The parameter distributions of the integer GPS model

 A parameter estimation theory is incomplete if no rigorous measures are available for describing the uncertainty of the parameter estimators. Since the classical theory of linear estimation does not apply to the integer GPS model, rigorous probabilistic statements cannot be made with reference to the classical results. The fact that integer parameters are involved in the estimation process forces a reappraisal of the propagation of uncertainty. It is with this purpose in mind that the joint and marginal distributional properties of both the integer and non-integer parameters of the GPS model are determined. These joint distributions can also be used to determine the distribution of functions of the parameters. As an important example, the distribution of the vector of ambiguity residuals is determined. Received: 30 January 2001 / Accepted: 31 July 2001     

The role of spatial effects in the empirical analysis of regional concentration

 Economists have recently devoted an increasing attention to the issue of spatial concentration of economic activities. However, surprisingly enough, most of the empirical work is still based on the computation of very basic statistical measures in which the geographical characteristics of data play no role. By making use of a series of empirical examples we show that spatial concentration consists of two different features that are rarely kept as separate in the statistical analysis: an a-spatial concept of variability which is invariant to permutations, and the concept of polarization that refers to the geographical position of observations. Received: 8 August 2000 / Accepted: 20 June 2001     

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     

A new method for computing the ellipsoidal correction for Stokes's formula

 This paper generalizes the Stokes formula from the spherical boundary surface to the ellipsoidal boundary surface. The resulting solution (ellipsoidal geoidal height), consisting of two parts, i.e. the spherical geoidal height N ₀ evaluated from Stokes's formula and the ellipsoidal correction N ₁, makes the relative geoidal height error decrease from O(e ²) to O(e ⁴), which can be neglected for most practical purposes. The ellipsoidal correction N ₁ is expressed as a sum of an integral about the spherical geoidal height N ₀ and a simple analytical function of N ₀ and the first three geopotential coefficients. The kernel function in the integral has the same degree of singularity at the origin as the original Stokes function. A brief comparison among this and other solutions shows that this solution is more effective than the solutions of Molodensky et al. and Moritz and, when the evaluation of the ellipsoidal correction N ₁ is done in an area where the spherical geoidal height N ₀ has already been evaluated, it is also more effective than the solution of Martinec and Grafarend. Received: 27 January 1999 / Accepted: 4 October 1999     

A model for adjustment of differential gravity measurements with simultaneous gravimeter calibration

 A mathematical model is proposed for adjustment of differential or relative gravity measurements, involving simultaneously instrumental readings, coefficients of the calibration function, and gravity values of selected base stations. Tests were performed with LaCoste and Romberg model G gravimeter measurements for a set of base stations located along a north–south line with 1750 mGal gravity range. This line was linked to nine control stations, where absolute gravity values had been determined by the free-fall method, with an accuracy better than 10 μGal. The model shows good consistence and stability. Results show the possibility of improving the calibration functions of gravimeters, as well as a better estimation of the gravity values, due to the flexibility admitted to the values of the calibration coefficients. Received: 15 November 1999 / Accepted: 31 October 2000     

A synthetic Earth for use in geodesy

 A synthetic Earth and its gravity field that can be represented at different resolutions for testing and comparing existing and new methods used for global gravity-field determination are created. Both the boundary and boundary values of the gravity potential can be generated. The approach chosen also allows observables to be generated at aircraft flight height or at satellite altitude. The generation of the synthetic Earth shape (SES) and gravity-field quantities is based upon spherical harmonic expansions of the isostatically compensated equivalent rock topography and the EGM96 global geopotential model. Spherical harmonic models are developed for both the synthetic Earth topography (SET) and the synthetic Earth potential (SEP) up to degree and order 2160 corresponding to a 5′×5′ resolution. Various sets of SET, SES and SEP with boundary geometry and boundary values at different resolutions can be generated using low-pass filters applied to the expansions. The representation is achieved in point sets based upon refined triangulation of a octahedral geometry projected onto the chosen reference ellipsoid. The filter cut-offs relate to the sampling pattern in order to avoid aliasing effects. Examples of the SET and its gravity field are shown for a resolution with a Nyquist sampling rate of 8.27 degrees. Received: 6 August 1999 / Accepted: 26 April 2000     

Spherical harmonic analysis and synthesis for global multiresolution applications

 On the Earth and in its neighborhood, spherical harmonic analysis and synthesis are standard mathematical procedures for scalar, vector and tensor fields. However, with the advent of multiresolution applications, additional considerations about convolution filtering with decimation and dilation are required. As global applications often imply discrete observations on regular grids, computational challenges arise and conflicting claims about spherical harmonic transforms have recently appeared in the literature. Following an overview of general multiresolution analysis and synthesis, spherical harmonic transforms are discussed for discrete global computations. For the necessary multi-rate filtering operations, spherical convolutions along with decimations and dilations are discussed, with practical examples of applications. Concluding remarks are then included for general applications, with some discussion of the computational complexity involved and the ongoing investigations in research centers. Received: 13 November 2000 / Accepted: 12 June 2001     

