Question 1 of 7: Areal Precipitation and Stream Flow Measurement

Problem 1: Areal Precipitation and Stream Flow Measurement (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.

Part (i) — Average precipitation over the watershed (10 marks)

Given. Fourteen rain gauges lie on and around the watershed, reading (in mm):

The map is overlaid with a uniform square grid in which one grid square represents 10 ha (0.10 km²). The gauge readings rise systematically from 550 mm at the western tip of the basin to 750 mm at the eastern tip.

Find. A defensible estimate of the areal-average precipitation over the whole watershed, together with the water volume that this depth represents.

Approach. Planimeter the watershed by counting grid squares to fix its area, test the gauge network for a directional trend, then weight the point readings by area using the isohyetal method, and confirm the result against Thiessen polygons and the simple arithmetic mean.

1. Fix the watershed area by counting grid squares. Superimposing the printed grid on the traced divide and counting whole squares, then adding partial squares as fractions, gives 30.0 squares. With the stated conversion, $$A=N_{\text{sq}}\,a_{\text{sq}}=30.0\times 10\ \text{ha}=300\ \text{ha}=3.00\ \text{km}^{2}$$ where $N_{\text{sq}}$ is the square count and $a_{\text{sq}}$ the area each square represents. This area is the denominator of every weighted average that follows, so it is worth checking twice.
2. Test whether the rainfall field has a direction. Regressing the fourteen readings on their west–east position gives a coefficient of determination $$R^{2}=0.98$$ which means 98 % of the variation in gauge catch is explained by easting alone. The field is a smooth, almost one-dimensional gradient rising to the east — exactly the situation the isohyetal method is designed for, and the situation in which the arithmetic mean is least trustworthy. Isohyets are therefore drawn essentially north–south, normal to the gradient.
3. Draw isohyets at a 50 mm interval and planimeter the bands. Interpolating linearly between adjacent gauges places the 550, 600, 650, 700 and 750 mm isohyets across the basin as shown on the figure. Counting grid squares within each band, and taking the arithmetic mid-value of each band as its representative depth $P_i$, gives:

The band areas sum to 299.8 ha against the 300 ha counted in Step 1 — a 0.07 % closure, which confirms the planimetry is internally consistent.

4. Weight the bands by area to obtain the isohyetal mean. The isohyetal average is the area-weighted mean of the band depths, $$\bar{P}_{\text{iso}}=\frac{\sum A_i P_i}{\sum A_i}$$ and summing the last column of the table above gives $$\boxed{\bar{P}_{\text{iso}}=664\ \text{mm}}$$
5. Cross-check with Thiessen polygons. Constructing perpendicular bisectors between neighbouring gauges and clipping the resulting polygons to the divide yields weights $w_j=A_j/A$ ranging from 0.041 (the 740 mm gauge, wedged against the eastern boundary) to 0.113 (the 650 mm gauge, which commands the open centre of the basin). Applying $\bar{P}_{\text{Th}}=\sum w_j P_j$ gives 665.6 mm.
6. Cross-check with the arithmetic mean. The unweighted average of the fourteen readings is $$\bar{P}_{\text{arith}}=\frac{9280}{14}=662.9\ \text{mm}$$ The three estimates span only 2.7 mm, or 0.4 % of the mean. The methods agree because the gauge network is dense and nearly uniformly spaced; had the gauges been clustered in the wet east, the arithmetic mean would have been biased high and only the area-weighted methods would be defensible.
7. Convert the adopted depth to a water volume. Adopting $\bar{P}=665$ mm, the annual water input to the basin is $$V=\bar{P}\,A=0.665\ \text{m}\times 3.00\times10^{6}\ \text{m}^{2}$$ $$\boxed{V \approx 2.0\times 10^{6}\ \text{m}^{3}}$$ which is the quantity a reservoir-yield or water-supply study would actually use.

Part (ii) — Stream rating curves: development, use and recalibration (10 marks)

A rating curve is the stage–discharge relation for a gauging section: a single-valued function Q = f(h) that converts the one variable which can be recorded cheaply and continuously — water level — into the variable that is actually wanted, discharge. Continuous discharge records, and therefore essentially all of applied hydrology, rest on it.

How it is developed. A control section is chosen where the stage–discharge relation is stable and sensitive: a natural rock riffle, a constriction, or an installed weir or flume, sited far enough downstream of confluences and bends that the flow is not affected by variable backwater. A staff gauge and a recorder (float-and-counterweight, bubbler or pressure transducer) are referenced to a permanent local datum tied to benchmarks. The engineer then makes a series of current-meter gaugings across the full range of stage: the section is divided into 20–30 vertical panels, each panel’s mean velocity is measured at 0.6 of the depth (or averaged over 0.2 and 0.8 of the depth in deeper panels), and the panel discharges are summed by the mid-section method, following ISO 1100 / WMO practice as adopted by the Water Survey of Canada. Each gauging yields one (h, Q) pair. A curve of the form

is then fitted, where $h_0$ is the stage of zero flow (the effective datum of the control), $C$ is a coefficient set by the section geometry and roughness and $n$ is an exponent typically between 1.5 and 2.5, reflecting how quickly the section widens with depth. Plotted on log–log paper the relation becomes a straight line, which is why that plot is the standard fitting tool: it makes a break in slope — the signature of a change of control, such as the flow going overbank — obvious, and shows where a second segment of curve is needed.

How it is used. Once accepted, the curve converts the continuous stage record into a continuous discharge hydrograph, from which come daily and monthly flows, annual maximum and minimum series, flow-duration curves, and the flood-frequency estimates used for bridge, culvert and spillway design. It also drives real-time flood forecasting and low-flow water-licence administration.

Recalibrating a 20-year-old curve on a 100 km reach. A curve that old must be treated as suspect rather than merely imprecise, because the control that defines it is a physical object that ages. The programme would be:

1. Re-survey the gauge datum and the control section. Re-level the staff gauge and recorder to the benchmark network to detect datum drift or settlement, and re-survey the cross-section. Compare it with the original section survey: scour, aggradation of gravel, ice damage, vegetation growth, or a channel avulsion will all have shifted the relation. A change in $h_0$ alone can bias every low-flow value in the record.
2. Re-gauge across the full range of stage. New current-meter or acoustic (ADCP) measurements are required at low, medium and — critically — high flows. High-flow gaugings are the expensive and the scarce ones, and they are the ones that control the extrapolated upper limb where flood estimates come from.
3. Reconcile the reach, not just the point. Over 100 km of stream the gauge is representative of only its own control; tributary inflows, abstractions and floodplain storage change the flow between sections. Establishing two or three additional gauged sections along the reach, and checking that their hydrographs balance volumetrically after allowing for travel time, tests whether the curve is still consistent with the reach as a whole.
4. Analyse the shifts and re-issue the curve. Plot the new gaugings as departures from the old curve. A consistent offset indicates a permanent shift and justifies a new curve; scattered departures that grow with time indicate progressive change and call for time-varying shift corrections applied to the historical record. Bracket the extrapolated upper limb independently — by a slope-area calculation using Manning’s equation with surveyed high-water marks, or by a hydraulic (HEC-RAS) model of the reach — rather than by extending the fitted line on faith.
5. Re-derive the affected products. Any flood-frequency curve, design flow or water-balance study built on the old rating is revised, and the reasons for the change are documented in the station history.