## Info

where N is the number of annual maxima in the sample. Then the upper and lower limits are calculated from:

where ta v are values of the t distribution obtained from standard statistical tables with a the probability limit required and v the degree of freedom.

The calculations for the 95 per cent confidence limits for the plotted example in Fig. 11.16 are set out in Table 11.9. The value of the t statistic is 2.06 for a = 100 — 95 per cent = 5 per cent and v = n — 1 = 23. The curves of the 95 per cent confidence limits are plotted on Fig. 11.16 for the range of selected T values.

### 11.7.4 The generalised extreme value distribution

The Gumbel distribution of the previous section is a special case of the generalised extreme value (GEV) distribution, which is described by the equations for discharge X at any value of F(X)

 T (years) 10 20 30 50 100 K(T) (from Table 11.8) 1.3 1.86 2.2 2.61 3.14 XX (equation 11.14) m3 s-1 449.7 504.1 537.1 576.9 628.4 SE(X) (equation 11.16) m3s-1 41.3 52.2 58.9 67.1 77.8 i5 23 SE(X) m3s-1 85.1 107.4 121.3 138.3 160.4 X97.5% m3 s-1 534.8 611.5 658.4 715.2 788.8 Xt 5% m3 s 1 364.6 396.7 415.8 438.6 468

The second form with k = 0 is the form of the Gumbel (EV1) distribution. If k = 0, then the sample of floods will not plot as a straight line on Gumbel probability paper such as that shown in Fig. 11.16. If it appears that there is a trend for the more extreme floods to curve upwards above a straight line, then k will be negative and the distribution is called an EV2 distribution. If it appears that there is a trend towards an upper limit below a straight line plot, then the distribution is called an EV3 distribution. Both have been used in past analyses of annual maximum floods, including in the UK Flood Studies Report (NERC, 1975).

One of the techniques used in comparing flood frequency distributions between different catchments is the growth curve. The growth curve is defined by normalising the floods observed at a site by dividing by the value of an index flood. In the FEH (Institute of Hydrology, 1999), the index flood is taken to be the median of the observed series (QMED; the value of X when F(X) = 0.5). The ratio of the observed discharge to the index flood provides a non-dimensional scale of discharge for comparing catchments but will not change the form of the frequency distribution. Defining x = X/QMED, the growth curve for the GEV distribution can then be defined by substitution in (11.28) and rearranging as:

The growth curve can also be related directly to return period, T, by substituting for F(x) such that:

11.8 Other statistical distributions used in flood frequency analysis

The statistical theory of extreme values suggests that the maximum values in independent sample series of fixed length taken from any fixed distribution should, as the number of samples increases, asymptotically approach the form of the GEV distribution, while peaks over threshold series should be approximately distributed as a generalised pareto (GP) distribution. A series of annual maximum series of flood discharges might also therefore approach the form of the GEV as the sample size increases. Why, therefore, are other distributions used in flood frequency analysis? This is because the number of samples is generally small, and we cannot be sure that there is a fixed underlying distribution for the occurrences of floods because of climate and hydrological variability.

In the USA, after comparing six different distributions, the log Pearson Type III was selected, although many factors, other than statistical, governed the final choice (Benson, 1968). The log-normal distribution, in which the logarithms of the peak flows conform to the Gaussian or normal distribution, also takes the form of a probability curve with a positive skew, as shown in Fig. 11.15b and is often applied successfully to annual maximum flow series.

In the UK, the GEV distributions were chosen as the basis for analysis of annual maximum flows in the Flood Studies Report (NERC, 1975). This has been replaced by the generalised logistic (GL) distribution in the FEH (see Chapter 13). With the longer series available in FEH, the GL distribution was selected as better representing annual maximum series. The advantage of the GL distribution over the GEV is that fitting the GEV more often results in a distribution with an upper bound (an EV3 distribution). It is perhaps debatable whether we should expect floods to have some upper bound on the basis of meteorological and hydrological conditions in any region. Reaching such a bound would normally occur at such a long return period that, for practical purposes, the assumption of an unbounded distribution can be taken as reasonable; hence the choice of the GL distribution in the FEH. Details of the GL and GP distributions may be found in Chapter 13 on the FEH procedures.

