The mechanical response of the gravimeter can be measured by applying artificial force to the superconducting sphere (Imanishi et al., 1996; GWR Instruments, 1997; Van Camp et al., 2000). The method of measurement is very simple; one applies an external signal (in voltage) \({V}_{in}\) through the feedback circuit of the gravimeter with the normal feedback control disabled, and record a resultant signal \({V}_{out}\) in the error output of the gravimeter (the channel called Gravity Balance) as well as the input signal \({V}_{in}\). Depending on the purpose of the measurement, the input signal \({V}_{in}\) can be an arbitrary function of time, including a step or a sinusoid.
Our experiment of measuring the instrumental response of CT #036 took place on June 7, 2018. We used an Agilent 33210A function generator to generate artificial signals and a Hakusan DATAMARK LS-8800 data logger to record both input and output signals. The sampling frequency was 200 Hz. Sinusoidal functions with 0.5 V amplitude at eight discrete frequencies from 0.001 to 0.2 Hz were applied (Table 1).
Table 1
Measurement results for the left-hand side of equation (46) and (47) for eight different frequencies.
Frequency (Hz)
|
Equation (46)
|
Equation (47)
|
0.001
|
2.086494 ± 0.000063
|
–0.207119 ± 0.000058
|
0.002
|
2.025367 ± 0.000073
|
–0.402342 ± 0.000074
|
0.005
|
1.691002 ± 0.000135
|
–0.841026 ± 0.000139
|
0.01
|
1.059860 ± 0.000178
|
–1.060019 ± 0.000175
|
0.02
|
0.412834 ± 0.000237
|
–0.856908 ± 0.000233
|
0.05
|
0.071902 ± 0.000215
|
–0.411178 ± 0.000215
|
0.1
|
0.006963 ± 0.000241
|
–0.210730 ± 0.000241
|
0.2
|
–0.003391 ± 0.000203
|
–0.109354 ± 0.000203
|
Figure 4 shows the recorded time series of the input and output signals. For each frequency, more than six cycles of oscillations were recorded. As seen in Figure 4(b), the output signal is affected by natural gravity changes, mostly the earth tides. We noticed that the power-line-cycle (60 Hz) noise became large in the output signal from the Gravity Balance when it exceeded ±3V. The cause of this noise is unknown.
We will divide the analysis of the measurement results into two stages. In the first stage, we adopt the linear model, corresponding to equations (17)–(21), to interpret the response of the gravimeter. The purpose of this treatment is to obtain an internally consistent set of parameters for CT #036, which will be needed to combine measured voltage with physical quantities. Then, in the second stage, we adopt the model in which nonlinearity is taken into account, corresponding to equations (22)–(33).
3.1 Case 1: Linear response
In this subsection, we treat the linear case (\({\beta }_{V}’=0\)). In this case, the input and output time-domain signals, \({V}_{in}\) and \({V}_{out}\), are linearly connected through an impulse response. In the frequency domain, this is described as
\({\tilde{V}}_{out}={\varphi }_{3} {\varphi }_{2} {\varphi }_{1}{\tilde{V}}_{in}\)
|
(35)
|
where \({\tilde{V}}_{in}\) and \({\tilde{V}}_{out}\) are the Fourier transforms of \({V}_{in}\) and \({V}_{out}\), respectively. \({\varphi }_{1}\), \({\varphi }_{2}\) and \({\varphi }_{3}\) are three stages of transfer functions as explained below.
