As an indispensable link in the soil-plant-atmosphere continuum, canopy transpiration (E_c) reflects the exchange and balance of material and energy between plants and environmental factors. Plant stoma not only transpire water into the atmosphere, but also uptake CO₂ into the plant. The opening and closing statuses of stoma are largely controlled by the surroundings and directly affect the extent of plant photosynthesis and the strength of E_c.

In practical applications, a more useful variable is the canopy conductance (g_c), which treats all stoma in parallel or considers the entire canopy as a big leaf. The g_c is commonly used in ecological, meteorological and air quantity simulation models. Its value is usually computed by a scale transformation method that extends the stoma conductance of a single leaf over the canopy. The disadvantages of this method are the difficulty of acquiring long-term observations, and large errors may occur in the scale conversion processes. The errors are introduced by the inhomogeneous distribution of sunlight at the canopy level and the different chlorophyll contents of older and younger leaves (Alves et al. 1998; Green et al. 2003).

Sap flow can now be continuously and accurately monitored by Granier' s thermal dissipation probe (TDP) method (Granier 1985; 1987). With recent advances in micro-meteorological observation technology, we can also compute the canopy conductance from sap flow and meteorological data by inverting the Penman-Monteith (PM) equation. This method is popular, because it easily and accurately obtains the long-term canopy conductance. For example, the g_cs of pine forest in Norfolk (Stewart 1988), Moso bamboo (Phyllostachys edulis) in western Japan (Komatsu et al. 2012), Douglas-fir forest in the Netherlands (Bosveld et al., 2001), and rain-fed citrus orchard under sub-humid tropical conditions (Oguntunde et al., 2007) have been determined by this method.

The opening and closing of g_c mainly depends on environmental factors such as solar radiation, air temperature, humidity, and soil moisture (Jarvis 1976; Komatsu et al. 2012; Oguntunde et al. 2007; Whitley et al. 2009). To predict g_c, multiple linear equations are advantaged by their simplicity, but non-linear Jarvis equations can better explain the mechanism of stomatal activity. However, most studies have focused on the applicability of g_c prediction methods or modified the pre-existing equations to improve the correlation between predictions and observations. The physical and mathematical rationalities of the modeling approaches have been largely ignored.

Qinghai spruce (Picea crassifolia) is one of the main constructive species that is widely spread throughout Qinghai, Gansu, Inner Mongolia, and Ningxia. It inhabits altitudes between 1 600 m and 3 800 m. Chang et al. (2014)investigated the relationships between g_c and environmental factors in a Qinghai spruce forest in the upper Heihe River Basin of arid northwestern China. However, classical g_c models have yet to be applied to Qinghai spruce. To explore the mechanisms by which environmental factors affect the water use of forest eco-systems at the canopy level and to reveal the adaptabilities of different canopy conductance models, we simulate the changing canopy conductance in Qinghai spruce forest by implementing the PM formula in two models: multiple linear regression and the non-linear Jarvis model. We also hope to find a model to simulate g_c simply and accurately, and to guide transpiration estimating and parameters calibrating with transportation models in high-cold region of western China.

1 Materials and Methods 1.1 Site description

Qinghai spruce was a dominant species in Datong County (Xining, Qinghai Province, China), but was largely destructed in the 1950s. The damaged areas have since been occupied by the pioneer species Betula platyphylla (white birch). To promote natural regeneration, numerous Qinghai spruce seedlings were planted under white birch or in forest clearings in the 1980s. The study site (latitude 37°10′N, longitude 101°34.5′E, altitude 2 860 m above sea level) is located in a 30-year-old Qinghai spruce plantation in the Baoku forestry farm, Datong County. The tree spacing is 3 m × 4 m, giving a density of 833 trees·hm^-2 at this site. The mean tree height is 11.35 m. The climate is a classic plateau continental climate characterized by abundant sunshine (2 553 h·a^-1), long cold winters, short cool summers, and distinct wet (May-September) and dry (October-April) seasons. The total annual rainfall and annual mean air temperature is 523.3 mm and 4.9 ℃, respectively.

