Application of the nonlinear optimisation in regional gravity field modelling using spherical radial base functions

The gravity field is a signature of the mass distribution and interior structure of the Earth, in addition to all its geodetic applications especially geoid determination and vertical datum unification. Determination of a regional gravity field model is an important subject and needs to be investigated and developed. Here, the spherical radial basis functions (SBFs) are applied in two scenarios for this purpose: interpolating the gravity anomalies and solving the fundamental equation of physical geodesy for geoid or disturbing potential determination, which has the possibility of being verified by the Global Navigation Satellite Systems (GNSS)/levelling data. Proper selections of the number of SBFs and optimal location of the applied SBFs are important factors to increase the accuracy of estimation. In this study, the gravity anomaly interpolation based on the SBFs is performed by Gauss-Newton optimisation with truncated singular value decomposition, and a Quasi-Newton method based on line search to solve the minimisation problems with a small number of iterations is developed. In order to solve the fundamental equation of physical geodesy by the SBFs, the truncated Newton optimisation is applied as the Hessian matrix of the objective function is not always positive definite. These two scenarios are applied on the terrestrial free-air gravity anomalies over the topographically rough area of Auvergne. The obtained accuracy for the interpolated gravity anomaly model is 1.7 mGal with the number of point-masses about 30% of the number of observations, and 1.5 mGal in the second scenario where the number of used kernels is also 30%. These accuracies are root mean square errors (RMSE) of the differences between predicted and observed gravity anomalies at check points. Moreover, utilising the optimal constructed model from the second scenario, the RMSE of 9 cm is achieved for the differences between the gravimetric height anomalies derived from the model and the geometric height anomalies from GNSS/levelling points.

Assessment of high-degree reference models and Recent Goce/Grace Global Geopotential Models over Sudan based on the GPS/Leveling data

Nowadays, Global Geopotential Models (GGMs) can be used as a reference to develop more detailed regional/local geoids, or they can be used to provide geoid heights on their own. Since 2000, several GGMs have been released, and they are mainly derived from satellite gravity measurements, satellite-only models, terrestrial gravimetry, altimeter-derived gravity data in marine areas, and airborne gravity data. With a precise geoid model, ellipsoidal heights obtained from GPS can be converted to orthometric heights, which is reasonably quite needed in Geodesy, Civil Engineering, etc. These heights reflect changes in topography as well as local variations in gravity. This paper evaluates some of the latest releases of high degree reference models and the satellite-only global gravity field model over Sudan using 19 GPS/Leveling stations. We have been selected 6 GGMs based on Gravity field Goce and Grace, and they released in 2020, 2019, 2014, 2008, and 1996 as shown in the International Centre for Global Earth Models website (ICGEM). The accuracy evaluation of the GGM models have been discussed, the accurate GGMs over Sudan are XGM2019e_2159 and GOCO05s, which have indicated -0.019 and 0.046 meters, respectively. The evaluation results produce valuable information to academia and geoid modeling research topics in Sudan, which shows the precise model from the selected GGMs in Sudan by using the available GPS/Leveling data.

Effects of New Level-1B Data on GRACE Temporal Gravity Field Models and Precise Orbit Determination Solutions

The quality of Gravity Recovery and Climate Experiment (GRACE) observation is the prerequisite for obtaining the high-precision GRACE temporal gravity field model. To study the influence of new-generation GRACE Level-1B Release 03 (RL03) data and the new atmosphere and ocean de-aliasing (AOD1B) products on recovering temporal gravity field models and precise orbit determination (POD) solutions, we combined the global positioning system and K-band ranging-rate (KBRR) observations of GRACE satellites to estimate the effect of different data types on these solutions. The POD and monthly gravity field solutions are obtained from 2005 to 2010 by SHORDE software developed by the Shanghai Astronomical Observatory. The post-fit residuals of the KBRR data were decreased by approximately 10%, the precision of three-direction positions of the GRACE POD was improved by approximately 5%, and the signal-to-noise ratio of the monthly gravity field model was enhanced. The improvements in the new release of monthly gravity field model and POD solutions can be attributed to the enhanced Level-1B KBRR data and the AOD1B model. These improvements were primarily due to the enhanced of KBRR data; the effect of the AOD1B model was not significant. The results also showed that KBRR data slightly improve the satellite orbit precision, and obviously enhance the precision of the gravity field model.

