## 3 Ground Motion Selection Techniques
### Selection Based on Spectral Values  
In a building code framework, spectral matching, which involves matching the response spectrum of candidate records with the target spectrum, is recommended. This target spectrum is characterized by a smooth design spectrum at different periods. It is possible to verify candidate response spectra with target records using different approaches ([PERSON] et al. [74], [PERSON] et al. [75]). [PERSON] et al. [74] proposed Equation (21) to verify the spectral compatibility of records. \(D_{rms}\) considers the difference of spectral values between a candidate and target spectrum. The thresholds for \(D_{rms}\) depend on the period range of interest and number of records required. Alternative approaches involve the scaling of spectral values at longer periods ([PERSON] and [PERSON] [76]). \[D_{rms}=\frac{1}{N}\sqrt{\sum_{i=1}^{N}\left(\frac{Sa_{O}(T_{i})}{ PGA_{O}}-\frac{Sa_{s}(T_{i})}{PGA_{s}}\right)^{2}} \tag{21}\]  
A building code's design spectrum is an approximate version of the Uniform Hazard Spectrum (UHS), which is more common in practice. The UHS is calculated in the PSHA framework using spectral accelerations at different periods for the same hazard levels. [PERSON] and [PERSON] [48], however, argue that UHS is ineffective for selecting ground motions, and instead propose a conditional mean spectrum (CMS). [PERSON] and [PERSON] [48] state that the UHS does not represent the spectrum of any individual ground motion; hence, it is unlikely that an individual ground motion will exceed the hazard level at all periods. Therefore, building designs based on UHS are more conservative. To propose CMS, [PERSON] and [PERSON] [48] used deaggregation analysis of PHSA. This information pertains to earthquake events (\(M_{w}\), \(R\) and \(\epsilon\)) that are most likely to produce a specific target spectral value over a particular period. The \(\epsilon\) value defines how much larger a target spectral value at a given period is compared to the predictions of a GMM (GMM of [PERSON] and [PERSON] [77] is used in the original study [48]). A fundamental principle of CMS is that the spectral value obtained from deaggregation of a given period does not necessarily apply to other periods. A constant value of \(\epsilon\) would represent a standard UHS curve. From a large suite of ground motions, [PERSON] [78] observed a high correlation between \(\epsilon(T=2s)\) and \(\epsilon(T=1s)\) (\(\rho=0.75\)), compared to \(\epsilon(T=0.2s)\) and \(\epsilon(T=1s)\) (\(\rho=0.44\)). We can use this correlation relationship to determine the trend over different periods, provided we know what the information was during a particular period. The target spectrum can be derived from mean values of \(\epsilon\), which are computed as a multiplication of \(\epsilon\) at a known period and correlation coefficients between \(\epsilon\) at known and unknown periods derived from a ground motion database. The procedure to compute CMS has been illustrated in [PERSON] [79]. Equations to compute mean \(\left(\mu_{\epsilon(T_{i})|\epsilon(T^{*})}\right)\) and median CMS values are given in Equations (22) and (23). \[\mu_{\epsilon(T_{i})|\epsilon(T^{*})}=\rho(T_{i}T^{*})\;\epsilon(T^{*}) \tag{22}\]  
\[\mu_{\ln Sa(T_{i})|\ln Sa(T^{*})}=\mu_{\ln Sa}(M,R,T_{i})+\rho(T_{i},\;T^{*}) \epsilon(T^{*})\sigma_{\ln Sa}(T_{i}) \tag{23}\]  
[PERSON] and [PERSON] [80] computed \(\rho\) based on statistical analysis of joint distributions of spectral accelerations. Ground motions are selected by matching the mean target response spectrum (Equation (23)), which is computationally inexpensive. The variance of target response spectrum can also be calculated and holds crucial information. Ground motion selection is then based on optimization schemes as those proposed by [PERSON] et al. [80]and [PERSON] and [PERSON] [81].