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Appendix A : Tests with frequency distributions 4.1 Introduction The General Extreme Value (GEV) distribution is detailed in Chapter 3. The Gamma distribution, another general distribution from which several other specific distributions derive, is outlined below. Then a computer program (FRAN) is described. This program was developed to enable an evaluation of different lrequency analysis methods on New Zealand flood data. It was found that the extreme value type I (EVl) distribution fitted by the Jenkinson method generally fitted the data well, but that in many cases the EVI distribution l'itted with Gumbel's method of leasl squares also gave satisfactory results. 4.2 Gamma distribution The three-parameter Camma distribution is the same as the Pearson Type 3 distribution. It has the pdl f(x) : I (x-xo¡r-le-(x-xo)/li .....A.1 0rr(r) which is defined for x > xo, where xo ù_ l)- I'(r): a location parameter, a scale parameter, a shape parameter, and the Gamma function, equal to ("y- 1)! for positive integer values of -y. lf "y : ¡, (x) describes an exponential distribution; and if xo = 0, f(x) describes a two-parameter Gamma distribution. Like the CEV distribution (section 3.1.3), Equation A.l describes a tämily of distributions, with each member characterised by the value of the shape parameter, in this case 7. This parameter is inversely related to the skewness of the variate, and as the skewness gets smaller, "y increases and Equation A.l tends to the Normal distribution (NERC 1975). When the skew is zero the symmetrical, two-parameter Normal distribution applies, with the pdf f(x) : I s- /tl(x- pl/ol' "F where ¡,r : a location parameter, and o : ascaleparameter. A2 The parameters ¡,t and o are, in fact, the population mean and standard deviation, respectively, of the variate x. An analogous situation to that described above applies tbr the three-parameter log-Gamma distribution. This distribution is the same as the log-Pearson Type 3 (LP3) distribution and has a pdf of the form f(x) : I (l'nx- xo)?- I e-(r)nx-xo)/B ..... 4.3 x0zf(r) which is defined for x > e"n. Like Equation A. I, Equation 4.3 describes a family of distributions, with each member being described by a particular value of 'y. When the skewness of the variate is zero, the two-parameter log-Normal distribution applies, with the pdf f(x) : I s- xoF /,1 (t'nx- p\/ øl' A4 which is defined for x > 0. The parameters p and r are now the population mean and standard deviation of the natural logarithms of the variate x. The df for Equations A.t to 4.4 must be calculated numerically. A.3 Methods used The GEV distribution described in section 3.1.3 and the Camma distribution outlined in section 4.2 have up to three parameters: a location, a scale and a shape parameter. These parameters must be estimatcd in the fitting ol a distribution to a data sample. The various techniques of parameter estimation, together with the choice of the parent distribution that may be used, give rise to the dilferent frequency analysis methods that are available. This study considered seven different frequency analysis methods. They were chosen on the basis of being the most common or the most useful, and they were incorporated in a computer program FRAN (Maguiness ef a/. in prep. a). (Another computer program FRANCES (section 3.1.7) was developed for use where historical information was available (Maguiness e! al. in prep.b).) The methods used in FRAN were the lollowing: (1) the three-parameter log-Camma or LP3 distribution fitted by the method of moments; (2', - the three-parameter log-Gamma or LP3 distribution, with an adjusted coefficient ol skew fitted by the method of moments' - (3) log-Normal distribution - fitted by the maximum likelihood method; (4) GEV distribution - fitted by the maximum likelihood method; Water & soil technical publication no. 20 (1982) (5) EVI distribution fitted by the maximum likelihood - method; (ó) EVI distribution - fitted by the least squares method; (7) EVI distribution - using the Jenkinson (1969) method. Each of the methods is briefly described below with reference to an annual series. In the case of methods I and 2, the distribution involved is subsequently referred to as the LP3 distribution. For a detailed explanation of the seven methods, refer to the report on FRAN by Maguiness e/ a/. (in prep. a). Method I This method was recommended by the United States Water Resources Council (1967) to be uniformly adopted in that country as the standard method for flood frequency analysis. The method in effect applies the threeparameter Gamma (Pearson) distribution (Equation A.l) to the logarithms of the annual series. The resulting frequency curve is a flexible one; it can plot concave upwards or downwards on log-Normal probability paper. It also incorporates the two-parameter log-Normal distribution, which plots as a straight line on the same paper. The fitting technique is the method of moments, which involves the calculation of the mean, standard deviation and the coefficient of skew of the logarithmically transformed series. These statistics are then used in the following equation to obtain the desired flood estimate. log'oX1 : X +K.S A5 where X1 : flood estimate for return period T, X : mean of the transformed series, S : standard deviation of the translormed series, and K : afrequencyfactor. 87
