**1. Introduction**

The Himalayan region has large river basins, and this categorizes it as having a dynamic hydrology which required scientific approaches to study the water resource management and planning at regional basis. Due to large-scale developmental activities along the river basin, the hydrology of the region is affected and hence the water quality. These development activities impact overall water balance of the region, which has a negative effect on various environmental factors like flora, fauna, soil, air, drinking water, and ultimately the human health. Water is a valuable natural resource which contains various suspended substances of organic matter and minerals.

Huge urbanization and drilling activities affect the land transformation, and hence surface water pollution impacts the quality of drinking water [1, 2]. In the indication of the water quality standards and the estimations of constituent concentrations, the laboratory processes and statistical methods are required, and worldwide researchers used the statistical analysis such as multiple linear regressions methods [3–6]. Researches emphasize to examine the physical properties of surface water, and the constituent concentrations can be used for the assessment of water quality situation or water balance with analyzing the parameter deviation and water quality standards [7–11]. Modeling of surface water quality is carried out that assesses numerical indices using earth observation datasets and laboratory methods such as X-ray diffraction technique over Allahabad district, India [12]. It is observed that the statistical methods are capable of comparing the numerical indices. Contaminated quantity separation in the groundwater samples were also studied with numerical index approach and found to be significant for water parameter characterization [13, 14]. Remote sensing and GIS-based approaches along with groundbased dataset have also been applied to study the water quality parameters [15, 16]. millions of people in South Asia. During the last few decades, the central Himalayan region is observing the cascading effects of the climate change including rise in temperature, receding of glaciers, erratic precipitation patterns, etc. [24]. In the aforementioned scenario, the Himalayan region is receiving global scientific attention for glacier and water resource studies. However, available climate models often have limitation in resolving the highly complex geographical topography in this region which directly or indirectly impacts the water balance over the region. Hence, timely and accurate relationship of water indices using statistical methods inheriting the relationship among water parameters complements the understanding of the available water in the fragile ecosystem of the state of Uttarakhand. The present study is carried out over the Tehri District of Uttarakhand. The detailed study area map (**Figure 1**) shows the location of the main rivers and drainages along the water quality parameters collected. The observations of water

*Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality*

*DOI: http://dx.doi.org/10.5772/intechopen.92865*

**Figure 1.**

**Figure 2.**

**23**

*(a) Data collection points and (b) drainage map of study area.*

*Box plot showing the water quality parameters over the study site.*

Fractal dimension (FD) and laboratory methods together have been used to study the water quality, and the indices are calculated as the weighted average of all observations of interest [17]. Fractal theory has been widely applied on diverse types of datasets in hydrology, geophysics, and climate as well as in other research areas to identify the patterns in time series datasets for describing the irregular and complex behaviors of dynamic systems [18–21]. The rainfall spatiotemporal variations are analyzed for flood seasons in China during 1958–2013 using Hurst exponent and concluded that the rainfall trends will persist in the future also having implications for the ecological restoration and farming operations [22]. Fractal approach has been applied to estimate the climatic indices for climatic variables (pressure, temperature, and rainfall) in the Himalayan foothill region [23]. Fractal dimension demonstrated significant variations from station to station with the values relatively closer to unity at high-altitude sites indicating better climate predictability than that of those over the low-altitude stations in the Himalayan foothills.

In this investigation fractal and statistical analysis is carried out to establish relationships among water parameters such as turbidity, chloride, ferrous, nitrate, pH, calcium, magnesium, fluoride, total dissolved salts (TDS), alkalinity, hardness, and sulfate and to get the significant understanding of WQI. Fractal analysis improves the understanding of WQI especially in the mountainous regions of the Himalayas where 3D models show limitations in resolving the highly complex geographical topography. Despite the advantage with statistical modeling to inherit the effects of terrain and correlated variations among various meteorological parameters, comprehensive investigations of such statistical relationships among observed water quality parameters are lacking over the Himalayan foothills and needs to be studied in detail.

This study intends to carry out the statistical and fractal analysis of groundwater parameters to establish the relationship among the various indices and to understand the behavior of water quality indices (WQI) with the predictability index (PI), Hurst exponent (H), and fractal dimension (D). This study may offer the basic understanding of the WQI of different water parameters regarding the regional hydrogeochemical processes with the laboratory testing methods.

