Which is a better proxy, site period or depth to bedrock, in modelling linear site response in addition to the average shear-wave velocity?

Abstract

This study aims to identify the best-performing site characterization proxy alternative and complementary to the conventional 30 m average shear-wave velocity V_S30, as well as the optimal combination of proxies in characterizing linear site response. Investigated proxies include T₀ (site fundamental period obtained from earthquake horizontal-to-vertical spectral ratios), V_Sz (measured average shear-wave velocities to depth z, z = 5, 10, 20 and 30 m), Z_0.8 and Z_1.0 (measured site depths to layers having shear-wave velocity 0.8 and 1.0 km/s, respectively), as well as Z_x-infer (inferred site depths from a regional velocity model, x = 0.8 and 1.0, 1.5 and 2.5 km/s). To evaluate the performance of a site proxy or a combination, a total of 1840 surface-borehole recordings is selected from KiK-net database. Site amplifications are derived using surface-to-borehole response-, Fourier- and cross-spectral ratio techniques and then are compared across approaches. Next, the efficacies of 7 single-proxies and 11 proxy-pairs are quantified based on the site-to-site standard deviation of amplification residuals of observation about prediction using the proxy or the pair. Our results show that T₀ is the best-performing single-proxy among T₀, Z_0.8, Z_1.0 and V_Sz. Meanwhile, T₀ is also the best-performing proxy among T₀, Z_0.8, Z_1.0 and Z_x-infer complementary to V_S30 in accounting for the residual amplification after V_S30-correction. Besides, T₀ alone can capture most of the site effects and should be utilized as the primary site indicator. Though (T₀, V_S30) is the best-performing proxy pair among (V_S30, T₀), (V_S30, Z_0.8), (V_S30, Z_1.0), (V_S30, Z_x-infer) and (T₀, V_Sz), it is only slightly better than (T₀, V_S20). Considering both efficacy and engineering utility, the combination of T₀ (primary) and V_S20 (secondary) is recommended. Further study is needed to test the performances of various proxies on sites in deep sedimentary basins.

1 Introduction

An earthquake recording site was simply delineated as either rock or soil sites in some early ground-motion models (GMMs) (e.g., Si and Midoriakwa 1999) before being explicitly characterized according to a piecewise site classification scheme (e.g., Atkinson and Boore 2003) or, more recently, a continuous site proxy (e.g., Boore et al. 2014). Undoubtedly, the time-averaged shear-wave velocity in the first 30 m, V_S30 (Borcherdt 1994) is the most widely used site or site class delineator.

However, many studies have shown that V_S30 alone is not adequate to distinguish the site effects at one site from those at another (e.g., Gallipoli and Mucciarelli 2009; Kotha et al. 2018). Thus, efforts have been made in the search for alternative or complementary site proxies (or their combinations) to V_S30. Based on that site period T₀ is used for site classification in Japan for engineering design practice (Japan Road Association 1990). Zhao et al. (2006) proposed to utilize T₀ and horizontal-to-vertical (H/V) response spectral ratios over a wide period range to classify K-NET stations in Japan and developed a GMM using the period-based classification scheme. Later, they found that T₀ was better than both V_S30 for soft soil sites (T₀ > 0.6 s) at long periods (above 1.0 s), and for other site classes and spectral period bands, they performed equally well (Zhao and Xu 2013).

Cadet et al. (2012) compared the misfits of five single site proxies V_Sz (shear-wave velocity averaged to a depth z, z = 5, 10, 20, and 30 m) and T₀ and four site proxy pairs (T₀, V_Sz) in modelling KiK-net surface-to-borehole amplification. T₀ and (T₀, V_S30) were found to be the best single proxy and proxy pair, respectively. Régnier et al. (2014) concluded that, in addition to V_S30, T₀ could reduce the site-to-site amplification variability of KiK-net sites in deep sedimentary basins within a specific V_S30 class. Applying neural network approach to KiK-net data, Derras et al. (2017) considered V_S30 and T₀ to be the best single-proxy for periods below and above 0.6 s, respectively.

Outside Japan, McVerry (2011) reported that, in New Zealand, site effects could be better characterized using T₀ than using V_S30 for oscillator periods of 0.5 s or longer. Hassani and Atkinson (2018a) concluded that T₀ was better than V_S30 in parameterizing sites in Central and Eastern North America. Meanwhile, for sites in California, Hassani and Atkinson (2018b) achieved an average 5% further reduction in the standard deviation of residuals by including T₀ after accounting for the site effects associated with V_S30 and Z_1.0 (depth to V_S = 1.0 km/s). For recording stations in Italy, Luzi et al. (2011) realized a significant reduction in standard deviation when V_S30 and T₀ are used together compared to a V_S30-based site classification. In addition to the empirical investigation, Stambouli et al. (2017) utilized the proxy pair (V_S30, T₀) to model numerical site responses of hundreds of global soil columns and achieved a reduction in intersite variance by over 60% compared to a model without site term.

Several previous research (e.g., Zhao and Xu 2013) focus on searching for a better alternative to V_S30. However, the period-dependency of site amplification determines that there is no such a single proxy that performs the best for all oscillator periods. Thus, to improve the site amplification estimation over the whole period range of engineering interest, it is more viable to use a combination of site proxies than to use a single predictor variable. Meanwhile, the efforts devoted to measuring or inferring V_S30 in the past decades, as well as the established status of V_S30 in current seismic regulations render the idea of replacing V_S30 unappealing. Therefore, to reduce the uncertainty associated with site characterization, it is deemed practical to find an additional site proxy to characterize the site effects that cannot be depicted by V_S30 alone, namely an easy-to-measure site proxy complementary to V_S30.

Site period T₀, as well as site depths Z_0.8 and Z_1.0, are all potential additional parameters, where Z_0.8 and Z_1.0 are the depths in meters to isosurfaces having shear-wave velocities 0.8 and 1.0 km/s, respectively. Then the question arises as to which parameter, is the most suitable proxy secondary to V_S30 in modelling site response. T₀ can be reliably obtained for many sites at relatively low costs using the horizontal-to-vertical (H/V) spectral ratio (HVSR) technique (Nakamura 1989; Lermo and Chávez-García 1993) and thus was proposed by Pitilakis et al. (2004, 2013, 2018) as a parameter for site classification specifically in Eurocode 8. However, the NGA-West2 GMMs (e.g., Boore et al. 2014, hereafter referred to as BSSA14) chose site depth as the secondary proxy, but there is no unanimous answer as to which depth parameters should be utilized. For instance, the revised European building code Eurocode 8 proposes to include Z_0.8, but BSSA14 opted for Z_1.0 in their site terms.

