PRECISION OF TRIANGULATION IN RELATION TO LENGTH OF SIDE

Abstract

Given a straight section of triangulation comprising a fairly large number of equilateral triangles, then, if the length of the section is held fixed but the size of the triangles is made to vary, the total displacement of the section is proportional to the root mean square error of the angular observations divided by the square-root of the length of a side of a triangle, Provided there is no pronounced antagonism between the triangular and length misclosures it will be sufficient to substitute the triangular misclosure ? for the root mean square error in the above statement.     

TWO SIMPLIFIED CHECKS

Abstract

The usual method employed is to plot or to compute the traverse from each end; the poin t having the same coordinates in each route is the station where the gross angular error occurred. There is, however, a method of finding the error by plotting the traverse one way only. Let us consider the traverse having the known terminals A B (see Fig.). Suppose that the error occurred at the point P and that the final point obtained (plotting the traverse from A) was B′ in place of the correct point B. We can easily see that the triangle PBB′is isosceles, and that therefore a straight line bisecting BB′at right angles will meet the traverse in the required point P.     

THE ORTHOMORPHIC PROJECTION OF THE SPHEROID—IV

Abstract

We now turn to a question which has received much attention of recent years; the possibility of transforming angular and linear field measures to an orthomorphic projection so that the results of a survey may be computed directly in plane Co-ordinates without having to go through the spheroid at all. Initially, orthomorphic projections were introduced into surveying practice for this very object. Over short lines they import so little distortion of angles that minor surveys, whose error of angular measurement is comparable with such distortion, may be reduced in the rectangular co-ordinate system of an orthomorphic projection just as if the earth were flat. But the present application goes far beyond that. We no longer ignore distortions of angles and lengths, but systematically introduce them into the field measures so that work of higher precision and of considerable extent may also be computed and adjusted in plane co-ordinates.     

PROGRESS OF THE GEODETIC SURVEY OF NEWFOUNDLAND

Abstract

The third season (1937) of the Geodetic Survey of Newfoundland has now ended, the second in which angular measurements were made. Approximately half of the basic project has been finished, and two more seasons will leave little of it incomplete.     

Fehlertheoretische Maxwell-Boltzmann-Verteilung

Summary The probability to find an error vector in multiples of the Helmert-Maxwell-Boltzmann point error σ² δ_ij(δ_ij Kronecker symbol) is calculated. It is found that the probability is for σ39%, for2 σ86% and for3 σ99% in two dimensions, for σ20%, for2 σ74% and for3 σ97% in three dimensions. The fundamental Maxwell-Boltzmann-distribution is tabulated0,02 (0,02) 4,50.      

ALTITUDES OF POINTS SEEN ON THE HORIZONTAL WIRE OF A THEODOLITE

Abstract

Many surveyors, even those of wide experience, would, if questioned, assert that, at any given pointing of a theodolite, all points in the field of view which lie on the horizontal wire are at equal angular altitudes above the horizon, provided that the horizontal wire is in correct adjustment to the trunnion axis.     

COMPENSATION OF TRAVERSES WHEN LENGTHS ONLY ARE CORRECTED

Abstract

If it be assumed that the linear measurements of traversing are alone afflicted with error, the method of distribution of error demands consideration. In the present paper the notation of Wright and Hayford will be followed (“Adjustment of Observations”, 1906 edn., p. 156).     

The angular characteristics of Moon-based Earth observations

ABSTRACT

The Moon, Earth’s only natural satellite, is a potential new platform for Earth observation. Moreover, with the wide applicability of the angular information from remote sensing data, it has been attracting increasingly more attention. Accordingly, this study focuses on the angular characteristics of Moon-based Earth observations. Using ephemeris DE430 and Earth orientation parameters, the position and attitude of the Sun, Earth, and Moon were obtained and their coordinates normalized to a single framework using coordinate transformations between the related reference systems. Then, an angular geometric model of Moon-based Earth observations was constructed, and the corresponding angular algorithms were presented. The results revealed the angular range and distribution characteristics of Moon-based Earth observations. For every point on the surface of the Earth, the view and solar zenith angles all vary widely, which decreases with increasing latitude. The view and solar zenith angles all vary widely with the largest range of values in the equatorial and polar regions and a smaller range of values in mid-latitudes. Furthermore, the range of solar angles of Moon-based Earth observations is the same as that of all-time solar angles, indicating the potential for monitoring and understanding large-scale geoscientific phenomena using Moon-based Earth observations.     

SOME NOTES ON THE ADJUSTMENT OF TRAVERSES

Abstract

It has now become a truism that angles in traverses are observed with a far greater accuracy than the sides, and this applies to all categories of traverse work: precise, transit-and-tape, tacheometric traverses, etc. The result of this is that the effect of angular errors on misclosures in co-ordinates is sometimes so small when compared with that of linear errors that it may be considered as negligible.     

