ECV201 Surveying I: Additional notes
Definitions:
1. Station: This is a point where a levelling staff is held for taking observations with
2. Height of the Instrument (HI): This has two meanings. It may mean height of the
a level.
instrument above the ground at the station where the instrument is placed. However,
usually it means elevation of the line of sight or line of collimation with respect to the
datum. Line of collimation is an imaginary line joining the optical centre of the objective
with the intersection of crosshairs and its continuation.
3. Back Sight: It is the first reading taken at a station of known elevation after setting up of
the instrument. This reading gives the height of the instrument (elevation of the line of
collimation) as...
Elevation of line of collimation = Known elevation + Back sight
As the name suggests these are readings taken between the 1st and
last reading before shifting the instrument to a new station.
5. Fore Sight (FS): This is the. Last reading taken before shifting an instrument to a new
station.
6. Turning Point or Change Point: For levelling over a long distance, the instrument has to be
shifted a number of times. Turning point or change point connects one set of instrument
readings with the next ser of readings with the changed position of the instrument. A staff
is held on the turning point and a foresight is taken before shifting the instrument. From
the next position of the instrument another reading is taken at the turning point keeping
the staff undisturbed which is known as back sight.
7. 7. Reduced Level (RL): Reduced level of a point is its height relative to the datum. The
level is calculated or reduced with respect to the datum.
4. Intermediate Sight (IS):
Different methods of levelling
Direct leveling: In levelling it is desired to find out the difference in level between two
points. Then if the elevation of one point is known. the elevation of other point can be
easily found out.
2. Trigonometric levelling: method that utilizes trigonometric relations in finding the
difference in elevation. It is used mainly to determine elevations of inaccessible points
such as mountain peaks. top of towers,
3. barometric levelling: the principle that pressure decreases with rise in elevation is used.
Hence it is possible to determine the difference in elevation between t\VO points by
measuring the pressure difference between the points by either mercury barometer or
aneroid barometer. As the aneroid barometer is strong and sturdy it is preferred to the
mercury barometer which is fragile and cumbersome. However. aneroid barometer is less
accurate compared to the mercury barometer.
1.
1
Direct levelling can be broadly classified as:
1) differential levelling:
2) check levelling:
the arithmetic checks carried out after each example above indicate only correctness of
arithmetical computations. They do not indicate that levels of the points are also correct. There
will always be errors in field work and it is always necessary to get an idea of the magnitude of
error. This can be obtained by taking the level back to the original benchmark or to another point
of known elevation or benchmark. It is advisable to make the length of foresight and backsight
equal to eliminate common instrumental errors. Figures 6.2(a) and 6.2(b) show the two types of
check. In Fig. 6.2(a) to check the level difference between A and B, the line of level is brought
2
back to the original station A. In such a case station B has to be made a change point, that is, after taking
foresight at B instrument should be changed to a new position and backsight taken.
In Fig. 6.2(b) line of le\'el is taken to another point P of known R.L. In both cases we can compute the error in
levelling. In the first case the level difference should be zero. In the second case, it should be known R.L. The
discrepancy represents the error of closure of the circuit and should be very small. If a large difference occurs
there must be some mistake in either (i) computation or, (ii) in reading of the rod or (iii) in entering the field
notes.
Errors can also be classified as (i) Instrumental errors, (ii) Personal errors, (iii) Natural errors.
i)
Instrumental errors: include
a.
b.
ii)
iii)
Level Out of adjustment causing collimation errors
Other instrumental errors are: (i) Sluggish bubble, (ii) Defective staff, (iii) Defective tripod, (iv)
Faulty focussing tube.
Personal errors: Bubble not properly centered, Parallax, non-verticality of the staff, and sighting error
Natural errors: curvature and refraction, wind vibration, temperature variation, settlement of tripod or
turning point
REDUCING ERRORS AND ELIMINATING MISTAKES LEVELLING
Errors in levelling can be reduced but never fully eliminated by systematic adjustment and manipulation of
both the level and staff. The following points should be kept in mind for an accurate levelling:
(i)
bubble should be checked before and after each reading,
(ii)
rod with circular bubble should be used:
(iii)
length of foresight and backsight should be made equal;
(iv)
usual field check should be done;
(v)
usual field book checks should be observed;
(vi)
telescope should be shaded from sun
(vii)
line of sight-should be at least 0.3 m above intervening terrain.
