## Depthdurationfrequency DDF curves

The analysis of rainfall frequencies is somewhat more complex than the analysis of river discharge frequencies to be considered in Chapter 11. In the case of rainfall extremes we have to consider not only the frequency of a single variable (e.g. flood peaks) but occurrences for which both magnitude and duration are important. Rainfall frequency clearly has an effect on discharge frequency (for both high and low flows) but the relationship between the two will depend on other variables, in particular the antecedent conditions and catchment scale. Small catchment floods, for example, will generally result from small intense rainstorms on already wetted ground. The 2004 Boscastle event in Cornwall was of this type. But for larger catchments, it is the magnitude of rainfall over longer periods that

7000

6000

2000 1000

10 15 20 25 30 35 40 45 50 Rainfall (mm)

July rainfall totals 1967-2007

14 s

505 257

505 257

05050 02570 22223

Rainfall (mm)

Annual rainfall totals at Hazelrigg 1966-2007

Rainfall (mm)

Fig. 9.17 Distributions of daily, July monthly and annual totals shown as histograms for the Hazelrigg site, Lancaster University (see Fig. 9.14 for time series). For the daily totals 40 per cent days with zero rainfall have been excluded from the plot. For the annual totals, a fitted normal distribution is shown. Both daily and monthly data show highly skewed distributions.

Rainfall (mm)

Fig. 9.17 Distributions of daily, July monthly and annual totals shown as histograms for the Hazelrigg site, Lancaster University (see Fig. 9.14 for time series). For the daily totals 40 per cent days with zero rainfall have been excluded from the plot. For the annual totals, a fitted normal distribution is shown. Both daily and monthly data show highly skewed distributions.

will be important, such as in the 2007 summer floods on the River Severn. Thus the concept of a DDF distribution is important in the analysis of recorded rainfalls.

Frequency analysis of any variable depends on certain statistical assumptions. In particular we assume that the occurrence of events of a given magnitude (here rainfall depths) over a long period of time can be represented as a statistical distribution function. Assuming a statistical distribution for extremes also (for the distributions usually used in the analysis of extremes, see Chapter 11) implies an assumption that any particular extreme value might be exceeded in the future (albeit with low probability).

There is evidence that on decadal timescales there has been variability in the frequency of heavy rainfalls. There is also speculation that these frequencies will change with future changes in climate (the statistics may not be stationary). We can, however, only fit a distribution to the data that is available at the current time. As many years of measurements as possible should be used to determine rainfall frequencies. With a record of only 20 years for example, the determination of a representative frequency pattern for annual rainfall, or for the annual maximum series for different durations, will be rather uncertain. The sample is simply too small. Similarly, the analysis of extreme rainfalls lasting for periods of less than a day, entails the abstraction of rainfall depths over specified durations, often from even shorter periods of record. We should expect, therefore some uncertainty in estimating frequencies of occurrence. The general form of the relationship of maximum accumulated rainfall, R, with duration, t, extracted from a recording rain-gauge record during a storm is shown in Fig. 9.18 for the extreme Martinstown Storm event (see Table 9.2 and Fig. 9.12 above). The shape of the average rainfall intensity curve shows how the average maximum rainfall intensity will decrease with increasing duration.

Return period (years)

Return period (years)

Fig. 9.19 Rainfall depth-duration-frequency curves for Leicester (from Flood Estimation Handbook, Vol. 2, loH, 1999, Copyright NERC CEH).

Duration (hours)

Fig. 9.19 Rainfall depth-duration-frequency curves for Leicester (from Flood Estimation Handbook, Vol. 2, loH, 1999, Copyright NERC CEH).

We can extend the analysis of a single storm to look at how rainfall amounts vary with duration over a large number of storms. This is shown in Fig. 9.19 for a recording rain gauge site at Leicester.

Fig. 9.19 represents the DDF in terms of contours of rainfall depths for different durations for different return periods. The return period is a way of expressing the expected frequency of a statistical variable. Any statistical distribution can be expressed in terms of a cumulative probability of occurrence. Thus for a distribution of rainfall depths r over a duration D, cumulative probability to a value of R is given by the integral

Jr=0

where p(r|D) is the probability of r given the duration D.

