16-Civ-B4 Engineering Hydrology · December 2013

Question 3 of 7: Point and Areal Precipitation, and Long-Term Streamflow Measurement

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Examinations, December 2013 — 98-Civ-B4 Engineering Hydrology. Three hours, closed book, one candidate-prepared two-sided 8½″ × 11″ aid sheet, and one approved Casio or Sharp calculator whose model must be declared. Seven problems are printed; page-1 Note 4 states that any five (5) questions constitute a complete paper and that only the first five answers appearing in the workbook are marked. Each problem carries twenty (20) points, so the examinable total is 5 × 20 = 100 points. The page-6 marking scheme breaks each problem into its sub-parts. All seven problems are solved here, because this set is a study resource rather than a timed sitting; the sub-part mark values shown below are the printed ones.

Reference texts. V. T. Chow, D. R. Maidment and L. W. Mays, Applied Hydrology (hydrologic cycle, unit hydrographs, routing, frequency analysis, infiltration); L. W. Mays, Water Resources Engineering, 3rd ed. (design application, rainfall–runoff, reservoir operation); W. Viessman and G. L. Lewis, Introduction to Hydrology, 5th ed. (measurement, areal precipitation, energy budget); V. T. Chow, Open-Channel Hydraulics (1959) (flood-wave propagation, dam-break and gradually varied unsteady flow); C. W. Fetter, Applied Hydrogeology, 4th ed. (Darcy’s law, hydraulic conductivity, recharge). Canadian practice references: Environment and Climate Change Canada / Water Survey of Canada HYDAT archive and the ECCC Engineering Climate Datasets (IDF curves and the IDF_CC climate-adjustment tool); the ISO 1100 / WMO Manual on Stream Gauging series as adopted by the Water Survey of Canada; the Canadian Dam Association Dam Safety Guidelines (inflow design flood and dam-break consequence classification); and the Transportation Association of Canada Guide to Bridge Hydraulics, 2nd ed.

Check — conventions used throughout this paper. A hydrologic year is taken as 365 days = 31 536 000 s unless a question says otherwise. Water density is 1000 kg/m³, gravitational acceleration is 9.81 m/s², and the latent heat of vaporisation of water is 2.45 MJ/kg at 20 °C. Several sub-parts ask for an explanation with an example rather than for the solution of stated data; in those cases a realistic Canadian data set is declared at the point of use and every number arising from it. Where the printed data are internally inconsistent — and Question 6(i) is such a case — the inconsistency is demonstrated arithmetically, the governing conservation requirement is stated, and the corrected reading actually used is declared, as page-1 Note 1 invites (“the candidate is urged to submit… a clear statement of any assumptions made”).

Question 3: Point and Areal Precipitation, and Long-Term Streamflow 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.

(i) Point and areal estimates of precipitation, and their reliability (10 marks)

Given. The watershed of the printed diagram is gauged by four recording stations G1–G4. To make the comparison concrete, take the watershed area as 120 km² and the following storm catch and Thiessen polygon areas.

Given data — illustrative storm at the four gauges
Gauge	Position in the watershed	Catch P_i (mm)	Thiessen area A_i (km²)
G1	upper left	85	30
G2	upper right	62	40
G3	lower left	110	20
G4	centre right	74	30
Total watershed area			120

Find. The point estimate at each gauge and the areal mean over the watershed by both the arithmetic and the Thiessen method, with a statement of which estimate is more reliable and why.

Figure 3.1 — The four gauges with perpendicular-bisector (Thiessen) polygons constructed around them, and the outflow gauging station at the watershed outlet. The polygon areas, not the number of gauges, carry the weighting.

Approach. Take each gauge catch as the point estimate at that location, then form the areal mean twice — unweighted, and weighted by the Thiessen polygon areas — and compare.

