# Bulletin of the Seismological Society of America

## Abstract

The decay parameter at the high frequency (kappa value, *κ*) introduced by Anderson and Hough (1984) is measured from 54 stations situated in and around the Taipei basin, northern Taiwan. Different frequency windows for the *S*‐wave amplitude spectra are used to measure the *κ*_{r} value for each individual record. The preferred *κ*_{r} values are determined by the least‐squared fitting with the best correlation coefficient. The site‐specific *κ*_{0} values, which are computed by removing the anelastic effect of regional geological structure by grouping *κ*_{r} into the crustal and subduction events according to the focal depth and site classification (i.e., class B, C, D, and E sites), are calculated to be in the range of 0.034–0.066 s. We correlate the *κ*_{0} values with the averaged shear‐wave velocity of the topmost 30‐m layer *V*_{S30} and find the independence of *κ*_{0} on *V*_{S30}. In addition, 28 stations lie on sediment in the Taipei basin. The gradient of *κ*_{0} versus the sedimentary thickness to the Tertiary base in the basin is calculated, and then used to estimate the effective quality factor *Q*_{ef} of the layer. *Q*_{ef} is 75.3 for the whole sedimentary layers (above the Tertiary base). Given one standard deviation, *Q*_{ef} are in the 41.9–376.7 range. *Q*_{ef} can provide an accurate referred parameter for the future ground‐motion simulation applications.

## Introduction

A filtered/high‐cut parameter (kappa) at the high frequency was first introduced by Anderson and Hough (1984), namely the kappa value, denoted by *κ*, which characterizes the behavior of spectrum decay with frequency for the shear waves due to the effect of several kilometers depth from surface. Common applications for *κ* are focused on the engineering seismology that has been suggested to be an important input of simulation of ground motion based on the stochastic model (Boore, 1983, 1996, 2003), a shape controller at high frequencies on the site amplification function for different site conditions (Boore and Joyner, 1997; Sokolov *et al.*, 2004; Huang *et al.*, 2005, 2007, 2009), and even as an adjustment factor taken into evaluation on ground‐motion prediction equations (GMPEs; Scherbaum *et al.*, 2006; Van Houtte *et al.*, 2011). A recent summary review of *κ* on the applications and restrictions is found in the study of Ktenidou *et al.* (2014).

In general, the spectral amplitude of acceleration recordings decays exponentially with frequency ∼*A*_{0}*e*^{−πκrf}, in which *A*_{0} are the spectral amplitudes and *f* is the frequency. The *κ*_{r} value can be directly measured by a linear fit on the spectra amplitudes in semilogarithmic scale at a frequency greater than a specific frequency *f*_{e}. The slope of linear fit is −*πκ*_{r}. The *κ*_{r} values are obtained from individual records of any epicentral distance. Anderson and Hough (1984) also concluded the terms of the distance‐corrected value *κ*_{r} and the site‐dependent value *κ*_{0} from the measured *κ*_{r} values for the individual site. The *κ*_{r} value represents the anelastic effect of regional geological structure assuming the dependency on distance between a site and the epicenter. Anderson and Hough (1984) and Anderson (1991) defined the distance‐ and site‐dependent *κ*_{r} value in the form of *κ*_{r}=*κ*_{0}+*κ*(*r*), in which *κ*(*r*) is formulated as *m*×*R*_{e} under assumption of dependency on the epicentral distance (*R*_{e}). The *κ*_{0} value is generally considered to vary with the site conditions whereas the seismic waves propagate through the near‐subsurface geology. The intercept also means the zero‐distance *κ*_{r} value; that is, a representative of a site. We decompose the measured *κ*_{r} value to obtain the site‐specific *κ*_{0} for comparison with the site indicator in a later section. The physics of the *κ*_{r} value is not completely understood and is still debated. Studies attribute it as source dependency (Papageorgiou and Aki, 1983; Purvance and Anderson, 2003); others support it as site dependency (Anderson and Hough, 1984; Anderson, 1991). In this study, we follow the assumption of Anderson and Hough (1984) to continue the investigation of *κ*_{r} on site dependence.

