## Abstract

Global warming has exerted a significant influence on the water cycle in global drainage basins, involving various aspects associated with ecology, society, economy and human activity. Therefore, accurate forecast and analysis of climatic change trends are the prerequisite to disaster alleviation and prevention. This paper selects a large-scale climate predictor by modeling steps and evaluation indicators for a statistical downscaling (SD) model and establishes a statistical relation between atmospheric circulation factor and precipitation and air temperature at eight stations in Wei River basin in Shaanxi Province for a global climate model output based on hydrological data for Wei River basin in Shaanxi Province. A study was made of the simulated precipitation and air temperature in Wei River basin by principal component analysis and multiple linear regression and achieved favorable results. The approach is better in physical significance and is characterized by accurate climate simulation, simple calculation for statistical downscaling and is convenient for application in various climate models.

## INTRODUCTION

As global warming has a great impact on the water circulation process and directly affects the ecology, society, economy and other aspects of human life, precise forecast and analysis is the important precondition to mitigate and/or prevent disasters (Clark *et al.* 2001; Stenseth *et al.* 2002; Wilby *et al.* 2002; Gao *et al.* 2017). In the early stage of the study, relative systematic work focused on the general circulation models (GCMs) to simulate integratively the upper atmosphere field, ground surface temperature and general atmospheric circulation. However, its spatial resolution was low and there was a lack of regional climate information, and so its application is limited to give an effective perspective of the regional climate.

Nowadays, there are two methods to make up for the short comings. One is to develop super-resolution GCMs, and the other is the downscaling method, which becomes the first priority without numerous computations of GCMs. The downscaling method can be divided into three categories, i.e. the dynamical downscaling (DD) method, the statistical downscaling (SD) method and the DD and SD method, with a common feature that they require the GCMs to provide grand scale climate information (Fan *et al.* 2005; Boé *et al.* 2007; Timbal & Jones 2008; Lu *et al.* 2016a). The DD method, taking advantage of the regional climate mode (RCM) coupled with the AOGCM to predict the future regional weather changes, is characterized with clear physical significance. It is suitable for various places and resolutions without observed data influences, but this method is time consuming with a large deal of computation (Timbal *et al.* 2009; Maraun *et al.* 2010; Sachindra *et al.* 2014). The AOGCM boundary conditions affect the performance of the parameters of the regional model and the regional coupling method should be adjusted when applying for different regions (Timbal *et al.* 2008; Chu *et al.* 2010; Wójcik 2015). The SD method, with multiple established techniques, has been widely used in predicting future climate changes.

There are mainly three categories of ways for downscaling the climate data using the SD method which include the regression method, the weather typing method and the weather generator method. The first method is based on statistics and in terms of mathematical function models, i.e. those of the linear and nonlinear transfer functions, with that of multiple linear regression equations being widely applied for calculating the grand weather field and climate variable field.

The second method is a weighted classification method of different climate factors such as sea-level pressure, geopotential height field, airflow index, wind direction and velocity as well as cloudage (Fowler *et al.* 2007; Guo *et al.* 2012; Evans *et al.* 2013). The common methods are the Lamb Weather Type, principal component analysis (PCA), average weighted grouping method, the method combining the PCA and the average weighted groupings, the artificial neural network classification method, etc. The weather typing method in the SD process has three basic steps, i.e. typing, calculation and generation of simulation values for future regional climate. The first step is to classify the atmospheric circulation, which is relevant to regional weather variants according to the data of existing grand atmospheric circulation and observations of the regional weather variants (Islam *et al.* 2013; Machetel & Yuen 2014; Tofiq & Guven 2014; Valverde *et al.* 2014). The second step is to calculate the average value of every type of circulation, occurrence frequency and variance distribution as well as the mean value of regional climate, occurrence frequency and variance distribution of different weather types. The last step is to generate simulation of value of the future regional climate by weighting the relative frequency of future circulation patterns to the regional climate. This paper illustrates the establishments of the multiple regression algorithm and the SD model, analyzing the principal predictors of each station and the monthly temperature data of eight hydrological stations in Shannxi in the Weihe River basin. The comparison results of the two methods have proved that the approach is better in physical significance and is characterized by accurate climate simulation, noncomplex calculation for statistical downscaling and application potentials.

