# Bulletin of the Seismological Society of America

## Abstract

Conditional mean spectrum (CMS) is increasingly being accepted as a more reasonable target spectrum for selecting earthquake records than uniform hazard spectrum. Construction of CMS requires the determination of correlation coefficients of spectral accelerations. This article presents the first investigation of correlation coefficients based on earthquake records from China. First, an original methodology to verify the appropriateness of a ground‐motion prediction equation (GMPE) to determine correlation coefficients is proposed and illustrated considering seismic hazard in China. A dataset of ground motions recorded from 2007 to 2014 in China, with magnitudes ranging from *M*_{s} 5.5 to 7.0 and epicentral distances less than 200 km, is then selected and analyzed to obtain correlation coefficients based on a newly developed GMPE. The obtained results indicate the same trends as the prediction model proposed by Baker and Jayaram (2008) based on Next Generation Attenuation‐West ground motions and GMPEs. The sensitivity of correlation coefficients to magnitude is also verified and the obtained results are provided in comprehensive tables. The results of this work provide the essential data required to develop CMS considering seismic hazard in China.

## Introduction

The shape of uniform hazard spectrum (UHS) is often considered to be unrealistic for a given site, especially when the spectral ordinates at different periods are governed by different earthquake scenarios. Furthermore, the spectral ordinates of UHS for long return periods are considered to be conservative (Naeim and Lew, 1995; Bommer *et al.*, 2000). To address these issues, Baker and Cornell (2006) introduced the conditional mean spectrum (CMS), which considers the condition that different spectral demands match target amplitudes at different periods. The difference between the target spectral acceleration and the spectral acceleration given by a ground‐motion prediction equation (GMPE) at the same period *T* is estimated by the parameter corresponding to the number of standard deviations associated with this GMPE. A particularly important parameter in the construction of CMS is the interperiod correlation coefficient between and values taken at two given periods *T*_{1} and *T*_{2}. This correlation coefficient provides the shape of the CMS in the period range of interest. A number of empirical equations have been proposed to estimate interperiod correlation coefficients considering seismic hazard in different regions (Inoue and Cornell, 1990; Goda and Atkinson, 2000; Baker and Cornell, 2005; Baker and Jayaram, 2008; Cimellaro, 2013; Lin *et al.*, 2013, Daneshvar *et al.*, 2015). One of the most commonly used correlation models was developed by Baker and Jayaram (2008) using four different Next Generation Attenuation (NGA)‐West GMPEs and considering the ground‐motion records from NGA‐West ground‐motion library.

China is an active seismic zone that experienced several major devastating earthquakes such as the 1970 Tonghai earthquake (*M*_{s} 7.3), the 1976 Tangshan earthquake (*M*_{w} 7.5), the 2008 Wenchuan earthquake (*M*_{w} 7.9), and the 2013 Lushan earthquake (*M*_{s} 7.0). In recent years, severe damage and loss have been caused by destructive earthquakes even though structures were designed according to the current seismic provisions. In this context, performance‐based seismic design and evaluation of rapidly expanding infrastructure are gaining wide popularity in China. Such approaches require tools for selection of the ground‐motion and structural response prediction, such as the CMS and relevant input parameters. However, the majority of the CMS‐related studies discussed above have been conducted considering the seismic activity of western North America (WNA) or regions other than China. In particular, there are currently no correlation models developed based on the specific seismic hazard in China. In the absence of correlation models that consider the specific characteristics of ground motions in this region, the applicability of available correlation models from other areas has not been addressed in the literature.

The main objective of this article is to estimate interperiod correlation coefficients for seismic analyses in China. First, an approach to evaluate the applicability of selected GMPEs to compute correlation coefficients is proposed. Correlation coefficients in the 0.01–2.0 s period range are then determined based on ground motions recorded in China between 2007 and 2014, with magnitudes 5.5≤*M*_{s}≤7.0 and epicentral distances less than 200 km. The obtained results are compared with the widely used prediction model of Baker and Jayaram (2008). The sensitivity of the obtained correlation coefficients to magnitude is also discussed.

