# A Markov Chain-Based Bias Correction Method for Simulating the Temporal Sequence of Daily Precipitation

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## Abstract

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## 1. Introduction

## 2. Study Area and Data Sources

#### 2.1. Study Area

#### 2.2. Data Sources

## 3. Methodology

#### 3.1. Quantile Mapping Bias Correction Method

#### 3.2. The First-Order Two-State Markov Chain

_{11}is the probability of a wet day following a wet day, and P

_{01}is the probability of a wet day following a dry day, as shown in the following formulas:

_{01}= Pr {precipitation on day t | no precipitation on day t − 1}

_{11}= Pr {precipitation on day t | precipitation on day t − 1}

_{01}, a wet day is generated; and otherwise, a dry day is generated. A similar process can also be used with P

_{11}.

#### 3.3. Hybrid Bias Correction Method

_{01}and P

_{11}must be adjusted according to the climate change signal simulated by the climate models. Previous studies showed that there are strong linear relationships between the mean monthly precipitation and the two conditional transition probabilities [16,19,30]. The linear relationship can be fitted using the observed precipitation time series. In other words, if we know the mean monthly precipitation of the validation period, two conditional transition probabilities for this period can be obtained based on the linear relationship established in the calibration period. As mentioned earlier, the QM method can provide an accurate simulation of the monthly precipitation amounts.

- (a)
- Taking January as an example, the total precipitation of this month over the whole period is sorted in ascending order and then separated into dry and wet groups. Then the mean monthly precipitation is calculated from each group. The daily series corresponded to the months in each group are gathered and used to calculate P
_{01}and P_{11}. With this step, a set of P01, P11, and the mean monthly precipitation is calculated for each group. - (b)
- The total precipitation of this month each year is also divided into two, even periods in chronological order. A set of parameters is also calculated for each period.
- (c)
- A linear equation is established for P
_{01}(and P_{11}) and the mean monthly precipitation based on the four samples calculated from step-(a) and step-(b).

_{01}and P

_{11}can be regressed using the established linear relationships for each calendar month. The first-order, two-state Markov chain is used to generate the sequence of precipitation occurrence. For a predicted wet day, the precipitation amount is generated by realigning the DBC-corrected precipitation amounts. Specifically, a set of random numbers is first generated for wet days simulated by the first-order, two-state Markov chain. The random numbers are then replaced by DBC-corrected precipitation amounts, according to the numerical value of the random number. In other words, the larger precipitation amount is given to the larger random number, and vice versa. If the first-order, two-state Markov chain is perfect for simulating the sequence of precipitation occurrence, then the simulated wet- and dry-day frequencies should be equal to the frequencies generated by the DBC. However, this may not be the case. Scaling the realigned daily precipitation for each month may be necessary to correct or offset the generation error of the first-order, two-state Markov chain. This is done by multiplying the ratio of the DBC-corrected mean monthly precipitation and MCBC-corrected mean monthly precipitation by the daily precipitation series of the MCBC-sampled for each specific month.

#### 3.4. Analysis Procedure And Statistics

## 4. Results

#### 4.1. Validation of the Daily Bias Correction

#### 4.2. Validation of the Linear Relationships

_{01}, P

_{11,}and mean monthly precipitation, which were calculated from the dry group, wet group, and two even groups of the calibration period, are plotted in Figure 3 and Figure 4 for January and July, respectively for the ten stations. The linear curves that were regressed using the four data points were used to interpolate the conditional transition probabilities in the validation period. For comparison, P

_{01}, P

_{11}, and the mean monthly precipitation were also calculated for the validation period and plotted in Figure 3 and Figure 4 as solid points. The names of the legend ‘cali’ and ‘vali’ represent calibration and validation, respectively.

