Flood Frequency Analysis of AMAX EventsContents1. Introduction2. Data Analysis3. Impact of Climate Change4. Conclusion5. References6. AppendixIntroductionThis report summarises the results of a flood risk assessment conducted using data obtained from a gauging station on the By Brook, upstream of Batheaston. The catchment area of 102km2 is predominantly rural with limited urbanisation in some areas (National River Flow Archive, 2018). This area has been assessed to identify potential sites for new residential developments to the east of Bath which have been planned in response to increased housing pressure within the city.Figure 1. Catchment area of the By Brook gauging station showing elevation (NERC (CEH), 2012)1.     Data Analysis2.1   Description of the dataThe data obtained from the By Brook gauging station gives the annual maximum peak flow for each water year from 1981 to 2015. River flow is recorded at 15-minute intervals, giving thousands of observations for each water year, but only the peak flow rate is used in the annual maximum series (AMAX). Although the AMAX data may be extreme compared to the flow rates recorded during the rest of each year, for flood engineering purposes these rates are the most important.The values for peak flow range from 5.88m­­3/s in 2011 to 17.67m3­­/s in 2005. The mean annual maximum is 9.52m3/s. Figure 2 shows that there is no apparent trend in the values for annual maximum peak flow over the observed period of time, with each maxima for successive years independent; these are common attributes of extreme event data (Stedinger, et al., 1993).Figure 2. Annual maximum peak flow from gauging station on the By Brook 2.2   The Gumbel distributionThe estimation of T-year events is vital when conducting a flood risk assessment. However, with only 35 water years in the observed AMAX, it is not possible to obtain reliable return period estimates of more serious events without extrapolating beyond the data by applying a suitable probability distribution. The Gumbel distribution is an extreme-value distribution (Upton & Cooke, 2014), widely used to represent the probability of environmental events such as earthquakes and flooding. For this reason, the Gumbel distribution has been used to describe the AMAX data from the By Brook in Figure 3 below.Figure 3. Gumbel distribution of AMAX data showing pdf and cdfThe cumulative distribution function (cdf) for the Gumbel distribution is(1)Fxx=e–e–x–µ?with the probability density distribution (pdf) defined as(2)fxx=1?e–x–??e–x–??The Gumbel distribution has 2 parameters, µ and ?. These can be estimated using(3)??=6?swhere s is the sample standard deviation, and(4)??=x?–0.5772??where 0.5772 is Euler’s constant.2.3   Goodness-of-fitThe Gumbel distribution is one of many other distribution functions that could also potentially provide good descriptions of the data, and so it is important to verify its utility. One way to do this is to calculate the probability, for each event in the AMAX series, of an event occurring that exceeds the event itself using the Gringorten plotting position formula(5)r–0.44n+0.12, r=1,…,nwhere r is the rank of observation, i.e. r = 1 for the greatest peak flow in the series, and n is the number of observations. The inverse of the plotting position gives the return period for each event. The Gumbel reduced variate, yT can then be calculated for each event using(6)yT= –ln?(–ln?1–1T)where T is the return period of the event.Figure 4. Relationship between peak flow and estimated return period.By plotting the peak flow against the Gumbel reduced variate, the data points should produce a straight line as shown in Figure 5. Only the Gumbel distribution will do this – other distributions will show a curve.Figure 5. Gumbel distribution fitted to By Brook AMAX data and plotted on Gumbel probability paperThe data in Figure 5 corresponds relatively closely to the straight trendline giving visual, graphical evidence that it is well described by the Gumbel distribution.Another method known as the ?2 test investigates the goodness-of-fit numerically. The data is divided into a number of bins k, each covering an equal range of probabilities, with the upper bin level defined as(7)xi(u)=?–?(ln?–ln?ik) The number of observations ni in each bin is then used to calculate the test statistic ?2.(8)?2=kn?i=1k(ni–nk)2The value for ?2 determined by the data was 4.4. This is then compared to the 90% quantile in a chi-square distribution with 4 degrees of freedom (calculated using the number of bins and Gumbel parameters). From statistical tables (Kokoska & Nevison, 1989, p. 59), the critical value ?0.902(4) = 7.7794 > 4.4, meaning the Gumbel distribution is accepted as a credible choice at a 10% level of significance.2.4   Design flood estimatesUsing the By Brook AMAX data, T-year flood events have been calculated from the Gumbel distribution. A level of uncertainty has been considered to allow for the limited range of AMAX data.The precision of a design event is its standard deviation. The range of uncertainty for a 95% confidence interval is defined as: (9)x?T–z1–q2sxT   to   x?T+z1–q2sxTwhere z(1-q/2) is the upper (q/2)100% percentile of the standard normal distribution. For q = 0.05 i.e. a 95% confidence interval, from standard statistical tables (Kokoska & Nevison, 1989, p. 56) it is found that z(1-q/2)= 1.96. The standard deviation of the T-year event sxT, can be derived as(10)sxT =?n1.11+0.52yT+0.62yT2The estimated peak flow rates and associated uncertainties for T-year events can be seen below in Table 1.T-year Event Return PeriodEstimated Peak flow(m3/s)Lower confidence interval peak flow (m3/s)Upper confidence interval peak flow (m3/s)10 years13.3511.6015.0950 years17.0914.4319.75100 years18.6715.6121.72200 years20.2416.7823.70Table 1. T-year event peak flow rates at