The Cumulative Distribution Function transform

CDF-t (Cumulative Distribution Function transform) is a statistical method based on Quantile Mapping (QM). QM methods are routinely applied to adjust climate model simulations compared to observations. When observations are of similar resolution as the climate model, QM can be viewed as a bias-adjustment method. On the other hand, when observations are of higher resolution, QM also attempts to bridge the scale mismatch and is then viewed as a downscaling method. 

Why CDF-t ?
Standard QM approaches rely on the hypothesis that model and observations distributions keep the same shape in the future. However, in a global warming context, this “stationarity” assumption is an hypothesis questioned by scientists and considered as a weak point of such methods.  
We use because CDF-t it doesn’t rely on the stationarity hypothesis: model and observational distributions can evolve and be different. The assumption is that the model and observational distributions can be inferred by a mathematical function (the “transform”) which remains the same for past and future distributions.

Is CDF-t the perfect method for downscaling and bias adjustment ?
There is no such thing as a perfect method. All methods have their pros and cons in terms of results and technical constraints. One important advantage of CDF-t is that it was extensively tested and referenced in dozens of research papers over the last 10 years. In parallel, we keep up to date with scientific and technical progress to adopt methodological advances in our processing chain.

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Main references

R Code available on rdocumention.com by M. Vrac

Key papers
Michelangeli, P.-A., M. Vrac, and H. Loukos (2009), Probabilistic downscaling approaches: Application to wind cumulative distribution functions, Geophys. Res. Lett., 36, L11708, doi 10.1029/2009GL038401. Over 300 citations according to Google Scholar.

Vrac, M., T. Noël, and R. Vautard (2016), Bias correction of precipitation through Singularity Stochastic Removal: Because occurrences matter, J. Geophys. Res. Atmos., 121, 5237–5258, doi 10.1002/2015JD024511. Over 75 citations according to Google Scholar.

Companion papers
Kallache, M., M. Vrac, P. Naveau, and P.-A. Michelangeli (2011), Nonstationary probabilistic downscaling of extreme precipitation, J. Geophys. Res., 116, D05113, doi: 10.1029/2010JD014892.

Vigaud, N., Vrac, M. and Caballero, Y. (2013), Probabilistic downscaling of GCM scenarios over southern India. Int. J. Climatol., 33: 1248–1263. doi 10.1002/joc.3509.

Vrac, M., P. Friedrichs (2013), Multivariate—Intervariable, Spatial, and Temporal—Bias Correction, J. of Clim. doi 10.1175/JCLI-D-14-00059.1

Vautard, R., Noël, T., Li, L. et al.  (2013), Climate variability and trends in downscaled high-resolution simulations and projections over Metropolitan France, Dyn 41: 1419. doi 10.1007/s00382-012-1621-8.

Famien, A. M., Janicot, S., Ochou, A. D., Vrac, M., Defrance, D., Sultan, B., and Noël, T. (2018) A bias-corrected CMIP5 dataset for Africa using CDF-t method. A contribution to agricultural impact studies, Earth Syst. Dynam. Discuss., under review, doi 10.5194/esd-2017-111. 

Galmarini, S., R. Bertalanic, A.J. Cannon, A. Ceglar, O.B. Christensen, N. de Noblet-Ducoudré, F. Dentener, F. J. Doblas-Reyes, A. Dosio, J. M. Gutierrez, H. Loukos, A. Maiorano, D. Maraun, S. Mcginnis, G. Nikulin, A. Riccio, E. Sanchez, E. Solazzo, A. Toreti, M. Vrac and M. Zampieri (2019) Climate Model Bias for Agricultural Impact Assessment: how to cut the mustard?, Cimate Services, doi 10.1016/j.cliser.2019.01.004.

Updated on: 19/01/2023