## Methodology

### Review of the Computation of Correlation Coefficients

Baker and Cornell (2006) incorporated the effect of in the CMS target spectrum, developing the CMS‐ that accounts for the spectrum shape. The target spectral acceleration SA(*T*^{*}) at the desired period *T*^{*}, usually taken as the first vibration period of the studied structure, is generally determined from the probabilistic seismic‐hazard analysis (PSHA) at a given site considering a target probability of exceedance. The mean (or the modal) value of magnitude *M*, as well as the epicentral distance *R* associated with SA(*T*^{*}) can be obtained from the deaggregation results of PSHA. The difference between the target SA(*T*^{*}) and the spectral amplitude predicted by a selected GMPE is evaluated by the number of standard deviations given by (1)in which the term *μ*_{logSA}(*M*,*R*,*T*^{*}) represents the mean value of log SA predicted by the selected GMPE at target period *T*^{*} and *σ*_{logSA(T*)} is the standard deviation in logarithmic units provided by the GMPE. Given the value of SA(*T*^{*}), the conditional mean value of SA at other periods *T* can be calculated as follows: (2)(Baker, 2011), in which *μ*_{logSA}(*M*,*R*,*T*) and *σ*_{logSA(T)} are, respectively, the mean and standard deviation of log SA(*T*) as predicted by the selected GMPE, and is the correlation coefficient for at periods *T* and *T*^{*}, that is, . The correlation coefficient at different periods *T*_{1} and *T*_{2} can be determined from empirical observations using the Pearson product moment as (3)in which *n* is the number of observations, and are the *i*th observations of and , respectively, and and are their mean values for the whole set of observations, respectively. This calculation is repeated for each period pair (*T*_{1}, *T*_{2}) of interest. The correlation coefficients can be tabulated and used through a lookup table. A corresponding empirical equation can also be developed as proposed by Baker and Jayaram (2008). Two important steps in the determination of correlation coefficients are the selection of ground‐motion records and GMPE, which will be discussed next.

### Selection of Ground‐Motion Records

The original set of ground‐motion records used in this study to determine correlation coefficients consists of 6636 group triaxial records, extracted from a dataset (2007–2014) provided by China National Strong Motion Networks Observation Center (see Data and Resources). The records are selected based on the following criteria.

The site is classified as soil.

The records are measured in the free field or the first story of a structure.

The magnitude of the earthquake event is greater than

*M*_{s}5.5.The epicentral distance is less than 200 km.

A majority of earthquake recording stations in China are classified as soil or rock without borehole data or detailed information. There are only 445 groups of records from 129 rock stations among all triggered 1290 stations in this dataset, and the number is more limited considering the constraints of magnitude and distance; therefore, soil sites are selected in this article. The lower‐bound magnitude of *M*_{s} 5.5 and upper‐bound epicentral distance of *R*=200 km are chosen to include only records with engineering‐significant effects on structures. Considering the period‐sensitive reliability of recorded ground motions at long periods (Boore and Akkar, 2003; Boore, 2005), only the records with high‐pass filter corner frequencies of less than 0.5 Hz were selected. Therefore, the values and correlation coefficients will be computed at periods *T* between 0.01 and 2.0 s.

Applying the previous selection criteria resulted in a total number of 767 pairs of two horizontal components; that is, 1534 records. All these records were sampled at a rate of 200 samples/s or higher, except for 88 records that were sampled at a rate of 100 samples/s. Considering that this study focuses on a period range between 0.01 and 2.0 s, we excluded the 88 records from the dataset used for computation of correlation coefficients. A list of the remaining 1446 records corresponding to 54 earthquake events is provided in Table 1. The selected ground motions are left oriented as recorded rather than rotated into fault‐normal and fault‐parallel components to effectively consider the random orientation with respect to fault direction (Baker and Cornell, 2006). Figure 1 shows the distributions of magnitudes *M*_{s} of the selected ground motions versus epicentral distances and the number of records for each magnitude bin. We note that the mainshock of Wenchuan earthquake (*M*_{s} 8.0) is excluded from the dataset to ensure that results are not excessively influenced by any peculiarities from this single major earthquake. The number of records within different magnitude windows shows that the distribution of selected ground‐motion records is generally uniform for 5.5≤*M*_{s}≤7.0.