## SD STUDY

Impacts of climate change on hydrology and water resources of a river have attracted great attention recently (Phatak *et al.* 2011; Tareghian & Rasmussen 2013; Lu *et al.* 2016b). At present, as the main means to estimate weather changes, the climate model also has a large deviation in simulation of the regional climate change. Its credibility has been doubted because of the rough GCM resolution ratio that cannot depict the complicated topography, surface conditions and some physical processes (Ghosh & Mujumdar 2008; Kumar *et al.* 2012; Lu *et al.* 2016c). This paper provides a rather ideal SD tool to analyze impacts of climate changes on the precipitation and temperature of the Shannxi section of the Weihe River. We aim to establish the statistical relationship between grand scale climate factors and monthly rainfall and temperature of the Shannxi section using the gradual regression method and a combination of principal component analysis and multiple regression. The two methods have been compared respectively in terms of advantages and disadvantages.

### Selection of study area and predictors

#### Study area

In order to set up the SD model, taking the Shannxi section of the Weihe River basin as the study area, this paper uses the monthly average data of global reanalysis provided by the US National Centers of Environmental Prediction (NCEP) as the observed climate data of grand scale, with spatial resolution 2.5 × 2.5° and grid number 2 × 2 longitude and latitude covering the Shannxi section. The monthly rainfall and temperature data measured in eight national weather stations thereof are used as the regional climate data of 1961–2000, of which the observed data from January 1961 to December 1990 are taken as resolution parameters and those obtained from January 1991 to December 2000 are the verification parameters. Figure 1 shows the locations of the eight stations and the NCEP grids.

#### Selection of grand scale predictors

The most important step in the SD method is the selection of predictors because it decides, to a large degree, the features of future climate scenarios. Normally, it follows four criteria (Tolika *et al.* 2008; Willems & Vrac 2011; Lu *et al.* 2017): (1) very strong correlation between the predictors and predictands, (2) representing surely the physical process and climatic variability of the grand weather field, (3) correct simulation of the predictors by the GCM, and (4) weak or no correlation between the predictors themselves. Very wide studies and research on selection of predictors lead to the application of predictors that have obvious physical significance.

According to the above principles, this paper discusses 12 selected predictors as sea level pressure, surface zonal wind velocity, surface warp-wise wind velocity, mean temperature at 2 m high from ground, 500/850 hpa zonal wind velocity, 500/850 hpa warp-wise wind velocity, 500/850 hpa geopotential height, and 500/850 hpa relative humidity.

### Establishment steps of SD model and evaluation indexes

This paper discusses the SD model that has the following procedure (Jeong & Adamowski 2016; Chapman & Darby 2016; Lim *et al.* 2017): (1) selection and determination of grand predictors and SD model, checked with NCEP measured data, (2) screening predictors with gradual regression method and establishment of downscale model, (3) obtaining principal components through standard treatment and principal component analysis, and setting up downscaling model with multiple linear regression algorithm, and (4) comparison between observed data and the downscaling methods.

*R*, and relative error of standard deviation , as evaluation indexes: in which is the average value of observed monthly precipitation, and is the standard deviation simulated with SD method.

_{mean}#### Relative error of standard deviation *Rsd*

in which is the standard error of observed monthly precipitation, and is the standard deviation of monthly precipitation simulated with the SD method.

### Method presentation

#### Multiple linear regression algorithm

*et al.*2009; Sohn

*et al.*2013; Sohn & Tam 2017). Assuming the linear relationship between the dependent ‘y’ and those independents ‘‘, ‘y’ has the multiple linear regression model as follows: in which are regression coefficients; and is the random error deferring to normal distribution .

The regression result is checked by statistical tests.

Prediction is made with the regression equation.

#### Principal component analysis

The principal component analysis (PCA) is also called the main component analysis, with the concept of dimensionality reduction that turns a multi-index to a few aggregative indicators (Yao *et al.* 2005; Fernández *et al.* 2018; Lu *et al.* 2018). Normally, in mathematical treatment, linear combination is performed for p indicators and new aggregative indicators are generated.

The PCA is to look for new variables and make them reflect key features of objects so as to reduce the original data matrix. The *r* variables are called ‘principal components’. Through PCA, data space has been pressed and the characteristics of multiple data have been expressed directly in the low dimensional space. The PCA calculation steps are demonstrated in the following.

is defined the contribution rate of the first principal component 称 is the accumulative contribution rate of r principal components.

If the accumulative contribution rate is larger than 90%, the r principal components include the original index information.

#### Inverse distance weighting

*p*and

*n*weighting parameters and number of consecutive points.

