Smoothing and spatial verification of global fields

Skok, Gregor; Kosovelj, Katarina

doi:https://doi.org/10.5194/gmd-18-7417-2025

Articles | Volume 18, issue 20

Article
Assets
Peer review
Metrics
Related articles

Articles | Volume 18, issue 20

https://doi.org/10.5194/gmd-18-7417-2025

Articles | Volume 18, issue 20

Article
Assets
Peer review
Metrics
Related articles

Methods for assessment of models

20 Oct 2025

Methods for assessment of models |

| 20 Oct 2025

Smoothing and spatial verification of global fields

Gregor Skok and Katarina Kosovelj

Gregor Skok

CORRESPONDING AUTHOR

gregor.skok@fmf.uni-lj.si

University of Ljubljana, Faculty of Mathematics and Physics, Jadranska Cesta 19, 1000 Ljubljana, Slovenia

Katarina Kosovelj

University of Ljubljana, Faculty of Mathematics and Physics, Jadranska Cesta 19, 1000 Ljubljana, Slovenia

Cited articles

Baddeley, A.: An error metric for binary images, in: Robust Computer Vision: Quality of Vision Algorithms, Wichmann verlag, Karlsruhe, 1992. a

Bentley, J.: Multidimensional binary search trees used for associative searching, Commun. ACM, 18, 509–517, https://doi.org/10.1145/361002.361007, 1975. a

Bentley, J.: Multidimensional Binary Search Trees in Database Applications, IEEE T. Software. Eng., SE-5, 333–340, https://doi.org/10.1109/TSE.1979.234200, 1979. a, b

Bi, K., Xie, L., Zhang, H., Chen, X., Gu, X., and Tian, Q.: Accurate medium-range global weather forecasting with 3D neural networks, Nature, 619, https://doi.org/10.1038/s41586-023-06185-3, 2023. a, b

Bouallègue, Z. B., Theis, S. E., and Gebhardt, C.: Enhancing COSMO-DE ensemble forecasts by inexpensive techniques, Meteorol. Z., 22, 49–59, https://doi.org/10.1127/0941-2948/2013/0374, 2013. a

Brown, B. G., Gilleland, E., and Ebert, E. E.: Forecasts of Spatial Fields, in: Forecast Verification, John Wiley and Sons, Ltd, Chichester, UK, 95–117, https://doi.org/10.1002/9781119960003.ch6, 2012. a

Brown, R. A.: Building a Balanced k-d Tree in O(knlog n) Time, J. Comput. Graph. Tech., 4, 50–68, http://jcgt.org/published/0004/01/03/ (last access: 13 October 2025), 2015. a

Buschow, S. and Friederichs, P.: SAD: Verifying the scale, anisotropy and direction of precipitation forecasts, Q. J. Roy. Meteor. Soc., 147, 1150–1169, https://doi.org/10.1002/qj.3964, 2021. a

Casati, B.: New Developments of the Intensity-Scale Technique within the Spatial Verification Methods Intercomparison Project, Weather Forecast., 25, 113–143, https://doi.org/10.1175/2009WAF2222257.1, 2010. a

Casati, B., Ross, G., and Stephenson, D. B.: A new intensity‐scale approach for the verification of spatial precipitation forecasts, Meteorol. Appl., 11, 141–154, https://doi.org/10.1017/S1350482704001239, 2004. a

Casati, B., Lussana, C., and Crespi, A.: Scale‐separation diagnostics and the Symmetric Bounded Efficiency for the inter‐comparison of precipitation reanalyses, Int. J. Climatol., 43, 2287–2304, https://doi.org/10.1002/joc.7975, 2023. a

Davis, C., Brown, B., and Bullock, R.: Object-Based Verification of Precipitation Forecasts. Part I: Methodology and Application to Mesoscale Rain Areas, Mon. Weather Rev., 134, 1772–1784, https://doi.org/10.1175/MWR3145.1, 2006a. a

