Institute of Geophysics, University of TehranJournal of the Earth and Space Physics2538-371X37420120121Accuracy evaluation and adjustment of the first order Leveling
Network of IranAccuracy evaluation and adjustment of the first order Leveling
Network of Iran1471632430710.22059/jesphys.2012.24307FAAbdolrezaSafariYahyaJamourAbdolrahmanMostafaeiJournal Article19700101In many countries, leveling networks are established for height determination which is one of the most important topics in geodesy. In these networks, the sum of the leveled height differences between A and B will not be equal to the difference in the orthometric heights HA and HB. The reason is that the leveling increment ?n, as we henceforth denote it, is different from the corresponding increment ?HB of HB , due to the nonparallelism of the level surfaces. Denoting the corresponding increment of the potential W by ?W, we have
??W = g ?n = g'?HB , (1)
where g is the gravity at the leveling station and g' is the gravity on the plumb line of B at ?HB. Hence,
(2)
There is, thus, no direct geometrical relation between the result of leveling and the orthometric height, since Equation (2) expresses a physical relation. If gravity g is also measured, then
(3)
is determined, so that we obtain
(4)
Thus, leveling combined with gravity measurements determines potential differences, which are, physical quantities.
It is somewhat more rigorous theoretically to replace the sum in Equation (4) by an integral, obtaining
(5)
Note that this integral is independent of the path of integration. In practical cases, it is better to use geo-potential numbers, which are calculated using Equation (6), instead of potential values.
(6)
Users usually like to work by the geometrical concept of the height. Therefore, the orthometric height of the point A is defined by
, (7)
where is mean value of the gravity along the plumb line between the geoid and the surface point A. According to potential differences, on the other hand, difference between orthometric height of two points is
(8)
Where is normal gravity for an arbitrary standard latitude.
So, determination of difference in orthometric heights between points is changed to determination of potential differences between them. Then, it is necessary to measure both height difference and gravity along the leveling lines.
Observations in leveling networks are under influence of random and systematic errors. Errors originating from instruments, ambient circumstances and observer, have such character that it is very difficult to remove them from observations, also assessment of leveling accuracy is not an easy task.
Unmodelled systematic effects in levelling may be revealed through autocorrelation function of discrepancies (Vanicek and Craymer, 1983) between the forward and backward running of levelling sections. Test results, conducted with simulated data indicate that autocorrelation function can be used as a diagnostic tool to detect systematic effects.
The aim of this study is accuracy estimation of the first order levelling network of Iran by the Lallemand’s and Vignal’s formulas as well as test for significant differences between lines caused by different sources of random and systematic errors. Then, computation of section and line discrepancies is explained and the random and systematic error computed by the Lallemand’s and Vignal’s formulas is portrayed. Next, the theory of analysis of variance is given in outline and practical computations are demonstrated. After that, various kinds of adjustment models for the levelling network adjustment are discussed. Finally, the weight matrix, which is estimated using covariance function, is applied to adjust the network. The obtained results, in this research, showed that there are considerable systematic errors in the levelling network of Iran.In many countries, leveling networks are established for height determination which is one of the most important topics in geodesy. In these networks, the sum of the leveled height differences between A and B will not be equal to the difference in the orthometric heights HA and HB. The reason is that the leveling increment ?n, as we henceforth denote it, is different from the corresponding increment ?HB of HB , due to the nonparallelism of the level surfaces. Denoting the corresponding increment of the potential W by ?W, we have
??W = g ?n = g'?HB , (1)
where g is the gravity at the leveling station and g' is the gravity on the plumb line of B at ?HB. Hence,
(2)
There is, thus, no direct geometrical relation between the result of leveling and the orthometric height, since Equation (2) expresses a physical relation. If gravity g is also measured, then
(3)
is determined, so that we obtain
(4)
Thus, leveling combined with gravity measurements determines potential differences, which are, physical quantities.
It is somewhat more rigorous theoretically to replace the sum in Equation (4) by an integral, obtaining
(5)
Note that this integral is independent of the path of integration. In practical cases, it is better to use geo-potential numbers, which are calculated using Equation (6), instead of potential values.
(6)
Users usually like to work by the geometrical concept of the height. Therefore, the orthometric height of the point A is defined by
, (7)
where is mean value of the gravity along the plumb line between the geoid and the surface point A. According to potential differences, on the other hand, difference between orthometric height of two points is
(8)
Where is normal gravity for an arbitrary standard latitude.
So, determination of difference in orthometric heights between points is changed to determination of potential differences between them. Then, it is necessary to measure both height difference and gravity along the leveling lines.
Observations in leveling networks are under influence of random and systematic errors. Errors originating from instruments, ambient circumstances and observer, have such character that it is very difficult to remove them from observations, also assessment of leveling accuracy is not an easy task.
Unmodelled systematic effects in levelling may be revealed through autocorrelation function of discrepancies (Vanicek and Craymer, 1983) between the forward and backward running of levelling sections. Test results, conducted with simulated data indicate that autocorrelation function can be used as a diagnostic tool to detect systematic effects.
The aim of this study is accuracy estimation of the first order levelling network of Iran by the Lallemand’s and Vignal’s formulas as well as test for significant differences between lines caused by different sources of random and systematic errors. Then, computation of section and line discrepancies is explained and the random and systematic error computed by the Lallemand’s and Vignal’s formulas is portrayed. Next, the theory of analysis of variance is given in outline and practical computations are demonstrated. After that, various kinds of adjustment models for the levelling network adjustment are discussed. Finally, the weight matrix, which is estimated using covariance function, is applied to adjust the network. The obtained results, in this research, showed that there are considerable systematic errors in the levelling network of Iran.https://jesphys.ut.ac.ir/article_24307_9f323f11e6618888c02343f5f8f9f146.pdf