Non-stationary noise in gravitational wave analyses: The wavelet domain noise covariance matrix

Gravitational wave detectors produce time series of the gravitational wave strain co-added with instrument noise. For evenly sampled data, such as from laser interferometers, it has been traditional to Fourier transform the data and perform analyses in the frequency domain. The motivation being that the Fourier domain noise covariance matrix will be diagonal if the noise properties are constant in time, which greatly simplifies and accelerates the analysis. However, if the noise is non-stationary this advantage is lost. It has been proposed that the time-frequency or wavelet domain is better suited for studying non-stationary noise, at least when the time variation is suitably slow, since then the wavelet domain noise covariance matrix is, to a good approximation, diagonal. Here we investigate the conditions under which the diagonal approximation is appropriate for the case of the Wilson-Daubechies-Meyer (WDM) wavelet packet basis, which is seeing increased use in gravitational wave data analysis. We show that so long as the noise varies slowly across a wavelet pixel, in both time {\em and} frequency, the WDM noise correlation matrix is well approximated as diagonal. The off-diagonal terms are proportional to the time and frequency derivatives of the dynamic spectral model. The same general picture should apply to other discrete wavelet transforms with wavelet filters that are suitably compact in time and frequency. Strategies for handling data with rapidly varying noise that violate these assumptions are discussed.