Groundwater closely interacts with surface water and even climate systems in most hydroclimatic settings. Fractal scaling analysis of groundwater dynamics is of significance in modeling hydrological processes by considering potential temporal long-range dependence and scaling crossovers in the groundwater level fluctuations. In this study, it is demonstrated that the groundwater level fluctuations in confined aquifer wells with long observations exhibit site-specific fractal scaling behavior. Detrended fluctuation analysis (DFA) was utilized to quantify the monofractality, and multifractal detrended fluctuation analysis (MF-DFA) and multiscale multifractal analysis (MMA) were employed to examine the multifractal behavior. The DFA results indicated that fractals exist in groundwater level time series, and it was shown that the estimated Hurst exponent is closely dependent on the length and specific time interval of the time series. The MF-DFA and MMA analyses showed that different levels of multifractality exist, which may be partially due to a broad probability density distribution with infinite moments. Furthermore, it is demonstrated that the underlying distribution of groundwater level fluctuations exhibits either non-Gaussian characteristics, which may be fitted by the Lévy stable distribution, or Gaussian characteristics depending on the site characteristics. However, fractional Brownian motion (fBm), which has been identified as an appropriate model to characterize groundwater level fluctuation, is Gaussian with finite moments. Therefore, fBm may be inadequate for the description of physical processes with infinite moments, such as the groundwater level fluctuations in this study. It is concluded that there is a need for generalized governing equations of groundwater flow processes that can model both the long-memory behavior and the Brownian finite-memory behavior.

Groundwater in both confined and unconfined aquifers is usually a complex and dynamic system that highly interacts with surface water and even climate systems in most hydroclimatic settlings due to its discharge to rivers and streams and its recharge affected by various related physical processes, such as precipitation, evapotranspiration, and infiltration (Green et al., 2011; Joelson et al., 2016; Li and Zhang, 2007; Rakhshandehroo and Amiri, 2012; Taylor et al., 2013). These processes, which take place over various spatiotemporal scales, add further complexity to groundwater systems. Groundwater level fluctuations dynamically reflect the responses of an aquifer to its diverse inputs and outputs. Consequently, groundwater level fluctuations are often nonstationary, rendering variabilities over different spatial and temporal scales and resulting in no dependence on single representative spatial and temporal scales. Therefore, groundwater level fluctuations are often characterized as scale-free processes and modeled as fractional Brownian motion (Hardstone et al., 2012; Yu et al., 2016). Although not necessarily totally random, groundwater level fluctuations may demonstrate long-range dependence through time, implying a power-law relationship over a variety of timescales, which can be represented by fractals (Yu et al., 2016).

Fractal analysis of both persistent and anti-persistent behavior has been extensively utilized to investigate possible relationships in variability among various scales (Blöschl and Sivapalan, 1995). Temporal fractal scaling analysis of groundwater dynamics can be essential to a better understanding of the modeling of hydrological processes by considering temporal correlations and scaling cascading issues, since groundwater closely links to surface water in hydrological modeling and hydrological models are built upon certain temporal and spatial scales (Blöschl and Sivapalan, 1995; Yu et al., 2016). Hence, fractal scaling analysis of groundwater level fluctuations can guide more representative modeling in hydrological models and in coupled land–atmosphere models. In fact, groundwater dynamics were found to provide a positive feedback to the memory of land surface hydrological processes in climate systems, and enhanced knowledge of fractal behavior in subsurface hydrological processes can help improve weather forecast and climate prediction on different temporal scales (Lo and Famiglietti, 2010). Furthermore, fractal scaling analysis of groundwater level fluctuations may help investigate extreme events and anthropogenic forcing in Earth systems (Yu et al., 2016).

