Introduction

On 2 December 2020 10:54 UTC a shallow earthquake rocked the hilly areas to the east of Thiva (Thebes), central Greece (Fig. 1; Valkaniotis and Ganas 2020; Elias et al. 2021; Kaviris et al. 2022). The epicentre was located 12 km east of Thiva, about 46 km to the NNW of Athens. The earthquake had a magnitude of MW = 4.6 according to the National Observatory of Athens (NOA 2020) and its maximum intensity was IV (ITSAK instrumental intensity map shown in Fig. S1). Using InSAR and GNSS data, both Valkaniotis and Ganas (2020) and Elias et al. (2021) found out that this moderate event caused an impressive permanent change on the ground surface, that is a permanent subsidence of several centimetres in amplitude. The effects (deformation, shaking) of the earthquake were disproportional to its moderate magnitude, likely because of its unusually shallow depth (centroid depth of 1.6 km; NOA 2020). Given the uniqueness of this event, we were motivated to investigate the rupture plane characteristics and the source parameters using high-quality geodetic data (GNSS and InSAR) and broadband seismic waveforms. Compared to the study by Elias et al. (2021), we have included in our processing two more GNSS stations in our processing, namely 014A (a HEPOS station; Gianniou 2010) and KALI (a NOA campaign station belonging to the KAPNET network; Ganas et al. 2007a; Marinou et al. 2015) in the vicinity of the epicentre (Fig. 1). Furthermore, the earthquake occurred within a region with a rich history of M6+ seismic events during the last two centuries (Ambraseys and Jackson 1990) where the NW–SE striking normal faults of the Gulf of Evia rift (Lemeille 1977; Rondoyanni-Tsiambaou 1984; Roberts and Jackson 1991; Ganas and White 1996; Ganas 1997; Kranis 1999; Ganas and Papoulia 2000; Goldsworthy and Jackson 2001; Sboras et al. 2006, 2010; Walker et al. 2010; Georgiou 2019) intersect (and vice-versa) the E–W striking normal faults of the Gulf of Corinth rift and its continuation towards southern Viotia (also written Beotia or Boeotia; Jackson et al. 1982; Papanikolaou et al. 1988; Stewart and Hancock 1991; Roberts and Koukouvelas 1996; Roberts and Ganas 2000; Ganas et al. 2005, 2007b; Kokkalas et al. 2007; Sakellariou et al. 2007; Tsodoulos et al. 2008).

Fig. 1
figure 1

Location map showing simplified geology, shaded relief, the focal mechanism (beachball; NOA solution) and the epicentre (yellow star) of the Kallithea 2 December 2020 earthquake. Triangles indicate GPS (GNSS) station locations of a local network (Ganas et al. 2007a). Orange stars indicate the locations of the 1893 and 1914 M6 epicentres. Black lines are active faults (NOAFAULTs database, Ganas 2022; Goldsworthy and Jackson 2001; Sboras et al. 2010). Fault names: KAPF (Kaparelli Fault), ERF (Erythres Fault), DAF (Dafni Fault), NEF (Neohoraki Fault), KAF (Kallithea Fault), ASF (Asopia Fault), TAF: (Tanagra fault), KIF (Kirikio Fault), LEF (Leontari Fault)

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Our research methods include the processing and analysis of geodetic (InSAR, GNSS) and seismological data. First, we present a detailed analysis of the geodetic data. The SAR images acquired by the SENTINEL-1 satellites are routinely distributed free of charge by the European Space Agency (ESA). The GNSS data originated from three sources: (a) from a permanent station west of the epicentre (station THIV) that was provided by the Greek private GNSS network HxGN SmartNet, (b) the permanent GNSS station 014A belonging to the national GNSS network of Greece HEPOS (https://www.hepos.gr/Map/SensorMap.aspx) and (c) the campaign GNSS station KALI that is part of the NOA local network KAPNET (Ganas et al. 2007a; Marinou et al. 2015). We then derive a co-seismic fault model from joint inversion of geodetic data (GNSS and InSAR) assuming that the earthquake can be modelled by the slip on a rectangular fault buried in an elastic and homogeneous half-space.

We also inferred kinematic source parameters by analysing waveforms recorded by the Hellenic Unified Seismological Network (HUSN) and the National Observatory of Athens Seismic Network. Our analysis aimed at the comparison of the two datasets (geodetic vs. seismological) given the very shallow depth of the slip centroid, as reported by NOA (2020). An additional aspect of our research concerns the rupture pattern of the active NW–SE striking normal faults in Viotia,, which continue southwards in Attica (e.g., Ganas et al. 2005; Grützner et al. 2016; Deligiannakis et al. 2018; Iezzi et al. 2021), despite their intersection with the E–W normal faults of the Gulf of Corinth that have propagated in this area during Quaternary (e.g., Roberts and Ganas 2000; Tsodoulos et al. 2008). Is it possible that the South Viotia NW–SE faults rupture partially in moderate magnitude earthquakes or do they rupture as whole segments? Are the ruptures shallow or at mid crustal depths? Are the normal fault segments single structures or comprise fault zones with several branches down-dip (vertical segmentation; e.g., Childs et al. 1996)? The unique case of the Kallithea M = 4.6 earthquake may shed light on some of those questions.

