Estimates of extreme rainfall are essential for the design of drainage works such as culverts and flood alleviation schemes. The main interest in the probable maximum precipitation (PMP) is for its use in estimating the probable maximum flood (PMF) in the design of reservoir spillways, where a serious under-design could lead to overtopping and dam break.
In the UK, the Flood Estimation Handbook (FEH) has revised estimates of extreme rainfall up to a return period of 10,000 years, without changing the estimates of PMP in the Flood Studies Report (FSR). However, the new estimates of the 10,000 year rainfall are in many areas of the UK in excess of the PMP as given in the FSR. This has raised questions about the possible underestimation of the PMP and therefore has implications for reservoir safety. It also raises questions about the FEH rainfall frequency analysis.
In this paper revised estimates of PMP for the whole of Great Britain are presented for the first time since publication of the FSR. The new estimates are based on statistical analysis, storm maximisation of historic storms, and are supported by a brief worldwide analysis of 24-hour storm rainfall in relation to perceptible water content. The results exceed FSR PMP by about 35% in northern Scotland, nearly 60% in southern Scotland, and about 50-67% in much of central and southern England.
Overtopping
About one-third of all dam failures are due to overtopping during a flood event (Charles, 1997). In Great Britain there has been 14 recorded dam failures, of which 12 have resulted in loss of life (Hughes et al 2000). Of these incidents, eight were ascribed to overtopping. There have been several near misses so far. Where overtopping is concerned there has been no loss of life since 1925, when Coedty dam was overtopped and 16 people killed.
During the 20th century the assessment of floods and reservoir safety has progressed from the Institution of Civil Engineers’ report (ICE, 1933) whose estimates of extreme but not maximum floods were revised upwards following the Lynmouth flood disaster in 1952. The Flood Studies Report was the first attempt to give an estimate of PMP for the UK (NERC, 1975). Later the Institution of Civil Engineers published its guide to floods and reservoir safety, making the important point that FSR extreme rainfall, though not PMP, was underestimated for parts of Somerset (Bootman & Willis, 1977), and advised engineers to use local data when designing dams in this area. Although this advice was maintained (ICE, 1996) in the third edition of the guide, there was no guidance on what design rainfall should be selected. This is especially important for category A dams which must have a spillway capable of safely passing the PMF, widely accepted as resulting from the PMP.
Others too questioned the validity of the FSR rainfall estimates (Stewart, 1989; Clark, 1991). Questions and new estimates have been made of the PMP (Collier & Hardaker, 1996), while for southwest England a map was produced showing new estimates of the PMP (Clark, 1995). These estimates tended to support the storm model approach of Collier & Hardaker (1995) in that the rainfall depth in 24 hours could be as high as 500mm as compared with 300mm in the FSR.
The FEH sought to revise estimates of rainfall up to the 10,000 year event. The FSR PMP map was retained (FEH, 1999). However, MacDonald & Scott (2000) compared FEH 10,000 year rainfall with FSR PMP for 12 reservoir sites in England and Wales, of which nine had rainfall depths greater than FSR PMP.
Following this study the Department for Environment Transport and Regions (DETR) asked consulting engineers Babtie to examine these differences further. Their results showed that as a whole FEH 10,000-year rainfall exceeded FSR PMP by an average of 14% (Babtie, 2000). The report concludes by stating: ‘The apparent differences in estimated design rainfall depth are highly significant to reservoir owners and panel engineers in terms of both safety and potential expenditure.’
It is with this background in mind that the present study was undertaken, noting at the same time that these recent concerns were not raised earlier following the PMP studies made by Collier & Hardaker (1995) and Clark (1995).
Analysis of daily rainfall data
The basic problem of estimating very rare events is a lack of good quality long term data. Rainfall has been measured on a daily basis at about 3000 sites in the UK, at some places for over 100 years. However, many sites have missing data, many have broken records (though complete for the years in which measurements were made) and very few extend beyond about 50 years.
The British Atmospheric Data Centre’s web site (www.badc.rl.ac.uk) was the main source of data for this study with some additional material being collected at the Meteoro-logical Office National Archive at Bracknell in Berkshire and also at Edinburgh. Only sites with 26 or more years of complete data were analysed. There was no missing data for the years analysed and the median length of record of the 81 sites studied was 35 years. Some 3746 station years were used in the study.
