重力位三阶梯度张量及其勘探能力初步分析
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摘要
在重力位三阶梯度张量基本原理的基础上,给出均匀球体重力位零阶至三阶梯度张量的解析表达式,结合理论分析与模型实验,对比各阶梯度张量对场源参数(质量与埋深、水平位置与几何形状)的敏感性。结果表明,重力位三阶梯度张量随场源与观测点之间距离的增大衰减最快。重力曲率张量具有如下特征:对场源埋深变化具有最高的敏感性;对浅部的剩余质量探测能力最强,对深部的剩余质量探测能力最弱;对场源的水平分辨能力及几何形态的区分能力随场源埋深的增大而减小,但重力曲率张量的分辨能力最强;三阶梯度张量的独立元素个数最多,场空间分布形态的类型也最多样,在相同测点分布情况下能够捕获更多的场与场源信息,尤其是重力场的短波成分与浅部的剩余质量。
We first introduce the basic theory of the third-order gradient tensor of the gravitational potential, which is also called the gravitational curvature tensor(GCT). Second, we provide analytic expressions of the third-order gradient tensor of the gravitational potential caused by the homogenous spheroid. Finally, through theoretical analysis and synthetic model tests, we discuss the sensitivity of the source's parameters(mass, buried depth, horizontal location and geometric shape) on the zero, first, second and third-order gradient tensors of the gravitational potential. Results show that, among the zero, first, second and third-order gradient tensors of the gravitational potential, the third-order gradient tensor of the gravitational potential decays most quickly as the distance between observation point and source increases. Therefore, the GCT has multiple characteristics, such as, highest sensitivity on the buried depth of the source; the best exploration capacity for shallow mass, and on the contrary the worst capacity for deep mass; and the highest sensitivity of the horizontal location of the source. This ability is weakened as the buried depth of the source increases. The type is also the most diverse, indicating that under the same distribution of points, the three-step tensor can capture more field and field source information, especially the short-wave component of the gravity field and the remaining mass of the shallow part.
We first introduce the basic theory of the third-order gradient tensor of the gravitational potential, which is also called the gravitational curvature tensor(GCT). Second, we provide analytic expressions of the third-order gradient tensor of the gravitational potential caused by the homogenous spheroid. Finally, through theoretical analysis and synthetic model tests, we discuss the sensitivity of the source's parameters(mass, buried depth, horizontal location and geometric shape) on the zero, first, second and third-order gradient tensors of the gravitational potential. Results show that, among the zero, first, second and third-order gradient tensors of the gravitational potential, the third-order gradient tensor of the gravitational potential decays most quickly as the distance between observation point and source increases. Therefore, the GCT has multiple characteristics, such as, highest sensitivity on the buried depth of the source; the best exploration capacity for shallow mass, and on the contrary the worst capacity for deep mass; and the highest sensitivity of the horizontal location of the source. This ability is weakened as the buried depth of the source increases. The type is also the most diverse, indicating that under the same distribution of points, the three-step tensor can capture more field and field source information, especially the short-wave component of the gravity field and the remaining mass of the shallow part.
引文
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[15] Elkins T A. The Second Derivative Method of Gravity Interpretation[J]. Geophysics, 1951, 16(1): 29-50
[16] 曾华霖. 重力场与重力勘探[M]. 北京: 地质出版社, 2005(Zeng Hualin. Gravity Field and Gravity Exploration[M]. Beijing: Geological Publishing House, 2005)
[17] Li X. Vertical Resolution: Gravity Versus Vertical Gravity Gradient[J]. The Leading Edge, 2001, 20(8): 901-904
[18] 王庆宾, 江东, 赵东明, 等. 重力垂直梯度测量探测能力的正演[J]. 测绘科学技术学报, 2011, 28(3): 262-264(Wang Qingbin, Jiang Dong, Zhao Dongming, et al. Underground Vacancy Detection Based on Vertical Gravity Gradient Measurements[J]. Journal of Geomatics Science and Technology, 2011, 28(3): 262-264)
[19] 王谦身, 张赤军, 蒋福珍. 微重力测量: 理论、方法与应用[M]. 北京: 科学出版社, 1995(Wang Qianshen, Zhang Chijun, Jiang Fuzhen. Microgravimetry: Theory, Method and Applications[M]. Beijing: Science Press, 1995)
[20] Liu Y X. Evaluation and Extraction of Weak Gravity and Magnetic Anomalies[J]. Applied Geophysics, 2007, 4(4): 288-293
