Influence of time variable geopotential models on precise orbits of altimetry satellites,global and regional mean sea level trends

During the last decade a significant progress has been reached in the investigation of the gravity field of the Earth. Besides static, also time variable geopotential models have been recently created. In this paper we investigate the impact of the recent time variable geopotential models on altimetry satellite orbits and such altimetry products based on these orbits, as global and regional mean sea level trends. We show that the modeling of time variable gravity improves the orbit solutions, at least for the GRACE period where time variable gravity is sufficiently accurately observed by this mission. Our analysis includes six geopotential models jointly developed by GFZ German Research Centre for Geosciences and Space Geodesy Research Group (CNES/GRGS) Toulouse: the stationary model EIGEN-GL04S, a stationary version of EIGEN-6S (EIGEN-6S_stat), a corrected version of EIGEN-6S and three enhanced versions of EIGEN-6S called EIGEN-6S2, EIGEN-6S2A and EIGEN-6S2B. By “stationary” we mean “containing periodic parameters such as annual and semi-annual variations, but no secular (drift) terms”. We computed precise orbits for the radar altimetry satellites ERS-1, ERS-2, TOPEX/Poseidon, and Envisat over 20 years between 1991 and 2011. The orbit, single-mission and multi-mission altimetry crossover analyses show that the time variable models EIGEN-6S_corrected, EIGEN-6S2 and its two precursors EIGEN-6S2A/B perform notably better than the stationary models for the GRACE period from 2003 onwards. Thus, using EIGEN-6S2 and EIGEN-6S2A/B we have got 3.6% smaller root mean square fits of satellite laser ranging observations for Envisat, as when using EIGEN-GL04S. However, for the pre-GRACE period 1991–2003, the stationary geopotential models EIGEN-GL04S and EIGEN-6S_stat as well as EIGEN-6S2 having no drift terms for degree 3–50 at this time interval perform superior compared to EIGEN-6S_correct and EIGEN-6S2A/B which contain drifts for this period. We found, that the time variable geopotential models have a low (0.1–0.2 mm/yr) impact on our results for the global mean sea level trend. However, we found strong East/West differences up to 3 mm/yr in the regional mean sea level trends when using orbits of all four satellites based on time variable and stationary geopotential models. We show that these differences are related to the relative drifts of the centers-of-origin between the orbit solutions based on the time variable and stationary geopotential models. From the results of our detailed study, we conclude that the final version of the time variable gravity field model EIGEN-6S2 performs best for the four satellites tested. This model provides the most reliable and mission-consistent sea level estimates for the whole time period from 1992 to 2010. This model is of maximum spherical harmonic degree and order 260 and contains time series for drifts as well as annual and semiannual variations of the spherical harmonic coefficients for degree 2–50.     

Semi-stochastic modification of second-order radial derivative of Abel–Poisson’s formula for validating satellite gravity gradiometry data

The geoid can be used to validate the satellite gravity gradiometry data. Validation of such data is important prior to their downward continuation because of amplification of the data errors through this process. In this paper, the second-order radial derivative of Abel–Poisson’s formula is modified stochastically to reduce the effect of the far-zone geoid and generate the second-order radial derivative of geopotential at 250 km level. The numerical studies over Fennoscandia show that this method yields the gradients with an error of 10 mE and when the long wavelength of geoid is removed from the estimator and restored after the computations (remove–compute–restore) the error will be in 1 mE level. We name this method semi-stochastic modification. The best case scenario is found when the degree of modification of the integral formula is 200 and the long wavelength geoid to degree 100 is removed and restored. In this case the geoid should have a resolution of 15′ × 15′ and the integration should be performed over a cap size of 3°.     

