Estimating ecosystem maximum light use efficiency based on the water use efficiency principle

Light use efficiency (LUE) defines the vegetation efficiency of converting radiative energy into biochemical energy through photosynthesis. Estimating the maximum LUE (ε max) is critical yet challenging for quantifying gross primary production (GPP) using LUE-based models. This study describes an analytical method for estimating ε max based on water use efficiency (WUE) as determined by plant water use and carbon gain. Unlike other complex parameterization schemes, this WUE-based method is simple and requires four variables relatively easy to acquire. The WUE-based ε max estimates compare favorably well with values based on traditional curve fitting method and that reported in the literature, and clearly distinguished ε max between C3 (1.48 ± 0.33 g C MJ−1 PAR ) and C4 (2.63 ± 0.21 g C MJ−1 PAR ) dominated ecosystems. The range in ε max estimates was narrow across different years and sites within a biome. The WUE-based ε max estimate is theoretically constrained by vegetation water use and can be directly incorporated into LUE models for GPP estimation across ecosystems.


Introduction
Vegetation converts solar energy into biochemical energy stored in carbohydrates through photosynthesis. At the ecosystem level, the efficiency of this process can be quantified as the ratio of gross primary production (GPP) to photosynthetically active radiation (PAR), commonly known as light use efficiency (LUE) (Monteith 1972(Monteith , 1977. The theoretical maximum value of LUE (ε max ) under optimal state (Bolton and Hall 1991) is usually downregulated due to biophysical and environmental constraints to obtain the actual amount of carbon assimilation, i.e. ε max = GPP/(PAR·f APAR ·f s ), where f APAR is the fraction of absorbed PAR determined by light conditions and canopy properties, and f s is the environmental stress factor representing limits induced by temperature, availability of resources, e.g. water and nutrient (Monteith 1972). This is a practical and widely adopted conceptual framework for GPP estimation across spatiotemporal scales (Potter et al 1993, Landsberg and Waring 1997, Veroustraete et al 2002, Running et al 2004, Xiao et al 2004, Yuan et al 2007, Yang et al 2013. The accuracy of such GPP models, however, relies heavily on the parameterization of ε max (Cramer et al 1999, Ruimy et al 1999, Wei et al 2017a.
Previously studies have shown that ε max values can vary by a factor of five based on experimental evidence (Choudhury 2000, Gitelson andGamon 2015). Early studies conducted by Monteith (1977Monteith ( , 1986 suggested an average ε max of about 1.5 g C MJ −1 for C 3 plants and 2.4 g C MJ −1 for C 4 plants. However, others found ε max values vary greatly from 0.24 to 4.82 g C MJ −1 for various plant types (Prince 1991). Consequently, discussions have centered on whether ε max is a consistent or variable parameter across species and biomes with different physical environments (Field 1991, Prince 1991, Ruimy et al 1994, Medlyn 1998, Kergoat et al 2008, Gitelson et al 2018. While consistent ε max is often interpreted as a result of an evolutionary plant response to resource availability (Monteith 1977, Field 1991, Gitelson et al 2018, evidence on the contrary can be explained by biotic and abiotic factors attributed to plant or ecosystem types (Gallagher and Biscoe 1978, Prince 1991, Landsberg and Waring 1997, Sinclair and Muchow 1999, Albrizio and Steduto 2005, light quality (Choudhury 2000, water (Passioura 1982, Shi et al 2014, nitrogen Horie 1989, Kergoat et al 2008) and atmospheric CO 2 concentration (Dewar et al 1998, Norby et al 2003 conditions, from both experimental and theoretical perspectives (Ruimy et al 1994, Medlyn 1998. It is important to understand how ε max varies across different species (i.e. C 3 and C 4 ) and biomes (e.g. forests to grasslands (GRAs)) to allow more robust estimation of GPP (Prince 1991, Gitelson and Gamon 2015, Wohlfahrt and Gu 2015.
