Survey of stochastic models for wind and sea-state time series”. 8 Copyright c○ 2006 by A
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Survey of stochastic modelsfor wind and sea state time seriesV.Monbet,P.Ailliot,M.Prevosto7th April2006AbstractThe knowledge of sea state and wind conditions is of central importance for many offshore or nearshore operations.In this paper,we make a complete survey of stochastic models for sea stateand wind time series.We begin the presentation with methods based on Gaussian processes and nonparametric resampling methods for time series are introduced followed by various parametric models.Finally we propose an original statistical method,based on Monte Carlo goodness-of-fit tests,formodel validation and comparison.The use of this method is illustrated by an example on wind speeddata in North Atlantic.keyword Sea state,Wind,Non linear time series,Simulation,Prediction,Reconstruction,Model validation.1IntroductionThe knowledge of sea state and wind conditions is of central importance for many offshore or nearshore operations.For instance,wind speed time series permit to evaluate the power values produced by wind turbine,or to investigate load matching and storage requirements(Brown et al.,1984),(Castino et al., 1998).Evolution of sea state and wind conditions are also determinant in coastal erosion(Waeles et al., 2004).And several questions concerning safety,reliability and feasibility of offshore activities(O’Caroll, 1984),(Monbet et al.,2001)as well as sea-transport(Baxevani et al.,2004),(Ailliot et al.,2003)are directly related to wave and wind parameters.Even if there is a huge amount of data collected on physical quantities related to ocean waves still these are sparse relative to the size of oceans(simply one can not observe everywhere at the same time).Hence estimates of risks for undesired scenarios for ocean operations are usually computed by means of mathematical models for sea surface dynamics.Often scenarios are defined as excursions above critical values of some responses of complicated systems and Monte Carlo can be the only way to derive probabilities of interest.In this paper,we make a survey of stochastic models and methods for sea state time series and we give a particular attention to simulation.We have chosen to focus on time series at the scale of the sea state (e.g.with time step from1to6hours)and at a given geographical point.As a consequence,events at the scale of waves are not modeled and no spatial information is taken into account.Several authors have proposed spatio-temporal models,see for instance(Ailliot et al.,2005),(Baxevani,2004)and references therein.In section2,after a short description of the data,we discuss the question”what is a good model?”and a brief presentation of methods to deal with the non stationarities present in metocean times series (trends,seasons,daily components,...).Then the most famous tools for time series modeling,e.g.the Box and Jenkins method(Box et al.,1976)and some extensions of it are introduced in section3.Non parametric resampling methods for time series are described in section4.These methods was rarely proposed in the literature for sea state processes but we think that they could be of high interest for a lot of applications.In section5,various parametric models are briefly presented.For each method,we indicate in which context it can be used and we discuss its performance.Finally,in section6,a validation method is proposed to measure the ability of a model to simulate realistic artificial sequences and it is illustrated on wind speed data in North Atlantic.12GeneralitiesIn this section,metocean data are briefly presented.Then,criteria to check the goodness of a model are proposed.Finally,methods to deal with the non stationarities present in metocean times series are described.2.1DataThe data used tofit models for sea state and wind time series are obtained from various sources:•In situ observations obtained from ships,buoys,satellite,...•Outputs of meteorological models:hindcast,nowcast,forecast.Some authors have compared these different data bases,see for instance(Woolf et al.,2002),(Caires et al.,2004),(Izquierdo et al.,2005)and their references.In buoy data,the time step is generally3 hours but it can vary from1hour to6hours.Their main drawbacks is that they often include missing and noisy data and the longer buoy time series are only about10years.Satellite data are recorded with very small time steps along the track but it generates sparse data in time for a given location.Generally, outputs of numerical models are easier to use because they are available on quite long period(until50 years)and they have non missing data.But their quality is not always good,in particular,they are known to be smoother as in situ data and to underestimate extreme events.Let us now introduce some notations for the synthetic parameters which are used throughout this paper.For more precise definitions,see(IAHR,1989).