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Increasing the Accuracy of the Orientation of Underground Topographic Works

Increasing the Accuracy of the Orientation of Underground Topographic Works Revista Minelor – Mining Revue ISSN-L 1220-2053 / ISSN 2247-8590 vol. 27, issue 1 / 2021, pp. 66-69 INCREASING THE ACCURACY OF THE ORIENTATION OF UNDERGROUND TOPOGRAPHIC WORKS 1 Larisa Ofelia FILIP University of Petroșani, Petroșani, Romania, larisafilip@yahoo.com DOI: 10.2478/minrv-2021-0009 Keywords: topo-geodetic measurements, topographic network, geodetic network Abstract: Geodetic and topographic works determine the necessary information base and of major importance in the design and management of underground construction works (mining, hydrotechnics, roads, etc.). The efficiency and safety of these investments with special financial implications depend on their quality. In this context, the geodetic orientation of the underground topographic works was the main and permanent subject of study and analysis on which this scientific paper refers. 1. Introduction Underground topographic works are related to a reference system defined on the surface in a system of representation or projection, adopted at national or local level (cylindrical, stereographic). The underground materialization of the surface reference system is achieved by topographic operations of topographic transmission on underground works: horizontal, inclined or vertical. The topographic element of the materialization is the orientation for which purpose different methods were used. The methods used are numerous and complex in the topographic transmission of the orientation on a vertical underground work. For this situation, specific analysis are required, with the related constructions [1]. 2. The junction of topographic elements If on the surface the reference system is defined by points𝐴 , 𝐵 located in the free area of the vertical work (fig.1) they are transmitted underground by their verticals through the points 𝐴 , 𝐵 [2]. Fig.1 Underground topographic works relate to orientation 𝜃 of the direction 𝐶𝐷 frequently obtained by the method of the connecting triangle (𝐶𝐴𝐵 ). Distances are measured in the connecting triangle 𝑎 , 𝑏 , 𝑐 and 𝛾 (through 𝛾 and 𝛾 ). It is known to be 1 2 determined: Corresponding author: Larisa Ofelia FILIP, Assoc.Prof.PhD.Eng., University of Petroșani, Petroșani, Romania, contact details (20, University str., 332006 Petroșani, larisafilip@yahoo.com) 𝐶𝐷 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 66-69 - The angles 𝛼 and 𝛽 with relationships: 𝑠𝑖𝑛 𝛼 = 𝑠𝑖𝑛 𝛾 (1) 𝑠𝑖𝑛 𝛽 = 𝑠𝑖𝑛 𝛾 - Orientation 𝜃 with relationships: 𝐶𝐷 𝜃 = 𝜃 + 𝛼 + 𝛾 ± 200 = 𝜃 − 𝛽 + 𝛾 ± 2 ∙ 200 (2) 𝐶𝐷 1 2 It is observed that the error with which the guidance is obtained𝜃 it is determined by the errors with which the distances are measured 𝑎 , 𝑏 , 𝑐 and with which the angles are determined 𝛼 , 𝛽 and 𝛾 . But: 𝑠𝑖𝑛 𝛼 = 𝑎 − 𝑐 + 𝑠𝑖𝑛 𝛾 (3) By differentiation they are obtained: 𝑑𝑎 𝑑𝑐 𝑑𝛾 = − + (4) 𝑎 𝑐 𝑡𝛾𝑔 and: 2 2 2 𝑚 𝑚 𝑚 𝑎 𝑐 𝑚 = ( ) + ( ) + ( ) (5) 𝑎 𝑐 𝑡𝛾𝑔 analogous: 2 2 2 𝑚 𝑚 𝑏 𝑐 𝑚 = ( ) + ( ) + ( ) (6) 𝑏 𝑐 𝑡𝛾𝑔 Errors 𝑚 and 𝑚 are minimal for: 𝛼 𝛽 𝛼 → 200 (fig. 2a) 𝛽 → 0 Fig. 2a or: 𝛼 → 0 (fig. 2b) 𝛽 → 200 Fig. 2b This is the case of a triangle with the surface 𝑆 ≅ 0 (with angle 𝛾 very small). The minimum conditions presented cannot be technically met in all situations and as a result errors must be taken into account. 