Measuring the spatiotemporal variation of ocean mass allows for partitioning of volumetric sea level change, sampled by radar altimeters, into mass-driven and steric parts. The latter is related to ocean heat change and the current Earth's energy imbalance. Since 2002, the Gravity Recovery and Climate Experiment (GRACE) mission has provided monthly snapshots of the Earth's time-variable gravity field, from which one can derive ocean mass variability. However, GRACE has reached the end of its lifetime with data degradation and several gaps occurred during the last years, and there will be a prolonged gap until the launch of the follow-on mission GRACE-FO. Therefore, efforts focus on generating a long and consistent ocean mass time series by analyzing kinematic orbits from other low-flying satellites, i.e. extending the GRACE time series.

Here we utilize data from the European Space Agency's (ESA) Swarm Earth
Explorer satellites to derive and investigate ocean mass variations. For this
aim, we use the integral equation approach with short arcs

Sea level rise, currently about 3

Swarm was successfully launched into a near-polar low Earth orbit (LEO) on
22 November 2013. The three identical satellites, referred to as Swarm A,
Swarm B, and Swarm C, were designed to provide the best-ever survey of the
geomagnetic field and its temporal variability. The attitude of each
satellite is measured by star trackers with three camera head units. For
precise orbit determination (POD), each spacecraft is equipped with an
8-channel dual-frequency GPS receiver

Swarm A and C fly side by side at a mean altitude of 450

In this study, we first compute a set of in-house time-variable gravity
fields from Swarm kinematic orbits to further derive a time series of ocean
mass change. To this end, we use the integral equation approach developed
earlier at the University of Bonn

This article is organized as follows: in Sect.

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,

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.

Utilized orbit and star camera data.

Background models used during the processing.

During gravity field recovery, we used the GOCO05c model

Drag modeling requires knowing the thermospheric density and temperature. In
this work, we make use of the empirical NRLMSISE-00 model

In order to address our central question of to what extent will Swarm enable
one to infer ocean mass change, we first compute time-variable gravity
fields from kinematic orbits, while considering different processing options.
Then, ocean mass is derived from the computed Stokes coefficients

In the following, we describe our modeling of the non-conservative forces
(Sect.

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.

Atmospheric drag is commonly taken into account by evaluating

Solar radiation is absorbed or reflected at the satellite's surface, leading
to an acceleration

Radiation emitted from the Earth's surface (ERP) is taken into account
similar to solar radiation pressure with the equation

Areas of investigation: ocean (OC), Amazon (AM), Mississippi (MI), Greenland (GR), Yangtze (YA), and Ganges (GA). The boundaries are taken from the Food and Agriculture Organization of the United Nations (FAO).

For gravity field estimation, we use the integral equation approach

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

We estimate the spherical harmonic coefficients from
degree 2 onward. As described in
Sect.

Parameterization for our monthly solutions and for our estimation of
CTAS signal terms. All results are subject to a 500

Parameterizations that have been tested in this study. This table
should not be read row-wise. It lists all possible choices for each heading.
One solution can consist of any combination of the entries, for example,
a monthly solution with an arc length of 60

As was mentioned already, we choose different regions for our investigation
(see Fig.

For computing smoothed basin mass averages, let

The integral is effectively evaluated for the smoothed area function

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

If not stated differently, we used the parameterization in
Table

Ocean mass from ITSG-Grace2016 and Swarm. GRACE data gaps are highlighted in gray.

Comparison of Swarm solutions from different institutes. Orbit product, computing method, and maximum d/o are provided.

Figure

Comparison of the variance (

An important issue in extending the ocean mass time series is the
accuracy of the trend. Table

Figure

Degree variances for GRACE and Swarm (solution for May 2016). Formal errors as well as the difference degree variance (GRACE minus Swarm) are shown with dotted lines.

Comparison of Swarm solutions from different institutes measuring trend,
amplitude, and phase. The values in parentheses indicate the results for the
exact same months that are available for GRACE, while the values without
parentheses are computed from the whole Swarm time series. The results are based
on the time series of
Fig.

Along-track acceleration of Swarm C. The black curve shows the
ACC3CAL_2_ product from

Effect of modeling of non-gravitational forces on ocean mass computation. IGG (mod.) is the monthly solution described in Table 3. The only difference in IGG (not mod.) is that non-gravitational accelerations were not modeled, but a constant value per arc was still co-estimated.

Figure

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.