Therefore, we dedicate this article in search of the best-performing site proxies not only alternative but also complementary to V_S30 in characterizing linear site response. We will first present the KiK-net dataset used in this study, including site data and ground-motion data. Then we will describe different techniques utilized in this investigation, including response-, Fourier- and cross-spectral ratio approaches, to evaluate site effects and then compare the site amplifications (AF) derived using these methods. This is followed by residual analyses in which site amplification is modelled as a function of various site proxies or their combinations. The performance of a site proxy or a combination is gauged based on the standard deviation of residuals between the observed amplification and the amplification predicted from the proxy or the combination.

2 Data selection

We use a KiK-net database which consists of about 157,000 ground-motion time-series recorded between October 1997 and December 2011. These records were processed by Dawood et al. (2016) following a stringent fully-automated processing protocol. Data processing was elaborated by Dawood et al. (2016), and thus we only briefly introduce the procedure.

Time-series were firstly baseline corrected, tapered and zero-padded. Then, a high-pass corner frequency f_c (0.04 Hz) was pre-selected to filter a record (three components for the surface station and three for the borehole station) using a high-pass, acausal, fourth-order Butterworth filter. The selection of f_c would then be confirmed if the filtered records passed the check on its final displacement, the ratio between the final and the maximum displacements, the slope of the trailing portion of the displacement and velocity time-histories, as well as the trend of the smoothed Fourier amplitude spectrum. Otherwise, the above filter process would be iterated using a higher f_c until a suitable f_c was found. Then, each component was checked on its signal-to-noise ratio (SNR ≥ 3) in the frequency range between 2f_c and 30 Hz and was flagged whether it passed the signal-to-noise ratio check or not. For more details about data processing, readers are encouraged to refer to Dawood et al. (2016).

From the processed database, we select ground motions with rupture distance (R_rup) up to 400 km, with f_c no lower than 0.12 Hz, passing the signal-to-noise ratio (SNR ≥ 3) check and from earthquakes with a moment magnitude (M_w) between 3.0 and 8.0. Thus, all selected records have a maximum usage period of at least 1/(2f_c) = 4.17 s. There are certain correlations between some parameters, e.g., short-period rock site spectral amplitude to source distance, and long-period rock site spectral amplitude to earthquake magnitude. However, magnitude and source distance exerted little influence on site amplification if the effect of nonlinear response was represented by using a function of rock site spectral values (Bazzurro and Cornell 2004). Thus, within the linear domain, site effects are often treated to be independent of earthquake magnitude and source-site distance in GMMs.

Therefore, to minimize the potential influences of magnitude and distance on site response, we limit our research on linear amplification. This is achieved by further selecting ground motions that are not significantly affected by soil nonlinearity based on its shear strain. Fujimoto and Midorikawa (2006) defined the minimum level of shear strain for a possible nonlinear response to be 3 × 10⁻⁴. The maximum shear strain within a velocity profile can be estimated using:

$$ \gamma_{eff}^{'} = 0.4PGV/V_{S30} $$

(1)

where $\gamma_{eff}^{'}$ is the maximum shear strain at an recording site, and PGV is the peak ground velocity (m/s). Ground motions with $\gamma_{eff}^{'}$ exceeding 3 × 10⁻⁴ are excluded from our dataset (Fig. 1a). Thus, we can assume a linear site response in this study.

Fig. 1

a Nonlinearity check; and b site conditions of borehole stations

In this study, site amplification at a surface station is referenced to its borehole sensor which is situated in layers at different depths and with various shear-wave velocities for different stations (Fig. 1b). To minimize the influence of inhomogeneous reference site condition on surface-to-borehole spectral ratios, we only utilize KiK-net sites with a shear-wave velocity at borehole (V_S,hole) above 800 m/s (Régnier et al. 2014). As shown in Fig. 1b, selected recording stations have V_S,hole in the range between 800 and 3300 m/s and are at least 100 m below the ground surface. As demonstrated by Oth et al. (2011), only a few KiK-net velocity profiles have an abrupt velocity contrast below the depth of 100 m, and on average site effects of velocity structures deeper than 100 m are insignificant. Thus, we do not correct the borehole records to a common depth.

For a site with multiple recordings, its intra-site (within-site) amplification variability can be partially attributed to the lateral inhomogeneity of near- and/or sub-surface velocity structures, thus, we only use stations that recorded at least three seismic events. The surface-to-borehole spectral ratios are then averaged over all records at a site to minimize the intra-site variability associated with the azimuthal effects of incoming waves on site response (Field et al. 1992). Then the deviation of the observed amplification at a specific site from the median amplification predicted using site proxy or proxies is treated as the site-to-site (inter-site) residual which arises from the inadequacy of site proxy or proxies, e.g., V_S30, (V_S30, T₀) or (V_S30, Z_0.8), in characterizing a site (e.g., Al Atik et al. 2010). The standard deviation of amplification residuals is taken as the inter-site standard deviation, which represents site-to-site amplification variability.

Though inter-site and intra-site residuals can be partitioned using a mixed-effect model (Abrahamson and Youngs 1992), we do not adopt this approach herein for two reasons. Firstly, the inter-site and intra-site residuals cannot be completely separated from each other even when a mixed-effect model is used (Zhao and Xu 2013). More importantly, the purpose of this study is to evaluate the performances of various site proxies in modelling site effects, and a good proxy or proxy combination is supposed to reduce both the total and inter-site variabilities, as shown by Derras et al. (2017). Therefore, a stringent partition between inter-and intra-site standard deviation is not implemented in this study.