𝑚 , 𝑚 , 𝑎 𝑏 𝑚 , 𝑚 . These errors are obtained using the theory of direct measurements of the same accuracy, if we 𝑐 𝛾 consider the quantities (𝑎 , 𝑏 , 𝑐 , 𝛾 ) independent [3]. However, these quantities are dependent (conditioned) by the existing relations in the triangle. ( ) ( ) ( ) ( ) For probable values 𝑎 , 𝑏 , 𝑐 , 𝛾 it can be written: ( ) ( ) ( ) 𝑎 𝑏 𝑐 = = (7) ( ) sin 𝛼 sin 𝛽 sin 𝛾 The number of conditions is given by the excess of measurements [4]. 4 sizes are measured and 2 sizes are required. Consequently, two conditions result. They are: (𝑎 ) 𝑠𝑖𝑛 (𝛾 ) = (𝑐 ) 𝑠𝑖𝑛 𝛼 (8) ( ) ( ) ( ) 𝑏 𝑠𝑖𝑛 𝛾 = 𝑐 𝑠𝑖𝑛 𝛽 According to the theory of conditional measurements, proceed as follows: ( ) ( ) ( ) 𝑎 + 𝑠𝑖𝑛 𝛾 = 𝑐 + 𝑠𝑖𝑛 𝛼 (9) (𝑏 ) + 𝑠𝑖𝑛 (𝛾 ) = (𝑐 ) + 𝑠𝑖𝑛 𝛽 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑡𝑔𝛽 𝑡𝑔𝛼 𝑑𝛼 𝑐𝑡𝑔𝛼 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝐶𝐷 𝐴𝐵 𝐴𝐵 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 66-69 Through series development results: − + + 𝜔 = 0 𝑎 𝑐 (10) − + + 𝜔 = 0 𝑏 𝑐 where: 𝜑 " 𝑎 sin𝛾 𝜔 = 𝑀 𝑐 sin𝛼 (11) 𝜑 " 𝑏 sin𝛾 𝜔 = 𝑀 𝑐 sin𝛽 M – factor of transformation of natural logarithms into decimal logarithms To specify: - The angles 𝛼 and 𝛽 it is calculated according to the sides 𝑎 , 𝑏 , 𝑐 measured. - The angle 𝛾 it is calculated according to the angels 𝛾 , 𝛾 measured. 1 2 Increases are considered to be errors and result in the system of equations of error, written as: 𝜌 𝜌 𝑣 − 𝑣 + + 𝜔 = 0 𝑎 𝑐 1 𝑎 𝑐 (12) 𝜌 𝜌 𝑣 𝑣 − 𝑣 + + 𝜔 = 0 𝑏 𝑐 2 𝑏 𝑐 From the system of normal equations, table 1 is formed: Table 1 Correction coefficients Correction 1/p a b 𝑣 𝑎 /2 0 𝑣 𝑏 /2 0 𝜌 𝜌 𝑣 𝑐 /2 − − 𝑐 𝑐 𝑣 1 𝑐𝑡𝑔𝛾 𝑐𝑡𝑔𝛾 The weights are: 𝑚 1 1 𝑝 = ( ) = ( ) = (13) 𝑚 𝑆 𝑆 or: 𝑝 = (14) 𝑚 2" 1 𝑝 = ( ) = ( ) = 𝑝 = 1 (15) 𝛾 𝛾 𝑚 2"√2 2 Literally the system of normal equations has the form: 𝑎 ′𝑎 ′ 𝑎 ′𝑏 ′ [ ] 𝐾 + [ ] 𝐾 + 𝜔 = 0 1 2 1 𝑝 𝑝 (16) 𝑎 ′𝑏 ′ 𝑏 ′𝑏 ′ [ ] 𝐾 + [ ] 𝐾 + 𝜔 = 0 1 2 2 𝑝 𝑝 𝑡𝑔𝛾 𝑡𝑔𝛾 𝑙𝑜𝑔 𝑙𝑜𝑔 𝑑𝛾 𝑐𝑡𝑔𝛾 𝑑𝑐 𝑑𝑏 𝑑𝛾 𝑐𝑡𝑔𝛾 𝑑𝑐 𝑑𝑎 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 66-69 By solving the system, the values of the correlates are obtained 𝐾 , 𝐾 , with which the values of the 1 2 corrections with the relations are obtained: 𝑣 = 𝐾 (17) 𝑎 1 𝑝 ∙𝑎 ′ 𝑣 = 𝐾 (18) 𝑏 2 𝑝 ∙𝑏 ′ ( ) 𝑣 = − 𝐾 − 𝐾 (19) 𝑐 1 2 𝑝 ∙𝑐 ′ 𝑣 = (𝐾 − 𝐾 ) (20) 𝛾 1 2 a', b', c' according to the sides of the weights are considered in m. The values of the determined corrections are added algebraically to the measured quantities obtaining the probable values with which the angles are calculated α and β [5]. 3. Conclusions Compared to the known methods, the processing of the measured quantities in order to obtain their probable values ensures the orientation of the topographic base in conditions of high precision, with special consequences regarding the accomplishment of the underground construction works. The additional volume of calculation that is required does not constitute a disadvantage compared to the requirements of making investments with special financial implications. References [1] Dima, N., 2005 Geodesy (in romanian), Universitas Publishing, Petroșani [2] Dima, N., Herbei, O., Vereș, I., 1999 Error theory and least squares method (in romanian), Universitas Publishing, Petroșani [3] Dima, N., ș.a., 1996 Mining topography (in romanian), Corvin Publishing, Deva [4] Filip, L., Dima, N., 2014 Special mining topography works, Universitas Publishing, Petroșani [5] Vereș, I., 2006 Automation of topo-geodetic works (in romanian), Universitas Publishing, Petroșani This article is an open access article distributed under the Creative Commons BY SA 4.0 license. Authors retain all copyrights and agree to the terms of the above-mentioned CC BY SA 4.0 license. 𝑚𝑚 𝑐𝑡𝑔𝛾 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mining Revue de Gruyter