•H s:significant wave height(m).For H s two different definition are used the average of the highest one-third of wave heights or4times the standard deviation of the sea surface elevation process.These definitions are equivalent for seas with narrow band spectra.•T:wave period(s).The most often used definition are the spectral peak wave period which is the inverse of the frequency at which the spectral density function of the elevation time series is ata maximum,and the zero crossing period which is the average period between zero downcrossingwaves.•Θm:mean direction to which waves are traveling(degree).•U:wind intensity(ms−1).It is the mean of the speed of the air particles at10meters over afixed time period(20minutes in general).•Φ:wind direction(degree).It is the mean direction from which the wind is blowing at10meters over afixed time period(20minutes in general).2.2What is a good model?First of all,the response to the question”What is a good model?”depends on for what the model is built.The context of use considered in this paper are:explanation of a physical phenomena,time series simulation,forecasting,time series reconstruction.However;a particular attention is paid to simulation and reconstruction.The word”reconstruction”can refer to two different problems:missing data reconstruction which uses the observed time series itself for missing values completion and cross-reconstruction where a time series is reconstructed given another one(for instance a wave time series can be approximately reconstructed given a wind time series(Monbet et al.,2005b).Once the context of use is specified,the choice of the model relies on a compromise between its ease of use and its accuracy in describing various features of the physical phenomenon.The ease of use can be measured qualitatively according to several criteria:•model robustness to the data sources(ship,buoy,satellite,meteorological model).As mentioned above,there may exist missing or aberrant data,the data may not be recorded at a constant time step and sometimes only descriptive statistics are available such as marginal distributions or persistence durations for instance(Vik,1981),(Hogben,1987).2•model robustness to the nature of the process.Indeed,each sea state parameter has specific characteristics of evolution.This evolution may also depend on the geographical area and on the season.Furthermore,depending on the considered application,the process may be univariate or multivariate,and in the second case a strong dependence may exist between the components as forH s and T or H s and U.This dependence has to be taken into account in the model.Finally,thestate space of the parameters introduced in the previous section is eitherfinite,real or the torus R/2πZ as in the case of circular processes such asθm andΦ).•mathematical properties:amount of data needed to accurately estimate the model,asymptotic properties of the estimators,existence of validation criteria...•necessary time for implementation of the algorithms,for running estimation,simulation or recon-struction.The accuracy of a model is usually evaluated by comparing statistics of the observed time series with those of the artificial realizations obtained with the model.Generally,only graphical comparison are performed,and it does not permit to measure the goodness offit.We propose in section6a formal method,based on Monte Carlo tests,to compare and validate the models.It measures,in particular,the ability to restore chosen features of the observed time series such as the marginal cumulative distribution functions(cdf),the distribution of annual extremes,the distribution of storms duration,the autocovari-ance functions,etc.2.3Modeling non-stationarityDepending on the considered time series,several types of non-stationary components can be identified. In particular,there may exist over-year trends,annual components due to the meteorological seasons and daily components due to the difference of temperature between day and night.The over-year trend is difficult to consider in statistical models because of the few amount of data with respect to the temporal scale of the events(Athanassoulis et al.,1995).At the opposite,the seasonal components are generally easy to observe on meteorological time series. Several models have been proposed to remove the seasonal components(Cunha et al.,1999).Two methods are commonly used:•If{Y t}denotes the sea state process,the following decomposition is used(Walton et al.(1990), Stephanakos(1999)):Y t=m(t)+σ(t)Y statt(1)where m andσare deterministic periodic functions with period one year which model respectively the seasonal variation of the mean and the standard deviation of the marginal distribution of theprocess.And{Y statt }is assumed to be a stationary process.Functions m andσcan be estimatedempirically and then smoothed(Monbet et al.,2001)or they can be modeled by a sum of cos and sin functions of period one year(Cunha et al.,1999).