1 Introduction

As is the case for many regions with limited observational ground motion databases, seismic hazard assessment in France is complicated by large epistemic uncertainty concerning the expected ground motion in future earthquakes. Thanks to the establishment in the past couple of decades of a reasonably dense national strong-motion network in the most seismically active parts of France (the Réseau Accélérometrique Permanent, RAP) many thousands of accelerometric records are now freely available (Péquegnat et al., 2008). Nevertheless, due to the relatively low earthquake occurrence rates in mainland France there are very few records from earthquakes with moment magnitude, M w , greater than 5.0. Due to recognized differences in magnitude- and distance-scaling of ground motions from small and large earthquakes (e.g. Bommer et al., 2007; Cotton et al., 2008, and references therein) it is currently not possible to develop robust, fully-empirical ground motion prediction equations (GMPEs) reliable for higher magnitudes based on these data. Three alternative methods for the estimation of earthquake ground motions in France could be applied: (1) assume that ground motions in France are similar to those in areas for which robust GMPEs (either empirical or simulation-based) have been proposed (e.g. California, Japan or Italy) (e.g. Cotton et al., 2006); (2) develop simulation-based GMPEs using input parameters derived from seismological analyses, as, for example, have been developed for eastern North America (e.g. Atkinson and Boore, 2006); or (3) adjust GMPEs developed for other regions to be more applicable to France through, for example, the hybrid empirical-stochastic technique (e.g. Campbell, 2003; Douglas et al., 2006) or the referenced empirical approach (Atkinson, 2008). Up until now method 1 has been used almost universally for France, probably due to the lack of sufficient strong-motion data from which to derive input parameters required for methods 2 and 3. Methods 2 and 3 generally require estimating various parameters characterizing the earthquake source (e.g. the stress drop parameter Δσ and the source spectral shape), the travel path (e.g. geometrical decay and Q) and the local site (e.g. shear-wave velocity profile and near-surface attenuation). Numerous previous studies have estimated one or more of these parameters for France or regions of France (e.g. Campillo et al., 1993; Drouet et al., 2005, 2008). However, we know of no published studies explicitly reporting estimates of κ as introduced by Anderson and Hough (1984). The site contribution to κ is commonly believed to be related to the attenuation (e.g. Q or damping) in the top couple of kilometers, although there is some evidence for decay of high frequencies due to source properties related to the size of a cohesive zone at the crack tip (e.g. Papageorgiou and Aki, 1983; Tsai and Chen, 2000). The effect of κ is to act as a high-frequency (>1 Hz) filter on ground motions and, therefore, it is a critical parameter for the accurate estimation of, for example, peak ground acceleration. Consequently, in this article we estimate κ using hundreds of accelerometric records from mainland France.

One motivation for this study was the finding of Douglas et al. (2009), who presented an approach to constrain the shear-wave velocity profile by making use of all available information on site conditions at a site of interest (e.g. soil type and depth to bedrock). They found that even when the shear-wave velocity profile is precisely known, the high-frequency site amplification is not. Douglas et al. (2009) attributed this, at first sight surprising, result to a lack of constraint on the near-surface attenuation. In their analysis they modelled attenuation by κ estimated using the empirical relationship of Silva et al. (1998) connecting κ and V s,30 (the average shear-wave velocity of the top 30 m); this relationship had a large associated standard deviation that led to the large uncertainty in the high-frequency site amplification. If κ could be better estimated at a given site then there is the potential to significantly reduce this uncertainty. Therefore, in this article we investigate κ to see whether it can be better constrained for French sites.

This article starts with a section describing the strong-motion data used; next we describe our method for the evaluation of κ from the Fourier spectra of these data including an approach to estimate the accuracy of the estimated κs due to the subjectivity of the adopted methodology; following that we investigate the dependence of κ on source-to-site distance, region and local site conditions; and finally we conclude.

2 Data Used

In order to concentrate on data of engineering interest and to limit the number of records analyzed, only records from earthquakes with magnitudes (any scale) larger than about 3.5 were downloaded from the RAP online strong-motion database (http://www-rap.obs.ujf-grenoble.fr/). Each acceleration time-history was then visually inspected and poor quality records (due to noise or severe baseline problems) on any of the three components were rejected from further consideration. In total, 263 triaxial records (i.e. 789 components) from 30 earthquakes and 83 different stations were retained for analysis (Table 1; Fig. 1). Note that κs were computed for all three components. Earthquake locations given in the RAP database were used here since these are from local networks (mainly the French national RéNaSS catalogue) and, in addition, most available data are from considerable source-to-site distances and, therefore, accurate hypocentral locations are not critical for this analysis.

