# Benchmarking Regridding Libraries Used in Earth System Modelling

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Regridding Benchmark

#### 2.1. The Benchmark Characteristics

#### 2.1.1. Grids

- torc: the ocean model NEMO (Nucleus for European Modelling of the Ocean) [17], rotated-stretched logically-rectangular grid with 182 × 149 points horizontally;
- nogt: the ocean model NEMO, rotated-stretched logically-rectangular grid with 362 × 294 points horizontally;
- bggd: the atmosphere model LMDz (Laboratoire de Météorologie Dynamique zoom), [18] regular latitude–longitude grid with 144 × 143 points horizontally;
- sse7: the atmosphere model ARPEGE (Action de Recherche Petite Echelle Grande Echelle) [19], Gaussian reduced T127 with 24,572 points horizontally (unstructured, described with up to 7 vertices per cell);
- icos: the atmosphere model Dynamico [20], low-resolution unstructured icosahedral grid with 15,222 points horizontally;
- icoh: the atmosphere model Dynamico, high-resolution unstructured icosahedral grid with 2,016,012 points horizontally.

#### 2.1.2. Analytical Functions

- (a)
- sinusoid: a slowly varying standard sinusoid over the globe;
- (b)
- harmonic: a more rapidly varying function with 16 maximums and 16 minimums in northern and southern bands;
- (c)
- vortex: a slowly varying function with two added vortices, one in the Atlantic and one over Indonesia;
- (d)
- gulfstream: the slowly varying standard sinusoid with a mimicked Gulf Stream.

#### 2.1.3. Regridding Algorithms

- 1.
- Nearest neighbour

- 2.
- First order non-conservative

- 3.
- Second order non-conservative

- 4.
- First order conservative with FRACAREA and DESTAREA normalisations

- 5.
- Second order conservative with FRACAREA normalisation

#### 2.1.4. Benchmark Metrics

- Ψ
^{s}: the analytical function on the source grid; - Ψ
^{t}: the analytical function on the target grid; - RΨ
^{s}: the source analytical function regridded on the target grid; - I
_{s}: the integral on the source grid; - I
_{t}: the integral on the target grid;

- mean misfit: mean (|RΨ
^{s}− Ψ^{t}|/|Ψ^{t}|); - maximum misfit: max (|RΨ
^{s}− Ψ^{t}|/|Ψ^{t}|); - RMS (root mean square) misfit: RMS (|RΨ
^{s}− Ψ^{t}|/|Ψ^{t}|); - L
_{min}: (min Ψ^{t}− min RΨ^{s})/max (|Ψ^{t}|) (A positive L_{min}detects an overestimate of the function minimum (i.e., it reinforces the minimum) while a negative L_{min}detects some smoothing of the function minimum); - L
_{max}: (max RΨ^{s}− max Ψ^{t})/max (|Ψ^{t}|) (A positive L_{max}detects an overestimate of the function maximum (i.e., it reinforces the maximum) while a negative L_{max}detects some smoothing of the function maximum); - Source global conservation: |I
_{t}(RΨ^{s}) − I_{s}(Ψ^{s})|/I_{s}(Ψ^{s}); - Target global conservation: |I
_{t}(RΨ^{s}) − I_{t}(Ψ^{t})|/I_{t}(Ψ^{t}).

#### 2.2. Implementation of the Regridding Benchmark for SCRIP, YAC, ESMF and XIOS

- SCRIP

- ESMF

- YAC

- XIOS

## 3. Benchmark Results

- the four analytical functions: sinusoid, harmonic, vortex, gulfstream (see Section 2.1.2);
- the six pairs of relatively low-resolution grids matching an ocean grid with an atmospheric grid: torc-bggd, torc-icos, torc-sse7, nogt-bggd, nogt-icos, nogt-sse7 in both directions (see Section 2.1.1); for the conservative remapping, we also analyse the regridding of the vortex function for icos-icoh and nogt-icoh in order to test the impact of that regridding on cases with large resolution difference (see Section 3.6);
- for the four regridding libraries: SCRIP (+SCRIP-L, i.e., with Lambert projection for conservative regridding), YAC, ESMF and XIOS;
- for all algorithms: nearest neighbour, 1st and 2nd order non-conservative, 1st and 2nd order conservative, except when the regridding library does not support the algorithm, such as, e.g., nearest neighbour for XIOS (see Section 2.1.3).

