WGS 84 koordinat sisteminden ülke koordinat sistemine dönüşümler

Abstract

Yeryuvarı veya yeryuvarının bir parçasının şeklinin belirlenebilmesi için yeryüzünde belirli referans noktaları seçilir. Bu noktaların oluşturduğu şekil jeodezik ağ olarak adlandırılır. Jeodezik ağlar üçe ayrılır: a) üç boyutlu ağlar, b) yatay kontrol ağları, c) düşey kontrol ağları. Jeodezide üç konum vektörünün eş zamanlı belirlendiği üç boyutlu ağlar ideal bir amaç olmasına rağmen, klasik ölçme teknikleriyle düşey açıların düşük presizyonla belirlenebilmesi nedeniyle jeodeziciler tarafından tercih edilmezler. Bu nedenle, jeodezik ağların, yatay ve düşey kontrol ağları olarak ayrı ayrı kurulması yoluna gidilmiştir. Yatay ve düşey kontrol ağları farklı referans yüzeylerine dayalıdırlar. Bunun yanında, genellikle yatay kontrol ağları görüş mesafesi yüksek ( örneğin tepelerde) yerlerde kurulurken, düşey ağlar ulaşılması kolay (karayolu, demiryolu yakınlarındaki) yerlerde tesis edilirler. Bu da yatay ağlarda, düşey konumun ya _ hiç bilinmemesine ya da yaklaşık değerlerle bilinmesine sebep olur. Üstelik çoğu ülkelerde (Türkiye'de olduğu gibi) düşey konumun referanslı olduğu jeoidin, referans elipsoidinden olan yükseklikleri (jeoit yükseklikleri) belirlenmemiştir. Bu sebeplerle, yatay ve düşey ağların birleştirilmesinden ortaya çıkacak üç boyutlu ağlar homojen yapıda olamamaktadır. Günümüzde uydu konumlama teknikleri üç boyutun eş zamanlı belirlendiği yüksek duyarlılıklı homojen ağlar kurulmasına olanak sağlamaktadır. Bunların arasında en geniş uygulama alanı bulanı Global Positioning System (GPS) olmaktadır. GPS ile elde edilen nokta koordinatları WGS84 elipsoidine referanslıdır. Yani, WGS84 datumundadır. Ancak, haritacılık faaliyetleri yıllardır klasik yersel ölçmelerle, yerel ya da ülke koordinat sistemindeki (datumundaki) kontrol ağlarıyla yürütülmektedir. Türkiye'de ülke nirengi ağlan ED50 (European Datum 1950) datumundadır. Bu bağlamda, kısa vadede yerel veya ülke koordinat sistemlerinden vazgeçmek mümkün değildir. Bu durum, "datum dönüşümü"nü güncel bir problem olarak ortaya çıkarmaktadır. Bu çalışmanın konusunu da datum dönüşümü ve değişik dönüşüm modelleri oluşturmaktadır. Konu çerçevesinde, bu modeller değişik yaklaşımlarla ele alınıp, datum dönüşümü için uygun çözümler aranacaktır.

