LEVELING

BY: ANGEL GABRIELLE G. CASPE, SO2

Table of Contents
➢Leveling
➢Definition of terms
➢Leveling Instruments
➢Mistakes in Leveling
➢Methods of Leveling
◦ Differential Leveling
◦ Double-rodded Leveling
◦ Three Wire Leveling
◦ Profile Leveling
◦ Reciprocal Leveling
◦ Trigonometric Leveling
◦ Cross-section Leveling
◦ Borrow-pit Leveling
 Leveling
the process of directly or indirectly
measuring vertical distances to
determine the elevation of points or
their differences in elevation.
Why do perform leveling?
▪to determine the topography of sites for design projects.
▪to set grades and elevations for construction projects.
▪to compute volumes of earthwork.
Leveling
Leveling operations are vital to surveying since it provides necessary data for
engineering design and construction, and the production of topographic maps.

Through the process of leveling, buildings, roads, canals, and other vertical and
horizontal structures can be designed and laid out to best conform to the
configuration of the ground.
DEFINITIONS OF TERMS
LEVEL SURFACE - curved surface where every point is perpendicular to
plumb line or direction of gravity. However, a level surface is not plane and
does not have regular form. To some effect, the direction of the gravity
depends on the distribution of the masses of the earth’s crust and on their
densities.
HORIZONTAL SURFACE - plane that is tangent to a level surface at a
particular point. It is also perpendicular to local direction of gravity.
• LEVEL LINE - 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 LINE - straight line in a horizontal plane which is tangent to the
level line, and perpendicular to the direction of the gravity at the point of
tangency.
VERTICAL LINE – vertical line parallel to the direction of the gravity. It is
exemplified by the direction taken by sting supporting a suspended plumb bob
passing through a point.
ELEVATION - vertical distance above or below mean sea level or any selected
datum; can be positive or negative elevations.
MEAN SEA LEVEL (MSL) - an imaginary surface of the sea which is midway
between high and low tides (determined by averaging the height of the sea’s
surface for all its tide stages over a long period of time)
✓taken as the reference surface to which most ground elevations are referred
✓not a steady frame of reference due to the melting of ice in the polar regions, the
effects of volcanic activity, and other influencing factors
✓The surface which is considered to be at zero elevation conforms to the spheroidal
shape of the earth and is perpendicular to the direction of gravity at every point.
DIFFERENCE IN ELEVATION (DE) - vertical distance between two level surfaces in
which the points lie.
DATUM - any convenient level surface coincident or parallel with the mean sea
level to which elevations of a particular area are referred
✓any surface may be used as a datum when relative elevations over a limited
area needs to be established
✓assumed elevation may be assigned to a reference point and the elevation of
other points may be determined with regard to this value
Old Datum of US: mean sea level (using 19 years record from 26 gauging stations at
Atlantic Ocean, Pacific Ocean and Gulf of Mexico)

New Datum US: NAVD88 (North American Vertical Datum of 1988) formerly
NGVD29 (National Geodetic Vertical Datum of 1929)
single tidal gauge benchmark located at Father’s Point,
Rimouski, Quebec, Canada

Local mean sea level: commonly used as datum

Luzon Datum of 1911 (defined by Station Balanacan at Marinduque) – became the
primary geodetic reference of all surveys in the Philippines.
- The Philippine Geodetic Network (PRN) was developed until 1946 and then
upgraded into Philippine Reference System of 1992 (PRS92)
PRS92 or the is a homogeneous national
network of geodetic control points (GCPs), marked by survey monuments or
mojons, that has been established using Global Positioning System (GPS)
technology.

By virtue of EO 45, PRS92 became the standard reference system for all surveying
and mapping activities in the Philippines. The order also mandated that all new
surveys and maps shall be referred to the new network and all old surveys shall be
integrated into it.

