- Article
- Open Access

# Peak ground motion predictions with empirical site factors using Taiwan Strong Motion Network recordings

- Jen-Kuang Chung
^{1}Email author

**65**:650090957

https://doi.org/10.5047/eps.2013.03.012

© The Society of Geomagnetism and Earth, Planetary and Space Sciences (SGEPSS); The Seismological Society of Japan; The Volcanological Society of Japan; The Geodetic Society of Japan; The Japanese Society for Planetary Sciences; TERRAPUB. 2013

**Received: **7 August 2012

**Accepted: **19 March 2013

**Published: **9 October 2013

## Abstract

A stochastic method called the random vibration theory (Boore, 1983) has been used to estimate the peak ground motions caused by shallow moderate-to-large earthquakes in the Taiwan area. Adopting Brune’s *ω*-square source spectrum, attenuation models for PGA and PGV were derived from path-dependent parameters which were empirically modeled from about one thousand accelerograms recorded at reference sites mostly located in a mountain area and which have been recognized as rock sites without soil amplification. Consequently, the predicted horizontal peak ground motions at the reference sites, are generally comparable to these observed. A total number of 11,915 accelerograms recorded from 735 free-field stations of the Taiwan Strong Motion Network (TSMN) were used to estimate the site factors by taking the motions from the predictive models as references. Results from soil sites reveal site amplification factors of approximately 2.0 ~ 3.5 for PGA and about 1.3 ~ 2.6 for PGV. Finally, as a result of amplitude corrections with those empirical site factors, about 75% of analyzed earthquakes are well constrained in ground motion predictions, having average misfits ranging from 0.30 to 0.50. In addition, two simple indices, *R*_{0.57} and *R*_{0.38}, are proposed in this study to evaluate the validity of intensity map prediction for public information reports. The average percentages of qualified stations for peak acceleration residuals less than *R*_{0.57} and *R*_{0.38} can reach 75% and 54%, respectively, for most earthquakes. Such a performance would be good enough to produce a faithful intensity map for a moderate scenario event in the Taiwan region.

## Key words

- Peak ground motion
- site factor
- stochastic method
- attenuation
- intensity

## 1. Introduction

Local geological conditions at specific sites are possibly the most critical factors in analyzing ground motions. One famous case is the widespread damage to buildings from strong ground motion following a particular pattern after the 1999 *M*_{w} 7.6 Chi-Chi earthquake in Taiwan (Shin et al., 2000; Shin and Teng, 2001). As shown in ~400 accelerograms recorded at free-field stations (Lee et al., 1999), stronger ground shaking appears clearly at regions well beyond the vicinity of the Chelungpu fault: for example, the Taipei metropolitan area, the Ilan plain, and some southwestern coastal regions. By comparing the response spectra of observed ground motions during this earthquake with predictions from commonly used ground motion prediction equations, Boore (2001) found that the observed motions differed from predicted by factors larger than expected from an earthquake-to-earthquake variation. The explanation could probably be attributed to specific site and propagation effects. Unless there is a clear understanding of the profile of the geological strata near the free surface, and seismograms recorded at ideal hard-rock sites are collected, it is generally difficult to isolate the site response from complicated overall effects that could mostly originate from the heterogeneity of the structure along the propagation path, and the variation of seismic energy radiated from the source. Besides carrying out geotectonic investigations over the study area, in practice a forward simulation of ground motion response seems to be an efficient method without a foreknowledge of every individual effect on the motion. However, from this perspective, the estimated ground motion can only be applied to generic sites located in a specific region and may fail to agree with observations to a large extent.

