1 Introduction

Prediction of strong ground motion is an important aspect in urban seismic risk analysis, urban development planning, land utilization and designing of significant constructions. If the local earthquake records are insufficient to determine the uniform hazard, i.e. regions with relatively low seismicity, then ground motion predictions are also based on studies of seismotectonic and earthquake characteristics. Ground motions are influenced by effects due to seismic source (e.g. radiation pattern, directivity, rupture model, stress drop etc.), wave propagation and site effect. Knowledge and specification of earthquake source characteristics and attenuation parameters are important issues in earthquake engineering and seismic hazard studies. The understanding of the variability and scaling laws appropriate for a region is critical to many seismological efforts, including simulation of strong ground motion produced by future large earthquakes. At the same time, scaling properties of earthquake population of a region provide important information on the rupture physics and characteristics of the surrounding media.

Many studies around the world, employing various methods, have been performed to assess an empirical determination of the kinematic and dynamic source parameters. A few studies have been done in the Eastern Mediterranean region. For example Hofstetter and Shapira (2000) used regional short-period seismic recordings to examine the scaling of radiated energy as function of rupture length and moment, based on the method of Kanamori et al. (1993), assuming simple elastic model. For the magnitude range 3.0 ≤ Mb ≤ 6.2 they obtained the energy-moment scaling relation of \(E \propto M_{0}^{1.19}\). Hofstetter (2003) analyzed the aftershock sequence of the Gulf of Aqaba earthquake recorded by short period and broad-band stations between November 1995 and December 1996 and found a constant stress drop for a large number of strong aftershocks. Recently, Ataeva et al. (2015) studied source parameters and moment-magnitude relations using 103 regional and local earthquakes with moment magnitude from 2.6 to 7.2, which were observed by the Israel Seismic Network.

The goal of this study is to accurately estimate source parameters of regional earthquakes, which can be applied in the assessment of seismic hazard. We separate the source spectra from the path and site effects, which may cause significant widening of the pulses and decreasing of the corner frequencies, and thus biasing the estimates. To obtain a better determination of the source parameters we correct the source spectra by using frequency-dependent Q factor and piecewise geometrical spreading function, unlike the above-mentioned studies. We also apply the Empirical Green’s Function (EGF) method as a consistency check of the corner frequency estimates. To identify possible differences in source scaling, which may depend on regional setting, the analysis is performed separately for several regional earthquake sequences that have occurred in Southern Lebanon, Dead Sea basin and Gulf of Aqaba. For these areas, we present relationships of source parameters such as corner frequency, seismic moment, stress drop and source radius. Analyzed sequences include small to moderate events with the duration magnitude in the range of 1.5–5.2. Finally, we test the self-similarity principle using spectra of regional earthquakes. This principle holds for the spectra of relatively small earthquakes decaying as \(\omega^{ - 2}\) at high frequencies, while it is inconsistent with some spectra in the dataset, mainly for moderate regional earthquakes. The latters exhibit complex spectra with substantial deviations from the \(\omega^{ - 2}\) spectral shape and a second corner frequency is apparent.

2 Seismotectonic framework of the study region

Seismic activity in Israel and its vicinity is mostly associated with the main tectonic elements along the Dead Sea fault (DSF) a main plate boundary between the Arabian plate on the east and the Sinai sub-plate on the west. The tectonic setting of the DSF was explored in a series of seismological, geophysical, geological and geodetic studies providing important information (several out of many, Garfunkel 1981; van Eck and Hofstetter 1990; Rotstein et al. 1991; ten Brink et al. 1993; Marco et al. 1996; Hofstetter et al. 2000, 2007, 2012, 2016; Aldersons et al. 2003; Mechie et al. 2005, 2009; Shamir et al. 2005; Al-Zoubi et al. 2007; Mohsen et al. 2011; Kaviani et al. 2013; Hofstetter and Dorbath 2014; Braeuer et al. 2012a, b, 2014; Masson et al. 2015; and references therein). The fault crosses a continental area and extends through the whole crust into the mantle (Kesten 2004; Kesten et al. 2008). The main slip displacement occurs along a system of left-lateral faults (for example, Garfunkel and Ben-Avraham 1996; Larsen et al. 2002; Ben-Avraham et al. 2008). The left-lateral motion is about 3–5 mm/year along the Arava segment, southern part of the DSF, which is consistent with the kinematics of the Arabian plate (Klinger et al. 2000a, b; Pe’eri et al. 2002; Le Beon et al. 2008, 2010, 2012). The total accumulated slip along these faults is about 105 km and the offset decreases as a result of branching-out faults, which splay off the transform as it approaches the collision zone of southeastern Turkey (Ben Menahem et al. 1976; Garfunkel 1981; Garfunkel et al. 1981; Ben-Avraham et al. 2008).

