Regional Precipitation Model Based on Geographically and Temporally Weighted Regression Kriging

: High-resolution precipitation ﬁeld has been widely used in hydrological and meteorological modeling. This paper establishes the spatial and temporal distribution model of precipitation in Hubei Province from 2006 through 2014, based on the data of 75 meteorological stations. This paper applies a geographically and temporally weighted regression kriging (GTWRK) model to precipitation and assesses the e ﬀ ects of timescales and a time-weighted function on precipitation interpolation. This work’s results indicate that: (1) the optimal timescale of the geographically and temporally weighted regression (GTWR) precipitation model is daily. The ﬁtting accuracy is improved when the timescale is converted from months and years to days. The average mean absolute error (MAE), mean relative error (MRE), and the root mean square error (RMSE) decrease with scaling from monthly to daily time steps by 36%, 56%, and 35%, respectively, and the same statistical indexes decrease by 13%, 15%, and 14%, respectively, when scaling from annual to daily steps; (2) the time weight function based on an exponential function improves the predictive skill of the GTWR model by 3% when compared to geographically weighted regression (GWR) using a monthly time step; and (3) the GTWRK has the highest accuracy, and improves the MAE, MRE and RMSE by 3%, 10% and 1% with respect to monthly precipitation predictions, respectively, and by 3%, 10% and 5% concerning annual precipitation predictions, respectively, compared with the GWR results. total of the predicted level data the multi-variable 1 × 1 monthly 1 x 1 km precipitation precision. precipitation monthly and of and the region model spatial GWR model in of predictive skill, accuracy, and residual autocorrelations. weighted regression an extension of the GWR model, spatial interpolation


Introduction
Precipitation is a critical flux in the water cycle [1,2]. It is, for this reason, imperative to study the spatial-temporal features of precipitation [3,4]. Precipitation data are usually derived from meteorological sites with limited spatial coverage and sensor-gathered data, such as remote sensing satellites and rainfall radars [5]. Meteorological site location observations yield local, discrete, and limited spatial data points, which cannot account for the spatial precipitation variability accurately [6,7]. The general spatial resolution of remote sensing precipitation data products is generally low, which does capture the precipitation distribution in small areas [8]. These problems constrain the application of precipitation data for multiple practical purposes. Thus, there is a need for further study on how to obtain continuous and accurate distributions of precipitation at regional scales.
Spatial interpolation of precipitation data falls into two categories: deterministic interpolation and spatial-temporal interpolation. Deterministic interpolation is further divided into two categories: global a study of the relation between total annual precipitation and elevation in the UK by employing the GWR model. Their results revealed that the rate of precipitation increased with elevation, and that the predicted sea level precipitation varied between 600 mm and 1250 mm. The precipitation data from the Tropical Rainfall Measuring Mission (TRMM) 3B43 products were fitted with a multi-variable GWR reduction method to obtain 1 km × 1 km precipitation data by Chen et al. [29]. The GWR method was compared with two other downscaling methods (single variable regression (UR) and multivariate regression (MR)). Chen et al. (29) concluded that the GWR method could predict annual and monthly TRMM 1 km x 1 km precipitation with high precision. The accuracy of TRMM precipitation products at the daily and monthly scales in the Qaidam Basin of China was evaluated by Lv et al. [30] with the GWR model. Their results indicate that the precipitation GWR model based on ground and satellite data reduced the error of TRMM products, which was of significance in the fields of hydrology and climate change. The vegetation and climate data (Normalized Vegetation Difference Index (NVDI) and rainfall) from 2002 through 2012 for the growing season (June-September) in the Sahel region of Africa was relied upon in the GWR model by Georganos et al. [31]. The results showed that the spatial pattern of the NDVI-rainfall relationship is characterized when selecting the appropriate scale. Their GWR model performs better than the OLS in terms of predictive skill, accuracy, and residual autocorrelations. With the further research of scholars, geographically weighted regression kriging (GWRK) [32], as an extension of the GWR model, appears in the spatial interpolation of temperature and soil properties. It has also been explored in the field of precipitation and prediction research, achieving excellent results. The GWRK model combines the GWR with the kriging method, and uses the kriging method to interpolate the residual part of the GWR model, which eliminates the influence of the spatial correlation of the residual on the model fit, and shows that it is masked by spatial non-stationary local variation.
The GTWR model focuses primarily on the time dimension, although it accounts for the characterization of spatial heterogeneity [33]. At a particular timescale, the GTWR handles the distribution of the time dimension in a manner dissimilar to that described by the first law of geography (Tobler [34]), and, thus, it can be improved. Ge et al. proposed the seasonal differential geographically and temporally weighted regression (seasonal-difference GTWR, SD-GTWR) [35]. The latter authors applied the SD-GTWR model to data for hemorrhagic fever with renal syndromes from Hubei Province to show that the SD-GTWR model is superior to the ordinary GTWR model. The SD-GTWR model relied on the results of incremental spatial autocorrelation when balancing the roles of space and time. Data from the Zhejiang coast (China) from 2012 through 2016 was employed by Du et al. [36] to propose a geographical and periodic time-weighted regression model (GcTWR) that unifies spatial distance and temporal distance. The results confirmed that the seasonal effects on coastal areas are related to an interannual effect.
Although the GWR model performs well in the spatial interpolation, precipitation not only has continuity in space, but also has strong continuity in time. However, few kinds of research have been carried out on the spatial-temporal interpolation of precipitation. Fortunately, the application of the GTWR model in different fields has gradually become mature, and it is possible to introduce it into precipitation interpolation. Therefore, to better understand the temporal and spatial heterogeneity of precipitation, this paper interpolates the spatial distribution from 2006 through 2014 of the monthly and annual rainfall in Hubei Province, based on the GTWR model. Then, this work adjusts the spatial and temporal weight, according to the temporal characteristics of precipitation. The Gaussian kernel model is selected as the spatial weight, and the exponential function model is chosen as the temporal weight. Meanwhile, this work introduces the kriging model to eliminate the influence of residual spatial correlation on model fitting, which can improve the interpolation accuracy.