COLLIMATION CORRECTION
It is not always possible in practice to make backsight equal to foresight. It is also not possible to
always ensure horizontal line of sight. Hence collimation error invariably occurs and a collimation
correction should be applied. This is also known as C-factor correction in which C represents the
inclination of the line of sight when the level bubble is centred. In Fig. 6.5(a) the line of
collimation is inclined upwards even if the bubble is central. The error is D tanα ≈ Dα as α is small
and the correction is ~Dα. This is often expressed as CD where C is the correction factor. In this
case C = - a. Hence if C is positive the line of sight is inclined downward. To determine C, set up
the instrument between A and B as shown in Figs. 6.5(b) and (c) The line of collimation is
assumed to be downward, i.e, C positive.
3
From the figures,
Once the C factor is known. this can be applied for necessary correction for unequal backsight
and foresight as shown in Fig. 6.5(d).
The correction to be applied as shown in the Fig. 6.5(d) is C factor times (∑F.S. intervals - ∑
B.S. intervals).
4
Example 6.7 A level setup in a position 30 m from peg A and 60 m from peg Breads 1.914 m on
a staff held at A and 2.237 m on a staff held at B, the bubble having been carefully brought to the
centre of its run before each reading. It is known that the reduced levels of the top of the pegs A
and B are 87.575 m and 87.279 O.D respectively (Fig. 6.6). Find (a) Collimation error; (b) The
readings that would have been obtained had there been no collimation error.
Solution:
Let us assume that the error is positive, i.e. the line of collimation is upward.
True difference of level between A and B
= 87.575 - 87.279
= 0.296
5
3) Fly levelling:
It is a quick but approximate method of levelling. Long distances are taken as sights. It is used
for reconnaissance oi an area or for approximate checking of levels.
4) Profile levelling:
As the name suggests, it shows a profile, that is, a line depicting ground elevations at a vertical
section along a survey line. This is necessary before a rail road, highway, transmission line. side
walk or sewer line can be designed. Usually a line of level is run along the centre line of the
proposed work as shown in Fig. 6.1. Level is taken every 15 m or 30 m interval. at critical points
where there is a sudden change of levels, at the beginning or end of curve. The basic objective is
to plot accurately the elevation of the points along the line of levels. The procedure is exactly the
same as in differential levelling as explained in Sec. 6.2. It is necessary to take staff readings
along the centre line, book them properly in the level book. compute the R.L.'s of different points
and apply suitable arithmetic checks; It is also necessary to start from a B.M. of known R.L. also
close with a known R.L. so that suitable field checks are applied. It is however not essential toput the instrument along the centre line. It can be pIaced anywhere if necessary, off the centre
line, so that large number of readiness can be token and foresights and backsights are made
approximately equal. It-is now necessary to plot the profile or longitudinal section. To show the
distortions of the ground the elevations are plotted on a much larger scale after taking a suitable
datum than the longitudinal distances. Based on the example given in Sec. 6.2 a typical
longitudinal section is shown in Fig. 6.8.
6
After the longitudinal or profile section is drawn. it is necessary to have a smooth surface: This is
known as grade line which is selected on various considerations like:
(i) minimum amount of cutting and filling of earth work;
(ii) balancing the cut and fill; and
(iii) keeping the slope within allowable limit.
If points A and G are joined by a Straight line the slope of the line becomes (99.705 99.535)/180 or 111059 which is very small and is within allowable limit. This may not, however,
ensure equal volumes of cut and fill and suitable adjustments of grade line may be necessary to
ensure this condition.
7
5) cross-sectional levelling:
For laying a pipeline or sewerline only longitudinal section is adequate because the width of the
line is small. In the case of roads and railways apart from longitudinal section, cross sections at
right angles to the centre line of the alignment are required at some regular intervals. This is
necessary to know the topography of the area which will be required for the roads and railways
and also to compute the volume of cut and fill for the construction work. Figure 6.9(a) shows the
plan. Figure 6.9(b) shews the cross section and the table shows the entry in the level book. Cross
section is usually plotted in the same horizontal and vertical scale
8
6) SIGHT RAILS AND BONING RODS
Sight rails and boning rods are used for excavation purposes associated with the grading of
drains and sewers. The sight roils are established at fixed points along the excavation line at a
height above the formation level equal to the length of the boning rod. The formation level
compared with the surface level gives the depth of excavation. When the boning rod is in line
with sight rails the excavation is correct depth (Fig. 6.10).