#### 3.1. Nearest Neighbour Regridding

#### 3.2. 1st Order Non-Conservative Regridding

#### 3.3. Second-Order Non-Conservative Regridding

_{max}for the different pairs of grids for these three regridding libraries for the gulfstream function.

_{max}.

_{max}for icos-torc and icos-nogt for the gulfstream function that do not appear for the other functions (not shown). Figure 8 shows 2D plots of the regridded field in the gulf stream region for ESMF and YAC. Indeed, it confirms that, compared to ESMF, which tends to smooth the local maximum with its patch averaging algorithm, YAC gives higher, but a priori non-anomalous, values in the centre of the gulf stream.

#### 3.4. First-Order Conservative Remapping with DESTAREA Normalisation

#### 3.5. 1st Order Conservative Remapping with FRACAREA Normalisation

^{−9}.

#### 3.6. Second-Order Conservative Remapping with FRACAREA Normalisation

_{min}and L

_{max}for the second-order conservative remapping with FRACAREA normalisation for the gulfstream function, which presents some outstanding values (the other functions do not present such outstanding values). XIOS shows a strong undershoot for torc-icos, as shown by L

_{min}, and a strong overestimate for bggd-nogt as shown by L

_{max}.

#### 3.7. Comparison of Regridding Algorithms

## 4. Discussion

- their sensitivity, as we perform the metric calculation for six pairs of grids in both directions and, in addition, for the icos-icoh and nogt-icoh pairs for the vortex function for second-order conservative FRACAREA remapping;
- their global conservation, as we provide the source and target global conservation metrics.

_{min}and L

_{max}metrics, our benchmark also allows a first analysis of the regridding library monotonicity and dissipation (or smoothing). We also started to evaluate the performances of the libraries with a first scalability analysis, not shown here but in [5], that will be completed in the coming months. However, we do not address the regridding library consistency, i.e., the preservation of discretization order and accuracy.

- For first-order non-conservative regridding, YAC, using an inverse distance weighting of the vertex values of the source polygon enclosing the target point, is less accurate on average than the SCRIP or ESMF bilinear schemes (Figure 5). For second-order non-conservative regridding, the ESMF patch algorithm gives slightly less accurate results than the SCRIP bicubic or the YAC spherical Bernstein–Bézier polynomial algorithms (Figure 6). For first- and second-order non-conservative regridding, all results for ESMF and YAC are reasonable, except for ESMF in the case of the gulfstream function for torc-bggd and torc-sse7 grid pairs, which show one anomalous value near the coast (Figure 7). For second-order non-conservative regridding in the case of icos-torc and icos-nogt for the gulfstream function, YAC also shows some higher, but a priori non-anomalous values, in the centre of the gulf stream (Figure 8).
- First-order conservative remapping with DESTAREA normalisation in YAC, ESMF, and XIOS show very similar and good results (Figure 9 and Figure 10), except for ESMF when nogt is the source grid if this grid is described with the SCRIP (structured) format (Figure 11 and Figure 12). For first-order conservative remapping with FRACAREA normalisation, YAC, ESMF, and XIOS show very similar and good results for all functions (Figure 13); this regridding raises no specific issues for any regridding library.
- YAC, ESMF, and XIOS show approximately the same behaviour with good global conservation for second-order conservative remapping with FRACAREA normalisation, implemented following [21] in the 3 libraries (Figure 14). One issue, however, is in ESMF when the source grid is the icosahedral one, icos, which shows a relatively high mean misfit for all functions, with an alternating positive and negative pattern (Figure 15). Another issue is present for XIOS, which shows a strong undershoot for the gulfstream function for torc-icos, with one clearly outstanding point near the coast.
- Unexpectedly, conservative algorithms do not always offer better global conservation than non-conservative ones (Figure 19b,d).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Analytical Functions

