Explicit Calibration and Simulation of Stochastic Fields by Low-Order Arma Processes

Abstract eng:
A simple framework for autoregressive simulation of stochastic fields is presented. The autoregressive format leads to a simple exponential correlation structure in the time-dimension. In the case of scalar processes a more detailed correlation structure can be obtained by adding memory to the process via an extension to autoregressive moving average (ARMA) processes. The ARMA format incorporates a more detailed correlation structure by including previous values of the simulated process. Alternatively, a more detailed correlation structure can be obtained by including additional ‘state-space’ variables in the simulation. For a scalar process this would imply an increase of the dimension of the process to be simulated. In the case of a stochastic field the correlation in the time-dimension is represented, although indirectly, in the simultaneous spatial correlation. The model with the shortest memory – the single-step autoregressive model – is analyzed in detail, and an efficient multi-step calibration procedure is developed. The calibration makes direct use of conditional correlations and means, expressed explicitly in terms of the zero and 𝑘-step correlation matrices of the stochastic field. The correlation structure of an isotropic turbulent wind field is developed from the generalized von K`arm`an spectrum in terms of the Airy function, and the simulation procedure is illustrated for turbulent wind with Airy or exponential function representation of the correlation structure. In spite of the basic simplicity of the simulation algorithm, the details of the transverse correlation, that should satisfy an integral condition for consistency, is represented well.

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