In Britain, rainfall frequency studies use annual maximum data, thereby discarding almost all of the storm data in each year even though in some years there could be storms not counted which were bigger than the largest storm in another year. While the use of annual maximum data is justified when undertaking a frequency analysis of river flow data (Gumbel, 1941), the daily rainfall catch does not influence the following days’ result since the gauge is emptied on day one, whereas the river flow on day one will have an effect on river flow the day after, and the problem then becomes of identifying independent events.
Although Stewart et al (1999) favoured the annual maximum (AM) approach to rainfall frequency analysis, their reasons for this choice are not in agreement with Madsen et al (1997) and Van Montfort (1986), who advocate the use of events over a threshold. The use of AM data will tend to lead to an underestimation of rainfall frequency. In view of these remarks all storms above a threshold of 15mm were included for analysis. Since the return periods of many of the storms were less than one year the modified reduced variate scale (Rakhecha & Clark, 2002) was used where the return periods were marked in days rather than years, and the reduced variate:-
y = – ln ln [ 1 – 1/T] [ 1.04133 T exp -0.03059]
where y = reduced variate
T = return period (days)
This modified scale has been found to yield useful results for both rainfall and runoff frequency analyses. The level of frequency was set at 106 years or 365.25 x 106 days. This is an acceptable level of risk (eg: Salmon & Hartford, 1995) although it is not such a high standard as in several other hazards (Hughes et al 2000).
The cumulative frequency in days for each storm class was plotted against the mid value of the class interval. The return period was calculated using:
Return period = N/ (M – 0.3)
where N = the sample size in days
M = rank order
The line of best fit was calculated using least squares with a condition that the correlation coefficient was at least 0.99 which rejects any variation from other than an almost perfect relationship. In practice for most sites this was the standard obtained where data from at least four classes of storm event were included. At some sites there was an inflection in the frequency plot at a return period of about 100 days or less, below which the storm frequency was more rare than expected. In these cases the smaller events were excluded from the analysis.
The resulting rainfall at the 106 years was then multiplied by 1.13 (WMO, 1986) since the original data were for one day and not 24 hours.
Some checks were made on the temporal variation of the rainfall and although some sites showed an increase in frequency, at others there was a decrease with no clear cut geographical pattern emerging.
The UK has a rich history of storm events, for which data have been collected from the network of rain gauges. A total of ten storms were identified for study including one from 1869.
Table 1 shows the details of these storms. The procedure of storm maximisation is based upon the fact that storm dew points are rarely at their maximum and that the ratio of perceptible water during the storm to the maximum perceptible water gives the moisture maximising factor (MMF) for that storm event from which the maximised storm rainfall is calculated:-
MR = R Mpw/Spw
Where MR = maximised rainfall (mm)
Mpw = maximum perceptible water
Spw = storm perceptible water
R = rainfall (mm)
The perceptible water is a function of the surface dew point. Data for wet and dry bulb temperature were extracted and the maximum 12-hour persisting dewpoint (WMO 1986) noted. Data from a nearby windward climatological station were extracted for the annual maximum 12-hour persisting dewpoint for the time period 14 days either side of the storm date for at least 20 years record. This data was converted to sea level temperature psuedoadiabatically and then subjected to a frequency analysis from which the 100 year persisting dewpoint was estimated (WMO 1986).
Worldwide maximum 24-hour rainfall
It is very unlikely that the maximum rainfall will have been measured in the UK. By comparing worldwide data of 24-hour rainfall with the depth of available moisture, the relationship of the British scene to the global pattern can be described. This will serve as an additional check on the previous two approaches to estimating PMP.
Data were collected from as many sites as possible, together with published estimates of PMP. Previously these data had been related to latitude (Clark, 1997) but a more realistic explanatory variable is perceptible water.
