2D magnetic resonance tomography applied to karstic conduit imaging

Available online at www.sciencedirect.com

ics 63 (2007) 103–116www.elsevier.com/locate/jappgeo

Journal of Applied Geophys

2D magnetic resonance tomography applied to karsticconduit imaging

J.-F. Girard a,⁎, M. Boucher a,b, A. Legchenko c,1, J.-M. Baltassat a

a Bureau de Recherches Géologiques et Minières (BRGM), 3, avenue C. Guillemin, BP 6009, 45060, Orléans Cedex 2, Franceb Institut des Sciences de la Terre d'Orléans (ISTO), UMR6113 CNRS/Université d'Orléans, Bâtiment Géosciences,

Rue de Saint Amand, BP 6759, 45067 Orléans Cedex 2, Francec Institut de Recherche pour le Développement (IRD), LTHE, BP53, 38041, GRENOBLE Cedex 9, France

Received 21 April 2006; accepted 22 August 2007

Abstract

Karstic conduits play a crucial role for water supply in many parts of the world. However, the imaging of such targets isgenerally a difficult task for most geophysical methods. Magnetic Resonance Sounding (MRS) is a geophysical method designedfor imaging of water bearing structures. Initially, MRS was developed for characterizing horizontally stratified aquifers. However,when applying a 1D MRS measuring setup to the imaging of 2D–3D targets, the size of which may be much smaller than the loop,the accuracy and the lateral resolution may not be sufficient. We have studied the possibility of simultaneously processing severalMRS aligned along a profile to perform a Magnetic Resonance Tomography (MRT). This work emphasizes the gain of resolutionfor 2D–3D imagery of MRT versus the interpolation of 1D inversion results of MRS along the same profile. Numerical modellingresults show that the MRT response is sensitive to the size and location of the 2D target in the subsurface. Sensitivity studies revealthat by using the coincident transmitting/receiving (TX/RX) setup and shifting the loop around the anomaly area, the depth, sectionand position of a single karstic conduit with a size smaller than the MRS loop size can be resolved. The accuracy of the resultsdepends on the noise level and signal level, the latter parameter being linked to the depth and volume of the karstic conduit and thewater content in the limestone matrix. It was shown that when applying MRT to the localization of 2D anomalies such as karsticconduits, the inclination of the geomagnetic field, the orientation of the MRT profile and the angle of crossover of the conduit bythe MRT profile must be taken into account. Otherwise additional errors in interpretation should be expected. A 2D inversionscheme was developed and tested. Both numerical and experimental results confirm the efficiency of the developed approach.© 2007 Elsevier B.V. All rights reserved.

Keywords: MRS; MRT; SNMR; PMR; Magnetic resonance sounding; Magnetic resonance tomography; Surface nuclear magnetic resonance;Proton magnetic resonance; Karstic conduit; Groundwater

1. Introduction

Magnetic Resonance Sounding (MRS) is a geophys-ical method designed for imaging and quantitative de-

⁎ Corresponding author. Tel.: +33 2 38 64 47 45; fax: +33 2 38 64 33 61.E-mail address: [email protected] (J.-F. Girard).

1 Tel.: +33 4 76 82 50 63; fax: +33 4 76 82 50 14.

0926-9851/$ - see front matter © 2007 Elsevier B.V. All rights reserved.doi:10.1016/j.jappgeo.2007.08.001

scription of water-bearing structures. It is based on thephenomenon of hydrogen proton magnetic resonance. Inthe subsurface, hydrogen is generally only present inwater molecules and, consequently, the MRS signal isspecifically linked with groundwater.