1.2 Sap flow measurements

Sap flow was monitored from June 1 to 30 in 2013 by a 32-stem flow measurement system (5 probes of 3 cm) based on Granier's TDP method (Dynamax Co., USA). Two cylindrical probes (single measurement set) were vertically implanted in the sapwood of the tree trunks. The downstream probe was continuously heated by a constant power source, and the unheated upstream probe provided a temperature reference. Temperature difference between the two probes was converted into a stem flow speed of the trunk by Eq. (1) (Granier 1985). The data logging internal was set to 15 min.

$ f = 0.011\;9{\left({\frac{{\Delta {T_{{\rm{max}}}} - \Delta T}}{{\Delta T}}} \right)^{1.231}} $

(1)

where f is stem sap flux density (cm cm^-2·s^-1), ΔT_max is the maximum temperature difference between the two probes (℃, measured over a > 24 h period), and ΔT is the current temperature between the two probes (℃).

Five Qinghai spruce trees with diameters near the mean tree diameter (12.5 cm) of the experimental plot were selected as representative samples. The trees are relatively uniform in age and size. The sapwood area of the samples was determined by a power-law relationship between sapwood area (A_s) and trunk diameter (D_t) (Eq. (2); r²=0.99, n=20). Canopy transpiration (E_c^m) was then calculated by Eq. (3).

$ {A_s} = 0.14D_t^{2.44} $

(2)

$ E_c^m = \frac{{3\;600 \cdot f \cdot {A_s}}}{{1\;000 \cdot {A_c}}} $

(3)

Where, E_c^m is measured in mm h^-1 and D_t, A_s and A_c denote the trunk diameter (cm), sapwood area and canopy projection area (m²) at the measured height (cm²), respectively.

1.3 Meteorological measurements

A portable automatic weather station (Weather link vantage pro2; Davis Co., US) was installed at a height of 2 m in a forest clearing and approximately 20 m from the sample area. The meteorological parameters, namely, rainfall, humidity, solar radiation (R, W·m^-2), air temperature (T, ℃), air pressure (P, kPa) and wind speed (U, m·s ^-1) were synchronously logged with the data of TDP.

1.4 Calculation of canopy conductance

The PM equation is popular because it clearly separates the influences of the atmosphere and the plant/soil system on transpiration process. Typically, g_c is calculated from sap flow and meteorological data by inverting the PM equation (Eq. (4)) (Amp et al. 1994; Buckley et al. 2012; Chang et al. 2014; Granier et al. 2000; Komatsu 2004; Whitley et al. 2009):

$ g_{\rm{c}}^{\rm{m}} = \frac{{\lambda E_{\rm{c}}^{\rm{m}}\gamma {g_{\rm{a}}}}}{{\Delta \left({{R_{\rm{n}}} - G} \right) + \alpha \rho {C_{\rm{p}}}D{g_{\rm{a}}} - \lambda E_{\rm{c}}^{\rm{m}}\left({\Delta + \gamma } \right)}} $

(4)

where g_c^m is the measured canopy conductance (m·s^-1), Δ is the slope of a saturation vapor pressure versus temperature plot (kPa·℃^-1), R_n is the net radiation above the forest canopy (MJ·m^-2 h^-1), and the soil heat flux G (MJ·m^-2 h^-1) is set to 0.1 R_n (Allen et al. 1998). α is a conversion factor (3 600 s·h^-1) that converts E_c^m to mm h^-1, ρ is the air density (kg·m^-3), C_p is the specific heat of air (1.013×10^-3 MJ·kg^-1 ℃^-1), γ is the psychometric constant (kPa·℃^-1), and λ is the latent heat of vaporization (MJ·kg^-1 ℃^-1). The aerodynamic conductance g_a from the canopy to the reference level of the scalar quantities (m·s^-1) is calculated by Eq. (5) (Rana et al. 2005; Kang et al. 1991).

