1 Introduction

Many forest dynamics models have been developed in the last decades. They generally present various kinds of organisation and structure that allow some classifications (see Porté and Bartelink 2002 for a review of forest model classifications). The particular organisation and structure of a given model depend on the purpose for which it has been developed.

Among these models, the Fagacées growth and yield model can be classified as a distance-independent tree-centered model. It has been mainly developed during the 1990s, first for the management of pure Beech stands (Dhôte 1991), then expanded to pure Oak stands (Dhôte 1999). Its development has been motivated by the lack of growth and yield model adapted for managing these species in France. Indeed, despite the fact that Sessile Oak (Quercus petraea Liebl.) and Beech (Fagus sylvatica L.) are respectively the first and third forest species in France (they represent respectively 12.3% and 9.3% of the productive forest area; Inventaire Forestier National 2007), no growth and yield model were developed for these species in France before. The main reference for Oak was a yield table published in 1962 (Pardé 1962) and for Beech a yield table published in 1973 (Décourt 1973). These tables have then been republished in 1984 (ENGREF et al. 1984). They contain reference trajectories for Oak and Beech silviculture and can be considered as ancestors of growth models for these species in France. The Fagacées growth model has been developed to provide the foresters a more flexible tool than growth and yield tables to manage even-aged high forest stands and coppice with standards stands, the forest structures that are the most common in France for Oak and Beech.

The main originality of the Fagacées model lies in its top–down organisation (Fig. 1): once the global growth of the stand has been computed, it is allocated among the trees within the stand. The model thus presents at the same time some properties of stand models and some properties of individual models.

Fig. 1
figure 1

General structure of the “Fagacées” growth model

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Unlike other classical tree-centered forest growth models used for European common species like SILVA (Pretzsch et al. 2002) or BWINPRO (Shröder et al. 2007), Fagacées is not a spatialised model. As for other distance-independent growth models like PROGNAUS (Monserud and Sterba 1996), competition between individuals is not explicitly computed but is assessed through global relations such as absolute or relative position within a histogram.

Although most parts of this model have already been published (generally in French, sometimes only in poorly accessible documents), it has never been fully presented. We present here the general structure of this model, the whole set of equations of the growth part and the corresponding parameters. We present also some additional procedures used to obtain a full simulator so that the growth simulator can be entirely rebuilt from the information presented in this paper.

Since Fagacées is a tree-centered model, individual trees are described by individual characteristics such as age, diameter at breast height (d 130) and height (h). This information can be further presented as a whole stem profile that will be used to compute volumes and wood qualities.

This paper contains many notations and formulae. In order to help the reader, Table 1 presents the main notations used.

Table 1 Main notations used in the paper
Full size table

2 Equations of the growth model

2.1 Origin of the data

Various samples have been used to establish or calibrate the equations presented in this paper.

The main data used were obtained from permanent stands that were established in plain forest of the northern part of France covering a climatic gradient from oceanic (Normandy region) to semi-continental (Alsace region). All of them were established between 1883 and 1935 and most of them were issued from natural regeneration. They are currently maintained by the French National Institute for Agronomical Research (INRA). They were established in order to study the influence of thinning intensity and frequency on wood production. Several plots were kept as control plots, with no silvicultural treatment. This point will be crucial for the adjustment of a self-thinning relation. Details on the permanent stands can be found in papers published by Pardé (1962), Oswald (1981) and Dhôte (1999).

For Oak, 11 sites contain 32 plots (several plots per site) between 0.2 and 1 ha and 339 forest inventories. The age classes vary from 27 to 277 years, but main data concern plots between 40 and 160 years old. The chosen sites mainly correspond to areas devoted to high-quality oak timber production and at least average site fertility index (top height at 100 years between 21 and 29 m).

For Beech, the main data represents five sites and 33 plots, with areas from 0.2 to 1 ha. Age classes vary from 35 to 130 years. As for Oak, the plots were chosen mainly within good fertility conditions (Dhôte 1997).

The top height curve for Oak was established from a sampling of 50 pure Oak plots (0.06 ha each) from five regions of the northern part of France. These plots were chosen so that they cover a large range of site fertility index. In each plot, the first, third and fifth largest dominant trees were sampled for a detailed stem analysis. This sampling was conducted at the French Forest Service (ONF) by Duplat and Tran-Ha (1997).

Relations for individual trees are issued partly from 82 mature sessile Oak trees measured in detail in five areas from Northern France sampled during a convention between INRA and ONF (Nepveu and Dhôte 1998) and partly from a sample of 3,000 individual conventional data collected in the permanent plots already mentioned (Dhôte et al. 2000).

Self-thinning curves, that are used to compute the Relative Density Index, have also been adjusted on data issued from the latter permanent plots (Dhôte 1997). Except for the self-thinning curves (see below), all the relations presented in this paper have been fitted using least squares methods.

2.2 The Relative Density Index

The notion of Relative Density Index (RDI) is based on Reineke’s observations (Reineke 1933) in very dense monospecific stands for several species from North America. Reineke defined a self-thinning line which represents the maximum number of stems a forest stand at a given diameter could support before self-thinning occurs. He defined a Stand Density Index to convert the number of trees in a stand to the number of trees the same stand would have for the same density with an average diameter of 10 in. (25 cm).