We study the scaling relations of earthquake sequences that occurred in the Gulf of Aqaba, Dead Sea basin, and Southern Lebanon (Fig. 1). The Gulf of Aqaba is divided into a series of several basins the Eilat, Aragonese, Dakar and Tiran basins (i.e., Ben-Avraham 1985; Ben-Avraham and Tibor 1993; Ben-Avraham et al. 2008). Studies show complex structure: left lateral strike-slip motion along the main axis of the gulf, and normal slip along the traversing faults. In the last three decades several sequences of intense seismic activity occurred including a strong earthquake with magnitude \(M_{W} = 7.2\) on November 22, 1995 (Pinar and Turkelli 1997; Klinger et al. 1999; Hofstetter 2003; Hofstetter et al. 2003).

Fig. 1
figure 1

Regional map showing earthquake sequences (large circles) used in analysis. Dashed lines present major faults: C. F.—Carmel F., R. F.—Roum F., Y. F.—Yammouneh F. and Rah. F.—Rachaya F

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Tectonically, the Dead Sea basin is divided into two major sub-basins, the southern and the northern, which is separated by the Lisan Peninsula, a large salt diapir. The seismic activity is mainly related to two main north–south faults along the eastern side of the northern basin (van Eck and Hofstetter 1989, 1990; Shamir et al. 2005; Hofstetter et al. 2008; Braeuer et al. 2012a, b, 2014; Hofstetter et al. 2012). Recent high-resolution analysis of seismic reflection data shows that the northern Dead Sea basin is comprised of a system of tectonically controlled sub-basins delimited by the converging EW boundary faults of the Dead Sea fault (Lazar et al. 2006).

The seismic pattern of Southern Lebanon is of particular interest since the tectonic framework of this region is variable. The northern part of the DSF splits here into several splays including the Yammouneh, Rachaya, and Roum faults (Fig. 1). Sinistral displacement along the north–south striking Roum fault decreases northward from 8.5 to 1 km and is transferred to a thrust fault along the continental margin (Daeron et al. 2004). Ginzburg and Ben Avraham (1987) and Ben-Avraham et al. (2002) proposed the existence of an ocean-continent boundary west of the Roum fault and close to the shore. Schattner et al. (2006) showed evidence of reactivation of the continental margin in the area between the Carmel structure and eastern Cyprian arc. Tectonically, this area is characterized by internal deformation resulting from shearing in a general NW–SE direction and acts as a buffer zone that absorbs the transformed energy from the Dead Sea fault. From 2008 to 2010 the several sequences of microearthquakes were recorded (more than 1300 events) in the coda magnitude range of 1.5–5.2 concentrated in Southwestern Lebanon, in a 40 km wide zone south of the Roum fault (Fig. 1). The largest earthquake in the Lebanon sequences occurred on 15/2/2008 at 33.327°N, 35.406°E at a depth of 3 km with Mw 5.1 (Meirova and Hofstetter 2013).

3 Methodology

We investigate the scaling properties of regional earthquakes in Israel and adjacent areas using the determination of spectra correction and estimation of the seismic source parameters. The estimation are based on the standard single-corner model of point source proposed by Brune (1970, 1971). This model has been extensively used in many studies (i.e., Boore 1983; Joyner 1984; Abercrombie and Rice 2005; Allmann and Shearer 2007).

In general the inelastic attenuation of seismic waves due to propagation has significant effect on widening the pulses and decreasing the corner frequencies (Madariaga 1976). To assess accurate estimates of regional source parameters we apply a path correction of the observed spectra. The displacement spectrum of the motion at the site Y, depends on the source spectra S (i.e., Joyner and Boore 1981), path propagation P and local site modification G, after removal of the instrument response, which is expressed as

$$Y(M_{0} ,r,f) = S(M_{0} ,f)P(r,f)G(f)$$
(1)

where \(M_{0}\) is the seismic moment. Assuming that the earthquake source is a point source, the displacement source term for a simple \(\omega^{ - 2}\) model is (Aki 1967; Brune 1970, 1971) is