➢visit the NAMRIA website for more information

BENCHMARK (BM) - a permanent or temporary fixed point of reference whose
elevation is either known or assumed; should be easily recognized and located
where they have the smallest likelihood of being disturbed
➢Permanent BM
➢Temporary BM
Permanent benchmark (PBM) - those which are established at intervals
throughout the country by the Philippine Coast and Geodetic Survey (PSGS) or
Bureau of Lands
✓serve as points of reference for levels in a given locality and their locations are
determined by precise leveling methods
✓ PCGS BM consists of bronze or brass disks which are permanently set in
concrete foundation
✓marked with the elevation above mean sea level, the year it was established
and its reference number
Temporary benchmark (TBM) - those set up by the surveyor for his own use in a
particular surveying project
✓may have assumed elevations
✓ should be stable and semi-permanent marks such as wooden peg in concrete,
nail or spike driven into a tree, an X mark in a bridge abutment or top of fire
hydrant
BACKSIGHT (BS) - a reading taken on a rod held over a point of known or assumed
elevation
✓ vertical distance from the established line of sight to the point sighted
✓always the first rod reading taken after the instrument has been set-up
✓referred to as plus sight (+S) since it is added to the ELEV of points being sighted
to determine the height of instrument (HI)

HI = ELEV + BS
FORESIGHT (FS) - a reading taken on a rod held over a point whose elevation is to
be determined
✓vertical distance from the line of sight of the instrument to the point being
observed
✓usually the last reading taken before the leveling instrument is moved to
another location
✓referred to as minus sight (-S) since it is subtracted from the HI to determine the
ELEV of the point

ELEV = HI - FS
BACKSIGHT DISTANCE (BSD) - measured from the center of the instrument to the
rod on which a backsight is taken
FORESIGHT DISTANCE (FSD) - measured from the center of the instrument to the
rod on which a foresight is taken
TURNING POINT (TP) - an intermediate point between two benchmarks upon
which foresight and backsight rod readings are taken
✓also referred to as change point (CP) and usually numbered consecutively
✓should be located on stable object like rock, stake driven into the ground or a
paint mark on the concrete pavement
HEIGHT OF INSTRUMENT (HI) - elevation of the line of sight of an instrument
above or below a selected reference datum
HI = ELEV + BS
LEVELING INSTRUMENTS
▪Spirit level or dumpy level or wye level - basic instrument used
▪Hand level, Alidade
▪Transit, Theodolite
▪Aneroid barometer
▪EDMs
 It is simple compact and stable. The telescope is rigidly fixed to its
support therefore cannot be rotated about its longitudinal axis. A long
bubble tube is attached to the top of telescope. The instrument is stable
and retains its permanent adjustment for long time. This instrument is
Dumpy Level commonly used.
 The wye level is very identical to the dumpy level. The only distinct difference
between these two instruments is in the manner by which their telescope are
attached to the supporting level bar. The wye level has a detachable telescope

Wye Level which rests in supports called wyes. It can be removed from the Y-shaped
supports and turned end for end during adjustment by releasing the two
clamping collars which fit across the tops of the Y’s.
 Self-leveling features are incorporated in automatic levels. This type of
Automatic level has become popular for conventional leveling work because of the
ease and speed of their operation. It does not use a level vial and its
ability to level itself depends upon the action of a complex pendulum-
Level and-prism device.
 LEVELING RODS
a graduated rod which is used for measuring the vertical distance
between the line of sight through a leveling instruments and the
point whose elevation is either required or known
made of wood, fiberglass or metal and have graduations in
meters and decimals which start from zero at the bottom and
extending upward to lengths of 3 or 4 meters
TYPES OF RODS
self-reading rod - can be read directly by the instrumentman.
target rod - has sliding target which is set and read by a
rodman at the position selected by the instrumentman.
READING AN E-TYPE LEVELING ROD

Read value at
the
horizontal
cross hair
1.932
1.930
1.920
1.910
1.900
1.133
1.130
1.120
1.110
1.100
MISTAKES IN LEVELING
Mistakes can be avoided by a well-arranged system of operation and by constant alertness by
the survey party members.
Checking, as described in some of the operations, will eliminate many possible areas of mistakes.
COMMON MISTAKES
▪Misreading the rod or reading the wrong mark - the figures on a rod may be obscured by brush
or may fall in a position in the field of view so that the instrumentman cannot see two
consecutive numbers
▪Not fully extending the rod for high readings
▪Touching the tripod during reading
▪Confusion between recording BS and FS entries into the field book
▪Moving turning points or not setting the rod on the same point for an FS and the following BS
▪Recording a reading in the wrong column
▪Using the wrong horizontal cross hairs - occurs on an instrument provided with stadia hairs
 METHODS IN LEVELING

