1 Introduction

Seismologists consider the seismic moment M 0 and moment magnitude M W (i.e., Kanamori 1977; Hanks and Kanamori 1979; Hanks and Boore 1984) as reliable measures of earthquake size that represents the total deformation at the source and correlates with the seismic energy release. The earthquake source parameters have been widely used for computing scaling laws applicable for seismic hazard assessments (Hanks and Wyss 1972; Thatcher and Hanks 1973; Rebollar 1985; Shapira and Hofstetter 1993; Baumbach and Bormann 1999, 2012; Tan and Helmberger 2007; Tusa and Gresta 2008; Tusa et al. 2012). Therefore, defining the seismic moment M 0 and other source parameters and developing a reliable relationship between them and magnitude are of fundamental importance.

In this study, we use earthquakes that occurred along the Dead Sea Fault and in the Eastern Mediterranean region (Fig. 1). The Dead Sea Fault, which is about 1200 km long, connects the Taurus-Zagros compressional front in the north to the extensional zone of the Red Sea in the south, accommodating the left-lateral motion between the Sinai subplate and the Arabian plate (several out of many, van Eck and Hofstetter 1990, Salamon et al. 1996, 2003; Ben-Avraham et al. 2008; Garfunkel 2013; and references therein). The Dead Sea Fault and the Eastern Mediterranean region are seismically active with several strong earthquakes, based on historical accounts and instrumental records. The earthquake on 11 July 1927, M 6.2, occurred in the northern part of the Dead Sea basin causing 285 deaths, 940 wounded, and extensive damage in many towns and villages on both sides of the Dead Sea Transform (Ben-Menahem et al. 1976; Shapira et al. 1992; Avni 1998). Several other widely felt earthquakes occurred along the Dead Sea Fault in 1903, 1928, 1956, 1970, 1979, 1995, and 2004, with magnitudes M ∼ 5.0–7.2, causing no or minor damage (i.e., Ben-Menahem et al. 1976; Arieh et al. 1982; Amiran et al. 1994; Salamon et al. 1996; Klinger et al. 1999; Ken-Tor et al. 2001; Hofstetter et al. 2003, 2008; Migowski et al. 2004; Al-Tarazi et al. 2006; Shamir et al. 2006).

Fig. 1
figure 1

Location of earthquakes and stations that are used in this study. Names of stations that are mentioned in the study are listed

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The use of P-wave data can be of particular interest when the S-wave data are clipped or when trying to estimate magnitudes in real time. Hanks and Wyss (1972) and Baumbach and Bormann (1999, 2012) demonstrated the use of P-wave spectrum for estimation of seismic moment M 0, corner frequency f 0, and source radius r. Watanabe et al. (1996) investigated the source characteristics from P and S waves for earthquakes with magnitudes ranging from 3.3 to 6.0 and confirmed that the seismic moments determined are consistent. Ottemoller and Havskov (2003) presented good results of the method to automatically determine the moment magnitude from the source spectra of P and S or Lg waves of local and regional earthquakes. Tusa and Gresta (2008) and Tusa et al. (2012) had used displacement spectra of P wave to estimate source parameters for events of magnitude less than 4.6 and for microearthquakes (M L < 3) with hypocentral distances in the range from 13 to 90 km and from 27 to 63 km, respectively. In this study, we investigate earthquake source parameters and seismic relations from spectral analysis of P and S waves for regional and local earthquakes that occurred from 1995 to 2012. In addition, source characteristics that have been obtained from P-wave spectra are compared to the analysis of S-wave spectra.

