Assessing the feasibility of a directional CRNS-sensor for estimating soil moisture

Cosmic Ray Neutron Sensing (CRNS) is a non-invasive tool for measuring hydrogen pools like soil moisture, snow, or vegetation. The intrinsic integration over a radial hectare-scale footprint is a clear advantage for averaging out small-scale heterogeneity, but on the other hand the data may become hard to interpret in complex terrain with patchy land use. This study presents a directional shielding approach to block neutrons from certain directions and explores its potential to gain a sharper view on the surrounding soil moisture distribution. 5 Using the Mont-Carlo code URANOS, we modelled the effect of additional polyethylene shields on the horizontal field of view and assessed its impact on the epithermal count rate, propagated uncertainties, and aggregation time. The results demonstrate that directional CRNS measurements are strongly dominated by isotropic neutron transport, which dilutes the signal of the targeted direction especially from the far field. For typical count rates of customary CRNS stations, directional shielding of halfspaces could not lead to acceptable precision at a daily time resolution. However, the mere statistical 10 distinction of two rates should be feasible.


Cosmic ray neutron sensing in environmental sciences
The use of CRNS in the environmental sciences has considerably increased in the past decade. Its main application is the measurement of soil water content (Zreda et al., 2008). Such measurement could serve a variety of purposes in both, research 15 and application, for example to close the water balance in atmospheric or hydrological models (Schreiner-McGraw et al., 2016;Dimitrova-Petrova et al., 2020), or to support irrigation management (Li et al., 2019;Franz et al., 2021) or snow cover analysis (Schattan et al., 2017). Furthermore, the use of CRNS to independently estimate biomass or even interception by vegetation has showed potential (Baroni and Oswald, 2015;Baatz et al., 2015).
Instead, directional measurements could be obtained by a partial enclosure ("shielding" or "collimator") of the sensor with a material that absorbs or slows down the incoming neutrons. Neutrons arriving from the shielded sides are therefore less likely 55 to be counted by the detector.
In planetary sciences, the directional measurement of neutrons helped to map water distribution on Moon (Feldman et al., 1999) and Mars (Mitrofanov et al., 2018). These spaceborne directional neutron detector use a directional shielding (collimator) made of PE, allowing only neutrons from the "collimation field of view (FOV)" to enter the detector. The very thin atmosphere on Mars, combined with the comparatively high energy level of the neutrons enabled a favourable "collimation efficiency" being the ratio between counts from the FOV N FOV and the total counts N total . This allowed mapping the Mars surface from approx. 400 km above the ground with a relatively high spatial resolution. For applications on Earth, however, such long-range measurements are unfeasible due to its much denser atmosphere. We will use this quantity η as the directional contribution being collected from the targeted field of view (FOV) as fraction of the total counts detected.

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In environmental sciences, Zreda et al. (2020) have recently patented a downward-looking CRNS sensor. While this approach clearly aims at retrieving the signal from short distances (i.e. the area directly below the downward-looking sensor), it likewise involves directional shielding by blocking neutrons reaching the sensor from other directions.
In 2018, we constructed a directional shielding as an add-on for a commercially available CRNS-sensor, a CRS 2000B (HydroInnova, see Fig. 1). Its purpose is to confine the measurement towards the direction of the area at the unshielded side 70 (FOV) by reducing the contribution from the part of the footprint outside the FOV. The directional shielding was also designed to allow a stepwise turning of the partly-shielded detector by a controlled stepper motor, and thus stepwise change of the FOV. This could be operated to cover the full 2π periphery, in flexible angular sections, and thus allow for a scanning mode producing time-series of count rates for different FOV in the footprint. The extent and thickness of the directional shielding were a compromise in order to keep size and weight in manageable proportions, and also its opening was large enough to obtain 75 still count rates and integration times in the range of standard CRNS applications. The shielding consists of 4.5 cm of layered boron-loaded polyethylene at three sides, top and bottom of the detector to moderate incoming neutrons. The inside of this chamber is additionally coated with 5 mm boron carbide to absorb the remaining thermalized neutrons. This configuration is used in the presented study as an example case for the theoretical development and neutron scattering simulations. Systematic practical tests will then be designed on the findings. However, when dealing with real measured neutron counts N , two sources deteriorating the signal must be considered: (1) statistical error of counts and (2) device noise / intended counts.
(1) Due to the stochastic nature of the counting process, the number of neutrons N (i.e. the realization of a measurement) is subject to a Poisson-type error σ. From the Poisson-characteristics of the counting process follows that

