## 4 Recommendation on Improving the Efficiency of Intensity Measures
### Analysis of Functional Forms
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Efficiency**} & \multicolumn{4}{c}{**Proficiency**} \\ \cline{2-10}
**IM(s)** & **Standard** & **Yeudy** & **Equation (25)** & **Equation (26)** & **\% Decrease** & **Standard** & **Yeudy** & **Equation (25)** & **Equation (26)** \\ \hline _PGA_ & 0.785 & 0.750 & 0.735 & **0.616** & **+21.5\%** & 0.776 & 0.966 & 1.001 & **0.839** \\ _PGV_ & 0.667 & 0.657 & 0.656 & **0.558** & **+16.3\%** & 0.717 & 0.715 & 0.717 & **0.610** \\ _PGD_ & 0.863 & 0.768 & 0.763 & **0.666** & **+22.8\%** & 1.390 & 1.309 & 1.401 & **1.224** \\ _ASI_ & 0.739 & 0.718 & 0.710 & **0.596** & **+19.4\%** & 0.707 & 0.836 & 0.868 & **0.728** \\ _VSI_ & 0.667 & 0.650 & 0.650 & **0.559** & **+16.2\%** & 0.725 & 0.738 & 0.725 & **0.624** \\ _Sa(T)_ & 0.876 & 0.796 & 0.783 & **0.481** & **+45.1\%** & 1.158 & 1.879 & 2.084 & **0.554** \\ _Tsig_ & 1.300 & 0.864 & 0.810 & **0.710** & **+45.4\%** & \(-3.248\) & \(-1.866\) & \(-1.253\) & \(-1.098\) \\ \(A_{\text{RMS}}\) & 0.766 & 0.741 & 0.725 & **0.605** & **+21.1\%** & 0.775 & 0.925 & 0.953 & **0.796** \\ \(I_{A}\) & 0.741 & 0.730 & 0.724 & **0.605** & **+18.4\%** & 1.392 & 1.512 & 1.572 & **1.313** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Efficiency and proficiency characteristics of various functional forms and IMs. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**IM(s)**} & \multicolumn{4}{c}{**Steadfastness Index**} \\ \cline{2-5}  & **Standard Form** & **Yeudy** & **Equation (25)** & **Equation (26)** & **\% Change** \\ \hline _PGA_ & 0.797 & 0.640 & 0.643 & 0.766 & \(-3.8\%\) \\ _PGV_ & 0.817 & 0.820 & 0.819 & 0.962 & **+17.7\%** \\ _PGD_ & 0.889 & 0.944 & 0.942 & 1.080 & +21.4\% \\ _ASI_ & 0.798 & 0.675 & 0.678 & 0.807 & +1.1\% \\ _VSI_ & 0.846 & 0.831 & 0.830 & 0.965 & +14.1\% \\ _Sa(T)_ & 0.498 & 0.307 & 0.306 & 1.045 & **+100.9\%** \\ _Tsig_ & 1.015 & 1.767 & 1.750 & 1.984 & \(-95.4\%\) \\ \(A_{\text{RMS}}\) & 0.791 & 0.663 & 0.665 & 0.795 & +0.5\% \\ \(I_{A}\) & 0.789 & 0.727 & 0.733 & 0.880 & +11.5\% \\ _CAV_ & 0.849 & 0.848 & 0.860 & 1.006 & **+18.5\%** \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Steadfastness analysis of different functional forms. With respect to _Vs30_, the IMs are sufficient for Equation (26) and insufficient for other functional forms. With respect to epicentral distance (\(R_{epi}\)), none of the functional forms show sufficiency. This observation contradicts [PERSON]'s findings that all the IMs are sufficient (\(p>0.05\)) when using their proposed IM-EDP relation (Equation (24)). This inconsistency may arise from the different near-field ground motion dataset used in their study. While [PERSON]'s dataset considers many small magnitude events (\(M_{w}<5\)) at distances less than 50 km, large events (\(M_{w}>5\)) dominate our dataset at least up to 100 km. The seismic energy radiated from these larger-magnitude earthquakes has different attenuation behavior with distance, contributing to variations in the predicted EDP. Small magnitude events produce ground motions that attenuate faster at shorter distances, while large events attenuate more gradually over larger distances. To verify, we divide our ground motion dataset into distance bins and calculate \(p_{R}\) values (Figure 14). When grouped into distance bins, we observe sufficiency for most IMs in the 30-100 km range. Figure 13: Sufficiency analysis in terms of \(p\) values of different functional forms and IMs. The dotted line marks the \(p=0.05\) threshold for classifying sufficient (\(p>0.05\)) and insufficient IMs (\(p<0.05\)). Very low \(p\) values (\(p<10^{-4}\)) are plotted at \(p=10^{-4}\) level. Equation (1) does not show sufficiency with any parameter for most of the analyzed IMs. Equation (24) shows sufficiency only with \(M_{w}\). Equation (26) shows sufficiency with respect to \(M_{w}\) and _Vs30_ for most IMs.