However, the values of this index can vary within a very large interval and the values obtained for different species or regions cannot be directly compared. The RDI has then been defined in order to be scaled within the [0, 1] interval. It is computed as the ratio of the actual number of stems in a stand and the theoretical number of stems this stand could support with the same DBH (Fig. 2):

Fig. 2
figure 2

Construction of the Relative Density Index (RDI) and introduction of self-thinning in the Fagacées model

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$$ {\text{RDI}} = \frac{N}{{{N_{{\max }}}}} $$
(1)

where, N is the current number of trees and N max the theoretical maximum number of trees obtained from the self-thinning equation.

For both sessile Oak and Beech, the self-thinning line is given by:

$$ \ln \left( {{N_{{\max }}}} \right) = a - b \cdot \ln \left( {\pi \cdot {D_g}} \right) $$
(2)

where, D g is the average quadratic diameter expressed in centimetres. However, this self-thinning relation is not always a straight line. In particular, the slope parameter b can increase as the forest grows (Zeide 2005). Nevertheless, this formulation is considered as a good approximation among a wide range of ages and fertility conditions.

Since the theoretical maximum value of the RDI is 1, Eq. 2 can be used in Eq. 1 to obtain a new computation of RDI:

$$ {\text{RDI}} = \frac{{N \cdot {{\left( {\pi \cdot {D_g}} \right)}^b}}}{{\exp (a)}} $$
(3)

The values of parameters a and b used in Fagacées are given in Table 2. They have been obtained by visually adjusting a straight line on the upper limit of the points [ln(D g); ln(N)] from the data of the permanent plots (Dhôte 1997).

Table 2 Parameters used for the computation of RDI
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Two main criticisms of this approach can be formulated. First, the chosen adjustment method is known to be highly dependent on the data quality. In particular, it is necessary that part of the stands used to adjust the relation is close from the self-thinning situation. Since there are some stands with no sivicultural treatment for decades in the data used, we consider that this condition is verified. Other methods exist to fit self-thinning relations such as the stochastic frontier analysis but they need additional assumptions to be properly applied (Zhang et al. 2005). Second, the parameters do not change according to the site fertility index. The fact that these parameters should change according to species or according to site fertility index is discussed (Jack and Long 1996; Bi et al. 2000; Morris 2003). In our case, if two different parameter estimations have been conducted for Oak and Beech, no significant relation with the site fertility index H 100 has been highlighted. However, it should be remarked that the values of the b parameter found here differ from those found by Pretzsch (2006) for the same species in central and southern Germany. It is higher (in absolute value) for Oak: the value used for the b parameter is 1.701 whereas it ranges from 1.222 to 1.628 according to site in Pretzsch et al. For Beech, the value used in Fagacées is lower: 1.574 against 1.723–1.873 in Pretzsch et al.

2.3 Stand initialisation

Natural regeneration is simulated to generate an initial stand. The word “recruitment” would actually be more appropriate than “regeneration” since the trees appear in the model only once they have reached a given state (circumference at 130 cm greater or equal than 1 cm). Several studies (Sano 1997; Fortin et al. 2006) showed that tree diameters are generally distributed according to an inverse j-shaped distribution. Among such distributions, the power distribution, the Weibull distribution and the exponential distribution are the simplest and are commonly used (Nord-Larsen and Johannsen 2007). Here, the initial individual diameters are randomly generated from an exponential distribution, which has been chosen for simplicity. Its unique parameter is chosen so that the average quadratic diameter of the stems equals a given value D g :

If the random variable D follows an exponential distribution with expected value equal to θ, then:

$$ E\left( {{D^2}} \right) = 2 \cdot {\theta^2} $$
(4)

The parameter θ is chosen so that \( \sqrt {{E\left( {{D^2}} \right)}} = {D_g} \) hence:

$$ \theta = \frac{{{D_g}}}{{\sqrt {2} }} $$
(5)

Simulations begin only when the growth model becomes valid, around 15 years old. Since the model is used to compare silvicultural scenarios, the same starting point is chosen for all of them. This starting point is defined by the initial RDI, RDI0, from which the initial number of stems N 0 to be generated is obtained according to Eq. 2:

$$ \ln \left( {{N_0}} \right) = \ln \left( {{\text{RD}}{{\text{I}}_0}} \right) + a - b \cdot \ln \left( {\pi \cdot {D_{{{g_0}}}}} \right) $$
(6)

Since a “do-nothing” scenario (corresponding to unmanaged forest from regeneration) belongs to the set of scenarios to be studied, the value of RDI0 has been fixed to 1.

Data analysis from forest inventories in several areas of the northern half of France showed the following fact. When the trees are about 15 years old, an oak stand in natural regeneration conditions has an average quadratic diameter D g approximately equal to 1 cm, regardless of the stand fertility index. So the reference D g0 = 1 cm at age A 0 = 15 years is taken as a starting point for the for regeneration simulations. The individual heights h are then obtained from the individual diameters and the stand top height using the allometric Eq. 24 that will be presented later in this paper.