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with σ being the standard deviation of all the values of N we would get when repeating the measurement in the given interval.
(2) A typical CRNS-detector is targeted at counting epithermal neutrons (0.5 eV < E < 10 5 eV, (Köhli et al., 2018)), because these neutrons show sensitivity to the the abundance of hydrogen within the footprint. However, the counting is affected by additional untargeted neutron counts N !epi , e.g. caused by ionizing particles from other energy levels and particles, terrestrial radiation and background radioactivity of detector materials . N !epi is associated with an error of σ !epi .

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Thus, the overall uncertainty σ total when measuring N total is a superposition of the two error sources.
The above-mentioned errors can be characterized by appropriate distributions and their respective parameters: Concerning neutrons counted of other energy levels N !epi , the measured signal contains a fraction of such "bycatch" flux.
For a typical gas detector, results from approx. 20 % thermal and 10 % counts from fast neutrons (Baatz et al., 2015;Köhli et al., 2021) in the measured signal (the precise values depend on ambient hydrogen pools and chosen cutoff thresholds).
Consequently, the actual number of counted neutrons N total is where is the fraction of bycatch neutrons.
Analogously to Eq. (3), the respective additional noise introduced with N !epi is According to our measurements, counts caused by radioisotope contamination of the detector walls are in the order of 0.5 counts h -1 per m 2 of the proportional counter (cf. Dębicki et al. (2011) for comparable tubes). This amounts to approx. 0.5 counts h -1 for a standard CRS1000. Like other counts of unintended ionizing radiation (detector gas, protons and muons, etc.) these can be effectively filtered out by appropriate detector threshold settings (Quaesta Instruments, pers. communication), and are thus ignored here. We also consider temperature effects (reported for neutron monitors, e.g. by Krüger et al., 2008) as negligible, as 110 we are dealing with a pairwise concomitant operation (see Fig. 3), in which such effects would be cancelled out. Noise from electronic components (e.g. from external fields) is in the order of < 10 counts h -1 for typical detectors, and can be effectively eliminated with appropriate threshold settings (Quaesta Instruments, pers. communication). Thus, it is also ignored here.

Specific challenges with directional neutron measurements in CRNS
Generally, CRNS-measurements are affected by the uncertainties described in section 1.3. For directional CRNS-measurements, 115 additional challenges arise: -Incoming cosmogenic fast neutrons are converted to epithermal neutrons by hydrogen pools within the footprint. In this context, we denote the location of this conversion as 'origin' as it constitutes the place for which we infer the information when measuring the neutron. However, most of these epithermal neutrons do not reach the sensor directly.
Instead, neutrons arriving at the sensor have usually experienced multiple elastic collisions, resulting in an irregular 120 trajectory. Consequently, the incidence angle of a neutron reaching the detector is only loosely correlated with its angle of origin. Thus, a directional shielding can only imperfectly filter epithermal neutrons for this direction of origin.
-A directional shielding blocks neutrons reaching the detector from certain angles. This blockage can be achieved by a sufficiently thick blocking material, e.g. layers of High-density polyethylene (HDPE). For practical reasons, however, compromises between blocking properties and size/weight constraints have to be made. Thus, the directional blocking 125 will be imperfect, allowing a certain fraction of neutrons to reach the detector also from the shielded side.
-Blocking neutrons arriving at the detector from certain angles effectively reduces the overall count rate. Consequently, longer integration times are required for obtaining the same number of counts in a given environment.
aggregation time n e u t r o n c o u n t r a t e s i g n a l c o n t r a s t Figure 2. "Tradeoff-triangle" in directional CRNS-measurements as ternary plot: for a given combination of two of these parameters, the third parameter must be adjusted accordingly to obtain the requested accuracy. The inner triangle illustrates the parameter space for an increase in required accuracy. The straight outline here is only chosen for the sake of simplicity and could be curved instead.
Directional neutron sensing aims to determine the neutron flux rate R 1 that is characteristic of the area A 1 in the field of view. The flux rate R 2 of the complementary area A 2 may also be of interest in itself, or effectively only be a factor influencing 130 the measurement. The signal contrast between the two (∆R, see Eq. (19)) is eventually determined by the different hydrogen inventories in the two areas.
Summarizing the general uncertainties (section 1.3) and the limitations specific to directional measurements, we may conceptualize the determination of these pools via directional neutron measurement of R 1 (and R 2 ) as a trade-off problem with the parameters count rate (R 1 ), the signal contrast and the aggregation time (∆t): To obtain measurements for R 1 at a given 135 accuracy, two of these parameters (e.g. count rate and signal contrast) determine the minimum of the third parameter (e.g. aggregation time, see Fig. 2). If we raise the required accuracy, the lower limits of the three parameters need to be increased (grayed-out areas in Fig. 2). The same applies if the relative noise of the measurement increases (e.g. by a higher device noise or a narrower shielding angle).
Systematically exploring these dependencies and limitations is a prerequisite for the design and application of directional 140 CRNS measurements. However, in situ experiments are impracticable, as they require well-defined setups with spatially-known neutron flux rates and multiple configurations of sensors and shieldings. Instead, we propose to use simulations of neutron scattering and neutron detection in order to better understand and quantify potentials and limitations of directional CRNS measurements. Our specific objectives are as follows: 1. Quantify directional specificity and reduction in neutron count of a directional CRNS-sensor setup Each of these points (A-C) can be addressed when looking at either (I) determining count rates at a chosen accuracy or (II) statistically distinguishing two count rates. These are somewhat different objectives with different requirements, which will be further demonstrated in the results section.

