Signal contribution of distant areas to cosmic-ray neutron sensors – implications on footprint and sensitivity

. This paper presents a new theoretical concept (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) approach (cid:58) to estimate the contribution of distant areas to the measurement signal of cosmic-ray neutron detectors for snow and soil moisture monitoring. The algorithm is based on the local neutron production and their transport mechanism, given by the neutron-moisture relationship and the radial intensity function, respectively. The purely analytical approach has been validated with physics-based neutron transport simulations for heterogeneous soil moisture 5 patterns, exemplary landscape features, and remote fields at a distance. We found that the method provides good approximations of simulated signal contributions in patchy soils with typical deviations of less than 1 %. Moreover, implications of this concept have been investigated for the neutron-moisture relationship, where the signal contribution of an area has the potential to explain deviating shapes of this curve that are often reported in literature. Finally, the concept (cid:58)(cid:58)(cid:58)(cid:58)(cid:58)(cid:58) method has been used to develop a new practical footprint definition to express whether or not a distant area’s soil moisture change is actually detectable in 10 terms of measurement precision. The presented concepts answer long lasting questions about the influence of distant landscape structures in the integral footprint of the sensor without the need for computationally expensive simulations. The new insights are highly relevant to support signal interpretation, data harmonization, and sensor calibration, and will be particularly useful for sensors positioned in complex terrain or on agriculturally managed sites.


Introduction
Cosmic-Ray Neutron Sensing (CRNS) is an established measurement technique for water content in soils and snow (Andreasen et al., 2017).The high integration depth and the large measurement footprint have been shown to provide an important advantage for field-scale applications compared to conventional point-scale sensors.However, the intrinsic integration over the whole footprint volume conceals the individual contributions of different patches and may result in biased observations (Franz et al., 2013;Schrön et al., 2018;Schattan et al., 2019).
The footprint has been initially characterized by its radius of around 300 m by Zreda et al. (2008) and Desilets and Zreda (2013) without significant dependency on soil moisture or air humidity.Later, Köhli et al. (2015) revisited the physical assumptions of the underlying neutron simulations and proposed a moisture-dependent footprint radius in the range from 130 m to 240 m.Besides the epithermal neutron transport, also thermal neutron footprints were investigated by Jakobi et al. (2021).
These studies take into account the high complexity of neutron transport physics, which usually can only be investigated with computationally expensive Monte-Carlo simulations.The accepted definition of the epithermal footprint radius, R 86 , covers the 1 − e −2 ≈ 86 % quantile of detected neutrons.This measure was introduced by Desilets and Zreda (2013) and has been inherited by Köhli et al. (2015) in order to maintain consistency.However, the definition involves four problems :::::::::: problematic :::::: aspects: (i) The radial intensity function, W r (h, θ), does not follow a simple exponential shape (Köhli et al., 2015;Schrön et al., 2017).Therefore, the 1 − e −2 limit may be misleading when it is used to draw conclusions about the intensity-radius relationship elsewhere.(ii) High-quantile values for strongly non-linear functions may overestimate the long-range influence of neutrons, regardless of how often and where they have probed the soil.(iii) The fact that the definition of the footprint has been developed for homogeneous situations increases the uncertainty and the interpretation of more heterogeneous and more complex terrain.And (iv), the definition hardly allows to investigate problems and questions that often arise during practical applications: Is the detector sensitive to remote soil moisture changes?Does a certain patch of the area influence the detector signal?By how much?
As a standard solution for such questions, neutron transport physics-based Monte Carlo codes could be employed with detailed modeling of the local conditions (as has been done by, e.g., Franz et al., 2013;Köhli et al., 2015;Schrön et al., 2018;Schattan et al., 2017;Li et al., 2019, among others).However, this technique is impractical for quick assessments and mostly limited to scientific applications.
