1 Introduction

Segments of the same or of different, neighboring faults are sometimes activated during seismic events that occur with a time difference of a few seconds to a few days. In such cases, geodetic data can provide much information on the cumulative surface deformation of the seismic events, but information on the dislocation field of each individual fault may not exist, or it may be too limited to permit dislocation modeling for each event separately. The reason is that the existing inversion algorithms are usually designed to accommodate observations of ground deformation corresponding to the cumulative effect of a single earthquake (e.g., Hreinsdóttir et al. 2003; Elliott et al. 2012; Douilly et al. 2015; Saltogianni et al. 2015) and any information on the dislocation field of individual faults associated with different earthquakes remains practically unexploited.

This was indeed a problem faced with the two earthquakes that hit in 2014, with a difference of a few days, the Cephalonia (Keffalinia, Kefalonia) Island, at the NW edge of the Aegean Arc (Ionian Sea) (Event 1: 2014.01.26, Mw6.1 and Event 2: 2014.02.03, Mw6.0; Sokos et al. 2015). Because of limitations in GPS and InSAR data, of the short time interval (8 days) between these two earthquakes and of the small SNR (signal-to-noise ratio) of the displacements, no complete or unambiguous finite fault models (FFM) for these earthquakes were possible.

For this reason, Merryman Boncori et al (2015) using InSAR data modeled only the second earthquake (Figure S1a). Sakkas and Lagios (2015) based on GPS survey data presented fault models for both earthquakes but a priori assuming that they originated from the same fault structure (Figure S1c). Briole et al (2015) used InSAR data to compute first a “composite” fault model for both earthquakes, and for the second earthquake separately, and then the first event was indirectly modeled (Figure S1d). Furthermore, in some of these studies, in order to reduce misfits between observations and model predictions, there were adopted complex fault patterns not justified by strong motion records, while none of the proposed models was correlated with geological structures. FFM from seismological datasets were also not without problems, and in fact it proved possible to compute finite fault models (FFM) only for the first earthquake (Figure S1b; Sokos et al. 2015).

Such results indicate that even using new data and techniques, modeling of earthquakes in this area remains a real challenge, especially because the wider Cephalonia area has been hit by two major earthquakes in the last 70 years [the 1953 M7.2 earthquake (Stiros et al. 1994) and the 1983 M7.0 earthquake (Scordilis et al. 1985)] and represents the seismically most active part of Europe (Yolsal-Çevikbilen and Taymaz 2012).

In this article, we try to respond to this computational challenge and present refined FFM of the 2014 earthquakes using new and published GPS data and an updated inversion algorithm. Results were validated by sensitivity analyses and comparison with InSAR data. Obtained FFM were then used as basis for two tasks.

First, to investigate the geometry of the 2014 faults in relation to the observed extreme seismic accelerations, which reached a PGA of 0.77 g during the second event (Theodoulidis et al. 2015); a value surpassed only by few other earthquakes in a global scale (for example, the Northridge earthquake, PGA=0.86 g; Bardet and Davis (1996), the 2008 Iwate–Miyagi earthquake, PGA = 4.02 g; Aoi et al. (2008) and the 2011 Christchurch earthquakes, PGA = 0.92–2.00; Fry et al (2011); Bradley et al. (2014)).

Second, to investigate the relation of the 2014 fault with the complex tectonic pattern of the area, and to shed light on the enigmatic juxtaposition of shear and compression at the NW edge of the Aegean Arc. The latter represents a young and active fold-and-thrust belt (Fig. 1c), while focal mechanisms of earthquakes show an unclear combination of compression and shear (Shaw and Jackson 2010). One of the questions arising is the significance of the recent series of strike-slip earthquakes (Mw6.0–6.6), propagating along the west shores of Cephalonia and Leucas (Lefkada, Lefkas) between 2003 and 2015 (Fig. 1a). Gradual propagation of ruptures in a fault system is clearly a topic of major interest; for example, in the North Anatolian Fault and its eastward continuation in the North Aegean Trough (Bohnhoff et al. 2013; Saltogianni et al. 2015), or the series of normal faulting earthquakes in central Italy (Cheloni et al. 2017). In this study, we examine also whether a string of strike-slip faults at the NW edge of the Aegean Arc reflects a transform fault (the Cephalonia Transform Fault, i.e., the plate boundary), possibly crossing Cephalonia, or if it represents a young strike-slip fault landwards of the plate interface (i.e., not a plate boundary), analogous to other strike-slip faults in the area (Fig. 1; Feng et al 2010; Stiros and Saltogianni 2016).