Detrended fluctuation analysis (DFA), originally used to analyze the long-range power-law correlations (i.e., persistent fractal scaling behavior) of time series, is considered a powerful method to quantify the scaling parameter or the Hurst exponent for its capacity in detecting nonstationarities and distinguishing seasonal oscillations from intrinsic fluctuations compared with conventional methods, such as R/S analysis or the variation method (Dubuc et al., 1989; Hardstone et al., 2012; Shang and Kamae, 2005). In order to characterize multifractal structures within complex nonlinear heterogeneous processes, multifractal detrended fluctuation analysis (MF-DFA; Kantelhardt et al., 2002) was developed based on the framework of DFA, which is mostly used to quantify monofractality. DFA and MF-DFA have been widely applied to evaluate the fractal scaling properties of rainfall and streamflow time series in hydrology (Kantelhardt et al., 2002; Koscielny-Bunde et al., 2006; Labat et al., 2011; Livina et al., 2003; Matsoukas et al., 2000; Zhang et al., 2008).

More specifically, DFA was first adopted in subsurface hydrology by Li and Zhang (2007) to systematically evaluate the fractal dynamics of groundwater systems. They analyzed 4 years of continuous hourly data from seven wells and found that groundwater level fluctuations are likely to follow fractional Brownian motion (fBm) and that temporal scaling crossovers exist in the fluctuations. These findings were later confirmed by Little and Bloomfield (2010), Rakhshandehroo and Amiri (2012), and Yu et al. (2016) with the application of DFA to hourly or 15 min interval data for up to 5 years from 7 wells, daily data for 6 years from 2 wells, and daily data from 22 wells that have more than 2500 records, respectively. Rakhshandehroo and Amiri (2012) further utilized MF-DFA to evaluate the multifractality of groundwater level fluctuations and concluded that the extent of multifractality in groundwater level fluctuations is stronger than that in river runoff.

Unlike the general finding of fBm-type behavior in groundwater level fluctuations (Li and Zhang, 2007; Little and Bloomfield, 2010; Rakhshandehroo and Amiri, 2012; Yu et al., 2016), Joelson et al. (2016) found persistent scaling behavior in the analysis of hourly groundwater level fluctuation time series for a 14-month duration and fit the fluctuation data with the Lévy stable distribution to account for the observed non-Gaussian heavy-tailed behavior.

Multiscale multifractal analysis (MMA) was proposed on the basis of MF-DFA, which normally analyzes time series with crossovers only on a predefined large or small scale to obtain the generalized Hurst surface, which simultaneously provides local fractal properties at various scale ranges (Gierałtowski et al., 2012; Wang et al., 2014). To the best of our knowledge, MMA has not yet been applied to analyze time series in hydrology or subsurface hydrology.

In this paper, DFA, MF-DFA, and MMA are applied to systematically
evaluate the temporal fractal scaling properties (monofractality and
multifractality) of groundwater level fluctuations in two confined
aquifer wells with daily data of 70 and 80 years in Texas,
USA. Long-term groundwater level data are used, since the Hurst
exponent estimated by a larger number of data points tends to be more
stable (Weron, 2002). We also check the variation in the estimated
Hurst exponent by DFA with different lengths of data and variable time
intervals, which is largely unexplored in the aforementioned
studies. The possible explanation of the existence of multifractality
is studied by MF-DFA and MMA. Furthermore, we investigate the
groundwater level fluctuation probability distribution by fitting the
data with the

Since the pioneering work of Hurst (1951) on the long-memory behavior (or
persistent fractal) of the storage capacity of reservoirs in the Nile
River, the Hurst exponent has been regarded as the best-known
estimator indicating the magnitude of long-range dependence in time
series and has been widely used to study fractal scaling behavior in
geophysical sciences, specifically for river flows and turbulence
(Nordin et al., 1972; Szolgayova et al., 2014; Vogel et al., 1998),
porosity and hydraulic conductivity in subsurface hydrology (Molz and
Boman, 1993), climate variability (Bloomfield, 1992; Franzke et al.,
2015; Koutsoyiannis, 2003), and sea level fluctuations (Barbosa
et al., 2006; Ercan et al., 2013). The Hurst exponent

Here, the Hurst exponent was adopted to quantify the scaling properties of groundwater level fluctuation time series. Many methods for the estimation of the Hurst exponent are used in the literature, and different methods may provide significantly different estimates. Detrended fluctuation analysis is chosen here due to its superior performance compared to conventional methods in detecting evolving nonstationarities, which can be very useful to investigate the fractal behavior of time series datasets with different time intervals and in differentiating seasonal trends from the inherent fluctuations of time series (Yu et al., 2016).