2 Data

The dataset consists of 1459 digital records of the Israel Seismic Network (ISN) from 103 local and regional earthquakes in the moment magnitude range from 2.6 to 7.2, which occurred in a distance range of 4.5 to 550 km during 1995–2012 (Fig. 1). The data include 855 vertical seismograms of Geotech S-13 short period seismometers; 272 vertical and 282 horizontal seismograms of STS-2 broadband seismometers, recorded with a sampling rate of 50 sps; and 50 vertical and horizontal components from strong motion accelerometers (ETNA and A-800) with a sampling rate of 200 sps. The horizontal and vertical components were used to obtain the empirical ratio between horizontal and vertical low-frequency spectral amplitudes of S waves. Figure 2 shows the distribution of the used vertical seismograms with respect to magnitudes and epicentral distances. It should be noted that there is a wealth of data of small and moderate earthquakes but only a few strong motion recordings. The database also includes broadband data of the Bar Giora (BGIO), Jerusalem (JER), and Eilat (EIL) regional stations that operated 2 years and recorded earthquakes of M W ≥ 5.0 which occurred during 1995–1997.

Fig. 2
figure 2

Distribution of duration magnitude, M d, as a function of epicentral distance, where broadband, short period, and accelerometric data (vertical component) are shown by red, black, and blue circles, respectively

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The selection of waveforms follows the following conditions: the earthquake was recorded by a number of calibrated seismic stations, signals are not clipped, and signal-to-noise ratio exceeds 2. In this respect, we selected events with M d ≥ 2.7, but tried to obtain a balanced coverage through a wide magnitude range and a wide range of epicentral distances. Most of these events occurred at depths in the range of 3–20 km that indicated activity in the upper crust.

3 Processing and calculations

3.1 Spectral analysis of weak and strong motion data

The analysis involved measuring the low-frequency spectral amplitudes of the displacements of P and S waves, Ω(P) and Ω(S), and corner frequencies f ′0 (P) and f ′0 (S) observed at the stations. These measured values are used to determine the seismic moment, M 0, in accordance with the Brune’s circular source model (Brune 1970, 1971). The spectral parameters have been manually defined approximating the spectrum by two lines which refer to constant low-frequency level and high-frequency decay. The corner frequency is defined as the intersection of low- and high-frequency asymptotes (Fig. 3c–f; Brune 1970; Baumbach and Bormann 1999, 2012; Kiratzi and Louvari 2001; Ashkpour Motolagh and Mostafazadeh 2008; Havskov and Ottemoeller 2010).

Fig. 3
figure 3

a Vertical component seismogram of the Dead Sea earthquake on 11 February 2004, M d = 5.2, recorded by the short period station MZDA located at a distance of 50 km from the epicenter. The analyzed time windows of P- and S-wave trains are marked by solid blue and red lines. Time is relative to the origin time. The background noise (about 2 s prior to P wave) was multiplied by 102; b P window length as a function of distance based on windows that were used in the analysis (blue circles) and the fit (red line); c, d calculated uncorrected P- and S-wave displacement spectra (m/s) of MZDA for the Dead Sea earthquake on 11 February 2004, M d = 5.2. The red lines indicate the low-frequency amplitude level and high-frequency decay. The measured f ′0 and \( \mathit{\mathsf{\varOmega}} \) are marked, and the dashed blue line shows f − 2 line; e, f the S-wave displacement spectra (m/s) of the 22 February 1995, M W = 7.2, Gulf of Aqaba earthquake calculated from the vertical component of broadband station BGIO and from the vertical component of accelerometer in ASK, located at distances of 332 and 319 km, respectively