Various methods have been used to estimate ε max and they fall into two broad categories: (a) fitting light response curves (LRCs) and (b) inverting the LUE model (i.e. ε max = GPP/(PAR·f APAR ·f s )). In the first category, ε max is a parameter based on the Michaelis-Menten LRC fitted to observed GPP and PAR data (Wang et al 2010, He et al 2014, Wei et al 2017a. The estimation is largely determined by the choice of either a linear or nonlinear model, with results varying by a factor of two (Ruimy et al 1995). The second is the most commonly used approach (Chen et al 2011, Sánchez et al 2015, Gitelson et al 2018, where ε max estimation relies on the parameterization of f APAR and f s for a given GPP and PAR using the LUE framework. Specifically, f APAR partitions available energy to vegetated and soil surfaces, and is usually derived from vegetation index (VI) (Mu et al 2007) and/or Beer's law using VI and light extinction coefficient (k PAR ) (Yuan et al 2010). f s is a multiplicative limiting factor varying from 0 to 1 to account for temperature, water (soil and atmospheric), and nutrient conditions. On an ecosystem scale, ε max can be calibrated against GPP using various schemes of f APAR and f s , often with (a) remotely sensed VI images (e.g. leaf area index (LAI), normalized difference vegetation index (NDVI) and enhanced vegetation index (EVI)) scaled to match flux site footprint (Li et al 2012, Yuan et al 2014 independently or in combination with constant k PAR (Zheng et al 2018), and with (b) f s as a multiplied limiting factor, based on water, temperature, and other limiting variables obtained in-situ or from remotely sensed data (Yuan et al 2007). Although this approach of employing VI and f s to estimate ε max is practical, main challenges remain to be overcome. First, there is a spatiotemporal discrepancy between the footprint of flux measurements (typically halfhourly, less than 1-3 km 2 ) (Chen et al 2012) and that of remotely sensed VIs (typically eight+ days, 10 1 -10 2 km 2 ) (Chen et al 2008, Fu et al 2014. Second, k PAR is not invariant, as it varies with canopy properties, seasonality, and vegetation type (Woodgate et al 2015, Zheng et al 2018. Therefore, f PAR can be biased when it is calculated from the remotely sensed VIs and/or constant k PAR , and may not accurately represent local vegetation profile. Third, the use of the multiplied limiting factor f s can be ambiguous. For example, soil moisture, vapor pressure deficit, and temperature are widely acknowledged as covariant variables, yet they are often multiplied as independent factors to formulate f s (Yang et al 2013). Additionally, the scale discrepancy similar to VIs also applies to f s when the limiting factors are obtained from remote sensing images. Thus, difficulties and uncertainties still remain in quantifying f as a combination of VIs (with/without k PAR ) and f s . Not surprisingly, a wide range of ε max values have been reported across species (e.g. C 3 to C 4 ) and biomes (e.g. forests to GRAs). This variability further introduces high uncertainties into model-derived GPP estimates, leading to difficulties for multi-model and multi-scale assessments (Ruimy et al 1999, Chen et al 2011, Yuan et al 2014, Zheng et al 2018. These challenges highlight the need for a better understanding of this parameter and a more robust method for ε max estimation. The objectives of this study were to develop a new method for estimating ε max that does not depend on the stress factor f, and to characterize the variability of ε max across different species and biomes. For the first objective, we derived and tested a simple method for estimating ε max from a coupled water and carbon perspective based on the water use efficiency (WUE) principle. For the second objective, we evaluated and characterized the variation of ε max at globally distributed eddy covariance flux sites for typical C 3 and C 4 species as well as major biome types. We also compared our WUE-based ε max estimates to that of traditional LRC fitting methods (supplementary, text S1 (available online at stacks.iop.org/ERL/16/104032/ mmedia)). Our results can be implemented into GPP models that are based on the LUE framework and will be useful to inform uncertainties in GPP simulations.

