Maximum entropy estimation of directional wave spectra from an array of wave probes

A procedure for estimating directional wave spectra from an array of wave probes based on the Maximum Entropy Method (MEM) is developed in the present paper. The MEM approach yields an angular spreading function at each frequency band consistent with the input cross-spectral density matrix. The method is evaluated using numerical simulations of directional sea states. The MEM is also used to analyze data obtained from the three-dimensional wave basin of the Hydraulics Laboratory, National Research Council of Canada. Finally, the MEM is compared with the Maximum Likelihood Method (MLM) and is shown to be a powerful tool for directional wave analysis.     

Directional spectrum methods for deterministic waves

This work discusses developments related to the generation and measurement of directional wave spectra in multi-directional wave tank using deterministic waves. The details of the generation method, based on the single summation method described by Jefferys (1987), are given and the capacity of the Edinburgh curved wave tank to generate such waves is assessed. The Maximum Likelihood Method (MLM) and one of its derivative, The Modified Maximum Likelihood Method (MMLM) (Isobe and Kondo, 1985), are adapted to the characteristics of deterministic waves. The methods are assessed both with simulated waves and real wave elevations from the Edinburgh curved tank. Both methods show very satisfactory results with very stable angular spreading estimates and good tracking of mean directions of propagation across frequencies. The adapted MLM compares favourably with the industry standard, the Bayesian Directional Method, while only taking a fraction of the time needed to the BDM to produce its spectral estimates.     

On the measurement of directional wave spectra at the southern Brazilian coast

The detailed reconstruction of the directional spectrum of wind waves from measurements of the wave field is an essential requirement for several applications, including the numerical modeling of wave evolution. Three reconstruction techniques that provide estimates of the directional distribution function D(f,θ), given the one-dimensional frequency spectrum, are compared using data from a coastal locality at the southern Brazilian coast. The techniques are the maximum entropy method (MEM), the Fourier Expansion Method using a cos² type function (FEM_cos) and the Fourier Expansion Method using a sech type function (FEM_sech). The main patterns of the wave climate at the study site are qualitatively assessed. Three main sea states, including swell, transition between local sea and swell, and directionally bimodal wind sea, are identified. Time series from three events associated with the main sea states provide test cases for inter comparison of the three reconstruction techniques. Maximum entropy estimates of D(f,θ) provide results that are more consistent than those obtained from the two FEM techniques in all cases considered.     

Some applications of maximum entropy spectral estimation to ocean waves and linear systems response in waves

In traditional spectral analysis the use of windows is required because the assumption is made that the data outside the sample are zero. A data window is simply equal to unity inside, and zero outside the sample. This is equivalent to the truncation of the autocovariance function beyond some lag after which zeros are added. In Fast Fourier Transform spectral estimation a spectral window with smaller or greater negative side lobes is used. Any use of windows results in a smearing or spectral leakage that limits the spectral resolution. Maximum entropy spectral estimation (MEM) is equivalent to an extrapolation of the autocovariance function being consistent with some model assumptions. The result of the extrapolation is an increased spectral resolution in the frequency domain. In applications of MEM no kind of windows are used. MEM is applied to describe wind wave scalar spectra. The technique can also be used in the plane to estimate directional spectra. Our results show that MEM is a powerful tool for estimation of scalar spectra and simulation of the sea surface. There are also theoretical reasons for assuming that MEM are superior to traditional methods when only short samples are available. Our preliminary results verify this assumption. We propose a data acquisition system based on MEM, and also show that the response of linear systems can be calculated and simulated by MEM.     

Question of the pitch-roll buoy response to ocean waves as a simple harmonic oscillator?

The assumption that the East and North deck slopes of a pitch–roll buoy respond to East and North sea slopes as simple harmonic oscillators is routinely made by the National Data Buoy Center (NDBC) and others producing directional wave data. Although directional wave data derived with this assumption usually appear to be of good quality, the validity of the assumption has not previously been more directly demonstrated. In this paper, a method is proposed to judge the validity of the assumption for any set of time series records of buoy angular motion. The proposed method is applied to 200 record sets taken by an NDBC buoy located at ocean station 46024, and to five record sets taken by another NDBC buoy at 46051. For the 46024 data, it was demonstrated that the simple harmonic oscillator assumption was near perfectly valid. For the smaller 46051 data set, the simple harmonic oscillator assumption was shown to be slightly less valid.     