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To perform the spectral analysis via FFT, we used instrumentally corrected vertical and horizontal seismic records. In addition, we applied baseline correction by removal of the mean and 5 % cosine taper. We manually defined an appropriate range of the P-wave time window length according to our ranges of epicentral distances and magnitudes, always starting at the P onset, disregarding the type of the first arriving P phase. The P-wave time window includes the first group (e.g., Pg, Pb, Pn), increasing with distance by adding 1 s for each 100 km, resulting in window sizes from 1.5 to 6.0 s. When the epicentral distance is short, the duration is limited by the arrival of the S waves. Each spectrum was visually inspected. For each event, we used at least six seismograms, which were recorded at different distances from the epicenter. The S/Lg-wave windows were picked manually, assuring that they include a significant part of the energy of the signal. Typical lengths of the windows are 10–30 s, depending on the magnitude M d and distance to the source. For larger events (M d ≥ 5.6), S-wave time windows of 50–80 s are used. Figure 3a presents time windows for P and S phases from recording of the Dead Sea earthquake on 11 February 2004, M d = 5.2, at station MZDA which is located at a distance of 50 km from the epicenter. Figure 3b presents P window length as a function of distance based on windows that were used in the analysis (blue circles) and the appropriate fit (red line). We note that for about the same distances, there are small variations in the window lengths from one event to another. Figure 3c, d shows examples of displacement P- and S-wave spectra of MZDA, respectively.

In addition, we have examined the applicability of similar spectral analysis using data from strong motion accelerometers of felt earthquakes. A comparison between the displacement spectra obtained from seismometers and accelerometers confirms the applicability of using such records. In the analysis of the M W = 7.2 Gulf of Aqaba earthquake on 22 November 1995, we use only S-wave displacement spectra which are determined from velocity data (BGIO) and from acceleration data (ASK, HAD, MAD, and HAC), at distances from 240 to 450 km. Figure 3e, f shows examples of displacement S-wave spectra from the broadband station BGIO and accelerometer ASK with time windows of 75 and 45 s for the seismogram and accelerogram, respectively.

3.2 Ratio between horizontal and vertical spectral amplitude levels

The empirical ratio between low-frequency spectral amplitudes of S waves recorded by vertical and horizontal seismographs was calculated from broadband stations. Using 141 records of earthquakes obtained by broadband seismometers and accelerometers in 2005–2011, we get Ω h/Ω v = 1.4 ± 0.32, where \( {\varOmega}_{\mathrm{h}}={\left[{\varOmega}_{\mathrm{EW}}^2+{\varOmega}_{\mathrm{NS}}^2\right]}^{\frac{1}{2}} \), and Ω EW and Ω NS correspond to recordings by the E-W and the N-S components, respectively. The observed average ratio agrees with 1.6 that was reported by Shapira and Hofstetter (1993). To correspond to Brune’s model, all spectral amplitudes of S waves for the vertical component have been corrected by a factor of 1.4.

3.3 Ratio between spectral amplitudes of S and P waves

We have investigated the relations between the low-frequency spectral displacements of S and P waves, arriving to the same station and its dependence on distance and magnitude. Figure 4a shows the observed Ω(S)/Ω(P) ratio as a function of the epicentral distance. Here we mark in red the data that correspond to measurements of earthquakes located in the Gulf of Aqaba. It is evident that those events, which were recorded at stations as far as 150 km or more, show significantly higher Ω(S)/Ω(P) ratios relative to earthquakes that occurred along the Dead Sea Fault and the Eastern Mediterranean region. All other spectral ratios are roughly of the same order. A careful check shows that the relatively high Ω(S)/Ω(P) ratios are due to significantly low Ω(P) observations (by a factor of 10 and more) in comparison with the spectral amplitudes of other earthquakes recorded at the same distances. Checked ratios Ω(S)/Ω(P) as a function of the duration magnitude we find that this anomaly was observed mainly on seismograms of larger magnitude earthquakes that are recorded at greater distances. The reason for the low Ω(P) observations is unclear. We may expect that it is due to a propagation path effect, which should be conducted in a separate study. Consequently, we have excluded Ω(P) observations obtained at epicentral distances greater than 150 km from earthquakes in the Gulf of Aqaba and have analyzed the relation between the low-frequency spectral displacements of S and P waves (Fig. 4b). Based on measurements from 702 seismograms, we derive the following relationship with standard deviation of 0.24 and coefficient of determination of 0.92

