PLOTTING FORMULA FOR PEARSON TYPE III DISTRIBUTION CONSIDERING
HISTORICAL
INFORMATION
VAN THANH-VAN NGUYEN and NOPHADOL IN-NA Department of Civil Engineering and Applied. Mechanics, McGill University, Montreal. Quebec, Canada H3A 21(6.
(Received August 1991) Abstract. Most of the existing plotting position formulas have been proposed for use in the analysis of systematic flood records, but little has been reported on the plotting formulas for historical or non-systematic flood samples. In particular, no previous investigations have specificallyexamined the probability plots for the Pearson type I II (P3) distribution in the analysis of historical flood information. The present paper suggests a new plotting position formula for the P3 distribution for use with both systematicand historical flood records. The proposed formula has a simple structure as do most existing formulas, but it is more flexiblebecause it can take explicitlyinto account the skewnesscoefficientof the underlyingdistribution. Further, results of graphical and numerical comparisons have demonstrated that the suggested formula provided the least bias in flood quantile estimation as compared with many available plotting formulas, including the well-known Weibull formula. Finally, results of a numerical example using actual flood data have indicated the practical convenience of the proposed plotting formula. It can be concluded that the formula developed in this study is the most appropriate for the P3 distribution in the analysis of flood records consideringhistorical information.
1. Introduction The use of probability plotting positions or probability plots in hydrologic frequency analyses has been popular with engineers and hydrologists for several decades following the publication of a study by Hazen in 1914 on the analysis of flood flow data. Since the appearance of this study, the subject of plotting positions have been widely discussed in the hydrological and statistical literature (see, e.g., Weibull, 1939; Gumbel, 1943; Blom, 1958, Kimball, 1960; Gringorten, 1963; Cunnane, 1978; Adamowski, 1981; Xuewu et al., 1984; Arnell et al., 1986; Nguyen et al., 1989; and In-na and Nguyen, 1989). In particular, most of the existing plotting formulas were proposed for use in the analysis of systematic flood records (i.e. complete flood samples which occurred during the period of systematic gauging). Such hydrometric records cover usually a relatively short period o f time. The probability estimates of rarer flood events is then poor. Such estimates could be significantly improved by the consideration o f h i s t o r i c a l f l o o d data (i.e. data on very large floods which occurred either outside or within the systematic gauging period) as shown by m a n y recent investigations (Condie and Lee, 1982; Stedinger and Cohn, 1986; Hirsch and Stedinger, 1987; Sutcliffe, 1987; Jin and Stedinger, 1989). However, few studies have been reported on the plotting position formulas for historical or non-systematic flood records (e.g., Zhang, 1982; Hirsch and Stedinger, 1987; Shi-Qian; 1987; Guang-Yan; 1987). In particular, no cited investigations have examined the probability plots for three-parameter distributions, such as the Pearson Type III (P3) distribution. In view of the important role of the P3 distribution in hydrologic frequency analyses, Environmental Monitoring and Assessment 23: 137-152, 1992. 9 1992Kluwer Academic Publishers. Printed in the Netherlands.
138
VAN
THANH-VAN
NGUYEN
AND
NOPHADOL
IN-NA
Nguyen et al. (1989) have proposed an unbiased plotting formula specifically for this distribution and for the analysis of systematically recorded flood series. The present study, a continuation of a previous one, emphasizes the development of a more general plotting position formula for the P3 distribution and for use with both systematic and historical flood records. Results of graphical and numerical comparisons have demonstrated that the plotting position formula proposed in this study can provide the least bias in flood quantile estimation as compared with several existing formulas. Note that only the bias in flood quantile estimates is examined in this study. The question of whether plotting positions should be unbiased in terms of exceedance probabilities or unbiased in term of flood estimated is not addressed here. This question should be best considered in the context of the use which is to be made of the results. Such consideration is beyond the scope of this paper. Nevertheless, the advantages related to the use of unbiased plotting positions in the estimation of flood quantiles have been shown in many previous works (see, e.g., Kimball, 1960; Cunnane, 1978). Further, the flood record length in the present study is assumed to be known. The effect of misspecification of the true record length on flood quantile estimates have been previously studied by Hirsch and Stedinger (1987), and Hirsch (1987).
2. Plotting Position Formulas for Historical Floods In this study, to describe the various plotting positions for historical flood records, some standard notations are introduced. Consider a historical period of Nyears in which a total of g floods were observed. This N-year period needs not be continuous and contains a systematic record period ors years during which all annual floods were recorded (s_< g < N). Among the g observed floods, k of them are known to be the k 'extraordinary floods' in a period of N years. Some of these k large floods, e, may have occurred during the systematic flood records (e _~ k and e m} is distribution-free. The conditional expectation E[Q,]P~,k] is estimated by Monte Carlo simulations using a total of 20 000 repetitions over the set of k values which have non-negligible probabilities. The expected value of the discharge for the given ['~ values is found by: E[F-'(1-P,,,)]= ~ k
F-'[1-~P,,,(k)]P{klk>_m}
(11)
tll
where F -~ [1 -Pm (k)] is the discharge with exceedance probability of P,, (k) which is estimated using a plotting position formula. In the following, comparison of various plotting position formulas for P3 distribution is performed by comparing the bias in the estimation of the largest flood discharge from a sample of size N = 150, a historical flood sample commonly available in practice. Further, the proposed E-P3 formula is compared only with those formulas that were recommended for use with the P3 distribution. Since the symmetrical normal (3' -- 0), and skewed exponential (7 = 2) distributions are special cases of the P3 distribution and, in particular, there exist probability papers and plotting formulas especially derived for these distributions, the normal and exponential distributions are selected for the comparison. More specifically, the E-B, based on Blom's (1958) formula, the E-A formula, given by Adamowski (1981), and the E-W, suggested by Hirsch and Stedinger (1987), are considered for this comparison. For 3 / = 2, it is more appropriate to replace the E-B formula by the E-C formula derived from Cunnane (1978) for the P3 distribution. Finally, the Weibull formula will be considered in the present comparison because of its popularity in engineering practice, even though this formula was not derived for the P3 distribution and for historical records. Results of the comparison of plotting position formulas are shown in Figures 1,2, and 3, respectively for normal distribution (3~ = 0), for exponential distribution (7 = 2), and for the case of 3, = 1. In all cases the coefficient of vriation is Cv = 0.50. Note that Figures 1-3 present E[Q~] as a function of E[k] where Elk] = NPe. Note also that the expected value of the largest flood discharge, E[Q~], estimated by Monte Carlo simulation procedure is represented by the curve marked 'true' in the above figures. For the normal distribution (',/= 0), it can be seen from Figure 1 that the E-B formula performs very well, as expected, especially for low values of Elk], because Blom's formula
142
V A N
T H A N H - V A N
N G U Y E N
A N D
N O P H A D O L
I N - N A
was specifically derived for the normal distribution. Further, as compared with the E-B, the proposed E-P3 formula performs equally well. Results obtained by these two formulas are better than those given by the E-A, E-W, and Weibull formulas. The well-known Weibull formula was found to be the most biased as compared with the others. ?.g II-.IL~
I'-JL . . . .
II-W 'Ira IBQ~Ir~
o 9
~
~
8.S
0
*
|
Fig. 1.