2.4 The stand level of the growth model

2.4.1 Top height curves

For Sessile oak, the top height depends only on stand age and on stand fertility index. The curve equation (Eq. 7 and Fig. 3), based on data from even-aged high forests available at the ONF, has been established by Duplat and Tran-Ha (1997). It is written as:

Fig. 3
figure 3

Top height curve for Sessile oak (after Duplat and Tran-Ha 1997)

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$$ \begin{array}{*{20}{c}} {{H_0} = f\left( {A,{H_{{100}}}} \right)} \\ { = {m_1} + \left( {{m_2} \cdot A + {m_3} \cdot {H_{{100}}} + {m_4}} \right){{\left[ {1 - \exp \left( { - {m_5} \cdot {A^{{{m_6}}}}} \right)} \right]}^{{{m_7}}}}} \\ { + \left( {{m_8} + {m_9} \cdot {H_{{100}}}} \right) \cdot A} \\ \end{array} $$
(7)

where,

  • H 0 is the top height of the stand (metres);

  • A is its age (years);

  • H 100 is the fertility index (metres).

Since the model has been designed for even-aged stands, the age of the stand is also the age of its trees. The site fertility index H 100 is defined by the top height of the stand at 100 years. The values of the parameters m 1 to m 9 are given in Table 3.

Table 3 Parameters of Duplat and Tran-Ha’s curve for Sessile oak top height
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This curve presents an oblique asymptote. Usually, growth equations present a horizontal asymptote. But even the oldest stands (around 220 years old) used to establish Eq. 7 were still growing. Since no high standard forest older than 250 years was available in the data, this curve is generally considered as acceptable until 250 years and may be extrapolated until 300 years, not after, due to the oblique asymptote.

This equation is mainly used in the model to compute the dynamics of stand height between two successive steps of simulation. The stand height is generally supposed to be independent from the silviculture for many timber species. However, this is not completely true. Hence in several forest models, for example the one from Nord-Larsen and Johannsen (2007), the height is estimated as an increasing function of the stand density. In the Oak version of the Fagacées model, the increment of stand height ΔH 0 between the ages A t and A t + 1 is computed as:

$$ \Delta {H_0} = \left\{ {\begin{array}{*{20}{c}} {{H_0}\left( {{A_{{t + 1}}}} \right) - {H_0}\left( {{A_t}} \right)} & {{\text{ if}}\;{\text{RDI}} \geqslant 0.5} \\ {2 \cdot {\text{RDI}} \cdot \left[ {{H_0}\left( {{A_{{t + 1}}}} \right) - {H_0}\left( {{A_t}} \right)} \right]} & {{\text{if}}\;{\text{RDI}} < 0.5} \\ \end{array} \begin{array}{*{20}{c}} {} \\ {{\text{ if RDI}} \geqslant 0.5} \\ \end{array} } \right. $$
(8)

The growth reduction factor due to low density can appear strong. It has been built so that the growth tends to 0 for very low density and to lead to continuous growth around RDI = 0.5. This reduction factor has been introduced mainly to give the model a behaviour for low density consistent with the theory. But, due to the lack of available data for very low stand densities, it has not been validated in this area.

For Beech, the current version of top height equation has been fitted by Bontemps (2006) on the Lundqvist–Matérn equation:

$$ {H_0}(A) = K \cdot \exp \left( { - {{\left[ {{{\left( {\ln \frac{K}{{{H_{{{A_0}}}}}}} \right)}^{{ - m}}} + \frac{{R \cdot m \cdot {C_m}}}{K}\left( {A - {A_0}} \right)} \right]}^{{ - \tfrac{1}{m}}}}} \right) $$
(9)

which is a classical sigmoid growth curve with horizontal asymptote. In this parameterisation, A is the current stand age, A 0 a reference age, H A0 the top height at the reference age, K the value of the horizontal asymptote and m a shape parameter. The quantity C m is a function of m adjusted to give to the parameter R the meaning of the growth speed at the inflexion point (taken as fertility indicator). For the Lundqvist–Matérn equation it can be written as:

$$ {C_m} = \exp \left\{ {\left( {1 + m} \right)\left[ {1 - \ln \left( {1 + m} \right)} \right]} \right\} $$
(10)

However, contrary to the Oak case, it has not been possible to give to Eq. 9 a single set of parameters for the whole northern part of France (Table 4).

Table 4 Parameters of Bontemps’ curve for Beech top height
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2.4.2 Basal area increment

The basal area increment between two steps of simulation is given by a potential-reductor equation:

$$ \Delta G = {f_1}\left( {{H_0},\Delta {H_0}} \right) \cdot {f_2}\left( {{\text{stand}}\;{\text{density}}} \right) $$
(11)

where f 1 is a function that expresses the potential stand growth, that depends only on the current top height H 0 and its variation ΔH 0 between two steps of simulation and f 2 is a function that expresses how the potential growth is reduced by the stand density.

