Distribution Of Significant Wave Height Research Articles

Stokes drift estimation based on long-term variation of wave conditions

This article provides a simple analytical method which can be used to give estimates of the Stokes drift based on long-term variation of wave conditions. This is achieved by providing bivariate distributions of significant wave height with surface Stokes drift as well as with volume Stokes transport. These Stokes drift parameters are defined in terms of significant wave height and characteristic wave periods. This article presents the mean value and the standard deviation of these Stokes drift parameters, that is, more precisely the conditional expected values and the conditional variances for given significant wave height, as well as examples of results corresponding to typical field conditions. Based on, for example, global wave statistics, the present analytical results can be used to make estimates of the Stokes drift.

Response to "Discussion of 'Modelling significant wave height distributions with quantile functions for estimation of extreme wave heights' [Ocean Engneering 54 (2012) 119–131]" by Zai-Jin You and David Callaghan

The discussers have commented on the reference paper, discussing would be used and compared as done for example in Guedes various questions that include the method of quantiles, the characteristics of the data sets used in the examples and the conclusion about the choice among the various quantile distributions. The authors agree that whenever there is a distribution that has an analytical expression for its distribution F(x), the quantile function Q(p) can be derived from it and both describe the probabilistic nature of the variable x. In fact, when both can be derived explicitly it is equivalent to use one or the other to extreme value predictions. It is then a matter of taste the choice of one or the other. This is already stated in the original paper, where it is indicated that the purpose of presenting the quantile functions is exactly to suggest an alternative and equivalent method to determine extremes. However in cases inwhich F(x) does not have a closed form, then Q(p) should be used for extreme value analysis as the discussers also state. This would then be an advantage of using the quantile functions because in most cases what is available is a data set from which inferences are made for a F(x). This means that in practice an analytical expression of F(x) is not available, although an analytical expression might be adopted to model the data. The discussers then go on to comment on the case studies used for the demonstration of the application of the method. They specifically comment on the choice of daily maxima, which was done to reduce the size of the data set and the correlation among data. The objective of the paper was to propose the quantile functions and to demonstrate how they could be applied. The objective was not to determine the extreme value significant wave heights at a particular location. If that would have been the case indeed more care would be given to choose a data set with less correlation between the data as discussed for example in Ferreira and Guedes Soares (2000) and also several methods

An “extended fetch” model for the spatial distribution of tropical cyclone wind–waves as observed by altimeter

An extensive database of altimeter measurements of wind speed and significant wave height is analysed to investigate the spatial distribution of significant wave height within tropical cyclones. The database includes transects of 440 tropical cyclones. As such, the data covers the full range of expected values of the velocity of forward movement, maximum wind velocity and radius to maximum winds. The data confirms previous measurements that JONSWAP scaling can be used to represent such waves. In addition, the data supports the concept of an extended fetch in such systems. The maximum waves occur when the wind direction and direction of propagation of the storm are aligned. In such cases, the waves move forward with the storm and experience an extended fetch. A parametric model based on JONSWAP scaling and representing the extended fetch is developed and optimized using the data. Combined with previous in situ measurements, this model can reproduce the spatial distribution of significant wave height, as well as estimate the full directional spectrum.

Discussion of “Modelling significant wave height distributions with quantile functions for estimation of extreme wave heights” [Ocean Eng. 54 (2012) 119–131

The authors had applied four quantile functions Q(p), the Davies, GEV, GP3 and Weibull, to estimate extreme wave heights from significant wave height data collected every 3h in Bengal Bay (1997–2005) and from daily maxima of significant wave data (1958–2001) hindcasted at four locations in the North Atlantic Ocean. The Davies quantile function in Eq. (5) of Muraleedharan et al. (2012) was selected by the authors as the best-fit function for estimation of extreme wave heights. This discussion is to comment on if the use of the Davies quantile function Q(p) can actually improve the accuracy of extreme wave heights estimated. For consistency, the notations used in this discussion will be the same as those defined by the authors.

