Prof. B.P. Parida University of Botswana P/Bag UB 00704, Gaborone, Botswana paridab@mopipi.ub.bw

1. Water Resources Assessment in Sub-Saharan Africa: Prediction in Ungauged BasinsRegionalisation Techniques Prof. B.P. Parida University of Botswana P/Bag UB 00704, Gaborone, Botswana paridab@mopipi.ub.bw WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

2. Global Perspective WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

3. BACKGROUND WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

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6. Regionalisation Concepts • Why and when necessary- • Elements for its success? • Some commonly used methods Station-Year Method Index Flood Method (USGS) Darlymple, 1960 Method based on Linear Mult. Regression Tech. • Newer Technique L-Moments WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

7. Forecasting (How much ? When?) Modelling on Real Time and/or Stochastic Modelling Rainfall-Runoff Models, UH-Type Models, AR/ARMA/ARIMA Models Prediction (How much ? How often ?) Statistical Modelling Gumbel (EV-1) ; Normal (2-P) GEV ; PT III ; Log-PTIII (3-P) Kappa (4-P) Wakeby (5-P) Operational Hydrologists WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

8. Statistical Models to produce Reliable Quantiles (XT) Depend on 2 aspects • Choice of an appropriate distribution (Descriptive Ability) 2-P, 3-P, 4-P, 5-P 2-P : produce results with less bias 3-P : produce results with less variability 4-P & 5-P additionally overcome problems of condition of separation (but require more data) WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

9. Choice of an appropriate parameter estimation technique ( Predictive Ability) MOM, MML, PWM MOM, MML : require large data to be unbiased PWM : overcomes problems of small sample even with outliers • Classical Study : Hoskings, Wallis and Wood (1991) Even with a wrong choice of distribution iff the method of Parameter Estimation was PWM – then one could obtain quantiles – which are least biased and also least variable • In real life - if addition the sample size could be increased the reliability of XT could increase (can be done through the technique of regionalisation which will also be useful for ungauged basins) WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

10. Why and Where necessary ? Atdata scare basins or for Ungauged basins! • where no flows are recorded • sites on a river which is gauged different locations upstream or downstream or gauged on some tributaries rather than the main river • even sites which have only few years of record or have half hazard records WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

11. To construct a regional /growth curve WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

12. Steps for success of Regionalsation 3 steps • Identification of a homogeneous region • Aggregation of data from all gauged sites within the identified region and identification of a common statistical distribution. (Development of regional curve) • Disaggregating the results to ungauged basins with in the region for estimation of XT . WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

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14. Some commonly followed methods • Station Year Method • Index Flood Method • Using Linear Multiple Regression Method Data requirement: Annual maximum flow series at all the gauged sites within the region (random, from similar climatic cond., free from measurement errors) WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

15. Station Year Method Say we have M stations in a region At say station 1 we have N years of record… which has average = xb. We can get standardised values at this station by dividing the observations with xb. xs1= x1/xb ; xs2= x2/xb and so on.. Similarly do this for station 2 which has say NN years of record and for all stations one by one.. Finally one gets a long record ( N+NN…+ ) of standardised values which can be subjected to statistical modelling to get a regional curve (xsT vrs T or Gumbel y) WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

16. Index Flood Method • For each station compute the average (xb) and standard deviation of say Flood data (xi, i=1,..N) ; hence the Gumbel parameters u,α • Using the above Gumbel parameters, at each station compute quantiles say x10, x50, x100, x200, x1000 • Then check for homogeneity of stations : - Find (x10/xb)i = βi at each station (i). - Find average of βi ; say βa - Find the changed x10, sayX10 at each station = βa . Xb - Find the change in Gumbel y = (X10 – u)/α; for each stn. - Plot the changed y vrs N at each stn and reject stations which fall outside the upper or lower limits of the curve WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

17. Homogeneity Test WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

18. Then compute (x50/xb)i , i.e. standardised quantile at each station (i); find the median value and plot this against Gumbel yT for T=50 yrs • Repeat this for all other T i.e 100, 200, 1000 WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

19. Linear Multiple Regression Technique • Choose the Dependent (Xb) and the independent variables (Area, Rainfall, Slope, Field Capacity, Stream Frequency, Forest Index…). • Assume a non-linear relationship say, Xb = c. Aa. Rr. FCf • To convert this equation to linear form – take logs of both sides of the equation, which now takes the form log Xb = log c + a log A + r log R + f log FC • This equation is now ready for Multiple Regression WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

20. Which or how many variables to be considered ? • For this: first construct a correlation matrix showing the correlation between the logs of variables • Look at the variables which have significant correlation coefficients with the dependent variables.( say A, FC) • Then check if these independent variables are truly independent ? ( Hint: They should not have significant correlation between themselves ) i.e correlation between A and FC which is -0.207 (not-significant; truly independent) • Then using SPSS undertake multiple regression between log Xb as the dependent and log A and log FC as the independent variables WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

21. Selecting the relationship(xb = 31.77 A-0.055.FC0.054) *Fcritical 2,20; **Fcritical 1,21both@5% significance WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

22. (xb = 31.77 A-0.055.FC0.054) • For ungauged basins, to estimate say xT ; we go to the regional curve and find the standardised quantile XS against the Gumbel y (equivalent of the chosen T) and then multiply this with the xb computed by the above equation to get the actual xT . • Similarly such type relationships can be developed separately for x10, x50, x100, x200 and x1000 and used when the basins in a region are ungauged and when the regional curve is unavailable. For this, however we need to obtain xT s at all gauged sites along with the basin characteristics. WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

23. Method of L-Moments 1= E [x] 2= (1/2)E [x(1:2) – x(2:2)] 3= (1/3)E [x(1:3) – 2.x(2:3) + x(3:3)] 4=(1/4)E[x(1:4)–3.x(2:4) – 3.x(3:4) – x(4:4)] Other ways of finding the L-Moments L-Coeff. of Variation (L-Cv), 2 = 2/1 L-Coeff. of Skewness(L-Sk), 3 = 3/2 L-Coeff. of Kurtosis (L-Ku), 4 = 4/2 • Homogeneity Test: H : <1; 1-2; >2 • Z stat. (using simulated and original data ) – a value close to 1.64 is a possible candidate (L-Skew /L-Kurt) WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

24. Other ways of finding the L-Moments Where, WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

25. L-Skew vs L-Kurt Diagram WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

26. Suppose one gets GEV as a plausible distn. Standardised Quartile (xS) = ξ + α [ 1 - {- log F (x)}k ] / k , for k ≠ 0 where F(x) = Probability of non-exceedence corresponding to recurrence interval T ; ξ = the location parameter, α = scale parameter and k = the shape parameter and xb = Average annual value . Then the desired rainfall quantile for a recurrence interval of T, xT = xS. xb WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

27. Regionalisation of Drought characteristics One can use annual maximum deficient volumes or maximum deficit durations to undertake regional analysis. Similarly one can use Low flow indices say (X1)95 WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

28. Thank You WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

29. Correlation Matrix using Logs of Variables(significant r = 0.413) WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008

30. The correlation coefficient WATER RESOURCES ASSESSMENT COURSE DAR-ES-SALAAM, 21-25 JAN 2008