Landsberg and Waring [8] initiated the 3-PG model as a simple process-based model, on the stand level, for forest growth. The model is composed of five sub-models: assimilation of carbohydrates; biomass partitioning to foliage, stem, and roots; determination of stem number; soil water balance; and the conversion of biomass into variables of interest to forest managers.

The main input parameters required are site factors and species-specific data of the species of interest (Table 1). The site factors include latitude, maximum available soil water (ASW_x: mm), soil texture (sandy, sandy loam, clay, or clay loam), and fertility rating (FR). Species-specific parameter values are adjusted for each species. If the parameter values for a species do not exist, parameterization of the new site-specific parameter is necessary. Another input is meteorological variables, including mean air temperature (T_a: °C), mean solar radiation (SR: MJ m⁻² day⁻¹), total precipitation (R: mm), frost days (d_F: days per month), and vapor pressure deficit (VPD: mbar). If VPD is not available, it can be calculated from the maximum and minimum temperatures (T_x and T_n: °C) as half the difference between the saturated vapor pressure at T_x and T_n. The meteorological inputs can be actual monthly data or long-term monthly averages.

The outputs are primarily foliage, root, and stem biomass (W_F, W_R, and W_S: t ha⁻¹), gross primary productivity (GPP: t ha⁻¹), and net primary productivity (NPP: t ha⁻¹). Other stand-level outputs can be leaf area index (LAI), specific leaf area (SLA), stem volume (V: m³ ha⁻¹), stem number (N: trees ha⁻¹), average stem diameter at breast height (B: cm), and mean annual stem-volume increment (MAI: m³ ha⁻¹ year⁻¹). 3-PG outputs can be estimated monthly or annually, as well as over longer periods.

In this study, we focused on parameterization of deciduous broadleaf forests for GPP. The species-specific parameters involved in estimating GPP are part of biomass partitioning and biomass production. Biomass partitioning is related to biomass allocation to the stem, foliage, and roots, each of which is influenced by the GPP. Biomass partitioning is controlled by the ratio of foliage to stem biomass allocation. Root biomass allocation is influenced by soil nutrition and available soil water. However, root data were not available in this study. The partitioning ratio between foliage biomass and stem biomass (pFS) is a function of stem diameter, as shown in the allometric relationship in Equation (1): (1) p F S = a p B n pwhere B is diameter at breast height (DBH), a_p = a_F n_F/a_s n_s, and n_p = n_F − n_s. Both a_p and n_p are expressed in terms of pFS₂ and pFS₂₀. Generally, pFS₂ and pFS₂₀ are extracted for B values of 2 and 20 cm, respectively [10].

To obtain a_s and n_s (Equation (2)), we determine the relationship between Ws and B by fitting the allometric relationship from the observed data. a_F and n_F (Equation (3)) are from the allometric relationship between W_F and B.

In biomass production, growth modifiers directly influence GPP. This modification accounts for environmental conditions affecting forest growth. The six growth modifiers are temperature (f_T), frost day (f_F), site nutrition (f_N), available soil water (f_θ), vapor pressure deficit (f_D), and stand age (f_AGE). The value of each modifier ranges from 0 to 1. As a modifier approaches 0, it has a stronger influence on forest growth [8,10]. Temperature and frost effects control biomass production in this study area. The equations describing each modifier are as follows. The other modifiers have been reported by Landsberg and Waring [8].

This parameter is given by the expression: (4) f T = [ ( T a − T n ) / ( T opt − T n ) ] [ ( T x − T a ) / ( T x − T opt ) ] ( Tx ‐ Topt ) / ( Topt ‐ Tn )where T_a is monthly mean temperature (the actual value for each month), and T_n, T_opt, and T_x are the minimum, optimum, and maximum temperatures for forest growth, respectively. If T_a ≤ T_n or T_a ≥ T_x, f_T becomes 0.