Return Value Estimation of Significant Wave Heights With Maximum Entropy Distribution

The maximum entropy distribution is proposed to fit the long term and extreme distribution of significant wave heights from which return value estimates are derived. The maximum entropy distribution is applied to data from two sites of different characteristics, namely from Japan characterized by the occurrence of typhoons and from the North Sea with continuous variation of sea state intensity. The compound distribution, Poisson-maximum entropy distribution, is described and adopted to model the data from these two locations. It is shown that in the case of continuous data from the North Sea, this model does not bring any advantage over the direct application of the maximum entropy distribution to adjust the significant wave heights larger than different thresholds. For this case the maximum entropy distribution provides good fits.

Some statistical aspects of wave-induced drift in sea states

The paper provides a simple analytical tool which can be used to give estimates of the mean value and the standard deviation of the Stokes drift for sea states. This is achieved by providing bivariate distributions of significant wave height with surface Stokes drift as well as with volume Stokes transport. These Stokes drift parameters are defined in terms of significant wave height and characteristic wave periods. Also bivariate distributions of spectral peak period and these two Stokes drift parameters are provided. The paper presents statistical aspects of the Stokes drift parameters, such as the conditional expected values and the conditional variances for given significant wave height, as well as examples of results corresponding to typical field conditions. The present analytical results can be used to make assessment of Stokes drift based on e.g. global wave statistics.

Numerical modelling of the nearshore wave energy resources of Shandong peninsula, China

In the present work, in order to investigate the nearshore wave energy resources, the third-generation wave model SWAN is utilised to simulate wave parameters of the Shandong peninsula in China for 16 years (1996–2011). The wind parameters used to simulate waves are obtained by the Weather Research & Forecasting Model (WRF). The modelling results of wave are validated by observation data. The spatial distributions of significant wave height and wave energy density are analysed under both extreme and mean wave conditions. The wave energy resources of the Chengshantou headland, with the highest wave energy density, the Langyatai headland and the Yellow River Delta are also studied in detail. For the above three sites, the mean month averaged wave energy is investigated, the wave energy resources are characterised in terms of wave state parameters, and wave energy roses are introduced. The values of extreme high and time-averaged nearshore wave energy density are 296 kW/m and 5.1 kW/m respectively.

Bivariate maximum entropy distribution of significant wave height and peak period

A bivariate maximum entropy distribution of significant wave heights and the corresponding peak periods is proposed. Two approaches are used to construct the bivariate maximum entropy distribution, the first of which is to adjust equivalent normal bivariate distributions to data transformed from the univariate maximum entropy distributions of wave height and period. The other method is the conditional approach of adjusting one maximum entropy marginal distribution to wave height and several log-normal distributions to the wave periods conditional on the wave height. The bivariate maximum entropy distribution is compared with other bivariate approaches using data from one location in the North Atlantic and a simulated sample by Monte Carlo method.

Modelling significant wave height distributions with quantile functions for estimation of extreme wave heights

This paper starts by introducing extreme wave height analysis using quantile functions, which are an alternative to the classical approaches to model long term maxima or extreme values. The long-term distribution of significant wave heights from four locations are modelled with Davies, 3-parameter Weibull, generalized extreme value (GEV) and generalized Pareto (GP3) quantile functions. Even though the 3-parameter Weibull and GP3 quantile functions are adequate wave height models in this study, the performance of the Davies quantile function for extreme wave analysis seems to be consistently good both temporally and spatially.

