DRAFT NON LINEAR SIMULATION OF MULTIVARIATE SEA STATE TIME SERIES

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1 Proceedings of OMAE th International Conference on Offshore and Artic Engineering Halkidiki, Greece, June 2-7, 25 OMAE DRAFT NON LINEAR SIMULATION OF MULTIVARIATE SEA STATE TIME SERIES Valérie Monbet IFREMER Hydrodynamics and metocean BP7 F-2928 Plouzané, France Pierre Ailliot Department of Applied Statistics University of South Britany CERYC F-56 Vannes, France Marc Prevosto IFREMER Hydrodynamics and metocean BP7 F-2928 Plouzané, France ABSTRACT In this paper, three non linear methods are described for artificially generating operational sea state histories. In the first method, referred to as Translated Gaussian Process the observed time series is transformed to a process which is supposed to be Gaussian. This Gaussian process is simulated and back transformed. The second method, called Local Grid Bootstrap, consists in a resampling algorithm for Markov chains within which the transition probabilities are estimated locally. The last models is a Markov Switching Autoregressive model which allows in particular to model different weather types. Introduction The knowledge of sea state and wind conditions is necessary for many offshore or near shore operations. In design studies for instance, the instantaneous joint distribution of the parameters may suffice. But in a lot of cases, it is also important to consider the evolution of the sea state conditions. 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., 984), (Castino et al., 998). Evolution of sea state and wind conditions can be determinant in the estimation of the profitability of maritime line (Ailliot et al., 2) or to study coastal dune erosion. In the same way, the succession of storms and calm weather induces different sediment transports. Address all correspondence to this author with post address: V. Monbet, UBS/SABRES, CERYC, 56 Vannes, France. The applications are generally complex so that, one of the most common method to study them consists in considering different scenario of sea state conditions and doing a statistical analysis. When the phenomena are studied on short time periods (some days to some months), the statistical analysis can be done using hindcast data which are available for 2 to 4 years. But the hindcast databases are in general no more sufficient when one have to consider the evolution of a system over several years. In these cases, artificial time series obtained by simulation permit to enrich the databases and consequently to improve the accuracy of the statistical studies. The most classical approach for modeling sea state or wind time series consists in applying the methodology developed by Box and Jenkins (see Box et al. (976)): a Box-Cox transformation is first applied to the original time series in order to get a time series with approximately Gaussian marginal distribution and then an ARMA model is fitted to this residual. It was shown that this type of model can be used successfully to generate synthetic wind time series with marginal distribution and second order structure close to those of the original time series, (Brown et al. (984), Daniel et al. (99), Nfaoui et al. (996)). Such models have also been used for significant wave height and peak period, (see O Carroll (984), Stephanakos (999), Cunha et al. (999), Yim et al. (22)). However, it is well known that ARMA models or more generally Gaussian processes can not catch all the non-linearities which exist in many time series. As concern the wind time series for instance, one source of nonlinearity is induced by the existence of weather type. Indeed, Copyright c 25 by ASME

2 in North Atlantic for instance, at some periods of time, when the conditions are anticyclonic for example, the wind intensity evolves slowly and is generally low, whereas when there are cyclonic conditions, groups of successive lows cross the studied area one after the other leading the wind speed to evolve more quickly and to reach high values (see also Toll (997)). In this paper, we have chosen to focus on three simulation methods which are complementary: the first one is based on Gaussian process simulation (Translated Gaussian Process), the second one is based on a Markov chain model (Local Grid Bootstrap) and the last one uses Markov Switching Autoregressive model. In the first section, these methods are described in a general framework and a method based on Monte Carlo statistical tests is introduced to validate the models and to compare them. In the second section, we apply the presented methods to generate artificial time series of sea state parameters. A particular attention is given to wind intensity and wind direction. 2 Models and methods In this section, we describe first three non linear models for stationary sea state processes. Then a statistical method is proposed for validation and comparison of the models. Above all we discuss shortly the modeling of the non-stationary components which can be observed in sea state time series. 