\chapter[Model of light interception]{Model of light interception by heterogeneous canopy} % Chapter title
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Radiation is fundamental in forest ecology. Besides being the main variable explaining regeneration growth (\autoref{ch:rege}), radiation drives plant photosynthesis and exchanges of mass and energy between soil, vegetation, and the atmosphere. As a consequence, the amount of energy and the spectrum of the radiation that is intercepted by trees have been widely studied, as it greatly affects the gross primary production of forests \citep{Duursma2007, Tian2010}, forest dynamics and competition between species \citep{Pacala1996}, individual tree growth \citep{Lieffers1999, Balandier2007}, and tree morphogenesis \citep{Balandier2006b, Galen2007, Niinemets2010}.

Measuring the amount of radiation, and photosynthetically active radiation (PAR) in particular, which is intercepted by the diverse components of a forest canopy, is tedious and time-consuming. Compared with PAR measurements in agricultural crops, and due to the complexity of multilayered forest canopy, the spatial and temporal variability of transmitted light beneath the forest canopy is substantial. Vegetation beneath a closed canopy indeed benefits from sunflecks or very brief increases in transmitted irradiance \citep{Messier1999}. Measurements must then be repeated at numerous points and for rather long periods of time to capture this variability. Moreover, measuring transmitted PAR at various canopy heights is almost impossible without installing a crane \citep{Mariscal2004}. Faced with these difficulties and given the cost of such measurements, modeling radiative transfer appears essential for supplementing field measurements and for a greater understanding of how radiation is partitioned among the components of forest ecosystems.

To explore how light regimes under complex canopies can be manipulated with silvicultural practices that could maintain the coexistence of species with contrasting shade tolerances (\autoref{ch:simulation}), the modeling approaches that have been proposed in the literature are firstly synthesized and discussed (\autoref{sec:review}). Second, one of these approaches was implemented to explore understory light conditions regarding forest structure and composition in uneven-aged mixed stands (\autoref{sec:4implementation}).

\section{Literature review} \label{sec:review}
Solar radiation and the attenuation of light through plant canopy were first modeled with physical laws by analogy of the Beer’s law in homogeneous medium. Consequently, they were firstly limited to homogeneous crops but were rapidly adapted to forest ecosystems using more complex laws. The general functioning of these latter models have already been presented in several literature reviews \citep{Sinoquet1993, Brunner1998, Lieffers1999} and have changed little since then. Briefly, they share a common structure that can be divided into three submodels (\autoref{fig:41chartflow}). Firstly, most models start by computing above canopy light both in magnitude and angular distribution using standard astronomical laws (``above canopy light model''). Secondly, the interception of light through the canopy (``radiative transfer model'') is computed. It depends mainly upon the geometric structure of the canopy and the mathematical formulation used to describe the fractions of light that are intercepted, absorbed, and transmitted by canopy components. Thirdly, a light reflection model can optionally be used (``scattering model'') to describe more precisely complex light trajectories within the canopy.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.7 \textwidth]{fig41chartflow}
\caption[Chart flow of forest radiative models.]{Chart flow of forest radiative models indicating the three submodels.}
\label{fig:41chartflow}
\end{figure}

Even though most forest radiative transfer models (FRTMs) share a common general framework, they have special features that rely on different assumptions and they are used to predict different output variables aggregated or not in time and space according to the objectives of the model. The costs of specific field measurements to calibrate the model, the increasing interest in complex forests composed of trees of different species and sizes, and the variety of study purposes have indeed stimulated modelers to adapt FRTMs to their needs. Most of these adaptations concerned the radiative transfer model, one of the submodels depicted in \autoref{fig:41chartflow}, which is the main topic of this review. For example, contrasted approaches have been used to model the distribution of foliage within the canopy. Some authors have used very detailed three-dimensional (3D) mock-ups of the canopy, which require intensive field measurements, whereas others describe the canopy as a single horizontal layer. The improvements brought to above canopy light models are very limited as only two standard algorithms (uniform overcast sky or standard overcast sky) have been used among all off the papers that we reviewed. Moreover, the use of one of them does not really affect model performance \citep{Brunner1998}, i.e., model ability to predict the measurements with a low level of error and bias. Similarly, the improvements brought to the scattering model affect model performance less than the improvements brought to FRTMs. Scattering depends upon the optical properties of leaves, wood and soil, and the considered radiation wavelength. Radiation scattering is especially low in PAR and in ultraviolet radiation (\SIrange{400}{700}{\nano\metre}) \citep{Sinoquet1993, Parker1997}. The reflectivity of a green leaf is approximately 0.2 and this value is even lower for an entire crown because of multiple scattering \citep{Brunner1998, Landsberg2010}. Moreover, scattering models require large numbers of parameters \citep{Kuusk1993} and much computation time \citep{Ross1981, Rey2008}. Consequently, the use of a scattering model has often been neglected in forestry assuming that leaves and ground behave like black bodies within the PAR waveband (i.e., showing no transmittance and no reflectance) \citep{Sinoquet1993} even though it might improve the precision of the prediction of transmitted irradiance \citep{Mariscal2004}.

Given the variety of applications and newly developed modeling approaches of FRTMs, it would be of great help for modelers to know which approach enables them to predict the variable of their interest with a given precision, a low bias, and accepted calibration efforts. Moreover, having insight of the expected precision and bias associated with a modeling approach is also precious information in assessing whether it would fulfill the study objectives. Similarly, knowing beforehand to which parameters the models are most sensitive and the order of magnitude of parameter values can speed up and improve the calibration work. This would allow for effort to be invested in estimating precisely the sensitive parameters and using simplification assumptions for the less sensitive parameters.

Unfortunately, as far as forests are concerned, the performance analyses and the sensitivity analysis of different modeling approaches have been reported separately and for very restricted sets of canopy conditions and study objectives. The information is therefore scattered and no general guideline has been formulated to help forest modelers to choose which modeling approach suits best his needs.

We synthesized both the performance analyzes and the sensitivity analyses of radiation transfer models through forest canopies. After classifying these approaches, we attempted to quantify the expected uncertainty and appraise the calibration efforts associated with most combinations of modeling approaches and model applications. We provided order of magnitude estimates for the main calibration parameters and attempted to report how sensitive models are to these parameters. Finally, we attempted to identify the modeling approaches that best suit to the different applications of FRTMs.

\subsection{Classification of FRTMs}
We identified four families of FRTMs that combine three approaches to describe the geometry of forest canopy and two approaches to compute the proportion of incident radiation intercepted by the canopy (\autoref{tab:41classification}). Below, we briefly describe and discuss each of these approaches; details can be found in previous articles \citep{Sinoquet1993, Cescatti1997a, Brunner1998, Lieffers1999}. 

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\begin{table}[htb]
\caption{Classification of FRTM modeling approaches.}
\footnotesize
\begin{tabularx}{\linewidth}{p{0.15 \linewidth}lXp{0.15 \linewidth}X}
\toprule
\textsc{frtm family} & \textsc{abbreviation} & \textsc{canopy} & \textsc{transmittance} & \textsc{main objectives} \\ 
\midrule
1D model  & 1D & Stand canopy is composed of one or several horizontal layers & Turbid medium & Ecophysiological processes at stand level \\ 
3D crown model with turbid medium  & 3D-TM & Tree crowns are composed of one or a set of geometric shapes  & Turbid medium  & Forest growth and yield; Dynamics of stand structure \\ 
3D crown model with porous envelope  & 3D-PE & Tree crowns are composed of one or a set of geometric shapes & Porous envelope & Forest growth and yield; Dynamics of stand structure  \\ 
3D surface model   & 3D-S & Trees are composed of surfaces representing the leaves, branches and stems & Porous envelope or radiative transfer theory & Tree architecture; ecophysiological processes at tree level\\ 
\bottomrule
\multicolumn{5}{l}{\footnotesize {\bf Note :} FRTM, forest radiative transfer model; 1D, one-dimensional; 3D, three-dimensional.} \\
\end{tabularx}
\label{tab:41classification}
\end{table}
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\subsubsection{Canopy geometry}
\paragraph{One-dimensional canopy}

The simplest approach for modeling the geometry of a forest canopy is to assimilate it to a single horizontal layer of vegetation without individualizing crowns or trees (one-dimensional (1D) model). Due to its simplicity, this approach is particularly appropriate modeling a homogeneous canopy such as the one of pure even-aged stands. This approach is more challenging with heterogeneous stands as additional assumptions and parameters are required \citep{Duursma2007, Kim2011} to offset the simplicity of the canopy description.
 
The model developed by \citet{Kim2011} illustrates well this approach. This model requires calibrating for each species three extinction coefficients (corresponding to the distribution of inclination angles of leaves, branches and stems), the one-sided area of leaves, branches and stems, the horizontal crown projection and tree density. Some of these parameters might also depend upon ray directions (i.e., ray zenith angle and ray azimuthal angle). Furthermore, the model assumes that branches and stems are randomly distributed, whereas the distribution of leaves depends upon a clumping factor. Simpler approaches with fewer variables and empirical parameters have been developed. For instance, \citet{Duursma2007} used a 1D model with four variables (leaf area, crown surface area, number of trees and extinction coefficient) and one empirical parameter.
 
Another solution to modeling the distribution of leaves consist of subdividing the canopy into several homogeneous regions, e.g., horizontal layers, on the condition that the vertical structure of the stand can be approximated. Such as multilayer models can then predict understory light at different canopy heights.
 
Moreover, due to the main hypothesis of considering continuous layer(s) of vegetation with the same properties for the whole forest stand we did not expect the 1D model to predict accurately the spatial variation of irradiance beneath the canopy. The 1D model is rather utilized to predict the temporal variation of irradiance \citep{Kim2011, Govind2013}.


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\begin{table}[htb]
\caption[Examples of 3D-geometric crown models.]{Examples of 3D-geometric crown models used in FRTMs. Most authors used relatively simple quadratic shapes whereas few tested more complicated shapes.}
\footnotesize
\begin{tabularx}{\linewidth}{lXXX}
\toprule
 & \textsc{quadratic shapes} & \textsc{combination of quadratic shapes} & \textsc{combination of nonquadratic shapes} \\ \midrule
 & \includegraphics[height=2.5cm]{crownshape1} & \includegraphics[height=2.5cm]{crownshape2}& \includegraphics[height=2.5cm]{crownshape3}\\
 No. of parameters & 3 & $\geq 4$ & 8-18 \\
 References & \citet{Pukkala1993, Canham1994, Koop1994, Bartelink1998b, Stadt2000, Pinno2001, Beaudet2002b, Courbaud2003, Mariscal2004, Beaudet2011, Paquette2008}
 & \citet{Gersonde2004}
 & \citet{Cescatti1997a,Brunner1998,Piboule2001,Groot2004,Piboule2005,DaSilva2012}
 \\
\bottomrule
\end{tabularx}
\label{tab:42crownshape}
\end{table}

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\paragraph{3D crown models}
Alternatively, 3D crown models (see 3D-TM and 3D-PE in \autoref{tab:41classification}) assimilate forest stands into a set of spatialized geometric shapes representing tree crowns and, optionally, trunks. Many different geometric shapes have been used (\autoref{tab:42crownshape}). The simplest are quadratic surfaces such as cylinder, cone, ellipsoid or paraboloid. Such shapes have the advantages of requiring few parameters and allowing analytical computation of the interceptions between light rays and crowns. In contrast, nonquadratic shapes (e.g., combinations of degenerated surfaces) fit well to real crowns but require numerous parameters and more complex numerical computations.

