Gravity field models may be derived from kinematic orbit positions of Low Earth Orbiting satellites equipped with onboard GPS (Global Positioning System) receivers. An accurate description of the stochastic behaviour of the kinematic positions plays a key role to calculate high quality gravity field solutions. In the Celestial Mechanics Approach (CMA) kinematic positions are used as pseudo-observations to estimate orbit parameters and gravity field coefficients simultaneously. So far, a simplified stochastic model based on epoch-wise covariance information, which may be efficiently derived in the kinematic point positioning process, has been applied.

We extend this model by using the fully populated covariance matrix, covering correlations over 50 min. As white noise is generally assumed for the original GPS carrier phase observations, this purely formal variance propagation cannot describe the full noise characteristics introduced by the original observations. Therefore, we sophisticate our model by deriving empirical covariances from the residuals of an orbit fit of the kinematic positions.

We process GRACE (Gravity Recovery And Climate Experiment) GPS data of April 2007 to derive gravity field solutions up to degree and order 70. Two different orbit parametrisations, a purely dynamic orbit and a reduced-dynamic orbit with constrained piecewise constant accelerations, are adopted. The resulting gravity fields are solved on a monthly basis using daily orbital arcs. Extending the stochastic model from utilising epoch-wise covariance information to an empirical model, leads to a – expressed in terms of formal errors – more realistic gravity field solution.

Various Low Earth Orbiting (LEO) satellites are equipped with a GNSS (Global Navigation Satellite System) receiver, which may not only be used for navigational purposes and Precise Orbit Determination (POD) but also for gravity field recovery. Since the CHAllenging Minisatellite Payload mission

We focus on kinematic positions derived from GRACE GPS data to reconstruct orbits and estimate gravity fields simultaneously by applying the Celestial Mechanics Approach

So far, a simplified stochastic model based on epoch-wise covariance information, derived in the
kinematic point positioning process, has been applied in the CMA. We extend this model by using the fully populated covariance matrix as

The paper is organised in six sections, where Sect. 2 briefly introduces the data and Sect. 3 the methods of gravity field recovery used in this study. In Sect. 4 stochastic noise modelling for kinematic orbit positions is explained together with the results for the different covariance matrices we have employed. First we discuss the simplified stochastic model based on epoch-wise covariances (Sect. 4.1.1), next the extension to covariances over-arching epochs (Sect. 4.1.2) and finally we adopt empirical covariances (Sect. 4.2). Section 5 gives an outline to the stochastic behaviour of undifferenced ambiguity fixed kinematic positions. Section 6 concludes the results of this study.

In order to test our methods of stochastic modelling, we use GRACE data collected in April 2007. We deliberately chose data of good quality to avoid issues with screening methods for a reliable derivation of empirical covariances. Kinematic positions are obtained from the GPS carrier phase observations using the Bernese GNSS Software

In the CMA gravity field recovery is based on the solution of a generalised orbit determination problem for each GRACE satellite

Models used for the full a priori force field.

In the second step the a priori orbits are introduced to set up normal equations along the orbit by using the same orbit parametrisation, and additionally, spherical harmonic (SH) coefficients representing the gravity field. The kinematic positions again serve as pseudo-observations and are weighted according to the covariance information. All daily normal equations are accumulated into one normal equation system covering one month, which is eventually solved for in order to obtain a solution for the correction of the orbit parameters and the SH coefficients. No regularisation is applied to compute the gravity field parameters.

To investigate the robustness and the dependencies of the approach, two types of orbit parametrisations were adopted for each type of covariance in orbit reconstruction and gravity field recovery:

A dynamic orbit represented by only six initial osculating (Keplarian) elements and nine additional parameters characterising the accelerometer, and

a reduced-dynamic orbit represented by the same parameters as the dynamic orbit and additional constrained Piecewise Constant Accelerations (PCA) set up over intervals of 15 min in radial, along-track and cross-track direction.

As the CMA is only one of several approaches to recover gravity field information from kinematic positions, we refer to

The stochastic noise modelling of kinematic orbit positions is a crucial part for high-precision gravity field recovery. The way the noise in the pseudo-observations is treated influences the quality of the result (e.g., in terms of formal errors) significantly, and it might also affect the solution itself. The correct characterisation of noise in the data retains the full signal content and separates signal from noise through a modelling of the latter – provided that the signal component is also adequately modelled.

In case of kinematic positions, which are introduced as pseudo-observations in the CMA, one has to consider the process of calculating these pseudo-observations in the stochastic modelling. The original observations are the GPS dual-frequency carrier phases. The kinematic positions of the LEO satellite are determined at the observation epochs of the GPS carrier phase measurements by a precise point positioning (PPP) approach

The kinematic positions are introduced as pseudo-observations in a standard least-squares adjustment to estimate orbit parameters and gravity field parameters in a joint adjustment. The least-squares adjustment yields

The process of calculating the pseudo-observations from the original carrier phase measurements allows for a covariance estimation. We derive the covariance matrix

The correlations in

The GPS carrier phase ambiguities are the only parameters in

The weight matrix

Introducing the pseudo-observations together with the full covariance matrix

The epoch-wise covariance information is a subset of the covariance matrix from the formal variance propagation, which may be easily derived in the kinematic point positioning process using pre-elimination and back-substitution techniques. Focusing only on the kinematic positions, the epoch-wise covariances, assembled in Eq. (

The epoch-wise covariance information mainly features the twice-per-revolution behaviour for polar orbiting satellites (orbital period for GRACE was

Independent components of the epoch-wise covariance matrix of GRACE-A for one arc, day 091, 2007, in radial (r), along-track (a) and cross-track (c) direction in the time

This information can only be derived from the formal variance propagation and cannot be neglected in the orbit reconstruction and gravity field recovery process without significantly degrading the recovered gravity field solutions using the classical CMA, see

The residuals of the reconstructed orbit (Fig.

