A method to validate gravimetric-geoid computation software based on Stokes's integral formula

A method is presented with which to verify that the computer software used to compute a gravimetric geoid is capable of producing the correct results, assuming accurate input data. The Stokes, gravimetric terrain correction and indirect effect formulae are integrated analytically after applying a transformation to surface spherical coordinates centred on each computation point. These analytical results can be compared with those from geoid computation software using constant gravity data in order to verify its integrity. Results of tests conducted with geoid computation software are presented which illustrate the need for integration weighting factors, especially for those compartments close to the computation point. Received: 6 February 1996 / Accepted: 19 April 1997     

Gravity field convolutions without windowing and edge effects

A new set of formulas has been developed for the computation of geoid undulations and terrain corrections by FFT when the input gravity anomalies and heights are mean gridded values. The effects of the analytical and the discrete spectra of kernel functions and that of zero-padding on the computation of geoid undulations and terrain corrections are studied in detail.Numerical examples show that the discrete spectrum is superior to the analytically-defined one. By using the discrete spectrum and 100% zero-padding, the RMS differences are 0.000 m for the FFT geoid undulations and 0.200 to 0.000 mGal for the FFT terrain corrections compared with results obtained by numerical integration.     

Ellipsoidal terrain correction based on multi-cylindrical equal-area map projection of the reference ellipsoid

An operational algorithm for computation of terrain correction (or local gravity field modeling) based on application of closed-form solution of the Newton integral in terms of Cartesian coordinates in multi-cylindrical equal-area map projection of the reference ellipsoid is presented. Multi-cylindrical equal-area map projection of the reference ellipsoid has been derived and is described in detail for the first time. Ellipsoidal mass elements with various sizes on the surface of the reference ellipsoid are selected and the gravitational potential and vector of gravitational intensity (i.e. gravitational acceleration) of the mass elements are computed via numerical solution of the Newton integral in terms of geodetic coordinates {,,h}. Four base- edge points of the ellipsoidal mass elements are transformed into a multi-cylindrical equal-area map projection surface to build Cartesian mass elements by associating the height of the corresponding ellipsoidal mass elements to the transformed area elements. Using the closed-form solution of the Newton integral in terms of Cartesian coordinates, the gravitational potential and vector of gravitational intensity of the transformed Cartesian mass elements are computed and compared with those of the numerical solution of the Newton integral for the ellipsoidal mass elements in terms of geodetic coordinates. Numerical tests indicate that the difference between the two computations, i.e. numerical solution of the Newton integral for ellipsoidal mass elements in terms of geodetic coordinates and closed-form solution of the Newton integral in terms of Cartesian coordinates, in a multi-cylindrical equal-area map projection, is less than 1.6×10^–8 m²/s² for a mass element with a cross section area of 10×10 m and a height of 10,000 m. For a mass element with a cross section area of 1×1 km and a height of 10,000 m the difference is less than 1.5×10^–4m²/s². Since 1.5× 10^–4 m²/s² is equivalent to 1.5×10^–5m in the vertical direction, it can be concluded that a method for terrain correction (or local gravity field modeling) based on closed-form solution of the Newton integral in terms of Cartesian coordinates of a multi-cylindrical equal-area map projection of the reference ellipsoid has been developed which has the accuracy of terrain correction (or local gravity field modeling) based on the Newton integral in terms of ellipsoidal coordinates.Acknowledgments. This research has been financially supported by the University of Tehran based on grant number 621/4/859. This support is gratefully acknowledged. The authors are also grateful for the comments and corrections made to the initial version of the paper by Dr. S. Petrovic from GFZ Potsdam and the other two anonymous reviewers. Their comments helped to improve the structure of the paper significantly.     

基于主成分变换模型的DEM格网聚合及其误差分析

利用主成分分析揭示变量之间关系的特性,进而提出一种既能保证较高精度又能较好地保持地形形态特征的DEM格网聚合方法。首先根据主成分变换模型推导DEM格网聚合数学公式,构建主成分聚合模型;然后以30m分辨率DEM转换为90m分辨率DEM为例,根据格网点属性间的权重关系聚合重构DEM。在此基础上,以均值聚合和双线性重采样聚合方法为比较对象,从聚合前后的检查点高程偏差的统计描述、空间分布与自相关性、地形形态保持程度方面分析3种聚合策略下重构DEM的误差特性。最后运用描述统计、半变异分析和等高线套合方法,定量评价主成分聚合重构DEM的质量效果。试验分析结果表明,同均值聚合和重采样聚合相比较,该方法重构的DEM既能保持较高精度,又能很好地保持地形形态特征。     