Direct
Indirect differ in
principle involved
types of instruments used
procedure involved
attainable degrees of precision
CONVENTIONAL AND TRADITIONAL LEVELING METHODS
▪Direct or spirit leveling
oDifferential leveling
oDouble-rodded leveling
oThree wire leveling
▪Reciprocal leveling
▪Trigonometric leveling
▪Profile leveling
▪Cross-section leveling
▪Borrow-pit leveling
NEWER AND MORE PRECISE LEVELING METHODS
▪total geodetic stations
▪airborne profile recorder
▪satellite doppler systems
▪inertial surveying systems
DIFFERENTIAL LEVELING
the process used in the establishment of differences in elevation between two
or more points with respect to a datum or mean sea level
 PRINCIPLE OF
DIFFERENTIAL LEVELING
The level, an instrument with
telescope, provides a horizontal line of sight
(collimation axis) that can be trained in any
direction. This optical line of sight is at the
same elevation as the telescope crosshair.
Read the graduated rod held vertical on a
point of known elevation (BM) to measure
the difference in elevation (DE).
 DIFFERENTIAL LEVELING
THEORY
▪Add rod readings (BS) to benchmark or known turning point elevations to get the elevation of the
line of sight (HI).
▪Subtract rod readings (FS) from the line of sight to establish elevations of unknown points.
▪Repeat over and over

HI = Elev + BS
Elev = HI - FS
PROCEDURE FOR DIFFERENTIAL LEVELING
TP 1
BM 2

BM 1
PROCEDURE FOR DIFFERENTIAL LEVELING
TP 2

TP 1
BM 2

BM 1
PROCEDURE FOR DIFFERENTIAL LEVELING
TP 2

TP 1
BM 2

BM 1
PROCEDURE FOR DIFFERENTIAL LEVELING
TP 2

TP 1
BM 2

BM 1

BM 2

BM 1
FIELD NOTES

STA sighted BS HI FS ELEV Remarks

BM1 1.100 100 concrete block supplementary BM (LAG-25)
TP1 0.750 drainage cover (metal)
FIELD NOTES

STA sighted BS HI FS ELEV Remarks

BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 0.750 100.350 drainage cover (metal)
FIELD NOTES

STA sighted BS HI FS ELEV Remarks

BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 1.245 0.750 100.350 drainage cover (metal)
TP2 0.350 big rock (with mark X)
FIELD NOTES

STA sighted BS HI FS ELEV Remarks

BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 1.245 101.595 0.750 100.350 drainage cover (metal)
TP2 0.350 101.245 big rock (with mark X)
 BM 2

BM 1

STA sighted BS HI FS ELEV Remarks

BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 1.245 101.595 0.750 100.350 drainage cover (metal)
TP2 1.355 0.350 101.245 big rock (with mark X)
BM 2 0.575 top of fire hydrant
 BM 2

BM 1

STA sighted BS HI FS ELEV Remarks

BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 1.245 101.595 0.750 100.350 drainage cover (metal)
TP2 1.355 102.600 0.350 101.245 big rock (with mark X)
BM 2 0.575 102.025 top of fire hydrant
Arithmetic check
STA sighted BS HI FS ELEV Remarks
BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 1.245 101.595 0.750 100.350 drainage cover (metal)
TP2 1.355 102.600 0.350 101.245 big rock (with mark X)
BM 2 0.575 102.025 top of fire hydrant
∑ 3.700 1.675

DE 1 = Elev BM1 - Elev BM2

= 100.000 - 102.025
= 2.025 (negative/positive sign indicates the
position of BM2 with respect to BM1)

DE2 = ∑BS - ∑FS

= 3.700 - 1.675
= 2.025

To check the elevation:

100.000
+ 3.700 ∑BS
- 1.675 ∑FS
102.025
DETERMINING ACCURACY of the survey
results:
TP 2

TP 1
BM 2

BM 1

TP 4

TP 3
STA sighted BS HI FS ELEV Remarks
BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 1.245 101.595 0.750 100.350 drainage cover (metal)
TP2 1.355 102.600 0.350 101.245 big rock (with mark X)
BM 2 0.575 102.025 top of fire hydrant
TP 3 0.475 101.755 1.250 101.280 land marker (concrete)
TP 4 0.700 101.340 1.115 100.640 side of the road (with mark X)
BM 1 1.300 100.040 concrete block supplementary BM(LAG-25)
 Error of Closure (EC)
Allowable Error of Closure (AEC)
Error of closure (EC) = Computed Elevation - Known Elevation
= BM 1’ - BM 1

