MODULE 3
MEASUREMENTS OF VERTICAL DISTANCES (1)
Introduction
The elevation of a point near the surface of the earth is its vertical distance above or below
an arbitrarily assumed level surface or curved surface every element of which is normal to the
plumb line. The level surface (real or imaginary) used for reference is called the datum. The datum
most commonly used in mean sea level, particularly the Mean Sea Level Datum established by the
United States government. (R.E. Davis Et. Al. Surveying Theory and Practice p. 118).
The determination of elevations using surveying techniques is a comparatively simple but
extremely important process. Leveling is a method of determining the difference in elevations
between series of points. If one point is at a known elevation, then the relative elevations of all of
the other points can be determined through leveling. The significance of relative elevations cannot
be exaggerated. They are so important that one cannot imagine a construction project in which
they are not critical. From terracing on a farm or the building of a simple wall to the construction
of drainage projects or the largest buildings and bridges, the control of elevations is of great
importance.(McCormac, Surveying p. 85)
Learning Outcomes
At the end of the session, learners of this program are expected to:
a. Define the different terms used in leveling.
b. Compute the elevation of points by the different methods of leveling
Lesson Proper/ Discussion
Measurements of Vertical Distances
Leveling-is the process of directly or indirectly measuring vertical distances to determine the elevation of
points or their differences in elevation.
Definition of terms used in Leveling
Level Surface- is a curved surface which is at any point perpendicular to the direction of gravity or the
plumb line.
Level Line- is a curved line in a level surface all points of which are normal to the direction of gravity and
equidistant from the center of the earth.
Horizontal Surface- is a plane that is tangent to a level surface at a particular point.
Horizontal Line-is a straight line in a horizontal plane which is tangent to a level line at one point.
Vertical Line- is a line parallel to the direction of gravity.
Mean Sea Level- is an imaginary surface of the sea which is midway between high and low tides. It is
taken as the reference surface to which most ground elevations are referred.
Datum- is any convenient level surface coincident or parallel with mean sea level to which elevations of a
particular point are referred.
Elevation- the vertical distance above or below mean sea level or any other selected datum.
Difference in Elevation is the vertical distance between the two level surfaces in which the points lie.
Leveling Methods
1. Direct or Spirit Leveling-is the commonly employed method of determinng the elevation of
points some distant apart by a series of set ups of a leveling instrument along a selected route.
2. Differential Leveling- is the process of determining the difference in elevation two or more points
some distance apart.
Definition of terms in Differential Leveling
1. Bench Mark (BM)- is a fixed point of reference whose elevation is either known or assumed.
2. Backsight (BS)- is a rod reading taken on a rod held on a point of known or assumed elevation.
Also known as “plus” sights.
3.
Foresight (FS)- a reading taken on a rod held on a point whose elevation is to be determined.
Also known as “minus” sights.
4.
Backsight Distance (BSD)- is the distance measured from the center of the instrument to the
rod on which the backsight is taken.
5.
Foresight Distance(FSD)- is the distance measured from the center of the instrument to the
rod on which the foresight is taken.
6.
Turning Point (TP)- is an intervening point between two bench marks upon which point foresight
and backsight rod readings are taken to enable a leveling operation to continue from a new
instrument position.
7. Height of Instrument (HI)- is the elevation of the line of sight of an instrument above or below
a selected reference datum.
Formula Used for Differential Leveling
HI= Elev BMA + BS
Elev TP1= HI – FS
Where:
HI= Height of Instrument
Elev BMA= Elevation of benchmark A
BS= Backsight
FS= Foresight
Elev TP1= Elevation of turning point 1
Arithmetic Check: Elevation of BMB = Elevation of BMA + Ʃ BS – Ʃ FS
Typical Set up for Differential Leveling
Instruments Used in Leveling
Enrichment Activities:
1. Complete the differential level notes shown below and perform the customary arithmetic check.
Level Notes from BM1 to BM4
Station (Sta)
BS
HI
FS
BM1
1.256 m
TP1
1.116 m
1.886 m
TP2
1.228 m
1.527 m
Elevation
127.133 m
BM2
1.189 m
2.246 m
BM3
1.070 m
2.017 m
TP3
1.831 m
2.656 m
BM4
Solutions: HI= Elev BM1 +BS1
Elev TP1= HI1 – FS1
HI2 = ElevTP1 + BS2
ElevTP2 = HI2 – FS2
2.765 m
HI1= 127.133m + 1.256 m
HI1= 128.389 m.