Contribution of satellite laser ranging to combined gravity field models

In the framework of satellite-only gravity field modeling, satellite laser ranging (SLR) data is typically exploited to recover long-wavelength features. This contribution provides a detailed discussion of the SLR component of GOCO02S, the latest release of combined models within the GOCO series. Over a period of five years (January 2006 to December 2010), observations to LAGEOS-1, LAGEOS-2, Ajisai, Stella, and Starlette were analyzed. We conducted a series of closed-loop simulations and found that estimating monthly sets of spherical harmonic coefficients beyond degree five leads to exceedingly ill-posed normal equation systems. Therefore, we adopted degree five as the spectral resolution for real data analysis. We compared our monthly coefficient estimates of degree two with SLR and Gravity Recovery and Climate Experiment (GRACE) time series provided by the Center for Space Research (CSR) at Austin, Texas. Significant deviations in C₂₀ were noted between SLR and GRACE; the agreement is better for the non-zonal coefficients. Fitting sinusoids together with a linear trend to our C₂₀ time series yielded a rate of (−1.75 ± 0.6) × 10⁻¹¹/yr; this drift is equivalent to a geoid change from pole to equator of 0.35 ± 0.12 mm/yr or an apparent Greenland mass loss of 178.5 ± 61.2 km³/yr. The mean of all monthly solutions, averaged over the five-year period, served as input for the satellite-only model GOCO02S. The contribution of SLR to the combined gravity field model is highest for C₂₀, and hence is essential for the determination of the Earth’s oblateness.     

The contribution of local gravimetric geoid models to the calibration of satellite altimetry data and an outlook of the latest GOCE GGM performance in Gavdos

The use of geoid heights has been one of the available methodologies utilized for the independent calibration/validation of altimeters on-board satellites. This methodology has been employed for long in the Gavdos dedicated cal/val facility (Crete, Greece), where calibration results for the Jason satellites have been estimated, both for ascending and descending passes. The present work gives a detailed overview of the methodology followed in order to estimate a high-resolution and accuracy gravimetric geoid model for the wider Gavdos area, in support of the on-going calibration work. To estimate the geoid model, the well-known remove-compute-restore method is used while residual geoid heights are estimated through least-squares collocation so that associated errors are determined as well. It is found that the estimated formal geoid errors from LSC along passes 018 and 109 of Jason satellites, used for the bias estimation, range between ±0.8–1.6 cm. The so-derived geoid heights are employed in the determination of the Jason-2 altimeter bias for all available cycles (cycles 1-114, spanning the period from July 2008 to August 2011) together with the RioMed DOT model. From the results acquired the Jason-2 bias has been estimated to be +196.1 ± 3.2 mm for pass 109 and +161.9 ± 5.1 mm for pass 018. Within the same frame, the GOCE/GRACE-based geopotential model GOCO02s has been used to estimate the mean dynamic ocean topography and the steady-state circulation in the area around Gavdos. The so-derived DOT model was used to estimate the Jason-2 bias in an effort to evaluate the performance of satellite-only geoid models and investigate whether their spatial resolution and accuracy provides some improvement w.r.t. traditional local gravimetric geoids. From the results acquired with geoid heights from GOCO02s, the estimated Jason-2 bias deviates significantly from that of the local gravimetric model, which can be attributed to a possible mean offset and the low resolution of GOCE-based GGMs. On the other hand, when the newly estimated GOCE-based DOT was employed with geoid heights from the local gravimetric geoid model, the Jason-2 bias has been estimated to be +185.1 ± 3.2 mm for pass 109 and +130.2 ± 5.1 mm for pass 018.     

