\chapter{Managing understory light} % Chapter title
\label{ch:simulation} % For referencing the chapter elsewhere, use \autoref{ch:examples} 

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\color{gray} {\slshape Nature never hurries: atom by atom, little by little, she achieves her work. The lesson one learns from yachting or planting is the manners of Nature; patience with the delays of wind and sun, delays of the seasons, bad weather, excess or lack of water.}
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Ralph Waldo Emerson
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As understory light is the main driver of regeneration growth (\autoref{ch:rege}), controlling understory light is a key factor in regenerating mixed stands \citep{Lieffers1999}. The control of understory light with partial cutting requires properly modifying stand structure and composition, in addition to solely managing stand density. To date, this question of how changes in stand structure and composition affect understory light has rarely been addressed, especially for heterogeneous broadleaf forests. Only a few field experiments have successfully defined levels of canopy openness that are suitable for the regeneration of mixed species \citep{VonLupke1998, Prevost2003}, and simulation studies have been limited to particular ecosystems. Cutting groups of spatially aggregated trees or creating gaps has been reported to drastically increase light availability for the regeneration in boreal mixedwoods \citep{Coates2003, Beaudet2011}, even-aged western hemlock or Douglas-fir forests \citep{Sprugel2009}, or in uneven-aged spruce forests \citep{Courbaud2001, Lafond2013}. Additionally, cutting understory poles and trees with branches immediately above the regeneration, or cutting from below in some way, has often been recommended for shelterwood systems. These poles and trees, unless they are removed, compete strongly with regeneration for nutrients, water and light resources \citep{Nyland1996}. Moreover, removing shade-tolerant species presumably increases understory light more effectively than removing trees randomly because shade-tolerant species usually have wider, deeper and denser crowns than less shade-tolerant species \citep{Coates2003, Beaudet2011}.

In this chapter, the radiative transfer model that was established in \autoref{ch:lightModel} is used to explore how silvicultural regeneration treatments that modify stand structure and composition can affect understory light and what is the best treatment to promote the regeneration of mixed species given the light requirements of two species with contrasting shade tolerances (\autoref{ch:rege}). In particular, the aims are:

\begin{itemize}
\item To compare different cutting scenarios by hypothesizing that at similar levels of harvest intensity, gap creation, cutting from below, removing shade-tolerant species (species-specific cutting), cutting randomly and cutting from above respectively induced high to low responses in transmitted light ($H_1$);
\item To test whether our first hypothesis is general or depends upon initial stand structure ($H_2$);
\item To identify the combinations of cutting scenarios that maximize the understory area receiving \SIrange{10}{20}{\percent} (levels favorable to regeneration of shade-tolerant species) or \SIrange{20}{40}{\percent} (levels favorable to regeneration of mid-tolerant species) and above \SI{40}{\percent} (little light limitation for most regeneration) of above canopy light.
\end{itemize}

\section{Methods}

\subsection{Study sites}
In the Belgian Ardennes, we selected 27 sites with varying stand structures and compositions, and with established regeneration of oak and (or) beech (\autoref{ch:studyArea}). These studied stands characterized the diversity of forest structures that can be found during forest succession from early-successional oak forests to late-successional pure beech forests (\autoref{fig:51structure}).
 
Every tree with a circumference greater than \SI{40}{\centi\metre} was mapped and measured. We measured the circumference at breast height, 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. Besides oak and beech, our data set contained \SI{7}{\percent} hornbeam (\textit{Carpinus betulus} L.), \SI{4}{\percent} small coniferous trees (\textit{Pseudotsuga menziessi} (Mirb.) Franco, \textit{Picea abies} (L.) Karst, and \textit{Pinus sylvestris} L.), \SI{2}{\percent} birches (\textit{Betula pendula} Roth, \textit{Betula pubescens} Ehrh.), and \SI{2}{\percent} other broadleaved species (\textit{Acer pseudoplatanus} L., \textit{Acer platanoides} L., \textit{Sorbus aucuparia} L, and \textit{Corylus avellana} L.).