### Precipitation simulation result

#### PCA accomplishments

It is believed that the predictor mean is the same as the data in the center of the grid, when NCEP has a spatial resolution of 2.5 × 2.5°. It can be seen from Figure 1 that the eight weather stations are located respectively within grids centered as (37.5°, 107.5°), (35°, 107.5°), (35°, 110°). The three grids have data of 36 dimensions (3 × 12), and each grid provides 12 factors including the downscaled predictors of precipitation and air temperature. The PCA method initially is used to reduce dimensions and to carry out filter processing of the NCEP predictors in order to compress and effectively reduce the dimensions of data sets, thus requiring fewer data and eliminating noise in respect of the downscaling model. For more correctness and finer precision, this paper carries out PCA of NCEP predictors of rainfall and temperature in terms of grids and months, and the results can be seen in Table 1.

Month | Statistics | 1 | 2 | 3 | 4 | 5 | Month | Statistics | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

January | Flag value | 4.12 | 3.48 | 1.74 | 1.02 | 0.64 | July | Flag value | 4.70 | 2.84 | 2.19 | 0.78 | 0.60 |

Variance contribution rate | 0.34 | 0.29 | 0.15 | 0.08 | 0.05 | Variance contribution rate | 0.39 | 0.24 | 0.18 | 0.07 | 0.05 | ||

Accumulated variance contribution rate | 0.34 | 0.63 | 0.78 | 0.86 | 0.92 | Accumulated variance contribution rate | 0.39 | 0.63 | 0.81 | 0.88 | 0.93 | ||

February | Flag value | 4.20 | 3.31 | 2.25 | 0.65 | 0.55 | August | Flag value | 4.79 | 2.75 | 1.95 | 1.43 | |

Variance contribution rate | 0.35 | 0.28 | 0.19 | 0.05 | 0.05 | Variance contribution rate | 0.40 | 0.23 | 0.16 | 0.12 | |||

Accumulated variance contribution rate | 0.35 | 0.63 | 0.81 | 0.87 | 0.91 | Accumulated variance contribution rate | 0.40 | 0.63 | 0.79 | 0.91 | |||

March | Flag value | 3.84 | 3.16 | 2.12 | 1.02 | 0.80 | September | Flag value | 5.65 | 3.17 | 1.24 | 0.83 | |

Variance contribution rate | 0.32 | 0.26 | 0.18 | 0.08 | 0.07 | Variance contribution rate | 0.47 | 0.26 | 0.10 | 0.07 | |||

Accumulated variance contribution rate | 0.32 | 0.58 | 0.76 | 0.84 | 0.91 | Accumulated variance contribution rate | 0.47 | 0.74 | 0.84 | 0.91 | |||

April | Flag value | 5.21 | 2.79 | 1.85 | 0.74 | 0.70 | October | Flag value | 5.01 | 2.78 | 2.18 | 0.77 | 0.50 |

Variance contribution rate | 0.43 | 0.23 | 0.15 | 0.06 | 0.06 | Variance contribution rate | 0.42 | 0.23 | 0.18 | 0.06 | 0.04 | ||

Accumulated variance contribution rate | 0.43 | 0.67 | 0.82 | 0.88 | 0.94 | Accumulated variance contribution rate | 0.42 | 0.65 | 0.83 | 0.89 | 0.94 | ||

May | Flag value | 5.19 | 3.49 | 1.43 | 1.01 | November | Flag value | 4.40 | 2.69 | 2.35 | 1.08 | 0.65 | |

Variance contribution rate | 0.43 | 0.29 | 0.12 | 0.08 | Variance contribution rate | 0.37 | 0.22 | 0.20 | 0.09 | 0.05 | |||

Accumulated variance contribution rate | 0.43 | 0.72 | 0.84 | 0.93 | Accumulated variance contribution rate | 0.37 | 0.59 | 0.79 | 0.88 | 0.93 | |||

June | Flag value | 6.08 | 2.04 | 1.59 | 1.04 | 0.47 | December | Flag value | 3.67 | 2.48 | 1.96 | 1.80 | 0.91 |

Variance contribution rate | 0.51 | 0.17 | 0.13 | 0.09 | 0.04 | Variance contribution rate | 0.31 | 0.21 | 0.16 | 0.15 | 0.08 | ||

Accumulated variance contribution rate | 0.51 | 0.68 | 0.81 | 0.90 | 0.94 | Accumulated variance contribution rate | 0.31 | 0.51 | 0.68 | 0.83 | 0.90 |

Month | Statistics | 1 | 2 | 3 | 4 | 5 | Month | Statistics | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

January | Flag value | 4.12 | 3.48 | 1.74 | 1.02 | 0.64 | July | Flag value | 4.70 | 2.84 | 2.19 | 0.78 | 0.60 |