Mapping mean lake surface from satellite altimetry and GPS kinematic surveys

Lake water height is a key variable in water cycle and climate change studies, which is achievable using satellite altimetry constellation. A method based on data processing of altimetry from several satellites has been developed to interpolate mean lake surface (MLS) over a set of 22 big lakes distributed on the Earth. It has been applied on nadir radar altimeters in Low Resolution Mode (LRM: Jason-3, Saral/AltiKa, CryoSat-2) in Synthetic Aperture Radar (SAR) mode (Sentinel-3A), and in SAR interferometric (SARin) mode (CryoSat-2), and on laser altimetry (ICESat). Validation of the method has been performed using a set of kinematic GPS height profiles from 18 field campaigns over the lake Issykkul, by comparison of altimetry’s height at crossover points for the other lakes and using the laser altimetry on ICESat-2 mission. The precision reached ranges from 3 to 7 cm RMS (Root Mean Square) depending on the lakes. Currently, lake water level inferred from satellite altimetry is provided with respect to an ellipsoid. Ellipsoidal heights are converted into orthométric heights using geoid models interpolated along the satellite tracks. These global geoid models were inferred from geodetic satellite missions coupled with absolute and regional anomaly gravity data sets spread over the Earth. However, the spatial resolution of the current geoid models does not allow capturing short wavelength undulations that may reach decimeters in mountaineering regions or for rift lakes (Baikal, Issykkul, Malawi, Tanganika). We interpolate in this work the geoid height anomalies with three recent geoid models, the EGM2008, XGM2016 and EIGEN-6C4d, and compare them with the Mean Surface of 22 lakes calculated using satellite altimetry. Assuming that MLS mimics the local undulations of the geoid, our study shows that over a large set of lakes (in East Africa, Andean mountain and Central Asia), short wavelength undulations of the geoid in poorly sampled areas can be derived using satellite altimetry. The models used in this study present very similar geographical patterns when compared to MLS. The precision of the models largely depends on the location of the lakes and is about 18 cm, in average over the Earth. MLS can serve as a validation dataset for any future geoid model. It will also be useful for validation of the future mission SWOT (Surface Water and Ocean Topography) which will measure and map water heights over the lakes with a high horizontal resolution of 250 by 250 m.     

Comparative evaluation of five global gravity models over a part of the Bay of Bengal

Several global gravity models (GGMs) are freely available in the public domain, which can be utilised to study the earth's gravity field in almost every part of the globe. The present study compared the free-air gravity anomalies calculated from the five GGMs EGM2008, EIGEN6C4, GECO, XGM2019e_2159, and SGG-UGM-2 archived by the International Centre for Global Earth Models (ICGEM) with respect to shipborne gravity in the Bay of Bengal. The average correlation coefficient and covariance are ～ 96 % and ～ 450mGal². The mean difference between the shipborne and the modelled gravity is ? 5 mGal. Relatively higher amplitude gravity differences observed at the continental-oceanic transition, the 85°E and Ninetyeast ridges, and the western basin are possibly due to high gradient, dominant density contrasts, and rugged topography. The average standard deviation and root-mean-square-error (RMSE) of the differences are ～ 6.5 mGal and ～ 7.5 mGal. A significantly lower standard deviation and RMSE found for the models generated at higher degree/order compared to lower degree/order is due to diminishing omission error of the GGMs with increasing degrees of truncation. The spectral analysis and coherence estimation of the modelled gravity demonstrate excellent correspondence for anomalies wider than ～ 25 km. The agreement between anomaly amplitudes and shapes and calculated statistics indicates that the accuracy and resolution of the modelled gravity data are certainly good enough for regional-scale studies aiming to map Moho topography and mantle structures. However, the delineation of shorter wavelength features originating from the shallow-level basement/sedimentary might be uncertain and requires further validations. The present study confirms that all five models show promising results in terms of their accuracy and resolution. Moreover, the SGG-UGM-2 and XGM2019e_2159 models compare favourably with respect to the GECO, EIGEN6C4 and EGM2008 models in the Bay of Bengal.     