_{01}from both the calibration and validation periods were fairly close to the regression lines. The correlation coefficient between mean monthly precipitation and transition probabilities was greater than 0.9 for 91 out of 120 cases (12 months × 10 stations) for P

_{01}, and for 75 out of 120 cases for P

_{11}. The reason why there was a stronger relationship for P

_{01}than P

_{11}may be because the occurrence of wet-following-wet events is lesser than that of wet-following-dry events, which contributed to greater variability and less reliability when estimating P

_{11}. As Zhang et al. (2012) presented in their study that a longer period should be used to obtain large samples of wet-following-wet events to improve the accuracy of the estimated P

_{11}.

#### 4.3. Performance of the MCBC Method

## 5. Discussions and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Chen, J.; Brissette, F.P.; Poulin, A.; Leconte, R. Overall uncertainty study of the hydrological impacts of climate change for a Canadian watershed. Water Resour. Res.
**2011**, 47. [Google Scholar] [CrossRef] - Jeong, H.; Bhattarai, R.; Hwang, S. How climate scenarios alter future predictions of field-scale water and nitrogen dynamics and crop yields. J. Environ. Manag.
**2019**, 252, 109623. [Google Scholar] [CrossRef] [PubMed] - Ramirez-Villegas, J.; Challinor, A.J.; Thornton, P.K.; Jarvis, A. Implications of regional improvement in global climate models for agricultural impact research. Environ. Res. Lett.
**2013**, 8, 024018. [Google Scholar] [CrossRef] - Wang, G. Agricultural drought in a future climate: Results from 15 global climate models participating in the IPCC 4th assessment. Clim. Dyn.
**2005**, 25, 739–753. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Leconte, R. Uncertainty of downscaling method in quantifying the impact of climate change on hydrology. J. Hydrol.
**2011**, 401, 190–202. [Google Scholar] [CrossRef] - Chen, J.; Chen, H.; Guo, S. Multi-site precipitation downscaling using a stochastic weather generator. Clim. Dyn.
**2017**, 50, 1975–1992. [Google Scholar] [CrossRef] - Martins, M.A.; Tomasella, J.; Dias, C.G. Maize yield under a changing climate in the Brazilian Northeast: Impacts and adaptation. Agric. Water Manag.
**2019**, 216, 339–350. [Google Scholar] [CrossRef] - Jakob Themeßl, M.; Gobiet, A.; Leuprecht, A. Empirical-statistical downscaling and error correction of daily precipitation from regional climate models. Int. J. Climatol.
**2011**, 31, 1530–1544. [Google Scholar] [CrossRef] - Troin, M.; Velázquez, J.A.; Caya, D.; Brissette, F. Comparing statistical post-processing of regional and global climate scenarios for hydrological impacts assessment: A case study of two Canadian catchments. J. Hydrol.
**2015**, 520, 268–288. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Chaumont, D.; Braun, M. Performance and uncertainty evaluation of empirical downscaling methods in quantifying the climate change impacts on hydrology over two North American river basins. J. Hydrol.
**2013**, 479, 200–214. [Google Scholar] [CrossRef] - Ines, A.V.; Hansen, J.W. Bias correction of daily GCM rainfall for crop simulation studies. Agric. For. Meteorol.
**2006**, 138, 44–53. [Google Scholar] [CrossRef] [Green Version] - Maraun, D.; Shepherd, T.G.; Widmann, M.; Zappa, G.; Walton, D.; Gutiérrez, J.M.; Mearns, L.O. Towards process-informed bias correction of climate change simulations. Nat. Clim. Chang.
**2017**, 7, 764–773. [Google Scholar] [CrossRef] [Green Version] - Manzanas, R.; Gutiérrez, J.M.; Bhend, J.; Hemri, S.; Doblas-Reyes, F.J.; Torralba, V.; Penabad, E.; Brookshaw, A. Bias adjustment and ensemble recalibration methods for seasonal forecasting: A comprehensive intercomparison using the C3S dataset. Clim. Dyn.
**2019**, 53, 1287–1305. [Google Scholar] [CrossRef] - Caya, D.; Laprise, R. A semi-implicit semi-lagrangian regional climate model: The canadian rcm. Mon. Weather Rev.
**1999**, 127, 341–362. [Google Scholar] [CrossRef] - Teutschbein, C.; Seibert, J. Regional Climate Models for Hydrological Impact Studies at the Catchment Scale: A Review of Recent Modeling Strategies. Geogr. Compass
**2010**, 4, 834–860. [Google Scholar] [CrossRef] [Green Version] - Zhang, X.C.; Chen, J.; Garbrecht, J.D.; Brissette, F.P. Evaluation of a weather generator-based method for statistically downscaling non-stationary climate scenarios for impact at point scale. Trans. ASAE
**2012**, 55, 1745–1756. [Google Scholar] [CrossRef] - Fowler, H.J.; Ekström, M.; Blenkinsop, S.; Smith, A.P. Estimating change in extreme European precipitation using a multimodel ensemble. J. Geophys. Res.
