# Correcting systematic biases across multiple atmospheric variables in the frequency domain

## Abstract

A procedure for correcting systematic biases across multiple variables is presented. This procedure operates in the frequency domain, using the cross-spectrum across variables to correct bias across each frequency band. The proposed approach is termed multivariate frequency bias correction or MFBC. The approach is illustrated using global climate model (GCM) simulations of multiple atmospheric variables, with variables selected based on recommended usage in downscaling applications. Results indicate clear benefits of using MFBC in representing both intra- and inter-variable dependence in corrected simulations. This has important implications in applications which require multiple atmospheric variables, and a need to correctly simulate both inter- and intra-variable dependence attributes. MFBC offers a mean to correct raw GCM atmospheric variables prior to downscaling or correct dynamically or statistically downscaled simulations prior to derived simulations of other variables of interest. Use of MFBC can have significant implications on derived hydrologic simulations, such as in sizing of storage reservoirs, or devising water sharing plans for the future.

## Keywords

Bias correction Downscaling Multivariate bias correction Frequency bias correction Multivariate frequency bias correction## Notes

### Acknowledgements

We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modelling groups (the two GCMs listed below) for producing and making available their model output. For CMIP the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. The GCM data is downloaded from. For this study, we used the two GCMs, including MIROC5 [Atmosphere and Ocean Research Institute (The University of Tokyo), National Institute for Environmental Studies, and Japan Agency for Marine-Earth Science and Technology, Japan] and IPSL-CM5A-MR IPSL-CM5A-MR (Institut Pierre Simon Laplace, France). We also acknowledge the reanalysis datasets obtained from the National Center for Environmental Prediction (NCEP) reanalysis2 dataset provided by the NOAA-CIRES Climate Diagnostics Centre, Boulder, Colorado, USA, from their web site at http://www.cdc.noaa.gov/. The first author would like to acknowledge the financial support from the Australian Awards Scholarships (AAS). Authors gratefully acknowledge funding support from the Australian Research Council for making this work possible.

