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Detection of field significant long-term monotonic trends in spring yields

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Abstract

Trend analysis is a frequently used tool in hydrology and climatology for the identification of long-term changes. However, studies are usually only oriented on local trends. This paper rather focuses on the spatial application of trend analysis in groundwater data. For this purpose, a modification of the Mann-Kendall test was developed, based on the trend-free pre-whitening approach. This method was successfully tested on 157 series of yields from headwater springs collected in Czechia during the 1971–2007 period. The analysis was done separately for year, each season and each month. Field significant trends in spring yields were identified in hydrogeological regions. The results showed that the field significant trends are outnumbered when cross-correlation is not taken into account. In the case of annual series, 4 of 18 hydrogeological regions investigated showed a significant decreasing trend after corrections for cross-correlation, compared to 12 regions with field significant trend when not considering cross-correlation.

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Acknowledgments

This work was partially supported by the Ministry of the Environment of the Czech Republic (Grant number SP/2e1/153/07).

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Appendix

Appendix

1.1 Presence of stationarity/nonstationarity in Czech spring yields

In order to evaluate if the processes generating Czech series of spring yields are stationary or nonstationary, the approach advised in Barbosa et al. (2008) or Fatichi et al. (2009) was adopted. Before analysis, the series should be deseasonalized. Here, similarly to Fatichi et al. (2009), only the classical technique was used: i.e., for all 157 series the mean and the variance of each month were estimated. Subsequently, the differences between monthly observations and corresponding means were divided by the corresponding standard deviations (for details see also Salas 1993; Grimaldi 2004). Then, such chronologically ordered standardized series, say Q t , consisting of 444 monthly values could be subjected to so-called unit root tests.

One of these two parametric tests, the Phillips-Perron (PP) test (Phillips and Perron 1988), is based on the model:

$$ Q_{t} = \alpha + \beta \cdot t + \pi \cdot Q_{t - 1} + \varepsilon_{t} , $$
(17)

where α and β are the parameters of a linear regression. The stationary process ε t can be serially correlated and heteroscedastic, since this issue is treated directly in the test statistic. The rejection of the null hypothesis, \( H_{0} :\pi = 1 \), in favour of the alternative hypothesis, \( H_{1} :\pi < 1 \), indicates that the generating process is not a random walk but the AR(1) process combined with a deterministic linear trend. Note that the test is one-sided because the case with \( \pi \le - 1 \) is not so frequent (Cipra 2008).

The basis of the second test, the Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test (Kwiatkowski et al. 1992), is the model:

$$ Q_{t} = \beta \cdot t + r_{t} + \nu_{t} , $$
(18)

with the parameter of a linear trend β, a random walk \( r_{t} = r_{t - 1} + \varepsilon_{t} ,\;\varepsilon_{t} \sim N(0,\sigma_{\varepsilon }^{2} ) \), and a stationary process ν t . The null hypothesis of the KPSS test is either \( H_{0} :\sigma_{\varepsilon }^{2} = 0,\beta = 0 \) (for the case of trend stationarity in the form of a constant level) or \( H_{0} :\sigma_{\varepsilon }^{2} = 0,\beta \ne 0 \) (for the general case of trend stationarity). The alternative hypothesis is \( H_{1} :\sigma_{\varepsilon }^{2} > 0 \).

It is recommended to employ these two tests jointly because their null hypotheses are set up complementarily. Depending on the results, the investigator can distinguish up to three categories of stochastic processes (Barbosa et al. 2008; Fatichi et al. 2009):

  • stationary processes with a deterministic trend if the null hypothesis of the PP test is rejected and the null hypothesis of the KPSS test is accepted;

  • unit root processes (including random walks) if the null hypothesis of the PP test is accepted and the null hypothesis of the KPSS test is rejected;

  • other nonstationary processes (maybe long memory processes) if both null hypotheses are rejected.

It is thus clear that not only the existence of stationarity or nonstationarity but also the presence of deterministic and stochastic trends may be determined by these tests.

On the other hand, there are some downsides as well. If both null hypotheses are accepted, the tests are powerless to discriminate due to insufficient information content in the time series. Note also that missing values pose another problem. In fact, the tests are not designed for series containing such sequences. Therefore, in the current study, only 63 spring yield series could be investigated. More than half of them (42) showed nonstationarity (32 due to a stochastic trend and 10 due to a unit root). A similar proportion could be expected if full series were available. The occurrence of unit roots is strange, since they are atypical in geophysical series (see Barbosa et al. 2008). Details of the analysis can be found in Table 4.

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Ledvinka, O., Lamacova, A. Detection of field significant long-term monotonic trends in spring yields. Stoch Environ Res Risk Assess 29, 1463–1484 (2015). https://doi.org/10.1007/s00477-014-0969-1

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