Space-Time Diffusion of Ground and Its Fractal Nature

We present evidences of the diffusive motion of the ground and tunnels and show that if systematic movements are excluded then the remaining uncorrelated component of the motion obeys a characteristic fractal law with the displacement variance dY^2 scaling with time- and spatial intervals T and L as dY^2 \propto T^(Alpha)L^(Gamma) with both exponents close to 1. We briefly describe experimental methods of the mesa- and microscopic ground motion detection used in the measurements at the physics research facilities sensitive to the 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.


Introduction
Motion of the ground was always of practical interest because of the scare of earthquake-induced damage and concerns about structural stability of buildings due large movements. In recent decades, development of large -scale facilities for scientific research also confronted the issue of very tight tolerances on element"s positions in the presence of microscopic motion of the ground. The most notable examples are gravitational wave detectors [1][2][3] and high energy particle accelerators [4][5][6][7]. In the gravitational wave detectors, the ground vibrations transferred to the motion of the mirrors in the arms of interferometers are one of the sources of noise limiting minimum detectable strain. In the accelerators, motion of numerous focusing magnets disturbs trajectories of tiny charged particle beams and, thus, affecting machine performance. Given the tight tolerances on positioning, quite sophisticated measurement, stabilization and correction/alignment systems are routinely employed there [8]. To design such systems one relies on certain phenomenological models of the ground motion which should predict the expected displacement of the ground Y(t,s) depends on the time interval t and distance between the points of control s. The spatial scales of interest L for these physics instruments range from several meters to dozens of km and the time intervals of interest T range from ms to years.
The instruments for the microscopic ground motion measurements have been originally developed for geophysics research, currently many of them are made easily applicable for other purposes and commercialized. Among widely used at the large physics facilities are optical interferometers, stretched wires and hydrostatic level systems (HLS) [9], laser position trackers [10], and geophones [11]. They are quite capable to detect the movements over the above noted scales of L and T even under very quiet conditions. Ambient ground motion has three distinct componentsperiodic motion (e.g. due to Earth tides, seasonal changes, etc), systematic drifts or trends (e.g due to temperature or air pressure variations, precipitation history, etc) and stochastic movements [12]. The stochastic component usually is less correlated in space, less persistent in time and less predictable than the first two while not necessarily smaller in amplitude, thus, often posing the biggest concern. Space-, time-or space-time variograms can be used to describe average characteristics of the motion Y(t,s) : (1) where the brackets <…> denote averaging over continuous or discrete time series and T and L are lags in time and space. Below we present and discuss evidences that the stochastic component of the ground motion can be described as diffusion in both time and space and has a characteristic fractal law variogram : with both exponents close to 1 (1) over wide ranges of time-and spaceintervals. Corresponding power spectral density (PSD) P(,k) in frequency =2f and spatial wave-number k=2/ for such a process scales as: with exponents =+1 and =+1 (detail discussion on mathematical methods of geophysical time series analysis can be found in [12]).
Power-law scaling of separate temporal or spatial variograms of the ground motion, i.e., dependencies of the type <dY 2 (T, L=const)>T  and <dY 2 (t=const, L)> L  have been long known to geophysicists, see, e.g. [13], [14], but it was high precision studies of dynamics numerous measurement points for large accelerators where simultaneous space-and-time diffusion was observed for the first time. An empirical ATL law [15] was proposed to summarize the experimental data, according to which the rms relative displacement dY of the points separated by a distance L grows with the time T as: where A is a site dependent constant of the order of 10 -5±1 m 2 /(sm). Such a wandering of the ground elements takes place in all directions. As long as the diffusive coefficient A is small the diffusion presents only a tiny contribution to the ground motion. For example, in the time period of 1 hour the amplitude of the absolute surface motion (e.g. measured by seismometer) could be as big as 100 m, while the ATL estimates relative displacement of about l m for the points 30 m apart. One would not worry about this contribution except it describes very important, at least for accelerators, uncorrelated background on top of the larger amplitude ground movements correlated in time and space. The later includes, but not limited to, low frequency seismic waves, tides, an ambient low-frequency ground motion generated by local sources such as wind, air pressure variation, temperature gradients, ground water, precipitation, etc. Obviously, the ATL law is a particular case of the more general equation (1). The PSDs of the ATL-type motion in the frequency and the wave-number domains scale as: This article reviews the evidences of the space-time diffusion of the ground surface or tunnel. In Section 2, we discuss the measurements made at the particle accelerators with use of standard alignment instrumentation, describe briefly the impact of misalignments on the beams in accelerators and present evidences of the beam orbit diffusion caused by diffusion of elements " positions. Section 3 contains review results of various geophysical studies made either at the accelerator facilities, or at the sites of future accelerators, or at the geophysics labs. We summarize all the measurements and discuss the limits of validity of the space-time ground diffusion laws in Section 4 and present a simple numerical model of the fractal ground motion which generated the landscape evolution according to the empirical law.

