| 06 Jun 2025
Centroids in second-order conservative remapping schemes on spherical coordinates
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.
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.
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