The Monte Carlo tool URANOS  is designed specifically for modelling neutron interactions within the environment in the framework of CRNS. The standard calculation routine features a ray-casting algorithm for a single neutron propagation and a voxel engine. Instead of propagating particle showers in atmospheric cascades, URANOS reduces the computational effort and makes use of the analytically defined cosmic-ray neutron spectrum by . The URANOS model can use setups with either open domains of at least 600 m or smaller sizes with periodic or reflecting boundary conditions.

The simulation geometry consists of a ground layer with a thickness of 1.3 m and a 1000 m air (buffer) layer. The soil consists of 50 %Vol solids and a scalable amount of H2O. The solids are composed of 75 %Vol SiO2 and 25 %Vol Al2O3 at a particle density of 2.86 gcm-3. The air medium consists of 78 %Vol nitrogen, 21 %Vol oxygen and 1 %Vol argon at a pressure of 1020 mbar. The air humidity is set to 7 gm-3 and soil moisture is set to 10 %Vol. In this study, three different simulation setups have been used:

In order to assess the effect the additional moderating effect of the directional shielding on the measured intensity, in a first setup, the detector with its actual dimensions has been placed inside a domain of 10 m lateral extension with periodic boundaries and a cosmic neutron source.

For the sake of simplicity, we use the simplest geometric setup imaginable for directional CRNS measurements: a plain divided into two homogeneous half-spaces (A1 and A2). Each half-space corresponds to a count rate R1 and R2, respectively, when measured far enough from its borders. At the border of both half-spaces, we implement a directional detector. When facing A1, it yields the count rate Rf1; when directed towards A2, it registers the count rate Rf2. Both count rates are measured and used in the computations. Consequently, the evaluation employs a FOV of 180∘ (π). This value has no direct relation to the geometry of the opening face of the shielding, which is apparently somewhat narrower. Instead, it is an arbitrary decision to target this area within the FOV with the measurement.

The total count rate Rtotal registered by a sensor results from non-epithermal and epithermal neutrons (see Eq. ).

The epithermal counts, in turn, are composed of counts from albedo neutrons having interacted with the hydrogen pools of interest (Ralb) and non-albedo neutrons (Rnon-alb) without such interaction. We denote the respective percentage as γ (see Fig. , left bar), so 6Ralb=(1-γ)⋅Repi. Note that γ differs between the omnidirectional and the directional detector, which will be shown later. Therefore, we distinguish in the following between a γD and a γ!D for the directional and the omnidirectional detector, respectively.

For the directional sensor placed entirely into the interior of one of the half-planes, A1 or A2, the directional shielding causes partial blockage of the neutrons arriving, reducing its count rate by the factor β to Rs (see Fig. , second bar): 7Rs,epi=β⋅Repi. Values for γ and β were obtained from neutron simulations as discussed in Sect. , setup 1.

When placing the directional sensor exactly between the two half-planes A1 and A2, the components of its counts (albedo and non-albedo) can be described as a mixing from the adjacent planes. The albedo neutrons mix according to the orientation of the shielding and the respective directional contribution η (Eq. ), i.e. the ratio between counts from the desired angle and total counts (see Fig. , third bar): 8aRf1,alb=η⋅Rs1,alb+(1-η)⋅Rs2,alb8bRf2,alb=η⋅Rs2,alb+(1-η)⋅Rs1,alb.

The directional detector may be oriented towards the left half-plane A1 or the right half-plane A2. Formally, there is no fundamental difference in the equations for each, and we exemplify the next step with the one oriented towards A1 and its count rate Rf1,alb.

Substituting Eq. () with Eqs. (), () and then () yields 9Rf1,alb=(1-γs)(η⋅Rs1,epi+(1-η)⋅Rs2,epi)=(1-γs)⋅β(η⋅R1,epi+(1-η)⋅R2,epi)=(1-γs)⋅β⋅(1-ϵ)(η⋅R1,total+(1-η)⋅R2,total).

