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Hydrological Sciences Journal

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Regional flood frequency analysis I: Homogeneitystatistics

S. E. WILTSHIRE

To cite this article: S. E. WILTSHIRE (1986) Regional flood frequency analysis I: Homogeneitystatistics, Hydrological Sciences Journal, 31:3, 321-333, DOI: 10.1080/02626668609491051

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Hydrological Sciences - Journal - des Sciences Hydrologiques, 31, 3, 9/1986

Regional flood frequency analysis I: Homogeneity statistics

S. E. WILTSHIRE Geomorphological Services Ltd, 1 Bassett Court, The Green, Newport Pagnell MK16 OJN, UK

ABSTRACT In the first of two papers the development of two statistical tests of region homogeneity for use in regional flood frequency analysis is described and the properties of the test statistics are examined. The preferred test is based on the distribution function of the regional parent and its power is shown to depend on region size, record length and choice of parent distribu­tion.

Analyse régionale de fréquences des crues I: études statistiques d'homogénéité RESUME Dans le premier de nos deux mémoires on décrit la mise au point de deux essais statistiques d'homogénéité d'une région pour utilisation dans l'analyse régionale de la fréquence de crues et on examine aussi les propriétés de ces tests statistiques. Le test qui a été adopté est basé sur la fonction de distribution de la population mère régionale et on démontre que sa puissance dépend de la superficie de la région, de la durée d'enregistrement et du choix de la distribution mère.

INTRODUCTION AND LITERATURE REVIEW

The estimation of a regional flood frequency distribution is a popular and practical means of providing flood information at sites with little or no local data. The derivation and use of a dimension-less flood distribution applicable to all drainage basins within a homogeneous region was first described by Dalrymple (1960). His index flood method is well known and the aspect of this procedure which is of interest here is the identification of a region that may be regarded as hydrologically homogeneous. The average dimension-less flood frequency curve can be defined reliably only if the component curves of the individual stations have comparable slopes. A test of homogeneity should therefore be a prerequisite of any regional flood analysis.

Dalrymple (1960) recommended a test which compares the varia­bility of 10-year flood estimates from each site in the region with that expected if sampling error alone were responsible for between-site differences. There have been several applications of this test: Dalrymple (1960) in Pennsylvania and Maryland, USA; Cole (1966) in England and Wales; Biswas & Fleming (1966) in Scotland;

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and Chong & Moore (1983) in Illinois. In these applications the regions being studied have been found in each case to be homogeneous suggesting either that the test may not be particularly powerful or that a wide variety of flood series and of basin types is consistent with homogeneity. Benson (1962) pointed to deficiencies in the test but despite these no alternative procedure has been advocated in the literature. Indeed it is the exception rather than the rule for the issue of region homogeneity to be tackled at all. In the UK Flood Studies Report (Natural Environment Research Council, 1975), for example, the British Isles is divided into 11 geographical regions prior to frequency analysis but with no statistical testing of the hydrological uniformity of the regions. Likewise Matalas et al, (1975) divided the United States into just 16 geographical regions for their demonstration of the so-called separation effect, in which regional collections of flood data show a much greater variability of skewness than would be expected from data which are distributed according to any one of a number of recognized flood distributions. It is not unreasonable to suppose that regions of such great size would be heterogeneous in terms of station skewness and, in the absence of a test of their homogeneity, comparison of these regional data sets with theoretical expectations may be misleading.

There have been some attempts to identify flood regions by methods other than geographical delimitation. These "alternative" techniques allocate basins to regions or groups by one of two general processes:

(a) classification by statistics of the basin flood frequency distribution;

(b) classification by basin characteristics. For example, in New Zealand, Mosley (1981) used cluster analysis to form groups of basins characterized by specific mean annual flood and coefficient of variation, Waylen & Woo (1984) have formed groups of basins in British Columbia on the basis of five different parameters of the peak-over-threshold flood series, and differen­tiated between them using discriminant analysis. The present writer has experimented with iterative search techniques which locate optimum divisions of the basin characteristic dataspace based on analysis of variance of flood statistics (Wiltshire, 1985, 1986a). Examples of the second approach to basin classification include the work of White (1975) who used a factor analysis of basin characteris­tic data to identify collections of physically similar basins in Pennsylvania. More recently, Acreman & Sinclair (1985) have used hierarchical cluster analysis on a matrix of basin characteristics to identify similar basins in Scotland.

