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Leveling

A First Look at Leveling

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A First Look at Leveling

A First Look at Leveling

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Outline• Terms• Methods of Leveling• Theory of Differential Leveling• Differential Leveling Notes• Note and Measurement Checks• Classification of Leveling• Leveling Equipment

Outline• Level Network Adjustment• Other Applications of Differential Leveling• Leveling Errors and Mistakes

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Terms

• Vertical Line– Line coincident with the direction of gravity,

e.g., a plumb line string• Horizontal Line

– Line perpendicular to the direction of gravity, e.g., line of sight through a level once the level is properly set up.

Terms Cont.

• Level Surface– Surface which at every point is normal or

perpendicular to the direction of gravity. A line on that surface is a level line.

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Terms Cont.• Elevation datum

– Level surface from which elevations are measured

– Standard• NGVD 1929• NAVD 1988

– Assumed• Elevation

– Vertical distance above an elevation datum

Terms Cont.• Benchmark (BM)

– Object of more or less permanent character for which the elevation is known, e.g., top of a concrete post marked with a metal disk.

• Turning point (TP)– Temporary benchmark, not always marked, but

is unique and distinguishable from immediate surrounding, e.g., top of a particular rock.

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Terms Cont.• Leveling

– Operation or process to obtain elevations or differences in elevation

• Vertical Control– Same as leveling.

Methods of Leveling

• Indirect– barometric– trigonometric

• Direct– taping– differential leveling

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Theory of Differential Leveling

• Equations– HI = Elev.1 + BS– Elev.2 = HI - FS

• Terms– HI = height of

instrument– BS = Backsight– FS = Foresight

Differential Leveling cont.

• Assumption– Horizontal line is

parallel with the elevation datum.

– Assumption is incorrect, but resulting error is small for short distances on the earth’s surface.

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Differential Leveling cont.

Fieldnote Form

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Differential Leveling Notes

Field Note and Measurement Checks

• Page check– check arithmetic

• Closure– check for gross blunder if compared with

permissible closure or maximum closure– to compute requires one of two conditions

• start and end on same point or• start on point of known elevation and end on point

of known elevation.

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Checks cont.

• Permissible closure or maximum closure– estimate of how random errors propagate– expression of form:

• a = estimate of random error• n = number of measurements

a n

Page Check

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Closure

Permissible Closure

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Classification of Leveling Operations

• Specifications• Standards of Accuracy ( permissible

closure)• Study table at end of Level Loop

Adjustment handout in notes

Leveling Equipment

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Leveling Instruments• Dumpy Level• Wye Level• Tilting Level• Automatic Level• Laser Level• Digital Level

Dumpy Level

• Work horse for over a century• Tubular level vial• 4 leveling screws

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Wye Level

• Similar to Dumpy level• Removable telescope – simplify instrument adj.• Clamps on wye permit removal

Tilting Level

• 3 leveling screws• Telescope is tilted to

center bubble• Coincidence Bubble

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Automatic Level

• 3 leveling screws• Compensator used in

place of tubular level vial to level the instrument.

Automatic Level Compensator

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Laser Level

• Sensor detects laser beam

• Rotating beam of laser light or single beam

• Electronic leveling technology, compensators, and leveling screws and bubbles used to level.

Digital Level

• Compensators used to level

• Bar-coded rod – like uniform product code on a box of Cheerios

• Distance measuring capability

• Electronic data recording.

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Hand Levels

Abney Level

Locke Level

Clinometer

Leveling Rods

• Philadelphia – 2 sections ≈ 13 ft.

extended• Chicago (Frisco)

– 3 sections ≈ 13 ft.• Telescopic

– 14 to 50 ft.• Others

– Pocket rod– Folding rule

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Leveling Rods continued

• Self reading rods• Target rods

Other Equipment

• Turning pins• Turning plates• Rod level

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Instrument Quality

• Spirit Levels (tubular level vial)– Sensitivity

• Angle (α) between two adjacent graduations on level vial (grad. 2 mm apart)

– Range (α) : 10” (best) to 90”

• Radius of curvature of vial.

Instrument Quality cont.– Radius of Curvature

cont.• α (radians) = S/R• R = S/ α (radians)

• Automatic Levels– Setting Accuracy:

± 0.3”, ±0.5”, and ±0.8”

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Analysis of Sensitivity Rating

• If the bubble is off center by one grad., the line of sight is elevated or depressed by one α, resulting in an error in the rod reading

Analysis of Sensitivity Cont.