Many refinements to 3D crown models have been described in the literature, and have been notably reviewed by \citet{Brunner1998}. For example, crowns can be divided into different sections characterized by different leaf densities. A combination of several shapes can be used to distinguish an illuminated surface (e.g., upper half ellipsoid) from a shaded surface. Similarly, a special foliage envelope can be constructed within a crown, for example with branches that have developed during the last 5 years. Missing parameters are often estimated using allometric relationships to diameter at breast height \citep{DaSilva2012} or relationships to species shade tolerance \citep{Canham1999, Beaudet2011}. Algorithms of crown reconstruction according to neighbor competition are also available \citep{Piboule2005}.

In brief, the recent improvements brought to 3D crown models attempted to model crown geometry as realistically as possible taking into account that trees often lean and crown shapes often deviate from simple geometric forms. Hence, they required adjusting a greater set of parameters, and some of them can hardly be measured in the field and must be estimated with additional models. 

Moreover, on one hand, the 3D crown model requires determining the position and crown dimensions of every tree whereas the 1D model requires only estimating stand features. On the other hand, measuring crown dimensions is easier and more commonly performed than estimating foliage aggregation within the stand canopy. Additionally, taking explicitly into account stand heterogeneity, 3D crown models can predict the spatial variability of transmitted light.

\paragraph{3D surface models}
3D surface models (3D-S) represent leaves, branches and stems as realistically as possible with surfaces or shells. This leads to very detailed 3D mock-ups (\autoref{fig:42mockup}) with components that are usually assimilated into opaque envelopes (\autoref{sec:4attenuation}). Due to the number of input parameters for these models, they are used to represent single trees \citep{Sinoquet2001}, orchard trees \citep{DaSilva2008}, or agroforestry systems \citep{Dauzat1997, Leroy2009} rather than entire forest stands. Homogeneous forest stands could be obtained by replicating one or a few model trees (\autoref{fig:42mockup}), but it seems unlikely to be able to gather all the necessary information to model every single tree of one heterogeneous stand. Hence, we believe that the use of 3D-S models in forestry is nowadays limited to applications involving tree architecture and ecophysiological processes.

\begin{figure}[htb]
\centering
\includegraphics[width=8cm]{fig42mockup}
\caption[3D plant mock-up.]{3D plant mock-up. Lateral view of a digitalized scene with six 3- to 4-year-old beech seedlings (in grey), scattered every \SI{50}{\centi\metre} from each other together with a systematic network of raspberry (in black) \citep{Balandier2009}.}
\label{fig:42mockup}
\end{figure}

\subsubsection{Radiation attenuation}\label{sec:4attenuation}
\paragraph{Turbid medium}
The first approach to computing the fraction of transmitted light through the canopy ($\tau$) consists of using an analogy to Beer’s law. Beer’s law describes the attenuation of a monochromatic ray within a turbid medium, i.e., a medium made up of small elements randomly scattered and presenting a homogeneous transparency \citep{Brunner1998}. Canopy is assimilated to such a turbid medium and $\tau$ is computed as a function of the density and the spatial distribution of canopy elements such as leaves and branches \citep{Sinoquet1993, Lieffers1999}. The turbid medium analogy can be applied to an entire stand canopy, a horizontal layer within a stand canopy, a crown envelope, or even a section of a crown. This approach relies on the assumption that canopies have the properties of such a homogeneous medium. Nevertheless, stand canopies or tree crowns are made up of various size elements often aggregated within crowns and vertices. Therefore, several correction coefficients have been developed to adapt Beer’s law to forest canopies. Briefly, the probability of beam interception ($1-\tau$) by canopy elements is a function of the canopy element density (leaf area density, LAD (\si{\square\metre\per\cubic\metre}), the path length of a ray through the canopy ($l$), the extinction coefficient ($k$) and the clumping factor ($\Omega$). $k$ and $\Omega$ depend upon canopy element inclination and spatial distribution, respectively (\autoref{eq:4lambert}). $\Omega$ describes the aggregative pattern of branches and leaves within the canopy. Many different mathematical expressions have been used to adapt the Beer’s law models to 1D and 3D models, but all somehow include these three parameters. \autoref{eq:4lambert} is commonly used to compute $\tau$, for a ray of zenith angle $\eta$ and azimuthal angle $\gamma$,with 3D crown models.

\begin{equation}
	\tau(\eta,\gamma) = \exp (-k \cdot \Omega \cdot LAD \cdot l(\eta,\gamma))
	\label{eq:4lambert} 
\end{equation}

The advantage of the turbid medium approach lies mainly in its mechanistic formulation and the use of the leaf area because leaf area is a key variable in forest ecology (e.g., used to model photosynthesis and transpiration). However, this approach often requires empirical estimation of $k$ and $\Omega$, and leaf area measurements that remain both difficult to obtain and imprecise \citep{Breda2003}. Moreover, this is the only approach that applies if the canopy geometry is described with a 1D model.

\paragraph{Porous envelope}
The second approach, called porous envelope, assumes that crowns or parts of crown are envelopes with one empirically estimated parameter, namely the crown openness ($p$, \autoref{eq:4porous}), which is the probability of a ray being intercepted by the foliage. This approach assumes that leaves do not transmit or reflect light (like black bodies). This assumption can be more easily verified for coniferous species with dark foliage than for broadleaved species with lighter foliage \citep{Williams1991}. Therefore, some authors have adjusted \autoref{eq:4porous} for the different radiation wavelength ranges \citep{Goudriaan1977}. The variable $p$ is independent of ray direction and path length ($l$) \citep{Canham1994, Groot2004, DaSilva2008, Boivin2011} and is also defined as the fraction of sky visible through a crown \citep{Canham1999}. 

\begin{equation}
	\tau(\eta,\gamma) = p
	\label{eq:4porous}
\end{equation}

In comparison with the previous approach, this one is less mechanistic but requires fewer parameters (usually one per species) and is therefore easier to calibrate. On the other hand, this approach describes the attenuation of radiation through crowns and not through stand canopy. Therefore it cannot be utilized with a 1D model.
 
Other simplifications of these submodels have also been proposed. For example, \citet{Koop1994} used an opaque crown model ($p=0$). Other authors have simplified the turbid medium submodel by removing $l$, $\Omega$ and LAD from \autoref{eq:4lambert}. In this way, they obtained a ``hit model'' with a formulation (\autoref{eq:4hit}) very close to \autoref{eq:4porous} with one empirical parameter, $k$, which they called the extinction coefficient \citep{Canham1994, Koop1994, Boivin2011}. 

\begin{equation}
	\tau(\eta,\gamma) = exp(-k)
	\label{eq:4hit}
\end{equation}

\subsection{Input variables}
Different input data and calibration parameters are required depending upon the chosen modeling approach. These parameters are measured, estimated or adjusted by model inversion. Below, we give an overview of model requirements in terms of input data and summarize their definitions and estimation methods.

\subsubsection{Stand and tree measurements}
Among all FRTM inputs, stand and tree measurements probably require the most workforce and consequently represent one of the major cost in utilizing FRTM. As far as stand and tree measurements are concerned 1D models obviously require less field data than 3D models and, hence, require less field work. 1D models depend mainly upon foliage density and distribution (see \autoref{sec:lad}, \autoref{sec:k} and \autoref{sec:omega}) which does not require carrying out intensive tree inventory. In most cases, measuring stand density is enough. However, 3D models often rely on an intensive inventory and mapping of every tree within a plot. For 3D crown models, at least three parameters are generally needed for each tree: height, crown base height and crown radius (\autoref{tab:42crownshape}). Additional measurements such as crown radii measured in different directions, height of maximum crown extension, and shape coefficients might be required to construct more complex shapes. For 3D surface models, a geometric description of leaves, branches, and stems can be obtained by vectorization \citep{Fournier1996}, digitalization \citep{Moulia1993}, or simulations of plant morphology \citep{Leroy2009}.

\subsubsection{Density of canopy elements}\label{sec:lad}
The stand leaf area index (LAI, \si{\square\metre\per\square\metre}) is the total one-sided foliage area per unit of soil surface. LAI is the primary descriptor of plant canopy and a key variable in studying plant physiological processes (e.g., photosynthesis, transpiration). In the field, stand LAI is directly assessed with litterfall traps or vertical line intercept sampling coupled with measures of leaf inclination. LAI can also be estimated indirectly with optical methods, mainly using hemispherical photographs or LAI-2000 devices (LI-COR Biosciences, Lincoln, Nebraska, USA) \citep{Jonckheere2004, Weiss2004}. These last two methods measure the gap fraction (i.e., the fraction of sky visible from the measuring point) and use the turbid medium analogy to infer leaf area. Optical approaches therefore rely on the same assumptions and parameters as \autoref{eq:4lambert}. Moreover, such methods give estimates of the effective LAI, which includes the area of branches and trunks \citep{Jonckheere2004}. Additionally, reference values and allometric relationships are available in the literature for the most common species.
 
Measurements of leaf area for individual trees are very difficult to replicate for every tree in a stand, even though individual values of leaf area are necessary with 3D-TM. Therefore, tree leaf area is often estimated for each species by the inversion of a turbid medium model \citep{Courbaud2003} or by measuring leaf area density (LAD, \si{\square\metre\per\cubic\metre}) on a sample of trees \citep{Stadt2000}. Nevertheless, as indicated by \citet{Nock2008}, intracrown leaf area decreases with tree age by up to \SI{40}{\percent}. It might therefore be preferred to use allometric relationships with, for example, tree diameter \citep{Bartelink1998, Gersonde2004}, tree height \citep{Essery2008}, sapwood area \citep{Gersonde2004}, or tapering equations \citep{Kim2011}.
 
Moreover, LAI is often used with a 1D model, whereas LAD is usually preferred with 3D crown models. LAD corresponds to the one-sided leaf area divided by the canopy element modeled volume \citep{Gersonde2004}. The reported values of LAD used with 3D-TM models range from \SI{0.3}{\square\metre\per\cubic\metre} to \SI{6}{\square\metre\per\cubic\metre} (\autoref{tab:43lad}). This variation is therefore substantial even for the same species. This encourages improving and harmonizing the methods used to estimate LAD.
 
Some simplifications used to estimate foliage density might appear rough to some readers but they reflect well the difficulty of measuring it; due to tree size and heterogeneity, its measure in forests is much more challenging than in crops or orchards. 

\begin{table}[htb]
\caption[Reported values of leaf area density.]{Reported values of leaf area density (LAD, \si{\square\metre\per\square\metre}) used in 3D crown models with the turbid medium analogy (3D-TM)}
\footnotesize
\begin{tabularx}{\textwidth}{lp{0.2 \textwidth}X}
\toprule
\textsc{species} & \textsc{lad} & \textsc{references} \\ 
\midrule
not specified & 1.86 & \citet{Kuuluvainen1991} \\ 
 &  &  \\ 
\textsc{softwood} &  &  \\ 
\quad \textit{Pinus ponderosa} & 0.03-0.36 & \citet{Law2001} \\ 
\quad \textit{Tsuga heterophylla} & 0.38 &\citet{Mariscal2004} \\ 
\quad \textit{Pseudotsuga menziesii} & 0.38; 0.96; 2.74; 6.00 & \citet{Webb1993, Brunner1998, Gersonde2004, Mariscal2004} \\ 
\quad \textit{Picea abies} & 0.40 & \citet{Courbaud2003} \\ 
\quad \textit{Pinus contorta} & 1.39 & \citet{Stadt2000} \\ 
\quad \textit{Picea glauca} & 1.80-1.88 & \citet{Stadt2000} \\ 
\quad \textit{Abies balsamea} & 1.98 & \citet{Stadt2000} \\ 
\quad not specified & 2.00 & \citet{Sprugel2009} \\ 
\quad \textit{Pinus ponderosa} & 2.01 & \citet{Gersonde2004} \\ 
\quad \textit{Pinus lambertiana} & 2.56 & \citet{Gersonde2004} \\ 
\quad \textit{Abies concolor} & 3.36 & \citet{Gersonde2004} \\ 
\quad \textit{Calocedrus sp.} & 4.51 & \citet{Gersonde2004} \\ 
 &  &  \\ 
\textsc{hardwood} &  &  \\ 
\quad \textit{Populus balsamifera} & 0.30 & \citet{Stadt2000} \\ 
\quad \textit{Populus tremuloides} & 0.44 & \citet{Stadt2000} \\ 
\quad \textit{Fagus sylvatica} & 0.66 & \citet{Piboule2001} \\ 
\quad not specified & 0.50-1.00 & \citet{Sprugel2009} \\ 
\quad \textit{Betula pendula} & 0.79 & \citet{Piboule2001} \\ 
\quad \textit{Betula papyrifera} & 0.80 & \citet{Stadt2000} \\ 
\quad \textit{Quercus kelloggii} & 1.32 & \citet{Gersonde2004} \\ 
\bottomrule
\end{tabularx}
\label{tab:43lad}
\end{table}

\subsubsection{Extinction coefficients}\label{sec:k}
The extinction coefficient, $k$, is a parameter of \autoref{eq:4lambert} and is therefore used whenever the radiation attenuation submodel corresponds to the turbid medium approach. It is usually a species-specific constant but it can also be computed as a function (the ``G-function''), which depends mainly upon the orientation and inclination of leaves as well as on the ray zenith angle ($\eta$). The parameter $k$ has been estimated by measuring leaf inclination and projected leaf areas \citep{Campbell1998, Kim2011}, computed with theoretical functions of leaf distribution \citep{Govind2013} or deduced from the relationship between transmittance and LAI measured at different canopy heights \citep{Landsberg2010}.
  