the By Brook for 10, 50, 100 and 200 year return periods with a 95% confidence interval.2.5   Risk during the lifetime of the developmentNew residential developments should have a design life L of 100 years (Ministry of Housing, Communites and Local Government, 2014). Assuming the structure can safely withstand a 100-year flood, the probability r that an event will exceed this and cause the structure to fail can be calculated from(11)r=1–(1–1T)Lwhere r is found to be 0.63. Although there is only a 1T = 0.01 probability that the structure will fail each year, it may not be considered sufficient that there is a 63% chance of failure within the design life. If a much lower chance of failure is desired, the structure would need to be designed for the appropriate T-year event seen below in Figure 6. However, the cost of this may exceed the benefits as, for example, a risk of failure of 25% would require design for a 348-year event, for which peak flow in the By Brook could be up to 25.27m3/s after considering uncertainty.Figure 6. Risk of failure of a structure within a 100-year design life when designed to accommodate a given T-year event3.     Impact of Climate Change3.1   The problemThe Gumbel distribution that has been applied to the AMAX data from the By Brook assumes a constant mean value and a constant variance. Climate change, however, is causing increased temperatures, and a warmer atmosphere can hold more moisture leading to more intense rainfall (Reynard, 2017). This means that the return periods of more extreme events may decrease, and so the predicted flow rates of T-year events must be adjusted to account for future flood risk.3.2    Impact on design flood magnitudesThe By Brook lies within the Severn River Basin District. If the proposed residential development is in Batheaston, there is a significant area encompassed by Zone 3a where the By Brook meets the river Avon, as seen by the dark blue zone in Figure 7. Zone 3a represents an area with “a 1 in 100 or greater annual probability of river flooding” (Ministry of Housing, Communites and Local Government, 2014).Figure 7. “Flood map for planning” for Batheaston showing the flood zones (Environment Agency, 2018)The Environment Agency defines buildings used for houses as “more vulnerable” (2017). Therefore, if situated in flood zone 3a, they fall under the allowance categories of Higher Central and Upper End. Using Table 2 below, assuming the category of Higher Central and a design life of 100 years, the allowance factor is 35%.Table 2. Peak river flow allowances by river basin district (Environment Agency, 2017)River Basin DistrictAllowance CategoryTotal potential change anticipated for the ‘2020s’ (2015 to 2039)Total potential change anticipated for the ‘2050s’ (2040 to 2069)Total potential change anticipated for the ‘2080s’ (2070 to 2115)SevernUpper end25%40% 70%Higher Central15%25%35%Central10%20%25%A 100-year event with a 35% allowance factor is therefore 25.20m3/s – up to 29.33m3/s with uncertainty (see Table 3 below.) Now considering climate change, if the structure has a design life of 100 years and the risk of failure in this time is desired to be as low as 25%, it would need to withstand a flood event of 34.12m3/s (with uncertainty).Table 3. Peak river flow adjusted for climate changeT-year Event Return PeriodEstimated Peak flow(m3/s)Peak flow adjusted for climate change (m3/s)Peak flow with uncertainty adjusted for climate change (m3/s)10 years13.3518.0220.3850 years17.0923.0726.66100 years18.6725.2029.33200 years20.2427.3331.994.     ConclusionThe effects of climate change can significantly affect design flood estimates, especially when the uncertainties caused by limited data are applied. The new residential developments near Batheaston should be designed to withstand an appropriate flood event based on preferred risk of failure and proximity to the flood Zones, taking all that has been discussed in this report into serious consideration when a specific site is to be chosen5.     ReferencesEnvironment Agency, 2017. Flood risk assessments: climate change allowances. [Online] Available at: https://www.gov.uk/guidance/flood-risk-assessments-climate-change-allowances#table-1[Accessed 5 November 2018].Environment Agency, 2018. Likelihood of flooding in this area. [Online] Available at: https://flood-map-for-planning.service.gov.uk/[Accessed 5 November 2018].Kokoska, S. & Nevison, C., 1989. Statistical Tables and Formulae. New York: Springer-Verlag.Ministry of Housing, Communites and Local Government, 2014. Flood risk and coastal change. [Online] Available at: https://www.gov.uk/guidance/flood-risk-and-coastal-change#flood-zone-and-flood-risk-tables[Accessed 5 November 2018].National River Flow Archive, 2018. 53028 – By Brook at Middlehill. [Online] Available at: https://nrfa.ceh.ac.uk/data/station/spatial/53028[Accessed 4 November 2018].NERC (CEH), 2012, Catchment area of the By Brook gauging station showing elevation. Available from: https://nrfa.ceh.ac.uk/data/station/spatial/53028 [Accessed 4 November 2018]Reynard, N., 2017. Is climate change causing more UK floods?. [Online] Available at: https://nerc.ukri.org/planetearth/stories/1849/[Accessed 5 November 2018].Stedinger, J., Vogel, R. & Foufoula-Georgiou, E., 1993. Frequency Analysis of Extreme Events. In: D. Maidment, ed. Handbook of Hydrology. New York: McGraw Hill Professional, p. 18.2.Upton, G. & Cooke, I., 2014. A Dictionary of Statistics. 3rd ed. Oxford: Oxford University Press.6.     AppendixGroup 37 AMAX data:Water YearPeak Flow (m3/s)198112.89198210.3819839.1319846.06198511.2719869.08198712.0819886.0919897.9119909.2919916.5419927.5519938.5719946.7219959.9419968.68199713.1819989.27199910.9320006.0820019.09200210.9520039.7220046.34200517.672006 6.63200716.6820089.8420099.9201014.7120115.8820128.5720137.9920147.14201510.31