$$S\left( f \right) = \frac{{R_{\varTheta \varPhi } Fd}}{{4\pi \rho \beta_{S}^{3} r}}\frac{{M_{0} }}{{1 + \left( {f/f_{0} } \right)^{2} }}$$

where \(f_{0}\) is the corner frequency, \(R_{\varTheta \varPhi }\) is the radiation pattern, usually averaged over a suitable range of azimuths and take-off angles, d represents the partition of total shear-wave energy into horizontal components, F is the effect of the free surface, \(\rho\) and \(\beta_{S}\) are the density and shear-wave velocity in the vicinity of the source, and \(r\) is the distance. The path propagation effect is \(P\left( {r,f} \right) = Z\left( r \right)\exp \left[ { - \pi fr/Q\left( f \right)\beta_{S} } \right]\) where \(Q = 132f^{0.96}\) following Meirova and Pinsky (2014), and Z(r) is the geometrical spreading. Site-specific effect \(G\left( f \right) = A\left( f \right)\exp ( - \pi \kappa f)\) reflects the generic rock site amplification A, where the average value is set to 1, and κ is the near-surface attenuation that has a direct influence on the scaling law description for small magnitudes. Previous studies of small-sized earthquakes (i.e., Anderson and Hough 1984; Prejean and Ellsworth 2001) showed that a trade-off exists between parameters κ and \(Q\), competing against each other in defining the observed spectral amplitudes at high frequency. Based on analysis of the observed spectra from small regional earthquakes the near-surface attenuation parameter κ is 0.015 (Meirova et al. 2008).

Many earthquake scaling studies adopt a model of regional propagation introduced by Joyner and Boore (1981), with a constant geometrical spreading for all distances (Shapira and Hofstetter 1993; Ataeva et al. 2015). Nevertheless, Raoof et al. (1999) and Malagnini (1999), among others, observed that geometrical spreading varies significantly as a function of distance. The approach of the latters allows the descriptions of geometrical spreading without any a priori assumptions of functional form of the geometrical spreading function. In this study the empirically based apparent geometrical spreading function is

$$Z\left( r \right) = \left\{ \begin{aligned} r^{ - 1.0}, {\rm for}\quad r < 40\;{\rm km} \hfill \\ r^{ + 0.1}, {\rm for} \quad 40\,{\kern 1pt} {\rm km} \le r < 70\,{\rm km} \hfill \\ r^{ - 0.4}, {\rm for} \quad 70{\kern 1pt} \,{\rm km} \le r < 80 \,{\rm km} \hfill \\ r^{ - 0.5}, {\rm for} \quad r \ge 80\;{\rm km} \hfill \\ \end{aligned} \right.$$

The exponents of the geometrical spreading function describe the geometry of the wavefront (e.g., spherical or cylindrical propagation, or even characterized by the appearance of supercritical reflections in the transitional distance ranges), which strongly depends on the velocity structure of the medium (Aki and Richards 2002). In our model the geometrical spreading at distances shorter than 40 km is related to the propagation of body waves with a spherical wavefront and, hence implicitly, a uniform crust. Between 40 and 70 km the direct wave is joined by the first post-critical reflection from internal crustal interfaces and the Moho discontinuity. For distances longer than 80 km the main contribution is assumed to be from surface waves, which are attenuated less rapidly than the intermediate field or far field body waves.

We determine corner frequency using two approaches. First, following Andrews (1986) and Snoke (1987), the corner frequency and the flat level of the displacement spectra \(\varOmega_{0}\) are

$$f_{0} = \frac{1}{{2\pi_{{}} }}\sqrt {\frac{{I_{v} }}{{I_{d} }}}$$
(2)

and

$$\varOmega_{0} = \sqrt {\frac{{2I_{d} }}{{f_{0} \pi }}}$$
(3)

where \(I_{v} = 2\int\nolimits_{0}^{\infty } {V(f)^{2} } df\,\), \(I_{d} = 2\int\nolimits_{0}^{\infty } {D(f)^{2} } df\,\) and \(V(f)\,\), \(D(f)\) are the velocity and displacement spectral amplitudes in the frequency domain, respectively.

The second approach to determine the corner frequency of the largest events is by means of the Empirical Green’s Function (EGF) method (e.g., Courboulex et al. 1996; Ide et al. 2003; Sonley and Abercrombie 2006). In this method the source spectrum is deconvolved with nearby smaller earthquake which acts as the medium transfer function. Thus, the deconvolution corrects for the path effect between the source and the receiver.

