Free-fall Absolute Gravimeter Research Articles

High-accuracy determination of the beam divergence error in free-fall absolute gravimeters

We investigate the bias due to the beam divergence of the collimator in a free-fall absolute gravimeter of type FG5X. First, we measure the beam parameters with a Shack-Hartmann sensor. Then, we use the parameters to simulate the relative gravitational acceleration error of an FG5X gravimeter, which employs an unbalanced Mach-Zehnder laser interferometer. This investigation we do with four different commercial collimators, providing different divergence angles. We compare the results to real gravity measurements using the same collimators. The larger the divergence angle, and the bigger the relative length error, the bigger is the bias in the gravity measurements. A good agreement between theory and experiment is found, resulting in a relative bias of −2.77(24)⋅10−9 ( −2.72(24) μGal) for our standard collimator of type Thorlabs TC25APC, which is usually used for free-fall acceleration determinations. The outcome is also important for the realization of the SI unit kilogram via Kibble balance experiments that, on one side, employ laser interferometers for velocity measurements, and, on the other side, require accurate values of the gravitational acceleration. For example, if this divergence error is not corrected in the Kibble balance, then the mass determination would be biased by 2.77(24) μg kg−1 (numbers are valid only for our gravimeter with our collimator and fiber).

A Method for Diagnosing the Performance of the Dropping Mechanism in the High-Precision Absolute Gravimeter

The absolute gravimeters used in gravity measurements have wide applications across disciplines. The precision of a classical free-fall absolute gravimeter is limited by the performance of its dropping mechanism. The mismatch between the linear bearings and guideways can give rise to vibrations in measurement and lead to errors in the gravity results. To detect this problem, this research develops a diagnostic approach based on the current analysis of the motor that drives the dropping mechanism. Several dropping chambers are tested under gravity measurement within 12 hours to calculate the uncertainty of the mean and be tagged as qualified or unqualified. The status of the dropping mechanism is assessed by applying four parameters of the current data in the time and frequency domains during steady motion and feature coefficients of transient current under wavelet packet decomposition (WPD). The proper ranges of the parameters are identified through comparative experiments. Empirical evidence confirms the feasibility of the parameters and coefficients used to determine whether a gravimeter dropping mechanism meets the requirement of precision around 1 micro Gal. In addition, the analysis based on continuous wavelet transform (CWT) proves its potential to locate the faults in the unqualified dropping mechanism. The proposed diagnostic approach can be applied within 20 times of motor current acquisition, which makes it practicable and helpful in improving and achieving high-precision gravity measurement.

Correction of vibration for classical free-fall gravimeters with correlation-analysis

In a free-fall absolute gravimeter, a laser interferometer is used to track the falling retro-reflector. To buffer the reference retro-reflector from seismic noise, a low-frequency vertical vibration isolator is traditionally used. However, an isolation device is usually complicated and expensive. A strap-down system using a seismometer to record the vibration and correct the measurement resolves the issue, but the actual recorded vibration cannot be directly used because of signal transfer delay and amplitude attenuation. Nevertheless, by quadratically fitting the trajectory of the falling retro-reflector and the motion of the reference retro-reflector, we find that their residuals are significantly correlated. Moreover, the transfer delay and the amplitude attenuation can be calculated using correlation analysis. With this capability, a vibration correction method for absolute gravimeters is proposed and demonstrated. The transfer delay and the gain attenuation are determined from data of only 25 drops, and can be used to correct subsequent measurements. The method is also applied in the T-1 absolute gravimeter. The standard deviation of the measurement results is improved by a factor of 20 after correction in a noisy environment, and improved by a factor of 5 in a quiet environment. Compared with vibration isolators, the strap-down system using this correction method is much more compact, enabling its use in field conditions or even dynamic environments not suitable for vibration isolators.

A general relativistic model for free-fall absolute gravimeters

Although the relativistic manifestations of gravitational fields in gravimetry were first studied 40 years ago, the relativistic effects combined with free-fall absolute gravimeters have rarely been considered. In light of this, we present a general relativistic model for free-fall absolute gravimeters in a local-Fermi coordinates system, where we focus on effects related to the measuring devices: relativistic transverse Doppler effects, gravitational redshift effects and Earth’s rotation effects. Based on this model, a general relativistic expression of the measured gravity acceleration is obtained.