Davis, C., Brown, B., and Bullock, R.: Object-Based Verification of Precipitation Forecasts. Part II: Application to Convective Rain Systems, Mon. Weather Rev., 134, 1785–1795, https://doi.org/10.1175/MWR3146.1, 2006b. a

Davis, C. A., Brown, B. G., Bullock, R., and Halley-Gotway, J.: The Method for Object-Based Diagnostic Evaluation (MODE) Applied to Numerical Forecasts from the 2005 NSSL/SPC Spring Program, Weather Forecast., 24, 1252–1267, https://doi.org/10.1175/2009WAF2222241.1, 2009. a

Dey, S. R. A., Leoncini, G., Roberts, N. M., Plant, R. S., and Migliorini, S.: A Spatial View of Ensemble Spread in Convection Permitting Ensembles, Mon. Weather Rev., 142, 4091–4107, https://doi.org/10.1175/MWR-D-14-00172.1, 2014. a

Dey, S. R. A., Plant, R. S., Roberts, N. M., and Migliorini, S.: Assessing spatial precipitation uncertainties in a convective‐scale ensemble, Q. J. Roy. Meteor. Soc., 142, 2935–2948, https://doi.org/10.1002/qj.2893, 2016. a

Dorninger, M., Gilleland, E., Casati, B., Mittermaier, M. P., Ebert, E. E., Brown, B. G., and Wilson, L. J.: The setup of the MesoVICT project, B. Am. Meteorol. Soc., 99, 1887–1906, https://doi.org/10.1175/BAMS-D-17-0164.1, 2018. a

Duc, L., Saito, K., and Seko, H.: Spatial-temporal fractions verification for high-resolution ensemble forecasts, Tellus A, 65, 18171, https://doi.org/10.3402/tellusa.v65i0.18171, 2013. a, b

Ebert, E. and McBride, J.: Verification of precipitation in weather systems: determination of systematic errors, J. Hydrol., 239, 179–202, https://doi.org/10.1016/S0022-1694(00)00343-7, 2000. a

ECMWF: IFS Documentation CY48R1 – Part III: Dynamics and Numerical Procedures, https://doi.org/10.21957/26F0AD3473, 2023a. a

ECMWF: IFS Documentation CY48R1 – Part IV: Physical Processes, https://doi.org/10.21957/02054F0FBF, 2023b. a

Faggian, N., Roux, B., Steinle, P., and Ebert, B.: Fast calculation of the fractions skill score, MAUSAM, 66, 457–466, https://doi.org/10.54302/mausam.v66i3.555, 2015. a

Friedman, J. H., Bentley, J. L., and Finkel, R. A.: An Algorithm for Finding Best Matches in Logarithmic Expected Time, ACM T. Math. Software, 3, 209–226, https://doi.org/10.1145/355744.355745, 1977. a, b

Gainford, A., Gray, S. L., Frame, T. H. A., Porson, A. N., and Milan, M.: Improvements in the spread–skill relationship of precipitation in a convective‐scale ensemble through blending, Q. J. Roy. Meteor. Soc., 150, 3146–3166, https://doi.org/10.1002/qj.4754, 2024. a, b

Gilleland, E.: A new characterization within the spatial verification framework for false alarms, misses, and overall patterns, Weather Forecast., 32, 187–198, https://doi.org/10.1175/WAF-D-16-0134.1, 2017. a

Gilleland, E., Ahijevych, D., Brown, B. G., Casati, B., and Ebert, E. E.: Intercomparison of Spatial Forecast Verification Methods, Weather Forecast., 24, 1416–1430, https://doi.org/10.1175/2009WAF2222269.1, 2009. a

Keil, C. and Craig, G. C.: A Displacement-Based Error Measure Applied in a Regional Ensemble Forecasting System, Mon. Weather Rev., 135, 3248–3259, https://doi.org/10.1175/MWR3457.1, 2007. a

Keil, C. and Craig, G. C.: A Displacement and Amplitude Score Employing an Optical Flow Technique, Weather Forecast., 24, 1297–1308, https://doi.org/10.1175/2009WAF2222247.1, 2009. a

