## 1 Introduction  
In the coming years, the use of autonomous vessels is expected to be significantly increased in both commercial and defense industries. For the safe and effective operation of those autonomous vessels, the real-time estimation of ocean waves is of high importance for various kinds of real-time controls and adjustments. Reliably estimated wave information can be shared among nearby vessels, offshore platforms, and marine facilities for their respective benefits. Estimating ocean wave characteristics in real time is also particularly crucial for decision making in certain marine operations, such as route optimization [1, 2] and seakeeping [3, 4]. Sea wave conditions can be predicted through global re-analysis models, such as those from the European Centre for Medium-Range Weather Forecasts (ECMWF) and the US-based National Centers for Environmental Prediction (NCEP) calibrated with their sparsely deployed buoy data. However, the forecasted data often include uncertainties and are limited by low resolution [5]. In this regard, researchers have developed vessel-onboard wave estimation methods to utilize on-site wave data to successfully carry out various marine missions in situ. Direct onboard measurement can be performed by X-band wave radar scanning [6], which is highly expensive, and its accuracy may be hampered by complex vessel motions. On the other hand, onboard motion sensors [7; 8] can be used as a much cheaper solution, i.e., inverse wave estimation can be achieved indirectly using low-cost motion sensors. The measured vessel-motion signal can be used to characterize ocean waves based on Bayesian modeling procedures [9; 10], the Kalman filter (KF) method [11; 12; 13], and machine learning (ML) models [14; 15; 16]. Bayesian modeling is computationally intensive for hyperparameter optimization, making real-time implementation challenging [13; 17]. To address this, more efficient algorithms have been developed by researchers [18; 19; 20]. In contrast, the KF and ML methods can be performed in real-time or near-real-time. Therefore, those real-time inverse wave estimation methods have a potential to significantly contribute to the control and adjustment of unmanned autonomous vessels and marine facilities, especially in areas that are not covered by global forecast models and oceanographic buoys. When using either wave-radar-scan or motion-sensor signals from a vessel with forward speed, there exists a Doppler frequency-shift effect, and the correspondingly obtained wave or motion time series and spectra are represented in the encounter frequency domain. Therefore, to recover the original wave spectrum, the conversion from the encounter-frequency domain to the original-frequency domain is needed, which cannot be performed straightforwardly in some cases, due to the non-uniqueness of the inverse problem. The recovery of the original wave spectrum is essential since many control systems and optimal route strategies are usually designed against original wave conditions. The mathematical relationship between these two frequency domains is well known and the conversion from the original wave frequency to the encounter wave frequency can be uniquely carried out. However, the conversion from the encounter frequency to the original frequency may face a non-uniqueness problem when the incident wave is from a following-wise direction [21]. This is because one encounter frequency can be mapped onto three different original frequencies under such conditions. Due to the 3-to-1 relation, the spectral energy of one encounter frequency should be distributed to that of three original frequencies, and the distribution ratio is not known in advance, making the spectral conversion problem underdetermined. Despite the practical importance of this inverse problem, relevant research resolving this inverse problem is surprisingly rare in the open literature.