Considering both rates Rf1,alb and Rf2,alb simultaneously in a vector Rf,alb, we may write in matrix notation: 10Rf,alb=Rf1,albRf2,alb=(1-γs)⋅β⋅(1-ϵ)⋅η1-η1-ηηR1,totalR2,total=(1-γs)⋅β⋅(1-ϵ)⋅A⋅Rtotal, where A is a symmetric matrix containing the coefficients for the mixing. For the non-albedo neutrons Rf,non-alb, the mixing is simply the average of the corresponding rates: 11Rf,!alb=12⋅J⋅RD,!alb=12⋅J⋅γsRD,epi=12⋅J⋅γs⋅βRepi=12⋅J⋅γs⋅β⋅(1-ϵ)Rtotal, where J is the matrix of ones.

Applying Eq. () for the non-epithermal counts leads to 12Rf,!epi=ϵ1-ϵ⋅Rf,epi=ϵ1-ϵ⋅Rf,alb+Rf,!alb=ϵ⋅β⋅(1-γs)⋅A+12⋅J⋅γsRtotal. Adding Eqs. (), () and () yields the total count rates of the directional detector Rf,total: 13Rf,total=Rf,alb+Rf,!alb+Rf,!epi=(1-ϵ)⋅β⋅((1-γs)⋅A+12⋅J⋅γs+ϵ1-ϵ(1-γs)⋅A+12⋅J⋅γs)Rtotal=A(β-βγs)+12βγsJRtotal=BRtotal, with B being a matrix summarizing all the operations.

For reconstructing the unshielded count rates in the half-spaces (Rtotal) from the two directional measurements (Rf1 and Rf2), we use the inverse operation: 14Rr,total=B-1Rf=1k1k2k2+2k2+2k2Rf,total, where 15k1=2β(2γsη-γs-2η+1),k2=(2γsη-γs-2η).

The subscript “r” in Eq. () denotes the reconstructed rates for estimating the true (and unknown) ones in the half-spaces (R1 and R2).

As described in Sect. , the errors in neutron counts follow a Poisson distribution. In this study, we exclusively consider large numbers for values of N (i.e. N≫20, which is consistent with practical CRNS applications). As a consequence of the central limit theorem, we can approximate the errors with a Gaussian distribution and corresponding standard deviations: 16P(λ)∼N(μ=λ,σ2=λ2).

As the considered errors in the two directional counts (Nf1 and Nf2) are independent, the superposition of these errors can be described based on Gaussian error propagation: 17σF=∂F∂x2σx2+∂F∂y2σy2, where F denotes a function combining the contributions of x and y. In our case, F would be the reconstruction of the count rates from the directional counts (i.e. Eq. ).

When reconstructing the rates in the half-spaces R1 and R2 as described in Eq. () and propagating the errors in both directional counts (see Eq. ), and performing substitutions (Eqs.  and ), the error in the reconstructed epithermal counts is 18σr,total=B-1∘2σf,total∘2∘=B-1∘2Nf,total∘∘2∘=B-1∘2BNtotal∘. (∘ denotes the Hadamard operations; i.e. the square and square root operation are applied element-wise on the vectors and matrices.)

As mentioned in Sect. , directional CRNS is motivated by two aims: I.

determining the neutron count rate for the area the detector is directed at (optionally, also the rate for the complementary area) and

Both aims can be pursued jointly or separately; however, achieving one does not necessarily guarantee achieving the other. For example, it might be possible to determine two count rates with high precision, but nevertheless it might be impossible to detect a significant difference between them, when they are (nearly) equal. Conversely, very dissimilar count rates can be discerned, even if their actual value cannot be determined very precisely. Therefore, we look at both of these aims separately. We formalize the determination of count rates (aim I) as being able to confine their 95 %confidence interval (CI) to a value smaller than a desired precision, expressed as a fraction of the actual value. We chose a value of 5 % for our illustrations, which roughly corresponds to an error of 2 %-points in volumetric water content for the conversion used by . For distinguishing two count rates (aim II), we propose they are significantly different, if their difference – regarding the associated uncertainties – is statistically different from 0 with a chosen p value (0.05 in our case).

