I. INTRODUCTION

A hydrostatic leveling system (HLS), based on the principle that the liquid (deionized water or purified water) in the HLS is always looking for an equipotential surface, uses connected reservoirs of water to determine elevation difference at the locations of HLS [1]. Fig. 1 briefly represents the measuring principle of HLS, among which h₁, h₂, h₃ represent the data collected from each sensor, thus the height difference $\Delta h_1$ and $\Delta h_2$ can be described as Eq. (1):

Fig. 1.Full-screen View

Schematic diagram of HLS.

HLSs are applied in CERN (Switzerland), ESRF (France), DESY (Germany), Fermilab (USA), BEPC II (China), SLAC (USA), and SSRF (China) [2-10]. Fig. 2 shows the structural diagrams of HLS vessel used in NSRL, BEPCII and SSRF [11].

Fig. 2.Full-screen View

Structural diagram of HLS vessel.

In precision engineering survey (micron scale), however, the tidal effects on HLS readings cannot be ignored. For instance, at CERN or SLAC, large particle accelerators were designed while taking earth tides into account for proper operation [12, 13]. Under the action of various forces, the Earth body undergoes complicated deformation. Any point on earth is affected by gravitation of the Sun and the Moon, besides the gravity and the rotational centrifugal force of the Earth. Then, the Earth deforms periodically under the action of tidal force, and this is the earth tide.

Figure 3 shows a DSQ water tube clinometer and its schematic diagram [14]. Technically, it works on the principle of liquid equipotential surface, too, but it differs from a conventional HLS in the sensors to measure the liquid level and in the structure parameters. With DSQ water tube clinometer, resolution of a hydrostatic leveling system is 0.001 arcsec and daily drift is less than 0.005 arcsec [15].

Fig. 3.Full-screen View

(Color online) DSQ water tube clinometer (a) and its schematics (b).

DSQ system has been used in tilt tide observation in seismostation. In this paper, we report the series harmonic analysis of the data and contrastive analysis of tidal monitoring observations between HLS and DSQ, carried out at the Jingxian seismostation, Anhui, China.

II. HARMONIC ANALYSIS OF HLS AND DSQ DATA

B. Harmonic analysis of HLS data

For harmonic analysis, Fig. 4 shows the height difference of sensor readings from 10/11/2012 to 26/01/2013.

Fig. 4.Full-screen View

(Color online) Height difference of sensor readings of the N-S (a) and E-W (b) segments.

1. Temperature compensation

Temperature compensation shall be made to the raw data. The temperature effect on the HLS readings can be compensated by Eqs. (2)–(4):

${\beta }_{\text{water}}=-\text{d}\rho /(\rho \text{d}T)\mathrm{,}$ (2) $\Delta V=({\beta }_{\text{water}}-{\beta }_{\text{pipe}})V\Delta T\mathrm{,}$ (3) $\Delta h=\Delta V/S\mathrm{,}$ (4)

where, β_water and β_pipe are thermal expansion coefficient of water in the HLS vessels and the connecting pipe, respectively; T is the water temperature; ρ is the water density; ΔV is the change in water volume; ΔT is the temperature change; Δh is the height change in liquid level in the HLS vessels; and S is the internal cross-section of the vessel. Typically, Δh = -0.1 ℃. The negative value is due to different thermal expansion coefficients of water (2.5×10^-4, 25 ℃), stainless steel (4.5×10^-5) and PE pipe (4.5×10^-4), respectively.

2. Data processing

Data handling, including linear and cubic interpolation, and multilinear regression, was done by using the T-soft code recommended by ICET (International Center for Earth Tides) [16]. The tilt angle is very small (Fig. 5), so the height difference could be converted into tilt angle using Eq. (5):

Fig. 5.Full-screen View

Conversion diagram.

$\theta =(D_2–D_1)\times {10}^{-3}\times 3600\times 180\times {10}^3/(L\text{π})\mathrm{,}$ (5)

where, D₁ and D₂ are the data (in millimeter) measured from each sensor, L is the distance between the two monitoring sites (in meter), and θ is the tilt angle (in mas, milli-arcsecond). The tilt angles of the E-W segment are shown in Fig. 6 using the T-soft code.

Fig. 6.Full-screen View

Tilt angles of the E-W segment from November 2012 to January 2013, obtained using the T-soft code.

3. Results of harmonic analysis

Harmonic analysis of HLS data was completed using the Eterna code, which is the standard software recommended by ICET [17]. The E-W and N-S tilt tidal factors of the semi-diurnal and diurnal tidal waves are given in Table 2. Fig. 7 shows the predicted tide from HLS data (N-S segment) after harmonic analysis. The influence of the tidal effects is about ± 1 μm in 18.1 m by converting the tilt angle to height difference.

TABLE 2.Full-screen View

Results of the harmonic analysis on HLS data

Tilt tidal factors (E-W)				Tilt tidal factors (N-S)
Semi-diurnal			Diurnal	Semi-diurnal			Diurnal
S2	0.50348	Q1	0.33916	S2	1.06434	Q1	0.68855
N2	0.42331	O1	0.41125	N2	0.49223	O1	0.76977
M2	0.72582	M1	0.4960	M2	0.63772	M1	0.87765
N2	0.78599	J1	1.0380	2N2	0.96087	J1	0.59780

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Fig. 7.Full-screen View

Predicted tide from HLS data (N-S segment).