Derivation of LUE from the WUE principle
The intrinsic coupling of carbon uptake and water loss through vegetation stomata is commonly quantified by WUE, which is the ratio of actual CO 2 assimilation (A) to transpiration (T). For modeling purpose, A and T are usually assumed to reach their theoretical maximum, as potential assimilation (PA) and potential transpiration (PT), respectively when they are not subject to resources constraints (i.e. soil water availability) (Hanks 1974). The relationship between actual and PA and PT has been extensively used in biomass prediction (Ritchie 1983, Monteith 1986, Sinclair and Horie 1989, as often given in the following form (de Wit 1958, Hanks 1974, 1983, Stewart et al 1977, Monteith 1986) where A and PA share the unit g C m −2 d −1 , T and PT share the unit mm d −1 . Above empirical relationship implies the assumption that a fractional change in assimilation is associated with a corresponding fractional change in transpiration, due to simultaneous stomatal control on both carbon and water fluxes (Monteith 1988). By rearranging equation (1), we can obtain an explicit expression of WUE as follows In order to derive ε max from this equation, a straightforward method is to express PA by applying the LUE framework using ε max . Specifically, the PA rate under non-stressed condition for a certain canopy can be calculated as where PAR is photosynthetic active radiation (MJ m −2 d −1 ), ε max is maximum LUE (g C MJ −1 PAR), and f APAR is the fraction of PAR that is intercepted by the canopy. For simplicity and consistency, PAR here is defined as the proportion of incident shortwave radiation (R s ) that falls into the 400-700 nm waveband, which is approximated as 0.45 × R s throughout this study (Monteith 1972, Britton andDodd 1976). Additionally, for the PA/PT term in equation (2), we assumed that the denominator PT for a given canopy could be approximated from potential evapotranspiration (PET) by attributing total PET to the vegetated area (Impens and Lemeur 1969, Childs et al 1977, Brisson et al 1993, Guan and Wilson 2009. Therefore, we have where f Rn is the fraction of net radiation (R n ) absorbed by the canopy (detailed elaboration of equation (4) is referred to text S2, supplementary). We further assumed that the fraction of intercepted PAR approximates the fraction of intercepted R n over the same canopy, i.e. f APAR = f Rn (see text S3, supplementary). Therefore, we have By substituting equation (3) and equation (4a) to equation (2), we have: To obtain ε max , we simply rearranged equation (5) as Equation (6) shows that ε max is linearly related to actual assimilation and transpiration (or WUE, equation (6a)) and is dependent on radiation (PAR) and atmospheric water demand (PET). This equation does not require additional information such as canopy profile (VIs, k PAR ), and environmental stress factors (f s ), or other variables needed by the LUE framework to parameterize f at flux sites. Alternatively, T/PET can be interpreted as f (equation (6a)), which corresponds to the effective fraction of PAR that is ultimately converted to biochemical energy stored in A. At an ecosystem scale, the use of T/PET is advantageous as it avoids the difficulty and uncertainty of calculating f APAR based on k PAR (fitted parameter with ambiguous mechanism) and VIs (e.g. NDVI, EVI, LAI) as well as f s (e.g. temperature, water, nutrient) derived from observational and/or remote sensing techniques as illustrated in section 1. Among the four variables required in equation (6), PET and PAR are relatively easy to obtain from readily available meteorological measurements at flux sites. PAR can be calculated from R s observations. As for PET, we adopted the radiation-based Priestley-Taylor equation (Priestley and Taylor 1972), which is one of the most widely used methods to account for atmospheric water demand in hydrological and land surface modeling ( Additionally, since A can be reasonably approximated by GPP at an ecosystem level, the only variable required to apply equation (6) is transpiration (T). It is widely acknowledged that T is difficult to measure directly and its simulation exhibits high uncertainties and large discrepancies based on the choice of methods (Wang et Zhou et al 2018). In this study we adopted the common assumption that ET is a reasonable approximation of T under conditions of high vegetation coverage and negligible soil evaporation during rain-free periods. Consequently, equation (6) becomes As a result, ε max can be estimated from GPP, ET, PAR and PET. For practical purposes, ε max can be estimated as the slope of the linear regression between GPP·PET and ET·PAR. This method for ε max estimation is herein referred to as the WUE-based method in the following discussions.