The seismic moment, \(M_{0}\) of an earthquake is

$$M_{0} = \frac{{4\,\pi \,\rho \,\beta_{S}^{3} \,r\,\varOmega_{0} }}{{R_{\varTheta \varPhi } F}}$$
(4)

Another important source parameter is the stress drop, which indicates the tectonic stress and material strength characteristics in the faulting region. In this study, we estimate the static stress drop following Snoke (1987), based on Brune’s model, as

$$\Delta \sigma \approx \frac{{0.55\,I_{v} \,\,\pi \,\rho }}{{R_{\varTheta \varPhi } F\,r\,\varOmega_{0} }}$$
(5)

Andrews (1986) and Snoke (1987) pointed out that if one were to assume that the P-wave contribution to the total radiated seismic energy is negligible and that the integrals of the power spectra have negligible angular dependence, then it results in a constant value for the ratio between the apparent stress and the Brune stress drop. The source radius (Brune 1970) is estimated by

$$R = \frac{{2.34\;\beta_{S} }}{{2\,\pi {\kern 1pt} f_{0} }}$$
(6)

using the circular source model.

4 Data

Our dataset comprises the records of three groups of regional earthquakes that have been occurred in Dead Sea basin, Gulf of Aqaba and southern Lebanon (Fig. 1). These groups include events in the duration magnitude range of \(1.5 \le M_{d} \le 5.2\) occurring at different tectonic settings. We select 170 earthquakes that occurred from 1992 to 2010 in the Dead Sea basin and Gulf of Aqaba region, and 120 from the swarm sequences of 2008–2010 in Southern Lebanon. Figure 2 shows the distribution of observations for the three regions as functions of the hypocentral distance and duration magnitude.

Fig. 2
figure 2

Distribution of observations as a function of hypocentral distance and duration magnitude of earthquakes that are used in the study

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All ground motion data are acquired from the database of Geophysical Institute of Israel, comprising short-period and broadband stations. A typical short period sensor consists of Mark Products L-4C and Geotech S-13 1-Hz vertical short-period seismometers digitized at 50 sps, providing output signals in the frequency range of 0.2–15 Hz. Broadband portable seismometers are STS-2 and Trillium Compact 120, with vertical and horizontal output signals in the frequency range of 0.001–100 Hz. All data are corrected for instrument response. Horizontal components are rotated to obtain SH-components of ground motion. Each waveform is visually examined to exclude noisy recordings and spikes. Earthquakes with poor signal-to-noise ratio are rejected from the analysis.

5 Spectral analysis

All velocity seismograms are baseline corrected by removing the mean and cosine tapered (5%). The amplitude Fourier velocity spectra for a window of 5–30 s starting with the S-wave onset are calculated for transverse and radial components. The observed amplitude spectrum at the each station is obtained by the summation of transverse and radial components. The displacement spectrum is obtained by integration of corrected for attenuation S-wave velocity spectra. We use the S-wave velocity of 3.4 km/s and density of 2.7 gr/cm3. We assume a simple form of the radiation pattern for the S waves equivalent to \(\sqrt {{2 \mathord{\left/ {\vphantom {2 5}} \right. \kern-0pt} 5}}\) (Andrews 1986), the amplification due to the free surface is F = 2.0, and the partitioning of energy into horizontal components is d = 0.707.

To test the accuracy of the adopted path propagation model we assume that the correct attenuation model compensates the attenuation of the flat level spectra as a function of distance. Since the attenuation is independent of the magnitude, then \(\varOmega_{0}\) values attenuate at a similar pattern with distance. Figure 3a demonstrates the uncorrected values of \(\varOmega_{0}\), where color of circles indicates the range of duration magnitude, varying in a step of 0.5 unit magnitude as a function of distance for events with duration magnitude from 1.5 to 5.1. The low frequency corrected values assure the reliable attenuation estimation in the distance range from 10 to 500 km (Fig. 3b).

