17. Regional flood frequency analysis - some less frequented corners

Max Beran Institute of Hydrology WALLINGFORD, OXON, UK

Abstract

The United Kingdom Flood Studies between 1970 and 1975 owed much to prior research in France. The work of Jacques Bemier in placing peak over threshold analysis on a solid theoretical footing and on quantifying risk were major inspirations. Other important work in France which greatly informed our British studies included France tieing the probability distribution of river flood to that of the causative rainfall, and the global syntheses of flood peak maxima. This paper describes four “niche” areas arising from these issues.

1. An examination of probabilistic connection between storm rainfall, antecedent conditions and consequent peak discharge leading to prior estimates of regression coefficients and limiting values of explainable variante. 2. Multisite analysis of flood peaks and plotting positions taking intersite correlation into account. 3. The evaluation of very frequent events is important for economic analyses. 4. An example of a risk analysis in which floods which occur close together in time have more serious consequences than when events of the same magnitude are separated in time.

Résumé

Les études de crues au Royaume Uni entre 1970 et 1975 se sont principalement basées sur les recherches effectuées en France dans ce domaine. Les travaux de Jacques Bemier, avec l’établissement d’une base théorique solide d’analyse statistique des crues dépassant une valeur de seuil et la quantification du risque, ont été une source d’inspiration importante. Un autre travail important en France qui a influencé les études Britanniques porte sur la combinaison statistique des distributions de probabilité des crues et de pluies maximales et sur les estimations des débits instantanés maxima. Cet article décrit quatre points issus de ces travaux:

1. Etude des relations probabilistes entre les épisodes de pluie, les conditions antécédantes et les débits de pointe résultant, pour estimer les coefficients de régression et les valeurs limites de la variante expliquable; 2. Analyse des débits maximum relatifs à différents sites et leur placement dans un graphique de distribution de fréquence en prenant en compte les correlations qui existent entre ces différents sites; 3. Estimation d’événements très fréquents, très importants pour les analyses économiques; 4. Exemple d’analyse de risque où l’on montre que les crues qui se succèdent dans un court laps de temps ont des conséquences plus graves que les événements de même amplitude mais qui seraient plus espacés dans le temps.

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17.1. Introduction

1s statistics servant or master to hydrology? It is true that the laws of probability are inviolate because they flow directly from those of logic. But “statistics” is not “probability”. Statistics is a branch of applied mathematics and cari therefore be bent to the problems thrown up by the real world. This is not to give carte blanche to sloppy thinking or bad maths; but it is to say that the needs of the user and the requirements of the problem determine the rules of battle.

What are the consequences of this line of thinking? First and foremost, it means that it is not for statisticians to say that a problem cannot be solved or is poorly posed, or to complain that the data fail to conform to constraints imposed by analytical ideals. What does matter is that the hydrologist needs an answer - the thousand year drought, the level of flood protection that balances risk against cost, even an estimate of the smallest flood that cari’‘’ happen - and, more often than not, at a locality with deficient data.

It is in this spirit that 1 believe Jacques Bernier has always approached his subject. His analytical capabilities are matched by very few, certainly not by this writer; but he knows that an answer to a problem is the bottom line. One of his roles has been to place on a solid footing the hazy ideas of hopeful empiricists whose analytical skills do not match their aspirations to apply statistics to hydrological problems. 1 hope he cari spare time to restore respectability to the selection of statistical applications presented below.

17.2. Outline

The United Kingdom Flood Studies Report (FSR) is probably the largest single flood analysis project ever conducted; certainly prior to 1975, when it was published, maybe since. It owed much to antecedent research in France and the United States. In France the work of Jacques Bernier in placing peak-over-threshold analysis on a solid theoretical footing was one major inspiration and led to a continuing link between himself and members of the Flood Studies team. Other important work in France which greatly informed our British studies included those of the Grenoble group at EZectticité de France tying the probability distribution of river flood to that of the causative rainfall, and ORSTOM’s global synthesis of flood peak maxima.

