two-dimensional distribution of relaxation time and water … · two-dimensional distribution of...

Two-dimensional distribution of relaxation time and water content fromsurface NMR

Raphael Dlugosch∗, Thomas Gunther∗, Mike Muller-Petke∗, and Ugur Yaramanci∗†

reprint of Dlugosch et al. (2013), Near Surface Geophysics

ABSTRACT

Development in instrumentation and data analysis ofsurface nuclear magnetic resonance has recently movedon from 1d soundings to 2d surveys, opening themethod to a larger field of hydrological applications.Current analysis of 2d data sets, however, does not in-corporate relaxation times and is therefore restrictedto the water content distribution in the subsurface.We present a robust 2d inversion scheme, based onthe qt approach, which jointly inverts for water con-tent and relaxation time by taking the complete dataset into account. The spatial distribution of relax-ation time yields structural information of the subsur-face and allows for additional petrophysical character-isation. The presented scheme handles separated loopconfigurations for increased lateral resolution. Assum-ing a monoexponential relaxation in each model cell,using irregular meshes, and gate-integrating the signal,the size of the inverse problem is significantly reducedand can be handled on a standard PC.A synthetic study shows that contrasts in both quanti-ties, water content and relaxation time, can be imaged.Inversion of a field data set outlines a buried glacial val-ley and allows distinguishing two aquifers with differentgrain sizes which can be concisely interpreted togetherwith a resistivity profile. The impact of the anisotropicweighting factor and subsurface resistivity on the inver-sion result are shown and discussed. A comparison ofthe results obtained by the previously used initial valueand time-step inversion approaches illustrates the im-proved stability and resolution capabilities of the 2d qtinversion scheme.

∗Leibniz Institute for Applied Geophysics (LIAG), Hannover,†Department of Applied Geophysics, Technical University of Berlin

INTRODUCTION

The method of nuclear magnetic resonance (NMR) hasproved to be a useful tool in geosciences since the firstlogging application in 1960s (Brown and Gamson, 1960).About 20 years later Russian scientists successfully ap-plied the technique of surface NMR (Semenov, 1987), whichallows probing the subsurface to depths of about 100 m byusing large loops placed at the surface (Weichman et al.,1999). Since then a growing community has been applyingNMR to hydrological problems.

The water content of media can be directly obtainedfrom the amplitude of the NMR signal which is a uniqueability among geophysical techniques. Additionally, therelaxation times of the signal T1 and T2 (s) are linked tothe inner surface of the material (Brownstein and Tarr,1979). This allows, after calibration, to conclude aboutpore sizes and permeability (Seevers, 1966; Kenyon et al.,1988). This procedure was transferred to predict hydraulicconductivities using surface NMR (Legchenko et al., 2002)and was recently revised for coarse-grained material (Dlu-gosch et al., 2013). Compared to laboratory or boreholeNMR, surface NMR struggles to detect water with veryshort relaxation times due to the long duration of theexcitation pulse, the significant instrumental dead time,and the low signal-to-noise ratio (Dlugosch et al., 2011).Although the capabilities of modern surface NMR instru-mentation are under constant development, this can stilllead to reduced water contents observed in silty or clay-rich materials.

The estimated relaxation time depends on the type ofthe NMR experiment. Commonly a free induction decay(FID) experiment is used in surface NMR applicationssince it can be conducted with a single excitation pulse.This saves energy and time which are both essential fac-tors for field measurements. The estimated relaxationtime from an FID experiment is called T ∗

2 . Compared

1

reprint of Dlugosch et al. (2013), Near Surface Geophysics 2

to T1 and T2, the link of T ∗2 to the inner surface of the

material can be significantly altered if heterogeneous mag-netic fields are present (Grunewald and Knight, 2011). Toovercome this limitations multi-pulse experiments similarto the ones conducted in the laboratory were transferredto the field to estimate T1 (Legchenko et al., 2004; Wal-brecker et al., 2011b) or T2 (Legchenko et al., 2009, 2010).However, the significantly greater effort still limits mostfield applications to T ∗

2 , especially for 2d surveys.In most surface NMR measurements a single coincident

(COI) transmitter (tx) and receiver (rx) loop is used andthe collected data are interpreted assuming 1d subsurfaceconditions. 2d targets may be investigated by combiningmultiple COI setups at different locations. However, torely exclusively on COI can be time-consuming becauseonly one data set is collected during one NMR experi-ment. Comprising separated tx and rx loop setups intoone survey, e.g. half overlapping (HOL) or edge-to-edge(E2E) setups, leads to increased resolution especially atshallow depth (Hertrich et al., 2005, 2007). An examplefor a comprehensive 2d data set consisting of four COIand six HOL loop setups is presented in Figure 1a. Thedevelopment of multi-channel instrumentation made theacquisition efficient since the data of several loop setupscan be collected simultaneously during one NMR experi-ment (Dlugosch et al., 2011).