FLUXNET data and VI
To test the WUE-based method proposed for determining ε max , meteorological and eddy covariance measurements at 52 flux sites (318 site-years of observation) were obtained from the FLUXNET database (http://fluxnet.fluxdata.org/). A brief description of the selected sites is presented in table S1 and figure S1. Half-hourly observations of latent heat flux (LE W m −2 ), R n (W m −2 ), R s (W m −2 ) and air temperature (T a • C) were used. Estimates of GPP derived from night time partitioning algorithm provided by FLUXNET were adopted as GPP observations in this study. PET was calculated from the Priestley-Taylor equation with the multiplication factor α PT set to 1.26 (Priestley and Taylor 1972). As stated earlier, PAR was taken as 45% of the total R s . The flux sites used in this study included seven plant functional types (PFTs) where high vegetation coverage was observed during the growing season, including evergreen broadleaf forest (EBF), deciduous broadleaf forest (DBF), evergreen needle leaf forest (ENF), GRA, woody savanna (WSA), wetland (WET) and cropland (CRO). Three crop sites with annual soybean (Glycine max L.) (C 3 ) and maize (Zea mays L.) (C 4 ) rotation were selected to represent C 3 and C 4 species in order to determine the effect of different photosynthetic pathways on ε max .
Original half-hourly data were filtered to retrieve reliable observations according to the standard quality control protocols (Reichstein et al 2005). Following criteria were used to further screening available data for ε max estimation: (a) only daytime observations with R s greater than 20 W m −2 were used; (b) negative entries of LE and GPP were eliminated; (c) data on rainy days (rainfall >0.5 mm) and the subsequent two days were excluded to focus only on transpiration (Beer et al 2009, Knauer et al 2018; (d) only growing season dates with high vegetation coverage present were used; (e) minimum of ten days of valid observations during each growing season were required for robust regression analysis. Growing season was defined according to Zhou et al (2014Zhou et al ( , 2016 as the days when daytime average GPP exceeded 10% of 95th percentile of daily GPP at a given site. High vegetation coverage was further ensured by high daily values of LAI in addition to GPP selection criterion. Observations during days when LAI exceeded a threshold of 2.0 were used to assure conditions where ET was a reasonable representation of T (Ritchie 1983, Zhou et al 2018. At each study site, LAI was determined using remotely sensed LAI (eight-day, 500 m resolution) from MODIS (MOD15A2 product) using observations within a 1 km × 1 km cell centered on the site coordinates (Gan et al 2018). Original eight day LAI time series were (a) filtered according to quality flags to select reliable entries, (b) smoothed using Savitzky-Golay filter, and (c) linearly interpolated to determine daily LAI (Zhang et al 2019). These daily LAI time series were then used to assist data screening during the growing season at each site.

Comparison of ε max estimates for C 3 and C 4 species
Long-term site-level estimate of ε max are illustrated in figures 1(a) and (b) for typical C 3 and C 4 species, respectively, at the site US-Ne2 (maize-soybean rotation, irrigated). Taking figure 1(a) as example, ε max of soybean was estimated using equation (7) as the linear regression slope between GPP·PET and ET·PAR (section 2.1), using data from the soybeangrown seasons over multiple years (section 2.2). The regression was forced to pass through the origin to be consistent with the fact that both photosynthesis and transpiration approach zero when stomata are closed. A strong linear correlation was found between GPP·PET and ET·PAR, with the coefficient of determination (R 2 ) being 0.96 (p < 0.001). The long-term ε max was then estimated as 1.47 g C MJ −1 PAR for soybean (C 3 ) at this site. To examine the applicability of the method for C 4 species, the same regression was performed during the maize-grown years as shown in figure 1(b). As expected, a clear difference was detected for the regression slope between soybean and maize. While exhibiting the same R 2 as soybean years, maize years presented a much higher ε max value of 2.53 g C MJ −1 PAR.