### GMPEs Used for Computation of Correlation Coefficients

The computation of correlation coefficients requires the use of a GMPE as described previously. However, there are currently only a few GMPEs directly derived from available ground motions recorded in China (Wang and Xie, 2009; Lu *et al.*, 2010; Zhang *et al.*, 2013). Most of these GMPEs used the recordings from some specific earthquake events; that is, Wenchuan and Lushan earthquakes. This is partly due to the sparse density of seismic stations over the territory and the relatively limited number of records. Some widely used GMPEs in China were proposed by Huo (1989) for southwest China, Yu and Wang (2006) for western China, Lei *et al.* (2007) for the Sichuan basin, and Yu *et al.* (2013) for the fifth‐generation Chinese seismic zonation map. These available GMPEs were developed using an intensity conversion method suitable for regions with no or insufficient ground‐motion data as China (Hu and Zhang, 1983). The GMPE proposed by Huo (1989; hereafter, HUO89) is selected in this work as it is the most widely used and validated GMPE for PSHA studies in China (Wen and Ren, 2014). To develop this GMPE, Huo (1989) used empirical conversion method of ground‐motion parameter from intensity with the records and intensity data of western America. The functional form of HUO89 is expressed as (4)in which *a*_{0} to *a*_{4} are regression coefficients. The obtained coefficients *a*_{0} to *a*_{4} are given in Table 2 as well as the standard deviation *σ*_{logSA(T)} for logSA(*T*).

HUO89 GMPE is widely used for seismic‐hazard assessment in China. However, in addition to being based on an empirical approach, it does not account for modern seismic data, namely ground motions recorded between 2007 and 2014. Therefore, a new GMPE fitted on modern records is proposed next to investigate possible bias in obtained correlation coefficients. For this purpose, 5282 records with the magnitude–distance distributions illustrated in Figure 2 are selected. This ground‐motion dataset is the same as described before except that records from magnitudes 3.0–5.5 are also included for a more complete assessment. The sites selected are classified as soil as previously mentioned.

The proposed functional form of the fitted GMPE is (5)in which coefficients *a*_{0} to *a*_{3} are derived using a regression based on the least‐squares method. The coefficients *a*_{2} and *a*_{3} are introduced to account for geometric attenuation and distance saturation. For simplicity, the source mechanism and site condition are neglected herein. The obtained coefficients *a*_{0} to *a*_{3} are given in Table 3 as well as the standard deviation *σ*_{logSA(T)} for logSA(*T*).

### Evaluation of the Distribution of Values Obtained Using Selected GMPEs

According to equation (1), when the predictions of the GMPE selected to compute correlation coefficients coincide with the records used, the distribution at a given period *T* can be characterized as a standard normal distribution with a zero mean (i.e., *μ*=0), and a unit standard deviation (i.e., *σ*=1) (Baker, 2011). When a different GMPE is used, the resulting values are then likely to be biased from a standard normal distribution. However, if the obtained values can still be modeled as a normal distribution, calculations show that the related correlation coefficients are not significantly affected. The reason behind this result stems from the insensitivity of Pearson’s product moment to linear transformations of random variables, that is, , in which *a*, *b*, *c*, and *d* are constants with *b*≠0 and *d*≠0. These results are illustrated next using data recorded from the 2013 *M*_{s} 7.0 Lushan earthquake (Wen and Ren, 2014), which occurred in Sichuan Province of China, as an example. For this purpose, a simplified *ad hoc* GMPE is developed based on 31 bins of two horizontal ground‐motion components recorded during Lushan earthquake within an epicentral distance of less than 200 km. The functional form of the simplified GMPE is selected as (6)in which coefficients *a*_{0}, *a*_{1}, and *a*_{2} are derived using a regression based on the least‐squares method, and in which the coefficient *a*_{1} is introduced to account for geometric attenuation. In what follows, this simplified GMPE will be referred to as the Lushan earthquake‐based GMPE.

