Static effects of horizontal acceleration
Let us consider that a superconducting sphere with a mass \(m\) is levitated inside the gravity sensor of the SG. The sphere is balanced at a position where the upward force from the supporting magnetic field is equal in magnitude to the downward gravity force. We take a cylindrical coordinate system \(\left( {r,\theta ,z} \right)\) whose origin \(O\) coincides with the average position of the sphere. Let \(z\) be upward positive, and let \(U\) be the potential sensed by the sphere. \(U\) takes a minimum at the balancing point. Assuming that the supporting magnetic field is cylindrically symmetric with respect to the vertical axis, \(U\) is expanded in a Taylor series around \(O\) as follows:
$$\begin{aligned} U\left( {r,z} \right) & = U\left( {0,0} \right) + r\left. {\frac{\partial U}{\partial r}} \right_{O} + z\left. {\frac{\partial U}{\partial z}} \right_{O} + \frac{1}{2}\left[ {r^{2} \left. {\frac{{\partial^{2} U}}{{\partial r^{2} }}} \right_{O} + 2rz\left. {\frac{{\partial^{2} U}}{\partial r\partial z}} \right_{O} + z^{2} \left. {\frac{{\partial^{2} U}}{{\partial z^{2} }}} \right_{O} } \right] \\ & \quad + \,\frac{1}{6}\left[ {r^{3} \left. {\frac{{\partial^{3} U}}{{\partial r^{3} }}} \right_{O} + 3r^{2} z\left. {\frac{{\partial^{3} U}}{{\partial r^{2} \partial z}}} \right_{O} +\, 3rz^{2} \left. {\frac{{\partial^{3} U}}{{\partial r\partial z^{2} }}} \right_{O} + z^{3} \left. {\frac{{\partial^{3} U}}{{\partial z^{3} }}} \right_{O} } \right] + \cdots \\ \end{aligned}$$
(1)
where the subscript \(O\) denotes the evaluation of the differential at the origin of the coordinate system. In a real instrument, a feedback control is in effect to hold the sphere in a fixed vertical position. Here, we consider only the mechanical properties, and the effect of feedback is discussed separately.
The superconducting sphere levitated inside the gravity sensor of the SG has rotational as well as translational degrees of freedom. By analyzing the instrumental noise inherent to the SG, Imanishi (2005, 2009) showed that the wellknown longperiod parasitic mode is likely the rotational motion of the sphere around its center of gravity. Although the existence of this mode implies a weak dependence of the potential on the rotational angles of the sphere, here we consider only translational degrees of freedom and neglect rotational ones.
In Eq. (1),
$$\left. {\frac{\partial U}{\partial r}} \right_{O} = 0$$
(2)
$$\left. {\frac{\partial U}{\partial z}} \right_{O} = 0$$
(3)
must be satisfied so that the sphere is balanced at \(O\) Additionally, by imposing a condition that \(U\) is infinitely differentiable at \(O\), the terms in the oddorder powers of \(r\) must vanish. Therefore, retaining the terms up to the thirdorder differentiation with respect to \(r\) and \(z\), we have
$$U\left( {r,z} \right) = U\left( {0,0} \right) + \frac{1}{2}\left( {\alpha_{\text{H}} r^{2} + \alpha_{\text{V}} z^{2} } \right) + \frac{1}{6}\left( {3\beta_{\text{H}} r^{2} z + \beta_{\text{V}} z^{3} } \right)$$
(4)
where
$$\alpha_{\text{H}} = \left. {\frac{{\partial^{2} U}}{{\partial r^{2} }}} \right_{O}$$
(5)
$$\alpha_{\text{V}} = \left. {\frac{{\partial^{2} U}}{{\partial z^{2} }}} \right_{O}$$
(6)
$$\beta_{\text{H}} = \left. {\frac{{\partial^{3} U}}{{\partial r^{2} \partial z}}} \right_{O}$$
(7)
$$\beta_{\text{V}} = \left. {\frac{{\partial^{3} U}}{{\partial z^{3} }}} \right_{O}$$
(8)
