Estimating photosynthetic capacity from leaf reflectance and Chl fluorescence by coupling radiative transfer to a model for photosynthesis

Summary In photosynthesis models following the Farquhar formulation, the maximum carboxylation rate V cmax is the key parameter. Remote‐sensing indicators, such as reflectance ρ and Chl fluorescence (ChlF), have been proven as valuable estimators of photosynthetic capacity and can be used as a constraint to V cmax estimation. We present a methodology to retrieve V cmax from leaf ρ and ChlF by coupling a radiative transfer model, fluspect, to a model for photosynthesis. We test its performance against a unique dataset, with combined leaf spectral, gas exchange and pulse‐amplitude‐modulated measurements. Our results show that the method can estimate the magnitude of V cmax estimated from the far‐red peak of ChlF and green ρ or transmittance τ, with values of root‐mean‐square error below 10 μmol CO 2 m−2 s−1. At the leaf level, the method could be used for detection of plant stress and tested against more extensive datasets. With a similar scheme devised for the higher spatial scales, such models could provide a comprehensive method to estimate the actual photosynthetic capacity of vegetation.


Introduction
Monitoring photosynthesis through remote-sensed signals leads to a better understanding of vegetation canopies and their interaction with the environment. At least two optical indicators have been shown to respond to photosynthetic processes dynamically: Chl fluorescence (ChlF) and photochemical reflectance index (PRI). Both ChlF and PRI are intrinsically linked to photosynthesis, and have both been established as good estimators of leaf light use efficiency (LUE) and photosynthesis (for reviews, see Garbulsky et al., 2011;A c et al., 2015). Developing methods for the estimation of photosynthesis from the two optical indicators is particularly relevant for the fast-developing field of precision agriculture (Tremblay et al., 2011;Wieneke et al., 2016), and for the European Space Agency's dedicated Fluorescence Explorer (FLEX) satellite mission (Drusch et al., 2017).
In order to develop such methods, it is key to understand how the enzyme kinetics of photosynthesis are reflected in the dynamic changes of plant optical properties. The solar energy, absorbed by the leaf, undergoes one of three possible pathways: it can be used in photochemistry, emitted as ChlF, or dissipated as heat. This conversion of energy can be detected as a dynamic optical signature in the visible and near-infrared part of the leaf spectrum.
The light is captured by the light-harvesting antennae, and the energy is transferred to the photosynthetic reaction centres. From the captured light, c. 2% of light is directly emitted as ChlF. The spectrum of ChlF has a typical double peak, and ranges from c. 650 to 850 nm. The rest of the captured energy follows a complex process of photochemistry, eventually resulting in the fixation of CO 2 .
However, in natural conditions, plants are often exposed to various stresses that lower their capacity to utilize the available light. Excess energy must then be effectively dissipated via one of the many photoprotective mechanisms of higher plants in order to prevent damage to the photosynthetic apparatus. For example, high light exposure causes rapid saturation of the photosynthetic reaction centres, and the fastest response of the photosynthetic membrane to excess light is to dissipate the surplus of energy as heat. This process is known as the nonphotochemical ChlF quenching (NPQ). It is closely related to the xanthophyll cycle, which involves an interconversion of three xanthophylls: violaxanthin via antheraxanthin into zeaxanthin, followed by a complex series of thylakoid conformational and pH changes, concluding with heat dissipation (Demmig-Adams & Adams, 1996;Ruban, 2016). These photoprotective mechanisms can be observed as dynamic changes in the green part of the visible spectrum, commonly expressed as the PRI (Gamon et al., 1992).
In order to interpret the remotely sensed data, models are needed. Simple indices using only one or two spectral bands, such as PRI, may be insufficient due to the contributions of various leaf pigments and canopy structure to the few selected spectral bands . Radiative transfer (RT) models can describe the light propagation within leaves and canopies based on biochemical and physical properties, enabling us to better decouple the contributions of individual parameters. Complementary to RT models, models for photosynthesis can explain the underlying biochemical processes. Coupling the two types of models would provide a unique insight into the connection between optical and biochemical properties of vegetation.
State-of-the-art models, such as the Soil-Canopy Observation of Photosynthesis and Energy balance (SCOPE) model (Van der Tol et al., 2009), are able to quantify both the variability of photosynthesis and spectral changes at different temporal and spatial scales by employing the biochemical and RT properties of vegetation.
