# Seismological and Engineering Demand Misfits for Evaluating Simulated Ground Motion Records
## 4 Engineering Demand Misfits  
In order to have an idea about the use of simulations in earthquake engineering practice, engineering demand misfits are defined next. These misfits are calculated in terms of the differences between the real and simulated values of different engineering demand parameters including Housner Intensity (\(HI=\int_{0.1}^{2.5}PSV(\xi=0.05,\,T)\ dT\)) [59], Acceleration Spectrum Intensity (\(ASI=\int_{0.1}^{0.5}S_{a}(\xi=0.05,\,T)\ dT\)) [60], Velocity Spectrum Intensity (\(VSI=\int_{0.1}^{2.5}S_{v}(\xi=0.05,\,T)\ dT\)) [60], both Elastic and Inelastic SDOF Acceleration Response Spectra using 5% viscous damping ratio (E-S\({}_{a}\) and IE-S\({}_{a}\), respectively) and seismic Input Energy for both Elastic (E-E\({}_{i}\)) as well as Inelastic (IE-E\({}_{i}\)) SDOF systems. Herein, PSV, \(S_{a}\) and \(S_{v}\) correspond to the Pseudo-Velocity Spectrum, Acceleration Spectrum and Velocity Spectrum, respectively. The terms \(T\) and \(\xi\) stand for the period and viscous damping ratio, respectively. Seismic input energy is calculated by taking the integral of the equation of motion for an either elastic or inelastic (elasto-plastic model with strength factor of 0.1 and damping ratio of 5% herein) SDOF system as stated in Uang and Bertero Uang and Bertero (1999):  
\[\int m\ddot{u}(t)\ du+\int\dot{cu}(t)du+\int\ f_{s}(u)\ du=-\int m\ddot{u}_{g}( t)\ du, \tag{3}\]  
where, the right-hand side term shows the input energy for an either elastic (E-E\({}_{i}\)) or inelastic (IE-E\({}_{i}\)) SDOF system. The three left-hand side terms, respectively, correspond to the kinetic energy, damping energy and absorbed energy (including the strain energy for the elastic system in addition to the hysteretic energy for the inelastic system). In this study, the spectral SDOF misfits in terms of spectral \begin{table}
\begin{tabular}{c c c} \hline \hline
**Case Study** & **Station** & **Seismological GOF Score** \\ \hline \multirow{3}{*}{Erzincan} & ERC & 68 \\  & REF & 49 \\  & TER & 54 \\  & DZC & 75 \\  & GYN & 47 \\  & IZN & 52 \\  & IZT & 46 \\  & AQA & 74 \\  & CLN & 58 \\  & GSA & 60 \\  & MTR & 64 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Station-wise seismological GOF scores for all case studies. acceleration (E-S\({}_{\rm a}\) and IE-S\({}_{\rm a}\)) and input energy (E-E\({}_{\rm i}\) and IE-E\({}_{\rm i}\)) are evaluated for different SDOF systems with varying period values between 0.1 and 4 s.  
Similar to the seismological misfits, the engineering demand misfits are defined in terms of both the log-scale misfits and GOF scores. Equation (2) is used for calculation of the log-scale misfits including HI, ASI and VSI, while the remaining log-scale misfits in terms of E-S\({}_{\rm a}\), IE-S\({}_{\rm a}\), E-E\({}_{\rm i}\) and IE-E\({}_{\rm i}\) are evaluated through the following equation:  
\[\mathit{Misfit}_{R}=\ \frac{1}{n_{T}}\sum_{T=0.1}^{4}\left|log\frac{R_{simulated}(T)}{R_{real}(T)}\right|, \tag{4}\]  
where, \(R_{simulated}\)\((T)\) and \(R_{real}\)\((T)\) take values corresponding to, respectively, the simulated and real parameters of E-S\({}_{\rm a}\), IE-S\({}_{\rm a}\), E-E\({}_{\rm i}\) or IE-E\({}_{\rm i}\) for a SDOF system with the period of \(T\). The term \(n_{T}\) is the total number of discrete periods between 0.1 and 4 s.  
Figures 7-9 present the station-wise engineering demand log-scale misfits for all case studies. For each case study, similar to the seismological log-scale misfits, the engineering demand log-scale misfits are defined based on the geometric mean of the engineering demand parameters from the observed records in NS and EW directions at every station. The results corresponding to the station-wise engineering demand log-scale misfits for all case studies reveal that the overall log-scale misfits are less than 1 unit in log-scale.