Distributions with greater numbers of parameters are also sometimes fitted to flood frequency data, including the four-parameter kappa distribution, and the five-parameter Wakeby distribution. The GEV and GL distributions are both special cases of the kappa distribution. Mixed distributions are also sometimes used where there is evidence that different events in the flood record are generated by different types of mechanism. Hydrological understanding would suggest that we might expect mixed distributions in some cases, such as the synoptic rainfall events and snowmelt or rain-on-snow events that make up the series of extremes in British Columbia (see Woo and Waylen, 1984); or the mixture of rainstorms and rare hurricanes, such as the destructive Hurricane Agnes in 1972, on the east coast of the USA. In all these cases, however, fitting distributions with more parameters will often result in higher uncertainty associated with the parameter estimates and wider confidence limits on estimates at longer return periods, unless very long sequences of extremes are available for analysis. Since this is rarely the case, the two- and three-parameter distributions remain the most commonly used in practice.

### 11.9 Using historical flood data

Whilst the FEH and other formal procedures generally rely on using gauged flow and rainfall records, there is often potential to supplement these estimates by making use of 'historical' flood information. Here the term 'historical' refers to information not contained within the gauged record, but to a mixture of quantitative and qualitative data such as flood marks on historic buildings, contemporary accounts of notable floods, photographs and engravings and records of changes in factors such as catchment land drainage or river channel cross-section shape.

The use of incomplete flood records in statistical approaches based on censored annual maximum series was discussed in the Flood Studies Report (NERC, 1975). There have been studies demonstrating the use of historical information since then, e.g. Acreman and Horrocks (1990) and Archer et al. (2007). However, historical flood data has not been incorporated routinely into statistical flood estimation procedures. Floods in the UK in the 1990s prompted a more systematic effort to look back to major historical events for comparison and to attempt to incorporate these into flood risk assessments. Prompted also by investigations into the potential effects of climate and land-use change (see Chapter 19), there has been a growing recognition that the mid-twentieth century might have been unusually 'flood sparse' and that statistical flood estimates might therefore underestimate flood risk. On the other hand, relatively short gauged records that include the floods in the 1990s onwards may overestimate the frequency of these events. The British Hydrological Society (BHS) Chronology of British Hydrological Events6 was launched in 1998 (Black and Law, 2004). It provides a readily accessible searchable database of hydrological events and reports including both floods and droughts.

Bayliss and Reed (2001) published guidance on using historic data in flood risk estimation, and recommended a graphical approach to include historical data on to a plot of the systematic gauged data and fitted frequency distribution based on adjusted plotting position formulae. Stedinger and Cohn (1986) have described an approach to fit probability distributions to a flood peak data series that combines continuous AMAX records from a gauging station with historical records of flooding, which are viewed as analogous to an incomplete AMAX record. The approach adopts a binomial distribution to model the probability of obtaining the number of historical peaks that have been observed. This is combined with a probability distribution for the gauged AMAX series (the GEV distribution above or the GL distribution described in Chapter 13 are suitable choices) and the resulting combined function fitted by the method of maximum likelihood. The method has the useful feature that, even if the magnitude of an historical flood flow is unknown, knowledge of the mere occurrence of a flood that exceeds a given threshold can be used to improve the statistical model, hence qualitative information about past events can be incorporated.

Fig. 11.17 shows a flood frequency analysis using these methods for the River Wansbeck at Morpeth, draining a predominantly lowland catchment in north-east England. The data comprises a combination of continuous records from gauging stations at Highford and Mitford, where the catchment area is 287 km2, and also historical information from nearby sites. The gauged record from 1963 to 2008 includes significant high-flow events in 2008, 1992, 1982, 1967 and 1963. Flood marks on

- - GEV fitted to gauged flows only

GEV fitted to gauged and historical data

* Gauged annual maximum data o Historical peak flow estimates

* Gauged annual maximum data o Historical peak flow estimates

1000 years

Fig. 11.17 Flood frequency curves for the Wansbeck at Morpeth based on fitting the generalised extreme value (GEV) distribution to gauged annual maximum flows and to a composite record including estimated flows for events in 1878, 1886 and 1898.