Orthometric corrections from leveling, gravity, density and elevation data: a case study in Taiwan

A new orthometric correction (OC) formula is presented and tested with various mean gravity reduction methods using leveling, gravity, elevation, and density data. For mean gravity computations, the Helmert method, a modified Helmert method with variable density and gravity anomaly gradient, and a modified Mader method were used. An improved method of terrain correction computation based on Gaussian quadrature is used in the modified Mader method. These methods produce different results and yield OCs that are greater than 10 cm between adjacent benchmarks (separated by 2 km) at elevations over 3000 m. Applying OC reduces misclosures at closed leveling circuits and improves the results of leveling network adjustments. Variable density yields variation of OC at millimeter level everywhere, while gravity anomaly gradient introduces variation of OC of greater than 10 cm at higher elevations, suggesting that these quantities must be considered in OC. The modified Mader method is recommended for computing OC.Acknowledgments.This study is supported by the Ministry of the Interior (MOI), Taiwan, under the project `Measuring gravity on first-order benchmarks'. The authors are grateful to F.S. Ning and his colleagues at BSB (Base Survey Battalion) for their precision work in collecting gravity data, and to R. Forsberg for the terrain correction program. They also thank the Institute of Agricultural and Forestry Aerial Survey for elevation data and MOI for leveling data. Dr. Will Featherstone and three anonymous reviewers are thanked for their constructive comments.     

Ellipsoidal geoid computation

Modern geoid computation uses a global gravity model, such as EGM96, as a third component in a remove–restore process. The classical approach uses only two: the reference ellipsoid and a geometrical model representing the topography. The rationale for all three components is reviewed, drawing attention to the much smaller precision now needed when transforming residual gravity anomalies. It is shown that all ellipsoidal effects needed for geoid computation with millimetric accuracy are automatically included provided that the free air anomaly and geoid are calculated correctly from the global model. Both must be consistent with an ellipsoidal Earth and with the treatment of observed gravity data. Further ellipsoidal corrections are then negligible. Precise formulae are developed for the geoid height and the free air anomaly using a global gravity model, given as spherical harmonic coefficients. Although only linear in the anomalous potential, these formulae are otherwise exact for an ellipsoidal reference Earth—they involve closed analytical functions of the eccentricity (and the Earths spin rate), rather than a truncated power series in e². They are evaluated using EGM96 and give ellipsoidal corrections to the conventional free air anomaly ranging from –0.84 to +1.14 mGal, both extremes occurring in Tibet. The geoid error corresponding to these differences is dominated by longer wavelengths, so extrema occur elsewhere, rising to +766 mm south of India and falling to –594 mm over New Guinea. At short wavelengths, the difference between ellipsoidal corrections based only on EGM96 and those derived from detailed local gravity data for the North Sea geoid GEONZ97 has a standard deviation of only 3.3 mm. However, the long-wavelength components missed by the local computation reach 300 mm and have a significant slope. In Australia, for example, such a slope would amount to a 600-mm rise from Perth to Cairns.     

Evaluation of approaches for AWiFS multi-date registration

AWiFS onboard IRS-P6 belongs to the category of high-repetivity sensors based on large swath, but with ground trace based on narrow-swath sensor (LISS-III). This is useful for cloud removal as well as vegetation phenology studies. Such multi-date analysis has a prerequisite of accurate multi-date registration. This study investigates the accuracy of multi-date registration over a mixed plain and hilly terrain in northern India (29–31°N latitude and 77.5–79.5°E longitude; 200–4000 m.a.s.l.). Simple polynomial rectification, multi-date registration using ortho-correction technique on standard product (level-2) and radiometric product (level-1) as a function of number of ground control points (GCPs) and external Digital Elevation Model (DEM) were investigated. The results indicated that ortho-rectification on level-1 product provided better accuracy in comparison to simple rectification and ortho-rectification on level-2 product.     