The WUE-based method also provided comparable estimates of ε max for C 3 and C 4 crops at two adjacent sites, namely US-Ne1 (continuous maize, irrigated) and US-Ne3 (maize-soybean rotation, rainfed), with strong linear correlation (R 2 ⩾ 0.95) and ε max estimated at 2.62 g C MJ −1 PAR for maize at US-Ne1 (irrigated), and 1.84 and 2.87 g C MJ −1 PAR for soybean and maize at US-Ne3 (rain-fed), respectively (table S1, supplementary). These estimates are reasonably close to that obtained at US-Ne2 shown in figure 1. In comparison with the long-term estimates, the robustness of the method was further examined on an annual scale, with ε max from each site-year calculated and summarized in table S2 (supplementary). For soybean, the annual average ε max across US-Ne1, US-Ne2 and US-Ne3 was about 1.6 g C MJ −1 PAR, with a standard deviation (SD) of 0.21 g C MJ −1 PAR and a coefficient of variation (CV) of about 13% across six site-years. For maize, the annual mean ε max was about 2.63 g C MJ −1 PAR, with a SD of 0.21 g C MJ −1 PAR and a CV of about 8% across 15 site-years. Annual average ε max for maize was significantly higher than that of soybean, which agrees well with the longterm value estimated at US-Ne2. The greater ε max estimates for C 4 species obtained here corresponds well with its genetically determined higher photosynthetic capacity compared with C 3 species, which demonstrates the applicability of the WUE-based method for distinguishing C 3 and C 4 dominated ecosystems.

Variation of ε max across different biomes
Site-specific ε max was also calculated using all available data during growing season at each of the 52 flux sites (table S1). Summary of cross-biome ε max estimates are presented in figure 2. ε max exhibited considerable variation within and across biome types. Among all natural ecosystems, forests (DBF, EBF, and ENF) presented a slightly higher photosynthetic capacity (greater ε max ) compared to nonforests (GRA, WET, and WSA). Broadleaf forests (DBF and EBF, 11 sites) with high vegetation coverage during the growing season had similar ε max estimates that fell in a narrow range between 1.50 and 1.58 g C MJ −1 PAR. These values were in good agreement with the C 3 crop (1.66 g C MJ −1 PAR). ENF presented the largest variation, with ε max ranging from about 0.9 to 2.4 g C MJ −1 PAR, and the average of 1.58 g C MJ −1 PAR was comparable to that of the C 3 crop. Non-forest GRA and WSA had much lower ε max (less than 1.2 g C MJ −1 PAR on average). In addition, WETs were the least efficient in terms of light use for photosynthesis with the lowest ε max of only about 0.83 g C MJ −1 PAR. For the four CRO sites under agricultural management (C 3 or C 4 species unspecified), ε max varied from about 1.3 to 2.0 g C MJ −1 PAR, with an average of about 1.62 g C MJ −1 PAR. All natural ecosystems dominated by C 3 vegetation had much lower ε max values than the C 4 crop. Although ENF and CRO had a few large values that exceeded 2.0 g C MJ −1 PAR, most C 3 dominant ecosystems showed ε max typically lower than 2.0 g C MJ −1 PAR. On the contrary, ε max for the C 4 crop was obviously larger than 2.4 g C MJ −1 PAR. These values were in good agreement with that suggested by Monteith (1977Monteith ( , 1986. Annual ε max was also calculated for each site-year to examine the interannual consistency of the WUEbased method and the results were summarized in table S2. The interannual variation of ε max estimates are small for most biome types, as indicated by low SD and CV. The largest CV was found for ENF and WET (about 29% and 35%, respectively), and the minimum CV was found for CRO-C 4 (about 8%). Overall CV was about 20% across all C 3 dominant ecosystems. High R 2 (0.91) also provided evidence of the robustness of this method across the annual scale for different biome groups. Biome-grouped εmax estimates based on the WUE method (blue boxes) at 52 eddy covariance flux sites. For each group, the mean and median εmax is shown as the triangle and the horizontal line within each box, respectively. The 25th and 75th quantiles are shown as the lower and the upper boundaries of each box, respectively. CRO-C3 εmax US-Ne1, US-Ne2, and US-Ne3 during soybean-grown years. CRO-C4 represents εmax estimates at the same sites during maize-grown years. 'n' represents the number of flux sites within each biome group. Abbreviations for plant functional types is the same as table S1. Horizontal dash line in blue and red represents the εmax value suggested by Monteith (1977Monteith ( , 1986 for C3 and C4 crops, respectively. εmax estimates derived from low light regression (LowL, orange boxes) and rectangular hyperbola light response curve (LRC, green boxes) are also presented for comparison.

