Matrix representation of lateral soil movements: scaling and calibrating CEDYNAM (v2) at a continental level
 ^{1}European Commission, Joint Research Centre, Sustainable Resources Directorate, Via E. Fermi 2749, I21027 Ispra, Italy
 ^{2}Laboratoire des Sciences du Climat et de l’Environnement, CEACNRSUVSQUPSACLAY, Gif sur Yvette, France
 ^{3}INRAE, UMR 1048 SADAPT, 16 rue Claude Bernard, 75231, Paris, France
 ^{4}LGENS (Laboratoire de géologie)  CNRS UMR 8538  École normale supérieure, PSL University  IPSL, Paris, France
 ^{5}Department of Science, Roma Tre University, 00146 Rome, Italy
 ^{1}European Commission, Joint Research Centre, Sustainable Resources Directorate, Via E. Fermi 2749, I21027 Ispra, Italy
 ^{2}Laboratoire des Sciences du Climat et de l’Environnement, CEACNRSUVSQUPSACLAY, Gif sur Yvette, France
 ^{3}INRAE, UMR 1048 SADAPT, 16 rue Claude Bernard, 75231, Paris, France
 ^{4}LGENS (Laboratoire de géologie)  CNRS UMR 8538  École normale supérieure, PSL University  IPSL, Paris, France
 ^{5}Department of Science, Roma Tre University, 00146 Rome, Italy
Abstract. Promoting sustainable soil management is a possible option for achieving netzero greenhouse gas emissions in the future. Several efforts in this area exist, and the application of spatially explicit models to anticipate the effect of possible actions on soils at a regional scale is widespread. Currently, models can simulate the impacts of changes on land cover, land management, and the climate on the soil carbon stocks. However, existing modeling tools do not incorporate the lateral transport and deposition of soil material, carbon and nutrients caused by soil erosion. The absence of these fluxes may lead to an oversimplified representation of the processes, which hinders, for example, a further understanding of how erosion has been affecting the soil carbon pools and nutrient through time. The sediment transport during deposition and the sediment loss to rivers create dependence among the simulation units, forming a cumulative effect through the territory. If, on the one hand, such a characteristic implies that calculations must be made for large geographic areas corresponding to hydrological units, on the other hand, it also can make models computationally expensive, given that erosion and redeposition processes must be modeled at high resolution and over long time scales. In this sense, the present work has a threefold objective. First, we provide the development details to represent in matrix form a spatially explicit processbased model coupling sediment, carbon, and erosion, transport and deposition processes (ETD) of soil material in hillslopes and valley bottoms (i.e., the CEDYNAM model). Second, we illustrate how the model can be calibrated and validated for Europe, where highresolution datasets of the factors affecting erosion are available. Third, we presented the results for a depositional site, which is highly affected by incoming lateral fluxes from upstream lands. Our results showed that the benefits brought by the matrix approach to CEDYNAM enabled the before precluded possibility of applying it to a continental scale. The calibration and validation procedures indicated: i) a close match between the erosion rates calculated and previous works on the literature at local and national scales; ii) the physical consistency of the parameters obtained from the data; and iii) the capacity of the model in predicting sediment discharge to rivers in locations observed and unobserved during its calibration (Model efficiency (ME) = 0.603, R^{2} = 0.666; and ME = 0.152, R^{2} = 0.438, respectively). The prediction of the carbon dynamics on a depositional site illustrated the model's ability to simulate the nonlinear impact of ETD fluxes on the carbon cycle. We expect that our work advances ETD models' description and facilitates its reproduction and incorporation in land surface models such as ORCHIDEE and DayCent. We also hope that the patterns obtained in this work can guide future ETD models at a European scale.
Arthur Nicolaus Fendrich et al.
Status: closed

RC1: 'Comment on gmd2022121', Yuanyuan Huang, 20 Jun 2022
The authors developed the processbased model that coupling carbon, erosion, transport and deposition processes. They brought in the lateral movement of carbon into processbased land carbon modelling. The community has been expecting this advancement, especially in large scale studies, for long, but with limited progresses partly due to the high computational cost. The authors presented a detailed and exciting case. The wellillustrated their model formulations, how they tackled on the computational bottleneck, the calibration and validation, with adequate discussion of limitations and future improvements. The study is welldesigned and informative. The writing is generally clear despite some part might be a little lengthy. My criticises are between major and minor. Please check below.
