1 Introduction

The Zagros Fold and Thrust Belt, known as one of the youngest and most active continental collision zones on Earth, is the result of the continued convergence between the Arabia and Eurasia plates (Snyder & Barazangi, 1986). The Zagros is located in the south and southwest of Iran, where a relatively large population is involved in tectonic events and activities. This region has three main faults: the Zagros Main Recent Fault (ZMRF), the High Zagros Fault (HZF), and the Zagros Mountain Front Fault (ZMFF). Most earthquakes are small, medium, and shallow. Of course, there have been a few exceptions. The most devastating earthquake of the Zagros occurred on January 23, 1909, with mb = 7.4, and produced over 40 km surface rupture (Tchalenko & Braud, 1974). Also, an unexpected and devastating earthquake occurred on November 12, 2017, with Mw = 7.3, in the west of Zagros (Sarpol-e Zahab near the Iran-Iraq border). Therefore, it is important to identify the various aspects of these events, including determining the effects of source, path, and site. According to the National Center of Broadband Seismic Network of Iran, the epicenter of this event was located at 34.88° N and 45.84° E, with a depth of 23 km (Fig. 1). Several studies indicate that this earthquake occurred on a shallow northeast dipping fault plane (e.g., Gombert et al., 2019; Durmus & Utkucu, 2021).

Fig. 1
figure 1

Left: Three major reverse faults in the Zagros region: ZMRF (Zagros Main Recent Fault), HZF (High Zagros fault), ZMFF (Zagros Mountain Front Fault). Also, the location of the main earthquake in Sarpol-e Zahab along with its focal mechanism is shown. Right: Magnified image of the blue box on the left; red circles: aftershocks, blue triangles: stations (the reference station is highlighted), black lines: records ray paths

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In this study, the effects of source, path, and site of the 2017 Sarpol-e Zahab earthquake were estimated using the generalized inversion technique (GIT). This method was first proposed by Andrews (1986) by recasting the method of spectral ratios into a generalized inverse problem. Other researchers used and developed this method (e.g., Boatwright et al., 1991; Castro et al., 1990; Hartzell, 1992; Oth et al., 2008, 2011; Parolai et al., 2000; Salazar et al., 2007). To achieve this, appropriate records of aftershocks were selected and their shear wave spectra were extracted for inversion. After determining the source spectrum, corner frequency and seismic moments were also estimated using the ω2-model (Brune, 1970, 1971). Also, seismic energy was determined, and after determining the moment magnitude, its relationship with the local magnitude was obtained. By determining the attenuation spectra, the best model for the quality factor of the area was obtained and compared with other studies. Finally, the structural effects were estimated and the results were compared by the horizontal-to-vertical spectral ratio (H/V) method.

2 Dataset

In order to perform inversion, the data of the accelerometer network of the Building and Housing Research Center (BHRC) were used. These data are related to the 2017 Mw = 7.3 Sarpol-e Zahab earthquake aftershocks, from which appropriate data were selected. The criteria for selecting the data included good signal-to-noise ratio (SNR(f) > 3), appropriate distance, recording of each event by at least two stations, recording of at least two events by each station, and so on. Based on these conditions, 87 records were selected from the available data (black line on Fig. 1). These records, which were recorded in 13 stations (Table 1), are related to 30 aftershocks (Table 2). Most records were recorded by stations ban and spz, with 15 and 14 data points, respectively. The magnitude range of ML is from 3.8 to 5.3 and the hypocentral distance range is from 9 to 70 km (Fig. 2). Of course, records with larger hypocentral distances were also available, but due to the low density of records at distances larger than 70 km, they were not selected for inversion.

Table 1 The accelerometer stations used in this study
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Table 2 The aftershocks used in this study
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Fig. 2
figure 2

Distribution of magnitude (ML) versus hypocentral distance

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First, a baseline correction and a Butterworth band-pass filter (from 0.2 to 30 Hz) were applied to each record. Since the major portion of the seismic energy is generated in the form of S-waves, the analysis should be performed on this part of the strong motion records (Zafarani et al., 2012). Husid’s method (Husid, 1967) was used to select the start time of the S-wave (t0) when the record energy reaches 5% of the total wave energy. Kinoshita’s method (Kinoshita, 1994) was also used to select the end time of the S-wave (t1). Kinoshita (1994) defined the acceleration envelop function [e(n)] and the cumulative acceleration function [c(k)] according to the following equations:

$$ e\left( n \right) = \left( {a^{2} \left( n \right) + H^{2} \left\{ {a\left( n \right)} \right\}} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$
(1)
$$ c\left( k \right) = \left( {k^{ - 1} \mathop \sum \limits_{n = 1}^{k} e^{2} \left( n \right)} \right)^{{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$
(2)