GRACE gravity field modeling with an investigation on correlation between nuisance parameters and gravity field coefficients

The GRACE (Gravity Recovery And Climate Experiment) monthly gravity models have been independently produced and published by several research institutions, such as Center for Space Research (CSR), GeoForschungsZentrum (GFZ), Jet Propulsion Laboratory (JPL), Centre National d’Etudes Spatiales (CNES) and Delft Institute of Earth Observation and Space Systems (DEOS). According to their processing standards, above institutions use the traditional variational approach except that the DEOS exploits the acceleration approach. The background force models employed are rather similar. The produced gravity field models generally agree with one another in the spatial pattern. However, there are some discrepancies in the gravity signal amplitude between solutions produced by different institutions. In particular, 10%–30% signal amplitude differences in some river basins can be observed. In this paper, we implemented a variant of the traditional variational approach and computed two sets of monthly gravity field solutions using the data from January 2005 to December 2006. The input data are K-band range-rates (KBRR) and kinematic orbits of GRACE satellites. The main difference in the production of our two types of models is how to deal with nuisance parameters. This type of parameters is necessary to absorb low-frequency errors in the data, which are mainly the aliasing and instrument errors. One way is to remove the nuisance parameters before estimating the geopotential coefficients, called NPARB approach in the paper. The other way is to estimate the nuisance parameters and geopotential coefficients simultaneously, called NPESS approach. These two types of solutions mainly differ in geopotential coefficients from degree 2 to 5. This can be explained by the fact that the nuisance parameters and the gravity field coefficients are highly correlated, particularly at low degrees. We compare these solutions with the official and published ones by means of spectral analysis. It is found that our solutions are, in general, consistent with others in the spatial pattern. The water storage variations of the Amazon, Chari and Ganges river basins have also been computed. The variations computed with the NPARB approach are closer to those produced by JPL and DEOS solutions, while the variations produced with the NPESS approach are in good agreement with those produced by the CSR and GFZ solutions. A simulation study is implemented with considering realistic noise and low-frequency error. The two approaches are used to recover the true model. The NPESS solution appears closer to the true one. Therefore we are inclined to estimate the nuisance parameters simultaneously with the geopential coefficients.     

Gravity field models from kinematic orbits of CHAMP,GRACE and GOCE satellites

The aim of our work is to generate Earth’s gravity field models from GPS positions of low Earth orbiters. Our inversion method is based on Newton’s second law, which relates the observed acceleration of the satellite with forces acting on it. The observed acceleration is obtained as numerical second derivative of kinematic positions. Observation equations are formulated using the gradient of the spherical harmonic expansion of the geopotential. Other forces are either modelled (lunisolar perturbations, tides) or provided by onboard measurements (nongravitational perturbations). From this linear regression model the geopotential harmonic coefficients are obtained.     

Modeling and analysis of Earth’s gravity field measurement performance by inner-formation flying system

The Earth’s gravity field can be measured with high precision by constructing the purely gravitational orbit of the inner-satellite in Inner-formation Flying System (IFS), which is independently proposed by Chinese scholars and offers a new way to carry out gravity field measurement by satellite without accelerometers. In IFS, for the purpose of quickly evaluating the highest degree of recovered gravity field model and geoid error as well as analyzing the influence of system parameters on gravity field measurement, an analytical formula was established by spectral analysis method. The formula can reflect the analytical relationship between gravity field measurement performance and system parameters such as orbit altitude, the inner-satellite orbit determination error, the inner-satellite residual disturbances, data sampling interval and total measurement time. This analytical formula was then corrected by four factors introduced from numerical simulation of IFS gravity field measurement. By comparing computation results from corrected analytical formula and the actual gravity field measurement performance by CHAMP, the correctness and rationality of this analytical formula were verified. Based on this analytical formula, the influences of system parameters on IFS gravity field measurement were analyzed. It is known that gravity field measurement performance is a monotone decreasing function of orbit altitude, the inner-satellite orbit determination error, the inner-satellite residual disturbances, data sampling interval and the reciprocal of total measurement time. There is a match relationship between the inner-satellite orbit determination error and residual disturbances, in other words, the change rate of gravity field measurement performance with one of them is seriously restricted by their relative size. The analytical formula can be used to quantitatively evaluate gravity field measurement performance fast and design IFS parameters optimally. It is noted that the analytical formula and corresponding conclusions are applied to any gravity satellite which measures gravity field by satellite perturbation orbit.     

