Journal of Applied Geophysics 78 (2012) 94–101

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Journal of Applied Geophysics

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Inference of the two dimensional GPR velocity field using collocated cokriging ofDirect Push permittivity and conductivity logs and GPR profiles

Erwan Gloaguen a,⁎, René Lefebvre a, Jean-Marc Ballard a, Daniel Paradis a,b,Laurie Tremblay a, Yves Michaud b

a Institut National de la Recherche Scientifique, 490, Rue de la Couronne, G1K 9A9, Québec, Canadab Geological Survey of Canada, 490, Rue de la Couronne, G1K 9A9, Québec, Canada

⁎ Corresponding author. Tel.: +1 418 654 2637.E-mail address: Erwan_Gloaguen@ete.inrs.ca (E. Glo

0926-9851/$ – see front matter © 2011 Published by Eldoi:10.1016/j.jappgeo.2011.10.015

a b s t r a c t

a r t i c l e i n f o

Article history:Received 7 December 2010Accepted 27 October 2011Available online 20 November 2011

Keywords:GPRCPTGeostatisticsMultivariate analysis

One of the main difficulties in processing GPR surface data is to infer the ground velocity models in order toconvert the radargram time scale into depth. Common conversion techniques usually fail if the ground lithologyis complex. Consequently, the ground velocity has to be guessed and generally assumed laterally invariant. In thispaper, we present a newgeostatistical approach to the velocity analysis of sandy aquifer based on the interpolationof the Direct Push relative permittivity and conductivity logs. The first step of the method consists in scaling theradargram to maximize the correlation with the logs. Then, collocated cokriging of electrical relative permittivityand conductivity and the scaled radargram information is computed. The cokriged fields are converted in velocityfield following EMhypothesis and the time to depth conversion of the original radargram is then applied using theinferred velocity field. Results on real data analysis show that themethodworkswell in fairly homogeneous sandyaquifer with intercalated dipping thin silt layers.

© 2011 Published by Elsevier B.V.

1. Introduction

GPR data analysis is of great importance in hydrogeological applica-tions since it gives spatial information of the structure of the shallowaquifers and on the electromagnetic (EM) wave velocity that can belinked to water content (Sénéchal et al., 2005; Topp et al., 1980). Themain difficulty in processing surface GPR data is to accurately infer thespatial variation of the EM velocity field in order to perform the timeto depth conversion and to convert the velocity field in hydrogeologicalparameters. The classical techniques are based on Common MidPoint analysis and/or on the picking of diffraction patterns (VanOvermeeren et al., 1997). In order to be accurate, the first methodimplies the presence of sub-horizontal homogeneous layers and thesecond one implies the presence of numerous diffracting objects ineach geological unit, more or less homogeneously distributed overthe studied area. Other techniques such as multiple-offset surveysallow for the estimation of the GPR velocity field (Perroud andTygel, 2005), but the low number of measurable offsets due to therapid EM attenuation at radar frequency is itsmain drawbacks. However,promising results were reported in the literature (Bradford, 2006, 2008;Fisher et al., 1992). In this paper, we propose a geostatistical methodthat integrates Direct Push (DP) technology and surface GPR data for

aguen).

sevier B.V.

both improving the estimation of DP logs between boreholes and tobetter estimate the EM velocity field along the GPR profile.

The use of geostatistics is not new in geophysics (Doyen, 1988;Dubrule, 2003). Early in the geostatistical development in the 1970s,it has been recognized that the high spatial sampling of geophysicaldata can help the estimation of variables of interest measured alongboreholes (Chilès and Delfiner, 1999). The first applications requiredthe full cokriging system to be solved (Dubrule, 2003). The full cokriginghas several drawbacks as numerical instability caused by the oversampled geophysical data and the modeling of the full model ofcoregionalization (variograms of all the variables and cross-covariancebetween variables). Several simplifications of the full cokrigingwere developed in the petroleum industry where secondary variableas 3-D seismic is always available. Particularly, the kriging with anexternal drift (KED) is used as a routine to interpolate the depth tothe top of reservoirs (measured at only a few wells) and constrainedby the seismic reflection picked on the 3-D seismic cube (Chilès andDelfiner, 1999). Inside the KED estimation, the seismic travel timesare linearly scaled to best describe the local mean of the primary variableand the residual is computed using simple kriging. KED requires that thesecondary variable (geophysical data) is a large-scale representation ofthe primary variable and that both variables are strongly linearly related.In our case, the GPR radargram cannot be considered as a smooth repre-sentation of the ground electrical conductivity and permittivity. Anothermethod, called collocated cokriging (CCK), allows taking advantage of afully sampled secondary variablewithout having the numerical instabilityof the full cokriging (Xu et al., 1992). Here, CCK is used to interpolate the

Fig. 1. Flow chart of the proposed method. 1) data, 2) data pre-processing, 3) CCK,4) 2D velocity field and 5) time to depth conversion.

95E. Gloaguen et al. / Journal of Applied Geophysics 78 (2012) 94–101

electrical properties measured with the DP along several holes andconstrained by the scaled radargram. The CCK method is first pre-sented. Then, the surveyed site and the data collection and analysis aredescribed.

2. Collocated cokriging

When the secondary variable (here the radargram) is much moredensely sampled than the primary one (here the DP logs), the fullcokriging system becomes numerically unstable because the correlationbetween close secondary data is much greater than the correlationbetween distant primary data (Goovaerts, 1997). Also, secondarydata that are exactly or very close to the location of a primary datatend to drive the estimate of the primary data at that location.Therefore, in the full cokriging estimation, the primary data dependsonly on one secondary variable but still needs to solve the entire system.Several techniques exist to take advantage of the over sampled second-ary data. Among them KED, simple kriging with local varying mean(SKLM) and CCK are the most used ones (Chilès and Delfiner, 1999).The CCK technique (Xu et al., 1992) considers only the collocatedsecondary variable at location where the primary variable has to beestimated. One of the main differences between the SKLM estimatesand CCK lies in how the secondary variable is taken into account. ForSKLM estimates, the secondary variable supports only informationabout the local mean of the primary datawhereas in CCK, the secondarydata influence directly the estimation of the primary one. Moreover,contrary to the kriging, in the CCK, the global linear correlation betweenvariables, as captured by the cross-covariance, is taken into account. Theequation of CCK estimate is:

Z�DP uð Þ ¼

Xn uð Þ

i¼1

λiZDP uið Þ þ λ2 Zradar uð Þ−mradar½ � þmDP ð1Þ

with the non bias condition:

Xn uð Þ

i¼1

λi uð Þ þ λ2 uð Þ ¼ 1 ð2Þ

where Zi is the value of the variable i, u is the location where ZDP has tobe estimated, n(u) is the number of measured ZDP, λi is the krigingweights for the primary variable and λ2 is the krigingweight for the sec-ondary variable, mi is the mean of variable i.

Unlike full cokriging and under the assumption of proportionalitybetween the covariances of the variables, CCK requires only the compu-tation and modeling of the covariance of the primary variable (C11(h))and of the cross-covariance between primary and secondary variable(C12(h)). Using the assumption of proportional covariances betweenprimary and secondary variables (Markov Model assumption), theCCK procedure can be further alleviated (Goovaerts, 1997):

C21 hð Þ ¼ C21 0ð ÞC22 0ð ÞC22 hð Þ ð3Þ

where C22(0) is the variance of secondary variable and C21(0) is thecorrelation between primary and secondary variables.

Therefore, the Markov Model only requires the computation of thecovariance of the secondary variable and the simple correlation betweencollocated primary and secondary data.

3. Case study

In this section the surveyed site and the data acquisition arepresented as well as the proposed methodology. In order to guide thereader, the flow chart of the proposed method is shown in Fig. 1.

3.1. Site description and data acquisition

The GPR and DP data were acquired in the Beaurivage's riverwatershed in the province of Québec, more specifically in the sub-watershed of a regional landfill located in St-Lambert-de-Lauzon, Québec,Canada (Fig. 2). The stratigraphy of the study area corresponds to a 10 mthick sand layer hosting an unconfined aquifer having a near-surfacewater table overlying clayey silts and tills. The sandy aquifer con-tains thin intercalated silty to clayey layers that show electricaland hydrogeological property contrasts compared with the sand.In this relatively flat area, natural streams as well as agriculturaland forestry drainage networks mainly influence the groundwaterflow. The geophysical and hydrogeological data sets are dedicatedto the hydrogeological and environmental characterization of thesub-watershed of the regional landfill. The full data set consists in21 km of GPR profiles, 5 km of surface electrical tomography, 35 DirectPush (DP) sites combining electrical, mechanical, and geochemical logsand 22 full-screened well installations with complete hydraulic logs.

