As a result of floods in the UK during the 1990’s and especially during the autumn of 2000 (Marsh & Dale, 2002), when the insurance industry paid out £1bn in claims, the need for flood risk assessment became acute. The water Resources Act 1991 instigated Section 105 surveys of areas at risk of flooding while PPG25 (DETR, 2001) formalised the risk assessment procedure. At the same time the Institute of Civil Engineers produced their final report on flood risk in England and Wales (Fleming, 2001). It stated ‘Historic floods produce the best flood plain records (level and extent) and are extremely valuable in indicating flood risk areas, assuming the actual discharge and flow frequency are known…There is a pressing need to update flood risk maps and to improve their accuracy.’ The Association of British Insurers (http://www.abi.org.uk/flooding) has produced its own review of the important issues regarding flood insurance, stating that ‘the minimum level of protection which would enable insurers to offer cover at normal terms for residential purposes, is at least a 0.5% probability (1 in 200 year return period) up to the year 2050, after taking climate change into account.’ (ABI, 2003). This is somewhat at odds with the view that properties with a moderate risk of flooding, that is at a risk of between 1 in 75 to 1 in 200, would still be insured after September 2007.

Recently the use of contoured surveys of the floodplain as tools to give a flood risk assessment (Thompson & Clayton, 2002) has been described. This approach would augment results produced by the standard Flood Estimation Handbook (IOH, 1999), but does not add to our knowledge of flood frequency. The Environment Agency (www.environment-agency.gov.uk) provides maps to show areas at risk of flooding at the 1 in 1000 or higher frequency, giving a threefold classification of slight 1 in 200 or less; moderate 1 in 75-1 in 200; and significant; greater than 1 in 75. These criteria belie an ignorance of the past that is not easy to unravel. There are six main reasons for this. First, there is a lack of detailed long term flood records in many places. Second, in some places there are no records at all, or what information existed has since been lost. Third, current methods of flood risk assessment have not taken advantage of the rich flood history lying buried in the National Archives, which could have been incorporated into FEH methodology and then applied to areas where no such data exist. Fourth, many small river basins do not have a flood plain, and the rapid change in topography makes flood risk mapping all the more difficult. Fifth, many flow records do not cover periods of more serious floods during the late 19th and early 20th century. Finally, a lack of knowledge of past floods has led to explanations of the occurrence of more recent floods as a result of climate change. However, we do not yet know the full range of variability under present conditions, and how this may have changed over a long enough time scale to make meaningful statements of flood risk at low probabilities.

**The value of the historic flood record**

The use of past flood events to make a realistic assessment of the flood risk is nicely illustrated by the case study of the village of Langtoft in East Yorkshire (Figure 1). It is the purpose of this paper to not only describe the evidence for previous floods at Langtoft but also to explain a new methodology of flood risk assessment at this site where there is limited data regarding flood history. The structure of the paper is as follows. First, the results of current methods of flood risk assessment at Langtoft will be briefly described. Second, the historic evidence of floods is presented. Third, the analysis of both field data and flow modelling are then used to produce estimates of the peak discharge of two historic events at Langtoft. Fourth, a new method of estimating the rarity of the flood events is described. Fifth, this method is applied to two rivers with contrasting bedrock: the river Lud which is a permeable catchment, and the river Brue a largely clay catchment. Sixth, the method is then applied to Langtoft and a flood frequency curve produced.

**Existing flood risk assessments at Langtoft, East Yorksire, UK**

The village of Langtoft in East Yorkshire has a catchment area of 6.6km^{2} at West Gate. This increases to 9.4 km^{2} in the village centre. The area forms part of the Yorkshire Wolds, with elevations ranging from 75m up to 163m. Slopes are moderate to steep. The underlying geology is chalk with some superficial valley deposits below Langtoft. The land use in 2005 was a mixture of arable land and chalk grassland. There is no surface drainage channel both in and above the village. Therefore no assessment of bankfull discharge is possible which can give important indications of flood frequency. In July 2005 Langtoft was shown on the Environment Agency website as having a narrow band of land at risk of flooding at a risk of 1 in 200 or greater, running from outside the western side of the village, through the village and downstream towards Kilham. In March 2006 this band was shown reduced in length and starting from the village centre, thereby excluding land to the west. The depth of flooding is not stated. Areas outside the flood risk zone are deemed to have a risk of flooding around 1 in 1000. It is not clear why there has been a reduction in estimated flood risk at Langtoft. Estimates of the flood risk at the west end of the village once known as West Gate, as given by the Flood Estimation Handbook (IOH, 1999) and ReFH (Kjeldsen et al 2005) are shown in Table 1.

