Time series of quality-screened, calibrated and corrected measurements are provided in the Swarm Level 1b products. The Swarm Satellite Constellation Application and Research Facility (SCARF, ) further processes Level 1b data and auxiliary data to Level 2 products. Here we use Level 2 kinematic orbits (see Table ) and Level 1b star camera data, which are required for transforming from the terrestrial to satellite reference frame. During the processing, the satellite reference frame needs to be referred to the inertial frame, which is achieved by multiplying the rotation matrix derived from the star camera data with the Earth rotation matrix .

For modeling non-conservative forces, we implemented a Swarm macro model consisting of area, orientation and surface material for 15 panels, supplemented with surface properties such as diffuse and specular reflectivity (ESA, Christian Siemes, personal communication, 2017) for computing solar radiation pressure and Earth radiation pressure consisting of measured albedo and emission.

During gravity field recovery, we used the GOCO05c model up to degree 360 as a mean background field. All time-variable background models (cf. Table ) are consistent with GRACE RL05 processing standards except for the atmospheric tides, which were chosen as such to be aligned with the Graz ITSG-Grace2016 solutions. The reason for this is that we compare our Swarm solutions to the monthly ITSG-Grace2016 solutions .

Drag modeling requires knowing the thermospheric density and temperature. In this work, we make use of the empirical NRLMSISE-00 model . NRLMSISE-00's database includes total mass density from satellite accelerometers and POD, temperature from incoherent scatter radar, and molecular oxygen number density collected under different solar activity conditions. The model is driven by the observed solar flux (F10.7 index) and geomagnetic index (AP). In we compare NRLMSISE-00 to GRACE-derived thermospheric density and derive an empirical correction for this model; this has not yet been applied here.

While all three Swarm satellites carry accelerometers intended to support POD and the study of the thermosphere, these data have unfortunately turned out as severely affected by sudden bias changes (“steps”) and temperature-induced bias variations.

developed a method to clean and calibrate the along-track acceleration of Swarm C. However, Swarm A and B (the former to a lesser extent than the latter), as well as the other C directions, are affected by serious issues and it is not clear whether these data can be used in gravity field applications. In the light of recent improvements of empirical thermosphere models and seeing that we require all three components of non-conservative acceleration amodel for gravity recovery, we decided instead to model them, using the well-known relation amodel=adrag+aSRP+aERP. amodel is the sum of atmospheric drag adrag, solar radiation pressure aSRP, and Earth radiation aERP. We will briefly summarize our implementation below.

For gravity field estimation, we use the integral equation approach . Kinematic orbits are partitioned into short arcs and each 3-D position r(τ) between the arc's beginning and end (rA and rB) can be expressed as r(τ)=rA(1-τ)+rBτ-T2∫01K(τ,τ′)f(τ′)dτ′, with normalized time τ and the integral kernel, as in K(τ,τ′)=τ′(1-τ)for τ′≤ττ(1-τ′)for τ′>τ. In other words, T2∫01K(τ,τ′)f(τ′)dτ′ in Eq. () represents the offset of the current position from a straight line connecting rA and rB, caused by gravitational and non-gravitational forces f(τ′). After discretization (sampling rate of kinematic orbits is 1 s after July 2014), one can write the above as an adjustment problem with two groups of solved-for parameters with the equation y=rArBaccperArcand x=c20c21s21⋮snnaccglobal. y contains all arc-related parameters, which can be eliminated from the normal equation system during the estimation. These include start and end position of each arc and additional parameters such as accelerometer bias or scale factors. The gravity field parameters are then collected in x. For more details of the integral equation approach, see and .