Variance contribution rate | 0.34 | 0.29 | 0.15 | 0.08 | 0.05 | Variance contribution rate | 0.39 | 0.24 | 0.18 | 0.07 | 0.05 | ||

Accumulated variance contribution rate | 0.34 | 0.63 | 0.78 | 0.86 | 0.92 | Accumulated variance contribution rate | 0.39 | 0.63 | 0.81 | 0.88 | 0.93 | ||

February | Flag value | 4.20 | 3.31 | 2.25 | 0.65 | 0.55 | August | Flag value | 4.79 | 2.75 | 1.95 | 1.43 | |

Variance contribution rate | 0.35 | 0.28 | 0.19 | 0.05 | 0.05 | Variance contribution rate | 0.40 | 0.23 | 0.16 | 0.12 | |||

Accumulated variance contribution rate | 0.35 | 0.63 | 0.81 | 0.87 | 0.91 | Accumulated variance contribution rate | 0.40 | 0.63 | 0.79 | 0.91 | |||

March | Flag value | 3.84 | 3.16 | 2.12 | 1.02 | 0.80 | September | Flag value | 5.65 | 3.17 | 1.24 | 0.83 | |

Variance contribution rate | 0.32 | 0.26 | 0.18 | 0.08 | 0.07 | Variance contribution rate | 0.47 | 0.26 | 0.10 | 0.07 | |||

Accumulated variance contribution rate | 0.32 | 0.58 | 0.76 | 0.84 | 0.91 | Accumulated variance contribution rate | 0.47 | 0.74 | 0.84 | 0.91 | |||

April | Flag value | 5.21 | 2.79 | 1.85 | 0.74 | 0.70 | October | Flag value | 5.01 | 2.78 | 2.18 | 0.77 | 0.50 |

Variance contribution rate | 0.43 | 0.23 | 0.15 | 0.06 | 0.06 | Variance contribution rate | 0.42 | 0.23 | 0.18 | 0.06 | 0.04 | ||

Accumulated variance contribution rate | 0.43 | 0.67 | 0.82 | 0.88 | 0.94 | Accumulated variance contribution rate | 0.42 | 0.65 | 0.83 | 0.89 | 0.94 | ||

May | Flag value | 5.19 | 3.49 | 1.43 | 1.01 | November | Flag value | 4.40 | 2.69 | 2.35 | 1.08 | 0.65 | |

Variance contribution rate | 0.43 | 0.29 | 0.12 | 0.08 | Variance contribution rate | 0.37 | 0.22 | 0.20 | 0.09 | 0.05 | |||

Accumulated variance contribution rate | 0.43 | 0.72 | 0.84 | 0.93 | Accumulated variance contribution rate | 0.37 | 0.59 | 0.79 | 0.88 | 0.93 | |||

June | Flag value | 6.08 | 2.04 | 1.59 | 1.04 | 0.47 | December | Flag value | 3.67 | 2.48 | 1.96 | 1.80 | 0.91 |

Variance contribution rate | 0.51 | 0.17 | 0.13 | 0.09 | 0.04 | Variance contribution rate | 0.31 | 0.21 | 0.16 | 0.15 | 0.08 | ||

Accumulated variance contribution rate | 0.51 | 0.68 | 0.81 | 0.90 | 0.94 | Accumulated variance contribution rate | 0.31 | 0.51 | 0.68 | 0.83 | 0.90 |

It can be seen from Table 1 that the front five principal components of January, February, March, April, November and December, which have a contribution rate up to 90% and the same rate of the front four principal components of the other four months, can replace all the original NCEP predictors and all the derived principal components can be used as inputs of the SD method. So, PCA has effectively compressed and dimensionally reduced the original data. Table 1 shows the PCA result with grid centering (N37.5°, E107.5°).

#### Calibration result

We separately calculated the relative error (R) and the correlation coefficient (R^{2}) between the measured rainfall data (1961–1990) and the SD simulated result using the multiple linear regression algorithms of the eight stations in Shannxi Province in the Weihe River basin, and the details are shown in Table 2. It can be seen from Table 2 that the mean values of simulated monthly rainfall series are well matched with the measured values, with the relative errors of 3%. However, the values of standard deviation between the simulated and measured results are not so well matched, with the relative error of −22.23– − 40.47%. Meanwhile, the R^{2} between the simulated values and measured values ranges from 0.45 to 0.7, indicating that the accuracy of the simulated results based on the SD method in this research is acceptable.