•An other approach consists in supposing that the process is piecewise stationary and tofit separate models for each month or for each season of the year(Ailliot et al,2004).In thefirst method,it is assumed that the standardized process Y statt is stationary it may be a toostrong assumption in many cases.However,its main advantage is that the estimation of m andσdoes not require a lot of data,generally3or4years of data provide accurate estimates.On the contrary,to apply the second method,more data may be needed since a different model isfitted each month.Furthermore, artificial ruptures of the model are induced between successive months.But a great advantage of this method over thefirst one is that the stationarity hypothesis are lower.Some sea state time series also exhibit daily components.One of the most common method to remove this components is to use the decomposition of equation(1)with m andσperiodic functions with period one day(Ailliot et al.,2004).In the sequel,we suppose that all the studied processes are stationary.33Models based on Gaussian processesThe time series of the synthetic parameters introduced in section2cannot in general be assumed to be Gaussian.For instance,the marginal distribution of these processes are often asymmetric with positive support and positive skewness.However,when the support of their marginal distribution function is continuous,it is possible to transform these time series into time series with Gaussian marginal distrib-utions.If the transformed time series is supposed to be Gaussian,we can then use one of the existing techniques to simulate it(ARMA models,exact simulation methods,...).Let us describe more precisely the simulation method in a general framework.Let{Y t}be a stationary process with values in R d.We assume that there exist a transformation f:R d→R d and a stationary Gaussian process{X t}such that Y t=f(X t).The procedure consists in three main steps:•Model calibration,which consists in determining the function f and the second order structure of the process{X t}.In practice,f is chosen in order that the marginal distributions and the second order structures of processes{X t}and{Y t}match.•Sample generation in which realizations of the process{X t}are generated given the second order structure estimated in model calibration step.•Mapping.In this step the generated samples of{X t}are transformed into samples of{Y t}using the transformation f.This general method includes in particular the method of Box and Jenkins(1976)and the Translated Gaussian Process method which are described more precisely hereafter.3.1Box and Jenkins methodThe method of Box and Jenkins(1976)is undoubtedly the most usual model for wind time series(Brown et al.,1984),(Daniel et al.,1991),(Nfaoui et al.,1996)and sea state time series(O’Carroll,1984), (Stephanakos,1999),(Cunha et al.,1999),(Yim et al.,2002).It is also usually used in many other applicationfields.Let usfirst consider the monovariate case for simplicity.Transformation g=f−1is selected in the family of the Box-Cox transformations(Cunha et al.,1999):g(y)=(yλ−1)/λfor0<λ≤1and g(y)=ln(y)forλ=0Parameterλis selected in such a way that the marginal distribution of X t=g(Y t)is roughly Gaussian. Various methods can be used to estimate the parameter(Brown et al.,1984),(Daniel et al.,1991). Generally,for H s,the transformation g(y)=ln(y)is used(O’Carroll,1984),(Stephanakos,1999),the marginal distribution of this process being generally well approximated by a lognormal distribution.For the wind intensity,one generally applies a Box-Cox transformation with0.5<λ<1(Brown et al., 1984),(Nfaoui et al.,1996).When the process is multivariate,a Box-Cox transformation is usually applied independently to the various components.Then,once the transformation f has been selected,an ARMA model is adjusted to the trans-formed time series.For the process H s,the following models were proposed:AR(1)(O’Carroll,1984), ARMA(2,2)(Stephanakos,1999),AR(20)(Cunha,1999)and ARMA(4,4)(Yim,2002).For bivariate time series(H s,T),Guedes Soares et al(2000)select AR(4)or AR(5)models depending on the location. For the wind process U,models AR(1)(Toll,1997),AR(2)(Daniel et al.,1991),(Nfaoui et al.,1996), (More et al.,2003)or AR(4)(Poggi et al.,2003)was generally used,more complex models giving no improvement.3.2Translated Gaussian ProcessFor sea state parameters,another approach,based on the same principle is also used.It is a non parametric method,in which the function f is selected on the basis of the normal score transformation and the Gaussian process is simulated by using exact simulation algorithms.This approach was proposed4initially by Walton et al.,(1990)in order to simulate realizations of the process H s.This method was then extended by Borgman et al.(1991)to multivariate time series(H s,T,θm)and it was also applied, for example,by Gioffre et al.