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WHU-SWPU-GOGR2022S: A combined gravity model of GOCE and GRACE

WHU-SWPU-GOGR2022S is a static gravity field model complete to spherical harmonic degree and order of 300 by combining GOCE and GRACE normal equations. Details of the processing procedures are as follows: (1) Details of the GOCE processing procedures: (1a) Input data: -- GOCE SGG data: EGG_NOM_2 (GGT: Vxx, Vyy, Vzz and Vxz) in GRF (9/10/2009-20/10/2013) -- GOCE SST data: SST_PKI_2, SST_PCV_2, SST_PRD_2 (9/10/2009-20/10/2013) -- Attitude: EGG_NOM_2 (IAQ), SST_PRM_2 (PRM) -- Non-conservative force: Common mode ACC (GG_CCD_1i) -- Background model: tidal model (solid etc.), third-body acceleration, relativistic corrections, ... (1b) Data progress strategies: -- Data preprocessing - Gross outlier elimination and interpolation (only for the data gaps less than 40s). - Splitting data into subsections for gaps > 40s -- The normal equation from SST data - Point-wise acceleration approach (PAA) - Extended Differentiation Filter (low-pass) - Max degree: up to 130 - Data: PKI, PCV, CCD -- The normal equation from SGG data - Direct LS method - Max degree: up to 300 - Data: GGT, PRD, IAQ, PRM - Band-pass filter: used to deal with colored-noise of GGT observations (pass band 0.005-0.100Hz ) - Forming the normal equations according to subsections - Spherical harmonic base function transformation instead of transforming GGT from GRF to LNRF -- Combination of SGG and SST - Max degree: up to 300 - The VCE technique is used to estimate the relative weights for Vxx, Vyy, Vzz and Vxz - Tikhonov Regularization Technique (TRT) is only applied to near (zonal) terms (m<20, n<=200) and high degree terms (n>200) - Strictly inverse the normal matrix based on OpenMP (2) Details of the GRACE processing procedures: (2a) Input data: -- GRACE L1B (JPL) data products: GNV1B RL02, ACC1B RL02, SCA1B RL03 and KBR1B RL03 -- AOD1B RL06 (GFZ) de-aliasing product -- Data period: 04/2002-05/2017 (2b) Data preprocessing: -- Splitting data of SCA1B into subsections for gaps > 120s and interpolation with polynomial for gaps <= 120s -- Splitting data of ACC1B into subsections for gaps > 5s and interpolation with polynomial for gaps <= 5s -- Gross outlier elimination ACC1B with a moving window of length 10 min, and interpolation with polynomial -- Pre-calibration of ACC1B with a-priori bias and scale Parameters provided by GRACE TN-02 (2c) Calculation method: - dynamic approach - numerical integrator: 8th-order Gauss-Jackson integrator - integrator step: 5 seconds - arc length: 24 hours (2d) Combination - GNV1B and KBR1B are combined with their a-priori precision, i.e. 2cm of GNV1B and 2um/s of KBR1B - The normal equations of different months are combined with variance components estimation (2e) Force models: - Earth's static gravity field: GGM05s up to d/o 180 - Solid earth tides: IERS 2010 - Ocean tides: FES2014b up to d/o 180 - Solid Earth pole tide: IERS 2010 - Ocean pole tide: Desai 2002 up to d/o 180 - N-body Perturbation: the Sun and Moon with JPL DE421 - atmospheric tides: Bode and Biancale model - AOD1B product: AOD1B RL06 model up to d/o 180 - General Relativistic effects: Schwarzschild terms of IERS 2010

