Groundwater Level Complexity Analysis Based on Multifractal Characteristics: A Case Study in Baotu Spring Basin, China

Groundwater resources are important natural resources that must be appropriately managed. Because groundwater level �uctuation typically exhibits non-stationarity, revealing its complex characteristics is of scienti�c and practical signi�cance for understanding the response mechanism of the groundwater level to natural or human factors. Therefore, employing multifractal analysis to detect groundwater level variation irregularities is necessary. In this study, multifractal detrended �uctuation analysis (MF-DFA) was applied to study the multifractal characteristics of the groundwater level in the Baotu Spring Basin and further detect the complexity of groundwater level variation. The main results indicate that groundwater level variation in the Baotu Spring Basin exhibited multifractal characteristics, and multifractality originated from broad probability density function (PDF) and the long-range correlation of the hydrological series. The groundwater level �uctuations in wells 358 and 361 exhibited a high complexity, those in wells 287 and 268 were moderately complex, and the groundwater level �uctuations in wells 257 and 305 were characterized by a low complexity. The spatial variability of hydrogeological conditions resulted in spatial heterogeneity in the groundwater level complexity. This study could provide important reference value for the analysis of the nonlinear response mechanism of groundwater to its in�uencing factors and the development of hydrological models.


Introduction
Hydrological processes are nonlinear, complex, dynamic, and widely dispersed, making it necessary to assess the behaviour of hydrological processes at different scales (Li and Zhang 2007, Rakhshandehroo and Mehrab Amiri 2012, Shang and Kamae 2005).Due to the notable in uences of natural and human factors, hydrological time series exhibit highly complex characteristics (Ma et al. 2019).To reveal these complex characteristics and reasonably describe the complexity of groundwater level uctuation constitutes the basis for assessing the impact of natural factors and human activities on hydrological processes.This could provide a solid scienti c basis for hydrological prediction, construction of hydrological models and water resource management.
Hydrologists have long recognized the importance of studying the complexity and scale evolution of hydrological processes.Hydrological time series are typical fractal objects with nonlinear, chaotic and fractal characteristics (Liu et al. 2015).Therefore, fractal theory is widely used in the eld of hydrology (Bhuyan et al. 2009, Yuan et al. 2014).Fractal theory attempts to explain complex processes by determining simple underlying processes.The study of fractal theory began with Hurst's discovery of the long-range correlation in runoff records (Hurst 1951).Hurst rst used rescaled range (R/S) analysis to study the long-range correlation in time-series records of natural phenomena, and with the use of the calculated Hurst exponent, R/S analysis could provide a process for the quanti cation of the memory of time series (Mandelbrot and Wallis 1969).Subsequently, many other studies have reported similar longterm correlated uctuating behaviour in nature (Chakraborty and Chattopadhyay 2021, Koscielny-Bunde 1998, Koscielny-Bunde et al. 2006, Weron 2002).Currently, the concept of fractal theory has been extended and applied to many disciplines, such as medicine, nance, geophysics, hydrology, remote sensing and social sciences, in addition to other studies of nonlinear or stochastic processes (Ihlen 2012, Labat et al. 2011).In hydrology, fractal theory has been employed to explore the variability of rainfall processes (Sivakumar 2001), scaling behaviour of precipitation runoff and sediments (Wu et al. 2018), complexity of the rock distribution in rivers (Dwyer et al. 2021), long-range correlation of runoff (Zhao et al. 2017), and particle deposition during arti cial groundwater recharge (Wang et al. 2021).
