ESTIMATION OF LOW RETURN PERIOD FLOODS. M.A. BERAN and M J. NOZDRYN-PLOTNICKI Institute of Hydrology, Wallingford, Oxon.

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1 Hydrological Sciences-Bulletin des Sciences Hydrologiques, XXII, 2 6/1977 ESTIMATION OF LOW RETURN PERIOD FLOODS M.A. BERAN and M J. NOZDRYN-PLOTNICKI Institute of Hydrology, Wallingford, Oxon. OXJ0 8BB, UK Received 8 June 1976, revised 20 December 1976 Abstract. A method was sought for estimating low return period floods (0.2-5 years) from annual maximum data. The theoretical relationship between the peak over threshold and annual maximum return periods is re-examined and an empirical relationship based upon United Kingdom flood data is proposed. Estimation des crues d'une basse période de retour Résumé. Nous avons cherché une méthode d'estimation des crues d'une basse période de retour (0.2-5 ans) utilisant des données des maximums annuels. Nous reconsidérons la relation théorique entre les périodes de retour pris au-dessus d'un certain^débit de base et la période de retour maximum annuel et nous proposons une relation empirique dérivée des données des crues en Royaume Uni. OBJECTIVES The objective of this paper is to show how low return period floods (five times per year to once in five years) may be estimated from the annual maximum (AM) series. The current procedure for doing this uses a relationship due to Langbein (199) which links the annual maximum return period (7A) with the return period (T p ) from the peaks over threshold (POT) series. Because of some restrictive conditions assumed in the derivation of the Langbein formula, 7A has been empirically related to 7J, using United Kingdom flood data. This led in turn to a re-evaluation of the Langbein relationship and suggestions for generalizing the relationship to allow for dependence between successive exceedance intervals. USE OF POT SERIES In economic evaluation of flood damage it is commonly the low return period flood events which carry the main weight. The peaks over threshold (POT) series provides an appropriate data set from which to estimate floods of a required recurrence interval. By contrast the annual maximum (AM) series yields a return period which is the mean interval between years containing an event, which by definition must exceed unity. Both AM and POT data were studied in depth in the Flood Studies Report (Natural Environment Research Council, 1975) but the more complete analysis was of AM data because of its greater computational convenience and avoidance of dependence problems. Langbein showed the annual maximum return period 7A could be deduced from the peak over threshold return period T p using 7 A =[l-exp(-l/7 p )]- 1 (1) 275

2 TABLE 1 Comparison of recurrence intervals of floods from AM and POT series using Langbein 's equation Peaks over threshold series [years] Annual maximum series [years] p>10 r A =7' P + o.5 Table 1 shows specimen values and Fig. 1 shows the relationship graphically. Langbein's derivation of the relationship used the appoximation which holds only when e is small compared with n, where e = expected number of times that a flood greater than the threshold will be exceeded in any one year; n = average number of potential independent flood events per year. Although Langbein regarded n as notionally an arbitrarily large number, in practical cases for many British rivers dependence between floods limits the size of «to at most five. The values of e associated with low return period floods are of similar magnitude and so the assumption n > e does not hold. Therefore, an investigation was carried out into the empirical validity of equation (1) for large values of e (low return period floods). EMPIRICAL PROCEDURE Records from 0 British gauging stations were used to test the relationship. The criteria for selection of these stations were that they should have at least 20 years of record and that SO' 0 0 >Ay^ *^A>^A < Peaks over threshold series return period - years 276 Fig. 1 - Relationship between return periods, peaks over threshold and annual maximum series.

3 there was an accurate flood rating up to the level of the mean annual flood. For each of the chosen stations the following procedure was carried out, as illustrated in Fig. 2 using data from the River Allt Leachdach at Intake. (1) The annual maximum data were plotted against the expected reduced variate for the Gumbel distribution [Fig. 2(a)]. The plotting position used for the ith lowest annual maximum innyears(iv= 2 in this case) wasy A = -log e [-log e (/-0.)/(7V+0.12)]. Fig. 2 - Method of comparing annual maximum with peaks over threshold series. 277