Assessment of wave energy resource of the Black Sea based on 15-year numerical hindcast data

In this study, the existence and variability of wave energy resource potential of the Black Sea based on 15-year hindcast data is described in detail. The hindcasts of wave parameters were carried out by using the third generation wave prediction model (Simulated WAves Nearshore – SWAN), which is one of the most popular numerical wave models and has been widely used for estimating ocean waves. The model was forced with the ECMWF ERA Interim wind fields and applied with a spatial resolution of about 0.0167°×0.0167° and a model time step of 6h to resolve efficiently offshore and nearshore wave conditions. The results were presented in the form of charts of the spatial distribution of significant wave height and wave power, on a monthly, seasonal and annual basis. Annual energy was calculated in the study region with the hindcast data set covering 15years (1995–2009). The areas with the highest wave energy resource were determined and the south west coasts of the Black Sea are suggested as the best site for the installation of a wave farm. It was determined the western parts of the Black Sea (especially the south-west) are exposed to energetic waves more than the eastern parts.

Parameter Estimation of the Maximum Entropy Distribution of Significant Wave Height

Dong, S.; Tao, S.; Lei, S., and Guedes Soares, C., 2013. Parameter estimation of the maximum entropy distribution of significant wave height. This paper compares the estimation of the four parameters of the maximum entropy distribution by different methods and applies them in two test cases with significantly different characteristics of variability. The moment method and the maximum likelihood method for the maximum entropy distribution with four parameters are formulated in the paper. These methods are compared with the moment method for the maximum entropy distribution with three parameters and an empirical curve-fitting method, both of which have been used earlier. These four estimation methods are applied to two test cases. One consists of hindcast wave heights at Weizhoudao hydrological station in the northern area of the South China Sea, which is subject to typhoon type of events. The other data set is hindcast wave heights at a location in the North Atlantic Ocean, which is subject to frequent storm weather. The maximum likelihood and the empirical methods appear to provide the most consistent results.

A joint distribution of significant wave height and characteristic surf parameter

The paper provides a joint distribution of significant wave height and characteristic surf parameter. The characteristic surf parameter is given by the ratio between the slope of a beach or a structure and the square root of the characteristic wave steepness in deep water defined in terms of the significant wave height and the spectral peak period. The characteristic surf parameter is used to characterize surf zone processes and is relevant for e.g. wave run-up on beaches and coastal structures. The paper presents statistical properties of the wave parameters as well as an example of results corresponding to typical field conditions.

Statistical trend analysis and extreme distribution of significant wave height from 1958 to 1999 – an application to the Italian Seas

Abstract. The study is a statistical analysis of sea states timeseries derived using the wave model WAM forced by the ERA-40 dataset in selected areas near the Italian coasts. For the period 1 January 1958 to 31 December 1999 the analysis yields: (i) the existence of a negative trend in the annual- and winter-averaged sea state heights; (ii) the existence of a turning-point in late 80's in the annual-averaged trend of sea state heights at a site in the Northern Adriatic Sea; (iii) the overall absence of a significant trend in the annual-averaged mean durations of sea states over thresholds; (iv) the assessment of the extreme values on a time-scale of thousand years. The analysis uses two methods to obtain samples of extremes from the independent sea states: the r-largest annual maxima and the peak-over-threshold. The two methods show statistical differences in retrieving the return values and more generally in describing the significant wave field. The r-largest annual maxima method provides more reliable predictions of the extreme values especially for small return periods (<100 years). Finally, the study statistically proves the existence of decadal negative trends in the significant wave heights and by this it conveys useful information on the wave climatology of the Italian seas during the second half of the 20th century.