The ReFH has not been calibrated for floods greater than 1 in 150 years. Overall these results suggest a minor risk of flooding and are certainly not suggestive of serious floods except at the probable maximum flood (PMF). In reality nothing could be further from the truth.

**The historic flood record at Langtoft**

There are three plaques at Langtoft which tell us that significant and damaging floods have taken place in the past. Figures 2 and 3 show two of these neither of which were erected soon after the events and therefore do not give a reliable estimate of the flood depth of each event. The best accounts of the floods come from Hood (1892) and the contemporary newspaper reports published in the East Riding Chronicle and the Driffield Times.

* 1657 – 10 April Flood marker set in Henry Stork’s house about 2.1m above ground level. The flood marker was discovered after the 1892 flood. No other details found.

* 1853 – May Thunderstorm; three horses killed by lightning, no flooding noted. (Hood, 1892).

* 1888 – 9 June Funnel cloud, accompanied by very heavy rainfall, torrent 40 feet (12m) wide flooded the village to a depth of 3-4 feet (0.9-1.2m). Hundreds of tons of soil, gravel and boulders carried with the flood. (Hood, 1892, East Riding Chronicle & Driffield Express, 16/6/1888,

* 1892 – 3 July Early evening thunderstorm leading to destruction of buildings. Two houses in Back Street were destroyed. Water 7 feet (2.1m) at West Gate and equaling the level in 1657 at Back Street where Henry Stork lived. Flood relief fund amounted to £1340. Driffield Times 9/7/1892, Hood (1892).

* 1910 – 20 May Great storm centered to the west of Langtoft causing much damage at Driffield 10km to the south. At Langtoft several houses and gardens were flooded causing considerable loss. Driffied Times 28/5/1910.

In spite of at least two summer storms in excess of 60mm taking place in 1948 and 1951 there does not appear to have been any flooding at Langtoft since 1910. Although accounts of the 1888 and 1892 events have been published (Hood, 1892, Lovel, 1893, Johnson, 1982, McEwan, 1991, and Wright, 2000), there have been no estimates of the rainfall, peak discharge or flood rarity. Although daily rainfall was measured at nearby Warter and Driffield in 1888 and 1892, no measurements were made in the catchment above Langtoft. Therefore any attempt to reconstruct these events must rely on a knowledge of: Soil permeability; Contemporary land use; Estimated duration and depth of the storm; Timing of the flood; Estimated depth of flooding; and Antecedent soil moisture deficit (SMD)

The duration and depth of the 1888 and 1892 floods were obtained from the description of these events in Hood (1892). The duration of the storm and the timing of the floods were also obtained from Hood (1892) and the contemporary newspaper reports. Contemporary land use was estimated from the Agricultural Statistics (PRO) for 1891, there being no survey for 1892, and the 1st Land Utilisation Survey in 1930. The catchment land use from the latter source was adjusted from the Parish data in the former source to give, for 1891: Arable 75%. Grassland 24%. Woodland 0.7%.

Soil permeability was measured directly in the field using the core method (Hollis & Woods, 1989). A total of 58 samples were measured on both arable and grassland in the same proportion as the land use. Figure 4 shows the results plotted as a histogram while Figure 5 shows the effect of rainfall intensity on percentage runoff via:-

% runoff = £ [ % 015 * R – (0 + 15)/2] + [ % 15-30 * R – (15 +30)/2] + …..

Where 0-15, 15-30 = percentage of soils with the given value of saturated hydraulic conductivity mm hr-1. R = rainfall intensity mm hr-1.

On arable land surface cracks were noted and an attempt was made to measure the hydraulic conductivity across the cracks since they could significantly affect the rate of surface runoff. It was found that below a depth of 6cm there was no difference in hydraulic conductivity between cracked and non-cracked soil. This was also checked in the field on Salisbury Plain where the same soil association (Andover 1) and land use exists.

The antecedent soil moisture deficit was estimated from empirical relationships between monthly mean temperature and open pan evaporation for the period 1986-2005, and the daily rainfall record measured at Driffield and Warter. The results showed that the SMD prior to the 1888 and 1892 events was 19mm and 7mm respectively.

**Estimation of the peak discharge of the 1888 and 1892 floods at Langtoft**

There is enough detail in the accounts of these two floods to enable a realistic estimate of the peak discharge to be made. In 1888 the highway at West Gate was about 7.5m wide. From a field survey the gradient of the valley at West Gate was 0.0158. The flood depth is stated as being 0.98-1.2m. Adopting a roughness value of 0.06 gives estimates of the peak discharge in the range 11.6-17.3m^{3}/sec. It is possible that some flood water may have passed around the northern side of the cottages at West Gate but it would have been very limited since there is an elevated embankment between Church Street and the village pond.

