卫星重力梯度数据处理理论与方法
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摘要
本文主要研究了利用卫星重力梯度观测数据恢复地球重力场的理论、方法和实用解算模型。作者在文中所做的主要工作和创新点有：
(1)研究了卫星重力梯度测量的相关基础理论。论述了卫星重力梯度测量的基本原理；给出了卫星重力梯度张量在不同曲线坐标系中的表述及其相互转换关系。
(2)研究了地球引力位、引力矢量和引力张量的球谐展开表示、轨道根数表示；建立了卫星重力梯度观测量的轨道面观测方程和球面观测方程,并推导得到了广义球谐函数及其相关积分的非奇异计算公式。给出了卫星重力梯度张量的球面调和分析和球谐综合公式,并对球面调和分析方法和球谐综合方法进行了模拟试算。
(3)深入研究并建立了利用卫星重力梯度张量构建地球重力场模型的广义轮胎调和分析方法。针对重力梯度张量数据,重点研究了球面到轮胎面的映射关系,球面上傅立叶分析与调和分析的关系,完善了轮胎调和分析方法,得到了能够处理重力矢量、重力梯度张量数据的广义轮胎调和分析方法。
(4)提出并建立了基于卫星重力梯度张量的点质量调和分析方法,进一步完善了点质量模型理论。巧妙运用球极坐标系关于计算点和流动点的微分运算关系,建立了基于重力梯度张量分量的全球点质量模型,研究并提出利用分块循环矩阵分解大型线性方程组的方法,解决了全球点质量模型构建中大型线性方程组的稳定解算瓶颈问题,给出了最小二乘解。通过点质量的球谐展开,得到了点质量调和分析方法的实用公式；提出了分频段点质量调和分析方法。
(5)提出了线质量调和分析方法,有效克服了单层点质量调和分析方法和分频段点质量调和分析方法的不足。
(6)研究并完善了基于重力梯度张量的最小二乘复数配置调和分析方法。完整的给出了重力梯度张量之间的协方差函数、重力梯度张量与引力位系数之间的协方差函数；通过扰动引力梯度观测数据为等经差规则格网数据的情况下,引力位与扰动引力梯度之间的协方差矩阵具有分块Toeplitz循环阵的结构,有效的利用FFT变换技术将其降阶；在复数配置的基础上,进一步建立了基于重力梯度张量的最小二乘矢量、张量配置调和分析方法。
(7)建立了利用重力梯度张量不变量来求解位系数的理论与方法,该方法能够有效的克服卫星姿态误差带来的影响。分别研究了球近似下的不变量观测方程和顾及J2项不变量观测方程及其相应的解算方法,最后运用广义轮胎调和分析方法等空域法进行了模拟试验,进一步验证了利用梯度张量不变量法的有效性和可行性。
This dissertation mainly performs research in the theory, methods and applied computation models of determining global gravity model from observations of satellite gravity gradients. The central ideas and new standpoints are lists as follows:
(1) With theories pertinent to satellite gradiometry, the principles of satellite gradiometry are dissertated in detail. Besides, the different expressions of satellite gravity gradients tensor in several curvilinear coordinate systems and the transformation methods between any two of them are given.
(2) The establishment of the observation equations both in orbit plane and spherical are based on the spherical harmonic and orbital element expressions of the earth potential, gravitation vector and gravitation tensor, which also makes great contributions to generalized spherical harmonics and non-singular formulae of correlated integrals. Then, the harmonic analysis and spherical harmonic synthesis formulae are obtained, and simulations are made.
(3) Generalized torus harmonic analysis is established based on the satellite gravity gradient tensor. With the support of gravity gradient data, the mapping relations from sphere to torus and that between Fourier analysis and harmonic analysis on sphere are emphasized. The torus harmonic analysis is modified, thus the generalized torus harmonic analysis which could deal with the gravitational vector and gradient tensor is obtained.
(4) The point mass harmonic analysis which is put forward and established based on satellite gravity gradient tensor improves the theory of point mass model. Through the special differential coefficient operation relation between the calculated point and the fluxional points in spheric polar coordinate system, global point mass model is established based on satellite gravitation gradient tensor components. The way of partitioned recurrence matrices which is put forward to divide huge linear equations solves the bottleneck problem in the stability solution of global point mass model. And then the least square solution is attained. The applied formula of point mass harmonic analysis is obtained with the spherical harmonic expressions of point mass. Besides, spectrum-based point mass harmonic analysis is also discussed.
(5)Line mass harmonic analysis, which could effectively overcome the disadvantages of spectrum-based point mass harmonic analysis and point mass harmonic analysis, is brought forward.
(6)Further research is made to modify the complex LSC harmonic analysis based on gravity gradient tensor. Complete covariance of gravity gradient tensors and that of tensors and gravity potential coefficients are presented. With the observational data of gravity gradient which are in grid form with the same longitude interval, covariance of gravitational potential and disturbed gravity gradient has partitioned Toeplitz recurrence structure, and then FFT is used to reduce its degree. With the support of complex collocation, the vector and tensor LSC harmonic analysis which is based on gravity gradient tensor is further discussed.
(7)The theory on the determination of the gravity model based on gravity gradient tensor invariant is established, which could overcome the bias caused by satellite attitude errors effectively. Furthermore, observational equations and the solutions of invariants both in sphere approximation and considering J2 item are discussed in detail. Finally, simulation experiments of space-wise such as generalized torus harmonic analysis are made to validate the efficiency and feasibility of gradient tensor invariants.
