The Stokes and Vening-Meinesz functionals in a moving tangent space

The regularized solution of the external sphericalStokes boundary value problem as being used for computations of geoid undulations and deflections of the vertical is based upon theGreen functions S ₁(₀, ₀, , ) ofBox 0.1 (R = R ₀) andV ₁(₀, ₀, , ) ofBox 0.2 (R = R ₀) which depend on theevaluation point {₀, ₀} S _R0 ² and thesampling point {, } S _R0 ² ofgravity anomalies (, ) with respect to a normal gravitational field of typegm/R (free air anomaly). If the evaluation point is taken as the meta-north pole of theStokes reference sphere S _R0 ² , theStokes function, and theVening-Meinesz function, respectively, takes the formS() ofBox 0.1, andV ₂() ofBox 0.2, respectively, as soon as we introduce {meta-longitude (azimuth), meta-colatitude (spherical distance)}, namely {A, } ofBox 0.5. In order to deriveStokes functions andVening-Meinesz functions as well as their integrals, theStokes andVening-Meinesz functionals, in aconvolutive form we map the sampling point {, } onto the tangent plane T₀S _R0 ² at {₀, ₀} by means ofoblique map projections of type(i) equidistant (Riemann polar/normal coordinates),(ii) conformal and(iii) equiareal.Box 2.1.–2.4. andBox 3.1.– 3.4. are collections of the rigorously transformedconvolutive Stokes functions andStokes integrals andconvolutive Vening-Meinesz functions andVening-Meinesz integrals. The graphs of the correspondingStokes functions S ₂(),S ₃(r),,S ₆(r) as well as the correspondingStokes-Helmert functions H ₂(),H ₃(r),,H ₆(r) are given byFigure 4.1–4.5. In contrast, the graphs ofFigure 4.6–4.10 illustrate the correspondingVening-Meinesz functions V ₂(),V ₃(r),,V ₆(r) as well as the correspondingVening-Meinesz-Helmert functions Q ₂(),Q ₃(r),,Q ₆(r). The difference between theStokes functions / Vening-Meinesz functions andtheir first term (only used in the Flat Fourier Transforms of type FAST and FASZ), namelyS ₂() – (sin /2)^–1,S ₃(r) – (sinr/2R ₀)^–1,,S ₆(r) – 2R ₀/r andV ₂() + (cos /2)/2(sin² /2),V ₃(r) + (cosr/2R ₀)/2(sin² r/2R ₀),, illustrate the systematic errors in theflat Stokes function 2/ or flatVening-Meinesz function –2/². The newly derivedStokes functions S ₃(r),,S ₆(r) ofBox 2.1–2.3, ofStokes integrals ofBox 2.4, as well asVening-Meinesz functionsV ₃(r),,V ₆(r) ofBox 3.1–3.3, ofVening-Meinesz integrals ofBox 3.4 — all of convolutive type — pave the way for the rigorousFast Fourier Transform and the rigorousWavelet Transform of theStokes integral / theVening-Meinesz integral of type equidistant, conformal and equiareal.     

Global height datum unification: a new approach in gravity potential space

The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential. The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector (from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of the offset of the zero point of the Iranian height datum from the geoid’s potential value W ₀=62636855.8 m²/s². According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid.     

The ellipsoidal correction to the Stokes kernel for precise geoid determination

Solving the geodetic boundary-value problem (GBVP) for the precise determination of the geoid requires proper use of the fundamental equation of physical geodesy as the boundary condition given on the geoid. The Stokes formula and kernel are the result of spherical approximation of this fundamental equation, which is a violation of the proper relation between the observed quantity (gravity anomaly) and the sought function (geoid). The violation is interpreted here as the improper formulation of the boundary condition, which implies the spherical Stokes kernel to be in error compared with the proper kernel of integral transformation. To remedy this error, two correction kernels to the Stokes kernel were derived: the first in both closed and spectral forms and the second only in spectral form. Contributions from the first correction kernel to the geoid across the globe were [−0.867 m, +1.002 m] in the low-frequency domain implied by the GRIM4-S4 purely satellite-derived geopotential model. It is a few centimeters, on average, in the high-frequency domain with some exceptions of a few meters in places of high topographical relief and sizable geological features in accordance with the EGM96 combined geopotential model. The contributions from the second correction kernel to the geoid are [−0.259 m, +0.217 m] and [−0.024 m, +0.023 m] in the low- and high-frequency domains, respectively.     

