Proper Choce of Data Used for the Estmaton of Datum Transformaton Parameters Hakan S. KUTOGLU, Turkey Key words: Coordnate systems; transformaton; estmaton, relablty. SUMMARY Advances n technologes and scence have progressvely allowed defnng better terrestral coordnate systems or ther reference frames used n surveyng. Ths crcumstance always keeps datum transformaton from one system, or one frame, to another up to date. Data used for the estmaton of datum transformaton parameters are the coordnate components of common ponts whose coordnates are known n both system of datum transformaton. Obtanng a relable set of the parameters requres a proper choce of the common ponts. What number of the common ponts should be used? Or, s ther dstrbuton sutable? These questons are too hard to decde by rule of thumb. Sometmes, some components of some common ponts may need to be elmnated whle some may be necessary for a relable soluton. In ths study, a rgorous method s suggested and useful remarks are made for the proper data choce n datum transformaton. 1/7
Proper Choce of Data Used for the Estmaton of Datum Transformaton Parameters 1. INTRODUCTION Hakan S. KUTOGLU, Turkey Advances n technologes and scence have progressvely allowed defnng better terrestral coordnate systems or ther reference frames used n surveyng. Ths crcumstance always keeps datum transformaton from one system, or one frame, to another up to date. As the terrestral coordnate systems are orthogonal coordnate systems a complete datum transformaton between them are establshed by the seven parameter smlarty transformaton (Seeber, 1993). However, data used for the estmaton procedure of the transformaton parameters are the coordnate components of common ponts whose coordnates are known n both system of datum transformaton, so alternatve transformaton methods n two or three dmenson such as affne and polynomals can also be appled n practce to mnmze the resduals of concdence between the common pont coordnates. It s fact that the common pont coordnates can be burdened wth dstortons due to some random and systematc errors and envronmental effects such as crustal movements (Kutoglu, 2004). These dstortons can be absorbed durng the parameter estmaton, and can cause the based transformaton parameters. The more the parameters are used for the transformaton the more the dstortons are absorbed. Obtanng a relable set of the parameters requres a proper choce of the common ponts. What number of the common ponts should be used? Or, s ther dstrbuton sutable? These questons are too hard to decde by rule of thumb. Sometmes, some components of some common ponts may need to be elmnated whle some may be necessary for a relable soluton. In ths study, a rgorous method s suggested and useful remarks are made for the proper data choce n datum transformaton. 2. METHOD FOR PROPER CHOICE OF COMMON POINTS In a parameter estmaton problem by the least square adjustment, redundancy numbers whch are the contrbuton of each data to the degree of freedom of the adjustment are computed by r T 1 T ( I A( A PA) A P) = (1) where A s the coeffcent matrx multpled by the unknown parameters vector of the adjustment, P s the weght matrx for the common pont coordnates composed of the precson estmatons of the coordnates, I s the unt matrx ( s the runnng ndex assocated wth the data number and represents the dagonal elements of the resultng matrx on the rght hand sde). Usng the redundancy numbers one can estmate possble dstortons n each coordnate component 2/7
v v = (2) r r and then the absorbed amount of the dstorton A 1 r = v (3) r where (Baarda, 1967 and 1976). v s the resdual of the data whle v s the dstorton mpact on the resdual As seen n the equatons ncreasng redundancy number decreases the possble dstorton and the amount of absorpton. Accordngly, a proper choce of common ponts to obtan relable transformaton parameters can be accomplshed by regardng the redundancy numbers (Kutoglu, 2006). 