# Using Stochastic Ray Tracing to Simulate a Dense Time Series of Gross Primary Productivity
## 1 Introduction  
Monitoring forest productivity not only widens our ecological understanding, but is increasingly used as a diagnostic tool for evaluating silvicultural practices and testing compliance with political regulations regarding sustainable resource use. From an ecological point of view, information about forest productivity is often indicative of biodiversity richness [1] and can be used to explain how anthropogenic influences and climatic changes affect forest health and functioning [2]. A key variable is the gross primary productivity (\(P_{g}\)) that defines the uptake of carbon from the atmosphere before any losses due to respiration are subtracted. Estimates of \(P_{g}\) are based on the amount of photosynthetically active radiation that is absorbed by the canopy (\(E_{ac}\)) and its pathways in photosynthesis [3; 4; 5]. The modelling of \(P_{g}\) is challenging, however, due to the size and variability of the global forest resources, the multitude of environmental drivers, and the lack of control over these drivers in the field. As a result, a wide range of models exists that operate at a wide range oftemporal and spatial scales. To validate these models, a hierarchical strategy is generally adopted where \(P_{g}\) models at broad scales are calibrated and validated using models and sampling techniques at finer scales. At broad spatial scales, such as regional to global, forest productivity models represent the canopy either as a single big leaf that extends across the landscape [3, 4, 6] or as a stack of foliage layers that provide for the modelling of vertical transfer of visible and thermal radiation [7, 8]. At these model scales, the net lateral transport of energy is assumed to be zero and the models are often parameterized using remote sensing estimates of incident short- and long-wave radiation and stand-level estimates of canopy structure, site nutrition, and hydrology [6, 9, 10, 11]. Paramount to the calibration and validation of these models is the eddy-covariance (EC) method that provides estimates of \(P_{g}\) at stand level (\(P_{gs}\)) from high-frequency measurements of the vertical wind velocity, and carbon dioxide (CO\({}_{2}\)) and water vapour (H\({}_{2}\)O) mixing ratios [12]. \(P_{gs}\) is obtained by subtracting daytime EC-measured net ecosystem exchange of CO\({}_{2}\) (NEE) from daytime ecosystem respiration (\(R_{e}\)) usually estimated using the relationship between nighttime EC-measured NEE and soil or air temperature. Currently, a global network of over 500 EC installations provides biome-specific, long-term information about variability in \(P_{gs}\) within footprints of around 10 ha. For homogeneous forest stands on horizontal terrain, the error bound on NEE obtained with the EC method is less than 50 g\(\cdot\)C\(\cdot\)m\({}^{2}\cdot\)year\({}^{-1}\) with long-term measurements and independent flux measurements of individual ecosystem components adding additional confidence [13]. For example, measurement of CO\({}_{2}\) exchange can be undertaken at scales ranging from leaf to tree level (using chambers or cuvettes) and can be used in support of EC-data acquisition. At these fine scales, the heterogeneity of the canopy structure requires a consideration of vertical as well as lateral transport of energy [14]. To compute the three-dimensional (3D) canopy radiation field, an analytic approach is widely used that represents the stand as a series of conical and ellipsoidal crown envelopes for which smooth, radial transitions in leaf density can be specified (e.g., [15, 16, 17, 15]. More recent approaches have adopted stochastic ray tracing to simulate the scattering of light within increasingly heterogeneous plant canopies, based on first principles [18, 19, 20, 21]. In ray tracing, the scattering and absorption of individual photons is simulated by computing intersections between light rays and the scene and by sampling light propagation from material-specific bi-directional reflectance (transmittance) distribution functions (BRDF and BTDF, respectively).