A GOCE only gravity model GOSG02S based on the SGG and SST observations

GOSG02S is a static gravity field model complete to spherical harmonic degree and order of 300 derived by using the Satellite Gravity Gradiometry (SGG) data and the Satellite-to-Satellite Tracking (SST) observations along the GOCE orbit based on least-squares analysis. Input data: -- GOCE SGG data: EGG_NOM_2 (GGT: Vxx, Vyy, Vzz and Vxz) in GRF (9/10/2009-20/10/2013) -- GOCE SST data: SST_PKI_2, SST_PCV_2, SST_PRD_2 (9/10/2009-20/10/2013) -- Attitude: EGG_NOM_2 (IAQ), SST_PRM_2 (PRM) -- Non-conservative force: Common mode ACC (GG_CCD_1i) -- Background model: tidal model (solid etc.), third-body acceleration, relativistic corrections, ... -- GOSG02S is a GOCE only satellite gravity model, since no priori gravity information was used in modelling procedure. Data progress strategies: -- Data preprocessing - Gross outlier elimination and interpolation (only for the data gaps less than 40s). - Splitting data into subsections for gaps > 40s -- The normal equation from SST data - Point-wise acceleration approach (PAA) - Extended Differentiation Filter (low-pass) - Max degree: up to 130 - Data: PKI, PCV, CCD -- The normal equation from SGG data - Direct LS method - Max degree: up to 300 - Data: GGT, PRD, IAQ, PRM - Band-pass filter: used to deal with colored-noise of GGT observations (pass band 0.005-0.100Hz ) - Forming the normal equations according to subsections - Spherical harmonic base function transformation instead of transforming GGT from GRF to LNRF -- Combination of SGG and SST - Max degree: up to 300 - The VCE technique is used to estimate the relative weights for Vxx, Vyy, Vzz and Vxz - Tikhonov Regularization Technique (TRT) is only applied to near (zonal) terms (m<20, n<=200) and high degree terms (n>200) - Strictly inverse the normal matrix based on OpenMP

The experimental gravity field model XGM2019e

XGM2019e is a combined global gravity field model represented through spheroidal harmonics up to d/o 5399, corresponding to a spatial resolution of 2’ (~4 km). As data sources it includes the satellite model GOCO06s in the longer wavelength area combined with terrestrial measurements for the shorter wavelengths. The terrestrial data itself consists over land and ocean of gravity anomalies provided by courtesy of NGA (identical to XGM2016, having a resolution of 15’) augmented with topographically derived gravity over land (EARTH2014). Over the oceans, gravity anomalies derived from satellite altimetry are used (DTU13, in consistency with the NGA dataset).The combination of the satellite data with the terrestrial observations is performed by using full normal equations up to d/o 719 (15’). Beyond d/o 719, a block-diagonal least-squares solution is calculated for the high-resolution terrestrial data (from topography and altimetry). All calculations are performed in the spheroidal harmonic domain.In the spectral band up to d/o 719 the new model shows over land a slightly improved behavior over preceding models such as XGM2016, EIGEN6c4 or EGM2008 when comparing it to independent GPS leveling data. Over land and in the spectral range above d/o 719 the accuracy of XGM2019e suffers from the sole use of topographic forward modelling; Hence, errors are increased in well-surveyed areas compared to models containing real gravity data, e.g. EIGEN6c4 or EGM2008. However, the performance of XGM2019e can be considered as globally more homogeneous and independent from existing high resolution global models. Over the oceans the model exhibits an improved performance throughout the complete spectrum (equal or better than preceding models).

The polar extended gravity field model TIM_R6e

TIM_R6e is an extended version of the satellity-only global gravity field model TIM_R6 (Brockmann et al., 2019) which includes additional terrestrial gravity field observations over GOCE's polar gap areas. The included terrestrial information consists of the PolarGap campaign data (Forsberg et al., 2017) augumented by the AntGG gravity data compilation (Scheinert et al., 2016) over the southern polar gap (>83°S) and the ArcGP data (Forsberg et al. 2007) over the northern polar gap (>83°N). The combination is performed on normal equation level, encompassing the terrestrial data as spectrally limited geographic 0.5°x0.5° grids over the polar gaps.

