NMR Image Reconstruction in Nonlinearly Varying Magnetic Fields: A Numerical Algorithm

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1 5430 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013 NMR Image Reconstruction in Nonlinearly Varying Magnetic Fields: A Numerical Algorithm Jochen A. Lehmann-Horn and Jan O. Walbrecker Commonwealth Scientific and Industrial Research Organisation, (CSIRO), NSW 2234, Australia Stanford University, Stanford, CA USA We propose a numerical algorithm for nuclear magnetic resonance (NMR) imaging in low magnetic fields that vary spatially in a nonlinear way. Since we operate at low frequencies 3 MHz, moderate electrically conductive materials in large sample volumes can be accessed. Frequency swept spin-echo pulse sequences are employed to control the excitation bandwidth. The resolution and sensitivity of this nonconventional imaging approach is studied. To localize the origin of the NMR signal in the sample volume we do not rely on linear magnetic field gradients as in modern high-field imaging devices, but utilize the generic spatial inhomogeneity of RF and DC fields generated by relatively large surface and volume coils. A numerical forward and inverse modeling example related to applications in the water and oil research is described. The algorithm may be applied to single-sided and borehole imaging systems in the future. Index Terms Frequency swept pulses, inverse theory, nuclear magnetic resonance imaging, porous media. I. INTRODUCTION I NDUSTRIES related to water, petroleum and gas are reliant on advanced and efficient sensing and imaging technologies. Analysis of water-content and porosity can improve exploration, characterization and separation of materials. In this context, nuclear magnetic resonance (NMR) is a powerful tool for nondestructive structural investigations of porous media, rocks and raw materials [1]. Traditionally, strong and homogeneous static magnetic fields are employed in NMR to increase the resonance signal. Because generally the signal scales with the square of static field amplitude, increasing the field amplitude gives access to small sample volumes. Strength and homogeneity of the field is improved by employing immobile superconducting magnets and shimming technologies [2]. In recent years, developments of pre-polarization [3], hyper-polarization [4], dynamic nuclear polarization [5] and SQUID detection [6] techniques have made the signal-to-noise ratio of many NMR applications less dependent on the magnetic field strength and therefore have enabled measurements in low fields [7]. In magnetic resonance imaging (MRI) much effort has been spent to control the homogeneity of static (DC) fields in larger volumes. By achieving a homogeneous, and adding controlled linear gradients to, each point in space is assigned a specific frequency [8]. This relation between the Larmor frequency and space coordinate is the centerpiece of conventional imaging principles (phase and frequency encoding) and used in a Fourier-Transform approach for image reconstruction. It is also important to generate oscillating excitation (RF) fields that are as uniform as possible in the imaging focus [2]. Because technically neither nor Manuscript received February 14, 2013; revised April 03, 2013; accepted June 02, Date of publication June 13, 2013; date of current version October 21, Corresponding author: J. A. Lehmann-Horn ( jochen. lehmann-horn@csiro.au). Color versions of one or more of the figures in this paper are available online at Digital Object Identifier /TMAG can be made arbitrarily homogeneous (especially in large sample volumes), adiabatic as well as composite pulses have been developed to compensate for such inhomogeneities [9]. In this contribution, we propose a numerical algorithm that enables NMR imaging in magnetic fields that are weak and vary spatially in a nonlinear way ( and ). The algorithm can be used to model arbitrary coil geometries and electrical conductivity structures, and calculate the corresponding spatial variations of and. Common amplitude and phase modulations of fields can be computed. The induced spin dynamics are described by the Bloch equations. To spatially locate the source of the NMR signals we do not rely on the traditional Fourier-Transform approach (classical phase and frequency encoding), and therefore high-precision linear magnetic field gradients are not required as in conventional MRI devices. Instead, an inversion routine is employed that utilizes the natural (nonlinear) variation of RF and DC fields generated by surface and volume coils for image reconstruction. We demonstrate the capability of the algorithm by simulating an unconventional low-field imaging setup. The goal of this conceptual setup is to image large sample volumes (1 10 litres) using minimal laboratory resources (without shimming techniques). Wideband spin-echo pulse sequences that stimulate NMR signals throughout a large sample volume are used in our simulated experiments. Image reconstruction is based on the spatial sensitivity of employed surface and volume coils. The resolution and sensitivity limit of this low-field imaging configuration is investigated. Our results are based on numerical experiments only; a prototype system has not been developed yet. Our algorithm could be combined with the computation of magnetic fields of permanent magnets to improve single-sided or borehole imaging concepts in highly inhomogeneous static fields ( 0.1 T). The simulation and reconstruction algorithm is based on C++ object oriented finite-element code. In Section II, we describe the theoretical details of our algorithm: computation of magnetic fields that are required in an NMR experiment in 2.1, simulating the spin dynamics in 2.2, pulse sequences in 2.3, and the IEEE

2 LEHMANN-HORN AND WALBRECKER: NMR IMAGE RECONSTRUCTION IN NONLINEARLY VARYING MAGNETIC FIELDS 5431 image reconstruction algorithm in 2.4. In Sections III and IV the setup for a conceptual study is described, and simulation results are shown, respectively. II. THEORY A. Magnetic Fields Computations Typical NMR experiments require two types of magnetic fields: a static magnetic field, and an oscillating magnetic field. The static magnetic field presets the (angular) Larmor frequency of the system as given by co-rotating component determines the interaction with the nuclear spins, the counter-rotating component determines the sensitivity of a given receiver coil to the nuclear spins [12]. B. Simulations Based on Bloch Equations The system we intend to image (a sample containing water or oil) is an ensemble of spin 1/2 protons and can be represented by a macroscopic magnetization vector. The dynamics of nuclear magnetization under the influence of a static magnetic field and a pulsed field can be described classically by the Bloch equation [8]: (1) where is the gyromagnetic ratio of the studied nuclei. In addition, instigates the slight tendency for spins to align along the axis of, giving rise to an equilibrium magnetization proportional and parallel to. For our imaging experiments, is fairly inhomogeneous. In addition, we apply controlled static gradient fields on top of the field. Each location in the sample has a designated Larmor frequency The well-defined spatial correspondence of frequency and location is used in conventional image reconstruction (phase and frequency encoding [8]). However, it is not an easy task to design a field or gradient that is homogeneous across the sample volume if the NMR sample is not small compared to the field source. For large samples, the field varies across the sample volume, depending on size and shape of the coil. The oscillating field isappliedinformofoneormore short pulses resonating at a reference frequency.theirrole is to excite the spin system out of its equilibrium state and prepare desired spin components by reorienting and/or refocusing for subsequent measurement. The excitation field can be computed by frequency-dependent Maxwell equations that take into account the electrical conductivity inside or surrounding the NMR sample. We apply a finite-element (FE) routine [10], extended to incorporate displacement currents. The FE method is based on solving the magnetic vector potential equation [11] from which the magnetic field can be obtained via. Alternatively, a variety of commercial FE solvers can be used to compute. An input current of the form leads to a field: The amplitude varies with position and coil geometry, the phase is a function of electrical conductivity. In most NMR applications the amplitude is much larger than. Therefore, the component of that is parallel to (i.e., the longitudinal component) is commonly neglected, and only the component of that is perpendicular to (i.e., the transverse component) is regarded to act on the nuclear spins. In its most general form, the transverse component is elliptically polarized. The transverse component is further decomposed into two circular components and, one co- and one counter-rotating at the reference frequency, and, respectively. The (2) (3) The quantities and are the longitudinal and transverse relaxation times, respectively. To solve (4), we first calculate,,and as described in the previous section. We then employ an implicit time-integrating solver [13] on a refined tetrahedral FE grid (number of cells is increased by a factor of 2 3) to obtain. The numerical integration method accounts for off-resonance effects [14], relaxation during the pulse [15], and the amplitude and phase modulated excitation pulses. A more detailed description of the time-integration method can be found in Appendix A. After excitation and refocusing pulses, a receiver coil is used to record the NMR response of the excited spin magnetization. The response is determined by the sum of the contribution of the spin magnetization in each unit volume (at location ) weighted by the local receiver sensitivity. To model the time-dependent NMR signal strength that is induced in a receiver coil by the spin magnetization of a unit volume at location, we apply the principle of reciprocity [12]: where is the complex magnetization in the positively rotating frame and the complex conjugate of the receiver fieldinthenegativelyrotating frame at unit current. Eq. (5) is given for a specific carrier frequency of the oscillating RF transmitter field and position in the nonlinearly varying static magnetic field.in the following, we perform numerical spin-echo experiments and predict the echo shapes and intensities generated by frequency swept pulses. C. Amplitude and Phase Modulated Pulse Sequence Commonexcitationpulsesfeatureanamplitude in (3) that is constant over time, corresponding to a rectangular pulse-envelope in the time domain. Such pulses have limited capability to affect nuclear spins in inhomogeneous static magnetic fields, where the involved Larmor frequencies deviate from the pulse carrier frequency. This off-resonance phenomenon can be described by a magnetic field with an amplitude equal to oriented along the static field axis. The effective field acting on the nuclear spins is then the vector sum of the longitudinal field and the transverse field component of [9]. The off-resonance performance of (4) (5)

3 5432 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013 Frequency swept pulse sequences enable the excitation of nuclear spins over a wide frequency band. Apart from this advantage, the power level is in some cases lower than for hard pulses as discussed in the following. The power level of a tuned circuit NMR experiment is [17] (10) Fig. 1. Amplitude (lines) and phase (dots) of an adiabatic double-pulse spinecho experiment. The experiment is defined by the pulse amplitude and phase, the pulse duration and the delay between the two pulses. a pulse is commonly expressed in terms of the pulse bandwidth, which to first order is given by the frequency transform of the pulse function. The pulse bandwidth can be improved by modulating amplitude and phase of the applied RF pulses, which is achieved by using time-dependent and in (3). When a pulse is modulated in such a way that the rate of change of the effective field is slower than the spin precession about the effective field, the pulse is called adiabatic. A popular adiabatic pulse, which has received much attention in the recent years, is the wideband uniform rate and smooth truncation (WURST) pulse [16]. In the following, we summarize the principle characteristics of the WURST pulse as discussed in [16] in detail. Considering the RF field the on resonance adiabatic condition can be quantified by the factor: The RF field is modulated by varying the amplitude (with the modulation parameters and ): and incrementing the RF pulse phase according to The bandwidth is given by (6) (7) with the sweep rate where the parameter defines the pulse duration of the first pulse. The maximal magnetic field at the center of the surface coil is limited by the available RF current and dimension of the coil. Magnetic field calculations are performed for a single frequency only, and the influence of on the frequency sweep of the adiabatic pulse is neglected. In Fig. 1, the amplitude and phase characteristics for the adiabatic spin-echo pulse sequence are shown. The pulse parameters of the second pulse are indicated by the index. In Fig. 2, the corresponding trajectories of the spin magnetization for the first -pulse [Fig. 2(a)] and the second -pulse [Fig. 2(b)] is shown. (8) (9) where is the quality factor of circuit and the parallel tuning capacitor in the circuit. The RF current (a linear scaling factor: ) in a solenoid using a wideband ( )pulseis (11) where is the RF fieldatunitcurrentand is set for example to to fulfill the adiabatic condition. In contrast, the RF current for a hard (H) pulse is given by (12) for a -pulse. Substitution of (11) into (10) and using we obtain the wideband pulse power relation, whereas the hard pulse power relation is (assuming ). In principle, the power level in broadband excitation applications, in which is large, can be reduced by employing wideband pulses with long pulse durations. In the example discussed in Section IV, the bandwidth is narrow (in the khz range) and conventional hard pulses could also be used. However, the power level becomes of significant importance for portable single-sided imaging devices or borehole NMR tools where static fields vary spatially over a wide range. D. Reconstruction Algorithm The forward and inverse problems are solved on a tetrahedral FE model grid. The model is a spatial 3-D distribution of nuclear spins in cells: (13) where are basis functions equal to unity within the cell and zero in all other cells and is the amount of nuclear spins in the cell. The forward problem can be expressed as [18] (14) The time-dependent voltages in a receiver coil can be predicted using the sensitivity function (signal strength) and a water-content model. If only one experimental configuration is used (i.e., a single carrier frequency, gradient field, and pulse bandwidth), (14) is severely underdetermined, and it is difficult to reliably solve the equation. To improve the situation, a number of different configurations can be used. Eq. (14) then describes the predicted NMR signal for configurations and time steps for each spin-echo measurement (see Fig. 1). The inverse problem consists of solving (14) for spin density (water content). We adapt the linear inverse problem to allow only plausible water-content values (0 100%) by using

4 LEHMANN-HORN AND WALBRECKER: NMR IMAGE RECONSTRUCTION IN NONLINEARLY VARYING MAGNETIC FIELDS 5433 Fig. 2. The evolution of spin magnetization during (a) an adiabatic half-passage from to, and (b) an adiabatic full passage from to. Gray unit-sphere is plotted as an eye-guide. a tangent-transformation ([18], [19]). Upon transformation, the sensitivity matrix that relates changes in the predicted data to perturbations in the water-content structure becomes (15) is in- To construct the inverse problem, an objective function troduced to be minimized: (16) where is the model objective function, isthedatamisfit function, and is a regularization parameter. The model objective function encourages solutions with smooth variation of the water content within the studied system, and is given by (17) The structure matrix is defined by,where is assembled using a finite-differences approximation. The distance between the centers of two neighboring cells is and a weighting function (here: )isused to relate all neighboring cells to each other. All tetrahedrons neighboring cell are linked by setting and.theindex runs from 1 to,where is the total number of cell neighbors. We define the data misfit between the observed and predicted data using the -norm: (18) with the data weighting matrix ; is the standard deviation of each noise observation, and is the number of observations (data points in recorded echo time series). We now solve the time-domain inverse problem by finding the model that minimizes the objective function. A minimum occurs where the first derivative of with respect to the model parameter is zero: (19) Fig. 3. The sensitivity function comprises data points per spin-echo time series, for each of the configurations, and each of the cells. The overall size of the matrix is. Following the concept in [20] we obtain the final equation for solving the inverse problem: (20) where we introduced and start with an initial value for the iteration step. A conjugate gradient least-square (CGLS) algorithm [21] is employed to solve (20). The matrices and are large (of the order of, ; Fig. 3), and sparse. They consist of a large number of time series (spin-echo signals) for three varying parameters: (i) the carrier frequency and (ii) the current in the x/y/z-gradient coils, and (iii) the bandwidth of the excitation pulse. III. DESCRIPTION OF SETUP The setup of our NMR imaging device is shown in Fig. 4. The instrument is based on a Helmholtz (H) coils pair, illustrated in Fig. 4(a), that operates at 67.6 A to generate a static magnetic field of at the center of the system. The H coils have a diameter of, a length of, and are separated by.thesimulated

5 5434 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013 Fig. 4. Configuration of the imaging device used in this study. (a) Helmholtz (H) coils pair, with coil axis z. radius of coil; coil pair separation; coil length. Inset shows field inhomogeneity due to the extent of the H coil. (b) saddle coil used as transmitter and receiver situated on cylindrical sample surface. (c) -gradient coils (black and blue, respectively). static magnetic field induced by the H coils pair is shown in Fig. 4(a) for a slice in the plane. The cylindrical sample of diameter and length is located within the spatially nonlinearly varying magnetic field of the H coils pair, as indicated in Fig. 4(b). We choose a single saddle coil surrounding the sample as transmitter (Tx) to excite the sample, and as receiver (Rx) to record the stimulated NMR signal [Fig. 4(b)]. Three additional arrangements of gradient coils surround the sample, one for each gradient direction,,and.the and -gradient coils are shown in Fig. 4(c) in black and blue, respectively. The -gradient system (not shown) is the same as for the -gradient, rotated by 90 degree around. The gradient coils have a diameter of. The length of the and gradient coils is, with a separation of. The gradient coils are used to generate three magnetic field gradients that add to the field of the H coils pair. The magnetic fields