Chapter 2: LevelingChapter 2: Leveling

Definitions

Leveling:

Determination of height differences for 2 or more points above the geoid.

Datum (datum surface):

A particular level surface chosen

Basis of all elevations in leveling work

MSL surface:

Most commonly adopted datum Makes international comparison of heights possible

Reduced level (RL):

Height of a point above the particular datum used

Benchmark (BM):

Point with previously determined RL Often constructed as permanent markers: See Fig. 2.1 (a) (stainless steel BM), (b) (survey nail).

(a) Close-up view (b)

 

 

Fig. 2.1 Benchmarks found on pavements & railroad platforms

Two datum surfaces used in Hong Kong: Principal Datum (HKPD)

Chart Datum (CD)

Note:

Mean sea level ~ 1.23 m above HKPD

CD: 0.146m below HKPD Used mainly in marine work

Level of lowest tides

Lands Department compiles records of (E, N) coordinates & RLs of various Hong Kong government benchmarks

Leveling:

Most commonly performed with automatic level

Basic components:

Telescope: providing a line of sight defined by its cross hairs

Adjustable mechanism including a circular (“bull’s eye”)

bubble: to make line of sight direction of gravity

Base: can be fastened to a tripod

When set horizontal, level is used to sightreadings on leveling staff (or leveling rod):

Graduated rod several cm wide

One piece/ telescopic / folding

0.5 or 1 cm graduation intervals, increasing from bottom plane (zero) up

Telescopic staff:

Extends to 4 or 5 meters in length

Circular bubble (staff level):

Ensures verticality; built-in / attached to staff’s straight edge by a rubber band

(a) Telescopic staffs

(b) A folding staff

(c) An one-piece

invar staff

(d) A staff level

(e) Readings

on a staff

Fig. 2.2

TheoryTheoryBasic Principle

To determine RLB:

Measure: B’s elevation (h)

above A (RLA is known)

Calculate: RLB = RLA + h

Instrument:

Set up & leveled at I, about

half-way between A & B:

Rodperson:

Hold leveling rod plumb with

its foot resting on A Fig. 2.3

Observer:

Turns telescope about vertical axis

Staff appears in center of view, read against horizontal crosshair (= a)

Staff moved to B; observer again directs telescope onto it & reads b

Fig. 2.4

Instrument correctly adjusted

Line of collimation truly horizontal

Difference in level between A & B, h = a – b, i.e.

h = BS – FS (2.1)

Where

BS (Backsight): always a sight taken on staff held on point of known height

FS (Foresight): always a sight taken on a point to determine its height

h > 0 rise = h;

h < 0 fall = | h |

(2.1) theoretically: height of instrument at I does not affect result of calculation

In reality: use higher line of sight whenever possible

Minimize bending of line of sight due to refraction.

A & B far apart / large elevation difference more than one instrument setting needed

In Fig 2.5:

Points 1, 2, 3: change points (CP’s) / turning points (TP’s) Backsights: taken at points A, 1, 2, 3, Foresights: taken at points 1, 2, 3, B

Elevation from A to B:

h = h1 + h2 + h3 + h4

= (BSA – FS1) + (BS1 – FS2) + (BS2 – FS3) + (BS3 – FSB)

Subscript on BS / FS: point where it is taken

General CaseGeneral Case

(N-1) change points between A & B (labeled 0 & N, respectively)

Elevation of B above A: h = (2.2)

where

hi = (BSi-1 – FSi ) = elevation of point i above point i–1 (2.3)

hii

N

1

Substituting (2.3) into (2.2),

h =

Or

h = (2.4)

∑ : either every BS (0, 1, 2, ... N-1), or every FS (1, 2, ... N).

RL of point B:

(2.5)

BS FSii

N

ii

N

1

1 1

N

ii

N

jj FSBS

1

1

0

RL RL BS FSB Aallall

Intermediate SightsIntermediate SightsBefore moving level for next set-up:

Can observe additional points (e.g. P & Q)

Intermediate sights (or intersights, IS)

Additional information about the land profile

Again, difference between adjacentreadings gives rise or fall:

BSA – ISP = rise from A to P;

ISP – ISQ = rise from P to Q, etc.

Inverted StaffInverted Staff RL of objects lying aboveline of sight & not on theground (e.g. underside ofbridge; ceiling):

Hold staff upside down

“Zero” plane flush on point of interest

Book staff reading with -ve sign in front (2.1) remains correct.

Effects due to Curvature of the Effects due to Curvature of the EarthEarth

Roundness of the earth: neglected so far

Ch.1: earth’s curvature may become important in determination of heights, even for a relatively small site at (say) 5 km by 5 km

R

L

L'B'

B

A

O

Horizontal plane Horizontal plane AB’: treated as curved level surface AB over arc length L

Leveling error BB’ h

Magnitude of h = ?

Fig. 2.9 Leveling Error due to the Earth’s Curvature

In right-handed triangle OAB’:

(2.6)

Substituting L’ = R tan ; canceling R2 on both sides of (2.6),

2Rh + (h)2 = (R tan )2

Hence where = L/R (2.7)

With R = 6371 km & L known quadratic equation for h

(or approximate answer by ignoring h in denominator; h << R).

