Stochastic source, path and site attenuation parameters and associated variabilities for shallow crustal European earthquakes

Abstract

We have analyzed the recently developed pan-European strong motion database, RESORCE-2012: spectral parameters, such as stress drop (stress parameter, Δσ), anelastic attenuation (Q), near surface attenuation (κ ₀) and site amplification have been estimated from observed strong motion recordings. The selected dataset exhibits a bilinear distance-dependent Q model with average κ ₀ value 0.0308 s. Strong regional variations in inelastic attenuation were also observed: frequency-independent Q ₀ of 1462 and 601 were estimated for Turkish and Italian data respectively. Due to the strong coupling between Q and κ ₀, the regional variations in Q have strong impact on the estimation of near surface attenuation κ ₀. κ ₀ was estimated as 0.0457 and 0.0261 s for Turkey and Italy respectively. Furthermore, a detailed analysis of the variability in estimated κ ₀ revealed significant within-station variability. The linear site amplification factors were constrained from residual analysis at each station and site-class type. Using the regional Q ₀ model and a site-class specific κ ₀, seismic moments (M ₀) and source corner frequencies f _c were estimated from the site corrected empirical Fourier spectra. Δσ did not exhibit magnitude dependence. The median Δσ value was obtained as 5.75 and 5.65 MPa from inverted and database magnitudes respectively. A comparison of response spectra from the stochastic model (derived herein) with that from (regional) ground motion prediction equations (GMPEs) suggests that the presented seismological parameters can be used to represent the corresponding seismological attributes of the regional GMPEs in a host-to-target adjustment framework. The analysis presented herein can be considered as an update of that undertaken for the previous Euro-Mediterranean strong motion database presented by Edwards and Fäh (Geophys J Int 194(2):1190–1202, 2013a).

1 Introduction

Estimation of regional seismological attributes such as source, path and site specific parameters is important in a wide range of applications in seismic hazard analysis. Usually, in engineering seismology and seismic hazard studies strong ground motions (specifically accelerations) are modeled in terms of a stochastic process (Hanks 1979; McGuire and Hanks 1980; Hanks and McGuire 1981) assuming that phases of high frequency waves (ground motions) are random. Typically, the stochastic (amplitude) model is characterized in terms of magnitude and stress parameter (Δσ) as being the source parameters; the geometrical spreading and anelastic attenuation (Q) of seismic wave describe the path effects. The site effects are modeled in terms of crustal amplification (Boore 2003; Joyner et al. 1981; Boore and Joyner 1997) and site-related attenuation parameter (κ ₀).

We have analyzed the recently developed pan-European strong motion database, RESORCE-2012 (Akkar et al. 2014a) in order to estimate these spectral parameters. In addition to estimate these seismological parameters that describe the amplitude and shape of Fourier spectra of strong ground motion, we also focus our discussion on variability and trade-off issues. More specifically, we analyze the following key issues related with the stochastic modelling of ground motion:

• (1) A European stochastic model has been derived by Edward and Fäh (2013a). This model was based upon a strong-motion database compiled more than a decade ago and, due to limited data, did not include regional variations in attenuation properties within Europe. The analysis of more recent strong-motion database indicates that regional variations of ground motions may be significant (Boore et al. 2014; Kotha et al. 2016; Kuehn and Scherbaum 2016) which is a motivation to evaluate the regional variations of stochastic ground motions parameters in Europe.

• (2) One of the major challenges in ground motion prediction is that motions need to be predicted for earthquakes without knowing the (future) stress-parameter (Δσ) associated with them. Therefore, usually blind predictions are made for an average value of Δσ with an associated standard deviation. Often, the average Δσ is assumed to be independent of magnitude (Aki 1967), however recent studies have indicated, that for small to moderate earthquakes (M _W  < 5) Δσ may increase with increasing magnitude and the average Δσ can also vary regionally (Malagnini et al. 2008; Edwards et al. 2008; Drouet et al. 2008, 2010; Yenier and Atkinson 2015a, b). Other than the magnitude dependence and regional variation, the spread (standard deviation) in estimated Δσ is also important in predicting ground motions for future scenarios (Cotton et al. 2013).

• (3) The decay of high frequency Fourier spectral amplitudes beyond the Brune’s corner frequency f _c is usually attributed to (an empirical parameter) κ (kappa) (Anderson and Hough 1984) that essentially captures the combined effect of whole path anelastic attenuation (Q) and near site attenuation κ ₀. Extrapolation of κ at near distances (≈0) is termed as κ ₀ and is believed to reflect near site attenuation. In a site-specific seismic hazard application, knowing κ ₀ beforehand at a target site is of crucial importance. As shown by Molkenthin et al. (2014) and Douglas and Jousset (2011) in a stochastic simulation framework, pseudo spectral accelerations (PSA) for small to moderate events at high oscillator frequencies >10 Hz are mainly controlled by κ ₀. However, measurement of a true site κ ₀ is not straightforward because the observed traces are recorded at a non-zero distance, which makes decoupling of path term (Q) and κ ₀ rather challenging without constraining any of the two a priori. In this context, regional variation of Q, as will be shown in this article, can also bias the estimation of κ ₀. Thus, the trade-off between Q and κ ₀ needs to be analyzed and accounted in the forward prediction as well.

• (4) Several studies have shown that κ ₀ varies significantly from site-to-site and regionally as well (e.g., Atkinson and Morrison 2009; Edwards et al. 2008; Campbell 2009; Edwards and Rietbrock 2009). The site-to-site/station-to-station variability of κ ₀ will be referred as between-station variability. The other component of variability in κ ₀ is related to the record-to-record (or within-station) variability. Within-station variability of κ ₀ can also have a strong impact on the site-to-site adjustment of GMPEs in the HTTA framework. However, this within-station variability has not been discussed much in past studies.

Assuming ω ² point-source model (Brune 1970, 1971)—which has been shown to be applicable to M _W  < 6 events globally, and a reasonable approximation for larger events at high frequencies (Chen and Atkinson 2002; Baltay and Hanks 2014)—a broadband (generalized) inversion technique (Edwards and Fäh 2013a; Bora et al. 2015) is performed to determine the source, path and site parameters.