Geographically and Temporally Weighted Regression Model
The GTWR model is expressed as Equation (1) [19,37]: where (u i , v i , t i ) denotes the spatial-temporal coordinates of the observed location i, which contains the location and temporal information; k denotes the index k = 1, 2 . . . n. β k (u i , v i , t i ) represents a set of values for the number n of parameters at point i, and ε i represents the random error of the predicted variable y i . The estimates of β k (u i , v i , t i ) are given by Formula (2): The spatial-temporal weight matrix W(u i , v i , t i ) is based on the definition of the spatial-temporal distance and its decay functions. Generally, the weight functions include the distance threshold method, distance inverse ratio method, Gaussian (Gauss) function method, and double square root (bi-square) kernel function method. The Gaussian and bi-square kernel function methods are commonly used in the GTWR model [38]. The Gaussian kernel function is given by Equation (3): where the element w ij of the weighting matrix is determined by the spatial-temporal distance d ST i j and spatial-temporal bandwidth h ST . The GTWR model sets the spatial-temporal distance d ST ij as a function of the temporal distance d T ij and the spatial distance, d S ij as expressed in Equation (4): λ and µ denote the spatial distance factor and the temporal distance factor, respectively, which balance the effect of the space and time dimension on parameter estimation. The w ij element of the spatial-temporal weight matrix is expressed by Equation (5): It is seen in Equation (5) that the spatial-temporal element or kernel function w ij equals the spatial kernel function w S ij multiplied by the temporal kernel function w T ij . h S and h T are the spatial and temporal bandwidths, respectively. One can determine the weight of the observed variable at a given Remote Sens. 2020, 12, 2547 5 of 19 location to the regressed variable at the same location at a specific time by the spatial-temporal kernel function. The spatial bandwidth b S and temporal bandwidth b T are decided by cross-validation (CV), and they are obtained by minimizing the expression on the right-hand side of Equation (6): when CV(λ, µ) is minimized, it yields the optimal spatial bandwidth and temporal bandwidth of the model. A method for determining spatial and temporal bandwidth in steps was proposed by Fotheringham et al. [21]. The principle of this method is that GTWR can be regarded as GWR for a period of time. In each time period of data, the spatial bandwidth is determined by the GWR model by minimizing CV(b S ), and then determining the time bandwidth that minimizes CV (b S1 , b S2 , . . . , b Sn , b T ). One obtains the optimal spatial bandwidth and temporal bandwidth when the CV is minimized. This paper relies on a step-by-step approach [8] to calculate the bandwidth. This approach considers the data at a specific time or a period at first. The GTWR became similar to the GWR model in this manner. One minimize the spatial data CV(b S ) of each period, and then obtain the appropriate time bandwidth by minimizing the CV (b S1 , b S2 , . . . , b Sn , b T ).

