Keywo

C

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

1. Introduction

The method of surface nuclear magnetic resonance(SNMR) is a relatively new geophysical technique that

exploits the NMR-phenomenon for a quantitative determi-nation of the sub-surface distribution of hydrogen protons,i.e. water molecules of groundwater resources, by non-intrusive means. The idea to employ NMR techniqueswithin the Earths magnetic eld to derive sub-surfacewater contents was rst proposed by Varian [1]. It wasE-mail address: marian@aug.ig.erdw.ethz.ch

e Sp5.1. Influence of the Earths magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2415.2. Influence of the sub-surface conductivity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241

6. Field data example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2437. Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2475. Limitations of the surface NMR method . . . . . . . . . . . . .3.1.1. Fixed geometry inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2333.1.2. Variable geometry inversion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2343.1.3. Reliability of water content estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

3.2. 2-D investigations: magnetic resonance tomography (MRT) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2364. Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.1. Relaxation processes in rocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2384.2. Acquisition of relaxation parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.3. Observed data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2394.4. Limitations of current schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2400079-6

doi:10.Inversion of surface NMR data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2323.1. 1-D investigations: magnetic resonance sounding (MRS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2332.2. The surface NMR signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2292.2.1. The magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2302.2.2. The vector spin magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.2.3. The NMR signal for surface loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2312.2.4. Isolating the integral kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

3.2.1. Basic principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2281. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2272. Surface NMR measurements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228ntsonterds: Earths eld; Geophysics; Tomography; Hydrology; Surface loopsReceived 18 October 2007; accepted 18 January 2008Available online 19 February 2008Marian Hertrich

ETH Zurich, Swiss Federal Institute of Technology, Schafmattstrasse 33, 8093 Zurich, SwitzerlandImaging of groundwater with nuclear magnetic resonanceProgress in Nuclear Magnetic Resonanc565/$ - see front matter 2008 Elsevier B.V. All rights reserved.1016/j.pnmrs.2008.01.002www.elsevier.com/locate/pnmrs

ectroscopy 53 (2008) 227248

Renot until the late 1970s that a group of Russian scientiststook up the idea and developed the rst eld-ready proto-type of surface NMR equipment in the 1980s. It allowedfor the rst time the recording of NMR signals fromgroundwater at considerable depths in the Earth [24].Numerous eld applications with the Russian Hydro-scope-equipment encouraged the ongoing technical devel-opments for about a decade [57]. They were supportedby several studies on the modeling, inversion and process-ing of surface NMR data [810]. Surface NMR becamebetter known to western scientists when the rst commer-cial equipment was launched by Iris Instruments (France)in 1996. A few groups worldwide actively pursued the fun-damental research and applications of surface NMR. Overthe past decade the continuous progress and experience hasbeen reported at periodic international workshops (Berlin1999, Orleans 2003, Madrid 2006), and followed-up pub-lishing special issues of peer reviewed journals devoted tosurface NMR [11,12]. Continuous technical developmentof surface NMR measurements has been carried out and,recently, two new suites of surface NMR hardware havebeen made commercially available [13,14]. The new systemsextend the available technical possibilities towardsimproved noise mitigation schemes and multi-channelrecording.

Major advances in the development of surface NMRwere triggered by a revision of the fundamental equationsproposed by Weichman et al. [15,16]. The improved formu-lation allows the correct calculation of complex-valued sig-nals of measurements on conductive ground [17] and thecalculation of surface NMR signals with separated trans-mitter and receiver loops. The latter feature has been stud-ied in detail by Hertrich and co-workers [18,19] whichrevealed that a series of measurements at multiple osetsalong a prole provides sucient sensitivity to allow forhigh resolution tomographic inversion. A fast and ecienttomographic inversion scheme has been developed thatprovides the correct imaging of 2-D sub-surface structuresfrom a series of surface measurements [20].

Various geophysical techniques, like geoelectrics, elec-tromagnetics, georadar and seismics, are routinely used ina structural mapping sense in hydrogeology, to delineatebedrock and sometimes determine depth of the water tableand other major geological boundaries. But surface NMRis the only technique that allows a quantitative determina-tion of the actual water content distribution in the sub-sur-face. Near surface aquifers are the major source of drinkingwater worldwide. Additionally, these aquifers might besubstantially aected by cultural pollution, mismanage-ment and natural retreat in the ongoing climate change.But also in many other environmental problems groundwa-ter plays a key role. Examples are unstable permafrost andhill-slope stability in the progressive global warming ordynamics of glaciers and ice-sheets. For those issues sur-face NMR may provide essential information in high reso-

228 M. Hertrich / Progress in Nuclear Magneticlution imaging of the sub-surface water contentdistribution and monitoring of groundwater dynamics.In conventional NMR applications (e.g. spectroscopy,medical imaging, non-destructive material testing) the exci-tation of the spin magnetization is in most cases inducedand recorded by uniform secondary magnetic elds suchthat the recorded signal amplitude can be calibrated bysamples of known spin density and the experiment can bedesigned such that perfectly controlled ip angles areobtained. By contrast, in surface NMR none of theserequirements can be met and the amplitude of the recordedsignal has to be quantitatively derived for non-uniformelds and the resulting arbitrary ip angles. Therefore, inthis review article, a comprehensive derivation of the sur-face NMR signal is given and the formulations of the prob-lem for 1-D and 2-D conditions are presented. State-of-the-art inversion techniques are needed to derive sub-surfacemodels of water content distribution from measured elddata state-of-the-art inversion techniques are needed andare applied with appropriate estimates of the reliability ofthose models given. The observed NMR relaxation timesmay in general provide additional information about theaquifer properties by their dependency on the pore spacegeometry, but their determination and interpretation aresomewhat limited compared to conventional laboratoryNMR techniques. A short account is given on possibleschemes of relaxation time determination and future direc-tions of surface NMR research. Since measured surfaceNMR signals substantially depend on the local settings ofthe Earths magnetic eld and the sub-surface resistivitydistribution, the dependency on these parameters isdescribed and their variability throughout the Earth isshown and accounted for in terms of likely response. Asan example of state-of-the-art surface NMR measure-ments, the inversion and interpretation of a real data setfrom a well-investigated test site is presented.

2. Surface NMR measurements

2.1. Basic principle

Exploration for groundwater using NMR techniquestakes advantage of the spin magnetic moment of protons,i.e. the hydrogen atoms of water molecules. In a zero exter-nal magnetic eld environment, the spin magnetic momentvectors are randomly oriented. In the presence of anapplied static magnetic eld, the vectors precess aboutthe magnetic eld and at thermal equilibrium between thewater molecules, the distribution of spin magnetic momentvectors has an alignment that results in a small net mag-netic moment along the eld direction (i.e. a longitudinalmagnetic moment; Fig. 1a). Within the Earth, the spinmagnetic moment vectors precess around the Earths mag-netic eld B0 at the Larmor frequency xL cp j B0 j,where the gyromagnetic ratio cp 0:26752 109 s1 T1.Worldwide values for j B0 j vary between 25,000 nT aroundthe equator and 65,000 nT at high latitudes, resulting in

sonance Spectroscopy 53 (2008) 227248Larmor frequencies of 0.93.0 kHz, i.e. signals in theaudio-frequency range. In surface NMR, an alternating

xcit

ndwtate. (c) Protons decaying (relaxing) back to their undisturbed state. The pulsee ofpro

c Resonance Spectroscopy 53 (2008) 227248 229current It is passed through the transmitter loop for aperiod of sp 40 ms, generating a highly inhomogeneousmagnetic eld B1 throughout the sub-surface. The energyinduced in the sub-surface is given by the moment of theintegral

q Z spt0

Ittdt; 1

which gives for a rectangular envelope of the pulse of con-stant It I0q I0sp: 2

Hence, the total energy emitted by the transmission pulseq is called the pulsemoment. The component ofB1 perpendic-ular to B0 imposes a torque on the precessing protons that

H-protons

e

H-protonsexcitedundisturbed

Earth'smagnetic

fieldlines

a bFig. 1. Simplied sketches showing the principle of surface NMR in grouEarths magnetic eld B0. (b) A small percent of the protons in an excited smoment controls the maximum penetration depth. The initial amplitudheavily inuenced by the water volume, and the shape of the decay curve

M. Hertrich / Progress in Nuclear Magneticauses them to tilt away from B0 (Fig. 1b). This results in areduced or zero net spin magnetization parallel to B0 andan enhanced net spin magnetization perpendicular to it.Upon switching o the current that generatesB1, the parallel(or longitudinal) and perpendicular (or transverse) compo-nents of spin magnetization relax exponentially to their ori-ginal states (Fig. 1c). The precession of the decayingtransverse component generates a small but perceptiblemac-roscopic alternating magnetic eld that can be detected andmeasured by the same or a second Faraday loop of wire (thereceiver loop) deployed at the surface.