ESA’s Release 6 GOCE gravity field model by means of the direct approach based on improved filtering of the reprocessed gradients of the entire mission (GO_CONS_GCF_2_DIR_R6)

"ESA’s Release 6 GOCE gravity field model by means of the direct approach based on improved filtering of the reprocessed gradients of the entire mission (GO_CONS_GCF_2_DIR_R6)" is a static gravitational model available via ICGEM (Ince et al., 2019)Model Characteristics----------------------GOCE Input Data:- Gradients: EGG_NOM_2 (re-calibrated release 2018, Siemes et al. 2019)- Orbits: SST_PRD_2 (reduced dynamic orbits)- Attitude: EGG_IAQ_2C- Data period: 20091009T000000-20131020T235959A-priori Information used:----------------------------The a-priori gravity field for the processing of the GOCE gravity gradients was the GOCE-model 5th release from the direct approach GO_CONS_GCF_2_DIR_R5 up to its maximum degree/order 300 (Bruinsma et al. 2014).Processing Procedures:----------------------The GOCE gravity gradients were processed without applying the external calibration corrections.The observation equations were filtered with a 0 - 125.0 mHz lowpass filter. Subsequently "SGG" normal equations to degree/order 300 have been computed separately for 46 continous time segments of approximately 1270 days totally (identified after the preprocessing of the data) and for each of the gradient components Txx, Tyy, Tzz and Txz.The Txx, Tyy, Tzz and Txz SGG normal equations were accumulated with the relative weight 1.0. But within the SGG components, all observation equations have been weighted individually according to its standard deviation estimated w.r.t. the a-priori gravity field.To overcome the numerical instability of the GOCE-SGG normal equation due to the polar gaps and to compensate for the poor sensitivity of the GOCE measurements in the low orders the following stabilizations were applied:1) The GOCE-SGG normal equation was fully combined with GRACE and SLR normal equations. Details about the latter contributions are given below.2) A spherical cap regularization in accordance to Metzler and Pail (2005) was iteratively computed to d/o 300 using the GRACE/SLR data mentioned below to degree/order 130.3) Additionally a Kaula regularization was applied to all coefficients beyond degree 180The solution was obtained by Cholesky decomposition of the accumulated normal equations.Details of the GRACE contribution:----------------------------------The GRACE part consists of 85 monthly normal equations to degree/order 200 out of the time span January 2007 till November 2014 from GFZ's GRACE Release 06 processing based on GNSS-SST and K-Band-Range-Rate data. For details of this GRACE release see Dahle et al. 2018.The following individual months are not covered by GRACE: 2001101, 201106, 201205, 201210, 201303, 201304, 201308, 201402 and 201407The harmonics of very-low degree, in particular degrees 2 and 3, cannot be estimated accurately with GRACE and GOCE data only. Therefore, normal equations from the following SLR missions were used in the combination in order to improve the gravity field solution:- LAGEOS-1/2, AJISAI, STARLETTE and STELLA from Jan. 2002 till Oct. 2018- LARES from Feb. 2012 till Oct. 2018The SLR tracking data were processed according to the GRACE Release 6 standards During the combination with GOCE, the GRACE contribution was taken only up to degree/order 130 and the SLR contribution only up to degree/oder 5As GRACE is sensitive for temporal variations in the Earth gravity field, the date 20100901 should be taken as reference epoch of this model. This date is mean of the included GRACE measurement time span by considering the mentioned missed months. This reference epoch is close to the mean of the measurement time span of the included SLR tracking data (20100701)Specific features of resulting gravity field--------------------------------------------The model is a satellite-only model based on a full combination of GOCE-SGG with GRACE and SLR tracking data, leading to both excellent orbit fits as well as GPS/leveling resultsProcessing details are presented in Pail et al. 2011.