Figure 1 (on p20) shows the distribution of the 81 sites for which data have been analysed and also the location of the 10 historic storm events. Also on p20, figure 2 shows rainfall frequency plots for two sites. Figure 3 shows the resulting estimates of 106 year rainfall which were related to the 1941-1970 average annual rainfall. The use of annual rainfall is justified as a possible explanatory variable since large and more frequent storms occur in wetter areas. It was apparent that a geographical division of the results could be made since there was a large scatter in the results. Since the UK stretches through nearly nine degrees of latitude it was considered that the more southerly areas would produce the higher estimates of extreme rainfall and this indeed proved to be the case. There is a clear latitudinal decline of extreme rainfall from south to north, a fact which is related to higher levels of solar radiation, dewpoint and therefore perceptible water. For a site with 1000mm average annual rainfall (AAR) the estimate of PMP varies from 475mm to 325mm, while for sites with 2000mm AAR the values range from 625mm to 400mm. It may be questioned as to whether or not the lines of best fit should be parallel. There are few sites in southern England which have over 2000mm rainfall. Therefore the results of the storm maximisation procedure are plotted onto the same diagram and shown in square boxes.
Historic storm maximisation
Table 1 gives the details of the storm maximisation. Taking the northern region, the three maximised storm rainfall agree well with the line of best fit. The central region only includes the Seathwaite storm, although the Norfolk 1912 storm is just outside the southern border of this region and therefore agrees fairly well with the frequency analysis. Results for the southern region which has the most severe storms suggest that the line of best fit is too high at the wetter end of the sample range, although the difference is only about 60 mm at an AAR of 2000mm. Storm number 8 was the Lynmouth event of 1952 and the initial storm depth of 267mm chosen was slightly more than the bucket catch of 254mm at Simonsbath but less than the more likely value of 280 mm. The lowering of the slope of the line of best fit means that the Martinstown storm falls above the line but it must be remembered that the peak rainfall measurement (Twort, Law & Crowley, 1985) was an unofficial value near the Hardy Monument and that the aerial extent of this value is unknown, although a depth covering 10km2 of about 330mm would reduce the maximised storm depth to 479 mm. Fortunately this result is supported by evidence from the Bruton storm. As a result
Figure 4 has been produced which shows that the trend of the data can be described by three parallel lines labelled according to latitude. This is more realistic since the results based solely on frequency analyses suggest that at very low latitudes the northern sites might receive higher depths of extreme rainfall than sites in the south. Values of PMP can now be estimated from a knowledge of AAR and latitude for any place in the UK:
R = 0.0982 AAR + 399.7 – [( 20.625 LAT) + ( 0.00552 AAR-500)] Eq 1
where R = PMP estimate (mm)
AAR = average annual rainfall (mm)
LAT = latitude – 50
Rainfall at higher frequencies, eg 10,000-year return period can also be calculated. This can be achieved by estimating the regression coefficients of the frequency plots. The relationship between AAR and the regression constant, can be described by the following equation:
Y = 0.0003102 AAR + 0.4561 Eq 2
r = 0.93
There was no distinction of regression constant based on latitude, and with such a high correlation this relationship can be used to estimate the constant. Figure 5 shows how the slope of the regression line is related to both AAR and latitude. Values of the regression slope can be calculated using the following relationship:
SR =[ -0.000020811 AAR + 0.18737] – LAT-50 [ -0.00000028 (AAR-500) + 0.0022]
Eq 3
where SR = regression slope
AAR = average annual rainfall (mm)
LAT = latitude
Results for worldwide 24-hour rainfall
Figure 6 shows the relationship between world maximum observed values of 24-hour rainfall as related to average perceptible water. Maximum observed rainfall rises rapidly with the perceptible water. It is also apparent that the estimates of PMP are similar to what the observed data would suggest. The storms in France and Norway are of relevance to Britain because they are in agreement with typical values from south to north of 550-450 mm. There will be some variation on this simple pattern according to orographic enhancement, which is associated with higher levels of AAR.
These results show that the three methods of estimating PMP are in broad agreement. Greater weight has been given to the storm maximisation procedure but the initial pattern of the relationship between PMP, AAR and latitude was established by the more extensive frequency analysis, although it was not able to exactly match the historic storm data.
PMP map
Figure 2 or equation 1 enables the PMP to be mapped for the whole of Britain. In practice the map produced (Figure 7 below) is generalised especially in areas where the AAR varies rapidly such as in Scotland. This is not a drawback because the storm events are bigger than the distance between areas of high rainfall; in other words the storms will overlap places which are close but which have markedly different AAR.
