Reading 3 - Leveling

7/31/2019 Reading 3 - Leveling

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Chapter 15 Leveling

Contents

1. Differential Leveling2. Leveling Equipment

3. Stadia4. 3-Wire Leveling5. Errors in Leveling6. Adjustment of Leveling Lines7. Profile Leveling8. Vertical Datums

Practice Problems

1. Differential Leveling

As the term implies, differential leveling is a process of measuring the difference in

elevation between points. A surveyors level and level rod are used in this process. Alevel rod is merely a piece of wood, fiberglass, or metal marked off in meters or feet and

fractional parts of those units. The level is a telescope equipped with cross hairs andattached to some type of device allowing the telescope to remain at the same height while

turning in a horizontal plane. Thus, the level establishes a line of sight in a horizontal

plane from which measurements can be made with the rod. As the observer focuses on

the level rod, the graduation of the rod coincident with the line of sight can be read thusreflecting the distance between the line of sight and the point on which the rod is resting.

Figure 15.1 illustrates the use of a level and rod to measure the difference between twopoints. In that illustration, bench mark (BM) 1 is a permanent object with a known

elevation of 436.27 feet above the datum being used. Turning point (TP) 1 is a temporaryobject, such as the top of a stake, for which an elevation is to be measured. For themeasurement, the level is set up in a location allowing visibility of both objects and the

following two-step process is followed:

1. With the rod on BM 1, a reading is made and recorded. This reading is known

as a backsight(BS) and is added to the elevation of the bench mark to determine

the height of instrument(HI) which is the elevation of the line of sight.

2. The rod is then placed on TP 1 and a reading made and recorded. This reading

is known as theforesight(FS) and is subtracted from the height of instrument to

determine the elevation of TP 1.


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Figure 15.1 Differential Leveling

(use Figure 13.2 from 5th Edition)

Using the process described above, the elevations of a continuing line of objects (or twoobjects separated by a greater distance or greater difference in elevation than can be

covered by one set up of the level) may be determined by moving the level between setups as illustrated in Figure 15.2 and the associated field notes. With that process, a leap

frog process is used between the rod and level. After a backsight is taken, the level rod

remains at its location while the level is moved ahead to the next set up and then abacksight is taken at the point that served as the previous foresight. The level then

remains at the new set up while the rod is moved to the next point for the foresight.

Figure 15.2 Continuous Differential Leveling

(use Figure 13.3 from 5th edition)


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Field Notes for Differential Levels

Point BS(+) HI FS(-) Elevation

BM 3 441.72

8.56 450.28 1.10

TP 1 449.18

10.34 459.52 1.37TP 2 458.15

7.75 465.90 1.83

TP 3 464.07

8.89 472.96 3.46

BM 4 469.50

2. Leveling Equipment

Level Rods As described previously, a level rod is merely a graduated rod used to

measure the distance between a point and the line of sight. The most commonly type of

level rod in current use is the Philadelphia rod. That device consists of two sliding partsand typically has graduations extending from zero to seven feet on the base part with the

ability to extend the rod on up to 13 feet with the sliding upper part. Most Philadelphia

rods are graduated in feet, tenths and hundreds with graduations running continuouslyfrom zero at the bottom to 13 feet at the top. Each full foot is marked with a red number

(white in Figure 14.3) and each tenth of a foot is marked with black numbers. The

hundreds graduations alternate from black to white. The black graduations are pointed ateach tenth of a foot (0.10, 0.20, 0.30, etc) as well as for each five hundredth of a foot

(0.05, 0.15, 0.25, etc.) for ease of reading. For greater precision, the thousandth of a footmay be interpolated. The rod is read at the point where the center cross hair of the

telescope intersects the rod. As examples, the correct rod reading for Figure 15.3(a) is5.130 feet while the correct reading for Figure 15.3(b) is 5.180 feet.

Figure 15.3 Graduations on a Philadelphia Rod

(use Figure 13.4 from 5th Edition)


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Philadelphia rods typically are constructed with a thin steel tape inserted in the face of awooden support. For projects requiring greater precision, rods with a graduated strip

made of invar is used. Invar is an alloy with a very low coefficient of expansion, thus

reducing imprecision due to contraction and expansion of the graduated strip withchanging temperature. Further, invar rods typically have the graduated strip supported at

a constant tension to standardize measurements. In addition, invar rods are typically inone continuous piece as opposed to having two sections as does the Philadelphia rod toeliminate any error due to the join. Typically, such rods are equipped with a level bubble

and prop poles (Figure 15.4) to insure that the rods are perfectly plumb when read.

