Rational function optimization using genetic algorithms

In the absence of either satellite ephemeris information or camera model, rational functions are introduced by many investigators as mathematical model for image to ground coordinate system transformation. The dependency of this method on many ground control points (GCPs), numerical complexity, particularly terms selection, can be regarded as the most known disadvantages of rational functions. This paper presents a mathematical solution to overcome these problems. Genetic algorithms are used as an intelligent method for optimum rational function terms selection. The results from an experimental test carried out over a test field in Iran are presented as utilizing an IKONOS Geo image. Different numbers of GCPs are fed through a variety of genetic algorithms (GAs) with different control parameter settings. Some initial constraints are introduced to make the process stable and fast. The residual errors at independent check points proved that sub-pixel accuracies can be achieved even when only seven and five GCPs are used. GAs could select rational function terms in such a way that numerical problems are avoided without the need to normalize image and ground coordinates.     

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.     

Hotine’s conjecture in differential geodesy

In this paper we present a critical examination of a conjecture ofMartin Hotine on the possibility of employing the geopotential function of the Earth’s gravitational field as a member of a triply orthogonal system of surfaces. If such a conjecture were valid, it would provide a natural triply orthogonal system of coordinates which would be of significance in mathematical geodesy. It is shown that Hotine’s arguments are inadequate to prove his conjecture, and finally that his conjecture is false. Dedicated to the memory of Martin Hotine (1898–1968)     

Approximate representation of thep-norm distribution

［1］Liu D J,Shi W Z,Tong X H,et al.Precision analysis and quality cont rol of GIS spatial data.Shanghai:Shanghai Publishing House of Scientific Documen ts,1999 ［2］Chen X R,Fang Z B,Li G Y,et al.Non_parameter statistics.S hanghai:Shanghai Publishing House of Science and Technology,1989 ［3］Li Q H,Tao B Z.Application of probability statistical theory in survey ing.Beijing:Beijing Publishing House of Surveying and Mapping,1982 ［4］Sun H Y.p_norm distribution theory and its application in surveyin g data processing:[Ph.D Thesis].Wuhan:Wuhan Technical University of Surveying and Mapping,1995     

Construction of Green's function for the Stokes boundary-value problem with ellipsoidal corrections in the boundary condition

Green's function for the boundary-value problem of Stokes's type with ellipsoidal corrections in the boundary condition for anomalous gravity is constructed in a closed form. The `spherical-ellipsoidal' Stokes function describing the effect of two ellipsoidal correcting terms occurring in the boundary condition for anomalous gravity is expressed in O(e <sup>2</sup> <sub>0</sub>)-approximation as a finite sum of elementary functions analytically representing the behaviour of the integration kernel at the singular point ψ=0. We show that the `spherical-ellipsoidal' Stokes function has only a logarithmic singularity in the vicinity of its singular point. The constructed Green function enables us to avoid applying an iterative approach to solve Stokes's boundary-value problem with ellipsoidal correction terms involved in the boundary condition for anomalous gravity. A new Green-function approach is more convenient from the numerical point of view since the solution of the boundary-value problem is determined in one step by computing a Stokes-type integral. The question of the convergence of an iterative scheme recommended so far to solve this boundary-value problem is thus irrelevant. Received: 5 June 1997 / Accepted: 20 February 1998     

High-resolution atmospheric angular momentum functions related to Earth rotation parameters during CONT08

Due to the temporal resolution of available numerical weather analyses, the effect of the atmosphere on Earth rotation at daily and sub-daily periods is usually investigated using 6-hourly atmospheric angular momentum (AAM) functions. During the period of CONT08, however, atmospheric analysis data were provided by the European Centre for Medium-Range Weather Forecasts (ECMWF) also on an hourly basis. In this paper, we, therefore, determine two sets of AAM functions from ECMWF data—one for CONT08 with hourly resolution and one for the year 2008 with 6-hourly resolution. The comparisons of the AAM functions to high-resolution Earth rotation parameters (ERP) from VLBI and GPS observations are carried out in the frequency domain. Special attention is paid to the preparation of the high-resolution data sets for the geodetic purposes, as there are jump discontinuities at 12 h intervals. Hence, the hourly AAM functions need to be concatenated. The revised functions yield much smaller amplitudes than their 6-hourly counterparts, as can be seen from the equatorial and the axial frequency spectra of atmospheric excitation in Earth rotation. This decrease of spectral power in the hourly AAM functions is found to be associated with a strong counteraction of pressure and wind terms, which originates from atmospheric circulation on short time scales. The results are compared to previous findings published by Brzeziński and Petrov (IERS Tech Note 28:53–60, 2000) based on the data from the U.S. National Centers for Environmental Prediction (NCEP).     

