Review of observations of ground diffusion in space and in time and fractal model of ground motion

We present numerous observations of the diffusive motion of the ground and tunnels for scientific instruments and show that if systematic movements are excluded the remaining uncorrelated component of the motion obeys a characteristic fractal law with the displacement variance $d{Y}^2$ scaling with time and spatial intervals $T$ and $L$ as $d{Y}^2\propto T^{\alpha }{L}^{\gamma }$ with both exponents close to 1 ($\alpha \approx \gamma \approx 1$). We briefly describe experimental methods of the mesoscopic and microscopic ground motion detection used in measurements at physics research facilities sensitive to ground motion, particularly large high energy elementary particle accelerators. A simple mathematical model of the fractal motion demonstrating the observed scaling law is also presented and discussed. This paper is a subsequent full detail publication to [V. Shiltsev, Phys. Rev. Lett. 104, 238501 (2010)].

Figure 1

Mean square difference of vertical orbit distortions in the HERA electron ring vs time interval duration obtained from data stored during 1993 operation data [18].

Figure 2

PSD of the HERA-proton orbit vertical motion normalized to a specific location of the ring. The dashed line corresponds to the ATL law [17, 18].

Figure 3

Changes of rms vertical and horizontal orbits in TRISTAN ring (from 19).

Figure 4

Variance of the TRISTAN orbit variations [17].

Figure 5

KEK-B circumference variations from March 1 to June 30, 2002 [20] (data courtesy of Katsunobu Oide of KEK).

Figure 6

Variance of the KEK-B circumference variations; the black line is for raw data, the red line is for the data with linear trend subtracted, and the dashed line is a linear fit.

Figure 7

Spectrum of the KEK-B circumference variations [21]. The dashed line is for the ATL-law scaling $8.2/f^2$; the solid red line is for a power-law fit $(1.74\pm 0.2)/f^{2.21\pm 0.07}$.

Figure 8

Horizontal and vertical orbit motion as measured by one of the beam position monitors in the Tevatron [21].

Figure 9

Tevatron proton orbit 2nd difference variance. Dashed lines are linear fits of the ATL-like component of the variance.

Figure 10

Variance of the LEP vertical orbit distortions vs time interval $T$ with effects of movements of the strongest focusing magnets removed (from Refs. 22, 24).

Figure 11

The rms vertical and horizontal LEP beam orbit drifts during 1999 operation. The $\sigma \propto T^{1/2}$ growth with time interval $T$ is visible (from Ref. 23).

Figure 12

Power spectra of orbit movement at 26 and 270 GeV in the SPS (from Ref. 23).

Figure 13

Elevations of the CERN’s LEP focusing magnets measured in 1993–1994 [26] vs cumulative distance along the ring (i.e., the point at 0 m placed close to the point at 26.7 km, data courtesy of Jean-Pierre Quesnel of the CERN’s Survey Group).

Figure 14

The variance of relative displacement of the CERN LEP magnets vs the distance between them $L$ (from Ref. 17).

Figure 15

Displacements of the CERN’s Super Proton Synchrotron magnets measured in 1985, 1988, and 1991 along the circumference ring; the point at 0 m placed close to the point at 6912 m (data courtesy of Jean-Pierre Quesnel of the CERN’s Survey Group).

Figure 16

The variance of the relative vertical displacement of the SPS magnets after various time intervals vs distance between the points of the position survey $L$: (a) three years (1985–1988), (b) three years (1988–1991), (c) six years (1985–1991), (d) 12 years (1976–1988) (from Ref. 28).

Figure 17

Vertical displacement of more than 200 “tie rods” in the Tevatron tunnel over the period of 2003–2005 and a 6 yr period of 2001–2007 (data courtesy of James Volk and Fermilab’s Alignment Group).

Figure 18

Variances of the averaged Tevatron tie rod vertical displacements over time intervals of one (multiplied by 6) and six years vs the distance $L$ (from Ref. 29, 30).

Figure 19

Variances of the Tevatron alignment rod displacements per unit distance vs the time interval between the measurements (see text, from Ref. 29, 30).

Figure 20

Variances of the accelerator magnet displacements per unit time vs distance for the SLC, UNK, PEP, and SLAC tunnel (see text, from Ref. 28).

