Centroids in second-order conservative remapping schemes on spherical coordinates

Saito, Fuyuki

doi:10.5194/egusphere-2025-1268

Preprints

https://doi.org/10.5194/egusphere-2025-1268

Preprints

06 Jun 2025

| 06 Jun 2025

Centroids in second-order conservative remapping schemes on spherical coordinates

Fuyuki Saito

Abstract. The transformation of data from one grid system to another is common in climate studies. Among the many schemes used for such transformations is second-order conservative remapping. In particular, a second-order conservative remapping scheme to work on the general grids of a sphere, either directly or indirectly, has served as an important base in a variety of studies.

In this study, the author describes a fundamental problem in the derivation of the method proposed by a pioneer study relating to the treatment of the centroid used as a reference point for the second-order terms in the longitudinal direction. In principle, use of the original formulation has a potential to cause damage to the entire remapping result. However, a preprocessing procedure on the longitude coordinate suggested in the algorithm for other objectives tends to minimize or even erase the error as a side effect in many, if not most, typical applications. In this study, two alternative formulations are proposed and tested and are shown to work in a simple application.

Received: 18 Mar 2025 – Discussion started: 06 Jun 2025

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Journal article(s) based on this preprint

24 Jun 2026

Rederivation of the centroid formulation in a second-order conservative remapping scheme on spherical coordinates

Fuyuki Saito

Geosci. Model Dev., 19, 5423–5438, https://doi.org/10.5194/gmd-19-5423-2026,https://doi.org/10.5194/gmd-19-5423-2026, 2026

Short summary

Fuyuki Saito

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-1268', Phil Jones, 03 Jul 2025

This paper is a re-submission of an earlier paper that describes a potential problem in the SCRIP high-order conservative remapping algorithm that requires a correction. This revision has a slightly better understanding of the underlying algorithm but still has some significant problems and the testing appears to be incorrect.
As in the first submission, the author attempts to derive a flux distribution from a Taylor series in section 2.1. As I wrote in the first review, this derivation is incorrect and the author should not attempt to derive this from a Taylor series. In particular, the constraints in equations 4,5 do not follow uniquely from 3 without additional assumptions. For example, it cannot always be true that a flux evaluated at c_n will be the mean. DK87 actually makes this point in the sentence referenced by the author in which DK87 says it is a Taylor series "only if the mean is _assumed_ to be located at the centroid". Instead, both DK87 and J99 use the distribution of the flux as
[can't seem to upload an equation image but latex form is: f_n = \overline{f}_n + \nabla_n f\cdot({\vec{r}} - \vec{r}_n) ]
which is a construction that automatically satisfies Eq 1 as long as r_n is the centroid so that the second term integrates to zero. By making the leap from his eq 3 to the constraints in 4,5, the author is essentially making the same assumption that the original DK87 and J99 approaches have done by construction, so it's better to start with that anyway and state it as a common constructions that satisifies eq 1. The only reason a Taylor-like expansion is used is so that DK87 can claim the distribution is second-order as long as the gradient is first-order. The author should simply present the distribution as is done in DK87 and J99 to make the assumption explicit and avoid the incorrect derivation in 2.1.
Section 2.2, line 132, the author calls phi_c, theta_c a reference point and mentions that J99 calls it a centroid. In fact, this is still required to be the centroid and it is not an arbitrary reference point.
The actual demonstration of the issue with longitudinal weights appears correct and the pivot fix seems reasonable as shown in Fig 1.
An additional proposed fix around line 250 should be removed and the C Scheme later also removed. There is a reason why metric terms are included in coordinate system transformations and relevant operators. Removing the cos(lat) weighting in this fix is likely to introduce significant error in more general meshes, especially near the polar singularities.
The biggest problem with the current paper is in the testing section. A couple things are simply oversights: The RLL mesh description in line 337 is wrong and needs to be corrected. Also, they refer to a Scheme O, which I assume was the original formulation which they have now named Scheme N in the previous section. The author needs to pick a consistent name across the sections.
More seriously, most of the tests use a high-res mesh as the source and perform remaps to coarser meshes. The high-order terms only impact a coarse-to-fine remap so these tests need to use the coarse meshes as the source mesh and the finer mesh as a destination. The fine-to-coarse operation in the tests presented are an averaging and remove most of the impact of the high-order terms. Indeed, in many cases where the overlap is complete, the high-order terms should integrate exactly to zero.
The global offset case does not fix the multi-value longitude issue. It only shifts the problem to a different longitudinal branch cut. So the high errors in these tests are more likely an incorrect correction of the multi-value longitude. If a 2-d map of the solution was shown, I would guess all of the errors would be along a branch cut in the domain and are separate from the error in the formulation that this paper is trying to address. In general, it would be a good idea to show a 2-d greyscale or colormap of the resulting field after a remap in addition to showing the global error norms to demonstrate there are no such artifacts. It's possible they would still not show up in these cases due to the fine-to-coarse averaging, but they would absolutely show up in a coarse-to-fine remap.
In the end, there are many issues with SCRIP, including probably the modification in this paper, but this paper requires significant revision to make that case.