- (A)
- sinusoid

- (B)
- harmonic

- (C)
- vortex

- (D)
- gulfstream

## Appendix B. List of CSV Files Containing Metrics Values

- “DISTWGT_1” for nearest neighbour
- “BILINEAR” for first-order non conservative
- “BICUBIC” for second-order non-conservative
- “CONSERV” for first-order conservative
- “CONS2ND” for second-order conservative

**Table A1.**Regridding benchmark files included in the tar file Regridding_Benchmark_metrics.tar available on Zenodo at https://doi.org/10.5281/zenodo.5343166 (accessed on 31 August 2021).

ESMF-820bs08-U_CONS2ND_FRACAREA_classic.csv | SCRIP_CONS2ND_FRACAREA_harmonic.csv |

ESMF-820bs08-U_CONS2ND_FRACAREA_gulfstream.csv | SCRIP_CONS2ND_FRACAREA_vortex.csv |

ESMF-820bs08-U_CONS2ND_FRACAREA_harmonic.csv | SCRIP_CONSERV_DESTAREA_classic.csv |

ESMF-820bs08-U_CONS2ND_FRACAREA_vortex.csv | SCRIP_CONSERV_DESTAREA_gulfstream.csv |

ESMF-820bs08-U_CONSERV_DESTAREA_classic.csv | SCRIP_CONSERV_DESTAREA_harmonic.csv |

ESMF-820bs08-U_CONSERV_DESTAREA_gulfstream.csv | SCRIP_CONSERV_DESTAREA_vortex.csv |

ESMF-820bs08-U_CONSERV_DESTAREA_harmonic.csv | SCRIP_CONSERV_FRACAREA_classic.csv |

ESMF-820bs08-U_CONSERV_DESTAREA_vortex.csv | SCRIP_CONSERV_FRACAREA_gulfstream.csv |

ESMF-820bs08-U_CONSERV_FRACAREA_classic.csv | SCRIP_CONSERV_FRACAREA_harmonic.csv |

ESMF-820bs08-U_CONSERV_FRACAREA_gulfstream.csv | SCRIP_CONSERV_FRACAREA_vortex.csv |

ESMF-820bs08-U_CONSERV_FRACAREA_harmonic.csv | SCRIP_DISTWGT_1_classic.csv |

ESMF-820bs08-U_CONSERV_FRACAREA_vortex.csv | SCRIP_DISTWGT_1_gulfstream.csv |

ESMF-820bs08_BICUBIC_classic.csv | SCRIP_DISTWGT_1_harmonic.csv |

ESMF-820bs08_BICUBIC_gulfstream.csv | SCRIP_DISTWGT_1_vortex.csv |

ESMF-820bs08_BICUBIC_harmonic.csv | XIOS_CONS2ND_FRACAREA_classic.csv |

ESMF-820bs08_BICUBIC_vortex.csv | XIOS_CONS2ND_FRACAREA_gulfstream.csv |

ESMF-820bs08_BILINEAR_classic.csv | XIOS_CONS2ND_FRACAREA_harmonic.csv |

ESMF-820bs08_BILINEAR_gulfstream.csv | XIOS_CONS2ND_FRACAREA_vortex.csv |

ESMF-820bs08_BILINEAR_harmonic.csv | XIOS_CONSERV_DESTAREA_classic.csv |

ESMF-820bs08_BILINEAR_vortex.csv | XIOS_CONSERV_DESTAREA_gulfstream.csv |

ESMF-820bs08_CONSERV_DESTAREA_classic.csv | XIOS_CONSERV_DESTAREA_harmonic.csv |

ESMF-820bs08_CONSERV_DESTAREA_gulfstream.csv | XIOS_CONSERV_DESTAREA_vortex.csv |

