An Algorithm for Adjustment of Geometric Levelling Networks

Abstract

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.