Comparison of ε max from different methods and previous studies
In addition to the WUE-based method as proposed in this study, we also calculated ε max estimates using the same dataset by fitting two traditional LRCs (text S1), as summarized in table 1 and figure 2 (details see table S1). Result show that linear LowL-based ε max estimates were similar to our estimates using the WUEbased method (the differences are 0.2 g C MJ −1 PAR for C 3 and 0.18 g C MJ −1 PAR for C 4 ). The mean estimates are the closest for CRO whereas the largest discrepancy is found for GRA, with the difference being 0.1 and 0.47 g C MJ −1 PAR, respectively. Much higher ε max estimates were obtained using the LRC method, which is 1.73 and 1.51 times larger than that of WUE and LowL on average. While the closest ε max difference is 0.51 g C MJ −1 PAR for CRO, the largest difference is over 1.4 g C MJ −1 PAR for EBF and GRA based on LRC compared to the WUE method. It is worth noting that LRC-based ε max estimates exhibit significantly higher variations (CV > 50%) across species and biomes compared to WUE (8%) and LowL (12%) method. The LRC-based ε max estimates also diverse greatly from values suggested by Monteith (1977Monteith ( , 1986 in general. It is also noted that the R 2 values are comparable between LowL and LRC but much lower than that of the WUE method, indicating a stronger correlation between GPP·PET and ET·PAR compared to GPP and PAR (regardless of linear or non-linear regression).
Apart from LowL and LRC methods, we also compared WUE-based ε max estimates to those reported in previous studies. Across species and biomes, our ε max estimates agree well with the results from experimental and modeling estimates reported in the literature. For C 3 and C 4 crops at the same sites (i.e. US-Ne1, US-Ne2, and US-Ne3), Gitelson et al (2018) reported similar but slightly lower ε max values for soybean and maize, i.e. 1.45 and 2.23 g C MJ −1 PAR (using green LAI and intercepted PAR measurements) compared to our estimates of 1.66 and 2.67 g C MJ −1 PAR, respectively (figure S2). Besides, our ε max estimates were almost identical to those obtained by Nguy-Robertson et al (2015) at the same sites for both soybean (1.72 g C MJ −1 PAR) and maize (2.42 g C MJ −1 PAR), and comparable to that of Chen et al (2011) for maize (1.95 g C MJ −1 PAR) using optimized LUE models. Zhang et al (2016Zhang et al ( , 2017, suggested that ε max should be less than 2 g C MJ −1 PAR for C 3 and close to 3 g C MJ −1 PAR for C 4 vegetation based on sun-induced chlorophyll fluorescence, which is also consistent with our ε max estimates. These comparisons showed that the WUEbased method provides reasonable ε max estimates for species with distinct photosynthetic capacity differences. For the other biomes, the WUE-based method also produced ε max estimates that were comparable to those reported in the literature ( figure S4). Our WUE-based ε max estimates were close to those from the EC-LUE model across most biomes (DBF, EBF, ENF, GRA, and WSA), and also similar to the results obtained with two other models, namely MODIS and CFix reported by Yuan et al (2014). However, small differences were found between the WUE-based ε max estimates here and the result of Yuan et al (2014) for broad leaf forests (DBF and EBF) when compared with other biome types on average, except for the model VPRM. Major differences for forest and crops were found when compared with other biomes from Wang et al (2010) and the MODIS look-up-table (Running and Zhao 2015). For further validation, we also compared WUEbased ε max to those determined from experimental studies. In the original LUE Monteith (1972Monteith ( , 1977 suggested a relatively consistent ε max of about 1.4 g C MJ −1 PAR for C 3 crops. However, lower ε max values that varied between 0.2 and 1.5 g C MJ −1 PAR were found for woody vegetation (Raymond 1994). Other experimental results presented a wider range of 0.2-4.8 g C MJ −1 PAR as summarized by Prince (1991). Our ε max estimates fell in these broad ranges yet showed a much narrower variation between 0.83a and 1.66 g C MJ −1 PAR for C 3 dominated ecosystems. Interestingly, the biomass measurements conducted by Cannell et al (1988) estimated ε max of 1.5-1.6 g C MJ −1 PAR for the broad-leaf forests (DBF and EBF), which is essentially the same as the WUE-based ε max estimates in this study (1.5-1.58 g C MJ −1 PAR).

