3.2.1. GCM-Selection
For assessing climate change impacts on water resources and possible subsequent planning and management adaptation strategies, the important future meteorological drivers, i.e., precipitation and temperatures, of the hydrological processes in the study basin must be determined. This is usually done by employing downscaled predictors of a Global Circulation Model (GCM) [
5,
22].
In this study, the Canadian Earth System Model (CanESM2) with a latitude x longitude grid resolution of 2.81° × 2.81° has been selected for daily min. and max. temperatures for a 24-year period, 2006–2029. This GCM output has been chosen for further downscaling (see below), owing to (1) previous experiences with downscaling of climate predictors, also in Iran, Refs. [
23,
24] and (2) to the long available predictor period (1961 to 2005).
For precipitation, several GCM model outputs were tested against the observed precipitation and, finally, one of the 5th-Coupled Model Intercomparison Project (CMIP5) - GCM model outputs, namely, the recent 1.86° × 1.87° grid MPI-ESM-LR (ECHAM-6) GCM of the MPI Hamburg under the (benevolent) radiative forcing scenarios RCP 2.6 as a very low forcing level scenario (leading to very low anthropogenic greenhouse gas concentrations), the (intermediate) emission scenarios, RCP 4.5 as medium stabilization scenario and RCP 8.5 as a very high baseline emission scenario (very high greenhouse gas concentrations) [
25] were employed. Representative Concentration Pathways (RCPs) are the latest generation of climate change mitigation scenarios, named according to their radiative forcing target level for year 2100.
Note that, for the selected model in the CMIP5 ensemble, this domain contains three grid boxes of which the bottom corner of each box is marked with a red circle as representative GCM points in
Figure 2b (nearest-neighbor interpolation). Therefore, the zonal averages of precipitation within the selected domain were extracted from output of the GCM. However, since the spatial resolution of these two GCMs is not fine enough for a regional analysis, the coarse-grid information of GCMs must be downscaled on to a finer grid.
3.2.2. SDSM- and QM - Downscaling of Climate Predictors
For the downscaling of the temperatures the well-known Statistical Downscaling Model (SDSM) [
26] was used, as previous publications [
7,
27] confirmed suitable performance of the SDSM for temperature downscaling. SDSM is a transfer- or multiple linear regression model which employs a few selected large-scale atmospheric predictors of the parent GCM to predict a climate variable on the local scale, once the regression model has been calibrated on observed local climate variables.
On the other hand, because of well-known problems of the SDSM with the downscaling of intermittent precipitation [
26,
28] and the poor correlations of the observed precipitation with the large scale atmospheric predictors, the more-recently-coming-to-the-fore Quantile Mapping (QM) bias correction technique, which has been found to do better for this climate variable than the other methods mentioned [
22,
29,
30], was employed for the downscaling of precipitation. QM has shown to be an effective method for removing some GCM- predictors’ biases with little computational expense [
22,
29,
30,
31] and appears to deliver much better correlations with the observed predictands (precipitation) at the high resolution local scale.
The above choice of using different GCM- models and downscaling methods for temperature and precipitation may appear somewhat unusual, but has been done solely for convenience of easy data handling of the temperature predictors of the CanESM2 GCM- model for use in the SDSM- downscaling technique. Moreover, it is generally agreed upon that differences in future temperature predictions between various GCM- models are only minor (all projecting increases), which means that future temperature uncertainties are usually small. In addition, as temperature is of less importance than precipitation for the purpose of the present hydrological study of streamflow prediction, it was important here to get the best downscaled precipitation predictors as possible, using the above mentioned combination of ECHAM-6 - GCM and QM- bias correction.
The basic idea behind QM- bias correction is to adjust the future climate predictors of a GCM by the error encountered by the GCM, when trying to predict the local climate observations in the historical reference period, i.e. the bias. QM has evolved since its inception more than a decade ago from simple distribution mapping under the assumption of stationarity of historical and future predictor series to variants that release this assumption [
32].