In the past, spatially (and temporarily :::::::: temporally) variable factors within the footprint influencing the neutron signal have often been identified as the source of unexplained features in the data.These discoveries sometimes boosted scientific insights on neutron transport, and even led to more reliable hydrological data (see, e.g., Bogena et al., 2013;Schrön et al., 2017Schrön et al., , 2018;;Schattan et al., 2019;Rasche et al., 2021).However, at most heterogeneous sites, CRNS calibration and validation remains a challenge, since the influence of the differing structures or patches in the footprint to the signal is usually not known (Coopersmith et al., 2014;Lv et al., 2014;Iwema et al., 2015a;Franz et al., 2016;Heistermann et al., 2021a).For this reason, many authors reported differing shapes of the neutron-moisture curve and conducted site-specific empirical re-parameterizations to fit their data (Rivera Villarreyes et al., 2011;Lv et al., 2014;Iwema et al., 2015b;Heidbüchel et al., 2016).Others developed directional sensors to focus the measurement only on specific parts of the landscape (Francke et al., 2022), which remains a technological challenge on its own.
One way to approach the estimation of signal contributions of different areas in the footprint is to use the radial intensity function W r .First attempts to realize this idea have been performed by Schrön et al. (2017), who improved the sensor calibration by applying different weights to areas depending on size, distance, and landuse class, and also by Schrön et al. (2018), who excluded the contribution of a concrete area around a grass land site in order to improve reliability of soil moisture dynamics measured by stationary CRNS.
In the present study we aim at generalizing this concept for typical combinations of heterogeneous land use and soil moisture patterns.Our hypothesis is that the contribution to the detector signal of various complex areas in the footprint can be estimated analytically based on the existing theories about neutron production and transport.The first section will describe the proposed concept ::::::: approach : and discuss its potential limitations.Then, the concept :::::: method will be evaluated by dedicated neutron transport simulations for various scenarios of different soil moisture patterns, land-use types, and geometries.We further aim at exploring two applications of this concept, first, to assess its explanatory power for the shape of the neutron-soil moisture relationship.And second, to provide a more practical footprint definition expressing whether or not a distant area's soil moisture change (e.g., by irrigation or rainwater interception, or faster drainage) is actually visible to the neutron signal in terms of measurement precision.

The radial intensity function
The sensitivity of a central detector to an infinitesimal ring at distance r has been described by Köhli et al. (2015) and refined by Schrön et al. (2017) as: which is a combination of two exponential functions with factors and slopes (F 1...4 (h, θ, P, H veg )) that represent the complex nature of neutron transport in humid ::::::::::: homogeneous : environments.This radial intensity function W r (see Fig. 1a) depicts the number of detected neutrons that originated in the soil at the distance r :: (in ::: m) under certain homogeneous conditions of air humidity h, (soil) water equivalent θ, air pressure P , and vegetation height H veg .It can also be expressed as W r * (h, θ) with r * = r(P, H veg ) being scaled by air pressure and vegetation influence (see Schrön et al., 2017, for the details).We use this simplified formulation in the following, while results can be easily transferred to other pressure and vegetation conditions by rescaling r as mentioned.
An alternative parameterization, W * r ≈ W r (h, θ), has been proposed by Schrön et al. (2017) as an approximation for average humidity and soil moisture conditions: This approximation can be evaluated in a computationally more efficient way, as it does not depend on humidity and soil moisture, but at the same time it is less accurate towards the extreme ends of dry or wet conditions.
The integral of W r (h, θ) over all radii r represents the total number of detected neutrons, N : In other words, the detectable neutron intensity at the center of the radial footprint is the sum of all the ring intensities, W r , across in the whole domain Ω.