Point estimates. Each gauge gives the depth at a point, and nothing more: 85, 62, 110 and 74 mm at G1, G2, G3 and G4 respectively. A point value is the most reliable number in the whole analysis for the location where it was measured — a properly exposed, Alter-shielded recording gauge is accurate to a few percent for rain — and the least reliable one anywhere else, because storm cells are far smaller than a 120 km² basin.
Areal mean by the arithmetic (station-average) method. The simplest areal estimator weights all gauges equally: $$\bar{P}_{\text{arith}} = \frac{1}{n}\sum_{i=1}^{n} P_i = \frac{85 + 62 + 110 + 74}{4} = \frac{331}{4} = 82.75\ \text{mm}$$ It is defensible only when the gauges are evenly distributed and the terrain is uniform.
Areal mean by the Thiessen polygon method. Perpendicular bisectors between neighbouring gauges divide the watershed so that every point is assigned to its nearest gauge; each gauge is then weighted by the area it represents: $$\bar{P}_{\text{Thiessen}} = \frac{\sum A_i P_i}{\sum A_i} = \frac{30(85) + 40(62) + 20(110) + 30(74)}{120}$$ $$\boxed{\bar{P}_{\text{Thiessen}} = \frac{2550 + 2480 + 2200 + 2220}{120} = \frac{9450}{120} = 78.75\ \text{mm}}$$
Compare, and read the difference as an error estimate. The arithmetic mean exceeds the Thiessen mean by $$\Delta = 82.75 - 78.75 = 4.00\ \text{mm}, \qquad \frac{\Delta}{\bar{P}_{\text{Thiessen}}} = 5.08\ \text{percent}$$ The arithmetic value is biased high because the wettest gauge, G3 at 110 mm, sits in the smallest polygon: it represents only a sixth of the watershed but carries a quarter of the unweighted average.

Reliability of the estimates. Three statements are needed, and an examiner is looking for all three. First, the point estimates are the raw material and their own errors are small but systematic: wind-induced undercatch, which is a few percent for rain but can reach 20–50 percent for unshielded snow in a Canadian winter, plus wetting and evaporation losses. Second, the areal estimate is far less certain than any point value, and its error scales with gauge density and storm type: with four gauges on 120 km² (one per 30 km²) a frontal, widespread rain is captured within roughly 5–10 percent, while a summer convective cell that falls between the gauges may be in error by a factor of two. Third, the choice of method matters most where the gauge network is irregular, which is exactly this network: Thiessen is preferred to the arithmetic mean here because it corrects for the uneven spacing, and the isohyetal method would be preferred to both if orographic effect were significant, because only isohyets let the analyst impose knowledge of where the rain actually falls. The practical recommendation is therefore: report the Thiessen value of 78.75 mm as the areal depth, quote the 4.00 mm spread against the arithmetic mean as an indication of the method uncertainty, and cross-check the whole estimate against weather-radar accumulation or the ECCC gridded product before using it for design.

Final results — Question 3(i)
Quantity	Basis	Result
Point estimates	gauge catches G1–G4	85, 62, 110, 74 mm
Areal mean, arithmetic method	ΣP_i / n	82.75 mm
Areal mean, Thiessen method	ΣA_iP_i / ΣA_i	78.75 mm (adopted)
Method spread	arithmetic − Thiessen	4.00 mm (5.08 percent)
Gauge density	120 km² / 4 gauges	1 gauge per 30 km²

(ii) Accurate streamflow measurement at the outflow over a 25-year period (10 marks)

A 25-year record is not a long series of measurements; it is one continuous stage record converted to discharge by a rating that must be maintained. The programme has four parts.

Establish a permanent gauging station at a hydraulically sound section. The outflow location is the right place because it integrates the whole watershed, but the section itself must be chosen for stability: a straight reach with a well-defined channel, no backwater from a downstream confluence or lake, and a natural or constructed control — a bedrock riffle, or a broad-crested weir or flume where the channel is small enough — that fixes a single-valued relation between stage and discharge. Install a stilling well with a float or a pressure/bubbler sensor logging stage continuously (15-minute interval is Water Survey of Canada practice), referenced to a set of permanent benchmarks so the datum survives bank erosion, ice damage and equipment replacement over 25 years.

