GPS/重力边值问题的求解及应用

从分析GPS技术在确定地球形状中的作用入手，论述了建立一类新的大地边值问题——GPS/重力边值问题的意义，给出了GPS/重力边值问题的定义及数学描述，推导出GPS/重力边值问题的逼近解式，并给出了应用GPS/重力边值问题确定（似）大地水准面、地面垂线偏差及外部重力场的基本公式. 对GPS技术用于物理大地测量的优势及有待解决的问题进行了简要归纳.     

GPS/重力边值问题实用公式推导及分析

基于GPS/重力边值问题的连续趋近解,推导了形式上更为简单,物理意义更加明确的实用公式.同时利用扰动重力的泰勒级数展开式直接证明了GPS/重力边值问题实用公式的正确性,并且进一步阐明其物理含义.在实用公式中,我们引入了扰动重力垂直梯度这个物理量,这在观测数据多元化的今天具有积极意义.     

应用GPS/重力数据确定(似)大地水准面

作为GPS/重力边值问题理论及方法的应用,本文在对GPS/重力方法确定（似）大地水准面的原理进行简要介绍与分析的基础上,利用收集到的N区的702个GPS重力数据以及52个高精度的GPS水准数据,计算出该区域的似大地水准面. 通过拟合法和系统差直接改正法进行的精度分析表明,应用GPS/水准方法确定的该地区似大地水准面的精度达到厘米级.     

O(T2)精度下椭球界面Dirichlet边值问题的积分解

研究了边界是参考椭球面的Laplace方程Dirichlet边值问题的求解，在O(ε4·T)精 度下给出了参考椭球界面上扰动重力位Dirichlet外问题的积分解式. 该结果理论上优于目 前常用的球近似下的积分解式，从而为研究物理大地测量中边值问题的求解提供了新的依据     

O(T2)精度下椭球界面Dirichlet边值问题的积分解

研究了边界是参考椭球面的Laplace方程Dirichlet边值问题的求解，在O(ε4·T)精 度下给出了参考椭球界面上扰动重力位Dirichlet外问题的积分解式. 该结果理论上优于目 前常用的球近似下的积分解式，从而为研究物理大地测量中边值问题的求解提供了新的依据     

GPS-重力边值问题

详细研究了在O(T²)精度下如何利用地面GPS和重力观测值确定地球外部重力场的问题, 以参考椭球面为界面建立起了完整的理论和相应的精度判断标准. 具体求解方法主要分成两个步骤: 地面重力观测值到参考椭球面上的调和延拓; 参考椭球面上Neumann外问题的积分解式. 得到的积分解式在O(T²)精度下包含了重力场频谱的全部信息; 得到的公式不仅实用, 而且理论上有所突破, 在O(T²)精度下解决了椭球界面边值问题的积分解.     

基于有限元方法的陆海大地水准面衔接

大陆上用重力数据和GPS水准数据确定(似)大地水准面,海洋上用卫星测高数据确定(似)大地水准面.由于沿海地区和近岸海域往往缺少完好的重力数据,近岸海域卫星测高数据质量相对较差,两类大地水准面在陆海相接区域精度偏低且存在拼合差.纯几何方法拟合陆海局部区域大地水准面,不能顾及大地水准面的物理特性,拟合结果不稳定.顾及到大地水准面的物理特性,依据其在局部所应满足的数学物理方程,拟合陆海局部区域大地水准面问题,转化为Laplace第一边值问题.讨论了有限元法衔接陆海局部区域大地水准面的数学思想,给出了相应的数学模型.     

超定大地边值问题的变分解及相关理论

借助于变分法中求泛函极小化的方法提出求解超定大地边值问题的新方法. 首先依据变分原理引入了超定边值问题的变分解; 其次从理论上证明了变分解的存在唯一性以及最佳逼近性质; 然后讨论了变分解的一般算法, 并在球边界情况下给出了变分解的表达式; 最后利用EGM96重力场模型给出了一个算例, 计算结果表明变分解法能够有效地提高处理重力数据的精度. 总之, 该方法有利于各种类型数据的有效使用, 并且具有严格的数学基础, 能将不同类型的观测数据纳入统一的模型中进行研究.     

基于改进Runge定理的调和延拓及其相应的边值问题

利用改进的Runge定理引入了外大地水准面和外重力位等新概念 ,建立了关于外扰动位的边值问题 ,该问题同时具备了Stokes问题和Mododensky问题的优点 .作为一个整体 ,讨论了外正高的计算和地面重力的归算 ,给出了在O(T2 )量级的精度下确定外大地水准面、地面以及地球外部重力场的方法 .     

