A data-adaptive network design for the regional gravity field modelling using spherical radial basis functions

A high-precision regional gravity field model is significant in various geodesy applications.In the field of modelling regional gravity fields,the spherical radial basis functions(SRBFs)approach has recently gained widespread attention,while the modelling precision is primarily influenced by the base function network.In this study,we propose a method for constructing a data-adaptive network of SRBFs using a modified Hierarchical Density-Based Spatial Clustering of Applications with Noise(HDBSCAN)algorithm,and the performance of the algorithm is verified by the observed gravity data in the Auvergne area.Furthermore,the turning point method is used to optimize the bandwidth of the basis function spectrum,which satisfies the demand for both high-precision gravity field and quasi-geoid modelling simultaneously.Numerical experimental results indicate that our algorithm has an accuracy of about 1.58 mGal in constructing the gravity field model and about 0.03 m in the regional quasi-geoid model.Compared to the existing methods,the number of SRBFs used for modelling has been reduced by 15.8%,and the time cost to determine the centre positions of SRBFs has been saved by 12.5%.Hence,the modified HDBSCAN algorithm presented here is a suitable design method for constructing the SRBF data adaptive network.

A High-Resolution Earth’s Gravity Field Model SGG-UGM-2 from GOCE,GRACE,Satellite Altimetry,and EGM2008

This paper focuses on estimating a new high-resolution Earth’s gravity field model named SGG-UGM-2 from satellite gravimetry,satellite altimetry,and Earth Gravitational Model 2008(EGM2008)-derived gravity data based on the theory of the ellipsoidal harmonic analysis and coefficient transformation(EHA-CT).We first derive the related formulas of the EHA-CT method,which is used for computing the spherical harmonic coefficients from grid area-mean and point gravity anomalies on the ellipsoid.The derived formulas are successfully evaluated based on numerical experiments.Then,based on the derived least-squares formulas of the EHA-CT method,we develop the new model SGG-UGM-2 up to degree 2190 and order 2159 by combining the observations of the Gravity Field and Steady-State Ocean Circulation Explorer(GOCE),the normal equation of the Gravity Recovery and Climate Experiment(GRACE),marine gravity data derived from satellite altimetry data,and EGM2008-derived continental gravity data.The coefficients of degrees 251–2159 are estimated by solving the block-diagonal form normal equations of surface gravity anomalies(including the marine gravity data).The coefficients of degrees 2–250 are determined by combining the normal equations of satellite observations and surface gravity anomalies.The variance component estimation technique is used to estimate the relative weights of different observations.Finally,global positioning system(GPS)/leveling data in the mainland of China and the United States are used to validate SGG-UGM-2 together with other models,such as European improved gravity model of the earth by new techniques(EIGEN)-6C4,GECO,EGM2008,and SGG-UGM-1(the predecessor of SGG-UGM-2).Compared to other models,the model SGG-UGM-2 shows a promising performance in the GPS/leveling validation.All GOCE-related models have similar performances both in the mainland of China and the United States,and better performances than that of EGM2008 in the mainland of China.Due to the contribution of GRACE data and the new marine gravity anomalies,SGG-UGM-2 is slightly better than SGG-UGM-1 both in the mainland of China and the United States.

WHU-Grace01s:A new temporal gravity field model recovered from GRACE KBRR data alone

A new temporal gravity field model called WHU-Grace01s solely recovered from Gravity Recovery and Climate Experiment （GRACE） K-Band Range Rate （KBRR） data based on dynamic integral approach is presented in this paper. After meticulously preprocessing of the GRACE KBRR data, the root mean square of its post residuals is about 0.2 micrometers per second, and seventy-two monthly temporal solutions truncated to degree and order 60 are computed for the period from January 2003 to December 2008. After applying the combi- nation filter in WHU-Grace01s, the global temporal signals show obvious periodical change rules in the large-scale fiver basins. In terms of the degree variance, our solution is smaller at high degrees, and shows a good consistency at the rest of degrees with the Release 05 models from Center for Space Research （CSR）, GeoForschungsZentrum Potsdam （GFZ） and Jet Pro- pulsion Laboratory 0PL）. Compared with other published models in terms of equivalent water height distribution, our solution is consistent with those published by CSR, GFZ, JPL, Delft institute of Earth Observation and Space system （DEOS）, Tongji University （Tongji）, Institute of Theoretical Geodesy （ITG）, Astronomical Institute in University of Bern （AIUB） and Groupe de Recherche de Geodesie Spatiale （GRGS}, which indicates that the accuracy of WHU-Grace01s has a good consistency with the previously published GRACE solutions.