1000 years

Gumbel reduced variate

Fig. 11.17 Flood frequency curves for the Wansbeck at Morpeth based on fitting the generalised extreme value (GEV) distribution to gauged annual maximum flows and to a composite record including estimated flows for events in 1878, 1886 and 1898.

nearby buildings at East Mill and Bothal Mill also allow flow rates to be estimated for floods that occurred in 1878, 1886 and 1898, using reconstructed rating curves. The results illustrate that the inclusion of historical events may lead to a reassessment of flood frequency estimates from the gauged record. Although the events of 1878, 1886 and 1898 rank highly in the composite record, the historical information effectively extends the record and places the largest events, including gauged peaks in 1963 and 2008, into a longer historical context (effectively shifting the plotting positions of the two largest AMAX values to the right). Here, the inclusion of this information dating back to the nineteenth century suggests that statistical estimates based on the gauged record alone may overestimate the frequency of high flows. Note that this is not a general result; underestimation may be apparent in other cases and a careful interpretation of the gauged and historical information is required in each individual study.

### 11.10 Droughts

Droughts as rainfall deficiencies have been considered earlier in Chapter 9 in terms of both long-term precipitation characteristics and in their effect on evaporation and transpiration losses to the atmosphere. Droughts are also important in terms of water supply. When there are deficiencies in river flows, it is the bulk users of water, namely agriculture, industry and the large urban concentrations of domestic consumers that begin to suffer. Water authorities and Water utilities whose duty it is to maintain public supplies of water are always concerned for their resources during periods of low flows and by the increase in demand that dry spells generally stimulate. An analysis of the expected minimum flows in those rivers providing water supplies is therefore essential. It is also important in other engineering design problems, such as low head hydropower installations. The economic viability of a design may be dependent on the expected durations of flows below some minimum threshold for power production.

In the UK, all major rivers are perennial, but some of the headwaters and small tributary streams rising in limestone or chalk country are intermittent or ephemeral. Their upper courses dry up in summer, and in drought periods the lack of water in the streams may extend further down the valleys, particularly when the flows are also affected by groundwater abstractions. Increasing attention is being paid to flow deficiencies, and in the perennial streams it is considered desirable to maintain a defined minimum discharge. The sustaining of a minimum discharge is particularly important in all rivers receiving waste water effluents in order to ensure required dilution of pollutants and the habitat conditions necessary for good ecological status within the requirements of the water framework directive (see Section 8.2.2 and Chapter 17).

In the analysis of low-flow conditions, it is preferable to have natural discharge records unaffected by major abstractions or sewage effluent discharge. In drought periods, hydrologists are encouraged to make extra gaugings on small tributary streams as well as at the established gauging stations. In the UK, the Environment Agency has carried out many such low-flow surveys on the major river networks. Such records provide more detailed information on changes in base flow conditions in relation to the geology of the catchment.

At gauging stations, where continuous river discharge records are available, several features of the data sets can be abstracted or computed to give measures of the characteristics of low flows. A variety of different indices have been used in the analysis of low flows derived from the flow-duration curve, consideration of low-flow spells, and the frequency analysis of the annual minimum series of low flows.

The flow-duration curve (Figs 11.11 and 11.12) was considered earlier and gives the duration of occurrence of the whole range of flows in the river. Selected points on the lower end of the curve can give measures of flow deficiency. The flow that is exceeded 95 per cent of the time, Q95, and the percentage of time that a quarter of the average flow is exceeded, are two suggested indices. In Fig. 11.12, the 95 per cent exceedance flow is 13.5 m3 s—1 and a quarter of the average flow (44 m3 s—1) 11 m3 s—1, is exceeded 99.45 per cent of the time. To compare low-flow values between catchment areas, the Q95 value may be expressed as a runoff depth over the catchment in millimetres per day. Hence for the Thames curve, Q95/A = 13.5 m3 s—V9870km2, which gives 0.118 mm day-1.