9
6
"
8
~
m
~
~
~
~
9
.
.
.
.
.
9
*
I0
20
.
.
~.(k) 80
Bias in discharge for normal distribution, 7 = 0.
In the exponential case (7 = 2), Figure 2 indicates clearly the best performance of the E-P3 formula. Moreover, it is noted that, although E - C formula was specifically recommended, for the P3 distribution, this formula performs slightly better than the E-A, E-W, and Weibull formulas, especially for low value of E[k]. Finally, results for the P3 distribution with ",/= 1 are similar to those for the exponential case as indicated in Figure 3. In summary, for a symmetrical normal distribution the E-P3 formula gave comparable performance as the E-B formula. However, for a skewed distribution the E-P3 formula performs much better than other existing formulas. Therefore, it can be concluded that the E-P3 formula developed in this study is the most appropriate for the P3 distribution for the analysis of historical flood data.
PLOTTING
F O R M U L A
FOR
PEARSON
DISTRIBUTION
143
8.8
. . . .
!1
......
W~
-:.-.-:-...:.....7..Z
-
Q Q Q
Q I
Q
Q O ~
O Q
m
n
~ 0
6~
4.El
*
J
|
Fig. 2.
9
9
,
Z(k)
9 9
10
IO
~10
Bias in discharge for exponential distribution, 3/= 2.
4. Application of Plotting Position Formulas The application of the proposed plotting formula to actual flood samples with historical information will be illustrated by using flood data of the Huangbizhuang river at Huangbizhuang, China (UNESCO, 1987). For this river, the record of annual maximum floods is available from 1794 to 1974, and there exist two very large floods occurred during the period of systematic gauging (1949-1974). Thus, this data set represents a good example for assessing the performance of'exceedance' plotting formulas (E-P3 and E-W) as compared to that given by their traditional counterparts (P3 and Weibull). Results of this comparison would indicate the importance of historical information in the estimation of floods. In the present study, to evaluate the performance of various plotting formulas, the flood quantiles estimated by these formulas are compared with those computed by some conventional distribution methods. For systematic flood series, the method of moments have been frequently recommended for use in the estimation of the of parameters of the P3 distribution (Bobee and Robitaille, 1977; UNESCO, 1987). In the case of historical flood records, the method of historical weighted moments which has been widely used in practice (see, e.g., Condie and Lee, 1982) is employed in the fitting of the P3 distribution to flood data.
l~
VANTHANH-VANNGUYENANDNOPHADOLIN-NA 8.5
B-'P~ r . . .c
E-".A . . . .
IB.-~ 9 ~EI B L U , ~
6.0
5.6
9
~
9
~ U
5o0
9 |
~
9
9
Q
O
O
D
Q
D
O
~.(t,)
m m 10
20
SO
Fig. 3. Bias in discharge for P3 distribution, ",/= 1.
4.1. APPLICATION OF PLOTTING FORMULAS TO SYSTEMATICFLOOD RECORDS As mentioned above, there exist two very large floods in the systematic flood record for the Huangbizhuang river. The purpose of the present example is to assess the performance of the P3 and Weibull formulas when they are applied to this particular data series. Figure 4 and Table I show, respectively, results of the graphical and numerical comparisons of flood quantile estimates. As compared to the Weibull formula, it is found that the P3 formula gives flood estimates closer to the values computed from the fitted P3 distribution. However, it can be seen from Figure 4 and Table I that, the P3 distribution did not give a good fit to the observed data, because the flood records are not censored in order to consider some extreme floods above a given threshold as historical floods. The bias is most pronounced at the upper end of the plot. Hence the P3 formula, which has been recommended for systematic flood records (Nguyen et al., 1989), is not appropriate for cases where historical information exists in the data samples. Therefore, the use of new exceedance-based formulas suggested in this study would significantly improve the fitting of theoretical distribution to the observed historical flood data, as will be shown in the following sections.
PLOTTING
FORMULA
FOR
PEARSON
0 (P3) O (WEIBULL)
I-1 O
145
DISTRIBUTION
~x
.x
CO E
CD
X (I/
>
L OJ 69 n 0 0
F 0
I 20
I 40
~
[]
I 60
I 80
I 100
I 120
140
O fitted (x i000 m3/s) Fig. 4. Quantile-quantile plot of P3 and Weibull formulas (Huangbizhuang River, Cs,, = 3.0, N = 25).
4.2. APPLICATION OF PLOTTING FORMULAS TO HISTORICAL FLOOD RECORDS One of the difficulties in the analysis of historical flood records is the determination of a perception threshold or base level Q0 above which all floods are considered as historical
floods. The base level Q0 is often selected as the smallest known historical flood (Hirsch and Stedinger, 1987). In the present example, using historical flood data of the
Huangbizhuang river, a sensitivity analysis is performed to illustrate the effects of selecting different base levels on the performance of the fitted distribution. In particular, the best fit condition is assessed based on the graphical comparison between empirical
(plotting position) and fitted probabilities, as well as on the numerical comparison between flood estimates from plotting formulas and fitted theoretical distributions in terms of the minimum root mean square error (RMSE) and the smallest absolute
difference. A close examination of the flood record suggests some different base levels (9000 m3/s, 1000 m3/s, and 13000 m3/s) which could be selected for the sensitivity analysis. Results of graphical and numerical comparisons for the E-P3 formula and the fitted P3 distribution are shown in Figures 5-7 and Table II, respectively. It can be seen that, with a base level selected at 9000 m3/s, the theoretical P3 distribution shows the best fit to the observed data (Figure 5). More specifically, at this base level, the RMSE and maximum absolute difference of flood quantiles estimated from the E-P3 formula and the fitted P3
146
VAN
THANH-VAN
NGUYEN
AND
NOPHADOL
IN-NA
TABLE I Numerical comparison of P3 and Weibull formulas (Csu ----3,0, N = 25). Huangbizhuang River
P3 distribution
m
Qj~t,ea (m3/s)
P3
Weibull
Q (P3) (m3/s)
Q (Weibull) (m3/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
200 398 550 750 760 920 950 950 980 1100 1160 1260 1300 1330 1370 1800 1850 2080 2450 2550 3350 3700 3820 12000 13100
0.061 0.100 0.138 0.176 0.214 0.252 0.291 0.329 0.367 0.405 0.443 0.481 0.520 0.558 0.596 0.634 0.672 0.711 0.749 0.787 0.825 0.863 0.901 0.940 0.978
0.038 0.077 0.115 0.154 0.192 0.231 0.269 0.308 0.346 0.385 0.423 0.462 0.500 0.538 0.577 0.615 0.654 0.692 0.731 0.769 0.808 0.846 0.885 0.923 0.962
100 150 200 300 400 550 600 650 700 750 780 800 850 900 1000 1500 1750 2000 2300 3000 3500 4500 5800 7500 11500
50 120 180 250 350 480 580 620 680 720 760 790 830 880 970 1450 1700 1750 2100 2700 3200 4000 5000 6500 9300
1090 4500
1398 5500
RMSE Max I Q~,,ed- Ojo,,,~,aI
d i s t r i b u t i o n were f o u n d to be smallest ( T a b l e II). N o t e t h a t the bias increases for h i g h e r values o f Q0 (or for smaller n u m b e r s o f historical floods k). This result is in a g r e e m e n t with t h o s e g i v e n by the M o n t e C a r l o e x p e r i m e n t d e s c r i b e d previously. In the following, the base level at 9000 m 3 / s will be selected to o b t a i n the best fit o f the P3 d i s t r i b u t i o n to the o b s e r v e d historical f l o o d data.
