Improving the Gross Primary Productivity Estimation by Simulating the Maximum Carboxylation Rate of Maize Using Leaf Age

Abstract

Although the maximum carboxylation rate (V_cmax) is an important parameter to calculate the photosynthesis rate for the terrestrial biosphere models (TBMs), current models could not satisfactorily estimate the V_cmax of a crop because the V_cmax is always changing during crop growth period. In this study, the Breathing Earth System Simulator (BESS) and light response curve (LRC) were combined to invert the time-continuous V_m25 (V_cmax normalized to 25 °C) using eddy covariance measurements and remote sensing data in five maize sites. Based on the inversion results, we propose a Two-stage linear model using leaf age to estimate crop V_m25. The leaf age can be readily calculated from the date of emergence, which is usually recorded or can be readily calculated from the leaf area index (LAI), which can be readily obtained from high spatiotemporal resolution remote sensing images. The V_m25 used to calibrate and validate our model was inversely solved by combining the BESS and LRC and using eddy covariance measurements and remote sensing data in five maize sites. Our Two-stage linear model (R² = 0.71–0.88, RMSE = 5.40–7.54 μmol m⁻² s⁻¹) performed better than the original BESS (R² = 0.01–0.67, RMSE = 13.25–18.93 μmol m⁻² s⁻¹) at capturing the seasonal variation in the V_m25 of all of the five maize sites. Our Two-stage linear model can also significantly improve the accuracy of maize gross primary productivity (GPP) at all of the five sites. The GPP estimated using our Two-stage linear model (underestimated by 0.85% on average) is significantly better than that estimated by the original BESS model (underestimated by 12.60% on average). Overall, our main contributions are as follows: (1) by using the BESS model instead of the BEPS model coupled with the LRC, the inversion of V_m25 took into account the photosynthesis process of C4 plants; (2) the maximum value of V_m25 (i.e., PeakV_m25) during the growth and development of maize was calibrated; and (3) by using leaf age as a predictor of V_m25, we proposed a Two-stage linear model to calculate V_m25, which improved the estimation accuracy of GPP.

1. Introduction

Gross primary productivity (GPP) refers to the carbon dioxide fixed by plants through photosynthesis, constituting the largest carbon exchange between terrestrial ecosystems and the atmosphere [1]. GPP plays a crucial role in regulating the terrestrial carbon budget, thus exerting a significant impact on climate change [2,3,4,5]. Croplands exhibit high primary productivity during the growing season, underscoring the importance of accurately characterizing crop physiology throughout the growing seasons to predict carbon exchange in agricultural systems [6,7]. Process-based models, such as terrestrial biosphere models (TBMs), are widely employed for estimating GPP at both regional and global scales [8,9,10]. These models typically incorporate photosynthesis modules based on photosynthesis models [11,12]. Central to these photosynthesis models is the parameter known as the maximum carboxylation rate (V_cmax), which characterizes the potential photosynthetic capacity of leaves [11,13]. Consequently, the accurate estimation of the V_cmax is paramount for enhancing the accuracy and performance of TBMs [14].

V_cmax fluctuates abruptly during the crop growth period; thus, taking the variation in V_cmax into account is crucial for accurately calculating GPP in models [15,16,17]. However, V_cmax cannot be directly measured, and it must be inferred from leaf gas exchange measurement, which is time-consuming, resulting in limited data availability across a wide range of conditions [18]. In most TBMs, V_cmax is assumed to be a fixed value (normalized to 25 °C, V_m25) based on plant functional type, disregarding temporal and spatial variations in V_m25 [7,19,20,21,22]. This assumption inevitably introduces significant bias in simulated photosynthesis, particularly in regions characterized by substantial seasonal fluctuations [15,23]. To minimize the bias, temporal variations in V_m25 are modeled by establishing relationships between V_m25 and more readily available plant traits [18,24]. Some studies have inverted V_m25 using eddy covariance measurements and remote sensing data [25,26,27,28,29,30]. Yuan used the ensemble Kalman filter (EnKF) to obtain the temporal variation in the V_m25 of maize [17]. Zheng and Xie inverted the time series of V_m25 by coupling Boreal Ecosystem Productivity Simulator (BEPS) and light response curve (LRC) based on eddy covariance observations in flux sites [31,32]. The results indicated that this approach can effectively optimize V_m25, but it has only been applied to C3 plants, and its applicability to C4 plants has not been studied yet [31,32,33].

