In this paper I present new methods for bias adjustment and statistical downscaling that are tailored to the requirements of the Inter-Sectoral Impact Model Intercomparison Project (ISIMIP). In comparison to their predecessors, the new methods allow for a more robust bias adjustment of extreme values, preserve trends more accurately across quantiles, and facilitate a clearer separation of bias adjustment and statistical downscaling. The new statistical downscaling method is stochastic and better at adjusting spatial variability than the old interpolation method. Improvements in bias adjustment and trend preservation are demonstrated in a cross-validation framework.

Bias adjustment in climate research is the adjustment of statistics of climate simulation data for the purpose of making them more similar to climate observation data. In many application cases, these climate simulation and observation data have different spatial resolution. In most of these cases, the climate observation data are more highly resolved. In any of these cases, bias adjustment requires bridging the resolution gap.

In previous phases of the Inter-Sectoral Impact Model Intercomparison Project

Commonly, the bulk of resources for the development of solutions to these problems is allocated to problem (i), and problem (ii) is solved by a mere spatial interpolation of the simulation data to the spatial resolution of the observation data prior to bias adjustment. For example, this approach was adopted in the ISIMIP Fast Track

These issues can be overcome by spatially multivariate bias adjustment or, as suggested by

In this paper, I present the bias adjustment and statistical downscaling methods to be used in phase 3 of ISIMIP. These methods have been developed following the paradigm of a clear separation of bias adjustment and statistical downscaling. In ISIMIP3, climate simulation data shall first be bias-adjusted at their original spatial resolution using spatially aggregated climate observation data. In a second step, their spatial resolution shall be increased using the original climate observation data and a stochastic statistical downscaling method.

In addition to this paradigm shift, the new bias adjustment method has been developed to work better than its predecessor in several respects. The following design decisions were taken in this context. While structurally different bias adjustment methods (including but not restricted to quantile mapping methods) were used for different climate variables in ISIMIP2b

The remainder of this paper is organized as follows. Climate simulation and observation data used in this study are described in Sect.

Climate simulation data are taken from the fifth phase of the Coupled Model Intercomparison Project

Climate variables considered in this study.

As observational reference data for bias adjustment and statistical downscaling I use the EartH2Observe, WFDEI and ERA-Interim data Merged and Bias-corrected for ISIMIP

The ISIMIP2b bias adjustment and statistical downscaling method is comprehensively described in

For pr, psl, rlds, sfcWind, and tas, monthly mean values are adjusted for the purpose of removing the bias in their historical multiyear mean value (for abbreviations for variables, see Table 1). This adjustment is done multiplicatively for pr, rlds, and sfcWind and additively for psl and tas. In order to preserve trends in multiyear monthly mean values, the same scaling factor or offset is used in all application periods. In a second step, day-to-day variability around the monthly mean value is adjusted using transfer functions derived for every calendar month from historical simulations and observations.

An indirect bias adjustment of tasmax and tasmin is done by adjusting

Bias adjustment of rsds is done by parametric quantile mapping using beta distributions with lower bounds of zero and upper bounds estimated by rescaled climatologies of downwelling shortwave radiation at the top of the atmosphere. For trend preservation, upper bounds, mean values, and variances of historical observations are modified using simulated trends prior to quantile mapping. Using beta distributions with fixed lower and upper bounds of 0 and 100 %, respectively, this method is also used to bias-adjust hurs. Bias-adjusted prsn values are obtained by multiplying bias-adjusted pr values with the original prsn-over-pr ratio. This ratio is therefore not bias-adjusted.

The newly developed ISIMIP3 bias adjustment and statistical downscaling method is comprehensively described in the following. It consists of a bias adjustment method that is applied at the spatial resolution of the climate simulation data and a statistical downscaling method that is applied to the bias-adjusted climate simulation data for the purpose of increasing its spatial resolution to that of the climate observation data. These two new methods are presented in the following two subsections.

