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.     

A simulation study on the uncertainty of environmental contours due to sampling variability for different estimation methods

Environmental contours are often used in design of marine structures to identify extreme environmental conditions that may give rise to extreme loads and responses. Recently, attention has been given to the fact that different methods exist for establishing such contours, and that in some cases significant differences may be obtained from the various methods.In this study, another aspect of the uncertainty related to the calculation of environmental contours is addressed, namely the uncertainty due to sampling variability when environmental contours are constructed based on metocean data of finite sample size. The uncertainty of environmental contours for the joint distribution of significant wave height and wave period for different sample sizes (10, 25 and 100 years of data) are investigated considering different underlying datasets and for different estimation methods for the joint distribution. Both cases where samples are drawn from a known joint distribution of wave height and periods and cases where samples are drawn from a real hindcast dataset and fitted to the joint distribution are considered. The uncertainty of the estimated contours is quantified and discussed in light of differences that can be anticipated from the different methods used to calculate the contours. Moreover, the potential bias from assuming different estimation methods is illustrated.     

Effect of surge uncertainty on probabilistically computed dune erosion

To assess the flood protection capacity of dunes in The Netherlands, a semi-probabilistic dune-erosion prediction method is currently in use in which uncertainties in input parameters of an empirical dune erosion model were taken into account, with the exception of the uncertainty in the extreme surge distribution. Previous research has shown that the surge is by far the most influential parameter affecting erosion in the currently used erosion model, which is due both to the influence of the surge level itself and to the conditional dependence of the wave height and period on the surge level in the probabilistic model used for the assessment. Furthermore, the distribution of extreme surge levels has been shown to contain large statistical uncertainty. The inclusion of uncertainty in input variables into probabilistic models results in more extreme events (in this case erosion) for the same exceedance probability, largely due to the incorporation of higher values of the input variables. The goal of the research described in this paper was to determine the impact of the inclusion of uncertainty in the extreme surge distribution on the estimate of critical erosion (erosion associated with an exceedance frequency of 10^− 5 per year). The uncertainty in the surge distributions was estimated and parameterized, and was incorporated into the probabilistic model. A reduction in uncertainty was subsequently imposed to estimate what value a reduction in uncertainty can offer, in terms of the impact on critical erosion. The probabilistic technique first-order reliability method (FORM) was applied to determine the relative contribution of the uncertainty in the surge distribution (as well as the remaining stochastic variables) to the critical erosion. The impact of the inclusion of uncertainty in the surge distribution on the critical retreat distance was found to be substantial with increases ranging from 34% to 93% of the original estimate at five locations along the Dutch coast. The reduced uncertainty showed a more subtle impact, with increases in critical retreat distance ranging from 10% to 26% of the original estimate. The relative importance analysis showed that the uncertainty in the surge distribution has a strong influence, with the relative importance ranging from 10% to 23% for an exceedance frequency of 10^− 5 per year.     

Scale-dependent effects of nutrient loads and climatic conditions on benthic and pelagic communities in the Gulf of Finland

Eutrophication and climate change are ranked among the most serious threats to the stability of marine ecosystems worldwide. The effects of nutrient loads and climatic conditions vary in direction, magnitude and spatial extent. To date the factors that are behind the scale-specific spatial and temporal variability are poorly known. In this study we assessed how variability in nutrient loads and climatic conditions at local, gulf and regional scales explained the spatial patterns and temporal trends of zooplankton and benthic invertebrates in the Gulf of Finland. In general both local and gulf scale environmental variability had an important effect on benthic invertebrate species and the variability was mainly due to local nutrient loading, gulf scale temperature and salinity patterns. Zooplankton species were equally affected by environmental variability at all spatial scales, and all nutrient load and climatic condition variables contributed to the models. The combination of variables at all spatial scales did not explain the substantially larger proportion in invertebrate variability than variables at any individual scale. This suggests that large-scale pressures such as nutrient loads and change of climatic conditions may define broad patterns of distribution but within these patterns small-scale environmental variability significantly modifies the response of communities to these large-scale pressures.     

Parametric modelling of joint probability density distributions for steepness and asymmetry in deep water waves

Traditional wave steepness s = H/L does not define steep asymmetric waves uniquely. Three additional parameters characterising single zero-downcross waves in a time series are crest front steepness, vertical asymmetry factor and horizontal asymmetry factor. Parametric models for joint probability density distributions for deep water waves are presented. The joint distributions are for crest front steepness-wave height, vertical asymmetry factor-wave height, total wave steepness-wave height and wave height-wave period. The parametric models are estimated from zero-downcross analysis of wave data obtained from measurements at sea on the Norwegian continental shelf. The results of the analysis presented here can be used in the estimation of the probabilities of occurrence of steep asymmetric waves and breaking waves in deep water. Thus the results are useful for the practical naval architect and ocean engineer who are considering unusual events in the sea, the associated accidents or responses and the probability of occurrence of such events.     