The effect of topography on the determination of the geoid using analytical downward continuation

The method of analytical downward continuation has been used for solving Molodensky’s problem. This method can also be used to reduce the surface free air anomaly to the ellipsoid for the determination of the coefficients of the spherical harmonic expansion of the geopotential. In the reduction of airborne or satellite gradiometry data, if the sea level is chosen as reference surface, we will encounter the problem of the analytical downward continuation of the disturbing potential into the earth, too. The goal of this paper is to find out the topographic effect of solving Stoke’sboundary value problem (determination of the geoid) by using the method of analytical downward continuation. It is shown that the disturbing potential obtained by using the analytical downward continuation is different from the true disturbing potential on the sea level mostly by a −2πGρh 2/p. This correction is important and it is very easy to compute and add to the final results. A terrain effect (effect of the topography from the Bouguer plate) is found to be much smaller than the correction of the Bouguer plate and can be neglected in most cases. It is also shown that the geoid determined by using the Helmert’s second condensation (including the indirect effect) and using the analytical downward continuation procedure (including the topographic effect) are identical. They are different procedures and may be used in different environments, e.g., the analytical downward continuation procedure is also more convenient for processing the aerial gravity gradient data. A numerical test was completed in a rough mountain area, 35°<ϕ<38°, 240°<λ<243°. A digital height model in 30″×30″ point value was used. The test indicated that the terrain effect in the test area has theRMS value ±0.2−0.3 cm for geoid. The topographic effect on the deflections of the vertical is around1 arc second.     

Surveying co-located space-geodetic instruments for ITRF computation

A new and comprehensive method is presented that can be used for estimating eccentricity vectors between global positioning system (GPS) antennas, doppler orbitography and radiopositioning integrated by satellites (DORIS) antennas, azimuth-elevation (AZ-EL) very long baseline interferometry (VLBI) telescopes, and satellite laser ranging (SLR) and lunar laser ranging (LLR) telescopes. The problem of reference point (RP) definition for these space-geodetic instruments is addressed and computed using terrestrial triangulation and electronic distance measurement (EDM) trilateration. The practical ground operations, the surveying approach and the terrestrial data processing are briefly illustrated, and the post-processing procedure is discussed. It is a geometrically based analytical approach that allows computation of RPs along with a rigorous statistical treatment of measurements. The tight connection between the geometrical model and the surveying procedure is emphasized. The computation of the eccentricity vector and the associated variance–covariance matrix between an AZ-EL VLBI telescope (with or without intersecting axes) and a GPS choke ring antenna is concentrated upon, since these are fundamental for computing the International Terrestrial Reference Frame (ITRF). An extension to RP computation and eccentricity vectors involving DORIS, SLR and LLR techniques is also presented. Numerical examples of the quality that can be reached using the authors approach are given. Working data sets were acquired in the years 2001 and 2002 at the radioastronomical observatory of Medicina (Italy), and have been used to estimate two VLBI-GPS eccentricity vectors and the corresponding SINEX files.     

Effects of detailed soil spatial information on watershed modeling across different model scales

Hydro-ecological modelers often use spatial variation of soil information derived from conventional soil surveys in simulation of hydro-ecological processes over watersheds at mesoscale (10–100 km²). Conventional soil surveys are not designed to provide the same level of spatial detail as terrain and vegetation inputs derived from digital terrain analysis and remote sensing techniques. Soil property layers derived from conventional soil surveys are often incompatible with detailed terrain and remotely sensed data due to their difference in scales. The objective of this research is to examine the effect of scale incompatibility between soil information and the detailed digital terrain data and remotely sensed information by comparing simulations of watershed processes based on the conventional soil map and those simulations based on detailed soil information across different simulation scales. The detailed soil spatial information was derived using a GIS (geographical information system), expert knowledge, and fuzzy logic based predictive mapping approach (Soil Land Inference Model, SoLIM). The Regional Hydro-Ecological Simulation System (RHESSys) is used to simulate two watershed processes: net photosynthesis and stream flow. The difference between simulation based on the conventional soil map and that based on the detailed predictive soil map at a given simulation scale is perceived to be the effect of scale incompatibility between conventional soil data and the rest of the (more detailed) data layers at that scale. Two modeling approaches were taken in this study: the lumped parameter approach and the distributed parameter approach. The results over two small watersheds indicate that the effect does not necessarily always increase or decrease as the simulation scale becomes finer or coarser. For a given watershed there seems to be a fixed scale at which the effect is consistently low for the simulated processes with both the lumped parameter approach and the distributed parameter approach.     