Fig. 1
figure 1

The 2014 Cephalonia earthquakes: summary of data and structural and geodynamic background. a Epicenters (stars), focal mechanisms and GPS-based Finite Fault Models (FFM) of the two main 2014 earthquakes (2014a, 2014b in red) derived in this study, the 2003 (2003a, 2003b) and the 2015 Leucas earthquakes (in black, after Saltogianni et al. 2017), as well as the 1983 event (in orange). Map fault projections are indicated by dotted lines, fault surface traces by solid lines, dashed when extrapolated. Focal mechanisms after Kiratzi and Langston (1991), Zahradník et al (2005), Sokos et al (2015) and Saltogianni et al (2017). Green circles indicate M > 3 aftershocks of the 2014 earthquakes for almost 1 month after the second event (after Sokos et al. 2015). Main thrusts in Cephalonia are after Underhill (1989) including those reactivated in 1953 (after Stiros et al. 1994). The 2003, 2014, 2015 earthquakes define a fault line along or close to the west coasts of the two islands, regarded as an immature, segmented strike-slip fault (Cephalonia–Leucas Fault, CLF), and not as a plate boundary (Cephalonia Transform Fault, CTF). Two lines of thrust and strike-slip faults offshore, derived from seismic profiles (redrafted after Cushing (1985)), indicate two scenarios for the eastward boundary of the plate interface, landwards of which is located the CLF and other strike-slip faults. b Aegean Arc, location map. Highlighting indicates areas in both the west and the east edge of the Aegean Arc (Rhodes area) in which compression and shear are superimposed (Saltogianni et al, 2017; Howell et al 2015). AEF: 2008 Achaia-Elia fault. c Schematic geologic cross section [AA′ in (a)], after Underhill (1989), showing imbricate thrust sheets and evaporite intrusions above an evaporite horizon. Faults reactivated during the 1953, M7.2 seismic sequence and those modeled for the 2014 earthquakes are marked

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2 Structural and Geodynamic Background

The wider region of the south Ionian Sea represents since Lower Pliocene a young and active fold-and-thrust belt, characterized mainly by imbricate carbonate thrust sheets above an evaporitic decollement (Fig. 1c; BP Co. Ltd 1971; Mercier et al. 1979; Bornovas and Rondoyanni 1983; Underhill 1988). This deformation is reflected to an impressive relief (anticlines next to major bathymetric escarpments, Fig. 1a) as well as in the distribution of uplifted Quaternary marine sediments (Sorel 1976; Stiros et al. 1994) and of coastal uplifts accompanying major earthquakes (Pirazzoli et al. 1996), especially the 1953 M7.2 Cephalonia earthquake (Stiros et al. 1994). This structural pattern is not readily reflected in focal mechanisms of earthquakes, which, however, clearly indicate a combination of thrusting and strike-slip (Shaw and Jackson 2010), while shear seems dominant in more recent and smaller earthquakes (Konstantinou et al. 2017).

A benchmark in the understanding of the geodynamic background of the area was the 1983 M7.0 oblique slip earthquake SW of Cephalonia, which led to the hypothesis of the Cephalonia Transform Fault (CTF, or KTF) controlling the tectonics at the NW edge of the Aegean Arc (Scordilis et al. 1985; Kiratzi and Langston 1991; Louvari et al. 1999; Sachpazi et al. 2000), although Anderson and Jackson (1987) and Baker et al (1997) suggest that it may have been a thrust event. The location and structure of CTF are, however, uncertain and a matter of debate (see Saltogianni and Stiros 2015). It is usually assumed to represent not a typical transform fault, but a segmented oblique slip fault running along the west coasts of Cephalonia and Leucas (Louvari et al. 1999) or to coincide with a major bathymetric escarpment > 15 km west of the coast (Cushing 1985; Clément et al. 2000; Chamot-Rooke et al. 2005; Shaw and Jackson 2010).

CTF has been associated with a series of recent essentially strike-slip earthquakes along the western coasts of Cephalonia and Leucas in 2003, 2014 and 2015 (Fig. 1a) and it has even been argued that it cuts through land in the Paliki peninsula, in western Cephalonia (Karakostas et al. 2015; Fig. 14 in Briole et al. 2015). Although the scenario of a transform fault offshore Cephalonia is consistent with the bending of the NW Aegean Arc (Fig. 1), it is not easy to accept that the structural pattern of the area, summarized in Fig. 1c, is controlled by a transform fault because it cannot produce significant vertical dislocations. In addition, it is not easy to accept that a rapidly deforming plate boundary crosses the young relief of Cephalonia for a few million years without having left any lithological or geomorphological traces. For this reason, in more recent studies, the 2014 Cephalonia faults have been separated from a typical transform fault (Avallone et al 2017).

An alternative hypothesis was presented by Saltogianni and Stiros (2015) and Saltogianni et al (2017), who proposed a shear zone containing recently reactivated strike-slip faults landward of the plate interface. This hypothesis explains the juxtaposition of thrusting (for example during the M7.2, 1953 earthquakes, Stiros et al. 1994) with strike-slip observed in the recent earthquakes.

A key, however, for the understanding of the structural pattern of the area is the tectonic pattern offshore Cephalonia and Leucas. This has first been studied by Cushing (1985), who analyzed seismic profiles collected for the needs of the petroleum industry and which show some major thrusts (Fig. 1a), but which have been rarely only examined so far (Clément et al. 2000; Kokinou et al. 2006). This evidence is examined in Sect. 5.4.

3 Methodology

Inversion is based on the TOPological INVersion algorithm (for full analysis, see (Saltogianni and Stiros (2015) and Saltogianni et al (2015, 2017)), but is expanded to include techniques to accommodate observations of dislocation of individual faults and of their cumulative effect, as well as a collocation-type analysis of observation uncertainties.

3.1 Observation Equations

Inversion of observations of surface deformation is based on the elastic dislocation model of Okada (1985). The physical model corresponds to a set of m observation equations

$$ f({\mathbf{x}}) - {\varvec{\ell }} = {\varvec{\upupsilon}} $$
(1)

where bold symbols indicate vectors, f typical Okada (1985) equations, x the unknown parameters of the activated fault (represented by a matrix of n unknown scalar variables), ℓ a vector of m scalar observations and υ a matrix of m unknown scalar observation errors, accompanied by a m × m diagonal variance-covariance matrix σ, with diagonal elements corresponding to variances σ2. For a single fault, n =9.