Investigation of the Tikhonov Regularization Method in Regional Gravity Field Modeling by Poisson Wavelets Radial Basis Functions

The application of Tikhonov regularization method dealing with the ill-conditioned problems in the regional gravity field modeling by Poisson wavelets is studied. In particular, the choices of the regularization matrices as well as the approaches for estimating the regularization parameters are investigated in details. The numerical results show that the regularized solutions derived from the first-order regularization are better than the ones obtained from zero-order regularization. For cross validation, the optimal regularization parameters are estimated from L-curve, variance component estimation（VCE） and minimum standard deviation（MSTD） approach, respectively, and the results show that the derived regularization parameters from different methods are consistent with each other. Together with the firstorder Tikhonov regularization and VCE method, the optimal network of Poisson wavelets is derived, based on which the local gravimetric geoid is computed. The accuracy of the corresponding gravimetric geoid reaches 1.1 cm in Netherlands, which validates the reliability of using Tikhonov regularization method in tackling the ill-conditioned problem for regional gravity field modeling.

Lithospheric Structure in the North China Craton Constrained from Gravity Field Model(EGM 2008)

A detailed knowledge of the thickness of the lithosphere in the North China craton（NCC） is important for understanding the significant tectonic reactivation of the craton in Mesozoic and Ce-nozoic.We achieve this goal by applying the newly proposed continuous wavelet transform theory to the Gravity Field Model（EGM 2008） data in the region.Distinct structural variations are identified in the scalogram image of profile Alxa-Datong（大同）-Qingdao（青岛）-Yellow Sea（profile ABC）,trans-versing the main units of NCC,which we interpret as mainly representing the Moho and lithosphere-asthenosphere boundary（LAB） undulations.The imaged LAB is as shallow as 60-70 km in the south-east basin and coastal areas and deepens to no more than 140 km in the northwest mountain ranges and continental interior.A rapid change of about 30 km in the LAB depth was detected at around the boundary between the Bohai（渤海） Bay basin（BBB） and the Taihang（太行） Mountains（TM）,roughly coincident with the distinct gravity decrease of more than 100 mGal that marks the North-South Grav-ity Lineament（NSGL） in the region.At last we present the gravity modeling work based on the spectral analysis results,incorporating with the observations on high-resolution seismic images and surface to-pography.The observed structural differences between the eastern and western NCC are likely associ-ated with different lithospheric tectonics across the NSGL.Combined with seismic tomography results and geochemical and petrological data,this sug-gests that complex modification of the litho-sphere probably accompanied significant litho-spheric thinning during the tectonic reactivation of the old craton.

Gravity field recovery from GOCE orbits using the energy conservation approach

The sum of the dissipative energy and energy constant of the GOCE satellite is found by a priori gravity field model at first, and the GOCE dissipative energy is obtained by computing the adjacent epoch difference via the differential method. Then, a gravity field model GOCE-ECPO1, which up to the degree and order 80, is recovered by the energy conservation approach from the 103-day precise orbital data of the GOCE satellite collected from November 1, 2009 to January 12, 2010. Finally, the model is compared with existing models EGM96, ITG-CHAMP05S, EIGEN-GRACE2010S, EIGEN-6C and GO CONS GCF 2 DIR_R3. The results show that at the same order and degree, the accuracy of model GOCE-EBPO1 is higher than those of models EGM96 and ITG-CHAMP05S, but lower than those of models EIGEN-GRACE2010S, EIGEN-6C and GO_CONS_GCF 2 DIR_R3, which is mainly caused by the pole gap.