For a single transmission with a xed pulse moment q,spins in certain regions in the sub-surface are tilted fromtheir equilibrium orientation. Increasing q increases thesub-surface volume within which spins are tilted. Further-more, spins exposed to relatively strong elds, i.e. closeto the loop, are tilted by large amounts, such that theymay go through multiple revolutions. In a highly inhomo-geneous secondary eld the spins that experience multiplerevolutions loose coherence and their eects mutually can-cel out. Hence, increasing q leads at one hand to a largervolume of investigation and on the other hand to a mask-ing of NMR signals in strong elds close to the loop. Bychoosing a suitable series of q values, the water content dis-tribution can be reliably determined for relatively largesub-surface volumes.

During each measurement, the exponentially decayingsignal is recorded (Fig. 2, black lines). Single- or multi-exponential decay curves are tted to the recorded time ser-ies (Fig. 2, green lines), which allows the derivation of ini-tial amplitudes V 0, FID relaxation constants T 2 and phaselags, f, relative to the current in the transmitter [9]. The ini-tial amplitudes V i are linked quantitatively to the trans-verse magnetization acquired by the tipping pulse and thenumber of protons (i.e. water molecules) in the investigatedsub-surface volume. From the dierent time series atincreasing q, amplitudes of the complete suite of measure-

the electromagnetic signal after the magnetic eld has been turned o isvides information on the pore sizes. Modied from Ref. [50].Receiver

Aquifer:freewater

Aquiclude:adhesivewater

H-protons

ing pulse

Transmitter

relaxation

response

cater exploration. (a) A small percent of the protons are aligned with thements (red line in Fig. 2) can be used to estimate sub-sur-face water content.

2.2. The surface NMR signal

The surface NMR signal, induced and recorded by thesurface loops, is the superposition of signals from all indi-vidual proton spins in the sub-surface. After making themeasurements, it is necessary to invert the data in termsof the distribution of spins (i.e. water content). Inversionof the data requires formulations that relate the inducingelectromagnetic elds to the NMR-phenomenon. It is par-ticularly important to account for the highly inhomoge-neous electromagnetic elds in the sub-surface created bythe surface loops.1 Their orientation and complex eldamplitudes throughout the sub-surface depend naturallyon the size and shape of the surface loop and distance from

1 To compute the signal response from a sub-surface volume into thereceiver loop, the eld of the receiver loop at this sub-surface volume iscomputed and reciprocity applied. Thus, the virtual eld of the receiver hasto be computed.

5ualme

Rethe loop but also on the electrical conductivity and mag-netic susceptibility distribution of the ground. Within theinvestigation volume, the elds may (i) vary over ordersof magnitude, (ii) be oriented in all directions and (iii) beaected by electromagnetic induction in conductiveground. The latter causes attenuation and elliptical polari-

0

100

200

300

400 0

0

200

400

600

800

recording time [ms]

sign

al a

mpl

itude

[nV]

Fig. 2. Data space of a surface NMR measurement. Time series for individ(green lines) and initial amplitude V 0 (red line). A suitable suite of pulse moprovide spatial water content distributions.1000

230 M. Hertrich / Progress in Nuclear Magneticzation [21] that aects the NMR-phenomenon strongerthan the induction-related attenuation. Since the quantita-tive computation of NMR signal amplitudes in non-homo-geneous elds is rarely done (if ever) for conventionalNMR applications the derivation is given below.

In its most general form, the surface NMR voltage V tis given by:

V t Z

d3r

Z 10

dt0BRr; t0 oMot r; t t0; 3

where, BR is a virtual magnetic eld caused by passing aunit current through the receiver loop andM is the nuclearmagnetization after excitation. The inner integral is theconvolution of the temporal variation of spin magnetiza-tion with the receiver eld as a function of position r. Thisrepresents the contribution of spin magnetization at r tothe total signal. The outer integral is the volume integralover the entire population of contributing spins. Both, BRandM are functions of the time-dependent electromagneticelds of the surface loops Bt, acting at r.

2.2.1. The magnetic elds

To evaluate the integral in Eq. (3), analytic expressionsfor the interaction of the spins with loop elds BT;R, gener-ated by the transmitter and receiver, respectively, have tobe derived. Because spins only absorb and emit energy atthe Larmor frequency xL, only this frequency needs to beconsidered in the computations. Hence, for a more conve-nient notation, the frequency dependency of the loop mag-netic elds and their components as they enter themathematical description of the surface NMR signal isdropped subsequently.

1015

2025

30

pulse moment q [As]

measured signalsfitted curvescombined measurement

recordings (black lines) are explained by the respective exponential curvesnts constitutes a surface NMR measurement curve that can be inverted to

sonance Spectroscopy 53 (2008) 227248Of the electromagnetic eld BT;Rr; t generated (in real-ity or virtually) by the surface loops at r, only the compo-nent perpendicular to the local magnetic eld of the Earth,B?T;Rr interacts with the spin system, whereB?T;Rr; t BT;Rr; t b0 BT;Rr; tb0: 4

In the following, bT;Rr; b?T;Rr denote the static unitvectors of the time-dependent elds BT;Rr; t;B?T;Rr; t,i.e. the unit vectors of the major axis of the polarizationellipse, and b0 denotes the unit vector of B0.

For an elliptically polarized excitation eld, which is thegeneral case in conductive media, its perpendicular projec-tion B? is also elliptically polarized unless B0 lies in theplane of B. The projected elliptical eld, determined byEq. (4), can be decomposed into two circular rotating partsthat spin clockwise and counterclockwise relative to thespin precession as follows:

B?T;Rr; t BT;Rr; t BT;Rr; t aT;Rrb?T;Rr cosxLt fT;Rr bT;Rrb0

b?T;Rr sinxLt fT;Rr; 5

where BT;R and BT;R are the circularly polarized co- and

counter-rotating components of the elliptically polarizedvector B?T;R, a and b are the major and minor axes of the

c Resonance Spectroscopy 53 (2008) 227248 231ellipse and f is chosen in such a way, that a and b are real.B and B can be written as

BT;Rr; t 1

2aT;Rr bT;Rr b?T;Rr cosxLt fT;Rr

b0 b?T;Rr sinxLt fT;Rr; 6

where

j BT;Rr; t j1

2aT;Rr bT;Rr 7

and the complex unit vectors

bT;Rr; tBT;Rr; tjBT;Rr; t j

8

b?T;RrcosxLt fT;Rrb0b?T;RrsinxLt fT;Rr 9

12b?T;Rr ib0b?T;RreixLtfT;Rr c:c:; 10

where c.c. is the complex conjugate of the precedingexpression.

2.2.2. The vector spin magnetization

For spin ensembles that generate a macroscopic magne-tization Mr, only an excitational force of a monochro-matic eld in direction b?T r will force it on aprecessional motion in the plane spanned by the vectorsb?T r and b0 b?T r. Its oscillation can be described interms of its spatial components as

Mr; t Mr Mkrb0 M?r b?T r sinxLt fTr

b0 b?T r cosxLt fTr; 11

where Mr is the spin magnetization at location r, and Mkand M? the components of the magnetic moment orientedin the directions of the static eld B0 and perpendicular toit, respectively. OnlyM? oscillates and, consequently, emitsan electromagnetic signal. The time derivative of Mr; t inEq. (3) is

otMr; t xLMrM?r b?T r cosxLt fTr b0 b?T r sinxLt fTr: 12

Clearly, only the term in brackets of the second line of Eq.(12) contributes. The expression in the brackets of Eq. (12)is the unit co-rotating part of the transmitter eld in Eq.(9). Hence, Eq. (12) simplies to

otMr; t xLMrM?rbT r; t: 13This explicitly demonstrates that the spin magnetizationMr; t, created by the co-rotating part of the tipping pulse,oscillates in a xed direction and with xed phase relativeto the transmitter eld.

2.2.3. The NMR signal for surface loops

M. Hertrich / Progress in Nuclear MagnetiWith the expressions for the magnetic eld componentsand the spin magnetization, the expression for the surfaceNMR signal can now be quantitatively evaluated. Substi-tuting Eq. (13) into Eq. (3) and including the spatial aspectsyields

V t xLZ

d3rMrM?rZ 10

dt0BRr; t0bT r; t t0:14

By substituting the complex expression for bT r; t fromEq. (10), Eq. (14) becomes

V t xLZ

d3rMrM?rZ 10

dt0BRr; t0

12b?T r ib0 b?T reixLtt

0fTr c:c: :15

Rewriting the exponential expression eixLtt0fTr as

eixLteixLt0eifTr, Eq. (15) can be rearranged to

V t 12xL

Zd3rMrM?r

eifTrb?T r ib0 b?T reixLt

Z 10

dt0BRr; t0eixLt0

c:c:

: 16

The integral in the third line simply represents a Fourier-integral that transforms the causal time-dependent eldamplitude BRr; t0 into a frequency dependent oneBRr,2 so that the expression can be changed into

V t 12xL

Zd3rMrM?r eifTrb?T r ib0

b?T reixLt BRr; t c:c:: 17The vector BR is multiplied by the static unit vectors b

?T and

b0 b?T in the plane normal to B0, so that all componentsof BR parallel to B0 vanish and only the perpendicular com-ponent of BR is physically meaningful. Thus, BR can beconveniently replaced by B?R. Using the relationships pro-vided by Eq. (5), Eq. (17) becomes

V t 12xL

Zd3rMrM?r

eifTrb?T r ib0 b?T reixLt

eifRraRrb?Rr ibRrb0 b?Rr c:c:: 18

Commonly the positive envelope of the signal is determined,digitally or by hardware lters [9]. This results in the eect ofthe Larmor frequency oscillations being removed, such thatthe signal simplies to its real and imaginary envelopes:

V 0 xLZ

d3rMrM?r eifTrb?T r ib0 b?T r eifRraRrb?Rr ibRrb0 b?Rr: 192 As for all other parameters BRr;x acts only at xL, the frequency isdropped for a more convenient notation.

