## 2 Formulation for Modeling
### Design of Modified TF to Suppress Overestimation in High-Frequencies  
The performance of the designed Kalman filter was examined when the frequency range is extended to 2.0 rad/s. In Figure 8, we start to see some problem i.e., excessive estimation of spectral amplitude in the high frequency region. No overestimation occurred when the sensor error was removed. In case there is sensor error and RAO is intentionally amplified, so that its magnitude is more than 1 in all frequency region, overestimation did not occur. Therefore, it is expected that the cause of the overestimation at the high frequency region is the combination of sensor errors and small values of RAO. In order to better understand the phenomenon, the internal variables, P and K for every frequency component during the operation of the filter were analyzed. Figure 9 shows a comparison of the changes of the internal variables P and the absolute value of K in the frequency domain during the Kalman filter operation for both cases i.e., (a) and (b) when the maximum frequency is 1.15 rad/s and (c) and (d) when the maximum frequency is changed to 2.0 rad/s. In the case of (a) and (b), the P and K values converge to proper values after initial transient stage. However, in the case of (c) and (d), the P and K values do not converge well and the errors in the high frequency region are particularly large. The convergence rates in the high frequency region are also very slow. This is due to Equation (7) of the 'Calculate P' step in the Kalman filter loop. P is decreasing each time step by the equation. The high-frequency region of \((I-K_{k}H_{k})\) is close to 1, the element of an identity matrix, since the H is small in the high-frequency zone due to the motion characteristics (RAO) of the vessel. Therefore, the high-frequency region of P, which is multiplied by \((I-K_{k}H_{k})\) every time step, slowly decreases. P is a variable that determines K in Equation (5) of the 'Calculate K' step. The slow decreasing of P also slows down the convergence rate of K. Therefore, in Figure 9d, the convergence speed of  
Figure 8: Generated and estimated spectrum/Evolution of parameters (Hs = 2 m, Tp = 7 s, Frequence: 0.1-2.0 rad/s). Figure 7: Generated and estimated spectrum/Evolution of parameters (Hs = 2 m, Tp = 7 s, Frequence: 0.1-1.15 rad/s). K also becomes slow. K is gain between the measured value and the predicted value in Equation (6) of 'Estimate x with measurement' step. The overestimated K in the high frequency range gives an excessive value of x, which is the estimated result. Essentially, the small response in the high frequency region of RAO with respect to the given sensor error causes an overestimation problem. First, as a potential solution to overcome the phenomenon of overestimation in the high frequency region, while using multi-motion signals was attempted i.e., Surge, Sway, Roll, Pitch, and Yaw were added so that a total of six motions can be used. It may be possible that the ill behavior of heave in the high frequency region can be compensated by sensor fusion. The test results (Figure 10) show that the unfavorable behavior is improved when compared to the heave only case, but overestimation still occurs i.e., the generalization of Kalman filter technique to include all 6 DOF motions helps the problem, but is not highly effective. Figure 10: Generated and estimated spectrum / Evolution of parameters using six motion (Hs = 2 m, Tp = 7 s, Frequency: 0.1–2.0 rad/s). Figure 9: Comparison of error covariance and Kalman gain for frequency extension. (a) P (to 1.15 rad/s); (b) K (to 1.15 rad/s); (c) P (to 2.0 rad/s); (d) K (to 2.0 rad/s). Meanwhile, the phenomenon of excessively amplified noise is mentioned as a problem of the inverse filter that is used in the image restoration field [19]. This problem can occur when the transfer function is zero or very small. If the measured signal has sensor error, it is more amplified at high frequencies. In the image restoration, the Wiener filter is used to solve this problem.