The coefficients of the secondorder terms of the potential \(\alpha_{\text{H}}\) and \(\alpha_{\text{V}}\) are the “spring constants” in the horizontal and vertical directions, respectively. \(\alpha_{\text{H}} > 0\) and \(\alpha_{\text{V}} > 0\) must hold so that \(U\) takes a minimum at the balancing position. The sphere is supported by a weak spring in the vertical direction and by a strong spring in the horizontal direction; otherwise, the sphere would very easily move away from the central axis of the coils. Therefore, \(\alpha_{\text{H}} \gg \alpha_{\text{V}}\). The (angular) eigenfrequencies for vibrations in small amplitude (as simple harmonic oscillators) are given by
$$\omega_{\text{H}} = \sqrt {\frac{{\alpha_{\text{H}} }}{m}}$$
(9)
$$\omega_{\text{V}} = \sqrt {\frac{{\alpha_{\text{V}} }}{m}}$$
(10)
in the horizontal and vertical directions, respectively. It follows from \(\alpha_{\text{H}} \gg \alpha_{\text{V}}\) that \(\omega_{\text{H}} \gg \omega_{\text{V}}\).
The coefficients of the thirdorder terms of the potential \(\beta_{\text{H}}\) and \(\beta_{\text{V}}\) denote the deviation from a purely harmonic potential. \(\beta_{\text{H}}\) is the coefficient of the term denoting the coupling between the horizontal and vertical components. \(\beta_{\text{V}}\) is the coefficient of the higherorder term for the vertical component. In an ideal gravity sensor, both \(\beta_{\text{H}} = 0\) and \(\beta_{\text{V}} = 0\) would be satisfied; in reality, however, these coefficients are finite.
The restoring forces on the sphere for this potential are given by
$$F_{\text{r}} =  \frac{\partial U}{\partial r} =  \alpha_{\text{H}} r  \beta_{\text{H}} rz$$
(11)
$$F_{z} =  \frac{\partial U}{\partial z} =  \alpha_{\text{V}} z  \frac{1}{2}\beta_{\text{H}} r^{2}  \frac{1}{2}\beta_{\text{V}} z^{2}$$
(12)
in the radial and vertical directions, respectively.
When we adjust the vertical orientation of the gravimeter, we seek an extremum of the gravity output by tilting the instrument so that the direction of the gravity sensor exactly coincides with the local vertical. Experimentally, the change in the gravity output \(\Delta g\) depends quadratically on the tilt angle, and thus,
$$\Delta g = c\varphi^{2}$$
(13)
where \(\varphi\) is the deviation angle from the optimal orientation and \(c\) is a constant. Whereas \(c\) is negative in usual springtype gravimeters, it is positive in the SG. Because the vertical restoring force as given by Eq. (12) decreases for finite \(r\), tilting the SG causes a horizontal displacement of the sphere from the central axis and therefore a downward motion of the sphere, i.e., an apparent increase in the gravity (GWR Instruments 1985). This means that \(\beta_{\text{H}} > 0\). When the gravimeter is tilted from the local vertical by a small angle \(\varphi\), the sphere experiences additional forces \(mg\sin \varphi\) and \(mg\left( {1  \cos \varphi } \right)\) in the horizontal and vertical directions, respectively, where \(g\) is the gravity acceleration. Then, the balance of the sphere is expressed as
$$ \alpha_{\text{H}} r  \beta_{\text{H}} rz + mg\varphi = 0$$
(14)
$$ \alpha_{\text{V}} z  \frac{1}{2}\beta_{\text{H}} r^{2}  \frac{1}{2}\beta_{\text{V}} z^{2} + \frac{1}{2}mg\varphi^{2} = 0$$
(15)
where we approximate \(\sin \varphi \sim\varphi\) and \(1  \cos \varphi \sim\left( {1/2} \right)\varphi^{2}\). Equation (14) indicates that \(r\) is on the first order of \(\varphi\). Additionally, as observed from Eq. (15), \(z\) is on the second order of \(\varphi\). Therefore, to the first order of \(\varphi\),