4.3. COMPARISON BETWEEN PLOTTING FORMULAS FOR SYSTEMATIC AND HISTORICAL FLOOD RECORDS As s h o w n a b o v e , it is a p p r o p r i a t e to analyze historical f l o o d r e c o r d s using p l o t t i n g f o r m u l a s (e.g. the P3 f o r m u l a ) w h i c h were derived for systematic f l o o d samples. In this e x a m p l e , g r a p h i c a l and
numerical comparisons
o f f l o o d estimates f r o m f o r m u l a
r e c o m m e n d e d for systematic records, a n d f r o m those suggested for c o m b i n e d historic a n d systematic d a t a are carried o u t in o r d e r to give general i m p r e s s i o n o n the m a g n i t u d e o f the
P L O T T I N G F O R M U L A FOR PEARSON DISTRIBUTION
3ooooll PEARSON TYPE 3
147
PROBABILITY PAPER, SKEWNESS COEFFICIENT
=
3.0
-]'3
27000 ~:0RErIC&L
F I ~TI:D
24000
/
/
/
21000
/
I
18000 /
m
\ 15000 v12000
" !!H 9000
/
,J
~V
/6
/
6000
,/ I
[]
3000 0 .t.5
Fig. 5.
.8
.9
.98
.99
.998
.999
.9998
.9999
Comparison of EP3 formula and fitted P3 distribution for a base level Q0 = 9000 m3/s (Huangbizhuang River).
PEARSON
TYPE
3
PROBABILITY
PAPER,
SKEWNESS
COEFFICIENT
= 3.0
30000 27000 24000 21000 1BOO0
-ff
~
v
15000 12000 9000 6000 3000
i I I I I
f /
/
0 .1.5
Fig. 6.
.5
.9
.98
.99
.998
.999
.9998
.9999
P =
Comparison of E-P3 formula and fitted P3 distribution for a base level Q0 = 10000 m3/s (Huangbizhuang River).
148
VAN
THANH-VAN
NGUYEN
AN[)
NOPHADOL
IN-NA
PEARSON TYPE 3 PROBABILITY PAPER, SKEWNESS COEFFICIENT : 3.0
i
H
ooo, 24000 I itlt
I( ~L FI ['TgD
z/
21000 I
/
m 18000 ! /
\ '%
15000 0P12000 9000 6000 [ ~1
/6
~
J
. ! .g
Fig. 7.
z
I
J
I
.8
I
.9
.98
.99
.998
.ggg
.g998
.9999
Comparison of E-P3 formula and fitted P3 distribution for a base level Q0 = 13 000 m 3/ s (Huangbizhuang River).
differences between the two results. The performance of P3 and E-P3 formulas is assessed using flood data of the Huangbizhuang River. Further, the base level selected for this comparison is Q0 -- 9000 m3/s as indicated in the previous section. Finally, due to its popularity in engineering practice, the E-W formula is also considered in this comparative study. Figure 8 and Table III show respectively the graphical and numerical comparisons between P3 and E-P3 formulas. It was found that the E-P3 formula provides the best performance as compared with the P3 and E-W formulas. In addition, the use of the E-P3 for historical flood data provides the minimum RMSE and the smallest bias in flood estimates as compared to those given by the P3 and E-W (Table III). In summary, a recognition of the fundamental difference between a flood record with historical and systematic data, and a record which is entirely systematic is very important. It is clear from the above example that the use of exceedance plotting formula (E-P3) rather than its traditional counterparts (P3) could significantly improve the flood quantile estimation. 5. Conclusions
The following conclusions can be drawn from the present study: (1) For practical applications, a simple plotting position formula, which can take into account both systematic and historical flood records, has been developed for
PLOTTING FORMULA FOR PEARSON DISTRIBUTION
149
TABLE II Sensitivity of P3 distribution for different base levels (Huangbizhuang River). Huangbizhuang River Q0 = 9000 m3/s
Q~,rea
E-P3
Qo = 10 000 mS/s Q(E-P3)
E-P3
(m3/s) 200 398 550 750 760 920 950 950 980 1100 1160 1260 1300 1330 1370 1800 1850 2080 2450 2550 3350 3700 3820 9650 11500 12000 13100 13500 14750 17150 23750
0.077 0.116 0.155 0.193 0.232 0.271 0.310 0.349 0.388 0.427 0.466 0.505 0.544 0.583 0.622 0.661 0.700 0.739 0.777 0.816 0.855 0.894 0.933 0.965 0.970 0.974 0.979 0.983 0.988 0.993 0.997
RMSE Maxl Qj~,,e~-Qj~rm,ml
100 400 500 600 700 800 850 900 950 980 1000 1200 1250 1300 1400 1800 1850 2000 3000 3200 4100 5500 7000 10000 11200 12000 12900 13000 15100 17000 21900
Q(E-P3)
Q0 = 13 000 m3/s E-P3
(mS~s) 0.077 0.116 0.115 0.195 0.234 0.273 0.312 0.351 0.390 0.430 0.469 0.508 0.547 0.586 0.625 0.665 0.704 0.743 0.782 0.821 0.860 0.899 0.938 0.970 0.975 0.979 0.984 0.988 0.993 0.997
779 3180
Q(E-P3)
(m3/s)
100 400 500 610 710 810 860 910 960 990 1050 1210 1260 1350 1450 1700 2000 2500 3100 3900 4900 6000 8000
0.078 0.117 0.156 0.196 0.235 0.275 0.314 0.353 0.393 0.432 0.471 0.511 0.550 0.590 0.629 0.668 0.708 0.747 0.786 0.826 0.865 0.905 0.944
100 400 500 610 710 810 860 910 960 990 1060 1220 1260 1360 1500 1750 2100 2600 3200 4000 5000 6100 8500
11200 12000 12900 13000 15100 17000 21900
0.975 0.980 0.984 0.989 0.993 0.997
12000 12900 13000 15100 17000 21900
1027 4180
1135 4680
(.) = Flood estimated from plotting position formula for the same non-exceedance probability. t h e P3 d i s t r i b u t i o n . It w a s f o u n d t h a t t h e f o r m u l a d e v e l o p e d in this s t u d y c a n p r o v i d e less bias in t e r m o f d i s c h a r g e s t h a n several existing f o r m u l a s . (2) A s c o m p a r e d w i t h existing p l o t t i n g p o s i t i o n s , the f o r m u l a s u g g e s t e d in this s t u d y is c o n c e p t u a l l y m o r e flexible a n d c o m p u t a t i o n a l m o r e c o n v e n i e n t b e c a u s e it c a n t a k e explicitly i n t o a c c o u n t t h e s k e w n e s s coefficient o f t h e u n d e r l y i n g d i s t r i b u t i o n . (3) T h e w e l l - k n o w n W e i b u l l a n d E - W
f o r m u l a s w e r e s h o w n t o b e b i a s e d f o r P3
150
VAN T H A N H - V A N N G U Y E N A N D N O P H A D O L IN-NA TABLE 3 Comparison of E-P3 and E - W formulas.