The inventoried plots had an oval shape of variable area because they surrounded fenced areas in which advanced regeneration has been studied (\autoref{ch:rege}). Trees were measured if they were located at a distance of less than \SI{20}{\metre} from the fence. Plot area ranged from \SI{2070}{\square\metre} to \SI{10540}{\square\metre} with an average of \SI{4340}{\square\metre}.

\begin{figure}[htb]
\includegraphics[width=\textwidth]{fig51structure}
\caption[Structure of the studied stands.]{Stand structure and composition of the 27 studied plots expressed as tree frequency by diameter class. The charts are sorted by decreasing proportion of oak. Plot basal area is reported next to the plot id number showing no trend between plot basal area and plot composition. The asterisks next to plot id numbers denote the 9 plots where initial mean percentage of above canopy light (PACL) was below \SI{20}{\percent}.}
\label{fig:51structure}
\end{figure}

\subsection{Light model settings}\label{sec:settings}
We set SamsaraLight to sample 130 diffuse and 81 direct ray directions for each month of the growing period (from April to October). Ray directions are sampled at regular increasing zenithal angles with a starting value of \ang{10} and an angle step of \ang{15}. For every direction, parallel rays are cast at ground level in either cell centers or any other specified locations (virtual sensor). Samsaralight then identifies the interceptions of light rays by tree crowns and computes radiation attenuation using Beer’s law (\autoref{eq:4lambert}).

SamsaraLight predicts transmitted light within a rectangular plot. Since our inventory plots were not rectangular, we developed an algorithm that added virtual trees in order to obtain a rectangular plot (\autoref{fig:52plot}). For each site, virtual trees were randomly drawn with replacement from the measured trees. Their location outside the inventoried area was then randomly generated. This process was repeated until the basal area of the rectangular plot equaled the basal area of the inventoried plot. The number of virtual trees created in each plot ranged between 0 and 68, and the area over which they were simulated represented on average \SI{28}{\percent} of the rectangular plot area. The dimensions and foliage density of the modeled crowns were measured and estimated as described in \autoref{sec:4foliage}.

SamsaraLight additionally required monthly meteorological records of total and diffuse irradiances in \si{\mega\joule\per\square\metre}. We computed such monthly averaged data from data recorded between 2007 and 2011 by the meteorological institute of Belgium in Humain (\ang{50;33;}N \ang{5;43;}E). Furthermore, we set SamsaraLight to predict percentages of above canopy light (PACL) at \SI{2}{\metre} above the forest floor at each intersection of a $7 \times 7$ rectangular grid (\autoref{fig:52plot}). We thus obtained 49 estimates of PACL for each simulation. 

\begin{figure}[htb]
\centering
\includegraphics[width=\linewidth]{fig52plot}
\caption[Simulated plot layout.]{Three dimensional visualizations of the different zones within a plot with and without trees. In order to evaluate the cutting scenarios, our model was set to predict the percentage of above canopy light at \SI{2}{\metre} above forest ground and at every intersection of a \SI{7}{\metre} $\times$ \SI{7}{\metre} grid within the center part of the plots.}
\label{fig:52plot}
\end{figure}

\subsection{Cutting scenarios}
For the 27 inventoried stands, we simulated 5 cutting types that reproduced 5 silvicultural regeneration strategies commonly practiced in forests of the Belgian Ardennes; namely: cutting from above, cutting from below, gap creation, species-specific cutting and uniform cutting (\autoref{tab:51algorithm}). Cutting from above harvests the most valuable trees. Typically, the trees with a diameter greater than the exploitable diameter are cut, i.e., these are diameter limit cuts. In contrast, cutting from below harvests small trees with low economic value. Such a strategy is typically applied to promote the growth of dominant trees, increase light for natural regeneration and mimics self-thinning. Cutting that creates gaps are especially used to increase light for a clump of saplings and promote regeneration. Species-specific cutting has been practiced recently because oak has become scarce. Therefore, foresters conserve oak seed trees even if, for example, their diameter exceeds the exploitable diameter or if they are wounded. Finally, we simulated uniform cuttings in which trees were randomly harvested. This scenario can be considered to be the control treatment.

The 5 algorithms of the 5 cutting types started computing $score_t$ (\autoref{tab:51algorithm}) for every tree and then cut the trees by order of decreasing $score_t$ until the harvest intensity level was reached. $score_t$ computed for cuttings from below and from above were on average greater for narrow and large trees, respectively. $score_t$ included a random component that ensured that tree selection differed between simulations. The weight given to this random contribution for the cuttings from above and from below was set to 0.2 so that the distribution of the diameter of cut trees followed a realistic normal distribution. Gap creation scenarios harvested trees around a random location that must lie within the central part of the plots (\autoref{fig:52plot}). Furthermore with cutting intensities greater or equal to 0.4, the gap radius was often greater than \SI{20}{\metre} which corresponded to the buffer distance between the plot boundary and the central part of the plots (\autoref{fig:52plot}). In such cases, the gap shape became a truncated circle. These algorithms are now available in Capsis \citep{DufourKowalski2012} for most individual tree growth models.