With 3D crown models, most authors assume that leaf inclination follows a theoretical distribution that is either spherical with $k=0.5$ \citep{Cescatti1997a, Brunner1998, Piboule2001, Courbaud2003} or ellipsoidal \citep{Stadt2000}. With 1D models, authors have sometimes preferred using empirical estimates of $k$ ranging between 0.2 and 0.6 \citep{Jarvis1983, Pierce1988, Bartelink1998b, Aubin2000}. 

\subsubsection{Clumping factors}\label{sec:omega}
The assumption of the random spatial distribution of canopy elements is rarely satisfied \citep{Sinoquet2005, DaSilva2008}, especially with 1D models, because the spatial distribution of trees, but also of canopy elements within trees, is rarely completely random. Moreover, leaves are gathered around branches, and branches are grouped around stems and sometimes grouped in vertices. Whenever the spatial distribution of canopy elements is not known or approximated (e.g., stand map, tree architecture), a clumping factor, $\Omega$, is required in \autoref{eq:4lambert}. Indeed, a canopy with regularly spaced elements ($\Omega>1$) transmits less light than a canopy with randomly scattered elements ($\Omega=1$). By contrast, a canopy with aggregated elements transmits much more light ($\Omega<1$) \citep{Niinemets2010}.
 
There is a lack of reference values for the clumping factor and this measurement is therefore often estimated by model inversion. Like the extinction coefficient, the clumping factor is rarely measured because measuring it for adult trees is very labor intensive. Consequently, it is often assumed to be constant. With 1D models, \citet{Chen1999} used values of the clumping factor of 0.5 and 0.7 for softwoods and hardwoods, respectively. These values were computed by measuring the canopy gap size distribution \citep{Chen1997}. \citet{Kim2011} estimated the clumping factor by harvesting pine shoots. They observed that the clumping factor varied between 0.2 (upper part of the canopy) and 0.9 (lower part of the canopy), while other studies reported values for long-needled species ranging between 0.4 and 0.9 \citep{Therezien2007}. In addition, some studies have demonstrated a variation in the clumping factor with stand density, crown diameter and ray zenith angle \citep{Wang1990, Govind2013}. Because tree aggregation is already taken into account with 3D crown models, the clumping factor in these models is usually assumed to be 1.

\subsubsection{Crown openness}\label{sec:p}
Crown openness ($p$) can be assessed by photographing isolated crowns \citep{Canham1994, Canham1999, Beaudet2002, Astrup2006, Paquette2008, Beaudet2011, DaSilva2012} or by model inversion \citep{Groot2004}. Photographs can be taken using either fish-eye lens \citep{Canham1994, Canham1999, Beaudet2002, Astrup2006, Beaudet2011} or classic lens \citep{DaSilva2012}. In comparison with fish-eye lenses, classic lenses allow smaller isolated parts of the crown to be photographed. Therefore, the use of a classic lens makes it easier to obtain a correct exposure even when sky conditions are not overcast. Subject crowns need to be isolated with no overlapping neighboring crowns. Photographs are processed to compute the proportion of sky pixels (i.e., the crown openness). \citet{Boivin2011} presented an algorithm that identifies the crown extent and argued that crown delimitation has a strong influence on porosity estimates. \citet{Canham1994} observed no variation in $p$ with zenith angle and tree dimensions. However, \citet{Astrup2006} observed that $p$ varied, to a limited extent, in spruce stands with tree diameter and by regions. Similar to the estimates of LAD, standard methods are required to uniformly estimate $p$ taking into account the variability among species and individuals.
 
Crown openness ($p$) and LAD are two related parameters. Indeed, \citet{DaSilva2012} bridged the two approaches proposing a method to compute $k \cdot \textnormal{LAD}$ from $p$. The two approaches of modeling light attenuation through canopy, namely the turbid medium and porous envelope approaches, can therefore be calibrated with field measurements of $p$. Even though it applied only to isolated trees, the measurement of $p$ is more convenient and less laborious than the measurement of LAD. Moreover, the reported variability of $p$ seems lower than that of LAD with values ranging from 0.03 to 0.29 and, an average around 0.1 (\autoref{tab:44openess}).

\begin{table}[htbp]
\caption[Reported values of crown openness.]{Reported values of crown openness used with 3D crown models (3D-PE).}
\footnotesize
\begin{tabularx}{\textwidth}{lp{0.2 \textwidth}X}
\toprule
\textsc{species} & \textsc{crown openness} & \textsc{references} \\  \midrule
\textsc{softwood} &  &  \\ 
\quad \textit{Abies alba} & 0.034 & \citet{DaSilva2012} \\ 
\quad \textit{Abies amabilis} & 0.060 & \citet{Canham1999} \\ 
\quad \textit{Tsuga heterophylla} & 0.080 & \citet{Canham1999} \\ 
\quad \textit{Pinus sylvestris} & 0.065 - 0.110 & \citet{DaSilva2012} \\ 
\quad \textit{Abies lasiocarpa} & 0.090 & \citet{Canham1999} \\ 
\quad \textit{Thuja plicata} & 0.090 & \citet{Canham1999} \\ 
\quad \textit{Picea glauca} & 0.109; 0.110; 0.130 & \citet{Canham1999, Astrup2006, Beaudet2011} \\ 
\quad \textit{Abies balsamea} & 0.111 & \citet{Beaudet2011} \\ 
\quad \textit{Pinus banksiana} & 0.124 & \citet{Beaudet2011} \\ 
\quad \textit{Pinus contorta} & 0.135 & \citet{Canham1999} \\ 
\quad \textit{Thuja occidentalis} & 0.144 & \citet{Beaudet2011} \\ 
\quad \textit{Picea mariana} & 0.070 - 0.290 & \citet{Groot2004} \\ 
 &  &  \\ 
\textsc{hardwood} &  &  \\ 
\quad \textit{Fagus sylvatica} & 0.048 & \citet{DaSilva2012} \\ 
\quad \textit{Fagus grandifolia} & 0.050 & \citet{Beaudet2002b} \\ 
\quad \textit{Malus sylvestris} & 0.052 &\citet{DaSilva2012} \\ 
\quad \textit{Quercus petraea} & 0.043 - 0.073 & \citet{DaSilva2012} \\ 
\quad \textit{Betula papyrifera} & 0.058 & \citet{Canham1999} \\ 
\quad \textit{Populus tremula} & 0.082 & \citet{DaSilva2012} \\ 
\quad \textit{Sorbus torminalis} & 0.049 - 0.120 & \citet{DaSilva2012} \\ 
\quad \textit{Betula alleghaniensis} & 0.097 & \citep{Beaudet2002b} \\ 
\quad \textit{Betula papyrifera} & 0.101 & \citet{Beaudet2011} \\ 
\quad \textit{Acer saccharum} & 0.108 & \citet{Beaudet2002b} \\ 
\quad \textit{Populus balsamifera} & 0.140 & \citet{Canham1999} \\ 
\quad \textit{Populus tremuloides} & 0.163; 0.183 & \citet{Astrup2006, Beaudet2011} \\ 
\quad \textit{Betula alleghaniensis} & 0.206 & \citet{Canham1999} \\ 
\quad \textit{Populus hybrid} & 0.300 & \citet{Paquette2008} \\ 
\bottomrule
\end{tabularx}
\label{tab:44openess}
\end{table}

\subsection{Model applications and output variables}
Because FRTMs are most often used to model forest ecosystem functioning, they are usually coupled with many other models describing, for example, photosynthesis, transpiration, tree growth, timber production, carbon sequestration, nutrient uptake, or hydrological balance. The applications of FRTMs are therefore numerous but they can be seen, for the sake of simplicity, as either ecophysiological applications or stand-dynamics applications.
 
\paragraph{Ecophysiological applications}
Ecophysiological studies uses FRTMs coupled mainly with process-based models that embody our current knowledge of the functioning of forest ecosystem. Such holistic models include, for instance, equations describing the fluxes of carbon, water, and energy between soil, vegetation, and atmosphere \citep{Govind2013}. Such studies aim at improving our understanding of forest ecosystem functioning, exploring forest productivity as a function of resource availability, or at exploring how variation of forest structure and composition can affect ecosystem functioning and dynamics. Forest productivity, standing forest biomass, nutrient uptake or hydrological balance as examples can then be predicted for particular environmental conditions and for forecasted modifications of these. Within this context, FRTMs are necessary to predict the irradiance (\si{\watt\per\square\metre}) and (or) the spectrum of the intercepted radiation at various scales, from a global scale to a leaf scale.

\paragraph{Stand-dynamics applications}
The second group of studies preferentially uses empirical models. They bring less concerns on ecosystem functioning and focus rather on the dendrometric data of interest for forest managers. They aim mainly at predicting the natural evolution of stand structure and composition and (or) predicting the effect of overstory management on understory light conditions \citep{Courbaud2001, Beaudet2002b, Sprugel2009, Beaudet2011}. Also, their objective is the best possible prediction of forest growth and the outcome of intra and inter specific competition. Such models can more precisely predict forest dynamics but, contrary to process-based models, their uses are more strictly limited to the conditions from where they were derived. On the other hand, using empirical models of tree growth allows focusing on light availability without calibrating and programming all of the physiological processes in relation to tree growth. This simplification has often been justified whenever light can be considered as the first limiting factor of vegetation growth (e.g., tree regeneration under closed canopy). Within this context, FRTMs are needed to predict tree irradiance, i.e., the amount of energy absorbed by trees \citep{Courbaud2003}, the proportion of radiation transmitted (transmittance) to the regeneration \citep{Pacala1996, Lieffers1999, Beaudet2002b, Sprugel2009, Beaudet2011}, the amount of light available for understory biodiversity \citep{Balandier2006b, Barbier2008}, or the vertical gradient of transmittance within the canopy \citep{Gersonde2004, Mariscal2004}. They are used mostly at plot or stand scales (from \SI{100}{\square\metre} to \SI{1}{\hectare}) with a simulated period of at least 1 year. 