Figure 15.4

Invar Rod with Prop Pole

Surveyors Levels Traditionally, surveyors levels were spirit levels (Figure 15-5).

Such an instrument is equipped with an elongated slightly curved glass tube filled withalcohol or other liquid which allows establishing a level horizontal line by centering a

bubble. Setting up such instruments can be time consuming since the instrument has to

be carefully adjusted to insure that it is level when pointed at any direction.

Figure 15.5 Spirit Level


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Today, self-leveling levels are in much more common use than spirit levels. Suchinstruments are equipped with a compensator and a simple bulls eye level rather than a

long spirit level. A compensator is a device consisting of a system of prisms and mirrors

suspended as a pendulum (Figure 15.6) within the level. As long as the level is roughlyleveled up, typically within +/- 10 of the true vertical, the compensator swings freely and

establishes a horizontal line of sight. As a result, a self-leveling level is light, easy tohandle and set up, and its operation s quick and accurate.

Figure 15.6 Compensator in a Self-Leveling Level

The current state of the art in levels is the digital level. A digital level (Figure 15.7)combines the advantages of the self-leveling level with a solid-state camera and

electronic image processing. Such instruments use a level rod with bar-code graduations.Thus, the rod readings are automatically recorded eliminating observer and recording

error.

Figure 15.7 Digital Level


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3. Stadia

Distances between the level and the level rods may be measured by use of the stadia

hairs in the telescope. One stadia hair is above and one below the principle horizontalcross hair. Distance measurement by this method is based on the principle that the

interval between the two stadia hairs is proportional to the distance between theinstrument and the rod. Most levels have a stadia constant of 100. Therefore, thedistance to the rod may be determined by multiplying the interval between the top and

bottom cross hair by 100. As an example, in Figure 15.8, the top stadia hair has a rod

reading of 5.248 (interpolated to third place). The bottom stadia hair has a reading of

5.013. Therefore, the distance to the rod is 100(5.248 5.013) or 23.5 feet.

Figure 15.8 Stadia

4. 3-Wire Leveling

For increased precision with leveling the process of 3-wire leveling may be used.

That process increases the precision by using the average of three thread observations

for each sighting. In addition, that process provides an ideal method for determiningif leveling shots are balanced as well as allowing post correction to eliminate

collimation error. The process involves observing rod readings for all three

horizontal cross hairs for each sighting. By comparing the average of the threereadings with the middle hair reading, bad readings may be detected immediately and

the readings repeated. For this process, notes are typically kept as shown in Figure

15.9.


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Figure 15.9 3-Wire Leveling Notes

BS FS Elev FS Dist BS Dist

10.000

top 3.000 1.500mid 2.500 0.995

bot 2.000 0.490

mean +2.500 -0.995 11.505 100 101

top 5.255 2.962

mid 4.555 2.254

bot 3.853 1.546

mean +4.554 -2.254 13.805 140 142

5. Errors in Leveling

Collimation Error The most common cause of errors in leveling is imperfectadjustment of the level. Almost all level lines will have at least a slight difference from a

truly horizontal line. That difference is called collimation error(Figure 15.10). This

error can be eliminated by insuring that the backsight and foresight distances are equal inlength which cancels the error. In addition, by frequently performing collimation checks

on the level and adjusting the instrument as needed, this error can be minimized even in

unbalanced shots.

Figure 15.10 Collimation Error

Collimation checks are made by setting two points about 200 feet apart. The level is first

set up about 20 feet from the first point on a line between the two points and rod readings

for all three horizontal crosshairs read and recorded. The level is then moved to a

location on the line between the two points about 20 feet from the second point and rod


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readings for all three horizontal crosshairs read and recorded. The correction for thecollimation error (C factor) may then be calculated as the sum of the short readings

minus the sum of the long readings divided by the sum of the long distances minus the

sum of the short distances. This may be expressed numerically as follows:

)()()()(

2121

2121

SSLL

LLSS

DDDDRRRRC++ ++=

(15.1)

where C = collimation correction in ft/ft or m/m

R = rod reading for middle hair

D = distance calculated by stadia

S = subscript for short readings or distances

L = subscript for long readings or distances

1 = subscript for readings or distances for position 1

2 = subscript for readings or distances for position 2

If the C factor and the distances to the rod for each set up known, post-correction ofleveling data to eliminate the collimation error can be accomplished. If the C factor is

large (generally considered to be over 0.01 ft/ft), the level should be adjusted. This can

be done while the level is still set up at position 2, by adjusting the alignment screw untilthe middle thread reading for the short sight reads the value calculated as follows:

))(( 22 SS DCRCCorrect += (15.2)

where the symbols are the same as Equation 15.1

Curvature Error Due to the curvature of the earth, a level reading on a distant rod isgreater than if leveling on a plane (Figure 15.11). The effect of curvature of the earth

may be calculated as 0.667 M2

feet (where M is distance in miles) or 0.785 K2

(where Kis distance in kilometers). Therefore, this is not a significant error for short sightings. It

can, however, become a factor for long sightings. This error can be eliminated by having

balanced backsights and foresights.

Figure 15.11 Curvature Error


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Refraction Error Refraction causes errors in leveling due to its bending of the lightrays due to differences in air temperature encountered in the line of sight. Refraction is

proportional to distance and is generally in the opposite direction from curvature error. It

therefore cancels out some of the curvature error. Refraction error may be minimized byavoiding sightings that pass close to the earth since temperature gradients are generally

greatest close to the earths surface. The general rule is to avoid sightings where the lineof sight passes closer than a half meter from the ground. Refraction error may also beminimized by having short, balanced sightings and taking the backsight and foresight

observations in quick succession.

Recent research by the National Geodetic Survey has resulted in a refraction modelingprocess to determine refraction error. Therefore, for precise geodetic leveling conducted

by or for that agency, tripods for levels are equipped with temperature sensors mounted

about 0.3 and 1.3 meters above the ground. Temperatures from those sensors arerecorded with each leveling sighting for correction purposes.

Rod Plumb Error One of the most common and yet most easily prevented errors is

that caused by the level rod not being plumb at the time of sighting. This error may be

prevented by the person holding the rod being especially alert to the plumb of the rod andby use of easily obtainable rod bubbles. As may be seen in Figure 15.11, the true reading

of an inclined level may be calculated as the product of the rod reading and the sine of the

angle from the vertical.

Figure 15.11 Level Rod Plumb Error


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Parallax Parallax is the apparent change in the position of the cross hair as viewedthrough the telescope. Because the reticle (the ring that holds the crosshairs in the

telescope) is stationary, the distance between it and the eyepiece must be adjusted to suit

the eye of each individual observer. The eyepiece is adjusted by turning it slowly untilthe crosshair is as black as possible. After the eyepiece is adjusted, the object viewed

should be brought into sharp focus by means of the focusing knob for the objective lens.If the crosshairs seem to move across the object when the viewer moves his eye slightly,parallax exists. It is eliminated by carefully adjusting the eyepiece and the objective lens.

If parallax is not eliminated, it can affect the accuracy of the rod readings.

Other Errors in Leveling In addition to those errors listed above, there are numerous

other potential sources of errors in leveling. Possibly the most common are blunders in

observing or recording of rod readings. The proliferation of digital levels is rapidlyeliminating that source. Another common source is the choice of turning points. Turning

points should be exceptionally stable and symmetrical on top so that the same elevation is

reflected for both the foresight and backsight on the point. Other errors are caused indexand expansion errors of the level rod. For precise leveling, this is typically corrected by

the calibration of the rod against a known standard.

6. Adjustment of Leveling Lines

Good leveling practice calls for either redundancy or closure in leveling lines to detectblunders in observations. A preferred method is for each segment in a leveling line to be

double run, once forward and once back, thus making two independent sets ofmeasurements. A second method is for the leveling line to be run in a complete circuit or

closed on a second bench mark of known elevation.

With the double run process, there are typically some differences in the forward and back

runs which allows determination of a closure error. In addition, since each segment in

the line is double run, this allows distribution of corrections to the appropriate segmentby averaging the forward and back runs. The process for determining error of closure

and abstracting double run levels is illustrated in Figure 15.12.