In the following sections 1 revisit some less frequented aspects of flood frequency analysis. Some arose from the needs of the FSR with its central objective of deriving procedures that are applicable at ungauged locations; others emerged later when applying the FSR to special circumstances :

0 Making regression respectable 0 Correlated data and regional flood analysis l Frequent flooding 0 Hydrological and financial risk.

17.3. Making regression respectable

17.3.1. Need for regionalization

Gauging station sites are generally selected for accuracy of measurement and relevance to water resources assessment. Hydrologists need flood estimates at localities where there is a need to protect against inundation. This mismatch of criteria means that the flood analyst frequently has to transfer information from gauged sites to the ungauged location

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at risk of flooding. The FSR procedures for transferring information from gauged to ungauged rivers included regressing the mean annual flood on catchment characteristics, an approach previously pioneered by the Geological Survey in the United States. Twenty years on, the procedure is commonplace, though not always applied with as much insight or caution as in the FSR.

Unfortunately a regression equation cari yield nonsense predictions, a danger which arises from extrinsic and intrinsic causes :

0 data deficiencies and mismatch between the calibration and the prediction set

0 the unconstrained nature of mode1 building by regression. TO elaborate a little on the latter intrinsic cause, there is no (hydro)logical basis for assuming a linear form of relationship, no “common currency” for a summarizing flood index across different catchments, no reason to expect any particular level of explained variante, indeed no a priori reason to expect a stable formula at all.

17.3.2. Strategy for safe flood estimation

A two-pronged approach to model-building was adopted in the FSR. At the same time as building the best predictive mode1 on statistical criteria, we also attempted to address some of the fundamental issues about regression analysis expressed above. The former task was achieved with linear regression on the logarithms of variables. The assumed “common currency” was the mean armual flood, though estimated in a variety of ways including arithmetic averaging, data extension, graphical interpolation, and armual maximum and peak-over-threshold analyses.

Mapping the regression residuals from the analysis of the pooled data revealed a patchwork of regions with consistent positive and negative departures. This suggests that the mean armual flood does not provide the holy grail of a common currency. An ANOVA on pre-defined regional subsets gave a family of equations which explained about 90 per cent of the variante, with a factorial standard error of estimate between 1.4 and 1.5.

TO help minimise concerns that a regression equation may throw up spurious flood estimates, the FSR recommends a second and independent flood estimation procedure based on a rainfall-runoff model. This type of flood estimate complements the regression approach in that it is hydrologically well-constrained, yielding flood peaks that are conformable to the type of catchment and storm event. On the other hand the rainfall : runoff approach is statistically only weakly constrained because there is no a priori

reason for any given combination of storm and catchment condition to generate a flood peak of a predetermined frequency of exceedance.

17.3.3. Simulating regression

We have emphasized the practical advantages for combining flood prediction techniques to take advantage of this complementarity of properties, giving confidence beyond what either approach - statistical and rainfall : runoff - is individually capable of. More to the point in the current context is the opportunity for putting regression on a hydrologically more secure foundation.

The FSR reported the first ever application of simulation for bridging the statistical and the rainfall:runoff approaches in the context of flood estimation. The basic mode1 is driven by four input variables :

l storm rainfall depth 0 storm rainfall duration 0 storm intensity profile within its duration a catchment wetness prior to the storm ,

and by sampling from across the probability densities of each input variable it was possible to generate the full flood frequency distribution of output flood peaks. The primary use of this was to identify stable combinations of inputs that would yield flood peaks of required return periods on output, again an FSR “first”.

This simulation was applied to 80 gauged catchments and provided a large data set of simulated and recorded flood statistics. The properties of the simulated frequencies reinforced the superiority of a heavy-tailed distribution such as EV2, over Gumbel, and it supported the regional trend in “tail heaviness”. Turning to measures of central tendency, correlation was high between recorded and simulated mean armual flood, 0.98 (0.96 in log space). Overall the simulated mean annual flood (MAF) underestimated the recorded value by 10 per cent, largely caused by underestimates of a few large rivers, and no longer apparent in the log domain. The same regional pattern of residuals seen for the regional regression on MAF (Section 17.3.2) was observed when comparing recorded and simulated MAF confirm ing that this is no artefact; a given set of catchment characteristics accompanies a considerably larger MAF in the south-west than in the east of the country.