Mathematical modelling of historical reconnaissance CORONA KH-4B Imagery

This paper describes three mathematical modelling methods for high-resolution declassified CORONA KH-4B images. Since CORONA images are collected with a panoramic camera, several types of geometric distortions are involved. Two methods use the modified collinearity equations, and the third involves the terrain-dependent rational function model (RFM) which is considered to be a generic sensor model. Comparative analysis of the three mathematical modelling methods is undertaken. The results show that a ± 1·5 pixels level of horizontal and vertical accuracy can be obtained. A digital elevation model (DEM) of a test site is also created.     

Topological vector spaces of harmonic functions and the trace operator

Many problems in physical geodesy can be formulated in terms of boundary-value problems (BVPs) for the gravitational potential; many of them can be ultimately formulated as a Dirichlet problem. For this reason, there is a flourishing literature of geodetic contributions to potential theory. In this paper, the authors pick up some classical arguments from the mathematical analysis of BVPs and show, by using only Hilbert spaces of harmonic functions, how they can be systematically cast into a functional scheme clarifying the role of duality when dealing with the harmonic subspaces of classical Sobolev spaces, of any real order. The analysis is here restricted to the case of functions harmonic in spherical domains to make the results transparent and more readable by geodesists. A further step is then taken showing how to generalize the Dirichlet problem for the space of all the functions that are harmonic outside a sphere, which exploits the more general theory of Fréchet topological spaces. Basically, the result is that any functions harmonic in the exterior of a sphere can be uniquely identified by a suitably defined trace on the sphere. The paper concludes with comments and discussion of future work.     

The solution of the Korn–Lichtenstein equations of conformal mapping: the direct generation of ellipsoidal Gauß–Krüger conformal coordinates or the Transverse Mercator Projection

The differential equations which generate a general conformal mapping of a two-dimensional Riemann manifold found by Korn and Lichtenstein are reviewed. The Korn–Lichtenstein equations subject to the integrability conditions of type vectorial Laplace–Beltrami equations are solved for the geometry of an ellipsoid of revolution (International Reference Ellipsoid), specifically in the function space of bivariate polynomials in terms of surface normal ellipsoidal longitude and ellipsoidal latitude. The related coefficient constraints are collected in two corollaries. We present the constraints to the general solution of the Korn–Lichtenstein equations which directly generates Gau?–Krüger conformal coordinates as well as the Universal Transverse Mercator Projection (UTM) avoiding any intermediate isometric coordinate representation. Namely, the equidistant mapping of a meridian of reference generates the constraints in question. Finally, the detailed computation of the solution is given in terms of bivariate polynomials up to degree five with coefficients listed in closed form. Received: 3 June 1997 / Accepted: 17 November 1997     

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     

Satellite Doppler solutions in terms of a single parameter

By an appropriate combination of the integrated doppler counts for a motionless ground station over two consecutive arcs, of a satellite path, it is possible to obtain a linear mathematical model relating the coordinates of the ground station to the observations. In this mathematical model, the involvement of the fourth unknown of the problem—the frequency off-set parameter, is, however, not linear. By application of the least squares technique, the solutions for the coordinates are obtained as analytical functions of the frequency off-set, parameter only. These, in turn, reduce the basic formula for the doppler count to be an implicit function of the same single variable. The value of the variable which provides the best fit of this function with the observed doppler counts, minimizing the sum of squares of the deviations for all involved pairs of satellite positions is the best value for the unknown frequency off-set parameter and an iterative technique is devised to compute this value. The desired values for the coordinates of the ground station can then be obtained by substitution of the best value of the frequency off-set parameter into the corresponding formula, and correcting for reducing the effects of the random noise in the observed dopple data.     

