1. INTRODUCTION:

Rivers are the life line of any country because River water gives life to its aquatic biological organisms, other animals and humans directly or indirectly. A number of studies are continuing on hydrochemistry and pollution of Rivers. The impact of hydrological conditions on biological community is studied by several researchers [1]. In India maintaining the River water quality is one of the greatest challenges because the urbanization and industrialization are affecting River water quality a lot. The River Ramganga is the main tributary of River Ganga and navigates through the large population of Uttarakhand and Uttar Pradesh. The data of dissolved oxygen, biochemical oxygen demand and total coliform are imported from website of Uttar Pradesh Pollution Control Board belonging to station Kannauj (code-1064) from month October 2015 to June 2020 and extended up to April 2021 through stationary wavelet transform.

Dissolved Oxygen (DO) is the amount of gaseous oxygen (O₂) which enters the water by direct absorption from the atmosphere, by rapid movement, or as a waste product of plant photosynthesis. This dissolved oxygen is the base of living of aquatic animals because they are unable to split oxygen molecule of water or other oxygen contained compounds. The rapidly moving water has more DO than stagnant water. Generally, the quantity of DO is inversely proportional to the temperature of River water. Biological oxygen demand (BOD) is the amount of DO required by biological organisms to break the organic matter present in water at a temperature in a specific time interval. Commonly, it is expressed in milligram of oxygen consumed per litre of water sample in 5 days of incubation at temperature 20⁰C [2]. It generally defines the organic pollution of water. Coliforms are commonly used to indicate the sanitary quality of water. Coliforms are frequently found in aquatic environment and faeces of warm blooded animals. Generally, the coliforms are not harmful but they indicate the presence of other harmful biological organisms like bacteria, viruses, protozoa and many multicellular parasites of fecal origin. The total coliform (TC) in water indicates that the water is contaminated and harmful biological organisms may be present. The fecal coliforms are the part of total coliforms and may cause serious diseases.

Fourier transforms is the inner product of a signal and exponential function. It provides frequency resolution of any signal but the time resolution is lost. Another drawback of Fourier transform is that this is unable to analyse the non-stationary and transient signals. The window Fourier transforms is the inner product of any signal with a window function for which that non-stationary signal can be treated as stationary. The resolution limit for window function is constant. The resolution in time and frequency can not be arbitrary small, because their product is lower bounded and restricted by Heisenberg uncertainty principle related to the time bandwidth product. The Fourier and window Fourier transforms are not sufficient to analyze non-stationary and transient signals. Wavelet transform is the inner product of a signal with wavelets which are generated by a single function called mother wavelet. The wavelet transforms of a signal captures the localized time frequency information of a signal. A wavelet is a wave like oscillation localized in the sense that it grows from zero reaches maximum and decreases back to zero. Thus it has a location where it maximizes, a characteristic time period and a scale where it amplifies and declines. The wavelet has become a new analytical tool for analyzing chaotic data to the Physicists, Mathematicians and Engineers. It allows detection and characterization of short-lived structures in data [3].

2. WAVELET BASICS:

A mother function is used to generate a whole family of wavelets using dilation and translation:

where is the dialation or scaling parameter, is the translation parameter and (t) is real valued. The collection of wavelets is used as orthonormal basis. The continuous wavelet transform of a function is defined as:

By taking and with , that is, the integers representing the set of discrete dilation and discrete translation, the discrete wavelet transform is defined as:

and discrete wavelets are defined as [4]:

2.1 Daubechies Wavelet:

I. Daubechies invented compactly supported orthonormal wavelets so that the discrete wavelet analysis has become more practicable [5]. The names of the Daubechies family wavelets are written as dbN, where N is the order and db represents to Daubechies. The db1 wavelet is the same as Haar wavelet. The db4 scaling function and wavelet are as following:

Figure 1: Daubechies4 (db4) scaling function and wavelet

2.2. Multiresolution Analysis:

A multiresolution analysis is introduced by Mallat [6-7] consists of a sequence , of closed subspaces of . The function is called scaling function of given MRA and a dilation equation is as following:

where is low pass filter and is defined as:

The wavelet functionis expressed as:

where, =is high pass filter (Kumar 2017).We can express a function in spaces as following:

Since

Where

And

Figure 2: Division of vector subspace

2.3 Decomposition of signal:

With help of MRA, any function or signal can be first order decomposed as:

By second order decomposition, it can be expressed as:

In general, for order decomposition, a signal can be expressed as,

Where p represents to the order of decomposition of signal or level of the wavelet transforms. Here,

are collectively known as approximation and detailed coefficients [8-9]. Thus a given signal takes place a new version such as,

Where

And

Here is approximation and is detail of signal at ^th level or time frames. Taking , i.e. we can write,

A signal can be decomposed as in simplest form (level 1):

Taking we can write,

2.4 Stationary wavelet transforms:

The stationary wavelet transforms (SWT) reconstructions result contains lower error values and faster convergence compared to discrete wavelet transforms (DWT). The SWT thresholding provides a translation-invariant basis [10]. For SWT, a redundant decomposition can be obtained as:

where ∈ {0, … . − 1} allows for all the possible shifts in a discrete setting. For decomposition to levels, 2 different orthogonal bases can be generated. Each node in binary tree is indexed by parameters (𝑗, ), to which the set of coefficients is associated. Each path from the root of the tree to a leaf corresponds to the set of functions,

where , form an orthogonal wavelet basis, resulting in a standard wavelet reconstruction [11].

3. RESEARCH METHODOLOGY:

The data of dissolved oxygen, biochemical oxygen demand and total coliform are imported from website of Uttar Pradesh Pollution Control Board (UPPCB) belonging to station Kannauj (code-1064) from month October 2015 to June 2020 and taken as input signal in the wavelet toolbox of software MATLAB. This signal is extended up to April 2021 using stationary wavelet transform, Daubechies4 (db4) wavelet, level-6 [12-13]. The signal in extended form is again decomposed into approximation and detail using the same wavelet. The highest scale value (lowest resolution) represents trend of the signal, it is the also the slowest part of the signal. That is, the highest approximation represents to the average behaviour of the signal and describes the coarse information of a signal.

The data analysis is performed by statistical parameters like average, skewness, kurtosis and standard deviation of the original and extended signal, while the spectral analysis is performed by wavelet transforms. The skewness represents the asymmetry of the signal about the mean value, kurtosis represents the peakedness of any signal, while standard deviation represents the spreading of data about the mean value [14]. The statistical results and wavelet analytical results are compared and discussed.

4. RESULTS AND DISCUSSION:

In figure 3, 5 and 7, the quantitative behaviour of observed DO, BOD and TC is confined in the red rectangle, while the behaviour of data in extended form is confined in the yellow rectangle. The extended signal is obtained using stationary wavelet transforms, wavelet db4, level-6.

Figure 3: Original and extended behaviour of dissolved oxygen

Figure 4: Trend of the dissolved oxygen

Figure 5: Original and extended behaviour of biochemical oxygen demand

Figure 6: Trend of the biochemical oxygen demand

Figure 7: Original and extended behaviour of total coliform

Figure 8: Trend of the total coliform

As the scale of approximation increases (Resolution decreases) we move towards a better estimate of the unknown signal. At the highest scale the average behaviour, that is, the trend of the signal is obtained. Figure 4, 6 and 8 represent the trend of the DO, BOD and TC of the Ramganga River from October 2015 to April 2021. The statistical parameters of original and extended signal are enlisted in the table 1.

