The detection of gravitational waves by resonant detectors must be pursued with more antennas working simultaneously. The data from all the antennas must be integrated in order to achieve a better filtering and to extract physical information on the wave.

Two different kinds of networks are considered: geographical networks and local arrays. The detection strategies are different in the two cases: for a geographical network, i.e. antennas scattered over the Earth surface, quite far from each other, non linear data analysis procedures must be used and only some parameters of the wave can be estimated. For local arrays linear techniques can be employed and the components of the metric tensor estimated. In this case it seems particularly promising the implementation of a local array of high frequency antennas for both practical operating reasons and for the scientific interest of the detection of gravitational waves in the upper kHz region.

The necessity of using more antennas in the detection of gravitational waves is well known. In fact only the coincident detection of more antennas can demonstrate that a signal is not due to local disturbances. For this goal often in the past parallel antennas were used.

There are other advantages in the use of an organized network of gravitational antennas:

- a single antenna can see only a limited set of sources

- the location of the source and the parameters of the detected pulse can be estimated

- the sensitivity can be enhanced by using some generalized coincidence
technique and by checking the conformity of the responses
of the antennas to the possible theoretical sources (*network filtering*
and * pattern rejection procedure*).

The development of a ``local" network (*array*)
constituted by a big number of small
antennas can be an advantageous substitute of a single big antenna. This
is true for
many reasons, the most important of which is the higher operative frequency.

In this work we present some ideas on the geographical networks and local arrays of gravitational antennas; in the case of the local arrays we propose a new type of antenna that has some advantages respect to the classical cylinders and the ``new" spherical detectors.

A geographical network is constituted by a certain number of antennas whose locations are distant from each other. The advantages of such an instrument are due to the optimum fulfillment of the assumption of independence of the noise in the various detectors and to the possibility to use the delay in the arrival time of the big gravitational pulses in the different antennas (which depends on the different distances from the source to the detectors). This phenomena can be used to locate univocally the source and this can be used as a proof of the extraterrestrial origin of the event. On the other hand, these delays exclude the possibility to use linear procedures in the network filtering for low SNR events (in which the delays cannot be precisely determined) and therefore cannot be used to enhance the sensitivity.

But even in the case the source location were known and one could combine the data samples corrected by the right time delay, linear procedures would be inadequate because only a limited control of the whole apparatus is possible: the antennas are run by different groups and a unitary management is very difficult to achieve. In practice, this results in a growth of the sources of uncertainty in the network data. Thus the information which can be extracted from it is reduced because it is necessary to define quantities which are less sensitive to the uncertainty factors. For example it is better-grounded to estimate the energy spectral density related to a candidate gravitational event than the dynamical h tensor at every instant of time. This kind of argument leads directly to antenna output processing which is not linear.

A ``detection algorithm'' has been developed for this type of data analysis. It allows to estimate the location of the source and some parameters which characterize the gravitational wave pulse, namely the percentage of linearly polarized energy, the polarization angle and the energy spectral density of the wave at the antenna frequency. The algorithm also performs a filtering procedure based on the matching between the different data outputs and the possible gravitational wave generated patterns. The performance depends on the signal to noise ratio, on the number of antennas and on their mutual orientation. It gives positive results even in low (about 3) signal to noise conditions.

One of the most interesting astrophysical events that can be detected by gravitational waves detectors is the stellar collapse to a black hole. During this process the quasi-normal modes of the black hole are excited and gravitational radiation is emitted. The energy spectral density of the radiation presents, in most cases, a maximum at frequencies a little smaller than 10kHz. What makes this phenomenon so appealing to gravitational astronomy is that it is the only one through which the existence of black holes can be established. In fact if gravitational radiation were detected at such high frequencies, there are no other likely candidate sources other than black holes.

In the region of a few kHz not only the laser beam antennas are not sensitive enough, but also the resonant antennas. In fact the cross section of a resonant antenna is proportional to its mass (and independent from the frequency) and so, in order to obtain high sensitivities, we must use big antennas, that means to operate at low frequencies.

A way to circumvent this problem is to set up a local array of small antennas, i.e. a set of some tens of identical detectors all located in the same place, with different orientations, to constitute a single instrument.

It is easy to see that if we take the average of the output of N parallel equal antennas, we obtain a SNR equal to that of an antenna with N times the cross section of a single one. For non parallel antennas the results in sensitivity are similar and informations on the direction of the source and the polarization of the wave can also be obtained.

But we have more by the array:

- local disturbance rejection. In fact it is very unlikely that the response of the antennas to a local disturbance can mimic the response to a gravitational pulse

- reliability, for two reasons. The first is due to the modularity of the array, so even if a certain number of antennas doesn't work, the performance of the array is not too much affected. The second is due to the size of the single antennas which makes the time to repair much lower than that of bigger antennas, because of the reduction of the thermal cycle time (that can be reduced from about one year in the case of a big resonant sphere to about one week)

- easier development and collaboration between groups; a gravitational antenna is composed of different parts (e.g. transducer, suspensions,...) developed in parallel by different groups, all of which can easily work on the real instrument were the piece will be mounted.

The classical resonant antenna is a cylinder more long than thick: e.g. in the case of the two big antennas of the Rome group (Explorer at CERN in Geneva and Nautilus at INFN in Frascati) the cylinders have the length of 3 m and the radius of 30 cm, the mass being 2300 kg. This geometry has some practical aspects, but has two drawbacks, namely given the operative frequency, the mass is low and there is a strong selection on the directions and polarizations of the sources.

Recently spherical or almost spherical resonant antennas have been proposed. A sphere of diameter d resonates at its five degenerate quadrupolar modes at about the same frequency of a cylinder of the same material, of length d, at its longitudinal mode. Such a sphere has a mass 16 times higher than an Explorer-like cylinder and is sensible to all the directions and polarizations. Anyway there is the drawback of using six transducers on it and the problem to deal with five degenerate (or not completely degenerate) modes.

A different geometry we propose is that of a stumpy cylinder (with length equal to the diameter), with three transducers: one on one end face to monitor the ``classical" longitudinal mode and the other two on the curved surface, at 45 degrees from each other to monitor the two-fold degenerate quadrupolar mode. Full omni-directionality is obtained with two stumpy cylinders whose axis are at 45 degrees from each other. Thus from each antenna of the array one would obtain three different signal lines. The cross-section of such a cylinder is about the same of that of the sphere with the diameter. A stumpy cylinder can be the ideal detector for an array.

A further advantage of the implementation of an array of small stumpy cylinders is that, due to the small dimensions of these, it is feasible to make them of materials which have a greater cross section to gravitational waves, such as molybdenum or chromium, with respect to aluminium which is the material commonly used in bar antennas. In this way about an order of magnitude on the cross-section can be gained.