Parolai and Bindi (2004) analyzed the effect of soil layer on the estimate of *κ*_{r} value and concluded that the fundamental resonance frequency and even the first‐mode frequency of harmonics would cause discrepancy between the measured *κ*_{r} value and its original one. They also mention that the wider spectral windows applied to estimate the *κ*_{r} value would reduce or average out the effect of local peaks that would bias the measured results to be more or less.

The aims of this study are to correlate the site‐specific *κ*_{0} value to the existing site indicator; that is, the averaged shear velocity of the topmost 30‐m layer *V*_{S30}, and to evaluate the effective *Q* value of the sedimentary layers in the Taipei basin. Prior to correlation and evaluation, more stable and precise *κ*_{r} values for each individual record are measured under the process of least‐squared fitting with varied frequency windows on spectral amplitudes for the shear wave and are determined by the correlation coefficient with the best result. The selected window of frequency on fitting the spectral amplitudes is subjective (Ktenidou *et al.*, 2013). To reduce the visual and manual errors on the spectral amplitudes, there are studies on selecting the start and stop of frequencies varied in the 1–20 and 11–70 Hz ranges, respectively, to minimize the fitting errors in obtaining the *κ*_{r} values (Ktenidou *et al.*, 2013; Edwards *et al.*, 2015). We follow and modify the conception of methodology on selecting frequency windows to obtain *κ*_{r}.

The near‐surface *V*_{S30} are commonly used to identify the site conditions under National Earthquake Hazards Reduction Program (NEHRP); that is, classes A, B, C, D, and E representing hard rock, rock, very dense soil and soft rock, stiff soils, and soft soils, respectively (Building Seismic Safety Council [BSSC], 2001; Boore, 2004; Holzer *et al.*, 2005). We do a comparison of the *κ*_{0}–*V*_{S30} couples in this study to those of relationships developed in other areas (Chandler *et al.*, 2006; Edwards *et al.*, 2011; Van Houtte *et al.*, 2011; Ktenidou *et al.*, 2015). In addition, the thickness to the known basement for each observation site was suggested to be related to *κ*_{0} (Campbell, 2009; Ktenidou *et al.*, 2015). We apply procedures to correlate *κ*_{0} with thickness to the base rock, as well as the thickness of the topmost soil layer to infer the effective *Q* values for the future applications on the simulations of ground motion. Finally, *κ*_{0} correlates to dominant/resonant frequency measured from microtremor arrays deployed inside the Taipei basin.

### Recording Sites and Data

The data are from the Geophysical Database Management System (GDMS; Shin *et al.*, 2013), an integrated information system regarding the geophysical observation in Taiwan area, developed by the Central Weather Bureau (CWB), Taiwan. The observation stations in GDMS belong to the network of the Taiwan Strong Motion Instrumentation Program (TSMIP), which has recorded the strong ground motions at free field since 1992. The details regarding the network can be found in Liu *et al.* (1999). Each station is equipped with triaxial force‐balanced accelerometers, with a flat frequency response from direct current to about 50.0 Hz. The seismograms of acceleration are selected from the period of 1993 to 2010 with local magnitude (*M*_{L}) ranging from 5.0 to 7.3 and focal depths in the 1.9–175.6 km range. The *M*_{L} is determined using accelerograms to simulate the Wood–Anderson seismograms (Shin, 1993). The total number of events and records for analysis at 54 stations are 85 and 1267, respectively. Figure 1 shows the distribution of the earthquakes, with three sizes of circles displaying the various magnitudes and colors indicating the focal depths.