Citation: https://doi.org/10.5194/egusphere-2025-1268-RC1
- AC1: 'Reply on RC1', Fuyuki Saito, 24 Dec 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1268/egusphere-2025-1268-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1268-AC1
RC2:
'Comment on egusphere-2025-1268', Anonymous Referee #2, 27 Nov 2025
This manuscript revisits the analytical foundation of the second-order conservative remapping scheme presented in Jones (1999, J99). The authors identify an inconsistency in J99’s longitudinal term and propose a corrected formulation that resolves the issue. This topic is of continued interest, and clarifying the original derivation is potentially helpful for users and developers of legacy tools derived from J99.
At the same time, I believe the manuscript would benefit from substantial revision to improve clarity of motivation, definitions, and scope. Many statements are broadly phrased and may give readers the impression of general geometric validity, even though the proposed corrections remain tied to the coordinate-specific structure of regular latitude–longitude grids. A clearer presentation would greatly strengthen the manuscript’s impact and accessibility.
I offer the following comments in the hope that they will assist the authors in presenting their valuable observations in the clearest possible way.

1. Clarifying the motivation and historical context
The manuscript motivates its analysis by stating (Lines 29–30) that J99 “extends the DK87 theory to spherical coordinates, offering an approach that can be applied to any type of grid on a sphere.” This wording may lead some readers to infer that J99 handles arbitrary spherical geometry correctly, whereas in practice:
SCRIP, which implements J99, computes weights using line integrals in latitude–longitude space,

and treats edges as straight lines in the coordinate chart,

rather than as great-circle arcs or geodesic boundaries on the sphere.

Thus, J99 is applicable to any grid in the sense of an algorithm that can be run on arbitrary inputs, but it does not necessarily handle geometric properties of arbitrary spherical grids appropriately. A more precise description of this distinction may strengthen the manuscript.
Similarly, the discussion of Climate Data Operators (CDO) on Lines 34–40 might be reconsidered. The phrase “have once included” is somewhat ambiguous, and current CDO implementations no longer rely on the SCRIP-based second-order scheme. Today, CDO’s first-order conservative remapping is based entirely on YAC. A second-order scheme was also planned, again using YAC, but has not been deployed so far because the available YAC implementation is parallel (MPI-based), and porting it into CDO’s primarily serial remapping framework has turned out to be technically nontrivial. This decision was driven by software-engineering constraints, not by any concerns about YAC’s geometric formulation. Thus, citing CDO as an example of active reliance on the J99 formulation may be less compelling than intended.
In addition, more recent work such as the GECoRe paper provides a geometrically exact remapping framework between regular latitude–longitude and cubed-sphere grids, together with closed-form expressions for the relevant integrals (e.g. its Appendix A). It would be helpful if the authors could clearly position the present manuscript relative to such geometry-exact approaches — for example, by explaining that their contribution is a correction within the J99/SCRIP analytical framework, rather than an alternative to fully geometric methods like GECoRe, and by briefly discussing what specific gap or practical advantage this correction offers in that broader context.
Nevertheless, because J99/SCRIP remain historically important and are still used in legacy workflows, there is value in clarifying the original analytical inconsistency. Presenting this as a pedagogically motivated correction, and clearly situating it with respect to later developments such as GECoRe and geometry-aware couplers (YAC, TempestRemap, etc.), may resonate better with modern readers.