**2007**, 112. [Google Scholar] [CrossRef] [Green Version] - Murphy, J. An Evaluation of Statistical and Dynamical Techniques for Downscaling Local Climate. J. Clim.
**1999**, 12, 2256–2284. [Google Scholar] [CrossRef] - Zhang, X.C. Verifying a temporal disaggregation method for generating daily precipitation of potentially non-stationary climate change for site-specific impact assessment. Int. J. Climatol.
**2013**, 33, 326–342. [Google Scholar] [CrossRef] - Cannon, A.J.; Sobie, S.R.; Murdock, T.Q. Bias Correction of GCM Precipitation by Quantile Mapping: How Well Do Methods Preserve Changes in Quantiles and Extremes? J. Clim.
**2015**, 28, 6938–6959. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Chaumont, D.; Braun, M. Finding appropriate bias correction methods in downscaling precipitation for hydrologic impact studies over North America. Water Resour. Res.
**2013**, 49, 4187–4205. [Google Scholar] [CrossRef] - Teutschbein, C.; Seibert, J. Bias correction of regional climate model simulations for hydrological climate-change impact studies: Review and evaluation of different methods. J. Hydrol.
**2012**, 456, 12–29. [Google Scholar] [CrossRef] - Hnilica, J.; Hanel, M.; Puš, V. Multisite bias correction of precipitation data from regional climate models. Int. J. Climatol.
**2017**, 37, 2934–2946. [Google Scholar] [CrossRef] - Rosenberg, E.A.; Keys, P.W.; Booth, D.B.; Hartley, D.; Burkey, J.; Steinemann, A.C.; Lettenmaier, D.P. Precipitation extremes and the impacts of climate change on stormwater infrastructure in Washington State. Clim. Chang.
**2010**, 102, 319–349. [Google Scholar] [CrossRef] [Green Version] - Schoof, J.T.; Shin, D.W.; Cocke, S.; LaRow, T.E.; Lim, Y.K.; O’Brien, J.J. Dynamically and statistically downscaled seasonal temperature and precipitation hindcast ensembles for the southeastern USA. Int. J. Climatol.
**2009**, 29, 243–257. [Google Scholar] [CrossRef] [Green Version] - Roosmalen, L.V.; Christensen, J.H.; Butts, M.B.; Jensen, K.H.; Refsgaard, J.C. An intercomparison of regional climate model data for hydrological impact studies in Denmark. J. Hydrol.
**2010**, 380, 406–419. [Google Scholar] [CrossRef] - Schmidli, J.; Frei, C.; Vidale, P.L. Downscaling from GCM precipitation: A benchmark for dynamical and statistical downscaling methods. Int. J. Climatol.
**2006**, 26, 679–689. [Google Scholar] [CrossRef] - Themeßl, M.J.; Gobiet, A.; Heinrich, G. Empirical-statistical downscaling and error correction of regional climate models and its impact on the climate change signal. Clim. Chang.
**2012**, 112, 449–468. [Google Scholar] [CrossRef] - Shen, M.; Chen, J.; Zhuan, M.; Chen, H.; Xu, C.-Y.; Xiong, L. Estimating uncertainty and its temporal variation related to global climate models in quantifying climate change impacts on hydrology. J. Hydrol.
**2018**, 556, 10–24. [Google Scholar] [CrossRef] - Chen, J.; Zhang, X.J.; Brissette, F.P. Assessing scale effects for statistically downscaling precipitation with GPCC model. Int. J. Climatol.
**2014**, 34, 708–727. [Google Scholar] [CrossRef] - Ines, A.V.; Hansen, J.W.; Robertson, A.W. Enhancing the utility of daily GCM rainfall for crop yield prediction. Int. J. Climatol.
**2011**, 31, 2168–2182. [Google Scholar] [CrossRef] [Green Version] - Maraun, D.; Wetterhall, F.; Ireson, A.M.; Chandler, R.E.; Kendon, E.J.; Widmann, M.; Brienen, S.; Rust, H.W.; Sauter, T.; Themeßl, M.; et al. Precipitation downscaling under climate change: Recent developments to bridge the gap between dynamical models and the end user. Rev. Geophys.
**2010**, 48. [Google Scholar] [CrossRef] - Rajczak, J.; Kotlarski, S.; Schär, C. Does Quantile Mapping of Simulated Precipitation Correct for Biases in Transition Probabilities and Spell Lengths? J. Clim.
**2016**, 29, 1605–1615. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P. Comparison of five stochastic weather generators in simulating daily precipitation and temperature for the Loess Plateau of China. Int. J. Climatol.
**2014**, 34, 3089–3105. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Leconte, R. A daily stochastic weather generator for preserving low-frequency of climate variability. J. Hydrol.
**2010**, 388, 480–490. [Google Scholar] [CrossRef] - Chen, J.; Brissette, F.P.; Leconte, R. Downscaling of weather generator parameters to quantify hydrological impacts of climate change. Clim. Res.
**2012**, 51, 185–200. [Google Scholar] [CrossRef] [Green Version] - Moon, S.-E.; Ryoo, S.-B.; Kwon, J.-G. A Markov chain model for daily precipitation occurrence in South Korea. Int. J. Climatol.
**1994**, 14, 1009–1016. [Google Scholar] [CrossRef] - Wilks, D.S. Multisite generalization of a daily stochastic precipitation generation model. J. Hydrol.
**1998**, 210, 178–191. [Google Scholar] [CrossRef] - Wilks, D.S. Interannual variability and extreme-value characteristics of several stochastic daily precipitation models. Agric. For. Meteorol.
**1999**, 93, 153–169. [Google Scholar] [CrossRef] - Wilks, D.S. Multisite downscaling of daily precipitation with a stochastic weather generator. Clim. Res.
**1999**, 11, 125–136. [Google Scholar] [CrossRef] [Green Version] - Wilks, D.S.; Wilby, R.L. The weather generation game: A review of stochastic weather models. Prog. Phys. Geogr.
**1999**, 23, 329–357. [Google Scholar] [CrossRef] - Apipattanavis, S.; Podestá, G.; Rajagopalan, B.; Katz, R.W. A semiparametric multivariate and multisite weather generator. Water Resour. Res.
**2007**, 43, 1973–1989. [Google Scholar] [CrossRef] - Gu, L.; Chen, J.; Xu, C.-Y.; Wang, H.-M.; Zhang, L. Synthetic Impacts of Internal Climate Variability and Anthropogenic Change on Future Meteorological Droughts over China. Water
**2018**, 10, 1702. [Google Scholar] [CrossRef] [Green Version] - Zhuan, M.-J.; Chen, J.; Shen, M.-X.; Xu, C.-Y.; Chen, H.; Xiong, L.-H. Timing of human-induced climate change emergence from internal climate variability for hydrological impact studies. Hydrol. Res.
**2018**, 49, 421–437. [Google Scholar] [CrossRef] - Mpelasoka, F.S.; Chiew, F.H. Influence of Rainfall Scenario Construction Methods on Runoff Projections. J. Hydrometeorol.
**2009**, 10, 1168–1183. [Google Scholar] [CrossRef] - Gabriel, K.R.; Neumann, J. A markov chain model for daily rainfall occurrence at tel aviv. Q. J. R. Meteorol. Soc.
**2010**, 88, 90–95. [Google Scholar] [CrossRef] - Li, Z.; Brissette, F.; Chen, J. Assessing the applicability of six precipitation probability distribution models on the Loess Plateau of China. Int. J. Climatol.
**2014**, 34, 462–471. [Google Scholar] [CrossRef] - Heim, R.R. A Review of Twentieth-Century Drought Indices Used in the United States. Bull. Am. Meteorol. Soc.
**2002**, 83, 1149–1166. [Google Scholar] [CrossRef] [Green Version] - Wilks, D.S. Adapting stochastic weather generation algorithms for climate change studies. Clim. Chang.
**1992**, 22, 67–84. [Google Scholar] [CrossRef] - Schoof, J.T.; Pryor, S.C. On the Proper Order of Markov Chain Model for Daily Precipitation Occurrence in the Contiguous United States. J. Appl. Meteorol. Climatolo.
**2008**, 47, 2477–2486. [Google Scholar] [CrossRef] - Zhang, X.C.; Garbrecht, J.D. Evaluation of CLIGEN precipitation parameters and their implication on WEPP runoff and erosion prediction. Tran. ASAE
**2003**, 46, 311. [Google Scholar] - Iman, R.L.; Conover, W.J. A distribution-free approach to inducing rank correlation among input variables. Commun. Stat. Simul. Comput.
**2007**, 11, 311–334. [Google Scholar] [CrossRef] - Li, Z. A new framework for multi-site weather generator: A two-stage model combining a parametric method with a distribution-free shuffle procedure. Clim. Dyn.
**2013**, 43, 657–669. [Google Scholar] [CrossRef] - Zhang, X.C. Generating correlative storm variables for CLIGEN using a distribution-free approach. Trans. ASAE
**2005**, 48, 567–575. [Google Scholar] [CrossRef] - Li, X.; Babovic, V. Multi-site multivariate downscaling of global climate model outputs: An integrated framework combining quantile mapping, stochastic weather generator and Empirical Copula approaches. Clim. Dyn.
**2019**, 52, 5775–5799. [Google Scholar] [CrossRef] - Li, X.; Babovic, V. A new scheme for multivariate, multisite weather generator with inter-variable, inter-site dependence and inter-annual variability based on empirical copula approach. Clim. Dyn.
**2019**, 52, 2247–2267. [Google Scholar] [CrossRef]