These two types of approach enshrine different classification philosophies although they employ the same algorithms; it is sufficient to say here that any practical technique for forming regions should embrace information drawn both from the flood data and from the hydrologist*s understanding of basin characteristics. Both approaches represent welcome alternatives to arbitrarily-defined geographical regions and are likely to produce "regions" of a more homogeneous nature, but a statistical test of homogeneity is still lacking. The following sections describe two approaches to developing homogeneity tests, and their properties and power are examined using simulation experiments.

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HOMOGENEITY TESTS

A homogeneous flood frequency region will contain annual maximum flood populations whose flood frequency relationships have similar slopes on a probability plot. Since the slope of a probability plot is related to the coefficient of variation (CV) of the standardized annual maximum flood series it is reasonable to develop a homogeneity test based upon the regional variability in the site CV's. Such a test would have the advantage of being relatively distribution-free and constitutes the first approach described here. A second, alternative, approach is to select a priori the type of parent distribution applicable to the region and to examine the scatter of the site data about the fitted region-average distribution.

CV-based test

Let nj denote the record length at site j and CVj the sample coefficient of variation at the jth site. Then U j , the sampling variation of CVj, is given by:

V u. = — (i) J nj

where V is the regional variance of CV. It is natural to express the total variation, S, of CV within a

region of N sites by:

N (CVj - CV.) 2

S = IT * — (2)

where CV is the weighted region-average value of CV given by:

N £ cv./u.

cv- = -Jh7u7jL (3)

J

The variance term, V, in equation (1) can be calculated as the average of the jack-knife variances computed for each site within the region. The jack-knife estimation of variance is described by Parr (1983) but modification is needed in the current case of multiple samples from the N sites.

Let the flood series at the jth site consist of nj independent and identically distributed floods, Q]_,...,Qn, and let CVn denote the coefficient of variation of this series. Let CVn_j denote the CV computed from a sample of size (n - 1) consisting of Qj_, . . . , Qi-1> Qi+ii•••>Qn> (i.e. the CV of the series with the ith obser­vation removed). The jack-knife variance estimate of CVJ is the variance of the set of n CV values, each value being calculated from the series with one different observation removed, i.e.

v = (n - 1) E.J [CV - (Z„J CV )/n.]/n, (4) 3 J i k n-1 3 3

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and the regional variance V across N sites is:

1 N V = ± Z. ,n.v. (5)

N 3=1 3 3

The statistic, S, of equation (2) has the form of a x2 statistic and S is expected to be distributed as x with (N - 1) degrees of freedom. If the value of S exceeds the critical value of x I o r

(N - 1) degrees of freedom for a particular level of significance then the hypothesis of a homogeneous region must be rejected and the region must be regarded as heterogeneous.

The assumption of a X2 distribution for S can be tested using Monte Carlo simulation. Data can be randomly generated from a known flood-like distribution for a region of N sites each with n years of data. If this is done hundreds of times, the mean and variance of S can be computed and compared with the expected x2 mean and variance of (N - 1) and 2(N - 1).

Table 1 shows the mean and variance of the S statistic computed

Table 1 Mean and variance of the S statistic observed in 1000 simulations for a homogeneous region

Record length (n)

5

10

20

50

Number of sites (N): 10 20

7.1 16.7

8.2 17.6

8.5 17.6

9.1 17.7

14.5 31.8

16.6 36.2

17.6 38.0

18.9 35.2

50

37.7 85.0

42.1 84.5

45.1 86.1

48.7 86.0

100

75.5 194.1

84.8 188.9

92.5 189.7

98.9 191.1

from 1000 realizations of regions with N = 10, 20, 50, 100 and n = 5, 10, 20, 50. Data were generated from an EV1 (Gumbel) distribution with parameters u = 10, a = 4, implying a CV of 0.417. The regional mean of S is the upper number in each pair of values in the body of the table. Examination of Table 1 reveals that the mean of S is consistently less than its expected value, (N - 1), and that it is asymptotic to this value as the record length increases. The variance of S is also consistently less than its expected value, the estimate apparently improving as the region size increases. These two effects indicate that the test is likely to be effective only for large regions having long record lengths and that for more typical situations the assumption of a x2 distribution would not be justified. The consequence of using the test would be to accept a hypothesis of homogeneity too frequently since the value of the calculated S statistic would consistently be too small. The reason for this poor result probably lies in the fact that the variance, V,