Similarly for the other flow-duration curve examples (Fig. 11.11), where the Q95 flows are: for the Eden, 9.90 m3 s—1 over a catchment of 2286 km2, equivalent to 0.374 mm day-1 or 30 per cent of the mean daily flow; and for the Miriam 0.218 m3 s—1 over a catchment of 133.8 km2, equivalent to 0.141 mm day-1 or 42 per cent of the mean daily flow. These catchments are therefore rather different in their baseflow characteristics, reflecting the geological differences between the two catchments: hard rock, partly overlain by glacial moraines, and chalk, respectively. The index Q95/A can be correlated with various catchment characteristics and useful regional patterns of drought properties can be identified.

The study of low-flow spells seeks to overcome the shortcomings of the flow-duration curve, which gives no indication of sequences of low flows. The most serious stress to water supply arises when there are periods of extended low flows. Such periods occurred in the UK, e.g. in 1976, 1995 and 2003. From the continuous record of daily mean discharges, the number of days for which a selected flow is not exceeded defines a low-flow spell. A Q95 flow may not be exceeded for a sequence of 10 days, thus giving a low-flow spell of 10 days' duration. The frequency or probability of occurrence of low-flow spells of different durations may be abstracted and assessed from the record. Hence D days, the duration of spells of low-flow < Q95, can be plotted against the percentage of low-flow spells > D. Catchments can again be compared by plotting individual results on the same graph. Where the volume of flow deficit below a threshold is also of interest, such as in assessing inputs to water supply reservoirs, the cumulative deficit can also be used as an index for frequency analysis.

In the UK, the estimation of low flows has been the subject of an extended study by the Institute of Hydrology and Centre for Ecology and Hydrology at Wallingford. This has resulted in a commercial software package, Low-Flows 20007 (Young etal., 2003), that can be used to estimate flow-duration curves and the low-flow characteristics for ungauged sites. More details on these studies are given in Chapter 13.

### 11.11 Frequency of low flows

Although the investigation of flood flows always attracts a great deal of research, the frequency analysis of low flows has not been neglected. In the UK a low-flows study was published by the Institute of Hydrology in 1980. Tallaksen and van Lanen (2004) provide a comprehensive overview of low-flow frequency analysis. Guidelines for frequency analysis of low flows in the UK were updated by Zaidman et al. (2002). Low-flow frequency analyses are carried out using a series of annual minimum ('AMIN') flow data. Low-flow periods can be prolonged and hence the annual minimum series should include a definition of the duration of interest, e.g. AMIN(30) refers to the series of annual minimum discharges of 30-day duration. In fitting a statistical distribution to annual minimum discharges, the principal requirements are that the distribution should be skewed, have a finite lower limit > 0 and, as in case of short flood records, have a small number of parameters. The EV3 distribution of the smallest value is one of the most reliable and is relatively simple to compute.

The minimum daily mean flows for 10 years (Table 11.10) are used to demonstrate the method. In practice, at least 20 years of flow data should be obtained in order to constrain uncertainty in the frequency estimates. The annual minimum values are ranked starting with the highest. The probabilities P(X), although calculated from the Gringorten and Weibull plotting positions as before, have a different meaning for the low-flow case. P(X) is now the probability that (Qannual minimum < X).

All the Weibull points and the two extremes of the Gringorten points are plotted in Fig. 11.18 on Gumbel extreme value paper. Such a plot is sufficient to demonstrate that the Gumbel distribution itself is not an adequate representation of low-flow frequency because it is not bounded. Thus, if a straight line were fitted by eye through the plotted points, it would give a probability of exceedance of zero flow of approximately 0.98 and thus apparently a probability of Q < 0 of 2 per cent. Since this is not the case in this catchment, which remains perennial in flow, the EV3 distribution will be more appropriate. Similar arguments would apply to the analysis of the total volume of flow deficit below some specified threshold discharge.

