Analytical methods for error propagation in planar space–time prisms

The space–time prism demarcates all locations in space–time that a mobile object or person can occupy during an episode of potential or unobserved movement. The prism is central to time geography as a measure of potential mobility and to mobile object databases as a measure of location possibilities given sampling error. This paper develops an analytical approach to assessing error propagation in space–time prisms and prism–prism intersections. We analyze the geometry of the prisms to derive a core set of geometric problems involving the intersection of circles and ellipses. Analytical error propagation techniques such as the Taylor linearization method based on the first-order partial derivatives are not available since explicit functions describing the intersections and their derivatives are unwieldy. However, since we have implicit functions describing prism geometry, we modify this approach using an implicit function theorem that provides the required first-order partials without the explicit expressions. We describe the general method as well as details for the two spatial dimensions case and provide example calculations.     

Impact of proximity-adjusted preferences on rank-order stability in geographical multicriteria decision analysis

This paper presents a new approach to deriving preferences assigned to evaluation criteria in geographical multicriteria decision analysis. In this approach, the preferences, expressed by numeric weights, are adjusted by distance measures derived from the explicit consideration of a locational structure. The structure is given by locations of decision options and high importance reference objects. The approach is demonstrated on the example of a house selection case study in San Diego, California. The results show that proximity-adjusted preferences for the evaluation criteria can alter significantly the rank order of decision options. Consequently, the explicit modeling of spatial preference variability may be needed in order to better account for decision-maker’s preferences.     

Increasing Accuracy of the GLONASS Time Service

GLONASS time is the reference for all time scales aboard GLONASS satellites. It is derived from the Central Synchronizer (CS), which consists of an ensemble of hydrogen clocks (HC) having a instability of (1...3) × 10⁻¹⁴ (averaging interval 24 hours). The CS time is compared to UTC(SU) in a differential mode using a TV channel of the Technical TV Center. Presently the error of GLONASS time is within 20...150 ns; however, there is a potential to decrease this value to 15 to 50 ns (3σ). The paper discusses the possible benefits of mutual phase comparisons between the HCs to determine the grouped frequency standard. These comparisons are averaged daily. The approach yields an accuracy of better than 10 ns (3σ) for CS, bringing the expected GLONASS time accuracy to the level of 15 to 50 ns (3σ). ? 2001 John Wiley & Sons, Inc.     

A computational approach for the analysis of changes in polygon distributions

 This paper develops a computational method for analyzing changes in polygon distributions. Unmovable polygons that change discontinuously without explicit functional linkage information are discussed. Six types of primitive events are used to describe the change: 1) generation, 2) disappearance, 3) expansion, 4) shrinkage, 5) union, and 6) division. The change of polygon distributions is decomposed into a combination of these events. A computational procedure for deducing a set of events from polygon distributions of two times is proposed. The method is applied to the analysis of the spatial competition between the major and small chains of convenience stores in Tokyo, Japan. Some empirical findings are shown. Received: 3 August 2000 / Accepted: 6 February 2001     

Upward and downward continuations in the geopotential determination

Summary The geopotential on and outside the earth is represented as a series in surface harmonics. The principal terms in it correspond to the solid harmonics of the external potential expansion with the coefficients being Stokes’ constantsC _nm andS _nm . The additional terms which occur near the earth’s surface due to its non-sphericity and topography are expressed in terms of Stokes’ constants too. This allows performing downward continuation of the potential derived from satellite observations. In the boundary condition which correlates Stokes’ constants and the surface gravity anomalies there occur additional terms due to the earth’s non-sphericity and topography. They are expressed in terms of Stokes’ constants as well. This improved boundary condition can be used for upward and downward continuations of the gravity field. Simple expressions are found representingC _nm andS _nm as explicit functions of the surface anomalies and its derivatives. The formula for the disturbing potential on the surface is derived in terms of the surface anomalies. All the formulas do not involve the earth’s surface in clinations.     

