Wyniki wyszukiwania - Biblioteka Nauki

Nowa wersja platformy, zawierająca wyłącznie zasoby pełnotekstowe, jest już dostępna.
Przejdź na https://bibliotekanauki.pl

Ograniczanie wyników

2 2024

2 Cvetkov V.

Znaleziono wyników: 2

Liczba wyników na stronie

Wyniki wyszukiwania

Sortuj według:

Ogranicz wyniki do:

An Algorithm for Adjustment of Geometric Levelling Networks

100%

Cvetkov V.

Inżynieria Mineralna

2024

tom Vol. 1, no. 1

187--194

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.

Geometric Levelling Data and Some Systematic Faults in Their Treatment

100%

Cvetkov V.

Inżynieria Mineralna

2024

tom Vol. 1, no. 1

293--300

The aim of this article is to illuminate some latent systematic faults in the mathematical treatment of precise levelling data. The first one is associated with the use of the average of both measurements of the height differences between the terminal benchmarks in levelling lines. Another weak point in the classical treatment of levelling data is the incomplete minimization of the impact of the spatial network configuration on the produced mean standard errors of the nodal benchmarks from the adjustment. Generating sixty random paired samples of size 1000, derived from three continuous distributions, e.g. Normal (0, 1), Uniform (-1.732, 1.732) and Gamma (1, 1), it was found that the average of two same distributed and ordered observations is very nearby to the theoretical expectation, in comparison to both observations, only in approximately 27-30% of all cases. Contrary, in other 70-74% of cases, either the “first” or the “second” observation is in close proximity to the expectation. The miss of this fact leads to a statistically significant deterioration of the final accuracy of the levelling networks. In the current study, it is also shown that the minimization of the standard errors of the adjusted normal heights of the nodal benchmarks in the Bulgarian Levelling Network 1980 cannot be achieved with the weights w=const.L-1, which are the most popular and used type of weights in the adjustment of geometric levelling networks. Finally, it is illustrated that taking into account the above marks and applying an appropriate adjustment algorithm, the mean of the standard errors of the adjusted heights of the nodal benchmarks in the analysed network is possible to be less than 1mm. The standard error of the adjusted height of the most remoted benchmark “Pushkarov”, which is 598 km far away from the datum point located in Varna, is equal to 1.40mm. The obtained from the adjustment mean standard error for the weight unit is estimated to be 0.164 mm/√km. In comparison, the adjustment mean standard error for the weight unit, but yielded by the classical approach of adjustment of the analysed network, is 1.289 mm/√km or almost 9 times higher. Despite being tedious and time-consuming, it is not on point of discarding the precise geometric levelling as a main geodetic method for solving of a couple of scientific and engineering tasks, where differences in heights have to be determined with the highest accuracy.