## 2 Formulation for Modeling
### Design of Modified TF to Suppress Overestimation in High-Frequencies
The results show substantial improvement (overestimation does not occur) when the proposed modified motion TF is applied to the FPSO, as shown in Figure 16.  
\((x_{2j-1}+ix_{2j})\) is an incoming wave of a complex number in Equation (10). For the two states \(x_{2j-1}\), \(x_{2j}\) estimated by the Kalman filter, if Equation (17) is used, the real-time estimation of the corresponding wave elevation (actual wave profile) at each time is conducted (Figure 17) and Figure 18 shows the results. The real-time wave profile that was estimated by the Kalman filter is compared to the theoretically reconstructed wave profile by using the random phase and the relation between the wave and motion via the phase of transfer function. Figure 17 describes the estimation process for wave elevation. In the case of the wave spectrum estimation, only the motion time series of the ship needs to be generated through the process of 1, 2 in the figure. In the case of the wave elevation estimation, we need to additionally generate the original wave elevation through the 3 calculation. This becomes the target for comparison with the estimated one. It should be noted that the motion and wave random pahses differ by the TF phase \(\varphi\), so that the random phases of incident wave can be assigned. In Figure 18, the original and inversely estimated wave profile agree very well. This means the the present Kalman filter technique applied to the inverse problem successfully reproduces not only the real-time wave spectrum, but also the real-time wave elevation. The latter is very important for the active control in many ocean-engineering applications. To the best knowledge of authors, the real-time inverse estimation of wave elevation by the Kalman filter cannot be found in the open  
Figure 16: Generated and estimated spectrum / Evolution of parameters using Modified TF (Hs = 2 m, Tp = 7 s, Frequence: 0.1\(\sim\)2.0 rad/s). Figure 7:0 Modified Heave response amplitude operator (RAO) (red dotted line) of FPSO. literature. In [7; 8; 23; 24], no time series of wave elevation was obtained although wave spectra were inversely estimated. Figure 7:1 Wave elevation estimation process using Kalman filter and Formula. Figure 7:2 **(a)** Input heave motion & **(b)** Estimated wave elevation using Modified TF (Hs = 2 m, Tp = 7 s, Frequency: 0.1–2.0 rad/s). For further checking of convergence, RMSE of P is represented from Equation (27). The 'trace' means sum of diagonal elements. Figure 19 confirms that P does not diverge. Figure 7:3  
The variations of the internal variables P and K in the frequency domain were examined as well in Figure 7:4 When compared with the '-2.0 rad/s' case of Figure 9c,d, it can be seen that both P and K converge near zero in the high frequency range at 500 s. The application of Modified TF suppresses K in the high frequency. Consequently, the overestimation of x does not occur in the high-frequency region when it is calculated by Equation (6). The measurements from a sensor include sensor error. When the sensor maker provides sensor error information, we can set a reference value for R according to the information. However, a potential problem is that the statistical properties of sensor errors are not the same during actual measurements. Accordingly, [25] proposed an adaptive Kalman filter for harmonic signals. Therefore, to take advantage of the adaptive R, Equation (14), e.g., [8] is applied with the modified TF in subsequent tests. In the equation, only the diagonal component was taken from the covariance matrix for the errors. By two 'diag' processes, we can construct a diagonal matrix, so that it matches the dimensions of the \(H_{k}P_{\overline{k}}H_{k}\)' matrix. To explain more about the dimensions of the \(H_{k}P_{\overline{k}}H_{k}\)' matrix, variable definition and dimensions are in Figure 7:5 The number of states x and measurements z are independent. However, the dimension of matrix \(\phi\), H, P, R, etc. are determined by the x and z [26].