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Chapter 2: Leveling Definitions Leveling: Datum (datum surface):

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Presentation on theme: "Chapter 2: Leveling Definitions Leveling: Datum (datum surface):"— Presentation transcript:

1 Chapter 2: Leveling Definitions Leveling: Datum (datum surface):
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

2 MSL surface: Reduced level (RL): Benchmark (BM):
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).

3 Fig. 2.1 Benchmarks found on pavements & railroad platforms
Close-up view (a) (b)   Fig. 2.1 Benchmarks found on pavements & railroad platforms

4 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

5 Leveling: Basic components:
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

6 When set horizontal, level is used to sight
readings 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

7 Fig. 2.2 (d) A staff level (e) Readings on a staff (c) An one-piece
invar staff (a) Telescopic staffs (b) A folding staff Fig. 2.2

8 Theory Basic Principle Fig. 2.3 To determine RLB: Instrument:
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

9 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

10 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 > rise = h; h < 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.

11 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

12 General 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– (2.3)

13 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)

14 Intermediate Sights Before 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 adjacent readings gives rise or fall: BSA – ISP = rise from A to P; ISP – ISQ = rise from P to Q, etc.

15 Inverted Staff RL of objects lying above line of sight & not on the
ground (e.g. underside of bridge; ceiling): Hold staff upside down “Zero” plane flush on point of interest Book staff reading with -ve sign in front (2.1) remains correct.

16 Effects due to Curvature of the Earth
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

17 In right-handed triangle OAB’: (2.6)
Horizontal plane AB’: treated as curved level surface AB over arc length L Leveling error BB’  h Magnitude of h = ? In right-handed triangle OAB’: (2.6) Substituting L’ = R tan ; canceling R2 on both sides of (2.6), 2Rh + (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). Fig Leveling Error due to the Earth’s Curvature

18 Spreadsheet Method Spreadsheet method to solve (2.7) as it stands: (“circular equation”) Excel’s iteration capabilities Useful for tackling other circular equations (not quadratic)

19 Type in values for R & L in cells A4 & B4
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 Leave the answer (cell E2) blank for now; this in effect makes it a zero value. Then define E4 as “= *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.

20 Finally, put the formula “=D4^2/F4” in E2. Excel will automatically
iterate until a solution is found, usually in split seconds. The results are 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, & (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.

21 Field Work Sources of Error & Precautions
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) Fig. 2.10

22 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 & foresight distances eliminates leveling error due to earth curvature Can perform computations as if leveling did take place on a flat earth

23 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 but horizontal lines are tilted upward (or downward) at both ends. Such tilting errors will cancel out if equal backsight & foresight distances are used

24 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.

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

26 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: 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

27 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

28 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)) Fig. 2.11

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


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