Finally, there are 1840 ground motions selected in this study. The distribution of selected records are presented in M_w-R_rup space in Fig. 2a, and the number of recordings at each period and at each station is illustrated in Fig. 2b and c, respectively. Average velocities (V_Sz) and site depths (Z_0.8 and Z_1.0) of recording stations are derived from available one-dimensional (1D) velocity profiles which are established from downhole PS-logging (Aoi et al. 2000; Okada et al. 2004). Though these PS-logging data might be unreliable at some sites (Kawase and Matsuo 2004; Pilz and Cotton 2019), they are the best information publically available at this stage.

Fig. 2

Selected ground motions aM_w versus R_rup; b number of recordings at each period; and c number of recordings per station

Wang et al. (2018) derived fundamental periods (T₀) of KiK-net sites using the horizontal-to-vertical spectral ratio (HVSR) technique. Individual HVSR curve was computed on 5%-damped response spectrum of earthquake ground motions (complete waveforms) and then was averaged over all recordings (at least 10) at a given site. Local maximum points on an average HVSR curve with amplitudes larger than both $1.48*\overline{HVSR}$ and 2.0 were considered as significant peaks, where $\overline{HVSR}$ is the average HVSR curve integrated over all usable periods. For sites with more than one significant peak, the longest period peak was taken as the fundamental mode and its period as T₀. However, T₀ was only assigned to a site if it passed a consistency-check, namely 0.5 < T₀/T_R < 2.0, where T_R is the theoretical fundamental period calculated from a velocity profile using the Rayleigh method (Biggs 1964). Site fundamental periods derived by Wang et al. (2018) are utilized in this study.

All site parameters used in this investigation, including V_Sz, T₀, Z_0.8 and Z_1.0, are obtained from site-specific measurements. The distributions of V_S30 against Z_0.8 and Z_1.0 are illustrated in Fig. 3. V_S30 is correlated with Z_0.8 and Z_1.0 to different extents. T₀ is depicted against V_Sz in Fig. 4, which shows that there are a few sites with T₀ larger than 1.0 s. The 5th, 50th and 95th percentile values of site data are also given in Table 1.

Fig. 3

Selected KiK-net recording stations aV_S30 vs. Z_0.8; bV_S30 versus Z_1.0

Fig. 4

Selected KiK-net recording stations aV_S10 and V_S30 versus T₀; bV_S20 and V_S5 versus T₀

Table 1 Percentile values of site data

3 Surface-to-borehole spectral ratios

3.1 Fourier spectral ratio (SSR_FAS)

In the frequency domain, amplitude spectrum A_ki(f) of a recording at a surface site (i) during an earthquake (k) can be represented by the convolution of the source, path and site effects:

$$ A_{ki} \left( f \right) = O_{k} \left( f \right) \cdot P_{ki} \left( f \right) \cdot S_{i} \left( f \right) $$

(2)

where O_k(f) is the source term of event k; P_ki(f) is the path term between station i and event k; and S_i(f) is the site term for station i.

Standard spectral ratio (SSR) technique is to calculate the ratio of the Fourier amplitude spectrum (FAS) of a recording at a site of interest (i) to that at a reference site (j). If the reference site has similar source and path effects to the site of interest and has negligible site response (ideally a flat transfer function with an amplitude of one), the spectral ratio is an estimate of site response at site i. Then the SSR_FAS of the recording at the station (i) to that at the reference station (j) can be simplified as follows:

$$ SSR_{k,ij} \left( f \right) = \frac{{A_{k,i} \left( f \right)}}{{A_{k,j} \left( f \right)}} = \frac{{O_{k} \left( f \right) \cdot P_{k,i} \left( f \right) \cdot S_{i} \left( f \right)}}{{O_{k} \left( f \right) \cdot P_{k,j} \left( f \right) \cdot S_{j} \left( f \right)}} = \frac{{S_{i} \left( f \right)}}{{S_{j} \left( f \right)}} \approx S_{i} \left( f \right) $$

(3)

3.2 Response spectral ratio (SSR_PSA)

SSR is conventionally calculated using FAS of an earthquake recording. However, response spectrum or pseudo-spectral acceleration (PSA) has also been used by some researchers (e.g., Zhao et al. 2006; Cadet et al. 2012) due to some of its merits. Firstly, response spectral amplitude reflects structural response (a series of oscillators of varying natural frequency) and thus is widely adopted in engineering-orientated research, although Fourier spectrum has an explicit physical meaning. In addition, the Fourier spectrum requires extra soothing to reduce the effects of noise on spectral ratios (Safak 1997). In contrast, the damping ratio (e.g., 5%) of the response spectrum has a uniform smoothing effect on all recordings (Zhao et al. 2006).

Cadet (2007) calculated spectral ratios of KiK-net sites using both FAS and PSA. They found that, in a statistical sense, the SSR_PSA was comparable to SSR_FAS in the period range between 0.07 and 2.0 s but was higher than SSR_FAS outside this range. However, Safak (1997) reported that SSR_PSA was only comparable to SSR_FAS at long periods (> ~ 1.0 s) but were higher than SSR_FAS at other frequencies. Given the conflicting results, both SSR_PSA and SSR_FAS are obtained for a surface-borehole station pair in this study.

3.3 Cross-spectral ratio (c-SSR_FAS)

SSR approach should be implemented with caution when the reference site is in a borehole rather than on surface-rock. This is because the downhole recording usually contains downgoing waves reflected from the free surface and other interfaces between sedimentary layers, as well as waves scattered from local inhomogeneity (e.g., Shearer and Orcutt 1987). These downgoing waves can interfere destructively with the upgoing incident waves at some frequencies, producing a notch in the FAS of the borehole recording and thus artificial peaks in surface-to-borehole spectral ratios (e.g., Steidl 1993).