Table 17.1 compares the results of regressing simulated mean and 10 year floods on catchment characteristics with recorded values.

TABLE 17.1 Comparison of logarithmic regressions on recorded and simulated mean annuel flood

Dependent Regression coefficient variable AREA RSMD SOIL SLOPE

R2 fsee

Recorded 1.01 1.47 1.39 0.28 0.91 1.51 Simulated 0.99 1.15 1.02 0.26 0.96 1.29

Notes : AREA is catchment area; RSMD is the 5 year return period effective rainfall; SOIL is an index of the soil’s rainfall acceptance; SLOPE is a chantre1 slope. R2 is the coefficient of multiple determ ination, fsee is the factorial standard error of estimate.

A number of points emerge from the comparisons of MAF regressions. The regression coefficients for simulated MAF approach unity and suggest an underlying “round-number” form of relationship :

Q = comt AREA.RAIN.SOIL.SLOPE’14 (17.1)

for which there may be some underlying rationale. Because SOIL is close to a standardized runoff coefficient, the first three terms represent net rainfall on the catchment. SLOPE114 is rem iniscent of catchment response time relations where unit hydrograph time to peak, TP, was proportional to SLOPE-“4, SO by setting

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j= RAIN/TP, equation (17.1) becomes :

Q =cAi (17.2)

the well-known rational formula! The improvement in explained sum of squares from 91 to 96 per cent indicates that

about half of the original unexplained sum of squares is due to error in the dependent variable and inappropriate form of model. Alternatively it may be said that regression is incapable of explaining more than around 95 per cent of variante in mean annual flood no matter how accurate the data or complete the list of independent variables.

17.4. Correlated data and regional flood analysis

17.4.1. Regional flood frequency curves

Typical gauging station records, say of 20 years duration, are far too short to estimate the 100 year flood. In almost a11 design circumstances, the FSR recommends that a regional flood frequency curve is to be preferred to the locally derived one, both for gauged and ungauged locations. The derivation of the regional curves, which express flood magnitude dimensionlessly as a multiple of the mean armual flood, combined the index-flood and the station-year methods applied within a framework of graphical flood frequency analysis.

In its simplest form, the station year approach regards m years of record at each of k stations as equivalent to a single sample of mk independent data points. Strictly this holds only if the annual maxima are uncorrelated, both through time and between stations. The truth of the converse is very evident if one considers the extreme case of inadvertent duplication of one station’s record. Perfect correlation between a pair of the k stations would reduce the value to at most m x (k-1) station years worth of data. The FSR attempted to side-step the effect of correlation by forming station-year samples from dissimilar and geographically remote stations. While subsequent studies by Hosking (1987) and by Reed and Stewart (1994) have approached the issue, few have considered the implications to the classic graphical approach and order statistics with their intuitive appeal.

17.46. Plotting positions for dependent data

The graphical treatment of an at-site flood record involves plotting the ordered flood magnitudes q(i) qt2) .“... plotting positions, y(‘),

qCrn), against corresponding plotting positions, y(‘), Y(~)... .Y(~). The are derived from the sampling distribution of order statistics of the

standardized form of the fitted distribution, F(y).

g(y’=‘) dy= n ( 1

I F(Y) (*-l) [l-F(y)] '"-"f(y) dy (17.3)

In the case of the largest flood in a sample of size m, JC(~) would be plotted against E(ytm)), the expected value of equation (17.3) for n =r=m. In the uncorrelated station- year case this is simply extended to the sample size n = mk, SO the plotting position associated with the largest flood peak in the region would be evaluated from E(‘ycmk)).