First, \({\varphi }_{1}\), in the unit of \(\text{m}{\text{s}}^{-2}{\text{V}}^{-1}\), translates voltage into acceleration. A signal (in voltage) of unit magnitude input to the feedback circuit of the gravimeter generates an acceleration applied to the superconducting sphere, whose magnitude is equal to \({\varphi }_{1}\). Therefore, \({\varphi }_{1}\) stands for the DC sensitivity of the gravimeter, and often called a scale factor. Calibration of the scale factor is usually made through parallel registration with an absolute gravimeter (e.g. Imanishi et al., 2002; Crossley et al., 2018). Absolute gravity measurements at Ishigakijima was performed in January 2015 (Miyakawa et al., 2020) for the purpose of calibrating instrumental drift as well as the sensitivity of CT #036. The resultant scale factor (for the normal 100 kΩ feedback resistor) was
\({\varphi }_{1}=\left(58.0 \pm 0.1\right)\times {10}^{-8}\text{ m}{\text{s}}^{-2}{\text{V}}^{-1}\)
|
(36)
|
which falls within the typical range (i.e. \(\left(50-100\right)\times {10}^{-8}\text{ m}{\text{s}}^{-2}{\text{V}}^{-1}\)) of sensitivity for an SG. In this study, we treat \({\varphi }_{1}\) as a known quantity.
Next, \({\varphi }_{2}\) is the mechanical response of the superconducting sphere against applied acceleration of unit magnitude. Neglecting higher-order terms in equation (14), \({\varphi }_{2}\) as the linear response of a damped harmonic oscillator is derived for angular frequency \(\omega\) as
\({\varphi }_{2}=\frac{1}{-{\omega }^{2}+2\eta i\omega +{\omega }_{V}^{2}}\)
|
(37)
|
This transfer function translates acceleration into a displacement of the sphere. The unit of \({\varphi }_{2}\) is s2. Note that among the three transfer functions \({\varphi }_{2}\) is the only frequency-dependent part. Note also that when \(\omega =0\), \({\varphi }_{2}\) is equal to \(1/{\omega }_{V}^{2}\).
Finally, \({\varphi }_{3}\) converts a displacement of the superconducting sphere of unit magnitude into voltage through the position detector. This has a unit of \(\text{V}{\text{m}}^{-1}\). In this paper, we do not take possible nonlinearity of the position detector into account, and assume that \({\varphi }_{3}\) is invariant within the range of displacement of the sphere in our experiment. As far as we are aware, this factor has gathered much less attention than the scale factor of the gravimeter among the users of SG. One of the purpose here is to calibrate \({\varphi }_{3}\) which will be used later to convert measured voltage into displacement of the sphere.
Putting together the three stages, the total transfer function is given by
\(\frac{{\tilde{V}}_{out}}{{\tilde{V}}_{in}}=\frac{\gamma }{-{\omega }^{2}+2\eta i\omega +{\omega }_{V}^{2}}\)
|
(38)
|
where |
\(\gamma ={\varphi }_{3} {\varphi }_{1}\)
|
(39)
|
is a new variable in the unit of \({\text{s}}^{-2}\). \({\omega }_{V}\), \(\eta\) and \(\gamma\) in the right-hand side of equation (38) are the parameters to be measured by our experiment.
For each frequency \(f\), the input signal \({V}_{in}\) is fit to the model function:
\({V}_{in}={A}_{0}^{in}+{A}_{1}^{in}\text{cos}\omega t-{B}_{1}^{in}\text{sin}\omega t\)
|
(40)
|
where \(\omega =2\pi f\), and \({A}_{0}^{in}\), \({A}_{1}^{in}\) and \({B}_{1}^{in}\) are the free parameters (in the unit of V) to be adjusted. If multiplied by \({\varphi }_{1}\), the right-hand side of equation (40) corresponds to the right-hand side of equation (16). In other words,
\(\left\{\begin{array}{c}\mu ={\varphi }_{1}{A}_{1}^{in}\\ \nu ={\varphi }_{1}{B}_{1}^{in}\end{array}\right.\)
|
(41)
|
The term \({A}_{0}^{in}\) in the right-hand side of equation (40) is included to account for a small offset of the DC component that may exist in actual experiments.
Similarly, the output signal \({V}_{out}\) is fit to the function:
\({V}_{out}={A}_{0}^{out}+{A}_{1}^{out}\text{cos}\omega t-{B}_{1}^{out}\text{sin}\omega t\)
|
(42)
|
where \({A}_{0}^{out}\), \({A}_{1}^{out}\) and \({B}_{1}^{out}\) are the free parameters (in the unit of V) to be adjusted. If divided by \({\varphi }_{3}\), the right-hand side of equation (42) corresponds to the right-hand side of equation (17), in other words,
\(\left\{\begin{array}{c}{a}_{1}=\frac{{A}_{1}^{out}}{{\varphi }_{3}}\\ {b}_{1}=\frac{{B}_{1}^{out}}{{\varphi }_{3}}\end{array}\right.\)
|
(43)
|
\({A}_{1}^{out}\) and \({B}_{1}^{out}\) must be corrected for the response of the analog lowpass filter for the Gravity Balance channel at the angular frequency \(\omega\).