GPS-only gravity field recovery with GOCE,CHAMP, and GRACE

Gravity missions such as the Gravity field and steady-state Ocean Circulation Explorer (GOCE) are equipped with onboard Global Positioning System (GPS) receivers for precise orbit determination (POD), instrument time-tagging, and the extraction of the long wavelength part of the Earth’s gravity field. The very low orbital altitude of the GOCE satellite and the availability of dense 1 s GPS tracking data are ideal characteristics to exploit the contribution of GPS high-low Satellite-to-Satellite Tracking (hl-SST) to gravity field determination. We present gravity field solutions based on about 8 months of GOCE GPS hl-SST data from 2009 and compare the results with those obtained from the CHAllenging Minisatellite Payload (CHAMP) and Gravity Recovery And Climate Experiment (GRACE) missions. The very low orbital altitude of GOCE significantly improves gravity field recovery from GPS hl-SST data above degree 20, but not for the degrees below 20, where the quality of the spherical harmonic coefficients remains essentially unchanged. Despite the limited time span of GOCE data used, the gravity field of the Earth can be resolved up to about degree 115 using GPS data only. Empirically determined phase center variations (PCVs) of the GOCE onboard GPS helix antenna are, however, mandatory to achieve this performance.     

Generation of vertical–horizontal and horizontal–horizontal gravity gradients using stochastically modified integral estimators

The Earth’s gravity field modelling is an ill-posed problem having a sensitive solution to the error of data. Satellite gravity gradiometry (SGG) is a space technique to measure the second-order derivatives of geopotential for modelling this field, but the measurements should be validated prior to use. The existing terrestrial gravity anomalies and Earth gravity models can be used for this purpose. In this paper, the second-order vertical–horizontal (VH) and horizontal–horizontal (HH) derivatives of the extended Stokes formula in the local north-oriented frame are modified using biased, unbiased and optimum types of least-squares modification. These modified integral estimators are used to generate the VH and HH gradients at 250 km level for validation purpose of the SGG data. It is shown that, unlike the integral estimator for generating the second-order radial derivative of geopotential, the system of equations from which the modification parameters are obtained is unstable for all types of modification, with large cap size and high degree, and regularization is strongly required for solving the system. Numerical studies in Fennoscandia show that the SGG data can be estimated with an accuracy of 1 mE using an integral estimator modified by a biased type least-squares modification. In this case an integration cap size of 2.5° and a degree of modification of 100 for integrating 30′ × 30′ gravity anomalies are required.     

Regional gravity field model in Pakistan area from the combination of CHAMP,GRACE and ground data using least squares collocation: A case study

This study describes a methodology of recovery of the Earth’s gravity field from CHAMP and GRACE satellites data in Pakistan using least squares collocation (LSC) based downward continuation technique. The CHAMP height anomalies and GRACE gravity disturbances derived from the observed satellite data have been used in combination solution using LSC with observed gravity values at the Earth surface. The combined covariance functions of height anomalies and/or gravity disturbances at satellite altitudes and observed gravity anomalies at Earth surface have been used as the basis for combination and downward continuation solution. The variance of predicted gravity anomalies from GRACE gravity disturbances is relatively lower than the corresponding results of gravity anomalies from CHAMP height anomalies. This fact may be attributed partly to the amplification of noise and partly to the unstable inverse transformation process of height anomalies to gravity anomalies. The impact of data error variance has been studied in the context of smoothing and noise reduction in the final solution of downward continuation using least squares collocation. The raising of data error suppresses the noise and as a result a smooth final solution is obtained. The prediction results appear to be dependent on the quality of data and goodness of combined covariance function, which are fairly comparable for the CHAMP and GRACE data. The recovered gravity field from satellite data appears to contribute mainly to medium and long wavelength parts of total gravity field spectrum. Due to flexibility of data handling in least squares collocation, this procedure is applicable to any observable of gravity field being at different altitudes and with different data spacing.     

