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     

STANDARDIZING EQUIPMENT OF THE GOLD COAST SURVEY DEPARTMENT

Abstract

The following account of the standardizing equipment of the Gold Coast Survey Department has been written, at the request of the Editor of the Review, because this equipment includes a completely enclosed standard of length 300 feet long which is believed to be one of the very few enclosed standards of this length in any of the Crown Colonies.     

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.     

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.     

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.     

A THREE-MILLION-ACRE TITLE SURVEY

Abstract

A Brief account of a survey, recently completed by me, of a block of land situated in Northern Rhodesia and some 3,000,000 acres in extent, may be of interest. The sketch-map opposite shows the geographical situation and.exterior boundaries of this block of land. It is seen that it stretches along the entire length of the Northern Rhodesia-Tanganyika Territory border, which is approximately 150 miles in length.     

THE GEODIMETER: AN INSTRUMENT FOR THE ACCURATE MEASUREMENT OF DISTANCES BY HIGH-FREQUENCY LIGHT VARIATIONS

Abstract

The constant K in equation (12) represents distance expended through time lags in the instrument itself, and, although the value of K can be calculated from electrical data, this would not be very satisfactory and it would be better to determine it directly by means of observations over a line of known length. In addition, the point from which K would be reckoned is not a convenient one for actual field measurements. Instead of this, it is more convenient to choose an index mark on the instrument itself and referall measurements to this and thence to the mark over which the instrument is set up.     

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.     

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.     

RE-MEASUREMENT OF THE OLD 10-FT.LENGTH STANDARDS O1 AND OI1 OF THE ORDNANCE SURVEY,AND SOME NOTES ON THE RELATIVE STABILITY OF CERTAIN STANDARDS OF LENGTH

Abstract

The old 10-ft. length standards of wrought iron, O₁ and OI₁, made for the Ordnance Survey in 1826 and 1856 respectively, are briefly described and some account is given of the purpose for which they were constructed.

Both these 10-ft. standards were measured in terms of the Yard in 1864, and one of them in terms of the Metre in 1906. They have recently been re-measured at the National Physical Laboratory, and it was found that, allowing for the known shortening of the Imperial Standard Yard since 1895, the 10-ft. Ordnance Survey standards have remained unchanged in length during the last 50 years or so. Furthermore, if it is assumed that the Imperial Standard Yard shortened rather more rapidly between 1853 and 1895 than it has since that date, then the 10-ft. standards can be said to have remained substantially unchanged in length for nearly a century.

Additional evidence for the change in the length of the Yard between 1853 and 1895 is provided by the results of measurements made in 1864 on some of the old Toise standards used for geodetic surveys on the Continent, and by some recent measurements made at the N.P.L. of another yard standard contemporary with the Imperial Standard.     

Statistical analysis of discrepancies in high precision levelling

An investigation was made of the behaviour of the variable (where ρ_ij are the discrepancies between the direct and reverse measurements of the height of consecutive bench marks and theR _ij are their distance apart) in a partial net of the Italian high precision levelling of a total length of about1.400 km. The methods of analysis employed were in general non-parametric individual and cumulative tests; in particular randomness, normality and asymmetry tests were carried out. The computers employed wereIBM/7094/7040. From the results evidence was obtained of the existence of an asymmetry in respect to zero of thex _ij confirming the well-known results given firstly by Lallemand. A new result was obtained from the tests of randomness which put in evidence trends of the mean values of thex _ij and explained some anomalous behaviours of the cumulative discrepancy curves. The extension of this investigation to a broader net possibly covering other national nets would be very useful to get a deeper insight into the behaviour of the errors in high precision levelling. Ad hoc programs for electronic computers are available to accomplish this job quickly. Presented at the 14^th International Assembly of Geodesy (Lucerne, 1967).     

