Ultra-sensitive space-borne accelerometers on board of low Earth orbit (LEO) satellites are used to measure non-gravitational forces acting on the surface of these satellites. These forces consist of the Earth radiation pressure, the solar radiation pressure and the atmospheric drag, where the first two are caused by the radiation emitted from the Earth and the Sun, respectively, and the latter is related to the thermospheric density. On-board accelerometer measurements contain systematic errors, which need to be mitigated by applying a calibration before their use in gravity recovery or thermospheric neutral density estimations. Therefore, we improve, apply and compare three calibration procedures: (1) a multi-step numerical estimation approach, which is based on the numerical differentiation of the kinematic orbits of LEO satellites; (2) a calibration of accelerometer observations within the dynamic precise orbit determination procedure and (3) a comparison of observed to modeled forces acting on the surface of LEO satellites. Here, accelerometer measurements obtained by the Gravity Recovery And Climate Experiment (GRACE) are used. Time series of bias and scale factor derived from the three calibration procedures are found to be different in timescales of a few days to months. Results are more similar (statistically significant) when considering longer timescales, from which the results of approach (1) and (2) show better agreement to those of approach (3) during medium and high solar activity. Calibrated accelerometer observations are then applied to estimate thermospheric neutral densities. Differences between accelerometer-based density estimations and those from empirical neutral density models, e.g., NRLMSISE-00, are observed to be significant during quiet periods, on average 22 % of the simulated densities (during low solar activity), and up to 28 % during high solar activity. Therefore, daily corrections are estimated for neutral densities derived from NRLMSISE-00. Our results indicate that these corrections improve model-based density simulations in order to provide density estimates at locations outside the vicinity of the GRACE satellites, in particular during the period of high solar/magnetic activity, e.g., during the St. Patrick's Day storm on 17 March 2015.

Recent gravimetric satellites, for example the satellite missions
CHAllenging Minisatellite Payload (CHAMP;

Nevertheless, satellite accelerometer measurements need to be calibrated
before being used in any applications such as solar terrestrial studies

In order to better understand and reconcile differing results in the
literature, three calibration procedures are applied to GRACE
accelerometer measurements in this study. The aim is to assess the impact of a specific
calibration method on the estimation of global thermospheric neutral
densities as will be discussed in what follows. (1) The first approach is here
called the multi-step numerical estimation (MNE), which is based on the
numerical differentiation of kinematic positions. The application of this
method is similar to that of

In recent decades, empirical and physical models of the atmosphere have gone
through considerable development, while reflecting the range of density
variability in response to solar and geomagnetic forcing. The Mass
Spectrometer and Incoherent Scatter (MSIS) empirical models of the neutral
atmosphere

In the following, data sets and models are introduced in
Sect.

We use GRACE Level-1B data

Satellite body fixed reference frames

Space-borne capacitive accelerometers such as the SuperSTAR accelerometer on board each GRACE twin satellites contain a proof mass, which is kept at the center of mass of a satellite by compensating the non-gravitational forces with induced electrostatic forces. The measured accelerations of the proof mass of the SuperSTAR accelerometer, which are proportional to the voltage needed to generate the compensating electrostatic forces, are labeled as ACC1B within the GRACE Level-1B data.

This accelerometer has a resolution of

Two star cameras on board each GRACE satellite provide its inertial
orientation in terms of quaternions, labeled as SCA1B. These data are used
(in Sect.

In this study, a macro model is required to model the non-gravitational accelerations acting on
the surface of the satellite (Sect.

Surface properties of the GRACE macro model

Each GRACE satellite is equipped with three GPS receivers, whose data are
used for precise orbit determination (POD) and which ensure the precise time
tagging of other on-board measurements. Level-1B GRACE satellite orbits
(GNV1B) are obtained from a dynamic POD procedure

In this study, neutral thermospheric densities derived from two empirical
density models JB2008

As we will show in Sect.

Gravitational force models.