\subsection{Model evaluation}
Most authors evaluate their model by comparing estimates and measurements of irradiance or transmittance taking into account either total incident light or only the incident diffuse light (e.g., indirect site factor). Light measurements are performed usually with hemispherical photographs and less frequently with light sensors. Hemispherical photography is an indirect method with an associated level of error that can occasionally be substantial. Moreover, this approach uses a radiative model to assess irradiance and transmittance. Using hemispherical photography to evaluate FRTMs therefore leads to a comparative evaluation of one radiative model with another. In addition, FRTMs have been evaluated for their performance in predicting irradiance or transmittance at the ground level, above the understory vegetation, or, occasionally, at different heights within a tree canopy \citep{Mariscal2004, Kim2011}. In the present study, we compiled published information on the performance of different models, i.e., their ability to predict the measurements with low levels of error and bias. We considered all published light models that were used to predict radiative transfer through forest canopies and for which the performance was evaluated. In \autoref{tab:45evaluation}, we summarized the reported performance of these models. Unfortunately, authors have used different approaches to assess model performance, making comparison difficult. They often adjusted linear models between estimates and observations and hence assessed model precision using the coefficient of determination ($R^2$) or the root mean square error (RMSE). Model bias was usually examined graphically but some authors have used the coefficients of the linear regression between measurements and estimates. Moreover, even similar indicators (e.g., $R^2$) could not be compared without caution. While most authors compared estimates and measurements carried out at different periods \citep{Kim2011, Govind2013} or at different positions within plots, others compared average estimates and measurements between sites of areas ranging from \SI{900}{\square\metre} to \SI{1600}{\square\metre} \citep{Stadt2000} or the average gap profile \citep{Mariscal2004}. In addition, others compared the statistical distribution of transmittance estimates and measurements \citep{DaSilva2012}. Finally, the numbers of plots and sites were also far from constant among model evaluations.
 
Most authors found good agreement between measurements and estimates when averaged at the stand scale \citep{Stadt2000, Mariscal2004}. In other words, most models predicted transmitted light poorly at a particular location (at sensor scale) beneath the canopy but rather well at the stand scale. After a good calibration, the relationship between measurements and predictions was found to be generally close to a 1:1 relationship and, hence, without unacceptable bias in the point-to-point comparison. Nevertheless, large deviations were often observed between estimates and measurements of transmittance (often up to \SI{20}{\percent} of above canopy light) even for low-light levels. Many authors argued that point-to-point variations beneath the forest canopy are rarely predicted with accuracy because minor errors in tree position and crown dimensions can lead to dramatic changes in simulations of transmittance \citep{Fournier1996, Groot2004, Mariscal2004, DaSilva2012}. On the other hand, site averages were better predicted \citep{Stadt2000, Pinno2001, Mariscal2004, DaSilva2012}. Similarly, model performance increased with the length of the simulated period (e.g., 1 year versus 1 hour) especially when direct radiation were taken into account \citep{Brunner1998, Groot2004}.
 
All of the different modeling approaches were reported to perform well but few authors compared the different approaches. \citet{Kim2011} found that their 1D model performed as well as 3D crown models when they took into account the aggregation of trees, leaves and branches with appropriate clumping factors. Similarly, \citet{Balandier2009} obtained very similar results with a 1D model and a 3D surface model in a mixed understory of European beech (\textit{Fagus sylvatica} L.) and European red raspberry (\textit{Rubus idaeus} L.) (\autoref{fig:42mockup}). 3D-TM (e.g., MIXLIGHT) and 3D-PE (e.g., SORTIE-ND) appeared to exhibit a very similar performance, although \citet{Canham1994} found that a hit model (i.e., 3D-PE with $p=0$) performed better than a 3D-TM. These authors interpreted this result as an effect of foliage scattering at the crown periphery, especially for shade intolerant species. \citet{Boivin2011} used a 3D-PE model and obtained good results in plantations of hybrid poplars, with $p=0$, whereas in the same study, this parameter value provided the worst results in young boreal mixedwoods.
 
Nevertheless, we noted that the relative performance depended upon the modeled description of canopy geometry (\autoref{tab:45evaluation}). The highest levels of model performance were obtained with very detailed 3D-TM models \citep{Cescatti1997b, Brunner1998}. These models used asymmetric crowns described by a combination of nonquadratic shapes and modeled the radiative transfer using a turbid medium approach with $k=0.5$ and $\Omega=1$. In addition, \citet{Brunner1998} restricted the foliage to shells within the crown but acknowledged that simplification of his canopy model would have altered the results only slightly. Moreover, he restricted his sample to a single even-aged coniferous stand with a large gradient of transmittance, which made model calibration and validation easier. The model of \citet{Cescatti1997b} differed from the model developed by \citet{Brunner1998} in its inclusion of an algorithm for scattering processes.
   
Additionally, model performance appears to be related to forest structure (\autoref{tab:45evaluation}). Model performance decreases from even-aged young stands to multilayered and mixed mature stands. This observation is well supported by pairs of publications in which the same model has been applied to different forest structures. For instance, the model FOREST was shown to provide excellent results in an 80-year-old spruce stand in Finland \citep{Cescatti1997b}, whereas it provided only moderate results in a multilayered forest of ponderosa pine in Oregon \citep{Law2001}. Similarly, using the model tRAYci, \citet{Brunner1998} successfully predicted the spatial distribution of the transmittance under a 20-year-old stand of Douglas fir, whereas \citet{Gersonde2004} obtained a lower model performance under a mixed conifer forest. MIXLIGHT successfully predicted point-to-point variation in transmittance under aspen-dominated stands aged from 1 to 30 years \citep{Pinno2001}, but it was less successful in predicting the average values of transmittance in various even-aged boreal stands aged from 69 to 159 years \citep{Stadt2000}.  Finally, \citet{Boivin2011} observed the same trend between young boreal forests and hybrid poplar plantations, and \citet{Koop1994} found that including measurement points located close to small trees reduced the $R^2$ from \SI{94}{\percent} to \SI{77}{\percent}.

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\begin{table}[p]
\caption[Summary of the published evaluations of FRTMs.]{Summary of the published evaluations of FRTMs. Models are classified by canopy model and type of forest stand forest structure. $R^2$ with an * symbol means that the validation was performed with predictions and measurements averaged for each sites rather than by point-to-point comparisons.}
\footnotesize
\renewcommand{\arraystretch}{0.8}% Tighter
\begin{tabularx}{\linewidth}{XXX}
\toprule
 \textsc{plantation}  & \textsc{even-aged pure stand} & \textsc{uneven-aged or mixed stands} \\ \midrule
\multicolumn{3}{l}{\textsc{1d canopy}} \\ 
 & no model name &   \\ 
 & \citet{Govind2013} &   \\ 
 & $R^2$ = 0.93; &   \\ 
  & RMSE = $30-70$ \si{\micro\mole\per\square\metre\per\second} &   \\ 
\addlinespace[2mm] 
 & no model name &    \\ 
 & \citet{Kim2011} &   \\ 
 & RMSE = $0.06-135$ \si{\si{\micro\mole\per\square\metre\per\second}} &    \\ 
\addlinespace[2mm] 
 & 4C-A-RTM &     \\ 
 & \citet{Bertin2011} &     \\ 
 & good relationship ($R^2$ = 0.92) with seedling growth &     \\ 
\addlinespace[2mm]
\quad MIXLIGHT & MIXLIGHT & SORTIE   \\ 
\quad \citet{Pinno2001}  & \citet{Stadt2000}  & \citet{Canham1999}    \\ 
\quad $R^2$ = 0.92 & $R^2$ = 0.74* & $R^2$ = 0.86   \\ 
\addlinespace[2mm] 
\multicolumn{3}{l}{\textsc{3d crown with quadratic shapes  }} \\ 
\quad SORTIE-ND & SORTIE-ND & SILVI-STAR   \\ 
\quad \citet{Paquette2008}  & \citet{Boivin2011}  & \citet{Koop1994}    \\ 
\quad $R^2$ = 0.92 & $R^2$ = 0.52 & $R^2$ = 0.77 – 0.94   \\ 
\addlinespace[2mm] 
\quad SORTIE-ND &  & SAMSARA   \\ 
\quad \citet{Boivin2011}  &  & \citet{Courbaud2003}   \\ 
\quad $R^2$ = 0.46 - 0.92 &  & RMSE = \SI{5.2}{\percent}   \\ 
\addlinespace[2mm] 
 &  & OLTREE \& SolTran   \\ 
 &  & \citet{Mariscal2004}    \\ 
 &  & $R^2$ = 0.94*   \\ 
  &   &     \\ 
\addlinespace[2mm]
 & MAESTRO & tRAYci   \\ 
 & \citet{Wang1990}  & \citet{Gersonde2004}    \\ 
 & RMSE = \SI{10}{\percent} & $R^2$ = 0.80   \\ 
\addlinespace[2mm] 
 & TASS/tRAYci & CORONA   \\ 
 & \citet{Brunner1998}  & \citet{Groot2004}    \\ 
 & $R^2$ = 0.97 & $R^2$ = 0.88   \\ 
\addlinespace[2mm]
\multicolumn{3}{l}{\textsc{3d crown with nonquadratic shapes}} \\ 
 & tRAYci & M$\mu$SLIM   \\ 
 & \citet{Piboule2001} & \citet{DaSilva2012}    \\ 
 & RMSE = \SI{2}{\percent} & “in the range of model using   \\ 
 &   & similar scales of description”   \\ 
\addlinespace[2mm]
 & FOREST & FOREST  \\ 
 & \citet{Cescatti1997b}  & \citet{Law2001}    \\ 
 & $R^2$ = 0.97 & $R^2$ = 0.62   \\ 
\addlinespace[2mm]
\multicolumn{3}{l}{\textsc{3d surfaces of leaves, branches, and stems}} \\ 
\quad AmapSim & no model name &   \\ 
\quad \citet{Leroy2009}  & \citet{Fournier1996}  &   \\ 
\quad no statistical difference in 3$/$4 sites & RMSE $\approx$ 10 – \SI{15}{\percent} &    \\ 
\addlinespace[2mm]
\quad RATP &   &     \\ 
\quad \citet{Sinoquet2001}  &   &   \\ 
\quad RMSE = $60-120$ \si{\si{\micro\mole\per\square\metre\per\second}} &   &    \\  \bottomrule
\end{tabularx}
\label{tab:45evaluation}
\end{table}
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\subsection{Model sensitivity}
We attempted to identify which model parameter was reported to have strong effect on FRTM performance. As many studies actually lack a common referenced method, we classified the studied parameters into three qualitative levels according to their impact on FRTM predictions: high, medium, and low. This classification was based mainly on the discussion by the respective authors as part of their studies. For example, given the results of \citet{Gersonde2004}, LAD and crown radius were classified respectively in the low and high categories. Indeed, these authors observed that replacing individual LAD with a species average did not reduce the model fit. On the other hand, in the same study, the same simplification applied to the crown radius reduced very significantly model fit ($R^2$ was reduced from about \SI{75}{\percent} to \SI{65}{\percent}). Next, we counted the number of publications of each combination of parameter and sensitivity level (\autoref{tab:46sensitivity}). Unfortunately, not all of the parameters had been tested in similar conditions or with the same model, as far as forest stands are considered. We therefore recognize that our analysis is incomplete, but underlines well the common trends between different studies.
 
As a general rule, we found that models appeared more sensitive to the parameters describing between-crown gaps than to those describing within-crown gaps or within-crown architecture. In most studies, a larger amount of light was transmitted between rather than through the crowns. Indeed, as shown by the estimates of crown openness in \autoref{tab:44openess}, only \SI{5}{\percent} to \SI{30}{\percent} of incident light passes through the crown. The parameters describing the between-crown gap geometry were therefore of primary importance, justifying the use of a hit model (3D-PE with $p=0$) for certain types of stands. Nevertheless, we found that authors often noted that FRTMs were more sensitive to other parameters such as tree clumping or leaf area in dense stands \citep{Wang1990, Boivin2011}.
 
