1 Introduction

Sustainable development in any region depends on the availability and renewability of freshwater resources. Groundwater provides about one-third of the global freshwater. The availability and sustainability of groundwater resources in many basins or plains in the world are in danger because of climatic stresses (Brekke 2009; Wang and Ding 2003; Famiglietti 2014; Xu et al. 2016; Fu et al. 2020) and excessive human water withdrawals (Moeck et al. 2016; Zhang et al. 1997; Crosbie et al. 2013; Panwar and Chakrapani 2013; Charlier et al. 2015; Zamanirad et al. 2018; Cui et al. 2018; Liu et al. 2018; Nourani et al. 2018; El Asri et al. 2019; Fu et al. 2020).

Iran is a country with an arid and semi-arid climate. Population growth and the government’s aim of food self-sufficiency have led to increasing irrigated crop production and exploitation of surface and groundwater resources in the last four decades. The country ranks among the top groundwater miners in the world. Many plains across Iran are in a critical state (Madani and AghaKouchak, 2016), so 66.50% (405 of 609) of the aquifers are in a critical situation and characterized by reduced groundwater levels (Forootan et al. 2014; Samani 2022). Due to water resources mismanagement and climate change, Iran has confronted many environmental disasters, from shrinking many lakes and rivers to land subsidence, floods, droughts, desertification, deteriorating water quality, and soil erosion.

Sarbalesh karstic aquifer is a strategic groundwater resource providing a significant part of the municipal water needs of Fars and Bushehr provinces but has experienced intensive depletion in the last three decades. The authorities responsible for water management blame climate change and try to dismiss (cover up) their mismanagement role in this regard. This study investigates the relative effects of climate change and human activities (excessive water withdrawal) on the Sarbalesh aquifer storage depletion. For this, 32 years of monthly and annual time series of precipitation at four stations, temperature at three stations, and groundwater level at two piezometers were subjected to a comprehensive multi-statistical analysis.

Statistical methods from simple linear regression to more advanced approaches have been used for hydro-climatological time series analysis in the recognition of climate change and anthropologic effects on water resources. Mann–Kendall (MK) test (Mann 1945; Kendall 1975; Gilbert 1987) is perhaps the most broadly used nonparametric test for detecting trends in hydro-climatological data (Hamed and Rao 1998; Nalley et al. 2012 and 2013; Malakar et al. 2018; Tootoonchi et al. 2020; Chinchorkar et al. 2015). MK test statistically evaluates if there is a monotonic upward or downward trend in the time series of interest over time. If a significant trend is found, the rate of change can be calculated using the Şen’s slope estimator (Helsel and Hirsch 1992; Mondal et al. 2012; Şen 2018) to establish the magnitude of a trend. Pettitt-Mann–Whitney (PMW) and cumulative sum analyses have widely been applied to detect change points in time series (Pettitt 1979; Hsu and Wu 2012; Mondal et al. 2015; Samani and Jamshidi 2017; Yang and Xing 2022). The sequential MK test is a rank-based test for detecting the change of trend with time as well as the change points in a given time series (Sneyers 1990). Wavelet transform (WT) has been applied as a useful tool for trend analysis. Wavelet theory was initiated by Grossman and Morlet in signal processing literature. Then, Meyer introduced the first orthogonal wavelet (continuous) in 1985. Research in this field has advanced rapidly since the construction of compactly supported wavelets by Daubechies (1988) and the multi-resolution framework of Mallat (1999). The WT decomposes the time series into unlike subcomponents (Ravansalar and Rajaee 2015; Nourani et al. 2018; Yang et al. 2017) and displays various aspects of datasets such as trend, periodicity, change points, and discontinuities that might not be easily recognized by other methods (Nalley et al. 2012; Nourani et al. 2015). Due to the irregular and asymmetric shapes of the mother wavelets with different window sizes used in WT, the method is appropriate for analyzing the hydro-climatological time series with abrupt changes and local discontinuities.

In contrast to previous hydro-climatology trend analyses that use one or two statistical methods, mentioned above, in this research, we employed a combination of several statistical methods. By employing the Mann–Kendall (MK) and modified MK tests we delineated the presence of trends in the time series. The rate of change was calculated using Şen’s slope estimator (Helsel and Hirsch 1992). By the combination of discrete wavelet transform (DWT), MK, and sequential MK analyses, we detected the long-term trends and the dominant time scales (periodicities) affecting the time series trends. The presence or absence of seasonal patterns and the state of autocorrelation in the time series that affect the result of trend detection were first examined to apply the most suitable MK test (i.e., MMK test) to the original time series and the series resulted from the DWT decomposition. Additionally, the time series were corrected for the border effect (often overlooked or missing in published trend detection) by either zero-padding (ZP), periodic (PD), or symmetrization (SYM) extension methods examining three criteria. The first criterion is the mean relative error (MRE), the second is the relative error (RE), and the third one proposed in this study is the normalized root mean square error (NRMSE), which proved to be a better indicator for the determination of the decomposition level and border extension to be used in the analyzed data. Using the Pettitt-Mann–Whitney (PMW) and cumulative sum methods, the change points in the time series were calculated. We also explored the coincidence of the change year in the Southern Oscillation Index (SOI) and rainfall series to justify the causes of climate change by comparing the average value of SOI and rainfall before and after the change point. Significant increasing trends in the temperature series indicated that the study region experiences climate change. A highly significant declining trend in the groundwater level series compared to that of the precipitation series showed that human intervention is the cause of aquifer depletion, and reduced precipitation as a result of climate change is intensifier. The wavelet transform coherence (WTC) justified these results. All statistical analyses were conducted at 95% confidence limits (p-value \(\le 0.05\)). We believe that our multi-statistical approach can serve as a baseline reference for the management of the aquifers that experience the same fate as the Sarbalesh aquifer because of over-exploitation or/and climate change. The paper is organized as follows: “Sect. 2” introduces the study area and the data used, the theoretical background and research methodology are presented in “Sect. 3,” “Sect. 4” describes and discusses the results obtained by each method, and “Sect. 5” presents the summary and conclusions of the research.

2 Materials

2.1 Study area

The study area in this research is the Sarbalesh karstic aquifer (Sarbalesh anticline) in the southwest of Fars province located in the southwestern part of Iran within the latitude of 29°–30° N and longitude of 51°–52° E (Fig. 1). The aquifer is a strategic source for supplying a part of the drinking water of Fars and Bushehr provinces. Sarbalesh anticline is located in the structural zone of the Zagros folded belt and with a length of 86 km extends from the northwest to the southeast of the Kazerun plain. In addition to 5 observation wells, some 30 pumping wells were drilled in the northeast of the anticline limb of which 15 are productive as shown in Table 1 and Fig. 1. The climate regime in the study area is dominantly arid and semi-arid. The average annual precipitation for a period of 32 years (1988–2019) is 530.2 mm. About 97% of the annual precipitation occurs from November to April. January and July are the wettest and driest months in the region, respectively. The average monthly temperature is 19.7°C, the maximum temperature is 35.9°C in July, and its minimum is − 1.7°C in February. The Mediterranean air mass (MedT) from the west and North Atlantic and the Black Sea cyclones (mP) from the northwest are the source of precipitation in the area. The topography varies from 400 to 1966 m above mean sea level (a.m.s.l). Parishan lake, a freshwater lake in the south of the Kazeun plain, which was mainly recharged by groundwater, dried up in 2004 due to groundwater over-abstraction around the lake (Samani and Jamshidi 2017).

