chapter 2: leveling definitions leveling: determination of height differences for 2 or more points...
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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