Applying Eq. (), e.g. using provided R code, the described analysis can be performed for any combination of parameters of interest, i.e. count rates (R1, R2) or their respective contrast (ΔR), temporal resolution (Δt), directional contribution (η), overall reduction of count rate due to the directional shielding (β) and the fraction of bycatch neutrons (ϵ). However, for illustrative purposes, in the results section (Sect. ), we provide some example plots trying to capture the typical ranges of the parameters involved. The parameters η, γs and β were set to extreme values and combined in two scenarios termed “favourable” and “unfavourable”, encompassing the most favourable and most adverse settings, respectively (see Table ). The other parameters are varied continuously along a range. This selection was guided by a range of diverse settings found in the literature summarized in Table . Detailed explanations are given in the following.

Fraction of non-epithermal counts (ϵ): in the examples, we illustrate the situation for an “favourable” scenario assuming a low fraction of non-epithermal counts (ϵ=0.1) and for a “unfavourable” setting (ϵ=0.3).

Directional contribution (η): the effect of the directional shielding is expressed by the directional contribution η, i.e. the ratio between the counts from the area targeted and total counts (see Eq. ). In the displayed examples, we only consider symmetrical half-planes (see Fig.  for the “favourable” scenario with the perfect shielding (η=0.72) and the “unfavourable” implementation of the actual shielding (η=0.61) as resulted from the neutron simulations (see Sect. , Table ).

Fraction of non-albedo neutrons in the directional detector (γs): the fraction of non-albedo neutrons, i.e. those without the interaction with the surface, is a function of the hydrogen inventory in the footprint. We estimate respective values based the neutron simulations (see Sect. ). For the “favourable” scenario, we choose a low value (γs=0.2) corresponding to very dry conditions; the “unfavourable” implementation (γs=0.31) mimics a very wet environment (see Sect.  and Table ).

Overall reduction of count rate due to the directional shielding (β): the directional shielding reduces the total count rate in the directional detector to the fraction β (see Eq. ). For the “favourable” scenario, we choose a high value (β=0.4) corresponding to very wet conditions; the “unfavourable” implementation (β=0.3) mimics a very dry environment (see Sect. , Table ).

The following parameters were varied along plausible ranges.

Count rate (R): count rates depend on site conditions (i.e. incoming neutrons and hydrogen pools) and detector sensitivity. We selected values of 500, 2000, 8000, 40 000 and 150 000 countsh-1 as example values for the count rates registered at the detectors (Rtotal; see Eq. ).

Contrast in count rates of the half-planes (ΔR): we denote the difference in the count rates in the two half-planes with ΔR, expressed as their difference relative to the lower of the two values: 19ΔR=R2-R1R1. Information on a realistic range of this value would ideally be obtained from spatially distinct sensor locations, e.g. from roving. As this information is unavailable for most considered examples, we use the temporal variation of the signal as a proxy, resulting in example values of 0.2, 0.5, 0.8 and 1.4.

Aggregation time Δt: suggests that the typical temporal resolution of standard stationary detectors is in the range of 3 to 12 h, depending on detector technology and site conditions. and use longer aggregation intervals of 24 h. For our study, we display results for values of 1, 6, 12 and 24 h. Longer aggregation times are not recommended from a hydrological perspective, since they would commonly imply too high a change of the observed variable during that interval (e.g. by rainfall or drying and respective change in R).

For a subsequent evaluation of the feasibility for a specific case, we use the setting listed in line 6 of Table , i.e. the measurement using the prototype directional detector described in Sect.  with R1=2100 countsh-1, ΔR=0.4.

Increasing the thickness of the polyethylene shielding not only reduces the neutron flux from the non-targeted direction, it also partially reflects neutrons from the other direction and changes the energy response. Furthermore, the moderator itself produces secondary neutrons, which can be regarded as an offset bias. Thus, with the main function of blocking neutrons from the non-targeted directions, a number of secondary effects come along, which slightly change the characteristics of the CRNS probe. Table  summarizes the effects of adding a directional shielding to a detector. It demonstrates that the fraction of non-albedo neutrons γ is higher for the directional detector than for the unshielded operation. Furthermore, it increases with the amount of hydrogen in the footprint.