The stations of TSMIP network are installed in and around the Taipei basin, where the geographical environment is surrounded by Tatun volcanoes in the north, Linkou Tableland in the east, and the western foothill in the southeast (SE), as shown in Figure 2a. The geological sediments in the basin can be recognized by four layers; that is, (from top to bottom) Sungshan Formation, Chingmei Formation, Wuku Formation, and Banchiao Formation, over a base rock of the Tertiary from shallow seismic reflection surveys and borehole‐drilling data (Wang *et al.*, 1996, 2004; Fig. 2b). The topmost Sungshan Formation is composed of unconsolidated sand, silt, and clay and is attributed to the lowest *S*‐wave velocity of 170, 230, and 340 m/s. The averaged *S*‐wave velocities are 450, 660, and 880 m/s, respectively, for the Chingmei Formation, Wuku Formation, and Banchiao Formation. Many studies show significant site effect induced by the geology and topography beneath the basin from analysis of seismic data (Wen *et al.*, 1995; Wen and Peng, 1998; Wang *et al.*, 2004; Sokolov *et al.*, 2009). The first analysis of downhole array on site amplification in the Taipei basin was given by Wen *et al.* (1995), which concluded that the larger amplification of peak ground acceleration (PGA) 60 m depth is due to the soft‐soil layer. Although Wang *et al.* (2004) compared the PGAs of 50 earthquakes in the Taipei basin with the *S*‐wave velocity structures, results showed high PGAs correlated with low *S*‐wave velocity. Wen and Peng (1998) analyzed the amplitude spectral ratio of TSMIP’s seismic data in the Taipei basin. The frequency bands of 0.2–1.0 Hz are correlated with the bathometry of Tertiary base rock, leading to an amplification factor of at least 1.5. Sokolov *et al.* (2009) characterized the large site amplification at frequencies 0.3–1.0 Hz inside the basin. Furthermore, seismic‐wave propagation simulations based on the structure of seismic‐wave velocity can show the amplifications of PGA to be a factor of 5 (Lee *et al.*, 2008). In addition, Miksat *et al.* (2010) give a maximum factor of 8 from ground simulations related to the rock site based on the structure of seismic‐wave velocity. The basin tectonics can trap the seismic energy and amplify the seismic waves that can be found from the spectral analysis of seismograms or from waveform simulations. Figure 2a shows the study area and the distribution of stations denoted by open and solid triangles. The solid triangles display the stations located inside the Taipei basin. As shown in Figure 3, the gray area is the subsurface topography of the Tertiary base rock, which tilts from east to west in depth distance span of about 13 km and reaches the deepest depth of ∼750 m at the northwest (NW) of the basin. The notes A and B correspond to those in Figure 2a. The details of the stations are listed in Table 1, in which *D*_{B} is the thickness of the sediment above the Tertiary base rock. The *M*_{L} versus *R*_{e} and focal depths, respectively, are shown in Figure 4. The *R*_{e} ranges from 18.6 to 205.0 km. Seismograms are from both the crustal and subduction events.

## Method

### Estimate of *κ*_{r} Value

The study of Anderson and Hough (1984) defined the decay shape for the Fourier spectra of *S* wave *A*( *f*) with frequencies greater than a specific one *f*_{e}, which is definitive after the corner frequency *f*_{c}, in the form of (1)The *κ*_{r} value is the decay parameter in second. This equation can be expressed as the overall and frequency‐independent attenuation within a chosen range of frequencies under assumption. The slope of equation (1) can be obtained from the fit on the semilogarithmic plot of spectral amplitudes over a certain window of frequencies starting at *f*_{e} and stopping before the spectra of the background noise or the flat response of instrument.

