16-Civ-B4 Engineering Hydrology · December 2017

Question 6 of 7: Frequency and Probability Analysis of Precipitation and Floods

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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.

6(i) — How an IDF curve integrates duration, frequency and intensity (6 marks)

An IDF curve is a compact statement of three quantities at once, and understanding it means understanding how the three are tied together. It is built as follows. From a long record of continuous rainfall at a station, a moving window of each duration of interest — 5, 10, 15, 30, 60, 120 minutes and so on — is passed over every year, and the largest depth found in that year for that duration is recorded. That yields, for each duration, an annual maximum series. Each of those series is fitted separately with an extreme-value distribution, usually Gumbel in Canadian practice, and the fitted distribution is inverted to give the depth expected to be exceeded once on average in $T$ years. Dividing that depth by the duration converts it to an intensity. Plotting intensity against duration, with one line for each return period, gives the family of curves.

The integration of the three variables is visible in the geometry. Along a curve, duration and intensity trade off: intensity falls with duration, because the longer the averaging window the more the intense core of the storm is diluted by the weaker rain around it. But the depth, being intensity multiplied by duration, rises with duration, so the longer storm delivers more total water at a lower rate. Between curves, frequency enters: a rarer event has a higher intensity at every duration, and the spacing between the curves reflects the shape of the fitted extreme-value distribution. Every point on the diagram is therefore a triple — a duration, an intensity, and a return period — and the design engineer picks the return period from the consequence of failure and the duration from the response time of the drainage area, and reads the intensity.

Given. A fitted 100-year IDF relation for a Canadian station, $i = a/(t_d + b)^c$ with $a = 1150$, $b = 8.0$ min and $c = 0.78$, giving $i$ in mm/h for $t_d$ in minutes. An urban catchment of 12.5 ha with runoff coefficient $C = 0.75$ and time of concentration $t_c = 20$ min.

Find. The design intensity, a demonstration that depth rises as intensity falls, and the resulting peak discharge.

1. Read the curve at two durations. $$i_{20} = \frac{1150}{(20+8)^{0.78}} = \frac{1150}{13.45} = \boxed{85.5\ \text{mm/h}}, \qquad i_{60} = \frac{1150}{(60+8)^{0.78}} = \frac{1150}{26.87} = 42.8\ \text{mm/h}$$
2. Convert both to depths. $$P_{20} = (85.5)\left(\tfrac{20}{60}\right) = 28.5\ \text{mm}, \qquad P_{60} = (42.8)(1) = 42.8\ \text{mm}$$ The intensity halved while the depth rose by half — the trade-off the curve encodes.
3. Select the design duration and apply the rational method. The critical duration for a peak is the time of concentration, because that is the shortest storm for which the whole catchment contributes simultaneously. With $t_d = t_c = 20$ min, $$Q = 0.00278\,C\,i\,A = (0.00278)(0.75)(85.5)(12.5) = \boxed{2.23\ \text{m}^3/\text{s}}$$

The constant 0.00278 converts millimetres per hour acting on hectares into cubic metres per second. Note the logic of step 3: a storm shorter than $t_c$ gives a higher intensity but never engages the whole catchment, and a longer storm engages it all but at a lower intensity, so the peak occurs at $t_d = t_c$. That is the single most important consequence of the IDF relation for design.

6(ii) — Three important parts of a statistical flood-frequency method (6 marks)

The question distinguishes transient (short-term) from annual flood events, and the distinction determines how the data are selected, which is the first of the three parts.

Part 1 — selecting the data series, and the corresponding definition of return period. Two selections are standard. The annual maximum series takes the single largest instantaneous peak in each water year, giving one value per year of record; it is simple, the values are almost automatically independent, and it is what the published Canadian frequency analyses use. Its weakness is that it discards the second and third largest floods of a wet year while retaining the trivial maximum of a dry one, which matters when the record is short or when short-term transient events are of interest. The partial-duration (peaks-over-threshold) series takes every independent peak exceeding a chosen threshold, so a wet year may contribute several values and a dry year none. It uses the data far more efficiently for frequent events and is the right choice for transient events with return periods below about ten years, but it requires a defensible independence criterion between successive peaks and a threshold high enough that the exceedances are genuinely extreme. The two series give slightly different answers at short return periods and converge above about ten years; the return period must be quoted consistently with the series used, since a partial-duration return period is an average recurrence interval rather than an annual exceedance probability.

Part 2 — choosing and fitting a probability distribution. A distributional form is selected on the basis of extreme-value theory and of what fits regionally: Gumbel (Extreme Value Type I) is the traditional Canadian choice for annual maxima, the generalised extreme value distribution is now often preferred because it lets the shape parameter be estimated rather than assumed, the log-Pearson Type III is mandated in the United States, and the generalised Pareto is the natural partner of a peaks-over-threshold series. Parameters are estimated by the method of moments, by L-moments (now standard, because L-moment estimators are far less biased in small samples), or by maximum likelihood. Fit is judged by a probability plot against the appropriate plotting position, by L-moment ratio diagrams, and by a goodness-of-fit statistic, and outliers are examined rather than deleted — a historical flood known from documentary evidence is valuable information and should be incorporated as such.