Tectonic setting: geology

Numerous structural studies have shown that central Greece is a young, extensional province of the Aegean area and one of the most actively deforming in the world (King et al. 1985; Roberts and Jackson 1991; Stewart and Hancock 1991; Collier et al. 1998; Roberts and Ganas 2000; Goldsworthy and Jackson 2001; Goldsworthy et al. 2002). Surface topography and geomorphology are clearly associated with seismic activity along large normal faults that have developed over the last 1–5 Ma (million years; Jackson et al. 1982; Roberts and Ganas 2000; Ganas et al. 2005, 2007b; Tsodoulos et al. 2008; Grützner et al. 2016; Deligiannakis et al. 2018; Iezzi et al. 2021). Extension is mainly directed N–S with increasing rates towards the west (Clarke et al. 1998; Briole et al. 2000; Avallone et al. 2004; Chousianitis et al. 2013; D’Agostino et al. 2020; Briole et al. 2021). The main extensional features are two E–W to NW–SE Pliocene–Pleistocene rifts, namely the Gulf of Corinth and the Gulf of Evia (Fig. 1; Ambraseys and Jackson 1990; Ganas and Papoulia 2000; Goldsworthy and Jackson 2001; Ganas et al. 2004, 2005; Walker et al. 2010; Fernández-Blanco et al. 2019). In between the two major rifts, there are smaller structures comprising mainly E–W striking active normal faults with mostly antithetic dip, i.e. to the south, that are capable of hosting events up to ~ M6.0 such as the 2013 Kallidromon Mountain sequence (Ganas et al. 2014) or have well-developed limestone scarps with thick colluvium on their hanging wall but with unknown post-glacial activity (e.g., Ganas et al. 2007b). Other significant normal faults are visible on the eastern margin of the Gulf of Corinth and along the Asopos river valley (Fig. 1; Papanikolaou et al. 1988; Ganas et al. 2005; Sboras et al. 2006, 2010; Tsodoulos et al. 2008). One such fault is the south-dipping Kaparelli fault that ruptured on 4 March 1981 (Jackson et al. 1982; Kokkalas et al. 2007). Another one is the Corini fault in south-west Viotia that is arranged en-echelon to the Kaparelli Fault and preliminary structural data were presented in Ganas et al., (2007b).

In the Thiva–Kallithea region (Fig. 1), Late Miocene–Pliocene normal faulting juxtaposed Upper Triassic–Jurassic carbonates, which were deposited in shallow-water environments onto syn-rift lacustrine-brackish deposits. Since at least Pliocene times, central Greece has been subjected to NE–SW (more recently N–S) oriented extension due to post-orogenic extension and back-arc stretching imposed by the African slab rollback (e.g., Jolivet and Brun 2010). At present, this extensional regime is mostly accommodated by seismic slip along active normal faults striking mainly WNW–ESE, which have been intersected by a younger generation of E–W faults, generating several Plio-Quaternary basins (e.g., Asopos, Thiva, Schimatari; Papanikolaou et al. 1988; Goldsworthy et al. 2002; Georgiou 2019; see Fig. 1 for locations). The 2020 event near Kallithea highlighted the structural complexity in this region of the Asopos Rift valley as the reactivation of the WNW–ESE structures indicates their significant role in strain accommodation and that they still represent a seismic hazard for this region.

Previous strong earthquake activity near Thiva includes the shallow events of 11 May 1893 and 17 October 1914 with magnitudes reaching or slightly exceeding 6.0 (Ambraseys and Jackson 1990; Kaviris et al. 2022; Fig. 1). Both events destroyed Thiva (intensities were assigned to 9 on the MSI scale). The damage reports of the 1914 event included villages to the east of Thiva such as Tanagra (Fig. 1) so the epicentre was located by Ambraseys and Jackson (1990) somewhere in between. Despite the M6.0 magnitudes, no surface ruptures were reported, either because of a lack of ground surveys or for the ruptures did not appear on the ground surface. A full account of past earthquakes near Thiva is given by Kaviris et al. (2022). The present-day stress field as determined by analyses of focal mechanism data is extensional (N–S) and is manifested by the orientation of the T (extensional) principal axis of the moment tensors being related to the S3 minimum principal axis of the stress tensor (Kapetanidis and Kassaras 2019).

Seismicity

The MW = 4.6 earthquake of 2 December 2020 10:54 UTC occurred to the east of the city of Thiva (Fig. 1), along an ESE–WNW striking normal fault as indicated by the moment tensor solutions of regional data (NOA 2020; also compiled by EMSC; https://www.emsc-csem.org/; see also Elias et al. 2021; Kaviris et al. 2022). The mapped active structures in this area dip to the southwest (Goldsworthy et al. 2002; Sboras et al. 2010; Ganas 2022); there are three normal faults NEF, KAF, ASF (Fig. 1) with lengths between 3 and 6 km, so it is reasonable to initially assume that one of these faults hosted the earthquake. We note that both Sboras et al. (2010) and Georgiou (2019) mapped two segments of the Kallithea fault, one abutting the homonymous village and a second segment, towards the SE, running along the “Mavrovouni” ridge (see the bedrock ridge above station ASOP in Fig. 1). The NOA epicentre (38.3272°N 23.4590°E) plots on the footwall of both KAF and ASF (about 3 km NE from Kallithea village), while the Aristotle University of Thessaloniki (AUTH; see the catalog at http://geophysics.geo.auth.gr/ss/CATALOGS/preliminary/finalcat.cat) AUTH epicentre location (38.315°N23.431°E; magnitude M = 4.5) plots in the immediate hanging wall of KAF (Fig. 1).