Strength of WUE method for estimating ε max
The WUE-based method proposed in this study provides a new perspective and a coupled approach for estimating ecosystem scale ε max . Compared to the traditional LRC fitting method that is highly sensitive to the choice of linear or non-linear model, i.e. LRC estimated ε max was 1.5 times greater than LowL accompanied by 4 times larger CV, the WUE-based method was less variant and provides more convergent ε max estimates (figure 2 and table 1). Although curve fitting is straightforward and requires only GPP and PAR, previous studies had a similar issue performing parameterization especially LRC which can lead to unrealistic and variant ε max estimates (Ruimy et al 1995, Ye 2007, Ye and Yu 2008. Compared to the LUE-reversing method, the WUE-based method requires only four variables that are relatively easy to acquire at site or on larger scales. A major limitation of deriving ε max by reversing the LUE models is the consideration of water (e.g. soil moisture) as an abiotic limit for carbon assimilation (He et al 2014, Yuan et al 2014, Wagle et al 2016, Hu et al 2018, implying that limited water availability constrains carbon assimilation. The intrinsic biophysical coupling between water loss and carbon gain, however, demonstrates that the two fluxes depend on each other and constrain each other interactively (Katul et al 2010, Medlyn et al 2011). Accordingly, the energy-converting efficiency of the carbon process should be related to WUE. The WUE-based method presents this coupling relationship explicitly, where ε max is a function of WUE (biophysical processes), environmental condition (atmospheric water demand, and energy availability). Therefore, the WUE-based method directly conceptualizes and presents the joint regulation of vegetation on both water and carbon processes.
Our result also demonstrates that the WUE-based method is simple and robust for ε max estimation. The much higher ε max for C 4 crops which have a different photosynthetic pathway than C 3 vegetation is clearly distinguishable. The method produced ε max estimates for C 3 and C 4 species that were comparable with estimates based on LowL and values reported in the literature (section 3.3, figure S3). The method was also capable of producing consistent ε max for different ecosystems under a wide range of climate showing good agreement with LowL and previous studies ( figure S4). In principle, ε max should be achieved at lower PAR, as calculated by LowL regression in this study. Hence, the relatively close ε max estimates obtained by WUE compared to LowL method may be considered as a reasonable verification of the reliability of our method, whereas the large uncertainties in LRC estimates require further investigation. The differences of ε max estimates based on reversing LUE model compared to WUE method (figure S4) can be partly attributed to the uncertainties when parameterizing the f factor (i.e. VIs, k PAR , f s ) using different LUE models (as illustrated in section 2.1), which contains scale discrepancy and covariant limiting factor issues. Others also argued that the uncertainty in parameterizing the f factor is a major limitation of the LUE model, and can result in inconsistent estimates of ε max (Yuan et al 2014, Zheng et al 2018. The slightly wider variation of annual ε max estimates for CRO at an annual scale (table S2) (CV = 22%) is likely due to possible changes in species grown within and across crop sites and years (possible C 3 and C 4 rotation), and this may require further information and investigation. For other ecosystems, the variation in the annual ε max estimates were also relatively small across different sites within a PFT (SD generally <25% of the mean ε max ). Previous studies have demonstrated that canopy characteristics (Sánchez et al 2015), water availability and nutrients (Christina et al 2015), as well as radiation conditions (Gitelson et al 2015, could contribute to variations in ε max . The possible interannual variation of ε max induced by such biophysical and environmental factors was explicitly presented in our method, shown by equation (6). A further analysis between the biophysical underlying WUE (uWUE a = GPP·VPD 0.5 /ET, Zhou et al 2014) and annual ε max (figure S5) showed that photosynthetic capacity co-varied with the vegetation water use and the atmospheric water demand.
