Dear Reviewer 1,

thank you very much for your positive review. We will briefly comment on your concerns in the spirit of the interactive discussion. (RC=reviewer comment, AR=author response)

# Main comments 

> RC1.1: "In my opinion, one of the strongest assumption that might be warranted is that neutron contribution of each sub-domain can be assessed individually (L149). As far as I have understood, this is also actually not challenged by the MonteCarlo simulations because also in these analyses the distance of the first contact with the land is used for the calculation (L179 and L212). In contrast, assuming a more diffuse transport process I expect more interactions and mixing. This might change significantly the results. I would say that for addressing this question, a different definition of detected distance in the MonteCarlo simulation should be tested. If this will not the case within the present study, I suggest at least to extend the discussion around that."

AR: This is indeed a fundamental assumption and has been discussed in lines 138-149. In fact, the question of the importance of secondary interactions beyond the individual area and the implication for the detector signal was one to which we paid particular attention in this context.  That's why we challenged this hypothesis in section 3.1 with very heterogeneous soil moisture patterns. It turned out that the prediction of the contributions is still very close to Monte-Carlo simulations, so the assumption, that these secondary interactions are of minor importance seems to hold. We believe this is one of the major findings of the study, so we will better clarify this the text in the revision.

Regarding the definition of origin in the simulations: Previous studies dedicated to the CRNS footprint were based on the same definition that we use in our study, i.e., on the distance between first soil contact (=origin) and detection. Ofcourse there are further interactions on the way (on average two to three land-atmosphere interface crossings). However, simulations from Köhli et al. (2015) have shown that the first contact is the most important for the neutron, i.e., the soil moisture at the first-contact location is the most dominant in the detected neutron signal.

> RC1.2: "I like the practical question that has been formulated, i.e., “At what distance are soil moisture changes still visible to the CRNS” (L325). This could help better understanding neutron signal and supporting agro-hydrological applications. However, I’m not convinced that this should be posted as an alternative definition of footprint size R86 (L324). As far as I understood, on the one hand, R86 refers to the average radius over all the directions from the sensor (as stated at L365) and it does not assume that no neutrons are coming from more remote areas (on the contrary to the statement at L367). On the other hand, the Authors nicely show how soil moisture changes at a field nearby can be detected only at different distance than R86 in case soil moisture where the sensor is placed remains constant. But this does not contradict R86, i.e., on average over all direction R86 is different then the new R.

AR: The statement in L367 says that the definition of R86 suggests that extreme changes of soil moisture might not be "...sensible much beyond the conventional footprint radius". We are not saying that "no" neutrons are coming from beyond that radius, in fact it is implicit that 14% of neutrons are coming from there. But the definition does not take strong soil moisture contrasts and asymmetric geometries into account. Thus, it will fail in extreme situations (as shown in Fig. 8b). Depending on the field geometry, in some directions the 86% limit might hold, but in some it might be completely off. That is why we believe that for many applications, R has a higher practical relevance than the "on average" quantity R86.

We would like to point out that the original formulation of the footprint definition as R86 has already sparked discussions in the community regarding its practical applicability. The exponential shape of the radial weighting function with its very long tails and its stong near-field sensitivity is not very intuitive. Therefore providing one number for the radius led to misunderstandings regarding the actual sensitivity of a CRNS instrument. Nevertheless, R86 has its qualities, too, but it is only a radially symmetric approximation. It is certainly useful for many applications, but it is less useful in highly asymmetric cases, such as adjacent fields. On the other hand, the new R is a good alternative definition for some specific cases, but certainly it might be less useful in other situations. In the revised manuscript we will make more clear that R is not a general alternative to R86, but rather more practical in certain scenarios. In general, however, the concept of defining the footprint in this, spatially more explicit way, and to include the sensor sensitivity, is applicable to any thinkable scenario.

> RC1.2 cnt.: "Moreover, I do not have anything against these showcases but I think these support the conclusion that CRNS is not suitable for supporting irrigation management at relative small farms. But this I think was already clear from first CRNS publications. In contrast, it should be acknowledged that in any other conditions we soil experiences wetting or drying soil moisture profiles even if at different degrees that is still detected (even if in a non-linear way) by the sensor. So overall, I see the new R as an additional indicator rather than an alternative footprint size, i.e., I would still like having an indicator that accounts for neutron intensity changes in all directions (L365)."

