Vibration error compensation algorithm in the development of the laser interference absolute gravimeter

Measurement error arising from vibration interference is recognized as the primary obstacle limiting the accuracy and stability of laser interference absolute gravimeters. The present work addresses this issue by proposing a global search optimization algorithm that determines the optimal absolute value of gravity based on the measured time-displacement coordinates of a falling body and 10 the signal obtained from the passive vibration isolation system of the inertial reference corner-cube in a laser interference absolute gravimeter. Results of numerical calculations conducted under vibration interference conditions with added white noise resulting in a signal-to-noise ratio of 40 dB demonstrate the following. (1) The accuracy and standard deviation of the gravimeter obtained using the proposed algorithm are −0.04 μGal (1 μGal = 1×10m/s2) and 0.24 μGal, respectively, while those values 15 obtained by the standard least-squares solution are 10.19 μGal and 154.11 μGal, respectively. (2) The resolution of the test results shows that the average response of the reference value of acceleration due to gravity superimposed by a disturbance of 1.00 μGal is 1.01 μGal using the proposed algorithm and 0.87 μGal using the standard least-squares solution.

obtained using the standard least-squares solution (LSS). The results demonstrate that the proposed algorithm provides a substantial anti-vibration capability and is worthy of applying within absolute gravimeter designs based on laser interferometry.

System analysis of the PVIS
The main component of a laser interference absolute gravimeter is a drop chamber under vacuum, inside which the test mass serving as a falling body (FB) is positioned within a cart mechanism that serves to raise, release, and catch the FB. There is no mechanical connection between the FB and the cart. The cart mechanism is driven by a servo motor installed on the outside of the drop chamber. After raising the FB 65 to its initial position, the cart is accelerated downward by the servo motor at an acceleration rate slightly greater than that due to gravity until the FB is freely falling. Then, the motion of the cart is matched with the free-falling motion of the FB, and the time-displacement coordinates of the FB relative to the reference corner-cube are measured. However, the random vibrations arising from the servo motor in the instrument and human activities within the measurement environment will be introduced during the 70 measurement process, which therefore couples vibration error with the calculation results for the absolute value of the acceleration due to gravity.
The PVIS used to reduce the influence of vibration interference on the inertial state of the reference corner-cube is illustrated schematically in Fig. 1, in which the reference corner-cube and the mass block of the PVIS assembly of mass M is connected with the ground by a spring of stiffness K and a damping 75 mechanism with a damping factor σ. Damping is produced by a mechanism employing a coil and magnetic steel structure that is commonly used in seismometers, where the value of σ is adjusted by varying the coil current. The displacement x(t) of the assembly relative to the ground is directly output by a precisely designed differential capacitance sensor and its corresponding circuitry. In addition, the absolute displacement of the ground is defined as y(t), and the absolute displacement of the assembly 80 with respect to the earth center of mass is defined as z(t). Finally, we note that the PVIS applies no closedloop feedback at any point. https://doi.org/10.5194/gi-2020-33 Preprint. Discussion started: 27 November 2020 c Author(s) 2020. CC BY 4.0 License.
( 2.3) The transfer function of the vibration isolation effect of the PVIS can be given as follows: where 2 = and the squared intrinsic frequency 0 2 = . The transfer function of the detection 95 effect of the PVIS can be given as follows: (2.5) Applying (2.5) in conjunction with (2.3) yields the following: ( ). (2.6) The attenuation amplitude versus frequency and phase-frequency characteristic curves of  1 ( ) and 100  2 ( ) are plotted in Fig. 2 for K = 50 N/mm, σ = 60, and M = 5 kg. It can be seen from the amplitude-https://doi.org/10.5194/gi-2020-33 Preprint. Discussion started: 27 November 2020 c Author(s) 2020. CC BY 4.0 License. frequency curve that the vibration isolation operation of the PVIS (i.e., Φ1(s)) attenuates signals in ( ) above the intrinsic frequency. The vibration isolation effect will become increasingly obvious as the frequency of the signals increases above the intrinsic frequency , but these high frequency signals after attenuation will still be coupled in the measurement system of the absolute gravimeter, and signals below 105 the intrinsic frequency will be directly coupled into the measurement system without attenuation. In terms of the signal detection operation of the PVIS (i.e., Φ2(s)), signals above the intrinsic frequency will be detected without any attenuation, but signals below the intrinsic frequency are attenuated. Because of the height restrictions of the drop chamber, the free-falling distance of the FB is controlled within 20 cm, and the free-falling period is about 0.2 s, which can be regarded as the maximum period of the vibration interference signal with a corresponding frequency of 5 Hz. If we consider that signals with amplitude attenuations greater than −60 dB are effectively attenuated, signals from approximately 115 0.05 Hz to the intrinsic frequency of the system will still exist in the output signal of the PVIS, although they will be attenuated. Meanwhile, signals below 0.05 Hz can be considered to vary linearly with respect to time during the measurement period of 0.2 s. Additionally, it can be seen from the phase-frequency curves in Fig. 2 that the phases of Φ1(s) and Φ2(s) maintain a consistent relationship. Therefore, the PVIS can be applied as both a vibration isolation device for the reference corner-cube and a detection 120 sensor of the vibration signal of the reference corner-cube simultaneously. Most of the useful signals related to the disturbance error of the absolute gravity measurement in Z(s) can be recovered by the synchronously output vibration signal X(s). https://doi.org/10.5194/gi-2020-33 Preprint. Discussion started: 27 November 2020 c Author(s) 2020. CC BY 4.0 License.