Cross-spectral ratio (c-SSR) is defined as the product of the spectral ratio estimate and the coherence function (Eqs. 4, 5), implicitly accounting for the record coherence in the formulation of site response estimate (e.g., Bendat and Piersol 1980). Thus, the c-SSR technique is recommended for surface-borehole recordings (Steidl 1993; Safak 1997; Assimaki et al. 2008). Since recording stations with V_S,hole < 800 m/s are excluded in this study, effects of downgoing waves on borehole recordings can be minimized but cannot be ruled out completely. Hence, the c-SSR approach is also implemented in this study.

$$ c - SSR_{xy} \left( f \right) = \frac{{P_{xy} \left( f \right)}}{{P_{xx} \left( f \right)}} $$

(4)

$$ C_{xy} \left( f \right) = \frac{{\left| {P_{xy} \left( f \right)} \right|^{2} }}{{P_{xx} \left( f \right)P_{yy} \left( f \right)}} $$

(5)

in which P_xx(f) and P_yy(f) are the power spectral densities of the surface (x) and borehole (y) recordings, respectively; P_xy(f) is the cross power spectral densities between x and y; C_xy(f) is the magnitude-squared coherence, which is a function of frequency with values ranging between 0 and 1.

All three techniques, including SSR_PSA, SSR_FAS and c-SSR_FAS, are utilized to derive surface-to-borehole spectral ratios in this study. SSR_PSA at each horizontal component is computed as the ratio of 5% damped pseudo-spectral acceleration of the waveform recorded at the ground surface to that recorded at the borehole. The geometrical mean of the SSR_PSA in each horizontal direction is used and then averaged over different events (≥ 3) recorded at the site. SSR_FAS and c-SSR_FAS are obtained in a similar manner to SSR_PSA. One difference lies in that SSR_FAS is calculated as the ratio of FAS between surface and borehole recordings, while c-SSR_FAS is derived as the ratio of the cross-power density spectrum between the surface and downhole recordings to the power density spectrum of surface recording using Welch’s method (Welch 1967). The other difference is that, for SSR_FAS and c-SSR_FAS, we take an extra step—smoothing before deriving the spectral ratio. Konno and Ohmachi (Konno and Ohmachi 1998) smoothing is utilized with a bandwidth coefficient b = 20.

3.4 Comparison of SSR_PSA, SSR_FAS, and c-SSR_FAS

Values of SSR_FAS are compared with those of SSR_PSA and c-SSR_FAS at T = 0.2 and 2.0 s in Fig. 5. SSR_FASs, SSR_PSAs and c-SSR_FASs at different spectral periods are given in Online Resource 1. It is worth noting that, in Fig. 5, amplifications at T = 2.0 s are much less notable than at T = 0.2 s. This is because that most selected KiK-net sites do not have very thick sediments. There are only 20 (or 10%) sites with Z_0.8 more than 100 m (Fig. 3) and also a few sites with T₀ above 1.0 s (Fig. 4). Thus, at these investigated KiK-net stations, site effects are not significant at relatively long periods (> ~ 1.0 s).

Fig. 5

Comparison between SSR_PSA, SSR_FAS and c-SSR_FAS at 0.2 (first column) and 2.0 s (second column)

Ratios between SSR_PSA and SSR_FAS, as well as between c-SSR_FAS and SSR_FAS at various periods are presented in Fig. 6. At T = 0.02 s, SSR_PSA deviates significantly from SSR_FAS (Fig. 6a). This is attributable to the peculiarity of response spectrum of which spectral content is not directly proportional to that of the corresponding Fourier spectrum at some oscillator periods (or frequencies). For instance, peak ground acceleration (PGA) is controlled by the entire Fourier spectrum (e.g., Bora et al. 2016). In addition, the strong scenario-dependency of response spectral ratios at short periods (Stafford et al. 2017) may explain the large scatter at T = 0.02 s in Fig. 6a. However, at T = 0.2 s, SSR_PSA is, on average, only slightly smaller than SSR_FAS with SSR_PSA/SSR_FAS = 0.94 (± 0.12) (Fig. 6a), and then the ratio between SSR_PSA and SSR_FAS increases gradually with the oscillator period, reaching 1.09 (± 0.35) at T = 4.0 s, suggesting an insubstantial divergence in the period range between 0.2 and 4.0 s. The trend of SSR_PSA/SSR_FAS with period (Fig. 6a) is compatible with the results of Cadet (2007). In Fig. 6b, the median value of c-SSR_FAS/SSR_FAS is systematically lower than unity in the period range from 0.1 to 4.0 s. For instance, the ratio between c-SSR_FAS and SSR_FAS is 0.71 (± 0.11) at T = 0.2 s. As pointed out by many researchers (e.g., Field et al. 1992; Steidl 1993), c-SSR does give a consistently lower amplification than SSR.

Fig. 6

Ratios of a SSR_PSA to SSR_FAS and b c-SSR_FAS to SSR_FAS at various spectral periods

As indicated by Assimaki et al. (2008) and Thompson et al. (2009), downgoing waves could be identified in KiK-net recordings but did not severely contaminate the borehole records. This is especially true for our KiK-net dataset because of the inclusion of only sites with V_S,hole ≥ 800 m/s. For these sites, abrupt velocity contrasts are more likely to be present above borehole sensors, resulting in the trapping of seismic waves between a velocity contrast and the free surface and thus inhibiting the surface reflections reaching downhole stations (e.g., Oth et al. 2011). However, destructive interferences at downhole stations, though deemed insubstantial here, still cannot be completely excluded and may contribute to the upward bias in SSR_FAS. This is evidenced by the shrinking aberration with the increase in spectral periods. Since for most selected KiK-net sites, T_S,ave at which fundamental mode of destructive interference between incident waves and surface reflections occurs is below 1.0 s (85^th percentile), as shown in Fig. 7, constructive interference has a negligible impact on the selected dataset at spectral periods above 1.0 s, e.g., c-SSR_FAS/SSR_FAS = 0.97 (± 0.07) at T = 2.0 s and 0.99 (± 0.08) at T = 4.0 s (Fig. 6b).

Fig. 7

Fundamental period of constructive interference between up-and down-going seismic waves at a borehole station. T_{S, ave} = 4H_Hole/V_S,hole, in which H_Hole is the distance and time-averaged velocity between the ground surface and downhole sensors

Another reason for the deviation between SSR_FAS and c-SSR_FAS is that noises in signals have different impacts on them. For noise-free data, SSR_FAS is identical to c-SSR_FAS. If noise is present in recordings, it will modify the SSR_FAS and c-SSR_FAS by factors [(1 + SNR⁻²_S)/(1 + SNR⁻²_B)]^0.5 and 1/(1 + SNR⁻²_B), respectively, where SNR_S and SNR_B represent signal-to-noise ratios of surface and borehole recordings, respectively. There is relatively little noise in downhole recordings compared with those at the ground surface (Field et al. 1992), thus SSR_FAS will be scaled up whereas c-SSR_FAS will remain relatively unchanged.

