The free nonlinear boundary value problem of physical geodesy

Summary Within potential theory of Poisson-Laplace equation the boundary value problem of physical geodesy is classified asfree andnonlinear. For solving this typical nonlinear boundary value problem four different types of nonlinear integral equations corresponding to singular density distributions within single and double layer are presented. The characteristic problem of free boundaries, theproblem of free surface integrals, is exactly solved bymetric continuation. Even in thelinear approximation of fundamental relations of physical geodesy the basic integral equations becomenonlinear because of the special features of free surface integrals.     

LONG LINES ON THE EARTH: THE DIRECT PROBLEM

Abstract

19. Formulae.—In Nos. 6, vol. i, and 9, vol. ii, pp. 259 and 156, there has been described a new method for dealing with long geodesics on the earth's surface. There the so-called “inverse” problem has claimed first attention: given the latitudes and longitudes of the extremities of a geodesic, to find its length and terminal azimuths. It remains to discuss the “direct” problem : a geodesic of given length starts on a given azimuth from a station of known latitude and longitude; to find the latitude and longitude of its extremity and the azimuth thereat. The solution of this direct problem demands a certain recasting of the formulae previously given. In order of working the several expressions now assume the forms below.     

TWO PROBLEMS OF RESECTION

Abstract

That admirable annual, The Surveyor (Ceylon), was generously forwarded to us some months ago. In this issue, vol. 2, no. 4, p. 93, there is given the solution of a question on resection in an examination paper. Since the solution appears rather laboured and the problem is interesting in itself and by no means valueless, it seems not out of place to attempt a simpler and more obvious answer.     

Terrain radiometric calibration of airborne UAVSAR for forested area

In the field of biomass estimation, terrain radiometric calibration of airborne polarimetric SAR data for forested areas is an urgent problem. Illuminated area correction of σ -naught could not completely remove terrain features. Inspired by Small and Shimada, this paper tested gamma-naught on one mountainous forested area using airborne Uninhabited Aerial Vehicle Synthetic Aperture Radar data and found it could remove most terrain features. However, a systematic increasing trend from far range to near range is found in airborne SAR cases. This paper made an attempt to use the relationship between distance to SAR sensor and γ-naught to calibrate γ -naught. Two quantitative evaluation methods are proposed. Experimental results demonstrate that variation of γ -naught can be constrained to a limited extent from near range to far range. Since this method is based on ground range images, it avoids complicated orthorectification.     

THE STEREOGRAPHIC SOLUTION OF THE SPHERICAL TRIANGLE

Abstract

At the suggestion of Mr. T. H. Corfield, who has himself given two solutions (E.S.R., No. 12, pp. 345–6) of Mr. A. J. Potter's problem, I venture to submit a third solution, which has at least the merit of simplicity.     

Mixed Integer-Real Valued Adjustment (IRA) Problems: GPS Initial Cycle Ambiguity Resolution by Means of the LLL Algorithm

In order to achieve to GPS solutions of first-order accuracy and integrity, carrier phase observations as well as pseudorange observations have to be adjusted with respect to a linear/linearized model. Here the problem of mixed integer-real valued parameter adjustment (IRA) is met. Indeed, integer cycle ambiguity unknowns have to be estimated and tested. At first we review the three concepts to deal with IRA: (i) DDD or triple difference observations are produced by a properly chosen difference operator and choice of basis, namely being free of integer-valued unknowns (ii) The real-valued unknown parameters are eliminated by a Gauss elimination step while the remaining integer-valued unknown parameters (initial cycle ambiguities) are determined by Quadratic Programming and (iii) a RA substitute model is firstly implemented (real-valued estimates of initial cycle ambiguities) and secondly a minimum distance map is designed which operates on the real-valued approximation of integers with respect to the integer data in a lattice. This is the place where the integer Gram-Schmidt orthogonalization by means of the LLL algorithm (modified LLL algorithm) is applied being illustrated by four examples. In particular, we prove that in general it is impossible to transform an oblique base of a lattice to an orthogonal base by Gram-Schmidt orthogonalization where its matrix enties are integer. The volume preserving Gram-Schmidt orthogonalization operator constraint to integer entries produces “almost orthogonal” bases which, in turn, can be used to produce the integer-valued unknown parameters (initial cycle ambiguities) from the LLL algorithm (modified LLL algorithm). Systematic errors generated by “almost orthogonal” lattice bases are quantified by A. K. Lenstra et al. (1982) as well as M. Pohst (1987). The solution point of Integer Least Squares generated by the LLL algorithm is = (L')⁻¹[L'◯] ∈ ℤ ^m where L is the lower triangular Gram-Schmidt matrix rounded to nearest integers, [L], and = [L'◯] are the nearest integers of L'◯, ◯ being the real valued approximation of z ∈ ℤ ^m , the m-dimensional lattice space Λ. Indeed due to “almost orthogonality” of the integer Gram-Schmidt procedure, the solution point is only suboptimal, only close to “least squares.” ? 2000 John Wiley & Sons, Inc.     