A discrimination test procedure for ambiguity resolution on-the-fly

Ambiguity validation tests include the acceptance test and discrimination test, which are both important steps in the Global Positioning System ambiguity resolution process. An ambiguity discrimination test procedure based on a test statistic which is constructed by the difference (not the ratio as used in current procedures) between the minimum and second minimum quadratic form of the residuals in ambiguity identification, and its standard deviation, is proposed. The distribution function of the proposed test statistic is theoretically identified as a standard normal distribution when the known a priori variance factor is used, or as a Student's t distribution when the estimated variance factor is used. With this procedure, the ambiguity discrimination test is based on a more rigorous test statistic whose critical value can be calculated with any chosen level of significance. Test results indicate that the proposed ambiguity discrimination test procedure is reliable for use in ambiguity resolution on-the-fly. Received: 17 December 1997 / Accepted: 30 June 1998     

The spherical horizontal and spherical vertical boundary value problem – vertical deflections and geoidal undulations – the completed Meissl diagram

 In a comparison of the solution of the spherical horizontal and vertical boundary value problems of physical geodesy it is aimed to construct downward continuation operators for vertical deflections (surface gradient of the incremental gravitational potential) and for gravity disturbances (vertical derivative of the incremental gravitational potential) from points on the Earth's topographic surface or of the three-dimensional (3-D) Euclidean space nearby down to the international reference sphere (IRS). First the horizontal and vertical components of the gravity vector, namely spherical vertical deflections and spherical gravity disturbances, are set up. Second, the horizontal and vertical boundary value problem in spherical gravity and geometry space is considered. The incremental gravity vector is represented in terms of vector spherical harmonics. The solution of horizontal spherical boundary problem in terms of the horizontal vector-valued Green function converts vertical deflections given on the IRS to the incremental gravitational potential external in the 3-D Euclidean space. The horizontal Green functions specialized to evaluation and source points on the IRS coincide with the Stokes kernel for vertical deflections. Third, the vertical spherical boundary value problem is solved in terms of the vertical scalar-valued Green function. Fourth, the operators for upward continuation of vertical deflections given on the IRS to vertical deflections in its external 3-D Euclidean space are constructed. Fifth, the operators for upward continuation of incremental gravity given on the IRS to incremental gravity to the external 3-D Euclidean space are generated. Finally, Meissl-type diagrams for upward continuation and regularized downward continuation of horizontal and vertical gravity data, namely vertical deflection and incremental gravity, are produced. Received: 10 May 2000 / Accepted: 26 February 2001     

中性大气折射的映射函数

在球对称大气模型下，我们导出了与余误差函数形式相联系的中性大气折射改正的母函数，并进一步讨论了它的几种数学展开形式。由此方法建立的映射函数可以对各类大气模型直接进行参数拟合。     

Green's function solution to spherical gradiometric boundary-value problems

 Three independent gradiometric boundary-value problems (BVPs) with three types of gradiometric data, {Γ _rr }, {Γ _{r θ},Γ _{r λ}} and {Γ_θθ−Γ_λλ,Γ_θλ}, prescribed on a sphere are solved to determine the gravitational potential on and outside the sphere. The existence and uniqueness conditions on the solutions are formulated showing that the zero- and the first-degree spherical harmonics are to be removed from {Γ _{r θ},Γ _{r λ}} and {Γ_θθ−Γ_λλ,Γ_θλ}, respectively. The solutions to the gradiometric BVPs are presented in terms of Green's functions, which are expressed in both spectral and closed spatial forms. The logarithmic singularity of the Green's function at the point ψ=0 is investigated for the component Γ _rr . The other two Green's functions are finite at this point. Comparisons to the paper by van Gelderen and Rummel [Journal of Geodesy (2001) 75: 1–11] show that the presented solution refines the former solution. Received: 3 October 2001 / Accepted: 4 October 2002     

Accounting for topography in the calculation of quasigeoidal heights and plumb-line deflections from gravity anomalies

A calculation of quasigeoidal heights and plumb-line deflections according to Molodensky formulae was carried out under elimination of the effect of topography from gravity anomalies. After the masses of topography had been removed a smoothed-out surface passing through astronomical and gravity stations was considered as representing the physical surface of the Earth. Thus it has been practically rendered possible to use the first-approximation formulae of Molodensky, and, in many cases, also the “zero-approximation” formulae analogous to the formulae of Stokes and Vening-Meinesz. The effect of the restored masses of topography was then added to the quantities found; the said effect was expressed as the effect of topography condensed on the normal equipotential surface passing through the point under investigation, plus a correction for condensation. Following some transformations, the resulting formulae (13) and (18) were obtained which formulae differ in their “zero-approximation” (15) and (20) from traditional formulas in that they contain terrait reductions added to free-air anomalies. Moreover, in the calculation of plumb-line deflections directly in mountain regions a correction for differing effects of topography before and after its condensation is to be introduced. A tentative expansion of terrain reduction in terms of spherical harmonics up to the third order is given; it can be seen therefrom that the Stokes series in its usual form is subject to a mean arror about 15–20%. It is also shown that the expansion of free-air anomalies in terms of spherical functions contains a first-order harmonic with a mean values about ±0.3 mgl. The said harmonic practically disappears in the expansion of the sum of free-air anomalies and terrain reductions.     