ElevTP1 = 128.389 – 1.886 m
ElevTP1 = 126.503 m
HI2 = 126.503 + 1.116 m
HI2 = 127.619 m
ElevTP2 = 127.619 m -1.527 m
ElevTP2 = 126.092 m
HI3 = ElevTP2 + BS3
HI3 = 126.092 + 1.228 m
HI3 = 127.32 m
ElevBM2 = HI3 – FS3
ElevBM2 = 127.32 – 2.246 m
ElevBM2 =125.074 m
HI4 = ElevBM2 + BS4
HI4 = 125.074 m + 1.189 m
HI4 = 126.263 m
ElevBM3 = HI4 – FS4
HI5 = ElevBM3 + BS5
ElevTP3 = HI5 – FS5
HI6 = ElevTP3 + BS6
ElevBM4 = HI6 – FS6
ElevBM3 = 126.263 – 2.017 m
ElevBM3 = 124.246 m
HI5 = 124.246 + 1.070 m
HI5 = 125.316 m
ElevTP3 = 125.316 – 2.656 m
ElevTP3 = 122.66 m
HI6 = 122.66 m + 1.831 m
HI6 = 124.491 m
ElevBM4 = 124.491 m – 2.765 m
ElevBM4 =121.726 m.
Tabulated Solutions:
Station
(Sta)
BM1
BS
HI
FS
1.256
m
128.389
m
TP1
1.116
m
127.619
m
1.886
m
126.503
m
TP2
1.228
m
127.32 m
1.527
m
126.092
m
BM2
1.189
m
126.263
m
2.246
m
125.074
m
BM3
1.070
m
125.316
m
2.017
m
124.246
m
TP3
1.831
m
124.491
m
2.656
m
2.765
m
122.66 m
127.133
m
BM4
ƩBS= 7.69 m
Arithmetic Check:
Elevation
121.726
m
ƩFS=13.097m
ElevBM4 = ElevBM1 + ƩBS – ƩFS
121.726m = 127.133 + 7.69m – 13.097m
121.726 = 121.726 (Check)
3. Profile Leveling-is the process of determining differences in elevation along a fixed line at
designated short measured intervals
Definition of terms used in Profile Leveling
1. Profile- is a curved line which graphically portrays the intersection of a vertical
plane with the surface of the earth.
2.
Stationing (Sta)- a numerical designation given in terms of horizontal distance
any point along a profile line and is away from the starting point.
3. Intermediate Foresights (IFS)- these sights, which are also known as ground rod
readings, are taken along the centerline of the proposed project to provide an
accurate representation of the ground surface.
4.
Full Stations- are points which are established along the profile level route at
uniformly measured distances.
5. Plus Stations- any other intermediate point established along a profile level route
which is not designated as a full station.
6. Vertical Exaggeration- is a process of drawing the vertical scale for a profile
much larger than the horizontal scale in order to accentuate the differences in
elevation.
Schematic Diagram of a profile leveling route and sketch of the profile of the ground surface
Enrichment Activities:
1.Complete the following set of profile level notes and show the customary arithmetic
check.
Sta
BS
HI
FS
IFS
Elev
BMA
0.95
125.50m
0+ 00
3.0
0+ 10
2.3
TP1
3.13
0.64
0+ 22
2.7
0+ 30.5
2.8
0+ 34.2
3.1
0+ 39.3
0.5
0+ 42.5
0.8
TP2
2.16
1.28
0+ 45
0.9
0+ 48.6
1.2
0+ 50
1.7
0+ 53.7
2.8
TP3
0.82
2.37
TP4
1.35
3.50
0+ 75
BMB
3.0
1.99
Solution:
a.) Computation of HI’s and Elevation of Stations.