Fig. 3
figure 3

a Attenuation of low frequency flat level values \(\varOmega_{0}\) with distance measured from calculated displacement spectra in the magnitude ranges of \(1.5 \le M_{d} \le 2\) (inverted triangles), \(2 < M_{d} \le 2.5\) (crosses), \(2.5 < M_{d} \le 3\) (circles), \(3 < M_{d} \le 3.5\) (triangles), \(3.5 < M_{d} \le 4\) (diamonds), \(4 < M_{d} \le 4.5\) (squares) and \(4.5 < M_{d}\) (stars). Dashed lines show the trends of attenuation for each group of \(\varOmega_{0}\) values; b low frequency flat level values \(\varOmega_{0}\) measured after spectra correction

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We compare the seismic moment estimates, obtained from the spectral analysis as above modeled, with those obtained from waveform inversion. We applied the code of Dreger and Helmberger (1993) for the seismic moment determinations of a few medium earthquakes, using three-component waveform broadband data (Fig. 1). The local velocity model (Gitterman et al. 2005) is applied to calculate Green’s functions. The seismic moment values obtained by the two methods, for a number of calibration earthquakes that occurred in the Southern Lebanon, Dead Sea basin and Gulf of Aqaba are in a good agreement (Table 1).

Table 1 Comparison of seismic moment estimates for a number of calibration earthquakes obtained by full waveform inversion (Dreger and Helmberger 1993) and Andrews’ method
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The integral method introduced by Andrews (1986) is used to estimate the zero-frequency corrected displacement amplitude \(\varOmega_{0}\), corner frequency, stress drop and seismic moment. Examples of source spectra for the earthquake recorded on February 11, 2004, in the Dead Sea basin with \(M_{d} = 5.2\) are shown in Fig. 4. As it appears, the corner frequency values obtained for this event are not uniform and display large discrepancies, for example values of 1.11 and 0.42 Hz in stations MMLI and KSDI, respectively. This may be due to several reasons such as: directivity effect, attenuation, site effect and uncertainty in the hypocentral depth estimates that may influence the radiation pattern.

Fig. 4
figure 4

Left: ground velocity seismograms of the Dead Sea earthquake on February 11, 2004, \(M_{d} = 5.2\), recorded at broadband stations; right: corrected displacement spectra for path effects. Corner frequencies are marked by short vertical lines following Andrews (1986). Resulting average corner frequency is 0.77 ± 0.28 Hz

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We also applied the Empirical Green’s Function (EGF) method for 40 selected events, as consistency check of the corner frequency estimates. The EGF method requires earthquakes that are close to each other sharing common path effects. A collocated smaller earthquake has to be 1–2 magnitudes smaller than the main event. The epicentral location of the Green’s function event is selected within a few hundred meters of the larger event. An additional criterion for selection of EGF pairs is the similarity of waveform shapes in time domain at common stations, indicating a similar source mechanism and depth. We selected 27 events and 13 events from the swarm sequences of 2008–2010 in the Southern Lebanon and the earthquakes in 1992–2009 in the Dead Sea basin, respectively. After applying a signal-to-noise cutoff, we have about 90 pairs of seismograms with duration magnitude between 1.9 and 5.2.

Using the deconvolved spectra we estimate the corner frequencies when it is possible (Fig. 5a). In practice, deconvolution instability, noise and nonlinear site effects may contaminate the higher frequencies of the EGF spectral ratio. Noise in the data introduces high‐frequency oscillations in the spectral ratio with amplitude drops, masking the corner frequency of the small earthquake and the low amplitude plateau. Therefore, we select events with corner frequencies that clearly lie within the data resolution limits and exclude from the EGF analysis the earthquakes for which corner frequencies are close to or above the frequency resolution limit (15 Hz), as it may be an artifact from instability in the deconvolution.

Fig. 5
figure 5

a An example of EGF corner frequency estimation. Upper part shows the data that were used for two events with \(M_{d}\) 3.7 and 1.9 recorded at station KMTI. Below are the displacement spectra calculated for these waveforms in a window of 5 s starting from the S arrival. Lower part shows the displacement spectrum of deconvolution and resultant estimate of \(f_{0}\). Intersection of the dotted lines indicates the EGF estimate of \(f_{0}\). b Comparison of corner frequencies following Andrews (1986) from corrected path propagation spectra (vertical axes) and EGF method. Andrews’ corner frequency is calculated at each station separately and represents a mean value for all possible event-station records. The vertical error bar presents the error of corner frequency estimate

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We compare the corner frequency estimates with those measured values from the corrected spectra (Fig. 5b), to explore a possible bias between the two methods. As can be seen the frequencies estimated using Andrews’ approach and EGF method are quite similar. However, there is a shift in the estimates that is more prominent at low frequencies. This shift could be due to overestimating of the attenuation correction term while evaluating the Andrews’ corner frequencies or the uncertainty in the radiation pattern. On the average, the difference between estimates of corner frequency by the two methods is small and EGF method confirms the results obtained by using the Andrews’ approach.

