Mean amplitudes of the signal in the seasonal and shorter period length of day time series computed by the frequency-dependent autocovariance

The frequency-dependent autocovariance (FDA) function is defined in this paper as the autocovariance function of a wideband oscillation filtered by the Fourier transform bandpass filter (FTBPF). It was shown that the FDA estimation is a useful algorithm to detect mean amplitudes of oscillations in a very noisy time series. In this paper the least-squares polynomial harmonic model was used to remove the trend, low frequency as well as the annual and semi-annual oscillations from the IERS eopc04R_IAU2000_daily length of day (LOD) time series to compute their residuals. Next, the mean amplitudes of the signal as a function of frequency were determined from the difference between the FDA of LOD residuals and FDA of power-law noise model similar to the noise present in LOD residuals.  Several power-law noise model data were generated with a similar spectral index and variance as the noise in LOD data to estimate the mean amplitude spectrum in the seasonal and shorter period frequency band.  It was shown that the mean amplitudes of the oscillations in LOD residuals are very small compared to the noise standard deviation and do not depend on the filter bandwidth of the FTBPF. These small amplitudes explain why LOD prediction errors increase rapidly with the prediction length.