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.     

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.     

基于Copula函数的联合概率法在海洋工程中的应用

本文介绍了Copula函数的定义、属性和几种特殊的Copula函数.将Copula函数应用到近海工程中,用其建立了最大有效波高和最大风速的联合分布函数并检验,结果表明:Copula函数能够比较好的模拟实际的联合概率分布。通过Copula方法,可以由边缘分布和一个连接它们的Copula函数来得出联合分布函数,发挥其描述相关性尤其是尾部相关性的优势。该方法在近海工程的工程可靠度、设计标准和失效概率的计算中具有广泛的应用前景。     

Adaptive model of plankton dynamics for the North Atlantic

Plankton ecosystems in the North Atlantic display strong regional and interannual variability in productivity and trophic structure, which cannot be captured by simple plankton models. Additional compartments subdividing functional groups can increase predictive power, but the high number of parameters tends to compromise portability and robustness of model predictions. An alternative strategy is to use property state variables, such as cell size, normally considered constant parameters in ecosystem models, to define the structure of functional groups in terms of both behaviour and response to physical forcing. This strategy may allow us to simulate realistically regional and temporal differences among plankton communities while keeping model complexity at a minimum.We fit a model of plankton and DOM dynamics globally and individually to observed climatologies at three diverse locations in the North Atlantic. Introducing additional property state variables is shown to improve the model fit both locally and globally, make the model more portable, and help identify model deficiencies. The zooplankton formulation exerts strong control on model performance. Our results suggest that the current paradigm on zooplankton allometric functional relationships might be at odds with observed plankton dynamics. Our parameter estimation resulted in more realistic estimates of parameters important for primary production than previous data assimilation studies.Property state variables generate complex emergent functional relationships, and might be used like tracers to differentiate between locally produced and advected biomass. The model results suggest that the observed temperature dependence of heterotrophic growth efficiency [Rivkin, R.B., Legendre, L., 2001. Biogenic carbon cycling in the upper ocean: effects of microbial respiration. Science 291 (5512) 2398-2400] could be an emergent relation due to intercorrelations among temperature, nutrient concentration and growth efficiency. A major advantage of using property state variables is that no additional parameters are required, such that differences in model performance can be directly related to model structure rather than parameter tuning.     

Maximum Entropy Distribution Function and Uncertainty Evaluation Criteria

Marine environmental design parameter extrapolation has important applications in marine engineering and coastal disaster prevention.The distribution models used for environmental design parameter usually pass the hypothesis tests in statistical analysis,but the calculation results of different distribution models often vary largely.In this paper,based on the information entropy,the overall uncertainty test criteria were studied for commonly used distributions including Gumbel,Weibull,and Pearson-III distribution.An improved method for parameter estimation of the maximum entropy distribution model is proposed on the basis of moment estimation.The study in this paper shows that the number of sample data and the degree of dispersion are proportional to the information entropy,and the overall uncertainty of the maximum entropy distribution model is minimal compared with other models.     

Probabilistic analysis of the DoB in axially-loaded tubular KT-joints of offshore structures

The degree of bending (DoB) characterizing the through-the-thickness stress distribution has a profound effect on the fatigue behavior of tubular joints commonly found in steel offshore structures and the determination of DoB values is essential for improving the accuracy of fatigue life estimation. Probability density functions of the involved random variables are necessary for the fatigue reliability analysis of offshore structures. The objective of present research was the derivation of probability density function (PDF) for the DoB in tubular KT-joints commonly found in jacket-type offshore platforms. A total of 162 finite element (FE) analyses were carried out on 81 FE models of KT-joints subjected to two types of axial loading. Generated FE models were validated using experimental data, previous FE results, and available parametric equations. Based on the results of parametric FE study, a sample database was prepared for the DoB values and density histograms were generated for respective samples based on the Freedman-Diaconis rule. Thirteen theoretical PDFs were fitted to the developed histograms and the maximum likelihood (ML) method was applied to evaluate the parameters of fitted PDFs. In each case, the Kolmogorov-Smirnov test was used to evaluate the goodness of fit. Finally, the Generalized Extreme Value model was proposed as the governing probability distribution function for the DoB. After substituting the values of estimated parameters, six fully defined PDFs were presented for the DoB in tubular KT-joints subjected to two types of axial loading.     