2. Clarifying terminology and coordinate dependence
(a) Use of “spherical coordinate system”
At Line 31, the manuscript refers to “the spherical coordinate system,” but the meaning of this term is not formally defined. In practice:
There is no unique “spherical coordinate system”; multiple conventions exist.

Modern conservative remapping frameworks (e.g., YAC, TempestRemap) deliberately avoid relying on latitude–longitude formulas and instead express all geometry in 3D Cartesian coordinates, ensuring rotational invariance.

The manuscript would be stronger if it were clarified early on that the derivation is performed entirely within the latitude–longitude coordinate chart, and that this choice is intentional.
(b) Inner product discussion (Lines 112–113)
The sentence
“The inner product on spherical coordinates is not simply a component-wise product as in Cartesian coordinates…”
may inadvertently suggest a misuse that most current geometry-aware remapping codes already avoid. Since $\theta$ and $\phi$ are angular parameters rather than Cartesian axes, they are never combined using componentwise Cartesian logic in modern implementations.
If the intention is to emphasize that the gradient operator in spherical coordinates includes metric factors and basis-vector rotation, it may help to express this more precisely. Equation (8) can in fact be written cleanly in terms of 3D vectors, making all inner products well-defined and coordinate-invariant.
(c) Misuse of the term “centroid”
Throughout the manuscript — including in the title — the term centroid is used without a consistent definition. Equation (7) defines the geometric mass centroid of a cell, which lies inside the sphere and might then need to be projected outward onto the spherical surface. However, none of the three schemes introduced later in the paper (Scheme N, Scheme P, or Scheme C) actually computes this geometric centroid (the interior point). Both the “pivot” point $(\theta_p,\phi_p)$ and the “centroid” point $(\theta_c,\phi_c)$ are coordinate-based constructions rather than true geometric centroids on the sphere. Referring to these quantities as centroids is therefore potentially misleading for readers who interpret centroid in the strict geometric sense (i.e. $\mathbf{c}_{\mathrm{geo}} = \tfrac{1}{A}\int_A \mathbf{x}, dA$, which again lies inside the sphere).
The authors might either need to justify why the flux formulation would remain valid if the reference point were taken to be the geometric centroid (which lies inside the sphere), or explicitly explain that this interior centroid must first be projected back to the spherical surface (e.g. by normalization) if a surface reference point is required.
(d) Closed-form expressions and relation to GECoRe
It may help readers if explicit closed-form formulas for the reference points in Schemes N, P, and C are included, especially since the manuscript’s title centers on “centroids.”
In this context, it may also be useful to refer to the GECoRe paper, which provides closed-form expressions (and derivations) in its Appendix A. The GECoRe paper also presents a geometrically exact regridding solution between regular latitude–longitude and cubed-sphere grids. It would be beneficial to include a brief comparison between the present manuscript and the GECoRe approach — for example, by pointing out the potential gap between the two methods and discussing the potential benefits of the present work relative to GECoRe.
3. Scope and generality
Two limitations of the current formulation deserve clearer acknowledgment:
(a) The corrections apply only within the coordinate framework of J99/SCRIP
Because SCRIP uses a latitude–longitude planar approximation, the corrected formulas presented here remain applicable only to the same class of structured grids. The method does not extend to arbitrary, non-uniform unstructured grids. A brief comment on this limitation may prevent misinterpretation.
(b) The proposed centroid/pivot definitions are meaningful only for regular RLL grids
Area-weighted averages of $\theta$ and $\phi$ (even with a $\cos\theta$ weight for longitude) do not correspond to geometric centers for general spherical polygons. Their interpretation becomes ambiguous when cells are skewed, large, or irregular.
These points do not invalidate the proposed correction — which is internally consistent within the J99 framework — but clarifying scope will improve the precision of the manuscript.

