We use GRACE Level-1B data

ftp://podaac-ftp.jpl.nasa.gov/allData/grace/L1B/JPL/RL02/; last access: 5 November 2017

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 . However, the choice of orbit data depends on their application. Since here we aim at calibrating accelerometer measurements for thermospheric density estimations, the kinematic precise orbits, which are free from accelerometer measurements, are applied. For this, kinematic orbits are processed by the Graz University of Technology

ftp://ftp.tugraz.at/outgoing/ITSG/tvgogo/orbits/GRACE/; last access: 5 September 2017

In this study, neutral thermospheric densities derived from two empirical density models JB2008 and NRLMSISE-00 are compared to accelerometer-derived densities from GRACE. The JB2008 model uses geophysical indices from an orbital drag database and accelerometer measurements. The NRLMSISE-00 model additionally makes use of the composition of the atmosphere provided by the Solar Maximum Mission. Both models are forced by a number of parameters, e.g., the geomagnetic planetary index Kp accounting for variations in geomagnetic activity, and the F10.7 index being a proxy for the solar electromagnetic radiation at a wavelength of 10.7 cm = 2.8 GHz. NRLMSISE-00 can be used for the entire space age (applied magnetic and solar indices start before 1957), and it predicts atmospheric temperature, density and composition. JB2008 is constructed using a combination of solar- and geomagnetic proxies and indices that have been available since 1998, thus the model cannot be used before that year . Both models account for spatial and temporal variations in the solar activity. Assessments of JB2008 and NRLMSISE-00 simulations indicate that JB2008 is closer to independent observations during average solar activity . The models show limited skill during the high solar/magnetic activity as model equations do not perfectly reflect timing of the heating transfer as demonstrated by .

As we will show in Sect.  and , gravitational forces must be known while performing the two calibration procedures of the multi-step numerical estimation (Sect. ) and the dynamic estimation (Sect. ). Here, these forces are accounted for using background models as listed in Table .

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 e.g.,. The idea is that by applying a second numerical differentiation on kinematic orbit positions one is able to obtain an estimate for the satellite's total acceleration atotal. To obtain such derivatives, one option is to apply central differentiation operators, for example using seven (or more) points for calculation e.g.,. This implementation, however, introduces three main problems: (1) unwanted phase differences because of the averaging kernel in the differentiation operator and assuming linearity of the path, (2) amplifying the noise because of random errors in the original data points and (3) amplifying temporally correlated noise because the differentiation operator convolves successive orbital positions within the filtering window. To mitigate these problems, the Savitzky–Golay filter is recommended in , which combines smoothing and differentiation operators. Here, we perform a synthetic experiment, where different numerical derivatives are applied to orbit positions sampled from a true orbit defined through an analytical representation (see Appendix , Eq. ). By definition, the main frequencies and amplitudes of the true orbit are known, and subsequently various numerical derivatives can be compared with the results of the analytical derivative. Our results (see Appendix ) confirm that the Savitzky–Golay filter is a suitable approach to estimate derivatives from kinematic orbits because the amplification of noise is limited and phase shifts are prevented.

The problem of designing the Savitzky–Golay filter is equivalent to finding coefficients cn,m, obtained from a least squares adjustment, which can be convolved with the time series of the kinematic positions to compute derivatives . Thus, the derivative of the orbit r, shown by r¨, can be obtained as r¨i=∑n=-n/2n/2cn,m⋅ri+n. Here, r¨ contains the total acceleration (a sum of both gravitation and non-gravitational forces). In Eq. (), i is the epoch index to which the filter is applied; the width of the window is denoted by n and m expresses the order of the fitted polynomial. From our numerical experiments (see Appendix ), combining a window length of n=11 data points with a polynomial of degree m=5 are found to be the best second derivative filter settings for the kinematic orbits used in this study, i.e., this filter adds less noise to the true derivative than others. The filtering requires equally spaced data, thus in this study the kinematic positions are interpolated to an interval of exactly 10 s using a cubic polynomial interpolation. We found that changing the interpolation technique to polynomial or a harmonic interpolation does not considerably change the final results. Data gaps are bridged by fitting a polynomial of degree 9 to one orbital revolution, where the chosen degree minimizes the difference between the true orbit and an interpolated orbit with simulated gaps.

