## 4 Discussion
In past studies, \(\epsilon\) has been assumed constant for all vegetation [56], fixed for each vegetation type (varying among different plant communities) [69; 73] or variable depending both on vegetation type and stress linked to environmental factors [24]. As has been discussed elsewhere [74], we would not normally expect very large variations in efficiency (\(\epsilon\)) over the season in grassland ecosystems. For this largely deciduous ecosystem, CO\({}_{2}\) fluxes can be reasonably approximated by \(f\)APAR\({}_{\text{green}}\) (estimated by NDVI alone) or by APAR\({}_{\text{green}}\), allowing us to assume a constant efficiency (\(\epsilon\)) over much of the growing season. This supports the hypothesis that, in these annual ecosystems, controls on ecosystem fluxes are largely related to canopy structure, with relatively little physiological control evident [35]. These findings are in agreement with a broad range of studies that note strong relationships between CO\({}_{2}\) fluxes and green canopy structure, expressed as APAR\({}_{\text{green}}\)), LAI, or total canopy chlorophyll [62; 75; 59; 37]. In such cases, modeling efforts should focus primarily on accurate estimation of the APAR\({}_{\text{green}}\) term in the LUE model. We note that not all ecosystems lend themselves to such simple analyses. In many cases, a simple APAR\({}_{\text{green}}\)-based model based may be insufficient, particularly for evergreen ecosystems where dynamic light-use efficiency can be an important determinant of CO\({}_{2}\) fluxes and may have to be included in the model [35; 74; 36]. Additionally, the exact definition and parameterization of the LUE model may itself affect observations regarding light-use efficiency. Many studies define LUE as the flux rate divided by PAR irradiance (incident PPFD) [71; 77], causing the efficiency term to be confounded with green canopy structure. In deciduous canopies, this definition of the model terms can lead to large seasonal variations in _apparent_ efficiency that may actually be caused by variations in green canopy structure (the APARgreen term of our model), not efficiency as defined in our analysis. As introduced, we defined APAR based on radiation _absorbed_ by _green_ canopies (APARgreen), not incident radiation or total absorbed radiation. This distinction and its consequences has been discussed in more detail elsewhere [34]. These differences in model definition and formulation make it difficult to directly compare results across sites and ecosystems, hindering meta-analyses, and confounding larger conclusions beyond the seemingly disparate results of different studies. Consequently, we recommend further attention be given to model formulation with the goal of a standardized definition and parameterization of model terms. Despite the generally good fit between a simple APARgreen-based model and measured CO2 fluxes, some unexplained features remain. In our study, we found that the fit between CO2 fluxes and both NDVI and APARgreen presented some hysteresis between the first and second halves of the growing season, a finding that agrees with other recent studies at similar annual-dominated sites [61; 78]. Using different equations for these two growth phases considerably improved the flux estimates over the use of a single equation. Such hysteresis could have several causes. It is likely that ecosystem carbon balance between photosynthesis and respiration shifts over the course of the season, contributing to the hysteresis in APARgreen and measured fluxes, and this could also cause a seasonal shift in the APARgreen-flux patterns that could be interpreted as a change in light-use efficiency [79]. In grasslands, the dead biomass that partially hides new green biomass in the early stage, which is the carry over effect of the previous season's growth, along with the gradually declining chlorophyll and nitrogen content of the leaves during the senescence phase, can also affect the optical-flux relationship.