It is not clear to me how the matrixrelevant techniques helped the current study. The equations for the lateral carbon fluxes are presented mostly in carbon balance equations (i.e., no need for the matrix form). Are the matrix techniques only used for constructing the ORCHIDEE emulator? If the ORCHIDEE output only an input to CEDYNAM, or any parameter changes that require a redo of model spinup that requires computation resources?
Parameter values for soil discretization are optimized. How about parameter values for other parts of the model? I might miss some part, but is there any table or supplementary information that documents values of relevant parameters used in this study?
Minor comments:
Line 111. Is the background for abstract a little lengthy?
Line 24. I would suggest DayCent. For people without background, they don’t know what DayCent refers to. DayCent is not mentioned in texts other than here. Besides, the CENTURY vs. DayCent is another layer that needs background. So DayCent here is not necessary to add more information to this already complex manuscript.
Line 88 add in before Table 1
Line 92. Is it better to write ORCHIDEE CENTURY as ORCHIDEE (CENTURYCarbon type) or other better rewordings. ORCHIDEE CENTURY is confusing.
Line 93. Is the first order kinetics necessary? Current application is with linearmodels due to computational cost. In theory, for example, if our studying region is small, the coupling with an nonlinear carbon model is possible, right?
Line 125. Could we write it as “ induced by the terrain slope (S[x,y]) and the flow accumulation (w[(x,y)])” to reduce confusing?
Table 1. Could you use other symbols for ω[(x,y)] vs w[(x,y)] (?)? They looks the same
Equation 6. Lines 200205, and across the manuscript. By erosion b – t, you mean from lower soil layer to the upper soil layer and the flux is there between adjacent soil layers? Please clarify, by b – t, it refers specifically from the third to the first layer, in your context. So it is not clear to me why “losses from the layers below must be added to the layers above”
Line 525. Is the breakdown of aggregates a transport process? Please clarify. Aggregates breakdown could happen without the transport process.
Line 525530. If the “halfway between….”a cause of the difficulty in finding the optimal resolution, or the computation cost and applications?
AC1: 'Reply on RC1', Arthur Nicolaus Fendrich, 15 Jul 2022
Dear Dr. Huang,
First, thank you for the thoughtful review, which will certainly improve our manuscript.
We answer the major questions below and will address the minor comments in the next round of manuscript revision.
Question: "It is not clear to me how the matrixrelevant techniques helped the current study. The equations for the lateral carbon fluxes are presented mostly in carbon balance equations (i.e., no need for the matrix form). Are the matrix techniques only used for constructing the ORCHIDEE emulator? If the ORCHIDEE output only an input to CEDYNAM, or any parameter changes that require a redo of model spinup that requires computation resources?"
Answer: The matrix techniques are used only for constructing the emulator, as the Reviewer asked. The ORCHIDEE output is used as an input to CEDYNAM, in such a way that if we run CEDYNAM without enabling erosion, transport, and deposition (ETD) modules, we recover the original ORCHIDEE results. However, when we enable these modules, the results change, and spinup calculations must be redone because a new equilibrium state is obtained when new processes are included. In CEDYNAM, this is done with the emulator by using the rates extracted from the original ORCHIDEE plus the new rates of ETD dynamics presented in the manuscript. Regarding how the matrix technique helped the study, we agree that all equations refer to carbon balance. However, the dependence between adjacent cells significantly impacts the time demanded to calculate the new equilibrium state and the model dynamics. As we mentioned in the paragraph L.480501, the matrix approach allowed us to overcome the existing barriers to the implementation of CEDYNAM at a continental scale, as we could precalculate the matrices for every simulation month (L.478479) and increase the number of parallel threads compared to the previous implementation (L. 485). In the next revision, we will improve this part of the text to clarify for the readers.
Question: "Parameter values for soil discretization are optimized. How about parameter values for other parts of the model? I might miss some part, but is there any table or supplementary information that documents values of relevant parameters used in this study?"
Answer: In this study, we intend to isolate the effects of ETD on the carbon cycle. Therefore, the parameters presented refer only to those introduced by the current formulation. We opted not to modify any of ORCHIDEE's default values, which can be found in other publications such as Krinner et al. (2005) [10.1029/2003GB002199].