In this study, the GPR data consists in a 450 m long profile (greenline in Fig. 2) acquired with a pulse EKKO IV system with 100 MHzantennas (Fig. 3). The spatial sampling is 25 cm. The data processingis conventional (Cassidy, 2009). It consists in applying a DEWOW, aDC-shift, first shift, a SEC gain and a bandpass filter (40–110 MHz).The roc topography and the thin intercalated clayey to silty layers(highlighting the mode of deposition) appear clearly in the radargram(Fig. 3). Also shown in Fig. 3, the EM wave attenuation shows highlateral variability from high (between 150–250 m and 400–450 m)to low elsewhere.

Based on the GPR information (Fig. 3), a Direct Push (DP) soundingcampaign was carried out over the entire Beaurivage's river watershed

Fig. 2. Location and hydrographic system of the study area. The blue arrows indicate the water flow direction and the red curve shows the border of the studied watershed. Thelandfill area is delimited with a black line. The radar survey location is indicated with a thick green line.

96 E. Gloaguen et al. / Journal of Applied Geophysics 78 (2012) 94–101

area. In the specific area where the present research focuses on, threesites were investigated with the DP system (Fig. 4) driven by the radar-gram reflection lateral heterogeneity (strong and weak attenuations).The DP system shown in Fig. 4 involves a cone penetrometer testingsystem including pore pressure measurement (CPTu) combined witha relative permittivity and resistivity (SMR) probe. The CPTu/SMRsoundings were conducted using a Geotech 605-D rig that is crawler-mounted for best all-terrain capability. A 15 cm2 penetrometer conewith a 60° conical tip was used in accordance with ASTM D3441standards (ASTM 2000). The penetrometer is pushed vertically intothe soil at a constant rate of 2 cm/s, though this rate must be reducedwhen compact layers are encountered. Inside the probe (Fig. 4), twoload cells independently measure the vertical resistance against theconical tip and the side friction along the sleeve. A pressure transducerin the cone is also used tomeasure the porewater pressure as the probeis pushed into the ground. Pore pressure is an indicator of the presence

Fig. 3. Radargram showing the water table (white line), the sand–roc interface (black line).the sandy aquifer correspond to thin clayey or silty layers. The lateral exaggeration is abou

of clay andwasused to correct tip stress data. The SMRprobe is composedof four electrodes that are connected directly behind the penetrometer(Shinn et al., 1998). The inner two rings are used to measure soil relativepermittivity. The relative permittivity probe operates at 100 MHz, there-by reducing the effects of soil type on the measurement and improvingthe ease to compare with GPR data. The instrument measures shifts inthe resonance frequency as the signal passes through the soil. This changein frequency is related to soil moisture content. Spacing between the twoinner rings is 3 cm. The resistivity measurement employs the outer tworings of the SMR probe to apply the current and to measure the voltagedrop. The outer electrodes are spaced 9 cm apart. The probe operatesat a frequency of 1000 Hz to avoid soil polarization effects. Fig. 5shows an example of the real time data given by the DP soundingsystem corresponding to the site P02. The vertical data sampling is3 cm. The permittivity variation indicates that the water table isabout 1 m. It is to be noticed that the thin layer permittivity variability

Here roc stands for everything that is below the aquifer. The oblique reflections insidet 30:1.

Fig. 4. Pictures of the DP system used in this study. On the left, we can see the data acquisition system that allows getting the physical property in real time during sounding. On theright, the DP probes are shown and the measuring tools are described.

97E. Gloaguen et al. / Journal of Applied Geophysics 78 (2012) 94–101

is not clearly visible in the raw data due to the scale. Also, DP permitsrapid installation of permanent wells without requiring the use ofgravel-pack. Three permanent full-screened wells were installed, onefor two of the three sites shown in Fig. 6. The DP electrical relativepermittivity and conductivity logs were measured along five holes(Fig. 6). The relative permittivity logs are almost constant over thearea except at the top of P02 and P03 where the relative permittivityis low and the resistivity is high. This indicates the presence of asuperficial low water content zone. In the case of the resistivitylogs, they confirm the lateral variability of the EM attenuation asobserved on the radargram i.e., P01 and P03 show lower resistivitythan P02. Water conductivity profiles measured along the full-screened wells confirmed that the resistivity changes are associatedwith variability of the water conductivity. One may suggest calculatingthe interpolation of the physical properties between the wells basedonly on the log data. However, the high short scale vertical variabilityon the logs, like the high lateral variability shown on the radargramrenders the lateral interpolation hazardous. Especially, the dippinglayers that can be shown in Fig. 3 cannot be guessed based only onthe logs.