Monthly gravity field solution from GRACE range measurements using modified short arc approach

In this paper we present a series of monthly gravity field solutions from Gravity Recovery and Climate Experiment（GRACE） range measurements using modified short arc approach,in which the ambiguity of range measurements is eliminated via differentiating two adjacent range measurements.The data used for developing our monthly gravity field model are same as Tongji-GRACEOl model except that the range measurements are used to replace the range rate measurements,and our model is truncated to degree and order 60,spanning Jan.2004 to Dec.2010 also same as Tongji-GRACE01 model.Based on the comparison results of the C_（2,0）,C_（2,1）,S_（2,1）,and C_（15,15）,S_（15,15）,time series and the global mass change signals as well as the mass change time series in Amazon area of our model with those of Tongji-GRACE01 model,we can conclude that our monthly gravity field model is comparable with Tongji-GRACE01 monthly model.

The construction of high precision geostrophic currents based on new gravity models of GOCE and satellite altimetry data

The new gravity field models of gravity field and steady-state ocean circulation explorer(GOCE),TIM_R6 and DIR_R6,were released by the European Space Agency(ESA)in June 2019.The sixth generation of gravity models have the highest possible signal and lowest error levels compared with other GOCE-only gravity models,and the accuracy is significantly improved.This is an opportunity to build high precision geostrophic currents.The mean dynamic topography and geostrophic currents have been calculated by the 5th(TIM_R5 and DIR_R5),6th(TIM_R6 and DIR_R6)release of GOCE gravity field models and ITSG-Grace2018 of GRACE gravity field model in this study.By comparison with the drifter results,the optimal filtering lengths of them have been obtained(for DIR_R5,DIR_R6,TIM_R5 and TIM_R6 models are 1°and for ITSG-Grace2018 model is 1.1°).The filtered results show that the geostrophic currents obtained by the GOCE gravity field models can better reflect detailed characteristics of ocean currents.The total geostrophic speed based on the TIM_R6 model is similar to the result of the DIR_R6 model with standard deviation(STD)of 0.320 m/s and 0.321 m/s,respectively.The STD of the total velocities are 0.333 m/s and 0.325 m/s for DIR_R5 and TIM_R5.When compared with ITSG-Grace2018 results,the STD(0.344 m/s)of total geostrophic speeds is larger than GOCE results,and the accuracy of geostrophic currents obtained by ITSG-Grace2018 is lower.And the absolute errors are mainly distributed in the areas with faster speeds,such as the Antarctic circumpolar circulation,equatorial region,Kuroshio and Gulf Stream areas.After the remove-restore technique was applied to TIM_R6 MDT,the STD of total geostrophic speeds dropped to 0.162 m/s.

Determination of Antarctic geoid by using global gravity field

With Chinese latest global gravity field model WDM94, the authors provide the geoid height and mean free air gravity anomaly of Antarctica (The range of latitude is from -60° to -90°). In order to conclude and analyze the characters of Antarctic geoid roundly, the authors collect the latest oversea global gravity field model OSU91 (to degree and order 360) and JGMOSU (to degree and order 360), get the corresponding geoid height and mean free air gravity anomaly. The results are compared with the results got from WDM94, thus we get the difference. The standard deviation of geoid height between WDM94 and OSU91 is ±1.90 m;the deviation of geoid between WDM94 and JGMOSU is ±2.09 m. The standard deviation of mean gravity anomaly are ±8.97 mGal and ±9.32 mGal respectively. [WT9.HZ]

A GOCE only gravity model GOSG01S and the validation of GOCE related satellite gravity models

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 analysis of spectral accuracy characteristics of the newly derived gravitational model GOSG01S and the existing models GOTIM04S, GODIR04S, GOSPW04S and JYY_GOCE02S based on their comparison with the ultrahigh degree model EIGEN-6C2 reveals a significant consistency at the spectral window approximately between 80 and 190 due to the same period SGG data used to compile these models. 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 suggests that all these models provide much more information on the gravity field than EGM2008 in areas with low terrestrial gravity coverage. And STDs of height anomaly differences in China for the selected truncation degrees show that GOCE has improved the accuracy of the global models beyond degree 90 and the accuracies of the models improve from 24 cm to 16 cm. STDs of geoid height differences in USA show that GOSG01S model has best consistency comparing with GPSleveling data for the frequency band of the degree between 20 and 160.