In a real experiment, special care must be taken in estimating \({A}_{0}^{out}\), \({A}_{1}^{out}\) and \({B}_{1}^{out}\), because the gravimeter is subject to the gravity changes of natural origin as well as the injected artificial signal during the experiment. Then, instead of equation (42), the formula to be used is
\({V}_{out}={A}_{0}^{out}+{A}_{1}^{out}\text{cos}\omega t-{B}_{1}^{out}\text{sin}\omega t-\frac{1}{C}{g}_{calc}\left(t\right)\)
|
(44)
|
where \({g}_{calc}\left(t\right)\) is the known gravity signals in the unit of acceleration. Here we take into account the theoretical tides and the effect of atmospheric pressure as known natural signals, and we assume that these are long-periodic enough to be treated as DC components. We also assume that there are no other agents of natural gravity signals affecting the measurements. The parameter \(C\) in the right-hand side of equation (44) is the “scale factor” of the gravimeter operated without feedback. It is given by
\(C=\frac{1}{{\left.{\varphi }_{3}{\varphi }_{2}\right|}_{\omega =0}}=\frac{{\omega }_{V}^{2}}{{\varphi }_{3}}=\frac{{\omega }_{V}^{2}}{\gamma }{\varphi }_{1}\)
|
(45)
|
where we have used equations (37) and (39). Because \(C\) is dependent on \({\omega }_{V}\) and \(\gamma\), determination of \(C\) as well as \({\omega }_{V}\), \(\eta\) and \(\gamma\) must be done in an iterative way as follows. First, a provisional value of \(C\), say \({C}_{0}\), is assumed to correct for the known natural signals in equation (44). Once the unknown parameters in the right-hand sides of equations (42) and (44) are adjusted, \({\omega }_{V}\), \(\eta\) and \(\gamma\) in equation (38) can be estimated. This is actually done in terms of the cosine and sine parts of the both sides of equation (38) for angular frequency \(\omega\) as
\(\frac{{A}_{1}^{in}{A}_{1}^{out}+{B}_{1}^{in}{B}_{1}^{out}}{{\left({A}_{1}^{in}\right)}^{2}+{\left({B}_{1}^{in}\right)}^{2}}=Re\left\{\frac{\gamma }{-{\omega }^{2}+2\eta i\omega +{\omega }_{V}^{2}}\right\}\)
|
(46)
|
and |
\(\frac{{A}_{1}^{in}{B}_{1}^{out}-{B}_{1}^{in}{A}_{1}^{out}}{{\left({A}_{1}^{in}\right)}^{2}+{\left({B}_{1}^{in}\right)}^{2}}=Im\left\{\frac{\gamma }{-{\omega }^{2}+2\eta i\omega +{\omega }_{V}^{2}}\right\}\)
|
(47)
|
Then, we can calculate \(C\) based on equation (45), which must reproduce the initial value \({C}_{0}\) so that the solution is internally consistent. After some trials, we found that the estimates of \({\omega }_{V}\), \(\eta\) and \(\gamma\), and therefore of \(C\), are highly invariant for a plausible range of the choice of \({C}_{0}\). Figure 5 shows the estimated value of \(C\) for different initial values of \({C}_{0}\). As a result, \(C=27.65\times {10}^{-8}\text{ m }{\text{s}}^{-2}{\text{V}}^{-1}\) was found to be the most appropriate value of \(C\). Comparing this value with the usual scale factor given in equation (36), we can say that the gravimeter without feedback control is approximately twice as sensitive as that under feedback control in the case of CT #036.