Minor Comments
Line 48: The phrase “serious damage to the remapping result” would benefit from a concrete explanation or example illustrating the numerical consequences.

Some expressions, such as those involving coordinate shifts ($\phi_{\text{ofs}}$), could be clarified to ensure readers understand exactly how invariance is being tested.

The manuscript might benefit from reorganizing Sections 2.1–2.2 to distinguish clearly between the flat Cartesian case and the spherical-coordinate case.

Reference:

Ullrich, P. A., P. H. Lauritzen, and C. Jablonowski, 2009: Geometrically Exact Conservative Remapping (GECoRe): Regular Latitude–Longitude and Cubed-Sphere Grids. Mon. Wea. Rev., 137, 1721–1741, https://doi.org/10.1175/2008MWR2817.1.

Kritsikis, E., Aechtner, M., Meurdesoif, Y., and Dubos, T.: Conservative interpolation between general spherical meshes, Geosci. Model Dev., 10, 425–431, https://doi.org/10.5194/gmd-10-425-2017, 2017.

Paul A. Ullrich and Mark A. Taylor, 2015: Arbitrary-Order Conservative and Consistent Remapping and a Theory of Linear Maps: Part 1. Mon. Wea. Rev., 143, 2419–2440, doi: 10.1175/MWR-D-14-00343.1

Paul A. Ullrich, Darshi Devendran and Hans Johansen, 2016: Arbitrary-Order Conservative and Consistent Remapping and a Theory of Linear Maps, Part 2. Mon. Weather Rev., 144, 1529-1549, doi: 10.1175/MWR-D-15-0301.1.

Moritz Hanke, René Redler, Teresa Holfeld und Maxim Yastremsky, 2016: YAC 1.2.0: new aspects for coupling software in Earth system modelling. Geoscientific Model Development, 9, 2755-2769. DOI:10.5194/gmd-9-2755-201.

Moritz Hanke und René Redler, 2019: New features with YAC 1.5.0. Reports on ICON, No 3. DOI: 10.5676/DWD_pub/nwv/icon_003.
Citation: https://doi.org/10.5194/egusphere-2025-1268-RC2
- AC2: 'Reply on RC2', Fuyuki Saito, 24 Dec 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1268/egusphere-2025-1268-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1268-AC2

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-1268', Phil Jones, 03 Jul 2025

This paper is a re-submission of an earlier paper that describes a potential problem in the SCRIP high-order conservative remapping algorithm that requires a correction. This revision has a slightly better understanding of the underlying algorithm but still has some significant problems and the testing appears to be incorrect.
As in the first submission, the author attempts to derive a flux distribution from a Taylor series in section 2.1. As I wrote in the first review, this derivation is incorrect and the author should not attempt to derive this from a Taylor series. In particular, the constraints in equations 4,5 do not follow uniquely from 3 without additional assumptions. For example, it cannot always be true that a flux evaluated at c_n will be the mean. DK87 actually makes this point in the sentence referenced by the author in which DK87 says it is a Taylor series "only if the mean is _assumed_ to be located at the centroid". Instead, both DK87 and J99 use the distribution of the flux as
[can't seem to upload an equation image but latex form is: f_n = \overline{f}_n + \nabla_n f\cdot({\vec{r}} - \vec{r}_n) ]
which is a construction that automatically satisfies Eq 1 as long as r_n is the centroid so that the second term integrates to zero. By making the leap from his eq 3 to the constraints in 4,5, the author is essentially making the same assumption that the original DK87 and J99 approaches have done by construction, so it's better to start with that anyway and state it as a common constructions that satisifies eq 1. The only reason a Taylor-like expansion is used is so that DK87 can claim the distribution is second-order as long as the gradient is first-order. The author should simply present the distribution as is done in DK87 and J99 to make the assumption explicit and avoid the incorrect derivation in 2.1.
Section 2.2, line 132, the author calls phi_c, theta_c a reference point and mentions that J99 calls it a centroid. In fact, this is still required to be the centroid and it is not an arbitrary reference point.
The actual demonstration of the issue with longitudinal weights appears correct and the pivot fix seems reasonable as shown in Fig 1.
An additional proposed fix around line 250 should be removed and the C Scheme later also removed. There is a reason why metric terms are included in coordinate system transformations and relevant operators. Removing the cos(lat) weighting in this fix is likely to introduce significant error in more general meshes, especially near the polar singularities.
The biggest problem with the current paper is in the testing section. A couple things are simply oversights: The RLL mesh description in line 337 is wrong and needs to be corrected. Also, they refer to a Scheme O, which I assume was the original formulation which they have now named Scheme N in the previous section. The author needs to pick a consistent name across the sections.
More seriously, most of the tests use a high-res mesh as the source and perform remaps to coarser meshes. The high-order terms only impact a coarse-to-fine remap so these tests need to use the coarse meshes as the source mesh and the finer mesh as a destination. The fine-to-coarse operation in the tests presented are an averaging and remove most of the impact of the high-order terms. Indeed, in many cases where the overlap is complete, the high-order terms should integrate exactly to zero.
The global offset case does not fix the multi-value longitude issue. It only shifts the problem to a different longitudinal branch cut. So the high errors in these tests are more likely an incorrect correction of the multi-value longitude. If a 2-d map of the solution was shown, I would guess all of the errors would be along a branch cut in the domain and are separate from the error in the formulation that this paper is trying to address. In general, it would be a good idea to show a 2-d greyscale or colormap of the resulting field after a remap in addition to showing the global error norms to demonstrate there are no such artifacts. It's possible they would still not show up in these cases due to the fine-to-coarse averaging, but they would absolutely show up in a coarse-to-fine remap.
In the end, there are many issues with SCRIP, including probably the modification in this paper, but this paper requires significant revision to make that case.

Citation: https://doi.org/10.5194/egusphere-2025-1268-RC1
- AC1: 'Reply on RC1', Fuyuki Saito, 24 Dec 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1268/egusphere-2025-1268-AC1-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1268-AC1
RC2:
'Comment on egusphere-2025-1268', Anonymous Referee #2, 27 Nov 2025
This manuscript revisits the analytical foundation of the second-order conservative remapping scheme presented in Jones (1999, J99). The authors identify an inconsistency in J99’s longitudinal term and propose a corrected formulation that resolves the issue. This topic is of continued interest, and clarifying the original derivation is potentially helpful for users and developers of legacy tools derived from J99.
At the same time, I believe the manuscript would benefit from substantial revision to improve clarity of motivation, definitions, and scope. Many statements are broadly phrased and may give readers the impression of general geometric validity, even though the proposed corrections remain tied to the coordinate-specific structure of regular latitude–longitude grids. A clearer presentation would greatly strengthen the manuscript’s impact and accessibility.
I offer the following comments in the hope that they will assist the authors in presenting their valuable observations in the clearest possible way.

1. Clarifying the motivation and historical context
The manuscript motivates its analysis by stating (Lines 29–30) that J99 “extends the DK87 theory to spherical coordinates, offering an approach that can be applied to any type of grid on a sphere.” This wording may lead some readers to infer that J99 handles arbitrary spherical geometry correctly, whereas in practice:
SCRIP, which implements J99, computes weights using line integrals in latitude–longitude space,

and treats edges as straight lines in the coordinate chart,

rather than as great-circle arcs or geodesic boundaries on the sphere.