AR: The reviewer is correct, that for irrigated patches much smaller than the footprint the corresponding signal strength might be weak. In general, we fully agree that the new R should not replace the conventional R86 characteristics, it wasn't meant to. The new R should rather be an additional quantity to be more useful for some practical questions. We will adapt the nuances of the text to make that more clear.

> RC1.3: "my final main comment is related to the fact that all the discussion is based on a forward operator N(theta) (eq. 11). I agree that the results of this contribution will support some practical questions, e.g., where to install the sensor or modelling applications, e.g., data assimilation. However, I believe that in several applications soil moisture is the targeted variable, i.e., we do not know soil moisture within the footprint. Thus, I think one could conclude that 1) it remains an ill-posed problem to try to resolve soil moisture variability within the footprint and 2) CRNS sensor should be strategically located to avoid difficulties with signal interpretation. I would encourage the Authors to extend on these."

AR: We agree that we cannot directly derive soil moisture patterns from the neutron signal alone, and ideally and in the sense of a good interpretability of the signals, one should always aim at operating the sensor in areas that are as homogeneous as possible. However, this requirement is often -- if not in the majority of cases -- not met, mainly for practical reasons. The presented method can help, firstly, to better assess the potential influence of spatial heterogeneity on the sensor signal, e.g. by forward modeling different variations of soil moisture, see how it influences the signal theoretically, and improve the uncertainty assessment of the signal. It has also the potential to support better understanding of certain features in the signal and thereby "detect" non-reported irrigation, for instance. Secondly, to drive inversion experiments (L413), i.e., to test the known variations of soil moisture patterns and find the best match to the actual soil moisture signal. This way, the subfootprint-patterns could be derived indirectly (see also Franz et al. 2015, 10.1002/2015gl063963). We will better clarify this in the revised manuscript.

# Specific comments

> "L2: Through the manuscript, the Authors use the term “concept” to refer to what they have developed and tested. A concept is in my opinion an abstraction or perception. So, more than a concept, the Authors have developed an “approach” or a “method”. I suggest using one of these terms instead of using “concept”." 

AR: Thank you, we will adapt the terms where necessary. In general, we believe that the "generation times transport" formula is still a theoretical concept, but the way how it is applied to CRNS might indeed better be described by "method".

> "L3. I understand that similar concept (or approach, see comment before) can be developed for snow. But the study focuses on soil moisture and the snow application is not addressed. I would remove from abstract and methods the snow applications and only refer to that as a possible extension o the study in the conclusions section.

AR: Besides irrigation, patchy snow cover is one of the most relevant fields where this method has a good potential for application. The reason is that the average descriptions of R86 and N(theta) mainly fail at the very extremes of soil moisture heterogeneity, which is one of the main take home messages of this paper. Patchy snow cover (=pure water) on rocks (=purely dry soil) hits this sweet spot, where new methods are necessary to describe the CRNS signal (see e.g., Franz et al. 2013, or Schattan et al. 2017, 2019).

> "L86. Why only humid?"

AR:  This word was misleading and will be removed.

> "L88. Vegetation height might be not a good proxy for biomass effect to the neutron signal. I guess it was selected for simplicity but would be nice to extend the discussion on the consequences of this approximation."

AR: The influence of vegetation height to the footprint has been discussed in Köhli et al. 2015. It will be interesting to better understand its influence on signal contributions in general, but this is out of scope of this study. 

> "L172. What is “drf”?"

AR:  It refers to the detector response function as explained in the end of this sentence. We will make that more clear in the revision.

> "L193. I lost from where these ranges come from."

AR: Sorry for the confusion. This is the Euklidian distance to the corners of a 500x500 or 200x200 square, respectively. We will make that more clear in the revision.