A MACHINE METHOD OF RESECTION

Abstract

With the modern calculating machine in easy reach of every computer, the problem of determining the position of an occupied point from which direction observations have been made to three or more known points has become quite simple. The method outlined below is quite elegant in form and exceedingly simple on the machine. Let A, B, C be the three points whose co-ordinates (X₁Y₁), (X₂Y₂), (X₃Y₃) are known, and let (XY) be the co-ordinates of the point P which we wish to fix.     

On the night errors in astronomical determinations

Summary The discrepancy between precision and accuracy in astronomical determinations is usually explained in two ways: on the one hand by ostensible large refraction anomalies and on the other hand by variable instrumental errors which are systematic over a certain interval of time and which are mainly influenced by temperature.In view of the research of several other persons and the author’s own investigations, the authors are of the opinion that the large night-errors of astronomical determinations are caused by variable, systematic instrumental errors dependent on temperature. The influence of refraction anomalies is estimated to be smaller than 0″.1 for most of the field stations. The possibility of determining the anomalous refraction from the observations by the programme given by Prof. Pavlov and Anderson has also been investigated. The precision of the determination of the anomalous refraction is good as long as no other systematic error working in a similar way is present.The results, which are interpreted as an effect of the anomalous refraction by Pavlov and Sergijenko, could also be interpreted as a systematic instrumental error. It is furthermore maintained thatthe latitude and longitude of a field station can be determined in a few hours of one night if the premisses given in [3, p.68]are kept. It has been deplored that the determination of the azimuth has not been given the necessary attention. It is therefore proposed to intensify the research on this problem. The profession has been called upon to acquaint itself better with the valuable possibilities of astronomical determinations and to apply them in a useful and appropriate manner. At the same time, attention has been called to the possibility of improving astronomical determinations with regard to accuracy as well as effectiveness.     

On non-linear low–low SST observation equations for the determination of the geopotential based on an analytical solution

In satellite data analysis, one big advantage of analytical orbit integration, which cannot be overestimated, is missed in the numerical integration approach: spectral analysis or the lumped coefficient concept may be used not only to design efficient algorithms but overall for much better insight into the force-field determination problem. The lumped coefficient concept, considered from a practical point of view, consists of the separation of the observation equation matrix A=BT into the product of two matrices. The matrix T is a very sparse matrix separating into small block-diagonal matrices connecting the harmonic coefficients with the lumped coefficients. The lumped coefficients are nothing other than the amplitudes of trigonometric functions depending on three angular orbital variables; therefore, the matrix N=B ^T B will become for a sufficient length of a data set a diagonal dominant matrix, in the case of an unlimited data string length a strictly diagonal one. Using an analytical solution of high order, the non-linear observation equations for low–low SST range data can be transformed into a form to allow the application of the lumped concept. They are presented here for a second-order solution together with an outline of how to proceed with data analysis in the spectral domain in such a case. The dynamic model presented here provides not only a practical algorithm for the parameter determination but also a simple method for an investigation of some fundamental questions, such as the determination of the range of the subset of geopotential coefficients which can be properly determined by means of SST techniques or the definition of an optimal orbital configuration for particular SST missions. Numerical results have already been obtained and will be published elsewhere. Received: 15 January 1999 / Accepted: 30 November 1999     

REPRODUCTION OF THE 1/1,250 AND 1/2,500 SCALE NATIONAL PLANS,OF GREAT BRITAIN

Abstract

Field work for the 1/1,250 scale re-survey of Great Britain was fully described in an article by Brigadier R. P.Wheeler in the April, 1948, issue of this Review (ix, 68, 234–247). The object of this article is to outline the method’ of reproduction of these plans and of the resultant 1/2,500 scale plans of urban areas. The 1/2,500 series covering rural areas is a separate problem, one of revision rather than re-survey. Experiments are in hand now to find out the best ways to provide field material and produce the final plans on National Grid sheet lines. The 1/1,250 scale series will contain about forty thousand plans and the 1/2,500 series of the same areas about nine thousand. It is therefore important that production methods should be straightforward and maintain an economical balance between the use of men and machines.     

CONDITION FOR THE CONSTRUCTION OF A CONFORMAL PROJECTION

Abstract

In the Empire Survey Review for October 1938 (iv, 30, 480) a simple demonstration of the condition to be satisfied for conformal representation was given. This condition may be expressed by the equation w = f(z), where w and z are complex variables representing corresponding points in the w-plane and z-plane respectively, and f(z) is an analytic function of z.     