Experimental assessment of the speed of light perturbation in free-fall absolute gravimeters

Precision absolute gravity measurements are growing in importance, especially in the context of the new definition of the kilogram. For the case of free fall absolute gravimeters with a Michelson-type interferometer tracking the position of a free falling body, one of the effects that needs to be taken into account is the ‘speed of light perturbation’ due to the finite speed of propagation of light. This effect has been extensively discussed in the past, and there is at present a disagreement between different studies. In this work, we present the analysis of new data and confirm the result expected from the theoretical analysis applied nowadays in free-fall gravimeters. We also review the standard derivations of this effect (by using phase shift or Doppler effect arguments) and show their equivalence.

Absolute ballistic gravimeter ABG-VNIIM-1 by D.I. Mendeleyev Research Institute for Metrology

A new free-fall absolute ballistic gravimeter ABG-VNIIM-1 was fabricated at D.I. Mendeleyev Research Institute for Metrology (VNIIM). The gravimeter ABG-VNIIM-1 was tested at the gravimetric site Lomonosov-1 in Lomonosov branch of VNIIM. The estimated total instrumental uncertainty of ABG VNIIM-1 was determined to be 2 × 10−8 m s−2. Typical residuals in the least square evaluation of the trajectory of the test body in a single drop at the site Lomonosov-1 are from 0.3 to 0.8 nm.

Measurement of the speed-of-light perturbation of free-fall absolute gravimeters

We report on a direct measurement of the relativistic Doppler shift with a commercial free-fall absolute gravimeter of the type FG5X. The observed Doppler shift, which is commonly called speed-of-light perturbation, can be well described by the relativistic Doppler formula, where the constant object velocity is replaced by a time-dependent velocity with constant acceleration. The observed speed-of-light perturbation stands in contrast to other publications, which predict a higher frequency shift. It has been measured with a relative uncertainty of 1.1 × 10−3.

Relativity, Doppler shifts and retarded times in deriving the correction for the finite speed of light: a comment on ‘Second-order Doppler-shift corrections in free-fall absolute gravimeters’

In the paper by Rothleitner and Francis (2011 Metrologia 48 187–95) the correction due to the finite speed of light in absolute gravimeters is analysed from the viewpoint of special relativity. The relativistic concepts eventually lead to the two classical approaches to the problem: analysis of the beat frequency, and introduction of the retarded times. In the first approach, an additional time delay has to be assumed, because the frequency of the beam bounced from the accelerated reflector differs at the point of reflection from that at the point of interference. The retarded times formalism is equivalent to a single Doppler shift, but results in the same correction as the beat frequency approach, even though the latter explicitly combines two Doppler shifts. In our comments we discuss these and other problems we found with the suggested treatment of the correction.

Reply to ‘Comment on second-order Doppler-shift corrections in free-fall absolute gravimeters’

In reply to the comment by Nagornyi et al we show that the introduction of a time delay after the application of a relativistic Doppler shift, as proposed by the authors, is equivalent to an introduction of a time dilation into the reference beam. This would partly cancel the effect of the Doppler shifts. We conclude that the introduction of such an additional time delay is incorrect.

Comment on ‘Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter’

Comment on ‘Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter’

Reply to the Comment on ‘Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter’

Reply to the Comment on ‘Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter’

Second-order Doppler-shift corrections in free-fall absolute gravimeters

In a free-fall absolute gravimeter usually a Michelson type interferometer is employed to track the trajectory of a freely falling retroreflector. The accelerated motion of the retroreflector produces a Doppler-shift in the laser wavelength. From the interference signal (beat signal) of the Doppler-shifted and the reference electromagnetic waves the relative motion of the freely falling retroreflector with respect to an inertial reference retroreflector is reconstructed. Considerations of second-order Doppler-shift terms lead to a correction in the acceleration due to gravity of several microgals (1 µGal = 10 nm s−2). This correction is commonly called speed of light correction. To date different correction formulae have been proposed, which differ by several microgals.In this paper we review several previous publications and show the reasons for the different results.