Stand density and stand edges are not model calibration parameters but they greatly affect transmitted light estimates \citep{Bartelink1998b, Piboule2001}. Moreover, it is even possible that stand density modifies model sensitivity to the inner-crown parameters. In dense and homogeneous stands, most rays were found to intercept crowns, whereas in low-density stands, the majority of transmitted radiation comes from canopy gaps. Models are thus found to be less sensitive to inner-crown parameters with low-density stands.

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\begin{table}[htbp]
  \caption[FRTM sensitivity to calibration parameters.]{FRTM sensitivity to calibration parameters. For each parameter, we counted the number of publications ($n$) analyzing its effect on model predictions. According to the authors’ conclusion, we classified their results into three categories: low, medium or high impact. The score corresponds to (high + 0.5 $\cdot$ medium)/$n$.}
    \footnotesize
    \begin{tabularx}{\linewidth}{p{5cm}rRRRrp{10cm}}
    \toprule
    \textsc{parameter} & $n$ & \textsc{low} & \textsc{medium} & \textsc{high}  & \textsc{score} & \textsc{references} \\
    \midrule
\multicolumn{7}{l}{\textsc{3d crown model}} \\ 
\quad crown radius & 7     &       &       & 7     & 1     & \citet{Beaudet2002,Brunner1998,Gersonde2004,Piboule2001, Cescatti1997b,DaSilva2012, Stadt2000} \\
\quad crown shape & 2     &       &       & 2     & 1     & \citet{Brunner1998, Piboule2001} \\
\quad stand density & 2     &       &       & 2     & 1     & \citet{Bartelink1998b, Essery2008} \\
\quad stand edge & 1     &       &       & 1     & 1     & \citet{Piboule2001} \\
\quad understory vegetation & 1     &       &       & 1     & 1     & \citet{Beaudet2002} \\
\quad tree height & 2     &       & 1     & 1     & 0.8   & \citet{Beaudet2002, Gersonde2004} \\
\quad crown length & 3     &       & 2     & 1     & 0.7   & \citet{Piboule2001, Stadt2000, Gersonde2004} \\
\quad crown openness & 2     & 1     &       & 1     & 0.5   & \citet{Beaudet2002, Boivin2011} \\
\quad LAD   & 5     & 2     & 2     & 1     & 0.4   & \citet{Bartelink1998b, Brunner1998, Gersonde2004, Piboule2001, Stadt2000} \\
\quad foliage clustering within crowns & 2     & 1     & 1     &       & 0.3   & \citet{Bartelink1998b, Cescatti1997b} \\
\quad trunk interception & 2     & 1     & 1     &       & 0.25  & \citet{Brunner1998, Fournier1996} \\
\quad crown max-width height & 1     & 1     &       &       & 0     & \citet{Piboule2001} \\
\quad foliage clustering around shoots & 1     & 1     &       &       & 0     & \citet{Bartelink1998b} \\
\quad foliage inclination & 1     & 1     &       &       & 0     & \citet{Stadt2000} \\
\quad foliage as a sub-shell of crown & 1     & 1     &       &       & 0     & \citet{Gersonde2004, Cescatti1997b} \\
\addlinespace[2mm]          
\multicolumn{7}{l}{\textsc{1d canopy model}} \\ 
\quad leaf distribution & 1     & 1     &       &       & 0     & \citet{Wang1990} \\
\quad crown surface & 1     &       &       & 1     & 1     & \citet{Duursma2007} \\
\quad leaf area & 3     &       &       & 3     & 1     & \citet{Kim2011, Bertin2011, Duursma2007} \\
\quad tree clumping & 3     &       & 1     & 2     & 0.8   & \citet{Kim2011, Bertin2011, Duursma2007} \\
\quad branch area & 2     &       & 2     &       & 0.5   & \citet{Kim2011, Bertin2011} \\
\quad foliage inclination & 1     &       & 1     &       & 0.5   & \citet{Kim2011} \\
\quad vertical distribution of leaves & 1     &       & 1     &       & 0.5   & \citet{Kim2011} \\
\quad stem area & 2     & 1     & 1     &       & 0.3   & \citet{Kim2011, Bertin2011} \\
\quad crown shape & 1     & 1     &       &       & 0     & \citet{Duursma2007} \\
    \bottomrule
    \end{tabularx}
  \label{tab:46sensitivity}
\end{table}

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\subsubsection{1D model}
The main source of variability of 1D models was found in the areas covered by crowns \citep{Duursma2007} and leaves \citep{Duursma2007, Bertin2011, Kim2011}. These parameters appeared to be highly influential in all the sensitivity analyses that studied their effects. These estimates were further corrected with clumping factors that took into account the aggregation of stems, branches, and (or) leaves. The clumping factor or the spatial structure of trees or crowns therefore played an important role in the corresponding studies. However, model sensitivity to these parameters decreased with stand density, stand LAI and leaf inclination \citep{Kim2011}.

Parameters of crown architecture played a secondary role. Only a few studies reported that changes in values of branch area, foliage inclination, and vertical distribution of leaves affected notably 1D model predictions. Similarly, crown shape and modeling stem interception had only a minor effect on model prediction. 

\subsubsection{3D crown model}
3D crown models appeared highly sensitive to the crown radius in all of the sensitivity analyses that studied the influence of this parameter \citep{Canham1994, Cescatti1997b, Brunner1998, Stadt2000, Piboule2001, Beaudet2002b}. It even appeared that estimating the crown radius using allometric relationships instead of measuring this parameter in the field individually for each tree significantly decreased model precision \citep{Piboule2001, DaSilva2012}. Indeed, crown dimensions largely depend upon neighboring competition and tree history which are two effects usually not included in allometric relationships and that should be taken into account \citep{Piboule2005b}.
 
Crown shape, tree height and height to crown base seemed to have less influence \citep{Brunner1998, DaSilva2012}. FRTMs appeared highly sensitive to these parameters in less than \SI{50}{\percent} of the sensitivity analyses. For instance, \citet{Brunner1998} tested two representations of crowns (with cones or asymmetric ellipsoids) and concluded that the two models gave similar results. Similarly, the height of maximum crown extension is needed to define the most complicated crown shapes and using such complex shape led to only minor improvements \citep{Piboule2001}.
 
The density of leaves and branches within crowns (expressed by LAD and crown openness) moderately affected the estimates of transmitted light beneath the forest canopy \citep{Bartelink1998b, Brunner1998, Lieffers1999, Stadt2000}. Interestingly, reasonable changes in LAD affected model predictions markedly in only one of the five corresponding sensitivity analyses \citep{Bartelink1998b, Brunner1998}. It is worth noting that good estimates of LAD (3D-TM) or crown openness (3D-PE) require the correct delimitation of crown volume. Errors in computing the crown volume would lead to, for example, errors in calculating total leaf area. Similarly, LAD, clumping factors and extinction coefficients are strongly correlated (\autoref{eq:4lambert}, \citet{Bartelink1998b}). This finding therefore justifies using fixed values of LAD per species.
 
Foliage distribution, clustering and inclination were studied to a slightly lesser extent, but these parameters appeared to have a small effect on 3D crown model predictions \citep{Wang1990, Cescatti1997b, Bartelink1998b, Stadt2000, Gersonde2004}. Foliage distribution likely depends upon species. For instance, \citet{Cescatti1997b} reported evidence of the aggregation of spruce needles around vertices. Nevertheless, the influence of this aggregation on FRTM predictions appeared weak \citep{Cescatti1997b, Bartelink1998b}, probably because 3D models are used mostly for stands with rather open canopy where gaps between crowns are more important than gaps within crowns. Similarly, modeling trunk interception was reported without significant effects in the two sensitivity analyses studying this parameter \citep{Fournier1996, Brunner1998}. With 3D crown model, using the default values of $k=0.5$ and $\Omega=1$ appeared therefore a reasonable simplification.

\subsection{Discussion}
This review illustrates and provides an overview of the variety of approaches used to represent a forest canopy and to estimate radiation intercepted by the canopy elements (\autoref{fig:43modeluse}). Even though more thoroughgoing conclusions could likely have been formulated with benchmarking of a wide set of FRTMs using a common data set with different stand structures, we were able to highlight the specificity of forest canopy in comparison with many agricultural crops and consequences in terms of modeling. The recent studies that we reviewed enabled us to quantify model uncertainty, to identify sensitive parameters, and to recommend the modeling approaches that best suit the different FRTM applications. 

\begin{figure}[tb]
\includegraphics[width=\textwidth]{fig43modeluse}
\caption[Overview of the end-uses of FRTMs.]{Overview of FRTMs according to their end-uses. Most FRTMs included in stand-dynamics models are 3D crown models, whereas most FRTMs used for ecophysiological studies are 1D models. LAD, leaf area density; $k$, extinction coefficient; $\Omega$, clumping factor; $p$, crown openness; 1D, one dimensional; 3D, three-dimensional.}
\label{fig:43modeluse}
\end{figure}

\subsubsection{Model uncertainty}
1D models accurately predict the temporal variations of irradiance (RMSE ranged roughly between \SI{30}{\micro\mole\per\square\metre\per\second} and \SI{120}{\micro\mole\per\square\metre\per\second}) within homogeneous stands \citep{Kim2011, Govind2013}. The vertical gradient of transmitted light beneath the canopy can be predicted reasonably well with multilayer models \citep{Landsberg2010}, but, the spatial variation in irradiance in the understory is not predicted by 1D models.

3D crown models accurately predict averaged measures of transmitted light in a forest plot. The prediction of transmittance at a particular location beneath the canopy is clearly less accurate with errors often up to \SI{20}{\percent} of transmittance \citep{Cescatti1997b, Brunner1998, Stadt2000}. 3D crown models are capable of capturing a certain spatial variability of transmitted light but not with a resolution as precise as the one using light sensors. Within forest dynamics models, small errors in the spatial distribution of understory light should be considered as being of minor importance compared with, for example, the prediction of the cumulated area of microsites favorable to regeneration growth, i.e., receiving an irradiance within the range corresponding to the needs of a given species.

Moreover, model uncertainty increases with the complexity of the canopy. Transmittance beneath young plantation is better predicted than beneath uneven-aged and mixed stands. Transmitted light under homogeneous canopy is usually well predicted with every modeling approach. On the contrary, transmitted light under heterogeneous canopy is supposed to be hardly predicted with 1D models. 3D crown model appeared so far the best solution to model the radiative transfer through such complex canopies.

Model uncertainty also decreases with increasing length of the simulated period especially if direct radiation is taken into account. As already pointed out for forest productivity models \citep{Landsberg2010}, predicting transmitted radiation for a brief period (e.g., 30 minutes) requires very accurate radiation and canopy measurements. Accurate monthly and yearly averaged predictions are however more easily obtained \citep{Essery2008, Landsberg2010}, probably because of errors compensation.
  
We found no clear evidence regarding the advantage of using the turbid medium approach (3D-TM) versus the porous envelope approach (3D-PE). Both methods provided satisfactory results. Besides, if the crown openness varies with crown dimensions, both method are somehow related because $k \cdot \textnormal{LAD}$ can be deduced from $p$ \citep{DaSilva2012}. If LAD and $p$ are species constant, then for every tree, independent of their size, the porous envelope approach affects transmittance to every crown in the same way, whereas it depends upon crown size and beam orientation with the turbid medium approach. In this latter case, model performance of both approaches likely differs with the variability of crown dimensions.