Exploring the long-term and interannual variability of biogeochemical variables in coastal areas by means of a data assimilation approach

Dynamic Harmonic Regression (DHR) models are applied here to the investigation of the interannual changes in the trend and seasonality of biogeochemical variables monitored in coastal areas. A DHR model can be regarded as a time-series component model, where the phases and amplitudes of the seasonal component, as well as the trend, are parameters that vary with time, reflecting relevant changes in the evolution of the biogeochemical variables. The model parameters and their confidence bounds are estimated by data assimilation algorithms, i.e. the Kalman filter and the Fixed Interval smoother. The DHR model structure is here identified by a preliminary spectral analysis and a subsequent minimization of the Bayesian Information Criterion, thus avoiding subjective choices of the frequencies in the seasonal component. The methodology was applied to the investigation of the long-term and interannual variability of ammonia, nitrate, orthophosphate and chlorophyll-a monitored monthly in the lagoon of Venice (Italy) during the years 1986–2008. It was found that the long-term evolutions of the biogeochemical variables were characterized by non-linear patterns and by statistically significant changes in the trend. The seasonal cycles of all the variables were characterized by a marked interannual variability. In particular, the changes in the seasonality of chlorophyll and nitrate were significantly related to the changes in the seasonality of water temperature at the study site and of nutrient concentrations in river discharges, respectively. These results indicate that the methodology could be a sound alternative to more traditional approaches for investigating the impacts of changes in environmental and anthropogenic forcings on the evolution of biogeochemical variables in coastal areas.     

Long-term abundance patterns of macroalgae in relation to environmental variables in the Tagus Estuary (Portugal)

Seasonal and inter-annual patterns of macroalgal abundance in a Tagus Estuary oyster reef are described. Macroalgal abundance was estimated as canopy percent cover by three permanent point intercept transects over a 7-year period. Four categories were defined, corresponding to bare substrate and three different macroalgal functional-form groups: (1) ULVA, foliose group, included Ulva spp.; (2) GRACIL, terete corticated macrophyte group, included only Gracilaria gracilis; and (3) FILAM, small (<10 cm) filamentous group, including eight species. A canonical correspondence analysis (CCA) showed that: (1) ULVA were associated with long and hot days, being usually dominant during spring and especially summer; (2) FILAM were associated with mild temperatures and relatively long days, abundant in spring but showed frequent peaks in summer; and (3) GRACIL were also favoured by spring season, although associated to lower temperature and less daylight hours than FILAM. GRACIL and FILAM were present throughout the year. On the contrary, ULVA were absent or with low cover during colder periods. A negative correlation between GRACIL and FILAM seems to indicate competition between the two categories. The applied models explained 23.3% of the temporal variance in category abundance. Rainfall negatively affected macroalgal cover, as indicated by the positive correlation between rainfall and bare substrate. Our conclusions are in agreement with previous studies that consider algae as excellent environmental integrators, even on a small scale, due to a strong link between the macroalgal communities and relevant environmental variables. It is also relevant that this study used open-access databases of environmental variables, which open up new possibilities for mining existing data resources in new ways. Due to large inter-annual variability, long-term studies are essential to understand population dynamics in estuarine phytobenthic communities.     

Effects of East China Sea shallow-water environment on acoustic propagation

Operational environmental acoustics experiments were conducted over the frequency range of 25 to 800 Hz in September 1997 in the East China Sea, where the water depth was about 100 m. Objectives of the data analysis reported here are to characterize this environment and to assess its complexities as they may impact acoustic propagation as measured by its transmission loss (TL). Conductivity-temperature-depths and expendable bathy-thermographs sampled the ocean, such that its spatial and temporal variability could be approximately separated. The sound-speed profiles are downward refracting, involve two water masses associated with the Kuroshio Current and Taiwan Warm Current, and have thermocline variations caused by internal tides. The bottom geoacoustic characteristics, presumed to be approximately horizontally isotropic, were based on data atlases and were estimated from the measured TL, for some interpretations. The TL data were obtained in octave bands from explosive signal underwater sound sources and sonobuoy receivers, both deployed at a depth of about 18 m. Tests were conducted in directions approximately normal and parallel to the bathymetric contours and the measured TL was, to zero order, independent of the direction of propagation. To higher order, directional differences in the TL were observed and ascribed to anisotropies in bottom properties. A state-of-the-art TL model was adopted, based on environmental idealizations typical of operational forecasting and compared with the measured TL. The comparison yields a probability density function that quantifies the uncertainty of such a TL model, caused by the stochastic variability of the environment, typically unknown a priori. For the model used, the pdf has a standard deviation of about 2 dB from 50 to 800 Hz and larger below 50 Hz.     