Fig. 1
figure 1

Geological map of the study area. Location of pumping and observation well and rainfall stations

Full size image
Table 1 Location and some specifications of pumping (W) and observation (P) wells 
Full size table

2.2 Data

To achieve the objective of the study, following data were collected from the archive of Fars-Regional Water Authority and Fars Province Meteorological Organization and analyzed:

  • Thirty-two years (1988–2019) of daily precipitation in 4 measuring stations: Boushigan, Dasht-Arjan, Kazerun, and Parishan stations (Fig. 1). By aggregating these data, the monthly and annual time series at each station were extracted.

  • Thirty-two years (1988–2019) of monthly mean temperature in Dasht-Arjan, Kazerun, and Parishan stations which were aggregated to extract the mean annual times series in each station.

  • Thirty-two years (1988–2019) of monthly water level in two observation wells (P1 and P2, Fig. 1) which were aggregated to generate the mean annual time series in each observation well. Other observation wells lake continuous water level recording.

3 Methodology

3.1 Mann–Kendall test

The purpose of the MK test (Mann 1945; Kendall 1975; Gilbert 1987) is to statistically evaluate if there is a monotonic upward or downward trend of the time series of interest over time. A monotonic upward (downward) trend means that the time series continuously increases (decreases) through time, but the trend may or may not be linear. The MK test is perhaps the most broadly used nonparametric test for detecting trends in hydro-climatological data. It does not assume the data to be normally distributed. The test presumes a null hypothesis, H0, of no trend and alternate hypothesis, Ha, of increasing or decreasing monotonic trend. In this method, first the difference between each observation and all subsequent observations is calculated and the parameter \(S\) is obtained according to the following equation: 

$$S=\sum_{k=1}^{n-1}\sum_{j=k+1}^{n}\mathrm{sgn}\left({x}_{j}-{x}_{k}\right)$$
(1)

where \(n\) is the number of series observations, and \({x}_{j}\) and \({x}_{k}\) are the data of \(j\) and \(k\) series, respectively and

$$\mathrm{sgn}\left({x}_{j}-{x}_{k}\right)=\left\{\begin{array}{c}1 if {x}_{j}-{x}_{k}>0\\ 0 if {x}_{j}-{x}_{k}=0\\ -1 if {x}_{j}-{x}_{k}<0\end{array}\right.$$
(2)

    

In the next step, the variance of \(S\) is calculated by one of the following equations:

$$\mathrm{Var}(S)= \frac{n\left(n-1\right)\left(2n+5\right)-\sum_{t=1}^{m}t\left(t-1\right)\left(2t+5\right)}{18} \mathrm{for} n>10$$
(3)
$$\mathrm{Var}\left(S\right)= \frac{n\left(n-1\right)\left(2n+5\right)}{18} \mathrm{for} n\le 10$$
(4)

where \(m\) represents the number of series in which there is at least one duplicate data. Finally, the presence of a statistically significant trend is evaluated by calculating the \(Z\) value of MK test (Adamowski and Bougadis 2003). A positive (negative) value of \(Z\) illustrate an upward (downward) trend:

$$Z=\left\{\begin{array}{c}\frac{S-1}{\sqrt{\mathrm{Var}(S)}} if S>0\\ 0 if S=0\\ \frac{S+1}{\sqrt{\mathrm{Var}(S)}} if S<0\end{array}\right.$$
(5)

To analyze for either an upward or downward monotone at α level of significance (α = 0.05), H0 is rejected if the value of \(-{Z}_{1-\frac{\alpha }{2}}\le Z\le {Z}_{1-\frac{\alpha }{2}}\), where ± Z1-α/2 is taken from the standard normal cumulative distribution tables. The magnitude and rate of the trend is calculated by Kendall’s tau statistic and Şen’s slope estimator (Helsel and Hirsch 1992) described in the Supplementary Materials.

In using the MK test, the existence or absence of a seasonal pattern and the state of autocorrelation in the time series which affect the result of trend detection should be examined (Hamed and Rao 1998; Basistha et al. 2009). The probability of rejecting null hypothesis increases, if there is a positive serial correlation in the time series and vice versa. The MMK test by Hirsch and Slack (1984) is used for time series showing a seasonality effect with a significant lag-1 autocorrelation coefficient; if the time series exhibits only the lag-1 autocorrelation coefficient, the MMK test by Hamed and Rao (1998) should be used, and if neither of seasonality pattern and lag-1 autocorrelation observed, the original MK test should be employed (Tabari et al. 2011a, b; Nalley et al. 2012). See Supplementary Materials for the details of MMK tests.

3.2 Sequential modified Mann–Kendall test

Sequential MMK test is a rank-based test for detecting the change of trend with time as well as the change points in a given time series (Sneyers1990). The series is arranged as ranked order and test statistic \({t}_{i}\) is calculated using:

$${t}_{i}=\sum_{1}^{j}{n}_{i}$$
(6)

The value of \({x}_{j}(j=1.\dots .n)\) are compared with \({x}_{k}\left(k=1.\dots .j-1\right)\) and the number of the cases with \({x}_{j}>{x}_{k}\) is counted as \({n}_{i}\).

The distribution of the test statistic is asymptotically normal with mean \(E({t}_{j})\) and variance \(\mathrm{Var}({t}_{j})\):

$$E\left({t}_{j}\right)=\frac{j(j-1)}{4}$$
(7)
$$\mathrm{Var}\left({t}_{j}\right)=\frac{j(j-1)(2j+5)}{72}$$
(8)

Consequently, the standardized time series with zero mean and unit standard deviation is named \(U\left({t}_{i}\right)\) and calculated as follows:

$$U\left({t}_{i}\right)=\frac{{t}_{j}-E({t}_{j})}{\sqrt{\mathrm{Var}({t}_{j})}}$$
(9)

This is the prograde sequential statistic \(U\left({t}_{i}\right)\). If the statistic is calculated in the same way but starting from the end of the series, the statistic is called retrograde sequential statistic and represented by \(\overset'U(t_i)\).

The intersection of the curves \(U\left({t}_{i}\right)\) and \(\overset'U(t_i)\) locates the approximate potential trend change point. If the intersection of curves occurs beyond the confidence level of the standardized statistic \({Z}_{\mathrm{sg}}\) (i.e., \(\pm\) 1.96), a detectable change at that point in the time series can be deduced (Lu et al. 2004; Partal and Kahya 2006; Tabari et al. 2011a, b).

3.3 Wavelet transform methods

The general theory behind WT is similar to that of the short time Fourier transform (STFT). WT permits for a completely flexible window function (named the mother wavelet), which can be changed over time based on the shape and compactness of the signal (Hernández and Weiss 1996). The key advantage of the WT compared to the Fourier transform is the ability to extract both local spectral and temporal information. The fundamental purpose of the mother wavelet is to provide a source function to generate other wavelets which are the shifted and scaled versions of the mother wavelet. The lower scales represent the compressed wavelet and correspond to the rapidly changing details or high-frequency components of the signal. The higher scales are identified from the stretched version of a wavelet and the corresponding components represent the slowly changing or low-frequency components of the signal. As the mother wavelet moves through the signal the original time series is decomposed to two subsidiary signal shape components (coefficients), called the approximation (A) and details (D) components, that illustrate the similarity between the signal and the mother wavelet (at any specific scale). In general, a mother wavelet function (ψ) can be expressed as follows:

$$\psi \left(\tau ,s\right)=\frac{1}{\sqrt{\left|s\right|}}\psi (\frac{t-\tau }{s})$$
(10)

where τ is the time step that the wavelet is translated along the series that often called the translation parameter and s (s > 0) is the scale that often called the dilation factor (Kunagu et al. 2013). Therefore, wavelet transforms investigate trends in time series by separating its short, medium, and long-period components (Drago and Boxall 2002). For hydro-climatological studies, Daubechies (db) and Morlet wavelets are most commonly applied (Nalley et al. 2013). The (db) wavelet family provides important support, particularly for localizing events in the signal analysis.

