Corrosion Process Monitoring Using Wavelet Analysis - Industrial

Localized corrosion is more difficult to predict and monitor. Localized corrosion processes can be divided into two main steps: the initiation step, i...
0 downloads 0 Views 529KB Size
Download PDF
1256

Ind. Eng. Chem. Res. 2000, 39, 1256-1263

Corrosion Process Monitoring Using Wavelet Analysis X. D. Dai, R. L. Motard, and B. Joseph* Chemical Engineering Department, Washington University, St. Louis, Missouri 63130

D. C. Silverman Solutia Inc., 10300 Olive Boulevard, P.O. Box 66760, Mail Zone WG1S, St. Louis, Missouri 63166-6760

The objective of this paper is to explore the use of wavelet transform methods in the characterization and identification of intensity and type of localized corrosion using electrochemical noise (in this case, fluctuations in potential). A series of experiments were conducted using two identical stainless steel electrodes inserted into known corrosive media. The fluctuations in corrosion potential (electrochemical noise) generated from these electrodes were monitored periodically (about every hour) over a few days. These fluctuations were stored in blocks of 1024 data points (collected at 1 Hz). They were analyzed by using conventional signal processing techniques in the time and frequency domains as well as by using wavelet techniques in the time-frequency phase plane. The results were compared with microscopic visual examination of the corroded surface to correlate the electrochemical noise signal with the type and extent of corrosion. The results show that wavelet transforms hold promise in decoding this noise pattern. A method to compute the corrosion intensity from the phase plane data is proposed and is in qualitative agreement with the experimental results. 1. Introduction 1.1. Wavelet Analysis. Consider a signal represented as a time series. In this representation, one can fairly easily distinguish the fluctuations but not so easily the intensity of various frequencies present in the signal or the time at which these fluctuations are present. A Fourier transform of the signal typically plotted as amplitude versus frequency reveals the intensity of the various frequency components averaged over time. The Fourier transform does not show how the frequency content varies with time. The basis functions (i.e., sine and cosine waves) onto which we project the signal to obtain the Fourier transformed signal are localized in the frequency but distributed over the entire time interval. Wavelet theory, i.e., wavelet transform and its extensions, has been proposed as a new time-frequency analysis method and has drawn widespread attention in recent years (Rioul and Vetterli,1 Jawerth and Sweldens,2 Daubechies,3 Wickerhauser,4 and Motard and Joseph5). Wavelets are an extension of the timefrequency distributions investigated by Wigner,6 Ville,7 and Page8 and reviewed by Cohen.9 This theory is a cousin of the short-time Fourier transform (STFT; Gabor10), which is used to analyze a signal by looking at its frequency distribution within a window moving in time. The wavelet transform overcomes some of the limitations of STFT by using a family of basis functions which are localized in both the time and frequency domains. A STFT can be used to probe the frequency content over limited time intervals. However, this approach has limitations because the time interval chosen is arbitrary. To overcome these limitations, wavelet transforms were proposed. In this case, the basis functions are * To whom correspondence should be addressed. E-mail: [email protected].

chosen to have localized energy in both the time and frequency domains. The result is that one can probe the signal to determine what frequencies were present at what instant. Wavelets were proposed as a tool for seismic signal analysis by the geophysicist Mallat.11 The mathematical basis was established firmly by Daubechies et al.12 Currently, a number of books are available on the subject including Combes et al.,13 Daubechies,3 Chui,14 Chui,15 Wickerhauser,16 Young,17 and Motard and Joseph.5 Matlab, a software package from Mathworks, provides a Wavelet Toolbox.18 In this work we employ the discrete form of the wavelet transform (DWT). Consider a family of basis functions generated from a mother wavelet, ψ(t)

ψm,n(t) ) R0-m/2ψ(a0-mt - nb0) m, n ∈ Z The DWT of a continuous function f(t) is defined by

wm,n ) 〈f(t), ψmin〉 ≡ a0-m/2

∫-∞+∞f(t) ψ(a0-mt - nb0)

