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Chapter 2: Leveling
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  1. Chapter 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

  2. 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).

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

  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: • 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, levelis 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. (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 • 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

  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 > 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.

  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–1(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. 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. 2.9 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. 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 “=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.

  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, 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.

  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. 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

  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