At the leaf level, SCOPE employs two models: a biochemical model, able to explain the relationship between ChlF, photosynthesis and NPQ under different environmental conditions (Van der Tol et al., 2014); and an RT model, FLUSPECT (Vilfan et al., 2016). Both of the models are well tested, computationally affordable, and can function separately, providing independent outputs.
The biochemical model follows Farquhar's 1980s photosynthesis formulation (Farquhar et al., 1980), in which the maximum carboxylation rate V cmax of leaves is a key parameter. V cmax determines the maximum photosynthesis rate of a plant under optimal conditions, and it has a great influence on the modelled photosynthesis.
FLUSPECT simulates leaf reflectance q, transmittance s, and ChlF spectra as a function of leaf pigment content and structure. Recently, FLUSPECT has been extended to simulate the dynamics of green q as a function of the xanthophyll de-epoxidation, an RT analogy to the PRI (Vilfan et al., 2018).
In this study, we couple the two leaf models via ChlF and photochemical q parameters, and effectively link spectral to biochemical properties. Attempts to create such links have been made before; however, in most cases, PRI and ChlF as proxies of photosynthesis were studied separately (for a review, see Grace et al., 2007), with a few exceptions (Cheng et al., 2013;Atherton et al., 2016). The link of q dynamics to photochemistry is generally addressed with the use of the PRI, and relations of spectral information to photosynthetic parameters have mostly been defined via regression models (Cheng et al., 2013;Serbin et al., 2015;Zhang et al., 2016;Dechant et al., 2017;Liu et al., 2017). With the introduction of dynamic xanthophyll q into FLUSPECT, we could avoid the use of PRI and approach changes in both q and ChlF in an RT-based manner. We developed a scheme that links leaf V cmax to q, s and ChlF. Such a model has not been published before.
The coupled model enabled us to devise a method for the retrieval of the V cmax from hyperspectral measurements of leaf ChlF and q or s. In this study, we describe the coupled model and the retrieval method. We test its performance against a unique dataset, with combined leaf spectral, gas exchange and pulse-amplitude-modulated (PAM) measurements. We evaluate the sensitivity of the method to ChlF and q, and we discuss whether combined they provide an even better constraint to the retrieval of leaf photosynthetic parameters.

Laboratory experiment
The experiment was conducted in the laboratories of Forschungszentrum J€ ulich in February and March 2014. Sugar beet (Beta vulgaris L.) and barley (Hordeum vulgare L.) plants were grown in pots under controlled conditions in a glasshouse in Forschungszentrum J€ ulich between December 2013 and March 2014. Owing to the limited winter light conditions, the natural light was complemented with artificial light from growth lamps for 15 h d À1 . The plants used in this experiment were grown under a light intensity of c. 1000 lmol m À2 s À1 (measured with a quantum sensor, LI-190SL; Li-Cor Inc., Lincoln, NE, USA). When the plants were fully grown, some of the pots grown under full light were exposed to water deficits. For a full description of growth conditions, see Vilfan et al. (2016). Measurements were collected on the same leaves in two separate experimental settings.
The first set-up, 'Chamber dataset', is presented in Fig. 1. It consisted of a portable gas-exchange system (LI-6400; Li-Cor Inc.) connected to a (1) clear top MiniPAM Adapter (6400-10; Li-Cor Inc.) housing the pulse-amplitude-modulated fluorescence system (Mini-PAM-II; Heinz Walz GmbH, Effeltrich, Germany); and (2) a spectroradiometer (FieldSpec 4; Analytical Spectral Devices, Boulder, CO, USA; 350-2500 nm, 3 nm visible and near-infrared spectral resolution). The gas-exchange chamber bottom was equipped with an airtight slot, fitting an optical fibre of the spectroradiometer. The top could not be adjusted to house the optical fibre due to technical limitations. The chamber was illuminated externally with a cold halogen lamp (KL 2500 LCD; Schott Benelux BV, Culemborg, Netherlands). The lamp was positioned to illuminate the chamber under an angle of c. 15°, ensuring that the whole leaf surface within the chamber was illuminated and not shaded by the PAM optical fibre. A short-pass filter can be slotted into the opening of the lamp, which cuts off the incoming light spectrum above c. 700 nm. This allowed for measurements of forward ChlF signal ('forward' referring to the emission being in same direction as the excitation radiation, which means in our set-up the ChlF emanating from the abaxial side of the leaf turned away from the light source above the leaf) from c. 700 to 850 nm. With this set-up, almost simultaneous measurements of passive and active ChlF, s, and gas-exchange measurements were taken under controlled conditions.