Despite technological advances in deriving V_m25, modeling V_m25 through plant traits remains challenging. Advances in remote sensing and hyperspectral imaging technology have proven useful for estimating photosynthetic capacity across large spatial and temporal scales [18,34,35,36,37,38]. The strong correlation between the leaf nitrogen content (N_Leaf) and V_m25 has garnered significant attention [39,40,41]. However, obtaining accurate N_Leaf through remote sensing data remains a challenge, which impedes the use of leaf nitrogen to parameterize V_m25 [8,24]. Chlorophyll plays a crucial role in photosynthesis by capturing photons and providing the biochemical energy necessary for carbon fixation reactions [42]. Since N_Leaf includes both photosynthetic and non-photosynthetic components, some researchers argue that leaf chlorophyll content (Chll) is a more accurate indicator of V_m25 than N_Leaf content [24,43]. Houborg summarized the semi-empirical relationship between Chll and V_m25 [19]. By parameterizing this relationship, it has been successfully applied to a variety of crops [18,44,45]. The practical limitation of using Chll to parameterize V_m25 at large spatial scales has been the lack of accurate remote sensing Chll products at regional or global scales. The Sentinel-2A satellite, launched in 2015, and the Sentinel-2B satellite, launched in 2017, carry multispectral imagers (MSIs) with red-edge bands sensitive to Chll and offer high temporal and spatial resolution. However, estimating Chll relies on complex models that typically require ground validation data for calibration [24,46]. The availability of the satellite observations of sun-induced fluorescence (SIF) offers a new perspective for monitoring crop V_m25 [19,20]. Chlorophyll fluorescence is widely regarded as a direct proxy for electron transport and, consequently, photosynthesis [47,48]. Studies have demonstrated a strong connection between SIF and V_m25, indicating that SIF could be helpful for improving the accuracy of V_m25 estimations at large spatial scales [42,49,50]. However, although SIF is effective and remote sensing products with 500–5000 m resolution are available, such as CSIF and GOSIF, the resolution of current SIF products is still relatively low compared to other vegetation parameter products, which can achieve a resolution of 30 m or higher [51]. The vegetation canopy structure directly influences various physical and biological processes, such as radiative transfer and photosynthesis [52]. Thus, in recent years, vegetation indices (VIs) have been employed to empirically estimate V_m25 [53]. Muraoka, et al. [54] divided the growth period into two stages and established distinct relationships between VIs and canopy-level V_m25 for each stage. However, Zhou highlighted that the relationship between traditional VIs and V_m25 was not universal across different sites [55]. LAI is a VI commonly used to characterize seasonal changes in V_m25 [56]. However, it is commonly observed that photosynthesis peaks earlier than the canopy structure indices [57].

Leaf age has been found to be well correlated with V_m25 and substantial evidence indicates biochemical differences between young and old leaves [58,59]. Recent studies have demonstrated that leaf age plays a crucial role in determining photosynthetic rates [60,61,62], with V_m25 exhibiting notable changes with the aging of leaves [8,16,63,64,65]. Generally, it is observed that the V_m25 of newly mature leaves tends to be higher compared to that of younger and older leaves [66]. As leaves undergo senescence, Rubisco gradually becomes inactivated, the electron transport rate decreases, and enzymes are deactivated. Consequently, both the rates of photosynthesis and respiration decline with leaf aging [58,59,67]. It is worth noting that most of the research on the relationship between V_m25 and leaf age has focused on trees [24,66], and there has been limited investigation into this relationship in crops [58,59,68]. Miner and Bauerle analyzed the seasonal changes in V_m25 for maize and sunflower through gas exchange experiments [6]. The study revealed that the V_m25 of maize decreases in a nearly linear trend from the mid-vegetative stage to the late senescence stage. For sunflowers, V_m25 remained relatively stable from the late vegetative stage to the early reproductive stage, then significantly decreased in the late reproductive stage. Li found that rice V_m25 exhibited a trend similar to leaf age under different experimental conditions and emphasized that a general formula for V_m25 changes with leaf age has not yet been established [65].

Despite the well-known, strong correlation between leaf age and V_m25 in crops, to our knowledge, this relationship has not yet been quantified. Therefore, the objectives of our study are (1) to derive time-continuous V_m25 for maize at five sites by considering the photosynthesis process of C4 plants; (2) to calibrate the maximum value of V_m25 (i.e., PeakV_m25) during the growing season of maize based on the inverted V_m25; and (3) to propose a Two-stage linear model that leverages leaf age to improve the accuracy of V_m25 and GPP estimation.

2. Data Availability

2.1. Eddy Covariance Data

Hourly or half-hourly GPP data obtained by eddy covariance (EC) observations were used to estimate V_m25 in five flux sites located in maize fields in United States and in China (

Table 1. Summary of flux sites information.

Site	Latitude	Longitude	Crop Type	Year	LAI
US-Ne1	41.165°N	96.477°W	Maize	2001–2012	—
US-Ne2	41.165°N	96.470°W	Maize–Soybean	2001–2012	Literature *
US-Ne3	41.180°N	96.440°W	Maize–Soybean	2001–2012	Literature *
Daman	38.853°N	100.376°E	Maize	2018, 2019, 2021	measured
Fenzidi	41.153°N	107.653°E	Maize	2017, 2018, 2020	measured

* the LAI data for the site were partially obtained from the literature [72].

). The hourly GPP data for the US-Ne1, US-Ne2, and US-Ne3 were obtained from the FLUXNET2015 database [69]. The US-Ne1 site was always planted with maize, whereas the US-Ne2 and US-Ne3 sites were rotating, planted with maize and soybeans. The half-hourly GPP data for the Daman were acquired from the National Tibetan Plateau Data Center [70,71]. We obtained the half-hourly GPP data for the Fenzidi ourselves, from our own flux site in the Hetao Irrigation District, Inner Mongolia Autonomous Region, China. The air temperature (T), relative humidity (RH), wind speed (WS), atmosphere pressure (Pa), carbon dioxide concentration (Ca) and the incoming solar radiation (SR) were also provided by the five flux sites.

2.2. Remote Sensing Data

The moderate-resolution imaging spectroradiometer (MODIS) surface reflectance data were obtained from MCD43A4.061. The reflectance values across all bands served as the input for the BESS model and for the calculation of LAI. It is worth noting that MODIS LAI tends to significantly underestimate crop LAI [73]. Measured LAI values were provided for the Fenzidi and Daman sites, while LAI values for certain years at US-Ne2 and US-Ne3 were extracted from the literature [72]. To estimate LAI values for other years, a machine learning model (ExtraTreesRegressor) was employed, demonstrating strong simulation performance (R² = 0.94) (Figure A1).