In the case of two faults x1 and x2, the vector x of the unknown fault parameters takes the form x = {x1, x2}, and n =18. In the case of observations ℓ1 and ℓ2 relating to only one of these faults, the observation system in (1) takes the form

$$ f({\mathbf{x}}^{1} , \, {\mathbf{0}}) - {\varvec{\ell }}^{1} = {\varvec{\upupsilon}}^{1} {\text{or }}f({\mathbf{0}},{\mathbf{x}}^{2} ) - {\varvec{\ell }}^{2} = {\varvec{\upupsilon}}^{2} $$
(2)

with x1 and x2 indicating vectors with 9 variables and 0 indicating a null vector with 9 variables.

In the case of an observation expressing the cumulative displacement ℓ1,2 of the two faults reactivated during two earthquakes, Eq. (1) yields

$$ f({\mathbf{x}}^{1} , \, {\mathbf{0}}) \, + \, f({\mathbf{0}},{\mathbf{x}}^{2} ) - {\varvec{\ell }}^{1,2} = {\varvec{\upupsilon}}^{1,2} $$
(3)

Hence, the system of observation Eq. (1) is modified to accommodate various combinations of ground displacement due to the activation of two or more faults (Eq. (2)) and of their cumulative effect (Eq. (3)). The adopted inversion algorithm can accommodate various types of observations of ground dislocation, as well as the fusion of different types of data. A requirement for the latter is that uncertainties are well defined (Eq. 4; Saltogianni and Stiros 2015).

3.2 Stochastic Equations and Constraints in the Inversion, a Unified Approach

Equation (1) includes the term υ which represents the unknown error of each observation and which is described by an a priori known matrix σ, assuming all elements of υ are free of gross errors. This term υ transforms observation equations to stochastic equations, in analogy to least squares (e.g., Mikhail 1976).

Introducing an unknown scalar parameter k, the observation Eq. (1) is transformed into the observation inequality (4), which practically defines confidence intervals for each observation:

$$ | \, f({\mathbf{x}}) - {\varvec{\ell }}| \, < k{\varvec{\upsigma}} $$
(4)

To exclude non-realistic solutions, additional constraints in the form of Eq. (4) are possible to be included in the observation system. These constraints may define threshold intervals (i.e., intervals of possible values) for the aspect ratio r and/or seismic moment Mo of a fault. These constraints are in the form:

$$ g({\mathbf{x}}) \, < {\mathbf{c}} $$
(5)

where c is a matrix of dimensions cq  × 1, and q is the number of constraints.

Hence, the overall physical problem is transformed into a system of inequalities containing a set of observations (4) and a set of constraints (5).

3.3 Solution

At first, we adopt a search space which determines the possible values of each unknown variable xi that defines the geometry and the slip of the fault(s). This search space is defined for each variable by an equation of the form:

$$ {\mathbf{x}}_{\hbox{min} } \le {\mathbf{x}} \le {\mathbf{x}}_{\hbox{max} } $$
(6)

The limits of the search space are selected using geological and seismological evidence and with the criterion to exclude only physically unreasonable values. The search space is then discretized to form an n-dimensional grid (hyper-grid) G with p points in total.

The compound system of inequalities (4) and (5) is then solved using exhaustive repeated searches for all points of grid G for different values of k based on Boolean logic. The optimal value of k, which represents the only unknown variable in this process, permits to identify one or more closed n-dimensional spaces which contain all the grid points which are within the confidence intervals defined by Eq. (4) and the constraints of Eq. (5).

Solutions and their variance–covariance matrices for each closed space are then determined as statistical moments. The computation of an optimal solution corresponding to the mean value of all possible solutions around the “true” solution derived from the “scanning” of the search space excludes the possibility this optimal solution to be trapped into local minima, as is the case with inversion techniques leading to point, minimum misfit solutions derived from sampling of the search space (cf. Sambridge and Mosegaard 2002; Menke 2012). In this study, misfit functions are however computed a posteriori, to further assess results.

3.4 Collocation-Type Accommodation of Observation Uncertainties

The above approach fully accommodates uncertainties in observations in analogy to least squares (cf. Mikhail 1976). According to the conventional collocation, if observations tend to somewhat deviate from the physical model, but not at a level to be classified as outliers, an additional unknown error, i.e., a new unknown variable, is introduced (Dermanis 1984; Kotsakis 2007). In our analysis, the unknown error was accommodated assuming higher uncertainty, through an empirically defined collocation factor CF. Details are explained in Supplementary material Text S1.

4 Data, Computations and Results

In the study area, both GPS and InSAR data are available. However, poor coverage by InSAR scenes (the critical westernmost part of the island is in many cases not covered, no couples of ascending and descending orbits available) in combination with the highly fragmented and anomalous relief of Cephalonia does not permit unambiguous estimations of LOS displacements. To overcome this problem, other investigators have constrained the pattern of radar fringes using GPS data (e.g., Briole et al. 2015), an approach which may introduce significant errors (Pritchard et al. 2002), exceeding the displacement signal especially in cases of low SNR values.

For this reason, to avoid possible bias from the fusion of GPS and of InSAR data, and to avoid using correlated data (GPS and GPS-constrained interferograms), we decided to include in the inversion only GPS data and to use the quasi-independent InSAR data to assess the results.