Wave direction analysis from data buoys

This paper describes a new procedure of directional wave analysis from pitch-roll buoy measurements. The two previous procedures adopted by the National Data Buoy Center (NDBC) [Steele, K. E., Lau, J. C. K. and Hsu, Y.-H. L. (1985) Theory and application of calibration techniques for an NDBC directional wave measurement buoy. IEEE Journal of Oceanic Engineering OE-10(4), 382-396; Steele, K. E., Teng, C.-C. and Wang, D. W. C. (1992) Wave direction measurements using pitch-roll buoys. Ocean Engineering 19(4), 349-375] are relevant to our formulations. In these two studies, an estimate for the total phase shift of the sea surface displacement/slope spectra from the measured buoy heave/pitch and heave/roll spectra was calculated either by a weighted average method or a maximum heave/pitch quad-spectrum method. These two formulations were based on a fundamental assumption of symmetric hull-mooring effect on pitch and roll motions, which will never be true in the oceans. In the present study we essentially incorporate the basic formulations of NDBC, but calculate two estimates for this total phase shift.Examples of directional wave analysis from data measured by a 3 m diameter discus buoy during Typhoon Herb are presented in this paper. This data set was also analyzed by the weighted average method of Steele et al. (1985) which yielded unsatisfactory results of wave directions during severe wave climates.     

Evaluating the efficacy of SVMs,BNs, ANNs and ANFIS in wave height prediction

Wave Height (WH) is one of the most important factors in design and operation of maritime projects. Different methods such as semi-empirical, numerical and soft computing-based approaches have been developed for WH forecasting. The soft computing-based methods have the ability to approximate nonlinear wind–wave and wave–wave interactions without a prior knowledge about them. In the present study, several soft computing-based models, namely Support Vector Machines (SVMs), Bayesian Networks (BNs), Artificial Neural Networks (ANNs) and Adaptive Neuro-Fuzzy Inference System (ANFIS) are used for mapping wind data to wave height. The data set used for training and testing the simulation models comprises the WH and wind data gathered by National Data Buoy Center (NDBC) in Lake Superior, USA. Several statistical indices are used to evaluate the efficacy of the aforementioned methods. The results show that the ANN, ANFIS and SVM can provide acceptable predictions for wave heights, while the BNs results are unreliable.     

Experiments on the reflection coefficients of a detached breakwater in a directional wave field

Results of estimating the directional wave fields in front of a detached breakwater are presented here in this paper. Two of non-phase-locked methods, i.e., the Maximum Likelihood Method (MLM) and the Extended Maximum Entropy Principle Method (EMEP), were adopted for the purpose. In general, the latter outperforms the former. It is shown that the reflection coefficients decrease with increasing distances away from the detached breakwater, and within four wavelengths from the detached breakwater, the rate of the decrease is faster for wave fields having larger directional spreads. When the measuring distance is more than four wavelengths away from the detached breakwaters, the reflection coefficients tend to reach to a constant value. It is shown that, with the use of the non-phase-locked EMEP method, the effective region can be extended, as compared with the results of Huntley and Davidson [J. Waterw. Port Coast. Ocean Eng. 124 (1998) 312].     

海浪双峰方向分布的一种物理解释

用 １８个波高计组成的直径为 ４０ｃｍ的圆形阵列在大型风浪槽内系统地测量了风浪和涌浪方向谱。用两种分辨力较高的方向谱估计方法最大似然法（ＭＬＭ）和贝叶斯方法（ＢＤＭ）分析的结果表明：风浪高频域出现的依赖于估计方法的双峰方向分布是一种物理假象;在较成长的涌浪低频域,得到跟传播方向对称、两峰间隔大约６０°－９０°非常规则的双峰方向分布,它跟频率和涌浪的成长状态有关,而跟估计方法无关,这种现象可以用非线性波－波相互作用过弱,在不同方向之间不能有效传递能量来解释。     

Stochastic time-series simulation of wave parameters using ship observations

A stochastic simulation technique was used with ship wave observations, which form the largest world-wide data base of wave information. Twenty years of wave parameter (height, period, and direction) observations from the Comprehensive Ocean–Atmosphere Data Set (COADS) were used as the input data. Simulations were compared to four years of wave parameters from a National Data Buoy Center (NDBC) data buoy near Monterey Bay, CA. The comparisons are satisfactory with differences mainly caused by biases between ship observations and buoy data. The stochastic simulation technique is attractive because it is computationally efficient and few decisions are required for its application. The applied techniques can be employed with global COADS data to simulate wave conditions at many world-wide locations where measurements and hindcasts by computer models do not exist.     