Allowable Error of Closure (AEC) = c √k

where: c - constant (4, 8.4 or 12 mm if 1st, 2nd, or 3rd order)
k - total loop distance along the line of sight, km

Adjustment of the elevation of BM 2

Correction = (a/k) (EC)
where: a - distance from BM1 to BM2

If EC ≤ AEC, the survey is acceptable!

STA sighted BS HI FS ELEV Remarks
BM1 1.100 101.100 100 concrete block supplementary BM (LAG-25)
TP1 1.245 101.595 0.750 100.350 drainage cover (metal)
TP2 1.355 102.600 0.350 101.245 big rock (with mark X)
BM 2 0.575 102.025 top of fire hydrant
TP 3 0.475 101.755 1.250 101.280 land marker (concrete)
TP 4 0.700 101.340 1.115 100.640 side of the road (with mark X)
BM 1 1.300 100.040 concrete block supplementary BM(LAG-25)
Sample Problem 1

Computation:
HI = Elev + BS
Elev = HI - FS

+
-
Sample
Problem 2

Reduce the set of

differential leveling
notes in the table and
perform the arithmetic
check.
HI AT BM-100: HI AT TP-2 = 481.14+2.51
HI= Existing elevation+BS = 483.65
HI=483.61+2.71 ELEVATION AT TP-3 = 483.65-2.81
= 486.32 = 480.84
ELEVATION AT TP-1 HI AT TP-3 = 480.84+3.17
ELEVATION= HI-FS = 484.01
= 486.32-4.88 ELEVATION AT TP-4 = 484.01-1.62
= 481.44 = 482.39
HI AT TP-1 = 481.44+3.62 HI = 482.39+1.47
= 485.06 = 483.86
ELEVATION AT TP-2 = 485.06-3.92 ELEVATION AT BM-101 = 483.86-1.21
= 481.14 = 482.65
SOLVED TABLE:
ARITHMETIC CHECK

ΣBS – ΣFS = LAST ELEVATION – FIRST ELEVATION

➢ (2.71+3.62+2.51+3.17+1.47) – (4.88+3.92+2.81+1.62+1.21) = 482.65 – 481.44
➢-0.96 = -0.96
THUS, VERIFIED.
DOUBLE RODDED LEVELING
▪the process used in the establishment of differences in elevation between two
or more points by employing two level routes simultaneously

TP 2H

TP 1H
BM 2

BM 1

TP 2L

TP 1L
DOUBLE RODDED LEVELING NOTES
STA sighted BS HI FS ELEV Remarks
BM 1
TP 1H
TP 1L
TP 2H
TP 2L
BM 2

- Compute the Elevation of BM2 using data from Route 1 (Highpoints)

- Compute the Elevation of BM2 using data from Route 2 (Lowpoints)
- Compute for mean elevation of BM2

- Mean elevation BM2 = ½ (BM2L + BM2H)

THREE WIRE LEVELING
▪the process used in the establishment of differences in elevation between two or more points
using equipment with stadia hairs
▪all the three horizontal hairs are read and the average is taken as the correct value

TP 2

TP 1
BM 2

BM 1
THREE WIRE LEVELING
▪the process used in the establishment of differences in elevation between two or more points
using equipment with stadia hairs
▪all the three horizontal hairs are read and the average is taken as the correct value

TP 2

TP 1
BM 2

BM 1
 a

c a b

HDHD== Ks
Ks+ C
+ C
 Sample Problem
A survey crew is performing
three-wire leveling using the
data below. Calculate the
missing data from the field
notes shown below. The stadia
interval factor is 100.