Mr 2M0f r: 22

ReThe factor of two arises from the chemistry of the watermolecule containing two hydrogen protons. Substitutingthe identities from Eqs. (21) and (22) into Eq. (20) yieldsthe formulation of the surface NMR initial amplitudes asintroduced by Weichman et al. [16]:

V 0q 2xLM0Z

d3rf r sincq j BT r j j BRr; t j eifTrfRr b?Rr b?T r ib0 b?Rr b?T r: 23

2.2.4. Isolating the integral kernel

The general forward functional of Eq. (23), can beexpressed as an integral with the water content distributionf r as the dependent parameter and a general data kernelKq; r:

V 0q Z

Kq; rf rdr; 24

with

Kq; r 2xLM0 sincq j BT r j j BRr j eifTrfRr b?Rr b?T r ib0 b?Rr b?T r: 25

This data kernel contains measurement-conguration-dependent information (e.g. loop conguration), the mag-nitude and inclination of the local Earth magnetic eld,pulse moment series, sub-surface resistivity distribution(contained in BT;R) and the various physical constants. Itis calculated for each measurement conguration and pulseseries. The spatial distributions of the electromagneticelds generated and received by the surface loops are calcu-By carrying out the multiplications and rearranging, Eq.(19) can be rewritten as

V 0 xLZ

d3rMrM?r aRr bRr eifTrfRr

b?Rr b?T r ib0 b?Rr b?T r: 20Here, M? can be expressed by the approximation for the

spin perturbation [22]

M? sinHT sincq j BT r j; 21where HT denotes the spin tipping angle. The tipping angleis determined by the spin nutation sinc j BT j, scaled bythe pulse moment q.

The expression in brackets in the second line of Eq. (20)is the absolute value of the counter-rotating part of thereceiver eld in Eq. (7). Furthermore, the magnetizationMr is the spin magnetization of the investigated waterprotons, which can be represented as the product of thespecic magnetization of hydrogen protons M0 and thewater content f r

232 M. Hertrich / Progress in Nuclear Magneticlated for conducting ground using Debye potentials in thespatial wave-number domain and then transferred to thespace domain (e.g. [23]). The sub-surface resistivity distri-bution has to be known a priori and is usually assumedto be independent of the water content distribution.

In Eq. (23), the magnetic eld components of the trans-mitter and receiver loops are represented only by their co-and counter-rotating parts, respectively. The fact that thecounter-rotating part of the receiver eld enters the equa-tion is a consequence of the reciprocity of mutual inductionbetween the loop and the spin magnetization.

The three lines of Eq. (23) can be interpreted as follows:

(1) The rst line contains the signal amplitude of the spinsystem emitting the NMR response. It includes thewater content term f r and the sinusoid of the tipangle, which is determined by the pulse moment qand the normalized amplitude of the co-rotating partof the transmitter eld.

(2) The second line describes the sensitivity of the recei-ver loop to a signal in the sub-surface; it is indepen-dent of the excitation intensity. It is a function ofthe hypothetical magnetic eld distribution associ-ated with the receiver loop and phase lags causedby electromagnetic attenuation.

(3) The nal line accounts for the possible separation ofthe transmitter and receiver loops. Whereas the rsttwo lines contain scalar quantities, this one includesinformation on the vectorial evolution of the mag-netic elds and is generally complex-valued. Theresulting phase shift, which is due to the geometryof the entire system, is in addition to phase phenom-ena associated with (i) excitation pulses o the Lar-mor frequency, (ii) signal propagation in conductivemedia and (iii) hardware related phases, e.g. resonantcircuits for the receiver loop.

3. Inversion of surface NMR data

Inversion techniques are required to derive models ofsub-surface water content from single or multiple surfaceNMR measurements. Whereas most NMR imaging meth-ods produce highly spatially selective data, the surfaceNMR techniques is rather integrative. Each measurementat a specic pulse moment q senses large regions of thesub-surface at individual sensitivity. From a series of mea-surements at varying pulse moments, that provide a suit-able coverage, a spatially resolved water content modelcan be obtained. Thus, models that explain the data in abest-t sense have to be determined, usually by incorporat-ing a priori model constraints.

Inversion is not a simple turnkey operation but gener-ally requires individually adopted processing steps whichin turn requires a basic understanding of inversion princi-ples. The layered model, the style of model discretization,the a priori information and the actual inversion scheme

sonance Spectroscopy 53 (2008) 227248all have a marked inuence on the inversion result. Addi-tionally, the derived models are neither exact nor unique

c Rewithin the nite measurement accuracies. In the followingthe basic inverse formulations for surface NMR data arederived and applied to 1-D and 2-D synthetic data. Forthe 1-D examples, two dierent, but complementaryapproaches are discussed in more detail.

The forward problem is given by Eqs. (23)(25). Eq. (23)is a common Fredholm integral equation of the rst kind, aform that is quite common in geophysical inverse methods[24]. Evaluating the integral in Eq. (23) for a range of qivalues yields a suite of readings V i V qi

V i Z

Kirf rdr; 26

with i 1; 2; . . . ;Nq the number of pulse moments for acomplete measurement. Further discretization in the spacedomain allows appropriate numerical modeling methods tobe employed. By approximating the continuous water con-tent to be piecewise constant for intervals Dr, Eq. (26)becomes

V i

X

Kirjf rjDrj; 27where j 1; 2; . . . ;Nr is the number of discretized spatialelements. Values for Kirj have to be determined eitherby a simple quadrature rule or by more accurate numericalintegration, depending on the size of the discretizationsteps. The system of equations in Eq. (27) can be conve-niently written in matrix notation as

V Kf; 28with the dimensions of the matrices as follows:V : 1 Nq, K : Nq Nr and f : Nr 1.

The aim of inversion is to determine a sub-surface watercontent model that explains the surface NMR measure-ments as follows:

f K1V: 29This problem is ill-conditioned and ill-posed in most

cases. It cannot be solved directly.Several schemes for solving the surface NMR inverse

problem have been published [10,2529]. They includeapproaches based on models with xed and variable geom-etry and dierent means of seeking optimum solutions (e.g.linearized least-squares, MonteCarlo, simulated anneal-ing). The following schemes employ least-squares tech-niques, which are very common in geophysics.

3.1. 1-D investigations: magnetic resonance sounding

(MRS)

In standard 1-D investigations, the sub-surface isassumed to be horizontally stratied and the water contentto vary only with depth. Measurements are performedusing the deployment of a single loop that generates thepulse and records the resultant NMR signal. The charac-teristics of this measurement conguration to achieve

M. Hertrich / Progress in Nuclear Magnetiincreasing depth penetration depth with increasing q isthe basis for the name (depth) sounding.For this 1-D problem using the coincident loop congu-ration, bTr and bRr are identical such that Eq. (23) sim-plies to

Kq; r 2xLM0 sincq j Br j j Br j ei2fr;xL:30

Writing the general forward problem from Eq. (30) inCartesian coordinates:

V i Z 10

Z 11

Z 11

Kix; y; zf x; y; zdxdy dz; 31

and assuming laterally homogeneous water content

of xox

of yoy

0; 32

the data kernel Kix; y; z can be pre-integrated in both hor-izontal dimensions x and y to give

Ki;1Dz Z 11

Z 11

Kix; y; zdxdy: 33

Thus, the forward problem to compute a series of syntheticmeasurements V i from a known water content model f zis given by

V i Z 10

Kizf zdz: 34

Here, the data kernel and water content are continuousfunctions with depth. To determine a 1-D water contentmodel we review two inversion schemes: one that assumesa large number of layers with xed boundaries and thendetermines the water content in each layer, and the otherthat determines thicknesses and water contents of a fewlayers.