In 1892 the flood depth at West Gate was 2.1m (Hood, 1892). Making the same assumptions as for the 1888 event gives a peak discharge of 40.2m^{3}/sec, a value that may be reduced to 34m^{3}/sec if there was a slight overestimation of the flood depth of 0.15m. Although there is always some uncertainty in these estimates it is clear from all the descriptions of this flood that it was far worse than in 1888. The account of Hood (1892) also allows an estimate of the flood at Back Street to be made. Here the catchment area is 9.4km^{2}. During the flood two of the cottages were destroyed while the last one of the row in which Henry and Mary Stork lived was badly damaged (Figure 6). This picture also shows the damage done to the outbuilding behind their cottage, proving that flood water passed between the two properties as well as along the main street. Reports of debris being brought down by the flood suggest that the roughness coefficient was about the same as at West Gate. The combined discharge along these two pathways was estimated as 62m^{3}/sec. Using the Flood Estimation Handbook a value of 34m^{3}/sec for the PMF at this site is obtained. Since the flood of 1892 was probably equalled in 1657 then the FEH estimate of the PMF is not only far too low, but even higher peak discharges are possible.

*Estimation of the rarity of the 1888 and 1892 floods*

The rarity of a flood depends on the combined effects of antecedent conditions, rainfall, and runoff processes that are influenced by land use. In standard flood frequency analysis (Stedinger, Vogel, & Foufoula-Georgiou, 1993; IOH, 1999) the flood record consists of either measured flows and or estimates of historic floods (Harrison, 1961; Williams & Archer, 2002; Payrastre, Gaume, & Andrew, 2005). The floods are ranked in order of magnitude and each given a return period. The results are then plotted onto extreme probability paper. Any additional flood for which a discharge has been assigned can be compared with the flood frequency curve in order to give it a rarity. In the absence of any measured or historic flood data a flood frequency curve can be produced using estimation methods as described in the Flood Estimation Handbook (IOH, 1999) or using rainfall records (Rahman, Weinmann, Hoang, & Laurenson, 2002; Calver, Lamb & Morris, 1999). Unfortunately this approach needs to be checked against rare floods about which very little is known. Recently the author (Clark, 2006) has proposed an SMD transfer function related to the ratio of the return period of the storm and flood. However, it cannot be applied in the case of a very sparse historic record. Therefore an alternative approach is called for.

**New approach to the estimation of flood rarity**

Floods occur according to the joint probability of rainfall depth and intensity, and soil moisture deficit. All else being equal, a storm that takes place on a dry catchment will produce a more moderate flood than if the same storm took place over a catchment at field capacity. During winter (October-March) when the soil is at field capacity, the return period of the flood is equal to the return period of the storm as shown by annual maximum data for October -March. During the summer months when there can be an appreciable SMD, it is necessary to show the dependence, if any, of rainfall depth on SMD. Should no dependence be established, then the probability (P) of the flood event will be the product of the probability of the effective rainfall (rainfall – SMD) and SMD. Flood rarity can also be expressed in terms of the return period of the same variables, where the return period = 1/P.

Fr_{p} = ERr_{p} * SMDr_{p} Where Fr_{p} = return period of the flood, ERr_{p} = return period of effective rainfall, (ER = rainfall – SMD), SMD_{rp} = return period of SMD.

*Effective rainfall*

Figure 7 shows the relationship between daily rainfall above 15mm and SMD as measured using weighing lysimeters at Charldon Hill Research Station (CHRS) in the UK. The absence of even a weak correlation shows that higher rainfall is not associated with either high or low SMD. Neither is the relationship curvilinear, which can also result in a low correlation. Since the highest daily rainfall in the sample was only 48mm, data from historic storms were drawn from the archives together with an estimate of the antecedent SMD. Table 2 shows the results.

When this data is added to the data presented in Figure 7 the correlation coefficient = 0.09 which is not significant at the 10% level. Therefore the joint probability model of effective rainfall and SMD can be applied to summer events.

The model requires an estimate of the flood producing rainfall and antecedent SMD. The estimate of rainfall is obtained by back analysis using the non-linear flow model of Clark (2004) which is briefly described below. The majority of the model follows that for the Upper Brue in East Somerset but using a local time to peak adjustment for Langtoft, while the delayed flow was designed for the Lud in Lincolnshire which is a chalk catchment as is the case at Langtoft.

Catchment area 6.6km^{2}

Slope = 5.6°

Time step = 0.166 hr

Quickflow is determined by the unit hydrograph ordinates corrected for the ratio of the sine of catchment slope, and net rainfall as used in the convolution of the unit hydrograph.