Determining the size of spatial clusters in focused tests: Comparing two methods by means of simulation in a GIS

 Based on a four-point evaluation system consisting of accuracy, consistency, power, and chance to commit type I errors, this study compares Tango's minimum p (MinP) and Stone's maximum relative risk (MaxRR) methods for detecting focused cluster size through simulations in GIS. It reveals that the MinP method is more effective than the MaxRR method. The MinP method exhibits higher levels of accuracy and consistency; and its power and chance to commit type I errors are similar to the MinP method. The MaxRR method has a tendency to underestimate the cluster size, while the MinP method tends to overestimate the cluster size, particularly when the clusters are relatively big and have high relative risk levels. In addition, the MinP method seems to be most effective in revealing the size of clusters when clusters are neither too strong nor too weak. The lowest detection rates for clustering occur when the clustering signal is relatively weak, which is easily understandable. In practice, it might be useful to use both the methods to estimate a range of possible cluster sizes, where the MaxRR method indicates the lower estimate, while the MinP method gives the higher estimate of the cluster size. Received: 24 August 2002 / Accepted: 20 December 2002     

Improvement of the separability of a survey scheme for monitoring crustal deformations in the area of an active fault

 Monitoring of the crustal movements along a tectonic fault is of particular importance in the study of the mechanism of an earthquake. There are several techniques to gauge crustal deformations, including terrestrial survey methods, space-positioning techniques and permanently installed geotechnical instruments. Each technique or method has its own advantages and limitations. Integration of the various techniques into a monitoring scheme is recommended. It is discussed how a proper integrated system can significantly improve the separability of a monitoring scheme at little additional expense. Separability is the ability of a monitoring scheme to distinguish among potential deformation models, and can be used for the optimum design of monitoring schemes. Discussion concentrates on the separability between a dislocation model and a rigid movement model in the area of an active fault. The addition of a few strain observations to a conventional terrestrial survey scheme can better distinguish between the above-mentioned models. A simulated example is presented to demonstrate the idea. Received: 4 November 1997 / Accepted: 9 July 2001     

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     

Evaluating the importance of accessibility to congestion response using a GIS-based travel simulator

 This paper examines the effect of accessibility on individual response to unexpected traffic delays and congestion. The dataset used was collected by means of a travel simulator developed within a geographic information system (GIS) environment. The simulator models a commute trip where congestion takes place, and subjects are asked to respond by making a choice among alternative courses of action. Available alternatives for dealing with the unexpected traffic delay include changing the location of the planned activities or changing the activities to be performed. Accessibility to the new locations and to the different activities is computed using a cumulative measure. Analysis using CHAID tree technique found that accessibility is a good predictor of subjects' choice when responding to unexpected traffic delays. Received: September 2002 / Accepted: January 2003 Support for this research by a grant from the Center of Intelligent Transportation Research of the Ohio State University is gratefully acknowledged. The author would also like to thank Joe Weber, Mei-Po Kwan and two anonymous reviewers for their valuable comments.     

Spatial analysis in a Markov random field framework: The case of burning oil wells in Kuwait

 This paper discusses a modeling approach for spatial-temporal prediction of environmental phenomena using classified satellite images. This research was prompted by the analysis of change and landscape redistribution of petroleum residues formed from the residue of the burning oil wells in Kuwait (1991). These surface residues have been termed “tarcrete” (El-Baz et al. 1994). The tarcrete forms a thick layer over sand and desert pavement covering a significant portion of south-central Kuwait. The purpose of this study is to develop a method that utilizes satellite images from different time steps to examine the rate-of-change of the oil residue deposits and determine where redistribution is are likely to occur. This problem exhibits general characteristics of environmental diffusion and dispersion phenomena so a theoretical framework for a general solution is sought. The use of a lagged-clique, Markov random field framework and entropy measures is deduced to be an effective solution to satisfy the criteria of determination of time-rate-of-change of the surface deposits and to forecast likely locations of redistribution of dispersed, aggraded residues. The method minimally requires image classification, the determination of time stationarity of classes and the measurement of the level of organization of the state-space information derived from the images. Analysis occurs at levels of both the individual pixels and the system to determine specific states and suites of states in space and time. Convergence of the observed landscape disorder with respect to an analytical maximum provide information on the total dispersion of the residual system. Received: 5 September 2000 / Accepted: 10 July 2001     

