## 3 Results  
### ANN Model Performance  
The performance of the trained ANN model was evaluated using the R2 score and model errors. Table 4 presents the results of the performance evaluation based on all validation cases, which were drawn from 20% of the dataset and were not included in the 80% used for training. The trained ANN models exhibited sufficient overall accuracy. It is noteworthy that Model 1 achieved high accuracy despite using only 1/10 of the dataset size of Models 2 and 3, due to the fixed peak enhancement factor. The performance of Model 2 was slightly improved when the additional input \(H\)s was employed. However, the performance of Model 3 was as good as that of Model 2, as demonstrated in Figure 8. In the case of \(S_{e}(\omega_{e})\) obtained through motion sensors, the estimation of Hs may be less robust compared to the present numerically evaluated \(H\)s from the theoretically generated \(S_{e}(\omega_{e})\). Model 3 is more practical and reliable compared to Model 2, as it does not require the real-time estimation of \(H\)s as an additional input. Therefore, from a practical perspective, Model 3 is more suitable, and its accuracy remains sufficiently high. For this reason, we have selected Model 3 for the subsequent discussions.  
Figure 9 illustrates the estimated wave spectrum using Model 3 for several environmental conditions. The selected validation cases for various \(H\)s, \(Tp\), \(\gamma\), \(\beta\), and \(V\) values show that the inversely predicted wave spectrum agrees very well with the original wave spectrum. In most conditions, the results demonstrate successful recovery of the original wave spectrum regardless of wave directions and vessel speeds. The wave heading is 40 degrees for this figure. Especially, the peak period is estimated very accurately. Figure 10 also shows the results of Model 3 where the wave heading is changing while \(V\) is fixed at 10 knots. This case shows even better performance compared to Figure 9. The small differences in wave spectrum (wave amplitude squared) become even smaller when the corresponding wave elevations are compared.  
Figure 11 presents the inversely estimated spectra for different \(\gamma\) values. The recovery of \(S(\omega)\) from \(S_{e}(\omega_{e})\) demonstrates strong robustness across various \(\gamma\) values, although slight errors are observed at lower \(\gamma\) values. We also checked many other cases while varying other wave parameters and the results are all similarly good, as presented in Figures 9-11.  
\begin{table}
\begin{tabular}{c c c c} \hline
**ANN Model Metrics** & **Model 1** & **Model 2** & **Model 3** \\ \hline MAE (Mean Absolute Error) & 0.046 & 0.052 & 0.059 \\ MSE (Mean Squared Error) & 0.025 & 0.025 & 0.045 \\ RMSE (Root-Mean-Squared Error) & 0.159 & 0.159 & 0.213 \\ R Square (R2) & 0.993 & 0.993 & 0.987 \\ \hline \end{tabular}
\end{table}
Table 4: Artificial neural network (ANN) model metrics.  
Figure 8: Comparison of estimated original wave spectra through Model 2 and Model 3.  
Figure 9: Estimated original wave spectra at different forward speeds \(V\) (Model 3, heading \(\beta=40\) deg, and peak enhancement factor \(\gamma=2\)).  
Figure 10: Estimated original wave spectra at different wave headings \(\beta\) (Model 3, forward speed \(V\) = 10 knots, and peak enhancement factor \(\gamma\) = 2).