Statistical properties of successive wave heights and successive wave periods

Two successive wave heights are modeled by a Gaussian copula, which is referred to as the Nataf model. Results with two initial distributions for the transformation are presented, the Næss model [Næss A. On the distribution of crest to trough wave heights. Ocean Engineering (1985);12(3):221–34] and a two-parameter Weibull distribution, where the latter is in best agreement with data. The results are compared with existing models. The Nataf model has also been used for modeling three successive wave heights.Results show that the Nataf transformation of three successive wave heights can be approximated by a first order autoregressive model. This means that the distribution of the wave height given the previous wave height is independent of the wave heights prior to the previous wave height. Thus, the joint distribution of three successive wave heights can be obtained by combining conditional bivariate distributions. The simulation of successive wave heights can be done directly without simulating the time series of the complete surface elevation.Successive wave periods with corresponding wave heights exceeding a certain threshold have also been studied. Results show that the distribution for successive wave periods when the corresponding wave heights exceed the root-mean-square value of the wave heights, can be approximated by a multivariate Gaussian distribution.The theoretical distributions are compared with observed wave data obtained from field measurements in the central North Sea and in the Japan Sea, with laboratory data and numerical simulations.     

由模拟波面统计分析波高－周期联合分布

阐述以实测或拟合海浪谱为靶谱，用等能量分割法作波面数值模拟，从而利用模拟波面统计分析波高（Ｈ）－周期（Ｔ）联合分布。对波候的Ｈ－Ｔ联合分布、长时段海浪连续记录的Ｈ－Ｔ联合分布以及风浪、涌浪和混合浪的Ｈ－Ｔ联合分布进行了讨论。结果表明，只要已知特征波高和周期，就可反演出模拟波面，进而估测Ｈ－Ｔ联合分布情况，这对了解与各种特征波高对应的周期问题及在海洋工程应用上有一定的参考意义。     

Short-term statistics of waves measured off Ratnagiri,eastern Arabian Sea

The short-term wave characteristics are required for design and operation of industrial facilities within the coastal areas. Water surface displacement measured using waverider buoy moored at 13 m water depth in the eastern Arabian Sea off the west coast of India have been analyzed to study the short-term statistics of waves covering full one year period. The study indicates that the values of the observed maximum wave height as a function of duration are not consistent with the theoretical expected value. There is significant variation (1.29–2.19) in the ratio between highest 1% wave and significant wave height compared to the theoretical value of 1.67. The data recorded at 13 m water depth indicates that the significant wave height is ∼8% lower than that predicted by the conventional Rayleigh distribution. The theoretical bivariate log-normal distribution represents the joint distributions of wave heights and periods for the study area.     

大洋波浪特征：西太平洋波浪调查研究报告

本文分析了西太平洋实测的波高分布、同期分布、波高和周期的联合分布,以及波谱的分析。分析结果指出,大洋波浪的波高和周期都大于近海波浪的波高和周期。分布特征也不同于近海。波浪多属混合浪,会出现典型的双峰谱。     

The difference between the joint probability distributions of apparent wave heights and periods and individual wave heights and periods

The joint distribution of wave heights and periods of individual waves is usually approximated by the joint distribution of apparent wave heights and periods. However there is difference between them. This difference is addressed and the theoretical joint distributions of apparent wave heights and periods due to Longuet-Higgins and Sun are modified to give more reasonable representations of the joint distribution of wave heights and periods of individual waves. The modification has overcome an inherent drawback of these joint PDFs that the mean wave period is infinite. A comparison is made between the modified formulae and the field data of Goda, which shows that the new formulae consist with the measurement better than their original counterparts.     

Hilbert变换在海浪波高与周期联合分布函数中的应用

Longuet-Higgins(1983)[1]导出了波高与周期的联合分布函数,此分布函数虽然与实际数据符合良好,但存在很大的缺陷,如:由此分布函数得出的波高分布为形式较为复杂的非Rayleigh分布,很难应用于工程计算中。孙孚(1988a)[2]应用射线理论导出了一种波高与周期联合分布,虽然弥补了Longuet-Higgins的一些缺陷,但推导过程过于复杂。本文在窄谱假定下通过应用Hilbert变换方法得出新的分布函数并与前两者比较,表明Hilbert变换的方法不但简便,而且完全克服Longuet-Higgins的不足,可以方便的应用于工程计算中。本文也为Hilbert变换的方法在工程中的应用提供了理论依据。     

Correlation between individual waves in a real sea state

The correlation between individual waves in a real sea state has a central role in existing theories of wave grouping. The attractive Kimura (1980) theory has two critical assumptions, that the sequence of individual wave heights follows a Markov process and that the joint distribution of consecutive wave heights follows a bivariate Rayleigh form. Analysis of measured water surface records suggests that sequences of individual waves can reasonably be described as a first order mixed autoregressive, moving-average or ARMA process, though a distinction among ARMA (1,0), ARMA (0,1) and ARMA (1,1) models was beyond the resolution of the data. These include the Markov or ARMA (1,0) model. The decisive detail, the joint distribution of consecutive wave heights in the sea state, was evaluated by a simulation methodology that is consistent with the Gaussian random wave model. The estimates are dependent on spectral shape and are consistently narrower and more sharply focussed at the peak than the corresponding bivariate Rayleigh estimate. The resulting predictions of run and group length statistics differ from the Kimura theory, though not by a sufficient margin to displace the Kimura theory as a pragmatic choice for wave grouping.     