4.1 Co-seismic Displacements

In our inversion, we included data from 22 continuous and selected survey stations, shown in Fig. 2. We used survey stations, which were transformed into continuous after the first event (stations ASSO and KAT1), and survey data from Sakkas and Lagios (2015) collected a few years before the first 2014 event and some weeks after the second. Therefore, some of the stations of Fig. 2 record well both events and some describe only their cumulative dislocations. Such inhomogeneous information was not possible to be accommodated in the past because of limitations of existing algorithms, but all the above data are exploited by our algorithm.

Fig. 2
figure 2

Uniform FFM (left column) derived from the inversion of GPS observations (black vectors with uncertainty ellipses, red vectors modeled dislocations). a Fault model of Event 1 (earthquake of 2014.01.26), modeled as a strike-slip fault reaching the surface (red line) and marked as 2014a. b Fault model of Event 2 (earthquake of 2014.02.03, marked as 2014b). The projection of the fault is shown as a rectangular with black dotted lines and its trace on the surface by a dashed red line. c Combination of the two fault models. The right column shows misfits between GPS observations and model predictions for the three axes

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Care was taken to cover the meizoseismal area (Cephalonia and south Leucas) with data of known accuracy and with a nearly uniform coverage. However, two survey stations in NW Cephalonia (stations 15 and 16 in Sakkas and Lagios 2015) were excluded as outliers because they indicate excessively large displacements (Briole et al. 2015), probably reflecting local ground instability in the vicinity of ground fissures.

We analyzed data collected with a sampling rate of 30 s and obtained daily solutions before and after each earthquake, or before and after the whole seismic sequence. In the absence of a major seismic event between 2010 to 2013 and because a nearly perfect linear secular trend characterizes the stations coordinates in the area before the earthquake period [Figure S2 and Hollenstein et al. (2008)], the estimation of the interseismic velocity of campaign stations ASSO and KAT1 was based on linear extrapolation (Figure S2) using the secular trend of four reference stations in the wider area (Figure S3). Additional details for data processing and analysis can be found in supporting information Text S2 (cf. Herring et al. 2010a, b; Altamimi et al. 2011). For the survey data of Sakkas and Lagios (2015), we adopted their reported displacements.

Data used have different error uncertainties, because of their style collection and of their analysis method. For this reason, we calculated uncertainties repeatedly applying the law of error propagation in each set of observations. However, this law covers only random errors and not colored noise which characterizes GPS displacements (Williams 2003; Moschas and Stiros 2013). To overcome this problem, we adopted flat errors for different classes of displacements. This permits to avoid circumstantial differential weighting while conditions in data collection and processing were uniform. In addition, empirical collocation factors (CF) making more realistic the uncertainties of some data were assigned to certain observations to account for local deviations from a uniform slip model. Details are provided in Text S1.

4.2 Post-Seismic Displacements

The available data do not give evidence of significant pre-seismic or post-seismic movements for the stations and intervals covered; the only exception was station ASSO (Figure S4). In this station, there is clear evidence of post-seismic vertical movement of 1.3 cm 3 days after the second main event of the seismic sequence; this displacement is about four times larger than the corresponding co-seismic displacement (Table S1; Figure S4). Such a step-type displacement cannot be assigned to a large aftershock of the seismic sequence (see Fig. 1a; also Karastathis et al. 2015), and cannot be regarded as conventional post-seismic motion because it is not characterized by a smooth, logarithmic or exponential pattern (Figure S4; Savage and Prescott 1978; Marone et al. 1991; Langbein et al. 2006; Kreemer et al. 2006; Ding et al. 2015). The likely explanation is that this motion may reflect ground deformation related to mobilization of evaporites by the strong motion, i.e., their transformation into a nearly viscous mass tending to rise to upper levels, especially through intrusion along faults (Stiros et al. 1994; De Paola et al. 2008; Davison 2009). As will be shown below, this result is consistent with the association of the second event (2014b) with a thrust rooting on a layer of Triassic evaporites (Fig. 1c). Evaporites indeed are widespread in the area both as detachment layers and as intrusions along faults (BP Co. Ltd 1971; Brooks and Ferentinos 1984; Underhill 1988; Stiros et al. 1994; Velaj et al. 1999) have an ambient seismic signature (Tselentis et al. 2006) and can be mobilized by a strong motion. However, their mobilization can be recorded with a significant time lag from the main shock (cf. Aftabi et al. 2010; Barnhart and Lohman 2013; Nissen et al. 2014).

4.3 Displacement Pattern

Computed co-seismic displacements are summarized in Figs. 2 and S4 and in Table S1. These stations indicate either cumulative displacements of the two seismic events or strictly co-seismic displacements of each of the two events. The overall displacement pattern indicates slip to SW except for stations KIPO, 17 and 18, on the west coast of Cephalonia which indicate (cumulative) slip to the north. Displacements are highly attenuated to the north (station PONT) and to the east (station VLSM; see Fig. 2 and Table S1). In combination with preliminary and detailed focal mechanisms (Sokos et al. 2015), this pattern clearly indicates that both events were essentially associated with strike-slip faulting and that stations to the west constrain the western edge of the horizontal projection of the fault. This pattern covers all stations, but a diverging east component of displacements at the near stations KAT1 and KEFA exists (Fig. 2). This was regarded as local perturbation in the displacement field, either because of secondary tectonic faulting or of near-surface ground instability effects, and was accommodated by empirical collocation factors CF>1 in the corresponding measurement uncertainties (see Text S1).

4.4 Uniform Slip Modeling

Strong motion records (Theodoulidis et al. 2015) indicate that each of the two events of the 2014 sequence can be assigned to a single fault, which, within the resolution of the available data, can be described by the Okada (1985) elastic dislocation model (uniform slip model). Strong motion data do not a priori justify more complicated pattern of faulting adopted only to reduce the misfit between model predictions and observations (Merryman Boncori et al. 2015; Briole et al. 2015).