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is a sample-based estimate and is correlated with the numerator of equation (2). In fact the calculated correlation between numerator and denominator from these simulation experiments varies between 0.4 and 0.6 which will clearly suppress both the variance and the mean of the S statistic. Whilst it would be possible to produce significance levels for this test from extensive simulations, it is clear that the CV test in its present form is unsuitable for use in assessing region homogeneity. The writer has, however, used this test in an earlier study (Wiltshire, 1986a), the results of which are not greatly diminished by the findings of the simulation experiments reported here.

Distribution-based test

The geometry of the cumulative distribution function (CDF) of a flood frequency distribution provides an alternative characteriza­tion of the flood data set. Its form depends on the relative frequency of floods of different magnitudes and also upon the parameters of the distribution from which the flood data are assumed to derive. One property of a CDF is that the marginal distribution of the non-exceedance probabilities is uniform with lower and upper limits of 0 and 1. Hence, a dimensionless regional frequency distribution fitted to a set of data from N sites will have a uniform distribution of non-exceedance probability. The relation­ship between the CDF, the probability density function (PDF) and this marginal distribution is summarized in Fig.l.

distribution of f(F)

f(F) q

Fig. 1 Relationship between the CDF, PDF and the distribution of non-exceedance probabilities.

Consider a random sample (x1,x2, . . . .x. , population. A value of non-exceedance probability

,xn) drawn from such a

G. = G(x.) J J

(6)

can be calculated for each sample point, x^, (i.e. each dimension-less annual maximum flood). These values, (Gj,G2,...,Gj,...,Gn), should have a uniform distribution apart from deviations caused by random sampling effects.

Conversely, now consider a random sample, (y^,y2>•.•,yn), which is not drawn from the same population as above. Let the same

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distribution function of equation (6) be used to calculate the non-exceedance probabilities:

G. = G(y.) (7) J 3

Since equation (6) does not represent the distribution function of the population from which the y sample is drawn, the distribution of the set of values, (Gj_,G2, . . . ,Gn) , will now differ from the uniform distribution by an amount greater than could be attributed to sampling effects alone. Figure 2 shows some possible consequences

Fig. 2 Distribution of f(F) of CDF when population may be either type (a), (b) or (c) but the population is assumed to be type (a).

of assuming the wrong parent distribution for a particular series of floods. The true PDF is (a), but if it is assumed to be (b) or (c) then a non-uniform distribution of the GJ'S results. Thus if a region is homogeneous, the dimensionless annual maximum series from each of the N sites in a region would all be expected to show a uniform distribution of non-exceedance probabilities (termed here the "G-points"), with deviations from this expected geometry due solely to sampling error. Here lies the basis for the second test of homogeneity.

The formulation of this test is similar to that of the CV-based test , and shares the same requirement of a single measure to describe (in this case) the set of G-points at each site. The mean G-point is an obvious candidate, and if a flood series did follow exactly the regional (parent) distribution, the site mean should be close to 0.5. However, flood series with a high CV relative to the parent would have a greater proportion of G-points near the limits of the 0-1 range but with the sample mean still likely to be 0.5. Similarly a series with a low CV relative to the parent would show a clustering of G-points around the 0.5 value and again the sample mean is likely to be at or near 0.5. In other words, the mean G-point cannot distinguish between high CV's, low CV s and CV's comparable to that of the parent. A simple transformation can overcome this problem. The non-exceedance probability of the kth flood at the jth site, Gik, can be transformed by the expression

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Regional flood frequency analysis I 327

j k = 2 G „ - 0 . 1 Jk

( 8 )