Increasing the Accuracy of the Orientation of Underground Topographic Works

Filip, Larisa Ofelia
Mining Revue , Volume 27 (1): 4 – Mar 1, 2021
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de Gruyter
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© 2021 Larisa Ofelia Filip, published by Sciendo
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2247-8590
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10.2478/minrv-2021-0009
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Abstract

Revista Minelor – Mining Revue ISSN-L 1220-2053 / ISSN 2247-8590 vol. 27, issue 1 / 2021, pp. 66-69 INCREASING THE ACCURACY OF THE ORIENTATION OF UNDERGROUND TOPOGRAPHIC WORKS 1 Larisa Ofelia FILIP University of Petroșani, Petroșani, Romania, larisafilip@yahoo.com DOI: 10.2478/minrv-2021-0009 Keywords: topo-geodetic measurements, topographic network, geodetic network Abstract: Geodetic and topographic works determine the necessary information base and of major importance in the design and management of underground construction works (mining, hydrotechnics, roads, etc.). The efficiency and safety of these investments with special financial implications depend on their quality. In this context, the geodetic orientation of the underground topographic works was the main and permanent subject of study and analysis on which this scientific paper refers. 1. Introduction Underground topographic works are related to a reference system defined on the surface in a system of representation or projection, adopted at national or local level (cylindrical, stereographic). The underground materialization of the surface reference system is achieved by topographic operations of topographic transmission on underground works: horizontal, inclined or vertical. The topographic element of the materialization is the orientation for which purpose different methods were used. The methods used are numerous and complex in the topographic transmission of the orientation on a vertical underground work. For this situation, specific analysis are required, with the related constructions [1]. 2. The junction of topographic elements If on the surface the reference system is defined by points𝐴 , 𝐵 located in the free area of the vertical work (fig.1) they are transmitted underground by their verticals through the points 𝐴 , 𝐵 [2]. Fig.1 Underground topographic works relate to orientation 𝜃 of the direction 𝐶𝐷 frequently obtained by the method of the connecting triangle (𝐶𝐴𝐵 ). Distances are measured in the connecting triangle 𝑎 , 𝑏 , 𝑐 and 𝛾 (through 𝛾 and 𝛾 ). It is known to be 1 2 determined: Corresponding author: Larisa Ofelia FILIP, Assoc.Prof.PhD.Eng., University of Petroșani, Petroșani, Romania, contact details (20, University str., 332006 Petroșani, larisafilip@yahoo.com) 𝐶𝐷 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 66-69 - The angles 𝛼 and 𝛽 with relationships: 𝑠𝑖𝑛 𝛼 = 𝑠𝑖𝑛 𝛾 (1) 𝑠𝑖𝑛 𝛽 = 𝑠𝑖𝑛 𝛾 - Orientation 𝜃 with relationships: 𝐶𝐷 𝜃 = 𝜃 + 𝛼 + 𝛾 ± 200 = 𝜃 − 𝛽 + 𝛾 ± 2 ∙ 200 (2) 𝐶𝐷 1 2 It is observed that the error with which the guidance is obtained𝜃 it is determined by the errors with which the distances are measured 𝑎 , 𝑏 , 𝑐 and with which the angles are determined 𝛼 , 𝛽 and 𝛾 . But: 𝑠𝑖𝑛 𝛼 = 𝑎 − 𝑐 + 𝑠𝑖𝑛 𝛾 (3) By differentiation they are obtained: 𝑑𝑎 𝑑𝑐 𝑑𝛾 = − + (4) 𝑎 𝑐 𝑡𝛾𝑔 and: 2 2 2 𝑚 𝑚 𝑚 𝑎 𝑐 𝑚 = ( ) + ( ) + ( ) (5) 𝑎 𝑐 𝑡𝛾𝑔 analogous: 2 2 2 𝑚 𝑚 𝑏 𝑐 𝑚 = ( ) + ( ) + ( ) (6) 𝑏 𝑐 𝑡𝛾𝑔 Errors 𝑚 and 𝑚 are minimal for: 𝛼 𝛽 𝛼 → 200 (fig. 2a) 𝛽 → 0 Fig. 2a or: 𝛼 → 0 (fig. 2b) 𝛽 → 200 Fig. 2b This is the case of a triangle with the surface 𝑆 ≅ 0 (with angle 𝛾 very small). The minimum conditions presented cannot be technically met in all situations and as a result errors must be taken into account. 𝑚 , 𝑚 , 𝑎 𝑏 𝑚 , 𝑚 . These errors are obtained using the theory of direct measurements of the same accuracy, if we 𝑐 𝛾 consider the quantities (𝑎 , 𝑏 , 𝑐 , 𝛾 ) independent [3]. However, these quantities are dependent (conditioned) by the existing relations in the triangle. ( ) ( ) ( ) ( ) For probable values 𝑎 , 𝑏 , 𝑐 , 𝛾 it can be written: ( ) ( ) ( ) 𝑎 𝑏 𝑐 = = (7) ( ) sin 𝛼 sin 𝛽 sin 𝛾 The number of conditions is given by the excess of measurements [4]. 4 sizes are measured and 2 sizes are required. Consequently, two conditions result. They are: (𝑎 ) 𝑠𝑖𝑛 (𝛾 ) = (𝑐 ) 𝑠𝑖𝑛 𝛼 (8) ( ) ( ) ( ) 𝑏 𝑠𝑖𝑛 𝛾 = 𝑐 𝑠𝑖𝑛 𝛽 According to the theory of conditional measurements, proceed as follows: ( ) ( ) ( ) 𝑎 + 𝑠𝑖𝑛 𝛾 = 𝑐 + 𝑠𝑖𝑛 𝛼 (9) (𝑏 ) + 𝑠𝑖𝑛 (𝛾 ) = (𝑐 ) + 𝑠𝑖𝑛 𝛽 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑡𝑔𝛽 𝑡𝑔𝛼 𝑑𝛼 𝑐𝑡𝑔𝛼 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝑙𝑛 𝐶𝐷 𝐴𝐵 𝐴𝐵 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 66-69 Through series development results: − + + 𝜔 = 0 𝑎 𝑐 (10) − + + 𝜔 = 0 𝑏 𝑐 where: 𝜑 " 𝑎 sin𝛾 𝜔 = 𝑀 𝑐 sin𝛼 (11) 𝜑 " 𝑏 sin𝛾 𝜔 = 𝑀 𝑐 sin𝛽 M – factor of transformation of natural logarithms into decimal logarithms To specify: - The angles 𝛼 and 𝛽 it is calculated according to the sides 𝑎 , 𝑏 , 𝑐 measured. - The angle 𝛾 it is calculated according to the angels 𝛾 , 𝛾 measured. 