Selection of the Electrical Generator for a Wave Energy Converter

This paper presents the analysis made to select the appropriate electrical generator for a new design of Wave Energy Converter (WEC), identifying the features that have influence on the WEC's performance, the efficiency and economical feasibility of the system. The proposed WEC is a near shore submerged converter that takes advantage of the oscillating water column created by the wave when it passes on the converter's platform. The wave column moves the platform, transmitting the force to a single acting cylinder, that pumps sea water to the coast, where is turbined generating electricity. A constant flow is obtained by means of accumulators located inside the module. The nominal power of the facility depends on the number of modules connected in parallel, producing 25kW per module. The facility size is between 100kW (4 modules) and 500kW (20 modules). After introduced the proposed WEC, a review of different solutions adopted by other WEC system is presented. There are two generator types suitable for variable speed operation: the synchronous permanent magnet generator (PMG) with the frequency inverter and the asynchronous induction generator. On the other hand, for constant speed operation the synchronous PMG and the squirrelcage induction generator fulfil the requirements. However, before selecting the generators, it is mandatory to study the operation of the generator and the status of the host power network of which the generator will be a component. The WEC control algorithm governs the optimal performance of the facility. The energy absorbed by the WEC is maximum when platform's movement is tuned to the incident wave frequency. The PLC acts on system pressure and generator's load torque in order to work in resonance. The paper includes a study about the influence of the generator system on the control. There are two options: to work with variable speed to take advantage of the highest efficiency, or work with constant speed, reducing the initial investment. In both cases the PLC sets the generators load with the available wave size information for the given rotation speed (variable or fixed). The paper analyses the energy output difference between variable and constant speed strategies. The study is based on the efficiency of the turbine for both dispositions, followed by a evaluation of the energy loss for a given location, applied as example on a hypothetical implantation in Bilbao Port. The WEC works with variable pressure to adapt the absorption to the actual wave size, and the efficiency of the Pelton turbine only depends on the working pressure. Then, the expected efficiencies can be obtained for different working pressures supposing constant speed (synchronism). The wave energy resource for a given location is defined by a bivariate distribution of significant wave height, Hs (m), and wave period, T, (s), counting the occurrences for each wave size and period during a given time period. Multiplying the occurrences by the power absorbed for each Hs-T combination, the result is a good estimation of the potential power of the given location. The lost power due to constant speed for the location can be defined because the efficiencies are defined for each wave condition. Beside the variable-constant speed strategy influence on the efficiency and expected energy output for a given location, the initial cost of the facility and the exploitation cost (maintenance and reactive energy requirements) are decisive when studying the feasibility of the investment. We have found necessary and valuable to accomplish this review and reflection about the selection of the generator. The influence of the generator on converter’s performance has been studied, as well as its implementation in the system s control. In addition, an example of feasibility study has been applied on a given disposition to evaluate the different generators with real data (four modules, 110kW rated power, located in Bilbao port). Each conversion system has shown advantages and disadvantages, year by year prices are more affordable and technologies are improving. The generator must be selected for each location in collaboration with the promoter.

On the use of discrete seasonal and directional models for the estimation of extreme wave conditions

Extreme value theory is commonly used in offshore engineering to estimate extreme significant wave height. To justify the use of extreme value models it is of critical importance either to verify that the assumptions made by the models are satisfied by the data or to examine the effect violating model assumptions. An important assumption made in the derivation of extreme value models is that the data come from a stationary distribution. The distribution of significant wave height varies with both the direction of origin of a storm and the season it occurs in, violating the assumption of a stationary distribution. Extreme value models can be applied to analyse the data in discrete seasons or directional sectors over which the distribution can be considered approximately stationary. Previous studies have suggested that models which ignore seasonality or directionality are less accurate and will underestimate extremes. This study shows that in fact the opposite is true. Using realistic case studies, it is shown that estimates of extremes from non-seasonal models have a lower bias and variance than estimates from discrete seasonal models and that estimates from discrete seasonal models tend to be biased high. The results are also applicable to discrete directional models.

Simulation of Significant Wave Height by Neural Networks and Its Application to Extreme Wave Analysis

Abstract The derivation of the long-term statistical distribution of significant wave heights (Hss) is discussed in this paper. The distribution parameters are estimated using artificial neural networks (ANNs) trained with the help of a simulated annealing algorithm and operated in an autoregressive mode. The ANNs were utilized in estimating the parameters of a conditional probability distribution related to a desired Hs given its preceding Hss, approximated by a proposed distribution called the hepta-parameter spline. The performance function during training was based on the likelihood function of the statistical method of maximum likelihood estimation (MLE). Given the observed dataset, the most probable weights and biases of the neural networks were determined in such a way that the performance function was optimized. The distribution could be used in the simulation and forecasting of Hss. This paper also presents an extreme wave analysis using the simulated Hss. The extreme analysis conducted in this study using the maxima method offers an alternative approach, avoiding the unrealistic hypothesis that annual Hss are identically distributed, as is conventionally assumed when using the Fisher–Tippet theorem.