Essentially, the inversion method remains the same as detailed in Bora et al. (2015) in which a full Fourier amplitude spectrum is used to estimate Brune’s source corner frequency (f _c ) and the whole path inelastic attenuation operator (t*) simultaneously. The inverted t* values are further analyzed to derive a frequency-independent Q ₀ model for the dataset and to analyze regional variation in Q ₀ within the selected dataset. The choice of a particular Q ₀ model is seen to have a large impact on the derived κ ₀ values. With respect to the regional Q ₀ models, we investigated two different components of variability in station κ ₀ in terms of station-to station variability (between-station) and record-to-record variability (within-station). After constraining the reference model by using the database M ₀ and regional inelastic attenuation Q ₀ models, average residuals are used to capture the linear part of the site amplification in the Fourier domain (Edwards et al. 2013; Drouet et al. 2010; Poggi et al. 2011). The final estimation of source f _c and M ₀ is obtained from site-effects corrected spectra, which are further analyzed to estimate stress parameter (Δσ) values.

2 Data

Different subsets of the RESORCE-2012 database have been used to derive several pan-European GMPEs (Akkar et al. 2014b; Bindi et al. 2014; Bora et al. 2014; Derras et al. 2014; Hermkes et al. 2014). The same dataset as used by Bora et al. (2015) is used in this study. This dataset is a subset of a larger parent database of RESORCE-2012 (Akkar et al. 2014a) and it contains strong motion recordings made across Europe, the Mediterranean region and the Middle East. To ensure the validity of the point-source (Brune model) approximation, we have discarded near distance traces from large magnitude earthquakes (Bora et al. 2015). For events with moment magnitude M _W  < 5.5 no trace was discarded; for events with 5.5 ≤ M _W  < 6.5 traces recorded at hypocentral distance R < 15 km were discarded; and for M _W  ≥ 6.5, traces recorded at R < 30 km were not considered. The dataset consists of 1200 (2400 including both the horizontal components) acceleration traces recorded at 350 stations from 365 earthquakes. Figure 1 summarizes the main metadata features of the selected dataset. Figure 1a depicts geographical distribution of epicenters from the selected dataset. The magnitude and distance range covered by the present dataset is M _W 4–7.6 and hypocentral distance up to 224 km respectively (Fig. 1b). The station Vs ₃₀ values range from 92 to 2165 m/s out of which 223 are measured and rest are inferred from different methods. All the events are shallow crustal events up to the focal depth of 30 km (Fig. 1c). For complete metadata information of the recordings, the reader is referred to the electronic supplement supplied with Bora et al. (2015). The processed database was disseminated to ensure a uniform data processing scheme for empirical ground motion prediction equation (GMPE) derivation.

Fig. 1

Metadata summary of the selected dataset. a Distribution of earthquake epicenters. b M _W -hypocentral distance distribution. c M _W -depth distribution. d Light-shade high-pass and dark-shade low-pass frequency. e Number of records per station versus number of stations

Additionally, for our analysis we use the Fourier amplitudes only between the useable frequency limits. We chose the filter high-pass and low-pass frequencies as the useable frequency limits, if a record is not assigned a low-pass frequency in the metadata then a flat low-pass frequency of 50 Hz was chosen. It is worth mentioning here that no smoothing is applied over the observed Fourier spectral amplitudes prior to the inversion except that the amplification curves are presented after smoothing (Konno and Ohmachi 1998) observed Fourier amplitude spectra. Figure 1d shows the distribution of high-pass and low-pass frequencies in the dataset, while Fig. 1e depicts the number of records per station against the number of stations.

3 Fourier spectral inversion

We use the Brune (1970, 1971) point source model with a single corner frequency (f _c ) to characterize the far field Fourier spectrum of acceleration records. In the stochastic modelling framework (Boore 1983, 2003), assuming that the high frequency ground motions of engineering interests are randomly distribution in phase the Fourier spectral amplitude Y at a frequency, f can be modeled using the following analytical relationship (Bora et al. 2015):

$$Y\left( f \right) = CM_{0} G\left( R \right)\left\{ {\frac{{\left( {2\pi f} \right)^{2} }}{{1 + \left( {\frac{f}{{f_{c} }}} \right)^{2} }}} \right\}e^{{ - \pi ft^{*} }} A\left( f \right).$$

(1)

In Eq. (1), M ₀ is the seismic moment in units of Nm and f_c is the corner frequency in Hertz, given by 0.4906β(Δσ/M ₀)^1/3 (Eshelby 1957; Brune 1970, 1971), in which Δσ is the stress parameter in Mega Pascal and β (=3500 m/s) is the shear wave velocity in the vicinity of the source. The constant C is generally taken as $\Theta _{\lambda \varphi } F\xi /\left( {4\pi \rho \beta^{3} } \right)$, in which $\Theta _{\lambda \varphi }$ (=0.55) is the average radiation pattern for S waves (Boore and Boatwright 1984), F (=2.0) is the near surface amplification, $\xi \left( { {=} 1/\sqrt 2 } \right)$ is a factor to account for the partition of total shear-wave energy into two horizontal components, and ρ (=2800 kg/m³) is the average density near the source (Boore 1983, 2003). G(R) is the geometrical spreading function representing a frequency-independent decay of amplitude as function of distance. Theoretically, G(R) is equal to 1/R at near distances (<~50 to 100 km) for an isotropic and homogenous whole space. However, the earth is not homogeneous and many studies have found it to be a complex function of distance (Campillo et al. 1984; Atkinson and Mereu 1992; Edwards et al. 2008; Atkinson and Boore 2011). To limit potential trade-off and bias, G(R) is constrained by using an earlier derived G(R) model from the same dataset in Bora et al. (2015) as:

$$G\left( R \right) = \left\{ {\begin{array}{*{20}l} {\left( {\frac{{R_{0} }}{R}} \right)^{1.14} } \hfill & {R \le 70} \hfill \\ {\left( {\frac{{R_{0} }}{{R_{1} }}} \right)^{1.14} \left( {\frac{{R_{1} }}{R}} \right)^{0.5} } \hfill & {R > 70} \hfill \\ \end{array} } \right.$$

(2)

In Eq. (2), R ₀ is assumed to be 1 km. Bora et al. (2015) derived the G(R) model from low-frequency (0.2–1 Hz) Fourier spectral amplitudes to minimize the trade-off resulting from high-frequency attenuation Q.

