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A Joint Probability Model for Environmental Parameters
Abstract
The joint probabilistic models (JPM) of the environmental parameters of wave, wind and current are nowadays extremely needed in order to perform reliability analyses of offshore structures. These JPM are also essential steps for the design of offshore structures based on long-term statistics and to perform dynamic response analysis of floating units that are strongly dependent on the directionality of the environmental actions, such as turret-moored FPSOs. Recently, some JPM have been proposed in the literature to represent the joint statistics of a reduced number of environmental parameters. However, it is difficult to find a practical and fully operational model taking into account the complete statistical dependence among all the environmental parameters intensities and their correspondent directions. In this paper, it is presented a straightforward methodology, based on the Nataf transformation, to create a JPM of the environmental parameters taking into account the dependence between the intensity and direction of all variables. The proposed model considers the statistical dependence of ten short-term variables: the significant wave height, peak period and direction of the sea waves, the significant wave height, peak period and direction of the swell waves, the amplitude and direction of the 1-h wind velocity and, finally, the amplitude and direction of the surface current velocity. The statistical dependence between them is evaluated using concepts of linear-linear, linear-circular and circular-circular variables correlation. Some results of the proposed JPM methodology are presented based on simultaneous environmental data gathered in a location offshore Brazil.
... Horn et al. [8] also went to higher dimensions for the CMA, including probabilistic modeling of the wind parameters together with wind generated sea, resulting on a multidirectional model. Sagrilo et al. [9] developed a joint probability function based on Nataf model for ten environmental parameters, including intensities and angular variables. Bitner-Gregersen [10] presented a joint met-ocean model capable of describing both wind-sea and swell waves. ...
Probability-based design of offshore structures usually require a joint distribution of significant wave heights and zero-up-crossing periods, or spectral peak periods, in order to describe the long-term wave scenario, and thus predict long-term extreme responses. Several joint statistical models have been proposed in the literature, often comprising environmental parameters aside from significant wave heights and zero-up-crossing or spectral peak periods, such as wind and current velocities, among others. However, since there is no theoretical environmental distribution function, the assessment of different models is done on an empirical basis for each available data set. The fitting of a joint distribution model on a set of long-term wave data, given the extensive number of available joint statistical models and the inherent extrapolations required for extreme conditions, can be extremely cumbersome. This paper proposes the application of Taylor diagram in the assessment of the fit of joint models. The aim is to find the models with the best performances through an easy and straightforward visual comparison. For simplification purposes, the joint statistical model considered comprises significant wave heights and spectral peak periods only. Results show that Taylor diagram is a useful tool for primarily evaluations of adjusted probability models. The indicatives for better model performances identified in the diagram were shown to correspond to other common identifiers, which demonstrates that the diagram can be a reliable indicator for good models. Some study cases are presented to demonstrate Taylor diagram application in joint probability distributions fittings for wave environmental parameters.
... 提出了三种固定式平 台的设计标准，其中单因素设计法，会造成平台响 应值过大；其他的设计标准比单因素设计法所得响 应值稍小，但仍然存在不足 [3] 。近年来，波高与风 速联合概率的研究取得了一定的进展 [4][5] ，风浪流 的三维分布的研究则相对较少，随机模拟技术 [3] 与 Nataf 变换 [6] 分别被用于多维概率分布的求解。随着 Copula 函数在金融、水文等领域的成功应用，其在 结构所受荷载的联合分布构造中有了应用 [7][8][9][10][11][12] 。多 维 Copula 有了一定的发展 [13 -15] ，最近 Plackett ...