6 Source parameters and scaling relations

We estimate the seismic moment, \(M_{0}\), corner frequency, \(f_{0}\), stress drop, \(\Delta \sigma\), and source radius, R, based on Brune’s model, using spectra of S-waves recorded by the short period and broadband seismic stations. The relationships of corner frequency and seismic moment estimates for earthquakes in Southern Lebanon, Dead Sea basin and Gulf of Aqaba are (see Fig. 6)

Fig. 6
figure 6

Relationships between the seismic moments and corner frequencies, along with error bars, estimated for earthquakes in the regions: a Southern Lebanon; b Dead Sea basin; c Gulf of Aqaba. Lines of constant static stress drops are indicated

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$${\text{Southern}}\;{\text{Lebanon}}:\log M_{0} = (15.55 \pm 0.21) - (5.36 \pm 0.38)\log f_{0} ,\quad \sigma = 0.5$$
(7)
$${\text{Dead}}\;{\text{Sea}}\;{\text{Basin}}:\log M_{0} = (15.46 \pm 0.25) - (4.36 \pm 0.38)\log f_{0} ,\quad \sigma = 0.5$$
(8)
$${\text{Gulf}}\;{\text{of}}\;{\text{Aqaba}}:\log M_{0} = (15.95 \pm 0.21) - (6.02 \pm 0.42)\log f_{0} ,\quad \sigma = 0.6$$
(9)

The corner frequency values \(f_{0}\) are in the range from 0.9 to 9.7 Hz, and the corresponding seismic moment estimates range from 1.580E+11 to 5.82E+16 Nm. The corner frequency values are decreasing with the seismic moment more rapidly for events in the Gulf of Aqaba and Southern Lebanon (Fig. 6). Plot of corner frequency versus seismic moment is often used as a test for self-similarity i.e. \(M_{0} \sim f_{0}^{ - 3}\). Figure 6 does not support such relations for all regions. Empirical \(M_{0} (f_{0} )\) trends are significantly steeper as compared with the simple relation, suggesting the scaling relationship of \(M_{0} \sim f_{0}^{ - (3 + \varepsilon )}\) with \(\varepsilon \approx\) 1.3–3 varying from region to region. Our scaling is close to results reported by Kanamori and Rivera (2004), Mayeda et al. (2005), Meirova and Hofstetter (2013), and Ataeva et al. (2015).

The scaling relationships of seismic moment with duration magnitude are (see Fig. 7)

Fig. 7
figure 7

The relationships between seismic moment and duration magnitude for earthquakes in the regions: a Southern Lebanon; b Dead Sea basin; c Gulf of Aqaba. Error bars present the error of seismic moment estimation

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$${\text{Southern}}\;{\text{Lebanon}}:\log M_{0} = (1.36 \pm 0.08)M_{d} + (9.57 \pm 0.03),\quad \sigma = 0.24$$
(10)
$${\text{Dead}}\;{\text{Sea}}\;{\text{Basin}}:\log M_{0} = (1.10 \pm 0.05)M_{d} + (9.97 \pm 0.12),\quad \sigma = 0.34$$
(11)
$${\text{Gulf}}\;{\text{of}}\;{\text{Aqaba}}:\log M_{0} = (1.11 \pm 0.03)M_{d} + (10.13 \pm 0.08),\quad \sigma = 0.25$$
(12)

Regression coefficients for Southern Lebanon are similar to those as obtained by Meirova and Hofstetter (2013) and the coefficients for the Dead Sea basin and Gulf of Aqaba are similar to those reported by van Eck and Hofstetter (1989), Hofstetter et al. (1996), Hofstetter (2003), and Ataeva et al. (2015). The correlation coefficients for the Dead Sea basin and the Gulf of Aqaba differ from those of Southern Lebanon, possibly due to interplate tectonic in the formers and intraplate tectonic in the latter (see Fig. 1).