A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions

 Spherical harmonic expansions form partial sums of fully normalised associated Legendre functions (ALFs). However, when evaluated increasingly close to the poles, the ultra-high degree and order (e.g. 2700) ALFs range over thousands of orders of magnitude. This causes existing recursion techniques for computing values of individual ALFs and their derivatives to fail. A common solution in geodesy is to evaluate these expansions using Clenshaw's method, which does not compute individual ALFs or their derivatives. Straightforward numerical principles govern the stability of this technique. Elementary algebra is employed to illustrate how these principles are implemented in Clenshaw's method. It is also demonstrated how existing recursion algorithms for computing ALFs and their first derivatives are easily modified to incorporate these same numerical principles. These modified recursions yield scaled ALFs and first derivatives, which can then be combined using Horner's scheme to compute partial sums, complete to degree and order 2700, for all latitudes (except at the poles for first derivatives). This exceeds any previously published result. Numerical tests suggest that this new approach is at least as precise and efficient as Clenshaw's method. However, the principal strength of the new techniques lies in their simplicity of formulation and implementation, since this quality should simplify the task of extending the approach to other uses, such as spherical harmonic analysis. Received: 30 June 2000 / Accepted: 12 June 2001     

Solution and econometric estimation of spatial dynamic models in continuous space and continuous time

 This paper presents and demonstrates a general approach to solving spatial dynamic models in continuous space and continuous time that characterize the behaviour of intertemporally and interspatially optimizing agents and estimating from discrete data the parameters of such models. The approach involves the use of a projection method to solve the models and a quasi-Newton algorithm to update quasi-FIML parameter estimates. Received: 26 July 2000 / Accepted: 31 January 2001     

Spatially restricted data distributions on the sphere: the method of orthonormalized functions and applications

 In many geoscientific applications data are irregularly distributed and not globally available, e.g. caps around the poles which are uncovered due to non-polar satellite orbits, or signals being defined solely on bounded regions on the globe. Starting from a sequence of base functions with global support, which in the present case is composed of spherical harmonics being initially non-orthogonal on a bounded subdomain, a set of functions is generated that constitutes an orthonormal basis. Different approaches to realize this transformation are studied and compared with respect to numerical stability and computational effort, and the corresponding effects on the coefficient recovery are investigated. A number of synthetic tests demonstrate the applicability, the benefit, but also the limitations, of this method. Received: 24 March 2000 / Accepted: 9 October 2000     

An integrated representation of spatial and temporal relationships between evolving regions

 The study of relationships between evolving regions within GIS still needs the development of operators that integrate the spatial and temporal dimensions. This paper introduces a new approach that combines topological relationships between regions in 2-dimensional space with temporal relationships between convex intervals in time. Resulting relationships are defined and visually presented within a 3-dimensional space that integrates the geographical space as a 2-dimensional space and the time line as the third dimension. Conceptual neighbourhoods are identified and extended by the concept of semi-transitions and transitions. Such a flexible framework presents the advantage of being derived from accepted relationships in both space and time. Its computational implementation is therefore compatible with current spatial and temporal GIS models. Received: May 2000 / Accepted: June 2001     

Improvement of satellite derived pollution maps with the use of a geostatistical interpolation method

 The small number of ground stations for the assessment of the spatial distribution of air pollutants motivates the search for methods that make use of satellite images. One such method, known as Differential Texture Analysis (DTA), is used to measure the Aerosol Optical Thickness in the Visible (AOTV), which correlates highly with air quality. With this method, the presence of clouds and/or land cover changes produce patches of missing values. In this paper we demonstrate that universal kriging can be used to obtain reasonable estimates for these missing values. The methodology was applied to a satellite derived AOTV map of the city of Brescia (Italy). Received: 17 July 2001 / Accepted: 11 December 2001     

Change detection thresholds for remotely sensed images

 The detection of change in remotely sensed images is often carried out by designating a threshold to distinguish between areas of change and areas of no change. The choice of threshold is often arbitrary however. The purpose of this paper is to offer a statistical framework for the selection of thresholds. The framework accounts for the facts that one carries out multiple tests of the null hypothesis of no change, when searching for regions of change over an image with a large number of pixels. Special attention is given to global spatial autocorrelation, which can affect the selection of appropriate threshold values. Received: 8 March 2001 / Accepted: 12 October 2001     

Short-arc analysis of intersatellite tracking data in a gravity mapping mission

 A technique for the analysis of low–low intersatellite range-rate data in a gravity mapping mission is explored. The technique is based on standard tracking data analysis for orbit determination but uses a spherical coordinate representation of the 12 epoch state parameters describing the baseline between the two satellites. This representation of the state parameters is exploited to allow the intersatellite range-rate analysis to benefit from information provided by other tracking data types without large simultaneous multiple-data-type solutions. The technique appears especially valuable for estimating gravity from short arcs (e.g. less than 15 minutes) of data. Gravity recovery simulations which use short arcs are compared with those using arcs a day in length. For a high-inclination orbit, the short-arc analysis recovers low-order gravity coefficients remarkably well, although higher-order terms, especially sectorial terms, are less accurate. Simulations suggest that either long or short arcs of the Gravity Recovery and Climate Experiment (GRACE) data are likely to improve parts of the geopotential spectrum by orders of magnitude. Received: 26 June 2001 / Accepted: 21 January 2002