Statistical estimation of extreme ocean environments: The requirement for modelling directionality and other covariate effects

With increasing availability of good directional data, provision of directional estimates of extreme significant wave heights, in addition to the omni-directional estimates, is more common. However, interpretation of directional together with omni-directional design criteria is subject to inconsistency, even in design guidelines. In particular, omni-directional criteria are usually estimated ignoring directional effects. In this article, for data which exhibit directional effects, we show that a directional extreme value model generally explains the observed variation significantly better than a model which ignores directionality, and that omni-directional criteria developed from a directional model are different from those generated when directionality is not accounted for. We also show that omni-directional criteria derived from a directional model are more accurate and should be preferred in general over those based on models which ignore directional effects. We recommend use of directional extreme value models for estimation of both directional and omni-directional design criteria in future, when good directional data are available. If effects of other covariates (e.g. time or space) are suspected, we similarly recommend use of extreme value models which adequately capture sources of covariate variability for all design analysis.     

Some statistical characteristics of large deepwater waves around the Korean Peninsula

The relationship between significant wave height and period, the variability of significant wave period, the spectral peak enhancement factor, and the directional spreading parameter of large deepwater waves around the Korean Peninsula have been investigated using various sources of wave measurement and hindcasting data. For very large waves comparable to design waves, it is recommended to use the average value of the empirical formulas proposed by Shore Protection Manual in 1977 and by Goda in 2003 for the relationship between significant wave height and period. The standard deviation of significant wave periods non-dimensionalized with respect to the mean value for a certain significant wave height varies between 0.04 and 0.21 with a typical value of 0.1 depending upon different regions and different ranges of significant wave heights. The probability density function of the peak enhancement factor is expressed as a lognormal distribution, with its mean value of 2.14, which is somewhat smaller than the value in the North Sea. For relatively large waves, the probability density function of the directional spreading parameter at peak frequency is also expressed as a lognormal distribution.     

Wave energy assessment based on trivariate distribution of significant wave height,mean period and direction

Based on the Vine copula theory, a trivariate statistical model of significant wave height, characterized wave period and mean wave direction was constructed. To maintain the properties of the different types of variables, a special copula function was derived from the model developed by Johnson and Wehrly based on the maximum entropy principle. It was then combined with the Archimedean copulas to construct the proposed model. An effective algorithm for generating corresponding joint pseudo-random numbers was also developed. Statistical analysis of hindcast data for the significant wave height, mean wave period, and direction, which were collected from an observation point in the North Atlantic every three hours from 1997 to 2001, was performed. The marginal distributions of the significant wave height and mean wave period were fitted by a modified maximum entropy distribution, and the mean wave direction was fitted by a mixture of von Mises distributions. It was shown that the proposed model is a good fit for the data. The seasonal wave energy resources in the target area were assessed using the model estimates. Histograms of the directional wave energy, wave energy roses, and scatter and energy diagrams were presented.     

Statistical Analyses of Wave Height Distribution for Multidirectional Irregular Waves over A Sloping Bottom

The main objective of this paper is to examine the influences of both the principal wave direction and the directional spreading parameter of the wave energy on the wave height evolution of multidirectional irregular waves over an impermeable sloping bottom and to propose an improved wave height distribution model based on an existing classical formula. The numerical model FUNWAVE 2.0, based on a fully nonlinear Boussinesq equation, is employed to simulate the propagation of multidirectional irregular waves over the sloping bottom. Comparisons of wave heights derived from wave trains with various principal wave directions and different directional spreading parameters are conducted. Results show that both the principal wave direction and the wave directional spread have significant influences on the wave height evolution on a varying coastal topography. The shoaling effect for the wave height is obviously weakened with the increase of the principal wave direction and with the decrease of the directional spreading parameter. With the simulated data, the classical Klopman wave height distribution model is improved by considering the influences of both factors. It is found that the improved model performs better in describing the wave height distribution for the multidirectional irregular waves in shallow water.     

Temporal significant wave height estimation from wind speed by perceptron Kalman filtering

Significant wave height estimates are necessary for many applications in coastal and offshore engineering and therefore various estimation models are proposed in the literature for this purpose. Unfortunately, most of these models provide simultaneous wave height estimations from wind speed measurements. However, in practical studies, the prediction of significant wave height is necessary from previous time interval measurements. This paper presents a dynamic significant wave height prediction procedure based on the perceptron Kalman filtering concepts. Past measurements of significant wave height and wind speed variables are used for training the adaptive model and it is then employed to predict the significant wave height amounts for future time intervals from the wind speed measurements only. The verification of the proposed model is achieved through the dynamic significant wave height and wind speed time series plots, observed versus predicted values scatter diagram and the classical linear significant wave height models. The application of the proposed model is presented for a station in USA.     