For this reason, we assumed that each of the two 2014 earthquakes (events) can be assigned to the rupture of an individual fault of uniform slip, defining thus a system of inequalities (Eq. 4) with n =18 unknown variables and 84 observation equations in total (six stations with 3 components of displacement for the two earthquakes, and sixteen stations describing cumulative displacements) and the computed weights (Table S1). Eight additional inequalities reflecting constraints for the minimum and maximum accepted values of the aspect ratio and of the seismic moment for each of the two faults were also introduced to discard invalid solutions (see Sect. 3). These constraints are for the aspect ratio r of the faults, with rmin = 1.2 and rmax = 4.0 (Nicol et al. 1996; Stock and Smith 2000), and the seismic moment Mo of the first event was constrained in the interval [1.0–2.6] × 1018 Nm and for the second in the interval [0.7–1.7] × 1018 Nm. These values were selected to limit earthquake magnitudes in the range of 5.8–6.3 as suggested by seismological evidence (Table S2).

For each fault, a broad search space was adopted covering all possible solutions based on seismological and geodetic evidence and excluding only non-plausible solutions (Table S2). This search space was transformed into Grid G1 in the R18 space. The resolution of Grid G1 was selected so that the total number of grid points was kept below the level of 1011, imposed by computational reasons.

After an exhaustive search in Grid G1 defining the search space for the inversion, the algorithm identified only a single cluster of possible solutions (a closed topological 18-dimensional space containing the “true” solution, see Text S1). Search was subsequently refined using a nested Grid G2 around the cluster of points-solutions derived in G1, but with smaller spacing in each axis, and a new cluster of successful points was identified (Figures S5, S6). Finally, from this cluster of gridpoints, we computed the first and second statistical moments corresponding to the optimal solution and its variance–covariance matrix (Figure S7).

Results are summarized in Fig. 2 and in Table S2, indicating a solution characterized by a small misfit (\( \chi_{v}^{2} \) = 11). The obtained solution suggests that Event 1 was associated with an about 10 km deep strike-slip fault at the SW coast of Paliki peninsula, and Event 2 with a blind shallow oblique slip fault, ~ 5 km deep (see Fig. 1).

4.5 Variable Slip Modeling

Because slip in a fault is proportional to dislocation, it is possible to adopt the overall geometry of uniform slip faults, and using the same observations, to compute at a second stage the distribution of slip in various parts of the faults, i.e., to compute variable slip faults. For this analysis, we used the software of Wang et al. (2012), which is based on least squares optimization (gradient method). Input was the fault geometry defined previously for both faults, divided into patches of 2 km × 2 km wide. For the estimation of variable slip models for the two uniform slip faults computed in Sect. 4.4, we used observations of cumulative co-seismic displacements; this was to exploit all the available information.

Results are summarized in Fig. 3 and indicate that for the first event (fault, 2014a earthquake) two areas of slip of the order of 1.10 and 1.00 m are identified at the south edge of the fault at depth of 6.5 km and at the north edge at depth of 1 km, respectively. For the second event (fault, earthquake 2014b), slip distribution is concentrated almost at the north/center part of the fault reaching up to 1.00 m slippage at a depth of 3.5 km.

Fig. 3
figure 3

Variable slip models of the two faults of the 2014 Cephalonia earthquakes. a Fault geometry derived from uniform slip inversion and used for the variable slip inversion. Observed displacements (black error with error ellipses) and modeled displacements for the variable slip model are shown in (a) and in (b). (c) Obtained variable slip on the fault surfaces of (a). (d) Computed (Briole et al. 2015) and predicted interferograms derived from the variable slip model of (c). (e) Location map of the GPS stations used in the inversion. Stations recording displacements of both events are denoted with rhombs and those recording cumulative displacement of the two events with circles

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4.6 Model Validation Using InSAR Data

In a second step, we validated our GPS-based dislocation model with interferograms computed by Briole et al. (2015) (Fig. 3d). The latter were derived from the analysis of radar data by RADARSAT-2 for the period 12 Dec 2011–11 Feb 2014, covering the whole seismic sequence and by COSMO-SkyMed for the period of 2 Feb 2014–10 Feb 2014 covering the second event only (Briole et al., 2015) and were constrained by GPS data because of the discontinuity of the land.

Using our computed variable slip fault models, we computed the predicted 3-D displacement fields. These displacements were projected on the plane of the radar line-of-sight (LOS) of the corresponding satellite passes and then were transformed into fringes of the corresponding radar signals in an idealized ground surface.

The patterns of both sets of interferograms are very much similar and observed differences are smaller than those typically expected (e.g., Pritchard et al. 2002), making this comparison meaningful. Observed differences are identified, especially in the NW part of Paliki where the GPS coverage is relatively poor.

4.7 Sensitivity Analysis

A previous study has indicated that small changes of the weights (adopted uncertainties of observations) may lead to significant changes in the model (Saltogianni and Stiros 2015). For this reason, we further investigated whether our proposed FFM for the 2014 Cephalonia earthquakes is stable and not depending on small variations of the observations and of their weights. We tested different datasets and weights and selected to show three alternative models A, B, C which are representatives of the wide classes of input data. These alternative models are summarized in Figure S8 and correspond to uniform slip models.