DWT transforms f(t) into a countable set of wavelet coefficients which correspond to points on a twodimensional grid or lattice of discrete points in the frequency-time domain indexed by m and n. Figure 1 schematically shows the lattice. Wickerhauser19 proposed a phase-plane representation to view the wavelet transform results. It is a twodimensional time-frequency representation. Each transform coefficient wm,n is associated with a rectangular patch on the time-frequency plane, and the phase plane is generated by using the power of the signal component present in each rectangle. Figure 2 compares four different types of bases for viewing a signal. At the top is the time domain graph of a Doppler signal which shows that the frequency present in the signal is

10.1021/ie990506u CCC: $19.00 © 2000 American Chemical Society Published on Web 03/31/2000

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1257

Figure 1. The lattice of points used in wavelet analysis. The points indicate localization of the basis functions.

Figure 2. The configuration of different bases on the timefrequency phase plane.

decreasing with time. At the bottom is the phase-plane plot (we use this terminology to be consistent with the current literature on wavelets) which shows the timefrequency decomposition outlined in Figure 1. Each rectangle on the phase plane represents a particular time and frequency interval. In this figure the gray scale coding indicates energy of the signal in that rectangle with darker bands representing higher energy. By using the wavelet basis, we can monitor the presence or absence of various frequencies at various times. The standard basis of time-sampled signal representation completely hides the frequency information, whereas the Fourier transform provides information about the frequency content but averages it over the entire time axis. STFT is unable to capture the lowfrequency trends. The wavelet basis captures both lowand high-frequency variations and the times at which they occur in the signal. Figure 3 shows an example signal and its phase-plane representation. Note how the occurrence of specific frequency events in the time domain appears on the phase plane. This study uses the phase plane to study the patterns hidden in fluctuations in the potential between two identical electrodes (electrochemical noise). It is the first phase of an effort to improve the ability to analyze such signals for the purpose of corrosion detection and identification. 1.2. Electrochemical Noise. Though corrosion has been characterized as having a number of forms, two

Figure 3. (a and b) Phase plane of a Doppler signal.

broad types have a significant impact on the process industries, general or uniform corrosion and localized corrosion in the form of crevice or occluded cell corrosion and pitting. Uniform corrosion, which attacks the alloy more or less evenly across the surface, is fairly easy to predict and monitor by such technologies as electrochemical impedance spectroscopy, electrical resistance probes, polarization resistance measurement, and direct immersion of alloy specimens (coupon immersion). Localized corrosion is more difficult to predict and monitor. Localized corrosion processes can be divided into two main steps: the initiation step, in which passivity is lost in a very localized region of the alloy surface, and the propagation step, in which the penetration proceeds at a more or less constant rate, sometimes rather rapidly. By the time the localized corrosion is propagating, damage has already appeared. Sometimes, such corrosion can initiate, propagate, and temporarily stop with the cycle repeating itself. Because the desire in corrosion monitoring is to capture events before damage is done, the desire here would be to detect events involved in the initiation phase of such corrosion. A technique that is sensitive to such initiation events that has been proposed for corrosion monitoring is analysis of electrochemical noise.20,21 Electrochemical noise is defined as the fluctuations of potential, current, or both, originating from uncontrolled variations in a corrosion process.22 Such an electrochemical signal is believed to be a rich source of

1258

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 Table 1. Corrosion Test Conditions of Experiments test run no.