• Error due to setting accuracy of automatic level results in a similar error.

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Analysis of Sensitivity Cont.

• What is the equation for computing the error (e) when the line of sight (LOS) is elevated or depressed by, say α, and the distance from the instrument to the rod = d ?

Analysis of Sensitivity Cont.

)tan(α⋅= de• e = error due to elevation or depression of LOS• d = distance from level to rod• α = angle the telescope is elevated or depressed

above or below a horizontal LOS.

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Analysis of Sensitivity Cont.

• Note: Sensitivity (α) is expressed in seconds of arc (“) and to use it with your calculator requires decimal degrees (°)

)("3600)(1)(")( °

⋅=° αα

Analysis of Sensitivity Cont.

• To find the radius (R) of curvature of a tubular level vial, the sensitivity must be expressed in radians (rad).

)(180)(

)("3600)(1)(")(

)(002.0

)(_

)(

°⋅

°⋅=

===

radrad

radm

radlengtharc

radSR

παα

ααα

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Find observation errors and radii of curvature. Assume 1 grad. off level

d = 100.00 ft.

90”30”10”

n/a0.8”n/a0.5”n/a0.3”Radius (m)e = error (ft.)α

Calculation Results

d = 100.00 ft.

4.58 m0.0436 ft.90”13.75 m0.0145 ft.30”41.25 m0.0048 ft.10”n/a0.0004 ft.0.8”n/a0.0002 ft.0.5”n/a0.0001 ft.0.3”Radius (m)e = error (ft.)α

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Find the comparable setting accuracy for a Laser Level

• A laser level specification for accuracy is given as 1/8 “ per 100 ft.

• What is the comparable setting accuracy in seconds of arc?

Comparable setting Accuracy Cont.

12000)(100

)(12)(1)(

81

arctan)(

arctan)(

′′′°=

⋅=°

=°

α

α

α

ftinftin

de

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Find the sensitivity of a level

• Make the level vial axis and axis thru two opposite leveling screws coincide.

• Obtain reading (r1) with bubble centered• Obtain reading (r2) with bubble off center, say 3

graduations. (Move bubble by adj. level. screws)

Finding Sensitivity Cont.

7218121003

84.127050.0arctan3

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′′=

′′=′′′°=

=∆

=

−=∆

αα

αftft

dr

rrr

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Level Net Adjustment

Adjusting for Random Errors

Level Net Adjustment Methods

• Rigorous least squares method– random error distributed or corrections made

with effort made to minimize the sum of the residuals (corrections) squared.

• Alternative methods– random error distributed or corrections made

with no effort made to minimize the sum of the residuals (corrections) squared.

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Assumptions of Alternative Method

• Random errors are proportional to the distances between bench marks , i.e., the longer the distance between benchmarks the more opportunities there are for error.

• Consequently, the longest lines should receive the greatest adjustment.

• The number of measurements between bench marks may also be used as a weighting factor in lieu of distance

Adjustment Techniques

• Adjust Differences in Elevation• Adjust Elevations

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Adjust

Differences in Elevation

Procedures to Adjust Differences in Elevation

• Reduce field notes, i.e., calculate elevations• Perform page check• Ascertain that closure is permissible• Extract from field notes diff. in elev. and

distances between bench marks• Calculate corrections• Add corrections to differences = adj. diff.• Use adjusted differences to calculate elev.

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Adjusting Differences in Elevation

Extracting Differences in Elevation From Field Notes

∆ Ei FH BM1 698 23 700 00177( & ) . ..

= − == −

ALSO

∆ Ei FH BM1 BS FS5 67 4 53 8 89 3 0810 20 1197 177

( & )( . . ) ( . . )( . ) ( . ) .

= − =

= + − + == − = −

∑∑

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Extracting Distances Between Bench Marks

d FH BM1100 90 200 200

590 ft

i ( )( )

.

− == + + + ==

Extract Differences in Elevation and Distances Between BMs.

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Extract Differences in Elevation and Distances Between BMs.

CALCULATION CHECK

• For a level loop, i.e., a leveling operation that starts and ends on the same point, the sum of the extracted differences will equal to the closure.

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Calculation Check

Note on Preventing Round off Error

• Generally, in surveying the intermediate calculations are carried (rounded) to, at least, one more decimal place than the number obtained from the measurement. For example, if elevations are measured to 0.01 ft., then intermediate calculations will be rounded to 0.001 ft.

