T

480 504

600 624

Rainfall DP AS GW

Fig. 11.8 Analysis of contributions to total Stream discharge of different sources in the Haute Menthue catchment using a three-component end-member mixing analysis (after Iorgulescu et al., 2005, with kind permission of John Wiley & Sons). DP, direct precipitation; AS, soil waters; GW, groundwater.

Fig. 11.9 Changing contributions of different sources of stream discharge in the Haute Menthue catchment with changing catchment storage (after lorgulescu et al., 2005, with kind permission of John Wiley & Sons). DP, direct precipitation; AS, soil waters; GW, groundwater.

Fig. 11.9 Changing contributions of different sources of stream discharge in the Haute Menthue catchment with changing catchment storage (after lorgulescu et al., 2005, with kind permission of John Wiley & Sons). DP, direct precipitation; AS, soil waters; GW, groundwater.

concentrations measured for each source also show some variability. Such variability should be expected to result in some uncertainty in the calculated proportions for each component. A number of different methods are available for assessing such uncertainty (see e.g. Joerin et al., 2002).

In some catchments, concentrations may be available for many different chemical constituents in waters from different sources. One way of trying to distinguish end members in this case is to use principal components analysis to determine linear combinations of the different concentration observations that provide separation between the end members. Burns et al. (2001), for example, used the first two principal components in an analysis of seven different chemical characteristics to allow a three-component separation of runoff sources in the Panola catchment, Georgia (in this case, hillslope runoff water, riparian groundwater and runoff from a bare granite outcrop in the catchment, Fig. 11.10).

11.4 Flow-duration curves

For many problems in water engineering, the hydrologist is asked for the frequency of occurrence of specific river flows or for the length of time for which particular river flows are expected to be exceeded. Thus frequency analysis forms one of the important skills required of a hydrologist. Estimates of the frequencies of floods of a particular magnitude are important in assessing flood risk. Estimates of the frequencies of low flows are important in assessing reservoir yields and the potential for low head hydropower schemes in rivers. In attempting to provide any answer to the questions of frequency, good reliable hydrological records are essential, and these must if possible extend beyond the expected life of the engineering scheme being considered.

From the basic assemblage of river flow data comprising the daily mean discharges and the instantaneous peaks, analysis of the daily mean flows will be considered first. Taking the n years of flow records from a river gauging station, there are 365n + leap year days of daily mean discharges. The frequencies of occurrence in selected discharge classes (groups) are compiled, starting with the highest values. The cumulative frequencies converted into percentages of the total number of days are then the basis for the flow-duration curve, which gives the percentage of time during which any selected

I 70

I 70

110 90

110 90

b) Hillslope runoff

~ Hillslope runoff

---Stream runoff rate

T

110 90

110 90

Fig. 11.10 Stream discharge and proportions provided from different runoff sources (hillslope runoff, runoff from a bare granite outcrop and riparian groundwater) in the Panola catchment, Georgia (after Burns etal., 2001, with kind permission of John Wiley & Sons).

Fig. 11.10 Stream discharge and proportions provided from different runoff sources (hillslope runoff, runoff from a bare granite outcrop and riparian groundwater) in the Panola catchment, Georgia (after Burns etal., 2001, with kind permission of John Wiley & Sons).

discharge may be equalled or exceeded. An example is demonstrated in Table 11.4, in which the daily mean discharges for 4 years for the River Thames at Teddington Weir are analysed. The flow-duration curve plotted on natural scales is seen in Fig. 11.11a. The area under the curve is a measure of the total volume of water that has flowed past the gauging station in the total time considered. For the reliable assessment of water supply, the flow-duration curves for the wettest and driest years of the record should be derived and plotted.

The representation of the flow-duration curve is improved by plotting the cumulative discharge frequencies on log-probability paper (Fig. 11.11b). (The abscissa scale is based on the normal probability distribution; if the logarithms of the daily mean discharges were normally distributed, they would plot as a straight line on the log-probability paper.) From the plot (Fig. 11.11b), it can be readily seen e.g. that for 2 per cent of the 4-year period, flows exceeded 290 m3s-1. At the other extreme, flows of less than 12 m3 s-1 occurred for the same proportion of the time. Alternatively it can be stated that for 96 per cent of that 4-year period, the flow in the River

Table 11.4 Flow frequencies

; over a 4-year period: River Thames at Teddington

Daily mean discharge (m3 s-1

) Frequency (days)

Cumulative frequency

Cumulative percentage

Over 475

3

3

0.21

420-475

5

8

0.55

365-420

5

13

0.89

315-365

8

21

1.44

260-315

25

46

3.15

210-260

36

82

5.61

155-210

71

153

10.47

120-155

82

235

16.08

105-120

52

287

19.64

95-105

42

329

22.52

85-95

50

379

25.94

75-85

58

437

29.91

65-75

83

520

35.59

50-65

105

625

42.78

47-50

72

697

47.71

42-47

75

772

52.84

37-42

73

845

57.84

32-37

84

929

63.59

26-32

103

1032

70.64

21-26

152

1184

81.04

16-21

128

1312

89.80

11-16

141

1453

99.45

Below 11

8

1461

100.00

Thames at Teddington is between 12 and 290 m3 s-1. The 50 per cent time point provides the median value (45 m3 s-1).

