CRNSs measure SM by detecting the environmental neutron density produced by cosmic radiation, which is inversely related to below- and aboveground hydrogen pools that surround the sensor. These environmental neutrons have different energies depending on the number and type of interactions that have occurred. Primary cosmic rays with energies around 1 GeV generate high-energy (larger than 20 MeV) neutrons in the atmosphere. By interacting with heavy atoms, these neutrons lose energy and become fast neutrons (0.5 to 20 MeV). The energy of these fast neutrons is further reduced by elastic collisions with lighter atoms (generally hydrogen), first to the epithermal regime (0.5 eV to 0.5 MeV) and finally to a thermal equilibrium (below 0.5 eV). CRNSs typically measure neutrons in the thermal to fast energy regimes (Köhli et al., 2021; Weimar et al., 2020). The measured neutron flux is affected by multiple hydrogen pools, such as soil water, water bodies, lattice water, and biomass. Typically, the CRNS signal is mainly controlled by SM variations, but the additional hydrogen pools can strongly influence the accuracy of the SM estimates (Baatz et al., 2015; Baroni et al., 2018; Iwema et al., 2021; Jakobi et al., 2020; Zreda et al., 2012). Generally, a CRNS is composed of one or more neutron detectors that can be bare (thermal–epithermal neutron detection) or moderated with HDPE (epithermal to fast neutron detection). More detailed information on the main detector components and physics can be found in Zreda et al. (2012), Schrön et al. (2018b), and Weimar et al. (2020).

The quantitative description of the horizontal area over which a CRNS measures is named “footprint”. Detected neutrons that had no contact with the ground (non-albedo neutrons) are, by definition, excluded from the footprint calculation. Thus, the footprint only depends on detected neutrons that had contact with the ground (albedo neutrons). Since CRNSs cannot detect the origin of a neutron and whether it had contact with soil nuclei, the footprint is generally obtained via neutron transport simulations. Although some studies suggested an asymmetric or “amoeba-like” footprint (Schattan et al., 2019; Schrön et al., 2022), most studies assume a simplified circular footprint that depends on the Euclidean distance between the points where neutrons had first contact with the ground and the point of detection. A quantile definition is widely used to define a distance within which most detected neutrons originate (Desilets and Zreda, 2013; Zreda et al., 2008). Commonly used radii are the one e-folding length (∼ 63 % of neutrons) and the two e-folding lengths (∼ 86 % of neutrons). The footprint varies depending on environmental conditions and instrument characteristics (Schrön et al., 2017). Monte Carlo simulations showed that the two e-folding lengths (R86) are ∼ 240 m in fully dry conditions and are reduced by up to 40 % with increasing SM and, to a lesser degree, with variations in humidity, vegetation, and other environmental variables (Köhli et al., 2015). The penetration depth of a CRNS also depends on SM and is higher below the instrument, where it ranges between 83 and ∼ 15 cm (Köhli et al., 2015) depending on SM. The characterization of the CRNS support volume in terms of footprint and measurement depth is a complex and ongoing research subject (Schrön et al., 2022), as shown by a range of recent simulation and field studies that further investigated the spatial sensitivity of SM determined with CRNSs (Badiee et al., 2021; Francke et al., 2022; Schrön et al., 2017) as well as the footprint of thermal neutrons (Jakobi et al., 2021).

The URANOS model, which is freely available online (http://www.ufz.de/uranos, last access: 10 November 2022), was used in this study. This model was first developed to address neutron-only interactions and was later adapted to the cosmic-ray neutron problem (Köhli et al., 2015). URANOS is based on a Monte Carlo approach for the simulation of neutron transport and interactions with matter (Köhli et al., 2018). It is tailored to the study of neutron transport in environmental science, and thus certain processes such as gamma cascades or fission are neglected or represented by effective models. This reduces the computational effort and generally allows the simulation of a larger number of neutrons, which results in more accurate simulations (Köhli et al., 2015). In URANOS, neutrons are emitted from point sources that are randomly distributed within a user-defined source layer with energies sampled from a realistic spectrum matching that of the Earth's atmosphere (Sato, 2015). Then, URANOS uses a standard calculation routine that features a ray-casting algorithm for single neutron propagation and tracks the relevant physical interactions for millions of neutrons (e.g. elastic collisions, inelastic collisions, absorption, and emission processes such as evaporation). URANOS follows the ENDF (Evaluated Nuclear Data File) database standard implementations from Romano and Forget (2013) with cross-sections, energy distributions, and angular distributions obtained from the datasets of Chadwick et al. (2011) and Shibata et al. (2011).