The fall-off of acceleration spectra at high frequencies is modeled by using a whole-path anelastic attenuation operator (t*). t* is alternatively named κ(R) (Anderson and Hough 1984) and κ _r (Ktenidou et al. 2014). The t* implies spectral decay at high frequencies due to path and site effects, while some authors (e.g., Kilb et al. 2012) argue contribution of source effects in t* as well. The combined effect of anelastic attenuation Q and site-related attenuation κ ₀ (Ktenidou et al. 2014) in t* is represented by the following equation:

$$t^{*} = \frac{R}{Q \beta } + \kappa_{0}$$

(3)

in which β (= 3.5 km/s) is the average shear wave velocity used to infer Q and R is the hypocentral distance. Some studies have suggested (Singh et al. 1982; Atkinson and Mereu 1992; Malagnini et al. 2000; Bay et al. 2003; Atkinson 2004; Drouet et al. 2008; Malagnini et al. 2011; Akinci et al. 2014) a Q model as a function of frequency as follows:

$$Q\left( f \right) = Q_{0} \left( {\frac{f}{{f_{0} }}} \right)^{\eta }$$

(4)

in which, η ranges from 0, for a frequency-independent Q, to 1 and Q ₀ is the reference Q value at f ₀ = 1 Hz. However, estimation of Q from spectra of observed recordings is strongly tied with the assumed geometrical spreading (e.g. Pacor et al. 2016). As shown by Edwards et al. (2008) a frequency-dependent Q function can lead to a strong trade-off with the geometrical spreading. Furthermore, Morozov (2008, 2009) have suggested that from a modeling perspective distinction between frequency-dependent Q and geometric attenuation is ambiguous. Thus, some studies (Anderson and Hough 1984; Hough et al. 1988; Edwards et al. 2008, 2011; Campbell 2009; Edwards and Fäh 2013a) in engineering seismology also use a frequency-independent Q = Q ₀ (constant) over the frequency-band it is measured. Thus in this article, we restrict our model formulation to a constant-Q model. It is beyond the scope of this article to investigate sensitivity of a chosen Q model (frequency-independent or constant) over the parameters derived here, however a constant-Q is expected to give lower κ ₀ value than that with a frequency-dependent Q. Nevertheless, the derived parameter values (with a constant-Q assumption) are consistent within the entire model framework (taking geometrical spreading, Δσ, Q ₀ and κ ₀ together). A(f) in Eq. (1) represents site amplification, which essentially captures the effect of impedance contrast during the wave propagation from the half-space through the upper soil layers to the station.

In our inversion scheme the observed spectra are inverted with respect to the natural log of the model described in Eq. (1) to determine M ₀, f _c and t* using a least squares fit in which the Newton’s method is used to linearize the nonlinear equation. To address the problem of two unresolved degrees of freedom, that is, M ₀ and A(f), in the first iteration of inversion the low-frequency spectral level is constrained by using M ₀ obtained from the database M _W (Hanks and Kanamori 1979). Thus, database M ₀ values and f _c and t* determined from the first iteration of inversion, essentially describe our reference model. This reference model along with a generic crustal amplification function defines the motion at the base of the soil column beneath the station. The (logarithmic) difference between the observed amplitude and the amplitude obtained by the combination (addition in log) of reference model and the generic crustal amplification is used to constrain the amplification A(f) at a given station (for details see section Site Amplification). The final estimates of f _c and M ₀ are obtained from site-corrected, A(f), spectra.

The inversion was performed over the full spectrum between the high-pass and low-pass frequencies of each record given in the metadata file. If a record is not assigned with a low-pass frequency then a flat low-pass frequency limit of 50 Hz is used. It is also known that site-effects can potentially bias the determination of seismological parameters from the surface recorded spectra. From records recorded at rock and hard rock site stations, determination of t* can be biased due to significant resonance effects at high frequencies (Parolai and Bindi 2004; Edwards et al. 2015). Similarly, at soft soil sites, resonance effects present at low frequencies can bias the determination of f _c . However, in the present dataset majority of the earthquakes are of small to moderate magnitudes (M _W 4–5.5), hence we believe that the corresponding f _c values will remain unaffected from the resonance effects. Furthermore, an event-wise (common for all records originated from an event) determination of f _c can limit the potential bias due to the site-effects (Edwards et al. 2008). In order to limit bias in the determination of t* due to crustal amplification, we correct all the spectra for a reference rock amplification function (Bora et al. 2015; Edwards et al. 2015). Also, fitting the entire shape (determined by f _c and t*) of the spectrum simultaneously limits the error in t* estimation that may arise due to the resonance peaks at high frequencies. The majority of the stations in the selected dataset are located over soil or stiff-soil sites (180 < Vs ₃₀ ≤ 750 m/s). The generic rock amplification of California (Boore and Joyner 1997) anchored at Vs ₃₀ 620 m/s was considered to be appropriate as reference rock amplification for the present dataset.

4 Attenuation parameters: t*, Q ₀ and κ ₀

As mentioned in the previous section, in the first step of our broadband inversion scheme we determine f _c and t* simultaneously based upon the model described in Eq. (1) while seismic moments are constrained from database M _W values (Akkar et al. 2014a) using the relationship of Hanks and Kanamori (1979). In our analysis, we use hypocentral distance as the preferred distance metric, following Edwards and Fäh (2013a). We obtain two t* values per record, one for each component, however in order to limit the scatter in data points, a mean t* (from both the components) is used here. Performing a least-squares linear fit using Eq. (3), over the dataset that contains t* values against R (hypocentral distance), we find dataset common Q ₀ (dividing the slope by β) and κ ₀ (the intercept) values as 1029 and 0.0361 s respectively. The 68% confidence interval of the best-fit slope corresponds to Q ₀ values 982 and 1080. Similarly, the 68% confidence interval of the intercept gives κ ₀ values of 0.035 and 0.0372 s.

In terms of comparison between Q ₀ values form this study and those from earlier studies, Edwards and Fäh (2013a) obtained the Q ₀ value as 619 from broadband fit using a smaller subset of the present dataset. In their study, Edwards and Fäh (2013a) used records only up to 100 km of hypocentral distance. Edwards et al. (2011) and Douglas et al. (2010) presented Q ₀ values 1216 and 1630 for Switzerland and France respectively using the records up to 300 km; while in this study we use records up to 224 km. Thus, this difference in Q ₀ values can be due to the differences arising from data-selection criteria, distance metric used and the actual regional difference in Q ₀. The choice of distance range in fitting t* − R data can also influence the estimated Q ₀ values.