Thus, J99 is applicable to any grid in the sense of an algorithm that can be run on arbitrary inputs, but it does not necessarily handle geometric properties of arbitrary spherical grids appropriately. A more precise description of this distinction may strengthen the manuscript.
Similarly, the discussion of Climate Data Operators (CDO) on Lines 34–40 might be reconsidered. The phrase “have once included” is somewhat ambiguous, and current CDO implementations no longer rely on the SCRIP-based second-order scheme. Today, CDO’s first-order conservative remapping is based entirely on YAC. A second-order scheme was also planned, again using YAC, but has not been deployed so far because the available YAC implementation is parallel (MPI-based), and porting it into CDO’s primarily serial remapping framework has turned out to be technically nontrivial. This decision was driven by software-engineering constraints, not by any concerns about YAC’s geometric formulation. Thus, citing CDO as an example of active reliance on the J99 formulation may be less compelling than intended.
In addition, more recent work such as the GECoRe paper provides a geometrically exact remapping framework between regular latitude–longitude and cubed-sphere grids, together with closed-form expressions for the relevant integrals (e.g. its Appendix A). It would be helpful if the authors could clearly position the present manuscript relative to such geometry-exact approaches — for example, by explaining that their contribution is a correction within the J99/SCRIP analytical framework, rather than an alternative to fully geometric methods like GECoRe, and by briefly discussing what specific gap or practical advantage this correction offers in that broader context.
Nevertheless, because J99/SCRIP remain historically important and are still used in legacy workflows, there is value in clarifying the original analytical inconsistency. Presenting this as a pedagogically motivated correction, and clearly situating it with respect to later developments such as GECoRe and geometry-aware couplers (YAC, TempestRemap, etc.), may resonate better with modern readers.

2. Clarifying terminology and coordinate dependence
(a) Use of “spherical coordinate system”
At Line 31, the manuscript refers to “the spherical coordinate system,” but the meaning of this term is not formally defined. In practice:
There is no unique “spherical coordinate system”; multiple conventions exist.

Modern conservative remapping frameworks (e.g., YAC, TempestRemap) deliberately avoid relying on latitude–longitude formulas and instead express all geometry in 3D Cartesian coordinates, ensuring rotational invariance.

The manuscript would be stronger if it were clarified early on that the derivation is performed entirely within the latitude–longitude coordinate chart, and that this choice is intentional.
(b) Inner product discussion (Lines 112–113)
The sentence
“The inner product on spherical coordinates is not simply a component-wise product as in Cartesian coordinates…”
may inadvertently suggest a misuse that most current geometry-aware remapping codes already avoid. Since $\theta$ and $\phi$ are angular parameters rather than Cartesian axes, they are never combined using componentwise Cartesian logic in modern implementations.
If the intention is to emphasize that the gradient operator in spherical coordinates includes metric factors and basis-vector rotation, it may help to express this more precisely. Equation (8) can in fact be written cleanly in terms of 3D vectors, making all inner products well-defined and coordinate-invariant.
(c) Misuse of the term “centroid”
Throughout the manuscript — including in the title — the term centroid is used without a consistent definition. Equation (7) defines the geometric mass centroid of a cell, which lies inside the sphere and might then need to be projected outward onto the spherical surface. However, none of the three schemes introduced later in the paper (Scheme N, Scheme P, or Scheme C) actually computes this geometric centroid (the interior point). Both the “pivot” point $(\theta_p,\phi_p)$ and the “centroid” point $(\theta_c,\phi_c)$ are coordinate-based constructions rather than true geometric centroids on the sphere. Referring to these quantities as centroids is therefore potentially misleading for readers who interpret centroid in the strict geometric sense (i.e. $\mathbf{c}_{\mathrm{geo}} = \tfrac{1}{A}\int_A \mathbf{x}, dA$, which again lies inside the sphere).
The authors might either need to justify why the flux formulation would remain valid if the reference point were taken to be the geometric centroid (which lies inside the sphere), or explicitly explain that this interior centroid must first be projected back to the spherical surface (e.g. by normalization) if a surface reference point is required.
(d) Closed-form expressions and relation to GECoRe
It may help readers if explicit closed-form formulas for the reference points in Schemes N, P, and C are included, especially since the manuscript’s title centers on “centroids.”
In this context, it may also be useful to refer to the GECoRe paper, which provides closed-form expressions (and derivations) in its Appendix A. The GECoRe paper also presents a geometrically exact regridding solution between regular latitude–longitude and cubed-sphere grids. It would be beneficial to include a brief comparison between the present manuscript and the GECoRe approach — for example, by pointing out the potential gap between the two methods and discussing the potential benefits of the present work relative to GECoRe.
3. Scope and generality
Two limitations of the current formulation deserve clearer acknowledgment:
(a) The corrections apply only within the coordinate framework of J99/SCRIP
Because SCRIP uses a latitude–longitude planar approximation, the corrected formulas presented here remain applicable only to the same class of structured grids. The method does not extend to arbitrary, non-uniform unstructured grids. A brief comment on this limitation may prevent misinterpretation.
(b) The proposed centroid/pivot definitions are meaningful only for regular RLL grids
Area-weighted averages of $\theta$ and $\phi$ (even with a $\cos\theta$ weight for longitude) do not correspond to geometric centers for general spherical polygons. Their interpretation becomes ambiguous when cells are skewed, large, or irregular.
These points do not invalidate the proposed correction — which is internally consistent within the J99 framework — but clarifying scope will improve the precision of the manuscript.