Rising limb ordinates: {[ INVLOG [ 2 (t – 0.7Tp)]} / {1 + [ INVLOG ( 2 (t – 0.7Tp )]} * Qp

Falling limb ordinates: {[INVLOG [ t1 – 0.7( TB – Tp )]} / {1 + [ INVLOG ( t1- 0.7 (TB – Tp )]} * Qp

The ordinates are corrected using (Sine catchment slope)/( Sine Upper Brue catchment slope). For Langtoft this becomes Sin 5.6/Sin 6.5 = 0.862.

Qp = 330/ Tp * Area/1000

TB = 2.52 Tp

t1 = (TB – T)

T = time in hours

Tp = time to peak (hours)

TB = time base

Time to peak (Tp) = [(2.4073R + 10.1005)/ R1.015] -1.3 where R = rainfall mm hr-1 lasting for 0.5 hours.

Percentage runoff is obtained from Figure 5. This is based on a survey of the hydraulic conductivity of soils in the Langtoft catchment area.

Slope runoff (Clark, 2004) only occurs when rainfall intensity >/= 8 mm hr-1 on chalk catchments.

Delayed flow ordinates: based on the relationship for the river Lud at Louth in Lincolnshire UK (Clark & Vetere Arellano, 2004).

Cf * [ 0.614 * (Log R + 0.1) exp ( r/r – 0.9)]

Where R = rainfall mm 1.0 hr-1, r = rainfall mm 0.5 hr-1, Cf = catchment area Langtoft/ catchment area Lud = 6.6/52. Correction factor for lag time and time to peak for delayed flow = mainstream length Langtoft/mainstream length Lud = 3.6/11; lag time for the river Lud = 1.0 hours, time to peak for delayed flow = 2.0 hours. Rising and falling limb ordinates linear.

Total flow is the sum of quickflow and delayed flow.

The model was then applied using a range of rainfall depths in order to estimate the likely depth of rainfall that produced the flood. The result was then used in the joint probability model to estimate the return period. Should the estimate of peak discharge be in error then the new rainfall would have a correspondingly higher or lower depth, which would in turn result in a revised return period. The key parameter in converting rainfall into runoff is the saturated hydraulic conductivity of the soil. A check on the results is provided by an analysis of the water balance.

The water balance for a flood event can be written:-

Rainfall = runoff + SMD + percolation

Where all units are in mm. Figure 8 shows the modelled hydrographs for the 1888 and 1892 floods. In 1888 the duration of the flood was about two hours. This refers to the time when water entered and left dwelling houses. In 1892 the discharge was 34-40m^{3}/sec but the duration of flooding was not given so this aspect of the simulation cannot be compared with the historic description. The water balance for the two events becomes:

1888 60 = 12.5 + 19 + 27.4 This is a shortfall of discharge and/or SMD of 1.1mm

1892 90 = 29.9 + 7 + 46.5 A shortfall of discharge and/or SMD of 6.6mm

These results show that the model has not overestimated the flood volume, while the flood peaks are within 20% of the true value. For the 1888 flood this equates to 13.4-20m^{3}/sec while for the 1892 flood a range of 32-48m^{3}/sec is suggested. These results compare well with the earlier estimates of peak discharge of 11.6-17.3 and 34-40m^{3}/sec for the 1888 and 1892 floods respectively.

The effective rainfall is the total rainfall derived from the non-linear flow model less the SMD. The return period of the effective rainfall is estimated from an analysis of local rainfall records and estimates of PMP for that area. A seasonal sine wave adjustment factor for the intercept in the rainfall frequency relationship is included:

SF = {[Sin (D – 122 – 0.01369D]+1}/2

Where SF = seasonal adjustment factor

D = day of the year

This factor is applied in:-

ISADJ = SF [Is – Iw] ) + Iw

Where ISADJ = seasonally adjusted intercept; Is & Iw = intercept of the summer and winter rainfall frequency curve respectively.

In deriving the sine adjustment factor two sets of observations were taken into account. The first is the monthly change in the maximum one-day rainfall recorded in Britain, which peaks in July/August, and the average monthly Central England Temperature, which has a similar monthly distribution and also peaks at the same time as the monthly rainfall maxima.

An allowance is also made for the areal reduction in rainfall, or ARF, (Keers & Westcott, 1977), since rainfall frequency analysis uses point data from raingauges whereas a catchment covers a much bigger area which has a lower areal average rainfall.

*Soil moisture deficit*

There is only one set of detailed SMD data in the UK which have been measured for over 10 years: at CHRS UK there are two weighing lysimeters which have been in service since 1995. Data from drainage lysimeters have been published in British Rainfall until 1968, but these cannot give evaporation on a daily basis or give detailed SMD data because only part of the water balance equation is provided. The lysimeters at the Rothamsted Experimental Station (now Rothamsted Research) have always been of the drainage type.

