Lokal üç boyutlu ve iki boyutlu dik koordinat sistemlerinde dengeleme ve karşılaştırmalar

Abstract

Jeodezinin başta gelen görevlerinden biri jeodezik kontrol ağları oluşturmak ve ağ noktalarının konumlarını bir koordinat sisteminde belirlemektir. Jeodezik problemlerin çözümü için gerekli olan jeodezik kontrol ağları : genelde, - Yatay kontrol (Nirengi) ağları, - - Düşey kontrol (Nivelman) ağları, - Gravite ağları, olarak adlandırılan, birbirinden bağımsız üç ayrı ağ halinde kurulagelmiştir. Bunlardan nirengi ağları iki boyutlu, nivelman ve gravite ağları bir boyutludur. Bu ayırımı yapmanın nedeni, uygulamada karşılaşılan güçlüklerdir. Birbirinden bağımsız üç ayrı ağ kurularak problem çözülmeye çalışılsa da bağımsızlığı tam olarak sağlamak mümkün değildir. Dolayısıyla yatay koordinatların belirlenebilmesi için düşey koordinatlara, düşey koordinatların belirlenmesi için de yatay koordinatlara ihtiyaç vardır. Bu yüzden problemin çözümü için, bütün ölçülmüş verilerin üç boyutlu bir matematik modelde ortaklaşa kullanılması gerekir. Bu çalışmada, doğrultu, kenar ve zenit açılarına dayalı olmak üzere, refraksiyon katsayısı bilinen ve bilinmeyen alınarak üç boyutlu ağ dengelemesi hesabı ve yine k refraksiyon katsayısı bilinen ve bilinmeyen alınarak trigonometrik nivelman dengelemesi hesabı yapılmıştır. Hesaplamalar İ.T.Ü. Elektronik Bilgi işlem Merkezinde IBM-4341 serisi bilgisayarlarında ve kısmen de Epson XT-PCe kişisel bilgisayarlarda yapılmış olup İ.T.U. Jeodezi Anabilim Dalı program arşivinden yararlanılmıştır. Ayrıca başlangıç noktasından geçen yatay düzlemde iki boyutlu ağ dengelemesi yapılmış, ve sonuçlar karşılaştırılmıştır.