Table 11.10 Frequency of low flows (annual minimum daily mean flows)

X(m3 s—')

Rank R

P(X) Gringorten

P(X) Weibull

0.408

1

0.055

0.091

0.351

2

0.154

0.182

0.315

3

0.253

0.273

0.256

4

0.352

0.364

0.238

5

0.451

0.455

0.222

6

0.549

0.545

0.210

7

0.648

0.636

0.187

8

0.747

0.727

0.152

9

0.846

0.818

0.074

10

0.945

0.909

Xm = 0.241

sm = °.°98

Return period

 \ \ \ \ \ X X

Probability

Probability

Reduced variable, y x WeibuN o Gringorten + Fitted points

Fig. 11.18 Frequency plot of annual minimum flows for data of Table 11.10.

1000

### 11.12 Low flows and water yield analysis

An important issue that arises in water supply management is the water yield of a catchment area. The utility of a water supply reservoir, for example, will depend critically on the yield under dry conditions. When there is a sufficiently long representative series of discharge records for analysis, the occurrence and frequencies of periods of critically low flows can be determined. Some regard must be paid both to the proposed life of the water resource scheme in accepting the representativeness of the data, and to the possibility of an increased yield being required in the future. The expected loss by evaporation from a reservoir must also be taken into account. Using the runoff time series, there are several straightforward methods of analysis.

Monthly data may be used for evaluating the amount of storage required in a reservoir to guarantee a given demand or supply rate. An initial determination of the reservoir capacity needed to ensure supplies over a low-flow period can be obtained by evaluating the cumulative volumes over that period. In the mass curve or Rippl method, the sequences of months from the historical record having the lowest flows are abstracted and for each sequence the cumulative amounts plotted against time (see Fig. 11.19). The testing of all drought periods is needed before deciding on a design drought on which to assess the yield given a specified storage. Some of the notable drought periods in the UK were given in Section 9.7. This method gives a good first estimate of yields and required reservoir capacities, but gives little information on the probabilities of reservoir failure to meet demand under extreme dry conditions.

Example: Calculation of water yield from monthly discharge data using the Rippl method

In Fig. 11.19, the mean monthly flows for a catchment of 150 km2 for the 5 years 1973-77 have been converted into volumes of water and the cumulative sums plotted.

The slope of the straight line OA represents an average catchment flow rate of 74 ML day-1 over the period from O to A.

To sustain this flow rate as a steady abstraction from a reservoir full at O, a storage of 13 000 ML would be required at the point of maximum deficit, X. Over this period, there are two minor dry spells, but from A a more severe dry spell develops.

The more gentle slope of the line AB represents only 53 ML day-1, average inflow rate over the period A to B, and if this rate is to be maintained as a steady abstraction, a storage of 12 000 ML is necessary, as seen at the point of maximum deficit Y.

On the evidence of what is known to be an exceptionally dry period 1975-76 (AB), a reliable yield of 53 ML day-1 would be assured with reservoir storage of 12 000 ML.

The residual mass curve method is an extension of the mass curve technique with the advantage of having smaller numbers to plot and hence increased accuracy on the ordinate scale. Each flow value in the record is reduced by the mean flow (mean monthly or mean annual according to the duration studied) and the accumulated residuals plotted against time (Fig. 11.20). A line such as AB drawn tangential to the peaks of the residual mass curve would represent a residual cumulative constant yield that would require a reservoir of capacity CD to fulfil that yield starting with the reservoir full at A and ending full at B. The largest deficit between this residual yield line and the residual mass curve gives the minimum storage required to maintain

Fig. 11.19 Cumulative volume of monthly discharge in ML for the years 1973-1977. See example for explanation of lines 0-A and A-B.

the yield. In Fig. 11.20, the residual mass curve has been plotted for the same data used in Fig. 11.19. The mean rate of flow over the 5-year period is 1855 ML per month. The slope of the residual yield line (AB) is -240 ML month-1, which when adjusted by the mean rate gives 1855 -240 = 1615MLmonth-1 or 52MLday-1 for the actual yield rate for the period A ^ D ^ B. The storage required is then 12 000 ML given by the line CD. This result compares favourably with the answer obtained in Fig. 11.19 for the necessary storage to sustain a 52 ML day-1 yield over the particularly dry period represented by the period A ^ D ^ B in Fig. 11.20.