Minor Comments
Line 48: The phrase “serious damage to the remapping result” would benefit from a concrete explanation or example illustrating the numerical consequences.

Some expressions, such as those involving coordinate shifts ($\phi_{\text{ofs}}$), could be clarified to ensure readers understand exactly how invariance is being tested.

The manuscript might benefit from reorganizing Sections 2.1–2.2 to distinguish clearly between the flat Cartesian case and the spherical-coordinate case.

Reference:

Ullrich, P. A., P. H. Lauritzen, and C. Jablonowski, 2009: Geometrically Exact Conservative Remapping (GECoRe): Regular Latitude–Longitude and Cubed-Sphere Grids. Mon. Wea. Rev., 137, 1721–1741, https://doi.org/10.1175/2008MWR2817.1.

Kritsikis, E., Aechtner, M., Meurdesoif, Y., and Dubos, T.: Conservative interpolation between general spherical meshes, Geosci. Model Dev., 10, 425–431, https://doi.org/10.5194/gmd-10-425-2017, 2017.

Paul A. Ullrich and Mark A. Taylor, 2015: Arbitrary-Order Conservative and Consistent Remapping and a Theory of Linear Maps: Part 1. Mon. Wea. Rev., 143, 2419–2440, doi: 10.1175/MWR-D-14-00343.1

Paul A. Ullrich, Darshi Devendran and Hans Johansen, 2016: Arbitrary-Order Conservative and Consistent Remapping and a Theory of Linear Maps, Part 2. Mon. Weather Rev., 144, 1529-1549, doi: 10.1175/MWR-D-15-0301.1.

Moritz Hanke, René Redler, Teresa Holfeld und Maxim Yastremsky, 2016: YAC 1.2.0: new aspects for coupling software in Earth system modelling. Geoscientific Model Development, 9, 2755-2769. DOI:10.5194/gmd-9-2755-201.

Moritz Hanke und René Redler, 2019: New features with YAC 1.5.0. Reports on ICON, No 3. DOI: 10.5676/DWD_pub/nwv/icon_003.
Citation: https://doi.org/10.5194/egusphere-2025-1268-RC2
- AC2: 'Reply on RC2', Fuyuki Saito, 24 Dec 2025
  
  The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2025/egusphere-2025-1268/egusphere-2025-1268-AC2-supplement.pdf
  
  Citation: https://doi.org/10.5194/egusphere-2025-1268-AC2

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Fuyuki Saito on behalf of the Authors (20 Feb 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (18 Apr 2026) by James Kelly

RR by Phil Jones (23 Apr 2026)

RR by Anonymous Referee #2 (18 May 2026)

Suggestions for revision or reasons for rejection

In Section 2.2, around line 130, the manuscript states that modern implementations formulate the relevant inner products in three-dimensional Euclidean space to obtain coordinate-invariant expressions, citing Ullrich et al. (2009). This discussion could be strengthened by also citing recent work on closed-form face-level geometry for spherical polygons. For example, Chen et al. (2026) provide formulas for the mass centroid, first moment, and area of arbitrary spherical polygonal faces, including boundary-integral expressions for great-circle polygons and correction terms for constant-latitude edges. This reference would help clarify the distinction between the coordinate-invariant mass centroid of a spherical face and the pivot longitude introduced later, which is a metric-weighted reference coordinate for the longitudinal second-order moment.