Recovering Marine Gravity Over the Gulf of Guinea From Multi-Satellite Sea Surface Heights

A regional gravity field product, comprising vertical deflections and gravity anomalies, of the Gulf of Guinea (15°W to 5°E, 4°S to 4°N) has been developed from sea surface heights (SSH) of five altimetry missions. Though the remove-restore technique was adopted, the deflections of the vertical were computed directly from the SSH without the influence of a global geopotential model. The north-component of vertical deflections was more accurate than the east-component by almost three times. Analysis of results showed each satellite can contribute almost equally in resolving the north-component. This is attributable to the nearly northern inclinations of the various satellites. However, Cryosat-2, Jason-1/GM, and SARAL/AltiKa contributed the most in resolving the east-component. We attribute this to the superior spatial resolution of Cryosat-2, the lower inclination of Jason-1/GM, and the high range accuracy of the Ka-band of SARAL/AltiKa. Weights of 0.687 and 0.313 were, respectively, assigned to the north and east components in order to minimize their non-uniform accuracy effect on the resultant gravity anomaly model. Histogram of computed gravity anomalies compared well with those from renowned models: DTU13, SIOv28, and EGM2008. It averagely deviates from the reference models by −0.33 mGal. Further assessment was done by comparing it with a quadratically adjusted shipborne free-air gravity anomalies. After some data cleaning, observations in shallow waters, as well as some ship tracks were still unreliable. By excluding the observations in shallow waters, the derived gravity field model compares well in ocean depths deeper than 2,000 m.

Joint analysis of selected GRACE monthly spherical harmonic solutions and monthly MASCON solutions

Study presented in this paper is focused on comparison and statistical assessment of differences between the selected Level 2 products of the satellite mission Gravity Recovery and Climate Experiment (GRACE). Global monthly gravity field models in terms of spherical harmonic coefficients produced by three institutes of GRACE Science Data System are compared with the partially independent MASCON global gravity field model. Detailed comparison and statistical analysis of differences is performed in 5 selected river basins: Amazon, Congo, Danube, Yenisei and Lena. For each spherical harmonic solution, 8 different filtrations available at International Center for Global Gravity Field Models (ICGEM) are tested over the time span from April 2002 to July 2016. Fischer test at two significance levels 10% and 5% has been performed in order to qualify the statistical significance between the particular solutions.

Towards an updated, enhanced regional gravity field solution for Antarctica

<p>In the frame of the IAG Subcommission 2.4f “Gravity and Geoid in Antarctica” (AntGG) a first Antarctic-wide grid of ground-based gravity anomalies was released in 2016 (Scheinert et al. 2016). That data set was provided with a grid space of 10 km and covered about 73% of the Antarctic continent. Since then a considerably amount of new data has been made available, mainly collected by means of airborne gravimetry. Regions which were formerly void of any terrestrial gravity observations and have now been surveyed include especially the polar data gap originating from GOCE satellite gravimetry. Thus, it is timely to come up with an updated and enhanced regional gravity field solution for Antarctica. For this, we aim to improve further aspects in comparison to the AntGG 2016 solution: The grid spacing will be enhanced to 5 km. Instead of providing gravity anomalies only for parts of Antarctica, now the entire continent should be covered. In addition to the gravity anomaly also a regional geoid solution should be provided along with further desirable functionals (e.g. gravity anomaly vs. disturbance, different height levels).</p><p>We will discuss the expanded AntGG data base which now includes terrestrial gravity data from Antarctic surveys conducted over the past 40 years. The methodology applied in the analysis is based on the remove-compute-restore technique. Here we utilize the newly developed combined spherical-harmonic gravity field model SATOP1 (Zingerle et al. 2019) which is based on the global satellite-only model GOCO05s and the high-resolution topographic model EARTH2014. We will demonstrate the feasibility to adequately reduce the original gravity data and, thus, to also cross-validate and evaluate the accuracy of the data especially where different data set overlap. For the compute step the recently developed partition-enhanced least-squares collocation (PE-LSC) has been used (Zingerle et al. 2021, in review; cf. the contribution of Zingerle et al. in the same session). This method allows to treat all data available in Antarctica in one single computation step in an efficient and fast way. Thus, it becomes feasible to iterate the computations within short time once any input data or parameters are changed, and to easily predict the desirable functionals also in regions void of terrestrial measurements as well as at any height level (e.g. gravity anomalies at the surface or gravity disturbances at constant height).</p><p>We will discuss the results and give an outlook on the data products which shall be finally provided to present the new regional gravity field solution for Antarctica. Furthermore, implications for further applications will be discussed e.g. with respect to geophysical modelling of the Earth’s interior (cf. the contribution of Schaller et al. in session G4.3).</p>