Groundwater level uctuations comprise the dynamic response of a given groundwater system to recharge and discharge, which are in uenced by numerous factors.For example, natural factors, such as precipitation, evapotranspiration, seepage, soil moisture, and topography, and human factors, such as mining, irrigation, and construction of hydraulic projects (Li and Zhang 2007, Rakhshandehroo and Mehrab Amiri 2012), could vary on different spatial and temporal scales.Hydrogeological condition refers to conditions relating to groundwater formation, distribution and variation patterns, including groundwater recharge, burial, runoff, discharge, water quality and quantity.According to the concept of hydrogeological condition, groundwater recharge and discharge characteristics also belong to the category of hydrogeological condition.Therefore, it is because of the different hydrogeological conditions that there are different groundwater recharge-discharge laws, which lead to the varied characteristics of groundwater level uctuations.Moreover, due to the in uence of climatic conditions, the uctuations in groundwater level time series may exhibit periodicity and seasonal cycles.These uctuations are usually not consistent, and ordinary linear and deterministic models cannot be adopted to simulate these uctuations (Yu et al. 2016).
Non-stationary uctuation in the groundwater level could be described as fractional Brownian motion and could be largely long-range correlated rather than completely random (Rakhshandehroo and Mehrab Amiri 2012, Yu et al. 2016).This suggests that monofractals (homogeneous variable processes) cannot comprehensively describe uctuations in groundwater levels, and multifractal (complex nonlinear heterogeneous processes) analysis is required to detect irregularities in time series (Stanley 1999).
Multifractal detrended uctuation analysis (MF-DFA) is a common and effective tool to determine the multifractal behaviour and can be adopted to quantify the complexity of physical mechanisms (Ihlen 2012).MF-DFA has proven to be a valuable tool in the multifractal characteristics analysis of time series (Adarsh et Zhu et al. 2020).In previous decades, due to excessive groundwater exploitation, groundwater resources in the Baotu Spring Basin were seriously threatened, and even spring water was dry up.Reasonable protection of karst water resources in the Baotu Spring Basin is very important.At present, there is no research to analyze the multifractal characteristics and complexity of groundwater level in this area.The main objective of this study is to quantitatively calculate the complexity of the physical mechanisms that control the groundwater level through the multifractal method.The lower complexity is associated with the simpler physical mechanism, on the contrary, the groundwater level is more di cult to predict successfully.Therefore, the complexity of groundwater level can re ect the complexity of recharge and discharge in the groundwater system and the response sensitivity of the groundwater system to recharge and discharge.
Multifractal analysis is applied in this study to detect the multifractal behaviors and complexity in the time series of the groundwater level in the Baotu Spring Basin.Theoretically, this study is important for improved analysis of the nonlinear mechanism of the groundwater response to natural and anthropogenic factors.In addition, in practical engineering, this study can provide a scienti c basis for the development of hydrological models.In this study, rst, the detrended uctuation analysis (DFA) method is used to determine whether data processing is required before multifractal analysis.Multifractal analysis of the groundwater level in different aquifers in the study area is conducted via the MF-DFA method.The complexity of the physical mechanisms controlling the groundwater level can be assessed based on the complexity index (CI) (Lana et al. 2020), so the CI of the groundwater level is calculated.Finally, the source of the multifractality of groundwater level time series is examined.