125.50 + 0.95 = 126.45
126.45 – 3.0 = 123.5
126.45 - 2.3 = 124.2
126.45 – 0.64 = 125.81
(HI1)
(Elev Sta 0+ 00)
(Elev Sta 0 + 10)
(Elev TP1)
125.81 + 3.13 = 128.94
128.94 – 2.7 = 126.2
128.94 – 2.8 = 126.1
128.94 – 3.1 = 125.8
128.94 – 0.5 = 128.4
128.94 – 1.28 = 127.66
(HI2)
(Elev Sta 0 + 22)
(Elev Sta 0 + 30.5)
(Elev Sta 0 + 34.2)
(Elev Sta 0 + 42.5)
(Elev TP2)
127.66 + 2.16 = 129.82
129.82 – 0.9 = 128.9
129.82 – 1.2 = 128.6
129.82 – 1.7 = 128.1
129.82 – 2.8 = 127.0
129.82 – 2.37 = 127.45
(HI3)
(Elev Sta 0
(Elev Sta 0
(Elev Sta 0
(Elev Sta 0
(Elev TP3)
127.45 + 0.82 = 128.27
128.27 – 3.50 = 124.77
124.77 + 1.35 = 126.12
126.12 – 3.0 = 123.12
126.12 – 1.99 = 124.13
(HI4)
(Elev TP4)
(HI5)
(Elev Sta 0 + 75)
(Elev BMB)
+
+
+
+
45)
48.6)
50)
53.7)
a) Determining Summations of Foresights and Backsights
ƩFS = -(0.64 + 1.28 + 2.37 + 3.50 + 1.99) = - 9.78
ƩBS = 0.95 + 3.13 + 2.16 + 0.82 + 1.35 = +8.41
b) Tabulated Solution
Sta
BMA
0+ 00
0+ 10
TP1
0+ 22
0+ 30.5
0+ 34.2
0+ 39.3
0+ 42.5
TP2
0+ 45
0+ 48.6
0+ 50
0+ 53.7
TP3
TP4
0+ 75
BMB
BS
0.95
HI
126.45
FS
IFS
3.0
2.3
3.13
128.94
0.64
2.7
2.8
3.1
0.5
0.8
2.16
129.82
1.28
0.9
1.2
1.7
2.8
0.82
1.35
128.27
126.12
2.37
3.50
3.0
ƩBS = 8.41
c) Arithmetic Check
1.99
ƩFS = -9.78
Elev
125.50m
123.5
124.2
125.81
126.2
126.1
125.8
128.4
128.1
127.66
128.9
128.6
128.1
127.0
127.45
124.77
123.1
124.13
Elev BMB = Elev BMa + ƩBS – ƩFS
124.13 = 125.50 + 8.41 – 9.78
124.13 = 124.13 (Check)
Self-Assessment Questions:
1. Complete the differential level notes shown below and perform the customary arithmetic check.
Sta
BS
HI
FS
Elev
BMW
2.08
85.50m
TP1
2.01
0.98
TP2
1.86
1.43
TP3
0.58
1.53
BMX
0.42
2.42
TP4
1.45
1.81
TP5
1.78
1.72
TP6
2.05
2.34
TP7
2.92
1.01
BMY
3.19
2.36
TP8
2.80
0.99
TP9
0.77
1.21
BMZ
0.60
2. Complete the following set of profile level notes and show the customary arithmetic check.
Sta
BS
HI
FS
IFS
Elev
BM1
1.24
TP1
1.11
100.00m
3.40
10 + 000
4.6
10 + 020
2.8
10 + 040
3.6
10 + 060
2.4
TP2
3.21
2.46
10 + 080
1.2
10 + 100
3.2
10 +120
2.8
TP3
BM2
3.06
2.74
2.40
MEASUREMENTS OF VERTICAL DISTANCES (2)
Introduction
Another method of measuring vertical distances is by Reciprocal Leveling. Reciprocal
Leveling is employed to determine the difference in elevation between two points when it is
difficult or impossible to keep backsights and foresights short and equal. Such a condition
usually exists when running a line of levels across wide rivers, lakes, or in rugged terrain
where deep canyons and ravines are encountered. In this method of leveling, two sets of rod
readings are observed. One set is taken with the instrument set up close to one point and
another with the instrument set up close to the other point.
Trigonometric levleling or indirect leveling is another method of determining vertical
distances. Trigonometric Leveling is defined as the detrmination of differences in
elevation from observed vertical angles and either horizotal or inclined distances. Vertical
measurements determined by trigonometric methods are only approximate unless the
observed vertical angles and the required distances are precisely measured. This method of
leveling is used extensively when undertaking topographic surveys over rugged or rolling
terrain since it provides a rapid means of determining vertical distances and elevation of
points.