Worked example:
Example 6.11 A. B. C. D; E and F are the sites. of manholes 100 m apart on a straight sewer. The
natural ground can be considered as a plane surface rising uniformly from A to F at a gradient of
1 vertically in 500 horizontally, the ground level at A being 31.394 m. The level of the sewer
invert is to be 28.956 m at A, the invert then rising uniformly at 1 in 200 to F. Sight rails are to
be set up at A, B, C, D, E and F so that a 3 m boning rod or traveller can be used. The backsights
and foresights were made approximately equal and a peg at ground level at A was used as datum.
Draw a level book showing the readings.
Solution:
9
10
7) Reciprocal levelling:
While crossing a river or ravine it is not possible to put the level midway so that the backsight
and foresight are equal. Sight distance, however, is long and errors due to (i) collimation, i.e,
inclined line of sight. (ii) curvature and refraction are likely to occur. To avoid these errors two
observations are made. As shown in Fig. 6.12 instrument is placed near station A and
observations are made on staffs at A and B. Similarly, instrument is placed near Band staff
readings are taken on B and A.
From first set of readings:
difference in level = d = BB1 = a1 + c + e - r – b1
= (a1 – b1) + (c - r) + e
From second set of readings:
difference in level d = AA1 = - (b2 + c + e - r – a2)
= (a2 – b2) - (c - r) - e
(- sign as difference is measured at A instead of at B)
By adding 2d = (a1- bl) + (a2 – b2)
11
or
d = 1/2 [(al – b1) + (a2 - b2)]
2(c - r + e) = [(a2 - b2) - (al - bl)]
c - r + e = 1/2 [(a2 - b2) - (al – b1)]
c =curvature error
r =refraction error
e =error due to collimation
If the combined error due to curvature and refraction are known error due to
collimation can be found out.
Subtracting
or
Here
12
ERROR, ADJUSTMENT AND PRECISION OF LEVEL
See attached notes
(Roy, 1999)
References
Roy, S. K. (1999). Fundamentals of Surveying. New-Delhi: Asoke K. Ghosh, Prentice-Hall of
India Private Limited.
13
\I
II
Levelling II 147
.
•
difference betweenthe upper and lower reading provides the staff intercept necessary
to calculate the sight distance and to check whether backsight and foresight are
equal. A typical page of the level book used for three wire levelling in given
below.
.
serr
station
(I)
"
Distance reading
B.S.
ES.
(2)
(3)
H.I.
(4)
R.L.
Remarks
-
B.S.
ES.
(5)
(6)
(7)
B~Il
CPI
CP2
CP3
B~12
Ariiametic check
6.15. ERROR, ADJUSTi\IENT AND
•
PRECISIO~
OF LEVEL
In levelling error is likely to be more when the length of the line is more' or the
number of set up of the instrument is more. Since standard error or probable error
is directly proportional to the square root of the length of a nne and weight is
inversely proportional to thesquare of probable error weight is inversely proportional
to the length of a line in a level circuit. Similarly weight is inversely proportional
to the number of instrument set ups. Weight also varies directly withthe number
of repetitions. Therefore adjustment orcalculation of most probable value of R.L.
of a point in a level line is based on the above principles. When misclosure of
level is known, l.e. when levelling ends at a point of known elevation or at the
starting point, discrepancy is adjusted in proportionto 'length from the starting
point. In multi loop circuits a benchmark should be made common to both the
circuits. Though theory of least square is the best method for adjusting such a .
. circuit, approxlmateadjustments can be made. The outer loop should be adjusted
first. The adjustment required forthecommon point is' found out. From theadjustment
necessary for the common point, the adjusted value of the starting point is found.
. out as also misclosure in the innerloop. This rnlsclosure is, again adjusted. Finally,
the outer loop is again adjusted based on new adjustment'. of the inner loop.
Examples 6.17 and 6.18 show the procedure.