4 (2) The peaks over threshold were plotted against the expected reduced variate for the exponential distributiony v, i.e. the ith lowest of the «TV values is plotted at i y p = X (Nn+l-jy 1 j=i () To obtain the return period in year units from y p it was necessary to transform to an equivalent annual maximum reduced variate,yj. Substituting the T:y relationship for the double exponential and exponential distributions [equations (2) and ()] into the Langbein relationship equation (1) it was found thatyj = y p - log e n and this is shown in Fig. 2(b) (n = 5 in this case). y A = -logec-log^l-t^1)] (2) y P = iog e T P () If equation (1) were correct then y A - yj and the two lines of Fig. 2(a) and 2(b) would be equivalent. () Figure 2(c) shows the comparison between the annual maximum plotting position from annual maximum data,^a) an( that from the peaks over threshold series data,j>x> ar >d is drawn as follows. For each annual maximum flood of Fig. 2(a) for which JA < 2.0 the corresponding yj was found by linear interpolation in Fig. 2(b). Figure 2(c) was constructed from the y^, yj line and the line of equality. The plotting position used is the expected value of y given that the sample is drawn from the Gumbel distribution. However, no assumptions are made of linearity between Q and_y on Fig. 2(a) and the axis is repeated in Fig. 2(c). Thus altering the plotting position or distributional assumptions would not materially affect the result. RESULTS The trend of the points of Fig. 2(c) indicates the nature of the departure from the Langbein relationship. There was no consistent tendency for a curvature, either convex or concave, within the range y = -1.5 to +1.5 so a least squares fit to the points was employed in order to use the intercept and slope as parameters. The results for Allt Leachdach were fairly typical with a positive intercept of y ^ for yj = 0 and a slope flatter than 5. In fact there was no case where the fitted line did not fall above the line of equality for small y and approach it as y increased. This result indicates that the Langbein transformation underestimates the return period on the annual maximum series. Attempts were made to correlate the parameters of the 'best fit' regression line on indices of catchment characteristics such as basin size, drainage, climate, slope and soil type. No significant regressions were found although the climate variables came nearest to being significant. There does exist a quite considerable negative correlation between regression slope and intercept and this may have obscured the relationship because it was observed that the catchments did group reasonably well by region. Thus the catchments were divided into four groups and Fig. 1 shows the implied T&:T p relationship from the average slope and intercept for each group while Table 2 shows values taken from the curves. Figure shows the locations of the regional groups and Table shows the groups in relation to the hydrometric region numbers that are used to reference British rivers and also the corresponding Flood Studies Report region numbers used to classify flood frequency relationships. 278

5 TABLE 2 Annual maximum return period T\ [years] for selected peak over threshold return periods T T p [years] 1 2 Grouf > Theoretical excl. 101,102) Hydrometric 28, are Hydrometric are ^ 102 ydrometric areas Hydrometric are 5-5 Fig. - Location of groups. TABLE Flood Studies Report region numbers, hydrometric areas and the groupings derived from the analysis Region number FSR) iydrometric jroup lumber , ^

6 THEORETICAL CONSEQUENCES OF RESULTS There are two main theoretical consequences which follow from the empirical results: (1) the relative closeness of the empirical to the theoretical 7A : 7p relationship despite the similarity of «and e; and (2) a model for the departure, such as it is. The observed departures from the theoretical line of Fig. 1 were not as marked as might have been supposed given the relative values of «and e and this prompted a re-examination of the way equation (1) was derived. This analysis, shown in appendix 1, revealed that, given the Poisson assumption, equation (1) is an exact result and does not depend upon the condition n > e. The property of the Poisson distribution that is used, i.e. that a random selection from a Poisson process is also a Poisson process, is well known in time series work and has appeared in the hydrological literature before (Dyck and Kluge, 197; Todorovic and Zelenhasic, 1970; Natural Environment Research Council, 1975, p ). The fact that the empirical T^:T p lines plot above the theoretical line on Fig. 1 is evidence for dependence between peak magnitudes or else between the time intervals between exceedances. Both these possibilities were investigated in a different context and it was found that although peak magnitudes were independent, the inter-event times, especially in winter, displayed some dependence (Natural Environment Research Council, 1975, pp ). The negative binomial distribution was considered as an alternative to the Poisson distribution in describing the annual number of threshold exceedances, but despite having two parameters, y and a, it did not yield a greatly improved fit. Values of the parameters in the corresponding 7A : Tp relationship can be found which yield values similar to Table 2. This was not pursued beyond confirming that the dry part of the country which showed the greatest departures from the Langbein curve also experienced the greatest preponderance of winter floods. PRACTICAL CONSEQUENCES OF RESULTS The resultant Tj^-.Tp relationship summarized in Fig. 1. and Table 2 can be used to calculate low return period floods from the annual maximum flood series (appendix 2). A great deal of work has been carried out to regionalize United Kingdom flood frequency relationships. These take the shape of region 'growth curves^ showing Q(T)/Q plotted against T, where Q(T) is the 7A year return period flood and Q is the mean annual flood. It is recommended that unless a_great deal of local data are available these curves should be used to scale the data based Q to calculate Q(T). The growth curves, are of the form 6/2 = u + a{\-q- ky )lk where u, a and k are parameters whose values have been estimated by pooling all flood data in a region (Natural Environment Research Council, 1975, pp ), andy is the Gumbel reduced variate. Table shows sample values of Q(T p ) for low return period floods based upon the region Q(T\) growth curves and the empirical 7A.:7p relationships. The correspondences between the regions and groups are shown in Table. Although in Fig. 1 the departures from the theoretical line appear slight the difference when converted to discharge terms is significant. For example for a region, group 1 station 280