The bivariate and trivariate distributions of wave height, wind velocity and current velocity are constructed with Plackett Copula, respectively. Different load combinations for these three variables are obtained in three ways: single variable design method, conditional distribution method (including both mode and mean value cases) and the trivariate joint distribution method. The base shearing force of a jacket platform is calculated and the joint return periods of different combinations are estimated. Results show that the computed base shear forces by the first method are obviously larger than others, and the corresponding return periods are larger than the joint return period. The results by the second method change with the choice of the predominant variable. Wave height contributes much more than both wind velocity and current velocity to the product of the platform base shear force. In the cases of considering current velocity or wind velocity as a predominant factor, the platform base shear is relatively smaller. The trivariate distribution constructed with Plackett Copula, considering the correlations among variables, can be used to estimate the design parameters of ocean environmental conditions. Compared with the single variable design method, the estimated design parameters can decrease and lead to lower platform construction cost, which will be suitable for the exploitation of a marginal oil field.
... Nataf transformation is a mathematical model for the transformation from correlated original space to mutually independent standard normal one. It requires the marginal cumulative distribution function (CDF) of each random variable and their correlation coefficient matrix (CCM), which are easy to be obtained in engineering applications[17][18]are available, a correlated standard normal variable (SNV) vector Y= [y 1 , y 2 , ..., y n ] T can be obtained by marginal transformation[19]1 ( ( )) 1, 2,..., ...
Probabilistic assessment of voltage stability margin (VSM) with existence of correlated wind speeds is investigated. Nataf transformation is adopted to establish wind speed correlation (WSC) model. Based on the saddle-node bifurcation transversality condition equations and Monte Carlo simulation technique, probability distribution of VSM is determined. With correlation coefficients range low to high value, the effect of WSC on VSM is studied. In addition, two risk indexes are proposed and the possible threat caused by WSC is evaluated from the viewpoint of risk analysis. Experimental results show that the presence of correlated wind speeds is harmful to safe and stable operation of a power system as far as voltage stability is concerned. The achievement of this paper gives a detailed elaboration about the influence of WSC on voltage stability and provides a potentially effective analytical tool for modern power system with large-scale wind power sources integration.
... ast China Sea. Wist et al. (2005) used the NATAF model to modeling successive wave heights and successive wave periods. The NATAF transformation , which transforms the initial distributions to normal distributions , is referred to as the Gaussian copula. Not like other bivariate models, the NATAF model can be easily extended to multivariate models. Sagrilo et al. (2011) presented a methodology to create a joint probability model of wave, wind, and current environmental parameters based on the NATAF transformation. The joint tails of multidimensional distributions must be considered in the load design of engineering design. The popular multivariate extreme theory assumes that the marginal variables are ...
... The 1-yr, 10-yr, 100-yr long-term extreme lateral motions of a circular-shaped monocolumn platform are analyzed, and the non-Gaussian (nonlinear) results are compared with those obtained by assuming the response as a Gaussian process. is the joint probability distribution of all parameters contained in S [6]. ...
In this paper, the long-term extreme response of a floating marine structure subjected to first and second order (slow-drift) wave effects is addressed. The proposed formulation is based on the long-term up-crossing rate, which is obtained by integrating the contribution of all short-term sea states. In order to speed up the evaluation of the long-term integral, an analytical expression for the short-term responses up-crossing rates was adopted. This expression is based on Hermite polynomials, and considers the response as a second order Volterra stochastic process. This formulation was applied to evaluate the 1-yr, 10-yr and 100-yr lateral motions of a circular-shaped monocolumn platform, and the non-Gaussian (nonlinear) results are compared with those obtained by assuming the response as a Gaussian process.
... Joint distributions of wind speed and other sea state characteristics have started to be used in applications in the last two decades. A joint fit of significant wave height and wind speed for the NW Australia location is presented in Bitner-Gregersen (2010) using the Conditional Modelling Approach (CMA) and for the Brazilian location by Sagrilo et al. (2011) applying the Nataf model. It should be noticed that the Nataf model should be used with care as it can give bias results (see Bitner-Gregersen and Hagen, 1999;Bitner-Gregersen, 2012). ...
... Joint distributions of wind speed and other sea state characteristics have started to be used in applications in the last two decades. A joint fit of significant wave height and wind speed for the NW Australia location is presented in Bitner-Gregersen (2010) using the Conditional Modelling Approach (CMA) and for the Brazilian location by Sagrilo et al. (2011) applying the Nataf model. It should be noticed that the Nataf model should be used with care as it can give bias results (see Bitner-Gregersen and Hagen, 1999;Bitner-Gregersen, 2012). ...