Initially, MRS was developed for characterizing hori-zontally stratified aquifers and has been used worldwidefor more than 15 years. MRS is also used to describe the

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http://dx.doi.org/10.1016/j.jappgeo.2007.08.001

104 J.-F. Girard et al. / Journal of Applied Geophysics 63 (2007) 103–116

lateral hydrological variation across a watershed. Forthat purpose, MRS results in vertical logs are interpo-lated from sounding to sounding to provide a cross-section. The efficiency of 1D (one dimensional) profilingfor imaging anomalies whose size is comparable to theloop size has already been studied byWarsa et al. (2002)and Legchenko et al. (2006). However, when applyingthis approach to the imaging of 2D–3D targets, the sizeof whichmay bemuch smaller than the loop (25 to 150mdiameter), the lateral resolution may be not sufficient.

Karstic conduits play a crucial role for water supply inmany parts in the world. However, the imaging of suchtargets is generally a difficult task for most geophysicalmethods. A karst network develops in limestone by dis-solution. The variability of karsts encountered in nature isso vast that each system is considered as unique: from thinand flat conduits developed at the top of an impermeablelayer to large galleries, several tens of metres high andwide. If a drill-hole hits the conduit it can provide a hugewater flow. However, since the hydraulic conductivity ofthe surrounding limestone matrix is several orders ofmagnitude less than the conduit itself, missing the karstconduit means that the borehole fails. Hence, highaccuracy is needed to locate the drill site.

It has been shown that, if the conduit is big enoughand full of water, standard 1D processing of MRS allowsunambiguous detection of such targets (Vouillamoz et al.,2003). However, lack of lateral resolution is the limitingfactor for siting a borehole usingMRS results. To improveMRS efficiency, we propose carrying out a simultaneousinversion of several MRS measurements along a profile.We propose to introduce the terminology of MagneticResonance Tomography (MRT) when several MRS areprocessed simultaneously. The gain in resolution withMRTcan be increased by using smaller distances betweentwo soundings: a constant larger step can be used alongthe whole profile and can be reduced around the detectedanomaly (down to one tenth of the loop diameter). Pleasenote that we consider only the case where the same loop isused as transmitting and receiving antenna (coincidentTX/RX loop setup).

We focused our numerical study on the response from2D water-filled cavities with a section smaller than theloop size. Based on the modelling results, we propose a2D inversion scheme specifically designed for prospect-ing water-filled conduits, and for detecting and estimat-ing their size and location.

2. Background

A detailed explanation of the method can be found inprevious papers (Weichmann et al., 2000; Legchenko

and Valla, 2002). An enhanced model has recently beenproposed (Legchenko, 2004) to consider the effect oftime-varying drift of the geomagnetic field duringmeasurements, especially in the presence of shallowwater.

In thermal equilibrium, groundwater has a macro-scopic spin magnetization vector aligned along thegeomagnetic field. In MRS, a quasi-static and homoge-neous geomagnetic field is assumed at the loop scale. Anexciting magnetic field is generated by a pulse ofoscillating current in the transmitting loop with a specificfrequency (the Larmor frequency). The duration (τ) andintensity (I0) of the pulse both characterize the pulsemoment q= I0 d τ (A ms).

In MRS, only the component of the exciting fieldperpendicular to the geomagnetic field contributes totilting the spin magnetization vectors away from theequilibrium direction. The amplitude of this effectivefield is proportional to the current amplitude. Accordingto Bloch's equations, the MRS signal reaches its max-imum when the tilt angle between the magnetizationvector and the geomagnetic field is 90° (Slichter, 1996).The signal (Eq. (1)) is recorded in the loop after thepower is turned off and is characterized by its initialamplitude E0 (nV), decay time T2⁎ (s), phase φ0 (rad)and pulsation ω0 (rad/s).

signal q; tð Þ ¼ E0 qð Þ cos x0 � t þ u0 qð Þð Þexpð� t

T 4app2 ðqÞÞ þ Noise tð Þ

¼Z

V

w rð Þ:K3D q; rð Þexpð� t

T 42 rð ÞÞ � dr

3 þ Noise tð Þ

ð1Þ

W(r) and T2⁎(r) are respectively the water content w(r)and decay time distributions in the ground, and r=r(x,y,z)the coordinate vector. The kernel function K3D(q,r) is theresponse of a unit volume dr3 at position r in the ground:

K3D q; rð Þ ¼ x0

I0B1 rð Þeiu0 rð ÞM8 q; rð Þ ð2Þ

B1(r) is the transmitted magnetic field componentperpendicular to the geomagnetic field, and M⊥(q,r) isthe transverse component of the spin magnetization forpulse q, which creates an alternating magnetic fieldmeasured after the pulse cut-off. The phase shift φ0(r)relative to the current in the loop is due to the electricalresistivity of the ground, the pulse shape, and thefrequency shift between the Larmor frequency and thepulse frequency (Legchenko, 2004). As shown inEq. (1), the MRS response is integrative. The responsesfrom all water molecules below the loop (25 to 150 mdiameter) are added together. The apparent decay timeT2⁎(q) is defined by fitting the recorded signal to a single

Fig. 1. MRS response of a 20 m thick aquifer (water content 5%) using a square loop (75 m side) with a 60° geomagnetic inclination.

Fig. 2. Synthetic MRS sounding curvesE0(Q) above a karstic limestone.

105J.-F. Girard et al. / Journal of Applied Geophysics 63 (2007) 103–116

decreasing exponential curve. Note that a multi-ex-ponential fitting approach can be used when the dataquality is sufficiently high (Mohnke and Yaramanci,2005, Lubczynski and Roy, 2003). The apparent T2⁎(q) isan average value of the true T2⁎(r) distribution weightedby the water content of each layer for a given pulsemoment q. For low pulse moments, the tilt angle ap-proaches 90° only for relatively shallow water. Whenincreasing q by increasing the current intensity (pulseduration is commonly fixed around 20–40 ms), then thetilt angle for shallow water is more than 90° and itsresonance signal decreases, whereas resonance increasesfor deeper water. Depth sounding behaviour of MRS(Q≈pseudo-depth) is illustrated in Fig. 1. If we considera dry environment and a tabular aquifer with 5% watercontent (saturated pore volume / total volume) between20 to 40m depth, we expect amaximum signal of 100 nV(Fig. 1) with a square 75-m-side loop (1 turn) for a 60°geomagnetic inclination, the mean value in westernEurope. The pulse moment for themaximum follows anydepth variation of the aquifer. In the field, the responsesfor a given set of q values are inverted to provide thewater content log (with depth). The direct link betweengroundwater at a certain depth with the pulse moment isthe main advantage of the MRS method.

The electrical conductivity properties of the soil in-fluence the MRS result in both penetration depth (de-creasing for high conductivities that can be roughlyevaluated from a skin depth estimation), and dephasingof the signal (Braun et al., 2005). It has been shownthat for electrical resistivities N100 Ω m, only minorvariation is observed in the MRS response (Valla andLegchenko, 2002; Hunter and Kepic, 2004). Con-sequently, working in electrically resistive rock likelimestone, MRS does not need to be accompanied byelectrical resistivity measurements. Note that the signalphase may also be influenced by the 3D distribution of

water, but only when using separated TX/RX loops(Hertrich et al., 2005). As we consider only a coincidentloop set-up, phase variations are not influenced by theconduit. We thus propose only using the amplitude of theMRS signal in inversion and a 1000-Ωm resistivity willbe considered in the all modelling in the rest of thearticle. This approach is particularly relevant for aconduit filled with fresh water, but should be checked inthe case of a saline-water filled conduit.

3. Focalization effect

The response from the limestone matrix can be de-scribed as a homogeneous half-space. Since decay timeof the MRS signal of fresh limestone is many times lessthan the decay time of free water in a cavity, where it canreach up to 1 s, this strong contrast makes MRS mea-surements specifically sensitive for detecting water-filled cavities (Vouillamoz et al., 2003). Because of theintegrative behaviour of MRS, the matrix and conduitsignals are added together, and the conduit signal may

Fig. 3. Interpolation of 1D inversion results above 2D water filled conduit.


generate only a few percent of the total response. Fig. 2presents the MRS response of a karstic limestone wherethe matrix is characterized by 1% water content whereasthe karstic conduit is a 5×5 m conduit full of watercentred at 13.75 m depth.

The standard 1D inversion scheme results in a log ofwater content and decay time (Legchenko and Shush-akov, 1998). 1D profiling is the result of interpolating the1D inversion results along a profile of MRS measure-ments. It provides a vertical cross-section that images thelateral variations of water content and permeabilitythrough an empirical law linking pore size toMRS decaytime (Legchenko et al., 2002). One can imagine thatreducing the distance between soundings necessarilyimproves the accuracy of imaging. We calculated the

Fig. 4. a) 1D kernel for a 75 m side square loop with a 60° geomagnetic incldepth for pulse Q=2000 A ms.

synthetic response of a 2D north–south conduit of5×8 m, full of water at 20 m depth, for 21 positions of a75-m-square loop with 10-m steps for a 60° geomagneticinclination. Then, in order to resolve the true position ofthe conduit, 1D logs of the water content were inter-polated (Fig. 3). However, although 1D profiling, i.e. alateral interpolation of 1D results, provides the depth ofthe conduit, the lateral resolution is very poor. An im-proved resolution is obtained through the use of a smallerloop which investigate a smaller lateral volume but at thecost of decreasing the investigation depth (Warsa et al.,2002). In addition, the signal-to-noise (S/N) ratio de-creases with smaller loop. Indeed, under good fieldmeasuring conditions, needed when searching for atarget like a karstic conduit, the noise level of the data

ination. b) Focalization below the loop: slice in the 3D kernel at 20 m

Fig. 5. The MRS signal is generated by the component of the exciting field perpendicular to the geomagnetic field.


from a large or a small loop can be reduced to ap-proximately the same level through filtering andstacking. The ambient noise is higher with a largeloop, but longer stacking will result in a final noise that islimited only by instrument noise at about 5 nV; however,

Fig. 6. West to east MRT profile with 75 m coincident TX/RX square loop peand 27.5 m depth (lower section) in the northern hemisphere and with 60° g

the MRS signal being stronger for a large loop than for asmall one, the S/N ratio will be higher with the largeloop.

The development of a 2D inversion routine aims atvastly improving the resolution of MRS when using a

rpendicular to N–S conduit 2.5×10 m at 12.5 m depth (upper section)eomagnetic inclination; 1% water content is assumed in the matrix.

Fig. 7. North–south MRT profile with 75 m coincident TX/RX square loop perpendicular to E–W conduit 2.5×10 m at 12.5 m depth (upper section)and 27.5 m depth (lower section) in the northern hemisphere and with 60° geomagnetic inclination; 1% water content is assumed in the matrix.


large coincident TX/RX loop compared to the target size.In the literature, examples are given of the application ofa separate TX and RX loop setup for the investigation ofshallow 2D–3D targets and promising results arereported (Hertrich et al., 2005). However, the use ofcoincident TX and RX loops allows maximum penetra-tion depth and a better S/N ratio.

4. 2D sensitivity of MRS

Numerical modelling was carried out assuming acoincident TX/RX square loop with 75-m sides, a 60°geomagnetic field inclination and a maximum pulsemoment of 10,000 A ms (easily attained with currentlyavailable equipment). The computed contribution ofdifferent layers of the subsurface is depicted in Fig. 4a.This response pattern is also referred to as the 1D kernelthat is used as a linear filter in most 1D inversionschemes (Legchenko and Shushakov, 1998,Mohnke andYaramanci, 2002). For a given pulse moment, the re-sponse of an infinite horizontal layer placed at varyingdepths is computed (unit normalized in nV/m). The

relationship between the depth of a layer and the pulsemoment for the maximum response is unique (dash-dotcurve, Fig. 4a). One sees that the contribution of waterdeeper than about 70 m is negligible (except in theextreme case of a very large volume of deep water), thusdefining the depth of investigation in this case.

Because the effective component of the transmittedfield (the component perpendicular to the geomagneticfield) varies significantly inside the volume affected bythe loop, the sensitivity pattern (Weichmann et al., 2000)below the loop is in fact non-uniform (Fig. 4b).The response at 20 m depth (normalized to unit volumein nV/m3) below the 75-m-side square loop was calcu-lated for a 2000 A ms pulse moment, which correspondsto the maximum response for 20 m depth. The resultsshow that the water contributing to the measured signal islocated inside a cylinder with a diameter of approximately1.5 times the side of the loop. Of particular note is the factthat the most sensitive zone below the loop is situated inthe southern half of the zone. This is easily explained, asthe effective part of the stimulating field is the componentperpendicular to the geomagnetic field. The magnetic

Fig. 8. Illustration of the poor sensitivity of the MRS profile to the shape of a 2D water filled conduit perpendicular to the MRS profile.


field generated by the loop laid on the surface is symmetricwith revolution around a vertical axis centred in the loop.In the northern hemisphere, the component perpendicularto the geomagnetic field is stronger in the southern partbelow the loop (Fig. 5a). In the southern hemisphere, thesensitivity pattern would be reversed (Fig. 5c), but itwould be symmetric at themagnetic equator (Fig. 5b) or atthe poles. In the case of 2D–3D target imaging, this highersensitivity below the southern part of the loop, forinstance, will generate a difference if the profile is orient-ated east–west (symmetric response) or north–south(asymmetric response) and this effect has to be studied.

5. Sensitivity study

MRT profiles across an elongated parallelepiped(2.5×10×600 m) full of water were modelled for ourtest conditions, defined by: northern hemisphere, mag-netic inclination 60°, 1000Ω m (characteristic electricalresistivity for dry limestone). The loop used for themodels was a 75-m-side square loop, 1 turn, with a2000 Hz Larmor frequency. These conditions are com-monly encountered in metropolitan France.

Profiles were computed perpendicular to shallow(12.5–15 m) and deep (27.5–30 m) conduits. The FID1amplitude (i.e. amplitude E(q) of the signal after the firstpulse and the instrumental dead time of 40 ms) wasinterpolated between 41 soundings made along a straightline with a refined step in the central part (5 m stepbetween −50 to 50 m, and 10 m step from −150 to+150 m). The same experiment was modelled for east–west (Fig. 6) and north–south (Fig. 7) profiles. Asexpected, the shallow model generates a stronger signaland the maximum response is obtained for a pulseq≈1100 A ms. The deep-model response presents asmoother variation and a maximum for pulse q≈2000 Ams. For the east–west profile (Fig. 6), the anomaly issymmetric and the maximum amplitude decreases fromN70 nV for the shallow case to 55 nV for the deeperconduit. The anomaly is assymmetric for the north–southprofile and the maximum amplitude decreases fromN70 nV for the shallow case to 65 nV for the deep conduit.

In all cases, a clear anomaly is centred on the cavitylocation. This anomaly is clearly symmetric for the E–Wcase, while the centre of the assymmetric anomaly forthe N–S profile is shifted northward. We call this a

Fig. 9. The relative objective function (in percent), with a 5×5 m conduit as reference.


focalization effect that can be easily understood byreferring to Fig. 4 (the maximum response is below thesouthern half part of the loop). Note that the two smallmaxima observed on the E–Wprofiles below the abscissa−25 and +25m are too low to be reliably distinguished infield data (below the instrumental-noise threshold). Itunderlines that the conduit can be detected in all fourcases, but that the MRT profile and conduit orientationshould be taken into account during interpretation for anaccurate localization. In addition, under a 2D assumption,one should note that if the profile orientation is notperpendicular to the conduit, the apparent section may bemisinterpreted (see later discussion about error introducedby a non-perpendicular profile and Fig. 13). Consequently,as for any 2D-imaging method, the best result is expectedwhen the MRT profile is perpendicular to the conduit.

