Daily meteorological data such as temperature or precipitation from climate models are needed for many climate impact studies, e.g., in hydrology or agriculture, but direct model output can contain large systematic errors. A large variety of methods exist to adjust the bias of climate model outputs. Here we review existing statistical bias-adjustment methods and their shortcomings, and compare quantile mapping (QM), scaled distribution mapping (SDM), quantile delta mapping (QDM) and an empiric version of PresRAT (PresRATe). We then test these methods using real and artificially created daily temperature and precipitation data for Austria. We compare the performance in terms of the following demands: (1) the model data should match the climatological means of the observational data in the historical period; (2) the long-term climatological trends of means (climate change signal), either defined as difference or as ratio, should not be altered during bias adjustment; and (3) even models with too few wet days (precipitation above 0.1 mm) should be corrected accurately, so that the wet day frequency is conserved. QDM and PresRATe combined fulfill all three demands. For (2) for precipitation, PresRATe already includes an additional correction that assures that the climate change signal is conserved.

Daily data from climate models are used for various applications, e.g., in hydrology, silviculture and for general climate risk studies

Simple methods that only correct the mean and/or the variance of the model data have been introduced

The distribution of meteorological variables can be described with empirical CDFs which is a non-parametric approach

One key feature of traditional QM is that it may alter the raw climate change signal (CCS) found in the model (i.e., the change of the arithmetic mean of a meteorological variable over time)

This means that the choice of a climate model with plausible weather patterns and a plausible CCS is crucial

Bias adjustment methods that do not alter the CCS implicitly assume time invariance

Note that the definition of the time-invariance assumption, sometimes also called stationarity assumption

EDCDFm and QDM are always capable of preserving the CCS in the median (and also at every quantile). If applied additively, this also holds true for the arithmetic mean. For precipitation, a multiplicative approach is more suitable.

Most of the bias-correcting methods correct a wet day bias of a climate model (i.e., the number of wet days above a specific precipitation threshold) only if the model has a positive wet day bias. However, in some rare cases, the model may have too few wet days. Often, a multiplicative bias adjustment is selected for precipitation

There is no single best bias-adjustment method that fits all needs. The advantages and disadvantages of the bias-adjustment methods mentioned here depend on the application.

The goal of this paper is to find a suitable quantile-based bias-adjustment method that could be used for climate impact modeling studies that are sensitive to the changes in means to thresholds effects. In the course of this, we will also show the systematical differences of several methods. We choose to focus only on quantile-based methods, because they usually outperform simpler methods, as described above. We posit that three important demands should be met:

The bias-adjusted data should match the observational data in the historical period in terms of arithmetic mean.

The CCS should not be altered during bias adjustment. In other words the mean change between historical and simulated future period from the raw model should be preserved. This should also hold true for the ratio of the CCS, if the bias adjustment is applied multiplicatively.

Models with too few wet days should be corrected reasonably, which means that a way has to be found to add wet days.

Grouping of some quantile-based bias-adjustment methods in two categories. Note that this list is not complete. The methods in bold are used in this work.

This study focuses on Austria which is located in Central Europe and is representative of a mountainous area in the middle latitudes. The topography is shown in Fig.

Area of interest with Austrian state borders. © European Union, Copernicus Land Monitoring Service 2020, European Environment Agency (EEA).

Austria has a large number of high-quality weather observation stations that are operated by Zentralanstalt für Meteorologie und Geodynamik (ZAMG). Also, gridded observational data sets called SPARTACUS for minimum temperature, maximum temperature and precipitation are available on a daily basis at a high spatial resolution of 1 km

For the observational data, SPARTACUS

To generate artificial data with loo few wet days and too little precipitation, the data were further manipulated. This was done by multiplying the precipitation of each day with a uniformly distributed random number between 0 and 1. Furthermore, a trend to even drier conditions was introduced by successively canceling more and more wet days going from 1961 to 2019.

To show that the bias-adjusted model data do not always match the observations in the historical period, we analyzed data sets in Austria from the projects ÖKS15

We calculated climatological annual precipitation sums for all models in ÖKS15 and STARC-Impact in the reference period 1971–2000 and for the observation data set GPARD1 for the same time period

Looking further into all the models used in ÖKS15 and STARC-Impact, we found that the largest errors occur in very dry models with a distinct negative wet day bias. Therefore, we focus on the bias adjustment of very dry climate models in this paper.