Dear Reviewer 2,

thank you very much for your positive review. We will briefly comment on your concerns in the spirit of the interactive discussion. (RC=reviewer comment, AR=author response)

# Specific comments 

> RC2.1: "L184: Please explain the terms house gas and tree gas and why those are used. A bit unclear upon first read. "

AR: Thanks for pointing out that we have been a bit unspecific in describing the materials used in the simulation. We refer to the material codes that are predefined in the URANOS model. Recently, the model code has been published on GitHub [1], and the corresponding detailed model description is now on GMD Discussions [2]. In the material codes description [3], house gas is described as as a soil-like material with 0.15 g/cm³ density and 10% water content, which roughly resembles cement. Tree gas is defined as Cellulose (H, O, and C molecules) with 3 kg/m³ density. We will add these details to the revised manuscript.

1. https://github.com/mkoehli/uranos
2. https://gmd.copernicus.org/preprints/gmd-2022-93/
3. https://github.com/mkoehli/uranos/blob/main/src/MaterialCodes.txt

Dear Reviewer 3,

thank you very much for your positive review. We will briefly comment on your concerns in the spirit of the interactive discussion. (RC=reviewer comment, AR=author response)

# Specific comments 

> RC3.1: "The authors assume throughout the paper that the soil moisture (and the variability thereof) within a given field is known, which is not usually the case. I understand that in this instance these heterogeneities are artificially created for the purpose of testing the authors’ newly developed methods, but it would be good for the authors to mention whether or not these applications are feasible without intensive soil moisture surveys."

It is true that soil moisture needs to be known before the application of the concept, but this is the nature of all forward models, like COSMIC, MCNP, or URANOS. The great benefit from these models is that they can be used to better understand neutron data, to infer hidden hydrological processess or irrigation events, and to assess the systematic uncertainty of the data. Here, intensive soil sampling is not needed, but one could play with the soil moisture variable in the model to gain a better understanding of potential influencing factors and their impact on the signal. This is already indicated in the conclusion (L413), but we will better communicate this in the revision. Please refer also to the reply to Reviewer 1 (RC1.3.)

> RC3.2: "Except for the last case study, the authors conduct their analyses under constant soil moisture conditions, but in reality soil moisture varies in time, which means the neutron contribution of each unique area will also change with time. This significantly complicates the cases of detected and delineated the effects of heterogeneous soil moisture patterns (section 3.1) and complex land use features (section 3.2). The authors might consider addressing this issue of temporal variability of soil moisture and whether or not the analysis carried out in the first two case studies is practical in reality."

The generalized analytical method is aimed at being practical in reality, while the presented examples were only used to validate the analytical model with Monta Carlo physics codes. Since the validation with these specific complex scenarios was successful, we can conclude that the analytical model can be applied to any other user-defined soil moisture condition very easily. The presented framework allows users to quickly assess the neutron signal for their specific and arbitrarily varying soil moisture pattern. The attached online-notebook even provides a user-friendly interface to do exactly that. For example, the user can run the same model for low and high soil moisture conditions to gain insights on the impact of spatial contributions for dry and wet days, respectively. Hence, we hope that the presented method is as practical as possible to emulate reality.

Dear Reviewer 4,

thank you very much for your positive review and the specific comments in the PDF. We will briefly comment on your concerns in the spirit of the interactive discussion and we will also respond to some key comments from the annotated PDF. (RC=reviewer comment, AR=author response)

# General comments 

> RC4.1: "The results have some practical implications for where to install (or avoid installing) cosmic-ray neutron detectors in heterogenous environments, depending on the intended purpose of the measurements. The manuscript would be strengthed if the text focused more on explaining those practical implications [...] The focus should be on choosing installation sites that minimize heterogeneity in the footprint. The results here could help inform such choices."

AR: We agree that the method can help to inform the choice of installation sites, which is one of the key results. Often enough, choosing fully homogeneous installation sites is not possible, particularly in complex terrain. In those cases, the method can help to better understand the observed neutron-soil moisture relationship, to assess the systematic uncertainties produced by inhomogeneities, and to help to better identify specific events, such as remote irrigation. This is another key outcome. But understanding signal contributions is not only useful for stationary sensors: mobile CRNS is often bound to accessible roads, where sensor placement is less important than the understanding of the signal. 

