Utilizing pre-polarization to enhance SNMR signals—effect of imperfect switch-off

Surface nuclear magnetic resonance (SNMR) is a well-established technique for the hydrogeological characterization of the subsurface up to depths of about 150 m. Recently, SNMR has been adapted to investigate also the shallow unsaturated zone with small surface loop setups. Due to the decreased volume, a pre-polarization (PP) field prior to the classical spin excitation is applied to enhance the measured response signal. Depending on the strength and orientation of the applied PP-field, the enhancement can often reach several orders of magnitude in the vicinity of the PP-loop. The theoretically achievable enhancement depends on the assumption of an adiabatic, that is perfect, switch-off of the corresponding PP-field. To study the effect of imperfect switch-off, we incorporate full spin dynamics simulations into the SNMR forward modelling. The affected subsurface volume strongly depends on the chosen PP switch-off ramp and the geometry of the loop setup. Due to the imperfect switch-off, the resulting SNMR sounding curves can have significantly decreased signal amplitudes. For comparison, the signal amplitudes of either a 1 ms exponential or linear switch-off ramp are reduced by 17 and 65 per cent, respectively. Disregarding this effect would therefore yield an underestimation of the corresponding subsurface water content of similar magnitude.


I. INTRODUCTION
until today OR-excitation is by far the most common excitation technique used in SNMR, whereas AP-excitation has been introduced to SNMR just within the past few years [24].
Another technique to increase the SNR of SNMR measurements is the pre-polarization (PP) of the net magnetization by applying an enhanced background static magnetic field prior to performing the actual measurement at a weaker field [28]. Therefore, it is also possible to conduct NMR measurements in the weak Earth's magnetic field if a proper prepolarization is applied [e.g. [29][30][31]. The effect of pre-polarization to SNMR measurements (SNMR-PP) with small loop setups (d pp ≈ 5 m) was conceptually shown by [32] for targets in the shallow vadose zone. The first field SNMR-PP measurement was conducted on a water reservoir by [33] with a 2 m square PP-loop and 1200 A effective DC. Quite recently, the application of SNMR-PP for underground tunnel measurements has also attracted interest within the SNMR community [34,35]. Furthermore, although not being a classical hydrogeophysical SNMR application in terms of hydrogeophysical aquifer characterization, the detection of oil under sea ice from a mobile, helicopter-borne NMR device also makes use of pre-polarization techniques to amplify the measured NMR signal [36,37].
A typical SNMR-PP measurement sequence is sketched in Fig. 1. Initially a strong PP B p -field is generated by energizing a PP-loop with a large DC current I p for a duration long enough to polarize the spins to a known level to enhance the subsequent signal amplitude.
The length of the PP-pulse (ms to s) depends on the T 1 relaxation time of the investigated medium as the enhanced magnetization builds up with T 1 . After switching-off the PP-field within the switch-off ramp time τ r , a pulse (Tx) is used to excite the hydrogen protons in the subsurface. Subsequently a NMR response, typically a free induction decay (FID), can be recorded for up to several hundreds of milliseconds depending on the particular relaxation times of the investigation target. By increasing either the length or amplitude of the Txpulse, it is possible to excite deeper and larger volumes in the subsurface and therefore gather quantitative, depth-resolved information about water content and relaxation time.
Common to the aforementioned SNMR-PP applications, is the assumption that the PP switch-off itself is carried out properly. This means that the PP-field has to be switched-off (or better ramped down -red line in Fig. 1) in a particular manner during the time span τ r , so that the enhanced magnetization is perfectly aligned with the Earth's magnetic field at the end of the switch-off. Two important things need to be considered here. Due to its spatial inhomogeneity, the magnitudes of the PP-field range over several orders of magnitude.