The specific points on the surface of the earth are selected in order to define the shape of the earth or some part of it. The sets formed by these points are denoted geodetic networks, and they are divided into three groups, namely 3-D networks, horizontal networks and vertical networks By means of classical survey techniques measurements which would be observed between control points define geometrical relations to each other. But they are not sufficient to determine their positions thorough the earth. For this aim and computations a reference surface that is appropriate to real shape of earth surface and could be expressed via simple mathematical models is necessary. As such a reference surface ellipsoid is best convenient. Thus, dimension of selected ellipsoid and its orientation thorough the earth determine geodetic datum. There are two approach to introduce position at the ellipsoid of a point on the physical surface; ellipsoidal geographical or Cartesian coordinates. Conversion from ellipsoidal to Cartesian is given below; X= (N+h)cosBcosL Y= (N+h)cosBsinL Z=((l-e2)N+h)cosB (l-a,b,c) Conversion from Cartesian to ellipsoidal are the followings; L= arctan(Y/X) B = arctan ( Z(l-f) + e2asin3n.(l-f)(Vx2+Y2acos3uJ (2-a,b) XI h= (X2+Y2)1/2 cosB+ZsinB-a(l-e2sin2B)1/2 (2-c) Where: H = Vx2+Y2 e2a (i-f)+- (2-d) Classical survey methods are strict with regional earth surface. Because of these reasons each country or groups of countries prefer determining a local ellipsoid which is best convenient to their region rather than a global ellipsoid. Our country has also followed this way, and European Datum 1950 (ED50) orientated through the Middle Europe has been selected as reference. Today all mapping studies has being continued on this reference. Today, the satellite positioning techniques provide a means to form high precision homogenous networks determined three dimensionally simultaneously. GPS leading these satellite techniques is widely used. The coordinates of the points obtained by GPS are referenced to the WGS84 ellipsoid. In a short term leaving local or national coordinate frames is not seemed probability. In this context, datum transformation arise as actual problem in geodesy. WGS84 and ED50 systems are also geodetic (ellipsoidal) coordinate frames, and referenced on Conventional Terrestrial Coordinate System. The geometrical structure of both is defined as follows;. Origin is center of reference ellipsoid. Z-axes coincide semi-minor axes of ellipsoid. X-axes lies on the intersection of Greenwich geodetic meridian and equator planes. Y-axes completes right-hand system As expressed above, both systems have the same geometrical structure, so linear transformation could be mentioned between two systems. The best general linear transformation is similarity transformation. There are condition of unchanged angles and the relations between systems are provided seven parameters ( three translations, three rotations, a scale). The general expression of similarity transformation is; Xll X2=X +(l+k)RXı (3) where; X2 : second system coordinates Xi : first system coordinates k : scale X° : translation matrix R : rotation matrix Where R matrix is; R= cosPcosy cosa sin y + sina sin P cosy sinasiny -cosa sin P cosy -cosPsiny cosacosy-sinasinpsiny sinacosy + cosa sin P sin y sinP -sinacosP cosacosP (4) If we assume rotations are very small, so that R can be written as follow; R = 1 y -y 1 P -a a 1 (5) There are many private approach of similarity transformation. The best known of them are Bursa- Wolf and Molodensky- Badekas models. In Bursa- Wolf model common points positions are assumed that they have random errors. General equation of Bursa- Wolf model is; X2 + V2 = X ° + Xi + V, + ( U + k I ) X, (6) where being different from general similarity equation; X1U I+U= (?) In Molodensky-Badekas model general equation is; X2 = X° + Xo + ( 1 +k ) ( I + U ) ( X i - Xo ) (8) where; Xo : the position vector of mass center of common points in first system There are only difference from Bursa- Wolf model; first system coordinates are shifted to mass center of common control point. But it is very important difference, because of influencing translations between two system. If we consider coordinate systems are distorted, afin transformation could be introduced. Its general equation is; X2= X°+Xo+MTR Ş ( Xi-Xo) (9) where; M : rotation matrix rotating first system into local system Ş : matrix introducing affinity property All transformation models mentioned above are efficiently appreciated that three dimensions are also present. In geodesy, although it is ideal that three dimensional networks determined by their three positions vectors together in a whole context, this technique has not been favored by many geodesist due to low precision vertical angle measurements. Thus, geodetic networks have been formed separately as horizontal and vertical networks. Horizontal and vertical networks have separate reference surfaces. Moreover, it is preferable that the horizontal network points are set up on the peaks of hills whereas easy access (low elevations such as near the roads and railways) is the main consideration for the vertical network points. This is why in the horizontal networks are never considered or known approximately. Furthermore, in many countries as Türkiye the geoid heights from the reference ellipsoid are not known. XIV For these reasons, the three dimensional networks formed by the combination of horizantal and vertical networks may not be homogenous. But this arise height problem in three dimensional transformations. To exceed this problem there are many approaches. The best known of them are to assume ellipsoidal heights are zero or ellipsoidal heights of local systems equals to orthometric heights. But, they include model errors. The topic of this study is the datum transformation and various transformation models. This datum problem is investigated in different aspects in order to find appropriate solutions. Furthermore different approaches are referred to exceed height problem.