Description and simulation of nonstationary processes based on Hilbert spectra

A new method is proposed for the description and simulation of nonstationary random processes based on Hilbert spectra of their sample observations. A sample of a random process is first decomposed into intrinsic mode functions (IMFs) by the method of empirical mode decomposition. The Hilbert transforms of the IMFs yield the instantaneous amplitude and frequency, from which the Hilbert spectrum can be obtained as a function of time and frequency. The average of the Hilbert spectra over the samples is then defined as the Hilbert spectrum of the process and used as the target in the simulation of the process. The method is also extended to vector random processes. Unlike current procedures such as those based on the evolutionary process, no assumptions of functional forms for the spectra are necessary which are unknown a priori; and no assumptions of piecewise stationarity and egodicity of the process are required in parameter estimation. Applications to spectral characterization and simulation of multivariate earthquake ground motions show that the Hilbert spectra give a clear description of the time-varying spectral content of the motions and the simulated samples represent an accurate statistical image of the records. The response spectra compare well with those of the records and retain the jagged look. The method has great potential for engineering applications when dealing with nonstationary, nonlinear random processes.