The 5 cutting types were applied to the 27 stands with 4 different levels of harvest intensity (\SI{10}{\percent}, \SI{20}{\percent}, \SI{40}{\percent} and \SI{60}{\percent} of initial plot basal area). Since the algorithms of every cutting type had stochastic components, we repeated the simulation 10 times. For each simulation, 49 estimates of PACL were computed according to the grid introduced in \autoref{sec:settings}. We therefore tested 20 cutting scenarios with \SI{5427}{} simulations and \SI{265923}{} computations of PACL$_{\textnormal{model}}$. 



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\begin{table}[htb]
\caption[Description of the 5 cutting types.]{Description of the 5 cutting types. With ($x_t$ ; $y_t$) the tree coordinates, ($x_g$ ; $y_g$) the random gap center coordinates that must be within the central part of the plot (i.e., within ($x_{min}$, $x_{max}$, $y_{min}$, $y_{max}$), dbh$_t$ the diameter of tree $t$, dbh$_{min}$ and dbh$_{max}$ the minimum and maximum of dbh, $u$ a random number generated between 0 and 1.}
\footnotesize
\begin{tabularx}{\linewidth}{lXp{0.3 \linewidth}}
\toprule
\textsc{cutting type} & \textsc{description} & $score_t$ \\
\midrule
Uniform cutting & Random harvest of trees & $score_t \sim u$ \\
\addlinespace[2mm] 
Species-specific cutting & Preferential harvest of beech and hornbeam ($\beta = 2$), then the other species ($\beta = 1$) and finally oak ($\beta = 0$) & $score_t = \beta + u$ \\
\addlinespace[2mm] 
Cutting from below & Preferential harvest of small trees & $ score_t = 0.2 \: u + 0.8 \: (1 - \frac{\textnormal{dbh}_t - \textnormal{dbh}_{min}}{\textnormal{dbh}_{max} + \textnormal{dbh}_{min} + 1})$ \\
\addlinespace[2mm] 
Cutting from above & Preferential harvest of large trees & $ score_t = 0.2 \: u + 0.8 \: (1 - \frac{\textnormal{dbh}_{max} - \textnormal{dbh}_{t}}{\textnormal{dbh}_{max} + \textnormal{dbh}_{min} + 1})$ \\
\addlinespace[2mm] 
Gap creation & Harvest of all trees around a gap center. The location of the gap center was determined at random but must be within the central part of a plot (\autoref{fig:52plot}). & 
$score_t = \sqrt{(x_t - x_g)^2 + (y_t - y_g)^2} $
\newline
$x_g \sim U_{[x_{min},x_{max}]}$  
\newline
$y_g \sim U_{[y_{min},y_{max}]}$ 
\\
\bottomrule
\end{tabularx}
\label{tab:51algorithm}
\end{table}

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\subsection{Statistical analyses}
In order to compare the different scenarios, we computed the differences between the average of the 49 estimates of transmitted light before and after harvesting ($\Delta$PACL). Then, we adjusted a linear mixed model (\autoref{eq:5model} with lme4 R package \citep{Bates2013} to quantify the relationship between $\Delta$PACL and cutting intensity ($I_i$). 

\begin{equation}
\begin{array}{rcl}
\Delta\textnormal{PACL} & = & (b_l + \beta_{jl}) \: I_i + \epsilon_{ijkl} \\
\beta_{jl} & \sim & N(0,\theta_\beta) \\
\epsilon_{ijkl} & \sim & N(0,\theta_\epsilon) \\
\end{array}
\label{eq:5model}
\end{equation}

With $i$, $j$, $k$, $l$ the indices corresponding to the cutting intensity, plot, the simulation run and the cutting type, respectively. $b_l$ was the fixed-effect parameter which was estimated for each $l$ cutting type. $\beta_{jl}$ was a random-effect parameter varying between plot and cutting type. Similarly to the residual term, $\beta_{jl}$ followed a centered normal distribution. This model assumed that $\Delta$PACL was proportional to $I_i$ and that the slope of this relationship ($b_l + \beta_{jl}$) varied with cutting type and initial site conditions. The hypothesis $H_1$ was tested computing the approximate confidence intervals of $b_l$. These confidence intervals were obtained from the likelihood profile of $b_l$ \citep{Bates2013}.