Since it is difficult to obtain ground-truth measurements for such a wide range of biome types globally distributed, we can infer from the above comparisons and discussions that the WUE-based method is reliable for ε max estimation in accordance with current knowledge.

Possible implication and limitation
The WUE-based ε max estimates can be directly incorporated with LUE models at local to regional scales according to species and biome types for GPP estimation. With a narrow range of ε max (0.83-1.62 g C MJ −1 PAR for C 3 dominated ecosystems, figure 2), uncertainty in GPP estimates is likely to be reduced using our ε max values across different ecosystems (compared with other parameterization schemes, figures 2, S3-S4). Moreover, this WUEbased method uses T/PET as a top-down limiting factor instead of the ambiguous f factor in the LUE model. This treatment not only simplifies the parameterization of ε max , but also avoids the possible uncertainties brought about by multiplying various limiting factors (e.g. LAI, VPD, soil water, etc) to quantify f. Additionally, because all variables required by this WUE-based method are readily available at flux sites (where the GPP models are calibrated to), local ε max estimates can be easily obtained with high credibility. Measurements from these flux sites are considered highly reliable and accurate as eddy covariance represents the state-of-the-art technique. As such, ε max estimates using the WUE-based method are likely to be more robust than estimates from traditional methods, and the use of this method can help to assess model uncertainty in GPP estimation. This is especially true when LUE model structure and data reliability require further evaluation, while the ε max value is theoretically and biophysically consistent.
A limitation of the WUE-based method for estimating ε max is the assumption that transpiration (T) can be approximated by total ET under high vegetation coverage. While this assumption is valid under complete or high vegetation coverage across the landscape, T could be less than ET in non-forest ecosystems (WET, GRA, and WSA). More accurate ε max estimates would be possible using transpiration measurements (rather than using ET as an approximation) based on sap flow and/or isotopic techniques (Roupsard et al 2006, Wei et al 2017b. Future studies could try to employ the latest SAPFLUXNET data (Poyatos et al 2016) to derive in-situ T to be implemented in our WUE-based method. Current site-based ε max could be extrapolated to regional scale via PFT look-up-table method as adopted in the MODIS GPP algorithm. For regions where reliable ET estimates are available, our method can also be used under high LAI to derive ε max . For simplicity, this study also assumed that the fraction of absorbed PAR is the same as that of absorbed R n . Ross (1981) showed that both shortwave and longwave radiation can be approximated by Beer's law with the same bulk extinction coefficient k. We consider such assumption is valid for big-leaf canopy as conceptualized in this study (text S3). Variations in k can be further explored considering different canopy characteristics (e.g. leaf inclination angle, nutrient status, phenology) and climate and radiation conditions (e.g. direct and diffuse radiation, atmospheric CO 2 concentration) (Turner et al 2003, Dai et al 2004). Yet such variation should not violate the fundamentals and conclusions of this study.

Conclusions
This study proposed a new method to estimate ε max based on the water and carbon coupling principle. The method was examined at 52 flux sites across different vegetation species and biome types. Results showed that the method was capable of distinguishing vegetation types with different photosynthetic pathways. The estimated ε max for seven main biomes distributed global-wide agreed well with experimental and modeling methods. The WUE-based method is simple and the ε max estimates could be incorporated into LUE models for better GPP simulation. The method could also be used to better understand the behavior and variation of ε max across species and biomes considering interrelated water and carbon processes.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: http://fluxnet.fluxdata.org/.