Box and whisker plot for the relative annual precipitation bias (%) of the ÖKS15 and STARC-Impact models (a total of 35 models) to the observational data set GPARD1 for the reference period 1971–2000. A positive bias indicates that the model is wetter than the observations.

This study focuses on implementing QDM and PresRATe to bias-correct data from climate models and compares it with two existing methods, namely QM and SDM. All methods are quantile-based bias-adjustment methods that adjust the climate model data to match the CDF of the observation. The daily data of each grid cell of the model are adjusted separately with the observations on a monthly basis. For the calibration data, a time period of 30 years is typical, since the statistical distribution of data of a shorter time period can be very noisy and a longer time period usually has pronounced climatological trends.

As discussed in the introduction, QM and QDM are systematically different when correcting future values. The model bias in QM is fixed on the quantiles from the calibration distribution, hence, it is fixed on absolute values. QDM and similar methods assume a stationary bias for each quantile but with the quantiles for a value from the future period derived from the future distribution.

We postulate that the bias of a climate model is correlated to the modeled weather pattern. In other words, we consider that the RCM is able to predict a ranked category of temperature or precipitation but not the value for this variable

A cold winter day in Austria is related to moderate northeasterly flows and usually high atmospheric pressure with low wind and clear sky conditions. Cold translates to a low quantile for temperature. In future, the error of the model in this weather situation is assumed to stay constant. This weather situation will still translate to a low quantile in the future distribution, however, the absolute temperature values are higher (respectively, the corresponding quantile calculated from the historical distribution is higher).

Consider the daily maximum temperatures for a grid point during a summer month in Europe, where three quantiles of the observations in the reference period are 20, 25 and 30

Schematic of bias adjustment for temperature data. CDFs are shown for following data: observational data (black), raw historical model (orange), raw future model (red), future model corrected with QDM (blue) and future model corrected with QM (purple). The arrows illustrate the bias-adjustment path for future model data. Panel

QDM

The mathematical description of QDM and PresRATe is similar to EDCDFm in Eq. (2) in

For variables that have a meteorologically meaningful zero value as a lower boundary, a multiplicative approach is more useful, e.g., for precipitation, wind speed or global radiation

Any desired time period for bias adjustment is selected (future or historical). It is possible to choose the calibration period itself. The time period to be chosen is usually a 30-year period, as for the calibration time period. For the final bias adjustment, the CVs are added (Eq. 4 for temperature and dew point) or multiplied (Eq. 5 for precipitation, global radiation and wind speed) to the selected (e.g., future) model data

The graphical solution for bias adjustment for temperature data is shown in Fig.

The observational data feature a mean of 10

The raw historical model (1981–2010) has a cold bias in the data with a mean of 8

The raw future model (2071–2100) is warmer with a mean of 12.4

QDM and PresRATe (Sect.

For a future time period, the CCS for precipitation for the raw model for one grid point is

All tested methods correct by default the number of wet days if the model has more wet days than the observational data by multiplying the lower parts of the model CDF by 0. However, a quantile-based bias adjustment cannot add wet days that are initially not in the model. This is already described in

If we use QDM without SSR, we call this QDMd (d for dry).

We compare the performance of QDM and PresRATe with other methods. One of them is the traditional QM in a non-parametric form, that is widely used

SDM is a further development based on QDM and is explicitely parametric

Tests showed that the fitting is sometimes defective and results in errors when the corrected model data are compared with the observations (see Fig.

Therefore, we generated several versions of SDM. For this work, we improved the fitting of the gamma functions by adding initial guesses to the fitting function. According to the methods of moments

QDM, PresRATe, QM and SDM are evaluated in terms of three demands expressed at the end of Sect.

Bias adjustment of precipitation data. The model is produced by smoothing OBS.

The four versions of SDM are compared with non-parametric QM and QDM/PresRATe, respectively. We already showed that biases can be introduced by the bias-adjustment methods themselves with the example of ÖKS15 and STARC-Impact data (Fig.