AC1: 'Reply on RC1', Arthur Nicolaus Fendrich, 15 Jul 2022

RC2: 'Comment on gmd2022121', Holger Metzler, 04 Jul 2022
The authors provide a spatially upscaled version of an eriosion, transport and deposition model (CEDYNAM) to European scale, while sticking to a high spatial resolution. This is an important scientific contribution because lateral C transport has largely been ignored so far in processbased models because usually models consider different grid cells as independent. Such lateral transports can have a significant effect and should be investigated because nonlinear effects could potentially lead to drastically new insights and better understanding compared to existing models.
The spatial upscaling was computationally feasible only because the authors emulated the original model by a matrix approach. This allows the application of sparse matrix models as well as an improved application of parallel computation methods. Another advantage of matrix models is that they allow a rigorous mathematical analysis, something the authors did not do in this manuscript (it was not their goal) but can be done in the future based on the matrix reformulation. This was not possible with the original implementation. Furthermore, I appreciate all the effort the authors showed in reimplementing an existing model with a matrix approach, emulating the original simulation results very vell. In this regard I like also the explanation of the matrix shape in Section 3.3. Nevertheless, I wished I could have "seen" a matrix, at least as a schematic block matrix after the authors speaking so much about matrices.
The presentation and the writing is clear, explaining the model calibration and simulation results as well as model limitiations and future opportunities very well. Sometimes though the text appears too lengthy in my opinion. This starts with the abstract and continues with quite some overlap in the sections about calibration, results, simulations and limitations. Furthermore, in particular in later sections I was overwhelmed by an extensive use of potentially unnecessary numbers.
I furthermore do have some issues about an easy reproducibility of the method, because to me it seems that in Section 2 some formulas are incorrect and notation is not precise. Well, either the formulas are incorrect or I understood them wrong, neither option is preferrable. In particular I consider the statement (p. 3, l. 71): "We expect our mathematical development of CEDYNAM to facilitate its reproduction and incorporation in LSMs such as ORCHIDEE, DayCent, and others." to be pretty bold. Under this point of view formulas and notation should show no flaws.
This starts with Table 1, which is in general very nice, but it is incomplete, some symbols that are used later on do not appear here. This was sometimes annoying for me while reading. For example I* was never properly introduced. On p.5, l. 132 it is stated that inputs are disturbed along D, which I do not understand.
Here my issues with notation and formulas, which should be thoroughly checked:
p. 8. l. 1: Should it be Δ_{i}? Is it used to compute d_{0}=0, d_{1}=d_{0}+Δ_{0} and so forth?
p. 8. l. 183: k_{e} is missing in Table 1. The "Total mass of soil" under the bracket refers to total mass of soild in hill slopes? Why is it decreasing with h? To me this looks like we have an infinite erosion if we do not have any hill slopes, isn't this going in the wrong direction?
p. 10, l. 202. If there is going to be a new input flux, should there also be a new output flux? The input must come from somewhere. Please also indicate where the new input flux goes, I do not want to guess here.
p. 10, l. 213. The way k_{τ} (missing in Table 1) is defined two lines later, it seems wrong to me to call it a flux. It is a rate (dimension 1/time). It will only become a flux once it is multiplied with a pool content. This leads to major confusions for me later on.
p. 11, l. 235: Please use other notation than (a, b) for the indexing of the sum, the two letters are already taken. It becomes very confusing this way. Again, I think that if k_{τ} is a rate rather than a flux, then k_{s }will be as well because all the other factors in Eq. 11 are dimensionless.
p. 10, l. 241 "all PFTs" should rather be "all PFTs but p_{0}", right?
p.12, l. 247: "Such input flux". I disagree again, same problem. It is not yet a flux because it is not yet multiplied with a carbon stock, which should be S_{[(a, b), ...]} here?
p. 12, l. 257: The "P" here looks different from the ones introduced on p. 10, l. 221. Furthermore, I am not sure whether it belongs here in the first place, well the absolute value is unecessary in any case. But why multiply by the number of nonzero PFTs in cell (a, b)? Shouldn't this be already included in k_{τ} already? Could you write it down explictly for yourself without the P but a second sum instead and check whether it is correcct this way?
p. 12, l. 260: In k_{s} I think that source and target are confused.
p.13, l. 265: What is k_{t}^{*}?
p. 13. l. 266. Should k_{s} with source (a, b) be multiplied with some stock indexed by (a, b) instead (x, y).