As hydrologists, we are most often interested not in cumulative probability itself but in the probability of exceedance, which is P(R|D) = {1 - F(R|D)}. This tells us how often we would expect a event larger than or equal to R to occur for that duration. The probability of exceedance is then directly related to return period, T as:

Thus, the return period is an estimate of the average time between the exceedances of R over a long period of data. For example, if a certain magnitude of daily rainfall has

Period |
Duration |
Rainfall total |
Return period |

(h) |
(mm) |
(years) | |

00:00 Jan 7- 12:00 Jan 8 |
36 |
206.9 |
173 |

19:00 Jan 7-01:00 Jan 8 |
6 |
64.4 |
21.7 |

03:15 -04:15 Jan 7 |
1 |
17.6 |
3.4 |

an annual probability of exceedance of 0.01, then its return period will be 100 years. A table of estimated return periods for the Wet Sleddale site shown in Fig. 9.4 is given in Table 9.5.

Note that the return period concept does not suggest that if a 1 in 100 year event occurs this year it will be another 100 years before it will occur again. We are treating occurrences of R as a statistical variable, which means that, whether or not there is an occurrence this year, there is finite probability that it will be exceeded next year (0.01 in the case of an event of 100-year return period; 0.1 in the case of a 10-year return period event). There is more discussion of frequency distributions for extreme events in Chapter 11.

In evaluating the frequency of intense rainfalls, mention should be made of two classic studies that established the framework for DDF analyses in the UK. The best known study of the frequency of short-period continuous rainfall in the UK is that by Bilham published in 1936 and reissued by the UK Meteorological Office in 1962. Bilham, who was a civil engineer, assembled 10 years of autographic rain-gauge data from 12 stations representative of lowland England and Wales, and from their analysis developed a formula to relate rainfall depths, durations and frequency of occurrence. A later study by Dillon (1954), then took 35 years of autographic recordings at Cork and identified the relationship of intensity and frequency of occurrence. Dillon proposed expressing the DDF as:

in which D is duration, T is return period and c, d and f are constants for a particular data set. This general form reflects the expectation that the higher the frequency of occurrence of a storm of given duration, the smaller the average intensity.

This approach has been extended in the UK Flood Estimation Handbook (IoH, 1999), building on the extensive analyses carried out for the earlier Flood Studies Report, by treating the DDF as a number of straight-line segments for each return period on a plot of lnR v. lnD (Fig. 9.20). The straight-line segments run from durations of 1-12 h, 12-48 h and above 48 h. The full DDF is then defined by six parameters, values of which are tabulated for every catchment area in the UK greater draining an area of greater than 0.5 km2. The DDF distributions have been fitted to rainfall data with durations from 1 h to 8 days and are extrapolated to return periods of up to 1000 years (probability of exceedance 0.001). Thus, the hydrologist in the UK is provided with the frequencies or return periods of rainfalls for any time period from

In 1 In 12 In 48 In (duration)

Fig. 9.20 Flood Estimation Handbook depth-duration-frequency model showing different linear segments. (Copyright NERC CEH.)

Fixed duration (days) |
Multiply by |
Fixed duration (h) |
Multiply by |

1 |
1.16 |
1 |
1.16 |

2 |
1.11 |
2 |
1.08 |

4 |
1.05 |
4 |
1.03 |

8 |
1.01 |
8 |
1.01 |

> 12 |
1.00 | ||

1 min to greater than 8 days for use in engineering design, though the more extreme return period estimates are necessarily somewhat uncertain.

In applying the DDF functions defined in this way, it is important to remember that they have been derived from an analysis of maximum rainfalls in sliding time periods determined from continuous recordings rather than fixed time increments. If the return period for a measured rainfall in a particular fixed time period is then required, the amount must be adjusted to account for the discretisation inherent in fixed period measurements (Dwyer and Read, 1995). The relevant adjustment factors are given in Table 9.6.

### 9.6.2 The concept of the design event

One of the applications of a rainfall frequency analysis is to provide an input to a design problem, such as determining the capacity of a culvert or storm water detention pond. Depth-duration-frequency analysis provides some standard tools, e.g. those in the UK Flood Estimation Handbook described above, for estimating the total storm volume and duration of rainfalls for different probabilities of exceedance. The way in which these are used to produce estimates of discharges with the required return period is explained in Section 13.5. We will only note here that, because of the complexity of runoff generation processes, particularly the effects of antecedent wetness of a catchment on runoff generation, it cannot be assumed that the return period of a design rainstorm will be the same as the discharge resulting from that event (see Fig. 13.4).

In 1 In 12 In 48 In (duration)

Fig. 9.20 Flood Estimation Handbook depth-duration-frequency model showing different linear segments. (Copyright NERC CEH.)

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