$$r = \frac{mg\varphi }{{\alpha_{\text{H}} }}$$
(16)
is the radial displacement of the sphere. Substituting Eq. (16) into Eq. (15), we obtain the vertical displacement as
$$z = \frac{mg}{{2\alpha_{\text{V}} }}\left( {1  \frac{{\beta_{\text{H}} mg}}{{\alpha_{\text{H}}^{2} }}} \right)\varphi^{2}$$
(17)
to the second order of \(\varphi\). The change in the vertical force (upward positive) sensed by the gravimeter is given by \(\alpha_{\text{V}} z\), and thus, the change in the upward acceleration \(\Delta a_{z}\) is
$$\Delta a_{z} = \frac{{\alpha_{\text{V}} z}}{m} = \frac{g}{2}\left( {1  \frac{{\beta_{\text{H}} mg}}{{\alpha_{\text{H}}^{2} }}} \right)\varphi^{2}$$
(18)
When a feedback control on the gravity is enabled, an additional magnetic field generated by the feedback coil exerts a force on the sphere so that it cancels any external forces and the sphere is consequently held in a fixed position. Because tilt adjustment deals only with static signals, both the external force and the feedback force are regarded as temporally constant, and thus, there is no vertical motion of the sphere. This feedback force is equal in magnitude to the change in gravity to be measured. The change in the gravity acceleration is given by
$$\Delta g =  \Delta a_{z} = \frac{g}{2}\left( {\frac{{\beta_{\text{H}} mg}}{{\alpha_{\text{H}}^{2} }}  1} \right)\varphi^{2}$$
(19)
Comparing this with Eq. (13), we have
$$c = \frac{g}{2}\left( {\frac{{\beta_{\text{H}} mg}}{{\alpha_{\text{H}}^{2} }}  1} \right)$$
(20)
The actual value of the coefficient \(c\) can be estimated experimentally. Figure 4 shows a result of the tilt adjustment for CT #036. The parameter Tilt Reset determines the null position of the tilt sensor. Note that a positive change in the output voltage corresponds to a negative change in the gravity for the SG. We search for the optimal setting of Tilt Reset for each of the two tilt sensors, which we shall call x and y. Fitting a quadratic function to the gravity changes with Tilt Reset as a free variable gives the estimate of \(c\) as the optimal coefficient of the secondorder term. The results are as follows:

$$c = 7.105 \pm 0.068 \times 10^{  6} \;{\text{V}}\;{\text{digit}}^{  2} \quad (x)$$

$$c = 8.137 \pm 0.018 \times 10^{  6} \;{\text{V}}\;{\text{digit}}^{  2} \quad (y)$$
where digit denotes the unit of Tilt Reset. The two values need not coincide with each other, because the effective sensitivities of Tilt Reset for x and y can be different. An additional file describes the calibration of sensitivity of Tilt Reset (see Additional file 1). The results of calibration are \(4.189 \pm 0.005 \times 10^{  6}\) and \(4.746 \pm 0.009 \times 10^{  6} \;{\text{rad/digit}}\) for x and y, respectively. Note that this calibration is different from the calibration of Tilt Power (Riccardi et al. 2009) with the automatic leveling function enabled. Using these results as well as the scale factor of this gravimeter (\(1 {\text{V}} = {}\,1134.6 \times 10^{  8} \,{\text{ms}}^{  2}\)), the above values of \(c\) are translated into

$$c = 4.594 \pm 0.044\,{\text{ms}}^{  2} \quad (x)$$

$$c = 4.099 \pm 0.009\,{\text{ms}}^{  2} \quad (y)$$
These estimates are different by approximately 12%, suggesting that the supporting magnetic field may be azimuthally asymmetric. Here, neglecting this difference, we assume that there is no dependence of the coefficient \(c\) on the azimuth. By taking a simple average of both x and y, we obtain
$$c = 4.35 \,{\text{ms}}^{  2}$$
(21)
as the estimate of \(c\) for this gravimeter.
