Lecture 7

Misclosure error As stated at the start of this section, there are two methods of checking a leveling survey. (i) The leveling begins and finishes on the same point via the same or a different route, in which case the difference in level should be zero. (ii) The leveling begins on one point of known level and finishes on another, in which case the observed difference in level should equal the known difference. In all probability there will be an error in the closure, which should not exceed the allowance misclosure of the circuit. The allowable misiclosure for normal build­ing surveys depends upon the number of times the instrument is set up during a leveling. The generally acceptable formula for calculating the misclosure Eis: E = (±S.fii) mm

Where n is the number of times the instrument is set up during a circuit. For four set-ups the allowance is there­fore ±(S x 2) = 10 mm. should there be an error, within the .misclosure allowance, it should be distributed equally to each of the set-up points. If the error is out­with the allowance, the circuit of leveling should be repeated.   In the example of Fig. 3.10 there are in total six set-up positions. The allowable misdosure is (±S.j6) mm = 12 mm. The actual error of the circuit is +6 mm, so is within the allowance. Each set-up position there­fore receives a correction of -1 mm cumulatively and each FS point is corrected by -1 mm. The correction to the 6th (final) foresight is therefore -6 mm.   In Fig. 3.10, the points X and Y are points where both foresights and backsights are observed. As already seen the instrument position is moved after the foresight and reset before the backsight. The points X and Y are called change points. The letters CP are often inserted in the remarks column but are not absolutely necessary.   Arithmetic check As in all" surveying operations, a check should be pro­provided on the arithmetic. This is shown on lines 8, 9 and 10 of Table 3.3.  

Fig. 3.11

A moment's thought will show that the last reduced level is calculated as follows: Last reduced level = first reduced level + all rises - all falls Therefore last reduced level = first reduced level + sum rises -sum falls   Each rise or fall, however, is the difference between its respective backsight and foresight. Therefore, the sum of the rises minus the· sum of the falls must equal the sum of the backsights minus the sum of the foresights. The complete check is therefore: (Last reduced level- first reduced level) = (sum rises - sum falls) = (sum BS - sum FS) (23.906 - 23.900) = (8.300 - 8.294) = (15.850 - 15.844) =0.006   This particular leveling is an example of flying leveling. The shortest route between the points A and B is chosen and as few instrument settings as possible are used. Since the last point is actually also the first point in a closed circuit, then the last reduced level minus the first reduced level ·should be zero. The sum of the BS column should therefore equal the sum of the FS column. In the field, the leveling can be very quickly verified by simply checking the sum of the BS column against that of the FS column.  

While in theory the difference between the sums of the columns should be zero, in practice this is unlikely to happen, due to the minor errors that inevitably arise during a leveling. 3-In Fig. 3.11, four pegs are spaced around a construction site. These pegs are to be used as temporary bench marks for the duration of the site. A flying leveling was made around the pegs as shown, in order to establish the reduced levels of the pegs. Book the readings and calculate the reduced levels of the pegs, assuming that peg A has an assumed level of 10.000 m AD.   Answer (Table 3.4). Exercise 3.1 Figure 3.12 shows the station points of the linear survey of Chapter 3 (Fig. 3.1) and Table 3.5 shows the results of a flying leveling of those stations carried out from a nearby Ordnance Survey bench mark. Calculate the reduced levels of the stations.

(c) Series leveling When the reduced levels of many points are required, the method known as series leveling is used.   Points observed from single instrument station In Fig. 3.13, the reduced levels of five points B to F are required relative to a temporary bench mark A. Since all points can be observed from one instrument station, they are simply sighted "in turn. A backsight is taken to TBM A, followed by intermediate sights to Fig. 3.12

Bs S FS Rise Fall Reduced level Remarks 2.596 10.000 A 3.294 1.384 1.212 11.212 B 1.512 0.859 2.435 13.647 C 0.357 4.035 2.523 11.124 D 1.481 1.124 7.759 3.647 -7.759 -3.647 -10.000 0.000 Bs Is Fs rise fall Reduced level Remarks 1.955 5.200 OS BM 1.315 2.030 Station C 1.243 0.885 Station 0 2.071 1.485 Station E 1.570 0.880 Station F 1.835 1.590 Station A 0.631 0.540 Station B 1.200 3.289 1.130

Points B, C, D and E and finally a foresight to point F Points B, C, D and E and finally a foresight to point F. The readings are entered as in Table 4.6. The reduction of levels calculation simply involves finding the differ­ence in level between each pair of sights, A to B, B to C, C to D, D to E and E to F. The resultant positive or negative figures are then added successively to the known starting level of point A to produce the reduced levels of the various station points. Table 3.6 shows the complete reduction. The arithmetic check is applied in the same manner as in Sec. (b). The arithmetical check provided in all leveling checks only that the observed readings entered in the field book have been correctly computed. It does not prove that the reduced level of any point is correct. Verification of each reduced level is required, so the fieldwork must be repeated from a different instrument station (Example 4, following) which in effect makes the total leveling a dosed circuit