#### **2. Study region and observational dataset**

The state of Uttarakhand located in the lap of the central Himalayan region has been identified as a hotspot of anthropogenic stress and one of the most vulnerable regions for climate-mediated risks. The region provides water resources supporting

### *Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality DOI: http://dx.doi.org/10.5772/intechopen.92865*

millions of people in South Asia. During the last few decades, the central Himalayan region is observing the cascading effects of the climate change including rise in temperature, receding of glaciers, erratic precipitation patterns, etc. [24]. In the aforementioned scenario, the Himalayan region is receiving global scientific attention for glacier and water resource studies. However, available climate models often have limitation in resolving the highly complex geographical topography in this region which directly or indirectly impacts the water balance over the region. Hence, timely and accurate relationship of water indices using statistical methods inheriting the relationship among water parameters complements the understanding of the available water in the fragile ecosystem of the state of Uttarakhand.

The present study is carried out over the Tehri District of Uttarakhand. The detailed study area map (**Figure 1**) shows the location of the main rivers and drainages along the water quality parameters collected. The observations of water

**Figure 1.** *(a) Data collection points and (b) drainage map of study area.*

**Figure 2.** *Box plot showing the water quality parameters over the study site.*

Huge urbanization and drilling activities affect the land transformation, and hence surface water pollution impacts the quality of drinking water [1, 2]. In the indication of the water quality standards and the estimations of constituent concentrations, the laboratory processes and statistical methods are required, and worldwide researchers used the statistical analysis such as multiple linear regressions methods [3–6]. Researches emphasize to examine the physical properties of surface water, and the constituent concentrations can be used for the assessment of water quality situation or water balance with analyzing the parameter deviation and water quality standards [7–11]. Modeling of surface water quality is carried out that assesses numerical indices using earth observation datasets and laboratory methods such as X-ray diffraction technique over Allahabad district, India [12]. It is observed that the statistical methods are capable of comparing the numerical indices. Contaminated quantity separation in the groundwater samples were also studied with numerical index approach and found to be significant for water parameter characterization [13, 14]. Remote sensing and GIS-based approaches along with groundbased dataset have also been applied to study the water quality parameters [15, 16]. Fractal dimension (FD) and laboratory methods together have been used to study

the water quality, and the indices are calculated as the weighted average of all observations of interest [17]. Fractal theory has been widely applied on diverse types of datasets in hydrology, geophysics, and climate as well as in other research areas to identify the patterns in time series datasets for describing the irregular and complex behaviors of dynamic systems [18–21]. The rainfall spatiotemporal variations are analyzed for flood seasons in China during 1958–2013 using Hurst exponent and concluded that the rainfall trends will persist in the future also having implications for the ecological restoration and farming operations [22]. Fractal approach has been applied to estimate the climatic indices for climatic variables (pressure, temperature, and rainfall) in the Himalayan foothill region [23]. Fractal dimension demonstrated significant variations from station to station with the values relatively closer to unity at high-altitude sites indicating better climate predictability than that of those over

In this investigation fractal and statistical analysis is carried out to establish relationships among water parameters such as turbidity, chloride, ferrous, nitrate, pH, calcium, magnesium, fluoride, total dissolved salts (TDS), alkalinity, hardness, and sulfate and to get the significant understanding of WQI. Fractal analysis improves the understanding of WQI especially in the mountainous regions of the Himalayas where 3D models show limitations in resolving the highly complex geographical topography. Despite the advantage with statistical modeling to inherit the effects of terrain and correlated variations among various meteorological parameters, comprehensive investigations of such statistical relationships among observed water quality parameters are lacking over the Himalayan foothills and needs to be studied in detail.

This study intends to carry out the statistical and fractal analysis of groundwater parameters to establish the relationship among the various indices and to understand the behavior of water quality indices (WQI) with the predictability index (PI), Hurst exponent (H), and fractal dimension (D). This study may offer the basic understanding of the WQI of different water parameters regarding the regional

The state of Uttarakhand located in the lap of the central Himalayan region has been identified as a hotspot of anthropogenic stress and one of the most vulnerable regions for climate-mediated risks. The region provides water resources supporting

hydrogeochemical processes with the laboratory testing methods.

