2.2. Method
The method of assessing the risk of the occurrence of groundwater drought consists in calculating the probability based on statistically matched theoretical distributions of data.
Selected sets of values of the minimum levels of groundwaters were the basis for drawing the distribution functions of the empirical and theoretical distributions of the minima. For the purposes of research on estimating the minimum levels of groundwaters, the probability distributions that are traditionally applied in hydrogeology, as presented in Table 1 (Cammalleri et al. 2022; Rutkowska and Ptak 2012). The estimation was preceded by a stationarity test. The ADF (Augmented Dickey–Fuller) test was used for this purpose (Said and Dickey 1984). In this way, it was assessed whether the factors that determine the course of the described phenomenon remain constant and unchanged at the given point during the study.
Table 1 Distribution functions of selected probability distributions
As the number of factors that influence the form of distribution of minimum groundwater levels is high, when conducting tests for various observation points, several distributions should be applied each time, in order to find the one that will best reflect the empirical distribution of the analysed parameter.
The selected models with the estimated parameters were then used to assess the risk of drought in specific periods on the analysed test points. The best matched distribution was selected based on the tests of conformity of the empirical distribution of semi-annual minimum levels of groundwaters with theoretical distributions. The tests conducted were the Anderson-Darling test (Anderson and Darling 1952a, 1954b;ntschi and Bolboacă 2018), the Akaike information criterion (AIC) (Akaike 1973), and the Bayes test (BIC) (Schwarz 1978). The Anderson-Darling test is different from the others, as it is more sensitive to the conformity of distributions in their tails, which define the extreme phenomena to a great extent.
Probability, which performs the function of a measure of risk of the occurrence of groundwater drought, was calculated from the formula:
$$P\left(M\right)=1-{G}_{min}\left({x}_{{h}_{kr}}\right)$$
where:
\(M\) = min (x1, x2, …, xi), x1, x2, …, xi – water level values observed once a week during a half-year period,
\({x}_{{h}_{kr}}\) – value of the level that exceeds the critical threshold of groundwater level for the analysed test point,
\({G}_{min}\) – theoretical distribution function of the minimum values of semi-annual groundwaters level.
For the purposes of the present study, five risk levels were distinguished: very high, high, moderate, low, and negligible (Table 2).
Table 2
Scale of levels for probabilistic measures of the risk of groundwaters drought (\({\text{P}}_{{\text{h}}_{\text{k}\text{r}}}\)-probability)
Value of risk measure\({P}_{{h}_{kr}}\)
|
Risk level
|
p ≥ 0.15
|
Very high risk
|
p [0.05; 0.15]
|
High risk
|
p [0.005; 0.05]
|
Moderate risk
|
p [0.002; 0.005]
|
Low risk
|
p < 0.001
|
Negligible risk
|
As a result, a procedure was developed to estimate the probabilistic measure of the risk of occurrence of groundwater drought. It consists of the following steps:
Step 1. Collecting measurement data.
Step 2. Selecting the minimum values of groundwater levels in predefined time horizons.
Step 2. Data stationarity assessment.
Step 3. Drawing and estimating the parameters of the empirical and theoretical distributing functions of the distribution of minimum values of the analysed parameter.
Step 4. Assessment of the match between the estimated theoretical distributions and empirical ones.
Step 5. Selection of the best matched theoretical distribution of the minimum values of groundwater levels.
Step 6. Determination of the critical level hkr of the analysed parameter (groundwater level), that will be used to estimate the measure of the risk of the occurrence of drought.
Step 7. Calculating the probabilistic measure of risk of drought based on the distribution selected in Step 5, with the use of the formula
$$p=1-G\left({h}_{kr}\right)$$
Step 8. Assessment of the probabilistic level of risk of the occurrence of groundwater drought for the given test point based on the scale of probabilistic levels of the risk of drought.
If the scope of data was long enough, the observations were divided into sub-sets corresponding to measurement decades: 1980–1989, 1990–1999, 2000–2009, and 2010–2019. For shorter observation periods, the authors also attempted to maintain the division into individual decades.