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 detectors 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 Virgos 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 Virgos 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.
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