Model-based analysis of the relationship between endmember variability and temporal unmixing performance

Temporal unmixing, an extension of traditional spectral unmixing in a multi-temporal context, leverages endmembers defined by their temporal signatures to decompose mixed pixel responses into fractional cover. Although numerous methods that account for endmember variability have been successfully applied, there is a notable scarcity of theoretical frameworks to quantitatively elucidate how endmember variability induced by various spectral-temporal feature combinations impacts unmixing accuracy. The primary objective of this study is to formulate and analyze the quantitative relationship between endmember variability and temporal unmixing performance. To this end, we developed corresponding endmember variability measures (EMvar) and unmixing error measures at both the bi-class confusion level and the one-class aggregation level. We comprehensively investigated their relationship by constructing univariate linear and exponential models alongside a bivariate nonlinear model that integrates dataset dimensionality as an explanatory factor. Evaluations were conducted using extensive real and synthetic multi-temporal Sentinel-2 datasets covering Spartina alterniflora-invaded mangrove forest. Results demonstrate a robust correlation between EMvar and unmixing error measures at both levels (0.4 ≤ |r| ≤ 0.8), and this correlation is notably more pronounced at the one-class aggregation level. Beyond EMvar, dataset dimensionality played a pivotal role in explaining unmixing accuracy. While linear models sufficed for datasets of fixed dimensionality (0.3 ≤ R2 ≤ 0.6), the exponential model provided a superior fit across varying dimensionalities (0.3 ≤ R2 ≤ 0.7). Significantly, the bivariate model outperformed the univariate counterparts and achieved high predictive accuracy (R2 ≈ 0.8). These results provide the first quantitative evidence that EMvar is a direct and predictable driver of unmixing accuracy. Consequently, our EMvar-unmixing error models establish a practical, predictive framework that shifts temporal unmixing from an empirical endeavor to a model-driven optimization process. This framework enables the a priori identification of optimal spectral-temporal features, paving the way for more efficient and accurate sub-pixel land cover monitoring.