Table 1 lists the left-hand sides of equations (46) and (47) calculated from the estimates of \({A}_{1}^{in}\), \({B}_{1}^{in}\), \({A}_{1}^{out}\) and \({B}_{1}^{out}\) obtained for the eight frequencies. These results are then used to estimate \({\omega }_{V}\), \(\eta\) and \(\gamma\) based on equations (46) and (47). Applying a weighted least squares method, we obtain
\({\omega }_{V}=\left(0.825\pm 0.004\right) {\text{s}}^{-1}\)
|
(48)
|
\(\eta =\left(6.52\pm 0.03\right) {\text{s}}^{-1}\)
|
(49)
|
\(\gamma =\left(1.43\pm 0.01\right) {\text{s}}^{-2}\)
|
(50)
|
From equations (6), (8) and (48) we have |
\({\alpha }_{V}’={\omega }_{V}^{2}=0.68 {\text{s}}^{-2}\)
|
(51)
|
Also, from equations (36), (39) and (50), we obtain |
\({\varphi }_{3}=\frac{\gamma }{{\varphi }_{1}}=2.5\times {10}^{6} \text{V}{\text{m}}^{-1}\)
|
(52)
|
It will be useful now to verify the physical magnitude of the applied acceleration and the resultant displacement of the sphere. We take the case of frequency = 0.001 Hz. For the input signal, \({A}_{1}^{in}=-0.0268 \text{V}\) and \({B}_{1}^{in}=-0.4984 \text{V}\) (Note that \({\left({A}_{1}^{in}\right)}^{2}+{\left({B}_{1}^{in}\right)}^{2}\cong {\left(0.5\right)}^{2}\)). Therefore, from equations (36) and (41), we have \(\mu =-0.16\times {10}^{-7} \text{m}{\text{s}}^{-2}\) and \(\nu =-2.89\times {10}^{-7} \text{m}{\text{s}}^{-2}\). For the output signal, \({A}_{1}^{out}=-0.1592 \text{V}\) and \({B}_{1}^{out}=-1.0344 \text{V}\) (after correction for the response of the Gravity Balance lowpass filter). From equations (43) and (52), we have \({a}_{1}=-0.64\times {10}^{-7} \text{m}\) and \({b}_{1}=-4.19\times {10}^{-7} \text{m}\). On the other hand, the theoretically-derived equation (20), with the present estimates of \({\alpha }_{V}’\), \(\eta\), \(\mu\) and \(\nu\), predicts \({a}_{1}=-0.73\times {10}^{-7} \text{m}\) and \({b}_{1}=-4.16\times {10}^{-7} \text{m}\). The close agreement between the theory and the measurement indicates that the present model describes well the dynamics of the magnetic suspension of the SG.
3.2 Case 2: Nonlinear response
Now we move on to the more general case where the third-order term of the potential is treated as finite. In this case, the linear transfer function \({\varphi }_{2}\), and therefore the relation (38), does not apply. The purpose here is to directly estimate the magnitude of the second harmonics of the sphere excited by a sinusoidal input. Knowledge of \({\varphi }_{3}\) is necessary to convert observed amplitude in voltage to physical quantities in terms of displacement.
Following equation (26), the function to be fit to the output signal \({V}_{out}\) is
\({V}_{out}={A}_{0}^{out}+{A}_{1}^{out}\text{cos}\omega t-{B}_{1}^{out}\text{sin}\omega t\)
\(+{A}_{2}^{out}\text{cos}2\omega t-{B}_{2}^{out}\text{sin}2\omega t-\frac{1}{C}{g}_{calc}\left(t\right)\)
|
(53)
|
where \({A}_{0}^{out}\), \({A}_{1}^{out}\), \({B}_{1}^{out}\), \({A}_{2}^{out}\) and \({B}_{2}^{out}\) are the free parameters to be adjusted. The parameter \(C\) is treated as known from the analysis of the linear case. Once these parameters are estimated, they are divided by the known parameter \({\varphi }_{3}\) to obtain \({a}_{0}\), \({a}_{1}\), \({b}_{1}\), \({a}_{2}\) and \({b}_{2}\) in the right-hand side of equation (22). In particular, the coefficients for the second harmonics are given by
\(\left\{\begin{array}{c}{a}_{2}=\frac{{A}_{2}^{out}}{{\varphi }_{3}}\\ {b}_{2}=\frac{{B}_{2}^{out}}{{\varphi }_{3}}\end{array}\right.\)
|
(54)
|
The coefficients \({A}_{2}^{out}\) and \({B}_{2}^{out}\) must be corrected for the response of the analog lowpass filter for the Gravity Balance channel at the angular frequency \(2\omega\).