Because of the integral nature of MRT results, wehave to deal with equivalent solutions. Indeed, responsesfrom two aquifers with the same water volume centred at

the same depth are nearly identical (for example 10-mthick with 5% water content and 20-m thick with 2.5%water content). The equivalence problem for cavitieswith the same water volumes but different shapes wasstudied. The MRT response for three conduits withrectangular (10×2.5 m), flat horizontal (1×25 m) orvertical (25×1 m) sections were compared with theresponse of a square (5×5 m) conduit, all centred at thesame depth (Fig. 8).

rms ref ; datað Þ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPIi¼1

PJj¼1

Erefi qjð Þ�Edata

i qjð Þð Þ2J

I

vuuutð3Þ

The rms (residual mean square) was calculatedover the whole cross-section (i=1,2,…,I soundings andj=1,2,..,J pulses). For all models used to compute the rms,I=31 soundings from X=−100 to +100 m with 10 m

Fig. 10. The relative objective function (in percent) with variable X varying from −60 to 60 m for four models.


distance between two consecutive loops and a refined 5 mspacing is used between X=−50 to +50 m. Pulse mo-ments vary exponentially from 60 to 5000 A ms. In thisnumerical study, J=40 values, but in practice fewer pulsemoments would be sufficient as the curve behaviour iswell rendered (16 pulse values or less are commonly usedin the field). For two depths with a dry and 1% water-content matrix, the rms remains b1.6 nV (Fig. 8). Sinceinstrument noise is commonly taken around 5 nV, con-duits with the same section at the same depth are undis-tinguishable by MRT measurements at present. Keepingin mind that MRS amplitude is directly linked to watervolume, if one assumes a single saturated conduit then thesection and position of the conduit can be resolved.

For modelling, we assumed that the water content inthe matrix is known and homogeneous (at least thevariations were assumed to be many times smaller thanvariations due to the conduit). In reality, a sounding farfrom the conduit (without any anomaly related to thesignature of the karstic conduit) can be used to estimatethe limestone matrix water content.

In our inversion scheme a model of a water-filledconduit is characterized by three parameters: positionalong profile (X), depth (Z) and section (S=thickness×width). From the variations of an objective function, wewant to learn about the sensitivity of the MRT responseto each of these parameters. The inversion scheme aimsat minimizing this objective function. We define theabsolute objective function (in nanovolts) as

absolute objective function in nV : rms MODEL X ; Z; Sð Þ; datað Þð4Þ

and the relative objective function (in percentage) as:

relative objective function ink :rms MODEL X ; Z; Sð Þ; datað Þ

mean amp� 100k

ð5Þwhere

mean amp ¼

PIi¼1

PJj¼1

Edatai qj

� �

I4J: ð6Þ

Fig. 11. The absolute objective function (in nanovolts) for all three parameters Z, S varying (from −80% to +80%) and X (from −60 m to +60 m).


Because the response of a water-filled conduit adds tothe matrix response, the rms error is a direct evaluation ofthe geophysical anomaly (in nV), but one should keep inmind how much and which part of the total signal thisanomaly represents, which is the goal of the relativeobjective function. In the case of several percent of watercontent in the matrix, recorded signals will be higher(and in a first approach easier to record for field geo-physics) but the need for measurement accuracy to re-solve the conduit location remains the same. This is thekey point of the sensitivity study.

The sensitivity study was done for a 5×5 m paral-lelepiped target centred at 13.75 m depth. Its section Sand depth Z vary from −80% to +80% and the horizontalposition X from −60 m to +60 m (Fig. 9). The choice ofparameters is very important andmay significantly affectthe robustness of the inversion. The three chosen param-eters appear equally sensitive and would thus be equallyresolved. One may notice that the rms minimum is not

zero. As in a real case, we used the equivalence of theconduit shape: we arbitrarily fixed the conduit width to10 m and adjusted the thickness. Because the real modelis 5×5 m we observed a small difference, but this wasbelow the detection threshold and hence could not bebetter resolved.

As expected, the dry matrix case presents strongervariations of the objective function than the 1% watercontent case. If there is some water in the matrix, then amuch of the signal would come from it and the responseof the conduit would be proportionally smaller.

