The slope
1. Introduction
Electrochemical noise (EN) is the term used to describe the spontaneous fluctuations of current or potential, which are generated during the corrosion processes. It has been investigated extensively since the 1968, and data has shown the use of EN measurements offers valuable sources of information about complex electrochemical reactions such as those in corrosion systems (Gabrielli et al., 1985; Bertocci & Huet, 1995).
Many methods can be used to analyze the data, such as the variance, standard deviation and root mean square in the time domain. Alternatively, the signal has been transformed from the time domain to the frequency domain using fast Fourier transform or the maximum entropy method, giving the power spectrum density (PSD). The technique of wavelet analysis may be used instead, where a set of wavelets of varying amplitude, duration, and location be constructed such that reproduces the signal of interest.
Wavelets based methods are modern mathematical tools for multiscale time frequency analysis and characterization of in general nonstationary EN signals. This work presents the short overview to usability and possibilities of wavelet transformation in comparison with classic analysis.
2. Corrosion processes and electrochemical noise
Corrosion can be defined as the deterioration of materials due to its interaction with its environment and is appearing in various forms: as localized corrosion and as general (uniform) corrosion. Localized corrosion results in the accelerated loss of material at discrete sites in a passive materials surface, leading to the perforation or other failure. Since the majority of the surface is unaffected the difficulty with localized corrosion is its detection and prediction. Uniform corrosion results from the sites that are distributed randomly over the surface regarding both the space and time. Uniform corrosion damage is manifested in dissolving and the progressive thinning of a metal. To prevent corrosion or to predict the outcome of a corrosion situation the knowledge of mechanism of various form of corrosion is fundamental. Thermodynamic principles can be applied to determine which processes can occur and how strong the tendency is for changes to take place. Kinetic laws then describe the rates of the reactions.
Corrosion of metals in aqueous environments is electrochemical in nature. It occurs when two or more electrochemical reactions take place on a metal surface, producing dissolved species of metal or solid corrosion products and thus lowering the energy of the system. The corrosion process has been written as two separate reactions occurring at two distinct sites on the same surface: the anode (metal dissolution site) and the cathode (site of the accompanying reduction reaction). At corrosion of iron in an acid solution containing dissolved oxygen, iron is oxidized to ferrous ion which passes into solution in the anode region
and H^{+} ion is reduced to hydrogen or O_{2} is reduced to water in the cathode region
The overall corrosion reaction is the sum of the anodic and cathodic partial reactions:
sum of the reactions (1) and (2)
and sum of the reactions (1) and (3)
Electrochemical corrosion processes can be investigated by observation of charge flows between the electrolyte and the corroding metal. The measure of the rate at which reducible or oxidizable species can gain or lose electrons is the current density,
where
where
In measurements procedure a potentiostat is often used and the electrochemical cell with tree electrodes: working electrode (WE) represents the interface of interest, the reference electrode (RE) acts as standard for potential measurements and the counter electrode acts as electron sink or source for reactions that occur on the surface of WE. The potential of the WE is controlled with respect to the RE at a constant value, and the current density
In our investigations the current noise was monitored by a lownoise batteryoperated potentiostat Jaissle IMP88 PCR at a sampling rate of 10 points per second. The cell assembly was put in a Faraday cage. An example of measured time series for two processes (pitting as signal
3. Stochastic processes and 1/f noise
ENsignals are generated from corrosion processes which are stochastic in their nature. A short theoretical overview of stochastic processes follows (Schroeder, 1991; Flandrin, 1992; Gao et al., 2007).
For understanding the stochastic processes it is essential to understand the concept of probability which is associated with random events. Often random events are presented by numbers, called a random variable. Let us denote a random variable by
The mean or average of
The p^{th} order moment of
The second moment represents the power of a random variable
The p^{th} order central moment of
Variance of
For a a given sample space
The activity of complex systems can usually be characterized by appearance of
where
the variance
and autocorrelation
By proving that
where
An example of selfaffine stochastic process is fractional Brownian motion process (fBm). It is nonstationary zero mean Gaussian process denoted as
where
Although fBm process is nonstationary, it has stationary increments, which means that the probability properties of the difference process
The other example of
It was reported, that
This is the reason for studying
Let
4. Classical statistical and Fourier analysis methods
In the past, the most common ENanalysis methods were statistical and Fourier methods. These methods assume the stationary or quasi stationary nature of processes and signals under consideration.
An early overview of different ENdata analysis methods was made in the work by R. A. Cottis (Cottis, 2001). Follows a little extended theoretical overview of classical methods (Orfanidis, 1996).
4.1. Background of statistical and Fourier methods
By analyzing random processes the statistical parameters as mean value and moments are defined by expectation operators, i.e. by statistical averaging of many realizations of stochastic process. In practice this is many times impossible and there is available only one block or array of
The p^{th} moment of sample
The first moment is mean value
