TRAVERSE ADJUSTMENT: WHY NOT LEAST SQUARES?

Abstract

In the above article by Mr H. L. P. Jolly published in a previous issue (E.S.R., vol. iv, no. 28) the author, after referring to the precision of the Nigerian traverses, makes the statement that measurements of the highest accuracy are worthy of the best possible methods of adjustment. But this argument cuts both ways. For in general the greater the accuracy of measurement the smaller will be the ultimate misclosure to be eliminated; so that different methods of adjustment will produce smaller and smaller variations in the corrections, until in the limit when there is no error we should obtain the same result however much latitude we permitted in the adjustment.     

MARGINAL SCALES OF LATITUDE AND LONGITUDE ON TRANSVERSE MERCATOR MAPS

Abstract

The problem of computing marginal scales of latitude and longitude on a rectangular map on the Transverse Mercator projection, where the sheet boundaries are projection co-ordinate lines, may be solved in various ways. A simple method is to compute the latitudes and longitudes of the four corners of the sheet, and then, assuming a constant scale, to interpolate the parallels and meridians between these corner values. Although it is probably sufficiently accurate for practical purposes, this method is not precise. It is not difficult to adapt the fundamental formulce of the projection to give a direct solution of the problem.     

LEAST-SQUARE ADJUSTMENTS OF TRIANGULATION DIRECTIONS VERSUS ANGLES

Abstract

IT has been assumed in the past that because angles for triangulation are usually observed by the direction method, therefore it must be more correct theoretically to perform the least-square adjustments by directions rather than by angles. It is fairly obvious that an adjustment of the same figure by directions will not give the same result as an adjustment by angles: the unknowns in each case are different and the number of directions is usually about 25 per cent. greater than the number of angles for the same figure. Strictly, the least square method is only applicable to observations from which all systematic errors have been eliminated, and in which the remaining errors are truly accidental. It is generally safe to assume that most survey errOlS consist of a random and a systematic part. Rarely, if ever, is it possible to state that all systematic error has been eliminated, strive how we may to take all precautions against it.     

ADJUSTMENT OF SIMPLE FIGURES—II

Abstract

3. Adjustment of a Polygon.—The adjustment of a polygon by the method of the preceding paragraph leads to results similar to that for a quadrilateral, although not quite so simple, since there are two correlatives, k₁ k₂ , for deducing the corrections.     

POINT TO POINT COMPUTATION OF GEOGRAPHICAL COORDINATES OF LONG LINES

Abstract

The computation of geographical coordinates in a geodetic triangulation is usually carried out using Puissant's method, in which the assumption is made the sphere radius ν (the radius of curvature of the spheroid perpendicular to the meridian) not only touches the spheroid along the whole small circle of latitude ?,but also, since ρ (the radius of curvature in meridian) is very nearly equal to ν it makes such close contact with the spheroid that the lengths of sides and angles of a geodetic triangle may be considered identical on both sphere and spheroid.     

DETERMINATION OF AZIMUTH AND LATITUDE FROM OBSERVATIONS OF A SINGLE UNKNOWN STAR BY A NEW METHOD

Abstract

The purpose of this paper is to describe a new and easy method of determining the (astronomical) latitude and azimuth at any place and to explain the line of approach and the formulae. It will be seen that the method should be useful to a wide circle of land surveyors. One of its principal advantages is that identification of the star is not necesssary and it can be used when no star chart or star catalogue is available.     

ON THE ORTHOMORPHIC PROPERTY OF SCHOLS ADJUSTMENT

Abstract

In vol. iv, nos. 29 and 30, of the E.S.R., there appeared an article by Mr. D. R. Hendrikz on the “Adjustment of the Secondary Triangulation of South Africa”. He shows that, in applying the Schols method of orthomorphic transmission to the adjustment of a secondary net to a primary triangle, the secondary sides suffer small displacements.     

Fast spherical collocation: theory and examples

 It has long been known that a spherical harmonic analysis of gridded (and noisy) data on a sphere (with uniform error for a fixed latitude) gives rise to simple systems of equations. This idea has been generalized for the method of least-squares collocation, when using an isotropic covariance function or reproducing kernel. The data only need to be at the same altitude and of the same kind for each latitude. This permits, for example, the combination of gravity data at the surface of the Earth and data at satellite altitude, when the orbit is circular. Suppose that data are associated with the points of a grid with N values in latitude and M values in longitude. The latitudes do not need to be spaced uniformly. Also suppose that it is required to determine the spherical harmonic coefficients to a maximal degree and order K. Then the method will require that we solve K systems of equations each having a symmetric positive definite matrix of only N × N. Results of simulation studies using the method are described. Received: 18 October 2001 / Accepted: 4 October 2002 Correspondence to: F. Sansò     

LEAST SQUARES ADJUSTMENTS OF TRIANGULATION

Abstract

Mr. Rainsford's paper in E.S.R. No. 78 establishes as thoroughly as one can wish by computational example that there is little to choose in relative accuracy between direction adjustment and angle adjustment, and that the deciding factor is really ease and speed of computation.     

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.     