Calibration result of all areass | |||||||
---|---|---|---|---|---|---|---|

Month | Measured rainfall | SD simulated rainfall | |||||

Mean value (mm) | Standard deviation (mm) | Mean value (mm) | R_{mean} (%) | Standard deviation (mm) | R_{sd}(%) | R^{2} | |

1 | 5.71 | 4.73 | 5.710902 | 0.00 | 3.61532374 | −23.60 | 0.64 |

2 | 9.90 | 7.43 | 9.901645 | 0.00 | 5.07979309 | −31.67 | 0.51 |

3 | 24.98 | 12.69 | 24.19834 | −3.13 | 7.8377521 | −38.21 | 0.42 |

4 | 47.36 | 20.71 | 47.35888 | 0.00 | 14.5527426 | −29.73 | 0.51 |

5 | 60.29 | 33.85 | 61.55285 | 2.09 | 25.4921378 | −24.69 | 0.60 |

6 | 59.65 | 27.10 | 59.65264 | 0.00 | 18.7121028 | −30.96 | 0.57 |

7 | 116.34 | 39.52 | 116.3437 | 0.00 | 25.7682664 | −34.80 | 0.47 |

8 | 105.15 | 45.70 | 105.1501 | 0.00 | 35.3399775 | −22.67 | 0.56 |

9 | 109.23 | 52.12 | 109.231 | 0.00 | 31.0318732 | −40.47 | 0.41 |

10 | 57.24 | 33.72 | 57.24088 | 0.00 | 24.2789739 | −27.99 | 0.54 |

11 | 22.82 | 14.98 | 22.81603 | 0.00 | 9.16553475 | −38.83 | 0.43 |

12 | 5.09 | 5.65 | 5.093359 | 0.00 | 4.3932503 | −22.23 | 0.65 |

Calibration result of all areass | |||||||
---|---|---|---|---|---|---|---|

Month | Measured rainfall | SD simulated rainfall | |||||

Mean value (mm) | Standard deviation (mm) | Mean value (mm) | R_{mean} (%) | Standard deviation (mm) | R_{sd}(%) | R^{2} | |

1 | 5.71 | 4.73 | 5.710902 | 0.00 | 3.61532374 | −23.60 | 0.64 |

2 | 9.90 | 7.43 | 9.901645 | 0.00 | 5.07979309 | −31.67 | 0.51 |

3 | 24.98 | 12.69 | 24.19834 | −3.13 | 7.8377521 | −38.21 | 0.42 |

4 | 47.36 | 20.71 | 47.35888 | 0.00 | 14.5527426 | −29.73 | 0.51 |

5 | 60.29 | 33.85 | 61.55285 | 2.09 | 25.4921378 | −24.69 | 0.60 |

6 | 59.65 | 27.10 | 59.65264 | 0.00 | 18.7121028 | −30.96 | 0.57 |

7 | 116.34 | 39.52 | 116.3437 | 0.00 | 25.7682664 | −34.80 | 0.47 |

8 | 105.15 | 45.70 | 105.1501 | 0.00 | 35.3399775 | −22.67 | 0.56 |

9 | 109.23 | 52.12 | 109.231 | 0.00 | 31.0318732 | −40.47 | 0.41 |

10 | 57.24 | 33.72 | 57.24088 | 0.00 | 24.2789739 | −27.99 | 0.54 |

11 | 22.82 | 14.98 | 22.81603 | 0.00 | 9.16553475 | −38.83 | 0.43 |

12 | 5.09 | 5.65 | 5.093359 | 0.00 | 4.3932503 | −22.23 | 0.65 |

Table 3 shows the simulated results with the calibrated model in combination of 1991–2000 predictor principal components and verification SD modeling of measured precipitation. The results show that the simulated values are not quite well matched with the measured values in terms of the mean value and the standard deviation for the monthly rainfall series using PCA with the multiple linear regression algorithm method, but the accuracy is acceptable. Meanwhile, the R^{2} values of the simulated results are not as good as that in the period of 1961–1990.