The minimum flows over various durations (e.g. 6, 12, 18, 24, 30, etc. consecutive months) can be abstracted from the runoff record and their flow volumes plotted against the corresponding durations (Fig. 11.21). A required yield line at the relevant slope is drawn from the origin of the plot, and a parallel line tangential to the plotted o o o

"O

6 Mean monthly flow 1855 ML/month Fig. 11.20 Residual mass curve analysis.

droughts curve identifies both the critical drought duration and the amount of storage, S, needed to provide the yield. In practice, it is recommended to add a year's required supply to the storage as a safety factor. An example of the yield assessment of Vyrnwy Reservoir in North Wales was given in Shaw (1989).

### 11.13 Some concluding remarks

This chapter has set out various methods for the analysis of river discharge data, including river regime plots, mixing models, flow-duration curves, flood frequency analysis, the analysis of low flows and yield estimation. It is often the case that such analyses reveal some data anomalies and missing data. These might be apparent errors of timing relative to recorded rainfalls, flows that are larger than the measured rainfalls, apparently flat-topped peaks, or strange sudden steps in the recorded flows. The hydrologist always needs to bear two things in mind when analysing flow data. The first is that, at most river flow-gauging sites, flow is not measured directly. Stage or water level is measured and then converted to flow using a rating curve. This introduces the potential for error, both in extrapolation beyond the range of available discharge measurements used to infer a rating curve and also for changes in the rating over time

Fig. 11.21 Reservoir yield from analysis of minimum flows.

(such as after major floods). Where possible, try to check the original measurements for the rating curve and any other available gaugings at a site to check for such anomalies, particularly at sites that do not have a proper flow measurement structure. Flood peak discharge estimates in particular should be looked at with some care (e.g. Costa and Jarrett, 2008).

The second issue is that very few rivers are now unaffected by the influence of man, either through abstractions of flow for water supply and agriculture, the addition of effluents, the operation of weirs or the storage of water behind dams and reservoirs. Sometimes a series of reconstructed 'naturalized' discharges are available from agencies, but the assumptions on which the naturalization has been carried out might vary from agency to agency or region to region. The effects of abstractions, effluent discharges and storages are expected to be greatest at low flows but in some catchments, flood detention basins and controls on flood plain storage might affect flood peaks. The lesson is always to examine available discharge data with some care.

A third issue is that new projects requiring estimates of river flows are rarely planned close to stream gauges with an observed discharge available. The vast majority of sites, e.g. for new low-head hydropower projects, for the design of a new bridge, or for the mapping of flood plain inundation, will be ungauged. Thus, procedures are required to estimate the flow characteristics at such sites, informed by the type of analyses of gauged sites described above. In hydrology, the process of estimating flows at ungauged sites is called regionalisation. In the UK, the FEH (Institute of Hydrology, 1999) and Low Flows 2000 (Young et al., 2003) provide the basis for the regionalisation of flood frequency and flow-duration curves, respectively. The use of the FEH methods is described later in Chapters 13 and 17.

Notes

1 Such as the Environment Agency for England and Wales (EA), the Scottish Environmental Protection Agency (SEPA), the Rivers Agency of Northern Ireland, the Office of Public Works (OPW) in Eire, or the United States Geological Survey, which makes its data freely available in real time on the internet through the National Water Information System; see http://waterdata.usgs.gov/nwis.

2 http://www.ceh.ac.uk/data/nrfa/index.html (This archive contains over 50 000 years of data from 1300 gauging stations using data provided by the EA, SEPA or the Rivers Agency of Northern Ireland.)

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