The model domain in URANOS was composed of six (or seven) layers: one (or two) soil and five atmospheric layers. The soil layer extended to 1.6 m depth. The atmospheric layers extended from the soil surface to 1000 m height. The thickness of the five layers was 2, 0.5, 47.5, 30, and 920 m from the bottom to the top layer. The fourth layer was the source layer (from 50 to 80 m above the soil surface, respectively). The pressure of the atmosphere and of the air in the porous media was set to 1020 hPa. A humidity of 3 g cm-3 and a composition of 78 % nitrogen, 21 % oxygen, and 1 % argon were assumed. The soil bulk density was set to 1.43 g cm-3, and the porosity was set to 50 %.

The simulation domain was 1200×1200 m (144 ha) with a resolution of 1 m. It was divided into two areas: (a) a square area at the centre of the domain with five different dimensions (i.e. 0.5, 1, 2, 4, and 8 ha) and (b) the surrounding area. The SM in the inner and in the outer areas was modified independently with increments of 0.05 cm3 cm-3. Additionally, a 1 ha circular area and a 1 ha rectangular area (142×70 m) were simulated with SM variations in the inner area from 0.05 to 0.35 cm3 cm-3. These SM variations were applied homogeneously (both vertically and horizontally) within each area. Finally, for a 1 ha square inner area, simulations with two soil layers were produced, where the SM variation was applied only in the top 10 cm or 30 cm of soil. As in the previous circular and rectangular area scenarios, SM in the inner area varied from 0.05 to 0.35 cm3 cm-3. Figure 1 conceptually describes the simulation strategy taking the 8 ha scenario as an example. A first simulation is performed with a uniform SM of 0.05 cm3 cm-3 in both inner and outer areas. Then, the SM is varied either in the inner area, the outer area, or both areas. This results in 100 simulations with different SM combinations for each scenario and a total of 500 simulations. For each simulation, 108 neutrons were used as this provided sufficient precision and a reasonable computation time (Köhli et al., 2015).

Each simulation provided information on the simulated neutrons that crossed the detector at the centre of the domain (e.g. energy at detection, coordinates of first soil contact). The detector was a vertical cylinder of 9 m radius positioned in the second atmospheric layer (2 to 2.5 m aboveground) and at the centre of the simulation domain. However, not all neutrons that pass through the detector tube are detected, whereby the probability of detection depends on neutron energy and direction, as well as on sensor characteristics such as the used conversion gas, geometrical configuration, and moderator type (Köhli et al., 2018). Different moderators are commercially available and are typically made of HDPE of various thicknesses. In this study, we investigated the use of 5, 10, 15, 20, 25, 30, and 35 mm HDPE moderators. Furthermore, an additional gadolinium oxide (Gd2O3) shield was investigated for a moderator composed of 25 mm HDPE. As was proposed by Desilets et al. (2010), a 25 mm HDPE moderator thickness was selected for the detector shielded with gadolinium. This is supported by Weimar et al. (2020), who found only small differences in response for moderator thicknesses of 20 to 27.5 mm HDPE with gadolinium shielding. Thus, the combination of gadolinium shielding with other moderator thicknesses was not considered. The energy-dependent response functions of the investigated moderators as reported by Köhli et al. (2018) are shown in Fig. 2. Detectors with thin HDPE moderators (e.g. 5 mm HDPE) are highly sensitive to thermal neutrons, while detectors with thicker moderators have a higher sensitivity to epithermal neutrons. The 25 mm HDPE moderator with gadolinium shielding has similar sensitivity to epithermal neutrons as the non-shielded variant but shows less sensitivity to thermal neutrons (Köhli et al., 2018; Weimar et al., 2020). In fact, such shielding can absorb and thus prevent the detection of more than 90 % of the incoming thermal neutrons (Ney et al., 2021), which have a smaller footprint and different sensitivity to SM than epithermal neutrons (Jakobi et al., 2021; Jakobi et al., 2022; Rasche et al., 2021). A cubic spline of each response function was applied to the output of each simulation, and a weight was assigned to each neutron depending on its energy at detection. Then, these weights were summed to obtain the number of detected neutrons and subsequently used in a weighted calculation of the R86. The effect of the angular distribution of the neutrons was not considered here as it was assumed to be negligible, which is reasonable given the absence of changes (e.g. vegetation, atmospheric pressure, humidity) in the vicinity of the detector.