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Note on Preventing Round off Error cont.

• Rounding to the proper number of decimal places is done at the end after all calculations are completed.

• Do not report results to a greater degree of measurement resolution than that obtained by the original measurement.

• NOTE: I may ask that you report intermediate results.

Calculate the Correction to Each Difference in Elevation

CCd

d correction to diff in elevDii

i= − ⋅ =∑

. .

Note: See notes for definition of variables

−⎛

⎝⎜⎜

⎞

⎠⎟⎟ =

∑Cd

Correction per fti

.

The correction per ft. is multiplied by the distance (di)between bench marks to calculate the correction to the difference in elevation between bench marks

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Sample calculation for correction between FH and BM1

C 0 08 ft

d 3285 ftd 590 ft

C FH to BM10 08

3285590

C FH to BM1 0 014 ft

i

i

Di

Di

= −

=

=

= −−

⋅

= +

∑. .

..

( ).

( ) . .

Sample calculation for correction between FH and BM1

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Calculate Corrections Between Bench Marks

CALCULATION CHECK

• The sum of the corrections should equal minus the Closure.

• The closure represents the sum of the random errors being distributed, and the sign of the correction should be opposite of the sign of the error.

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CALCULATION CHECK

Minus the Closure = Sum of Corrections

Calculate the AdjustedDifference Between Bench

MarksNote: See notes for definition of variables

A CEi Ei Di∆ ∆= +

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Sample calculation for adjusteddifference between FH and BM1

A CEi Ei Di∆ ∆= +

∆

∆∆

Ei

Di

Ei

Ei

177 ftC 0 014 ftA 177 0 014 1756 ftA 1756 ft

= −= += − + = −= −

. .. .

. . . .

. .

Sample Calculation of AdjustedDiff. in Elevation: FH to BM1

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Calculate the Adjusted Differences in Elevation Between Bench Marks

CALCULALTION CHECK

• For a level loop, the sum of the adjusted differences in elevation should equal zero.

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Calculate Elevations Based on Adjusted Differences in Elev.• The beginning elevation (from field notes) does not change

• The current elevation equals the previous elevation plus the adjusted difference between the previous and the current bench marks

E E Ai i 1 i 1 i= +− −∆ ( , )

Calculate Elevation of BM1

E E Ai i 1 i 1 i= +− −∆ ( , )

E 700 000A 1756

E Elevation of BM1E 700 000 1756E 698 244 ft

i 1

i 1 i

i

i

i

−

−

== −

== + − ==

..

. ( . )

. .

( , )∆

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Calculate Elevation of BM1

Calculate Elevations of BMs

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Adjust

Elevations

Procedures to Adjust Elevations• Reduce field notes, i.e., calculate elevations• Perform page check• Ascertain that closure is permissible• Extract from field notes elevations and

Accumulated distances from the beginning to the BM of interest.

• Calculate corrections to the BM elevations• Add corrs. to the unadjusted BM elevations

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Adjusting Elevations

Extract Elevations and Accumulated Distances

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Extract Elevations and Accumulated Distances

Extract Elevations and Accumulated Distances

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Calculate the Correction to Each Bench Mark Elevation

Note: See notes for definition of variables

CCd

dEii

ai= − ⋅∑

−⎛

⎝⎜⎜

⎞

⎠⎟⎟ =

∑Cd

Correction per fti

.

The correction per ft. is multiplied by the accumulateddistance (dai) from the beginning to the BM of interest in order to calc. the corr. to the BM elevation of interest.

Sample calculation for correction to Elevation at BM1 and BM2

C0 08

3285590 0 014 ft

C0 08

32851035 0 025 ft

E BM1

E BM2

= −−

⋅ = +

= −−

⋅ = +

.. .

.. .

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Calculate the Correction to Each Bench Mark Elevation

Calculate the Correction to Each Bench Mark Elevation

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Calculate the Adjusted BM Elevation

Note: See notes for definition of variables

AE E Ci i Ei= +

NOTE:The starting elevation is not corrected.It is considered a constant with no error.

Calculate the Adjusted Elevation For BM1

AE E Ci i Ei= +

AE E C

AE 698 23 0 014

AE 698 244 ft

BM 1 BM 1 E BM 1

BM 1

BM 1

= +

= + + =

=

. ( . )

. .

