16-Civ-B4 Engineering Hydrology · May 2016

Question 6 of 7: Frequency and probability analysis of precipitation and floods

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

Notes on this paper

Paper format. 98-Civ-B4 Engineering Hydrology, May 2016 — a three-hour closed-book examination; a candidate-prepared two-sided aid sheet and one approved Casio or Sharp calculator are permitted. The cover page states that “any five(5) questions constitute a complete paper” and that “each question is equally weighted at twenty (20) points”, so the seven printed Problems each carry 20 marks towards a 100-mark paper. The page-6 Marking Scheme confirms the sub-part split for all seven. Note 1 invites the candidate to state any assumptions made where a question is open to interpretation; this sitting needs that licence twice, and both places are flagged in a callout below. All seven Problems are worked here, because this set is a study resource rather than a timed sitting.

Reference texts. V. T. Chow, D. R. Maidment and L. W. Mays, Applied Hydrology (hydrologic cycle, unit hydrograph theory, Horton infiltration, level-pool and Muskingum routing, frequency analysis); W. Viessman and G. L. Lewis, Introduction to Hydrology, 5th ed. (areal precipitation, hydrograph analysis, conceptual watershed models); P. B. Bedient, W. C. Huber and B. E. Vieux, Hydrology and Floodplain Analysis, 5th ed. (rating curves, reservoir and river routing, urban design storms); R. S. Gupta, Hydrology and Hydraulic Systems, 4th ed. (groundwater recharge and discharge areas, streamflow measurement); L. W. Mays, Water Resources Engineering, 3rd ed. (Rational Method, IDF design practice). For the Canadian frame: Environment and Climate Change Canada IDF curve files, the Water Survey of Canada Hydrometric Manual (mid-section gauging to ISO 748), and the Transportation Association of Canada Drainage Manual for design-storm and runoff-coefficient practice.

Check — two source-data issues and one declared convention.

(1) Problem 1(iii) gives the IDF relation as i = 6.0 − 0.3 t_d without stating the units of i. The paper's own Problem 6(iii) figure plots rainfall on an axis labelled “Rainfall and Infiltration, mm/h” with a peak near 13, so mm/h is adopted and the alternative reading is carried through in the answer as a sensitivity.

(2) Problem 7(i) as printed cannot be satisfied: the stated area and river discharge fix the runoff depth at 5045.76 mm/a, which is 63 times the 80 mm/a of rain the question supplies, so the residual evapotranspiration comes out large and negative. The answer boxes the runoff depth, demonstrates that the balance cannot close, and then adopts a declared corrected precipitation. This is the response Note 1 asks for.

(3) Problems 3(iii), 5(iii), 6(i) and 7(iii) ask for method, not arithmetic; each is worked on a small dataset that is the solver's own representative example, clearly labelled as illustrative. Every number in those examples, and every number taken from the real source data.

Question 6: Frequency and probability analysis of precipitation and floods (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) Using an IDF curve to predict the 100-year peak runoff (6 marks)

Given. An IDF family for the site gives rainfall intensity as a function of duration for each return period; the illustrative watershed and IDF coefficients below are the solver's own representative values, used to show the procedure end to end.

Table 6.1 — Illustrative 100-year design case (solver's own example data)
Quantity	Symbol	Value
Watershed area	A	120 ha
Time of concentration	t_c	30 min
100-year IDF fit (t_d in minutes)	i	1800 / (t_d + 12)^0.85 mm/h
Composite runoff coefficient	C	0.55

Find. The 100-year peak runoff, and the assumptions on which the prediction depends.

Approach. Select the 100-year curve from the IDF family, enter it at the duration equal to the time of concentration, and combine the resulting intensity with the area and a composite runoff coefficient in the Rational formula.

Select the correct curve and duration. An IDF plot is a family of curves, one per return period. Choosing the 100-year curve makes the rainfall a 1-per-cent-annual- probability event. Enter that curve at $t_d = t_c$, the duration at which the whole watershed first contributes: $$t_d = t_c = 30\ \text{min}$$ For a Canadian site the curve itself comes from the Environment and Climate Change Canada IDF file for the nearest long-record station, adjusted for elevation or transposed if necessary.
Read the design intensity. Substituting into the fitted relation, $$i = \frac{1800}{(30 + 12)^{0.85}} = \frac{1800}{23.97} = \boxed{75.1\ \text{mm/h}}$$
Compute the peak with the Rational formula. With a composite C obtained by area-weighting the runoff coefficients of the land-use classes present, $$Q_{100} = \frac{C\,i\,A}{360} = \frac{0.55 \times 75.1 \times 120}{360} = \boxed{13.8\ \text{m}^3\text{/s}}$$ For a watershed above roughly 100 to 200 ha the same intensity would instead be applied through a design-storm hyetograph and a unit hydrograph, because the Rational Method's uniform-contribution assumption no longer holds.
State the assumptions the prediction rests on. Five matter: (a) the rainfall intensity is uniform over the watershed and constant for the whole duration; (b) the critical duration equals the time of concentration, which requires the IDF curve to fall monotonically with duration — true of every real IDF family; (c) the runoff coefficient is constant and independent of storm magnitude, whereas in reality C rises as the soil saturates, so C for a 100-year event should be taken at the top of the tabulated range; (d) the return period of the peak flow equals the return period of the rainfall, which requires antecedent conditions, snow cover and any storage in the system to be typical rather than extreme; and (e) the IDF curve is stationary, so a long historical record represents the design life — an assumption now routinely adjusted upward for climate change in Canadian municipal practice.
Check sensitivity to the land-use assumption. If the composite coefficient rises from 0.55 to 0.69 as the watershed develops further, the peak becomes 17.3 m³/s, an increase of 25 per cent for no change in the storm at all — which is why the runoff coefficient, not the IDF curve, is usually the dominant uncertainty in an urban design flow.