222 ')( LRhR

hR

Rh

2

tan22

Spreadsheet MethodSpreadsheet Method Spreadsheet method to solve (2.7) as it stands: (“circular

equation”)

Excel’s iteration capabilities

Useful for tackling other circular equations (not quadratic)

1. Type in values for R & L in cells A4 & B4. Then put the formula “=B4/A4” in cell C4, & then put “=A4*tan(C4)” in D4. Note that Excel trigonometric functions use radians as input, so no conversion to degrees is needed for the argument C4

2. Leave the answer (cell E2) blank for now; this in effect makes it a zero value. Then define E4 as “=1000000*E2” to get a Dh that is in mm (it would be zero for now). Then, in F4, put in the denominator of (2.7)’s RHS, i.e. “=2*A4+E2”. At this point in time, Excel would treat E2 as a zero if it were needed in a calculation

Note:

E2: intentionally left blank

if formula (2.7) were placed there too early error (formula would reference F4, which refer back to E2 itself “circular reference”

To activate Excel’s ability to handle such circular references: Tools – Options from pull-down menu, check “Calculation” tab & select Iteration - OK.

3. Finally, put the formula “=D4^2/F4” in E2. Excel will automaticallyiterate until a solution is found, usually in split seconds. The resultsare shown in Table 2.1

Values in B4: try 0.1, 0.5, 1, 2, 5 (km), etc., respective errors are 0.8, 19.6, 78, 313.9 & 1962.0 (mm), etc.

Ordinary leveling instruments: can detect height differences to a few mm

Earth’s curvature cannot be neglected in leveling.

Effect on leveling calculations presented in 2.2.1? “Negligible” if good field procedures (next section) are

followed.

Field WorkField Work Sources of Error & Precautions

Curvature Effects of the Earth:Curvature Effects of the Earth:

• Fig. 2.10: level’s horizontal line of collimation will deviate from level surface as it travels far

• True level difference between points A & B:

(a’ – b’),

• Using field staff readings: (a – b)

error = (a – b) – (a’ – b’) = (a – a’) – (b – b’) (2.8)

A

B

a'

a

b'

bhorizontal line

Level surface passing B

Level surface passing A

Level surface passing instrument

C

O

Fig. 2.10

• However, if level is placed at (about) mid-way between A & B, Arcs OA = OC, thus

(a – a’) = (b – b’)

(using OA = OC = “L” in (2.7))

• Using (approximately) equal backsight & equal backsight & foresight distancesforesight distances eliminates leveling error due to earth curvature

• Can perform computations as if leveling did take place on a flat earth

Instrument not being in adjustment:Instrument not being in adjustment:

• A level should be in proper adjustment when used

• Otherwise, line of sight is not truly horizontal

– Sweeps out a cone rather than a horizontal plane as telescope is rotated about vertical axis

– Similar to situation in Fig. 2.10 but horizontal lines are tilted upward (or downward) at both ends. Such tilting errors will cancel out if equal backsight & foresight distances are used

Differential settlement of staff or tripod:Differential settlement of staff or tripod:

• Use firm, stable & well-defined turning points

• Leveling over soft ground: can use a base plate (or change plate): triangular metal plate with corners folded down, & a dome raised at center. When placed on ground & stamped firm, central dome provides a stable point to place staff on

• Tripod: if on soft ground, ensure metal shoes are firmly planted into soil.

Tilting of staff sideways:Tilting of staff sideways:

• Always attach staff level ( “bull’s eye” bubble) for fast & correct staff plumbing

• Observer: check staff’s coincidence with vertical crosshair, and signal staffperson for any correction necessary

Leaning of staff towards or away from observer:Leaning of staff towards or away from observer:

• Use staff level; also look from side of staff & line it up with vertical objects

Bubble not being central:Bubble not being central:

• Observer & staffperson: make sure circular bubbles (on level & staff) both centralized before measurement begins

• Attach 2 or more bubbles to staff if available (can detect malfunctioning bubble)

Incorrect reading of staff:Incorrect reading of staff:

• Have a second observer double-check reading

• Spend time beforehand to familiarize with staff & examine it close-up

• Useful (time-consuming) technique: “rocking”: staffperson to slowly wave staff top towards & away from observer; min. reading = correct

Mishandling of staff:Mishandling of staff:

When extending telescopic staff:

• Lower sections first

• No section left partially extended (like having a kink in a tape)

• Don’t let a staff get too high that it catches overhead power cables: staff holder could get electrocuted

Setting staff on sloping ground:Setting staff on sloping ground:

• Fig. 2.11(a): correct way: staff bottom plane ( “zero”) flush against point of interest

• Some mistakenly think: staff should be “centered” over the point offset error (Fig. 2.11(b))

Point of interest

Bottom plane of staff

Bottom plane of staff

Point of interest

Error

(a) (b)

Fig. 2.11

Parallax:Parallax:

• Parallax: relative movement between image & cross hairs as eye moves

• Rotate eyepiece until cross hairs appear sharp, & focus on staff until image is clear & such relative movement is eliminated

Adverse weather conditions:Adverse weather conditions:

• Bring an umbrella to protect level from extended exposure to sun or unexpected showers