### Datum Definition in the Solution R-MINOS of the Adjustment of a Free Trilateration Network

José L. Vacaflor

#### Abstract

Free trilateration networks in the 2D and 3D space, ranging from a local to a global scale, are continuously designed and established with a wide variety of objectives such as: cartographic, geodynamics, civil engineering, cadastral among others, using – for example - the Global Positioning System (GPS) among others Global Navigation Satellite Systems (GNSS). To evaluate the quality of the adjustment of a free trilateration network it is very useful to characterize properly the datum definition involved. The geodetic datum definition is the set of all conventions, algorithms and constants necessaries to define and realize the origin, orientation, scale and their time evolution of a Geodetic Reference System (GRS), in such a way that these attributes are accessible to the users through occupation, direct or indirect observation. In this work, we deal with the adjustment of a twodimensional free trilateration network constituted by physical points, where distances between these points have been observed. For the network adjustment, it is used a coordinate based formulation in a no stochastic linear model through a underdetermined consistent system of indirect linear observational equations. The network point positions are defined in a Geodetic Reference System of Cartesian Coordinate (x,y): GRS(x,y). The GRS(x,y) is characterized by : a) right-handed convention is adopted for the axis; b) the origin is a point not specified of the Earth ; c) the ox (+) and oy(+) axis do not have specified orientations; d) the scale or length defined of the unit vectors along ox and oy is the meter (SI), and it is realized by the observed distances of the trilateration network. The lack of definition in the origin and orientation of the GRS(x,y) cause a datum defect and a rank-deficiency in the design matrix. The solution of the adjustment of the free trilateration network called Minimum Norm Solution with respect to the R–seminorm (R-MINOS) of the underdetermined consistent system of indirect linear observational equations is obtained based in an optimum criterion, which resolves the datum problem. The set of the physical points of the trilateration with coordinates given by the RMINOS is the Geodetic Reference Frame of Cartesian Coordinate (x,y): GRF(x,y)-(R-MINOS). In this work, the GRF(x,y)-(R-MINOS) is characterized when R is the identity matrix (I) and when R is a positive semidefinite matrix. It is shown that, for R equal to I, the realization of the origin and orientation of the GRS(x,y) is given through the fulfillment of the conditions “No Net Translation” (NNT) and “No Net Rotation” (NNR) respectively. As a numerical example, the GRF(x,y)-(RMINOS) is characterized in the adjustment of a free two-dimensional trilateration network with six points when R is the identity matrix (I) and when R is a positive semidefinite matrix.

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