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TO understand the adjustment necessary to deal with correlated data, it helps to visualise sampling distributions such as equation (17.3) as the outcome of a simulation experiment in which a drawing is made at random from a parent population. For station- year data, the parent population is defined in a mk-dimensional space with identical marginal distributions, f(y). Every point in the hyperspace represents a single set of station-year data. In the uncorrelated case there is no tendency for any value along one axis to predispose the value along another and the relative frequency of a given sample is the product of the marginal frequencies. The positively correlated case is represented by local increases in density reflecting a tendency for a value on one axis to accompany a more restricted range of values along another axis.

One envisages a simulation experiment in which samples are drawn repeatedly from the hyperspace. The samples are treated as if they were station-year data, first ordering, then assembling histograms and evaluating moments of ranked data.

17.4.3. Correlated station-year data

Station-year data give rise to a blocked correlation structure as illustrated in figure (17.1). Identical blocks down the leading diagonal emerge from within-year, inter-station correlation. The zero values elsewhere arise from an absence of between-year correlation. Of course, where such serial correlation is present, as in drought analysis, it cari be included in the correlation structure. The structure cari also be modified to allow for unequal record lengths, ie mi c > mj.

1 .5 .5 .5 .5 1 .5 .5 .5 .5 1 .5 .5 .5 .5 1

\ 0

1 .5 .5 .5 1 .5 .5 .5 1 .5 .5 .5

Fig. 17.1: Correlation matrix for 80 station years - 4 stations and 20years with common interstation correlation of 0.5. Van.able inter-station correlations would replace “0.5” values

Visualising a system is one matter, obtaining solutions is quite another. In fact analytical results are available only for a very restricted class of correlation structure, most notably for the equi-correlated, Normal distribution. Most distributions cari be transformed to approximate Normal$ SO this is not a serious limitation, and despite these restrictions, powerful approximations have been obtained (Stevens, pers comm).

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For example the expectation of order statistics for the equi-correlated case (correlation matrix R with a11 off-diagonal elements, p) cari be obtained from the uncorrelated case as

Jw (“k);R) = (l-p)‘/2 E(Y’“~);~) (17.4)

Equation (17.4) shows that probability plots Will still be linear even if correlation is neglected, but the slope Will be biassed by the factor ( 1-p)‘j2 thus overestimating return periods. Equation (17.4) has been extended to the block correlation (station-year) structure by setting p in equation (17.4) to the average correlation across the entire matrix (Rawlings, 1976), (m-lj/@zk-I)p,,, where p,, is the average within-block (inter- station) correlation. As a rule of thumb, SO long as this average is less than 0.02 the error in return period is less than 10 per cent. We also examined the effect of variability in the correlation structure, postulating a further adjustment c(l-p)ii2 in equation (17.4) suggested by the theory underpinning equation (17.4). Table (17.2) shows the relationship between c and the standard deviation among the elements of R This led to a second rule of thumb that equation (17.4) cari be used without serious error SO long as m » 20pw

TABLE (17.2) Bias in equation (17.4) due to variability among inter-station correlations

st dev (p) 0 0.1 0.2 0.3 0.4 0.5 C 1 0.99 0.965 0.915 0.845 0.70

TO exemplify these findings consider a regional dataset comprising k = 20 stations with m = 10 years of record each, and with an average inter-station correlation of 0.5. The plotting position of the largest value from an independent sample of 200 is 2.746. Allowing for the effect of correlation using equation (17.4) the plotting position is reduced to 2.68, corresponding to an independent sample of 165. For this combination of m and p one ought to make the further adjustment based upon the standard deviation of correlation coefficients, which reduces the effective sample size to 142. The net effect therefore is to reduce the information content of the sample from the 10 nominal years down to about 7 years per station.

17.5. Frequent flood events

17.5.1. Background to frequent flood estimation

The evaluation of rare floods is the “glamorous” end of flood hydrology, SO, mututis mutandis, estimating very frequent events must be the unglamorous end. Yet when we evaluate the economics of many flood protection schemes, it is the elimination of just these frequent flood events that contributes most to the post-project benefit stream. Events in this category may include multiyear occurrences such as the flood exceeded two or three times per year.

