Relative Vertical Tolerance → Point Standard Deviation and Canadian Levelling Order: 18-Geom-A1 Surveying, May 2019, Q2

Question 2: Relative Vertical Tolerance → Point Standard Deviation and Canadian Levelling Order (10 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. Relative vertical tolerance between any two points $T = \pm 5$ mm; longest distance between points $K = 2.8$ km.

Find. (i) An interpretation of the tolerance; (ii) the standard deviation $\sigma$ of a single survey point; (iii) the Canadian levelling order to specify.

Assumptions (stated as the paper asks). (1) The ±5 mm is a 95 % confidence tolerance, the convention used by Canadian control-survey accuracy standards. It is a one-dimensional (height) quantity, so the coverage factor is $k = 1.96$. (2) Errors are normally distributed. (3) Every point in the network is established with the same, independent standard deviation $\sigma$, so the difference in height between two points has variance $2\sigma^2$.

Approach. Convert the 95 % relative tolerance to a relative standard deviation, share it equally between the two points, then compare the required closure over 2.8 km with the Canadian levelling-order allowances $C\sqrt{K}$.

Interpretation. The tolerance says that for any pair of marks, the error in their computed height difference must fall inside ±5 mm 95 times in 100. It is a relative requirement, so it controls the precision of height differences along the tunnel rather than absolute datum heights.
Relative standard deviation. $$ \sigma_{rel} = \frac{T}{1.96} = \frac{5.0}{1.96} = 2.55\text{ mm}. $$
Point standard deviation. With $\sigma_{rel}^2 = \sigma^2 + \sigma^2 = 2\sigma^2$, $$ \sigma = \frac{\sigma_{rel}}{\sqrt{2}} = \frac{2.551}{\sqrt{2}} \;\Rightarrow\; \boxed{\sigma \approx 1.8\text{ mm}} $$ (If the tolerance were read as a $2\sigma$ bound instead, $\sigma_{rel} = 2.5$ mm and $\sigma = 1.77$ mm, which makes no practical difference.)
Required closure constant. Canadian specifications state each order as an allowable discrepancy $C\sqrt{K}$ (mm, $K$ in km) between forward and backward levellings. The project needs $$ C \le \frac{T}{\sqrt{K}} = \frac{5}{\sqrt{2.8}} = 2.99\text{ mm}/\sqrt{\text{km}}. $$
Compare the orders over 2.8 km. $$ \text{Special }(3\sqrt{K}) = 5.02\text{ mm},\quad \text{First }(4\sqrt{K}) = 6.69\text{ mm},\quad \text{Second }(8\sqrt{K}) = 13.39\text{ mm}. $$ Only Special Order matches the ±5 mm requirement. Its allowance of 5.02 mm equals the tolerance to within rounding, while First Order would already allow 6.7 mm.

Recommendation. Specify Special Order levelling for the tunnel vertical control. In practice that means a precise digital level with invar bar-code staffs, double-run (forward and back) sections, balanced and short sight lengths, and staff and instrument calibration before and during the work.

Quantity	Value
Interpretation of ±5 mm	95 % relative (height-difference) tolerance
Relative standard deviation, $\sigma_{rel}$	$2.55$ mm
Point standard deviation, $\sigma$	$\approx 1.8$ mm
Required $C = T/\sqrt{K}$	$2.99$ mm/√km
Recommended Canadian levelling order	Special Order ($3\sqrt{K}$ = 5.02 mm)

Concept & Theory

Topic. Turning a relative tolerance into a per-point precision, and choosing a Canadian vertical-control order.

Key relations. $\sigma_{rel} = T/1.96$ (95 %, 1-D); $\sigma_{rel}^2 = 2\sigma^2$; allowable discrepancy $C\sqrt{K}$ with $C$ = 3 (Special), 4 (First), 8 (Second), 24 (Third) and 120 (Fourth) mm.

Why this works. A tolerance on height differences constrains the combined error of two marks. Variance propagation shares it between them, and the $\sqrt{K}$ law, under which levelling error grows with the square root of distance, turns it into a closure constant that can be read against the order table.

Common pitfall. Setting the point standard deviation equal to the whole tolerance, which forgets both the 1.96 and the $\sqrt{2}$. Another is quoting US FGCS classes when the question asks for the Canadian order.

Source. Surveys and Mapping Branch (EMR Canada), Specifications and Recommendations for Control Surveys and Survey Markers (1978), vertical-control orders; Ghilani, Adjustment Computations, 6th ed., Ch. 4 (error propagation).

Question 2 of 9: Relative Vertical Tolerance → Point Standard Deviation and Canadian Levelling Order

Question 2: Relative Vertical Tolerance → Point Standard Deviation and Canadian Levelling Order (10 marks)