A review of the recent progress made regarding ocean environmental data and models with uncertainties in focus is given in the perspective of marine structures׳ design and operations. Uncertainties of wind, waves, current, sea water level and ice data and models are discussed and recent achievements leading to improvement of their accuracy are addressed. The study includes mathematical, probabilistic, empirical and statistical models. Particular attention is given to rogue waves. Challenges related to wind data and models due to the increase of offshore wind energy installations are presented focusing on wind forecast, information about the wind profile and extreme wind events. Attention is also given to ice which is of particular importance due to the projected climate changes opening new opportunities in the Arctic regions for the marine industry. Impacts of improved accuracy of environmental description on design and operations of ships and offshore structures are highlighted.
... Thus, the Nataf transformation provides a way to derive multivariate probability distribution models consistent with prescribed marginal distributions and a correlation structure (Liu and Der Kiureghian, 1986). In a number of studies, models for the joint probability distribution of metocean variables, involving significant wave height, wind speed, peak period, mean zero up-crossing period, and surface current velocity, have been developed based on Nataf distribution (Ditlevsen, 2002;Wist et al., 2004;Fouques et al., 2004;Sagrilo et al., 2011), showing that it is not only relatively easy to implement but also gives reasonably well fitting joint distribution models. ...
This paper reviews state‐of‐the‐art numerical tools for the operation and maintenance (O&M) of offshore wind farms, focusing on decision support models for maintenance scheduling and the consideration of human and environmental uncertainty. In this review, various factors that can influence the successful conduct of maintenance operations will be examined and special attention will be paid to the most significant ones. Data‐driven technologies for improved offshore asset management are also examined and the most used data‐driven methods for modeling and optimizing turbine operation and maintenance are presented. A focus will be placed on the choice of maintenance strategy, which is the basis for the planning of operations and thus the optimization problem discussed. As offshore maintenance is a complex operation whose efficiency and safety depend on human and environmental factors, special attention will be paid to the planning strategy that minimizes the risks involved while maximizing efficiency by considering these factors. The choice of planning technique for turbine maintenance and better consideration of uncertainties are crucial areas of improvement as they can lead to better overall efficiency, higher profit margins, better safety, and improved sustainability of offshore wind farms. The paper covers the application of digital technologies for offshore wind O&M planning and the associated challenges. The paper also highlights the various environmental and human factors to be considered for the operation and maintenance of wind turbines.
In order to perform a more accurate analysis of marine structures, joint probability distributions of different metocean parameters have received an increasing interest during the last decade, facilitated by improved availability of reliable joint metocean data. There seems to be no general consensus with regard to the approach of estimating joint probability distributions of metocean parameters and a general overview of recent studies exploring different joint models for metocean parameters is presented. The main objective of this article is twofold: first to establish a joint distribution of significant wave height and current speed and then to assess the possible conservatism in the Norwegian design standard by applying this joint distribution in a simplified load case. Based on NORA10 wave data and simulated current data, a joint model for significant wave height and current speed at one location in the northern North Sea is presented. Since episodes of wind-generated inertial oscillations are governing the current conditions at this location, a joint conditional model with current speed conditional on significant wave height is suggested. A peak-over-threshold approach is selected. The significant wave height is found to be very well modelled by a 2-parameter Weibull distribution for significant wave height exceeding 8 m, while a log-normal distribution describes the current speed well. This model is used to Monte-Carlo simulate joint significant wave heights and current speeds for periods corresponding to the ultimate and accidental limit states (ULS and ALS), i.e. 100 and 10 000 years. The possible conservatism in the Norwegian design standard is assessed by a simplified case study. The results give a clear indication that the Norwegian design standard in not necessarily conservative, neither at ULS nor ALS level.
