Bias correction capabilities of quantile mapping methods for rainfall and temperature variables

This study aims to conduct a thorough investigation to compare the abilities of QM techniques as a bias correction method for the raw outputs from GCM/RCM combinations. The Karkheh River basin in Iran was selected as a case study, due to its diverse topographic features, to test the performances of the bias correction methods under different conditions. The outputs of two GCM/RCM combinations (ICHEC and NOAA-ESM) were acquired from the CORDEX dataset for this study. The results indicated that the performances of the QMs varied, depending on the transformation functions, parameter sets, and topographic conditions. In some cases, the QMs’ adjustments even made the GCM/RCM combinations’ raw outputs worse. The result of this study suggested that apart from DIST, PTF:scale, and SSPLIN, the rest of the considered QM methods can provide relatively improved results for both rainfall and temperature variables. It should be noted that, according to the results obtained from the diverse topographic conditions of the sub-basins, the empirical quantiles (QUANT) and robust empirical quantiles (RQUANT) methods proved to be excellent options to correct the bias of rainfall data, while all bias correction methods, with the notable exceptions of performed PTF:scale and SSPLIN, performed relatively well for the temperature variable.


INTRODUCTION
Simulations of the present and future climate conditions often require the use of general circulation models (GCMs). Thus, employing GCMs as tools for long-term climate modeling has become common practice in most climate change studies (Zolghadr-Asli et al. a). However, using these coupled atmosphere-ocean general circulation models is computationally expensive, and resultantly, it could potentially increase the computational cost and, in turn, limit the horizontal resolution of the models' projections (Plummer et al. ; Romera et al. ). Needless to say, high-resolution predictions are of great importance for climate change impact assessment studies. To address this issue, downscaling techniques are employed by researchers to post-process and recalibrate the raw projections of GCMs. Based on the conceptual and mathematical frameworks used in these methods, there are two broad downscaling techniques: statistical downscaling and dynamic downscaling (Faiz et al. ). The statistical downscaling approaches attempt to link the large-scale atmospheric predictor variables to local-or regional-scale meteorological time series (Dibike & Coulibaly ). For more information about these approaches, readers can refer to Busuioc et al. On the other hand, dynamic downscaling approaches use finer gridded models, technically referred to as regional climate models (RCMs), to redefine the climate change predictions at a regional scale from the coarse projection of the GCMs (e.g. Jha et al. ; Tadross et al. ; Kay et al. ). As RCMs operate at a relatively finer horizontal resolution, they can provide localized and high-resolution detailed hydro-climatic information that can be of great importance, especially for the regions with complex topographies. As a result, utilizing RCMs is a common practice to project hydro-climatic variables such as rainfall which is particularly sensitive to the horizontal resolution, given its strong correlation with surface topography and physiography (Samuelsson et al. ).
Unlike GCMs, where the ensemble projected results would be coordinated to highlight the underlying uncertainties of the outcomes (e.g. the atmospheric model intercomparison project (AMIP) or the coupled model intercomparison projects (CMIP)), RCMs projections may suffer from the lack of a vastly accepted coordinating framework.
Resultantly, most RCM-oriented studies have been isolated and tied to a specifically targeted objective, and thus a comprehensive picture of regional climate change projections based on dynamic downscaling methods is not as readily available as it should be (Giorgi et al. ). To overcome this significant issue, the coordinated regional climate downscaling experiment (CORDEX)  Their results also showed the ability of this technique to improve the statistics of the simulated results that depended strongly on the temporal sequence of the original fields, such as the number of consecutive dry days and the total amount of rainfall following heavy rainfall events, as well as improve the probability distribution functions (PDF) of the simulated rainfall and temperature variables. In another case, Teutschbein & Seibert () compared the performance of some bias correction techniques including linear scaling, local intensity scaling, power transformation, variance scaling, distribution transfer, and the delta-change approach to adjust the RCMs' rainfall and temperature projections, indicating that while all of these methods would, more or less, improve the RCMs' predictions, some performed better. Rajc- While it seems that the QM techniques, in general, can be considered as potentially valid techniques to correct the bias, it should be noted that the QM is referred to a family of methods. It is worth investigating which of the QMs can be the most suitable one for a given domain. Thus, it is crucial to conduct a thorough comparison over the effectiveness of the QM methods in the bias correction procedure, especially for the climatic variables (e.g. rainfall and temperature) that play the most crucial role in water resources impact assessment. Consequently, this study aims to cover a comprehensive set of potentially essential QM methods and provide an insight into their roles in the bias correction procedure.
Identifying suitable bias correction techniques for the CORDEX database for different domains is an important topic (Jacob et al. ). The main objective of the CORDEX project is to coordinate the RCM-oriented climate change studies at regional scales. Therefore, investigating the role of the QM methods in bias correction of the gathered dataset can be the next milestone along the project development line. This could be crucially important for Iran, a country that has already faced a series of mild to severe water crises and, accordingly, the situation may even get worse due to the climate change phenomena (Zolghadr-Asli et al. a). Therefore, the Karkheh River basin in Iran has been selected as a representative and strategically important case study. Furthermore, the Karkheh River basin is known for its diverse range of topographic features (e.g. both flat plains and mountains), which proved to be challenging characteristics for both GCMs and RCMs.
Thus, this basin is ideal to test whether the bias correction techniques are able to cope with the challenges for such a topographically diverse landscape. To the best of our knowledge, the CORDEX dataset has never been implemented in this region, and thus, the results can also verify the validity of this dataset. According to the CORDEX, Iran is located in the domain of Middle East and North Africa (MENA) and also the domain of South Asia, but this study concentrates on the MENA domain only. The results obtained from two GCM/RCM combinations (ICHEC (Irish center for high-end computing) and NOAA-ESM (national oceanic and atmospheric administration Earth system models)) with a spatial resolution of 0.22 and a monthly temporal resolution are used. It should be noted that for the MENA region, the CORDEX datasets only provide two GCM/RCM combinations, both of which are employed in this study.

