A bilinear approximation of the surface relief in terrain correction computations

A new method is presented for the computation of the gravitational attraction of topographic masses when their height information is given on a regular grid. It is shown that the representation of the terrain relief by means of a bilinear surface not only offers a serious alternative to the polyhedra modeling, but also approaches even more smoothly the continuous reality. Inserting a bilinear approximation into the known scheme of deriving closed analytical expressions for the potential and its first-order derivatives for an arbitrarily shaped polyhedron leads to a one-dimensional integration with – apparently – no analytical solution. However, due to the high degree of smoothness of the integrand function, the numerical computation of this integral is very efficient. Numerical tests using synthetic data and a densely sampled digital terrain model in the Bavarian Alps prove that the new method is comparable to or even faster than a terrain modeling using polyhedra.     

Unfolding the Earth: Myriahedral Projections

Abstract

Myriahedral projections are a new class of methods for mapping the earth. The globe is projected on a myriahedron, a polyhedron with a very large number of faces. Next, this polyhedron is cut open and unfolded. The resulting maps have a large number of interrupts, but are (almost) conformal and conserve areas. A general approach is presented to decide where to cut the globe, followed by three different types of solution. These follow from the use of meshes based on the standard graticule, the use of recursively subdivided polyhedra and meshes derived from the geography of the earth. A number of examples are presented, including maps for tutorial purposes, optimal foldouts of Platonic solids, and a map of the coastline of the earth.     

Grid point search algorithm for fast integer ambiguity resolution

A Grid Point Search Algorithm (GRIPSA) for fast integer ambiguity resolution is presented. In the proposed algorithm, after the orthogonal transformation of the original ambiguity parameters, the confidence ellipsoid of the new parameters is represented by a rectangular polyhedron with its edges parallel to the corresponding axes. A cubic grid covering the whole polyhedron is then identified and transformed back to the original coordinate system. The integer values of the corresponding transformed grid points are obtained by rounding off the transformed values to their nearest integer values. These values are then tested as to whether they are located inside the polyhedron. Since the identification of the grid points in the transformed coordinate system greatly reduces the search region of the integer ambiguities, marked improvements are obtained in the computational effort. Received: 13 October 1997 / Accepted: 9 June 1998     

The computation of aliasing effects in local gravity field approximation

The aliasing effects in local gravity field computations are presented in this study. First the relation between the power spectral density of a 2-D continuous signal and its corresponding sampled version is derived. Then the power spectral density of the aliasing errors related to non band-limited signals is derived. Finally the variance of these aliasing errors is computed using gravity anomalies at different grid spacings. This computation prerequires some known gravity anomaly power spectral density model. The model used in this study corresponds to a second-order Gauss-Markov covariance function for the anomalous potential. Editor’s notice: Comments on this paper will follow in the next issue of Bulletin Géodésique.     

Second-order derivatives in a local frame developed in spheroidal and spherical coordinates

Second-order derivatives of a general scalar function of position (F) with respect to the length elements along a family of local Cartesian axes are developed in the spheroidal and spherical coordinate systems. A link between the two kinds of formulations is established when the results in spherical coordinates are confirmed also indirectly, through a transformation from spheroidal coordinates. IfF becomesW (earth's potential) the six distinct second-order derivatives—which include one vertical and two horizontal gradients of gravity—relate the symmetric Marussi tensor to the curvature parameters of the field. The general formulas for the second-order derivatives ofF are specialized to yield the second-order derivatives ofU (standard potential) and ofT (disturbing potential), which allows the latter to be modeled by a suitable set of parameters. The second-order derivatives ofT in which the property ΔT=0 is explicitly incorporated are also given. According to the required precision, the spherical approximation may or may not be desirable; both kinds of results are presented. The derived formulas can be used for modeling of the second-order derivatives ofW orT at the ground level as well as at higher altitudes. They can be further applied in a rotating or a nonrotating field. The development in this paper is based on the tensor approach to theoretical geodesy, introduced by Marussi [1951] and further elaborated by Hotine [1969], which can lead to significantly shorter demonstrations when compared to conventional approaches.     

