动力法反演地球重力场模型研究
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摘要
卫星重力测量技术可以快速获取全球重力场中长波静态信息及时变信息,已成为地球物理、大地测量、海洋等学科研究的一种高技术手段,因此利用卫星测量技术反演地球重力场模型成为近年来国际大地测量领域研究的热点。动力法反演地球重力场模型是一类最早得到应用的传统的反演地球重力场模型的方法,最初的和目前得到广泛应用的地球重力场模型大多是采用这种方法获取的。随着CHAMP、GRACE重力卫星计划的成功实施,一些新的方法(例如能量法等)迅速得到研究和重视,但动力法也还是最基础的一种方法。本文主要研究了在新的观测资料的条件下进行动力法反演地球重力场模型的原理,深入研究了卫星重力观测数据特点,并利用CHAMP数据进行实际计算。同时,由于目前卫星重力测量数据的大量获取,不同科研机构建立的地球重力场模型的数量也越来越多,这些模型在应用中与实际地面重力测量数据比较,相差并不太大,但其对应的系数却不完全相同(特别是高阶项,互差值可能是系数值本身的30%～100%,有的甚至更大),即存在地球重力场模型系数的稳定性问题。因此本文从近现代相关学科的基本原理出发,研究了动力法反演地球重力场模型的问题。
本文的主要内容如下:
1.给出了卫星重力测量所涉及的基础理论知识,详细阐述了各个摄动力模型,并给出了实用计算公式。
2.利用数值方法研究了地心运动对卫星轨道的影响。
3.给出了引力位在地固坐标系中的球谐表达式和轨道根数表达式以及在卫星重力测量实用计算中的直角坐标系下的引力位、引力向量和引力梯度张量的表达式。
4.讨论了地球引力位系数的一般特性、低阶项的物理意义,以及重力场功率谱约束等;分析了地球引力位系数排列的多种方式对法方程结构的影响,确定了卫星重力测量解算中最优的地球引力位系数的排列方式。
5.研究了动力法精密定轨的基本原理,给出了不同观测资料的偏导数,针对卫星重力测量SST-HL和SST-LL模型分别给出动力法解算地球重力场模型的具体数学模型和计算过程。
6.研究分析了CHAMP星载加速度计数据、轨道数据的特点和先验重力场模型、积分弧长等问题,利用CHAMP数据进行了实际计算,计算得到模型G-ModelS1(50阶),并采用与GGM02S、DQM2000d、UGM05和EGM2008四种模型计算相同重力场元的互差的方式进行了精度评估。
7.在进一步分析了地球重力场模型位系数作为常数的物理意义的基础上,给出了利用“点质量”思想表示的地球引力位系数的表达式,分析了C n 0, C nk ,S nk与地球全球质量积分的关系。
8.利用带谐函数的性质研究了C n 0, C nk ,S nk的特点,得出:不同的C n0对于n个纬度位置上的一层质量及其变化完全不敏感;当n→∞时,C n0趋向于仅包含两个极点质量的积分值或互差值(具体由奇偶性决定),由于数值会非常小,近似也可以认为C n0→0。对C nk ,S nk也有相类似的结论。
9.从相对论的基本原理出发,定性分析了动力法反演地球重力场模型存在的问题,得出:根据广义相对论,我们习惯用的牛顿引力位的地球重力场模型是平直空间中地球引力场的精细描述,但却是实际地球周围弯曲空间的近似描述,由于度规的不同,不考虑后牛顿效应动力法解算的地球重力场模型(假定无误差)在地球平均半径ae处计算的大地水准面的极限平均精度约为10-9。
10.基于混沌学讨论了动力法反演地球重力场模型的近可积非线性动力系统的本质和存在的内在随机性问题(注:这里的内在随机性表现为不同模型的各个系数的数值有偏差,但不同模型总体精度却相差不大)。
With the application of satellite gravimetry, static as well as time-variant information of middle and long wavelength gravity field of the Earth can be fast acquired, which has made satellite gravimetry one high-tech tool in the research of geophysics, geodesy and oceanography. Therefore, the recovery of the Earth's gravity field using satellite techniques has become a hot issue for the research of geodesy in recent years. Dynamic method is one of the earliest applied methods for the recovery of the Earth's gravity field. The earliest and current widely used gravity field models were mostly acquired using the Dynamic method. With the successful implementation of satellite gravity missions like CHAMP, GRACE, some new methods, say, the Energy method, etc., attract more attention, however the Dynamic method is still one basic method that plays an important role. The dissertation covered the principle of the recovery of the Earth's gravity field using the Dynamic method with new types of observations, discussed the characteristics of satellite gravimetry observations and made practical computations using CHAMP data. At the same time, due to the enormous increasing of satellite gravimetry observations, there are more and more gravity field models of the Earth that were established by a lot of research institutions. The models make less difference when compared with real ground gravity observations, but their coefficients differs a lot, especially the higher degree, for which the inter-difference reaches 30%-100% of the coefficient itself and some to a higher extent. Therefore, the stability problem of gravity field model coefficients arises. From the fundamental of related contemporary sciences, the dissertation studied the recovery of the Earth's gravity field using the Dynamic method.