In order to verify whether this behaviour is a generalfeature, we calculated the objective function variations forfour models: 25 m2 centred at 13.75 m, 25 m2 centred at28.75 m, 40 m2 centred at 20 m, and 50 m2 centred at12.5 m. For demonstration purposes, we present only thesensitivity to the horizontal position (Fig. 10) for a dry and1% water content matrix. A similar continuous and singleminimum trend is observed for all three parameters. It is

Fig. 12. Inversion algorithm to localize a water-filled cavity.


noticeable that, if the matrix is dry and only the responsefrom the conduit is measured, then a scale invariance ofthe objective function is observed (Fig. 10, black lines):

Fig. 13. Error introduced by a poor a priori orientation of the conduit: alperpendicular to the profile.

the parameters are equally resolved for the “deep”, “shal-low” and “middle” models. However, in the general casewhere some signal comes from the matrix, the shallower

l profiles were inverted using an a priori orientation of the conduit


and bigger the target, i.e. the higher the contrast betweenthe conduit andmatrix water contribution to the signal, thebetter its position is resolved (Fig. 10, grey line).

Because of the cumulative property of theMRTsignal,the matrix response adds to the conduit response. If thesame conduit with a dry and 1% water-content matrix isconsidered, then the objective function in nanovolts ismore or less unchanged (Fig. 11). The rms variation of 5%does not have the same significance for dry and wetmatrix cases (Fig. 10) whereas a 5 nV variation has thesame meaning. The absolute objective function is a betterindicator than the relative objective function. Necessarily,

Fig. 14. 2D inversion of field data: measured amplitude of 10 MRS data alosingle minimum RMS below 7 nV (bottom).

the noise level needs to be evaluated carefully and shouldbe lower than the conduit signature.

6. Inversion scheme

We tested the continuity and single minimum of theobjective function. Following this, several processes forapproximating the parameter sets that minimize thegiven objective function could be used. Although a widespectrum of minimization methods exists, it was foundthat for the set of parameters chosen in our scheme,the gradient method provides fast and robust solutions.

ng the profile (top) and best model (middle). Solutions space shows a


Gradient methods use information about the slope of thefunction to dictate a search direction where the mini-mum is thought to lie. The simplest of these is themethod of steepest descent in which a search is per-formed in a direction towards the minimum of theobjective function (Tarantola, 2005).

Optimal states are always defined with respect toa given neighbourhood in the search space. Given theaccuracy of the measurements, because the objectivefunction is evaluated in terms of amplitude, we can usethis threshold (i.e. 5 nV) to estimate the range of un-certainty of the best solution.

When looking for a water-filled conduit, if a longer T1and higher amplitude anomaly is observed in the data ofaMRS profile, we propose an inversion scheme based ona 2D assumption to process such a dataset, summarizedin Fig. 12. First, the effect of the surrounding envi-ronment (e.g. limestone matrix) has to be characterized.A saturated limestone matrix with a water content ofseveral percent generates an MRS signal that adds to theconduit response. An MRS sounding far from the con-duit is used for this purpose. Under exploration con-ditions, the absence of any conduit is characterized whenthere is no amplitude or long T1 anomaly (Boucher et al.,2005).

The depth of the conduit can be estimated even if noinformation on its size or orientation is known. To do this,we can use a sounding made near the centre of theanomaly. Depending on the conduit orientation, an errorcan be introduced in the depth estimation (Figs. 6 and 7).Otherwise, the depth can be inverted simultaneously withthe two other parameters. For the inversion, an initial setof parameters is chosen, section S (m2) of the conduit,position X (m) along the profile, and depth Z (m) of theconduit if not evaluated separately. An iterative processwith gradient optimization is applied and leads to the bestfit. The accuracy of the position and size of the conduitcan be evaluated from the accuracy of the measurementwith respect to the noise level. Current instrumental levelis 5 nV but a 2 nV instrumental limit is also shown inFig. 11. If we consider a 5 nV threshold for the 25 m2

section target, then a 20% accuracy is expected for centredepth (12.5±3 m), 30% in section (25±8 m2), andhorizontal position of centre ±15 m. One should notethat if the threshold is lowered to 2 nV, then the ac-curacy increases to 10 and 15% (X±5 m, S±4 m2 andZ±1.5 m). Considering that the development of equip-ment with greater accuracy is underway, a better ac-curacy can be expected for future devices.

