We use Superconducting Gravimeter (SG) and Global Positioning System (GPS) measurements from Ny-Ålesund, Svalbard, to infer changes in ice mass loss between September 1999 and September 2010. We find that during this period, the gravity rate and vertical crustal velocities are changing with time, adding to evidence about varying rates of ice mass loss. The gravity rate varies through 10 years of observation; −0.23 μGal/yr in 2000–2002, −3.22 μGal/yr in 2002–2005 and −1.10 μGal/yr in 2005–2010. The gravity changes agree well with the observed uplift rates measured by GPS, which are 4.4, 11.3 and 7.4 mm/yr, over the same periods. In addition, we generate model predictions which account for past and present-day ice mass variation. We find that the models under predict both the observed uplift rates and gravity changes.
 Ongoing vertical land motion across the Svalbard archipelago is attributable to ice mass variation from the Last Glacial Maximum up until the present day. Past studies [Sato et al., 2006a; Kierulf et al., 2009a; Altamimi et al., 2007], have shown that geodetic observations from Ny-Ålesund can largely be explained by a combination of the viscoelastic response of the Earth to past ice mass variation and the instantaneous elastic response driven by contemporary ice mass loss. However, the geophysical models tend to underpredict the signal. A general thinning of the ice masses due to Present Day Ice Melt (PDIM) during the the last 40 years has been observed [Nuth et al., 2010, 2007]. Liestøl  indicates that the glaciers have a general retreat since 1869. During the last 15 years [Kohler et al., 2007; Kääb, 2008] an acceleration in ice mass loss has been observed. Laser altimetry observations from ICESat show a decrease in the ice mass loss during the later years [Moholdt et al., 2010].
 In this paper we present an analysis of both geometric and gravity measurements, spanning 11 years of data, between September 1999 and September 2010. We analyze the trend and rate of change of both GPS and SG data from Ny-Ålesund, Svalbard, using different regression models and conclude on the best representation of the available observations. We discuss our results and compare them to the uplift and gravity rates estimated from the viscoelastic response of earlier ice mass variation (Glacial Isostatic Adjustment (GIA)) and the elastic response of PDIM.
 Accurate knowledge of GIA, in-situ mass balance and gravity changes are important since it allows decontamination of the land uplift signal from the changes in the gravity field observed from satellite missions such as Gravity Recovery and Climate Experiment (GRACE) and Gravity field and steady-state Ocean Circulation Explorer (GOCE).
2.1. Superconducting Gravimeter Measurements
 We use gravity measurements from the SG C039 located at Ny-Ålesund, Svalbard, Norway, to estimate gravity change. We have approximately 347 million gravity measurements spanning 11 years, from September 1999 to September 2010. The original gravity measurements have a spacing of 1 second.
 The 1 second gravity and co-located air pressure measurements are resampled every 1 minute using a symmetric numerical Finite Impulse Response (FIR) zero phase low-pass filter with cut off at 120 seconds [Wenzel, 1996]. A symmetric FIR filter of order N is a weighted sum of the current and N+1 previous and subsequent values of the input, where N = 360 in our study. The 1 minute gravity data was cleaned by substituting outliers and data affected by earthquakes with a synthetic signal based on Tamura's  tide model. We corrected the one minute gravity data for the effect of air pressure by using the value obtained by Sato et al. [2006b] of −0.422 ± 0.004 μGal/hPa.
 During Liquid Helium (LHe) refilling of the SG, where LHe is used to maintain the superconductivity of the instrument, we observe large and abrupt changes in the measured gravity value. We correct for this as is done in similar studies [Sato et al., 2006b; Mémin et al., 2011], by adjusting the gravity values so that they match before and after filling of LHe. Note that, in 2002, the computer failed and a replacement was installed two months later. We have therefore assumed no change in gravity during this period of autumn 2002.
 The SG is known to have an instrumental drift [Van Camp and Francis, 2007] of as much as 2–4 μGal a year. This drift may be estimated by comparing SG data with Absolute Gravimeter (AG) data. We estimated the linear drift (using unweighted least squares) to be −2.37 ± 0.32 μGal/yr based on 6 comparisons performed after the SG was installed (2000, 2001, 2002, 2004, 2007 and 2010). An AG measurement from 1998 exists, but this was before measurements with SG began. In addition the gravity building was not completed in 1998, i.e., no roof and walls, adding more uncertainties to this value.
 The gravity data was then resampled again, but to every 1 hour using a symmetric FIR zero-phase filter, and then resampled one further time to obtain daily values using a filter with uniform weights, i.e., a flat filter.
 There are several permanent GPS recorders at Ny-Ålesund. Two of them, NYAL and NYA1, are operated by the Norwegian Mapping Authority and contribute to the International GNSS Service (IGS) network. We opt to use only the data from NYA1 as this station has a slightly better quality performance [Kierulf et al., 2009a].
 The data was analyzed with the GPS Inferred Positioning System/Orbit Analysis and Simulation Software (GIPSY/OASIS-II) in Precise Point Positioning (PPP) mode [Zumberge et al., 1997]. Recomputed fiducial free orbits and clock products come from Jet Propulsion Laboratory (JPL). We use an elevation cut off angle of 10°, the Vienna Mapping Function (VMF1) [Boehm et al., 2006] and ocean loading coefficients [Scherneck, 1991] using FES2004. Absolute phase center variations were used [Schmid et al., 2007]. The results are realized in the ITRF2005 reference frame [Altamimi et al., 2007].
Kierulf et al. [2009a, 2009b] found systematic differences between the GPS solution using the GPS Analysis Software of MIT (GAMIT) and GIPSY/OASIS-II. The secular uplift rate derived from the GIPSY/OASIS-II solution was 2–3 mm/yr larger than the uplift rate from the GAMIT solution. It was also shown that the GAMIT results showed good agreement with observations from the Very Long Baseline Interferometry (VLBI) antenna co-located at Ny-Ålesund. The GIPSY/OASIS-II uplift results presented in this paper are in accordance with both the GAMIT and VLBI results.
 We believe that the differences between our GIPSY/OASIS-II solutions and those from the earlier studies [Kierulf et al., 2009a, 2009b] is likely due to the use of absolute rather than, before, relative antenna phase center variation. At high latitudes where we have no satellites near zenith, poorly modeled satellite antenna phase center may give a bias, and changes in satellite constellation may lead to a rate bias.
3. Time-Series Analysis
 The SG and GPS data time-series are analyzed using a least squares-fit solution. In the below, we describe several different methods we used for the regression analysis of the data. We discriminate between the quality of these different methods by applying an adjusted R2 and F-test.
 As our first option, we use the same method as those of Sato et al. [2006b] and Kierulf et al. [2009b], by fitting our data to a linear trend assuming a white noise model. For the period 16 June 2000 to 15 June 2010, we obtain secular rates of −1.77 μGal/yr for gravity (SG) and 8.5 mm/yr for uplift (GPS). The uncertainty is estimated as standard error of regression assuming a white noise model and is ±0.013 μGal/yr for gravity and ±0.036 mm/yr for uplift. From previous GPS analyses by Williams et al. , it is known that the white noise assumption is unrealistic due to correlation, and they recommend a combination of white noise and flicker noise as appropriate for most GPS sites. White and flicker noise has a power spectral density of the form S(f) ∝ 1/fα, where f is frequency and α = 0 and 1, respectively. If we include the flicker noise we get an 1 σ uncertainty of ±0.58 mm/yr, a factor 16 higher than the white noise model only (CATS software [Williams, 2003]). Using the same noise model for the gravity data we get an 1 σ uncertainty of ±0.12 μGal/yr, a factor of 9 higher than white noise model only. It is worth noting that the CATS software indicates no significant presence of white noise in the gravity time-series of daily values. The effect of noise on the AG and the SG is discussed by Van Camp et al. .
 A visual inspection of both the uplift and gravity time-series indicate a change in trend in 2002 and 2005, as well a clear annual signal (see Figures 1 and 2). Hence, our second option is to fit the data to a piecewise linear trend. The annual signal is captured using four harmonic constituents. We have computed the piecewise linear trend for the three periods 16 June 2000 to 15 June 2002, 16 June 2002 to 15 June 2005 and 16 June 2005 to 15 June 2010. The measured uplift, meas, and the measured gravity changes, meas, are given in Table 2. The uncertainties in Table 2 are based on a white noise model. Including flicker noise in the noise model increases the uncertainties by a factor of 5 to 10. The factor is largest for the longest periods.
Table 2. Modeled (GIA and PDIM), Measured and Unmodeled (Measured - GIA - PDIM) Values of Piecewise Linear Trends for Gravity and Uplifta
The / ratio is given for the unmodelled, GIA and PDIM parts of the signal. The linear trend for the whole period is given in the last line.
 To determine which model best fits the gravity and uplift data, we calculate the adjusted R-Square (Radj2) [see, e.g., Draper and Smith, 1998], given as Radj2 = 1 − (1 − R2)*(n − 1)/(n − p − 1). Where p is the number of regressors in the model (not counting the constant term), and n is the number of observations. R2 is given as 1 − ∑n(yn − n)2/∑n(yn − )2, where yn, and n is the observed, mean and modeled value, respectively. R2 is a statistic that will give some information about how well the model provides fit to the data. Unlike R2, the adjusted R2 increases only if the new term improves the model more than would be expected by chance.
 From the adjusted R2 results we find that the piecewise linear model gives improved fit when compared to the linear trend (regardless of whether we use the 4 harmonic constituents or not). For the uplift, we find an increase from 0.940 for the linear model to 0.947 for the piece-wise linear model (or an improvement of 0.68%) and, for the gravity, an increase from 0.848 to 0.878 (an improvement of 3.54%). We also found that adding 4 harmonic constituents gave an improvement both for uplift and gravity.
 To verify that the model using a piecewise trend and 4 constituents is a significant improvement compared to the other options we conduct an F-test [Larsen and Marx, 1998]. The F-test shows that, for both the uplift and gravity data, the improvement obtained is statistically significant at a very high confidence level (99.95%). This indicates that the changes of slope in 2002 and 2005 are realistic.
 The adjusted R2 and the F-test assume uncorrelated observations. As pointed out above geodetic time series suffer from correlation. Pending on the decorrelation time, averaged data may reduce or remove the presence of correlation. To reduce the influence of correlation we have repeated the Radj2 calculation and F-test using monthly, quarterly and yearly averaged data. For the monthly data 0, 1, 2 and 4 harmonic constituents were tested. Due to the reduced number of data points, only the 0 and 1 harmonic constituents were applied for the quarterly data. For yearly data no harmonic constituents were included. In all cases the Radj2 increased when we change from a linear trend to a piecewise linear trend. The improvement was between 0.39% and 0.79% for uplift and between 3.47% and 5.68% for gravity. The F-test showed that the improvement was significant, at confidence level of 99.5% in all cases.
 As an independent test to locate the best-fit model, and in addition to the Radj2, we have used the Akaike Information Criterion (AIC) [Akaike, 1974] and AIC corrected [Hurvich and Tsai, 1989]. In all, but one case, both the AIC and AIC corrected shows that a piecewise linear trend is better than a linear trend. For the yearly data, the AIC corrected indicates that a linear trend is better.
4. Ice Models and Uplift Predictions
Kierulf et al. [2009b] estimate uplift at Ny-Ålesund due to PDIM as 3.2 mm/yr for the period 1993–2008. The modeled uplift was calculated using the approach of Farrell , with an elastic and spherically symmetric Earth model based on the Gutenberg-Bullen A. A calculation using an Earth model based on the Preliminary Reference Earth Model (PREM) [Dziewonski and Anderson, 1981] gave a similar results and indicates that the results are not very dependent on the adopted Earth model. Kierulf et al. [2009b] used a detailed ice and mass balance model for Svalbard which takes the spatial pattern of present day ice mass loss into account as forcing for the elastic uplift modelling. This model correspond to a mean ice mass loss of 0.37 m water equivalent (mweq)/yr across Svalbard. The ratio between uplift and PDIM change was reported to be 8.7 mm/mweq [Kierulf et al., 2009b], this is the equivalent of 0.238 mm/Gt. This implicity assumes the pattern of ice mass changes to be proportional for all time spans. Kierulf et al. [2009b] showed a very good agreement between measured and modeled uplift rates (correlation coefficient 0.8). This indicates that the reported ratio is a good estimate of the relation between uplift and ice mass changes also for yearly results. Kierulf et al. [2009b] estimated the modeled uplift rates by using the reported ratio of 8.7 mm/mweq multiplied with mass balance values of nearby glaciers obtained by the Norwegian Polar Institute [Kohler et al., 2007]. We use the same approach here to compute uplift values for the different time periods. For the time span 16 June 2000 to 15 June 2010 we predict uplift of 3.1 mm/yr due to PDIM, the corresponding ice mass loss is 13.0 Gt/yr. The uplift rates for the other periods are included in Table 2.
 An observed uplift of 8.5 mm/yr at Ny-Ålesund is much larger than expected from models of GIA. A mean of existing GIA model predictions suggests uplift at Ny-Ålesund of 1.6 ± 0.3 mm/yr [Kierulf et al., 2009b]. However, note that this value only include deglaciation in the last ice age, and that Holocene ice mass changes are not included. Modeled uplift from PDIM is 3.1 mm/yr at Ny-Ålesund and, thus, the total contribution from both GIA and PDIM variation is 4.7 mm/yr. When compared to the observed value, the models therefore under predict uplift by ∼3.8 mm/yr.
 Gravity changes are an independent measure of land uplift. However, gravity also depends on the mass redistribution. For instance, the mass redistribution of the Earth interior is related to PDIM change and GIA differently. In addition, PDIM variation also affects the direct Newtonian attraction from the nearby glaciers. Wahr et al.  developed a method to discriminate GIA from PDIM when both the gravity change signal and the geometrical uplift was known. They obtained a proportional factor of 6.5 between the viscous portion of uplift rate and gravity anomaly. Adding the free-air contribution (−0.308 μGal/mm) to the inverse of 6.5 gives a ratio between gravity change and uplift of / = −0.15 μGal/mm, for a viscoelastic process like GIA. Whereas, de Linage et al.  derived a theoretical ratio between gravity and uplift of / = −0.26 μGal/mm for elastic deformation, when the station is outside the loading mass. Sato et al. [2006a] found the same ratio for the response of PDIM at Ny-Ålesund. More recently, Mémin et al.  showed that using more realistic and detailed elevation and ice location data, the ratio changed significantly. In the rest of our discussion, we use the value by Sato et al. [2006a] and de Linage et al. . Using this value and the ratio between uplift and ice mass loss derived in Section 4 (0.238 mm/Gt), we find a value of −0.062 μGal/Gt between gravity change in Ny-Ålesund and ice mass loss across Svalbard.
 Using the ratio of Wahr et al. , modeled uplift from GIA of 1.6 mm/yr corresponds to a gravity change of −0.24 μGal/yr. For the modeled PDIM variation, we predicted 3.1 mm/yr uplift which, using the ratio of Sato et al. , corresponds to a gravity change of −0.81 μGal/yr. In total, the modeled gravity change is −1.05 μGal/yr and the difference between observed and modeled gravity change is −0.72 μGal/yr.
 The ratio between unmodeled (observed minus modeled values) gravity changes and unmodeled uplift is −0.19 μGal/mm for the period 2000–2010. This is close to the expected value of −0.15 μGal/mm for a viscoelastic effect like GIA. Suggesting that the source of the unmodeled uplift and gravity change are to be found in the GIA models, either in the ice history or the local Earth structure.
 As mentioned above, the GIA models used in this analysis do not include ice mass variations during the Holocene period (∼10 ka). Geomorphological observations from the Ny-Ålesund area suggest that glaciers retreated since 1869 [Liestøl, 1988] and so this may contribute to the uplift. The work of Ivins and James  has shown that present-day uplift rates are potentially very sensitive to late Holocene ice mass changes. Studies from Patagonia [Dietrich et al., 2010] and Alaska [Larsen et al., 2005] also show a possible increased uplift due to viscoelastic response for late Holocene ice mass reduction.
 The discrepancies may also have other reasons, underestimated the mass loss from the surrounding glaciers, mismodelled spatial pattern of ice mass changes, unknown tectonic motions and/or a possible drift in the geodetic reference frame. We address these issues below.
 The deglaciation of the surrounding glaciers may be larger than the model we are using. However, a recent study of Moholdt et al.  using ICESat laser altimetry does not indicate this. Missmodelled spatial pattern of the ice mass changes, especially the Newtonian gravitational effect due to the topography of the glaciers close to Ny-Ålesund, may influence the results. Neither can we eliminate the possibility of some neotectonic activity. Ny-Ålesund is located only 150 km away from the Mid-Atlantic ridge and, in the past, the area has undergone significant tectonic activity [Birkenmajer, 1981; Blythe and Kleinspehn, 1998]. As oppose to the gravity measurements, the uplift results depend on a reference frame. Several studies which have compared the reference frame ITRF2005 with geophysical models [Argus, 2007] and gravity measurements [Teferle et al., 2009] have argued that the older reference frame ITRF2000 is in better agreement with the geophysical evidence. The geocenter Z drift of the reference frame has been especially questioned. In Ny-Ålesund, the difference between ITRF2005 and ITRF2000 correspond to a difference in uplift of 1.3 mm/yr.
 In Table 2 the modeled and measured uplift rates and gravity trends for the complete time-series as well as the individual periods are shown. As the table illustrates we observes variations in uplift and gravity rate. This indicates changes in PDIM and this is verified by ice mass balance observations by Kohler et al. . We see that the uplift data fit very well, while the gravity also fits quite well, however larger deviations are observed. Local mass changes of hydrological origin will especially affect the gravity signal and, for shorter time scales, this may lead to large deviations. The 2000–2002 gravity difference is probably also affected by the large drift usually occurring during the first months after start up of a new SG instrument.
 A seasonal signal is clearly visible in the gravity data. This is mainly due to loading effects and Newtonian gravitational effect of local mass changes of hydrological origin, such as rain, snow, and changes in groundwater level.
 We showed that a detailed analysis of the GPS and SG measurements at Ny-Ålesund, Svalbard, reveals variations in gravity and uplift. We have verified that the change of rate in 2002 and 2005 are significant by using a F-test. This indicates variations in present day ice melt rate at Svalbard. The modeled uplift and gravity change, based on in-situ PDIM measurements, given in Table 2, is consistent with measured values. An increase in ice mass loss rate is present until 2005, while there is a decrease after 2005. This demonstrate that geodetic techniques are able to recapture changes in PDIM.
 The presences of non-linear features in the geodetic time-series from Ny-Alesund make it necessary to use consistent time-periods when you make geophysical interpretations. Time-series from different time-span will contain different geophysical signals and are consequently not comparable.
 We have also showed that the geophysical GIA/PDIM models under predict the observed changes in uplift and gravity change. The differences are possible due to unmodeled changes in ice mass loss since 1869.
 Continuous observations of uplift and gravity change will be crucial for constraining geophysical GIA/PDIM models, and may give useful proxy for regional glacier mass balance. Continuous observations are also important as they function as in-situ measurements for calibration of GRACE/GOCE type of satellite missions.
 We thank J. Hinderer, A. Memin (EOST), O. Francis (ECGS) and R. Falk (BKG) for providing AG data, and J. Kohler for providing updated mass balance values. We thank T. Sato and Y. Tamura, Japan, for all the effort they have put into the SG at Ny-Ålesund.
 The Editor thanks two anonymous reviewers for their assistance in evaluating this paper.