method_ecdfm.md

Equi-distant/ratio CDF matching (ECDFm)

Overview

The first step in any bias correction prodcedure involves establishing a statistical relationship or transfer function between model outputs and observations over a reference (i.e. historical/training) time period. The established transfer function is then applied to the target model data (e.g. future model projections) in order to produce a "bias corrected" model time series.

Many bias correction procedures are quantile based, meaning the model data are corrected on a quantile by quantile basis. In equidistant cumulative density function matching (EDCDFm; Li et al, 2010), the transfer function represents the distance (i.e. arithmetic difference) between the observations and model for each quantile of the training period. Those differences are then added to the target model data according to the quantile each target data point represents over the target period. For instance, if a target temperature of $25^{\circ}$ Celsius corresponds to the 0.5 quantile (i.e. the median) in the target data, the difference between the median value in the observations and reference model data is added to the target value in order to obtain the bias adjusted value. The underlying assumption is that the distance between the model and observed quantiles during the training period also applies to the target period, hence the name equidistant. The reference to CDF matching is clear from the mathematical representation of the method: $$x_{m-adjust} = x_{m,p} + F_{o,h}^{-1}(F_{m,p}(x_{m,p})) - F_{m,h}^{-1}(F_{m,p}(x_{m,p}))$$

where $F$ is the CDF of either the observations ($o$) or model ($m$) for a historic training period ($h$) or target period ($p$). That means $F_{o,h}^{-1}$ and $F_{m,h}^{-1}$ are the quantile functions (inverse CDF) corresponding to the observations and model respectively. Returning to our target median value of $25^{\circ}$ (i.e. $x_{m,p} = 25$), the corresponding CDF would return a value of 0.5 (i.e. $F_{m,p}(25) = 0.5$). The difference between the observed ( $F_{o,h}^{-1}(0.5)$ ) and reference model ( $F_{m,h}^{-1}(0.5)$ ) median values would then be added to the target value of $25^{\circ}$ to get a bias corrected value.

For variables like precipitation, multiplicative as opposed to additive bias correction is preferred to avoid the possibility of getting bias corrected values less than zero. In this case, equiratio CDF matching (EQCDFm; Wang and Chen, 2013) is used:

$$x_{m-adjust} = x_{m,p} \times (F_{o,h}^{-1}(F_{m,p}(x_{m,p})) \div F_{m,h}^{-1}(F_{m,p}(x_{m,p})))$$

Methodological decisions

There are a number of choices to make when applying EDCDFm or EQCDFm:

• Parametric or non-parametric: It is generally accepted that non-parametric quantile mapping is best for bias correction, so EDCDFm and EQCDFm is usually applied without fitting a parametric distribution to the data first. Our qqscale software takes a non-parametric / empirical approach.
• Downscaling (when and how): Model data is usually on a coarser spatial grid than observations. Some authors downscale the model data first (via simple spatial interpolation, statistical downscaling or dynamical downscaling) and then perform bias correction. Others upscale the observations, perform the bias correction on the model grid and then downscale the bias corrected model data. Our qqscale software does the former and downscales the model data first using bilinear interpolation.
• Time grouping: It is common to bias correct individual seasons or months separately to avoid conflating different times of the year (e.g. spring and autumn temperatures often occupy the same annual quantile space but may be biased in different ways). The qqscale software allows the user to specify what type of time grouping to apply. We commonly use monthly time grouping for EDCDFm and EQCDFm (i.e. process each month separately). We've found that something like a 30-day running window is far more computationally expensive and produces similar results to monthly grouping.
• Qunatiles: Our qqscale software allows the user to specify the number of quantiles to calculate. We've found that it's best to have approximately 10 data values between each quanite. If you're processing 30 years of daily data, that means 100 quantiles if the time grouping is monthly or 1000 quantiles if no time grouping is applied.
• Adjustment factor smoothing: By default, the bias correction applied to each target data point is the closest value from the array of adjustment factors. For example, it might be a 12 (months) by 100 (quantiles) array of adjustment factors. Linear or cubic interpolation / smoothing of the adjustment factors can be optionally applied along the time (e.g. month) axis. For example, the adjustment factor for a target data point from 29 July that corresponds to the 0.651 quantile could be a linear combination of the adjustment factors for the nearest quantile (0.65) from both July and August. We've found that linear and cubic interpolation along the time axis produce slighty better results than no smoothing at all but can be more computationally expensive.
• Singularity stochastic removal (Vrac et al 2016) is used to avoid divide by zero errors in the analysis of precipitation data. All near-zero values (i.e. values less than a very small positive threshold value) are set to a small random non-zero value prior to data processing, and then after the bias correction process is complete any values less than the threshold are set to zero.