4 Site proxies alternative to V_S30

After obtaining site amplifications using SSR_PSA, SSR_FAS and c-SSR_FAS (Online Resource 1), we first model the observed amplification factor (AF) at a given spectral period using a single site proxy, i.e., V_Sz, T₀, Z_0.8 and Z_1.0. Fig 8a–c depict the AF at T = 0.4 s derived using c-SSR_FAS against V_S30, T₀, Z_0.8 and Z_1.0, respectively. AF decreases with the increase in V_S30 at T = 0.4 s (Fig. 8a) and all other spectral periods except for 0.02 and 0.1 s at which AF scales positively with V_S30. Kawase and Matsuo (2004) also found an increase of AF with V_S30 at very short periods (e.g., 0.06 and 0.08 s) using K-NET, KiK-net and JMA stations.

Fig. 8

Site amplification (c-SSR_FAS) at T = 0.4 s versus aV_S30, bT₀, cZ_0.8, and dZ_1.0 in ln–ln space. Dots represent sites with 800 < V_s,hole < 1800 m/s whereas circles are sites with 1800 < V_s,hole < 3300 m/s. The dotted line in each plot represents the regression to all data points, and R² is the coefficient of determination

For a simple configuration of layered sediments underlain by bedrock, surface-to-bedrock amplification is governed by two competing phenomena, impedance contrast and attenuation (e.g., intrinsic material damping, geometric spreading and scattering). The increase of AF with V_S30 at short periods may be caused by the fact that the detrimental impact of short-period (high-frequency) attenuation (e.g., Van Houtte, et al. 2011) on amplification outweigh the incremental effect of impedance contrast. For a site with a large V_S30, high-frequency wave energies tend to be attenuated to a lesser extent than they are at a softer site, resulting in a higher AF at a stiffer site.

In the linear range, many amplification models or site terms in GMMs (e.g., BSSA14) introduce a limiting velocity beyond which amplification no longer scales with V_S30. This limiting velocity (V_c in BSSA14) is period-dependent and decreases with the spectral period. For example, the value of limiting velocity V_c in BSSA14 is as large as 1500 m/s at 0.02 s, reducing to 844 m/s at 4.0 s. Since only a few sites investigated in this study have a V_S30 higher than 800 m/s (Fig. 3), a limiting velocity is considered unnecessary in this research, and a simple linear function Eq. 6 is used for V_S30. However, as shown in Fig. 8b–d, AF does not scale linearly with T₀, Z_0.8 and Z_1.0. The relationships of AF with site period and depth can be represented by a piecewise linear function, as shown in Fig. 9a and b, respectively. Hassani and Atkinson (2018a, b) modelled the trend of AF with T₀ using a similar function as shown in Fig. 9a and fixed a₁ and a₂ to 0.05 and 2.0 s, respectively. However, our results show that all these coefficients (e.g., a₁ and b₁) are period-dependent. With current dataset, it is difficult to constraint all these period-dependent coefficients.

Fig. 9

Trends of amplification with aT₀ and bZ_0.8 (or Z_1.0) at a given spectral period

However, based on visual inspection, the relationship between AF and T₀ can also be described by a polynomial function. Given the purpose of the present research is to evaluate the efficacy of site proxies, rather than proposing a robust AF estimation model, we thus depict the trend using a polynomial function. The function with fewest terms that are statistically significant is a quadratic function. Hence a second-order polynomial (Eq. 7) is adopted to characterize the variations of AF with T₀, Z_0.8 and Z_1.0. We calculate the amplification residual Res₁ (Eq. 8) of observation about the regression line (Eqs. 6 or 7), as well as the standard deviation of amplification residuals $\sigma_{Res1}$ according to Eq. 9. The performance of each site parameter is assessed based on $\sigma_{Res1}$, which is displayed in Fig. 10 for each amplification model using a single proxy.

Fig. 10

Standard deviations of intersite amplification residuals of AF models using a single site proxy

$$ \ln \left[ {AF\left( {V_{S30} } \right)} \right] = a_{0} + a_{1} \ln (V_{S30} ) $$

(6)

$$ \ln \left[ {AF\left( Y \right)} \right] = b_{0} + b_{1} ln\left( Y \right) + b_{2} \left[ {ln\left( Y \right)} \right]^{2} $$

(7)

$$ Res_{1} = ln\left( {AF_{obs} } \right) - ln\left( {AF_{pre} } \right) $$

(8)

$$ \sigma_{Res1} = \sqrt {\frac{{\mathop \sum \nolimits_{1}^{N} \left[ {ln\left( {AF_{obs} } \right) - ln\left( {AF_{pre} } \right)} \right]^{2} }}{N}} $$

(9)

where AF_obs—Observed site amplification at a certain period using either SSR_PSA, SSR_FAS or c-SSR_FAS; AF_pre—Predicted site amplification, including AF(V_S30) and AF(Y); AF(V_S30) and AF(Y)- Site amplifications predicted using V_S30 and Y, respectively; Y—Site period or depths, e.g., T₀, Z_0.8 or Z_1.0; Res₁—Amplification residual; σ—Standard deviation of amplification residuals; N—Number of sites.

Figure 10 shows that, among V_S30, T₀, Z_0.8 and Z_1.0, there is no one single-proxy which is the best for all spectral periods. However, it is obvious that, regardless of the techniques used to derive amplification, T₀ has the best overall performance, especially for oscillator periods between 0.1 and 4.0 s. V_S30 performs relatively well for periods from 0.2 to 0.7 s and is only second to T₀ in this period range, but for periods higher than 0.7 s, V_S30 is among the worst-performing indexes. Site depth Z_0.8 exhibits nearly identical performance to T₀ at periods over 0.5 s, but site depth to a stiffer layer Z_1.0 does not further improve amplification estimation. Comparing the efficacies of V_Sz (z = 5, 10, 20 and 30 m), shear-wave velocity averaged to a larger depth can lead to better overall performance, and the improvement manifests in a period range from 0.2 to 0.7 s.

