<abstract xmlns="http://www.w3.org/1999/xhtml">

Alignment of an engineering object project in the field is always conducted at the points of the geodetic control network, the coordinates of which are determined on the basis of the results of its elements survey and with connection to the national spatial reference system. The points of the national spatial reference system determined on the basis of previous surveys have specified coordinates with adequate accuracy, which is included in their covariance matrix. The coordinates of the geodetic control network points are determined more accurately than the points of the national spatial reference system and this means that the results of surveys of the geodetic control network have to be adequately incorporated into the coordinates of the reference points. In order to perform this incorporation, it may be assumed that the coordinates of the reference points are random, that is, they have a covariance matrix, which should be used in the process of adjusting the results of the geodetic control network observation. This research paper presents the principles for the estimation of the Gauss-Markov model parameters applied in case of those geodetic control networks in which the coordinates of the reference points have random character. On the basis of the observation equations δ + <bold>AX</bold> = <bold>L</bold> for the geodetic control network and using the weighting matrix <bold>P</bold> and the matrix of conditional covariances (<bold>P</bold>−1 + <bold>AC</bold><italic>X</italic><bold>A</bold><italic>T</italic>) for the observation vector <bold>L</bold>, the parameter vector <bold>X</bold> is estimated in the form of the derived formula 
<inline-formula>
<alternatives>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_rgg-2021-0001_ieq_001.png"></inline-graphic>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mover accent="true"><mtext mathvariant="bold">X</mtext><mo>^</mo></mover><mo>=</mo><msup><mrow><mrow><mo>(</mo><mrow><msubsup><mtext mathvariant="bold">C</mtext><mi>X</mi><mrow><mo>−</mo><mn>1</mn></mrow></msubsup><mo>+</mo><msup><mtext mathvariant="bold">A</mtext><mi>T</mi></msup><mtext mathvariant="bold">PA</mtext></mrow><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><msup><mtext mathvariant="bold">A</mtext><mi>T</mi></msup><mtext mathvariant="bold">P</mtext><mo>⋅</mo><mtext mathvariant="bold">L</mtext></mrow></math>
<tex-math>{\bf{\hat X}} = {\left( {{\bf{C}}_X^{ - 1} + {{\bf{A}}^T}{\bf{PA}}} \right)^{ - 1}}{{\bf{A}}^T}{\bf{P}} \cdot {\bf{L}}</tex-math>
</alternatives>
</inline-formula>
. The verification of these estimation principles has been illustrated by the example of a fragment of a levelling geodetic control network consisting of three geodetic control points and two reference points of the national spatial reference system.
The novel feature of the proposed solution is the application of covariance matrices of the reference point coordinates to adjust the results of the survey of geodetic control networks and to determine limit standard deviations for the estimated coordinates of geodetic control network points.
</abstract>

Alignment of an engineering object project in the field is always conducted at the points of the geodetic control network, the coordinates of which are determined on the basis of the results of its elements survey and with connection to the national spatial reference system. The points of the national spatial reference system determined on the basis of previous surveys have specified coordinates with adequate accuracy, which is included in their covariance matrix. The coordinates of the geodetic control network points are determined more accurately than the points of the national spatial reference system and this means that the results of surveys of the geodetic control network have to be adequately incorporated into the coordinates of the reference points. In order to perform this incorporation, it may be assumed that the coordinates of the reference points are random, that is, they have a covariance matrix, which should be used in the process of adjusting the results of the geodetic control network observation. This research paper presents the principles for the estimation of the Gauss-Markov model parameters applied in case of those geodetic control networks in which the coordinates of the reference points have random character. On the basis of the observation equations δ + AX = L for the geodetic control network and using the weighting matrix P and the matrix of conditional covariances (P−1 + ACXAT) for the observation vector L, the parameter vector X is estimated in the form of the derived formula 



X^=(CX−1+ATPA)−1ATP⋅L
{\bf{\hat X}} = {\left( {{\bf{C}}_X^{ - 1} + {{\bf{A}}^T}{\bf{PA}}} \right)^{ - 1}}{{\bf{A}}^T}{\bf{P}} \cdot {\bf{L}}


. The verification of these estimation principles has been illustrated by the example of a fragment of a levelling geodetic control network consisting of three geodetic control points and two reference points of the national spatial reference system.
The novel feature of the proposed solution is the application of covariance matrices of the reference point coordinates to adjust the results of the survey of geodetic control networks and to determine limit standard deviations for the estimated coordinates of geodetic control network points.

Gauss-Markov model with random parameters to adjust results of surveys of geodetic control networks

Reports on Geodesy and Geoinformatics

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Alignment of an engineering object project in the field is always conducted at the points of the geodetic control network, the coordinates of which are...

k\(1 − α)	0.99	0.95	0.90	0.80	0.60
k = 2	10.0	4.4	3.1	2.1	1.4
k = 3	5.1	2.9	2.3	1.7	1.3
k = 4	3.7	2.4	1.9	1.6	1.2
k = 5	3.0	2.1	1.8	1.5	1.2
k = 6	2.6	1.9	1.6	1.4	1.2
k = 7	2.3	1.8	1.6	1.4	1.1
k = 8	2.2	1.7	1.5	1.3	1.1
k = 9	2.1	1.6	1.5	1.3	1.1
k = 10	2.0	1.6	1.4	1.3	1.1

Gauss-Markov model with random parameters to adjust results of surveys of geodetic control networks

Article Category: Original Article

Publié en ligne: 25 juil. 2021

Pages: 1 - 6

Reçu: 01 avr. 2021

Accepté: 25 juin 2021

DOI: https://doi.org/10.2478/rgg-2021-0001

Mots clés
Estimation of models with random parameters, establishing geodetic control networks

© 2021 Marek Banaś et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Figure 2

Coefficients to determine limit standard deviations for the estimated parameters (coordinates of points of the geodetic network

Gauss-Markov model with random parameters to adjust results of surveys of geodetic control networks

Article Category: Original Article

Publié en ligne: 25 juil. 2021

Pages: 1 - 6

Reçu: 01 avr. 2021

Accepté: 25 juin 2021

DOI: https://doi.org/10.2478/rgg-2021-0001

Mots clésEstimation of models with random parameters, establishing geodetic control networks

© 2021 Marek Banaś et al., published by Sciendo

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Figure 1

Figure 2

Coefficients to determine limit standard deviations for the estimated parameters (coordinates of points of the geodetic network

Mots clés
Estimation of models with random parameters, establishing geodetic control networks