Based on this definition, Köhli et al. (2015) derived the hitherto accepted CRNS footprint radius, R 86 , as the distance within which 86 % of all the detected neutrons originated:

The concept of signal contributions from sub-domain areas
Let A i ∈ Ω be a set of sub-domain areas with water content θ i , constituting the whole domain Ω (see Fig. 2 for an exemplary illustration).We propose that the total measured neutron intensity at the center, or effective neutron intensity N , is the sum of all the neutrons which were generated in A i , weighted by their ability to reach the sensor, i.e.,

Signal from area
110 This quantity can be expressed as the product of the locally generated neutrons, N i = N (θ i ), and the radial intensity weight of its area, w i .The total signal then is: , where In a homogeneous domain, where N i = N j ∀ i, j, the integral weight w i of an arbitrary subset area A i directly represents the area's contribution to the measured neutron signal, only depending on size and distance.In an inhomogeneous scenario, 115 the contribution depends also on the local count rate N i .For example, the effective signal of a symmetrical domain containing two identical half-spaces with a sensor in the center would be an equal combination of the individual intensities, N = 0.5 N 1 + 0.5 N 2 .

Potential limitations and remarks
The simplified concept ::::::: approach : cannot claim to be able to simulate the complex physics of neutron transport for every possible scenario.Just from the way it is formulated, some potential limitations of this approach can be already expected.The performance tests in the present study aim at challenging some of these objections with synthetic examples and evaluations by Monte Carlo simulations.
A direct limitation is that W r has been initially defined as a radially symmetric function, assuming equal contribution at distances r in all directions.However, most heterogeneous regions are not radially structured, such that highly variable soil moisture patches would lead to asymmetric weights in different directions, giving the corresponding footprint radius an amoeba-like shape (see, e.g., Köhli et al., 2015;Schattan et al., 2019, Figs. 9 and 9, respectively).
The W r function has also been derived for homogeneous conditions, and thus sharp borders of soil moisture patterns may not be resolved adequately.This is particularly true in regions of high sensitivity, such as the first few meters below and around the detector.These cases could lead to road-effect-like biases (Schrön et al., 2018) :::::::::::::::::::::: (Schrön et al., 2017(Schrön et al., , 2018) ) and should be avoided -while they rarely occur -in realistic applications.
Moreover, the actually detected neutron flux at the center not only depends on the neutron response of all the individual fields, but also on secondary interactions with water and soil between the detector and the remote field.These intermediate fields may influence the neutron's travel path and moderation probability since neutrons typically undergo several interactions with the soil on their way to the detector (Köhli et al., 2015).
The concept ::::::: approach : also treats each area individually and cannot reproduce spatial interaction effects.They can occur when neutrons generated in an area typically diffuse to nearby areas, influencing their apparent local neutron intensity (see, e.g., Schrön et al., 2018).For this reason, scenarios in this study will exhibit sufficient space between distinct areas such that their neutron contribution can be assessed individually.
When it comes to evaluation of the presented concept ::::::: approach : with measurement data, it should be noted that results obtained from W r -based approaches as well as from URANOS particle origins only represent detected neutrons with preceding soil contact, while a considerable fraction of the CRNS-measured signal is direct radiation from incoming neutrons (see Schrön et al., 2016, Fig. 3).This additonal :::::::: additional : signal component is usually rather constant, but could lead to a slightly lower magnitude of signal contributions in real-world examples.
It is briefly noted that similar analysis could also be conducted for vertical footprints, i.e., the sensor's penetration depth.
For example, the question, at which depth groundwater rise is visible to the CRNS, could be answered by similar methods as described above, using the depth-weighting function W d instead of W r (Schrön et al., 2017).However, neutrons undergo much more interactions in the soil on their way to the detector, in strong contrast to horizontal transport, such that we suspect this endeavour to be less promising, especially for strong vertical soil moisture profiles.
3 Results and Discussion

Heterogeneous soil moisture patterns
In order to provide a reliable representation of the average soil moisture in an heterogeneous domain, it is necessary to consider the specific soil moisture conditions of each individual area.We challenge the presented concept :::::::: approach with complex soil moisture patterns that are designed to cover difficult aspects of neutron transport for the test.The results of this theoretical calculation were compared in the last step with the results of dedicated URANOS simulations.
