Observations of Random Walk of the Ground in Space and Time

We present results of micron-resolution measurements of the ground motions in large particle accelerators over the range of spatial scales $L$ from several meters to tens of kilometers and time intervals $T$ from minutes to several years and show that in addition to systematic changes due to tides or slow drifts, there is a stochastic component which has a “random-walk” character both in time and in space. The measured mean square of the relative displacement of ground elements scales as $d{Y}^2\approx ATL$ over a broad range of the intervals, and the site dependent constant $A$ is of the order of ${10}^{-5\pm 1}(\mu {\mathrm{m}}^2/\mathrm{s})/\mathrm{m}$.

Figure 1

Vertical displacement of more than 200 “tie rods” in the Tevatron tunnel over the period 2003–2005 and a 6 yr period of 2001–2007.

Figure 2

Variances of the Tevatron “tie-rod” vertical displacements over time intervals of 1 yr (multiplied by 6) and 6 yr vs the distance $L$.

Figure 3

Variances of the Tevatron alignment “tie-rods” displacements per unit distance vs the time interval between the measurements.