The WT analysis may be performed in two schemes, the continuous wavelet transform (CWT) and discrete wavelet transform (DWT). The CWT is an integral transform and decomposes signals at all scales (Mallat 1989). The CWT of a signal \(x(t)\) is determined by the following equation:

$$\mathrm{CWT}\left(\tau ,s\right)=\frac{1}{\sqrt{\left|s\right|}}{\int }_{-\infty }^{+\infty }x(t){\psi }^{*}(\frac{t-\tau }{s})\mathrm{dt}$$
(11)

where, \(\psi\) denotes the mother wavelet and (*) is the complex conjugate of the mother wavelet. Therefore, the concept of frequency is exchanged by scale, which can characterize the variation in the signal \(x(t)\), at a given time scale. The CWT produces a high amount of redundant information, and the coefficients are correlated spatially and temporally (Percival 2008; Nalley et al. 2012). Rashid et al. (2015) utilized the CWT to identify trends in annual, seasonal, and monthly rainfall at 13 stations in the Onkaparinga catchment in South Australia during 1950–2010. They found it intricate to interpret the CWT results for the monthly data (due to a large amount of data). Therefore, they used four specific levels for the decomposition of precipitation data. For the same reason, the result of Malaekeh et al. (2022) CWT analysis did not show a well-defined periodicity but a range of periodicity for annual data only after plotting the wavelet power spectrum (Malaekeh et al. 2022, Figs. 13–14). It is also preferred to choose the DWT over CWT because CWT does not produce information in the form of a time series, but rather in a two-dimensional format (Percival 2008; Nally et al. 2012). The DWT simplifies the process of transformation and reduces the amount of work; yet, it still produces a very efficient and accurate analysis (Partal and Küçük, 2006). This is because the DWT is normally based on the dyadic calculation of the position and scale of a signal (Chou, 2007; Nalley et al. 2012; Nourani et al. 2018), so that the wavelet coefficients \(\mathrm{DWT}\left(\tau ,s\right)\) of the DWT become (Partal and Küçük 2006):

$$\mathrm{DWT}\left(\tau ,s\right)=\frac{1}{({2)}^\frac{s}{2}} \sum_{t=0}^{n-1}x\left(t\right).\psi (\frac{t}{{2}^{s}}-\tau )\mathrm{dt}$$
(12)

where \({2}^{s}\) represents the dyadic scale of the DWT. The selection of mother wavelet and decomposition level (DL) are two essential issues in dyadic DWT. According to the theory of dyadic DWT (Chui and Heil 1992), there are several recommended methods to determine the most appropriate number of decomposition levels, of which one of the most commonly used is given by (de Artigas et al. 2006):

$$\mathrm{DL}=\frac{\mathrm{Log}(\frac{n}{2\nu -1})}{\mathrm{Log}(2)}$$
(13)

where \(\mathrm{DL}\) is the maximum number of decomposition levels, n is the number of data points in the time series, and v is the number of vanishing moments of the db mother wavelet. In MATLAB, v is equal to the type number of the db (de Artigas et al. 2006). The theoretical minimum decomposition level can be computed by (Aussem 1998):

$$M=\left[\mathrm{Log}({L}_{x\left(t\right)})\right]$$
(14)

where [] means taking the integer part of real variable in square bracket. \({L}_{x\left(t\right)}\) is the length L of series \(x\left(t\right)\). To determine the maximum and minimum decomposition levels, the numbers obtained from Eqs. (13–14) are rounded up. WT has been extensively used for the analysis of hydro-climatological time series (Adamowski et al. 2013; Shih et al. 2010; Nalley et al. 2012, 2013; Nicolay et al. 2010; Nourani et al. 2014; Prokoph and Patterson 2004; Sang et al. 2018; Sang et al. 2018; Wang et al. 2011; Araghi et al. 2015). One of the main reasons to apply wavelet-based methods in hydrological studies is because of its robust property—it does not involve any possibly incorrect assumptions of distribution and parametric testing protocols (Kisi and Cimen 2011).

Besides the determination of the DLs, the time series should be corrected for the border effect (Nalley et al. 2012). At the borders of a finite dataset, the wavelet analysis extends into a region with no data, and the convolution processes cannot be done, or transform values close to the border of the signal get corrupted by the unavailable data of the signal edge. Therefore, the WT will require the computation of non-existent values outside these boundaries (Su et al. 2014), and we need to deal with the boundary problems (border effect). To reduce border distortion in WT, thereby allowing for the computation of the wavelet coefficients near the edges (start and end) of the signal, zero-padding (ZP), periodic extension (PD), and symmetrization (SYM) are the most commonly used extension methods (Nalley et al. 2013) that we employ here.

In order to determine the extension mode to be used in the data analysis for each data type and dataset, three criteria are examined:

The first criterion used was proposed by de Artigas et al. (2006) and is based on the mean relative error (MRE):

$$\mathrm{MRE}=\frac{1}{n}\sum_{j=1}^{n}\frac{\left|{a}_{j}-{x}_{j}\right|}{\left|{x}_{j}\right|}$$
(15)

where \({x}_{j}\) is the original data value of a signal whose number of records is \(n\), and \({a}_{j}\) is the approximation value of \({x}_{j}\).

The second criterion is based on the relative error (RE), (Nalley et al. 2012; Araghi et al. 2014):

$${R}_{E}=\frac{\left|{Z}_{a}-{Z}_{o}\right|}{{Z}_{o}}$$
(16)

where \({Z}_{a}\) and \({Z}_{o}\) are the MK Z values of the last approximation for the decomposition level used, and the MK Z-value of the original dataset, respectively.

The third criterion is the one proposed in this study and is based on the normalize root mean square error (NRMSE):

$$\mathrm{NRMSE}=\sqrt{\frac{\sum_{t=1}^{n}{\left[x\left(t\right)-\widehat{x}(t)\right]}^{2}}{\sum_{t=1}^{n}{x}^{2}(t)}}$$
(17)

where \(x(t)\) is the original data value of a signal and \(\widehat{x}(t)\) is the approximation value of a signal.

We examined each extension modes for the smooth db4 wavelets and the different decomposition levels and calculated the value of the above three error criteria. The extension mode with the lowest error value was selected to extend the time series for border conditions before subjecting them to further DWT analysis.

3.3.1 Procedures of data analysis by DWT

DWT was conducted on the time series by taking the following steps:

  1. (a)

    The presence (or lack) of seasonality effect and lag-1 autocorrelation in each time series are assessed by plotting its correlogram, as described in Supplementary Materials.

  2. (b)

    Based on step (a) the appropriate MK-test is selected.

  3. (c)

    Decomposition level is calculated for each time series by Eqs. (13) and (14).

  4. (d)

    Each time series is decomposed into sub-signals using the multilevel 1-D wavelet decomposition function in the MATLAB software toolbox and the Daubechies mother wavelet db4 (Daubechies N = 4).

  5. (e)

    The border extension mode is adapted using the relative error criterion between the original data and the approximation component at the last decomposition level as described in “Sect. 3.3” and the most appropriate border extension is selected.