Calibration of the accelerometer measurements

In this study, the calibration equation is written as

This calibration method makes use of a 2-fold numerical differentiation of
kinematic orbit positions, a procedure that has been often applied and tested
in gravity retrieval studies

The problem of designing the Savitzky–Golay filter is equivalent to finding
coefficients

The total acceleration

Outliers in each direction of the non-gravitational accelerations

The non-gravitational accelerations in Eq. (

The MNE procedure applied here differs from the calibration procedure of

Accelerometer calibration parameters can also be estimated within a dynamic POD procedure. In dynamic POD, orbits are estimated from observables, while accounting for the forces acting on the satellite including the non-gravitational forces from accelerometer measurements. In our implementation, kinematic orbits are the observables.

In the variational equation approach

Assuming that the non-gravitational accelerations are realistically modeled,
e.g., using an empirical approach, calibration of the accelerometer
measurements could be done by adjusting the on-board measurements to the
modeled values

The atmospheric drag is caused by the interaction of the particles within the
atmosphere with the surface of the satellite. The impact of drag on the
satellite's surface is derived from

Both visible and infrared radiation of the Sun interact with the surface of
LEO satellites in terms of reflection or absorption, which accelerate the
satellite due to the solar radiation pressure (SRP;

The Earth emits thermal radiation and reflects a fraction of the incoming sunlight back into space, where both radiations interact mainly with the nadir-pointing surface of the satellite in terms of reflection or absorption. This causes an acceleration due to the Earth radiation pressure (ERP), which decreases with an increasing distance of the satellite to the Earth, and cannot be neglected in force modeling for LEO satellites.

Accelerations due to ERP acting on the GRACE satellite are usually modeled
following

After several investigations, we conclude that the polynomial fit used in the
original model of

Finally, the calibration parameters from the above methods are applied to the
raw accelerometer measurements

In the following, we compare the calibration parameters, represented in the
SRF, obtained from the three calibration procedures during three individual
months of the current (24th) solar cycle with varying solar activity. As
already mentioned in Sect.

Scale factors

In general, the estimated scales for GRACE accelerometer are close to 1,
which is in agreement with the expected behavior. Especially in the radial
direction, the scale of 0.94 (GRACE A) and 0.93 (GRACE B) derived from DE is
similar to the scales obtained during a POD by

Daily biases for GRACE A during November 2008, February 2014 and March 2015
corresponding to low, medium and high solar activity obtained from the three
calibration procedures using constant scales (Table

Comparison of daily biases of acceleration measurements of GRACE A
during low (November 2008), high (February 2014) and medium (March 2015)
solar activity. In these figures, MNE represents the multi-step numerical
estimation (blue), DE indicates the dynamic estimation (red) and EMA stands
for the empirical model approach (green). Results are presented in the
along-track (

The magnitude of biases (Fig.

Based on the numerical results, one can see that calibration parameters
obtained from MNE and DE are fairly similar in the along-track and
cross-track directions. The offset between along-track calibration parameters
obtained from EMA, compared to other methods, is caused by the dependency of
its results on the density derived from empirical models

Mean values and standard deviations of biases

The mean and standard deviation of the calibration parameters of March 2015 are provided in
Table

By applying daily biases and the constant scale factors on raw accelerometer
measurements, corresponding calibrated time series are computed. The
calibrated accelerometer measurements obtained from the three applied
calibration procedures (Sect.

Daily average of along-track densities during November 2008 (low),
February 2014 (medium) and March 2015 (high solar activity). Density obtained
from calibrated accelerometer measurements: multi-step numerical estimation
(MNE, blue), dynamic estimation (DE, red) and empirical model approach (EMA,
green). Density obtained from empirical density models are shown as
NRLMSISE-00 (MSIS, black) and Jacchia–Bowman 2008 (JB, yellow). Density sets
by

Daily averages of along-track densities during three particular months with
high, medium and low solar activity are presented in Fig.

In addition to daily mean densities, the along-track densities on
1 November 2008 are presented in Fig.

