for International

International Symposium On

Managing Water Supply for Growing Demand

Bangkok, Thailand 16-20 October 2006

Proceedings

Edited by Sacha Sethaputra, Kitchakarn Promma

IHP Technical Documents in Hydrology No.6

U N E S C O office , Jakarta 2006

nd

X

International S y m p o s i u m o n

M a n a g i n g W a t e r Supply for G r o w i n g D e m a n d

In conjunction with the

14

Regional Steering C o m m i t t e e Meeting for

U N E S C O - I H P Southeast Asia a n d T h e Pacific

The Grand Hotel Bangkok, Thailand 1 6 - 2 0 October 2006

Sponsored by

U N E S C O Office, Jakarta

anese Ministry of Education, Culture, Sports, Science and Technology (MEXT) Department of Water Resources, Thailand

Organized by

Thai National Committee for IHP-UNESCO Thailand National Commission for U N E S C O Ministry of Natural Resources and Environment, Thailand

Preface

The IHP Symposium on Managing Water Supply for Growing D e m a n d was held in conjunction with the 3rd A P H W Conference on Water Resources Management towards Sustainable Growth and Poverty Reduction during October 16-20, 2006 at the Grand Hotel, Bangkok , Thailand. It was aimed to strengthen the cooperation a m o n g the Asia - Pacific countries to solve the problem of limited and fragile water resources in order to respond to the ever-lasting demand of water. It also provided the international arena for all stakeholders in the region to exchange experiences and information on hydrological sciences in order to achieve both solutions for better life as well as to consolidate a regional commitment so as to upgrade knowledge on water resources management and increase the capacity of relevant stakeholders.

As the I H P Symposium w a s organized in parallel with the 3rd A P H W Conference, it provided the opportunities for all counties to establish the solid cooperation and to m a k e the utmost benefit from experiences and information exchange to tackle with the water-related problem in their particular countries under the framework of U N E S C O ' s International Hydrological Programme onwards.

The Proceedings published for the IHP Conference comprise 26 technical document papers, in which various issues and their interesting analysis are included. All papers were presented in the IHP Symposium sessions and were accumulated as the U N E S C O technical papers both in forms of hard copy and C D - R O M . The organizing committee do hope mat this technical paper will be of great benefit to the readers and could contribute to the improvement of water resources management and hydrological knowledge in the future.

Finally, the organizing committee would like to render our great appreciation and thanks to the U N E S C O Office in Jakarta and the Japanese Ministry of Education, Culture, Sports , Science and Technology ( M E X T ) for their financial supports. Without these supports, the organization of the IHP Symposium and the publishing of the Proceedings would not be possible.

CONTENT

Preface iii Content v A Application of a N e w Flood Stochastic Simulation M o d e l Developed

by Russian Hydrologist in the Basin of Qiantang River

YUANFANG CHEN, GUOXINCHEN, WENPENG WANG, SHUJIAN LI, AOMI CHEN 1.

*- Statistical Experiment Study of Design Annual Runoff Estimated b y Curve-Fitting Method for Pearson Type- III Distribution

Chen Yuan-fang, Zhao Li-hong, Xu Sui, Ma Bing-xun, Wang Wen-peng 9-

= Exact and approximate analytical solutions for steady seepage inflows to horizontal wells

L. Q. Liongson 1 5 , 4 Effective management of water resources via demand management: S o m e examples from Southeast

N. W. Chan, V. Nittvattananon 2 3 - G Managing Increased Water D e m a n d in China: A Great Challenge

Z.X.XU.J. Y.LI 39- fc? Water Distribution Network Analysis for D M A Design of Ladprao Branch, Bangkok, Thailand

A. Pornprommin, S. Lipiwattanakarn, S. Chittaladakorn 45 y Water Rights and water allocation in Andean basins

J. Molina, E. Villarroel, J. Alurralde, A. Apaza,F.Soria 51 O M e m b r a n e Filtration Removing Salts and Arsenic in Drinking Water for Rural Areas

Pikul Wanichapichart, Wiriya Duangsuwan, Darunee Bhongsuwan, Pusadee Mohamad,

and Porntip Sridang, 59 Cf Basin water use accounting method with application to the M e k o n g Basin

Mac Kirby, Mohammed Mainuddin, Geoff Podger, Lu Zhang 67, A 0 A Framework to Assess Model Structural Stability through a Single-Objective

Global Optimization Method

GihaLEE, Yasuto TACHIKAWA, Kaoru TAKARA 79 A\ Derivation of Rainfall Intensity-Duration-Frequency Relationships for Short-Duration Rainfall from Daily Data

LeMinhNHAT, Y. Tachikawa, T. Sayama, K. Takara 89 A ¿_ Stochastic Modeling of Rainfall Maxima Using Neyman-Scott Rainfall Model

Carlo Mondoñedo, Yasuto Tachikawa, Kaoru Takara 97

\ 2) Effect of Moving Storm Rainfall on Soil Erosion and Sediment Transport from Watersheds