\subsubsection{Sensitive input parameters}
Calibration efforts must focus primarily in the description of between-crown gaps or crown radius, which are the most sensitive parameters of FRTMs. This result is perhaps not intuitive but probably reflects that the forest canopy consist of deep crowns with dense foliage. Therefore, crowns intercept a large amount of light whatever their specific features. As a result, the proportion of transmitted light that is not intercepted by tree crowns, i.e., transmitted through gaps between crowns, is fundamental.

With 1D models, the key calibration parameters are the crown surface and (or) the LAI, which might further be corrected with extinction and clumping factors. On the other hand, 3D models require overall an accurate tree map and individual measurements of the crown radius, especially in order to characterize heterogeneous canopies. Crown shape appeared of lesser importance than crown radius and we did not find that it was worth using complex crown shapes. Furthermore, standard values of the extinction coefficient and clumping factors ($k=0.5$ and $\Omega=1$, respectively) were often used with 3D crown model and frequently gave a good approximation. Therefore, 3D models appear better suited for long simulation periods than 1D model because 3D model usually do not imply predicting changes of extinction and clumping factors.

\subsubsection{Applications}
We identified the advantages and drawbacks of the FRTM modeling approaches which allowed us to identify the approaches, that best suit to different potential applications of FRTM (\autoref{tab:47synthesis}). Overall, the potential applications depend mainly upon how the canopy is modeled.

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\begin{table}[htb]
\caption[Synthesis of FRTM advantages and drawbacks.]{Synthesis of model advantages and drawbacks with suggested scale of uses, forest structures, and examples of applications.}
\footnotesize
\begin{tabularx}{\linewidth}{lp{3cm}p{3cm}XXX}
\toprule
\textsc{model}  & \textsc{scale of uses} & \textsc{forest structure} & \textsc{advantages} & \textsc{disadvantages} & \textsc{applications} \\ 
\midrule
1D model & from stand to global, from hours to one year & even-aged stand, with one or two species & low number of field data; easy to couple to stand level data and models (canopy gas transfers, canopy cover, Leaf Area Index). & model parameters (foliage distribution) are hardly measured in the field; lack of knowledge to model adequately changes of foliage distribution parameters through time and after thinning; do not take explicitly into account spatial structure. & net primary production; carbon uptake; nutrient cycling; photosynthesis rate \\ 
\addlinespace[2mm]
3D-M or 3D-PE & from plot to stand, years & even-aged or unven-aged stand, pure and mixed stand & translate directly changes of stand structure (tree density, species, sizes, spatial distribution) into changes of canopy;
allow the calculation of radiation interception by each tree; allow the calculation of maps of irradiance under canopy. & require information on tree location (obtained either from field measures or from theoretical distributions); require information on crown characteristics (obtained either from field measures or from allometric relationships) & timber production; understory growth; forest dynamics \\ 
\addlinespace[2mm]
3D-S & from leaf to tree, from hours to years & isolated tree, agroforestry system, homogeneous stand & allow the most precise modeling of radiation interception; can be coupled to plant structure-function models. & require information on plant architecture (either measured or modeled); high computing complexity limiting the number of simulated trees.  & tree physiology; tree morphogenesis; 
tree architecture \\ 
\bottomrule
\end{tabularx}
\label{tab:47synthesis}
\end{table}

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1D models predict well the irradiance of canopy with a low number of input data and parameters. They are sensitive to the parameters describing foliage density and distribution ($k$ and $\Omega$, respectively). They are ideally coupled with process-based of photosynthesis rate in order to explore forest productivity, carbon uptake, or nutrient cycling at the stand or greater scale. They are, however, limited to stands with one or two species and a relatively homogeneous (even-aged) structure. Moreover, the simulated forest should have a stable structure or the simulated time period should be limited because the calibration parameter depends strongly upon stand structure and composition. In most cases, 1D models hardly predict light conditions after silvicultural operations such as partial harvests. Silvicultural operations often modify not only stand height, and density but also foliage features and distribution because some categories of trees are preferentially cut or because gaps are created. Through time, tree growth, self-thinning and self-pruning additionally modify the spatial distribution of foliage. In particular, as trees become mature, foliage tends to accumulate in fewer and bigger crowns. Next, as several regeneration cohorts develop, foliage becomes aggregated in crowns of varied dimensions.
 
3D crown models (3D-TM and 3D-PE) are better used in studies focusing on timber production and stand dynamics because, in many cases, those processes are generally not explicitly related to photosynthesis, transpiration, or nutrient cycling. They accommodate to heterogeneous stand structure. They include explicitly stand spatial structure, which enables studying the impact of forest structure (e.g., testing different silvicultural scenarios) on forest production and yield. This is an interesting feature with the general increasing interest devoted to uneven-aged, shelterwood or close-to-nature silviculture in many countries. Moreover, single tree based models such as 3D crown models offer the best opportunity to analyze forest management strategies \citep{Porte2002}. Indeed, partial harvest modifies the average dimensions of trees and stand spatial structure. The increase in transmitted light is therefore complex and does not depend only upon changes of stand density. 3D crown models are sensitive to canopy spatial structure and, hence, require precise mapping and crown measuring. Currently, the cost of these measurements limits the size of the modeled stand but the use of airborne technologies is promising \citep{Essery2008}.

3D surface models require acquiring large amounts of field data to reproduce tree architecture but they enable studying light capture at plant organ to tree levels \citep{Rey2008}. Whenever trees can realistically be duplicated in a forest model, 3D surface models offer good opportunity for studying the effects of tree architecture on tree physiological processes. Future technologies, such as terrestrial LiDAR can also help in acquiring the field data required to build 3D surface models for more complex stands.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Implementing a radiative transfer model}\label{sec:4implementation}

Exploring how forest managers can manipulate understory light requires a model that is capable of predicting light interception by heterogeneous canopies, especially in uneven-aged forests that are composed of several broadleaved species, and that takes directly into account the structure and density of the simulated stands. According to our literature review, the most efficient strategy for achieving this goal consists in utilizing a 3D crown model. Crowns are best modeled with combinations of quadratic shapes to reproduce accurately the variability of crown dimensions and to compute analytically (i.e., faster than with numerical computations) the interception of radiation by crowns. The turbid medium approach is preferred over the porous envelope approach, where we have hypothesized that this approach might better predict how much radiation is intercepted by crowns of various sizes.
 
Such a model is implemented in the forest simulation platform Capsis \citep{DufourKowalski2012} in a library named SamsaraLight. The full description of the functioning of this library is beyond the scope of this thesis. A first version of SamsaraLight was described by \citet{Courbaud2003} and further documentation can be found on the Capsis website \citep{samsaralight}.

Hereafter, I introduce how a heterogeneous canopy was modeled, i.e., the 3D reconstruction of a stand canopy with models of crown geometry (\autoref{sec:4crown}), crown dimensions (\autoref{sec:4crownradius}, \autoref{sec:4treeheight} and \autoref{sec:4crownbase}), and how foliage features were characterized (\autoref{sec:4foliage}). Models of crown dimensions and foliage density were adjusted with the data that were gathered in the 27 studied sites (\autoref{ch:studyArea}). Next, I report on an evaluation of model predictions (\autoref{sec:4validation}).

\subsection{Crown geometry}\label{sec:4crown}
We modeled crown geometry with one or several sections of asymmetric ellipsoids (\autoref{fig:44crownshape}). The choice of one of these options depends mainly upon the tree measurements that are available. For example, in the next chapter, we opted to model crowns with single asymmetric ellipsoids because we did not measure the height to the largest crown cross-section or the 4 crown radii of every tree.

The crown geometry is defined by 4 crown radii, which are measured along the cardinal directions, together with tree height and height to the base of the crown. These measurements enable us to compute the parameters of the equation of an ellipsoid or sections of an ellipsoid (\autoref{eq:4ellipsoid}):

\begin{equation}
	\frac{(x-x_0)^2}{a}+\frac{(y-y_0)^2}{b}+\frac{(z-z_0)^2}{c}=1
	\label{eq:4ellipsoid}
\end{equation}

where ($x_0$, $y_0$, $z_0$) are the coordinates of the ellipsoid center, and $a$, $b$, and $c$ are the semi-axes that are respectively oriented from West to East, South to North, and from bottom to top. 

Furthermore, to estimate missing measurements of crown dimensions and to develop an algorithm describing the evolution of crown dimensions, we adjusted a set of allometric relationships for the six groups of species that were used in our simulation program. In \autoref{sec:Ahlce}, we additionally introduce how the height to the largest crown extension can be estimated to define crown shapes with upper and lower sections of different length (different c\textsubscript{up} and c\textsubscript{down} in \autoref{fig:44crownshape}).

\begin{figure}[htb]
\myfloatalign
\subfloat[1 ellipsoid]{\includegraphics[width=0.3 \textwidth]{fig44crownSection1}} \quad
\subfloat[2 sections of ellipsoid]{\includegraphics[width=0.3 \textwidth]{fig44crownSection2}} \quad
\subfloat[8 sections of ellipsoid]{\includegraphics[width=0.3 \textwidth]{fig44crownSection3}} 
\caption[Geometric models of tree crown.]{Geometric models of tree crown with the parameters of the ellipsoid equation (\autoref{eq:4ellipsoid}).}
\label{fig:44crownshape}
\end{figure}

\subsection{Crown radius}\label{sec:4crownradius}
Crown radii (CR) were best modeled with power function of tree diameter (dbh) in a manner similar to that used by \citet{Beaudet2011}. The model was adjusted with nonlinear least-squares methods \citep{R2013}. The power function models of crown radius fitted well the 6 groups of species. Root mean square error (RMSE) was less than \SI{81}{\centi\metre} (\autoref{tab:48crownradius}). For comparable tree dbhs, individuals of the shade-tolerant species (e.g., hornbeam and beech) had wider crowns than individuals of the less shade-tolerant species (e.g., oak and birch). 

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\begin{table}
\caption[Parameters of the model of crown radius.]{Parameter estimates and their \SI{95}{\percent} confidence intervals with $\alpha =0.05$ (CI) for the power function model between crown radius (CR) and tree diameter (dbh): CR $= a \cdot \textnormal{dbh}^b$. Also presented are the number of measured trees ($n$), the ranges of measured dbh, the ranges of measured CR, and the root mean square error (RMSE).}
\footnotesize
\centering
\begin{tabularx}{\linewidth}{lrRRRRRRR}
\toprule
\textsc{species} & $n$ & dbh & CR & \multicolumn{2}{c}{$a$} & \multicolumn{2}{c}{$b$} & RMSE \\ 
\cmidrule(lr){5-6}\cmidrule(lr){7-8}
 &  & \si{\centi\metre} & \si{\metre} & Estimate & CI & Estimate & CI & \si{\metre} \\ 
\midrule
Oak 			& 314 & 2.4;92.3 & 0.5;7.7 & 0.310 & 0.240;0.396 & 0.698 & 0.634;0.764 & 0.808 \\ 
Beech 			& 475 & 2.4;80.5 & 0.3;10.4 & 0.742 & 0.679;0.808 & 0.516 &0.492;0.541& 0.755 \\ 
Hornbeam 		& 67 & 2.4;42.0	& 0.8;5.1 & 0.854 & 0.682;1.054 & 0.503 & 0.425;0.583 & 0.632 \\ 
Birch 			& 40 & 2.4;49.7 & 0.6;3.8 & 0.536 & 0.457;0.621 & 0.493 & 0.444;0.545 & 0.266 \\ 
Other hardwoods & 43 & 2.5;19.8 & 0.7;3.4 & 0.960 & 0.616;1.436 & 0.325 & 0.144;0.515 & 0.576 \\ 
Other softwoods & 45 & 2.4;21.0& 0.6;2.5 & 0.516 & 0.441;0.601 & 0.509 & 0.440;0.579 & 0.179 \\ 
\bottomrule
\end{tabularx}
\label{tab:48crownradius}
\end{table}