C. Harmonic analysis of data from DSQ

DSQ water tube clinometers are installed on bedrock, which is very stable. The data before and after multilinear regression are partially shown in Fig. 8(a) to reflect the tidal effects on the clinometers (in mas), with just the data of N-S segment. Fig. 8(b) shows the predicted tide from DSQ data (N-S segment) after harmonic analysis. The E-W and N-S tilt tidal factors are listed in Table 3.

TABLE 3.Full-screen View

Result of the harmonic analysis on DSQ water tube clinometers

Tilt tidal factors(E-W)				Tilt tidal factors (N-S)
Semi-diurnal			Diurnal	Semi-diurnal			Diurnal
S2	0.82839	Q1	0.50892	S2	0.78760	Q1	1.03943
N2	0.56528	J1	0.57366	N2	0.46880	J1	0.91027
M2	0.80333	J1	0.42540	M2	0.63136	J1	1.01894
N2	0.58420	J1	1.29475	2N2	0.52474	J1	1.02653

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Fig. 8.Full-screen View

Partial data before and after multilinear regression (a) and the predicted tide from DSQ data (b), for the N-S segment.

III. CONTRASTIVE ANALYSIS OF THE TILT TIDAL FACTOR

B. Relations between damping constant and the frequency response of the instrument

Frequency response Q(T) of HLS and DSQ can be described as Eq. (10), where ξ₀ is the damping constant of the liquid surface vibration; α=T₀/T is the cycle ration, T₀ is duty cycle of the instrument, and T is the cycle of ground motion.

$Q(T)=\frac{1}{\sqrt{{(1-{\alpha }^2)}^2+4{\xi }_0^2{\alpha }^2}}\mathrm{.}$ (10)

Figure 9 shows the frequency response of the instrument. According to the design parameters of HLS at Jingxian seismostation, duty cycle of the instrument is 67 s [1], which is smaller than the cycle of earth tide, the α value is almost zero, so Q(T)=1 in Fig. 8(a), the tidal signal is not distorted at all. But if an earthquake or ground effect happened, sympathetic vibration would happen when the cycle of the disturbing signal is close to the duty cycle of the instrument. HLS differs from DSQ in their duty cycle and ξ₀, hence the different frequency response of the instruments. From the raw data recorded by the two instruments, the sympathetic vibration phenomenon happened during the period, which might lead to different frequency domain signal distortion. This is another reason to explain the different results of harmonic analysis.

Fig. 9.Full-screen View

Frequency response curve.

IV. TIDAL CORRECTION TO THE HLS DATA USED IN SPRING-8 STORAGE RING

A validation study based on measurement data of the HLS used in particle accelerator for tidal correction on the HLS reading is necessary. A set of HLSs were installed in the SPring-8 storage ring, in Hyogo, Japan, as shown in Fig. 10(a). The two HLSs, linked with pipes, were 146.4 m apart. Two HLS readings during 1/1/2006 to 11/1/2006 are shown in Fig. 10(b). The changing trend of the readings is well reflected. Through harmonic analysis to the data of HLS in SPring-8 storage ring, the local tilt tide model was calculated and predicted by the Eterna code. The results are shown in Fig. 10(c). The tidal effects on the height difference between the two HLSs are shown in dashed line in Fig. 11(a), and subtracting the residual signal after tidal correction from the height difference, the ground deformation is obtained, as shown in Fig. 11(b).

Fig. 10.Full-screen View

(Color online) The HLS locations (a), the HLS readings (b) and the predicted tide (c) at SPring-8 in 2006.

Fig. 11.Full-screen View

(Color online) Height difference and the relative tidal correction (a), and residual signal after tidal correction (b).

It is known that the earth tide does not affect the straightness of accelerator alignment, but it changes inclination of the whole system. So, the tidal effects on the HLS readings must be corrected, otherwise it will cause alignment errors in the particle accelerator. According to Fig. 11(a), the maximum amplitude of tilt tide model is ±0.038 mm, in other words, the absolute influence of the tidal effects is 0.076 mm in 146.4 meters, hence the non-negligibility of the tidal effects on the HLS readings. Due to the research on the tidal effects on HLS, related components of the accelerator are working properly, and accuracy of the alignment is improved.

V. SUMMARY

It is known that the main goal for the processing of the HLS measurements is therefore to isolate the local ground deformations from the raw measurement signal. Two different instruments to determining the ground deformations have been tried. Based on the data recorded by HLS and DSQ, and the contrastive analysis of the harmonic results, the tilt tide can be well reflected in the data of the two instruments. So the method discussed in this paper to handle the tidal effects on the readings of HLS can be proved to be feasible and reliable, the local ground deformations are obtained by deleting the tide effects on the readings of the instruments. Then the validation study based on the measurement data of the HLS used in Spring-8 storage ring for tidal correction on the reading of HLS is completed, and the maximum amplitude of the tidal effects on the readings of HLS is obtained. Obviously, the accuracy of alignment with using HLS is improved. It can be concluded that HLS application in micron-level precision alignment of particle accelerators is achievable.