Wave runup on a concentric twin perforated circular cylinder

The regular wave interaction with a twin concentric porous circular cylinder system consisting of an inner impermeable cylinder and an outer perforated cylinder was studied through physical model and numerical model studies. The experiments were carried out on the twin concentric cylinder model in a wave flume to study the wave runup and rundown at the leading and trailing edges of the perforated cylinder. It was found that the maximum wave runup on the perforated cylinder is almost same as the incident wave height. The experimental results were used to develop the predictive formulae for the wave runup and rundown on the perforated cylinder, which can be easily used for design applications. The wave runup profiles around the perforated cylinder for different values of ka and porosities were studied numerically using Green's Identity Method. The results of the numerical study are presented and compared with the experimental measurements.     

Multidirectional wave transformation around detached breakwaters

The performance of the new wave diffraction feature of the shallow-water spectral model SWAN, particularly its ability to predict the multidirectional wave transformation around shore-parallel emerged breakwaters is examined using laboratory and field data. Comparison between model predictions and field measurements of directional spectra was used to identify the importance of various wave transformation processes in the evolution of the directional wave field. First, the model was evaluated against laboratory measurements of diffracted multidirectional waves around a breakwater shoulder. Excellent agreement between the model predictions and measurements was found for broad frequency and directional spectra. The performance of the model worsened with decreasing frequency and directional spread. Next, the performance of the model with regard to diffraction–refraction was assessed for directional wave spectra around detached breakwaters. Seven different field cases were considered: three wind–sea spectra with broad frequency and directional distributions, each coming from a different direction; two swell–sea bimodal spectra; and two swell spectra with narrow frequency and directional distributions. The new diffraction functionality in SWAN improved the prediction of wave heights around shore-parallel breakwaters. Processes such as beach reflection and wave transmission through breakwaters seem to have a significant role on transformation of swell waves behind the breakwaters. Bottom friction and wave–current interactions were less important, while the difference in frequency and directional distribution might be associated with seiching.     

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.     

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.     

Prediction of wave parameters by using fuzzy logic approach

The purpose of this study is to investigate the relationship between wind speed, previous and current wave characteristics. It is expected that such a non-linear relationship includes some uncertainties. A fuzzy inference system employing fuzzy IF–THEN rules has an ability to deal with ill-defined and uncertain systems. Compared with traditional approaches, fuzzy logic is more efficient in linking the multiple inputs to a single output in a non-linear domain. In this paper, a sophisticated intelligent model, based on Takagi–Sugeno (TS) fuzzy modeling principles, was developed to predict the changes in wave characteristics such as significant wave height and zero up-crossing period due to the wind speed. Past measurements of significant wave height values and wind speed variables are used for training the adaptive model and it is then employed to predict the significant wave height amounts for future time intervals such as 1, 3, 6 and 12 h. The verification of the proposed model is achieved through the wave characteristics time series plots and various numerical error criterias. Also the model results were compared with classical Auto Regressive Moving Average with exogenous input (ARMAX) models. For the application of the proposed approach the offshore station located in the Pacific Ocean was used.     

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.     

Evaluation of the wave transformation in an open bay with two spectral models

An evaluation of two state of the art phase averaged wave models for the transformation scale, SWAN and STWAVE, is carried out in the present work. The target area is the Obidos Bay located in the central part of the Portuguese continental nearshore. The wave input for the two models is provided by an offshore buoy. In order to compare the nearshore outputs of the wave models against in-situ measurements, a directional buoy and an ADCP, operating in intermediate water depth, are used. The wave parameters considered for comparisons are significant wave height, peak period and wave direction. Sensitivity analyses studies and evaluations in the spectral and geographical spaces concerning the results of the two models are also carried out in both intermediate and shallow water. The present study provides some information on the performances of the two wave models in different forcing conditions as well as on their sensitivity in relationship with various input parameters and some physical processes. STWAVE appears to be faster and more robust than SWAN, which on the other hand has more options and flexibility. In statistical terms the results are comparable.     

Examination of empirical formulas for wave shoaling and breaking on steep slopes

Estimation of the wave height transformation of shoaling and breaking is essential for the nearshore hydrodynamics and the design of coastal structures. Many empirical formulas have been well recognized to the wave height transformation, but most of them were only applicable for gentle slopes. This paper reports the experimental results of wave shoaling and breaking over the steep slopes to examine the applicability of the previous empirical formulas. Two steep bottom slopes of 1/3 and 1/5, and one gentle slope of 1/10 were conducted in the present experiments. Experimental results show that the shoaling distance of steep slopes become short and the surface waves may be partially reflected from the steep bottom, thus the estimation of wave shoaling using the well-known previous formula did not conform completely to the experimental results. The previous empirical formulas for the wave breaking criteria were also examined, and the modified equations to the steep beaches were proposed in this work. A numerical model was finally adopted to calculate the wave height transformation in the surf zone by introducing the modified breaking index.     