Table 1 Earthquakes and stations used to compute κ
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Fig. 1
figure 1

Earthquake (circles) and station (triangles) locations and travel paths (lines) of the records used for this study. 1 Alps and Côte d’Azur (southern part of map) and 2 Pyrenees

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Most of the records selected were recorded by stations in the three most seismically active regions of France: the Pyrenees (109 records), the Alps (88 records) and the Côtes-d’Azur (50 records) (although sometimes the earthquake recorded occurred in a different region). Possible regional dependence of κ between these different areas is examined in Sect. 5. Other regions contribute few records and therefore they are not examined separately.

According to the classifications given on the RAP website, 178 of the selected records are from rock stations, 75 are from soil stations and 10 are from borehole stations. Note that, although κs were estimated from borehole records they were not used to derive the following models. Possible dependence of κ on the local site conditions is investigated in Sect. 5. A number of stations have recorded multiple earthquakes, which allows a station-specific κ model to be established. Specifically, in Sect. 5 we develop such models for 11 stations that have recorded more than five earthquakes amongst those selected: OGAN (6 records), OGMO (8), OGMU (7), OGSI (6), PYAD (9), PYAT (9), PYFE (7), PYLO (9), PYLS (11), PYOR (8) and PYPR (8).

3 Method Used to Estimate κ

In this study the classic method of estimating κ developed by Anderson and Hough (1984) is used. It is slightly modified [as done by Hough et al., (1988) for a comparable dataset] due to the use of high-quality digital records from small events rather than analogue records from moderate and large earthquakes as used by Anderson and Hough (1984). Each component of a triaxial record is processed individually. The first step is to remove the mean and plot the acceleration time-history. Time-histories that are too noisy or have other problems are rejected. Next, the pre-event, P-wave and S-wave portions of the time-history are selected by eye. Then the Fourier amplitude acceleration spectra of each of these three portions are computed and plotted on the same graph with a logarithmic y-axis (amplitude) and a linear x-axis (frequency). Based on the S-wave spectra two frequencies are selected by visual inspection: f E , the start of the linear downward trend in the acceleration S-wave spectrum, and f X , the end of the linear downward trend or when the S-wave spectrum approaches the noise spectrum (i.e. when the signal-to-noise ratio becomes too small for the spectral amplitudes to be reliable). Figure 2 shows an example of a spectra with a clear high-frequency linear trend and low noise levels and the f E and f X frequencies chosen for this record by one of the analysts. We find that f E is generally around 3 Hz but with a large scatter [within the 2–12 Hz range used by Anderson and Hough (1984)]. Thanks to the high resolution and low noise levels of the selected records f X is generally in the range 20–50 Hz. The final step in the procedure is to fit, using standard least-squares regression, a line fitting the acceleration spectrum between f E and f X , from whose slope κ is given by κ =  − λ/π where λ is the slope of the best-fit line. Generally there is a sufficient frequency range between f E and f X to give a robust estimate of κ.

Fig. 2
figure 2

Example of direct shear-wave and noise spectra computed from a record that shows a clear high-frequency linear trend. Also shown are the intervals used to estimate the pre-event noise and the direct shear-wave spectra (black parts of acceleration time-history) and the frequencies f E and f X chosen by one of the analysts (the other analysts chose similar f E and f X for records such as this)

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A non-automatic procedure for estimating κ was adopted because we noted that the frequency, f E , at which the acceleration spectral amplitudes show a decline varied significantly from record to record and therefore assuming a constant f E , such as has been done in some previous studies, could lead to biased estimates for κ. Similarly, due to varying signal-to-noise ratios (visually inspected), f X shows large variations and therefore it was not possible to use a constant value for all records. Since the procedure followed here is non-automatic, it is quite time-consuming and also subjective because analysts can have different views on the selection of the pre-event, P-wave and S-wave portions of the record and on the selection of f E and f X , which can lead to some variations in κ between analysts. We found that differences in picking of the pre-event, P-wave and S-wave portions did not significantly affect the κs obtained.

A semi-automatic procedure to choose the intervals used to compute the direct shear-wave spectra and noise spectra was also applied. Since both P- and S-wave arrival times had been previously picked, time windows of 5 s for the pre-event noise and direct S-wave were used to compute the Fourier spectra. Various lengths of time windows from 1 to 10 s were also tested with similar results, so a standard length of 5 s was finally chosen. The time series were processed using a Hanning taper of 5%. The resulting Fourier spectra were then smoothed by a Konno and Ohmachi (1998) filter (filter bandwidth of 40), and only data having a signal-to-noise ratio greater than three were used to compute κ. The values of f X and f E used to compute κ in this procedure were chosen by the analyst, as in the completely manual approach described above. In the next section we present the approach we took to quantify the subjectivity and precision of the obtained κs.

