### Terrestrial Reference Frame in the Datum Definition of a Free Trilateration Network

#### Abstract

Terrestrial Reference Frames (TRFs) are very useful tools in geodesy, surveying, navigation by land, sea or air, cartography, cadastral works, engineering applications and geophysical investigations. Nowadays, the TRFs are obtained using a wide range of techniques including the Global Positioning System (GPS) among others. The datum problem is one of the central issues in the solution of the inverse problems related with the adjustment of geodetic networks. Many contributions to the study of the geodetic datum problem or zero order design problem (Grafarend, 1974) has been realized since the fundamental works of Meissl (1965,1969) and Blaha (1971), followed by Grafarend and Schaffrin (1974,1976).The relation between minimal constrained solutions have been well established by means of the S-transformation (Baarda, 1973). The concepts and terminology of the Terrestrial Reference System of coordinates and frames were redefined in the 1980s by the astronomical and geodetic communities (Kovalevsky et al., 1989) , and incorporated in the geodetic datum analysis in works such as : a) a review of the algebraic constraints in TRF datum definition by Sillard, P, and Boucher, C. (2001), b) the datum definition in the combination of several particular TRFs from Geodetic Space Techniques for the release of the International Terrestrial Reference Frame by Altamimi, Z ; Sillard, P and Boucher, C. (2002) and, c) the datum definition of geodetic networks within an adjustment model known as Singular Gauss-Markov Model (SGMM) by Vacaflor, J.L. (2008,2010,2011) where the structure of the weights and the selection matrix as well as the role of the Helmert transformation parameters which may have in the datum minimum constraints equations were established. It is (c), the specific context, where a new advance: two representations of the Terrestrial Reference Frame chosen in the datum definition “TRFd” were found for a planar geodetic network and are presented here. Indeed, the main goal of this work is to obtain an explicit and implicit

representations of the Terrestrial Reference Frame TRFd(x,y) chosen in the datum definition of a free two-dimensional trilateration network, when the datum is defined by means of three linear conditions equations (minimum constraints), in order to define and realize the origin and the orientation of the Terrestrial Reference System of coordinates TRS(x,y) . The research is realized within the specific adjustment model known as Singular Gauss-Markov Model (SGMM) and in relation with the Restricted LEast Square Solution (RLESS) of the inverse problem: the estimation of the planar

coordinates (x,y) of the network points based in observed distances. The explicit representation obtained of the TRFd(x,y) is given by the coordinates (xd,yd) of the chosen datum points and used in the minimum constraints, while the implicit representation is given by four parameters of a plane coordinate Helmert transformation : two translation ,one differential rotation and one scale factor, with respect to a known “a priori” Terrestrial Reference Frame TRFd(xo,yo) chosen in the datum definition.

representations of the Terrestrial Reference Frame TRFd(x,y) chosen in the datum definition of a free two-dimensional trilateration network, when the datum is defined by means of three linear conditions equations (minimum constraints), in order to define and realize the origin and the orientation of the Terrestrial Reference System of coordinates TRS(x,y) . The research is realized within the specific adjustment model known as Singular Gauss-Markov Model (SGMM) and in relation with the Restricted LEast Square Solution (RLESS) of the inverse problem: the estimation of the planar

coordinates (x,y) of the network points based in observed distances. The explicit representation obtained of the TRFd(x,y) is given by the coordinates (xd,yd) of the chosen datum points and used in the minimum constraints, while the implicit representation is given by four parameters of a plane coordinate Helmert transformation : two translation ,one differential rotation and one scale factor, with respect to a known “a priori” Terrestrial Reference Frame TRFd(xo,yo) chosen in the datum definition.

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Güemes 3450

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**Asociación Argentina de Mecánica Computacional**Güemes 3450

S3000GLN Santa Fe, Argentina

Phone: 54-342-4511594 / 4511595 Int. 1006

Fax: 54-342-4511169

E-mail: amca(at)santafe-conicet.gov.ar

**ISSN 2591-3522**