(1)研究了卫星重力梯度测量的相关基础理论。论述了卫星重力梯度测量的基本原理；给出了卫星重力梯度张量在不同曲线坐标系中的表述及其相互转换关系。
(2)研究了地球引力位、引力矢量和引力张量的球谐展开表示、轨道根数表示；建立了卫星重力梯度观测量的轨道面观测方程和球面观测方程,并推导得到了广义球谐函数及其相关积分的非奇异计算公式。给出了卫星重力梯度张量的球面调和分析和球谐综合公式,并对球面调和分析方法和球谐综合方法进行了模拟试算。
(3)深入研究并建立了利用卫星重力梯度张量构建地球重力场模型的广义轮胎调和分析方法。针对重力梯度张量数据,重点研究了球面到轮胎面的映射关系,球面上傅立叶分析与调和分析的关系,完善了轮胎调和分析方法,得到了能够处理重力矢量、重力梯度张量数据的广义轮胎调和分析方法。
(4)提出并建立了基于卫星重力梯度张量的点质量调和分析方法,进一步完善了点质量模型理论。巧妙运用球极坐标系关于计算点和流动点的微分运算关系,建立了基于重力梯度张量分量的全球点质量模型,研究并提出利用分块循环矩阵分解大型线性方程组的方法,解决了全球点质量模型构建中大型线性方程组的稳定解算瓶颈问题,给出了最小二乘解。通过点质量的球谐展开,得到了点质量调和分析方法的实用公式；提出了分频段点质量调和分析方法。
(5)提出了线质量调和分析方法,有效克服了单层点质量调和分析方法和分频段点质量调和分析方法的不足。
(6)研究并完善了基于重力梯度张量的最小二乘复数配置调和分析方法。完整的给出了重力梯度张量之间的协方差函数、重力梯度张量与引力位系数之间的协方差函数；通过扰动引力梯度观测数据为等经差规则格网数据的情况下,引力位与扰动引力梯度之间的协方差矩阵具有分块Toeplitz循环阵的结构,有效的利用FFT变换技术将其降阶；在复数配置的基础上,进一步建立了基于重力梯度张量的最小二乘矢量、张量配置调和分析方法。
(7)建立了利用重力梯度张量不变量来求解位系数的理论与方法,该方法能够有效的克服卫星姿态误差带来的影响。分别研究了球近似下的不变量观测方程和顾及J2项不变量观测方程及其相应的解算方法,最后运用广义轮胎调和分析方法等空域法进行了模拟试验,进一步验证了利用梯度张量不变量法的有效性和可行性。
This dissertation mainly performs research in the theory, methods and applied computation models of determining global gravity model from observations of satellite gravity gradients. The central ideas and new standpoints are lists as follows:
(1) With theories pertinent to satellite gradiometry, the principles of satellite gradiometry are dissertated in detail. Besides, the different expressions of satellite gravity gradients tensor in several curvilinear coordinate systems and the transformation methods between any two of them are given.
(2) The establishment of the observation equations both in orbit plane and spherical are based on the spherical harmonic and orbital element expressions of the earth potential, gravitation vector and gravitation tensor, which also makes great contributions to generalized spherical harmonics and non-singular formulae of correlated integrals. Then, the harmonic analysis and spherical harmonic synthesis formulae are obtained, and simulations are made.
(3) Generalized torus harmonic analysis is established based on the satellite gravity gradient tensor. With the support of gravity gradient data, the mapping relations from sphere to torus and that between Fourier analysis and harmonic analysis on sphere are emphasized. The torus harmonic analysis is modified, thus the generalized torus harmonic analysis which could deal with the gravitational vector and gradient tensor is obtained.
(4) The point mass harmonic analysis which is put forward and established based on satellite gravity gradient tensor improves the theory of point mass model. Through the special differential coefficient operation relation between the calculated point and the fluxional points in spheric polar coordinate system, global point mass model is established based on satellite gravitation gradient tensor components. The way of partitioned recurrence matrices which is put forward to divide huge linear equations solves the bottleneck problem in the stability solution of global point mass model. And then the least square solution is attained. The applied formula of point mass harmonic analysis is obtained with the spherical harmonic expressions of point mass. Besides, spectrum-based point mass harmonic analysis is also discussed.
(5)Line mass harmonic analysis, which could effectively overcome the disadvantages of spectrum-based point mass harmonic analysis and point mass harmonic analysis, is brought forward.
(6)Further research is made to modify the complex LSC harmonic analysis based on gravity gradient tensor. Complete covariance of gravity gradient tensors and that of tensors and gravity potential coefficients are presented. With the observational data of gravity gradient which are in grid form with the same longitude interval, covariance of gravitational potential and disturbed gravity gradient has partitioned Toeplitz recurrence structure, and then FFT is used to reduce its degree. With the support of complex collocation, the vector and tensor LSC harmonic analysis which is based on gravity gradient tensor is further discussed.
(7)The theory on the determination of the gravity model based on gravity gradient tensor invariant is established, which could overcome the bias caused by satellite attitude errors effectively. Furthermore, observational equations and the solutions of invariants both in sphere approximation and considering J2 item are discussed in detail. Finally, simulation experiments of space-wise such as generalized torus harmonic analysis are made to validate the efficiency and feasibility of gradient tensor invariants.
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