The potential stand growth f 1 has the following expression:

$$ \left\{ {\begin{array}{*{20}{c}} {{\text{Oak:}}} & {{f_1}\left( {{H_0},\Delta {H_0}} \right) = {p_1} + {p_2} \cdot \Delta {H_0} - {p_3}{H_0}} \\ {{\text{Beech:}}} & {{f_1}\left( {{H_0},\Delta {H_0}} \right) = {p_1} + {p_2} \cdot \Delta {H_0}} \\ \end{array} } \right. $$
(12)

The reductor function takes a different form for Oak and Beech. For Oak, the stand density is expressed by the RDI index:

$$ {f_2}\left( {\text{RDI}} \right) = \frac{{\left( {1 + r} \right) \cdot RDI}}{{r + RDI}} $$
(13)

whereas for Beech, it uses the sum of diameters exceeding a given threshold Σ σ :

$$ {f_2}\left( {{\Sigma_{\sigma }}} \right) = 1 - \exp \left[ { - \left( {{r_1} + {r_3}} \right) \cdot {\Sigma_{\sigma }}} \right] $$
(14)

The stand density index Σ σ used for Beech is computed as

$$ {\Sigma_{\sigma }} = \left( {\frac{1}{\text{stand area}}} \right) \cdot \sum\limits_{{{d_{{130}}} \geqslant \sigma }} {{d_{{130}}}} $$
(15)

The diameter threshold σ cannot be determined at this step. It will be numerically adjusted to equilibrate stand growth and individual growth later in this paper (Eq. 16).

For both Oak and Beech, the values of the parameters used for functions f 1 and f 2 can be found in Table 5. They have been empirically established and adjusted on existing datasets (Dhôte 1995; Nepveu and Dhôte 1998).

Table 5 Parameters used for the increment of basal areal
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In the global Eq. 11, the growth has two independent components (for Oak at least if RDI ≥ 0.5): the stand height curve that depends only on age and fertility index and the stand density that depends on silviculture.

The first component (Eq. 12) has the general shape of the derivative of a classical growth curve: first increasing, reaching a maximum then decreasing. However, it should not be used after 300 years which is the limit of validity for the model.

The second component (Eq. 13 for Oak, Eq. 14 for Beech) expresses how the increment of basal area given by 12 is depressed if the stand density is not at its maximum value. This is the only way by which the silviculture acts on growth in the model. It is an increasing and concave function of stand density with values increasing from 0 to 1. It expresses that the speed of growth increases with stand density, but relatively more for low density values than for high densities.

2.5 From stand level to tree level

Fagacées is a tree-centered model. Within the stand, each tree growth is individually computed as follows. Once the global growth is computed, it is allocated among all the individuals (Fig. 1). The individual growth is first computed for the increment of the individual basal area. The individual height is then adjusted by allometry.

The transition from stand level to tree level for the basal area increment is given by Eq. 16:

$$ \sum\limits_{\text{trees}} {\delta g\left( \sigma \right) = \Delta G \cdot S} $$
(16)

where, δg is the individual basal area increment and S is the total stand area (ha). The parameter σ is an intermediate quantity used in the computation of individual basal area increment (Eq. 17 below). It is adjusted to make individual basal area growth sum up to stand growth as determined by the stand growth compartment of the model.

2.6 Growth of an individual tree

2.6.1 Increment of individual basal area

Dhôte (1999) has shown that in oak stands, individual tree growth is favoured for trees belonging to the biggest trees of the stand. At the opposite, the competition for a small tree can be so strong that its growth is almost stopped. For that reason, in the Fagacées model, the increment of basal area for a given tree is given by:

$$ \delta g = \gamma \cdot \max \left( {{d_{{130}}} - \sigma; 0} \right) $$
(17)

where,

  • δg is the individual increment of basal area (square centimetre per year);

  • γ is a maximal efficiency for growth (centimetre per year), see below for explanations;

  • d 130 is the current tree diameter at 130 cm (centimetre);

  • σ is a threshold which defines a minimum diameter for radial growth to occur (centimetre).

Only the trees with a diameter exceeding σ will grow (Fig. 4). This threshold σ has no fixed value but is numerically adjusted at each step of the simulation according to Eq. 16.

Fig. 4
figure 4

Individual growth of basal area

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In this model, the growth is expressed in terms of basal area which is linked to the diameter at 130 cm (d 130) by the relation:

$$ g = \pi \cdot d_{{130}}^2/4 $$
(18)

If expressed in terms of diameter, the increment can be rewritten as:

$$ \delta {d_{{130}}} = \sqrt {{d_{{130}}^2 + \frac{4}{\pi }\delta g}} - {d_{{130}}} = {d_{{130}}} \cdot \left( {\sqrt {{1 + \frac{4}{\pi } \cdot \frac{{\delta g}}{{d_{{130}}^2}}}} - 1} \right) $$
(19)

To interpret the γ parameter of Eq. 17, suppose that the σ parameter is neglectable compared to the biggest diameters in the stand. In this case, from the fact that \( 2\frac{\gamma } {{{\text{d}}_{{130}} }} < < 1 \), the first-order Taylor series of relation (19) gives for the high values of d 130:

$$ \delta {d_{{130}}} = {d_{{130}}} \cdot \left( {\sqrt {{1 + \frac{4}{\pi } \cdot \frac{{\gamma \left( {\pi \cdot {d_{{130}}} - \sigma } \right)}}{{d_{{130}}^2}}}} - 1} \right) \approx {d_{{130}}} \cdot \left( {1 + 2\frac{\gamma }{{{d_{{130}}}}} - 1} \right) = 2\gamma $$
(20)

The γ term represents the theoretical maximum increment of tree radius, so that it is called “maximum efficiency” of the growth.

For sessile Oak, it has been found to be linked to the fertility index of the stand H 100 by the relation:

$$ \gamma = 0.2007 + 0.0314 \cdot {H_{{100}}} $$
(21)

where, γ is expressed in centimetres per year and H 100 is expressed in metres.

For Beech, this coefficient depends on the variation of top height ΔH 0 but also on the stand density expressed with the index Σ σ defined in Eq. 15:

$$ \gamma = {f_1}\left( {\Delta {H_0}} \right) \cdot \left[ {{r_3} + {r_1} \cdot \exp \left( { - {r_2} \cdot {\Sigma_{\sigma }}} \right)} \right] $$
(22)

The parameters of this equation are given in Table 5. The diameter threshold σ is determined by numerically solving the equation

$$ \left( {{\Sigma_{\sigma }} - \sigma \cdot {N_{\sigma }}} \right) \cdot \left[ {{r_3} + {r_1} \cdot \exp \left( { - {r_2} \cdot {\Sigma_{\sigma }}} \right)} \right] = {f_2}\left( {{\Sigma_{\sigma }}} \right) $$
(23)

where, N σ is the number of trees per hectare with diameter exceeding σ, so that global growth given by (16) equals total individual growth given by (17).

The γ coefficient depends only on fertility index for Oak and on both fertility index and stand density index for Beech. This can be related to the fact that Beech is more shade-tolerant than Oak. Hence, it reacts more progressively than Oak to stand density variations.

2.6.2 Increment of individual height

Once the tree diameter d 130 (cm) is computed, the tree height h (m) is computed using an allometric relation (Dhôte and de Hercé 1994):

$$ h = 1.30 + \frac{{\alpha \left( {{d_{{130}}}} \right) - \sqrt {{{\alpha^2}\left( {{d_{{130}}}} \right) - 4 \cdot \pi \cdot {q_1} \cdot {q_2} \cdot \left( {{H_0} - 1.30} \right) \cdot {d_{{130}}}}} }}{{2 \cdot {q_2}}} $$
(24)

where,

$$ \alpha \left( {{d_{{130}}}} \right) = {H_0} - 1.30 + \pi \cdot {q_1} \cdot {d_{{130}}} $$
(25)

This is a hyperbolic curve, that crosses the point (d 130 = 0 cm, h = 1.30 m) and presents a horizontal asymptote at h = H 0. The values of parameters q 1 and q 2 are different for sessile Oak and Beech (Table 6). This relation is considered as robust within a wide range of ages and site conditions (Dhôte and de Hercé 1994). But since it is only an allometry, the individual height consistency depends on those of the top height growth and of the individual basal area growth.

Table 6 Parameters of the allometric relation for individual height
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2.7 Robustness and sensitivity analysis

The quality of the model can be studied at three levels: the general organisation of the different model components, the general form of the equations that describe these components and the parameters associated to these equations.

The Fagacées model can be considered as robust due to its top–down organisation. The main equations have been established at the stand scale, that is known to be generally more stable and reliable than the tree scale (Peng 2000).

The form of these equations has been chosen so that they respect theoretical considerations regarding the phenomenons they are supposed to represent. For example, the RDI Eq. 3 is derived from the linear relationship between the maximum number of stems and the stand diameter in a log–log scale as expressed by Reineke’s self-thinning rule. Due to these constrained forms, the model cannot have an aberrant behaviour. However, these forms may be appropriate only within a limited domain. For example the top hight curve used for Oak that presents an oblique asymptote cannot be extrapolated to very old stands. It is however often difficult to precisely define such domain limits. There is in general no difficulty to establish the validity of a relation in parts of the domain where observations are available. The question is how far from the observed area these relations can be extrapolated. An answer may be given by expert’s advice, but this answer cannot be very precise.

Except for self-thinning, the values of the parameters associated to the different relations have been obtained from observations using least squares methods. In most cases, the stability of the estimations has been studied by cross-validation, bootstrap or other resampling methods. These results have then been used to study the sensitivity of the response to variations of the parameters, at the scale of the different relations. In all cases, the numerical outputs obtained from variations of the parameters were comparable to the observed data.

In a general way, all available data have been used. The robustness of the behaviour at the model scale has always been privileged by the choice of the form of the equations and the way they have been arranged. In several cases, it would have been possible to use equations more adapted to the available data. But these equations were more sensitive to variations of parameters or to the bounds of the validity domain. With the chosen formulation, there is a slight loss of precision in terms of numerical outputs, but a more stable behaviour of the equations.

3 Complementary modelling for the evaluation of a silvicultural scenario

3.1 Stem profile

3.1.1 Stem taper curve

In Fagacées the trees are individualized. So each stem of the stand has its own characteristics. At the end of a growth step, each stem is allocated a diameter (d 130) and a total height h. These two variables are then used along with allometric relations to deliver a stem profile. The stem profile is a function of the height z in the tree at which the diameter is computed. For both Beech and Oak, this function has the same general form (Eq. 26 and Fig. 5).

Fig. 5
figure 5

Stem profile in the Fagacées model: compartments of the stem and parameters of the stem taper curve

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The stem profile can be assimilated to a succession of two cones, one for the stem above the crown basis, the other for the stem under the crown basis (Hatsch 1997). The latter has to be corrected at the bottom of the stem to take the butt influence into consideration.

For 0 ≤ z ≤ h, the diameter of the stem above bark (in centimetre) at height z (in metre) is given by:

$$ d(z) = \left\{ {\begin{array}{*{20}{c}} {{\mu_2} \cdot \left( {{\mu_1} - z} \right)} & {{\text{if}}\;{\mu_3} \leqslant z \leqslant {\mu_1}{\text{(within}}\,{\text{the}}\,{\text{crown)}}} \\ {\left[ {{\mu_4} \cdot \left( {{\mu_3} - z} \right) + {\mu_5} \cdot {\mu_2} \cdot \left( {{\mu_1} - {\mu_3}} \right)} \right]\frac{{1 + \lambda (z)}}{{1 + \lambda \left( {{\mu_3}} \right)}}} & {{\text{if}}\;0 \leqslant z \leqslant {\mu_3}{\text{(under}}\,{\text{the}}\,{\text{crown)}}} \\ \end{array} } \right. $$
(26)

where, λ (z) is a function to represent the variation of diameter at the butt:

$$ \lambda (z) = {\mu_6} \cdot \exp \left[ { - \frac{z}{{{\mu_7}}}\ln \left( {100 \cdot {\mu_6}} \right)} \right] $$
(27)

with:

  • μ 1 is the height of the apex, it is then equal to the total height h (metre);

  • μ 2 is the decrease of diameter per metre in the crown (centimetre per metre);

  • μ 3 is the height of the crown basis (metre);

  • μ 4 is the decrease of diameter per metre under the crown (centimetre per metre);

  • μ 5 is the discontinuity of the diameter at the crown limit (no dimension, μ 5 ≥ 1);

  • μ 6 represents the correction for the butt at the basis of the stem (no dimension);

  • μ 7 represents the height at which the influence of the butt falls to 1% of the stem diameter (metre).

All the parameters from μ 1 to μ 7 are computed from the individual diameter d 130, the individual height h and top height of the stand H 0 using allometric relations. Details on these computations can be found in Hatsch (1997) and Dhôte et al. (2000). These relations do not correspond to dynamic models: the parameter values at a given iteration of the simulation do not depend on their values at the previous iterations.

3.1.2 Volume within the stem

The main application of the stem taper Eq. 26 is the computation of volume between two heights. Since the modelled stem is a solid of revolution, the volume between two heights is then computed as:

$$ V\left( {{z_1},{z_2}} \right) = \frac{\pi }{{40000}}\int\limits_{{{z_1}}}^{{{z_2}}} {{d^2}(z)dz} $$
(28)

Again, the coefficient used in this formula is due to the fact that the diameter is expressed in centimetres whereas the height is expressed in metres. The volume is then expressed in m 3. This computation can be conducted either numerically or analytically. In order to be adapted to further modifications of the stem taper Eq. 26, the numerical computation has been chosen in the current version of the model.

3.1.3 Compartments within the stem

From the bark to the pith, three compartments are considered in the stem: bark, sapwood and heartwood. In addition, the evolution of the crown basis during growth helps to define two parts in the sapwood and in the heartwood: knotty and clear wood (see Fig. 5). The limits between all these compartments are based on allometric relations that can be found in Dhôte et al. (2000).

They provide detailed results on the geometry of each stem: volume of sapwood and heartwood, ring width. These characteristics can then be used to predict various properties such as wood density that are crucial to estimate some characteristics of the stem such as its wood quality or its carbon content. Various indicators can then be derived to process a multicriteria evaluation of the simulated stand (Bucket et al. 2005).

3.2 Thinning and self-thinning

Self-thinning occurs after the growth step, but only when RDI exceeds 1 (Fig. 2). In this case, the smallest trees systematically die until RDI falls to 1. Thinning is the main intervention included in Fagacées. It starts when the stand reaches a given stage (i.e. age, average diameter, RDI). In the original version of the model, it was manually conducted. Its technical characteristics (number of stems or basal area to harvest in each diameter class) were a consequence of the user’s decision. In the current version, another possibility using a parameterised rule has been defined. This rule allows the programme to run in “batch mode” without user intervention. It is used for studies that need a lot of simulations.

It has been designed so that the thinned trees are preferably, but not systematically, the smallest of the stand. It could however be modified to allow thinning from above. But in that case, the top height growth would be modified, so the current Eq. 8 should be replaced by another one, more adapted to this situation.

The thinning procedure depends on two user-defined parameters. The first one is the thinning intensity (expressed in terms of basal area of RDI to reach), the second one defines the kind of thinning. It takes its value in the [0;1] interval. The 0 value corresponds to the case where the thinning can occur until the maximum diameter of the stand, the 1 value to a thinning strictly from below.

The thinning procedure is a two-step one: the first defines the classes of diameter that may be thinned, the second one is the thinning itself.

