Determining evolutionary spectra from non-stationary autocorrelation functions

Abstract

For non-stationary stochastic processes, the classic integral expression for computing the autocorrelation function from the evolutionary power spectral density (evolutionary spectrum) developed by Priestley is not invertible in a unique way. Thus, the evolutionary spectrum cannot be determined analytically from a given autocorrelation function. However, the benefits of an efficient inversion from autocorrelation to evolutionary spectrum are vast. In particular, it is more straightforward to estimate the autocorrelation function from measured data, yet efficient simulation depends on knowing the evolutionary spectrum. This work examines the existence and uniqueness of such an inversion from the autocorrelation to the evolutionary spectrum under a certain set of conditions. It is established that uniqueness of the inversion is likely although it is not proven. A methodology is presented to determine the evolutionary spectrum from a prescribed or measured non-stationary autocorrelation function by posing the inversion as a discrete optimization problem. This method demonstrates the ability to perform the inversion but is computationally very expensive. An improved method is then proposed to enhance the computational efficiency and is compared with some established optimization methods. Numerical examples are provided throughout to demonstrate the capabilities of the proposed methodologies.