基于偏微分方程的小比例尺晕渲图地貌综合方法研究

基于高精度DEM数据的晕渲图中,破碎的地形细节破坏了晕渲法的立体塑造能力,但是采用传统的均值滤波法进行地貌综合时,在消除地形细节的同时,模糊了地貌特征。为了克服消除地形细节和保持地貌特征这一矛盾,利用偏微分方程各向异性的特点,采用基于经典总变分模型的偏微分方程法对地貌进行综合。实验结果证明偏微分方程法在综合地貌时能较好地保持地貌结构特征。     

Analysing topographical changes in open cast coal-mining region of Patratu,Jharkhand using CARTOSAT-I Stereopair satellite images

This study provides an assessment of changes in the terrain topography due to opencast coal mining in the Patratu region of Jharkhand state during the period of 1962–2007. It demonstrated the potential of digital elevation model (DEM) differencing technique using Cartosat-I satellite (2007) derived DEM with reference to DEM derived from contours obtained from Survey of India topographical map (1962). The topographical changes through DEM differencing revealed positive relief changes (up to 49 m) due to coal-mining overburden dumps. The dumping of overburden near the banks of perennial Damodar River also caused positive topographic changes (up to 20 m) indicating adverse effects on its hydrological regime. Negative relief changes are represents by deep depressions (up to 66 m) created within coal mines due to the extraction of coal. These depression areas within the abundant mines generally become the zones of water accumulation causing wastage of surface and ground water resources.     

Topographic effects by the Stokes–Helmert method of geoid and quasi-geoid determinations

The topographic potential and the direct topographic effect on the geoid are presented as surface integrals, and the direct gravity effect is derived as a rigorous surface integral on the unit sphere. By Taylor-expanding the integrals at sea level with respect to topographic elevation (H) the power series of the effects is derived to arbitrary orders. This study is primarily limited to terms of order H ². The limitations of the various effects in the frequently used planar approximations are demonstrated. In contrast, it is shown that the spherical approximation to power H ² leads to a combined topographic effect on the geoid (direct plus indirect effect) proportional to H˜² (where terms of degrees 0 and 1 are missing) of the order of several metres, while the combined topographic effect on the height anomaly vanishes, implying that current frequent efforts to determine the direct effect to this order are not needed. The last result is in total agreement with Bjerhammar's method in physical geodesy. It is shown that the most frequently applied remove–restore technique of topographic masses in the application of Stokes' formula suffers from significant errors both in the terrain correction C (representing the sum of the direct topographic effect on gravity anomaly and the effect of continuing the anomaly to sea level) and in the term t (mainly representing the indirect effect on the geoidal or quasi-geoidal height). Received: 18 August 1998 / Accepted: 4 October 1999     

CURRENT PROBLEMS IN OCEAN MAPPING

The author, in a review of papers presented at the First All-Union Conference on the Geography and Mapping of the Oceans, outlines the present status and current trends of marine cartography in the USSR. Issues warranting increased attention include development of a systems approach for the mapping of marine ecosystems; improvement of methods and principles for the mapping of coastal zones, bottom relief, and shelf topography; and accelerated production of economic/resource maps of the oceans. Work now underway on a comprehensive, multi-volume A tlas of the Oceans reflects a Soviet concern with these issues. Translated from: Geodeziya i kartografiya, 1984, No. 12, pp. 38-42.     

New analytical and numerical approaches for geopotential modeling

 The standard analytical approach which is applied for constructing geopotential models OSU86 and earlier ones, is based on reducing the boundary value equation to a sphere enveloping the Earth and then solving it directly with respect to the potential coefficients _n,m . In an alternative procedure, developed by Jekeli and used for constructing the models OSU91 and EGM96, at first an ellipsoidal harmonic series is developed for the geopotential and then its coefficients _n,m ^e are transformed to the unknown _n,m . The second solution is more exact, but much more complicated. The standard procedure is modified and a new simple integral formula is derived for evaluating the potential coefficients. The efficiency of the standard and new procedures is studied numerically. In these solutions the same input data are used as for constructing high-degree parts of the EGM96 models. From two sets of _n,m (n≤360,|m|≤n), derived by the standard and new approaches, different spectral characteristics of the gravity anomaly and the geoid undulation are estimated and then compared with similar characteristics evaluated by Jekeli's approach (`etalon' solution). The new solution appears to be very close to Jekeli's, as opposed to the standard solution. The discrepancies between all the characteristics of the new and `etalon' solutions are smaller than the corresponding discrepancies between two versions of the final geopotential model EGM96, one of them (HDM190) constructed by the block-diagonal least squares (LS) adjustment and the other one (V068) by using Jekeli's approach. On the basis of the derived analytical solution a new simple mathematical model is developed to apply the LS technique for evaluating geopotential coefficients. Received: 12 December 2000 / Accepted: 21 June 2001     