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By applying the neutron simulation model URANOS (section 2.1.1), we assess the characteristics of angular shielding (section 2.1.2). As these simulations are computationally demanding, we then generalize the findings in an analytical approach (section 2.2) in order to outline the possible range of application of directional CRNS measurements. The Monte-Carlo tool URANOS (Köhli et al., 2015) is designed specifically for modeling 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 Sato (2015). The URANOS model uses setups with either open domains of at least 600 m or smaller sizes with periodic or reflecting boundary conditions. 165

Scenarios simulated with URANOS
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 H 2 O. The solids are composed of 75 % Vol SiO 2 and 25 % Vol Al 2 O 3 at a compound density of 2.86 g cm -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 g m -3 and soil moisture is set to 10 % Vol . In this study three different simulation setups 170 have been used: 1. In order to assess the effect of adding a further moderator (i.e. 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. 2. Secondly, the to-scale-model of the detector has been used to calculate the response function (Köhli et al., 2018) of each 175 face of the detector by probing the response to a series of monoenergetic neutrons released perpendicular to its face.
3. Thirdly, a virtual detector has been placed in a large-size domain. This entity was equipped with the beforehand found response functions and had, in order to collect enough statistics, a geometry which is slightly larger than the physical instrument. For the purpose of investigating the distance dependent angular response it can be regarded as a suitable approximation. This spherical virtual detector has been placed at a height of 1.75 m with a radius of 1.25 m within a 180 domain size of 800 m × 800 m. Cosmic neutrons are released at a height of 50 m using a circular source with a radius of 400 m. Thermal neutron transport was disabled for reasons of computational speed. This configuration leads to a homogeneous neutron flux distribution within the innermost 400 m × 400 m. Origin data can be retrieved for a radius of approximately 300 m.

Generalisation of interpreting directional measurements 185
Aiming to generalize the the findings of the neutron simulations (sections 3.2 and 3.3), we conceptualize the directional measurement as a linear mixing and analytically express the corresponding propagation of errors for an simplistic geometric setup.

Description of idealized geometric setup
For the sake of simplicity, we use the simplest geometric setup imaginable for directional CRNS-measurements: A plain 190 divided into two homogeneous half-spaces A 1 and A 2 . Each half-space corresponds to a count rate R 1 and R 2 , respectively, when measured far enough from its borders. At the border of both half-spaces, we implement a directional detector. When facing A 1 , it yields the count rate R f1 ; when directed towards A 2 , it registers the count rate R f2 . Both count rates are measured and used in the computations. Consequently, the evaluation employ a FOV of 180 • (π).

