4.2 The model for the regular (even-aged) stands
4.2.1 The model
4.2.1.1 The ~ r i n c i ~ l e s
Static description of each domain
At a date t, the domain is described as an age table, i.e. as a vector AREA(t), whose elements AREA(class,t) are the area in different (age, volume) classes. These classes are arbitrarily chosen, and are defined as:
with age(i) and vol(vc) being the limits of the classes (volume classes are noted vc).
Age classes are those previously defined (see section 3.3. I), except that an age class is added to account for the case of "forest bare landn (forest areas which have been recently clearcut and are going to be regenerated in the near future). This leads to a total number of 13 age classes.
Similarly, a special volume class was introduced for "bare land" (with volume equal to zero !).
Other volume classes are described below (see section 4.2.2).
This static description of the wood resource in a domain is based on a double discretization relative to both and volume. It may be, completed by some further -variables like:
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the mean sawtimber volume in each class: SAW(age,vc,t).-
the mean annual volume increment in each class over the past 5 years: GRO(age,vc,t).Description of dynamics
Forest dynamics is viewed as transitions between the different (age.volume) classeg. During the basic time step, i.e. between dates t and t + 1, it is assumed to be represented by:
where:
CUT(t) is a cutting transition matrix accounting for clearcuttings and thinnings between t and t + l ;
I is the identity matrix;
INC(t) is an increment transition matrix accounting for natural growth of stands between t and t + l .
This method of representing the dynamics is quite simple and is based on some form of direct analysis of what happens between 2 successive dates. It also implies a time discretization which is commonly accepted for forest dynamics (time step will usually be 5 years).
Restrictions to the model
Some restrictions were imposed to the model (actually to the matrices INC(t) and CUT(t)) to ensure that the parameters of the model can be estimated by using the NFIS data sent to IIASA (these restrictions could be avoided
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at least some of them-
if more detailed data were available, see section 4.2.1).The trajectory of a stand is assumed to be the following:-
First, at the beginning of the time step, stands may:either be clearcut: i.e. they move from their current (age,vc) class to the bare land class (0,O);
or be thinned down to the neighboring lower volume class (from (age,vc) down to (age,vc-1): their age does not change);
or remain in the same (age,vc) class.
Mortality due to any reason (pest, air pollution, storms, drought,.
.
.) could be introduced at this stage.-
Second, they are matured (by the model) and may:either move to the next volume class or remain in the same volume class;
either move to the next age class or remain in the same age class;
be regenerated in the case of bare land (i.e. the stands move from bare land -i.e. class (0,O)- to 1st age class and 1st volume class 4.e. class (1,l)-).
Growth reducers like air pollution effects, climatic cycles, etc..
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could be introduced at this point.It is further assumed that volume transition is inde~endent from age transition. This is a strong and unrealistic hypothesis since age-class width is superior or equal to 20 years and the time step is only 5 years, so that the oldest stands of an age class (those which are going to move to next age class at next time step) are probably among those which have the highest volume and increment in the age class.
4.2.1.2 Comments
About the the hypothesis of stationarity
At this point, the model does not contain any hypothesis about the stationarity of CUT(t) and INC(t). These matrices are simply assumed to depend on time. The assumptions of growth stationarity (i.e. that INC(t) is constant over time) and of independence of growth and cutting level (i.e. INC(t) is not affected by the value of CUT(t) and reciprocally) are introduced later (see section 4.2.3) in order to "enable" the extrapolation of past growth to future.
In the current version of the programs, CUT(t) cannot be modified from one time step to another during the course of a simulation. This could easily be changed. However, the lack of detail would prevent any attempt to model the feed-back relationship between growth and harvesting level.
About the sense of classes: the "standlplot problem"
The definition of the classes may be an illusion, since it is well known that classes assessed according to lot or to stand characteristics have not the same meaning (see Hagglund, 1983, for a discussion of this topic).
This criticism does not concern the age classification because it is both a stand and plotwise classification (due to the definition of even-aged stands).
But it concerns the volume classification. Since forest management is organized at the stand level, the volume classes should be standwise classes. But, since NFIS is using small plots, the data put into AREA(t) are actually plotwise data. For this reason, the assessment of sivicultural treatments according to plotwise volume classification may not correspond to the intentions of the modeller.
About the stochastic nature of the model
This model may either be viewed as deterministic (in that case the values in INC(t) and CUT(t) are just ratios) or as stochastic (in that case the parameters in INC(t) and CUT(t) are probabilities).
Following the trajectory of a stand (or a plot) from one state to another, leads to the probabilistic point of view. But, considering the different classes as global compartments linked by flows of stands leads to the deterministic point of view.
These two conceptions are not independent: the deterministic point of view may be deduced from the probabilistic description by using mathematical expectations conditional to the initial state. But it must be stressed that the basic foundations are stochastic (the construction of the
About the restrictions to the model
The assumptions concerning the possibility for a stand to make only small moves (only one volume-class jumps, except for regeneration) could be avoided (at least partly) if permanent plots data were available, or if the past history of each stand or plot were reconstructed (to carry out this last possibility, it would be necessary to go back to the basic data
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around 100000 plots for even-aged stands in France).About the way of representing clearcuttings
Another major drawback of the model is that the way of simulating clear-cuttings is not fully consistent with the traditionnal way of regenerating the broadleaf stands (i.e. several successive cuttings with a short time interval over a 10 to 20 years period).
About volume discretization
The traduction of volume discretization may have (at least) two forms, which indicate a lack of consistency of the model:
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either all the stands (or plots) of the same volume class are assumed to have the same volume, equal to the mid-volume of the class: this form is used to get estimates of transition probabilities (see section 4.2.3).-
or these stands are assumed to have a uniform volume distribution, within each volume class: this form is used to get estimates of the coefficient of variation of volume distribution within each age class (see 4.2.3).About the separation of cuttings and growth
The division of time step into two logical sub steps ([I.] cuttings, [2] growth) may be considered as a further time discretization of the model.