Guillermo Q. Tabios III, PhD 109

> | £ Status of Disaster Database and Limitations for Planning

T. Merabtene, J. Yoshitani, A. Pathirana 125

\ k I C H A R M ' s Research Strategy toward Effective Implementations of Flood Forecasting and Warning System in Asia

K. Fukami, H. Inomata, P. Hapuarachchi, R. Oki 133 f 7 Modified Remote Sensing Information Model of Water Erosion on Hillslopes

D.Y. Shen, K. Takara 139

\ 8 Capacity Building - Application of Geoinformatics for Disaster

Lai Samarakoon, Takashi Moriyama, Chu Ishida 145 A°\ Scope of Flood Hazard Mapping in Developing Countries

Shigenobu Tanaka, Rabindra Osti, Toshikazu Tokioka 153 L 0 The Prediction of Flooding Area in the Pasak River Basin by Using Mathematical Model :

A Case Study on Land Use

N. Hungspreug, A. Penghuaro 159 Combating Reservoir Sedimentation: A Challenge for Sustainability

T. Tingsanchali, N.M. Khan 169 /Lt. Impact of 2004-tsunami Natural Disaster on Water District-10 Banda Aceh of

Krueng Aceh River Basin Development

Masimin, Zouhrawaty A. Ariff. 179 Surface Water Treatment with Microfiltration at The Village of Pranon, Nakhon Sawan Province, Thailand

William R. Sellerberg, P.E., Voravuthi RakTae-ngan 187 /L.¿^, Overview of United States Drinking Water Regulations

Anthony M. Wachinski, PhD, PE 193 /]_ g Effect of E N S O on Southeast Asian Rainfall

Tsing-Chang (Mike) Chen 201 /)(-, Tropical storms and associated flood risk on Grande Terre, N e w Caledonia

Ray A. Kostaschuk, James P. Terry, Geoffroy Wotling 207

Application of a N e w Flood Stochastic Simulation Model Developed by Russian Hydrologist in the Basin of Qiantang River

Y U A N F A N G C H E N , GUOXIN C H E N , W E N P E N G W A N G , SHUJIAN LI, A O M I C H E N Department of Hydrology and Water resources, Hohai University, Nanjing 210098, China

E-mail: vfchen@,mail.edu.cn

ABSTRACT

In order to simulate flood process better, the study focus on the application of a n e w flood stochastic simulation model developed b y Russian hydrologist Sambotsky, which could describe the characteristics of rapid rise and slow fall for flood hydrograph, in the basin of Qiantang river in the southern region of China.

In the case study, a lot of stochastic tests have been done for the n e w model, including the test of annual m a x i m u m peak flow, annual flood volumes. The results show that flood process simulated m a y pass all tests.

Meanwhile, the n e w model is also compared with the conventional models, including seasonal A R ( 1 ) model and the Disaggregation model by statistical test for simulation flood process. The results show that the n e w model is better than the other two models at least in this basin. S o it's said that the n e w model is feasible to simulate the flood process for the basin.

K e y w o r d s : Flood stochastic simulation , Russia, A R ( 1 ) model disaggregation model, Qiantang river, A p - plication, Statistical test

1 INTRODUCTION

Traditional flood stochastic model is often based on different kinds of correlativities in time sequence and spatial sequence of daily flow or shorter-time interval flow. A n d it pays m u c h attention to statistical characteristics of aggregate and constituent. This kind of model includes seasonal A R ( 1 ) model and the Disaggregation model and so on. Generally these conventional models could reflect main characteristics of daily flow in time sequence well. Therefore they are widely used nowadays w h e n w e simulate daily flow. However as w e k n o w that flood process is non-reversible in time sequence for daily flow.

For example, there is obvious distinction between the growth curve and regression curve of one flood daily flow process in mountain river basin in southern China. Consequently, in order to describe characteristic of daily flow in time sequence more clearly, w e ' d better take this non-reversible characteristic into account.

Seasonal A R ( 1 ) model and the Disaggregation model is difficult to simulate the flood natural process of rapid rise and slow fall. Though shot noise models m a y reflect diversity of characteristic between the growth curve and regression curve, it's still not a good w a y to describe main characteristics of daily flow. This point of view w a s put forward by O'Connell in 1979. Meanwhile, for the sake of simplifying calculation shot noise models assume regression exponential as a constant, which is not conform

practical regression order. In fact, regression exponential is nonlinearity andchanged along with magnitude of discharge. Recently, a n e w flood stochastic simulation model w a s developed by Russian hydrologist, which could describe the characteristics of rapid rise and slow fall for flood hydrograph. Here 3 2 years(1958-1989) of hydro- logic observations in Quxian station in the basin of Qiantang river in flood season are selected as a sample, then our study probe into the feasibility, simulation effect, advantage and disadvantage and so on of this n e w model used in China stream simulation.

2 M O D E L DESCRIPTION

The stochastic model of the hydrograph of a flood season describes the temporal variance of water discharge during a flood season as realization of a stochastic process. The time is considerate with a step per one day within an interval ' where ~~ is date of the earliest

t = T

beginning and is a date of the latest season.