Economic convergence or divergence? Modeling the interregional dynamics of EU regions, 1985–1999

In this paper we start from a continuous time framework derived from the classical predator-prey model in order to analyze the recent dynamics of regional evolution in the EU. The model describes a system of interrelated units obeying a complex functional dynamics that at any moment may encompass divergent forces. After briefly reviewing the modeling framework presented elsewhere (Arbia and Paelinck 2002), we consider the dynamics of per-capita income in 119 NUTS-2 European regions in the years 1985–1999, and fit the model using Simultaneous Dynamic Least Squares (Paelinck 1996b). The model is shown to fit the empirical data well. The – potential divergence – leads up to a formidable coordination task for public authorities responsible for regional policies, as they are supposed – if their aim is absolute convergence in an economic sense, meaning equalizing incomes per head – to produce at any time a consistent set of regional policies for all the regions concerned, and not only for the less developed ones.     

基于双线性内插规则格网DEM地形误差模型

双线性DEM表面模型的精度包括:使用双线性建模方法从格网量测数据传递过来的误差和地形表面的线性表达导致的精度损失。从理论上推导并证明了地形线性表达的误差极值是格网分辨率与地形坡度乘积的常数倍。并根据地形说明了,在不同地形时常数c的不同取值。     

基于地形特征的采煤沉陷盆地构建与水平移动信息智能提取方法

在西部黄土高原复杂地形地貌下构建采煤沉陷盆地和提取水平位移的难度较大,传统地表沉降监测手段只能获取线状数据,效率低,而重复轨道合成孔径雷达干涉测量技术在大梯度形变区域易出现失相干现象,难以达到矿区地表沉降监测精度要求。提出了一种基于无人机载激光雷达点云数据构建沉陷盆地和提取水平位移的方法。结合多地形因子构建深度神经网络（deep neural network,DNN）模型,提取沉陷盆地构建过程中受地形影响较小的特征稳定区,利用较优插值算法对稳定区进行拟合,得到完整沉陷盆地。为了提取采煤地表水平移动信息,将二进制形状上下文特征描述算子与多地形因子融合起来,以改进特征匹配算法。基于此设计地表水平移动提取方案,提取主断面水平移动信息,同时对水平移动提取误差与点云密度、地形因子进行定量分析。榆神矿区结果表明,利用结合地形因子的DNN模型能有效提取特征稳定区,在复杂地貌下减小了沉陷建模误差,为构建采煤沉陷盆地提供了一种新方法;利用融合地形特征的改进特征匹配算法提取的水平移动曲线符合采煤沉陷水平移动基本规律,与水平移动偏差相关性较强的地形因子可用于衡量改进特征匹配算法对水平移动提取误差的大小。     

三维电子沙盘构建方法与实现

介绍了应用分层建模法构建三维电子沙盘的过程,包括数据处理、地形三维建模、规则建筑物提取、复杂实体三维建模、地形与地物的融合等技术内容。分析了该方法的优缺点,并给出中南大学三维电子沙盘构建的应用实例,提出了下一步需要研究解决的技术问题。     

关于地形复杂地区地形改正的FTT算法

大面积地形改正的快速计算,对现代大地测量实践具有重大意义,尤其是在那些地形复杂地区。本文通过理论分析选出了一种适于此类地区地形改正计算的FFT算法,并给出了用于实际计算的数学模型。试算结果表明,该模型用于特大山区能够获得与双三次样条棱柱算法精度相当的结果。     

全球经纬度网格的雷达探测范围真三维建模

针对雷达探测范围会受到复杂地形因素影响,单一地依靠电磁传播模型可视化雷达探测区域很难真实有效地展现电磁覆盖情况的问题,该文提出一种基于GeoSOT-3D空间剖分的雷达探测范围的计算与表达一体化的真三维模型。GeoSOT-3D剖分框架将上至地球50 000km,下至地心的全部地球空间划分成多层次、多尺度空间体元,体元可承载多源异构空间数据。依据GeoSOT-3D的剖分流程,将地形数据、雷达探测空间数据离散化。最后,开发了雷达探测范围可视化验证平台。实验结果表明,该建模方法对复杂的地形适应性强,可视化表现效果真实直观。     

Role of satellite remote sensing in the geographic information economics in France