For several engineering applications, joint met-ocean probabilities are required. Recently increasing attentions have been given to importance of inclusion wind-sea and swell components in the joint description. Presence of wind-sea and swell will affect design and operability of fixed and floating offshore structures as well as LNG terminals. The study presents a joint met-ocean model which can be applied for design and operations of marine structures, including LNG platforms. The model is fitted to hindcast data from four locations: Southern North Sea, West Shetland, and Northwest Shelf of Australia and off coast of Nigeria. Uncertainties related to the proposed fits are examined focusing on location specific features of the wave climate and an adopted partitioning procedure for the wave components. Implications of the uncertainties on design and operation criteria of LNG platforms are discussed and demonstrated by examples.
This work focuses on a methodology to predict the fatigue life of flexible pipes with wires broken in their tensile armors. Initially, the mechanical behavior of these pipes is discussed. Relying on this discussion, a simple set of equations is proposed in order to calculate the stresses in the armors of these pipes. These equations employ pre-estimated linear coefficients to convert forces and moments that act on the pope into stresses. These stresses are then processed by well-known cycle counting methods and S-N curves are finally used to evaluate fatigue damage at several points in the pipe’s cross section. The use of this methodology is exemplified by the assessment of the fatigue life of a 6” flexible pipe in which 0 up to 5 wires of its outer tensile armor are broken. The results indicate a substantial reduction in the fatigue life of the pipe with the increasing number of wires broken.
Response based approaches are not common in riser design. Due to the high computational costs associated to these methodologies, it is usual to replace the calculation of extreme long term responses by the calculation of responses to a few number of artificial sea states, supposed extreme. However, this hypothesis may not always be applicable. The extreme response of a riser is influenced by several factors. For instance, vessel response motions resonance can occur for waves of periods lower than the ones associated to the desired long term period. In this way, this work has two main objectives. The first is to propose a computationally feasible methodology to calculate long term extreme responses; the second is to calibrate loading conditions, based on the long term responses, to be used when designing catenary risers. The parameter selected to represent the response is the centenary (100y) riser top tension. The utilization of the proposed methodology is illustrated by a case study where three possible positions for a turret in a FPSO hull were compared. The obtained results indicate that this methodology can contribute to substantial changes in the way risers are designed, focusing on the response instead of on the occurrence of extreme sea states.
This paper focuses on a theoretical approach to access the fatigue life of flexible pipes. This methodology employs functions that convert forces and moments obtained in time-domain global analyses into stresses in their tensile armors. The stresses are then processed by well-known cycle counting methods, and
S-N
curves are used to evaluate the fatigue damage at several points in the pipe’s cross-section. Finally, Palmgren-Miner linear damage hypothesis is assumed in order to calculate the accumulated fatigue damage. A study on the fatigue life of a flexible pipe employing this methodology is presented. The main points addressed in the study are the influence of friction between layers, the effect of the annulus conditions, the importance of evaluating the fatigue life in various points of the pipe’s cross-section, and the effect of mean stresses. The results obtained suggest that the friction between layers and the annulus conditions strongly influences the fatigue life of flexible pipes. Moreover, mean stress effects are also significant, and at least half of the wires in each analyzed section of the pipe must be considered in a typical fatigue analysis.
SUMMARY A coefficient to measure association between angular variables is discussed, its asymptotic distribution found, and its properties developed. Comparisons with other statistics in current use are made, and some examples given.
S ummary
Directional data analysis is emerging as an important area of statistics. Within the past two decades, various new techniques have appeared, mostly to meet the needs of scientific workers dealing with directional data. The paper first introduces the two basic models for the multi‐dimensional case known as the von Mises–Fisher distribution and the Bingham distribution. Their sampling distribution theory depends heavily on the isotropic case and some developments are discussed. An optimum property of an important test for the von Mises–Fisher case is established. A non‐parametric test is proposed for the hypothesis of independence for observations on a torus. In addition to some numerical examples on the preceding topics, five case studies are given which illuminate the power of this new methodology. The case studies are concerned with cancer research, origins of comets, arrival times of patients, navigational problems and biological rhythms. Some unsolved problems are also indicated.