Two steps are adopted to measure the *κ*_{r} values from the seismic records. The first step is that the *S* wave of recordings for the horizontal components are computed from waveform in time series to the frequency spectra. To speed up the computation based on the technique of fast Fourier transformation, two time windows are selected for comparison after manually picking the arrival of *P* and *S* waves. The length of *S*‐wave window for *M*_{L}>5.0 in Taiwan can be obtained using Lee *et al.* (2015). The strong‐shaking duration, which is commonly considered as the *S*‐wave time window, is in terms of earthquake magnitude (associated with *f*_{c}), distance from source, and site condition (*V*_{S30}). We denote the time windows w1 and w2 as the pre‐*P*‐ and *S*‐wave time windows, respectively. The w1 can be considered as background noise and is equal to the length of w2. Figures 5a and 5b display the waveform of the north–south (N–S) component and those of the east–west (E–W) component, respectively. The gray lines show the pre‐*P*‐ and *S*‐wave windows regarding w1 and w2. The time series were processed using a cosine taper of 5% at both ends of the windows. The dark and gray lines in Figure 5c,d show the smoothed spectral amplitudes of pre‐*P* and *S* waves for the time windows (i.e., w1 and w2). In Figure 5c,d, the spectral amplitudes of w2 (less than 23 Hz) are larger than those of w1, which is due to more spectral amplitude of the background noise. The amplitude spectral ratio of w2 over w1 at 23 Hz is ∼3, which is defined as signal‐to‐noise ratio (SNR). In our cautious picking of the *P* and *S* arrival and the inspection of time window w1 and w2, the time window w2 can contain the main energy of *S* wave for most of the events to estimate the *κ*_{r} value for each single record.

The second step is to determine the frequency windows and obtain the *κ*_{r} value automatically by moving windows. Equation (1) performs a linear relationship between the frequencies and spectral amplitudes, and describes a negative correlation indicating that the spectral amplitudes decrease with frequencies. To reduce the deviation of measuring the *κ*_{r} values, we apply the parameter of correlation coefficient under the process of the linear fit. The correlation coefficient *R* is one of the measures for goodness of fit describing the discrepancy between two variables, and the value is expected to be in the range of −1.0 and 1.0. An example shown in Figure 6 is the amplitude spectra for both of the horizontal components. As mentioned earlier, the *f*_{e} should be above the *f*_{c}, which depends on the seismic moment and stress drop of the seismic source (Aki, 1967; Brune, 1970). The regional‐dependent relationship between seismic moment (*M*_{0}) and the *f*_{c} was proposed by Huang and Wang (2009), in which they analyzed 22 aftershocks with 5.1≤*M*_{L}≤6.5 of the 1999 Chi‐Chi, Taiwan, earthquake (*M*_{L} 7.3). They concluded that the scaling law between *M*_{0} and *f*_{c} is log(*M*_{0})=−3.65×log(*f*_{c})+23.36. Wu *et al.* (2001) formulated the relationship between *M*_{L} and *M*_{w} from 32 events in the Taiwan area as *M*_{L}=4.53×ln(*M*_{w})−2.09 for 5.0≤*M*_{L}≤7.1 and 4.8≤*M*_{w}≤7.6, in which *M*_{w} is the moment magnitude. Substituting the minimum *M*_{L} 5.0 gives *M*_{w} 4.8. The *M*_{0} can be calculated from *M*_{w} and is equal to 1.88×10^{16} N·m (Hanks and Kanamori, 1979). The *f*_{c} is 1.05 Hz according to the *M*_{0}–*f*_{c} relationship with a few extrapolations. Meanwhile, the dominant frequency of resonance effect is about 1.0 Hz due to sediments in the Taipei basin based on the method of amplitude spectrum ratio (Wen and Peng, 1998). The dominant frequencies determined by the horizontal‐to‐vertical (H/V) spectral ratio at a site from microtremor array surveyed in the Taipei basin are in the 0.5–5.0 Hz range (Huang, 2009). To reduce the effects of *f*_{c} and dominant frequency of resonance effect on measuring the *κ*_{r} value, we select *f*_{1}(=*f*_{e}) at 5, 6, 7, 8, 9, and 10 Hz, and the *f*_{2} varies from 20 to 50 Hz with an increase of 1 Hz for iterations. In addition, the iteration will select the *κ*_{r} with the best correlation coefficient and SNR≥3. As shown in Figure 6b, the solid lines in the deepened gray colors with leading notes, that is, *f*_{1}=5, 6, 7, 8, 9, and 10, show that the correlation coefficients vary with frequencies. The best fit with *R*=−0.87 (denoted by an open square) is to select *f*_{1}=5 Hz and *f*_{2}=23 Hz for this recording. Meanwhile, the measured *κ*_{r} values also display for the condition of *f*_{1}=5 Hz and different *f*_{2} from 20 to 50 Hz range. The dashed lines shown in Figure 5c,d are the fitting results.