Alternative models A and B are based only on (semi-) continuously operating GPS stations reported in this study (i.e., excluding all survey data of Sakkas and Lagios, 2015). The two models A and B differ as far as the observations uncertainties are concerned (weights of observations and collocation factor CF, see Table S1). These two models share the same structural pattern with our preferred model, though with small variations, mainly in the strike and dip of the first event (Figure S8). These two models indicate that our preferred model of Fig. 2 is stable.

The input of model C on the contrary was selected to be compatible with that of previously presented models, which are characterized by absence of control of ground displacement (absence of InSAR coverage or of GPS stations) at the westernmost part of the island. This corresponds to an inversion with input similar with those in models A and B but omitting station KIPO.

The impact of the change in the configuration of observables in this model is quite clear and leads to shifting of faults to the east, as can be derived from Figure S8c. Interestingly, fault model C is somewhat similar to some previously proposed models, which also have no good control of the deformation at the westernmost part of Paliki (see Figure S1).

5 Discussion

5.1 Model Assessment

Certain lines of evidence indicate that the proposed FFMs for the 2014 Cephalonia earthquakes are reliable and precise within their uncertainty limits clearly defined in Figures S6–S8.

First, in the modeling care was taken so that observations are significant and optimally distributed in space. Hence, statistically significant estimates of the unknown variables are derived (Figs. 2 and S7, Table S2). Perhaps a more complicated model, for example a double-fault model corresponding to a listric fault for the second event would have been more realistic (cf. Briole et al. 2015) and could reduce the mean misfit error, but it is not justified by the existing data.

Second, our preferred model was validated by a sensitivity analysis showing its overall stability (Sect. 4.7) and through comparison with InSAR data (Sect. 4.6).

Third, the fault model derives from an exhaustive search and corresponds to the mean value of nearly ~1300 distinct fault solutions satisfying observations within their error limits (Table S2) and not from sampling for point solutions, which are frequently trapped in local minima (Sambridge and Mosegaard 2002; Menke 2012). This is because misfits do not guarantee an optimal solution (see also Saltogianni et al. 2017) since they are functions of observations which contain uncertainties and, hence, are also characterized by uncertainties, as is dictated by the theory of error propagation (cf. Mikhail 1976).

Fourth, the high quality of the model (Fig. 2) is confirmed by the small spread of estimates for values for each variable and the corresponding values of the elements of the genuine variance-covariance matrices (Figure S7). In fact, as Figures S5 and S6 indicate, the only variables not very well controlled are the length and width of the fault of the first event and the strike and rake of the second event. Still, these uncertainties are relatively small, and much smaller than in previous models.

5.2 Fault Pattern of the 2014 Earthquakes

In summary, the first earthquake (Event 1) of the 2014 Cephalonia seismic sequence seems to have been associated with an about 11 km deep strike-slip fault close to the west coast of Paliki peninsula, probably cutting through the entire upper plate (cf. Clément et al. 2000). Although this fault is well controlled, the uncertainty mainly in its length and width may indicate no significant statistical difference from the model of Sokos et al (2015) (Figure S1b) which is derived from analysis of teleseismic data. No clear geomorphic evidence for this fault exists, although it is possible that it corresponds to lineament abutting to the north to a fault-controlled embayment (Fig. 3). This is not, however, a problem because the landscape of Paliki is dominated by easily eroding soft sediments in a region with high rainfall record, while strike-slip faults cutting pre-existing thrusts are not unusual in the wider area (for example, BP Co. Ltd 1971; Stiros and Saltogianni 2016).

The second earthquake (Event 2), on the other hand, is modeled as an oblique slip, shallow low-dip fault. This fault correlates excellently with a major thrust rooted on a layer of Triassic evaporites (Fig. 1c), possibly a listric fault. Hence, the predicted surface trace of its extension in Fig. 1a may only indicate its western boundary. Such faulting seems to have played a significant role in the shaping up of the landscape in the area, since uplift of Upper Quaternary marine sediments are most probably related to repeated reactivation of this thrust. Marine quaternary uplifted sediments in the Vardiani Islet, south of Paliki peninsula, are consistent with the extension of the fault southwards (Fig. 4b).

Fig. 4
figure 4

Finite Fault Models (FFM) proposed in this study for the two main events of the 2014 Cephalonia seismic sequence (a) FFM (projections) superimposed on a Digital Elevation Model (b) and on map with main tectonic characteristics of Cephalonia (after Bornovas and Rondoyanni 1983; Underhill 1988; Stiros et al. 1994). Late Quaternary uplifts (based on Sorel 1976 and Stiros et al. 1994) are shown. Focal mechanisms after Sokos et al. (2015). Triangles indicate location of strong motion sites. (c) Geologic cross section AA′ in (b), showing also recent faulting (modified after Underhill 1988). The fault of event 2 (2014b) may be listric, with its surface extension farther east than that modeled assuming a planar fault

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5.3 The Cephalonia–Leucas Fault (CLF)

The 2014 Cephalonia faults, so far, have been viewed in the context of the recent earthquakes occurred along the west coasts of Leucas and Cephalonia between 2003 and 2015 and have been explained either as segments of the Cephalonia Transform Fault (Karakostas et al. 2015; Briole et al. 2015) or of a manifestation of a shear zone (Saltogianni and Stiros 2015; Sokos et al. 2015; Saltogianni et al. 2017). Alternatively, they can also be regarded as segments of a conventional strike-slip predicted by Cushing (1985) to explain the structural pattern of SW Leucas.