(2000)to simulate the wind pressure on a building and by DelBalzo et al. (2003)to simulate(H s,T,θm)using buoy and ship observations.In the sequel,it will be denoted TGP (Translated Gaussian Process).The normal score transformation of a variable Y into a Gaussian variable X,is given byx=N−1F Y(y)where F Y is the cumulative distribution function(cdf)of Y and N is the standard normal cdf.As concern multivariate variables Y=(Y1,···,Y n),afirst approach consists in independently applying the transformation on the various components.One chooses then(x1,···,x d)=f(y1,···,y d)=(N−1F Y(1)(y1),···,N−1F Y(d)(y d))with F Y(i)the cdf of{Y(i)}.This method is used for example in(Borgman et al.,1991)for the process (H s,T,Θm)and in(Gioffre et al.,2000).However,it was shown in(Monbet et al.,2001)that,when a strong dependence exists between the components of the process,this transformation does not allow to restore the marginal joint distribution.The Rozenblatt transformation(2)is then used.f:(y1,···,y d)→(2) N−1F Y(1)(y1),N−1F Y(2)|Y(1)=y1(y2),···,N−1F Y(d)|Y(1)=y1,···,Y(d−1)=y d−1(y d)where F Y(1)denotes the cdf of Y(1)and F Y(k)|Y(1)=y1,···,Y(k−1)=y k−1that of the conditional distributionP(Y(k)|Y(1)=y1,···,Y(k−1)=y k−1).In(Monbet et al.,2001),the case of(H s,T)was given as example. In(Ailliot et al.,2001)a transformation is proposed for the process(H s,Θm)and in(Ailliot et al.,2004) a transformation was given for(U,Φ).These transformations take into account the specificity of the circular parametersΘm andΦ.Generally,the cdf used to defined the transformation f are estimated empirically.Parametric models can be used to approximate the tails of distribution(Walton et al.,1990).Once the initial process is transformed into a process which is assumed Gaussian,the second order structure can be estimated as in(Borgman et al.,1991)and(Monbet et al.,2001).It may occur that the autocorrelation functions of the generated time series are significantly different from those of the initial sequence(Ailliot et al.,2004).Gioffre et al.(2000)have proposed a solution to this problem in the particular case where the normal score transformation is applied independently on the different components of the time series.Thefinal step consists in simulating realizations of the stationary Gaussian process{X t}whose marginal distribution is the standard Gaussian distribution and whose autocorrelation function is known. Various methods of exact simulation were proposed(Borgman et al.,1991),(Popescu et al.,1998).3.3General discussionThe Box and Jenkins method allows to model the marginal distribution and the second order structure of the time series.This method is also convenient to make simulation,forecasting and reconstruction (see for instance Stephanakos(1999),Guedes Soares et al.(1996),Ho et al.(2005)).Until now,TGP method has only be used for simulation,however it could also be applied for reconstruction and forecast since the underlying Gaussian process is completely characterized.Both,Box and Jenkins and TGP methods are robust to noisy data(Monbet et al,2001a).These methods are easy to adapt and they have been used for various metocean parameters,as it is shown in the references cited above.Their main limit is the dimension of the time series:it is difficult to apply these methods to time series with high dimension when the relation between the different components is complex.Indeed in this case it may be hard to transform the original time series to Gaussian ones.For Box and Jenkins method,statistical criteria exist to help in the choice of the model parameters such as the order of ARMA models.Moreover statistical properties of these models are well known(see Brockwell et al.,1991).As far as we know,convergence properties of TGP have not been studied.When5the underlying models are parametric they do not need a large amount of data for the estimation.In TGP,the autocorrelation function and eventually the marginal functions are estimated empirically and it requires larger data sets as in the case of parametric models.The algorithms used in Box and Jenkins and TGP methods are extensively used in many application fields so that they are implemented in quite a lot of softwares.And they run quickly.It has been shown that these methods successfully restore the marginal distribution and the auto-correlation function.However,they cannot restore some non linearities that exist in many observed time-series:for example,a monovariate stationary Gaussian process with0mean,the time durations of the sojourns above a threshold s0has the same distribution as the time durations of the sojourn below threshold(−s0).The transformed time series have similar characteristics as the Gaussian time series and thus can not succeed in reproducing both calm and storm durations if they have different characteristics, as it is mostly the case for sea state time series(see the example in section6.2).4Resampling methodsResampling methods have only seldom been used to study the impact of the evolution of meteorological parameters on dynamic systems.The principle is simple since it consists in randomly sampling in the data.These methods are generally used for bootstrap estimation and it was proved that it allows to estimate the distribution of a wide range of estimators in the case of independent observations(Efron et al.,1993)and of time series(H¨a rdle et al.,2003).We describe here briefly standard methods for time series