In order to further explore the distance dependent estimation of Q ₀, we plot median of sorted (by distance) t* − R data in each 10 km distance bin, and as can be observed from Fig. 2a a clear trend indicating distance-dependent attenuation, Q ₀, (varying slope of t* − R relationship) is visible. As a cross-check, we also analyzed the t* values obtained from high-frequency linear fit method (Anderson and Hough 1984). The lower limit of the high-frequency range was selected (automatically) such that it is sufficiently above than the source f _c for each record for an assumed Δσ of 10 MPa and database M _W . The upper limit of the frequency range was either fixed to the low-pass frequency given in the metadata information or to 50 Hz for the records that are not assigned a low-pass frequency. Finally, only those t* values were selected which were measured over a band of at least 10 Hz. The high frequency fit t* values are plotted against distance in Fig. 2b and a binning scheme similar to the one in Fig. 2a was applied on the t* − R data. A similar trend to that in Fig. 2a can also be observed in Fig. 2b indicating that the t* − R relationship from the selected dataset shows a distance-dependent slope (hence Q ₀). This observation was validated further from least square fit on the actual t* − R data (without binning and averaging them) assuming a bilinear relationship with a slope-transition distance of 40 km. It can also be observed from Fig. 2a, b that, the straight-line fit of t* − R data gives different zero-distance intercept, κ ₀, whereas the values are rather identical for a bilinear fit. Hence a single linear fit model of t* − R relationship might overestimate the κ ₀ values. Anderson (1991) has also suggested that if a straight line fit does not fit the data well, any other smooth functional form can be chosen. Selection of a bilinear form for t* − R relationship against more complicated (e.g., quadratic R) forms was based upon a compromise between simplicity and effectiveness of the model in capturing the observed trend. Assuming an average β = 3.5 km/s, the Q ₀ values at R ≤ 40 and R > 40 km were obtained as: 610 and 1152 from broadband fit t* and 542 and 2493 from high-frequency fit t* values.

Fig. 2

t*-distance relationship from the entire dataset. a From broadband inversion method. b From high-frequency linear fit method (Anderson and Hough 1984). The empty circles indicate the individual data points while the empty squares indicate the median of t* − R data in each 10 km distance bin. The extent of vertical bars indicate the t* values corresponding to 16 and 84 percentiles in each distance bin

In order to investigate the regional variations in inelastic attenuation we split the t* − R data in three regional categories based upon the station locations: (1) stations located in Turkey, (2) stations located in Italy and (3) the remaining stations in the dataset. Although the physical properties are not expected to follow the political boundaries, our criterion for selecting data in different nation-regions is rather simple and based upon a similar criterion used in recent NGA-West2 GMPEs, e.g., Abrahamson et al. (2014) and Boore et al. (2014). The t* − R data for the three region categories is plotted in Fig. 3. As can be noted from Fig. 3, the majority of the data belongs to Turkey with 667 data points, followed by 372 from Italy and the remaining 161. Out of the three, only data from Turkey shows a uniform distribution with distance, the data from Italy and remaining stations is mostly concentrated at <70–80 km. Similar to Fig. 2, the t* − R data is binned in 10 km distance bins and corresponding median values and spread in each bin is also plotted over the actual distribution of the data. Although, one may observe a distance dependent variation in slope, a straight line fit over t* − R data from Turkey captures the observed trend reasonably well. The Italian data is mostly concentrated at smaller distances. Nevertheless, the straight line fit captures the observed trend well over the full distance range. Due to the very limited data points beyond 80 km, the fit was performed only up to 80 km for the remaining t* − R data (Fig. 3c). It is worth mentioning that, the fitted lines shown in Fig. 3 are obtained from the fitting of actual data points. Using β = 3.5 km, the Q ₀ values and associated variabilities for the three region categories are shown in Fig. 4a. Figures 3 and 4a clearly indicate that within the selected dataset and at large in the RESORCE database, there are strong regional variations in anelastic attenuation. The data from Turkey exhibits rather low attenuation (high Q ₀) in comparison to that for Italy, an observation that was also noted by Boore et al. (2014) and Kotha et al. (2016) in their empirical models.

Fig. 3

t*-distance model for: a Turkey, b Italy, c and remaining dataset. Empty circles indicate individual data points while empty squares represent the median of t* − R data in each 10 km distance bin, while the extent of vertical bars indicates the t* values corresponding to 16 and 84 percentiles in the bins

Fig. 4

Regional variations in inelastic attenuation parameter Q ₀ (a) and site-related attenuation κ ₀ (b). The squares in (b) indicate average κ ₀ values when the respective Q ₀ values of (a) are used while the discs indicate the κ ₀ values when the database based bilinear Q ₀ model shown in Fig. 2a is used. The extent of the vertical bars corresponds to the 68% confidence interval of each parameter estimate

Estimation of κ ₀ using Eq. (3) is strongly biased upon the assumed Q model. Consequently, and as can also be observed in Fig. 4b, a higher Q ₀ gives a higher κ ₀ and similarly lower κ ₀ is obtained for a lower Q ₀. A smaller variability in Q ₀ and κ ₀ (Fig. 4b) for Italy in comparison to that for Turkey can be attributed to that the Italian dataset is mainly concentrated at smaller distances. Hence, the ray paths are mostly sampling similar (shallower) depths in the subsurface, thus fewer variations due to anelastic attenuation. For the remaining dataset, the large uncertainty can be explained due to its regional heterogeneity. Therefore, depth variations in Q ₀ can be expected as well as the regional variations. In order to further investigate the regional variations in κ ₀, we constrained Q ₀ from the common (database) bilinear model shown in Fig. 2a to estimate the average κ ₀ in Turkey and Italy. Additionally, we limit the data only up to 50 km for this analysis to constrain the bias from Q ₀ variations. As expected due to the coupling of Q ₀ and κ ₀, one can observe in Fig. 4b that the κ ₀ values are different than that when we use regional Q ₀ values (Fig. 4a) for Turkey and Italy. However, the rather important observation is that using a common Q ₀ value for the two regional datasets also indicates a significant variation in attenuation properties for Turkey and Italy due to surficial layers as near distance earthquakes are expected to sample near-surface structure of the subsurface. Hereafter, regional Q ₀ values for Turkey and Italy will be used for further analysis. While for the remaining dataset the database based distance-dependent Q ₀ model (Fig. 2a) will be used.