Example 6.14 . The difference in level between two points A and B was found
by three routes-(l) vla C and D, (2) v la E, F and C, (3) via H, distances being
as follows:
Route 1 AC = 180 m
Route 2 AE 1~4 m
Route 3 AH = 26~ .m
=
=
HB =369 m
CD 282 m
EF = 156 m
DB =228 m
FG = 32~ m, CS= 270 m
The sections on Route 1 were levelled eight times, those on Route 2 twice and
148 Fundamentals of Surveying
those on Route 3 four times and the differences in level were found to be 8.2iS
m, 8.292 m and 8.285 m respectively, If the probable error in any section for a
single levelling is proportional to the square root of the length of thatsection, find
the most probable value of the difference in level between A and B. (Salford)
Solution Most probable value is the weighted mean of the observed values.
Weight is inversely proportional to the square of probable error and directly
proportional to the number of repetitions.
wI : w2 :
where
lilt "2
w)
=
11\11 1 : 112/12 : lIil).
and 11) = no. of repetitions
=corresponding lengths
Here 111 =8
112 =2
113 =4
1\ =180 + 282 + 228 =MOm
12 = 144 + 156 + 324 + 270 =894 m
I) = 264 + 369 = 633 m
II, 12 and I)
Most probable value =Weighted mean
_ \"\-'"\ + II'~X2 + \\'3-'")
\\'\ + \\'2 + \\'3
_
(8.275) (~) + 8.292
690
-
(2-)
+ 8.285 (.i-.)
894
633
824
-+-+­
690
894
633
i
=8.279 m
Example 6.15 Ina topographical survey, the difference in level between two
points A and B is found by three routes-via C, D, E, and F, via G and H and via
I, the distances being
=
Route 1 AC = 120 m CD = 162 m DE = 300 m EF 258 m FB = 132 m
Route 2 AG = 240 m GH =306 m HB = 384 m
Route 3 AJ = 294 m JB = 462 m
The sections on Route 1 are each levelled four times, those on Route 2 eight
times, and on Route 3 twice; the established differences in level thus obtained
being 30.81 m, 30.57 m and 31.08 m respectively. If the probable error in any
section at each levelling is proportional to its 'length, and the usual laws for
combination of readings hold, find the most probable value for the difference in.
level between A and B.
(L.U., B.Sc.,)
Solution
From the given data,
= 120 + 162 + 300 + 258 + 132 = 972 m
[2 = 240 + 306 + 384 = 930 m
[) = 294 + 462 = 756 in
[I
WI .: »'2 :. w)
=4/9ii. : 8/930: 2rl56
,.
Levelling II ' 149
Most probable value ::: Weighted mean '
••
4
8"
'2
','
(30.81) + - (30.57) + - (31.08)
_ 972
'
'930
756
'
8, 2
-+-+­
-
,4
972, 930
756
_ 0.1268 + 0.2630 + 0.0822
- 0.00411 + 0.0086 + 0.0026
=30.72 m
Example 6.16 A line of levels' is carried from B.M.A whose elevation is
146.522 rn, to a new B.M.P requiring 10 set ups. The measured difference in
elevation is - 3..436 rn, A line is carried from B.M.B whose elevation is 146.851 m
to B.M.P requiring six set ups. The measured difference in elevation is - 3.755 m.
A line is carried from B.M.e whose elevation is 132.768 m to B.M.P requiring
four set' ups. The measured difference in elevation is + 10.312 m. Compute the
weighted elevation of B.M.P and the standard error of this elevation. r;"'Ioffit)
Solution Standard error is proponional to the square root of the number
of setups required. Hence standard errors oflines 1, 2 and 3 are proportional to
.JfO•.[6 and .J4 respectively. Weights are inversely proportionalto square of
standard errors, hence,
... .:., •• v -' I . 1 . 1
•• \ • '~2 •• 3 -
10' '6 . '4
=
Level of B.M.P.• by -1st route 146,522 -: 3.436
;, 143.086 ITi
by ,2nd route = 146.851 - 3.755 .
= ' 143.096 m .
by3rd route
=132.768 + 10.312
=143;080 m
Weighted elevation'
_to (143.086) + i (l~3.096) +*(143.08)
-
1..+1+1
10 6
= 143.08632 m
Standard error of the weighted mean
v\
= 143.086 -
143.08632 = - .00032
1.'2
= 143.096 -
143.08632 = + .00968
u3
= 143.080 - 143.08632 = - .00632
uf = 1.024 X10-7. P;l.lf = 1.024(l0-S)
4
150 Fundamentals oj Surveying
u~
=9.37 x 10-5 P:.u;- = 1.562(10-5)
vi.