Comparison Models with Different Spatial-Temporal Parameters
The spatial-temporal distance calculation method of the GTWR model is similar to the extension method, which adds the temporal distance as the third dimension to the distance calculation. Previous studies have generally neglected the shortcomings of the extension method, whereby the uncertainty of the units (say, m or km for spatial units) introduces uncertainty of the spatial-temporal interpolation results when calculating the spatial-temporal distance [13]. The calculated results differ substantially depending on the adopted units, such as the unit of spatial distance being in meters or kilometers, and the time unit being the year, month, day, minute, or second. This paper chooses km as the spatial distance unit, whereas the timescale may be annual, monthly, and daily, according to the experimental data scale.
The temporal weighting formulas may be inconsistent with the spatial weighting formulas whenever the spatial-temporal weight is decomposed into the product of the spatial weight and the temporal weight. This work proposes a time weight in the form of an exponential function, as shown in Formula (7), in agreement with the last line on the right-hand side of Equation (5): Furthermore, the distribution of variables or objects in the time dimension is not entirely governed by the previously cited first law of geography. There are apparent cyclical changes in the four seasons of the year. The physical characteristics of the climate at a location differ in the winter compared with summer, but they have statistical similarities for the same season in different years. Concerning monthly precipitation it is known that there may be statistical similarities between a given month's precipitation and non-adjacent month precipitation in the same quarter. Thus, it is necessary to improve the calculation method of the spatial-temporal distance to capture such similarities. The temporal distribution of precipitation exhibits periodicity, therefore, this paper relies on the sinusoidal function to calculate the temporal distance. It selects the exponential function model as a temporal weight formula. The calculation formula of the periodic temporal distance is given by Equation (8), where T denotes the period of the function: Remote Sens. 2020, 12, 2547 6 of 19

Geographically and Temporally Weighted Regression Kriging
It is a spatial interpolation method based on geostatistics of the kriging method which fully considers the characteristics of the spatial variability of the sample points. It has the advantages of strong applicability and high prediction accuracy, and is currently most widely used in the fields of meteorology, ecology, and soil. The kriging algorithm can achieve optimal linear unbiased values, improving the accuracy of estimation to a certain extent. Moreover, for regionalized variables, it reveals its spatial structure well.
The GTWR model embeds temporal information and geographic location into the model, making full use of the spatial and temporal characteristics of data, and has a useful application in regional regression analysis. Geographically and temporally weighted regression kriging (GTWRK) is a hybrid method based on the GTWR model. First, we use the GTWR method to establish the regression relationship between precipitation and auxiliary information. Second, we use the kriging method to interpolate the residuals ε of the GTWR model. Finally, we add the interpolation result of residuals and the GTWR regression estimation value to obtain the GTWRK estimation result. Therefore, the GTWRK method considers the relationship between precipitation and influencing factors and the spatial autocorrelation of precipitation. The GTWR model is given by Equation (9): where: represents the residual interpolation result of GTWR regression obtained by ordinary kriging (OK) interpolation. The variogram must be selected when the kriging method is used to interpolate the residuals. This work chooses an exponential variogram for ordinary kriging interpolation based on exploratory analysis of the precipitation data.
The flow chart of the GTWRK model is shown in Figure 1.

