the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Truly Conserving with Conservative Remapping Methods
Abstract. Conservative mapping of data from one horizontal grid to another should preserve certain integral or mean properties of the original data. This may be essential in some model applications, including ensuring realistic exchange of energy and mass between coupled model components. It can also be essential for certain types of analysis, such as evaluating how far a system is from an equilibrium state. For some common grids, existing remapping algorithms may fail to perfectly represent the shapes and sizes of grid cells, which leads to errors in the remapped fields. A procedure is presented here that enables users to rely on the mapping weights generated by remapping algorithms but corrects for their deficiencies. With this procedure, for a given pair of source and destination grids, a single set of remapping weights can be applied to remap any variable, including those with grid cells that are partially or fully masked.
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Karl E. Taylor
Status: open (until 30 Oct 2023)

CC1: 'Comment on gmd2023177', Moritz Hanke, 11 Sep 2023
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Hello,
You hopefully do not mind, if I give some comments and remarks on your paper.
Thank you very much for this paper! It truly helps with implementing coupling software for climate models and gives great guidelines for users on how to use conservative remapping.My experience with conservative remapping is limited to ESMF, SCRIP, and XIOS, whose implementations I read at some point. In addition, I implemented the conservative remapping algorithm in YAC.
Regarding the misrepresentation of true grid cell shapes:
 To my knowledge it is true, that ESMF represents all edges using great circles, which leads to the issues described in this paper.
 In SCRIP, vertices are connected using rhumb lines, which results in the correct representation of edges located on circles of longitude and latitude, but is wrong for edges on great circles. Due to the way intersections and areas are computed in SCRIP, accuracy can become very low towards the pole. Therefore, the library has the option to project cells, which are close to the pole, towards the equator. This in turn changes the representation of the edges to something else. It also produces a lot of other issues (see [1] and [2]).
 As for ESMF, XIOS also only uses great circles to represent edges. However, it approximates edges on circles of latitudes (and small circles, if I remember correctly) using small great circle segments, which should significantly reduce the error in the misrepresentation of the grid cell shapes.
 YAC supports edges on great circles and circles of longitude/latitude. Therefore, it should not misrepresent the shapes of grid cells, unless the edges are on different circles than ones already mentioned.
Since, half of the implementations for conservative interpolation, which I personally know, more or less correctly represent the the true grid cell shapes for most common grid types, I would not agree with the statement: “many (perhaps all) of these packages slightly misrepresent the true grid cell shapes for certain common grids”.
Line 7578:
This is true for ESMF. SCRIP will assume the correct shape for regular lon/lat grids, if no tranformation for cell close to the poles is applied. XIOS and YAC will also assume the correct shape, if the grid is defined as a regular lon/lat grid (also works for Gaussian reduced grids).Line 110119:
Great quantification of error in the misrepresent of the grid shape.Line 150:
It could be noted that the computation of the global mean, may be time consuming and have bad scaling behavior. However, since regular lon/lat grids are in my experience seldomly used for high resolutions runs, this might not be an issue.Line 243:
Recent releases of YAC contain a feature, which allows the user the provide fractions along with the source field for the interpolation (see [3]). These fractions are taken into account as described in formula (16) of the paper. My implementation was based on the work of the OASIS development team. A similar feature will most probably be part of the next OASIS release.Line 354 “2) the unmasked fractions on the destination grid are defined by (15)”:
This sums up an an important point for users that set up the grids and masks for two coupled models!Paragraph starting at line 360:
If the remapping algorithm supports concave cells, an alternative approach would be to add additional vertices along the edges located on latitude circles (as done automatically by XIOS). This should produce the same results as using subcells, but directly produces weights that can be used without the need for any postprocessing of the weights. A drawback of this approach may be, that the remapping algorithm might support concave cells by triangulation, which could reduce accuracy.Paragraph starting at line 471:
Alternatively, if the source fractions are changed, the source model can multiply the source field with the source fractions and interpolate the source fractions along with the source fields. The target model can then adjust the received field using the received fractions.Line 495497:
What about second order conservative remapping (implemented in ESMF, XIOS, and YAC based on [4]).List entry starting at line 512:
In CDO, only the implementation of conservative remapping is based on YAC.List entry starting at line 518:
MCT itself does not have any weight computation functionality, therefore it probably does not fit in this list.List entry starting at line 524:
OASIS either gets its weights from offline weight computation tools (e.g. ESMF) or it can generate weights for conservative interpolation using SCRIP. MCT is used for the parallel sparse matrix vector multiplication, but not for the weight computation itself.List entry starting at line 528:
ESMF, XIOS, and YAC use a significantly different algorithm from SCRIP. Therefore, I would not agree that “most remapping packages have been based on SCRIP”.General remarks:
 CERFACS did a benchmark for regridding libraries, which might be interesting (see [5]).