Books

Abstract

This is a summary of the problems which are involved in the apparently trivial task of measuring the length of a sinuous line on a map. It represents an extended review of the publication Cartometric Measurements, by H. Kishimoto. It is concerned with three basic problems: (1) the sorts of errors which may result from using different instruments and methods of measurement and how these may be corrected: (2) the sorts of errors which may occur in the map and how these may be corrected: 3) the fundamental problem of what is 'length'. Extensive use is made of East European literature on these subjects.     

Ability to detect and locate gross errors on DEM matching algorithm

Abstract

Digital elevation model (DEM) matching techniques have been extended to DEM deformation detection by substituting a robust estimator for the least squares estimator, in which terrain changes are treated as gross errors. However, all existing methods only emphasise their deformation detecting ability, and neglect another important aspect: only when the gross error can be detected and located, can this system be useful. This paper employs the gross error judgement matrix as a tool to make an in-depth analysis of this problem. The theoretical analyses and experimental results show that observations in the DEM matching algorithm in real applications have the ability to detect and locate gross errors. Therefore, treating the terrain changes as gross errors is theoretically feasible, allowing real DEM deformations to be detected by employing a surface matching technique.     

THE STEREOGRAPHIC SOLUTION OF THE SPHERICAL TRIANGLE

Abstract

In the course of his stimulating and suggestive paper in your recent issue, No. ro, pp. 226–38, Mr. A. J. Potter writes on p. 233 “but there is no simple construction by which X can then be found”, and again on p. 237 “a direct construction, if there be such”. This cheerful challenge invites the construction of a circle centred on a given line, passing through a given point thereon, and touching a given circle, and I have found the lure of Mr. Potter's gauntlet as irresistible as its recovery has proved delicate. In order to shoulder responsibility and by no means to claim highly improbable originality, let me confess that the problem is new to me and the two constructions I offer are my own; I venture to hope that Mr. Potter may consider one or other of them not unworthy of his epithet “simple”, though I freely admit the aptitude of his empiric procedure to its purpose. The proofs are not long, but for fear of overshooting my welcome I offer them to anyone for the asking; and for the same reason my diagrams are small and therefore mere.     

Sonifying data uncertainty with sound dimensions

ABSTRACT

The communication of data uncertainty is a crucial problem in data science, information visualization, and geographic information science (GIScience). Effective ways to communicate the uncertainty of data enables data consumers to interpret the data as intended by the producer, reducing the possibilities of misinterpretation. In this article, we report on an empirical investigation of how sound can be used to convey information about data uncertainty in an intuitive way. To answer the research question How intuitive are sound dimensions to communicate uncertainty? we carry out a cognitive experiment, where participants were asked to interpret the certainty/uncertainty level in two sounds A and B (N = 33). We produce sound stimuli by varying sound dimensions, including loudness, duration, location, pitch, register, attack, decay, rate of change, noise, timbre, clarity, order, and harmony. In the stimuli, both synthetic and natural sounds are used to allow comparison. The experiment results identify three sound dimensions (loudness, order, and clarity) as significantly more intuitive to communicate uncertainty, providing guidelines for sonification and information visualization practitioners.     

MAP MAINTENANCE,POLICY AND TECHNIQUE

Abstract

Introduction.—Although much has been written on the various branches of surveying, few text-books do more than give the problem of Map Maintenance a passing mention. It is true that probably the major problem of the surveyor, when considering the world as a whole, is the urgent need of any sort of accurate map for most of its surface, yet at the same time quite considerable areas have already been surveyed, and the maintenance of these areas forms a very formidable problem in itself. Moreover, it is not generally realized that problems of maintenance necessitate a complete reorientation of technical outlook in comparison with questions of an initial survey. Clearly, when maps are being made in the first instance, the primary requirement is an adequate framework, whose main essentials are accuracy; rigidity and permanence. The filling in of the topographical detail and the final reproduction are matters of necessarily secondary importance. Furthermore, as the time required for the field work is large in comparison with the office stages, there is little actual or apparent gain in reducing the time taken on drawing and reproduction. In these circumstances the problem of an initial survey is essentially mathematical in its technique, little attention being required for the final stages of drawing and reproduction, which can be carried out, by craftsmen, at the unhurried pace which it has been their privilege to maintain in the past.     

CORRECTION TO BEARING FOR HEIGHT OF SIGNAL

Abstract

When a beacon B _h stands on a mountain of height h, the bearing of B _h as seen from another station A is in general affected by its elevation. The correction never exceeds one second of arc, but in primary triangulation it is not always negligible.     

INTERPOLATING FROM TRAVERSE TABLES TO SECONDS

Abstract

In the Empire Survey Review, no. 4, 1932, Mr. Clendinning has described a method of interpolating from traverse tables to seconds. Below is another method, due to Prof. Nekrassov, for use with traverse tables published by him. The method is described in The Geodezist, Moscow, 1936, no. I, pp. 47–52.