Detection of nonlinear waves and their contribution to ocean wave spectra Part II: Observation

This is a Part II of a paper of nonlinearities of wind waves in the deep open ocean. As shown in Part I, bound waves in deep sea are detectable by extracting secondorder Doppler spectra from the Doppler spectra of HF (high-frequency) radio waves scattered from the sea surface. There is a remarkable agreement between the calculated and measured Doppler spectra, considering the noise levels in measured Doppler spectra and the uncertainties in directional properties. The theoretical expression for bound waves is thus verified. Furthermore, the upper limit in calculating the Doppler spectra for the second-order approximation is presented from field observations, although we cannot conclude that it is equivalent to the limitation of the second-order bound wave theory. It is shown that analysis of radio wave scattering by the sea surface is one useful means of understanding the nonlinear properties of ocean waves.     

Comparison of inversion algorithms for HF radar wave measurements

All ocean wave components contribute to the second-order scattering of a high-frequency (HF) radio wave by the sea surface. It is therefore theoretically possible to estimate the ocean wave spectrum from the radar backscatter. To extract the wave information, it is necessary to solve the nonlinear integral equation that describes the relationship between the backscatter spectrum and the ocean wave directional spectrum. Different inversion techniques have been developed for this problem by different researchers, but there is at present no accepted “best” method. This paper gives an assessment of the current status of two methods for deriving sea-state information from HF radar observations of the sea surface. The methods are applied to simulated data and to an experimental data set with sea-truth being provided by a directional wave buoy     

Extraction of Ocean Wave Spectra From Simulated Noisy Bistatic High-Frequency Radar Data

An algorithm is developed for the inversion of bistatic high-frequency (HF) radar sea echo to give the nondirectional wave spectrum. The bistatic HF radar second-order cross section of patch scattering, consisting of a combination of four Fredholm-type integral equations, contains a nonlinear product of ocean wave directional spectrum factors. The energy inside the first-order cross section is used to normalize this integrand. The unknown ocean wave spectrum is represented by a truncated Fourier series. The integral equation is then converted to a matrix equation and a singular value decomposition (SVD) method is invoked to pseudoinvert the kernel matrix. The new algorithm is verified with simulated radar Doppler spectrum for varying water depths, wind velocities, and radar operating frequencies. To make the simulation more realistic, zero-mean Gaussian noise from external sources is also taken into account     

CODAR wave measurements from a North Sea semisubmersible

CODAR, a high-frequency (HF) compact radar system, was operated continuously over several weeks aboard the semisubmersible oil platform Treasure Saga for the purpose of wave-height directional measurement and comparison. During North Sea winter storm conditions, the system operated at two different frequencies, depending on the sea state. Wave data are extracted from the second-order backscatter Doppler spectrum produced by nonlinearities in the hydrodynamic wave/wave and electromagnetic wave/scatter interactions. Because the floating oil rig itself moves in response to long waves, a technique has been developed and successfully demonstrated to eliminate to second order the resulting phase-modulation contamination of the echo, using separate accelerometer measurement of the platform's lateral motions. CODAR wave height, mean direction, and period are compared with data from a Norwegian directional wave buoy; in storm seas with wave heights that exceeded 9 m, the two height measurements agreed to within 20 cm RMS, and the mean direction to better than 15° RMS     

An analysis of the second-order Doppler return from the ocean surface

A theoretical analysis of the scattering of high-frequency (HF) electromagnetic waves from rough surfaces is proposed. The analysis, when applied to a model for the ocean surface, leads to new predictions of its second-order backscattered radar cross section in addition to those provided by existing theories. Some of the predictions of the theory have already been verified experimentally.     

Wave crest and trough distributions in a broad-banded directional wave field

It is well established that the modulational instability enhances the probability of occurrence for extreme events in long crested wave fields. Recent studies, however, have shown that the coexistence of directional wave components can reduce the effects related to the modulational instability. Here, numerical simulations of the Euler equations are used to investigate whether the modulational instability may produce significant deviations from second-order statistical properties of surface gravity waves when short crestness (i.e., directionality) is accounted for. The case of a broad-banded directional wave field (i.e. wind sea) is investigated. The analysis is concentrated on the wave crest and trough distribution. For completeness a comparison with a unidirectional wave field is presented also. Results will show that the distributions based on second-order theory provide a good estimate for the simulated crest and trough height also at low probability levels.     

