## 2 Formulation for Modeling
### Design of Modified TF to Suppress Overestimation in High-Frequencies
\[\mathrm{G(u,v)}\ =\ \mathrm{F(u,v)}\ \mathrm{D(u,v)}\ +\ \mathrm{N(u,  
In fact, the Wiener filter is used to minimize the mean square error between the estimated signal and the desired signal. It produces an estimate of the desired signal of a measured noisy signal, in principle, while assuming that original signal and noise are known [21]. However, it is common to use an approximation when the power spectra are not known. It means that the ratio \(S_{n}(u,v)/S_{f}(u,v)\) can be substituted by a constant C. The C can be chosen by logistic trial and error [19; 22] as Equation (25). In this paper, the original signal i.e., wave spectrum is assumed to be unknown. Accordingly, C can be determined by trial and error.  
In Figure 12, through comparison with inverse filter (1/D), we see the effect of Wiener filter W(u, v). The Wiener filter suppresses the gain by a certain amount to suppress amplified noise in the frequency region where the magnitude of Inverse filter is too large. To see the effect of Wiener filter with respect to transfer function D, the inverses of both Inverse filter and Wiener filter are plotted in Figure 13; 1/W(u, v) increases the very small gain of D in the high-frequency region.  
Figure 11: Schematic diagram of noise amplification.  
Figure 12: Effect of Wiener filter.  
When comparing the wave estimation with the image restoration, F corresponds to wave, G to ship motion, and D to motion TF or H, respectively. Unlike the Inverse filter, the Kalman filter is the recursive filtering algorithm finding more accurate state with more measurements. However, at a particular time step, the Kalman filter also computes the original signal in the inverse direction from the measurement with sensor noise, while using the pre-programmed output equation between the state and measurement.  
We applied the principle of Wiener filter to the present Kalman-filter technique for the inverse wave estimation in the frequency region where the magnitude of Inverse filter is too large. The issue is that the inverse form of transfer function like (1/D) cannot directly be applied to the Kalman filter. Rather, it is not in inverse form, but TF is included in H as equation (11). Therefore, we had to devise a way to modify the TF while keeping the Wiener's effect the same. As mentioned earlier, the effect of the Wiener filter is to increase the gain of D by a certain amount in the frequency region where the magnitude is 0 or too small.  
If Wiener filter W is used instead of Inverse filter (1/D) in Equation (24), noise term N/D is eliminated, and G \(\times\)W = F is established. Subsequently, G/F = 1/W. Let us put this as Equation (a). Meanwhile, in the Equation (6) in the Kalman filter, it is used as Z-HX. If we change these as variables of image restoration, it is G-DF. Assuming that there is noise, it is not 0 but G-DF = N. Afterwards, D = G/F-N/F and let us put this as Equation (b). When we input 1/W to D in the Equation (b), the Equation (b) becomes G/F = G/F-N/F by the Equation (a) and N/F becomes 0. That is, if we use 1/W instead of D in Kalman filter, we can obtain the effect of Wiener filter and eliminate the noise. In conclusion, the modified TF as Equation (26) was applied to the Kalman filter in the inverse form of Wiener filter. For easy comparison, Figure 14 is given.  
\[\frac{(\left|\text{TF}\right|_{2}+\text{C})}{\text{TF}^{*}} \tag{26}\]  
where  
TF\({}^{*}\): complex conjugate of TF  
Through trials and errors, C was found to be 2.5 \(\times 10^{-5}\), when the sensor noise (error) standard deviation is 0.023 m. Figure 15 shows the Modified RAO (TF) when Equation (26) is applied. We also  
Figure 14: Comparison of Image Restoration and Wave Estimation. tested the formula for various wave conditions and two different floating platforms, such as FPSO and semi-submersible case, and the results are equally satisfactory. This is covered in detail in Section 3.2.