On the use of maximum entropy/autoregressive spectrum in harmonic analysis of time series

The maximum entropy (ME) spectrum, or its equivalent form of the autoregressive (AR) spectrum, has been used as a tool for harmonic analysis of time series in geophysics. This paper critically examines its usage in estimating the amplitude and the exponential decay rate of a harmonic function. The argument is based upon Prony's relation, which relates a complex-conjugate pair of poles for the AR model of the time series on one hand, to the complex frequency of one harmonic component in the time series on the other. It is found that: (i) the ME spectrum can be used as an estimator for the decay rate in a way similar to the Fourier spectral analysis; (ii) the ME spectrum contains no information whatsoever about the amplitude, contrary to what has been claimed and practiced in geophysical applications.