Annals 47, 1, 2004, 01/07def 39 ANNALS OF GEOPHYSICS, VOL. 47, N. 1, February 2004 Key words self-potential signals – wavelet analysis 1. Introduction Environmental conditions act on self-poten- tial variability driving non-stationary patterns which can highly distort background behav- iours (for a discussion of self-potential electro- chemical theory see e.g., Pham et al., 2001; and references therein). Such disturbances severely limit the possibility of studying the electrical variability of pure geophysical origin and cor- rectly using self-potential measures for the monitoring of some geophysical phenomena. In particular, the relationship between electrical variability and seismic or volcanic activity, which might be useful in prediction studies of earthquakes and volcanic eruptions (e.g., Sobolev, 1975; Massenet and Pham, 1985; Varotsos et al., 1993; Cuomo et al., 1996; Di Bello et al., 1996; Park, 1996; Varotsos et al., 1996; Uyeda et al., 2000), can be strongly hid- den by environmental disturbances such as me- teorological variability and anthropic activities. Anthropic influences can be reduced by choos- ing monitoring sites in zones which are suffi- ciently far from industrial or urban areas, high power lines, and any source of human distur- bance. This is possible just in theory, since for applied purposes we have to take into account that risk areas generally do not satisfy these re- quirements. In any case, even with optimal in- strumental and ambient conditions, electric sig- nals are sensitive to meteorological variability Wavelet analysis as a tool to characterise and remove environmental noise from self-potential time series Domenico Chianese (1), Gerardo Colangelo (1), Mariagrazia D’Emilio (2) (4), Maria Lanfredi (1) (3) (4), Vincenzo Lapenna (1), Maria Ragosta (2) (4) and Maria Francesca Macchiato (3) (4) (1) Istituto di Metodologie per l’Analisi Ambientale (IMAA), CNR, Tito Scalo (PZ), Italy (2) Dipartimento di Ingegneria e Fisica dell’Ambiente, Università degli Studi della Basilicata, Potenza, Italy (3) Dipartimento di Scienze Fisiche, Università degli Studi di Napoli «Federico II», Italy (4) Istituto Nazionale per la Fisica della Materia (INFM), Genova, Italy Abstract Multiresolution wavelet analysis of self-potential signals and rainfall levels is performed for extracting fluctua- tions in electrical signals, which might be addressed to meteorological variability. In the time-scale domain of the wavelet transform, rain data are used as markers to single out those wavelet coefficients of the electric sig- nal which can be considered relevant to the environmental disturbance. Then these coefficients are filtered out and the signal is recovered by anti-transforming the retained coefficients. Such methodological approach might be applied to characterise unwanted environmental noise. It also can be considered as a practical technique to remove noise that can hamper the correct assessment and use of electrical techniques for the monitoring of geo- physical phenomena. Mailing address: Dr. Maria Lanfredi, Istituto di Meto- dologie per l’Analisi Ambientale (IMAA), CNR Area della Ricerca di Potenza, Contrada S. Loja, 85050 Tito Scalo (PZ), Italy; e-mail: lanfredi@imaa.cnr.it 40 Domenico Chianese et al. that can alter electrochemical phenomena. Therefore, studies related to the assessment of soil electric activity as a signature of geophysi- cal phenomena for monitoring purposes cannot neglect the presence of inherent meteorological disturbances. Self-potential signals are good examples of inhomogeneous signals containing both regular- ities and isolated singularities in the form of pulses, jumps, power or deltalike singularities. In order to single out unwanted interferences, our first task is to analyse electrical fluctuations retaining information on the localization of dis- continuities and transient variations. To this pur- pose, wavelet analysis is a useful tool, able to carry out multiresolution studies and to enhance local features against long term dynamic struc- tures. Formalization of wavelet theory was actu- ally initiated by works on seismic signals (Goupillaud et al., 1984; Grossmann and Mor- let, 1984). Environmental perturbations in elec- trical signals might be singled out in the time- scale domain of the wavelet transform more eas- ily than in the physical space. By transforming meteorological data, we locate the time-scale re- gions where the environmental stress acts. Then we transform the self-potential signal and, in the previously selected regions, we extract the ex- cited wavelet coefficients. Such coefficients ac- count for local fluctuations which are candidates for describing the electrical responses to the ex- ternal disturbance. Excited coefficients can be filtered out and the signal can be recovered by anti-transforming the retained coefficients. In this exploratory work, we focus on hourly electrical variability observed during rainy peri- ods using hourly rainfall levels as support data. These levels are proxy data for the triggering ac- tion of rainfalls, since they are related to transient variations in the soil water content. The informa- tion stored in rain data concerns the strength of the external forcing and its duration but it does not provide indications on the evolution of possi- ble subsequent long term soil responses. In dy- namic terms, our support data allow us to investi- gate those short range