This section describes the new bias adjustment method. Before diving into details I will explain the concept of the method and outline how the new unified bias adjustment framework can be customized for an application to the climate variables listed in Table

The ISIMIP3 bias adjustment method is a parametric quantile mapping method that has been designed to (i) robustly adjust biases in all percentiles of a distribution and (ii) preserve trends in these percentiles. It is applicable for bias adjustment of different kinds of climate variables including those listed in Table

In order to overcome the multitude of approaches to bias adjustment used for different variables in ISIMIP2b, the new method features a unified framework, which can be customized for an application to one particular climate variable. Customization specifications for the variables considered here are listed in Table

Specification of the ISIMIP3 bias adjustment method for all climate variables considered in this study. Where a lower (upper) bound is specified, no values smaller (greater) than this bound will occur in the bias-adjusted data. For every lower (upper) bound, a lower (upper) threshold is defined, which is only slightly greater (smaller) than the bound. The lower (upper) threshold is used to adjust the relative frequency of values smaller (greater) than the threshold. Note that the units of

In the following, I will describe the unified framework of the ISIMIP3 bias adjustment method in detail. In this context, let

The bias adjustment algorithm with inputs

(For rsds only.) Scale values in

(For prsnratio only.) Replace missing values in

(For psl, rlds, and tas only.) Detrend

(For bounded variables only.) Randomize values beyond thresholds in

(For all variables.) Transfer the simulated climate change signal for every distribution quantile from

(For all variables.) Use parametric quantile mapping to adjust the distribution of values in

(For psl, rlds, and tas only.) Add trend subtracted from

(For rsds only.) Scale values in

Steps 1 and 8 are only applied to rsds and reflect that this climate variable has a physical upper bound which varies over the annual cycle. In order to fit this case into the unified framework, which at its core assumes constant bounds and thresholds, rsds values are scaled to the interval

Step 2 is only applied to prsnratio and reflects that values of this variable are missing on days of zero precipitation because on these days the ratio

Steps 3 and 7 are only applied to psl, rlds, and tas and reflect that these variables can have significant trends not only between but also within training period and application period. In order to prevent a confusion of these trends with interannual variability during quantile mapping (steps 5 and 6), linear trends within

Step 4 is only applied to bounded variables, i.e., variables which have either a lower bound (and threshold) or an upper bound (and threshold) or both (see Table

Step 5 generates pseudo future observations, which are needed for parametric quantile mapping in step 6. These pseudo future observations are generated such that trends in all quantiles between any two application periods are approximately the same before and after quantile mapping. This makes the bias adjustment method trend-preserving in all quantiles. Different kinds of trends are preserved for different climate variables (Table

Pseudo future observations for one specific future time period are generated by transferring simulated climate change signals between the historical and the future time period to the historical observations. This transfer is done quantile by quantile using a nonparametric kind of quantile mapping. In the following, I will describe the transfer for additive, multiplicative, mixed, and bounded trend preservation. Figure

Schematic of climate change signal transfer from simulations to observations for wet-day precipitation. Empirical cumulative distribution functions of historical and future simulations and observations are displayed using a linear precipitation scale in

In what follows, let

Multiplicative trend preservation is achieved by a multiplicative climate change signal transfer, i.e., in this case,

In order to prevent generating such unrealistically large

Function

For climate variables with both lower bound

Step 6 is the core of the new unified bias adjustment framework. For unbounded climate variables, it consists of a parametric quantile mapping of

Frequencies of values beyond thresholds (see Table

All other values in

The parametric quantile mapping method used in ISIMIP3 is inspired by the scaled distribution mapping method introduced by

Equations (

Note that in contrast to all other climate variables, the likelihood of individual events is not adjusted for tas. Instead, in this case, Eqs. (

This section describes the new statistical downscaling method. Before diving into details I will explain the concept of the method, reveal its algorithmic origin, and elaborate on prerequisites and best practices of its application.