I obviously do have some confusions about the firsorder description, where sources and targets seem not to match, at least in my head. So would like to encourgae the authors to carefully check the notatian and the formulas again, along with their implementation.
Small issues:
 In general units are sometimes italic and sometimes not, sometimes with a space between the number and the unit, sometimes without.
 The use of singular and plurar gets mixed up quite often.
 p. 3, l. 86: "approach" > "an approach"?
 p. 3, l. 88: Table 1 > (Table 1)?
 Table 1:
  Description sometimes ends with a period, sometimes not.
  The adimensional respiration rates of carbon k_{r} actually do have a dimension: 1/day.
  I^{[c_j, c_j]}: One of the j's should be an i. Probably the first one, then please also adapt the description, to make it consistent with k_{t}.
  Then it is a little unfortunate to use ω and w for the depth to bedrock and the flow accumulation, respectively.
 p. 7, l. 145: I am not sure if such a procedure necessarily converges to a pullback attractor (given there is one in the first place). The pullback attractor is reached when starting the simulation earlier and earlier, basically moving toward an infinite simulation history.
 p.7, l. 152: Why now change the notation from S, I, and k to x, B, and A?
 p. 10, l. 243: "soi"
 p.14, l. 280: The seventhlargest what?
 p. 15, l. 308: u_{i} instead of p_{i}?
 p. 15, l. 312: "P is Panagos", what does this mean?
 p.15. unnamed formula: what is r(y, m)?
 p. 18, l. 398, 399: What are Cfactor and "R factor"? Please also note the different way of writing them.

AC2: 'Reply on RC2', Arthur Nicolaus Fendrich, 15 Jul 2022
Dear Dr. Metzler,
We would like to thank you for the very relevant comments about our work.
We answer below the comments, and we will fix all the issues in the next round of manuscript revision.Reviewer: "In this regard I like also the explanation of the matrix shape in Section 3.3. Nevertheless, I wished I could have "seen" a matrix, at least as a schematic block matrix after the authors speaking so much about matrices."
Answer: We agree that it would be interesting to visualize the matrix structure. We will add this image and information in the next manuscript version.
Reviewer: "Sometimes though the text appears too lengthy in my opinion. This starts with the abstract and continues with quite some overlap in the sections about calibration, results, simulations and limitations. Furthermore, in particular in later sections I was overwhelmed by an extensive use of potentially unnecessary numbers."
Answer: It was a common point in both Reviewer's comments that some text sections are lengthy. In order to fix this problem and improve readability, we will rewrite several passages in the next version of the manuscript, with extra attention to the aforementioned unnecessary numbers.
Reviewer: "I furthermore do have some issues about an easy reproducibility of the method, because to me it seems that in Section 2 some formulas are incorrect and notation is not precise. Well, either the formulas are incorrect or I understood them wrong, neither option is preferrable (...) Under this point of view formulas and notation should show no flaws. (...) This starts with Table 1, which is in general very nice, but it is incomplete (...). So would like to encourgae the authors to carefully check the notatian and the formulas again, along with their implementation."
Answer: We thank very much the Reviewer for pointing out all the problems with the notation adopted. We strongly agree that this is critical for adopting and disseminating the matrix approach and for future developments of the method. In order to fix all issues, all the formulas of the manuscript will be carefully reviewed and rewritten to prioritize precision, conciseness, and clarity. We hope the new notation will be precise and resolve any confusion that may have arisen....
Reviewer: "This allows the application of sparse matrix models as well as an improved application of parallel computation methods. Another advantage of matrix models is that they allow a rigorous mathematical analysis something the authors did not do in this manuscript (it was not their goal) but can be done in the future based on the matrix reformulation."
Answer: We truly appreciate this suggestion and agree that it would be an interesting work to develop in the future.
Status: closed

RC1: 'Comment on gmd2022121', Yuanyuan Huang, 20 Jun 2022
The authors developed the processbased model that coupling carbon, erosion, transport and deposition processes. They brought in the lateral movement of carbon into processbased land carbon modelling. The community has been expecting this advancement, especially in large scale studies, for long, but with limited progresses partly due to the high computational cost. The authors presented a detailed and exciting case. The wellillustrated their model formulations, how they tackled on the computational bottleneck, the calibration and validation, with adequate discussion of limitations and future improvements. The study is welldesigned and informative. The writing is generally clear despite some part might be a little lengthy. My criticises are between major and minor. Please check below.
