1. Booking & Calculations – Rise & Fall Method
Staff readings: usually recorded in level book / booking form printed for that purpose
Readings: have to be processed to find RL’s (usually carried out in the same book)

2. Recommended: hand-held calculator / notebook computer with spreadsheet: avoid hand calculations & potential mistakes
Rise & fall method: one of most common booking methods
all rise/falls computed & recorded on sheet
RL of any new station: add rise to (or subtract fall from) previous station’s RL, starting from known BM.

3. Example 1. Rise & fall method (staff readings in Fig. 2.12):
Table 2.2:
2.518
3.729
4.153
0.556
4.212
0.718 B
CP2
CP1
Fig. 2.12
Table 2.2 BM

4. From (2.3), (2.4) & (2.5),
= Total rise – total fall = Last RL – first RL
Equalities checked in last row of Table 2.2.
Any discrepancy  existence of arithmetic mistake(s), but has nothing to do with accuracy of measurements.

5. Fig. 2.13 BS & FS Observed at Stations A - D
Example 2. BS, FS (& IS) readings in Fig are booked
as shown in Table 2.3:
0.595
1.522
2.234
3.132
2.587
1.985
1.334
TBM
2.002
58.331m
above MSL B A C
Fig BS & FS Observed at Stations A - D D
Table 2.3
Using rise & fall method, a spreadsheet can be written to deduce RLs of points A through D as shown in Table 2.4. (use IF & MAX in Excel): you are encouraged to reproduce Table 2.4 on Excel.

6. Table 2.4
Last row of Table 2.4
= Total rise – total fall = Last RL – first RL no mistake with arithmetic.

7. Closure Error Definition of misclosure & allowable values
Whenever possible: close on either starting benchmark or another benchmark to check accuracy & detect blunders. Misclosure (evaluated at closing BM):
 = measured RL of BM  correct RL of BM (2.9)
If  acceptable: corrected for so that closing BM has correct known RL

8. Max. acceptable misclosure (in mm):
E =  C
where K = total distance of leveling route (in number of kilometers)
C = a constant: typically between 2 mm (precise leveling work of highest standards) & 12 mm (ordinary engineering leveling)

9. Somewhat empirical values; can be justified by statistical theory; Bannister et al. (1998).
Construction leveling: often involves relatively short distances yet a large number (n) of instrument stations. In this case, an alternative criterion for E can be used:
E =  D (2.10)
5 mm & 8 mm: commonly adopted values for D.

10. LS Adjustment of Leveling Networks Using Spreadsheets
Surveyors: often include redundancy
Fig. 2.15: leveling network & associated data
Arrowheads: direction of leveling; e.g.
Along line 1: rise of m from BM A to station X, i.e. RLX – RLA = 5.102,
Along line 3: fall of m from B to Z, i.e. RLZ – RLB = –1.253.
(unknown) RLs of stations X, Y, Z: lower-case letters x, y, z.
Fig. 2.15

11. Common practice in leveling adjustments: observations assigned weights inversely proportional to (plan) sight distances L:
wi = (2.11)
i = 1, 2, …, 7.
Objective: determine x, y, z.
Many different solutions
(e.g. by loop A-X-Y-Z-A, or B-Z-Y-X-B),
probably all differ slightly  random errors in data.

12. Utilize all available data & weights: least squares analysis.
Note:
7 observed elevation differences: vector
[x – , – x, z – , – z, y – x, y – , z – y]T

13. This vector can be decomposed into a matrix product as follows:
(2.12)

14. Separate unknowns from constants  re-write leveling information
Ax + k1 ~ k2
where
A = coefficient matrix of 0’s & 1’s on RHS of (2.12),
k1 = last vector in (2.12) containing benchmark values,
k2 = [5.102, 2.345, , , , , 1.703]T.
Problem now in “Ax ~ k” form,
where k = k2 – k1,
weight matrix W = Diag [1/40,1/30,1/30,1/30,1/20,1/20,1/20]

15. Problem treated in Ch.1:
Solution: (1.5) numerical matrix computations
Spreadsheet method:
fast, easy to learn, highly portable
instant, automatic recalc. if #s in problem changed (common situation in surveying updating of control coordinates, discovery of mistakes, etc.).

16. Spreadsheet: shown in Table 2.6. Note:
computed #s in Table 2.6: do not necessarily show all d.p.  paper space limitations (all computations: full accuracy).
Format – Cells – Number – Decimal places to display only desired number of d.p. (computations always carry full accuracy).
Select any cell in matrix  ctrl - *  whole matrix selected (matrix must be completely surrounded by blank border)
See Table 2.6 steps to be carried out on spreadsheet:

18. Table 2.6 Performing LS Adjustment of Leveling Network on a Spreadsheet
Most probable RL’s for stations X, Y, Z: m, m, m, respectively.