The set of G'-points are thus the transformed G-points formed by "folding" the distribution to the left of the 0.5 point onto the right-hand part of the distribution. The expected result for sites with the same distribution as the parent is another identical uniform distribution. At sites whose data do not derive from the parent, systematic departures of G-points from the expected horizon­tal uniform distribution will be effectively amplified by the folding process. Figure 3 shows the result of folding five different

ffGi

f!Gl

flGl

flGj

(IG)

POPULATION

D 1

HIGH CV

- ^ _ ^ -) 1

LOW CV

^ ^ - " \ )

HIGH + ve SKEW

1

HIGH - ve SKEW

f'G'.

fiG'

^

f'G'i

fIG'l

••£>

f'G'i

Fig. 3 The effect of applying the transformation of equation (8) on the distribution of G-points.

hypothetical sets of G-points. In Fig.3(a) floods follow the same distribution as the parent and so the set of G points and G'-points are also uniform. The mean value of the G-points at site j , Gj _, is indicated by the arrow in Fig.3(a) and would be expected to be close to 0.5. In Figs.3(b) and 3(c) the effect of applying the trans­formation of equation (8) is respectively to increase and to decrease the value of Gj , indicating that the folded G-points can now distinguish between high and low CV data. Figures 3(d) and 3(e) show that the folding process will represent positively skewed and negatively skewed data in the same way, i.e. larger than 0.5.

Within a region of N sites, the variability, R, of the Gj_ values can be written in the same form as equation (2), i.e.:

with a value of G'-

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H(G'. - G' ) 2

R = ry~^ •-*— o) 3 ui

where u, is the sampling variance of G'. , given by the equivalent of equation (2), in which V is now a constant and equal to one-twelfth. The region-average value of G' is defined as:

G' = EN n. G' /EN n. (10) 3 3 3 - 3 3

The R statistic is expected to be distributed as x2 with (N - 1) degrees of freedom. As with the CV test, the degree to which R behaves as a x2 statistic can be examined by simulation experiments. Table 2 shows the mean and variance of R computed from 1000 realiza­tions of each of 16 different configurations of regions. In contrast

Table 2 Mean and variance of the 12 statistics observed in 1000 simulations for a homogeneous region

Record length (n)

5

10

20

50

Number of 10

9.6 17.8

9.5 19.7

9.0 17.2

8.7 16.0

sites (N): 20

19.9 34.9

19.3 40.2

18.6 32.1

18.6 35.2

50

51.1 96.7

49.5 90.9

48.4 91.0

48.0 95.0

100

103.8 195.9

100.4 197.9

98.0 197.7

97.7 197.1

to the results of the CV experiments of Table 1, it can be seen that the calculated values of the mean and variance follow closely the expected values of (N - 1) and 2(N - 1) across the range of region configurations. The mean value of R appears to be biased slightly upward for shorter record lengths, and the variance shows some downward bias for small regions, but overall this performance is much superior to that of the CV-based statistic and the R statistic may be more reliably assumed to have an x distribution.

POWER OF THE R-STATISTIC TEST

A test statistic of region homogeneity has been defined in equation (9), but so far it is not clear how powerful this test will be when applied to real data. In other words, how heterogeneous must a collection of data be before the computed R-statistic indicates rejection of the null hypothesis of homogeneity? The power of the test may be explored through the use of simulation experiments, and

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in particular a power curve can be constructed which plots the rate of H0 rejection against a measure of region heterogeneity. Produc­tion of a power curve from simulation experiments uses the following procedure:

(1) For a region of N sites and n years of record at each site generate flood data from two parent distributions of known CV. Data for half of the sites come from the first distribution, with CV = CV^ and data at the other sites are generated from the second distribution with CV = CV^.

(2) Initially CV]_ = CV2 so the region is homogeneous. If a 5% level of significance is adopted, then in a large number of generated data sets, 5% of the R values should exceed the critical (X2) value.

(3) Subsequently, CVj is held constant and CV2 is incremented, to produce a heterogeneous region characterized by the ratio CV3/CV2. Data are generated from these two parents and the frequency with which R exceeds the critical value from repeated realizations is plotted against the ratio CV1/CV2.

(4) Stage (3) is repeated several times with successive incre­ments in CV2 and hence an increasing degree of regional heteroge­neity.