The liequency factor K is a function of the coefficient of skew and the return period and may be obtained from tables (e.g., Harter 1969; USWRC 1967). The form of Equation 4.5, which is based on the use of a frequency factor, is preferred to the more formal type of LP3 equation (e.g., Equation A.3) for the method of moments fitting technique, as it makes the computations very much easier. The frequency factor idea has heen propounded by Foster (1924) and Chow (1951), and the derivation of the factor for the LP3 distribution is explained by NERC (1975, pp. 39-40) and Kite (1976, pp. 198-204,2291. Method 2 This method is the same as Method l, except that an adjustment is made to the computed skew coefficient. An adjustment is warranted because the computed skew value is likely to be unreliable for a data sample of typical size. Indeed, it has been suggested (Beard and Frederick 1975) that at least 100 sample items are needed to obtain a skew value that is representative of the population statistic. Since most hydrological data samples are much smaller than this, various efforts have been made to improve the reliability of the computed skew value through the use of generalised skew coeflficients (Beard 1977). One example is the use of a regional skew value taken from isolines of computed skew values. ln the early stages of this New Zealand study, computed skew values lbr flow stations in the top half of the South Island were plotted on a map to determine if there was any pattern in the skew coefficient. None was evident and, hence, the possibility of using generalised skew coefficients in this study was not pursued. Instead, the following tactor Fu, recommended by Bobée and Robitaille (1975), was used to adlust for the bias in the skew value that is due to the length of the data sample. Fo= where CS = the computed skew coefficient, and n : the number of sample items. The adjustment is made by multiplying the computed skew coefficient by Fu, but only when Equation 4.6 is applicable i.e., for samples with 20 or more items. Mefhod 3 This method is often referred to as the log- Normal method and uses the two-parameter log-Normal distribution, as distinct from the three-parameter one (see Kite 1976). The method has long been advocated for use in hydrological frequency analysis (e.g., Hazen l9t4), and appeals because of its simplicity the fitted frequency distribution plots - as a straight line on log-Normal probability paper. The application of the method involves the same computations as for Method l, except that the coefficient of skew of the logarithms of the series is set to zero. Method 4 This uses the maximum likelihood (ML) method to fit the CEV distribution to a data sample. This method of fitting is generally recognised as the most efficient for estimating the distribution parameters, and its use is recommended when the design events must be extracted from a small or irregular series (WMO 1969). However, the ML method involves equations that have no explicit solution. The solution is complex and requires the use of an iterative numerical scheme, and is only worthwhile attempting with the aid of a computer. Method 5 Although the GEV distribution incorporates EVI as a special case, only rarely will the application of Method 4 result in the EVI distribution being fitted to a data sample. To ensure that a fit was obtained with the EVI distribution, this distribution was fitted separately (by the ML method) to the sample by setting the shape parameter k in the CEV distribution to zero. 88 ¡ 16-11- *ry i.i,* .Tì"... ou Method ó This method is often called the "Gumbel method" after Gumbel (1941, 1954) and is probably the one most commonly employed in hydrology. lt has had wide use in New Zealand and was the method adopted by the New Zealand Meteorological Service (Robertson l9ó3) when determining rainfall depth-duration-f'requency relationships from New Zealand data. Melhod 7 This method follows the procedure devised by Jenkinson (1955, 1969) and also described by Samuelsson (1972). The method emphasises the extreme part of annual series and as shown by Samuelsson, it can be applied as the standard one to extreme values which belong to several different kinds of frequency distribution. A larger series of S-year maxima is produced from the annual series by considering all possible combinations of items of five in the original series. The EVI distribution is then fittcd to rhe series of 5-year maxima by the ML method. If an annual series is used in a lrequency analysis instead of a series of 5-year maxima, it is quite possible that the series may be non-homogeneous in that, for example, the smaller items may belong to one distribution (e.g., EV2) and the larger ones to another (e.g., EV3). Further, it can be shown mathematically (WMO 1969) that the lower parr (37V0) of the series may not even belong to the extreme value distribution as it is defined. The advantage of the Jenkinson method is that it generally overcornes this problem of non-homogeneity of data. The use of 5-year maxima can be thought of an increasing by fivefold the degree ol independence in the data, so that these maxinra should then form a homogeneous set of data that confbrms to EV theory. 