The model is based on the approximation of die hydrograph by function.

Q(t) = q

%{t) + Y

q

<p

(t-t

)

;=i

(l) W h e r e ,

: an average discharge of a base flow of a flood season of the given year;

: a number of the flood peaks of this year;

'i h

9 i , - , ?t

the dates of their passage;

: m a x i m u m discharge of these floods, independently formed without an influence of base flow and previous floods;

: a dimensionless function of a hydro- graph of base flow;

<Pj(t-tj)

: a dimensionless function of hydro- graph of a flood , which describes a growth of

J '-'J

-flood w h e n , and a regression curve

t>tj

w h e n

For the most of the investigated rivers formula (1) can be used with

io

* > , ( ' - ' , ) =

=1 and:

if t<ij-rl; (2)

1 + - « - ' , ) , ' / tl-xj<t<tj,

A s figure 1 shows, where ' is a duration of the

i aj

main growth of -flood; is an intensity of it's exponential regression after '

The statistical analysis has given the following result.

1. The m a x i m u m s " ' can be considered independent random variables with the unified probability distribution for the whole flood season in a case of it's homogeneity.

T - a

2. The variables ' and ' are independent and in most there is not statistically reliable depend-

q

ence between them and ' . For a whole flood of

T ' " T

every year the variables " ' * are independent and have the unified probability distribution. The same conclusion is true for the variables

3. The sequence of flood peaks of all investigated rivers can satisfactorily described by a model of Poisson process. In particular, the annual number of peaks have Poisson probability distribution, and with the fixed for a flood period of a k concrete years the dates " ' ' behave like the elements of a ranked sample form, the uniform distribution over interval [0,T]

The variables , ° and ' , ' , ' are the elements of the stochastic model of the hydrograph of flood season. In order to obtain the observations of elements the separa- tion of observed hydrograph is necessary to de- scribe a stochastic process ß ('} . in a case of n

nka

years of hydrologie observations

observations of the element , , can be used for the estimation of their function of distri- bution.

Fig. 1 Hydrograph of -flood

3 M O D E L APPLICATION

In our study, w e choose flood data from Quxian station at Qujiang river in Qiantang river basin during whole flood season. Sampling procedure listed below:

(1) According to general condition of basin and climate factor, flood season total duration T is ascertained as 90 days.

(2) In order to reflect the feature of flood process sufficiently and efficiently, here w e adopt 8 hours as time interval Ai. Therewith, the amount of time interval is 90 days * 8 hours = 270, that says the amount of kerf is 270.

rameter values in model formula, that is , ° ,

1, *, a, O = ! , • • • , * )

(D Extract annual number of the flood peaks W e meet with two kinds of flood process plotted in figure 2 . It is generally agreed that, if the height of growth curve is greater than or equal to one- third of the height of peak, w e m a y consider mis hydrograph as a whole flood process. Accord- ingly, the left figure is regarded as superposition of two flood processes and the right figure is re- garded as an independent flood process.

Q (rrf/s)

Q (rr?/s)

t 0 Fig. 2 T w o kinds of flood process

® Extract base flow

Because a base flow of a flood season is very small relative to corresponding flood discharge, it's reasonable to take m i n i m u m annual dry- weather flow as base flow of the given year.

i Extract m a x i m u m discharge ^1J

The maximal discharge of the kerf subtracts

J

base flow and previous flood recession flow, thus

9j is gained.

© Extract duration of growth

Model generalizes growth part as a zooming line, which can reflect characteristic of rapid growth.

However the real growth curve usually isn't a regular line plotted in figure 3. Since it is a gener- alization, therefore so long as the upper and nether segmented area formed b y the fitting line, base flow line and original growth curve are equal, then w e deem the extraction of the beginning of rise

point and duration of the main growth is reasonable.

Fig. 3Extraction of duration of the main growth

© Extract regression exponential J oc.

Q. i j Real flow '"' of kerf of receding -flood process is composed of base flow

previous flood regression discharge kerf * .

<¡0

and

Q.J

at

a. i,j

can be figured out through formula

a,,, =(1110,-Inß,,,)/A/

according to r e c e d i n g f l o o d f l o w f o r m u l a

Q,,j =qJ*exp(-aIJ*At)

At rj

Where: is the duration from to the calculating kerf , =1,2,..., , is the amount of date of regression curve. Average of

a. a.

serves as value of the -flood re- ceding process.

2. Distribution test and estimation of parameters

probability-weighted moments method. B y statistical test w e found that w h e n the level of significance =0.05, rate of pass is 100%, the above hypotheses is acceptable.

(3) Estimation of

In distributed function test the N U L L hypothesis

is that the sequence under consideration is uniform distribution. O n test law, when =0.25, CC

y2( 3 )

= 4.108>2.6740, therefore the N U L L hypothesis is acceptable. 2 parameters of uniform distribution are gained from maximal and minimal value of observed base flow.