This technique can be used to examine the effect on the R-test of the length of record, the number of sites in a region and on the choice of frequency distribution which is fitted to the regional data. In this study data are generated in 1000 repetitions from a pair of EV1 distributions with CV calculated from:

CV = ™ ^ f - (11) u + 0.5772a

The parameter u is fixed at a value of 10.0 for both distribu­tions; a is set to 4.0 for the first distribution but is incre­mented to produce increasing values of CV2 in the second distribu­tion. A regional EV1 distribution is fitted to the generated data using the method of probability-weighted moments (PWM's) as described by Greis & Wood (1981). A G-point is calculated from the EV1 distribution function for each generated data point as:

G(y) = exp(-e-y) (12)

where y is the reduced variate given by:

y = (x - u)/a (13)

where x ., is the kth flood at the jth site. Figure 4 shows power curves for four hypothetical regions, all

with 20 sites but with respective record lengths of 5, 10, 20 and 50 years at each site. The four curves all pass through the 5% point on the y-axis, indicating that when the two parent distributions in the region are equivalent (CV]/CV2 = 1) there is only a 5% chance of rejecting the null hypothesis. It can be seen that the R-test is more powerful when applied to a region containing sites with longer records, e.g. when CV2/CVj = 0.75, p(reject HQ) is 90% for the n = 50 years region, but only 12% for the n = 5 years region. This is qualitatively the expected result as the greater sampling errors

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n = 50 n=20

n= record length EV1 worlds; EV1 (pwrnj fit

Each region has N= 20 sites

Ho: region is homogeneous

Fig. 4 Power curves for the R test: variable record length.

or "noise" associated with the shorter records would be expected to obscure the genuine heterogeneity or "signal" within the region.

In producing the power curve of Fig.5 the record lengths in each region are held constant at n = 20 years, and the four regions differ in terms of the number of their constituent sites. Again the

N=50 ,N=20

EV1 worlds; EV1 fpwm] fit

Each site has n = 20 years of record

Ho: region is homogeneous

Power curves for the R test: variable region size.

intuitive result is seen, namely that a given degree of heterogeneity is more readily detected in a region containing a greater number of sites. A comparison of power curves in Figs 4 and 5 based on regions having the same total number of station-years (N = 50, n = 20 and N = 20, n = 50; N = 20, n = 10 and N = 10, n = 20) shows that the information content of the data is greater when it is deployed in a small number of long records rather than in a large number of short records. Two other conclusions can be drawn from these sets of curves. Firstly, if the series available for a regional analysis are typically 10 years or less in length then there will rarely be sufficient information present to detect heterogeneity even when

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there are gross differences among the constituent flood series. Secondly, for a typical region with N = 50 and n = 20, the R-test is of only moderate power, although alternative types of test are unlikely to do any better. The application of the test to real data should therefore be made with the power of such tests in mind, and the interpretation of the R-statistic as a significance test should not be too literal.

A further set of simulation experiments can be performed to examine the effect on the power of the R-test of different dis­tributional assumptions. In Figs 4 and 5 a two-parameter EV1 distribution was fitted to the simulated regional data using the PWM fitting technique but more flexible distributions might be preferred. The three-parameter General Extreme Value (GEV) dis­tribution has been widely used in the study of flood and rainfall maxima, and Hosking et al. (1985) have shown how its parameters may be fitted by PWM's. The distribution function of the GEV is:

G(y) = exp[-(l - ky)1/k] (14)

where y is the reduced variate of equation (13) and k is a shape parameter. The five-parameter Wakeby distribution can also be fitted using PWM's (Landwehr et al., 1979). The Wakeby distribution function must be expressed in inverse form:

a[l - (1 - G) ] - c[l - (1 - G) u] (15)

so a technique such as Newton-Raphson iteration must be used to solve for G given the flood, x. Figure 6 shows power curves for the R-test, fitting regional EV1, GEV and Wakeby distributions to the two-parent EV1 regions. The EV1 curve is the same as the N = 20, n = 20 curves in Figs 4 and 5. The GEV and Wakeby power curves lie well to the right of the EV1 curve, signifying a lower H0 rejection rate than the EV1 power curve for a given degree of heterogeneity. With more parameters with which to characterize the data, the GEV and Wakeby distributions are more tolerant of heterogeneous effects

10

Fig. 6

o* S

EV1 worSds; pwm fits

Region has 20 sites with 20 years of record

Ho: region is homogeneous 0-9 0-8 0-7 0-6

Power curves for the R test as a function of distribution type

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in the data and so the region is more likely to appear homogeneous.