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The total count rate R total registered by a sensor results from non-epithermal and epithermal neutrons (see Eq. (4)).
The epithermal counts, in turn, are composed of counts from albedo neutrons having interacted with the hydrogen pools of interest (R alb ) and non-albedo neutrons (R non-alb ) without such interaction. We denote the respective percentage as γ (see

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Please note that γ differs between the omni-directional and the directional detector, which will be shown later. Therefore we distinguish in the following between a γ s and a γ no for the directional and the omni-directional detector, respectively. and Rs2 symbolize the operation with directional shielding (creating a "directional CRNS sensor"). Rf1 and Rf2 denote the count rates the directional sensor registers when placed on the border and pointed towards A1 or A2, respectively. For the directional sensor placed entirely into the interior of one of the half-planes, A 1 or A 2 , the directional shielding causes partial blockage of the neutrons arriving, reducing its count rate by the factor β to R s (see Fig. 4, second bar): 205 Values for γ and β were obtained from neutron simulations as discussed in section 2.1.2, setup 1.

Mixing and un-mixing of signals from directional sensors
When placing the directional sensor exactly between the two half-planes A 1 and A 2 , 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. (1)), i.e. ratio between counts from the desired angle and total 210 counts (see Fig. 4, third bar): While the directional detector may be oriented towards the left half plane A 1 or the right half plane A 2 , there is no fundamental difference between the two and we exemplify the next step with the one oriented towards A 1 and its count rate 215 R f1,alb .
Substituting Eq. (8a) with the Eqs. (6), (7) and then (4) yields Considering both rates R f1,alb and R f2,alb simultaneously in a vector R f,alb , we may write in matrix notation:

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where A is a symmetric matrix containing the coefficients for the mixing. For the non-albedo neutrons R f,non-alb , the mixing is simply the average of the corresponding rates where J is the matrix of ones.
Applying Eq. (4) for the non-epithermal counts leads to Adding Eqs. (10), (11), and (12) yields the total count rates of the directional detector R f,total with B being a matrix summarizing all the operations.
For reconstructing the unshielded count rates in the half-spaces (R total ) from the two directional measurements (R f1 and R f2 ), 230 we use the inverse operation: where The subscript "r" in Eq. (14) denotes the reconstructed rates for estimating the true (and unknown) ones in the half-spaces 235 (R 1 and R 2 ).

Description of error propagation
As described in section 1.3, 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: As the considered errors in the two directional counts (N f1 and N f2 ) are independent, the superposition of these errors can be described based on Gaussian error propagation: where F denotes a function combining the contributions of x and y. In our case, F would be the reconstruction of the count 245 rates from the directional counts (i.e. Eq. (14)).
When reconstructing the rates in the half-spaces R 1 and R 2 as described in Eq. (14) and propagating the errors in both directional counts (see Eq. (17)), and performing substitutions (eqs. (4) and (13)), the error in the reconstructed epithermal counts is: Both aims can be pursued jointly or separately, however, achieving the 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 260 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 percent-points in volumetric water content for the conversion used by Fersch et al. (2020). For distinguishing two count rates (aim II), we propose they are significantly different, if their difference -regarding 265 the associated uncertainties -is statistically different from 0 with a chosen p-value (0.05 in our case).

Selected example values chosen in the analysis
Applying Eq. (18), e.g. using provided R-code, the described analysis can be performed for any combination of parameters of interest, i.e. count rates (R 1 , R 2 ) or their respective contrast (∆R), temporal resolution (∆t), directional contribution (η) and the fraction of bycatch neutrons ( ). However, for illustrative purposes, in the Results section (3.4) we provide some example 270 plots trying to capture the typical ranges of the parameters involved. This selection was guided by a range of diverse settings found in the literature summarized in Tab. 2.

Fraction of non-epithermal counts ( )
In the examples, we illustrate the situation for an "bestcase" scenario assuming a low fraction of non-epithermal counts ( = 0.1) and for a "worstcase" setting ( = 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. (1)). In the displayed examples we only consider symmetrical half-planes (see Fig. 3 for the "bestcase" scenario with the perfect shielding (η = 0.72) and the "worstcase" implementation of the actual shielding (η = 0.61) as resulted from the neutron simulations (see section 3.3, Tab. 3).

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Fraction of non-albedo neutrons in the directional detector (γ s ) 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 section 3.3). For the "bestcase" scenario, we choose a low value (γ s = 0.2) corresponding to very dry conditions; the "worstcase" implementation (γ s = 0.31) mimics a very wet environment (see section 3.3 and Tab. 3).