Four focal mechanisms are available for the Kallithea event with fault strike determinations N106°E–N137°E, i.e., in broad agreement with geological data. Until 31 December 2020 more than 110 aftershocks (with 0.8 ≤ ML ≤ 3.5) were recorded by NOA (Fig. 2). The depth distribution of the aftershocks is notably shallower than background seismicity (NOA catalog data; Fig. 3) within a 20-km radius of Kallithea, indicating a shallow earthquake sequence. We also note that the moment tensor solutions of the mainshock (see Table 1) indicate shallow normal faulting (four out of five solutions indicate depths between 1 and 4 km). The aftershock sequence extends within a radius of 5 km of the village of Kallithea with most of events occurring to the WSW of the mainshock (Fig. 2). Elias et al. (2021) relocated the seismicity data and presented an aftershock distribution mostly to the southwest of Kallithea and deeper than the centroid of the mainshock (aftershock depths between 2 and 5 km) without a down-dip alignment. As the location accuracy of the seismological solutions may reach up to a few kilometres, we also used additional geodetic data analysis in order to map the surface displacement field and better locate the seismic fault and its relationship to the Kallithea normal fault. An additional point of concern is the dip-angle of the seismic fault, i.e., a low angle as suggested by Elias et al. (2021) or a moderate–high angle as indicated by the focal mechanisms in Table 1.

Fig. 2
figure 2

Relief map showing earthquake epicentres in a 20-km area surrounding the 2 December 2020 event (red star) for the period 1 January 2020–1 December 2020 (data source NOA). Red circles indicate aftershock locations, violet circles background seismicity, respectively. Thin black lines are faults from Ganas (2022)

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Fig. 3
figure 3

Graph showing the frequency—focal depth distribution of earthquakes for the background period (1/1–1/12/2020; cyan colour) and for the Kallithea seismic sequence (N = 80; 2/12/2020–24/12/2020; blue colour), respectively. Data source: NOA catalogue of revised events

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Table 1 List of focal mechanism (moment tensor) solutions of the mainshock determined by NOA (National Observatory of Athens), OCA (Géoazur), USGS (United States Geological Survey) UOA (University of Athens), Elias et al. (2021; row E21) and this study (inversion of geodetic data). Column depth indicates centroid depth
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Geodetic data and methods

Sentinel-1 interferograms

We produced co-seismic interferograms using Sentinel-1 radar imagery for both the ascending (track 102) and the descending (track 007) orbits. The two pairs of SAR acquisitions were processed with the SNAP v8.0 ESA software (https://step.esa.int). For both tracks, the pre-event and post-event acquisitions are on 27 November 2020 and on 3 December 2020, respectively. The interferograms were produced by complex multiplication of the primary (pre-event) image with the complex conjugate of the secondary (post-event) image. The phase of such a complex image (interferogram) is related to the ground displacement between the two images. The obtained interferograms are shown in Fig. 4a and b; in both cases, we observed about two fringes each (coloured from blue to red) corresponding to a motion away from the satellite. Since the LOS displacement has a similar amplitude and pattern in both ascending and descending tracks, we can conclude that a vertical ground motion is predominant.

Fig. 4
figure 4

The co-seismic interferograms (wrapped phase; cropped swath; unwrapped) over the Kallithea area. One cycle of colour corresponds to 28 mm of displacement along the line of sight. a Wrapped phase showing the ascending image pair, b wrapped phase showing the descending image pair, c vertical displacement obtained by decomposing the ascending and descending data and showing co-seismic subsidence with WNW–ESE orientation, d cross-section orthogonal to fault’s strike showing vertical displacement in m (see profile trace in c)

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Phase unwrapping was carried out by using SNAPHU (Chen and Zebker 2002). We then applied a decomposition procedure to the ascending and descending unwrapped interferograms (Wright et al. 2004) in order to obtain the vertical displacement map shown in Fig. 4c. We found that 7 cm of subsidence occurred in the hanging-wall of the Kallithea normal fault, which can be interpreted as the result of a co-seismic motion along either (a) the Kallithea normal fault itself, running ESE–WNW and dipping to the south (fault code GR0608 in the NOAFAULTs database; Ganas 2022) or (b) along a ‘blind’ normal fault, parallel and synthetic to the Kallithea Fault whose geometry is unknown. This hypothesis will be investigated by geodetic inversion modelling in “Fault model” section.

In summary, the InSAR data fit the overall neotectonic pattern of the broader Thiva region as the epicentral area is characterised by an extensional structural setting, with fault structures trending approximately NW–SE (Fig. 4b) dipping either towards southwest or towards northeast. However, coseismic deformation has only been detected in the region of the village of Kallithea where it trends about N115°E (Fig. 4c; see also Elias et al. 2021, their Fig. 6). The surface deformation derived from the interferometric measurements (acquired along ascending and descending orbits) reveals a “spoon-like” geometry, elongated in the NW–SE direction, compatible to the orientation and south dip-direction of the Kallithea fault (or to a fault sub-parallel and synthetic to Kallithea F.), with an extent of about 8 km2.

Co-seismic motion of the GNSS stations

We analysed the daily data files (dual-frequency observations at 30-s sampling interval) of two permanent GNSS stations, station THIV belonging to the HxGN SmartNet and the station 014A belonging to the HEPOS Greek state network. Both stations are located to the west of the epicentre (Fig. 1). Station 014A is located about 5 km from the village of Kallithea (the location with the highest macroseismic intensity; see Fig. S1). The processing was done using the GIPSY—Precise Point Positioning (PPP) software and Fig. 5a, b shows the time series of the two stations. At THIV the position time series is 10-year long and stable. With this quality of data, the threshold of detectability is around 1 mm, therefore GNSS certifies that the displacement at this location is below this value (a similar result was obtained by Elias et al. 2021). On 16 December 2020, we also re-occupied a GNSS benchmark of the KAPNET local network (station KALI; Ganas et al. 2007a; Marinou et al. 2015) which is located about 2 km to the northwest of Kallithea (Fig. 1). Figure 5c shows the time series of coordinates from station KALI, detrended for a secular velocity that we assumed to be the same as THIV. We did not see any offset in the time series providing evidence of a co-seismic deformation, so this observation provides a useful constraint on the zero displacement on the InSAR displacement field. The GNSS data confirmed that the ground displacements at the three sites were very small, below noise level of the measurement so they were not used in the geodetic model inversion.

