Figure 1 Disturbance modelling at stand level

Selection table 

The selection table (S-TAB) comprises occurrence frequencies of injurious agents (ŠČ) with respect to forest eco-regions (OL) and tree species (DR). The frequency is given in per cents, and is based on the investigations of statistical data. The injurious agents are divided into the following categories:

• wind

• snow

• icing 

• bark miners and timber borers 

• defoliators and other insects 

• wood-destroying fungi

• air pollutants 

• drought 

• fire 

• illegal cutting 

Modifying the percentage of an injurious agent by the percentage of dead trees 

If in the previous 5-year period any dead trees occurred in the stand, the frequency of some injurious agents is modified by the multiplier:

The coefficient a is greater than 1 for bark miners (a=2), wood-destroying fungi (a=1.2) and fire (a=1.5). For other injurious agents, the coefficent is equal to 1.

Selection of an injurious agent from the selection table

An injurious agent is selected from the selection table (S-TAB) using random sampling with the probability equal to the frequency of an injurious agent (so called PPS sampling, i.e. Probability Proportional to Size). The selection is performed in several steps:

• injurious agents are arranged according to their frequencies, 

• cumulative frequencies are calculated, 

• a random number with uniform distribution from the interval <0;maximum cumulative frequency) is generated,

• the injurious agent with cumulative frequency equal to or the nearest greater than the generated number is selected. 

Classification table

The classification table (C-TAB) comprises the risk of occurrence of an injurious agent, average deadwood volume AVG(%V) and its standard deviation SD(%V). The values are dependent on a forest ecoregion (OL), tree species (DR), site category (STANOV), and a type of an injurious agent (ŠČ). They are valid for average stand conditions defined by 8 characteristics: vegetation zone (1), aspect (2), slope (3), age of tree species (4), stand density (5), proportion of tree species (6), absolute height site index of tree species (7), and height/diameter ratio (8). The classification table is derived from the survey of statistical data. 

Modifying the risk by real stand conditions 

The risk is modified by the model derived from the factor analysis. Eight characteristics of stand conditions are aggregated into factors. For each tree species and each injurious agent, a different number of factors (k=2..4) was derived by Vaculčiak (2007). The factors are linear combinations of the characteristics of stand conditions transformed into normal values (using their mean and standard deviation). The factors are calculated for average stand conditions taken from the classification table (F_i^mean) as well as for real stand conditions (F_i^real) using the actual characteristics of stand conditions: 

The modification of the risk is based on multipliers that are derived from the frequency of incidental cutting related to factor´s value. The frequencies of every factor were smoothed by regression analysis. The frequencies of average and real factors are calculated using Weibull regression function. Finally, the real risk is calculated by modifying the average risk derived from the classification table as follows:

Modifying the risk due to climate anomalies

The real risk is further modified as:

where s_i are real site variables and s_i^norm are standard site variables generared from the regionalised climate. The variables s_i^min and s_i^max are minimum and maximum values of a particular site variable for a particular tree species within its ecological amplitude. If a dummy variable a_i is equal to 0, the site variable has no effect, if a_i is equal to +1, the site variable has a positive influence, while the value -1 represents the negative influence of the site variable on the risk value. The values of dummy variables are given in Fabrika (2007).

   Activating the injurious agent 

The derived risk is compared with the random number from uniform distribution <0;100). If the risk is greater than the random number, the injurious agent selected in the previous step from the selection table is activated in the coming 5-year growth period.

Modifying the percentage of dead trees 

The percentage of dead trees AVG(%V) derived from the classification table is modified using the average multiplier. The volume of dead trees depends on the value of a particular factor. The relationships are smoothed by a polynomial regression function. The particular multiplier represents the ratio of the amount of dead trees in real stand conditions to their amount in average stand conditions:

Generating the amount of dead trees

The percentage of dead trees is generated from Gauss distribution defined by arithmetic mean AVG(%V) and standard deviation SD(%V).

Figure 2 Modelling disturbances at tree level 

a) Scattered (individual) propagation of dead trees 

The selection of the trees is either fully random or targeted depending on tree parameters (tree diameter, tree height, height/diameter ratio, crown width, crown length, coefficient of crown shape, tree vitality, bio-sociological tree status, competition pressure, tree quality, score of existence). These parameters are called selectors (x_i), while for every injurious agent a different group of selectors is defined (Table 1). Using fuzzy functions, the selectors are transformed into fuzzy values (fuzzy_i):   

 and they are connected with operators AND/OR. The operator AND selects the minimum fuzzy value from the set of the transformed selectors: 

AND = min[SET(fuzzy_i)]

while the operator OR selects the maximum value: 

OR = max[SET(fuzzy_i)]

Fuzzy functions are sensitive to individual parameters (through coefficients a, b), while their influence can be either positive or negative. The negative effect is derived from the positive one using the operator NOT:

NOT=1-fuzzy

The trees with the highest final fuzzy values are selected as dead trees. The number of trees for both propagation types (random and targeted) is specified by the required amount AVG(%V). The amount is evenly distributed among the individual years of the generated sub-interval within the next 5-year period. Targeted propagation is automatically activated for the following injurious agents: snow, icing, drought, wood-destroying fungi, and air pollutants. Fully random propagation is automatically activated in the case of illegal cutting. 

      Table 1 Default model setting for individual propagation of dead trees (+ positive influence, - negative influence)

b) Spotted propagation of dead trees

The spotted propagation is expanded from the local point. The trees are selected on the base of the generated parameters of individual spots (focuses). The number of spots is randomly generated from uniform distribution <1;5). The spotted propagation is defined by the propagation period (usually 3 years). The size of spots is specified by the required amount of dead trees AVG(%V), and continually it is linearly expanded during the individual years. All trees inside the spots are dead. The spotted propagation is automatically activated for bark miners and timber borers.

  c) Local propagation of dead trees

The local propagation represents sudden mortality of trees on a sub-plot. The trees are selected on the base of the generated parameters of the sub-plot. The area of the sub-plot is specified by the required amount of dead trees AVG(%V). A particular injurious agent is assigned a typical shape of the sub-plot (circle, strip, ellipse). Afterwards, the position of the sub-plot is randomly generated. All trees inside the sub-plot are considered dead. The year of the occurrence of the disturbance sub-plot is randomly generated from the next 5-year period. The local propagation is automatically activated for wind (in strips), defoliators (in ellipses), and fire (in ellipses).