Impact of Ground Motion on Operation of Accelerators
For the purposes of this study, particle accelerators can be considered as sequence of linear focusing elements (magnetic lenses) arranged either in a circle (circular accelerators) or in a line (linear accelerators). In an ideal accelerator with perfectly aligned magnetic elements, the beam orbit passes through the centers of the lenses magnets. Any alignment error results in the beam orbit distortion. If the distortions are large compared to apertures of the lenses or the size of the vacuum chambers or the size of a linear focusing field areas, then they become an obstacle for successful operation of the machine and must be correctedeither with use of electromagnetic orbit correctors or by means of mechanical realignment which brings the centers of the focusing lenses back to their ideal positions [16]. In large accelerators, such as 6.3-km circumference proton-antiproton Tevatron Collider (Fermilab, Batavia, IL, USA), 27-km circumference proton-proton Large Hadron Collider (LHC at CERN, Switzerland), 6.3 km circumference proton-electron collider HERA at DESY (Hamburg, Germany), and 25-50 km long future electron-positron Linear Colliders, which have many hundreds of magnetic elements, the motion of the ground and corresponding displacements of the magnets are the most important source of the beam orbit distortions. It has to be noted that the biggest effect is produced by uncorrelated relative motion of the neighboring focusing elements while very longwavelength movements are practically unimportant, and, for example, accelerators are not sensitive to their global displacements as a whole [6], [7]. Orbit distortions from numerous uncorrelated sources add in quadrature and, thus, the rms distortion of the beam orbit due to the ATL-law type ground motion (4) in a circular accelerator with circumference C can be approximated as [17]: that shows that larger orbit drifts are expected at larger accelerators. The numerical factor in (6) 2-5 depends on the design of the beam focusing optics. Typically, the ground motion effects start to be of a serious concern for accelerators at the amplitudes of the uncorrelated motion from a fraction of a micron to a dozen of microns, depending on the accelerator parameters and types. For accelerators which collide tiny size beams the final focusing magnet stability tolerances could be as tight as microns to few nanometers [7]. Because of the concerns with the magnet position stability, large accelerators are usually been installed inside deep concrete-and-steel enforced tunnels (typical diameters/sizes of the order of 5-8 m at the depth from 10 to 100 meters) at the location with known good and stable geology.

Orbit Drifts in Large Accelerators
To a greater or lesser extent long-term orbit drifts are seen at all accelerators and machine operators or/and automatic correction systems counteract the drifts. As large colliding beam facilities are particularly sensitive to the orbit motion, some extended investigations of the issue have been carried out there.
In this section we present observations of the beam orbit drifts in several large accelerators -HERA (Germany), TRISTAN (Japan), Tevatron (US) and LEP (Switzerland). Detailed parameters of these machines can be found in corresponding references below.