The p^{th} central moment of sample is defined as:
The second order central moment is the well known signal variance. The square root of variance is standard deviation, which is usually used for describing the amplitude of noise signals.
Classical spectral analysis bases on Fourier transform. The Fourier Transform of the deterministic continuous time signal
The Discrete Time Fourier Transform (DTFT) of the deterministic sampled signal with
where
where
where
EN signals are of stochastic nature; therefore sampled EN signals are random sequences. To obtain smooth spectra an ensemble averaging should be introduced and the spectrum calculated over autocorrelation function. The autocorrelation function of a zero mean random signal is defined as:
where
where
EN measurements cannot often be repeated to obtain smoothed spectra by ensemble averaging. One can compute an estimate of expected or true value by socalled sample autocorrelation using time average:
for
The DTFT of
Using the above equations we can express the periodogram also as:
where
There are some problems with such classical Fourier spectral analysis method. To achieve high statistical reliability, very long signal sequences should be used. But long signal sequences can no longer be stationary. However, the main shortcoming is the averaging the futures over the whole time domain.
This have lead researchers to find and develop of an advanced signal analysis methods. Recently wavelet based methods for signal analysis found to be useful for nonstacionary signals. Therefore in this overview chapter we will consider waveletbased methods for ENsignals analysis.
4.2. Overview of works using classical methods
In individual systems, the correlations between noise measurements and corrosion processes have been reported by many authors but only some can be mentioned here. The EN data for a passive system (SS 316L/Ringer′s solution) and several active systems (mild steel/NaCl, brass/NaCl, Al 6061/NaCl and Al 2024/NaCl) have been analyzed in the frequency domain using power spectral density (PSD) and spectral plots, obtained from the ratio of PSD plots of the potential and current fluctuations. Comparisons of spectral noise spectra with traditional impedance spectra have been made and good agreement has been observed for all systems after trend removal (Lee & Mansfeld, 1998; Mansfeld et al., 2001). Current fluctuation during general corrosion was analyzed upon a simple model, derived on the assumption that elementary fluctuation sources are related to the fluxes of electrons that are transferred from the metal to electronacceptor ions in solution. The number of successful electron transfers obeyed a Gaussian distribution, from which the corrosion current density and transfer coefficients were determined (Petek et al., 1997; Petek & Doleček, 2001). The timeseries noise patterns of the steel in bicarbonate solution (the simulated geological environment) were transformed into frequency domain by fast Fourier transformations, and then their power spectral densities at a frequency were determined to be compared with the corrosion rate (Haruna et al., 2003). Two new indices (S_{E} and S_{G}) were derived to evaluate pitting corrosion by dimensional analysis of three parameters of PSD, the slope of high frequency linear region, the critical frequency and the lowfrequency plateau level. As shown, the value of S_{E} can be related to the fluctuation velocity, which can represent the pitting corrosion rate and S_{G} should contain some information about slow corrosion processes (Shi et al., 2008). PSD had been employed to analyze EN data associated with corrosion behavior of A291D magnesium alloy in alkaline chloride solution. Three corrosion stages, the anodic dissolution process companying with the growth, absorption and desorption of hydrogen bubbles, the development of pitting corrosion, and the inhibition process by protective MgH_{2} film could be distinguished. However, the results obtained only from PSD was insufficient for better understanding the corrosion mechanism of alloy during the immersion and the wavelet transform was carried out (Zhang et al., 2007).
4.3. Our applications of classical methods
EN signal (Fig. 1) is represented as a time series, where one can easily distinguish the fluctuations but not the intensity and frequencies of fluctuations. In the paper (Planinšič & Petek, 2003) we analyzed EN corrosion signals also with some classical methods, which use correlation functions and histograms. Figure 2 shows estimated autocorrelation functions of two corrosion signals
Noise data were transformed into frequency domain using FFT algorithm and presented as PSD in Figure 3. PSD of current noise data for pitting process exhibited two parts: a lowfrequency plateau and highfrequency part, and the rolloff frequency, which is the frequency to separate the two parts of PSD. PSD plot of general corrosion can be characterized by “white noise” which is independent of frequency.
Amplitude distribution was studied using normalized histograms. As demonstrated by Figure 4, a current noise amplitude distribution of general corrosion is Gaussian.
5. Wavelet multiresolution analysis methods
The assumption of stationary behavior of corrosion processes and random signals is not always correct. Corrosion signals are a non stationary in general.
When we are interested on how signal frequency components vary with time, we should use joint timefrequency analysis. For this purpose we can use Window Fourier Transform (WTF), also called Short Time Fourier Transform (STFT) or spectrogram. It is known that the STFT can be considered as the filter bank, consists of Finite Impulse Response Filters (FIR) with equal bandwidth or equal frequency resolution. Therefore it is difficult to meet sharp localization in time and frequency simultaneously. For this reason, this technique is not always appropriate for analyzing natural signals or phenomena, where in the signal exist long duration low frequency components and short high frequency components at the same time. This problem can be elegantly solved using modern multiresolution time frequency analysis methods based on wavelets. It was shown that the Discrete Wavelet Transform can be viewed and realized as multirate filter bank with octave, also called constant