The Earth's gravity field as seen by the GOCE satellite - an improved sixth release derived with the time-wise approach (GO_CONS_GCF_2_TIM_R6)

The static gravitational model GO_CONS_GCF_2_TIM_R6 Is the 6th release of the GOCE gravity field model by means of the time-wise approach.GOCE Input Data:- Gradients: EGG_NOM_2 (re-calibration, released 2018, version 0202)- Orbits:-- SST_PKI (kinematic orbits); SST_PCV (variance information of kinematic orbit positions),-- SST_RNX (original RINEX orbit data)- Attitude: EGG_IAQ_2C- Non-conservative accelerations: EGG_CCD_2C- Data period: 09/10/2009 - 20/10/2013No static a-priori gravity field information applied (neither as reference model, nor for constraining the solution)Processing procedures:- Gravity from orbits (SST):- short-arc integral method applied to kinematic orbits, up to degree/order 150- orbit variance information included as part of the stochastic model, it is refined by empirical covariance functions- Gravity from gradients (SGG):- parameterization up to degree/order 300- observations used: Vxx, Vyy, Vzz and Vxz in the Gradiometer Reference Frame (GRF)- realistic stochastic modelling by applying digital decorrelation filters to the observation equations; estimated separately for individual data segments applying a robust procedure- Combined solution:- addition of normal equations (SST D/O 150, SGG D/O 300)- Constraints:* Kaula-regularization applied to coefficients of degrees/orders 201 - 300 (constrained towards zero)* observation equations for zero gravity anomaly observations in polar regions (>83°) to constrain polar gaps towards zero (degree 11 to 300)- Optimum weighting (SST, SGG, constraints) based on variance component estimationSpecific features of resulting gravity field:- Gravity field solution is independent of any other gravity field information- Constraint towards zero starting from degree/order 201 to improve signal-to-noise ratio- Related variance-covariance information represents very well the true errors of the coefficients- Solution can be used for independent comparison and combination on normal equation level with other satellite-only models (e.g. GRACE), terrestrial gravity data, and altimetry- Since in the low degrees the solution is based solely on GOCE orbits, it is not competitive with a GRACE model in this spectral region is available via this data publication and via ICGEM (Ince et al., 2019). Link to ICGEM Website: http://icgem.gfz-potsdam.de- The reference epoch is 2010-01-01 (MJD 55197)Further processing details can be found in Brockmann (2014), Brockmann et al. (2014) Mayer-Gürr et al. (2005) and Pail et al. (2014).

The satellite-only gravity field model GOCO06s

GOCO06s is a satellite-only, global gravity field model up to degree and order 300, with secular and annual variations up to degree and order 120. It was produced by the GOCO Team (Technical University of Munich, University of Bonn, Graz University of Technology, Austrian Academy of Sciences, University of Bern) and is based on 1,160,000,000 observations from 19 satellites. The contributing satellite mission are: GOCE (TIM6 gradiometer observations), GRACE (ITSG-Grace2018s), kinematic orbits from Swarm A+B+C, TerraSAR-X, TanDEM-X, CHAMP, GRACE and GOCE, and SLR observations to LAGEOS, LAGEOS 2, Starlette, Stella, AJISAI, LARES, LARETS, Etalon 1/2 and BLITS. The combination of the individual data sources is performed on the basis of the full systems of normal equations, where the relative weighting between each constituent is determined by variance component estimation. In order to account for the polar gap of GOCE, the solution is Kaula-regularized after degree and order 150.The model is available via the ICGEM Service (Ince et al., 2019)

A GOCE only gravity model GOSG01S based on the SGG and SST observations

We compile the GOCE-only satellite model GOSG01S complete to spherical harmonic degree of 220 using Satellite Gravity Gradiometry (SGG) data and the Satellite-to-Satellite Tracking (SST) observations along the GOCE orbit based on applying a least-squares analysis. The diagonal components (Vxx, Vyy, Vzz) of the gravitational gradient tensor are used to form the system of observation equations with the band-pass ARMA filter. The point-wise acceleration observations (ax, ay, az) along the orbit are used to form the system of observation equations up to the maximum spherical harmonic degree/order 130. The GOCE related satellite gravity models GOSG01S, GOTIM05S, GODIR05S, GOTIM04S, GODIR04S, GOSPW04S, JYY_GOCE02S, EIGEN-6C2 and EGM2008 are also validated by using GPS-leveling data in China and USA. According to the truncation at degree 200, the statistic results show that all GGMs have very similar differences at GPS-leveling points in USA, and all GOCE related gravity models have better performance than EGM2008 in China.This new model was developed by School of Geodesy and Geomatics (SGG) of Wuhan University (WHU) and Institute of Geodesy of University of Stuttgart. More details about the gravity field model GOSG01S is given in our paper “A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models ” (Xu X, Zhao Y, Reubelt T, et al. Geodesy and Geodynamics. 2017, 8(4): 260-272. http://dx.doi.org/10.1016/j.geog.2017.03.013). This work is supported by the National Key Basic Research Program of China (973 program, grant no.: 2013CB733301), the Major International (Regional) Joint Research Project (grant no.: 41210006).