19. Contours
Contour lines: best method to show height variations on a plan
Contour line drawn on a plan:
a line joining equal altitudes
Elevations: indicated on plan
“tidemarks left by a flood” that fell at a discrete contour interval.

20. Fig. 2.16: plan & section of an island
contour line of 0 meter value: “tidemark left by the sea”
Ascending at 10 m contour intervals: a series of imaginary horizontal planes passing through island  contours with values of 10 m, 20 m, 30 m, & 40 m, at their points of contact with island.

21. Fig. 2.16

22. Fig. 2.16 gradient of the ground between A & C:
Gradient along AC = = 1 in 6
Similarly,
Gradient along DE = = 1 in 3
regions where contours are more closely packed have steeper slopes
a contour line is continuous & closed on itself, although the plan may not have sufficient room to show.
Height of any point: unique  two contour lines of different values cannot cross or meet, except for a cliff / overhang.

23. Contouring: laborious. One direct method:
BM ( m above HKPD) sighted, back sight = m  height of instrument (HI) = m.
Staff reading = m  staff’s bottom at 30-m contour level
Staff then taken throughout site, and at every m reading, point is pegged for subsequent determination of its E, N coordinates by another appropriate survey technique  30-m contour located.
Similarly, a staff reading = m  a point on 9-m contour & so on.
Tedious & uneconomical for large area
Suitable in construction projects requiring excavation to a specific single contour line.

24. Trigonometric Leveling
Discussion so far: differential leveling: may not be practical for large
elevations (e.g. tall building’s height)
trigonometric leveling ( “heighting”): basic procedure: P
vertical line
Tall Building Z V
horizontal line B
A (instrument center) G
Fig. 2.17 B’ h

25. rough estimate of h, e.g. residential buildings: h  (number of stories  3 m).
Useful for checking result later, also a good separation (if possible) between instrument & building (why?).
If taping: horizontal distance AB from instrument to building obtained directly. Alternatively: EDM at A + reflector at some point B’ directly above / below B slope distance AB’ & zenith angle  AB & B’B computed. Also, vertical distance BG (or prism height B’G) to base of building: by a staff / tape.

26. Raise telescope to sight building top, measure v precisely.
Note: most theodolites give zenith angle z, vertical angle = v = 90 – z.
Height of building: PG = AB tan v + BG (where BG = B’G – B’B if EDM was used).

27. Modern Instruments
Many total stations: built-in Remote Elevation Measurement (REM) mode expedites trigonometric leveling:
Sight point B’ (Fig. 2.17) once; distance & zenith angle measured & stored.
As one raises / lowers telescope  corresponding height of new sighted point calculated & displayed automatically.
Reflector to be placed at B’ (usually: prism on top of a held pole)

28. Difficulties: People walking outside base of building may block prism:
Reflectorless total station: EDM laser beam can be reflected back from suitable building surfaces (e.g. white walls) w/o prism. Fig. 2.18(b)  can sight any convenient point B’ along PG (see Fig. 2.17) w/o prism,
Only limitations: laser’s maximum range (typically ~ 100 m) & type of building’s surface (certain absorbing/ dark surfaces may not work).

29. Sighting top of tall building  steep vertical angles  telescope points almost straight up  reading eyepiece becomes difficult to view:
Diagonal eyepiece: provides extension of eyepiece & allows comfortable viewing from the side: Fig. 2.18(a).
(a) A Diagonal Eyepiece (b) Nikon NPL-820 Reflectorless Total Station
Fig. 2.18
Leveling fieldwork: time-consuming & error-prone, especially for staff reading by eye.

30. Digital levels (DL):
capable of electronic image processing.
Require specially made staffs with bar codes on one side & conventional graduations on the other.
Observer directs telescope onto staff’s bar-coded side & focuses on it, as done in conventional leveling.
By pressing a key: DL reads bar codes & determines corresponding staff reading, displaying result on a panel.
Eliminate booking errors & expedite leveling work
Can be used in conventional way also.

31. Standard error for DL: typically < 1 mm at a sighting distance = 100 m
Observation range: typical upper limit ~ 100 m, lower limit ~ 2 m.
(a) Topcon DL-103 Digital Level (b) Bar-coded side of a staff
Fig. 2.19