  Fluctuations of gravitational forces cause so-called Newtonian noise (NN) in
gravitational-wave (GW) detectors which is expected to limit their
low-frequency sensitivity in upcoming observing runs. Seismic NN is produced by
seismic waves passing near a detector's suspended test masses. It is predicted
to be the strongest contribution to NN. Modeling this contribution accurately
is a major challenge. Arrays of seismometers were deployed at the Virgo site to
characterize the seismic field near the four test masses. In this paper, we
present results of a spectral analysis of the array data from one of Virgo's
end buildings to identify dominant modes of the seismic field. Some of the
modes can be associated with known seismic sources. Analyzing the modes over a
range of frequencies, we provide a dispersion curve of Rayleigh waves. We find
that the Rayleigh speed in the NN frequency band 10 Hz - 20 Hz is very low
($\lesssim$100\,m/s), which has important consequences for Virgo's seismic NN.
Using the new speed estimate, we find that the recess formed under the
suspended test masses by a basement level at the end buildings leads to a 10
fold reduction of seismic NN.