Konca-Kedzierska, K., Wibig, J., and Gruszczyńska, M.: Comparison and combination of interpolation methods for daily precipitation in Poland: evaluation using the correlation coefficient and correspondence ratio, Meteorol. Hydrol. Wate., https://doi.org/10.26491/mhwm/171699, 2023. a

Lam, R., Sanchez-Gonzalez, A., Willson, M., Wirnsberger, P., Fortunato, M., Alet, F., Ravuri, S., Ewalds, T., Eaton-Rosen, Z., Hu, W., Merose, A., Hoyer, S., Holland, G., Vinyals, O., Stott, J., Pritzel, A., Mohamed, S., and Battaglia, P.: Learning skillful medium-range global weather forecasting, Science, 382, 1416–1421, https://doi.org/10.1126/science.adi2336, 2023. a, b

Lang, S., Alexe, M., Chantry, M., Dramsch, J., Pinault, F., Raoult, B., Clare, M. C. A., Lessig, C., Maier-Gerber, M., Magnusson, L., Bouallègue, Z. B., Nemesio, A. P., Dueben, P. D., Brown, A., Pappenberger, F., and Rabier, F.: AIFS – ECMWF's data-driven forecasting system, arXiv [preprint], https://doi.org/10.48550/arXiv.2406.01465, 2024. a, b

Ma, S., Chen, C., He, H., Wu, D., and Zhang, C.: Assessing the Skill of Convection-Allowing Ensemble Forecasts of Precipitation by Optimization of Spatial-Temporal Neighborhoods, Atmosphere-Basel, 9, 43, https://doi.org/10.3390/atmos9020043, 2018. a, b

Malardel, S., Wedi, N., Deconinck, W., Diamantakis, M., Kuehnlein, C., Mozdzynski, G., Hamrud, M., and Smolarkiewicz, P.: A new grid for the IFS, ECMWF Newsletter, 23–28, https://doi.org/10.21957/ZWDU9U5I, 2016. a

Markou, M. and Kassomenos, P.: Cluster analysis of five years of back trajectories arriving in Athens, Greece, Atmos. Res., 98, 438–457, https://doi.org/10.1016/j.atmosres.2010.08.006, 2010. a

Marzban, C., Sandgathe, S., Lyons, H., and Lederer, N.: Three Spatial Verification Techniques: Cluster Analysis, Variogram, and Optical Flow, Weather Forecast., 24, 1457–1471, https://doi.org/10.1175/2009WAF2222261.1, 2009. a

Mittermaier, M., North, R., Semple, A., and Bullock, R.: Feature-Based Diagnostic Evaluation of Global NWP Forecasts, Mon. Weather Rev., 144, 3871–3893, https://doi.org/10.1175/mwr-d-15-0167.1, 2016. a

Mittermaier, M. P.: Using an intensity‐scale technique to assess the added benefit of high‐resolution model precipitation forecasts, Atmos. Sci. Lett., 7, 36–42, https://doi.org/10.1002/asl.127, 2006. a

Mittermaier, M. P.: Is there any skill in daily global precipitation forecasts over the Maritime Continent?, Q. J. Roy. Meteor. Soc., https://doi.org/10.1002/qj.4877, 2025. a, b, c

Necker, T., Wolfgruber, L., Kugler, L., Weissmann, M., Dorninger, M., and Serafin, S.: The fractions skill score for ensemble forecast verification, Q. J. Roy. Meteor. Soc., 150, 4457–4477, https://doi.org/10.1002/qj.4824, 2024. a

Roberts, N.: Assessing the spatial and temporal variation in the skill of precipitation forecasts from an NWP model, Meteorol. Appl., 15, 163–169, https://doi.org/10.1002/met.57, 2008. a, b

Roberts, N. M. and Lean, H. W.: Scale-Selective Verification of Rainfall Accumulations from High-Resolution Forecasts of Convective Events, Mon. Weather Rev., 136, 78–97, https://doi.org/10.1175/2007MWR2123.1, 2008. a, b, c, d, e

