Title

1. Title The Development of a Web-based Rainfall Atlas for Southern Africa Günther Kratz, Walter Zucchini, Oleg Nenadić, Institut für Statistik und Ökonometrie, University of Goettingen, Germany.

2. toc The Development of a Web-based Rainfall Atlas for Southern Africa contents: Development and Calibration of a daily rainfall model Interpolation of the daily rainfall model parameters Generating artificial rainfall sequences Information transfer

3. Purpose • Purposes of the Rainfall Atlas: A water resources decision support system

4. 1 – dev/calib i. Development and Calibration of a daily rainfall model: Joint distribution of the rainfall amount for site x • discrete part (“does it rain“) • continuous part (“if it rains, how much?“) Sample observed daily rainfall amounts for site x.

5. 1 – discrete part i. Development and Calibration of a daily rainfall model: • discrete part of the daily rainfall model: • first-order (seasonal)Markov Chain with two states W (wet) and D (dry): W D Observed daily rainfall amounts for a given site Estimated probabilities and conditional probabilities. First-order Markov Chain • Since pW+pD=1, pW|W+pD|W=1 and pD|D+pW|D=1 it is sufficient to consider only pW , pW|W andpW|D

6. i. Development and Calibration of a daily rainfall model: 1 – seas in discrete part • seasonality of the discrete part: l(T) e.g. pW|D(T): (Probability that rainoccurs on period T ,given that T-1 was dry) p>0.5 0 p<0.5 Seasonality of the probability for a wet day given that the day before was dry • Instead of using 365 Parameters (probabilities are estimated for each day), an approximation with the first two terms of the Fourier Series representation is used. • To avoid inadmissible estimates (estimated probabilities exceeding the interval [0,1]) logits (l=log(p/(1-p))) instead of probabilities are used.

7. January July i. Development and Calibration of a daily rainfall model: 1 – continuous part • continuous part of the daily rainfall model: Empirical distribution of rainfall-amount on rainy days Fitted Weibull distribution with Parameters a(T) (scale parameter) and b (shape parameter) • The mean m(T) exhibitsseasonal behavior, while theCVremainsconstant:

8. i. Development and Calibration of a daily rainfall model: 1 – model summary • the daily rainfall model: Parameters:(discrete part) seasonal ... t 1 2 3 365 d d d d State 1: “dry day“ Discrete part ... w w w w State 2: “wet day“ Amount of rainfall on wet days ... Continuous part Scale parameter: seasonal AMMU (0) AMMU (1) AMMU (2) PHMU (1) PHMU (2) CV Parameters:(continuous part) Weibull Distribution Shape parameter: non-seasonal

9. ii. Interpolation of the daily rainfall model parameters: 2 – interpolation gradient aspect Parameters for the calibrates sites roughness Calibrated sites (5070) kriging with externaldrift exposure Topological features 1.5 km Interpolated parameters ... ... resulting in a resolution of 1 square mile

10. ... t 1 2 3 365 d d d d State 1: “ dry day “ Discrete ... part ? w w w w State 2: “ wet day “ W | D ... ? Continuous Amount of rainfall W | W on wet days part 0.6 0.6 0.4 0.5 0.4 0.5 0.3 0.4 0.4 0.3 dweibull(x, 3, 2) 0.3 dweibull(x, 3, 2.5) dweibull(x, 3, 2.8) dweibull(x, 3, 2) 0.3 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.0 0.0 0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10 x x x x iii. Generating artificial rainfall sequences: 3 – generating Artificial rainfall sequence (5000 yrs) Model 16 Parameters 1,825,000 times(≙ 5000 years) Empirical statistics (~950) Site x Simulation for each of the 424,646 sites • ~ 1 GB zip-compressed ASCII-Files • 950 statistics for each of the 425,000 sites Database

11. Database iv. Information transfer: 4 – inftrans • Image database: • data preparation: 126x100 px 255x203 px 1065x842 px 443x355 px 693x550 px • ~ 5000 maps in 5 different sizes • Site database: • Since information is required in two forms – as maps (e.g. for ecologists) as well as in numerical form (e.g. for engineers) – the database was used to construct an Image and a Site Database • extraction of statistics for single sites into separate, small files

12. 1daymax 2daymax 5daymax 1daymax 2daymax 5daymax 1% 5% 10% 25% 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% 50% 75% 90% 95% 99% AMDW0 AMDW1 AMDW2 PHDW1 PHDW2 AMWW0 AMWW1 AMWW2 PHWW1 PHWW2 AMMU0 AMMU1 AMMU2 PHMU1 PHMU2 CV iv. Information transfer: 4 – data struc • Data Structure: Mean SD CV PercentilesRain per Rainday Stormdays Storm PercentageStormrain ANNUAL 1% 5% 10% 25% 50% 75% 90% 95% 99% Mean SD CV PercentilesRain per Rainday Stormdays Storm PercentageStormrain Exc. prob. MONTHLY 10mm 25mm 50mm 75mm100mm 125mm 150mm 200mm DAILY Dry run probability 5 days 10 days 15 days 20 days 25 days 30 days PARAMETERS Cities Rivers Altitude Max.mean Max.prob. Min.mean Max.mean SI-mean SI-SD OTHER DATA

13. iv. Information transfer: 4 – image db • Image Database: Selection of the desired data types (according to the scheme previously described) The resulting maps are displayed in the lower frame (with an option to view the maps in bigger size)

14. iv. Information transfer: 4 – image db 4 – site db • Site Database: 3.) 1.) The site is selected either by clicking on the imagemap or by entering coordinates. 2.) PHP-Script The required statistics can be selected via a form-element. The input is parsed to a PHP-script which reads the appropriate data and displays the output in tabular form. Site database

15. v. Conclusion: summary • It is shown, that it is possible to make a large repository of map- and numerical data available over the internet without heavyweight database-systems. • Currently, work is done in creating a Java-based applet, which simulates rainfall-sequences (on annual, monthly or daily basis). • Possibilites to expand the rainfall atlas to other countries are evaluated. • Preliminary results are located on http://134.76.173.220/rainfall/ (feedback, critism and suggestions are welcome!)