# Seismological and Engineering Demand Misfits for Evaluating Simulated Ground Motion Records
## 3 Seismological Misfits  
The observed and simulated records at the selected three study areas and the corresponding stations are evaluated through definition of misfits in terms of alternative seismological measures. In this study, for evaluation of the seismological misfits, seismological parameters are considered to be Fourier Amplitude Spectrum (FAS), PGA, PGV, PGV to PGA ratio (PGV/PGA), Cumulative Absolute Velocity (\(CAV=\int_{0}^{I_{d}}\!\!\left|a(t)\right|dt\)) [56], Arias Intensity (\(I_{a}=\frac{\pi}{2\xi}\int_{0}^{I_{d}}a(t)^{2}\ dt\)) [57] and Significant Duration (\(t_{eff}\)) [58] (determined as the time interval of 5-95% of the accumulated \(I_{a}\)). In the aforementioned formulas, T\({}_{\text{d}}\) corresponds to total duration of ground motion record, \(a(t)\) is acceleration time series, and \(\left|a(t)\right|\) shows the absolute value of \(a(t)\) at time. Figure 3: Selected stations along with the epicenter of the 2009 L’Aquila event where the rectangular box shows the fault projection. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Station**} & \multirow{2}{*}{**Code**} & **Latitude** & **Longitude** & **Site Class** & **R\({}_{\text{epi}}\)** & **PGA-EW** & **PGA-NS** & **PGA-Sim.** & **PGV-EW** & **PGV-NS** & **PGV-Sim.** \\  & & (\(\gamma\)) & (\(\gamma\)) & (\(\mathbf{ECS}\)) [54] & (\(\mathbf{dom}\)) & (\(\mathbf{c}\))/(\(\mathbf{c}\)) & (\(\mathbf{c}\)/(\(\mathbf{c}\))) & (\(\mathbf{c}\)/(\(\mathbf{c}\))) & (\(\mathbf{c}\)/(\(\mathbf{c}\))) & (\(\mathbf{c}\)/(\(\mathbf{c}\))) & (\(\mathbf{c}\)/(\(\mathbf{c}\))) & (\(\mathbf{c}\)/(\(\mathbf{c}\))) \\ \hline V. Aretro-F. & AQA & 42.376 & 13.339 & B & 4.2 & 350.46 & 347.59 & 254.94 & 29.86 & 24.07 & 22.87 \\ Celano & CLN & 42.085 & 13.5207 & A & 31.79 & 73.49 & 76.57 & 75.9 & 4.61 & 6.56 & 10.21 \\ Gran Sasao & GSA & 42.021 & 13.5194 & B & 14.15 & 131.88 & 139.02 & 157.36 & 9.63 & 7.41 & 15.03 \\ Montereale & MTR & 42.524 & 13.2448 & A & 22.13 & 42.17 & 51.65 & 54.66 & 3.25 & 3.09 & 5.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Information on the selected stations corresponding to the 2009 L’Aquila (Italy) event (Mw = 6.3). In this study, seismological misfits are defined in terms of both log-scale misfits and GOF scores. For calculation of FAS log-scale misfits, the following function is used:  
\[Misfit_{FAS}=\ \frac{1}{n_{f}}\sum_{f=0.1}^{10}\left|log\frac{FAS_{simulated}(f)}{FAS_{real}(f)}\right| \tag{1}\]  
where, \(n_{f}\) corresponds to the total number of discrete frequencies varying between 0.1 and 10 Hz. For the remaining single-value parameters defined above, log-scale misfits are calculated using the following formula:  
\[Misfit_{R}\ =log\left|\frac{R_{simulated}}{R_{real}}\right| \tag{2}\]  
where, \(R\) stands for the seismological parameters defined including PGA, PGV, PGV/PGA, CAV, \(I_{a}\) and \(t_{eff}\). For calculation of all seismological log-scale misfits, the geometric mean of the seismological parameters from the observed records in NS and EW directions at every station for each case study is considered. The results of station-wise seismological log-scale misfits for all study regions are illustrated in Figures 4-6. These illustrations reveal that for all case studies, the overall log-scale misfits are less than 1, in log-scale. However, depending on the study area and station, there is variability in terms of seismological log-scale misfits corresponding to alternative seismological measures. Next, for each station in each study area, the average seismological log-scale misfits are calculated by averaging the individual log-scale misfits in terms of \(FAS\) and \(R\) (including PGA, PGV, PGV/PGA, CAV, \(I_{a}\) and \(t_{eff}\)). The results are presented in Table 4. For Erzincan, as the first case study, among all selected three stations, the least log-scale misfit is obtained for station ERC which is located in the closest distance to the fault plane.