### Strong Datum and Terrestrial Reference Frames in the Adjustment of a Free Geodetic Network

#### Abstract

The free geodetic network considered in this work is a free two-dimensional trilateration network which is a set of physical points accessible through occupation, direct or indirect observation that provides to the users of parameters that allows us to know the shape and size of the Earth. The goal of this work is to find a LEast Square Solution (LESS) to the inverse problem of estimate the network points coordinates using the observed distances in an adjustment model of type Gauss- Markov Model (GMM). The positions of the network points are defined in a Local two-dimensional Geodetic Reference System (LGRS) using only Cartesian or plane coordinates (x,y). The LGRS is defined using attributes of a topocentric Terrestrial Reference System TRS described by Boucher,C. (2001) as one of the major types of TRS in use, hence, the LGRS is designated here as Terrestrial Reference Cartesian Coordinate System TRS(x,y).The origin “o” of the TRS(x,y) is on the Earth’s surface and the orientation of the axis ox and oy uses the local vertical direction in “o”. Right-handed convention is adopted for the axis. The scale or length defined of the unit vectors along ox and oy of the TRS(x,y) is the meter (SI), and it is realized by the observed distances of the trilateration network. The TRS(x,y) has not defined its origin and orientation in a given epoch. Since the available observation (distances),do not carry the necessary information to realize completely these attributes of the coordinate system (origin and orientation) ,there will be a datum defect – datum problem- and therefore a rank-deficient Singular Gauss-Markov Model (SGMM) in the adjustment of the network.

Hence, to estimate the network points coordinates (unknown parameters), the datum problem must be solved. It can be done, by introducing in the SGMM the necessary information not contained in the observations, i.e. definition and realization of the origin and orientation in a given epoch of the TRS(x,y) in the form of independent linear equations on the unknown parameters which are known as “constraints”.

In this work, the datum of the two-dimensional free trilateration network is defined by introducing in the adjustment model (SGMM) more constraints than the minimum required or necessaries – so called “Strong Datum” – namely, more linear equations than the network datum defect (which leads to an “over-constrained” adjustment problem) using a chosen Terrestrial Reference Frame TRFd(x,y) given by the coordinates (xd,yd) of all selected datum points according to Vacaflor, J.L (2012) and four parameters of a plane coordinate transformation : two translation , one differential rotation and one scale factor, with respect to a known “a priori” Terrestrial Reference Frame TRF(xo,yo).

The Weighted LEast Square Solution (W-LESS) of this “over-constrained” adjustment problem is developed following the general methodology given by Schaffrin, B (1985).

Hence, to estimate the network points coordinates (unknown parameters), the datum problem must be solved. It can be done, by introducing in the SGMM the necessary information not contained in the observations, i.e. definition and realization of the origin and orientation in a given epoch of the TRS(x,y) in the form of independent linear equations on the unknown parameters which are known as “constraints”.

In this work, the datum of the two-dimensional free trilateration network is defined by introducing in the adjustment model (SGMM) more constraints than the minimum required or necessaries – so called “Strong Datum” – namely, more linear equations than the network datum defect (which leads to an “over-constrained” adjustment problem) using a chosen Terrestrial Reference Frame TRFd(x,y) given by the coordinates (xd,yd) of all selected datum points according to Vacaflor, J.L (2012) and four parameters of a plane coordinate transformation : two translation , one differential rotation and one scale factor, with respect to a known “a priori” Terrestrial Reference Frame TRF(xo,yo).

The Weighted LEast Square Solution (W-LESS) of this “over-constrained” adjustment problem is developed following the general methodology given by Schaffrin, B (1985).

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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**