Joint modelling of extreme ocean environments incorporating covariate effects

Characterising the joint distribution of extremes of ocean environmental variables such as significant wave height (H_S) and spectral peak period (T_P) is important for understanding extreme ocean environments and in the design and assessment of marine and coastal structures. Many applications of multivariate extreme value analysis adopt models that assume a particular form of extremal dependence between variables without justification. Models are also typically restricted to joint regions in which all variables are extreme, but regions where only a subset of variables is extreme can be equally important for design. The conditional extremes model of Heffernan and Tawn (2004) provides one approach to overcoming these difficulties.Here, we extend the conditional extremes model to incorporate covariate effects in all of threshold selection, marginal and dependence modelling. Quantile regression is used to select appropriate covariate-dependent extreme value thresholds. Marginal and dependence modelling of extremes is performed within a penalised likelihood framework, using a Fourier parameterisation of marginal and dependence model parameters, with cross-validation to estimate suitable model parameter roughness, and bootstrapping to estimate parameter uncertainty with respect to covariate.We illustrate the approach in application to joint modelling of storm peak H_S and T_P at a Northern North Sea location with storm direction as covariate. We evaluate the impact of incorporating directional effects on estimates for return values, including those of a structure variable, similar to the structural response of a floating structure. We believe the approach offers the ocean engineer a straightforward procedure, based on sound statistics, to incorporate covariate effects in estimation of joint extreme environmental conditions.     

On the long-term joint distribution of characteristic wave height and period and its application

By analysing the scatter diagrams of characteristic the wave height H and the period T on the basis of instrumental data from various ocean wave stations, we found that the conditional expectation and standard deviation of wave period for a given wave height can be better predicted by using the equations of normal linear regression rather than by those based on the log- normal law. The latter was implied in Ochi' s bivariate log-normal model(Ochi. 1978) for the long-term joint distribution of H and T. With the expectation and standard deviation predicted by the normal linear regression equations and applying proper types of distribution, we have obtained the conditional distribution of T for given H. Then combining this conditional P(T / H) with long-term marginal distribution of the wave height P(H) we establish a new parameterized model for the long-term joint distribution P(H,T). As an example of the application of the new model we give a method for estimating wave period associated with an extreme w     

Experimental Investigation on the Extreme Waves Induced by Single Wave Packets in Finite Water Depth

Single Gaussian wave groups with different initial wave steepness ε_0 and width N are produced in laboratory in finite depth to study the nonlinear evolution, the extreme events and breaking. The results show that wave groups with larger ε_0 will evolve to be several envelope solitons(short wave groups). By analyzing geometric parameters, a break in the evolution of the wave elevation and asymmetric parameters after extreme wave may be an indicator for the inception of refocus and the maximal wave moving to the middle, namely, wave down-shift occurs. The analysis of the surface elevations with HHT(Hilbert-Huang Transform), which presents the concrete local variation of energy in time and frequency can be exhibited clearly, reveals that the higher frequency components play a major role in forming the extreme event and the contribution to the nonlinearity. Instantaneous energy and frequency in the vicinity of the extreme wave are also examined locally. For spilling breakers, the energy residing in the whole wave front dissipates much more due to breaking, while the energy in the rear of wave crest loses little, and the intra-wave frequency modulation increases as focus. It illustrates that the maximal first order instantaneous frequency f_1 and the largest crest tend to emerge at the same time after extreme wave when significant energy dissipation happens, and vice versa. In addition, it shows that there is no obvious relation of the CDN(combined degree of nonlinearity) to the wave breaking for the single Gaussian wave group in finite water depth.     

A methodology for deriving extreme nearshore sea conditions for structural design and flood risk analysis

Extreme sea conditions in the nearshore zone are required for coastal flood risk analysis and structural design. Many multivariate extreme value methods that have been applied in the past have been limited by assumptions relating to the dependence structure in the extremes. A conditional extremes statistical model overcomes a number of these previous limitations. To apply the method in practice, a Monte Carlo sampling procedure is required whereby large samples of synthetically generated events are simulated. The use of Monte Carlo approaches, in combination with computationally intensive physical process models, can raise significant practical challenges in terms of computation. To overcome these challenges there has been extensive research into the use of meta-models. Meta-models are approximations of computationally intensive physical process models (simulators). They are derived by fitting functions to the outputs from simulators. Due to their simplified representation they are computationally more efficient than the simulators they approximate.Here, a methodology for deriving a large Monte Carlo sample of extreme nearshore sea states is described. The methodology comprises the generation of a large sample of offshore sea conditions using the conditional extremes model. A meta-model of the wave transformation process is then constructed. A clustering algorithm is used to aid the development of the meta-model. The large sample of offshore data is then transformed through to the nearshore using the meta-model. The resulting nearshore sea states can be used for the probabilistic design of structures or flood risk analysis. The application of the methodology to a case study site on the North Coast of Spain is described.     