Following the Chi-Chi event, earthquake risk prevention in Taiwan remains an important issue. To achieve this goal, ground motion prediction is strategically used to establish an adequate building code for earthquake engineering and hazard assessment, based on the possible peak ground motion at a given distance from a large scenario earthquake. To find a link between the ground shaking and seismic damage, the peak ground acceleration (PGA) and peak ground velocity (PGV) induced by seismic waves in a frequency range of 0.5 ~ 10 Hz are generally simulated by statistical methods instead of theoretical or numerical modeling, since three-dimensional heterogeneities and non-uniform fault rupturing make seismograms more complex. Based on a great number of available recordings, regression techniques have been usually used to obtain prediction equations for peak ground motions in the Taiwan area (e.g. Tsai and Bolt, 1983; Tsai et al., 1987; Ni and Chiu, 1991). In such results, the behavior in attenuation of peak amplitude with distance may have a poor correlation with local site effects because the data used at that time was limited to a specific condition. In those studies, the common characters of the data used have revealed a lack of generalization of analyzed earthquakes and the localization of the recording area, without strictly considering the site categories. Even with a particular concern about the local-site response of metropolitan regions in some improved studies (e.g., Tan and Loh, 1996; Shin, 1998; Liu, 1999), the prediction equations still dominate over the regression of peak ground motions for the entire Taiwan area. Basically, this implies that the averaging effects on the source pattern and seismic attenuation along the propagation path seem to be an essential drawback for a more precise prediction of ground motion for any scenario event.

Liu and Tsai (2005) derived new ground motion prediction equations for crustal earthquakes in Taiwan using a traditional regression method. The earthquake magnitude and hypocentral distance are the only two parameters during regressions for deriving their attenuation relationships. However, another popular approach to predicting the ground motion requires a source model and a predictive relationship which allows the estimation of a specific ground motion parameter as a function of magnitude, distance, source parameters, and frequency. The predictive relationship is characterized by a geometric spreading function, a frequency-dependent crustal *Q* function, and a function describing the effective duration of the ground motion. Following the approaches described in the papers of Raoof et al. (1999) and Malagnini et al. (2000), the regional attenuation of seismic waves in Taiwan is firstly investigated in this study. The predictive relationships here were built up by a great number of strong motion data collected by the Taiwan Strong Motion Network (TSMN) operated by the Central Weather Bureau (CWB) under the Taiwan Strong Motion Instrumentation Program (Shin, 1993). Essentially, the advantage of their method over the classical regression analysis is to take into account the duration parameter as a function of frequency and distance, which is more effective in structural engineering applications.

A stochastic method called the random vibration theory was then used to estimate the peak motions of a set of random time histories. Finally, comparing the predicted peak motions with the observations, the generalized local site effect can be examined to correct the peak motions at every single station for a realistic scenario earthquake.

## 2. Procedure to Derive Attenuation Relationship

*a*(

*r, f*) is the peak amplitude of the ground acceleration recorded at the hypocentral distance

*r*and bandpass filtered at a central frequency

*f*.

*E*(

*r*

_{ref},

*f*) represents the seismic source term scaled to a fixed reference distance

*r*

_{ref}.

*P*(

*r*,

*r*

_{ref},

*f*) describes the crustal attenuation in the region, and

*S*(

*f*) is the local site term. The instrument factor is mostly eliminated when the motions are recorded by seismometers having the same response in the frequency range of interest. To obtain an empirical estimate of the path term in Eq. (1), a coda normalization technique (Aki, 1980; Frankel et al., 1990) can be used to eliminate the source and site terms by taking the ratio of the peak amplitude carried by

*S*or

*L*

_{g}waves in filtered time histories to the root-mean-square (

*rms*) amplitude of the stable seismic coda. The coda-normalized amplitude can be written in the form:where

According to Parseval’s theorem, in practice, a proper time window within the coda waves is chosen for calculating the *rms* amplitude. A better criterion indicates a length of window longer than the reciprocal of the lowest frequency of interest. The preliminary estimates of the empirical path term account for the intrinsic and scattering attenuation, geometrical spreading, and the general increase of duration with distance due to wave propagation and scattering at each hypocentral distance for each frequency. Based on the results of the coda normalization method, therefore, the *Q* quality factor model of the *S*/*L*_{g} waves propagating through the crust will be inverted by giving a proper geometrical spreading model.