In the original traditional QM- method, the adjusted (corrected) future predictor value for a particular month (
) of the original future value (quantile)
xm-p is found by inverting the cumulative distribution function (CDF)
Fm-c of the current GCM-predictor series with the CDF
Fo-c of the observed (current) series, i.e.,
where
x is the climate variable of interest, either observation (
o) or model (
m) for the current climate (
c) in the reference period, or future projection period (
p) and
F is the CDF and
F−1 is its inverse.
As mentioned, the underlying assumption in the application of Equation (1) that the climate variable of interest has a similar future distribution as that of the reference period and the form of this CDF, i.e., its variance and skew remains the same in the future, but only its mean, defining the bias, is allowed to change, whereas the mean may change and is then adjusted as part of the bias correction. On the other hand, because climate variables’ time series are often nonstationary, this assumption has been put into question by Milly et al. [
33] and, consequently, Li et al. [
32] proposed an improved QM-method, called equidistant CDF matching (EDCDFm), wherefore the bias corrected values are computed as:
However, this method may provide some negative values for the bias-corrected variable which is unacceptable for the precipitation. For a remedy, Miao et al. [
30] proposed a modified nonstationary CDF-matching (CNCDFm) technique, where Equation (2) is replaced by:
with
and
Based on the equations above, the bias correction of the monthly precipitation by means of the CNCDFm- method comprises then the following steps which were coded in the MATLAB© programming environment (version, Manufacturer, City, US State abbrev. if applicable, Country).
Firstly, the monthly precipitation model for each grid box and each month of the year of the 36 years (1977–2012) long dataset is calibrated and then validated by a repeated cross validation process for 30 times, using two non-parametric conversion function options, namely the Empirical Cumulative Distribution Function (ECDF), as advocated by Themeßl et al. [
22] for quantile mapping, and the Kernel Density Function (KDF) to estimate the CDFs and the inverse CDFs for the precipitation data for each month of the year (e.g., January, …, December) [
30]. For the calibration of the QM-model, 20 years out of 36 years of monthly precipitation sampled (i.e., 1977 to 1996) and the 16 years as the rest of the period (i.e., 1997 to 2005) are used for the validation of the model. This procedure is repeated 30 times and the average of the bias correction success rate and percentage of the improvement of the biases computed.
Secondly, the best validated option is then selected to assess the biases of raw GCM series and corrected values after the bias correction by computing the
QBt (quantile bias percentile
t)– index, for different percentiles (25%, 50% and 75%), of the corresponding CDF.
QBt is defined as [
34]
where the
PCPmodel and
PCPobserved are the quantiles of the monthly precipitation of the raw GCM and the observed series of the validation period above the threshold t, i.e., the percentile level (25%, 50% and 75%). Obviously, the
QBt index is equal to 1 when there is no bias in the prediction, whereas a value above or below 1 indicates that the amounts of the precipitation above the specific threshold are overestimated or underestimated, respectively.
Thirdly, the daily future precipitation of the future period at each station is predicted from the average of 10 ensembles created based on the validated model of the previous step. The ensembles are generated based on the statistics of the monthly (bias-corrected) precipitation time series using a two-state first-order Markov chain process for [
35,
36], wherefore four different transitions dry–dry dry–wet, wet–wet and wet–dry, denoted by the observationally determined transition probabilities p
00, p
01, p
11, p
10, respectively, are possible. Using this approach, the synthetic daily time series
xt (
t = 1, …, 30,31) for a particular month is generated, beginning at day 1 with x
1, either dry (=0) or wet (=1), depending on whether a random number drawn from a uniform [0,1] distribution is less than the stationary, long-term probability π1 of the occurrence of precipitation, which in turn depends on p
01 and p
11 [
35,
36]. The situation of the subsequent day (t = 2) and after depends then on the value of a new random number, relative to these two transition probabilities p
01 or p
10, respectively.
Finally, the synthetic daily wet-and-dry series for a month is scaled to get the appropriate bias-corrected daily precipitation amounts of the wet days, while making sure at the same time that the monthly aggregated GCM-predicted precipitation amounts are guaranteed.