Leaf optical properties models

page updated on March 24th, 2006

While experimental measurements of leaf optical properties were progressing, deterministic approaches based on diverse representations of light interactions with plant leaves were also developed. These models are distinguished by the underlying physics and by the complexity of the leaf. The simplest ones consider the blade as a single scattering and absorbing layer. In the most complicated ones, all the cells are described in detail (shape, size, position, and biochemical content). Whatever the approach, these models have improved our understanding of the interactions of light with plant leaves. They can be categorised into different classes, arranged in order of increasing complexity:

Plate models

The first plate model was introduced by Allen et al. (1969) who represented a leaf as an absorbing plate with rough surfaces giving rise to Lambertian diffusion. Parameters here are an index of refraction and an absorption coefficient. This model was successful in reproducing the reflectance spectrum of a compact corn leaf characterised by few air-cell wall interfaces. The same authors rapidly extended the model to non-compact leaves by regarding them as piles of N plates separated by N-1 air spaces (Allen et al., 1970). The solution of such a system, provided in the last century by Stokes (1862), has been extended to N being a real number: this is the so-called generalised plate model. This additional parameter N actually describes the leaf internal structure and plays a role similar to that of the scattering coefficients in the Kubelka-Munk model. Now in widespread use in the remote sensing community, the PROSPECT model (Leaf Optical Properties Spectra) has been designed this way (Jacquemoud and Baret, 1990). It was among the first radiative transfer codes to accurately simulate the hemispherical reflectance and transmittance of various plant leaves (monocots, dicots or senescent leaves) over the solar spectrum from 400 nm to 2500 nm. Several versions have been made widely available in the community (Fourty et al., 1996; Jacquemoud et al., 1996; Baret and Fourty, 1997; Fourty and Baret, 1998; Jacquemoud et al., 2000) and validated on different datasets.

For more information about PROSPECT, please contact Stéphane Jacquemoud, Equipe de Géodésie et Gravimétrie, Institut de Physique du Globe de Paris, Paris (France).

N-flux models

These models derived from the Kubelka-Munk theory (Kubelka and Munk, 1931) consider the leaf as a slab of diffusing (scattering coefficient s) and absorbing (absorption coefficient k) material. The N-flux equations are a simplification of the radiative transfer theory: the solution of these equations yields simple analytical formulae for the diffuse reflectance and transmittance. A two-flux model (Allen and Richardson, 1968) and a four-flux model (Fukshansky et al., 1991; Martinez von Remisowsky et al., 1992; Richter and Fukshansky, 1996) have been successfully used in the forward mode to calculate the s and k optical parameters of plant leaves. Yamada and Fujimura (1991) later proposed a more sophisticated version in which the leaf was divided into four parallel layers: the upper cuticle, the palisade parenchyma, the spongy mesophyll, and the lower cuticle. The Kubelka-Munk theory is applied with different parameters in each layer, and solutions are coupled with suitable boundary conditions to provide the leaf reflectance and transmittance as a function of the scattering and absorption coefficients. But these authors went further, interpreting the absorption coefficient determined in the visible region in terms of chlorophyll content. By inversion, their model became a nondestructive method for the measurement of photosynthetic pigments. The leaf biochemistry has been introduced by Conel et al. (1993) who used a two-flux model to study the influence of water, protein, cellulose, lignin, and starch on leaf middle infrared reflectance. However they did not validate it. Finally, a very simple model, directly issued from the expression of the reflectance, has been used to estimate the chlorophyll content of wheat leaves (Andrieu et al., 1988).

Compact spherical particle models

None of these models are adapted to needle-shaped leaves due to a lack of experimental data available on such targets. Dawson et al. (1998) adapted Melamed's theory of light interaction with suspended powders (Melamed, 1963) and designed the LIBERTY model (Leaf Incorporating Biochemistry Exhibiting Reflectance and Transmittance Yields) specifically to calculate the optical properties of both dried and fresh stacked conifer (particularly pine) needles. By treating the leaf as an aggregation of cells, with multiple radiation scattering between cells, output reflectance and transmittance spectra are a function of three structural parameters (cell diameter in mm, intercellular air space, leaf thickness) and the combined absorption coefficients of leaf biochemicals (chlorophyll concentration in mg m-2, water content in g m-2, lignin and cellulose content in g m-2, and nitrogen content in g m-2). To date, LIBERTY remains the only model used for this purpose.