Estimating the Effects of Map Error on Habitat Delineation for the California Spotted Owl in Southern California

The reliability of habitat maps that have been generated using Geographic Information Systems (GIS) and image processing of remotely sensed data can be overestimated. Habitat suitability and spatially explicit population viability models are often based on these products without explicit knowledge of the effects of these mapping errors on model results. While research has considered errors in population modeling assumptions, there is no standardized method for measuring the effects of inaccuracies resulting from errors in landscape classification. Using landscape‐scale maps of existing vegetation developed for the USDA Forest Service in southern California from Landsat Thematic Mapper satellite data and GIS modeling, we performed a sensitivity analysis to estimate how mapping errors in vegetation type, forest canopy cover, and tree crown size might affect delineation of suitable habitat for the California spotted owl (Strix occidentalis occidentalis). The resulting simulated uncertainty maps showed an increase in the estimated area of suitable habitat types. Further analysis measuring the fragmentation of the additional patches showed that they were too small to be useful as habitat areas.     

Mathematical considerations of the modified normal distribution

In this paper, Romanowski’s modified normal distribution is considered from the mathematical standpoint. Initially, the characteristic function is computed. This function is then utilized to (1) compute moments, (2) demonstrate normal tendency as the modulating (“a”) parameter tends to infinity, (3) to formulate the modified normal distribution in terms of well known classical functions. To complete this mathematical consideration, the probability distribution is formulated in terms of tabulated functions. The paper is concluded with a discussion on the direction for further research.     

Evaluation of deflections of the vertical on the sphere and the plane: a comparison of FFT techniques

This paper presents a set of efficient formulas to evaluate the deflections of the vertical on the sphere using gridded data. The Vening-Meinesz formula, the topographic indirect effect on the deflections of the vertical as well as the terrain corrections are expressed as both 2D and 1D convolutions on the sphere, and consequently can be evaluated by the 2D and the 1D fast Fourier transform (FFT). When compared with the results obtained from pointwise integration, the use of the 1D FFT gives identical results, and therefore these results were used as control values in this paper. The use of the spherical 2D FFT improves significantly the computational efficiency with little sacrifice of accuracy (0.6^″ rms difference from the 1D FFT results). The planar 2D FFT, which is as efficient as the spherical 2D FFT, gives worse results (1.2^″ rms difference from the 1D FFT results) because of the extra approximations. Received: 27 February 1996; Accepted: 24 January 1997     

Satellite geodesy on curved space–time manifolds. Earth-fixed charts

The reformulation of geodetic measurement processes within the framework of general relativity is discussed. The metric tensor plays an important role in general relativity and has to be represented with respect to a set of appropriate charts. Almost every quantity of interest in geodetic or geophysical applications refers to a geocentric, Earth-fixed coordinate system (chart), therefore they are of great importance in geodesy and geophysics. The space–time metric with respect to an Earth-fixed chart is derived at first post-Newtonian order. The field equations determining the terrestrial gravitational field are derived and its explicit representation is outlined. The impact of the results on the modelling of geodetic measurement processes including space–time positioning scenarios as well as the high-precision gravitational field estimation is outlined. Received: 7 January 1998 / Accepted: 17 August 1999     

＂Digital Region＂ in the Context of the ＂Grid Computing＂

This paper is to construct a “digital local, regional, region“ information framework based on the technology of “SIG“ and its significance and application to the regional sustainable development evaluation system. First, the concept of the “grid computing“ and “SIG“ is interpreted and discussed, then the relationship between the “grid computing“ and “digital region“ is analyzed, and the framework of the “digital region“ is put forward. Finally, the significance and application of “grid computing“ to the “region sustainable development evaluation system“ are discussed.     

Construction of spherical harmonic series for the potential derivatives of arbitrary orders in the geocentric Earth-fixed reference frame

The derivatives of the Earth gravitational potential are considered in the global Cartesian Earth-fixed reference frame. Spherical harmonic series are constructed for the potential derivatives of the first and second orders on the basis of a general expression of Cunningham (Celest Mech 2:207–216, 1970) for arbitrary order derivatives of a spherical harmonic. A common structure of the series for the potential and its first- and second-order derivatives allows to develop a general procedure for constructing similar series for the potential derivatives of arbitrary orders. The coefficients of the derivatives are defined by means of recurrence relations in which a coefficient of a certain order derivative is a linear function of two coefficients of a preceding order derivative. The coefficients of the second-order derivatives are also presented as explicit functions of three coefficients of the potential. On the basis of the geopotential model EGM2008, the spherical harmonic coefficients are calculated for the first-, second-, and some third-order derivatives of the disturbing potential T, representing the full potential V, after eliminating from it the zero- and first-degree harmonics. The coefficients of two lowest degrees in the series for the derivatives of T are presented. The corresponding degree variances are estimated. The obtained results can be applied for solving various problems of satellite geodesy and celestial mechanics.     