Secondly, the total count rate reduction by adding the shield β is at least 30 %. For the wetter conditions, it is even higher, reaching β=40 % for θ=50 % (see Table ).

The CRNS method relies on the principle that the detected neutrons have interacted with the soil of the footprint – usually several times – and thus carry information about its hydrogen inventory. Due to these atmosphere–ground interface crossings, the correlation between neutron origin and field of view of the probe is diluted. Most neutron scatterings before detection are located in the direct vicinity of the detector, which to some extent dissociates the vector of detection and the vector to the origin of the neutron. For this reason, the shielding does not as effectively filter neutron vectors from remote origins, but its directional specificity is much more pronounced for neutrons with origins in the near range.

From Fig. , which shows the case of a detector in the “favourable” scenario, we can conclude the following: the largest part of the neutron flux remains undetected (black), due to insufficient energy or being scattered off the detector material itself. Most neutrons counted entered the instrument from a viewing angle corresponding to the open side face (orange). However, their origin only partially lies in that direction. While often being transported “geometrically” (i.e. directly) to the detector when being released from the soil in the direct vicinity of the instrument (yellow line), more distant origins tend to incur much more directional changes of the neutron. This leads to a flattened angular distribution (orange and brown lines).

In the “unfavourable” scenario, the insufficient shielding of MeV neutrons to the sides leads to field-of-view contamination of approximately 10 % of the signal, mostly due to the limited thickness of the HDPE shielding; see Fig. . Compared to the ideally shielded detector in Fig. , the contamination causes an increase of roughly 50 % in the plateau region of undesired angles.

In order to quantitatively assess the directional sensing capabilities, we use the directional contribution (see Eq. ). We chose as representative target FOVs 90∘ (π/2) and 180∘ (π). The results are summarized in Table . For a rather narrow viewing angle (i.e. 90∘), even in an optimistic case, less than half of the signal originates from that direction. If the field-of-view limitation is set to a full half-space, 60 % of the signal is representative of information from those angles. As the actual detector suffers from a partial leaking-in of MeV neutrons, a near-field blur effect appears: as most neutrons from the direct vicinity are fast, the signal contamination due to insufficient shielding is to a large extent carrying information about the local area of the sensor.

Figure  confirms that for higher count rates R1 and R2, less time is required to obtain a robust value from the directional measurements, i.e. confining the CI to less than 5 % of the actual value. The respective contrast between R1 and R2 is of relatively small effect for higher count rates but makes a difference for lower ones. For these, somewhat counterintuitively, a higher contrast requires longer aggregation times. This may be explained by the fact that these higher count rates are associated with larger absolute errors. When mixed with the weaker signal (see Eq. ), they deteriorate its robust reconstruction. Consequently, a higher contrast in the rates aggravates the reconstruction of the lower one.

Conversely, Fig.  demonstrates that the statistical difference between R1r and R2r is easier to detect with a higher contrast between the two rates. This phenomenon effectively equates to the dilemma that the contrast in the count rates has the opposite effect, depending on whether we look at the precise determination of R1r (benefits from low contrasts) or the statistical distinction between the two rates R1r and R2r (benefits from high contrasts). This distinction is apparently much more feasible within reasonable aggregation times: in the “favourable” scenario, hourly resolution can be achieved from count rates of above approximately 3000 countsh-1 even for low contrasts. The “unfavourable” scenario increases the aggregation times by about a factor of 7.

For our example case, we can conclude that the precise determination of the count rates is hardly feasible even for the “favourable” scenario. Aggregation times exceeding at least 36 h are beyond the typical requirements in applications. Aggregation times of 24 h and less can only be achieved with count rates of more than 3100 or 20 700 countsh-1 for the “favourable” or “unfavourable” scenario, respectively. Distinguishing the rates in the two half-planes, however, could be possible for aggregation times on the order of hours with even higher potential for stronger contrasts.