In order to further analyze how initial stand structure affected the response ($H_2$), we fitted five additional models that included the effects of 5 different stand structure parameters (denoted by $P_j$ in \autoref{eq:5model2}). These parameters were stand basal area, quadratic mean diameter, basal area proportion of oak, standard deviation of dbh, Clark-Evans aggregation index and basal area of trees with dbh smaller than 25 cm. With the exception of the latter parameter, they have commonly been used to describe stand density and structure in similar studies \citep{Sprugel2009, Beaudet2011, Lafond2013}. These indices describe the stand density and structure before harvest. The Clark-Evans aggregation index (\autoref{eq:5CE}) gives values greater than 1 for regular tree distributions and lower than 1 for aggregated tree distributions. The basal area of small trees was added because poles and small trees were sometimes abundant in the understory and were expected to capture a high proportion of transmitted radiation. We tested the addition of the corresponding fixed-effect parameter $c_l$ with the log likelihood ratio test \citep{Bates2013}.

\begin{equation}
\Delta\textnormal{PACL} = (b_l + c_l \cdot P_j + \beta_{jl}) \cdot I_i + \epsilon_{ijkl}
\label{eq:5model2}
\end{equation}

\begin{equation}
CE = \frac{\bar{r}}{0.5 \: \sqrt{A/N}}
\label{eq:5CE}
\end{equation}

where $\bar{r}$ is the mean distance between trees and their nearest neighbor, $A$ the plot area and $N$ the number of trees within the plot.

Next, we created four classes of PACL. These classes had PACL values ranging between \SIrange{0}{10}{\percent}, \SIrange{10}{20}{\percent}, \SIrange{20}{40}{\percent}, and \SIrange{40}{100}{\percent}. They corresponded respectively to light levels that are unfavorable to natural regeneration of tree species, favorable to beech sapling growth, favorable to beech and oak sapling growth and above light saturation point (\autoref{ch:rege}). 

Furthermore, we computed the average frequency of the predictions by PACL classes and cutting intensity. In order to provide a guide to forest managers, we repeated these computations replacing the harvest intensity by the resulting post-harvest basal area. We restricted these analyses to the 9 plots where the mean PACL before cutting was less than \SI{20}{\percent}, i.e., where cutting was necessary to promote the natural regeneration of less shade-tolerant species.

\section{Results}
\subsection{Tree inventory}
According to the field data, stand composition and stand density varied considerably among study sites (\autoref{fig:51structure} and \autoref{tab:42stands}). The proportion of oak in the overstory ranged between \SI{0}{\percent} and \SI{98}{\percent}. High proportions of oak occurred mostly in stands with high basal area ($r=0.389$, $P=0.045$) and low quadratic mean diameter ($r=-0.523$, $P=0.004$). Additionally, in some sites, the distribution of tree diameters followed an inverted j-shaped curve while in other sites it approximated a bell-shaped curve (\autoref{fig:51structure}). Tree aggregation was greater in stands with complex vertical structure. Clark-Evans aggregation index was indeed negatively correlated with the standard deviation of tree diameter ($r=- 0.500$, $P=0.008$). 