Study Site And Data Collection
The Baotu Spring Basin is located in the midwestern part of Jinan city, Shandong Province (Fig. 1), and covers an area of 1730 km 2 .Karst water in the Baotu Spring Basin is widely distributed, and pore water mainly occurs in the northwestern part of the spring basin (Niu et al. 2021).
The northern part of the western boundary of the Baotu Spring Basin is the Mashan Fault, and the northern part of the eastern boundary is the Dongwu Fault.The northern boundary comprises Carboniferous-Permian igneous rocks, and the rest of the boundary comprises surface divides.
Controlled by the stratum dip and topography, the overall direction of karst groundwater runoff is from southeast to northwest (Qian et al. 2006).The hydrogeology in the Baotu Spring Basin is shown in Fig. 1, and the data is obtained from the hydrogeological investigation report of this area.According to the geological structure, the spring basin can be divided into magmatic rock, Cambrian and Ordovician karst carbonate rock strata, igneous rock, and Quaternary strata.The strata in the middle of the Baotu Spring Basin mainly include Cambrian and Ordovician karst carbonate strata, and the Cambrian strata are characterized by interbedded limestone and shale.The Ordovician strata comprise thick limestone, argillaceous limestone, and dolomitic limestone, with well-developed karst ssures and a high permeability, which facilitates groundwater recharge, runoff, and discharge.
Although there are many groundwater level observation wells in the Baotu Spring Basin, most of these wells are located in the north of the spring basin due to the mountainous area in the south of the spring basin.Moreover, most of these wells exhibit a short record time or suffer a large amount of missing data, and interpolation cannot be applied to obtain missing data.Considering the data availability and completeness, six groundwater level observation wells were selected in this study (Table 1), and the distribution of these wells in the study area is shown in Fig. 1.Wells 257, 358, and 361 are karst water level monitoring wells, and wells 305, 268, and 287 are pore phreatic water level observation wells.Wells 358 and 361 are located in the Cambrian-Ordovician limestone distribution area, and the limestone in this area is exposed on the surface.The area has well-developed karst pores, ssures and conduits, which provide huge space and channels for groundwater storage and transportation.Wells 257,268,287,305 are located in the limestone lie concealed region, overlying the Quaternary.According to the availability of data, the research time ranges from 1992 to 2012, and groundwater level data are every ten-day monitoring data, with a total of 756 measurements in 21 years.Step 1: In regard to time series x k (k = 1, …, N) to be analysed, the cumulative deviation of the time series is determined, also referred to as the 'pro le': Here, N is the length of the time series, and < x > denotes the average of {x k }.
Step 2: For each given scale s, is divided into N s nonoverlapping segments of equal length.
The setting of scale(s) in DFA and MF-DFA is the most noteworthy, the minimum scale should preferably be greater than 10 and the maximum scale should not be too large (Ihlen 2012).Therefore, in this study, when the calculation object is the groundwater level time series from 1992-2012, the minimum value of the scale is set to 16 and the maximum scale is 128 through debugging; when the calculation object is the groundwater level from 1992-2002, 2003-2012 or the precipitation time series from 2010-2019, because the time series length is reduced, the minimum value of the scale is set to 16 and the maximum scale is set to 64.
Step 3: Local trends are calculated for each segment via least squares tting.The variance can be determined as: Here, is the local variance for every one of the N s segments, and is the tting polynomial of any appropriate order within segment v (v = 1, 2, …, N s ).In this study, linear polynomials was employed Step 4: The qth-order uctuation function can be obtained as follows: where q denotes any real value other than zero.For q = 2, the standard DFA procedure is retrieved.In order to avoid large numerical errors, the selection of q-order should avoid large Step 5: The scaling behaviour of uctuation functions can be analysed by log-log plots of F q (s) versus s for each q.
Where H q is the slope of the curve of logF q (s) ~ log(s), and is referred to as the q-order Hurst exponent.
Before MF-DFA application, it is necessary to use DFA to calculate H 2 , and H 2 is called the generalized Hurst exponent.The time series persists when 0.5 < H 2 < 1 or H 2 > 1.5.Persistence represents long-range correlated process, that means if a time series increases (or decreases) over time, it is likely to continue to increase (or decrease).When 0 < H 2 < 0.5 or 1 < H 2 < 1.5, the time series is anti-persistent.Anti-persistence usually indicates that the trend of change of the time series is opposite to that of the previous period (Sun et al. 2019).Moreover, if 0 < H 2 < 1 then the signal is fGn (fractional Gaussian noise, which is a stationary process).If 1 < H 2 < 2 then it is fBm (fractional Brownian motion, which is nonstationary process) with some range of uncertainty in between (Eke 2002).When H 2 varies between 0.2 and 1.2, the time series resembles noise, at which point MF-DFA can be directly applied to the time series without any transformation, and conversely, the data should be processed before application (Ihlen 2012).Considering a monofractal time series, H q is independent of q.If there occurs multifractal behaviour, H q signi cantly depends on q.
Step 6: The level or strength of multifractals can be represented by the Hölder exponent spectrum (or singularity spectrum) f(α), and α is the Hölder exponent or singularity strength.α and f(α) can be obtained as: ∆α, α 0 and R are three important multifractal parameters.The spectrum width can be de ned as ∆α = α max -α min , ∆α is a measure of the α range, and the larger ∆α is, the richer the structure of the physical process and the stronger the multifractal.α 0 is the critical (central) Hölder exponent, which corresponds to the maximum value of f(α).The larger α 0 is, the more irregular the underlying process.The asymmetry of the multifractal spectrum can be quanti ed as R=(α max − α 0 )/(α 0 − α min ), with a right-skewed spectrum (R > 1) indicating that the physical process exhibits a ne structure and a left-skewed shape (R < 1) indicating a more regular or smoother structure.
The complexity index (CI) of groundwater level can be obtained by summing three normalized multifractal parameters, i.e., α 0 , ∆α, and R. Considering sequences with higher α 0 values, the width of the spectrum f(α) is larger, and right-skewed shapes (R > 1) are more complex than those with opposite features (Lana et al. 2020).In general, a higher CI value indicates that a more signi cant complexity on the physical mechanisms governing groundwater level uctuation and suggest a di cult success predictability, while a lower CI value indicates that is easier to predict.
Results And Discussion

