Trend analysis of annual and seasonal precipitation data in Arcadia region (Greece)

Abstract

The purpose of the present study is to analyze the annual and seasonal precipitation trends in the prefecture of Arcadia, in Greece. Data series are based on records of 10 ground-based meteorological stations plus the available CRU TS 4.05 data series for Greece and for the geographic region of Peloponnese. The study contains a 30-year reference period from 1980 to 2009 and the last decade from 2010 to 2020. Homogeneity and serial correlation tests are considered in the analysis. Mostly negative trends are observed in annual and seasonal (except summer and irrigation period) series of reference period, with the majority of the significant trends to be in irrigation and autumn periods. The observed negative trends of the reference period tended to be transformed into positive trends over the last decade. Regarding the CRU TS 4.05 precipitation data, weak negative annual trends were observed during the reference period. However, significant positive trends were observed during autumn and irrigation periods. In the last decade, significant positive trends were detected during summer and irrigation periods.

Appendices

Appendix 1. Homogeneity tests

1.1 Alexandersson’s standard normal homogeneity test (SNHT)

For an annual series X_i (i is the year from 1 to n) with mean “X” and standard deviation “s,” a statistic T(k) to compare the mean of first “k” years of the record with that of last “n − k” years is described as

$$T\left(k\right)=k\overline z_1^2+\left(n-k\right)\overline z_2^2\;for\;k=1,2,\dots n$$

(1)

where

$${\overline{z} }_{1}=\frac{1}{k}\sum\nolimits_{i=1}^{k}\frac{\left({X}_{i}-\overline{X }\right)}{s} and {\overline{z} }_{2}=\frac{1}{\left(n-k\right)}\sum\nolimits_{i=k+1}^{n}\frac{\left({X}_{i}-\overline{X }\right)}{s}$$

(2)

If a break is located at year “K,” then T(k) reaches a maximum near the year k = K. The test statistic T₀ is defined as

$${T}_{0}=\underset{1\le k\le n}{\mathrm{max}}T\left(k\right)$$

(3)

The null hypothesis is rejected when T₀ is above the critical value, which depends on the sample size. Probability p was computed using the Monte Carlo method (using 2 × 10⁵ simulations).

1.2 Buishand’s range test

In Buishand’s range test for an annual series X_i (i is the year from 1 to n) with mean “X” and standard deviation “s,” the adjusted partial sums are defined as

$$\begin{array}{cc}S_0^\ast=0andS_k^\ast=\sum_{i=1}^k\left(X_i-\overline X\right)&\mathrm{where}\;k=1,2,\dots,n\end{array}$$

(4)

For a homogeneous series, the value of S ∗ k fluctuates around zero, as no systematic deviations of the X_i values with respect to their mean will appear. When a break point is present in the series, the S ∗ k value reaches a maximum value (negative shift) or minimum value (positive shift) near the year k = K. The (S ∗ k/s)/√n is depicted in the graphs representing the results of this test. The significance of the shift can be tested using “rescaled adjusted range” denoted as R, which is the difference between the maximum and minimum of the S ∗ k values scaled by the sample standard deviation.

$$R=\left(\underset{0\le k\le n}{\mathrm{max}}{S}_{k}^{*}-\underset{0\le k\le n}{\mathrm{min}}{S}_{k}^{*}\right)/s$$

(5)

R/√n values are compared with the critical values of Buishand (1982) to test for significance.

1.3 Pettitt’s test

Pettitt’s test is a nonparametric rank test. The ranks r₁, r₁, …, r_n of the series X₁, X₂, …, X_n are used to calculate the test statistics as

$$\begin{array}{cc}Y_k=2\sum\nolimits_{i=1}^kr_i-k\left(n+1\right),&\mathrm{where}\;k=1,2,\dots,n\end{array}$$

(6)

When a break occurs at year K, the statistic is maximum or minimum at year k = K,

$${Y}_{k}=\underset{1\le k\le n}{\mathrm{max}}\left|{Y}_{k}\right|$$

(7)

The value of Y_k is compared with critical values given by Pettitt (1979) to test for statistical significance.