Final results — Problem 6(i) illustrative case
Quantity	Value
Design duration (= t_c)	30 min
100-year design intensity, i	75.1 mm/h
100-year peak runoff, Q₁₀₀	13.8 m³/s
Same watershed at C = 0.69	17.3 m³/s (+25 per cent)

(ii) How a "100-year flood" can occur twice in fifteen years (6 marks)

Given. A river has experienced two floods of the 100-year magnitude within a 15-year window; the agent believes this is a contradiction.

Find. An explanation of the return-period concept, with the probability of exactly this outcome, that settles the confusion.

Approach. Restate the return period as an annual exceedance probability, model the years as independent Bernoulli trials, and compute the binomial probability of two or more exceedances in fifteen years.

Restate what "100-year flood" means. The phrase is a statement of probability, not of schedule. A 100-year flood is the discharge whose annual exceedance probability is $$p = \frac{1}{T} = \frac{1}{100} = 0.01$$ that is, a flood with a 1 per cent chance of being equalled or exceeded in any given year. The return period is the average interval between exceedances over a very long record; it is not a countdown, and nothing resets after an event occurs. The clearest way to say this to a non-technical audience is to call it the “1 per cent annual chance flood”, which is the term now preferred in Canadian and American flood-mapping practice precisely because it prevents this misunderstanding.
Model the record as independent annual trials. Annual maximum floods in successive years are treated as independent (each year's weather is a fresh draw), so the number of exceedances $X$ in $n$ years is binomial: $$P(X = k) = \binom{n}{k}\,p^{k}\,(1 - p)^{\,n-k}$$
Compute the probability of the agent's experience. With $p = 0.01$ and $n = 15$: $$P(X = 0) = (0.99)^{15} = 0.8601, \qquad P(X = 1) = 15(0.01)(0.99)^{14} = 0.1303$$ so the chance of two or more 100-year floods in a 15-year window is $$P(X \ge 2) = 1 - 0.8601 - 0.1303 = \boxed{0.0096\ \text{, about }1.0\text{ per cent}}$$ Two exceedances in fifteen years is therefore an unusual outcome, but at roughly one chance in a hundred it is entirely unsurprising that it happens somewhere — on a river network with hundreds of gauges, it should be expected to occur at several of them in every fifteen-year period. Nothing about it contradicts the 100-year designation.
Put the risk in a form the client can act on. The same binomial gives the risk of at least one exceedance during a design life: $$P(X \ge 1) = 1 - (1 - p)^{n}$$ which is 14 per cent over 15 years, and 63 per cent over a 100-year design life. A property owner in the floodplain therefore has better than even odds of seeing a “100-year” flood if they hold the property for a working lifetime — the argument that sells flood insurance, and the argument for freeboard above the design level.
Add the two caveats an honest answer needs. First, a short record makes the estimate of the 100-year discharge itself uncertain: with only 30 or 40 years of data, the fitted 1-per-cent quantile has a wide confidence interval, so what was called a 100-year flood may in truth be a 40-year flood, and two occurrences may be evidence that the frequency curve needs revising rather than evidence of bad luck. Second, the independence assumption fails if the watershed or the climate has changed — urbanisation, forest harvest, a dam removal, or a trend in extreme rainfall all make the recent years non-stationary, and then the two floods are a signal, not a coincidence. The professional response is to re-analyse the annual maximum series including the new events and to test for trend, not to defend the old label.

Final results — Problem 6(ii)
Quantity	Value
Annual exceedance probability of the 100-year flood	p = 0.01 (1 per cent per year)
P(no exceedance in 15 years)	0.8601
P(exactly one in 15 years)	0.1303
P(two or more in 15 years)	0.0096 (about 1 per cent, or 1 in 104)
P(at least one in 15 years)	0.1399 (14 per cent)
P(at least one in a 100-year design life)	0.6340 (63 per cent)

(iii) How precipitation, retention and infiltration determine runoff (8 marks)

The figure printed with the question plots two things on the same axes of rate against time: the rainfall hyetograph, in mm/h, and the falling infiltration capacity curve of the soil. Everything about the runoff follows from the geometry of those two lines, and the figure below reproduces that geometry.