Detrended fluctuation analysis (DFA), also known as variance of the regression residuals, was proposed by Peng et al. (1994). The method is briefly summarized as follows.

First, the original time series

Then,

Multiscale multifractal analysis (MMA) is a generalization of multifractal detrended fluctuation analysis (MF-DFA), which is developed from DFA (Gierałtowski et al., 2012). In contrast to MF-DFA, which requires the presumption of scaling ranges, MMA is capable of concurrently characterizing different fractal properties (monofractality or multifractality) of time series over a wide range (both small and large) of temporal scales. MMA can be specified as follows.

Based on DFA, the

The strength of multifractality may be further measured by the
Hölder spectrum or singularity spectrum (Feder, 1988). The
Hölder exponent

The above estimators show the formulation of MF-DFA. After the
calculation of all

The capability of MMA, which is inherited from DFA and MF-DFA, is that
it can effectively detect observational noise and nonstationarities in
time series. Similar to MF-DFA, the results of

The

The stability index

Stable distributions are heavy tailed, and the tails of these
distributions demonstrate asymptotical power law behavior with

Statistics and geophysical properties of studied wells in Texas, USA.

Note: mean represents the mean groundwater level (hydraulic head) depth below land surface.

Groundwater level time series data of

Since the Lévy

Autocorrelation function (ACF) for all groundwater level
datasets

Two confined aquifer wells with long groundwater level records (70 and
80 years long) were chosen in this study to perform fractal scaling analysis
(see Appendix A for the selection procedure). Groundwater level time series
data of the two wells were obtained from the Water Data for Texas website
(

Power spectra of

The autocorrelation function (ACF) in Fig. 2 shows very slow decay in both datasets, and the dataset of Well 1 decays more slowly than that of Well 2. In fact, it takes several years (more than 1000 days) for Well 1 to become decorrelated, while it takes a couple of years (more than 500 days) for Well 2. Moreover, the ACF plots greatly vary in different 20-year intervals of the two datasets (bottom left and right panels in Fig. 2), which may imply that the long-range dependence characteristics of the two wells would vary through time.

The power spectra of Well 1 (1945–2014) and Well 2 (1935–2014) groundwater levels are presented in Fig. 3. The power-law exponents are estimated as 2.44 and 2.08 for Well 1 and Well 2 groundwater levels, respectively, indicating the existence of fractals in both datasets. Hurst exponents can be deduced from the power-law exponents (Heneghan and McDarby, 2000) as 0.72 and 0.54 for Well 1 and Well 2 groundwater levels, respectively. Furthermore, the Kwiatkowski–Phillips–Schmidt–Shin (KPSS) test (Kwiatkowski et al., 1992) is conducted to test the stationarity of the data. The null hypothesis for the KPSS test is that a time series is stationary, and the alternative is that data are nonstationary. The estimates of the KPSS statistic are 5.6357 and 1.8012 for Well 1 and Well 2 groundwater levels, respectively, and both reject the null hypothesis at a 1 % significance level, which suggests that the two time series are nonstationary. These results provide reference for the quantification of the Hurst exponent later by DFA.

The Hurst exponents of groundwater level fluctuation data, quantified
by the DFA approach over different time intervals, are investigated
here. The evolution of the Hurst exponent

The Hurst exponent

Figure 5 presents the Hurst exponents of groundwater level data estimated with different moving time windows (5, 10, and 20 years). Daily data were used in different time windows: 5-year moving window (i.e., 1945–1949, 1946–1950, …, 2010–2014), 10-year moving window (i.e., 1945–1954, 1946–1955, …, 2005–2014), and 20-year moving window (i.e., 1945–1964, 1946–1965, …, 1995–2014). Figure 5a and b show that the Hurst exponents vary greatly in different time windows (i.e different length of groundwater level fluctuation data) and also do not remain constant even with the same time window when the time window moves in time. Moreover, the results in Fig. 5a and b demonstrate that the Hurst exponent tends to be stable as the time window increases, which is consistent with the results in Fig. 4.

