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The combined geodetic network adjusted on the reference ellipsoid – a comparison of three functional models for GNSS observations

Roman Kadaj
Geodesy and Cartography | 2016 | vol. 65 | No 2 | 229-257 | DOI: 10.1515/geocart-2016-0013
Keywords combined geodetic network hybrid geodetic network GNSS vectors on the ellipsoid network adjustment on the ellipsoid
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Abstract

The adjustment problem of the so-called combined (hybrid, integrated) network created with GNSS vectors and terrestrial observations has been the subject of many theoretical and applied works. The network adjustment in various mathematical spaces was considered: in the Cartesian geocentric system on a reference ellipsoid and on a mapping plane. For practical reasons, it often takes a geodetic coordinate system associated with the reference ellipsoid. In this case, the Cartesian GNSS vectors are converted, for example, into geodesic parameters (azimuth and length) on the ellipsoid, but the simple form of converted pseudo-observations are the direct differences of the geodetic coordinates. Unfortunately, such an approach may be essentially distorted by a systematic error resulting from the position error of the GNSS vector, before its projection on the ellipsoid surface. In this paper, an analysis of the impact of this error on the determined measures of geometric ellipsoid elements, including the differences of geodetic coordinates or geodesic parameters is presented. Assuming that the adjustment of a combined network on the ellipsoid shows that the optimal functional approach in relation to the satellite observation, is to create the observational equations directly for the original GNSS Cartesian vector components, writing them directly as a function of the geodetic coordinates (in numerical applications, we use the linearized forms of observational equations with explicitly specified coefficients). While retaining the original character of the Cartesian vector, one avoids any systematic errors that may occur in the conversion of the original GNSS vectors to ellipsoid elements, for example the vector of the geodesic parameters. The problem is theoretically developed and numerically tested. An example of the adjustment of a subnet loaded from the database of reference stations of the ASG-EUPOS system was considered for the preferred functional model of the GNSS observations.
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Authors and Affiliations

Roman Kadaj

Empirical methods of reducing the observations in geodetic networks

Roman Kadaj
Geodesy and Cartography | 2016 | vol. 65 | No 1 | DOI: 10.1515/geocart-2016-0001
Keywords cartographic mapping reducing of geodetic observations empirical methods of reducing the observations mapping of long geodesics
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Abstract

The paper presents empirical methodology of reducing various kinds of observations in geodetic network. A special case of reducing the observation concerns cartographic mapping. For numerical illustration and comparison of methods an application of the conformal Gauss-Krüger mapping was used. Empirical methods are an alternative to the classic differential and multi- stages methods. Numerical benefits concern in particular very long geodesics, created for example by GNSS vectors. In conventional methods the numerical errors of reduction values are significantly dependent on the length of the geodesic. The proposed empirical methods do not have this unfavorable characteristics. Reduction value is determined as a difference (or especially scaled difference) of the corresponding measures of geometric elements (distances, angles), wherein these measures are approximated independently in two spaces based on the known and corresponding approximate coordinates of the network points. Since in the iterative process of the network adjustment, coordinates of the points are systematically improved, approximated reductions also converge to certain optimal values.
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Authors and Affiliations

Roman Kadaj

New algorithms of GPS post-processing for multiple baseline modelsand analogies to classical geodetic networks

Roman Kadaj
Geodesy and Cartography | 2008 | vol. 57 | No. 2 | 61-79
Keywords GPS post-processing carrier phase differences triple-differences multiplebaseline session elimination of nuisance parameters BETA method diagonal weightmatrix
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Abstract

The paper presents new differencing algorithms for post-processing GPS data, using double or triple carrier phase differences and multiple baseline sessions. The characteristic feature of the new algorithms is, that they use full sets of Schreiber's type observation differences with theoretically proved diagonal weight matrices. The proposed estimation models are equivalent to the least squares estimation applied to the original system of un-differenced observation equations. The theoretical ground of the algorithms are the theorems on the properties of differencing equations of Schreiber's type. The theorems become practically useful mainly in case of functional models with triple-differences. In a classical approach, this task was simplified for the sake of necessity of inverting non diagonal covariance matrix, usually of a large dimension. Diagonal weight matrix is also obtained in case of multiple point observation session where correlation of the GPS vectors forces in practice the use of the simplified stochastic models. The proposed method eliminates also the problem of selection of a reference satellite. It is very important especially in case of long observation sessions. The algorithms are applied in professional software for GPS relative positioning.
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Authors and Affiliations

Roman Kadaj
ORCID: ORCID

Conformal and empirical transformation between the PL-ETRF89 and PL-ETRF2000 reference frames using the new adjustment of the former Polish I class triangulation network

Roman Kadaj
Geodesy and Cartography | 2021 | vol. 70 | No 2 | article no. e05 | DOI: 10.24425/gac.2021.136680
Keywords European reference frames empirical transformation interpolation grid network adjustment 3D Helmert transformation
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Abstract

The European reference frame ETRF2000 was introduced on the territory of Poland on 1 July 2013, named PL-ETRF2000, as a result of the appropriate measurement campaign 2008-2011. The new PL-ETRF2000 reference frame has replaced the previously used PL-ETRF89 frame, which had more than 10 years of history in Poland until 2013, implemented in almost all geodetic and cartographic “products”, in geodetic networks, economic map systems and databases. The relationship of the new reference frame with the previously used PL-ETRF89 frame has become an important practical issue. Currently, all position services of the ASG-EUPOS (Active Geodetic Network – EUPOS) system use only the PL-ETRF2000 reference frame, which also results from the relevant legal and technical regulations. The relationships between the frames was considered in two aspects: “theoretical”, expressed by conformal (Helmert, 7-parameter) transformation, and “empirical”, based on an interpolation grid that allows to take into account local distortions of the PL-ETRF89 frame. The estimation of the parameters of the conformal transformation model was based on 330 points of the POLREF network, while to create an interpolation grid approximately 6500 points of the old triangulation network were additionally used, after new adjustment in PL-ETRF200 reference frame. Basic algorithms for the transformation between two frames and mapping systems are implemented in the new version of the TRANSPOL program, which is available on the web ( www.gugik.gov.pl).
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Authors and Affiliations

Roman Kadaj
1
ORCID: ORCID
  1. Rzeszów University of Technology, Rzeszów, Poland

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