Precision Evaluation
Because of the large amount of data generated in the interpolation comparison, it would be burdensome to display all the results. The results corresponding to precipitation fitting in July 2008 and May 2013 are representative of the monthly scale data, and the results of precipitation fitting in 2010 and 2012 are representative of the annual scale data. The evaluation of the interpolation models relied on several performance indices, namely the mean absolute error (MAE), mean relative error (MRE), and the root mean square error (RMSE). The smaller the values of MAE, MRE, and RMSE, the better the interpolation effect.
Remote Sens. 2020, 12, 2547 (12) in which n denotes the sample size, Y i represents the ith a sample value,Ŷ i denots the sample estimates.

Study Area and Data
Hubei Province is located between northern latitudes 29 • 05 and 33 • 20 and eastern latitudes 108 • 21 and 116 • 07 within China. The province covers an area of 185,900 square kilometers, and includes a variety of topographical regions, including mountains, plains, and transitional topographic zones. The topography of Hubei Province exhibits the highest elevation of 3090 m in its western region, and lowest towards its eastern area, with a lowest elevation of −142 m [39]. Hubei Province lies within a north-south transitional climatic zone, belonging to the northeastern Asian monsoonal region, except for the higher elevations of the western mountainous areas. Most of the province features a subtropical monsoonal, humid climate. The warm temperature coincides with the rainy period, providing abundant rainfall to support agroforest production [39]. The spatial distribution of meteorological sites and the digital elevation model (DEM) are shown in Figure 2.
Remote Sens. 2020, 12, 2547 8 of 19 in which n denotes the sample size, represents the ith a sample value, denots the sample estimates.

Study Area and Data
Hubei Province is located between northern latitudes 29°05′ and 33°20′ and eastern latitudes 108°21′ and 116°07′ within China. The province covers an area of 185,900 square kilometers, and includes a variety of topographical regions, including mountains, plains, and transitional topographic zones. The topography of Hubei Province exhibits the highest elevation of 3090 m in its western region, and lowest towards its eastern area, with a lowest elevation of −142 m [39]. Hubei Province lies within a north-south transitional climatic zone, belonging to the northeastern Asian monsoonal region, except for the higher elevations of the western mountainous areas. Most of the province features a subtropical monsoonal, humid climate. The warm temperature coincides with the rainy period, providing abundant rainfall to support agroforest production [39]. The spatial distribution of meteorological sites and the digital elevation model (DEM) are shown in Figure 2.
in which n denotes the sample size, represents the ith a sample value, denots the sample estimates.

Study Area and Data
Hubei Province is located between northern latitudes 29°05′ and 33°20′ and eastern latitudes 108°21′ and 116°07′ within China. The province covers an area of 185,900 square kilometers, and includes a variety of topographical regions, including mountains, plains, and transitional topographic zones. The topography of Hubei Province exhibits the highest elevation of 3090 m in its western region, and lowest towards its eastern area, with a lowest elevation of −142 m [39]. Hubei Province lies within a north-south transitional climatic zone, belonging to the northeastern Asian monsoonal region, except for the higher elevations of the western mountainous areas. Most of the province features a subtropical monsoonal, humid climate. The warm temperature coincides with the rainy period, providing abundant rainfall to support agroforest production [39]. The spatial distribution of meteorological sites and the digital elevation model (DEM) are shown in Figure 2.  Many factors influence the spatial distribution of precipitation. Previous studies have demonstrated that topographic characteristics are the main controlling factor [40,41]. The Normalized Differential Vegetation Index (NDVI) is influenced by precipitation, and vice versa [42,43]. This work employs the Digital Elevation Model (DEM) Shuttle Radar Topography Mission (SRTM) data with 90 m resolution. The NDVI data are the MODND1M for China with 500 m resolution NDVI monthly synthesis products. All of the above data are available from the Geospatial Data Cloud website (http://www.gscloud.cn).

