SIMPLE RESECTION

Abstract

The figure which follows shows the geometrical solution of Simple Resection by Cassini in 1669, two years before the Collins solution. It is clearly the geometrical illustration of the Delambre (1786) solution; for d_b = b cosec β, d_c = c cosec γ and the angle QAR is known, being BAC + β + γ ? 180°. Hence the Delambre solution-that in most common use to-day—reduces to a triangle in which two sides and the contained angle are given, as has been mentioned elsewhere.     

TRANSVERSE MERCATOR FORMULAE

Abstract

The Transverse Mercator Projection, now in use for the new O.S. triangulation and mapping of Great Britain, has been the subject of several recent articles in the “Empire Surpey Review. The formulae of the projection itself have been given by various writers, from Gauss, Schreiber and Jordan to Hristow, Tardi, Lee, Hotine and others—not, it is to be regretted, with complete agreement, in all cases. For the purpose for which these formulae have hitherto been employed, in zones of restricted width and in relatively low latitudes, the completeness with which they were given was adequate, and the omission of certain smaller terms, in the fourth and higher powers of the eccentricity, was of no practical importance. In the case of the British grid, however, we have to cover a zone which must be considered as having a total width of some ten to twelve degrees of longitude at least, and extending to latitude 61 °north. This means, firstly, that terms which have as their initial co-efficients the fourth and sixth powers of the longitude ω (or of y) will be of greater magnitude than usual, and secondly that tan² ? and tan⁴ ? are likewise greatly increased. Lastly, an inspection of the formulae (as hitherto available) shows a definite tendency for the numerical co-efficients of terms to increase as the terms themselves decrease—e.g. terms in η⁴, η⁶, etc.     

LONG LINES ON THE EARTH: VARIOUS FORMULAE

Abstract

Extensions were given for all these formulae, so that precise results may now be obtained even for lines of 500 miles in latitudes above 45°. The present instalment gives the extension of the Clarke approximate (sic) formulae to lines of 500miles, with a practical example and general conclusions: the great advantage of the method is that 8-figuretables sufficeto give rigorous results.     

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.     

A TRIGONOMETRICAL UPSET

Abstract

If the geographical co-ordinates, Φ₀, L 0, and the azimuth A ₀ at a station O of a triangulation undergo corrections, ?Φ₀, ?L ₀ and ?A ₀, the geographical co-ordinates, Φ, L, and the azimuth A have to be re-computed for all the vertices throughout the whole triangulation. This is a tedious operation. It may be vastly simplified, however, by the employment of differential formulae. The derivation of these formulae would consume considerable space, so that the results alone are given here.     

A comparison of different mass elements for use in gravity gradiometry

Topographic and isostatic mass anomalies affect the external gravity field of the Earth. Therefore, these effects also exist in the gravity gradients observed, e.g., by the satellite gravity gradiometry mission GOCE (Gravity and Steady-State Ocean Circulation Experiment). The downward continuation of the gravitational signals is rather difficult because of the high-frequency behaviour of the combined topographic and isostatic effects. Thus, it is preferable to smooth the gravity field by some topographic-isostatic reduction. In this paper the focus is on the modelling of masses in the space domain, which can be subdivided into different mass elements and evaluated with analytical, semi-analytical and numerical methods. Five alternative mass elements are reviewed and discussed: the tesseroid, the point mass, the prism, the mass layer and the mass line. The formulae for the potential, the attraction components and the Marussi tensor of second-order potential derivatives are provided. The formulae for different mass elements and computation methods are checked by assuming a synthetic topography of constant height over a spherical cap and the position of the computation point on the polar axis. For this special situation an exact analytical solution for the tesseroid exists and a comparison between the analytical solution of a spherical cap and the modelling of different mass elements is possible. A comparison of the computation times shows that modelling by tesseroids with different methods produces the most accurate results in an acceptable computation time. As a numerical example, the Marussi tensor of the topographic effect is computed globally using tesseroids calculated by Gauss–Legendre cubature (3D) on the basis of a digital height model. The order of magnitude in the radial-radial component is about  ± 8 E.U. Electronic supplementary material  The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

THE STEREOGRAPHIC SOLUTION OF THE SPHERICAL TRIANGLE

Abstract

At the suggestion of Mr. T. H. Corfield, who has himself given two solutions (E.S.R., No. 12, pp. 345–6) of Mr. A. J. Potter's problem, I venture to submit a third solution, which has at least the merit of simplicity.     

Determination of complexity factor and its relationship with accuracy of representation for DEM terrain

Based on the estimating rule of the normal vector angles between two adjacent terrain units, we use the concept of terrain complexity factor to quantify the terrain complexity of DEM, and then the formula of terrain complexity factor in Raster DEM and TIN DEM is deduced theoretically. In order to make clear how the terrain complexity factor E _CF and the average elevation h affect the accuracy of DEM terrain representation RMSE _Et , the formula of Gauss synthetical surface is applied to simulate several real terrain surfaces, each of which has different terrain complexity. Through the statistical analysis of linear regression in simulation data, the linear equation between accuracy of DEM terrain representation RMSE _Et , terrain complexity factor E _CF and the average elevation h is achieved. A new method is provided to estimate the accuracy of DEM terrain representation RMSE _Et with a certain terrain complexity and it gives convincing theoretical evidence for DEM production and the corresponding error research in the future.     

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.     

