Frequency Analysis for Non‐stationary Flood Series

Prepared By:Narendra Kumar Goel, Sunil Poudel

and R.B. Jigajinni

Indian Institute of Technology, Roorkeegoelhy@gmail.com

Presented By:Sunil Poudel

INTRODUCTION

What is stationarity and non‐stationarity?

Factors causing non‐stationarity?

Standard approaches to flood frequencyanalysis is assumption of stationarity.

Estimate for a design flood quantile is stillrequired for a river that demonstrates non‐stationarity.

STUDY AREA AND DATA AVAILABILITY

ZONE 3 MAP Bordered by red Line

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Sub-Zone River BasinNumber of

SitesCatchment Area

(km2)

3 (a) Mahi and Sabarmati 4 30.1- 1094

3 (b) Lower Narmada and Tapi 9 17.2- 284.9

3 (c) Upper Narmada and Tapi 12 41.8- 2110.83 (d) Mahanadi 17 30-11503 (e) Upper Godavari 8 31.3 – 2227.43 (f) Lower Godavari 15 35 – 8243 (g) Indravati 03 (h) Krishna and Penner 11 31.72 – 16903 (i) Kaveri 0

Table 1: Summary of annual flood data used

CONTD...

StateNumber of rain gauge

stations

Andhra Pradesh 24Bihar 6

Gujarat 25Karnataka 27

Kerala 10Maharashtra 46

Orissa 15Tamilnadu 31

West Bengal 21

Table 2: Summary of rainfall data used in the study

INVESTIGATION OF NON‐STATIONARITY

Non‐stationarity due to short term dependence and longterm dependence have been examined.

Short‐term dependence was examined using:(i) Median crossing test (Fisz, 1963);(ii) Turning point test (Kendall and Stuart, 1976);(iii) Rank difference test (Meacham, 1968);(iv) Kendall’s rank correlation test (Kendall, 1970);(v) Run test (Guttman et al.,1971);(vi) Linear regression test ( Kottegoda, 1980);(vii) Wald‐Wolfowitz test (Wald and Wolfowitz, 1943);

CONTD..

(viii) Runs above and below the median test (Shiau andCondie, 1980);

(ix) Rank Von Neumann ratio test (Madansky, 1988);(x) Von Neumann ratio test (Madansky, 1988); and(xi) Auto correlation test (Yevjevich, 1971)

Long‐term dependence was examined using:Hurst Coefficient (K) has been examined includingbootstrapping approach.

ANALYSIS OF ANNUAL FLOOD SERIES AND ANNUAL DAILY MAXIMUM RAINFALL SERIES

Variable AFS ADMRSr1 –0.476 to 0.715 0.435 to 0.65

Hurst’s K 0.451 to 0.938 0.466 to 0.952Short-term dependence 9.21% series 9.76% series

Long-term dependence 17.10% series 11.22% series

The probabilities are fairly high and there is no reason todisregard long term dependence.

If a series shows short-term independence, one should stillinvestigate for long-term dependence.

IMPACT OF NON‐STATIONARITY ON FREQUENCY ESTIMATES

The impact of dependencies in extreme flow and rainfall estimateswere examined using synthetic sequences.

Data sequences were generated using a mixed noise model (Booyand Lye, 1989).

(d)

t(d)

1tdd

(c)t

(c)1tcc

(b)t

(b)1tbb

(a)t

(a)1taat

Xw

XwXwXwX

where, wa , wb , wc and wd are weights , and a ,b , c and d areserial correlation coefficients, t

(a) , t(b) , t

(c) , t(d) are independent

processes , having zero mean and variances (1-a2), (1-b

2), (1-c2),

(1-d2) respectively.

CONTD…

Parameters of mixed-noise model i.e. wa, wb, wc , wd , a, b, cand d are computed as per procedure given by (Booy and Lye,1989).

In this study, two types of data sets are generated.•1000 samples of 100 years length having r1 and K of original series.•1000 samples of 100 years length having r1 =0.0 and K=0.5

This modelling approach is designed to reproduce, on average,the lag one serial correlation coefficient and the Hurst coefficient, K.

Quantiles for 50 years, 100 years, 200 years return period arecomputed using General Extreme Value distribution and probabilityweighted moments (PWM) method.

The expected values of flood quantiles for return periods of 50,100 and 200 years i.e. E (Q50), E (Q100), E (Q200) are computed forgenerated series.

CONTD…

In annual flood series, average Hurst’s K for series, havinglong-term dependence is 0.8324.

For annual daily maximum rainfall series average Hurst’s Kis 0.7630.

Datasets with long term 

dependence

Underestimation on

50 years return period

100 years return period

200 years return period

Annual flood series 44.28% 54.20% 64.26%

Annual daily maximum rainfall

series

30.88% 42.28% 54.83%

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It is quite evident that on the independence assumption,when the series is in fact non-stationary leads tounderestimation of quantiles. This underestimationincreases with the increase in return period. This has alsobeen found to be directly related with the Hurst coefficient.

The under estimation due to independence assumptionwere obtained as per the procedure explained in theprevious section. For a return period of 100 years theunderestimation has been found to be linearly varying withHurst coefficient (K) as follows:

Y= 53.495 X- 27.89; r=0.69

Where, Y is % underestimation in 100 years return periodquantile, X is Hurst’s coefficient (K) and r is coefficient ofcorrelation.

CONCLUSIONS

Before estimation of flood quantiles, the data series areinvestigated not for short term and long-term dependence.

The long-term dependence should be taken into account asit may significantly increase the risk associated with futurepeak flows.

Data sequences are generated using mixed noise model.The intent with mixed noise model is to preserve in thegenerated data sequences both short-term and long-termdependence observed in the original series.

Long-term dependence, if present in a data series,increases degree of uncertainty associated with extreme flowquantiles.

THANK YOUFor Your Kind Attention 

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