On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms

The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−E _Y = (X−E _X ) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, E _Y and E _X . Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition. For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion matrix of the estimated parameters.     

THE APPLICATION OF TRANSFORMATION FACTORS TO THE ADJUSTMENT OF AIR PHOTOGRAPHS COVERING NARROW LAND AREAS AND COASTLINES

Abstract

1. This new method of adjusting air photographs numerically to a uniform scale on a final map grid has already been described by the writert, where it concerns a large block-adjustment. The method to be described in this paper is the application of the same principle to the adjustment of photographic strips, covering narrow land areas and coastlines.     

Maps > Representation

Abstract

Colour is considered a key means of expression for use in cartographic works. This is because colours and the relations among them influence not only the aesthetic impression a map creates but also its overall utility. In addition to Newton’s spectral colour theory, today theories with origins in artistic technique are gaining ground in cartography. This article introduces J. Itten’s colour theory (first published in 1961 in The Art of Colour [Kunst der Farbe]) with special attention given to his concept of seven colour contrasts. The article also discusses the suitability and unsuitability of their application in practical cartography, and it contributes original examples employing thematic maps, a discipline with broad possibilities for the application of these inventive methods by today’s mapmakers.     

THE PENGGAWAS' ROADS OF KELANTAN

Abstract

The survey of roads in the densely settled areas of the Kelantan plain is a problem the solution of which is interesting, more because it illustrates one of the facets of the surveyor's life than because it is likely to have any general application.     

A NOTE ON CLARKE'S FORMULAE

Abstract

An elegant proof of Clarke's Formulae for lines of medium length has been given by G. T. McCaw. It is based on the properties of polar triangles in spherical trigonometry, and on the application of Legendre's Theorem.     

THE CONFORMAL TRANSFORMATION OF A NETWORK OF TRIANGULATION

Abstract

In a recent issue of this Review, an example is given of the conformal transformation of a network of triangulation using Newton's interpolation formula with divided differences. While the application of the method appears to be new, attention should be drawn to the fact that Kruger employed Lagrange's interpolation formula in a discussion and extension of the Schols method in a paper which was published in the Zeitschrift für Vermessungswesen in 1896. A reference to this paper was given at the end of the paper, “Adjustment of the Secondary Triangulation of South Africa”, published in a previous issue of the E.S.R. (iv, 30, 480).     

THE DEFLECTION PRISM

Abstract

The optical principle of the deflection prism may not be known to all readers. Though there is nothing of very modern origin in the optics of a prism, as a wedge of very acute angle it possesses special powers which admit of useful adaptation to certain types of instrument. This application is a development of comparatively recent times. It will not be out of place, therefore, to make some reference to it in this Review.     

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     

RELAXATION AND THE CO-ORDINATE METHOD OF TRIANGULATION ADJUSTMENT

Abstract

Few, most certainly, will dispute the value of Mr Black's paper describing a method of “Systematic Relaxation”, which appeared in a previous number of this Review. At the same time, however, it seems to the writer to be only fair to readers to point out that the application of the method to triangulation adjustment is really a treatment, from a slightly different aspect, of methods that have long been established.     

THE SCALE ERRORS OF AIRY'S PROJECTION BY BALANCE OF ERRORS,WITH SPECIAL REFERENCE TO THE BRITISH ISLES

Abstract

THE formula for the projection is based upon the spherical assumption. To calculate it for the spheroid might be very complicated and would not be worth while. The projection is suitable for very large areas as a compromise between the Zenithal Equal-area projection on the one hand and the Zenithal Equidistant or Zenithal Orthomorphic on the other. Its application to an area as small as the British Isles would not serve any useful purpose. An analysis of its errors in the general case reveals some unexpected simplicities. This analysis is given below, followed by its application to the particular case of the British Isles on the ten-mile scale. This is done merely to find out what changes would have occurred if the supposed drawing of that map on Airy's projection had been real.     

Scree Representation on Topographic Maps

Abstract

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Scree patterns are an important element of mountain maps in Swiss style. The size and density of scree dots vary with the exposition towards a source of illumination, which makes the dots extremely labour intensive to map without specialized algorithms. This paper identifies design principles for the symbolisation of scree fields on mountain slopes and presents a digital method for the quick placement of dot symbols requiring only minimal interventions by a cartographer. When digitally produced scree is combined with a shaded relief and a rock drawing, the terrain appears as a continuous three-dimensional surface to the reader. The described method is implemented in Scree Painter, a specialized free open-source software application. Scree patterns produced with Scree Painter match the quality standards of manually generated scree representations.     