Figure 4. Experimental setup.

information on the interfacial electrochemical corrosion process. The measurement and analysis of this signal has received considerable attention, especially in more recent years, and a substantial amount of literature has appeared on the subject. For example, ref 2 contains 177 references itself dating back to the early work of Iverson in 1968.23 The measurement of the fluctuations is very straightforward. The potential, current, or both is sampled at constant time intervals. Current is often sampled between two identical electrodes coupled through a zero resistance ammeter. Voltage is sampled relative to a reference electrode or relative to an electrode identical with itself. Many times, current and voltage are sampled simultaneously. The reader should consult the cited literature for examples of the measurement techniques available. The problem is that while the measurement is simple, the analysis of that signal and ability to reduce that analysis to a practical, routine predictive tool is extremely difficult. While the simplicity of the measurement makes this technique enticing for routine corrosion monitoring, the difficulties in the analysis have prevented its widespread use as a practical, corrosion prediction and monitoring tool. The results of a recent round robin on the technique demonstrates the difficulties that can arise when trying to compare results in different laboratories if variables such as sampling time, filtering, on-line data manipulation, and baseline noise level are not the same.24 Beyond these problems is the difficulty in trying to analyze the data in a monitoring situation in which the corrosion phenomenon is being estimated from the signal in the absence of corroborative information. Good correlation has not been established between the noise features and the corrosion process across broad ranges of corrosion systems. Attempts are being made to provide a theoretical basis for these types of measurements.25 Traditionally, fluctuations in current and potential have been analyzed by statistical means assuming that the basic process is stochastic or random in nature. Such variables as the mean, variance, skewness, kurtosis, standard deviation, and root mean square have been used either separately or in conjunction with each other (refs 1 and 2 and many of the references contained therein). Alternatively, the signal has been transformed from the time domain to the frequency domain usually using either fast Fourier transforms or the maximum entropy method.26 The power spectrum density is estimated from these transformations. Though the analyses are fairly easily done, the problem has remained in how to routinely relate these signals to the corrosion process automatically in the absence of detailed knowledge of the corrosion process itself and without manual intervention. More recently, attempts have been made to combine the statistical results with artificial neural networks to

electrode

pH

248 630 631

316ss 304ss 304ss

632

304ss

633 635 636 637 641 643 645 647 649 652

304ss 304ss 304ss 304ss 304ss 304ss 304ss 304ss 304ss 304ss

654 656

304ss steel

natural

658 660 662 664

304ss 304ss platinum steel

7 7 2 natural

666

steel

natural

668 670

304ss 304ss

2 2

2.5-3.0

2.5-3.0 3 2 2.5-3.0 2.5-3.0 2.5-3.0 2 2.25

solution

T (°C)

plant condensate 0.001 M FeCl3 0.001 M Fe3SO4 + 0.003 M NaCl 0.01 M Fe3SO4 + 0.03 M NaCl 0.001 M FeCl3 0.5 M NaCl 0.1 M FeCl3 0.001 M FeCl3 0.001 M FeCl3 0.001 M FeCl3 0.01 M FeCl3 0.01 M FeCl3 0.01 M FeCl3 0.01 M Fe2(SO4)3 + 0.03 M NaCl process stream 200 ppm NaNO2, 3 wt % NaCl phosphate buffer phosphate buffer 0.01 M FeCl3 200 ppm NaNO2, 3 wt % NaCl added on day 2 200 ppm NaNO2, 3 wt % NaCl added on day 2 0.01 M FeCl3 0.01 M FeCl3

35 60 60 60 90 60 60 60 60 60 60 60 60 60 80 60 60 60 60 60 35 95

try to generalize the interpretation.27 Alternatively, wavelet transformations have been proposed for the interpretation of electrochemical noise signals.28-30 This study is the first phase of an effort to explore the application of wavelets to practical yet somewhat intractable signal processing problems. From the discussion above, the analysis of electrochemical noise is obviously one. The goal of this thrust was to explore if wavelet transformations could provide enhanced discriminating power to differentiate among several types of corrosion. 2. Experimental Studies Though electrochemical noise studies tend to include both current and voltage signals, this study concentrated on voltage signals only. At this stage, the purpose was to determine if the wavelet transform could detect gross differences in corrosion. The procedure was as follows. Two identical electrodes immersed in the solution under study were connected to a Hewlett-Packard 3497A digital voltmeter. This device had a sensitivity of 1 µV. It was connected to a Hewlett-Packard 9816S microcomputer which was used to sense and record the voltage. The setup is shown in Figure 4. Voltage was sampled in packets of 1024 data points at 1 point/s. Measurements were spaced so that they were taken about 18 times over each 24 h period for up to about 6 days. Noise across a 1000 Ω resistor inserted in place of the electrochemical cell was (1 µV when measured on the most sensitive scale. Alloy/environment combinations examined were UNS S30400 and UNS S31600 in various concentrations of FeCl3 with and without NaCl at a pH of 2, (Fe)2(SO4)3, (Fe)2(SO4)3 with NaCl, NaCl acidified with sulfuric acid to a pH of 2, and a sodium phosphate buffer at a pH of 7. Deionized water was used to dilute all of the above

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1259

Figure 7. Power spectral density plots for B1, C1, C2, and P1 calculated with a Hanning window of length ) 64.

Figure 5. Original time series data sets of B1, C1, and C2, with 1024 samples taken at t ) 25 h at 1 Hz.

a

chloride especially when acidified and ferric sulfate especially with added sodium chloride would be expected to promote localized corrosion. The phosphate buffer solution would be expected to be benign with respect to stainless steel and was used to provide “baseline” information, e.g., the fluctuations that might appear in the absence of localized corrosion. UNS G10180 steel was examined in sodium nitrite and sodium nitrate plus sodium chloride at various concentrations. The addition of sodium chloride to the sodium nitrite would promote pitting of the passivated steel. The electrode configuration was such that crevice corrosion was isolated to the region between the electrode and Teflon holder. Temperatures were 35, 60, and 95 °C. The electrodes were bullet shaped and fully immersed in the fluid. Table 1 summarizes the experimental conditions. The electrodes were examined after immersion under a stereomicroscope for crevice corrosion especially at the end of the electrode that contacted the holder and pitting on the exposed surface. Broad, qualitative indices of the severity of corrosion were devised for the observed corrosion so that the degree of attack could be ranked. 3. Analysis Using Time and Frequency Domain Techniques

b

Electrochemical noise is commonly analyzed by using differences in the mean, standard deviation, etc., of the actual signal in the time domain or differences in the power spectrum in the frequency domain. These types of analyses were applied to four test cases: B1:

C1: C2: P1:

Figure 6. (a and b) Mean and standard deviations as a function of time.

solutions. Several in-plant process streams were included as well. The environments containing sodium

baseline tests: phosphate buffer solution at 60 °C, 304 SS electrodes these runs do not show any corrosion activities in the test specimens 0.001 M FeCl3 at 60 °C, 304 SS electrodes the specimens here showed mild crevice corrosion between the holder and the electrode 0.01 M FeCl3 at 60 °C, 304 SS electrodes this specimen showed severe crevice corrosion between the holder and the electrode 0.001 M FeCl3 solution at 60 °C, 304 SS electrodes this specimen showed pitting corrosion attack at the end of the run

The same test material (304SS) is used at the same temperature (60 °C) in the above experiments. When the concentration is varied, the corrosion intensity is changed. One snapshot of the original data at time )

1260

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

Figure 8. (a and b) Power spectral density plot for B1, C1, C2, and P1 computed after fitting models of orders 1 and 2, respectively.

25 h is shown in Figure 5. B1 is least noisy, and C2 is most noisy. Parts a and b of Figure 6 show the mean and standard deviation of the data over a 24 h period. The mean does not show any trend. The standard deviation does show an increase from B1 to C1 to C2. However, the variation is small with considerable scatter. Corrosion in the P1 environment included pitting on the exposed surface, whereas corrosion in the other two was confined between the holder and electrode (crevice corrosion). Figure 7 shows the power spectral density (PSD) plots of the above cases. A Hanning window of length 64 was used to smooth the PSD plots. While the baseline B1 clearly stands out, the others show some overlap. These plots are consistent with the observations of some earlier workers such as Searson and Dawson,31 Mansfield and Xiao,32 and Hladkey,33 but the similarity of the PSD plots makes distinguishing between the various types of corrosion present in the sample difficult. Another attempt was made to refine the PSD plots by employing a parametric model based technique of fitting these data to an autoregressive (AR) model of various orders. Parts a and b of Figure 8 show typical results. This technique appears to be less conclusive as the pitting sample P1 is sandwiched between the two crevice corrosion samples, C1 and C2. Also the results are dependent on the model order chosen for the AR model. This method of analysis is not recommended.

Figure 9. (a-d) Signal patterns of runs B1, C1, C2, and P1 as they appear on the phase plane.

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1261

a

c

b

d

Figure 10. (a-d) Feature vectors extracted from the planse plane for runs B1, C1, C2, and P1.

4. Application of Wavelet Analysis to Corrosion Data Wavelet transforms were next used to analyze the electrochemical noise. The first step was to visually examine the phase planes generated by the wavelet transform. These patterns are more easily distinguished using color to map the energy intensity in each timefrequency rectangle. These color coded maps were compared with the microscopic examination of the electrode specimens to discover if a pattern existed. Comparison of numerous phase-plane plots indicated that a relationship existed between the energy distribution on the phase plane and the actual corrosion observed. Parts a-d of Figure 9 present the signal patterns for the test data sets B1, C1, C2, and P1. These are snapshots of data collected during the hourly sampling. Figure 9a shows that the baseline test B2 has almost no activity on the phase plane, except for the very low frequency part of the signal pattern. In parts b and c of Figure 9, the signal patterns cover the entire phase plane. When signal patterns in b and c were carefully compared, they were discovered to share similar time-frequency structures but to have different intensities. Lighter patches indicate stronger intensity. Figure 9d in which pitting was a dominant surface feature had a completely different signal pattern. Similar signal patterns obtained from over 1500 hourly observations on 23 different electrode/solution

systems were compared with the optical observations made at the end of each of these 23 runs. These comparisons lead to the following conclusions: (i) The signal patterns painted on the phase plane are characteristic of the type of corrosion present in the sample. Crevice corrosion is characterized by a widely distributed noise pattern on the phase plane whereas pitting corrosion is characterized by more intense episodes with much lower intensity time periods in between. (ii) The intensity or energy exhibited at various timefrequency slots increases with the intensity of corrosion. This conclusion is also supported by PSD graphs shown earlier. While the visual patterns are useful, ideally one would like to quantify the extent and type of corrosion using the features extracted from the phase-plane representation. Numerous ways exist to extract feature vectors for pattern recognition purposes. One such technique is presented below. It is not claimed to be the best way but is intuitive and seemed to work for this situation. The visual representation is obtained by associating a color (or shade of gray) with the strength of the signal present in each of the rectangular slots on the phase plane. Let Pij denote this strength. Let P ) {Pij} denote a signal pattern on the phase plane.

1262

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000

Table 2. Results for Severe, Baseline, and Corrosion Tests a. Comparison of Predicted and Observed Results for Severe Corrosion Testsa test run no.

average intensity

calculated results

physical observation

632a,b 636a,b,c,d 645a,b,c,d,e,f 647a,b,c,d,e,f 649a,b,c, d 652a,b,c,d

877, 707 1451, 1202, 1309, 1250 852, 717, 620, 636, 601, 632 894, 1058, 1152, 1147, 1235, 1080 758, 679, 736, 723 633, 901, 1214, 1338

severe C severe C severe C, very minor P severe C severe C severe C, minor P

severe C severe C severe C, very minor P severe C, very minor P severe C, very minor P severe C, minor P

b. Results for Baseline Corrosion Tests test run no.

average intensity

calculated results

physical observation

658a,b,c 660a,b,c 662a,b,c,d

63, 32, 37 26, 10, 15 87, 39, 22, 23

baseline, very minor P baseline baseline

baseline, very minor etch on one electrode baseline baseline

c. Results for Minor Corrosion Tests

a

test run no.

average intensity

calculated results

physical observation

248a,b 630a,b 631a,b,c,d,e 633a,b 635a,b,c 637a,b,c,d 641a,b,c,d,e 643a,b,c,d,e 654a,b,c,d 668a,b,c,d,e 670a,b

112, 71 451, 304 1163, 1334, 719, 51, 34 339, 330 275, 230, 71 483, 414, 243, 336 404, 287, 265, 157, 144 282, 201, 154, 77, 27 215, 184, 115, 169 366, 208, 163, 194, 213 279, 426

very minor P pitting, minor C severe C, pitting crevice, very minor P crevice, pitting crevice, pitting crevice, pitting minor C, pitting pitting, minor C crevice, pitting crevice, pitting

very minor P pitting, minor C crevice, minor P crevice, minor P crevice crevice crevice, pitting crevice, pitting minor P, minor C crevice crevice

C, crevice corrosion; P, pitting corrosion.

Step 1. Let

pb1 ) max {Pij} pb2 ) min {Pij} ∆p ) (pb1 - pb2)/m where m ) an arbitrary integer representing the number of energy bands that we would like to recognize. Step 2. Divide P ) {Pij} into subsets PSk (k ) 1, ..., m), where

PSk ) {Pij; pb2 + ∆p(k - 1) e Pij < pb2 + ∆pk}, k ) 1, ..., m Count the number of elements in each subset PSk

V(k) ) n{PSk}, k ) 1, ..., m Essentially, this procedure builds a histogram of the energy distribution on the phase plane but using a scale dependent on the spread of variations. Because the electrochemical noise has a wide range of energy distributions, the variations about the mean rather than an absolute scale are used for the construction of the histogram. For this analysis we set m ) 14 which is sufficiently large to contain detailed information about the features present. Parts a-d of Figure 10 show the variations in the extracted feature V(k) as a function of time for runs B1, C1, C2, and P2. A time-voltage data set was taken every hour, and V(k) is computed for the phase plane generated from each data set. From these and other experiments the following are observed: (1) The base case of no corrosion shows very little changes in V on the phase plane. (2) Crevice corrosion shows V(1)...V(5) to be large and then tapering off for higher values of k. (3) C2 shows larger values for V(1)...V(5) than C1.

(4) Pitting corrosion shows rather uniform but small variations on the V surface. The distinct pattern of crevice corrosion is absent in this picture. (5) If localized corrosion occurs, it is detectable using the phase-plane analysis, by comparison with the baseline corrosion phase plane. (6) The more severe the corrosion, the greater the intensity detected in the phase plane. (7) Minor pitting and minor crevice corrosion are detectable but difficult to distinguish. More than 1500 feature vectors belonging to 23 test runs were examined and classified using the nearestneighbor classification method. This classification was ambiguous only in those cases where the corrosion was very minor. Parts a-c of Table 2 show the extent of corrosion identified from the feature vector along with the actual results. (a-f in the test run nos. refer to various hourly data sets that were used.) Table 2a summarizes severe corrosion trials. This set shows only one misclassification for run 647, which actually showed some very minor pitting. Probably the severity of the crevice corrosion masked the signal sufficiently so that wavelet analysis could not detect the minor pitting in this case. Table 2b summarizes baseline (no corrosion) experiments. Again there is agreement between optical observation and wavelet-based classification except for the experiment labeled run 658. A surface etch was found on one of the electrodes from this experiment which suggests some general corrosion. The cause is believed to be slight contamination of the solution used because all alloy specimens were derived from the same heat. This anomaly was detected by the wavelet analysis as well but was classified as very minor pitting. This experiment was repeated as run 660 with the same conditions as run 658. The specimen exhibited no corrosion as expected, and the wavelet analysis also classified it correctly. Table 2c summarizes minor crevice and pitting corrosion trials. The latter shows some misclassification

