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.     

SOME SURVEY ODDITIES FROM BRITISH NORTH BORNEO

Abstract

A Survey gang recently built a hardwood beacon on Bukit Toodan, 1,600 feet, overlooking the sleepy hollow of Kota Belud, and an observer on returning to the station a week later found that one of the four legs of the beacon had been torn to shreds by lightning and hurled many feet away, while the concrete slab marking the exact point was discovered to have been shattered to fragments. A week or two later an observer reported the Pulau Gaya hardwood trig. beacon to be lying at an angle from the perpendicular, and on observation with a powerful telescope it was disclosed that the whole beacon was covered with flying foxes roosting upside down, whose talons had in the course of time severed one of the beacon legs–rather a shock to any ordinary observer. But that is the kind of thing that surveyors in the East have to contend with quite frequently.     

AN ASPECT OF ATTRACTION

Abstract

Measured deviations of the vertical have been used in support, or in destruction, of such pleasant little diversions as the theory of isostasy. They have also been used to adjust a triang~lation for swing, by methods which may fairly be criticized; but they have not, as far as I know, been used for reducing the horizontal measures of a triangulation to the standard conventional level of the spheroid of reference. In most cases such corrections would, of course, be too small to worry about, but it by no means follows that they are always small. In the case of a continental arc of meridian traversing a very disturbed mountainous region exhibiting certain constant tendencies, it should at least be demonstrated that they are small before the question can be considered finally settled.     

EXAMPLES OF CURVATURE AND REFRACTION CORRECTIONS TO VERTICAL ANGLES

Abstract

The correction to observed vertical angles for curvature and refraction can be found by adding a log factor to the log distance, which gives the log of the correction in seconds. This factor is 8·144 for metres and 7·628 for feet. The examples will make this clear. For machine computation, the correction in seconds can be obtained by multiplying the length in metres by 0.0139 or the length in feet by 0·00425. Alternatively, this can be done on the slide. rule by dividing the distance in metres by 72 or in feet by 235. The mean coefficient of refraction is taken as 0·07.     

MEAN SQUARE ERRORS OF POINT DETERMINATIONS

Abstract

Considering first of all the general case of a triangle ABC with base AB and apex angle C, the assumption is made that at A and at B only one orienting ray has been observed at each point besides the forward rays to C and that at Conly B and A have been observed. In other words, the three angles of the triangle have been observed. The adoption of General Schreiber's rule with regard to the weights of forward and back directions is valid here, since it may easily be shown that, when the weight of a forward ray is one-half of the weight of a back ray, the adjusted angles of a triangle conform to the condition of least squares.     

THE GEODETIC LEVELLING OF CEYLON (1926–1929)

Abstract

The Net.—The total length of the lines of the level-net is roughly 2400 miles. The net comprises 27 circuits with perimeters varying between 74 and 268 miles, and is generally closer in the wet zone than in the sparsely populated and undeveloped dry zones. In 12 circuits there are differences of level exceeding 1000 feet. The highest point reached in the net is 6572 feet, and a branch line runs from Nuwara Eliya to the summit of Pidurutalagala, the highest mountain in the island (8282 feet).     

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.     

THE TRANSVERSE MERCATOR PROJECTION OF THE SPHEROID

Abstract

In January 1940, in a paper entitled “The Transverse Mercator Projection: A Critical Examination” (E.S.R., v, 35, 285), the late Captain G. T. McCaw obtained expressions for the co-ordinates of a point on the Transverse Mercator projection of the spheroid which appeared to cast suspicion on the results originally derived by Gauss. McCaw considered, in fact, that his expressions gave the true measures of the co-ordinates, and that the Gauss method contained some invalidity. He requested readers to report any flaw that might be discovered in his work, but apparently no such flaw had been detected at the time of his death. It can be shown, however, that the invalidities are in McCaw's methods, and there seems no reason for doubting the results derived by the Gauss method.     

Approximation Theory Applied to DEM Vertical Accuracy Assessment

Existing research on DEM vertical accuracy assessment uses mainly statistical methods, in particular variance and RMSE which are both based on the error propagation theory in statistics. This article demonstrates that error propagation theory is not applicable because the critical assumption behind it cannot be satisfied. In fact, the non‐random, non‐normal, and non‐stationary nature of DEM error makes it very challenging to apply statistical methods. This article presents approximation theory as a new methodology and illustrates its application to DEMs created by linear interpolation using contour lines as the source data. Applying approximation theory, a DEM's accuracy is determined by the largest error of any point (not samples) in the entire study area. The error at a point is bounded by max(|δ_node|+M₂h²/8) where |δ_node| is the error in the source data used to interpolate the point, M₂ is the maximum norm of the second‐order derivative which can be interpreted as curvature, and h is the length of the line on which linear interpolation is conducted. The article explains how to compute each term and illustrates how this new methodology based on approximation theory effectively facilitates DEM accuracy assessment and quality control.     

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.     

Preliminary results of the establishment of a marine geodetic point in the Pacific Ocean

In November 1968, a marine geodetic control point was established in the Pacific Ocean at a water depth of6,200 feet. The control point (reference point) consists of three underwater acoustic transponders, two of which are powered with lead-acid batteries and the third with an underwater radioisotope power source “URIPS” with a10- to20- year life expectancy. Four independent measuring techniques (LORAC airborne line-crossing, satellite, ship inertial, and acoustic techniques) were used to measure and determine the coordinates of the control point. Preliminary analysis of the acoustic and airborne data indicates that high accuracies can be achieved in the establishment of geodetic reference points at sea. Geodetic adjustment by the method of variation of coordinates yielded a standard point error of±50 to±66 feet in determining the unknown ship station. The original location of the ship station as determined by shipboard navigation equipment was off by about1,600 feet. Paper previously published in the Proceedings of the Second Marine Geodesy Symposium of the Marine Technology Society.     