Compared to laboratory or borehole application of NMR,the volume probed by surface NMR is huge. The sensi-tivity of a NMR experiment in the subsurface is summa-rized by the 3d kernel function K3d (V/m3). For a 2dinversion, K3d is integrated over the y-dimension to K(V/m2) (Fig. 1b). It primarily depends on direction andstrength of the Earth’s magnetic field, coil geometry andstrength of the applied excitation pulse, but is also af-fected by a electrically conductive subsurface (Weichmanet al., 2000). For calculating K3d of arbitrary loop geome-tries, 3d subsurface conductivity models or topography werefer to Lehmann-Horn et al. (2011). Because of the com-plex spatial sensitivity of a surface NMR experiment, in-version is necessary to determine the distribution of watercontent and relaxation time in the subsurface (Fig. 1c).

There are several techniques for inverting FID experi-ments. The initial value inversion (IVI) after Legchenkoand Shushakov (1998) reduces all FIDs to their initial am-plitudes, e.g. using a monoexponential fit. It can there-fore only derive the spatial water content distribution. Toobtain information about the relaxation time, time-stepinversion (TSI) was introduced by Legchenko and Valla(2002). A TSI consists of several IVIs conducted for dif-ferent time steps of the FIDs, followed by a fitting in themodel domain to obtain relaxation times. The inversionof surface NMR data was significantly improved by theintroduction of the qt inversion (QTI) scheme (Mueller-Petke and Yaramanci, 2010), which inverts the data of allmeasured FIDs simultaneously and thus increases bothspatial resolution and stability of the inverse problem.

Current 2d IVI, using coincident loop setups (Girard

et al., 2007; Hertrich et al., 2007) or incorporating sep-arate tx and rx loop setups (Hertrich et al., 2009; Dlu-gosch et al., 2011), are limited to estimate the spatialwater content distribution. Under fully saturated con-ditions it is therefore difficult to distinguish between fineand coarse-grained layers which is often essential for hy-drological questions, e.g. when looking for an optimumwell location. To obtain the spatial distribution of relax-ation times and thus pore properties a 2d TSI was firstpresented by Walsh (2008).

We present a 2d QTI approach which jointly estimatesthe spatial distribution of water content and relaxationtime (Fig. 1). We demonstrate its successful applicationon a synthetic data set and present results from a field casemeasured at a site with a shallow buried glacial valley anddeep aquifers with different grain sizes. We compare theresults obtained from QTI with IVI and TSI, discuss thechoice of regularization parameters, and show the impactof subsurface resistivity.

MODELLING AND INVERSION

Forward calculation

Surface NMR exploits the intrinsic magnetic moment (spin)of hydrogen protons present in the groundwater. A spinprecesses at a characteristic Larmor frequency (1-3 kHz)in the Earth’s magnetic field. When excited by a sec-ondary magnetic field, the spin axis is tilted from thethermal equilibrium. In surface NMR this secondary mag-netic field is generated by an alternating current passingthrough a large loop laid out on the Earth’s surface. Wemodelled the magnetic field based on an analytic solutionfor a circular loop over a layered half-space after Wardand Hohmann (1988). The strength of the excitation isgiven by the pulse moment q = Iτ (As), where I (A) isthe current and τ (s) the duration of the excitation pulse.Increasing q leads to an increasing flip angle for a pro-ton at a given location in the subsurface and allows theexcitation of protons at greater depths.

After the excitation pulse, the relaxation of the protonsback to their equilibrium is observed. Due to the preces-sion of the spins during their relaxation they emit smallmagnetic fields which induce an alternating voltage in asurface loop. Commonly a quadrature detection at theLarmor frequency is used to estimate the envelope vobs

(V) of this signal. For 2d conditions and ignoring theimpact of electromagnetic noise, a synthetic signal vsyn

(V) can be described at a time t (s) after the start of therelaxation by (Mohnke and Yaramanci, 2008)

vsyn(q, t) =∫K(x, z, q)

∫wpart(x, z, T

∗2 )e−t/T∗

2 dT ∗2 dxdz . (1)

The partial water content wpart (m3/m3) comprises thewater distribution spatially (x, z) in the subsurface and fora range of specific T ∗

2 as described by Mueller-Petke andYaramanci (2010). The signals vobs and vsyn are generally


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Forward modelling

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Figure 1: Sketch of the presented 2d inversion approach. (a) Data domain consisting of subsets of COI and HOL datasets, presented as a matrix with tx over rx loop position. (b) 2d kernel function linking data and model domain forinverse and forward modelling. (c) Loop positions and model domain consisting of spatial distribution of water contentand relaxation time.

complex values showing a phase shift between transmittedand received signal. This can be due to (i) low resistiv-ity in the subsurface (Braun et al., 2005), (ii) the use ofseparated tx and rx loops (Weichman et al., 2000), (iii)off-resonant excitation (Walbrecker et al., 2011a), and (vi)instrumental effects.

For 2d surveys, several NMR experiments with differentq, and different loop positions/setups p are conducted toretrieve information about the spatial distribution of wpart

and T ∗2 in the subsurface. To simplify the notation we

merge p and q into an NMR experiment or recorded FIDindex l running from 1 . . . L.

