimates as described by Masson et al. (2019). In Fig. 9, the dispersions on the horizontal and vertical components are illustrated.

Fig. 9
figure 9

Scatter plot of GPS-derived uncertainties for horizontal and vertical velocities, and color palette represents the vertical uncertainties

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It was interesting to notice that the uncertainties in vertical component were about three times larger in comparison with each horizontal component, due to seasonal effects that have mainly impact on vertical direction as known sources of tidal effects, surface mass distributions from the atmosphere, oceans, snow, and soil moisture. That reflects the larger uncertainties on the vertical component than the horizontal plain, with mean value \(0.6 \mathrm{mm}/\mathrm{yr}\) and \(0.4 \mathrm{mm}/\mathrm{yr},\) respectively.

Comparison with previous selected solutions

In this section, we perform a validation and quality assurance by comparing our estimated geodetic velocities with those obtained from previous studies. This comparison serves to verify the reliability of our analysis and ensure the consistency among the solutions and robustness of velocity field. We select previously published velocity solutions with the precondition that the various datasets complement one another. In the frame of EPN, three related Working Groups (WGs) provide geodetic velocities: European Dense Velocities Working Group (Chair: E. Brockmann), EPN Densification Working Group (Chair: A. Kenyeres), and Deformation Models Working Group (Chair: M. Lidberg). Since 2016, our research findings have been actively contributing to the European Dense Velocities Working Group and the Deformation Models Working Group, augmenting their efforts in understanding crustal movements and developing accurate geodetic velocity fields within the European region. The evaluation includes five velocity fields (England et al. 2016; Brockmann et al. 2019; Bitharis 2021; Briole et al. 2021; Piña-Valdés et al. 2022), which are briefly described below. We conduct a comparative analysis by juxtaposing our results with those from previous studies that have employed diverse techniques in GNSS processing. Specifically, we consider studies that have utilized the precise point positioning (PPP) method and combined solutions derived from different processing schemas. By examining and contrasting these approaches, we aim to gain valuable insights into the strengths and limitations of our own methodology while enriching the understanding of geodetic phenomena in the context of various processing techniques.

  • GJI-2021 This velocity solution was derived from Briole et al. (2021) and was carried out using PPP method. The coordinate time series of 282 permanent GNSS stations located in Greece and 47 in surrounding countries are calculated and analyzed. The studied period is 2000–2020, with mean time series length \(6.5\mathrm{ yr}\). The geodetic velocities and their uncertainties are given in ITRF2014, in topocentric coordinate system. In order to compare the velocities with the other studies, we use the official transformation parameters to express the geodetic velocities wrt European plate, in ETRF2000. We should note that in the GJI-2021 dataset, the GNSS station (KRIN), located near the village of Krini, exhibits a distinct kinematic behavior compared to the dominant one in this area (see Fig. 10b). This deviation is attributed to an active landslide, as described by Briole et al. (2021). This assessment is further corroborated by Tsironi et al. (2022) through InSAR time series analysis, confirming that the observed displacement rate is a result of the active landslide in the area and is not linked to tectonic motion.

  • EUREF-EDV This solution is provided from European Terrestrial Reference Frame (EUREF) Technical Working Group—European Dense Velocities (Brockmann et al. 2017), where more than 60,000 GNSS stations are included to dataset. The main scope of the TWG is to provide a dense velocity field in the European region, exploiting the existing velocity estimates provided by the contributing national agencies, following the simple “classical approach” combining the individual data sets using a HT model.

  • JGR-2016 This study focuses on the Anatolian and Aegean regions providing the geodetic horizontal velocities at 346 GNSS sites (England et al. 2016). The final solution was performed as a combination of 24 previously published studies, with the selection criterion being the level of velocity uncertainties.

  • EPOS-UGA The solution (doi: http://dx.doi.org/10.17178/GNSS.products.all) provides vertical and horizontal velocities with their respective uncertainties for 1092 GNSS stations, most of them located in metropolitan France and Italy as well as sparse information in UK, Scandinavia, Germany, Greece, and the Iberic peninsula. The velocity solution was processed using double difference with GAMIT software, and then, velocities were estimated through a statistical analysis of the position time series using the MIDAS software. The velocity field is given with respect to the stable Eurasian plate as defined by ITRF2014. The solution is provided through the European Plate Observing System (EPOS).