The shape of the flow-duration curve gives a good indication of a catchment's characteristic response to its average rainfall history. An initially steeply sloped curve results from a very variable discharge, usually from small catchments with little storage where the stream flow directly reflects the rainfall pattern. Flow-duration curves that have a very flat slope indicate little variation in flow regime, the resultant of the damping effects of large storages. Groundwater storages are provided naturally by extensive chalk or limestone aquifers, and large surface lakes or reservoirs may act as runoff regulators either naturally or controlled by man. Examples of some different flow-duration curves are given in Fig. 11.12. Hydrographs for these same sites can be seen in Fig. 11.1. The comparisons are simplified by plotting the logarithms of the daily mean discharges as percentages of the overall daily mean discharge. The Cumbria Eden drains the Eastern Lake district and northern Pennines, whereas the catchment of the Mimram, a tributary of the Lee is nearly all chalk.

The comparison of flow-duration curves from different catchments can also be used to extend knowledge of the flow characteristics of a drainage area that has a very limited short record. At least 1 or 2 years of records are required overlapping with those of a long-term well-established gauging station on a nearby river, whose flow-duration curve for the whole length of its record may be taken to represent long-term flow conditions. For satisfactory results, the two catchments should be in the same hydrological region and should experience similar meteorological conditions.

Fig. 11.11 Flow-duration curves for the Thames at Teddington: (a) discharge v. per cent of time exceeded; (b) log discharge v. per cent of time exceeded on normal probability scale.

Jan-Dec Dec-Mar Jun-Sep

Jan-Dec Dec-Mar Jun-Sep

% of time flow exceeded Jan-Dec Dec-Mar Jun-Sep

% of time flow exceeded

Fig. 11.12 Seasonal flow-duration curves for: (a) Eden at Sheepmount (076007); and (b) the Mimram at Panshanger Park (038003). (Reproduced from the National River Flow Archive, Centre for Ecology and Hydrology. Copyright NERC CEH.)

% of time flow exceeded

Fig. 11.12 Seasonal flow-duration curves for: (a) Eden at Sheepmount (076007); and (b) the Mimram at Panshanger Park (038003). (Reproduced from the National River Flow Archive, Centre for Ecology and Hydrology. Copyright NERC CEH.)

The method for supplementing the short-term record is to construct its long-term flow-duration curve by relating the overlapping short-period flow-duration curves of both catchments, as shown in Fig. 11.13. The available data are plotted and flow-duration curves (in full lines) are drawn in Fig. 11.13a and b. Sa and Sy are flow-duration curves for the short overlapping records; La is the flow-duration curve for the long-term neighbouring record. Selected percentage discharge values from the two short-period flow-duration curves are plotted in Fig. 11.13c, and a straight-line relationship drawn. Time percentage discharge values for La are converted to corresponding percentage values for Ly via the Sa and Sy relation. The derived long-term duration curve for the short-period station, Ly, is shown by a broken line in Fig. 11.13. By these means, the variation in the flow characteristics embodied in the long-term flow-duration curve has been translated to the short-period station.

Flow-duration curves from monthly mean and annual mean discharges can also be derived, but their usefulness is much less than those constructed from daily mean

(a) Long-term station

(b) Short-period station

5 102040 50 60 80 90 95 % time

(b) Short-period station

\ Derived Lh

5 102040 50 60 80 90 95 % time

(c) Correlation of Sa and Sh

La5%

\ Derived Lh

510 2040 50 60 80 90 95 % time

(c) Correlation of Sa and Sh

La5%

/

- /

/

/ "95%

10 Lh5% 100

Fig. 11.13 Method for supplementing a flow-duration curve for a station with only short-term record, given a flow-duration curve for a similar station with a long-term record.

flows, since the extreme high and low discharges are lost in the averaging. It should be stressed that no representation of the chronological sequence of events is portrayed or enumerated in flow-duration curves. For assessing water resources, the frequency of sequences of wet or dry months can be evaluated, given a suitably long record. Hydrograph sequences can also be modelled, given a suitably calibrated rainfall-runoff model for a site and either measured or model generated rainfalls. Such models are considered in later chapters.