It is not clear to me how the matrixrelevant techniques helped the current study. The equations for the lateral carbon fluxes are presented mostly in carbon balance equations (i.e., no need for the matrix form). Are the matrix techniques only used for constructing the ORCHIDEE emulator? If the ORCHIDEE output only an input to CEDYNAM, or any parameter changes that require a redo of model spinup that requires computation resources?
Parameter values for soil discretization are optimized. How about parameter values for other parts of the model? I might miss some part, but is there any table or supplementary information that documents values of relevant parameters used in this study?
Minor comments:
Line 111. Is the background for abstract a little lengthy?
Line 24. I would suggest DayCent. For people without background, they don’t know what DayCent refers to. DayCent is not mentioned in texts other than here. Besides, the CENTURY vs. DayCent is another layer that needs background. So DayCent here is not necessary to add more information to this already complex manuscript.
Line 88 add in before Table 1
Line 92. Is it better to write ORCHIDEE CENTURY as ORCHIDEE (CENTURYCarbon type) or other better rewordings. ORCHIDEE CENTURY is confusing.
Line 93. Is the first order kinetics necessary? Current application is with linearmodels due to computational cost. In theory, for example, if our studying region is small, the coupling with an nonlinear carbon model is possible, right?
Line 125. Could we write it as “ induced by the terrain slope (S[x,y]) and the flow accumulation (w[(x,y)])” to reduce confusing?
Table 1. Could you use other symbols for ω[(x,y)] vs w[(x,y)] (?)? They looks the same
Equation 6. Lines 200205, and across the manuscript. By erosion b – t, you mean from lower soil layer to the upper soil layer and the flux is there between adjacent soil layers? Please clarify, by b – t, it refers specifically from the third to the first layer, in your context. So it is not clear to me why “losses from the layers below must be added to the layers above”
Line 525. Is the breakdown of aggregates a transport process? Please clarify. Aggregates breakdown could happen without the transport process.
Line 525530. If the “halfway between….”a cause of the difficulty in finding the optimal resolution, or the computation cost and applications?
AC1: 'Reply on RC1', Arthur Nicolaus Fendrich, 15 Jul 2022
Dear Dr. Huang,
First, thank you for the thoughtful review, which will certainly improve our manuscript.
We answer the major questions below and will address the minor comments in the next round of manuscript revision.
Question: "It is not clear to me how the matrixrelevant techniques helped the current study. The equations for the lateral carbon fluxes are presented mostly in carbon balance equations (i.e., no need for the matrix form). Are the matrix techniques only used for constructing the ORCHIDEE emulator? If the ORCHIDEE output only an input to CEDYNAM, or any parameter changes that require a redo of model spinup that requires computation resources?"
Answer: The matrix techniques are used only for constructing the emulator, as the Reviewer asked. The ORCHIDEE output is used as an input to CEDYNAM, in such a way that if we run CEDYNAM without enabling erosion, transport, and deposition (ETD) modules, we recover the original ORCHIDEE results. However, when we enable these modules, the results change, and spinup calculations must be redone because a new equilibrium state is obtained when new processes are included. In CEDYNAM, this is done with the emulator by using the rates extracted from the original ORCHIDEE plus the new rates of ETD dynamics presented in the manuscript. Regarding how the matrix technique helped the study, we agree that all equations refer to carbon balance. However, the dependence between adjacent cells significantly impacts the time demanded to calculate the new equilibrium state and the model dynamics. As we mentioned in the paragraph L.480501, the matrix approach allowed us to overcome the existing barriers to the implementation of CEDYNAM at a continental scale, as we could precalculate the matrices for every simulation month (L.478479) and increase the number of parallel threads compared to the previous implementation (L. 485). In the next revision, we will improve this part of the text to clarify for the readers.
Question: "Parameter values for soil discretization are optimized. How about parameter values for other parts of the model? I might miss some part, but is there any table or supplementary information that documents values of relevant parameters used in this study?"