Seasonal Shift in Community Structure of Periphytic Ciliates in Estuarine Waters in the Northern Bay of Bengal,Bangladesh

To investigate the seasonal heterogeneity of the periphytic ciliate communities, a 1-year baseline survey was conducted in the Karnaphuli River estuary, northern Bay of Bengal, Bangladesh. A total of 54 ciliate species were recorded, including seven common and 14 dominant species. Maximum species number was in autumn whereas maximum abundance was in winter; the minimum for both occurred in summer. Multivariate analyses, i.e., canonical analysis of principal coordinates (CAP) and principal co-ordinates analysis (PCoA), revealed a clear seasonal heterogeneity of community structure and environmental variables. Multivariate correlation analysis (RELATE) demonstrated that the community structure of the periphytic ciliate communities was significantly correlated with environmental variables, and best matching analysis (BIOENV) indicated that heterogeneity of community patterns was mainly driven by water temperature, pH, dissolved oxygen, total dissolved solids and nutrients. Species richness and diversity peaked in autumn whereas species evenness peaked in summer. These results suggest that environmental conditions shape periphytic ciliate community structure, which is a potentially useful bio-indicator of estuarine water quality.     

Modelling uncertainty in long-term predictions of significant wave height

This paper deals with the application of different methods to estimate the occurrences of high sea states. A case study of the Norwegian continental shelf is considered. The Annual Maxima Method and the Peak Over Threshold Method are used to obtain return values for 25, 50 and 100 years. Several parametric models are used to fit the long-term distribution of significant wave height and to obtain predictions of extremes values. It is shown that the prediction of these sea states depends very much on the tail behaviour of the fitted distribution.     

A Tactical Approach to Environmental Uncertainty and Sensitivity

The issues of acoustic sensitivity to environmental parameters and the uncertainty in performance prediction due to lack of accurate environmental inputs are addressed. Special emphasis is given to the current state of affairs and to the importance of communicating in a simple, robust means the uncertainty to the sonar operator. A multistaged approach to estimating sensitivity, computing acoustic propagation uncertainty due to environmental variability is presented. A heuristic approach to estimating the most sensitive environmental parameters is developed. A measurements-based statistical approach is applied to environmental data taken in the Mediterranean Sea to estimate performance uncertainty due to sound speed and geoacoustics uncertainty     

表面增暖形态决定了全球水汽加温度直减率反馈的不确定性

气候反馈通常是利用与全球平均表面温度升高相关的辐射效应变化来评估的。但是反馈的不确定性不仅仅是全球平均表面温度升高的函数。全球气候模式的预测表明,海表面温度变化的地域不同,给区域尺度上的大气环流和降水响应带来了重大不确定性。在本文中我们证明了表面增暖的空间格局是造成水汽-直减率综合反馈的不确定性的主要因素。基于运用辐射核理论计算的31个气候模型中,气温和相对湿度变化的全球平均辐射效应证明了这一观点。我们的结果突出了区域气候变化对气候反馈不确定性的重要贡献,并确定了世界上限制表面增暖形态对提高气候预测能力最有效的区域。     

Characterizing impulsive wave-in-deck loads on coastal bridges by probabilistic models of impact maxima and rise times

When waves impact a seawall, a vertical breakwater, an exposed jetty, a pier or a coastal bridge, they abruptly transfer their momentum into the structure. This energy transfer can be very violent and its duration exceptionally short. In the case of coastal bridges, whose spans are designed to have very short vibration period, wave impacts might have duration comparable to the natural period of oscillation of the structure, which therefore becomes prone to damage and failure. Previous forensic studies have documented the relative importance of impulsive loads on deck suspended structure, demonstrating the need to assess the effect of wave impacts on both the stability and the integrity of structural members since the early stages of the design. This requires the estimation of the dynamic characteristics of the loading pattern, and in particular the wave impulse and corresponding impact maxima and rise times. Based on the conservation of momentum, functional relationships between these parameters have been identified since pioneering work dating back to the late '30s of the 20th century. The complexity of the loading process, however, results in a significantly large variability of wave impact maxima and rise times even under similar conditions, suggesting the need for a probabilistic approach to the definition of the relationship between these two variables, to be applied when estimating the dynamic properties of wave for use in structural analysis of coastal structures. In the recent past, some effort has been made to identify functional relationships between such quantities; these require the assessment of the conditional quantiles (or similarly the conditional distribution) of wave impact maxima given the rise times. In this paper, we compare three different statistical methods proposed in the literature to accomplish this task, in order to assess the reliability of the approach and suggest guidelines for practical applications. A copula-based method, Generalized Additive Models for Location, Scale and Shape (GAMLSS), and quantile regression are applied to measurements from large-scale 3-dimensional physical model tests. The investigation suggests that quantile regression gives the simplest results to be used in practice; copula approach and GAMLSS are possible alternative when semi-parametric or fully parametric modeling is needed.