Hence, the implication on sensor placement is only one of many interesting aspects of this study. We will consider improving the discussion on these practical implications in the revised manuscript. 

> RC4.2: "[...] and less on framing the distances calculated here as "a new practical footprint definition" for this type of detector."

AR: We agree that the footprint definition should not be the main focus of the study, that is why we allocated only one of the five subsections in the Results to this topic. As discussed also with Reviewer 1, we do not intend to suggest a replacement of the conventional footprint with this new definition, but rather to suggest this new footprint concept for some specific scenarios where it could be more informative than symmetrical footprints. We will clarify this more clearly in the revision.

> RC4.3: "Because the detectors are strongly biased toward areas of dry soil, users need to understand that they do not inherently provide an accurate areal average of heterogenous footprints, as the results here show."

AR: Thank you for pointing this out again in a concise way. This is a take-home message of our study and we might consider using it as an inspiration to clarify the conclusions.

# Specific comments (excerpt)

• Eq 5: "Explain how this [N(theta_i)] is being estimated and the assumptions involved. It seems that the local neutron "production" rate for a portion of the landscape could likely be a different function than the functions often used for relating the "effective neutron intensity" to the soil moisture in the sensor's footprint (e.g. the Desilets 2010 function or similar). Please elaborate and justify any assumptions being made."
  
  
  
  AR: We agree that this assumption is not trivial, but one of the outcomes of this study is that it seems to work out. The assumption that the "effective neutron intensity" of a completely homogeneous footprint behaves similar to the neutron production of a portion of the landscape seems natural, though. We tried to indicate that this is just an assumption in L105 ("We propose that ..."), but we agree that this topic needs clarification, which will done in the revised manuscript.
• L129: "Explain what Wr and r mean exactly in the context of a grid cell. Are you making any assumptions about the size of the grid cell relative to the size of the inhomogeneous areas or to the size of the total footprint?"
  
  
  
  AR: Thanks for asking, r is the distance to the center of a grid cell and W_r is the radial intensity at this distance. As for all numerical approximations, the size of the grid cell should be small compared to relevant structures in the footprint. Although this concept and the details already go back to Schrön et al. 2017, we agree that it deserves better explanation and we will elaborate on it more clearly in the revised text.
• L178: "Why so large? That encompasses the majority of the sensitive zone as shown in Fig. 1."
  
  
  
  AR: We chose 9 m radius for the detector to speed up simulations, as a reduction of the radius of the detector would drastically decrease the area of exposure to the neutrons and thus increase the simulation time. This is a standard procedure in detector simiulations. If no structures were to be present below 9 m radius, the results would equally well emulate a smaller detector. For this reason, we used heterogeneous structures in our examples only beyond 9 meters. This limitation has also been discussed in L180-182. It is true that this model setup would be insufficient for the scenario in Figure 1, but this is only an illustrative plot which has not been simulated in our study. Nevertheless, we will add a note to the revised manuscript to clarify this.
• L267: "I think most of the text in this section [3.3] above this point should be moved to the Methods section."
  
  
  
  AR: The reshaping procedure is one of the results of this study and has therefore been placed in the results section. This is similar to the footprint functions in section 3.5. We believe that both these aspects are *implications* or *consequences* of the methodological concept, rather than the concept itself. So we would like to keep them outside the methods section.
• Table 1: "The chosen 5% difference in soil water content here is comparable to the magnitude of spatial variability that is commonly observed in "homogenous" fields. I think a table like this for a 10% difference in soil water content would be more instructive/educational. Differences of 10% would indicate to me an important but plausible level of inhomogeneity."
  
  
  
  AR: Thanks for sharing your opinion about typical soil moisture changes. The results for 10% difference are already visible in the supplement tables. We will consider also showing them in the main manuscript as suggested.
• Appendix A: "I tried running the code in Binder on Google Chrome. The code seemed to run without errors, but the figures shown below were not produced or not displayed. The sliders were visible, but not the actual figures."
  
  
  
  AR: Thanks for testing the online notebook! The figures appear as soon as the sliders are used/moved. Can you please retry and feed back whether it worked on your system?