$ {g_{\rm{a}}} = \frac{{{k^2}{u_z}}}{{{\rm{ln}}\left({\frac{{z - d}}{{{z_0}}}} \right){\rm{ln}}\left({\frac{{z - d}}{{{h_c} - d}}} \right)}} $

(5)

where k is the Karman constant (0.41), u_z is the wind speed at the reference height (m·s^-1), h_c is the height of the measured tree (m), and z is the reference height (m). The roughness length z₀ and zero-plane displacement d were set to 0.1 h_cm and 0.67 h_c, respectively.

1.5 Prediction of canopy conductance

From the main meteorological factors, we computed g_c by two common methods: (1) multiple linear regression analysis and (2) the non-linear Jarvis regression model. The multiple linear regression formula is given by Eq. (6) (Oguntunde et al. 2007).

$ {\rm{ln}}\left({\mathit{g}_{\rm{c}}^{\rm{p}}} \right) = {\beta _0} + {\beta _1}\mathit{D} + {\beta _2}\mathit{T} + {\beta _3}\mathit{R} $

(6)

where g_c^p is the predicted value of g_c, and the parameters β₀-β₃ are determined by multiple linear regression of D, T and R versus g_c^m.

Jarvis model is among the most widely used non-linear methods in g_c predictions. The original Jarvis model is a product of five response functions with independent inputs of photon flux density, leaf water potential, vapor pressure deficit, leaf temperature and ambient CO₂ concentration (Jarvis, 1976). To estimate g_c from the commonly measured meteorological factors, researchers have revised Jarvis equation several times. In this study, we multiply the maximum canopy conductance (g_cmax) by the response functions of vapor pressure deficit, air temperature and solar radiation (Oguntunde et al., 2007; Stewart, 1988). The values of the response functions range from 0 to 1, and the model is given by Eq. (7):

$ g_{\rm{c}}^{\rm{p}} = {g_{{\rm{cmax}}}}f\left(D \right)f\left(T \right)f\left(R \right) $

(7)

where f(D) is computed as

$ f\left(D \right) = {{\rm{e}}^{ - {k_1}D}} $

(8)

and f(T) is given by Eqs.(9) or (10).

$ f\left(T \right) = \frac{{\left({T - {T_{\rm{L}}}} \right)\left({{T_{\rm{H}}} - T} \right){^{\frac{{{T_{\rm{H}}} - {k_2}}}{{{k_2} - {T_{\rm{L}}}}}}}}}{{\left({{k_2} - {T_{\rm{L}}}} \right){{\left({{T_{\rm{H}}} - {k_2}} \right)}^{\frac{{{T_H} - {k_2}}}{{{k_2} - {T_L}}}}}}} $

(9)

$ f\left(T \right) = {{\rm{e}}^{ - {k_3}{T^2}}} $

(10)

Given that temperature can influence g_c and E_c in the long-term, the optimal temperature (T_opt) was incorporated into Eq. (10). The revised formula is given by Eq. (11):

$ f\left(T \right) = {{\rm{e}}^{ - {k_4}{{\left({T - {T_{opt}}} \right)}^2}}} $

(11)

and f(R) is given by Eq. (12) or (13):

$ f\left(R \right) = \frac{R}{{{R_{\rm{m}}}}}\frac{{{R_{\rm{m}}} + {k_5}}}{{R + {k_5}}} $

(12)

$ f\left(R \right) = {k_6} + {k_7}R + {k_8}{R^2} $

(13)

where the high and low temperatures T_H and T_L are fixed at 45 ℃ and 0 ℃, respectively, and R_m is the radiation constant. Typically, R_m is arbitrarily assigned between 1 000 and 2 000 W·m^-2 (Oguntunde et al., 2007). The maximum local R throughout the study period was measured as 1 145 W·m^-2. Therefore, to constrain f(T) between 0 and 1, we set R_m to 1 200 W·m^-2. The undetermined parameters g_cmax, T_opt and k₁-k₈ can be determined by multivariate nonlinear regression of the measured g_c and the environmental factors.