Measurements were taken on attached leaves of intact plants using one or two healthy and fully developed leaves per pot. The leaf was positioned in the chamber with its adaxial side facing the light source. The gas-exchange system was set to a constant leaf temperature of 25°C, constant air humidity, and a CO 2 concentration of 400 ppm. We measured both light and CO 2 curves for
Each measurement consisted of the following sequence of recordings: gas exchange, followed by the transmitted radiance recorded with the spectroradiometer, the filtered transmitted radiance recorded with the spectroradiometer after slotting the filter into the opening of the lamp, and the active PAM measurement at the top of the leaf after removing the filter. The procedure was repeated for each change of either illumination or CO 2 conditions. Before each recording, we waited for the gasexchange conditions to stabilize, 5 min on average, and up to 20 min for a change in the illumination conditions.
A standard reflectance panel (Spectralon; Labsphere, North Sutton, NH, USA) was used separately to estimate the incident and the filtered incident radiation of the lamp and the exact transmittance of the filter for each light intensity. The panel was positioned at the same distance from the light source as the leaf.
Each radiance measurement was the average of five individual measurements over the region of 350-2500 nm, using a 136 ms integration time. Transmittance and forward ChlF spectra were calculated using the standard formulas as described in Vilfan et al. (2016). It should be noted that the measurements of filtered radiance below 700 nm (and consequentially the red peak of ChlF) are unreliable due to the cut-off filter's characteristics. Moreover, the relative positioning of the reflectance panel, as well as the shape and characteristics of the gas-exchange chamber, might have contributed to additional scattering and inaccuracies in the calculated ChlF spectra. Furthermore, many hyperspectral measurements of barley had to be excluded from the study because the leaves were too small to cover the surface of the entire leaf chamber, contaminating the measurements with additional illumination.
LUE was calculated from gas-exchange data as: where A (lmol CO 2 m À2 s À1 ) is the assimilation rate and iPAR the incoming photosynthetically active radiation (PAR; lmol m À2 s À1 ) (Peñuelas et al., 1995;Barton & North, 2001;Gitelson et al., 2015). From s, we calculated the PRI as (R 531 À R 570 )/(R 531 + R 570 ). For a better comparison of the spectral measurements of different leaves, we normalized both the PRI and ChlF at the far-red peak (F 740 ) to their respective unstressed reference values (PRI 0 and F 0 740 ). By subtracting PRI 0 from all values of PRI and F 0 740 ) from F 740 , we obtained DPRI and DF 740 , respectively. For the light responses, PRI 0 and F 0 740 ) were obtained from the spectrum measured at the lowest illumination (50 lmol m À2 s À1 ), and for CO 2 curves from the spectrum measured at 1200 ppm CO 2 .
For PAM measurements, standard procedures were followed (instruction manual for MINI-PAM-II; Maxwell & Johnson, 2000). During each measurement, a short, intense pulse of light was given, from which the quantum yield of photosystem II (PS-II) Φ PSII and electron transport rate (ETR) were calculated. ETR is automatically calculated by the accompanying software of the instrument, assuming an absorption coefficient of 0.84. Φ PSII reflects the proportion of light absorbed by PS-II that is used for photochemistry and was calculated as: where F s is the steady state ChlF and F 0 m is light-adapted maximum ChlF yield.
In the early morning, before the start of the measurement setup, PAM measurements were taken on dark-adapted leaves. This allowed for the calculation of minimal and maximal dark-adapted ChlF, F o and F m , respectively, and subsequently the NPQ: where F m represents the maximal dark-adapted ChlF yield. Because the information of PAM experiments is contained in the ratios, we normalized the signals to F o . Since ETR, Φ PSII and NPQ are also outputs of the photosynthesis model, these measurements provided important additional insights into the model. The second set-up, the 'FluoWat dataset', is described in detail in Vilfan et al. (2016). Measurements of bidirectional leaf q, s, and ChlF were collected with the FluoWat leaf-clip, coupled to the same spectroradiometer that was used in the first set-up. The leaf clip has two openings for the fibre optics and one light entrance, fitting both a light source at a 45°angle and a shortpass filter (< 650 nm, TechSpec; Edmund Optics GmbH, Mainz, Germany). For more details on the FluoWat leaf clip design, see Van Wittenberghe et al. (2013). The samples were illuminated with the same cold light lamp as in the first set-up. We used measurements taken under three different light intensities, with iPAR of c. 200, 500 and 700-800 lmol m À2 s À1 .