Data Preprocessing
All data were consolidated under the WGS-84 geographic coordinate system. The grid resolution employed was 0.1° × 0.1°. Precipitation constitutes the dependent variable; longitude, latitude, DEM, and the NDVI are the independent variables. The timescale of precipitation data is divided into monthly and annual categories. The monthly data represent the time series data from January 2006 through December 2014. The yearly data denote the time series data from 2006 through 2014.
The interpolation accuracy of the GTWR models was evaluated by cross-verification (CV). The 75 meteorological sites were randomly divided into 60 sites (80% of the total number of sites) as the modeling set for interpolation; the remaining 15 sites (20% of the total number of sites) were chosen Many factors influence the spatial distribution of precipitation. Previous studies have demonstrated that topographic characteristics are the main controlling factor [40,41]. The Normalized Differential Vegetation Index (NDVI) is influenced by precipitation, and vice versa [42,43]. This work employs the Digital Elevation Model (DEM) Shuttle Radar Topography Mission (SRTM) data with 90 m resolution. The NDVI data are the MODND1M for China with 500 m resolution NDVI monthly synthesis products. All of the above data are available from the Geospatial Data Cloud website (http://www.gscloud.cn).

Data Preprocessing
All data were consolidated under the WGS-84 geographic coordinate system. The grid resolution employed was 0.1 • × 0.1 • . Precipitation constitutes the dependent variable; longitude, latitude, DEM, and the NDVI are the independent variables. The timescale of precipitation data is divided into monthly and annual categories. The monthly data represent the time series data from January 2006 through December 2014. The yearly data denote the time series data from 2006 through 2014.
The interpolation accuracy of the GTWR models was evaluated by cross-verification (CV). The 75 meteorological sites were randomly divided into 60 sites (80% of the total number of sites) as the modeling set for interpolation; the remaining 15 sites (20% of the total number of sites) were chosen as the validation set, which was used to evaluate the models' accuracies. This paper employs the GTWR models corresponding to several timescales listed in Table 1.  as the validation set, which was used to evaluate the models' accuracies. This paper employs the GTWR models corresponding to several timescales listed in Table 1.      The average error maps of MAE, MRE, and RMSE for monthly scale data are shown in Figure 6.

Monthly Data
(a) MAE distribution chart of monthly scale data It can be seen in Figure 6 and Table 2 that the optimal timescale of the GTWR model is daily for the monthly scale data. The MAE, MRE, and RMSE decrease by 36%, 56%, and 35%, respectively, when choosing GTWR(D) instead of GTWR(M). The GTWR(E) improves the accuracy of the results compared to GTWR(D), by reducing the MAE, MRE, and RMSE by 0.7%, 1.1%, and 0.6%, respectively. The fitting accuracy of GTWR(E) and GWR are similar, with a difference of about 3% for the monthly scale data results shown in Table 2. The GTWR(C) has a lower accuracy compared with GWR, and increased MAE, MRE, and RMSE by 25%, 45%, and 24%. The GTWRK has the highest accuracy compared with GWR, and decreased MAE, MRE, and RMSE by 3%, 10%, and 1%, respectively. Figure 7 shows the MAE, MRE, and RMSE of monthly GWR and GTWRK models in different seasons. According to the character, the fitting accuracy of the GTWRK model is higher than that of the GWR model as a whole, especially in spring, autumn, and winter.  It can be seen in Figure 6 and Table 2 that the optimal timescale of the GTWR model is daily for the monthly scale data. The MAE, MRE, and RMSE decrease by 36%, 56%, and 35%, respectively, when choosing GTWR(D) instead of GTWR(M). The GTWR(E) improves the accuracy of the results compared to GTWR(D), by reducing the MAE, MRE, and RMSE by 0.7%, 1.1%, and 0.6%, respectively. The fitting accuracy of GTWR(E) and GWR are similar, with a difference of about 3% for the monthly scale data results shown in Table 2. The GTWR(C) has a lower accuracy compared with GWR, and increased MAE, MRE, and RMSE by 25%, 45%, and 24%. The GTWRK has the highest accuracy compared with GWR, and decreased MAE, MRE, and RMSE by 3%, 10%, and 1%, respectively. Figure 7 shows the MAE, MRE, and RMSE of monthly GWR and GTWRK models in different seasons. According to the character, the fitting accuracy of the GTWRK model is higher than that of the GWR model as a whole, especially in spring, autumn, and winter.

