  Newtonian noise from seismic fields is predicted to become a sensitivity
limiting noise contribution of the gravitational-wave detectors Advanced LIGO
and Virgo in the next few years. It also plays a major role in the planning of
next-generation detectors, which might be constructed underground, as planned
for the Einstein Telescope, mostly to suppress Newtonian noise. Coherent noise
cancellation using Wiener filters provides a way to mitigate Newtonian noise.
So far, only the cancellation of Newtonian noise produced by seismic surface
waves has been studied in detail due to its relevance for Advanced LIGO and
Virgo. However, seismic body waves can still contribute significantly to
Newtonian noise in surface detectors, and they might be the dominant source of
gravity fluctuations in underground detectors. In this paper, we present the
first detailed analysis of coherent cancellation of Newtonian noise from body
waves. While the required number of seismometers to achieve a certain level of
noise suppression is higher than for seismic surface waves, we show that
optimal seismometer arrays can greatly reduce body-wave Newtonian noise. The
optimal array configurations and achieved residuals depend strongly on the
composition of the seismic field in terms of average compressional-wave and
shear-wave content. We propose Newtonian-noise cancellation to achieve the
ambitious low-frequency target of the Einstein Telescope.