ON THE ELIMINATION OF THE ERRORS OF PAIRING AZIMUTH OBSERVATIONS

Abstract

The perfect pairing of east and west observations for azimuth, put briefly, should consist of combinations of simultaneous observations of stars of the same altitude and of the same declination. Needless to say, this is a counsel of perfection, and in practice the surveyor has usually to rely on some sort of an approximation to this ideal. It is only on very rare occasions that the necessary time is available and the atmospheric conditions favourable enough to obtain this perfect harmony of observations. Assuming that the latitude of the azimuth station and the atmospheric refraction are accurately known, the necessity of pairing would not arise, and the grouping of the observations into two sets of east and west stars with a varying discrepancy between the members of the individual pairs would be quite unnecessary. In general one might say, when an azimuth observation is taken, neither the latitude nor the atmospheric refraction is known accurately, and the question arises as to whether there is any simple method of eliminating or reducing these two causes of error.     

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.     

LEAST SQUARES ADJUSTMENTS OF TRIANGULATION: DIRECTIONS VERSUS ANGLES

Abstract

This paper continues the discussion started in an article of the same title (E.S.R., ×, 78, :353-66), on which a further letter was written in October, 1952. The amount of computation required originally was very considerable, and it was obviously impossible to publish it all. The recent letter was necessary to answer the suggestion that the agreement between errors put in and corrections obtained from the L.S. solution was not very close. It seemed sufficient to give the list of errors and corrections, leaving readers to judge for themselves. The correlation coefficient from the two sets of figures was 0.78, which looked quiteg90d. Unfortunately, it was not realized before that corrections from a L.S. solution cannot, legitimately, be compared with errors put in on directions unless a station correction is first applied to the errors to make the sum of the errors at each station equal to zero. This is one of the points about the direction method of adjustment which is not very easy to understand.     

A NOTE ON THE TRIGONOMETRICAL SURVEY OF SOUTHERN RHODESIA

Abstract

After the completion of Simms's Geodetic Chain in 1901 and the publication of the results in 1905—Volume iii of the Geodetic Survey of South Africa—nothing further of a geodetic nature was done until 1928 when a short chain was run westwards from Simms's chain, at about latitude 17° 10′, to fix the Copper Queen mining area. The Eastern Circuit was commenced shortly after this; it runs from Salisbury eastwards to the Portuguese Boundary, southwards through Umtali to about latitude 20° and then westwards, joining Simms's chain again to the east of Bulawayo. Another chain running north from Simms's work has been commenced near Bulawayo. The several series are exhibited on the outline map attached.     

COMPUTATION OF DISTANCES OF LONG ARCS FOR RADIO PURPOSES

Abstract

In the E.S.R., No. 59, Vol. VIII, of January, 1946, I gave a formula which I had worked out to give a rapid and easy means of computing the lengths of long arcs, up to 1000 kilometres, between two points whose latitude and longitudes are known on a definite figure of the earth.     

THE FIXATION OF MINOR TRIANGULATION ON THE TRANSVERSE MERCATOR PROJECTION

Abstract

The fixation of Minor Triangulation in a Primary system does not, in general, warrant rigorous adjustments of figures; less laborious methods are desirable. For Secondary work a least square adjustment to approximate coordinates is quite sufficient, while, for Tertiary, graphical solutions are amply accurate. Apart from that, cases may arise to which a figure adjustment is not applicable, as in the small net shown in Fig. 2, p. 464. The line BC cannot be equated to the line AB in the ordinary way since it is not the side of a triangle. In this case an adjustment to approxima te coordina tes will overcome the difficulty.     

THE METHOD OF SYSTEMATIC RELAXATION APPLIED TO SURVEY PROBLEMS

Abstract

This paper reviews and elaborates the application of the “Method of Systematic Relaxation of Constraints”, originally devised by R. V. Southwell for the calculation of the forces in frameworks, to the problem of the adjustment of survey networks to conform to the condition of least squares. The method is illustrated by the solution of a simple example of a triangulation network, and its application to larger surveys is discussed.     

THE DERIVATION OF CERTAIN PROJECTION FORMULAE FOR THE SPHERE

Abstract

The purpose of this article is to make available to readers of the Review a simple method of derivation from first principles of the projection farmulae for same of the more impartant normal conical projections.     

THE TRANSVERSE MERCATOR PROJECTION

Abstract

I. Introduction.—Map projection is a branch of applied mathematics which owes much to J. H. Lambert (v. this Review, i, 2, 91). In his “Beyträge zum Gebrauche der Mathematik und deren Anwendung” (Berlin, 1772) he arrived at a form of projection whereof the Transverse Mercator is a special case, and pointed out that this special case is adapted to a country of great extent in latitude but of small longitudinal width. Germain (“Traité des Projections”, Paris, 1865) described it as the Projection cylindrique orthomorphe de Lambert, but he also introduced the name Projection de Mercator transverse or renversée; he shows that Lambert's treatment of the projection was remarkably simple.     

CONNECTION OF INDIA,SIAM AND MALAYA TRIANGULATIONS

Abstract

In an article in the Review of October 1938, iv, 30,450-457, under the heading “Geographical Positions in Malaya and Siam”, Mr. A. G. Bazley gives a comparison of the Indian and Siamese, and Siamese and Malaya, triangulations at common points and discusses the possibility of an error in the longitude of the datum of the Malayan system. In the Review of April, 1939, v, 32, 112-113, he has elaborated certain points, and remarks in connection with the doubt in the longitude of the Malayan datum that connection of the F.M.S. network with that of Siam and India, and some more latitude and longitude observations by the F.M.S. Survey, are essential to a satisfactory solution of this rather involved problem. Since the above article was written, a lot more infornlation has become available about the Indo-Siamese triangulation connections and a firm connection between the triangulations of Sianl and Malaya has been established in 1946. It is hoped that a review of the present position would be of interest, especially as the various links effected open up a definite possibility of a continuous chain of triangulation from India to Australia.