In this relationship, a is the record of acceleration and H is the Hilbert transform. In this method, the end time of the S-wave is equal to the time at which the cumulative envelope function begins to decline (following the strategy of Kinoshita, 1994). In the next step, the Fourier spectrum of the S-wave must be calculated. However, the sudden interruption of the wave has a great impact on the Fourier spectrum. Therefore, after selecting the time of the beginning and end of the S-wave, a cosine taper was applied to it.

The abovementioned steps should be done for both horizontal components, and then the root mean square of both horizontal components should be calculated. Due to sudden changes in the Fourier spectrum, a smoothing function must be applied to it. One of the suitable methods for this work is the method of Konno and Ohmachi (1998). In this method, a window is applied to the signals according to the following equation:

$$ W\left( {f,f_{c} } \right) = \left[ {\frac{{\sin \left( {\log \left( {\frac{f}{{f^{\prime } }}} \right)^{b} } \right)}}{{\log \left( {\frac{f}{{f^{\prime } }}} \right)^{b} }}} \right]^{4} $$
(3)

where f′ represents the central frequency around which the smoothing is performed and supposed b = 20. This window has constant width on the logarithmic frequency axis. The frequency range in this study is 0.5–25 Hz, from which the values of the acceleration spectrum at 40 points with the same logarithmic distances are interpolated from this range.

3 Generalized Inversion Technique

The generalized inversion technique is based on the ground motion convolution model (multiplication in the frequency domain):

$$ U_{ij} \left( {f,R} \right) = S_{i} \left( f \right). P\left( {f,R} \right).I_{j} \left( f \right) $$
(4)

In this formula, Uij (f, R) is the Fourier amplitude spectrum (in this study, the horizontal component of the observed acceleration) from the ith source and the jth station at the hypocentral distance R. Also, Si(f) is the source function, P(f, R) is the attenuation function along the path, and Ij(f) is the local amplification of the site. By applying logarithms to both sides of this equation, the problem becomes linear:

$$ \log U_{ij} \left( {f,R} \right) = \log S_{i} \left( f \right) + \log P\left( {f,R} \right) + \log I_{j} \left( f \right) $$
(5)

If we have N records, the above equation will become a system of linear equations with N relation (Ax = b). Like most geophysical problems, it is overdetermined. Therefore, by choosing an appropriate approach to inversion such as singular value decomposition (SVD) (Menke, 1989) we will have:

$$ x = \left( {A^{T} A} \right)^{ - 1} A^{T} b $$
(6)

These steps were first described by Andrews (1986). He worked on data with low hypocentral distance, as well as data from which the effect of geometric spreading had been removed. If we have a suitable attenuation model (geometric distribution, quality factor, etc.) from other studies, and NE is the number of sources (events) and NS is the number of sites (stations), the number of unknowns will be NS + NE. The unknowns will be calculated at each frequency separately. The coefficient matrix will consist of zeros and ones only. In other studies such as Hartzell (1992), Hartzell et al. (1996), and Parolai et al. (2001), the effects of the attenuation function on the records were also corrected.

As Andrews (1986) and other researchers have pointed out, there is a degree of freedom in relation (5), so that by multiplying one number in each sentence and dividing the same number in another sentence, there will be no change in the acceleration spectrum. This can also be problematic in the SVD method, with zero singular values for the coefficient matrix. To prevent this, we can consider a default value for the source (e.g., Salazar et al., 2007) or site. A common assumption for the site is that the site response (I) in rock sites is independent of frequency, and its value is considered to be 1 (e.g., Hartzell, 1992; Hartzell et al., 1996; Parolai et al., 2000).