In this research, it was investigated to compare with classical methods and adjust in 3-D for measurements done for a determination analysis. In the classical approach, the problem of 'the determi nation of positions' is considered in two groups as the two dimensions on a reference surface and the distance of the point to this surface. Three dimensional model has some advantages on conventional models. No reductions of measurements are needed and it may be developed with little mathematical knowledge as an exact combination of traditional methods such as trigonometric levelling and triangulation. The progress of geodetic instruments and computers has made solution of difficult geodetic prob lems possible, e.g., continuing exact geodetic control and determining crustral motions in areas of high moun tains. These problems are solved on the basis of tradi tional astrogeodetic terrestrial measurements with the main stress on measured distances as precise and as inclined as possible. The purpose of geodetic surveying is to determine fundamental control networks and positions of the points of the net in a coordinate system. Therefore, it's necessary to measure various quantities (distances, angles, height differences) on the surface of the Earth. These measurements are made under natural conditions and are affected by temporal variations (atmospheric, meteorogical effects, movements of the Earth). If the reference surfaces are used in applicating positions, measurements must be reduced to these surfaces. As known, an ellipsoid is accepted as a reference surface due to similarity between ellipsoid and geoid. In the classical way, first an ellipsoid is chosen then, the problem of the location and orientation of network is taken into account (datum problem). Thus, the control VI points were built on physical surface. The data are obtained on the Earth's surface. Measurements are transformed from natural to the corresponding reference system (reductions). So, the observation between points on the Earth's surface must be reduced to the horizontal plane for computation of the coordinates of the points on the horizontal plane. Therefore, all the computations were made on the horizontal plane which is at the height of point 1. Consequently, the coordinates are obtained by means of the adjustment of the network. Although, three dimensional geodesy is not new, it is commonly used nowadays. At first, it was considered by VILLARCEAU in 1868 and BRUNS in 1878. Three dimensional geodesy had not been applied for along time because of refraction problem and low accuracy in the measurements of vertical angles. But some geodesist have been interested in this subject and suggested different adjustment models. These models are based on geometrical, physical and Doppler measurements. Geometrical and physical measurements are used in integrated geodesy. Nowadays, interactive network design has an important role on the problem of geodetic network in terms of working time and economy. In the second chapter, the coordinate systems were investigated as 3-D. Various coordinate systems are used to solve geodetic problems. The terrestrial coordinate systems are used in three dimensional geodesy. They can be classified in two main groups as geocentric and topocentric systems. There are two kinds of topocentric systems as local astronomic system and local geodetic system. In the third chapter, local cartesian coordinate system was explained. A local cartesian coordinate system will prove smaller extent, established for preparing projects of more extensive construction in high- mountain regions, for evaluting tectonic and other motions at separate localities, for accurate height determination in vertical gravity comparison bases, etc. A three dimensional geodetic coordinate system is suitable for treating terrestrial network of larger and smaller extents, the main purpose of which is to determine the coordinates in currently used geodetic, horizontal and vertical systems. It is very convenient particularly for processing three-dimensional nets, the main purpose of which is to establish geodetic control in high-mountain areas. Vll In upgrading geodetic control it is necessary to establish and treat high-mountain nets like three-dimen sional triangulation and to employ the original coordinates as approximate coordinates of the three dimensional adjustment. In the local cartesian coordinate system, the z-axis is the normal to the reference ellipsoid and the x-axis runs horizontally in the direction of the sight line starting at point P. An approximate orientation of the network with an accuracy of 0.5 is sufficient to determine the mutual positions of the vertexes accurately. If the network contains lines of sight mostly shorter than 4-5 km, the simultaneous determina tion of the refraction and of the deflections of the vertical from the zenith angle is not accurate enough and with lines of sight shorter than 3 km practically impossible. The horizontal angles and zenith angles, observed at the origin P of the coordinate system, are not transformed. The horizontal angles and zenith angles observed at the other points of the network, e.g., at point P, are transformed in to the cartesian coordinate system by leaving point P in its original position and rotating its normal to the ellipsoid of reference through the central angle Zy, thus rendering it parallel with the z-axis, i.e. with the normal to the reference ellipsoid at orijin P. In the fourth chapter, an application was done. Taşkesti Triangulation Network in Bolu was adjusted as 3-D. In this adjustment, point 1 was accepted as a constant point. In other words, the coordinates of this point was accepted as X=0, Y=0, Z=0. Firstly, an average Gauss's radius of curvature R for the whole net and refraction coefficient k was calculated. After, k was figured out as 0.16, approximate heights differences of the points were also computed by the following formula. AH.. = 1.. * cos Z.. + ** * l2. ij ij ij 2R ij Vlll And then, the approximate heights of the points were computed by helping the height of point 1. The measured horizontal directions and zenith angles were reduced on the horizontal plane by means of the following formulas. m m Z., = Z., + 2v.. cos a,. Jk J* J-J Jk m mm c c a.. = a.. - 2y.. cot Z.. sin a.. jk jk 'ij jk jk A computer programming was prepared in FORTRAN called VEDAT. FORTRAN. And the net was adjusted on the horizontal plane with two models in three dimensions ( k is known and unknown). Then, the results of adjustment were compared. Different parameters can be chosen for adjustment according to the measurements. Various adjustment models were suggested and applied for this purpose. Ludvik Hradilek was suggested a new mathematical model for adjusting trigonometric leveling nets. The ellipsoidal heights and deflections of the vertical can be calculated from vertical angles in this model. The observation equations of zenith anggles in indirect adjustment are very suitable for computer programming. Another adjustment computer programming was also prepared in FORTRAN named TRIG02 to compute ellipsoidal heights. This adjustment was also done with two models. These are: 1 The refraction coefficient can be accepted a constant value as k=0.16 (first model). And, it is also first adjustment in three dimensions. 2 The refraction coefficient can be accepted in adjustment as an unknown for all points in the network (second model). And, it is also second adjustment in three dimensions. Then, the network was adjusted in these models with the 2enith angles which were used in three dimensional adjustment. And the results were compared with the results of three dimensional adjustment. IX After this, the same net was adjusted as 2-D. And the results of adjustments compared with the results of three dimensional adjustment. In conclusion, these results shown that measured zenith angles are very important in adjustment. And in order to obtain better results, the approximates values of the unknown parameters can be determined with sufficient accuracy.