Moho depth inversion in the Tibetan Plateau from high-precision gravity data

The Tibetan Plateau(TP)is the youngest orogenic belt resulting from a continental collision on the Earth.It is a natural laboratory for studying continental dynamics,such as continental convergence,plate subduction,and plateau uplift.Investigating the deep structure of the TP has always been a popular issue in geological research.The Moho is the boundary between the crust and the mantle and therefore plays a crucial role in the Earth’s structure.Parameters such as depth and lateral variation,as well as the fine structure of the crust-mantle interface,reveal the lithospheric dynamics in the TP.Two methods are generally employed to study the Moho surface:seismic detection and gravity inversion.Seismic detection has the characteristic of high precision,but it is limited to a few cross-sectional lines and is quite costly.It is not suitable for and cannot be carried out over a large area of the TP.The Moho depth over a large area can be obtained through gravity inversion,but this method is affected by the nature of gravity data,and the accuracy of the inversion method is lower than that of seismic detection.In this work,a high-precision gravity field model was selected.The Parker-Oldenburg interface inversion method was used,within the constraints of seismic observations,and the Bott iteration method was introduced to enhance the inversion efficiency.The Moho depth in the TP was obtained with high precision,consistent with the seismic detection results.The research results showed that the shape of the Moho in the TP is complex and the variation range is large,reaching 60−80 km.In contrast with the adjacent area,a clear zone of sharp variation appears at the edge of the plateau.In the interior of the TP,the buried depth of the Moho is characterized by two depressions and two uplifts.To the south of the Yarlung Zangbo River,the Moho inclines to the north,and to the north,the Moho depresses downward,which was interpreted as the Indian plate subducting to the north below Tibet.The Moho depression on the north side of the Qiangtang block,reaching 72 km deep,may be a result of the southward subduction of the lithosphere.The Moho uplift of the Qiangtang block has the same strike as the Bangong−Nujiang suture zone,which may indicate that the area is compensated by a low-density and low-velocity mantle.

Construction of nonsingular formulae of variance and covariance function of disturbing gravity gradient tensors

When the computational point is approaching the poles, the variance and covariance formulae of the disturbing gravity gradient tensors tend to be infinite, and this is a singular problem. In order to solve the problem, the authors deduced the practical non-singular computational formulae of the first- and second-order derivatives of the Legendre functions and two kinds of spherical harmonic functions, and then constructed the nonsingular formulae of variance and eovarianee function of disturbing gravity gradient tensors.

An iterative Wiener filtering method based on the gravity gradient invariants

How to deal with colored noises of GOCE （Gravity field and steady - state Ocean Circulation Explorer） satellite has been the key to data processing. This paper focused on colored noises of GOCE gradient data and the frequency spectrum analysis. According to the analysis results, gravity field model of the optima] degrees 90-240 is given, which is recovered by COCE gradient data. This paper presents an iterative Wiener filtering method based on the gravity gradient invariants. By this method a degree-220 model was calculated from GOCE SGG （Satellite Gravity Gradient） data. The degrees above 90 of ITG2010 were taken as the prior gravity field model, replacing the low degree gravity field model calculated by GOCE orbit data. GOCE gradient colored noises was processed by Wiener filtering. Finally by Wiener filtering iterative calculation, the gravity field model was restored by space-wise harmonic analysis method. The results show that the model＇s accuracy matched well with the ESA＇s （European Space Agency） results by using the same data,

Recovery of the gravity field from GOCE data by using the invariants of gradient tensor