Orbit residuals of GRACE-A using epoch-wise covariances for day 091, 2007, for the reduced-dynamic orbit

The K-band validation of the fitted dynamic orbit is smoother than of the fitted reduced-dynamic orbit (see Table

K-band RMS over all days using epoch-wise covariances in the orbit reconstruction for April 2007.

The gravity field solutions based on the epoch-wise covariance information when using the reduced-dynamic and the dynamic orbit parametrisations are shown in Fig.

Monthly GRACE GPS-only gravity field solution (April 2007) based on a reduced-dynamic orbit (red) and a dynamic orbit (blue). The solid lines depict the difference degree amplitudes to GOCO05s, the dashed lines denote the formal errors from the least-squares adjustment.

In the dynamic and the reduced-dynamic case the formal errors (dashed lines) of the solution do not reflect the accuracy assessed by the differences to the (superior) GOCO05s

Apart from the epoch-wise covariance information from the formal variance propagation, the covariances connecting a certain number of epochs may be taken into account as well. As discussed at the beginning of Sect. 4.1, their behaviour reflects the presence of the carrier phase ambiguities as solely these parameters connect different epochs in a kinematic point positioning. They comprise the observation geometry, thus the influence of the constellation on the correlation between epochs.

The computation of the covariances is in principle straightforward, however, depending on the number of correlated epochs that are of interest, memory consumption may become an issue (e.g., for one day of 10

Covariance function over 50

The jumps in the covariance function occur due to the setup of new ambiguities (changes in the observed GPS constellation).

For both, the dynamic and the reduced-dynamic orbit, covariances covering correlations up to 50

Orbit residuals of GRACE-A using covariances from the formal variance propagation over a period of 50

The K-band range residuals for the reduced-dynamic orbit are significantly lower than in the classical approach of weighting (cf. Tables

K-band RMS using covariances over 50

Weighting the kinematic positions including 50

GRACE GPS-only gravity field solution for April 2007. Covariances considering correlations over 50

The residuals of an orbit fit with respect to the kinematic positions reflect all model (functional and stochastic) and data deficiencies. Consequently, deriving covariances from the residuals leads to a stochastic description of the entire physical system. The residuals

The covariance function for a certain time interval

The correlations between the axes are obtained through a cross-correlation between the residuals of the respective axes

We do not want to model short term variations within an arc, therefore, we use the residuals of a whole month to derive a mean covariance function. The use of empirical covariances relies on the quality and the shape of the residuals, in particular outliers will affect the empirical covariances heavily. This leads to an interdependency between the orbit parametrisation and the pseudo-observations. The dependency on the a priori gravity field is mitigated as much as possible by first estimating an independent gravity field solution and obtaining the covariance function from post-fit residuals. Consequently, the time until the correlations become negligible is highly dependent on these two factors. Even under the assumption that there are no outliers (a prerequisite) in the data, the parametrisation of the underlying orbit affects the magnitude and time the correlations take to vanish significantly, see Fig.

We introduce the epoch-wise covariance information from the formal variance propagation into the a priori gravity field and orbit recovery, on which the empirical covariance function is built, because the epoch-wise covariances transport information about the tracking scenario (inferior constellation over polar regions), which cannot be easily recovered by orbit dynamics only. However, introducing white noise as most simple assumption to weight the kinematic positions in the a priori gravity field and orbit recovery, leads to the same gravity field solution, but requires at least one additional iteration.

Memory consumption is not an issue for the usage of empirical covariances since they are valid over one month and are fully described by one function from which the variance-/covariance matrix can be compiled according to Eq. (

We expect the nature of the residuals in a LOF to be more stationary, as periodic behaviours due to the satellites' orbit will be reflected much cleaner and easier to interpret, thus, we derive the empirical covariances in a LOF. For the reduced-dynamic orbit the largest correlations over epochs occur in the radial direction, which can be attributed to the simultaneous estimation of kinematic positions and receiver clock corrections, while the other directions play a minor role. This is different for the dynamic parametrisation, where the correlations in all three axes are about the same magnitude. Noteworthy is the large and long lasting correlation in cross-track and between the radial and the along-track axis. Thus, the three axis are not independent from each other, even over several epochs.

