In this paper, we present a novel approach for the prediction of rogue waves in oceans using statistical machine learning methods. Since the ocean is composed of many wave systems, the change from a bimodal or multimodal directional distribution to unimodal one is taken as the warning criteria. Likewise, we explore various features that help in predicting rogue waves. The analysis of the results shows that the Spectral features are significant in predicting rogue waves. We find that nonlinear classifiers have better prediction accuracy than the linear ones. Finally, we propose a Random Forest Classifier based algorithm to predict rogue waves in oceanic conditions. The proposed algorithm has an Overall Accuracy of 89.57% to 91.81%, and the Balanced Accuracy varies between 79.41% to 89.03% depending on the forecast time window. Moreover, due to the model-free nature of the evaluation criteria and interdisciplinary characteristics of the approach, similar studies may be motivated in other nonlinear dispersive media, such as nonlinear optics, plasma, and solids, governed by similar equations, which will allow for the early detection of extreme waves
Powerful rogue ocean waves have been objects of fascination for centuries. Elusive and awe-inspiring, with the ability to inflict catastrophic damage, rogue waves remain unpredictable and imperfectly understood. To gain further insight into their behavior, we analyzed 3,441,188,683 ocean surface waves in order to determine the statistical height distribution of the largest waves. We found that the distribution of rare events which resolves the St. Petersburg paradox also describes the relative height distribution of the largest waves. This result will contribute to the modeling of ocean surfac…
Prediction is a central goal and a yet-unresolved challenge in the investigation of oceanic rogue waves. Here we define a horizon of predictability for oceanic rogue waves and derive, via extensive computational experiments, the first statistically-converged predictability time-scale for these structures. We show that this time-scale is a function of the sea state as well as the strength (i.e. overall height) of the expected rogue wave. The presented predictability time-scale establishes a quantitative metric on the combined temporal effect of the variety of mechanisms that together lead to th…
We review the study of rogue waves and related instabilities in optical and oceanic environments, with particular focus on recent experimental developments. In optics, we emphasize results arising from the use of real-time measurement techniques, whilst in oceanography we consider insights obtained from analysis of real-world ocean wave data and controlled experiments in wave tanks. Although significant progress in understanding rogue waves has been made based on an analogy between wave dynamics in optics and hydrodynamics, these comparisons have predominantly focused on one-dimensional nonlin…
Ocean rogue waves are large and suddenly appearing surface gravity waves, which may cause severe damage to ships and other maritime structures. Despite years of research, the exact origin of rogue waves is still disputed. Linear interference of waves with random phase has often been cited as one possible explanation, but apparently does not satisfactorily explain the probability of extreme events in the ocean. Other explanations therefore suggested a decisive role of a nonlinearity in the system. Here we show that linear interference of a finite and variable number of waves may very well expla…
This work presents an analysis of ocean wave data including rogue waves. A stochastic approach based on the theory of Markov processes is applied. With this analysis we achieve a characterization of the scale dependent complexity of ocean waves by means of a Fokker-Planck equation, providing stochastic information of multi-scale processes. In particular we show evidence of Markov properties for increment processes, which means that a three point closure for the complexity of the wave structures seems to be valid. Furthermore we estimate the parameters of the Fokker-Planck equation by parameter…
In this study we discuss the shapes and statistics of the rogue (freak) waves emerging due to wave-current interactions. With this purpose, we use a simple governing equation which is a nonlinear Schrodinger equation (NLSE) extended by R. Smith (1976). This extended NLSE accounts for the effects of current gradient on the nonlinear dynamics of the ocean surface near blocking point. Using a split-step scheme we show that the extended NLSE of Smith is unstable against random chaotic perturbation in the current profile. Therefore the monochromatic wave field with unit amplitude turns into a chaot…
Recent results of numerical simulations of fully nonlinear evolutionary equations for long-crested deep-water waves are discussed, where formation of extreme waves was observed. Several examples demonstrate that three-dimensionality of the fluid motion has an essential influence on the process of rogue wave formation. In particular, in the presence of elongate wave groups, the most tall extreme waves occur when in an initial state the wave fronts were oriented obliquely to the direction of the group. An "optimal" angle, resulting in the highest rogue waves, depends on initial wave amplitude an…
We propose a new conceptual framework for the prediction of rogue waves and third-order space-time extremes of wind seas that relies on the Tayfun (1980) and Janssen (2009) models coupled with Adler-Taylor (2009) theory on the Euler characteristics of random fields. Extreme statistics of the Andrea rogue wave event are examined capitalizing on European Reanalysis (ERA)-interim data. A refinement of Janssen's (2003) theory suggests that in realistic oceanic seas characterized by short-crested multidirectional waves, homogeneous and Gaussian initial conditions become irrelevant as the wave field…
In this paper we propose an extended Kundu-Eckhaus equation (KEE) for modeling the dynamics of skewed rogue waves emerging in the vicinity of a wave blocking point due to opposing current. The equation we propose is a KEE with an additional potential term therefore the results presented in this paper can easily be generalized to study the quantum tunneling properties of the rogue waves and ultrashort (femtosecond) pulses of the KEE. In the frame of the extended KEE, we numerically show that the chaotic perturbations of the ocean current trigger the occurrence of the rogue waves on the ocean su…
In this work we improve forecasting of Sea Surface Height (SSH) and current velocity (speed and direction) in oceanic scenarios. We do so by resorting to Random Forests so as to predict the error of a numerical forecasting system developed for the Santos Channel in Brazil. We have used the Santos Operational Forecasting System (SOFS) and data collected in situ between the years of 2019 and 2021. In previous studies we have applied similar methods for current velocity in the channel entrance, in this work we expand the application to improve the SHH forecast and include four other stations in t…