7 TABLE Regional values ofq/q corresponding to varying r p Region T P 1.0 [years] / the 2 per year flood would be given by Q/Q = 0.68 using the empirical result, and 0.65 using the theoretical result, a per cent underestimate. For a region 5, group station the empirically derived Q/Q is 0.72 and the theoretical result gives 0.6, a 1 per cent underestimate. CONCLUSIONS It has been shown that despite the stated restrictions to the validity of the Langbein relationship between annual maximum and peak over threshold return periods it does provide reasonable if slightly underestimated values of low return period floods. This underestimation can be removed by using empirically derived relationships for the United Kingdom although the procedure would be applicable to most countries. Acknowledgements. Especial thanks are due to P.P. Lynn who advised on the analysis and suggested the regional grouping. Other colleagues who suggested improvements were R.T. Clarke and Dr J.V. Sutcliffe, also the referees suggested several important points. Acknowledgements are also due to the Ministry of Agriculture, Fisheries and Food (the funding body) and to the Director of the Institute of Hydrology for permission to publish. REFERENCES Dyck, S. and Kluge, C. (197) Investigations on the structure of frequency distributions. In Mathematical Models in Hydrology (Proceedings of the Warsaw Symposium, July 1971), vol. 1, pp. 9-6: IAHS Publ.no Natural Environment Research Council (1975) Flood Studies Report, 5 volumes: NERC, London. Langbein, W.B. (199) Annual floods and the partial duration flood series. Trans. Amer. Geophys. Un. 0, no. 6. Todorovic, P. and Zelenhasic, E. (1970) A stochastic model for flood analysis. Wat. Resour. Res. 6,

8 Appendix 1 THEORETICAL DERIVATION OF THE T A :T p RELATIONSHIP Consider a base threshold Q 0 exceeded on average by n independent exceedances per year, and a higher threshold Q exceeded on average by e exceedances per year. In N years there will therefore, be Nn <2o exceedances. The probability that any Q 0 exceedance is also a Q exceedance is en/nn = e/n. To calculate the probability PA(Q) that the annual maximum flood exceeds Q, it is necessary to consider, for k = 0, 1,2,,... exceedances of Q 0 in a year, the probability that the maximum of the k exceedances also exceeds Q. Assume the Q 0 exceedance rate is Poisson, the probability of k such exceedances in a year is therefore P(k) = e-"nk/k\ The probability that the maximum of the k exceedances (which is therefore the annual maximum flood for the year considered) exceeds Q is then Since P A (2 fc) = 1 - (1 - /«)* PA(Q) = I PA(Q\V P(k) -.?.{'- '} " " '" = l-e _ * and since, by definition, the return period 7\ of the annual maximum flood is given by T A = 1/P A (G) we have,, T, TA = {l-e- }- 1 0rT A = {l-e-»^}- 1 since the mean length of time interval between Q exceedances is 2p = 1/e. In previous proofs it was assumed that the expected value of i^ is generated from the year containing the expected number of Q 0 exceedances, n, instead of by considering all types of years with number of Q 0 exceedances = 0, 1,2,... and the probability of their occurrence. EXAMPLE Appendix 2 The twice a year flood on the AUt Leachdach at Intake is required. This station is in hydrometric area 91 i.e. Flood Studies Report region 1 and group. Using Table 2 the 7X corresponding to Tp = 0.5 years is 1,25 years. So, once the annual maximum flood frequency curve has been derived, the required flood may be read off directly as 5.m [s [see Fig. 2(a)]. Alternatively^ if no data were available, Table gives a value of Q/Q = 0.72 for region 1 when Tp = 0.5. Q can be estimated from catchment characteristics as described in section of the Flood Studies Report and is found to be 10.0 m /s. Therefore, 0(0.5 years) = 0.72 X 10.0 = 7.2 m /s, an overestimate of per cent.