This parameter is calculated by the expression: (5) f F = 1 − k F ( d f / 30 )where k_F is the number of days of production lost for each frost day, and d_f (day month⁻¹) is the mean number of frost days per month. If frost does not occur at the site, then f_F = 1. Note that the number shown above as 30 changes according to the number of days in each month.

GPP is calculated as shown in Equation (6), by converting absorbed photosynthetically active radiation (APAR, ϕ_pa) to GPP using the canopy quantum efficiency (α_C), which is constrained by the growth modifiers [8,10]: (6) α C = α C x φ f T f F f Nwhere α_Cx is the maximum canopy quantum efficiency, and f_T, f_F, and f_N are the temperature, frost day, and site nutrition modifiers, respectively. φ (Equation (7)) is a physiological modifier that is determined by the minimum vapor pressure deficit (f_D) and available soil water modifiers (f_θ) multiplied by the stand age modifier (f_age):

Then, GPP is estimated using Equation (8): (8) GPP = f T f F f N α C x φ φ p awhere ϕ_pa is APAR calculated using Beer's Law using intercepted solar radiation based on LAI for photosynthesis. NPP is a constant fraction of GPP multiplied by 0.47 ± 0.4 [17,18].

We collected data at two study sites, Hitsujigaoka and Takayama. Hitsujigaoka is located in northern Japan at 42°59′13″ N and 141°23′7″ E and is 180 m above sea level. Takayama is in central Japan at 36°8′46″ N and 137°25′23″ E and is 1,420 m above sea level. At each site, a flux tower collected meteorological data and ecological characteristics. Averaged over five years, from 2001 to 2006 (Figure 1), the maximum temperatures at Hitsugaoka and Takayama were 24.9 ± 1.8 °C and 22.4 ± 1.6 °C in August, while the minimum temperatures were −12.4 ± 1.7 °C and −11.7 ± 2.1 °C in February, respectively. The mean annual precipitation and mean annual solar radiation at Hitsujigaoka were 1,047 ± 174 mm and 12.25 ± 0.4 MJ m⁻² day⁻¹, respectively, whereas at Takayama they were 2,275 ± 165 mm and 12.43 ± 0.3 MJ m⁻² day⁻¹, respectively. Both sites experience snow cover in the winter (December to April) to a depth of 1 m.

The main vegetation type was deciduous broadleaf natural forest, and the dominant species was Betula platyphylla. The stand basal area for this species accounted for around 81% and 63% of the total basal area at Hitsujigaoka and Takayama, respectively. Tree height ranged from 14 to 28 m. Overstory trees at Hitsujigaoka (around 100 years) were older than those at Takayama (around 50 years; [19,20]). The mean DBH was larger at Hitsujigaoka (23.0 cm) than at Takayama (11.4 cm; [19]). We note that the flux tower at Hitsujigaoka was compromised and the forest around the flux tower was severely damaged by a typhoon in 2003. Data from Takayama were used for calibration, while data from Hitsujigaoka were used for validation. The different locations did not significantly influence the model output because the two sites are located in the same climatic zone and are characterized by the same canopy species. Site data is summarized in Table 2.

Each site has two flux towers. The first tower continuously monitored meteorological data. The second monitored only ecological properties. Meteorological data, including air temperature, precipitation, global solar radiation, humidity, wind speed, and soil temperature were collected every 30 min. The 3-PG model requires data on air temperature, precipitation, and solar radiation. Figure 1 shows monthly meteorological data, which were used as input parameters, for Hitsujigaoka and Takayama from 2001–2006. Vapor pressure deficit was calculated based on maximum and minimum temperatures. Days when the temperature was below 2 °C were counted as frost days. CO₂ flux was measured by the eddy covariance method [21]. These data were collected in 30-min increments and used to validate the results. CO₂ flux was used to estimate net ecosystem exchange (NEE). Then, GPP was estimated by NEE and ecosystem respiration. Canopy respiration was not observed in the study, therefore NPP is not available.