2. Modeling of non stationary components Depending on the considered time series, several types of non-stationary components can be identified such as the overyear trend, the annual components due to the meteorological seasons and the daily components due to the difference of temperature between day and night. The over-year trend is difficult to be considered in statistical models because of the few amount of data with respect to the temporal scale of the events (Athanassoulis et al., 995). At the opposite, the seasonal components are generally easy to be observed on sea state time series and several models have been proposed to remove the seasonal components. Two methods are commonly used:. If {Y t } denotes the sea state process, the following decomposition is generally used (Walton et al. (99), Stephanakos (999)): Y t = m(t)+σ(t)y stat t () where m and σ are deterministic periodic functions with period one year. These functions model respectively the variation of the mean and the standard deviation of the marginal distribution of the process. And {Yt stat } is assumed to be a stationary process. 2. The second approach consists in supposing that the process is piecewise stationary and to fit separate models for each month or for each season of the year. Some sea state time series have also daily non stationary component. One of the most common method used to removed this non-stationarity is to apply the decomposition of equation () with m and σ periodic functions with period one day. 2.2 Translated Gaussian Process The idea of the Translated Gaussian Process (TGP) simulation method is to transform the observed time series into a realization which can be assumed to be Gaussian. Several transformations can be applied. This method has been used by Scheffner et al. (992) and Monbet et al. (2a) to simulate sea-state parameters, by Gioffre et al. (2) to simulate wind pressure fluctuations on a building (see also reference therein) and also by Ailliot et al. (2) to simulate significant wave height coupled with the mean direction of wave propagation. Let {Y t } be a stationary process with values in R d. We assume that there exists a transformation f : R d R d and a stationary Gaussian process {X t } such that Y t = f (X t ). This procedure has three main steps: Model calibration, which consists in determining the function f and the second order structure of the process {X t }. Sample generation in which realizations of the process {X t } are simulated given the second order structure estimated in model calibration step. Several algorithms have been developed in order to simulate realizations of stationary Gaussian processes given the second order structure. In this work, the method described by Scheffner et al. (992) is used. Mapping. In this step the generated samples of {X t } are transformed into samples of {Y t } using the transformation f. In order to estimate the second order structure of the process, the empirical covariance function of ĝ(y t ) is calculated where y t represents a realization of {Y t } and ĝ the empirical estimate of g = f. Other calibration methods are proposed for instance in (Gioffre et al., 2). 2.3 Local Grid Bootstrap Local Grid Bootstrap (LGB) algorithm is a non parametric bootstrap procedure for continuous space discrete time stationary Markov chains (Monbet et al., 24). Recently, several approaches have been proposed for bootstrapping stationary time series (Bühlman, 22). The algorithm presented here is named Local Grid Bootstrap because the probability density functions are estimated locally as in Local Bootstrap (see for instance Paparoditis et al., 2) and the local area is discretized to form a 2 Copyright c 25 by ASME

3 grid around the current state. The last feature allows to sample unobserved states as in a smoothed bootstrap method. Let {Y t } be a stationary Markov chain of order in R d. Let P(y,A) be its transition kernel and {y t } t T be a realization of this time series. The idea of LGB consists in generating a Markov chain with a transition kernel ˆP(y, A) which is a non-parametric estimate of P(y, A). This empirical kernel assigns probabilities to a finite subset of convenient states. More precisely, at each step, if Y denotes the current state, ˆP(y,y )= ˆp(y,y ) x Gy ˆp(y,x) G y (y ) (2) where ( ) ( ) ˆp(y,y y )= i Iy K y i+ d h T K y yi d h T I y = {τ {,...,T } dist(y τ,y) < σ T,y } with dist(.,.) a distance in R d and σ T,y a constant which may depend on the density of the data around y. G y = G S {y τ+ τ I y }, with G a grid which is defined as a discretization of the neighborhood of the current state. This discretization is a computational way to implement bootstrap smoothing in multidimensional space. K d is a probability density kernel on R d which satisfies usual conditions and {h T } is a sequence of positive numbers such that h T = d i= s ii d ( ) 4 /(d+4) h (3) (2d + )T where s ii denotes the marginal variance of component i of x and h a constant. This algorithm can be straightforwardly be generalized for higher order and cyclostationary Markov chains (Monbet et al., 24). 