In this study, we consider two different ways of parameterizing the gravity field: (1) to be consistent with GRACE, we estimate monthly spherical harmonic coefficients complete to varying low degrees. (2) We use the CTAS solution: as we aim at a long and stable time series, we additionally parameterize a set of trends and semi-annual harmonic amplitudes to the constant part for each Stokes coefficient in a single adjustment with the equation cnm(t)=c‾nm+c˙nm(t-t0)+cnmc1cos⁡2πt-t0yr+cnms1sin⁡2πt-t0yr+cnmc2cos⁡4πt-t0yr+cnms2sin⁡4πt-t0yr,snm(t)=s‾nm+s˙nm(t-t0)+snmc1cos⁡2πt-t0yr+snms1sin⁡2πt-t0yr+snmc2cos⁡4πt-t0yr+snms2sin⁡4πt-t0yr.

We estimate the spherical harmonic coefficients from degree 2 onward. As described in Sect. , we derive non-gravitational accelerations from models, which we then use in the gravity field estimation as a proxy for accelerometer measurements. Due to the presence of errors, e.g. those caused by uncertainties in the density model or errors in the macro model, the resulting non-gravitational accelerations might not always reflect the truth. To prevent these errors from propagating into the gravity field estimates, it is common to introduce additional parameters. Here we co-estimate an “accelerometer bias” per arc and per axis, either as a constant value or with an additional trend parameter. While we found this usually sufficient, we also performed tests with an additional global scaling factor per axis. Another possibility that is also evaluated in this paper is to co-estimate the bias globally. The influence of this “accelerometer parameterization” will be evaluated in the course of this paper, yet one needs to bear in mind that these parameters measure force model inconsistencies and should not be mixed up with instrument errors. We furthermore investigate the influence of different arc lengths, which affects the temporal acceleration parameterization, as well as the effect of spherical harmonic truncation.

As was mentioned already, we choose different regions for our investigation (see Fig. ), but our focus is on the total ocean in order to test the hypothesis that Swarm can bridge the GRACE ocean mass time series.

For computing smoothed basin mass averages, let Fλ,Φ be the equivalent water height (EWH) derived from the spherical harmonics . The smoothed region average F‾OW, considering the smoothing kernel W (here a 500 km Gaussian filter) over the region O, can be expressed as F‾OW=1O‾W∫ΩOW(λ,θ)Fdw.

The integral is effectively evaluated for the smoothed area function OW(λ,θ) as OW(λ,θ)=∑n=0∞∑m=-nnO‾nmWY‾nm(λ,θ)=14π∫ΩW(λ,θ,λ′,θ′)O(λ′,θ′)dw′.

Some postprocessing needs to be applied to the estimated gravity fields, depending on the application. As we compare our results to the monthly GRACE solutions, we test replacing the c20 coefficient with those derived from satellite laser ranging (SLR) . While replacing c20 leads to a workflow more in line with GRACE, keeping the Swarm-derived c20 would answer the question whether Swarm alone is able to measure mass change relative to a reference (here GOCO05c). In the next step, we add all degree 1 coefficients to correct for geocenter motion , which cannot be detected with the current GRACE and Swarm processing. We apply a correction for glacial isostatic adjustment following , but as long as we apply the same correction to GRACE, the comparison between Swarm and GRACE will be independent of this choice. We employed an ocean mask that includes the Arctic ocean and does not have a coastal buffer zone.

Figure  shows monthly ocean mass change in mm EWH derived from GRACE as a reference and from different Swarm time-variable gravity (TVG) solutions from AIUB, ASU, IfG and the Institute of Geodesy and Geoinformation (IGG) in Bonn (processing details can be found in Table ). The IGG time-variable gravity field was computed with an arc length of 30 min, modeled non-gravitational accelerations, and a constant bias per arc and direction being co-estimated, which leads to our best solution. All Swarm time series show a behavior similar to the GRACE solution, but they appear overall noisier, as can be seen from the variances in Table . The quality of all solutions improves after the global navigation satellite system (GNSS) receiver update in July 2014. The impact of tracking loop updates on gravity field recovery is discussed in . It is furthermore interesting to compute the RMSE of all solutions when we assume the GRACE solution to be the truth (first row of Table ). The ASU time series has the lowest RMSE, with 2.8 mm; it is closest to GRACE. The IGG solution has the second lowest RMSE, with 4.0 mm. To assess the spread between the different Swarm solutions, we compute the RMSE for each combination (off-diagonal of Table ) which is of the same magnitude as the RMSEs of GRACE and Swarm.