CONCLUSIONS

A significance test of regional homogeneity has been developed based on the folded non-exceedance probabilities of individual flood maxima. The power of the test is seen to improve with larger numbers of sites and longer record lengths and this will be true of any homogeneity test. The test is distribution-specific, but this is not seen as a disadvantage since a choice of regional flood distribution is ultimately necessary and is generally made indepen­dently of the choice of regions. The following paper (Wiltshire, 1986b) describes the application of this test to regional data sets and to regions formed by clustering basins within a flood statistic dataspace.

ACKNOWLEDGEMENTS This work was conducted while the writer was employed at the Institute of Hydrology, Wallingford, and forms a part of a Ministry of Agriculture, Fisheries and Food project on regional flood statistics. The writer is grateful to David Jones for his assistance in formulating the test statistic described above, and to the anonymous reviewer whose informed comments and suggestions helped greatly in the preparation of this manuscript.

REFERENCES

Acreman, M.C. & Sinclair, C D . (1985) Regional flood frequency analysis from basin characteristics in Scotland. Manuscript in preparation.

Benson, M.A. (1962) Evolution of methods for evaluating the occurrence of floods. US Geol. Surv. Wat. Supply Pap. 1550-A.

Biswas, A.K. & Fleming, G. (1966) Floods in Scotland: magnitude and frequency. Wat. Wat. Engng, 246-252.

Chong, S. & Moore, S.M. (1983) Flood frequency analysis for small watersheds in southern Illinois. Wat. Resour. Bull. 19(2), 277-282.

Cole, G. (1966) An application of the regional analysis of flood flows. In: Symposium on River Flood Hydrology, Session B, 39-57. Institution of Civil Engineers, London.

Dalrymple, T. (1960) Flood frequency analyses. US Geol. Surv. Wat. Supply Pap. 1543-A.

Greis, N.P. & Wood, E.F. (1981) Regional flood frequency estimation and network design. Wat. Resour. Res. 17(4), 1167-1177.

Hosking, J.H.M., Wallis, J.S. & Wood, E.F. (1985) Estimation of the Generalised Extreme Value distribution by the method of probabi­lity weighted moments. Technometrics 27(3), 251-261.

Landwehr, J.M., Matalas, N.C, & Wallis, J.R. (1979) Estimation of parameters and quantiles of Wakeby distributions. Wat. Resour. Res. 15, 1361-1372.

Matalas, N . C , Slack, J.S. & Wallis, J.E. (1975) Regional skew in search of a parent. Wat. Resour. Res. 11(6), 815-826.

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Mosley, M.P. (1981) Delimitation of New Zealand hydrological regions. J. Hydrol. 49, 173-192.

Natural Environment Research Council (1975) Flood Studies Report, vol.1. NERC, London.

Parr, W.C. (1983) A note on the jack knife, the bootstrap and the delta method estimators of bias and variance. Biometrika 70(3), 719-722.

Waylen, P.R. & Woo, M. (1984) Régionalisation and prediction of floods in the Frazer river catchment, B.C. Wat. Resour. Bull. 20(6), 941-949.

White, E.L. (1975) Factor analysis of drainage basin properties: classification of flood behaviour in terms of basin geomorphology. Wat. Resour. Bull. 11(4), 676-686.

Wiltshire, S.E. (1985) Grouping basins for regional flood frequency analysis. Hydrol. Sci. J. 30(1), 151-159.

Wiltshire, S.E. (1986a) Identification of homogeneous regions for flood frequency analysis. J. Hydrol. (in press).

Wiltshire, S.E. (1986b) Regional flood frequency analysis II: multivariate classification of drainage basins in Britain. Hydrol. Sci. J. 31(3), 335-346.

Received 21 June 1985; accepted 24 January 1986.

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