重力和形变资料联合反演地壳密度时间变化的一种方法

基于单层位理论，利用动态大地测量基本方程，选取Ｂｊｅｒｈａｍｍａｒ虚拟球作球近似同时将地面观测数据δｇｔ（重复重力）和δＨｔ（重复水准）等解析延拓至Ｂｊｅｒｈａｍｍａｒ球面上，得到求解地壳内部密度时间变化及相关重力位场时间变化的一组公式，并给出了相应的迭代求解方法。     

Update of the precision geoid determination in Korea

From the late 1990s, many studies on local geoid construction have been made in South Korea. However, the precision of the previous geoid has remained about 15 cm due to distribution and quality problems of gravity and GPS/levelling data. Since 2007, new land gravity data and GPS/levelling data have been obtained through many projects such as the Korean Land Spatilaization, Unified Control Point and Gravity survey on the Benchmark. The newly obtained data are regularly distributed to a certain degree and show much better improvement in their quality. In addition, an airborne gravity survey was conducted in 2008 to cover the Korean peninsula (South Korea only). Therefore, it is expected that the precision of the geoid could be improved. In this study, the new South Korean gravimetric geoid and hybrid geoid are presented based on land, airborne, ship‐borne, altimeter gravity data, geopotential model and topographic data. As for the methodology, the general remove‐restore approach was applied with the best chosen parameters in order to produce a precise local geoid. The global geopotential model EGM08 was used to remove the low‐frequency components using degree and order up to 360 and the short wavelength part of the gravity signal was dealt with by using the Shuttle Radar Topography Mission data. The parameters determined empirically in this study include for Stokes’ integral 0.5° and for Wong‐Gore kernel 110–120°, respectively and 10 km for both the Bjerhammar sphere depth and attenuation factor. The final gravimetric geoid in South Korea ranges from 20–31 m with a precision of 5.45 cm overall compared to 1096 GPS/levelling data. In addition, the South Korean hybrid geoid produces 3.46 cm and 3.92 cm for degrees of fitness and precision, respectively and a better statistic of 2.37 cm for plain and urban areas was achieved. The gravimetric and hybrid geoids are expected to improve further when the refined land gravity data are included in the near future.     

Regional gravimetric quasi-geoid model and transformation surface to national height system for Turkey (THG-09)

Turkish regional geoid models have been developed by employing a reference earth gravitational model, surface gravity observations and digital terrain models. The gravimetric geoid models provide a ready transformation from ellipsoidal heights to the orthometric heights through the use of GPS/leveling geoid heights determined through the national geodetic networks. The recent gravimetric models for Turkish territory were computed depending on OSU91 (TG-91) and EGM96 (TG-03) earth gravitational models. The release of the Earth Gravitational Model 2008 (EGM08), the collection of new surface gravity observations, the advanced satellite altimetry-derived gravity over the sea, and the availability of the high resolution digital terrain model have encouraged us to compute a new geoid model for Turkey. We used the Remove-Restore procedure based on EGM08 and applied Residual Terrain Model (RTM) reduction of the surface gravity data. Fast Fourier Transformation (FFT) was then used to obtain the residual quasigeoid from the reduced gravity. We restored the individual contributions of EGM08 and RTM to the whole quasi-geoid height (TQG-09). Since the Helmert orthometric height system is adopted in Turkey, the quasi-geoid model (TQG-09) was then converted to the geoid model (TG-09) by making use of Bouguer gravity anomalies and digital terrain model. After all we combined a gravimetric geoid model with GPS/leveling geoid heights in order to obtain a hybrid geoid model (THG-09) (or a transformation surface) to be used in GPS applications. The RMS of the post-fit residuals after the combination was found to be ± 0.95 cm, which represents the internal precision of the final combination. And finally, we tested the hybrid geoid model with GPS/leveling data, which were not used in the combination, to assess the external accuracy. Results show that the external accuracy of the THG-09 model is ± 8.4 cm, a precision previously not achieved in Turkey until this study.     

