LEVELING I. Definitions: 1. Leveling is the procedure used to determine differences in elevation between points that are remote from each other. 2. Elevation is the vertical distance above or below a reference datum. 3. Mean Sea Level (MSL) is the universally employed reference datum. It is assigned an elevation of 0.000m. 4. Vertical line is a line from the surface of the earth to the earth’s center. It is also referred to as a plumb line. 5. A level surface is a curved surface parallel to the mean surface of the earth 6. A level line is a line in a level surface. 7. A horizontal line is a straight line perpendicular to a vertical line. II. Theory of Differential Leveling Differential leveling is used to determine differences in elevation between points that are remote from each other by using a level

instrument together with a graduated measuring rod.

Leveling Operations Definitions 1. Benchmark (BM) is a permanent point of known elevation. They are bronze disks established using precise leveling techniques. 2. Turning Point (TP) is a point temporarily used to transfer an elevation. 3. Backsight (BS) is a rod reading taken on a point of known elevation in order to establish the elevation of the instrument line of sight. 4. Height of Instrument (HI) is the elevation of the line of sight. 5. Foresight (FS) is a rod reading taken on a point whose elevation is to be determined.

The leveling procedure is used to determine the elevations of selected points with respect to a point of known elevation. In every set-up, the level is set up midway between the BM and the point whose elevation is to be determined, and rod readings are taken at both locations. The HI and elevation of TP are then determined as follows: HI= Elevk + BS ElevD= HI – FS These two equations completely describe the differential leveling process.

MSL-mean sea level

ElevD

HI

FS

BS

Elev K

SINGLE SET-UP LEVELING

Example: Given the plan of a line of levels: A. Write down the given data in field notes form. B. Compute all elevations. C. Show arithmetic check on the elevation of BM-19. D. Compute error of the circuit.

E. By BS and FS, compute the difference in elevation of BM-19 and BM-20. Differential Leveling (Single-Rodded) FIELD NOTES:

STATIONM BS HI FS Elev.

BM-19 1.902 13.467 11.565

1.364 13.896 0.935 12.532

1.493 14.533 0.856 13.040

1.562 15.131 0.964 13.569

BM-20 0.966 14.272 1.825 13.306

1.054 13.472 1.854 12.418

0.402 12.239 1.635 11.837

BM-19 0.669 11.570

BS= 8.743 FS =8.738

C. Arithmetic check to computed elevation of BM-19 Elev BM-19 (Given) = 11.565

+ BS = 8.743 SUM = 20.308 Elev BM =-19 (Comp) = 11.570 D. Error of the circuit

1. From the BS and FS: Error = 8.743-8.738 =0.005 2. From the given computed elevations of BM-19 and BM-20 by

BS and FS:

BS =1.902+1.364+1.493+1.562=6.321

FS = 0.935+0856+.964+1.825=4.580 Difference in elevation =6.321-4.580=1.741m (BM-20 is higher than BM-19)

BM 19

Elev. 11.565

m

1.364

1.902 1.562

1.493

0.669

0.402 1.054 0.966

1.635 1.854

1.825

0.964 0.856

0.935

BM - 20

TP-5

TP-4

TP-3

TP-2 TP-1

DOUBLE-RODDED LEVELING

Double-rodded differential leveling is a method of determining differences in elevation between points by employing two level routes simultaneously. This method differs from conventional differential leveling in that two turning points are established such that at each setup of the leveling instrument, two sets of independent backsights and foresights are taken. This method has an advantage of providing a continuous check on the process of determining ground elevations while the work is in progress. It is extremely useful when there is an urgent need to undertake diffential leveling in a short period of time where no established bench marks are available for checking results. Double-rodded leveling is also useful when running a long line of levels which do not close back on the initial bench mark. Complete the following differential level notes for a double-rodded line from BM-1 to BM-2. Show the customary arithmetic check:

STA BS HI FS ELEV

BM1 1.584 1.584

198.365

TP1 L TP1 H

2.287 2.565

1.124 1.467

TP2 L TP2 H

1.385 1.785

1.864 2.204

TP3 L TP3 H

0.328 0.786

2.993 3.448

BM2 2.824 2.824

LECTURE PROBLEMS ON LEVELING 1. In leveling across a river the following conditions are given: a. Elevation of BM-1 832.47 ft b. BS is 9.24 at a distance of 100 ft c. FS is 3.28 on BM-2 at a distance of 1200 ft.