Leaf models in SCOPE
FLUSPECT FLUSPECT is an RT model for the leaf, based on the model PROSPECT (Jacquemoud & Baret, 1990), where the absorption is a function of the concentrations and specific absorption coefficients (SACs) of pigments and water. FLUSPECT computes q and s spectra from 400 to 2500 nm, as well as ChlF spectra from 640 to 850 nm. It is implemented in MATLAB and published under GNU General Public License at https://github.com/christiaanvandertol. Input parameters, together with their definitions and standard values, are provided in Table 1. For a published full description of the model (FLUS-PECT-B), see Vilfan et al. (2016).
In this study, we use the latest version of FLUSPECT, called FLUSPECT-CX (Vilfan et al., 2018), which is able to simulate changes in green q from c. 500 to 570 nm, as a function of the xanthophyll de-epoxidation parameter C x , an RT analogy to the PRI. Moreover, in FLUSPECT-CX we adopted the SAC for anthocyanins as well as recalibrated SACs for chlorophylls and carotenoids from PROSPECT-D (F eret et al., 2017).
Changes in ChlF spectra can be simulated by varying g I and g II : the fluorescence quantum efficiency parameters for PS-I and PS-II, respectively. In analogy to SACs, FLUSPECT uses two spectra for the probability density function φ to describe the spectral distribution of emitted ChlF: φ I and φ II , one for each of the PS-I and PS-II. The two functions were adopted from Franck et al. (2002) and can be linearly mixed. However, owing to systematic discrepancies between measured and simulated ChlF spectra, φ has recently been recalibrated into a single spectrum for φ, with a single fluorescence quantum efficiency parameter g (C. Van der Tol et al., unpublished), used in this study.
FLUSPECT can be inverted and its parameters estimated from measured spectra. However, FLUSPECT cannot explain the underlying processes of photosynthesis related to the dynamic parameters, such as C x and g. They can, however, be estimated indirectly with the biochemical model that describes the relationship between ChlF, photosynthesis, and NPQ.
The biochemical model The biochemical model used in SCOPE simulates the photosynthetic rate and fluorescence quantities as measured with PAM, as a function of absorbed light, leaf temperature, relative humidity and the concentrations of CO 2 and oxygen (O 2 ). It follows Farquhar's formulation (Farquhar et al., 1980), where the assimilation of CO 2 depends on electron transport and carboxylation, and the actual rate of assimilation is determined by the most limiting of these processes. Maximum carboxylation rate per leaf area under light-saturated conditions V cmax is the key parameter in this model. Input parameters of the biochemical model are provided in Table 1. Only parameters relevant for this study are shown.
We used the empirical relationship between photochemical and fluorescence yield for unstressed conditions described in Van der Tol et al. (2014), a nonlinear relationship between the relative light saturation of photosynthesis and nonradiative energy dissipation in plants of different species calibrated to active ChlF measurements. To calculate the probability of the different fates of the excitation energy, it uses rate coefficients K: K p and K n for the photochemical ChlF quenching (PQ) and NPQ, respectively, and K d and K f for heat dissipation and fluorescence, respectively.

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Two outputs are particularly relevant for this study: the fluorescence emission efficiency e, expressed as the ratio of the steady state F s to the dark-adapted fluorescence yield F o ; and K n , which is considered to be equivalent to NPQ, and is calculated as follows: where K 0 n , c and b are fitting parameters, and x is a measure for the light saturation of photosynthesis, calculated as: Eqn 5 / p and / o p represent the photochemical yield and the photochemical yield of dark-adapted state, respectively.