Because both of f(T) and f(R) are not uniquely expressed, we investigated six Jarvis models with different forms of the response functions, as shown in Tab. 1.

The decoupling factor (Ω) quantifies the coupling between the evaporating surface and free airstream conditions. It varies between 0 (perfect coupling) and 1 (complete isolation) (Jarvis et al., 1986; Kumagai et al., 2004), as apparent from Eq. (14):

$ \Omega = \frac{{1 + \frac{\Delta }{\gamma }}}{{1 + \frac{\Delta }{\gamma } + \frac{{{g_a}}}{{{g_c}}}}} $

(14)

1.6 Time lag

Although the activity of leaf stoma is controlled by the external environment, it does not respond immediately to changes in the external environmental factors (Hu, 2010). In this paper, g_c was calculated from sap flow data. Therefore, when cross-correlating E_c/g_c with the external environment factors (D, T, R, E_p), we advanced or delayed the sap flow by 15-step intervals, up to the essential time lag (τ, ± 120 minutes). The correlation was calculated as the square of the Pearson coefficient (r²):

$ {r^2} = \left\{ {\frac{{{\rm{Cov}}\left[ {X\left(t \right), Y\left({t + \tau } \right)} \right]}}{{{\delta _x}\delta _y}}} \right\}^2 $

(15)

In Eq.(15), Cov; ] denotes the covariance function, X and Y are series of environmental factors and g_c (or E_c), respectively, and δ_x and δ_y represent the standard deviations of X and Y, respectively. t is the current time, and τ is the lag time (in minutes) between X and Y.

1.7 Statistics and validation

To validate the selected model, we divided the whole dataset into odd days (Database A) and even days (Database B), depending on their day date. The selected models were then fitted to Database A and validated on Database B (Han et al., 2012). The predicted g_c (g_c^p) and E_c (E_c^p) were separately compared against their own measured values.

Environmental factors such as rainfall, morning dew and fog will wet the leaves, causing errors in the calculated canopy conductance. To avoid such effects, we input only measurements collected from 10:00 to 18:00 (UTC+8). Data with R below 120 W·m^-2 (the lowest R measured under sunny conditions by a standard meteorological station), data with obvious abnormalities (such as those introduced by voltage instability when replacing the TDP batteries), and data collected on overcast and rainy days, were also excluded.

The discrepancies between the predicted and the measured values were analyzed by statistical parameters, namely, the root mean square error (RMSE), mean absolute error (MAE) and mean relative error (MRE, %), respectively calculated by Eqs. (16), (17) and (18).

$ {\rm{RMSE}} = \sqrt {\frac{1}{{n - 1}}\sum {{{\left({{y_i} - {x_i}} \right)}^2}} } $

(16)

$ {\rm{MAE}} = \frac{1}{n}\sum {\left| {{y_i} - {x_i}} \right|} $

(17)

$ {\rm{MRE}} = \frac{1}{n}\sum {\frac{{\left| {{y_i} - {x_i}} \right|}}{{{y_i}}}} $

(18)

Here, y_i and x_i denote the i^th measured and predicted values, respectively, and n is the number of sample pairs.

To optimize the unknown coefficients, we ran 10 computations in 1^st Opt software (Chinese version; the English version is Auto2Fit). For the numerical optimization calculation, we selected a software platform that solves non-linear regression and other mathematical problems without guessing the initial start values (http://www.7d-soft.com/en). We applied the Levenberg-Marquardt algorithm (Marquardt 1963), and the regression parameters that maximized r² and minimized the RMSE. Most of the remaining statistical analyses were performed in Microsoft Excel, and all figures and fitted curves were generated by SigmaPlot software.

