Differential Leveling
Differential leveling is the process of finding the difference in elevation between two or more points. When the two points are within the sight limits of the instrument, two readings are taken. The difference in rod readings represents the difference in elevation between the two points. When one or more of the points are beyond the range of the instrument turning points are used.
One of the most common applications of differential leveling is to run a circuit of sights to determine the elevations of one or more bench marks relative to a previously established bench mark. The procedure for differential leveling will be described using this type of circuit, illustrated schematically in Figure 15.9. The diagram shows that three instrument setups were made in traveling from BM1 to BM2. Also note that a "return check" was made between BM2 and BM1, and that three more setups were made in this phase of the survey.
The survey begins with the instrument person going forward a convenient distance (not exceeding the limits of the instrument) and setting up the level, following the procedure previously described. The instrument person sights on the rod while it is held on the top of BM1 by the rod person, and notes a center crosshair reading of 3.03 ft. This is the backsight, so the 3.03 ft rod reading is added to the BM1 elevation (assumed 100.00 ft), resulting in a height of instrument (HI) of 103.03 ft. The rod person then moves forward past the instrument and selects a turning point, TP1. The FS rod reading of this TP1 is 3.86 ft. The foresight is subtracted from the HI, and the elevation of TP1 is found to be 99.17 ft. The instrument can now be moved forward and set up at a new position. A backsight rod reading of 2.60 ft is observed on TP1 and added to the TP1 elevation of 99.17 ft, giving an instrument height of 101.77 ft. Again a new turning point, TP2, is selected, and a FS rod reading of 4.53 ft is recorded. This rod reading is subtracted from the HI of 101.77 ft, giving the elevation of 97.24 ft for TP2. This process is repeated a third time, and the elevation of BM2 is found to be 95.30 ft. We now know that
the difference in elevation between BM1 and BM2 is 100.00 ft minus 95.30 ft, or 4.7 ft, assuming no errors were made. In summary, the procedure for differential leveling is as follows:
1. Set up the instrument.
2. Take the BS reading on BM1.
3. Establish the TP, and take the FS reading.
4. Move the instrument, and set up again.
6. Establish the next TP, and take the FS reading.
7. Move the instrument, and set up again.
8. Repeat steps 5 to 7 until a foresight is taken on the last station.
The data for the differential leveling survey in Figure 15.9 are recorded in the data table, Table 15.1. The lefthand page of a standard surveying field notebook contains five columns. These columns are needed for the column headings of station (STA), backsight (BS), height of instrument (HI), foresight (FS), and elevation (ELEV). Additional columns may be added as needed to record additional information, for example, the distance for each sight.
STA 
BS 
HI 
FS 
ELEV 
BM1 
3.03 
103.03 
100.00  
TP1 
2.60 
101.77 
3.86 
99.17 
TP2 
4.22 
101.46 
4.53 
97.24 
BM2 
6.43 
101.73 
6.16 
95.30 
TP3 
3.85 
101.14 
4.44 
97.29 
TP4 
5.11 
103.29 
2.96 
98.18 
BM1 
3.30 
99.99 
15.10.1. Error Control
In surveying, it is important to control as many errors as possible. For differential leveling surveys three error checks should be completed: close the loop, note check, and calculation of the allowable error of closure. The survey is closed to provide the information for the other two checks. The note check is conducted to catch any mathematical errors in the notes. For checking notes, the absolute value of the sum of the foresights minus the absolute value of the sum of the backsights, should equal the absolute value of the difference (A) in elevation for BM1 (beginning and closure elevation). Expressed mathematically:
Problem: Are the field notes in Table 15.1 accurate?
Solution:
EBS EFS
3.86 
3.03 
4.53 
2.60 
6.16 
4.22 
4.44 
6.43 
2.96 
3.85 
3.30 
5.11 
25.25 
25.24 
£FS  £BS =  A Elevation BM1  A Elevation BM1 = BM1beginnmg  BMlending 25.25  25.241 = 1100.00  99.99 0.01 = 0.01.
£FS  £BS =  A Elevation BM1  A Elevation BM1 = BM1beginnmg  BMlending 25.25  25.241 = 1100.00  99.99 0.01 = 0.01.
The note check equation is true, this means the 0.01 ft difference in the elevation of BM1 is not caused by a math error in the data table. An error that is discovered after the fact cannot be corrected; therefore the next step is to determine if the error is acceptable.
The third check is for the error of closure. The allowable error of closure procedure was established because surveyors realized that it was impossible to survey without some error. When perfection is not possible, limits of acceptability must be established. The equation for allowable error of closure sets the acceptability limits for surveys.
where AE is the allowable error, K is a constant ranging from 0.01 to 1.0, and M is the distance traveled (mi).
The constant K is determined by the level of survey. A high order survey, one with very little allowable error, would use a K value of 0.01. A very low order survey, one where more error is acceptable, might use a K value of 1.0. The order of survey must be established before the survey is started because it will determine the quality of the equipment and the procedures that must be followed to collect the data. A K value of 0.10 or 0.05 is acceptable for most general construction and agricultural surveys.
Problem: Is the closure error of 0.01 ft for the differential leveling survey in Figure 15.9 acceptable if the total distanced surveyed, out and back, was 3,600 ft and a K value of 0.10 is acceptable?
Solution:
0.01 < 0.08 The closure error is acceptable.
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