## References

- Argüeso D, Evans JP, Fita l (2013) Precipitation bias correction of very high resolution regional climate models. Hydrol Earth Syst Sci 17:4379–4388CrossRefGoogle Scholar
- Bennett JC, Grose MR, Corney SP, White CJ, Holz GK, Katzfey JJ, Post DA, Bindoff NL (2014) Performance of an empirical bias-correction of a high-resolution climate dataset. Int J Climatol 34:2189–2204CrossRefGoogle Scholar
- Boé J, Terray L, Habets F, Martin E (2007) Statistical and dynamical downscaling of the Seine basin climate for hydro-meteorological studies. Int J Climatol 27:1643–1655CrossRefGoogle Scholar
- Cannon AJ (2016) Multivariate bias correction of climate model output: matching marginal distributions and intervariable dependence structure. J Clim 29:7045–7064CrossRefGoogle Scholar
- Cannon AJ (2017) Multivariate quantile mapping bias correction: an N-dimensional probability density function transform for climate model simulations of multiple variables. Clim Dyn:1–19Google Scholar
- Cannon AJ, Sobie SR, Murdock TQ (2015) Bias correction of GCM precipitation by quantile mapping: how well do methods preserve changes in quantiles and extremes? J Clim 28:6938–6959CrossRefGoogle Scholar
- Chatfield C (2004) The analysis of time series: an introduction. CHAPMAN & HALL/CRC Press, New YorkGoogle Scholar
- Chen J, Brissette FP, Lucas-Picher P (2015) Assessing the limits of bias-correcting climate model outputs for climate change impact studies. J Geophys Res Atmos 120:1123–1136CrossRefGoogle Scholar
- Christensen JH, Boberg F, Christensen OB, Lucas-Picher P (2008) On the need for bias correction of regional climate change projections of temperature and precipitation. Geophys Res Lett 35:L20709CrossRefGoogle Scholar
- Eden JM, Widmann M, Grawe D, Rast S (2012) Skill, correction, and downscaling of GCM-simulated precipitation. J Clim 25:3970–3984CrossRefGoogle Scholar
- Ehret U, Zehe E, Wulfmeyer V, Warrach-Sagi K, Liebert J (2012) HESS opinions “should we apply bias correction to global and regional climate model data?” Hydrol Earth Syst Sci 16:3391–3404CrossRefGoogle Scholar
- Emery WJ, Thomson RE (2001) Chapter 5—time-series analysis methods. In: Data analysis methods in physical oceanography. Elsevier, AmsterdamGoogle Scholar
- Geckinli N, Yavuz D (1978) Some novel windows and a concise tutorial comparison of window families. IEEE Trans Acoust Speech Signal Process 26:501–507CrossRefGoogle Scholar
- Gudmundsson L, Bremnes JB, Haugen JE, Engen-Skaugen T (2012) Technical note: downscaling RCM precipitation to the station scale using statistical transformations—a comparison of methods. Hydrol Earth Syst Sci 16:3383–3390CrossRefGoogle Scholar
- Haerter JO, Hagemann S, Moseley C, Piani C (2011) Climate model bias correction and the role of timescales. Hydrol Earth Syst Sci 15:1065–1079CrossRefGoogle Scholar
- Hagemann S, Chen C, Clark DB, Folwell S, Gosling SN, Haddeland I, Hanasaki N, Heinke J, Ludwig F, Voss F, Wiltshire AJ (2013) Climate change impact on available water resources obtained using multiple global climate and hydrology models. Earth Syst Dynam 4:129–144CrossRefGoogle Scholar
- Hegge BJ, Masselink G (1996) Spectral analysis of geomorphic time series: auto-spectrum. Earth Surf Proc Land 21:1021–1040CrossRefGoogle Scholar
- Hempel S, Frieler K, Warszawski L, Schewe J, Piontek F (2013) A trend-preserving bias correction—the ISI-MIP approach. Earth Syst Dyn 4:219–236CrossRefGoogle Scholar
- Johnson F, Sharma A (2012) A nesting model for bias correction of variability at multiple time scales in general circulation model precipitation simulations. Water Resour Res 48:W01504Google Scholar
- Kottegoda NT (1980) Analysis of hydrologic time series. Stochastic water resources technology. Palgrave Macmillan, LondonCrossRefGoogle Scholar
- Li H, Sheffield J, Wood EF (2010) Bias correction of monthly precipitation and temperature fields from Intergovernmental Panel on Climate Change AR4 models using equidistant quantile matching. J Geophys Res Atmos 115:D10101CrossRefGoogle Scholar
- Li C, Sinha E, Horton DE, Diffenbaugh NS, Michalak AM (2014) Joint bias correction of temperature and precipitation in climate model simulations. J Geophys Res Atmos:119:13153–13162CrossRefGoogle Scholar
- Macadam I, Argüeso D, Evans JP, Liu DL, Pitman AJ (2016) The effect of bias correction and climate model resolution on wheat simulations forced with a regional climate model ensemble. Int J Climatol 36:4577–4591CrossRefGoogle Scholar
- Mao G, Laux VOGLS, Wagner PS, Kunstmann H (2015) Stochastic bias correction of dynamically downscaled precipitation fields for Germany through Copula-based integration of gridded observation data. Hydrol Earth Syst Sci 19:1787–1806CrossRefGoogle Scholar
- Maraun D (2016) Bias correcting climate change simulations—a critical review. Curr Clim Change Rep 2:211–220CrossRefGoogle Scholar
- Maraun D, Wetterhall F, Ireson AM, Chandler RE, Kendon EJ, Widmann M, Brienen S, Rust HW, Sauter T, Themeßl M, Venema VKC, Chun KP, Goodess CM, Jones RG, Onof C, Vrac M, Thiele-Eich I (2010) Precipitation downscaling under climate change: recent developments to bridge the gap between dynamical models and the end user. Rev Geophys 48:RG3003CrossRefGoogle Scholar
