Real-Time Inverse Estimation of Ocean Wave Spectra from Vessel-Motion Sensors Using Adaptive Kalman Filter  
[PERSON]  
1 Department of Ocean Engineering, Texas A&M University, Haynes Engineering Building, 727 Ross Street, College Station, TX 77843, USA 1  
[PERSON]  
1 Department of Ocean Engineering, Texas A&M University, Haynes Engineering Building, 727 Ross Street, College Station, TX 77843, USA 1  
[PERSON]  
1 Department of Ocean Engineering, Texas A&M University, Haynes Engineering Building, 727 Ross Street, College Station, TX 77843, USA 1  
Footnote 1: email: {[PERSON], 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19  
Although multiple methods exist for wave-spectrum estimation, such as radar [4], satellite [5], or buoy [6], they require expensive equipment or time-consuming post-processing. As an alternative, one can consider a method of estimating incoming ocean waves from motion sensors on board. The responses occur when wave loads act on a vessel. Conversely, the method of estimating the ocean wave by measuring the vessel motion is a kind of 'inverse problem' i.e., estimating the input from the response output. The procedure for estimating wave spectra in this way can be classified into 'parametric' and 'non-parametric'. The non-parametric method is general. The Kalman filter can be regarded as a non-parametric method that can be applied to any sea states. The conventional methods for solving the inverse problem involve substantially heavy computations, such as the Monte-Carlo method, for resolving the uniqueness problem, which means that multiple different inputs can result from a single output. When compared to those costly and time-consuming methods, the use of Kalman filter can be an alternative inexpensive solution to achieve the real-time on-board estimation of incoming waves. The Kalman filter is fast enough to measure in real time, easy to include the forward speed of the ship, and capable of compensating sensor noises and model errors. The research using the Kalman filter for the inverse estimation of ocean waves is very rare until recently despite the powerful advantages of the Kalman filter. Two papers [7; 8] are representative. The Kalman filter in [8] was applied to ocean wave estimation while using synthesized motion data. In [7], the motion data of the actual field from a real ship was used. Nevertheless, this research is still in the development phase and it needs to be elaborated [9]. Figure 1 demonstrates the overall process of the wave spectrum estimation process while using the Kalman filter. The Kalman filter is difficult to estimate high-frequency waves where the vessel's response is weak. Thus, in the case of large vessels, additional high-accuracy sensors may be utilized or other responses sensitive to high frequencies may be used. More research is needed on this issue and the Kalman filter setting alone may be the solution [7]. Figure 2 shows the actual sea conditions at North West Atlantic. They represent significant wave height, Hs (a) and spectral peak period, Tp (b) at the same time, respectively. It can be seen that high-frequency waves with a Tp less than 5 s (or \(\omega>1.257\) rad/s) occupy a large portion in the figure. Therefore, estimating those wide-spreading high-frequency waves is of importance. Accordingly, it is important to design the Kalman filter, so that the high-frequency waves can also be estimated in a robust manner. Figure 1: Wave spectrum estimation process using Kalman filter. In [8], the frequency range was limited up to 1.15 rad/s since higher frequencies can cause problems when using Kalman filter. However, in this paper, the estimation range was extended to 2.0 rad/s, so that those high-frequency waves can be recovered. The cause of overestimation at the high-frequency zone and its remedy are investigated and new algorithms that suppress those overestimations are proposed. The numerical test results are promising for various wave conditions, vessel types, and sensor errors.