## 2 Theory and Methodology  
The ocean wave spectrum can inversely be estimated by utilizing the on-board measurement of vessel motion [11; 12]. In this method, the 6-DOF motion measurements are related to the characteristics of incident waves through the motion transfer function, which is frequently called the response amplitude operator (RAO). Then, the Kalman filter (KF) algorithm can recover the incident wave elevation and spectrum through minimizing differences between the measured and predicted vessel motions. Figure 1 shows an example of a simulated motion time series of a floating vessel and the estimated unidirectional wave spectrum using the motion data and the Kalman filter method, for both cases, with and without the forward speed of the vessel. If the vessel has a forward speed, the motion signals and estimated wave spectra are measured with respect to the encounter frequency due to the Doppler frequency-shift effect. To obtain the original wave spectrum in the absolute wave-frequency domain, it is essential to convert the estimated encounter-frequency-based wave spectrum to the original wave spectrum. Figure 1: Estimated unidirectional wave spectrum using Kalman filter with simulated motion time series with and without forward speed (significant wave height \(H\)s = 5 m, peak periods \(Tp\) = 9 s, peak enhancement factor \(\gamma\) = 2, and wave heading \(\beta\) = 0\({}^{\circ}\)): (**a**) no forward speed; (**b**) forward speed \(V\) = 5 knots. Figure 2 shows the relationship between the original wave frequency \(\omega\) and the encounter wave frequency \(\omega_{e}\), as given by Equation (5). When \(\omega_{e}>\frac{g}{4V\cos\beta}\), there is a one-to-one relation between the encounter and original frequencies. When \(\omega_{e}<\frac{g}{4V\cos\beta}\), the given encounter frequency is mapped onto three original frequencies \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) in areas 1, 2, and 3, respectively. The three areas are determined by  
Area 1: \(\omega_{1}<\frac{g}{2V\cos\beta}\),  
Area 2: \(\frac{g}{2V\cos\beta}<\omega_{2}<\frac{g}{V\cos\beta}\),  
Area 3: \(\frac{g}{V\cos\beta}<\omega_{3}<\frac{g\left(1+\sqrt{2}\right)}{2V\cos\beta}\). In this paper, we utilize the JONSWAP wave spectrum as the original wave frequency spectrum assuming long-crested unidirectional irregular waves. The parameterized wave spectrum in the original-frequency domain is defined by [25]  
\[S(\omega)=[1-0.287\text{ln}(\gamma)]\frac{5}{16}H_{s}^{2}\omega_{p}^{4}\omega ^{-5}\text{exp}\left(-\frac{5}{4}\left(\frac{\omega}{\omega_{p}}\right)^{-4} \right)\gamma^{\text{exp}\left(-0.5\left(\frac{\omega-\omega_{p}}{\omega_{p} }\right)^{2}\right)}, \tag{11}\]  
where \(\omega_{p}\) is the spectral peak angular frequency, \(\gamma\) is the non-dimensional peak shape (or overshoot) parameter, and \(\sigma\) is the spectral width parameter. The value of \(\sigma\) is 0.07 for \(\omega\leq\omega_{p}\) and 0.09 for \(\omega>\omega_{p}\). The original wave spectrum \(S(\omega)\) can be converted uniquely to the encounter-frequency-based spectrum \(S_{e}(\omega_{e})\) by using Equations (4), (9), and (10). However, its inverse process, i.e., conversion from \(S_{e}(\omega_{e})\) to \(S(\omega)\), is not uniquely determined in Equation (9). This is because the single value of \(\omega_{e}\) is mapped into three frequencies \(\omega_{1}\), \(\omega_{2}\), and \(\omega_{3}\) when \(\omega_{e}<\frac{g}{4V\cos\beta}\), as shown in Figure 2. In this case, there exists no one-to-one relation, which means that the conversion from \(S_{e}(\omega_{e})\) to \(S(\omega)\) cannot be performed in a straightforward manner. In certain cases, we may be able to use a reasonably devised ad hoc approximation method. Figure 3b shows the original wave spectrum \(S(\omega)\) with respect to the original frequency (black solid line) and the corresponding \(S_{e}(\omega_{e})\) with respect to the encounter frequency (green solid line).