The clumping index (Ω) is another vegetation canopy structure index, quantifying the degree of deviation in leaf spatial distribution from a random pattern in BESS [74,75]. The Ω values were estimated using the MODIS BRDF product at a resolution of 500 m [76,77]. Ω values range from 0 to 1, where 1 indicates a randomly distributed canopy. Smaller Ω values indicate a more clustered canopy structure.

3. Method

Our approach is depicted in Figure 1. Firstly, the BESS model and LRC were combined to derive the time-continuous V_m25. The BESS model distinguishes the PAR absorbed by the sunlit leaves and shaded leaves. By integrating the BESS model and LRC, canopy GPP is separated into contributions from sunlit and shaded leaves. V_cmax is inverted utilizing Collatz’s model based on the GPP from sunlit leaves, and then normalized to V_m25 using a temperature function (Section 3.1). Next, the Two-stage linear model was employed to simulate V_m25 based on leaf age (Section 3.2). Finally, V_m25 was simulated using our Two-stage model and validated against the inverted V_m25, with further verification through GPP simulation results (Section 3.3).

3.1. Inversely Solving V_m25 by Coupling the BESS and LRC

Zheng and Xie inverted the time series of V_m25 by coupling Boreal Ecosystem Productivity Simulator (BEPS) with light response curve (LRC), utilizing EC observations data and remote sensing data from flux sites [31,32]. This method effectively optimizes V_m25 by separating the contributions of sunlit leaves and shade leaves. However, the BEPS model does not account for the physiological process for C4 plants, such as maize [33]. What is more, BEPS employs a fixed V_m25 value. The Breathing Earth System Simulator (BESS) model not only differentiates between sunlit and shaded leaves [56], but also distinguishes between C3 and C4 plants. Additionally, BESS incorporates LAI to account for seasonal variation in V_m25. Therefore, we substituted the BEPS model with the BESS model.

3.1.1. The Separation of Sunlit and Shaded GPP by BESS

In the BESS model, the photosynthesis rate at leaf-level is calculated by Farquhar’s and Collatz’s model [11,12], while a “two-leaf” canopy model is employed to upscale the leaf-level photosynthesis rate to canopy GPP. The total GPP of the canopy is determined by summing the contributions from sunlit leaves and shaded leaves as follows:

$$\mathrm{G}\mathrm{P}\mathrm{P}={\mathrm{G}\mathrm{P}\mathrm{P}}_{\mathrm{s}\mathrm{u}\mathrm{n}}+{\mathrm{G}\mathrm{P}\mathrm{P}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}$$

(1)

where GPP_sun and GPP_shade are the GPP of sunlit and shaded leaves, respectively. GPP can be calculated from photosynthesis rate and the corresponding LAI; therefore, Equation (1) can be expressed as:

$$\begin{array}{c}\mathrm{GPP}={\mathrm{A}}_{\mathrm{s}\mathrm{u}\mathrm{n}}{\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{u}\mathrm{n}}+{\mathrm{A}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}{\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}\end{array}$$

(2)

where A_sun and A_shade are the photosynthesis rates per units of sunlit and shaded leaves, respectively; LAI_sun and LAI_shade are the LAI of sunlit and shaded leaves, which can be calculated by:

$${\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{u}\mathrm{n}}=2\cos \mathsf{\theta }\left(1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{0.5\mathsf{\Omega }\mathrm{L}\mathrm{A}\mathrm{I}}{\cos \mathsf{\theta }}}\right)\right)$$

(3)

$${\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}=\mathrm{L}\mathrm{A}\mathrm{I}-{\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{u}\mathrm{n}}$$

(4)

where θ is solar zenith angle, Ω is the clumping index, and LAI is the leaf area index of the whole canopy.

3.1.2. Photosynthesis Rates of Sunlit and Shaded Leaves Estimated by LRC

Light response curve is a commonly utilized tool to depict the correlation between the photosynthetic rate and photosynthetically active radiation (PAR). The rectangular hyperbola is employed to characterize the shape of the light response curve [78]:

$$\begin{array}{c}\mathrm{A}={\displaystyle \frac{\mathsf{\alpha }\mathrm{I}\mathrm{P}}{\mathsf{\alpha }\mathrm{I}+\mathrm{P}}}\end{array}$$

(5)

where A is the gross photosynthesis rate; α is the maximum light use efficiency (LUE) obtained by the initial slope of the curve; I is the absorbed photosynthetically active radiation (APAR) of the leaf per unit; P is the gross photosynthetic rate under saturated radiation.