Radiophysical investigations of sea roughness (radiooceanography) at the Ukrainian Academy of Sciences

Experiments on the scattering of radio waves in the range 200 m to 3 cm from a rough sea surface are described. Amplitude, frequency, and space-time characteristics of scattered radio signals at different states of the sea surface are presented. It is shown that the problem of the short and medium wave scattering from the sea can be solved by the perturbance method. In this case the mechanism of scattering is of "resonant" character. The intensity of the backscatter signals is proportional to the density of the spatial spectrum on the half-length of the radio waves. The high frequency radio wave scattering is well described by a two-scale model of the scattering surface, "ripple on the large wave." The intensity of scattered radio signals is also proportional to the spectrum density of "ripples" whose length is approximately equal to half a radio wave. The effect of the large waves is to modulate the amplitude of a scattered radio signal and to broaden its frequency spectrum. Methods of solution of the reverse problem were considered. This allowed determination of parameters of sea roughness by characteristics of scattered radio signals. The principles of design of the corresponding equipment are described.     

Rogue waves during Typhoon Trami in the East China Sea

As concluded from physical theory and laboratory experiment,it is widely accepted that nonlinearities of sea state play an important role in the formation of rogue waves;however,the sea states and corresponding nonlinearities of real-world rogue wave events remain poorly understood.Three rogue waves were recorded by a directional buoy located in the East China Sea during Typhoon Trami in August 2013.This study used the WAVEWATCHⅢmodel to simulate the sea state conditions pertaining to when and where those rogue waves were observed,based on which a comprehensive and full-scale analysis was performed.From the perspectives of wind and wave fields,wave system tracking,High-Order Spectral method simulation,and some characteristic sea state parameters,we concluded that the rogue waves occurred in sea states dominated by second-order nonlinearities.Moreover,third-order modulational instabilities were suppressed in these events because of the developed or fully developed sea state determined by the typhoon wave system.The method adopted in this study can provide comprehensive and full-scale analysis of rogue waves in the real world.The case studied in this paper is not considered unique,and rules could be found and confirmed in relation to other typhoon sea states through the application of our proposed method.     

Difference-frequency wave loads on a large body in multi-directional waves

The second-order difference-frequency wave forces on a large three-dimensional body in multi-directional waves are computed by the boundary integral equation method and the so-called FML formulation (assisting radiation potential method). Semi-analytic solutions for a bottom-mounted vertical circular cylinder are also developed to validate the numerical method. Difference-frequency wave loads on a bottom-mounted vertical cylinder and stationary four legs of the ISSC tension-leg platform (TLP) are presented for various combinations of incident wave frequencies and headings. These force quadratic transfer functions (QTF) can directly be used in studying slowly varying wave loads in irregular short-crested seas described by a particular directional spectrum. From our numerical results, it is seen that the slowly varying wave loads are in general very sensitive to the directional spreading function of the sea, and therefore wave directionality needs to be taken into account in relevant ocean engineering applications. It is also pointed out that the uni-directionality of the sea is not necessarily a conservative assumption when the second-order effects are concerned.     

Measurement of oceanic wind speed from HF sea scatter by skywave radar

Remote measurements of the spatial mean ocean wind speeds were obtained using Doppler spectra resolved to 0.08 Hz from high-resolution HF skywave-radar backscatter measurements of the ocean surface. A standard deviation of 2.4 m/s resulted from the correlation of observed winds over the ocean and the broadening of the Doppler spectra in the vicinity of the higher first-order Bragg line. This broadening, for Doppler spectra unperturbed by the ionospheric propagation, is proportional to the increase in power caused by higher order hydrodynamic and electromagnetic effects in the vicinity of the Bragg line and inversely proportional to the square root of the radio frequency. A lower bound on the measure of wind speed was established at 5 m/s by the low resolution spectral processing and low second-order power. An upper limit is suggested by the steep slope in the region of the sea backscatter spectrum outside the square root of two times the first-order Bragg line Doppler.     

Nonlinear wave loads on a vertical cylinder

Nonlinear contributions due to elevation of the free surface, the dynamic head, and the second-order velocity potential on the wave loads are presented in closed-form expressions. Such nonlinearities resulting from large-amplitude ocean waves are associated with the irrotational flow interacting with a fixed bottom-mounted vertical cylinder piercing the surface. These are expressed in the form of dynamic, waterline and quadratic forces all of which depend on the square of the wave amplitude. The appropriate modifications are made to both the classical Morison equation and the well-known linear diffraction theory of MacCamy and Fuchs for accounting the second-order effects.A limited comparative study is performed to verify the present theoretical derivations. In general, satisfactory agreements have been obtained with the test results from various laboratory studies by different researchers. However, under certain environmental conditions, some discrepancies still exist with the measured results.     