Furthermore, the spatial distribution of relative orientations between the Earth's magnetic field and the PP-field varies with the inclination of the Earth's magnetic field and locally ranges between 0 • and 180 • . For practical considerations this means that a particular switchoff ramp (with a fixed switch-off time τ r ) employed in a certain SNMR-PP device, needs to be optimized for all combinations of PP-field amplitudes and relative orientations. Considering this rather large parameter space it is very unlikely that the maximal theoretically possible enhanced magnetization is established over the entire affected subsurface after the PP switchoff [e.g. [38][39][40]. Furthermore, when targeting the shallow subsurface (e.g. vadose zone), which features T 1 relaxation times even shorter than about 100 ms, the time span between PP switch-off and FID record should be as short as possible to avoid signal loss due to T 1 relaxation. This time span consists of wait time τ w , Tx-pulse excitation and dead time τ d (cf. Fig. 1), where τ w and τ d are generally device-dependent and the Tx-pulse length depends on the measurement protocol and / or target depth. Depending on these parameters it may be possible to prolong the switch-off time τ r to increase the adiabatic switch-off performance.
In any case, one needs to evaluate the trade-off between switch-off performance due to longer τ r and magnetization decrease due to short T 1 relaxation.
The objective of this work is to quantify the effect of an imperfect PP switch-off on the resulting NMR signal due to the application of different PP switch-off ramps. To do so, we employ full spin dynamics simulations of the PP-switch-off within the calculation of the three-dimensional SNMR forward kernel. This means that for every point in the subsurface where the B-field is calculated, we need to solve a set of differential equations to determine the magnetization after the PP switch-off. Considering the B-field grid discretization (nr × nϕ×nz = 71×361×192) later used in section II, the number of points, and hence the number of needed calculations is almost 5 million. If for a given SNMR-PP setup and PP switch-off ramp, the effect of the PP switch-off on the SNMR signal is negligible, one could omit this extensive and time consuming procedure for any further SNMR kernel calculations. The surface loop sizes exemplarily used in this work are d = 2 m and are comparable to the sizes used in the works of [33] and [41]. Consequently, this size roughly holds for any geophysical SNMR-PP setup currently in use and is constrained by the technically applicable DC current that drives the PP-loop (which is in the order of 20 A to 100 A per turn and an effective DC of about 1000 A). If available, we utilize PP switch-off ramp shapes that have been applied by other groups either for geophysical SNMR-PP [33,41] or alternative applications that also employ PP for NMR signal enhancement [38,40]. The intention of this work is not to present or propose a particularly optimized switch-off ramp for a certain special application.
In contrast, we want to raise the awareness that the non-consideration of the PP switch-off might lead to unexpected and undesired results.
Our study was partially inspired by the works of [38][39][40] and uses some of their results as benchmark cases for our numerical implementation (cf. appendix A). We like to point out that the application of SNMR-PP significantly differs from the works of the aforementioned groups. While [29,38] and [39] were using NMR with PP in the Earth's magnetic field to study the relaxation of liquids similar to laboratory NMR relaxometry measurements, [36] and [37] use NMR with PP to qualitatively detect oil on water under a layer of sea ice.
Essentially, and in contrast to SNMR, the latter application is a detection method with additional quantification capability. Because SNMR applications always aim to exactly quantify the amount of water in a particular depth-resolved subsurface volume, they need to employ the extensive forward modeling described above. This paper is structured as follows: first, we introduce the basic concepts of SNMR including the difference of OR-and AP-excitation. Then, we show the spatial sensitivity of an ideal enhancement of the SNMR signal due to a perfect PP-pulse switch-off. Subsequently, we focus on the imperfect PP-pulse switch-off and carry out a parameter study to evaluate the dependence of the PP-effectiveness of different switch-off ramps on the control param-eters. We finalize the results part by comparing the influence of different switch-off ramps on forward modeled SNMR signal amplitudes.

