In this study, five different ground water resources located in the vicinity of Yalova (at three study areas; Taşköprü, Çiftlikköy and Altınova) in Turkey were investigated (Fig. 1).The content of potassium (K+), sodium (Na+), calcium (Ca+2) and magnesium (Mg+2) were examined for the relation to amount of rainfall. This work was accomplished analyzing five different ground water samples every month in the period of over all the 10 years (June 2010-August 2020).
In the research area, annual mean temperature is 23.6 0C and annual mean of precipitation is 808.4 mm year−1. In the long period years (1975-2020), total annual potential evaporation is 540.5 mm water year−1 and total annual real evaporation of research area is 476.9 mm water (TSM, 2019; TSM, 2020).
Standard sampling methods were used in the work, and the samples were analyzed according to standard methods (APHA, 1985). Chemical analysis were used of potassium (K+), sodium (Na+), calcium (Ca+2) and magnesium (Mg+2).
Having received considerable attention since it was theoretically developed, the wavelet analysis is a sophisticated tool employed in signal processing (Grossmann and Morlet, 1984). As an alternative to the Fourier transform in the preservation of local, non-periodic, and multi-scaled phenomena, it has been increasingly used in communications, image processing and optical engineering applications. Wavelet transforms, which can generally be used in exploring, denoising, and smoothening time series, are of help in forecasting and other empirical analyses (Sehgal, et.al. 2014; Sang, et.al. 2015; Shafaei, 2016).
In wavelet analysis, which breaks up a signal into shifted and scaled versions of the original (or mother) wavelet, the signal-cutting problem is solved through the use of a fully scalable modulated window. By shifting the window along the signal, the spectrum for each position is calculated. After repeating this process many times using a slightly shorter (or longer) window for each new cycle, a collection of time-frequency representations of the signal, all of which will have different resolutions, will be obtained. This collection of representations allows us to speak of a multi-resolution analysis. Both the dominant modes of variability and how those modes vary in time can be determined through the decomposition of a time series into time-frequency-space. Proven to be a powerful tool for the analysis and synthesis of data from long memory processes, wavelets are strongly associated with such processes as the same shapes repeat in different orders of magnitude. By simultaneously localizing a process in time and scale domain, wavelets enable the representation of many dense matrices in a sparse form (Ramana et al. 2013; Parmar and Bahrdwaj, 2012; Sing and Rashmi, 2014).
A sufficient condition for f(t) to qualify as a mother wavelet is given as below (Meyer, 2000; Siddiqi, 2010; Dökmen and Aslan, 2013; Nourani, 2015; Daubechies, 1992).
The Fourier transform F of f(t) is defined as
A function ψ(t) satisfying the following condition is called a continuous wavelet:
Here a is a scaling parameter, b is a location parameter and ψa,b(u) is often called continuous wavelet (or daughter wavelet) while ψ(u) is the mother wavelet. For 1D continuous wavelet analyses the Mexican hat wavelet `upside-down', with a central trough (top of the hat) and two symmetric bumps on either side (the curled rim) was used in (Fig. 2).
In this paper as a wavelet function, f(t) temporal variations of potassium (K+), sodium (Na+), calcium (Ca+2) and magnesium (Mg+2) have been considered.
Continuous wavelet transform using Db Wavelet at level 8 have been analyzed for each water parameter. Continuous Wavelet analyses analysis decomposes each data in 10 parts namely, s, a3, d1, d2, d3, d4, d5, d6, d7, d8. The first part “s” represents signal or raw data, second part “a3” correspond to amplitude of signal for Db Waelet at level 3. d1.d8 represents details of signal or raw data at eight different levels. 1D Continuous wavelet analysis has three parts; first part on top represents the analyzed signal or raw data, second part contains the scalogarms value.