(4) Estimation of

and

t a}

The N U L L hypotheses of and are type

x) a

P-III distribution. In the case of , when

*a(12)

=0.050, =21.03>19.0835, assuming

k *j

(1) Estimation of and

Model theory has indicated that the sequence of the annual number of flood peaks k have Poisson probability distribution, and with the fixed for

k

a flood period of a concrete years the dates

1 ' ' ' ' » k

behave like the uniform distribution over interval [0,m], and in our study m = 2 7 0 . (2) Estimation of ^%

In model theory, the series of are supposed 1j

as independent random variables. In order to prove the validity of the hypotheses, independence

* * f ^> • • A- ui Ex Cv Cs test of is indispensable.

of peak flow sequence can be estimated by

to be type P-III distribution is acceptable. In the case of , the same conclusion is drawn.

a

Ex Cv

Cs

mated by fitting a curve method and probability- weighted moments method respectively.

3. Modeltest

In order to further judge this model to be good or not, it is necessary to analyze and test applicability for the reproduced series. So in this study, 500 groups of 32 annual flood season series were reproduced as inference population. The main statistical property tests are how confidence intervals of simulative sample parameters and time interval flow contain those of measured sample. Testing methods include long term series and short term series. The detail results as follow:

The testing for measured sample parameters falls into certain confidence interval of corresponding simulative ones.

Table 1 Testing for measured sample parameters Parameter

k

9o

TJ

aj

<ij

Ex Cv Cs Ex Cv Cs Ex Cv Cs Ex Cv Cs Ex Cv Cs

Measured value 8.1 0.29 0.08

10.8 0.52 0.13 3.93 0.48 1.28

0.3 0.36 0.79 1790 0.70 1.45

Long series 8.1 0.36 0.35 10.5 0.58 0.02 3.93 0.48 1.27 0.3 0.36

0.8 1780 0.69

1.40

Short series 8.1 0.35

0.3 10.5 0.58 0.01 3.93 0.48 1.24

0.3 0.36 0.78 1780 0.69 1.36

Short series mean square deviation 0.5

0.05 0.41

1.1 0.07 0.27 0.1 0.03 0.28

0 0.20 0.21 81.0 0.04 0.28

This model only gives distribution function of the n u m b e r of peak , so that the other parameter

k

distribution only can be identified by experiential try. After m a n y time comparisons this research eventually recognized these distributions, and from current hypotheses and test results can infer that the selection of distribution is reasonable.

Meanwhile w e should concern that our study is just base on single station and sample size is limited, so h o w to select proper distribution is still a question.

(2) Testing of time interval flow statistical properties

Because of real data discontinuousness, the length of time interval is not too long or at most not m o r e than the duration of one flood process.

According to the measured data 10 time interval (length of each interval is 8 hours) were properly selected from Quxian station.

F r o m above table, it shows that only of doesn't pass through test in case of one m e a n square deviation while others preserve

Cs k

statistical properties well. T h o u g h of

Cs

pass the test, oscillation of is a little big. The cause exists in the reproduced series incredibly preserves second or third m o m e n t of real series for w h e n real series is fitted by use of m e a n

k

k Qj Tj

for Poisson distribution. Whereas

aj

and series are all fitted by use of three parameters for P — I H distribution, their testing result are of course better.

T a b l e 2 Testing of time interval flow statistical properties ( short series ) Parameter

Ex

Cv

Cs

Time interval ( 8 hours) Observed data Simulated data M e a n square

deviation Observed data Simulated data M e a n square

deviation Observed data Simulated data M e a n square

deviation

1 1.07 1.17 0.08 0.348 0.362 0.054 0.457 0.681 0.501

2 2.01 2.09 0.15 0.366 0.366 0.055 0.514 0.688 0.501

3 2.77 2.87 0.20 0.376 0.371 0.055 0.566 0.697 0.504

4 3.42 3.54 0.25 0.385 0.375 0.056 0.645 0.704 0.511

5 3.98 4.11 0.29 0.386 0.380 0.057 0.609 0.712 0.514

6 4.46 4.60 0.33 0.392 0.384 0.057 0.665 0.720 0.518

7 4.91 5.02 0.37 0.401 0.388 0.058 0.714 0.726 0.519

8 5.35 5.38 0.40 0.408 0.393 0.058 0.694 0.733 0.519

9 5.78 5.70 0.42 0.419 0.397 0.059 0.719 0.737 0.520

10 6.15 5.98 0.45 0.423 0.401 0.059 0.705 0.742 0.520

T a b l e 3 Testing of time interval flow statistical properties (long series) Parame-

ter

Ex Cv Cs

Time interval ( 8 hours) Observed data

Simulated data Observed data

Simulated data Observed data

Simulated data

1 1.07 1.17 0.348 0.368 0.457 0.844

2 2.01 2.09 0.366 0.373 0.514 0.858

3 2.77 2.87 0.376 0.377 0.566 0.870

4 3.42 3.54 0.385 0.381 0.645 0.886

5 3.98 4.11 0.386 0.386 0.609 0.898

6 4.46 4.60 0.392 0.391 0.665 0.908

7 4.91 5.02 0.401 0.395 0.714 0.915

8 5.35 5.38 0.408 0.400 0.694 0.922

9 5.78 5.70 0.419 0.404 0.719 0.928

10 6.15 5.98 0.423 0.408 0.705 0.934

Carefully analyzing table 2 and table 3, w e find two phenomena as follow.