Radiative transfer theory

Compared with canopy level, only few models directly use the radiative transfer equation at leaf level. The poor information we have on leaf internal structure and biochemical distribution leads to strong simplifications which make such an approach less efficient as compared to more robust formulations. In Ma et al. (1990), the leaf is described as a slab of water with an irregular surface containing randomly distributed spherical particles. In LEAFMOD (Leaf Experimental Absorptivity Feasibility MODel), it is compared to a homogeneous mixture of biochemicals which scatter and absorb light (Ganapol et al., 1998). Each model was able to provide a faithful simulation of leaf optical properties.

Stochastic models

 

Tucker and Garatt (1977) proposed an original stochastic model, LFMOD1, where the radiation transfer is simulated by a Markov chain. A black maple leaf is partitioned into two independent tissues, a palisade parenchyma and a spongy mesophyll. Four radiation states (solar, reflected, absorbed, and transmitted) are defined, as well as the transition probabilities from one radiation state to another, between the different compartments. These probabilities are set on the basis of the optical properties of the leaf material. Starting with an initial state vector representing the incident radiation, the steady state is computed by iteratively applying the one-step transition matrix, and yields both the reflectance and transmittance.

The SLOP (Stochastic model for Leaf Optical Properties) model (Maier et al., 1999; Maier, 2000) is an improved version of the stochastic model, which differs in that the leaf is partitioned into four different tissues.

 

 

Ray tracing models

Among various approaches, only ray tracing techniques can account for the complexity of internal leaf structure as it appears in a photomicrograph. They require a detailed description of individual cells and their unique arrangement inside tissues. The optical constants of leaf materials (cell walls, cytoplasm, pigments, air cavities, etc.) also have to be defined. Using the laws of reflection, refraction, and absorption, it is then possible to simulate the propagation of individual photons incident on the leaf surface. Once a sufficient number of rays have been simulated, statistically-valid estimates of the radiation transfer in a leaf may be deduced. The technique has been applied with a number of variants. The first studies were performed at the cell level (Haberlandt, 1914; Gabrys-Mizera, 1976), in particular with epidermal cells the shape of which might influence the path of the incident beams. Research efforts were also directed toward understanding the transmission path of light through entire leaves: Allen et al. (1973), and afterwards Brakke and Smith (1987) modeled an albino maple leaf by 100 circular arcs and of two media: intercellular space air and cell walls characterised by their indices of refraction. The model was used to test the specular and the diffuse nature of the reflection at the cell walls. Simulations led to an underestimation of the reflectance and an overestimation of the transmittance in the near-infrared plateau, which was demonstrated shortly afterwards by Kumar and Silva (1973) who found that the actual reflectance and transmittance could be better reproduced by adding two more media into the model, cytoplasm and chloroplasts, thereby increasing the internal diffusion. Whatever the approach, the absorption phenomena that characterise leaf optical properties outside the near-infrared plateau has been ignored. Moreover in all these models, leaves were always described as two-dimensional objects although the three-dimensional structure of these organs is very important to their physiological function.

For this reason, Govaerts et al. (1996) used a three-dimensional ray tracing model, RAYTRAN (Govaerts and Verstraete, 1998), on a virtual 3D dorsiventral leaf, to characterise the light environment, including absorption, scattering and transmission, within and between cells: Cells of variable size, cell wall thicknesses, chemistry and air spaces were modeled and implications for absorption profiles, light harvesting, and photosynthesis were successfully investigated (Ustin et al., 2001). Finally, in a completely different domain, namely image synthesis, modeling of light interaction with biological tissues such as plant leaves recently emerged with ABM (Algorithmic BDF Model) (Baranoski and Rokne, 1997, 1999).

Despite decades of research, much more work is required before we will accurately model leaf optical properties. Models are nevertheless essential to understand how electromagnetic radiation interacts with leaf elements, but also to directly relate observed optical properties to leaf biophysical attributes.

References