Differential-Levelling Field Notes and Page Check: 18-Geom-A1 Surveying, May 2014, Q6

Question 6: Differential-Levelling Field Notes and Page Check (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A differential-levelling circuit from BM1 (Elev $=88.00$ ft) through three turning points to BM2:

Station	BS (ft)	FS (ft)
BM1	2.45	—
TP1	5.43	6.53
TP2	3.18	4.91
TP3	4.22	7.42
BM2	—	6.11

Find. The completed field-note table (HI and Elevation at every station), the elevation of BM2, and the arithmetic page check.

Figure 6 — Differential-levelling run BM1 → TP1 → TP2 → TP3 → BM2; each set-up takes a backsight (BS) to the known point and a foresight (FS) to the next.

Approach. March the height of instrument (HI $=$ Elev $+$ BS) and elevation (Elev $=$ HI $-$ FS) alternately down the line, then verify with the page check $\Sigma\text{BS} - \Sigma\text{FS} = \text{Elev}_{\text{last}} - \text{Elev}_{\text{first}}$.

Complete the field notes. Starting from BM1 (Elev $=88.00$, HI $=88.00+2.45=90.45$) and applying HI $=$ Elev $+$ BS and Elev $=$ HI $-$ FS in turn:

Station BS HI FS Elevation (ft)

BM1 2.45 90.45 — 88.00

TP1 5.43 89.35 6.53 83.92

TP2 3.18 87.62 4.91 84.44

TP3 4.22 84.42 7.42 80.20

BM2 — — 6.11 78.31

$\Sigma$ 15.28 24.97
Elevation of BM2. The last elevation carried down the line is $$\text{Elev}_{\text{BM2}} = \text{HI}_{\text{TP3}} - \text{FS}_{\text{BM2}} = 84.42 - 6.11 = \boxed{78.31\ \text{ft}}$$
Page check. The sums of the backsights and foresights must reproduce the net elevation change: $$\Sigma\text{BS} - \Sigma\text{FS} = 15.28 - 24.97 = -9.69\ \text{ft}$$ $$\text{Elev}_{\text{BM2}} - \text{Elev}_{\text{BM1}} = 78.31 - 88.00 = -9.69\ \text{ft}\ \checkmark$$ The two agree, so the arithmetic of the field notes is verified.

Station	BS	HI	FS	Elevation (ft)
BM1	2.45	90.45	—	88.00
TP1	5.43	89.35	6.53	83.92
TP2	3.18	87.62	4.91	84.44
TP3	4.22	84.42	7.42	80.20
BM2	—	—	6.11	78.31
$\Sigma$	15.28		24.97

Quantity	Value
$\Sigma$ Backsights	$15.28$ ft
$\Sigma$ Foresights	$24.97$ ft
Net elevation change	$-9.69$ ft
Elevation of BM2	$78.31$ ft
Page check	$-9.69 = -9.69$ ✓

Concept & Theory

Topic. Differential-levelling reduction (height-of-instrument method) with the page-check arithmetic control.

Key relations. HI $=$ Elev $+$ BS; Elev $=$ HI $-$ FS; page check $\Sigma\text{BS}-\Sigma\text{FS}=\text{Elev}_{\text{last}}-\text{Elev}_{\text{first}}$.

Why this works. A backsight raises the line of sight to the height of instrument above a point of known elevation; a foresight then drops from that HI to the next point. Summing all rises (BS) and falls (FS) must equal the overall change in elevation — this is a bookkeeping identity, so any disagreement signals an arithmetic error in the notes (it does not detect field blunders such as a mis-read rod, which need a closed loop or a second run).

Common pitfall. Adding the FS to the HI instead of subtracting, or applying the page check to detect field error — it only checks the note arithmetic. Turning-point elevations are working values, not final control.

Source. Wolf & Ghilani, Elementary Surveying §5 (leveling — theory, field notes & the page check); Canadian orthometric heights referenced to CGVD2013. Units are US feet as printed.

Question 6 of 7: Differential-Levelling Field Notes and Page Check

Question 6: Differential-Levelling Field Notes and Page Check (20 marks)