[21] Stavrev P, Reid A. Degrees of Homogeneity of Potential Fields and Structural Indices of Euler Deconvolution[J]. Geophysics, 2007, 72(1): L1-L12
[22] 邱峰, 杜劲松,陈超. 矩形棱柱体重力位三阶梯度张量正演计算公式[J]. 大地测量与地球动力学, 2019,39(3):313-316(Qiu Feng, Du Jinsong, Chen Chao. Forward Modeling Formulae for Third-Order Gradient Tensor of Gravitational Potential Caused by the Right Rectangular Prism[J]. Journal of Geodesy and Geodynamics, 2019,39(3):313-316)
[2] Camp M, Viron O, Watlet A, et al. Geophysics from Terrestrial Time-Variable Gravity Measurements[J]. Reviews of Geophysics, 2017, 55(4): 938-992
[3] Moddlemiss R P, Samarelli A, Paul D J, et al. Measurement of the Earth Tides with a MEMS Gravimeter[J]. Nature, 2016, 531(7 596): 614-617
[4] Balakin A B, Daishev R A, Murzakhanov Z G, et al. Laser-Interferometric Detector of the First, Second and Third Derivatives of the Potential of the Earth Gravitational Field[J]. Izvestiya Vysshikh Uchebnykh Zavedenii, Seriya Geologiyai Razvedka, 1997(1): 101-107
[5] Brieden P, Muller J, Flury J, et al.The Mission OPTIMA—Novelties and Benefit[R]. Geotechnologien, Potsdam, 2010
[6] Fantino E, Casotto S. Methods of Harmonic Synthesis for Global Geopotential Models and Their First-, Second- and Third-Order Gradients[J]. Journal of Geodesy, 2009, 83(7): 595-619
[7] Fukushima T. Recursive Computation of Oblate Spheroidal Harmonics of Second Kind and Their First-, Second-, and Third-Order Derivatives[J]. Journal of Geodesy, 2013, 87(4): 303-309
[8] Rosi G, Cacciapuoti L, Sorrentino F, et al.Measurements of the Gravity-Field Curvature by Atom Interferometry[J]. Physical Review Letters, 2015, 114(1)
[9] Ghobadi-Far K, Sharifi M A, Sneeuw N. 2D Fourier Series Representation of Gravitational Functionals in Spherical Coordinates[J]. Journal of Geodesy, 2016, 90(9): 871-881
[10] Sharifi M A, Romeshkani M, Tenzer R. On Inversion of the Second- and Third-Order Gravitational Tensors by Stokes’ Integral Formula for a Regional Gravity Recovery[J]. Studia Geophysica et Geodaetica, 2017, 61(3): 453-468
[11] Deng X L, Shen W B. Evaluation of Optimal Formulas for Gravitational Tensors up to Gravitational Curvatures of a Tesseroid[J]. Surveys in Geophysics, 2018, 39(3): 365-399
[12] Rothleitner C, Francis O. Measuring the Newtonian Constant of Gravitation with a Differential Free-Fall Gradiometer: A Feasibility Study[J]. Review of Scientific Instruments, 2014, 85(4): 469-526
[13] 陈钊, 李辉, 邢乐林, 等. 重力垂直梯度非线性变化研究[J]. 大地测量与地球动力学, 2014, 34(2): 27-30(Chen Zhao, Li Hui, Xing Lelin, et al. Study on Non-Linear Variation of Vertical Gradient of Gravity[J]. Journal of Geodesy and Geodynamics, 2014, 34(2): 27-30)
[14] DiFrancesco D, Meyer T J, Christensen A, et al.Gravity Gradiometry—Today and Tomorrow[C]. 11th SAGA Biennial Technical Meeting and Exhibition, Swaziland, 2009
[15] Elkins T A. The Second Derivative Method of Gravity Interpretation[J]. Geophysics, 1951, 16(1): 29-50
[16] 曾华霖. 重力场与重力勘探[M]. 北京: 地质出版社, 2005(Zeng Hualin. Gravity Field and Gravity Exploration[M]. Beijing: Geological Publishing House, 2005)
[17] Li X. Vertical Resolution: Gravity Versus Vertical Gravity Gradient[J]. The Leading Edge, 2001, 20(8): 901-904
[18] 王庆宾, 江东, 赵东明, 等. 重力垂直梯度测量探测能力的正演[J]. 测绘科学技术学报, 2011, 28(3): 262-264(Wang Qingbin, Jiang Dong, Zhao Dongming, et al. Underground Vacancy Detection Based on Vertical Gravity Gradient Measurements[J]. Journal of Geomatics Science and Technology, 2011, 28(3): 262-264)
[19] 王谦身, 张赤军, 蒋福珍. 微重力测量: 理论、方法与应用[M]. 北京: 科学出版社, 1995(Wang Qianshen, Zhang Chijun, Jiang Fuzhen. Microgravimetry: Theory, Method and Applications[M]. Beijing: Science Press, 1995)
[20] Liu Y X. Evaluation and Extraction of Weak Gravity and Magnetic Anomalies[J]. Applied Geophysics, 2007, 4(4): 288-293
[21] Stavrev P, Reid A. Degrees of Homogeneity of Potential Fields and Structural Indices of Euler Deconvolution[J]. Geophysics, 2007, 72(1): L1-L12
[22] 邱峰, 杜劲松,陈超. 矩形棱柱体重力位三阶梯度张量正演计算公式[J]. 大地测量与地球动力学, 2019,39(3):313-316(Qiu Feng, Du Jinsong, Chen Chao. Forward Modeling Formulae for Third-Order Gradient Tensor of Gravitational Potential Caused by the Right Rectangular Prism[J]. Journal of Geodesy and Geodynamics, 2019,39(3):313-316)