四种改进积分法的低空扰动引力计算

针对Stokes积分方法计算扰动引力中计算点从空中趋近地面时存在积分奇异和不连续的问题,该文提出了去中央奇异点法、奇异点积分值修正法、中央格网加密算法和改进积分式法4种改进Stokes积分的计算公式,并进行了实验计算。计算结果表明:近地空间范围内,4种改进算法都能在一定程度上改进原始积分的奇异性问题;相同条件下,奇异点积分值修正法和改进积分式法计算精度最高,适宜于低空计算;改进积分式法通过理论推导,得到了从球外部到球面统一、连续且无奇异的改进Stokes积分公式,理论严谨。     

低阶修正的Hotine截断核函数的频谱分析与应用

传统截断核函数存在谱泄露问题,且实测数据在移去恢复频段的利用率低。本文以Hotine核函数为例引入了一种高低阶均修正的截断核函数,在其基础上进一步提出了仅低阶修正的截断核函数,具体包含余弦修正和线型修正两种类型。修正核函数能够有效地控制截断核函数存在的谱泄露问题,并且增大了实测数据在修正频段对高程异常的贡献率。试验结果表明,当低阶修正带宽一定时,低阶修正核函数计算的似大地水准面精度优于传统截断核函数计算的似大地水准面精度,并且与高低阶均修正的核函数的解算精度相当。但在计算效率上,低阶修正核函数明显优于高低阶均修正的核函数。本文的试验证实了在基于Helmert第二压缩法的边值问题(Stokes-Helmert或Hotine-Helmert边值问题)中低阶修正核函数是一种比较有效的核函数。     

地图空间认知的数学原理

根据非退化区间为等势集合的原理,论证了从制图区域到二维地图平面间的映射与复合映射是双一一函数;主体意图对制图物体性质的确定,使地图符号既满足视觉感受的要求又满足制图物体Ax与地图符号pgf(Ax)之间的双一一函数关系。根据集合的连续定理,证明了地图符号pgf(Ax)是制图物体Ax的同胚象,从而为地图的空间认知提供了数学基础和理论支持。     

战场数字化建设与军事测绘新技术集成

本文阐述了战场数字化的基本概念及其对军事测绘的要求，分析了地理信息系统（ＧＩＳ）、遥感（ＲＳ）和全球定位系统（ＧＰＳ）等军事测绘新技术及其集成的功能，以及它们在战场数字化建设中的作用。     

Research on ground deformation monitoring method in mining areas using the probability integral model fusion D-InSAR,sub-band InSAR and offset-tracking

With the aim of addressing the problem of accurately monitoring complete deformation fields over mining areas by means of Synthetic Aperture Radar (SAR), this paper proposes a solution to obtain complete deformation fields using the probability integral model to fuse deformation data derived from Differential Interferometric SAR (D-InSAR), sub-band InSAR and offset-tracking. This method is used for small-scale, medium-scale and large-scale deformation monitoring using D-InSAR, sub-band InSAR and offset-tracking, respectively. Finally, the probability integral model is utilized to integrate the three deformation fields, and a complete deformation field with high-accuracy over the study area can be obtained. The method is tested on 13 TerraSAR-X (TSX) images from December 2, 2012 to April 24, 2013 of the working face 52,304 of the Daliuta mining area in Shaanxi province, China. The complete deformation field of the working face during the 113-day mining period is obtained. The results show that during the process of working face advancing, the subsidence basin has been expanding along the direction of excavation. The relationship between the average maximum subsidence rate and the advancing distance of the working face can be described by a quadratic polynomial. It has been also observed that, when the underground mining reaches the full mining condition, the maximum subsidence value does not increase further. The accuracy of the proposed method is verified against the global positioning system field survey data. The root mean square errors in the strike and dip directions are 0.134 m and 0.105 m, respectively. Due to the support provide by the reserved coal pillars, the subsidence value above the reserved coal pillars is smaller.     

The nullfield method for the ellipsoidal Stokes problem

The ellipsoidal Stokes problem is one of the basic boundary-value problems for the Laplace equation which arises in physical geodesy. Up to now, geodecists have treated this and related problems with high-order series expansions of spherical and spheroidal (ellipsoidal) harmonics. In view of increasing computational power and modern numerical techniques, boundary element methods have become more and more popular in the last decade. This article demonstrates and investigates the nullfield method for a class of Robin boundary-value problems. The ellipsoidal Stokes problem belongs to this class. An integral equation formulation is achieved, and existence and uniqueness conditions are attained in view of the Fredholm alternative. Explicit expressions for the eigenvalues and eigenfunctions for the boundary integral operator are provided. Received: 22 October 1996 / Accepted: 4 August 1997     