Integrating NGS GRAV-D gravity observations into high-resolution global models

<p>Within this contribution we present a method that allows a smooth integration of in-situ ground gravity observations into high-resolution global models up to d/o 5400 (2’ global resolution). The functionality is shown on the example of the airborne GRAV-D gravity dataset which is integrated into a global satellite-topographic spherical harmonic model.</p><p>Conceptually, the method is divided into three steps: firstly, since the processing is based on residuals, a precursor model needs to be identified which is used for reducing the observations. In the actual example a combination between a satellite-only model (GOCO06s) and topographic model (EARTH2014) is chosen (named SATOP2) to ensure independency to the observations. Secondly, the previously reduced (GRAV-D) observations are gridded onto a regular geographic grid making use of the recently developed partition-enhanced least squares collocation approach (PE-LSC). PE-LSC allows an efficient collocation of virtually arbitrary large datasets using a partitioning technique that is optimized for computational performance and for minimizing fringe effects. As a third and last step, the obtained regular grid gets analyzed and combined with a satellite-only model (GOCO06s) on the normal equation level up to d/o 5400. This can be achieved efficiently by using a so-called kite-normal equation system which emerges when combining dense and block-diagonal normal equation systems (assuming equal accuracies for the ground gravity grid).</p><p>The herby obtained global gravity field model, named SGDT, is dominated by the satellite information in the lower frequencies (up to d/o 200), by GRAV-D in the mid-frequencies (d/o 200-2000) and by the topographic information in the high frequencies (above d/o 2000). The main purpose of the SGDT model is to validate the method itself and to allow a comparison of GRAV-D observations to pre-existing ground-gravity data by synthesizing SGDT to actual observation sites.</p>

On frequency response of Stokes and Hotine-Koch integral transforms in calculation of height anomaly in the local area by means of SRBF

<p>The problem of determining the height anomaly in a local area of the radius ψ<sub>0</sub> using gravity disturbances and gravity anomalies is discussed. The influence of the far zone, as usually, is approximately taken into account using the global gravity field model and the truncation coefficients Q<sub>n</sub> (ψ<sub>0</sub>) introduced by M.S. Molodensky [1]. The modification Q<sub>n</sub><sup>0</sup>(ψ<sub>0</sub>) by O.M. Ostach [2] of these coefficients is described. They provide - in contrast to the original coefficients - the continuity of the used integral transform kernel Ker<sup>0</sup> (ψ) in the whole its definition domain. As a consequence, the modified coefficients decrease faster compared to the original ones with an increase of the degree n (frequency). It reduces the error of the far zone influence. Coefficients are interpreted as Fourier coefficients of the outer part of the kernel when it is decomposed into the orthogonal system of nonnormalized Legendre polynomials. The relationship between Q<sub>n</sub> (ψ<sub>0</sub>) and Q<sub>n</sub><sup>0</sup>(ψ<sub>0</sub>) is indicated. In the frequency domain, the expression for the truncated kernel ΔKer<sup>0</sup> (ψ) of the integral transform used (Stokes or Hotine-Koch) differs from the corresponding full kernel by a multiplier, which is proposed to be called the frequency characteristic of the kernel truncation operator onto the inner zone of radius ψ<sub>0</sub>.</p><p>In local modeling, when describing the details of the "useful signal", it is advisable to use approximation by means of spherical radial basis functions (SRBF) instead of traditional integration due to their good spatial localization [3, 4]. The procedure of constructing scaling functions and corresponding wavelets is briefly described. New scaling functions, based on the above-mentioned concept of frequency characteristic of the kernel truncation operator onto the inner zone of the radius ψ<sub>o</sub>, are proposed. To prove the effectiveness of these scaling functions, numerical experiments were conducted. Both gravity anomalies Δg and disturbances δg were used as input data. The results of the calculations showed a high accuracy of recovering height anomalies from gravity anomalies. Besides, introduction of frequency characteristic of kernel truncation of corresponding integral transform onto the inner zone allows to cut off implicit influence of far zone. Known scaling functions that do not use this frequency characteristic lead, as experiments have shown, to biased results.</p><p><strong>References:</strong></p>