self-potential fluctuations which strictly follow significant gradients of the soil water content. Of course, with additional support data, our methodological approach can be useful to pick up long range features as well. 2. Observational data We analyse self-potential signals measured by means of a new geoelectrical monitoring net- work installed in a seismic active area of the Southern Apennine Chain (Italy). We focused on the analysis of self-potential data measured by the remote station prototype developed at the be- ginning of 1999 at the Institute of Methodologies for Environmental Analysis of the National Re- search Council (IMAA/CNR) Geophysical Lab- oratory, located in Tito Scalo (PZ). For further details on the monitoring network and the de- scription of the experimental equipment, we refer to Balasco et al. (2002). Pluviometric measures were supplied by the Istituto Idrografico of Ca- tanzaro. Figure 1a,b shows two examples of self-po- tential time series recorded in rainy periods. The signal in fig. 1a is very erratic on short time scales and does not exhibit significant long range trends. Similarly to earthquakes, pluviometric measures can be regarded as realizations of a «point pro- cess» (Cowpertwait, 1994) that is a succession of discrete events we will call «rain events». Rain events in fig. 1a induce evident perturbations which seem to follow the triggering action of the rainfalls rather strictly. The first three consecutive small events induce a slight variation in the mean electric potential values (a trend over a period of ∼ 1 day), whereas a cluster of more important events drives a rather sharp and large variation. This last perturbation persists for a time which is almost equal to the rainy period. In fig. 1b, a cluster of rain events located at a time distance of a very few hours is likely to be responsible for the concomitant strong po- tential variation shown in the plot (∼ 100 mV). Differently from the previous example, this transient is superimposed on a non stationary background. After that the rainfalls stop, the signal decrease is followed by an increase pre- sumably due to the progressive drying process. The increase asymptotically restores the signal morphologies observed before the rainfalls but with mean values which are greater than those observed immediately before the rainfalls. Er- ratic short range variability with anti-persistent features is ubiquitous, as already highlighted in recent works (Cuomo et al., 2000; Colangelo et 41 Wavelet analysis as a tool to characterise and remove environmental noise from self-potential time series al., 2001). In these two cases, although corre- spondence between rain and signal anomalies appears rather evident, the strength and direc- tion of the variations as well as duration of transients seem to be concerned with local dy- namics. 3. The problem and the analysis rationale 3.1. The dynamic problem The physical-statistical investigation of ob- servational signals, aimed to reveal their un- derlying dynamics, is constrained by many fac- tors, the main one being the nature itself of the dynamics. The presence of complex mecha- nisms related to many sources, possible non- linear interactions among different perturba- tions, noises, make it very hard to solve the problem with simple tools. Unfortunately, we cannot a priori assume that disturbances in- duce simple effects. Ideally we can globally decompose the signal into smooth and fast components. In this picture, we split the dy- namics on the basis of the frequency content: low frequencies account for long range varia- tions whereas high frequencies account for generally erratic short range fluctuations (noise). Stationary noises in the context of lin- ear dynamics are negligible because their ef- fects have no significant consequences of dy- Fig. 1a,b. Two examples of hourly measures of self-potential and contextual hourly rainfall levels: a) measures recorded from 20.11.1999 to 03.12.1999; b) from 08.05.1999 to 29.05.1999. The vertical axis on the left side in the plot refers to self-potential while the axis on the right side refers to rain. a b 42 Domenico Chianese et al. namic value. Differently, externally induced abrupt changes and transient phenomena com- plicate the understanding of the electrical dy- namics and its sources. The local character of these effects can alter the global dynamic out- look of the phenomenon, short and long time scales can interact and the simple global pic- ture described above does not work well. 3.2. The signal processing problem Mathematical transformations are usually applied to experimental data to obtain dynamic information that does not clearly appear in raw data. It is well known that pure traditional tools, such as the standard Fourier transform, are not suited for the analysis of non stationary signals. The Fourier transform of a signal with a local disturbance spreads the information concerning this singularity in all its coeffi- cients so any filter will distort both the spec- trum and the recovered signal. We need instead analysis tools able to carry out both global and local investigations allowing us to single out features at various temporal scales retaining in- formation on the localization of discontinu- ities. Starting from traditional methods, some techniques have been developed, such as the short time Fourier transform (see e.g., Portnoff, 1981), to carry out scale and time analysis si- multaneously. In order to account for non sta- tionary patterns, a temporal window is shifted along the series and the signal fluctuations are analysed separately within the series segments selected by the window. The width of the win- dow is independent of the time scale so these methodologies are single-resolution. Tran- sients in self-potentials may generally exist at different scales. When one cannot previously estimate the duration of the local patterns, a flexible window width is recommended. Wavelet analysis overcomes such problems since it is able to provide multiresolution time and frequency characterisation. It breaks up a signal in waveforms of duration matched to the scale. Such waveforms may be irregular and asymmetric, so providing a wide collection of functions to investigate shape, duration and ar- rival time of transients. 3.3. Rationale We use the wavelet transforms of support time series as independent «markers» to select those time-scale regions where we expect that important features of the signal are mainly driv- en by environmental forcing. In the wavelet transform of the signal, such features are repre- sented by anomalous (usually high absolute val- ue) wavelet coefficients which we consider ex- cited by external forcing. Owing to the mul- tiresolution character of the wavelet analysis, the complexity of the patterns described in the physical space is separated in a multi-layer sim- pler descriptor. At any scale and time, we search for those excited coefficients that match signifi- cant coefficients of the marker. At this prelimi- nary stage, we do not focus on the extensive study of electrical responses to different rainfall dynamics or on filtering strategies which can be more or less sophisticated depending on the specific needs. We merely aimed to evaluate the possibility of discriminating in a simple way wavelet coefficients accounting for meteorolo- gy induced morphological distortions by fol- lowing the fingerprints of suited support data. We simply filtered the selected wavelet coeffi- cients substituting their values with values ran- domly extracted form the distribution of the co- efficients estimated at the scale concerned im- mediately before the rainfalls. This procedure is only mildly invasive because it preserves all the other coefficients and therefore all those details that, at any scale and time, cannot be directly as- sociated with the given forcing, also during transients. 4. Some basic concepts of wavelet analysis At present, there is an extensive literature concerning wavelet analysis and its applica- tions, from classical papers (e.g., Mallat, 1984; Daubechies,1992) to more recent textbooks (e.g., Percival and Walden, 2000). Here we lim- it ourselves to some background concepts use- ful for understanding this paper. Wavelet analysis uses functions (wavelets), which are localised both in time and in fre- quency (or scale, in the rigorous wavelet for- 43 Wavelet analysis as a tool to characterise and remove environmental noise from self-potential time series malism). In practice, the only basic require- ment for regarding a function as a plausible wavelet is that it is an oscillating function which has limited duration, zero mean with an inherent pass-band like spectrum. The wavelet transform takes a one-dimensional time signal into a two-dimensional transformed function of time and scale. Therefore it resolves features at various scales without losing information on time localization. Such intrinsic ability to zoom in on short-lived high frequency phenomena is simply achieved by introducing a scale param- eter s which adapts the width of the wavelet kernel to the resolution required (thus chang- ing its frequency contents) at locations which are determined by a parameter t0. Dilated (changing s) and translated (changing t0) ver- sions are obtained from a wavelet prototype ψ (mother wavelet). By iteratively dilating the wavelet and shifting it along the signal we ob- tain a multiscale information of the temporal properties of the signal. Dilated and translated versions of the mother wavelet are obtained by the action U ,U s t t s t t 0 0 = -} }^ ] ah g k where s, t0 ∈ R and s > 0 for the Continuous Wavelet Transform (CWT). The wavelet trans- form Wf of a function f (t) is then obtained pro- jecting the function on the dilated and translat- ed wavelets , ,W s t s f t U s t t dt 1 f 0 0 ) = }#_ ^ _ ^i h i h where (*) denotes complex conjugation. CWT is obtained by continuously shifting scalable functions which do not form an orthogonal ba- sis. For many applied purposes we use the Dis- crete Wavelet Transform (DWT, Daubechies, 1992). Discrete wavelets are not continuously scalable and translatable but are scaled and shifted in discrete steps. In the DWT formalism the wavelet representation is .t s t kt s ,j k j j 0 = - } }] cg m The parameters j and k are integers and s > 1 (s = = 2 for the usual dyadic sampling) is a fixed di- lation step. The translation parameter t0 de- pends on this dilation step: for t0 = 1 we have a dyadic sampling of the time axis as well. Wavelet series decomposition transforms sig- nals in series of wavelet coefficients. In discrete wavelet analysis, the signal is decomposed in approximation and details. The process can be iterated. The original signal is decomposed in successively lower resolution components so producing a wavelet decomposition tree. Figure 2 shows the generic form of a one-dimensional (1D) wavelet transform. The signal is passed through H (a low-pass filter) and G (a high-pass filter), and then down-sampled by a factor of two. This process is considered one level of the transform and can be repeated for a finite num- ber of levels n. The resulting outputs, Di (i = 1, 2, …, n) and Ln, are called wavelet coefficients, and are details and approximation respectively. When we use orthogonal wavelet basis func- tions, we can reconstruct an arbitrary signal by anti-transforming its wavelet coefficients. In this work, a dyadic wavelet decomposition is carried out down to level n = 3 that corre- sponds to a scale 23 = 8 h (the coarsest time res- olution of our analysis). This time scale appeared sufficient for investigating relatively fast varia- tions in the given signals. We used the daub- echies 3 (asymmetric) wavelet. Nevertheless tri- als with different wavelet functions did not imply remarkable differences in our analysis. 5. Results Figure 3a,b shows the rain and self-potential wavelet coefficients estimated for the signal of Fig. 2. Multilevel wavelet transform. The symbol ↓2 indicates down-sample by two. 44 Domenico Chianese et al. Fig. 3a,b. Wavelet coefficients (a) for rainfall measures and (b) for self-potentials relative to the signal report- ed in fig. 1a. a b 45 Wavelet analysis as a tool to characterise and remove environmental noise from self-potential time series fig. 1a. Correspondence between significant rain wavelet coefficients (fig. 3a) and excited signal coefficients (fig. 3b) is generally very good. We need no particular comparison criterion; a naked eye view is sufficient to understand that rain data are suited to locate the time-scale re- gions where the self-potential variability is al- tered. It is interesting to observe the signal co- efficients (fig. 3b) at level 1 (scale = 2 h) in cor- respondence with the cluster of rainfalls (100- 150 h). If we exclude a slighthy excited coeffi- cient associated with the largest events, the co- efficients at this scale do not appear particular- ly sensitive to rainfalls. This suggests that, when triggered, the rain effects are played on time scales longer that two hours. If a smooth- ing filter is used to remove the meteorological anomaly, these fine scale details may be aver- aged out. Our procedure instead retains these details. Figure 4 shows a zoom in representation of the original signal (fig. 1a) and the recovered signal. The two signals differ just during the rainfalls. It is interesting to see that no remark- able difference can be detected between the re- constructed signal during the external event and that of the real fluctuations observed before and after it. As stressed above, fine scale details are not synthetic. Our filter removes just excited coefficients so restoring the background behav- iour. This fact highlights the mildly invasive character of our reconstruction. The problem for the signal of fig. 1b is slightly more difficult because hourly rainfall levels in this case are not optimal support data. Figure 5a,b shows the estimated self-potential and rain wavelet coefficients. By comparing the coefficients at the same level of approximation or detail, we again observe a good agreement, but the excited regions for the signal are slight- ly longer that those detectable in the wavelet Fig. 4. Original (black line) and recovered (bold line) signals for the time series of fig. 1a. 46 Domenico Chianese et al. Fig. 5a,b. Wavelet coefficients (a) for rainfall measures and (b) for self-potentials relative to the signal report- ed in fig. 1b. a b 47 Wavelet analysis as a tool to characterise and remove environmental noise from self-potential time series transform of the rainfalls. As already observed in Section 2, the triggering action of the rain- falls seem to be followed by slower responses of the soil. In this case, our support data do not provide sufficient information to understand where the transient ends. In order to recover the signal, we made the operational hypothesis that the excited coefficients relative to the 24 h after the end of the rainfalls were relevant to the me- teorological stress as well. As shown in fig. 6, the morphology of the recovered signal in the rainy period traces patterns which are similar to those observed in dry periods. About 24 h after the end of the rainfalls, the recovered signals adapts itself to intersect the observational data. 6. Conclusions The main idea we present in this paper is that the joint multiresolution wavelet analysis of self-potential signals and support data relat- ed to environmental forcing can represent a suitable basis to extract environment-induced electrical fluctuations. Wavelet transform trans- lates the complexity of mixed global behav- iours and transient patterns described by the electrical signals in simpler time sequences of coefficients over several resolutions or scales. We focused on hourly self-potential variability driven by rainfalls by exploiting hourly rainfall level measures as support data. Such data, transformed in the wavelet domain, were used to mark the time-scale regions where rainfalls act influencing self-potential variability. We showed that in these regions excited wavelet co- efficients of the signal are detectable and they well account for local alterations ascribable to the rain. 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