**2. Study region and observational dataset**

**22**

the low-altitude stations in the Himalayan foothills.

*Fractal Analysis - Selected Examples*

quality parameters used for the analysis are obtained through the website of the Ministry of Drinking Water & Sanitation, New Delhi, India.

self-similarity within themselves which was coined by Benot Mandelbrot in 1975. Fractals are characterized by self-similarity property having similar characteristics when analyzed over a large range of scales, and individually a single entity will have similar characteristics to that of the whole fractal [26, 27]. Fractal dimension estimation from a fractal set has various methods due its simplicity and automatic computability. The box counting is one of the major categories of fractal analysis and the most used technique to analyze image features such as texture segmentation, shape classification, and graphic analysis in many fields [28, 29]. The variance and spectral methods are two other major categories of fractal dimension analysis of a time series that recognize the determinism and randomness in data [30]. To study the naturally complex features such as

*Fractal Analysis for Time Series Datasets: A Case Study of Groundwater Quality*

*DOI: http://dx.doi.org/10.5772/intechopen.92865*

coastlines, river boundaries, mountains, and clouds, the fractal dimension analysis has also provided a mathematical model as a fractal geometry [31, 32]. The glacial and fluvial morphologies are distinguished by using an automated

approach (i.e., multifractal). In previous study, a multifractal detrended fluctuation analysis (MFDFA) has been carried out to estimate the variation of elevation profile of glacial and fluvial landscapes [33]. It has been observed that glacial landscapes reveal more complex structure than that of the fluvial landscapes as indicated by fractal parameters, such as degree of multifractality, asymmetry index, etc. The basic definition of fractal dimension is the Hausdorff dimension; however, box counting or box dimension is another popular definition which is

Hurst exponent (H) is used as a measure of long-term memory of time series and a real-valued time series defined as the exponent in the asymptotic scaling relation [30, 34]. The Hurst exponent and fractal dimension are also directly related to each other and indicate the roughness of a surface. The Hurst exponent's value lies in a time series as persistent (0.5 < H ≤ 1) or anti-persistent (0 ≤ H < 0.5), and when the data are not intercorrelated, then H = 0.5 which implies that the series is unpredictable. This approach is used in various complex engineering fields as it

In terms of asymptotic scaling relation, the Hurst exponent of real-valued time

where C is a constant, angular brackets h i ⋯ denote expected value, S(n) is the standard deviation of the first "n" data of the seriesf g *X*1, *X*2, ⋯, *Xn* , and R(n) is

*R n*ð Þ¼ max f g *X*1,*X*2, ⋯,*Xn* � min f g *X*1, *X*2, ⋯,*Xn*

If f(t) is a self-affine random process, "t" a position parameter (time or distance), a > 0 is a scale (dilatation) parameter, w(t) is a mother wavelet, and

be computed from wavelet method for the time seriesf g *X*1,*X*2, ⋯, *Xn* .

The Hurst exponent H is calculated from rescaled range technique and can also

*S n*ð Þ <sup>¼</sup> <sup>C</sup>*n<sup>H</sup>*, as'*n*'approaches to infinity (1)

easy to calculate.

series is defined as:

their range:

**25**

**3.3 The Hurst exponent**

provides statistical self-similarity relationship.

*R n*ð Þ

*3.3.1 Estimate of the Hurst exponent: Wavelet approach*

In this study, we compute the Hurst exponent, fractal dimensions, and the predictability index (PI) of water quality parameters such as (1) turbidity, (2) chloride, (3) ferrous, (4) nitrate, (5) pH, (6) calcium, (7) magnesium, (8) fluoride, (9) total dissolved salts **(**TDS), (10) alkalinity, (11) hardness, and (12) sulfate, at high-altitude Tehri stations in the Himalayan foothills using the fractal theory. **Figure 2** shows the box plot of all the aforementioned 12 water parameters obtained for the study site. The irregular pattern in the WQI can be used in prediction purposes by analyzing its dynamic flow (i.e., chaotic, random, or deterministic structural pattern). Proper identification, classification, and mapping of water parameters of high-intensive and complex nature require frequent monitoring of these datasets especially in the context of drinking water.