Evaluation of Harbor Tranquility with Considering Effect of Double-peaked Offshore Spectrum

Generally, a joint distribution of significant wave height, period and direction outside a harbor is analyzed with assumption that a single-peaked spectrum is observed in each time. However, the significant wave periods should be underestimated when double-peaked spectra are observed, and that seems to affect wave height exceedance within the harbor. In this paper, the occurrence frequency of the double-peaked spectrum is estimated by using newly proposed method. Moreover, the estimation of the probability of wave height exceedance within the harbor is carried out with considering both the separated significant periods of wind wave and swell from the double-peaked spectrum. On a berth in the target harbor, it reduces 0.1% when the number of underestimation of significant wave periods is 2% in whole observed spectra.

A generalized approach for long-term modelling of extreme crest-to-trough wave heights

This paper deals with the long-term modelling of high sea waves. The solution is given for the return period of sea storms during which an arbitrary chosen number of waves, with crest-to-trough heights exceeding a fixed threshold, occur. This return period is derived starting from the Equivalent Triangular Storm (ETS) model, which associates a triangle to each actual storm and thus represents a significant wave height time series at a fixed location by means of a sequence of triangular storms. The short-term statistics is then applied to investigate the occurrence of large crest-to-trough wave heights during a given storm. Finally, by combining the statistical distribution of significant wave heights, the ETS model and the short-term wave statistics, the solution is given for the return periods R N and R ⩾ N of a sea storm in which N or at least N waves higher than a fixed threshold occur. The values of R N are then calculated, starting from data of two buoys moored in the Pacific Ocean and in the Mediterranean Sea.

Links of the significant wave height distribution in the Mediterranean sea with the Northern Hemisphere teleconnection patterns

Abstract. This study analyzes the link between the SWH (Significant Wave Height) distribution in the Mediterranean Sea during the second half of the 20th century and the Northern Hemisphere SLP (Sea Level Pressure) teleconnection patterns. The SWH distribution is computed using the WAM (WAve Model) forced by the surface wind fields provided by the ERA-40 reanalysis for the period 1958–2001. The time series of mid-latitude teleconnection patterns are downloaded from the NOAA web site. This study shows that several mid-latitude patterns are linked to the SWH field in the Mediterranean, especially in its western part during the cold season: East Atlantic Pattern (EA), Scandinavian Pattern (SCA), North Atlantic Oscillation (NAO), East Atlantic/West Russia Pattern (EA/WR) and East Pacific/ North Pacific Pattern (EP/NP). Though the East Atlantic pattern exerts the largest influence, it is not sufficient to characterize the dominant variability. NAO, though relevant, has an effect smaller than EA and comparable to other patterns. Some link results from possibly spurious structures. Patterns which have a very different global structure are associated to similar spatial features of the wave variability in the Mediterranean Sea. These two problems are, admittedly, shortcomings of this analysis, which shows the complexity of the response of the Mediterranean SWH to global scale SLP teleconnection patterns.

A method for spatial calibration of wave hindcast data bases

In the last years, new tools have been developed to describe wave climate. In particular, wave hindcast models allow a very detailed description in time and space. However, the data bases generated from these models require calibration with instrumental observations. In this work, a methodology for spatial calibration of wave hindcast data bases is developed. It is based on an Empirical Orthogonal Function decomposition and a non-linear transformation of the spatial–time modes. The method is applied to monthly long-term distribution of significant wave height in the Western Mediterranean. The calibration (transformation of the modes) is carried out using existing buoys in the area. After calibration, the validation with satellite data shows that this methodology is useful to better define wave climate from hindcast data bases.