Figure

For comparison, the model data were corrected with other methods such as QM and QDM/PresRATe where the error is close to zero. This is because these methods calculate empirical CDFs of both model and OBS which produces very accurate results in the reference period.

Running means of temperature data of detrended observations, the raw model and three different bias-correcting methods (QM, SDM and QDM). SDM and QDM are almost identical. On top: average linear trends in

Demand (2) for the bias adjustment is that the CCS of the raw climate model should not be altered. As stated by

For precipitation, the CCS is defined as a relative value as shown in Eqs. (

As before, SDM underestimates the CCS (Fig.

Error of CCS compared to CCS of raw model (Eq.

As already discussed in relation to Fig.

Climatological annual precipitation (mm) sum in the historical period for dry model data.

Figure

The parametric SDM(2) produces too many new precipitation days (Fig.

Wet days per year (

Statistical bias-adjustment methods are widely used to improve direct model output from climate models but cannot fully remove all model errors. The adjusted data are often used as input for climate impact studies where biases can significantly alter the impact analysis, so one has to be aware of the limitations of the bias-adjustment methods. We compared different methods (Empirical QM, SDM and QDM/PresRATe) that all adjust the statistical distribution of meteorological variables. We evaluate the methods with the three demands formulated in the introduction for synthetic climate data to show that errors can originate from the bias-adjustment method and not only from climate models.

Table

Demand (1): QDM/PresRATe and QM are capable of statistically correcting the model's past climate to fit the observations accurately. This is mostly due to the fact that they are non-parametric methods, i.e., they use empirical distribution functions instead of fitted functions (for variables like temperature, precipitation etc.) which allows the CDF to follow any possible shape (Table

Demand (2): QDM/PresRATe and SDM barely enhance or suppress the mean CCS in contrast to traditional QM; i.e., they explicitly reproduce the same CCS as in the raw model. For additive QDM and SDM (e.g., for temperature), this is valid without any limitation (Fig.

Demand (3): QM is not able to correct models with too few wet days, if applied multiplicatively. SDM(0), SDM(1) and SDM(2) interpolate the wet days to the expected number of wet days which should correct the bias. Figure

As in Table

A good performance of the corrected data in any of the three demands is crucial, as it is used as input for further impact studies. Impact models (e.g., plant growth models) are often calibrated with bias-corrected historical meteorological data from a climate model. The focus of impact studies often lies on the CCS. If an impact model is calibrated with inaccurate meteorological data in the historical period, the impact of climate change can lead to wrong conclusions even if the CCS is accurate.

To sum up, QDM and PresRATe are able to reproduce the observation's statistical distribution, are able to preserve the raw model's CCS and can add wet days if necessary because of a supplementary algorithm.

QDM along with many other methods corrects each grid cell independently and therefore belongs to the group of univariate bias-adjustment algorithms. We showed that the spatial patterns of the corrected data match the observations for long-term means, which is a significant improvement over the raw model data. However, for spatial patterns of smaller timescales (e.g., a season, a month or a single day), some univariate methods are still able to improve spatial patterns compared to raw model data

Some authors introduce methods to correct the temporal autocorrelation across several days, weeks or months

Other authors find the results of univariate methods for spatial precipitation patterns on specific days in the model unsatisfactory

The code has been developed in Python and is available from the corresponding author on reasonable request.

All data used in this study are publicly available. The primary data source is SPARTACUS which is available from

All three authors contributed to the study conception and design. FL is the corresponding author who has contributed to the code, produced the figures, written the draft and reviewed all sections of the document. IN contributed to writing and testing the software code. HF reviewed previous versions of the paper and approved the final paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was partially supported by the research project FORSITE (Waldtypisierung Steiermark – FORSITE – Erarbeitung der ökologischen Grundlagen für eine dynamische Waldtypisierung). The precipitation data set SPARTACUS was generously provided by ZAMG. We also thank Copernicus Land Monitoring Service as part of the European Environment Agency (EEA) for the topography data. Finally, we thank Douglas Maraun for his helpful comments.

This research has been supported by the federal province of Styria, the federal government of the Republic of Austria and the European Union (grant no. ABT10-185835/2016-115).

This paper was edited by Chris Forest and reviewed by one anonymous referee.