A joint environmental model for a long term response calculations is proposed. The model is based on experience gained from measurements and hindcast data from the Norwegian Continental Shelf. The model is an extension of an existing joint environmental model for Haltenbanken (off central Norway). The present model includes the following environmental parameters: 1-hour mean wind speed, current speed, significant wave height, spectral peak period, main wave direction (wind and current are assumed to be collinear with the main wave direction), and water level. Additionally, a simultaneous description of wind sea and swell is also introduced. The model is expected to be a useful tool for long term reliability calculations of offshore structures, especially as dynamics and/or other frequency dependent loading mechanisms become important. The model accounts for the lack of full correlation among the various environmental processes and it can be used for assessing the conservatism of the traditional design approach. Herein, as an illustration, the model is applied in connection with a reliability analysis of an idealized structural system, where the degree of dynamics is varied artificially. A first order reliability method (FORM) is used for this purpose.
This paper presents the results of a comparison of the fit of three bivariate models to a set of 14 years of significant wave height and peak wave period data from the North Sea. One of the methods defines the joint distribution from a marginal distribution of significant wave height and a set of distributions of peak period conditional on significant wave height. Other method applies the Plackett model to the data and the third one applies the Box-Cox transformation to the data in order to make it approximately normal and then fits a bivariate normal distribution to the transformed data set. It is shown that all methods provide a good fit but each one have its own strengths and weaknesses, being the choice dependent on the data available and applications in mind.
An extension of joint omnidirectional probabilistic model including combined seas and possibility of environmental parameters approaching from different direction is presented. The following parameters are considered: mean wind speed, wind direction, significant wave height (sea and swell), spectral peak period (sea and swell), wind sea direction, swell direction, current speed and current direction. Two approaches for wind sea and swell modelling are proposed. The joint description is developed for three locations at the Norwegian Continental Shelf and for the West Shetland. Uncertainties related to it are discussed. The model is meant to be applied in reliability analysis of offshore structures, in particularly FPSO units, for which a proper account of directional effects and combined seas is important.
Preface; 1. The purpose of the book; 2. Survey of contents; 3. How to
use the book; 4. Notation, terminology and conventions; 5.
Acknowledgements; Part I. Introduction: Part II. Descriptive Methods:
2.1. Introduction; 2.2. Data display; 2.3. Simple summary quantities;
2.4. Modifications for axial data; Part III. Models: 3.1. Introduction;
3.2. Notation; trigonometric moments; 3.3. Probability distributions on
the circle; Part IV. Analysis of a Single Sample of Data: 4.1.
Introduction; 4.2. Exploratory analysis; 4.3. Testing a sample of unit
vectors for uniformity; 4.4. Nonparametric methods for unimodal data;
4.5. Statistical analysis of a random sample of unit vectors from a von
Mises distribution; 4.6. Statistical analysis of a random sample of unit
vectors from a multimodal distribution; 4.7. Other topics; Part V.
Analysis of Two or More Samples, and of Other Experimental Layouts: 5.1.
Introduction; 5.2. Exploratory analysis; 5.3. Nonparametric methods for
analysing two or more samples of unimodal data; 5.4. Analysis of two or
more samples from von Mises distributions; 5.5. Analysis of data from
more complicated experimental designs; Part VI. Correlation and
Regression: 6.1. Introduction; 6.2. Linear-circular association and
circular-linear association; 6.3. Circular-circular association; 6.4.
Regression models for a circular response variable; Part VII. Analysis
of Data with Temporal or Spatial Structure: 7.1. Introduction; 7.2.
Analysis of temporal data; 7.3. Spatial analysis; Part VIII. Some Modern
Statistical Techniques for Testing and Estimation: 8.1. Introduction;
8.2. Bootstrap methods for confidence intervals and hypothesis tests:
general description; 8.3. Bootstrap methods for circular data:
confidence regions for the mean direction; 8.4. Bootstrap methods for
circular data: hypothesis tests for mean directions; 8.5. Randomisation,
or permutation, tests; Appendix A. Tables; Appendix B. Data sets;
References; Index.