Optimizing the forward problem

According to the QTI approach to handle the whole dataat once, the matrix K has the size (data domain size ×model domain size) and can easily reach 104 × 105 en-tries. For 2d applications this is due to the multitude ofloop configurations and positions p in the data domainand the added x-dimension in the model domain. Thismakes handling K with the memory of a personal com-puter difficult and even a forward modelling is very slow.To reduce the size of K we apply the following steps.

1. To minimize the size of the model domain we reduceit to monoexponential relaxation for each discretelocation in the subsurface as already done for 1dcases (e.g. Gunther and Muller-Petke, 2012). Thusthe forward problem as given in equation (1) for an

experiment l reads:

vsynl (t) =∫ ∫Kl(x, z)w(x, z)e−t/T∗

2 (x,z)dxdz . (2)

It is now an explicit formulation of total water con-tent w (m3/m3) and a single T ∗

2 instead of wpart

distributed over a fixed range of T ∗2 bins. The signal

vsynl can still be multiexponential because it inte-grates over the whole subsurface.

2. An irregular triangle mesh with cell sizes growingwith depth (Hertrich et al., 2007) and an index crunning from 1 . . . C is used during the inversion.Because of the highly oscillating characteristic of Kclose to the surface the kernel is calculated on a finermesh and integrated over the cells of the coarser in-version mesh.

3. The collected complex FID voltages are gate-integrated(Behroozmand et al., 2012) and reduced to a numberof about N = 40 bins with associated mean timestn.

To simplify the notation, we use T instead of T ∗2 in the

following equations. Using equation (2) and an index no-tation, the voltage vsynln at a specific time gate n can bewritten as the sum of signals from all cells c with respec-tive water contents wc and relaxation times Tc

vsynln =∑c

Kclwce−tn/Tc . (3)

When applying the described reductions, the size of Kis decreased significantly. The factor is specific for each


application but is in the range of 10 for both model anddata domain which leads to a size of K of approximately103 × 104.

Inversion

For inversion the measured and modelled complex volt-ages vln are reduced to real-valued amplitudes dln (V) asdiscussed later. Thus the data vector becomes

d = [|v11|, |v12|, . . . , |v1L|, |v21|, . . . , |vNL|]T . (4)

The desired model vector consists of the w and T ∗2 values

of all cells lined up in a single vector

m = [w1, . . . , wC , T1, . . . , TC ]T. (5)

To restrict w (i.e. 0-0.5) and T ∗2 (i.e. 0.01-0.8 s) to rea-

sonable values, a cotangent transformation is applied (seeAppendix).

We use the inversion approach as described in detail byHertrich et al. (2007) and adapted it to the qt scheme afterMueller-Petke and Yaramanci (2010). The dimensionlessobjective function Φ to be minimized consists of terms fordata misfit Φd and model roughness Φm

Φ = Φd + λΦm = ‖Wd∆d‖22 + λ‖Wmm‖22 → min . (6)

The matrix Wd = diag(1/εln) holds the error estimateεln (V) which is calculated for each data point dln. Ac-cording to Gunther and Muller-Petke (2012), first the er-ror of each measured voltage vl(t) is estimated from thestandard deviation of the corresponding set of stacks, i.e.the voltage error before gating. Then, to account for thegating, this voltage error is divided by the square rootof the number of readings within the gate, which leadsto the gate and data point specific error εln. The vector∆d = dobs−dsyn (V) contains the misfit between observedand forward calculated data. The first-order flatness ma-trix Wm ensures a smoothness-constrained solution on theunstructured mesh (Gunther et al., 2006; Hertrich et al.,2007). To account for an anisotropic regularisation we usea weighting factor rb for each cell boundary b accordingto (Coscia et al., 2011)

rb = 1 + (rz − 1)nb · ez , (7)

where nb is the normal vector on the boundary and ez isthe unit vector in z-direction. Small values of rz lead to adecreased penalty for gradients in a vertical direction andthus predominantly layered models. The dimensionlessfactor λ weights between minimal data misfit and modelroughness term. We selected the highest λ, i.e. least struc-tured model, which fully explains the data. This is gener-ally indicated by leaving no structures in the misfit and achi-squared value (χ2 = Φd/(NL)) close to 1.

Applying Gauss-Newton minimization we iteratively de-rive a model update ∆mk in each iteration k from solvingthe regularized normal equation

(JTWTd WdJ + λWT

mWm)∆mk =

JTWTd Wd∆dk − λWT

mWmmk , (8)

using a dedicated conjugate gradient solver (Gunther et al.,2006). An explicit line search procedure (Nocedal andWright, 2006) is used to optimize step length and thusconvergence. The Jacobian matrix J is a function of themodel parameters and needs to be recalculated for eachiteration as described in the Appendix.

Other inversion approaches

We compare the 2d results obtained using the QTI withthe IVI (Legchenko and Shushakov, 1998) and the TSI(Legchenko and Valla, 2002; Mohnke and Yaramanci, 2008)approaches. A detailed comparison and review of thetheory was done for 1d by Mueller-Petke and Yaramanci(2010).