4 Results

4.1 Source Effect

One of the unknowns in the inversion procedure is the effect of the seismic source, which is obtained as a spectrum for each source (event). Since 30 events were considered in the present study, 30 spectra of the source are obtained in this section. For example, four source spectra are shown in Fig. 3. The spectrum obtained from generalized inversion is marked with a full line, and its standard deviation is marked with a gray color (shadow). Also, the origin time of each event and its depth and moment magnitude are written on each spectrum.

Fig. 3
figure 3

The source spectrum of four events obtained by GIT

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From the source spectra, we can have an analysis of other parameters such as seismic moment (M0), corner frequency (fc), and stress drop (Δσ). Based on the ω2 source model (Brune, 1970, 1971) we will have:

$$ S\left( f \right) = \left( {2\pi f} \right)^{2} \cdot \frac{{R_{\theta \phi } V F}}{{4\pi \rho_{s} \beta_{s}^{3} R_{0} }} \cdot \frac{{M_{0} }}{{1 + \left( {f/f_{c} } \right)^{2} }} $$
(7)

where Rθϕ = 0.55 is the mean value of the S-wave radiation pattern, and the value of V = 1/√2 is considered to include the ratio of the division of the total energy into two horizontal components. F = 2 is free surface amplification. ρs = 2800 gr/cm3 and βs = 3.7 km/s are considered as the average for the path density and velocity of the S-waves, respectively. Corner frequency and seismic moment are our unknowns that will be obtained using nonlinear regression. By placing these two parameters in the source equation, the theoretical source spectrum is obtained (dashed lines in Fig. 3). The logarithmic diagram of the corner frequency in terms of the seismic moment is shown in Fig. 4a. Other quantities called source radius (r) and stress drop (Δσ) in this source model are defined as follows:

$$ r = \frac{{2.34 \beta_{s} }}{{2\pi f_{c} }} $$
$$ \Delta \sigma = \frac{{7 M_{0} }}{{16 r^{3} }} $$
(8)
Fig. 4
figure 4

a Corner frequency (fc) versus seismic moment (M0). The relationship between fc and M0 for three various constant stress drops (1, 10, and 30 Mpa) is represented by solid lines. These lines come from Eq. 8. b Mw obtained in this study versus ML reported by the ISC catalog, along with the best-fitted line. c Radiated energy (Es) versus seismic moment along with the best-fitted line. d the scatter diagram of stress drop versus moment magnitude

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In order to determine the moment magnitude (Mw) based on the seismic moment, the equation of Hanks and Kanamori (1979) has been used:

$$ \log M_{0} = 1.50 M_{w} + 9.05 $$
(9)

The diagram of Mw versus local magnitude (ML) is shown in Fig. 4(b). The Mw range is obtained from 3.8 to 5.2. The best-fitted line for this diagram is estimated to be Mw = 0.71 ML + 1.27, and its regression coefficient was R2 = 0.73.

Also, the apparent stress σa can be calculated by σa = μEs/M0. where μ and Es are the rigidity modulus (\(\mu = \rho_{s} \beta_{s}^{2}\)) and the radiated energy of the S-wave, respectively. In this study, the equation of Izutani and Kanamori (2001) is used to estimate the energy:

$$ E_{s} = \frac{4\pi }{{5\rho_{s} \beta_{s}^{5} }} \mathop \int \limits_{0}^{\infty } \left| {\frac{{f M_{0} }}{{1 + \left( {\frac{f}{{f_{c} }}} \right)^{2} }}} \right|^{2} df $$
(10)

These parameters for all 30 sources are shown in Table 3. The corner frequencies vary from 0.8 to 3.8 Hz. The minimum and maximum seismic moments are 5.3 × 1014 and 7.9 × 1016 N m, respectively. As shown in Fig. 4a, the corner frequency decreases as the magnitude of the earthquake increases. Seismic moment and the cube of corner frequency are inversely related to each other, so we can write: M0fc3 = constant. In this study, the constant is equal to 3 × 1016 N m s−3. In other words, logM0 = 16.48 − 3logfc with regression coefficient 0.80. The ranges of stress drop and apparent stress are from 0.98 to 30.18 MPa and from 0.23 to 7.03 MPa, respectively. The relationship between these two parameters is estimated as Δσ = 4.29σa with a regression coefficient of 0.99. In this study, no significant relationship was observed between stress drop (or apparent stress) and moment magnitude (or seismic moment). This can also be seen in Fig. 4d. The S-wave radiated energy varies from 5.96 × 109 to 4.41 × 1012 J. It is clear that energy is directly related to the magnitude of the earthquake (or seismic moment) (Fig. 4b), so that in this study the relationship between them is as follows (with regression coefficient R2 = 0.79):