2.4 Markov Switching Autoregressive model In Markov Switching Autoregressive Model the observed process {Y t } is modeled by an autoregressive model of order r with parameters depending on a non observable process {S t } which is a Markov chain. An important subclass of MS-AR model are the Hidden Markov Models (HMM), which correspond to the case r =. In these models the conditional distribution of Y t given {Y t } t <t and {S t } t t only depends on S t :two successive values of the observed process are assumed to be conditionally independent given the values of the hidden process. HMM have already been used for modeling the existence of weather type in meteorological time series like rainfall or wind direction, see MacDonald (997). However, this model is in general not able to catch the strong dependency existing between successive values sea state time series and we found that a MS- AR model of order r > suits sea state data better. A MS-AR process is a bivariate process {S t,y t } such that {S t } is a Markov chain on a finite space S = {,...,M} with M > the number of regimes. Conditionally to {S t }, {Y t } is a non-homogeneous Markov chain of order r ony R d. More precisely, we assume that the conditional distribution of Y t given {Y t } t <t and {S t } t t only depends on S t and Y t where Y t denotes the random vector {Y t } t r t t. In MS-AR model, different types of autoregressive models can be used in order to describe the evolution of the observed process {Y t } in the different regimes. Generally, a functional autoregressive model of the form (4) is used. Y t = f (S t) (Y t )+Σ (S t) E t (4) where {E t } represents a white noise which distribution has a density with respect to Lebesgue measure and such that E t is independent of Y t for t < t. The functions f (s) can be linear (see Hamilton (989) and Krolzig (997)) or not (see Rynkiewicz (2)). Models of the form (4) may not suit to describe time series which take their values in a given subset of R d as it is the case for most sea state parameters, for which Y = R + for instance. It is then more natural to specify directly the conditional distribution P(Y t Y t = y t,s t = s t ). For example, a Markov Switching Gamma Autoregressive Models (MS-γAR) can be used, in which the conditional distributions are gamma distributions with mean µ (s t) (y t )= p i= a(s t) i y t i + b (s t) standard deviation σ (s t) The constraints a (s) i, b (s) > and σ (s) > are added, for s S and i {...r}, in order the model to be well defined. The gamma distribution has been chosen because it is defined on R + and its density can be easily expressed from its first two moments. Other distributions, like the log-normal or Weibull distributions, could also be used, but there exists no physical evidence in general nor statistical criteria that permits to choose the shape of this conditional distribution. Non-homogeneous hidden Markov (NHHMM) models have also been used to describe meteorological time series. In Hugue et al. (999), it is used in order to relate broad scale atmospheric circulation variables (which play the role of the covariate) to local rainfall (the observed process). In this model, the hidden process is also interpreted as a weather type. In MacDonald et al. (997), a NHHMM is proposed to model seasonal and daily components existing in time series of wind direction in Koeberg 3 Copyright c 25 by ASME

4 (South Africa). In their model, the transition matrix of the hidden process is a (deterministic) function that depends only on the time. We also found that a NHMS-γAR model can be used to describe the daily components that exist in wind time series during the summer (Ailliot, 24). In the present paper, a NHMS-γAR model is fitted to jointly model wind intensity and wind direction. 2.5 Model validation and comparison We focus now on the problem of model validation. The most widespread method consists in comparing certain statistics calculated from the observations with the corresponding theoretical values. In general, several criteria are considered, such as the distribution function of the marginal distribution or the autocorrelation function. And, the authors often only perform visual comparisons. Here, we propose a method based on Monte Carlo statistical tests in order to decide whether the observed differences are significant or not. In most envisaged applications, it is convenient that the model restores the marginal distribution of the process, its autocovariance function or the distribution of the time duration of sojourn above given levels, etc. Let us consider as example the particular case of distribution functions. We want to test H : F = F versus H : F F (5) where F denotes the marginal distribution of the true process and F that of the considered model. To make this test is equivalent to give a reject rule for hypothesis H and a fixed risk. For this, we generate with the model a large amount of time series. The simulations permit to estimate very accurately the theoretical