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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. (7)). For the "bestcase" scenario, we choose a high value (β = 0.4) corresponding to very wet conditions; the "worstcase" implementation (β = 0.3) mimics a very dry environment (see section 3.3, Tab. 4 ).
Count rates depend on site conditions (i.e. incoming neutrons and hydrogen pools) and detector sensitivity. We selected values of 500, 2000, 8000, 40000, and 150000,counts h -1 as example values for the count rates registered at the detectors (R total , see (4)).

Contrast in count rates of the half-planes (∆R)
We denote the difference in the count rates in the two half-planes with ∆R, expressed as their difference relative to the lower 295 of the two values: 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. For a subsequent evaluation of the feasibility for a specific case, we use the setting listed in line 2 of Tab. 2, i.e. the measurement using the prototype directional detector described in section 1.2, i.e. R 1 =2100 counts h -1 , ∆R=0.4.

Results and Discussion
3.1 Neutron simulation 310

Effect of shielding on count rates
Increasing the thickness of the polyethylene shielding does not only reduce 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 315 CRNS probe. Tab. 3 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. Table 3. Effect of directional shielding on neutron counts. Simulated counts for different configurations (volumetric soil moisture ("theta") and shieldings ("no shield "/"directional")). The counts are differentiated in those with / without surface interaction ("albedo"/"non-albedo").
configuration counts fraction of "total" fraction of "no shield" 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 325 specificity is much more pronounced for neutrons with origins in the near-range.
From Fig. 5, which shows the case of a detector in the "bestcase" 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 330 being released from the soil in the direct vicinity of the instrument (light blue line), more distant origins tend to incur much more directional changes of the neutron. This leads to a flattened angular distribution (dark blue line).

Shielding effect
In the "worstcase" 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 . Neutron origin angles for the designed directional detector (shielded to all sides except the front which points to −π/2). The range groups are defined by the distance of neutron origin to the detector, except 'Last interaction', which refers to the last scattering before detection, typically in air. The distribution 'not detected' refers to the total flux through the instrument, please note the different scale.
ideally shielded detector in Fig. 5, 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. (1)). We chose as representative target FOVs 90 • (π/2) and 180 • (π). The results are summarized in Tab. 4. 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 340 is set to a full half-space 60 % of the signal is representative for 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.

Feasibility of directional CRNS-measurements
Based on the presented findings of the neutron simulations, the following analysis has been made to assess the feasibility of 345 the directional detector.  Figure 7. Example for 95 %-confidence intervals for the count rates obtained within the half-spaces (R1,total = 2100 counts h -1 , R2,total = 2520 counts h -1 , "bestcase scenario", moderate contrast ∆R = 0.2), by the directional CRNS-sensors (Rf1,total, Rf2,total) and the rates reconstructed from the directional sensors (R1r,total, R2r, total). The vertical arrows indicate the minimum required aggregation time to confine the CI within 5 % of the true value (∆t determ min ). Figure 7 illustrates an example simulation (R 1,total = 2100 counts h -1 , R 2,total = 2520 counts h -1 , = 0.3). It clearly shows how the rates of the directional sensor ( R f1,total , R f2,total ) are considerably lower due to the blocking effect of the directional shielding. Concerning the determination of rates, the confidence intervals narrow with increasing aggregation time. Notably, the CI for the reconstructed rates R 1r,total and R 2r,total is always considerably wider than their directly measured counterparts 350 R 1,total and R 2,total . So, how much aggregation time is required to determine a count rate precisely? We define ∆t determ min (arrows in Figure 7) as the time when the CI gets smaller than the chosen precision of 5 % of the true value, i.e. 5 % relative precision.
We use this time as an indicator for the time beyond which the determination of a count rate becomes reasonably accurate. As our focus is on the rates reconstructed from the directional measurements, the following statements refer to R 1r, total , the lower of the two reconstructed count rates, as it has the larger value of ∆t determ min (green arrow in Figure 8 at 36 h).