Annual Data
The interpolation distribution maps corresponding to 2010 and 2012 are shown in Figures 8 and  9, respectively.

Annual Data
The interpolation distribution maps corresponding to 2010 and 2012 are shown in Figures 8  and 9, respectively.  It is seen in Figure 10 and Table 3 that the optimal timescale of the GTWR model is daily for the annual scale data. The MAE, MRE, RMSE decrease by 13%, 15%, and 14%, respectively, when  It is seen in Figure 10 and Table 3 that the optimal timescale of the GTWR model is daily for the annual scale data. The MAE, MRE, RMSE decrease by 13%, 15%, and 14%, respectively, when It is seen in Figure 10 and Table 3 that the optimal timescale of the GTWR model is daily for the annual scale data. The MAE, MRE, RMSE decrease by 13%, 15%, and 14%, respectively, when choosing the GTWR(Y) over the GTWR(D). In the results, the accuracy error of GTWR(E) can be reduced by about 0.2%, compared to GTWR(D). When the GTWRK is compared to the GWR model, MAE, MRE, and RMSE decrease by 3%, 10%, and 5%, respectively.
choosing the GTWR(Y) over the GTWR(D). In the results, the accuracy error of GTWR(E) can be reduced by about 0.2%, compared to GTWR(D). When the GTWRK is compared to the GWR model, MAE, MRE, and RMSE decrease by 3%, 10%, and 5%, respectively.

Discussion
The performances of GWR and GTWR models are relatively similar at the annual and monthly scales according to the reductions in average MAE, MRE, and RMSE. However the application of the GTWR model is not effective with respect to all the indicators, which reflects the extreme uncertainties of rainfall in time.
It is preferable to use a daily time scale to calculate spatial weights. Precipitation is formed by precipitation processes, and the duration of each process is usually several days. Therefore, as the timescale goes from yearly to daily the fine temporal details of precipitation gradually become prominent. Figure 11 shows the precipitation process in August 2008. The average monthly precipitation stands for the rainfall of the whole month. Consequently, the performance of the GTWR(D) model with a daily time scale is much better than the GTWR(M) and GTWR(Y) models' at the monthly (Table 2) and annual (Table 3) scales. The periodicity of precipitation has some impact on improving the accuracy of interpolation. As shown in Figure 6 shows the GTWR(C) model performs better than GTWR(M). However, the frequency and amplitude of each precipitation cycle are very different, as shown in Figure 3 and Figure 11. The calculation of the periodic function needs further study.
The introduction of kriging is reasonable in the improvement of interpolation accuracy. Table 4 shows the normality test of residuals between the results of the GTWR model and the actual precipitation. The residuals of most months fitted normal distribution (significance > 0.05), while other months, such as May, June, September, and December, fitted approximate normal distribution (kurtosis < 10 and skewness < 3). Compared with the GTWR model, the GTWRK model performed well in terms of average MAE, MRE, and RMSE (Tables 2 and 3). The GTWRK model produces a more accurate spatio-temporal precipitation. The GTWRK model's performance varies performs through the seasons. This work represent summer as June, July, and August. It seen in Table 4, the residuals in these months all fitted the approximately normal distribution, which affects the accuracy of the GTWRK model to some extent.