The map shows both the effect of relief and hence high AAR, and latitude. The highest values are for north Wales, parts of south Wales, and southwest England. For southwest England these results can be compared with those of Clark (1995) who showed that the drier areas had a PMP in excess of 500mm, while wetter areas such as Dartmoor the value was about 450mm. The present results show the same spatial pattern but the differences between wet and dry sites have been reversed, with the wetter areas having the high values, although the general depths are very similar.
For the Peak District of Derbyshire the results of Collier & Hardaker (1995) extrapolated to 24 hours give a value of 500mm. This compares with a value of 475mm shown on Figure 7. Further north, the Lake District and the Southern Uplands of Scotland have PMP values of over 500mm and 475mm respectively. For many parts of the Grampians and northwest Highlands, PMP is about 400 mm. Along the east coast of Scotland and England the values are lower, but still reach 325mm in northeast Scotland and over 400mm in parts of East Anglia.
In comparison with the FSR PMP, these results are considerably higher, with a ratio of 1.67 in SW England and 1.71 in Snowdon, where the highest FEH/FSR ratio (Babtie, 2000) was reported. In the remainder of England the present results are about 1.5 times greater than the FSR.
Comparison with the FEH
There are several important points which need highlighting in any assessment of the results of rainfall frequency and PMP estimation.
Firstly, PMP estimates should merge with T year rainfall estimates. In the course of the present study the author prepared a map showing the return period of FSR PMP as assessed using the FEH. The result (not shown here) is startling. Return periods vary from a low of 247 years in northwest Scotland to a high of 98,000 years in the Southern Uplands. Furthermore, there are very steep gradients of the return periods such as from 5000-80,000 over a distance of 40km on a windward slope, where the average annual rainfall only changes by about 100mm. Similarly, while the FEH gives a return period of 9000 years for 300mm in Hampshire, in parts of Somerset it rises to over 33,000 years even though the latter area has witnessed some of the most severe storms ever recorded.
The corollary of such unusual or at least unexpected estimates of rarity are reflected in a map of 24-hour 10,000 year rainfall as estimated by the FEH (Figure 8 below). This map is based on a minimum of 16 determinations of extreme rainfall in each 100 x 100km grid square. Although there is a gradual variation in the depth of rainfall in central and eastern England, there are large and unexpected changes in parts of western Scotland – as much as 240mm in less than 40km.
At the same time values in mid Wales seem lower than expected for such a wet area, while the result of 505mm in the northern Highlands of Scotland as compared with 337mm on a nearby windward slope seems hard to accept.
Secondly, there are methodological reasons why the estimates of rainfall frequency using the FEH FORGEX method are too low, at least below a return period of 200 years. The drawback of the station year method, as was used in the FSR (NERC, 1975), was highlighted long ago by Clarke-Hafstead (1942), and although the FORGE (Dales & Reed, 1989) later FORGEX (FEH, 1999) method was intended to calculate the effective number of independent rain gauges. However, there have been several notable storm events which have affected places over 100km apart (1768, 1968, 1969) and which would effectively reduce the number of independent rain gauge sites.
In the FORGE/FORGEX method the effective area which is spanned by the rain gauge network can sometimes include significant areas of sea in which there are no rain gauges: this leads to an overestimation of the effective area and hence to an overestimation of the effective rain gauge record and therefore return periods of storm events.
Perhaps the most telling reason is in the use of the horizontal distance between two or more frequency plots as a measure of station independence (Reed & Stewart, 1994). Consider the following two sets of annual maximum rainfall data (Table 2). It is clear that the annual maxima differ by an average of 12mm and the annual maxima differ quite markedly in some years, but when the data are plotted onto frequency graph paper the two stations are seen to be identical or totally dependent. Thus to use such a crude measure of independence is unjustified.
Thirdly, the return periods of events using the FORGEX method and the expected or median return periods are sometimes considerably different.
Figure 9 shows the results for six widely spaced sites in southwest England, and also shows the results obtained using the present method. These sites were not close to the centre of any of the historic storms which have taken place in this region.
Fourth, the discrepancy between the number of events from an annual maximum series and the recorded number is shown for two sites (Table 3). At Dale Head Hall the frequency of an event with 90 mm is 1.8 times greater than that suggested by the FEH. At Kindrogan there are similar discrepancies. In general the differences are greatest for wetter sites because there is often more than one large rainfall event in each year. The FEH only uses the biggest event whereas the present study has included all such events. The advice given in the FEH is that local data should not be used to modify the results of a rainfall frequency estimate. This was the advice also given in the FSR (NERC, 1975) and later disregarded by ICE (1996) for Somerset.