The changes in R86 due to SM variations in the inner and outer areas as well as the use of different moderators were analysed. Although a footprint description that relies on a single value can be misleading (Badiee et al., 2021; Schrön et al., 2022), the R86 represents a standard in CRNS applications and was thus selected to investigate the simulations of this study. Here, the initial hypothesis is that a relatively small R86 is beneficial when monitoring irrigation in small fields as a lower contribution from the surrounding area could be expected. To analyse the Monte Carlo simulation runs, the detected neutrons in each simulation were divided into two types: albedo and non-albedo neutrons. Albedo neutrons carry environmental SM information and were further divided into neutrons that originated within the inner area and neutrons that originated in the outer area. Here, it was assumed that the neutrons that originate within the inner irrigated field carry the bulk of the information of interest. Although neutrons that originate outside the irrigated field can have occasional within-field interactions before reaching the CRNS (Köhli et al., 2015), these were considered of secondary importance for the scope of this study. In a following step, the relative changes in detected neutrons with SM variations were investigated. For a given dimension of the inner area, moderator type, and SM of the outer area (10 simulations), the highest simulated neutron count was set to 100 %. Then, the results of the other nine simulations were scaled to that count rate. The influence of the moderator type, the dimension of the irrigated area, and the SM in the outer area were then compared. Here, larger changes in detected neutrons were considered beneficial as this leads to higher accuracy of the SM estimates.

Next, the sensitivity to irrigation events was assessed in more detail for the moderator type that provided the largest relative changes in detected neutrons. Starting from homogeneous SM conditions in the simulated domain, an irrigation event was assumed to increase the SM in the irrigated area by either 0.05 or 0.10 cm3 cm-3. Such SM variation was applied to the entire soil profile. For the 1 ha square area, the SM variation was separately applied to the top 10 and 30 cm and to the entire soil profile. The initial homogeneous SM conditions were between 0.05 and 0.25 cm3 cm-3 as this is the relevant range for irrigation applications. The selected detection thresholds, shown in Table 1, were based on the relative error in the simulations: 1σ=100⋅1/N1+1/N2, where N1 and N2 are the neutron counts obtained with the initial and final SM conditions, respectively. In addition to σ, a value of α=1 % was included in each threshold to represent a generic detection uncertainty limit for a detector that can achieve ∼ 1000 counts per hour and aggregation times of <12 h that are relevant in SM monitoring (Schrön et al., 2022).

Lastly, relative changes in detected neutrons due to homogeneous SM variations within the simulation domain were compared with those due to SM variations that occur in the inner or in the outer areas only. Again, the moderator type that had provided the largest relative changes in detected neutrons was selected. For each dimension of the irrigated area (100 simulations), the highest simulated neutron count was set to 100 %. The reduced sensitivity of the detection–SM relationship in the case of SM variations that occur only in the inner area was assessed. Here, a lower reduction in sensitivity was considered beneficial for irrigation monitoring. Also, the influence of the SM in the outer area on the count rate was compared to that of SM variations in the inner area. A strong influence of the SM in the outer area was considered disadvantageous for irrigation monitoring.

The analysis of the CRNS footprint can be useful to understand the general distance from which the measured neutrons originate. The boxplots of R86 for all moderators and for different dimensions of the irrigated field are shown in Fig. 3. In general, the footprint increases when a thicker HDPE moderator is used and increases further when gadolinium shielding is added. Additionally, there is a trend towards a larger variability in the footprint size with SM variations when larger irrigated areas are considered. Generally, the largest R86 values are obtained using a 25 mm HDPE moderator with gadolinium shielding (e.g. min = 147, med = 196, and max = 273 m in the 8 ha scenario), whereas the lowest values are obtained with a 5 mm HDPE moderator (e.g. min = 127, med = 162, and max = 238 m in the 0.5 ha scenario). Nonetheless, for each dimension of the irrigated area, the difference in R86 obtained with different moderators and shielding is rather small and on the order of 10 to 20 m only. Such differences in R86 may thus have limited influence on CRNS measurements of small, irrigated fields.