K = 100
K = interval factor

THREAD INT = THREAD INTERVAL

Solution

THREAD INT is THREAD INTERVAL

THREAD INT = (BSupper – BSlower) x 100
ELEV = ELEVfirst + BSlast – FSlast
 TWO-PEG TEST
All instruments are subject to errors.
The checking of the instrument
(level) is therefore important. The
main error is where the line of sight is
not parallel to the horizontal line of
collimation. In this case your levels
will not be correct. A test for
checking the level is known as the
two peg test. This test determines the
amount of error and if an error occurs
notify the technician (the level must
be serviced).
Process:
1. Establish two points (A and B) approximately 50m apart on level ground, and
put the level staffs in each point, and the level half way between two point.
2. Take readings on both pegs, and find the difference in elevation.
3. Move the level as close as possible to one of the peg (5 meters). Take the two
staff readings again.
4. If the difference in height is the same, the level is okay. If not, the instrument
needs to be serviced.
EXAMPLE

A two-peg test is done with the following results: b1 = 1.543 m,

d1 = 1.586m,b2 = 1.529 m, d2 = 1.588 m, X=50.000 m. Compute the error in
mm per m. Is the error accepted. Compute the adjusted d2 value.

Solution
• Error e = [(1.543-1.586)-(1.529-1.588)] /2 = 0.008 m = 8 mm per 50 m
• Error per 30 m=30(8/50)=4.8 mm > 2 mm per 30 m (Adjustment is needed)
• Adjusted d2 rod reading = 1.588-3(0.008) = 1.564 m.
PROFILE LEVELING
▪method of determining the elevations of series of points (or difference in elevation between
points) at measured intervals along a line such as the centerline of a proposed ditch or road or
the centerline of a natural feature such as a stream bed
PROFILE LEVELING
an extension of differential leveling
Elevations are determined in the same manner
The same definitions define the concepts and terms involved
The same types of mistakes and errors are possible
A closure check should be done if the profile line runs between bench marks
PROFILE LEVELING
THEORY
Add rod readings (BS) to benchmark or known turning point elevations to get
the elevation of the line of sight (HI). HI = ELEV + BS

Subtract rod readings (FS) from the line of sight to establish elevations of
unknown points.
ELEV = HI – FS

Take any number of IFS readings at points along the line until it is necessary to
establish a turning point to move the level.

Repeat as required.
PROFILE LEVELING
DEFINITION OF TERMS
PROFILE
▪a curved line which graphically portrays the intersection of the imaginary
vertical plane with the ground surface
▪it should give an accurate and useful representation of the existing ground
configuration
PROFILE LEVELING - DEFINITION OF TERMS
STATIONING
▪a numerical designation indicating the horizontal distance from the starting
point of any point along a profile line
➢the line along which the profile is required must be properly marked by
stakes on the ground
➢the choice of intervals between stakes will depend largely on the desired
accuracy and terrain to be traversed
PROFILE LEVELING - DEFINITION OF TERMS

FULL STATION
▪points established along the profile level route at uniformly measured distances
▪usually made in multiplies of 1000, 100, 50, 30, 20 or 10 meters
 PROFILE LEVELING - DEFINITION OF TERMS
PLUS STATION
▪any intermediate points between full stations
▪taken at breaks in the ground surface and critical points (e.g., location of
culverts, bridges, structures)
 PROFILE LEVELING - DEFINITION OF TERMS
INTERMEDIATE POINTS (IFS)
▪rod readings taken along the centerline of the proposed projects to provide an
accurate representation of the ground surface
▪should be taken at regular intervals and at points where there are abrupt
changes in elevation
PROFILE LEVELING - DEFINITION OF TERMS
VERTICAL EXAGGERATION
▪a process of drawing the vertical scale for a profile much larger than the
horizontal scale in order to accentuate the differences in elevation

ELEV
(m)

DISTANCE (km)
PROFILE LEVELING - DEFINITION OF TERMS

PROFILE PAPER
▪a special heavy grade graphing paper used for plotting profiles
Example (STATIONING)
Write the distance 325 as:
1. 100 m stationing
2. 50 m stationing
3. 20 m stationing
Solution
➢Distance 325 m as 100 m stationing: 3+25
➢Distance 325 m as 50 m stationing: 6+25
➢Distance 325 m as 20 m stationing: 16+05
EXAMPLE OF PROFILING
RECIPROCAL LEVELING
▪the procedure to determine the difference in elevation between two points
when it is difficult or impossible to keep backsights and foresights short and
equal

▪Usually employed when running a line of levels across wide rivers, lakes or
rugged terrain where deep ravines are encountered