## 3. Strong Ground Motion Data

*V*

_{S30}) at these rock sites, which were mostly located in the Central Range and the northern mountain area, could be summarized to be in the range 760 ~ 1,500 m/s from the

*P*-

*S*logging profiles. Strictly speaking, these sites cannot be considered as reference stations which, by definition, must have a nearly flat transfer function with an amplitude of one. Steidl et al. (1996) have shown that surface-rock sites can have a site response of their own. However, such a concern is not of relevance in this study, because the site response due to near-surface weathering of the rock can be almost extracted through a correction factor in taking the ratio of the in-situ measurement to the prediction.

List of used earthquakes selected from the CWB database.

No. | Origin Time (UT) | Epicenter | Depth (km) | Magnitude ( | Distance* (km) | Record No. | ||
---|---|---|---|---|---|---|---|---|

Lon. (°E) | Lat. (°N) | |||||||

1 | 1993/12/15 | 21:49 | 120.524 | 23.213 | 12.50 | 5.70 | 65.1 | 143 |

2 | 1994/03/28 | 8:11 | 120.687 | 22.985 | 14.37 | 5.41 | 59.6 | 84 |

3 | 1994/05/24 | 4:00 | 122.603 | 23.827 | 4.45 | 6.17 | 175.9 | 164 |

4 | 1994/06/05 | 1:09 | 121.838 | 24.462 | 5.30 | 6.20 | 100.6 | 220 |

5 | 1994/10/05 | 1:13 | 121.720 | 23.156 | 31.28 | 5.83 | 161.2 | 203 |

6 | 1995/02/23 | 5:19 | 121.687 | 24.204 | 21.69 | 5.77 | 101.4 | 293 |

7 | 1995/04/03 | 11:54 | 122.432 | 23.935 | 14.55 | 5.88 | 144.8 | 137 |

8 | 1995/05/27 | 18:11 | 121.465 | 23.008 | 19.73 | 5.26 | 105.5 | 141 |

9 | 1995/06/25 | 6:59 | 121.669 | 24.606 | 39.88 | 6.50 | 103.6 | 324 |

10 | 1995/07/07 | 3:04 | 121.090 | 23.893 | 13.07 | 5.30 | 77.2 | 162 |

11 | 1995/07/14 | 16:52 | 121.851 | 24.320 | 8.79 | 5.80 | 91.8 | 195 |

12 | 1995/10/31 | 22:27 | 120.359 | 23.291 | 10.65 | 5.19 | 45.9 | 122 |

13 | 1996/03/05 | 14:52 | 122.362 | 23.930 | 6.00 | 6.40 | 142.3 | 222 |

14 | 1996/11/26 | 8:22 | 121.695 | 24.164 | 26.18 | 5.35 | 91.8 | 191 |

15 | 1998/07/17 | 4:51 | 120.662 | 23.503 | 2.80 | 6.20 | 77.4 | 246 |

16 | 1998/11/17 | 22:27 | 120.790 | 22.832 | 16.49 | 5.51 | 60.8 | 150 |

17 | 1999/09/20 | 17:57 | 121.044 | 23.912 | 7.68 | 6.44 | 93.6 | 345 |

18 | 1999/09/22 | 0:14 | 121.047 | 23.826 | 15.59 | 6.80 | 106.8 | 385 |

19 | 1999/09/25 | 23:52 | 121.002 | 23.854 | 12.06 | 6.80 | 100.8 | 375 |

20 | 1999/10/22 | 2:18 | 120.426 | 23.515 | 16.64 | 6.40 | 101.0 | 340 |

21 | 1999/10/22 | 3:10 | 120.431 | 23.533 | 16.74 | 6.00 | 83.7 | 269 |

22 | 1999/11/01 | 17:53 | 121.725 | 23.358 | 33.53 | 6.90 | 143.4 | 407 |

23 | 1999/11/17 | 7:35 | 120.643 | 24.019 | 9.55 | 5.29 | 53.2 | 110 |

24 | 2000/02/15 | 21:33 | 120.740 | 23.316 | 14.71 | 5.59 | 66.5 | 214 |

25 | 2000/05/17 | 3:25 | 121.098 | 24.193 | 9.74 | 5.59 | 70.1 | 121 |

26 | 2000/06/10 | 18:23 | 121.109 | 23.901 | 16.21 | 6.70 | 103.7 | 432 |

27 | 2000/07/28 | 20:28 | 120.933 | 23.411 | 7.35 | 6.10 | 91.2 | 263 |

