2.3.1. Delta Change Factor

The raw climate simulations, especially for precipitation time series, are highly biased as mentioned in many studies [6,35,36]. Thus, an additional post-processing step (e.g., bias correction of climatic variables) is a standard procedure for related climate change studies. In this study, the delta change method is adopted due to its simple and common use as described in Olsso et al. [37], Lenderink et al. [38], Teutschbein and Seibert [36] and Maraun [35]. This approach does not adjust the output of climate models, but uses observations and the change signal of regional climate models forced by global climate models to generate future data. Also, this approach is to avoid considerable variability in the day-to-day change signals and changes in extremes are linearly scaled with changes in the mean. The core of this method is that the historical observations are transformed into future projections using

monthly average correction factors that derived from the regional climate models forced by global climate model outputs for the baseline and future climate. It is expressed by the equation:

$$P\_{OBS.f} = \frac{\overline{P\_{GCM.f}}}{\overline{P\_{GCM.b}}} . P\_{OBS.b} \tag{1}$$

where *PGCM*. *<sup>f</sup>* is the monthly precipitation from the future climate. *PGCM*.*<sup>b</sup>* is the monthly precipitation from the baseline climate. *POBS*. *<sup>f</sup>* is future projections and *POBS*.*<sup>b</sup>* is historical observations.

With this approach, the correlation structure of downscaled future data in spatio-temporal terms are physically reasonable because it reflects observed conditions. The drawback of this approach, however, is that the future and baseline scenarios just differ in terms of their means and intensity, but other statistics of the data (e.g., skewness or structure of dry and wet days) are mostly unchanged. Also, the sample is limited to the length of the observed record. It should be noted that for a near-term future (2021–2050), the changes in the dry/wet days are probably trivial. Sun et al. [39] used the lag 1 autocorrelation to investigate the stationary land-based gridded annual precipitation (1940–2009). The results showed a stationary annual precipitation over an area of about 80% of the global land surface. Wilks and Wilby [40] show that calculating the autocorrelation of non-zero precipitation amounts is essential at time step of hourly (or sub-hourly) rather than a daily time step. Meanwhile, the autocorrelation between successive nonzero precipitation amounts is usually of little practical importance and quite small. At a time step of monthly precipitation, thus, a stationary assumption in the temporal correlation is made in this study. In other words, rescaling the precipitation time series observed during the baseline could lead to not realistic results when future scenarios that preserve the observed autocorrelation in time are not considered.