calculated for our setup are shown in an -slice plane for the H coil in Fig. 4(a), and for the gradient coilsinfig.5.the field of the H coil in Fig. 4(a) is nonuniform, because of the large dimensions of the sample. This inhomogeneity of is highlighted by the contour lines in Fig. 4(a), representing deviations of 0.5 to 5.0% from the nominal.in Fig. 5, we illustrate the magnetic field lines (contours) and intensities (color) induced by passing a unit current through (a) the -gradient coil, (b) the saddle coil and (c) the -gradient coil. The shown field strengths scale linearly with the amplitude of the inducing current. To excite the spin system, we employ a frequency swept spinecho double-pulse sequence (Fig. 1). The delay between the two pulses is. The relaxation time T2 is set to 100 ms. The first pulse duration is two times the second pulse duration (i.e., ). The amplitude of the second pulse is twice the amplitude of the first pulse (i.e., ). The sweep rates are set according to. Two different sweep rates are employed ( and ) to obtain a spin echo signal with a corresponding bandwidth of and, respectively. We assume the pulse durations and echo times to be short enough to neglect diffusion effects. IV. NUMERICAL STUDY In this numerical study, four samples representing hypothetical water-content distributions are considered. The samples are cylinders of homogeneous, well-connected porous rock saturated with water. The aim is to image the location of water within the porous medium. The studied sample is placed in the center of the coil system, seefig.4.wesimulatethemagneticfields of the H coils for a current of 67.7 A, and the magnetic fields of the gradient coils for 14 combinations of DC currents (see Appendix B). For each field combination, we simulate spin-echo NMR data using the pulse sequence described above. Data are recorded at 19 combinations of carrier frequency (11 settings, ranging between ), and pulse bandwidths (2 settings, 8 khz and 32 khz). In Fig. 6, we exemplarily show the real parts of the complex sensitivity function (in V) in a cross section cutting through the center of the sample volume, obtained at gradient current coil strengths of 2 A (100 turns). Each image (a-h) shows the sensitivity for one specific setting of carrier frequency and bandwidth. The images illustrate that for each setting a different part of the sample is explored, which helps disambiguate the image reconstruction. The four synthetic samples used for our experiment are presented in the first column of Fig. 7, depicting -slice planes through of an irregularly shaped 3-D water-content distribution contained in the cylindrical sample volume. Dark to light colors indicate areas of low (minimum 0%) to high (maximum 30%) water content, respectively. NMR spin-echo time series data based on (5) are computed for different combinations of (i) 14 settings of gradient coil currents, (ii) 11 settings of carrier frequency, and (iii) 2 bandwidths

6 LEHMANN-HORN AND WALBRECKER: NMR IMAGE RECONSTRUCTION IN NONLINEARLY VARYING MAGNETIC FIELDS 5435 Fig. 5. Intensity (contour lines and color code) of the magnetic field of (a) the -gradient, (b) saddle coil, and (c) -gradient coil. Fig. 6. Sensitivity function (slices in, -plane) for 8 carrier frequencies and bandwidth (in V). of the frequency swept pulse sequence. To test the performance of our image reconstruction in the presence of noise, three levels of Gaussian noise were added corresponding to a signal-to-noise ratio (SNR) of (i) 15, (ii) 6, and (iii) 2. We inverted the synthetic data employing a starting model with uniform water content of 10%. The inversion results for the 5 samples and 3 noise levels are displayed in Fig. 7(e) (p). For SNR of 15 and 6, the inversion scheme reconstructs the important features of the original models (compare Fig. 7(a) (d), to Fig. 7(e) (h) and Fig. 7(i) (l), respectively). In general, features near the surface of the cylindrical sample are less well resolved. For the low SNR of 2, the inversion results are only partly representations of the respective synthetic models. We consider this SNR the lower limit for image reconstruction. These numerical experiments illustrate that it is possible to utilize nonlinearly varying static magnetic fields for imaging. We demonstrate the theoretical feasibility to visualize the distribution of hydrogen protons (i.e., water or oil) in porous media. The time a simulated experiment would require in a practical realization is mainly determined by the number of measurements, which is in the example discussed above. In practice, each of these 266 measurements will require an acquisition time of 100 ms 1 s (dependent on the T1 relaxation time), accumulating to a time demand of about minutes for each image. If repeat measurements are required for stacking to achieve a, recording time increases accordingly by the repetition factor.