The total acceleration r¨ at a specific time t is related to the force f=ma acting on the satellite via the equation of motion r¨(t)=at,r,r˙,p. The acting acceleration a depends on the time t, the satellite's position r, velocity r˙, and force model parameters denoted by p consisting of a gravitational and a non-gravitational part. To remove gravitational accelerations agrav from Eq. (), models of Table  are used and the desired non-gravitational accelerations acting on the spacecraft are estimated as ang=r¨-agrav. In order to allow a physical interpretation of the non-gravitational accelerations, these are transformed from the celestial to the science reference frame by applying a rotation matrix that is derived from the entries of the star camera quaternions. The resulting non-gravitational accelerations ang can then be used to calibrate the on-board non-gravitational accelerometer measurements araw.

Outliers in each direction of the non-gravitational accelerations ang, which are caused by the constant time step interpolation of kinematic orbits, are detected and removed using the one-dimensional statistical test that considers the modified standard deviation σa=(∑i(angai-ãnga)2)/(r-1) for each axis (a=1, 2, 3) with the number of data r based on the median ãnga instead of the mean a¯nga. Accelerometer observations araw for the same epochs are discarded as well to keep the time domain of the data sets in agreement. Based on the data snooping procedure , the outlier detection is iteratively repeated until the standard deviation is found to be smaller than the global standard deviation computed on the basis of days when no gap-filling interpolation was needed. The threshold in the along-track and cross-track directions is found to be 1×10-5 m s-2, whereas the threshold in the radial direction is equal to 2×10-5 m s-2.

The non-gravitational accelerations in Eq. () can be used to calibrate GRACE accelerometer measurements (Eq. ). Since errors of ang are not Gaussian distributed (as a result of the numerical differentiation), an ordinary least squares estimation (OLS) does not provide the optimal solution. Therefore, we apply a generalized least squares (GLS) method e.g.,, which allows us to jointly estimate the calibration parameters along with an iterative fit of an autoregressive process AR(q) to account for correlated errors of ang. The optimal order of the AR process is determined from the Akaike information criterion , which provides the relative quality of various AR models used in the GLS procedure. Our numerical experiments indicate that AR(q) with q=20 removes the correlations sufficiently and avoids over-parameterization. The AR coefficients are then used to decorrelate the input data of the OLS and the calibration parameters are estimated again. In an iterative procedure, an autoregressive process is fitted to residuals of the GLS again until convergence.

The MNE procedure applied here differs from the calibration procedure of in the way it handles autocorrelation in ang. applies an inverse operator of the second derivative filter on ang as well as on accelerometer measurements to recover the orbit and estimates the calibration parameters by comparing these orbits. Applying this procedure introduces new numerical errors while converting accelerations to orbital positions. Since the application of the GLS together with the AR process has already reduced the correlation errors, we directly find the calibration parameters from Eq. ().