The FFMs of the 2014 earthquakes and the corresponding fault models of the 2003 and 2015 earthquakes, presented in Fig. 5, permit to zoom into the tectonics of the Ionian Sea in the last years. The fault segments that ruptured the last decades in the Ionian Sea seem to correspond to a typical quasi-linear, multi-segment conventional fault (cf. Figure 36 in Battaglia et al. 2013) for two main reasons.

Fig. 5
figure 5

Summary of the detailed FFM of recent earthquakes in the Ionian Sea. a Perspective view of the FFM of the 2003 Leucas, 2014 Cephalonia and the 2015 Leucas earthquakes, using a homogeneous variable slip scale. The FFM of the 1983, M7.0, earthquake based on teleseismic data is also shown. b Spatio-temporal pattern of faulting in (a) projected in an SN oriented profile, testifying to nearly similar fault-lengths, overlaps and a possible seismic gap

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First, their pattern is similar to that excepted in the case of linkage between small/medium scale fractures in a major nascent/young fault (Moir et al. 2010; Manighetti et al. 2015) and is different from that characterizing typical, multi-segment transform faults (Rodríguez-Pérez and Ottemöller 2014).

Second, in the NW Aegean Arc, a boundary rapidly deforming since Lower Pliocene (see Sect. 2), a lithological offset of several kilometers is expected, but has not been noticed in the detailed mapping of Cephalonia (BP Co. Ltd 1971; Bornovas and Rondoyanni 1983).

Therefore, we regard this fault line as a conventional young or nascent strike-slip fault, thereafter Cephalonia–Leucas Fault (CLF).

The kinematics of CLF are interesting. The segments of CLF seem to have been activated with a time lag ranging between a dozen of seconds (during the 2003 seismic sequence; Benetatos et al. 2005; Zahradník et al. 2005) and a dozen of years (2003 and 2014 earthquakes), as schematically summarized in Fig. 5b, while the overall activity of CLF is characterized by an overall SW progression between 2003 and 2015, but also with some delayed backward advancement necessary to rupture a strong asperity between the two 2003 segments (Fig. 5; cf. Sokos et al. 2015).

Fault segments activated between 2003 and 2015 are in their majority characterized by similar length and high dip, they are rooted at a depth of approximately 12 km and they are also characterized by invariability in the fault-plane slip distributions (Perrin et al. 2016). The only exception is Event 2 (2014b) of the 2014 Cephalonia seismic sequence which corresponds to a low-angle fault and it seems to accommodate shear deficit of layers above a major evaporite detachment surface (Fig. 1c).

Segmentation of CLF is probably controlled by ruptures of pre-existing faults in a highly fragmented crust with asperities limiting rupture propagation. This is probably also the likely cause of gaps, such as that filled by the 2015 Leucas earthquake (Fig. 5b; Sokos et al. 2015; Saltogianni et al. 2017). This may suggest a possible gap NW of Cephalonia (Fig. 4b; cf Avallone et al 2017). However, the possibility of a “soft” segment, with strain released through small aftershocks of the 2014 event (Fig. 1a) cannot be ruled out. The NE limit of CLF is well constrained by the relief, but not its SW edge; for this reason, further progression of faulting to SW in the vicinity of the poorly constrained 1983 fault cannot be excluded (Fig. 5b).

5.4 Shear and Plate Interface in the NW Aegean Arc

The 1983 strike-slip earthquake inspired the concept of the Cephalonia Transform Fault (CTF), of a transform fault boundary at the NW edge of the Aegean Arc (Scordilis et al. 1985; Kiratzi and Langston 1991; Louvari et al. 1999; Sachpazi et al. 2000; Shaw and Jackson 2010). The trace of this fault was, however, unclear, and its trace was shown at a distance of a few km west of Leucas (i.e., to correspond to what is recognized here as CLF) to > 15 km west of Leucas (see Introduction and Saltogianni and Stiros 2015). This last alternative was more likely, because this boundary was derived from a 3 km bathymetric low (Cushing 1985; Chamot-Rooke et al. 2005; Shaw and Jackson 2010) (see Fig. 1a).

To shed light to this problem and put constraints on the plate boundary in the area, we followed the same approach with Cushing (1985) and examined the available seismic profiles from the petroleum industry that he published. From these seismic profiles, occasionally only examined so far (Kokinou et al. 2006; Clément et al. 2000), we selected and plotted in Fig. 1a the main east-dipping thrust at the westernmost part of each profile. Thrusts 1–6 in Fig. 1a correspond to profiles Monopolis and Bruneton 3 (1), OLIVIA north segment (2), AUROUX 30 (3), MEDOR 19 (4), OLIVIA south segment (5) and MEDOR 18 (6). The line connecting these thrusts is likely to represent the eastern limit (envelope) of the plate boundary in the area and is in agreement with bathymetric data (Fig. 1a). An obvious conclusion from this analysis is that CLF is located > 15 km east (landwards) of the plate interface and, hence, may not directly conform with local plate boundary kinematics, in some analogy to the Darfield–Christchurch earthquake faulting in New Zealand (Sibson et al. 2011).

Apart of CLF, other active rather immature strike-slip faults are found landwards of the plate interface, for example, the strike-slip fault of 2008 earthquake in NW Peloponnese (AEF in Fig. 1b; Feng et al. 2010) and the Palairos strike-slip fault (Fig. 1a; Stiros and Saltogianni 2016). These faults seem to define an about 100 km wide shear zone along the Aegean Arc. This shear zone seems to be superimposed on the compression accommodated by the system of imbricate NNW-SSE-trending thrusts and folds (Fig. 1c). This situation is somewhat different from the cases of mature strike-slip faults oblique to the direction of compression (Avouac et al. 2014).