In this range the concept of return period has to be carefully handled. While return period is always defined as the average recurrence interval between events, the event in

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question differs according to the data collected. In the case of annual maximum data, the event is “a year which contains a flood”. The corresponding return period is therefore the average inter-val between years containing a flood of at least the given magnitude. This distribution is bounded at one year and capable only of assuming integer values. On the face of it then, armual maximum analysis is inappropriate for treating multiyear events. On the other hand, peak-over-threshold (Bernier, 1967) or POT analysis is designed to tope with multiyear events. In this formulation it is legitimate to talk of a six month return period flood.

17.52. The Langbein approximation

Notwithstanding these conceptual problems, AM data are often used for the entire range of flood frequencies primarily because they are much easier to extract than POT data. TO overcome the restriction on return period hydrologists employ a theoretical relationship due to Langbein (Langbein, 1949) to infer POT return periods from AM frequency analysis :

(17.5)

SO if an estimate is required of the flood which occurs twice a year (TPOT =0.5), it is usual to fit a distribution to the annual maxima and estimate Q(TPOT=.5) from WAM = 1.d

In his original paper, Langbein (1949) makes use of an asymptotic expression for the exponential function :

(I-E/n)” = emE

to derive equation (17.5). He warns that the formula holds only when wz where notionally n is the number of occurrences above a low threshold, and E the number of occurrences above the threshold of interest. In practice it is seldom possible to extract more than five events per year except for very flashy small catchments because of the difficulty of discerning independent events. As we have seen, flood design may concern thresholds corresponding to g as low as two or three per year; a combination that certainly invalidates equation (17.6).

These concerns led Beran and Nozdryn-Plotnicki (1977) to investigate the Langbein relationship empirically. They compared AM and POT data from 40 UK gauging stations. Table (17.3) presents results for the region that departed most markedly from the relationship; a11 other data showed better agreement. Even SO the departure is barely significant in practical terms - estimates based on equation (17.6) could underestimate the true flood by up to 13 per cent. However it was intriguing that the formula could provide this degree of accuracy where the conditions for the approximation are SO clearly not met, and this led to a theoretical examination of the derivation of equation (17.6).

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TABLE (17.3) Relationship between TAM and T,, for one UK region

T POT years Empirical

T AM - years Iangbein approximation

0.2 1.06 1.01 0.5 1.36 1.16 1.0 1.90 1.58 2.0 2.89 2.54 5.0 5.53 5.52

Note : Data are shown for the “worst fitting” region. Other areas gave equivalences for Ta doser to equation (17.5).

17.5.3. Theoretical derivation of TAM : Tm relationship

Consider a base threshold, Q,, exceeded on average n times per year; also a higher variable threshold, Q, exceeded E times per year. Over the course of N years there are Nn exceedances of Q, and the probability that an exceedance of Q, Will also exceed Q is N.$Nn = ~/n. TO calculate the probability that the armual maximum flood exceeds Q, it is necessary to consider k = 0,1,2,3 etc exceedances of Q, per year, and SO determine the probability that the maximum of the k exceedances also exceeds Q

Assume that Q, exceedances form a Poisson process SO that the armual number of exceedances, k, is distributed as :

P(k) = emn n”/ k! ; k = 0,1,2,3... (17.7)

For a given value of k, the probability that the maximum of the k exceedances exceeds Q is :

Since : PAM(Q ( k) = I - [l- E/nlk (17.8)

P*(Q) = s/,dQ bP(k) (17.9)

=c {I-(1- E/n) k}e-nnk/k! (17.10)

= l-emE (17.11)

which, since TAM = W’/.odQ, and TpOT = l/~, reduces to equation (17.6). The significance of equation (17.8) through equation (17.11) is the demonstration that the Langbein relation follows exactly from the Poisson assumption. It involves no approximation SO agreement as close as Table (17.3) is therefore unsurprising. Departures arise because of the inadequacy of the Poisson assumption, for example a tendency for exceedances to cluster in time or for peak magnitudes to be serially correlated (Beran and Nozdryn- Plotnicki, 1977).

