In the framework of the COmbination Service for Time-variable Gravity fields (COST-G) gravity field solutions from different analysis centres are combined to provide a consolidated solution of improved quality and robustness to the user. As in many other satellite-related sciences, the correct application of background models plays a crucial role in gravity field determination. Therefore, we publish a set of data of various commonly used forces in orbit and gravity field modelling (Earth's gravity field, tides etc.) evaluated along a one day orbit arc of GRACE, together with auxiliary data to enable easy comparisons. The benchmark data is compiled with the GROOPS software by the Institute of Geodesy (IfG) at Graz University of Technology. It is intended to be used as a reference data set and provides the opportunity to test the implementation of these models at various institutions involved in orbit and gravity field determination from satellite tracking data. In view of the COST-G GRACE and GRACE Follow-On gravity field combinations, we document the outcome of the comparison of the background force models for the Bernese GNSS software from AIUB (Astronomical Institute, University of Bern), the EPOS software of the German Research Centre for Geosciences (GFZ), the GINS software, developed and maintained by the Groupe de Recherche de Géodésie Spatiale (GRGS), the GRACE-SIGMA software of the Leibniz University of Hannover (LUH) and the GRASP software also developed at LUH. We consider differences in the force modelling for GRACE (-FO) which are one order of magnitude smaller than the accelerometer noise of about

The correct application of background models plays a crucial role in many satellite-related sciences, such as orbit and gravity field determination. Nowadays not only one software is used to perform such tasks but various institutions around the world have set up their own processing schemes based on their in-house developed software packages. In order to minimise systematic differences caused by a diverse handling of well established models, for example, background force models, the effort of comparing software implementations is picked up regularly. Software comparisons are always a useful tool for detecting inconsistencies or even bugs in software implementations.

In the framework of the COST-G

Modelling background forces is vital to gravity field determination, especially when the time variable part is considered. Therefore, we publish a set of data of various commonly used forces in orbit and gravity field modelling evaluated along a one day orbit arc of GRACE, together with auxiliary data to enable easy comparisons. This data set is intended to be used as a reference data set and provides the opportunity to test the implementation of these models in various software packages.
The COST-G consortium consists at the time of writing of ACs and further candidate ACs. Thus, we took the opportunity to test several software packages available within the COST-G consortium, which are the Bernese GNSS software

The paper is organised in five sections, where Sect. 2 introduces the data set used and published in this study. Section 3 summarises the evaluation of each force model, intended to be easily comprehensible for programming. We discuss the result of a force comparison for one software package in more detail in Sect. 4 and give the summarised outcome of the comparisons for all COST-G ACs and candidate ACs. Finally, Sect. 5 concludes the results of this study and highlights the importance beyond the COST-G group.

The benchmark data set was compiled at the Institute of Geodesy (IfG) at Graz University of Technology. It consists of several accelerations a spacecraft experiences, evaluated along a given orbit, which are commonly used in orbit and gravity field determination. The models that describe the accelerations are evaluated along a one day GRACE orbit arc (integrated for 3 July 2008 using Encke's method, see

Accelerations modelled in the benchmark data set. The magnitude is a rough estimation for the given GRACE orbit, the list is ordered according to the magnitude of the forces.

All resulting accelerations are expressed in the CRF, the gravity field is additionally given in TRF. For each acceleration one file is provided. Accelerations consisting of several components, such as ocean tides, are provided in separate files for the largest constituents. Each file provides time stamps in GPS time (expressed in Modified Julian Date [MJD]) and the respective

Norm of the benchmark accelerations.

Additional data set

The main goal of the data is to provide a reference for basic software comparisons. It enables a comparison of the background force model implementations by evaluating the models at the given orbital positions in different software packages, and may serve as a reference for the handling of celestial and terrestrial reference frames. The most straight forward approach of comparison is to evaluate the force models with a given space geodesy software package and to print the resulting accelerations for each model. By subtracting the obtained accelerations from the benchmark data, differences may be revealed

In the following we also use the maximum absolute deviation from the reference (

In this section we give a few notes on each force in the benchmark data set. All formulae correspond with the IERS 2010 conventions

The gravity field model used for the benchmark data set is EIGEN-6C4

The accelerations and the potential are related by

The influence of other celestial bodies (denoted by the subscript cb) than the Earth is computed with positions derived from the JPL DE421

Solid Earth tides are computed according to IERS 2010 conventions using the anelastic model. Besides fundamental quantities of the Earth, the solid Earth tides depend on the position of the Sun, the Moon and the load Love numbers. They affect the spherical harmonic spectrum up to degree four.

According to the IERS 2010 conventions the computation is divided into two steps. Step 1 computes the coefficients due to the tide generating potential for degree 2 and 3, as well as the effect of degree 2 on degree 4 coefficients. Step 1 is frequency independent, whereas step 2 states frequency dependent corrections for degree two.