A comprehensive framework is set forth for the analysis of structural reliability under incomplete probability information. Under stipulated requirements of consistency, invariance, operability, and simplicity, a method is developed to incorporate in the reliability analysis incomplete probability information on random variables, including moments, bounds, marginal distributions, and partial joint distributions. The method is consistent with the philosophy of Ditlevsen's generalized reliability index and complements existing second-moment and full-distribution structural reliability theories.
In-situ measurements over a three year period have been extensively analysed from a statistical point of view. Distributions of wind and current (intensity and direction), together with waves systems - main swell, secondary swell and wind sea - (peak period, significant wave height, spectral width, direction) are analysed jointly, looking for possible correlation between the variables (principal component analysis, statistical tests, ...). When correlation was proven, joint distributions have been estimated and modelled by analytical distribution laws. As distribution of extreme values may differ from distribution for intermediate values, matching of the two solutions is performed. Finally, I -FORM contours are derived based on the analytical laws. For coherence with extreme value prediction, some constraints are imposed so that 100 year extreme value estimation for the significant wave height, current velocity and wind velocity is consistent
Several types of joint distribution function for significant wave height and zero-upcrossing period are compared with reference to measured wave data from the Norwegian Continental Shelf. The comparison is based on the utility of the distribution functions for predictions of extreme response of offshore structures. Logarithmic contour plots and contour plots of normalised deviations between the data and the fitted distribution are used in the comparison. A joint distribution combining a marginal 3-parameter Weibull distribution for significant wave height with a conditional log-normal distribution for zero-up-crossing period is recommended on the basis of this investigation.
The characteristic environmental load effect for the design of mooring systems of floating units can be defined by means of three procedures: (a) the one associated to an extreme sea state with a given return period, (b) the worst one from a set of sea states on a contour line associated to a return period or (c) the extreme one based on response statistics for a long-term period. This work presents the result of a reliability-based partial safety factor calibration study for a LRFD mooring line design criteria considering the three approaches mentioned above. The calibration exercise is applied to three FPSOs considering North Sea environmental conditions and different water depths: 200, 800 and 3000 m. The mooring systems investigated take into account mooring lines made up of chains and polyester ropes. It is shown that, among all cases investigated, the design procedure based on the long-term response is the one that presents less scattered reliability indices around the target level.
This book describes the stochastic method for ocean wave analysis. This method provides a route to predicting the characteristics of random ocean waves - information vital for the design and safe operation of ships and ocean structures. Assuming a basic knowledge of probability theory, the book begins wuith a chapter describing the essential elements of wind-generated random seas from the stochastic point of view. The following three chapters introduce spectral analysis techniques, probabilistic predictions of wave amplitudes, wave height and periodicitiy. A further four chapters discuss sea severity, extreme sea state, directional wave energy spreading in random seas and special wave events such as wave breaking and group phenomena. Finally, the stochastic properties of non-Gaussian waves are presented. Useful appendices and an extensive reference list are included. Examples of practical applications of the theories presented can be found throughout the text. This book will be suitable as a text for graduate students of naval, ocean and coastal engineering. It will also serve as useful reference for research scientists and engineers working in this field.
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- Det Norske
- Veritas
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Ditlevsen, O., and Madsen, H. O., 1996, Structural Reliability Methods, Wiley,
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- E Bitner-Gregersen
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Distribution for Significant Wave Height and ZeroCrossing Period", Applied Ocean Research, Vol.12,
No.2, pp. 93-103.
Offshore Standard DnV OS F201-Dynamic Risers
- Det Norske
- Veritas
Det Norske Veritas, 2001, "Offshore Standard DnV OS F201-Dynamic Risers," Norway.
Joint Probability Model for Reliability Calculations
- E M Bitner-Gregersen
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Bitner-Gregersen, E. M., and Haver, S., 1991, "Joint Probability Model for
Reliability Calculations," Proceedings of ISOPE Conference, Edinburgh, UK.