Data series subtraction with unknown and unmodeled background noise

LISA Pathfinder (LPF), the precursor mission to a gravitational wave observatory of the European Space Agency, will measure the degree to which two test masses can be put into free fall, aiming to demonstrate a suppression of disturbance forces corresponding to a residual relative acceleration with a power spectral density (PSD) below $(30\mathrm{fm}/{\mathrm{s}}^2/\sqrt{\mathrm{Hz}}{)}^2$ around 1 mHz. In LPF data analysis, the disturbance forces are obtained as the difference between the acceleration data and a linear combination of other measured data series. In many circumstances, the coefficients for this linear combination are obtained by fitting these data series to the acceleration, and the disturbance forces appear then as the data series of the residuals of the fit. Thus the background noise or, more precisely, its PSD, whose knowledge is needed to build up the likelihood function in ordinary maximum likelihood fitting, is here unknown, and its estimate constitutes instead one of the goals of the fit. In this paper we present a fitting method that does not require the knowledge of the PSD of the background noise. The method is based on the analytical marginalization of the posterior parameter probability density with respect to the background noise PSD, and returns an estimate both for the fitting parameters and for the PSD. We show that both these estimates are unbiased, and that, when using averaged Welch’s periodograms for the residuals, the estimate of the PSD is consistent, as its error tends to zero with the inverse square root of the number of averaged periodograms. Additionally, we find that the method is equivalent to some implementations of iteratively reweighted least-squares fitting. We have tested the method both on simulated data of known PSD and on data from several experiments performed with the LISA Pathfinder end-to-end mission simulator.

Figure 1

Schematic of LPF. The figure shows the reference TM, TM2, and the two laser interferometers—represented by their respective laser beam paths—that measure $x_1$ and $x_{12}$, respectively. The $x$ axis, shown in the figure, is parallel to the line joining the centres of mass of the two TMs. The $z$ axis, normal to the figure, points toward the Sun. Also shown are the electrodes used to apply the forces to TM2, necessary to keep it at nominally fixed distance from the reference TM. Similarly, the picture shows a pair of $\mu N$ thrusters that are used to force the satellite to stay at a nominally fixed position relative to the reference TM. Not shown in the figure are the electrodes and the $\mu N$ thrusters used to control TM and satellite along degrees of freedom other than $x$.

Figure 2

Result of noise projection with simulated data. Upper noisy red line: the square root PSD of the base data series $g[n]$. Lower noisy black line: the square root PSD of the residuals after fitting $g[n]$, with ${\alpha }_1\frac{d^2\delta a}{d{t}^2}[n]+{\alpha }_2\delta g[n,\tau ]$. The solid blue line represents the theoretical value of the expected PSD.

Figure 3

Marginalized histograms for the three fitting parameters, obtained with the MCMC method and the LL.

Figure 4

Deviation of the best fit parameter obtained at each step of the IRLS procedure, from those obtained with the MCMC mapping of the LL. Deviations are divided by the standard deviation obtained from the MCMC chain of the LL. Colors mark the different parameters. Red circles: A. Blue triangles: $-{\omega }_2^2$. Black diamonds: $-({\omega }_1^2-{\omega }_2^2)$. Brown stars: $\tau$.

Figure 5

PSD of various data series. The upper, blue, noisy continuous line is the PSD of $a[n]$ before the fit. The lower, black, noisy continuous line, barely visible behind the red dashed line, is the PSD of the residuals of the fit with the LL. The red dashed line is the PSD of the residuals of the IRLS fit.

Figure 6

The effect of subtraction of $\mathrm{\Delta }{F}_z$, $\mathrm{\Delta }{F}_y$, and $N_x$. The dotted line represents the PSD of the data series before subtraction. The solid line represents the PSD of the data series after subtraction, i.e. the series of the fit residuals.

Figure 7

The spectral window for the Blackman-Harris plotted for different values of k, spaced by $k_o=8$.

Figure 8

The absolute value of the correlation coefficient defined in Eq. (a9), as a function of the DFT coefficient spacing $\mathrm{\Delta }k$. Values are calculated for the Blackman-Harris window in the case of white noise.