To ensure that the results of all approaches are compa-rable, we use a similar data processing, i.e. data clippingand gate integration, and used the same objective func-tion and solver for the normal equation. The initial val-ues vl(t = 0) necessary for IVI are estimated by fitting acomplex monoexponential function to the FIDs and rotat-ing these values (Fukushima and Roeder, 1981) to obtaincorrected and real-valued amplitudes (Mueller-Petke andYaramanci, 2010). Different from QTI and TSI, IVI usesthe covariances for vl(t = 0) obtained from this fit as anerror estimate εl. To cover the whole data set during TSIwe apply an individual IVI on every time gate.

In the model domain we use the same mesh for all in-version schemes and apply the same model anisotropy rzand bounds for the tangent transformation when estimat-ing w. Since ε differ for each inversion scheme, we adaptλ individually for each IVI and selected one global λ foreach TSI. In analogy to QTI we therefore select the high-est λ which fully explains the data within the noise level.To estimate a spatial distribution of T ∗

2 from TSI, a mo-noexponential function is fit to the w results versus timestep for every cell.

SYNTHETIC STUDY

We use a synthetic study to show the successful applica-tion of the presented 2d QTI approach and its capabil-ity to resolve structures that show only a contrast in onesubsurface parameter T ∗

2 or w. The setup consists of fourhalf-overlapping loops with a diameter of 80 m placed ona profile orientated magnetic W-E (Fig. 2a). The com-bination of HOL and COI loops was chosen to comple-ment a high lateral resolution close to the surface with alarge maximum penetration depth. The subsurface model(Fig. 2a and b) consists of two horizontal layers. Bothhave a T ∗

2 of 0.1 s but a changing w from 0.35 at the topto 0.1 below 60 m. A shallow structure, striking perpen-dicular to the profile, is 50 m wide and 25 m deep andthus significantly smaller than the loop size. This struc-ture exhibits no contrast in w but a significantly longerT ∗2 of 0.4 s.


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Figure 2: Comparison of different 2d inversion approaches using a synthetic study. Synthetic model of w (a) and T ∗2

(b) consisting of a layered subsurface and a shallow structure surveyed by four loops. (c) Synthetic data set of COI andHOL as a function of loop positions (tx over rx). 2d inversion results using the IVI (d, e), TSI (g, h), and QTI (j, k)approach. Respective error-weighted data misfit as line or image plots with χ2 values of [4.4, 1.1, 1.0] for [f, i, l].


Forward modelling

For the kernel calculation, the propagation of the mag-netic field is calculated to a distance of more than fourtimes the loop radius and scaled to 20 pulse momentsspaced logarithmically between 0.01 and 15 As. We as-sumed a resistive Earth with a magnetic field strength of49300 nT and an inclination of 68 to calculate K on afine mesh. For the forward calculation of the NMR signalswe created a coarse irregular mesh using distmesh (Pers-son and Strang, 2004) and integrated K over each cell.This mesh ranges two loop radii beyond the presentedprofile and has cell sizes of about 5 m with cell bound-aries adapted to the outlines of the anomalies. We re-duced the complex synthetic signals to real-valued ampli-tudes, added Gaussian noise with a standard deviation of35 nV, clipped the time series to an effective dead time of30 ms and a maximum of 0.5 s, and gate-integrated themto N=39 bins. Figure 2c shows the resulting dsyn con-sisting of four COI soundings and six HOL setups. It alsoillustrates that signals from HOL loops can exhibit ampli-tudes comparable to COI setups, particularly for higherpulse moments.

Inversion

For the inverse modelling we created a new irregular meshwith cell boundaries not adapted to the outlines of theanomaly and calculated K accordingly. The cell sizesare about 5 m at the surface and coarser towards depth.This ensures a reasonable inversion speed and capabilityto image the demanded coarse structures. As a startingmodel we chose a homogeneous subsurface with w=0.1and T ∗

2 =0.1 s. The smoothness anisotropy of rz=0.2 waschosen to prefer horizontal layers and is in the range usedin electrical resistivity tomography (ERT) for aquifer char-acterization by Coscia et al. (2011). The resulting spatialdistributions of w and T ∗

2 using the different inversion ap-proaches are presented in Figure 2.

It shows that IVI can well resolve the w contrast (Fig. 2d)at the layer boundary at 60 m. The shallow structure,which only shows a contrast in T ∗

2 but not w, is traced byan area of artificially low w values. This is the result of themonoexponential fitting of multiexponential signals in thedata space which leads to an underestimation of the initialamplitudes as described by Mueller-Petke and Yaramanci(2010). The achieved χ2 values of IVI are > 1 for allλ. This indicates that the covariances from the FID fitsare underestimating the real data error. We therefore ig-nored χ2 ≈ 1 but chose λ to achieve the smoothest modelwhich leaves no unexplained structures in the data misfit(Fig. 2f).

TSI resolves the contrast in both model parameter w(Fig. 2g) and T ∗

2 (Fig. 2h). But to achieve a result whichexplains the data within the noise level (Fig. 2i), TSI tendsto under-regularise the result leading to a erratic pattern.Additionally the layer below 60 m with low w values isimaged with artificially long T ∗

2 times. The latter is the

result of the lower boundary of the tangent transformationof the model space forcing positive w values and penaliz-ing the approximation of w to zero for all time gates. Thisleads to increased w(t) values for late gates which overes-timates T ∗

2 when fitted monoexponentially.Similar to TSI, QTI can resolve the contrast in w (Fig. 2j)

and T ∗2 (Fig. 2k). But because of the native implemen-

tation of model smoothness constraining w and T ∗2 simul-

taneously QTI avoids the erratic patterns in w and T ∗2 .

For both TSI and QTI, λ can be chosen using the χ2 ≈ 1criterion leaving no structure in the data misfit (Fig. 2l).

FIELD CASE

Site information

The presented data were acquired in 2009 at the Eddel-storf site located close to Luneburg in Northern Germany.The survey is part of a data set which has already beenpresented in Dlugosch et al. (2011), but inverted for wa-ter content only using IVI. In this work we focus on thepart that ranges over the buried valley. Geology is charac-terized by Pleistocene glacial sediments of sand and loamlayers. The result of a coinciding ERT profile is presentedin Figure 3a including a simplified interpretation compris-ing information from a borehole located 200 m south of theprofile. A gravel layer, tracked by a comparably resistiveregion (300 Ωm), is present below 65 m. It is covered by a25 m thick electrically conductive loam layer (25 Ωm). Thesand layer above (500 Ωm) is approximately 30 m thick.A groundwater table at a seasonally changing depth isreported within this layer from a close by borehole butcannot be resolved by the ERT profile. The sand layer isinterrupted by a 120 m wide buried glacial valley crossingthe profile. The ERT profile separates the valley into alower (100 Ωm) and an upper (25 Ωm) part. As reportedfrom other glacial valleys commonly found in this area,the filling most likely consists of sand and silt interbed-dings. Ten meters of loam (25 Ωm) finally covers the wholeprofile.

Surface NMR survey

On a part of the ERT profile we conducted a surface NMRsurvey consisting of four circular loops extending over theSW edge of the buried glacial valley. The loops have di-ameters of 80 m and overlap each other by 40 m (Fig. 3c).Data were collected using the GMR device from VistaClara Inc. with pulse moments ranging from 0.1 to 13.5 As(Fig. 3d). The effective dead time (Dlugosch et al., 2011)of the data after processing is 50 ms. This includes in-strumental dead time (10 ms) and half the pulse length(τ/2 = 20 ms) to account for relaxation during pulse (Wal-brecker et al., 2009). Additionally, the first 20 ms of thelow-pass filtered (500 Hz) data were removed because ofartefacts. They are probably the result of instrumentaleffects occurring in the early version of the GMR whichwere reduced later by cycling the phases of the transmit-


c)

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Figure 3: Results of the Eddelstorf field data set. ERT profile (a), pseudo-2d resistivity model used for kernel calculation(b), loop layout and 2d QTI result for w (c) and T ∗

2 (e), measured data set (d), and error-weighted data misfit (f) witha χ2 value of 1.1. Interpretation of the ERT profile including borehole information is indicated by dashed black lines.


ter pulses allowing for a significantly shorter effective deadtime (Walsh et al., 2011).

Inverse modelling

The ERT profile from the Eddelstorf site (Fig. 3a) showslarge electrically conductive regions which have a signifi-cant impact on the electromagnetic field propagation. Be-cause our code to calculate magnetic fields can only handle1d resistivity conditions, we apply a pseudo-2d approachas an approximation. Therefore we split the kernel cal-culation into loop setups lying mainly inside and outsideof the buried valley and use respective 1d resistivity mod-els as approximation (Fig. 3b). Two HOL configurationswith one loop on either side of the edge were thereforeexcluded from the data set (Fig. 3d). If not otherwisestated, rz was set to 0.2 during the inversion. As a start-ing model we chose a homogeneous subsurface with w=0.1and T ∗

2 =0.1 s.

Qt inversion

The spatial distribution of w and T ∗2 obtained by the QTI

approach is presented in Figure 3c and e, respectively. Theerror-weighted data misfit (Fig. 3f) shows that the inver-sion result explains the data within the noise level. Theburied valley is traced by a rise in the NMR-visible watertable from about 30 to 18 m. The top loam and the upperpart of the sand layer exhibit only little detectable waterwith T ∗

2 <0.2 s. This is likely due to partial saturation butcan also be the result of the long effective dead time of theNMR data set preventing the detection of fast relaxing sig-nals from water in small pores. The part of the expectedburied valley shows a pattern of low w values but still longT ∗2 up to 0.4 s. Below 30 m, w rises to 0.4 and does not

trace any layering. In contrast, the T ∗2 distribution en-

ables to distinguish the loam layer with T ∗2 ≈ 0.15 s from

the underlying gravel layer with T ∗2 >0.4 s. The depth of

this transition is in good agreement to borehole informa-tion and is supported by the ERT survey.