2 Results 2.1 Time lag and environmental conditions

Because E_c modeling is subject to hysteresis the time lags (τ) between E_c and potential transpiration (E_p), estimated with the procedure outlined in FAO 5 (Allen et al., 1998), and other atmosphere variables (D, T and R) were determined by cross-correlation analysis. As shown in Fig. 2, E_c lagged R and E_p by 15 minutes (r²=0.49 and 0.41, respectively), D and T by 120 minutes (although the correlations with these variables were low; r²=0.17 and 0.22 respectively). Thus, when modeling the canopy conductance in the next subsection, we imposed a time lag of τ=15 behind the fluctuations in the meteorological parameters.

As shown in Fig. 3, the g_c (which varied from 0.003 8 m·s^-1 to 0.020 2 m·s^-1) was a clearly decreasing function of vapor pressure deficit (D), which fluctuated from 0.48 to 2.59 kPa, and air temperature (T), which varied from 12.7 ℃ to 25.6 ℃. In contrast, g_c was not obviously related to R.

The E_c displays opposite trends; that is, no significant correlations exist between E_c and D/T, but E_c clearly increases with R (Fig. 4).

Figure 5 plots the Ω values of the Qinghai spruce throughout the time series. The Ω vales ranged from 0.01 to 0.19, which indicated that the leaf surfaces were very strongly coupled to the surrounding air rather than to R.

2.2 Simulation of canopy conductance

The values of g_cmax, T_opt and k₁-k₈ in the different Jarvis models were optimized by a non-linear least-squares algorithm performed with 1stOpt software. Table 2(A) listed the obtained parameter values and their statistics. The six non-linear Jarvis functions explained at least 91% (0.91 < r² < 0.92) of the variations, with an overall error of approximately 0.001 0 m·s^-1. This indicates that all six equations can accurately simulate g_c. The MREs vary from 7.86% (model 5) to 8.43% (model 1). In addition, the r² calculations indicate that f(D) is best modeled by Eq. (10) and least well-modeled by Eq. (11). Eq.(9) yields an intermediate performance, whereas Eq. (13) better computes the f(R) than Eq. (12).

Parameters of the multiple linear model (Eq. (6)), which predicts the g_c from meteorological variables, are presented in Tab. 2(B). This function yields a lower correlation (r²=0.90) and higher RMSE (0.001 1 m·s^-1) among the three variables than the six Jarvis models, but the MRE (8.28%) is the third-highest of the seven evaluated functions. Therefore, the multiple linear model can also estimate g_c with high accuracy.

2.3 Validation of canopy conductance models

The parameters of the g_c functions determined from database A were validated on database B. As shown in Figure 6, all of the regression lines between the measured and predicted g_c approximate the line y=x. In addition, the RMSE values of all seven models approximated 0.003 8 m·s^-1 and the MREs varied from 7.17% (Jarvis model 5) to 8.42% (multiple linear model) (Tab. 3). Again, the best and worst predictors of f(D) were Eqs. (10) and (11), respectively, and Eq. (9) showed intermediate performance. Based on the fluctuations in the MRE values, Eq. (13) more accurately predicted the f(R) than Eq. (12).

An objective of the g_c models is to accurately estimate the E_c. Computations on database B reveal no statistical differences between the measured and predicted E_c (P > 0.01; Student's t-test), and the points on the predicted versus measured E_c plots distribute around y=x (Fig. 7). The errors between E_c^p and E_c^m are also presented in Tab. 3. The seven models estimated the 1/4-hourly E_c and daily E_c with MREs varying from 7.87% to 9.47% and from 5.46% to 7.39% respectively. The E_c prediction accuracies of the six Jarvis models followed the same trend as the g_c prediction accuracies. The multiple linear method gave the second-worst performance in the 1/4 hourly E_c prediction, but the best performance (tying with Jarvis model 6) in the daily E_c prediction.

3 Discussion

In this study, E_c (or measured sap flow) lagged behind R and E_p by 15 minutes. Similarly, Allen et al. (1998) reported a 30 minute lag of the E_c response to environmental factors in a rain-fed citrus orchard under sub-humid tropical conditions. Their data were logged at 30-minute intervals. Green et al. (2003) expected a response delay in apple trees at the Massey University research orchard near Palmerston North, New Zealand and at the Nelson Research Centre, Motueka in summer. However, Granier et al. (2000) reported no time lag in two beech stands inhabiting a rainy climate.