A sensitivity analysis was carried out to examine how parameter values influenced the output. The parameters examined were a_s and n_s, biomass partitioning ratio at B = 2 and 20 cm, maximum canopy quantum efficiency (α_Cx), T_n, and T_opt. These are the main parameters of biomass partitioning and biomass production, and the most influential for GPP estimation. We determined the effects of changes in these parameters on W_s, W_f, and GPP. Additionally, sensitivity to meteorological data was analyzed with the GPP results. The parameter values changed by −20%, −10%, +10%, and +20% from the default values, except for temperature, which was changed to degrees Celsius (−2 °C, −1 °C, +1 °C, and +2 °C) since changes in degrees Celsius usually showed a more significant effect than those in percentage.

Biomass can be divided into three parts: stem, foliage, and root biomass. We did not examine root biomass because empirical data were not available. Figure 6 shows the estimated stem and foliage biomass versus the observed biomass at each site. Both relationships were close to one-to-one linear relationships. Linear regression showed a high correlation between the estimated and observed values (r² = 0.89 and 0.94 for stem and foliage biomass, respectively). These values were close to the one-to-one line, in which the estimated values should be equal to the observed values. Foliage and stem biomass at Takayama were slightly overestimated compared to those at Hitsujigaoka. This difference was likely because the canopy at Takayama is younger than that at Hitsujigaoka. A summary of regression statistics is given in Table 4.

The GPP estimation at Hitsujigaoka and Takayama is presented in Figure 7a and 7b, respectively. The monthly estimated GPP was compared with the observed GPP from 2000 to 2004. The results for both locations showed a good correlation in terms of the linear regression, with r² = 0.85 and 0.97 at Takayama and Hitsujigoaka, respectively. The 3-PG model indicated that inter-annual GPP peaked around July and August and was 0 in winter (November to April). GPP was lower at Takayama than at Hitsujigaoka because of the smaller LAI at Takayama. Thus, the interception of radiation was lower at Takayama. The GPP at Takayama was underestimated because of the mixed age of the trees in the plot and the influence of the temperature modifier. The low value of the latter at the beginning of the growing season resulted in the low estimate for GPP (low points in the graph). At Hitsujigaoka, the GPP was overestimated because the large tree leaves could absorb more solar radiation for photosynthesis. The average GPP over 5 years was 973.30 g C m⁻² (Takayama) and 1255.21 g C m⁻² (Hitsujigaoka). Table 4 compares the estimated and observed values of GPP in terms of root mean square error.

Changing selected parameter values from −20 to 20% had different effects on the estimated output. Figure 8 shows the effects of changes in a_s, n_s, pFS2, pFS20, α_Cx, T_n, T_opt, W_F, and W_s. Changes in a_s (Figure 8a) and n_s (Figure 8b) primarily affected W_F; when a_s and n_s were changed by −20%, W_F differed by around 16% and 6%, respectively. However, the selected outputs were less sensitive to changes in pFS20 (Figure 8d) than pFS2 (Figure 8c). GPP was highly sensitive to α_Cx (Figure 8e) and to T_opt (Figure 8f). W_s was also sensitive to both of those parameters, but not as sensitive as GPP. T_n (Figure 8g) most affected GPP when it was changed by 20 °C. From these results, we recommend that α_Cx, T_opt, and T_n be carefully measured if accurate results are desired, especially in GPP calculations.

Figure 9 shows the effects of meteorological data on GPP. Solar radiation (Figure 9b) and T_n (Figure 9a) had significant impacts on GPP, with differences around ± 3 g C m⁻², but T_x had only a slight effect (Figure 9a). Increased solar radiation and T_n increased GPP. However, GPP was not sensitive to frost days or precipitation, since the study sites are not limited by water. Snow melt increases the soil water content in the summer. The effect of frost days on GPP was large in the winter, as large as the effect of temperature. However, frost days did not affect GPP in the summer, during which GPP was most affected by temperature and solar radiation.