▪the procedure followed is known as the methods of reversion (unsed in two-peg

test) and it is done by reading, throwing out of level, releveling and reading
again (minimum of two repetitions)
 TP 1 TP 3

BM 1

BM 100 TP 2
RECIPROCAL LEVELING

Direction of the survey

 A
DE 1 = (a-b)
DE 2 = (a’-b’)
TDE = (DE 1 + DE 2) / 2
where:
B
TDE - true difference in
elevation

a’ b’

If the value of TDE is a b

negative, it only tells us
that point is higher than
pt B
Sample problem 1 (Reciprocal)

In leveling across a deep and wide river, reciprocal level readings were taken between tow
points, X and Y, as follows:
❑a) With instrument set up near X, the rod readings on X are 1.283 and 1.285 meters; on the
distant point Y, the rod readings are 2.618, 2.619, 2.621, and 2.622 meters.
❑b) With instrument set up near Y, the rod readings on Y are 3.478 and 3.476 m; on the distant
point X, the rod readings are 2.143, 2.140, 2.146, and 2.144 m.
❑Determine the true difference in elevation between the two points and the elevation of Y if the
known elevation of X is 290.082 m.
Solution
a) determining the difference in elevation
At 1st set-up
am = (1.283+1.285)/2 = 1.284 (mean rod reading on point X)
bm = (2.618 + 2.619 + 2.621 + 2.622)/4
= 2.620 m (mean rod reading on point Y)
DEx = (am-bm) = (1.284-2.620)
= -1.336 (diff in elevation bet X and Y with instrument set up near X)
At 2nd set-up
am’ = (2.143 + 2.140 + 2.146 + 2.144)/4 = 2.143 (mean rod rdng at point X)
bm’ = (3.478 + 3.476)/2 = 3.477 m (mean rod reading at point Y)
DEy = (am’-bm ’) = (2.143 – 3.477) = -1.334 m (Diff in elev with instr. near Y)
Calculating True difference in Elevation and Elevation of Y
TDE = (DE x + DEy)/2 = (-1.366 +(-1.334))/2 = -1.335 (True diff in ELE bet X & Y)

Elev of Y = Elev of X + TDE = 290.082 + (-1.335) = 288.747 m

(Note: a negative value of TDE means point Y is lower than X, hence TDE is
subtracted from the elevation of point X to determine the elevation of point Y.
Sample problem 2 (Reciprocal)
In leveling across a wide river, a reciprocal level readings were taken between two points A and
B as shown in the accompanying tabulation. Determine the following:
a) difference in elevation between the two points
b) elevation of B if the elevation of A is 951.750m.
Instrument setup near A Instrument setup near B
STA BS FS STA BS FS
1.283 1.478
1.284 1.480
a a’
1.286 1.476
1.283 1.478
0.675 2.143
0.674 2.140
b 0.677 2.145
b’
0.674 2.142
0.677 2.143
0.678 2.146
Sum Sum
Mean Mean
Solution
a) Determining mean rod readings and difference in elevation

am = (1.283 + 1.284 + 1.286 + 1.283)/4 = 1.284 m

bm = (0.675 + 0.674 + 0.677 + 0.674 + 0.677 + 0.678)/6 = 0.0676 m

am’ = (2.143 + 2.140 + 2.145 + 2.142 + 2.143 + 2.146)/6 = 2.143 m

bm’ = (1.478 + 1.480 + 1.476 + 1.478)/4 = 1.476 m

DE1 = (am – bm) = (1.284 – 0.676)
= +0.608 (diff in elevation between A and B with instrument set
up near A)
DE2 = (am’ – bm’) = (2.143 – 1.478)
= + 0.665 (difference in elevation between A and B with instrument set-up near B)