\begin{table}
\caption[Parameters of the model of tree height.]{Parameter estimates and their \SI{95}{\percent} confidence intervals with $\alpha =0.05$ (CI) of Mitscherlich’s model between tree height and tree diameter (dbh): height $= m \cdot (1-\exp(-(\textnormal{dbh}-a)/b)$. Also presented are the number of measured trees ($n$), the ranges of measured dbh, the ranges of measured tree height, and the root mean square error (RMSE).}
\footnotesize
\centering
\begin{tabularx}{\linewidth}{lrRRRrRrRrR}
\toprule
\textsc{species} & $n$ & dbh & height & \multicolumn{2}{c}{$a$} & \multicolumn{2}{c}{$b$} & \multicolumn{2}{c}{$m$} & RMSE \\ 
\cmidrule(lr){5-6}\cmidrule(lr){7-8}\cmidrule(lr){9-10}
 &  & \si{\centi\metre} & \si{\metre} & Estimate & CI & Estimate & CI & Estimate & CI & \si{\metre} \\ 
\midrule
Oak   & 706 & 2.4;94.9 & 3.1;35.8 & -1.629 & -1.93;-1.33 & 32.733 & 29.95;35.51 & 31.524 & 30.38;32.67 & 2.565 \\
Beech & 1271 & 2.4;84.4 & 3.3;39.5 & -1.835 & -2.15;-1.52 & 36.298 & 33.44;39.15 & 36.424 & 34.88;37.97 & 2.626 \\
Hornbeam & 175 & 2.4;42.0 & 4.1;25.8 & -2.441 & -3.74;-1.14 & 32.776 & 14.31;51.25 & 32.083 & 20.43;43.74 & 2.522 \\
Birch & 69 & 2.4;49.7 & 3.6;27.3 & -0.762 & -1.44;-0.08 & 19.122 & 13.17;25.07 & 28.617 & 24.40;32.84 & 1.619 \\
Other hardwoods & 73 & 2.5;76.7 & 3.3;29.3 & -1.134 & -2.53;0.26 & 50.156 & 22.60;77.71 & 42.830 & 27.53;58.13 & 2.354 \\
Other softwoods & 60 & 2.4;21.0 & 3.1;14.1 & -1.822 & -4.76;1.12 & 42.926 & -49.86;135.72 & 29.273 & -18.46;77.01 & 1.207 \\
\bottomrule
\end{tabularx}
\label{tab:49height}
\end{table}

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\subsection{Tree Height}\label{sec:4treeheight}
Tree height was best modeled with Mitscherlich’s model of tree diameter (dbh) (\autoref{tab:49height}). The model fitted well every group of species and was adjusted with nonlinear least-squares methods \citep{R2013}. With the exception of coniferous trees, root mean square error was less than \SI{2.6}{\metre}. However, the models that we have presented should only be used within the range of observed dbh. In particular, we did not measure large coniferous trees and the validity of the model is doubtful for such trees.

\subsection{Height to the base of the crown}\label{sec:4crownbase}
The height to the base of the crown ($H_{cb}$) was estimated with the species averaged ratio between crown length and tree length (\autoref{tab:410crownbase}). The use of a power model, as was proposed by \citet{Beaudet2011}, did not substantially improve the predictions and required the estimation of one additional parameter for each species. However, the parameters of the adjustment of the power model were also reported (\autoref{tab:411crownbase}) because they can be used to run simulations with another simulation program of stand dynamics: SORTIE-ND \citep{Canham2011}. The parameter estimates underscore our observations that shade-tolerant species have deeper crowns than less shade-tolerant species (greater $a$ estimate in \autoref{tab:410crownbase}).


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\begin{table}
\caption[Parameters of the model of the height to the base of the crown.]{Parameter estimates and their \SI{95}{\percent} confidence intervals with $\alpha =0.05$ (CI) of the model between the crown length ($CL=H-H_{cb}$) and tree height ($H$): $CL = a \cdot H$. Also presented are the numbers of measured trees ($n$), the ranges of measured tree diameters (dbh), tree height and height to the crown base ($H_{cb}$), and the root mean square error of the crown length model.}
\footnotesize
\centering
\begin{tabularx}{\linewidth}{lrRRRRRR}
\toprule
\textsc{species} & $n$ & dbh & $H_{cb}$ & height & \multicolumn{2}{c}{$a$} & RMSE \\ 
\cmidrule(lr){6-7}
 &  & \si{\centi\metre} & \si{\metre} & \si{\metre} & Estimate & CI & \si{\metre} \\ 
\midrule
Oak   & 702   & 2.4;94.9 & 0.4;23.2 & 3.1;35.8 & 0.673 & 0.66;0.69 & 4.344 \\
Beech & 1083  & 2.4;84.4 & 0.1;19.6 & 3.3;39.0 & 0.794 & 0.78;0.80 & 3.356 \\
Hornbeam & 154   & 2.4;42.0 & 0.2;11.9 & 4.1;25.8 & 0.813 & 0.79;0.83 & 1.910 \\
Birch & 65    & 2.4;49.7 & 0.4;18.3 & 3.6;27.3 & 0.669 & 0.63;0.71 & 2.691 \\
Other hardwoods & 57    & 2.5;76.7 & 0.2;20.5 & 3.4;29.3 & 0.674 & 0.63;0.72 & 3.311 \\
Other softwoods & 29    & 2.4;20.7 & 0.1;1.9 & 3.1;12.6 & 0.774 & 0.72;0.82 & 0.864 \\  
\bottomrule
\end{tabularx}
\label{tab:410crownbase}
\end{table}

\begin{table}
\caption[Parameters of the power model of the height to the base of the crown.]{Parameter estimates and their \SI{95}{\percent} confidence intervals with $\alpha =0.05$ (CI) of the power model between the crown length ($CL=H-H_{cb}$) and tree height ($H$): $CL = a \cdot H^b$. Also presented are the numbers of measured trees ($n$), the ranges of measured tree diameter (dbh), tree height and height to the crown base ($H_{cb}$), and the root mean square error of the crown length model.}
\footnotesize
\centering
\begin{tabularx}{\linewidth}{lrRRRRRRRR}
\toprule
\textsc{species} & $n$ & dbh & $H_{cb}$ & height & \multicolumn{2}{c}{$a$} & \multicolumn{2}{c}{$b$} & RMSE \\ 
\cmidrule(lr){6-7}\cmidrule(lr){8-9}
 &  & \si{\centi\metre} & \si{\metre} & \si{\metre} & Estimate & CI & Estimate & CI & \si{\metre} \\ 
\midrule
Oak   & 706   & 2.4;94.9 & 0.4;23.2 & 3.1;35.8 & 1.162 & 0.82;1.50 & 0.827 & 0.74;0.92 & 4.300 \\
Beech & 1271  & 2.4;84.4 & 0.1;19.6 & 3.3;39.0 & 1.063 & 0.94;1.18 & 0.900   & 0.87;0.94 & 3.285 \\
Hornbeam & 175   & 2.4;42.0 & 0.2;11.9 & 4.1;25.8 & 1.098 & 0.85;1.35 & 0.885 & 0.80;0.97 & 1.853 \\
Birch & 69    & 2.4;49.7 & 0.4;18.3 & 3.6;27.3 & 1.298 & 0.88;1.72 & 0.719 & 0.61;0.83 & 1.952 \\
Other hardwoods & 73    & 2.5;76.7 & 0.2;20.5 & 3.4;29.3 & 0.773 & 0.32;1.22 & 0.931 & 0.74;1.12 & 3.206 \\
Other softwoods & 60    & 2.4;21.0 & 0.1;1.9 & 3.1;12.6 & 0.547 & 0.47;0.63 & 1.233 & 1.16;1.30 & 0.441 \\
\bottomrule
\end{tabularx}
\label{tab:411crownbase}
\end{table}

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\subsection{Crown openness and leaf area density}\label{sec:4foliage}
In addition to information about crown geometry, SamsaraLight requires the estimation of either crown openness or crown leaf area density (LAD). We measured both parameters with photographs of isolated crowns in a manner similar to the method that was used to estimate crown openness \citep{Canham1999, Beaudet2002, Astrup2006}. This method is rapidly executed relative to previously reported methods that use leaf samples or leaf traps \citep{Bartelink1997, Jonard2006} or vertical line-intersect sampling methods \citep{Nock2008}, but it applies only to trees with relatively isolated crowns.
 
We took 112 photographs of isolated crowns of 21 oaks, 13 beeches, 8 birches, and 4 hornbeams. The photographs were processed with PiafPhotem \citep{Adam2006} to compute the gap fraction, i.e., crown openness. Additionally, for every photographed crown, we recorded 4 crown radii, crown base height and tree height.
 
We computed the path length, $l$ (\autoref{eq:4lambert}), as the distance between the intersections between the modeled crown ellipsoids and the photograph direction (\autoref{fig:45LAD}). Computing the intersections between an ellipsoid and a line requires solving a second-degree equation (\autoref{eq:4seconddreee}).

\begin{equation}
\label{eq:4seconddreee}
A \: L^2 + B \: L + C = 0
\end{equation}

{\small
\begin{align*}
A &= \frac{\cos^2(\theta) \: \cos^2(\alpha)}{a^2} + \frac{\cos^2(\theta) \: \sin^2(\alpha)}{b^2} + \frac{\sin^2(\theta)}{c^2} \\
B &= -\frac{2 \: x_1 \: \cos(\theta) \: \cos(\alpha)}{a^2} - \frac{2 \: y_1 \: \cos(\theta) \: \sin(\alpha)}{b^2} - \frac{2 \: z_1 \: \sin(\theta)}{c^2} \\
C &= \frac{x_1^2}{a^2} + \frac{y_1^2}{b^2} + \frac{z_1^2}{c^2} - 1	
\end{align*}
}		

where ($x_1$, $y_1$, $z_1$) are the coordinates of the ellipsoid center in the same coordinate system as the photograph direction. The distance between the intersections, i.e., the path length of light within the crown \autoref{fig:45LAD}, is given by the difference between $L$ solutions. 

Photograph direction was computed from the recorded photograph elevation angle and the estimated distance to the trunk. The equation of this direction within a polar coordinate system where the origin corresponds to the camera position is given in \autoref{eq:4direction}.

\begin{align}
\label{eq:4direction}
x &= L \: \cos(\theta) \: \cos(\alpha) \\
y &= L \: \cos(\theta) \: \sin(\alpha) \nonumber \\
z &= L \: \sin(\theta) \nonumber
\end{align}

where $L$ is the distance along the direction from the camera position, $\theta$ is the direction elevation angle, and $\alpha$ is its azimuthal angle (\autoref{fig:45LAD}). Because the distance between the photographer and the tree trunk was not recorded in the field, we estimated it by considering that the photographer was \SI{1.7}{\metre} tall and always aimed the camera at the mid-height of the targeted crown. The distance $d$ is estimated as (\autoref{eq:4distance}):

\begin{equation}
\label{eq:4distance}
d = \frac{0.5 \: (H-H_{cb}) + H_{cb} - 1.7}{\tan \theta}
\end{equation}  

\begin{figure}[htb]
\includegraphics[width=\textwidth]{fig45LAD}
\caption[Measurements required to estimate LAD.]{Illustration of the different measurements involved in the computation of crown LAD.}
\label{fig:45LAD}
\end{figure}

where $H$ is tree height, $H_{cb}$ is height to the base of the crown, and $\theta$ is the recorded elevation angle (\autoref{fig:45LAD}).

