NOTE ON THE REDUCTION OF CIRCUMMERIDIAN ALTITUDES TO THE MERIDIAN

Abstract

The method of reducing circummeridian altitudes or zenith distances to the meridian, using the factors m and n as tabulated by Chauvenet, is well known. The following method, which does not use these factars, has been faund both more convenient and more accurate in practice. The formula can be easily obtained by expanding m and n in powers of t, but far the sake af campleteness the derivatian is here given from the beginning.     

EXPANSION OF PAPER AS AFFECTING BEARING

Abstract

The problem of the influence of the paper itself on bearings taken off maps or plans has been recently discussed by G. T. M. in this Review (iii, 22, 479). The following alternative method of solving the problem may be of interest; the same figure and symbols are used as in the article mentioned.     

A NOTE ON AZIMUTH DETERMINATION

Abstract

The purpose of this note is twofold; first, to criticize the “azimuth” section of the paper “Some Notes on Astronomy as Applied to Surveying”, by R. W. Pring (E.S.R., July 1952, xi, 85, 309–318),and secondly, out of these criticisms to develop an alternative method of making observations for azimuth. It will be apparent that this method owes much to the ideas put forward by Mr. Pring.     

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.     

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 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 CLARKE FORMULÆ FOR LATITUDE,LONGITUDE AND REVERSE AZIMUTH,WITH CORRECTIVE TERMS FOR USE ON VERY LONG LINES

Abstract

This extremely simple and elegant method of computing geographical co-ordinates, given the initial azimuth and length of line from the standpoint, was published by Col. A. R. Clarke in 1880. There is no other known method giving the same degree of accuracy with the use of only three tabulated spheroidal factors. Clarke himself regarded this as an approximate formula (vide his remark in section 5, p. 109, “Geodesy”); but as this article demonstrates, it is capable of a high degree of precision in all occupied lati tudes when certain corrections are applied to the various terms. These corrections are comparatively easy to compute, require no further spheroidal factors, and some of them may be tabulated directly once and for all.     

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.     

INTERPOLATING FROM TRAVERSE TABLES TO SECONDS

Abstract

In the Empire Survey Review, no. 4, 1932, Mr. Clendinning has described a method of interpolating from traverse tables to seconds. Below is another method, due to Prof. Nekrassov, for use with traverse tables published by him. The method is described in The Geodezist, Moscow, 1936, no. I, pp. 47–52.     

Terrain radiometric calibration of airborne UAVSAR for forested area

In the field of biomass estimation, terrain radiometric calibration of airborne polarimetric SAR data for forested areas is an urgent problem. Illuminated area correction of σ -naught could not completely remove terrain features. Inspired by Small and Shimada, this paper tested gamma-naught on one mountainous forested area using airborne Uninhabited Aerial Vehicle Synthetic Aperture Radar data and found it could remove most terrain features. However, a systematic increasing trend from far range to near range is found in airborne SAR cases. This paper made an attempt to use the relationship between distance to SAR sensor and γ-naught to calibrate γ -naught. Two quantitative evaluation methods are proposed. Experimental results demonstrate that variation of γ -naught can be constrained to a limited extent from near range to far range. Since this method is based on ground range images, it avoids complicated orthorectification.     

NORMAL EQUATIONS RESOLVED BY APPROXIMATION

Abstract

We are indebted to Professor R. V. Southwell for the approximate method of computation known as the systematic relaxation of constraints. In an article to the Empire Survey Review, 1938, Mr A. N. Black showed how Southwell's ideas could be applied to the adjustment of the co-ordinates of a point.     

LONG LINES ON THE EARTH

Abstract

The Arc of the Geodesic.—In the first part of this paper a method was given for computing the azimuth of a geodesic. The method gives the convergence of the geodesic correctly up to the second power of e the eccentricity. The formula (9), however, also depends on the assumption that σ, the arc-length of the geodesic, can be obtained with sufficient accuracy from the Supplemental Dalby Theorem, that is to say, by a purely spherical computation. It is, therefore, needful to show that this supposition is justifiable; a means must in fact be indicated for verifying the assumption.     

TRIANGULATION: BEARINGS VERSUS ANGLES

Abstract

There has always been a marked difference of opinion on the relative merits of the methods of bearings and of angles as applied to triangulation, though it is probable that the majority of writers prefer the method of bearings for first-order work. The subject was mentioned in a recent issue of this Review (vii, 47, 19).     

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     

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.     

MACHINE METHOD OF INTERSECTION

Abstract

The following method will be found better and quicker than the usual logarithmic process in computing the co-ordinates of intersected points in minor triangulation and traverse work. Let A and B be two stations whose co-ordinates (x ₁ y ₁), (x ₂ y ₂) are known. Let P be an intersected point whose co-ordinates (x, y) we wish to determine. Let α and β be the observed angles at A and B respectively.     

THE ORTHOMORPHIC PROJECTION OF THE SPHEROID–III

Abstract

In the last instalment we were able to obtain most of the surveyor's projections in common use by applying simple scale conditions to the meridians and parallels. This method of approach naturally suggests that results of some value might be obtained by applying similar conditions to the plane co-ordinate lines. If we do so, we are immediately led to consider curves on the surface known as geodesics, which are the nearest approach to straight lines it is possible to draw on a curved surface. Accordingly, we give some account of these curves for the benefit of surveyors who have not hitherto made their acquaintance.     

FURTHER NOTES ON RANGE-FINDING WITH A THEODOLITE

Abstract

Since writing the article which appeared in E.S.R., no. 36, p. 364, the writer has used this method to obtain thousands of spot heights with differences of elevation varying up to 7,500 feet in a single shot. The results of this experience and the modifications introduced in the method may be of some iilterest.     

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).     

On the night errors in astronomical determinations

Summary The discrepancy between precision and accuracy in astronomical determinations is usually explained in two ways: on the one hand by ostensible large refraction anomalies and on the other hand by variable instrumental errors which are systematic over a certain interval of time and which are mainly influenced by temperature.In view of the research of several other persons and the author’s own investigations, the authors are of the opinion that the large night-errors of astronomical determinations are caused by variable, systematic instrumental errors dependent on temperature. The influence of refraction anomalies is estimated to be smaller than 0″.1 for most of the field stations. The possibility of determining the anomalous refraction from the observations by the programme given by Prof. Pavlov and Anderson has also been investigated. The precision of the determination of the anomalous refraction is good as long as no other systematic error working in a similar way is present.The results, which are interpreted as an effect of the anomalous refraction by Pavlov and Sergijenko, could also be interpreted as a systematic instrumental error. It is furthermore maintained thatthe latitude and longitude of a field station can be determined in a few hours of one night if the premisses given in [3, p.68]are kept. It has been deplored that the determination of the azimuth has not been given the necessary attention. It is therefore proposed to intensify the research on this problem. The profession has been called upon to acquaint itself better with the valuable possibilities of astronomical determinations and to apply them in a useful and appropriate manner. At the same time, attention has been called to the possibility of improving astronomical determinations with regard to accuracy as well as effectiveness.