Along-track densities on 1 November 2008. Density obtained from
calibrated accelerometer measurements: multi-step numerical estimation (MNE,
blue), dynamic estimation (DE, red) and empirical model approach (EMA,
green). Density obtained from empirical density models: NRLMSISE-00 (MSIS,
black) and Jacchia–Bowman 2008 (JB, yellow). Densities by

Along-track densities during the St. Patrick's Day storm (17 and 18 March 2015). Density obtained from calibrated accelerometer measurements: multi-step numerical estimation (MNE, blue), dynamic estimation (DE, red) and empirical model approach (EMA, green). Density obtained from empirical density models: NRLMSISE-00 (MSIS, black) and Jacchia–Bowman 2008 (JB, yellow).

On 17 and 18 March 2015, thermospheric densities reach a maximum due to a
strong solar event (St. Patrick's Day storm). Along-track densities during
these days are presented in Fig.

In the following, the accelerometer-based densities

NRLMSISE-00 daily mean empirical corrections (emp. corr.) of GRACE A during November 2008 (low), February 2014 (medium) and March 2015 (high solar activity). Densities obtained from calibrated accelerometer measurements of the multi-step numerical estimation (MNE, blue) and the dynamic estimation (DE, red).

In November 2008, the MNE-based empirical corrections are less reliable due
to less stable calibration parameters resulting from this approach during
periods of a low signal-to-noise ratio. Due to the similarity of the
calibration parameters derived from MNE and DE during high and medium solar
activity, the density corrections obtained from both approaches are found to
be similar with mean values of 1.11 (MNE) and 1.12 (DE) in March 2015, and in
February 2014 the mean values are found to be 0.99 (MNE) and 0.97 (DE).
During the St. Patrick's Day storm (17 and 18 March 2015), the mean empirical
corrections increase on 17 March, and decrease afterwards due to the delayed
and weakened maximum in the empirical thermospheric densities (see also
Fig.

The empirical model NRLMSISE-00 underestimates the thermospheric neutral
density with values of up to 28 % of simulated densities during high
solar activity. It also overestimates neutral densities, which are found to
be on average 22 % of simulated densities during low solar activity. Only
during medium solar activity, the empirical corrections are approximately
close to 1. Overestimated model densities during low solar activity have also
been reported in

Since the necessity of correcting empirical thermospheric neutral density
models is evident, we provide global empirical corrections on a daily
basis, which can be used
to scale model-derived neutral density estimations for the altitude of

NRLMSISE-00 empirical corrections along the orbit of GRACE A on 2 March 2015. Densities obtained from calibrated accelerometer measurements of the multi-step numerical estimation (MNE, blue) and the dynamic estimation (DE, red).

NRLMSISE-00 empirical corrections of GRACE A during 2 March 2015.

Additionally, a spatial representation of the corrections, which are
estimated for the NRLMSISE-00 empirical model along the orbit of GRACE A on
2 March 2015, are shown on the left column of Fig.

In order to derive global patterns of differences between GRACE densities and
model output, as GRACE does not exactly repeat its daily tracks, daily
density scales (

From the average cross-track spacing, we found that a spherical harmonic
degree of 11 could be resolved, i.e., fixing the maximum degree to 10 is
appropriate. The daily spherical harmonic coefficients are estimated using a
least squares estimation. Numerical problems are not expected, since the
analysis of the normal equation matrix yields a condition number below

In this study, the measurements of the SuperSTAR accelerometers on board the
GRACE satellites are calibrated using three procedures. The multi-step
numerical estimation approach is based on the numerical differentiation of
kinematic orbits, where the main challenges are the noise amplification and
the temporal correlation after applying a numerical differentiation operator.
Here, similar to

The three accelerometer calibration procedures are applied successfully using constant scale factors and are found to provide largely comparable biases particularly in the along-track and cross-track directions. The calibration parameters computed using the dynamic estimation yields the most realistic calibration parameters and thermospheric neutral densities, likely due to the physical consistency of this approach. Results obtained with the multi-step numerical estimation are similar to the dynamic estimation during high and medium solar activity.