THE SURVEY FRAMEWORK OF NIGERIA

Abstract

Angular and Linear Errors.—In the results which follow, the number of stations quoted on the chains is exclusive of all the stations on the base-extension nets at Minna, Rijau, Chafe, and Naraguta. The total number of stations can therefore be obtained by adding the numbers on all the chains and the base-extension nets and deducting the number of points common to other chains. The quoted fractional misclosures in length are the actual misclosures obtained and the theoretical fractional misclosures which might be expected to be developed through the chains from the probable errors of the adjusted angles.     

LINEAR UNITS,OLD AND NEW

Abstract

Units of length are tools of the surveyor. Even should he himself, as is unlikely, evince no interest in their origin, he may reckon on being consulted at some time by someone who is interested. And if to such an inquiry he replies that his business ends with a knowledge of their use, it is to be feared that knowledge of their present use would not be held to excuse ignorance of their past history.     

EARLY MEASUREMENTS OF ARC IN ‘IRAQ

Abstract

In a tale, delightfully told in vol. ii, no. 8, pp. 121–2 of this Review, Mr. Kitching revealed how he was called upon to assist in determining the length of an Arab mile. In the notes following the story a reference was made to measurements of arc carried out in the ninth century A.D. The following notes add some more details to those already given. The full table of the Arab units of length is given below; in the reference above, one step was left out.     

A DRAFTING MACHINE FOR CIRCULAR ARCS OF ANY RADIUS

Abstract

1. Theory.—In Figs. 1. a and 1. b the letters R represent in each case four equal links of any length forming a freely jointed rhombus. L, L are two equal links of any length. (n ± 1), n are respectively a fixed and a rotating link whose lengths are in the ratio (n ± 1)/n.     

A NOTE ON CLARKE'S FORMULAE

Abstract

An elegant proof of Clarke's Formulae for lines of medium length has been given by G. T. McCaw. It is based on the properties of polar triangles in spherical trigonometry, and on the application of Legendre's Theorem.     

THE RECTIFICATION OF AIR PHOTOGRAPHS

Abstract

The application of the formulae for the rectification of air photographs, with or without change of scale, becomes greatly simplified when the tilt of the photographs is small. Usually this tilt does not exceed 2°, and the rectification may take place in a comparatively simple form of camera and with hardly any computations, since, as will be shown, variations of the camera settings are then accurately linear in relation to the tilt. It will also be shown that in the result the errors due to small differences in the focal length of the air camera lenses, to distortion from the pressure plate, and to average shrinkage of the film, can be automatically eliminated.     

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.     

LONG LINES ON THE EARTH: THE DIRECT PROBLEM

Abstract

19. Formulae.—In Nos. 6, vol. i, and 9, vol. ii, pp. 259 and 156, there has been described a new method for dealing with long geodesics on the earth's surface. There the so-called “inverse” problem has claimed first attention: given the latitudes and longitudes of the extremities of a geodesic, to find its length and terminal azimuths. It remains to discuss the “direct” problem : a geodesic of given length starts on a given azimuth from a station of known latitude and longitude; to find the latitude and longitude of its extremity and the azimuth thereat. The solution of this direct problem demands a certain recasting of the formulae previously given. In order of working the several expressions now assume the forms below.     

THE APPLICATION OF RADAR TO SURVEYING

Abstract

Radar can be applied to surveying for precise measuren1ent of long lines, and as a navigational aid and position-fixing device for an aircraft performing a photographic survey. Trials of the radar method have recently been carried out in Australia using a modified “Shoran” equipment. The results of a large number of radar measurements of six distances, varying from 160 to 310 miles in length, indicate that an accuracy of 7 parts in 10⁵ can be achieved. Equipment errors constitute the immediate limit to accuracy, but reasonable modifications would yield a figure of 2 parts in 10⁵. Radar measurements can be completed in a fraction of the time required by normal ground survey methods, since a measurement of upwards of a hundred miles is made in a single step.

As an aid to photographic surveying a straight-line track indicator actuated by data from the “Shoran” equipment has been designed and flight tested. Its performance enabled a pilot taking aerial photographs to keep the aircraft to within an average departure of less than 0.02 mile from any desired straight-line flight path.