**1. 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 interpolation and local interpolation [9–11]. Spatial–temporal interpolation includes two categories: subtraction and extension [12]. Spatial–temporal interpolation methods for spatial–temporal irregular dataset interpolation and missing data patching include the spatial–temporal inverse distance weighting method, the spatial–temporal kriging method, and collaborative spatial-temporal kriging, among the main ones [13–15]. It has become a common practice to explore the distribution of precipitation employing spatial statistical analysis to cope with the spatial-temporal non-smoothness of precipitation. The geographically weighted regression (GWR) model was proposed for the study of spatial relations and spatial correlation, based on the common linear regression model by Fotheringham et al. [16]. The GWR model prescribes parameter estimation based on the location function expressing the non-stationary spatial features of precipitation. The regression coefficients in the GWR model capture the locational attribute. They can, therefore, take into account the influence of spatial heterogeneity, thus significantly improving the ability to analyze the variation in spatiotemporal characteristics of precipitation. This means the GWR model has attracted wide attention regarding quantitative precipitation estimation, as well as other spatial variables [17,18]. However, the GWR model only considers the spatial characteristics of precipitation data, while ignoring time characteristics of precipitation. The geographically and temporally weighted regression (GTWR) model was proposed in 2010 by Huang et al., and incorporates the time dimension into the model formulation [19]. On the one hand, the GTWR model has the basic characteristics of a general variable coefficient model and exhibits the high fitting skill of the local regression model, which captures the differences in spatial position and takes into account the spatial heterogeneity of precipitation. On the other hand, the model adds the time series traits, synthesizes the time dimension distribution information of the sample points, and embeds the spatial–temporal characteristics into the model [20].

GTWR performs wells in predicting spatial–temporal heterogeneity, and many studies in a variety of fields of science have proven the effectiveness of the GTWR model in spatial economic analysis, atmospheric sciences, population analysis, and other social and economic fields. The GTWR model was applied to model housing price data in London by Fotheringham et al. [21], which validated the proposed method and its superiority over the traditional GWR method while highlighting the importance of time explicit spatial modeling. The GTWR model was applied to assess the spatial–temporal differences in the influence of each driving factor on the scale of carbon emissions and the intensity of carbon emissions in China by Xiao et al. [22]. Liu et al. studied housing price data and related factors in Beijing from 1980 to 2016 [23], to propose a calculation method for travel distance, applying the GWR. The GTWR model was employed to study the influencing factors on housing prices, and it was concluded that the GTWR model is suitable for identifying effective real estate management policies. The fire record data from 2002 to 2010 in Hefei, China, was reviewed by Song et al. [24], using the linear model (LM), GWR, and GTWR to model urban fire risk. The latter authors concluded that road density and commercial spatial distribution have the most significant influence on fire risk. GTWR can detect small changes in variable spatial–temporal heterogeneity of diverse phenomena. The performance of the GTWR model was verified with particulate matter ≤2.5 μm (PM2.5) concentration data in the Xuzhou area, China, and compared with ordinary least squares (OLS), GWR, and time-weighted regression (TWR) models by Bai et al. [25]. The results indicate that the regression coefficient of the GTWR model was the highest, and its interpolation skill was optimal. The GTWR model was applied to estimate the ground concentration of nitrogen dioxide (NO2) in central China by Qin et al. [26], and cross-validation results proved that the fitting results of the GTWR model were better than those of the OLS, GWR, and TWR models. Five models, including GTWR, were implemented to analyze the relationships between PM2.5 and other criteria of air pollutants by Wei et al. [27], and GTWR showed great advantages over the other three models in terms of higher model R<sup>2</sup> and more desirable model residuals, and only slightly less than TWR.

Precipitation has a high causal correlation both in space and time. Therefore, it is intuitively logical to use the GTWR model to fit precipitation data. At present, the use of spatial statistical analysis to fit precipitation interpolation is mainly represented by the GWR model. Brunsdon et al. [28] reported 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.

#### **2. Materials and Methods**