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\begin{table}[htbp]
  \footnotesize
  \caption[Stand structure and composition in the studied sites.]{Stand structure and composition in the studied sites. The presented parameters are density (N), basal area (BA), quadratic mean diameter (Dg), the Clark-Evans aggregation index (CE) and the minimum (dbh$_{\textnormal{min}}$) and maximum (dbh$_{\textnormal{max}}$) of tree diameter. Plots are ordered by increasing oak proportion.}
\begin{tabularx}{\linewidth}{RRRRRRRRRRRRR}
    \toprule
      & \multicolumn{4}{c}{\textsc{stand}} & \multicolumn{4}{c}{\textsc{oak}} & \multicolumn{4}{c}{\textsc{beech}} \\
    \cmidrule(lr){2-5}
    \cmidrule(lr){6-9}
    \cmidrule(lr){10-13}
       Site & N   & BA   & Dg    & CE    & BA   & Dg    & dbh$_{\textnormal{min}}$ & dbh$_{\textnormal{max}}$ & BA & Dg & dbh$_{\textnormal{min}}$ & dbh$_{\textnormal{max}}$ 	 \\
    \midrule
    19    & 98.3  & 14.94 & 43.98 & 0.96  &       &       &       &       & 14.94 & 43.98 & 12.73 & 84.35 \\
    25    & 127.2 & 21.84 & 46.75 & 1.05  &       &       &       &       & 21.84 & 46.75 & 13.05 & 84.35 \\
    27    & 35.3  &  7.82 & 53.13 & 1.05  &       &       &       &       & 6.45  & 52.97 & 12.73 & 72.57 \\
    28    & 51.4  &  7.47 & 43.04 & 1.19  & 0.28  & 47.42 & 41.70 & 52.52 & 7.19  & 42.89 & 12.73 & 64.94 \\
    26    & 61.7  & 14.60 & 54.89 & 1.27  & 1.94  & 75.76 & 75.76 & 75.76 & 12.63 & 53.62 & 14.32 & 78.30 \\
    3     & 113.6 & 18.10 & 45.03 & 1.07  & 3.17  & 39.42 & 28.01 & 60.16 & 14.93 & 46.57 & 14.32 & 65.57 \\
    18    & 46.7  &  7.52 & 45.29 & 0.97  & 1.40  & 62.32 & 54.43 & 64.30 & 6.06  & 45.33 & 12.73 & 79.90 \\
    1     & 87.2  & 11.15 & 40.34 & 1.18  & 2.86  & 44.33 & 27.06 & 65.25 & 7.90  & 39.35 & 12.73 & 71.62 \\
    2     & 113.9 & 16.33 & 42.73 & 0.97  & 4.35  & 49.30 & 16.23 & 59.21 & 11.95 & 41.35 & 12.73 & 72.26 \\
    14    & 124.5 & 21.21 & 46.57 & 1.28  & 6.70  & 65.64 & 54.43 & 82.76 & 14.51 & 42.00 & 12.73 & 69.71 \\
    20    & 143.9 & 19.49 & 41.52 & 1.04  & 6.97  & 71.87 & 55.70 & 92.31 & 12.52 & 35.46 & 12.73 & 80.53 \\
    29    & 127.0 & 20.56 & 45.41 & 1.20  & 11.14 & 55.42 & 40.43 & 68.12 & 9.42  & 38.53 & 12.73 & 65.25 \\
    13    & 133.8 & 15.37 & 38.25 & 1.09  & 8.69  & 60.21 & 46.15 & 72.89 & 6.68  & 28.70 & 12.73 & 69.71 \\
    5     & 217.8 & 20.51 & 34.62 & 1.03  & 11.95 & 53.87 & 17.83 & 71.30 & 8.56  & 25.67 & 12.73 & 68.12 \\
    12    & 156.6 & 22.65 & 42.92 & 1.16  & 13.61 & 53.24 & 39.47 & 80.85 & 9.04  & 34.74 & 12.73 & 75.76 \\
    4     & 207.3 & 24.16 & 38.52 & 0.97  & 16.70 & 57.73 & 38.20 & 72.26 & 7.46  & 25.72 & 12.73 & 84.03 \\
    22    & 216.5 & 15.31 & 30.00 & 1.12  & 11.07 & 29.02 & 13.69 & 49.66 & 3.62  & 43.36 & 17.51 & 77.99 \\
    21    & 120.3 & 19.54 & 45.48 & 0.97  & 14.71 & 73.06 & 57.61 & 94.86 & 3.91  & 31.51 & 12.73 & 76.39 \\
    23    & 121.5 & 16.31 & 41.35 & 1.00  & 12.41 & 43.87 & 14.64 & 75.76 & 2.76  & 45.31 & 14.32 & 77.03 \\
    15    & 190.5 & 24.69 & 40.62 & 1.06  & 19.80 & 57.21 & 32.15 & 71.62 & 0.06  & 13.05 & 12.73 & 13.37 \\
    17    & 188.5 & 26.89 & 42.62 & 1.31  & 22.27 & 54.85 & 24.51 & 73.85 & 1.76  & 29.13 & 12.73 & 49.34 \\
    6     & 107.5 & 20.79 & 49.62 & 1.12  & 17.38 & 66.59 & 28.97 & 89.76 & 2.51  & 27.71 & 12.73 & 49.02 \\
    10    & 135.2 & 21.65 & 45.15 & 1.29  & 19.29 & 48.99 & 16.55 & 59.52 & 1.06  & 30.41 & 29.60 & 31.19 \\
    8     & 215.3 & 19.06 & 33.58 & 1.08  & 17.31 & 40.77 & 16.55 & 63.03 & 0.23  & 21.35 & 13.05 & 30.56 \\
    24    & 111.9 & 11.80 & 36.64 & 1.30  & 10.85 & 40.35 & 20.05 & 55.07 &       &       &       &  \\
    11    & 114.6 & 14.44 & 40.05 & 1.00  & 13.51 & 48.22 & 12.73 & 66.85 & 0.23  & 19.69 & 17.51 & 21.65 \\
    9     & 122.5 & 12.74 & 36.40 & 1.25  & 12.46 & 37.54 & 13.24 & 69.71 &       &       &       &  \\
    \bottomrule
    \end{tabularx}
  \label{tab:42stands}
\end{table}
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\subsection{Cutting scenarios}
The different cutting scenarios harvested the same quantities of basal area but caused very different modifications to stand structure. 
Firstly, post-harvest density varied notably between cutting scenarios. Cutting from below harvested the greatest number of trees while cutting from above harvested the fewest number of trees. The other cutting scenarios harvested an intermediate number of trees.
 