  6. (f)

    Having established the DLs and the most appropriate border extension mode, each time series is decomposed to A and Ds components by DWT, and step (a) above was carried out on all the components to select the appropriate MK test.

  7. (g)

    The MK Z value is calculated for the original, Ds, A, and A + Ds series produced by the wavelet decomposition, and the periodicities are determined.

  8. (h)

    The most important (prominent) periodicities affecting the trends are determined by considering the following criteria:

    • The approximate equality of the MK Z values of the original data and their respective A + Ds components.

    • The correlation coefficient between the original and A + Ds series

    • The similarity of sequential MK graph of the original time series and A + Ds components.

One of the paired series with the highest correlation coefficient, the highest similarity of sequential MK graph, and the closet Z-value is selected as the prominent periodicity.

3.4 Pettitt-Mann–Whitney (PMW) test for the detection of change point

The change point detection is an important aspect to assess the period from where significant change has occurred in a time series. PMW and cumulative sum analysis widely have been applied to detect change points. These methods assume that the datasets are stationary time series (Coulibaly 2006).

In this research to detect change point in time series, PMW and cumulative sum tests were applied. Mann–Whitney-Wilcoxon test was also used for identifying the statistical significance of change points. Thus, change points in each time series was identified with the highest probability.

In PMW test, every time series \(\left\{{X}_{1},\dots ,{X}_{T}\right\}\) with length \(T\) is considered as two parts: \(\left({X}_{1}, {X}_{2}, \dots , {X}_{T}\right)\) and \(\left({X}_{t+1}, {X}_{t+2}, \dots , {X}_{T}\right)\). The indices \(U(t)\) and \(V(t)\) is then calculated from:

$${V}_{t,T}=\sum_{j=1}^{T}\mathrm{sgn }({X}_{t}-{X}_{j})$$
(18)
$${U}_{t,T}={U}_{t-1,T}+{V}_{t,T} \mathrm{for} t=2,T$$
(19)
$${U}_{1,T}={V}_{1,T}$$
(20)
$$\mathrm{sgn}\left(x\right)=\left\{\begin{array}{c}+1 if ({X}_{t}-{X}_{j})>0\\ 0 if ({X}_{t}-{X}_{j})=0\\ -1 if ({X}_{t}-{X}_{j})<0\end{array}\right.$$
(21)

The main significant change point is considered as the point where the value of \(\left|{U}_{t,T}\right|\) is maximum.

The estimated significant probability \(p(t)\) for a change point is given as (Pettitt 1979):

$$p\left(t\right)=1-\mathrm{exp}\lceil\frac{-6{U}_{t,T}^{2}}{{{T}^{3}+ T}^{2}}\rceil$$
(22)

The significant change point takes place at maximum value of \(p\left(t\right)\) or absolute value of \({U}_{t,T}\). In addition to \(p (t)\), the cumulative sum method was used to determine changes in the time series and to determine the change point, which is calculated as follows.

$${S}_{k}={\sum }_{j=1}^{k}({X}_{j}-\overline{X })$$
(23)

In which, k is the kth record of time series and \(\overline{X }\) is the average value of time series. The possible change point occurs at maximum \({S}_{k}\) value. The Mann–Whitney-Wilcoxon test used to explore if the identified \(p\left(t\right)\) and \({S}_{k}\) values are significant (α = 5%). and is briefly described in the Supplementary Materials.

4 Results and discussion

4.1 Results of modified Mann–Kendall test analysis of the monthly time series

The correlograms of the time series are presented in Fig. S1 (Supplementary Materials). The correlograms with sinusoidal variations illustrate the presence of periodicities (seasonal effects). Notably, the correlogram of the water level series at P1 and P2 show sinusoidal patterns after they were detrended. The positive value of lag-1 autocorrelation coefficient (ACF) given in these figures indicates the serially correlated data in all the time series. Due to the presence of the serial correlation and the seasonal variation in the time series, as argued above, the modified Mann–Kendall trend test due to Hirsch and Slack (1984) was used to evaluate the trend of the monthly time series. On running the MMK trend test, the results given in Table 2 were obtained for the nine time series. As presented in Table 2, the values of \({Z}_{\mathrm{sg}}\) of groundwater time series \((\mathrm{i}.\mathrm{e}., -20.655\mathrm{ and }-22.131)\) fall beyond the confidence level of MK Z value (i.e., ± 1.96) at \(p\le 0.05\), and hence the null hypothesis is rejected implying that the groundwater level series show a significant decreasing trend. The negative trend is backed with the values of Şen’s slope β and Kendall’s tau (τ) coefficient which are (− 0.111, − 0.753) and (− 0.123, − 0.807) for the piezometer P1 and P2, respectively.

Table 2 Modified MK trend test results of the monthly time series. \(\left|{Z}_{\mathrm{sg}}\right|\ge 1.96\) indicates significant trend at 95% confidence limits (p-value \(\le 0.05\))
Full size table

The values of \({Z}_{\mathrm{sg}}\) are negative but statistically insignificant (i.e., they fall within ± 1.96) and \(\beta =0.0\) for all precipitation time series and null hypothesis H0 is accepted and thereby indicating that no trend can be seen in the monthly precipitation data (Table 2).

For the temperature data, the (\({Z}_{\mathrm{sg}}\)) values show that there is an increasing trend for Kazerun (\({Z}_{\mathrm{sg}}=8.982\)) and Dasht-Arjan station (\({Z}_{\mathrm{sg}}=1.46\)), but a decreasing trend for Parishan station (\({Z}_{\mathrm{sg}}=-1.936\)). The nonsignificant trend of the temperature time series is backed with their respective very low value of \(\tau\) and \(\beta\) compared to those of groundwater time series (Table 2).

4.2 Results of modified Mann–Kendall test analysis of the annual time series

For the nine annual time series, the MMK test by Hamed and Rao (1998) was applied because they possessed significant lag-1 autocorrelation coefficients and lacked seasonality pattern (Fig. S2, Supplementary Materials). The values of the MMK Z coefficient, Şen’s slope, Kendall’s tau (τ), and the significance level p(t) are summarized in Table 3. Similar to the monthly times series, the values of \({Z}_{\mathrm{sg}}\) of the annual groundwater time series (− 6.288 and − 6.417) fall behind the confidence level of MK Z value (i.e., ± 1.96) at \(p\le 0.05\), and hence the annual groundwater level series show a significant decreasing trend. The negative trend is supported by the values of Şen’s slope β and Kendall’s tau (τ) coefficient which are (− 1.280, − 0.778) and (− 1.463, − 0.827) for the piezometer P1 and P2, respectively.

Table 3 Modified MK trend test results of the annual time series. \(\left|{Z}_{\mathrm{sg}}\right|\ge 1.96\) indicates significant trend at 95% confidence limits (p-value \(\le 0.05\))
Full size table

In contrast to the monthly time series, the values of the statistic (\({Z}_{\mathrm{sg}}, \beta , \mathrm{and} \tau\)) indicate significant decreasing trends for annual rainfall time series at Boushigan \((-2.660, -4.923,\mathrm{ and}-0.333\)), Kazerun \((-3.616, -10.851,\mathrm{ and}-0.45)\), Parishan (\(-3.778, -9.831,\mathrm{ and}-0.47)\), and Dasht-Arjan (\(-4.450, -14.874, -0.387)\) stations at \(p\le 0.05\) (Table 3).