Figs. 3c and 4c show the simulated relative contribution of each area to the overall signal, where red crosses indicate the origin of neutron particles that have later hit the virtual detector.In most areas, the spatial contributions are in very good agreement with the theoretical estimations.In a few cases, the contribution of remote dry areas are underestimated which may be due to the uniform soil moisture condition of θ ≈ 20 vol.% assumed for anchoring the radial intensity function W r (h, θ).While 230 the long-range transport is slightly underestimated in this case, typical scenarios are probably not as complex such that a better choice of W r could be made.Moreover, the contribution of wet fields that are arranged behind dry areas are slightly overestimated by the analytical approach, which is an effect of intermediate scattering of those neutrons on their way to the detector.While this effect is replicated in the Monte Carlo simulation, it cannot be resolved by the analytical approach.
-400-300-200-100 0 100 200 300 400  Overall, the concept :::::: method : of estimating the signal contributions of different areas in and beyond the CRNS footprint shows a good agreement and might be helpful for the assessment of measurement sites without rigorous neutron transport modeling.Although higher-order corrections for interactions of the neutrons across different fields cannot be resolved with this analytical approach, results in Figs.3-4 indicate good overall accuracy of the estimated contributions.Where precision matters, and under highly heterogeneous conditions (e.g., patchy snow cover), more accurate estimations may be tackled with Monte Carlo simulations.

Complex land-use features
Many field sites are not only characterized by heterogeneous soil moisture patterns, but also exhibit complex land-use types, such as tree groups, water bodies, and even urban structures (Lv et al., 2014;Iwema et al., 2015a;Schrön et al., 2018;Fersch et al., 2020).A general view on such conditions will be provided with the following example.
This exemplary scenario consists of four regions of equal area and distance from the detector, and a fifth reference region with the same soil moisture content as the remaining field (θ = 20 vol.%).The five regions span over a distance from 50 m to 100 m and over a 45°arc, while they are separated by a 25°arc space.The land use features represented in this example are: soil (reference area, θ 1 = 20 vol.%), concrete pavement (equivalent to θ 2 = 10 vol.%), a forest (θ 3 = 30 vol.% plus 20 m tree gas), a water body (θ 4 → ∞), and a building-like structure (θ 5 = 10 vol.% plus 20 m house gas to mimic the height of the building).
Results shown in Fig. 5 indicate an estimated contribution of the reference area of c 1 ≈ 2.1 % (panel b), which is well matched by the simulation, 2.3 ± 0.3 % (panel c).The building and the concrete pavement exhibit the same dry material composition in the ground and thus lead to similar estimated contributions, c 2 = c 5 = 2.5 %.In contrast, the simulation shows much higher contribution of the building, 3.2 ± 0.4 %, since it accounts also for the above-ground house material.The same holds for the forest area, c 3 = 1.9 %< 2.7 ± 0.3 %.As expected, the water body shows the lowest contribution, c 4 = 0.98 %< 0.75 ± 0.17 %.
In general, the analytical approach seems to provide good performance throughout different land use regimes, with minor deviations at the pure-water end of the soil moisture spectrum.Significant limitations of the purely ground-driven approach are evident for above-ground objects, such as forests or buildings.In these cases, however, the concept :::::: method : might still be applicable by defining a "soil moisture equivalent" of those land use types.For example, setting θ * 3 = 9 vol.% and θ * 5 = 4 vol.% would lead to a perfect match with the simulations for the forest and the building, respectively.While these values certainly depend on the specific material composition and distance of the actual building or forest, future studies may show whether the contribution of these very special land use types can be generalized. -
The application of such a correction will be explained by the following theoretical example.Consider an area of interest A 1 ∈ Ω, for which the soil moisture dynamics are to be measured by a neutron detector.A second area, A 2 = Ω \ A 1 , which does not respond to soil moisture changes, e.g., a concrete pavement, a building, a water body, a swamp, or rocky terrain lies within the sensor footprint.This area will generate a constant, invariant stream of neutrons, N (θ 2 = const.),and thus dampen the total effective neutron measurement as a function of θ 1 .