A more in-depth analysis of how R86 changes with SM variation is shown in Fig. 4 for a 5 mm HDPE moderator and for a 25 mm HDPE moderator with gadolinium shielding. The value of R86 depends on the SM of both the inner and outer area. Similar patterns in R86 variation with SM are found when other moderators are used (not shown). For example, for a 5 mm HDPE moderator, the highest R86 value in the 0.5 ha scenario is ∼ 234 m for high SM in the inner and low SM in the outer area (bottom right in Fig. 4a). The R86 decreases to ∼ 220 m when the SM in the inner area is reduced to 0.05 cm3 cm-3. A more pronounced decrease in R86 to ∼ 132 m occurs when the SM in the outer area is increased up to 0.50 cm3 cm-3. When the size of the inner area increases (Fig. 4b), the general trends in R86 with SM remain rather constant except for simulations where SM is high in the inner area and low in the outer area (bottom right corners). For these conditions, R86 tends to increase with increasing size of the inner area. For example, R86 is ∼ 258 m in the 8 ha scenario (Fig. 4b), which is almost 15 m larger than the R86 in the 0.5 ha scenario (∼ 234 m) for a SM of 0.50 cm3 cm-3 in the inner area and 0.05 cm3 cm-3 in the outer area. Similar considerations can be drawn for a 25 mm HDPE moderator with gadolinium shielding (Fig. 4d–f).

In previous studies, the smallest R86 was often assumed to occur for the highest SM in the CRNS surroundings. This is true for homogeneous SM conditions but may be different for heterogeneous SM distributions (Schrön et al., 2022). When the SM of the inner area is low and that of the outer area is high, the outer area becomes a less important source of neutrons, and thus the footprint is reduced. An opposite effect occurs when the inner SM is high and the outer SM is low. In this case, the inner area is a smaller source, and the neutrons from the outer area become more important, resulting in a larger footprint. For example, with a 5 mm HDPE moderator (Fig. 4a) the R86 of a 0.5 ha field reduces by 39.9 % when SM increases from 0.05 to 0.50 cm3 cm-3 in the outer area, whereas the same SM increase in the inner area enlarges R86 by 6.4 %. For a 25 mm HDPE moderator with gadolinium shielding, these percentages are 37.2 % and 9.9 %, respectively.

For each dimension of the simulated inner area and for each moderator type, Table 2 shows the minimum and maximum percentages of detected (albedo plus non-albedo) neutrons that originate in the inner area depending on SM conditions. This percentage represents the detected neutrons that originate within an irrigated field and thus carry the bulk of the information of interest in case of irrigation applications. Considerable differences are found between simulations depending on the SM in the inner and outer area. The dimension of the irrigated area also influences the results. In particular, 27.7 % to 61.1 % of the detected neutrons originate from within a 0.5 ha irrigated field, whereas larger fields show higher percentages (e.g. 50.7 % to 85.3 % from within an 8 ha field). In addition, thinner moderators generally show a higher percentage of detected neutrons from the inner area.

Figure 5 shows the variation in the percentage of detected albedo neutrons that originate in the inner or in the outer area for the 0.5 ha scenario as a function of the SM of both areas for a detector with 25 mm HDPE moderator and gadolinium shielding, which shows the lowest percentages in Table 2. Non-albedo neutrons are not considered in Fig. 5, and the reader is referred to Appendix A for a more detailed description of such detected non-albedo neutrons. Clearly, the fraction of neutrons originating in the inner and outer area strongly depends on SM. The percentage of neutrons originating in the inner area is smallest when the inner area is wet and the outer area is dry (31.4 %). This percentage increases up to 58.0 % if the SM in the inner area is reduced to 0.05 cm3 cm-3 and then to 69.6 % when the SM of the outer area increases to 0.50 cm3 cm-3. The percentage of neutrons originating in the outer area shows the opposite trend. From these results, it is clear why CRNS applications focused on irrigation of small fields can face considerable challenges. Especially in small irrigated fields, the number of detected neutrons that originate outside the irrigated area can be higher than the number of neutrons that originate inside of the irrigated area. This is especially true when the SM outside the irrigated area is relatively low, which is often the case when the inner area is irrigated. Figure 5b presents the same analysis for an 8 ha field. Here, the percentage of neutrons from the inner area is higher than in the 0.5 ha scenario and shows a lower overall variation. Again, the lowest percentage of detected neutrons originating from the inner area (67.6 %) is found when the inner area is wet and the outer area is dry, whereas the highest value (90.4 %) is found with reversed SM conditions.

