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Polynomial regression model for translation between GPS and local terrestrial network  Lü Zhongshu Information institute of Photogrammetry and Remote Sensing Yunnan Bureau of Surveying and Mapping, Kunming 650034 P. R. China Shi Kun Department of Geomatics Kunming University of Science and Technology, Kunming 650093 P.R.China shikun@263.net
Abstract
The polynomial regression model in which the precision information of satellite and local terrestrial networks is considered is applied to carry out translation between GPS and local terrestrial systems, and the geometrical meaning of translation parameters is explained tentatively. In order to overcome the overparameters and choose the effectual translation parameters, the translation parameters in the polynomial regression model are analyzed based on the theory of reliability, and determinability of parameters is applied to the translation model to select the the effective parameters. Finally, in order to test the model, an observed network in north of China has been calculated and analyzed with this model and translation parameters have been selected with determinability. Introduction The model selection for translation between GPS and terrestrial network has been an important problem in modern geodesy. As well known, it is important to select the optimum translation model within a limited area of a certain local terrestrial network for the departments such as engineering surveying, mineral, geology and so on. For the purpose, the problem has been studied widely and deeply.[2],[3],[4],[5].Traditional translation model, 7-parameters model, between the two systems mainly focused on the solution of seven translation parameters, i.e. three shifts Dx Dy Dz)of the two systems ;three components of rotation about X,Y,Z axes or Eular rotation elements(ex ey ez) and one scale element K. However, that this model is not suitable in a limited small area has been proved in reality. Considering the systematic error of local terrestrial network and the merits and demerits of existed main translation model between the two systems, the polynomial regression model is applied to the translation for the two systems, and the geometrical meaning of translation parameters is explained tentatively. In order to overcome the overparameters and choose the effectual translation parameters, the translation parameters in the polynomial regression model are analyzed based on the theory of reliability,. Translation approach with polynomial regression model The realization of translation between GPS and terrestrial network is usually based upon the coordinate differences at the same point that has the coordinates of both satellite frame and terrestrial frame. According to Weierstrass[ ],the coordinate differences lR can be approximated by polynomial with certain order term, i.e. for the I point of data in translation model, considering the second order term of the polynomial, the coordinate differences between the two systems for the same vector can be expressed as follows, (Rt) i=(Rg) i+(dR) i (1) (dR) i=(dx dy dz) i (2) and 
In the formula,(Rt) i is the vector of i point in geocentric system or GPS frame; (Rg) i is the vector of i point in geodetic system or local terrestrial frame; (dR) i is the difference of the same vector of in the tow systems And subscript k is the reference point of terrestrial network; m0,n0,l0 are the shifts of the original point; m1,n1,l1 are the scale elements of the directions of X,Y,Z axes; (m,n), (m,l), (n,l) are the rotation about X,Y,Z axes,…,m,n,…n,l,…l are the systematic errors that can not be distinguished. Data Processing Data processing in this work is based on the principle of adjustment of correlated observations[7]. For the i point in network, the observation equation can be established as following, (3) 
In the formula, X is the correction of the coordinate of i point; Y is the parameters for translation between two systems; A and B are the coefficients matrix relative to X and Y; B is the approximate of geodetical latitude of the i point; L is the approximate of geodetical longitude of the i point; H is the approximate of geodetical height of the i point.
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