Differential Level Errors
We discussed the Canadian datum used in leveling and leveling methods and equipment in the field
practice. However, there are a number of errors that we should consider in more detail when
determining vertical distances from one point to another. These include: i) the earth’s curvature, ii)
atmospheric refraction of light, iii) collimation error
Curvature Error: (Section 4.4)
(R + C )
2
= K 2 + R2
R 2 + 2RC + C 2 = K 2 + R 2
K
C(2R + C) = K 2
Horizontal
C=
K2
K2
2R + C 2R
Vertical
R
Ve
rti
ca
l
Since R >> C then 2R + C
R
Le
ve
l
C
Ckm =
K2
km2
= 0.0000785K 2
= 0.0000785K 2 km
2(6370)
km
O
Cm = 0.0785K 2 m
R = Mean Radius of Earth = 6370 km
where K is in kilometers
Example: Determine Curvature Error for K=1.8 km
Cm = 0.0785(1.8)2 = 0.254 m
ENGI 3703
Surveying and Geomatics
Topic
Leveling Errors
Instructor: Prof. Ken Snelgrove
Lect 5 - Sept 19/07 Slide 1 of 7
Refraction Error: (Section 4.4) Light waves are bent as they move from one medium to another.
The atmosphere cools and has lower pressure as we rise in the atmosphere. This tends to refract
light upward away the surface. This makes objects (such as a rod readings!) appear higher then they
actually are. In an atmosphere of constant density there would be no refraction. While the effect is
temperature, humidity and pressure controlled, it is approximated as 1/7 of the curvature of the earth
acting in the opposite direction of curvature
1
(0.0785)K 2
7
= 0.011K 2
Rm =
The two effects are often combined to give :
hm = 0.0785K 2 0.011K 2
= 0.0675K 2 m
where K is in kilomters
Example: Determine Refraction and
Combined Error for K=1.8 km
Rm = 0.011(1.8)2 = 0.036 m
hm = 0.0675(1.8)2 =0.218 m
ENGI 3703
Surveying and Geomatics
Topic
Leveling Errors
Instructor: Prof. Ken Snelgrove
Lect 5 - Sept 19/07 Slide 2 of 7
Collimation Error: (Section 4.15.5) While we assume that our level instrument, when leveled,
shoots along a horizontal line, there is no guarantee of this. The cross hairs many be out of
alignment, the leveling tube may not be in line with the optics of the instrument (figure 4.11), the
automatic compensation (figure 4.15) mechanism may be sticking. Each of these many result in a
collimation error. A test for this is known as the peg test.
The peg test involves setting two points (A & B) and taking level reading from two positions (1&2). If
the collimation error (e) is zero the apparent elevation of points will not be influenced by the distance
from them. If an error exists it will double if the distance from the shot is doubled.
rA e = RB 2e Set - up 1
RA 2e = rB e Set - up 2
solving both for e :
e = RB rA
e = RA rB
adding the two equations gives :
2e = RB rA + RA rB
or
R r r + RA
e= B A B
2
Note : Example 4.3 p.98
ENGI 3703
Surveying and Geomatics
Topic
Leveling Errors
Instructor: Prof. Ken Snelgrove
Lect 5 - Sept 19/07 Slide 3 of 7
Correcting Curvature, Refraction and Collimation Errors (Section 5.4)
A procedure for negating these errors, is to balance foresights and backsights in differential leveling.
Since these errors are dependent on distance and differential errors subtracts foresights and
backsights then these errors would tend to cancel. Of course this assumes that: i) the atmospheric
conditions are the same, ii) the curvature of the earth is constant, and iii) the collimation error is
constant.
Leveling Precision (Section 5.5)
Differential leveling is run to a bench mark and back to the original reference. The difference in the
sum of foresights and the sum of backsights is the “closure error”. Allowable error depend on the
application and Canadian Specification follow US (FGCS) specifications with different naming
convention. Both compute allowance misclosure as a function of distance covered in the loop as:
C = ±m K
ENGI 3703
Surveying and Geomatics
Topic
Leveling Errors
where: C is allowable misclosure (mm)
m is an integer 3, 4, 8, 24, 120
K is loop length in (km)
Instructor: Prof. Ken Snelgrove
Lect 5 - Sept 19/07 Slide 4 of 7
Canadian Spatial Reference System
Specifications and Recommendations for Control Surveys and Survey Markers
http://www.geod.nrcan.gc.ca/stand/survey_specs_e.pdf
Type
Allowable
Misclosure
(m)
Total Difference
between backsight and
foresight distances
Special Equipment
Required
Longest
foresight /
backsight
Special Order
3 mm
5m
40X Zoom
10"/2 mm level
1mm Invar rod
Steel pin T.P.
Sun Shade
50m
First Order
4 mm
10 m
32X Zoom
10"/2 mm level
1cm Invar rod
Sun Shade
110m
(three wire)
Second Order
8 mm
10 m
20"/2 mm level
110m
(three wire)
Third Order
24 mm
unspecified
20"/2 mm level
unspecified
Fourth Order
120 mm
unspecified
50”/2 mm level
unspecified
ENGI 3703
Surveying and Geomatics
Topic
Leveling Errors
Instructor: Prof. Ken Snelgrove
Lect 5 - Sept 19/07 Slide 5 of 7
Converting Distance Closure to Number of Setups (Section 5.5)
If we know the average foresight / backsight distance in the level loop we can determine the closure
error per set-up rather than total distance.
Example: For first order closure (4 mm per ilometer), determine the allowable error per set-up
with the maximum foresight / backsight distance (110m) was used.
Solution: Set ups are spaced 220 m apart therefore 1000m/220m = 4.55 setups/km would be
required.
ENGI 3703
Surveying and Geomatics
C = ±4
K
n
n
C = ±4
1
n = ±1.88 n
4.55
Topic
Leveling Errors
Instructor: Prof. Ken Snelgrove
Lect 5 - Sept 19/07 Slide 6 of 7
Leveling Errors - Summary (Section 5.12)
Systematic (S)
Random (R)
Instrument Error
Collimation (S,R)
Horizontal Cross Hair (R)
Short/Long Rod (S)
Natural Error
Refraction & Curvature (S,R)
Temperature (R)
Wind (R)
Settlement (S)
Personal Error
Failure to Fine Level (R)
Parallax (focus) (R)
Rod Reading (R)
Rod Handling (R)
Level Mistakes: Rod not extended, poor turning point locations, misreading rod by 1 m, waving rod on
flat surface, transposing figures in note, reading bias on rod, touching instrument and legs.
Error Reducing Leveling Techniques
1.
Three-Wire Leveling (averaging 3 readings)
2.
Reciprocal Leveling (when FS and BS do not balance)
3.
Rod Plumbing and Rocking
4.
Repeated Measurements and Distribute Error
Other Leveling Methods
1.
Barometric (+/- 1m)
2.
Trigonometric Leveling (OK for short shots [i.e. big vertical angle change per unit height diff.])
ENGI 3703
Surveying and Geomatics
Topic
Leveling Errors
Instructor: Prof. Ken Snelgrove
Lect 5 - Sept 19/07 Slide 7 of 7