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17.6. Financial risk of flood inundation

17.6.1. Background

The fourth and final unfrequented area again relates closely to one of Jacques Bernier’s interests - risk evaluation (Bernier, 1987; Ulrno and Bernier, 1973). Risk is sometimes understood as the combination of a probability of a hazardous event and the magnitude of the hazard (Royal Society Study Group, 1992). Other definitions express risk as the total probability of exposure to a hazard over a design lifetime.

A feature of most hydrological applications of risk theory is that the contribution to risk of a hazardous event, like a flood, does not depend on the point in time when it occurs. The applicationto be described here is unusual in that the position in time of the hydrological event is highly significant : floods which occur close together in time have more serious consequences than the same magnitudes of events separated in time, and floods which occur early in the life of a scheme are riskier than those which occur later on. An additional point of interest is the prospect for using the concept to quant@ perceived risk in an objective fashion, SO building a bridge between hydrology and psychology.

17.6.2. Design of offstream storage

Offstream storages have become increasingly popular for the protection of urban areas from river flooding. An area of bankside land is bunded in order to confine the inundation within a designated flood zone, and equipped with inlet and outlet structures. The storage is filled by diverting flood water into it according to an operating rule that contains downstream discharge below a critical value. The storage is emptied when the discharge drops below the critical value. There are several operational advantages of providing flood detention storage close to the protected area and, particularly important in the UK context, they avoid the need for permanent reservoirs in the headwaters. Another important advantage is that the land remains available for use for recreation or agriculture.

However they also carry some disadvantages. The valley geometry means that the storage generally cannot be deep, SO land-take is likely to be higher than in a headwater reservoir. A second issue is that landowners need to be paid for the use of their land. The crux of this risk analysis is the need to find an equitable basis for compensating occupants for the dirninished value of their land.

In a recent application, the storage-area occupied farmland which had originally been part of the floodplain, but, thanks to past flood protection projects, was nowadays seldom flooded. Because of this protection, and the high fertility, the land developed into highly productive agriculture and horticulture. ‘I?re river through the urban area downstream was capable of passing the 10 year flood within bank, SO the operating rule for the flood storage was that it was brought into use at the 10 year return period level. Larger floods would inundate successively larger areas of the detention storage, and it was designed to fill, ie inundate the total designated flood area, at the 100 year level.

17.6.3. Flood fund concept

A fair system of compensation was sought for farmers occupying the affected land. In a conventional risk analysis, compensation is based on the present value (Pu) of future

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losses :

PV = 100 D/Tr (17.12)

where D is the average damage due to inundation, T is the return period of flooding, and r is the discount rate. Because PV is inversely proportional to the frequency of flooding, farmer A, occupying land in the lowest portion of the inundated area, would experience 10~s ten times as frequently as his upslope neighbour, B, an occupant of land in the highest area, and SO receive ten times as much compensation. This ratio does not conform to the perception of flood plain occupants who a11 share some sense of blight, and was rejected. There is also an important distinction between benefits and costs in this evaluation. Benefits are considered to accrue to society at large whereas costs fa11 on particular individuals.

A revised formulation was based on the flood fund concept. In this the farmer’s compensation is regarded as the starting capital of a “flood fighting fund”. The relevant expression of risk then shifts from the hydrological event to the financial one - is the flood fund sufficient to fight the losses due to flooding? TO evaluate this risk it is necessary to assume that the fund is treated independently of the working capital of the farm and invested for the sole pur-pose of paying for lost production when the land is deliberately inundated. A natural measure of risk then becomes the financial one of the probability that the fund becomes extinct - a variant on “gambler’s ruin” on stochastic process theory.

We cari then focus on the performance of the fund through time as it earns real interest of r per cent per annum and is sporadically drawn upon for payment of losses. This calculation has some similarities with actuarial calculations carried out by insurers where prerniums are paid into a fund which is drawn on to pay for claims (Beard et al, 1984). It cari be appreciated that calls on the fund during the early years, before it bas had the opportun@ to accrue interest, are more serious than later occurrences. A r-un of events is also disproportionately damaging to the ftmd than the same total number well separated in time.