Huangbizhuang River Systematic
Systematic+Historic
Qo = 9000m3/s
Q,,,,,,,,
P3
Q(P3)
(m3/s) 200 398 550 750 760 920 950 950 980 1100 1160 1260 1300 1330 1370 1800 1850 2080 2450 2550 3350 3700 3820 9650 11500 12000 13100 13500 14750 17150 23750
E-P3
Q(E-P3)
(m3/s) 0.061 0.100 0.138 0.176 0.214 0.252 0,291 0.329 0.367 0.405 0.443 0.481 0.520 0.558 0.596 0.634 0.672 0.711 0.749 0.787 0.825 0.863 0.901 0.940 0.978 -
100 150 200 300 400 550 600 650 700 750 780 800 850 900 1000 1500 1750 2000 2300 3000 3500 4500 5800 7500 11500 -
RMSE Max[ O/~t,,a-Qy~rmu~]
E-W
(m3/s) 0.077 0.116 0.155 0.193 0.232 0.271 0.310 0.349 0.388 0.427 0,466 0.505 0.544 0.583 0.622 0.661 0.700 0.739 0.777 0.816 0.855 0.894 0.933 0.965 0.970 0.974 0.979 0.983 0.988 0,993 0.997
100 400 500 600 700 800 850 900 950 980 1000 1200 1250 1300 1400 1800 1850 2000 3000 3200 4100 5500 7000 10000 11200 12000 12900 13000 15100 17000 21900
1090 4500
Q(E-W) (m3/s)
0.040 0.080 0.119 0.159 0.199 0,239 0.279 0.319 0.358 0.398 0.438 0,478 0.518 0.558 0.597 0.637 0,677 0.717 0.757 0.797 0.836 0.876 0.916 0.961 0.966 0.971 0.975 0.980 0.985 0.990 0.995
779 3180
50 150 420 550 650 750 820 880 930 970 990 1100 1220 1280 1350 1500 1820 1900 2900 3100 4000 5000 6600 9800 10000 11200 12100 12900 14000 15600 18500 1212 5250
Q(') = Flood estimated from plotting position f o r m u l a f o r t h e same non-exceedance probability. distribution. Therefore, they should be used only for the uniform distribution for which they were specifically derived. (4) F i n a l l y , r e s u l t s o f a n u m e r i c a l adequacy
and
the practical
example using actual flood data have indicated the convenience
developed in this study. In summary,
of the new
plotting
position
formula
it c a n b e c o n c l u d e d t h a t t h e p r o p o s e d
plotting
PLOTTING
FORMULA
FOR
PEARSON
DISTRIBUTION
151
PEARSON TYPE 3 PROBABILITY PAPER, SKEWNESS COEFFICIENT = 3.0 30000 27000
?IC kL FI ~T: D 24000
21000 18000 I ",, 115000 2000 t /g
9000 6000 1
U
/ D
3000
.1.5
.8
.9
.98
,99
.996
,999
Fig. 8. Comparison of E-P3 and E-W formulas (Huangbizhuang River, C,~
.9998
.9999
3.0, N = 181).
position formula is the most approriate for the P3 distribution in the analysis of historical flood records.
Acknowledgments Financial support for the research project from the Natural Sciences and Engineering Research Council Canada is gratefully acknowledged.
References Adamowski, K.: 1981, 'Plotting Position Formula for Flood Frequency', WaterResour. Bulletin, 7(2), 197-201. Arnell, N. W., Beran, M., and Hosking, J. R. M.: 1986, 'Unbiased Plotting Position for the General Extreme Value Distribution'. J. Hydrol., 86, 59-69. Bobee, B., and Robitaille, R.: 1977, 'The Use of the Pearson Type 3 and Log Pearson Type 3 Distributions Revisited', Water Resour. REs., 13(2), 427-443. Blom, G.: 1958, Statistical Estimates and Transformed Beta- Variable, John Wiley and Sons, New York, N.Y. Condie, R. and Lee, K.: 1982, 'Flood Frequency Analysis with Historic Information', J. Hydrol., 58, 47-61. Cunnane, C.: 1978, 'Unbiased Plotting Positions - A Review', J. Hydrol., 37, 205-222. Gringorten, I.I.: 1963, 'A Plotting Rule for Extreme Probability Paper', J. Geophys. Res., 68(3), 813-814. Guang-Yan, J.: 1987, 'Problem is Statistical Treatment of Flood Series', J. Hydrol., 96, 173-184. Gumbel, E.J.: 1943, 'On the Plotting of Flood Discharges', Trans. Am. Geophys. Res., 68(3), 813-814. Hazen, A.: 1914, 'Storage to be Provided in Impounding Reservoirs for Municipal Water Supply', Trans. Am. Soc.Civ. Eng., 77,1547-1550. Hirsch, R. M.: 1987, 'Probability Plotting Position Formulas for Flood Records with Historical Information', J. Hydrol., 96, 185-199.
152
VAN THANH-VAN NGUYEN AND NOPHADOL IN-NA
Hirsch, R. M., and Stedinger, J. R.: 1987, 'Plotting Positions for Historical Floods and their Precision', Water Resour. Res., 23(4), 715-727. In-na, N., and Nguyen, V. T. V.: 1989, 'An Unbiased Plotting Position Formula for the General Extreme Value Distribution', J. Hydrol., 106, 193-209. Jin, M. and Stedinger, J.R.: 1989, 'Flood Frequency Analysis with Regional and Historical Information', Water Resour. Res., 25(5), 925-936. Kimball, F.: 1960, 'On the Choice of Plotting Positions on Probability Paper', J. Am. Stat. Assoc., 55, 546-560. Nguyen, V. T. V., In-na, N., and Bobee, B.: 1989, 'New Plotting Position Formula for Pearson Type III Distribution', J. Hydraul. Eng., ASCE, 115(6), 709-730. Shi-Qian, H.: 1987, 'A General Survey of Flood-Frequency Analysis in China', J. Hydrol., 96, 15-25. Stedinger, J. R., and Cohn, T.: 1986, 'Flood Frequency Analysis with Historical and Paleoflood Information', Water Resour. Res., 22(5), 785-793. Sutcliffe, J. V.: 1987, 'The Use of Historical Records in Flood Frequency Analysis', J. Hydrol., 96, 159-171. Thomas, W. O., Jr.: 1985, 'A Uniform Technique for Flood Frequency Analysis', J. Water Resour. Plan. and Man., 111(3), 321-337. UNESCO: 1987, Casebook of Methods for Computing Hydrological Parameters for Water Projects, Paris, France. Weibull, W.: 1939, 'A Statistical Theory of Strength of Materials', Ing. Vet. Ak. Handl., 151, Generalstabens Litografiska Anstals Forlag, Stockholm, Sweden. Xuewu, J., Jing, D., Shen, H. W., and Salas, J. D.: 1984, 'Probability Plots for Pearson Type III Distribution', J. Hydrol., 74, 1-29. Zhang, Y.: 1982, 'Plotting Positions of Annual Flood Extremes Considering Extraordinary Values', Water Resour. Res., 18(4), 859-864.