II. THEORETICAL BASICS OF SNMR
The nuclear spins of hydrogen protons precess around the Earth's magnetic field B 0 (with where s 0 is the initial FID amplitude, T * 2 is the free induction decay time and φ s is the phase of the signal with respect to the transmitted Tx-pulse. By varying the pulse moment q = I ac τ , with Tx-pulse current amplitude I ac and pulse length τ , a 1D sounding curve can be obtained. The measured amplitudes and relaxation times can be inverted for depthresolved subsurface water content and pore size information, respectively [e.g. 21,43,44]. The forward operator corresponding to eq. (1) is [8] s 0 (q) = K(q, r)f (r)dr , with water content distribution f (r) and sensitivity kernel K(q, r). For the general case of separated Tx-and Rx-loops K(q, r) is given by where B + T and B − R are the co-rotating and counter-rotating parts of the Tx-and Rx-field, respectively. The exponential term in eq. (3) accounts for phase lags associated with subsurface conductivity structures. The unit vectors b ⊥ T , b ⊥ R and b 0 in the third row of eq. (3) account for the relative orientations of transmitter, receiver and Earth's magnetic field, respectively.
Two general excitation schemes exist in SNMR. When the Tx-pulse is generated by an alternating current I ac at Larmor frequency ω L , the excitation is called on-resonant (OR) excitation [45]. This is the general case for most SNMR applications. More recent approaches use adiabatic fast passage pulses (AP), where the Tx-pulse current I ac and its corresponding frequency are modulated in a particular manner to provide a more homogeneous excitation in the subsurface [23,24]. In general M ⊥ can be determined for both excitation schemes where the flip angle Θ T describes the orientation of the magnetization after the Tx-pulse. Before studying the effect of PP on SNMR measurements and for introductory reasons, we first compare the effect of OR-and AP-excitation without additional PP. Figure 2a+b exemplarily shows the corresponding sensitivity kernels and Fig. 2e the associated sounding the Larmor frequency f L with a hyperbolic tangent. The amplitude modulation for the AP-pulse is given implicitly according to a typical surface NMR coil response with a quality factor of Q = 30 [24]. The pulse current I ac in Fig. 2 refers to the constant AC current in the OR-case and to the maximum current at the end of the Tx-pulse in the AP-case.
The OR-kernel ( Fig. 2a) shows the typical banded sensitivity structure, where an increase in I ac yields a corresponding deeper depth focus. Here, the deepest band represents the first perfect excitation, i.e. a flip angle of 90 • . In contrast to this, the AP-kernel (Fig. 2b) exhibits a more homogeneous sensitivity structure that has its maximum close to the surface. The sensitivity structure of both kernels also gets reflected in the corresponding sounding curves.
Both sounding curves are calculated for a constant water content of 30 %. For smaller pulse currents I ac < 0.5 A, the OR sounding curve (red) has larger amplitudes compared to the AP-case (green). For larger I ac and due to the increasing homogeneous excitation with depth, the AP-amplitudes increase up to a factor of 4 compared to the OR-values at similar For a more detailed discussion on the implementation of AP-excitation in SNMR and its particular characteristics we refer the reader to [24].

III. SNMR-PP WITH PERFECT PRE-POLARIZATION SWITCH-OFF
To further increase the net magnetic moment M 0 , and thereby increase the detectable signal amplitude, it is possible to temporarily apply a strong pre-polarizing magnetic field B p prior to the actual excitation pulse [28]. During the presence of the PP-field B p , the spins will align with the resulting field B = B 0 + B p and hence, the magnetic moment M 0 will increase to the spatially varying M p (r) ∝ |B 0 + B p | [29][30][31][32]. Switching off the polarizing field B p adiabatically causes the spins to reorient towards B 0 and at the same time M p exponentially decreases towards M 0 with relaxation time constant T 1 (cf. eq. 6). In case of an adiabatic switch-off, M p is perfectly aligned with B 0 after B p has vanished and M 0 in eq. 3 can be replaced by M p to calculate the forward kernel. The adiabatic condition is satisfied if the frequency γ |B(t)| remains much larger than the rate of reorientation of B where α is the angle between B and B 0 (cf. Fig. 7a), for the entire duration of the switch-off.
