### 4.10 Network of detectors

Several gravitational-wave detectors can observe gravitational waves from the same source. For example a network of bar detectors can observe a gravitational-wave burst from the same supernova explosion, or a network of laser interferometers can detect the inspiral of the same compact binary system. The space-borne LISA detector can be considered as a network of three detectors that can make three independent measurements of the same gravitational-wave signal. Simultaneous observations are also possible among different types of detectors, for example a search for supernova bursts can be performed simultaneously by bar and laser detectors [17].

We consider the general case of a network of detectors. Let be the signal vector and let n be the noise vector of the network of detectors, i.e., the vector component is the response of the gravitational-wave signal in the th detector with noise . Let us also assume that each has zero mean. Assuming that the noise in all detectors is additive the data vector can be written as

In addition if the noise is a stationary, Gaussian, and continuous random process the log likelihood function is given by
In Equation (87) the scalar product is defined by
where is the one-sided cross spectral density matrix of the noises of the detector network which is defined by (here E denotes the expectation value)
The analysis is greatly simplified if the cross spectrum matrix is diagonal. This means that the noises in various detectors are uncorrelated. This is the case when the detectors of the network are in widely separated locations like for example the two LIGO detectors. However, this assumption is not always satisfied. An important case is the LISA detector where the noises of the three independent responses are correlated. Nevertheless for the case of LISA one can find a set of three combinations for which the noises are uncorrelated [7880]. When the cross spectrum matrix is diagonal the optimum -statistic is just the sum of -statistics in each detector.

A derivation of the likelihood function for an arbitrary network of detectors can be found in [39], and applications of optimal filtering for the special cases of observations of coalescing binaries by networks of ground-based detectors are given in [483275], for the case of stellar mass binaries observed by LISA space-borne detector in [60], and for the case of spinning neutron stars observed by ground-based interferometers in [33]. The reduced Fisher matrix (Equation 43) for the case of a network of interferometers observing spinning neutron stars has been derived and studied in [79]

A least square fit solution for the estimation of the sky location of a source of gravitational waves by a network of detectors for the case of a broad band burst was obtained in [43].

There is also another important method for analyzing the data from a network of detectors – the search for coincidences of events among detectors. This analysis is particularly important when we search for supernova bursts the waveforms of which are not very well known. Such signals can be easily mimicked by non-Gaussian behavior of the detector noise. The idea is to filter the data optimally in each of the detector and obtain candidate events. Then one compares parameters of candidate events, like for example times of arrivals of the bursts, among the detectors in the network. This method is widely used in the search for supernovae by networks of bar detectors [14].