In the absence of the high-frequency decay quantified here by κ Fourier amplitude spectra should be flat above the corner frequency, f c , of the source. When fitting the best-fit lines to determine κ it is necessary that f E (the frequency chosen as the start of the best-fit line) is greater than f c otherwise the κ estimates can be biased. When using strong-motion data from moderate and large earthquakes (M w  ≥ 5.5) as done by Anderson and Hough (1984) f c is generally lower than 1 Hz hence bias in κ due to f c is not a problem. However, in this study where we are using data from earthquakes with 3.4 ≤ M ≤ 5.3 f c is generally between 1 and 6 Hz, using Fig. 8 of Drouet et al. (2008) showing the relation between magnitude and f c . The f E values are selected here to be above f c based on visual inspection (Figs. 2, 3) and, therefore, most best-fit lines will be minimally affected by f c , especially since f X (the frequency up to which the line is fitted) is usually greater than 30 Hz.

Site amplification curves, relative to reference sites displaying little amplification, for some of the stations considered here are provided by Drouet et al. (2008). Some of these curves show peaks in the site amplifications at high frequencies where they could complicate the estimation of κ (e.g. >5 Hz), e.g. PYAD and PYBA. Although not investigated by Drouet et al. (2008), records from QUIF also show a high-frequency site effect (see Fig. 3). In our analysis we attempted to compensate for the peak in the Fourier amplitude spectra from such stations to avoid biasing the obtained κs (as done by Anderson and Hough (1984) for Santa Felicia Dam, with a similar high-frequency amplification). The relative site amplification curves for 49 stations provided by Drouet et al. (2008) could be used to correct the observed spectra as done by Margaris and Boore (1998), for example, but this has not been attempted for simplicity, and in order to be consistent between all records, even those from stations not analyzed by Drouet et al. (2008). Parolai and Bindi (2004) conduct simulations assuming a 1D single sedimentary layer overlaying a bedrock half-space and earthquakes with 2 ≤ M w ≤ 6 to examine the effect of local site amplification on κ estimates. They find that in the presence of strong site amplifications at frequencies greater than 4 Hz, it is necessary to fit the best-fit line to determine κ over a wide frequency band (e.g. 10–34 Hz) in order to obtain accurate κs. Thanks to the high resolution (24 bits) and low noise levels on the digital accelerograms used in this study we are generally able to extend the fitting of the best-fit lines to 30 Hz or higher. Therefore, it is likely that most κ estimates found here are not biased by high-frequency site effects. However, the combination of high-frequency site effects and higher noise levels at some RAP stations means that some κ values obtained in this study may be too high (see Sect. 5).

Fig. 3
figure 3

Example of direct shear-wave and noise spectra computed from a record used in the analysis that shows a high-frequency site effect seen on all records from this station (and hence it is difficult to estimate a reliable κ from this record). Also shown are the intervals used to estimate the pre-event noise and the direct shear-wave spectra (black parts of acceleration time-history) and the frequencies f E and f X chosen by one of the analysts (f E and f X for records such as this varied between analysts)

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3.1 Variability in κ Estimates

The first three authors of this article independently processed (the first two using the non-automatic procedure and the third the semi-automatic technique) the 263 records and their estimated κs were compared. It was found that for most records the estimated κs of the three analysts were similar (within 10–20% of one another) but for some records with no clear linear amplitude dependence on frequency the measured κs vary greatly (up to 50%). After discussion, some of these large differences were reduced by one or two analysts reprocessing the problem records. However, there remains a subjective aspect to the estimation of κ. Therefore, due to the variability in κ among individual accelerograms, we believe that robust estimates of κ at seismic stations should be based on a large number of individual observations. Therefore, in this study we only seek conclusions based on many records. Note that, as discussed above, if the best-fit lines are estimated over a frequency band affected by high-frequency site amplifications or high corner frequencies, κ estimates could be biased either upwards or downwards. In this situation, whatever method is used to average the estimates from each analyst the κs obtained will not be correct. As stated above we do not think that this is the situation for the vast majority of the records we analyzed.

