  We report correlations in underground seismic measurements with horizontal
separations of several hundreds of meters to a few kilometers in the frequency
range 0.01Hz to 40Hz. These seismic correlations could threaten science goals
of planned interferometric gravitational-wave detectors such as the Einstein
Telescope as well as atom interferometers such as MIGA and ELGAR. We use
seismic measurements from four different sites, i.e. the former Homestake mine
(USA) as well as two candidate sites for the Einstein Telescope, Sos Enattos
(IT) and Euregio Maas-Rhein (NL-BE-DE) and the site housing the MIGA detector,
LSBB (FR). At all sites, we observe significant coherence for at least 50% of
the time in the majority of the frequency region of interest. Based on the
observed correlations in the seismic fields, we predict levels of correlated
Newtonian noise from body waves. We project the effect of correlated Newtonian
noise from body waves on the capabilities of the triangular design of the
Einstein Telescope's to observe an isotropic gravitational-wave background
(GWB) and find that, even in case of the most quiet site, its sensitivity will
be affected up to $\sim$20Hz. The resolvable amplitude of a GWB signal with a
negatively sloped power-law behaviour would be reduced by several orders of
magnitude. However, the resolvability of a power-law signal with a slope of
e.g. $\alpha=0$ ($\alpha=2/3$) would be more moderately affected by a factor
$\sim$ 6-9 ($\sim$3-4) in case of a low noise environment. Furthermore, we
bolster confidence in our results by showing that transient noise features have
a limited impact on the presented results.