distribution F and to estimate the distribution of the test statistical and consequently to deduce the reject rule. Let us illustrate the idea by an example. On Figure, we Figure. Cumulative distribution function of the wind speed. MS-γAR model with M = 2 (left) and M = 3 (right). Solid line: observed, dashed line: simulated, dotted line: 95% interquantile range. have plotted F obs : the observed distribution function corresponding to the time series of wind speed which is used in section 3. F : the theoretical distribution function corresponding to the fitted MS-γAR model. It has been estimated using a large number of simulated sequences. For all x, an interval I(x), which will be called 95% interquantile range. It is defined by P H (F obs (x) I(x)) = 95%. It is added to help the visual comparison of the distribution functions F obs (x) and F (x). It has been estimated using the empirical quantiles at 2.5% and 97.5% corresponding to a large number of simulated sequence. Let us first comment the figure on the left. It corresponds to the MS-γAR model with M = 2 regimes. The corresponding observed test statistics w obs is equal to, while the reject rule for the null hypothesis is w obs.4 for a risk equal to 5%, so that we conclude that there is a significant difference between the observed and theoretical distribution functions. The tests have been run with N = synthetic time series, each of them having the same length as the original one. We use the first 5 ones to estimate the distribution of ˆF(x) under H and the 5 last ones to compute the cut-off value w α of the critical area. The simulated samples are also used to compute the 95% interquantile range plotted on the figures. In Figure, we can see that the observed distribution function is above the 95% interquantile range, which means that the model does not simulate enough wind of low intensity. As we have w obs =, this means that the observed distribution function is above all the simulated ones for some value of x. If we look at the figure on the right, which corresponds to the case M = 3, we can see that the observed distribution function stays in the 95% interquantile range, which implies that the observed test statistics is bigger than 5% and that the null hypothesis is accepted. 3 Applications The method presented above can be used to generate artificial time series for different vector of sea state parameters. Hereafter, the application to wind simulation is studied in details and then short discussion are given for other sea state parameters. 3. Wind intensity and direction Wind intensity time series are of practical use for example for assessment of wind power or prediction of coastline evolution due to erosion. In this part, the presented models are fitted on hindcast data, produced by Oceanweather, for a point of coordinates (46.25N,.67E) located near the French Atlantic coast. It consists in 22 years of data with a record every 6 hours. Y denotes the wind speed and Φ the wind direction. It is well known that the wind data are non-stationary. There exists generally daily and seasonal components, and possibly an over-year trend that is neglected here. Seasonal non-stationarity 4 Copyright c 25 by ASME

5 In particular, we assume that the evolution of the wind speed is conditionally independent of the wind direction given the hidden process. We will see below, in the validation part, that the model is able to catch the complex relation which exists between the wind speed and direction, what justifies this assumption. TGP - When the bivariate wind process {Z t } = {Y t,φ t } is considered, the transformation g of TGP is defined by g = g 2 g with Figure 2. Joint probability density function of (u, v) is removed by fitting a separate model for each month. We concentrate our study on the January month. For this month, there is no daily components in our data, and they were also neglected. Finally, the 22 months of January available in our data base will be assumed to be 22 realizations of a stationary process. There exists a complex relation between these two processes as we can see on Figure 2, which represents the bivariate marginal distribution of this process. This distribution is bimodal, with a first peak corresponding to cyclonic conditions and the other one to anticyclonic conditions. 