Part 3 — estimating the quantile and stating its uncertainty. The fitted distribution is inverted at the required exceedance probability. In the frequency-factor form used throughout hydrology, $Q_T = \bar{Q} + K_T s$, where $K_T$ depends only on the distribution and the return period. The estimate must then be qualified: the standard error of a quantile grows rapidly as $T$ exceeds the record length, and a 100-year flood estimated from 45 years of record is an extrapolation whose confidence interval is typically wide enough to matter for design. Regional frequency analysis — pooling data from hydrologically similar gauged basins by the index-flood method — is the standard remedy, and the record must also be tested for trend and change point before any of this is legitimate.

Given. An annual maximum series of $n = 45$ years with mean $\bar{Q} = 425$ m³/s and standard deviation $s = 138$ m³/s, to be fitted with a Gumbel distribution.

Find. The 100-year and 2-year peak discharges.

1. Evaluate the Gumbel frequency factor. Chow’s form is $$K_T = -\frac{\sqrt{6}}{\pi}\left[0.5772 + \ln\!\left(\ln\frac{T}{T-1}\right)\right]$$ For $T = 100$, $\ln(100/99) = 0.01005$ and $\ln(0.01005) = -4.600$, so $$K_{100} = -0.7797(0.5772 - 4.600) = 3.137$$
2. Apply the frequency-factor equation. $$Q_{100} = \bar{Q} + K_{100}s = 425 + (3.137)(138) = \boxed{858\ \text{m}^3/\text{s}}$$
3. Repeat for the 2-year flood. With $K_2 = -0.164$, $$Q_2 = 425 + (-0.164)(138) = 402\ \text{m}^3/\text{s}$$ which is close to the mean, as it should be for a distribution whose median lies a little below its mean.

6(iii) — Why distributions are used, with two hydrologic variables and their distributions (8 marks)

Why probability distributions at all. Hydrologic variables are the outcome of an atmospheric and land-surface system that is deterministic in principle but unpredictable in practice beyond a few days, and whose governing conditions vary in ways no engineer can enumerate. The result is that no design question of the form “how large will the flood be?” has a deterministic answer. What can be answered is the probabilistic question: with what frequency is a given magnitude exceeded? A probability distribution is the object that answers it, and it earns its place for four specific reasons.

First, it encodes the whole record in a few parameters, so that a fifty-year series becomes two or three numbers that can be compared across sites, mapped, and pooled regionally. Second, it interpolates and extrapolates in a controlled way: no gauge record contains a 200-year flood, but a fitted distribution supplies an estimate of one, with a stated basis and a computable standard error, which arbitrary graphical extension does not. Third, it converts hydrology into risk, which is the form in which engineering decisions are actually made. If the annual exceedance probability is $p = 1/T$, the probability that the event occurs at least once in $n$ years is

so for the 100-year flood over a 25-year design life,

a result that surprises most non-specialists and is the single most useful thing frequency analysis produces. Fourth, it makes the level of protection an explicit, defensible choice that can be weighed against cost and consequence, which is what the CSA and provincial design standards require and what a professional engineer must be able to justify.

Variable 1 — annual maximum flood peak, characterised by an extreme-value distribution. Annual maxima are, by construction, the largest of many independent events within each year, and extreme-value theory shows that the maximum of a large sample converges to the generalised extreme value family regardless of the parent distribution. That theoretical backing is why the Gumbel (Type I) distribution, the light-tailed member of that family, has been the traditional Canadian choice, and why the full GEV, which allows a heavier or bounded tail through its shape parameter, is increasingly preferred. The log-Pearson Type III is used where a skewed fit is needed and is mandated in United States federal practice. The same reasoning applies to annual maximum rainfall depth at each duration, which is why the IDF curves of 6(i) are Gumbel-fitted.

Variable 2 — low flow, characterised by the Weibull or log-normal distribution. The seven-day minimum flow with a ten-year return period, the 7Q10, governs licensing of water withdrawals and effluent-dilution requirements across Canada. Low flows are minima rather than maxima, and the extreme-value theory for minima of a bounded-below variable gives the Weibull distribution, which is the standard choice; the three-parameter log-normal is also widely used because it is bounded at zero and fits the observed skew well. Two further variables and their conventional distributions are worth naming for completeness: annual runoff volume and monthly streamflow, which are sums of many contributions and are therefore commonly log-normal by appeal to the central limit theorem acting on logarithms; and the number of storm events exceeding a threshold in a season, which is a count of rare independent events and is therefore Poisson — the distribution that underpins the peaks-over-threshold model of 6(ii).