Finally, LAD values were estimated by inverting Beer’s law (\autoref{eq:4lambert}), with the common assumption of spherical distribution of leaves ($k = 0.5$) \citep{Phattaralerphong2006, DaSilva2012}. We could have used $k \cdot \textnormal{LAD}$ instead of LAD. Although this would have allowed us to assume that the distribution of leaves was not spherical, it would not have allowed us to compare LAD values with other published work.

We finally modeled the LAD estimates for beech and oak. We obtained the best fits with a polynomial function of dbh that was adjusted according to ordinary least-squares methods \citep{R2013}. For the other species we did not collect enough data to model LAD as a function of dbh. Nevertheless, \citet{Courbaud2003} mentioned that SamsaraLight sensitivity to leaf area density was low between \SI{0.3}{\square\metre\per\cubic\metre} and \SI{0.9}{\square\metre\per\cubic\metre}. We assumed LAD to be \SI{0.6}{\square\metre\per\cubic\metre} for all other species as 0.6 corresponded to the LAD average of all measured trees. Trunks were modeled as cylinders of dbh diameter and crown base height and do not transmit light.

Estimates of crown LAD varied noticeably from tree to tree and even from photograph to photograph of the same tree. Foliage biomass is usually assumed to depend upon sapwood area (pipe model) \citep{Shinazaki1964} and, hence, also upon tree dbh. However, tree dbh explained only \SI{48}{\percent} and \SI{30}{\percent} of LAD variability for oak and beech trees, respectively (\autoref{tab:412LAD}). LAD decreased with tree dbh and more so for beech than for oak. Additionally, LAD stopped decreasing for beech at around \SI{50}{\centi\metre} dbh and then started to increase gently. Our LAD estimates matched previously reported values of leaf area for beech and oak \citep{Bartelink1997, Jonard2006} and were in the range of reported LAD values for broadleaved species \citep{Stadt2000, Piboule2001, Gersonde2004, Sprugel2009}.


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\begin{table}
\caption[Parameters of the polynomial model of LAD.]{Parameter estimates and their \SI{95}{\percent} confidence intervals (CI) of the polynomial model between crown leaf area density (LAD) and tree diameter (dbh) : LAD $= a + b \cdot (\textnormal{dbh} \cdot \pi) + c \cdot (\textnormal{dbh} \cdot \pi)^2$. Also presented are tree number ($n_t$), photograph number ($n_p$), ranges of measured dbh (in \si{\centi\metre}), ranges of estimated LAD (in \si{\square\metre\per\cubic\metre}), and the model root mean square error (RMSE). We assumed LAD to be \SI{0.6}{\square\metre\per\cubic\metre} for all other species.}

\footnotesize
\centering
\begin{tabularx}{\linewidth}{lrRRRRRRRRRR}
\toprule
\textsc{species} & $n_t$ & $n_p$ & dbh & LAD & \multicolumn{2}{c}{$a$} & \multicolumn{2}{c}{$b$} & \multicolumn{2}{c}{$c$} & RMSE \\ 
\cmidrule(lr){6-7}\cmidrule(lr){8-9}\cmidrule(lr){10-11}
 &  & &  &  & Estimate & CI & Estimate & CI & Estimate & CI &  \\ 
 
 &  & &  \si{\centi\metre} & \si{\square\metre\per\cubic\metre} &   &  & \scriptsize{$\cdot 10^{-2}$} & \scriptsize{$\cdot 10^{-2}$} & \scriptsize{$\cdot 10^{-5}$} & \scriptsize{$\cdot 10^{-5}$} & \si{\square\metre\per\cubic\metre}\\  
\midrule
Beech	&13	&71	& 7.6;71.0 & 0.23;2.38 & 1.720 & 1.450;1.990 & -1.88 & -2.41;-1.35 & 6.27 & 4.21;8.33 & 0.326 \\
Oak		&21	&112& 5.1;71.9 & 0.18;2.25 & 1.207 & 0.972;1.441 & 1.04  & -1.47;-0.61 & 3.09 &	1.34;4.85 &	0.292 \\
Hornbeam &4 &23 & 7.6;72.2 & 0.29;0.84 & 0.521 & 0.468;0.574 & 	     &             &      &           & 0.125 \\
Birch	&8	&42	& 12.7;34.4& 0.42;0.90 & 0.595 & 0.562;0.628 &  	 &	 	 	 	& 	 	& 	    & 0.107 \\
\bottomrule
\end{tabularx}
\label{tab:412LAD}
\end{table}

\end{landscape}
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} 

\subsection{Model evaluation}\label{sec:4validation}
A first version of SamsaraLight was initially validated in one heterogeneous stand of Norway spruce (\textit{Picea abies} (L.) H. Karst.) \citep{Courbaud2003}. Nevertheless, we further evaluated model predictions in mixed broadleaf forests. Additionally, we compared the predictions of SamsaraLight with the predictions of another radiative transfer model (\autoref{sec:comparison}).

\subsubsection{Method}
In mid-July 2010, we took 307 hemispherical photographs in 19 sites in heterogeneous forests of oak and beech. The photographs were taken just before sunrise and above the regeneration layer of the plots that had been installed every \SI{4}{\metre} following a square grid. The number of photographs per site therefore depended upon the area of each site and ranged between 6 and 39. We then computed the percentage of above canopy light (PACL$_{\textnormal{photo}}$) that is transmitted through the canopy between 1\textsuperscript{st} April and 31\textsuperscript{st} October 2012.
 
To predict understory light with SamsaraLight, we measured and mapped every tree with a circumference greater than \SI{40}{\centi\metre}. We measured the circumference at breast height (\SI{1.3}{\metre}), total height, and height to the base of the crown for each tree. On 13 sites, we also measured at least 4 crown radii for every tree. The inventoried plots are the same as those that were used in \autoref{ch:rege} and had an oval shape of variable area because they surrounded fenced areas in which advanced regeneration has been studied. Trees were measured if they were located at a distance of less than \SI{20}{\metre} from the fence. Due to local abundance of the understory, we additionally measured and mapped smaller trees near the points where hemispherical photographs were taken. Two different measurement protocols were applied because of a lack of time and resources. In 10 sites, we measured and mapped every tree with a circumference that was greater than \SI{7.4}{\centi\metre} and \SI{20}{\centi\metre}, which were respectively located less than \SI{7}{\metre} or less than \SI{15}{\metre} from the points where hemispherical photographs were taken. In the 9 other sites, we measured trees with a circumference that was greater than \SI{20}{\centi\metre} and which were located less than \SI{7}{\metre} from the points where hemispherical photographs were taken. Plot area ranged from \SI{2070}{\square\metre} to \SI{10540}{\square\metre}, with an average of \SI{4340}{\square\metre}.   

We compared the hemispherical photograph light estimates (PACL$_{\textnormal{photo}}$) with SamsaraLight predictions (PACL$_{\textnormal{model}}$) that were computed for the same period and at the same locations. We adjusted linear models with the ordinary least-squares method between PACL$_{\textnormal{model}}$ and PACL$_{\textnormal{photo}}$. Next, we computed the confidence intervals ($\alpha = 0.05$) of model coefficients to estimate the deviation of the modeled relationship from a $1:1$ relationship. Using a protocol similar to \citet{DaSilva2012}, we also compared the cumulative distribution function (CDF) of PACL$_{\textnormal{photo}}$ and PACL$_{\textnormal{model}}$ for every site. We computed a Kolmogorov-Smirnov statistic (K-S test) to quantify and test the distance between the two CDFs, against the null hypothesis that the samples are drawn from the same population. 

\subsubsection{Results and discussion}
There was a good linear relationship between the modeled PACL$_{\textnormal{model}}$ and the measured PACL$_{\textnormal{photo}}$ ($R^2$ = \SI{68}{\percent}) although our model tended to overestimate PACL$_{\textnormal{photo}}$ (intercept significantly greater than 0 and the slope not significantly different from 1). The model predicted better PACL values when averaged at the site level ($R^2$ = \SI{87}{\percent}, with intercept not significantly different from 0 and the slope slightly significantly greater than 1) (\autoref{fig:46validation}).
 
Removing small trees from the dataset significantly altered the relationship between PACL$_{\textnormal{model}}$ and PACL$_{\textnormal{photo}}$. For example, removing every tree with a circumference smaller than \SI{20}{\centi\metre} reduced $R^2$ to \SI{65}{\percent}, while removing every tree with a circumference smaller than \SI{40}{\centi\metre} reduced the $R^2$ to \SI{58}{\percent}.
  
According to K-S tests, the distributions of PACL$_{\textnormal{model}}$ and PACL$_{\textnormal{photo}}$ differed significantly ($P < 0.05$) from one another for 7 of the 19 plots (\autoref{fig:47validation2}). Nevertheless, the differences were noteworthy for only 5 plots in which PACL$_{\textnormal{model}}$ clearly overestimated PACL$_{\textnormal{photo}}$. These plots were characterized by an abundance of small beech and hornbeam trees that covered the regeneration layer where PACL$_{\textnormal{photo}}$ measurements were taken. Yet even in these cases, the variation between the distributions of PACL$_{\textnormal{model}}$ and PACL$_{\textnormal{photo}}$ was less than \SI{15}{\percent}.

Overall, our model captured the variability of PACL independently of stand structure. The agreement between PACL$_{\textnormal{photo}}$ and PACL$_{\textnormal{model}}$ was satisfactory and within the range of previously reported studies \citep{Law2001, Boivin2011}. For some plots, PACL$_{\textnormal{model}}$ slightly overestimated PACL$_{\textnormal{photo}}$. Such over-estimation mostly occurred in the presence of dense understory trees \citep{Beaudet2011}. The predictions that were made by the PACL$_{\textnormal{model}}$ could be affected by these understory trees, because some had a circumference smaller than our inventory circumference threshold, or had a crown that was not correctly modeled by sections of ellipsoids. Moreover, the precision of PACL$_{\textnormal{photo}}$ estimates was poorer in the presence of such dense understories. Nevertheless, the bias was of limited magnitude.

\begin{figure}[htb]
\includegraphics[width=\textwidth]{fig46validation}
\caption[Predictions versus measures of PACL.]{Relationships between predicted percentages of above canopy light (PACL$_\textnormal{model}$) and the percentages of above canopy light estimated from hemispherical photographs (PACL$_\textnormal{photo}$): point-to-point comparison of all PACL$_\textnormal{model}$ and PACL$_\textnormal{photo}$ (a) and comparison of the PACL$_\textnormal{model}$ and PACL$_\textnormal{photo}$ averaged by site (b). The dotted lines show the 1:1 relationships, whereas the full lines correspond to the linear least-squares regressions.}
\label{fig:46validation}
\end{figure}

\begin{figure}[htb]
\includegraphics[width=\textwidth]{fig47validation2}
\caption[Cumulative distributions of predictions and measures of PACL.]{Cumulative distributions of modeled (dashed lines) and measured (solid lines) percentages of above canopy light (PACL). The number of measurements and predicted PACL values (n), K-S test statistics (D) and associated P-values (P) are also reported for each plot. Our model captured the variability of PACL, except in sites with an abundant understory. The plots were ordered by decreasing oak proportions showing no effect of stand composition on model performance.}
\label{fig:47validation2}
\end{figure}

\subsubsection{Conclusion}
After synthesizing and discussing the approaches that have been reported in the literature, we implemented a mechanistic and relatively simple radiative model. Compared to other forest radiative models \citep{Cescatti1997b, Brunner1998, Courbaud2003, Gersonde2004}, we utilized a model with a low number of parameters that did not require calibration by model inversion.
 
The model captured well the variability of PACL in stands with varied density, structure and composition ranging from early-successional oak forests to late-successional beech forests. Therefore, we have confidence in the model’s robustness for exploring how the understory light regime can be affected by overstory density, overstory structure, overstory composition, and topography. However, caution should be exercised regarding the exactitude of single model predictions, especially where understory is abundant.