As described in the introduction, the ISIMIP3 bias adjustment method shall be applied at the spatial resolution of the climate simulation data using spatially aggregated climate observation data. Since the resulting data can be considered to have unbiased distributions of daily values per climate variable, grid cell, and calendar month, their subsequent statistical downscaling should be done using a method which preserves values at the aggregated spatial resolution. The ISIMIP3 statistical downscaling method has this property. Since the new method is based on the MBCn algorithm by

The MBCnSD algorithm is independently applied to every climate variable and calendar month. It requires the coarse grid of the climate simulation data and the fine grid of the climate observation data to be compatible in the sense that every fine grid cell is entirely contained in one coarse grid cell. For example, that is the case if the coarse and fine grid are the global

The MBCn algorithm by

The MBCn algorithm applies a series of univariate nonparametric quantile mappings along randomly chosen axes. Mathematically, this is achieved by repeatedly rotating the climate simulation and observation data using random

Two-dimensional illustration of one iteration of the modified MBCn algorithm used for statistical downscaling in ISIMIP3 (MBCnSD), which at its core consists of two steps. In the first step, data point

The MBCn algorithm cannot be used as it is to solve the downscaling problem at hand because it does not have the required preservation property. The preservation of values at the aggregated spatial resolution translates to a preservation of the weighted sum of all time series contained in one coarse grid cell. With MBCnSD, this is achieved by an additional conservation step following the

Statistical downscaling of artificial two-dimensional climate data with the original MBCn algorithm by

If the resolution gap between climate simulation and observation data is large, then statistical downscaling can be done in one big step or in multiple small steps.

Statistical downscaling of precipitation from

In the following, I will describe the MBCnSD algorithm in detail. In this context, let

(For all variables.) Bilinearly interpolate

(For bounded variables only.) Randomize values beyond thresholds in

(For all variables.) Apply the core of the MBCnSD algorithm independently to every coarse grid cell

(For bounded variables only.) De-randomize values beyond thresholds in

Step 1 broadcasts the previously bias-adjusted climate simulation data to the fine grid. This is done using bilinear instead of conservative interpolation, which in this case would be equivalent to setting

Statistical downscaling of precipitation from

Steps 2 and 4 are only applied to variables which have either a lower bound (and threshold) or an upper bound (and threshold) or both. The bounds and thresholds used for statistical downscaling are identical to those used for bias adjustment (Table

Step 3 is the core of the MBCnSD algorithm and is applied independently to every coarse grid cell

The core of the MBCnSD algorithm proceeds in three sub-steps, which I will refer to as 3a, 3b, and 3c in the following. Sub-step 3a adjusts

Generate a

Set

Do a nonparametric quantile mapping of

In sub-step 3b, the following three steps are repeated either a fixed number of times or until

Generate a random

For all

Project

In sub-step 3c, all data are rotated back to the original axes. A last quantile mapping along these axes ensures that there are no values out of bounds in the resulting data. Mathematically, sub-step 3c proceeds as follows.

Set

For all

Results obtained with the ISIMIP2b and ISIMIP3 bias adjustment and statistical downscaling methods will be compared in Sect.

I will begin with results of bias adjustment applied at

I will then compare results obtained with the ISIMIP2b and ISIMIP3 statistical downscaling methods. To that end, both downscaling methods are combined with the ISIMIP3 bias adjustment method. These bias adjustment–statistical downscaling method combinations are abbreviated LI

Results at

Secondly, I will compare results at

For

The comparison of methods with regard to their ability to adjust biases and spatial variability is done in a cross-validation framework. This is done to prevent different extents of overfitting by different methods to dominate differences in results. I first use odd-numbered years from the time period 1980–2015 for training and even-numbered years from the same time period for application. Secondly, I swap these training and application years. Finally, I merge the results of application to odd-numbered and even-numbered years to arrive at bias-adjusted and statistically downscaled data for cross-validation which fully cover the 1980–2015 time period. Compared to the more common use of consecutive and nonoverlapping time periods (here 1980–1997 and 1998–2015) for training and validation, the division into even-numbered and odd-numbered years reduces the influence of climate trends on cross-validation results

The metrics introduced above (RMSD, dry-day frequency, percentiles) are calculated independently for every data set (climate observations, original climate simulations, climate simulations bias-adjusted with the ISIMIP2b/ISIMIP3 method, climate simulations bias-adjusted and statistically downscaled with the LI

In the following I will first present results obtained with the ISIMIP2b and ISIMIP3 bias adjustment methods applied at

The goodness of bias adjustment and trend preservation by the ISIMIP2b and ISIMIP3 bias adjustment methods is assessed based on Fig.