Orbit Drifts in HERA Proton-Electron Collider
HERA is a high energy accelerator in Hamburg (Germany), which was in operation as proton-electron collider in 1992-2007. The circumference of HERA is 6.3 km. The facility is located in an underground tunnel in a depth of approximately 25 meters below the surface. It consisted of two independent accelerators-storage rings for 30 GeV electrons and 820 GeV (since 1998 -920 GeV) protons installed in the same tunnel (the height difference between electron and proton beam is 0.8 m, focusing optics lattice are very different).    [18,17].
Analysis of the vertical motion in the other (proton) ring is summarized in the Power Spectral Density (PSD) shown in Fig.2. The squares at lower frequencies represent the Fourier spectra of the proton orbit differences from different running periods of the accelerator [18]. The procedure was to measure the closed orbit position at all 131 BPMs in the HERA-proton machine and subtract the result from a previous one to obtain the difference orbit, indicating any eventual orbit drift. The analysis of difference orbits was limited to time intervals of about 5 days maximum during which no intentional change of the closed orbit occurred. Continuous line represents the Fourier spectrum of readings from one specific beam position monitor in the accelerator [17]. As continuous observations were performed repetitively within several hours of the proton beam lifetime, the lowest frequency of this spectrum is about 0.5 mHz. Series of peaks in the spectrum above 1 Hz are due to cultural seismic noise which is quite prominent in a big city like Hamburg. The dashed line in Fig.2 shows the PSD scaling P orbit (f)>=810 -4 [m 2 s]/f 2 as expected from the ATL law with the constant A HERAp 810 -6 m 2 /s/m which fits very well the data in the range of frequencies from 210 -6 Hz to about 210 -2 Hz. In time domain such a PSD corresponds to irregular noisy "random walk"-like proton orbit drifts over the time intervals few some minutes to several days. The PSD power-law fit results in the exponent of =1.95±0.2. Mechanical motion of the focusing magnets was found to be the reason of the HERA orbit drifts, as other sources -long term drifts of orbit corrector strengths and low-frequency noises of the BPMs-were negligible.

Orbit Drifts in TRISTAN and KEK-B Positron-Electron Colliders
TRISTAN is a high energy accelerator in Tsukuba (Japan), which was in  Note that the horizontal COD is smaller than the vertical one. At large orbit distortions, the beam current circulating in the accelerator degraded significantly so that a correction of the orbit was needed toward the "ideal" orbit (sharp drops at points D, E, H, and some others in Figure 3).   PSD of the circumference change is presented in Fig. 7 and shows distinctive peaks at frequencies of ~2/day (some 15 m changes due to ground expansion due to solar and lunar tides) and some 30 m peak due to daily temperature changes. The circumference also found changing due to air pressure variation, especially during the time when a typhoon hit the area (not in Fig.3). At very low frequencies less   can compute the variance of the second differences <ddY 2 (T)> which is equal to :

Orbit Drifts in Tevatron Proton-Antiproton Collider
It is easy to see that contrary to variance of the (first) difference (1), effectively filters linear trends and slow periodic variations out. Indeed, for the process which contains a linear trend, a periodic component, a diffusive ATL-like component and truly uncorrelated noise (e.g. due to measurement errors) dY(t)>=Et+Fsin(t)+(ATL-like diffision)+(noise with rms of G) one gets : The result of such analysis for the Tevatron orbit drift data is shown in Fig.10