## Results

Figure 7 shows that the *κ*_{r} values of the two horizontal components increase gradually with *R*_{e}. We continue to compute *κ*(*r*)’s according to the site conditions and the focal depths. The site classification based on *V*_{S30} was constructed by the National Center for Research on Earthquake Engineering and the CWB in Taiwan to perform well loggings in 2000 for the engineering applications. The majority of the boreholes have a drilling depth of ∼30 m. The United States’ criteria for classifying sites following the NEHRP’s provisions are based on the following shear velocities: *V*_{S30}>1500 m/s for class A sites, *V*_{S30}=760–1500 m/s for class B sites, *V*_{S30}=360–760 m/s for class C sites, *V*_{S30}=180–360 m/s for class D sites, and *V*_{S30}<180 m/s for class E sites. In the Taipei area, 54 stations have site classifications based on the evaluation of *V*_{S30} (Kuo *et al.*, 2011, 2012), which are listed in Table 1. The *κ*_{r}–*R*_{e} couples are classified into four categories according to site conditions (Kuo *et al.*, 2011, 2012). In addition, datasets from the four site conditions are separated into two groups: crustal and subduction events, according to the focal depth of 40 km (Ustaszewski *et al.*, 2012). The class B, C, D, and E sites are shown in Figure 7a, 7b, 7c, and 7d, respectively. As shown in Figure 7, the solid circles denote data from crustal events, whereas the open circles denote those from subduction events. The increasing trend of fitting lines is displayed by a solid line and a dashed line, respectively, representing the crustal events and subduction events. The least‐squared technique of robust linear model (RLM) is utilized to obtain the linear fitting instead of the commonly used method, ordinary linear fitting. The advantage of RLM is demonstrated in the study by Ktenidou *et al.* (2013). Ordinary linear fitting would have been influenced easily by the outliers in the data, which is not the case for RLM.

We observe that the slopes of crustal events are larger than those of subduction events for the four site classes. Similar results are also found in data from Japan (Van Houtte *et al.*, 2011) and Greece (Ktenidou *et al.*, 2013). The slopes of four site classes vary from 0.00010 to 0.00022 s/km for the crustal event, whereas the slopes are in the 0.000061–0.00015 s/km range for the subduction events. The slope (*m*) is considered as the regional effect of seismic‐wave lateral propagation. Then, the site‐specific *κ*_{0} value can be calculated by minimizing the errors by applying the regional effect of *m*, accordingly, provided by the four site classes and the crustal and subduction events for each of the stations and following the equation *κ*_{r}=*κ*_{0}+*m*×*R*_{e}. The estimated *κ*_{0} values for the 54 stations are also listed in Table 1.