The results of the present study would indicate that the relevant issue is not whether local data should be ignored, but that sites with at least 25 years data must be analysed taking peaks over a threshold and that the results over a sample of sites averaged in order to obtain meaningful results. The FEH has not only ignored all but the annual maxima of the rainfall record, but has also taken about half of its sample from stations with less than 20 years record, with the further drawback that many of these cover the same time period. By itself this can only give some bias to the results. Rainfall frequency curves show considerable variation because rainfall varies greatly in time and space. To accept a locally high or low frequency curve might lead to an over or under design standard.
The ICE (1996) guide to floods and reservoir safety recommended that for category A dams where a dam-break incident would lead to a loss of life, the spillway should be capable of passing the PMF. For category B dams this standard is relaxed to the 1 in 10,000 year flood. Oddly enough, the most recent advice for risk management for reservoirs (Hughes et al 2000), does not discuss any of the previous concerns relating to rainfall frequency and dam safety. Whilst no structure can ever be 100% safe given that circumstances might arise in the future which could not be foreseen, it behoves the engineering profession to set safety standards in such a way that the general public can accept them.
A recent paper (anon, 2000) which questions the benefits of modifying dams to safely pass the PMF, makes the point that dam professionals do not want to be associated with dam failure which results in death. If this is the case then the risk of serious overtopping as a result of adopting the present PMP estimates could be quoted as one in one million.
If a lower design standard is adopted then this should be made known to the public at large. Higher standards of living have resulted in the expectation of higher safety standards.
If in the future there is an increase in temperature, then the depth of perceptible water in the atmosphere will also increase. In the summer a change of 0.5oC from 20.5-21.0oC will result in an increase of 2.3mm precipitable water. When the storm efficiency is in the region of 10-15 then this increased water content will have a significant impact on storm depth. During the winter, with lower temperatures, the same temperature change will result in a smaller increase in storm depth.
The overall implications for dam safety cannot be stressed enough. In one study the resulting PMF was more than double the design value (Clark, 1997) thereby putting the dam at serious risk from erosion and consequent major flood.
Future work
In recent years there has been an unwillingness to discuss the discrepancy in the estimates of extreme rainfall. The mismatch of the FEH and FSR estimates has brought that discussion more into the open.
There are two weaknesses of the rain gauge network which have received little if any mention in the past. The first is that if the 3000 or so rain gauges were placed side by side they would cover an area less than the size of a football field: much of the area of Britain has no gauge at all and it is unknown how representative the results from each gauge are of the area close by. Second, and more critical is that upland areas have very few if any gauges present, most are situated in valley bottoms where the rainfall tends to be much lower than the nearby hills. These two facts suggest that the heaviest falls in Britain have been missed and the aerial extent of many storms underestimated. Clearly more upland sites are needed.
All the evidence presented in this paper suggests that FSR PMP estimates are too low. The PMP map presented here is the first revision for the whole of Britain and it has been largely based on the analysis of all storms over the threshold of 15 mm. Although the daily data are not ideal the correspondence between the results of this data and other independent data are too close to be coincidence.
The present study has restricted itself to one day data converted to 24-hour rainfall using a factor of 1.13. There is a need to extend the study to a duration of 2 and 3 days. More historic storms should be studied. One problem with historic events is that the maximisation process is limited by the rain gauge data available. Once radar data can give accurate and detailed results at extreme rainfall intensities then the picture will become clearer.
Another aspect of extreme rainfall is the choice of storm profile, which has not been considered here. The FSR has divided storms into those with a summer or winter profile and this has been maintained in the FEH (1999). In the present study all storms north of East Anglia took place in winter, but it is possible, as at Louth in May 1920 that a serious storm might take place in the summer.
Although our understanding and knowledge of extreme rainfall has improved in recent years, there is still room for improvement. However, the present picture shows that estimates of PMF at dam sites in Britain need to be revised and the implications of those new estimates acted upon.
TablesTable 1 – Details of the ten storms used in the study Table 2…Annual maximum rainfall (mm) from two sites Table 3. Frequency of events… from annual maximum (AM), peaks over a threshold (POT), and estimated number of events