Orbit Drifts in CERN's Large Electron-Positron Collider (LEP) and Super-Proton Synchrotron (SPS)
Large As for other accelerators we considered above, stability of the beam orbit was essential for successful operation of the collider. Motion of few very strong superconducting focusing magnets correlated with temperature variations at the magnet support structure was found to be main source of ~3 mm vertical beam orbit movements [22]. Employment of local orbit correctors allowed to reduce this effect by an order of magnitude. The residual orbit motion was found variance growing  of movements of the strongest focusing magnets removed (from Refs. [22,24]). Similar analysis has been extended for 30,000 orbits were recorded while LEP was colliding beams for its experiments in 1999 [23]. The orbit data was analyzed to reconstruct the orbit drifts that were compensated by the LEP slow orbit feedback and to remove the effects due to the earth tides, motion of few very strong superconducting focusing magnets mentioned above and other known intentional corrections implemented to optimize the accelerator operation.   We believe that one of such effects which was not properly accounted in Ref. [23] is regular periodic orbit distortions due to the Earth tides. The above considered Tevatron orbit variations - Fig.9 set an example which shows the tides, if not properly excluded from the data, can increase formally calculated diffusion coefficient by a factor of 2 to 10. It was reported in [25] that the tidal deformations of the Earth"s crust do cause a 1 mm variation in the circumference of LEP.
Variations of the orbit distortions over the time intervals of about 3 hours (considered in the Fig.11 data) can be as big as 10-30% of that, thus, possibly dominating the rms orbit analysis. In addition to the periodic tidal variations, slow systematic seasonal changes of the LEP circumference of 2 mm have been observed.
These movements might also affect the orbit analysis. They are particularly pronounced after important rainfall and might be produced by an expansion of the earth or by a pressure due to underground water levels (sponge effect) [25].  Figure 12 from Ref. [23] shows power spectra of the vertical beam motion of a 270GeV and 26GeV beams that was sampled by a monitor with about 2 μm r.m.s resolution (seen as white noise above 0.1Hz). The 26GeV data are thought to be dominated by slow drifts of the magnetic fields rather than by ground motion. The 270 GeV data shows characteristic ATL-law spectrum scaling of 1/f 2 . Using a pre-calculated vertical orbit sensitivity factor κ for the SPS and fitting the observed orbit drifts spectra, the following SPS ground motion coefficient estimate can be obtained A SPS =(6.3±3)10 -6 m 2 /s/m.

Ground Diffusion in the Accelerators Alignment Data
Despite having sophisticated orbit correction systems, all accele rators undergo regular realignment of the magnets positions back their ideal values.
That allows to reduce greatly the dependence on the correction systems and helps to maintain stable operation of the facilities over periods of many years.
Modern commercial instruments, e.g. laser trackers, for geodetic survey and alignment allow to achieve accuracies of a fraction of a mm over distances of a km and their description can be found elsewhere (see, e.g., Ref. [8]). In this section we present analysis of long term ground motion drifts as observed during the realignment of large accelerators.

Long-Term Motion of LEP Magnets
Several times a year, positions of more than 700 focusing magnets of the LEP were measured and restored back to their prescribed values to follow an ideal smooth curve" . Results of the LEP magnets elevations measurements in 1993-1994 [26] are shown in Fig.13. 2) as one is not interested in the smooth spatial curves, the lowest five Fourier harmonics were subtracted from the data. Now, the variances of the first difference <dY 2 (L)>=< (dY(l)-dY(l+L)) 2 > have been calculated as where brackets < . . . > denote averaging over all possible pairs of the magnets distanced by L. The results are presented in Fig.12 where the straight lines represent liner fits : One can see that for L < 1000m, the variances for just-realigned accelerator <dY 2 (L)> I and <dY 2 (L)> III are 1.5-2 times less than what is measured after several months without alignment. It has to be noticed that the variance grows linearly with L even right after the alignment. That is because of the method of survey when the alignment is made sequentiallyone segment of the machine after anotherand the random errors of the position measurement of a given magnet with respect to the previous one add up like a random walk.
Such a random walk error can be estimated by the closure errors of about 2 mm over the entire circumference (measured at different periods) that is equivalent to 0.14mm 2 /kmin a good agreement with the analysis shown in which are remarkably close to each other. Therefore, the LEP alignment data demonstrate that the variance of the relative displacements in time scales proportionally to the distance between the points. Six-year elevation changes of the LEP magnets in 1993-1999 have been analyzed in Ref. [27]. It was shown that after exclusion of the linear trends and systematic drifts from the data, the remaining random diffusion can be described by the ATL law with coefficient A LEP =(2.9±0.6)10 -6 m 2 /s/m.

Motion of CERN's Super Proton Synchrotron Magnets
The noted above CERN"s Super Proton Synchrotron (SPS) was constructed in mid-1970s and has 6.9 km circumference. There are 744 bending magnets and N=216 focusing magnets placed practically uniformly over the ring.  These data were processed the way similar to the one as for the LEP alignment data discussed above, so, for example, the values for several magnets around 600 m and few were not taken into considerations as these magnets were intentionally displaced during the period.  1985-1988, 1988-1991, 1985-1991 and 1976-1988, correspondingly. It has to be emphasized that the time intervals vary from 3 years to 12 years, and nevertheless the diffusive constants are almost the same. An average value of the coefficient for the SPS data is thus A SPS =(14±5)10 -6 m 2 /s/m. Note, that a power-law fit <dY 2 (L) >L  with exponent  less than 1 might better describe the variances than the linear fit.