The light response curves are employed to estimate A_sun and A_shade, respectively [31,33]. Then, the total canopy GPP of Equation (2) can be expressed as:

$$\mathrm{G}\mathrm{P}\mathrm{P}={\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{u}\mathrm{n}}{\displaystyle \frac{\mathsf{\alpha }{\mathrm{I}}_{\mathrm{s}\mathrm{u}\mathrm{n}}{\mathrm{P}}_{\mathrm{s}\mathrm{u}\mathrm{n}}}{\mathsf{\alpha }{\mathrm{I}}_{\mathrm{s}\mathrm{u}\mathrm{n}}+{\mathrm{P}}_{\mathrm{s}\mathrm{u}\mathrm{n}}}}+{\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}{\displaystyle \frac{\mathsf{\alpha }{\mathrm{I}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}{\mathrm{P}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}}{\mathsf{\alpha }{\mathrm{I}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}+{\mathrm{P}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}}}$$

(6)

where I_sun and I_shade are PAR absorbed by sunlit leaves and shade leaves (APAR_Sun and APAR_Sh), respectively, and are calculated using the two-leaf canopy radiative transfer model in BESS (Appendix B); and P_sun and P_shade are given by:

$${\mathrm{P}}_{\mathrm{s}\mathrm{u}\mathrm{n}}={\displaystyle \frac{{\mathrm{P}}_0\mathrm{k}\left(1-{\mathrm{e}}^{-\mathrm{k}\mathrm{L}\mathrm{A}\mathrm{I}-{\mathrm{k}}_{\mathrm{n}}\mathrm{L}\mathrm{A}\mathrm{I}}\right)}{\left(\mathrm{k}+{\mathrm{k}}_{\mathrm{n}}\right)\times \left(1-{\mathrm{e}}^{\mathrm{k}\mathrm{L}\mathrm{A}\mathrm{I}}\right)}}$$

(7)

$${\mathrm{P}}_{\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{d}\mathrm{e}}={\displaystyle \frac{{\mathrm{P}}_0\left(\left(2\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }{\mathrm{k}}_{\mathrm{n}}\mathrm{k}-\left({\mathrm{k}}_{\mathrm{n}}+\mathrm{k}\right){\mathrm{e}}^{\mathrm{k}\mathrm{L}\mathrm{A}\mathrm{I}}\right){\mathrm{e}}^{-\mathrm{k}\mathrm{L}\mathrm{A}\mathrm{I}-{\mathrm{k}}_{\mathrm{k}}\mathrm{L}\mathrm{A}\mathrm{I}}-\left(\left(2\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }{\mathrm{k}}_{\mathrm{n}}-1\right)\mathrm{k}-{\mathrm{k}}_{\mathrm{n}}\right)\right)}{\left({\mathrm{k}}_{\mathrm{n}}^2+{\mathrm{k}}_{\mathrm{n}}\mathrm{k}\right)\left(\mathrm{L}\mathrm{A}\mathrm{I}-2\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }\left(1-{\mathrm{e}}^{-\mathrm{k}\mathrm{L}\mathrm{A}\mathrm{I}}\right)\right)}}$$

(8)

where P₀ is P on the top of the canopy; k_n describes the rate at which leaf nitrogen content decreases with increasing depth into the canopy, and is taken as 0.3 following the previous study [21].

$$\mathrm{k}={\displaystyle \frac{0.5\mathsf{\Omega }}{\mathrm{c}\mathrm{o}\mathrm{s}\mathsf{\theta }}}$$

(9)

Here, α and P_o represent the two unknown parameters, both of which are contingent on the biological conditions of the leaves. During periods of low incoming radiation, such as in the morning or afternoon, the photosynthetic rate is presumed to be primarily influenced by radiation, leading to the maximum LUE at these times. To determine α for the day, GPP data are selected with incoming PAR less than 350 μmol m⁻²s⁻¹ for regression analysis against APAR. The resulting slope of this regression is considered as α for the given day. The daily value of P₀ is then obtained through the optimization of eddy covariance GPP data after determining α.

3.1.3. The Inversion of V_cmax from Sunlit GPP

According to Collatz’s model:

$$\mathrm{A}=\mathrm{m}\mathrm{i}\mathrm{n}\left({\mathrm{A}}_{\mathrm{j}},{\mathrm{A}}_{\mathrm{c}}\right)-{\mathrm{R}}_{\mathrm{d}}$$

(10)

where A is the gross photosynthesis; A_j represents the rate of photosynthesis limited by radiation and A_c represents the rate of photosynthesis restricted by Rubisco; R_d is the rate of dark respiration.

When estimating V_cmax through the reversal of Collatz’s method, it is essential that the photosynthetic rate is limited by V_cmax, rather than radiation. In shaded leaves, the photosynthetic rate is primarily regulated by radiation. Conversely, sunlit leaves receive more radiation than shaded leaves; thus, their photosynthetic rate is not controlled by radiation but by V_cmax when incident radiation levels are high. The separation of sunlit and shaded leaves enables the inversion of V_cmax from the GPP of sunlit leaves. For sunlit leaves, when incident radiation levels are low, such as in the early morning, the photosynthetic rate is dependent on radiation. However, as incident radiation increases, it gradually becomes limited by Rubisco. A threshold of 900 μmol m⁻²s⁻¹ is utilized for radiation levels. When APAR exceeds this threshold, the photosynthetic rate is predominantly limited by V_cmax. Consequently, V_cmax is derived from the GPP of sunlit leaves under such conditions [31].