GSFC DORIS contribution to ITRF2008

The NASA GSFC DORIS analysis center has provided weekly DORIS solutions from November 1992 to January 2009 (839 SINEX files) of station positions and Earth Orientation Parameters for inclusion in the DORIS contribution to ITRF2008. The NASA GSFC GEODYN orbit determination software was used to process the orbits and produce the normal equations. The weekly SINEX gscwd10 submissions included DORIS data from Envisat, TOPEX/Poseidon, SPOT-2, SPOT-3, SPOT-4, SPOT-5. The orbits were mostly seven days in length (except for weeks with data gaps or maneuvers). The processing used the GRACE-derived EIGEN-GL04S1 gravity model, updated modeling for time-variable gravity, the GOT4.7 ocean tide model and tuned satellite-specific macromodels for SPOT-2, SPOT-3, SPOT-4, SPOT-5 and TOPEX/Poseidon. The University College London (UCL) radiation pressure model for Envisat improves nonconservative force modeling for this satellite, reducing the median residual empirical daily along-track accelerations from 3.75 × 10⁻⁹ m/s² with the a priori macromodel to 0.99 × 10⁻⁹ m/s² with the UCL model. For the SPOT and Envisat DORIS satellite orbits from 2003 to 2008, we obtain average RMS overlaps of 0.8–0.9 cm in the radial direction, 2.1–3.4 cm cross-track, and 1.7–2.3 cm along-track. The RMS orbit differences between Envisat DORIS-only and SLR & DORIS orbits are 1.1 cm radially, 6.4 cm along-track and 3.7 cm cross-track and are characterized by systematic along-track mean offsets due to the Envisat DORIS system time bias of ±5–10 μs. We obtain a good agreement between the geometrically-determined geocenter parameters and geocenter parameters determined dynamically from analysis of the degree one terms of the geopotential. The intrinsic RMS weekly position repeatability with respect to the IDS-3 combination ranges from 2.5 to 3.0 cm in 1993–1994 to 1.5 cm in 2007–2008.     

Determination of the mean dynamic ocean topography model through combining multi-source gravity data and DTU15 MSS around China's coast

Mean dynamic ocean topography (or MDT) is closely related to ocean circulation and global climate change. It has important scientific significance and application value for the development and utilization of marine resources in China's coastal areas. Based on the terrain gravity, marine gravity, and SRTM 3?s data, an algorithm to reduce the problem of gravity data gaps between land and sea is proposed. A consistent land-sea gravity model is established based on point-mass fusion method. Then geoid model, which accuracy was estimated to be 8.5?cm through the verification of 348 GNSS/level data from the coastal provinces, of China's coastal areas was calculated through remove-restore technique. Connecting the above geoid model with DTU15 MSS model to establish a MDT model in China's coastal areas using the direct method in space domain. The effect of gravity field model, dominant factors of sea surface topography, and low pass filter are analyzed. Taking Bohai Sea and Yellow Sea as an example, and comparing MDT with the two international models CNES_2013_MDT and DTU15_MDT. The results show that the MDT has the potential to construct a vertical datum of the ocean and carry out related scientific research and application.     