Fig. 5
figure 5

Position time series (E, N, Up) of GNSS station THIV (a) 014A (b) and campaign KALI (c); see location in Fig. 1. The thin grey line indicates the time of the main shock. The deformation signal is not large enough to be separated from the measurement error (noise)

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Fault model

We jointly modelled the line-of-sight (LOS) displacement field retrieved from both ascending and descending InSAR data with a finite dislocation fault in an elastic and homogeneous half-space (Okada 1992) with a typical Poisson’s ratio value of 0.25. We also applied a topography compensation (to relate the source depth to the actual ground surface) and assessed possible residual offsets and ramps in the InSAR measurements. Moreover, the InSAR data were preliminarily down-sampled to a 70-m regular grid in order to reduce the computational load of the inversion process.

Our source modelling strategy is based on a well-established two-step approach capable of computing the distribution of co-seismic slip over the fault plane (Atzori et al. 2008, 2009; Atzori and Salvi 2014); in particular, we use the implementation of the ENVI SARscape® software. The first step is carried out by fitting a uniform slip model to the data (via a nonlinear inversion) to constrain the geometry of the fault plane; in particular, it provides the position of one point of the plane and the values of strike and dip angles. This step also computes the uniform slip value (and its rake) and the fault plane dimensions (length and width). However, these parameters are refined in the second step; in fact, we compute (via a linear inversion; see Fig. 6) the slip distribution over the inferred fault plane by partitioning it into a number of small patches. The distribution of the patches with a non-zero slip value finally defines the extension of the actual slipping part of the plane (rupture area).

Fig. 6
figure 6

Map (top) and 3-D view of the coseismic slip distribution of the Kallithea earthquake rupture. The retrieved model parameters are listed in Table 2. The distributed slip over a mesh of 100 × 100 m2 is displayed as a projection in a map view (top panel) and in 3D view (bottom panel)

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Based on this approach, we performed the preliminary uniform slip modelling without constraints; our best solution is characterised by one fault plane oriented N117.2E° and dipping 53.8° southwest (see the parametric analysis of uncertainties in Fig. S2). Note that these values are consistent with the moment tensor solution found by USGS and NOA (see Table 1). The retrieved parameters highlighted a normal faulting mechanism with a slight strike-slip component (left-lateral). Our nonlinear model (for uniform slip) shows that the slipped area corresponds to a plane with an area of 2.1 × 0.7 km2 that does not reach the ground surface (depth to top fault is 795 m below ground surface). In order to calculate the slip distribution by using a linear inversion, we considered patches of 100 × 100 m2 covering a plane that includes the previously computed uniform slip fault. The final slip distribution over the modelled fault plane is shown in Fig. 6 in map view (top) and 3D view (bottom). We suggest that the maximum slip value is 0.35 m, which is reached at a depth of about 1100 m. Moreover, assuming a value of 0.01 m as the threshold for non-slipping areas—chosen according to the uncertainty resolution of the available data—we measured an actual slipped area of about 4.6 × 106 m2 or about three times the size of the modelled uniform slip fault plane. The retrieved slip distribution is centered on the area of maximum slip but it is elongated in the direction NW–SE, i.e., parallel to the fault’s strike with a deepening towards SE.

In Fig. 7, we present the comparison between the observed geodetic (InSAR) data and the LOS-projected modelled results (non-uniform slip model) for both ascending (first row) and descending (second row) orbits. The first and second columns show the maps (data and models), respectively, while the third column presents their difference (residuals). It can be seen that the retrieved source parameters are able to reproduce the observed signal with excellent approximation, since the residuals are less than 1.5 cm, with large areas showing even much lower values.

Fig. 7
figure 7

Maps showing the modelling of the InSAR data: Left) Data, Middle) modelling results, and left) residuals for ascending (ASC) and descending (DESC) orbits of InSAR interferograms. The middle panel shows Line of sight (LOS) projected displacement maps computed from the retrieved analytical model for the Sentinel-1 interferograms. The outline of the retrieved rupture plane is indicated by the white rectangle, while the magenta lines represent the mapped faults of the area retrieved from the NOAFaults v4.0 database (Ganas 2022). The white star indicates the epicentre (AUTH) of the main shock. Uniform slip modelling results are presented in Fig. S3 in the Supplementary Material

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All the retrieved fault parameters are summarised in Table 2. We also note that the retrieved seismic source has a geodetic moment of 1.42 × 1016 Nm, corresponding to a moment magnitude of 4.7, which is fairly consistent with the value estimated by the NOA MT (MW = 4.6).

Table 2 List of parameters of the seismic fault from linear inversion of the geodetic data
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Kinematic source parameters, seismic energy and radiation efficiency

We collected waveforms recorded at 54 stations managed by the Hellenic Seismological Network (HUSN) and by the National Observatory of Athens Seismic Network (see Acknowledgments section for network details). Based on the maximum epicentral distance—set to 230 km—and the signal-to-noise ratio (SNR), we inferred source parameters at the 15 stations indicated with blue triangles in Fig. S4.