Calibration results of all areas | |||||||
---|---|---|---|---|---|---|---|

Month | Measured rainfall | SD simulated rainfall | |||||

Mean value (mm) | Standard deviation (mm) | Mean value (mm) | R_{mean} (%) | Standard deviation (mm) | R_{sd} (%) | R^{2} | |

1 | 5.67 | 3.98 | 6.656635 | 17.42 | 2.94969082 | −25.81 | 0.19 |

2 | 7.20 | 6.01 | 8.327076 | 15.63 | 3.94840369 | −34.31 | 0.27 |

3 | 26.79 | 15.05 | 28.16123 | 5.13 | 7.9631557 | −47.08 | 0.62 |

4 | 40.64 | 18.58 | 42.1432 | 3.70 | 10.2332616 | −44.93 | 0.31 |

5 | 52.80 | 40.16 | 51.54762 | −2.36 | 24.2796912 | −39.55 | 0.53 |

6 | 71.17 | 39.55 | 60.63022 | −14.81 | 21.2697672 | −46.22 | 0.74 |

7 | 98.74 | 39.43 | 105.3381 | 6.69 | 16.8436109 | −57.28 | 0.56 |

8 | 98.01 | 37.34 | 89.85505 | −8.32 | 44.6923034 | 19.68 | 0.50 |

9 | 63.11 | 29.80 | 87.5861 | 38.79 | 26.6013173 | −10.75 | 0.11 |

10 | 53.59 | 25.50 | 40.76011 | −23.94 | 21.8451358 | −14.35 | 0.68 |

11 | 20.17 | 15.68 | 17.90833 | −11.22 | 7.80902636 | −50.19 | 0.47 |

12 | 4.00 | 5.18 | 5.867537 | 46.55 | 3.81224965 | −26.34 | 0.52 |

Calibration results of all areas | |||||||
---|---|---|---|---|---|---|---|

Month | Measured rainfall | SD simulated rainfall | |||||

Mean value (mm) | Standard deviation (mm) | Mean value (mm) | R_{mean} (%) | Standard deviation (mm) | R_{sd} (%) | R^{2} | |

1 | 5.67 | 3.98 | 6.656635 | 17.42 | 2.94969082 | −25.81 | 0.19 |

2 | 7.20 | 6.01 | 8.327076 | 15.63 | 3.94840369 | −34.31 | 0.27 |

3 | 26.79 | 15.05 | 28.16123 | 5.13 | 7.9631557 | −47.08 | 0.62 |

4 | 40.64 | 18.58 | 42.1432 | 3.70 | 10.2332616 | −44.93 | 0.31 |

5 | 52.80 | 40.16 | 51.54762 | −2.36 | 24.2796912 | −39.55 | 0.53 |

6 | 71.17 | 39.55 | 60.63022 | −14.81 | 21.2697672 | −46.22 | 0.74 |

7 | 98.74 | 39.43 | 105.3381 | 6.69 | 16.8436109 | −57.28 | 0.56 |

8 | 98.01 | 37.34 | 89.85505 | −8.32 | 44.6923034 | 19.68 | 0.50 |

9 | 63.11 | 29.80 | 87.5861 | 38.79 | 26.6013173 | −10.75 | 0.11 |

10 | 53.59 | 25.50 | 40.76011 | −23.94 | 21.8451358 | −14.35 | 0.68 |

11 | 20.17 | 15.68 | 17.90833 | −11.22 | 7.80902636 | −50.19 | 0.47 |

12 | 4.00 | 5.18 | 5.867537 | 46.55 | 3.81224965 | −26.34 | 0.52 |

From Tables 2 and 3, we can conclude that the accuracy of the simulated results in the verification period (1991–2000) is not as good as that in the calibration period. The relative error (R) and the correlation coefficient (R^{2}) between the simulated values and the measured values in the verification period are both unsatisfactory when comparing the result in the calibration period. The above conclusion demonstrates that the simulation for extreme rainfall event using the method in this study is not satisfactory.

### Result analysis of temperature simulation

#### Calibration result of PCA plus multiple regression algorithm

The multiple regression algorithm is essential in establishing an SD model (Huang *et al.* 2007; Hsu 2012; Honti *et al.* 2014; Inam *et al.* 2017), which uses the principal components from PCA temperature factors and monthly observed data of the eight stations. In detail, as shown in Table 4, the multiple regression algorithm is a best estimation method to simulate the monthly rainfall series, almost without any error except for those of April, November and December; while it is not so satisfactory in the simulation of standard deviation, with a relative error −47– − 6%. With respect to , simulations are excellent for January, February, March, October, November and December, but not so good for the remaining months. In short, the multiple regression method works well in simulating temperature for the months as mentioned above and vice versa for the remaining months.