For the modelling of errors introduced by non-perpendicular crossover, a site location is assumed in thenorthern hemisphere with a 60° inclination. In the in-

version scheme, the profile was considered perpen-dicular to the conduit in every situation. Hence, wecomputed the effect of this false a priori informationon the inversion of the MRT profile across a 5×8 mrectilinear conduit at 20 m depth (Fig. 13). When profileand conduit are truly perpendicular, the inversion exactlyresolves themodel. In the case of N–S or E–Worientatedconduits, an error up to 45° in deviation would result in amaximum 4 m error toward the north for position X, andan 8-m2 over evaluation for section S for an E–Wconduit, and a ±2 m for X and 4 m2 over evaluation for Sin case of a N–S conduit,. These errors remain lower thanthe uncertainties due to instrument noise. One may con-clude that if the measuring profile crosses the conduitwith an angle less than 45°, the perpendicular hypothesisdoes not adversely affect the efficiency of the method.

7. Field results

An MRS field study aiming to detect a limestonekarstic conduit was made near Rocamadour (France)using our proposed methodology. Eleven MRS sound-ings were performed along NW–SE profile crossing aknown karstic conduit. The data and 2D-inversionresults are shown in Fig. 14. This site has the doubleadvantage of a low EM noise level and the existence of alarge, shallow, water-filled conduit. Nevertheless, forfiltering purposes a 25 m square loop (3 turns) using aremote loop for analogical noise filtering was used toobtain a good S/N ratio: amplitude of anomaly variesfrom 30 to 70 nV whereas the maximum of matrixresponse is below 30 nVand the noise after stacking wasreduced by around 8 nV. The solutions space shows asingle minimum of 6 nV. A homogeneous limestonematrix was assumed, characterized by the 1D inversionof the southern MRS sounding apparently outside thedetected anomaly. But small variations inside the matrixthat are not integrated in our scheme and residual ex-ternal EM noise in the data could easily explain this bestfit that is slightly larger than the instrument noise. Theposition of the E–W conduit, 20 m deep with a 42 m2

section (Fig. 14) has been confirmed by speleologistdivers and the use of on-site transponders (see detailedfield study by Boucher et al., 2006).

8. Conclusions

We have shown the possibility of using standardMRS, with a coincident transmitting-receiving loop, forthe imaging of 2D conduits with a section size muchsmaller than the loop size. In such situations, interpola-tion of 1D inversion results is not suitable because of the


lack of lateral resolution (even if the depth resolution isgood). Thus, we propose using a 2D model. Based on asingle-conduit approximation, a 2D inversion allows thedepth, section and position of a conduit to be resolved.The most favourable conditions are a low noise leveland a dry matrix, i.e. a high parametrical contrast of theanomaly.

Practically, an MRT profile with refined 20-m stepsand overlapping loops can be made and inverted in 2D toprovide precise imaging. A field case under favourableconditions is described and has proven its feasibility forcharacterizing a 20-m-deep and 42-m2 section conduit.

The theoretical limitations of this method werestudied numerically. As with most surface geophysicalmethods, the resolution decreases with depth and, due tothe decreasing contribution of the conduit signal to thetotal response, the smallest size of a detectable conduitincreases with depth. Because we have shown the signalfrom a conduit to be weak, the electromagnetic site noiseand internal noise are the limiting factors. Any progressin instrument accuracy would greatly improve the ac-curacy of results.

Acknowledgements

The presented research results were partly fundedby the French national research program FNS-ECCO/WATERSCAN.

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2D magnetic resonance tomography applied to karstic conduit imaging