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 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.     

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.     

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 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.     

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.     

ANGULAR CORRECTIONS FOR THE LAMBERT ORTHOMORPHIC CONICAL PROJECTION

Abstract

In E.S.R., viii, 56, 70, Brigadier K. M. Papworth has given expressions for the angular corrections, known as (t – t) corrections, in the Lambert NO.2 Projection, derived from empirical considerations based on actual detailed calculations. Apparently some difficulty has been experienced in offering a proof. In view of the widespread use of the Lambert Projection in World War II, it is hoped that the following proof will be found to be of more than academic interest.     

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.     

TRAVERSE ADJUSTMENT: WHY NOT LEAST SQUARES?

Abstract

In the above article by Mr H. L. P. Jolly published in a previous issue (E.S.R., vol. iv, no. 28) the author, after referring to the precision of the Nigerian traverses, makes the statement that measurements of the highest accuracy are worthy of the best possible methods of adjustment. But this argument cuts both ways. For in general the greater the accuracy of measurement the smaller will be the ultimate misclosure to be eliminated; so that different methods of adjustment will produce smaller and smaller variations in the corrections, until in the limit when there is no error we should obtain the same result however much latitude we permitted in the adjustment.     

DISTORTING A NETWORK TO FIT CONTROL POINTS: AN ALTERNATIVE EXPRESSION FOR THE SCHOLS ADJUSTMENT

Abstract

Another form of Mr. Lauf's expression for a conformal adjustment of a system of coordinated points may be of interest. These are assumed to be already in harmony with i control points and are to be brought into agreement with j further points. (Mr. Lauf deals explicitly in his paper with the special case i = 2, j = 1, but he adumbrates a general solution.)     

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.     

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.     

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.     

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 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.     

THE PROJECTIVE TRANSFORMATION IN AERIAL SURVEYING

Abstract

4.3. Having found s we may proceed to obtain the coordinates of the air station. This is the vertex of a pyramid, the base edges being of lengths a, b, c and the opposite edges measuring u, v, w respectively. Let h be the distance of the vertex from the plane of the base, and suppose d, e₁ e₂, e₃ are respectively twice the areas of the base and of the oblique faces in order.     

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.     

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.     

THE ORTHOMORPHIC PROJECTION OF THE SPHEROID

Abstract

Chesterton did not, of course, intend this gibe to be taken literally. But the more we consider what he would doubtless have called the “Higher Geodetics”, the more we must conclude that there is some literal justification for it. Not only are straight lines straight. A sufficiently short part of a curved line may also be considered straight, provided that it is continuous (i.e. does not contain a sudden break or sharp corner), and provided we are not concerned with a measure of its curvature. Similarly a square mile or so on the curved surface of the conventionally spheroidal earth is to all intents and purposes flat. We shall achieve a considerable simplification, without any approximation, in the treatment of the present subject by getting back to these fundamental glimpses of the obvious, whether the formalists and conformalists accept them or not.