Secondly, harvests affected stand composition. Species-specific cutting increased the proportion of oak trees. Moreover, because the understory was mainly composed of shade-tolerant species (beech and hornbeam), cutting from below also tended to increase the proportion of oak trees. Cutting from above tended to harvest more oak trees than the other scenarios.

Thirdly, stand spatial structure was little affected by harvests except by gap creation. Gap creation increased the aggregation of trees as indicated by a reduction in the Clark-Evan aggregation index. Even at low harvest intensity, large gaps were created and remaining trees were aggregated along gap periphery. For example, removal of \SI{10}{\percent} and \SI{20}{\percent} stand basal area using gap harvesting led to an average opening of \SI{475}{\square\metre} and \SI{1182}{\square\metre}, respectively.

\subsection{Simulation}
The increase in understory light levels ($\Delta$PACL) varied significantly between the cutting scenarios as illustrated by \autoref{fig:55meanPACL} and the results of the adjusted model (\autoref{tab:53model}). In agreement with our first hypothesis ($H_1$), the cutting types ordered by decreasing $\Delta$PACL response were: gap creation, cutting from below, species-specific cutting, uniform cutting and cutting from above. This hypothesis was verified by ordering the slopes of the relationship ($b_l$) between changes in PACL and cutting intensity for all the scenarios. The slope of this relationship was statistically different among cutting treatments except between cutting from below and species-specific cutting.
 
We found no evidence that the relationship between $\Delta$PACL, cutting type and cutting intensity depended upon the initial stand structure. All likelihood ratio tests indicated that adding any stand parameter $P_j$ in the model (\autoref{eq:5model2}) did not significantly improve it. Within the conditions of our sampled study sites, our first hypothesis appeared rather general and independent of initial stand conditions. Furthermore, the between-site variability of $\Delta$PACL response ($\theta_\beta$ ranged from 0.172 to 3.870) was limited in comparison to the within-site variability ($\theta_\epsilon = 4.150$). $\Delta$PACL response depended more likely upon the conditions of the immediate surroundings of the measurement point rather than on general stand structure.

\begin{figure}[tb]
\centering
\includegraphics[width=0.6 \textwidth]{fig55meanPACL}
\caption[Mean increase of PACL by cutting type and intensity.]{Mean increase of transmitted light levels ($\Delta$PACL) by cutting scenario and by intensity. The adjusted mixed model indicated that the differences between cutting types are significant (\autoref{tab:53model}) and accentuated as cutting intensity increased. Gap creation induced the greatest $\Delta$PACL responses.}
\label{fig:55meanPACL}
\end{figure}

\begin{table}
\caption[Parameters of the model of post-harvest PACL.]{Fixed-effect estimates, $b_l$, of the adjusted model (\autoref{eq:4lambert}) with approximate confidence intervals with $\alpha=0.05$ (CI) and standard errors. Our model assumed that any removal of \SI{1}{\percent} of stand basal area induced an increase in PACL of $b_l$. additionally, $\theta_\beta$ is the standard deviation of the random effect and indicated the variability of this relationship between plots.}
\footnotesize
\begin{tabularx}{\textwidth}{lRRR}
\toprule
\textsc{cutting type} & \multicolumn{2}{c}{$b_l$} & $\theta_\beta$ \\
\cmidrule(lr){2-3}
	&	Estimate & CI & \\
\midrule
gap creation & 0.835 & 0.76;0.90 & 1.580 \\
cutting from below & 0.615 & 0.57;0.67 & 2.635 \\
species-specific cutting & 0.579 & 0.53;0.63 & 3.148 \\
uniform cutting & 0.459 & 0.43;0.48 & 3.870 \\
cutting from above & 0.349 & 0.30;0.40 & 0.172 \\
\bottomrule
\end{tabularx}
\label{tab:53model}
\end{table}

For the 9 plots where initial mean PACL was below \SI{20}{\percent} (\autoref{fig:51structure}), we analyzed the percentage of understory area receiving PACL ranging between \SIrange{0}{10}{\percent}, \SIrange{10}{20}{\percent}, \SIrange{20}{40}{\percent} or \SIrange{40}{100}{\percent}. Contrary to $\Delta$PACL, changes in percentage of microsites above a given light level depended noticeably upon an interaction between cutting type and harvest intensity (\autoref{fig:56cuttingIntensity}) or post-harvest basal area (\autoref{fig:57targetBasalArea}).
 
As most plots received an average of more than 10 PACL before harvest, the proportion of understory area receiving less than 10 PACL decreased with harvest intensity (\autoref{fig:56cuttingIntensity}a). The proportion of microsites receiving \SIrange{10}{20}{PACL} also decreased rapidly with harvest intensity. Nevertheless, harvesting only \SI{10}{\percent} of stand basal area did not significantly reduce the proportion of these microsites except with gap creation (\autoref{fig:56cuttingIntensity}b). A cutting intensity of \SI{10}{\percent}, with all cutting types but gap creation, maintained basal area around \SIrange{15}{20}{\square\metre\per\hectare} (\autoref{fig:57targetBasalArea}).