Comparison to other inversion approaches

The impact of different inversion approaches on the ob-tained spatial distribution of w and T ∗

2 is presented inFigure 4. The result of QTI, already presented in Fig-ure 3c and e, is plotted for comparison (Fig. 4g and h).

The w distribution obtained from IVI (Fig. 4a) agreeswell with the QTI result. It traces the buried valley by arise in the NMR-visible water table but cannot distinguishthe loam from the gravel layer below 40 m. In analogy tothe synthetic study, the χ2 value is significantly higherthan 1 indicating that the used εl underestimates the realdata error.

The spatial distribution of w obtained from TSI (Fig. 4d)agrees well with the IVI and QTI results. The T ∗

2 image(Fig. 4e), to some degree, can resolve the boundary be-tween the gravel and the loam layer at 65 m. However due

to the lack of spatial regularization the T ∗2 image appears

erratic. In contrast to QTI, TSI cannot resolve the buriedvalley by a specific T ∗

2 . Additionally, the top loam and theupper part of the sand layer with very low w values showvery long T ∗

2 times. This is in analogy to the syntheticstudy and can be the result of the tangent transforma-tion of w during the inversion. The remaining structuresin the data misfit (Fig. 4f) show that the presented TSIresult cannot fully explain the measured data. Reducingλ improves the data misfit only slightly but leads to veryerratic images of w and T ∗

2 .

Impact of regularisation anisotropy

We show the impact of different regularization anisotropiesrz on the QTI result with values ranging from 1 to 0.03(Fig. 5). We adopted λ appropriately to reach χ2 ≈ 1,i.e. all presented results equivalently explain the mea-sured data within the error level. A rz value of 1 leads toround anomalies in the model, while smaller values prefera horizontal layering. Because all models explain the dataequally well, additional information, e.g. from a boreholeor other geophysical data, or assumption is required todecide which rz to chose. We decided for rz=0.2 whichseems to be a reasonable balance to trace the structurespresent at the profile, especially for T ∗

2 .

Impact of electrical resistivity

The impact of the subsurface resistivity on surface NMRis comprised in the magnetic field calculation and thusthe kernel K. This impact is independent of the inver-sion scheme, but essential for a successful application ofsurface NMR (Braun and Yaramanci, 2008). We showhow three simple 1d resistivity models affect the 2d QTIresult (Fig. 6). The resistivity models were chosen to ap-proach the real distribution of subsurface resistivities inthree steps.

As already shown for 1d by Braun and Yaramanci (2008),ignoring true low subsurface resistivities (Fig. 3a) and as-suming a resistive half-space (Fig. 6a), leads to (i) a corre-lation of low w values with regions of low resistivity, and(ii) a shift of subsurface structures to greater depth. Botheffects are visible in w (Fig. 6b) and T ∗

2 (Fig. 6c), e.g. atthe loam layer between 40 and 60 m.

Using a 1d resistivity model according to the resistiv-ity distribution outside of the buried valley (Fig. 6d) (i)increases the overall w values and (ii) removes the cor-relations of w with regions of low resistivity outside theburied valley (Fig. 6e). Inside the buried valley, a smallarea with lower w values persists. The boundary betweenthe loam and gravel, still clearly visible in the T ∗

2 image(Fig. 6f), is slightly raised.

Finally we adapted the resistivity model to the pseudo-2d approach (Fig. 6g) to account for the lower resistivi-ties inside the buried valley. This (i) removes the area oflower w values (Fig. 6h), (ii) further rises the loam-gravelboundary below the buried valley (Fig. 6i), and (iii) helps


sand / silt

loam

loam

gravel

sand

a)p1 p2 p3 p4 p1 p2 p3 p4 rx p1 p2 p3 p4c)

-100

-80

-60

-40

0

-20

Dep

th (

m)

-100

-80

-60

-40

0

-20

Dep

th (

m)

w ( )0 0.1 0.2 0.3 0.4 0.5

*T (s)2

0 0.1 0.2 0.3 0.4 0.5

-100

-80

-60

-40

0

-20

Dep

th (

m)

d)

g)

Pul

se m

omen

t (A

s)

p1

p2

p3

p4

-10 50

p1

p3

p4

p1

p2

p3

p4

0 40 80 120 160 200Profile (m)SW Profile (m)SW

Profile (m)SW Profile (m)SW

Profile (m)SW Profile (m)SW

NE

NE

NE

NE

NE

NE

f)

i)0.2

13.5

Pul

se m

omen

t (A

s)

0.2

13.5

-2

0

2

50 51050 510

50 51050 510

Time (ms)