According to equation (1), the values computed using the Lushan earthquake‐based GMPE are expected to correspond to a standard normal distribution as it is directly fitted to the observed records. This result is confirmed in Figure 3a, showing that the distribution of the obtained has a zero mean *μ*=0 and a unit standard deviation (i.e., *σ*=1). For comparison purposes, the distribution of values computed based on the HUO89 GMPE and our proposed fitted GMPE (equation 5) are presented in Figures 3b and 3c, respectively. It is clearly seen that the mean *μ* and standard deviation *σ* of the obtained distributions have different degrees of bias from zero and unity, respectively, for practically all periods in the range of interest.

Next, the Kolmogorov–Smirnov (K‐S) goodness‐of‐fit test is proposed to verify the null hypothesis that the computed values can be represented by normal distributions. The *p*‐values obtained from various GMPEs described above are depicted in Figure 3, as well as the contour lines of correlation coefficients. The resulting high *p*‐values for a 0.01–2.0 s period range show that the above null hypothesis cannot be rejected at a significant level *α*=1%. The results also confirm that, as justified before, the correlation coefficients are practically the same for the three GMPEs over the period range of interest. We note that this conclusion was also reached when studying other recorded events in China besides Lushan earthquake.

Finally, the distribution of values is computed for the 1446 records described previously using HUO89 GMPE and our proposed GMPE (equation 5) at periods *T* between 0.01 and 2.0 s. The resulting values are shown in Figure 4 along with the corresponding *p*‐values from K‐S goodness‐of‐fit test to evaluate whether data can be represented by a normal distribution. The obtained *p*‐values are higher than the significance level *α*=1%, except around a period of 0.07 s for HUO89 GMPE. Although this slight discrepancy would have no significant effect on the computed correlation coefficients, the proposed GMPE in equation (5) will be used next to determine correlation coefficients corresponding to the earthquake database presented in the Selection of Ground‐Motion Records section.

## Obtained Correlation Coefficients and Discussions

### Observed Correlation Coefficients and Comparison with Predictions

In this section, the correlation coefficients are computed for the 1446 records described before using the proposed fitted GMPE (equation 5) at periods *T*_{1} and *T*_{2} between 0.01 and 2.0 s. The obtained results are provided in Figure 5. As mentioned previously, one of the most commonly used correlation models was proposed by Baker and Jayaram (2008; hereafter, BJ08), using four different NGA‐West GMPEs and corresponding shallow crustal ground‐motion records from NGA‐West ground‐motion library. The period range of the BJ08 correlation model is from 0.01 to 10.0 s. Its applicability to regions other than WNA was investigated by (1) Jayaram *et al.* (2011) using records from Japan, including subduction ground motions, (2) Cimellaro (2013) using records from Pan‐European earthquake database, and (3) Daneshvar *et al.* (2015) using records typical of eastern North America. In these studies, the BJ08 correlation model was shown to provide satisfactory results for the different regions considered. To the authors’ knowledge, however, there is no published work investigating the applicability of BJ08 correlation model to records from China. Such investigation is conducted next by comparing the predictions of BJ08 model, illustrated in Figure 5b, to those obtained previously considering the 1446 records using the proposed fitted GMPE (Fig. 5a).

Comparison of the results in Figure 5 shows that the correlation coefficients obtained using the BJ08 model are quite similar to those computed based on fitted GMPE. In both cases, it is observed that the correlation coefficients approach one when the two periods *T*_{1} and *T*_{2} are nearly equal, whereas they decrease as the periods become further apart from each other. Such observation was also reported in other studies (Baker and Cornell, 2006; Daneshvar *et al.*, 2015).

To further compare these results, Figure 6a and 6b shows, respectively, observed correlation coefficients and predictions of the BJ08 model for a selected set of periods *T*_{2}, plotted as a function of period *T*_{1} varying between 0.01 and 2.0 s. A close agreement is observed between both types of results.

To visually compare correlation coefficients for different combinations of periods, the contour map of the difference between observed data and predictions of the BJ08 model is obtained by Kriging interpolation, as illustrated in Figure 7. In the 0.01–2.0 s period range, the overall absolute difference is between −0.15 and 0.15, and more than two‐thirds of it is within ±0.05. Globally, no statistically significant differences are identified as the confidence interval is wider for low correlation values (Kutner *et al.*, 2004). In other words, a difference of 0.1 or 0.2 in this region has little impact on the final result considering the large number of selected records in this article.