28 | 2000/08/23 | 0:49 | 121.635 | 23.636 | 27.48 | 5.57 | 99.8 | 154 |

29 | 2000/09/10 | 8:54 | 121.584 | 24.085 | 17.74 | 6.20 | 87.2 | 228 |

30 | 2000/12/10 | 19:30 | 120.226 | 23.116 | 12.02 | 5.35 | 69.3 | 165 |

31 | 2000/12/29 | 18:03 | 121.884 | 24.361 | 6.96 | 5.26 | 53.5 | 85 |

32 | 2001/03/01 | 16:37 | 120.997 | 23.838 | 10.93 | 5.80 | 75.8 | 206 |

33 | 2001/06/14 | 2:35 | 121.928 | 24.419 | 17.29 | 6.30 | 101.3 | 284 |

34 | 2001/06/19 | 5:16 | 121.077 | 23.177 | 6.58 | 5.41 | 68.9 | 116 |

35 | 2001/06/19 | 5:43 | 121.098 | 23.197 | 11.70 | 5.22 | 82.9 | 141 |

36 | 2001/12/18 | 4:03 | 122.652 | 23.867 | 12.00 | 6.70 | 169.5 | 238 |

37 | 2002/02/12 | 3:27 | 121.723 | 23.741 | 29.98 | 6.20 | 123.1 | 385 |

38 | 2002/03/31 | 6:52 | 122.191 | 24.140 | 13.81 | 6.80 | 143.6 | 421 |

39 | 2002/05/28 | 16:45 | 122.397 | 23.913 | 15.23 | 6.20 | 142.8 | 173 |

40 | 2002/08/28 | 17:05 | 121.372 | 22.261 | 12.03 | 6.03 | 90.7 | 66 |

41 | 2002/09/06 | 11:02 | 120.729 | 23.890 | 28.82 | 5.30 | 64.3 | 156 |

42 | 2002/09/30 | 8:35 | 120.614 | 23.328 | 8.12 | 5.04 | 62.2 | 174 |

43 | 2003/06/09 | 1:52 | 122.023 | 24.370 | 23.22 | 5.72 | 109.7 | 332 |

44 | 2003/06/09 | 5:08 | 121.851 | 24.380 | 2.36 | 5.03 | 57.6 | 115 |

45 | 2003/06/10 | 8:40 | 121.699 | 23.504 | 32.31 | 6.48 | 136.6 | 494 |

46 | 2003/06/16 | 18:33 | 121.654 | 23.542 | 28.26 | 5.38 | 104.2 | 231 |

47 | 2003/12/10 | 4:38 | 121.398 | 23.067 | 17.73 | 6.42 | 131.5 | 480 |

48 | 2004/05/19 | 7:04 | 121.370 | 22.714 | 27.08 | 6.03 | 117.4 | 284 |

49 | 2004/11/08 | 15:54 | 122.760 | 23.795 | 10.00 | 6.58 | 181.1 | 237 |

50 | 2004/11/11 | 2:16 | 122.158 | 24.312 | 27.26 | 6.09 | 97.3 | 132 |

51 | 2005/01/20 | 7:47 | 120.826 | 23.505 | 19.16 | 5.04 | 66.3 | 160 |

52 | 2005/02/18 | 20:18 | 121.674 | 23.340 | 15.28 | 5.60 | 95.5 | 149 |

53 | 2005/09/26 | 18:50 | 121.401 | 23.232 | 21.34 | 5.29 | 57.3 | 70 |

54 | 2006/07/28 | 7:40 | 122.658 | 23.966 | 27.97 | 6.02 | 152.7 | 104 |

## 4. Characterization of Propagation in the Crust

Each accelerogram was first narrow-bandpass filtered around each of the following frequencies: 0.2, 0.25, 0.33, 0.4, 0.5, 1.0, 2.0, 3.0, 4.0, and 5.0 Hz. The bandpass filter used here is defined by a 10th-order Gaussian function in the frequency domain. The peak amplitude of each filtered time history, carried by direct *S* or *L*_{g} waves windowed at a length of 5 ~ 10 s depending both on distance and magnitude, was then picked out. It has been pointed out that the general form of coda can be established after 2 ~ 3 times the *S*-wave travel time from the origin time of an event (Rautian and Khalturin, 1978; Spudich and Bostwick, 1987). Due to a shortening of the record length caused by the preset of the trigger mode, the length of the time window, expediently starting 20 s after the onset of an *S*-wave, for calculating the *rms* amplitude of coda waves, was set to be 10 s in this study.

*r*is the hypocentral distance.

*Q*

_{0}is the

