## 1 Introduction
Multifractal theory presents a powerful tool for quantifying scale-invariant structures in complex natural systems, and is particularly applicable to phenomena such as precipitation and vegetation growth, which are characterized by intermittency and aggregation [21; 22]. These factors directly affect the WUE, which usually exhibits high variability at different temporal and spatial scales. Multifractal analysis reveals hidden scale invariance patterns and hierarchical structures in WUE data that are difficult to capture using traditional methods, especially in regions with significant seasonal WUE variations [23; 24]. Multifractals have been widely applied in the analysis of rainfall, precipitation indices, soil types, drought, and other areas, and offer a better description of the intensity distribution and frequency of extreme events in rainfall time series [25]. The approach can reveal multiscale behaviors and intrinsic patterns in rainfall processes, rendering it a powerful tool for assessing nonlinear characteristics in the time series of complex regions [26; 27; 28]. Fractal models have also been applied in studying the spatial distribution of soil properties [29]. The standardized precipitation index (SPI) has been proven to exhibit multifractality, and has been used in precipitation anomaly detection and complexity analysis [30]. Multifractal analysis has significant advantages for the anomaly detection and complexity analysis of time series [31], and WUE fluctuates over time and shows strong spatial heterogeneity. Precipitation and other factors related to WUE that exhibit multifractal characteristics have been widely studied in various regions [28; 32; 33]; however, the multifractal theory has not as yet been applied to WUE. Multifractal analysis has certain advantages for the analysis of changes on both macroscales and microscales, with the multifractal theory capturing the complexity of phenomena across multiple scales [34]. The multifractal spectrum can identify variation patterns and self-similarity at different scales, providing insights for understanding and analyzing the spatiotemporal variability of WUE [35]. Traditional analysis methods often struggle to simultaneously account for complex interactions on different scales, and multifractals can systematically compare and analyze differences between regions through detailed fractal spectra, revealing both microscale details and overall macroscale characteristics [35; 36]. Therefore, in this study, multifractal theory was combined with spatiotemporal change analysis methods to analyze the characteristics of WUE and validate the feasibility of using multifractal theory [37], not only deepening the understanding of WUE variation with climate change, but also providing a scientific basis for future ecosystem management and climate change adaptation strategies [38; 39]. This study aims to analyze the spatiotemporal variation characteristics and factors influencing WUE across different temporal and spatial scales using MODIS data from 2001 to 2022, exploring the effectiveness of multifractal theory in analyzing WUE changes at various spatiotemporal scales. Most current research focuses on WUE changes on temporal scales such as single years and months or spatial scales such as the vegetation type, and a detailed analysis of WUE changes across different scales is lacking. Based on the spatiotemporal scale variation analysis of WUE, multifractal theory is thus introduced and the temporal scales meticulously divided into years, seasons, and months, and the spatial scales into terrain and soil types. The capability of multifractal theory is analyzed and evaluated in the context of spatiotemporal WUE changes.