The gravity acceleration at the gravimeter pier of the VERA Ishigakijima station measured using an absolute gravimeter is \(g = 9.79002515\,{\text{ms}}^{  2}\) (Miyakawa, personal communication). From Eqs. (20) and (21), we have
$$\frac{{\beta_{\text{H}} m}}{{\alpha_{\text{H}}^{2} }} = 0.193\, {\text{m}}^{  1} \,{\text{s}}^{2}$$
(22)
Dynamic effects of horizontal acceleration
Thus far, we have dealt with the static equilibrium of forces in the gravity sensor. Now, we will consider dynamic cases where external acceleration is applied to the gravimeter pier so that inertial forces are exerted on the superconducting sphere. The potential in Eq. (4) can be rewritten in the rectangular coordinate system as
$$U\left( {x,y, z} \right) = U\left( {0,0,0} \right) + \frac{1}{2}\left[ {\alpha_{\text{H}} \left( {x^{2} + y^{2} } \right) + \alpha_{\text{V}} z^{2} } \right] + \frac{1}{6}\left[ {3\beta_{\text{H}} \left( {x^{2} + y^{2} } \right)z + \beta_{\text{V}} z^{3} } \right]$$
(23)
Let \(X,Y, Z\) be the displacement of the pier in the inertial frame, and \(x,y, z\) the displacement of the sphere with respect to the gravimeter. Then, the equations of motion of the sphere are
$$m\ddot{x} + 2mh_{\text{H}} \omega_{\text{H}} \dot{x} + \alpha_{\text{H}} x =  m\ddot{X}$$
(24)
$$m\ddot{y} + 2mh_{\text{H}} \omega_{\text{H}} \dot{y} + \alpha_{\text{H}} y =  m\ddot{Y}$$
(25)
$$m\ddot{z} + 2mh_{\text{V}} \omega_{\text{V}} \dot{z} + \alpha_{\text{V}} z + \frac{1}{2}\beta_{\text{H}} \left( {x^{2} + y^{2} } \right) + \frac{1}{2}\beta_{\text{V}} z^{2} =  m\ddot{Z}$$
(26)
where we have retained the terms up to the first order in the horizontal components and the terms up to the second order in the vertical component. In addition, we assume that the sphere is subject to frictional forces due to eddy currents in the gravity sensor whose magnitude is proportional to the velocity of the sphere. The coefficients of the friction are \(2mh_{\text{H}} \omega_{\text{H}}\) and \(2mh_{\text{V}} \omega_{\text{V}}\) for the horizontal and vertical components, respectively. Here, \(h_{\text{H}}\) and \(h_{\text{V}}\) are dimensionless positive constants. Whereas \(h_{\text{V}}\) can be directly measured as described later, the value of \(h_{\text{H}}\) is unknown. Here, we assume that the friction has no dependence on the direction of the sphere velocity, and thus, \(2mh_{\text{H}} \omega_{\text{H}} = 2mh_{\text{V}} \omega_{\text{V}}\). Given this, Eqs. (24)–(26) can be rewritten as
$$\ddot{x} + 2\eta \dot{x} + \omega_{\text{H}}^{2} x =  \ddot{X}$$
(27)
$$\ddot{y} + 2\eta \dot{y} + \omega_{\text{H}}^{2} y =  \ddot{Y}$$
(28)