To get the maximum yield out of PP, two things need to be considered when switching off the polarizing field B p . First, the adiabatic condition itself (eq. 5) has to be satisfied, meaning that the switch-off has to be slow enough, so that M p is "locked" to B as it decreases in amplitude and reorients to B 0 [38,39]. If the switch-off is too fast, most of the magnetization will not follow the reorienting B and will not be aligned in parallel to B 0 at the end of the switch-off. Consequently, these magnetization components do not get coherently excited by a subsequent Tx-pulse and hence, do not contribute to the recorded NMR signal. Second, the switch-off has to be fast enough (much faster than T 1 ) to get the maximum gain out of the increased M p . If the switch-off is not much faster than T 1 , then a considerable amount of magnetization will decay due to T 1 -relaxation before the actual Tx-pulse is applied. The critical part when switching off B p is at the end of the switch-off when B p becomes small and the precession frequency of M around B 0 is small compared to the rate of change of reorientation of B. This reorientation rate needs to slow down in order to maintain the inequality in eq. 5.
Due to technical limitations of current available SNMR-PP devices and depending on the length of the Tx-pulse, it can take about 50 ms after the PP switch-off until the actual NMR signal is recorded (cf. Fig. 1). Therefore, it is undesirable to have a long PP switch-off (long in respect to the relatively short T 1 ) that adds magnetization losses due to T 1 -relaxation.
Reported switch-off times for recent SNMR-PP devices are about 1 ms [33,41,47] and are therefore about two to three orders of magnitude smaller than typical T 1 -values for saturated porous media.  to the cases without PP and the applied PP-field corresponds to the one shown in Fig. 3a.
Note that with PP, the color bar values in Fig. 2c+d  Although the amplitudes slightly decrease again for larger I ac , they are still more than one order of magnitude higher compared to the AP-case without PP. Due to the much more homogeneous AP-excitation the undesirable kernel oscillations have also vanished.
Both SNMR-PP-kernels show a high sensitivity at the shallow subsurface (bright yellow and white colors) due to the large B p -field magnitudes close to the PP-loop and the strong decreasing effect of the B p -field with increasing depth (cf. also Fig. 3a). This basic comparison demonstrates the advantages of SNMR-PP for shallow subsurface measurements in terms of signal enhancement, especially when combined with AP-excitation.

IV. SNMR-PP WITH IMPERFECT PRE-POLARIZATION SWITCH-OFF
Now, we want to study the effect of an imperfect PP switch-off on the enhancement of the final NMR signal. But before doing so, we compare different switch-off ramps and how the particular ramp shape and ramp time τ r influence the switch-off characteristics. In order to reliably quantify a spatially varying subsurface water content distribution it is essential to determine the actual magnetization M p after the switch-off. Considering the practical applicability of SNMR-PP devices, we need to evaluate to what extent a particular switch-off ramp influences the recorded NMR signal. Here, our objective is not to propose or develop a particular switch-off ramp but rather to familiarize the reader with the consequences that may arise when using one of the presented ramps.
To evaluate different PP switch-off ramps, we need to study the full NMR spin dynamics of the magnetization M p during the switch-off of the B p -field. To this end, we developed the software BLOCHUS (Bloch Universal Simulator), a set of Matlab TM tools including a graphical user interface (GUI) that implements the governing equation of motion, the Bloch equation [46], in a laboratory frame of reference with longitudinal and transversal relaxation times T 1 and T 2 . By controlling the amplitude of the PP-current I p (t) during switch-off, any arbitrary ramp shape and its corresponding effect on B p , and hence M p , can be modeled. Internally, BLOCHUS solves eq. (6) with Matlab's "ode45" routine, which employs a fifth-order Runge-Kutta method with adaptive time stepping [48]. To use the extensive modeling capabilities of MRSmatlab [21], BLOCHUS also contains an interface to the kernel calculation routines provided by MRSmatlab. As mentioned above and without loss of generality, eq. (6) can also be used to model the magnetization M during a Tx-pulse (either OR or AP) and therewith implicitly consider effects like relaxation during pulse [49]. In appendix A, we provide two benchmark scenarios to validate our implementation of eq. (6).

