摘要: The high-accuracy and high-resolution regional gravity modeling based on heterogeneous data sets is a focused issue in physical geodesy. With the abundant multi-sources data, including satellite-only global gravity models, and airborne, shipboard as well as terrestrial gravity data sets, the regional gravity field could be further improved. However, these heterogeneous data sets have different spatial coverage and resolutions, various error characteristics as well as different spectral contents. Thus, how to make use of these heterogeneous data sets remains an unsolved problem.
Under the framework of remove-compute-restore methodology, only the residual disturbing potential is parameterized by using Poisson wavelets radial basis functions (RBFs). While, the long- and short-wavelength part of the gravity field is recovered by global gravity model (GGM) and residual terrain model (RTM), respectively. The functional model could be formulated based on the relationship between the disturbing potential and observations. By using the Monte-Carlo variance component estimation, the proper weight of the disjunctive observation group could be determined. The unknown coefficients of the RBFs are computed based on least squares principle. The network design of the RBFs is one of the critical issues, where the depth and spatial resolution of the RBFs are the two key elements. Given the potential drawbacks of the generalized cross validation and multipole analysis, we focus on minimizing the STD differences between the predicted and observed values at GPS/leveling points. Different depths and spatial resolutions of the RBFs are combined to formulate various network parameterizations, based on which the corresponding gravimetric quasi-geoids are computed. The combination of depth and number of RBFs that obtains the best fit to the GPS/leveling data is considered as the optimal parameterization of the RBFs' network. In addition, the topographic masses, which play a crucial role in determining the high-frequency part of the regional gravity field, affecting the network design of RBFs as well as the accuracy of the regional gravity field models. We focus on the RTM based on tesseroids in recovering the gravity field signal at short scales. The so-called "two-step" method, which neglects the effect of topography, and the "three-step" method, which uses the RTM model in smoothing the high-frequency gravity field signal, are compared with each other to illustrate the effect of topography on the RBFs' network design as well as the accuracy of the solutions.
With the incorporation of RTM reduction, the residual gravity field is much smoother, and the most significant improvement is made in mountainous areas. The standard deviation of the terrestrial residual gravity anomalies reduces from 15.924 mGal to 10.652 mGal, with roughly 33% improvement. Due to the poor quality and low resolution of the bathymetry model, the standard deviation of shipboard residual gravity anomalies only reduces from 13.994 mGal to 12.403 mGal, with approximately 11% improvement. The airborne residual gravity disturbance is nearly unchanged for the similar reason as most of the airborne measurements are located at several kilometers above the sea. As an example, we model the regional gravimetric quasi-geoid based on airborne, shipboard and terrestrial gravity data sets, where the long- and short wavelength parts are recovered by a GGM called DGM1S and RTM based on high-accuracy and high-resolution digital terrain model (DTM), respectively. To get the proper network parameterization of RBFs in different regions with various topographical characteristics, we use a flat region called Omega(A) and mountainous area called Omega(B) as the target area, respectively. Numerical experiments show the optimal depth as well as the number of RBFs in region Omega(A) obtained from the "two-step" and "three-step" methods are identical, i.e., the optimal depth is 40 km and the number of RBFs is 7394. As the area Omega(A) is relatively flat, where the RTM corrections are quite small. Thus, the residual gravity field is barely smoothed, and the optimal network design of RBFs is unchanged with the incorporation of RTM reduction. However, the accuracy of the quasi-geoid is improved by 0.004 m, the standard deviation fit decreases from 0.015 m to 0.011 m. While, in the mountainous area Omega(B), the optimal depth increases from 25 km to 35 km after RTM corrections are incorporated, and the optimal number of RBFs shows approximately 18% reduction, which reduces from 14518 to 11894. Moreover, the accuracy of quasi-geoid shows roughly 51% improvement, the standard deviation fit decreases from 0. 097 m to 0. 048 m. The topography over Omega(B) shows relatively strong variation, and the local high-frequency part of the signal is dominated by the local topography undulation. The regional gravity field is significantly smoothed after incorporating RTM corrections, which leads to a reduction of the number of RBFs. As the gravity signal at a very short scale could be represented well by RTM, the residual part of the gravity signal could be better recovered by RBFs, and the quality of quasi-geoid is improved. Moreover, the frequency range of the residual gravity signal is changed correspondingly as the local high-frequency part of the gravity signal is smoothed by RTM, thus the optimal depth of RBFs shifts to a deeper value. Based on these results, we formulate optimal network parameterization over different regions by using various parameters, and a regional gravimetric quasi-geoid is computed over the whole target area. The accuracy of the gravimetric quasi-geoid is 1.12 cm, 2.80 cm and 2.92 cm at Netherlands, Belgium and Germany, respectively.
The main conclusions can be drawn as follows: (1) The network design of RBFs is of great importance for regional gravity field modeling, as the effect of topographical masses and the optimal network parameterization over various regions may be different. (2) Due to the limitation of the spatial resolution of gravity observations, the aliasing problems occur if the effect of topographical masses is neglected. After the incorporation of RTM reduction, the residual gravity field is significantly smoothed and the optimal network design is simplified. In the meanwhile, the accuracy of the gravimetric quasi-geoid model is improved by 4 mm in flat areas. While in mountainous areas, more significant improvement is obtained if RTM correction is incorporated with the magnitude of 5 cm. (3) Overall, the accuracy of regional gravimetric quasi-geoid modeled by radial basis functions based on the "three-steps" method is at cm level. The accuracy of the gravimetric quasi-geoid is 1.12 cm, 2.80 cm and 2.92 cm at Netherlands, Belgium and Germany, respectively.
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