comparison_metrics.tex

\section{Comparison Metrics}

In this study response spectra is the most important factor to choose the best model (see Khoshnevis and Taborda 2018 a,b). As we discussed the equivalent linear method is a linear process, therefore, the final displacement will not be accurate specially when we have a permanent displacement. Therefore, acceleration and response spectra would be the best options. A simple method to compute the response spectra residuals is subtracting two data and divide them by one of them. 


\begin{equation}
Residual_{Sa} = mean[\frac{Sa_{nonlinear} - Sa_{equivalent linear}}{Sa_{nonlinear}}],
\end{equation}

According to this equation, if nonlinear simulation results is much higher than equivalent linear, then the residual will be close to one, however, if it is much lower, the residual will be a negative number. The problem with this method is its weakness in representing higher residuals in more than 1 for the positive residuals.  Other approaches are mentioned in the literature to compare the response spectra of two waveform. \citet{Anderson_2004_Proc} uses,

\begin{equation}
S_{sa} = mean[S(SA_1(f_j),SA_2(fj))],
\end{equation}

where the average is over all frequencies at which SA is computed in the frequency band being considered. And $S $ defined as,

\begin{equation}
	S( p_1, p_2) = 10 exp \{ - [ \frac{(p_1 - p_2)}{ \min( p_1, p_2 ) }]^2 \},
\end{equation}

The equation, which is used for GOF score, put the scores from comfortable range of values (0-10) regarding the poorest fit and excellent fit. This is a very good scale, however, it can not represent which simulation had a higher values. Because the results are always positive.  \citet{Assimaki2008quantifying} used cumulative normalized error between observation and prediction:

\begin{equation}
e_{SA}=\frac{1}{n}\sqrt{\sum_{i=1}^{n}[\frac{SA_o(T_i)-SA_p(T_i)}{SA_o(T_i)}]^2}
\end{equation}

This metric also represents positive residual values. 

\citet{Assimaki2012} and \citet{Carlton2016comparison} (and also broadband platform!!?) used the following equation to measure of misfit between two $SA$s:

\begin{equation}
e_{SA}^{non}=\mu(e_{SAi})=\mu(log(\frac{SA_{i}^{non}}{SA_{i}^{eq}}))
\end{equation}

This metric can represent a good variation of residuals and does not saturate. Also positive residuals represent higher values for nonlinear and negative values represent higher values for equivalent linear. Fig.~\ref{fig:response_spectra_sensitivity} shows the relationship between these metrics. In this study we use logarithmic residuals. 


\begin{figure}[H]
    \centering
    \includegraphics[width=\textwidth]{figures/pdf/response_spectra_sensitivity.pdf}
    \caption{a,b,c) Different methods to estimate the residuals. d) Shows how simple residual is easily saturated}
    \label{fig:response_spectra_sensitivity}
\end{figure}