{"references": ["M. Shinozuka and C.M. Jan, Digital Simulation of Random Processes\nand its Applications. Journal of Sound and Vibration, 25(1), 111-128,1972.", "G. Solari and F.Tubino, A turbulence Model based on Principal Components.\nProbabilistic Engineering Mechanics, 17, 327-335, 2002.", "A. Kareem, Numerical simulation of wind effects: A probabilistic perspective. Journal of Wind Engineering and Industrial Aerodynamics,\n96, 1472-1497, 2008.", "X. Chen,A. Kareem, Aeroelastic analysis of bridges under multicorrelated\nwinds: integrated state-space approach. Journal of Engineering Mechanics ASCE, 127 (11), 1124-1134, 2001.", "J.C. Kaimal, J.C. Wyngaard and Y. Izumi, O.R. Cote, Spectral Characteristics\nof Surface-Layer Turbulence. Quarterly Journal of the Royal\nMeteorological Society, 98, 1972.", "M. Shiotani and Y. Iwayani, Correlation of Wind Velocities in Relation to\nthe Gust Loadings. Proceedings of the 3rd Conference on Wind Effects\non Buildings and Structures, Tokyo, 1971.", "E. Samaras, M. Shinozuka and A. Tsurui, ARMA representation of random processes. Journal of Engineering Mechanics ASCE, 111(3), 449461, 1985.", "A. Papoulis, Probability, Random Variables and Stochastic Processes,\n2nd Ed. Mc Graw-Hill, 1984.", "W. Gersch and J. Yonemoto, Synthesis of multivariate random vibration\nsystems: A two-stage least squares AR-MA model approach. Journal of\nSound and Vibration, 52(4), 553-565, 1977.\n[10] Y. Li and A. Kareem, ARMA systems in wind engineering. Probabilistic\nEngineering Mechanics, 5(2), 50-59, 1990.\n[11] P. Van Overschee and B. De Moor, Subspace Identification for Linear\nSystems: Theory-Implementation-Applications, Dordrecht, Netherlands:\nKluwer Academic Publishers, 1996.\n[12] H. Akaik, Stochastic theory of minimal realization, IEEE Transactions\non Automatic Control 19, 667-674, 1974.\n[13] T. Katayama, Subspace Methods for System Identification, first ed.,\nSpringer, 2005.\n[14] H. Akaik, Markovian representation of stochastic processes and its\napplication to the analysis of autoregressive moving-average processes,\nAnnals of the Institute of Statistical Mathematics 26(1), 363-387, 1974."]}

Turbulence of the incoming wind field is of paramount importance to the dynamic response of civil engineering structures. Hence reliable stochastic models of the turbulence should be available from which time series can be generated for dynamic response and structural safety analysis. In the paper an empirical cross spectral density function for the along-wind turbulence component over the wind field area is taken as the starting point. The spectrum is spatially discretized in terms of a Hermitian cross-spectral density matrix for the turbulence state vector which turns out not to be positive definite. Since the succeeding state space and ARMA modelling of the turbulence rely on the positive definiteness of the cross-spectral density matrix, the problem with the non-positive definiteness of such matrices is at first addressed and suitable treatments regarding it are proposed. From the adjusted positive definite cross-spectral density matrix a frequency response matrix is constructed which determines the turbulence vector as a linear filtration of Gaussian white noise. Finally, an accurate state space modelling method is proposed which allows selection of an appropriate model order, and estimation of a state space model for the vector turbulence process incorporating its phase spectrum in one stage, and its results are compared with a conventional ARMA modelling method.