Answer: In this study, we intend to isolate the effects of ETD on the carbon cycle. Therefore, the parameters presented refer only to those introduced by the current formulation. We opted not to modify any of ORCHIDEE's default values, which can be found in other publications such as Krinner et al. (2005) [10.1029/2003GB002199].

AC1: 'Reply on RC1', Arthur Nicolaus Fendrich, 15 Jul 2022

RC2: 'Comment on gmd2022121', Holger Metzler, 04 Jul 2022
The authors provide a spatially upscaled version of an eriosion, transport and deposition model (CEDYNAM) to European scale, while sticking to a high spatial resolution. This is an important scientific contribution because lateral C transport has largely been ignored so far in processbased models because usually models consider different grid cells as independent. Such lateral transports can have a significant effect and should be investigated because nonlinear effects could potentially lead to drastically new insights and better understanding compared to existing models.
The spatial upscaling was computationally feasible only because the authors emulated the original model by a matrix approach. This allows the application of sparse matrix models as well as an improved application of parallel computation methods. Another advantage of matrix models is that they allow a rigorous mathematical analysis, something the authors did not do in this manuscript (it was not their goal) but can be done in the future based on the matrix reformulation. This was not possible with the original implementation. Furthermore, I appreciate all the effort the authors showed in reimplementing an existing model with a matrix approach, emulating the original simulation results very vell. In this regard I like also the explanation of the matrix shape in Section 3.3. Nevertheless, I wished I could have "seen" a matrix, at least as a schematic block matrix after the authors speaking so much about matrices.
The presentation and the writing is clear, explaining the model calibration and simulation results as well as model limitiations and future opportunities very well. Sometimes though the text appears too lengthy in my opinion. This starts with the abstract and continues with quite some overlap in the sections about calibration, results, simulations and limitations. Furthermore, in particular in later sections I was overwhelmed by an extensive use of potentially unnecessary numbers.
I furthermore do have some issues about an easy reproducibility of the method, because to me it seems that in Section 2 some formulas are incorrect and notation is not precise. Well, either the formulas are incorrect or I understood them wrong, neither option is preferrable. In particular I consider the statement (p. 3, l. 71): "We expect our mathematical development of CEDYNAM to facilitate its reproduction and incorporation in LSMs such as ORCHIDEE, DayCent, and others." to be pretty bold. Under this point of view formulas and notation should show no flaws.
This starts with Table 1, which is in general very nice, but it is incomplete, some symbols that are used later on do not appear here. This was sometimes annoying for me while reading. For example I* was never properly introduced. On p.5, l. 132 it is stated that inputs are disturbed along D, which I do not understand.
Here my issues with notation and formulas, which should be thoroughly checked:
p. 8. l. 1: Should it be Δ_{i}? Is it used to compute d_{0}=0, d_{1}=d_{0}+Δ_{0} and so forth?
p. 8. l. 183: k_{e} is missing in Table 1. The "Total mass of soil" under the bracket refers to total mass of soild in hill slopes? Why is it decreasing with h? To me this looks like we have an infinite erosion if we do not have any hill slopes, isn't this going in the wrong direction?
p. 10, l. 202. If there is going to be a new input flux, should there also be a new output flux? The input must come from somewhere. Please also indicate where the new input flux goes, I do not want to guess here.
p. 10, l. 213. The way k_{τ} (missing in Table 1) is defined two lines later, it seems wrong to me to call it a flux. It is a rate (dimension 1/time). It will only become a flux once it is multiplied with a pool content. This leads to major confusions for me later on.
p. 11, l. 235: Please use other notation than (a, b) for the indexing of the sum, the two letters are already taken. It becomes very confusing this way. Again, I think that if k_{τ} is a rate rather than a flux, then k_{s }will be as well because all the other factors in Eq. 11 are dimensionless.
p. 10, l. 241 "all PFTs" should rather be "all PFTs but p_{0}", right?
p.12, l. 247: "Such input flux". I disagree again, same problem. It is not yet a flux because it is not yet multiplied with a carbon stock, which should be S_{[(a, b), ...]} here?
p. 12, l. 257: The "P" here looks different from the ones introduced on p. 10, l. 221. Furthermore, I am not sure whether it belongs here in the first place, well the absolute value is unecessary in any case. But why multiply by the number of nonzero PFTs in cell (a, b)? Shouldn't this be already included in k_{τ} already? Could you write it down explictly for yourself without the P but a second sum instead and check whether it is correcct this way?
p. 12, l. 260: In k_{s} I think that source and target are confused.
p.13, l. 265: What is k_{t}^{*}?
p. 13. l. 266. Should k_{s} with source (a, b) be multiplied with some stock indexed by (a, b) instead (x, y).
