An Algorithm for Adjustment of Geometric Levelling Networks

• Autorzy: Cvetkov, Vasil

Warianty tytułu

PL

Algorytm regulacji geometrycznych sieci niwelacyjnych

• Języki publikacji: EN

Abstrakty

EN

The natural way to reduce the duration of measurement of a levelling network is to cut down on the number of levelling lines without damaging the quality of the final results. The main objective of the study is to demonstrate that this is possible without any lack of accuracy, if some mathematical facts regarding the average of both measurements of the line elevations are taken into account. Based on 60 paired random samples of size 1000, derived from different continuous distributions, e.g., N (0, 1), U (-1.732, 1.732) and Gamma (1, 1), each of them with theoretical standard deviation σ=1, it was found that the averages of each pair form new distribution with standard deviation σ≈0.707. However, the samples, which were formed by selecting the nearest to the known theoretical expectation from both measurements and their average have distributions, which standard deviations tend to σ≈0.53, σ≈0.46 and σ≈0.43 for the U (-1.732, 1.732), N (0, 1) and Gamma (1, 1) distributions, respectively. Therefore, if we choose the more appropriate value from the “first”, the “second” measurement and their average, we will increase the accuracy of the network almost √2 times in comparison to the accuracy, yielded by the only use of the averages. If our network contains n lines, the process of finding of these elevation values, which leads to the best fit of the network, is based on 3n single adjustments of the network. In addition, we can minimize the impact of the shape of the network on the final standard errors of the adjusted heights or geopotential numbers of the nodal benchmarks in the network, if we apply some iterative procedures, e.g., Inverse Distance Weighting (IDW), Inverse Absolute Height Weighting (IAHW), etc. In order to check the above explained algorithm, the Second Levelling of Finland network was adjusted in three variants. In the first variant, the whole network was adjusted as a free one. The classical weights w=L-1 were used. In the second variant, the network was separated into two parts. Applying 312 and 314 independent adjustments, the selection of the best fitted values of line elevations was done and the network was adjusted by using them. The IDW and IAHW with power parameter p=5 were finally applied. In the third variant, the network was separated in four parts. Applying 313, 312, 316 and 312 independent adjustments, the new selection of the line elevations was done and the network was adjusted by them. The IDW (p=6.5) and IAHW (p=6) were executed. Comparison of the standard errors of the adjusted geopotential numbers in the separate variants revealed that there was no statistically significant difference between the results, yielded in the second and the third variant. However, these variants produced 3-5 times increase of the accuracy in comparison to the classical first variant. The best results were obtained in the second variant with IAHW, where the mean value of the standard errors of the adjusted geopotential numbers is below 1.4 mgpu.

Słowa kluczowe

PL

algorytm; adiustacja; geometryczne sieci niwelacyjne; dokładność

EN

algorithm; adjustment; geometric levelling networks; accuracy

• Czasopismo: Inżynieria Mineralna
• Rocznik: 2024
• Tom: Vol. 1, no. 1
• Strony: 187--194
• Opis fizyczny: Bibliogr. 12 poz., tab., wykr.

Twórcy

autor

Cvetkov, Vasil

• University of Architecture, Civil Engineering and Geodesy, Geodetic Department, 1 Hristo Smirnenski Blvd., 1164 Sofia, Bulgaria,

Bibliografia

• 1. M. Sacher, J. Ihde, G. Liebsch and J. Mäkinen, “EVRF2007 as Realization of the European Vertical Reference System”, Presented at the Symposium of the IAG Sub-commission for Europe (EUREF) in Brussels, June 18–21 2008.
• 2. Ł. Borowski, B. Kubicki and J. Gołąb, “Implementation of the EVRF2007 height reference frame in Poland” Journal of Applied Geodesy, 2023, https://doi.org/10.1515/jag-2023-0020.
• 3. S. Gospodinov and K. Stereva, “Determining of areas on the territory of R Bulgaria with a low intensity of the recent vertical movements of the Earth's crust”, 20th International Multidisciplinary Scientific Geo Conference SGEM 2020, Vol. 20, Issue 2.2, https://doi.org/10.5593/sgem2020/2.2/s09.006.
• 4. K. Kowalczyk and J. Rapinski, “Evaluation of levelling data for use in vertical crustal movements model in Poland”, Acta Geodyn. Geomater., 10, 4 (172), 401-410, 2013, https://doi.org/10.13168/AGG.2013.0039.
• 5. L. Borowski, J. Kudrys, B. Kubicki, M. Slámová and K. Maciuk, “Phase Centre Corrections of GNSS Antennas and Their Consistency with ATX Catalogues.” Remote Sens. 2022, 14, 3226. https://doi.org/10.3390/rs14133226.
• 6. Y. Tanaka and Y. Aoki, “A Geodetic Determination of the Gravitational Potential Difference Toward a 100-km-Scale Clock Frequency Comparison in a Plate Subduction Zone”. In: International Association of Geodesy Symposia. Springer, Berlin, Heidelberg, 2022, https://doi.org/10.1007/1345_2022_147.
• 7. A. Angelov, “Geodetic methods in the study for deformation process of high buildings and engineering facilities”, Monographic, Second Edition, 2022, https://uacg.bg/filebank/att_22864.pdf, (in Bulgarian).
• 8. E. Kääriäinen, “The Second Levelling of Finland in 1935–1955”, Publications of the Finnish Geodetic Institute No. 61, 1966, Helsinki, http://hdl.handle.net/10138/347072.
• 9. V. Cvetkov, “Two adjustments of the second levelling of Finland by using nonconventional weights” Journal of Geodetic Science, vol. 13, no. 1, 2023, https://doi.org/10.1515/jogs-2022-0148.
• 10. V. Cvetkov and S. Gospodinov, “Inverse Absolute Height Weighting in the Highest Order Levelling Networks”, EGU General Assembly 2023, Vienna, Austria, 24–28 Apr 2023, EGU23-4219, https://doi.org/10.5194/egusphere-egu23-4219 , 2023.
• 11. V. Saaranen, P. Lehmuskoski, M. Takalo and P. Rouhiainen, “The Third Precise Levelling of Finland”, FGI Publications No. 161, Kirkkonummi, 2021, The Third Precise Levelling of Finland (helsinki.fi).
• 12. Ю.Т. Кузнецов, “Altitude support. State leveling network (main high-rise base)”, Gravimetry and Geodesy, 2010, Moscow, (in Russian).

Uwagi

Opracowanie rekordu ze środków MNiSW, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2024).

Identyfikatory

DOI

10.29227/IM-2024-01-20