OMAE ANALYSIS AND PROBABILISTIC MODELING OF THE STATIONARY ICE LOADS STOCHASTIC PROCESS WITH LOGNORMAL DISTRIBUTION

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1 Proceedings of the ASME rd International Conference on Ocean, Offshore and Arctic Engineering OMAE014 June 8-13, 014, San rancisco, California, USA OMAE ANALYSIS AND PROBABILISTIC MODELING O THE STATIONARY ICE LOADS STOCHASTIC PROCESS WITH LOGNORMAL DISTRIBUTION Petr Zvyagin St. Petersburg State Polytechnical University, St. Petersburg, Russia; Kirill Sazonov Krylov State Research Centre St. Petersburg, Russia Krylov State Research Centre, St. Petersburg, Russia ABSTRACT Until recent times researchers who investigated ice loads stochastic processes usually stated the fact of normal distribution for them. In the paper the model of a stationary stochastic process with a lognormal distribution for ice loads is offered. This model relates to the strain gauge transducer ice loads measurements as well as to some examples considered in different papers that were published earlier. or this model dependencies of the autocorrelation function were found that allows to simulate the ice loads process relatively easily. The procedure of such a simulation is described in details and the example of the analysis and simulation ice loads measurements is provided. INTRODUCTION The problem of ice-structure interaction probabilistic modeling can be investigated by several approaches. One of the approaches is to model discrete ice features parameters and then recalculate ice loads on the structure that are caused by interaction with these ice features. Within this approach a researcher should operate with a number of discrete interacting events [1]. The other approach is to consider ice loads as a continuous stochastic process with constant characteristics. This approach can be used if the interaction with a homogeneous ice feature is considered, for example, with a level ice without hummocks. By the level ice we mean a homogeneous ice feature, but its structural characteristics, such as thickness and strength, vary and are considered as random ones in the papers devoted to a simulation problem []. That is due to the fact that ice loads that are caused by the mentioned above feature can be considered as a stochastic process as well. It is reasonable to consider ice loads as a stochastic process [3-5], at least because autocorrelation of field ice loads measurements appears to be usually non-zero. To make reasoning about spectral properties more simple we suppose that the ice loads process is continuous despite the fact that we have discrete observations of this process usually made at equidistant time moments. In general, if we draw a sample of all available observations of ice loads, we find that the most appropriate distribution can vary depending on experiment parameters, sampling rate etc. In this paper the lognormal distribution for the ice loads process is chosen due to two reasons: - a number of practical ice loads measurements give lognormal distribution; - the process with this distribution has properties that are important for both a simulation and analysis. or some cases when lognormal distribution does not provide the best fit for the available data, this stochastic process still can be taken as a rough estimation, or a process that explains the significant part of data dynamics. LOGNORMAL ICE LOADS PROCESS AND ASSUMPTION O ITS STATIONARITY In some papers devoted to probabilistic modeling [6, 1] the lognormal distribution is assumed for random ice loads. On the graph published by Guo [4] we can see that the histogram of the crushing ice loads measured on the panel of Norstromsgrund lighthouse is asymmetric and can be related 1 Copyright 014 by ASME Downloaded rom: on 11/6/014 Terms of Use:

2 more to the lognormal distribution than to the normal one, which the author deals with. Let us consider time series of measurements of the strain gauge transducer positioned on the bow frame of scientific vessel Academic iodorov while heading through the solid ice. The measurements were made in 005 by AARI and were investigated by Zvyagin [7-8]. In this example the autocorrelation function takes large values for every two close observations (up to 1 sec time lag). As elements of the set to study let us take every 70 th measurement of this gauge transducer, because the absolute autocorrelation value is about 0.0 for the lag of 70 (approximately.7 sec). Thus we get a set X 70 of uncorrelated observations. urther reasoning concerning the lognormal origin of the general population for this set will allow us to state that these observations can be considered also as independent. igure 1. Histogram of vessel s bow frame strain gauge transducer independent measurements. The histogram of the set X 70 with the fitted lognormal distribution curve is presented on the ig. 1. Measurements of the strain gauge transducer can be recalculated into the force using the linear relation. So it is possible to state that local ice loads also can be -described as the lognormal stochastic process. Speaking about the stationarity of the stochastic process we shall assume the stationarity in wide sense [9] with autocorrelation function depends only on the time lag, and constant mathematical estimation and variance. In practice, it is relatively easy to check if available data fit all the requirements mentioned, or not. Let us presume that the continuous ice force process t is the stationary (in wide sense) stochastic process with the lognormal distribution. Thus, we have where t t is a normal process. Remark 1. If t e (1) t is the lognormal process with independent t states, then is the normal process with independent states, and vice versa. t is the stationary process in wide sense, t is stationary in wide sense, and vice versa. Corollary 1. If then Proof. The Process have where functions t is stationary in wide sense, so we Mt M, t D D, M t and t estimation and variance of the process M and D are constants. D are a mathematical or the mathematical estimation and variance of have: D t t respectively, t we 1 D Mt ln M ln 1 a const, M D ln 1 const M. () Let us consider the covariance function t 1, t K K for t : t, t M t t Mt Mt r t t (3) , r is an autocorrelation function for t where t 1,t t, t r. Due to the stationarity of t we have t t M t t Mt M t K K 1, 1 1,, Copyright 014 by ASME Downloaded rom: on 11/6/014 Terms of Use:

3 where t t1. rom the properties of normal and lognormal random variables we can derive M so we obtain and finally r t t exp a 1 rt t K, 1 1, a exp rt, t exp K t 1, t ln 1 r (4) exp a and that, with relation to (3), proves the first statement of Corollary 1. The second statement has the similar proof. Corollary. If states of stationary lognormal process wide sense are uncorrelated then they are independent. That follows just from (1) and the Corollary 1. t in AUTOCORRELATION UNCTION AND SPECTRAL DENSITY UNCTION OR THE ICE LOADS PROCESS While investigating measured ice loads time series of the stationary stochastic process t we can calculate a spectral density function s s using its definition: 0 K cos d. (5) Karna et al. [3] in their paper investigated ice loads time series measured on the panels of Norstromsgrund lighthouse, and offered to use the following expression for the spectral density: D s, (6) where 0 is a constant parameter, D is a constant variance of the stationary ice loads process. Originally in the mentioned above paper the dimensionless form of the spectral density was used. Based on a number of empirical ice loads spectral densities (igure ), Karna et al. numerically estimated constant parameter values of the function (6) that can be considered as the average spectral density. igure. Non-dimensional spectral density functions of local forces measured on all force panels [3]. The spectral density (6) with respect to (5) corresponds with the covariance function provided below: K De, 0, 0. (7) The expression (7) is relatively simple. Having an empirical vector i, K i and estimation for variance D, it is possible to find an estimation for using, for example, a LSQ method: n ln K i i i1 i1 n i i1 n i ln D The covariance function, that corresponds to the spectral density presented in the igure 3, has , and because of that the correlation coefficient for the lag is 1: 1 K r 1 e D That is consistent with the empirical value of the correlation for two neighboring observations that were found directly from the available time series of strain gauge transducer measurements.. 3 Copyright 014 by ASME Downloaded rom: on 11/6/014 Terms of Use:

4 igure 3. Spectral density for bow frame strain gauge transducer time series. SIMULATING LOGNORMAL STOCHASTIC PROCESS WITH THE GIVEN SPECTRAL DENSITY UNCTION The method of ice loads process simulation with the given spectral density, used by Guo [4], yields to a normal stochastic process. That happens according to the Central limit theorem. Every state of such a stochastic process is a random variable with normal distribution. It means that observed outcomes which are larger than the mathematical estimation will be encountered approximately as often as outcomes which are smaller than the mathematical estimation. In reality ice loads time series outcomes usually do not show such a symmetry, and this peculiarity can be described by the lognormal distribution of the stochastic process. The lognormal stationary process t simulating problem, in respect to (1), can be reduced to the power normal process t t is simulating problem. If the spectral density of t can be found given, then the autocorrelation function for using (4) as it follows: s cos d 1 0 r ln 1. expa Using the spectral density for t of the type (6), and therefore, the covariance function (7), we can obtain 1 lnd exp a r 1. (8) urther to simulate equidistant observations of t with a constant time lag t we shall apply the algorithm to normal Markov process simulating while using only the value of r t, calculated by (8). This algorithm consists of the steps represented below: 1) Get value x as a value of normal random variable t 0 with the mathematical estimation a and variance ) Calculate r t using the formulae (8).. 3) Get value y as a value of a normal random variable with the mathematical estimation a x a a yx yx 1 t 0 t which is correlated with coefficient is ). and variance. Here y is a value of random variable t 0 (the correlation 4) Set t 0 t 0 t and take x y, then go back to the point 3. The point 3 is based on the fact that the conditional probability density for states of the stationary process with normal distribution is: 1 y a x a exp 1 1 f t y x Random numbers for x and y with the specified normal distribution can be simulated by the algorithms offered by Knuth [10] or got by a built-in function of the statistical program, such as SPSS, STATISTICA, etc. Having the value of process t, the corresponding value of the t can be found by (1). Here M t a, t D, see also (). 4 Copyright 014 by ASME Downloaded rom: on 11/6/014 Terms of Use:

5 igure 4. a) Strain gauge transducer measurements; igure 5. Histogram for simulated time series presented in igure 4 b). igure 4. b) Simulated time series with the same characteristics. In the igure 4 a) time series of strain gauge transducer observations is presented. This transducer was mounted on the bow frame of the vessel Academic iodorov and observations were made while heading through the ice. In the igure 4 b) t with we represent the simulated time series approximately the same distribution law and autocorrelation function. The correlation coefficient for two neighbor states of the power process t calculated by (8) occurred to be r It can be compared with the correlation for neighbor states of t. igure 6. Graphs for autocorrelation functions: 1) trend estimation e ; ) for the time series presented in igure 4 a); 3) for the time series presented in igure 4 a). The histogram for simulated ice loads (igure 5) with a lognormal probability density curve on it verifies the lognormal origin of modeled data and can be compared with the histogram in igure 1. One can compare autocorrelation functions for simulated (igure 6, curve 3), field (igure 6, curve ) time series and exponential trend fitted to the field time series autocorrelation function (igure 6, curve 1). OUTCOMES It is evident that data with asymmetric histogram cannot be described reasonably by the symmetric normal process. Asymmetric lognormal distribution of the process seems to be more convenient for that. The examples discussed in the paper 5 Copyright 014 by ASME Downloaded rom: on 11/6/014 Terms of Use:

6 justify that the lognormal distribution is applicable for the ice loads description. While simulating the lognormal process, one can encounter the problem how to derive the autocorrelation function for power, being aware about the autocorrelation function for the entire lognormal ice loads process. This problem is solved in the paper. If the sampling rate of ice loads measurements is high, then the significant correlation between two neighbor observations can be expected. The example of ice loads strain gauge transducer measurements shows that the correlation of two observations of ice loads can decrease like an exponent function with negative power while increasing the time lag between them. The spectral density function (6), already used for ice loads by other authors, gives the same autocorrelation function. The simulating algorithm for the correlated lognormal process is described in the paper in details. This algorithm operates with mentioned above type of the autocorrelation function. The given theoretical model and simulating algorithm provide fast way for probabilistic modeling lognormally distributed stationary ice loads process. [5] Zvyagin, P. Sazonov, K., 013, Statistically Based Method of Representing Ice Signal as a Sum of Several Uncorrelated and Stationary Processes, Proc. th POAC Conf. Espoo, inland, 7pp. [6] Karna, T., Kamesaki, K.,Tsukuda, H., 1999, A Numerical Model for Dynamic Ice-Structure Interaction, Computers and Structures, 7, pp [7] Zvyagin, P., 007, Application of uzzy Neural Networks for the Data Clusterization Problem, Optical Memory and Neural Networks Journal (Information Optics), Vol. 16, No., pp [8] Zvyagin, P., 011, Models of structural data analysis (in Russian), Lambert Academic Publishing, 131 pp. [9] Pugachev, V.S., 1965, Theory of Random unctions and its Application to Control Problems, Pergamon (Translated from Russian), 833 pp. [10] Knuth, D. E., 1968, The Art of Computer Programming, Addison-Wesley, 634 pp. ACKNOWLEDGMENTS Authors wish to thank Prof. Oleg Timofeev from St. Petersburg Krylov Research Center for his support and provided field observation materials. Authors are grateful to anonymous reviewers of this paper. Petr Zvyagin wishes to thank Dr. Kari Kolari from VTT (inland) for his vision of the problem and helpful discussions. REERENCES [1] Bekker, A., Sabodash, O., Kovalenko, R., 013, Probabilistic Modeling of Extreme Values Distributions of Ice Loads on MOLIKPAQ Platform for Sakhalin-II Project, Proc. 3th OMAE Conf., Nantes, rance, OMAE [] Su, B., Riska, K., Moan, T., 011, Numerical Simulation of Ship Operating in Level Ice, Proc. 1th POAC Conf., Montreal, Canada, POAC [3] Karna, T. Qu, Y. Kuhnlein, W. Yue, Q., Bi, X., 007, A Spectral Model of Ice orces due to Ice Crushing, Journal of Offshore Mechanics and Arctic Engineering, 19, pp [4] Guo,., 01, A Spectral Model for Simulating Continuous Crushing Ice Load Proc. 1st IAHR International Symposium on Ice, Dalian, China, pp Copyright 014 by ASME Downloaded rom: on 11/6/014 Terms of Use: