Corrosion Research Group

University of Cádiz. Department of Material Science, Metallurgical Engineering and Inorganic Chemistry

New mathematical methods to analyse

electrochemical noise signals

APPLICATION OF WAVELET TRANSFORM TO ANALYSE ENM DATA

Introduction

In the literature, three different main approaches have been proposed for analyse data from electrochemical noise measurement (ENM): the statistical, the spectral and the Chaos Theory-based methods. However, these methods find serious limitations to deal with corrosion processes in which diverse kinds of corrosion events occur simultaneously and the overall time records result from the composition of those elemental signals. We are working to investigate the validity of wavelet analysis as alternative procedure to process electrochemical noise records (ENR), especially those in which different signals are superposed (1 ,2).

Wavelet transfom underground

The wavelets essential approach consists in representing a time record x_r (r=1, 2,...,N) using a linear combination of basis function f_j,n and Y_j,n (3):

where s_J,n, d_J,n,...,d_1,n are named wavelet coefficients, n=1,2,...N/2^j and j=1,2,...,J; J is often a small natural number which depends mainly on N, f_j,n and Y_j,n. The basis functions are generated from a pair of functions (the father wavelets f and the mother wavelets Y) through scaling and translation.

Father and mother wavelets:

The wavelet coefficient values can be computed from:

Therefore, each wavelet coefficient informs just about the segment of the studied signal where the corresponding basis function is not zero. The wavelet coefficient increases with both the fluctuation amplitude of the studied signal and the similarity between the shape of the signal and the basis function in the segment. So, the signal is studied segment by segment by changing the values of n in the basis functions. This study is performed taking into account only the features of the signal with a determined scale. This scale is given by the value of j in the basis functions. So, after a wavelet transform, it is obtained different series of coefficients, which are named crystals d_j=(d_j,1,d_j,2,...d_j,n), that provide a description of the studied signal (segment by segment) at a certain scale giving by:

Example of analysis

Consider a signal like the one in Fig. 1 containing two components which act at different time-scale:

Figure 1: Current record corresponding to a sample of 304SS after 10 hours of immersion in a 0.001M FeCl3 solution.

The fast fluctuation can be related to uniform corrosion while the transients can be related to localized corrosion. Like in this case, time records are seldom stationary. Therefore, the corresponding PSD does not offer useful information. Observe in Fig.2 that it is not possible to note that there are two processes acting simultaneously.

Figure 2: Power spectral density corresponding to Fig. 1.

If a wavelet transform is performed over the signal in Fig. 1 the coefficientes represented in Fig. 3:

Figure 3: Wavelet coefficients arranged in crystals corresponding to Fig. 1.

The fast fluctuation component, which appears through the original signal, is reflected mainly in the d₁-d₃ crystals throughout the studied time. However, some of these coefficients, the largest ones, are related to other component: the transients. In fact, since the transient shape has sharp and smooth features, the transients are reflected in all the d crystals. However, the exact position of the transients must be determined by looking for the largest coefficient in the d₁ crystal. It is possible to take advantage of this to count and localise transients automatically and to classify them by their time-scale.

To summarize the wavelet results, it is possible to represent the relative energy for each crystal E^s_J and E^d_j like in Fig. 4.

Figure 4: Energy distribution plot corresponding to Fig. 1.

Fig. 4 measures the relative weight of every process: there are two maximums. The maximum at d₁ corresponds to the fast fluctuations and the one at s₈ to the transients.

Another possibility is to use the wavelets transform to obtain a non-parametric regression and smoothing. The signal in Fig. 1 can be considered to be formed by two components so that:

x_r=l_r+e_r where l_r is a discrete signal and e_r are independent and identically distributed normal errors: e_r»N(0,s²).Thus, it is possible to estimate the two components of the signal x_r and e_r by using the wavelets shrinkage estimation decomposition (4). That methodology involves three steps:
1) applying the wavelet transform
2) assigning the higher coefficients to the component l_r and the remaining coefficients to the component e_r.
3) reconstructing each component by applying the inverse wavelet transform to the corresponding set of wavelet coefficients.

The result of a wavelets shrinkage estimation decomposition applied over the signal in Fig.1 is shown in Fig. 5:

Figure 5: Wavelets shrinkage estimation decomposition corresponding to Fig. 1 (above l_r, below e_r).

Conclusions

Although further research in this field is needed, this study suggests that using wavelet transform offers the following new possibilities:

	Improving the corrosion type identification with regard to results coming from PSDs.
	Detecting if there are different corrosion types acting simultaneously and, if so, quantifying the relative weight of them and even separating the corresponding signals.
	Counting and localising transients automatically and classifying them by their time-scale.

References

1.- A. Aballe, M. Bethencourt, F.J. Botana, M. Marcos, Wavelet Transform-Based Analysis for Electrochemical Noise, Electrochemistry Communications, 1(7) (1999), 266.
2.- A. Aballe, M. Bethencourt, F.J. Botana, M. Marcos, Using Wavelets Transform in the Analysis of Electrochemical Noise Data, Electrochimica Acta (in press).
3.- A. Bruce, H-Y. Gao, Applied Wavelet Analysis with S-Plus, Springer, New York (1996).
4.- D. L. Donoho, I. M. Johnstone, Ideal spatial adaptation via wavelet shrinkage, Biometrika, 81 (1994), 425.