3.2.1 First step of batch mode for thinning: determination of an upper diameter for thinning

When the thinning occurs, the stand can be described in terms of number of stems (N), average diameter at (D), average quadratic diameter (D g ) or density index (RDI). The lowest and the upper diameters (respectively D min and D sup) can also be defined. A thinning will be realised by removing stems until a given state is reached. Very often, it is formulated in terms of a value of basal area to remove or a given targeted RDI (lower than the starting one!). The simplest way to reach this goal would be to systematically remove the smallest trees until the objective is achieved, but it does not correspond to the real silvicultural practice. A procedure has then been designed to define an upper diameter D U below which thinning can occur.

A thinning more intensive than the aimed one is used. If the goal is to decrease to the value RDIobj, a thinning to the value φ ⋅ RDIobj will be considered (alternatively, if the goal is expressed in terms of basal area to remove, so that the basal area of the stand after thinning is G obj, the value φ ⋅ G obj will be considered), where φ is an additional parameter chosen within the [0, 1] interval. The value of D U is the diameter below which a systematic thinning of the smallest trees should occur to reach the φ ⋅ RDIobj value.

In practice, a function called “partial RDI” is used to compute D U : Let Ω(D) be the set of trees with diameter (d 130) greater than or equal to D, N Ω(D) and D Ω(D), respectively, the number of stems and the average quadratic diameter of Ω(D). The partial RDI RDIΩ(D) is the value of RDI computed from N Ω(D) and D Ω(D) using Eq. 3.

The partial RDI is a decreasing function of diameter, with values varying from the RDI of the whole stand when D is less than or equal to D min , to 0 when D is greater than or equal to D sup . The value D U is the lowest diameter for which RDIΩ(D) ≤ φ ⋅ RDIobj (Fig. 6). For clarity of illustration, the data used for Fig. 6 have been obtained by simulating a stand with exaggerated range of considered diameters.

Fig. 6
figure 6

The partial RDI function and the determination of the D U parameter used for batch thinning

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3.2.2 Second step of batch mode for thinning: the thinning itself

Once D U has been defined, each stem of the stand is associated to a probability of being thinned, which is a function of the stem diameter d 130. This probability is taken as proportional to:

$$ \tau = \max \left( {\frac{{{D_U} - {d_{{130}}}}}{{D - {D_{{\inf }}}}};0} \right) $$
(29)

so that the small trees have a greater probability to be thinned than the big ones and no tree with d 130 ≥ D U will be thinned.

In practice, each tree is then associated to a random variable X defined as:

$$ X = \tau \cdot U $$
(30)

where, U is randomly taken in an Uniform law [0, 1].

The trees are then ordered by decreasing X value. They are thinned from the beginning of the list until the targeted RDI or basal area is reached. Figure 7 presents an example of result. The chosen value of φ in this example is 0.7.

Fig. 7
figure 7

Example of result of the batch rule for thinning

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3.3 Batch scenarios

The first versions of the Fagacées model were designed to be manually run by the user who had to take the silvicultural decisions at each step. This approach still exists in the current version of Fagacées available through the CAPSIS platform (de Coligny 2005). However, since some studies based on this model need intensive simulations, a batch mode has also been developed. This mode needs batch silvicultural scenarios to be defined.

These batch scenarios are defined by four elements:

  • a reference trajectory to follow in the age × RDI space;

  • a tolerance around this reference trajectory;

  • a rule for the kind of thinning;

  • a stop condition for the simulation.

The principles of the batch scenarios are the following.

At the beginning, nothing special happens after regeneration. The trees grow with self-thinning as the only cause of mortality. Due to the fact that the Fagacées model has not been calibrated on very young stands, it is not worth to define precise instructions in the model for early stages.

The scenario itself begins at a given age (more than 15 years old). The silvicultural interventions will aim to make the stand follow a pre-defined trajectory in the age × RDI space. Of course, if this trajectory were to be exactly followed, an intervention would be needed at each step of the simulation (3-year steps in general). It is not realistic. So a tolerance envelope has been defined around the reference trajectory. Each time the stand reaches the upper limit of the envelope, an intervention is carried out to bring the stand RDI down to the lower limit. An additional constraint can be defined to forbid a too short interval between two consecutive thinnings. In that case, the second thinning will be delayed until this constraint is respected.

The previous procedure defines the date and the intensity of the thinnings. The thinnings themselves are conducted according to the thinning procedure detailed in Section 3.2. The φ parameter value to use is given by a function of the stand current state.

The scenario is run until a defined stop condition is encountered. This stop condition can be any logical rule based on the stand state (current age, average diameter).

Figure 8 illustrates an example of scenario for Oak where the main trajectory is defined by three straight segments to define three phases of the silvicultural schedule (preparation of the stand, designation of the crop trees and maturation). During the first two phases, the reference RDI varies linearly with the age of the stand. These two phases have pre-defined lengths. During the last phase, the reference RDI is maintained constant. This phase continues until the stop condition is encountered, so its length is not in general defined in advance. Here the final age has been set to 250 years just for illustration. The width of the tolerance envelope in this example has been fixed to a constant value (0.05 above and under the reference). The φ parameter used for thinning cannot be shown on this figure. It has also been set to a constant value (φ = 0.7). At the end of the scenario, all the remaining trees are harvested so the RDI falls to 0.

