Multitaper spectral estimation of power law processes

In many branches of science, particularly astronomy and geophysics, power spectra of the form f^-β, where β is a positive power-law exponent, are common. This form of spectrum is characterized by a sharp increase in the spectral density as the frequency f decreases toward zero. A power spectrum analysis method that has proven very powerful wherever the spectrum of interest is detailed and/or varies rapidly with a large dynamic range is the multitaper method. With multitaper spectral estimation, a set of orthogonal tapers are applied to the time series, and the resulting direct spectral estimators (“eigenspectra”) are averaged, thus, reducing the variance. One class of processes with spectra of the power-law type are fractionally differenced Gaussian processes that are stationary and can model certain types of long-range persistence. Spectral decay f^-β can be modeled for 0<β<1. Estimation of the spectral slope parameter by regression on multitaper spectral ordinates is examined for this class of processes. It is shown that multitapering, or using sine or Slepian tapers, produces much better results than using the periodogram and is attractive compared with other competing methods. The technique is applied to a geophysical estimation problem