Long-Term Hydrologic Trends in the Main Greek Rivers: A Statistical Approach

Abstract

The scope of this research effort was to examine the effect of water management practices and land use changes on river flow over the last 3 decades, to identify the dominant trends in the discharge and precipitation time series and to examine the interrelationship between these two parameters. In order to accomplish these aims, the annual discharge time series of seven (7) major rivers in Greece were compared to the annual precipitation of the corresponding watersheds. This comparison was achieved through trend analysis of each time series, which involves the determination of basic statistical characteristics (normality, homogeneity, stationarity). Due to lack of satisfactory discharge time series at the downstream parts of each catchment examined, the results from E-HYPE pan-European hydrological model was used (European – HYdrological Predictions for the Environment). The main outcome of this work concludes that there is no consistent, single trend for the entire study period for any of the investigated rivers, while there are some wet and dry periods in the data which are very clear in all catchments and coincide at a temporal level. The main dry periods were at the end of the 1980s and the beginning of the 2000s. There is also a prolonged wet period during the last decade for all study catchments.

Annexes

1.1 Annex I

Statistical criteria of E-HYPE model performance by comparing the observed (o) and predicted (p) data of a sample size (n)

Mean error ME: $\mathrm{ME}=\frac{\Sigma_{i=1}^n\left({o}_i-{p}_i\right)}{n}$

Mean absolute error MAE: $\mathrm{ME}=\frac{\Sigma_{i=1}^n\left|{o}_i-{p}_i\right|}{n}$

Mean absolute percentage error MAPE: $\mathrm{ME}=\frac{\Sigma_{i=1}^n\left|\frac{o_i-{p}_i}{o_i}\right|}{n}\times 100$

Root mean squared error RMSE: $\mathrm{ME}=\sqrt{\frac{\Sigma_{i=1}^n{\left({p}_i-{o}_i\right)}^2}{n}}$

Pearson’s correlation coefficient R: $R=\frac{\Sigma_{i=1}^n\left({p}_i-\overline{p}\right)\left({o}_i-\overline{o}\right)}{\sqrt{\Sigma_{i=1}^n{\left({p}_i-\overline{p}\right)}^2\ }\sqrt{\Sigma_{i=1}^n{\left({o}_i-\overline{o}\right)}^2}}$

Squared correlation coefficient R ²: R ² = R ²

Nash–Sutcliffe coefficient of efficiency Nr: $\mathrm{Nr}=1-\frac{\Sigma_{i=1}^n{\left({o}_i-{p}_i\right)}^2}{\Sigma_{i=1}^n{\left({o}_i-\overline{o}\right)}^2}$

1.2 Annex II

1.2.1 Normality Tests

Kolmogorov–Smirnov (KS) test: KS = sup _x |F*(x) − Fn(x)|, where sup stands for supremum, Fn(x) is theoretical cumulative distribution function of the normal distribution function and F*(x) is the normal empirical distribution function of the data, with known mean μ and standard deviation σ.

Lilliefors (LF) test: LF = max _x |F*(x) − Sn(x)|, where Sn(x) is the sample cumulative distribution function of the normal distribution function and F*(x) is the empirical distribution function, with the sample mean $\mu =\overline{x}$ and the sample variance s ² defined with denominator n − 1.

Shapiro–Wilk (SW) test: $\mathrm{SW}=\frac{{\left({\Sigma}_{i=1}^n{a}_i{x}_i\right)}^2}{\Sigma_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}$, where x _i stands for ordered (increasing ordered) sample values and a _i stands for constants generated from the means, variances and covariances of the order statistics of a sample of size n from a normal distribution.

1.2.2 Homogeneity Tests

von Neumann test: $N=\frac{\Sigma_{i=1}^{n-1}{\left({x}_i-{x}_{i+1}\right)}^2}{\Sigma_{i=1}^n{\left({x}_i-\overline{x}\right)}^2}$, where x _i is the hydrologic variable constituting the sequence in time, n is the total number of hydrologic records and $\overline{x}$ is the average of x _i .

Cumulative deviations test: Sensitivity to the departures from homogeneity is defined by the following statistic:

$Q={\mathrm{max}}_{0\le k\le n}\left|{S}_k^{\ast \ast}\right|$, where S ^* * _k is the rescaled adjusted partial sums.

${S}_k^{\ast \ast }={S}_k^{\ast }/{D}_x$, k = 1, 2, …, n, where ${S}_k^{\ast }={\Sigma}_{i=1}^k\left({x}_i-\overline{x}\right)$, k = 1, 2, …, n, and D _x the sample standard deviation.

High values of Q are an indication for non-homogeneity.

The homogeneity can also be tested with the following statistic:

$$ R={\mathrm{max}}_{0\le k\le n}\left|{S}_k^{\ast \ast}\right|-{\mathrm{min}}_{0\le k\le n}\left|{S}_k^{\ast \ast}\right| $$

Bayesian Test: $U=\frac{1}{n\left(n+1\right)}{\Sigma}_{k=1}^{n-1}{\left({S}_k^{\ast \ast}\right)}^2$, for p _k independent of k.

