Simulation of Nonstationary Stochastic Processes by Spectral Representation

This paper presents a rigorous derivation of a previously known formula for simulation of one-dimensional, univariate, nonstationary stochastic processes integrating Priestly’s evolutionary spectral representation theory. Applying this formula, sample functions can be generated with great computational efficiency. The simulated stochastic process is asymptotically Gaussian as the number of terms tends to infinity. This paper shows that (1) these sample functions accurately reflect the prescribed probabilistic characteristics of the stochastic process when the number of terms in the cosine series is large, i.e., the ensemble averaged evolutionary power spectral density function (PSDF) or autocorrelation function approaches the corresponding target function as the sample size increases, and (2) the simulation formula, under certain conditions, can be reduced to that for nonstationary white noise process or Shinozuka’s spectral representation of stationary process. In addition to derivation of simulation formula, three methods are developed in this paper to estimate the evolutionary PSDF of a given time-history data by means of the short-time Fourier transform (STFT), the wavelet transform (WT), and the Hilbert-Huang transform (HHT). A comparison of the PSDF of the well-known El Centro earthquake record estimated by these methods shows that the STFT and the WT give similar results, whereas the HHT gives more concentrated energy at certain frequencies. Effectiveness of the proposed simulation formula for nonstationary sample functions is demonstrated by simulating time histories from the estimated evolutionary PSDFs. Mean acceleration spectrum obtained by averaging the spectra of generated time histories are then presented and compared with the target spectrum to demonstrate the usefulness of this method.