$$\ddot{z} + 2\eta \dot{z} + \omega_{\text{V}}^{2} z + \frac{1}{2}\frac{{\beta_{\text{H}} }}{m}\left( {x^{2} + y^{2} } \right) + \frac{1}{2}\frac{{\beta_{\text{V}} }}{m}z^{2} =  \ddot{Z}$$
(29)
where we define
$$\eta = h_{\text{H}} \omega_{\text{H}} = h_{\text{V}} \omega_{\text{V}}$$
(30)
The mechanical eigenfrequencies of the massspring system are given by
$$f_{\text{H}} = \frac{{\omega_{\text{H}} }}{2\pi }$$
(31)
$$f_{\text{V}} = \frac{{\omega_{\text{V}} }}{2\pi }$$
(32)
for the horizontal and vertical directions, respectively. \(\omega_{\text{H}} \gg \omega_{\text{V}}\) means \(f_{\text{H}} \gg f_{\text{V}}\). The eigenfrequencies are dependent on the useradjustable coil currents and therefore are not intrinsic to the particular instrument. Typically, \(f_{\text{V}}\) is on the order of 0.1 Hz, and it can be measured by applying external vertical forces on the sphere (Imanishi et al. 1996; Van Camp et al. 2000). Meanwhile, there is no easy way to directly measure \(f_{\text{H}}\) experimentally.
Before the SG was moved to Ishigakijima, we measured its openloop and closedloop transfer functions at Tsukuba University (Ikeda et al. 2013). Sinusoidal functions with 1 V amplitude at nine discrete frequencies from 0.002 to 1 Hz were applied to the feedback coil of the SG to measure the responses. The vertical eigenfrequency and dissipation parameter obtained from this experiment are \(f_{\text{V}} = 0.120\,{\text{Hz}}\) and \(h_{\text{V}} = 6.41\). Although we have not performed a similar experiment at Ishigakijima, we can estimate the present value of the eigenfrequency as follows. When adjusting the supporting magnetic field, we measure the displacement of the levitating sphere in response to a constant vertical force generated by a 10 mA current. This is called the “magnetic gradient,” a parameter that corresponds to the inverse of the stiffness of the spring. The final values of the magnetic gradient were 1.613 V/10 mA at Tsukuba and 1.834 V/10 mA at Ishigakijima. Since the magnetic gradient is inversely proportional to \(\alpha_{\text{V}}\), it follows from the final gradients that the value of \(\alpha_{\text{V}}\) at Ishigakijima is equal to 0.879 times that at Tsukuba. Considering that the eigenfrequency is proportional to the square root of \(\alpha_{\text{V}}\), \(f_{\text{V}}\) at Ishigakijima should be equal to 0.112 Hz. Also, considering that the dissipation is provided by the currents induced in the nonsuperconducting parts of the gravity sensor, we assume that the parameter \(\eta\) is independent of the magnetic gradient. Therefore, the value at Tsukuba
$$\eta = 2\pi h_{\text{V}} f_{\text{V}} = 4.833\,{\text{s}}^{  1}$$
(33)
also applies to Ishigakijima.