Ind. Eng. Chem. Res., Vol. 39, No. 5, 2000 1263

as expected. More investigation is required to determine if this misclassification is a limitation of the sensitivity of the method. 5. Conclusion Wavelet analysis provides a promising procedure for analyzing signals rich in information generated by complex phenomena such as electrochemical noise from corrosion on surfaces. The experiments and analysis show that the discrimination power can be enhanced by transforming the signal into the wavelet domain phase plane and then assembling feature vectors from this phase plane. In corrosion monitoring, measurement of the intensity as well as the type of corrosion, namely, pitting vs crevice attack, is important. These two types of localized corrosion led to distinct patterns (“fingerprints”) on the time-frequency phase plane. A feature vector for automated pattern recognition was proposed and verified. Compared to conventional procedures using time domain or frequency domain analysis, the proposed method shows promise to be more robust and discriminatory. The intensity and pattern of the signal in the phase plane show reasonable consistency with the optical observations on the corrosion occurring on electrode specimens. A highly corrosive environment produces a high-intensity pattern. One area that requires further work is determining the sensitivity of the technique to detecting mildly corrosive conditions and distinguishing among types of corrosion under these conditions. Acknowledgment Support provided by NSF Grant CTS-95-29578 is gratefully acknowledged. Nomenclature a0 ) dilation factor used with wavelet transform b0 ) translation factor used with wavelet transform B1 ) baseline corrosion test C1, C2 ) crevice corrosion tests m ) bin size used for classification Pij ) power of the signal at the lattice point (i, j) in the phase plane P1 ) pitting corrosion test pb1 ) max {Pij} pb2 ) min {Pij} PSk ) subset containing elements of Pij V(k) ) count of the number of elements in each subset PSk wm,n ) wavelet coefficients ι(t) ) mother wavelet function

Literature Cited (1) Rioul, O.; Vetterli, M. Wavelets and signal processing. IEEE Signal Process. Mag. 1991, Oct, 14-38. (2) Jawerth, B.; Sweldens, W. An overview of wavelet based multiresolution analysis. Research report 1993:1; Department of Mathematics, University of South Carolina, Columbia, SC, 1993. (3) Daubechies, I. Ten Lectures in Wavelets Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992. (4) Wickerhauser, M. V. Adapted wavelet analysis from theory to software; A. K. Peters: Wellesley, MA, 1993. (5) Motard, R. L., Joseph, B., Eds. Wavelet Applications in Chemical Engineering; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1994. (6) Wigner, E. P. On the quantum correction for thermodynamic equilibrium. Phys. Rev. 1932, 40, 749-759. (7) Ville, J. Theory and Applications of the Concept of Analytical Signals. Cables Transm. 1948, 2A, 61-74.