Tevatron Alignment Data Analysis
Alignment system of the Tevatron Collider employs more than 200 geodetic "tie rods" installed in the concrete tunnel wall all over the ring , approximately 30 m apart.

Alignment Data on Ground Motion in Other Accelerators
The   [30]. It has to be noted also, that for all the data presented in Fig.17 the exponent  of a power-law fit <dY 2 (L) >L  varies between 0.7 and 1.0.

Geophysics Measurements Data on Ground Diffusion
Evidences of the ground diffusion either in space or in time or simultaneously in space and time have been reported in geophysics studies of various types. Below we present many of these results, classifying them by the method of the measurements: made with optical and laser interferometers, stretched wire and several types of HLSs.

Strain Measurements in PFO
Horizontal motion of massive near surface monuments emplaced in competent, weathered granite has been made by laser interferometers ("optical anchors") at Pinon Flat Observatory (PFO) in southern California [9]. The data on the optical path difference dL over the distance L=732 m have been normalized in the units of strain ε=(dL/L) and its power spectral density is shown in Fig.21 from [31]. The peaks in the spectrum around multiples of 1 cycle/day are caused by earth tides and temperature effects; the peak at high frequencies of ~0.1 Hz is caused by microseisms ("7-second hum"). Except for these peaks, the spectrum is very well fit by the power law 1/f 2 .
The diffusion is very small compared to any examples we considered abovethat is no surprise given that the PFO has been located in a very stable area with hard granite bed-rock suitable for very precise geophysics observations.

Laser Beam Measurements in the SLAC Tunnel
Several measurements of slow ground motion were performed using laser alignment system [32] installed in the SLAC 2-mile linear accelerator tunnel. This system consists of a light source, a detector, and about 300 targets, one of which is located at each point to be aligned over a total length of 3050 m. The target is a rectangular Fresnel lens which has pneumatic actuators that allow each lens to be flipped in or out. The targets are installed in a 2-foot diameter aluminum pipe which is the basic support girder for the SLAC linear accelerator. The schematic of the measurements with just one of the lenses exactly the middle of the system is shown in Fig.22. In such configuration, the laser spot position in the detector is equal to x 1 +x 3 -2x 2 (for either vertical or horizontal planesee  In the other series of measurements, reported in Ref. [34], it was found that the amplitudes diffusive motion in vertical and horizontal planes are about the same, see Fig. 24, and the excess in the vertical plane is often correlated with the atmospheric pressure variations.

Motion of the CERN PS Pillar
Yet another manifestation of the ground diffusion is the movement of central CERN Proton Synchrotron (PS) pillar over period of more than 2 years shown in Fig.25 from Ref. [35].

HLS Measurements in Japan
Below we review several slow ground motion measurements made with HLS sensors made in various locations in Japan: in geophysics laboratories, in accelerator facilities and in several tunnels. More detail descriptions of the conditions and instruments can be found in the cited References.  [17]. The results are presented in    The diffusion studies in several more tunnels in Japan confirmed that the ATLlaw scaling Eq.(5) offers a very good fit to most of the data, and concluded that the diffusion parameter A is influenced dominantly by the earth and rock properties [41,42]. The observed parameter A is smaller in the tunnel in a solid rock than in the broken rock. The excavation method of the tunnel also affects significantly the diffusion: e.g. , a tunnel made by dynamite blasting had A=510 -6 m 2 /s/m while a tunnel in a similar rock bored by a tunnel-boring-machine had A=110 -6 m 2 /s/m.

(T)>=<((t)-(t+T)) 2 > calculated in
Such a difference was attributed to artificial fragmentation of the rock occurred during the construction. Values of the diffusion coefficients measured in various Japanese tunnels will be presented in Table 1 below.