3. CASE STUDY For conductng a sample applcaton, 20 control ponts have been determned n Istanbul Metropoltan Area, Turkey (Fg. 1). These ponts have coordnates both n ED50 datum and ITRF94. Let us thnk these ponts are coordnated only n ED50. Anyone who wants to produce datum parameters between ED50 and ITRF can use all these ponts after obtanng ther coordnates n ITRF by an observaton campagn carred out on the feld. On the other hand, someone else can decde to choose the proper ponts applyng the above-mentoned method. Such a proper set of the ponts proposed by the method are llustrated n Fg 2. Durng ths desgnaton, the transformaton method assumed s the three dmensonal smlarty transformaton and the redundancy numbers are taken care of beng homogenous and not below 0.50. Table 1: Datum parameters obtaned from both approaches Parameter Usng all the ponts Usng the proper ponts t x (m) 87.359±0.017 87.390±0.027 t y (m) 91.181±0.017 91.224±0.027 t z (meter) 127.494±0.017 127.541±0.027 ε x (arcsecond) -3.808±0.212-3.658±0.212 ε y (arcsecond) 0.132±0.325-0.102±0.380 ε z (arcsecond) 1.550±0.190 1.666±0.220 k (ppm) 3.245±0.827 3.170±0.952 The transformaton parameters are estmated for both approaches stated above. Table 1 summarzes the obtaned results. As seen the table, the parameters obtaned from both approach are qute close to each other. 3/7
BlackSea 1 11 18 19 2 6 5 4 7 3 8 10 12 14 13 17 9 Istanbul 41.0 N 15 16 Marmara Sea 20 28.5 E 29.0 E Fgure 1. Common ponts n Istanbul Metropoltan Area In the frst approach, the redundancy numbers change between 0.70-0.95 (Fg. 3). The redundancy numbers over 0.90 belong to the ponts heaped up n the center area. These values show that there s over control whch means the majorty of the ponts n ths area do not have an mportant contrbuton to the soluton. The man dsadvantage of that stuaton s that observng such ponts for producng common ponts wll not be cost-effectve. Secondly, such heaped ponts may localze a soluton. Black Sea 1 11 18 19 2 8 Istanbul 41.0 N 15 Marmara Sea 20 28.5 E 29.0 E Fgure 2. A proper choce of common ponts After applyng the proposed method, almost all of the ponts n the centre area have been elmnated, and the number of the common ponts has been reduced from 20 to 8 whose redundances range between 0.55-0.84 (Fg. 2 and 4). However, the remanng ponts have 4/7
stll produced smlar parameters to the prevous ones. Also, use of the method provdes some further nformaton about the data such as possble dstorton and absorpton (Fg. 5). These further nformaton allow practtoners to make more rgorous analyss for the pont selecton. Redundancy number 1.00 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Pont No Fg. 3. Redundancy numbers n the case that all the ponts are used Redundancy number 1.00 0.80 0.60 0.40 0.20 0.00 Pont No Fg. 4. Redundancy numbers for the selected ponts 5/7
(n centmeters) 0.3 0.2 0.1 0-0.1 X axs -0.2 Pont No (n centmeters) 0.1 0.05 0-0.05-0.1 Y axs -0.15 Pont No (n centmeters) 0.3 0.2 0.1 0-0.1-0.2-0.3 Z axs Pont No resdual pos.dst. absorpton Fg. 5. Resduals, possble dstorton and absorpton n coordnates of the selected ponts 4. CONCLUSION In ths study, a useful method s proposed for the common pont selecton n datum transformaton. Applyng ths method, one can 6/7
- avod unrelable transformaton parameters caused by mproper number and dstrbuton of the common ponts, - avod producng redundant common ponts, and thus provdes cost-effectve solutons, - provde further nformaton, such as possble dstorton and absorpton, about the common ponts, so one can carry out a more rgorous analyss for the pont selecton. For nstance, some components of some common ponts may need to be elmnated whle some may be kept for a relable soluton. REFERENCES Baarda, W.: Statstcal concepts n geodesy Publcaton on Geodesy, New Seres, 2(4), Netherlands Geodetc Comsson, 1967. Baarda, W.: Relablty and precson of networks, n: Proc., VII. Int. Kurs für Ingeneurmessungen Hoher Prazson, Darmstadt, Germany, 1976. Kutoglu, H.S.: Fgure condton n datum transformaton Journal of Surveyng Engneerng, 130(3), 138-141, 2004. Kutoglu, H.S. and Ayan, T.: The role of common pont dstrbuton n obtanng relable parameters for coordnate transformaton Appled Mathematcs and Computaton, 176, 751-758, 2006. Seeber, G.: Satellte geodesy Walter de Gruyter, New York, USA, 1993. BIOGRAPHICAL NOTES, Assocate Professor n Department of Geodesy and Photogrammetry Engneerng at Zonguldak Karaelmas Unversty. CONTACTS Assoc. Prof. Zonguldak Karaelmas Unversty Engneerng Faculty Geodesy & Photogrammetry Engneerng ZKU Muhendslk Fakultes 67100 Zonguldak TURKEY Tel. + 90 3722575126 Fax + 90 3722574023 Emal: kutogluh@hotmal.com Web ste: http://jeodez.karaelmas.edu.tr/lnkler/akademk/kutoglu/kutoglu.htm 7/7