An important issue in extending the ocean mass time series is the accuracy of the trend. Table  shows the trends as well as the amplitude and phase of the Swarm solutions. The trend of the IGG solution (3.3 mmyr-1) is the closest to GRACE (3.5 mmyr-1). While the trend over three years itself cannot be considered as representative for the GRACE era due to interannual variability of barotropic modes, this suggests that Swarm data could be used to bridge a gap between GRACE and GRACE-FO.

Figure  shows the degree variances and the difference degree variances of GRACE and our IGG solution for May 2016 with respect to our reference field GOCO05c. Obviously, the higher the degree, the higher is the discrepancy between GRACE and Swarm. The difference (dotted gray line) indicates that for this particular month Swarm is only reliable for degrees up to about 10, which is due to the much lower precision of the GPS data compared to the GRACE inter-satellite K-band ranging. Since the formal errors (dotted black line) are not calibrated, they are too optimistic and always lower than the difference between GRACE and Swarm. This will be addressed in the future by including realistic covariance information of the kinematic orbits. As Fig.  only shows the degree variances for one particular month, we investigate different maximum degrees in the following (see Table ). We evaluate our monthly fields until d/o 10, 12 or 14. Even though higher degrees do not contribute a reasonable time-variable signal, we estimate the monthly fields until d/o 20 or 40, because high degrees can absorb errors that would otherwise propagate in the lower degrees. For our CTAS solution, we estimate a static part (c‾nm and s‾nm in Eq. ) until d/o 20, 40, or 60, while the time-variable part is estimated until d/o 10, 12, or 14.

Figure  compares modeled non-gravitational accelerations (see Sect. ) to the ACC3CAL_2_ product from , who removed sudden bias changes from the accelerometer measurements and corrected the low-frequencies with POD-derived non-gravitational accelerations. Both time series are very close together, which supports our use of the modeled non-gravitational accelerations for gravity field estimation. Small systematic deviations can be compensated for by co-estimating additional bias or scale parameters.

Modeling non-gravitational accelerations from the Swarm satellites within TVG recovery provides an ocean mass time series significantly closer to the one from GRACE (see Fig. ), and it also improves the trend estimate as can be seen in Table . This means that errors caused by neglecting non-gravitational accelerations would propagate in the spherical harmonic coefficients.

Figure  shows (1) monthly Swarm solutions compared to (2) ocean mass derived with a CTAS signal for each spherical harmonic coefficient. Obviously, the second approach fits much better to the GRACE time series, depicted in red: the RMSE decreases from 4.0 mm for (1) to only 1.7 mm for (2). Furthermore, we find a trend estimate of 3.5 mmyr-1, which is surprisingly close to GRACE (see Table ). In other words, directly parameterizing CTAS terms for each harmonic coefficient, instead of computing the usual monthly solutions, leads to solutions which are much closer to GRACE. The reason for this is that the estimation of CTAS terms from the whole Swarm period (December 2013 to December 2016) is more stable than estimating a set of spherical harmonic coefficients for each month. To our knowledge, this has not been investigated for Swarm, prior to this study.

We investigated the effect of different arc lengths of 30, 45, and 60 min on ocean mass estimates (see Fig. ). The remaining parameters have been chosen according to our best results. For the CTAS approach, the solution with 30 min differs most from GRACE and the other two solutions, while 45 min provide the lowest RMSE (1.7 mm) and the best trend estimate (3.5 mmyr-1). When considering monthly solutions, 30 min provide the best result (RMSE: 4.0 mm and trend: 3.3 mmyr-1).

In addition to modeling the non-gravitational forces, which are introduced in the gravity estimation process as accelerometer data, we carried out several tests, as listed in Table , concerning the co-estimation of “accelerometer bias and scale factors” (see Sect. ). For Figs. a and b we find that a global scaling factor per axis only has a minor influence.