I obviously do have some confusions about the firsorder description, where sources and targets seem not to match, at least in my head. So would like to encourgae the authors to carefully check the notatian and the formulas again, along with their implementation.
Small issues:
 In general units are sometimes italic and sometimes not, sometimes with a space between the number and the unit, sometimes without.
 The use of singular and plurar gets mixed up quite often.
 p. 3, l. 86: "approach" > "an approach"?
 p. 3, l. 88: Table 1 > (Table 1)?
 Table 1:
  Description sometimes ends with a period, sometimes not.
  The adimensional respiration rates of carbon k_{r} actually do have a dimension: 1/day.
  I^{[c_j, c_j]}: One of the j's should be an i. Probably the first one, then please also adapt the description, to make it consistent with k_{t}.
  Then it is a little unfortunate to use ω and w for the depth to bedrock and the flow accumulation, respectively.
 p. 7, l. 145: I am not sure if such a procedure necessarily converges to a pullback attractor (given there is one in the first place). The pullback attractor is reached when starting the simulation earlier and earlier, basically moving toward an infinite simulation history.
 p.7, l. 152: Why now change the notation from S, I, and k to x, B, and A?
 p. 10, l. 243: "soi"
 p.14, l. 280: The seventhlargest what?
 p. 15, l. 308: u_{i} instead of p_{i}?
 p. 15, l. 312: "P is Panagos", what does this mean?
 p.15. unnamed formula: what is r(y, m)?
 p. 18, l. 398, 399: What are Cfactor and "R factor"? Please also note the different way of writing them.

AC2: 'Reply on RC2', Arthur Nicolaus Fendrich, 15 Jul 2022
Dear Dr. Metzler,
We would like to thank you for the very relevant comments about our work.
We answer below the comments, and we will fix all the issues in the next round of manuscript revision.Reviewer: "In this regard I like also the explanation of the matrix shape in Section 3.3. Nevertheless, I wished I could have "seen" a matrix, at least as a schematic block matrix after the authors speaking so much about matrices."
Answer: We agree that it would be interesting to visualize the matrix structure. We will add this image and information in the next manuscript version.
Reviewer: "Sometimes though the text appears too lengthy in my opinion. This starts with the abstract and continues with quite some overlap in the sections about calibration, results, simulations and limitations. Furthermore, in particular in later sections I was overwhelmed by an extensive use of potentially unnecessary numbers."
Answer: It was a common point in both Reviewer's comments that some text sections are lengthy. In order to fix this problem and improve readability, we will rewrite several passages in the next version of the manuscript, with extra attention to the aforementioned unnecessary numbers.
Reviewer: "I furthermore do have some issues about an easy reproducibility of the method, because to me it seems that in Section 2 some formulas are incorrect and notation is not precise. Well, either the formulas are incorrect or I understood them wrong, neither option is preferrable (...) Under this point of view formulas and notation should show no flaws. (...) This starts with Table 1, which is in general very nice, but it is incomplete (...). So would like to encourgae the authors to carefully check the notatian and the formulas again, along with their implementation."
Answer: We thank very much the Reviewer for pointing out all the problems with the notation adopted. We strongly agree that this is critical for adopting and disseminating the matrix approach and for future developments of the method. In order to fix all issues, all the formulas of the manuscript will be carefully reviewed and rewritten to prioritize precision, conciseness, and clarity. We hope the new notation will be precise and resolve any confusion that may have arisen....
Reviewer: "This allows the application of sparse matrix models as well as an improved application of parallel computation methods. Another advantage of matrix models is that they allow a rigorous mathematical analysis something the authors did not do in this manuscript (it was not their goal) but can be done in the future based on the matrix reformulation."
Answer: We truly appreciate this suggestion and agree that it would be an interesting work to develop in the future.
Arthur Nicolaus Fendrich et al.
Model code and software
Source codes Fendrich et al. https://doi.org/10.5281/zenodo.6553890
Arthur Nicolaus Fendrich et al.
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