Carlton and Abrahamson (2014) found that correlation coefficients might be sensitive to the high‐frequency content of ground‐motion records. They proposed *T*_{amp1.5}, the shortest period at which SA(*T*) reaches 1.5 times the peak ground acceleration to approximate the peak period (*T*_{p}) of the response spectrum. *T*_{amp1.5} is used as a measure to determine the period at which the effects of high‐frequency content on spectral amplitudes would be predominant. At periods smaller than *T*_{amp1.5}, the epsilon values will be more correlated with the epsilon values of *T*_{p}, and hence more correlated with epsilon values of periods greater than *T*_{p}. Carlton and Abrahamson (2014) also concluded that a satisfactory agreement is obtained between predictions of BJ08 model and correlation coefficients obtained from earthquake sets having *T*_{amp1.5} close to 0.1 s, which corresponds to the *T*_{amp1.5} of the ground‐motion database used to develop the BJ08 correlation model. In the opposite case (i.e., *T*_{amp1.5} of studied ground records different from 0.1 s), Carlton and Abrahamson (2014) proposed modifications to the BJ08 model to improve the predictions. In the present case, the *T*_{amp1.5} of the studied records is computed as 0.09 s, which is close to *T*_{amp1.5}=0.1 s. Accordingly, no modifications are required to apply the BJ08 correlation model in this region.

### Magnitude Dependence of Correlation Coefficients

The BJ08 correlation model assumes that correlation coefficients are independent of the magnitude of ground motions, based mainly on the observations by Baker and Cornell (2005). However, Daneshvar *et al.* (2015) investigated the applicability of this assumption to typical ground motions of eastern North America, and concluded that records of lower magnitude demonstrated higher correlations at short periods for longer conditioning periods *T*.

Next, a mathematical derivation is provided to investigate the influence of magnitude variations on correlation coefficients. For simplicity, the proposed fitted GMPE in equation (5) is used to compute spectral correlations as follows. For a given earthquake event *i*=1,2,…, the spectral predictions at an arbitrary period *T*_{1} for *n* sites logSA(*T*_{1})_{ij}, *j*=1…*n*, with source‐to‐site distances *R*_{ij}, can be expressed according to equation (1) as (7)or in condensed form, (8)in which denotes the number of standard deviations for event *i* at source‐to‐site distance , and *σ*_{logSA}(*M*^{(i)},*T*_{1}) is the standard deviation provided by the GMPE for this same event. Similar relationships can be written at another arbitrary period *T*_{2}: (9)The mean values of logSA at periods *T*_{1} and *T*_{2} can be approximately related by the linear equations (10)Equation (9) can then be rewritten as (11)For convenient notation, we introduce the scalars and , as well as vectors (12)Equations (8)–(11) can then be simplified as (13)The correlation coefficients for spectral accelerations can be expressed as (14)in which the notations *D*(.) and *E*(.) are used to denote variance and expectation, respectively.

As mentioned before, the values of correspond to a standard normal distribution with a zero mean and a unit standard deviation when the predictions of the GMPE selected to compute correlation coefficients coincide with the records used. Assuming that the GMPE in equation (5) satisfies such conditions yields and for *i*=1,2,…. Considering that and are independent of , that is, , equation (14) can then be rewritten as (15)Because is usually considered to approximately equal (Baker and Cornell, 2006), equation (15) yields (16)Equation (16) clearly shows that the correlation coefficient generally depends on magnitude with corresponding parameters *k*^{(i)}, , and . This equation can also be used to evaluate the degree of dependency between correlation coefficient and magnitude as follows. When *T*_{1} is close to *T*_{2}, the ratio *k*^{(i)} corresponding to different magnitudes approaches one and the corresponding parameters and are approximately equal. Therefore, earthquake magnitude does not have a marked impact on in the high‐correlation region illustrated in Figure 5. On the opposite, when *T*_{1} is far from *T*_{2} (i.e., low‐correlation region in Fig. 5), the parameter *k*^{(i)} departs from unity for events with significantly different magnitudes and the corresponding parameters and are not necessarily equal. In this case, the correlation coefficient is expected to be more sensitive to magnitude. Although the equations are derived based on a simple GMPE model, these conclusions can be extended to more complex attenuation models. In the next section, the earthquake data described previously are subdivided into different magnitude groups to evaluate the effect on correlation coefficients.