$A={\Sigma}_{i=1}^{n-1}{\left({Z}_k^{\ast \ast}\right)}^2$, k = 1, 2, …, n, for p _k proportional to [k(n − k)]⁻¹. Z ^** _k is the weighted rescaled partial sums, ${Z}_k^{\ast \ast }=\left[{\left\{k\left(n-k\right)\right\}}^{-1/2}{S}_k^{\ast}\right]/{D}_x$.

1.2.3 Stationarity Tests

t-test: To apply this test, the annual time series is divided into two (or more) subseries of size n ₁ and n ₂ (n ₁+n ₂ = n):

${t}_s=\frac{\left|{\overline{x}}_2-{\overline{x}}_1\right|}{S\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$, $S=\sqrt{\frac{\left({n}_1-1\right){s}_1^2+\left({n}_2-1\right){s}_2^2}{n-2}}$, where ${\overline{x}}_1$, ${\overline{x}}_2$, s ²₁ and s ²₂ are the estimated means and variances of the first and the second subseries, respectively.

Mann–Whitney test: To apply this test, the annual time series n _t is divided into two (or more) subseries of size n ₁ and n ₂ (n ₁+n ₂ = n), and a new series z _t (t = 1, 2, …, n) is defined by arranging the original data (n _t ) in increasing order of magnitude:

$u=\frac{\Sigma_{t=1}^{n_1}R\left({n}_t\right)-{n}_1\left({n}_1+{n}_2+1\right)/2}{{\left[{n}_1{n}_2\left({n}_1+{n}_2+1\right)/12\right]}^{1/2}}$, where R(n _t ) is the rank of the observation n _t in ordered series z _t .

1.2.4 Trend

Mann–Kendall test: The Mann–Kendall statistic S compares each value of the series (x _t ) with all subsequent values (x _t+1) and is defined as

$S={\Sigma}_{t^{\prime }=1}^{n-1}{\Sigma}_{t={t}^{\prime }+1}^n\operatorname{sgn}\left( xt-{x}_{t^{\prime }}\right)$, where sgn is the signum function, $\operatorname{sgn}\left( xt-{x}_{t^{\prime }}\right)=\left\{\begin{array}{c}\hfill 1,\mathrm{if}\ {x}_t>{x}_{t\hbox{'}}\hfill \\ {}\hfill 0,\mathrm{if}\ {x}_t={x}_{t\hbox{'}}\hfill \\ {}\hfill -1,\mathrm{if}\ {x}_t<{x}_{t\hbox{'}}\hfill \end{array}\right.$

Based on Mann [41] and Kendall [40], when n ≥ 8, the statistic S is approximately normally distributed with the mean m and the variance V as follows: E(S) = 0, $V(S)=\frac{1}{18}\left[n\left(n-1\right)\left(2n+5\right)-{\Sigma}_{i=1}^g{e}_i\left({e}_i-1\right)\left(2{e}_i+5\right)\right]$, g is the number of tied groups, and e _i is the number of data in the ith tied group.

The standardised test statistic Z is defined as $Z=\frac{S+m}{\sqrt{V(S)}}.$

Spearman’s Rho: The Spearman’s Rho D statistic is defined as

$D=1-\frac{6{\Sigma}_{1=1}^n{\left[R\left({X}_i\right)-i\right]}^2}{n\left({n}^2-1\right)}$, where R(X _i ) is the rank of ith observation X _i in the sample size n.

Under the null hypothesis that the time series has no trend, it can be shown that the statistic t _s has a Student’s t-distribution with n–2 degrees of freedom. Here, t _s is defined as ${t}_{\mathrm{s}}=D\sqrt{\frac{n-2}{1-{D}^2}}.$

Sequential Version of the Mann–Kendall Test (Mann–Kendall Rank Correlation Test): The sequential version of the Mann–Kendall test is calculated so that rank (x _i ) > rank (x _j ) (i > j). The t statistic is calculated as $t={\Sigma}_{i=1}^n{n}_i$. The distribution of t is assumed to be asymptotically normal with the following expectation: $E(t)=\mu =\frac{n\left(n-1\right)}{4}$ and $\mathrm{Var}(t)={\sigma}^2=\frac{n\left(n-1\right)\left(2n+5\right)}{72}.$

The null hypothesis that there is no trend is rejected for high values of the reduced variable |u(t)|, which is calculated as $u(t)=\frac{t-E(t)}{\sqrt{\mathrm{Var}(t)}}$. The statistic u′(t) is computed backwards starting from the end of the time series.

CUSUM Test: The test statistic Vk is defined as ${V}_k={\Sigma}_{i=1}^k\operatorname{sgn}\left({x}_i-{x}_{\mathrm{median}}\right)$, k = 1,2, …, n, where x _median is the median value of the x _i data set and sgn(x).

Sen’s Slope Estimator: The Sen’s slope estimation test is defined for a season g as $\beta =\mathrm{Median}\left(\frac{x_i-{x}_j}{i-j}\right)$, i < j, where Q is the slope between points x _i and x _j , x _i is data measurement at time i and x _j is data measurement at time j.

It is defined as the estimator β which is the median overall combination of record pairs for the whole data set and is resistant/robust to the extreme observations or outliers. The positive value of the β connotes the slope of the upward trend and negative value for the downward trend [29].