The gravity field models GUCAS_EGM and GUCAS_EGM_DL are established from GOCE data (GOCE Level 2 Products from Nov. 1 to Dec. 31, 2009) based on the method of the invariants of the gravity gradient tensor, where GUCAS_EGM is derived after GOCE gravity gradient data are filtered with FIR, and GUCAS_EGM_DL is computed with an additional Durbin-Levison arithmetic apart from FIR. Since this method, different from current programs dealing with GOCE data, is introduced for the first time, some new problems are required to be discussed in advance; for example, how to filter GOCE gravity gradient data, how to compute the invariants of the gradient tensor, and how to deal with the pole gap and so on. In addition, by comparing our models with ones recommended by ESA, it can be seen that the variations of GUCAS_EGM and the models recommended by ESA to EGM08 are almost equivalent, and the variation of GUCAS_EGM_DL to EGM08 is obviously less than ones of the recommended models.

GRACE-based estimates of water discharge over the Yellow River basin

As critical component of hydrologic cycle, basin discharge is a key issue for understanding the hydrological and climatologic related to water and energy cycles. Combining GRACE gravity field models with ET from GLDAS models and precipitation from GPCP, discharge of the Yellow River basin are estimated from the water balance equation. While comparing the results with discharge from GLDAS model and in situ measurements, the results reveal that discharge from Mosaic and CLM GLDAS model can partially represent the river discharge and the discharge estimation from water balance equation could reflect the discharge from precipitation over the Yellow River basin.

Recent Status of Taiji Program in China

The Taiji-1 satellite is a pilot satellite mission of Taiji program,which is used to verify Taiji’s key technology and also to testify the feasibility of Taiji roadmap.Taiji-1 was launched on 31 August 2019 and its designed mission was successfully completed.The in-orbit scientific achievements of Taiji-1 satellite in the first stage have been published and now it has entered the extended task phase.Taiji-2 will prepare all the technology needed by Taiji-3,and remove all the technical obstacles faced by Taiji-3.

GOCE kinematic orbit adjustment for EGM validation and accelerometer calibration

The main principle and mathematical model of GOCE kinematic orbit adjustment for Earth gravity field model （EGM） validation and accelerometer calibration are presented. Based on 60 days GOCE kinematic orbits with 1-2 cm accuracy and accelerometer data from 2009-11-02 to 2009-12-31, the RMS-of-fit （ROF） of them using EGM2008, EIGEN-SC, ITG- GRACE2010S and GOCO01S up to 120, 150 and 180 degree and order （d/o） are evaluated and compared. The scale factors and biases of GOCE accelerometer data are calibrated and the energy balance method （EBM） is performed to test the accuracy of accelerometer calibration. The results show that GOCE orbits are also sensitive to EGM from 120 to 150 d/o. The ROFs of EGMs with 150 and 180 d/o are obviously better than those of EGMs with 120 d/o. The ROFs of GOCO01S and ITG-GRACE2010S are almost the same up to 120 and 150 d/o, which are about 3.3 cm and 1.8 cm, respectively. They are far better than those of EGM2008 and EIGEN-SC with the same d/o. The ROF of GOCO01S with 180 d/o is about 1.6 em, which is the best one among those EGMs. The accelerometer calibration accuracies （ACAs） of ITG-GRACE2010S and GOCO01S are obviously higher that those of EGM2008 and EIGEN-SC. The ACA of GOCO01S with 180 d/o is far higher than that of EGMs with 120 d/o, and a little higher than that of ITG-GRACE2010S with 150 d/o. I t is suggested that the newest released EGM such as GOCO01S or GOCO02S till at least 150 d/o should be chosen in GOCE precise orbit determination （POD） and accelerometer calibration.

Precise Relative Orbit Determination of Twin GRACE Satellites

When formation flying spacecrafts are used as platform to gain earth oriented observation, precise baselines between these spacecrafts are always essential. Gravity recovery and climate experiment （GRACE） mission is aimed at mapping the global gravity field and its variation. Accurate baseline of GRACE satellites is necessary for the gravity field modeling. The determination of kinematic and reduced dynamic relative orbits of twin satellites has been studied in this paper, and an accuracy of 2 mm for dynamic relative orbits and 5 mm for kinematic ones can be obtained, whereby most of the double difference onboard GPS ambiguities are resolved.