Whole area results in calibration time | |||||||
---|---|---|---|---|---|---|---|

Month | Measured temperature | SD simulated temperature | |||||

Mean value (°C) | Standard deviation (°C) | Mean value (°C) | R_{mean} (%) | Standard deviation (°C) | R_{sd} (%) | R^{2} | |

1 | −3.37 | 1.19 | −3.37 | 0.00 | 1.04 | −12.35 | 0.85 |

2 | −0.88 | 1.93 | −0.88 | 0.00 | 1.83 | −5.05 | 0.94 |

3 | 5.00 | 1.28 | 5.00 | 0.00 | 1.19 | −7.52 | 0.90 |

4 | 11.48 | 1.03 | 11.48 | 0.00 | 0.90 | −12.68 | 0.85 |

5 | 16.80 | 0.89 | 16.80 | 0.00 | 0.69 | −21.88 | 0.69 |

6 | 21.40 | 0.97 | 21.40 | 0.00 | 0.80 | −17.97 | 0.83 |

7 | 23.08 | 0.85 | 23.08 | 0.00 | 0.67 | −20.65 | 0.69 |

8 | 22.03 | 1.02 | 22.03 | 0.00 | 0.90 | −11.33 | 0.82 |

9 | 16.37 | 0.86 | 16.37 | 0.00 | 0.65 | −23.45 | 0.72 |

10 | 10.94 | 1.01 | 10.94 | 0.00 | 0.89 | −12.46 | 0.82 |

11 | 3.99 | 1.29 | 3.99 | −0.05 | 1.22 | −5.60 | 0.94 |

12 | −1.93 | 1.60 | −1.93 | 0.00 | 1.53 | −4.47 | 0.95 |

Whole area results in calibration time | |||||||
---|---|---|---|---|---|---|---|

Month | Measured temperature | SD simulated temperature | |||||

Mean value (°C) | Standard deviation (°C) | Mean value (°C) | R_{mean} (%) | Standard deviation (°C) | R_{sd} (%) | R^{2} | |

1 | −3.37 | 1.19 | −3.37 | 0.00 | 1.04 | −12.35 | 0.85 |

2 | −0.88 | 1.93 | −0.88 | 0.00 | 1.83 | −5.05 | 0.94 |

3 | 5.00 | 1.28 | 5.00 | 0.00 | 1.19 | −7.52 | 0.90 |

4 | 11.48 | 1.03 | 11.48 | 0.00 | 0.90 | −12.68 | 0.85 |

5 | 16.80 | 0.89 | 16.80 | 0.00 | 0.69 | −21.88 | 0.69 |

6 | 21.40 | 0.97 | 21.40 | 0.00 | 0.80 | −17.97 | 0.83 |

7 | 23.08 | 0.85 | 23.08 | 0.00 | 0.67 | −20.65 | 0.69 |

8 | 22.03 | 1.02 | 22.03 | 0.00 | 0.90 | −11.33 | 0.82 |

9 | 16.37 | 0.86 | 16.37 | 0.00 | 0.65 | −23.45 | 0.72 |

10 | 10.94 | 1.01 | 10.94 | 0.00 | 0.89 | −12.46 | 0.82 |

11 | 3.99 | 1.29 | 3.99 | −0.05 | 1.22 | −5.60 | 0.94 |

12 | −1.93 | 1.60 | −1.93 | 0.00 | 1.53 | −4.47 | 0.95 |

#### Verification result of PCA plus multiple regression algorithm

The same method has been employed in calculating SD simulation results of the verification period, as shown in Table 5, which shows that the simulated monthly average temperature is good in accuracy judging from the values of relative errors. However, the standard deviation of the simulated monthly temperature is not well matched with the measured values, especially in May, July, August and September. Generally, the correlation coefficients (R^{2}) between the simulated and measured values are satisfactory.

All areas verification results | |||||||
---|---|---|---|---|---|---|---|

Month | Measured temperature | SD simulated temperature | |||||

Mean value (°C) | Standard deviation (°C) | Mean value (°C) | R_{mean} (%) | Standard deviation (°C) | R_{sd} (%) | R^{2} | |