In order to correct for the damping effect, we propose to rescale the amplitude of the neutron counts by the signal contribution c 1 from area A 1 , because only this fraction will be able to stimulate neutron dynamics: where ) is a stationary reference offset (i.e., an invariant neutron stream from area A 2 ) around which the amplitude will be stretched in order to make sure that the correction sustains identity for θ 1 = θ 2 .If θ 2 is not known, the mean observed neutron counts could be a first-order approximation, as has been done in an urban terrain by (Schrön et al., 2018).
The concept ::::::: approach is tested in an exemplary scenario with a central area of 20 m radius and variable soil moisture, θ 1 , surrounded by an area of constant soil moisture of θ 2 = 10 vol.% (Fig. 6a).The signal contribution of the inner area varies from 30 % to 41 % depending on θ 1 , with a mean of c 1 ≈ 33 % (Eq.7). Figure 6b shows the simulated effective neutron intensity of the central detector as a function of θ 1 (blue points), the standard N (θ) relationship (black line, see also Fig. 1b), and the function N resc which was rescaled by the factor c 1 (Eq.12) to resemble the damping effect (blue dashed line).The shaded area represents the mentioned range of c 1 and demonstrates the robustness of the approach if the signal contribution cannot be determined precisely.
In summary, this application of the signal contribution theory offers an explanation for site-specific parameterizations of the N (θ) relationship, which could be tested with existing and future CRNS data sets.
-   1b).However, the function can be rescaled using c1 and Eq. ( 12) to resemble the damping effect (blue dashed line).The shaded area represents the range of c1 (30-41 %) depending on soil moisture.

Remote field at a distance
In this section, the signal estimation concept :::::::: approach is challenged with a more simplistic scenario, but this time without radial symmetry, in order to represent typical land use geometries.The investigated domain is split in two half spaces with different soil moisture, like two agricultural fields neighbouring each other or like partly irrigated land.
In a first exemplary scenario, the soil moisture of the two fields is set to θ 1 = 10 vol.% and θ 2 = 30 vol.%.According to Eqs. ( 6)-( 7), the dry area contributes c 1 ≈ 58 % and the wet area c 2 ≈ 42 % of the total neutron count rate, while the apparent soil moisture average is θ ≈ 15.2 vol.% (Eq.10).This value is substantially lower than the naive mean, θ < 20.0 vol.% due to the non-linearity of the θ(N ) relationship (Fig. 1b).URANOS simulation results confirm this approach with θsim ≈ 14.8 vol.%.
In order to extend the analysis to arbitrary domain splits, we now consider a scenario that consists of two areas split at the distance R from the center, where A 1 (x < R) is the field around the central detector and A 2 (x > R) is the remote field.The interface between the two fields is a straight line orthogonal to the x-axis as illustrated in Fig. 7.The total neutron count rate can be described following Eq.( 6): where The weight w in the Cartesian geometry is expressed in radial coordinates to avoid any corner effects, since the nature of neutron transport and detection usually follows radial symmetry.The term π −1 arccos R/r represents the length of an arc within the circle area constrained by x > R (see also Fig. 8a).It can be derived from the opening angle of the sector, cos α = R/r, where 2α is the same fraction of 2π as is the arc of the total circumference 2π r.
In general, a purely radial geometry, where soil moisture changes in the whole region defined by r > R, would be a more simple scenario to calculate.However, we consider these radial field arrangements to be a rather rarely encountered situation compared to the much more typical straight field geometries.In cases where circular fields and the corresponding soil moisture differences are relevant (e.g., for Pivot irrigation Finkenbiner et al., 2019), the integrand can simply be solved without the arccos R/r term.