It is to be noticed that GPR resolution and scale of support at100 MHz frequency is of the order of several decimeters. DP soundingshave a resolution of 3 cm but a scale of support of the order of the sizeof the tool, that is 15 cm. Due to difference between DP and GPR datascales and resolution, DP logs have to be upscaled at GPR resolution.Consequently, DP sounding was integrated on a 30 cm with a movingwindow every 15 cm.

3.2. Two-dimensional velocity model estimation combining DP and GPRinformation

First, the DP electrical relative permittivity (εr) and conductivity(σ) are converted into EM velocity using

α ¼ ωc

12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2r þ

σωε0

� �2s

−εr

24

35

12

; ð4Þ

and

β ¼ ωc

12

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2r þ

σωε0

� �2s

þ εr

24

35

12

ð5Þ

v ¼ ω=β ð6Þ

where,α is the attenuation in (Np/m),ω (rad), c is the EM velocity infree space (m/s), εr is the relative permittivity of the probed material,σ is the electrical conductivity of the probed medium (S/m), ε0 is thepermittivity of free space (F/m), β is the phase constant (rad) and v isthe velocity of the probed medium (m/s).

These EM velocities permit to convert DP depths into pseudo EMtravel time space (Step 2 in Fig. 1). This procedure renders the super-position of the radargram and relative permittivity logs possible in thetime domain and permits to avoid the possible errors in depth domaindue to inappropriate velocity field on the GPR data (Fig. 7). In Fig. 7, itis obvious, that the agreement of both data in the time space is goodbut due to the different scales of the radargram and log data, theircorrelation is low (0.3). This is mainly due to the nonlinearity of therelation. In the previous section, it has been shown that computationof CCK requires the relation between logs and radargram to be linear.Consequently, the radar reflections were iteratively scaled by locallyadjusting themean and variance until the correlation between collocatedvariables is maximized. It consists in a simple variable change as used inLe Ravalec-Dupin (2005). This process is not fully automatic and requiresseveral trials and errors. Eq. (4) allows guiding the scaling by givingan idea of the attenuation. Also, it is to be noticed that the employedtechnique works only in the presence of small reflecting layers in aquasi-homogeneous background. If this is the case, the reflectionsare collocated with the laterally varying thin layers. After severaliterations, the correlation between logs and the collocated radar datareaches the value of 0.62. The paradigm behind the scaling step issupported by the fact that the scaled radargram acts like a structuralguide for the interpolation of themeasured log data. After the processing,the radargram is considered as an image that shares a statistical relationwith the DP data. This step of themethodology allows inputing the petro-physical knowledge into the processing but also requires the user toensure that he is not biasing the results.

In addition, to compute CCK, the experimental variogram of theprimary variable has to be calculated and modeled. The modeledvariogram that best fit the experimental one is a spheric withranges of 24 and 150 in Y and X directions, respectively. Then,CCK of relative permittivity (Step 3 in Fig. 1) is computed usingthe modeled variogram, the measured permittivity and the scaledradargram (Fig. 8). As the electrical properties are interpolated onthe entire area, they are converted in terms of velocity usingEqs. (5) and (6) (Step 4 in Fig. 1). The data interpolated in the traveltime space are converted into depth using the 2D velocity field (Step5 in Fig. 1). This procedure (Dix, 1955) consists in multiplying thevelocity field and the GPR travel times column-by-column startingfrom the top to bottom. Finally, the corresponding permittivities

relativepermittivty

ResistivityOhm.m

Fig. 5.Measured parameters during one run of DP acquisition. From left to right: sleeve stress (kPa), tip stress (kPa), ratio of sleeve and tip stresses (%), pore pressure (kPa), relativepermittivity, resistivity (Ohm.m), and inclination and classifications of soil type using Eslami and Robertson classification based on the stresses.

98 E. Gloaguen et al. / Journal of Applied Geophysics 78 (2012) 94–101

are resampled along the vertical axis on a 5 cm spaced grid. Thecokriged image (Fig. 8) respects both the amplitude of the mea-sured logs and the spatial structure of the radargram reflections.It is to be noticed, that the relative permittivity in high conductive

zones tends to be higher compared to the original radargram. Theexplanation is that the changes in amplitude in the radargram areprincipally originating from changes in conductivity. The same pro-cedure was applied for the conductivity field but the description

Fig. 6. Relative permittivity and resistivity measured by the Direct-Push system: (a) permittivity; (b) logarithm of the resistivity.