### Correlation Coefficients for Different Magnitude Groups

The ground motions are classified based on corresponding magnitude as three groups: **M** 1 (5.5≤*M*_{s}<6.0), **M** 2 (6.0≤*M*_{s}<6.5), and **M** 3 (6.5≤*M*_{s}≤7.0). The applicability of the GMPE in equation (5) is verified for each magnitude group using the proposed approach. The distribution and *p*‐value of K‐S test for different magnitude groups are shown in Figure 8. The *p*‐value for different magnitude groups are all above a significant level of 0.01. Therefore, the set of values obeys the normal distribution and the results computed using the proposed GMPE are reliable.

The correlation coefficient for each group is computed separately as shown in Figure 8. In regions with high correlation, that is, , there is no noticeable difference. However, differences are more significant in regions with lower correlation, especially for the **M** 2 group. To clearly compare the difference between the three magnitude bins, Figure 9 shows observed correlation coefficients for different magnitude groups and results of the BJ08 model for a selected set of periods of *T*_{2}, plotted versus *T*_{1} values between 0.01 and 2.0 s. The global trend is basically consistent with predictions of the BJ08 and different magnitude bins, as shown in the separated magnitude group. When *T*_{1} and *T*_{2} are close to each other, there is no significant difference between magnitude groups. The dependency of obtained magnitude is pronounced as one of the two periods *T*_{1} or *T*_{2} and is shifted toward the longer period or shorter period. The correlation coefficients from ground motions in **M** 2 bin are smaller than the predictions of the BJ08 model, whereas the results of **M** 3 bins are larger in the low correlation region. The results of **M** 1 bins are the closest to the predictions of BJ08 over the whole period range studied. The good agreement can partly be related to the number of records in **M** 1 bin being approximately half of the records in the whole database selected. The observed correlation coefficients for different magnitude bins are provided in the Appendix. Such information could be efficiently used in construction of CMS and ground‐motion selection in China.

## Conclusions

This article presented the first investigation of correlation coefficients considering seismic hazard in China. An original approach was proposed to verify the appropriateness of using a given GMPE to determine correlation coefficients. The proposed procedure was illustrated using ground motions from China, as well as existing and newly developed GMPEs. Correlation coefficients in a 0.01–2.0 s period range were then determined using records between 2007 and 2014, with magnitudes 5.5≤*M*_{s}≤7.0 and epicentral distances less than 200 km. The observed correlations were compared with the widely used BJ08 prediction model proposed by Baker and Jayaram (2008), indicating the same trends over the period range studied. Based on the global observed satisfactory agreement between predicted and computed correlation coefficients, we conclude that the BJ08 model can be used to develop CMS in China. A mathematical approach was proposed to investigate the sensitivity of correlation coefficients to magnitude, which is also verified by comparing observed correlation coefficients corresponding to different magnitude bins. The magnitude sensitivity was found to be pronounced in low‐correlation regions where periods *T*_{1} and *T*_{2} are far apart. The final correlation results for different magnitude bins were provided in the Appendix and can be used for ground‐motion selection and construction of CMS in China. The conclusions of this work are drawn based on current available ground motions recorded in China and they need to be validated in light of future observations.

## Data and Resources

Strong‐motion recordings in this article were obtained in China National Strong Motion Networks Observation Center (http://www.csmnc.net or http://222.222.119.9; last accessed February 2017).

## Appendix

The computed correlation coefficients for different magnitude bins using Chinese records (Tables A1–A4).

## Acknowledgments

This work has been supported by the Nonprofit Industry Research Project of China Earthquake Administration under Grant Number 201508005, Chinese National Natural Science Fund Grant Number 51308515, and Science Fund of Heilongjiang Province Number LC2015022.

- Manuscript received 15 September 2016.