Empirical vegetation indices derived from spectral reflectance data are widely used in remote sensing of the biosphere, as they represent robust proxies for canopy structure, leaf pigment content, and, subsequently, plant photosynthetic potential. Here, we generalize the broad family of commonly used vegetation indices by exploiting all higher-order relations between the spectral channels involved. This results in a higher sensitivity to vegetation biophysical and physiological parameters. The presented nonlinear generalization of the celebrated normalized difference vegetation index (NDVI) consistently improves accuracy in monitoring key parameters, such as leaf area index, gross primary productivity, and sun-induced chlorophyll fluorescence. Results suggest that the statistical approach maximally exploits the spectral information and addresses long-standing problems in satellite Earth Observation of the terrestrial biosphere. The nonlinear NDVI will allow more accurate measures of terrestrial carbon source/sink dynamics and potentials for stabilizing atmospheric CO2 and mitigating global climate change.