A new combined wavelet methodology: implementation to GPR and ERT data obtained in the

Montagnole experiment

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J. Geophys. Eng. 10 (2013) 025017 (17pp) doi:10.1088/1742-2132/10/2/025017

A new combined wavelet methodology:implementation to GPR and ERT dataobtained in the Montagnole experimentL Alperovich1, L Eppelbaum1, V Zheludev2, J Dumoulin3, F Soldovieri4,M Proto5, M Bavusi5 and A Loperte5

1 Department of Geophysical, Atmospheric and Planetary Sciences, Tel Aviv University,Ramat Aviv 69978, Tel Aviv, Israel2 School of Computer Sciences, Tel Aviv University, Ramat Aviv 69978, Tel Aviv, Israel3 LUNAM Universite, IFSTTAR, MACS, F-44340, Bouguenais, France4 Consiglio Nazionale delle Ricerche, Istituto per il Rilevamento Elettromagnetico dell’Ambiente(IREA), Naples, Italy5 Consiglio Nazionale delle Ricerche, Istituto di Metodologie per l’Analisi Ambientale (IMAA),Tito Scalo (PZ), Italy

E-mail: levap@post.tau.ac.il

Received 21 May 2012Accepted for publication 19 February 2013Published 10 April 2013Online at stacks.iop.org/JGE/10/025017

AbstractGround penetrating radar (GPR) and electric resistivity tomography (ERT) are well assessedand accurate geophysical methods for the investigation of subsurface geological sections. Inthis paper, we present the joint exploitation of these methods at the Montagnole (French Alps)experimental site with the final aim to study and monitor effects of possible catastrophicrockslides in transport infrastructures. The overall goal of the joint GPR–ERT deploymentconsidered here is the careful monitoring of the subsurface structure before and after a seriesof high energetic mechanical impacts at ground level. It is known that factors such as theambiguity of geophysical field examination, the complexity of geological scenarios and thelow signal-to-noise ratio affect the possibility of building reliable physical–geological modelsof subsurface structure. Here, we applied to the GPR and ERT methods at the Montagnole site,recent advances in wavelet theory and data mining. The wavelet approach was specificallyused to obtain enhanced images (e.g. coherence portraits) resulting from the integration of thedifferent geophysical fields. This methodology, based on the matching pursuit combined withwavelet packet dictionaries, permitted us to extract desired signals under differentphysical–geological conditions, even in the presence of strongly noised data. Tools such ascomplex wavelets employed for the coherence portraits, and combined GPR–ERT coherencyorientation angle, to name a few, enable non-conventional operations of integration andcorrelation in subsurface geophysics to be performed. The estimation of the above-mentionedparameters proved useful not only for location of buried inhomogeneities but also for a roughestimation of their electromagnetic and related properties. Therefore, the combination of theabove approaches has allowed us to set up a novel methodology, which may enhance thereliability and confidence of each separate geophysical method and their integration.

Keywords: wavelet analysis, integration, coherency, complex Gaussian function, GPR, ERT

(Some figures may appear in colour only in the online journal)

1742-2132/13/025017+17$33.00 © 2013 Sinopec Geophysical Research Institute Printed in the UK 1

J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

1. Introduction

Geophysical investigations of the subsurface are notoriouslycomplicated by numerous factors of different origin. The mostcommon forms of noise affecting near-surface investigationsare depicted in figure 1. The calculation of all disturbingfactors is not always possible (Eppelbaum et al 2010). Undersuch situations the role of advanced mathematical methods forretrieval of useful information about the subsurface targets isincreased.

Geological media as a whole or as compositions ofsmall-scale structures (objects) are constantly affected bydifferent internal and external impacts. Therefore, it isimportant to understand and estimate the processes thatdecrease the stability of these structures/media, and theirsubsequent effects, especially for man-made constructionssuch as railways, roads, bridges and landing strips in airports.

Geophysical investigation by the use of groundpenetrating radar (GPR) and electric resistivity tomography(ERT) is frequently performed for subsurface investigation(e.g. Neal (2004), Kofman et al (2006), Ezersky (2008), Piegariet al (2009), Soldovieri and Orlando (2009), Papadopouloset al (2010), Persico et al (2010), Clement et al (2011)). At thesame time, the ambiguity of geophysical field examination,complex geological media and low useful signal–noise ratiosaffect the possibility of achieving reliable physical–geologicalmodels of the geological sections investigated.

The informational approach for the analysis of singlegeophysical fields and their integration (e.g. Khesin andEppelbaum (1997), Eppelbaum et al (2011a)) enables usto detect a few anomalous geological targets in complexenvironments, but the estimation of quantitative parameters ofthese targets is usually beyond the scope of this methodology.

In this study, we move in the framework to investigate themost effective mathematical tools to extract the maximumamount of retrievable geological–geophysical informationfrom geophysical surveys. In this context, wavelet single andcombined analysis is now recognized as a comprehensiveprocessing tool for geophysical data examination (e.g.Grinsted et al (2004), Wapenaar et al (2005), Hu et al (2007),Eppelbaum et al (2011a), (2011b)).

Thus, the paper focuses on the detailed and combinedGPR–ERT measurements carried out at the Montagnoleexperimental polygon (which is situated in a complex tectono-structural setting in the French Alps) before and after droppingon the ground a massive iron metallic ball. This experimentwas performed as one of the activities in the framework ofthe EU-FP7 ‘Integrated System for Transport Infrastructuressurveillance and Monitoring by Electromagnetic Sensing’(ISTIMES) project (e.g. Proto et al (2010)); the experimentaimed monitor, by means of the integration of differentelectromagnetic sensing techniques, the damage to a concretebeam 16 m long affected by the impact of falling blocks.

With reference to the Montagnole experiment, we haveanalysed and tested complex wavelets (including complexGaussian transform) and combined geophysical coherencyorientation angle as useful tools to achieve sensor integrationwith the aim of detecting and characterizing subsurfacechanges.

2. Short description of the Montagnole geologicalsection

The Montagnole site is located at the north termination of theChartreuse massif (French Alps). This massif belongs to thesub-Alpine massifs. It is made of secondary and tertiarysedimentary rocks deposited in an oceanic context that haveundergone a west–east shortening, thus forming north–southfolds and faults (Gidon 1963).

Malatrait (1981) provides a detailed stratigraphicdescription of terrains located in the Montagnole quarry (i.e.the actual test site). According to this work, the quarry groundis made of an alternation of Portlandian limestone and marls(J2, 20 m in thickness), massive Portlandian limestone (J3, 20 min thickness), Berriasian dark marls (N1, 20 m in thickness) andan alternation of Berriasian clayey limestone and marls (N2,30 m in thickness). The combined GPR–ERT profile crossesJ2–J2 geological associations (i.e. Portlandian limestone andmarls). The upper part of these rocks is weathered. BothGidon (1963) and Malatrait (1981) indicate a complex tectono-structural pattern for this area. It was recognized that theporosity of the Portlandian limestone is relatively high (9–19%). The porosity of marls ranges in between 9–19%. Themarls are far more plastic compared with the Portlandianlimestone.

From a geophysical, the upper geological section of theMontagnole site presents a shallow layered structure formedby a low resistive landfill of up to 0.3–0.4 m followed by amedium resistive layer of up to 1.0–1.20 m and high resistivebedrock showing a very strong resistive nucleus. Moreover, afault can be inferred at 24 m (figure 3) about where an abruptchange in resistivity occurs.

3. A brief description of the Montagnole experiment

The Montagnole test site was initially designed to certifymetallic protection net systems that are used in mountains toprevent catastrophic rockslides over transport infrastructures.In the ISTIMES project, the test site was set up in a purelyresearch-oriented facility to verify the performance of non-invasive electromagnetic sensing techniques to monitor theprogressive damage to a purpose-built concrete beam structure.In particular, the experiment shows the progressive damage,at different stages, to a concrete beam and its foundations bymeans of falling blocks, thus allowing the different techniquesto be tested in the presence of hazards (Dumoulin et al 2011).

The test site of Montagnole is owned by IFSTTAR. Itenables one to drop heavy loads, up to 20 tons, from the top ofa cliff (figure 2(A)) down to structural systems in order to testresistance to big shocks. When the 20 ton block impacts thetested structure, its velocity reaches 35 m s−1 and the energyreleased during the impact may be as large as 16 000 kJ. Thehorizontal distance between the trajectory of the falling blockand the cliff is included in an arc of a circle of 12 m radius.This allows one to test large-scale structures that may evenbe slanted. In front of the cliff, observation platforms uphillenable monitoring of the experiments under safe conditions.

Figure 2 shows global views of the falling block test bedfacilities. To avoid damaging the crane structure during a block

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

Figure 1. A generalized scheme of noise appearing in near-surface geophysical studies (modified after Eppelbaum et al (2010)).

drop due to dynamical effects, a dropping hook was designedwith a special system (figure 2(B)). This consists of a reversedmass which can be adapted to the dropped block and it is huangwith the block. A wireless remote control facility is included inthis specific hook to liberate the block on request. Moreover,it is very important to pay attention to the repeatability ofresults concerning new devices for experiments. Regardless ofthe height of the fall, the accuracy of the impact was at least50 cm.

The radio-controlled system allows triggering monitoringand dropping at the same time. Due to bounce risk withthe dropped block, the safety of personnel is ensured bystrict operating rules. An observation platform has beenpositioned on an embankment along the test site in order tofollow experiments without risk (figure 2(A)). This observationplatform could also be used for remote control of sensortechnologies during the falling block phase. Figure 2(C) showsan overview of the test structure acquired from the top ofthe crane. The ERT and GPR measurements were carried outalong a line parallel to the beam. The mechanical solicitationat ground level was made at 5 m from the first foundation (seethe red circle in figure 2(D)). There is a distance of 70–80 m

between the experimental zone and the location where otherinstrumentations can be allocated.

The experiment shows the progressive damage, atdifferent stages, of a structure (compounded of a reinforcedconcrete beam, bearing systems, pillars fixed on dedicatedfoundations) by means of indirect and direct impacts of thefalling steel blocks; the damage was monitored and assessedby means of the different sensing techniques exploited in theISTIMES project (Proto et al 2010, Dumoulin et al 2011).

Steel blocks of 2.5, 5 and 10 tons have been used and twofalling actions were studied.

• The first falling action was used to study and diagnosestructural damage induced by an indirect impact of thefalling ball on the ground close to the structure. Theapproach leans gives a progressive energy release.

• The second falling action was used to study and diagnosestructural damage induced by a direct impact of the ballon the armed concrete beam. In this second configuration,due to the high risk of rapidly breaking the concrete beam,only the 2.5 ton steel ball was used and also tested with aprogressive energy release approach.

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

(A) (B)

(C )

(D)

Figure 2. (A) View from the observation platform. (B) View of the dropping hook hanging the 10 ton steel ball. (C) View of the reinforcedbeam structure from the Montagnole test site crane. (D) Schematic representation of the GPR and ERT line of observation and view at theMontagnole test site.

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Table 1. Parameters of the different launches (elevation and mass ofthe steel ball used).

Launch number Block mass (kg) Fall height (m)

1 2500 52 2500 83 2500 124 2500 125 2500 166 2500 207 10 000 58 10 000 159 10 000 30

10 10 000 4511 10 000 60

In this paper, only the study of ground evolution after thefirst falling action is investigated through the analysis of ERTand GPR measurements carried out before and after a seriesof 11 mechanical impacts (table 1).

Figure 2(D) shows the distribution of ERT electrodes over48 m on a line parallel to the beam where GPR measurementswere also carried out following this line.

During the Montagnole experiment (e.g. Dumoulin et al(2011)) GPR and ERT surveys were carried out before and24 h after finishing a series of mechanical impacts.

It should be noted that the process of microfault(microcrack) generation (without doubt, a series of strongmechanical impacts could produce microfaults at least in somepart of the studied domain) was studied in detail in Aleinikovet al (1999). It was concluded that the process of destructionin geological rocks may be considered as a phase transition.However, the exact mathematical estimation of this transitiondemands obtaining some physical parameters (coefficients ofsurface tension, stress and deformation tensors, etc) which areusually not observed. Therefore, the phase transition effectsmay be discovered through the use of modern data processingmethodologies.

4. The ERT and GPR surveys

Figure 3 demonstrates the preliminary results of the applicationof the GPR and ERT methods, respectively before any impactand after the series of 11 impacts on the ground as detailedin table 1. Recalculated GPR images before and after impactswith correct limiting times are presented in figures 7(A) and 9,respectively.

The ERT survey was carried out along a 47 m long surveyline close and parallel to the beam using an IRIS Syscal R2system equipped with a multicore cable and 48 electrodesspaced at 1 m with a Wenner–Schlumberger array.

Two-dimensional (2D) multi-electrode arrays provide a2D vertical picture of the sounded medium. The current andvoltage electrodes are maintained at a regular fixed distancefrom each other along a measurement line at the soil surface. Ateach step, one measurement is recorded by injecting currentsbetween two electrodes and measuring voltage between twoother electrodes. The set of all these measurements at this firstinter-electrode spacing gives a profile of resistivity values.

Then, the inter-electrode spacing is increased by a factor ofn = 2, and a second measurement line is carried out. Thisprocess (increasing the factor n) is repeated until the maximumspacing between electrodes is reached. Hence, increasing thevalue of n increased the depth of investigation (Samouelianet al 2005). The data are then arranged in a 2D ‘pseudo-section’plot that gives a simultaneous display of both horizontal andvertical variations in resistivity (Edwards 1977). An image ofthe ERT data is given under a trapezoid geometry with a largebasis in the lowest part of the cross-section and a short basisin the deepest part.

The Montagnole underground resistivity spatial distribu-tion is presented on a logarithmic scale for the sake of a clearervisualization (figure 3). The ERT data obtained in the experi-ment consist of resistivity values M(z, x) at a number of sub-surface points, where the range of the depth coordinate was0.519 m � zk � 7.434 m (11 levels). The range of the spatialcoordinates xk changes from [1.5, 45.5] m at depth z = 0.519 m(45 data points) to [1.5, 27.55] m at depth z = 7.434 m (27data points).

All ERT data have been inverted using the commercialsoftware Res2DInv (Loke and Baker 1996). This softwarepackage performs an automatic 2D inversion starting fromthe apparent resistivity data; such an inversion approach isbased on the smoothness constrained least-squares inversion(Sasaky 1992) implemented by a quasi-Newton optimizationtechnique.

From top to bottom (see figure 3), two ERT results suggestthe presence of a layered shallow structure formed by a lowresistive landfill up to 0.3–0.4 m followed by a mediumresistive layer up to 1.0–1.20 m and a high resistive bedrockshowing a very strong resistive nucleus (R) in the left part ofthe image. Moreover, a possible fault is located at an abscissaof about 24 m, where changes both in the GPR and in the ERTdata occur.

ERT data obtained after the last impact on the groundshow strong differences with respect to the previous ERTdata, especially at the left portion of tomography. Simplevisual analysis of the ERT image indicates a highly significantdecrease of the rocks’ resistivity values in the left part of thesection, where the series of impacts (at an abscissa of 10 m)was carried out. Some variations of the ERT portrait in thecentral part of the ERT section can also be easily outlined.Results of damage in the right part of the ERT section may bedetectable, but they are not sizeable.

The change in the resistivity pointed out by the ERT couldbe explained by: (1) change in the electrode contact resistanceand (2) the possible redistribution of water in the soil due tothe strong impact shocks where the fault acts as a permeabilityboundary.

The GPR survey was carried out using a GSSI SIR 3000system equipped with a 400 MHz antenna along a surveyline of 47 m (coincident with the ERT survey); markershave been put at each metre. Processing of ground GPR dataincluded trace cut, trace normalization, gain correction andfk-filtering.

By moving the antenna system along a selected profile(line) above the ground surface a 2D reflection profile

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

Figure 3. Preliminary GPR and ERT images after the series of impacts.

(radargram) is obtained in which for each location of theantenna system a trace is achieved where the amplitude andthe delay time of the recorded echoes (that can be related tothe depth of the underground reflectors) are drawn.

To pass from delay time t of propagation to the effectivepenetrating depth in metres h, we used the velocity of the

electromagnetic wave in the subsurface given by c = c0/√

εr

where c0 is the electromagnetic wave velocity in free spaceand εr is the relative dielectric permittivity of the soils so that

h = c × (t/2).

For the Montagnole site, we estimated εr = 12;accordingly, the depth reached by the electromagnetic wave is

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defined by the longest time onset. Note that these limiting timesare different: before the series of impacts t = 179.65 ns andafter the series t = 119.77 ns. It means that the electromagneticwave can reach 7.8 m before the impacts and about 5.2 mafter all mechanical impacts. This effect can be caused bythe increase in electromagnetic attenuation of the wave afterimpacts that in turn may be due to microcracks having beenfilled by conductive pore liquid.

Thus, we can suggest that the studied medium wassignificantly changed (phase transition as a result of theseries of impacts). It may be due to two reasons. Oneis the appearance of empty cracks and scattering of theelectromagnetic wave on them (wave cannot reach the previousdepth). The second is the strong absorption in the mediumtriggered by the appearance of conductive liquid in the cracks.In this investigation we tried to select the most reasonablehypothesis.

5. Integrated analysis of ERT and GPR dataobserved in the Montagnole experiment

As shown in figure 1, geophysical investigation of thesubsurface is accompanied by different kinds of noise. Forthe extraction of useful anomalies under conditions of lowsignal-to-noise ratio, various probabilistic and informationmethodologies can be applied (e.g. Khesin and Eppelbaum(1997)). First, an attempt was made to combine the GPRand ERT data by the use of informational methodology(Eppelbaum et al 2011b). However, practical realization ofthis approach was only able to mathematically join the dataand to reveal some anomalies at different slices (without anysuggestions about the nature of these anomalies).

The main point of the paper is to represent differentgeophysical observations (in our case GPR and ERT) in aunique dimensionless framework. The proposed methodologycan be applied to all the data formats obtained in the noisyenvironment as curves and maps, images and in any possibleappearance. In this work, we illustrate this methodology forthe results of GPR and ERT measurements obtained during theMontagnole experiment.

The flow chart in figure 4 depicts the main steps of theproposed method.

The first step is a presentation of the GPR and ERTmeasurements as geophysical 2D images. In the case of GPR,we have the radargram representing the scattered echoes inthe scene, where the horizontal coordinate is the location ofthe transmitter–receiver antenna and the vertical coordinateis the two-travel time. For ERT, the image depicts thedistribution of apparent resistivity in Cartesian coordinates,where x is the horizontal distance and z is the depth.

After that, the GPR and ERT images are presented ina ‘vector form’. At this stage, the methodology exploits thecomputation of the spatial gradients for each point of theimages; in this way we obtain an array of magnitudes anddirections of the gradients, separately for ERT and GPR. Thenwe select some arbitrary point in the image and determineamplitude and direction of the largest gradient in the vicinityof this point. We fill using these parameters all the image’s

space and store their directions and magnitudes. In such amanner we find a number of vectors having fixed directionsand magnitudes and this is the basis for histogram construction.This process is described in detail in section 6.

We compute separately the histograms of magnitudesand directions, i.e. compute the number of vectors withdefinite directions remembering their magnitudes. In thisway, we obtain two histograms for each sensing technique:a histogram of magnitude coherency (MC) and a histogram ofdirection coherency (DC). After normalizing these histogramswith respect to their maximum, we can combine them byconstruction of the phase GPR–ERT MC plane before andafter impacts (see below).

Accordingly, for the Montagnole experiment, wecomputed eight histogram functions based on the measuredGPR and ERT data. Initially, the GPR histograms of MC andDC before any impact and the same pair of GPR histogramsafter a series of impacts were computed. In a similar manner,we have four histograms for the ERT data before and afterimpacts.

These histograms were exploited to build a ‘phase plane’,where the horizontal axis accounts for the GPR DC histogramand the vertical axis is relative to the ERT DC histogram.In that way, for each direction we have a vector in the GPR–ERT coordinates. One can study the main characteristics of thecurve generated by the extreme of the vector with the completerevolution of the vector from 0 to 2π angle.

We have applied 2D wavelet analysis to these curves usingthe complex Gaussian wavelets. In this way, we constructedpolarograms that enabled us to find the extent of the mainaxis of the polarization ellipse in GPR–ERT coordinates oneach wavelet scale. The same procedure is applied also to thehistograms of MC.

The algorithm can be applied to the whole cross-sectionor to some intervals (parts) of it. For example, in the lastcase we divided the whole cross-section into 13 thin layersand performed ‘layer-by-layer’ analysis to extract a layermost sensitive to the mechanical impacts. This procedure wasapplied both before and after all impacts.

It should be stressed that this strategy may be employedwith any geophysical method for subsurface investigation.We applied wavelet analysis both to analysis of 2D imagesand ‘time’ series. As mentioned before, the main point ofthe analysis is based on the presentation of the ERT–GPRimages in vector form. Then, the magnitudes of the vectorfield correspond to the values of densities of the image andthe directions of the vectors coincide with the directions of themaximal gradients at the point of location of the vector.

6. Computing coherence characteristics

As the ‘signature’ of an image we use the histogram of thedistribution of the DC in the image. The coherency measureis widely used in seismic processing for the delineationand rendering of the subsurface seismic layers (e.g. Marfurtet al (1999)). We start with the calculation of the coherencymeasures in four directions for each pixel xi, j in the (GPRor ERT) image. Let zi, j denote the value of the (GPR, ERT)

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

Figure 4. A block-scheme of the integrated geophysical data processing in the Montagnole experiment.

quantity related to the pixel xi, j. The directions are: D0—horizontal; D1—45◦ slope; D2—vertical; D3—135◦ slope(figure 5(A)). Let r be a prescribed radius of the coherency.We denote the respective coherency measures in a pixel xi, j asC0, C1, C2, C3, which are defined as follows:⎧⎪⎨⎪⎩

C0i, j = (

∑rl=−r zi+l, j)

2

(2r+1)∑r

l=−r z2i+l, j

, C1i, j = (

∑rl=−r zi+l, j+l )

2

(2r+1)∑r

l=−r z2i+l, j+l

,

C2i, j = (

∑rl=−r zi, j+l )

2

(2r+1)∑r

l=−r z2i, j+l

, C3i, j = (

∑rl=−r zi+l, j−l )

2

(2r+1)∑r

l=−r z2i+l, j−l

.

⎫⎪⎬⎪⎭

(1)

The coherency measure is a proper tool to catch edges inan image or in a 3D volume, which corresponds to subsurfacelayers, rock fracture or other spatial non-homogeneities.Assume, for example, that the values zi, j in the horizontaldirection are constant: zi, j+l = zi, j, for l = −r, . . . , r. Then,C0

i, j = 1. In contrast, the more variable the values zi, j are inthe horizontal direction, the closer C0

i, j is to zero. The same istrue for C1

i, j,C2i, j and C3

i, j.Typically, neither of the four coherency measures Cm

i, j isclose to 1. Then, to detect edges and evaluate their direction,we calculate the coherency measures Cm

i, j, m = 0, 1, 2, 3, foreach pixel xi, j. Assume for a pixel (i, j) the C3

i j measure is thesmallest for the given pixel. Then, we define the second-orderpolynomial (parabola) y = p (x), which interpolates the threeremaining values of coherency measures. To be specific, wedraw the parabola through the points {(xk, yk)}, k = 0, 1, 2,

where x0 = 0, x1 = 45, x2 = 90, y0 = C0, y1 = C1, y2 = C2

(figure 6).Assume that the parabola y = p(x) achieves its maximal

value y = y when x = x. Then, we define the direction, whoseslope is x, as the DC Di, j in the pixel zi, j (see figures 5(A) and(B)). The ‘strength’ of the edge is determined by the magnitudey of the coherency vector at the given pixel. The point isregarded as belonging to an edge if the magnitude y at thispoint exceeds a predetermined threshold. The threshold may becomputed by different methods (according to user experienceand the image characteristics). It may be, for instance, someaverage value for the entire image, or for a horizontal or verticalband, or for some defined domain within the image.

Configuration of the edges can hence characterize animage (e.g. Averbuch et al (2010a)). To practically implementsuch a characterization, we calculate the values Di, j for allthe pixels of the image. In other words, we present the initialimage as a vector map where for each point we have a vectorwith some MC and DC.

After that, this procedure enables us to split any initialimage into two sub-images. The first is a map of directionsand the second is a map of magnitudes. Figure 7 shows mapsof distribution of GPR DC images obtained in the Montagnoleexperiment before (figure 7(A)) and after the series of impacts(figure 7(B)). The horizontal and vertical coordinates designate

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

(A)

(B )

Figure 5. (A) Simplified example of computing directions (x-axis > i, y-axis > j). (B) Direction of the coherency. The parabola achieves itsmaximum at x = 99◦.

0C

1C2C

3C

,i jx

SUBSURFACE LAYER

COHERENCY VECTOR

Figure 6. Sketch of the construction of the coherency vector.

distance (in m) and two-travel time (in ns), respectively.The colour in the map indicates the directions of vectors ateach point (see also the scale on the right-hand side of thefigure). Analogously, the maps of coherency magnitudes (MC)are obtained as results of the above-mentioned procedure.

Figures 8 and 9 display the GPR MC before and after anyimpacts, respectively. We must note that all the followinganalysis was carried out using both MC and DC images (bothfor GPR and ERT data). Figures 10(A) and (B) show ERTDC before and after the series of impacts, respectively. The

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(A)

(B )

Figure 7. (A) GPR coherency directions before the series of impacts. (B) GPR coherency directions after the series of impacts.

computed angle distributions characterize some geologicalpeculiarities (inhomogeneities) within the images. We set thethreshold based on the magnitude, let us say, 10% from the

peak vector magnitude of an image. It means that we ignoredsmall vectors with magnitudes less than 10% of the maximumamplitude. Figures 11(A) and (B) illustrate ERT MC maps

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

Figure 8. GPR coherency magnitudes before any impact.

Figure 9. GPR coherency magnitudes after all impacts.

before and after the series of impacts. Comparing figures 11(A)and (B) we can see the ‘reconstruction’ of the medium as awhole, especially in the place of impact. At the same time wemust note that ERT MC images are more informative comparedto the ERT DC ones. Nevertheless, we utilized all the availabledependences (eight sections).

The next step includes computation of direction andmagnitude histograms. The number of lots can be chosen

arbitrarily. In this study, we build the DC histograms with36 lots (5◦ sectors). The MC histograms comprise the samenumber of lots. Thus, the whole range of magnitude fromthe maximum value to the threshold is divided into 36 lotsand in each lot the relative magnitude with respect to thepeak value is calculated. The distributions of the values Di, j

and the vector magnitudes among lots of the histograms areconsidered as signatures of the GPR image before and after the

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(A)

(B )

Figure 10. ERT coherency directions before (A) and after (B) the series of impacts.

(A)

(B)

Figure 11. ERT magnitude directions before (A) and after (B) the series of impacts.

series of impacts. By such a procedure, we reduce sub-imagesto non-dimensional dependences as (1) a function of numberof directions versus angle, and (2) number of vectors withinthe chosen lots versus magnitude in the case of the magnitudehistogram.

Figure 12 show the GPR DC histograms. Thecorresponding GPR MC histograms were also computed.Analogously, the histograms of ERT coherency for directionsand magnitudes (ERT DC and MC) were computed (we do not

show the histograms for reasons of space). After normalizingall histograms to their maximal values we obtained fourcoupled coherency GPR–ERT (directions and magnitudesbefore and after) histograms. The histograms enable us tostudy the dependence of ERT as a function of GPR.

The main point of the proposed approach is itsapplicability to any curves, maps, images and cross-sectionsresulting from the surveys using geophysical tools (optical,physical, geochemical, etc). Integration of different fields

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(A)

(B)

Figure 12. (A) GPR coherency direction histogram before the series of impacts. (B) GPR coherency direction histogram after the series ofimpacts.

(images) should have clear physical (geological, geochemical)substantiation (Eppelbaum et al 2011b).

7. Computing complex Gaussian functions

The complex Gaussian and Morlet wavelets were utilized.It appears that the results are similar, therefore we briefly

describe just the Gaussian wavelet and the main resultsof the method. The complex Gaussian functions have beenapplied to the eight histograms obtained from both GPR andERT observations. Each histogram has been normalized toits maximal value. Then we obtain eight scalograms, i.e.distribution of the wavelet coefficients on different scales(frequencies) as a function of direction angles (time).

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

Figure 13. Complex Gaussian wavelet of order 4.

The complex Gaussian function is expressed as

cGaun(x) = Cn · ∂n(e−ixe−x2)

∂xn, (2)

where Cn is a constant factor which depends on the derivativeorder n and is computed to normalize the Gaussian wavelet.The real and imaginary parts of the Gaussian wavelet of order4 are shown in figure 13.

Computation of the coherency maps and their waveletanalysis give us eight scalograms, i.e. distribution of thewavelet coefficients on different scales (frequencies) as afunction of direction angles and magnitudes (time). Thewavelet coefficients account for the ‘inner product’ of theinvestigated functions (the histograms) with the wavelet: largevalues of the wavelet coefficients mean good ‘similarity’ ofthe investigated function with the wavelet. The large valuesof the scalograms coincide with the singularities of the testedhistograms.

Use of the complex Gaussian function enables us toperform a polarization analysis of the dependence of ERTversus GPR. The scale-by-scale ellipticity and the ellipseorientation (polarization angle) were computed based on awavelet coherence matrix J (e.g. Schekotov et al (2007)):

J =∥∥∥∥

AxA∗x AxA∗

y

AyA∗x AyA∗

y

∥∥∥∥ , (3)

where Ax and Ay are the wavelet coefficients of the first andsecond geophysical methods, respectively.

Wavelet decomposition Ax = AGPR(DA) and Ay =AERT(DA) are found at each level Lj, i.e. Ax(DA) ⇒AGPR(DA, Lj) = Aw

x and Ay(DA) ⇒ AERT(DA, Lj) = Awy .

The dimension of Awx and Aw

y is the number of wavelet levelsused in the analysis. Assume that we have a histogram of DC inthe range of the direction angles from 1◦ to 180◦ with sampling1◦ rate. We apply the ‘cgau4’ wavelet; then for the frequencyrange from the lowest frequency 1/180 to the frequency 1/5 wefind that the wavelet scale range contains 126 corresponding

scales from 1 to 90. A similar procedure has also been appliedto the maps of the MC with the same number of lots (15)arranged from the maximal magnitude to the lowest thresholdvalue which has been chosen experimentally as 10% from themaximal magnitude.

The complex vectors Awx and Aw

y allow us to calculate theparameter of ellipticity β and orientation angle (α) (Fowleret al 1967, Born and Wolf 1980, Schekotov et al 2007):

β(Lj) = 1

2sin−1

2Im(Aw

y Aw∗x

)√(

Awy Aw∗

y − Awx Aw∗

x

)2 + 4∣∣(Aw

y Aw∗x

)∣∣2 ,

(4)

α(Lj) = 90

πtan−1

Re(Aw

y Aw∗x

)

Awy Aw∗

y − Awx Aw∗

x

. (5)

Here an asterisk means a conjugate variable.The polarization angle is assumed to be the angle between

the main axis of the polarization ellipse and the horizontal GPRaxis with counter clockwise defined as the positive direction.Finally, we find the values of α and β angles averaged over allscales Lj, i.e.

{α, β} = 1

NL

NL∑j=1

{α, β}(Lj).

Here, Lj is the index of the Lth level (1 � Lj � 90) andNL is the number of the levels. In our case NL = 126. Weconsider parameters α and β as the integral characteristics ofthe coupled GPR and ERT methods. In the described example,these are the DC of the GPR and ERT images. Finally, the twoscalewise average values allow us to get two characteristicsof the GPR–ERT couple. The first is the polarization angleα in the GPR–ERT coordinate system, and the second is theellipticity β.

Figure 14 shows the angle dependence of the polarizationangle in the GPR–ERT plane before and after impacts. First

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

Table 2. Average polarization angle (α) and average ellipticity (β) before and after impacts.

Before After

α (deg) β (deg) α (deg) β (deg)

Coherency Coherency Coherency Coherency Coherency Coherency Coherency Coherencydirection magnitude direction magnitude direction magnitude direction magnitude(DC) (MC) (DC) (MC) (DC) (MC) (DC) (MC)

23.1 −3.2 −0.03 0.02 26.4 16.1 0.2 0.07

Figure 14. Calculated combined GPR–ERT ellipse orientation angleof the MC.

of all, two curves are shifted with respect to each other insome angle ranges at 30% (∼120◦–140◦). The vector rotatescounter clockwise, that is, to the ERT axis. The averagechange in the orientation angles is about 20%. We proposethat it means that the conductive properties of the wholecross-section are changed noticeably because of the series ofimpacts. The values of average polarization angle (α) andaverage ellipticity (β) before and after mechanical impactsare shown in table 2. One can see that the most sensitivecharacteristic is the polarization angle of the coherencymagnitude. The vector of MC rotated at about 20◦ counterclockwise. It may be interpreted as a general attenuationof the received electromagnetic wave intensity in the GPRmethod and increasing integral electric resistance of the wholecross-section. In turn, it can be produced by microfracturesinduced by the mechanical impacts. On the one hand, thescatter microcracks the high frequency electromagnetic waves,on the other hand, the microcrack domain can lead to bothincreasing and decreasing of the effective resistance of therocks comprising this section. It depends on the filling ofcracks. If the cracks are filled with highly conductive, let’s

Figure 15. Calculated combined GPR–ERT MC versus depth performed using a layer-by-layer procedure for the cases before any impact(blue solid line) and after all impacts (red dashed line).

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J. Geophys. Eng. 10 (2013) 025017 L Alperovich et al

Figure 16. The same as in figure 15, but for the coherency directions (DC).

say, pore fluid, then the effective resistance goes down andvice versa decreases in the case of empty cracks.

The same analysis has been applied not only to thewhole cross-section, but also to separate layers composingthis section. Figures 15 and 16 demonstrate the polarizationangle of MC (figure 15) and DC (figure 16) versus depth. Onecan see that the assumed parameter (polarization angle) mightbe used not only for ascertaining the changes in geologicalmedium, but also for revealing some local variations in thegeological section. Figure 15 shows that the most significantchange is fixed at the depth of 2–3.5 m. We can proposethat the revealed tendency (turning to large angles) mayindicate, for instance, faults appearing in the interval of 2–3.5 m filled by a conductive liquid. We can assume that24 h (time interval from the last impact to the beginning ofgeophysical measurements) is sufficient time for the fracturesto fill with conductive underground water. We should also notethe comparatively high porosity of the Portlandian limestoneand marls composing the Montagnole geological site (seesection 2).

We suggest that the interpreting methodology presentedmay have significant perspective for the integrated geophysicalmonitoring of the subsurface geological medium. Somelimitations of this approach may be associated with thedifferent resolutions associated with concrete geophysicalmethods (for instance, the GPR resolution greatly exceedsthe ERT resolution) and discrepancy in obtaining images (forexample, GPR and ERT images essentially misfit in the lowerpart of the geological section).

8. Conclusion

The wavelet approach was used for the development ofenhanced (e.g. coherence portraits) and combined images ofgeophysical fields observed in the Montagnole experiment(French Alps). Application of such parameters as a complexGaussian wavelet combined scalogram and combined GPR–ERT coherency orientation angle allows non-conventionaloperations of geophysical method integration and correlationin subsurface geophysics to be performed. The main point ofconducted analysis is based on the presentation of the ERT–GPR images in vector form, obtaining an array of magnitudesand directions of the gradients separately for ERT and GPR,and utilization of the coherency measures. In order to combinethe two methods we utilized a wavelet transform based on thecomplex Gaussian function. The developed methodology wasapplied not only to the whole cross-section, but also to separatelayers of this section.

Analysis of the combined GPR–ERT images before andafter impacts testifies to a significant change in the electricand electromagnetic properties of the geological section. Inparticular, a general reduction of intensity of the receivedelectromagnetic waves in the GPR method and increasing ofthe integral electric resistance (ERT method) of the wholecross-section was fixed. We suggest that the interpretingmethodology has an essential perspective for integratedmonitoring of the subsurface geological medium (as well asfor the development of combined deep physical–geologicalmodels). Despite the fact that this methodology might be

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applied to any geophysical field integration, the differencein resolution of the employed methods should be taken intoaccount.

Acknowledgments

The authors would like to thank two anonymous reviewerswho thoroughly reviewed the manuscript, and their criticalcomments and valuable suggestions were very helpful inpreparing this paper. The research leading to these resultshas received funding from the European Community’sSeventh Framework Program (FP7/2007-2013) under GrantAgreement no 225663.

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