The discussion around line 270 is difficult to follow because it moves from the RLL-grid case to unstructured source grids without explaining the connection. The preceding discussion focuses on regular latitude-longitude cells, where the mid-longitude offset can coincide with the pivot longitude and reduce or eliminate the inconsistency. Unstructured source grids are a different geometric setting, often represented by general spherical polygons with great-circle-arc edges. If the intended point is that CDO users cannot encounter this issue for unstructured source grids because REMAPCON2 is unavailable, this should be stated directly. Otherwise, the paragraph may suggest that the RLL-specific cancellation also applies to unstructured grids, which would require justification.

Around line 290, the manuscript mentions possible inaccuracies in J99/SCRIP intersection computation, but the type of inaccuracy is unclear. Please clarify whether this refers to the geometric treatment of edge types, such as great-circle versus constant-latitude edges, or to floating-point robustness in edge-edge intersections. This distinction matters because the experiments are intended to isolate sensitivity to inconsistent reference longitudes. If overlap-construction errors are also present, it may be difficult to attribute the observed errors to the longitude-offset issue alone. Chen et al. (2026) may be useful here, as they discuss robust spherical edge-edge intersections for different edge types and common sources of numerical error.

As a minor comment, the manuscript could be more compact. The contribution is a focused correction to the J99/SCRIP second-order conservative remapping formulation, but the central point only becomes fully clear in Sections 2.2 and 2.3. Stating the main goal earlier and shortening some background or derivational material would make the structure clearer and better highlight the proposed correction.

Reference

Chen, H., Ullrich, P. A., Panetta, J., Marsico, D., Hanke, M., Jain, R., Zhang, C., and Jacob, R. L.: Accurate and Robust Geometric Algorithms for Regridding on the Sphere, EGUsphere, 2026, 1–40, https://doi.org/10.5194/egusphere-2026-636, 2026.

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ED: Publish subject to minor revisions (review by editor) (18 May 2026) by James Kelly

AR by Fuyuki Saito on behalf of the Authors (26 May 2026) Author's response Author's tracked changes Manuscript

ED: Publish subject to minor revisions (review by editor) (04 Jun 2026) by James Kelly

AR by Fuyuki Saito on behalf of the Authors (07 Jun 2026) Author's response Author's tracked changes Manuscript

ED: Publish as is (10 Jun 2026) by James Kelly

AR by Fuyuki Saito on behalf of the Authors (11 Jun 2026) Author's response Manuscript

Journal article(s) based on this preprint

24 Jun 2026

Rederivation of the centroid formulation in a second-order conservative remapping scheme on spherical coordinates

Fuyuki Saito

Geosci. Model Dev., 19, 5423–5438, https://doi.org/10.5194/gmd-19-5423-2026,https://doi.org/10.5194/gmd-19-5423-2026, 2026

Short summary

Fuyuki Saito

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https://doi.org/10.5194/egusphere-2025-1268-supplement

Data sets

Resources of Saito (submitted to GMD) — software and experiment data archives Fuyuki Saito https://doi.org/10.5281/zenodo.10892796

Model code and software

SCRIP-p (p is for pivot) Fuyuki Saito https://github.com/saitofuyuki/scrip-p

Fuyuki Saito

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Month	HTML	PDF	XML	Total
Jun 2025	455	85	25	565
Jul 2025	185	45	30	260
Aug 2025	1,968	35	9	2,012
Sep 2025	8,210	39	5	8,254
Oct 2025	290	60	10	360
Nov 2025	193	90	30	313
Dec 2025	113	70	50	233
Jan 2026	566	40	25	631
Feb 2026	121	45	25	191
Mar 2026	308	124	10	442
Apr 2026	82	30	1	113
May 2026	23	9	1	33
Jun 2026	7	3	5	15
Jul 2026	6	6	2	14

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A second-order conservative remapping is a common and standard method for transformation of data from one grid system to another in climate studies. The author describes a fundamental problem in the derivation of the method in spherical coordinates proposed by a pioneer study, which is left unresolved and unrealized. It has a potential to damage the remapping, however no significant impacts are expected in the past studies, because of a formulation introduced for the other objective.


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