3.. Simulation methods description Let us first describe briefly how the methods are applied to simulate wind time series. MS-AR model - At first, we propose an extension of the MSγAR model discussed in the previous section. We have shown that there exists a strong relation between the hidden process and the wind direction, and that the different weather types are more likely to be associated to wind blowing from different directions. To describe this relation, we will assume that the hidden process is a non-homogeneous Markov chain whose transition matrix depends on the wind direction. In this section, the wind direction is considered as a covariate. More precisely, the NHMS-γAR model we propose is such that: the evolution of the hidden variable {S t } is a nonhomogeneous Markov chain with transition probabilities q (t) θ (i, j) q i, j exp(κ j cos(φ t Φ j )) (6) The constraints M j= q i, j = are imposed in order to ensure the identifiability of the parameters. The choice of this parametrization is justified hereafter. the evolution of the wind speed in the different regimes is described by a gamma autoregressive model with parameters indexed by {S t } as in the MS-γAR model. g : (y,φ) (R F Y Φ=φ (y),f Φ (φ)) g 2 : (y,φ) (ycos(2πφ),ysin(2πφ)) where F Φ, R and F Y Φ=φ denote respectively the distribution functions of the marginal distribution of {Φ t }, the Rayleigh distribution and the conditional distribution of Y t given Φ t = φ. In practice, the distribution functions which appear in the definition of g are estimated using the usual empirical estimates and the second order structure of the process {X t } using the empirical covariance function of ĝ(y t ) where y t represents a realization of {Y t } and ĝ the empirical estimate of g. LGB - LGB is not applied directly to {Z t } but to the cartesian components u = Y cos(φ) and u = Y sin(φ) of the wind. In practice, a Markov chain model of order 2 is retain and Gaussian kernels are used. The algorithm parameters σ T,y and h are chosen given the data. LGB method is also used here to generate the covariate {Φ t } for the MS-γAR simulations Model interpretability It is possible to give a physical interpretation for the MS-γAR model but it is not the case for TGP and LGB. According to the Bayes Information Criterion, the MS-γAR model of order r = and with M = 3 regimes is selected. The parameters which govern the evolution of {Y t } in the different regimes are given below: regime : a () =.85[.], b () =.9[.29], σ () =.8[.] regime 2: a (2) =.58[.2], b (2) =.85[.64], σ (2) =.6[.] regime 3: a (3) =.7[.2], b (3) = 4.[.279], σ (3) =.8[.7] According to these values, we can propose the following meteorological interpretation to the different regimes. First, we can see that σ (2) and σ (3) are higher than σ (). This means that the second and third regimes correspond to periods when the wind speed evolve more quickly, as it is the case in presence of stormy conditions, whereas the first one is associated to slowly evolving wind, 5 Copyright c 25 by ASME

6 N E S W N N E S W N N E S W N Figure 3. Conditional transition probabilities φ P(S t = j S t = i,φ t = φ). i = on the top, then i = 2 and i = 3 below, j = on the left, j = 2 in the middle and j = 3 on the right. and thus corresponds to anticyclonic conditions. Parameters a () and a (2) are closed to each other but b (2) is higher than b (). This implies that the mean of the stationary distribution corresponding to the second regime (which is given by b (2) /( a (2) ))is higher than the mean of the first regime: we find again that the wind is usually higher in cyclonic conditions. The parameters γ i, j,κ j and Φ j are more difficult to interpret directly and are not given here. Instead, we have plotted the conditional transition probabilities φ P(S t = j S t = i,φ t = φ) on Figure Statistical validation To validate and compare the models, the statistical validation method is used. According to the envisaged applications, the statistics listed below are used. F Y : distribution function of Y F Φ : distribution function of Φ F (Y,Φ) : distribution function of (Y,Φ) F extr : distribution of monthly maxima of Y F [Y>2/3] : distribution of time duration of sojourn above level 2/3max(Y obs ), with max(y obs ) the largest wind speed in the observed time series F [Y<2/3] : distribution of time duration of sojourn under level 2/3max(Y obs ) F [Y</3] : distribution of time duration of sojourn under level /3max(Y obs ) C u : autocovariance function of the zonal component u = Y cos(φ) C v : autocovariance function of the meridional component v = Y sin(φ) The autocovariance functions C u and C v are computed instead of the autocorrelation functions of Y and Φ because the autocovariance function of the circular time series is difficult to interpret. As in the homogeneous model, the results obtained with this model have been compared to those corresponding to TGP and LGB. Table shows that TGP well restores the marginal and joint distribution functions of the process. It may be surprising to ob- TGP LGB MS-γAR F U.662 [.2].432 [.2].44 [.2] F Φ.74 [.8].392 [.4].392 [.4] F (Y,Φ).542 [.2].94 [.2].224 [.2] F extr.668 [.4].82 [.2].292 [.8] F [Y>2/3].3 [.8].32 [.6].46 [.6] F [Y<2/3].6 [.6].54 [.6].56 [.2] F [Y</3]. [.4].8 [.4].52 [.6] C u. [.6].68 [.6].28 [.2] C v. [.2]. [.8]. [.8] Table. Numerical results. The first value is the observed statistic w obs and the value in bracket the cut-off value w α with α =.5. The null hypothesis is rejected at the level α if w obs < w α Figure 4. Distribution functions of calm durations. Solid: observation, dashed: simulated with TGP, dotted 95% interquantile interval. The time in abscissa is expressed in days serve that TGP fails to reconstruct the covariance functions. Indeed, in TGP, the covariance function of the