-
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: [email protected]
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
2
4
6
8
10
12
14
Amplitude [nV]
pulse
mom
ent q
[As]
0 10 20 30
0
10
20
30
40
50
60
70
80
90
100
water content [%]
dept
h [m
]
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
0
2
4
6
8
10
12
14
Amplitude [nV]
pulse
mom
ent q
[As]
a
0 10 20 30
0
10
20
30
40
50
60
70
80
90
100
water content [%]
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
50
100
150
epth
[m]
9.2
As 0
50
100
150 75 0 75
50
100
150
150 75 0 75
0
50
100
150
0
50
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
100
150
150 75 0 75 150
0
50
100
150
150 75 0 75 150
0
50
100
150
W distance [m] E150 75 0 75 150
0
50
100
150
150 75 0 75 150
50
100
150
150 75 0 75 150
0
50
100
150
150 75 0 75 150
0
50
100
150
W distance [m] E150 75 0 75 150
0
50
100
150
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
50
0
5060
40
20
0
y [m]
P4
P3
P2
P1
z [m
]
dept
h [m
]
distance [m]
P1 P2 P3 P4
60 40 20 0 20 40 60
01020304050
wate
r con
tent
[%]
0
5
10
15
20
25
30
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
2
4
6
8
10
amplitude V [nV]
P2
100 300 5000
2
4
6
8
10
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
10
20
30
40
50
water content [%]
P1
dept
h [m
]
0 10 20 300
10
20
30
40
50
water content [%]
P2
0 10 20 300
10
20
30
40
50
water content [%]
P3
0 10 20 300
10
20
30
40
50
water content [%]
P4
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
t/T2*
a
timea
mpl
itude
V0 V0
d
b
time
am
plitu
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.
01000
20003000
4000
05
1015
200
500
1000
1500
delay [ms]
pulse moment q [As]
am
plitd
e V 0
o
r V 0,
[nV
]
0 5 10 15 200
200
400
600
800
1000
pulse moment q [As]
rela
x. ti
mes
T1,
T 2*
[m
s] T1
T2*
a
b
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
-
25000
30000
30000
30000
35000
35000
35000
40000
4000040000
45000
4500045000
50000
50000
50000
55000
55000
180 240 300 0 60 120 180
60
30
0
30
60a
60
60
40
40
20
20
0
0
20
20
4040
6060
180 240 300 0 60 120 180
60
30
0
30
60b
0.4
0.4
0.6
0.6
0.6
0.80.8
1
1
1
1.2
1.2
1.2
180 240 300 0 60 120 180
60
30
0
30
60c
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
c
g
c Reincl=0de
pth
[m]
0 5 10 15
0
50
100
150
200
incl=30
0 5 10 15pulse mom
0
5
10
15
20
incl=0
dept
h [m
]
incl=30
a b
e f
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
5 10 15
incl=90
q [As]0 5 10 15
sen
sitiv
ity [n
V/m]
0
100
200
300
400
500
incl=60 incl=90
25000nT35000nT45000nT55000nT65000nT
d
h
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
dept
h [m
]re
al p
art
0 5 10
0
25
50
75
100
100
0 5
a bA
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
h [m
]im
agin
ary
part
0 5 10
0
25
50
75
1000 5 1
pulse m
pulse m
se0 20
0 2000 4000
0
5
10
15
1000 m
pulse
mom
ent q
[As]
0 2 4
0
5
10
15
pulse
mom
ent q
[As]
0 2000 40
100 m
0 20 4
amp
conductive kern
pha
e f
i j
m n
B
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
0 5 10
1 m
0 5 10
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
oment q [As]
oment q [As]0 5 10
nsitivity [nV/m]40 60 80 100
00
0
0 2000 4000
10 m
0 100 200
0 2000 4000
1 m
litude [nV]
el insulative kernel
200 0 200
0
5
10
15
se [deg]
g h
k l
o p
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.
0 500 1000
0
2
4
6
8
10
12
14
16
18
20
Amplitude [nV]
pulse
mom
ent q
[As]
0 20 40
0
10
20
30
40
50
60
70
80
90
100
water content [%]
dept
h [m
]
measured dataresamplesbootstrapsmedian fit
bootstrapsmedian modelbootstrap std
0 500 1000
0
2
4
6
8
10
12
14
16
18
20
Amplitude [nV]
pulse
mom
ent q
[As]
0 20 40
0
10
20
30
40
50
60
70
80
90
100
water content [%]
dept
h [m
]
measured dataresamplesbootstrapsmedian fit
bootstrapsmedian modelbootstrap std
a
b
Fig. 16. Bootstrap analysis of the data set in Fig. 15 for (a)
the xed layerinversion scheme and (b) the variable layer inversion
scheme.
0 500 1000
0
5
10
15
20
Amplitude [nV]
pulse
mom
ent q
[As]
0 50 100
0
5
10
15
20
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
0 100 200 300
0
5
10
15
20
Relax. Time T2*
[ms]2042 2044 2046
0
5
10
15
20
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].
0 20 40
0
10
20
30
40
50
60
70
80
90
100
water content [%]
dept
h [m
]
fixed boundary modelvariable boundary modelaquiferconfinig
bedaquifer(assumed)
0 50
natural g
gamma
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
a
I]0 5
neutron-neutron
NN short102
resistivity
short normal
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