HLS Measurements in Luxembourg
Yet another example of the power-law ground drifts is measurements with a 43 m long floatless water-tube tiltmeter which has been in operation since 1997 at the Walferdange Underground Laboratory for Geodynamics in the Grand Duchy of Luxembourg [43]. The instrument "s very low noise level and its high resolution up to the long-period seismic band (where for instance the resolution is better than 5×10 -12 rad) allow the successful recording of miniscular drifts as well as rarely observed grave toroidal and spheroidal free oscillations of the Earth excited by major earthquakes. In the environmental conditions of its installation (in a gypsum mine at 100 m depth), the instrument shows a high degree of reliability and a very low drift rate (<0.005 microrad/month). The observed spectrum of the tilt is shown in Fig. 30 and has distinct power-law scaling at frequencies below 0.0001 Hz PSD1/f 2.2 (red dots); effective ATL diffusion constant at the lowest frequency of f=210 -7 Hz can be found from Eq.(5) to be about A=0.110 -6 m 2 /s/m. Fig. 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. [43]).

Measurements with Multi-probe HLS systems in Illinois
The examples of the HLS measurements considered above provide confirmation of the ground diffusion in time as in all of them only two HLSs were used. To study the diffusion in space or spatial correlations of the ground motion, a series of extensive studies with systems of connected HLS probes has been performed in various locations in Illinois. High precision HLS probes developed for these studies (see  A pair of the probes set side-by-side shows the differential noise level of σ 2 =(0.09m) 2 + 1.25210 -7 m 2 /s T (more details can be found in Ref. [44]). In a typical measurement arrangement, six to 20 of such probes installed in the same water level system spaced 15 to 30 meters apart usually along the line as shown in Fig.32. Once a minute, a PC based data acquisition system collects not only the water level data (averaged over the minute), but also all probe"s temperature readings for correction, readings from one or two air pressure sensor for monitoring.

Studies in the Proton West (PW) tunnel on site of the Fermi National
Accelerator laboratory had been carried out in 1999-2000 [45]. This is an unused beam line for fixed target experiments with a shallow (5 m depth) tunnel built by "cut-and-cover" method in 1970"s. It has flat concrete floor that made quite easy the installation of 6 HLSs over total length of 180 m (30+30+60+30+30 meter apart).  [45]).
An important drawback of the tunnel was that it was not sealed and there were large temperature variations from one end to the other sometimes by few o C a day causing large of changes in the water level readingssee in Fig.33. The ground tilts due to earth tides occured twice times a day with some 20 μm peak-to-peak amplitude in the level difference Y 2 -Y 6 between two probes #2 and #6 150 m apart but practically absent in the second difference SD 2446 =Y 2 -2Y 4 + Y 6 Fig.37 shows a snapshot of the magnet elevation changes after 23 days of observations. One can see that the differential movements over the ~600 m section of tunnel could be as big as 30-50 m.    [29].
To remove the systematic effects due to the tides, the FFT of the 1 month long record of the level difference Y 0 -Y 3 data has been calculated (see Fig.40).
The power law fit 1/f indicated by the red line in Fig.40 corresponds to the ATL diffusion coefficient of A MINOS =0.1810 -6 m 2 /s/m [29].  [13][14][15]2000 in Aurora mine, IL [46]. There were no blasts over weekends as well as sometimes the temperature does not change much as well, so one can use such records for analyzing "natural" ground diffusion at shorter time scales. For example, on a quiet weekend of Oct. [13][14][15]2000, the temperature variation was less than 0.05 o C. The 2 days record analysis is presented in Fig.42 which shows the variance of the second differences