(8) Page, C. H. Instantaneous power spectra. J. Appl. Phys. 1952, 23, 103-106. (9) Cohen, L. Time-frequency distributionssA review. Proc. IEEE 1989, 77 (7), 941-981. (10) Gabor, D. Theory of communication. J. Inst. Electr. Eng. 1946, 93, 429-441. (11) Mallat, S. A theory for multiresolution signal decomposition: the wavelet representation. IEEE Trans. Pattern Anal. Mach. Intell. 1989, 11, 674-693. (12) Daubechies, I.; Grossmann, A.; Meyer, Y. Painless nonorthogonal expansions. J. Math. Phys. 1986, 27, 293-309. (13) Combes, J. M., et al., Eds. Wavelets, Time-Frequency Methods and Phase Space; Springer-Verlag: Berlin, 1989. (14) Chui, C. K. An Introduction to Wavelets; Academic Press: Boston, 1992. (15) Chui, C. K., Ed. Wavelets: A tutorial in theory and applications; Academic Press: Boston, 1992. (16) Wickerhauser, M. V. Best-adapted wavelet packet bases. In Different Perspectives on Wavelets; Daubechies, I., Ed.; American Mathematical Society: San Antonio, TX, 1993; pp 155-171. (17) Young, R. K. Wavelet theory and its application; Kluwer Academic Publishers: Hingham, MA, 1993. (18) Mathworks, Inc. Wavelet Toolbox for Matlab User Manual; The Mathworks Inc.: Natick, MA, 1996. (19) Wickerhauser, M. V. Acoustic signal compression with wavelet packets. WaveletssA Tutorial in Theory and Applications; Chui, C. K., Ed.; Academic Press: Boston, 1992; pp 679-700. (20) Dawson, J. L.; Eden, D. A.; Hladky, K. The Use of Electrochemical Noise for Corrosion Monitoring. In On-Line Monitoring of Continuous Process Plants; Butcher, D. W., Ed.; Society of Chemical Industry, Ellis Horwood Ltd.; London, 1983; Chapter 8. (21) Eden, D. A. Electrochemical NoisesThe First Two Octaves. Presented as paper 386 at CORROISON/98, San Diego, CA, 1998 (reprint available from NACE International). (22) ASTM. Standard Terminology Relating to Corrosion and Corrosion Testing; ASTM Standard G-15; American Society of Testing Materials: Philadelphia, PA, 1998; Vol.03.02. (23) Iverson, W. P. J. Electrochem. Soc. 1968, 115, 617. (24) Goellner, J.; Burkert, A.; Heyn, A. Evaluation of a Round Robin Experiment on Electrochemical Noise. Presented as Paper 385 at CORROSION/98, San Diego, CA, 1998 (reprint available from NACE International). (25) Cottis, R. A.; Turgoose, S. Mater. Sci. Forum 1995, 192194, 663. (26) Bertocci, U.; Frydman, J.; Gabrielli, C.; Huet, F.; Keddam, M. J. Electrochem. Soc. 1998, 145 (8), 2780. (27) Reid, S.; Bell, G. E. C.; Edgemon, G. L. The Use of Skewness, Kurtosis, and Neural Networks for Determining Corrosion Mechanism from Electrochemical Noise Data. Presented as Paper 176 at CORROSION/98, San Diego, CA, 1998 (reprint available from NACE International). (28) Stevens, K. J.; Wells, D. B.; Burnell, J. B. Crack nucleation and short crack growth in sensitized type 316 stainless steel. Proceedings of the 13th International Corrosion Congress; Australasian Corrosion Assoc.: Clayton, Australia, 1996; Papers 222/1-222/7. (29) Whitham, T.; Huizinga, S. Evaluation of electrochemical noise analysis as an online monitoring tool to distinguish between biofilm-associated localized corrosion and oxygen corrosion. Eur. Fed. Corros. Publ. 1997, 22 (Microbially Induced Corrosion), 9-102. (30) Tachibana, K.; Matsuki, K. In situ impedance measurement using the electrochemical noise generated during cyclic voltammetry. Proc.sElectrochem. Soc. 1997, 96 (16), 57-64. (31) Searson, P. C.; Dawson, J. L. Analysis of electrochemical noise generated by corroding electrodes under open-circuit conditions. J. Electrochem. Soc. 1988, 38 (2), 1908-1914. (32) Mansfield, F.; Xiao, H. Electrochemical noise analysis of iron exposed to NaCl solutions of different corrosivity. J. Electrochem. Soc. 1993, 140 (8), 2205-2209. (33) Hladkey, K. Corrosion Monitoring. U.S. Patent 4575678, 1986.

Received for review July 12, 1999 Revised manuscript received February 21, 2000 Accepted February 23, 2000 IE990506U