ESMF-820bs08_CONSERV_DESTAREA_harmonic.csv | XIOS_CONSERV_FRACAREA_classic.csv |

ESMF-820bs08_CONSERV_DESTAREA_vortex.csv | XIOS_CONSERV_FRACAREA_gulfstream.csv |

ESMF-820bs08_DISTWGT_1_classic.csv | XIOS_CONSERV_FRACAREA_harmonic.csv |

ESMF-820bs08_DISTWGT_1_gulfstream.csv | XIOS_CONSERV_FRACAREA_vortex.csv |

ESMF-820bs08_DISTWGT_1_harmonic.csv | YAC_BICUBIC_classic.csv |

ESMF-820bs08_DISTWGT_1_vortex.csv | YAC_BICUBIC_gulfstream.csv |

SCRIP-L_CONS2ND_FRACAREA_classic.csv | YAC_BICUBIC_harmonic.csv |

SCRIP-L_CONS2ND_FRACAREA_gulfstream.csv | YAC_BICUBIC_vortex.csv |

SCRIP-L_CONS2ND_FRACAREA_harmonic.csv | YAC_BILINEAR_classic.csv |

SCRIP-L_CONS2ND_FRACAREA_vortex.csv | YAC_BILINEAR_gulfstream.csv |

SCRIP-L_CONSERV_DESTAREA_classic.csv | YAC_BILINEAR_harmonic.csv |

SCRIP-L_CONSERV_DESTAREA_gulfstream.csv | YAC_BILINEAR_vortex.csv |

SCRIP-L_CONSERV_DESTAREA_harmonic.csv | YAC_CONS2ND_FRACAREA_classic.csv |

SCRIP-L_CONSERV_DESTAREA_vortex.csv | YAC_CONS2ND_FRACAREA_gulfstream.csv |

SCRIP-L_CONSERV_FRACAREA_classic.csv | YAC_CONS2ND_FRACAREA_harmonic.csv |

SCRIP-L_CONSERV_FRACAREA_gulfstream.csv | YAC_CONS2ND_FRACAREA_vortex.csv |

SCRIP-L_CONSERV_FRACAREA_harmonic.csv | YAC_CONSERV_DESTAREA_classic.csv |

SCRIP-L_CONSERV_FRACAREA_vortex.csv | YAC_CONSERV_DESTAREA_gulfstream.csv |

SCRIP_BICUBIC_classic.csv | YAC_CONSERV_DESTAREA_harmonic.csv |

SCRIP_BICUBIC_gulfstream.csv | YAC_CONSERV_DESTAREA_vortex.csv |

SCRIP_BICUBIC_harmonic.csv | YAC_CONSERV_FRACAREA_classic.csv |

SCRIP_BICUBIC_vortex.csv | YAC_CONSERV_FRACAREA_gulfstream.csv |

SCRIP_BILINEAR_classic.csv | YAC_CONSERV_FRACAREA_harmonic.csv |

SCRIP_BILINEAR_gulfstream.csv | YAC_CONSERV_FRACAREA_vortex.csv |

SCRIP_BILINEAR_harmonic.csv | YAC_DISTWGT_1_classic.csv |

SCRIP_BILINEAR_vortex.csv | YAC_DISTWGT_1_gulfstream.csv |

SCRIP_CONS2ND_FRACAREA_classic.csv | YAC_DISTWGT_1_harmonic.csv |

SCRIP_CONS2ND_FRACAREA_gulfstream.csv | YAC_DISTWGT_1_vortex.csv |

## Appendix C. List of Metric Plots

**Table A2.**Regridding benchmark plots included in the tar file Regridding_Benchmark_metrics_plots.tar available on Zenodo at https://doi.org/10.5281/zenodo.5347696 (accessed on 31 August 2021).

plot_remap_metrics_BICUBIC_classic.pdf |

plot_remap_metrics_BICUBIC_gulfstream.pdf |

plot_remap_metrics_BICUBIC_harmonic.pdf |

plot_remap_metrics_BICUBIC_vortex.pdf |

plot_remap_metrics_BILINEAR_classic.pdf |

plot_remap_metrics_BILINEAR_gulfstream.pdf |

plot_remap_metrics_BILINEAR_harmonic.pdf |

plot_remap_metrics_BILINEAR_vortex.pdf |

plot_remap_metrics_CONS2ND_FRACAREA_classic.pdf |

plot_remap_metrics_CONS2ND_FRACAREA_gulfstream.pdf |

plot_remap_metrics_CONS2ND_FRACAREA_harmonic.pdf |

plot_remap_metrics_CONS2ND_FRACAREA_vortex.pdf |

plot_remap_metrics_CONSERV_DESTAREA_classic.pdf |

plot_remap_metrics_CONSERV_DESTAREA_gulfstream.pdf |

plot_remap_metrics_CONSERV_DESTAREA_harmonic.pdf |

plot_remap_metrics_CONSERV_DESTAREA_vortex.pdf |

plot_remap_metrics_CONSERV_FRACAREA_classic.pdf |

plot_remap_metrics_CONSERV_FRACAREA_classic_sansconservationMTR.pdf |

plot_remap_metrics_CONSERV_FRACAREA_gulfstream.pdf |

plot_remap_metrics_CONSERV_FRACAREA_harmonic.pdf |

plot_remap_metrics_CONSERV_FRACAREA_vortex.pdf |

plot_remap_metrics_DISTWGT_1_classic.pdf |

plot_remap_metrics_DISTWGT_1_gulfstream.pdf |

plot_remap_metrics_DISTWGT_1_harmonic.pdf |

plot_remap_metrics_DISTWGT_1_vortex.pdf |

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**Figure 1.**Illustration of the types of grids included in the benchmark: (

**a**) rotated-stretched logically-rectangular (torc, nogt), (

**b**) regular latitude–longitude (bggd), (

**c**) Gaussian-reduced (sse7), and (

**d**) icosahedral (icos, icoh).

**Figure 2.**The four analytical functions defining the coupling field: (

**a**) sinusoid, (

**b**) harmonic, (

**c**) vortex, (

**d**) gulfstream.

**Figure 3.**(

**a**) mean, (

**b**) rms and (

**c**) maximum misfit for the different pairs of grids for the harmonic function for nearest neighbour algorithm for ESMF, SCRIP and YAC.

**Figure 4.**Maximum misfit for the different pairs of grids for the different functions sinusoid, vortex, harmonic, gulfstream for the nearest neighbour algorithm for ESMF, SCRIP and YAC.

**Figure 5.**(

**a**) mean, (

**b**) rms, and (

**c**) maximum misfit for the different pairs of grids for the vortex function for first-order non-conservative regridding for ESMF, SCRIP and YAC.

**Figure 6.**(

**a**) mean, (

**b**) rms, and (

**c**) maximum misfit and (

**d**) L

_{max}for the different pairs of grids for the gulfstream function for second-order non-conservative algorithms for ESMF, SCRIP and YAC. The red circles identify anomalous regriddings detailed in the text.

**Figure 7.**Misfit (%) for the gulfstream function in the gulf stream region for torc-bggd regridded with the second-order non-conservative algorithm for (

**a**) YAC HCSBB and (

**b**) ESMF PATCH. The red circle identifies the anomalous value near the coast for the ESMF patch algorithm discussed in the text.

**Figure 8.**Misfit (%) for the gulfstream function in the gulf stream region for icos-torc regridded with the second-order non-conservative algorithm for (

**a**) YAC HCSBB and (

**b**) ESMF PATCH. The red circle identifies the highest, but a priori non-anomalous, value in the centre of the gulf stream for YAC discussed in the text.