T₀ is shown to be the best-performing single-proxy among V_Sz, T₀, Z_0.8 and Z_1.0 in modelling linear site response at KiK-net sites. We then depict in Fig. 11a the $\sigma_{Res1}$ reduction brought by replacing V_S30 with T₀. Compared with the conventional parameter V_S30, using T₀ can reduce the site-to-site amplification variability by up to 17%, 24% and 27% for SSR_PSA, SSR_FAS and c-SSR_FAS, respectively. The reductions are negligible for spectral periods below 0.1 s, suggesting the inability of T₀ in improving amplification prediction in this range. There are troughs in the period range between 0.2 and 0.7 s (Fig. 11a) since V_S30 performs relatively well at these periods, as shown in Fig. 11b which illustrates the Pearson correlation coefficient (R²) between AF and V_S30. Fig 11b is nearly identical to Fig. 23 in the paper by Kawase and Matsuo (2004). At 0.1 s, AF scales poorly with V_S30, but at periods from 0.2 to 0.7 s, AF is well correlated with V_S30. Thus, for periods between 0.2 and 0.7 s, substituting T₀ for V_S30 can only induce a limited reduction. The decrease in intersite variability is most pronounced at relatively long periods (> 0.7 s), implying the efficiency of T₀ in this period band (Fig. 11a).

Fig. 11

a Reduction in intersite variability of model AF(T₀) relative to model AF(V_S30); b Pearson correlation coefficient between AF and V_S30

5 Site proxies (secondary) complementary to V_S30 (primary)

It is known that V_S30 alone cannot account for all aspects of site effects. However, given the widespread use of V_S30 in current seismic provisions, replacing V_S30 with a new site proxy appears to be unappealing for many. Considering that the V_S30-corrected residual amplification Res₁ shows a certain degrees of dependency on T₀, as well as site depths Z_0.8 and Z_1.0, it is desirable to search for a site proxy complementary to V_S30 in classifying a site or parameterizing site effects. Then the question arises as to which site index among T₀, Z_0.8 and Z_1.0 is the best parameter secondary to V_S30 in modelling the residual amplification. To address this issue, we first model Res₁ as a quadratic function (Eq. 10) of a secondary site proxy (Y), including T₀, Z_0.8 and Z_1.0. Then we derive the residual Res₂ (Eq. 11) between Res₁ and proxy-based prediction Res₁(Y), as well as its standard deviation $\sigma_{Res2}$ (Eq. 12). The difference between σ_Res1 and σ_Res2 is then utilized to gauge the efficacy of each secondary site proxy. The best-performing candidate is considered to be the one that can produce the largest reduction.

$$ Res_{1} \left( Y \right) = b_{0} + b_{1} ln\left( Y \right) + b_{2} \left[ {ln\left( Y \right)} \right]^{2} $$

(10)

$$ Res_{2} = Res_{1} - Res_{1} \left( Y \right) $$

(11)

$$ \sigma_{Res2} = \sqrt {\frac{{\mathop \sum \nolimits_{1}^{N} \left( {Res_{1} - Res_{1} \left( Y \right)} \right)^{2} }}{N}} $$

(12)

Figure 12 depicts the standard deviation (Eq. 12) and its relative reduction brought by adding an additional site proxy (secondary) on top of V_S30 (primary). It shows that, for the investigated oscillator periods, all secondary parameters (T₀, Z_0.8 and Z_1.0) can reduce intersite amplification variability to different extents. At periods around 0.4 s, adding a secondary parameter into AF(V_S30) leads to a very limited reduction in variability. This is because, at periods around 0.4 s, V_S30 performs well (Fig. 11b), and thus introducing an extra predictor variable can only induce an inconsequential improvement. However, for periods between 0.7 and 2.0 s, these additional variables are rather effective in improving amplification prediction.

Fig. 12

Standard deviation of intersite amplification residuals (first column) of each amplification model and its relative reduction (second column) compared with that of model AF(V_S30) using SSR_PSA, SSR_FAS, and c-SSR_FAS

The most remarkable reduction is achieved by T₀. The induced reductions are as much as 20% (T = 1.0 s), 27% (T = 2.0 s), and 26% (T = 2.0 s) for SSR_PSA, SSR_FAS and c-SSR_FAS, respectively. T₀ is followed by Z_0.8, including which into the model AF(V_S30) leads to reductions up to 16% (T = 1.0 s), 17% (T = 1.0 s), and 15% (T = 0.7 s) for SSR_PSA, SSR_FAS and c-SSR_FAS, respectively. In general, Z_1.0 is more difficult to obtain than Z_0.8, but the former does not instigate a more apparent reduction in site-to-site variability than the latter (Fig. 12). The percentages of decrease pertaining to Z_1.0 are 12% (T = 1.0 s), 17% (T = 2.0 s), and 15% (T = 2.0 s) for SSR_PSA, SSR_FAS and c-SSR_FAS, respectively.

A three-dimensional (3D) subsurface velocity structure model, the Japan Seismic Hazard Information Station (J-SHIS) model, was constructed by Fujiwara et al. (2009) for the whole Japan (see Data and Resources). Regional velocity models are also available for some areas outside Japan, i.e., the 3D Community Velocity Model CVM-S4 (Kohler et al. 2003) and CVM-H1.1.0 (Süss and Shaw 2003) for the Southern California, as well as the 3D Velocity Model of the Bay Area (Boatwright et al. 2004) for the Northern California. These regional velocity models are developed primarily to model the propagation of long-period ground motions (> ~ 1.0 s). However, it is tempting to extract site parameters from these existing velocity models in a site-specific investigation. For instance, in the NGA-West2 project (Ancheta et al. 2014), subsurface models for Japan and California were queried to establish the site depth database. Thus, it is intriguing to examine to what extent the depth parameters extracted from a regional velocity model can improve site-response estimation if they are used as complementary proxies to V_S30.

