An algorithm to simulate nonstationary and non-Gaussian stochastic processes

H P Hong,X Z Cui,D Qiao

doi:10.1186/s43065-021-00030-5

Abstract

We proposed a new iterative power and amplitude correction (IPAC) algorithm to simulate nonstationary and non-Gaussian processes. The proposed algorithm is rooted in the concept of defining the stochastic processes in the transform domain, which is elaborated and extend. The algorithm extends the iterative amplitude adjusted Fourier transform algorithm for generating surrogate and the spectral correction algorithm for simulating stationary non-Gaussian process. The IPAC algorithm can be used with different popular transforms, such as the Fourier transform, S-transform, and continuous wavelet transforms. The targets for the simulation are the marginal probability distribution function of the process and the power spectral density function of the process that is defined based on the variables in the transform domain for the adopted transform. The algorithm is versatile and efficient. Its application is illustrated using several numerical examples.

Highlights

Observed time histories of the seismic ground motions [31], wind velocity [48], wave height [32], etc. fluctuate randomly in time and space
The steps in the iterative power and amplitude correction (IPAC) algorithm in a pseudocode form are shown in the flowchart depicted in Fig. 1 and are described as follows: I) Prescribe the targets and initiate the simulation process: Sample {u(jΔt)}N based on a random number generation algorithm for a uniformly distributed random variable between 0 and 1
Summary and conclusions We elaborated on the concept of defining a modulated and intensity function adjusted (MODIF) stochastic process in the transform domain

Summary

Introduction

Observed time histories of the seismic ground motions [31], wind velocity [48], wave height [32], etc. fluctuate randomly in time and space. The definitions lend themselves to an understandable and almost trivial algorithm to simulate stochastic processes: A) Sample Gaussian white noise, w(t), and calculate the normalized coefficients of w(t) in the transform domain (e.g., eiθ F ðwðtÞÞ, or eiθWðwðtÞÞ, or eiθSðwðtÞÞ if FT, or WT, or ST is used, respectively). A critical issue of applying the MODIF process with prescribed target energy distribution is that the energy distribution of the sampled signals for given intensity function may not be readily established, except for the case where FT is used (i.e., transforms with non-redundant representation). This is because unlike the FT, both WT and ST provide redundant representation. JyWðs; τÞjeiθWðwðtÞÞ and jySðs; τÞjeiθSðwðtÞÞ are not equal to xWðs; τÞ 1⁄4 WT ðIWT ðjyWðs; τÞjeiθWðwðtÞÞÞÞ and xS ð f ; τÞ 1⁄4 ST ðIST ðjySðs; τÞjeiθSðwðtÞÞÞÞ, respectively

To see the impact of this inequality on the simulated

Consider that we simulate the MODIF process using

IAAFT algorithm

Set xAC ð jΔtÞ