Determination of the potential of homogeneous polyhedral bodies using line integrals

For the determination of the potential of irregular inhomogeneous bodies they can be decomposed into (polyhedral) parts of homogeneous density. Efficient formulas for the computation of the gravitational potential (and its first and second derivatives) of homogeneous polyhedral bodies are presented. They are obtained using a transformation of the volume integral into line integrals. The most important property of the solution is that all ten quantities under consideration (potential, 3 components of the gravitation vector, 6 components of the tensor of the second derivatives) can be represented by using only two different line integrals. Furthermore, all coordinate transformations needed in the evaluation are chosen in such a way that they do not appear in the final result. The consequence, favorable for efficient programming, is that the same transcendental expressions along each edge of the polyhedron are needed for all ten quantities; even the same linear combinations of them for individual surfaces are appearing in different formulas. The expressions obtained are probably the simplest possible, which is also reflected in the fact that for the special case of a right rectangular prism they may easily be specialized to the usual well-known formulas. Received 28 Juni 1994; Accepted 13 September 1996     

Integrating terrain and vegetation indices for identifying potential soil erosion risk area

The present paper offers an innovative method to monitor the change in soil erosion potential by integrating terrain and vegetation indices derived from remote sensing data. Three terrain indices namely, topographic wetness index (TWI), stream power index (SPI) and slope length factor (LS), were derived from the digital elevation model. Normalized vegetation index (NDVI) was derived for the year 1988 and 2004 using remote sensing images. K-mean clustering was performed on staked indices to categorize the study area into four soil erosion potential classes. The validation of derived erosion potential map using USLE model showed a good agreement. Results indicated that there was a significant change in the erosion potential of the watershed and a gradual shifting of lower erosion potential class to next higher erosion potential class over the study period.     

Lidar Elevation Data for Surface Hydrologic Modeling: Resolution and Representation Issues

This paper is concerned with the application of high spatial resolution elevation data derived from light detection and ranging technologies (lidar) to surface hydrologic modeling. In recent years, airborne lidar technology has been employed to develop high accuracy digital elevation models (DEMs) with horizontal resolution on the order of a few meters. As with any spatial data product there are limits to the lidar's practical use that vary with the intended application. This paper considers potential issues and challenges for the use of lidar-derived DEMs in surface hydrologic modeling applications, such as characterizing flow direction and power, identifying sub-basins in a watershed, and calculating upstream contributing area and other variables. We compare results using conventional 30m DEMs and 6m lidar for a high relief study area and a low relief study area. Results are more comparable between these data sources, regardless of hydrologic operation, for the high relief area, while the similarity of results in the low relief area is dependent upon the particular operation. Post-processing on the lidar data successfully removed such flow obstacles as bridges that might have artificially impeded surface flow. An exploration of the effect of spatial resolution on results suggests that cell size is a more significant factor than production method.     

A formula for computing the gravity disturbance from the second radial derivative of the disturbing potential

 A formula for computing the gravity disturbance and gravity anomaly from the second radial derivative of the disturbing potential is derived in detail using the basic differential equation with spherical approximation in physical geodesy and the modified Poisson integral formula. The derived integral in the space domain, expressed by a spherical geometric quantity, is then converted to a convolution form in the local planar rectangular coordinate system tangent to the geoid at the computing point, and the corresponding spectral formulae of 1-D FFT and 2-D FFT are presented for numerical computation. Received: 27 December 2000 / Accepted: 3 September 2001     

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.     

A MAP OF THE AVERAGED RELIEF OF THE RUSSIAN PLAIN

The authors outline a methodology for the small-scale hypsometric mapping of plains areas based on the averaging of relief—this for the purpose of portraying the most important details of local relief (small erosional landforms) without obscuring larger, major relief forms or compromising the readability of the map. An example map (25-m contour interval) of the Soviet portion of the East European Plain at a scale of 1:5,000,000 is included. The elevations portrayed on this map were averaged, according to a multi-stage procedure, from grid cells on a 1:2,500,000 map before transference onto the smaller-scale map base. The new map makes possible a revised estimate of the average elevation of the Russian Plain and depicts well the effects of both endogenous and exogenous factors on relief formation. Translated from: Geomorfologiya, 1987, No. 1, pp. 22-29.     

