## 3 Testing
### Results and Discussion  
Figures 22-28 present the results of the inversely estimated real-time spectra, Hs and Tp with time for the seven sea states when applying the proposed modified TF and adaptive R (Equation (14)) of [8]. It is seen that the developed Kalman filter technique works well for various sea states, which opens the door to a continuous inverse estimation of ocean waves from heave-motion sensor, regardless of sea states during service or voyage without adjusting the wave-frequency range and Kalman-filter parameters.  
\begin{table}
\begin{tabular}{c c c} \hline
**Sea State No.** & **Hs (m)** & **Tp (s)** \\ \hline
1 & 1.5 & 6 \\
2 & 5 & 9 \\
3 & 9 & 11 \\
4 & 13 & 13 \\
5 & 17 & 15 \\ \hline
6 (NP area in Figure 2) & 1.5 & 18 \\
7 (NA area in Figure 2) & 13 & 11 \\ \hline \end{tabular}
\end{table}
Table 2: Simulated sea states and their parameters.  
Figure 21: Comparison of estimated spectra averaged over 700–1000 s for the different number of frequency components.  
Figure 23: Generated and estimated spectrum/Evolution of parameters (Sea sate 2, Hs = 5 m, Tp = 9 s).  
Figure 24: Generated and estimated spectrum/Evolution of parameters (Sea sate 3, Hs = 9 m, Tp = 11 s).  
Figure 22: Generated and estimated spectrum/Evolution of parameters (Sea sate 1, Hs = 1.5 m, Tp = 6 s).  
Figure 28: Generated and estimated spectrum/Evolution of parameters (Sea sate 7, Hs = 13 m, Tp = 11 s).  
Figure 27: Generated and estimated spectrum/Evolution of parameters (Sea sate 6, Hs = 1.5 m, Tp = 18 s).  
Figure 26: Generated and estimated spectrum/Evolution of parameters (Sea sate 5, Hs = 17 m, Tp = 15 s).  
Figure 25: Generated and estimated spectrum/Evolution of parameters (Sea sate 4, Hs = 13 m, Tp = 13 s).  
Next, we tested the developed modified Kalman-filter algorithm for another vessel (OC4 semisubmersible) to see whether the developed algorithms are sensitive to vessel types. In this regard, Figure 29 provide the motion TF of OC4 semisubmersible ([28,29]). Sea state No. 1, 3, and 5 were tested (Figures 30-32) for the OC4. The estimation of real-time wave spectra, Hs, and Tp was good.  
Up to this point, the developed Kalman-filter algorithm has been applied to the stationary sea state. However, sea states continue to change in actual sea, so, it is of interest to inversely estimate waves for non-stationary seas. The sea state sequence is set as shown in Table 3, which is taken from a series of sea states along the dotted line from Jacksonville, Florida to the 'NA' point in Figure 2. It is assumed that the sea state is changing accordingly. The sensor error's standard deviation was set as 2.3 cm and the modified TF was applied to the FPSO. Simulation was performed assuming that the vessel's motion continuously varies due to the changes in sea state. The superposed motion time series for each sea state were then connected in order (Figure 33).  
The algorithm was tested while using both modified TF and adaptive R. As a result (Figure 34), the real-time wave spectra are well estimated, even with drastic changes in sea states. Figure 34b shows the change in R and Q (fixed) during the estimation and Figure 35 is the RMSE of P, which is converging with more measurements.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sequence** & **No. 1** & **No. 2** & **No. 3** & **No. 4** & **No. 5** \\ \hline Hs (m) & 3 & 2 & 4 & 8 & 13 \\ Tp (s) & 9 & 8 & 10 & 11 & 11 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Simulated sea state sequence along the route.  
Figure 33: Time series of heave motion during sea state change.