CNES/GRGS 10-day gravity field models (release 2) and their evaluation

GRACE, designed to monitor temporal variations in the fluid mass at the surface of the Earth, is still operating and providing invaluable data 7 years after launch. One hundred and ninety-nine satellite-only geopotential solutions to degree and order 50 were recomputed per 10-day interval for the period 29 July 2002–27 May 2008 using an improved data editing and solution regularization procedure. These release 2 solutions are significantly improved compared to release 1 solutions, the noise over deserts and oceans in the form of North–South striping being reduced by 20–40%. This is thanks to the tailored regularization of each individual Stokes coefficient applied in the solution procedure, and to a time-variable reference model containing mean annual, semiannual and secular variations for degrees 2–50 towards which the variations per 10-day solution are constrained. It may attenuate signals of the order of a few percent, whereas this always occurs when applying a Gaussian smoother even with a half-width smoothing radius as small as 300 km. The uncertainty of an individual point in the time series of a basin expressed in equivalent water height inferred from the 10-day solutions is approximately 20 mm. Comparison of these 10-day solutions to monthly GRACE project solutions (CSR, GFZ and JPL) shows substantial differences. Even for the largest basin, the Amazon, a 15% difference in annual amplitude is found between CNES release 2 and CSR versus GFZ and JPL. The mass-loss estimates for East and West Greenland vary by 100%. Sometimes clear outliers are detected in the GFZ and JPL solutions when a particular basin is studied, which have to be eliminated. In view of the large differences detected between the time series for specific basins, it is hazardous to draw conclusions based on a single solution.     

Improvement of the GRACE star camera data based on the revision of the combination method

The new release of the sensor and instrument data (Level-1B release 02) of the Gravity Recovery and Climate Experiment (GRACE) had a substantial impact on the improvement of the overall accuracy of the gravity field models. This has implied that improvements on the sensor data level can still significantly contribute to arriving closer to the GRACE baseline accuracy. The recent analysis of the GRACE star camera data (SCA1B RL02) revealed their unexpectedly higher noise. As the star camera (SCA) data are essential for the processing of the K-band ranging data and the accelerometer data, thorough investigation of the data set was needed. We fully reexamined the SCA data processing from Level-1A to Level-1B with focus on the combination method of the data delivered by the two SCA heads. In the first step, we produced and compared our own combined attitude solution by applying two different combination methods on the SCA Level-1A data. The first method introduces the information about the anisotropic accuracy of the star camera measurement in terms of a weighing matrix. This method was applied in the official processing as well. The alternative method merges only the well determined SCA boresight directions. This method was implemented on the GRACE SCA data for the first time. Both methods were expected to provide optimal solution characteristic by the full accuracy about all three axes, which was confirmed. In the second step, we analyzed the differences between the official SCA1B RL02 data generated by the Jet Propulsion Laboratory (JPL) and our solution. SCA1B RL02 contains systematically higher noise of about a factor 3–4. The data analysis revealed that the reason is the incorrect implementation of algorithms in the JPL processing routines. After correct implementation of the combination method, significant improvement within the whole spectrum was achieved. Based on these results, the official reprocessing of the SCA data is suggested, as the SCA attitude data are one of the key observations needed for the gravity field recovery.     

An alternative computation of a gravity field model from GOCE

GOCE is the first satellite with a gravitational gradiometer (SGG). This allows to determine a gravity field model with high spatial resolution and high accuracy. Four of the six independent components of the gravitational gradient tensors (GGT) are measured with high accuracy in the so-called measurement band (MB) from 5 to 100 mHz by the GOCE gradiometer. Based on more than 1 year of GOCE measurements, two gravity field models have been derived. Here, we introduce a strategy for spherical harmonic analysis (SHA) from GOCE measurements, with a bandpass filter applied to the SGG data, combined with orbit analysis based on the integral equation approach, and additional constraints (or stabilization) in the polar areas where no observation is available due to the orbit geometry. In addition, we combined the GOCE SGG part with a set of GRACE normal equations. This improves the accuracy of the gravity field in the long-wavelength parts, due to the complementarity of GOCE and GRACE. Comparison with other models and with external data shows that our results are rather close to the GPS-levelling data in well-selected test regions, with an uncertainty of 4–7 cm, for truncation at degree 200.     