We denote depths inferred from the J-SHIS model as Z_{x_infer} (x = 0.8, 1.0, 1.5 and 2.5 km/s). The subscript “infer” is to distinguish them from the measured depths. Following the same procedure as Z_0.8 and Z_1.0, we gauge the performances of Z_{x_infer} based on the decreases in site-to-site variability due to their incorporations into the amplification model AF(V_S30). Fig 13 displays the percentages of reduction in standard deviation. It can be seen that Z_{0.8_infer}, Z_{1.0_infer}, Z_{1.5_infer} and Z_{2.5_infer} can secure reductions, but the reductions do not exceed 8% (T = 0.7 s), 8% (T = 0.7 s), 10% (T = 1.0 s) and 9% (T = 1.0 s), respectively. Comparing measured (Fig. 12) with inferred depths (Fig. 13), the formers (Z_0.8 and Z_1.0) are advantageous over their inferred counterparts (Z_{0.8_infer} and Z_{1.0_infer}). This is mainly attributable to the different levels of uncertainty in inferred and measured depth data. Inferred depths from the J-SHIS model are biased and have a substantial amount of uncertainty when compared with site-specific depth measurements, as indicated by Zhu et al. (2019).

Fig. 13

Relative reduction in the standard deviation of intersite amplification residual due to the incorporation of inferred site depth Z_{x_infer} (x = 0.8, 1.0, 1.5 and 2.5 km/s) compared to model AF(V_S30) using SSR_FAS

Although Z_{x_infer} (x = 0.8, 1.0, 1.5 and 2.5 km/s) perform less well than their measured counterparts, introducing Z_{x_infer} into AF(V_S30) can lead to a noticeable improvement in prediction. Thus, one can exploit an available regional velocity model for purposes other than long-period ground motion simulation. However, comparison with measured depths warrants a further improvement to the regional velocity model J-SHIS. As reported by Dhakal and Yamanaka (2013), there are evident discrepancies between the J-SHIS model and other subsurface models for the same region. For regions outside Japan, the 3D Community Velocity Model CVM (Version 2, Magistrale et al. 2000) for southern California was also found to have a significant bias (Stewart et al. 2005) and, as pointed out by Graves and Aagaard (2011), further refinement to the CVM was needed.

6 Site amplification as a function of T₀ (primary) and V_Sz (secondary)

In engineering practice, V_S30 has already been derived at many recording stations and is widely adopted to categorize sites in current seismic codes (e.g., European code EC8) or to describe site effects in many GMMs (e.g., NGA-West2 GMMs). However, V_S30 entails higher costs than T₀ which can be readily acquired using the HVSR technique on either ambient noises (Nakamura 1989) or earthquake recordings (Lermo and Chávez-García 1993). More importantly, T₀ is proved to be the best-performing single-proxy (Fig. 10) among V_Sz, T₀, Z_0.8 and Z_1.0 in depicting linear site effects. Thus, considering both the performance and engineering utility of T₀, as well as the established status of V_Sz, especially V_S30, we parameterize site effects using both parameters with T₀ as the primary and V_Sz as the secondary site indicator.

Figure 14 shows the site-to-site variabilities of each site-effect model, including AF(T₀) and AF(T₀, V_Sz), as well as the percentages of reduction in variability due to the incorporation of V_Sz into AF(T₀). Adding V_Sz as secondary site proxies can decrease estimation uncertainty. The reductions brought by V_S5 and V_S10 are insignificant, but the reduction increases with z (z = 5, 10, 20 and 30 m). V_S20 and V_S30 can lead to apparent variability reduction in the period range from 0.2 to 0.7 s, suggesting that, besides V_S30 (Fig. 11b), V_S20 performs also well in describing linear amplification in this period range. In comparison, the reduction as a result of the inclusion of V_S30 into AF(T₀) is less significant than the reduction due to the incorporation of T₀ into AF(V_S30) (Fig. 12), manifesting that T₀ alone can account for most of the site effects.

Fig. 14

Standard deviations of residuals of site-effects models using T₀ as the primary site proxy with or without V_Sz as the secondary predictor (left column); and relative reductions in the standard deviation of residuals of models AF(T₀, V_S5), AF(T₀, V_S10), AF(T₀, V_S20), and AF(T₀, V_S30) compared with AF(T₀) (right column)

Figure 15a compares the intersite variabilities of the amplification (SSR_PSA) model AF(T₀, V_S30) with that of the model AF(V_S30, T₀). The former uses T₀ as the primary and V_S30 as the secondary predictor whereas the latter utilizes V_S30 as the primary and T₀ as the secondary variable. The level of estimation uncertainty associated with AF(T₀, V_S30) is lower than that of AF(V_S30, T₀), implying that the sequence of predictors entering the model affects the model efficacy. If both T₀ and V_S30 were to be included in an amplification model, using T₀ as the primary indicator is preferable than the other way around, which is consistent with the finding of Hassani and Atkinson (2018a). Comparing with the configuration of V_S30 (primary) and T₀ (secondary), Fig. 15b shows that the combination of T₀ (primary) and V_S30 (secondary) can reduce model uncertainty by up to 12% (SSR_PSA), 7% (SSR_FAS) and 12% (c-SSR_FAS).

Fig. 15

a Standard deviations of residuals of amplification models of AF(T₀, V_S30) and AF(V_S30, T₀) using SSR_PSA; b reduction in the uncertainty of model AF(T₀, V_S30) relative to model AF(V_S30, T₀)

7 Discussion and conclusions

In an effort to pinpoint the best-performing site proxy or optimal combination of site proxies in modelling linear site response, we selected 1840 ground-motion recordings from a KiK-net database processed by Dawood et al. (2016). Site effects were estimated using surface-to-borehole spectral ratios. T₀ was found to be the best-performing single-proxy among V_Sz (z = 5, 10, 20 and 30 m), T₀, Z_0.8 and Z_1.0. Substituting T₀ for V_S30 in an amplification model could induce a significant reduction in site-to-site amplification variability, especially for spectral periods at 0.7 s or longer. There seems to be a consensus that T₀ has a better overall performance than V_S30 in parameterizing site effects (e.g., Zhao and Xu 2013; Cadet et al. 2012; McVerry 2011; Derras et al. 2017; Stambouli et al. 2017; Hassani and Atkinson 2018a). T₀ is also recommended as one of the main proxies in the draft of revised EC8.