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 e.g.,, the dynamic POD is estimated by solving the equation of motion (see Eq. ) together with the associated variational equations. The force model parameters p in the equation of motion consist of gravitational and non-gravitational accelerations including accelerometer calibration parameters. The partial derivatives of the equation of motion with respect to the force model parameters p can be used to build the variational equations written as d2dt2∂r∂p=∂at,r,r˙,p∂r∂r∂p+∂at,r,r˙,p∂r˙ddt∂r∂p+∂at,r,r˙,p∂p, and the partial derivatives of the equation of motion with respect to the initial state x0=[r0r˙0]T as d2dt2∂r∂x0=∂at,r,r˙,p∂r∂r∂x0+∂at,r,r˙,p∂r˙ddt∂r∂x0. In Eqs. () and (), at,r,r˙,p corresponds to the equation of motion introduced in Eq. (). The system of equations is built using the partial derivatives of the equation of motion r¯(t)=∂r∂p|r¯Δp+∂r∂x0|r¯Δx0+r̃(t), where r¯(t) is the given kinematic orbit and r̃(t) states the computed orbit using approximate values of p and x0 . According to Eq. (), the parameters, which include the satellite's position r, velocity r˙ and accelerometer calibration parameters b and S, are improved iteratively in a least-squares sense. Convergence is reached as soon as the starting position changes less than 1 mm, since beyond this threshold the calibration parameters will not change significantly.

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 . For this, ang in Eq. () is empirically modeled as ang=adrag+aSRP+aERP. In this formulation, we consider ang to consist of atmospheric drag adrag as well as accelerations due to radiation pressure of the Sun aSRP and the Earth aERP, which are presented in Sects. , , and , respectively. These non-gravitational forces also depend on the surface properties of the satellite, which are considered here using GRACE's satellite macro model (see Sect. ).

Finally, the calibration parameters from the above methods are applied to the raw accelerometer measurements araw to obtain calibrated accelerometer measurements according to Eq. () as acal=b+Saraw. Thermospheric neutral densities based on the calibrated accelerometer data acal are estimated following Eq. 6 ρcal=-2macal-aSRP-aERP⋅rACD|vrel|vrel⋅r, where the equation of atmospheric drag adrag is solved for the density and the drag force is replaced by the relation of non-gravitational forces acting on the satellite according to Eq. (). In Eq. (), r denotes the position of the satellite.

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. , the calibration parameters are estimated iteratively similar to to avoid highly (anti-)correlated biases and scales. To keep the calibration methods comparable, constant scales are assumed for all three approaches. We use the scales derived from the dynamic estimation (DE), since the (anti-)correlation dominates especially in the multi-step numerical estimation (MNE) and the empirical model approach (EMA) causing unrealistic scales not close to 1, most pronounced in the cross-track and radial directions (not shown). The mean and standard deviation of the scales of accelerometer measurements derived from DE using data between August 2002 and July 2016 are provided together with similar statistics from other studies for the GRACE satellites in Table .

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 and during the estimation similar to MNE by . In the along-track and cross-track directions, the scales of both satellites resulting from this study are about 0.03 (along track) and 0.06 (cross track) smaller than those from other methods. This variation is likely influenced by the period of data used to estimate the calibration parameters, which is at most 7 years in other studies, whereas the scales here are derived from 14 years of data. When using data between August 2002 and March 2009, the along-track scale of GRACE A obtained from DE is found to be 0.951, which is close to the scale of 0.960 published by . We conclude that the scales of accelerometer measurements vary with the length of data used as well as with the method of estimation as already stated 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 ) are presented in Fig. .

The magnitude of biases (Fig. ) is confirmed to be different for the along-track, cross-track and radial directions. Along-track biases vary between -1.4×10-6 and -1×10-6 m s-2, where biases from EMA differ from those of DE and MNE particularly during high and medium solar activity. In the cross-track direction, biases (2.7×10-5 to 3×10-5 m s-2) are one order of magnitude larger than in the along-track direction. Cross-track biases of the three approaches are similar until the second decimal digit. Radial biases vary between -8×10-7 and -4×10-7 m s-2. Especially in the radial direction, biases show larger variations with time while comparing to the along-track and cross-track directions. Moreover, an offset is found between the calibration parameters obtained from MNE and other approaches; however, the reason for this difference remains unclear.

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 . This empirical approach will likely become more realistic by introducing a horizontal wind model to the calculation of the satellite's relative velocities within drag estimations (Eq. ; ). Moreover, found that an improved understanding of the gas–surface interactions impacts the drag coefficient and hence yields more realistic neutral density for the GRACE satellites. The results obtained in this study confirm the order of magnitude of biases in all three directions, which are derived by Fig. 14 for 1.5 years at the beginning of the mission.