A short theoretical overview of wavelet methods follows (Daubechies, 1992; Burrus, 1992; Fladrin,1992, 1993; Radolphe, 1994; Wornell, 1996; Mallat, 1998; Dai et al., 1994; Palawajjhalla et al., 1994).
5.1. Background of wavelet methods
Wavelets are waves which construct basis of signal decomposition in wavelet transforms, similar as trigonometric functions with different frequencies in Fourier Transform. Wavelets are scaled and shifted versions of the so called mother or primary wavelet function
where parameters
where
Admissibility implies that a wavelet must be an oscillatory decaying function with zero mean. There are also additional other desirable properties for a function to be a useful wavelet, as smoothness, good time and frequency localization, number of vanishing moments. These properties suggest that wavelets are bandpass filters.
In the contrast with Fourier analysis where basis functions are trigonometric functions, by waveletbased analysis different kind of mother wavelet function can be used, appropriate for particular application. There are many types of wavelet transforms according to input signals, time and scaling parameters, used wavelet functions, namely continuous, discrete, biorthogonal and semiorthogonal and orthonormal bases version.
However wavelet transform can be broadly classified into Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT). CWT of a function
Discrete wavelet transform involves discretization of parameters,
where
are called frames (Daubechies,1992) and form Riesz basis. Discrete wavelets can be further classified into orthogonal, semiorthogonal or nonorthogonal.
To obtain orthonormal basis, one can chose samples on dyadic grid (base 2):
Orthonormal bases and orthonormal wavelet transform, play an important role in theory and practice of multiresolution analysis. The DWT can be further classified into Wavelet Series Transform (WST), when analyzed signal is continuous (
where
where the finite sequence
The multiresolution analysis form the theoretical basis for fast Discrete Wavelet Transform (fast DWT), using discrete signals
where
For ENsignal originating from corrosion process, wavelet transform decompose it into approximation and detail signal components at different scales and locations. The wavelet transform is therefore convenient tool to analyze the selfsimilarity of 1/f time series.
The orthonormal wavelet transform based methods were used for estimating slope
where
5.2. Overview of works using wavelets
Wavelets have found many applications in different natural scientific disciplines, among them also in chemical engineering (Radolphe et al., 1994; Banjanin et al., 2001). The use of wavelets to study electrochemical noise transients was reported by Aballe (Aballe et al., 1999; 2001). The wavelet analysis of electrochemical noise signals, where the signal was decomposed into waveletsubbands was used for the characterization of pitting corrosion intensity (Smulko et al., 2002). Wharton et al. demonstrated how the wavelet variance exponent can be used to evaluate corrosion behavior for variety of stainless steels in chloride medium, i.e. be able to discriminate between various corrosion processes covering a wide range of EN signals (Wharton et al., 2003). Wavelet analysis based on the fractional energy contribution of smooth crystals and the lowest frequency detail crystal can provide information on the type and onset of corrosion (general corrosion, metastable pitting, stable pitting) in performed potentiostatic critical pitting temperature test for a superduplex stainless steel (Kim, 2007). In study of the copper anode passivation by electrochemical noise analysis using wavelet transforms it has been found that during active dissolution the electrode surface is dominated by long time scale process and the change of the position of the maximum relative energy from D7 to D8 could be an indication of future passivation (Lafront et al., 2010). It was shown, that electrochemical potential noise analysis of CuBTA system using wavelet transformation can be used to achieve the inhibition efficiency (Attarchi et al., 2009).
Some other authors also reported about the fractal nature of corrosion processes and corresponding electrochemical noise signals. The electrochemical potential and current noise originating from the corrosion of carbon steel in distilled water was analyzed using multifractal analysis. The multifractal spectra are found to be qualitatively different for different temporal stages of the corrosion process (Muniandy et al., 2011).
5.3. Our applications of wavelets methods
Our applications of wavelets transformation or combination with classical methods for the electrochemical current noise analysis were reported for different corrosion processes in several publications (Planinšič & Petek, 2003; 2004; 2007; 2008). For little more detailed insight the short overview of this research is as follows.
Daubechies wavelets ‘’db2’’ were used to transform the EN signal from Fig. 1. The discrete wavelet transform (DWT) decomposition of signal into on joint time (position) and frequency (scale level) depended amplitudes are presented with color lightness in timefrequency plane in Fig. 6.
Next, the DWT multiresolution decomposition of processes on 5 levels are shown in Fig. 7. The crystals from
Events with small time constants are taken into account by the fine scale coefficients, details
Variances of details were calculated to detect the intensity of particular signal components on level
For the time series
After short computation the value for Hurst parameter
The obtained slope
The obtained results indicating the presence of fractional Brownian motion in pitting corrosion with adjacent values in the time series being strongly correlated and fractional Gaussian noise in general corrosion, with adjacent values in the time series being weak correlated. The Hurst parameter in case of pitting is greater than ½, indicating also the persistence, i.e. a dependence of new values on old values. A summary of the wavelet – based fractal analysis is given in Table 1.