Figure 6.1 — Reproduction of the figure accompanying Problem 6(iii). The rainfall that lies below the infiltration capacity curve enters the soil; the early rainfall consumed by interception and surface storage is retention; only the rainfall above the capacity curve, in the second burst, becomes the surface runoff that produces downstream flooding.

Precipitation supplies the water, and its rate is what matters. The hyetograph is the whole supply, and its ordinate at any moment is the rate at which water arrives at the surface. Total depth alone determines nothing about flooding: the same 60 mm delivered over three days infiltrates almost entirely, while the same depth delivered in two hours produces a flood. It is the comparison of rate against the soil's capacity, moment by moment, that decides the outcome.

Retention removes the first part of the storm and delays everything else. Retention — the initial abstraction — is the water held by interception on leaves and roofs, by wetting of the surface, and in depression storage in hollows, ruts and ponds. It must be satisfied before overland flow can begin, so it acts as a threshold: the first few millimetres of the storm produce no runoff at all, and the hydrograph does not start to rise until it is filled. In the figure this is the early shaded portion of the first burst, which is why the first burst generates no surface runoff even though its intensity is the highest of the whole event. Retention is also the part of the budget that urbanisation destroys most completely — a paved surface has almost no depression storage and no interception, so the threshold nearly vanishes and small storms that once produced nothing begin to produce flood peaks.

Infiltration sets the capacity that the rainfall rate must beat, and it declines during the storm. The infiltration capacity curve starts high, at the dry-soil value $f_0$, and decays towards the saturated value $f_c$ as the surface pores fill, the wetting front advances and the hydraulic gradient across it falls — the behaviour Horton's equation describes. While the rainfall rate lies below the curve, all the rain enters the soil and there is no surface runoff; the water becomes soil moisture, and later either recharge or evapotranspiration. The crossing point is the moment of ponding.

The runoff is the residual, and that is what floods downstream. Formally, rainfall excess is $$R = \int \max\bigl[\,i(t) - f(t),\,0\,\bigr]\,dt \;-\; \text{(retention not yet satisfied)}$$ which is the hatched area above the capacity curve in the figure. Three features of the figure explain the flood risk. The first burst, though intense, falls on dry soil with a high capacity and unsatisfied retention, so it is almost entirely absorbed — it produces little runoff but it does the damage, because it exhausts the retention and drives the capacity curve down to near $f_c$. The second burst, of lower intensity, then arrives on a soil whose capacity has fallen to about 2 mm/h with no retention left, so nearly all of it becomes surface runoff. That is the general and practically important lesson: a moderate storm on a wetted watershed produces far more runoff than an intense storm on a dry one, and antecedent moisture is therefore as much a part of a flood forecast as the rainfall itself. Finally, because the runoff is generated over several hours and then routed to the outlet, the flood peak downstream is the convolution of that hatched area with the basin's response — so a longer wet burst covering the whole watershed is more dangerous than a brief cloudburst over part of it.

Topic. IDF-based design flows, return period as annual exceedance probability with binomial risk, and the rainfall-retention-infiltration partition that generates runoff.

Key relations. IDF fit $i = a/(t_d + b)^{c}$; Rational $Q = C i A/360$; return period $T = 1/p$; binomial $P(X = k) = \binom{n}{k}p^k(1-p)^{n-k}$; risk of at least one event $P = 1 - (1-p)^n$; rainfall excess $R = \int \max[\,i - f,\,0\,]\,dt$.

Why this works. The binomial model follows from exactly two assumptions — each year is an independent trial, and the exceedance probability is the same every year — and those two assumptions are the entire content of the phrase “return period”. Once they are stated, the apparent paradox disappears: repeated rare events in a short window are not evidence against the label, because the label was never a promise about spacing. The runoff partition works the same way: it is not a correlation but an identity, since water that arrives must either be retained, infiltrate, or run off, and the hyetograph-versus-capacity plot is simply that identity drawn to scale.

Common pitfall. Telling the client that a 100-year flood “happens once every 100 years”, or worse, that having had one means the next is 100 years away. On the partition question, the frequent error is to compare rainfall depth with infiltration rate, or to treat the infiltration capacity as a constant — the decline of the capacity curve during the storm is the mechanism that makes the second burst dangerous.

Source. Chow, Maidment and Mays, Applied Hydrology, Ch. 12 (frequency analysis) and Ch. 15 (design storms); Bedient, Huber and Vieux, Hydrology and Floodplain Analysis, Ch. 3 (frequency) and Ch. 1 (abstractions); Environment and Climate Change Canada IDF curve documentation; Viessman and Lewis, Ch. 4 (infiltration).

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