# Using Stochastic Ray Tracing to Simulate a Dense Time Series of Gross Primary Productivity
## 3 Results  
The geometrically explicit model of canopy structure provided estimates of \(E_{al}\) by numerically integrating down-welling diffuse and direct PAR. Figure 4 shows a 2D histogram of \(E_{al}\) for elements \(p_{l}\) within 5 m of the plot centre vertical axis, for a clear and a cloudy day in May 2009, using \(GF_{\perp}=15\%\). The graph shows a large variation in \(E_{al}\) throughout the canopy and negligible absorption of radiation in the lower canopy for which two main causes can be identified: First, the forest plot was modelled using template crowns that exhibit no decay of foliage due to shading. However, when trees grow in stands, some leaves fall below the compensation point where gross CO\({}_{2}\) uptake equals respiration and foliage is shed; this phenomenon was not included in scene construction (Section 2.3). Second, a more technical concern, relates to the sampling density of photons in the Monte Carlo ray tracing simulations. Accurate sampling of light absorption in the lower canopy requires the simulation of a larger number of photons; however, as the larger contribution to \(P_{gs}\) is made by the upper third of the canopy, the effects of restricted sample sizes on \(P_{gs}\) estimates are limited [(59; 21)]. Figure 5 shows the flux-tower derived \(P_{gs,EC}\)_versus_ simulated \(P_{gs,sim}\) without the use of modifier functions for down-regulation (_i.e._, case 1) and using \(GF_{\perp}=15\%\). The scatter plot shows a positive bias among a selection of the simulation results, and an overall low fraction of explained variation (\(r^{2}=0.66\)). To investigate this bias, data points were stratified by \(T\) and \(h\) (Figure 6). Plots along the horizontal are stratified by \(T\) and along the vertical by \(h\). For \(T\) and \(h\) close to the optimum for photosynthesis, _i.e._, towards cooler temperatures and a relative humidity between 75% to 100%, a clear linear relationship between simulated and eddy-covariance derived \(P_{gs}\) was observed, whereas with distance from this meteorological optimum the relationship between simulated and measured productivity weakens and eventually becomes absent (_i.e._, for \(T\) around 25 \({}^{\circ}\)C and \(h<50\%\)). Figure 4: Simulation results of absorbed PAR for 6:00 PST (**a**) and 13:00 PST (**b**) on the cloudy day of 3 May 2009; and for 6:00 PST (**c**) and 16:00 PST (**d**) on a sunny day (8 May 2009) plotted as a 2D heat map. The gap fraction, \(GF_{\perp}\), of photosynthetic scene elements, \(p_{l}\), was set to 15%. The \(x\)-axis is incident PAR (\(\mu\)mol photons m\({}^{-2}\). s\({}^{-1}\)) and the \(y\)-axis is height above the ground level at the plot centre. Colours represent observation frequency for an arbitrary, constant bin size. Figure 7 shows the behaviour of the modifier functions (\(y\)-axis) as they deviate from the optimum environmental condition (e.g., optimal temperature or relative humidity) expressed along the \(x\)-axis, and a scaling parameter that controls the pace of down-regulation with distance from the optimal condition. Figure 5: Model inversion for case 1 (2094 observations, no cross validation): absence of down-regulation functions. Shown is (**a**) the correlation with eddy-covariance derived gross primary productivity estimates; (**b**) the distribution of measured and simulated estimates of gross primary productivity against incident photosynthetically active radiation; (**c**) average diurnal responses. See Table 1 and text for an overview of parameters used for the different cases and Table 2 for an overview of parameter estimates obtained through model inversion. Figure 6: A stratification of the data points shown in Figure 5 into temperature and relative humidity classes.