Dark-adapted fluorescence yield is then computed as: and the steady state fluorescence as: Here, it should be noted that F s , and consequently e, are related to K n through the calculation of F 0 m : Coupling the leaf models FLUSPECT and the biochemical model have parameters related to photosynthesis in common, and this makes it possible to retrieve photosynthesis from the measured spectra of ChlF and s. The two dynamic input parameters of FLUSPECT, g (the emission efficiency of fluorescence) and C x (the absorption SAC for the xanthophyll cycle dynamics) are related to the outputs e (F s /F o ) and K n (the rate coefficient for NPQ) of the biochemical model, and the simplest possible relation is a linear one.
To couple C x to K n , we adopt the relation of Vilfan et al. (2018): Similarly, g and e are coupled as: where ς is a scaling factor, calculated as a ratio of a typical value of g to a typical value of e, with a value of 0.007, which represents the quantum efficiency of fluorescence in a dark-adapted leaf. This scaling is necessary because ɛ = F s /F o is a relative value, whereas g is an absolute emission efficiency.

Retrieving maximum carboxylation capacity
V cmax was retrieved in three ways, notably using gas exchange (method 1), PAM data (method 2), and hyperspectral measurements (method 3). Each method was applied to both the CO 2 and the light response curves, resulting in six sets of values for V cmax . Method 1 (gas exchange) is the traditional way of retrieving V cmax ; therefore, we used the values retrieved with this method to validate the other two methods. Fig. 2 provides an overview of these methods, Table 1 default parameters of the biochemical model, and Table 2 presents the retrieved parameters, their boundaries and constraints. For method 1, we retrieved V cmax by inverting the biochemical model only, by minimizing the squared difference between measured and simulated assimilation rates A, following Kosugi & Matsuo (2006)  Measured values of PAR, leaf temperature, CO 2 and water vapour were used to force the model. In method 2, we minimized the quadratic difference between modelled and (PAM) measured F s and NPQ (Fig. 2a). In order to achieve an accurate fit of measured vs modelled F s and NPQ, we fitted not only V cmax but also the three empirical parameters of Van der Tol et al. (2014) for NPQ, notably K 0 n , c and b (Eqn 4). With this method we could test the potential of using steady-state ChlF and NPQ data to retrieve photosynthesis from the biochemical model, in the absence of the informative measurements with the saturating flashes F m and F 0 m ). In method 3, we used only the spectroscopy measurements to retrieve V cmax , which was our ultimate aim (Fig. 2b). This method enabled us to test whether V cmax in the combined radiative transfer and biochemical model can be sufficiently constrained by passive ChlF and s data. A narrow band of ChlF at the far-red peak (730-750 nm) and of green s (525-545 nm) were selected as the constraint for the model inversion. We retrieved V cmax and the additional parameters K 0 n and ς (Table 2) by minimizing the squared difference between the simulated and measured spectra (Eqn 11) for the wholelight-or CO 2 -response curve at once. In this retrieval, the other parameters of FLUSPECT, notably the pigments and leaf structure parameter N, were kept to leaf-specific values. These values were retrieved once per leaf before the retrieval of V cmax , K 0 n and ς, and were assumed not to change during the lightand CO 2 -response curves (Table 1). In this way, we attributed the dynamic changes to the s spectrum to the xanthophyll cycle, and thus to C x and K n (Eqns 9, 10).
In all three methods, a trust-region algorithm was applied in MATLAB using the built-in function lsqnonlin to minimize a cost function: where M is the measured data and S the corresponding simulation (for the three methods: gas exchange, PAM data, and hyperspectral data). For method 3, M and S were matrices of multiple spectra: one spectrum for each of the 10 points on the CO 2 -response curve or the eight points on the lightresponse curve. These spectra included s and forward ChlF. Because method 3 only uses spectra as input (and no gasexchange or PAM data), it was also applied to the additional FluoWat leaf clip dataset. The advantage of the FluoWat leaf clip data is that it provided not only the forward measurements (transmittance s and ChlF at the shaded side), but also the backward measurements (reflectance q and ChlF at the illuminated side). However, due to some nonisotropic scattering and specular reflectance present in the q of some samples, we had to normalize the spectra of each light curve by subtracting the q at 565 nm. At this wavelength, the xanthophyll cycle ceases to have an effect on the spectra simulated by the FLUSPECT model. We fitted q and backward ChlF, s and forward ChlF, and both q and s and forward and backward ChlF together to test the dependence of the performance on the side of the leaf that is measured.

Evaluating the model inversion
For method 1, we compared the V cmax retrieved from the lightresponse curve with those retrieved with the CO 2 -response curves. These retrievals were then used to validate the other methods. We evaluated the goodness-of-fit of V cmax , A, and other variables by calculating the root-mean-squared error (RMSE) and Pearson's correlation coefficient R 2 . The number of data points is slightly different among the three methods due to unrealistic values caused by occasional human error in the measurements, such as light pollution, vibrations during measurements, or incorrect measurement light and pulse settings of the PAM system.
The retrieval accuracy was evaluated for the chamber dataset with the relative RMSE. We further investigated the sensitivity of s and ChlF spectra to the relevant parameters, the sensitivity of the retrievals to the starting values of the trust-region algorithm, and the ill-posedness of the retrieval. We calculated the Jacobian matrices J for s and ChlF of the model for one representative sample. To obtain comparable values of the sensitivities and to normalize J, we multiplied J by the span of each parameter (see Table 2).
Error propagation in the retrieved parameters due to the measurement noise was estimated as: where the standard deviations of the retrieved parameters r p are found as the square roots of the diagonal elements of this matrix. r r is the SD of the measurements due to the measurement noise. For a full derivation of Eqn 12, refer to Vilfan et al. (2018).

Optical and physiological measurements
In Fig. 3 we display different physiological and optical responses of leaves to variations in CO 2 concentrations and illumination. Assimilation increases with both increasing CO 2 and illumination intensity, until a plateau is reached. The initial slope of the assimilation curve represents the maximum LUE of the leaf, whereas the plateau signifies the light-saturated rate of photosynthesis (Bj€ orkman, 1981).
For the case of CO 2 curves, the indicators of photosynthetic capacity (A, LUE and Φ PSII ) continuously increase until the external concentration of c. 700 ppm CO 2 , when the maximum assimilation and efficiency are reached. NPQ and PRI respond similarly, with the highest response at the lowest CO 2 concentrations. The ChlF response, however, is negligible. For the case of light-response curves, assimilation and ChlF increase with increasing light, whereas LUE, DPRI and Φ PSII decrease .
When comparing measured spectral responses of DPRI and DF 740 to physiological variables (Fig. 4), it is immediately evident that the two types of leaf response curves do not generate the

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Retrievals of V cmax
V cmax and assimilation values estimated with the three methods are presented in Fig. 5, with supporting statistical information.
In Fig. 5(b) we compare V cmax retrieved from CO 2 -and lightresponse curves of the same leaves with method 1. The data display a high level of correlation (R 2 = 0.82 and RMSE % 6), with  Fig. 2 The schematics for three methods of V cmax retrieval. (a) In methods 1 and 2, the biochemical model is inverted by constraining the inversion with either the assimilation rate A curves (method 1) or with nonphotochemical quenching (NPQ) and steady-state fluorescence F s curves (method 2). (b) In method 3, the combined model is inverted. First, the PROSPECT parameters (see Table 1) are retrieved once per leaf. Next, the biochemical model is initialized with standard input values. Parameters for photochemical reflectance C x and fluorescence quantum efficiency g are prescribed as functions of NPQ and fluorescence emission efficiency e, following Eqns 9 and 10, respectively. C x and g, together with the estimated PROSPECT parameters, are provided as inputs of FLUSPECT. The difference between the FLUSPECT simulation and measured spectra is minimized by a cost function, resulting in the optimization of the chosen parameters. very accurate predictions of A ( Fig. 5a; R 2 = 0.98 and RMSE % 1). The two other methods (Fig. 5e,h) capture the span of V cmax for both types of response curves, with accurate predictions of A (Fig. 5d,g, with R 2 > 0.80 and RMSE < 4.6). Retrievals of V cmax from PAM measurements provide the highest error (RMSE % 20-23 lmol m À2 s À1 ). Where hyperspectral data are used to constrain the model (Fig. 5h), for both CO 2 and light spectral responses, only the magnitude of V cmax can be estimated (RMSE < 10), not the variation among leaves. Predictions of A are accurate, with R 2 > 0.85 and RMSE < 4.6.
The residuals (retrieved values minus the values retrieved from light curves with method 1) for the three methods are also shown (Fig. 5c,f,i). The smallest range of differences occurs with method 1, with unreliable estimations from method 2. Methods 2 and 3 show a consistent overestimation of V cmax , especially at high V cmax values.