The different cutting scenarios provided very different proportions of microsites receiving \SIrange{20}{40}{PACL} which is the range of light conditions that promotes the less shade-tolerant oak (\autoref{fig:56cuttingIntensity}c and \autoref{fig:57targetBasalArea}c). Gap creation maximized this proportion at \SI{10}{\percent} harvest intensity (target basal area \SIrange{20}{25}{\square\metre\per\hectare}) but, at higher harvest intensities, very little area was in the \SIrange{20}{40}{PACL} range. Cutting from below and species-specific harvesting maximized the proportion of microsite receiving \SIrange{20}{40}{PACL} at about \SI{20}{\percent} of harvest intensity (target basal area of \SIrange{15}{20}{\square\metre\per\hectare}) but they provided more than \SI{40}{\percent} of the area in the \SIrange{20}{40}{PACL} range in all but the most intense harvesting scenario. Uniform cutting maximized the area receiving \SIrange{20}{40}{PACL} at about \SI{40}{\percent} of harvest intensity (target basal area of \SIrange{10}{15}{\square\metre\per\hectare}) whereas cutting from above provided about \SI{40}{\percent} of the area in the \SIrange{20}{40}{PACL} range independently of harvest intensity.

Cutting from above maintained a remarkably high proportion of microsites with less than 20 PACL at all cutting intensities and only created a low proportion of microsites with more than 40 PACL (\autoref{fig:56cuttingIntensity}d). In contrast, the proportion of microsites with more than 40 PACL was the greatest for all cutting intensities with gap creation. The difference in microsite with greater than 40 PACL between gap-harvesting and the other cutting types was particularly notable at harvest intensity of \SI{20}{\percent}.

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\caption[Post-harvest understory light and cutting intensity.]{Frequency of microsites with percentage of above canopy light (PACL) ranging between \SIrange{0}{10}{\percent}, \SIrange{10}{20}{\percent}, \SIrange{20}{40}{\percent} and \SIrange{40}{100}{\percent}. These frequencies were computed by harvest intensity in the 9 plots where initial mean PACL was below \SI{20}{\percent}. The proportion of microsites with PACL of \SIrange{0}{10}{} (a) or \SIrange{10}{20} (b) was high prior to cutting (white bar on the left). This proportion decreased the most rapidly with gap creation in contrast with cutting from above. High proportions of microsites receiving \SIrange{20}{40}{PACL} (c) were obtained with \SI{10}{\percent} gap creation, \SI{20}{\percent} cutting from below, \SI{20}{\percent} species-specific cutting, or \SI{40}{\percent} uniform cutting. }
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\caption[Post-harvest understory light and target basal area.]{Frequency of microsites by class of percentage of above canopy light (PACL) and post-harvest basal area. The frequencies were computed from the 9 plots where initial mean PACL was below \SI{20}{\percent} and, next, averaged by classes of post-harvest basal area. Cutting scenarios that maximized the understory area receiving \SIrange{10}{20}{PACL} (b) reduced stand basal area to about \SIrange{15}{20}{\square\metre\per\hectare}. Cutting From above only slightly affected the proportion of microsites with \SIrange{20}{40}{PACL} (c). High proportions of microsites with \SIrange{20}{40}{PACL} were obtained with cutting that created gaps that reduced basal area to \SIrange{20}{25}{\square\metre\per\hectare}, with cutting from below that reduced basal area to \SIrange{15}{20}{\square\metre\per\hectare}, or with uniform cutting and cutting from above that reduced basal area to \SIrange{10}{15}{\square\metre\per\hectare}.}
\label{fig:57targetBasalArea}
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\section{Discussion}
\subsection[Mean light response]{Mean light response to the different cutting scenarios}
The different cutting types led to different increases in mean PACL that were ordered according to our first hypothesis ($H_1$) independently of initial stand structure and composition (in contradiction with $H_2$). On average, harvesting \SI{10}{\percent} of stand basal area increased mean PACL by about \SI{8.4}{\percent} with gap creation, \SI{6.2}{\percent} with cutting from below, \SI{5.8}{\percent} with species-specific cutting, \SI{4.6}{\percent} with uniform cutting and \SI{3.5}{\percent} with cutting from above (\autoref{tab:53model} and \autoref{fig:55meanPACL}).