3.1.1. Fixed geometry inversionFor xed geometry inversions, the models are dened

by many layers with xed (and known) boundaries. Onlythe water content in each layer is allowed to vary duringthe inversions. The water content distribution f is there-fore the only dependent variable of the forward problem.From Eq. (28), we see that for a measurement, thesurface NMR signal V is linearly related to the watercontent distribution f, such that the kernel K is the sen-sitivity or Jacobian matrix of the inverse problem. Thedata functional to be minimized by least-squares analysisis

UdV XPi1

V i Kijfii

2

kDV Kfk22; 35

where mists between the measured data V i and model-predicted data Kijfi, based on water content estimates fi,are weighted by their errors i, which is equivalent to thedata weighting matrix D. As in many geophysical appli-cations, the number of layers required to provide su-

sonance Spectroscopy 53 (2008) 227248 233cient resolution (i.e. the number of model parameters),often exceeds the number of data points. Consequently,

3.1.2. Variable geometry inversion

The variable geometry inversion scheme is based on theassumption that sub-surface water content can be repre-sented by a small number of discrete boundaries and watercontents of which can be determined during the inversionprocess. Such a scheme is useful if geological or other a pri-ori information indicates that simple layered models areappropriate representations of the sub-surface.

Assuming that three layers with water contentsf f1; f2; f3T and depths z 0; z1; z2; zmax are sucient,then Eq. (34) can be rewritten as

V iZ z10

Kizf1zdzZ z2z1

Kizf2zdzZ zmaxz2

Kizf3zdz:

40

The Jacobian matrix of this forward operator in respect tothe water content f is given by

Gfij oV iofj

41

Z z10

Kizdz;Z z2z1

Kizdz;Z zmaxz2

Kizdz T

42

K1i;K2i;K3iT; 43

where K1i;K2i;K3i are rows of the Jacobian denoting thesensitivity of the solution to the water contents in the

Rethe system of equations is under-determined, such thatadditional model constraints are required. Plausiblemodel constraints include demanding the model to besimple (damping) by minimizing the Euclidean lengthof the model parameters fT f, or demanding minimumvariations between adjacent model parameters (smooth-ing) by applying Tikhonov regularization [24]. The modelfunctional then gives

Umf kCmfk22; 36where Cm is the a priori model covariance matrix, that con-tains the appropriate model constraint. The complete func-tional to be minimized is given by

U Ud kUm ! min; 37with k a regularization weighting factor (also known astrade-o or damping factor).

The system of equations to be solved is linear andcould be solved in a direct fashion. Unfortunately, suchinversions may result in ridiculous results withf > 100% or f < 0% water content estimates. To avoidthese patterns, it is necessary to apply constraints onthe range of water content. Constraining the water con-tent in the inversion scheme by modifying the Jacobianmatrix changes the system to be slightly non-linear, thusrequiring an iterative approach. Starting from an arbi-trary initial model, the model vector is updated duringthe lth iteration by

fl1 fl mlDfl; 38where ml is the line search parameter. The model update Dfl

is determined by solving the GaussNewton system ofequations [30]

Dfl KTC1d K kC1m KTC1d V Kfl1 kC1m fl1;39

where C1m CTmCm and C1d DTD are the model anddata covariances. Fig. 3a shows surface NMR data fora simple three layer model in which water content is sig-nicantly higher in the middle layer. The true model isshown in Fig. 3b by the gray line. To simulate realisticconditions, 10 nV Gaussian noise (corresponding to about5% of the simulated data with median amplitudes of some200 nV) is added to the data. Application of the xedgeometry inversion scheme produces the smooth model,represented by the dashed line in Fig. 3b. Although theprincipal features of the true model are reproduced bythis inversion, the smoothness constraints lead to smear-ing of the layer boundaries and a slight overestimationof the maximum water content within the second layer.The advantages of the xed geometry inversion schemeis that no a priori information is required and evencomplex models with an unknown number of layers orsmooth water content variations can be reliably recon-structed. However, accurate derivations of layer bound-

234 M. Hertrich / Progress in Nuclear Magneticaries and water contents may be systematically limitedby this approach.0 200 400 600

0

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[As]

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dept

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]

a b

Fig. 3. Comparison of the two inversion schemes: one with xed geometryof numerous layers in which the water content of each layer is allowed tovary and one with three layers in which the layer boundaries and watercontent are allowed to vary. The true model (gray line in (b)) was used togenerate a synthetic data set aected by 10 nV of Gaussian noise (circles in(a)). Inversion using the xed-boundary approach and smoothnessconstraints on the model yields a reasonable model of water content(dashed in (b)), but with both boundaries smoothed rather than sharp andthe maximum water content of the middle layer is slightly overestimated.Inversion for variable three layer model (dash-dotted in (b)) accuratelyrepresents all important details of the true model. Both models t the dataequally well.

sonance Spectroscopy 53 (2008) 227248respective layers. Determining the Jacobian for the layerboundaries z1; z2 yields

is the practice of estimating median values and standarddeviations of the model when measuring those propertieswhen sampling from an approximating distribution of thedata.

For a bootstrap analysis probability density functions(PDF) for each data point are determined, with the datapoint itself as a mean value and the residuals of the mea-sured data and data predicted from a best-t model asthe standard deviation. In a next step, numerous replicasof the measured data set are then generated with randomnumbers for each data point within its PDF. Thus, inver-sion of the replicated data sets yields a suite of models thatall explain the measured data within their measurementaccuracy. Variations in the model parameters can be ana-lyzed, thus providing the median values and the standarddeviations of model parameters.

For a more robust delineation of outliers studentizedresiduals are usually employed. These are the residuals of

M. Hertrich / Progress in Nuclear Magnetic Resonance Spectroscopy 53 (2008) 227248 235Gzij oV iozj

44

f2 f1Kiz1; f3 f2Kiz2T: 45The total Jacobian matrix is then given by

Gij Gzij;GfijT 46 K1i;K2i;K3i; f2 f1Kiz1; f3 f2Kiz2T: 47For surface NMR inversion with variable boundaries, thematrix entries for the depths are dependent on the watercontents of the respective layers. Thus, the inversion is ingeneral non-linear and has to be solved in an iterative fash-ion. The inverse problem can then be written as [24]

Dfl GTl C1d Gl kC1m gGTl V Kfl1; 48where Dfl is the model update for the lth iteration, C

1m and

C1d are again the model and data covariances, k is the reg-ularization factor and V Kf are the residuals of themeasured data V and the water model fl1 from the l 1stiteration. For the given inversion scheme, the number ofdata points is larger than the number of model parameters.Thus, no additional model constraints need to be applied.However, constraint matrices Cm and/or Cd should beimplemented to avoid a badly conditioned inverse ofGTG and therefore stabilize the inversion.

Application of the variable geometry inversion usingthree layers to the synthetic data of Fig. 3a yields a near-per-fect reconstruction of the true model (black dash-dotted linein Fig. 3b), compared to the initial model (gray line).Clearly this inversion scheme is capable of providing highlyaccurate results as long as there are a limited number of dis-crete layers and the number of layers is known beforehand.Of course, if the number of layers is unknown and/or theboundaries are gradational rather than sharp, the resultsof applying the variable-boundary scheme will be awed.

3.1.3. Reliability of water content estimates

The reliability of the sub-surface water content modeldepends on the data quality. Usually, a range of modelscan be found that explain the observed data equally wellwithin the measurement errors. Inversion may yield themodel with the closest t to the data, but other models withquite dierent parameters might be just as valid. In addi-tion to determining the best-t model, estimating its reli-ability and evaluating model ambiguity are importantcomponents of a surface NMR investigation.

For linear problems and for data contaminated byGaussian distributed noise, model uncertainty can be esti-mated from the model covariances [31]. In surface NMR,the inverse problem is non-linear and the measurementscan include both Gaussian noise and non-Gaussian mea-surement errors. A powerful tool for estimating modeluncertainty in this case is the method of bootstrap resam-pling [32], based on studentized residuals.3 Bootstrapping3 As their name implies, studentized residuals follow Students tdistributions.the measured data and data predicted from a best-t modelweighted by their individual importance [31]. As an alter-native, studentized residuals can be determined by repeat-ing the inversion by the number of data points anddetermine the residual of each data point with respect tothe model built after discarding this observation from thedata set.

Fig. 4 shows the result of applying a bootstrap analysisto the simulated data of Fig. 3a. Studentized residualsbased on the best-t model were used to generate 32 replicadata sets, which were then inverted. From the resultantsuite of models (gray lines in Fig. 4), median model param-eters (dashed line in Fig. 4) and their standard deviations(dash-dotted lines in Fig. 4) were computed (Table 1).

The bootstrap analysis demonstrates that the upperboundary of the high water content layer occurs at25:5 m 1 m (true value 25 m), whereas the lower bound-ary is less well resolved at 49.7 m 4.8 m (true value50 m). The water contents of the three layers were deter-

0 200 400 600

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pulse

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[As]

a

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dept

h [m

]

b

synthetic dataresampled databootstrapsmedian fit

true modelbootstrapsmedian modelbootstrap stdFig. 4. Results of applying a bootstrap analysis to the same data as shownin Fig. 3. A model with variable geometry was used for the inversion.

mined to variable degrees of accuracy with the error for themiddle layer being the highest.