1 2 Increases are considered to be errors and result in the system of equations of error, written as: 𝜌 𝜌 𝑣 − 𝑣 + + 𝜔 = 0 𝑎 𝑐 1 𝑎 𝑐 (12) 𝜌 𝜌 𝑣 𝑣 − 𝑣 + + 𝜔 = 0 𝑏 𝑐 2 𝑏 𝑐 From the system of normal equations, table 1 is formed: Table 1 Correction coefficients Correction 1/p a b 𝑣 𝑎 /2 0 𝑣 𝑏 /2 0 𝜌 𝜌 𝑣 𝑐 /2 − − 𝑐 𝑐 𝑣 1 𝑐𝑡𝑔𝛾 𝑐𝑡𝑔𝛾 The weights are: 𝑚 1 1 𝑝 = ( ) = ( ) = (13) 𝑚 𝑆 𝑆 or: 𝑝 = (14) 𝑚 2" 1 𝑝 = ( ) = ( ) = 𝑝 = 1 (15) 𝛾 𝛾 𝑚 2"√2 2 Literally the system of normal equations has the form: 𝑎 ′𝑎 ′ 𝑎 ′𝑏 ′ [ ] 𝐾 + [ ] 𝐾 + 𝜔 = 0 1 2 1 𝑝 𝑝 (16) 𝑎 ′𝑏 ′ 𝑏 ′𝑏 ′ [ ] 𝐾 + [ ] 𝐾 + 𝜔 = 0 1 2 2 𝑝 𝑝 𝑡𝑔𝛾 𝑡𝑔𝛾 𝑙𝑜𝑔 𝑙𝑜𝑔 𝑑𝛾 𝑐𝑡𝑔𝛾 𝑑𝑐 𝑑𝑏 𝑑𝛾 𝑐𝑡𝑔𝛾 𝑑𝑐 𝑑𝑎 Revista Minelor – Mining Revue vol. 27, issue 1 / 2021 ISSN-L 1220-2053 / ISSN 2247-8590 pp. 66-69 By solving the system, the values of the correlates are obtained 𝐾 , 𝐾 , with which the values of the 1 2 corrections with the relations are obtained: 𝑣 = 𝐾 (17) 𝑎 1 𝑝 ∙𝑎 ′ 𝑣 = 𝐾 (18) 𝑏 2 𝑝 ∙𝑏 ′ ( ) 𝑣 = − 𝐾 − 𝐾 (19) 𝑐 1 2 𝑝 ∙𝑐 ′ 𝑣 = (𝐾 − 𝐾 ) (20) 𝛾 1 2 a', b', c' according to the sides of the weights are considered in m. The values of the determined corrections are added algebraically to the measured quantities obtaining the probable values with which the angles are calculated α and β [5]. 3. Conclusions Compared to the known methods, the processing of the measured quantities in order to obtain their probable values ensures the orientation of the topographic base in conditions of high precision, with special consequences regarding the accomplishment of the underground construction works. The additional volume of calculation that is required does not constitute a disadvantage compared to the requirements of making investments with special financial implications. References [1] Dima, N., 2005 Geodesy (in romanian), Universitas Publishing, Petroșani [2] Dima, N., Herbei, O., Vereș, I., 1999 Error theory and least squares method (in romanian), Universitas Publishing, Petroșani [3] Dima, N., ș.a., 1996 Mining topography (in romanian), Corvin Publishing, Deva [4] Filip, L., Dima, N., 2014 Special mining topography works, Universitas Publishing, Petroșani [5] Vereș, I., 2006 Automation of topo-geodetic works (in romanian), Universitas Publishing, Petroșani This article is an open access article distributed under the Creative Commons BY SA 4.0 license. Authors retain all copyrights and agree to the terms of the above-mentioned CC BY SA 4.0 license. 𝑚𝑚 𝑐𝑡𝑔𝛾

Journal

Mining Revue – de Gruyter

Published: Mar 1, 2021

Keywords: topo-geodetic measurements; topographic network; geodetic network

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