The main work is as follows:
1. The basic theory for satellite gravimetry was systematically presented. Perturbing forces that affect satellite gravimetry were categorized and discussed in detail and practical formulae were also presented.
2. The effect of the Earth center movement on satellite orbit was studied using numerical methods.
3. The spherical harmonic expressions of gravitational potential within the Earth Fixed System was provided. For the practical computations of satellite gravity, the expressions of gravity potential, gravity vector and gravity gradient tensor were also provided.
4. The features of the Earth's gravity potential as well as the physical meanings of lower-degree coefficients were discussed. Power spectral constraint of the gravity field was also analyzed. The influence of the arrangement of gravity potential model coefficients on the structure of normal equations was analyzed, from which the optimal arrangement of model coefficients was determined that can be applied in satellite gravity solution.
5. Precise Orbit Determination (POD) using the Dynamic method was researched, in which the partial derivatives for different observations were obtained. The specific math model and procedure for the Earth's gravity field solution were presented aiming at SST-HL and SST-LL model in current satellite gravimetry.
6. Data of space-borne accelerometer as orbit of CHAMP were analyzed, along with the a priori gravity field model and integral arc length. Based on CHAMP data, practical computations were made, from which gravity field model G-Model S1(up to 50 degree) was derived. And then the model was assessed by making comparisons in computing gravity field elements with four models, i.e. GGM02S, DQM2000d, UGM05 and EGM2008.
7. When used as constants, the gravity field model coefficients have certain physical meanings, which were analyzed in the dissertation. Based on the physical meanings of model coefficients, the expression of model coefficients in terms of point masses was presented, and the relation between C n 0, C nk ,S nk and the global mass integral was analyzed.
8. C n 0, C nk ,S nk were studied based on the characteristics of zonal harmonics. From the analysis, different C n0 is totally insensitive to the mass layer and mass variation of n latitudes, and when n→∞, C n0 converges to the integral or inter-difference that includes two polar masses. Due to the infinitesimally small value, approximately C n0→0 can be taken. Similar points holds for coefficients C nk ,S nk.
9. From the principle of the Theory of Relativity, the problem that gravity field recovery using the Dynamic method encounters was qualitatively analyzed. According to the Generalized Theory of Relativity, the Earth's gravity field model of Newton gravitational potential that people are used to is the subtle description of the gravitational field within 3-D Cartesian system; however, it is the approximation of the curved space that surrounds the Earth. Due to the different scale, the limit mean accuracy reaches 10-9 for the geoid computed at the mean radius of the Earth using the Earth's gravity field model that is solved with no post-Newton effect taken into account.
10. The nature of the near integrable nonlinear dynamic system as well as its internal randomness was discussed from the viewpoint of the Chaos Theory. The randomness manifests that there exist bias between coefficients of different gravity field models, but the total accuracy between different models make less difference.
本文的主要内容如下:
1.给出了卫星重力测量所涉及的基础理论知识,详细阐述了各个摄动力模型,并给出了实用计算公式。
2.利用数值方法研究了地心运动对卫星轨道的影响。
3.给出了引力位在地固坐标系中的球谐表达式和轨道根数表达式以及在卫星重力测量实用计算中的直角坐标系下的引力位、引力向量和引力梯度张量的表达式。
4.讨论了地球引力位系数的一般特性、低阶项的物理意义,以及重力场功率谱约束等;分析了地球引力位系数排列的多种方式对法方程结构的影响,确定了卫星重力测量解算中最优的地球引力位系数的排列方式。
5.研究了动力法精密定轨的基本原理,给出了不同观测资料的偏导数,针对卫星重力测量SST-HL和SST-LL模型分别给出动力法解算地球重力场模型的具体数学模型和计算过程。
6.研究分析了CHAMP星载加速度计数据、轨道数据的特点和先验重力场模型、积分弧长等问题,利用CHAMP数据进行了实际计算,计算得到模型G-ModelS1(50阶),并采用与GGM02S、DQM2000d、UGM05和EGM2008四种模型计算相同重力场元的互差的方式进行了精度评估。
7.在进一步分析了地球重力场模型位系数作为常数的物理意义的基础上,给出了利用“点质量”思想表示的地球引力位系数的表达式,分析了C n 0, C nk ,S nk与地球全球质量积分的关系。
8.利用带谐函数的性质研究了C n 0, C nk ,S nk的特点,得出:不同的C n0对于n个纬度位置上的一层质量及其变化完全不敏感;当n→∞时,C n0趋向于仅包含两个极点质量的积分值或互差值(具体由奇偶性决定),由于数值会非常小,近似也可以认为C n0→0。对C nk ,S nk也有相类似的结论。
9.从相对论的基本原理出发,定性分析了动力法反演地球重力场模型存在的问题,得出:根据广义相对论,我们习惯用的牛顿引力位的地球重力场模型是平直空间中地球引力场的精细描述,但却是实际地球周围弯曲空间的近似描述,由于度规的不同,不考虑后牛顿效应动力法解算的地球重力场模型(假定无误差)在地球平均半径ae处计算的大地水准面的极限平均精度约为10-9。
10.基于混沌学讨论了动力法反演地球重力场模型的近可积非线性动力系统的本质和存在的内在随机性问题(注:这里的内在随机性表现为不同模型的各个系数的数值有偏差,但不同模型总体精度却相差不大)。