In the homogeneous case with soil moisture θ 1 = θ 2 , the apparent average soil moisture also equals θ = θ 1 and the total neutron count rate results to N (θ 1 , θ 2 ) = N (θ 1 ).If the remote area changes from θ 1 to θ 2 , however, the weighting function of the total domain changes slightly.The influence of this change on W r (h, θ 1 → θ) is usually marginal for small changes or high distances, but the calculation could be re-iterated along updates of θ if precision matters.
Interesting to note that the effectively measured soil moisture in this example is θ = 5.1 vol.%.Although the remote field is very close to the outer margins of the radial footprint R 86 , it still contributes 3 % to the total neutron intensity and thereby increases the CRNS-average soil moisture by 0.1 volumetric percent.The dry-bias can be explained by the large distance of the wetter field (which is even larger than R at all but one point), as well as by the strong non-linearity of W r towards lower r (Fig. 1a), and the non-linearity of the θ(N ) relationship (Fig. 1b).
In general, an exact understanding of the weighting function W r (h, θ) plays an essential role for precise estimation of farfield influences.This sensitivity can be illustrated using the approximation W * r (Eq.( 2)).It is usually less accurate due to the missing dependency on air humidity and soil moisture.Considering an exemplary scenario with θ 1 = 5 vol.% and a field at R = 57 m distance with θ 2 = 10 vol.%, the "actual" signal contribution of the remote field is c 2 = 15.3 % calculated by URANOS.The accurate theoretical estimation using W r (h, θ) yields c 2 = 14.7 %, while the simplified W * r approach only yields 11.9 %.

A practical footprint definition based on field distance and detector sensitivity
This section proposes an alternative and : a : more practical definition of the footprint size , by answering the :: for :::::::::: rectangular :::: field As : it has been shown in the previous sections, the intensity distribution around the sensor, W r (h, θ), weights different regions of the footprint in a highly inequal way.Therefore, a new approach is suggested to interpret the footprint as the distance R, to which a remote change of soil moisture is still visible in the detector signal.
In order to assess this sensitivity, we reject all neutron intensity changes below a certain significance level of the sensor.
The relative stochastic precision of a neutron detector, σ N = 1/ √ N , highly depends on the count rate N (Zreda et al., 2012;Weimar et al., 2020).It is a function of detector volume, its efficiency, atmospheric conditions, soil moisture, and temporal aggregation (see e.g., the concept of N 0,base in Schrön et al., 2021).
For average conditions, usual CRNS detectors with an average count rate of N ≈ 1000 cph can achieve a precision of σ N = 3.2 % per hour, or σ N / √ 24 = 0.6 % per day.With regards to increasingly improving detectors and to the generally relevant time scales of 6-12 hours, we will condition our analysis on the σ N = 1 % uncertainty limit.I.e., we will consider CRNS detectors sensitive to a certain environmental change if the induced relative change of the count rate exceeds 1 %.Note that this definition implicates that the practical footprint R may be different for different detectors, site conditions, and temporal Table 1.Analytical results for the minimal footprint distance R, such that soil moisture changes of a remote field (θ2 = θ1 + 5 :::::::::: θ2 = θ1 + 10 vol.%) in an initially uniform domain (θ2 = θ1, h = 5 g/m 3 ) become visible by the CRNS (see Fig. 7 for an illustration with R = 207 m).Cases consider CRNS measurement precisions ::::::: precision ::::: values of σN = 1 % and 2 %, more cases for σN , h, and ∆θ are presented in the Supplement S1.Conventional footprints R86 are displayed for comparison, soil moisture is displayed in vol.%.
The effectively apparent soil moisture θ is dry-biased due to the non-linearity of θ(N ).aggregations.Yet, as current commercial stationary systems are limited to N ≈ 5000 cph, this approach can be regarded as relevant to all existing installations.
Following the above concept of the remote fields, changes of remote soil moisture conditions will only be measureable if the difference between the total neutron count rates before, N (θ 1 , θ 1 ), and after the change, N (θ 1 , θ 2 ), is larger than the accuracy ::::::: precision : limit: Equations ( 13) and ( 14) can be solved for R numerically, while an analytical solution is not straight forward due to the complexity of W r (h, θ).To facilitate easy application of this approach for scientists and CRNS users, an interactive online tool has been developed and briefly presented in the Appendix A. For the change θ 1 → θ 2 we suggest to use ∆θ = 5 ::::::: ∆θ = 10 vol.%, which is a good compromise between typical artificial or natural variations of soil moisture that are of interest for hydrologists.
Figure 8b shows the calculated ranges for h = 5 g/m 3 , σ N = 1 % and three different soil moisture changes of the remote field, ∆θ = +5 . . .20 vol.%.For arid regions between 1 and 5 percent of soil water content, the changes of +5 vol.% are visible to the detector at distances above 150 m.In humid climate up to 25 vol.%water content, wetter remote fields beyond 20 m distance will not show significant contribution to the detector.In wetland areas, +5 vol.% changes of soil moisture are hardly measureable even if this area covers almost half of the footprint.Higher soil moisture changes of ∆θ = 10 vol.% and 20 vol.% (e.g., during irrigation) are much more prominent in the neutron signal and thus allow the remote field to be at a larger distance from the sensor.
The figure also indicates the conventional footprint radius R 86 (dashed line) based on Köhli et al. (2015) and Schrön et al. (2017).Under most conditions, R 86 is much larger than R, as it accounts for neutron intensity changes in all directions (not only a one-sided remote field), and it has not been restricted to the mentioned accuracy limits and soil moisture changes.The radial footprint definition also fails to explain that extensive irrigation of a remote field in arid regions can be sensible much beyond the conventional footprint radius.
Similar to the wetting of the remote field, ∆θ > 0, also the case of soil drying, ∆θ < 0 has been investigated.Due to the non-linearity of N (θ), the neutron production in dry areas is disproportionally higher than the neutron reduction in wet areas.
This leads to a strong influence of distant dry fields under otherwise wet conditions, which consequently manifests in longer maximal distances R by factors of 1.5 (under wet) to 2.0 (under dry conditions).For example, in an area with θ = 20 vol.%, a dry-out by −10 vol.% can be detected from a field at up to 140 m distance, while a wetting of that field by +10 vol.% is only detectable from up to 80 m distance.The corresponding tabulated data is provided in the supplements.
Using a numerical fit, the new practical footprint distance R can be expressed relative to the conventional footprint radius (solid lines in Fig. 8b): where θ = θ 1 in units of m 3 /m 3 , ∆θ = ±0.05m 3 /m 3 , and σ N = 1 %.Practical functions for higher soil moisture changes ∆θ and higher measurement uncertainty σ N could be determined using the presented calculation procedure or the provided data in the supplements.
The relative formulation based on R 86 already accounts for most dependencies on air pressure, air humidity and other factors.
The equation has been tested for various air humidity conditions, for instance, and indicated good performance (not shown).
If the radial footprint radius is not known, an even further simplified approximation for average air humidity h, standard air pressure P , ∆θ = ±0.05m 3 /m 3 , and σ N = 1 % would be: While these relationships may be useful to quickly assess the potential influence of distant fields on the sensor signal, we strongly encourage researchers to perform experiments (e.g., strategic irrigation) that could appropriately falsify the presented theory.
Based on this concept, two applications for the CRNS signal interpretation have been a investigated.First, ::: we found that knowledge about the signal contribution of the area of interest could help to explain seemingly site-specific shapes of the N (θ) relationship.The area's signal contribution value could be used to rescale the neutron-moisture relationship, such that the damping effect of invariant landscape features can be excluded from the signal (Fig. 6).Second, a new footprint definition has been proposed which represents the maximum orthogonal distance to a remote field (Fig. 8a) such that its soil moisture changes are still visible in the measured neutron signal.In the presentation of the results, a typical detector precision of σ N = 1 % and positive soil moisture changes of ∆θ = +5 to +20 vol.% have been chosen, while the approach is adaptable to any combination of parameters.The resulting practical footprint distances for wetting remote fields are 1-90 m (wet climate), 18-180 m (humid), and 100-255 m (arid), showing the strong dependence on the initial soil moisture conditions in the field.In contrast, the dryout of remote fields (∆θ < 0) is usually easier to detect due to the non-linearity of the neutron-water relationship, leading to 1.5-2.0×larger distances.
to substantial improvement of the manuscript.