Multifractal detrended uctuation analysis (MF-DFA)
To determine whether data conversion is required before MF-DFA application, DFA was applied to the groundwater time series.As shown in Fig. 3a, the generalized Hurst exponents were 1.2602, 1.7801, and 0.9291 for wells 257, 358, and 361, respectively.As shown in Fig. 3b and very close to 1, which is inconsistent with the non-stationary of groundwater level uctuations, but is also plausible to some extent due to the range of uncertainty between fGn and fBm.
Fluctuation functions F q (s) with different q values for the groundwater level time series were calculated via MF-DFA.For each q, a straight line was tted to the plots of log 2 (F q (s)) versus log 2 (s), and the slope of this line represents H q for that speci c q value.Figure 4 shows curves of the q-order Hurst exponents H q ~q.The curves of the q-order Hurst exponents H q versus q of the time series of the groundwater level in the observation wells show that H q nonlinearly decreases with increasing q, indicating that the time series exhibit multifractal characteristics.In Fig. 4, compared to the other karst water level monitoring wells, H q varied the most gently with q in well 257, and the intensity of the H q uctuation with q in well 358 varied between those in wells 257 and 361.Moreover, the drastic change in H q in well 361 indicate that the multifractal features of the groundwater level uctuation were highly complex.Among the pore water level observation wells, H q changed with q in well 268 to a similar extent as that in well 287, while in well 305, H q changed very slightly in well 305 for q < 1.
Figure 5 shows the multifractal spectra for the groundwater level records of the six observation wells.If a higher level of multifractality is observed in the signal, the f(α) spectrum widens, but this width converges to one point for a purely monofractal signal.Therefore, Fig. 5 shows that the groundwater level time series of these six observation wells were characterized by multifractality.As shown in Fig. 5, the structure and width of the multifractal spectra of the karst water level time series greatly differed, so the multifractal characteristics of the groundwater level time series in the study area exhibited very obvious spatial heterogeneity.