粒子群算法在概率积分法沉陷预计模型参数反演中的应用

针对开采沉陷预计模型参数反演所存在的算法复杂、计算量大等缺陷,将粒子群算法引入到概率积分法开采沉陷预计模型参数反演中。研究粒子群算法反演概率积分法预计模型参数的基本原理、编码方法及适应度函数的构造方法,同时结合河北省某煤矿的实测数据,以下沉拟合值与实测值的中误差作为反演精度的评价标准对算法进行实例验证,对提高开采沉陷预计的精度有一定的参考实用价值。     

北京市1:1万地形图建库及修测更新一体化工艺研究

为了加快数字北京建设的步伐,更好的满足社会、政府部门的应用需求,北京市测绘设计研究院建立了覆盖全北京市域的1:1万DLG空间数据库。数据库采用Oracle9i进行存储,通过ArcSDE8.3空间数据引擎对数据库进行管理访问。本文主要对前期数据的加工方法,后期的数据库维护以及增量更新方案的确定与应用进行了研究、讨论。     

无障碍设施空间布局核密度热点探测和空间自相关分析——以北京市核心区室外公共空间为例

城市无障碍设施在区域空间中往往呈现聚集分布的特征,通常利用核密度估计方法分析总体空间分布形态,研究区域空间分布的数量差异,探测分布热点;同时通过分析无障碍设施空间自相关性特征,反映无障碍设施服务的聚集特点。将空间分析方法引入到无障碍环境评估当中,可以优化无障碍环境发展空间。结果表明,北京市核心区无障碍设施总体呈现出"多核分布"的态势。无障碍设施的总体分布存在空间差异性,局部无障碍设施空间分布存在聚集特性。     

The Map of the Future May Not Be a Map!

Abstract

Although studies concerned with the communication of information and of power-relations have been central to the development of cartographic theory, as hermeneutics they do not seem to have fully explored the relationship between maps and aesthetics. In this short paper, I will briefly discuss the reasons why aesthetics belongs in the development of cartographic theory and suggest possible avenues for further research.     

Performance of three types of Stokes's kernel in the combined solution for the geoid

When regional gravity data are used to compute a gravimetric geoid in conjunction with a geopotential model, it is sometimes implied that the terrestrial gravity data correct any erroneous wavelengths present in the geopotential model. This assertion is investigated. The propagation of errors from the low-frequency terrestrial gravity field into the geoid is derived for the spherical Stokes integral, the spheroidal Stokes integral and the Molodensky-modified spheroidal Stokes integral. It is shown that error-free terrestrial gravity data, if used in a spherical cap of limited extent, cannot completely correct the geopotential model. Using a standard norm, it is shown that the spheroidal and Molodensky-modified integration kernels offer a preferable approach. This is because they can filter out a large amount of the low-frequency errors expected to exist in terrestrial gravity anomalies and thus rely more on the low-frequency geopotential model, which currently offers the best source of this information. Received: 11 August 1997 / Accepted: 18 August 1998     

双移动载体GPS精密相对定位研究

GPS双移动载波相位差分技术的关键,是如何实现可变基线条件下整周模糊度的快速解算。为此,本文详细地讨论了其确定整周模糊度的方法,并设计了检验该技术精度的实验方案;实算结果表明,用该技术获得了厘米级的相对定位精度。     

Estimating regional deformation from a combination of space and terrestrial geodetic data

We discuss an approach for efficiently combining different types of geodetic data to estimate time-dependent motions of stations in a region of active deformation. The primary observations are analyzed separately to produce loosely constrained estimates of station positions and coordinate system parameters which are then combined with appropriate constraints to estimate velocities and coseismic displacements. We define noninteger degrees of freedom to handle the case of finite constraints and stochastic perturbation of parameters and develop statistical tests for determining compatibility between different data sets. With these developments, we show an example of combining space and terrestrial geodetic data to obtain the deformation field in southern California. Received: 23 January 1997 / Accepted: 30 September 1997     

菲涅耳积分计算公式及其应用

菲涅耳积分是物理光学、微波技术与天线等多学科共用的特殊函数,其形式为上限可变而又不能积出的两个定积分,通常将其按变量u的不同区间分别用正或负幂级数计算。本文在原有公式的基础上研究出了简单、精密且变量复盖实数域的计算公式及其计算机程序。内容包括:(1)最佳分区点u₁的选择;(2) u≤u_t时菲涅耳积分的变比级数表达式;(3) u>u₁时菲涅耳积分的三角函数表达式;(4)程序流程图;(5)抽样运行数据及检验结果。