For the identification of stable points using invariant analysis the height differences from adjustment are analyzed. By analyzing the changes in the height differences the following results are presented in tab. 3. The height differences from point 10( in bold) exceed the allowable value for height differences.
The
From |
to |
1 |
epoch Heigh difference [ m ] |
Table 3. Allowable difference 2 epoch
Height difference
[ m ]
Mean square error [ mm ]
m1
Mean square error [ mm ]
m2 difference 2 epoch –
1 epoch [ mm ]
Allowable difference Δh
2 epoch – 1 epoch
[ mm ]
10 |
11 |
-0.3276 |
0.1 |
-0.3252 |
0.1 |
2.4 |
0.3 unstable |
10 |
12 |
-0.1886 |
0.1 |
-0.1861 |
0.1 |
2.5 |
0.3 unstable |
10 |
13 |
-0.4190 |
0.1 |
-0.4167 |
0.1 |
2.3 |
0.3 unstable |
11 |
12 |
0.1390 |
0.1 |
0.1391 |
0.1 |
0.1 |
0.3 stable |
11 |
13 |
-0.0914 |
0.1 |
-0.0915 |
0.1 |
-0.1 |
0.3 stable |
12 |
13 |
-0.2304 |
0.1 |
-0.2306 |
0.1 |
-0.2 |
0.3 stable |
formula( 1) is used to compute the allowable height difference Δh. |
( 1) Δ h = 2. 0 m 2 2 1 + m 2,
where 2.0 is the 95 % confidence level expansion factor, m 1 and m 2 are the mean square errors for the first and second epoch deformation surveys( tab. 3).
The procedure for detection of deformations using the Penev method uses free network adjustment results from the first and second epochs. Deformation analysis of a leveling network is performed using the principle of maximal displacement. The point with maximal displacement is removed, and the network is readjusted until there aren’ t statistically significant displacements. P. Penev focuses on the fact that if the elevation of one point in the network changed with Δh, other points’ elevations changed with Δh / n. For this reason, it is advisable to remove unstable benchmarks one by one, starting with the point that has the largest height difference. Comparing the elevation differences results of the first and second epoch adjustments shows the maximal displacement of station 10( tab. 4). Later, we apply the check to the sum of the heigh differences, and this sum is equal to zero( tab. 4). The conclusion is that point 10 changed his elevation, and the displacements of other points are due to this change.
Point
1 Epoch elevation [ m ]
Table 4. Network adjustment 1 Epoch 2 Epoch standard elevation deviation [ m ]
[ mm ]
2 Epoch standard deviation [ mm ]
Height difference Δh [ mm ]
10 |
500.6001 |
0.1 |
500.5983 |
0.1 |
-1.8 |
11 |
500.2725 |
0.1 |
500.2731 |
0.1 |
0.6 |
12 |
500.4115 |
0.1 |
500.4122 |
0.1 |
0.7 |
13 |
500.1811 |
0.1 |
500.1816 |
0.1 |
0.5 ∑ Δh = 0.0 |
To identify stable benchmarks using the Milev method, leveling benchmarks one by one is accepted as stable and the heights and mean square errors of the remaining points are calculated. This leads to homogenization and minimization of mean square errors, because several mean square errors are obtained for each individual benchmark. The minimal mean square error of every point is accepted in the analysis. From the selected datum point at least one stable point( better two) from the group of previously known reference points must be found after deformation analysis. By applying a constrained adjustment with a fixed height at point 10, points 11, 12 and 13 are determined to be unstable. By applying a constrained adjustment with a fixed height at point 11, points 12 and 13 are determined to be stable, while point 10 is unstable. Since the condition for finding other stable point( s) is satisfied for point 11 the deformation analysis ended with the conclusion that point 10 is unstable. The decision of which point is most stable can also be made from a free network adjustment. The minimum and maximum displacements from free network adjustment are the starting points for further analyses. Point 13 has the minimum displacement( Tab. 4) and can be accepted as the“ most stable”. Once again, by
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