® All of the parameters pass through test on occasion of confidence interval is one mean square deviation. However, of simulated

Cs

data fluctuates in a relatively large scope. A s w e k n o w , the estimation of high-order m o m e n t of stochastic variables by use of samples with small capacity is unreliable, therefore it's not strange to meet this phenomenon.

® The accuracy of extracting information is influenced greatly by model maker. W e find the phenomenon that from interval 0 to 8 real mean value is less than simulation's and following situation is on the contrary, although simulative result is satisfying.

A s figure 1 show, w h y leading to this phenome- non is that current flood process is influenced not only by former flood but also by next one, as re- sults in identifying initial time of flood rising more difficult. In order to extract receding exponent easily, effects of die next flood rising are neglected, so mat

a

is less than actual value, which induces the simulative flood discharge inclination to increase at the beginning. In order to achieve a reasonable simulative result, was corrected by multiply-a, ing corrective coefficient ( >1, in this study c c

C =1.2).

(3) Parameter parsimony test

During selecting model, it is necessary to take the number of parameters into account in terms of

R = 10 31

finite information. In our study, , Based on Chinese hydrology data, the range of

D

is ascertained from 5 to 10 or so. In this case this model parameter number meets requirement.

(4) Comparison with and Disaggregation Model (Dis)

Because of the model firstly applied to China, it is indispensable to compare simulation result of this new model with seasonal model and Dis- aggregation model, judging whether superior to these two conventional models or not

Based on measured data, time intervals from 1 to 3 0 (each time interval is 8 hours) were determined as modeling sample series, containing a maximal individual flood process of every year season flood. The reason is (D discontinuous real data

AR(X)

don't permit and Disaggregation models to simulate each year whole flood season hydro- graph,

® Disaggregation model generally applies to

simulate a individual flood process. , ,

Cs

of maximal 1, 2 , 3 total time interval dis- charge were selected to compare with each other.

Comparative results as follows:

Table 4 Comparison about time interval flow statistical properties (short series)

Parameter

Ex

Cv

Cs

Time interval ( 8 hours)

Model Measured data Simulated data M e a n square

deviation Measured data Simulated data M e a n square

deviation Measured data Simulated data M e a n square

deviation

1 New

1.07 1.12 0.08 0.348 0.362 0.054 0.457 0.681 0.501

Dis 1.07 0.95 0.07 0.348 0.468 0.071 0.457 0.882 0.625

Sea- sonal AR(1) 1.07 0.95 0.09 0.348 0.537 0.140 0.457 1.367 0.956

2 New

2.01 2.09 0.15 0.366 0.366 0.055 0.515 0.688 0.501

Dis 2.01 1.88 0.13 0.366 0.462 0.070 0.515 0.845 0.630

Sea- sonal AR(1) 2.01

1.85 0.17 0.366 0.557 0.142 0.515 1.370 0.951

3 New

2.77 2.87 0.20 0.37 6 0.37

1 0.05

5 0.57

9 0.69

7 0.50

4

Dis 2.77 2.62 0.17 0.37 6 0.45

8 0.07

0 0.57

9 0.81

2 0.62

1

Sea- sonal AR(1) 2.77 2.55 0.24 0.376 0.564 0.140 0.579 1.335 0.936

Table 5 C o m p a r i s o n about time interval flow statistical properties (long series)

Parame- ter

EX Cv Cs

Time interval ( 8 hours)

Model Measured data Simulated data Measured data Simulated data Measured data Simulated data

1 New

1.07 1.12 0.348 0.362 0.457 0.681

Dis 1.07 0.95 0.348 0.468 0.457 0.882

Sea- sonal AR(1) 1.07 0.95 0.348 0.537 0.457 1.367

2 New

2.01 2.09 0.366 0.366 0.515 0.688

Dis 2.01 1.88 0.366 0.462 0.515 0.845

Sea- sonal AR(1) 2.01 1.85 0.366 0.557 0.515 1.370

3 New

2.77 2.87 0.376 0.371 0.579 0.697

Dis 2.77 2.62 0.376 0.458 0.579 0.812

Sea- sonal AR(1) 2.77 2.55 0.376 0.564 0.579 1.335

After analyzing table 4 and table 5, w e find that the simulation effect of n e w model is better than

AR(\)

and Disaggregation models.

4 CONCLUSION

Through investigation and application to the n e w flood stochastic simulation model developed by Russian hydrologist, the following primary conclusions were drawn:

1. In this research, this n e w flood stochastic model was firstly exploratory application for Quxian station simulating flood process, and the simulation results show that it is feasible to reproduce the flood process for the basin.

2. Distinguish characteristic of this model is that simulated flood hydrograph describes the character of rapid rise and slow fall, which well fits South China mountain rivers flood process.