Fig. 4
figure 4

a The ratio between low-frequency spectral displacements from S and P waves for all events versus the epicentral distances. The red circles mark earthquakes that occurred in the Gulf of Aqaba region; b the relation between the low-frequency spectral displacements, Ω (m/s) of S and P waves along with the regression line; c, d distance correction is D = R α e δR for the low-frequency spectral amplitude displacement, Ω, from P and S waves, as a function of distance R, where the best-fitted lines correspond to D = Re 0.0042R (P wave) and D = R 0.8333 e 0.00365R (S wave), respectively

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$$ \log \varOmega \left(\mathrm{S}\right)=\left(0.49\pm 0.07\right) + \left(0.95\pm 0.01\right) \log \varOmega \left(\mathrm{P}\right) $$

3.4 Regional attenuation of the low-frequency spectral displacements

To obtain the source spectrum from the displacement spectrum of P and S waves, we study the distance effect on the low-frequency spectral amplitude, Ω, in a similar way to the procedure presented by Shapira and Hofstetter (1993). They described the distance correction D = R α e δR of S wave in terms of geometrical spreading R α and the inelastic attenuation e δR, i.e., where R is the epicentral distance. We calculate the distance corrections for sets of 74 and 85 earthquakes that are recorded by at least six stations (usually 8–11 stations) for P wave and S wave, respectively. For the geometrical attenuation of P waves, we fix α = 1, and for S wave, an airy phase α = 5/6 and then estimate δ by searching the lowest sum of squared discrepancies between observed and predicted values. The obtained optimal values are δ = 0.0042 km− 1 for P waves and δ = 0.00365 km− 1 for S waves with coefficients of determination 0.87 and 0.97, respectively. Figure 4c, d shows the distance corrections versus the distance R for P and S waves. The obtained distance attenuation coefficients are almost identical to those for S wave of Shapira and Hofstetter (1993) and roughly the same for P and S waves. The maximum differences between the observed and predicted for P and S waves are ±1.5 and ±0.8 in logarithmic units, respectively, with distance dependence.

3.5 Distance effect on the measured corner frequency and ratio between corner frequencies obtained from S and P waves

To estimate the corner frequency at the source from the P- and S-wave displacement spectrum at distance R, we need to obtain the distance scaling parameters. We assume that the distance effect is characteristic of the medium and can be empirically determined. We use the equation f 0 = f 0 e γR for each station for a given event, where f 0 is the corrected corner frequency at the source and f ′0 is the corner frequency at the station at distance R. We executed this procedure for the measured corner frequencies from 714 and 900 records for P and S waves, respectively. Figure 5a presents S-wave corner frequencies at stations as a function of the epicentral distances for selected events in the magnitude range of 2.7–5.5. The lines are essentially parallel with no dependency of their shape on the magnitude. Then applying a search technique, we find γ = 4.3 × 10− 4 and γ = 4.4 × 10− 4 km− 1 with coefficients of determination 0.67 and 0.80 for P and S waves, respectively (see Fig. 5c, d), being in good agreement with Shapira and Hofstetter (1993). The maximal differences between the observed and predicted values are ± 0.2 and ± 0.15 in logarithmic units for P and S waves, respectively. The corner frequency of the event is the geometrical average of f 0 estimates from the available seismograms of the event with standard deviations in logarithmic units of 0.03 and 0.02 for P and S waves, respectively.

Fig. 5
figure 5

a S-wave corner frequency at stations as a function of the epicentral distances for selected events. The change in colors relates to different earthquakes in the magnitude range of 2.7–5.5; b corner frequency determined from P-wave analysis f 0(P) versus f 0(S) determined from S-wave analysis; c, d the ratio between f ′0 and f 0 as a function of epicentral distance R. The solid lines represent the relations ln(f ′0 /f 0) = − 0.00043R and ln(f ′0 /f 0) = − 0.00044R for P-wave and S-wave corner frequencies, respectively

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It is expected that f 0(P) > f 0(S), following the studies of Archambeau (1966), Hanks and Wyss (1972), Molnar et al. (1973), Watanabe et al. (1996), and Tusa and Gresta (2008). Hanks and Wyss (1972) claimed that f 0(P) should be shifted by a factor of V P/V S relative to f 0(S), where V P and V S are P-wave and S-wave velocities, respectively, but, as they suggested, the validity of that assumption has to be empirically checked. A comparison between corner frequencies estimated from P and from S waves is presented in Fig. 5b. Our results based on 93 earthquakes for which the spectral parameters are derived both from P and S waves show an average ratio of f 0(P)/f 0(S) = 1.24 ± 0.14 that is in good agreement with this assumption.