*Qs*value at the reference frequency

*f*

_{ref}of 1 Hz and

*η*is the frequency-dependence factor. The reference hypocentral distance

*r*

_{ref}used here is 40 km and the average shear-wave velocity

*β*in the source regions is supposed to be 3.5 km/s according to the layered structure in Taiwan (Yeh and Tsai, 1981).

*g*(

*r*) is a piecewise linear geometrical spreading function. Similarly to the multi-segment models assumed in Atkinson and Mereu (1992) and Atkinson and Boore (1995), when considering dominant wave types at various distances, a bilinear geometrical spreading function was used herein. This is in the form 1/

*r*

^{ b }, where

*b*= 1.0 for

*r*< 50 km,

*b*= 0.0 for 50 ≤

*r*< 200 km. This model was used for the Taiwan region (Sokolov et al., 2000) and was also adopted by Roumelioti and Beresnev (2003), based on data averaging of a variety of site conditions from rock sites to soft soils of different thicknesses. From Eq. (4) and the assumed geometrical spreading function

*g*(

*r*), the average

*Qs*values can be estimated by giving the distances and the coda-normalized amplitudes at frequencies. These discrete anelastic attenuation factors varying with frequency were fitted with the linear function shown in Fig. 5. Therefore, the frequency-dependent

*Q*model generalized for the whole of Taiwan, as the input parameter of simulation using the random vibration theory, can be expressed as:

The quality factor increases with a frequency-dependence exponent of about 0.70, which is smaller than that derived only from reference stations between the Taipei basin and the Ilan plain for earthquakes in northeastern Taiwan (Chung, 2007). This discrepancy implies that the crustal structure in central and southern Taiwan could be more fractured than in northern Taiwan. Several previous studies (Wang, 1987; Chen, 1998; Chung et al., 2009) were reviewed and compared to outline variations in common. In general, there is no significant difference between their results and ours. However, the regionalization for the frequency-dependent *Q* factor in the Taiwan area, as shown in Chung et al. (2009), is worth considering for modeling the attenuation function in the future.

## 5. Peak Ground Motions at Reference Sites

In this study, a stochastic method was used to simulate ground motion by building functional descriptions of the ground motion’s amplitude with a random phase spectrum, as represented by Eq. (1). Then, by using the random vibration theory as initially proposed by Cartwright and Longuet-Higgins (1956), the extremes of transient earthquake ground motions could be obtained giving the Fourier amplitude spectra of the time history and signal duration. The essential aspects of this theory is described in Appendix A. A computer code (Boore, 2000) was therefore used in this study, which demands several input parameters including the seismic source spectrum, a crustal attenuation function and a distance-dependent duration function.