X-波段雷达近海海浪频谱反演的神经网络模型

X-波段雷达作为国内海浪观测的一种新工具,在海浪频谱获取和有效波高反演方面仍存在较多问题.本文利用非线性回归方法,将现场实测浮标数据频谱和雷达一维图像谱分别与标准频谱模型进行拟合,发现浮标频谱和一维图像谱具有标准频谱的特征,能够较准确地获取相应的谱参数.提出了建立由雷达一维图像谱参数反演海浪频谱参数的神经网络模型,同时在模型中加入影像序列信噪比,进而反演有效波高,并将反演结果与现场实测数据和传统算法(建立影像序列信噪比与有效波高之间的线性回归方程)进行了对比,结果表明,获取谱参数的误差和反演有效波高的平均误差在20％以内,而传统算法计算有效波高平均误差在20％以上.     

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.     

A global wave parameter database for geophysical applications. Part 1: Wave-current–turbulence interaction parameters for the open ocean based on traditional parameterizations

Ocean surface mixing and drift are influenced by the mixed layer depth, buoyancy fluxes and currents below the mixed layer. Drift and mixing are also functions of the surface Stokes drift U_ss, volume Stokes transport T_S, a wave breaking height scale H_swg, and the flux of energy from waves to ocean turbulence Φ_oc. Here we describe a global database of these parameters, estimated from a well-validated numerical wave model, that uses traditional forms of the wave generation and dissipation parameterizations, and covers the years 2003–2007. Compared to previous studies, the present work has the advantage of being consistent with the known physical processes that regulate the wave field and the air–sea fluxes, and also consistent with a very large number of in situ and satellite observations of wave parameters. Consequently, some of our estimates differ significantly from previous estimates. In particular, we find that the mean global integral of Φ_oc is 68 TW, and the yearly mean value of T_S is typically 10–30% of the Ekman transport, except in well-defined regions where it can reach 60%. We also have refined our previous estimates of U_ss by using a better treatment of the high frequency part of the wave spectrum. In the open ocean, U_ss ≃ 0.013U₁₀, where U₁₀ is the wind speed at 10 m height.     

海洋工程设计风速与波高的联合分布

选取我国渤海某处21a的风暴过程后报资料,考虑风暴发生频次的影响,提出泊松二维逻辑分布,并且将其用于海洋石油工程设计中极值风速与波高的联合概率计算,给出了计算海域的风浪设计参数,并与传统的设计标准进行了比较.计算结果表明,新的统计模式适用于受风暴影响海区的海洋工程结构设计,特别是边际油田的开发建设.     

A comparative analysis of models for dynamic boundary layer of atmosphere

This paper considers the currently available approaches to constructing numerical models describing the dependence of parameters of the atmospheric boundary layer on waving parameters (dynamic boundary layer). The models proposed in [1, 2] are characterized by detailed numerical algorithms, numerical calculations, and comparisons of the resistance coefficient C _d as functions of the parameters of the waving surface, the state of which is given by the model two-dimensional wave spectrum as represented in [3]. For the same spectrum, the calculation results obtained by different models are shown to yield estimates for the value of C _d with a more than twofold discrepancy; however, the trends in the dependence of C _d on wave age and wind strength are close to one another and to observational data. Possible shortcomings of both approaches are analyzed, and ways to eliminate them are proposed. The requirements for setting special experiments needed to verify theoretical models of the dynamic boundary layer are discussed.     

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.     

Environmental contours using copulas

A procedure is proposed for constructing environmental contours using copula theory. Copulas are functions that define the multivariate probability distribution of a random vector or a set of random variables, and, thus, also determine their dependence structure. Constructing environmental contours requires knowledge of the joint probability distribution of the environmental variables. In many practical applications, the available statistical data is used to estimate the marginal distributions and the linear correlation matrix, and then the Nataf distribution model is employed to obtain the multivariate probability distribution. It turns out that such an approach implies a particular model of dependence structure defined by a Gaussian copula, which might not always be the appropriate one. In this work, some classes of bivariate copulas are considered for modeling the dependence structure of the environmental variables. We examine measures of association, rank-based methods for estimation of copulas, goodness of fit tests for copulas, and copula selection criteria, and apply them to metocean data from hindcasts of tropical storms and extra-tropical events in the Gulf of Mexico. A formulation is proposed for expressing the variates that define the environmental contours as functions of copulas. It is then applied for computing environmental contours of significant wave height, peak spectral period and wind velocity using the estimated copula models.     

