Geodetic Datum and No Net Translation (NNT) - No Net Rotation (NNR) Conditions from Transformation Parameters, a Reference Frame and a Selection Matrix

Abstract

The definition of the geodetic datum is a fundamental issue in the solution of the inverse problems related with the adjustment of free geodetic networks. Taking into account the conventions given by the International Earth Rotation and Reference System Service (IERS) on the definition and realization of a Terrestrial Reference System (TRS), it is considered here that a geodetic datum is the set of all conventions, algorithms and constants necessaries to define and realize the origin, orientation, scale and their time evolution of a TRS in such a way that these attributes be accessible to the users through occupation, direct or indirect observation.
In this work, we deal with the adjustment of a two-dimensional trilateration network using coordinate based formulations within a Gauss-Markov Model (GMM), where the point positions are defined by means of coordinates (x,y) in a local Terrestrial Reference Cartesian Coordinate System TRS(x,y), which has not defined its position and orientation in a given epoch, causing a datum defect and a rank-deficient Singular Gauss-Markov Model (SGMM).
It is developed here, within this stochastic linear model for the adjustment of a twodimensional free trilateration network of the type SGMM with datum defect, three linear condition equations namely minimum constraints: two “ No Net Translation” (N NT) and one “ No Net Rotation” (N NR) which define the datum in a given epoch respect to the position and orientation of the TRS(x,y) respectively, based in: a) three zero values conventionally adopted of three parameters of a plane coordinate Helmert transformation : two translation and one differential rotation, b) a Terrestrial Reference Frame TRF(xo,yo) kno wn “ a priori” considered “ free of error” and c) a selection matrix S , which allows to choose or exclude simultaneously the coordinate increments (dx,dy) of specifics network points.