Discussion
The performances of GWR and GTWR models are relatively similar at the annual and monthly scales according to the reductions in average MAE, MRE, and RMSE. However the application of the GTWR model is not effective with respect to all the indicators, which reflects the extreme uncertainties of rainfall in time.
It is preferable to use a daily time scale to calculate spatial weights. Precipitation is formed by precipitation processes, and the duration of each process is usually several days. Therefore, as the timescale goes from yearly to daily the fine temporal details of precipitation gradually become prominent. Figure 11 shows the precipitation process in August 2008. The average monthly precipitation stands for the rainfall of the whole month. Consequently, the performance of the GTWR(D) model with a daily time scale is much better than the GTWR(M) and GTWR(Y) models' at the monthly (Table 2) and annual (Table 3) scales.
The periodicity of precipitation has some impact on improving the accuracy of interpolation. As shown in Figure 6 shows the GTWR(C) model performs better than GTWR(M). However, the frequency and amplitude of each precipitation cycle are very different, as shown in Figures 3 and 11. The calculation of the periodic function needs further study.
The introduction of kriging is reasonable in the improvement of interpolation accuracy. Table 4 shows the normality test of residuals between the results of the GTWR model and the actual precipitation. The residuals of most months fitted normal distribution (significance > 0.05), while other months, such as May, June, September, and December, fitted approximate normal distribution (kurtosis < 10 and skewness < 3). Compared with the GTWR model, the GTWRK model performed well in terms of average MAE, MRE, and RMSE (Tables 2 and 3). The GTWRK model produces a more accurate spatio-temporal precipitation. The GTWRK model's performance varies performs through the seasons. This work represent summer as June, July, and August. It seen in Table 4, the residuals in these months all fitted the approximately normal distribution, which affects the accuracy of the GTWRK model to some extent.

Conclusions
In this work, a GTWRK model combined with the GTWR and kriging model was introduced to interpolate the spatial distribution of monthly and annual precipitation from 2006 through 2014 in Hubei Province. The main conclusions are as follows: (1) GTWRK obtains a better average interpolation accuracy, compared to the GWR model. In the comparison between the GTWRK and GWR, the MAE decreased from 101.64 to 98.27. Consequently, we conclude that it is an improvement to extend GTWR with kriging.
(2) The optimal timescale for interpolating precipitation data with the GTWR model is daily. The fitting accuracy is improved when the timescale is converted from yearly to daily. Compared with the GTWR(M) model, the average MAE, MRE, and RMSE of the monthly scale data decreased by

Conclusions
In this work, a GTWRK model combined with the GTWR and kriging model was introduced to interpolate the spatial distribution of monthly and annual precipitation from 2006 through 2014 in Hubei Province. The main conclusions are as follows: (1) GTWRK obtains a better average interpolation accuracy, compared to the GWR model.
In the comparison between the GTWRK and GWR, the MAE decreased from 101.64 to 98.27. Consequently, we conclude that it is an improvement to extend GTWR with kriging. (2) The optimal timescale for interpolating precipitation data with the GTWR model is daily.
The fitting accuracy is improved when the timescale is converted from yearly to daily. Compared with the GTWR(M) model, the average MAE, MRE, and RMSE of the monthly scale data decreased by 36%, 56%, and 35%, respectively, when using daily data. The same indices for the annual data reduced by 13%, 15%, and 14% when using daily data, respectively. (3) The temporal weight based on an exponential function improved the GTWR model at the monthly and annual data. It reduced the accuracy difference of the monthly scale between GTWR and GWR by about 3%. For the yearly scale data, the years with improved accuracy account for about 55%. Especially in 2008,2009,2010,2011, and 2013, the accuracy was improved significantly. Meanwhile, the GTWRK improves the accuracy as measured by the MAE, MRE, and RMSE by 3%, 10%, and 1%, respectively, of monthly precipitation prediction, and by 3%, 10%, and 5%, respectively, of annual precipitation predictions. (4) The proposed model could be applied to manage similar phenomena with a large historical dataset. Meanwhile, the GTWR model takes into account the spatial and temporal heterogeneity of precipitation and produces better estimates of the residuals.