Signal-to-noise enhancement by environmental cross-correlation

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There are many situations of interest in which data are contaminated by the environment. Often this contamination is understood, and by monitoring the environment it is possible to ``clean up" or ``reduce" the data, by subtracting the effects of the environment from the signal or signals of interest. In the case of the data stream from an interferometric gravitational radiation detector, the signal of interest is the differential displacement of suspended test masses. This displacement arises from gravitational waves but also has contributions arising from other contaminating sources, such as the shaking of the optical tables (seismic noise) and forces due to ambient environmental magnetic fields. The key point is that the gravitational waves are not correlated with any of these environmental artifacts.

The method implemented here works by estimating the linear transfer function between the IFO_DMRO channel and specified environmental channels on the basis of the correlations over a certain bandwidth in Fourier space. The method is explained in detail in the paper `Automatic cross-talk removal from multi-channel data' (WISC-MILW-99-TH-04)²Here we will just give a very brief overview to introduce the quantities calculated.

We denote the channel of interest, normally the InterFerOmeter Differential Mode Read-out (IFO_DMRO), by $X$ or $Y_1$. The other sampled channels consist of environmental and instrumental monitors which we denote $Y_2,\dots,Y_N$. We assume that all fast channels have been decimated so that all channels are sampled at the same (slow) rate, $986.842\cdots\$Hz for the November 1994 40-meter data.

We assume that the contribution of channel $i$ to channel $1$ is described by an (unknown!) linear transfer function $R_i(t-t')$. The basic idea of the method is to use the data to estimate the transfer functions $R_i$. For the reasons discussed in the paper, we work with the data in Fourier space. The transfer function is estimated by averaging over a frequency band, that is a given number of frequency bins. The number of bins in any band is denoted by $F$ in the cited paper and correlation_width in the programs below. The method assumes that $\tilde R_i$ can be well approximated by a complex constant within each frequency band, in other words that the transfer function does not vary rapidly over the frequency bandwidth $\Delta f = F/T$ where $T$ is total time of the data section under consideration. The choices 32, 64 and 128 appear most appropriate for $F$ for the 40-meter data.

Within a given band, $b$, the Fourier components of the field may be thought of as the components of an $F$-dimensional vector, ${\bf Y}_i^{(b)}$. Correlation between two channels (or the auto-correlation of a channel with itself) may be expressed by the standard inner product $({\bf Y}_i^{(b)},{\bf Y}_j^{(b)}) = {\bf Y}_i^{(b)}{\cdot}{\bf Y}_j^{(b)*}$ (no summation over $b$). Our assumption that $\tilde R_i$ is constant over each band means that the `true' channel of interest (the IFO_DMRO channel with environmental influences subtracted) can be written

$$\bar {\tilde {\bf x}}^{(b)} = {\tilde {\bf X}}^{(b)} - \sum_{j=2}^N r^{(b)}_j {\tilde {\bf Y}_j}^{(b)}.$$ (5.1.27)

where $r^{(b)}_j$, $j=2,\dots,N$ are constants. The fundamental assumption is that the best estimate of the transfer function in the frequency band $b$ is given by the complex vector $(r^{(b)}_2,\dots,r^{(b)}_N)$ that minimises $\vert\bar {\tilde {\bf x}}^{(b)}\vert^2$. To measure the `improvement' in the signal we define $\vert\rho\vert^2$ by

$$\vert\bar {\tilde {\bf x}}^{(b)}\vert^2 = \vert {\tilde {\bf X}}^{(b)}\vert^2 \left( 1 - \vert\rho\vert^2 \right) .$$ (5.1.28)

denoted by rho2 in the programs below. By definition $0 \leq \vert\rho\vert^2 \leq 1$. If any of the environmental channels are strongly correlated with the channel of interest, a significant reduction in $\vert\bar {\tilde {\bf x}}^{(b)}\vert^2$ is obtained, that is, $\vert\rho\vert^2$ will be close to 1.

To understand the origin of the `improvement' it is also convenient to study the best estimate that can be obtained using any given single environmental channel. Thus we define

$$\bar {\tilde {\bf x}}^{(b)}_i = {\tilde {\bf X}}^{(b)} - {r'}^{(b)}_i {\tilde {\bf Y}_i}^{(b)}$$ (5.1.29)

and choose the complex number ${r'}^{(b)}_i$ to minimise $\vert\bar {\tilde {\bf x}}^{(b)}_i\vert^2$. Of course, in general this will not correspond to the $i$th component of the vector used in the multi-channel case. The corresponding improvement $\vert\rho_i\vert^2$ given by

$$\vert\bar {\tilde {\bf x}}^{(b)}_i\vert^2 = \vert{\tilde {\bf X}}^{(b)}\vert^2 \left( 1 - \vert\rho_i\vert^2 \right)$$ (5.1.30)

is denoted by rho2_pairwise in the programs below. By definition $0 \leq \vert\rho_i\vert^2 \leq 1$. If the $i$th environmental channel is strongly correlated with the channel of interest, a significant reduction in $\vert\bar {\tilde {\bf x}}^{(b)}_i\vert^2$ is obtained, that is, $\vert\rho_i\vert^2$ will be close to 1.