4 Determination and analysis of source parameters

4.1 Seismic moment and moment magnitude

We use P- and S-wave spectrum for estimation of the seismic moment M 0. Following the source model of Brune (1970, 1971), the seismic moment, M 0 (N m), is

$$ {M}_0=4\pi \varOmega \left(\mathrm{P},\mathrm{S}\right)\rho D{V}_{\mathrm{P},\mathrm{S}}^3/(CF) $$

where Ω(P, S) is the measured low-frequency level of the displacement spectrum of the P or S wave (m/s); ρ is the density (2700 kg/m3); D = R α e δR is the distance correction, derived for P and S waves; V P,S is the average velocity of P wave 6200 m/s or S wave 3,600 m/s in accordance with the velocity model used for earthquake location when the events occurred in the upper crust at depths less than 20 km; C is the free surface amplification, assumed to be equal to 2; and F is the P- and S-wave radiation pattern correction factor, i.e., F P = 0.64 (Baumbach and Bormann 2012) and F S = 0.18 (Hanks and Boore 1984).

The seismic moments are estimated for P and S wave and then geometrically averaged for all stations that recorded the event. The values of the seismic moment, M 0, and the corresponding moment magnitude, M W, are in the range of 1.0 × 1013 ≤ M 0 ≤ 7.7 × 1019 N m and 2.6 ≤ M W ≤ 7.2. The standard deviation for determining the moment magnitude, M W, is defined for each event, then it is averaged over all events being 0.1 (P wave) and 0.05 (S wave). Figure 6a compares the seismic moment estimated from P waves, M 0(P), and that from S waves, M 0(S). The good agreement demonstrates the reliability of M 0(P) estimations.

Fig. 6
figure 6

a Seismic moment M 0(P) versus seismic moment M 0(S). The straight line represents M 0 (P) = M 0 (S); b, c the relations between the seismic moments M 0, determined from P waves (93 events) and S waves (99 events), and the duration magnitudes, M d, excluding the largest earthquake with M W = 7.2. The red and black lines represent the linear equations for two magnitude ranges (2.7 ≤ M d ≤ 4.0, dashed red line; 4.0 ≤ M d ≤ 5.6, solid red line) and polynomial quadratic equations (see Table 2)

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It should be noted that for the Gulf of Aqaba earthquake on 22 November 1995, we obtain M 0 = 7.67 × 1019 N m, using velocity and acceleration data as described above. This result is in very good agreement with the estimates obtained from waveform inversion techniques, i.e., M 0 = 7.21 × 1019 N m by Harvard’s centroid moment tensor (CMT) solution, M 0 = 3.8 × 1019 N m by Pinar and Turkelli (1997), M 0 = 7.42 × 1019 N m by Klinger et al. (1999), and M 0 = 7.7 × 1019 N m by Hofstetter et al. (2003). Table 1 lists estimates of M 0 and M W obtained from P- and S-wave spectral analysis in this study, for which waveform inversion studies were also carried out. All these estimates are in agreement with each other.