Nearshore directional wave measurements by surface-following buoy and acoustic Doppler current profiler

Directional energy spectra of nearshore surface waves were measured for a 3-year period (2004–2007) at a site with mean depth 14 m and mean tidal range 2.1 m. Triaxys surface-following wave buoys reported hourly directional wave energy spectra and wave parameters near the offshore end of the Savannah River Entrance Channel, Georgia, USA. An acoustic Doppler current profiler (ADCP) was located beside the wave buoy for 3 months. Directional and non-directional surface wave energy spectra and the corresponding bulk wave parameters (height, period, and direction) are compared for the two systems. Most parameters derived from the spectra agree closely; the most significant differences were found at the upper and lower frequency measurement limits, where signal-to-noise ratios were lower. The wave buoy consistently reports a small amount of energy below 0.05 Hz that does not appear in the ADCP-derived spectra and does not appear to be related to the mooring system. This leads to larger mean and peak periods reported by the buoy. All directional spectra were computed using the Maximum Entropy Method for both instruments, but the buoy, with spectra derived from six independent time series, provides lower directional resolving power than the ADCP, which utilizes twelve time series. Both systems gave similar results defining mean and peak wave directions, with the primary difference being that the ADCP indicates energy to be more tightly concentrated around the peak direction.     

Theory and application of calibration techniques for an NDBC directional wave measurements buoy

The National Data Buoy Center (NDBC) of the National Oceanic and Atmospheric Administration (NOAA) deployed a 10-m-diameter discus-type hull in the Pacific Ocean some 185 km southwest of Los Angeles, California, in April 1984. Aboard this hull was an electronic system capable of acquiring, processing, and transmitting to shore directional wave measurements. For this system to produce accurate data, a number of factors had to be taken into account. These factors included noise, amplitude and phase alterations due to mechanical and electrical components, and magnetic fields arising from the hull. Comprehensive calibration and verification techniques were developed and applied to ensure data quality. The system configuration is described with emphasis on the methods used in the data processing to correct for the various factors. Examples of the resulting corrected data are given.     

推算波浪多年一遇波高的新方法

采用卫星遥感推算多年一遇波高的新方法，对中国近海多年一遇波高进行推算研究。运用卫星遥感散射计得到的风资料推算了整个渤海区域波浪多年一遇的波高分布。通过与有实测资料4个点推算的结果比较表明：卫星资料和实测计算的误差随着重现期的增大而减小，最大误差百年和50年一遇波高为20cm；而百年和50年一遇波高正是工程中最为关心的，说明运用卫星遥感散射计风资料推算渤海多年一遇波高可行且结果合理。本研究方法为进一步运用和完善卫星遥感资料推算中国近海多年一波高场提供了一条新途径。     

海浪方向谱估计方法的比较

分别利用数值模拟和实测资料对目前被认为分辨力较高的最大似然法(MLM)、扩展本征矢方法(EEV)、扩展最大滴方法(EMEP)以及贝叶斯方法(BDM)等四种海浪方向谱估计方法的可靠性进行了分析,从不同频率、不同噪声水平和不同方向集中度三个角度检验其再现性、稳定性和实用性,结果表明MLM、EEV和BDM大致给出相同的方向分布,其中BDM的再现性最好,但实用性逊于MLM和EEV,EMEP由于稳定性差,不适用于实测资料的分析.     