0.2

13.5

0.2

13.5

0.2

13.5

0.2

13.5-10 100

-10 100 -10 100

0.2

13.5

sand / silt

loam

loam

gravel

sand

sand / silt

loam

loam

gravel

sand sand / silt

loam

loam

gravel

sand

0 40 80 120 160 200

0 40 80 120 160 200 0 40 80 120 160 200

0 40 80 120 160 200 0 40 80 120 160 200

tx

0.2

13.5

p2

0.2

13.5

Pul

se m

omen

t (A

s)

0.2

13.5

-2

0

2

50 51050 510

50 51050 510

Time (ms)

0.2

13.5

0.2

13.5

Φ ( )d

b)

e)

h)

sand / silt

loam

loam

gravel

sand sand / silt

loam

loam

gravel

sand

Δd/

ε (

)Δ

d/ε

( )

Δd/ε ( )

Figure 4: Comparison of different 2d inversion approaches on the Eddelstorf site. Results of w and T ∗2 using IVI (a, b),

TSI (d, e), and QTI (g, h). All inversions use a pseudo-2d resistivity model and rz = 0.2. Respective error-weighteddata misfit as line or image plots with χ2 values of [3.5, 1.6, 1.1] for [c, f, i].


a)p1 p2 p3 p4

-100

-80

-60

-40

0

-20

Dep

th (

m)

-100

-80

-60

-40

0

-20

Dep

th (

m)

-100

-80

-60

-40

0

-20

Dep

th (

m)

c)

e)

0 40 80 120 160 200Profile (m)SW NE

p1 p2 p3 p4

Profile (m)SW NE

0 40 80 120 160 200

sand / silt

loam

loam

gravel

sand sand / silt

loam

loam

gravel

sand

*T (s)2

0 0.1 0.2 0.3 0.4 0.5w ( )

0 0.1 0.2 0.3 0.4 0.5

sand / silt

loam

loam

gravel

sand sand / silt

loam

loam

gravel

sand

sand / silt

loam

loam

gravel

sand sand / silt

loam

loam

gravel

sand

b)

d)

f)

Figure 5: Impact of regularisation anisotropy on the QTI. Images of w and T ∗2 for different values of rz [0.03, 0.2, 1] for

[(a, b), (c, d), (e, f)]. All inversions use a pseudo-2d resistivity model and achieve χ2 values of [1.17, 1.14, 1.16].

to outline the NE of the buried valley due to T ∗2 . Be-

cause of the imperfect implementation of the subsurfaceresistivity using a pseudo-2d approach, the raise of theloam-gravel boundary below the buried valley seems tobe slightly higher than expected and the lower loam layerappears interrupted at the edge of the valley. A structuraljoint inversion of resistivity and surface NMR data willhelp to reduce the ambiguities of both methods (Guntherand Muller-Petke, 2012). While different resistivity mod-els lead to a relocation of anomalies, their T ∗

2 values seemmore robust than their w values.

DISCUSSION

Benefit of the 2d qt inversion

A comparison between IVI, TSI, and QTI has been treatedin detail by Mueller-Petke and Yaramanci (2010) for 1dlayering. They concluded that because the complete dataset is taken into account jointly, the available informationcontent of surface NMR measurements is extracted withincreased spatial resolution and stability. For the 2d appli-cation the used concepts do not change and the presentedexamples support this conclusion. Although further adap-

tations to the presented IVI and TSI schemes might im-prove the inversion results, QTI solves these problems nat-urally. It allows using the stacking error of the data as anerror estimate and jointly inverts for w and T ∗

2 enablinga proper model regularisation. This leads to a simplifiedinversion and improved results compared to the TSI.

Monoexponential model space

To minimize the size of the inverse problem for QTI wedecreased the model space to a monoexponential relax-ation for each model cell. Arguments that support thissimplification are that (i) surface NMR exhibits long effec-tive dead times and low signal-to-noise ratio. Both makesthe detection of very fast relaxing signals difficult (Dlu-gosch et al., 2011). This can remove the necessity to ac-count for clay bound water in the model which is a majorsource for clearly multiexponential signals of unconsoli-dated material, e.g. when using laboratory NMR. Addi-tionally, (ii) pores in unconsolidated material are likelynot isolated (James and Ehrlich, 1999). For material withseveral pore radii within the diffusion length of a protonduring its relaxation, NMR allows to identify only a mean


a)p1 p2 p3 p4 p1 p2 p3 p4

-100

-80

-60

-40

0

-20

Dep

th (

m)

-100

-80

-60

-40

0

-20

Dep

th (

m)

Resistivity (Ωm) *T (s)2

0 0.1 0.2 0.3 0.4 0.5

-100

-80

-60

-40

0

-20

Dep

th (

m)

d)

g)

0 40 80 120 160 200Profile (m)SW Profile (m)SWNE NE

sand / silt

loam

loam

gravel

sand

0 40 80 120 160 200

p1 p2 p3 p4

Profile (m)SW NE

0 40 80 120 160 200

w ( )0 0.1 0.2 0.3 0.4 0.5

sand / silt

loam

loam

gravel

sand

25 100 300 1k

b)

e)

h)

sand / silt

loam

loam

gravel

sand sand / silt

loam

loam

gravel

sand

sand / silt

loam

loam

gravel

sand sand / silt

loam

loam

gravel

sand

c)

f)

i)

Figure 6: Impact of different resistivity models used for the kernel calculation on the 2d QTI result of w (b, e, h) and T ∗2

(c, f, i) for the Eddelstorf site. Resistive half-space (a), 1d background resistivity (d), and pseudo-2d resistivity model(g). All inversions use rz = 0.2 and achieve a χ2 value of 1.1.

inner surface and therefore a narrow relaxation spectrum(Ramakrishnan et al., 1999). Finally, (iii) the monoexpo-nential assumption is the simplest model which sufficientlyexplains our measured data.