**Figure 9.**(

**a**) mean and (

**b**) maximum misfit for ESMF, SCRIP, SCRIP-L, YAC, and XIOS for the first-order conservative remapping with DESTAREA normalisation for the different pairs of grids for the harmonic function. For ESMF, nogt and torc are described with the unstructured format.

**Figure 10.**Source global conservation (%) for ESMF, SCRIP, SCRIP-L, YAC and XIOS for the 1st order conservative remapping with DESTAREA normalisation for the different pairs of grids for the 4 functions: (

**a**) sinusoid, (

**b**) harmonic, (

**c**) vortex, and (

**d**) gulfstream.

**Figure 11.**(

**a**) Mean and (

**b**) maximum misfit for ESMF, SCRIP, SCRIP-L, YAC, and XIOS for the first-order conservative remapping with DESTAREA normalisation for the different pairs of grids for the harmonic function. For ESMF, nogt and torc are described with the SCRIP structured format. The red oval shapes identify ESMF regriddings showing anomalous mean and maximum misfits when nogt is the source grid. These regriddings are discussed in the text.

**Figure 12.**Two-dimensional plot of the misfit of the harmonic function regridded with ESMF first-order conservative remapping for nogt-bggd with nogt described (

**a**) with the SCRIP format and (

**b**) with the unstructured format. The red oval shape identifies in (

**a**) the grid points linked to the north fold of the NEMO grid showing anomalous misfit when the nogt grid is described in the SCRIP format.

**Figure 13.**Maximum misfit for ESMF, SCRIP, SCRIP-L, YAC, and XIOS for the first-order conservative remapping with FRACAREA normalisation for the different pairs of grids for the 4 functions: (

**a**) sinusoid, (

**b**) harmonic, (

**c**) vortex, (

**d**) gulfstream.

**Figure 14.**(

**a**) Mean, (

**b**) rms, (

**c**) maximum misfit, and (

**d**) source global conservation for the different pairs of grids for the harmonic function for second-order conservative remapping with FRACAREA normalisation. The red circles identify anomalous regriddings for ESMF when the source grid is the icosahedral one (icos) detailed in the text.

**Figure 15.**Misfit (%) on the target grid nogt for the icos-nogt second-order conservative remapping with FRACAREA normalisation for ESMF, with a zoom on the left.

**Figure 16.**L

_{min}and L

_{max}for the different pairs of grids for the gulfstream function for 2nd order conservative remapping with FRACAREA normalisation.

**Figure 17.**Misfit in the gulf stream region for the second-order conservative remapping of the gulfstream function for torc-icos for XIOS, YAC and ESMF. The red circles identify the grid point near the coast showing an outstanding value for XIOS.

**Figure 18.**Misfit (%) for the (

**a**) first-order and (

**b**) second-order conservative remapping (both with FRACAREA normalisation) of the vortex function from the low-resolution icosahedral grid icos to the high-resolution icosahedral grid icoh with YAC.

**Figure 19.**Mean misfit (%) for the different regridding algorithms in (

**a**) ESMF and (

**c**) YAC; source global conservation (%) for the different regridding algorithms in (

**b**) ESMF and (

**d**) YAC.

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Valcke, S.; Piacentini, A.; Jonville, G. Benchmarking Regridding Libraries Used in Earth System Modelling. *Math. Comput. Appl.* **2022**, *27*, 31.
https://doi.org/10.3390/mca27020031

**AMA Style**

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/mca27020031

**Chicago/Turabian Style**

Valcke, Sophie, Andrea Piacentini, and Gabriel Jonville. 2022. "Benchmarking Regridding Libraries Used in Earth System Modelling" *Mathematical and Computational Applications* 27, no. 2: 31.
https://doi.org/10.3390/mca27020031