To model the path-independent loss of energy for ground motions at high frequencies, the spectral amplitudes predicted by source models have to be modified by multiplying a high-cut filter. A simplified diminution function exp (−*πκ*_{0}*f*), where *f* is the frequency, representing the regional average of the combined effects for rock site amplification and anelastic attenuation in a mountain region, can therefore be introduced as a high-frequency attenuation operator to describe the exponential decay of the seismic spectrum (Anderson and Hough, 1984). The *kappa* values have been determined to be in the range 0.03 ~ 0.07 s for the generic site in Taiwan (Tsai and Chen, 2000). However, a fixed *kappa* value of 0.025 s was used in this study, after analyzing the records from reference stations. This value is similar to the one inferred from California ground motion records on rock sites (Boore et al., 1992). Instead of using the Haskell matrix method, the frequency-dependent amplifications relative to the source were calculated by the root-impedance approximation (see Boore, 2003, for details) outlined in Appendix B. It needs a layered velocity model and a density model referred to the shallow crustal structure in central Taiwan, which was proposed by Chen (1995) using a three-dimensional inversion of *P*- and *S*-travel times from the CWB seismic network.

The duration of strong shaking is a function of the path, as well as the source duration that is related to the inverse of the corner frequency. The definition of the duration of ground motion in this study is given as the width of the time window that limits the 5% ~ 75% portion of seismic energy following *P*-wave arrival (Trifunac and Brady, 1975). Consequently, for our dataset, the duration increases gradually from 3.5 s for zero distance to 10.5 s at about 100 km of recording distance.

As regards the source spectrum in Eq. (1), the typical *ω*-square model (Aki, 1967; Brune, 1970) with a single-corner frequency was adopted in this study. The radiation coefficient averaged over some portion of the focal sphere is assumed to be 0.55 if the partition factor of the *S*-wave energy is taken to be
(Boore and Boatwright, 1984). The stress drop of moderate seismic sources in the range from 10–100 bar was derived by Chiang (1994). For some moderate-to-large earthquakes, a wider range (3~180 bar) of stress drops was suggested through a waveform inversion analysis (Wu, 2000). In Wu’s study, an average stress drop of 30 bar has been obtained, which was therefore chosen as the input parameter in the predictive model here.

*M*

_{W}= 0.99

*M*

_{L}+ 0.052 (Wu, 2000), the predictions of the peak horizontal ground motions (PGA and PGV) at the reference sites for local magnitudes

*M*

_{L}= 4.0, 5.0, 6.0, and 7.0, respectively, are shown in Fig. 7. The change at a hypocentral distance of 50 km in these curves is attributed to the change at this distance defined in the bilinear geometrical spreading function. The predicted motions for shear waves, particularly at distances greater than 30 km, are much lower than previous simulations which were based on the statistical regression of peak values taken from all the recordings made at generic stations (e.g., Chang et al., 2001; Liu and Tsai, 2005). It is to be expected that the differences would be reduced if only data from rock sites had been used in these studies, even for a different set of earthquakes. This shows that the site amplification effects on free-field ground could predominate in the predictions. In addition, the peak motions used in these studies were probably picked from surface waves generated from some very shallow major seismic sources. Readings not arising from shear waves cannot essentially be modeled by a general source spectrum as used here. In order to obtain recorded PGVs, the time histories of ground-particle velocity were derived, using integration in the frequency domain and a bandpass filtering of 0.1 ~ 20 Hz, from recorded accelerograms. For evaluating the proposed model, the residual values, which are defined as the natural logarithm of the ratio of the observed value to that predicted, are illustrated as a function of the hypocentral distance (Figs. 8(a) and 8(c)). Their distribution patterns confirm that the parameters related to the attenuation behavior of seismic waves are appropriate in the prediction.

## 6. Distribution of Site Factors

With the advantage of island-wide deployment of free-field accelerometers under the 6-year instrumentation program conducted by the CWB since 1992, the amplification of seismic waves at each station due to soil subsurface layers can be obtained based on the number of observations and our prediction model shown in Fig. 7. In this study, the amplification of a seismic wave, strictly speaking, is defined by a relative factor depending on the reference derived from the empirical modeling, and can be properly called the site factor. It is defined as the average of the ratios of the observed peak value to that predicted for all used data. In addition, they are not linked to any particular ground motion frequency, because their calculation is made without considering the frequencies of PGA or PGV. By multiplying the predicted values by the corresponding site factors, therefore, the residuals reduce to a lower level with a zero mean (Figs. 8(b) and 8(d)).