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Figure 15.12

Abstract of Double Run Levels

BM Direction Dif in Elev Avg Elev

1 10.000F 1.092

B -1.089 1.091

2 11.091

F 5.336

B -5.340 5.338

3 16.429

F -2.786

B 2.779 -2.783

4 13.646

F -5.443

B 5.447 -5.445

5 8.201

Total Fwd Run -1.801

Total BackRun 1.797

Closure 0.004

For single run closed leveling circuits, closure error may be determined by the difference

between the measured final elevation and the known elevation of the closing bench mark.

Since the location of the error is not known as in double run levels, an alternate methodmust be used to distribute the error. Since random errors in leveling usually accumulate

in proportion to the number of instrument setups, it follows that the error of closure

should be distributed within the various segments of the level line in proportion to thenumber of setups in each segment. The process for adjusting and abstracting a typical

closed leveling circuit closing on beginning bench mark is illustrated in Figure 15.13. As

may be seen, the correction that is applied to each segment is calculated as follows:

.).(.

.elevendingelevbeginning

setupsofnototal

segmentinsetupsofnocorrection = (15.3)


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Figure 15.13

Abstract of Closed Level Circuit

SegmentDif inElev Elev Setups Correction

AdjustedElev

10.000 10.000

BM 1 - BM 2 1.092 3 -0.006

11.092 11.086

BM 2 - BM 3 5.336 2 -0.004

16.428 16.418

BM 3 - BM 4 -2.786 3 -0.006

13.642 13.626

BM 4 - BM 1 -3.618 4 -0.008

10.024 10.000

Total

Closure = 10.024 - 10.000 = 0.024 12

The leveling adjustments discussed above deal with simple closed leveling lines or loops.When dealing with a more complex leveling network, a more complex approach to

adjustment must be used to distribute any error since there are typically multiple

solutions. Figure 15.14 illustrates such a complex network. With this type of network,the best approach is a least squares adjustment.

Figure 15.14 Typical Complex Leveling Network


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Least squares adjustment of a leveling network utilizes matrix algebra. (Chapter 12covers the basic matrix algebra operations that are used in a least squares adjustment.)

The network illustrated in Figure 15.14 will be used to illustrate the adjustment process.

The first step is to create an observation equation is created for each line in the network

which defines the unknown points in terms of the known elevations and elevation

differences. This process is illustrated as follows:

Line 1: A = (100 + 5.10) = 105.10

Line 2: A = (107.50 - 2.34) = 105.16

Line 3: C = (107.50 1.25) = 106.25Line 4: C = (100 + 6.13) = 106.13

Line 5: A - B = 0.68

Line 6: B = (107.50 - 3.00) = 104.50Line 7: B - C = - 1.70

The second step is to use the above equations to create three matrices, typically calledA, L, and X. Matrix A contains a column for each unknown point in the network

(A, B, and C) and a row for each observation equation. Each element in the first column,

which represents point A, is the coefficient for A from the corresponding equation.

Therefore, the elements are either 1 or 0. There is also a corresponding column for Band C. Matrix L has only one column and also has a row for each observation equation

which contains the values from the right hand side of the equations. Matrix X containsone column with a row for each unknown point in the network.

=

=

=

C

B

A

XLA

70.1

50.104

68.0

13.106

25.106

16.105

10.105

110

010

011

100

100

001

001

The adjustment process involves solving Equation 15.3 provided below for matrix X.Typically, this is accomplished by use of a calculator with matrix algebra capability or acomputer program.

LAAAXTT 1)( = (15.3)


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When solved, equation 15.3 used with data from this illustration yields the matrix Xwhich contains the adjusted or most probable values for A, B, and C, the three points in

the illustration with unknown elevations.

=

=20.106

49.104

14.105

C

B

A

X

An additional important step with a least squares adjustment is an evaluation of the

residuals from the adjustment. This allows the spotting of blunders as well as providingan idea of the size of the random errors in the data. Using the same A, X and L matrices

as in the adjustment itself, a residual matrix may be generated using the following

equation:

LAXV = (15.4)

For the sample data used above in the adjustment, the calculation and the resulting

residuals are illustrated below:

=

==

005.0

017.0

022.0

058.0

062.0

019.0

041.0

70.1

50.104

68.0

13.106

25.106

16.105

10.105

20.106

49.104

14.105

110

010

011

100

100

001

001

LAXV

The above process assumes that all level lines in the network have an equal weight for thedistribution of errors. A possibly more valid approach is to distribute errors weighted to

the length of the various lines, or to the number of setups in each line. This may be

accomplished for the illustrated network by creating a weight matrix W based on thenumber of setups tabulated in Figure 15.14. Weights in a leveling network are inverselyproportional to the lengths of lines or the number of setups. Therefore, the elements for

each line in matrix W are entered as 1 divided by the number of setups. Note that matrixW has the same number of columns and rows with both equal to the number of

observational equations and that the table is a diagonal matrix with the weights entered in

a diagonal line and all other elements being zero.