Similar to the findings of numerous studies \citep{Canham1994, Cescatti1997a, Brunner1998, Stadt2000, Beaudet2002, Beaudet2011, DaSilva2012}, the main factor limiting understory light was the absence of gaps between crowns. Consequently, at similar cutting intensities, harvests that create gaps strongly increased understory light. Additionally, we only considered immediate post-harvest conditions while the evolution of understory light several years after cutting may lead to increased or reduced differences between treatments. In particular, the increase in understory light due to the creation of large gaps would be expected to last longer than the effects of the other cutting types since larger openings would take longer to close \citep{Sprugel2009}.
 
Another factor that strongly limits light availability for regeneration is the presence of a sublayer of shade-tolerant species. We already noticed this influence of a dense understory when evaluating the performance of our light model (\autoref{ch:lightModel}). In addition, our simulation confirmed that understory trees might intercept a large proportion of light and that cutting from below can increase significantly understory light levels and can therefore be essential to promote the regeneration of less shade-tolerant species.
 
Preferentially harvesting shade-tolerant species, i.e., species-specific cutting, increased more transmitted light than harvesting trees randomly. Trees of shade-tolerant species intercept more light than trees of less shade-tolerant species since they have wider, deeper and denser crowns (\autoref{tab:48crownradius}, \autoref{tab:410crownbase} and \autoref{tab:412LAD}). Additionally, shade-tolerant species are usually more abundant in the understory than less shade-tolerant species. Species-specific cutting therefore tend to harvest a high proportion of poles and small trees similarly to the cutting from below. 

\subsection{Optimum cutting scenario} 
The optimum cutting scenario within the context of this study maximized the understory area that is favorable to the natural regeneration of less shade-tolerant species, i.e., the area receiving \SIrange{20}{40}{PACL}. Contrary to mean PACL, the area receiving \SIrange{20}{40}{PACL} depended upon the interactions between cutting type and cutting intensity. High proportions of microsites with \SIrange{20}{40}{PACL} were obtained by either harvesting few trees or by harvesting more than \SI{50}{\percent} of stand basal area.

Creating gaps appears particularly promising to promote small clumps of oak regeneration with limited reduction of stand stocking. Harvesting the few trees located within and around these clumps largely increase the proportion of microsites with \SIrange{20}{40}{PACL}. In our simulations, the gaps that maximize this proportion of microsites are about \SI{470}{\square\metre} in size which corroborates the recommendations by \citet{VonLupke1998} and \citet{Bruciamacchie2005} as well as the observations of \citet{Rugani2013} in old growth beech forests. These authors reported that oak regeneration was possible in gaps of at least \SI{500}{\square\metre} created by harvesting \SIrange{4}{5}{} mature trees \citep{Bruciamacchie2005}. Larger gaps increase the proportion of microsites with more than 40 PACL and should likely be avoided during the first stages of regeneration development because such conditions are favorable to the rapid development of competitive herbaceous species \citep{Gaudio2008, Gaudio2011}.

Cutting from below and cutting preferentially shade-tolerant species were the best techniques to promote the recruitment of less shade-tolerant regeneration especially if saplings were uniformly spread in the understory as it happens after generalized masting. For the studied stands, the optimum harvest intensity was about \SI{20}{\percent} which corresponded approximately to a target basal area of about \SIrange{15}{20}{\square\metre\per\hectare}.
 
Randomly cutting trees requires a greater harvest intensity to maximize the proportion of microsites with \SIrange{20}{40}{PACL} than gap creation, cutting from below and species-specific cutting. We obtained an optimum number of microsites with \SIrange{20}{40}{PACL} with a harvest intensity of \SI{40}{\percent} which corresponded to a target basal area of about \SIrange{10}{15}{\square\metre\per\hectare} and agrees with the results obtained by \citet{Balandier2006} in even-aged oak stands.
 
Cutting from above maintained large understory areas receiving less than \SI{40}{\percent} full light. It maintained a more asymmetric right-skewed distribution of PACL \citep{Beaudet2011} than the other cuttings and, hence, a high proportion of microsites in shady conditions even after harvesting up to \SI{60}{\percent} of stand basal area. By preferentially eliminating large overstory oaks and maintaining low-light levels in the understory, this treatment can be expected to quickly lead to a successional transition to dominance by shade-tolerant species.
  
In conclusion, promoting less shade-tolerant species can be achieved with various regeneration treatments. Forest managers should consider whether the seedlings of less shade-tolerant species are aggregated or uniformly spread, whether a small reduction in stand stocking is more appropriate, and what is the desired composition of the different tree layers after harvest. The results from this study provide foresters with the necessary tools to evaluate how silvicultural treatments can be manipulated to create or maintain favorable conditions for the regeneration of species of different shade tolerances. 