TDE = (DE1 + DE2)/2 = (0.608 + 0.655)/2 = + 0.637 (true difference in elevation between the two
benchmarks
Instrument setup near A Instrument setup near B
STA BS FS STA BS FS
1.283 1.478
1.284 1.480
a a’
1.286 1.476
1.283 1.478
0.675 2.143
0.674 2.140
b 0.677 2.145
b’
0.674 2.142
0.677 2.143
0.678 2.146
Sum 5.138 4.055 Sum 12.859 5.912
Mean 1.284 0.676 Mean 2.143 1.478
b) Elevation of B = Elev at A + TDE
= 951.750 + 0.637
= 952.387 m
(Note: The TDE is added to the elevation of A since B is higher than A. In the solution of DE1 and
DE2 above, a positive value is determined which shows that B is higher that A. If the value were
negative, B would have been lower that A)
TRIGONOMETRIC LEVELING
▪the procedure that involves observing vertical (or zenith) angle and slope
distance between two points
▪the difference in elevation can be calculated using trigonometric equations
The vertical distance V can
B can be determined:

V = d tanα or
V =s sinα

The difference in elevation

between A and B can be
determined by:

DEab = d tanα + HI – RR or
DEab = s sinα + HI – RR

If the elevation of A is known, the elevation of B can then be determined as:

Elev B = Elev A + DEab

Note: This method of determining differences in elevation should be limited only

to horizontal distances not exceeding 300 m. When distances are much longer,
the combined effects of earth’s curvature and refraction much be considered
and applied in the calculation of vertical distances.
Assuming long sights are involved, the difference in elevation between points A
and B would be:
DEab = d tanα + HI – RR + 0.0675 (d/1000)2 or
DEab = s sinα + HI – RR + 0.0675 (d/1000)2
When using trigonometric methods to establish accurate elevations, the
following must be taken into consideration:
▪Due to effects of curvature and refraction, the instrument to target distance
must be kept relatively short. A good rule of thumb is not to exceed 300 m.
▪Make sure you understand your equipment’s capabilities. Instruments that can
measure zenith angles and slope distances to a high order of accuracy will
produce good trigonometric elevations.
▪Set-up and level your instrument and target carefully. Measure the height of
instrument and height of target accurately.
▪Measure several slope distances and use representative or mean value.
▪Measure Direct and Reverse zenith angles, and use the adjusted value for your
calculations.
▪For longer line of sight, correct for curvature and refraction.

Modern total station instruments have built in capabilities to reduce and

display trigonometric elevations.
▪The effects of the curvature can be eliminated if vertical angle observations are made
simultaneously at both ends of the such asuch as at points A and B.
▪This technique is termed reciprocal vertical angle observations. When applied, it is no longer
necessary to take into accounts the effects of curvature and refraction.
▪DEab = d/2 (tanα + tabβ) or
▪DEab = s/2 (sinα + sinβ)
Sample Problem 1:
▪A vertical angle of +13045’ is read to a target 1.23m above point B. The measured inclined
distance, s, is 823.29m and the elevation of point A is 123.65m above datum.
▪ If the height of the instrument at A is 1.35 m, determine the difference in elevation between A
and B and the elevation of B, considering the effects of curvature and atmospheric refraction.
Solution
sinθ = v/s; v = s sinθ = 823.29 sin (13045’)
v = 195.68m (vertical component of the inclined distance)

hcr = 0.0675 (d/1000)2; where d = s cosθ = 823.29 cos (13045’) = 799.70m

hcr = 0.0675 (799.70/1000)2
= 0.04 m (combined effects of curvature and refraction)

DEab = HI + V – RR + hcr
= 1.35 + 195.68 – 1.23 + 0.04
= 195.84 m (diff in elevation between A and B considering curvature
and refraction)
Elev B = Elev A + DEab = 123.65 + 195.84
= 319.49 m (required elevation and point B)
Sample Problem 2

▪A vertical angle of -12025’ is measured to the top of a water tank from an instrument set up on a
hill 585.00 m away from it.
▪The telescope of the instrument is 1.45 m above the ground whose elevation is 462.73 m.
▪Making due allowance for the earth’s curvature and atmospheric refraction, determine the
elevation of the base of the water tank if the tank in 32.0 m high.
Solution
TanФ = v/d
v = d tanФ = 585 tan (12025’)
= 128.80 m (vertical distance from the horizontal line of sight to the top of water tank)

hcr = 0.0675 (d/1000)2

= 0.02 m (combined effects of curvature and refraction)
DEpb = Ht + V – HI – hcr
= 32.0 + 128.80 -1.45 – 0.02
= 159.33 m (diff in elevation between points P and B making due
allowance for curvature and refraction)
Elev B = Elev P – DEpb
= 462.73 – 159.33
= 303.40 m (elevation of water tank’s base)
Sample Problem 3:

▪Let A be a point of elevation 130.48 m above datum, and let B and C be points of unknown
elevation. By means of an instrument set 1.22 m above B, vertical angles are observed, that to A
being -14o45’ and that to C being +8o32’. If the horizontal distance AB is 547.20m and the
horizontal distance BC is 923.95m, determine the elevations of B and C, making due allowance
for earth’s curvature and atmospheric refraction.
Solution
a) determining difference in elevation between A and B
Vab = (AB) tanα = 547.20 tan (14o45’)
= 144.07m (vertical distance from horizontal line to point A)
hcr = 0.0675 (AB/1000)2 = 0.0675 (547.20/1000)2
= 0.02 m (combined effects of curvature and refraction)
DEab = Vab – HI –hcr = 144.07 – 1.22 – 0.02
= 142.83m (diff in elevations bet A and B, making due
allowance for the curvature and refraction)
b) Determining difference in Elevation between B and C

Vbc = (BC) tanϕ = 923.95 tan (8o32’)

= 138.63 m (vertical distance from horizontal line to point C)
hcr = 0.0675 (BC/1000)2 = 0.0675 (923.95/1000)2
= 0.06 (combined effects of curvature and refraction for the sight to C
DEbc = HI + Vbc + hcr = 1.22 +138.63 + 0.06
= 139.91 m (diff in elevation bet B and C, making due allowance for
curvature and refraction)
c) Determining elevations of B and C

Elev B = Elev A + DEab = 130.48 + 142.83

= 273.31 m (elev of B above datum)
Elev C = Elev B + DEbc = 273.31 + 139.91
= 413.22 m (elev of C above datum)
Cross Section Leveling
▪short profile taken perpendicular to the centerline of projects (e.g. highways,
railroad, canal, sewer line)
▪taken at regular intervals (usually full stations and some plus stations) and
where there’s abrupt changes in the profile of the centerline
▪Prolonged up to the limits of right of way or point where possible earthwork will
be undertaken
 3 + 00

2 + 00 f

1 + 50
1 + 25
1 + 00 e
d CL
c
0 + 00 b

a
1.2
1.2 2.4
2.0
BM 1
3.0 3.0

section a-a
Cross-sections leveling
▪Obtained in a manner similar to that of profile leveling
▪At any suitable position, first take a BS on BM or any point of know Elev to detemine HI.
▪To accurately determine the shape of the ground surface, rod readings are then taken at
sufficient number of points along a cross-section.
▪A steel tape is used to determine the distance of a ground point from the center line station.
▪The elevations of ground are taken at nearest decimeter, however, when BM or TP are observed,
rod reading are usually taken at nearest cm.
▪In the field notes, the recorded values (shown as fractions) are nothing but two
recorded quantities separated by a horizontal line.
▪The value indicated above the line is the rod reading on the ground point and
the value below the line is the measured distance from the centerline to the
point on which the rod reading is taken.
▪The computed ground elevations are recorded and enclosed in a parenthesis or
written italics in slanting position.
▪The elevations determined by cross-section leveling are plotted on sheets of
graph paper using identical horizontal and vertical scales.
▪A planimeter is often used in determining the cross-sectional areas. The areas
are in turn used in computing the volume of earthwork for a given project.
Borrow pit Leveling
▪method of determining the relative elevation of points in borrow pit excavations for the
purpose of calculating volumes of earthwork

BORROW PIT - an open area which is usually adjacent to a construction project

where suitable fill material is excavated

▪employed in the construction of structures and excavations of borrow pits

▪Two methods can be used
Grid Leveling
Radial Line Leveling
PROCEDURE IN GRID:

1 2 3 4 5

▪Grid leveling is a method for locating A BM 1

contour lines and topographic features by
stacking an area in squares of 5, 10, 50,
100 m, or more depending on the project B TP 1
extent, ground roughness, and accuracy
required C

E
PROCEDURE IN RADIAL LINE:

▪This method for locating contour lines and topographic features is simpler to
perform compared to grid leveling, and it requires less time.
▪The level instrument is set up in the middle of the field and the rod person
moves along radial lines from the Instrument. Radial lines are spaced at equal or
unequal central angles.