Figure 21

(top) PSD of the earth strain at Pino Flat Observatory in southern California; (bottom) the solid line is rms wander of the earth computed from the full spectrum, and that computed if the “7-second” microseism peak is filtered out, from 32.

Figure 22

Schematic of SLAC linear accelerator laser measurement system.

Figure 23

Variance of the vertical laser spot movement in the SLAC laser system (from Ref. 34).

Figure 24

Diffusion coefficient $A$ as measured from the spectra laser spot vertical and horizontal movements in the frequency band 0.000 24 to 0.015 Hz (from Ref. 35).

Figure 25

Horizontal movement of the PS central pillar in 1965–1968 (from Ref. 36).

Figure 26

Calculated diffusion constant $A$ as a function of the time interval $\Delta T$ in the ATL rule. The three different curves refer to the horizontal (solid) and vertical (dotted) data of Sec. 1 and the horizontal data of Sec. 2 (dashed line). The $A$ constant was determined over a distance of about twice 15 m (from Ref. 38).

Figure 27

Secular tilting motion measured at the Esashi station in 1979–1994 (from Ref 39, original data records courtesy of Professor Shigeryu Takeda of KEK, Japan).

Figure 28

Variance of the tilt elevation vs time interval (from Ref. 17).

Figure 29

A spectrum of ground motion in Sazare mine (Japan). The straight line indicates $1/f^2$ (from Ref. 40).

Figure 30

The PSD of the 6-years-long record in Walferdange (with and without the instrumental response correction—black and gray curve, respectively) and the low tilt noise reference model from Ref. [9] (red dots). Shaded rectangles pinpoint the frequencies ranges of the Earth tides and the Earth free modes (from Ref. 44).

Figure 31

SAS-2 HLS sensor used in the ground motion studies in Illinois (from Ref. 48).

Figure 32

Schematics of the systems of HLS sensor used in the studies in Illinois.

Figure 33

91 days data records starting November 12, 1999 from the PW studies: (top to bottom) the level difference between probes #2 and #6 120 (vertical scale of about $150\mu \mathrm{m}$), mean temperature in the tunnel (vertical scale of $3.5°\mathrm{C}$), the second level difference ${\mathrm{SD}}_{2446}$ (see in the text, scale $240\mu \mathrm{m}$), and variance of the second level difference ${\mathrm{SD}}_{2446}$ for intervals of up to 91 days (from Ref. 45).

Figure 34

Diffusion coefficient $A$ calculated for all possible combination of the probes distanced by $L$ from 15 to 135 m from 1 month data records in MI8 tunnel [46].

Figure 35

HLS probe on Tevatron accelerator focusing magnet.

Figure 36

One week record of elevation difference of two neighbor focusing magnets in the Tevatron tunnel as measured by HLS (starts midnight February 7, 2004; Ref. 30).

Figure 37

Change of the elevations of 20 Tevatron magnets after 23 days of observations (January 7–February 1, 2004; from Ref. 30).

Figure 38

Dependence of the growth rate of the variance of the 2nd difference vs distance between the HLS probes in the Tevatron tunnel, the week of February 7, 2004 (from Ref. 30).

Figure 39

January 2006 record of elevation difference for two HLS probes 90 m apart in the FNAL MINOS hall [30].

Figure 40

FFT of the elevation difference for HLS probes 90 m apart as measured in the Fermilab’s MINOS hall [30].

Figure 41

Slow ground motion in 120 m deep dolomite mine (Aurora, IL) in December 2000. Top to bottom (a) to (e), see comments in the text [47].

Figure 42

Variance of the vertical relative ground motion for the points 30 and 90 m apart, measured on October 13–15, 2000 in the Aurora mine in Illinois [46].

Figure 43

Coefficients of the ground diffusion from Table  vs the depth of the observations. The dashed line represents the average value for operational accelerators.

Figure 44

Fractal set of ground blocks (see text).

Figure 45

Elevations of the set points after 16 000 steps with parameters of the model $\lambda =\mu =1$. Coordinate $z$ is given in the units of the minimal block size.

Figure 46

Variance of the displacements vs distance between points for 128, 1024, 4096, and 32 000 steps.

Figure 47

Variance of the displacements vs distance between points after 16 000 steps for different scaling exponents $\lambda$ and $\mu$ (see text).