Distributions of freak wave heights measured in the North Sea

We present a statistical analysis of some of the largest waves occurring during 793 h of surface elevation measurements collected during 14 severe storms in the North Sea. This data contains 104 freak waves. It is found that the probability of occurrence of freak waves is only weekly dependent on the significant wave height, significant wave steepness and spectral bandwidth. The probability does show a slightly stronger dependency on the skew and kurtosis of the surface elevation data, but on removing the contribution to these measures from the presence of the freakwaves themselves, this dependency largely disappears.Distributions of extreme waves are modelled by fitting Generalised Pareto distributions, and extreme value distributions and return periods are given for freak waves in terms of the empirical fitted parameters. It is shown by comparison with these fits that both the Rayleigh distribution and the fit of Nerzic and Prevosto severely under-predict the probability of occurrence of extreme waves. For the most extreme freak wave in our data, the Rayleigh distribution over-predicts the return period by about 300 times when compared to the fitted model.     

响水近岸海域波浪特性研究

基于响水波浪站累计一整年的现场观测资料，分析了波高和波周期的年内变化特性，研究了波浪的统计特性和波谱特性，并总结归纳了该海域各特征波要素之间以及各波谱参数之间的转换关系。结果显示:响水海域全年有效波高的变化幅度在0.10~2.80 m之间，年平均值为0.56 m；最大波高的变化幅度在0.15~5.58 m之间，年平均值为0.93 m；平均波周期的变化范围为1.91~9.02 s，年平均值为3.90 s。夏季大波高发生频率明显要小于冬、春季节，波浪季节性变化较为显著。就波高和波周期分布而言，通过拟合得出的Weibull分布较为适合本海域实测波高分布和波周期分布。波谱特性方面，本海域双峰谱占到总数的62.5%，且低频谱峰值普遍高于高频谱峰值，其中低频谱峰出现在0.04 Hz左右，高频谱峰则出现在0.15~0.20 Hz之间，分别为本海域涌浪和风浪所集中的频率区间。采用回归分析方法进一步分析了各特征波要素之间以及各波谱参数之间的关系，发现多数波参数之间存在显著的相关性，但受波浪浅水变形影响，各参数之间的比值与理论深水关系有所区别。本文的研究成果可为沿海建筑物的设计以及防灾减灾提供参考和依据。     

宁波北仑海域风浪统计性质研究

根据宁波北仑海域的连续3a的实测风浪资料,讨论了在不同风速条件下波高分布和周期分布,以及两者的联合分布;通过与理论结果的比较,得出波高分布与Rayleigh分布基本符合,但有一些差异,周期分布与孙孚的理论周期分布较符合,而波高与周期的联合分布除了图形的形状以及大波对应的无因次周期的值与孙孚的理论值有差异外,两者吻合较好。     

双峰谱型海浪的波高和周期分布

本文利用收集到的实测双峰谱型海浪过程资料，把这些资料以波高和周期的相关系数为参数分成５组，讨论每组双峰谱型下被高、周期的统计分布．并探讨了相关系数对波高分布和周期分布的影响。     

Extreme waves for Kuwaiti territorial waters

The extreme significant wave heights and the corresponding wave periods were predicted for return periods of 12, 25, 50, 100 and 200 yr for 19 different locations in Kuwaiti territorial waters. Though the total coast length of Kuwait is only about 500 km including all islands and the total area of the Kuwaiti territorial water is about 7611 km², the extreme significant wave height vary from 1.86 to 4.02 m for 100 yr return period, among these 19 locations. In general Weibull distribution is found to fit the data well compared to the Gumbel distribution. The input wave data for the present work is obtained by hind casting waves using a WAM model. Wave data is hindcasted for a total period of 12 yr, starting from 1 January 1993 to 31 December 2004. From the joint probability of wave height and wave period, a simple polynomial relationship is obtained between the significant wave height and mean period for all the 19 locations. It is found that the wave period for wave heights of 100 yr return period cannot exceed 6.5 s. A large number of coastal projects are in progress and many new projects are planned for the near future in the Kuwaiti territorial waters. The results of the present study will be highly useful for optimal design of these projects.     