### Site‐Specific *κ*_{0} versus *V*_{S30}

The numbers of class B, C, D, and E sites are 3, 14, 32, and 5, respectively. Figure 8 shows the 54 couples of *κ*_{0} and *V*_{S30} denoted by circles with one standard deviation bars. The dotted lines are the thresholds for the site classifications (i.e., class B, C, D, and E sites with notations at the bottom). The *κ*_{0}–*V*_{S30} couples perform a slightly decreasing *κ*_{0} with increasing *V*_{S30}. The existing relations between *κ*_{0} and *V*_{S30} provided by Silva *et al.* (1998), Chandler *et al.* (2006), Edwards *et al.* (2011), Van Houtte *et al.* (2011), and Ktenidou *et al.* (2015) from worldwide data are displayed by the thin dashed, the thin solid, the dashed‐dotted, the thick dashed, and the thick solid lines, respectively. Silva *et al.* (1998) first introduced the relation using data for *V*_{S30} greater than 300 m/s in California. Chandler *et al.* (2006) provided the relationship in eastern North America for *V*_{S30} in the range of about 300–3000 m/s, for which the site conditions belong to class A, B, and C sites. Edwards *et al.* (2011) developed the relationships in Switzerland for class A, B, and C sites, namely *V*_{S30} greater than 400 m/s. Van Houtte *et al.* (2011) compiled the various results from the United States, Taiwan, France, and Switzerland to construct the *κ*_{0}–*V*_{S30} relation for *V*_{S30} greater than 500 m/s. Ktenidou *et al.* (2015) provided the *κ*_{0}–*V*_{S30} relation from a downhole array in Greece for *V*_{S30} greater than 180 m/s. These three relationships state that *κ*_{0} decreases with increasing *V*_{S30} in different degrees of inclination, in which the relationships of Silva *et al.* (1998), Chandler *et al.* (2006), and Van Houtte *et al.* (2011) describe a larger inclination than those of Edwards *et al.* (2011) and Ktenidou *et al.* (2015). The discrepancies among these relationships diminish with increasing *V*_{S30} for the hard rock and the rock sites. The *κ*_{0}–*V*_{S30} couples show that *κ*_{0} is independent of *V*_{S30}, which differ from the existing *κ*_{0}–*V*_{S30} relationships, and imply the inadequacy of *V*_{S30} as a proxy for site attenuation in this area. Boore (2003) provided a useful way to estimate the averaged effective quality factor *Q*_{ef} within a sedimentary layer from site conditions of classes B and C. The value of *Q*_{ef} is assumed to be independent of frequency, which is supported by the evidences provided in Anderson and Hough (1984). The formula can be expressed as follows: (2)in which *H* is the thickness and *V*_{S} is the averaged *S*‐wave velocity. *κ*_{0} is inversely proportional to *Q*_{ef} for the specific *H* and *V*_{S}. In the frequency range of 2.0–6.0 Hz, Wang (1993) obtained *Q*=126, for the *S* waves in northern Taiwan. Fletcher and Wen (2005) investigate the *Q*‐value of the coda wave, *Q*_{c}, within a narrow frequency band of 0.67–1.0 Hz in the Taipei basin and surrounding area from the records of the 1999 Chichi earthquake in Taiwan. The *Q*_{c}’s vary from 29 to 324. Using the spectral ratio method, Wen *et al.* (2004) calculate the *Q*‐value of the shallow structure to a depth of ∼140 m from a downhole array in Taipei basin. A time window after the *S*‐wave arrival for computation is 10 s, indicating the minimum analytic frequency to be 0.1 Hz. Four *Q* models as functions of frequency are proposed in various depths as follows: *Q*( *f*)=3.6*f*^{0.96} for a 0–30 m depth; *Q*( *f*)=7.2*f*^{0.99} for a 30–60 m depth; *Q*( *f*)=10.2*f*^{1.17} for a 60–90 m depth; and *Q*( *f*)=40.7*f*^{1.24} for a 90–141 m depth. The average *Q*‐values are 32.5, 70.2, 156.4, and 745.8 when computing the four *Q* models in the frequency band at 0.1–20.0 Hz. Despite their suitable use of frequency, *Q* increases with diminishing depths/areas. The high *Q* results in the low *κ*_{0}. It is suggested that the *Q* for depth greater than 30 m affects the *κ*_{0}. Many sites with low *V*_{S30} located inside the basin display a similar *κ*_{0} to those with high *V*_{S30}. Furthermore, Ktenidou *et al.* (2014) compiled the existing *κ*_{0}–*V*_{S30} couples and questioned if *V*_{S30} represented the attenuation of *κ*_{0} due to the effect of regional *Q*. They also suggest that *κ*_{0} correlates with the deeper structure as well; that is, thickness of sediment above the base. The thickness of the sediment and dominant/resonant frequency overlying the Tertiary base against *κ*_{0} will be investigated in The Effective *Q* for the Alluvial Sediment and The Correlation between *κ*_{0} and the Dominant/Resonant Frequency in the Basin sections. Besides, the *κ*_{0} values of class B sites maintain an average of ∼0.05 s because of stabilization, which is meant to be a limited effect of hysteretic damping, nonlinearity, and scattering in softened sediment (Ktenidou *et al.*, 2015).