For the CHRS data set, values of SMD at the end of April-September for the period 1996-2005 were ranked and subjected to an extreme frequency analysis from which the probability of antecedent SMD can be assessed for any date during that period. Figure 9 shows the results of this procedure for May and July. Equations for all six months are given in Table 3.

Since there is no comparable SMD data for other parts of the UK which could then be used directly to estimate the rarity of floods the CHRS data have been compared to that published in Smith & Trafford (1976). They give median values of SMD for June-September as well the date of return and end of field capacity for 66 agroclimatic areas and a range of annual average rainfall in each of those areas which cover the whole of England and Wales. CHRS is located in area 35 of Smith & Trafford (1976) and the lysimeter data is compared in Figure 10 with the data in Smith & Trafford (1976) under the same average annual rainfall at CHRS during 1996-2005. The monthly distribution of rainfall will affect the values of SMD. Table 4 shows the monthly average rainfall for Area 35 for April-September, with the same median monthly rainfall at CHRS. The median values were chosen because the average rainfall can be affected by a single large or small value in such a small sample. With the exception of September the values are very comparable. For this month the values of SMD were corrected for the difference in rainfall. The higher rainfall at CHRS in April only makes a small difference to SMD value in the summer because field capacity ends at about 25 April.

In order to correct the data in Smith & Trafford (1976) for other areas of England and Wales two corrections are necessary. The first is to apply the ratio of SMD at CHRS:Area 35, derived from Figure 10 at a given date to the value in the area of interest using local SMD values in Smith & Trafford (1976) at the same annual average rainfall as CHRS (880 mm). Second, a correction is needed for the difference in rainfall at CHRS and the study area. This is calculated from the local date of the end of field capacity to the date of the flood, accounting for differences in local rainfall as interpolated in Smith & Trafford (1976) and at CHRS. The median values of rainfall from April to September are given in Table 4.

The return period of the antecedent SMD is calculated from linear interpolation of the regression slopes of equations given in Table 3, according to the date in question. The value of median SMD is then substituted into the equation in order to calculate the regression intercept. The resulting equation allows the return period of the SMD to be estimated.

**Example of the flood of 29 May 1920 on the river Lud in Lincolnshire, and the Upper Brue, Somerset, UK **

*The river Lud*

1. SMD for 28 May = 26 mm. This is based on daily rainfall at Louth and empirical relationships between temperature and measured pan evaporation at CHRS.

2. The Lud catchment is area 17E (Smith & Trafford, 1976) and the median SMD for 29 May at an annual rainfall of 700mm and interpolated from the data = 28mm. The Correction factor for 29 May = 16/30, this being the ratio of SMD CHRS/SMD Smith & Trafford for 29 May. Hence the corrected median SMD = 14.9mm.

3. The annual average rainfall for the Louth catchment = 700mm. The median date for the end of field capacity for an annual average rainfall of 700 mm is 15 April. Therefore the correction for rainfall:

(15/30) * (CHRS_{APRIL}/ ST_{APRIL}) = (15/30) * (68 – 47).

4. For May the correction is: 29/31 (65 – 58).

5. Total corrected median SMD for 29 May = 14.9 + 10.5 + 6.5 = 31.9mm.

6. The value of 38.4mm is plotted against a return period of 2 years (y = 0.189)

7. The equation for the frequency of SMD at the end of May (Table 3) is adjusted for 29 May: slope of regression:

[(-18.027 – – 33.662)* 29/31] -33.662 = -19.035.To obtain the intercept substitute the median SMD (31.9), and the value of y at a return period of 2.0 years = 0.189 into the equation to obtain the intercept:

SMD = Regression slope * 0.189 + intercept

Which in the present case gives SMD = -19.035y + 35.497

8. The return period of a SMD = 26mm can now be obtained = y = 0.4989. Using equation 1 gives a return period of 2.4 years.

9. The rainfall of 29 May 1920 is estimated as lasting for 3 hours (Robinson, 2000, Clark & Arellano, 2004). The catchment average rainfall is estimated as being 150mm, a result obtained from the non-linear flow model and assuming a peak discharge of 150-160m^{3}/sec.

10. The effective rainfall = 150 -26 = 124mm. This would be expected to have taken place in about 2.5 hours (124/150) * 3. To obtain the return period of this rainfall annual maximum daily rainfall data for Cadwell (NGR TF 283814) for the period 1973-1999 were subjected to a frequency analysis. The result was adjusted to give an estimate of 24 hour rainfall (Faulkner, 1999) and the equation for 24 hour rainfall calculated assuming a PMP of 460 mm which is close to the value for this area of 400mm (Clark, 2002). The result was Log R = 0.1130y + 1.6018. To obtain the result for rainfall of 2.5 hour duration the 2.5 hour PMP was calculated from R = 220.075 LogD + 156.249. This result is obtained from a regression analysis of 24 hour PMP = 460mm, and 0.5 hour PMP = 90mm.