Box plots of the Hurst exponents under different moving time
windows for

Additionally, the correlation coefficient

Figure 6 further investigates the variation in the Hurst exponents with
box plots for 5-, 10-, 20-, 30-, 40-, and 50-year moving time windows. Unlike the
inconsistency of the linear correlation between

The multifractal results obtained by MF-DFA in Fig. 7 include log–log plots
of

The singularities of the processes in the groundwater levels of Well 1
and Well 2 are revealed in Fig. 7d. The width of the singularity
spectrum,

Two types of rationale are used to account for multifractality in time series (Kantelhardt et al., 2002). The first type is that a broad probability density function of time series data, which cannot be represented by a regular distribution with finite moments, causes multifractality. The second type is that multifractality is caused by long-range correlations of small and large fluctuations (Kantelhardt et al., 2002; Rakhshandehroo and Amiri, 2012). To distinguish these two types of multifractality, the corresponding randomly shuffled dataset is analyzed. The multifractality will vanish if it is totally due to the second type and will remain otherwise. If the multifractality is due to both types, the shuffled data will present weaker multifractality than the original data (Kantelhardt et al., 2002).

Therefore, a shuffling procedure was conducted to investigate the
types of the multifractality for Well 1 and Well 2 groundwater
levels. The corresponding multifractality results are shown in
Fig. 8. This figure clearly shows that multifractality still exists in
the shuffled groundwater level data of Well 1, since dependency
between

Multiscale multifractal analysis for

The results in Fig. 8 reveal that different types of multifractality exist in Well 1 and Well 2 groundwater level time series. For Well 1, the multifractality is clearly due to the combined effect of a broad probability density function and temporal correlations in diverse magnitudes of fluctuations. Meanwhile, the multifractality is almost purely caused by long-range temporal correlations in small and large fluctuations for Well 2 groundwater levels.

Since the Hurst exponent varies for different time intervals of the
groundwater level time series of Well 1 and Well 2 (Figs. 4, 5, and
6) and the small and large fluctuations of temporal
correlations contribute to the multifractality of both datasets (Fig. 8),
multiscale multifractal analysis (MMA) is adopted to investigate
the fractal behavior at different temporal scale ranges in detail, as
demonstrated in Fig. 9. It is noted that the generalized Hurst
surfaces for the original datasets of both Well 1 and Well 2
groundwater levels (top images in Fig. 9) are far from flat
(hill-like shape), which clearly suggests that different fractal scaling
exponents are needed to represent fractal behavior at multiple
temporal scales for both datasets. In addition, the generalized Hurst
exponents at

Histograms and normal probability plots for various time intervals of groundwater levels of Well 1.

Histograms and normal probability plots for various time intervals of groundwater levels of Well 2.

Probability density function and cumulative distribution function (first two rows and last two rows, respectively) of groundwater level fluctuation time series data of Well 1.

Probability density function and cumulative distribution function (first two rows and last two rows, respectively) of groundwater level fluctuation time series data of Well 2.

Multifractal analysis suggests that multifractality is partially due to a broad probability density distribution that may have infinite moments. However, fBm (fractional Brownian motion), which has been identified as an appropriate model to characterize groundwater level fluctuations (Li and Zhang, 2007; Little and Bloomfield, 2010; Rakhshandehroo and Amiri, 2012; Yu et al., 2016), is Gaussian with finite moments. Therefore, fBm may be inappropriate for the description of physical processes with infinite moments, such as the groundwater level fluctuations in this study. Histograms and normal probability plots for Well 1 and Well 2 groundwater levels in six selected durations of varying length apparently indicate that the Gaussian distribution may not be suitable to represent the groundwater level processes of both wells, especially for Well 1 (Figs. 10 and 11, in which the probability curve would lie on the straight red line if the data are normally distributed). Well 1 groundwater levels clearly show a heavy tail, and Well 2 groundwater levels demonstrate right-skewed behavior. As such, the Lévy alpha-stable distribution, which is non-Gaussian with a heavy tail and has infinite variance, was adopted to fit the groundwater datasets. Moreover, to obtain a relatively comprehensive picture of the underlying probability distribution, the Gaussian distribution, gamma distribution, and lognormal distribution were also used to fit the datasets (the Statistics and Machine Learning Toolbox in Matlab is used for this purpose). The fitting procedure is conducted continuously with the data starting from 1945 for Well 1 and from 1935 for Well 2 and moves forward year by year with the same end year, 2014, for all the fitting durations (at least 15 years of daily data are used to ensure a good characterization of the data distribution). Results for the six selected durations are presented in Figs. 12 and 13 for Well 1 and Well 2 groundwater levels, respectively.