HISTORICAL
INFORMATION
VAN THANH-VAN NGUYEN and NOPHADOL IN-NA Department of Civil Engineering and Applied. Mechanics, McGill University, Montreal. Quebec, Canada H3A 21(6.
(Received August 1991) Abstract. Most of the existing plotting position formulas have been proposed for use in the analysis of systematic flood records, but little has been reported on the plotting formulas for historical or non-systematic flood samples. In particular, no previous investigations have specificallyexamined the probability plots for the Pearson type I II (P3) distribution in the analysis of historical flood information. The present paper suggests a new plotting position formula for the P3 distribution for use with both systematicand historical flood records. The proposed formula has a simple structure as do most existing formulas, but it is more flexiblebecause it can take explicitlyinto account the skewnesscoefficientof the underlyingdistribution. Further, results of graphical and numerical comparisons have demonstrated that the suggested formula provided the least bias in flood quantile estimation as compared with many available plotting formulas, including the well-known Weibull formula. Finally, results of a numerical example using actual flood data have indicated the practical convenience of the proposed plotting formula. It can be concluded that the formula developed in this study is the most appropriate for the P3 distribution in the analysis of flood records consideringhistorical information.
1. Introduction The use of probability plotting positions or probability plots in hydrologic frequency analyses has been popular with engineers and hydrologists for several decades following the publication of a study by Hazen in 1914 on the analysis of flood flow data. Since the appearance of this study, the subject of plotting positions have been widely discussed in the hydrological and statistical literature (see, e.g., Weibull, 1939; Gumbel, 1943; Blom, 1958, Kimball, 1960; Gringorten, 1963; Cunnane, 1978; Adamowski, 1981; Xuewu et al., 1984; Arnell et al., 1986; Nguyen et al., 1989; and In-na and Nguyen, 1989). In particular, most of the existing plotting formulas were proposed for use in the analysis of systematic flood records (i.e. complete flood samples which occurred during the period of systematic gauging). Such hydrometric records cover usually a relatively short period o f time. The probability estimates of rarer flood events is then poor. Such estimates could be significantly improved by the consideration o f h i s t o r i c a l f l o o d data (i.e. data on very large floods which occurred either outside or within the systematic gauging period) as shown by m a n y recent investigations (Condie and Lee, 1982; Stedinger and Cohn, 1986; Hirsch and Stedinger, 1987; Sutcliffe, 1987; Jin and Stedinger, 1989). However, few studies have been reported on the plotting position formulas for historical or non-systematic flood records (e.g., Zhang, 1982; Hirsch and Stedinger, 1987; Shi-Qian; 1987; Guang-Yan; 1987). In particular, no cited investigations have examined the probability plots for three-parameter distributions, such as the Pearson Type III (P3) distribution. In view of the important role of the P3 distribution in hydrologic frequency analyses, Environmental Monitoring and Assessment 23: 137-152, 1992. 9 1992Kluwer Academic Publishers. Printed in the Netherlands.
138
VAN
THANH-VAN
NGUYEN
AND
NOPHADOL
IN-NA
Nguyen et al. (1989) have proposed an unbiased plotting formula specifically for this distribution and for the analysis of systematically recorded flood series. The present study, a continuation of a previous one, emphasizes the development of a more general plotting position formula for the P3 distribution and for use with both systematic and historical flood records. Results of graphical and numerical comparisons have demonstrated that the plotting position formula proposed in this study can provide the least bias in flood quantile estimation as compared with several existing formulas. Note that only the bias in flood quantile estimates is examined in this study. The question of whether plotting positions should be unbiased in terms of exceedance probabilities or unbiased in term of flood estimated is not addressed here. This question should be best considered in the context of the use which is to be made of the results. Such consideration is beyond the scope of this paper. Nevertheless, the advantages related to the use of unbiased plotting positions in the estimation of flood quantiles have been shown in many previous works (see, e.g., Kimball, 1960; Cunnane, 1978). Further, the flood record length in the present study is assumed to be known. The effect of misspecification of the true record length on flood quantile estimates have been previously studied by Hirsch and Stedinger (1987), and Hirsch (1987).
2. Plotting Position Formulas for Historical Floods In this study, to describe the various plotting positions for historical flood records, some standard notations are introduced. Consider a historical period of Nyears in which a total of g floods were observed. This N-year period needs not be continuous and contains a systematic record period ors years during which all annual floods were recorded (s_< g < N). Among the g observed floods, k of them are known to be the k 'extraordinary floods' in a period of N years. Some of these k large floods, e, may have occurred during the systematic flood records (e _~ k and e m} is distribution-free. The conditional expectation E[Q,]P~,k] is estimated by Monte Carlo simulations using a total of 20 000 repetitions over the set of k values which have non-negligible probabilities. The expected value of the discharge for the given ['~ values is found by: E[F-'(1-P,,,)]= ~ k
F-'[1-~P,,,(k)]P{klk>_m}
(11)
tll
where F -~ [1 -Pm (k)] is the discharge with exceedance probability of P,, (k) which is estimated using a plotting position formula. In the following, comparison of various plotting position formulas for P3 distribution is performed by comparing the bias in the estimation of the largest flood discharge from a sample of size N = 150, a historical flood sample commonly available in practice. Further, the proposed E-P3 formula is compared only with those formulas that were recommended for use with the P3 distribution. Since the symmetrical normal (3' -- 0), and skewed exponential (7 = 2) distributions are special cases of the P3 distribution and, in particular, there exist probability papers and plotting formulas especially derived for these distributions, the normal and exponential distributions are selected for the comparison. More specifically, the E-B, based on Blom's (1958) formula, the E-A formula, given by Adamowski (1981), and the E-W, suggested by Hirsch and Stedinger (1987), are considered for this comparison. For 3 / = 2, it is more appropriate to replace the E-B formula by the E-C formula derived from Cunnane (1978) for the P3 distribution. Finally, the Weibull formula will be considered in the present comparison because of its popularity in engineering practice, even though this formula was not derived for the P3 distribution and for historical records. Results of the comparison of plotting position formulas are shown in Figures 1,2, and 3, respectively for normal distribution (3~ = 0), for exponential distribution (7 = 2), and for the case of 3, = 1. In all cases the coefficient of vriation is Cv = 0.50. Note that Figures 1-3 present E[Q~] as a function of E[k] where Elk] = NPe. Note also that the expected value of the largest flood discharge, E[Q~], estimated by Monte Carlo simulation procedure is represented by the curve marked 'true' in the above figures. For the normal distribution (',/= 0), it can be seen from Figure 1 that the E-B formula performs very well, as expected, especially for low values of Elk], because Blom's formula
142
V A N
T H A N H - V A N
N G U Y E N
A N D
N O P H A D O L
I N - N A
was specifically derived for the normal distribution. Further, as compared with the E-B, the proposed E-P3 formula performs equally well. Results obtained by these two formulas are better than those given by the E-A, E-W, and Weibull formulas. The well-known Weibull formula was found to be the most biased as compared with the others. ?.g II-.IL~
I'-JL . . . .
II-W 'Ira IBQ~Ir~
o 9
~
~
8.S
0
*
|
Fig. 1.