The site factors estimated in the mountain regions, including the Longitudinal Valley, on the other hand, are in the range of 0.9 to 1.5 for PGA and 0.8 to 1.3 for PGV, respectively. It is obvious that constrained observations in the broad Central Range were limited at the reference sites, through a lack of information regarding the site response is not critical for hazard mitigation in such sparsely populated areas.

## 7. Residual Analyses with Site-Factor Correction

_{ln}) of a set of residuals can be defined as follows:where

*N*is the number of residuals for one earthquake. The geometrical mean of NS- and EW-component peak values was taken to compare with the predicted value (

*A*

_{0}) for each record.

Seismic intensity at local areas is undoubtedly the information of most concern for the public. Another way to evaluate the reliability of peak ground motions predicted here is to take count of recording stations, whose seismic intensities can be exactly predicted, for determining the performance rate in the dissemination of intensity information. In the CWB’s intensity scale, a seismic intensity (*I*) of under 7 was defined by the relationship (Hsu, 1979), log *A*_{
I
} = (*I*/2) − 0.6, which approximately correlated with the threshold of peak ground acceleration (*A*_{
I
}) in gals recorded by the accelerograph. Additionally, PGAs larger than 400 Gals were categorized into intensity 7. From a statistical viewpoint, a probability between 50% to 100% of a correct intensity prediction could be expected based on the assumption that the PGA residuals have an equal probability of appearance within the considered error ranges. For example, even for a PGA residual close to zero, only a slightly larger probability than 50% could be expected to obtain the correct intensity using the predicted PGA, when the observed PGA is just the threshold value crossing two consecutive intensity values, say 80 Gals, that defines the boundary between intensity 4 and 5. If the observed PGA, say 140 Gals (classified to intensity 5, whose PGA ranges from 80 gals to 250 gals), is in the middle of two adjacent threshold values of intensity classification, however, up to a probability of 100% could be expected as long as the PGA residual is small enough. It is thus clear that the probability of a correct intensity prediction obviously depends both on the observed PGA and the prediction residual.

*R*

_{0.57}and

*R*

_{0.38}, were therefore proposed in this study to evaluate the intensity predictions as follows:where the intensity

*I*ranges from 1 to 5. A reliable intensity map for the observed area can objectively be obtained when all the PGA residuals are smaller than the qualified ones. In practice, either qualified residual can be used as a reference to assess the results of intensity prediction. That is totally a probabilistic subject. In this study, the percentages of qualified stations, whose absolute values of the residual of peak accelerations are smaller than the qualified residuals, are shown in Fig. 12(b). Neglecting the data from a few earthquakes having a misfit larger than 0.65, the average percentages of qualified stations for peak acceleration residuals less than

*R*

_{0.57}and

*R*

_{0.38}can attain (75 ± 8)% and (54 ± 10)%, respectively, for most events. Such a performance would be good enough to produce a faithful intensity map for a moderate scenario earthquake in the Taiwan region. An example is shown in Fig. 13. For the 1999 Chiayi earthquake (

*M*

_{L}= 6.40), PGA residuals smaller than

*R*

_{0.57}were obtained at 282 stations (about 83% of a total of 340 triggered stations). In those qualified stations, 217 correct intensity predictions were performed which is 77%. Due to the high percentage of qualified stations, the isoseismal map derived from predicted PGA values is very similar to that contoured using observed PGA values.