5 Station and site-class specific κ ₀

In stochastic simulations (Boore 2003) as well as in HTTA adjustments of empirical GMPEs a prior measurement of κ ₀ at a given site is required (e.g., Campbell 2003; Van Houtte et al. 2011; Edwards et al. 2016). In order to obtain a station-specific κ ₀ estimate, we correct all the individual record t* (both the components individually) for the slope in t* − R straight-line fit corresponding to regional Q ₀ values, that is, 1462 for Turkey and 601 for Italy. For the remaining dataset, the database based, the two-slope t* − R model presented in Fig. 2a is used. Subsequently, the median of all record κ ₀ values at a station is presented as the station κ ₀. Table 1 presents estimated κ ₀ values for 45 stations recording at least 14 component-records.

Table 1 Station specific κ ₀ estimation for stations recording at least seven records

Figure 5 depicts variation of station κ ₀, and associated variability with Vs ₃₀ for stations with Vs ₃₀ > 360 m/s and with minimum ten records (including both the components). Figure 5a depicts the plot only for stations that have recorded earthquakes located at a distance ≤40 km. As mentioned earlier, regional variation in Q models can bias the estimation of κ ₀; thus Turkish stations are observed to exhibit consistently higher κ ₀. An important observation from Fig. 5 is that, there is a rather large record-to-record (within-station) variability (the vertical bars) in κ ₀, which in many cases is comparable to the station-to-station (between-station) variability (horizontal dashed-lines) of κ ₀. The between-station variability is mainly affected by regional variations in Q ₀. Although we did not observe a clear correlation between κ ₀ and Vs ₃₀ (Fig. 5b), the between-station variability can also increase since softer sites may exhibit higher κ ₀ (Chandler et al. 2006; Van Houtte et al. 2011; Edwards and Fäh 2013a). On the other hand, the within-station variability is due to the fact that Q ₀ is not homogeneous with respect to depth (Edwards et al. 2008; Edwards et al. 2011). Hence estimating κ ₀ from near as well as distant earthquakes using a homogenous Q model can also inflate the within-station variability. As can also be noted from Fig. 5b: the Italian stations depict less within-station variability in comparison to the Turkish stations. Additionally, a possible source component in κ ₀ (Kilb et al. 2012) can also contribute to the larger within-station variability. Figure 6 demonstrates the effect of within-station variability in κ ₀ by showing plots of spectra obtained from actual fit and that from regional Q ₀ and station κ ₀, vis-à-vis observed spectra. The spectra are shown at a station in Turkey with Vs ₃₀ = 747 m/s and for earthquakes at less than 40 km distance from the station.

Fig. 5

Station κ ₀ plotted against Vs ₃₀ values for Vs ₃₀  > 360 m/s: a when earthquakes located at <40 km (from a station) are used, b when all the earthquakes recorded at a station are used. Markers (empty circles, disks and empty squares) indicate the median while the extent of vertical bars indicates the values corresponding to 16 and 84 percentiles at each station, i.e., within-station variability. The horizontal solid line indicates the median value of all station κ ₀ in the sample, while two dashed lines indicate 16 and 84 percentile values in the sample, i.e. between-station variability. In both cases stations which have recorded at least 10 records (including both the components) are used

Fig. 6

Within station variability of κ ₀ . Spectra for acceleration traces recorded at a station (Stn. Id. 2322) in Turkey with a Vs ₃₀ of 747 m/s. The regional Q ₀ 1462 for Turkey is used along with the station κ ₀ 0.0468 s

In order to cover a broad range of station sites, we also estimate site class specific κ ₀ values, which can be used as a first order approximation for stations not having endemic measurements of κ ₀. In addition to the regional classification based upon Q ₀, stations were classified in different site-classes based upon their Vs ₃₀ values as: very soft soil as Vs ₃₀ ≤ 180 m/s, 180 < Vs ₃₀ ≤ 360 m/s as soft soil, 360 < Vs ₃₀ ≤ 750 m/s as stiff soil and Vs ₃₀ > 750 m/s as rock sites. Subsequently the site class κ ₀, in each regional subset, is computed as the median of all record κ ₀. The site class specific κ ₀ and corresponding variabilities are presented in Table 2. A rather large κ ₀ for rock sites in Turkey can be a sampling issue with only eight data points. For Italian sites we observe a decreasing κ ₀ from very soft soil sites to rock sites. In the remaining dataset, there were no stations corresponding to the very soft soil site condition. For a combined (data from all regions), but significantly limited subset of this dataset, Edwards and Fäh (2013a) obtained κ ₀ as 0.0326, 0.0375, 0.0303 and 0.0241 s for very soft soil, soft soil and stiff soil and rock site respectively. The values of site class κ ₀ for Turkish dataset from this study are comparable with the findings of Askan et al. (2014) with κ ₀ values 0.0377 and 0.0455 s in stiff and soft soil category.

Table 2 Site-class specific estimates of κ₀ in each regional subset

6 Station and site-specific amplification

We invert for a reference model using a priori seismic moments and geometrical spreading from Bora et al. (2015). The station (Fourier) site amplification factors (AF _Fourier) are estimated with respect to this reference model from a residuals analysis (Edwards et al. 2008; Drouet et al. 2010; Edwards and Fäh 2013a). For the reference model, M ₀ is used from database M _W and t* values were fixed to a value that is obtained from a combination of regional Q ₀ and station κ ₀ derived in the previous section. In order to obtain the site AF _Fourier at one station, we take mean of all (log) residuals (at each frequency) with respect to the reference model. Out of total 350, only 223 stations characterized with measured Vs ₃₀ estimates are used in the site amplification analysis.