Up to this point we have in effect been assuming that when the instrument is leveled, its
line of sight will represent a level line. In this setion you will learn that this is not the case,
due to the Earth’s Curvature. The line of sight of the telescope will actually be
perpendicular to a plumb line at one point only –that will be at the instrument location. You
may then decide that the telescope line of sight is a horizontal line, but you will learn that
this is not correct either, beacause of atmospheric refraction. (McCormac, Surveying p.
105)
Learning Outcomes
At the end of the session, learners of this program are expected to:
a. Compute the elevation of points by reciprocal leveling.
b. Compute the elevation of points by trigonometric levlenig.
c. Compute for the effects of curvature and refraction of the earth’s surface.
Lesson Proper/ Discussion
4. Reciprocal Leveling- is employed to determine the difference in elevation between two points
when it is difficult or impossible to keep back sights and foresights short and equal.
Formula:
DE1 = (a – b )
DE
2
= ( a’ – b’ )
TDE = (DE1 + DE2) / 2
Where: DE = Difference in Elevation
TDE = True Difference in Elevation
Typical Set Up in Reciprocal Leveling
Enrichment Activities:
1. A reciprocal leveling is employed across a deep creek and rod readings were taken at two
points A and B with the data tabulated below. Detrmine te following: a) Differece in
elevation between the two points. b) Elevation of B if the elevation of A is 150.45 m.
Instrument Set-Up Near A
BS
FS
1.283
1.284
1.286
1.283
B
0.675
0.674
0.677
0.674
0.677
0.678
Sum
Mean
Sta
A
Instrument Set-up Near B
BS
FS
1.478
1.480
1.476
1.478
a’
2.143
2.140
2.145
2.142
2.143
2.146
Sum
Mean
Sta
b’
SolutionS:
a) Determining Mean Rod Readings and Difference in Elevation.
am = (1.283 + 1.284 + 1.286 + 1.283) / 4
am = 1.284m
bm = (0.675 + 0.674 + 0.677 + 0.674 + 0.677 + 0.678)/6
bm = 0.676 m
a’m = (2.143 + 2.140 + 2.145 + 2.142 + 2.143 + 2.146)/6
a’m = 2.143 m
b’m = (1.478 + 1.480 + 1.476+ 1.478)/4
b’m = 1.478m
DE1 = (am – bm)
DE1 = 1.284 – 0.676
DE1 = +0.608m (Difference in elevation between A and B with Instrument set up near A)
DE2 = (a’m – b’m)
DE2 = 2.143 – 1.478)
DE2 = +0.665m (Difference in elevation between A and B with Instrument set-up near B)
TDE = (DE1 + DE2)/2
TDE = (0.608 + 0.665)/2
TDE = + 0.637 m (True difference in elevation between the two benchmarks.
Tabulated Solution:
Instrument Set-Up Near A
Sta
BS
A
1.283
Instrument Set-up Near B
FS
Sta
BS
b’
FS
1.478
1.284
1.480
1.286
1.476
1.283
1.478
B
Sum
5.136
0.675
0.674
0.677
0.674
0.677
0.678
4.055
Mean
1.284
0.676
a’
Sum
2.143
2.140
2.145
2.142
2.143
2.146
12.859
5.912
Mean
2.143
1.478
b) Elev of B = Elev of A ± TDE
Elev of B = 150.45 + 0.637
Elev of B = 151.087m
Note: The TDE is added to the elevation of A since B is higher than A. In the solution for DE 1 and
DE2 above, a positive value is determined which shows that B is higher than A.
5. Trigonometric Leveling-also called indirect leveling, is defined as the determination of
difference in elevation from observed vertical angles and either horizontal or inclined distances
Basic formula for Trigonometric Leveling
1)
2)
V= d*Tan Ø
V= s*Sin Ø
3) DEab= d*Tan Ø + HI – RR
4) DEab= s* Sin Ø + HI – RR
5) Elev B = Elev A + DEab
Typical setup for Trigonometric Leveling
Enrichment Activities:
1. From the top of a cliff, a rod reading was recorded to be 1.625m. At a point 185.45m away from
the foot of the cliff, a vertical angle of 38 045’ was read from an Engineer’s transit. A backsight of
1.725m was taken at a Bench Mark whose elevation is 50.00m. above mean sea level. Find the
elevation of the point below the stadia rod.(see figure above).
