## 2 Theory and Methodology
Then, the green line \(S_{e}(\omega_{e})\) is converted back to \(S(\omega)\), as marked by the blue cross sign in Figure 3b. This is the case for which the conversion cannot be performed theoretically, due to \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\). In this case, we initially assume \(S(\omega_{2})=S(\omega_{3})=0\), and the conversion was performed by using \(S(\omega_{1})\) only in Equation (9), the result of which looks acceptable compared to the original wave spectrum. In this case,  
Figure 2: Relationship between original and encounter frequencies for following/quatering sea. the original wave spectrum has most of its spectral energy in area 1 and, thus, the recovered wave spectrum (blue cross) was possible by using only \(S(\omega_{1})\). However, if the original wave spectrum is widely distributed over areas 1, 2, and 3, this kind of simple approach does not work. Figure 4 illustrates that kind of case, for which the above simple method of using only \(S(\omega_{1})\) is not applicable, since the original wave spectrum is evenly distributed along areas 1 and 2. Then, we may try \(S(\omega_{1}):S(\omega_{2}):S(\omega_{3})=1:1:0\), which compares much better against the actual original-frequency-based spectrum. Then, the following question arises: \"how can we distribute energy among \(S(\omega_{1})\), \(S(\omega_{2})\), and \(S(\omega_{3})\) without knowing the original spectral shape a priori?\". Without knowing the shape of the original wave spectrum in advance, it is not possible to properly distribute the proportion among \(S(\omega_{1})\), \(S(\omega_{2})\), and \(S(\omega_{3})\) so that the original wave spectrum can be recovered, as was demonstrated in the above example. In this regard, a more robust ML-based method to solve the above non-unique inverse problem (converting \(S_{e}(\omega_{e})\) to \(S(\omega)\) when \(\omega_{e}<\frac{g}{4V\cos\beta}\)) is explained in the next section. According to Equation (9), the encounter wave spectrum exhibits a singularity when the denominators become zero, leading to an unwanted singularity in the spectrum. This occurs when the frequency approaches \(\omega_{e}=\frac{g}{4V\cos\beta}\), as indicated by the green line in Figure 3b. However, this singularity may not be captured in the measured spectrum, due to processing steps such as smoothing or insufficient resolution. To represent this case and observe its consequence, we also included the ANN training/validation cases without the sharp peak. It has been observed that a sharp singularity does not significantly affect the  
Figure 4: Encounter wave frequency conversion (\(H\)s = 5 m, \(Tp\) = 9 s, \(\beta\) = 30\({}^{\circ}\), and \(V\) = 15 knots): (**a**) original vs. encounter frequency; (**b**) recovered spectrum. Figure 3: Encounter wave frequency conversion (\(H\)s = 5 m, \(Tp\) = 9 s, \(\beta\) = 30\({}^{\circ}\), and \(V\) = 7 knots): (**a**) original vs. encounter frequency; (**b**) recovered spectrum. performance of the ANN model, as illustrated in Figure 5. This may be due to the fact that the singularity is difficult to detect in the discretized \(S_{e}(\omega_{e})\). In the ensuing sections, only the results without those singularities are presented.