- Maraun D, Shepherd TG, Widmann M, Zappa G, Walton D, Gutierrez JM, Hagemann S, Richter I, Soares PMM, Hall A, Mearns LO (2017) Towards process-informed bias correction of climate change simulations. Nat Clim Chang 7:664–773CrossRefGoogle Scholar
- Mehrotra R, Sharma A (2012) An improved standardization procedure to remove systematic low frequency variability biases in GCM simulations. Water Resour Res 48:W12601Google Scholar
- Mehrotra R, Sharma A (2015) Correcting for systematic biases in multiple raw GCM variables across a range of timescales. J Hydrol 520:214–223CrossRefGoogle Scholar
- Mehrotra R, Sharma A (2016) A multivariate quantile-matching bias correction approach with auto- and cross-dependence across multiple time scales: implications for downscaling. J Clim 29:3519–3539CrossRefGoogle Scholar
- Meyer JDD, Jin J (2016) Bias correction of the CCSM4 for improved regional climate modeling of the North American monsoon. Clim Dyn 46:2961–2976CrossRefGoogle Scholar
- Mudelsee M (2014) Climate time series analysis: classical statistical and bootstrap methods. Springer, BerlinCrossRefGoogle Scholar
- Nahar J, Johnson F, Sharma A (2017) Assessing the extent of non-stationary biases in GCMs. J Hydrol 549:148–162CrossRefGoogle Scholar
- Nguyen H, Mehrotra R, Sharma A (2016) Correcting for systematic biases in GCM simulations in the frequency domain. J Hydrol 538:117–126CrossRefGoogle Scholar
- Nguyen H, Mehrotra R, Sharma A (2017) Can the variability in precipitation simulations across GCMs be reduced through sensible bias correction? Clim Dyn 49:3257–3275CrossRefGoogle Scholar
- Percival DB, Walden AT (1993) Spectral analysis for physical applications: multitaper and conventional univariate techniques. Cambridge University Press, New YorkCrossRefGoogle Scholar
- Piani C, Haerter JO (2012) Two dimensional bias correction of temperature and precipitation copulas in climate models. Geophys Res Lett 39:L20401CrossRefGoogle Scholar
- Piani C, Haerter JO, Coppola E (2010) Statistical bias correction for daily precipitation in regional climate models over Europe. Theoret Appl Climatol 99:187–192CrossRefGoogle Scholar
- Pierce DW, Cayan DR, Maurer EP, Abatzoglou JT, Hegewisch KC (2015) Improved bias correction techniques for hydrological simulations of climate change. J Hydrometeorol 16:2421–2442CrossRefGoogle Scholar
- Prichard D, Theiler J (1994) Generating surrogate data for time series with several simultaneously measured variables. Phys Rev Lett 73:951–954CrossRefGoogle Scholar
- Rodríguez-Iturbe I, Nordin CF (1969) Some applications of cross-spectral analyses in hydrology: rainfall and runoff. Water Resour Res 5:608–621CrossRefGoogle Scholar
- Sippel S, Otto FEL, Forkel M, Allen MR, Guillod BP, Heimann M, Reichstein M, Seneviratne SI, Thonicke K, Mahecha MD (2016) A novel bias correction methodology for climate impact simulations. Earth Syst Dyn 7:71–88CrossRefGoogle Scholar
- Sperna Weiland FC, van Beek IPH, Kwadijk JCJ, Bierkens MFP (2010) The ability of a GCM-forced hydrological model to reproduce global discharge variability. Hydrol Earth Syst Sci 14:1595–1621CrossRefGoogle Scholar
- Teutschbein C, Seibert J (2012) Bias correction of regional climate model simulations for hydrological climate-change impact studies: review and evaluation of different methods. J Hydrol 456–457:12–29CrossRefGoogle Scholar
- Thomson RE, Emery WJ (2014) Chapter 5—time series analysis methods. In: Emery RETJ (ed) Data analysis methods in physical oceanography (third edition). Elsevier, BostonGoogle Scholar
- Troin M, Velázquez JA, Caya D, Brissette F (2015) Comparing statistical post-processing of regional and global climate scenarios for hydrological impacts assessment: a case study of two Canadian catchments. J Hydrol 520:268–288CrossRefGoogle Scholar
- Vrac M, Friederichs P (2014) Multivariate—intervariable, spatial, and temporal—bias correction. J Clim 28:218–237CrossRefGoogle Scholar
- Vrac M, Noël T, Vautard R (2016) Bias correction of precipitation through singularity stochastic removal: because occurrences matter. J Geophys Res Atmos 121:5237–5258CrossRefGoogle Scholar
- Welch P (1967) The use of fast Fourier transform for the estimation of power spectra: a method based on time averaging over short, modified periodograms. IEEE Trans Audio Electroacoust 15:70–73CrossRefGoogle Scholar
- White RH, Toumi R (2013) The limitations of bias correcting regional climate model inputs. Geophys Res Lett 40,:2907–2912CrossRefGoogle Scholar
- Wilby R, Charles S, Zorita E, Timbal B, Whetton P, Mearns L (2004) Guidelines for use of climate scenarios developed from statistical downscaling methods. IPCC Task Group on Data and Scenario Support for Impact and Climate Analysis (TGICA). http://ipcc-ddc.cru.uea.ac.uk/guidelines/StatDown_Guide. pdf. Accessed 1 Apr 2017
- Wood AW, Leung LR, Sridhar V, Lettenmaier DP (2004) Hydrologic implications of dynamical and statistical approaches to downscaling climate model outputs. Clim Change 62:189–216CrossRefGoogle Scholar