Build and maintain the stage–discharge rating curve. Discharge is measured directly by the velocity–area method — current meter or acoustic Doppler profiler, the section divided into subsections with $Q = \sum a_i v_i$ and no subsection carrying more than about 5 percent of the total — at a range of stages, and a rating is fitted:

$$Q = a\,(H - H_0)^b$$

where H is the recorded stage, $H_0$ the stage of zero flow at the control, and a and b are fitted constants. For a section calibrated as $a = 12$, $H_0 = 0.25$ m and $b = 2.1$, a recorded stage of 2.35 m gives

$$Q = 12\,(2.35 - 0.25)^{2.1} = 12\,(2.10)^{2.1} = 57.0\ \text{m}^3/\text{s}$$

The rating must be re-checked with fresh gaugings at least a few times a year and always after a major flood, because scour and deposition shift the control; a shifted rating applied unknowingly is the single largest source of error in a long record. High flows beyond the gauged range are extrapolated by a slope–area (Manning) computation on surveyed high-water marks rather than by extending the fitted curve blindly.

Handle the Canadian-specific problems: ice and missing record. For four to five months a year the control may be ice-affected and the open-water rating is simply invalid; discharge for that period is estimated by the standard backwater-correction procedures using periodic under-ice gaugings, and the record is published with the ice-affected flag. Every long record also has gaps from instrument failure; these are infilled by correlation with a neighbouring long-term Water Survey of Canada station and flagged as estimated. Both practices must be documented, because a user 20 years later cannot otherwise tell a measurement from an inference.

Add the water-quality dimension and use the record. Quantity alone does not preserve a watershed. Pair the hydrometric station with continuous sensors for temperature, conductivity and turbidity, and with an automatic sampler triggered on the rising limb, because the great majority of the annual sediment and nutrient load moves during a small number of high-flow days. Load is then computed as $L = \int C(t)\,Q(t)\,dt$. Over 25 years the resulting series supports exactly the decisions the question points at: a flood-frequency analysis for design, a low-flow and 7Q10 analysis for setting water-taking limits and in-stream flow needs, an annual water balance for the basin, a trend test on load and on low flow to detect the effect of land-use change, and the calibration record without which no conceptual model of this watershed can be built.

Topic. Areal precipitation estimation from a point network, and continuous discharge determination by the stage–rating method.

Key relations. Arithmetic mean $\bar{P} = \frac{1}{n}\sum P_i$; Thiessen mean $\bar{P} = \sum A_i P_i / \sum A_i$; rating curve $Q = a(H-H_0)^b$; velocity–area discharge $Q = \sum a_i v_i$; constituent load $L = \int C Q\,dt$.

Why this works. A rain gauge measures a depth over about 200 cm² and the answer required is a depth over 10⁸ cm²; every areal method is therefore an interpolation rule, and the rules differ only in what they assume about the field between the gauges. Thiessen assumes the nearest gauge is the best predictor, which is a defensible default on flat terrain with an irregular network. The rating curve rests on a different idea: at a stable hydraulic control the depth of flow determines the discharge uniquely, so measuring the easy quantity continuously and the hard quantity occasionally yields a continuous discharge record.

Common pitfall. Treating the rating curve as permanent. Ratings shift with the bed, and an unchecked rating produces a record that looks continuous and is quietly wrong — the error is invisible in the data and appears only as an unbalanced annual water budget. On the precipitation side, the equivalent pitfall is recomputing Thiessen weights from an old polygon map after a gauge has been moved or discontinued; the polygons must be reconstructed whenever the network changes.

Source. Viessman & Lewis, Introduction to Hydrology, Ch. 2–3 (precipitation measurement and areal averaging); Chow, Maidment & Mays, Applied Hydrology, Ch. 6 (streamflow measurement); WMO/ISO 1100 stream-gauging practice as applied by Water Survey of Canada.

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