11. Therefore the 2.5 hour all year rainfall frequency equation: Log R = 0.1130y + 1.2458. Using the 2-year 2.5 hour winter rainfall depth = 11.7mm (Kjeldsen et al 2005), the 2.5 hour winter rainfall equation: LogR = 0.1130y + 1.0468.

12. The seasonal sine wave correction factor for 29 May = 0.710. Therefore the 2.5 hour rainfall frequency equation becomes Log R = 0.1130y + 1.18809. Apply an areal reduction factor (Keers & Wescott, 1977) or ARF = 0.88 for a catchment area 52 km^{2} and storm duration of 2.5 hours to give a point rainfall of 141mm. Therefore an areal depth of 141 mm in 2.5 hours has a return period of 81580 years.

13. The rarity of the 1920 Louth flood then becomes = return period of rainfall * return period of SMD = 81580 * 2.4 = 196440 years.

By itself this result clearly contains a great deal of uncertainty. Therefore, to place it into the wider picture of flood frequency, the flow model was used to estimate floods of much lower return periods. Generally speaking most floods on chalk streams take place during the winter. The procedure for winter floods includes the use of a winter rainfall frequency analysis, and assuming an SMD = 0. The return period of the flood is then equal to the return period of the rainfall. This approach was applied to the river Lud to give estimates of floods with return periods of 200, 50, 10, and two years. The results are shown in Figure 11, which also has the measured flows at Louth for 1967-1993. There is a good comparison for flows up to the five year event but beyond this the measured flood flows are much lower than the model based result suggests. However, it is widely recognised that the peak discharge of the 1920 flood was about 150m^{3}/sec (Crosthwaite, 1921), or even higher, (Clark & Vetere Arellano, 2004). Since the rainfall for that event was below the PMP then higher peak discharges are possible. Furthermore, the regression line calculated using measured data for the Lud below a return period of two years gives a good comparison with the higher and more rare events at this site given by the flow model. It would seem that in the recent past the measured flood flows with moderate return periods have been much less than what the long term modelled flows would suggest. Once again the analysis of the historic flood of May 1920 at Louth places the measured flows into a more clearly defined context. Furthermore, the FEH summer PMF is estimated at 133m3/sec, while floods with a return period of 150 years or less are also underestimated using the ReFH. This latter result is obtained in spite of using a time to peak of three hours instead of the FEH PMF time to peak of 4.6 hours.

*The Upper Brue*

The new method described above was used on the upper Brue in East Somerset and the results compared with those of the well documented flood history at the same site. Table 5 shows the results while Figure 12 gives a comparison with the flood frequency curve based on a 235 year record (Clark, 2003). Whilst there is a slightly lower estimate of floods with a higher frequency than 1 in 50, the overall comparison is good and well within the level of uncertainty for this type of flood frequency analysis. It also shows that the estimation of return periods of rare floods using standard methods can give somewhat lower estimates than the present results suggest.

**Results for Langtoft floods of 1888 and 1892**

From the description of the 1888 event the rainfall lasted about 1 hour, the flood waters rose rapidly and then receded just as quickly. The peak discharge was estimated as about 15m^{3}/sec. Using the rainfall record at Driffield the antecedent SMD was estimated as 19mm. Applying a rainfall of 60mm in one hour gives the following unit hydrograph parameters:

Time to peak = 1.122 hours

Peak ordinate = 1.941 m3 s-1 mm-1 net rainfall

Time base = 2.52 * 1.122 = 2.827 hours

0.7 time to peak = 0.785

0.7(TB – Tp) = 1.193

Peak delayed flow ordinate = 0.139

Lag time and time to peak of delayed flow = Lag time and time to peak for River Lud * mainstream length at Langtoft, (km)/ (mainstream length of Lud at Louth, km) = 3.6/11.

Applying the non linear flow model gives a peak discharge of 16.7m^{3}/sec which is within the range 11.6-17.3.

*Estimation of the return period of the 1888 flood with a peak discharge of 16.7m ^{3}/sec*

Effective rainfall = storm rainfall – SMD, = 60 – 19 = 41mm in 41/60 hours. 24 rainfall frequency for Langtoft based on all year local rainfall data:

Log R = 0.12187y + 1.5332.

This assumes a 24 hour PMP of 475 mm. To obtain the 0.68 hour PMP use the equation R = 230.399Log D + 157.000 where D = storm duration in hours and R = PMP (mm). This result is based on a 24 hour PMP = 475mm and a 1 hour PMP of 157mm. Therefore 0.68 hour PMP = 118mm.