By analyzing the three κ estimates from a single record it was found that the error in the measurement of an individual κ were multiplicative rather than additive, i.e. κ estimates from each analyst were higher or lower than the average κ by a certain percentage (e.g. 20%) rather than by an absolute amount (e.g. 0.005 s). Assuming multiplicative errors also has the benefit of excluding the possibility of predicting negative κs. Therefore, the logarithms of the six κ estimates for an individual record (from three analysts and for the two horizontal components) were computed and the mean and standard deviations computed from these six logarithms were used in the subsequent analysis. By averaging κs for both horizontal components we make the assumption that κ is the same for both components and hence it is independent of the azimuth of the incoming waves. The mean κs and associated standard deviations were then used to undertake weighted regression analysis using diagonal weighting matrices derived from the inverse variances of each κ estimate (e.g. Draper and Smith, 1998). Since the variances are derived from the logarithms of the κs but the regression was performed on the untransformed κs (to be consistent with previous studies), the weighting matrices are slightly incorrect with respect to the regression performed, but we do not believe that this significantly affects the results. A traditional, non-weighted least squares regression was also computed in order to see the effect of the uncertainty measured on the κ values. Both regressions are quite similar. The results of these regression analyses are reported below.

3.2 κ Estimates from Vertical Components

κ was computed for the three components of ground motion but only the horizontal components were used to develop the κ models. Figure 4 shows the relation between the κ values computed using the horizontal and vertical components. The error bar for each measurement has also been plotted. This figure shows that vertical estimates are slightly smaller than the horizontal ones but, in general, the estimates are similar. In absence of three-component stations, κ values obtained from vertical components may be helpful for a first estimate of this parameter, although a slight adjustment of κs from vertical components could be required.

Fig. 4
figure 4

Comparison of κ values computed from vertical and horizontal components. ±1 standard deviation bars are also plotted. The dashed line represents the 1:1 relation

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4 Distance Dependence

The first-order model that is often fitted to κ estimates is: κ = κ 0 + m κ r epi, where r epi is epicentral distance and κ 0 and m κ are constants (e.g. Anderson and Hough, 1984). κ 0 is believed to be station-dependent and related to the near-surface attenuation in the top couple of  kilometers under the site whereas m κ is believed to be region-dependent and related to the regional attenuation. As mentioned above in this study we have used the estimated standard deviations of each κ value to apply weighted regression analysis to find κ 0 and m κ for our data. The results from non-weighted regression are also shown in the legend of the corresponding figures for completeness.

As a first step regression analysis is performed for all surface records (263 records) using the form: κ = κ 0,rock S rock + κ 0,soil S soil + m κ r epi, where S rock = 1 for rock stations and 0 otherwise and S soil = 1 for soil stations and 0 otherwise. By using this functional form we allow near-surface attenuation at rock stations to be different from that at soil stations but we assume that the regional attenuation is the same since a common m κ is used for rock and soil sites. The estimated κs with respect to r epi and site class are shown in Fig. 5 along with the fitted lines. The equations of the best-fit lines are:

$$ \begin{aligned} \kappa_{{\rm soil}} = & 0.0270 + 0.000175 r_{{\rm epi}}\\ \kappa_{{\rm rock}} = & 0.0207 + 0.000175 r_{{\rm epi}}. \end{aligned} $$
(1)
Fig. 5
figure 5

Distance dependence of κ for all stations for soil and rock conditions. The vertical lines represent the standard deviation of six independent measurements of κ (three estimates, two horizontal components). Results for κ 0 and m κ from standard (solid line) and weighted (dashed line) least squares regressions are shown in the legend of each plot

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Note that if these models are used in SMSIM (Boore, 2005), for example, then it is not also necessary to apply Q attenuation since this is already included in these κ models. However, it is standard practice when using SMSIM to use only the κ 0 terms and model the regional attenuation through a Q model.

Hough et al. (1988) present equations for estimating a two-layer Q model from κ 0 and m κ values. Their approach has not been applied here because the values found using this method assume that Q is independent of frequency, which has not previously been found in France (e.g. Campillo et al., 1985; Drouet et al., 2008). The Q tomography technique of Hough and Anderson (1988) has not been attempted since the distribution of data with respect to distance is insufficient and, in addition, there is not enough resampling of travel paths.

5 Regional Dependence

There are sufficient records from the Pyrenees (109 records), the Alps (88 records) and the Côte d’Azur (50 records) to derive individual best-fit equations for these regions. Figure 6 shows the κ values for these three regions for both soil and rock conditions. The regional m κ values are relatively close to each other. However, we find that the Pyrenees area has slightly lower attenuation than the Alps, and both are less attenuating than the Côte d’Azur. These results are in agreement with regional attenuation studies in France using the isoseismal distribution from historical earthquakes (e.g. Baumont and Scotti, 2006) and previous Q estimates by Drouet et al. (2008), who find lower Q values for the Alps (322), i.e. faster attenuation, than for the Pyrenees (376).