On the Development of a Second-Order Bistatic Radar Cross Section of the Ocean Surface: A High-Frequency Result for a Finite Scattering Patch

The development of a model for the second-order bistatic high-frequency (HF) radar cross section on an ocean surface patch remote from the transmitter and receiver is addressed. A new approach is taken that allows a direct comparison with existing monostatic cross sections for finite regions of the ocean surface. The derivation starts with a general expression for the bistatically received second-order electric field in which the scattering surface is assumed to be of small height and slope. The source field is taken to be that of a vertically polarized dipole, and it is assumed that the ocean surface can be described, as is usually done, by a Fourier series in which the coefficients are zero-mean Gaussian random variables. Subsequently, a bistatic cross section of the surface, normalized to patch area, is derived. The result is verified by the following two means: 1) the complete form of the bistatic HF radar cross section in backscattering case is shown to contain an earlier monostatic result that has, itself, been used extensively in radio oceanography applications; and 2) the bistatic electromagnetic coupling coefficient is shown to reduce exactly to the monostatic result when backscattering geometry is imposed. The model is also depicted and discussed based on simulated data     

一种实体嵌入的三维海浪可视化方法

针对实体嵌入对海面的随机干扰问题,提出一种海浪三维可视化方法。该方法采用扰动的思想改进了海浪的数学模型,并利用信息更加全面的方向谱描述海浪,进而将方向谱和改进海浪模型共同应用于海浪的三维可视化,形象展示实体嵌入时对海浪的随机性影响,并给出了三维实体的建模方法。通过多组实体嵌入的海浪可视化仿真试验的比较与分析,验证了本文提出方法的有效性。     

Deterministic decomposition of pitch-and-roll buoy measurements

Wave-gauge arrays, current meters and pitch-and-roll buoys are widely used for the recording of directional properties of ocean waves. For the determination of directional spectra the traditional stochastic procedure usually includes the selection of a parameterized spreading function. The present theory, which is illustrated below for a pitch-and-roll buoy, decomposes the information into frequencies, amplitudes, directions, and also phases. Furthermore, this procedure requires no assumption of any function describing the expected form of the directional spread. The theory of this deterministic decomposition is described and compared to the traditional stochastic principles. Only for reasons of this comparison and presentation, the deterministically obtained directional distributions are fitted to normal distributions.Measurements taken by pitch-and-roll buoys and by current meter/wave gauge are presented and discussed. The remarkable tendency in the variation of the directional spread as a function of frequency is found for two quite different locations. To quantify the directional spread obtained from the deterministic method normal distributions are fitted, and the mean values and variances are plotted and discussed.     

HF radar data quality requirements for wave measurement

HF radar wave measurements are presented focussing on theoretical limitations, and thus radar operating parameters, and quality control requirements to ensure robust measurements across a range of sea states. Data from three radar deployments, off the west coast of Norway, Celtic Sea and Liverpool Bay using two different radar systems, WERA and Pisces, and different radio frequency ranges, are used to demonstrate the wave measurement capability of HF radar and to illustrate the points made. Aspects of the measurements that require further improvements are identified. These include modifications to the underlying theory particularly in high sea states, identification and removal of ships and interference from the radar signals before wave processing and/or intelligent partitioning to remove these from the wave spectrum. The need to match the radio frequency to the expected wave peak frequency and waveheight range, with lower radio frequencies performing better at higher waveheights and lower peak frequencies and vice versa, is demonstrated. For operations across a wide range of oceanographic conditions a radar able to operate at more than one frequency is recommended for robust wave measurement. Careful quality control is needed to ensure accurate wave measurements.     

Estimation Approaches of ηUV,PUV and UV Directional Wave Spectra and Their Comparison

The observation and estimation of directional spectra of sea waves is one of the essential subjects of study of oceanic dynamics. On the basis of the irregular linear wave theory, estimation methods for i/UV, PUV and VV directional wave spectra are derived. By using ij and PUV data measured in-situ, directional wave spectra are estimated, meanwhile the virtues and defects of various spectra are comparied. This method provide a basis for the observations of sea waves.     