Figure 12 shows that, in general, the Lévy stable distribution fits the groundwater level fluctuation time series of Well 1 over different durations very well. Meanwhile, the other distributions, i.e., normal, gamma, and lognormal, cannot satisfactorily capture the behavior of the groundwater levels of Well 1. This verifies the finding that the irregular distribution of Well 1 groundwater levels contributes to the multifractality. For Well 2 groundwater levels, Gaussian distribution adequately fits the data, except at the peak values (Fig. 13). Furthermore, the fitted stable, gamma, and lognormal distributions converge to the Gaussian distribution. This may imply that fBm may partially represent the behavior of Well 2 groundwater levels, which has the Hurst exponent fluctuating between 0.48 and 0.52 (Fig. 14b).

Furthermore, the stability index

The results indicate that fBm, which has Gaussian characteristics, may be a reasonable model for representing groundwater level fluctuations under certain conditions, such as in the case of the dataset of Well 2, which has the Hurst exponent fluctuating close to around 0.5. However, fBm may be an insufficient model for capturing the behavior of groundwater fluctuations in other cases, for example in the case of the groundwater levels of Well 1, in which a non-Gaussian distribution, such as a heavy-tailed stable distribution (Lévy motion), is needed instead. In the presence of long memory, fractional Lévy motion may be more appropriate to model and forecast the groundwater dynamics.

It is important to note that the results obtained so far are limited to the analysis of temporal correlations of the groundwater level fluctuations at certain locations. The properties of the groundwater levels at the two wells, such as their fractal behavior and underlying distributions, are highly different from each other, which confirms that the results are site specific. Well 1 and Well 2 are chosen because the groundwater level fluctuation records of these two wells are long and complete. In addition, these two wells are very representative in terms of fractal scaling behavior and underlying probability density distribution.

Groundwater dynamics in aquifers result from multiple complex dynamic processes, such as hydrologic processes (precipitation, river runoff, etc.), the hydraulic properties of soil and aquifers, and anthropogenic perturbations (such as the construction of reservoirs and pumping of water). These processes and properties vary at different spatiotemporal scales, which directly or indirectly affect groundwater systems. The two confined aquifer wells analyzed in this study are located at the same type of aquifer but present drastically different dynamics of groundwater level fluctuations. The results obtained herein may be attributed to the time–space heterogeneity of aquifer characteristics, hydrometeorological conditions, and even anthropogenic forcing, but detailed research, such as the employment of time–space analysis, needs to be conducted to justify this and to account for the effect of heterogeneity on fractal behavior at different temporal scales. The non-Gaussian fractal property of the groundwater system in Well 1 that demonstrates long memory provides further insight for the resulting transport processes in the porous medium, which may also present non-Gaussian features with memory similar to the non-Gaussian behavior that is found in the precipitation time series in other studies (Joelson et al., 2016; Lovejoy and Mandelbrot, 1985). Unlike Well 1 groundwater levels, the origin of multifractality for the Well 2 groundwater levels is difficult to explain due to the very weak multifractality after the shuffling. An intuitive explanation may be that it is due to noise. However, the fractal structure is not affected by dynamical noise (Serletis, 2008). Additionally, Gaussian distribution may partially represent the dataset of Well 2 groundwater levels, but it fails to capture the peak of the skewed distribution of Well 2 groundwater levels, which may imply that an irregular distribution that also holds certain Gaussian characteristics may be needed to fully characterize the groundwater dynamics of Well 2.