1.4 Von Neumann’s ratio test

For an annual series X_i (i is the year from 1 to n) with mean “X,” the Von Neumann ratio “N” is defined as the ratio of the mean square successive (year to year) difference to the variance, given as

$$N=\sum\nolimits_{i=1}^{n-1}{\left({X}_{i}-{X}_{i+1}\right)}^{2}/\sum{\left({X}_{i}-\overline{X }\right)}^{2}$$

(8)

If there is a break in the series, the value of N tends to be lower than the expected value. If there is a rapid variation in the mean, the values of N may rise above two. This test does not indicate the exact location of the break year.

Appendix 2. Trend detection methods

2.1 Autocorrelated Mann–Kendall test

The Mann–Kendall test is a nonparametric test for monotonic trend detection. It does not assume the data to be normally distributed and is flexible to outliers in the data. The test assumes a null hypothesis, H₀, of no trend and alternate hypothesis, H_a, of increasing or decreasing monotonic trend. For a time series X_i = x₁, x₂, …, x_n, the Mann–Kendall test statistic S is calculated as

$$S=\sum\nolimits_{i=1}^{n-1}\sum\nolimits_{j=i+1}^{n}sign\left({x}_{j}-{x}_{i}\right)$$

(9)

where n is the number of data points, x_i and x_j are the data values in time series i and j (j > i), respectively, and sign (x_j – x_i) is the sign function as

$$sign\left(x_j-x_i\right)=\left\{\begin{array}{c}-1\;if\;\left(x_j-x_i\right)<0\\0\;if\;\left(x_j-x_i\right)=0\\1\;if\;\left(x_j-x_i\right)>0\end{array}\right.$$

(10)

Statistics S is normally distributed with parameters E(S) and variance V(S) as given below:

$$V\left(S\right)=\frac{n\left(n-1\right)\left(2n+5\right)-\sum_{k=1}^{m}{t}_{k}({t}_{k}-1)(2{t}_{k}+5)}{18}$$

(11)

where n is the number of data points, m is the number of tied groups, and t_k denotes the number of ties of extent k. Standardized test statistic Z is calculated using the formula below:

$$Z=\left\{\begin{array}{ccc}\frac{S-1}{\sqrt{V\left(S\right)}}&if&S>0\\0&if&S=0\\\frac{S+1}{\sqrt{V(S)}}&if&S<0\end{array}\right.$$

(12)

To test for a monotonic trend at an α significance level, the alternate hypothesis of trend is accepted if the absolute value of standardized test statistic Z is greater than the Z_1-α/2 value obtained from the standard normal cumulative distribution tables. A positive sign of the test statistic indicates an increasing trend and a negative sign indicates a decreasing trend.

According to variance correction approach proposed by Hamed and Ramachandra Rao (1998), recently used by Patakamuri et al. (2020), when the time series exhibit a significant serial correlation, the following modified variance V(S)^* of the Mann–Kendall test was used

$${V}^{*}\left(S\right)=V\left(S\right)*CF$$

(13)

The proposed correction factor CF is

$$CF=1+\frac{2}{n(n-1)(n-2)}\sum\nolimits_{k=1}^{n-1}\left(n-k\right)\left(n-k-1\right)\left(n-k-2\right){r}_{k}^{R}$$

(14)

where r_k and r_k^R are the lag − k serial correlation coefficients of data and ranks of data respectively and n is the total length of the series. In the case of correction factor (CF), only significant correlation coefficients are used.

2.2 Sen’s slope estimator

The Sen’s slope method (Sen 1968) is a robust nonparametric method of estimating the magnitude of trend slope. For a given time series X_i = x₁, x₂, …, x_n, with N pairs of data, the slope is calculated as given below:

$$\begin{array}{ccc}\beta_i=\frac{x_j-x_k}{j-k},\forall\;\mathrm k\leq\mathrm j&\mathrm{and}&i=1,2,\dots,N\end{array}$$

(15)

Median of N values of βi gives the Sen’s estimator of slope β

$$\beta=\left\{\begin{array}{ccccc}\frac{\beta_{\mathrm N+1}}2&if&N&is&odd\\\frac{\left(\frac{\beta_{\mathrm N}}2+\frac{\beta_{\mathrm N+2}}2\right)}2&if&N&is&even\end{array}\right.$$

(16)