The typical relationship between the rate of photosynthesis (and, in turn, CO_{2} uptake) by C_{3} and C_{4} species and the incident PAR is often described as a nonrectangular hyperbola (Lambers *et al*., 2006). As light intensity increases, photosynthesis levels off towards the asymptote, especially in C_{3} plants. The inversion methodology used in this paper is based on the concept that at a light intensity below the saturation point, there is a near linear relationship between incidence PAR and CO_{2} uptake. Therefore, the implementation of the inversion approach requires a maximum light limit to be imposed on photosynthesis. This limit can be adjusted to suit the climatology of the area under investigation and the photosynthetic capacity of plants in that ecosystem. The NEE (μmol m^{−2} s^{−1}) estimated using the eddy covariance techniques can be represented as follows (Hanan *et al*., 2002):

- (Eqn 1)

where *α*_{e} (mol mol^{−1}) represents the ecosystem LUE (the number of moles of CO_{2} fixed per mole of PAR incident on the canopy) and *R*_{eco} (μmol m^{−2} s^{−1}) is the whole-ecosystem respiration. For PAR below the light saturation point, (Eqn 1) can be used to estimate *R*_{eco} and *α*_{e} by regression of measured CO_{2} flux against incident PAR. In this regression, the slope of the relationship has been shown to represent the ecosystem LUE and the intercept (PAR = 0) represents the whole-ecosystem respiration (Suyker & Verma, 2001; Hanan *et al*., 2002). The ecosystem LUE term (*α*_{e}) can further be divided into a physiological component (*α*, the intrinsic quantum yield) that can be estimated and an unknown structural component which represents the efficiency of absorption by photosynthetic components of canopy (i.e. the FAPAR_{ps}; Hanan *et al*., 2002). The relationship in (Eqn 1) can be rewritten as:

- (Eqn 2)

where *α*_{a} (mol mol^{−1}) is the ‘actual quantum yield’ (the number of moles of CO_{2} fixed per mole of PAR absorbed by photosynthetic elements in the canopy). The first part of the equation (i.e. PAR × FAPAR_{ps} × *α*_{a}) represents gross primary productivity (GPP). In C_{4} plants where Rubisco oxidation (i.e. addition of oxygen to the Rubisco enzyme through photorespiration, hence reducing photosynthesis efficiency) is minimal, the actual quantum yield is similar to the intrinsic yield of photosynthesis. In the C_{3} plants where Rubisco oxidation occurs, the actual quantum yield depends on temperature and leaf internal CO_{2} concentrations (Ehleringer *et al*., 1997; Hanan *et al*., 2002). Therefore, in mixed canopies, the NEE can be written as:

- (Eqn 3)

where PC_{3} is the proportion of C_{3} species in the canopy, *α*_{3} is the intrinsic quantum yield of C_{3} species, Ψ_{e} (unitless) is a function of temperature and leaf internal CO_{2} concentration in C_{3} species, and *α*_{4} is the intrinsic quantum yield of C_{4} species. In both the C_{3} and C_{4} plants, the quantum yield is often influenced by VPD, with the effect being more pronounced during the hours around midday (Farquhar & von Caemmerer, 1982; Tezara *et al*., 1999). The mechanism for this influence is not agreed on, but it could be because evaporation at high VPD causes water stress (Shirke & Pathre, 2004). The water stress could affect several components of photosynthetic metabolism (e.g. electron transport, ATP synthesis, light dissipation, Rubisco, and carbohydrate metabolism), hence reducing photosynthesis efficiency (Lawlor & Cornic, 2002). The influence of VPD on quantum yield can be represented as follows:

- (Eqn 4)

where VPD (KPa) is the instantaneous (e.g. hourly) value. This formulation is based on measurements of stomatal conductance in tropical, temperate and boreal ecosystems on crops, grasses, shrubs deciduous forests and evergreen forests (Aber & Federer, 1992; Hollinger *et al*., 1994; Leuning, 1995). It has been shown that high VPD often only occurs for a short period around the midday hours and hence its stress effects may not persist during the rest of the day (Monteith, 1995; Shirke & Pathre, 2004). As such, the influence of VPD on the quantum yield was restricted to the hours around midday (10:00–14:00 h) and the mean VPD data at these times of the day was used in parameterizing the ƒ_{D} term. To include effects of VPD, (Eqn 3) is rewritten as follows:

- (Eqn 5)

The maximum intrinsic quantum yield terms for *α*_{3} (i.e. 0.08 mol mol^{−1}) and *α*_{4} (i.e. 0.06 mol mol^{−1}) were used to parameterize *α* in (Eqn 6) (Collatz *et al*., 1991, 1992; Hanan *et al*., 2002). Laboratory measurements and kinetic analyses using chloroplast suspensions have shown that these values accurately describe the maximum intrinsic quantum yield of C_{3} and C_{4} plants, respectively (Collatz *et al*., 1991, 1992). The Ψ_{e} term in Eqns 3, 5 and 6 describes the influence of leaf temperature (*T*_{1}) and leaf chloroplast CO_{2} partial pressure (*C*_{i}) on actual quantum yield in C_{3} plants. The Ψ_{e} term is derived as a function of the response of C_{3} photosynthesis to CO_{2} partial pressure and compensation point, and the leaf temperature as follows (Ehleringer & Bjorkman, 1977; Collatz *et al*., 1991; Hanan *et al*., 1998):

- (Eqn 7)