Figure 7 depicts AF _Fourier plots for selected stations, which have recorded at least ten horizontal records (five earthquakes), except for a station (station Id. 2498) in Greece. Stations with station Ids 131 and 3690 indicate resonance effects present at such sites, which are consistent with the notion that stiff soil/rock sites may indicate resonance peaks at high frequencies while at softer soil sites such effects are mostly dominant at lower frequencies. It is worth to note here that, for determining AF _Fourier, we additionally excluded the records recorded at R ≤ 60 km (from earthquakes with M _W  ≥ 6.5) to avoid the nonlinear soil response effects. Consistent with our site-class specific κ ₀ estimates, we also present site-class specific amplification factors AF _Fourier for the four site classes for each regional subset in Fig. 8. Such plots also provide guidance in defining the amplification at stations for which direct measurements of Vs-profiles are not available. Although, resonance peaks are not apparent in the site-class AF _Fourier curves due to the broad site classification, the very soft and soft soil site indicate a large amplification at lower frequencies, and a deamplification at large frequencies may indicate (residual) non-linear site effects. Whereas, the stiff soil and rock sites indicate amplifications almost independent of frequency. Regional variations in site-class average site amplification factors are not apparent from Fig. 8. Rock motions show an amplification close to one, which indicates that the reference is well calibrated. Also, the overall slight-deamplification for rock (Vs ₃₀ > 750 m/s) in Turkey is consistent with respect to the chosen reference amplification of California (Boore and Joyner 1997) anchored at Vs ₃₀ 620 m/s. However, the similar AF _Fourier curves for stiff soil and rock conditions in Italy indicate towards misclassification for some of the stations (Lucia Luzi, personal communication). It is worth emphasizing again that these amplification curves are obtained with respect to a crustal reference amplification of Boore and Joyner (1997) by removing it from the observed spectra. Therefore, Boore and Joyner (1997) crustal amplification curve should be used along with these AF _Fourier curves in a forward prediction application.

Fig. 7

Station-specific Fourier amplification curves for selected stations. The thick curve indicates mean amplification and the gray shaded bands indicate extent of the standard deviation. It may be noted that the site amplification curves are presented only for the stations, which are characterized by a measured Vs ₃₀ estimate

Fig. 8

Region-wise site class-specific Fourier amplification. The thick curve indicates the mean amplification and the gray shaded bands indicate the extent of one standard deviation

7 Source parameters M _W and Δσ

After estimating the site amplification curves, AF _Fourier, we refit the site-corrected spectra to determine event-specific f _c as well as seismic moments. We use site-class specific AF _Fourier curves (Fig. 8) to correct the observed spectra for site amplification effects. The site-class AF _Fourier curves represents the site amplification effects over a broad range of stations; hence obviously they may not capture the detailed amplification characteristics of a single station, rather reflecting a typical feature. Nevertheless, they allow including the stations, which have recorded fewer earthquakes and also limiting the bias due to those fewer recordings.

In this iteration, other than the geometrical spreading function, the high frequency slope t* is fixed to the value that is a combination of regional Q ₀ models and a regional site-class κ ₀ (Table 2). The fitting is focused to fit the low frequency spectral level of the acceleration spectrum, i.e., f ≤ 10 Hz. In order to avoid the trade-off between M _W (magnitude) and Δσ at frequencies beyond f _c , we do not include the high frequency spectral amplitudes in fitting at this stage. The choice of 10 Hz is rather subjective and is based on the assumption that this can be the highest f _c in the dataset as most of the earthquakes are of low-to-moderate magnitudes (M ≥ 4). In Brune’s (1970, 1971) source model for far-field spectrum of displacement motion, the spectral amplitude Y (plateau) below f _c is related to M ₀ as:

$$Y \left( {f \ll f_{c} } \right) = \frac{{M_{0} \xi \theta F}}{{4\pi \beta^{3} \rho }}.$$

(5)

In Eq. (5), the values, of near source density (ρ), near source average shear wave velocity (β), average radiation coefficient for S _H waves (θ), energy partition coefficient (ξ) and free surface amplification factor (F) remain the same as used in Eq. (1). The estimated M ₀ is used to compute the inverted magnitude M _W using the Hanks and Kanamori (1979) relation. Almost 1:1 correlation can be observed between database and inverted M _W in Fig. 9. However, the over prediction of M _W values from uncorrected spectra illustrates the challenge in estimating M _W values from observed Fourier spectra in presence of significant site-effects.

Fig. 9

Comparison of inverted M _W with that from database. Disks when inverted M _W are obtained from site-class specific amplification corrected; and empty triangles when inverted M _W are obtained from uncorrected spectra. Events that have been recorded at least at three stations (six records including both the components) are shown

The inverted f _c and M _W are used to compute the stress parameters (Δσ) using the following relationship:

$$\Delta \sigma = M_{0} \left( {\frac{{f_{c} }}{0.4906\beta }} \right)$$

(6)

(Brune 1970, 1971; Eshelby 1957) where β is the near-source shear wave velocity assumed to be 3500 m/s. Figure 10a depicts the variation of Δσ values with respect to database M _W and Fig. 10b illustrates the same variation with focal depth. To obtain a robust estimate of inverted f _c and M _W , events recorded on at least three stations are used for this analysis. As found for the previous database (Edwards and Fäh 2013a) Fig. 10a does not show any magnitude dependency of Δσ. Thus, assuming a constant Δσ model, the Δσ (in MPa) using inverted M _W is obtained as:

$$\log_{10}\Delta \sigma = \log_{10} 5.75 \pm 0.43 .$$

(7)

If we constrain M ₀ to a value from database M _W and invert for f _c only, the median Δσ is obtained identical to that in Eq. (7) with smaller (lognormal) standard deviation as:

$$\log_{10}\Delta \sigma = \log_{10} 5.65 \pm 0.33$$

(8)

The median Δσ values are slightly smaller than those obtained by Edwards and Fäh (2013a) as 8.8 and 7.4 MPa from inverted and database M _W respectively, while the standard deviations are comparable. The Δσ variability obtained in this study is also comparable with that inferred (Cotton et al. 2013) from between-event variability in the GMPEs of Akkar et al. (2014a), Boore et al. (2014) and Bindi et al. (2014) as: 0.43, 0.42 and 0.41 respectively.