3.2. 2-D investigations: magnetic resonance tomography

(MRT)

At locations where water content varies laterally, 2-D or3-D data acquisition and inversion are required. Examplesfor these structures are isolated groundwater occurrences,called perched water lenses, or water-lled caverns or cav-ities like Karst structures. For 2-D situations in whichwater content is a function of x (prole direction) and z(depth), but not the y-coordinate, Eq. (33) becomes

Ki;2-Dx; z Z 11

Kix; y; zdy 49

and Eq. (34) becomes

V i Z 10

Z 11

Kix; zf x; zdxdz: 50

Individual values are thus determined by multiplying the2-D water content distribution with a 2-D data kernel.

To acquire information to estimate 2-D distributionsof water content requires measurements along the prole.For this purpose, a coincident transmitterreceiver loopcan be incrementally shifted along the prole and/orseparate transmitter and receiver loops can be movedsystematically relative to each other along the prole.Separate loop measurements provide additional spatialsensitivity, in particular for the shallow sub-surface[18,19].

Fig. 5 displays the eective sensitivities for a coincidentloop and three separate loop congurations for three typi-cal pulse moments and for the sum of 16 pulse moments.Increasing the pulse moments for coincident loops (leftcolumn) simply increases the probed volume. By contrast,varying the loop separation provides increased sensitivitiesat shallow depth below the receiver loop. Consequently, byacquiring data using a range of pulse moments and variabletransmitterreceiver loop separation, it is possible toincrease depth (volume) penetration and sensitivity to lat-eral changes in the shallow sub-surface.

Table 1Results of the bootstrap analysis for the noise-contaminated three layerdata set in Fig. 4 based on 32 inversions of the resampled data

True values Median model l Standard deviation r

f1 5% 4.8% 0.2%f2 20% 20.1% 1.5%f3 10% 10.1% 0.9%z1 25 m 25.5 m 1.0 mz2 50 m 49.7 m 4.8 m

]s 0

dq =

150

150

0

150

150150

0 0

236 M. Hertrich / Progress in Nuclear Magnetic Resonance Spectroscopy 53 (2008) 227248150 75 0 75 150

] 0

150 75 0 75

0dept

h [m

q =

0.4

A

150 75 0 75 150

50

100

150

dept

h [m

]q

= 3.

1 As

150 75 0 75 150

0

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150

epth

[m]

9.2

As 0

50

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150 75 0 75

50

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150 75 0 75

0

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100de

pth

[m

W distance [m] E150 75 0 75 150

50

100

150

W distance [m] E150 75 0 75 150

50

100

150

2Dkern4 3 2

Fig. 5. Contour representation of 2-D data kernels for a coincident-loop conseparations (middle and right columns; see sketches shown to the top of themoments, the bottom row shows the sum of all 16 pulse moments for a typiintensity B0 48,000 nT, inclination I 60, azimuth 45, half-space resistiv150 75 0 75 150

50

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150 75 0 75 150

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150 75 0 75 150

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W distance [m] E150 75 0 75 150

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el [nV/m2]1 0 1 2

guration (left column) and for congurations with three dierent loopgure). The rst three rows show the sensitivities for three selected pulse

cal measurement. Data kernels are calculated for: Earths magnetic eldity 100 Xm.

High sensitivities may extend well outside of the loopboundaries. For separate loop measurements the sensitivi-ties are in general non-symmetric and are conned to smallvolumes below the receiver loop. Additional asymmetry,including measurements in a coincident loop conguration,is a result of the dipping (dipolar) nature of the Earthseld or may be caused by induction eects that lead toasymmetric splitting of BT and B

R in Eq. (23) [19]. Conse-

quently, the distance and orientation of an isolated volumeof water relative to the data acquisition loops aect theamplitude versus pulse moment curves.

Inversion of data acquired along a prole using coinci-dent and/or separate loops along a prole yields 2-D watermodels. The principles of 2-D inversion or magnetic reso-nance tomography (MRT), are essentially the same asthose described for 1-D inversion in Section 3.1. The datakernels for the measurements are determined by Eq. (49),and the 2-D sub-surface models are represented by a gridof water content values.

The inuence of the perched water lens in Fig. 6 on aseries of coincident loop measurements made along a pro-le is shown in Fig. 7. The background water content is 5%

by volume, whereas the water content within the lens is25%. For the four loop positions P1P4, two series of syn-thetic measurements are simulated: one for the true 2-Dwater lens model (solid lines in Fig. 7) and one for the lat-erally homogeneous 1-D case with layer thicknesses andwater contents equal to those of the water content modelvertically below the coil centers. For P1 and P4, the watercontent beneath the coil center is uniformly 5%. However,the loop with a radius of 24 m extends partly across the2-D lens at these locations and is thus aected by its anom-alous water content. At P2 and P3, the loop is entirelyacross the lens. Comparison of data simulated for the sim-plied 1-D and full 2-D models demonstrates the stronginuence of the perched water lens (Fig. 7). At P1 andP4, synthetic data have lower amplitudes for the simplied1-D assumption (neglecting the lens) than for the true 2-Dmodel. At P2 and P3, the reversed pattern are found. Theasymmetry of the simulated 2-D data is again the resultof the Earths magnetic eld being inclined.

The result of applying the xed geometry 1-D inversionalgorithm to the 2-D synthetic data (i.e. solid lines) ofFig. 7 are presented in Fig. 8. Dierences between the left

500

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P3

P2

P1

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]

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60 40 20 0 20 40 60

01020304050

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r con

tent

[%]

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a b

ject)ncrurroclin

de V30

6. Ds. So

M. Hertrich / Progress in Nuclear Magnetic Resonance Spectroscopy 53 (2008) 227248 23750x [m]

Fig. 6. Sketch of a 2-D model of a perched water-lens model (gray obmeasurements (black loops in (a)) centered along lines P1P4. Coil center ibetween coils). Water content in the lens is 25% by volume and that of the spulse moments 10 As, Earths magnetic eld intensity B0 48,000 nT, in

amplitude V [nV]

pulse

mom

ent q

[As]

P1

100 300 5000

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10

amplitude V [nV]

P2

100 300 5000

2

4

6

8

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amplitu1000

2

4

6

8

10

a b c

Fig. 7. (ad) Synthetic surface NMR amplitudes at positions P1P4 in Fig.equal to those of the 2-D model vertically below the respective coil center

synthetic measurements are not symmetric about the center of the lens; the obliqkernel.and the locations of four synthetic coincident transmitter/receiver coilement is 24 m and the diameter of the 2 turn coil is 48 m (i.e. 50% overlapundings is 5%. Synthetic data shown in Fig. 7 are calculated for: maximumation I 60, azimuth 45; half-space resistivity q 100 Xm.

[nV]

P3

0 500amplitude V [nV]

P4

100 300 5000

2

4

6

8

10

true 2D amplitudesidealized 1D amplitudes

d

ashed lines are the signals for 1-D models with water content distributionslid lines are the signals for the 2-D water-lens model. Note, that the 2-D

ue inclination of the Earths magnetic eld results in asymmetry of the 2-D

tial to provide other useful groundwater-related informa-

0 10 20 300

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1D inversiontrue values

(solr coes orefe

238 M. Hertrich / Progress in Nuclear Magnetic Resonance Spectroscopy 53 (2008) 227248and right pairs of models reect the asymmetries in the 2-Ddata. All inversion results correctly predict the 5% watercontent above the lens and good estimates of the depthof the top of the lens are provided at P2 and P3. For looppositions P1 and P4 the inversions overestimate the volumeof water immediately below the loop center. At P2 and P3,the depths to the lens lower boundary, its total water con-tent, and water content below the lens are signicantlyunderestimated. From these results, we conclude that thelateral sensitivity or footprint of the individual measure-ment series exceeds the coil dimension.

A full 2-D tomographic inversion of the four surfaceNMR measurements reproduces well the boundaries andwater content of the perched water lens (Fig. 9). None ofthe inversion artifacts seen in the 1-D models are evidentin the 2-D model. The lens boundary is correctly shownto be relatively sharp and water content within the lensbody and surroundings is accurately reconstructed. The2-D model is practically symmetric, verifying that the

Fig. 8. 1-D inversion results of the four synthetic measurements P1P4inversion (gray areas and color bars). Dashed black lines show the true wateof water directly below the centers of coils P1 and P4 (see Fig. 6); the apicinterpretation of the references to color in this gure legend, the reader istomographic inversion scheme has properly accounted forthe asymmetry of the data caused by the inclination ofthe Earths magnetic eld. The gradual transitions at thetwo ends of the reconstructed lens in Fig. 9 are a directresult of the applied smoothing.

distance [m

dept

h [m

]

P1 P260 40 20 0

10

20

30

40

50

Fig. 9. Two-dimensional inversion result of the four coincident loop SNMRsynthetic data are contaminated with 10 nV Gaussian noise (5%), comparablblack dashed lines delineate the original model in Fig. 6, whereas the four distion. From borehole investigations of hydrocarbonreservoir rocks, empirical relations have been derived thatrelate NMR porosity and relaxation constants to hydraulicconductivity [33,34], a crucial parameter in hydro-geologi-cal studies. Although the range and type of data availablefrom surface NMR techniques are more limited than thoseprovided by borehole methods, the possibility of estimatinghydraulic conductivities from surface NMR data is worthexploring.