$$ \log E_{s} = 1.37\log M_{0} - 10.09 $$
(11)
Table 3 Some of the source parameters obtained in this study such as corner frequency (fc), seismic moment (M0), moment magnitude (Mw), source radius (r), stress drop (Δσ), apparent stress (σa), and radiated S-wave energy (Es)
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As mentioned, the relationship between moment magnitude and local magnitude is estimated as Mw = 0.71 ML + 1.27. In order to compare this relationship with previous studies, five relationships were considered (Table 4):

Table 4 Some of the relationships between Mw and ML considered in this paper
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The relationship obtained in this study agrees well with other relationships, especially with the relationship of Zare et al. (2014) and Shahvar et al. (2013), which has the least differences with them. In order to better represent and compare these relationships, their diagrams for the local magnitude range 3.6–5.5 are shown in Fig. 5.

Fig. 5
figure 5

Comparison of the relationship between moment magnitude and local magnitude obtained in this study and the results of other studies

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4.2 Attenuation Effect

One of the results of inversion is nonparametric attenuation functions, which are estimated at different frequencies. For example, Fig. 6 shows three attenuation functions at frequencies of 1.84, 9.17, and 20.46. As can be seen, the attenuation function changes smoothly and decreases with increasing distance. In order to better understand the attenuation function curve, its variations in all 40 selected frequencies along with curve 1/R are shown in Fig. 7. Due to the choice of R0 = 9 km as the reference distance, the log(A, R) in this distance is zero.

Fig. 6
figure 6

The attenuation spectra versus hypocentral distance in three selected frequencies. The gray crosses and shaded area represent the attenuation and the standard deviation around the mean, respectively

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Fig. 7
figure 7

Attenuation spectra versus hypocentral distance for all frequencies

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Considering the attenuation equation (Boore, 2003):

$$ A\left( {f,R} \right) = \left( {\frac{{R_{0} }}{R}} \right)^{b} \exp \left( {\frac{ - \pi f }{{Q\left( f \right) \cdot \beta }}\left( {R - R_{0} } \right)} \right) $$
(12)

in this study, \(\left( {\frac{{R_{0} }}{R}} \right)^{b}\) is a simple model of geometrical spreading, where b is the geometrical spreading component. By applying the logarithm to the sides of Eq. (12) and arranging its sentences we will have:

$$ \log A\left( {f,R} \right) - \log \left( {\frac{{R_{0} }}{R}} \right)^{b} = \frac{ - \pi f }{{Q\left( f \right) \cdot \beta }}\left( {R - R_{0} } \right) $$
(13)

According to the necessary condition of \(\log A\left( {f, R} \right) - \log \left( {\frac{{R_{0} }}{R}} \right)^{b} < 0\), we seek out a maximum b to meet this condition for each frequency, indicating the strongest geometrical spreading (Wang et al., 2018). By drawing the diagram to the left of Eq. 13 versus distance (for each frequency) and calculating the slope, the Q factor will be determined. On the other hand, we considered an additional condition (in addition to the above condition) so that by selecting it, the regression coefficient of Eq. (13) becomes at least 0.80. Values of 0.6 to 1 were tested for all frequencies, and the appropriate value is shown in Fig. 8a. As can be seen, in most frequencies the values 1 and 0.9 were selected, and only a few of them are 0.8 and 0.7. The values of Q at different frequencies are shown in Fig. 8b. A line (Q = Q0fN) was traditionally fitted to it. In this study, the quality factor was estimated as 78 f0.98 by the least-square solution. The value of Q0 is small, and the power of the frequency is close to 1, which is characteristic of an active tectonic environment.

Fig. 8
figure 8

a The geometrical spreading exponent for all frequencies. b The Q values at each frequency (0.5–25 Hz) along with their error bars. The solid line represents the best-fitted Q-model for this study

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4.3 Site Response

As mentioned, in order to eliminate the degree of freedom of Eq. 5, the response of the rock site can be considered equal to 1 (independent of frequency). In this study, because the type of sites is unknown, previous studies have been used. Boore and Joyner (1997) estimated the site response with Vs30 = 620 m/s (Fig. 9). According to Table 1, among our sites, the spz station has Vs30 = 619 m/s. Therefore, the site response of the spz station was assumed to be known (i.e., equal to the values obtained by Boor and Joyner (1997) for Vs30 = 620 m/s).