In national and international economics, geographic information plays a role which is generally acknowledged to be important but which is however, difficult to assess quantitatively, its applications being rather miscellaneous and indirect. Computer graphics and telecommunications increae that importance still more and justify many investments and research into new cartographic forms.As part of its responsibility for participating in the promotion of those developments, by taking into account needs expressed by public or private users, the National Council for Geographic Information (C.N.I.G.) has undertaken a general evaluation of the economic and social utility of geographic information in France. The study involves an estimation of the cost of production and research activities, which are probably about 0.1% of the Cross National Product—similar to many other countries. It also devised a method of estimating “cost/advantage” ratios applicable to these “intangible” benefits.Within that framework, remote sensing emphasizes particular aspects related both to the increase of economic performances in cartographic production and to the advent of new products and new ways of utilization. A review of some significant sectors shows effective earnings of about 10–20%, or even 50% or 100% of the costs, and these are doubtless much greater for the efficacy in the exploitation of products. Finally, many applications, entirely new result from extensions in various fields which would have been impossible without remote sensing: here the “cost advantage” ratio cannot even be compared with previous processes.Studies were undertaken in parallel for defining different types of products derived from satellite imagery, as well as those domains where development effort is required in order to make new advances.     

TERRA: Terrain Extraction from elevation Rasters through Repetitive Anisotropic filtering

Over the past decades, several filters have been developed to derive a Digital Terrain Model (DTM) from a Digital Surface Model (DSM), by means of filtering out aboveground objects such as vegetation. In this filtering process, however, one of the major challenges remains to precisely distinguish sharp terrain features, e.g. ridges, agricultural terraces or other anthropogenic geomorphology such as open-pit mines, riverbanks or road ramps. Hence, loss of elevation data around terrain edges (and consequent smoothing) is very common with existing algorithms. In terraced landscapes, the preservation of precise geomorphology is of key importance in digital terrain analyses, such as hydrologic and erosion modelling, or automatic feature recognition and inventorying. In this work, we propose a new filtering method called TERRA (Terrain Extraction from elevation Rasters through Repetitive Anisotropic filtering). The novelty of the algorithm lies within its usage of terrain aspect to guide the anisotropic filtering direction, therefore maximising the preservation of terrain edges. We derived six DTMs from DSMs using UAV Structure from Motion (SfM) photogrammetry, laser altimetry and satellite sources (grid resolutions ranging from 0.1–1.0 m). The results indicated a close agreement of DTMs filtered using the TERRA algorithm and reference DTMs, while terrace risers were well preserved even under thick canopies of vines and trees. Compared to existing filtering approaches, TERRA performed well in minimising Type I errors (false ground removal), while Type II errors occurred locally where vegetation was covering the terrace edges. Given the promising filtering performance, and supported by the minimal requirements of parameterisation and computation, the TERRA algorithm could be a useful tool in DTM preparation for digital terrain analysis of agricultural terraces and similar hillslopes characterised by a complex mosaic of sharp terrain and non-terrain features.     

Thin-plate spline quadrature of geodetic integrals

Thin-plate splines — well known for their flexibility and fidelity in representing experimental data — are especially suited for the numerical evaluation of geodetic integrals in the area where these are most sensitive to the data, i.e. in the immediate vicinity of the computation point. Quadrature rules that are exact for thin-plate splines interpolating randomly spaced data are derived for the inner zone contribution (to a planar approximation) to Stokes's formula, to the formulae of Vening Meinesz and to theL ₁ gradient operator in the analytical continuation solution of Molodensky's problem.The quadrature method is demonstrated by calculating the inner zone contribution to height anomalies in a mountainous area of Lesotho and carrying out a comparison with GPS-derived heights. Height anomalies are recovered with an accuracy of 6 cm.     

顾及非线性地形因子的地表面积计算

研究地表面积统计数学模型及其影响因素,消除不同分辨率DEM计算所得地表面积的差异,对综合利用多尺度DEM数据精确统计和监测地表面积具有重要意义。本文研究提出了一种顾及复杂地形因子的地表面积统计方法。该方法首先利用泰勒级数逼近原理对微观地形因子进行最小二乘估计,然后利用这些地形因子对DEM和多边形区域边界进行加密,最后利用加密后的DEM和多边形边界构建地表三角网统计地表面积。试验表明,在局部地形因子显著的山区或丘陵地区,使用不同分辨率DEM所统计的地表面积存在较大的差异,而顾及地形因子的地表面积统计方法可明显提高低分辨率DEM地表面积统计精度。