4-The leveling shown in Fig. 3 4-The leveling shown in Fig. 3.13 was re-observed from a second instrument station. The observations were as follows: Enter these results in a leveling table, then calculate and check the reduced levels of the stations from the temporary bench mark A (reduced level 1 07.520 m). The re-leveling shows that there is a 5 mm difference in the reduced levels of points C and F, which is acceptable. Answer (Table 3.7) A. 0.240 D.0.650 B.3.450 E.2.290 C. 0.655 F. 1.955 Fig. 3.13

SS IS FS Rise Fall Reduced level Distance Remarks 0.510 107.520 A. (TBM) 3.720 3.210 104.310 B. Foundation level 1 0.920 2.800 107.110 C. Foundation level 2 D. Foundation level 2 2.560 1.640 105.470 E. Foundation level 3 2.220 0.340 105.810 ~ Foundationlevel4 3.140 4.850 -4.850 -107.520 -1.710

Table 7 SS IS FS Rise Fall Reduced level Remarks 0.240 107.520 A(TBM) 3.450 3.210 104.310 8 0.655 2.795 107.105 C 0.650 0.005 107.110 2.290 1.640 105.470 E 1.955 0.335 105.805 F 3.135 4.850 -1.955 -4.850 -107.520 -1.715 1.715

Calculate the reduced levels of the stations Exercise 3.2 1 Figure 3.14 shows the station points of the linear survey of Fig. 3.1. Table 4.8(a) shows the results of a leveling of these points. Calculate the reduced levels of the stations BS IS FS Remarks 2.650 OSBM (5.200m) 2.727 Station C 2.292 Station D 2.537 Station E 1.346 Station F 1.370 Station A 0.065 Station B Fig. 3.14

Points observed from multiple instrument stations Table 4.8(b) shows the results of a check leveling of the stations from a different instrument set-up position. Calculate the reduced levels of the stations, as a check on the survey, and determine which, if any, stations have been wrongly observed. Table 3.8(b) SS IS FS Remarks 2.763 OS BM 2.840 StnC 2.385 Stn 0 2.650 Stn E 1.459 Stn F 1.463 StnA 0.178 Stn B Points observed from multiple instrument stations Figure 3.15 shows a small site where reduced levels are required at various points of detail, fences, manholes, etc. Since there are buildings on the site, it is not pos­sible to observe all of the points of detail from one instrument position. It is also probable that some of the points may be too high or too low or simply too far distant to permit a sight to be taken

The first set-up station is chosen to allow the bench mark (RL 35 The first set-up station is chosen to allow the bench mark (RL 35.27 m), to be sighted as a backsight. This is followed by sights to as many points as are possible or practicable. Thus, points A, B and C become intermedi­ate sights and point D becomes a foresight (change point I), since the next point E is too far away.   The second set-up station is chosen such that point D can be re-observed as a backsight. Points E and Fare observed as intermediate sights and, since point G is Fig. 3.14

Table 9 SS IS FS Rise Fall Reduced level Distance Remarks 1.56 35.27 Not Bench mark 1.43 0.13 35.40 required A. Manhole 0.59 0.84 36.24 B. Fence 1.07 0.48 35.76 C. Corner of building 2.35 1.09 0.02 35.74 D. Corner of building change point 1 2.48 35.61 E. Fence 1.98 0.50 36.11 F. Fence 0.95 1.76 0.22 36.33 G. Corner of building change point 2 1.50 0.74 0.21 36.54 H. Corner of building change point 3 1.35 0.15 36.69 I. Fence J. Fence 1.63 36.41 K. Corner of building 2.76 1.13 35.28 6.36 6.35 2.05 2.04 -6.35 -2.04 -35.27 =0.01 = 0.01

the only remaining visible station, it is observed as a foresight (change point 2). The procedure is repeated at a third instrument station. A backsight is taken to G and a foresight to H (change point 3).   The fourth and final set-up station is chosen such that point H is visible as a backsight, point I, J and K as intermediate sights, with a final foresight observed to the BM in order to close the leveling. The field results of this leveling are shown in Table 3.9. The reduction of levels follows the pattern detailed in the previous sections (b) and (c) but since multiple instrument stations are the norm in leveling, the reduction is now given in detail.   (i) Calculate the rise or fall between each pair of points in set-up no. 1. (BS. BM) - (IS. A) = 1.56 -1.43 = 0.13 (rise) (IS. A) - (IS. B) = 1.43 - 0.59 = 0.84 (rise) (IS. B) - (IS. C) = 0.59 - 1.07 = -0.48 (fall) (IS.C) - (FS.D) = 1.07 - 1.09 = -0 02 (fall)   (il) Calculate the rise or fall between each pair of points in set-up no. 2 (BS. D) - (IS. E) = 2.35 - 2.48 = -0.13 (fall) (IS. E) - (IS. F) = 2.48 - 1.98 = 0.50 (rise) (IS. F) - (FS G) = 1.98-1.76=0.22 (rise)