用多种数据构建2008年汶川特大地震同震位移场

本文主要以GPS、精密水准观测和卫星SAR遥感图像分析2008年汶川特大地震同震位移特征.GPS数据包括:(1)四川盆地和川西高原地区各类国家等级GPS网点复测;(2)沿破裂带国家天文大地网GPS复测.前者推算的同震位移测定精度优于2 cm,后者6~8 cm.SAR遥感资料包括:(1)ALOS 卫星升轨相位干涉图像,精度优于8 cm;(2)ALOS和ENVISAT卫星影像合成的三维位移图,精度优于0.5 m.同震位移场显示,断层下盘(四川盆地)变形总体呈扇形集中指向震中,断层上盘(龙门山)变形总体上呈逆时针旋转态势,最大的实测水平位移5.5 m.汶川、理县、茂县等地测站位移指向破裂带方向,而平武、青川等地测站逐渐转变为平行,乃至远离破裂带方向,与汶川地震逆冲兼走滑的破裂特征一致.断层上盘大幅隆升,下盘靠近断层的区域以下沉为主,远场表现为幅度很小的隆升,垂直升降区域间,有一条与龙泉山断裂带平行的升降过渡带,调节龙泉断层的应力状态.用实测变形场检验多个地震波破裂模型表明,近场(距离断层50 km) 模型形变准确度可达40~50 cm, 远场精度优于5 cm.     

Gravity field determination using boundary element methods

The Boundary Element Method (BEM), a numerical technique for solving boundary integral equations, is introduced to determine the earth's gravity field. After a short survey on its main principles, we apply this method to the fixed gravimetric boundary value problem (BVP), i.e. the determination of the earth's gravitational potential from measurements of the intensity of the gravity field in points on the earth's surface. We show how to linearize this nonlinear BVP using an implicit function theorem and how to transform the linearized BVP into a boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation using the single layer representation. A Galerkin method is used to transform the boundary integral equation into a linear system of equations. We discuss the major problems of this approach for setting up and solving the linear system. The BVP is numerically solved for a bounded part of the earth's surface using a high resolution reference gravity model, measured gravity values of high density, and a 50 50 m² digital terrain model to describe the earth's surface. We obtain a gravity field resolution of 1 1 km² with an accuracy of the order 10^–3 to 10^–4 in about 1 CPU-hour on a Siemens/Fujitsu SIMD vector pipeline machine using highly sophisticated numerical integration techniques and fast equation solvers. We conclude that BEM is a powerful numerical tool for solving boundary value problems and may be an alternative to classical geodetic techniques.     

Airborne gravimetry using a strapped-down LaCoste and Romberg air/sea gravity meter system: a feasibility study

Marine gravimeters mounted on stabilized platforms are commonly used in aircraft to perform airborne gravity measurements. The role of the stabilized platform is to level the sensor mechanically, whatever the aircraft attitude. However, this compensation is generally insufficient due to the sensitivity of modern gravity sensors. Correcting the offlevel error requires that an offlevel correction calculated from positioning data be added to gravimeter measurements, which complicates not only the processing, but also the assessment of precision and resolution. This paper is a feasibility study describing the levelling of a completely strapped‐down LaCoste and Romberg gravimeter for airborne gravimetry operation, by means of GPS positioning data. It focuses on the calculation of the sensor offlevel correction needed for the complete gravity data processing. The precision of the offlevel correction that can be achieved, in terms of GPS data precision and gravity wavelengths, is theoretically studied and estimated using the gravity and GPS data acquired during the Alpine Swiss French airborne gravity survey carried out in 1998 over the French Western Alps. While a 1 cm precision of GPS‐determined baseline coordinates is sufficient to achieve a 5 mGal precision of the offlevel correction, we maintain that this precision has to reach 1 mm to ensure a 1 mGal precision of the offlevel correction at any wavelength. Without a stabilized platform, the onboard instrumentation becomes significantly lighter. Furthermore, the correction for the offlevel error is straightforward and calculated only from GPS data. Thus, the precision and the resolution of airborne gravity surveys should be estimated with a better accuracy.     

我国1′×1′高程异常模型的建立及应用

高精度地确定我国陆海任意点的似大地水准面高（或称高程异常），其基础是先建立相应区域的高精度、高分辨率的高程异常数字模型，然后在此基础上通过内插软件确定.本文阐述了建立我国陆海1′×1′高程异常数字模型的数据、方法和步骤等.分别采用全国854个（一、二级网和A、B级网点）和75个（地震监测站点）高精度GPS/水准点作为外部检核点，对该数字模型进行了精度估计，结果表明：由该数字模型确定全国任意点高程异常的精度，东部地区分别为±0.18 m和±0.13 m，西部地区分别为±0.30 m和±0.22 m，全国总体精度分别为±0.23 m和±0.17 m.比规划的±0.5 m精度指标提高了1倍多.少部分区域达到了厘米级精度.由1′×1′高程异常数字模型及相应软件确定任意点结果的速度一般在2s内.     