We discuss the possible usage of the compressive sampling for the early detection of rogue waves. One of the promising techniques for the early detection of the oceanic rogue waves is to measure the triangular Fourier spectra which begin to appear at the early stages of their development. For the early detection of the rogue waves it is possible to treat such a spectrum as a sparse signal since we would mainly be interested in the high amplitude triangular region located at the central wavenumber. Therefore compressive sampling can be a very efficient tool for the rogue wave early warning syst…
Here, we show through a statistical analysis that rogue waves in broadband non-breaking seas are spatially asymmetric. In addition to the top-down asymmetry due to nonlinear effects, we show that the two troughs adjacent to the rogue wave crest are generally of different depths, which is unlike the conventional picture of rogue waves with symmetric fore and aft troughs often obtained from model equations. The rogue-wave trough asymmetry is demonstrated for sea states 4 to 6 on Douglas Sea Scale. Considering the deepest trough leads to approximately 10% increase in the calculation of the mean r…
Nonlinear instability and refraction by ocean currents are both important mechanisms that go beyond the Rayleigh approximation and may be responsible for the formation of freak waves. In this paper, we quantitatively study nonlinear effects on the evolution of surface gravity waves on the ocean, to explore systematically the effects of various input parameters on the probability of freak wave formation. The fourth-order current-modified nonlinear Schrödinger equation (CNLS4) is employed to describe the wave evolution. By solving CNLS4 numerically, we are able to obtain quantitative predictions…
The occurrence of extreme (rogue) waves in the ocean is for the most part still shrouded in mystery, as the rare nature of these events makes them difficult to analyze with traditional methods. Modern data mining and machine learning methods provide a promising way out, but they typically rely on the availability of massive amounts of well-cleaned data. To facilitate the application of such data-hungry methods to surface ocean waves, we developed FOWD, a freely available wave dataset and processing framework. FOWD describes the conversion of raw observations into a catalogue that maps characte…
The issue of rogue wave lifetimes is addressed in this study, which helps to detail the general picture of this dangerous oceanic phenomenon. The direct numerical simulations of irregular wave ensembles are performed to obtain the complete accurate data on the rogue wave occurrence and evolution. The simulations are conducted by means of the HOS scheme for the potential Euler equations; purely collinear wave systems, moderately crested and short-crested sea states have been simulated. We join instant abnormally high waves in close locations and close time moments to new objects, rogue events, …
The mechanism of a rogue water wave is still unknown. One popular conjecture is that the Peregrine wave solution of the nonlinear Schrödinger equation (NLS) provides a mechanism. A Peregrine wave solution can be obtained by taking the infinite spatial period limit to the homoclinic solutions. In this article, from the perspective of the phase space structure of these homoclinic orbits in the infinite dimensional phase space where the NLS defines a dynamical system, we exam the observability of these homoclinic orbits (and their approximations). Our conclusion is that these approximate homoclin…
As an example for complex systems with extreme events we investigate ocean wave states exhibiting rogue waves. We present a statistical method of data analysis based on multi-point statistics which for the first time allows grasping extreme rogue wave events in a statistically highly satisfactory manner. The key to the success of the approach is mapping the complexity of multi-point data onto the statistics of hierarchically ordered height increments for different time scales for which we can show that a stochastic cascade process with Markov properties is governed by a Fokker-Planck equation.…
In this essay we give an overview on the problem of rogue or freak wave formation in the ocean. The matter of the phenomenon is a sporadic occurrence of unexpectedly high waves on the sea surface. These waves cause serious danger for sailing and sea use. A number of huge wave accidents resulted in damages, ship losses and people injuries and deaths are known. Now marine researchers do believe that these waves belong to a specific kind of sea waves, not taken into account by conventional models for sea wind waves. This paper addresses to the nature of the rogue wave problem from the general vie…
We review recent progress in modeling the probability distribution of wave heights in the deep ocean as a function of a small number of parameters describing the local sea state. Both linear and nonlinear mechanisms of rogue wave formation are considered. First, we show that when the average wave steepness is small and nonlinear wave effects are subleading, the wave height distribution is well explained by a single "freak index" parameter, which describes the strength of (linear) wave scattering by random currents relative to the angular spread of the incoming random sea. When the average stee…
We consider the problem of large wave prediction in two-dimensional water waves. Such waves form due to the synergistic effect of dispersive mixing of smaller wave groups and the action of localized nonlinear wave interactions that leads to focusing. Instead of a direct simulation approach, we rely on the decomposition of the wave field into a discrete set of localized wave groups with optimal length scales and amplitudes. Due to the short-term character of the prediction, these wave groups do not interact and therefore their dynamics can be characterized individually. Using direct numerical s…
Generative Lagrangian data assimilation for ocean dynamics under extreme sparsity
Department of Applied Mathematics, University of California, Santa Cruz, CA 95064, USA（应用数学系，加州大学圣克鲁兹分校）；Department of Electrical and Computer Engineering, University of California, Santa Cruz, CA 95064, USA（电气与计算机工程系，加州大学圣克鲁兹分校）；Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, Raleigh, NC 27695, USA（海洋、地球与大气科学系，北卡罗来纳州立大学）
The numerical simulation of the nonlinear dynamics of the sea surface has shown that the collision of two groups of relatively low waves with close but noncollinear wave vectors (two or three waves in each group with a steepness of about 0.2) can result in the appearance of an individual anomalous wave whose height is noticeably larger than that in the linear theory. Since such collisions quite often occur on the ocean surface, this scenario of the formation of rogue waves is apparently most typical under natural conditions.