4.4 Evaluation of the frequency analys¡s methods 4.4.1 General Prior to the development of the regional curves, two evaluation tests were carried out on 42 flood records altogether, using the seven frequency analysis methods described in section 4.3 and contained in the computer program FRAN. The purpose of the tests was twofold: (¡) to observe, and to indicate to users of FRAN, the relative merits of the seven different methods on individual New Zealand flood records; (ii) to assist in the selection of a frequency distribution that would adequately describe the regional curves. This section describes the tests and discusses the results obtained. 4.4.2 Evaluation criteria and method Most studies that have attempted to discriminate between frequency analysis methods have relied, at least to some extent, on objective goodness-of-fit indices. Recent examples of such studies are those carried out by Benson (1968), Beard (1974), Kite (1976) NERC (t975), Kopiuke ef ø/. (1976) and Bobée and Robitaille (1977). However, as is generally acknowledged (e.g., Benson 1968), the classical goodness-of-fit indices such as Chi-square and Kolmogorov-Smirnov are not sufficiently sensitive or powerful enough, because of the small samples found in hydrology, to distinguish between the worth of different frequency analysis methods. Moreover, NERC (1975) found that other goodness-oi-fit indices had major weaknesses and concluded that, because of the deficiencies ol goodness-of-fìt indices, a visual inspection must be made of the probability plots. The judgement on the performance of a method is then a subjective one, "... but the objective tests that are available are so ineffective that their objectivity alone is insufficient to recommend them" (NERC 1975). In the evaluation tests, much more emphasis was placed on the probability plots than on the Chi-square value, which the computer program calculated. F-ollowing an ex- Water & soil technical publication no. 20 (1982)
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WATER & SOIL TECHNICAL PUBLICATION
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Regional flood est¡mat¡on in New
Page 5 and 6:
Tables 1.1 Risk ofexceedence for sp
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Preface Water & soil technical publ
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annual flood Q (the mean of the ann
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Doto ovoiloble on TIDEDA Figure 2'1
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SOUTHEBN REDUCEO Y VRBIßTE NETURN
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SOUTH ISLFND HEST COFST I]ÊTR èo
Page 35 and 36:
SOUTH CÊNTERBURY DRTÊ NEOUCEO Y V
Page 37 and 38: COMB I NED ,,IE5T CORST DRTR o rt
Page 39 and 40: BÊY OF PLENTY DRTF o o o NEOUCEO Y
Page 41 and 42: NOBTH ISLRND ßEGIONÊL CUBVES 1.50
Page 43 and 44: data for this area are required bef
Page 45 and 46: NELSON REGIONFL CURVE BEOUCEO Y VRR
Page 47 and 48: 46 EÊSTEFìN NEI,,I ZEFLÊND DÊTI
Page 49 and 50: Table 3.9 The grouping and the grou
Page 51 and 52: allowing curves rather than just st
Page 53 and 54: 3.5.5 Extension method The NERC (19
Page 55 and 56: Table 4.1 South lsland catchment ch
Page 57 and 58: I 2 t l¡ 5 5 7 I 9 l0 t1 ,12 I5 1{
Page 59 and 60: decided instead to use estimates of
Page 61 and 62: \Í l{ I t, ìj( Fþurc 4.4 Distrib
Page 63 and 64: 62 Fþurc 4,6 Logarithmic rcsidual
Page 65 and 66: Table 4.6 Final equations for South
Page 67 and 68: I ++ ++ * E 4.6 Analysis of North l
Page 69 and 70: Non Pumice = õ = 3-24xto'6 AREA'7
Page 71 and 72: Northlond f C"ro^ondel f Eost Cope
Page 73 and 74: Table 4.11 Comparable equations for
Page 75 and 76: Poo led cv Pooled o.40 F¡guro 4.12
Page 77 and 78: mates of Cp typically range between
Page 79 and 80: -J æ 'll o Eo !¡ Assemble llood p
Page 81 and 82: Year 1958 1959 1960 l96r t962 1963
Page 83 and 84: The estimate of Q from the regional
Page 85 and 86: Water & soil technical publication
Page 87: Maguiness, J.A.; Blackwood, P.L.; B
Page 91 and 92: e.9., Linsley et ol. (1915); lrish
Page 93 and 94: Maguiness, J.A.; Blackwood, P.L.; B
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Page 99 and 100: S ITI 29202 RuNttiltcl I ¡t tllEtt
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Page 103 and 104: 9100r GNE? R ¡T DOBSO| s¡18 93207
Page 105 and 106: 6800 1 sBlrri R tT SrITEC¡,tttS 6I
Page 107 and 108: APPENDIX C Summary of tho data for
Page 109 and 110: oæ ON LAT. STATION NA ATION LAT. N
Page 111 and 112: o NUI,IBER HHAKAIIARU 854801 38 L7'
Page 113 and 114: l9 AT NU¡IB ËR AR',tËtlANA ÂRAI
Page 115 and 116: A ON LAT. STATION NA NUMBER AHUNA 5
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Page 119 and 120: æ TATI ¡rouNT sor.tERS 7L740? 43
Page 121 and 122: N) o STATIoN NAr{gmLfï3---ffij NUI
Page 123 and 124: APPENDIX E Compadson of Regional Fl
Page 125 and 126: Table F.2 Flood peak data. NEW OTAG
Page 127 and 128: F.3 Analysis and results tatively p
Page 129 and 130: ét êt REOUCEO VßFIHTE 2. 33 5 l0
Page 131 and 132: SOLITH I EA l') COAST soUTH CANTERB
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Page 135: Water & soil technical publication
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