**Figure 2.**Relative errors (REs) of the mean monthly precipitation for all 12 calendar months at all ten stations. The red boxplots present the ensemble of the raw GCM simulations, and the yellow boxplots present the DBC corrected results.

**Figure 3.**Relationships between the conditional precipitation occurrence probabilities of wet-following-wet (P

_{11}) and wet-following-dry (P

_{01}) and mean monthly precipitation amounts in January for all stations.

**Figure 4.**Relationships between the conditional precipitation occurrence probabilities of wet-following-wet (P

_{11}) and wet-following-dry (P

_{01}) and mean monthly precipitation amounts in July for all stations.

**Figure 5.**The corrected and observed cumulative frequencies of wet spell lengths for stations 1, 3, 7, and 9. The black line represents the cumulative frequencies of observation. M1-M10 represents 10 GCMs corresponding to the order in Table 2. The X-axis is on a logarithmic scale.

**Figure 6.**The corrected and observed cumulative frequencies of dry spell lengths for stations 1, 3, 7, and 9. The black line represents the cumulative frequencies of observation. M1-M10 represents 10 GCMs corresponding to the order in Table 2. The X-axis is on a logarithmic scale.

**Figure 7.**The bias on the frequency of spell lengths with (

**a**) ≥3, (

**b**) ≥5, and (

**c**) ≥consecutive wet days. The bias on the frequency of spell lengths with (

**d**) ≥3, (

**e**) ≥5, and (

**f**) ≥7 consecutive dry days. The bias is represented as the ratio of the simulated series to observations. The range of the color represents the ensemble of 10 simulations. M1-M10 represents 10 GCMs corresponding to the order in Table 2.