Interestingly, a somewhat similar situation is observed at the eastern edge of the Aegean Arc (Fig. 1b), where major Holocene and long-term uplift of Rhodes is associated with thrusting correlating with a deep basin offshore (Kontogianni et al. 2002; Stiros and Blackman 2014), while shear is accommodated by strike-slip faults or a shear zone landwards of the plate interface (Howell et al. 2015).

A final question arising is why focal mechanisms of earthquakes which occurred in the last years selectively show strike-slip faulting (Konstantinou et al. 2017) and only minor evidence of thrusting exists (Shaw and Jackson 2010). A likely explanation is differences in recurrence intervals between usually smaller strike-slip earthquakes, characterized by short recurrence intervals (the 1983, M7.0; see Fig. 1, may indeed be an exception) which produce limited damage and larger thrust-events with larger recurrence intervals, which produce major destruction. Such events have not been recorded during the last decades of detailed instrumental studies and the most recent one was the 1953 M7.2 seismic sequence which devastated the Ionian Islands (Stiros et al. 1994; Papazachos and Papazachou 1997). For the main event of this seismic sequence, a dubious thrust-fault mechanism was presented in the past and compression focal mechanisms have recognized in the area (Jackson et al. 1981), while raised shorelines along the east Cephalonia coast testify to major compressional earthquakes recurring probably every 500 years (Stiros et al. 1994; Pirazzoli et al. 1996).

5.5 Fault Pattern and Peak Ground Accelerations

The 2014 Cephalonia earthquakes were marked by unusually high accelerations (see Sect. 1) and have been the focus of various investigations and even the possibility of revision of the Seismic Codes has been discussed (Bouckovalas 2014; Pitilakis et al. 2014; Theodoulidis et al. 2015; Garini et al. 2015). At least for the second (2014b, 2014.02.03) smaller event, near-fault effects were inferred (Garini et al. 2015), while the overall evidence inspired more detailed studies of the soil dynamic effects in Cephalonia (Theodoulidis et al. 2018).

The detailed and well-constrained FFM of the two 2014 Cephalonia earthquakes presented in this study permit a preliminary investigation of the relationship between faulting and PGA values. A plot of the observed maximum PGA in an axis roughly normal to the two 2014 faults is shown in Fig. 6. These graphs are representative of the maximum intensities in the wider area, because they correlate with damage, especially in specific structures of similar dynamics, for example, cemetery tombstones (Garini et al. 2015), and with the distribution of seismic intensities (Papadopoulos et al. 2014; Lekkas and Mavroulis 2015; Papathanassiou et al. 2017), although the latter are highly depending on lithological conditions (Stiros et al 1994).

Fig. 6
figure 6

Recorded PGA values in relation to the modeled faults for (a) Event 1 (2014a; earthquake of 2014.01.26) and (b) Event 2 (2014b; earthquake of 2014.02.06) of the 2014 seismic sequence. Strong motion stations are in a direction nearly normal to faults (ΒΒ′ profile and site locations in Fig. 3b). Strong motion data after Pitilakis et al (2014) and Theodoulidis et al (2015). Mark the apparent correlation between PGA and the 2014b seismic fault, probably indicating amplification of accelerations above a blind shallow thrust

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Hence, a strong correlation between PGA values and faulting can be derived. Especially for the second event, the observed high PGA are consistent with the predictions of increased PGA above shallow and low-dip thrusts (Allen et al. 1998; Ma et al. 1999; Shi and Brune 2005; Radiguet et al. 2009), mainly blind thrusts (Pitarka et al. 2009).

Of course, the role of local soil conditions had played an important role in the recorded PGA, especially because peak vertical accelerations are slightly smaller than horizontal ones, possibly implying specific excitations, perhaps trampoline-type effects (cf. (Aoi et al. 2008; Fry et al. 2011). Such investigations, however, are beyond the scope of this study.

6 Conclusions

New GPS data and analysis permitted to overpass difficulties in finite fault modeling in the NW Aegean Arc and to present well-constrained uniform slip and variable slip models for the two main events of the 2014 Cephalonia sequence. Our models were validated using InSAR data and a sensitivity analysis which showed their stability and that the absence of control of ground deformation in the westernmost part of the island tends to shift faulting to the east. This explains a main difference between our preferred FFM and previous models, not well constrained to westernmost part of Cephalonia

Modeled faults seem to represent segments of the CLF, of a line of essentially strike-slip faults very close to the west coasts of Leucas and Cephalonia. This segmented, essentially strike-slip fault is located landwards of the plate interface and, hence, does not represent a plate boundary; it represents one of the faults bounding a shear zone accommodating shear at the west part of arc, an active thrust and fold belt, in analogy to the eastern edge of the Aegean Arc (Rhodes area; cf. Howell et al. 2015).

A rather unique 3-D visualization of a multi-segment strike-slip fault was also presented. Composite FFM of various fault segments have also been shown in various cases (for example Elliott et al. 2012; Douilly et al. 2015), but the string of faulting along the Leucas-Cephalonia coast is quite unique, because it covers a series of homogenous FFM of five fault segments activated in three earthquakes which occurred during a period of about 12 years. The examples of Central Italy (Cheloni et al. 2017) and of the North Aegean Trough, at the continuation of the North Anatolian Fault (Saltogianni et al. 2015) indicate that such seismic sequences may be more usual than previously assumed.