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.     

“Mission Geography” and the Use of Satellite Imagery in K‐12 Geographic Education — A NASA ‐ GENIP Partnership

Abstract

Mission Geography is a collaboration between the United States' National Aeronautic and Space Administration (NASA) and the American “Geography Education National Implementation Project” (GENIP), created to provide for and develop curriculum supplementary material for teachers of grades Kindergarten through High School (grade twelve). Mission Geography uses NASA imagery and scientific inquiries to teach about Earth. This paper provides an example of how geographic concepts concerning volcanoes and volcanic hazards can be taught at the American middle school level (grades 5‐8) using NASA resources     

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.     

NOTE ON AMERICAN PLANE SURVEYING AND METHODS OF INSTRUCTION FOR ENGINEERS

Abstract

While at first sight there appears to be considerable difference in surveying practice between American and British engineers, the underlying principles are of course the same. Perhaps the major contrast is in terminology; for example, the term transit is invariably used for the (transit) theodolite, stadia for tacheometry, spiral for transition, elevation for reduced level, whilst the telescope is plunged, not transitted. These and other differences in nomenclature are at first misleading, but are soon in regular use by the visitor to the States.     

INTERVISIBLE FEATURES

Abstract

It is frequently required to find whether a feature A of height h ₀ will interrupt the view between two other features A₁ and A₂, of heights h ₁ and h ₂ respectively. Suppose that the right line from A₁ to A₂, whose zenith distance is ζ at A₁, has a height h at A; it is then obvious that no more is necessary than to compute h and compare it with the known height h ₀ of the feature A.     

THE EFFECT OF ALTITUDE ON TRAVERSE ERRORS

Abstract

I. These notes are the results of following up in some detail the well-known fact that the horizontal distance between two points at altitude h is greater, by an amount proportional to h, than the distance between the corresponding points at sea-level. Traverses based on rectangular coordinates are considered, with special reference to the residual errors left after adjusting the misclosures of such traverses without first eliminating errors due to altitude.     

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).     

A VLBI Delay Model for Radio Sources at a Finite Distance

A relativistic delay model for Earth-based very long baseline interferometry (VLBI) observation of sources at finite distances is derived. The model directly provides the VLBI delay in the scale of terrestrial time. The effect of the curved wave front is represented by using a pseudo source vector K = (R ₁ + R ₂)/(R ₁ + R ₂), and the variation of the baseline vector due to the difference of arrival time is taken into account up to the second-order by using Halley’s method. The precision of the new VLBI delay model is 1 ps for all radio sources above 100 km altitude from the Earth’s surface in Earth-based VLBI observation. Simple correction terms (parallax effect) are obtained, which can also adopt the consensus model (e.g. International Earth Rotation and Reference Frames Service conventions) to finite-distance radio source at R > 10 pc with the same precision. The new model may enable estimation of distance to the radio source directly with VLBI delay data.     

THE SEVEN-LENS AIR-SURVEY CAMERA

Abstract

By kind permission of the Editor of the Royal Engineers Journal, this paper has been adapted from the article by the Author in the issue of June 1937 (Vol. li, pp. 217–223). We are also indebted to the Editor for placing generously at our disposal the blocks used to illustrate the paper.     

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.     

DEVIATION OF THE VERTICAL. DISCUSSION

Abstract

The following is a report of the discussion on the paper by Mr. A. R. Robbins on “Deviation of the Vertical” which was read at a meeting of the Land Surveying Division of the Royal Institution of Chartered Surveyors held on Tuesday, 12th December, 1950, and which was published in the January issue of this Review (xi, 79, 28–36).     

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.     

EVALUATION OF THE COEFFICIENT OF TERRESTRIAL REFRACTION

Abstract

In two previous articles (E.S.R., vol. iv, nos. 23 and 25) it was shown that, at the time of maximum diurnal temperature in the tropics, a definite relationship exists in the lower layers of the atmosphere between the magnitude of the coefficient of terrestrial refraction at a point and the height of that point above plain level, provided the weather is fine and clear. In fact the coefficient K increases with the height h, within certain limits which are probably defined by the condensation layer.     

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.     

A note on frame transformations with applications to geodetic datums

Rigorous equations in compact symbolic matrix notation are introduced to transform coordinates and velocities between ITRF frames and modern GPS-based geocentric geodetic datums. The theory is general but, after neglecting higher than second-order terms, it is shown that the equations revert to the formulation currently applied in most major continental datums. We discuss several examples: the North American Datum of 1983 (NAD83), the European Terrestrial Reference System of 1989 (ETRS89), the Geodetic Datum of Australia of 1994 (GDA94), and the South American Geocentric Reference System (SIRGAS). Electronic Publication     

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.     

FIGURES OF REFERENCE FOR THE EARTH

Abstract

1. The object of this note is to clear up what I believe to be some misconceptions regarding the use of a reference system by a surveyor of the earth's surface. In his article “An Aspect of Attraction”, E.S.R., No. 7, pp. 24–8, Major M. Hotine expressed doubts as to the validity of the process usually followed. I may say at once that I consider these doubts are unfounded.     

THE APPLICATION OF A RED FILTER TO SURVEYING INSTRUMENTS

Abstract

I regret that my note on this subject (vol. ii, no. 7,PP· 32-3) should have apparently provoked so little comment or interest. I feel that the silence may be either that accorded to my Impetuous Inrush or that produced by alarm at the possible complications to be overcome before theory can be reconciled with practice. If the latter, then my original title is perhaps in some measure responsible. It is really very simple; let two facts suffice.     

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.