Multifractal parameters and complexity
To measure the multifractal strength and complexity of the groundwater level time series for the six observation wells, speci c parameters were calculated via MF-DFA, as listed in Table 2.The complexity of the physical mechanism controlling the time series could be quanti ed via the CI.The CI values of wells 358 and 361 were 3.50 and 3.48 respectively, which were the highest among all wells, and the difference between the two wells was small, so wells 358 and 361 were classi ed as high complexity.Similarly, wells 257 and 305 were classi ed as low complexity.The CI values of wells 287 and 268 were − 0.77 and − 1.46, respectively, signi cantly lower than CI with high complexity and higher than low complexity.Therefore, wells 257 and 305 were classi ed as moderate complexity.As shown in Fig. 5 and Table 2, the reason for the low complexity of wells 257 and 305 is that the shape of the multifractal spectrum of both wells is left-skewed (R < 1) and Δα of both wells is small.In addition, the high complexity of well 358 is due to the large width of the multifractal spectrum and the high complexity of well 361 is due to the large width of the multifractal spectrum and its right-skewed structure.Therefore, it is more di cult and uncertain to predict the groundwater level in wells 361 and 358, while the groundwater level of wells 305 and 257 exhibited a simple structure and could be more easily predicted.).In this study, the groundwater depths are all greater than 3m (Table 1) and groundwater evaporation is minimal and is not considered in this study.
(1) Functional areas of the Baotu Spring Basin From the perspective of the spring basin functional area where the observation well is located, according to its storage space, hydraulic characteristics and functional characteristics, the Baotu Spring Basin can be divided into a discharge area, direct recharge area and indirect recharge area, as shown in Fig. 6.
The direct recharge area of the Baotu Spring Basin is the Cambrian Chaomidian Formation-Middle Ordovician limestone area widely distributed in the hilly area in the south of the spring basin and the piedmont plain.Karst groundwater can directly accept both atmospheric precipitation and surface water leakage recharge in the direct recharge area.Because wells 361 and 358 are located in the direct recharge area, they respond acutely to precipitation and surface water leakage recharge, which is the reason for the more notable uctuation in the groundwater level in wells 361 and 358.Moreover, wells 358 and 361 are located in low mountainous areas where stepped terrain exists and the slope of the strata is high, providing favorable conditions for groundwater ow.
The discharge area of the Baotu Spring Basin contains a at terrain and a large thickness of Quaternary surface strata, which is conducive to receiving precipitation recharge.Most of the precipitation directly replenishes the pore water in the Quaternary aquifer.Due to the shallow groundwater depth in wells 268 and 287, and the shallow aquifer responds more dynamically to phenomena such as precipitation, evapotranspiration, vegetation, discharge, and soil capillarity, which may account for the higher complexity of pore water levels in wells 268 and 287 compared to karst water level in well 257.In addition, wells 257, 305, 268 and 287 are located in the plain area, with a small stratum gradient.
Compared with the mountain area, the groundwater ow rate is small.According to borehole information, there is a clay layer separates karst aquifer from Quaternary pore aquifer, and the precipitation can be recharged to the karst water only after passing through the Quaternary aquifer and clay layer, so the precipitation signal will be weakened by the ltering of the Quaternary aquifer and clay layer.Moreover, the karst water of Well 257 is not easily affected by other surface factors.Therefore, the groundwater level in well 257 remains stable. (

2) Precipitation
The four precipitation stations closest to the groundwater level observation wells are shown in Fig. 6.Since it is very di cult to obtain detailed precipitation data in the early stage, in this study, we only use the precipitation time series after 2010 to analyze the complexity of precipitation in different regions.This calculation result cannot fully represent the complexity of precipitation in the period of the groundwater level series used for calculation.However, it can re ect the complexity characteristics of precipitation in this region to a certain extent, and provide some reference for analyzing the complexity of groundwater time series.The CI was calculated for the cumulative daily precipitation time series of 1220 days during the 2010-2019 ood season (June, July, August, September), and the results are shown in Table 3.There is little difference in precipitation between the rainfall stations during the ood season, and the precipitation time series in descending order of complexity are Qiujiazhuang, Yanzishan, Donghongmiao, and Changqing.In addition, all six groundwater level observation wells are located in locations where the land use type is arable rather than impermeable, and therefore precipitation can recharge groundwater in the vertical direction.The Qiujiazhuang station, nearest to well 358, and the Yanzishan station, nearest to well 361, have higher precipitation time series complexity than the other stations, and the complexity of the time series of precipitation in ltration recharge obtained for wells 358 and 361 is therefore higher than for the other wells.2) shows that its uctuation pattern differs from that of the other wells in that the groundwater level in well 358 (Fig. 2b) declines sharply in 1998 and 2002 and then rises rapidly thereafter.It can therefore be speculated that there was irregular and very intense groundwater extraction as well as arti cial recharge activity in the vicinity of well 358, which has affected the multifractal results of groundwater levels.In summary, with approximately the same amount of precipitation recharge, the complexity of the precipitation time series near wells 358 and 361 is the highest, and karst water can receive direct recharge from precipitation through the limestone.In addition, wells 358 and 361 are located in a layer with a high slope and a high degree of karst development, which provides suitable conditions for groundwater transport.Although well 257 is also a karst well, the precipitation time series in the vicinity is less complex and the precipitation signal is weakened by the overlying strata, furthermore the groundwater ow rate at the location is low and therefore the water level is more stable.Wells 305, 368, 287 also have stable groundwater levels due to factors such as precipitation and the nature of the aquifer.