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Calculate the Adjusted Elevation For BM1

Calculate the Correction to Each Bench Mark Elevation

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Calculate the Correction to Each Bench Mark Elevation

End Results Are The SameAdjusting

Differences in ElevationAdjustingElevations

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Other Applications of Differential Leveling

• Profile Leveling– includes intermediate foresights (IFS), i.e.,

more than one FS from each instrument setup.– See example of note form in notes– page check

• BS FS last HI first elev∑ ∑− = − .

Other Applications cont.

• Cross Section Leveling– horizontal note form - see example in notes– profile form of notes - see example in notes– no page check

• Many other special applications– geodetic leveling, reciprocal leveling, three-

wire leveling etc.

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Leveling Errors

And Mistakes

Mistakes – Gross Blunders

• Incorrect Use of Long Rod –– “slipping” of top section– failure to fully extend for high readings– Old Philly rod, read 7 for 13

• Holding rod in different place for FS and subsequent BS

• Incorrect note recording• reading the wrong cross wire

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Gross Blunders Cont.

• Reading a foot too high

• “Waving” the ordinary flat-bottomed rod on a flat surface– At first thought, the

smallest rod reading represents the reading on a plumb rod

Tilted Rod Analysis

• r = plumb rod reading• α = angle rod is tilted• w = width of rod• r b = rod reading while

tilted backwards• error = r b - r

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Sources of Error in Leveling

• Instrumental– Instrument adjustment

• Horizontal cross wire must lie in horizontal plane• Axis of level vial must be perpendicular to vertical

axis• Axis of line of sight must be parallel with the level

vial axis. (collimation error – detect via 2-peg test)

– Rod not standard length– Tripod legs loose

2-Peg test

• Test for Collimation error

• Requires 2 level setups - one such that: d1= d2and the other very close to one of the rods or pegs.

• If d1= d2 then e1= e2

• rod readings are r1, r2, r3, and r4

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2-Peg Test Cont.

• bs = r1 - e1 = r1 - e• fs = r2 - e2 = r2 - e• ∆ elev. = bs - fs• ∆elev.=(r1 - e) - (r2 -e)• or ∆elev.= r1 - r2 = ∆elev. free of collimation error.

2-Peg Test Cont.

• Therefore, if (r1 - r2) = (r3 - r4) , then the level is free of collimation error. That is the line of line of sight is parallel with the tube level axis.

• If collimation error is present, how can you minimize its effects?

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Sources of Error in Leveling

• Instrumental– Instrument adjustment

• Horizontal cross wire must lie in horizontal plane• Axis of level vial must be perpendicular to vertical

axis• Axis of line of sight must be parallel to axis of level

vial (collimation error – detect via 2-peg test)

– Rod not standard length– Tripod legs loose

Sources of Error Cont.

• Natural Errors– Earth Curvature– Refraction– Temperature variations– Wind– Settlement of instrument/rod

• Personal Errors– Bubble not centered

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Earth Curvature(R C) R DR 2RC C R D

CD2R

C2R

C2R

0

R 20,906,000ft

dD

1000D d 1000

C(d 1000)

2 209060000.0239 d

2 2 2

2 2 2 2

2 2

2

22

+ = +

+ + = +

= −

≈

=

=

= ⋅

=⋅

⋅= ⋅

D = distance from level to rod

R = Earth radius

C = correction (ft.) for curvature d = distance in thousands of ft.

Refraction

• Correction (ft.) for curvature and refraction

C 0 0239 86 dC 0 021 d

c r2

c r2

+

+

= ⋅ ⋅

= ⋅

. (. )

.

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Sources of Error Cont.

• Natural Errors– Earth Curvature– Refraction– Temperature variations– Wind– Settlement of instrument/rod

• Personal Errors– Bubble not centered

Sources of Error Cont.

• Personal Errors cont.– Parallax - apparent movement of object due to

movement of ref. frame from which the object is viewed.

– Faulty rod readings – due to interpolation, environmental conditions not 7 for 13 or 1 ft. too high.

– Rod handling– Target setting

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Summary• Terms• Methods of Leveling• Theory of Differential Leveling• Differential Leveling Notes• Note and Measurement Checks• Classification of Leveling• Leveling Equipment

Summary• Level Network Adjustment• Other Applications of Differential Leveling• Leveling Errors and Mistakes