Interface depth used in a two-layer model of nonlinear internal waves

A two-layer model includes three parameters: interface depth h ₁, upper layer density \(\rho_{1}\) , and lower layer density \(\rho_{2}\) . Many theoretical and laboratorial studies of internal waves, as well as most numerical models, are based on the two-layer assumption. However, these three parameters cannot be directly measured because a pycnocline in the real ocean has finite thickness, and the densities in both the mixed layer and the deep ocean are not constant. In the present study, seven different methods are used to determine the interface depth of the two-layer model and compared with the depth of maximum vertical displacement: the depth of maximum buoyancy frequency (Ν _max), the depth where the first mode eigenfunction has its maximum (Φ_max), the depth where the lowest mode temperature empirical orthogonal function has its maximum, the depth where either the two-layer Korteweg–de Vries (KdV) or Benjamin–Ono equation has closest coefficients with their continuously stratified counterparts, and the same KdV approach with stratification replaced by two idealized distributions. The multi-ship measurement conducted near the Luzon Strait is used for deep ocean comparison, and two measurements conducted in the east of Dongsha Atoll are used for shallow water comparison. The results show that in the deep ocean, the KdV approach with idealized type I stratification gives the interface closest to the depth of maximum vertical displacement. In shallow waters, the KdV approach agrees with the measurement best.     

Multiple thermohaline states due to variable diffusivity in a hierarchy of simple models

The effect of variable vertical diffusivity is investigated in dynamically reduced models of the thermohaline circulation (THC) in a rectangular basin. In a simple box model, sufficiently strong variation of the diffusivity κ_v with stability G can lead to the existence of two stable equilibria. Related behaviour is found in well-resolved frictional geostrophic (FG) models. A hierarchy of under-resolved FG models is constructed, the simplest of which is an 8-cell cube, to connect the two extremes of resolution. Multiple solutions in low-order models are found to correspond to the formation of high-gradient layers which are unlikely to be resolved by current ocean models. Physical arguments show that layering and multiple solutions require κ_v to decrease more rapidly than 1/G and sensitivity experiments suggest that, in addition, κ_v must vary by a factor of 10–100. In two-hemisphere runs with salinity forcing included, the dependence of diffusivity on stratification is found to marginally favour equatorially symmetric states. Finally, such variation is shown to have a profound effect on the periodic, flush-collapse cycle under strong saline forcing; specifically, if diffusivity is taken to be a function of stratification rather than depth, regime transitions can occur much more easily. It will therefore be important for climate modelling to determine which is more realistic.     

Prediction of berm geometry using a set of laboratory tests combined with teaching–learning-based optimization and artificial bee colony algorithms

Understanding sediment movement in coastal areas is crucial in planning the stability of coastal structures, the recovery of coastal areas, and the formation of new coast. Accretion or erosion profiles form as a result of sediment movement. The characteristics of these profiles depend on the bed slope, wave conditions, and sediment properties. Here, experimental studies were performed in a wave flume with regular waves, considering different values for the wave height (H₀), wave period (T), bed slope (m), and mean sediment diameter (d₅₀). Accretion profiles developed in these experiments, and the geometric parameters of the resulting berms were determined. Teaching–learning-based optimization (TLBO) and artificial bee colony (ABC) algorithms were applied to regression functions of the data from the physical model. Dimensional and dimensionless equations were found for each parameter. These equations were compared to data from the physical model, to determine the best equation for each parameter and to evaluate the performances of the TLBO and ABC algorithms in the estimation of the berm parameters. Compared to the ABC algorithm, the TLBO algorithm provided better accuracy in estimating the berm parameters. Overall, the equations successfully determined the berm parameters.     