The relationships between the stress drop and seismic moment are (Fig. 8):

Fig. 8
figure 8

The relationship between seismic moment and Brune stress drop with error bars for earthquakes in: a Southern Lebanon; b Dead Sea basin; c Gulf of Aqaba

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$${\text{Southern}}\;{\text{Lebanon}}:\log \Delta \sigma = (0.60 \pm 0.03)\log M_{0} - (8.80 \pm 0.39),\quad \sigma = 0.30$$
(13)
$${\text{Dead}}\;{\text{Sea}}\;{\text{Basin}}:\log \Delta \sigma = (0.56 \pm 0.37)\log M_{0} - (8.00 \pm 0.47),\quad \sigma = 0.34$$
(14)
$${\text{Gulf}}\;{\text{of}}\;{\text{Aqaba}}:\log \Delta \sigma = (0.57 \pm 0.02)\log M_{0} - (8.64 \pm 0.36),\quad \sigma = 0.25$$
(15)

The scatter is due to a combination of effects, such as simplified assumptions about the source model or heterogeneities of physical properties along the fault. In general stress drop values increase with large events. Various authors have reported an increase of stress drop with depth (e.g., Jones and Helmberger 1996; Venkataraman and Kanamori 2004). In southern California, Shearer et al. (2006) observed a median stress drop increase from 0.6 MPa near the surface to 2.2 MPa at 8 km depth, assuming a constant rupture velocity. We do not observe such increase of stress drop values with depth, based on our dataset. The stress drop is increasing with source radius (Fig. 9). However, the dependence of the source radius on seismic moment shows a smaller increase than it would be expected for the constant stress-drop model. For instance, with respect to the scatter of the resolved \(f_{0}\) and \(M_{0}\) the small events with seismic moment \(M_{0} < 10^{14}\) Nm have source radius ranging from ~100 to ~1600 m.

Fig. 9
figure 9

The relationships between source radius with error bars and Brune’s stress drop Δσ for earthquakes in: a Southern Lebanon; b Dead Sea basin; c Gulf of Aqaba

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7 Discussion and conclusions

We studied the source properties and scaling relationships of small and moderate earthquakes occurring in Israel and adjacent areas. The Andrews’ approach (1986) and EGF method are used to examine the seismic moment, corner frequency, source radius and range of the stress drop for local and regional earthquakes. We explore possible differences in source scaling depending on regional features by separately analyzing three regional earthquake sequences that occurred in Southern Lebanon, Dead Sea basin and Gulf of Aqaba. We assume that the attenuation properties are independent of the investigated regions and source to station azimuth. For individual estimates of source parameters the displacement spectra were corrected for path propagation. Regional propagation is modeled by piecewise geometrical spreading function, a frequency dependent attenuation parameter \(Q\) (following Meirova and Pinsky 2014), and kappa parameter κ, characterizing near-surface attenuation. Corner frequency values obtained by using Andrews’ approach are quite similar to those obtained by EGF technique.

Based on the Brune source model, the corner frequencies of regional events range from 0.9 to 9.7 Hz and the seismic moments from 1.58E+11 to 5.82E+16 Nm. Scaling of corner frequencies \(f_{0}\) with \(M_{0}\) in the magnitude range of \(1.5 \le M_{d} \le 5.2\) for Southern Lebanon, Dead Sea basin and Gulf of Aqaba are \(f_{0} \sim M_{0}^{ - 0.12}\), \(f_{0} \sim M_{0}^{ - 0.16}\) and \(f_{0} \sim M_{0}^{ - 0.13}\), respectively, which are weaker scaling than it can be expected for a constant stress drop model. The results show similar tendency to reported scaling of \(f_{0} \sim M_{0}^{ - 0.16} - M_{0}^{ - 0.25}\) by Kanamori and Rivera (2004) and Mayeda et al. (2005), and in general do not confirm the self-similarity principle. The scaling relationships between the duration magnitudes and seismic moments for seismic sequences in Dead Sea area and Gulf of Aqaba are similar. The correlation coefficients for the Southern Lebanon sequences somewhat differ from correlation coefficients of the formers, possibly due to interplate tectonic in the formers and intraplate tectonic in the latter. Stress drop values for regional earthquakes are in the range 0.005–19 MPa, increasing with seismic moment as \(\Delta \sigma \propto M_{0}^{0.56}\), \(\Delta \sigma \propto M_{0}^{0.6}\), and \(\Delta \sigma \propto M_{0}^{057}\) for the Dead Sea basin, Southern Lebanon and Gulf of Aqaba, respectively. Stress drop for regional earthquakes slightly increases with source radius. Dependence of source radius on seismic moment shows a smaller increase than the expected value for the constant stress-drop model.