4.1. Relaxation processes in rocks

In a bulk uid a precessing proton transfers energy to itssurrounding. This causes the proton to relax with the time4. Relaxation

In addition to supplying estimates of sub-surface watercontent surface NMR relaxation properties have the poten-

id lines in Fig. 7). Results are shown for a xed-boundary least-squaresntent directly below the coil centers. There is no anomalous concentrationf the lens at these locations are delineated by arrows in (a) and (d). (Forrred to the web version of this article.)constant T 1 into its low-energy state in which the protonprecesses around an axis parallel to B0. The relaxation ofthe transverse magnetization, with the time constant T 2,is additionally aected by diusion. Thus, T 2 is in generalsmaller than T 1, but since diusion is of minor importance

]

P3 P4

RMS=8.8

20 40 60

wate

r con

tent

[%]

0

5

10

15

20

25

30

data sets simulated for the perched water lens model (see Fig. 6). Thee to that encountered under favorable eld measurement conditions. Thecrete columns are the 1-D smooth inversions.

c Rein homogeneous magnetic elds, T 2 approximately equalsT 1 [3336].

If water is captured in a porous rock, additional relaxa-tion processes take place at the pore wall and both T 1 andT 2 are dramatically decreased. Protons close to the grainsurface encounter fast magnetic relaxation with high prob-ability. The cause of this relaxation is the presence ofstrong, highly localized magnetic elds, generated byunpaired electrons in paramagnetic atoms such as manga-nese and iron, which are attached to the negatively chargedgrain surface in many natural settings. As water moleculesconstantly diuse through the pore space driven by Brown-ian motion, unrelaxed spins are delivered to the surface,while relaxed spins are moved from the surface into theopen pore space. If the self-diusion process is fast com-pared to the surface-induced relaxation, the overall relaxa-tion will be averaged to a uniform mono-exponential decaythroughout the pore [37].

Naturally, the relaxation strongly depends on the poresurface relaxivity, i.e. type and concentration of paramag-netic ions, but also on the pore size. In general, small poresare characterized by higher relaxation rates, i.e. shorterrelaxation times, than large ones. A rock hosting adistribution of isolated pores of dierent size will exhibita multi-exponential decay, due to the superposition ofNMR signals in the record. In well coupled pore systems,however, as present in unconsolidated sediments, diusionwill transport molecules across several pores in relevanttime-scales, leading to eectively averaged relaxation con-stants and thus to mono-exponential decay [38].

In practice, water molecules throughout the investigatedvolume may also be aected by slightly dierent macro-scopic and microscopic magnetic elds independent of poregeometry, leading to slightly dierent Larmor frequencies(i.e. inhomogeneous spectral broadening in nuclear mag-netic spectroscopy). The diering Larmor frequencies causea substantial loss of phase coherence in the spin ensemble,and in consequence lead to a decreased transverse decayparameter T 2. These biased values of the transverse relaxa-tion time decay parameter are referred to as T 2.

4.2. Acquisition of relaxation parameters

NMR receiver coils pick up the superimposed magneticeects of all transverse decay and dephasing mechanisms:the so-called free induction decay (FID). For this reasonT 2 is the decay parameter most easily extracted fromNMR observations (Fig. 10a). Magnetic eld gradientscausing such dephasing occur at dierent scales, rangingfrom those from large geological or anthropogenic objectswith high susceptibility, through gradients from featuresat the granular scale (e.g. from weathered hard rock), tointernal gradients at the micro-scale caused by magnetizedcoating of inner pore surfaces. Such magnetic gradientshave no inuence on hydraulic properties. Consequently,

M. Hertrich / Progress in Nuclear MagnetiFID-determined T 2 values are rarely good proxies forhost rock pore space properties and it is recommendableto restrict oneself to T 1 and/or T 2 data when aiming forthose targets.

In laboratory and borehole investigations, T 1 and T 2 canbe determined indirectly by invoking various appropriatesequences of pulses at the Larmor frequency, for examplethe inversion recovery method [36]. Unfortunately, onlyapproximate estimates of T 1 can be determined for sub-sur-face water protons with currently available surface NMRequipment [39,40].

The determination of relaxation constants by surfaceNMR is restricted by physical limitations: pulse sequencesfor T 1 and T 2 determination, that are well established inlaboratory and borehole applications, are realized withsecondary elds B1, inducing tip angles of 90

and 180.In surface NMR B1 is highly inhomogeneous throughoutthe investigated volume, making it dicult to achieveuniform tip angles for a large ensemble of spins, and thuspreventing the use of sophisticated pulse sequences. Onlyan approximate estimate of T 1 can be determined forsub-surface water protons by applying a pseudo saturationrecovery sequence: the spin magnetization V 0, generated bythe initial electromagnetic pulse will gradually relaxtowards its equilibrium state (Figs. 1 and 10a). Duringthe decay of the signal generated by the rst pulse, a secondpulse is applied (in-phase or with arbitrary phase delay).The spin magnetization is tilted again and a second FID-determined V 00 is measured. Under ideal conditions, whenthe rst pulse is on-resonant and tips the spin vectors uni-formly by 90, the application of a second 90-pulse leadsto the same results as a common saturation recoverysequence (for the latter, see [34]). The transverse magneti-zation is not being saturated at all during this sequence.The longitudinal component, however, having recoveredaccording to T 1 between two pulses, will be mapped intothe (detectable) transverse plane by the second pulse,whereas the decayed transversal component will bemapped into the (non-detectable) longitudinal direction.The signal amplitude after the second pulse will carry infor-mation on T 1, in dependence of the delay time between thepulses sd, obeying to V 00 1 expsd=T 1.

The T 1 recovery curve is then estimated from just threepoints: (i) for a hypothetical pulse delay of zero the spintransverse magnetization is assumed to be fully saturated,i.e. V 00 0, (ii) the initial amplitude V 00 at the delay timesd, (iii) the amplitude V 0 that has been measured after a sin-gle pulse is assumed to be equal to the asymptotic value ofthe relaxation process V 10 V 0.

4.3. Observed data

Fig. 11a shows experimental data of V 00 for a range of qvalues and three delay times of 380, 480 and 680 ms, respec-tively (black, blue and red lines). V 10 is determined fromthe asymptotic values at a delay time of 3600 ms (greenline). Fig. 11b shows the results of tting exponential

sonance Spectroscopy 53 (2008) 227248 239curves to the various suites of recorded data. The T 2estimates shown by dashed lines are based on three

Re240 M. Hertrich / Progress in Nuclear Magneticindependent suites of V 0 values. The solid lines are T 1 esti-mates based on single second pulse measurements, at only

whereas T 1 values vary between 600 and 900 ms over the

V0V0e

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itude

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de

Fig. 10. Sketch of the NMR pseudo saturation recovery sequence. (a) From aand initial amplitude V 0 can be determined. (b) Amplitudes V 00 of individual fdelays to (c) achieve a recovery functional that increases as 1 expt=T 1.

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Fig. 11. (a) Surface NMR data recorded at the Haldensleben test site inGermany. For a range of pulse moments q: the green curve is theamplitude V 0 in Fig. 10 and the black, blue and red curves are amplitudesV 00 in Fig. 10 for three dierent delay times sp 380, 480 and 680 ms. Foreach q value, the dashed line is the best-t exponential function1 expt=T 1 to the four measured amplitudes (i.e. the points deningthe black, blue, red and green curves) and the assumed zero amplitude atzero delay time. (b) For the same data set, the lower black, blue and reddashed lines are independent T 2 estimates obtained for a range of pulsemoments q. From the upper, the solid purple line represents the best-t T 1values determined from the dashed lines in (a). The solid black, blue andred lines are as for the purple line, but instead of using three dierent delaytimes in the construction of the exponential functions, only amplitudes fora single delay time are employed.range of pulse moments, indicating variations in sub-sur-face pore properties.

Several hydro-geological assessments have shown agood correlation of surface NMR T 1-based hydraulic con-ductivities with those determined by pumping tests [40,41].

4.4. Limitations of current schemes

Currently, only one single second pulse at a single spe-one sd each (black, blue and red lines), and estimates basedon all three pulse delays (magenta line). The T 2 values areroughly constant at 300 ms for all pulse moments,

pulse delay d

am

plitu

de 1et/T1

c

single free induction decay (FID) experiment only the T 2 relaxation timeree induction decay (FID) experiments are determined at increasing pulseModied from Ref. [51].

sonance Spectroscopy 53 (2008) 227248cic delay time is employed in surface NMR. This schemeonly provides useful estimates of T 1 for mono-exponentialsignals and a high signal-to-noise ratio. To better constrainthe exponential T 1 recovery curve for a multi-exponentialanalysis or limited data quality a series of V 00 values mustbe recorded by repetition of the scheme for several delaytimes (Fig. 10).