Once the GPP of sunlit leaves has been separated, the photosynthesis rate of sunlit leaves can be calculated as:

$${\mathrm{A}}_{\mathrm{c}}={\displaystyle \frac{{\mathrm{G}\mathrm{P}\mathrm{P}}_{\mathrm{s}\mathrm{u}\mathrm{n}}}{{\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{s}\mathrm{u}\mathrm{n}}}}$$

(11)

For C4 plants, according to Collatz’s model [12], when the CO₂ fixation is controlled only by Rubisco, V_cmax is calculated as:

$${\mathrm{V}}_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}={\mathrm{A}}_{\mathrm{c}}$$

(12)

3.1.4. Normalizing V_cmax to 25 °C

The inverted V_cmax can be normalized to 25 °C (V_m25) using the temperature function [79]:

$${\mathrm{V}}_{\mathrm{m}25}={\displaystyle \frac{{\mathrm{V}}_{\mathrm{c}\mathrm{m}\mathrm{a}\mathrm{x}}}{\mathrm{f}\left(\mathrm{T}\right)}}$$

(13)

$$\mathrm{f}\left(\mathrm{T}\right)={\left(1+\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\left(-\mathrm{220,000}+710\left(\mathrm{T}+273\right)\right)}{\left(\mathrm{R}\left(\mathrm{T}+273\right)\right)}}\right)\right)}^{-1}$$

(14)

where T is the leaf temperature; and R is the gas constant.

3.2. Two-Stage Linear Model

Crops typically experience rapid growth in the early stages, reaching a peak before gradually aging. Thus, we divided the crop growth period into an ascending phase and a descending phase (Figure 2). According to FAO56 guidelines [80], the rapid growth period of maize typically spans from 35 to 50 days, counting from the date of emergence. Therefore, within this date range, we identified the day with the smallest change in LAI as the cut-off point between the ascending phase and descending phase, denoted as D_Peak. We discovered that V_m25 exhibits a stronger correlation with leaf age than with LAI, particularly during the descending phase (Figure 2).

Building upon this discovery, we proposed a Two-stage linear model that utilizes leaf age to simulate V_m25 (Figure 3). In this model, V_m25 is linearly fitted separately for the ascending and descending phases (Equations (16) and (17)). Our Two-stage linear model is as follow:

$${\mathrm{V}}_{\mathrm{m}25}=\left\{\begin{array}{c}{\mathrm{k}}_1\times \mathrm{D}+{\mathrm{V}}_{\mathrm{m}25,\mathrm{i}\mathrm{n}\mathrm{i}} , {\mathrm{D}}_{\mathrm{i}\mathrm{n}\mathrm{i}}<\mathrm{D}<{\mathrm{D}}_{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}}\\ {\mathrm{k}}_2\times \left(\mathrm{D}-{\mathrm{D}}_{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}}\right) , {\mathrm{D}}_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}\le \mathrm{D}<{\mathrm{D}}_{\mathrm{e}\mathrm{n}\mathrm{d}}\end{array}\right.$$

(15)

where D_ini represents the leaf age when the crop initiates rapid growth, recorded as 0; D_Peak represents the leaf age when the V_m25 reaches maximum; D_end denotes the ending leaf age of the crop growth period, typically when the GPP value observed by the flux sites approaches 0; and D represents the current leaf age. k₁ and k₂ denote the slopes of the ascending and descending stages of crop development, respectively.

$${\mathrm{k}}_1={\displaystyle \frac{{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{V}}_{\mathrm{m}25}-{\mathrm{V}}_{\mathrm{m},\mathrm{i}\mathrm{n}\mathrm{i}}}{{\mathrm{D}}_{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}}}}$$

(16)

$${\mathrm{k}}_2={\displaystyle \frac{{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{V}}_{\mathrm{m}25}}{{\mathrm{D}}_{\mathrm{e}\mathrm{n}\mathrm{d}}-{\mathrm{D}}_{\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}}}}$$

(17)

Leaf age is commonly used to describe the growth status and developmental stage of plants. Its specific definition and calculation methods can vary depending on the research objectives and plant species. In our study, leaf age refers to the physical age of the plant, meaning the actual time from emergence to the current leaf development stage, measured in days. The emergence dates for US-Ne1, US- Ne2, and US-Ne3 can be obtained from the literature. For the Daman and Fenzidi, the emergence dates can be inferred from the changing trends in LAI values.

3.3. Model Validation

3.3.1. Calibration and Validation of V_m25

Data from one-third of the years were utilized for calibrating the PeakV_m25 (Section 3.2), while data from the remaining two-thirds of years were employed to validate the accuracy of the Two-stage linear model. It is worth mentioning that the estimation of V_m25 for leaves operating under low radiation conditions is typically less accurate. The cumulative effect of errors in various underlying assumptions can lead to a relatively low signal-to-noise ratio [81]. Thus, V_m25 values corresponding to low radiation were excluded.

The PeakV_m25 was calculated for the three US sites (US-Ne1, US-Ne2, US-Ne3) and the two Chinese sites (Daman, Fenzidi) separately, to account for the difference in maize species. To determine the value of the PeakV_m25, data from the first third of each year at each site were selected. Linear regression between V_m25 inverted by LRC and leaf age was performed for both the ascending phase and descending phase, respectively. The average PeakV_m25 value for the first third years of the three US sites and the two Chinese sites are 65 and 38 μmol m⁻²s⁻¹ (rounded to an integer), respectively. Note that V_m25,ini is difficult to estimate when the GPP is close to zero or negative at the beginning of the growth period. Thus, the V_m25 at the beginning of the growth period is set as 10 μmol m⁻²s⁻¹, the median of the LRC inverted V_m25 during this period (5–15 μmol m⁻²s⁻¹).