GOCE orbit analysis: Long-wavelength gravity field determination using the acceleration approach

The restricted sensitivity of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) gradiometer instrument requires satellite gravity gradiometry to be supplemented by orbit analysis in order to resolve long-wavelength features of the geopotential. For the hitherto published releases of the GOCE time-wise (TIM) and GOCE space-wise gravity field series—two of the official ESA products—the energy conservation method has been adopted to exploit GPS-based satellite-to-satellite tracking information. On the other hand, gravity field recovery from data collected by the CHAllenging Mini-satellite Payload (CHAMP) satellite showed the energy conservation principle to be a sub-optimal choice. For this reason, we propose to estimate the low-frequency part of the gravity field by the point-wise solution of Newton’s equation of motion, also known as the acceleration approach. This approach balances the gravitational vector with satellite accelerations, and hence is characterized by (second-order) numerical differentiation of the kinematic orbit. In order to apply the method to GOCE, we present tailored processing strategies with regard to low-pass filtering, variance–covariance information handling, and robust parameter estimation. By comparison of our GIWF solutions (initials GI for “Geodätisches Institut” and IWF for “Institut für WeltraumForschung”) and the GOCE-TIM estimates with a state-of-the-art gravity field solution derived from GRACE (Gravity Recovery And Climate Experiment), we conclude that the acceleration approach is better suited for GOCE-only gravity field determination as opposed to the energy conservation method.     

The Nonlinear Model of the Response of Airglow to Gravity Waves

In this paper, we develope a time-dependent, nonlinear, photochemical-dynamical 2-D model which is composed of 3 models: dynamical gravity wave model, middle atmospheric photochemical model, and airglow layer photochemical model. We use the model to study the effect of the gravity wave propagation on the airglow layer. The comparison between the effects of the different wavelength gravity wave on the airglow emission distributions is made. When the vertical wavelength of the gravity wave is close to or is shorter than the thickness of the airglow layer, the gravity wave can make complex structure of the airglow layer, such as the double and multi-peak structures of the airglow layers. However, the gravity wave that has long vertical wavelength can make large scale perturbation of the airglow emission distribution.     

An improved altimeter-derived gravity anomaly from shipborne gravity based on the mean sea surface height constraint factors method

Coastal marine gravity modeling faces challenges due to the degradation of the quality and poor coverage of altimeter data in coastal regions. The effective fusion of shipborne gravity data and altimeter-derived marine gravity data can make shipborne gravity data more useful for the accurate estimation of altimeter-derived coastal marine gravity. A mean sea surface height constraint factor (MSSHCF) method based on the ordinary kriging method and the remove-restore technique is proposed to fuse altimeter-derived gravity model with shipborne gravity data. In this method, all data are standardized during the interpolation process to reduce the error and mean sea surface as a vertical variable is added to the semi-variance function in ordinary kriging to obtain the residual shipborne gravity as corrected data source. The coastal marine gravity models V2.1 and V3.1 which fused altimeter-derived gravity data with shipborne gravity data and V1.1 without shipborne gravity data at a spatial resolution of 1′×1′ can be obtained. Validation experiments show that the accuracy of the gravity model V3.1 obtained by the MSSHCF method more closely agrees with the validated gravity model DTU17 and SS V31 than the model V2.1 obtained by the ordinary kriging interpolation method and the V1.1 model. Our results were validated against shipborne gravity data; the accuracy of model V3.1 was 4.95 % higher than the model V1.1 in South China Sea area A and 2.48 % higher in South China Sea area B. Meanwhile, the accuracy of model V3.1 was 2.07 % higher than model V2.1 in South China Sea area A and 2.42 % higher in South China Sea area B. The effects of distance from the coast and sea depth on the marine gravity model were also evaluated. The results show that the gravity model V3.1 has higher accuracy with the change in ocean distance and depth than the V2.1 and V1.1 gravity models. Thus, our study shows that the MSSHCF method effectively refines coastal altimeter-derived gravity using shipborne gravity data.