The annual temperature time series at Kazerun station shows a significant (\(p\le 0.05)\) positive trend with \({Z}_{\mathrm{sg}}, \beta , \mathrm{and} \tau\) values of 6.0, 0.054, and 0.512, respectively and the null hypothesis is rejected. The annual temperature series at Parishan and Dash-Arjan stations show a non-significant negative and positive trend, respectively (Table 3).

Based on the above findings, It may be concluded that the declining Sarbalesh aquifer water level is mainly due to over-extraction and the reduced precipitation as a result of climate change has intensified it.

4.3 Result of DWT analysis of the monthly time series

As shown in “Sect. 4.1,” the monthly time series possess seasonality and showed a significant lag-1 autocorrelation coefficient (step a, “Sect. 3.3.1”). Therefore, in the WT analysis of data MMK test of Hirsch and Slack (1984) was employed (step b, “Sect. 3.3.1”). The minimum and maximum composition levels were calculated for all the time series by Eqs. 13–14 and found equal to 6 and 7, respectively (step c, “Sect. 3.3.1”). Using the decomposition levels 6 and 7, each time series was decomposed into sub-signals using the multilevel 1-D wavelet decomposition function and the Daubechies mother wavelet db4 (Daubechies N = 4) in the MATLAB software toolbox (step d, (Sect. 3.3.1. The border extension modes (i.e., ZP, SYM, and PD) were examined using the error criteria (i.e., MRE, RE, and NRMSE, “Sect. 3.3”) between the original data and the approximation component at the last decomposition level (step e, “Sect. 3.3.1”). The values of error criteria are presented in Tables 4, 5, and 6. Table 4 demonstrates that (MRE) values do not differ significantly among two border extension modes (ZP and SYM) for the six and seven level of decomposition. In contrast, except for the groundwater time series, the value of MRE is much higher when the periodic extension mode was used. As a result, the periodic extension mode is not a good choice for border condition correction.

Table 4 The mean relative error (MRE) resulting from the application of the three border extension modes at two decomposition levels on the monthly time series
Full size table
Table 5 The relative error (RE) resulting from the application of the three border extension modes at two decomposition levels on the monthly time series
Full size table
Table 6 The normalize root mean square error (NRMSE) resulting from the application of the three border extension modes at two decomposition levels on the monthly time series
Full size table

Table 5 tabulates the values of RE for the DLs of 6 and 7 by the three boundary extension modes. The bolded values show the lowest value of RE, suggesting different border extension methods and decomposition levels for various time series. In other words, no unique mode is distinguished appropriate for time series border extension.

Table 6 presents the values of NRMSE resulting from three border extension modes and the two decomposition levels. Interestingly, the values computed for the symmetric extension mode at decomposition level 6 are the lowest. Comparing the values of error criteria in Table 6 with those of Tables 4–5, we conclude that the DWT analysis of all the time series may be best performed by using the symmetric extension mode for border corrections and choosing the decomposition level 6 to produce the approximate and detailed components.

Figures 2 depicts the original, approximation (A) and detailed (D1–D6) components of the time series P1, rainfall series at Boushigan, and temperature series at Kazerun (step f, “Sect. 3.3.1”). For the sake of brevity, those of other time series are presented in the Supplementary Materials, Figs. S3-S4.

Fig. 2
figure 2

The original, approximate (A), and detailed components (D1–6) of the monthly time series resulted from DWT decomposition: rainfall at Boushigan station (left), temperature at Kazerun Station (center), and groundwater level at P1 (right)

Full size image

Table 7 tabulates lag-1 autocorrelations and seasonality of the original monthly time series, details (D1–D6), approximations (A), and a set of combination of the details and their respective approximation (A + Ds). Star sign indicates the presence of seasonality cycle and boldface number indicates significant lag-1 serial correlations at a 5% significance level.

Table 7 Lag-one autocorrelations and seasonality of the original monthly time series, details (D1–D6), approximations (A), and a set of combination of the details and their respective approximation (A–Ds) (*Indicates the presence of seasonality cycle, bold number indicates significant lag-1 serial correlations at a 5% significance level)
Full size table

Based on the result of Table 7, the MMK test by Hirsch and Slack (1984) was used for series that exhibited seasonality patterns and had significant lag-1 autocorrelations. The MMK test by Hamed and Rao (1998) was applied to series which showed lag-1 autocorrelations only (i.e., the A and the (D1) of all the decomposition series), and the original MK test was used to datasets that did not show significant autocorrelation and seasonality (i.e., A + D1 of the temperature series at Kazerun station).

Table 8 shows the MK Z values for the original series, their detail components (Ds), approximations (A), and the combination of the Ds with the approximation added to them (Ds + A). The MK Z values closer to the \(Z\) values of the original time series are highlighted in boldface from which the effective periodicities on the trend are inferred. According to the \({Z}_{\mathrm{sg}}\) values of the original rainfall series, it can be concluded that all the MK Z values (− 0.52, − 1.18, − 0.99, and − 0.42 for Boushigan, Kazerun, Parishan, and Dasht-Arjan, respectively) show negative trend, but none significant. Also, it can be seen in Table 8 that none of the MK Z values of the different individual details (D1–D6), except for the D1 components of Boushigan (Z =  − 5.32) and the D5 component of Dasht-Arjan (Z = 2.37), is statistically significant. However, for the approximation series and combinations of the details plus their corresponding approximations, the trends are negatively significant. A very interesting finding is that after the approximation components (A) were added to the different details, the trend values became statistically significant. It is clear that the approximation components have a profound effect on the trend of original data and these approximations carry the most portion of potential trend.

Table 8 MK Z values of the original data, details components (D1–D6), approximations (A), and a set of combination of the details and their respective approximation (A + Ds). The most effective periodic components affecting the time series trend are indicated in boldface
Full size table

With respect to the results of WTMK test on temperature time series (Table 8), except for Parishan station with Z =  − 1.94, positive trend observed in the other two stations. Only Kazerun with \({Z}_{\mathrm{sg}}\) =  + 8.98 shows statistically significant positive trend. It can also be seen in Table 8 that except for the D6 component of Parishan (\({Z}_{\mathrm{sg}}\) = 2.84) and Dasht-Arjan (\({Z}_{\mathrm{sg}}\) = 2.48) stations, none of the \({Z}_{\mathrm{sg}}\) values of the different individual details (D1–D6) is statistically significant, even for the station whose original series showed significant MK values (i.e., Kazerun station). For the Kazerun station—whose original MK \({Z}_{\mathrm{sg}}\) values are significant, the trend of its approximation is also positively significant.

The values of the \({Z}_{\mathrm{sg}}\) statistics obtained through the MMK test for the groundwater levels time series are also shown in Table 8. Significantly drastic declining trends, at 5% significance level, in the groundwater levels are witnessed in both piezometers with \({Z}_{\mathrm{sg}}\) values of − 20.66 and − 22.13 for P1 and P2, respectively. Table 8 shows that none of the individual detail components alone showed statistically significant MK \({Z}_{\mathrm{sg}}\) values for groundwater series. Again, after the addition of the approximation to their respective details, the MK \({Z}_{\mathrm{sg}}\) values became significant, emphasizing that the approximation signal carries the most portion of potential trend in the both time series.

The prominent periodicities: To define the most prominent periodic variations (periodicities) affecting the trend of the studied time series, the result of WTMK analysis (Table 8) were examined with the three criteria given in step h, “Sect. 3.3.1.” So that, based on these criteria the combination (A + Ds) that provide the highest correlation, the closest \({Z}_{\mathrm{sg}}\) value, and the most similar sequential MK curve with their original time series is selected as the most effective periodicity causing the trend.