In this study, the g_c^m is a clearly decreasing function of vapor pressure deficit (D), while it was not obviously related to R. Similar results were presented by Chang et al. (2014), who investigated a Qinghai spruce forest in the upper Heihe River Basin in arid northwestern China. In a conifer study, Morén (1999) also reported a stronger dependence of canopy conductance on D than on R. The E_c displays opposite trends; that is, no significant correlations exist between E_c and D/T, but E_c exponentially increases with R(Fig. 3). The light appeared to saturate around 700 W·m^-2, meaning that a further increase in R yielded no further increase in E_c^m. Similarly, Oguntunde et al. (2007) reported that light saturates at approximately 400 W·m^-2 under sub-humid tropical conditions. In summary, the E_c of the Qinghai spruces is mainly driven by R, but the opening and closing of the g_c is mainly controlled by both D and T.

The Ω vales range from 0.01 to 0.19, similar to those obtained in two natural rainforests in French Guiana (NW of Kourou) during two successive dry seasons (0-0.2) and in the state forest of Hesse, France (0.05-0.2) (Granier et al., 1996). The Hesse forest comprises 90% beech trees, and experiences an annual precipitation and average annual temperature of 820 mm and 9.2 ℃, respectively (Granier et al. 2000). Ω values between 0 and 0.4 were reported in a natural forest in Lambir Hills National Park, which has a tropical climate (Kumagai et al. 2004). Jarvis et al. (1986) pointed out that Ω is very low in Pineceae plantations (normally below that of broad-leaved vegetation), indicating that the leaf surfaces are very strongly coupled to D rather than to R.

This study newly applied linear and non-linear g_c prediction models to Qinghai spruce, and evaluated their accuracies. As demonstrated above, the non-linear Jarvis methods conferred no spectacular advantage over the multiple linear model. The multiple linear model was the best performer in the daily E_c prediction. Various forms of g_c equations have been extensively evaluated. Although Oguntunde et al. (2007) also evaluated the accuracies of the Jarvis method and the multiple linear equation, they compared only the measured and predicted g_c values in a citrus orchard. They concluded that the Jarvis model performs slightly better than the multiple linear equation, but both methods accurately determined the g_c in their setting. The accuracy of E_c prediction was not verified. In a Populus pruinosa forest in Xinjiang of China, Wang et al. (2016) reported that the Jarvis method significantly outperforms the Ball model, a linear model advanced by Ball et al. (1987) and improved by Leuning (1995). Conversely, Wu et al. (2007) found that the Ball model outperformed the Jarvis method under urban conditions in north China in 2007. A similar conclusion was reached by Wang et al. (2012), who studied the winter wheat Lu 23 on the North China Plain.

For this Qinhai spruce, the canopy conductances (g_c) obtained from the sap flow rates were up-scaled to represent the transpiration rates based on the quarter-hourly temporal scale. Their values ranged from 0.003 81 m·s^-1 to 0.020 2 m·s^-1, with a mean of (0.010 1 ± 0.003 7)m·s^-1. Estimates of the same magnitude have been reported in other spruce stands. In 1998, Oltchev et al. (1998) reported a maximum canopy conductance of slightly under 0.001 2 m·s^-1 for 114-year-old spruce (Picea abies) on a large plateau in the Soiling Hills in Central Germany. Chang et al. (2014) obtained 0.003-0.057 m·s^-1 of g_c for Qinghai spruce in the upper Heihe River Basin of arid northwestern China in 2014. Broad-leaved trees have also yielded g_c values from 0 to 70 mm·s^-1; for example, Oguntunde et al. (2005) obtained g_c values of 0.004 0-0.021 2 mm·s^-1 in young cashew trees, with a mean of (0.008 0 ± 0.003 3) mm·s^-1. In western Japan, Komatsu et al. (2012) reported g_c values from 0.005 m·s^-1 to 0.015 m·s^-1 and from 0.002 m·s^-1 to 0.007 m·s^-1 in bamboo and cypress forests, respectively. Kumagai et al. (2004) obtained a g_c range of 0 mm·s^-1 to 70 mm·s^-1 in a lowland mixed dipterocarp forest in Sarawak, Borneo.