Artificial neural network to translate offshore satellite wave data to coastal locations

Owing to the spatial averaging involved in satellite sensing, use of observations so collected is often restricted to offshore regions. This paper discusses a technique to obtain significant wave heights at a specified coastal site from their values gathered by a satellite at deeper offshore locations. The technique is based on the approach of Artificial Neural Network (ANN) of Radial Basis Function (RBF) and Feed-forward Back-propagation (FFBP) type. The satellite-sensed data of significant wave height; average wave period and the wind speed were given as input to the network in order to obtain significant wave heights at a coastal site situated along the west coast of India. Qualitative as well as quantitative comparison of the network output with target observations showed usefulness of the selected networks in such an application vis-à-vis simpler techniques like statistical regression. The basic FFBP network predicted the higher waves more correctly although such a network was less attractive from the point of overall accuracy. Unlike satellite observations collection of buoy data is costly and hence, it is generally resorted to fewer locations and for a smaller period of time. As shown in this study the network can be trained with samples of buoy data and can be further used for routine wave forecasting at coastal locations based on more permanent flow of satellite observations.     

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.     

Non-stationary extreme value models to account for trends and shifts in the extreme wave climate due to climate change

The extreme values of wave climate data are of great interest in a number of different ocean engineering applications, including the design and operation of ships and offshore structures, marine energy generation, aquaculture and coastal installations. Typically, the return values of certain met-ocean parameters such as significant wave height are of particular importance. There exist many methods for estimating such return values, including the initial distribution approach, the block maxima approach and the peaks-over threshold approach. In a climate change perspective, projections of such return values to a future climate are of great importance for risk management and adaptation purposes. However, many approaches to extreme value modelling assume stationary conditions and it is not straightforward how to include non-stationarity of the extremes due to for example climate change. In this paper, various non-stationary GEV-models for significant wave height are developed that account for trends and shifts in the extreme wave climate due to climate change. These models are fitted to block maxima in a particular set of wave data obtained for a historical control period and two future projections for a future period corresponding to different emission scenarios. These models are used to investigate whether there are trends in the data within each period that influence the extreme value analysis and need to be taken into account. Moreover, it will be investigated whether there are significant inter-period shifts or trends in the extreme wave climate from the historical period to the future periods. The results from this study suggest that the intra-period trends are not statistically significant and that it might be reasonable to ignore these in extreme value analyses within each period. However, when it comes to comparing the different data sets, i.e. the historical period and the future projections, statistical significant inter-period changes are detected. Hence, the accumulated effect of a climatic trend may not be negligible over longer time periods. Interestingly enough, such statistically significant shifts are not detected if stationary extreme value models are fitted to each period separately. Therefore, the non-stationary extreme value models with inter-period shifts in the parameters are proposed as an alternative for extreme value modelling in a climate change perspective, in situations where historical data and future projections are available.     

Anisotropy of wind and wave regimes in the Baltic proper

The directional distribution of moderate and strong winds in the Baltic Sea region is shown to be strongly anisotropic. The dominating wind direction is south-west and a secondary peak corresponds to north winds. North-west storms are relatively infrequent and north-east storms are extremely rare. Angular distribution of extreme wind speed also has a two-peaked shape with maxima corresponding to south-west and north winds, and a deep minimum for easterly winds. The primary properties of the anisotropy such as prevailing winds, frequency of their occurrence, directional distribution of mean and maximum wind speeds coincide on both sides of the Baltic proper. The specific wind regime penetrates neither into the mainland nor into the Gulf of Finland or the Gulf of Riga.Properties of the saturated wave field in the neighbourhood of proposed sites of the Saaremaa (Ösel) deep harbour are analysed on the basis of the wave model WAM forced by steady winds. The directional distribution of wave heights in typical and extreme storms is highly anisotropic. Remarkable wave height anomalies may occur in the neighbourhood of the harbour sites.     

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.     

Three-dimensional investigations of wave overtopping on rubble mound structures

To study the influence of wave obliquity and directional spreading on wave overtopping of rubble mound breakwaters a total of 736 three-dimensional model tests were carried out at Aalborg University. The results of these tests are presented and analysed in this paper yielding a new empirical reduction factor to describe the influence of wave obliquity and directional spreading on the average wave overtopping discharges. The study shows that perpendicularly incident, long-crested waves result in conservative values of the overtopping discharge for the tested cross-section.