## 1. Introduction

[2] To observe secular land movements of the order of a few millimeters per year, a very precise instrument with long-term stability is required. This can be achieved using absolute gravimeters [*Williams et al.*, 2001]. Vertical land movements would modify the gravity at a rate of about −10 nm s^{−2} (1 μGal) for 5 mm of uplift [*Ekman and Mäkinen*, 1996]. The deformation rate is given by the slope of a linear trend fitted to the repeated time series.

[3] Usually, the regression estimator used is some form of least squares adjustment and the measurement errors are assumed to be normal (Gaussian) and statistically uncorrelated from one another (white noise). However, many geodetic data sets have now provided evidence for error sources that introduce large temporal correlations into the data [*Agnew*, 1992]. One common statistical model for many types of geophysical signal (which may contribute to the noise) may be described as a power law process [*Mandelbrot*, 1983; *Agnew*, 1992]. The stochastic process is such that its power spectrum has the form

where *f* is the spatial or temporal frequency, *P*_{0} and *f*_{0} are normalizing constants, and κ is the spectral index [*Mandelbrot and Van Ness*, 1968]. For κ = 0, we have classical white noise and for κ ≠ 0 we refer to colored noise. If −3 < κ < −1, we have “fractional Brownian motion”; if κ = −1 we have “flicker” or “pink” noise and, if κ = −2, we have random walk. When the random walk spectrum flattens toward low frequencies, we have the first-order Gauss Markov (FOGM) noise, such that the power spectrum has the form

where β is approximately equivalent to the cross over frequency and σ_{rw} is the scaling for the random walk part of the model.

[4] The power law process has been observed in geodetic time series such as continuously recording strain meters [*Agnew*, 1992], GPS [*Zhang et al.*, 1997; *Mao et al.*, 1999; *Williams et al.*, 2004] and sea level changes [*Harrison*, 2002]. Accounting for the type of noise is very important when estimating the related uncertainties but does not influence the slope estimate significantly [*Williams*, 2003].

[5] The spectrum of a time series of measurements can be modeled as the sum of a white noise and colored noise. The frequency at which the colored and white noises are equal is the crossover frequency. For an ordinary globally referenced GPS position time series, the length of data series required to detect the Random Walk noise in the power spectrum is about 8 years in the best cases [*Johnson and Agnew*, 2000]. We show here that this is reduced to less than 1 week combining SG and AG measurements.

[6] The expected power spectra of gravity measurements, based on previous studies of seismic noise, were discussed by *Lambert et al.* [1995]. In particular, they supposed power law noise toward low frequencies. They already pointed out the need to combine absolute gravity and superconducting gravity measurements to achieve the optimum noise characteristics of both instruments. With this aim in view they realized that more work had to be done in characterizing the spectra of actual AG and SG data. *Francis et al.* [1998] published a first noise estimate for the FG5 absolute gravimeter. Because of the lack of available data, the errors were assumed statistically uncorrelated (white noise process). This underestimated the noise level at frequencies lower than about 0.1–1 cycle per day (cpd). A spectral comparison of actual AG and SG time series was presented by *Crossley et al.* [2001]. Three years of real data were used to compare with the *Lambert et al.* [1995] theoretical spectrum and negligible differences were found for periods longer than 1 day. The power law noise and the AG instrumental white noise at periods shorter than one day were both obvious in the spectrum. However, neither the influence of the power law noise on the uncertainties of estimated deformation rates or the AG setup noise were discussed.

[7] The aim of this paper is to present a more detailed analysis of the noise affecting absolute gravity measurements. This is done by comparing superconducting gravimeter (SG) and AG data individually and by applying the statistical method presented by *Zhang et al.* [1997] and *Williams* [2003]. Because of mechanical wear, absolute gravimeters are not well suited for continuous measurements lasting longer than a few days or a few weeks. The SG provides continuous data between episodic AG measurements over many years and allows easy computation of the geophysical noise spectrum from 0.1 Hz (seismic surface waves and Earth's free oscillations) to less than 1 cycle per year (Chandler wobble) [*Crossley et al.*, 1999]. SG data, however, have the disadvantage of drifting with time, which can be evaluated and removed by performing regular side-by-side AG measurement [*Francis et al.*, 2004a]. This is done assuming the AG setup-dependent offsets consist of a Gaussian white noise that should not influence the measurement of the long-term geophysical trend. Then, comparing drift-free SG data with AG time series provides information on the geophysical noise affecting AG at low frequencies (except DC) and on the uncertainties due to the setup of the AG instrument.

[8] With an improved understanding of the noise, we estimate the ability of AG to monitor vertical crustal deformations. In particular, we provide more realistic uncertainties of the geophysical trend observed at the Membach station [*Francis et al.*, 2004a]. Moreover, using this experience, we evaluate the uncertainties that can be expected when carrying out repeated AG campaigns at other stations. This study can be very useful considering the repeated AG measurement campaigns undertaken since the 1990s to measure crustal deformation (intraplate and interplate tectonic deformation [*Zerbini et al.*, 2001; *Van Camp et al.*, 2002; *Hinderer et al.*, 2003], postglacial rebound [*Lambert et al.*, 2001; *Williams et al.*, 2001], anthropogenic subsidence [*Van Camp*, 2003a] or ice-mass and water-mass changes [*van Dam et al.*, 2000]). Finally we provide advice for measuring Absolute Gravity in a noisy environment.