  • JGR-2022 In this study, a 3D secular velocity field that covers Eurasia is presented, including 4.863 GNSS stations (Piña-Valdés et al. 2022). The final velocity field is expressed in Eurasian fixed plate by combining 10 deferent datasets which all are aligned to EPN Densification (EPND) solution (Kenyeres et al. 2019) by applying a 6-parameter HT model to provide a harmonized combined solution. This method followed because the 10 individual datasets were in different reference frames.

Fig. 10
figure 10

Residual of the horizontal velocity fields in Greece between a the present study as derived from Bitharis (2021) and the previous studies: b GJI-2021 solution (Briole et al. 2021), c EUREF-EDV solution (Brockmann et al. 2017), d JGR-2022 solution (Piña-Valdés et al. 2022), e JGR-2016 solution (England et al. 2016), and f) EPOS-UGA solution (Deprez et al. 2019)

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The geodetic velocity fields that we select to compare are given in Fig. 10, as expressed with respect to Eurasian fixed plate, in ETRF2000 reference frame. A more comprehensive analysis of the geodetic velocity comparison between common GNSS stations from our present study and previously published results reveals the critical role of reference frame realization in influencing the compatibility of geodetic velocities. Figure 10 presents the discrepancies in geodetic velocity fields as residuals between our current study and the published results, underscoring the significance of reference frame considerations for accurate assessments of geodetic velocity agreement. Systematic differences were observed between our present study and two solutions: GJI-2021 solution (Briole et al. 2021) and JGR-2016 solution (England et al. 2016). These differences were particularly prominent in the southern part of Greece. The velocity differences exhibited a consistent northeastern direction, with mean magnitudes of 1.4 mm/yr and 2.3 mm/yr in the horizontal plane for the two solutions, respectively. The discrepancies identified in our study fall within an acceptable range for long-term analyses of the geodetic velocity field, rendering them suitable for application in tectonic and geodetic studies. However, it should be noted that to achieve harmonization among various studies and create a synthetic/combined velocity field, the implementation of Helmert transformation formulas and cross-correlation analysis is highly recommended. These techniques are essential for ensuring compatibility and coherence across different datasets, facilitating the creation of a unified and comprehensive geodetic velocity representation. The supplementary materials encompass the results pertaining to the residuals of horizontal velocity fields in Greece, facilitating a comparative examination between the findings of the current study, as derived from Bitharis (2021), and those of prior solutions.

In order to ensure the consistency of the velocity fields, we cross-compare the geodetic velocities between each study. As shown in Fig. 11, the differences of the horizontal velocities between the common GNSS sites for the selected previous studies are at the level of millimeter per year. Therefore, all previously published results are compatible with each other, providing reliable results of the geodynamic behavior in Greek area.

Fig. 11
figure 11

Histogram of horizontal velocity differences at common GNSS stations between previous studies. Upper triangle denotes differences in east component. Lower triangle denotes differences in north component

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Conclusions

Our study involved the analysis of GPS data collected from a highly concentrated GNSS permanent network comprising of 227 European GNSS stations, of which 186 were geographically located within the territory of Greece, the processed data collected by research institutes, universities, and private companies. The derived results were used to produce a new high accurate velocity field taking into analysis procedure important seismic events. A high accurate velocity field was very important on time-depending transformation between the geodetic datum, providing a more stable reference frame which needed in many geodetic, cartographic, and cadastral purposes. Additionally, an expanded spatial and geographic coverage, along with a long-time span of daily GPS position time series, enables us to present a comprehensive view of temporal crustal deforming areas and strain rate fields. Furthermore, we have emphasized realistic uncertainty estimation to ensure the reliability of our results, employing robust trend estimators. A comparison of geodetic velocities with previous published studies indicates the agreement of our results, improving the understanding of the geodynamic processes, in a complex and highly active tectonic region, such as Greece. The dataset will be made available to a broad spectrum of users, comprising scientists engaged in the study of earthquakes and tectonics but also researchers involved in geodesy and cartography.