Spectral modelling

We removed the instrumental response and bandpass filtered the original waveforms in the range 0.1–20 Hz by using a 4 poles Butterworth filter. We picked the S phase (on the three components) and used a variable time window to select the signal portion to be inverted. The windowed signal is then tapered with a cosine taper function. We applied the multitaper approach (Prieto et al. 2009) to obtain the amplitude velocity spectra. This approach has been proven to reduce the leakage effect (Prieto et al. 2009). The velocity spectra are then converted into displacement spectra by integration in the frequency domain. The final spectrum is obtained from the vector composition of the three components.

We adopted the Boatwright spectral model (Boatwright 1980) in order to estimate the spectral level at low frequency \({\Omega }_{0}\), the corner frequency (fc) and the quality factor (Q):

$$\Omega \left( f \right) = \frac{{\Omega_{0} e^{{ - \frac{\pi fT}{Q}}} }}{{\left[ {1 + \left( {\frac{f}{{f_{{\text{c}}} }}} \right)^{\gamma } } \right]^{\frac{1}{2}} }},$$
(1)

where f is the frequency, T is the travel time of the seismic phase considered, the exponential function takes into account for the anelastic attenuation, while the parameter \(\gamma\) controlling the spectral decay at high frequency is assumed to be equal to 4. Note also that for the anelastic attenuation, we have assumed a frequency independent quality factor Q.

Following Zollo et al. (2014), we applied the approach of Ben-Menahem and Singh (1981) to account for the geometrical spreading. To this aim, we used the crustal model of Ganas et al. (2014) to compute the take-off angle for each station.

To infer the model parameters, a spectral fitting was implemented via a grid search approach. A weight is assigned to each spectral point depending on the logarithm of the SNR. The noise corresponds to the pre-P noise computed on the vertical component. In a first step, we obtained a preliminary estimate of \({\Omega }_{0}\) by fitting only the flat part of the displacement spectrum. Using the best-fit value \({\Omega }_{0}^{{{\text{best}}}}\), we then inverted for the other two parameters, i.e., fc and Q. We also refine the estimate of \({\Omega }_{0}\) by exploring the range \(\Omega_{0}^{{{\text{best}}}} \pm \sigma_{{\Omega_{0} }}\) obtained in the first step.

To illustrate the results of this modelling procedure, we show in Fig. 8 four examples of observed and computed displacement spectra at stations indicated in each panel, whose locations are given in Fig. S4 and whose corresponding waveforms are shown in Fig. S5 to S8. Furthermore, we computed the average values of those inferred parameters at each station. By using the inferred \({\Omega }_{0}\), we obtained M0 = 1.06 × 1016 (7.63 × 1015, 1.47 × 1016) N m, which corresponds to MW = 4.6 (4.5, 4.7), while we obtained fc = 1.35 (1.05, 1.73) Hz and Q = 371 ± 94.

Fig. 8
figure 8

Observed (black lines) and modelled (coloured lines) displacement spectra at four stations. The modelled spectra are colour coded according to the logarithm of the SNR. The locations of the stations are given in Fig. S4 while corresponding waveforms are shown in Fig. S5 to S8. The grey dashed line represents the pre-P noise selected on the vertical component. The blue line corresponds to the H/V spectral ratio. For each station the best-fit parameters are also reported together with the final misfit, which is shown in bold

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Kinematic source parameters

From the selected model and the obtained model parameters, we computed the static stress drop using the Brune’s (1970) model as \({\Delta }\sigma = 0.44M_{0} /r^{3}\), where r is the source radius. For a circular source model, this latter can be obtained from the corner frequency (\(r = 0.37 v_{{\text{s}}} /f_{{\text{c}}}\), where vS is the S-wave velocity, assumed here as 3228 m/s and corresponding to the mean value between the velocity at the source and that at the receiver). Similarly, we computed the mean slip as \({\Delta }u = M_{0} /\mu \pi r^{2}\), where \(\mu\) is the shear modulus assumed here as 3.3 × 1010 Pa.

We obtained r = 1414 (1102, 1813) m, Δσ = 11.3 (4.8,26.8) MPa and Δu = 0.16 (0.08, 0.30) m. For all the considered parameters, the uncertainties, corresponding to the 95% confidence intervals, have been computed by using the technique proposed by Prieto et al. (2007). It is worth noting that the value of M0 and the mean slip are in agreement with the results obtained from the geodetic data inversion modelling (reported in Table 2).

Seismic energy and radiation efficiency

Seismic energy (ES) is measured from the integral of the square of the ground motion velocity computed in the frequency domain, Ic (Boatwright and Fletcher 1984):

$$E_{{\text{s}}} = \frac{{4\pi \rho v_{{\text{S}}} R^{2} }}{{F^{2} }}I_{c} = \frac{{4\pi \rho v_{{\text{S}}} R^{2} }}{{F^{2} }}\frac{1}{\pi }\mathop \smallint \limits_{0}^{\infty } \omega^{2} \left| {U\left( \omega \right)} \right|^{2} {\text{d}}\omega,$$
(2)

where R is the hypocentral distance, ρ the density and F is the free surface coefficient. We computed the displacement spectrum U(ω) from the previously obtained best-fitting spectral model (Zollo et al. 2014) corrected for the frequency band limitation (e.g., Ide and Beroza 2001). Then, the seismic energy is used to compute the apparent stress τa = μEs/Mo (Wyss and Molnar 1972) where μ is the crustal shear modulus (3.3 × 1010 Pa), and the radiation efficiency as ηSW = τa/Δσ. We obtained ES = 1.81 (0.62, 5.28) 1012 J, τa = 4.24 (1.78, 10.11) MPa and ηSW = 0.375.