For imperfect pulses (tilt angles others than 90), thetransverse magnetization after the second pulse is a morecomplex combination of T 1 and T 2. However, since themajor constituent of the recorded signal is induced by spinsexcited at or close to 90, a rst order approximation of thetrue T 1 can be made based on the acquired relaxation con-stants. Nevertheless, to derive robust estimates of poreproperties from the surface NMR relaxation parameters,more reliable T 1 determination schemes are required anda better understanding of the eects of sub-surface inhomo-geneities and resulting non-90 tipping angles is needed.Furthermore, the basic assumption that T 1 can be calcu-lated using a simplied saturation recovery formula islikely to be insucient for separate transmitter and receiverloops. Fortunately, ongoing projects, involving moresophisticated modeling and inversion schemes, are

c Reexpected to yield improved determinations of T 1 in thenear-future.

5. Limitations of the surface NMR method

5.1. Inuence of the Earths magnetic eld

Signicant variations in the magnitude and inclinationof the Earths magnetic eld substantially inuence theapplication of surface NMR methods worldwide. Prior toconducting a survey at any location, Larmor frequenciesand induced magnetization levels can be estimated fromthe following relationships

xL cp j B0 j; 51

M j B0 j c2h2q04kT

; 52both of which are linear functions of j B0 j. Eq. (52) isknown as Curies Law for the spin magnetization, with cthe gyromagnetic ratio, h Plancks constant, q0 the spindensity, k Boltzmans constant and T the absolute temper-ature in degrees Kelvin. According to Eqs. (23), (51) and(52), the surface NMR signal scales with the square ofthe Earths magnetic eld strength j B0j2. The inclinationenters the integral equation in a more complex way. Itdetermines the perpendicular projection of the secondaryeld on the Earths magnetic eld direction, as it aectsboth the tip angle in the sine term of Eq. (23) and the sen-sitivity of the receiving eld.

A map of estimated mean surface NMR signal responsebased on the global magnetic parameters (Fig. 12a and b) isshown in Fig. 12c. The values shown in this map are for astandard coincident loop measurement and are relative to avalue in central Europe (B0 48; 000 nT, inclination=65).Fig. 12c demonstrates that amplitudes in south Americacan be as small as 25% of that in central Europe, whereasthose typical in high northern and southern latitudes can be150% of the mid-European value.

Fig. 13ad show the dependency of the data kernel for a45; 000 nT Earths magnetic eld amplitude with inclina-tion angles ranging from 0 (equator) to 90 (poles). Thedistribution of sensitivities throughout the range of pulsemoments changes signicantly. For low inclinations, thesensitivities are high at very low q values, but they decayrapidly as q increases. By contrast, for high inclination sites(polar regions) the sensitivities are more uniform through-out the range of q values.

Simulated sounding curves for a 100% water contentthroughout the sub-surface for magnetic inclinations 090 are displayed in Fig. 13eh. Each gure shows graphsfor the Earth magnetic eld strength ranging from 25,000to 65,000 nT. Two important features can be observedfrom the series of amplitude versus q curves:

The amplitude of the signal versus q curves in each g-

M. Hertrich / Progress in Nuclear Magnetiure scales with increasing Earths eld in a quadraticfashion as predicted from Eqs. (49) and (50). The pattern of the curves changes with inclination. Theshape of the curves is a result of the pattern of the datakernels from Fig. 13ad. At low inclinations the highsensitivities at low values of q cause a prominent peakat corresponding low q. With increasing inclination themore uniform data kernels lead to more uniform mod-eled amplitude versus q curves with a less pronouncedpeak at low values of q.

From Fig. 13ad it is clear that local values of theEarths magnetic eld are important for expected surfaceNMR signals at any site worldwide. Employing the meanvalue of the amplitudes for a standard series of pulsemoments allows one to determine the possibility of obtain-ing usable surface NMR data.

5.2. Inuence of the sub-surface conductivity distribution

Electromagnetic elds generated by surface loops are, ingeneral, aected by induction due to the conductivity of theground. In earlier publications this resistivity inuence hasbeen related to the skin depth d at the local Larmor fre-quency [16,42]. However, the skin depth is not appropriatefor describing the induction eects in the near-eld of sur-face NMR transmitter loops, since it is dened for planewaves incident at the Earths surface [23], whereas propa-gation of the transmitted electromagnetic eld in the vicin-ity of a loop is dominated by geometric spreading.Comprehensive modeling shows that the resistivity inu-ence decreases with decreasing loop size, but for commoncombinations of loop size in the range of 5150 m andground resistivities between some few and several hundredXm, the resistivity inuence is considerable and needs to betaken into account [43,44].

Real and imaginary parts of the data kernels for homo-geneous ground resistivities in the 10001 Xm range aredisplayed in Fig. 14ah for a 100 m-diameter loop. At highresistivity of 1000 Xm and 100 Xm (Fig. 14a, b, e and f),there is no signicant dierence, either in the real or inthe imaginary part. At a sub-surface resistivity of 10 Xm(Fig. 14c and g) the real part of the data kernel is signi-cantly attenuated in amplitude and depth penetration,while the imaginary part becomes quite large. For resistiv-ity as low as 1 Xm (Fig. 14d and h) these eects becomeeven more pronounced. Data kernels are attenuated downto about 20% in depth penetration compared to their valueat 1000 Xm; real and imaginary parts have comparableamplitudes.

The corresponding simulated measurement curves for awater content of 100% are shown in amplitude and phaseform in Fig. 14ip (solid lines and circles). The simulatedmeasurements for purely insulative ground are shown(dashed lines) for comparison. With decreasing resistivity,progressively lower amplitudes and higher phase anglesare observed. At very low resistivities of 1 Xm the signal

sonance Spectroscopy 53 (2008) 227248 241amplitudes for pulse moments larger than 5 As are attenu-ated down to unmeasurably small values. The resistivity

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Fig. 12. Global maps showing the worldwide distribution of (a) the intensity B0 and (b) the inclination I of the Earths magnetic eld. (c) The estimatedworldwide surface NMR signal for coincident loop measurements normalized to mid-European conditions (B0 = 45,000 nT, I = 60). (Maps compiledfrom WMM-2005 magnetic data, US-National Oceanic and Atmospheric Administration (NOAA), http://www.ngdc.noaa.gov/seg/geomag).

242 M. Hertrich / Progress in Nuclear Magnetic Resonance Spectroscopy 53 (2008) 227248

0ent

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M. Hertrich / Progress in Nuclear Magnetiinuence on surface NMR measurements becomes evenmore complex for inhomogeneous resistivity distributionsin the sub-surface, either for 1-D resistivity stratication[45] or for 2-D or 3-D anomalous resistivity structures[46]. Thus, a priori knowledge of the sub-surface resistivitydistribution is essential for the correct computation of thedata kernel. Incorrect assumptions on the resistivity distri-bution can signicantly aect the water content model thatis derived by inversion with an incorrect kernel [47].

The imaginary part of the surface NMR signal exhibitsinteresting and complementary characteristics to the realpart. For example it normally increases with decreasingresistivity, and generally has its maximum sensitivity atgreater depth than does the real part. Therefore, by invert-ing complex surface NMR data involving complex kernels,there exists the potential for providing more completeinformation about the sub-surface [17].

6. Field data example

1-D surface NMR depth soundings have been acquiredat a test site approximately 70 km east of Berlin in Ger-many. Here, the near-surface geology is represented byQuaternary glacial sediments consisting of alluvial sands,marls and glacial tills. At this test site the sedimentarystratication is well known from a nearby borehole andcomplementary near-surface geophysical measurements

0 5 10 15 0 5 10 15 0amplitude V0 [

Fig. 13. (ad) Surface NMR data kernels for an Earths magnetic eld intensitmeasurements for 100% water content at inclinations I = 0, 30, 60 and 90 anincl=60

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25000nT35000nT45000nT55000nT65000nT

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sonance Spectroscopy 53 (2008) 227248 243[48]. A surface NMR measurement has been realized witha circular coincident loop for both transmitter and receiverwith a diameter of 100 m, 1 turn, and a suite of 24 logarith-mically spaced pulse moments ranging from 0.25 to18.5 As. The recorded data displayed in Fig. 15 show signalamplitudes in the range of 6001200 nV with a maximumat a pulse moment of about 1 As. The corresponding T 2relaxation constants range from below 200 ms for smallpulse moments and increase up to 300 ms for larger pulsemoments. The signal phase shows a pattern that cannotbe explained by induction eects of the loop magneticelds. The obvious correlation with the signal-derived Lar-mor frequency indicates that the major eect of the phase iscaused by o-resonant excitation of the proton spins ando-resonant recording in the resonance-tuned receiverloop. These technically induced phase shifts in the recordedsignal make the complex signal basically unusable forinversion. The amplitude, however, is not aected by theseo-resonance eects and the subsequent assessment ofinversion schemes on this data set is based on amplitudedata alone.