99E. Gloaguen et al. / Journal of Applied Geophysics 78 (2012) 94–101

and the comparison with a profile of electrical tomography acquired atthe same location are discussed in the next section (Fig. 9b).

3.3. Comparison with surface electrical tomography and water contentestimation

For high resistive ground, the radar reflexions are mainly drivenby the contrasts of permittivity. However, lateral variations of theamplitudes are associated with variation in the attenuation that ismainly driven by changes in resistivity. In this case, the proposedhypothesis is that the radargram supports structural informationabout the resistivity. To validate the results, the proposed methodis compared with a profile of electrical resistivity acquired at thesame location as the radar data using a SYSCAL Pro from IRIS In-struments (Fig. 9a). The mode of acquisition was a dipole–dipolearray with 2 m electrode spacing leading to 8394 measured apparentresistivity data. The total length of the roll-along profile is 480 m.Fig. 9a shows the results of the inversion of the measured apparentresistivity. The inversionwas conducted using Res2Dinv from geotomosoftware, version 3.55. The parameters of the Gauss–Newton inversion

Fig. 7. Superposition of the relative permittivity DP logs and the radargram. Depths of the mea

were robust constraints, with optimized smoothing factor, Jacobiancalculation and a squared inverse grid size of 1 m-edge length. Thefinal RMS error between calculated andmeasured apparent resistivitiesafter 5 iterations is 17%. The range of the resistivity of the inversemodelwas constrained to be in the range of the measured DP resistivity. Asthese constraints are soft constraints and because they are in contradic-tion with the measured surface apparent resistivity, the values of theinversed model exceed the range of the upper and lower constraints.As the paper does not dealwith electrical inversion, the reader is invitedto consult specific references on this topic if more information is needed(Butler, 2005).

Fig. 9b shows the results of the proposed method along the sameprofile. In order to compare the two techniques, resistivities of theproposed method were average on the same cell size as the electricaltomography, i.e. 1 m-edge length squares, to take into account thesupport effect (Chilès and Delfiner, 1999; Sénéchal et al., 2005). Itcan be seen, that globally, the large features are in good agreementwith Fig. 9a. However, there are 1) local and 2) global differences.1) Locally, there are differences because DP data were available only atsome locations and the radargram only supports structural information

sured DP electrical properties are converted into travel times using low-loss assumption.

Fig. 8. Results of the CCK relative permittivity estimates and measured DP relative permittivity.

100 E. Gloaguen et al. / Journal of Applied Geophysics 78 (2012) 94–101

of the resistivity field (anomaly in Fig. 9a at 0 to 1 m depth and 400 to450 m along profile is not reproduced in Fig. 9b) and the resolution indepths of the electrical inversion is not accurate (anomaly in Fig. 9a at10 to 12 m and 300 to 350 m along profile is not measured with theDP at this depth; 2) the average resistivity is higher in the electricaltomography image because of the averaging inherent to the surfaceelectrical tomography acquisition mode and to the inversion pro-cess itself. However, the aspect of the resistivity image is in goodagreement with the physics of the phenomenon. Indeed, after waterconductivity logs measured in the full-screened wells, the high conduc-tive zones proved to correspond to high conductive water.

4. Conclusion

As a first attempt in defining an optimal environmental charac-terization methodology, the use of surface GPR data helps the choiceof optimal DP sounding location and well installation. Geostatisticaldata fusion of DP and GPR technologies allows estimating the DPdata between boreholes respecting the GPR structure and allowsfinding an accurate EM velocity models between the holes. GPRradargram guides the spatial structure of the interpolation of themeasured physical data between several DP logs. The main

Fig. 9. a) Log of the inverse of the surface electrical tomographic apparent resistivity and thresistivity.

limitation of the technique, in its present form, is that it is only appli-cable in geological contexts where the GPR reflections are due tothin layers in a quasi-homogeneous background. We recommendthe use of forward modeling in order to handle different lithologicalcontexts.

Acknowledgments

The authors want to acknowledge the “Régie intermunicipale degestion des matières résiduelles (LET, Saint-Lambert-de-Lauzon)”,the professor Gloaguen's NSERC (#355856) and FQRNT (#125228)grants for their financial support.

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