基于实测数据的台湾海峡中部波浪特征分析 *

利用台湾海峡中部2号大浮标2017年全年的实测波浪资料, 对海浪的基本波要素及其与风的相关性、波谱特性进行统计分析, 得出了重要特征波参数之间的回归关系和适合台湾海峡中部的海浪谱形式。研究结果显示: 1) 台湾海峡中部的常浪向是NE向, 强浪向是NNE向, 月均有效波高的变化范围为0.87~2.98m, 7月波高最小, 12月波高最大, 波周期与波高有着相似的月际变化趋势; 2) 主要波浪类型是以风浪为主的混合浪, 谱型上以单峰为主, 波高与风速整体上呈正相关关系, 大浪主要由台风和强劲的东北季风引起; 3) 波浪的平均周期与大部分特征波周期之间具有良好的线性相关性, NNE、NE方向的波浪有效波高和有效波周期线性相关性较强; 4) 相比于Jonswap谱, 规范谱一是更符合本区域的海浪谱模式, 给出了基于有效波高和谱峰周期拟合的规范谱一形式。这些研究成果可为海洋工程设计和波浪数值模拟提供参考。     

Statistical characteristics of long waves nearshore

The random long wave runup on a beach of constant slope is studied in the framework of the rigorous solutions of the nonlinear shallow water theory. These solutions are used for calculation of the statistical characteristics of the vertical displacement of the moving shoreline and its horizontal velocity. It is shown that probability characteristics of the runup heights and extreme values of the shoreline velocity coincide in the linear and nonlinear theory. If the incident wave is represented by a narrow-band Gaussian process, the runup height is described by a Rayleigh distribution. The significant runup height can also be found within the linear theory of long wave shoaling and runup. Wave nonlinearity nearshore does not affect the Gaussian probability distribution of the velocity of the moving shoreline. However the vertical displacement of the moving shoreline becomes non-Gaussian due to the wave nonlinearity. Its statistical moments are calculated analytically. It is shown that the mean water level increases (setup), the skewness is always positive and kurtosis is positive for weak amplitude waves and negative for strongly nonlinear waves. The probability of the wave breaking is also calculated and conditions of validity of the analytical theory are discussed. The spectral and statistical characteristics of the moving shoreline are studied in detail. It is shown that the probability of coastal floods grows with an increase in the nonlinearity. Randomness of the wave field nearshore leads to an increase in the wave spectrum width.     

Probability distribution of wave heights attenuated by salt marsh vegetation during tropical cyclone

Zero-crossing wave heights, obtained from the field measurement of random waves propagating through salt marsh vegetation (Spartina alterniflora) during a tropical storm, were analyzed to examine their probability distribution. Wave data (significant wave heights up to 0.4 m in 0.8 m depth) were collected over a two-day period along a 28 m transect using three pressure transducers sampling at 10 Hz. Wave height distribution was observed to deviate from the Rayleigh distribution. The observed probability densities of the larger wave heights were reduced significantly by vegetation, producing wave heights lower than those predicted by the Rayleigh distribution. Assuming Rayleigh distributed wave heights for the incident waves to the vegetation patch, existing vegetation-induced wave attenuation formulations are used to derive a special form of two-parameter Weibull distribution for wave heights in the inundated wetland. The scale parameter of the distribution is theoretically shown to be a function of the shape parameter, which agrees with the measurements, effectively reducing the proposed distribution to a one-parameter type. The derived distribution depends on the local parameters only and fits well to the observed distribution of wave heights attenuated by vegetation. Empirical relationships are developed to estimate the shape parameter from the local wave parameters.     

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.     

Application of double bounded probability density function for analysis of ocean waves

Ocean waves and forces induced by them on offshore structures are random in nature. Experience has shown that short term statistics of wave heights can be described by the Rayleigh distribution for narrow band spectra (Longuet-Higgins, 1952) and that the long term statistics or the evaluation of design wave is based on certain well known extreme value distribution such as mixed Frechet distribution (Thom, 1973a, b).This paper presents a new application of the double bounded probability density function to describe the ocean wave statistics. The prime importance is to estimate the most probable maximum wave height for offshore structural designs.     

Wave height distribution in shallow water

Probability distribution of shallow water wave heights, obtained from a pressure type recorder, are examined. It is tested with the theoretical distributions of (a) Rayleigh, (b) Weibull, (c) Gluhovski, (d) Ibrageemov and (e) Goda. The best fit is shown by the Gluhovski probability density function with a correlation coefficient greater than 0.8. The functions of Weibull, Ibrageemov and Goda fit only half of the tested cases. The role of wave steepness in the wave height distribution is found to be negligible.     

ON THE JOINT DISTRIBUTION OF THE PERIODS AND HEIGHTS OF SEA WAVES

Based on the ray theory of waves and the linear model of sea waves, a joint distribution of periods and heights is derived. The theoretical expression has the same merits as those of Lon-guet-Higgins', but the wave height distribution is still Rayleigh's.