What is the elevation of BM-2 corrected for the curvature of the earth and refraction? Ans. 838.46 ft

2. In testing an Engineer's level the following readings were taken: With the instrument midway between stakes A and B, the reading on A, 3.84 and B, 6.22 ft. With the instrument at A, the reading was

4.156. What is the computed correct reading on B? Ans.: 6.54 ft What is the amount and direction of the inclination of the line of sight ifthe reading form A ion B is 6.58 ft? Ans.: Upward 0.04 ft. 3. In leveling across a river, reciprocal level readings were taken between two benchmarks A and B as follows: instrument near A, rod on

A, 4.21, rod on B, 2.21, instrument near B, rod on B 4.85, rod on A, 7.03. The elevation of A was 2465.73 ft. Find the elevation of benchmark B. Ans. 2467.82 ft.

4. If the distance between the benchmarks A and B of Prob 3 is 1600 ft, what is the error in the adjustment of the instrument? Ans. The line of sight is inclined upward 0.04 ft in the distance between the benchmarks.

5. Benchmark -1 of elevation 126.54 m above MSL is 10 km from BM-2. Differential leveling was run from BM-1 to BM-2. On the average, BS distances were 150 m and FS distances were 50 m at every set-up. The computed elevation of BM-2, as deduced from the Fn., was 355.12m above MSL. It was found after test that the line of sight of the instrument was inclined upward by 0.015m for every 50m, line of sight. Find the corrected elevation of BM-2. Ans. 353.62m

CURVATURE AND REFRACTION

The effects of earth 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. Due to the earth’s curvature, a horizontal line departs from a level line by 0.0785m in one kilometer. Atmospheric refraction varies with atmospheric conditions. Under ordinary conditions it is approximately equal to 0.011m in one km, also varying directly as the square of the length of the line.

The combination of the earth’s curvature and atmospheric refraction causes the telescope’s line of sight to vary form a level line by approximately 0.0785 minus 0.0110 or 0.0675 m in one km, varying as the square of the sight distance in kilometer.

This may be presented by a mathematical equation as follows.

20675.0 Khcr

where: hcr= departure of a telescope line of sight from a level line = elevation correction due to earth’s curvature and refraction (m) K = length of the line of sight (km) = level distance from point of tangency to the observer 0.0675 = coefficient of refraction

Earth Curvature: Due to the curvature of the Earth, the line of sight at the instrument will deviate from a horizontal line as one moves away from the level:

Ideally one would like the line of sight to be a curved line which is everywhere perpendicular to the direction of gravity. The error in staff reading due to Earth curvature is given by:

2R

s e

2

c

Point of Tangency

Level line

Refracted line

Horizontal line

Vertical line

Distance

B

C

D

A

where s is the sight length and R is the radius of curvature of the Earth. For a sight length of 100m the effect is only 1mm. As with collimation error, the effect is eliminated by using equal sight lengths for fore- and backsights. Refraction: The variable density of the Earth's atmosphere causes a bending of the ray from the staff to the level. The effect is illustrated in the sketch below:

The light ray is bent in a path which has a curvature less than that of the Earth's surface, and the combined effect is smaller than that due to Earth curvature alone:

2r s.

2R

k - 1 e

Here, k is the coefficient of refraction and represents the ratio of the radius of curvature of the Earth to the radius of curvature of the light path. An average value of k is 0.13, from which:

er = 0.068.10-3.s2

where s is in metres and er in millimetres. For example, for s = 100m, er = 0.7mm. The effect of refraction is almost totally eliminated by using equal fore- and backsights (because atmospheric conditions along the fore- and backsights will not be completely identical, there will be a small residual error).

Example: In leveling across a river the following observations were obtained: Elevation of BM-1 = 32.481m BS on BM-1 at a distance of 30m =1.24m FS on BM-2 at a distance of 400m = 2.32 Find the elevation of BM-2 corrected for C & R Problem: 1. A woman standing on a beach can just see the top of a lighthouse 24.140 m km. Away. If her line of sight is above the sea level at

1.738m, determine the height of the lighthouse above the sea level. Neglecting the effect of tide and waves, determine how far out to sea a boat will be when a light on its mass 60 m above the water disappears froM the sight of a man on shore whose eye level is 1.583m above the water.