7 5436 IEEE TRANSACTIONS ON MAGNETICS, VOL. 49, NO. 11, NOVEMBER 2013 Fig. 7. Slices (y, z plane) through the cylindrical sample volume showing the water content distribution for four phantoms (rows). The synthetic water-content model is presentedinthefirst column (a d); the inversion results (e p) are shown for a SNR of 15 (second column, e h), 6 (third column, i l), and 2 (fourth column, m p). Computations are based on the setup shown in Fig. 4 using 14 gradient coil current combinations, 11 (and 8) carrier-frequencies for 2 pulse bandwidths. V. CONCLUSION We presented a numerical algorithm for NMR imaging in static magnetic fields that are weak and vary spatially in a nonlinear way. Simulations of excitation pulses are based on the Bloch equations, and can account for pulse modulations (amplitude and phase). To solve the inverse problem, we used (i) the whole spin-echo time series, (ii) the magnetic fields induced by x/y/z-gradient coils, (iii) several different carrier frequencies, and (iv) different bandwidths of the frequency-swept pulse sequence. To demonstrate the algorithm s capabilities, a low-field imaging setup was designed that requires minimal experimental resources: since the proposed image reconstruction does not require the static field to be spatially homogeneous, large sample volumes can be imaged without shimming coils. Based on numerical experiments, we showed that it is possible to resolve the water-filled porosity for four different phantoms, representative of simple porous structures of about 20 cm diameter. The

8 LEHMANN-HORN AND WALBRECKER: NMR IMAGE RECONSTRUCTION IN NONLINEARLY VARYING MAGNETIC FIELDS 5437 algorithm can be modified to enable single-sided or borehole imaging in highly inhomogeneous static fields in the future. APPENDIX A We employ an implicit time-integration method to solve (4). In general, (4) can be written as (A-1) where is a matrix and is a vector. Applying the Crank- Nicolson time integration method [13], we obtain the following approximation: (A-2) where is a weighting coefficient (e.g. )and the time interval. Eq. (A-2) can be simplified to (A-3) In each time step a linear equation system is solved, from which the time evolution of the spin magnetization in the center of each tetrahedron cell in the FE grid is obtained. APPENDIX B Helmholtz coil current:. Gradient coil current combinations: (See (B-1) at the bottom of the page.) Transmitter coil current: (9.88 A). Sweep rates. Excitation pulse frequency offset and bandwidth combinations (where two bandwidths are given, both combinations were used with the respective pulse frequency, e.g. 100Hzand8kHz,and 100 Hz and 32 khz): (See (B-2) at the bottom of the page.) ACKNOWLEDGMENT JLH was supported by the capability development fund (CFD) within the Postdoctoral Fellowship Program of CSIRO. JW was supported by a grant to R. Knight from the National Science Foundation (grant no ), with additional support from a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD). JLH thanks Dr. David Miljak for many helpful discussions. REFERENCES [1] K. J. Dunn, D. J. Bergman, and G. A. Latorraca, Nuclear Magnetic Resonance Petrophysical and Logging Applications. New York: Pergamon, [2] S.StapfandS.I.Han,NMRImaginginChemicalEngineering. New York: Wiley, [3] S. Appelt, H. Kuhn, F. W. Hasing, and B. Blumich, Chemical analysis by ultrahigh-resolution nuclear magnetic resonance in the earth s magnetic field, Nature Phys., vol. 2, pp , [4] S.Appelt,F.W.Hasing,H.Kuhn,J.Perlo,andB.Blumich, Mobile high resolution xenon nuclear magnetic resonance spectroscopy in the earth s magnetic field, Phys. Rev. Lett., vol. 94, [5] D. M. TonThat, M. P. Augustine, A. Pines, and J. Clarke, Low magnetic field dynamic nuclear polarization using a single-coil two-channel probe, Rev. Sci. Instrum., vol. 68, pp , [6] R. McDermott, A. H. Trabesinger, M. Muck, E. L. Hahn, A. Pines, and J. Clarke, Liquid-state NMR and scalar couplings in microtesla magnetic fields, Science, vol. 295, pp , [7] B. Bluemich, F. Casanova, and S. Appelt, NMR at low magnetic fields, Chemical Physics Letters, vol. 477, pp , [8] P. T. Callaghan, Principles of Nuclear Magnetic Resonance Microscopy. Oxford, U.K.: Oxford University Press, [9] A. Tannus and M. Garwood, Adiabatic pulses, NMR Biomed., vol. 10, pp , [10] J. D. Jackson, Klassische Elektrodynamik. Berlin, Germany: de Gruyter, [11] J. A. Lehmann-Horn, M. Hertrich, S. A. Greenhalgh, and A. G. Green, Three-dimensional magnetic field and NMR sensitivity computations incorporating conductivity anomalies and variable-surface topography, IEEE Trans. Geosci. Remote Sens., vol. 49, pp , [12] D. I. Hoult, The principle of reciprocity in signal strength calculations A mathematical guide, Concepts Magn. Reson., vol. 12, pp , [13] J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat-conduction type, Adv. Comput. Math., vol. 6, pp , [14] J. O. Walbrecker, M. Hertrich, anda.g.green, Off-resonanceeffects in surface nuclear magnetic resonance, Geophysics, vol. 76, pp. G1 G12, [15] J.O.Walbrecker,M.Hertrich,andA.G.Green, Accountingforrelaxation processes during the pulse in surface NMR data, Geophysics, vol. 74, pp. G27 G34, [16] E. Kupce and R. Freeman, Adiabatic pulses for wide-band inversion and broad-band decoupling, J. Magn. Reson. A, vol. 115, pp , [17] J. Mispelter, M. Lupu, and A. Briguet, NMR Probeheads for Biophysical and Biomedical Experiments. London, U.K.: Imperial College Press, [18] M. Hertrich, M. Braun, T. Guenther, A. G. Green, and U. Yaramanci, Surface nuclear magnetic resonance tomography, IEEE Trans. Geosci. Remote Sens., vol. 45, pp , [19] J.A.Lehmann-Horn,J.O.Walbrecker,M.Hertrich,G.Langston,A.F. McClymont, and A. G. Green, Imaging groundwater beneath a rugged proglacial moraine, Geophysics, vol. 76, pp. B165 B172, [20] C. G. Farquharson and D. W. Oldenburg, Non-linear inversion using general measures of data misfit and model structure, Geophys.J.Int., vol. 134, pp , [21] T. Guenther, C. Ruecker, and K. Spitzer, Three-dimensional modelling and inversion of DC resistivity data incorporating topography II. Inversion, Geophys. J. Int., vol. 166, pp , (B-1) (B-2)