Monte Carlo method based radiative transfer simulation of stochastic open forest generated by circle packing application

Abstract

Monte Carlo Ray Tracing (MCRT) method is a versatile application for simulating radiative transfer regime of the Solar - Atmosphere - Landscape system. Moreover, it can be used to compute the radiation distribution over a complex landscape configuration, as an example like a forest area. Due to its robustness to the complexity of the 3-D scene altering, MCRT method is also employed for simulating canopy radiative transfer regime as the validation source of other radiative transfer models. In MCRT modeling within vegetation, one basic step is the canopy scene set up. 3-D scanning application was used for representing canopy structure as accurately as possible, but it is time consuming. Botanical growth function can be used to model the single tree growth, but cannot be used to express the impaction among trees. L-System is also a functional controlled tree growth simulation model, but it costs large computing memory. Additionally, it only models the current tree patterns rather than tree growth during we simulate the radiative transfer regime. Therefore, it is much more constructive to use regular solid pattern like ellipsoidal, cone, cylinder etc. to indicate single canopy. Considering the allelopathy phenomenon in some open forest optical images, each tree in its own ‘domain’ repels other trees. According to this assumption a stochastic circle packing algorithm is developed to generate the 3-D canopy scene in this study. The canopy coverage (%) and the tree amount (N) of the 3-D scene are declared at first, similar to the random open forest image. Accordingly, we randomly generate each canopy radius (r_c). Then we set the circle central coordinate on XY-plane as well as to keep circles separate from each other by the circle packing algorithm. To model the individual tree, we employ the Ishikawa’s tree growth regressive model to set the tree parameters including DBH (dt), tree height (H). However, the relationship between canopy height (Hc) and trunk height (H_t) is unclear to us. We assume the proportion between Hc and Ht as a random number in the interval from 2.0 to 3.0. De Wit’s sphere leaf angle distribution function was used within the canopy for acceleration. Finally, we simulate the open forest albedo using MCRT method. The MCRT algorithm of this study is summarized as follows (1) Initialize the photon with a position (r₀), source direction (Ω0) and intensity (I₀), respectively. (2) Simulate the free path (s) of a photon under the condition of (r', Ω, I’) in the canopy. (3) Calculate the new position of the photon r=r +sΩ’. (4) Determine the new scattering direction (Ω)after collision at, r and then calculate the new intensity I = Υ_L(Ω_L,Ω'→Ω)I'.(5) Accumulate the intensity I of a photon escaping from the top boundary of the 3-D Scene, otherwise redo from step (2), until I is smaller than a threshold. (6) Repeat from step (1), for each photon. We testify the model on four different simulated open forests and the effectiveness of the model is demonstrated in details.