The contribution of Very Long Baseline Interferometry to ITRF2005

The contribution of the International VLBI Service for Geodesy and Astrometry (IVS) to the ITRF2005 (International Terrestrial Reference Frame 2005) has been computed by the IVS Analysis Coordinator’s office at the Geodetic Institute of the University of Bonn, Germany. For this purpose the IVS Analysis Centres (ACs) provided datum-free normal equation matrices in Solution INdependent EXchange (SINEX) format for each 24 h observing session to be combined on a session-by-session basis by a stacking procedure. In this process, common sets of parameters, transformed to identical reference epochs and a prioris, and especially representative relative weights have been taken into account for each session. In order to assess the quality of the combined IVS files, Earth orientation parameters (EOPs) and scaling factors have been derived from the combined normal equation matrices. The agreement of the EOPs of the combined normal equation matrices with those of the individual ACs in terms of weighted root mean square (WRMS) is in the range of 50–60 μas for the two polar motion components and about 3 μs for UT1−UTC. External comparisons with International GNSS Serive (IGS) polar motion components is at the level of 130–170 μas and 21 μs/day for length of day (LOD). The scale of the terrestrial reference frame realized through the IVS SINEX files agrees with ITRF2000 at the level of 0.2 ppb.     

Real time estimation of position and the gravity vector with an inertial survey system

Based on sine functions for fitting horizontal velocity errors in an inertial survey system and a step or slope function approximation of the vertical deflection between stops, a positioning accuracy of 10 m (CEP) can be achieved with a car-borne system – a modified -21 aircraft navigation system, with a 10 min travel period between Zero Velocity Update Times (ZUPTs) and a total 1.5 hour survey time. A rough estimation of the vertical deflection can also be expected in real time out of the approach. Also a forward-backward-forward run is presented which demonstrates determination of the gravity anomaly. Received 10 March 1995; Accepted 8 July 1996     

Least-squares prediction of horizontal coordinate distortions in Canada

A least-squares prediction method is described to estimate horizontal coordinate distortions at lower order points of a network using known coordinate differences (NAD27 coordinate distortions Δϕ′s and Δλ′s) at higher order points between NAD27 coordinates and coordinates derived from a recent (MAY 76), relatively distortion free, adjustment of these points. Empirical autocovariance functions of Δϕ and Δλ and crosscovariance function between Δϕ and Δλ are derived from some 5,250 data points and modelled using series of exponential functions. Empirical mean square values of Δϕ and Δλ, which are a measure of the distortions in NAD27 ϕ and λ, are 0.051 and 0.645 arcsecs² respectively. The corresponding mean value of the product ΔϕΔλ, which is a measure of the correlation between Δϕ and Δλ, is 0.056 arcsecs². The accuracy obtainable for predicted Δϕ and Δλ at an arbitrary point (e.g., lower order station) is a function of the accuracy and configuration of known Δϕ′s and Δλ′s in the surrounding area. Accuracies obtainable for various types of data configuration are given. Under favorable conditions taking place in about 60% of cases, accuracies in terms of ms agreement with known values of 0″.02 (0.6 m) and 0″.01 (0.2 m along parallel at latitude 50°) for the predicted latitude and longitude distortions are obtainable. Finally, a comparison with a method based on the use of complex polynomials is made. Presented at International Symposium on Geodetic Networks and Computations, Munich, August–September 1981.     

临界值可变的抗差估计等价权函数

抗差估计等价权函数一般由正态分布统计量构造,其临界值(或称准则)一般由实际经验确定.首先分析了正态分布统计量和学生化残差统计量的区别,然后分别讨论了基于这两种统计量构造的等价权函数的区别.研究表明,利用学生化残差统计量构造的等价权函数以及顾及误差显著性水平确定的临界值,不仅考虑了观测误差的大小,而且还可以顾及了实际观测的图形强度和多余观测数,可以克服人为确定临界值可能带来的参数估计的有效性和抗差性方面的风险.