In order to obtain a model independent estimate of ηSW, we computed the total energy ET (by neglecting the thermal energy). We used the obtained value for the radiated energy Es while calculating total fracture energy EG from the stress drop map. For this purpose, we converted slip to stress using the approach proposed by Ripperger and Mai (2004). Specifically, the slip map obtained from the geodetic modelling is transformed into the 2D wavenumber (k) domain and multiplied by the static stiffness function K(k) in the same domain to obtain the 2D transform Δσ(k) of the stress drop. Finally, Δσ(k) is transformed in the along strike, along dip space to map the static stress drop on the fault plane.

The Δσ—distribution result is shown in fault-plane view (Fig. 9) and indicates that the maximum stress drop is about 25 MPa, and there is also a stress increase in the upper right (SE) part of the fault-slip map where future events may occur.

Fig. 9
figure 9

Static stress drop map in MPa obtained from the distributed slip map

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The same technique allowed us to compute the specific energy ESG = 2.3 × 105 J/m2. Selecting as the actual slipped fault area 4.61 × 106 m2 (obtained by considering only slip values greater than 0.01 m), the mean stress drop value obtained from the map is 4.5 MPa, which corresponds to the lower limit of the values obtained from the spectral inversion. The fracture energy is EG = 1.11 × 1012 J, providing a total energy Es + EG, of 2.92 × 1012 J and a radiation efficiency \(\eta\) = (ER/(ER + EG)) of 0.62. This result suggests that most of the available energy was radiated during the earthquake. Thus, in addition to the shallow depth, the high efficiency of the earthquake may have contributed to the disproportional observed effects (damage to Kallithea village) compared to the relatively moderate magnitude.

Discussion

Surface deformation and earthquake magnitude (M4.6)

Very shallow earthquakes can induce ground deformation even at magnitudes as low as MW = 4.1 (case of the 14 April 2019 event in Utah; Mesimeri et al. 2021). The mainshock of the 2020 Kallithea sequence is another case of a very shallow earthquake with moderate magnitude. In fact, it is the first time in Greece that InSAR has imaged surface deformation associated with a MW = 4.6 earthquake. We suggest that this is due to the shallow depth of the maximum slip (1.1 km) as provided by the inversion of the geodetic data (Fig. 6). We also mapped an asymmetric pattern of surface displacements across the Kallithea normal fault, i.e., the deformation was mapped entirely on the hanging wall of the Kallithea Fault (Fig. 4). The maximum hanging wall subsidence is estimated at 7 cm (on the decomposed vertical component). The strike-parallel pattern of surface deformation (Fig. 4c) also showed that the deformation resulting from co-seismic motion along the south-dipping normal fault confirms a strong relationship with present-day topography, as the maximum co-seismic subsidence correlates with topographic lows. The depth to the top of the fault is about 800 m (Table 2), indicating that either this fault is a young “blind” structure accommodating crustal extension or the 2020 event was a partial rupture of the Kallithea normal fault that did not reach the ground surface.

This case is interesting in view of very shallow earthquakes that occur from time to time in various places around the world (e.g., Messinia 2011 swarm, Kyriakopoulos et al. 2013; Niger 2017 event, Craig and Gibbons 2022; Le Teil 2019, De Novellis et al. 2020; Sparta earthquake in the eastern US, Figueiredo et al. 2022; the 2007 events in western Australia, Dawson et al. 2008), including places that are supposed to have low seismicity because of low strain rates, such as Le Teil in south France. The occurrence of such shallow earthquakes might be accelerated by human activity (Le Teil), but this does not seem to be the case for Kallithea even though there is industry activity (quarrying) located about 6 km to the NE of the Kallithea fault (i.e., in its footwall). In fact, this activity is not located in the very near field of the fault, which could have played a role assuming a mechanism of normal stress reduction, leading to unclamping (unlocking) of the fault plane.

It is also interesting to examine whether very shallow events are associated with particular conditions of ground water level. Groundwater level changes may increase the pore pressure along the fault plane, thereby facilitating slip. For Kallithea, Kaviris et al. (2022) examined rainfall patterns in relation to the ground deformation time series of station THIV (vertical component) and found a good correlation. It should be noted that rainfall patterns may reflect on seasonal trends in groundwater level, which induces a small-scale (a few cm) of seasonal variations in GNSS data (see also Argyrakis et al. 2020), therefore its use as a proxy may be justified in cases where there are no data from boreholes or wells in the vicinity of earthquakes.

Unfortunately, no water level data are available in the vicinity of Kallithea, making it difficult to prove a link to this earthquake using the GNSS vertical component time series as a proxy. In any case, we do not observe changes more than ± 1.5 cm on the vertical component of the two GNSS stations during the last 9 years preceding the earthquake (THIV and 014A; Fig. 5). In addition, using optical remote sensing (Sentinel-2 data), we have not detected any significant changes in land-use in recent years that could reflect a potential impact of groundwater loading and unloading effects. There may be a yet unknown deep source of fluids that could circulate along the Kallithea fault plane and reached the 2020 rupture plane, but we have no data to support this hypothesis. A seismic swarm that occurred about 10 km west of Kallithea during 2021 (July to October) was associated with deep fluids (Kaviris et al. 2022) but the depth range of these events was between 8 and 12 km. Therefore, we are inclined to suggest that the 2020 Kallithea earthquake resulted from regular tectonic loading since the seismic fault is located close (5–7 km; Fig. 1) to a crustal block boundary (a site of strain localisation) in central Greece, the E–W Asopos rift, bridging the Quaternary rifts of Corinth and South Gulf of Evia (Reilinger et al. 2006; Marinou et al. 2015; Briole et al. 2021; see Fig. S9 for a structural cross-section).

