  Terrestrial gravity fluctuations produce so-called Newtonian noise (NN) which
is expected to limit the low frequency sensitivity of existing
gravitational-waves (GW) detectors LIGO and Virgo, when they will reach their
full potential, and of next-generation detectors like the Einstein Telescope.
In this paper, we present a detailed characterization of the seismic field at
Virgo's West End Building as part of the development of a Newtonian noise
cancellation system. The cancellation system will use optimally filtered data
from a seismometer array to produce an estimate of the Newtonian-noise
generated by the seismic field, and to subtract this estimate from the
gravitational-wave channel of the detector. By using an array of 38 seismic
sensors, we show that, despite the influence of the complexity of Virgo's
infrastructure on the correlation across the array, Wiener filtering can still
be very efficient in reconstructing the seismic field around the test-mass
location. Taking into account the division of the building's foundations into
separate concrete slabs, and the different properties of the seismic field
across them, we conclude that the arrays to be used for the Newtonian-noise
cancellation at Virgo will require a relatively large number of seismometers
per test mass, i.e. significantly more than 10. Moreover, observed variations
of the absolute noise residuals over time, related to the daily evolution of
anthropogenic noise, suggest that the Wiener filter will need to be updated
regularly, probably more often than every hour, to achieve stationarity of the
background level after subtraction.