Determining the height of Mount Everest using the shallow layer method

<p>We proposed an alternative method to determine the height of Mount Everest (HME) based on the shallow layer method (SLM), which was put forward by Shen (2006). We use the precise external global Earth gravity field model (i.e., EGM2008 and EIGEN-6C4 models) as input information, and the digital topographic model (i.e., DTM2006.0) and crust models (i.e., CRUST2.0 and CRUST1.0 models) to construct the shallow layer model. There are four combined strategies：(1) EGM2008 and CRUST1.0 models, (2) EGM2008 and CRUST2.0 models, (3) EIGEN-6C4 and CRUST1.0 models, and (4) EIGEN-6C4 and CRUST2.0 models, respectively. We calculate the HME by two approaches: first approach, the HME is directly calculated by combining the geoid undulation (N) and geodetic height (h); second approach, we calculate the HME by the segment summation approach (SSA) using the gravity field inside the shallow layer determined by the SLM. Numerical results show that for four combined strategies, the differences between our results and the authoritatively released value 8848.86 m by the Chinese and Nepalese governments on December 8, 2020 are 0.448 m, -0.009 m, -0.295 m, and -0.741 m using first approach and 0.539 m, 0.083 m, -0.214 m, and -0.647 m using second approach. The combined calculation of the HME by the terrain model and gravity field model is more accurate than that by the gravity field model alone. This study is supported by the National Natural Science Foundations of China (NSFC) under Grants 42030105, 41721003, 41804012, 41631072, 41874023, Space Station Project (2020)228.</p>

The earth’s gravity field recovery using the third invariant of the gravity gradient tensor from GOCE

Due to the independence of the gradiometer instrument’s orientation in space, the second invariant $$I_2$$ I 2 of gravity gradients in combination with individual gravity gradients are demonstrated to be valid for gravity field determination. In this contribution, we develop a novel gravity field model named I3GG, which is built mainly based on three novel elements: (1) proposing to utilize the third invariant $$I_3$$ I 3 of the gravity field and steady-state ocean circulation explorer (GOCE) gravity gradient tensor, instead of using the $$I_2$$ I 2 , similar to the previous studies; (2) applying an alternative two-dimensional fast fourier transform (2D FFT) method; (3) showing the advantages of $$I_3$$ I 3 over $$I_2$$ I 2 in the effect of measurement noise from the theoretical and practical computations. For the purpose of implementing the linearization of the third invariant, this study employs the theory of boundary value problems with sphere approximation at an accuracy level of $$O(J_2^2\cdot T_{ij})$$ O ( J 2 2 · T ij ) . In order to efficiently solve the boundary value problems, we proposed an alternative method of 2D FFT, which uses the coherent sampling theory to obtain the relationship between the 2D FFT and the third invariant measurements and uses the pseudo-inverse via QR factorization to transform the 2D Fourier coefficients to spherical harmonic ones. Based on the GOCE gravity gradient data of the nominal mission phase, a novel global gravity field model (I3GG) is derived up to maximum degree/order 240, corresponding to a spatial resolution of 83 km at the equator. Moreover, in order to investigate the differences of gravity field determination between $$I_3$$ I 3 with $$I_2$$ I 2 , we applied the same processing strategy on the second invariant measurements of the GOCE mission and we obtained another gravity field model (I2GG) with a maximum degree of 220, which is 20 degrees lower than that of I3GG. The root-mean-square (RMS) values of geoid differences indicates that the effects of measurement noise of I3GG is about 20% lower than that on I2GG when compared to the gravity field model EGM2008 (Earth Gravitational Model 2008) or EIGEN-5C (EIGEN: European Improved Gravity model of the Earth by New techniques). Then the accuracy of I3GG is evaluated independently by comparison the RMS differences between Global Navigation Satellite System (GNSS)/leveling data and the model-derived geoid heights. Meanwhile, the re-calibrated GOCE data released in 2018 is also dealt with and the corresponding result also shows the similar characteristics.