Source spectra of small to moderate earthquakes are normally determined using the classical Brune one-corner spectral model, which includes a flat plateau at low frequencies and decreasing as \(\omega^{ - 2}\) at high frequencies (Fig. 10a). The spectra of the short period (PRNI) and broadband (HRFI) stations showing similarity with \(f_{01} \approx f_{02}\) for the same event with \(M_{d}\) 1.9. However, some spectra in the dataset are complex and a second corner frequency is apparent (Fig. 10b, c), exhibiting substantial deviations of individual source spectra from the ideal \(\omega^{ - 2}\) spectral shape. Figure 10c demonstrates similar spectra for short-period and broadband stations for the same event.

Fig. 10
figure 10

Examples of amplitude displacement spectra corrected for path propagation illustrating: a spectra of the short period (PRNI) and broadband (HRFI) stations showing similarity with \(f_{01} \approx f_{02}\) for the same event with \(M_{d}\) 1.9; b spectra with \(f_{01} < f_{02}\) for events with \(M_{d}\) 3.5–3.7; c spectra of the short period (HMDT) and broadband (KSDI) stations showing similarity with \(f_{01} < f_{02}\) for the same event with \(M_{d}\) 3.3

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At small magnitudes the corner frequencies are similar (\(f_{01} \approx f_{02}\)) and then the spectra is in agreement with the common \(\omega^{ - 2}\) spectral model. For larger magnitudes the scaling of \(f_{01}\) is qualitatively different from that of \(f_{02}\). Figure 11 shows examples of corrected displacement spectra of regional earthquakes. Similar types of spectra are noted by Brune (1970). Hanks (1982) and Papageorgiou (1988) identified the change in the spectra as f max. For the Dead Sea basin van Eck and Hofstetter (1989) reported f max values that are slightly lower than 10 Hz. The scaling of \(f_{01}\) with seismic moment is consistent with the conventional hypothesis of similarity and close to \(M_{0} \sim f_{01}^{ - 3}\). This kind of scaling corresponds to the assumption of geometrical and kinematic similarity of sources with different sizes, thus \(M_{0} \propto L^{3} \propto T^{3} \propto f_{01}^{ - 3}\), where L is source size, and T is duration of the source (Gusev 2013). The observed cutoff in the displacement spectrum located at \(f_{02}\) cannot represent the Brune’s one-corner spectral model. Rather, this kind of spectra behavior is an example of a specific feature of the spectral shape. Possible reasons for such specific spectra are: broadening of the front of propagating rupture, the assumed path attenuation correction is too small, and enhancement of the observed record energy by short-period surface waves in the frequency range above \(f_{01}\) (Gusev 2013).

Fig. 11
figure 11

Corrected displacement spectra based on data from regional earthquakes. Scaling for \(f_{01}\) is close to \(f_{01} \sim M_{0}^{{ - {1 \mathord{\left/ {\vphantom {1 3}} \right. \kern-0pt} 3}}}\) and for \(f_{02}\) it is close to \(f_{02} \sim M_{0}^{ - 0.16} - M_{0}^{ - 0.25}\)

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The self-similarity model predicts that the shape of spectra of small and large events on log–log plot is identical with an offset along a line of \(\omega^{ - 2}\). Figure 11 demonstrates that the spectral shapes for regional earthquakes are not self-similar and cannot be simultaneously fit with the \(\omega^{ - 2}\) model. We observe that the spectral shapes change with increasing magnitudes of events (Fig. 11). Shifting spectra along the \(\omega^{ - 2}\) line in our case does not satisfy the self-similarity conditions. The observed behavior of high frequency spectral decay means that a certain portion of regional spectra cannot be interpreted as having \(\omega^{ - 2}\) source spectral structure. It implies that we may obtain incorrect values for corner frequencies and seismic moments. The complexity of regional earthquake sources and propagation effects should be further explored.

Several studies (Boore 1986; Di Bona and Rovelli 1988; Ide and Beroza 2001; etc.) discussed the influence of finite bandwidth on the estimation of source parameters. It was shown that a limited bandwidth of seismic recording can introduce an apparent scaling of source parameters, including the relation of apparent stress with seismic moment. In our study the bandwidth of records ranges from 0.5 Hz to ~15 that may indicate the possibility of underestimation of our corner frequency. On the other hand, the small differences between the estimates of corner frequency using two independent methods confirm our findings. The effect of bandwidth limit should be further explored before drawing a final conclusion on the earthquake’s self-similarity in our dataset.