To obtain the intercept of the 0.68 hour rainfall frequency curve substitute into equation 2 using a value of y = 9.382 which has a return period of 106 years:-

Log 119.8 = 0.12187 * 9.382 + intercept.

Therefore the intercept = 0.9284. Equation for 0.68 hour rainfall:

Log R = 0.12187y + 0.9284.

The winter rainfall frequency using 0.68-hour 2-year rainfall (Kjeldsen et al 2005) and same slope of the frequency curve:

Log R = 0.12187y + 0.7551

Seasonal correction factor for 9 June = 0.796

Therefore 9 June 0.68 hour rainfall: Log R = 0.12187y + 0.8930

ARF = 0.91

Areal effective rainfall = 41/0.91 = 45mm

Return period of 45mm in 0.68 hours = 1095 years

Estimate the return period of SMD = 19mm. Correct median SMD of 36mm in area 13 (Langtoft) with an annual average rainfall of 736mm using the correction factor for 9 June: CHRSSMD/Area 35SMD = 22/36 = 22mm. Correct for differences in rainfall at CHRS and Langtoft. Average annual rainfall of Langtoft = 736mm. From Smith & Trafford (1976) the median date of the return to field capacity, by interpolation = 22 April. Therefore the correction for rainfall = 8/30 (68-48) + (65-58) + 9/30 (55-55) = 12.3. Total correction = 22 + 12.3 = 34.3 mm.

Calculate the SMD frequency equation for 9 June by interpolation from equations in Table 3. Slope of regression = [(-29.327 – – 18.027) * 8/30] – 18.027 = -21.417. Intercept: substitute median SMD and modified reduced variate = 0.189 for a return period of 2.0 years in the equation SMD = -21.417y + intercept. SMD = -21.417 * 0.189 + intercept. Intercept = 38.347. Therefore SMD = -21.417y + 38.347.

Return period of 19mm on 9 June = 3.2 years.

Return period of flood = return period of SMD * return period of effective rainfall

= 3.2 * 1095

= 3504 years

*The flood of 1892 at Langtoft*

The estimated peak flow for the 1892 flood was 34-40m3/sec. From the description of the event given by Hood (1892) the storm lasted for about 1 hour. The rate of rise was more rapid than in the flood of 1888. Using daily rainfall at Driffield the antecedent SMD was estimated as 7mm. The non linear flow model was used in a back analysis of the likely rainfall needed to produce the flood. A storm of 90mm in 1 hour produced a flood with a peak discharge of 39.4m3/sec.

Time to peak 1.055 hours

Peak unit hydrograph ordinate = 2.064 m3 s-1 mm-1 net rainfall

0.7 time to peak = 0.738 hours

Time base = 2.52 * 1.055 = 2.658

0.7(TB – Tp) = 1.122

Peak delayed flow ordinate = 0.152

Lag time and time to peak of delayed flow were taken as the same as the 1888 flood

*Estimation of the return period of the 1892 flood*

The effective rainfall = rainfall – SMD = 90 -7 = 83mm in 83/90 hours. The 0.92 hour PMP is calculated from: R = 230.399 Log D + 157.0. Therefore 0.92 PMP = 150.7mm. Calculate the 0.92 rainfall frequency equation by substitution of 0.92 hour PMP and y = 9.382 into the equation Log R = 0.12187y + intercept, give Log R + 0.12187y + 1.0298.

ARF = 0.92. Therefore 83/0.92 = 90mm.

Winter 0.92-hour 2-year rainfall (Kjeldsen et al 2005) = 7.0mm.

Therefore 0.92 hour frequency equation: Log R = 0.12187y + 0.8030

Seasonal correction factor for 3 July = 0.932

Therefore 0.92-hour rainfall frequency for 3 July: Log R = 0.12187y + 1.01566

90mm in 0.92 hours over 6.6 km^{2} has a return period = 13868 years

Estimate the median SMD for 3 July: CHRSSMD/Area 35SMD for 3 July = 50/66. Median SMD for the Langtoft area 3 July for annual rainfall of 736mm = 61mm (from interpolation in Smith & Trafford, 1976). Therefore corrected SMD = 61 * 41/66 = 38mm.

Correction for differences in rainfall. Date of the end of field capacity for annual average rainfall of 736 mm = 22 April. Therefore correction: 8/30 (68-48) + (65-58) + (55-55) + 3/31 (66-67) = 12.2. Therefore median SMD corrected for differences in evaporation and rainfall = 38 + 12.2 = 50.2mm

SMD equation for 3 July: slope of regression line = [3/31 (-26.528 – – 29.327)] -26.528 = -29.056 The regression intercept for the equation: SMD = -29.056y + intercept, where SMD = 50.2 and y = 0.189 (the value for return period = 2.0 years). Therefore the intercept = 56.691. Thus SMD = -25.649y + 56.691. The return period of SMD = 7mm then becomes = 7.4 years.