1 | −5.65 | 1.67 | −5.65664 | 0.12 | 1.64969082 | −1.22 | 0.81 |

2 | 1.2 | 3.43 | 1.327076 | 10.59 | 1.84840369 | −46.11 | 0.86 |

3 | 6.79 | 3.58 | 8.16123 | 20.19 | 1.6931545 | −52.71 | 0.66 |

4 | 11.64 | 1.45 | 12.1432 | 4.32 | 1.6324616 | 12.58 | 0.81 |

5 | 17.8 | 0.67 | 16.54762 | −7.04 | 1.6927805 | 152.65 | 0.76 |

6 | 21.17 | 0.89 | 20.63022 | −2.55 | 1.1256903 | 26.48 | 0.71 |

7 | 25.74 | 3.05 | 25.3381 | −1.56 | 1.215683 | −60.14 | 0.70 |

8 | 24.01 | 3.15 | 25.85505 | 7.68 | 1.264553 | −59.86 | 0.74 |

9 | 17.11 | 1.78 | 17.5861 | 2.78 | 0.6013173 | −66.22 | 0.80 |

10 | 13.59 | 1.05 | 12.76011 | −6.11 | 1.4813865 | 41.08 | 0.78 |

11 | 4.17 | 1.34 | 5.90833 | 41.69 | 1.089745 | −18.68 | 0.52 |

12 | −2.6 | 1.56 | −2.86754 | 10.29 | 1.8322245 | 17.45 | 0.79 |

All areas verification results | |||||||
---|---|---|---|---|---|---|---|

Month | Measured temperature | SD simulated temperature | |||||

Mean value (°C) | Standard deviation (°C) | Mean value (°C) | R_{mean} (%) | Standard deviation (°C) | R_{sd} (%) | R^{2} | |

1 | −5.65 | 1.67 | −5.65664 | 0.12 | 1.64969082 | −1.22 | 0.81 |

2 | 1.2 | 3.43 | 1.327076 | 10.59 | 1.84840369 | −46.11 | 0.86 |

3 | 6.79 | 3.58 | 8.16123 | 20.19 | 1.6931545 | −52.71 | 0.66 |

4 | 11.64 | 1.45 | 12.1432 | 4.32 | 1.6324616 | 12.58 | 0.81 |

5 | 17.8 | 0.67 | 16.54762 | −7.04 | 1.6927805 | 152.65 | 0.76 |

6 | 21.17 | 0.89 | 20.63022 | −2.55 | 1.1256903 | 26.48 | 0.71 |

7 | 25.74 | 3.05 | 25.3381 | −1.56 | 1.215683 | −60.14 | 0.70 |

8 | 24.01 | 3.15 | 25.85505 | 7.68 | 1.264553 | −59.86 | 0.74 |

9 | 17.11 | 1.78 | 17.5861 | 2.78 | 0.6013173 | −66.22 | 0.80 |

10 | 13.59 | 1.05 | 12.76011 | −6.11 | 1.4813865 | 41.08 | 0.78 |

11 | 4.17 | 1.34 | 5.90833 | 41.69 | 1.089745 | −18.68 | 0.52 |

12 | −2.6 | 1.56 | −2.86754 | 10.29 | 1.8322245 | 17.45 | 0.79 |

It can be seen from Tables 4 and 5 that the verification simulations are not as good as the simulated results in the calibration period. Especially, the accuracy of the forecasted standard deviation for the monthly temperature both in the calibration period and the verification period is not satisfactory, which demonstrates that the downscaling method in this study performs poorly to estimate the extreme temperature.

## CONCLUSIONS

The SD method has been widely used for describing the impact of weather changes to hydrological resources of a river basin (Li & Simonovic 2010; Dawadi & Ahmad 2012; Shang *et al.* 2016), because it has relative concise computation process compared to the DD method and broad applications to different weather models. With great physical significance in global weather models, it is precise in simulating climate information and downscales from grand to a basin scale through statistical relations. This paper promotes the development of the SD model, which is based upon the statistical relations between the general atmospheric circulation factors derived from establishment of a global weather model and the precipitation and temperature data of eight stations chosen from the Weihe hydrological data. The research results show that the accuracy of simulated monthly temperature is much better than that of monthly rainfall. Moreover, the estimation for the extreme climate conditions are not good in accuracy. Comparing the rainfall and temperature simulations with PCA plus multiple regression method and the gradual linear regression algorithm method, the two have similar results in rainfall but are very different in temperature correlations, perhaps because rainfall has complicated factors and a nonlinear relationship may exist between the atmospheric circulation factors and the rainfall of the stations. As a consequence, it is most important to select appropriate factors of atmospheric circulation for precise downscaling of climate information and climatic condition forecasting.

## CONFLICTS OF INTEREST

The authors declared that they have no conflict of interest in this work.

## ACKNOWLEDGEMENT

The paper was funded by the National Natural Science Foundation of China (Grant No. 51379219), Zhejiang province Funds for Distinguished Young Scientists (Grant No. LR15E090002), the Social Science Foundation of Zhejiang province (Grant No. 14YSXK02ZD-1YB) and the National Key Research and Development Program of China (2016YFC0401301 and 2016YFC0400204) for their support of this study. The study was also supported by the PhD Start-up Fund of Northwest A&F University (2452016215).