Schwartz, C. S., Kain, J. S., Weiss, S. J., Xue, M., Bright, D. R., Kong, F., Thomas, K. W., Levit, J. J., Coniglio, M. C., and Wandishin, M. S.: Toward Improved Convection-Allowing Ensembles: Model Physics Sensitivities and Optimizing Probabilistic Guidance with Small Ensemble Membership, Weather Forecast., 25, 263–280, https://doi.org/10.1175/2009WAF2222267.1, 2010. a

Skok, G.: Analysis of Fraction Skill Score properties for a displaced rainy grid point in a rectangular domain, Atmos. Res., 169, 556–565, https://doi.org/10.1016/j.atmosres.2015.04.012, 2016. a

Skok, G.: A New Spatial Distance Metric for Verification of Precipitation, Appl. Sci.-Basel, 12, 4048, https://doi.org/10.3390/app12084048, 2022. a, b, c, d

Skok, G.: Precipitation attribution distance, Atmos. Res., 295, https://doi.org/10.1016/j.atmosres.2023.106998, 2023. a, b

Skok, G.: Snapshot of the Smoothing on Sphere package, Zenodo [code], https://doi.org/10.5281/zenodo.15100264, 2025. a

Skok, G. and Hladnik, V.: Verification of Gridded Wind Forecasts in Complex Alpine Terrain: A New Wind Verification Methodology Based on the Neighborhood Approach, Mon. Weather Rev., 146, 63–75, https://doi.org/10.1175/MWR-D-16-0471.1, 2018. a

Skok, G. and Lledó, L.: Spatial verification of global precipitation forecasts, Q. J. Roy. Meteor. Soc., https://doi.org/10.1002/qj.5006, 2025. a, b, c, d

Skok, G. and Roberts, N.: Analysis of Fractions Skill Score properties for random precipitation fields and ECMWF forecasts, Q. J. Roy. Meteor. Soc., 142, 2599–2610, https://doi.org/10.1002/qj.2849, 2016. a

Skok, G. and Roberts, N.: Estimating the displacement in precipitation forecasts using the Fractions Skill Score, Q. J. Roy. Meteor. Soc., 144, 414–425, https://doi.org/10.1002/qj.3212, 2018. a

Smith, S. W.: The Scientist and Engineer's Guide to Digital Signal Processing, California Technical Publishing, ISBN 0-9660176-7-6, 1999. a

Wernli, H., Paulat, M., Hagen, M., and Frei, C.: SAL – A novel quality measure for the verification of quantitative precipitation forecasts, Mon. Weather Rev., 136, https://doi.org/10.1175/2008MWR2415.1, 2008. a

Wernli, H., Hofmann, C., and Zimmer, M.: Spatial forecast verification methods intercomparison project: Application of the SAL technique, Weather Forecast., 24, https://doi.org/10.1175/2009WAF2222271.1, 2009. a

Weyn, J. A., Durran, D. R., and Caruana, R.: Improving Data-Driven Global Weather Prediction Using Deep Convolutional Neural Networks on a Cubed Sphere, J. Adv. Model. Earth. Sy., 12, https://doi.org/10.1029/2020MS002109, 2020. a

Wilks, D. S.: Statistical Methods in the Atmospheric Sciences, Elsevier, ISBN 9780128158234, https://doi.org/10.1016/C2017-0-03921-6, 2019. a

Woodhams, B. J., Birch, C. E., Marsham, J. H., Bain, C. L., Roberts, N. M., and Boyd, D. F. A.: What Is the Added Value of a Convection-Permitting Model for Forecasting Extreme Rainfall over Tropical East Africa?, Mon. Weather Rev., 146, 2757–2780, https://doi.org/10.1175/MWR-D-17-0396.1, 2018. a, b

Zacharov, P. and Rezacova, D.: Using the fractions skill score to assess the relationship between an ensemble QPF spread and skill, Atmos. Res., 94, 684–693, https://doi.org/10.1016/j.atmosres.2009.03.004, 2009. a