3.3.2. Comparison with the V_m25 Obtained by BESS

In the BESS model, seasonal variation in V_m25 is taken into account [56,82]. It is assumed that the seasonal pattern of V_m25 followed the seasonal pattern of LAI [83]. V_m25 experiences a rapid increase during leaf development, reaching its peak in early leaf maturity, followed by a decline during senescence, irrespective of species. The date corresponding to the peak LAI value is identified, and the V_m25 for that date was quantified (PeakV_m25). Subsequently, V_m25 over the season is calculated as:

$${\mathrm{V}}_{\mathrm{m}25}={\mathrm{V}}_{\mathrm{m}25,\mathrm{m}\mathrm{i}\mathrm{n}}+{\displaystyle \frac{\mathrm{L}\mathrm{A}\mathrm{I}}{{\mathrm{L}\mathrm{A}\mathrm{I}}_{\mathrm{m}\mathrm{a}\mathrm{x}}}}\times \left({\mathrm{P}\mathrm{e}\mathrm{a}\mathrm{k}\mathrm{V}}_{\mathrm{m}25}-{\mathrm{V}}_{\mathrm{m}25,\mathrm{m}\mathrm{i}\mathrm{n}}\right)$$

(18)

where LAI_max and LAI represent the maximum and current leaf area index, respectively, throughout the growing period. In the original BESS model, the PeakV_m25 value of C4 crops was uniformly set as 33 μmol m⁻²s⁻¹, without distinction among crop types. V_m25,min = 0.3 × PeakV_m25.

In the original BESS model (Equation (18)), the PeakV_m25 value of C4 crops was uniformly set at 33 μmol m⁻²s⁻¹, without differentiation among crop species. In our model, the calibrated PeakV_m25 value was substituted into the original BESS model to calculate V_m25 (hereafter referred to as “BESS_P”), and then the performances of the original BESS, BESS_P and our Two-stage linear model in simulating V_m25 were compared against the V_m25 inversely solved by coupling the BESS and LRC using EC data.

3.3.3. Comparison with the GPP Obtained by BESS

Then, the original BESS model, BESS_P model, and the BESS model coupled with the Two-stage linear model (hereinafter referred to as “BESS_TL”) were used to calculate GPP. These simulated GPP values were then compared with the GPP values measured at the flux sites to evaluate the performance of each model. This comparison helps in assessing the accuracy and reliability of each method in simulating GPP.

4. Results

4.1. Calibration and Validation of V_m25

To assess the performance of the Two-stage linear model (Equation (15)), the simulated V_m25 was compared with the V_m25 inversely solved by coupling the BESS and LRC using EC data (Figure 4,

Table 2. The performance of the original BESS, BESS_P and our Two-stage linear model at simulating the V_m25 inversely solved by coupling the BESS and LRC using EC data at the five flux sites.

Site	BESS		BESS_P		BESS_TL
	RMSE	R2	RMSE	R2	RMSE	R2
US-Ne1	18.93	0.23	25.07	0.23	7.54	0.85
US-Ne2	16.58	0.52	22.67	0.52	6.82	0.88
US-Ne3	14.70	0.67	19.46	0.67	6.82	0.87
Daman	14.55	0.02	16.99	0.02	6.44	0.71
Fenzidi	13.25	0.01	15.30	0.01	5.40	0.82

). Notably, the simulated V_m25 aligns closely with that from LRC for both the calibration and validation samples at each site (Figure 4, Table 2). The performance of the Two-stage linear model was exceptional for the US-Ne2 and US-Ne3 sites, yielding R² values of 0.88 and 0.87, RMSE values of 6.82 μmol m⁻²s⁻¹ for both, whereas the performance was not as good for Daman (R² = 0.71; RMSE = 6.44 μmol m⁻²s⁻¹), due to the pronounced fluctuation in V_m25 during the early stages of the growth period at Daman (Figure 5y, z).

4.2. Comparison with the V_m25 Obtained by BESS

The V_m25 calculated by BESS, BESS_P and our Two-stage linear model were compared in simulating the time series of V_m25 inversely solved by coupling the BESS and LRC using EC data at the five flux sites (Figure 5). Unfortunately, the flux data for Daman in 2020 and Fenzidi in 2017 are incomplete; and due to the partial lack of remote sensing and meteorological data, the V_m25 inverted by LRC for US-Ne1 and US-Ne2 sites in 2001 is also incomplete. The BESS model consistently underestimates V_m25 throughout the growth period due to the fixed low PeakV_m25 value, with the discrepancy becoming particularly noticeable during the middle stages of the growth period. Conversely, towards the end of the growth period when V_m25 values are low, BESS tends to overestimate V_m25. BESS_P, which employs the same modeling approach as BESS but adjusts the PeakV_m25 parameter, exhibiting a somewhat improved performance in estimating the trend of V_m25 during the early growth phase. However, during the declining stage of V_m25, BESS_P tends to significantly overestimate the values. Due to the same formula structure, the R² between the calculated V_m25 of both methods and the verified values remains consistent, ranging from 0.01 to 0.67 (Table 2). The RMSE of the BESS model ranges from 13.25 to 18.93 μmol m⁻²s⁻¹ and the RMSE of the BESS_P ranges from 15.30 to 25.07 μmol m⁻²s⁻¹. Despite the adjustment in PeakV_m25, both BESS and BESS_P demonstrate poor performance in fitting the trend in V_m25 across all sites. The observations reveal a distinct pattern of V_m25 initially rising before declining, and our Two-stage linear model effectively captures this seasonal variation in V_m25, with R² ranges from 0.71 to 0.88 and RMSE ranges from 5.40 to 7.54 μmol m⁻²s⁻¹. During the early stages of crop growth, V_m25 generally exhibits an upward trajectory, albeit with significant diurnal fluctuations. Conversely, in the descending phase, V_m25 tends to display a more linear decline with reduced fluctuation.