### The Effective *Q* for the Alluvial Sediment

One of the aforementioned issues involves correlating *κ*_{0} with the layer thickness to the base rock (Tertiary base rock with *V*_{S} is ∼1500 m/s). Wang *et al.* (2004) utilize the seismic reflections and then integrate several sets of borehole data to develop an image of the subsurface tomography of sediments underneath the Taipei basin. Herein, we calculate the thickness (in meters) to Tertiary base (*D*_{B}), according to the subsurface tomographic data for 28 stations, which belong to class C, D, and E sites, as listed in Table 1.

As shown in Figure 9, the *κ*_{0} values increase mildly with *D*_{B}. The open circles with one standard deviation denote the *κ*_{0}–*D*_{B} couples, which mostly distribute with *D*_{B} less than 350 m. We attempt to depict the tendency according to a linear fit of the data with *D*_{B}<350 m. The slope (*m*_{DB}) and intercept (*y*_{DB}) of a linear fit are 0.000025 s/m and 0.050 s, respectively. The standard deviations of *m*_{DB} and *y*_{DB} are 0.000020 s/m and 0.003 s. We observe that *y*_{DB}=0.050 s is close to *κ*_{0} of the three class B sites (i.e., TAP067, TAP071, and TAP086, namely 0.050, 0.049, and 0.050 s, respectively). These class B sites are located outside the basin. The increasing trend of *κ*_{0} against *D*_{B} agrees with the results of Campbell (2009) in North America and Ktenidou *et al.* (2015) in northern Greece. The two estimates provided by Campbell (2009) that include site classes B and C (table 2 and equations 22 and 23 in his study) are displayed, respectively, by a gray solid and a dashed‐dotted lines. The *κ*_{0}–*D*_{B} couples distribute greatly above the estimates of Campbell (2009) because of the site classes in use. Furthermore, the dashed line shows the tendency of Ktenidou *et al.* (2015) from a downhole array, in which *V*_{S}’s are greater than 180 m/s.

Solving *Q*_{ef} from equation (2), the results can be . Campbell (2009) gives the expression, following the approaches of Hough and Anderson (1988) and Liu *et al.* (1994), for interpreting the sediments as a uniform layer over a half‐space to be , in which *b* is the slope of the relationship between *κ*_{0} and *H*.

Ktenidou *et al.* (2015) give a similar concept after Hough and Anderson (1988) in terms of *κ*_{0sur}=*κ*_{0DH}+*t*^{*}, in which *κ*_{0_sur} and *κ*_{0_DH} are the measured *κ*_{0}, respectively, at the surface and downhole, and *t*^{*} as the attenuation of the layer in between. In their model, *t*^{*} is computed and integrated using horizontal stratified layers (*i*) with parameters, such as thickness (*H*_{i}), *V*_{si}, and *Q*_{i} by following the equation *t*^{*}=*H*_{i}/(*V*_{si}×*Q*_{i}). Herein, we assume a single geological layer with various thicknesses as a representative of the sediment overlying the Tertiary base rock. The effects regarding scattering and nonlinearity from the sediment are excluded from the sediment. Therefore, the equation can be rewritten as , in which *Q* can be considered as *Q*_{ef} and *H*/*t*^{*} denotes the aforementioned *b*^{−1}.

The *H*‐values can correspond to *D*_{B}’s in this study. Table 2 lists the *V*_{S} structure of the four sedimentary layers underneath the Taipei basin. The thickness of the sediment varies increasingly from the NW part to the SE part. The averaged *V*_{S} values are calculated by the thickness of the layer divided by the travel time, under the assumption that the seismic wave propagates vertically through the layer, and are listed in the last column of Table 2. An average shear‐wave velocity is 530.9 m/s. Substituting the and *m*_{DB}(=0.000025 s/m) for *Q*_{ef}(*z*), a computed result gives 75.3. Furthermore, given plus and minus one standard deviation on *m*_{DB}, namely 0.000045 and 0.000005 s/m, results in two values of *Q*_{ef} as 41.9 and 376.7. The calculated *Q*_{ef} values fall in reasonable ranges compared with the aforementioned estimation of Wang (1993), Wen *et al.* (2004), and Fletcher and Wen (2005), despite their suitable use of frequency. We suggest that *Q*_{ef} can represent ground‐motion simulation in the Taipei basin.