Step 1 – corresponds with

Step 2: Corrections on

The Doodson angle argument reads as

In the tidal frequency vector

The final coefficients

Two different models are provided for the ocean tides: EOT11a

To complete the tidal spectrum, admittances between the major tides can be computed using linear interpolation

Relativistic corrections are computed according to the IERS 2010 conventions for General Relativity using

AOD1B RL06

“The pole tide of the solid Earth is generated by the centrifugal effect of polar motion”

Atmospheric tides are modelled using AOD1B RL06 product and contain all twelve tidal constituents from degree 2 to 180. Additionally, the accelerations caused by S1 tide only are given separately. The evaluation of the atmospheric tides follows

The model is given in ICGEM-format as well. Using this format, the application follows the method described for the ocean tides (see Eq.

Similar to the solid Earth pole tide, the ocean pole tide is a result of the centrifugal effect of polar motion on the oceans. The implementation for the benchmark data set follows the IERS 2010 conventions employing the Desai model

The benchmark data set was created, used and examined within the COST-G initiative, where AIUB, GFZ, GRGS and IfG are currently acting as COST-G ACs. LUH is a candidate AC. To augment the combination effort and in particular to rule out large systematic differences in the implementation of background force models, all contributing groups performed a comparison with the benchmark data using their own software packages. This includes the Bernese GNSS software (AIUB), EPOS (GFZ), GINS (GRGS), GROOPS (IfG), GRACE-SIGMA (LUH) and GRASP (LUH). The intention of this comparison is to test the implementation and handling of the background models in each software package. Every software tries to reproduce the reference accelerations as closely as possible by employing the provided rotation between TRF and CRF. The GROOPS software serves as a reference as it was used to compute the benchmark data set.

The software packages follow different approaches of modelling gravity fields from satellite data. Even though data is treated differently, we expect a high level of agreement with the benchmark data for background model handling. The following sub-sections give a brief introduction to each package.

The Bernese GNSS software is a scientific software package, used by more than 700 institutions around the world. It features space geodetic applications, mainly high-precision, multi-GNSS data processing for ground networks

The Earth Parameter and Orbit System (EPOS

The GINS (Géodésie par Intégrations Numériques Simultanées) software is developed and maintained by the GRGS of the French space agency. It is a multi-technique space geodetic software, capable of processing data from GNSS, SLR, VLBI, DORIS and inter-satellite ranging. It is used for operational processing of all space geodetic observation techniques.

The GRACE-SIGMA (GRACE-Satellite orbit Integration and Gravity field analysis in MAtlab) software is a recent development specifically designed for the processing of GRACE and GRACE-FO data. The software is developed at Institut für Erdmessung (IfE) of LUH. It is written entirely in MATLAB and uses strongly vectorised modules for modelling of disturbing forces, orbit propagation and orbit improvement. The integration of satellite ephemerides and state and sensitivity matrices is performed using an efficient in-house developed numerical integration approach

The GRAvity Satellite Processing engine (GRASP) software is dedicated to gravity field recovery from kinematic positions of satellites

The Gravity Recovery Object Oriented Programming System (GROOPS) used at IfG is a software suite for geodetic applications.
Its feature set includes the determination of GNSS orbits, clocks and ground station networks

To show the results of the software comparison for all provided force models, we evaluate all force models with the Bernese GNSS software and compute the norm of the difference vector between the benchmark accelerations and the accelerations from Bernese. We expect the magnitude of the differences to be small as the handling of the background forces is supposed to be generally compatible. Figure

Norm of the difference between the benchmark data and the evaluation with the Bernese GNSS software.

Maximum absolute deviation of the difference between the reference and the evaluation by the respective software of the COST-G ACs for each force considered in the benchmark data set.

Each of the above mentioned institutions performed the software comparison with the benchmark data. EPOS and GINS currently contribute only with a subset of forces, whereas all others accomplished the comparison for all accelerations provided in the reference data set. The limit for the difference in evaluating the models along the reference orbit was set to

We test and publish benchmark data of forces commonly used for gravity field and orbit determination purposes. The data set consists of orbital positions and accelerations evaluated along a one-day GRACE orbital arc. It is intended to enable fundamental software comparisons and bug detection and shall serve as a valuable contribution to the community. It is potentially interesting to any user of space geodetic software but especially to those groups where a solution is created from individual software packages by combination. The benchmark data was examined with the software packages currently used in the COST-G service. These packages agree with each other in the usage of the background models at a level of less than

The data is available via ftp from

TMG and AK compiled and provide the dataset, ML, KHN, JML, IK and MW performed the computational work of the comparison with the respective software package. 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 2020, EGU Geodesy Division”. It is a result of the EGU General Assembly 2020, 4–8 May 2020.

The study was performed in the framework of COST-G and the corresponding international team that is receiving support from the International Space Science Institute (ISSI) in Bern, Switzerland.

This research has been supported by the Swiss National Science Foundation (SNSF) (grant no. 200021_175942).

This paper was edited by Katrin Bentel and reviewed by two anonymous referees.