Table 1 M 0 and M W estimates obtained from spectral analysis of P and S waves and waveform inversion studies (Hofstetter et al. 2003; Hofstetter et al. 2008; Meirova and Hofstetter 2012), Harvard’s CMT solution (bold), GFZ MT solution (italics), and data from the European-Mediterranean RCMT catalog (bold italics)
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To correlate the seismic moment M 0 with the duration magnitude, M d, determined for the Israel Seismic Network, we apply regression analysis separately for P- and S-wave data (see Fig. 6b, c and Table 2). We also obtained the polynomial quadratic equation for the whole magnitude range. In general, these relations are in agreement with those developed previously from S-wave displacement spectra by van Eck and Hofstetter (1989), Shapira and Hofstetter (1993), Hofstetter et al. (1996, 2008), Hofstetter and Shapira (2000), and Hofstetter (2003), for events along the Dead Sea Fault and the Eastern Mediterranean region.

Table 2 The relationship log M 0 = aM d + b between the seismic moment M 0 [N m] and the duration magnitude M d. We also assume a quadratic relationship log M 0 = aM 2d  + bM d + c (bottom)
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5 Discussion and conclusions

In this study, we determine the source parameters, corner frequency, and seismic moment of 103 local and regional earthquakes of magnitudes ranging between 2.6 and 7.2, which were estimated from the P-wave and S-wave displacement spectra. Distance correction attenuation parameters for the low-frequency amplitude and for corner frequency were obtained, and the seismic moment, M 0, and moment magnitude, M W, were determined. We assume that this distance effect is characteristic of the medium and can be empirically determined. Figure 5a presents S-wave corner frequencies at stations as a function of the epicentral distances for selected events in the magnitude range of 2.7–5.5. The almost parallel lines show no dependency on the magnitude, site effect, size of the data window, or azimuths of the earthquakes, which support our assumption. Determination of Q values is beyond the scope of this study. However, the relatively small scattering as presented in Figs. 4, 5, and 6 suggests that our empirical determination of the distance effect is valid, and the Q effect in our study has no major effect as it is implicitly included in our distance correction. Thus, we argue that our empirical method provides reliable results similar to those of Brune’s method. The source parameters estimated from P-wave spectra and from S-wave spectra are consistent. We note that the source parameters determined from analyzing P waves show higher scatter relative to those determined using S waves. We find that the scaling relationships between seismic parameters of the study are in agreement with those obtained previously from the analysis of S waves by Shapira and Hofstetter (1993), Hofstetter (2003), and Hofstetter et al. (2008) for events along the Dead Sea Fault or the Eastern Mediterranean region, demonstrating the stable and systematic relationships of the earthquake source parameters in our region.

5.1 Stress drop

Uncertainties in measuring stress drop can be quite large; nevertheless, based on the above-mentioned analysis, we can estimate stress drop values. The seismic moment M 0 and the corner frequencies f 0 at the source were used to assess the stress drop, Δσ, in a similar way to Brune’s model (1970), separately for P- and S-wave data

$$ \varDelta \sigma \left(\mathrm{P},\mathrm{S}\right)=8.47{M}_0\left(\mathrm{P},\mathrm{S}\right){\left(\frac{f_0}{V_{\mathrm{S}}}\right)}^3 $$

where V S is the S-wave velocity near the source, V S = 3600 m/s; f 0 is the corner frequency at the source and, in the case of P wave, also is divided by a factor of 1.24; and Δσ is in pascal (kg m−1 s−2). The event’s stress drop Δσ is the geometrical average of individual determinations of Δσ by all the stations that recorded the event. Figure 7c presents the stress drop estimated from P-wave spectra and S-wave spectra. We find a good agreement between Δσ(P) and Δσ(S) estimates.

