Statistical Bias Correction On Sub-Daily/Daily Precipitation

1. Introduction

Statistical Bias Correction (BC) is a key technique to post-process and calibrate the direct output of the numerical modeling such as GCM/RCM, even the outputs from classic statistical downscaling (SDSM). Generally, the BC methods are applied for model-based precipitation and involve some form of transfer function derived from cumulative distribution functions (CDFs) of observations and model simulations. These approaches are also known as Model Output Statistics (MOS; Maraun et al., 2010). 

With the increasing skill of regional climate models (RCMs) and the availability of RCM scenarios, MOS methods are becoming popular (Maraun et al., 2010). The scheme of different downscaling approaches and their combinations is represented in the following figure.

There are many methods that can be used to carry out bias correction, such as pattern scaling (e.g. Mitchell 2000, 2003; Wood et al., 2004; Ines and Hansen, 2006; Piani et al.,  2010; Li et al., 2010; Yin, 2011).

2. Statistical bias correction methods for sub-daily/daily precipitation

Here, take daily precipitation as an example. The procedure is also suitable for sub-daily precipitation.

The daily precipitation event is intermittent in nature, especially for dry regions. Consequently, the separation of daily precipitation into frequency (fraction of precipitation days, or wet days) and intensity (rainfall per wet day) in model simulations allows for a more accurate precipitation modeling evaluation. Therefore correcting any bias of the two rainfall components will also correct the monthly total rainfall itself. A two-step BC procedure is therefore proposed to simultaneously adjust the two components of modeled rainfall to make it approximate the long-term observed distribution at each target grid (or site). For convenience, and consistency with the convention of updating model forecasts monthly, the proposed BC procedure was carried out monthly.

2.1 Correcting precipitation frequency

To perform a BC operation, the observed precipitation was firstly truncated using the threshold value of 0.1mm (  Xbar ) to obtain an ideal frequency distribution.  Then, the downscaled daily precipitation during the baseline period was fitted into an empirical frequency distribution. By finding a threshold, this distribution  was truncated so as to make sure that its frequency above the threshold would approximate the observed precipitation frequency (Ines and Hansen, 2006). The threshold was calculated from the empirical observed and modeled cumulative precipitation distribution as,

The procedure can be illustrated as follows.

2.2 Correction of precipitation intensity

After correcting precipitation frequency, the intensity distribution of the truncated modeled daily precipitation could be further adjusted by the rescaling method or the quantile-based mapping methods (CDF matching) (Panofsky and Brier, 1968; Law and Kelton, 1982). For the latter, a typical procedure is to map the distribution of modeled daily precipitation and there are two kinds of method can implement such a task:

• BC based on CDF and Replacing(CDFR)

To bias correct downscaled values for a future period, the method needs firstly to find the corresponding percentile values for these future (modeled) values on the CDF of the modeled values during the baseline period and then search for the observed values on the CDF of the observations at the same found percentile locations. Thus, the original downscaled results were totally replaced by those values found on the CDF of the observations. Because, this BC procedure was based on CDF and replacing, it hereafter was termed as CDFR. The following figure (A) illustrated how it worked.

The above BC method has been successfully used in hydrologic and crop simulations as well as many other climate impact studies (e.g., Ines and Hansen, 2006; Cayan et al., 2008; Piani et al., 2010). The significant characteristics of the method is that it adjusts all moments (i.e., the entire distribution matches that of the observations for the baseline period), while maintaining the rank correlation between modeled results and observations. However, the method was based on an important assumption; namely, the precipitation distribution does not change over time. In other words, the future projection will still follow the same statistical characteristics of the observed precipitation (e.g., the variance and skew) during the baseline period and only the mean would change. However, some studies have shown that the precipitation characteristics would change over time (e.g., Benestad et al., 2008; Meehl et al., 2007; Milly et al., 2008). 

• BC based on Equal Distance-based CDF mapping (EDCDF )

An equal distance-based CDF mapping method (EDCDF) was proposed by Yin (2011) to correct the modeled precipitation intensities. This method was different from the above traditional method in that the information from the CDF of the modeled projection was also incorporated into the BC procedure. For a given percentile, it was assumed that the difference between the modeled and observed value during the baseline period also applies to the future period. See the following figure (B).

As can be seen from the figure, the difference between the modeled and observed values during the baseline period at each percentile was considered as the systematic error to be superimposed upon the CDF of the modeled values at the corresponding percentile during the future period. This was the reason why the method was termed the equidistant CDF matching method. Compared with CDFP, the difference between the CDFs for the future and baseline periods was also taken into account in EDCDF. However, the two methods will generate an identical BC result if the distribution for the future climate is the same as that for the baseline period. Moreover, if the changes in variability are small, results from both methods will be close to each other.

3. Discussions

The above two BC methods are all based on CDF and rely on the statistical relationships between the two CDFs (observed, modeled for the baseline period) to correct the third one (modeled for the future period). Indeed, the differences between the modeled results of the future and baseline period could also be superimposed onto the observed CDF to construct new projections.

The above two methods involved the operation of taking the inverse of the CDFs. In practice, the CDFs can be either empirical (i.e., sorted arrays of observations) or fitted to some theoretical distribution such as the gamma distribution (e.g., Ines and Hansen, 2006; Piani et al., 2010; Li et al., 2010), Log-Logistic Distribution (e.g., Shouki et al., 1988) or exponential distribution (e.g., Madi and Raqab, 2007).

However, a single theoretical distribution does not always work well for regions with very complicated climate types and topographic features such as the case study area (Vl?ek and Huth, 2009). Moreover, fitting a distribution often needs long-term observed precipitation data.

4. More

Bias Correction (BC) techniques have became popular in the last decade due to their simplicity and straightforward applications. However, they have their own problems/limitations (White and Toumi, 2013; Maraun, 2013):

• Do not preserve the original temporal correspondence of the model output;
• Mostly operate on a single variable;
• Presume the biases are stationary in the future.

5. Applications

The bias correction procedure is mainly used for producing grid-to-grid time series for cropping or hydrology modelling. Now there are also trials to apply the bias-corrected time series to extreme precipitation analysis. Under such a case, the EDCDF method should outperform the CDFR method. However, it should be better to directly abstract extreme precipitation changes from GCMs/RCMs cross a area instead of a single grid.

References

Ines, A. V. M., & Hansen, J.W. (2006). Bias correction of daily GCM rainfall for crop simulation studies. Agricultural and Forest Meteorology, 138, 44-53.

Maraun, D., et al. (2010). Precipitation downscaling under climate change: Recent developments to bridge the gap between dynamical models and the end user, Rev. Geophys., 48, RG3003, doi:10.1029/2009RG000314.

Maraun, D. (2013) Bias Correction, Quantile Mapping, and Downscaling: Revisiting the Inflation Issue, Journal of Climate, 26, 2137–2143, doi:10.1175/JCLI-D-12-00821.1, https://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-12-00821.1.

Piani, C., Haerter, J. O., & Coppola, E. (2009). Statistical bias correction for daily precipitation in regional climate models over Europe, Theor. Appl. Climatol., 99, 187–192.

White, R. H. and Toumi, R. (2013) The limitations of bias correcting regional climate model inputs, Geophysical Research Letters, 40, 29072912, doi:10.1002/grl.50612, https://onlinelibrary.wiley.com/doi/10.1002/grl.50612/abstract.

Yin, C. (2011). Applications of Self-Organizing Maps to Statistical Downscaling of Major Regional Climate Variables (Doctoral dissertation, University of Waikato).