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.     

NOTES ON THE OBSERVATION OF ALTITUDE FOR TIME

Abstract

Observations of the altitude of stars is a common method for the determination of local time. When the time is required with no great accuracy and when, therefore, simple instruments are employed, the method is particularly suitable. In the conduct of the observation with a theodolite there arise certain questions to which some reference may be made here with a view to lending some assistance to the student.     

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.     

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.     

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.     

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.     

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.     

THE USE OF “EVEN ANGLES” IN STADIA SURVEYING

Abstract

While there is no standard method in stadia surveying of taking a set of readings for distance and altitude, the following may be regarded as the conventional, text-book method. The telescope is directed at the staff and the apparent lower hair brought to bear exactly on some convenient foot-mark. The readings of all three hairs are then taken and recorded (4.00, 5.41, and 6·83). Finally the vertical circle is read—to the nearest minute or to a fraction of a minute according to circumstances—and the result entered in the field-book (6° 31′ or 6° 31′ 20″). This method has its merits. It is straightforward and flexible and there is a simple (numerical) check on the accuracy of the staff-readings. Nevertheless it is by no means a perfect method.     

AIR SURVEY: THE RAND METHOD OF HEIGHTING FROM NEAR VERTICAL PHOTOGRAPHS

Abstract

The method outlined below is a simple and rapid method, requiring no expensive equipment, of obtaining true heights from pairs of near vertical air photographs with a minimum of ground control. It is a direct application of the Fourcade theorem and, as it was finally developed at the University of the Witwatersrand in 1948-49, it has been given the name “Rand method” in the hope that this will connect it with Dr. Fourcade's country.     

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.     

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).     

THE ADJUSTMENT OF A FIRST-ORDER LEVELLING NETWORK

Abstract

In the E.S.R. January and April numbers of 1955, Vol. xiii, Nos. 95 and 96, Mr. Hsuan-Loh Su described the “Adjustment of a Level Net by Successive Approximations and by Electrical Analogy”. It does not seem to be as generally known as it should be that the rigid least square solution can be greatly simplified by utilizing the electrical analogy and solving by Kirchhoff's method. The method as detailed below has been in use for over 40 years.     

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.     

ALMUCANTAR POSITION LINES

Abstract

This article discusses the observation equations which may be solved graphically by plotting position lines using the method of zenith distance intercepts, or solved analytically by the method of least squares.

The general observation equation is modified for the particular case in which zenith distance is made equal to assumed co-latitude, thus simplifying the reduction of the observations.

Adaptation of the theory to use with a theodolite is discussed together with the effects of sources of error and the methods which are proposed for their elimination.

A routine of reduction is proposed and an example is given.     

A RESECTION METHOD

Abstract

This is a variation of the well-known device of successive approximations. It was first used by the writer about 15 years ago on a Seismic (Geophysical) Survey when resections were continually employed to locate Shot Points and Geophonestations set on arcs of 10 to 15 miles radius. Speed was an important factor, and the normal text book methods of resection were very tedious. As far as is known the method is original. It is now used by many surveyors, and the writer trusts it will be of use to others. The method is easy to remember as it has no set formula and does not involve any elaborate geometrical construction.     

RANGE-FINDING WITH A THEODOLITE IN TOPOGRAPHICAL SURVEYING

Abstract

The method to be described consists of the measurement of a short base and the computation of the distances to various points, to which rays have been drawn on a plane table. The angles at the two ends of the base are observed with a theodolite. This method will be referred to as the “Short Base Method”.     

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     

The use of atmospheric dispersion in optical distance measurement

The development of lasers, new electro-optic light modulation methods, and improved electronic techniques have made possible significant improvements in the range and accuracy of optical distance measurements, thus providing not only improved geodetic tools but also useful techniques for the study of other geophysical, meteorological, and astronomical problems. One of the main limitations, at present, to the accuracy of geodetic measurements is the uncertainty in the average propagation velocity of the radiation due to inhomogeneity of the atmosphere. Accuracies of a few parts in ten million or even better now appear feasible, however, through the use of the dispersion method, in which simultaneous measurements of optical path length at two widely separated wavelengths are used to determine the average refractive index over the path and hence the true geodetic distance. The design of a new instrument based on this method, which utilizes wavelengths of6328 ? and3681 ? and3 GHz polarization modulation of the light, is summarized. Preliminary measurements over a5.3 km path with this instrument have demonstrated a sensitivity of3×10 ⁻⁹ in detecting changes in optical path length for either wavelength using1-second averaging, and a standard deviation of3×10 ⁻⁷ in corrected length. The principal remaining sources of error are summarized, as is progress in other laboratories using the dispersion method or other approaches to the problem of refractivity correction.     

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.