9
6
"
8
~
m
~
~
~
~
9
.
.
.
.
.
9
*
I0
20
.
.
~.(k) 80
Bias in discharge for normal distribution, 7 = 0.
In the exponential case (7 = 2), Figure 2 indicates clearly the best performance of the E-P3 formula. Moreover, it is noted that, although E - C formula was specifically recommended, for the P3 distribution, this formula performs slightly better than the E-A, E-W, and Weibull formulas, especially for low value of E[k]. Finally, results for the P3 distribution with ",/= 1 are similar to those for the exponential case as indicated in Figure 3. In summary, for a symmetrical normal distribution the E-P3 formula gave comparable performance as the E-B formula. However, for a skewed distribution the E-P3 formula performs much better than other existing formulas. Therefore, it can be concluded that the E-P3 formula developed in this study is the most appropriate for the P3 distribution for the analysis of historical flood data.
PLOTTING
F O R M U L A
FOR
PEARSON
DISTRIBUTION
143
8.8
. . . .
!1
......
W~
-:.-.-:-...:.....7..Z
-
Q Q Q
Q I
Q
Q O ~
O Q
m
n
~ 0
6~
4.El
*
J
|
Fig. 2.
9
9
,
Z(k)
9 9
10
IO
~10
Bias in discharge for exponential distribution, 3/= 2.
4. Application of Plotting Position Formulas The application of the proposed plotting formula to actual flood samples with historical information will be illustrated by using flood data of the Huangbizhuang river at Huangbizhuang, China (UNESCO, 1987). For this river, the record of annual maximum floods is available from 1794 to 1974, and there exist two very large floods occurred during the period of systematic gauging (1949-1974). Thus, this data set represents a good example for assessing the performance of'exceedance' plotting formulas (E-P3 and E-W) as compared to that given by their traditional counterparts (P3 and Weibull). Results of this comparison would indicate the importance of historical information in the estimation of floods. In the present study, to evaluate the performance of various plotting formulas, the flood quantiles estimated by these formulas are compared with those computed by some conventional distribution methods. For systematic flood series, the method of moments have been frequently recommended for use in the estimation of the of parameters of the P3 distribution (Bobee and Robitaille, 1977; UNESCO, 1987). In the case of historical flood records, the method of historical weighted moments which has been widely used in practice (see, e.g., Condie and Lee, 1982) is employed in the fitting of the P3 distribution to flood data.
l~
VANTHANH-VANNGUYENANDNOPHADOLIN-NA 8.5
B-'P~ r . . .c
E-".A . . . .
IB.-~ 9 ~EI B L U , ~
6.0
5.6
9
~
9
~ U
5o0
9 |
~
9
9
Q
O
O
D
Q
D
O
~.(t,)
m m 10
20
SO
Fig. 3. Bias in discharge for P3 distribution, ",/= 1.
4.1. APPLICATION OF PLOTTING FORMULAS TO SYSTEMATICFLOOD RECORDS As mentioned above, there exist two very large floods in the systematic flood record for the Huangbizhuang river. The purpose of the present example is to assess the performance of the P3 and Weibull formulas when they are applied to this particular data series. Figure 4 and Table I show, respectively, results of the graphical and numerical comparisons of flood quantile estimates. As compared to the Weibull formula, it is found that the P3 formula gives flood estimates closer to the values computed from the fitted P3 distribution. However, it can be seen from Figure 4 and Table I that, the P3 distribution did not give a good fit to the observed data, because the flood records are not censored in order to consider some extreme floods above a given threshold as historical floods. The bias is most pronounced at the upper end of the plot. Hence the P3 formula, which has been recommended for systematic flood records (Nguyen et al., 1989), is not appropriate for cases where historical information exists in the data samples. Therefore, the use of new exceedance-based formulas suggested in this study would significantly improve the fitting of theoretical distribution to the observed historical flood data, as will be shown in the following sections.
PLOTTING
FORMULA
FOR
PEARSON
0 (P3) O (WEIBULL)
I-1 O
145
DISTRIBUTION
~x
.x
CO E
CD
X (I/
>
L OJ 69 n 0 0
F 0
I 20
I 40
~
[]
I 60
I 80
I 100
I 120
140
O fitted (x i000 m3/s) Fig. 4. Quantile-quantile plot of P3 and Weibull formulas (Huangbizhuang River, Cs,, = 3.0, N = 25).
4.2. APPLICATION OF PLOTTING FORMULAS TO HISTORICAL FLOOD RECORDS One of the difficulties in the analysis of historical flood records is the determination of a perception threshold or base level Q0 above which all floods are considered as historical
floods. The base level Q0 is often selected as the smallest known historical flood (Hirsch and Stedinger, 1987). In the present example, using historical flood data of the
Huangbizhuang river, a sensitivity analysis is performed to illustrate the effects of selecting different base levels on the performance of the fitted distribution. In particular, the best fit condition is assessed based on the graphical comparison between empirical
(plotting position) and fitted probabilities, as well as on the numerical comparison between flood estimates from plotting formulas and fitted theoretical distributions in terms of the minimum root mean square error (RMSE) and the smallest absolute
difference. A close examination of the flood record suggests some different base levels (9000 m3/s, 1000 m3/s, and 13000 m3/s) which could be selected for the sensitivity analysis. Results of graphical and numerical comparisons for the E-P3 formula and the fitted P3 distribution are shown in Figures 5-7 and Table II, respectively. It can be seen that, with a base level selected at 9000 m3/s, the theoretical P3 distribution shows the best fit to the observed data (Figure 5). More specifically, at this base level, the RMSE and maximum absolute difference of flood quantiles estimated from the E-P3 formula and the fitted P3
146
VAN
THANH-VAN
NGUYEN
AND
NOPHADOL
IN-NA
TABLE I Numerical comparison of P3 and Weibull formulas (Csu ----3,0, N = 25). Huangbizhuang River
P3 distribution
m
Qj~t,ea (m3/s)
P3
Weibull
Q (P3) (m3/s)
Q (Weibull) (m3/s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
200 398 550 750 760 920 950 950 980 1100 1160 1260 1300 1330 1370 1800 1850 2080 2450 2550 3350 3700 3820 12000 13100
0.061 0.100 0.138 0.176 0.214 0.252 0.291 0.329 0.367 0.405 0.443 0.481 0.520 0.558 0.596 0.634 0.672 0.711 0.749 0.787 0.825 0.863 0.901 0.940 0.978
0.038 0.077 0.115 0.154 0.192 0.231 0.269 0.308 0.346 0.385 0.423 0.462 0.500 0.538 0.577 0.615 0.654 0.692 0.731 0.769 0.808 0.846 0.885 0.923 0.962
100 150 200 300 400 550 600 650 700 750 780 800 850 900 1000 1500 1750 2000 2300 3000 3500 4500 5800 7500 11500
50 120 180 250 350 480 580 620 680 720 760 790 830 880 970 1450 1700 1750 2100 2700 3200 4000 5000 6500 9300
1090 4500
1398 5500
RMSE Max I Q~,,ed- Ojo,,,~,aI
d i s t r i b u t i o n were f o u n d to be smallest ( T a b l e II). N o t e t h a t the bias increases for h i g h e r values o f Q0 (or for smaller n u m b e r s o f historical floods k). This result is in a g r e e m e n t with t h o s e g i v e n by the M o n t e C a r l o e x p e r i m e n t d e s c r i b e d previously. In the following, the base level at 9000 m 3 / s will be selected to o b t a i n the best fit o f the P3 d i s t r i b u t i o n to the o b s e r v e d historical f l o o d data.