4.3. Comparison with the GPP Obtained by BESS

To evaluate the role of the Two-stage linear model in improving GPP simulation, we compared the GPP simulation results using BESS, BESS_P and our Two-stage linear model for calculating V_m25, and validated them with flux site GPP data (Figure 6,

Table 3. The performance of the original BESS, BESS_P and BESS_TL in estimating observed GPP at the five flux sites.

Site	BESS		BESS_P		BESS_TL
	RMSE	R2	RMSE	R2	RMSE	R2
US-Ne1	3.69	0.66	4.07	0.70	2.29	0.86
US-Ne2	3.48	0.80	3.53	0.83	2.03	0.90
US-Ne3	2.99	0.83	3.32	0.85	2.04	0.89
Daman	2.96	0.53	3.02	0.53	2.32	0.82
Fenzidi	3.35	0.63	3.64	0.63	2.06	0.87

). The results indicate that the original BESS model consistently underestimates GPP across all five sites. On average, GPP is underestimated by 12.60%, with the most significant underestimation observed at US-Ne2 (16.29%). Similarly, BESS_P generally overestimates GPP, averaging 20.73% across all sites, with the most significant overestimation recorded at US-Ne1 (24.05%). In contrast, the GPP simulated by BESS incorporating our Two-stage linear model exhibits a strong correlation with the flux site’s observed GPP data, with RMSE ranging from 2.03 to 2.32 gC m⁻²d⁻¹ and R² ranging from 0.82 to 0.90. On average, GPP is only underestimated by 0.85% across all sites. The most substantial underestimation occurs at Daman (16.38%), while the most significant overestimation is at Fenzidi (4.60%). The simulation result of BESS_TL significantly outperform the BESS and BESS_P in GPP estimation.

To better reveal the role of the Two-stage linear model in improving GPP simulation, the discrepancy of the annual GPP estimation by BESS, BESS_P and BESS_TL were compared (Figure 7 and Figure 8). The mean GPP estimation discrepancy by BESS, BESS_P and BESS_TL are −1.56, 2.39 and 0.12 gC m⁻²d⁻¹ in US-Ne1, respectively, with the corresponding standard deviations of 3.35, 3.30 and 2.29 gC m⁻²d⁻¹ (

Table 4. Mean and standard deviation of the GPP results by the three methods in each site (gC m⁻²d⁻¹).

Site	BESS		BESS_P		BESS_TL
	Mean	Std	Mean	Std	Mean	Std
US-Ne1	−1.56	3.35	2.39	3.30	0.12	2.29
US-Ne2	−1.69	3.03	2.40	2.59	−0.02	2.03
US-Ne3	−1.56	2.56	2.31	3.32	0.29	2.02
Daman	−0.43	2.92	0.22	3.01	−1.42	1.82
Fenzidi	−0.06	3.29	−0.13	3.49	0.05	2.00

). Except for the Daman site, the means and standard deviations of GPP discrepancy of BESS_TL estimation are closer to 0 for most years compared to the BESS and BESS_P estimation (Table 4). At the Daman site, although the absolute value of the mean GPP estimation discrepancy of BESS_TL is greater than that of BESS, the standard deviation of BESS_TL GPP estimation discrepancy remains the smallest. Thus, it can be concluded that BESS_TL effectively enhances the simulation accuracy of GPP.

To provide a clearer view of the simulation results of the three methods, the seasonal variation in the GPP difference between the simulated GPP and EC observations is represented in Figure 9 and Figure 10. Figure 9 displays the annual simulation results for each site, while Figure 10 presents the multi-year average simulation outcomes across all sites. Significant seasonal differences are evident in the simulation outputs of all three methods. The original BESS model tends to slightly overestimate GPP towards the end of the growth period and underestimates during other times, particularly in the mid-growth period, where the underestimation is pronounced. Conversely, BESS_P demonstrates satisfactory simulations only during the early growth period, followed by a gradual overestimation of GPP. BESS_TL outperforms the other two methods in capturing seasonal shifts in GPP. The discrepancy between the simulated GPP and EC values remains relatively close to zero. GPP is slightly underestimated during the early and late growth stages and slightly overestimated during the mid-growth stage. Overall, BESS_TL demonstrates a more balanced performance across different growth stages compared to BESS and BESS_P.

5. Discussion

5.1. Foundation of V_m25 Estimation

This study highlights the importance of considering seasonal changes in the photosynthetic capacity of crops. However, estimating the temporal changes in V_m25 remains challenging. The method to inverting the time series of V_m25 by coupling the BEPS model and LRC has already been proposed. However, the BEPS model does not account for C4 plants [31,32]. In our research, we addressed this challenge by inverting V_m25 for five maize sites through the coupling of the BESS model with LRC. This approach enabled us to estimate V_m25 without relying on extensive gas exchange experiments. To achieve this, we utilized field measurements of EC data and remote sensing data. The BESS model, which replaces the original BEPS model, incorporates specific considerations for the photosynthesis of C4 plants like maize [56]. After removing the data from periods of low radiation, V_m25 exhibited a significant trend of initially increasing and then decreasing (Figure 5). Furthermore, V_m25 showed a strong correlation with leaf age when the growing period was divided into two phases (Figure 2). This relationship provides the basis for our research.