Comparison of fault parameters derived from spectral modelling of seismological data and the geodetic modelling

An interesting result of our study is the similarity of the main kinematic parameters independently obtained by the InSAR data modelling and seismic waveforms analysis (with consistent elastic parameters), as can be seen from the comparison of Tables 2 and 3. In particular, the estimates of moment magnitude and mean slip values are consistent; this suggests the occurrence of a rather simple rupture process involving a single fault and a smooth slip distribution, which is reasonable for a MW = 4.6 magnitude event. Also, the rupture area (RA) values are similar, and are consistent with the Wells and Coppersmith (1994) relationships which, for an event size of MW = 4.6 and a normal fault mechanism, provide RA = 7.9 ± 3.3 km2. The consistency of the two mean slip values may also suggest that almost all the accumulated strain energy was released during the earthquake. This result is indeed confirmed by the relatively high value of the radiation efficiency \(\eta\) = 0.62. Moreover, the high radiation efficiency and high stress drop (mean value of about 11 MPa) could have contributed to the disproportionate effects observed (damages to Kallithea village) compared to the relatively moderate magnitude.

Table 3 Source of the kinematic source parameters inferred from the spectral inversion. The numbers in parenthesis correspond to the 95% confidence interval
Full size table

Origin of shallow rupture

As noted above, the Kallithea earthquake that occurred at very shallow depth and, despite its moderate magnitude, produced noticeable surface deformation. This is not unusual as similar effects have been observed for other events such as the 1986 Mw = 5.2 Cusco earthquake, the 2010 Mw = 5.0 Pisayambo earthquake occurred in Ecuadorian Andes (Champenois et al. 2017), the 2017 Mw = 3.9 Ischia earthquake in Italy (De Novellis et al. 2018; Calderoni et al. 2019), the 2019 Mw = 4.9 Le Teil earthquake in France (e.g., De Novellis et al. 2020) and the 2018 M5.3 Lake Muir earthquakes in western Australia (Clark et al. 2020).

Only for two of the five earthquakes mentioned above stress drop estimates are available. In particular, for the Ischia earthquake Calderoni et al. (2019) have found a stress drop value of 0.01 MPa, while for the Le Teil earthquake De Novellis et al. (2020) reported a stress drop value of 1.3 MPa. These values are significantly lower than the 11.3 MPa inferred for the Kallithea earthquake, which may indicate some difference between these earthquakes.

A common feature characterising those events is the fact that the slip is concentrated in the upper part of the fault. For the 2010 Mw = 5.0 Pisayambo earthquake, Champenois et al. (2017) ascribed this result to a low rigidity medium. A similar argument has been proposed by Ritz et al. (2020) for the Le Teil earthquake. In both studies, a low rigidity value was also used to reconcile the seismic moment estimate obtained from InSAR data and seismological data. However, for the case of the Kallithea earthquake, the seismic moments inferred from the DInSAR data and from seismological data are very consistent, so a low rigidity medium is not a necessary condition.

We propose here two additional arguments that may help to explain the shallow location of the slip. The first one is that there may be a structural barrier at 1.5–2 km depth due to an intersection with a neighbouring fault, such as the known Kallithea fault (Fig. 10) that did not allow the rupture to propagate deeper. This scenario is possible assuming a 54° dip for the 2020 normal fault and a ~ 45° dip for the Kallithea fault (Fig. 10) but our capability of resolving between the two faults is at the limit of the accuracy of the method and data available. The second point relates to the limited energy budget available for the rupture propagation. Specifically, as reported in the previous section, we estimated a specific energy value of ESG = 2.3 105 J/m2. Assuming a linear slip-weakening model and a zero value for the strength excess (i.e., the initial stress is equal to the yielding stress), we can compute the characteristic slip-weakening distance (Dc) over which the energy is dissipated. Indeed, by using the previous assumption, the fracture energy is ESG = ½ Δσ·Dc (Andrews 1976; Guatteri and Spudich 2000) providing a Dc value of 0.04 m, which is a consistent with the expected value for a MW = 4.6 event (Ohnaka 2000; Colombelli et al. 2014). Such critical slip distance corresponds to a critical size of the nucleation zone Lc (= μ Dc /Δσ) of about 117 m. Thus, according to the nucleation model proposed by Ohnaka (2000) the rupture of such a small patch can lead to a small earthquake that prevents the rupture of the deeper and probably stronger fault plane.

Fig. 10
figure 10

Relief map (top) and cross-section (bottom) depicting normal fault planes near village Kallithea. The cross section is shown by a thin black line on the map. The green plane is the 2020 modelled seismic fault, the red plane is the Kallithea SW-dipping fault (continuous line is with 60° dip; dashed line is with 45° dip)

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Rupture plane and segmentation pattern around Kallithea

We further explore the location of our modelled fault (see its surface projection in Fig. 10; rectangle in green colour) and discuss its relationship with the known structures in the area, such as the Kallithea fault (KAF in Fig. 1). We can do that through the analysis of a cross-section A–A′ orthogonal to the strike of both faults (Fig. 10). This shows that our modelled fault and the Kallithea fault are almost parallel and synthetic in terms of kinematics. However, their traces are separated,, by an across-strike distance of about 905 m (Fig. 10; distance between points T and F, i.e. between the line where the projection of our model top-fault plane reaches the surface and the Kallithea fault trace) However, in terms of top-fault depth this distance reduces to 430 m so it is within the expected position uncertainty of the inversion modelling (see error bars in cross-section of Fig. 10 and see Fig. S2 east & north uncertainties), thus making it difficult to declare whether these two faults are separated or coincident.