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In the context of distinguishing the two count rates, Figure 8 shows that the p-value for comparing two count rates decreases with increasing aggregation time. As one would expect, a statistically significant difference between two rates is more discernible with longer aggregation times. For the directly-measured rates R 1,total and R 2,total , this distinction is possible even for R 1, total vs. R 2, total R 1r, total vs. R 2r, total R f1, total vs. R f2, total p thresh = 0. 05 ∆t | p < p thresh the lowest aggregation times, while it takes longer for the reconstructed rates R 1r,total and R 2r, total . (Somewhat surprisingly, distinguishing between R f1,total and R f2,total is apparently even harder. The reconstruction of these rates using Eq. (14) evidently 360 increases their signal-to-noise ratio.) How much aggregation time do we need to distinguish two count rates statistically? We choose the time ∆t 05 min as an indicator for the time beyond which the difference between two rates R 1r and R 2r is significant at the 5 %-level (brown arrow in Figure 8).
So while an aggregation time of at least ∆t determ min is needed to pinpoint one reconstructed value, we require ∆t 05 min to distinguish two rates. These two objectives are different, and we will show that both indicators differ accordingly. 3.4.1 A: What temporal resolution can be obtained? Figure 9 confirms that for higher count rates R 1 and R 2 , 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 R 1 and R 2 is of relatively small effect for higher count rates, but makes a difference for lower ones. For these, somewhat counter-intuitively, 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. (14)), they deteriorate its robust reconstruction.
Consequently, a higher contrast in the rates aggravates the reconstruction of the lower one.
Conversely, Figure 10 demonstrates that the statistical difference between R 1r and R 2r 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   Figure 11. Maximum possible contrast ∆R in reconstructing the count rates R1r from the two directional measurements with a CI smaller than 5 % of the true value.
opposite effect, depending on whether we look at the precise determination of R 1r (benefits from low contrasts) or the statistical 375 distinction between the two rates R 1r and R 2r (benefits from high contrasts). This distinction is apparently much more feasible within reasonable aggregation times: in the "bestcase" scenario, hourly resolution can be achieved from count rates of above approx. 3000 counts h -1 even for low contrasts. The "worstcase" scenario increases the aggregation times by about factor 7.
For our example case, we can conclude that the precise determination of the count rates is hardly feasible even for the "bestcase" scenario. Aggregation times exceeding at least 36 h are beyond the typical requirements in applications. Aggregation 380 times of 24 h and less can only be achieved with count rates of more than 3100 / 20700 counts h -1 for the "bestcase"/"worstcase" scenario. Distinguishing the rates in the two half planes, however, could be possible for aggregation times in the order of hours with even higher potential for stronger contrasts.
3.4.2 B: What spatial contrast in the count rates can be resolved? Figure 11 again illustrates the phenomenon mentioned in the previous section: With higher contrast in the signal, estimating 385 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 (∆R = 1.4) in daily resolution, count rates R 1 > 4000 counts h -1 are required for the "bestcase" scenario. For the "worstcase" scenario, R 1 must be larger than 40000 counts h -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. 12). According to the "worstcase" scenario, even relatively low contrasts (∆R=0.2) can be detected 390 with the lowest considered count rates, if aggregation times are slightly higher than 24 h.  Figure 12. Minimum contrast ∆R required to statistically separate the reconstructed count rates R1r and R2r from the two directional measurements with p-value < 0.05.
For our example case, we have already shown that the determination of the count rates is hardly feasible even for the "bestcase" scenario. However, with aggregation times of 24 h, distinguishing rates with contrasts lower as 0.2 could be possible even for the "worstcase" scenario.
3.4.3 C: What count rates are required to yield robust estimates? 395 Figure 13 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 "bestcase" scenario at the daily resolution, the minimum count rate must be well above 3000 counts h -1 for contrasts lower than 0.2. For the "worstcase" scenario, this value increases to over 20800 counts h -1 . As noted before, higher contrasts require even higher count rates (i.e., > 4500 / 31000 counts h -1 for the cases above) or longer aggregation times.

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Concerning the statistical discernability of R 1r and R 2r , Figure 14 suggests that already low count rates allow the two reconstructed count rates to be distinguished. For the "worstcase" scenario at the aggregation interval of 1 h, count rates of at least 15600 counts h -1 already allow resolving contrasts as low as 0.2. For the higher contrasts (∆R = 1.4), count rates of roughly 490 counts h -1 would suffice.
Thus, for our example case, we would require at least 3100 counts h -1 to successfully reconstruct both rates at 24-h-405 resolution. The current count rate of 2100 counts h -1 calls for aggregation times of at least 36 h. Merely distinguishing the rates R 1r and R 2r is possible with the given count rate at 1-h-aggregation time.