3. B y use of this model, assumption population was tested statistically and flood stochastic simulation w a s investigated. Results have shown that it can efficiently use the given hydrologie data on the basis of the account of the particular feature of a flood flow, and preserve various statistical properties.

4. The advantages of this model are that it has clear physical concept, simple structure, proper parameter and general applicability.

R E F E R E N C E S

Ding Jing, Deng Yuren, Stochastic Hydrology, Chengdu University of Science and Technology Publish House.

Ding Jing, The Progress of Stochastic Hydrology, Journal of Chengdu University of Science and Technology, N o 4, 1986, pl33-134.

Ding Jing, The Application of Disaggregation Model in Flood Simulation, Journal of Chengdu University of Science and Technology, N o 4, 1986, pl41-148.

Ding Jing, D e n g Yuren, Y a n g Rongfu, O n T h e Stochastic Approach in Flood Simulation, Papers in fifth Conference on Hydrology in China, Science and Techology Publish House, 1992, pi66-170.

Chen Yuanfang, The Application of Monte-Carlo Method, Heilongjiang People's Publish House.

H u Kangping, Verification and Validation of Stochastic Streamflow Models, Engineering Journal of W u h a n University, N o 1, 1987, pll-17.

Huang Zhengping, Hydrologie Statistics, Hohai University Publish House.

Statistical Experiment Study of Design Annual Runoff Estimated by Curve-Fitting Method for Pearson Type- III Distribution

Chen Yuan-fang Zhao Li-hong X u Sui M a Bing-xun W a n g W e n - p e n g (Dept. of Water Resources and Hydrology, Hohai University, Jiangsu Nanjing, 210098)

email: yfcheniSimail.edu.en

ABSTRACT

The curve-fitting method is a c o m m o n w a y to estimate design annual runoff values under different high probabilities in water resource assessment. However, there's nearly no theoretical analyses done about exactitude of the design values calculated under different curve-fitting criterion. Therefore, it's still a question that h o w these different criterion influence results. Meanwhile whether w e should use all the collected data to do the curve- fitting or not is also a question. In the paper, the above questions are discussed by Monte-Carlo experiments with consideration Pearson-III as population distribution. The criteria for evaluation are the un-biasness and efficiency of the parameter and design value with given probability, M a n y calculations show that the estimation result is better w h e n the proportion of data for curve-fitting from 5 0 % to 6 0 % than that w h e n the proportion less than 2 0 % , even slightly better than traditional curve-fitting method which w e used all the data, and the curve-fitting by absolute criterion is better than that of square criterion.

K e y words: curve-fitting method; criterion of curve-fitting; partial data curve-fitting; un-biasness; efficiency;

annual runoff ; P-III distribution

1 INTRODUCTION

In the planning and design of all water conservancy works, hydrological frequency analysis and calculation should be m a d e in order to determine the quantiles or design values of rain- storm or flood under different low probabilities including 1%,0.1% et al. Because the exactitude of the quantiles is very important to both project investment and security, a lot of scholars have been carrying out m u c h research for it and received plentiful and satisfactory results. A t present, Pearson-III distribution is adopted as the population distribution for the annual m a x i m u m flood peak &

flood volume and rainfall in China, and the curve- fitting methods (including a curve-fitting b y eye estimation and a curve-fitting b y mathematical optimal calculation) are also recommended to estimate their distribution parameters in Chinese Regulation for calculating design flood of water resources and hydropower projects^[,]. However, in water resources assessment, it's necessary to esti- mate hydrological design values, such as design annual runoff, under s o m e high probabilities (p equals to 7 5 % , 8 5 % or 95%),. In general, it is con- sidered to use directly above-mentioned method of the design flood to estimate these parameters. S o there are nearly no theoretical researches about the

exactitude of design values of annual runoff and h o w to estimate it better.

There are two kind of problems to be solved promptly in the application of the curve-fitting method to estimate the design annual runoff. O n e is that the exactitude of design values estimation and the effect of different criterion of the curve-fitting on estimation result are not clear. T h e other one is that there is still a disputed question whether w e should use all the collected (or observed) data to do the curve-fitting or not. N o w a d a y s some hydrolo- gists often use only partial lowest-collected data to do the curve-fitting. Is the result of this treatment better than that using all the collected data? If yes, what is the best proportion data to be used to do the curve-fitting?

Based on above view of point, the exactitude difference of the parameters and design value esti- mated b y different curve-fitting methods is c o m - puted and analyzed b y Monte-Carlo experiments with consideration Pearson-III as population distri- bution. T h e criteria for evaluation are the u n - biasness and efficiency of the parameter and design value with given probability. Through these, the above-mentioned questions could be solved , and it's useful for annual runoff frequency analysis in the future.