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The adjustment process using weights involves solving the following equation for matrix

X:

))( 1 WLAWAAXTT = (15.5)

For the illustrated network, a comparison of the results of unweighted and weighted

adjustments are as follows:

Point Unweighted Weighted

A 105.14 105.15

B 104.49 104.49

C 106.20 106.21

=

4

1000000

04

100000

004

10000

0006

1000

00004

100

000006

10

0000008

1

W


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7. Profile Leveling

Profile leveling is the process of determining elevations of points at measured intervals

along a fixed line. This is a common process in the planning of highways, canals,pipelines and other linear projects where a vertical section of the earth is needed to

determine the location of the centerline of the project. Profile leveling is similar todifferential leveling except that multiple foresight readings are typically taken from onebacksight reading.

At each setup of the level, foresight readings are taken along the centerline of the profile

at each full station and at any break of the topography. A breakis where there is asudden significant change in slope. The elevation of each of these points may then be

determined by subtracting the readings from the height of instrument at the setup. Figure

15.15 illustrates typical field notes for the profile leveling process. The resulting profilefrom the notes is illustrated in Figure 15.16. Note that the sightings on the bench mark

and turning point are read to a higher order of precision than sightings on the ground.


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Figure 15.15 Profile Leveling Notes

Station BS HI FS ElevationBM 4 478.26

4.87 483.13

32+00 11.5 471.633+00 9.4 473.7

+75 10.1 473.034+00 8.2 474.935+00 3 480.1

+15 1.9 481.2+70 2.3 480.8

36+00 5.2 477.9+50 6.8 476.3

37+00 5.9 477.238+00 13.3 469.8

TP 10.72 472.41

4.54 476.9538+60 13.2 463.8

39+00 12.0 465.040+00 3.9 473.1

41+00 1.2 475.842+00 0.8 476.2

+70 0.7 476.3

+80 1.5 475.543+00 0.4 476.6

BM 5 0.17 476.78

Figure 15.16 Profile Plotted from Level Notes in Figure 15.15

(use Figure 13.7 from 5th

Edition)


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8. Vertical Datums

To allow comparison with other geospatial data, surveys are typically referenced to a

common framework or datum. Typically, horizontal coordinates for points in a surveyare referenced to a datum based on an ellipsoid of revolution approximating the figure ofthe earth. Unlike horizontal coordinates, elevations have traditionally not been related to

the ellipsoid. Rather, they have been more typically referenced to an average stage of the

tide such as mean sea level although the use of a tidal datum creates challenges with use

for a large area because of the geographic variability of tidal datums. This challenge hasbeen dealt with by use of a horizontal surface shaped by the earths gravity force that

approximates the elevation of stilled sea level around the world. That surface is called

the geoid. It represents an elevation with equal gravitation force and is thus is a conceptideally suited for a datum for elevations.

In the United States, the geoidal datum used prior to 1991 was established by holding asfixed the elevations of mean sea level at 26 locations along the coastlines of the United

States and Canada and using a least squares adjustment of the level lines connecting those

points. That datum was known as the National Geodetic Vertical Datum of 1929 (NGVD

1929), formerly called the sea-level datum of 1929. With increased understanding of thegeographic variability of sea level, the current geoidal datum was established using a

different approach. That current datum, the North American Vertical Datum of 1988

(NAVD 88), was established by use of a minimal constraint adjustment holding mean sealevel at just one point, Father Point in Quebec, Canada, as fixed. A similar geoidal datum

has been established on the island of Puerto Rico, known as the Puerto Rican VerticalDatum of 2000 (PRVD 2000) by holding the elevation of mean sea level at San Juan as

fixed.

Elevations referenced to the geoid are considered to be orthometric heights and are

defined as the distance between the geoid and the point as measured along a plumb line

passing through the point. Currently in the United States, most elevations used intopographic mapping, geodetic surveys, engineering studies, construction surveys, and

geographic information systems are orthometric heights and are referenced to NAVD 88.