Retrievals from the light-response curve have a better goodness-of-fit (R 2 = 0.45) than retrievals from CO 2 -response curves (R 2 = 0.081). We evaluated other state variables besides V cmax and A; Fig. 6 shows that whereas ETR, Φ PSII , and K n are estimated rather well, simulated F s is poorly correlated to PAM-measured F s . This holds for both the CO 2 -and light-response curves, although the latter have a better goodness-of-fit.
The accuracy of spectral fit after optimization between measured and simulated s or ChlF is presented in Fig. 7. The spectral fit of s and forward ChlF at the selected wavelengths is similarly good, with error close to 0% on average, with a maximal deviation of 4% for s and 17% for ChlF. The measurement error of the spectra used for V cmax retrieval had a negligible effect on the estimated V cmax (3%, not shown). Most of the uncertainty in V cmax is due to the sensitivity of V cmax to the spectra and the model representation. This is illustrated in Fig. 8, showing the RMSE used as the cost functions for the first and third methods as a function of V cmax for one representative leaf. Both the RMSE of fluorescence and transmittance have a single minimum, indicating their sensitivity to V cmax . The RMSE of fluorescence has a sharp and deep minimum, which confirms the sensitivity of fluorescence to V cmax , but the location of the minimum differs between the CO 2 -and lightresponse curves, which is indicative of a model representation error. The RMSE of transmittance shows a less sharp minimum, but the location of the minimum agrees between the CO 2 and light curves, and with the value retrieved from the

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We further investigate the separate contributions of the fitting parameters used in method 3 (V cmax , K 0 n and ς) to the spectra simulated by the combined model in Fig. 9. s is influenced most by V cmax , less by K 0 n , and -obviously-ς (the scaling of ChlF) has no effect on s while ChlF is influenced mostly by ς, less by K 0 n , and the effect of V cmax is still relatively small despite the clear sensitivity shown in Fig. 8.
Retrieving V cmax from FluoWat data For the FluoWat data (Fig. 10), we obtained similar results as for the chamber dataset: the correct range of V cmax values is retrieved (RMSE c. 13 lmol m À2 s À1 ), but with low goodness-of-fit. The residuals have the same span as obtained for the chamber dataset (Fig. 5h), albeit reversed: there is an increase in underestimation with increasing V cmax values. There are no substantial differences in results when transmittance or reflectance of the leaf are used for the retrieval. It should be emphasized that we used the V cmax values obtained earlier in the chamber for validation, because the FluoWat leaf clip does not allow for gas-exchange measurements.

Discussion
Both the transmittance and the fluorescence spectra (method 3) contain sufficient information to constrain V cmax , as demonstrated by the clear minima in their RMSE with respect to varying V cmax (Fig. 8). The results show that, for our dataset, with limiting span of V cmax , the magnitude but not the variability of V cmax among leaves can be estimated from the coupled model, by using the combined information of hyperspectral ChlF and green q or s. The RMSE for the estimated V cmax is nevertheless below 14 lmol CO 2 m À2 s À1 , which complies with the error determined by similar studies of leaf level responses (Serbin et al., 2012;Dechant et al., 2017) and by a study on airborne data by Serbin et al. (2015). Both

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The response of ChlF to changing CO 2 is small, but nevertheless meaningful (see dashed line in Fig. 8c). This response is limited due to the fact that in a CO 2 -response curve, a reduction of PQ is compensated by an increase in NPQ and vice versa, with limited net effect on ChlF. The responses of NPQ and DPRI are unambiguous, but tuning V cmax cannot bring the RMSE of transmittance close to zero (Fig. 8b): the effect of NPQ on transmittance is small compared with the accuracy by which we can reproduce the overall transmittance spectrum.
Potential errors in the V cmax retrieval with method 3 include measurement errors, the performance of the optimization method, and the biochemical and radiative transfer model representation. We investigated the potential effects of measurement errors and accuracy of spectral fit on the retrievals of V cmax , but these effects were minimal: up to 3% of the retrieved V cmax values. Crucial is the model representation.