The response of this dynamic system to an external acceleration in the x direction is described in the frequency domain as
$$\frac{{\tilde{x}}}{{\tilde{X}}} = \frac{{\omega^{2} }}{{  \omega^{2} + 2i\eta \omega + \omega_{\text{H}}^{2} }}$$
(34)
where \(\omega\) is the angular frequency and \(\tilde{x}\) and \(\tilde{X}\) are the Fourier transforms of \(x\) and \(X\), respectively. A similar equation holds also for the y component. Note that there is no feedback control in the horizontal directions. The sphere is displaced horizontally according to the response in Eq. (34), resulting in a change in the mean vertical position of the sphere through the term containing \(\beta_{\text{H}}\) in Eq. (26). Denoting a temporal average as \(\langle \;\rangle\) we have
$$\langle z\rangle =  \frac{1}{2}\frac{{\beta_{\text{H}} }}{{\alpha_{\text{V}} }}\left[ {\langle{x^{2}}\rangle + \langle{y^{2}}\rangle } \right]$$
(35)
where we neglect the higherorder term of \(z\) (the term containing \(\beta_{\text{V}}\)). Because \(\alpha_{\text{V}} > 0\) and \(\beta_{\text{H}} > 0\), the righthand side of Eq. (35) is negative, and thus, the mean position of the sphere moves downward. This is observed as an increase in the DC component of the gravity acceleration given by
$$\Delta g =  \frac{{\alpha_{\text{V}} }}{m}z = \frac{1}{2}\frac{{\beta_{\text{H}} }}{m}\left[ {\langle{x^{2}}\rangle + \langle{y^{2}}\rangle} \right]$$
(36)
This implies that the meansquared horizontal displacements of the sphere are proportional to the apparent change in the gravity acceleration. In particular, if the input acceleration has a single angular frequency \(\sigma\), then
$$\frac{x}{X} = \frac{{\sigma^{2} }}{{  \sigma^{2} + 2i\eta \sigma + \omega_{\text{H}}^{2} }}$$
(37)
holds in the time domain, and therefore, we have
$$\Delta g = \frac{1}{2}\frac{{\beta _{{\text{H}}} }}{m}\left {\frac{{\sigma ^{2} }}{{  \sigma ^{2} + 2i\eta \sigma + \omega _{{\text{H}}}^{2} }}} \right^{2} \left[ {\left\langle {X^{2} } \right\rangle + \left\langle {Y^{2} } \right\rangle } \right]$$
(38)
We use the horizontal acceleration data caused by the motion of the VLBI antenna to verify whether this theoretical prediction is correct. Figure 5 shows a typical example of the ground vibrations induced by the antenna and the resultant disturbances in the gravity recordings. The seismometer used to record the ground vibrations is an L4C 1.0Hz three components velocity transducer, manufactured by Mark Products, that was placed on the gravimeter pier (Ohtaki and Nawa 2013). The raw data shown in Fig. 5 were converted into ground displacements by deconvolving the transfer functions of the sensors. We can see from Fig. 5 that the north–south and east–west components have similar magnitudes of power, although they are variable with time, whereas the power of the up–down component is only 1% of the magnitudes of the horizontal components. Figure 6 shows the power spectra of the three components. The spectral peaks at 4.92 Hz and at its harmonics found in all the components are due to the movements of the antenna. It is unknown which part of the antenna is responsible for the observed frequencies. The power ratio for the peaks at 4.92 and 9.84 Hz is approximately 117 and 179 for the north–south and east–west components, respectively. Therefore, we regard the vibration as monochromatic in the following analysis.
The positive steps in the gravity signals during the period of antenna movement as seen in Fig. 5 are collected and compared with the ground displacements in Fig. 7. The ground displacements are displayed in terms of timedomain variances. Note that the abovementioned x and y of the tilt sensors of the SG are not oriented in the east–west and north–south directions. For each north–south, east–west and up–down component (Fig. 7a, b, d), respectively), ground displacements and gravity changes appear to be correlated with linear correlation coefficients 0.78–0.90. Meanwhile, the linear correlation is clearer with the correlation coefficient as high as 0.97 in the case of the sum of the north–south and east–west components, i.e., the horizontal component (Fig. 7c). This fact provides evidence that the assumptions for our theoretical model are appropriate. In the following, we consider only the horizontal components of ground vibrations.