Figure  again illustrates the phenomenon mentioned in the previous section: with higher contrast in the signal, estimating the weaker signal with the required precision gets more difficult (i.e. requires longer aggregation times or higher count rates). For reproducing values with high contrast (ΔR=1.4) in daily resolution, count rates R1>4000 countsh-1 are required for the “favourable” scenario. For the “unfavourable” scenario, R1 must be larger than 40 000 countsh-1 for this purpose.

Conversely, higher contrasts allow the statistical distinction of the two reconstructed rates also for lower count rates and aggregation times (see Fig. ). According to the “unfavourable” scenario, even relatively low contrasts (ΔR=0.2) can be detected with the lowest considered count rates, if aggregation times are slightly higher than 24 h.

For our example case, we have already shown that the determination of the count rates is hardly feasible even for the “favourable” scenario. However, with aggregation times of 24 h, distinguishing rates with contrasts lower as 0.2 could be possible even for the “unfavourable” scenario.

Figure  again stresses the need of high count rates to reconstruct the target count rates with the chosen precision. Specifically, to compute those rates in the “favourable” scenario at the daily resolution, the minimum count rate must be well above 3000 countsh-1 for contrasts lower than 0.2. For the “unfavourable” scenario, this value increases to over 20 800 countsh-1. As noted before, higher contrasts require even higher count rates (i.e. > 4500/31 000 countsh-1 for the cases above) or longer aggregation times, which may exceed those required for practical applications (see Sect. , last paragraph).

Concerning the statistical discernibility of R1r and R2r, Fig.  suggests that already low count rates allow the two reconstructed count rates to be distinguished. For the “unfavourable” scenario at the aggregation interval of 1 h, count rates of at least 15 600 countsh-1 already allow resolving contrasts as low as 0.2. For the higher contrasts (ΔR=1.4), count rates of roughly 490 countsh-1 would suffice.

Thus, for our example case, we would require at least 3100 countsh-1 to successfully reconstruct both rates at 24 h resolution. The current count rate of 2100 countsh-1 calls for aggregation times of at least 36 h. Merely distinguishing the rates R1r and R2r is possible with the given count rate at 1 h aggregation time.

This analysis is based on the geometries of a specific directional detector prototype. This prototype was designed with pragmatic considerations. So far, we have just been able to operate the prototype in a setting with sufficiently high contrast to practically indicate discrimination of count rates could be successful. Other designs (e.g. larger vertical planar shielding blocking off one half-space) may provide superior characteristics, namely η, which could be assessed by further neutron simulations.

The presented computations assume constant neutron flux rates. Evidently, this assumption is less realistic with increasing aggregation times: on the one hand, incoming neutron flux is subject to variation, though usually moderate; on the other hand, changes in the hydrogen pool within the footprint, namely due to hydrological processes, will add more variability to the signal, effectively increasing its error and aggravating the determination and discrimination of count rates.

The directional contributions obtained from the numerical neutron simulations apply to a setting with selected values for fixed hydrogen inventories. Increasing this inventory would decrease the sensor footprint. As the angular specificity decreases with distance to the sensor, we might expect somewhat improved directional contributions. However, in practice, an increased hydrogen inventory (i.e. from soil moisture and biomass) will usually also incur higher air humidity (deteriorating angular specificity) and reduced count rates, which counteract this effect. This is similar to the adverse effect of road-construction material on roving CRNS measurements . In-depth neutron simulations need to be used to clarify this issue.

Likewise, our setup also assumed spatially homogeneous hydrogen pools. Deviations from this assumptions, especially close to the sensor, will have a pronounced effect on the recorded signal due to the high sensitivity of the sensor in the short range. These effects will be detrimental to the reconstruction of representative count rates for the half-planes, unless the interpretation is restricted to this very proximity of the sensor.

With disparate flux rates R1 and R2 two additional concurrent effects will occur: if R1 increases (e.g. less moisture in A1), we would also expect more A2 neutrons to be scattered back to the detector from A1, because thermalization is reduced there. This deteriorates the directional contribution η for the reconstruction of Rr1 as we are getting more neutrons from the “wrong” direction. Conversely, the increased R1 will tend to be more directional when coming from the area of less hydrogen (i.e. less scatter), in turn increasing the directional contribution for Rr1. Further neutron simulations are required to clarify which of these effects dominate and if they would notably influence η.