With the application of satellite gravimetry, static as well as time-variant information of middle and long wavelength gravity field of the Earth can be fast acquired, which has made satellite gravimetry one high-tech tool in the research of geophysics, geodesy and oceanography. Therefore, the recovery of the Earth's gravity field using satellite techniques has become a hot issue for the research of geodesy in recent years. Dynamic method is one of the earliest applied methods for the recovery of the Earth's gravity field. The earliest and current widely used gravity field models were mostly acquired using the Dynamic method. With the successful implementation of satellite gravity missions like CHAMP, GRACE, some new methods, say, the Energy method, etc., attract more attention, however the Dynamic method is still one basic method that plays an important role. The dissertation covered the principle of the recovery of the Earth's gravity field using the Dynamic method with new types of observations, discussed the characteristics of satellite gravimetry observations and made practical computations using CHAMP data. At the same time, due to the enormous increasing of satellite gravimetry observations, there are more and more gravity field models of the Earth that were established by a lot of research institutions. The models make less difference when compared with real ground gravity observations, but their coefficients differs a lot, especially the higher degree, for which the inter-difference reaches 30%-100% of the coefficient itself and some to a higher extent. Therefore, the stability problem of gravity field model coefficients arises. From the fundamental of related contemporary sciences, the dissertation studied the recovery of the Earth's gravity field using the Dynamic method.
The main work is as follows:
1. The basic theory for satellite gravimetry was systematically presented. Perturbing forces that affect satellite gravimetry were categorized and discussed in detail and practical formulae were also presented.
2. The effect of the Earth center movement on satellite orbit was studied using numerical methods.
3. The spherical harmonic expressions of gravitational potential within the Earth Fixed System was provided. For the practical computations of satellite gravity, the expressions of gravity potential, gravity vector and gravity gradient tensor were also provided.
4. The features of the Earth's gravity potential as well as the physical meanings of lower-degree coefficients were discussed. Power spectral constraint of the gravity field was also analyzed. The influence of the arrangement of gravity potential model coefficients on the structure of normal equations was analyzed, from which the optimal arrangement of model coefficients was determined that can be applied in satellite gravity solution.
5. Precise Orbit Determination (POD) using the Dynamic method was researched, in which the partial derivatives for different observations were obtained. The specific math model and procedure for the Earth's gravity field solution were presented aiming at SST-HL and SST-LL model in current satellite gravimetry.
6. Data of space-borne accelerometer as orbit of CHAMP were analyzed, along with the a priori gravity field model and integral arc length. Based on CHAMP data, practical computations were made, from which gravity field model G-Model S1(up to 50 degree) was derived. And then the model was assessed by making comparisons in computing gravity field elements with four models, i.e. GGM02S, DQM2000d, UGM05 and EGM2008.
7. When used as constants, the gravity field model coefficients have certain physical meanings, which were analyzed in the dissertation. Based on the physical meanings of model coefficients, the expression of model coefficients in terms of point masses was presented, and the relation between C n 0, C nk ,S nk and the global mass integral was analyzed.
8. C n 0, C nk ,S nk were studied based on the characteristics of zonal harmonics. From the analysis, different C n0 is totally insensitive to the mass layer and mass variation of n latitudes, and when n→∞, C n0 converges to the integral or inter-difference that includes two polar masses. Due to the infinitesimally small value, approximately C n0→0 can be taken. Similar points holds for coefficients C nk ,S nk.
9. From the principle of the Theory of Relativity, the problem that gravity field recovery using the Dynamic method encounters was qualitatively analyzed. According to the Generalized Theory of Relativity, the Earth's gravity field model of Newton gravitational potential that people are used to is the subtle description of the gravitational field within 3-D Cartesian system; however, it is the approximation of the curved space that surrounds the Earth. Due to the different scale, the limit mean accuracy reaches 10-9 for the geoid computed at the mean radius of the Earth using the Earth's gravity field model that is solved with no post-Newton effect taken into account.
10. The nature of the near integrable nonlinear dynamic system as well as its internal randomness was discussed from the viewpoint of the Chaos Theory. The randomness manifests that there exist bias between coefficients of different gravity field models, but the total accuracy between different models make less difference.
引文
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