For the CTAS solutions, parameterizing the bias as a linear function leads to a smaller RMSE with respect to the GRACE solution than a constant value per axis or not estimating it at all. The reason for this might be the large number of observations (10 s sampling for 37 months) compared to the low number of parameters. The additional parameters per arc give room for improving not only the modeled non-gravitational accelerations but also the gravity field parameters. Looking at monthly solutions, we find that a constant bias per axis has a smaller RMSE with respect to GRACE than a linear function or not estimating a bias.

For (a) and (b) we also introduced the bias as a constant value or a polynomial of degree 4 for the whole time span of either 37 months (a) or 1 month (b). The two solutions do not differ much, but they are of a minor quality compared to other solutions.

In this study, we combine the information from the three spacecrafts by simply accumulating the normal equations. For reasons of interpretation and validation, it makes sense to also investigate the single-satellite solutions. Figure  compares ocean mass change derived from the individual solutions, from the combined solution, and from GRACE for (a) the CTAS solutions and (b) the monthly solutions. It is expected that Swarm A and Swarm C provide similar solutions as they fly side by side. This is the case for the CTAS solutions, but it is not always true for the monthly solutions. One possible explanation is that the receivers have different settings, which were activated at different times .

Even though we concentrated on ocean mass in this study, we also derived river basin mass estimates to validate our TVG results in land regions. We investigated the same parameterizations that we used to derive ocean mass changes (see Table ). To assess the solutions with regard to their quality, we compared our results to those derived from the GRACE mission. We decided to not only compute the RMSE, but also to compute the ratio of the variance (VAR) of the GRACE time series to the RMSE. By using this method we can also compare the quality of the solutions in the different areas. The RMSE will be calculated with respect to the available GRACE data (27 out of 37 months from December 2013 to December 2016). This will give a kind of signal-to-noise ratio (SNR), expressed as SNR=VARF‾OW,GRACE(t)RMSEF‾OW,Swarm(t). Figure  shows the 100 best CTAS solutions (considering SNR for the ocean), while Fig.  shows an equal number of the best monthly solutions. To get an idea of the signals in the different basins, Fig.  shows the EWH derived from GRACE.

In general, the quality of the time series of the EWH derived from kinematic orbits of Swarm will be affected by (1) the basin size (see Fig. ) and (2) the signal strength (see Fig. ). As expected, the ratio of VAR / RMSE is highest for the ocean, followed by the Amazon basin, which means that these results are the most reliable. The reason for the good performance of Swarm is the large basin size for the ocean and the large signal combined with a large area for the Amazon basin. For the Greenland and Ganges mass estimates there are some CTAS solutions with VAR / RMSE >1 and in general, the time series of these two basins have a higher VAR / RMSE than those for the Mississippi and Yangtze basins, both for the CTAS and the monthly solutions. Considering monthly solutions, modeling non-gravitational accelerations provides better results than not modeling them. This can be seen in Fig. , where only very few solutions with no modeled non-gravitational accelerations are present. The best CTAS solutions for the ocean also have modeled non-gravitational accelerations, whereas for solutions 13 to 15 only empirical accelerations were co-estimated. These have been obtained with a higher VAR / RMSE for the Amazon, Mississippi, Greenland, and Ganges basins. The estimation of a bias is mandatory as both the best CTAS and monthly solutions always have a bias co-estimated. The best monthly solution was computed until d/o 40 and both GRACE and Swarm were evaluated until d/o 12. This is followed by solutions that were evaluated until d/o 10. The time-variable part of the best CTAS solution is even estimated and evaluated until d/o 14. In general, the results confirm what has been evaluated in Sects.  to .

