3rd-order Spectral Representation Method: Simulation of multi-dimensional random fields and ergodic multi-variate random processes with fast Fourier transform implementation

Abstract

This paper introduces a generalized 3rd-order Spectral Representation Method for the simulation of multi-dimensional random fields and ergodic multi-variate stochastic processes with asymmetric non-linearities. The formula for the simulation of general d-dimensional random fields is presented and the method is applied to simulate 2D and 3D random fields. The differences between samples generated by the proposed methodology and the existing classical Spectral Representation Method are analysed. The formula for the simulation of multi-variate random processes is subsequently developed. An important feature of the methodologies is that they can be implemented efficiently with the Fast Fourier Transform (FFT), details of which are presented. Computational savings are shown to grow exponentially with dimensionality as a testament of the scalability of the simulation methodology. Examples highlighting the salient features of these methodologies are also presented.