Throughout the study period (10:00-18:00 in June, 2013), the air temperature varied from 12.7 ℃ to 25.6 ℃. Consequently, the three temperature functions f(T) (Eqs. (9)-(11)) demonstrated a similar decline with increasing T. Equation (11) is modified from Eq. (10) and includes an additional unknown parameter T_opt, defined as the optimum temperature of g_c opening. In fact, T_opt is a good correction because Eq. (10) is a monotonically decreasing function of temperature, whereas g_c linearly increases or presents an inverted U-shaped curve with increasing air temperature. In this paper, T_opt was approximated as 10 ℃. Moreover, Eq. (10) performed slightly better than Eq. (11) in our study.

Equation (13) yielded a slightly better f(R) than Eq. (12). Under the parameters listed in Table 2, Eq. (12) is an increasing function of R, whereas Eq. (13) presents an inverted-U quadratic curve with a maximum at R ≈ 700 W·m^-2. This curve better fits the relationships between g_c and R and between E_c and R, as shown in Figure 4, respectively.

Although the multiple linear equation yielded a slightly greater MRE than most of the Jarvis methods in the g_c and 1/4-hourly E_c predictions, it produced the best result in the daily E_cpredictions. Consequently, the Jarvis and multiple linear models are both suitable for estimating the g_c of Qinghai spruce. In the selected study site and period, Jarvis model 5 achieved the minimum MRE and maximum r². However, the over fitting phenomenon precludes a unique solution for the optimal parameters in this model. Specifically, Eq. (13) in the Jarvis models includes three unknown parameters. For example, Jarvis model 6 estimates g_c from Eqs. (11) and (13), so there are seven undetermined parameters (g_cmax, T_opt, k₁, k₄, k₆, k₇ and k₈) in this function. In the curve-fitting by 1st Opt software, diverse solutions were obtained with r² ≈ 0.92. An alternative set of fitted results is listed in Table 4. Apart from k₁, the parameters are very different from the earlier case, but the validation results are as strong as those displayed in Table 3.

When too many parameters were input the multiple non-linear model, multiple solutions were output, which is undesirable. The overfitting phenomenon has two main causes (Hawkins, 2004); the application of an excessively flexible model; and a model with irrelevant components. In this study, the Jarvis models with too many undetermined parameters (introduced through Eq. (13)), were suboptimal for computing g_c, owing to the over fitting problem.

4 Conclusions

This study predicted the g_c values in a Qinghai spruce plantation in northwest China. The g_c is a critical parameter not only in E_c, simulations, but also in carbon cycle and other ecological models (Battaglia et al., 2004; Landsberg et al., 1997). Therefore, the study results could guide the transpiration estimating and parameters calibrating with transportation models in this region.

In the Qinghai spruce forest, the canopy transpiration lagged behind the meteorological factors by 15 minutes. R was the major influencer of E_c, but D and T chiefly controlled the opening and closing of stoma. The g_c exhibited a declining trend with increasing D and T.

Comparing the measured and estimated g_c, most of the Jarvis methods slightly outperformed the multiple linear model, but the linear model yielded the best predictions of daily E_c. Furthermore, no significant differences between the measured and predicted g_c/E_c were observed, indicating that all of the methods accurately predicted the g_c. However, when many variables were input into the Jarvis model, the overfitting problem occurred. Therefore, Jarvis model 1, Jarvis model 2 and the multiple linear method are suitable for evaluating the g_c of Qinghai spruce. However, the multiple linear model is more recommended for its simplest expression and high accuracy.