Though the basic concept appears simple it turns out that the combination of geometric growth through the compound interest formula

F(i + I) = F(i)(l +r/lOO) (17.13)

and even a rectangular distribution of losses, L, is not algebraically tractable. It cari be seen that the time series, F(i), has absorbing boundaries at zero, when the fund is extinguished, and at IOOF/r > L,,, at which point the interest exceeds the maximum possible withdrawal. Between times the probability density of F behaves in a non- stationary fashion gradually migrating towards one or other modes.

In practice the computation is further complicated by the fact that the lost production consequent on an inundation varies greatly with the time of year, and hence there is a need to consider the relative probability of inundation month by month. In land used for agriculture, losses are heaviest following summer flooding, and drop sharply after harvest and prior to land preparation. One must also consider the effect of more than one inundation in a year, though this may be simplified by the likelihood that no more than one reseeding is possible,

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17.6.4. Results

A simulation program was written to mimic the performance of a range of compensation levels for return period zones within the flood storage area. The random element consisted of Bernoulli trials of flood events conditioned by the return period of flooding, with a further random drawing for month of flooding, based on the empirical data of the region. The output from the program displayed histograms of the state of the fund after a specified elapsed time. These included ten years, which was thought to be the planning horizon for a landowner, and 30 years which was designated as the period at which the flood protection authority was due to re-evaluate the compensation.

In inspecting the results of the simulations with r=5 per cent, the tria1 discount rate for public sector schemes, special attention was paid to the viability of the fund after 10 years. The starting point was the behaviour of the fund capitalized with PV. This displayed a variable pattern of extinction from less than 10 per cent to more than 20 per cent within the 10 years. Such risk levels were regarded as too high and also the variation from zone to zone was undesirable. After studying sensitivity to interest rate and balancing profitability after 30 years against risk of extinction within 10 years, it was felt that a 5 per cent risk of extinction within 10 years provided an equitable basis. Table (17.4) shows the results which provided a more acceptable basis for acceptance by the land occupants.

TABLE (17.4) Consequences of compensation in flood zones

Return period Present Capitalisation zone value for 5% risk years &/ha £/ha

Profit after 30 years Probability Expectation

£/ha

10 - 15 1255 1630 0.71 1600 15 - 30 664 1090 0.79 1600 30 - 50 363 700 0.76 1200 50 - 70 239 620 0.83 1300 70 - 100 168 510 0.85 1200

17.7. Concluding remarks and apologia

This paper, dedicated to Jacques Bernier, has been a tour through several disparate regions of flood hydrology. They share very little, other than each is an attempt to bend statistics to a mould whose shape was determined by a real hydrological problem. While some of the applications are admittedly not recent, subsequent advances in hydrology bave passed by some of the issues. Perhaps the most notable exception is the issue of regional correlation. But even here modern solutions tend to approach the issue in an analytical fashion that does not lend itself to an intuitive grasp of the factors controlling 10~s of information due to correlation. However for a11 its popularity as a working tool no one seems to ask fundamental questions about regression analysis as a regionalization tool or the link between statistical and rainfall : runoff approaches to flood estimation. The question of a stable dependent variable for statistical analysis and standardization

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ought to be a “hot topic”. 1 am also aware that the mathematical treatment presented in Section (17.5) is

standard for deriving distributions of maxima for a wide class of generating processes. Nevertheless it merited a mention here because it had not been applied in the context of frequent flooding, nor to explore the generality of Langbein’s relationship between return periods.

The application of which 1 am proudest is that in Section (17.6). The reason is not the mathematical content; the need to simulate precluded much analysis. It is that it epitomizes the central point that the issue determines the approach. Also 1 have a feeling that, by expressing risk in a way that affects the pocket of the exposed group, one may obtain a more honest than usual appraisal of our tolerance to flooding. While the context was one of deliberate flooding it may provide salutory insight into protection levels in more conventional circumstances. Instead of asking “how much would you like society to spend on protecting you from flooding”, the alternative question could be posed, “how much would you accept as a one-off payment for society not to protect you from flooding”.

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