Far-zone effects for different topographic-compensation models based on a spherical harmonic expansion of the topography

The determination of the gravimetric geoid is based on the magnitude of gravity observed at the surface of the Earth or at airborne altitude. To apply the Stokes’s or Hotine’s formulae at the geoid, the potential outside the geoid must be harmonic and the observed gravity must be reduced to the geoid. For this reason, the topographic (and atmospheric) masses outside the geoid must be “condensed” or “shifted” inside the geoid so that the disturbing gravity potential T fulfills Laplace’s equation everywhere outside the geoid. The gravitational effects of the topographic-compensation masses can also be used to subtract these high-frequent gravity signals from the airborne observations and to simplify the downward continuation procedures. The effects of the topographic-compensation masses can be calculated by numerical integration based on a digital terrain model or by representing the topographic masses by a spherical harmonic expansion. To reduce the computation time in the former case, the integration over the Earth can be divided into two parts: a spherical cap around the computation point, called the near zone, and the rest of the world, called the far zone. The latter one can be also represented by a global spherical harmonic expansion. This can be performed by a Molodenskii-type spectral approach. This article extends the original approach derived in Novák et al. (J Geod 75(9–10):491–504, 2001), which is restricted to determine the far-zone effects for Helmert’s second method of condensation for ground gravimetry. Here formulae for the far-zone effects of the global topography on gravity and geoidal heights for Helmert’s first method of condensation as well as for the Airy-Heiskanen model are presented and some improvements given. Furthermore, this approach is generalized for determining the far-zone effects at aeroplane altitudes. Numerical results for a part of the Canadian Rocky Mountains are presented to illustrate the size and distributions of these effects.     

Positioning Disaster Relief Teams Given Dynamic Service Demand: A Hybrid Agent‐Based and Spatial Optimization Approach

During extreme events, like hurricanes, it is essential to position relief services for non‐evacuees throughout the affected region in optimal locations. While previous research has explored a variety of spatial optimization models to accomplish such a task, most work assumes that the population with demand is relatively static. However, this assumption neglects to account for potential feedbacks in the relief distribution system. For example, a population's behavior can both affect and be affected by the placement of relief services, resulting in a dynamic spatial distribution of demand. This article presents a hybrid modeling approach that utilizes GIS data with agent‐based and spatial optimization models to position emergency relief teams in Bay County, Florida during a hurricane event. Non‐evacuating household agents choose to remain at home or seek shelter, while relief team agents are periodically repositioned to account for changes in the spatial distribution of household agents. A total of 220 simulations are run to explore a variety of scenarios. Results show different repositioning timing strategies and the magnitude of the feedback effect drastically changes the level of access households have to relief teams. Ultimately, this work demonstrates the importance of accounting for behavioral and spatial dynamics in disaster relief distribution systems.     

Airborne LiDAR and Terrestrial Laser Scanning Derived Vegetation Obstruction Factors for Visibility Models

Research presented here explores the feasibility of leveraging vegetation data derived from airborne light detection and ranging (LiDAR) and terrestrial laser scanning (TLS) for visibility modeling. Using LiDAR and TLS datasets of a lodgepole pine (Pinus contorta) dominant ecosystem, tree canopy and trunk obstructions were isolated relevant to a discrete visibility beam in a short‐range line‐of‐sight model. Cumulative obstruction factors from vegetation were compared with reference visibility values from digital photographs along sightline paths. LiDAR‐derived tree factors were augmented with single‐scan TLS data for obstruction prediction. Good correlation between datasets was found up to 10 m from the terrestrial scanner, but fine scale visibility modeling was problematic at longer distances. Analysis of correlation and regression results reveal the influence of obstruction shadowing inherent to discrete LiDAR and TLS, potentially limiting the feasibility of modeling visibility over large areas with similar technology. However, the results support the potential for TLS‐derived subcanopy metrics for augmenting large amounts of aerial LiDAR data to significantly improve models of forest structure. Subtle LiDAR processing improvements, including more accurate tree delineation through higher point density aerial data, combined with better vegetation quantification processes for TLS data, will advance the feasibility and accuracy of data integration.