**Figure 8.**RE statistics of the wet- and dry spell lengths generated by the two bias correction methods for all ten stations. (

**a**) Mean wet spell, (

**b**) mean dry spell, (

**c**) standard deviation of wet spell, (

**d**) standard deviation of dry spell, (

**e**) longest wet spell, and (

**f**) longest dry spell.

Station | Name | Latitude (°N) | Longitude (°E) | Elevation (m) | Mean Annual Precipitation (mm) | Wet Day Frequency |
---|---|---|---|---|---|---|

1 | Wuqia | 39.72 | 75.25 | 2176 | 185 | 0.18 |

2 | Bayinbuluke | 43.03 | 84.15 | 2458 | 277 | 0.32 |

3 | Alashanzuoqi | 38.83 | 105.67 | 1561 | 209 | 0.15 |

4 | Langzhong | 31.58 | 105.97 | 383 | 1028 | 0.37 |

5 | Fengshan | 24.55 | 107.03 | 487 | 1526 | 0.44 |

6 | Anyang | 36.05 | 114.40 | 63 | 562 | 0.20 |

7 | Xianyou | 25.37 | 118.70 | 800 | 1565 | 0.39 |

8 | Dalian | 38.90 | 121.63 | 92 | 623 | 0.21 |

9 | Sunwu | 49.43 | 127.35 | 210 | 540 | 0.32 |

10 | Xinerbahuyouqi | 48.67 | 116.82 | 554 | 241 | 0.18 |

No. | Model Name | Modelling Centre | Source | Spatial Resolution (Longitude × Latitude) |
---|---|---|---|---|

1 | ACCESS-1.0 | CRIRO-BOM | Commonwealth Scientific and Industrial Research Organization (CSIRO) and Bureau of Meteorology (BOM), Australia | 1.875° × 1.25° |

2 | CESM1-CAM5 | NSF-DOE-NCAR | National Science Foundation, Department of Energy National Center for Atmospheric Research | 1.25° × 0.94° |

3 | CMCC-CM | CMCC | Centro Euro-Mediterraneo per I Cambiamenti Climatici | 0.75° × 0.75° |

4 | CSIRO-Mk3-6.0 | CSIRO-QCCCE | Commonwealth Scientific and Industrial Research Organization in collaboration with Queensland Climate Change Centre of Excellence | 1.90° × 1.90° |

5 | FGOALS-G2 | LASG-CESS | LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences and CESS, Tsinghua University | 2.81° × 3.00° |

6 | GFDL-ESM2G | NOAA GFDL | NOAA Geophysical Fluid Dynamics Laboratory | 2.50° × 2.0° |

7 | GFDL-ESM2M | |||

8 | MIROC5 | MIPOC | Atmosphere and Ocean Research Institute (The University of Tokyo), National Institute for Environmental Studies, and Japan Agency for Marine-Earth Science and Technology | 1.40° × 1.40° |

9 | MRI-CGCM3 | MRI | Meteorological Research Institute | 1.10° × 1.10° |

10 | MRI-ESM1 | 1.13° × 1.13° |

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## Share and Cite

**MDPI and ACS Style**

Liu, H.; Chen, J.; Zhang, X.-C.; Xu, C.-Y.; Hui, Y.
A Markov Chain-Based Bias Correction Method for Simulating the Temporal Sequence of Daily Precipitation. *Atmosphere* **2020**, *11*, 109.
https://doi.org/10.3390/atmos11010109

**AMA Style**

Liu H, Chen J, Zhang X-C, Xu C-Y, Hui Y.
A Markov Chain-Based Bias Correction Method for Simulating the Temporal Sequence of Daily Precipitation. *Atmosphere*. 2020; 11(1):109.
https://doi.org/10.3390/atmos11010109

**Chicago/Turabian Style**

Liu, Han, Jie Chen, Xun-Chang Zhang, Chong-Yu Xu, and Yu Hui.
2020. "A Markov Chain-Based Bias Correction Method for Simulating the Temporal Sequence of Daily Precipitation" *Atmosphere* 11, no. 1: 109.
https://doi.org/10.3390/atmos11010109