The relationship between neutron count rate and SM has been mostly investigated for homogeneous SM conditions. Here, we also explore the effect of SM variations that occur only within an inner area that surrounds the instrument (e.g. through irrigation). We considered sizes of the inner irrigated area from 0.5 to 8 ha and two opposite wetness situations for the outer area (i.e. SM of 0.05 and 0.50 cm3 cm-3). In addition, we considered the effect of the thickness of the moderator and additional shielding on the number of detected neutrons. In Fig. 6 and in Fig. 7, the neutron counts of the different simulations are scaled to the case with the highest count rates (SM of 0.05 cm3 cm-3 everywhere in the simulated domain). Generally, neutron count rates show a non-linear negative relationship with increasing SM in the irrigated area. The detector with a 5 mm HDPE moderator shows the smallest relative changes in neutron counts (Fig. 6). Thicker HDPE moderators (i.e. 15, 25, and 35 mm) result in larger relative changes. The largest difference in relative changes is observed between a 5 and a 15 mm HDPE moderator, whereas the differences between a 25 and a 35 mm HDPE moderator are rather small. The highest relative change in number of detected neutrons occurs for a 25 mm HDPE moderator with gadolinium shielding. Nonetheless, the described sensitivities are lower than with a homogeneous SM change in the entire domain (grey circles and squares in Fig. 6). The relatively high sensitivity of the 25 mm HDPE moderator with gadolinium shielding seems to contradict the relatively large R86 values as well as the relatively low percentage of detected neutrons that originate within the irrigated area for such a moderator. This is attributed to small relative variations in the R86 between different moderators and to the low number of detected thermal neutrons (< 5 %), which only contain limited SM information (see Appendix B). The SM in the outer area also has an influence on the results, although to a lesser degree. For example, a SM of 0.50 cm3 cm-3 in the outer area (Fig. 6b) leads to greater variations in detected neutrons than a SM of 0.05 cm3 cm-3 in the outer area (Fig. 6a).

The neutron count rates obtained with a 25 mm HDPE moderator with gadolinium shielding for irrigated areas ranging from 0.5 to 8 ha are shown in Fig. 7. An increased size of the irrigated area results in larger relative changes in the number of detected neutrons with SM variations, especially when the outer area is dry (Fig. 7a). Higher SM of 0.50 cm3 cm-3 in the outer area (Fig. 7b) results in larger relative changes in the number of detected neutrons, although this is mostly the case for relatively small dimensions of the irrigated area. For the 8 ha scenario, the impact of the SM in the outer area is very limited.

Generally, larger relative changes in detected neutrons with SM variations lead to an improved performance of a given CRNS in a certain environment. Thus, our analysis shows that the CRNS performance in an irrigated environment depends on both the moderator type and the SM outside of the irrigated area. However, the effect of different moderator types on the performance is stronger. A relatively thin HDPE moderator will generally result in low changes in the neutron count rate with SM variations, independent from the dimension of the irrigated field. A thicker HDPE moderator will provide higher neutron count changes and thus better sensitivity to SM changes in the inner area. Overall, a 25 mm HDPE moderator with gadolinium thermal shielding achieves the best results.

The sensitivity of a CRNS is further analysed to assess if it is possible to detect irrigation events in fields between 0.5 and 8 ha in size (Fig. 8). A 25 mm HDPE moderator with gadolinium shielding is used as it provides larger relative changes in the neutron count rate compared to the other investigated moderators. An irrigation event is assumed to increase the SM of the irrigated area by 0.05 or by 0.10 cm3 cm-3 starting from a homogeneous SM condition within the domain. The detection thresholds (see Fig. 8) are based on the relative error (σ) of the simulations (Eq. 1), which varied between 2.3 % and 3.4 % depending on the number of detected neutrons.