A. Parameter Study
As we have shown in Fig. 3a+b, the amplitudes B p and the relative orientation θ strongly vary in magnitude and spatial extent. Therefore, and depending on the shape and duration of the switch-off ramp, the PP switch-off is not perfect in every point of the subsurface volume. To quantify this, we determine the "adiabatic quality" p as a function of amplitude B p and angle θ for various switch-off ramps (cf. Fig. 4). respectively and would lead to a decreased Tx-excitation. In the following, we will use p as an effective measure to evaluate the quality of the PP switch-off.
We exemplarily employ four different ramp shapes: • EXP -exponential [33,47] • LINEXP -linear & exponential [e.g 40] • HCOS -half cosine [41] • LIN -linear [e.g 38] which have been published before and are partly in use in current SNMR-PP devices. Each ramp shape was calculated for three different switch-off times τ r of 0.1 ms, 1 ms and 4 ms where the ramp time τ r is defined as the total time span that is used to ramp the B pfield down to zero. Note that similar to a real SNMR-PP measurement, τ r is fixed for all combinations of B p and θ within the corresponding column of Fig. 4. For all calculations we neglected T 1 -and T 2 -relaxation.
With the chosen set of ramp shapes and ramp times, we cover a reasonable range of parameters for technical applicable SNMR-PP measurements. A ramp time of τ r = 1 ms is reported in [47] and [41] and the two other ramp times serve as "boundary" cases to evaluate the dependence of p onto τ r . We note that the short ramp time of τ r = 0.1 ms might be challenging to implement in a real SNMR-PP device, especially when considering the strong currents involved. The ramp shapes are chosen to cover a range of shapes that show distinct differences at early and late times, respectively. Considering the adiabatic criterion in eq. 5, obviously using the EXP-ramp seems to be a generally favorable choice as it releases a lot of energy over a short period of time (and early) while it significantly slows down at later ramp times. On the contrary, the HCOS-ramp releases the energy much more slowly and uniformly and will therefore violate eq. 5 over a larger period of time. However, as this ramp is also in practical use, we want to evaluate its effect on the switch-off quality. The other two shapes are the benchmark cases from appendix A, where for the cases presented here the switch-over field strength for the LINEXP-ramp is fixed to B * = B p /10. In Fig. 4 the adiabatic quality p is color coded from white (p = 1, perfect adiabatic switch-off) over yellow, green (p = 0) to blue (p = −1). Fig. 4, several features can be recognized.

By basic inspection of the p-distributions in
First and in line with the adiabatic criterion in eq. 5, for each ramp shape (rows in Fig. 4) the global adiabatic quality increases with increasing ramp time τ r . By "global" we refer to the whole range of B p and θ values considered here. However, the increase of the global adiabatic quality itself with increasing τ r is significantly different for the individual ramp shapes. For the case of τ r = 0.1 ms there seems to be no significant difference between the individual ramp shapes. In this particular case, all ramp shapes show an equally poor performance when compared to corresponding results for longer ramp times. Within the relevant PP-parameters, a switch-off time of 0.1 ms is simply too fast to allow for an adiabatic reorientation if the initial angle between B p and B 0 is already larger then θ 45 • . For a ramp time of τ r = 1 ms the global adiabatic quality p shows a distinct dependence on the ramp shape. For the EXP-ramp, the switch-off is perfect up to an angle of θ 115 • (Fig. 4b).
For the LINEXP, the HCOS and the LIN ramps this onset of low adiabatic quality happens continuously "earlier" at smaller angles θ (Fig. 4e,h,k). Furthermore, while for the EXPramp the distribution of p is rather independent on the magnitude of B p , the dependence on B p is continuously increasing for the LINEXP, the HCOS and the LIN ramps. For the latter ones, the global adiabatic quality increases with decreasing B p . The results for τ r = 4 ms are along the same line of arguments as for the results for τ r = 1 ms. The main difference is that a high adiabatic quality can be achieved over a larger range of θ values. These findings seem to confirm that for a large range of B p and θ values, the EXP-ramp outperforms the LINEXP, HCOS and LIN ramps and therefore should yield the maximum PP-effect in a real SNMR-PP measurement.