The mean and standard deviation of the calibration parameters of March 2015 are provided in Table . In general, the standard deviation of biases in the along-track direction (5.95×10-10 to 1.15×10-8 ms-2) are smaller for MNE and DE than those of the cross-track and radial directions. This is also observed for the scale factors with a standard deviation of 0.002 in the along-track direction (see Table ). The relatively larger standard deviations of MNE biases may be more realistic compared to other techniques, since we estimate the impact of autocorrelation through the generalized least squares method. In the other techniques, autocorrelation is neglected, which results in lower error estimations.

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. , , and ) are used to compute thermospheric neutral density profiles in the along-track direction (Sect. ). In addition, the derived densities are compared to two different empirical models NRLMSISE-00 and Jacchia–Bowman 2008 . For evaluating the accelerometer-based densities resulting from this study, density sets by , and more recently estimated densities from , both available between 2002 and 2010, are taken into account

http://tinyurl.com/densitysets; last access: 16 May 2018

In the following, the accelerometer-based densities ρcal are used to estimate empirical corrections for NRLMSISE-00 densities ρ in terms of scale factors s=ρcalρ-1 similar to . Mean empirical corrections during November 2008, February 2014 and March 2015 along the orbit of GRACE A are presented in Fig. . Since densities derived from EMA coincide with NRLMSISE-00 densities (see Fig. ), empirical corrections are not estimated for EMA.

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 , and we confirm that empirical density models perform well only during moderate conditions.

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 ∼ 400 km. The corrections can likely be used for the whole altitude range of ∼ 300–600 km, since the altitude-dependent changes in neutral density is close to linear. Empirical corrections on 2 March 2015, along the orbit of GRACE A, are presented in Fig. . Due to the similarity of the calibration parameters derived from MNE and DE, the density corrections obtained from both approaches are found to be similar on this day with mean values of 1.21 (MNE) and 1.24 (DE).

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. . The empirical corrections derived from MNE and DE indicate that NRLMSISE-00 neutral densities need to be increased by 22 % on average on this day.

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 (s=ρcalρ-1) are first estimated by scaling the GRACE-derived densities along its orbit by the corresponding NRLMSISE-00 neutral density simulations. These daily fields are then converted to spherical harmonic coefficients, which allow spectral filtering to retain only low degree differences between GRACE and model estimations. To estimate global empirical correction maps, we consider a trade-off which resembles the problem of estimating time-variable gravity: longer analysis intervals (currently 1 day) would allow us to accumulate more data, with better spatial coverage, at the expense of temporal resolution. The spherical harmonic coefficients smooth out real variability at timescales below 1 day (see neighboring tracks in Fig. ), but some signal may blend into the daily estimates. Shorter analysis intervals would enable better temporal resolution, but with less dense data coverage and thus a loss of spatial resolution. Indeed, we find daily analysis to be an appropriate balance between temporal and spatial resolution. However, we have not carried out a formal optimization, since it was out of the scope of the present paper.

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 103. The scale correction fields are then synthesized on a 1∘ × 1∘ grid using the coefficients of up to degree and order 7, which are presented on the right column in Fig. . The degree variance with a minimum of degree 7 suggests a synthesis of the same degree and an analysis of slightly higher degrees to avoid a loss of information. The pattern in the global corrections resulting from the three accelerometer calibration approaches is generally similar; however, the method of calibrating accelerometer data has a visible impact on the resulting densities. The global empirical corrections provide a measure of how much accelerometer data can improve empirically modeled densities obtained from NRLMSISE-00. According to the along-track empirical corrections in the left column, the global expansions (right column) indicate that NRLMSISE-00 neutral density simulations need to be corrected by 22 % on average during 2 March 2015.