Noise spectral estimation methods and their impact on gravitational wave measurement of compact binary mergers

Estimating the parameters of gravitational wave signals detected by ground-based detectors requires an understanding of the properties of the detectors’ noise. In particular, the most commonly used likelihood function for gravitational wave data analysis assumes that the noise is Gaussian, stationary, and of known frequency-dependent variance. The variance of the colored Gaussian noise is used as a whitening filter on the data before computation of the likelihood function. In practice the noise variance is not known and it evolves over timescales of dozens of seconds to minutes. We study two methods for estimating this whitening filter for ground-based gravitational wave detectors with the goal of performing parameter estimation studies. The first method uses large amounts of data separated from the specific segment we wish to analyze and computes the power spectral density of the noise through the mean-median Welch method. The second method uses the same data segment as the parameter estimation analysis, which potentially includes a gravitational wave signal, and obtains the whitening filter through a fit of the power spectrum of the data in terms of a sum of splines and Lorentzians. We compare these two methods and conclude that the latter is a more effective spectral estimation method as it is quantitatively consistent with the statistics of the data used for gravitational wave parameter estimation while the former is not. We demonstrate the effect of the two methods by finding quantitative differences in the inferences made about the physical properties of simulated gravitational wave sources added to LIGO-Virgo data.

Figure 1

Example results for a high mass BBH system (top), low mass BBH system (middle), and BNS system (bottom). The left column shows the amplitude spectral density of the interferometer data plus signal injection (gray), the off-source spectral estimation (green), and the median on-source noise model (orange). The right column shows the difference between the cumulative distribution of the percentiles ($P$) and expected credible level ($p$) versus the expected credible level ($p$) of the whitened Fourier amplitudes, assuming they are drawn from $\mathcal{N}(0,1)$. The gray ellipses represent expected fluctuations at the 1, 2, and $3\sigma$ level given the number of samples in the data.

Figure 2

Cumulative distribution of the square of the AD statistic for our high mass BBH (top), low mass BBH (middle), and BNS (bottom) injections. The black dashed line shows the theoretical expectation for random samples drawn from a normal distribution with a mean of 0 and a standard deviation of 1. The grey shaded regions show 1,2, and $3\sigma$ error regions.

Figure 3

Histogram of the real and the imaginary part of the whitened data obtained with the on-source and the off-source whitening filter for the same system as the bottom row of Fig. 2. The x axis is in units of standard deviation. For comparison we also plot a zero-mean, unit-variance Gaussian distribution. The off-source whitening is done on the data before the simulated injections are added. The on-source whitening has the signals included, but BayesWave fits any non-Gaussian noise with wavelets and we whiten the residuals.

Figure 4

KS statistic between the posteriors for selected parameters computed with the off-source and the on-source median noise variance for our high mass BBH (top), low mass BBH (middle), and BNS (bottom) injections.

Figure 5

Posterior distributions for the chirp mass obtained with the off-source and the on-source whitening filter for the two high mass BBH (top), low mass BBH (middle), and BNS (bottom) systems with the highest KS statistic from Fig. 4. The true chirp mass is represented by the black vertical line.

Figure 6

Posterior distributions for the mass ratio obtained with the off-source and the on-source whitening filter for the two high mass BBH (top), low mass BBH (middle), and BNS (bottom) systems with the highest KS statistic from Fig. 4. The true mass ratio is represented by the black vertical line.

Figure 7

Posterior distributions for the effective spin obtained with the off-source and the on-source whitening filter for the two high mass BBH (top), low mass BBH (middle), and BNS (bottom) systems with the highest KS statistic from Fig. 4. The true effective spin is represented by the black vertical line.

Figure 8

Posterior distributions for the effective precession obtained with the off-source and the on-source whitening filter for the high mass BBH (left), low mass BBH (right) systems with the higher KS values from Fig. 4. The true effective precession parameter is represented by the black vertical line.

Figure 9

Posteriors distributions for the effective tidal parameter obtained with the off-source and the on-source whitening filter for the BNS systems with the higher KS values from Fig. 4. The true effective tidal parameter is represented by the black vertical line.

Figure 10

KS statistic for the chirp mass of two high mass BBH events injected at $t_0$ and analyzed with whitening filters computed with the BayesLine on-source methodology from data at $t-t_0$ as a function of the time delay $t-t_0$. We find a generally increasing trend of the KS with the time distance between the signal and the data used to compute the whitening filter.