Comparative study on two methods for calculating the gravitational potential of a prism

The determination of the gravitational potential of a prism plays an important role in physical geodesy and geophysics. However, there are few literatures that provide accurate approaches for determining the gravitational potential of a prism. Discrete element method can be used to determine the gravitational potential of a prism, and can approximate the true gravitational potential values with sufficient accuracy (the smaller each element is, the more accurate the result is). Although Nagy’s approach provi...     

不等式约束最小二乘问题的解及其统计性质

对不等式约束最小二乘平差问题,借助非线性规划中的凝聚约束方法把多个不等式约束转化为一个等式约束,采用拉格朗日极值法求解,解与贝叶斯解或单纯形解一致。其优点在于该解能够表示为观测的明显表达式,由此解的统计性质与最优性可以确定。给出演示该方法的GPS单点定位算例。     

Diurnal atmospheric forcing and temporal variations of the nutation amplitudes

A 29-year time-series of four-times-daily atmospheric effective angular momentum (EAM) estimates is used to study the atmospheric influence on nutation. The most important atmospheric contributions are found for the prograde annual (77 μas), retrograde annual (53 as), prograde semiannual (45 as), and for the constant offset of the pole (δψsinɛ₀=−86 as, δɛ=77 as). Among them only the prograde semiannual component is driven mostly by the wind term of the EAM function, while in all other cases the pressure term is dominant. These are nonnegligible quantities which should be taken into account in the new theory of nutation. Comparison with the VLBI corrections to the IAU 1980 nutation model taking into account the ocean tide contribution yields good agreement for the prograde annual and semiannual nutations. We also investigated time variability of the atmospheric contribution to the nutation amplitudes by performing the sliding-window least-squares analysis of both the atmospheric excitation and VLBI nutation data. Almost all detected variations of atmospheric origin can be attributed to the pressure term, the biggest being the in-phase annual prograde component (about 30 as) and the retrograde one (as much as 100200 as). These variations, if physical, limit the precision of classical modeling of nutation to the level of 0.1 mas. Comparison with the VLBI data shows significant correlation for the retrograde annual nutation after 1989, while for the prograde annual term there is a high correlation in shape but the size of the atmospherically driven variations is about three times less than deduced from the VLBI data. This discrepancy in size can be attributed either to inaccuracy of the theoretical transfer function or the frequency-dependent ocean response to the pressure variations. Our comparison also yields a considerably better agreement with the VLBI nutation data when using the EAM function without the IB correction for ocean response, which indicates that this correction is not adequate for nearly diurnal variations. Received: 10 September 1997 / Accepted: 5 March 1998     

Transformation method of exterior orientation angular elements obtained via position and orientation system under Gauss-Kruger projection coordinate system

Data obtained via airborne position and orientation system (POS) is in WGS 84 global geocentric reference frame, while the national coordinate reference system for topographic mapping in China is generally Gauss-Kruger projection coordinate system. Therefore, data obtained via a POS must be transformed to national coordinate system. Owing to the effects of earth curvature and meridian deviation, there are some errors in the process of angle transformation from roll, pitch, and heading (Φ ,Θ ,Ψ )obtained dir...     