Given: RR @ top of cliff= 1.625m
BS @ BM
= 1.725m
Elev of BM
= 50.00m
Horizontal Distance, d= 185.45m
Vertical Angle 𝜙 = 38045’
Vertical Distance, V = Unknown
Elev. Of point below stadia rod = Unknown
Solution: V = d* tan𝜙
V= 185.45m * tan 38045’
V = 148.84m Answer
HI= Elev BM + BS
HI = 50.00m + 1.725m
HI = 51.725 m Answer
Elev of point below stadia rod = HI + V –RR
Elev of point below stadia rod = 51.725m + 148.84m – 1.625m
Elev of point below stadia rod = 198.94 m Answer
6. Effects of Earth’s Curvature and Atmospheric Refraction.
The effects of curvature and atmospheric refraction are taken into account in leveling work since
the measurements are made in vertical planes and these effects all occur in the same plane. The combined
effects are represented (see figure below). Due to the earth’s curvature, a horizontal line departs from a
level line by 0.0785 m in one kilometer, varying as the square ofthe length of the line. This expression for
earth curvature is based on the mean radius of the earth which is about 6371 km. In the given figure the
vertical distance between the horizontal line and thelevel line is a measure of the earths curvature.
The combination of the earth’s curvature and atmospheric refraction causes the telescope’s line of
sight to vary from a level line by approximately 0.0785 minus 0.0110 or 0.0675 m in one kilometer, varying
as the square of the sight distance in kilometers. This may be represented by a mathematical equation as
follows:
h’ = 0.0675 K2
where h’ is the departure of a telescope line of sight from a level line (in meters) and K
is the length of the line of sight (in kilometers) The value 0.0675 is called the coefficient of
refraction. (La Putt, Elementary Surveying 3rd ed. P. 174)
Figure of Curvature and Refraction
Enrichment Activities:
1. Two points, A and B, are 525.85 meters apart. A level is set up on the line between A and B and
at a distance distance of 240.50m from A. Ifthe rod reading on A is 3.455m and that of B is 2.806m,
determine the difference in elevation between the two points, taking into account the effects of
curvature and atmospheric refraction.
Solution: h’a = 0.0675 Ka2 = 0.0675 ( 240.50/1000)2
h’a = 0.004 m (effect of curvature and refraction on the sight at A)
h’b = 0.0675 Kb2 = 0.0675 ( 525.85-240.50/1000)2
h’b = 0.005 (effect of curvature and refraction on the singht at B)
a’ = a – ha’
= 3.455m – 0.004m
a’ = 3.451 m (corrected rod reading on A)
b’ = b – hb’
b’ = 2.801m
DE = a’ – b’
DE = 0.65 m
and refraction)
= 2.806m – 0.005m
(corrected rod reading on B)
= 3.451m – 2.801m
(difference in elevation between Aand B, considering the effect of curvature
Self-Assessment Questions:
1. Determine the combined effect of curvature and refraction on a level sights of 30, 50, 100,
250, 300, 500, 100 and 2000m.
2. An observer standing on shore can just see the top of a lighthouse 16.5 km. Away. If the eye
height oftheobserver above sea level is 1.735 m, determine the height of the lighthouse above
sea level, neglecting the effect of waves.
3. By taking a sight across a lake 24 km wide through a pair of binoculars, determine the height
of the shortest tree on the opposite shore whose tip the observer can see. Assume that the
eye of the observer are 1.675m above the shoreline on which he stands.
4. In reciprocal levelingacross a deep ravine the following sets of rod readings were taken:
Instrument Near A: BS on A= 1.719m, Ave. FS on B = 2.578 m
Instrument Near B: Ave BS on A= 1.335m, FS on B = 2.176 m
Point A is on one side of the ravine and has a known elevation of 197.428m. Point B is on the other
side. Determine the elevation of point B.
5. Two points A and B are 3,134.50 meters apart. From a third point, C, on the line between A
and B and 1,992.25 meters from A, the measured vertical angle to A is 35 028’ and that to B is
-15014’. Determine the difference in elevation between A and B making due llowance for the
effectcs of curvature and atmospheric refraction.
References:
1.
Elementary Surveying 3rd edition by Juny Pilapil La Putt
2.
Surveying by Jack McCormac Et. Al.
3.
Surveying Theory and Practice By R.E Davis, F. S. Foote
4.
The principles and Practice of Surveying by C. B. Breed; G.L. Hosmer and A. J. Bone