Base on the first criterion, the rainfall data at Kazerun and Parishan stations show 32 months, and at Dasht-Arjan and Boushigan stations 64-month periodicities, respectively (the A + Ds series showing the closest Z-value to the Z-value of their original series, boldface number in Table 8). Temperature at Kazerun station with significant rising trend shows 8-month periodicity (A + D3 temperature series shows the closest Z-value to the Z-value of its original series, boldface number in Table 8). Groundwater level series at the both piezometers show 64-month periodicity (i.e., A + D6 series shows the closest Z-value to the Z-value of its original series, boldface number in Table 8).

To test the second criterion, the correlation coefficient values between the original monthly time series and their (A + Ds) components were calculated and tabulated in Table 9. The numbers in boldface indicate the highest correlation between the two respective time series and show the most prominent periodicity in the original time series. Therefore, D6 (64 months), D3 (8 months), and D3 (8 months) are the main deriving components causing the trend in the groundwater, rainfall, and temperature time series, respectively.

Table 9 The correlation coefficient values between the original monthly time series and their (A + Ds) components
Full size table

To check the third criterion, the sequential MK curves of the original time series versus their A + Ds components are plotted in Fig. 3 for the rainfall series at Boushigan, the temperature series at Kazerun, and the P1 series. Here again for the sake of brevity those of the other series are presented in the Supplementary Materials, Figs. S5-S6. As it is observed in all the figures, the sequential MK curve of A + D6 of groundwater series and A + D3 of rainfall and temperature series show the highest similarity with their original series. These observations indicate that, like the results of Table 8, the most prominent periodicity causing the trend of the groundwater level, rainfall, and temperature time series are 64-month, 8-month, and 8-month periodicities, respectively.

Fig. 3
figure 3

Sequential Mann–Kendall graphs for the original monthly time series versus their (A + Ds) components: rainfall at Boushigan station (left), temperature at Kazerun station, and groundwater level at P1 (blue solid line: progressive trend for the original time series; gray solid line: progressive trend for the A + Ds; black solid line: 95% confidence interval limits)

Full size image

The combined MK and DWT analyses conducted on the study data in this section demonstrated that the rainfall data show an insignificant decreasing trend, and the 8-month periodicity is the cause. The temperature data show a significant increasing trend only at Kazerun station with an 8-month periodicity. In contrast to climate variables (rainfall and temperature), the groundwater level data exhibit a drastic and highly significant decreasing trend dominated by the D6 component (64-month periodicities). Low frequencies (long-term variation) indicate the occurrence of relatively stable and persistent phenomena (continuous longe-term pumping). Therefore, we conclude that the cause of the declining groundwater level in the Sarbalesh karstic aquifer is the continuous over-exploitation of water storage in this aquifer in the last 32 years. Like in the rainfall time series, we observed seasonality patterns in the groundwater time series (Fig. S1). Since the seasonality pattern of rainfall (as the only recharge source of the aquifer) transfers to groundwater level, we may conclude that the decreasing trend of rainfall and increasing trend of temperature (both as indicators of climate change) enhanced the declining trends of the groundwater.

4.4 Results of DWT analysis of the annual time series

As shown in “Sect. 4.2” the annual time series possessed significant lag-1 autocorrelation coefficients and lacked seasonality pattern (step a, “Sect. 3.3.1”). Therefore, in the WT analysis of data, MMK test of Hamed and Rao (1998) was applied (step b, “Sect. 3.3.1”). The minimum and maximum composition levels were calculated for all the time series by Eqs. (13–14) and found equal to 3 and 4, respectively (step c, “Sect. 3.3.1”). Using the decomposition levels 3 and 4, each time series was decomposed into sub-signals using the multilevel 1-D wavelet decomposition function and the Daubechies mother wavelet db4 (Daubechies N = 4) in the MATLAB software toolbox (step d above). The border extension modes (i.e., ZP, SYM, and PD) were examined using the error criteria (i.e., MRE, RE, and NRMSE, “Sect. 3.3”) between the original data and the approximation component (step e, “Sect. 3.3.1”).

Significantly, for the annual time series, the calculated NRMSE values were the lowest compared to the other error criteria. Table 10 shows the results of examining this error at different decomposition levels with different border extensions on the time series. As it is observed, the range of the NRMSE values for the annual rainfall, temperature, and groundwater level are 0.038–0.044, 0.004–0.009 and 0.001, respectively which were produced by the SYM border extension at the decomposition level-3. Note that the values of the other two error criteria are reported in Tables S1-S2 in the Supplementary Materials.

Table 10 The normalize root mean square error (NRMSE) resulting from the application of the three border extension modes at two decomposition levels on the annual time series
Full size table

Comparing the values of error criteria in Table 10 with those of Tables S1-S2, we conclude that the DWT analysis of all the annual time series may be best performed by using the symmetric extension mode for border corrections and choosing the decomposition level-3 to produce the approximate and detailed components. At the decomposition level-3 (dyadic dilation a = 8), there are three details components: the 2-year (D1), 4-year (D2), and 8-year (D3) periodicity. Therefore, each annual time series was decomposed into three levels (step f, “Sect. 3.3.1”). Figures 4 depict the approximation (A) and detailed (D1–D3) components of the annual time groundwater series P1, rainfall series at Boushigan, and temperature series at Kazerun (step f, “Sect. 3.3.1”). For the sake of brevity those of other annual rainfall and temperature time series are shown in the Supplementary Materials, Figs. S7-S8.

Fig. 4
figure 4

The original, approximate (A), and detailed components (D1–D3) of the annual time series resulted from DWT decomposition: rainfall at Boushigan station (left), temperature at Kazerun station (center), and groundwater level at P1 (right)

Full size image

Table 11 tabulates lag-1 autocorrelations and seasonality of the original annual time series, their details (D1–D3), approximations (A), and a set of (A + Ds). As it is observed, most of the original annual time series and their respective transformed series exhibited lag-1 autocorrelation without seasonality effect. Therefore, the modified MK test by Hamed and Rao (1998) was used and MMK Z values as presented in Table 12 are calculated. For few series, in Table 11 that did not show seasonality pattern and a significant lag-1 autocorrelation the original MK test was used (i.e., the original temperature time series at Parishan and Dasht-Arjan stations, the A + D1 series at Boushigan station).

Table 11 Lag-1 autocorrelations of the original annual time series and those of their transformed series. Bold numbers indicate significant lag-1 autocorrelation at a 95% confidence limit
Full size table
Table 12 MK Z values of the annual original data, details components (D1–D3), approximations (A), and a set of combination of the details and their respective approximation (A + Ds). \({Z}_{\mathrm{sg}}\) value of combinations closer to that of corresponding original time series are highlighted in boldface
Full size table

The WTMK test was applied to the original series, their detail components (Ds), approximations (A), and (Ds + A), and the \({Z}_{\mathrm{sg}}\) values are presented in Table 12. In this table, the combinations with the \({Z}_{\mathrm{sg}}\) value closer to the \({Z}_{\mathrm{sg}}\) value of original time series are highlighted in boldface.

The original annual rainfall time series at Boushigan (\({Z}_{\mathrm{sg}}\) = − 2.66), Kazerun (\({Z}_{\mathrm{sg}}\) = − 3.62), Parishan (\({Z}_{\mathrm{sg}}\)= − 3.78) and Dasht-Arjan (\({Z}_{\mathrm{sg}}\)= − 4.45) stations exhibit significant decreasing trends. Ds series lack significant trend, but as their respective approximation series are added to them, their trends become statistically significant all with negative \({Z}_{\mathrm{sg}}\) values.