11.5 Flood frequency

One of the most important hydrological analyses is the assessment of the frequency with which discharges of a given magnitude are exceeded at a site and one of the most important data sets is the measured instantaneous flood peak discharges estimated from the stage records at a site. The longer a record continues, homogeneous and with no missing peaks, the more its value is enhanced. Even so, it is very rare to have a satisfactory record long enough to match the expected life of many engineering works required to be designed (which may be 50 years or more) or to assess flood risk for the 0.01 annual exceedance probability (AEP, also referred to as 1 per cent AEP or 1 in 100 year) event, used in the UK as a standard for flood risk mapping as required under Section 105 of the Water Resources Act 1995.

As many peak flows as possible are needed in assessing flood frequencies but care should be taken in evaluating estimates of the higher peaks. A detailed review of the most extreme flood peaks recorded in the USA, for example, revealed that the discharges that had been estimated for a number of floods needed some revision (Costa and Jarrett, 2008). Very often flood discharge estimates are based on an extrapolation of the stage-discharge rating curve based on measurements at much lower discharges, if only because of the logistical and safety difficulties of getting measurements of discharge at the highest flows. Thus, where possible, the uncertainty in the flood peak estimates should be assessed as part of a flood frequency analysis.

The hydrologist defines two data series of peak flows: the annual maximum series (often called the AMAX series) and the partial duration or peaks over threshold series. These may be understood more readily from Fig. 11.15. The annual maximum series takes the single maximum peak discharge in each year of record so that the number of data values equals the record length in years. For statistical purposes, it is necessary to ensure that the selected annual peaks are independent of one another. This is sometimes difficult, e.g. when an annual maximum flow early in January may be related to an annual maximum flow at the end of the previous December. For this reason, it is generally advisable to use the water year rather than the calendar year; the definition of the water year depends on the seasonal climatic and flow regimes. In humid temperate areas, such as the UK, it is often taken to be October to September (in October, soils should normally have rewetted following the summer dry period). The partial duration series takes all the peaks over a selected threshold level of discharge. Hence the partial duration series is often called the peaks over threshold (POT) series. There will be more data values for analysis in this series than in the annual series, but there is more chance of successive peaks being related such that the assumption of statistical independence of the data might be less valid.

In Fig. 11.14, Pj, P2 and P3 form an annual series and Pj, pj, P2, P3 and p3 form a POT series. It will be noted that one of the peaks in the POT series, p3, is higher than the maximum annual value in the second year, P2. For sufficiently long records it may be prudent to consider all the major peaks and then the threshold is chosen so that there are N peaks in the N years of record, but not necessarily one in each year. This is called the annual exceedance series, a special case of the POT series.

Flood frequency analysis entails the estimation of the peak discharge, which is likely to be equalled or exceeded on average once in a specified period, T years. This is called the T-year event and the peak, QT, is said to have a return period or recurrence interval of T years. The return period, T years, is the long-term average of the intervals between successive exceedances of a flood magnitude, QT, and is effectively a shorthand way of referring to a probability with which QT might be expected to be exceeded in any 1 year (see next section). It is important not to misunderstand the return period concept. The intervals with which QT is exceeded might vary considerably around the

Pi

Pi

A Threshold

P3

P3 f\

Fig. 11.14 Definition of annual maximum discharges (upper case) and peaks over threshold (upper and lower case) peaks over a 3-year period.

average value T. Thus a very long record could show 10-year events, Qiq> occurring at intervals much greater or much less than 10 years, and even in successive years.

The annual series and the partial duration series of peak flows form different probability distributions but, for return periods of 10 years and more, the differences are minimal and the annual maximum series is the one most usually analysed. Both types of analysis can also be used to estimate the probability of exceedance or return period of a particular event in the historical record at a site.

11.6 Flood probabilities

When a series of annual maximum flows is sub-divided by magnitude into discharge groups or classes of class interval AQ, the number of occurrences of the peak flows (f) in each class can be plotted against the discharge values to give a frequency diagram (Fig. 11.15a). It is convenient to transform the ordinates of the diagram in two ways: with respect both to the size of the discharge classes (AQ) and the total number, N, of events in the series. By plotting f/(N.AQ) as ordinates, it is seen that the panel areas of the diagram will each be given by f/N, and hence the sum of those areas will be unity, i.e.

where n is the number of discharge classes.

If the data series is now imagined to be infinitely large in number and the class intervals are made infinitesimally small, then in the limit, a smooth curve of the probability density distribution is obtained (Fig. 11.15b), the area under this curve being unity, i.e.

0 0

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