THE RE-TRIANGULATION OF GREAT BRITAIN IV—BASE MEASUREMENT

Abstract

Accuracy and consistency in surveying are almost synonymous. It is true that no very great degree of precision in angular or linear measurement can be utilized in our final paper publications, even on the largest scales: all we need is an assurance that the results of further work, in the same or in adjoining areas, shall be in sympathy with antecedent measurement, so as to avoid the necessity for assigning several positions to the same point.     

THE NEW GEODETIC TAVISTOCK THEODOLITE

Abstract

Work on the original Geodetic Tavistock Theodolite was commenced in the autumn of 1931, and after suitable tests this instrument was sent out to East Africa and used on the East African Arc. Bt Major M. Hotine, R.E., writing in the E.S.R. of April 1935 (no. 16, vol. iii), stated: “The Tavistock instrument, although a first model, gave uniformly satisfactory service throughout and was used for over half the main angular observations.”     

ANGULAR CORRECTIONS FOR THE LAMBERT ORTHOMORPHIC CONICAL PROJECTION

Abstract

In E.S.R., viii, 56, 70, Brigadier K. M. Papworth has given expressions for the angular corrections, known as (t – t) corrections, in the Lambert NO.2 Projection, derived from empirical considerations based on actual detailed calculations. Apparently some difficulty has been experienced in offering a proof. In view of the widespread use of the Lambert Projection in World War II, it is hoped that the following proof will be found to be of more than academic interest.     

FURTHER NOTES ON FIGURES OF REFERENCE

Abstract

Major Hotine (E.S.R., No. II, pp. 264–8) still finds the location of a reference spheroid to offer insuperable difficulties. I confess that my difficulty is to see his! In my previous article (E.S.R., No. 8) at the foot of page 76, I used the word “coincidence” in error for “parallelism”. This harmonizes the article and I am glad that Major Hotine has directed attention to the error.     

A statistical filter for geodetic observations

A method for filtering of geodetic observationwhich leaves the final result normally distributed, is presented. Furthermore, it is shown that if you sacrifice100.a% of all the observations you may be (1−β).100% sure that a gross error of the size Δ is rejected. Another and, may be intuitively, more appealing method is presented; the two methods are compared and it is shown why Method 1 should be preferred to Method 2 for geodetic purposes. Finally the two methods are demonstrated in some numerical examples.     

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.     

Truncation error formulas for the geoidal height and the deflection of the vertical

A general formula giving Molodenskii coefficientsQ _n of the truncation errors for the geoidal height is introduced in this paper. A relation betweenQ _n andq _n, Cook’s truncation function, is also obtained. Cook (1951) has treated the truncation errors for the deflection of the vertical in the Vening Meinesz integration. Molodenskii et al. (1962) have also derived the truncation error formulas for the deflection of the vertical. It is proved in this paper that these two formulas are equivalent.     

LENGTH AND AZIMUTH OF LONG LINES ON THE EARTH

Abstract

In the E.S.R., viii, 59, 191–194 (January 1946), J.H. Cole gives a very simple formula for finding the length of long lines on the spheroid (normal section arcs), given the coordinates of the end points. In the course of the computation the approximate azimuth of one end of the line is found, the error over a 500-mile line being of the order of 3″ or 4″. If the formula is amended so that the azimuth at the other end of the line is used in computing the length of the arc, the error is then less than 0″·1 over such a distance. An extra term is now given which makes this azimuth virtually correct over any distance. Numerical tests show that Cole's formula for length and the new one for azimuth are very accurate and convenient in all azimuths and latitudes.     

CALIBRATION OF PRISMATIC COMPASSES

Abstract

Apart from “stickiness” of the suspension and looseness of the sights, prismatic compasses are subject to three internal sources of error:- <list list-type="roman-lower"> <list-item>

Collimation error. This may be caused by <list list-type="alpha-lower"> <list-item>

magnetic axis not being parallel to the zero line of the graduated circle;</list-item> <list-item>

line of sight not passing through the axis of rotation.</list-item> </list> It is unnecessary to aftempt to distinguish between the above faults, which introduce constant errors into the compass readings.</list-item> <list-item>

Eccentricity error. This is caused by the axis of rotation failing to pass through the centre of the graduated circle. This introduces an error into the compass readings of E sin θ cosec I°/R where E is the eccentricity, R the radius of the graduated circle and θ the angle between the line of sight and the line joining the centre of the circle to the axis of rotation. Eccentricity error is completely eliminated by observing both forward and back bearings, but this is not always practicable.</list-item> <list-item>

Irregular division of the graduated circle. This error is negligible in any modern compass.</list-item> </list>     

NOTE ON THE EVANS RESECTION METHOD

Abstract

IN certain classes of work such as extended reconnaissance in new country frequent fixing of position by resection is a useful and economical method to use. The three-point resection, with safeguards against gross error, occurs frequently. Its main disadvantage is the repetition of a laborious computation.