METHODOLOGY
As aforementioned, the main objective of this study is to explore the potential capabilities of different QM methods under various conditions. The flowchart of this study is illustrated in Figure 1. The following section provides the theoretical background behind the tested QM methods, as well as the statistical performance measurement criteria that are used in this study.

Quantile mapping
In general, the QM methods implement statistical trans- observed ones using a mathematical function, which can be mathematically expressed as (Piani et al. b): Given that the QM methods use the quantile-quantile relation to converge the simulated variables' distribution function to the observed one, one should note that with the CDFs of both observed and simulated variables' time series, their quantile relation can also be determined, as shown below (Ringard et al. ): which is technically referred to as the quantile function.
As stated earlier, various frameworks have been proposed to form the transformation function. Hence, different QM methods, as detailed below, are used in this study for bias corrections.

Parametric transformation functions (PTF)
This QM method is featured with a parameter-oriented framework, which employs parametric transformation functions to form the quantile-quantile relation. A series of predefined transformation functions are proposed for this purpose, and they can be mathematically expressed as (Maraun et al. ): in which P o and P m ¼ probability of the observed and modeled variables, respectively; a, b, c, y, and τ ¼ methodrelated parameters that are subject to calibration. Equations  (5)), PTF:power (Equation (6)), PTF:power. × 0 (Equation (7)), and PTF:expasympt. × 0 (Equation (8)

Performance evaluation criteria
Like any other modeling process, the QMs outputs also need to be validated by using the observed data. The Taylor diagram enables one to evaluate various results by using three different performance evaluation criteria, including linear correlation coefficient (R) (Equation (9)), normalized standard deviation (σ Ã ) (Equation (10)), and centralized root mean squared deviation (CRMSD) (Equation (11) in which σ o and σ m ¼ standard deviation of the observed and modeled datasets, respectively; and N ¼ number of the data. R is a dimensionless criterion that ranges from -1 to

CASE STUDY
The Karkheh River basin, with an area of 51,000 km 2 , is located in the south-western region of Iran (latitude: As shown in Table 1

RESULTS AND DISCUSSION
As stated earlier, the bias correction techniques can play a crucial role in adjusting the RCMs' results. This study, in particular, aims to shed light on the potential of the QM methods to mitigate the embedded bias of the CORDEX dataset for rainfall and temperature through a topographically diverse case study in the Karkheh River basin, Iran.
The related modeling and computations were conducted by using R programming language. As for the CORDEX dataset, the results of two GCM/RCM combinations (i.e. ICHEC and NOAA-ESM, hereafter simply referred to as GCM/RCM combination I and N, respectively) were obtained with a resolution of 0.22 (MNA22i). Figure 5 shows the performances of the QM methods to adjust the rainfall bias. The Taylor diagrams in Figure 5 revealed that the performances of the QM methods in all sub-basins and for both GCM/RCM combinations followed a similar pattern. DIST was the worst rainfall bias correction method, which made the results even worse compared to the non-corrected data. While the rest of the tested results were relatively more satisfactory, QUANT and RQUANT seemed to be the best options to correct the bias of the rainfall variable, for they not only improved the R and CRMSD criteria, but generated a value of σ Ã close to 1 (desirable value). Further analysis of the evaluation criteria also indicated that while PTF:scale and PTF:linear performed significantly better than DIST, the rest of the methods outperformed them.

Rainfall
In addition to R, σ Ã , and CRMSD, which were covered by the Taylor diagram, the abilities of the models to adjust the average of monthly rainfall time series were evaluated.
As shown in Figure 6     Note: R.S. ¼ number of stations that actively monitor rainfall; and T.S. ¼ number of stations that actively monitor temperature.
In the Qarasou sub-basin, for example, the simulated rainfall by GCM/RCM combination N and using SSPLIN as the bias correction method was 36.1 mm. This value is significantly closer to the observed value compared to the non-corrected condition, which signifies the role of the QM methods in adjusting the raw data of GCM/RCM  not only may PTF:scale be the worst available option for this purpose, but it seems to change the direction of the bias in all tested cases. DIST was also one of the methods that did not perform satisfactorily.

Temperature
The abilities of QMs to adjust the raw temperature projections of the CORDEX were also evaluated. Figure 8 shows the simulated temperature values in both non-corrected and corrected forms which were compared against the observed dataset using R,σ Ã , and CRMSD. According to     Figure 9, the performance of the SSPLIN method was satisfactory for the remaining sub-basins. Evidently, the QM methods were able to overcome this issue.

SUMMARY AND CONCLUSIONS
Although nowadays advanced regional climate models