In this study, the fractal scaling properties of groundwater level
fluctuations of two confined aquifer wells with 70 and 80 years of
daily data were analyzed. Detrended fluctuation analysis (DFA) was
utilized to quantify the Hurst exponent and monofractality. The DFA
results indicated that fractals exist in the groundwater level time series
of both wells, and it was shown that the Hurst exponent is highly
dependent on the length and specific time period of the time
series. A persistent scaling pattern was found for all investigated time
periods of Well 1 groundwater levels (Hurst exponent,

Multifractal detrended fluctuation analysis (MF-DFA) and multiscale
multifractal analysis (MMA) were adopted to examine the
multifractality and multifractal behavior at different temporal scales
for confined groundwater levels. Although the MF-DFA results showed
that a relatively high level of multifractality exists for the
groundwater levels of both wells, a stronger multifractality was observed for the
dataset of Well 1 compared to that of Well 2. The observed
multifractality was postulated to originate from the combined effect
of the underlying irregular probability distributions and different
magnitudes of fluctuations in multiple long-range temporal
correlations for Well 1 groundwater levels and mostly long-range
temporal correlations in small and large fluctuations for Well 2
groundwater levels. Moreover, the MMA results confirmed the existence
of multifractality and diverse correlations of groundwater levels over
different timescales. For Well 1, the Hurst exponent by

Furthermore, the underlying probability distribution of groundwater level fluctuations for Well 1 represented mainly long-memory characteristics, which were fitted reasonably well by the Lévy stable distributions for various time periods. On the other hand, those of Well 2 represented mainly Gaussian characteristics, which sometimes failed to capture the peaks of the skewedprobability distributions of Well 2 groundwater levels. Time series analysis of groundwater level fluctuations of the two wells demonstrated that the observed fractal behavior is site specific, and there is a need for generalized governing equations of groundwater flow processes that can model both the long-memory behavior and the Brownian finite-memory behavior (Kavvas et al., 2017; Tu et al., 2017).

The data used in this article can be accessed by contacting the corresponding author.

As of May 2016, there are 257 monitoring wells, both active and inactive, reported on the web page of Water Data for Texas
(

The longest dataset has more than 81 years of record with an approximate 2.6 % missing rate, and the second-longest dataset includes more than 72 years of record with an approximate 3.6 % missing rate. The third-longest dataset has more than 10 % missing measurements and has about one-third of the length of the second-longest dataset. Therefore, the first- and second-longest groundwater level records were analyzed in this study. Record lengths (in days) and percentage of missing data for the 20 longest groundwater level records reported by Water Data for Texas are presented in Fig. A2.

It can be seen from Fig. A2 that only the first two records are of excellent data quality with respect to length and completeness. Therefore, the groundwater level fluctuations of these two confined aquifer wells are analyzed in this study. The results indicate two different behaviors of the groundwater level fluctuations, i.e., Gaussian and non-Gaussian, which are not reported or compared in previous studies on the fractal scaling analysis of groundwater level fluctuations. Therefore, the results of this behavior with respect to these two confined aquifer wells are reported in the paper. These two wells are both located at the Edwards (Balcones Fault Zone) aquifer, which primarily consists of partially dissolved limestone. However, the dynamics of the groundwater level fluctuations in these two wells behave drastically differently, which may imply rather different climate-related and anthropogenic perturbations in these two wells. Unfortunately, due to the lack of high-quality datasets and detailed information about the aquifers in this area, further discussion on the regionalization of the fractal properties is difficult.

Spatial distribution of the confined aquifer wells in Texas, USA reported by Water Data for Texas. The two red stars denote the locations of the wells that have the first- and second-longest records; yellow solid circles denote the locations of the other 75 wells.

Record lengths (in days) and percentage of missing data for the 20 longest groundwater level records reported in Texas by Water Data for Texas.

The authors declare that they have no conflict of interest.

This article is part of the special issue “Hydro-climate dynamics, analytics and predictability”. It does not belong to a conference.

The authors would like to thank the two anonymous reviewers and the handling editor, Naresh Devineni, for reviewing this article. Edited by: Naresh Devineni Reviewed by: two anonymous referees