Fig. 8
figure 8

An example of trajectory followed by a stand using the batch scenario

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Since the functions for the reference trajectory, the tolerance envelope width and the φ parameter are not constrained to a particular form and the stop condition can be any logical rule based on the stand state, the procedure for batch scenarios was designed to study a very large set of silvicultural schedules. The only condition for the functions and the stop condition are to be consistent with what they are supposed to represent.

4 Discussion

Even if the equations can differ for Oak and Beech, the general approach used in the Fagacées growth model for these two species is the same. The model has been designed for high forest and coppice with standards forest management at a relatively large scale (a whole forest, a little region). According to Passioura (1973) cited by Battaglia and Sands (1998): “An appropriate model structure for management application is a transparent one: that it has few enough parameters so that each can be measured, directly or indirectly, and is composed of submodels corresponding to readily identifiable and experimentally accessible systems.”. With its top–down organisation and the privileged choice of robust equations, Fagacées does respect these general principles.

This top–down approach is not so common in forest modelling despite the fact it could lead to models useful for forest managers (Zeide 2003). It has been adopted by Corona et al. (2002) for the same reasons as Fagacées but in a much simpler situation: clonal plantation of poplar.

Although it is a tree-centered model, its main growth equation (Eq. 11) has been calibrated at the stand level. Indeed, the models adjusted at the scale of the stand are known to be generally more reliable for growth and yield than the models adjusted at the scale of the tree (Peng 2000).

The growth is mainly directed by the top height curve that is independent from silviculture as long as the stand density remains high enough. This result is generally admitted for a large range of initial spacing and thinning intensities as long as no thinning from above is performed (Skovsgaard and Vanclay 2008). Hence, in the current version, the model cannot be used to simulate silvicultures with thinnings from above.

If the stand density becomes very low, the stand height growth of Oak is depressed. However, due to the lack of available data for such stand densities, only an empirical correction has been introduced in the model. The model should then not be used to simulate silviculture with very low densities.

As a stand growth model, Fagacées is a classical one since the growth equations regulate stand height and basal area (Eqs. 7, 9 and 11). It becomes a tree-centered model by the way the global growth of basal area is allocated among individual trees (Eqs. 16 and 17). After this step, the individual heights and the whole stem profiles are simply rebuilt from allometries.

Both global and individual growths are expressed in terms of variation of basal area. They are consistent with differential equations that are classically used in process-based models. These equations also have a process-based interpretation whereas the individual height and the stem profile equations are purely descriptive relations (Dhôte 1996).

None of the growth equations contains random components. The Fagacées growth model is a purely deterministic one. As a consequence, due to Eq. 17, the hierarchy of trees will never change during stand development. If a given tree is smaller than another one at a given step of the simulation, it will remain smaller in all further steps, as long as none of them disappear due to thinning or self-thinning.

Another consequence is that the variability of diameters within a stand at a given age depends on the variability within the initial stand. It can only be modulated by the thinning rules the user may apply. However, the initial variability (expressed as diameter variance for example) is not enough to predict the evolution of diameter variability through growth. The whole diameter distribution is needed. Fagacées is a tree-individualised model and cannot be reduced to a population model nor to a stand model without loss of information.

If the growth equations have been calibrated so that they lead to unbiased predictions of basal area and global wood volume, the weakness of the variability representation may lead to biased prediction of wood quality. Indeed, wood quality clusters are based on conventional dimension limits for log diameter and ring width. Best qualities are defined by highest or lowest values for these dimensions. If the dimension variability is badly represented, the proportion of wood that verifies the wood quality conditions will be biased. In that case, alternative scenarios can be compared to predict which one produces the best quality in average, but not to predict precisely the wood quantities in each quality cluster. Hence the representation of the inter-tree variability is a point that could be improved in further versions of the model. This could be done for example by including random components for individual growth in the Eqs. 17 and 24.

5 Conclusion

The Fagacées model was built to simulate the consequences of a wide variety of silvicultures for even-aged high forests of Oak or Beech. Due to its top–down organisation, it combines the properties of a stand growth model (robustness and reliability) and some of tree-centered models (individual stem profiles). Only a very limited set of equations defines the growth model part and each of these equations reflects very simple principles. Thus the model can easily be improved if necessary. Several directions for that are considered: improvement of the growth equations, improvement of the stem taper profile, and designing of a model for Oak and Beech mixed stands based on the same concepts (Hein and Dhôte 2006).

The outputs of the growth model are used as inputs for modelling stem taper in order to obtain an estimation of the wood properties. However, the individual variability is badly represented. In addition, some defaults of Oak wood such as cracks or epicormic branches, that have a strong influence on wood quality, are not or insufficiently modelled. This can be a source of bias for some applications, related to wood quality. In this case, the model outputs have to be used only for comparisons of silvicultural scenarios. The under-representation of wood defects is due to a lack of knowledge on the deterministic mechanisms related to them. So it is still too early to include them in the model.

Despite these relative insufficiencies, this model is largely used by French forest managers and in forest research. It is still the only one available for Oak or Beech pure stands in the pedo-climatic conditions of northern France. It has been distributed to forest modellers and to forest managers, mainly through the CAPSIS project (de Coligny 2005) available at http://www-capsis.cirad.fr/home. It has contributed to the development of new standards of silviculture for the French Oak forest by the French National Forest Service (Jarret 2004; Sardin 2008).