Discussion of the results
Several conclusions can be made from the results presented above. First of all, the diffusive motion of the ground is often just a background to much more powerful processes, like ground expansion due to temperature changes, or bending due to atmospheric pressure variation or winds, long-term settlement drifts or Earth tides.
Special data processing is often needed to separate diffusive noise from systematic or periodic signals: in time-or space-domains, that can be achieved with use of digital filters, like the first or the second difference methods employed above; in the frequency-or wavelength-domains, Fourier analysis of windowed data sets (e.g. with Hanning window) makes visible the power-law component of the spectrum. Table I   where geophysical observatories are set at). Japanese data indicate that even the tunneling method may affect the diffusion rate.
One can also see that the ATL approximation is not always the best, and in general, the exponents in the fit <dY 2 (T,L)>T  L  can significantly differ from 1. It has to be noted that some of our observations show that at small time intervals T and large spatial separations L m , the motion of two points naturally independent and, therefore, the exponent  tends to be close to 0.
With the limited number of data sets, we can not explore in detail the boundary L m (T) beyond which the independence (or significant loss of correlation) occurs while it is a very important phenomena [ 48] which definitely calls for more studies.
The observations reviewed above cover time intervals from hours to several years and spatial scales from dozen meters to a dozen of kilometers (the largest accelerators). There are some evidences of the diffusion at much larger T or L intervals. For example, 50 years observation  of sea levels in 12 Japanese ports distanced by as much as 800 km [49], showed that besides daily and seasonal changes, the level variation has a long-term "random walk" component <dY 2

(T)> T with computed diffusion coefficient
A of about 3510 -6 m 2 /s/m [17]. It is long known to geophysicists, that Earth"s topography is fractal, and its power spectral density scales with the wave number as S(k) k -2 that corresponds to <dY 2 (L)>L over distances 100 km to 6000 km (see, e.g. Fig.17.19 in Ref. [14] and corresponding discussion). What this paper adds to previously known results is the notion that the diffusion takes place both in time and in space (at least, over the scales indicated in the Table   I and characteristic for high energy physics accelerators).

Modeling Diffusive Ground Motion
The fractal objects and time series are one of the favorite subjects for modern studies on geophysics, geomorphology, hydrology, landscape evolution, etc, and a variety of models have been proposed and studied in great detail (see e.g. [12,13,14,50] and references therein). To simulate the "ATL law" in computer codes for accelerator design, several algorithms that produce the required space and time dependencies have been developed. In the case of a linear system (points of the ground are equally distributed along a straight line) it could be a straightforward to apply the "random walk" procedure: for a given time step k it is only necessary to start at one end, giving each point a random displacement Δ m k with respect to the previous point Y i k = Y i k-1 + Σ i m=0 Δ m k [6,18]. It is easy to see that the variance of resulting relative displacement of any two points separated by L is given by ATL-law Eq.(4). With a bit more cumbersome mathematics, the method can be extended to any one dimensional geometry shape (e.g. circle) on a two dimensional surface [51].     We should note here, that widely accepted Langevin-type stochastic equation for the geological landscape evolution always consider, besides smoothing diffusion and erosion terms, an external stochastic noise source uncorrelated in both space and time and with finite variancesee Ref. [52] for detailed review and discussion. Of course, under these assumptions, the resulted variance scales σ 2 (T, L) T in the case of no smoothing and no erosion, leaving off any dependence on the distance between the observation points. We believe that such an ansatz is basically incorrect as the ground motion noise clearly shows its non-stationary character, certain correlation laws in both space and time and scaling. Besides the ATL-law observations, the fractal statistics of earthquakes [53] repudiates the notion of the stationary uncorrelated noise as the source of the observed ground motion.
- Fig.46: Variance of the displacements vs distance between points after 16,000 steps for different scaling exponents  and  (see text).

Summary
Numerous observations and analysis of the data on slow ground motion presented above reveal the phenomena of simultaneous ground diffusion in space and in time. The diffusion obeys a characteristic fractal law with t he ground displacement variance dY 2 scaling with time-and spatial intervals T and L as dY 2 T  L  with both exponents close to 1 (1). The most suitable instruments for studying such a diffusion are arrays of high precision instruments, e.g., Hydrostatic Level Sensors connected by common water pipe and spread over significant area or regular laser tracking of numerous alignment monuments installed in large underground facilities like high energy accelerators. Non-random, systematic movements do often dominate the ground motion but the diffusion components still can be clearly indentified using filtering methods. We believe that present landscape evolution models which assume random stochastic uncorrelated noise as a source of the ground motion are, therefore, incomplete.