Fig. 7
figure 7

a Stress drop versus seismic moment obtained from P- (blue) and S-wave spectra (red), including the Gulf of Aqaba earthquake on 22 November 1995, M W = 7.2; b moment magnitude, M W, versus the source area, A = πr 2, in square kilometers, where r is the source radius of earthquakes, obtained from P-wave (blue circles) and S-wave (red circles) spectra, including the 1995 Gulf of Aqaba earthquake; c P-wave stress drop, Δσ(P), versus S-wave stress drop, Δσ(S). The straight line presents Δσ(P) = Δσ(S); d relationship between the seismic moment, M 0, and the corner frequency, f 0, obtained from P waves (blue circles) and S waves (red circles). The solid lines indicate the expected relation when the stress drop is constant

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The stress drop values, Δσ, vary from 0.1 to 44 MPa and are consistent with stress drop values determined by Shapira and Hofstetter (1993) and Hofstetter (2003) for events in the seismic moment range of 1012–1016 N m. The latter suggested for the Dead Sea Fault a constant stress drop of 90–100 bar (9–10 MPa). The constant stress drop level is reached asymptotically at strong magnitude events. This has been pointed earlier by Hanks and Thatcher (1972, 1973), Kanamori and Anderson (1975), Ide and Beroza (2001), Atkinson (1993, 2004), Kanamori and Brodsky (2004), Imanishi and Ellsworth (2006), and Allmann and Shearer (2009). It should be noted that some studies find several anomalous events with high or low stress drop (Baltay et al. 2011) or observe a systematic increase of stress drop with seismic moment or magnitude (Takahashi et al. 2005; Walter et al. 2006; Hofstetter et al. 2008; Tusa and Gresta 2008; Drouet et al. 2011).

The stress drop values, Δσ, versus the seismic moment, M 0, as obtained from P- and S-wave recordings for all the events are shown in Fig. 7a. The observations show that the stress drop increases with the increasing seismic moments in the range of 1013–1016 N m. It appears that events with seismic moments of 1016 N m or more asymptotically reach a stress drop of about 40–45 MPa. Figure 7d illustrates the relationship between the corner frequency f 0 and the seismic moment M 0. The corner frequency decreases with increasing seismic moment, where for reference the solid lines represent stress drop values of 0.1, 1, 10, and 100 MPa, based on f − 3 scaling. The corner frequencies obtained from P wave, f 0 (P) are divided by a factor of 1.24. Not assuming constant stress drop model, our observations suggest a relation of ∼ f − 4.7. Recently, several studies, i.e., Mayeda et al. (2007), Malagnini and Mayeda (2008), van Eck and Hofstetter (1989), Hofstetter et al. (2008), and Meirova and Hofstetter (2012), for the Hector Mine sequence, California; San Giuliano sequence, Italy; Dead Sea basin; and southern Lebanon, respectively, reported that the scaling of seismic moment and corner frequency does not follow f − 3 and is consistent with f − 4, suggesting that it is an evidence for the idea of non-self-similarity.

5.2 Source radius and source area

The source radius, r, in kilometers, was estimated from P- and S-wave records on the basis of the kinematic rupture model of Brune (1970, 1971)

$$ r\left(\mathrm{P},\mathrm{S}\right)={K}_{\mathrm{P},\mathrm{S}}{V}_{\mathrm{S}}/2\pi {f}_0\left(\mathrm{P},\mathrm{S}\right) $$

where K P = 3.36, K S = 2.34, V S = 3.6 km/s, and f 0(P, S) is the corner frequency measured on the P- and S-wave displacement spectrum (Baumbach and Bormann 1999, 2012). For the analyzed earthquakes, the source radius ranges from 0.26 to 9.2 km, and accordingly, the source areas, A = πr 2, vary from 0.2 to 264 km2. Figure 7b shows moment magnitude, M W, versus the source area, A = πr 2 (km2), for both P and S waves. There is an obvious increase of the source area with increasing M W and the good accordance between P- and S-wave data.

This study confirms the applicability and reliability of determining earthquake source parameters by analyzing the spectra of the first few seconds of P-wave train recorded by the stations of the ISN. Moreover, one can use the P-wave spectrum for rapid determination of the seismic moment and other parameters when trying to estimate magnitudes in real time. We should, however, pay attention to the observed anomaly of the spectral amplitudes from P waves from earthquakes that occur in the Gulf of Aqaba and recorded at distances greater than 150 km.