Directional wave information from the SeaSonde

This paper describes methods used for the derivation of wave information from SeaSonde data, and gives examples of their application to measured data. The SeaSonde is a compact high-frequency (HF) radar system operated from the coast or offshore platform to produce current velocity maps and local estimates of the directional wave spectrum. Two methods are described to obtain wave information from the second-order radar spectrum: integral inversion and fitting with a model of the ocean wave spectrum. We describe results from both standard- and long-range systems and include comparisons with simultaneous measurements from an S4 current meter. Due to general properties of the radar spectrum common to all HF radar systems, existing interpretation methods fail when the waveheight exceeds a limiting value defined by the radar frequency. As a result, standard- and long-range SeaSondes provide wave information for different wave height conditions because of their differing radar frequencies. Standard-range SeaSondes are useful for low and moderate waveheights, whereas long-range systems with lower transmit frequencies provide information when the waves are high. We propose a low-cost low-power system, to be used exclusively for local wave measurements, which would be capable of switching transmit frequency when the waveheight exceeds the critical limit, thereby allowing observation of waves throughout the waveheight range.     

Space–time measurements of oceanic sea states

Stereo video techniques are effective for estimating the space–time wave dynamics over an area of the ocean. Indeed, a stereo camera view allows retrieval of both spatial and temporal data whose statistical content is richer than that of time series data retrieved from point wave probes. We present an application of the Wave Acquisition Stereo System (WASS) for the analysis of offshore video measurements of gravity waves in the Northern Adriatic Sea and near the southern seashore of the Crimean peninsula, in the Black Sea. We use classical epipolar techniques to reconstruct the sea surface from the stereo pairs sequentially in time, viz. a sequence of spatial snapshots. We also present a variational approach that exploits the entire data image set providing a global space–time imaging of the sea surface, viz. simultaneous reconstruction of several spatial snapshots of the surface in order to guarantee continuity of the sea surface both in space and time. Analysis of the WASS measurements show that the sea surface can be accurately estimated in space and time together, yielding associated directional spectra and wave statistics at a point in time that agrees well with probabilistic models. In particular, WASS stereo imaging is able to capture typical features of the wave surface, especially the crest-to-trough asymmetry due to second order nonlinearities, and the observed shape of large waves are fairly described by theoretical models based on the theory of quasi-determinism (Boccotti, 2000). Further, we investigate space–time extremes of the observed stationary sea states, viz. the largest surface wave heights expected over a given area during the sea state duration. The WASS analysis provides the first experimental proof that a space–time extreme is generally larger than that observed in time via point measurements, in agreement with the predictions based on stochastic theories for global maxima of Gaussian fields.     

Directional spectrum analysis and statistics obtained from ERS-1 SAR wave images

This work examines ERS-1 (the first European Remote Sensing Satellite) SAR (Synthetic Aperture Radar) water surface wave images over Hualien of Taiwan, indicating that the variation of SAR signals in space domain is similar to in situ wave data's in time domain. Some statistical properties of SAR data are investigated. The Rayleigh distribution function closely corresponds with the histogram of wave heights, but the Gaussian one cannot for water surface displacements. Evidence reveals that SAR wave signals do not respond well to actual ocean waves effectively. As wave spectral analysis of available SAR data reveals, the appropriate sample size of SAR wave image, sampling average, and moving average should be taken carefully to accurately confirm directional power spectra. Moreover, SAR spectra are compared with in situ ones, confirming that peak frequencies correlate well and wave directions approximately agree with each other. Some differences between both spectral shapes remain somewhat unclear and require further study. Nevertheless, in this study, ERS-1 SAR power spectra verified the feasibility of deriving an appropriate dominant wave direction and peak frequency.     

Three-dimensional nonlinear random wave groups in intermediate water depth

High waves at ocean occur during a complex space–time evolution of wave groups. In this paper the nonlinear structure of three-dimensional sea wave groups at intermediate water depth is investigated. To this purpose, the Boccotti's Quasi-Determinism theory is firstly applied to describe the linear wave groups when a given exceptionally high crest occurs. Then, the second-order correction to the linear solution is derived for the general condition of three-dimensional wave groups, at a finite water depth. Several numerical applications, finally, have been carried out in order to show how both the spectral bandwidth and the directional spreading modify the nonlinear high waves at different water depth.