Fig. 6
figure 6

Distance dependence of κ values for three regions in mainland France. The top plots present the results for stations located on soil. The bottom plots show the results for stations located on rock

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These differences between regions are also observed on the κ 0 values for stations located on rock but the stations in the Alps show a larger attenuation than the other two regions for stations located on soil. This may be explained by the fact that some of the stations are located in the sedimentary Grenoble basin where the deep soil layer could lead to large attenuation.

Figures 7 and 8 show κ estimates and fitted linear relations for 11 stations located in the Alps and the Pyrenees. Two sets of fits were made: one in which the slope (m κ ) and the intercept (κ 0) were unconstrained and one in which m κ was fixed to the value obtained from the regional analysis reported in Fig. 6 and a corresponding κ 0 found. Considering the unconstrained fits, for stations located in the Alps, all of which are sited on rock, m κ shows modest variations of about 50% around the (m κ ) estimated for this region (Fig. 6), whereas in the Pyrenees (Fig. 8), the variability of m κ κ is larger. Concerning κ 0 the values for stations in the Alps (Fig. 7) present similar values to those obtained for the whole region (Fig. 6). An interesting exception is station OGMU whose κ 0 value for the unconstrained fits is quite close to the soil estimate in this region. This could be due to a site effect at about 10 Hz for this station (Drouet et al., 2008), which could bias upwards the estimates of κ (Parolai and Bindi, 2004). Stations located in the Pyrenees present a larger variation of κ 0 with respect to the value computed for the whole region. This variability may come from structural differences beneath each station or perhaps from statistical variations in small sample sizes. Given the variation in the distribution of records with respect to distance between stations, the fits in which m κ is constrained to its regional value are probably more reliable. These fits suggest that κ 0 for some Pyrenean rock stations (e.g. PYAT, PYLS and PYOR) is lower than at the average rock station.

Fig. 7
figure 7

κ estimates and their ±1 standard deviations for stations located in the Alps. Also shown are the fitted linear relations. Four best-fit lines were fitted for each station: two (using standard and weighted regression) in which m κ was allowed to vary (black) and two (using standard and weighted regression) in which m κ was constrained to the value from the regional analysis shown in Fig. 6 (grey)

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Fig. 8
figure 8

κ estimates and their ±1 standard deviations for stations located in the Pyrenees. Also shown are the fitted linear relations. Four best-fit lines were fitted for each station: two (using standard and weighted regression) in which m κ was allowed to vary (black) and two (using standard and weighted regression) in which m κ was constrained to the value from the regional analysis shown in Fig. 6 (grey)

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6 Conclusions

In this article we have estimated the high-frequency attenuation parameter κ (Anderson and Hough, 1984) from 263 high-quality triaxial accelerograms from the French RAP strong-motion network. Furthermore, we have investigated the dependence of κ on distance, region and site conditions to develop simple κ models for use in seismic hazard assessment for mainland France. We have found that the three studied regions (the Pyrenees, the Alps and the Côtes-d’Azur) present different yet relatively similar dependency of κ on epicentral distance. The influence of local geology is slight yet noticeable.

The values obtained here are reasonably consistent with, although larger than (meaning higher attenuation), the 0.015 and 0.0125 s values obtained for Switzerland by Bay et al. (2003) and Bay et al. 2005, respectively, and the 0.012 s value for the western Alps found by Morasca et al. (2006), using a different technique. This could be attributed to more competent rock (higher shear-wave velocities) in Switzerland than in France. In contrast our average κ κ κ 0 is lower than the 0.05 s value found by Malagnini et al. (2000) for central Europe (mainly Germany).

Based on these results, in terms of near-surface attenuation it seems that mainland France lies between WNA (where κ has been found to be around 0.04 for rock sites) and ENA (where κ has been found to be much lower, 0.006 is a commonly used value). Similarly Campillo et al. (1985) concluded that their Q model situates France between ENA and WNA in terms of regional attenuation. This seems reasonable with respect to the seismotectonics of France (mainly a stable continental region but with areas of active tectonics, the Pyrenees and the Alps) and its geology (quite hard bedrock sites). Therefore, seismic hazard assessments for France could be conducted by adjusting κ contributions to GMPEs from active regions downwards or κ for GMPEs from stable continental regions upwards to the intermediate κ values we estimated.