It can be seen from equations (29) and (31) that the parameters \({a}_{0}\), \({a}_{2}\) and \({b}_{2}\) (in other words, \({A}_{0}^{out}\), \({A}_{2}^{out}\), and \({B}_{2}^{out}\)) contain information on the desired coefficient \({\beta }_{V}’\). However, estimating \({\beta }_{V}’\) based on the measured value of \({a}_{0}\) is not practical, because it may be seriously affected by possible DC offsets in the measurement of both input and output signals as well as contamination of tidal and other long-periodic geophysical signals. Instead, one can determine \({a}_{2}\) and \({b}_{2}\) in a more robust way, from which \({\beta }_{V}’\) can be estimated.
Let \({a}_{2}\pm \varDelta {a}_{2}\) and \({b}_{2}\pm \varDelta {b}_{2}\) be the estimates thus obtained (\(\varDelta {a}_{2}\) and \(\varDelta {b}_{2}\) are the uncertainties). The theoretical predictions of \({a}_{2}\) and \({b}_{2}\), given by equation (31), can be rewritten as
\({a}_{2}={\beta }_{V}’\stackrel{-}{A}\)
|
(55)
|
and |
\({b}_{2}={\beta }_{V}’\stackrel{-}{B}\)
|
(56)
|
where |
\(\stackrel{-}{A}=-\frac{1}{4}\frac{1}{{\left({\alpha }_{V}’-{4\omega }^{2}\right)}^{2}+16{\eta }^{2}{\omega }^{2}}\left[\left({\alpha }_{V}’-{4\omega }^{2}\right)\left({a}_{1}^{2}-{b}_{1}^{2}\right)+8\eta \omega {a}_{1}{b}_{1}\right]\)
|
(57)
|
and |
\(\stackrel{-}{B}=-\frac{1}{4}\frac{1}{{\left({\alpha }_{V}’-{4\omega }^{2}\right)}^{2}+16{\eta }^{2}{\omega }^{2}}\left[-4\eta \omega \left({a}_{1}^{2}-{b}_{1}^{2}\right)+\left({\alpha }_{V}’-{4\omega }^{2}\right)2{a}_{1}{b}_{1}\right]\)
|
(58)
|
Both \(\stackrel{-}{A}\) and \(\stackrel{-}{B}\) can be calculated from the parameters already obtained in the linear case. Therefore, we could estimate \({\beta }_{V}’\) in two ways by
\({\beta }_{V}’=\frac{{a}_{2}}{\stackrel{-}{A}}\)
|
(59)
|
for the cosine part and |
\({\beta }_{V}’=\frac{{b}_{2}}{\stackrel{-}{B}}\)
|
(60)
|
for the sine part. Considering that the relative estimation errors of \({a}_{2}\) and \({b}_{2}\) are much larger than those of the other parameters, the estimation error of \({\beta }_{V}’\), \(\varDelta {\beta }_{V}’\), for equations (65) and (66) may be obtained by
\(\varDelta {\beta }_{V}’=\frac{\varDelta {a}_{2}}{\stackrel{-}{A}}\)
|
(61)
|
and |
\(\varDelta {\beta }_{V}’=\frac{\varDelta {b}_{2}}{\stackrel{-}{B}}\)
|
(62)
|
respectively. However, estimation by equations (59) and (60) can be unstable because \({a}_{2}\) and \({b}_{2}\) depends on the initial phase of the applied oscillations. So, a more robust estimate of \({\beta }_{V}’\) can be obtained by taking a weighted mean of the two results as
\({\beta }_{V}’=\frac{\frac{\stackrel{-}{A}}{{\left(\varDelta {a}_{2}\right)}^{2}}{a}_{2}+\frac{\stackrel{-}{B}}{{\left(\varDelta {b}_{2}\right)}^{2}}{b}_{2}}{{\left(\frac{\stackrel{-}{A}}{\varDelta {a}_{2}}\right)}^{2}+{\left(\frac{\stackrel{-}{B}}{\varDelta {b}_{2}}\right)}^{2}}\)