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 ORTHOMORPHIC PROJECTION OF THE SPHEROID—V

Abstract

To complete the picture, the reader is invited to plough through the derivation of point-to-point working formulae for the remaining projections in common use. The entire field of modern geodetic practice in this subject will then have been covered, and at least as simply, without the usual recourse to the square root of negative numbers or other complex notions.     

LEAST SQUARES ADJUSTMENTS OF TRIANGULATION: DIRECTIONS VERSUS ANGLES

Abstract

This paper continues the discussion started in an article of the same title (E.S.R., ×, 78, :353-66), on which a further letter was written in October, 1952. The amount of computation required originally was very considerable, and it was obviously impossible to publish it all. The recent letter was necessary to answer the suggestion that the agreement between errors put in and corrections obtained from the L.S. solution was not very close. It seemed sufficient to give the list of errors and corrections, leaving readers to judge for themselves. The correlation coefficient from the two sets of figures was 0.78, which looked quiteg90d. Unfortunately, it was not realized before that corrections from a L.S. solution cannot, legitimately, be compared with errors put in on directions unless a station correction is first applied to the errors to make the sum of the errors at each station equal to zero. This is one of the points about the direction method of adjustment which is not very easy to understand.     

Predictor stabilisation of reduction formulae for observations of heavily-damped simple harmonic motions

The paper summarises attempts which have been made to derive general, simple and stable formulae of adequate precision for the reduction of observations to the extremal positions of oscillations which are simple harmonic or near-simple harmonic. A fallacy in the thirty-five year old Schuler Mean is pointed out and general formulae are offered for the reduction of observations for any constant damping, within the range0<f≤1.     

Generalyzed variance-covariance propagation law formulae and application to explicit least-squares adjustments

Standard formulae overlook the contribution of a number of terms in the derivation of variance-covariance matrices for parameters in nonlinear least squares adjustment. In a large class of nonlinear mathematical models, these terms can contribute to an important error in the estimation of parameter variances. Improved formulae are derived. A numerical example is given and the use of our improved formula in the case of least-squares adjustment in the explicit case (L=F(X)) is fully documented.     

TLS-bridged co-prediction of tree-level multifarious stem structure variables from worldview-2 panchromatic imagery: a case study of the boreal forest

ABSTRACT

In forest ecosystem studies, tree stem structure variables (SSVs) proved to be an essential kind of parameters, and now simultaneously deriving SSVs of as many kinds as possible at large scales is preferred for enhancing the frontier studies on marcoecosystem ecology and global carbon cycle. For this newly emerging task, satellite imagery such as WorldView-2 panchromatic images (WPIs) is used as a potential solution for co-prediction of tree-level multifarious SSVs, with static terrestrial laser scanning (TLS) assumed as a ‘bridge’. The specific operation is to pursue the allometric relationships between TLS-derived SSVs and WPI-derived feature parameters, and regression analyses with one or multiple explanatory variables are applied to deduce the prediction models (termed as Model1s and Model2s). In the case of Picea abies, Pinus sylvestris, Populus tremul and Quercus robur in a boreal forest, tests showed that Model1s and Model2s for different tree species can be derived (e.g. the maximum R²?=?0.574 for Q. robur). Overall, this study basically validated the algorithm proposed for co-prediction of multifarious SSVs, and the contribution is equivalent to developing a viable solution for SSV-estimation upscaling, which is useful for large-scale investigations of forest understory, macroecosystem ecology, global vegetation dynamics and global carbon cycle.     

DUALITY IN BI-PROJECTIVE CORRESPONDENCE

Abstract

Bi-projection connotes the method of constructing plastic models of an object by projecting two photographs taken from different stations. The variables representing the relative orientation of the two plates and others representing the direction of the base are shown to satisfy symmetrical bi-linear equations, from which it follows that solutions always occur in pairs, and when one solution of a pair is known the other can be written down immediately. It is unlikely that any doubt will ever arise in practice on which solution should be taken. The equations are well suited for the numerical solution of the biprojective problem using the cartesian coordinates of corresponding points on the plates.     

THE (t – T) CORRECTION FOR THE LAMBERT No. 2 (CONICAL ORTHOMORPHIC) PROJECTION

Abstract

I. The present writer has been trying for the past two years to get reasonably easy expressions for the (t – t) correction and the scale factor in the Lambert NO.2 Projection. He succeeded in obtaining a formula for (t – t) in terms of eastings and northings, but, like the author of the articles on “Grid Bearings and Distances on the Conical Orthomorphic Projection” which were published recently in this Review, he came to the conclusion that any such formula is too unwieldy for ordinary use. He then tried to get an expression for (t – t) in terms of latitude and longitude, and has now obtained one by purely empirical methods which seems to work in practice. No proof of this formula is offered, or is available at present, as this is a matter which the writer is content to leave to others with greater mathematical interests and attainments than he possesses.     

THE NAMING OF THE STARS

Abstract

The order of the constellations in the following catalogue is that of the list in Ptolemy's “Almagest, except that the modern groups in the northern hemisphere are inserted after Ptolemy's northern and before his zodiacal constellations, while the modern groups in the southern hemisphere follow the last of his list. The constellation-name in italic capitals is the Latin form in general use. It is followed by the English translation where necessary and by the French and German versions. The Greek and Arabic names, with their authorities, are then given and translated where they differ in meaning from the Latin.     

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.