Results further suggest that in most calendar months and grid cells, trends in psl, rlds, and tas are considerably better preserved by the ISIMIP3 method than by the ISIMIP2b method. Trends in hurs, rsds, sfcWind, tasmax, and tasmin are mostly better preserved by the ISIMIP3 method than by the ISIMIP2b method, yet there are a few exceptions of slightly better trend preservation by the ISIMIP2b method for these climate variables. For pr, results suggest that the ISIMIP3 method is much better at preserving trends in dry-day frequency, while both methods are similarly good at preserving trends in the 50th percentile of wet-day precipitation, and the ISIMIP2b method is a bit better at preserving trends in the 95th percentile of wet-day precipitation. Trends in prsn are generally better preserved by the ISIMIP2b method, presumably because the prsn

Goodness of bias adjustment (

The goodness of bias adjustment and trend preservation by the LI

Differences between LI

At

Same as Fig.

Biases at

In order to assess the goodness of bias adjustment across spatial scales, the

Same as Fig.

The goodness of spatial variability adjustment by LI

Spatial variability within staggered

Goodness of adjustment of spatial variability within regular (

The ISIMIP3 bias adjustment and statistical downscaling methods outperform their predecessors in several respects. The new trend-preserving parametric quantile mapping method used for bias adjustment preserves trends and adjusts biases in distribution quantiles more accurately than the ISIMIP2b bias adjustment method. The new stochastic method used for statistical downscaling prevents the variability inflation caused by spatial interpolation in ISIMIP2b.

A major fraction of the bias adjustment gains can be attributed to the newly introduced adjustment of the likelihood of individual events. This new feature effectively corrects for the imperfections of the distribution fits that are the basis of parametric quantile mapping. In addition, it simplifies the confinement of extreme values to the physically plausible range.

Trend preservation works better with the new methods because they apply it to all distribution quantiles compared an application to only distribution mean values for most climate variables in ISIMIP2b. In addition, the new approach of bias adjustment at the spatial resolution of the climate simulation data followed by statistical downscaling to the spatial resolution of the climate observation data ensures that trends are preserved at the spatial resolution at which they were simulated.

The new approach also better adjusts spatial variability at the spatial resolution of the climate observation data than the old approach of a bilinear interpolation of climate simulation data to the spatial resolution of the climate observation data followed by bias adjustment of these interpolated data. Overall, the results presented in this paper can be regarded as a proof of concept of the new paradigm of a clear separation of bias adjustment and statistical downscaling.

The next version of the ISIMIP3 bias adjustment and statistical downscaling method (ISIMIP3BASD) is already under development. In order to improve inter-variable consistency, bias adjustment in ISIMIP3BASD v2.0 will be done in a multivariate manner. In particular, the MBCn algorithm will be employed for an adjustment of the inter-variable copula. This additional adjustment step will be inserted between steps 4 and 5 of the bias adjustment algorithm presented herein. Apart from that, ISIMIP3BASD v2.0 and ISIMIP3BASD v1.0 will be identical: the bias adjustment method for the marginal distributions of all climate variables as well as the statistical downscaling method will remain unchanged.

The ISIMIP3 bias adjustment and statistical downscaling code is publicly available at

The author declares that no competing interests are present.

The author is grateful to Alex Cannon, Matt Switanek, Douglas Maraun, Simon Treu, Jens Heinke, and Katja Frieler for various helpful discussions at different stages of this work. He appreciates the constructive comments on the discussion paper version of this paper by Stefan Hagemann and one anonymous reviewer. He acknowledges the World Climate Research Programme's Working Group on Coupled Modelling, which is responsible for CMIP, and he thanks the climate modeling groups for producing and making available their model output.

This work has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 641816 Coordinated Research in Earth Systems and Climate: Experiments, kNowledge, Dissemination and Outreach (CRESCENDO)The article processing charges for this open-access publication were covered by the Potsdam Institute for Climate Impact Research (PIK).

This paper was edited by Heiko Goelzer and reviewed by Stefan Hagemann and one anonymous referee.