A new ellipsoidal gravimetric, satellite altimetry and astronomic boundary value problem, a case study: The geoid of Iran

A new gravimetric, satellite altimetry, astronomical ellipsoidal boundary value problem for geoid computations has been developed and successfully tested. This boundary value problem has been constructed for gravity observables of the type (i) gravity potential, (ii) gravity intensity (i.e. modulus of gravity acceleration), (iii) astronomical longitude, (iv) astronomical latitude and (v) satellite altimetry observations. The ellipsoidal coordinates of the observation points have been considered as known quantities in the set-up of the problem in the light of availability of GPS coordinates. The developed boundary value problem is ellipsoidal by nature and as such takes advantage of high precision GPS observations in the set-up. The algorithmic steps of the solution of the boundary value problem are as follows:

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and of the ellipsoidal centrifugal field for the removal of the effect of global gravity and the isostasy field from the gravity intensity and the astronomical observations at the surface of the Earth.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the gravity intensity and the astronomical observations at the surface of the Earth the effect of the residual masses at the radius of up to 55 km from the computational point.

- Application of the ellipsoidal harmonic expansion complete up to degree and order 360 and ellipsoidal centrifugal field for the removal from the geoidal undulations derived from satellite altimetry the effect of the global gravity and isostasy on the geoidal undulations.

- Application of the ellipsoidal Newton integral on the multi-cylindrical equal-area map projection surface for the removal from the geoidal undulations derived from satellite altimetry the effect of the water masses outside the reference ellipsoid within a radius of 55 km around the computational point.

- Least squares solution of the observation equations of the incremental quantities derived from aforementioned steps in order to obtain the incremental gravity potential at the surface of the reference ellipsoid.

- The removed effects at the application points are restored on the surface of reference ellipsoid.

- Application of the ellipsoidal Bruns’ formula for converting the potential values on the surface of the reference ellipsoid into the geoidal heights with respect to the reference ellipsoid.

- Computation of the geoid of Iran has successfully tested this new methodology.

Use of EGM08 model and shuttle radar topography mission data for geoid computation in the State of Rio de Janeiro,Brazil: A case study with Voronoi/Delaunay discretisations

The EGM08 geopotential model complete to degree and order 2159 was used in a remove-compute-restore (RCR) method for the geoid computation in the State of Rio de Janeiro, Brazil. Terrain and indirect effect corrections were computed using a 6-arcsec resolution DTE, derived from the TOPODATA Project (Shuttle Radar Topography Mission data) raised by the National Institute for Space Research. INPE, Brazil. We applied Voronoi/Delaunay discretisations for discrete Stokes integration. In these schemes, target area is partitioned into polygons/triangles, respectively, and the computation is carried out by point-wise numerical integration and no gridding is mandatory. For both procedures, the cells were produced using either observed gravity data combined with gridded Bouguer derived information. Particularly in Delaunay scheme, as the gravity anomalies are interpolated into the triangular cells, and geoid undulations are computed for their vertices, Stokes function singularity was gone. Externally estimated errors resulting from a comparison with GPS/leveling data were presented for both the schemes and classical ones, as well as for the EGM08 undulations. They yielded RMS differences equal to 0.105 m, 0.110 m, 0.110 m, 0.115 m and 0.228 m, respectively, for Voronoi, Delaunay, Voronoi/Delaunay with gridded-data alone and EGM08, computed between 32 GPS/leveling points.     

珠峰地区似大地水准面精化与珠峰顶正高的确定

本文使用珠峰及其周边地区的重力数据与SRTM3、1∶50000 DEM、GTOPO30地形数据,以该地区的44个GPS水准点为控制,选择国内外的EGM96、WDM94、IGG05B、DQM2000D和CG03C作为参考重力场模型,采用移去-恢复技术,首次完成了珠峰地区分辨率为2.5′×2.5′高精度似大地水准面,其精度达到±9 cm,并据此推算了珠峰顶高程异常值. 再利用登山线路上的新测重力点与珠峰地形数据,依据严格的重力归算理论及移去-恢复技术,完成了珠峰顶似大地水准面与大地水准面差值计算,即珠峰正常高与正高的换算.