Three-dimensional plant architecture and sunlit-shaded patterns: a stochastic model of light dynamics in canopies

Abstract

Background and aims: Diurnal changes in solar position and intensity combined with the structural complexity of plant architecture result in highly variable and dynamic light patterns within the plant canopy. This affects productivity through the complex ways that photosynthesis responds to changes in light intensity. Current methods to characterize light dynamics, such as ray-tracing, are able to produce data with excellent spatio-temporal resolution but are computationally intensive and the resulting data are complex and high-dimensional. This necessitates development of more economical models for summarizing the data and for simulating realistic light patterns over the course of a day.

Methods: High-resolution reconstructions of field-grown plants are assembled in various configurations to form canopies, and a forward ray-tracing algorithm is applied to the canopies to compute light dynamics at high (1 min) temporal resolution. From the ray-tracer output, the sunlit or shaded state for each patch on the plants is determined, and these data are used to develop a novel stochastic model for the sunlit-shaded patterns. The model is designed to be straightforward to fit to data using maximum likelihood estimation, and fast to simulate from.

Key results: For a wide range of contrasting 3-D canopies, the stochastic model is able to summarize, and replicate in simulations, key features of the light dynamics. When light patterns simulated from the stochastic model are used as input to a model of photoinhibition, the predicted reduction in carbon gain is similar to that from calculations based on the (extremely costly) ray-tracer data.

Conclusions: The model provides a way to summarize highly complex data in a small number of parameters, and a cost-effective way to simulate realistic light patterns. Simulations from the model will be particularly useful for feeding into larger-scale photosynthesis models for calculating how light dynamics affects the photosynthetic productivity of canopies.

Fig. 1.

Quantifying sunlit and shaded dynamics. (A) Reconstructed wheat plant from Burgess et al. (2015). (B) Set-up for the ray-tracer (Song et al., 2013). Red arrows show direct light rays; once a ray hits the boundary of the bounding box, it is moved to the opposed vertical face of the box. (C) Shading will occur when a ray is obstructed by other leaves or a stem. (D) Construction of canopies was done in two ways: putting the bounding box just outside the plant (red rectangle) or putting plants on a 3 × 3 grid at a distance d apart and putting the bounding box through the centres of boundary plants (blue rectangle). (E) Diurnal dynamics of light, in µmol m⁻² s⁻¹, at a particular patch showing ray-tracer simulation (solid black curve), light amplitude envelope (solid red curve) and inferred shaded periods (horizontal grey lines). Time resolution is 1 min. One of the shaded periods is extended to (C) to indicate schematically the occlusion caused. (F) Sunlit–shaded patterns for the patches comprising the leaf shown in blue in (A); each row corresponds to an individual patch, with patches ordered by the height of their centroids. The row shown in red corresponds to the particular patch shown in (E). (G) The two-state sunlit–shaded model: switching on (from shaded to sunlit) occurs at rate ${\lambda }_h^{\text{on}}\left(t\right)$ and switching off (from sunlit to shaded) at rate ${\lambda }_h^{\text{o}ff}\left(t\right)$.

Fig. 2.

Illustration of the notation for the model, showing the four possible combinations of states at the beginning and end of the interval $\left[0,T\right]$. At time $t=0$, a sunlit state is indicated by $x_1<0$ and a shaded state by $x_1>0$; at time $t=T$ a sunlit state is indicated by $y_n>T$ and a shaded state by $y_n<T$. The different sections are coloured to indicate how they contribute to the (log) likelihood functions: red denotes a contribution to on-switching functions [eqns (7) and (9)] and yellow to off-switching function [eqns (8) and (10)].

Fig. 3.

A simulated realization of Model 1, and maximum likelihood estimation from fitting the model to the realization. (A) A realization with $\lambda \left(t\right)=3+0.05t-0.075{\left(t-6\right)}^2$. (B) A plot of this true $\lambda \left(t\right)$ together with estimates of it based on data with various numbers of realizations, n, showing convergence of the estimates to the true $\lambda \left(t\right)$ as n increases.

Fig. 4.

Plants, canopies and fitted models: (i) Reconstructed plants (A–O); (ii) cumulative leaf area index as a function of depth; (iii) principal component analysis of fitted parameters; and (iv) relationship between the first principal component and LAI. Images of original plant (A); original plant rotated 90° (B), 180° (C) and 270° (D); original plant randomly rotated and positioned at distances 200 mm (E), 150 mm (F), 125 mm (G) and 100 mm (H); replica of a plant from the same line (I); plants from two different wheat lines (J, K); three plants of Bambara groundnut 39 d after sowing (L, M, N) and 80 d after sowing (O). For a detailed description of the lines and reconstructions, see Burgess et al. (2015, 2017a).

Fig. 5.

Distributions of the duration of sunlit (i) and shaded (ii) periods. Results shown are from ray-tracer (red) and stochastic model (blue) simulations for canopy (E) with plants at distance 200 mm (A), canopy F with plants at distance 150 mm (B), canopy G with plants at distance 125 mm (C) and canopy H with plants at distance 100 mm (D). Parameters are as given in Table 1.

Fig. 6.

Using model simulations to study photoinhibition. (A) A diffused light profile for a patch at the top of a canopy. (B) Scaling of diffused light as a function of normalized height (grey dots) and fitted spline. (C) Light–response curves for the top (red), middle (blue) and bottom (green) layers (Burgess et al., 2017b). (D) Simulated light patterns showing Photosynthetic Photon Flux Density (PPFD). (E) Calculated daily carbon gain based on ray-tracer and emulator [with colours matching (C)]; grey shows the 1:1 line and orange is the fitted LOESS curve. (F) The predicted effect of photoinhibition on carbon gain based on ray-tracer (RT) data from Burgess et al. (2015) and the stochastic model (SM).