Even though the geoid is is a somewhat irregular surface as compared to the predictable,

smooth, and mathematically perfect shape of the ellipsoid (Figure 15.17), elevations,referenced to the geoid, are generally preferable to those referenced to the ellipsoid for

most applications. One reason is perception since in some places in the United States, the

ellipsoid is 50 meters or more above sea level. The most important reason is that a datumbased on gravity is preferable since water can theoretically run uphill in terrain

measured relative to the ellipsoid in areas where the geoidal separation is changing.

An understanding of the difference between ellipsoidal and orthometric elevations is

especially important when dealing with elevations derived from GPS observations. Since

the GPS orbital information is referenced to the ellipsoid, elevations determined by GPSobservations are relative to the ellipsoid. For use for most applications such as


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topographic mapping, such elevations must be adjusted to orthometric heights bysubtracting the geoidal separation (parameterNin Figure 15.17) at that location using a

model of the geoid. Within the United States, models of the geoid have been developed

using a combination of geodetic leveling, GPS observations, and gravity observations. Inless developed areas of the world, precise geoidal models may not exist.

Figure 15.17 Orthmetric vs Ellipsoidal Heights

In addition to the vertical datums discussed above, local tidal datums, such as mean high

water and mean lower low water are used for some applications. Typical applicationsinclude water boundary and hydrographic surveys. See Chapters 24 and 25 for

discussions on these applications.

Practice Problems

1. The term height of instrument, as used in leveling, is which of the following?

(A) distance from the ground to the axis of the telescope(B) elevation of the line of sight above the datum being used(C) height of the line of sight above a bench mark(D) all of the above


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2. A distant point observed through a level telescope appears to be lower than itactually is because of which of the following?

(A) collimation error(B) refraction

(C) curvature of the earth(D) both (B) and (C)

3. Balancing distances to backsights and foresights eliminates which of thefollowing?

(A) curvature of the earth errors(B) collimation errors(C) rod plumb errors(D) both (A) and (B)

4. At a leveling setup, the monument on which the rod is set for the backsight has anelevation of 10.000 feet. The back sight reading is 2.456 and the foresight

reading is 4.232. What is the elevation of the foresight point?

(A) 7.544 ft(B) 8.224 ft(C) 12.456 ft(D) 16.688 ft

5. What is the curvature of the earth error associated with a leveling shot of 1000feet?

(A) 0.024 ft(B) 0.032 ft(C) 0.035 ft(D) 0.547 ft

6. Assuming a typical stadia constant, what is the

distance between the level and the rod in this illustration

if the rod is in feet?

(A) 23 ft(B) 46 ft(C) 59 ft(D) 67 ft


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7. What is an advantage of an invar leveling rod?

(A) low coefficient of expansion(B) more visible graduations(C) top portion of rod can be detatched(D) better resolution

8. What is the rod reading for a typical backsight in this illustration?

(A) 0.930(B) 1.000

(C) 1.100

(D) 1.270

9. For a typical modern level, what device defines the line of sight?

(A) spirit level bubble(B) bulls eye bubble

(C) compensator

(D) plumb bob

10.What is the closure error for the closed leveling run having the measurementsprovided in the table below?

(A) 0.005

(B) 0.006

(C) 0.007(D) 0.008

setup BS FSBM 1 - BM 2 2.543 5.222

BM 2 - BM 3 3.278 4.362

BM 3 - BM 2 4.135 3.053

BM 2 - BM 1 5.047 2.359


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11. What is the closure error for the closed leveling run having the measurementsprovided in the table below?

(A) 0.009 ft

(B) 0.011 ft(C) 0.012 ft(D) 0.015 ft

12.With a level rod reading of 6.000 feet, what is the corrected reading if the levelrod is 5 off vertical?

(A) 5.977 ft(B) 5.986 ft

(C) 6.002 ft

(D) 6.023 ft

13.What is the foresight rod reading necessary to find a point on the ground with anelevation of 2.00 feet if you sight on a bench mark with an elevation of 5.25 andread a backsight of 2.52?

(A) 2.73(B) 5.25

(C) 5.77

(D) 7.77

14.What is the elevation of bench mark B using the 3-wire field notes provided in thetable below?