Empirical covariance function derived from the reduced-dynamic orbit residuals

As we aim to reconstruct the weighting for the observations from estimated residuals

First minutes of the correctional term for the reduced-dynamic orbit parametrisation

Orbit residuals for day 091, 2007, of GRACE-A using empirical covariances derived for a correlation length of 50

The magnitude of the corrections is much lower than that of the covariance function computed from the residuals (compare graphs in Figs.

The covariance function of the residuals on the other hand reflects all model and data deficiencies, and consequently, its magnitude is significantly larger. Neglecting the correctional term provides the benefit of obtaining a nondegenerate covariance matrix by the auto- and cross-correlations. This simplification might only be allowed for kinematic positions, but not for significantly more precise observables such as ultra-precise K-Band observations.

The weight matrix is obtained through Eq. (

For both orbit parametrisations a weight matrix derived from empirical covariances covering correlations up to 50

Using the empirical covariances to weight the kinematic positions in the gravity field recovery process, one can find formal errors much closer to the degree variance differences (Fig.

GRACE GPS-only gravity field solution for April 2007. Empirical covariances based on the observation residuals are introduced in the gravity field determination process. These covariances are determined as a mean function over one month and include correlations over 50

Empirical covariances perform better than formally propagated ones because knowledge about model and data deficiencies is both transferred into the least-squares adjustment. Degrees 10 to 26 are slightly degraded compared to the classical solution (Fig.

In this section we give a short outlook on the use of undifferenced ambiguity fixed kinematic positions for gravity field determination. The process of undifferenced ambiguity fixing can be found in

Introducing such kinematic positions potentially helps to improve the gravity field determination
process as indicated by the performance of the K-band validation of a classical reduced-dynamic orbit (see Fig.

K-band range validation for one month of reduced-dynamic orbit fits of GRACE kinematic positions using epoch-wise covariances.

In the processing of carrier phase observations as many ambiguities as possible are resolved to their integer values before the least-squares adjustment for the kinematic positions takes place, and only the unresolved ambiguities remain in the system. Thus, the number of ambiguity parameters is significantly reduced by more than 90 %. This implies that the kinematic positions are almost
uncorrelated in time, which is shown in Fig.

Covariance function over 50 min for undifferenced ambiguity fixed kinematic positions (first 50 min of day 091, 2007 expressed in Earth-fixed reference frame).

In theory, these short correlations in time carry the whole information on the stochastic behaviour. However, for a gravity field solution it is not sufficient to introduce only these correlations in the least-squares adjustment to obtain a reasonably realistic description of the stochastic behaviour; the system behaves almost like employing epoch-wise covariances, which could be expected from the shape of the covariance function and the K-band validation results. The formal errors of the gravity field solution are at the same level as Fig.

Using undifferenced ambiguity fixed positions together with empirical covariances yields results at the same level as float ambiguities and empirical covariances (not shown). Concerning the signal content in the kinematic positions for gravity field recovery, currently the use of ambiguity fixed positions does not outperform results from a float solution.

Kinematic positions are widely used pseudo-observations in gravity field determination and their stochastic behaviour plays a crucial role when assessing the quality of gravity field solutions. Kinematic positions are correlated in time due to the ambiguities in the original GPS phase observations and not considering these correlations degrades a subsequent gravity field solution. Taking these correlations into account can be achieved either by the covariance information of the kinematic positions over longer time spans or by empirical covariances derived from residuals of an orbit fit with respect to the kinematic positions. However, these empirical covariances demand a stationary process and are, therefore, highly dependent on the quality and shape of the residuals.

Formally propagated covariances over-arching epochs are in case of a dynamic orbit parametrisation able to absorb parts of the errors caused by a deficient force field, piecewise constant accelerations have similar capabilities for reduced-dynamic parametrisations. Empirical covariances support the estimation of physically meaningful formal errors and absorb errors caused by a deficient force field in case of dynamic and reduced-dynamic orbit descriptions.

Undifferenced ambiguity fixed kinematic positions feature different stochastic characteristics than ambiguity float kinematic positions – they are almost uncorrelated – however, the possibility of describing the deficiencies that are still in the processing chain is lost through the ambiguity fixing process. Currently, an improvement of a gravity field solutions by using ambiguity-fixed kinematic positions as pseudo-observations cannot be substantiated.

Further steps to investigate the described methods of stochastic modelling include the diversification to different satellite missions, such as Swarm or GOCE, and an appropriate treatment of deteriorated data (affected by outliers etc.), for empirical covariances.

Data for GRACE was accessed via GFZ's ISDC data centre

ML performed the computational work of gravity field determination, DA contributed the kinematic positions. Concept, framework and interpretation of the study was accomplished by all authors.

The authors declare that they have no conflict of interest.

This article is part of the special issue “European Geosciences Union General Assembly 2019, EGU Geodesy Division Sessions G1.1, G2.4, G2.6, G3.1, G4.4, and G5.2”. It is a result of the EGU General Assembly 2019, Vienna, Austria, 7–12 April 2019.

We thank reviewers for insightful comments which helped to greatly improve the manuscript.

This research has been supported by the Swiss National Science Foundation (SNSF) (grant no. 200021_175942 Assessment of Noise Models for GRACE and GRACE-FO).

This paper was edited by Annette Eicker and reviewed by Akbar Shabanloui and one anonymous referee.