4.3. COMPARISON BETWEEN PLOTTING FORMULAS FOR SYSTEMATIC AND HISTORICAL FLOOD RECORDS As s h o w n a b o v e , it is a p p r o p r i a t e to analyze historical f l o o d r e c o r d s using p l o t t i n g f o r m u l a s (e.g. the P3 f o r m u l a ) w h i c h were derived for systematic f l o o d samples. In this e x a m p l e , g r a p h i c a l and
numerical comparisons
o f f l o o d estimates f r o m f o r m u l a
r e c o m m e n d e d for systematic records, a n d f r o m those suggested for c o m b i n e d historic a n d systematic d a t a are carried o u t in o r d e r to give general i m p r e s s i o n o n the m a g n i t u d e o f the
P L O T T I N G F O R M U L A FOR PEARSON DISTRIBUTION
3ooooll PEARSON TYPE 3
147
PROBABILITY PAPER, SKEWNESS COEFFICIENT
=
3.0
-]'3
27000 ~:0RErIC&L
F I ~TI:D
24000
/
/
/
21000
/
I
18000 /
m
\ 15000 v12000
" !!H 9000
/
,J
~V
/6
/
6000
,/ I
[]
3000 0 .t.5
Fig. 5.
.8
.9
.98
.99
.998
.999
.9998
.9999
Comparison of EP3 formula and fitted P3 distribution for a base level Q0 = 9000 m3/s (Huangbizhuang River).
PEARSON
TYPE
3
PROBABILITY
PAPER,
SKEWNESS
COEFFICIENT
= 3.0
30000 27000 24000 21000 1BOO0
-ff
~
v
15000 12000 9000 6000 3000
i I I I I
f /
/
0 .1.5
Fig. 6.
.5
.9
.98
.99
.998
.999
.9998
.9999
P =
Comparison of E-P3 formula and fitted P3 distribution for a base level Q0 = 10000 m3/s (Huangbizhuang River).
148
VAN
THANH-VAN
NGUYEN
AN[)
NOPHADOL
IN-NA
PEARSON TYPE 3 PROBABILITY PAPER, SKEWNESS COEFFICIENT : 3.0
i
H
ooo, 24000 I itlt
I( ~L FI ['TgD
z/
21000 I
/
m 18000 ! /
\ '%
15000 0P12000 9000 6000 [ ~1
/6
~
J
. ! .g
Fig. 7.
z
I
J
I
.8
I
.9
.98
.99
.998
.ggg
.g998
.9999
Comparison of E-P3 formula and fitted P3 distribution for a base level Q0 = 13 000 m 3/ s (Huangbizhuang River).
differences between the two results. The performance of P3 and E-P3 formulas is assessed using flood data of the Huangbizhuang River. Further, the base level selected for this comparison is Q0 -- 9000 m3/s as indicated in the previous section. Finally, due to its popularity in engineering practice, the E-W formula is also considered in this comparative study. Figure 8 and Table III show respectively the graphical and numerical comparisons between P3 and E-P3 formulas. It was found that the E-P3 formula provides the best performance as compared with the P3 and E-W formulas. In addition, the use of the E-P3 for historical flood data provides the minimum RMSE and the smallest bias in flood estimates as compared to those given by the P3 and E-W (Table III). In summary, a recognition of the fundamental difference between a flood record with historical and systematic data, and a record which is entirely systematic is very important. It is clear from the above example that the use of exceedance plotting formula (E-P3) rather than its traditional counterparts (P3) could significantly improve the flood quantile estimation. 5. Conclusions
The following conclusions can be drawn from the present study: (1) For practical applications, a simple plotting position formula, which can take into account both systematic and historical flood records, has been developed for
PLOTTING FORMULA FOR PEARSON DISTRIBUTION
149
TABLE II Sensitivity of P3 distribution for different base levels (Huangbizhuang River). Huangbizhuang River Q0 = 9000 m3/s
Q~,rea
E-P3
Qo = 10 000 mS/s Q(E-P3)
E-P3
(m3/s) 200 398 550 750 760 920 950 950 980 1100 1160 1260 1300 1330 1370 1800 1850 2080 2450 2550 3350 3700 3820 9650 11500 12000 13100 13500 14750 17150 23750
0.077 0.116 0.155 0.193 0.232 0.271 0.310 0.349 0.388 0.427 0.466 0.505 0.544 0.583 0.622 0.661 0.700 0.739 0.777 0.816 0.855 0.894 0.933 0.965 0.970 0.974 0.979 0.983 0.988 0.993 0.997
RMSE Maxl Qj~,,e~-Qj~rm,ml
100 400 500 600 700 800 850 900 950 980 1000 1200 1250 1300 1400 1800 1850 2000 3000 3200 4100 5500 7000 10000 11200 12000 12900 13000 15100 17000 21900
Q(E-P3)
Q0 = 13 000 m3/s E-P3
(mS~s) 0.077 0.116 0.115 0.195 0.234 0.273 0.312 0.351 0.390 0.430 0.469 0.508 0.547 0.586 0.625 0.665 0.704 0.743 0.782 0.821 0.860 0.899 0.938 0.970 0.975 0.979 0.984 0.988 0.993 0.997
779 3180
Q(E-P3)
(m3/s)
100 400 500 610 710 810 860 910 960 990 1050 1210 1260 1350 1450 1700 2000 2500 3100 3900 4900 6000 8000
0.078 0.117 0.156 0.196 0.235 0.275 0.314 0.353 0.393 0.432 0.471 0.511 0.550 0.590 0.629 0.668 0.708 0.747 0.786 0.826 0.865 0.905 0.944
100 400 500 610 710 810 860 910 960 990 1060 1220 1260 1360 1500 1750 2100 2600 3200 4000 5000 6100 8500
11200 12000 12900 13000 15100 17000 21900
0.975 0.980 0.984 0.989 0.993 0.997
12000 12900 13000 15100 17000 21900
1027 4180
1135 4680
(.) = Flood estimated from plotting position formula for the same non-exceedance probability. t h e P3 d i s t r i b u t i o n . It w a s f o u n d t h a t t h e f o r m u l a d e v e l o p e d in this s t u d y c a n p r o v i d e less bias in t e r m o f d i s c h a r g e s t h a n several existing f o r m u l a s . (2) A s c o m p a r e d w i t h existing p l o t t i n g p o s i t i o n s , the f o r m u l a s u g g e s t e d in this s t u d y is c o n c e p t u a l l y m o r e flexible a n d c o m p u t a t i o n a l m o r e c o n v e n i e n t b e c a u s e it c a n t a k e explicitly i n t o a c c o u n t t h e s k e w n e s s coefficient o f t h e u n d e r l y i n g d i s t r i b u t i o n . (3) T h e w e l l - k n o w n W e i b u l l a n d E - W
f o r m u l a s w e r e s h o w n t o b e b i a s e d f o r P3
150
VAN T H A N H - V A N N G U Y E N A N D N O P H A D O L IN-NA TABLE 3 Comparison of E-P3 and E - W formulas.