As shown in Fig. 8, an irrigation event that leads to a 0.05 cm3 cm-3 increase in SM can be detected with CRNSs (relative change in detected neutrons higher than 3σ+α) when the initial SM of the simulation domain is 0.05 cm3 cm-3. This is the case for all five investigated dimensions of the irrigated area. In the case of 4 and 8 ha fields, detection of this type of irrigation event is also achieved with an initial SM of 0.10 cm3 cm-3. For initial SM larger than 0.05 cm3 cm-3, the chance of detecting irrigation events depends on the dimension of the irrigated area, with larger irrigated fields yielding higher detection chances. Overall, the detection of a 0.05 cm3 cm-3 irrigation event is uncertain (relative change in detected neutrons lower than σ+α) when the initial SM is 0.20 cm3 cm-3 or higher in a 0.5 ha field (Fig. 8a) or 0.25 cm3 cm-3 in 1 ha field (Fig. 8b). Irrigation events that induce larger SM variations of 0.10 cm3 cm-3 result in larger relative changes in detected neutrons and thus have a higher chance of detection. Here, the CRNS detects irrigation events with initial SM up to 0.10 cm3 cm-3 in 0.5, 1, and 2 ha fields and with initial SM up to 0.20 cm3 cm-3 in 4 and 8 ha fields. If the investigation of a 0.10 cm3 cm-3 irrigation event is extended to higher initial SM (i.e. 0.30 to 0.40 cm3 cm-3), the only uncertain detection (lower than σ+α) is that of an initial SM of 0.40 cm3 cm-3 in a 0.5 ha field (not shown).

In fields between 0.5 and 8 ha, a CRNS is not only sensitive to the irrigation of the target field but also to SM variations in the outer area (e.g. due to irrigation of neighbouring fields). It is thus important to assess the impact of SM variations that occur outside the field of interest. Figure 9 shows the relationships between SM variations that occur in the inner area, in the outer area, or in both areas as well as the resulting relative changes in detected neutrons for a 25 mm HDPE moderator with gadolinium shielding. SM variations that are limited to the irrigated field show a reduced sensitivity of the relative change in neutron count rate compared to the case where SM variations occur homogeneously in the entire domain, which is due to the influence of the SM in the outer area. The strongest reduction in the sensitivity of this relationship is found for the 0.5 ha field (Fig. 9a). The sensitivity increases with larger dimensions of the irrigated area, and the observed reduction is lowest in the case of the 8 ha field (Fig. 9e).

In the case of a 0.5 ha irrigated field, SM variations in the outer area generally induce a higher relative change in detected neutrons compared to SM variations that occur within the irrigated area. For example, a SM variation from 0.05 to 0.10 cm3 cm-3 in the outer area (Fig. 9a) induces a larger relative change (-11.8 %) compared to the same SM variation in the inner area (-8.6 %). With larger irrigated areas, the influence of the outer area is reduced. In the 8 ha scenario, a SM variation from 0.05 to 0.10 cm3 cm-3 in the inner area induces a larger relative change in detected neutrons than that of a SM variation from 0.05 to 0.50 cm3 cm-3 in the outer area. These results suggest that it may be necessary to obtain SM information from the surroundings of a target field in real-world applications of CRNS-based irrigation monitoring.

It is also of interest to reduce this analysis to a smaller range of SM in the outer area (i.e. 0.05 to 0.15 cm3 cm-3) as this is generally the case in irrigated agricultural environments (shaded red area in Fig. 9). In the case of 0.5 and 1 ha fields, a SM variation of 0.10 cm3 cm-3 in the outer area (red error bars in Fig. 9a–b) produces a relative change in the neutron count rate that is higher than that of an irrigation event of 0.05 cm3 cm-3. This is not the case when the initial SM of the domain is 0.05 cm3 cm-3 in the 2 ha scenario (Fig. 9c), 0.05 to 0.15 cm3 cm-3 in the 4 ha scenario (Fig. 9d), and 0.05 to 0.20 cm3 cm-3 in the 8 ha scenario (Fig. 9e). Overall, even if the range of the SM in the outer area is reduced, it can still have a considerable impact on the CRNS. This again underlines the importance of obtaining information on the SM in the outer area, especially in the case of relatively small irrigated fields.