Fig. 10

Stress parameters (Δ σ) (obtained from site-class specific amplification corrected spectra) plotted against database M _W in panel (a) against depth in panel (b). Disks indicate the Δ σ values when both f _c and M _W were obtained from inversion while the empty circles indicate those when only f _c was obtained from inversion keeping the M ₀ fixed from database M _W . Again, events recorded at least at three stations (six records including both the components) are shown

Additionally, we estimated Δσ for the M _W 4.8 St. Die earthquake, as it has been widely investigated and discussed in the literature (e.g., Scherbaum et al. 2004). Although, the station-specific Vs ₃₀ values are not available for the stations recording St. Die earthquake, the soil type information of those stations was obtained from RESIF seismic data portal (http://seismology.resif.fr/). Out of the nine stations, three stations were classified in the EC (Eurocode)-8 soil type E, two in soil type B and the remaining four were classified in soil type A. We did not apply any corrections to empirical Fourier spectra to account for the local site amplification effects except correcting for the crustal amplification related with the generic rock amplification of California (Boore and Joyner 1997). However, the t* values were fixed using the two separate κ (or t*) models (i.e., for soil and rock) of Douglas et al. (2010), as some of the stations (which recorded St. Die earthquake) are included in their analysis as well. Fixing the low frequency spectral level by the M ₀ obtained from database M _W gives the Δσ value as 49.2 MPa, while inverting for both f _c and (M ₀) magnitude gives Δσ as 32.3 MPa corresponding to the fitted M _W 4.96. Such high values are consistent with high ground motion amplitudes observed for this event.

The values of Δσ determined in this study are compared in the context of recent studies involving Δσ determination for mainland Europe (Edwards and Fäh 2013a, b) in Fig. 11. Edwards and Fäh (2013a) involves earthquakes from all over Europe and the Mediterranean, which is essentially a subset of the present dataset, while the analysis of Edwards and Fäh (2013b) is based upon the earthquake from Swiss Alps and Swiss Foreland basin. Δσ values from present study are observed to be in good comparison to the other studies except that the earthquakes from Swiss Alps are exhibiting lower Δσ values.

Fig. 11

Δ σ comparison with the previous studies from the same region. Events recorded at least at seven stations (fourteen records including both the components) are shown in this figure. The encircled markers indicate the Δσ values (left and right) corresponding to St. Die and Friuli earthquakes respectively

Apart from St. Die earthquake, that is exhibiting a relatively larger Δσ, we did not observe discernable regional pattern in Δσ (from the present dataset) as suggested by some recent studies (Malagnini et al. 2008; Drouet et al. 2010; Yenier and Atkinson 2015b; Goertz-Allmann and Edwards 2014). The Friuli earthquake M _W 6, 1976 also indicates a large Δσ of 13.38 MPa with inverted M _W 5.79, while using the database M _W gives Δσ as 8.6 MPa. In Fig. 11, earthquakes recorded at least at seven stations (fourteen records including both the components) are shown. For the earthquakes shown in Fig. 11, inverted f _c , M _W , Δσ and the associated uncertainties are given in Table 3.

Table 3 Stress parameter values for earthquakes recorded on at least seven stations

8 Discussion

From the present analysis, we observed regional variations in anelastic attenuation Q ₀ and κ ₀ from shallow active crustal earthquakes recorded across Europe and Mediterranean. Although, estimation of κ ₀ is strongly linked with how one constrains Q ₀, our analysis also indicates that it may also vary significantly between Turkey and Italy. As some studies have investigated correlation of κ ₀ with deeper structure (Campbell 2009; Ktenidou et al. 2015), there is a possibility that κ ₀ has regional component (Ktenidou et al. 2015), which depends on varying crustal properties.

Furthermore, within a single region, significant, record-to-record (within-station) variability in κ ₀ is observed, which in many cases is comparable to the station-to-station (between-station) variability. Large within-station variability can be expected when a station records earthquakes over a range of distances. Thus the waves reaching at the station may encounter different anelastic attenuation regimes due to sampling deeper layers as well as the shallower layers in the subsurface. Essentially this variability is entering in κ ₀ through the depth variation of Q ₀. For the estimation of κ ₀ therefore it is recommended to use records from near station earthquakes in addition to account for regional differences in Q ₀. Large within-station variability in κ ₀ can also be contributed by the source-component present in κ ₀ (Kilb et al. 2012). From an application perspective, for example in stochastic simulations and HTTA adjustment of GMPEs, a linked (or combined) Q and κ ₀ model should be used to maintain the consistency. Finally, an important consequence of larger within-station variability (in κ ₀) from the GMPE adjustment perspective is that it can hinder the effect of site (station) corrections made in κ ₀ to account for site-to-site variability. This article presents broad site-class based κ ₀ measurements for the regional subsets as well as the station-specific κ ₀. We did not observe a clear relationship between κ ₀ and Vs ₃₀ as suggested by some other studies.

8.1 Comparison with previous studies

A meaningful comparison between the estimated parameters with previous studies can only be possible when underlying assumptions (e.g., mainly geometrical spreading) and method of estimation is the same amongst the studies. Nevertheless, anelastic attenuation Q, Δσ and κ ₀ for different regions across Europe and Mediterranean from some representative studies are given in Table 4 along with the assumed geometrical spreading function. To facilitate comparison with a frequency-independent Q ₀ from our study, we have fixed Q at 10 Hz from the studies involving frequency dependent Q. As expected a significant variation amongst the studies can be seen in Table 4. The Q ₀ values determined in this study agree with the general trend that Turkey and Greece exhibit higher Q values (lower attenuation) in comparison to that in Italy. Our Q ₀ estimates for Turkey are consistent with the findings for the northwestern part (Kurtulmus and Akyol 2013; Askan et al. 2014), which is expected as the majority of our Turkish records come from this region. Similarly, the κ ₀ value for Turkey from our study is in good agreement with the value of 0.045 s found by Akinci et al. (2013) for Anatolian region in Turkey. Moreover, recent GMPEs (Boore et al. 2014; Kotha et al. 2016; Kuehn and Scherbaum 2016) have also indicated regionally varying anelastic attenuation terms indicating a higher Q in Turkey and lower Q in Italy. The present dataset does not permit to investigate regional variations in Δσ. The average median Δσ value of 5.65 MPa from our analysis for the entire region is broadly consistent with the previous studies, except with the very high values of 20 and 60 MPa from Umbria-Marche and northeastern regions in Italy (Malagnini and Herrmann 2000; Malagnini et al. 2000). However, as stated earlier, comparisons amongst the parameter estimates should be made relative to the geometrical spreading function rather than treating them as absolute values.