Multifractal source analysis
In general, the multifractality of time series can classi ed into two types (Tu et al. 2017, Ye et al. 2017).
The rst one is caused by the broad probability density function (PDF) of the time series, and the other one is due to the long-range correlation on different scales.Since randomly shu ed sequences can destroy the long-range correlation, the above two types can be distinguished by performing multifractal analysis of the randomly shu ed time series.If the multifractality is entirely attributable to the second type, it will disappear; otherwise, it will be retained.Moreover, the shu ed data could exhibit a lower multifractality than that of the original data when the multifractality is due to both types (Ihlen 2012, Yuan et al. 2014).
Figure 7 shows the multifractal spectra for the shu ed groundwater level data.As shown in Fig. 7, the width of the multifractal spectra was narrower than that of the original data, but this width did not converge to a point.As indicate in Table 5, Δα of the shu ed data was smaller than that of the original data.The larger Δα is, the higher the multifractality of the time series.As shown in Fig. 7 and Table 5, the multifractality diminished but did not disappear for all time series.This result shows that the multifractality of the groundwater level time-series could not be removed by shu ing, so the multifractality originated from both the long-range correlation and broad PDF of the groundwater level series.The high spatial variability of the groundwater level complexity is caused by highly uneven hydrogeological conditions.Therefore, by analysing the complexity of the physical mechanism controlling the groundwater level, we could further analyse the nonlinear mechanism of the groundwater response to natural and human factors, and the nonlinear response mechanism is the source of groundwater level complexity.In future research, complexity analysis of the groundwater level and nonlinear response mechanism analysis of groundwater to various factors could be combined by collecting more detailed hydrogeological data, which is an effective method to better understand the complexity of regional groundwater level change.

Declarations Figures
Hydrogeological

MethodMF-
DFA is a common mathematical tool for the detection of the scale characteristics and multifractality of time-series datasets.DFA can detect the non-stationarity of evolution, effectively eliminate the apparent long-range correlation attributed to external effects and identify uctuations only related to the characteristics of aquifers.In this study, MF-DFA was utilized to analyse the multifractal properties of non-stationary time series, and ultimately, the obtained information on the scale behaviour and parameters could be useful for multifractal modelling.The DFA and MF-DFA procedure comprises the following main steps (Adarsh et al. 2020, Gao et al. 2019, Kantelhardt 2002, Zhang et al. 2014):
negative and positive values (Little and Bloom eld, 2010).Therefore, referring to other studies (Labat et al. 2011, Little and Bloom eld 2010, Sun et al. 2019, Yu et al. 2016), MF-DFA was employed with the range of q values is [-5, 5], and the step of q used in MF-DFA is 0.1.
map of the Baotu Spring Basin Page 18/22