To further evaluate these results in regard of a real SNMR-PP measurement, we calculate the spatial distribution of adiabatic quality p for the EXP and LIN-ramps with τ r = 1 ms as they differ most significantly from each other in terms of global p at this particular τ r (cf. Fig. 4). We use the amplitudes B p and angles θ from the PP-field presented in Fig. 3a+b and determine M p and therewith also p based on the Bloch equation eq. (6). Figure 5 shows the corresponding central slices of the 3D distribution of p when either an EXP-ramp (Fig. 5a+b) or LIN-ramp (Fig. 5c+d) is used. For each ramp, we determine p for two different inclinations: 60 • (Fig. 5a+c) and −60 • (Fig. 5b+d). This not only illustrates the spatial variability of the switch-off quality depending on the relative orientation of B 0 and B p , but once more demonstrates that it might be necessary to account for these effects already in the forward calculation.
Despite the general better global performance of the EXP-ramp compared to the LINramp (cf. Fig. 4), this ramp yields a larger volume of low p-values (Fig. 5). By a detailed evaluation of Figs. 3,4+5 it becomes apparent that for values smaller than about B p < 5 (dashed line in Fig. 4b+k), the LIN-ramp (Fig. 4k) shows indeed a higher adiabatic quality over a wider range of angles θ compared to the EXP-ramp (Fig. 4b). Whereas for amplitudes B p > 10, the adiabatic quality of the LIN-ramp significantly decreases. This can be seen e.g. in Fig. 5c below and inside the loop where the p-values are smaller than 1. For the case of −60 • inclination (Fig. 5b+d) the situation is similar to the one described before.
The total volume of reduced p-values is larger for the EXP-ramp than for the LIN-ramp.
Additionally, due to the reversed inclination a large amount of magnetization is oriented antiparallel below the center of the loop. In this case a substantial amount of magnetization would not contribute to the NMR signal. A simple way to deal with this kind of situation is to reverse the direction of the PP-current [40] and therewith establish the p-distribution as shown for the 60 • inclination case (Fig. 5a+c). However, depending on the particular inclination of the Earth's magnetic field, there will be points in the subsurface where p < 1 and therefore also the usable magnetization M p will be reduced correspondingly.   the other hand, the amplitude loss is much more dramatic with only 15 % (low I ac ) to 55 % (high I ac ) compared to the perfect case. The large difference between the two sounding curves partly originates from the individual switch-off performance of the corresponding ramp at large PP-field strengths present inside / below the PP-loop (cf. Fig. 4b+k) and partly from the high sensitivity at the shallow subsurface for this particular loop layout and AP-excitation (cf. Fig. 2a bottom right). Although the volume having reduced p-values is larger for the EXP-ramp (cf. Fig. 5), the resulting sounding curve deviates much less from the perfect case compared to the LIN-ramp. This is due to the better performance of the EXP-ramp for larger B p amplitudes at smaller angles θ (cf. Fig. 4)  It is not surprising that ramps which fulfill the adiabatic condition much better during switch-off (EXP, LINEXP) perform much better than e.g. the HCOS and LIN ramps.
The observed differences between the individual ramps might be significantly decreased for loop layouts where the Tx/Rx-loops are placed in a particular manner, so that their sensitive region is further away from the PP-loop and the relative orientations between B 0 and B p are more favorable. However, this always comes at the cost of reduced magnetization enhancement and penetration depth [32]. The cases presented here, should therefore be regarded as a kind of worst case scenarios in terms of underestimating the NMR signal amplitude if the particular PP switch-off ramp is not considered in the forward calculation.
More importantly, any signal loss, as presented in Fig. 6, is directly misinterpreted as reduced water content when performing the inversion of the SNMR-PP data.