Applying the xed-boundary inversion scheme to thedata set in Fig. 15 and using the bootstrapping procedurefor assessment of the reliability of the model gives the suiteof inversions in Fig. 16a. The models indicate a shallowaquifer from 2 m down to approximately 15 m with a watercontent of around 25% above a layer of low water content

5 10 15 0 5 10 15 V]

y B0 = 45,000 nT and inclinations I = 0, 30, 60 and 90. (eh) Syntheticd the Earths magnetic eld intensity varying from 25,000 nT to 65,000 nT.

m1

Re1000 m

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]re

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244 M. Hertrich / Progress in Nuclear Magneticfrom 15 to 30 m. At 30 m the water content rises to a sec-ond series of two aquifers with maximum water contents of18% and 22% at depth of 35 m and 62 m, respectively.These aquifers are interbedded with a layer of slightlylower water content at a depth of 45 m. Bootstrapping ofthe data, based on studentized residuals as introduced inSection 3.1.3, gives only slight variations in the series ofinverted water content models.

Using the inversion scheme of variable-boundary inver-sion for the same data set with a model geometry derivedfrom the xed-boundary inversion to have six layers yields

dept

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Fig. 14. (A) Surface NMR data kernels showing real and imaginary parts for(B) Synthetic measurements 100% water content within ground with homogecircles) and corresponding measurements for perfectly resistive ground (dashe0

10 m

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c d

sonance Spectroscopy 53 (2008) 227248the inversion results in Fig. 16b. The models show a similarsub-surface water content distribution, with three units ofhigh water content, enclosed within conning beds of lowwater content. The six layer model exhibits some dier-ences to the xed-boundary model: (i) whereas the xed-boundary model shows a thin layer of low water contentclose to the Earths surface, the variable-boundary modelcannot resolve this, (ii) the aquifer at 3040 m depth inthe variable-boundary model shows signicantly higherwater content than the equivalent one in the model of xedgeometry, (iii) water contents of the conning beds are gen-

0 0 5 10

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ground of homogeneous sub-surface resistivities ranging from 10001 Xm.neous sub-surface resistivities ranging from 10001 Xm (black lines andd lines).

c ReM. Hertrich / Progress in Nuclear Magnetierally lower in the variable geometry model than in thexed-boundary one.

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Fig. 16. Bootstrap analysis of the data set in Fig. 15 for (a) the xed layerinversion scheme and (b) the variable layer inversion scheme.

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Phase []Fig. 15. Recorded surface NMR data at the test site east of Berlin. The compphase (b). Additionally the derived relaxation time constant T 2 (c) and the varvariations of the Earths magnetic eld is shown.The bootstrapped results of the variable-boundarymodel show a higher variability and consequently a higherstandard deviation than the xed-boundary model. How-ever, the series of bootstrapped inversion yields consistentresults indicating that the data are tted best by theassumed model geometry with six layers.

Results of both inversion schemes are compared inFig. 17. The following observations can be made: (i) thesynthetic measurements based on the two median modelsof variable and xed-boundary inversion t the measureddata equally well, (ii) the main structures of the derivedmodels basically coincide within their systematic limita-tions and show the segmentation into three individual aqui-fers, interbedded with conning beds. The xed-boundaryinversion, however, heavily smoothes thin layers in partic-ular and makes a quantitative interpretation of aquiferproperties concerning boundaries and water contents

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Larmor frequency [Hz]lex signal amplitude versus pulse moment is plotted in amplitude (a) andiation of the Larmor frequency (d) during the measurement due to diurnal

sonance Spectroscopy 53 (2008) 227248 245impossible. The variable-boundary inversion shows a muchsharper distinction of aquifer boundaries and allows quan-titative estimates of aquifer water contents.

Comparison of the inversion results with the aquiferstructure (dark gray patch plots in Fig. 17b) that arederived from the borehole logs at the right-hand side, givesa fairly good correlation. Both inversion results delineatethe complex aquifer structure to a satisfactory degree.The estimated water contents are in good agreement withexpected aquifer properties in these Quarternary glacialdeposits. The borehole extends down to a depth of 60 m.At about 54 m after a loss of core material for some 2 m,Tertiary sediments have been found that consist of well-sorted marine sands. Hence, no signicant change of aqui-fer properties is found at this geological boundary, but theTertiary sediments are assumed to continue as a quitehomogeneous layer. The lower boundary of the third aqui-fer interpreted from surface NMR inversion at a depth of81 m is below the extent of the borehole and thus cannotbe conrmed. Additionally, the sensitivity of the surfaceNMR method does not allow a reliable prediction of the

amm

[APig.f th

Reboundaries and/or water contents at this depth. Eventhough a fairly good reproduction of this boundary withboth inversion schemes is given, the existence and reliabil-ity of this layer boundary should be treated with care.

Note that throughout the resistivity log the regions ofhigh water content are characterized by increased valuesof resistivity, but variations are too small to be resolvedby means of surface geoelectrical or electromagnetic meth-ods [49].

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Fig. 17. (a) Median models of water content distribution with depth from Fto interpreted aquifer stratication from borehole data. (b) Borehole logs ocenter.

246 M. Hertrich / Progress in Nuclear Magnetic7. Summary and conclusions

This review has provided information about the variousaspects of surface NMR. In the rst section the most basicformulation is presented which reveals the complexity ofthe forward functional. For the quantitative descriptionof the surface NMR signal the interaction of the spin sys-tem with the non-uniform, non-perpendicular and ellipti-cally polarized secondary eld is taken into account.Furthermore, for congurations with non-coincident trans-mitter and receiver loops, the vectorial relation of the spinmagnetization to the these elds was considered. Thisallows a complete forward functional with suitable formu-lations for 1-D and 2-D conditions to be derived by inte-grating the data kernel of the forward functional to therespective dimensions. In the second section these data ker-nels are the basis for the inversion of surface NMR mea-surements to reconstruct models of sub-surface watercontent distribution. Besides a least-squares inversion ofa model with a large number of layers and variable butconstrained water content, which is most common in geo-physical data inversion, a novel scheme with a small num-ber of discrete layers whose boundaries are allowed to varyin depth is presented. Both schemes provide comparablemodels. Comparing these two approaches, the variable-boundary model gives a better quantication of the depthsof layer boundaries and estimates of layer water contentthan does the model with xed boundaries. But inversionwith such a scheme can only provide useful sub-surfaceinformation if the variation of the water content in thesub-surface is sharp rather than gradational, and if an esti-mate of the number of geological units is known before-hand. Inversion of eld data is in general ambiguous.

100

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I]0 5

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NN short102

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16, obtained by xed and variable-boundary inversion schemes comparede research drill-hole in about 150 m distance from the measurement loop

sonance Spectroscopy 53 (2008) 227248This is particularly so in cases where there are considerableuncertainties in the measured data, which often occurs forthe weak surface NMR signals in the presence of strongambient background noise. A bootstrapping schemeapplied to surface NMR inversion is introduced in Section3. It provides a suitable tool to assess the ambiguity of themodel of water content distribution and allows the assign-ment of condence intervals for the shown example. Inmany NMR applications the relaxation time is the majorsource of information on properties of the object underinvestigation. Also in NMR applied to geo-materials therelaxation time can be a useful measure to estimate struc-tural parameters, but determination of the sub-surface dis-tribution of the relaxation constants is physically limitedand technically challenging for surface NMR. In Section4 the available techniques used for surface NMR areexplained. It is demonstrated that in sedimentary environ-ments the most easily accessible relaxation time T 2 is rarelya valuable measure for host rock properties. In any casequantitative formulations for the derivation of T 1 relaxa-tion from surface NMR measurements are not yet avail-able. The two inversion schemes and the bootstrappingtechniques are applied to a real data example in Section6. From both inversion schemes a consistent model of1-D aquifer stratication is obtained. The comparison to

c Reborehole data from a nearby research drill-hole shows thecapability and almost unique potential of the surfaceNMR technique in resolving discrete water-bearing zones.A model of similar spatial resolution of the water contentdistribution can only be interpreted by the combinationof a series of borehole measurements, but can denitelynot be obtained by any other surface geophysicaltechnique.

The development of surface NMR has undergone arapid progress over the last two decades. Nowadays ithas reached a mature phase in terms of available hardware,theoretical description of the forward functional andadvanced inversion techniques. However, major topicsfor further research lie in (i) forward calculation of the loopmagnetic elds in more complex environments such as var-ied topography or spatially complex resistivity distributionwithin the sub-surface; (ii) the quantitatively more preciseformulation of the spin dynamics in the weak magneticeld of the Earth and the non-uniform loop elds and(iii) establishing reliable correlations of surface NMRdetermined relaxation times and hydro-geological parame-ters. A major drawback of applying surface NMR togroundwater studies is the presence of background noise.Typical signal amplitudes of surface NMR measurementsare very weak and cannot be easily increased relative tothe ambient noise level by technical means. Hence, surfaceNMR measurements are nowadays not feasible in manyenvironments. Ongoing research by several groups world-wide, aimed at understanding, describing and recordingsurface NMR signals oers promise of further