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PART II

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PART II

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  1. INTRODUCTORY STUDY : WATER INDICATORS AND STATISTICAL ANALYSIS OF THE HYDROLOGICAL DATA EAST OFGUADIANA RIVERby Nikolas Kotsovinos ,P. Angelidis , V. Hrissanthou , and A. PechtelidisDemocritus University of Thrace -School of Engineering Greece

  2. PART II SOME STATISTICAL ANALYSIS

  3. AMARELEJA STATION We choose one station (AMARELEJA) to test rigorously the standardization procedure (probability transformation) assuming normal, log–normal, and gamma statistics for precipitation.

  4. AMARELEJA station: Annual precipitation for the period 1932-2005. Mean annual precipitation = 523 mm

  5. AMARELEJA station: Mean monthly precipitation and standard deviation for the period 1932-2005 The standard deviation is very high

  6. AMARELEJA station: The ratio of (Standard deviation / Mean monthly precipitation) for the period 1932-2005. The ratio varies from 0.65 to 2.5.

  7. Data and theoretical cumulative probability distributions for monthly precipitation at AMARELEJA station. Period 1932-2005. We tested three theoretical cumulative probability distributions: GAMMALOG-NORMALNORMAL The fit is not very good for the above three theoretical distributions

  8. PART III COMPUTING SPI

  9. The SPI is computed by fitting a probability density function to the frequency distribution of precipitationsummed over the time scale of interest. Each probability density function isthen transformed into the standardized normal distribution. Calculation:The monthly precipitation time series are modelled using different statistical distributions. 1. The first is the gamma distribution, whose probability density function is defined as where The calculations are quite complicated 2. Another possibility is the log–normal distribution. It has the advantage of simplicity since it is just a logarithmic transformation of the data, i.e. Y = ln(x) (for x > 0), with the assumption that the resulting transformed data are described by a Gaussian distribution.

  10. 3. Normal distribution. • The central limit theorem suggests that, as we move to extended time periods in excess of 6 months, the resultant time averaging will tend to shift the observed probability distributions towards normal. Because the gamma distribution tends towards the normal as the shape parameter α tends to infinity, it would be computationally more efficient to standardize the data directly from a fitted normal distribution. • We computed the multi-temporal SPI values by modelling the precipitation data with three different statistical distributions. • GAMMA • LOG-NORMAL • NORMAL • We concluded, that we can use for simplicity LOG-NORMAL or NORMAL distribution instead of GAMMA producing almost the same results

  11. SPI OF 24 MONTHS

  12. AMARELEJA STATION. Period 1932 -2005 Data and three theoretical cumulative probability distributions for the running sum of 24 months. The fit seems quite good for all the three theoretical distributions.

  13. AMARELEJA STATION. Period 1932 -2005 SPI 24 for GAMMA, LOG-NORMAL and NORMAL distribution. The results are very close

  14. AMARELEJA STATION. Period 1932 -2005 Comparison of SPI 24 for GAMMA, LOG-NORMAL and NORMAL distribution. The results are essentially coincided

  15. AMARELEJA STATION: Comparison of SPI 24 for GAMMA distribution: • Period 1932-2005 • Period 1932-1968 • Period 1969-2005 • Trend in SPI values indicates that the drought conditions has • changed significantly during last decades

  16. SPI OF 12 MONTHS

  17. AMARELEJA STATION. Period 1932 -2005 Data and three theoretical cumulative probability distributions for the running sum of 12 months. The fit seems quite good for all the three theoretical distributions

  18. AMARELEJA STATION. Period 1932 -2005 SPI 12 for GAMMA, LOG-NORMAL and NORMAL distribution. The results are very close

  19. AMARELEJA STATION. Period 1932 -2005 Comparison of SPI 12 for GAMMA, LOG-NORMAL and NORMAL distribution. The results are essentially coincided

  20. AMARELEJA STATION:Comparison of SPI 12 for GAMMAdistribution: • Period 1932-2005 • Period 1932-1968 • Period 1969-2005 • Trend in SPI values indicates that the drought conditions has • changed significantly during last years

  21. SPI OF 6 MONTHS

  22. AMARELEJA STATION. Period 1932 -2005 Data and three theoretical cumulative probability distributions for the running sum of 6 months. The fit seems quite good for all the three theoretical distributions

  23. AMARELEJA STATION. Period 1932 -2005 SPI 6 for GAMMA, LOG-NORMAL and NORMAL distribution. The results are to close, especially for GAMMA and LOG-NORMAL distributions

  24. AMARELEJA STATION. Period 1932 -2005 Comparison of SPI 6 for GAMMA, LOG-NORMAL and NORMAL distribution. The results are essentially coincided, especially for GAMMA and LOG-NORMAL distributions

  25. SPI OF 3 MONTHS

  26. AMARELEJA STATION. Period 1932 -2005 Empirical and three theoretical cumulative probability distributions for the running sum of 3 months. The fit is not very good for Log-normal distribution.

  27. AMARELESA STATION. Period 1932 -2005 SPI 3 for GAMMA, LOG-NORMAL and NORMAL distribution. The results for GAMMA and LOG-NORMAL are very close

  28. CONCLUSIONS We concluded, that we can use for simplicity LOG-NORMAL or NORMAL distribution instead of GAMMA producing almost the same results. We had analyzed a long data series of monthly precipitation for the period 1932 – 2005. Are the above conclusion valid for sorter data series? In order to answer this question, we made the same analysis: a) For the half data series (1932-1968) b) For the 10% of the data series, e.g. 1932 – 1939.

  29. LETS TAKE THE HALF DATA SETS1932 - 1968

  30. AMARELEJA STATION. Period 1932 -1968 Data and three theoretical cumulative probability distributions for the running sum of 24 months. The fit seems quite good for all the three theoretical distributions

  31. AMARELEJA STATION. Period 1932 -1968 Comparison of SPI 24 for GAMMA, LOG-NORMAL and NORMAL distribution. The SPI values are essentially coincided

  32. AMARELEJA STATION. Period 1932 -1968 Data and three theoretical cumulative probability distributions for the running sum of 12 months. The fit is good enough for all the three theoretical distributions

  33. AMARELEJA STATION. Period 1932 -1968 Comparison of SPI 12 for GAMMA, LOG-NORMAL and NORMAL distribution. The SPI values are essentially coincided

  34. LETS TAKE ONLY 10% OF THE DATA SETS 1932 - 1939

  35. AMARELEJA STATION. Period 1932 -1939 Data and three theoretical cumulative probability distributions for the running sum of 12 months. The fit is not very good for all the three theoretical distributions

  36. AMARELEJA STATION. Period 1932 -1939 SPI 12 for GAMMA, LOG-NORMAL and NORMAL distribution. The SPI values are essentially coincided

  37. CONCLUSIONS We can use for simplicity LOG-NORMAL or NORMAL distributions instead of GAMMA, producing almost the same results, especially with GAMMA and LOG-NORMAL. This conclusion was verified not only for very long data series, but also for sorter ones. The verification was made for SPI values of 24, 12, 6 and 3 months. ?????/////