= 3.99 x 10-5
p~ui
. . = 9.990(10~)
L pv 2 =2.562 x 10-5
Standard error of the weighted mean
2.562 x 10-5
0.5167 x 2
=
=± .004979
Standard error of each of the measured elevation
•
/2.562 x 10-5
- ~
2·
= ± .003579
=± (.003579)(10) = ± .03579
0"2 = ± (.003579)(6) =± .02147
0"3 =± (.003579)(4) =± .01432.
0"\
Example 6.17 Figure 6.15 shows a closed circuit of levels with difference of
levels betweendifferent points and the length between them. Compute the adjusted
values of the levels of different points.
. Solution
8
?>.~?,
~
d~~~
'\~
R.L. 100.00
A
c
'\P
'"
co
~
~
o
3
3
E
o
Fig. 6.15 Example 6.17.
.. 1
Levelling11 151
Misclosure = +' 3.52 of: 2.15 - 1.05 - 2.67 - 2.10
••
=- 0.15 m ov~ra length of 6.05 km
Elevation adjustment at B = 100.00.+ 3.52 + O.l~ ~_1.6
=103.559 m
? ' 2 0.15(2.6)
3 5- + .1;) + 6.05'
:It C = 100 +.
= 105.734 m
· ., ? 15 1 0·' 0.15(3.85)
at D = 100 + 3.;)- + _. -.;) + 6.05
= 10-1-.6-1-5 m
.
.
2 1- 1 O· ? 67' 0.15(5.25)
E 100 + ".J
at:::;
".;)- + .;) ":" .;) - -. + . 6.05
= 102.08 m
at A = 100 + 3.52 + 2.15 - 1.05 .,. 2.67 ., 2.10
+
0.15(6.05)
'6.05
= 100 m
Example 6.18 A level line is run from B.M.A and closes on the same point as'
shown in Fig. 6.16. B.M.B is included in both the loops. In the outer loop there
are five change points while in the inner loop there are four change points. The
elevations observed at different points are as follows:
Elevation observed
·B.M.A
C.Pl
. B.lvlB(I)
C.P.2
C.P.3
C.PA
C.P.5
B.M.B(2)
C.P.6
B.M.A
..
= 100.00
= 101.85
= 104.35
= 103.75
1st correction
2nd correction
= 101.50
= 105.45
= 103.45
= 102.15
= 101.30
= 100.14.
101.85
. 104.29
103.63
105.27
103.21
. 102.85
101.00
99.8-1-
101.93
100.00
Solution First the outer circuit, Le. Loop 2 is adjusted. Starting with
B.'t'.I.B(l) as 101.85 it closes on B.M.(2) as 1O:!.15 giving a mlsclosure of
+ 0.30 m. In Loop 2 there are 5 instrument set ups and hence a correction of
- 0.30/5 = - 0.06 rn/set up. B.~ 1(1) is to be reduced by 0.30 m. As C.P.6 and
B.M.A is based' onB.M.B as reference they are also reduced by the same amount
151 Fundamentals of Surveying
..
Loop 2
'3
Fig.6.16 Example 6.18.
giving R.L. ofB.M.A as'99,84 m. This gives a misclosure in loop 1 as - 0,16 and
is equal to 0.16/4 = 0,04 m per set up. The values in loop 2 are now adjusted.
B,M.A is adjusted by four instrument set ups and B.M.B by 2 instrument set ups.
Details are shown as 1st correction and 2nd correction in the example itself.
PROBLEMS
6.1 Draw a page of a typical levelling 'field book and explain how the readings
are recorded.
.
.
6.2 Describe in detail the methods of reduction of levels and explain their merits
and demerits.
.
.
6.3 Name the different sources of errors in levelling and explain how they can
be eliminated or minimized.
6.4 (a) Derive an expression for the combined effect of earth's curvature
and atmospheric refraction in levelling, given the diameter of earth as
12,740 krn,
'
(b). The following notes refer to reciprocal levels:
Instrument near
P
Q
Staff readings on
P
Q
1,850
2,850
1,000
2.200
Remarks
PQ = 1055 m
R.L of P = 126.100
Determine
(i) the true R.L. of Q (ii) thecombined correction for curvature and refraction
and (iii) the angular error, if any, in the collimation adjustment of
instrument.
. [AMIE, Sec B, Winter 1984]
L
."