Despite controlled experimental conditions, the values of V cmax estimated from the assimilation rates (method 1) differ up to 25% between light and CO 2 responses (Fig. 5c). A study by Miao et al. (2009) has similarly shown that significant differences exist between different methods of V cmax retrieval from A curves. This could be attributed to limitations of the Collatz model (Collatz et al., 1991) used by Van der Tol et al. (2014), which does not use the maximum electron transport capacity J max of the original Farquhar model as an additional parameter. The parameterization of photorespiration may also contribute to this difference; photorespiration was not suppressed in our experiment, which was carried out under ambient O 2 . Comparison of measurements (broken line, black) and retrieval accuracy (solid line, red) after fitting with method 3 in the selected bands of transmittance (left panels) and forward Chl fluorescence (ChlF, right panels) spectra. RRMSE is the relative rootmean-square error of retrieval accuracy, and the shaded area represents the SD of the mean (broken line). Differences in the optimized values from transmittance and fluorescence (Fig. 8b,c) may originate from the prescribed relations of g and C x to e and K n , which were reconciled by using a single cost function for both transmittance and fluorescence, and allowing variations in these relationships.
We established the link between the radiative transfer model FLUSPECT and the photosynthesis model via parameter g (the absolute ChlF quantum yield) to e = F s /F o , and C x to K n via a calibrated coefficient (Vilfan et al., 2018). This introduces uncertainty, because PAM F s cannot be reproduced well by the model. The poor correlation of PAM F s to simulated F s can at least  Fig. 10 Values of maximum carboxylation capacity V cmax , retrieved for the FluoWat dataset with method 3, vs control values for barley (red circles) and sugar beet (blue squares) leaves, with the residuals (measured minus estimated V cmax ) shown in the bottom panels. V cmax , K 0 n (fitting parameter for K n ; Eqn 4), and the scaling factor ς (links the Chl fluorescence (ChlF) efficiencies of the two leaf models in Eqn 10) were optimized to best reproduce measured (a) reflectance and backward ChlF, (b) transmittance and forward ChlF, or (c) all four simultaneously. Pearson's correlation coefficient (R 2 ) and RMSE are shown for all data combined. Note, that the control values of V cmax (V cmax , Method 1 ) were obtained by applying method 1 to the Chamber dataset. Number of samples for barley and sugar beet per retrieval is n a = 23/10, n b = 23/10, and n c = 14/5. New Phytologist (2019) 223: 487-500 Ó 2019 The Authors New Phytologist Ó 2019 New Phytologist Trust www.newphytologist.com partly be explained by the compensation of the effects of PQ and NPQ, resulting in a relatively small range of F s , which is then prone to uncertainties. This was also noted in the development of the extended biochemical model ( Van der Tol et al., 2014, Fig. 9).
In Van der Tol et al. (2014), two different parametrizations are given for K n for different datasets. Obviously, K 0 n , a parameter for the xanthophyll pool size, may vary. In a first attempt to fit s and ChlF of the light-response curves, we found that retrieving only V cmax while keeping K 0 n at the default value of Van der Tol et al. (2014) did not provide a satisfactory fit (not shown here). It was necessary, therefore, to include K 0 n in the retrieval. Inclusion of ς was necessary in order to translate F s in arbitrary units to absolute values of ChlF yield.
Indeed, the sensitivity analysis (Figs 8, 9) reveals that s simulated with the combined model depends primarily on V cmax , whereas the magnitude of ChlF is most affected by the hitherto unconstrained ς, and K 0 n has a similar effect on both indicators. These results indicate that both s and ChlF contribute to the V cmax estimations.
Potential ill-posedness could be reduced by introducing prior information on V cmax ; for example, based on vegetation indices or pigment concentrations, of which Chl concentration is a most valid candidate (Houborg et al., 2015;Gitelson et al., 2016). A more mechanistic modelling could potentially reduce the uncertainty in other parameters, such as K 0 n and ς. For example, the description of energy partitioning to NPQ used in this study could be replaced with MD12, whereas models developed by Zaks et al. (2013) and Matuszy nska et al. (2016) may help constrain ς using fluorescence kinetics.
The residuals of two datasets (chamber vs leaf clip) used with method 3 (Figs 5i, 9) have the same span, which is encouraging, as it shows that similar results can be achieved with different types of spectral measurements (i.e. reflectance or transmittance). In general, predictions of A are at least 10% more accurate for estimations from light-response curves compared with CO 2 responses; and similarly, light-response curves provide a higher accuracy of V cmax retrieval.
Our study showed that a quantification of photosynthesis from transmittance or reflectance and ChlF during light-and CO 2response curves is possible and very promising.