SGG-UGM-1: the high resolution gravity field model based on the EGM2008 derived gravity anomalies and the SGG and SST data of GOCE satellite

SGG-UGM-1 is a static gravity field model based on EGM2008 derived gravity anomalies and GOCE Satellite Gravity Gradiometry (SGG) data and the Satellite-to-Satellite Tracking (SST) observations up to degree and order 2159. Block-diagonal normal equation system up to degree and order 2159 are formed with EGM2008 gravity anomaly data using block-diagonal least squares method. Fully occupied normal equation system up to degree and order 220 are formed by GOCE SGG data and the SST observations along the GOCE orbit based on least-squares analysis. The diagonal components (Vxx, Vyy, Vzz) of the gravitational gradient tensor are used to form the system of observation equations with the band-pass ARMA filter. The point-wise acceleration observations (ax, ay, az) along the orbit are used to form the system of observation equations up to the maximum spherical harmonic degree/order 130. SGG-UGM-1 is resolved by combination of the two normal equation systems using least squares method. It is the first generation of high-resolution gravity model in ICGEM developed by School of Geodesy and Geomatics (SGG), Wuhan University (WHU). More details about the determination of the model are given in our paper “The determination of an ultra high gravity field model SGG-UGM-1 by combining EGM2008 gravity anomaly and GOCE observation data” (Liang W, Xu X, Li J, et al. Acta Geodaeticaet Cartographica Sinica. 2018, 47(4): 425-434. DOI:10.11947/j. AGCS.2018.20170269) and “A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models ” (Xu X, Zhao Y, Reubelt T, et al. Geodesy and Geodynamics. 2017, 8(4): 260-272. http://dx.doi.org/10.1016/j.geog.2017.03.013). The work is supported by the Natural Science Foundation of China (Nos. 41774020, 41210006 and 41404020

Using real polar terrestrial gravimetry data to overcome the polar gap problem of GOCE - the gravity field model IGGT_R1C

With the successful completion of ESA's PolarGAP campaign, terrestrial gravimetry data (gravity anomalies) are now available for both polar regions. Therefore, it is now possible to overcome the GOCE polar gap by using real gravimetry data instead of some regularization methods. But terrestrial gravimetry data needs to become filtered to remove the high-frequency gravity information beyond spher. harm. degree e.g. 240 to avoid disturbing spectral leakage in the satellite-only gravity field models. For the gravity anomalies from the Arctic, we use existing global gravity field models (e.g., EGM2008) for this filtering. But for the gravity anomalies from Antarctica, we use local gravity field models based on a point mass modeling method to remove the high-frequency gravity information. After that, the boundary-value condition from Molodensky's theory is used to build the observation equations for the gravity anomalies. Finally, variance component estimation is applied to combine the normal equations from the gravity anomalies, from the GOCE GGs (e.g., IGGT_R1), from GRACE (e.g., ITSG-Grace2014s) and for Kaula's rule of thumb (higher degree/order parts) to build a global gravity field model IGGT_R1C without disturbing impact of the GOCE polar gap. This new model has been developed by German Research Centre for Geosciences (GFZ), Technical University of Berlin (TUB), Wuhan University (WHU) and Huazhong University of Science and Technology (HUST).Parametersstatic model modelname IGGT_R1Cproduct_type gravity_fieldearth_gravity_constant 0.3986004415E+15radius 0.6378136460E+07max_degree 240norm fully_normalizedtide_system tide_freeerrors formal

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