Fig. 9
figure 9

The site amplification diagram for sites with Vs30 = 620 m/s (Boore & Joyner, 1997)

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The horizontal-to-vertical spectral ratio technique (the H/V or HVSR method), introduced by Lermo and Chavez-Garcia (1993), indicates some effects of the site (mainly the fundamental frequency, if it exists). This method has been widely used by various researchers. In this method, it is assumed that the local effects of the site do not have a significant effect on the vertical component records. In this section, the effects of the site obtained from GIT and the H/V method were compared. The diagrams of site amplification from GIT (blue line) and the H/V method (red line) in the frequency range of 0.5–25 Hz is shown in Fig. 10.

Fig. 10
figure 10

The site response of S-waves obtained by the GIT in this study and the corresponding S-wave HVSR

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4.4 Residuals Distribution

In order to evaluate the stability and accuracy of the inversion results, first, the residuals from the synthetic results (generated by GIT) and observations were calculated at each frequency. For example, the residual diagrams in three frequencies of 2.04, 8.29, and 18.50 are shown in Fig. 11 (first row). This diagram is plotted in terms of hypocentral distance. Given the small amount of residual and also the independence of the distance, it can be concluded that the results of the inversion are reliable and close to reality. In the next step, values of 1 to 10% of the observations were added to or subtracted from them (randomly). These changes were made in the observations to firstly assess the sensitivity of the inversion to the input data and secondly not to claim that the input data is 100% correct. GIT was then applied to these modified data, and no significant changes appeared in the results, so that the remaining values have changed slightly (Fig. 11, second row). Since the differences of the two residuals are not noticeable, in order to better understand the values of the differences, a diagram of the difference between them is also drawn (Fig. 11, third row).

Fig. 11
figure 11

Top: residual diagram of GIT for three frequencies (f = 2.04 Hz [first column], f = 8.29 Hz [second column] and f = 16.74 Hz [third column]). First row: residual with main observed data, second row: residual with changed observed data, third row: differences between first and second rows. Bottom: all the residuals of all the frequencies together in a plot (red dot) vs the hypocentral distance, and the average value (blue dot) and the corresponding standard deviation

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5 Discussion and Conclusion

The source, attenuation, and site parameters of the 2017 Sarpol-e Zahab aftershocks were investigated using the non-parametric generalized inversion method. Eighty-seven acceleration records were selected from 30 sources for inversion.

The range of the corner frequency and the seismic moment varies from 0.8 to 3.8 Hz and from 5.3 × 1014 to 7.9 × 1016 N.m, respectively. This range is consistent with other studies. For example, Zafarani et al. (2012) estimated the corner frequency of Alborz as a range of 0.05–4 Hz. Their data ranged in moment magnitude from 4 to 7.4 and seismic moment from 1.26 × 1015 to 1.41 × 1020 N.m. The ranges of stress drop and apparent stress are estimated from 0.98 to 30.18 MPa and from 0.23 to 7.03 MPa, respectively. We also suggested a relationship between moment and local magnitude as Mw = 0.71 ML + 1.27 for study area.

The attenuation curve decays almost uniformly with increasing distance. The decay rate is higher for higher frequencies than for lower frequencies. At most frequencies, this curve decays faster than 1/R. Also, the relationship of the quality factor for this region is estimated as Q = 78 f0.98.

We compared the site effects calculated from the GIT with the HVSR computed from the same S-wave dataset. In most cases, relatively good agreement between them regarding the estimation of the fundamental frequencies and the general average amplification at all the sites was observed. The small difference between them is mainly due to the constraints and assumptions of these two methods, because in the H/V method, the site effects on the vertical component are completely eliminated, and on the other hand in the GIT, the site effects for the reference station (in this study spz) are default values.

The successful applicability of the GIT implementation in this study is supported by (i) the good agreement of the source factors with those obtained by the other mentioned studies; (ii) the relative good agreement of the site characteristics with those retrieved by the HVSR, also presented in this study; and (iii) the relatively low standard deviation of the residuals.