370

m

30m

BM-2

2.32 1.24

BM-1

PROFILE LEVELING

Profile levels are taken along the centerline of a proposed road, sewer pipeline, canal etc. to determine elevations along the centerline route at even intervals, usually 20meters. Rod readings taken at every 20-m interval are called intermediate Foresights (IFS) and are usually read to closest 0.01m FIELD NOTES:

STATION BS HI FS IFS ELEV

BM-46 2.868 53.578 50.710

0+000 2.06 51.52

020 1.17 52.41

040 1.63 51.95

060 1.62 51.96

080 1.41 52.17

0+100 1.01 52.57

TP-1 1.977 54.573 0.982 52.596

0+120 1.73 52.84

140 1.70 52.87

160 1.89 52.68

180 1.67 52.90

0+200 1.60 52.97

0+220 1.45 53.12

RECIPROCAL LEVELING

In the river or valley crossings, it is not always possible to balance BS and FS distances. In this case, reciprocal leveling is carried out. The level is set-up and readings are taken at both sides of the river or valley. The level is then moved to the other side of the river or valley and the process repeated. The difference in elevation thus obtain, is averaged to obtain the final result. The averaging process will eliminate instrumental errors and such natural errors as curvature. Example:

1. In leveling across a deep and wide river, reciprocal level readings were taken between two points as follows: a) With instrument set up near A, the rod readings on A are 1.625 m and 1.623 m, on a distant B, the rod readings are 3.175m,

3.178m, 3.176m and 3.179m. b) With the instrument set up near B, the rod readings on B are 4.248m and 4.246m; on the distant point A, the rod readings are

2.691m, 2.693m, 2.690m and 2.694m. c) Determine the true difference in elevation the two points and the elevation of B if the known elevation of A is 160.321m above

mean sea level. 2. Reciprocal leveling was done to determine the difference in elevation between two points in each bank of a river. The level was

kept near C and the readings obtained were 2.605m and 2.300m. The level was then shifted to Sta. D and the readings were again taken on D and C, respectively. The readings were 1.750m and 2.205m. If the elevation of Sta.C is 175.850m, find the elevation of Sta. D.

0+000

0+060

0+100

0+160 0+220

Tp-1

2.868

BM-46

Elev. 50.710m

ANGLES AND DIRECTION The purpose of a survey is to determine the relative location of points on or near the surface of the earth. The location of a point is

fixed if measurements are made of a) its direction and distance from a known point b) its direction from two known points c) its distance from two known points or d) its distance from one known point and its direction from another. If the relative locations of points as seen in horizontal projections are desired, the field operations involve the measurement of

horizontal distances, and the determination of direction in the horizontal plane. If the relative elevation of points is required, they are determined by one of the methods of leveling.

For horizontal projection or plan, the direction of any line (as fixed by two points is defined by the horizontal angle between the line and some reference line. For vertical projection, the direction of one point with respect to another is defined by the vertical angle between the plane of the horizon and the line joining the two points. In general, therefore, the angular measurements of surveying are either horizontal or vertical, or approximately.

When the angle between two points is mentioned, it is understood to mean the horizontal angle, or angle between the projections in the horizontal plane of two lines passing thorough the two points and converging at a third point.

DEFINITION OF TERMS 1. Meridian – the line of reference passion the observer and the north-south directions. It is called TRUE if the poles used are the

geographic poles; MAGNETIC if the poles used are the magnetic poles. 2. Bearing – the angle which a line makes with either the south or north end of the reference line. It is called TRUE if the meridian

is true; MAGNETIC, if the meridian is magnetic. The range of a bearing angle is from 0˚ to 90˚. Bearing 0˚ is either due north or due south; bearing 90˚ is either due east or due west.

Examples:

AB=N3500'E CD= DUE EAST FG= S4010'E

HI= DUE SOUTH JK= S7825'30" LN= N 1500'W 3. Azimuth - the angle which a certain line makes with the south end of the reference line measured clockwise, magnetic or

assumed. English azimuth is one that is measured from the north end of the reference line also in the direction clockwise. The range of

an azimuth angle is from 0 to 360, and no direction is required to define it.