THE NOMENCLATURE OF MAP PROJECTIONS

Abstract

Map Projections.—A matter that should have been mentioned in the original article under this title (E.S.R., vii, 51, 190) is the definition of a map projection. In the list of carefully worded “Definitions of Terms used in Surveying and Mapping” prepared by the American Society of Photogrammetry (Photogrammetrie Engineering, vol. 8,1942, pp. 247–283), a map projection is defined as “a systematic drawing of lines on a plane surface to represent the parallels of latitude and the meridians of longitude of the earth or a section of the earth”, and most other published works in which a definition appears employ a somewhat similar wording. This, however, is an unnecessary limitation of the term. Many projections are (and all projections can be) plotted from rectangular grid co-ordinates, and meridians and parallels need not be drawn at all; but a map is still on a projection even when a graticule is not shown. Objection could be raised also to the limitation to “plane surface”, since we may speak of the projection of the spheroid upon a sphere, or of the sphere upon a hemisphere. Hence, it is suggested that “any systematic method of representing the whole or a part of the curved surface of the Earth upon another (usually plane) surface” is an adequate definition of a map projection.     

THE TRANSVERSE MERCATOR PROJECTION: A CRITICAL EXAMINATION

Abstract

In January 1938 the writer decided against holding up for more years some work on the Transverse Mercator Projection (E.S.R., 27, 275). The extension to the spheroid was not then complete, nor is the present paper to be regarded as a logical continuance. It is first proposed to show the results of “transplanting” orthomorphically upon the spheroid a spherical configuration forming a graticule.     

A NOVEL METHOD OF MEASURING THE COEFFICIENT OF THERMAL EXPANSION OF INVAR

Abstract

In a letter published in a recent issue of Nature, Prof. L. F. Bates and Mr J. C. Wilson, of University College, Nottingham, have described a new and novel method of determining the coefficient of thermal expansion of invar. Although this method is hardly likely to be applied to the measurement of the coefficient of expansion of long invar tapes, such as are used by surveyors, yet it is so novel and ingenious in itself that a short reference to it may not be out of place in this Review. One extremely interesting thing about it is that no measurements of a length, or of changes of length, are involved.     

THE CALCULATION OF THE HEIGHT OF KILIMANJARO

Abstract

The observations to height Kilimanjaro were made from two ground stations, Domberg (5,081·6 ft.) and Lelatema (5,323.1 ft.) and from a point called Kibo near Kaiser Wilhelm Spitze which is regarded as the highest point on the crater rim. It was originally intended to include a third ground station, Kifaru, but it was discovered that the ice cap obstructed observations between this point and the top.     

GRID BEARINGS AND DISTANCES ON THE CONICAL ORTHOMORPHIC PROJECTION

Abstract

Scale Correction Factor at a Point in Terms of X and Y.—Let dσ be a small line element of the curve ACB on the plane and ds the corresponding line element on the spheroid.     

EVALUATION OF THE COEFFICIENT OF TERRESTRIAL REFRACTION

Abstract

In a previous article on this subject (Empire Survey Review, January 1937) the writer sought to show that for trigonometrical observations of vertical angles made near noon in the Tropics the coefficient of refraction depends chiefly on height above ground level in the case of stations sited within a few hundred feet above the general level of the ground surface. Indeed, the computed values of the coefficient K show a definite and appreciable increase with “h”, the height of the observing station above ground level; it is usually assumed that K decreases with increase in height above the Mean-Sea-Level surface. From analysis of the results obtained by varying h but holding the heights above Mean Sea Level fixed the writer came to the conclusion that the variations in K could only be due to abnormal values of dt/dh and d²t/dh², “t” denoting the air temperature. Now it is generally recognized by meteorologists that abnormal lapse-rates of temperature do frequently occur in the lower air layers in the Tropics; but up to the present time no temperature soundings in Nigeria are available. Recently, however, the writer came across the results of the aerological soundings made by an expedition in East Africa during the year 1908. The results of many of the soundings were of no use for the purpose of this paper; many of the observations were not taken at or near noon, and in others counterlapses of temperature in the lower layers indicated that conditions were not normal. A set of observations taken at Mombasa between 10 and 11 a.m. were eventually chosen as offering an example of what might reasonably occur in the lower layers of the atmosphere.     

Uniform network transformation for points pattern analysis on a non-uniform network

In the real world, there are many kinds of phenomena that are represented by points on a network, such as traffic accidents on a street network. To analyse these phenomena, the basic point pattern methods (i.e. the nearest neighbour distance method, the quadrat method, the K-function method and the clumping method) defined on a plane (referred to as the planar basic point pattern methods) are extended to the basic point pattern methods on a network (referred to as the network basic point pattern methods). However, like the planar basic point pattern methods, the network basic point pattern methods assume a uniform network and this assumption is hard to accept when analysing actual phenomena. To overcome this limitation, this paper formulates a transformation, called the uniform network transformation, that transforms a non-uniform network into a uniform network. This transformation provides a simple procedure for analysing point patterns on non-uniform networks: first, a given non-uniform network is transformed into a uniform network; second, the network basic point pattern methods (which assume a uniform network) are applied to this transformed uniform network. No modification to the network basic point pattern methods is necessary. The paper also shows an actual application of this transformation to traffic accidents in Chosei, Japan.     

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.