V. SUMMARY & CONCLUSIONS
The combined use of pre-polarization (PP) and adiabatic excitation pulses (AP) is a promising approach to significantly enhance the signal quality of SNMR measurements that target shallow subsurface water content distributions. The nowadays available SNMR-PP devices allow investigations down to depths of about 2 m depending on loop size and available PP-current. A typical SNMR-PP excitation consists of a strong PP-pulse that aligns the enhanced magnetization with the prevailing magnetic field. After the PP-field is switched off adiabatically, the magnetization aligns with the Earth's magnetic field and gets excited by a subsequent Tx-pulse. In theory, the amplified magnetization is properly aligned with the Earth's magnetic field at the end of the PP switch-off. Due to the inhomogeneity of the PP-field and the generally fixed settings of the particularly employed PP switch-off ramp, there are always regions in the subsurface where the PP switch-off is non-adiabatic, i.e. imperfect. The spatial distribution of these imperfect regions mainly depends on the relative orientation between the Earth's magnetic field and the PP-field and therefore of course implicitly on the inclination of the Earth's magnetic field.
To quantify the effect of an imperfect PP switch-off on SNMR-PP signals, we developed a numerical scheme that allows to study arbitrary ramp shapes and incorporates the PP switch-off simulation directly into the well-established SNMR modeling framework MRSmatlab. For parameters typically used in current SNMR-PP applications the enhancement performance varied between 15 % and 83 % of the perfect PP-case, depending on the chosen ramp parameters and Tx-pulse current (cf. Fig. 6). This is particularly interesting because until now, all SNMR-PP modeling studies assume adiabatic, i.e. perfect, switch-off conditions for the PP-field [32][33][34]47]. Depending on the severity of the performance loss, this effectively leads to an underestimation of the inverted water content.
The presented modeling framework directly incorporates the PP switch-off into the SNMR-PP forward calculation. This allows to correctly account for the PP switch-off in already available SNMR-PP devices. Furthermore, it enables the user to evaluate and optimize different PP/Tx/Rx-loop layouts and PP/Tx-excitation parameters during the development of a new generation of SNMR-PP devices. Inherently, the performance of any switch-off ramp will depend on the available power electronics of the particular SNMR-PP device. Regarding the maximal possible magnetization enhancement, it seems advantageous to use exponentially shaped ramps that fulfill the adiabatic condition almost during the entire switch-off time. However, in our opinion a generalization on the optimal switch-off ramp shape and time is difficult to make and depends on the particular application. Our future work will focus on the development of optimal PP/Tx/Rx measurement setups and parameters, in order to qualitatively and quantitatively target shallow subsurface water content distributions. to switch-off. The angle α is the angle between B 0 and B = B 0 + B p that varies during switch-off as B moves from its initial orientation towards B 0 . The polar and azimuthal angles φ p and φ a describe the final orientation of M p with regard to B p .
In the first example, the initial angle is fixed at θ = 90 • and the amplitude of the PP-field is B p = 100 · B 0 with B 0 = 50 µT. Fig. 7b shows the angles φ p and φ a as a function of the dimensionless switch-off rate 100/ω L /τ r . [38] found that the larger the switch-off rate is (shorter ramp time), the larger is the deviation of the final orientation of M p with regard to B 0 , and hence, the switch-off is no longer adiabatic. The gray symbols in Fig. 7b refer to the results of [38] and the black symbols are modeled with BLOCHUS. We see an excellent agreement over the whole range of switch-off rates.
In the second example the B p -field amplitude is B p = 50 · B 0 with B 0 = 50 µT and the switch-off ramp time is fixed to τ r = 10 ms. We plot the adiabatic quality p as a function of initial angle θ ∈ [0 • , 200 • ] (Fig. 7c). The gray curves in Fig. 7c refer to the results of [40] for different cross-over field strengths B * . They show that the larger the cross-over field strength is, the better is the adiabatic quality over a wider range of initial angles θ. For this particular set of parameters, this means that the earlier the switch to the exponential regime happens, the more adiabatic is the switch-off. Again, our results (black dashed curves) show