From Eq. (38) and considering the proportionality shown in Fig. 7c, we obtain
$$\frac{1}{2}\frac{{\beta_{\text{H}} }}{m}\left {\frac{{\sigma^{2} }}{{  \sigma^{2} + 2i\eta \sigma + \omega_{\text{H}}^{2} }}} \right^{2} = 1.36 \times 10^{4} \, {\text{m}}^{  1} \,{\text{s}}^{  2}$$
(39)
where \(\sigma = 2\pi \times 4.92 = 30.9\, {\text{s}}^{{{}1}}\). By defining two new variables \(\alpha_{\text{H}}^{{\prime }}\) and \(\beta_{\text{H}}^{{\prime }}\) as
$$\alpha_{\text{H}}^{{\prime }} = \frac{{\alpha_{\text{H}} }}{m}$$
(40)
$$\beta_{\text{H}}^{{\prime }} = \frac{{\beta_{\text{H}} }}{m}$$
(41)
Eqs. (22) and (39) can be rewritten as
$$\frac{{\beta_{\text{H}}^{{\prime }} }}{{\left( {\alpha_{\text{H}}^{{\prime }} } \right)^{2} }} = 0.193\, {\text{m}}^{  1} \,{\text{s}}^{2}$$
(42)
$$\frac{1}{2}\beta_{\text{H}}^{{\prime }} \left {\frac{{\sigma^{2} }}{{  \sigma^{2} + 2i\eta \sigma + \alpha_{\text{H}}^{{\prime }} }}} \right^{2} = 1.36 \times 10^{4} \,{\text{ms}}^{  1}$$
(43)
Together with Eq. (33), Eqs. (42) and (43) can be solved for \(\alpha_{\text{H}}^{{\prime }}\) and \(\beta_{\text{H}}^{{\prime }}\) as
$$\alpha_{\text{H}}^{{\prime }} = 2.88 \times 10^{2} \,{\text{s}}^{  2}$$
(44)
$$\beta_{\text{H}}^{{\prime }} = 1.60 \times 10^{4} \,{\text{m}}^{  1} \,{\text{s}}^{  2}$$
(45)
Therefore,
$$\omega_{\text{H}} = \sqrt {\alpha_{\text{H}}^{{\prime }} } = 17.0\,{\text{s}}^{  1}$$
(46)
$$f_{\text{H}} = \frac{{\omega_{\text{H}} }}{2\pi } = 2.70\,{\text{s}}^{  1}$$
(47)
$$h_{\text{H}} = 0.285$$
(48)
In the above result, there is no assumption regarding the value of \(m\).
The most uncertain parameter in the above analysis is \(\eta\), for which we have assumed the same value (\(\eta = 4.833\,{\text{s}}^{  1}\)) as measured at Tsukuba. Figure 8 shows the dependence of the estimates of \(\alpha_{\text{H}}^{{\prime }}\) and \(\beta_{\text{H}}^{{\prime }}\) on the variable \(\eta\). Both \(\alpha_{\text{H}}^{{\prime }}\) and \(\beta_{\text{H}}^{{\prime }}\) increase monotonically as \(\eta\) increases. In the extreme case where there is no dissipation (\(\eta = 0\)), we obtain \(\alpha_{\text{H}}^{{\prime }} = 2.70 \times 10^{2} \,{\text{s}}^{  2}\) and \(f_{\text{H}} = 2.61\,{\text{s}}^{  1}\). If the dissipation is approximately twice as large (\(\eta = 10\)) as our previous assumption, we have \(\alpha_{H}^{{\prime }} = 3.43 \times 10^{2} \,{\text{s}}^{  2}\) and \(f_{\text{H}} = 2.95\,{\text{s}}^{  1}\). Therefore, we can conclude that \(f_{\text{H}}\) of this gravimeter is approximately 3 Hz for a practically plausible range of the dissipation parameter.
Thus, we have successfully modeled the effects of horizontal acceleration due to the motion of the VLBI antenna on the SG quantitatively. As a byproduct of the model analysis, the horizontal eigenfrequency \(f_{\text{H}}\) of the massspring system was indirectly estimated. It is noted that the precondition for the analysis (\(f_{\text{H}} \gg f_{\text{V}}\)) is satisfied because \(f_{\text{V}}\) is on the order of 0.1 Hz.