E W

S

N

A

B

E W

S

N

D C E W

S

N

F

G

E W

S

N

H

I

E W

S

N

J

K

E W

S

N

L

N

AB= 4000' CD= 9000' FG= 15120'

HI= 18000' JK =21515' LM = 27000' 4. Astronomical azimuth- that azimuth which is obtained from direct astronomical observations dependent upon the plumb line. Deflection angle - angle which a line makes with the prolongation of the preceding line and is indicated with a magnitude not

exceeding 180, the direction of which is either right or left

B= 3700' E= 4000' 6. Secular variation - an extremely slow swing of the magnetic needle, periodic in nature and extending over a long period of time

due to the shifting of the earth's magnetic interior 7. Local Attraction- In the presence of magnetic bodies, the magnetic needle is attracted and deflected from its true position. The

bearings obtained are not correct but the angle computed from the bearings is correct. 8. The Surveyor's Compass - an instrument for determining the horizontal direction of a line with reference to the direction of a

magnetic needle. The needle is balanced at its center on a pivot so that it swings freely in a horizontal frame. The pivot is at the center of a

horizontal circle, which is graduated to degrees and half degrees, and numbered from two opposite ero-points each way to 90. The zero

points are marked with N and S and the 90 points with E and W. The circle is covered with a glass plate to protect needle and graduations. A screw is provided for raising the needle from pivot by means of a lever. The needle should always be raised when compass is lifted and varied to prevent dulling the pivot point; a dull pivot point is a source of error. Both circle and pivot are secured to a brass frame. On which are tow vertical sight so place that the plane through them also passes through the zero points of the circle. This frame rests on a tripod and is fastened to it by means of a ball-and-socket joint. On the frame are two spirit levels at right angles to each other, which afford a means of leveling the instruments. This ball-and-socket joint is connected with a frame by means of a spindle, which allows the compass head to be revolved in a horizontal plane or to be clamped in any direction. The magnetic needle possesses the property in pointing a fixed direction, the magnetic Meridian. The horizontal angle between the direction of this meridian and any other line may be determined by means of the graduated circle, and this angle is called the Magnetic bearing of the line. If the bearings of two lines are known the angle between them may be computed.

N

S

A

B

N

S

C D

N

S

F

G

N

S

H

I N

S

J

K N

S

M L

A

F

E

D

C

B

A

METHODS OF OBTAINING MAGNETIC BEARING In the case of a surveyor’s compass or of the transit, both are set upon and leveled on some point on the north-south set beneath the

needle. Tap glass cover, if needle is not free to move. If it appears to cling to glass (glass being electrified) place moistened finger over the glass.

Bearings are usually read to the nearest quarter of a degree although; it is possible to estimate somewhat closer.

Since the needle stands still and the box turns under it, the letters E and W in the box must be reversed from their true position so

that the direct reading of the needle will give not only the angle but also the proper quadrant. Rule: When the north point of the compass box is toward the point whose bearing is desired, read the north end of the needle. When the

south point of the box is toward the point, read the south end of the needle. If a bearing of the line is taken looking in the opposite direction it is called reversed or back bearing.

In reading the compass needle, electric currents are a great source of disturbance to the needle. In cities, the compass is useless.

Vehicles passing or parked nearby will attract the needle. In reading the compass needle the surveyor should take care to read the farther end of the needle, not across it. By looking at the needle sidewise it is possible to make it appear to coincide with a graduation, which is really at one side of it. This error is called parallax.

COMMON SOURCES OF ERROR IN COMPASS WORK 1. Iron or steel or electric current near compass. 2. Not letting needle down on pivot.

3. Reading the wrong side of the 10th degree; Example: reading 61instead of 59 4. Reading from the wrong end of the box, as from the W or E ends instead of from N (+) or S (-) ends. 5. Failure to change the direction as in the case of NE to NW when reading bearings with compass where the E and W are not

interchanged. Note: Since iron and steel near instrument affect the direction of the needle, great care should be taken that the tape, axe, marking

pins, etc. are not left near the compass. Small pieces of iron on the person such as keys, bracelet, rings etc. affect readings from the compass.

Compass Problem: 1. A pentagonal lot is determined of its interior angles beginning with corner A measuring the angle at each corner counter clockwise,

to corner E as follows. A= 10059’, B=11031’, C=9549’, D=12021’ and E=11210’. If bearing of AB is N5026’W, find the bearing of each of the sides.

N

E W

S

OBJECT

EYE

OBJECT

N 3000’E

EYE

OBJECT

N 5000’ W