Figure 7 Multifractal
Figure 7 al. 2020, Ihlen 2012), image analysis applications and remote sensing (Aleksandrowicz et al. 2022, Krupiński et al. 2020).It is of theoretical and practical signi cance to better understand the statistical characteristics of groundwater table series according to multifractal parameters (Labat et al. 2011, Little and Bloom eld 2010, Sun et al. 2019, Yu et al. 2016), but there are few studies assessing the complexity of groundwater in different types of aquifers.The Baotu Spring Basin is one of the typical karst systems and the important water sources in North China.Moreover, the karst springs in the Baotu Spring Basin are of historical and tourism value (Gao et al. 2020, Kang et al. 2011, Mandelbrot and Wallis 1969, Xing et al. 2018,

Table 1 .
Basic information on the observation wells The presence of periodic uctuations in meteorological and hydrological time-series data could signi cantly affect the MF-DFA results(Li and Zhang 2007, Lu et al. 2021, Yuan et al. 2014, Zhang et al. 2019).Therefore, the spectral analysis is required to eliminate the periodic component of the groundwater level data (Lu et al. 2021).Groundwater level uctuations are shown in Figure 2.
(Yu et al. 2016)d Hurst exponents of wells 305, 268 and 287 were 1.2584, 1.2164 and 1.5166, respectively.The groundwater level time series of wells 358 and 361 behaves as persistence, and the groundwater level time series of other wells show anti-persistent correlation.The calculated generalized Hurst exponent ranged from 0.9291-1.7801.Because MF-DFA is best applied when the Hurst exponent varies between 0.2 and 1.2, the groundwater level time series should be converted via a one-order difference(Ihlen 2012).Although the generalized Hurst exponent of well 361 ranged from 0.2 to 1.2, the groundwater level time series of well 361 also requires one-order difference in order to ensure the consistency of the input data.The groundwater level time series of wells 358, 361, 287 behaves as persistence, and the groundwater level time series of other wells show anti-persistent correlation.Groundwater level uctuation is usually non-stationary and should be characterized by Brownian motion(Yu et al. 2016).In this study, the generalized Hurst exponents of wells 257,358,305,268,287 are all greater than 1.Thus, the uctuations of groundwater levels at these ve wells are fractional Brownian motion (fBm).The generalized Hurst exponent of well 361 is less than 1

Table 2
Multifractal parameters of the groundwater level time-series Yu et al. 20162011uctuations are the dynamic responses of aquifers to recharge and discharge.The high spatial variability of the complexity of groundwater level uctuations is due to the coupled effects of conditions affecting recharge and discharge, such as aquifer characteristics, precipitation, and anthropogenic disturbances, evaporation, etc(Labat et al. 2011, Rakhshandehroo and Mehrab Amiri 2012,Yu et al. 2016

Table 3
Since 2003, large-scale karst water exploitation has been prohibited in the Baotu Spring Basin.Therefore, the groundwater level time-series was divided into 1992-2002 and 2003-2012.As shown in Table4, after 2003, the CI of wells 257, 361, 305, 268 and 287 decreased slightly, indicating the complexity of groundwater level uctuations in these wells decreased, while the CI of well 358 increased.The groundwater level time series for well 358 (Fig.

Table 4
The CI of the groundwater level time-series

Table 5
Multifractal parameters of the shu ed groundwater level time series (Δα(shu ed) is calculated from the time series of groundwater level after random shu ing, Δα (original) is calculated from the original time series of groundwater level without random shu ing) This study applied MF-DFA to investigate the multifractal characteristics of the groundwater level in the Baotu Spring Basin, and the groundwater level complexity was analysed.The results indicate that the Baotu Spring Basin groundwater level time series exhibited multifractal characteristics, and each multifractal spectrum differed.The multifractality of the groundwater level time series could not be removed by shu ing, so the multifractality originated from both the long-range correlation of the groundwater level time series and broad PDF.The CI was calculated based on the multifractal parameters.The CI values of wells 358 361 were 3.50 and 3.48, respectively (high complexity), those of wells 287 and 268 were − 0.77 and − 1.46, respectively (moderate complexity), and those of wells 257 and 305 were the lowest, at -2.32 and − 2.43, respectively (low complexity).This result shows that the CI of the groundwater level indicated spatial heterogeneity.