process  persistence  

3.0092  1.0046  0.9954  non stationary  strong 

0.7700  0.1150  2.1150  stationary  weak 
We proposed also a new way for determination of persistence nature of the electrochemical noise on the basis of correlation coefficients between original signal and details




0.0188  0.4196 

0.0482  0.5303 

0.1128  0.5097 

0.2048  0.3648 

0.3103  0.3138 
Correlation coefficients between successive details




0.0006  0.0006 

0.0000  0.0004 

0.0001  0.0004 

0.0012  0.0120 
Additionally, DWT with 3decomposition levels was made using different kinds of wavelet functions, from Daubechie’s fractallike wavelet “db2” (Massopust, 1994) to smoother wavelet functions, as Daubechie’s wavelet function “db5” and symmetrical Coiflet wavelet function “coif5”. After decomposition the coding gain (
where


entropy, 
entropy, white noise  
db2  1.3486  1.0090  4.5384  4.9304 
db5  1.3411  1.0076  4.5384  4.9304 
coif5  1.3411  1.0116  4.5384  4.9304 
The chose of different wavelets did not influence on the obtained
To study the approximation properties of DWT’s using different wavelet basis functions, a synthesis (inverse DWT) to different approximation levels was made. We expected better results with ‘’db2’’ assuming the fractal–like shapes of ENsignals. However, no significant differences were found, a smaller approximation error was obtained even using smoother wavelets, what confirms the approximation theory.
6. Conclusion
The most attractive prospective benefit of EN measurement is the ability to obtain information about the type of corrosion that is occurring, but there is much less agreement about the optimum analysis method for obtaining such information (Cottis, 2006). Wavelet transform has been developed over a number of years and only recently has been applied to electrochemical noise analysis. The main advantage of wavelet analysis of EN is the detection of transients which are localized in both the time and frequency domain and shows promise to be discriminatory for the intensity as well as the type of corrosion.
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