resampling,more details can be found in (H¨a rdle et al.,2003)and references therein.4.1Block resamplingResampling by block is one of the well known method for implementing bootstrap for time series.For a time series{y t},blocks are defined as follows:B i={y ti ,y ti+1,···,y ti+l i}Times t i and block lengths l i are sampled randomly.The resampled time series is the sequence of blocks:{B1,···,B N}The blocks may be overlapping or not.The length of the blocks can for instance be sampled from the geometric distribution.Refer to Lahari(2003)for a survey and exhaustive references.Block bootstrap was used to derive the statistical properties of many estimators(mean,variance, spectral density,etc.)for time series under quite low assumptions.But,this method may not be appropriate for applications which involve persistence statistics.Indeed,if the blocks are small the probabilities of sojourns in a set as well as autocorrelation functions may not be well reproduced.If the blocks are long the new time series will tends to be an exact replication of the reference time series and no innovation is brought by the simulated time series.At the opposite,several authors have shown that the block bootstrap is one of the most convenient method to estimate confidence intervals for any function of the empirical mean(see(H¨a rdle et al.,2003)and references therein).As far as we know this method has never been used to resample sea state time series.But Hogben et al.(1987)proposed a sampling method where observed time durations of sojourn over given levels are arranged at random.The main advantage of this method is that it can be applied when only persistence data are available.For a discussion on duration statistics of sea state parameters see for instance(Soukissian et al.,2001)and(Jenkins,2002).4.2Resampling of Markov chainsThe principle of the resampling method for Markov chain consists in building a non parametric estimator of the transition kernel,and then simulating realizations of the chain with this empirical kernel.Generally, the transition kernel is estimated locally using nearest-neighbor estimators.6In the meteorological field,nearest-neighbor resampling was first proposed by Young (1994)to simulate daily minimum and maximum temperatures and precipitations.Independently,Lall et al.(1996)used an analog method to generate hydrological time series,and Buishand et al.(2001)used basically the same method for multi-site generation of artificial meteorological time series.Nearest-neighbor resampling for time series {y t }is based on a very simple idea.Let us consider that we have already generated y sim 1,···,y sim t −1,the t −1first values of the simulated time series.We search in the data the k nearest neighbors of y sim t −1.One of these neighbors is randomly selected and the observed value for the date subsequent to the selected point is adopted as the simulated values at time t (Buishand et al.,2001).The nearest-neighbor search is generally time consuming.In Monbet et al.(2004a),a kdtree algorithm was used to perform a fast search.Various discrete probability distributions or kernels may be used to select randomly 1of the k nearest ll et al.(1996)suggested to use a kernel which assigns a higher probability to the closer points,as for instance p j =1/j k i =11/i ,j =1,···,k where the point j =1is the closest point and j =k the most distant.Monbet et al.(2003)suggested a more sophisticated method which permits to sample points which have not being observed as in smoothed bootstrap methods (Efron et al.,1993).This method,named Local Grid Bootstrap (LGB),was also used to simulate realizations of multivariate sea state processes:(H s ,T ),(H s ,T,U )(Monbet et al.,2004a)and (U,Φ)(Ailliot et al.,2004).It was shown that this method well restore most features of the data such as the marginal distribution,the distribution of storm durations and inter-arrivals as well as the covariance functions.An example of synthetic wind time series obtained with LGB is treated in section 6.2.4.3General discussionAs mentioned above,resampling methods can be used for simulation.Some of them can also be adapted for forecast and reconstruction.For instance,Ailliot et al.(2003)used a nearest neighbor method to simulate time series of (H s ,T,Θm )at several locations (x 0,···,x L )along a line to study the profitability of a maritime line in Aegean sea.Caires et al.