## 1 Introduction  
Water is integral to maintaining the normal functioning of terrestrial ecosystems [1]. Within vegetative ecosystems, the hydrological cycle exerts a direct influence on plant physiological and ecological processes by modulating interactions among the atmosphere, soil, and vegetation components [2; 3]. Specifically, water deficits inhibit vegetation growth, reduce transpiration rates, and diminish photosynthetic efficiency [4], thereby compromising the capacity for carbon sequestration. Gross primary productivity (GPP), which quantifies the amount of carbon dioxide assimilated by vegetation through photosynthesis, serves as a vital metric for assessing vegetation carbon sinks [5]. Understanding the spatiotemporal variability of GPP is essential for elucidating terrestrial carbon cycling dynamics. Variations in terrestrial carbon sinks across the Northern Hemisphere are largely attributed to climate change [6]. Adequate temperature and water availability constitute fundamental conditions for vegetation growth. Through photosynthesis, vegetation absorbs atmospheric carbon dioxide while simultaneously utilizing water via transpiration [7; 8]. In the context of global climate change, temperatures have exhibited a pronounced upward trend [9], summer surface soil moisture (RSM) has shown a declining pattern [10], and precipitation displays considerable spatial heterogeneity [11]. Alterations in precipitation patterns significantly influence vegetation carbon uptake [12], particularly in arid and semi-arid areas where water availability is a limiting factor [13]. GPP is influenced by a range of environmental factors, including temperature, solar radiation, RSM, precipitation (PRE), and soil characteristics [14]. The mechanisms governing GPP regulation differ among various ecosystems. In arid and semi-arid ecosystems, despite their sparse vegetation cover and relatively low annual productivity, research has highlighted their significant role in the global carbon cycle [15; 16]. This regulatory function is largely mediated by the pronounced effect of soil moisture on carbon sink capacity [17]. Notably, in these dry regions, RSM imposes a more substantial constraint on vegetation productivity than temperature [18]. This limitation manifests primarily in two ways: first, as a critical factor for plant photosynthesis, increased soil moisture markedly enhances GPP [19]; second, by influencing ecosystem water use efficiency--defined as the ratio of GPP to evapotranspiration (E)--RSM governs its spatial distribution patterns [17; 20]. Within the arid and semi-arid areas of China, vegetation productivity displays distinct spatiotemporal variability. [PERSON]'s research identified an asymmetric pattern in vegetation productivity across this area, with this asymmetry intensifying over time. RSM exerts a more pronounced effect than PRE on the spatiotemporal characteristics of this asymmetry [21]. Moreover, RSM significantly impacts both the mean values and variability of long-term net carbon and water fluxes. The dynamics of RSM are crucial in mediating the trade-off between carbon assimilation and water resource availability, while also modulating fluctuations in terrestrial PRE minus E [22]. Temporally, the most substantial influences on GPP dynamics occur during summer--the peak GPP period--and spring [23]. The association between RSM and GPP is particularly robust, as evidenced by several key observations: precipitation primarily affects GPP through its modulation of RSM [23]; interannual variations in RSM and GPP exhibit strong coherence [24]; and deficits in RSM result in disproportionately large declines in GPP [24]. This seasonal response pattern underscores the importance of RSM variability as a critical factor for elucidating the coupling mechanisms between carbon and water cycles in arid and semi-arid ecosystems.