Instantaneous global spatial interaction? Exploring the Gaussian inequality, distance and Internet pings in a global network

The Internet has been publicly portrayed as a new technological horizon yielding instantaneous interaction to a point where geography no longer matters. This research aims to dispel this impression by applying a dynamic form of trip modelling to investigate pings in a global computer network compiled by the Stanford Linear Accelerator Centre (SLAC) from 1998 to 2004. Internet flows have been predicted to have the same mathematical operators as trips to a supermarket, since they are both periodic and constrained by a distance metric. Both actual and virtual trips are part of a spectrum of origin–destination pairs in the time–space convergence of trip time-lines. Internet interaction is very near to the convergence of these time-lines (at a very small time scale in milliseconds, but with interactions over thousands of kilometres). There is a lag effect and this is formalised by the derivation of Gaussian and gravity inequalities between the time taken (Δt) and the partitioning of distance (Δx). This inequality seems to be robust for a regression of Δt to Δx in the SLAC data set for each year (1998 to 2004). There is a constant ‘forbidden zone’ in the interaction, underpinned by the fact that pings do not travel faster than the speed of light. Superimposed upon this zone is the network capacity where a linear regression of Δt to Δx is a proxy summarising global Internet connectivity for that year. The results suggest that there has been a substantial improvement in connectivity over the period with R ² increasing steadily from 0.39 to 0.65 from less Gaussian spreading of the ping latencies. Further, the regression line shifts towards the inequality boundary from 1998 to 2004, where the increased slope shows a greater proportional rise in local connectivity over global connectivity. A conclusion is that national geography still does matter in spatial interaction modelling of the Internet.     

The Bayesian approach applied to GPS ambiguity resolution. A mixture model for the discrete–real ambiguities alternative

 The problem of phase ambiguity resolution in global positioning system (GPS) theory is considered. The Bayesian approach is applied to this problem and, using Monte Carlo simulation to search over the integer candidates, a practical expression for the Bayesian estimator is obtained. The analysis of the integer grid points inside the search ellipsoid and their evolution with time, while measurements are accumulated, leads to the development of a Bayesian theory based on a mathematical mixture model for the ambiguity. Received: 29 March 2001 / Accepted: 3 September 2001     

The activities of the Ionosphere Working Group of the International GPS Service (IGS)

This article is based on a position paper presented at the IGS Network, Data and Analysis Center Workshop 2002 in Ottawa, Canada, 8–11 April 2002, and introduces the IGS Ionosphere Working Group (Iono_WG). Detailed information about the IGS in general can be found on the IGS Central Bureau Web page: http://igscb.jpl.nasa.gov. The Iono_WG commenced working in June 1998. The working group's main activity currently is the routine production of ionosphere Total Electron Content (TEC) maps with a 2-h time resolution and daily sets of GPS satellite and receiver hardware differential code bias (DCB) values. The TEC maps and DCB sets are derived from GPS dual-frequency tracking data recorded with the global IGS tracking network. In the medium- and long-term, the working group intends to refine algorithms for the mapping of ionospheric parameters from GPS measurements and to realize near–real–time availability of IGS ionosphere products. The paper will give an overview of the Iono_WG activities that include a summary of activities since its establishment, achievements and future plans. Electronic Publication     

基于整周模糊度概率特性的有效性检验

准确确定载波相位整周模糊度是快速高精度GPS定位的关键,已有的检验GPS整周模糊度有效性的方法几乎均是基于其为非随机常量建立的,因而都存在一定的缺陷。本文在研究整周模糊度概率特性的基础上,提出一种基于LABMBAD算法的整周模糊度概率分布函数的检验方法。实际演算表明该方法简单有效,统计概念明确。     

On the approximation of external gravitational potential with closed systems of (trial) functions

Summary Let S be the (regular) boundary-surface of an exterior regionE _e in Euclidean space ℜ³ (for instance: sphere, ellipsoid, geoid, earth's surface). Denote by {φ_n} a countable, linearly independent system of trial functions (e.g., solid spherical harmonics or certain singularity functions) which are harmonic in some domain containingE _e ∪ S. It is the purpose of this paper to show that the restrictions {ϕ_n} of the functions {φ_n} onS form a closed system in the spaceC (S), i.e. any functionf, defined and continuous onS, can be approximated uniformly by a linear combination of the functions ϕ_n. Consequences of this result are versions of Runge and Keldysh-Lavrentiev theorems adapted to the chosen system {φ_n} and the mathematical justification of the use of trial functions in numerical (especially: collocational) procedures.