2 DISTRIBUTION FUNCTION A N D PARAMETERS

The density function of Pearson-III distribution is as follows [2]:

/ w = 4 ^ ( ^ - ^a o r ^ ^{( i} " f l o ) ( ; c - ^a »>(i)

v 7

r(«)

Where a ß a0

are the parameter of density function, and > 0 , > 0 . The relationship a ß between them and three population statistical parameters E X , C^v, C^s is listed as following:

4

ß = „„„„ (2) Ci

2 EXCvCs

EX ! ~ 2 Cv Ci

3.1 T h e curve-fitting with all the collected (observed) data

(1) Square criterion (p^x

» , .²

Objective function: A = £ v C - *P°„, ) ( 4 )

m =1

(2) Absolute criterion ç2

" i i

Objective function: A = ^ J * » - *p. I ( 5 ) Where, x°_ is the population design value with

frequency P™

While m e parameters E X , Cv, and Cs are given, the corresponding objective function can be calculated. The parameters , " , ' Three statistical parameters E X , Cv, Cs can be

a ß

estimated from certain observed data, then , , ° can be obtained by using equation (2), in this case the design values estimated from the observed data under different frequency could be finally estimated.

3 DIFFERENT CURVE-FITTING METHODS In China, the curve-fitting method is a c o m m o n parameters estimation w a y for Pearson-III distribution. It is recommended by the Ministry of Water Resources [1] to use the curve-fitting method with eye estimation usually, and the curve-fitting method by mathematical optimum calculation could be also used. The plotting position formula of the expectation value is also recommended to calculate empirical frequency. D u e to difficulty to consider the curve-fitting method with eye estimation for Monte-Carlo experiments, so the main study focus on the curve-fitting by mathematical optimum calculation ,including by absolute criterion and square criterion.

Suppose the simple hydrological sample

x,,x„---,x„ , x'(m = \,2,--,n)

, range them to

according to the descending order. Plotting position equation is as following:

with which tends to the m i n i m u m are the final estimation results of parameters by the curve-fitting with certain criterion.

3.2 The curve-fitting with the partial collected data The curve-fitting with partial collected data means that it doesn't use all collected data to do the curve-fitting, but it uses partial data just in the right side of Haisen frequency curve to do it.

Let k indicate the percentage of right side data used for curve-fitting to all collected data. The value of k m a y be 10%, 2 0 & , 30%, etc. It's obvious that w h e n k = 100% it is the curve-fitting with all the collected data. The objective function m a y be rewritten as follows:

(1) Square criterion Ç>^

Objective function: ^A y /

(2) Absolute criterion ç> 2

Objective function : ^A _

m = n- nk +1

m =n-nk +1

(6)

(7)

If a certain design value under a given frequency is less than 0, it's put to zero.

p

= 7 0»=l,2,-n)

n+l

10

4 THE STANDARD OF EVALUATION

The standard for evaluation are the un-biasness and efficiency of the parameter and design value with given probability'31.

The un-biasness and efficiency of the parameters are expressed by the average and the m e a n square error respectively. If the average is very close to the population m e a n , it can be thought to be un-biased estimation. T h e smaller the m e a n square error is, the better the efficiency is.

The un-biasness and efficiency of the design value are expressed by the average of the relative error B ^ a n d the relative error of the root m e a n square Sxp as follows:

J

^I

^{ix' -x°]}

B =i^i *100% (9)

X

P N x

s p

W h e r e p is the design guaranteed rate (or x°p

design probability); is the population design value corresponding to p ; is the design value x' estimated by the i* stochastic sample corresponding to p ; Ns is the total number of stochastic samples for statistical experiment.

If B,tp > 0 , it means that the design value is positively biased, else negatively biased. The larger

| Bxp | is, m e more serious the bias of the design value is. W h e n | B ^ , | isn't exceeded 3 % , the design values can be considered to be un-biased. T h e smaller S ^ is, the better the efficiency is.

5 MONTE-CARLO EXPERIMENT SCHEMES 5.1 Scheme design of two curve-fitting criteria

The curve-fitting methods with absolute criterion and square criterion are considered respectively, and the curve-fitting with all collected data is studied. T h e other parameters values are as follows :n= 3 0 , 5 0 , 100; E X0 = 1.0; C v0= 0.3, 0.5, 1.0; C S Q / C V O = 2 , 3.There are 36 schemes in total.

5.2 Scheme design of the curve-fitting with different data proportion

n= 30, 5 0 , 100; E X0 = 1.0; C v0= 0.3, 0.5, 1.0;

Cso/Cvo = 2 , 3, 4 . A n d the partial data proportion takes 0.1, 0.2, 0.3, 0.5, 0.6, 0.7 and 1.0, in fact k = 10,20,30,50,60,70,100 respectively. There are

189 schemes in total. T h e curve-fitting method with absolute criterion is only considered for this case.

For all the above statistical experiments, Ns= 2000 and P = 0.50, 0.75,0.85, 0.95.