(A) 7.078 ft

(B) 8.921 ft

(C) 8.922 ft(D) 9.812 ft

Published elevation for BM 1 = 10.000 ft

Published elevation for BM 5 = 4.291 ft

setup BS FS

BM 1 - BM 2 3.124 5.667

BM 2 - BM 3 3.487 4.223

BM 3 - BM 4 5.213 5.673

BM 4 - BM 5 2.413 4.392

BS FS Elev

BM A 10.000

top 3.477 4.566

mid 2.965 4.044

bot 2.456 3.521

BM B


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15.Calculate the collimation correction ( C factor) for determining collimation error

given the following field notes for a C test. The rod readings are in feet.

(A) 0.002 ft/ft(B) 0.008 ft/ft(C) 0.011 ft/ft

(D) 0.015 ft/ft

16.Which of the following practices should be followed to minimize leveling error

due to refraction?

(A) not allowing line of sight to be close to ground

(B) balancing backsights and foresights

(C) taking backsights and foresights in quick succession(D) all of the above

Solutions to Practice Problems

1. In leveling, the term height of instrument means the elevation of the lineof sight above the datum being used.

The answer is (B).

2. A distant point appears to be lower than it actually is due to curvature of theearth.

The answer is (C).

3. Balancing distances to backsights and foresights eliminateserrors for both curvature of the earth and collimation.

The answer is (D).

setup short long

4.621 4.236

1 4.516 3.320

4.412 2.403

4.400 3.929

2 4.302 3.010

4.199 2.091


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24

4. The elevation of the foresight point may be calculated as follows:

BS elev 10.000 ft+ BS +2.456

HI 12.456-FS -4.2328.224 ft

The answer is (B).

5. The curvature of the earth may be calculated as follows:

ftMcurvature

mimift

ftft

024.0)189.0(667.0667.0

189.0/5280

10001000

22===

==

The answer is (A).

6. The top stadia hair is 6.13 and the bottom stadia hair is 5.67, so the stadiadistance is 100(6.13 5.67) = 46 feet.

The answer is (B).

7. An advantage of an invar leveling rod is that it has a low coefficient ofexpansion and thus reduces error due to temperature differences.

The answer is (A).

8. The rod reading is 1.100 ft.

The answer is (C).

9. Most modern levels us a compensator to define the line of sight.

The answer is (C).


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25

10.The error of closure may be calculated as shown below.

setup BS HI FS elev

BM 1 10.0002.543 12.543 5.222

BM 2 7.321

3.278 10.599 4.362

BM 3 6.237

4.135 10.372 3.053

BM 2 7.319

5.047 12.366 2.359

BM 1 10.007

Error of Closure = 10.007 10.000 = 0.007

The answer is (C).

11.The error of closure may be calculated as shown below.

setup BS HI FS elev

BM 1 10.000

3.124 13.124 5.667

BM 2 7.457

3.487 10.944 4.223

BM 3 6.7215.213 11.934 5.673

BM 2 6.261

2.413 8.674 4.392

BM 5 4.282

Error of Closure = 4.291 4.282 = 0.009 ft

The answer is (A).

12. True reading = (rod reading) cos(angle) = (6.000 ft)cos(5) = 5.977 ft

The answer is (A).


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26

13. BM Elev 5.25+ BS 2.52

HI 7.77

To find a contour of 2.00, FS = 7.77 2.00 = 5.77

The answer is (C).

14.The elevation may be calculated as illustrated below:

BSavgBS HI FS

avgFS Elev

BM A 10.000

top 3.477 4.566

mid 2.965 2.966 12.966 4.044 4.044

bot 2.456 3.521

BM B 8.922

The answer is (C).

15. The C factor may be calculated as below.

setup shortavg

reading dist long avg

4.621 4.236

1 4.516 4.516 20.9 3.320 3.320 183.34.412 2.403

4.400 3.929

2 4.302 4.300 20.1 3.010 3.010 183.8

4.199 2.091

sums 8.817 41.0 6.330 367.1

ftft

DDDD

RRRRC

SSLL

LLSS /008.0

1.3670.41

330.6817.8

)()(

)()(

2121

2121 =

=

++

++=

The answer is (B)


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16.To minimize error due to refraction in leveling, observers should not allowthe line of sight to be close to ground, balance backsights and foresights,

and take backsights and foresights in quick succession.

The answer is (D).

Date post:	04-Apr-2018
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Reading 3 - Leveling

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