As GRACE has met the end of its lifetime, we make efforts here to close the gap until GRACE-FO provides data. We study as well the possibility to fill monthly gaps, which are usually bridged by interpolating the previous and subsequent monthly solutions. To find out whether Swarm TVG should be preferred to interpolating GRACE data, we assume that existing monthly solutions are missing, such that we are still able to compare to the actual solutions. In Fig. a we assumed each individual monthly GRACE solution to be missing at one time. We then estimated a harmonic time series consisting of CTAS terms from all solutions except for the one that is considered to be missing. After having carried out the regression for each month, this leads to the blue curve. When comparing the interpolated GRACE time series to the Swarm solution, we find that they are both very close to the real GRACE solution, which offers two possibilities for bridging monthly gaps in the GRACE time series. For most months, the interpolated GRACE time series is closer to the real GRACE solution, which means that it is more reliable to close monthly gaps by interpolating than by using the Swarm solutions.

In case of a longer gap between GRACE and GRACE-FO, ocean mass estimates from Swarm will become more important than considering missing monthly solutions. Figure b shows what would happen if the last 6 months of GRACE were missing. We use the same procedure as before by estimating a harmonic signal (CTAS) from the remaining GRACE data. This leads to the blue curve, which is further offset from GRACE than our Swarm solution. Over 3 years, this would also lead to a degradation of the trend estimate (GRACE and Swarm: 3.5 mmyr-1 and interpolated GRACE: 4.0 mmyr-1), indicating that Swarm is useful to bridge longer gaps, which will be investigated in the following paragraph.

We simulated all possible gaps with a duration between 1 month and 18 months in the time series from December 2013 to December 2017 and tested all gap-filling methods (interpolating GRACE, using monthly Swarm solutions, and using CTAS Swarm solutions). For example, when we assumed a gap of three months, we investigated gaps from December 2013 to February 2014 until October 2016 to December 2016, which created 35 possibilities. The mean RMSE, with respect to the real GRACE data, is shown in Table . It is obviously better to use our CTAS solution to fill gaps instead of using monthly solutions. However, for a gap of three months, we get a mean RMSE of 1.1 mm for interpolating existing GRACE solutions compared to 1.5 mm for the CTAS solution, which indicates that in most cases of a three-month gap, interpolating the remaining GRACE solutions is closer to GRACE than using the Swarm solutions. For a prolonged gap of 18 months, our Swarm solution would, however, be closer to GRACE in 80 % of all cases.

With the Swarm accuracy as discussed in Table , the next logical question would be to ask what kind of sea level signal could be detected with Swarm. During the time span investigated here (December 2013 to December 2016), ocean mass evolves rather regularly, i.e. without apparent interannual variation. Therefore, we decided to look into data from the past.

and showed that the 2010/11 La Niña event led to a 5 mm drop in global mean sea level (GMSL). This has been derived from satellite altimetry as well as from a combination of GRACE and Argo data. As most of the anomaly has been shown to be caused by mass changes, it is reasonable to ask whether we would have been able to observe the drop in ocean mass with Swarm (or to observe a similar event in the future). A simple computation shows that with an RMSE of 4.0 mm for monthly Swarm solutions, we would be able to detect a 6-month drop of 4.0mm/6=1.6 mm. As the 2010/11 drop was both larger and lasted longer, we conclude that we should have been able to detect La Niña events with Swarm, therefore making it likely to be able to do so in the future.

We have conducted another simulation experiment with simulated ocean mass data from 1993 to 2004 taken from (see Fig. a). Using the Wenzel and Schröter time series as a basis here, we then generate 1000 simulated Swarm time series by adding white noise with a variance of 4.0 mm (pink area). When comparing the filtered (moving average of 1 year) time series shown as black line with standard deviation of 1.2 mm derived from the simulated Swarm time series (gray), we can clearly identify the drop between 1998 and 2000 standing out against the noise floor. To recap, strong La Niña events such as they occurred in the past could be observed with Swarm, which will be of special importance in case of a prolonged gap between GRACE and GRACE-FO.

So far, ocean mass has been shown without adding back the GAD product from the German Research Centre for Geosciences to our previous time series since our focus is on comparing estimates and the GAD product has a trend of zero for the ocean basin. Here, for better interpretation, we show ocean mass from GRACE from 2004 to 2014 (Fig. b) and ocean mass from Swarm from 2014 to 2016 (Fig. c) with the GAD product added back.