## 8. Discussion and Conclusions

Strong motion prediction is an important objective of seismic hazard assessment, particularly in the Taiwan region. As shown in previous historical data, however, peak earthquake ground motions can vary over a large range due to significant effects in terms of source, path, and site. Taking into account the advantage of flexibility in establishing a model for simulation, a stochastic method, which has been verified to be efficient, has been applied in this study to estimate the peak ground motion parameters of random time histories. Moreover, parameter analyses concerned with spectral amplitude, if necessary, can be carried out resulting in a greater understanding of ground motion.

First of all, for constructing a functional spectrum of ground motion, the generalized frequency-dependent *Q* model (Eq. (5)), for describing the anelastic attenuation of seismic waves with distance for the crust in Taiwan, has been obtained using a coda normalization method. It is basically in good agreement with that proposed by Chung et al. (2009). Although the coda *Q* from their investigation appears to be strongly dependent on the geology and tectonic features, it is still ambiguous as to how the spatial variations of the *Q* value act on the amplitude and frequency content of seismic waves propagating through different materials. Such a problem related to path behavior would be very time-consuming to resolve in detail even if we knew how to achieve this. In the current stage, an averaged *Q* model is reasonably satisfactory when applied for the rapid publishing of a preliminary map of strong shaking.

Secondly, the derived prediction models show acceptable results regarding the estimation of peak ground motions for shear waves at selected rock sites. Small biases relative to reference values may also be well corrected by empirical site factors for reducing the residuals to a generally lower level compared with previous results (e.g., Liu and Tsai, 2005). Some reference stations, however, exhibit a larger amplification in ground motion, probably due to the weathering condition of surface rock or near-site path effects characterized by a kappa value much smaller than 0.025 used in the model. For those known anomalous sites situated on hard ground or soft alluvium, more detailed investigations are necessary to determine their local area anomaly.

Based on reference ground motions, the site factors in mountain regions, including the Longitudinal Valley, are in the ranges 0.9 to 1.5 for PGA and 0.8 to 1.3 for PGV. The amplification factors expectably increase to about 2.0 ∼ 3.5 for PGA, and 1.3 ∼ 2.6 for PGV, at soil sites located in the Taipei area, the Ilan plain, and western coastal regions having large populations. Using these factors for modifying the predictions, a relatively smaller misfit range from 0.30 to 0.50 can be determined for most of the 54 analyzed shallow earthquakes. These results basically indicate that a preliminary isoseismal map for rapid reporting could be produced by the procedures implemented in this study. However, a large misfit, having an obvious systematic deviation of ground motion, can be shown in the predictions for some of the events. The possible reason causing such a type of residual pattern might be attributed to the stress drop of a seismic source which could be very different from the value of 30 bar used in this model. It appears to be difficult to determine the stress drop in a short time for simulating a better prediction of ground motion for the purpose of the rapid publication of the assessment of ground shaking hazards.

The proposed predictive model is reliable for moderate earthquakes (*M*_{L} < 7) with a point source mechanism. However, two major uncertainties of peak ground motion prediction arise from the presence of strong surface waves, as well as a large earthquake with an extended rupturing source. This apparently implies that distant, or very shallow, earthquakes which can usually excite strong surface waves dominated by energy of a relatively low frequency must both be of large magnitude and long fault rupturing (Chung, 1995; Chung and Yeh, 1997; Chang et al., 2002). Such problems, therefore, are certainly subjects to be considered in the next stage of investigations relating to ground motion predictions.

## Declarations

### Acknowledgments

I would like to thank Dr. C.-H. Chang of the Central Weather Bureau of Taiwan for archiving the accelerograms used in this study. This article has been considerably improved from its original form as a result of insightful reviews by two anonymous reviewers. Simulation of the ground motion was carried out using the **SMSIM** program ver. 2.18 written by Dr. D. M. Boore of U.S. Geol. Surv., which is available at his website http://www.daveboore.com/software_online.htm (last accessed July 2005). This research was primarily supported by the National Science Council, Taiwan, under the grants NSC 97-2745-M-231-001 and NSC 98-2119-M-231-002. I would also like to acknowledge the support of the Central Weather Bureau via Grant MOTC-CWB-98-E-09.

## Authors’ Affiliations

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