(2005)proposed a non parametric regression method to correct outputs of meteorological models.And another method,based on a non-parametric Hidden Markov model is described in (Monbet et al.,2005)to reconstruct H s time series given wind time series (see also (Marteau et al.,2004a)).However,one of the main drawbacks of non parametric methods are that their descriptive power is about null.All the mentioned methods can be used with noisy data.Indeed missing data will not affect the nearest neighbor search,and aberrant data may have a positive probability to be resampled but this probability can be estimated.The main advantage is that they can be easily adapted for various time series (wind,wave,direction,...)because of the few assumptions required.For circular data,a particular attention has to be paid to the choice of the distance used for nearest neighbor search and kernel estimation.If simulation is combined with cross-reconstruction,the dimension of the time series is not a difficulty (Ailliot et al.,2003).When the dimension is high,simulation can be performed in two steps:the time series of a subset of components are generated first,then the other components are simulated given the first time series.Some asymptotic convergence properties of resampling and reconstruction methods have been studied (see (Monbet et al.,2004b),(Monbet et al.,2005)and references therein).In practice,it is known that a large amount of data is needed for estimation in non parametric methods.A well known drawback is the difficulty to choice the smoothing parameters such as the bandwidth parameters of the probability density function estimators.As concern the computational aspects,algorithms for resampling are generally very simple to imple-ment but they can be time consuming.A solution consists in working with algorithms based on tree structures for the nearest neighbor search (Monbet et al.,2004).5Parametric modelsIn this section,we gathered various parametric models,all Markovian,which were proposed for sea state and wind time series.Let us note,first of all,that it is physically reasonable to suppose that these7processes are Markov chains of small order.For instance,a linear autoregressive model of order1or2is generally sufficient to describe the second order structure of the wind speed.The use of linear autoregressive models for sea state time series was already discussed in section3. Hence,we focus now,on one hand,onfinite state space Markov chain models and,on the second hand, on non linear autoregressive models.A short parenthesis is also made to present autoregressive models for circular data.5.1Finite State Space Markov chainsLet us briefly describe the general principle of the methods considered in this section.•when the process is defined on a continuous state space,as it is generally the case for sea state parameters,this space is discretized in afinite number of classes,which allows to use a representation as afinite state space process.•this new process is then supposed to be a Markov chain whose transition matrix is estimated from the observed sequence.•new realizations of the process arefinally simulated given the estimated transition matrix.The success of this method is undoubtedly mainly due to its simplicity.For example,Sahin et al.(2001) propose to use afirst-order Markov chain,with8states,to generate artificial wind speed time series.One of the limitations of this method is the number of parameters to be estimated in the case where no assumption is made on the shape of the transition matrix.Various models were then proposed to reduce the number of parameters.For instance,in(Vik,1981),modeled H s by afirst-order Markov chain with tridiagonal transition matrix,which means that only the transitions to the closest states are allowed.The transition matrix is estimated in such a way that the stationary distribution of the Markov chain and the average durations of persistence above certain levels match with those of the database.In(Ailliot et al.,2001)another approach is proposed for(H s,θm)where the data are preprocessed before using afirst-order Markov chain model.The preprocessing consists in detecting the slope change times byfitting a linear spline curve H lin to the H s time series.A marked point process(H,τ)is associated to H lin.Then a Markov model is proposed for the process(H,τ,θm).This model permits to reproduce the cdf of H s andθm as well as the mean duration of storms at different severities.Finite state space Markov chain models are generally well adapted to describe circular data such as the wind direction.Indeed,many databases provide this parameter directly in the form of discretized data:it is usual for the meteorologists to gather the wind directions in”sectors”(usually16classes are used).Moreover,there exists relatively few models making it possible to describe circular time series directly.MacDonald et al.(1997)proposed a second order Markov chain to describe the evolution of the wind direction.In order to limit the number of parameters,they used the Raftery’s model in which the transition matrix are given byP(Φt=j|Φt−1=i1,Φt−2=i2)=λ1q i1,j +λ2q i2,jwith Q=(q i,j)a stochastic matrix andλ1andλ2some positive parameters such thatλ1+λ2=1.This model was tested on wind data at Koeberg(South Africa).5.2Autoregressive model for circular dataCircular data can also be modeled by autoregressive models,see for example(Breckling,1989)and(Fisher et al.,1994).Let{Φt}be a stationary process with values in R/2πZ.Mainly three types of autoregressive models were proposed:•Models obtained by”wrapping”a real valued process(Wrapped Autoregressive model).One sup-poses thatΦt=Y t modulo2πwith{Y t}a real valued process which follows an autoregressive model.8。