6 RESULTS AND ANALYSIS

6.1 Effect of two different criterion in the curve-fitting to the estimation result of parameters and quantités

The partial results are listed in table 1. For the results of the parameters, Both estimation of C v and C s by two criteria are positively-biased slightly, and the un-biasness and efficiency of the parameters by absolute criterion are better m a n that of square criterion in most schemes, especially for the latter it is positively-biased seriously w h e n the population values of C v and C s are large. For die results of quantités, both design values by two criteria are less m a n the population value slightly in most x°

schemes. |Bxp| b y absolute criterion is less than 5 % for almost all schemes, so it m a y be considered as un-biased. H o w e v e r , the result by square criterion is unstable, because the design values are negatively- biased seriously w h e n Üre population values of C v and C s are large, but in s o m e case ,it's positively- biased seriously. The efficiency of the design values by two criteria are almost equivalent in total, but that of absolute criterion is better slightly and more stable.

Table 1 partial result of parameters and quantiles for the curve-fitting methods with t w o criteria

Population N 5 0 S 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0

n 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0

parameters Cv 0.30 0.30 0.30 0.30 0.50 0.50 0 50 0.50 1.00 1.00 1.00 1.00

Cs 0.60 0.60 0.90 0.90 1.00 1.00 1.50 1 50 2.00 2.00 3.00 3.00

Criteria

<Pi

<P¡

<Pi

<Pi

<Pi

<Px

<Pi

<P\

<Pi ECv

0.32 0.31 0.32 031 0.53 0.51 0.54 051 1.12 1.03 1.17 1.02

ECs 0.66 061 0.99 0.89 1.11 0.99 1.67 1.50 2.28 2.03 3.40 3.05

SCv

0.04 0.03 0.04 0.04 0.07 0.06 0.1 0.06 0.24 0.13 0 3 6 0.16

SCs 0.43 0.39 0.47 0.40 0.49 0.41 0.64 0.40 0.85 0.43 1.24 0 53

Bxpi

-0.58 -0.18 -0.74 -0.04 -1.64 -0.27 -2.38 -0.27 -8.12 -1.35 -8.82 -0.87

Bxp2

-1.34 -0.64 -1.36 -0.45 -3.05 -109 -2.86 -0.68 -11.4 -1.90 -9.07 -0.05

Bxp3

-1.66 -0.93 -1.50 -0.74 -348 -1.76 -2.39 -0.91 -12.06 -2.33 -10.37 -0 65

Bxp4

-206 -1.54 -1.43 -1.39 -3.58 -379 -1.28 -1 76 20.16 606 -1325 -2.50

Sxpl

498 4.82 5.03 4.77 8.90 8 30 9.19 7.88 24.03 18.94 21.68 15 98

Sxp2

5.75 5 5 3 5.23 4.97 10.68 10.05 864 7.94 30 26 26.53 15.03 8.00

Sxp3

640 6.35 5.36 5.33 11.92 11.77 8.30 7.94 43.73 34.91 17.60 6.65

Sxp4

9.73 10.15 7.63 7.98 21.33 21.82 13 49 11.60 157 00 106 30 23.36 10.34

Note: P¡=50%, P2=75%, P3=85%, P4=95%, EX0=1.0, it is same in the table 2.

Therefore, the curve-fitting b y absolute 6.2 Effect of the curve-fitting with different data criterion is better m a n that of square criterion proportion to the estimation result of parameters and according to the results of Monte-Carlo quantiles

experiments.

Table 2 partial result of parameters and quantiles estimated by the curve-fitting method with the different data proportion

Population parameters N

0

30

30

30

30

30

30

30

30

30

n

30

30

30

30

30

30

30

30

30

30

Cv

0 50

30.50

0.50

0 50

0.50

0 50

0 50

1 00

1 0 0

1 00

Cs

1.00

1,00

100

1 00

1.00

1.00

1 00

2 0 0

2.00

200

data pro- porti on 0 1

02

03

0 5

06

0.7

1 0

0 1

02

03

ECv

0,53

0,52

0 52

051

0 51

051

0.52

102

103

102

ECs

1 18

1 00

0 97

0 96

0 9 4

0 94

0 9 9

2 08

2 06

2 05

SCv

0.09

0.08

0.08

0 07

0 07

0 07

0 08

0 17

0 16

0.16

SCs

0 34

0-52

0 53

0 53

0.52

0 52

0,54

0 39

0 4 4

0 43 Bxpl

-2 31

-0.66

-0 33

-0.06

0 17

0 18

-0 46

-2 04

-1 90

-1 31 B*p2

-3 13

-1.56

-1 00

-0 57

-0.33

-0 27

-1.62

0.61

-1 22

-0 20 Bxp3

-2.54

-2 14

-1 59

-1.19

-107

-0.99

-2 59

6 45

1 06

2 15 Bxp4

1.47

-3 94

- 3 7 9

-3 81

^130

-4.13

-5 80

37 70

10 71

11 18 Sxpl

10 05

10 62

1060

1055

10 57

10 54

1075

22 64

22 69

22.44 Sxp2

13 4

13 49

13 21

12 72

12 63

1258

13 02

33.50

32.27

31 86 Sxp3

15.94

15.51

15 17

14 78

14 92

1504

15 33

42 95

40 46

39 98 Sxp4

23 11

24 97

25 46

26.94

28 29

28 92

28 56

86 19

94.06

95.18

12