5.2. The Advantages of Leaf Age as a V_m25 Predictor

Annual plants complete their entire life cycle within one growing season. Starting from seeds, they progress through stages of germination, growth, flowering, fruiting, and ultimately die, all within a single year. In contrast, perennials continue to grow and reproduce for several years or more. Compared to perennial plants, using annual plants as research subjects eliminates the need to consider interannual influences and allows for easier identification of their developmental stages. This simplification makes it more straightforward to study the relationship between leaf age and V_m25.

Our Two-stage linear model takes the advantage of leaf age and effectively captures the V_m25 seasonal variation with the crop growth processes. Maize, being annual plants, undergo significant changes in photosynthetic capacity throughout their growth cycle due to various physiological characteristics. Typically, V_m25 exhibits only one peak value during the entire crop growth period [33]. This uniqueness of the crop is the basis for our proposed model. In our model, the basic assumption is that the photosynthetic capacity increases during leaf development, reaching a maximum in spring or early summer, stabilizing or gradually decreasing in summer, and further decreasing during senescence [6,84]. Comparably, our model, in which V_m25 is quantified with leaf age, can perform well due to the strong correlation between V_m25 and leaf age (Figure 2). Now, it is understandable that the performance of the existing empirical relationships established between V_m25 and other variables, such as N_Leaf, Chll, and photoperiod, is basically determined by the similarity of that variable to the leaf age [7,8,40,85]. Also, the photosynthetic capacity and environmental factors are often mismatched [57].

For the estimation of V_m25, our Two-stage linear model outperforms both the original BESS model and BESS_P. The main difference between the original BESS model and BESS_P is the PeakV_m25: PeakV_m25 in the original BESS model is fixed, whereas the PeakV_m25 in the BESS_P is the same as the PeakV_m25 in our Two-stage linear model. Thus, solely improving the PeakV_m25 cannot solve the problem. In the original BESS model and BESS_P, LAI is used to seasonalize V_m25 [56]. However, it is proven that the correlation between V_m25 and LAI is not as strong as leaf age in the later half stage of crop growth (Figure 2) because the leaves turn yellow and senesce with the senescence of crops, but the change in LAI is not significant. Furthermore, the BESS model coupled with the Two-stage linear model (BESS_TL) has significantly improved the simulation of GPP. Across the five maize sites, BESS_TL only underestimated GPP by an average of 0.85%. In contrast, the original BESS model underestimated GPP by an average of 12.60%, and the BESS_P model overestimated GPP by an average of 20.73%. The superior performance of our Two-stage linear model underscores the importance of considering PeakV_m25 and leaf age in accurately modeling V_m25.

5.3. Readily Available Leaf Age

Leaf age, an important indicator of crop physiological characteristics, is usually relatively easy to obtain. The emergence date can be directly observed in the field, allowing for straightforward counting to determine leaf age [62]. Additionally, remote sensing technologies offer a powerful tool for estimating leaf age across large areas of farmland [86,87]. By calculating vegetation indices from remote sensing images, we can indirectly assess the growth stage of leaves. These indices exhibit specific patterns of change as the leaves progress from young to mature and eventually to senescent stages. By analyzing these patterns, we can accurately infer leaf age.

5.4. Model Limitations

Due to data availability, our Two-stage linear model was only validated with maize data from five flux sites. Its applicability to other sites relies on the quality of their own flux data. Also, our Two-stage linear model is derived from maize sites, and its applicability to other vegetation types requires further validation. While other vegetation types have also demonstrated a close correlation between V_m25 and leaf age [8,58,63,64], extending this method to other vegetation types, such as perennial plants, presents additional challenges. Perennials often have canopies that consist of leaves at various stages of development, ranging from young and fully functional to older and senescent leaves [63]. This variation necessitates a more nuanced approach to modeling V_m25 in perennials, one that can account for the complex age structure of their canopies [58].

6. Conclusions

As numerous studies have highlighted, V_m25 is a crucial parameter for calculating the photosynthesis rate. The accurate estimation of V_m25 is essential for the regional and global modeling of ecological systems. Although various indicators have been developed to characterize V_m25, accurately quantifying its dynamic changes remains challenging.

(1) 

V_m25 inversion: Considering the special photosynthetic process of C4 plants, we replaced the BEPS model with the BESS model coupled with the LRC to invert V_m25 at five maize sites. This method allowed us to obtain continuous V_m25 values throughout the growth period, enabling a detailed study of V_m25 variation trends.

(2) 

Two-Stage Linear Model Development: We developed a new Two-stage linear model to determine the dynamics changes in maize V_m25. This method divides V_m25 into two stages during the growth process and uses leaf age as the key variable in the simulation, effectively capturing the seasonal variation characteristics of V_m25. Additionally, compared to using a fixed value, this method allows for the calibration of the PeakV_m25 value, thereby enhancing model accuracy.

(3) 

Model Performance and Comparison: The Two-stage linear method more accurately simulated the variation trend of V_m25 compared to the V_m25 of the original BESS model. Furthermore, implementing this method significantly improved the simulation results of GPP. BESS_TL outperforms the other two methods in this study, showing higher R² and lower RMSE at each site. The GPP simulated by BESS_TL at both interannual and seasonal levels deviated less from the EC GPP. Overall, the developed V_m25 estimation method enhances the accuracy of farmland GPP simulation.