Prof. Dr. techn. Hussein A. Abd-Elmotaal Page 1 of 5
Computation of the levels of points
There are two methods to compute the level of points in a surveying levelling. They are:
1. Collimation Method 2. Rise and fall method 1. Collimation Method
The crest level CL (level of the horizontal line of sight) is computed by (cf. Fig. 1) b
A
CL=( )+ (1)
Fig. 1: Theory of surveying levelling.
Accordingly, the level of point B can be computed by f CL B)= −
( (2) Table 1 shows the computation of the levels of points using the collimation method for the levelling shown in Fig. 2.
Prof. Dr. techn. Hussein A. Abd-Elmotaal Page 2 of 5 Fig. 2: Example of a levelling work.
Table 1: computation of the level of points using the collimation method readings
point
BS IS FS CL levels remarks
A 1 I = (A)+1 (A) Benchmark
B 3 2 II = (B)+3 (B) Rotation point
C 5 4 III = (C)+5 (C) Rotation point
D 6 (D)
E 7 (E)
S 1+3+5 6 2+4+7 (A)+(B)+(C)+(D)+(E)
where
(B) = I-2 = (A)+1-2 (C) = II-4 = (B)+3-4 (D) = III-6 = (C)+5-6 (E) = III-7 = (C)+5-7 Computational checks
The computational checks are made to assure that the calculations were done in a correct way.
It doesn’t verify that the levels of points are correct as this depends on the levelling observations. Note that if you wrongly arrange the levelling observations into the levelling table, this will lead to wrong levels of the levelling points regardless of the correctness of the computational checks.
For the collimation method, we have three computational checks; they are:
Prof. Dr. techn. Hussein A. Abd-Elmotaal Page 3 of 5 1. Number of backsights equals number of foresights.
2. Inserting Eq. (1) into Eq. (2) gives:
f b A B)−( )= −
( (3) which can be interpreted as (note that Fig. 1 already shows a complete levelling of only one position of the level):
level of last point – level of first point = sum of backsights – sum of foresights
3. Sum of levels of all points except the first one + sum of foresights + sum of intermediate sights = sum of the crest levels times number of its usage to compute level of new points.
To illustrate the usage of the computational checks of the collimation method, let us perform them on the levelling example given in Table 1.
1. Number of backsights = number of foresights = 3.
2. (E)-(A) = (C)+5-7-(A) = (B)+3-4+5-7-(A) = (A)+1-2+3-4+5-7-(A) = (1+3+5)-(2+4+7) sum of backsights – sum of foresights = (1+3+5)-(2+4+7)
\ level of last point – level of first point = sum of backsights – sum of foresights = (1+3+5)-(2+4+7)
3. sum of levels of all points except the first one + sum of foresights + sum of intermediate sights = (B)+(C)+(D)+(E)+(2+4+7)+(6) = [(A)+1-2]+[(B)+3-4]+[(C)+5-6]+[(C)+5-7]+(2+4+7+6)
= (A)+(B)+2*(C)+(1+3+5+5)-(2+4+6+7)+(2+4+7+6) = (A)+(B)+2*(C)+(1+3+5+5) sum of the crest levels times number of its usage to compute level of new points =
= I*1 +II*1+III*2 = [(A)+1]*1+[(B)+3]*1+[(C)+5]*2 = (A)+(B)+2*(C)+(1+3+5+5)
\ sum of levels of all points except the first one + sum of foresights + sum of intermediate sights = sum of the crest levels times number of its usage to compute level of new points = (A)+(B)+2*(C)+(1+3+5+5)
2. Rise and Fall Method
Equation (3) can be interpreted as:
Level of next point – level of previous point = staff reading at the previous point – staff reading at the next point (cf. Fig. 1).
Note that both observations should refer to the same position of the level. This is the basis of the rise and fall method.
Prof. Dr. techn. Hussein A. Abd-Elmotaal Page 4 of 5 Table 2 shows the calculation steps of the same levelling example given in Fig. 2.
Table 2: computation of the level of points using raise and fall method readings
point
BS IS FS rise/fall levels remarks
A 1 (A) Benchmark
B 3 2 1-2 (B) Rotation point
C 5 4 3-4 (C) Rotation point
D 6 5-6 (D)
E 7 6-7 (E)
S 1+3+5 6 2+4+7 (1+3+5)-
(2+4+7) (A)+(B)+(C)+(D)+(E) where
(B) = (A)+1-2 (C) = (B)+3-4 (D) = (C)+5-6 (E) = (D)+6-7
Note: verify that the levels computed by the collimation method equal those computed by the rise and fall method.
Computational checks
For the rise and fall method, we have three computational checks; they are:
1. Number of backsights equals number of foresights.
2. From Eq. (3), we have
level of last point – level of first point = sum of backsights – sum of foresights 3. Equation (3) can also be interpreted as:
level of last point – level of first point = sum of rise and fall
To illustrate the usage of the computational checks of the rise and fall method, let us perform them on the levelling example given in Table 2.
1. Number of backsights = number of foresights = 3.
2. (E)-(A) = (C)+5-7-(A) = (B)+3-4+5-7-(A) = (A)+1-2+3-4+5-7-(A) = (1+3+5)-(2+4+7)
Prof. Dr. techn. Hussein A. Abd-Elmotaal Page 5 of 5 sum of backsights – sum of foresights = (1+3+5)-(2+4+7)
\ level of last point – level of first point = sum of backsights – sum of foresights = (1+3+5)-(2+4+7)
3. sum of rise and fall = (1+3+5)-(2+4+7)
\ level of last point – level of first point = sum of rise and fall = (1+3+5)-(2+4+7)
Common sources of errors in levelling
1. Instrument not correctly levelled.
2. Telescope not correctly focused.
3. The wrong cross-hair reading recorded (e.g., top instead of middle).
4. Staff incorrectly read or not held vertical.
5. Staff incorrectly booked.
All the above are mistakes (blunders) and cannot be corrected unless the field work is repeated.
Allowable error in levelling
The surveying works in general need always some filed checks. This is done by taken more observations than necessary, e.g., observe another benchmark at the end of the levelling route.
Accordingly one can compute the existing error in the levelling network, which is usually denoted by the closing error. Ideally, the closing error would vanish. This is not practically possible due to the errors inherent in the field work as given in the chapter of “Theory of Errors”, i.e., the closing error ∫ 0. The allowable error errallowin millimeter is given generally by
l c
errallow = (4) where l stands for the total distance of the levelling route in km and c is a constant depending on the levelling type, where
c = 5 for the precise levelling, c = 10 for the longitudinal levelling, c = 20 for the ordinary levelling.
If the closing error is smaller than or equal the allowable error, the levelling work can be adjusted. One method of adjustment is the equal weight adjustment, where all points get the same corrections in such a way to lead to zero closing error. If the closing error is larger than the allowable error, the field work should be repeated.