基于浮游植物吸收的海洋初级生产力模型的不确定性分析

Satellite-derived phytoplankton pigment absorption(a_(ph)) has been used as a key predictor of phytoplankton photosynthetic efficiency to estimate global ocean net primary production(NPP). In this study, an a_(ph)-based NPP model(Ab PM) with four input parameters including the photosynthetically available radiation(PAR), diffuse attenuation at 490 nm(K_d(490)), euphotic zone depth(Z_(eu)) and the phytoplankton pigment absorption coefficient(a_(ph)) is compared with the chlorophyll-based model and carbon-based model. It is found that the Ab PM has significant advantages on the ocean NPP estimation compared with the chlorophyll-based model and carbonbased model. For example, Ab PM greatly outperformed the other two models at most monitoring sites and had the best accuracy, including the smallest values of RMSD and bias for the NPP estimate, and the best correlation between the observations and the modeled NPPs. In order to ensure the robustness of the model, the uncertainty in NPP estimates of the Ab PM was assessed using a Monte Carlo simulation. At first, the frequency histograms of simple difference(δ), and logarithmic difference(δ~(LOG)) between model estimates and in situ data confirm that the two input parameters(Z_(eu) and PAR) approximate the Normal Distribution, and another two input parameters(a_(ph) and K_d(490)) approximate the logarithmic Normal Distribution. Second, the uncertainty in NPP estimates in the Ab PM was assessed by using the Monte Carlo simulation. Here both the PB(percentage bias), defined as the ratio of ΔNPP to the retrieved NPP, and the CV(coefficient of variation), defined as the ratio of the standard deviation to the mean are used to indicate the uncertainty in the NPP brought by input parameter to Ab PM model. The uncertainty related to magnitude is denoted by PB and the uncertainty related to scatter range is denoted by CV.Our investigations demonstrate that PB of NPP uncertainty brought by all parameters with an annual mean of5.5% covered a range of –5%–15% for the global ocean. The PB uncertainty of Ab PM model was mainly caused by a_(ph); the PB of NPP uncertainty brought by a_(ph) had an annual mean of 4.1% for the global ocean. The CV brought by all the parameters with an annual mean of 105% covered a range of 98%–134% for global ocean. For the coastal zone of Antarctica with higher productivity, the PB and CV of NPP uncertainty brought by all parameters had annual means of 7.1% and 121%, respectively, which are significantly larger than those obtained in the global ocean. This study suggests that the NPPs estimated by Ab PM model are more accurate than others, but the magnitude and scatter range of NPP errors brought by input parameter to Ab PM model could not be neglected,especially in the coastal area with high productivity. So the improving accuracy of satellite retrieval of input parameters should be necessary. The investigation also confirmed that the SST related correction is effective for improving the model accuracy in low temperature condition.     

Wave model predictions in the Black Sea: Sensitivity to wind fields

This paper evaluates the impact of using different wind field products on the performance of the third generation wave model SWAN in the Black Sea and its capability for predicting both normal and extreme wave conditions during 1996. Wind data were obtained from NCEP CFSR, NASA MERRA, JRA-25, ECMWF Operational, ECMWF ERA40, and ECMWF ERA-Interim. Wave data were obtained in 1996 at three locations in the Black Sea within the NATO TU-WAVES project. The quality of wind fields was assessed by comparing them with satellite data. These wind data were used as forcing fields for the generation of wind waves. Time series of predicted significant wave height (H_mo), mean wave period (T_m02), and mean wave direction (DIR) were compared with observations at three offshore buoys in the Black Sea and its performance was quantified in terms of statistical parameters. In addition, wave model performance in terms of significant wave height was also assessed by comparing them against satellite data.The main scope of this work is the impact of the different available wind field products on the wave hindcast performance. In addition, the sensitivity of wave model forecasts due to variations in spatial and temporal resolutions of the wind field products was investigated. Finally, the impact of using various wind field products on predicting extreme wave events was analyzed by focussing on storm peaks and on an individual storm event in October 1996. The numerical results revealed that the CFSR winds are more suitable in comparison with the others for modelling both normal and extreme events in the Black Sea. The results also show that wave model output is critically sensitive to the choice of the wind field product, such that the quality of the wind fields is reflected in the quality of the wave predictions. A finer wind spatial resolution leads to an improvement of the wave model predictions, while a finer temporal resolution in the wind fields generally does not significantly improve agreement between observed and simulated wave data.     

Wave groupiness and spectral bandwidth as relevant parameters for the performance assessment of wave energy converters

To date the estimation of long-term wave energy production at a given deployment site has commonly been limited to a consideration of the significant wave height H_s and mean energy period T_e. This paper addresses the sensitivity of power production from wave energy converters to the wave groupiness and spectral bandwidth of sea states. Linear and non-linear systems are implemented to simulate the response of converters equipped with realistic power take-off devices in real sea states. It is shown in particular that, when the converters are not much sensitive to wave directionality, the bandwidth characteristic is appropriate to complete the set of overall wave parameters describing the sea state for the purpose of estimating wave energy production.