Competing interests.M. Köhli holds a CEO position at StyX Neutronica GmbH.

Figure 1 .
Figure 1.Basic functions of CRNS theory.a) The radial intensity function, Wr(θ = 0.10, h = 5), representing the intensity contribution of all points at distance r to the detector signal.b) The conversion function, N (θ, h = 5), for a typical stationary CRNS sensor.

Figures 3
Figures3 and 4show soil moisture distributions at 1000 m and 500 m scales, respectively.The different areas are arranged such that fields that would theoretically contribute equally to the sensor (due to same size, distance and water content) still require their neutrons to pass other fields on their way to the detector that have much different soil moisture.The two different scales of the domain are also chosen to investigate the long-range ( ::::::distance :: to :::::: corner: : r < 707 m) and short-range (r < 354 m) performance of the analytical approach.Figures3a and 4aindicate the soil moisture pattern at the different domain scales, while the conventional footprint radius R 86 is indicated by a dashed line.Based on these hypothetical distributions of soil moisture, the individual contributions c i of each area to the neutron signal at the center (0,0) has been calculated following Eq.(7), with the results presented in in Figs.3b and 4b.It is clearly visible that the area with θ = 20 vol.% has the highest contribution to the signal, since it covers the direct vicinity of the detector in the center and also most of the remaining fields.As expected and in accordance with the theory, highest contribution is evident for areas that are closer to the detector and dryer than others.We briefly showcase the calculation of the contribution of an exemplary area A, e.g., the bent field with θ A = 50 vol.% in the upper right quadrant of Fig.4.To compute the weight of the area, one can either weight each grid cell i of the matrix with W ri /r i and sum it up, or integrate W r from radii r 1 = 98 m to r 2 = 167 m and from angles ϑ 1 = 2 • to ϑ 2 = 88 • .The last option is easier for radial geometries.The integration over the radii gives 0.118 for the radial weight of a full circular ring (relative to the total weight of the domain, Ω W r ), while the angular weight of the circular section equals 86 • /360 • ≈ 0.239.This results in the normalized spatial weight of w A ≈ 0.118 × 0.239 = 2.7 %.It would already be the sought contribution to the detector signal if the domain was homogeneous.In this heterogeneous example, however, the spatial weight needs to be multiplied with the neutrons produced by this area, N A = N (θ A = 50 %) ≈ 651 cph, and normalized by the effective count rate measured in the centre, N ≈ 860 cph, resulting in a contribution of c A = w A N A / N ≈ 2 % to the detector signal.

Figure 3 .Figure 4 .
Figure 3. Scenario 1000 × 1000 m with a) a complex soil moisture pattern (greyscale), see also Fig. 7 for details.b) Contribution to the detector signal estimated with the analytical method, and c) simulated with URANOS.
Figure 5. a) Exemplary scenario with different land use features (see sec.3.2), b) analytically estimated signal contributions based only on the soil moisture, and c) URANOS simulated signal contributions including 3D features (house, trees).

Figure 6 .
Figure 6.Exemplary scenario to demonstrate the impact of signal contributions on the N (θ) relationship.a) Scenario with variable soil moisture, θ1, in a central area of 20 m radius, surrounded by constant soil moisture of θ2 = 10 vol.% (e.g., concrete pavement).The signal contribution of the inner area is c1 ≈ 33 %.b) The simulated effective neutron intensity of the central detector as a function of θ1 (blue points) appears damped compared to the standard relationship (black line, see also Fig.1b).However, the function can be rescaled using