With respect to the results of WTMK test on annual temperature time series (Table 12), except for Parishan station with \(Z\)=  − 1.68, positive trend observed in the other two stations (\({Z}_{\mathrm{sg}}\) = + 6.00 at Kazerun station and \(Z\) =  + 0.93 at Dasht-Arjan station). Only Kazerun station shows statistically significant positive trend. It can be seen, none of the \({Z}_{\mathrm{sg}}\) values of the different individual details (D1–D3) is statistically significant at all the stations even for stations whose original series showed significant \({Z}_{\mathrm{sg}}\) values (i.e., Kazerun station). However, a remarkable consequence is that after the approximation components (A) were added to the different details, many of the trend values became statistically significant.

The values of the \({Z}_{\mathrm{sg}}\) statistics obtained through the MMK test for the groundwater levels are presented in Table 12. As shown, significantly declining trends, in the groundwater levels are observed in both piezometers with \({Z}_{\mathrm{sg}}\) values of − 6.29 and − 6.42 for P1 and P2, respectively. The \({Z}_{\mathrm{sg}}\) values at detail sub-signals are low. With the addition of the approximation to their respective details (Ds + A), the MK \({Z}_{\mathrm{sg}}\) values became greater and significant, emphasizing that the approximations carry most of the trend component.

The prominent periodicities: To determine the most important periodicities affecting the trends of the studied annual time series, the result of WTMK analysis (Table 12) were examined with the three criteria given in step h, “Sect. 3.3.1.” Based on these criteria the combination (A + Ds) that provide the closest \({Z}_{\mathrm{sg}}\) value, the highest correlation and most similar sequential MK curve with their original time series is selected as the most dominant periodic components affecting the production of trends in the annual time series.

According to the first criterion, the multi-resolution analysis (WTMK) of the rainfall shows (Table 12) that the combination A + D2 provides the nearest \({Z}_{\mathrm{sg}}\) value of (\(-4.01\)) (\(-4.07\)) to the original rainfall series at Dasht-Arjan and Parishan stations, respectively, and the combination A + D3 (8-year periodicity) provides the nearest \({Z}_{\mathrm{sg}}\) values of (\(-3.51 ) \mathrm{and} (-2.83\)) to the observed rainfall series at Kazerun and Boushigan stations, respectively. Therefore, the 4-year periodicity at Dasht-Arjan and Parishan stations and 8-year periodicity at Kazerun and Boushigan stations are dominant.

The multi-resolution analysis of the temperature series shows that the combinations A + D2 (4-year periodicity), A + D3 (8-year periodicity), and A + D1 provide the nearest \({Z}_{\mathrm{sg}}\) values to those of original temperature series at the Kazerun, and Dasht-Arjan and Parishan stations, respectively. Generally, the dominant periodic components of temperature series vary from one station to another, which could be due to the different elevation and humidity characteristics of these three stations.

The multi-resolution analysis of the groundwater level time series shows that the combinations A + D3 (8-year periodicity) and A + D2 (4-year periodicity) with the (\({Z}_{\mathrm{sg}}\) = − 6.23) at P1 and (\({Z}_{\mathrm{sg}}\)= − 5.39) at P2 shows the nearest value to the original groundwater level fluctuations.

Considering the correlation coefficient values (the second criterion) between original time series and their components at annual scale (Table 13), the component D1 (2 years), D2 (4 years), and D3 (8 years) are the main deriving components causing the trend in the temperature, rainfall, and groundwater level time series, respectively.

Table 13 The correlation coefficient values between original time series and their (A + Ds) components at the annual scale
Full size table

The sequential MK curves of the combination (A + Ds) that provide the highest similarity with the original time series were plotted to check the third criterion. See Fig. 5 for the rainfall series at Boushigan, the temperature series at Kazerun, and the P1 series. Here again for the sake of brevity those of the other series are presented in the Supplementary Materials, Figs. S8-S9. As it is observed in all the figures the sequential MK curve of A + D3 of the groundwater series, A + D2 of the rainfall series and A + D1 of the temperature series show the highest similarity with their original series. These observations indicate that, like the results of Table 13, the most prominent periodicity causing the trend of the groundwater level, rainfall, and temperature time series are 8-year, 4-year, and 2-year periodicities, respectively.

Fig. 5
figure 5

Sequential Mann–Kendall graphs for the original annual time series versus their (A + Ds) components: rainfall at Boushigan station (left), temperature at Kazerun station (center), and groundwater level at P1 (right). (blue solid line: progressive trend for the original time series; gray solid line: progressive trend for the A + Ds; black solid line: 95% confidence interval limits)

Full size image

The combined WT, MK, and sequential MK analyses conducted on the annual time series data in this section demonstrated that the annual rainfall data show a significant decreasing trend with 4-year periodicity, and the annual temperature data show a significant increasing trend only at Kazerun station with a 4-year periodicity. In contrast to climate variables (rainfall and temperature), the annual groundwater level data exhibit a drastic and highly significant decreasing trend dominated by the 8-year periodicity. The low frequency (long-term variation of 8 years) indicate the occurrence of relatively stable and persistent phenomena (i.e., continuous long-term pumping). Therefore, we conclude that the cause of the declining groundwater level in the Sarbalesh karstic aquifer is the continuous over-exploitation of water storage in this aquifer in the last 32 years. Since the periodicity of the groundwater series is different from those of climate series (rainfall and temperature), we may conclude that the significant decreasing trends of groundwater level has been caused mainly by human activities and the reduced rainfall enhanced the process.

4.5 Results of PMW test for the change point detection

In this section, the annual time series are subjected to the nonparametric PMW analysis to detect the existence of any abrupt shift (change point). Table 14 tabulate values of \({{\varvec{Z}}}_{{\varvec{c}}}\) statistic and \({\varvec{p}}\left({\varvec{t}}\right)\), see “Sect. 3.4.” The values of \({{\varvec{Z}}}_{{\varvec{c}}}\) for all the time series is less than − 1.96 at a high value of significant probability (≥ 0.99) indicating a significant trend shift in all series. The abrupt change in the precipitation and temperature time series is good evidence for the climate change and in the groundwater time series is an indication for climate change/excessive water withdrawal or both. To clear this point, the curves of cumulative sum (\({S}_{k}\)) and significant probability (corresponding to the absolute value of \({(U}_{t,T}\)) function (Eq. 19) are plotted in Fig. 6. The intersection of the curves at the maximum significance probability shows the most probable change year points which are tabulated in Table 14 and shown with arrows in Fig. 6 and with boldface number in Table 14. In the all series, another change point is observed with less significant probability (numbers in parentheses in columns 3 and 5, Table 14). Interestingly, the significance probability curves of \({U}_{t,T}\) function for the two piezometers exhibit a flat trend indicating that the abrupt decreasing sift is repeated annually (excessive water withdrawal) and their intersection points with the cumulative sum curve are the most probable change year point. Similar phenomenon is observed in the rainfall series of Kazerun and Parishan stations as well as temperature series of Kazerun station. The change year in the temperature series at Parishan station is 2004 which coincide with the drying up year of Parishan lake as reported by Samani and Jamshidi (2017).