Sea state characterisation of the test site of an offshore wave energy plant

The Archimedes Wave Swing (AWS) pilot plant, rated at 2 MW, was tested offshore at Póvoa de Varzim in northern Portugal in 2004, gathering a large team of engineers and scientists. The Mechanical Engineering Department of Instituto Superior Técnico was involved, following previous work done for the project. The present paper describes one of the studies conducted. It aims to characterise the sea state at the actual location of the pilot plant using the available pressure sensors. Two approaches were performed: a first one purely based in linear wave theory, neglecting the presence of the device, and a second one, based on the results from a Boundary Element Method (BEM) code named AQUADYN, originally developed at École Centrale de Nantes (France), which allowed a detailed quantification of the effects of the presence of the plant on the wave profile directly above it. Comparisons with a Datawell Waverider buoy located at a certain distance from the plant also played an important role in the current study.     

分析海浪方向谱的扩展本征矢方法Ⅰ.方法的导出

基于交叉谱矩阵可以按本征值划分为信号和含有噪音部分的思想，提出了一种自直接测量数据估计海浪方向谱的方法。该方法称为扩展本征矢方法（EEV），可应用于单点测量系统、仪器阵列以及由二者构成的复合阵列。现有的某些估计方法（如最大似然方法及其扩展形式等）仅是此方法的某种特例。     

Measured and modeled wave overtopping on a natural beach

The rate of wave overtopping of a barrier beach is measured and modeled. Unique rate of wave overtopping field data are obtained from the measure of the Carmel River, California, lagoon filling during a time when the lagoon is closed-off with no river inflow. Volume changes are based on measured lagoon height changes applied to a measured hypsometric curve. Wave heights and periods are obtained from directional wave spectra data in 15 m fronting the beach. Beach morphology was measured by GPS walking surveys. Three empirical overtopping models by Van der Meer and Janssen (1995), Hedges and Reis (1998) and Pullen et al. (2007) with differing parameterizations on wave height, period and beach slope and calibrated using extensive laboratory data obtained over plane, impermeable beaches are applied in a quasi-2D manner and compared with the field observations. Three overtopping events are considered when morphology data were available less than 2 weeks prior to the event. The models are tuned to fit the data using a reduction factor to account for beach permeability, berm characteristics, non-normal wave incidence and surface roughness influence. In addition, the run-up model by Stockdon et al. (2006) based on field data is examined and found to underestimate run-up as the calculated values were too small to predict any of the observed overtopping. The three overtopping models performed similarly well with values of 0.72–0.87 for the two narrow-banded wave cases, with an average reduction factor of 0.78. The European model (Pullen et. al., 2007) performed best overall and in particular for the case of the broad-banded, double peaked wave spectrum.     

利用SWAN波浪模式研究北部湾海域台风浪场的波浪方向谱特征

以高精度再分析风场为驱动,利用SWAN模式模拟了台风“达维”Damrey（2005）经过北部湾海域时的波浪场。通过与实测的风和波浪实测对比发现,波浪后报结果与实测结果符合较好。文章给出了台风浪期间波高、周期、波长和波向等要素的分布特征,讨论了以台风眼为中心不同海域的波浪方向谱特征。本文最后分析了台风期间实测波浪能谱的变化特征。     

On the stability of the moments of the maximum entropy wind wave spectrum

The stability of some current wind wave parameters as a function of high-frequency cut-off and degrees of freedom of the spectrum has been numerically investigated when computed in terms of the moments of the wave energy spectrum.From the Pierson-Moskovitz wave spectrum type, a sea surface profile is simulated and its wave energy spectrum is estimated by the Maximum Entropy Method (MEM). As the degrees of freedom of the MEM spectral estimation are varied, the results show a much better stability of the wave parameters as compared to the classical periodogram and correlogram spectral approaches. The stability of wave parameters as a function of high-frequency cut-off result the same as obtained by the classical techniques.     

分析海浪方向谱的扩展本征矢方法

采用Ｌｏｎｇｕｅｔ－Ｈｉｇｇｉｎｓ形式的方向分布函数作为已知谱，用模拟数据检验了作者是所提出的估计方法ＥＥＶ合理性，并与扩展最大似然方法（ＥＭＬＭ）及Ｌｙｇｒｅ等（１９８６）的最在熵方法（ＭＥＭ）作了比较，在验证和比较中，使用纵摇－横摇浮标，星形阵形和ＣＥＲＣ阵列作为复合阵列，计算表明，ＥＥＶ优于ＥＭＬＭ和ＥＭＥ。最后将ＥＥＶ和ＥＭＬＭ两种方法应用于仪器阵列的外海观测数据，得到了比较合理的海浪方向