### The Correlation between *κ*_{0} and the Dominant/Resonant Frequency in the Basin

We proceed to correlate the *κ*_{0} with the dominant/resonant frequency (hereafter denoted as *f*_{r}) in the Taipei basin at 28 locations (Huang, 2009). The *f*_{r} values are measured using the microtremor array installed in the Taipei basin according to the technique of the H/V spectral ratios (Nakamura, 1989). The *f*_{r}(=*V*_{S}/4*H*) indicate where the amplitude spectral ratios are largest and are greatly correlated with the thickness of sediment above the base rock. Figure 10 shows that *κ*_{0} varies with *f*_{r}. The open circles with one standard deviation bar display the *κ*_{0}–*f*_{r} couples. The *f*_{r} values range from 0.21 to 0.99 Hz, whereas the *κ*_{0} values vary from 0.036 to 0.066 s. The black line displays the least‐squared fitting of both logarithmic axes given a slope of −0.097, which implies a slight correlation between *κ*_{0} and *f*_{r}. Additionally, the similar tendency of Van Houtte *et al.* (2011) and Ktenidou *et al.* (2015) are depicted by a dotted line and a dashed line, respectively.

## Conclusion

The high‐frequency decay parameter *κ*_{r} measured from the semilogarithmic amplitude spectra of the *S* wave is determined automatically by moving frequency windows. We apply varied frequency windows in the 5–10 Hz range at the start and 20–50 Hz when stopping. Each calculation routine for the least‐squared fitting can obtain a correlation coefficient over a frequency window of more than 10 Hz (10–20 Hz). The *κ*_{r} values are determined by the best correlation coefficient. The site‐specific *κ*_{0} values at 54 stations are computed using the anelastic effect of regional geological structure by grouping data into crustal and subduction events according to the focal depth and site classification (i.e., class B, C, D, and E sites). *κ*_{0}’s are in the 0.034–0.066 s range. The anelastic effects are larger for the crustal events than for the subduction events. The *κ*_{0} values are compared with the *V*_{S30}, displaying an independency of *V*_{S30} on *κ*_{0} and implying the inadequacy of *V*_{S30} as a proxy for site attenuation in this area. Notably, *κ*_{0} is ∼0.05 s for the rock site because of stabilization. The relationships of the *κ*_{0}–*D*_{B} couples provide a useful method for estimating the *Q*_{ef} value in a sedimentary layer with averaged *V*_{S}. According to the estimate of methodology, *Q*_{ef} is 75.3 for the whole sedimentary layer (above the Tertiary base). Given one standard deviation on the relationships of the *κ*_{0}–*D*_{B} couples, the resultant *Q*_{ef} values are in the 41.9–376.7 range. Finally, the *κ*_{0} values are also compared with the dominant/resonant frequency regarding the thickness of a sediment above the base rock. A mildly decreasing trend in *κ*_{0} with increasing dominant/resonant frequency can be observed. The tendency indicates that *κ*_{0} correlates with the several hundred meters beneath the site.

## Data and Resources

The seismic events and records used in this study are available at http://gdms.cwb.gov.tw/ (last accessed January 2014), which are a part of the database managed by the Central Weather Bureau of Taiwan.

## Acknowledgments

The authors thank the Central Weather Bureau, Taiwan, for providing high‐quality earthquake data. This study was sponsored by National Science and Technology Center for Disaster Reduction and the Ministry of Science and Technology (under Grant Number MOST 104‐2116‐M‐865‐001). The authors give thanks to Associate Editor Hiroshi Kawase and two anonymous reviewers for their valuable comments on this article.

- Manuscript received 4 March 2016.