Table 14 The results of PMW test for the time series. The most probable change year point is shown in boldface. The numbers in parenthesis are the 2nd most probable change year point
Full size table
Fig. 6
figure 6

The significance probability of \({U}_{t,T}\) function and cumulative sum curves for the annual time series (left: rainfall series, center: temperature series and right: piezometer series)

Full size image

Figure 7 illustrates the average monthly variation (increase or decrease) of the hydro-climate parameters. As it is observed, the rainfall and groundwater time series show a considerable reduction after the change year points. Similarly, the temperature series experience an increase which is less noticeable at Parishan station.

Fig. 7
figure 7

Average monthly variation of rainfall (left), temperature (center), and groundwater (right)

Full size image

To delineate the cause of rainfall reduction after the change year, the SOI (Southern Oscillation Index) data as the ENSO indicator was extracted from the web site of the Australian Bureau of Meteorology (www.bom.gov.au), Table S3, and subjected to the PMW analysis and the change year was defined. The result is illustrated in Fig. 8a. The cumulative sum and the significant probability curves follow the same pattern with almost the same change year point (1998 and 2005) as those of the rainfall series (Fig. 6). Therefore, ENSO could be considered as the external mechanism for the climate change, and hence the La Niña phenomenon after change year caused a decrease in rainfall after the change point in the study area (Nazemosadat et al. 2006 reported ENSO as the mechanism affecting Fars province climate). Additionally, we plotted the average monthly variation of the SOI before and after change year in Fig. 8b. As it can be seen, the average monthly SOI after and before the change year is positive and negative, respectively. This again implicates that the La Niña phenomenon after the change year causes a considerable decrease in precipitation from November to April in the study area.

Fig. 8
figure 8

a the significant probability of \({U}_{t,T}\) function and cumulative sum curves for the annual SOI time series, and b average monthly variation of the SOI before and after the change year

Full size image

4.6 Wavelet transform coherence test

To further justify the findings, we conducted a casual inference testing to establish the cause-and-effect relationships between the hydro-climate variables. Maraun and Kurths (2004) recommend continuous wavelet coherence (WTC) for testing the causal relation between two time series. WTC ranges from zero to one, measures the correlation and phase shift between two time series as a function of frequency, and is interpreted as a correlation coefficient between the two time series. Figure 9a depicts the results of WTC analysis of rainfall time series at Kazerun station and groundwater level time series at piezometer P1. The relationship at 95% confidence intervals is shown within the area surrounded by the thick line. Arrows indicate the phase difference between the two series, horizontal arrows pointing left to right signify in-phase, and arrows with the opposite direction imply anti-phase. The coherence result indicates common periodicities between rainfall and groundwater level, specifically in the 8–16-month frequency band. Arrows pointing left to right in Fig. 9a indicate a direct correlation between rainfall and groundwater level fluctuations. However, it is important to note that this correlation gradually diminished and eventually became negligible from 2004 to 2009. Therefore, the decrease in groundwater level during this time period (2004–2009) is not influenced by rainfall. Interestingly, during this period the annual rainfall shows a decreasing trend (Fig. 2). In contrast, Fig. 9b shows an anti-phase relation between temperature and rainfall (arrows pointing right to left), indicating that rainfall reduces as temperature increases. The wavelet coherence graphs between temperature/rainfall time series at Kazerun station are shown in Fig. 9c. An antiphase direction of the arrows illustrates that with increasing temperature, rainfall were decreased without time lag. For the sake of brevity coherency graphs of other time series are presented in the Supplementary Materials, Figs. S11-13, which show the same result as Fig. 9.

Fig. 9
figure 9

Wavelet coherence between rainfall/temperature series at Kazerun station and groundwater level in piezometer P1. The sections marked by black color contours are the significant sections indicating high coherence between two time series. The cone of influence indicates the area affected by the boundary assumptions. X-axis is time, and Y-axis is period. red and blue represent the peaks and valleys of the energy density, respectively

Full size image

5 Conclusions

To assess the anthropogenic and climate change impact on the strategic Sarbalesh aquifer in Fars province, Iran, the rainfall (at 4 stations) and temperature series (at 3 stations) as climate variables and the groundwater level time series (at 2 observation wells) as hydrologic variables are subjected to trend analysis for a period of 32 years from 1988 to 2019. The MK test, MMK test, the combination of DWT, MK, and sequential MK tests, and the combination of PMW and cumulative sum methods were employed. The MK and MMK tests showed a highly significant decreasing trend at monthly and annual groundwater time series. Although the monthly precipitation data lacks any significant trend, the annual series showed a significant decreasing trend. For the temperature time series, at Kazerun station in particular, a significant positive trend was observed which may be due to the drying up of the Parishan lake. The negative/positive trends were backed by the values of Şen’s slope β and Kendall’s tau (τ) coefficient in all the time series.

Using the DWT analysis, the original time series decomposed into their details (Ds) and approximate (A) components, and the dominant time scales that have the highest effect on the production of the general trends in the data were identified. Different error criteria were taken into account to determine the best Daubechies (db) wavelet type, border extension mode, and optimum decomposition level in the DWT trend analysis. The border extension modes (i.e., zero-padding, symmetrization, and periodic) were examined using the three error criteria (i.e., MRE, RE, and NRMSE) between the original data and the approximation component at the last decomposition level. Remarkably, the proposed NRMSE defined levels 6 and 3 as the most appropriate decomposition level for the monthly and annual time series, respectively with the symmetrization border extension mode. After determining the DLs and the best border extension mode, each time series was decomposed to A and Ds components through DWT. The periodicities of each dataset that satisfied the three criteria (i.e., the approximate equality of the MK Z values, the correlation coefficient, and the similarity of the sequential MK graph of the original time series and their respective A + Ds components) assigned as the prominent periodicities in producing the trend. The most effective and prominent periodicities in the monthly and annual rainfall and temperature data were found to be 8 months and 4 years, respectively. For the monthly and annual groundwater level data, the most common periodic components effective in producing the significant decreasing trends are 64 months and 8 years, respectively. The low frequency (long-term variation of 8-year in groundwater level series) indicates the occurrence of relatively stable and persistent phenomena (i.e., continuous and long-term pumping). Based on this aggregation methodologies, we conclude that the cause of the over 30-m decline in groundwater level of the Sarbalesh karstic aquifer is the continuous over-exploitation of water storage from this aquifer in the last 32 years and reduced rainfall and increased temperature (as the climate change indicators, particularly in the annual series) has enhanced the declining trend. Contrary to individual trend tests of the hydro-climatologic data, we recommend the aggregation methodology presented in this paper.

Based on the PMW test and cumulative sum test, the groundwater level series experienced a continuous declining shift, and the most probable change years in the annual rainfall and temperature series happened in 2005–2006 and 1997, respectively, and in the groundwater level series of piezometer P1 and P2 in 2002. The same change year point in rainfall series and SOI and the existence of negative correlations between rainfall and SOI and El Niño cycles indicated that the climate change, and hence the La Niña phenomenon increased SOI after the change year leading to a decrease in precipitation from November to April in the study area. The results were further justified by the WTC analysis as a cause-and-effect relationship testing.

Overall, the results of this study provide an elaborate view of the historical rainfall, temperature, and groundwater trends and the dominant periodicities over the region. This information can be integrated into the methods/models aiming to investigate how natural climatic fluctuations and human activities would affect groundwater trends. This research also contributes to appropriate decision-making programs to preserve groundwater storage in the Sarbalesh aquifer and many other karst aquifers that experience the same fate as the Sarbalesh aquifer in Iran and elsewhere.