Comparison of three digital fringe signal processing methods in a ballistic free-fall absolute gravimeter

This paper reports results of comparison of three digital fringe signal processing methods implemented in the same free-fall absolute gravimeter. A two-sample zero-crossing method, a windowed second-difference method and a method of non-linear least-squares adjustment on the undersampled fringe signal are compared in numerical simulations, hardware tests and actual measurements with the MPG-2 absolute gravimeter, developed at the Max Planck Institute for the Science of Light, Germany. The two-sample zero-crossing method realizes data location schemes that are both equally spaced in distance and equally spaced in time (EST) along the free-fall trajectory. The windowed second-difference method and the method of non-linear least-squares adjustment with complex heterodyne demodulation operate with the EST data. Results of the comparison verify an agreement of the three methods within one part in 109 of the measured gravity value, provided a common data location scheme is considered.

Development of new free-fall absolute gravimeters

The design and first results of two free-fall absolute gravimeters are reported: a stationary gravimeter is designed and can be used as a reference system and a portable gravimeter is aimed at field measurements.The determination of the acceleration due to gravity is done interferometrically in both instruments. The whole fringe signal is digitized by a high-speed analogue-to-digital converter, which is locked to a rubidium frequency standard. This fringe recording and processing is novel as compared with commercial free-fall gravimeters, which use an electronic zero-crossing discrimination. Advantages such as the application of a zero-phase-shifting digital filter to the digitized data are depicted. The portable gravimeter's mechanics deviate from the conventional type. Springs are used to accelerate and decelerate the carriage supporting the falling object.A detailed uncertainty budget is given for both gravimeters. The combined standard uncertainty for the portable and for the stationary gravimeter is estimated at 38.8 µGal and 16.6 µGal, respectively. The corresponding statistical uncertainties are 1.6 µGal (over one day of measurement) and 0.6 µGal (over one month of measurement).The different designs and dimensions of the new free-fall gravimeters can help to reveal unknown or so far underestimated systematic effects. The assessments of the uncertainties due to seismic noise and shock vibrations, and electronic phase shifts give validity to this assumption.

Precise definition of the effective measurement height of free-fall absolute gravimeters

For up-to-date absolute gravimeters, the trajectory of the test mass during a free-fall experiment (drop) is about 20 cm along the vertical, and the corresponding gravity change is about 60×10−8 m s−2. The reference height of the derived free-fall acceleration g has to be defined with an accuracy of 1 mm to 2 mm within the dropping distance to preserve the accuracy of the measurement system (e.g. FG5: 1×10−8 m s−2 to 2×10−8 m s−2). The equation of motion comprises a vertical gravity gradient to take the height dependence of g into account. In general, a linear vertical gravity gradient γ is introduced that has been measured by relative gravimeters. In that case, the g-value refers to the origin of the coordinate system (z = 0), which is normally the starting position of the drop.In the case of an unknown or uncertain gradient we recommend an alternative approach. A simple parabolic equation (assumption γ = 0) can be used to evaluate the time/distance data pairs, and later these g-determinations have to be corrected for the vertical gravity gradient using the effective measurement height. The solution presented is not restricted to low initial velocities. It considers time/distance measurements equally spaced in distance. Also, in extreme cases, unknown non-linearities within the vertical gravity gradient do not significantly affect the result of the absolute gravity determination.

Using absolute gravimeter data to determine vertical gravity gradients

The position versus time data from a free-fall absolute gravimeter can be used to estimate the vertical gravity gradient in addition to the gravity value itself. Hipkin has reported success in estimating the vertical gradient value using a data set of unusually good quality. This paper explores techniques that may be applicable to a broader class of data that may be contaminated with "system response" errors of larger magnitude than were evident in the data used by Hipkin. This system response function is usually modelled as a sum of exponentially decaying sinusoidal components. The technique employed here involves combining the x0, v0 and g from all the drops made during a site occupation into a single least-squares solution, and including the value of the vertical gradient and the coefficients of system response function in the same solution. The resulting non-linear equations must be solved iteratively and convergence presents some difficulties. Sparse matrix techniques are used to make the least-squares problem computationally tractable.