translated observed process is used as base for simulation. The mismatch reported in table is due to the fact that the translated process is only approximately Gaussian, probably because of the complexity of the marginal distribution of this process. This error may be reduced using a more sophisticated method to estimate the covariance structure of the Gaussian process, see Gioffre et al. (2). As concern time durations of storm and calm periods, we find again a lack of fit. There are too many short calm durations in the TGP simulated data (see Figure 4). This may be due to the fact that the time durations of the sojourns above a threshold s has the same distribution that the time durations of the sojourn below threshold ( s ) for a monovariate stationary Gaussian process with mean. The translated Gaussian process have similar characteristics and thus can not succeed to reproduce both calm and storm durations if they have different characteristics as it is the case in the considered time series. The NHMS-γAR and LGB models restore all the selected criteria, excepted the autocorrelation function of the v compo- 6 Copyright c 25 by ASME

7 nent. Indeed, as shown on figure 5, NHMS-γAR underestimates the high correlation which exists at 3-4 days in the data. However, the physical interpretation of these high correlations is not clear and should be further investigated Figure 5. Autocovariance function C v. Solid: observation, dashed: simulated with NHMS-γAR, dotted 95% interquantile interval. The time in abscissa is expressed in days. To check the chronology of the simulated time series of wind direction, we can for instance compare the mean time duration of sojourn in intervals [Φ Φ,Φ + Φ] (see also (Breckling, 989)). Figure 6 shows the mean time duration of sojourn for the observed wind direction time series and for time series generated by LGB. There is a good agreement between the two time series for this statistics. Significant wave height and Peak period - For the couple (significant wave height, peak period), Monbet et al. (2a) propose a TGP method where the instantaneous joint distribution of the bivariate process is estimated non parametrically. The method was tested on real data of the North sea. The results show a good agreement between the observed and the generated time series for several statistics including instantaneous marginal and joint distributions of the process and, distribution of time duration of storms. Monbet et al. (2b) use LGB approach for the same couple and also for (significant wave height, peak period, wind speed). They obtain again satisfactory conclusions. It is not straightforward to fit MS-AR models to (significant wave height, peak period) and as far as we now it has not been done until now. Significant wave height and Mean direction of wave propagation - In (Monbet et al., 24), LGB method is applied to simulate artificial time series of couple (significant wave height, mean direction of wave propagation) from hindcast data. It is shown that the generated time series match the observed time series for the statistics listed in this paper in the validation method section above. An application is proposed which consists in estimating the mean time necessary to perform an offshore operation given the date of the beginning of the operation. Ailliot et al. (2) use the TGP method to generate time series of (significant wave height, mean direction of wave propagation) on the basis of hindcast data Figure 6. Mean time duration of sojourn in intervals [Φ.25,Φ +.25]. Solid: observation, dashed: simulated 3.2 Other sea state parameters Models introduced in section 2 have also been fitted for other multivariate sea state parameters. More than two parameters - If we have to consider processes including more than two parameters, parametrical modeling is mostly difficult and non parametric methods such as LGB become computationally expensive. In (Ailliot et al., 23), the authors need to simulate time series of (significant wave height, peak period, mean direction of wave propagation) at several geographical points along a line to study the profitability of a maritime line in Aegean sea. To do that, they first simulate wind time series at a point x using a MS-γAR model. Then, the sea state parameters corresponding to the simulated wind at time t are deduced in searching the nearest neighbor of vector (W sim (t,x ),SS sim (t,x ),,SS sim (t,x L )) in a hindcast database. Here W sim denotes the generated wind and SS sim the generated sea state parameter vector. Following a similar idea, Marteau et al (24) propose a method where the time series of two parameters are jointly simulated by LGB and then the time series of other parameters are reconstructed given the first one by a Viterbi algorithm. Such an algorithm also allows to reconstruct waves times series given observed wind (Monbet et al, 25) or to propagate a sea state information from a geographical point to another (Marteau et al, 24). 7 Copyright c 25 by ASME

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