Huangbizhuang River Systematic
Systematic+Historic
Qo = 9000m3/s
Q,,,,,,,,
P3
Q(P3)
(m3/s) 200 398 550 750 760 920 950 950 980 1100 1160 1260 1300 1330 1370 1800 1850 2080 2450 2550 3350 3700 3820 9650 11500 12000 13100 13500 14750 17150 23750
E-P3
Q(E-P3)
(m3/s) 0.061 0.100 0.138 0.176 0.214 0.252 0,291 0.329 0.367 0.405 0.443 0.481 0.520 0.558 0.596 0.634 0.672 0.711 0.749 0.787 0.825 0.863 0.901 0.940 0.978 -
100 150 200 300 400 550 600 650 700 750 780 800 850 900 1000 1500 1750 2000 2300 3000 3500 4500 5800 7500 11500 -
RMSE Max[ O/~t,,a-Qy~rmu~]
E-W
(m3/s) 0.077 0.116 0.155 0.193 0.232 0.271 0.310 0.349 0.388 0.427 0,466 0.505 0.544 0.583 0.622 0.661 0.700 0.739 0.777 0.816 0.855 0.894 0.933 0.965 0.970 0.974 0.979 0.983 0.988 0,993 0.997
100 400 500 600 700 800 850 900 950 980 1000 1200 1250 1300 1400 1800 1850 2000 3000 3200 4100 5500 7000 10000 11200 12000 12900 13000 15100 17000 21900
1090 4500
Q(E-W) (m3/s)
0.040 0.080 0.119 0.159 0.199 0,239 0.279 0.319 0.358 0.398 0.438 0,478 0.518 0.558 0.597 0.637 0,677 0.717 0.757 0.797 0.836 0.876 0.916 0.961 0.966 0.971 0.975 0.980 0.985 0.990 0.995
779 3180
50 150 420 550 650 750 820 880 930 970 990 1100 1220 1280 1350 1500 1820 1900 2900 3100 4000 5000 6600 9800 10000 11200 12100 12900 14000 15600 18500 1212 5250
Q(') = Flood estimated from plotting position f o r m u l a f o r t h e same non-exceedance probability. distribution. Therefore, they should be used only for the uniform distribution for which they were specifically derived. (4) F i n a l l y , r e s u l t s o f a n u m e r i c a l adequacy
and
the practical
example using actual flood data have indicated the convenience
developed in this study. In summary,
of the new
plotting
position
formula
it c a n b e c o n c l u d e d t h a t t h e p r o p o s e d
plotting
PLOTTING
FORMULA
FOR
PEARSON
DISTRIBUTION
151
PEARSON TYPE 3 PROBABILITY PAPER, SKEWNESS COEFFICIENT = 3.0 30000 27000
?IC kL FI ~T: D 24000
21000 18000 I ",, 115000 2000 t /g
9000 6000 1
U
/ D
3000
.1.5
.8
.9
.98
,99
.996
,999
Fig. 8. Comparison of E-P3 and E-W formulas (Huangbizhuang River, C,~
.9998
.9999
3.0, N = 181).
position formula is the most approriate for the P3 distribution in the analysis of historical flood records.
Acknowledgments Financial support for the research project from the Natural Sciences and Engineering Research Council Canada is gratefully acknowledged.
References Adamowski, K.: 1981, 'Plotting Position Formula for Flood Frequency', WaterResour. Bulletin, 7(2), 197-201. Arnell, N. W., Beran, M., and Hosking, J. R. M.: 1986, 'Unbiased Plotting Position for the General Extreme Value Distribution'. J. Hydrol., 86, 59-69. Bobee, B., and Robitaille, R.: 1977, 'The Use of the Pearson Type 3 and Log Pearson Type 3 Distributions Revisited', Water Resour. REs., 13(2), 427-443. Blom, G.: 1958, Statistical Estimates and Transformed Beta- Variable, John Wiley and Sons, New York, N.Y. Condie, R. and Lee, K.: 1982, 'Flood Frequency Analysis with Historic Information', J. Hydrol., 58, 47-61. Cunnane, C.: 1978, 'Unbiased Plotting Positions - A Review', J. Hydrol., 37, 205-222. Gringorten, I.I.: 1963, 'A Plotting Rule for Extreme Probability Paper', J. Geophys. Res., 68(3), 813-814. Guang-Yan, J.: 1987, 'Problem is Statistical Treatment of Flood Series', J. Hydrol., 96, 173-184. Gumbel, E.J.: 1943, 'On the Plotting of Flood Discharges', Trans. Am. Geophys. Res., 68(3), 813-814. Hazen, A.: 1914, 'Storage to be Provided in Impounding Reservoirs for Municipal Water Supply', Trans. Am. Soc.Civ. Eng., 77,1547-1550. Hirsch, R. M.: 1987, 'Probability Plotting Position Formulas for Flood Records with Historical Information', J. Hydrol., 96, 185-199.
152
VAN THANH-VAN NGUYEN AND NOPHADOL IN-NA
Hirsch, R. M., and Stedinger, J. R.: 1987, 'Plotting Positions for Historical Floods and their Precision', Water Resour. Res., 23(4), 715-727. In-na, N., and Nguyen, V. T. V.: 1989, 'An Unbiased Plotting Position Formula for the General Extreme Value Distribution', J. Hydrol., 106, 193-209. Jin, M. and Stedinger, J.R.: 1989, 'Flood Frequency Analysis with Regional and Historical Information', Water Resour. Res., 25(5), 925-936. Kimball, F.: 1960, 'On the Choice of Plotting Positions on Probability Paper', J. Am. Stat. Assoc., 55, 546-560. Nguyen, V. T. V., In-na, N., and Bobee, B.: 1989, 'New Plotting Position Formula for Pearson Type III Distribution', J. Hydraul. Eng., ASCE, 115(6), 709-730. Shi-Qian, H.: 1987, 'A General Survey of Flood-Frequency Analysis in China', J. Hydrol., 96, 15-25. Stedinger, J. R., and Cohn, T.: 1986, 'Flood Frequency Analysis with Historical and Paleoflood Information', Water Resour. Res., 22(5), 785-793. Sutcliffe, J. V.: 1987, 'The Use of Historical Records in Flood Frequency Analysis', J. Hydrol., 96, 159-171. Thomas, W. O., Jr.: 1985, 'A Uniform Technique for Flood Frequency Analysis', J. Water Resour. Plan. and Man., 111(3), 321-337. UNESCO: 1987, Casebook of Methods for Computing Hydrological Parameters for Water Projects, Paris, France. Weibull, W.: 1939, 'A Statistical Theory of Strength of Materials', Ing. Vet. Ak. Handl., 151, Generalstabens Litografiska Anstals Forlag, Stockholm, Sweden. Xuewu, J., Jing, D., Shen, H. W., and Salas, J. D.: 1984, 'Probability Plots for Pearson Type III Distribution', J. Hydrol., 74, 1-29. Zhang, Y.: 1982, 'Plotting Positions of Annual Flood Extremes Considering Extraordinary Values', Water Resour. Res., 18(4), 859-864.
