5.1 Noise spectrum estimation
Spacecraft Doppler tracking shares attributes with all other real observations: The noise is
non-stationary, there are low-level systematic effects, and there are data gaps. The noise can usually be
regarded as effectively statistically stationary for at most the interval of a tracking pass (
8 hours). The
non-stationarity of (what may be fundamentally Gaussian but with time-variable variance) noise is a
complication. For example in matched filtering for bursts, discussed below, the noise spectrum has to be
estimated locally [60
] for use in deriving locally-valid matching functions from the assumed GW
waveforms [58].
Noise characterization has historically been done via standard spectral and acf analysis techniques [67].
Spectra of various data sets have been presented in [4, 18, 25, 9, 19]. The data have typically been
analyzed with varying time-frequency resolution to assess the fidelity of the spectral estimates
and to provide local (in Fourier frequency) estimates of the underlying noise spectral density
for sinusoidal and chirp signal searches (below). Running estimates of the variance and third
central moments have been used as guides for identifying intervals of stationarity in pilot studies.
Bispectra
were computed for early data sets looking for non-linear, non-Gaussian effects. Bispectral analysis seemed to
have limited utility, however; the Doppler noise is close to Gaussian and the slow convergence of higher
statistical moments makes the bispectrum hard to estimate accurately over the length of a stationary data
interval.