In our calculations, for the “favourable”/“unfavourable” scenario, we use fixed values for the fraction of “bycatch” count (ϵ) at 10 %/30 %. This assumption implies that these adjacent energy levels (thermal and fast neutrons) show a similar sensitivity to hydrogen. In reality, this sensitivity is considerably smaller and different but has been exploited in some studies (e.g. ), effectively reducing uncertainties by increasing the rate of usable counts Repi. However, measuring with only single moderated detectors does not allow for such a separation of the signal in terms of energy levels. Instead, the actual value of ϵ, being a function of hydrogen pools and chosen energy cutoff thresholds, poses another considerable source of uncertainty, which we neglect completely in this study by assuming a fixed value. A similar effect can be expected for the fraction of non-albedo neutrons γ: its range was considered in the scenarios; however, its actual functional dependency on the hydrogen inventory ignored in the analysis.

Figure  suggests a setup of two independent detectors facing opposite directions. In this case, an imperfect calibration of their sensitivity can constitute another substantial device-specific error source, further aggravating the separation of the signals. Alternatively, a single sensor with temporally varying orientation (i.e. as presented in Sect. ) could be used. While this eliminates the issue of imperfect calibration, it implies that all the computed aggregation times must be doubled, as the single sensor can cover each direction only half of the time.

The directional contribution η depends on the transport characteristics of the neutrons from the measured object to the sensor and the properties of the shielding. The former is governed by the surrounding medium (i.e. air pressure and humidity) and is beyond control in monitoring situations. The shielding, however, can be modified without theoretical limitations. Although obtaining a narrow FOV is still unfeasible because of the above-mentioned transport characteristics, hemispherical blocking could be increased to a large extent. It is constrained, however, by practical issues such as size, weight and price. Directional shielding larger than a few metres would hardly be practicable (transport, visual impact, wind stress); extending the shielding below the soil surface is certainly unfeasible, and a freely rotating setup poses even stronger limits. On the other hand, a concomitant use of two sensors could potentially use the same shielding. An enlarged version of the shielding would presumably also have higher directional contribution. The presented methodology could help to find reasonable compromises.

In the choice of example values, we also included some count rates obtained with setups of roving CRNS. These setups consist of a larger number of counting tubes. Consequently, they are considerably bulkier and could not be equipped with a shielding in the described dimensions, unless advances in sensor technology provide significantly smaller detectors. Even if such a shielding could be scaled up, the resulting weight would pose a severe challenge for realizing a practicable rotating sensor platform, thus calling for two complementary, non-rotating sensors instead, as mentioned above.

As a more angle-specific option for the shielding, micro-channel plates have shown favourable directional characteristics . However, their limitation to thermal neutrons, considerable costs and the remaining problem of non-geometric transport make them unfeasible for CRNS.

Materials with low hydrogen content and a high scattering-to-absorption ratio with respect to the nuclear cross sections, for example, graphite or quartz, can be used as reflectors towards the inside for directions to be blocked. Such reflectors can be used either on the insides or outsides of the moderator or in “shark” geometry between HDPE plates.

We presented examples for a field of view of 180∘. While smaller angles would be desirable for higher spatial resolution, such signals from smaller angles will be more difficult to resolve, due to the reduction of the count rates (smaller β). Although the above-mentioned methodology remains the same, the matrices A and B are no longer symmetric.

This study exclusively considered count rates as the target observation variable. However, for application, count rates are merely a proxy which need to be converted to the actual quantity of interest, namely soil moisture, snow or biomass. The relationships used in such conversion (e.g. ) are not linear; instead, they saturate with increasing hydrogen pools and low count rates. This translates to an increased sensitivity of the target variable (e.g. soil moisture) for these low rates. This behaviour, therefore, amplifies the characteristics demonstrated in this study: considerably poorer applicability of directional CRNS measurements for lower count rates. Combining the presented approach with the one of  would allow the direct quantification of uncertainty for the target variable.