In order to gain more insights into this point, we re-modelled the InSAR data by performing a linear inversion after constraining all the geometric parameters (position, strike and dip) to be those of the Kallithea fault, using a dip-angle scenario of 60°. The obtained results (see Table S1 and Fig. S10 in Supplementary Material) show that, although the modelled slip is reasonably concentrated around a region at the centre of the fault plane, the corresponding ground displacement fits the InSAR data very poorly. This allows us to conclude that either the observed displacement could be attributed to a new, previously unknown fault different from the Kallithea fault, or the modelling of the Kallithea fault parameters (for example the dip-angle of 60°) needs to be updated.

With the data and modelling currently available, the question of the activated fault plane remains open and certainly worthy of further investigation. In the following, we present some speculative reasoning on the implication of the presence of a newly identified fault.

In terms of dip-angle, Georgiou (2019) reports field measurements for the Kallithea fault between 44° and 52°, so it is geometrically doable that the possible new fault joins the Kallithea one at the depth range of ~ 950–1500 m (Fig. 10). If this is the case, then the 2020 rupture plane may branch off the Kallithea fault. In an alternative scenario, the two faults run parallel until the depth of ~ 1500 m when the 2020 rupture stops. Without more data we cannot exclude either scenario although the kinematic arguments presented in "Origin of shallow rupture" section seem to favor the branching one. On a smaller (local) scale such a fault architecture resembles the “step-up” structures of Stewart and Hancock (1991); however, given that the fault tip is found at a depth of 795 m (under Neogene deposits), it is not possible to observe vertical fractures due to “shattering” (Stewart and Hancock 1990). We note that the size of the 2020 rupture (roughly 2.1 by 0.71 km; according to the results of the nonlinear inversion) and its location would be indicative of a hanging wall migration process and abandoning of the neotectonic Kallithea fault (Goldsworthy and Jackson 2001; Ganas et al. 2004; Palyvos et al. 2005). Another possible explanation could be that the normal fault (Kallithea proper) found on the footwall of the 2020 seismic fault (red line in Fig. 10) is also active, but it is now sharing accumulated strain release with the new fault. The tectonic strain may now accumulate faster on this new, synthetic structure which is of Late Quaternary age and it comprises a single segment.

In the pre-InSAR era, an earthquake of M4.6 near the village of Kallithea would almost certainly be assigned to the Kallithea south-dipping normal fault. With the use of InSAR data and modelling, we end up that this is not necessarily the case and, as such, it offers a structural clue on the accommodation of strain inside young, cross-cutting rifts. The use of the geodetic data and modelling call for a possible presence of two different fault planes that keep separated at least up to a depth of about 950 m (Fig. 10); however, due to the close proximity of the two fault traces (Fig. 10; Kallithea fault and the 2020 rupture plane), and the uncertainties involved in geodetic data inversion there is a possibility that a gentler Kallithea fault geometry may be able to fit the data. In that case, the Kallithea fault ruptured in an unusual way in the sense that it released only part of the accumulated energy without rupturing its entire length (6 km). More data are necessary to resolve the geometry of the active structures near Kallithea such as high-resolution fault zone observations (e.g., Karastathis et al. 2007; Pucci et al. 2016; Porras et al. 2022).

Conclusions

  1. 1.

    We used Sentinel-1 interferograms to map ground deformation due to the Mw = 4.6 December 2, 2020 shallow earthquake near Thiva (central Greece). Because of the shallow depth of the main event and the good coherence of the radar signals, the ground subsidence (which has a maximum value of 7 cm) is well visible with InSAR.

  2. 2.

    Using dislocation modelling, we identified that the seismic fault is striking N117°E, has a dip angle of ~ 54° towards the south, its rupture area is roughly 4.6 km2, it slipped by a maximum amount of 0.31 m with a slip centroid depth of 1.1 km.

  3. 3.

    Whether this fault is separated from or coincident to the well-known neotectonic Kallithea normal fault remains debated due to the accuracy limitations associated with the inversion model and data used; however, forward modelling obtained by fixing a high-angle Kallithea fault geometry fails to fit the data, thus indicating that either there is a new (undetected) normal fault south of the Kallithea fault, or the Kallithea fault parameters need to be investigated using high-resolution techniques.

  4. 4.

    A third possible interpretation is that the two normal faults are separated and merge at depths ~ 950–1500 m, so that the 2020 fault plane branches off the Kallithea fault.

  5. 5.

    The geodetically based model results are in agreement with the source parameters obtained by the analysis of broadband seismological data.

  6. 6.

    The earthquake was characterised by a radiation efficiency \({\varvec{\eta}}\) = 0.62, which indicates that most of the available energy was radiated during the earthquake contributing to the disproportional observed effects compared to the relatively moderate magnitude.

  7. 7.

    Our study confirmed that, also in Greece, moderate (4.5 ≤ Mw ≤ 4.7) shallow events can be traced in InSAR studies and can produce surface displacements useful for fault modelling.

  8. 8.

    The 2020 shallow event near Kallithea highlighted the structural complexity in this region of the Asopos Rift valley as the reactivation of the WNW–ESE structures indicates their significant role in strain accommodation and that they still represent a seismic hazard for this region of central Greece.