INTRODUCTORY STUDY : WATER INDICATORS AND STATISTICAL ANALYSIS OF THE HYDROLOGICAL DATA EAST OF GUADIANA RIVER by Nikolas Kotsovinos,P. Angelidis, V. Hrissanthou, and A. Pechtelidis Democritus University of Thrace -School of Engineering Greece

PART II SOME STATISTICAL ANALYSIS

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

AMARELEJA station: Annual precipitation for the period Mean annual precipitation = 523 mm

AMARELEJA station: Mean monthly precipitation and standard deviation for the period The standard deviation is very high

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

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

PART III COMPUTING SPI

The SPI is computed by fitting a probability density function to the frequency distribution of precipitation summed over the time scale of interest. Each probability density function is then 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.

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

SPI OF 24 MONTHS

AMARELEJA STATION. Period 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.

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

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

AMARELEJA STATION: Comparison of SPI 24 for GAMMA distribution: a)Period b)Period c)Period Trend in SPI values indicates that the drought conditions has changed significantly during last decades

SPI OF 12 MONTHS

AMARELEJA STATION. Period 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

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

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

AMARELEJA STATION:Comparison of SPI 12 for GAMMAdistribution: a)Period b)Period c)Period Trend in SPI values indicates that the drought conditions has changed significantly during last years

SPI OF 6 MONTHS

AMARELEJA STATION. Period 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

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

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

SPI OF 3 MONTHS

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

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

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 – 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 ( ) b) For the 10% of the data series, e.g – 1939.

LETS TAKE THE HALF DATA SETS

AMARELEJA STATION. Period 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

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

AMARELEJA STATION. Period 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

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

LETS TAKE ONLY 10% OF THE DATA SETS

AMARELEJA STATION. Period 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

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

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. ?????/////