Besides, T₀ was also found in this study to be most descriptive of the residual amplification after V_S30-correction among T₀, Z_0.8, Z_1.0 and Z_{x_infer} (x = 0.8, 1.0, 1.5 and 2.5 km/s). In addition to V_S30 (primary), adding T₀ (secondary) into an amplification model could reduce site-to-site amplification variability by up to 20–27% at relatively long periods (> 0.7 s). In site classification, Luzi et al. (2011) also achieved a significant reduction in variability when T₀ was included as a complimentary proxy to V_S30. This suggests that T₀ can be incorporated into current site-response models (or site terms) and site classification schemes to better account for site effects.

When T₀ was used as the primary proxy to model site response, T₀ alone could capture most of the site effects. Adding V_S5 or V_S10 (secondary) could only introduce a very limited further reduction in intersite amplification variability. In contrast, incorporating V_S20 or V_S30 (secondary) could apparently decrease variability at periods between 0.2 and 0.7 s but by no more than 10% and 14%, respectively. Furthermore, using T₀ as the primary and V_S30 as the secondary predictor variables is better than the configuration of V_S30 (primary) and T₀ (secondary), which confirms the results of Hassani and Atkinson (2018a).

Figure 16 compares the reductions in standard deviations of residuals between amplification observations and T₀-based predictions relative to the conventional model AF(V_S30). These relative reductions are given in Table 2. All T₀-based amplification models have less amount of uncertainty than AF(V_S30), especially for periods over 0.7 s. Adding V_S10, V_S20 or V_S30 into the model AF(T₀) can further improve model performance, especially for spectral periods between 0.2 and 0.7 s. Although models AF(T₀, V_S30) and AF(V_S30, T₀) utilize the same site proxies, the former performs better than the latter due to the difference in the sequence of variables entering the model. For the same reason, AF(T₀, V_S10) can achieve at least comparable overall performance to AF(V_S30, T₀). Figure 16 also demonstrates that AF(T₀, V_S30) is the best-performing proxy pair but is only slightly better than AF(T₀, V_S20). Given that V_S20 is strongly correlated with V_S30 (e.g., Boore et al. 2011) and that V_S30 may entail higher costs than V_S20, thus V_S20 (secondary) may be adequate for engineering use.

Fig. 16

Relative reductions in the standard deviation of residuals of models AF(T₀), AF(T₀, V_S10), AF(T₀, V_S20), AF(T₀, V_S30) and AF(V_S30, T₀) compared with AF(V_S30)

Table 2 Relative reductions in the standard deviation of residuals compared with AF(V_S30)

T₀ is shown to be preferable in site characterization. However, our previous study (Zhu et al. 2019) found that there were discrepancies in T₀ derived using the HVSR technique by different teams. For sites with prominent 2D or 3D features, HVSR is not always reliable in determining T₀ (Gueguen et al. 2007). This may be one source of uncertainty in T₀ and will inevitably affect its efficacy. Besides, T₀ is identified as corresponding to a significant peak on HVSR, but significant peaks are often defined rather subjectively, which may be another source of uncertainty in T₀. Thus, one should take due consideration of these factors when detecting T₀ using the HVSR method. In this study, we used only KiK-net sites which are often located on weathered rocks or thin sediments (Aoi et al. 2000). This is compatible with the maximum usable period of our selected recordings, i.e., at least 4.17 s since there are no prominent site effects beyond this period. However, there exist sites with rather thick sediments, especially those in deep sedimentary basins, e.g., the Kanto (Tokyo) basin in Japan and the Los Angeles basin in California. At these sites, there are significant amplifications at periods longer than 4.17 s. Also, basin-generated surface waves are likely to have a strong presence in the recordings. Thus the best-performing site proxy or combination of proxies should be further tested on deep basin sites and in a broad period range preferably up to ~ 10.0 s. Meanwhile, it needs to be stressed that site amplifications in this research are referenced to downhole bedrocks with different velocities (larger than 800 m/s). Since all site proxies are gauged on the same dataset, inhomogeneous reference site conditions will not affect the results. However, amplification should be normalized to a common reference for the development of empirical prediction models. In addition, we limit our study to the linear domain, but soft soil sites exhibit nonlinear behavior during strong ground shakings. Thus, the efficiencies of various site parameters in depicting nonlinear site response need to be investigated in future studies.

What distinguishes this work from many preceding ones is that we include many competitive site proxies all in one study. In addition to V_Sz (z = 5, 10, 20 and 30 m) and T₀, we also consider various depth parameters with differentiation of measured (Z_0.8 and Z_1.0) from inferred depths (Z_x-infer, x = 0.8 and 1.0, 1.5 and 2.5 km/s). Though Z_1.5 and Z_2.5 are also candidate proxies utilized in some research, e.g., Z_1.5 in the amplification model by Choi et al. (2005) and Z_2.5 in the site term of Campbell and Bozorgnia (2014), we do not include them here since there are not adequate sites with measured Z_1.5 and Z_2.5 available. For instance, there are only 42 KiK-nets sites of which borehole drillings penetrate 2.5 km/s horizon.

In summary, our results show that T₀ is the best-performing single-proxy among V_Sz, T₀, Z_0.8 and Z_1.0 in modelling linear site response at KiK-net sites. Thus, T₀ can be used as a substitute for V_S30. Meanwhile, T₀ is also the best-performing proxy among T₀, Z_0.8, Z_1.0 and Z_x-infer complementary to V_S30 in capturing the V_S30-corrected residual amplification. Hence, T₀ can be utilized as an add-on to calibrate existing V_S30-based amplificaiton models or site terms. Besides, T₀ alone can capture most of the site effects and should be utilized as the primary site proxy. (T₀, V_S30) is found to be the best-performing proxy pair among (V_S30, T₀), (V_S30, Z_0.8), (V_S30, Z_1.0), (V_S30, Z_x-infer) and (T₀, V_Sz) but is only slightly better than AF(T₀, V_S20). Given that V_S30 may entail higher costs than V_S20, the configuration of T₀ (primary) and V_S20 (secondary) is considered to be the optimal combination.