(iii) Calculate the rise or fall between pairs of points in set-up no (iii) Calculate the rise or fall between pairs of points in set-up no. 3. (BS. G) - (FS. H) = 0.95 - 0.7 4 = 0.21 (rise)   (iv) . Calculate the rise or fall between each pair of points in set-up no. 4. (BS.H) - (IS. I) = 1.50 - 1.35 = 0.15 (rise) (IS. I) - (IS. J) = 1.35 - 1.50 = -0.15 (fall) (IS. J) - (IS.K) = 1.50 - 1.63 = -0.13 (fall) (IS.K) - (FS.BM) = 1.63 - 2.76 =-1.13 (fall)       (v) Calculate the reduced levels of the points by suc­cessively adding the rises or falls to the BM value.

(vi) Apply the arithmetic check. RLBM RLA RLB RLC RLD RLE RLF RLG RLH RLI RLJ RLK = 35.27 m = 35.27 + 0.13 = 35.40 m = 35.40 + 0.84 = 36.24 m = 36.24 - 0.48 = 36.76 m = 35.76 - 0.02 = 35.74 m =35.74-0.13= 35.61 m = 35.61 + 0.50 = 36.11 m = 36.11 + 0.22 = 36.33 m = 36.33 + 0.21 = 36.54 m = 36.54 + 0.15 = 36.69 m = 36.69 - 0.15 = 36.54 m = 36.54 - 0.13 = 36.41 m = 36.41-1.13 = 35.28 m (vi) Apply the arithmetic check. Sum BS Col- Sum FS Col = 6.36 - 6.35 = 0.01 Sum Rises - Sum Falls = 2.05 - 2.04 = 0.01 Last RL - First RL = 35.28 - 35.27 = 0.01

The arithmetic check shows that the levels have been correctly calculated and also shows that an error of 10 mm has been made in the fieldwork. The normal acceptable limit of error is ±10 mm for this leveling. It must be pointed out that it is unusual to compute the results in parallel with the fieldwork for the simple reason that if any observation is wrongly read the cal­culated results must also be wrong. It is therefore wise to complete the fieldwork and verify that the leveling closes before calculating any results. The logical order in any leveling calculation is therefore: 1. Complete “the fieldwork. 2. Check that the· sum BS column = sum FS column within the acceptable limit of error. If the error exceeds these limits there is no point in continuing the calculation. 3. Calculate rises and falls. 4. Check that (sum rise column - sum fall column) = (sum BS column - sum FS column). 5. Calculate reduced levels. 6. Check that (last RL - first RL) = (sum rise column- sum fall column).

It is worth noting, at this stage, a common fallacy amongst surveyors, many of whom are unaware that the arithmetic check only proves that the field book notes, as written, are correctly computed. It does not, in fact, prove that the survey is correct. In Example 5 the check proves that the change point level only is correct. All of the intermediate sights could have been observed wrongly and the check would still work. In order to fully check the level of each point, the com­plete leveling must be repeated     Figure 3.16 shows the positions of a level and staff set up to observe levels along the time of a proposed drain. Draw up a page of a leveling book and reduce the readings, applying the appropriate arithmetic,. Check Fig. 3.15

BS IS FS Rise Fall Reduced level Distance Remarks 1.185 10.560 OS BM (10.560) 2.435 1.250 9.310 Drain 2.505 0.070 9.240 30 2.950 3.035 0.530 8.710 60 1.655 1.295 10.005 90 1.995 0.340 9.665 120 1.645 0.350 10.015 Site BM (10.015) 4.135 4.680 2.190 -4.680 -2.190 -10.560' -0.545

Exercise 3.3 1 Figure 3.17 shows the station points of the linear survey of Chapter 2 (Fig. 2.1) and Table 3.11 shows the results of a leveling of those stations from multiple set-up points. Calculate the reduced levels of the stations. Table3.11 BS IS FS Rise Fall Reduced level Remarks 1.256 5.200 OSBM 1.330 C 1.100 0.906 D 1.332 E 0.146 F 1.875 0.166 A 0.200 0.579 B 2.780 OS8M