多种类型高分辨率重力场数据的不断增加，使得在局部范围内精化重力场模型成为了可能。本文采用Abel-Poisson核将重力场量表示成有限个径向基函数线性求和的形式，对局部区域的多种重力场数据进行联合建模。为了提高运算速度，运用了基于自适应精化格网算法的最小均方根误差准则（RMS）来求解径向基函数平均带宽。以南海核心地区为例，联合两种不同类型、不同分辨率的重力场资料（大地水准面起伏6'×6'、重力异常2'×2'），构建了局部区域高分辨率的重力场模型。所建模型表示的重力场参量达到了2'×2'的分辨率，对原始的重力异常数据（2'×2'）拟合的符合程度达到±0.8×10－5m/s2。结果表明，利用径向基函数方法进行局部重力场建模，避免了球谐函数建模收敛慢的问题，有效提高了模型表示重力场的分辨率。
With the increasing number of various types of high-resolution gravity observations, earth gravity models can be regionally refined. We use Abel-Poisson kernel to represent the gravity as the linear summation of finite radial basis functions and combine the multiple gravity data to build a regional gravity model with high resolution. The minimum root mean square criterion based on the data adaptive algorithm is proposed to calculate the base function, which promote the speed of computation significantly. Taking the central South China Sea as an example, two different types of gravity data, namely geoid undulations with resolution of 6'×6' and gravity anomaly with resolution of 2'×2', are used to construct the high-resolution regional gravity model. The model has a resolution of 2'×2', and has a great agreement with original gravity anomaly, reaching to ±0.8×10-5m/s2.Our results show that using radial basis functions to construct the regional gravity field can avoid the problem of slow convergence of spherical harmonic functions, and can improve the resolution remarkably.
[1] 彭富清, 陈双军, 金群峰. 卫星测高误差对海洋重力场反演的影响[J]. 测绘学报, 2014, 43(4): 337-340. DOI: 10.13485/j.cnki.11-2089.2014.0050. PENG Fuqing, CHEN Shuangjun, JIN Qunfeng. Influence of Altimetry Errors on Marine Geopotential Recovery[J]. Acta Geodaetica et Cartographica Sinica, 2014, 43(4): 337-340. DOI: 10.13485/j.cnki.11-2089.2014.0050.
[2] 钟波, 罗志才, 李建成, 等. 联合高低卫-卫跟踪和卫星重力梯度数据恢复地球重力场的谱组合法[J]. 测绘学报, 2012, 41(5): 735-742. ZHONG Bo, LUO Zhicai, LI Jiancheng, et al. Spectral Combination Method for Recovering the Earth's Gravity Field from High-low SST and SGG Data[J]. Acta Geodaetica et Cartographica Sinica, 2012, 41(5): 735-742.
[3] HEIKKINEN M. Solving the Shape of the Earth by Using Digital Density Models[R]. Helsinki: Finnish Geodetic Institute, 1981.
[4] HOLSCHNEIDER M, CHAMBODUT A, MANDEA M. From Global to Regional Analysis of the Magnetic Field on the Sphere Using Wavelet Frames[J]. Physics of the Earth and Planetary Interiors, 2003, 135(2-3): 107-124.
[5] CHAMBODUT A, PANET I, MANDEA M, et al. Wavelet Frames: An Alternative to Spherical Harmonic Representation of Potential Fields[J]. Geophysical Journal International, 2005, 163(3): 875-899.
[6] SCHMIDT M, FABERT O, HAN S C, et al. Gravity Field Determination Using Multiresolution Techniques[C]//Proceedings of the 2nd International GOCE User Workshop. Frascati: GOCE, 2004.
[7] SCHMIDT M, FABERT O, SHUM C K. On the Estimation of a Multi-resolution Representation of the Gravity Field Based on Spherical Harmonics and Wavelets[J]. Journal of Geodynamics, 2005, 39(5): 512-526.
[8] SCHMIDT M, FENGLER M, MAYER-GÜRR T, et al. Regional Gravity Modeling in Terms of Spherical Base Functions[J]. Journal of Geodesy, 2007, 81(1): 17-38.
[9] EICKER A. Gravity Field Refinement by Radial Basis Functions from In-situ Satellite Data[D]. Bonn: Universitt Bonn, 2008.
[10] BENTEL K, SCHMIDT M, GERLACH C. Different Radial Basis Functions and Their Applicability for Regional Gravity Field Representation on the Sphere[J]. GEM-International Journal on Geomathematics, 2013, 4(1): 67-96.
[11] WITTWER T B. Regional Gravity Field Modelling with Radial Basis Functions[D]. Delft: Delft University of Technology, 2009.
[12] KLEES R, TENZER R, PRUTKIN I, et al. A Data-driven Approach to Local Gravity Field Modelling Using Spherical Radial Basis Functions[J]. Journal of Geodesy, 2008, 82(8): 457-471.
[13] PANET I, KUROISHI Y, HOLSCHNEIDER M. Wavelet Modelling of the Gravity Field by Domain Decomposition Methods: An Example Over Japan[J]. Geophysical Journal International, 2011, 184(1): 203-219.
[14] FENGLER M J, FREEDEN W, KOHLHAAS A, et al. Wavelet Modeling of Regional and Temporal Variations of the Earth's Gravitational Potential Observed by GRACE[J]. Journal of Geodesy, 2007, 81(1): 5-15.
[15] GRAFAREND E W, KRUMM F. The Abel-Poisson Kernel and the Abel-Poisson Integral in a Moving Tangent Space[J]. Journal of Geodesy, 1998, 72(7-8): 404-410.
[16] TENZER R, KLEES R. The Choice of the Spherical Radial Basis Functions in Local Gravity Field Modeling[J]. Studia Geophysica et Geodaetica, 2008, 52(3): 287-304.
[17] KOCH K R, KUSCHE J. Regularization of Geopotential Determination from Satellite Data by Variance Components[J]. Journal of Geodesy, 2002, 76(5): 259-268.
[18] REUTER R. Über Integralformeln der Einheitssphre und Harmonische Splinefunktionen[M]. Rheinisch: Hochschule Aachen, 1982.
[19] SCHMIDT M, HAN S C, KUSCHE J, et al. Regional High-resolution Spatiotemporal Gravity Modeling from GRACE Data Using Spherical Wavelets[J]. Geophysical Research Letters, 2006, 33(8). DOI: 10.1029/2005GL025509.
[20] KOCH K R. Parameter Estimation and Hypothesis Testing in Linear Models[M]. Berlin Heidelberg: Springer, 2013.
[21] PANET I, KUROISHI Y, HOLSCHNEIDER M. Flexible Dataset Combination and Modelling by Domain Decomposition Approaches[M]//SNEEUW N, NOVK P, CRESPI M, et al. VII Hotine-Marussi Symposium on Mathematical Geodesy. Berlin: Springer, 2012: 67-73.
[22] 汪海洪. 小波多尺度分析在地球重力场中的应用研究[D]. 武汉: 武汉大学, 2005. WANG Haihong. Research on Applications of Wavelet Multiscale Analysis in the Earth's Gravity Field[D]. Wuhan: Wuhan University, 2005.
[23] 王虎彪, 王勇, 陆洋, 等. 用卫星测高和船测重力资料联合反演海洋重力异常[J]. 大地测量与地球动力学, 2005, 25(1): 81-85. WANG Hubiao, WANG Yong, LU Yang, et al. Inversion of Marine Gravity Anomalies by Combining Multi-altimeter Data and Shipborne Gravimetric Data[J]. Journal of Geodesy and Geodynamics, 2005, 25(1): 81-85.