Limitations and Outlook
-This analysis bases on the geometries of a specific directional detector. This prototype was designed with pragmatic considerations. Other designs (e.g. larger planar shieldings) may provide superior characteristics, namely η, which could 410 be assessed by further neutron simulations.
- Fig. 3 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. actual "scanning" as presented in section 1.2) could be used. While this eliminates the issue of imperfect calibration, it implies that all the computed 415 aggregation times must be doubled, as the single sensor can cover each direction only half of the time.
-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 420 rates.
-The directional contributions obtained from the 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 (deterio-425 rating 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 (Schrön et al., 2018a). 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 430 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 R 1 and R 2 two additional concurrent effects will occur: If R 1 increases (e.g. less moisture in A 1 ), we would also expect more A 2 neutrons to be scattered back to the detector from A 1 , because thermalization is reduced there. This deteriorates the directional contribution η for the reconstruction of R r1 as we are getting more 435 neutrons from the "wrong" direction. Conversely, the increased R 1 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 R r1 . Further neutron simulations are required to clarify which of these effects dominate and if they would notably influence η.
lution, such signals from smaller angles will be more difficult to resolve, due to the reduction of the count rates (smaller 440 β). Although the abovementioned methodology remains the same, the matrices A and B are no longer symmetric.
-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 445 rotating sensor platform, thus calling for two complementary, non-rotating sensors instead.
-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. Desilets et al. (2010)) 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 450 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 Jakobi et al. (2020) would allow the direct quantification of uncertainty for the target variable.
-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 455 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 hardly feasible, a freely rotating setup poses even 460 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 our calculations, for the "bestcase"/"worstcase" scenario we use fixed values for the fraction of "bycatch" count ( ) at 10 %/30 %. This assumption implies that these adjacent energy levels (thermal and fast neutrons) show a similar 465 sensitivity to hydrogen. In reality, this sensitivity is considerably smaller and different, but has been exploited in some studies (e.g. Baatz et al. (2015); Tian et al. (2016)), effectively reducing uncertainties by increasing the rate of usable counts R epi . 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 , 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.
-As more angle-specific option for the shielding, micro-channel plates have shown favourable directional characteristics (Tremsin et al.). However, their limitation to thermal neutrons, considerable costs and the remaining problem of non-geometric transport makes them unfeasible for CRNS.

475
-Materials with low hydrogen content and a high scattering/absorption ratio with respect to the nuclear cross sections, for example graphite or quartz, can be used as reflectors towards the inside for directions, which are not of interest. Such reflectors can be used either on the insides or outsides of the moderator or in 'shark' geometry between HDPE plates.

Conclusions
This study combined neutron simulations with an analytical assessment of the directional specificity of epithermal neutrons, 480 and the potential for directional measurements in environmental monitoring.
The neutron simulations revealed the relatively low correspondence of incidence angle and angle to origin of the neutrons arriving at a CRNS-detector. This is a direct effect of the non-geometrical (i.e. not direct) path of the neutrons, being subject to multiple collisions. Consequently, the correlation of these angles gets lower with increasing distance of the origin (i.e. the location of its conversion to epithermal). Consequently, blocking certain incidence angles of a sensor only yields a limited Based on these findings, the subsequent analytical analysis focused on the feasibility of determining and statistically distinguishing the count rates from two adjacent half-spaces by reconstruction from measurements of directional sensors. While the 490 former benefits from a low contrast in count rates, the latter is aggravated by it. Both aspects profit from high count rates and longer aggregation intervals.
With the analyzed setup and reasonable count rates, the accurate reconstruction of the two count rates is hardly feasible with less than 24 h of aggregation time, given detectors with conventional sensitivity. Thus, it seems of little value in environments where variability needs to be resolved at this time scale. While a substantial increase in detector sensitivity might address this 495 issue, such an increase typically comes with higher costs and much larger detector sizes and hence an unfavorable increase in the dimensions of the required shielding.
The mere distinction of two rates, however, is more feasible and, even for moderate count rates and contrasts, perceivable at a resolution of a few hours. The effort of directional measurements for the mere purpose of distinction might appear somewhat incommensurate. Yet, the gain in information might be very relevant from a hydrological point of view, e.g. at the borders 500 between grassland and forest which might experience a reversal of horizontal sol moisture gradients in periods of drying.