The Principles of Selection

Abstract

The fundamental of cartographic generalisation, the reduction of the amount of information which can be shown on a map in relation to reduction in scale, is examined. The Principle of Selection was first proposed by Töpfer in 1961, and is expressed as an equation relating the number of occurrences of a particular feature at source map scale and at derived map scale. The application of this to small-scale maps involves the introduction of two constants to represent symbolic exaggeration and symbolic form. Examples are given and illustrated. The paper is accompanied by explanatory notes by Dr D. H. Maling, who presented the paper on behalf of the authors at the XX International Geographical Congress London 1964, Section IX, Cartography.     

A high-level architecture for a Geomatics Informatization Technology System: the Chinese case

Abstract

China will, as a component of ‘Digital Earth,’ establish a Geomatics Informatization Technology System (GITS) which is characterized by real-time acquisition, automatic processing, networking service, and socialized application with fundamental geographical information. The basic composition of GITS is proposed. GITS covers four layers and six platforms. The four layers are data acquisition, processing, management, and application and services. The six platforms are informatic geodetic datum, high-precision geo-spatial data acquisition, automatic geo-spatial data processing, grid-based geo-spatial information management, comprehensive geo-spatial information sharing and service, and geo-spatial information integration and application. The informatic geodetic datum platform provides a carrier for all the four layers and a base for the other five platforms. The high-precision geo-spatial data acquisition platform belongs to the acquisition layer. The automatic geo-spatial data processing platform belongs to the processing layer. The grid-based geo-spatial information management platform belongs to the management layer and is a bridge connecting geo-spatial data acquisition and spatial information sharing service and integrated applications. The comprehensive geo-spatial information sharing and service platform belongs to the application and service layer. Finally, this paper presents thoughts for constructing GITS.     

Population Enumeration on a Grid Square Basis: The Census of Scotland,a Test Case

Abstract

After examination of the existing Census organisation, the problems of making full use of the data are investigated, in particular the disparity of units on the ground, changes over time, and variations in the nature of the published information. Proposals are made and explained for the adoption of a grid square enumeration system, which in this case is tested by application to an urban and a rural area. The problem of conflict between grid square and local authority boundary is examined. The more general usefulness of a grid square enumeration system is demonstrated in relation to comparability of data, mechanisation of processing and graphic output, and the data bank concept. Further testing by means of a sample census is being currently undertaken.     

CORRECTION OF PLANE-TABLE HEIGHTS FOR CURVATURE OF THE EARTH AND ATMOSPHERIC REFRACTION

Abstract

Most text-books on surveying limit their discussion of the correction of vertical angles for curvature of the earth and atmospheric refraction to the correction of angles taken with a theodolite during triangulation and omit any reference to those taken with a clinometer. This is rather illogical, as in well-observed triangulation, with all vertical angles measured in both directions, no correction for these effects is necessary, whilst in plane-tabling on small scales where sketching at considerable distances is frequently employed the application of corrections for these effects is essential.     

A generalisation of the least-squares method for the adjustment of free networks

Summary The system of normal equations for the adjustment of a free network is a singular one. Therefore, a number of coordinates has to be fixed according to the matrix. The mean square errors and the error ellipses of such an adjustment are dependent on this choice. This paper gives a simple, direct method for the adjustment of free networks, where no coordinates need to be fixed. This is done by minimizing not only the sum of the squares of the weighted errorsV ^T PV=minimun but also the Euclidean norm of the vectorX and of the covariance matrixQ X ^T X=minimum trace (Q)=minimum This last condition is crucial for geodetic problems of this type.     

AIR SURVEY: THE RAND METHOD OF HEIGHTING FROM NEAR VERTICAL PHOTOGRAPHS

Abstract

The method outlined below is a simple and rapid method, requiring no expensive equipment, of obtaining true heights from pairs of near vertical air photographs with a minimum of ground control. It is a direct application of the Fourcade theorem and, as it was finally developed at the University of the Witwatersrand in 1948-49, it has been given the name “Rand method” in the hope that this will connect it with Dr. Fourcade's country.     

THE DETERMINANTAL METHOD OF SOLVING NORMAL EQUATIONS

Abstract

In the treatises on the method of least squares published so far in the English language the application and comparison of different methods of solving normal equations has received but slight notice.