Return period of flood = return period of SMD * return period of effective rainfall

= 7.4 * 13868

= 102623 years

*Flood frequency curve for Langtoft at West Gate*

Table 5 shows the results of applying the above procedure for a range of conditions both in winter and summer. Values of return period and peak discharge can now be combined to produce a flood frequency curve (Figure 13).

In calculating the return period of rainfall the ARF is taken into account. The result for 133mm in one hour with the soil at field capacity clearly has a rarity in excess of 106 years, which is the accepted rarity of the PMF. Thus the PMF at West Gate is about 50m3/sec, which is more than double the value obtained using the FEH. The results show that if there had been a significant error in the estimated peak flow of either the 1888 or 1892 flood then there would have been a corresponding higher or lower return period associated with the new peak discharge. The flood of 1892 could not have been predicted by the FEH, even allowing for a significant overestimation of its magnitude in this paper. Furthermore, higher rainfall depths would cause an even greater flood. The fact that the flood of 1657 was of similar magnitude to that in 1892 also suggests that neither of these events were equal, though not far short of the PMF as assessed in this paper.

**Discussion**

The aim of this paper has been to show the value of historic floods in estimating flood risk at a site in East Yorkshire. A new method has been described which allows the rarity of events to be estimated. A central problem with the assessment of rare events is the uncertainty of the results. At the same time the floods of 1657 and 1892 are proof that serious and life threatening floods have occurred in the past in a small chalk catchment. The method has produced realistic results when compared with results from a measured river flow gauging station at Louth, as well as the historic record at Langtoft and Bruton which are on contrasting bedrock. This lends support to the belief that the basic approach is valid. The main reason why the FEH methodology does not produce such a high flood risk at Langtoft is in the way in which rainfall is converted into runoff. In the present method local soils data have been used to estimate percentage runoff from rainfall intensity data. A central problem with permeable catchments is the treatment of delayed flow. A separate delayed flow model has been included for permeable catchments which was calibrated for the river Lud at Louth. Further work needs to be carried out to validate the model in other areas. As the database of saturated hydraulic conductivity measurements increases in size, it should be possible to extend the method to areas that have the same soil associations as those in areas that have been surveyed in the field. In this way the model may become more widely applicable such as in areas which have no documented historic floods.

The chief finding of the present study is that there are more properties at risk from fluvial flooding than has previously been believed. Whether this applies to chalk catchments in general is as yet unknown, but application of the method elsewhere should prove instructive. Whilst insurance cover and the implementation of flood alleviation schemes help to reduce the risk of flooding, a more realistic assessment of the hazard may call both these methods of adjustment to the flood hazard into question. This should also lead to higher margins of safety when new housing and dam spillways are planned or designed. Blaming climate change on recent floods only serves to highlight our ignorance of the past when it can be shown that even worse floods have occurred at a time when greenhouse gases were much lower than at present.

Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Figure 1: Location of the catchment at Langtoft, East Yorkshire UK Figure 1 Figure 2: Commemoration plaque of the 1657 and 1892 floods Figure 2 Figure 3: Flood marker for the 1892 flood Figure 3 Figure 4: Frequency distribution of saturated hydraulic conductivity for soils in the Langtoft catchment area Figure 4 Figure 5: Relationship of rainfall intensity and % runoff under saturated conditions Figure 5 Figure 6: Flood damage caused by the 1892 flood Figure 6 Figure 7: The relationship of daily rainfall and SMD at CHRS Figure 7 Figure 8: Estimated hydrographs for the 1888 and 1892 floods Figure 8 Figure 9: Magnitude frequency relationship for SMD at CHRS for May and July Figure 9 Figure 10: Changes in median SMD at CHRS and area 35 (Smith & Trafford, 1976) Figure 10 Figure 11: Flood frequency analysis for the river Lud at Louth UK (small circles). Large circles: estimated 2, 10, 50, and 200 year flood events, and the 1920 flood at Louth Figure 11 Figure 12: Flood frequency estimates for the Upper Brue at Bruton, UK Figure 12 Figure 13: Estimated flood frequency curve for Langtoft. Open circles: data from Table 5 Figure 13 Author Info:

Colin Clark, Charldon Hill Research Station. Email: colin4chrs@hotmail.com.

The author would like to thank Professor Paul Hardaker for his comments on an earlier draft of this paper.

Tables

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6