Ocean mass from GRACE and Swarm. The monthly solution is shown in
black while the CTAS solution is shown in blue. The parameterizations for the
two solutions can be found in
Table

Comparison of different IGG Swarm solutions. IGG: best monthly IGG solution. IGG (not mod.): same parameterization as IGG, but non-gravitational accelerations are not modeled. IGG (CTAS): IGG solution with an estimated constant, trend, annual and semiannual signal per spherical harmonic coefficient. The values in parentheses indicate the results for the exact same months that are available for GRACE, while the values without parentheses are computed from the whole Swarm time series.

Effect of varying the arc length.

Effect of co-estimating bias and scale factors for the
non-gravitational accelerations. The
numbers indicate the degree of the polynomial.

Figure

We investigated the effect of different arc lengths of 30, 45, and
60 min on ocean mass estimates
(see Fig.

Influence of individual satellites on the combined solution.

Evaluation of methods (CTAS solutions).

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

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

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

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

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.

Evaluation of methods (monthly solutions).

EWH derived from GRACE (d/o 12, 500

Mean RMSE (mm) of the gap-filler methods with respect to existing GRACE data. The columns indicate the number of missing months. The percentage of Swarm (CTAS) solutions with a lower RMSE than GRACE (interpolated) solutions is indicated in parentheses. To derive the value in parentheses, we counted the number of CTAS solutions with a lower RMSE than GRACE (interpolated) and computed the relation to the absolute number of CTAS solutions. The number of investigated solutions decreases from left to right, as the time span becomes longer.

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.

Bridging gaps with Swarm. Our IGG Swarm solution (black) is compared
to the monthly GRACE solutions (red) as well as to interpolated values when
we assume a part of the GRACE time series to be missing.

Time series of ocean mass (red). A variance of 4.0

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

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

With the Swarm accuracy as discussed in Table

We have conducted another simulation experiment with simulated ocean mass
data from 1993 to 2004 taken from

So far, ocean mass has been shown without adding back the GAD product from
the German Research Centre for Geosciences

Swarm-derived ocean mass estimates show
the same behavior as those from GRACE, but they appear overall noisier, as
expected. IGG monthly solutions have an RMSE of 4.0

In a second approach we estimated CTAS terms for each spherical harmonic
coefficient and for the whole period of time under study (December 2013 to
December 2017). We find that this significantly improves the agreement with
GRACE regarding ocean mass trend estimates; here we obtain an RMSE of
1.7

We validated TVG results by computing river basin mass estimates and
comparing them to GRACE. We found that the VAR

We tested three different methods for filling the gap that now will occur between GRACE and GRACE-FO, as well as for reconstructing missing single months in the GRACE time series: (1) interpolating existing monthly GRACE solutions, (2) using monthly Swarm solutions, (3) using the CTAS Swarm solution. As expected, (3) provides better results than (2) and whether (1) or (3) is better depends on the length of the gap and on the presence of episodic events and interannual variability. In the (short) Swarm period where ocean mass displayed little variability beyond the annual cycle, we found that for reconstructing either single months or three-month periods (1) may work slightly better than (3), whereas in case of a long 18-month gap, (3) should be preferred.

We showed that La Niña events like those from 2010–2011 and 1998–2000 could have been identified with Swarm, which is of special importance for the future after the termination of the GRACE mission.

In future work, we will concentrate on improving our ocean mass estimates
from Swarm by allowing the trend to change over time as shown, for example, in

The GRACE spherical harmonic coefficients that were used for comparison can be found at

The Swarm spherical harmonic coefficients from IfG Graz can be found at

The Swarm spherical harmonic coefficients from ASU Prague can be found at

The employed CERES data can be found at

The authors declare that they have no conflict of interest.

This article is part of the special issue “Dynamics and interaction of processes in the Earth and its space environment: the perspective from low Earth orbiting satellites and beyond”. It does not belong to a conference.

This study is supported by the Priority Program 1788 “Dynamic Earth” of the German Research Foundation (DFG) – FKZ: KU 1207/21-1. The authors are grateful for the Swarm macro model as well as the calibrated accelerometer data from Christian Siemes (ESA). We also would like to thank Christoph Dahle for sending us the Swarm gravity fields from AIUB (Bern).

We appreciate the work of Jose van den IJssel, whose kinematic orbits are available on the ESA FTP server. Thanks to Torsten Mayer-Gürr and his colleagues from IfG Graz and Aleš Bezděk (ASU Prague) for providing their gravity solutions online. Edited by: Simon McClusky Reviewed by: three anonymous referees