 Regridding libraries usually provide two normalization options for the first order conservative remapping (destarea and fracarea). If I understood the paper correctly, all formulas assume that destarea is being used for the weight computation. You could also discuss the implications of the fracarea normalization method.
 Is it possible to also apply the findings of this paper to the second order conservative remapping algorithm described in [4]?
Thanks again for the great paper!
With best regards,
Moritz Hanke[1]: Valcke, S.; Piacentini, A. Analysis of SCRIP Conservative Remapping in OASIS3MCT—Part A, Technical Report TR/CMGC/19129, CERFACS, France. 2019. Available online: https://oasis.cerfacs.fr/wpcontent/uploads/sites/114/2021/08/GLOBC_TR_ValckeSCRIP_CONSERV_TRNORM_partA_2019.pdf (accessed on 18 January 2022).
[2]: Jonville, G.; Valcke, S. Analysis of SCRIP Conservative Remapping in OASIS3MCT—Part B, Technical Report TR/CMGC/19155, CERFACS, France. 2019. Available online: https://oasis.cerfacs.fr/wpcontent/uploads/sites/114/2021/08/GLOBC_TR_JonvilleSCRIP_CONSERV_TRNORM_partB_2019.pdf (accessed on 18 January 2022).
[3]: https://dkrzsw.gitlabpages.dkrz.de/yac/d0/daa/frac_mask_desc.html
[4]: 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/gmd104252017, 2017.
[5]Valcke S, Piacentini A, Jonville G. Benchmarking Regridding Libraries Used in Earth System Modelling. Mathematical and Computational Applications. 2022; 27(2):31. https://doi.org/10.3390/mca27020031Citation: https://doi.org/10.5194/gmd2023177CC1 
RC1: 'Comment on gmd2023177', Anonymous Referee #1, 13 Sep 2023
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Karl Taylor has presented us with a very carefully written and clearly explained treatise on the problems we encounter when shifting Earth system modeling data from one grid to another. It is a valuable contribution to Geoscientific Model Development and can be published almost as is. Personally, I would like to see Dr. Taylor address some of the issues that I raise below in his text, not necessarily with new recipes. For example, a very important use for this work (not noted here) is in remapping emissions from some standard regular latlong grid onto the model grid. (One could argue that radiation is an emission.) It is essential that this remapping conserve emissions, e.g., of greenhouse gases, because otherwise the radiative forcing and climate trajectories will diverge based on model grid. This is a longstanding problem that many have solved by some but still plagues CESM versions and its related E3SM codes.
L3: Excellent and agreed; yet, there is a more difficult problem of the convergence/divergences that must be preserved when winds or fluxes are remapped. Do you have a solution for that?
L31: The assumption that there is a single r_s or r_d is dangerous. The Earth is not a sphere and more accurate models may need to take that into account. Moreover, surface orography is standard in most models  how is that treated in this equation? If you are regridding a quantity at the top of the atmosphere, a geometric atmosphere (spherical expansion, g varies with altitude) should probably be accounted for.
Is this problem accounted for somehow in discussion re equation 8?
L141ff: The analysis now gets into some very interesting problems. It is nicely explained how rescaling to get global conservation gets one into trouble in so many ways. That this problem is still with is indeed disconcerting. When we were developing tracer transport algorithms many decades ago, conservation was an important but most often violated condition. The approach of many was simply to evaluate the numerical loss and add it back into the system – but of course, there was no information on where to put it, and uniform replacement created unphysical variability in the chemical tracer. This seems to be going on here. The other test we used, similar to the one here was to transport a uniform tracer and show that artificial variations did not occur.
L205: This case and the problems with the remapping are clearly stated, thanks. Do you have other examples of problems with cubedsphere or polygonal grids that are being used so much in highresolution models?
L265ff: Indeed there are many cases re emissions and chemistry where this would be a show stopper. I am not sure that the chemistry community would be happy with these mu factors (eqn 19): they seem arbitrary and would have to be tuned for every species and variable (emissions, concentrations, ….). Would we just be adding many other parameters to be tuned?
L306ff: OK, the recipes are clear. Do you have any recommendations if this had to be applied 100+ items at a time?
L370: Yes, indeed, there is much to do with column integrals being remapped (e.g., ozone hole) in additions to the mass in each layer. Any example under (g) below? OK, I see this comes in Section 5 later.
L453ff: What if we finally go to geometric grids where g is the true value, function of R+z, and is not constant. Also, how do you average horizontally using cells with different lower boundaries? Is that obvious from above?
L487: This is great, but if you moved emissions from equator to pole, you can truly make a mess of the results. Can you invoke regionally conservative mapping, not just global?
Citation: https://doi.org/10.5194/gmd2023177RC1
Karl E. Taylor
Karl E. Taylor
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