Table 1: Statistical parameters of original and extended signal

S. No.	Statistical parameter	Original signal			Extended signal
S. No.	Statistical parameter	DO	BOD	TC	DO	BOD	TC
1	Average	8.061404	4.850877	62.33333	8.032813	4.885938	62.875
2	Skewness	-0.20571	-0.04672	3.291132	-0.20602	-0.13413	3.29877
3	Kurtosis	-1.24133	-0.20177	15.6638	-1.17752	-0.01567	16.47949
4	Standard Deviation	1.751811	1.023147	25.11	1.701732	0.979774	23.96591

The average value of DO is slightly decreasing while of BOD and TC slightly increasing. It indicates that in the future the pollutants in the Ramganga River are going to be slightly increased. The value of skewness for DO and BOD are negative and for TC is positive. The negative and positive values of skewness indicate that data are skewed left and right respectively. The kurtosis of the DO and BOD are negative increasing with time while of TC it is positive and also increasing. The skewness represents the intermittency of the data. The standard deviation for all DO, BOD and TC is slightly decreasing, which represents the slightly less spreading of data about mean. The statistical results are in the favour of quantities behaviour trend of Ramganga River water quality parameters of observed and extended signal as per spectral analysis of wavelet transforms.

5. CONCLUSION:

Slightly decreasing trend of DO and slightly increasing trend of BOD and TC in Ramganga River water are obtained from statistical and spectral analysis by stationary wavelet transforms. Our results indicate that there is a slight increment in pollutants in the Ramganga water with time. The wavelet transforms provides a simple and accurate framework for modelling the quantitative behaviour of the observed data and extended data of River water quality. Obviously, the statistical analysis provides strong agreement with the wavelet spectral analysis.

6. REFERENCES:

1. JJ Lin, Characterization of water-soluble ion species in urban ambient particles, Environment International. 2002; 28: 55–61.

2. S Kumar, A Kumar, R Kumar, JK Pathak and M Alam, Spectral analysis of bio-chemical oxygen demand in River water: An analytical approach of discrete wavelet transform, American Journal of Mathematics and Statistics, 2014; 4(2): 107-112.

3. JC Goswami and AK Chan, Fundamentals of Wavelets Theory, Algorithms and Applications, Willey-Interscience, New York, 1999.

4. JP Antoine, Wavelet analysis: A new tool in Physics, Wavelets in Physics, JC Van Den Berg, ed., Cambridge University Press, Cambridge, 2004, pp. 9-21.

5. I Daubechies, Ten lectures on wavelets, CBS-NSF, Regional Conference in Applied Mathematics, SIAM Philadelphia, 1992; 61: 278-285.

6. SG Mallat, A Wavelet Tour of Signal Processing, Academic Press New York, 1998.

7. SG Mallat, A theory for multiresolution signal decomposition: The Wavelet Representation, IEEE Trans. On Pattern Analysis and Machine Intelligence, 1989; 11: 674-693.

8. A Cohen, I Daubechies and P Vail, Wavelets on the interval and fast algorithms, Journal of Applied and Computational Harmonic Analysis, 1993; 1: 54-81.

9. RR Coifman and MV Wickerhauser, Entropy based algorithms for best basis selection, IEEE Trans. Inform. Theory, 1992; 38 (2): 713-718.

10. GP Nason, and BW Silverman, The stationary wavelet transform and some statistical applications, Wavelet and Statistics: Lecture Note in Statistics, Springer, New York, 1995; 103: 281-299.

11. H Ye, G Wang and XS Ding, A new parity space approach for fault detection based on stationary wavelet transform. IEEE Transactions on Automatic Control, 2004; 49; 281-287.

12. H Wong, W Ip, Z Xie and X Lui, Modeling and forecasting by wavelets, and the application to exchange rates. Journal of Applied Statistics, 2003; 30(5): 537-553.

13. L Xiaohong, Y Shujuan and L. Li, Prediction of temperature time series based on wavelet transform and support vector machine. Journal of Computers, 2012; 7(8): 1911-1918.

14. M Rockinger and E Jondeau, Entropy densities with an application to autoregressive conditional skewness and kurtosis. Journal of Econometrics, 2002; 106(1): 119-142.

Receivedon 15.08.2020 Acceptedon 24.09.2020

Int. J. Tech. 2020;10(2):122-128.

DOI: 10.5958/2231-3915.2020.00023.1