Complex signal

After quadrature detection the envelope vobs of the recordedNMR signal is complex. Mueller-Petke and Yaramanci(2010) discussed the use of real-valued amplitudes (|vobs|)or corrected amplitudes obtained by rotating vobs for theinversion. In contrast to their conclusion of rotating vobs,we decide for using amplitudes because we observe arte-facts when rotating clearly multiexponential surface NMRsignals. The main drawback of using amplitudes is the in-creased noise level at late times leading to artificially longT ∗2 . By gate-integrating and clipping vobs to t <0.5 s these

artefacts are reduced. However, we encourage to fully ex-ploit the complex vobs during the inversion when instru-mental phases are sufficiently studied in the future. Thispotentially reduces model ambiguities and improves depthresolution (Weichman et al., 2002; Braun et al., 2005).

CONCLUSIONS

The presented 2d qt inversion scheme is a valuable tool toimage water content and relaxation time in the subsurface.The applied optimizations significantly reduce the size ofthe inverse problem. This enables solutions using a per-sonal computer with computation times (excluding kernelcalculation) of only a few minutes. An anisotropic weight-ing factor governs the smoothness of the inversion resultswhich enables additional structural information, e.g. pre-dominantly horizontal layering known from a resistivitysurvey to be included. The native implementation of dataerror and model regularisation into the qt inversion leadsto a simplified and stable inversion and improved resultscompared to the time-step inversion.

Imaging water content and relaxation time allows forenhanced subsurface characterization. For the presentedfield case this helps to outline a buried glacial valley andto characterize and distinguish a layer of fine sand andcoarse gravel. An appropriate implementation of the sub-surface resistivity is essential for the interpretation of sur-face NMR data. This avoids the dislocation of structuresand artefacts of low w tracing low resistivity regions. In


contrast to their spatial distribution there is no indica-tion that the values of T ∗

2 are affected by the subsurfaceresistivity which helps to outline and identify structures.

While we present a pseudo-2d approach to account forsurface resistivity, a full 3d approach seems advisable andwill improve the results. Further developments can be (i)complex inversion, (ii) implementations of T1 experiments,and (iii) joint inversion with resistivity methods.

ACKNOWLEDGMENTS

We are grateful for many helpful comments from the edi-tor and the reviewers, which greatly improved the clarityof the paper. We like to thank Robert Meyer and Wolf-gang Sudekum for their help to collect the field data. Thisresearch was supported by the German Research Founda-tion (grant Mu 3318/1-1).

APPENDIX

Structure of the Jacobian matrix

The partial derivative with respect to the intrinsic modelparameter w and T ∗

2 for every synthetic data point dlnobtained from the respective voltage vln is given by:

∂dln∂wc

= KAcle

−tn/Tc and (A-1)

∂dln∂Tc

= (tn/T2c )KA

clwce−tn/Tc . (A-2)

Because we use real-valued amplitudes, KAcl is the trans-

formed kernel which yields |dl| = KAclmc according to

Mueller-Petke and Yaramanci (2010). For a fixed timegate n, matrices Gn and Hn referring to the derivativesof w and T ∗

2 can be written as:

Gn =

∂d1n

∂w1

∂d1n

∂w2· · · ∂d1n

∂wC

∂d2n

∂w1

. . ....

.... . .

...∂dLn

∂w1· · · · · · ∂dLn

∂wC

; (A-3)

Hn =

∂d1n

∂T1

∂d1n

∂T2· · · ∂d1n

∂TC

∂d2n

∂T1

. . ....

.... . .

...∂dLn

∂T1· · · · · · ∂dLn

∂TC

. (A-4)

Combining Gn and Hn of all time gates and merging themleads to:

J =

G1 H1

G2 H2

......

GN HN

. (A-5)

Transformation of the model domain

In order to restrict each of the water content and relax-ation time values within a lower bound ml and an up-

per bound mu, we use a cotangent transform similar toMueller-Petke and Yaramanci (2010)

mcot = − cot

(m−ml

mu −mlπ

). (A-6)

It additionally has the advantage of combining two phys-ical quantities w and T ∗

2 into a unit-less measure. Thetransformed Jacobean Jcot is then computed by means oftotal differentiation

Jcot = J/∂mcot

∂m

= Jmu −ml

πsin2

(m−ml

mu −mlπ

). (A-7)

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