## 1 Introduction  
The derivation of remotely-sensed vegetation indices is based on absorbance of red light and reflectance of near infrared light by chlorophylls mostly lying in plant leaves [1]. Vegetation indices have been widely used as efficient tools to monitor spatiotemporally and consistently the dynamics of living leaf biomass, leaf area index, land surface phenology, and ecosystem structures and functions such as carbon sequestration from local to global scales [2; 3; 4; 5]. As an important recent finding using vegetation indices, about one-third of the global vegetated land has been getting greener since 2000, and China and India are leading the greening mostly due to land-use management such as afforestation and intensified agriculture [6]. The dynamics of vegetation greenness have been reported to be driven by climate change, land-use/cover change, elevated atmospheric CO\({}_{2}\) concentration, and altered soil nutrients such as those through nitrogen deposition [2; 7]. Climate affects vegetation greenness by regulating water availability, carbon assimilation, partition of assimilated products to leaf, and onset and ending of growing season. In the long-term, vegetation greenness is also affected by climate-induced land-cover changes such as woody plant encroachment to grassland. Temperate deciduous forests and grasslands feature the strongest seasonality of greenness with strong climate modulation than other vegetation types. Temperate forests and grasslands occupy 10.4 and 15.0 \(\times\) 10\({}^{6}\) km\({}^{2}\) land, respectively, represent 19% of global ice-free land, share an essential portion of global ecosystem functions, and provide important ecosystem services, such as timber and forage [8]. The net primary production of temperate forests and grasslands is 7.6 and 5.3 Pg C yr\({}^{-1}\), respectively, accounting for 22% of the global ecosystems. As the amount of leaf area of vegetation is mostly determined by available soil moisture [8], the dynamics of greenness of temperate forests and grasslands should be primarily controlled by the balance of moisture input to and output from the ecosystems. Ample soil moisture promotes primary production and allometric leaf growth of temperate forests and grasslands by enhancing the plant physiological activities and the soil nutrient availability [9; 10; 11; 12]. Thus, effective precipitation is important in driving vegetation greenness of temperate forests and grasslands. In addition, rainfall during summer/autumn tends to delay autumn dormancy in temperate deciduous forests and to extend the growing season, and winter and spring precipitation allows the well-developed root systems of woody vegetation to hold enough soil moisture for plant early leaf-out in spring, thus to enhance the greenness of temperate vegetation [13; 14; 15]. Temperature is also crucial for green leaf production of temperate vegetation and regulates the physiological activities of plant and soil microbes [10; 16]. Compared to rainfall effects on greenness, temperature is more determinative for the length of growing season [14; 17]. Additionally, moisture output from an ecosystem via evapotranspiration is sensitive to temperature [18]. Global warming and its diurnal and seasonal patterns affect photosynthesis and respiration differently [19; 20; 21]. Nighttime warming is much stronger than daytime warming [22] and the asymmetric warming greatly affects vegetation leaf growth and vegetation greenness. Furthermore, nighttime warming in early spring or late autumn reduced the chances of frost damage and/or accelerated the thermal-time accumulation for leaf-out, and therefore advanced the start but delayed the end of growing season, both of which render longer growing seasons [23; 24; 25]. Modeling of vegetation greenness with various statistical and data science approaches aims to answer the question of how climate and other biophysical variables interact to determine the spatiotemporal dynamics of vegetation greenness and related ecosystem functions.