Mathematical model of the vibration error
In absolute gravimetry, the general method for treating a vertical gradient of the earth's surface is that 125 the vertical gravity gradient of the measurement point is first assumed to be 0, and then the absolute acceleration of the effective height of the instrument (Zumberge., 1981;Niebauer., et al, 1989;Timmen., 2003) is obtained using the absolute gravity measurement instrument. In addition, at least two sets of relative gravimeters are used to measure the vertical gravity gradient of the measurement point either before or after conducting the absolute gravity measurement. Finally, the absolute gravity measurement Here, 0 and 0 are the initial displacement and velocity of the FB, 0 is the absolute value of gravity to be determined from the measurement point, ( ) is the vibrational interference coupled to the measurement system, and e is additive white noise conforming to a normal distribution with a mean of 0 and a variance of 1. Here, ( ) is the signal of interest that must be uncoupled from the measurement. 140 For this purpose, we express the signal ( ) output from the PVIS as where , , and are the amplitude, angular frequency, and initial phase of the kth harmonic signal, respectively. In addition, the second term on the right is the trend component with as the coefficient, where is 0 = 1, 1 = , and 2 , while = 0, 1, 2. 145 Extracting ( ) from ( ) requires that we consider the following two important aspects.
①The voltage signal ( ) derived from the PVIS has not been converted into a displacement via sensitivity calibration. However, the calibration error is on the order of 10-7 m generally, and the error in the displacement measurement of a microgravity absolute gravimeter must be less than 10-11 m at least (Christian., 2004). Accordingly, the sensitivity calibration error is much greater than the allowable 150 error range of a microgravity absolute gravimeter. https://doi.org/10.5194/gi-2020-33 Preprint. Discussion started: 27 November 2020 c Author(s) 2020. CC BY 4.0 License. ② As described in Section 2.1, the vibration signal ( ) is not the absolute displacement of the reference corner-cube ( ), but is rather the result of the system transfer function, as described in equation (2.6), with which the low frequency signals are subject to attenuation.
These issues are addressed to extract ( ) from ( ) by introducing comprehensive coefficients 155 , , and influencing , , and to obtain the following (Akaike., 1980): .

Analysis of the vibration error compensation
Comparing ( In contrast, if relationship (2.17) is satisfied, the values of 0 , 0 , and can be obtained without any 180 vibration errors. In addition, the satisfaction of (2.17) can be considered as a correlation coefficient of 0 ( ) and ( ) obtaining a maximum value after de-averaging as follows: (2.18)

Algorithm design 185
The solution of the VECA is conducted based on a genetic algorithm. The computational flow of the VECA for the ith iteration is illustrated in Fig. 3. The relevant parameters employed in the simulation process are set as follows.
①Interference fringe signal simulation: the fringe signal ( ) can be simulated according to the mathematical form of a fringe signal given by Murata (Murata., 1978)  at a sampling frequency of 100 MHz. Therefore, the sampling interval is +1 − = 1 × 10 −8 s.
In addition, the absolute value of the acceleration due to gravity 0 is assumed to be 980,110,343.0 μGal in the simulation of the VECA. This term is used to generate the fringe signal required by the simulation of the VECA and is also used as a reference value to determine the accuracy of the results of the VECA and LSS calculations. 200 The anti-interference ability of the VECA is verified by replacing ( ) in (2.7) with the two following vibration interference models in turn:  (Svitlov., et al, 2014;Zumberge., et al, 2004]: (3.4) 215 ③Initialize the ergodic range of : first, an initial value of 2 is obtained by LSS with ( ) defined by (3.4). It is known that the vibration error range introduced by 2 2 will not exceed ±1000 μGal under the action of the PVIS in actual measurements [6]. Therefore, we ensure the rationality of the calculation results by setting the ergodic range of as 2 ± 5000.
④Set the parameters of the genetic algorithm: the maximum number of iterations is set as 150, the 220 number of variables is 1, the number of individuals is 80, the number of binary coding digits of the variable is 50, and the generation gap is 0.9.
⑤Design the objective function: the objective function is based on the correlation coefficients between 0 ( ) and ( ) given by (2.18), which is employed to complete the reinsertion, population restoration, and fitness allocation of the genetic algorithm.

Accuracy and standard deviation of the VECA results
The accuracy of the VECA is calculated as 230

Resolution testing
The resolution of the algorithm was tested by first calculating the theoretical values of the earth tide ∆ 245 at each minute = 1, 2, … ,1440 in Beijing on May 1, 2020 (Munk., et al, 1966). Then, ∆ is added to approximate the effect of ∆ on the absolute value of acceleration due to gravity in the regional gravity field at the same measurement period. The values of gi − g0 obtained at each minute by the VECA 250 and LSS are presented in Fig. 5(a)  superimposed by a disturbance of 1 μGal is 1.0085 μGal. In contrast, the system deviation of LSS is −99.0424 μGal, and the average response of the reference value of acceleration due to gravity is 0.8736 μGal, which therefore fails to reflect changes in the acceleration due to gravity under the influence of the earth tide. Accordingly, we can conclude that the proposed VECA significantly improves the resolution of the measurement results relative to the LSS. 260

Conclusion
Using the PVIS to suspend the reference corner-cube in a laser interference absolute gravimeter is a simple and feasible technology for reducing random vibration error in gravimetry measurements. A system analysis of the PVIS demonstrated the feasibility of applying the PVIS for both vibration isolation and signal detection simultaneously to improve the measurement accuracy of the absolute gravimeter. 265 Next, a reference corner-cube vibration interference error model was developed and applied to construct the newly proposed VECA for improving the accuracy, precision, and resolution of the measured value  https://doi.org/10.5194/gi-2020-33 Preprint. Discussion started: 27 November 2020 c Author(s) 2020. CC BY 4.0 License.