## 1 Introduction  
Climate change impacts significantly on the structure and function of terrestrial ecosystems [1, 2]. Against this background, interactions between the water and carbon cycles have received increasing attention. Vegetation water use efficiency (WUE), defined as the ratio of gross primary productivity (GPP) and evapotranspiration (ET), is one of the most direct indicators of water-carbon coupling [3]. Studying the spatial and temporal changes in WUE helps us understand the coupling relationship between water and carbon in vegetation and further deepens our understanding of the impact of climate change on vegetation. The feedback mechanism between climate change and the water-carbon cycle of ecosystems is extremely intricate. Traditional biomass growth surveys and the study of eco-hydrological processes are not suitable for short-term analysis because of consuming time and lack of precision [3]. New techniques, such as the eddy covariance method, can improve these studies. The eddy covariance method, a micro-meteorological technique, could directly measure the transport of carbon and water on a time scale of seconds [4]. Nevertheless, the observations are limited to the adjacent region and are unsuitable for macroscopic monitoring [5]. Remote sensing (RS) offers an excellent technological tool for WUE research. Extensive research has shown that the quantitative retrieval of WUE based on RS data allows for the continuous monitoring of WUE over a vast area [6; 7; 8]. These studies focused on the RS estimation of WUE [7; 9], the spatial distribution pattern and change tracking [10; 11], the response of WUE to climate change and its sensitivity [12; 13], and the effect of land cover change on WUE [14]. Compared with other high temporal resolution satellite data, such as Advanced Very High Resolution Radiometer (AVHRR), SPOT VEGETATION, and HuanJing-1B, Moderate Resolution Imaging Spectroradiometer (MODIS) provided extensive quantitative RS products. In addition, Landsat imagery with high spatial resolution suffered from a lack of data due to the long acquisition cycle and cloud coverage compared to MODIS datasets [15], making it difficult to monitor WUE changes accurately. MODIS datasets are currently the most suitable data for monitoring WUE [5; 11] and understanding vegetation responses to climate change [11]. Given this, we utilized MODIS GPP and ET products to examine WUE variations. Previous studies have focused on the spatio-temporal trend of WUE and have shown that the global WUE exhibited a general upward trend but with significant spatial heterogeneity [11; 16; 17]. Although a general linear regression model can estimate the average growth rate, cyclical variation is retained in the residuals, thus leading to a bias of model uncertainty [18]. The process of WUE change over time cannot be clearly shown. As illustrated in Figure 1, the inter-annual trend may be influenced by seasonal oscillations, which obscure long-term trends. The STL decomposition method is a seasonal-trend decomposition procedure that allows fast computation due to a simple design that comprises a sequence of loss smoothers and robustly estimates the trend and seasonal components that are not distorted by outliers [19]. Thus, this method has been used successfully to analyze time series for leaf area index (LAI) [20], normalized difference vegetation index (NDVI) [18], and chlorophyll-a [21], but not for the WUE time series before this study. Recently, an increasing number of studies have analyzed the relationship between bio-climatic conditions and WUE temporal trend [13]. For example, [PERSON] et al. [22] used multiple linear regression analysis to examine the response of regionally averaged WUE to temperature, precipitation, and LAI in China. However, the WUE change is complex and driven by various factors. The relationship between WUE variation and each factor cannot be described by a simple linear equation. The influence of various factors and their interactions on the spatial variation of WUE is still uncertain.