Table 4 Seismological parameters derived from different studies

8.2 Stochastic model predictions

We validate the model parameters derived in this study by comparing the model predictions against recorded data. The comparison is performed in terms of graphical comparisons of Fourier and response spectra in Figs. 12 and 13 respectively, while Fig. 14 depicts comparison of response spectral variability with the regional GMPEs. For graphical comparison in Figs. 12 and 13, we have chosen August 17, 1999 Kocaeli earthquake M _W 7.6. This choice of earthquake will also allow reader to appreciate the consistency of the point source model in simulating ground motions from rather large ruptures. In addition to the use in synthesizing ground motions for low seismicity regions the model parameters such as Δσ, geometrical spreading, Q and κ ₀ are also used to represent the source, path and site attributes of empirical GMPEs in their HTTA (Host-to-Target Adjustment) framework. To that end, as depicted in Fig. 13, a good comparison of the response spectra obtained from our stochastic model with the regional GMPEs of Akkar et al. (2014b), Bora et al. (2015) and Bindi et al. (2014) warrants the use of the present model in such exercises.

Fig. 12

Example of Fourier spectral fits to the observed recordings from M _W 7.6 Kocaeli earthquake (August 17, 1999). The model predictions (heavy line) are shown for inverted M _W 7.5, Δ σ = 9.1 MPa and station-specific κ ₀ and amplifications

Fig. 13

Pseudo spectral acceleration (PSA) for Kocaeli records corresponding to the plots shown in Fig. 12. PSA from Akkar et al. (2014b), Bindi et al. (2014) and Bora et al. (2015) are also shown for comparison

Fig. 14

Comparison of response spectral variability from the derived stochastic model (using median Δ σ = 5.65 MPa, regional Q along with station-specific κ ₀ and amplifications) with that from regional empirical models. a Residual bias, b between-event variability (τ), c within-event variability (Φ), d total variability (σ). The variability from Bora et al. (2015) corresponds to the curve (in their Fig. 17) that is obtained with event-specific Δ σ and station-specific κ ₀

Figure 14 depicts comparison of response spectral variability obtained from the present stochastic model with the regional empirical models (Akkar et al. 2014b; Bindi et al. 2014; Bora et al. 2015). Response spectral residuals were obtained for an average Δ σ = 5.65 MPa and site amplification curves for the stations recording at least four component-records, with measured Vs ₃₀ measurements along with the regional Q ₀ and corresponding station-specific κ ₀ values. The residuals were decomposed into between- and within-event components by performing a linear (intercept-only) mixed-effects regression (lme4 package, Bates et al. 2015) on the total residuals. Figure 14a depicts the mean residual bias at each oscillator frequency; Fig. 14b depicts between-event standard deviation (τ); Fig. 14c depicts within-event standard deviation (Φ) and Fig. 14d depicts the total standard deviation (σ). Significant reduction in within-event variability (Φ) is mainly due to capturing the between-station variability in terms of site amplification and κ ₀ in the model itself. Also, capturing the regional variability in Q ₀ has also led to further reduction in Φ.

9 Conclusions

This study was aimed at providing measurements of regional source, attenuation and site parameters for Europe and the Mediterranean region based upon a subset-dataset (Bora et al. 2015) of RESORCE-2012 database. Estimation of source, path and site parameters is based upon the far-field spectral representation of strong ground motion phases. In which, the source is represented by a Brune (1970) single corner frequency (f _c ) model. The path effects are modeled using simple geometrical and inelastic attenuation models. The inelastic attenuation is parameterized in terms of a frequency-independent Q ₀ model (Edwards et al. 2008; Campbell 2009; Edwards and Fäh 2013a). As detailed in Bora et al. (2015), to enable the validity of point source model, near distance records from moderate and large magnitude earthquakes have been discarded. In the first stage, we fit for source-corner frequency f _c and attenuation operator t* by fixing the M ₀ to a value obtained from database M _W (Hanks and Kanamori 1979).

The inverted t* values from the full dataset indicate a Q ₀ model varying with distance that essentially captures depth-dependent Q structure (Edwards et al. 2008, 2011). However, a clear regional variation in inelastic attenuation (Q ₀) is observed with a large value of Q ₀, i.e., 1462 (smaller attenuation) for Turkey and smaller Q ₀ (larger attenuation) of 601 for Italy. We also investigated κ ₀ variability in terms of between-station and within-station (record-to-record) components indicating a large contribution in within-station variability through regional and depth variations of Q. Station-specific average Fourier site amplification factors were obtained with respect to a reference site with shear wave velocity (Vs) profile of California (Boore and Joyner 1997) anchored at Vs ₃₀ 620 m/s. The base (or hard-rock) model is described in terms of seismic moments obtained from database M _W , a priori geometrical spreading function and regional Q models along with the station κ ₀. To cover a broader range of stations, site-class specific site amplification factors were also estimated.

In the second stage of inversion, the observed Fourier spectra were corrected for site-class specific amplification factor and subsequently further inverted to obtain the Brune’s corner frequency f _c and M ₀. The fitting procedure involves all the records from an event where the high frequency shape of each record was constrained by using the regional Q models along with the site-class κ ₀. Despite having very few events with a large number of multiple recordings, the inverted M _W values were found broadly consistent with the database M _W . The estimated Δσ from inverted f _c and M ₀ did not exhibit dependence over magnitude. The database common Δσ was estimated to be 5.75 and 5.65 MPa using inverted and database M _W respectively. The Δσ values obtained in this study were observed to be in good comparison with the previous studies involving earthquakes from the same region. Although, we did not observe a clear regional dependence of Δσ mainly because of the limited dataset, the St. Die earthquake M _W 4.8 located at French–German border was found to be exhibiting a relatively high Δσ of 32.3 MPa. We would like to mention here, what is already noted by Atkinson and Beresnev (1997), that the Δσ values obtained in this way do not represent the actual drop in stress (before and after the earthquake) one would expect during an earthquake rupturing, but rather a measure of the proportion of radiated high-frequencies.

Finally, we note that the inversion of spectral parameters in Bora et al. (2015) was focused on a record-wise fitting of each spectrum to enable a consistent extrapolation beyond the filter high-pass and low frequencies. While, this study is aimed at discussing current challenges that are associated with stochastic modelling of ground motion, thus providing robust estimates of regional stochastic model parameters for Europe and Mediterranean region. Furthermore, the present study is believed to facilitate an updated reference stochastic model, presented in Table 5, for Europe and Mediterranean regions.

Table 5 Seismological parameters for Europe and Mediterranean stochastic model

10 Data and resources

Data was taken from the RESORCE database (Akkar et al. 2014a). Figures were prepared using the program Mathematica except Fig. 1a was prepared using GMT (The Generic Mapping Tools). The linear mixed effects regression was performed using R-package lme4 (Bates et al. 2015).