|
(63)
|
with its uncertainty given by |
\(\varDelta {\beta }_{V}’=\sqrt{\frac{1}{{\left(\frac{\stackrel{-}{A}}{\varDelta {a}_{2}}\right)}^{2}+{\left(\frac{\stackrel{-}{B}}{\varDelta {b}_{2}}\right)}^{2}}}\)
|
(64)
|
Table 2 summarizes the results of fitting for frequencies \(f\) = 0.001, 0.002, 0.005 and 0.01 Hz. Note that for each frequency the magnitude of the \(2f\) terms (\({A}_{2}^{out}\) and \({B}_{2}^{out}\)) is smaller than those of the \(f\) terms (\({A}_{1}^{out}\) and \({B}_{1}^{out}\)) by three to four orders of magnitude. Here, \({AIC}_{1}\) and \({AIC}_{2}\) are the AIC (Akaike, 1972) for the models based on equations (44) and (53), respectively. For frequencies \(f\) = 0.001 and 0.002 Hz, \({AIC}_{2}\) is smaller than \({AIC}_{1}\), meaning that existence of the second harmonics in the measured data is statistically significant from the viewpoint of AIC. We shall exclude the frequencies \(f\) = 0.005 and 0.01 Hz from the following discussion, because \({AIC}_{2}\) is larger than \({AIC}_{1}\) for these frequencies.
For \(f\) = 0.001 and 0.002 Hz, we have obtained the estimates of \({\beta }_{V}’\) for the cosine and sine parts based on equations (59) and (60), as listed in Table 2. Note that all the four estimates of \({\beta }_{V}’\) are positive. Also, for each frequency, the estimates of \({\beta }_{V}’\) from the cosine and sine parts are marginally consistent with each other within the estimation error. Using equations (63) and (64), our final estimate is
\({\beta }_{V}’=2124\pm 386\)
|
(65)
|
for \(f\) = 0.001 Hz, and |
\({\beta }_{V}’=1902\pm 526\)
|
(66)
|
Table 2
Measurement results for the four lowest frequencies.
\(f\) (Hz)
|
\({AIC}_{1}\)
|
\({AIC}_{2}\)
|
\({A}_{1}^{out}\)
|
\({B}_{1}^{out}\)
|
\({A}_{2}^{out}\)
|
\({B}_{2}^{out}\)
|
\({\beta }_{V}’\)
Equation (59)
|
\({\beta }_{V}’\)
Equation (60)
|
0.001
|
–2862099.7
|
–2862128.1
|
–0.159234
± 0.000060
|
–1.034528
± 0.000065
|
0.000325
± 0.000061
|
–0.000115
± 0.000061
|
2473 ± 463
|
1327 ± 699
|
0.002
|
–2118572.1
|
–2118581.4
|
0.019691
± 0.000074
|
1.030543
± 0.000073
|
0.000237
± 0.000074
|
–0.000124
± 0.000074
|
2010 ± 625
|
1639 ± 978
|
0.005
|
–622784.2
|
–622782.2
|
–0.669290
± 0.000135
|
0.663900
± 0.000140
|
0.000166
± 0.000137
|
–0.000105
± 0.000138
|
|
|
0.01
|
–357834.4
|
–357833.6
|
0.529555
± 0.000178
|
–0.528527
± 0.000175
|
0.000095
± 0.000177
|
–0.000300
± 0.000176
|
|
|
for \(f\) = 0.002 Hz. These results from the two frequencies are also consistent with each other within the estimation error. Thus, we have obtained statistically significant results indicating that \({\beta }_{V}’\) is positive and its magnitude is around \(2\times {10}^{3}\) in the case of CT #036.