Furthermore, thermospheric neutral densities derived from calibrated
accelerometer measurements in the along-track direction of GRACE are compared to densities
obtained from the empirical models NRLMSISE-00 and Jacchia–Bowman 2008. The
results suggest that accelerometer-derived densities provide more reliable
results, especially on short timescales and during strong solar events, for
example during the St. Patrick's Day storm on 17 March 2015. Hence,
accelerometer-derived densities allow for the improvement of empirical density
models as already stated by

Empirical density corrections of the empirical model NRLMSISE-00 are computed along the GRACE orbit. The results suggest that it is necessary to apply corrections to model densities depending on the solar activity. Due to overestimated empirical model densities during low solar activity, empirical corrections of 22 % on average need to be applied on NRLMSISE-00 densities during quiet periods. In contrast, empirical corrections of up to 28 % are required during high solar activity, since the model underestimates neutral densities. The spherical harmonic expansion of these corrections on a global grid provides a measure indicating to what extent GRACE-derived thermospheric density estimation can improve simulations of empirical density models on a daily basis. These findings encourage the use of these factors to improve empirical density models.

Further efforts in satellite drag modeling will improve the empirical model approach to calibrate accelerometer measurements, as well as the thermospheric neutral densities estimated from the three methods. Moreover, including a horizontal wind model in the empirical model approach is expected to yield more realistic densities which might improve the consistency of the results. The multi-step numerical estimation method may be further developed through modeling the temporal correlations of accelerometer measurements.

In further studies, the empirical corrections derived from calibrated
accelerometer measurements of GRACE A could be used to model densities in
order to simulate non-gravitational accelerations acting on GRACE B, which
contributes to filling data gaps during months where only one satellite
provides accelerometer measurements. Other methods on transferring
non-gravitational accelerations of a satellite to a co-orbiting one are
discussed in

The calibration procedures are applicable to other satellite missions
carrying space-borne accelerometers as well. Combining the thermospheric
neutral densities derived from different calibrated accelerometers allows
further improvement of empirical density models. For example, the empirical
density corrections at different altitudes can be used to obtain altitude
profiles to correct empirical density models, which could then be used to
derive accurate drag predictions for other satellites which are not equipped
with an accelerometer, restricted to the period when the corrections are
available. Besides, the assimilation of calibrated accelerometer measurements
of various satellite missions into physical thermosphere/ionosphere models
would likely enable an improved representation of physical processes in the
atmosphere, e.g., following

The density data that were used for comparison can be found at

In this synthetic experiment, we apply different numerical differentiation
operators to an analytical orbit. We design an analytical orbit

Differences between the analytical second derivative and three
numerical second derivative filters of the

The difference between the analytical derivative and the smoothing
differentiation filter (black) shows that this filter introduces an unwanted
phase shift. Therefore, the smoothing differentiation filter is not suitable.
In comparison, the Savitzky–Golay filter (red and green) prevents phase
shifts, and the application of different filter settings clarifies that
the difference between the analytical and the numerical derivative is minimized,
i.e., the amplification of noise is limited, when using the settings

ERP is caused by albedo

Along-track ERP accelerations on 1 November 2008. Knocke model (red) and new model with spherical harmonics (green).

of the Earth. In order to calculate

Here, we replace the

ERP acceleration in the along-track direction on 1 November 2008, modeled
using Knocke and derived form the model using spherical harmonics, are
presented in Fig.

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 is not associated with a conference.

Kristin Vielberg thanks the German Academic Exchange Service (DAAD) for the PROMOS scholarship to conduct a part of this research at Cardiff University. We thank Aleš Bezděk for his generous comments on the computational steps of this study. The authors are grateful to the research grant through the D-SAT project (FKZ.: 50 LZ 1402) and the TIK project (FKZ.: 50 LZ 1606) supported by the German Aerospace Center (DLR). We also acknowledge the topical editor Eelco Doornbos and the two reviewers for their helpful remarks and suggestions. The topical editor, Eelco Doornbos, thanks two anonymous referees for help in evaluating this paper.