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)

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.

Example 1. Rise & fall method (staff readings in Fig. 2.12): Table 2.2:

CP2B

CP1

BM

4.212 0.718

3.7292.518

0.5564.153

Fig. 2.12

Table 2.2

Station BS FS Rise Fall RL Remarks BM 4.212 23.918 CP1 4.153 0.718 3.494 27.412 CP2 2.518 0.556 3.597 31.009 B 3.729 1.211 29.798 Total = 10.883 5.003 7.091 1.211 29.798 minus 5.003 1.211 23.918

= 5.880 5.880 5.880

From (2.3), (2.4) & (2.5),

allall

FSBS

= Total rise – total fall = Last RL – first RL

i. Equalities checked in last row of Table 2.2.

ii. Any discrepancy existence of arithmetic mistake(s), but has nothing to do with accuracy of measurements.

Example 2. BS, FS (& IS) readings in Fig. 2.13 are booked as shown in Table 2.3:

0.595

3.132 2.587 1.522 1.3342.234 1.9852.002TBM

58.331mabove MSL A B

CDFig. 2.13 BS & FS Observed at Stations A - 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.

BS IS FS Remarks 0.595 TBM 58.331 m 2.587 3.132 A (CP) 1.565 A-I1 1.911 A-I2 0.376 A-I3 2.234 1.522 B (CP) 3.771 B-I1 1.334 1.985 C (CP) 0.601 C-I1 2.002 D (BM 56.460 m)

Table 2.4

Last row of Table 2.4

allall

FSBS = Total rise – total fall = Last RL – first RL no mistake with arithmetic.

Station BS IS FS h Rise Fall RL

TBM 0.595 58.331 A 2.587 3.132 -2.537 2.537 55.794

A-I1 1.565 1.022 1.022 56.816 A-I2 1.911 -0.346 0.346 56.47 A-I3 0.376 1.535 1.535 58.005

B 2.234 1.522 -1.146 1.146 56.859 B-I1 3.771 -1.537 1.537 55.322

C 1.334 1.985 1.786 1.786 57.108 C-I1 0.601 0.733 0.733 57.841

D 2.002 -1.401 1.401 56.440

6.75 8.641 5.076 6.967

S

- FS = -1.891 Last RL – First RL = -1.891

Rise

- Fall = -1.891

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

Closure Error

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)

K

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.

n

LS Adjustment of Leveling Networks Using Spreadsheets

X

Y

Z

1 2

34

5

6

7

BM Bat

207.500 m

BM Aat 200.000 m

Surveyors: often include redundancy

Fig. 2.15: leveling network & associated data

Arrowheads: direction of leveling; e.g.

Along line 1: rise of 5.102 m from BM A to station X, i.e. RLX – RLA = 5.102,

Along line 3: fall of 1.253 m from B to Z, i.e. RLZ – RLB = –1.253.

(unknown) RLs of stations X, Y, Z: lower-case letters x, y, z.

Line

Observed Elevation Difference

(m)

Distance,

L (m)

1 5.102 40 2 2.345 30 3 –1.253 30 4 –6.132 30 5 –0.683 20 6 –3.002 20 7 1.703 20

Fig. 2.15

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.

iL

1

Utilize all available data & weights: least squares analysis.

Note:

7 observed elevation differences: vector

[x – 200.000, 207.500 – x, z – 207.500, 200.000 – z, y – x, y – 207.500, z – y]T

This vector can be decomposed into a matrix product as follows:

0

5.207

0

200

5.207

5.207

200

110

010

011

100

100

001

001

5.207

200

5.207

5.207

200

z

y

x

yz

y

xy

z

z

x

x (2.12)

Separate unknowns from constants re-write leveling information

Ax + k1 ~ k2where 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, -1.253, -6.132, -0.683, -3.002, 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]

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.).

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:

Table 2.6 Performing LS Adjustment of Leveling Network on a Spreadsheet

Most probable RL’s for stations X, Y, Z: 205.148 m, 204.482 m, 206.188 m, respectively.

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.

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.

01020

3040

D E A C

D EA

BC

0

20

40

60

Section along XY

60 m30 m

X Y

Fig. 2.16

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.

60

10

BC

AB

30

10

Contouring: laborious. One direct method:

1. BM (30.500 m above HKPD) sighted, back sight = 0.500 m height of instrument (HI) = 31.000m.

2. Staff reading = 1.000 m staff’s bottom at 30-m contour level

3. Staff then taken throughout site, and at every 1.000 m reading, point is pegged for subsequent determination of its E, N coordinates by another appropriate survey technique 30-m contour located.

4. Similarly, a staff reading = 2.000 m a point on 9-m contour & so on.

5. Tedious & uneconomical for large area

6. Suitable in construction projects requiring excavation to a specific single contour line.

Trigonometric LevelingDiscussion so far: differential leveling: may not be practical for large elevations (e.g. tall building’s height)

trigonometric leveling ( “heighting”): basic procedure:

P

vertical lineTall Building

Z

V

horizontal lineB A (instrument center)

G

Fig. 2.17

B’

h

• 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.

• 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).

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)

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).

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.

Digital levels (DL):

capable of electronic image processing.

Require specially made staffs with bar codes on one side & conventional graduations on the other.

1. Observer directs telescope onto staff’s bar-coded side & focuses on it, as done in conventional leveling.

2. 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.

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