# An Area-Based Matrix Model for Uneven-Aged Forests

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Material and Methods

#### 2.1. Data

- stem number per hectare;
- growing stock in m³ per hectare;
- dominant species;
- altitude above sea level in m; and
- ownership.

^{2}of basal area per hectare. Since NFI data was used, all necessary diameter measurements were available. Thus, the basal area of every sample tree is known and an estimate of the number of stems per hectare is easily calculated by dividing 4 m

^{2}by the basal area of the sample tree and then summing up over all trees in the sample plot:

_{i}is the basal area in m

^{2}of the ith tree in the plot.

#### 2.2. Model

#### 2.2.1. State-Space

#### 2.2.2. Transitions

_{ij}, the area matrix model is:

#### 2.2.3. Activities

#### 2.3. Estimation of Transition Probabilities

#### 2.4. Calamities

#### 2.5. Setting up the Simulations

**Table 1.**The settings for the activity probabilities in two simulations named “No harvest” and “Harvest” with different harvesting intensities for thinning alternatives “High thinning”(thHigh) and “Low thinning” (thLow), final felling (fF) and calamities (cc).

Simulation | Activity Probability | |||
---|---|---|---|---|

Run | thHigh (%) | thLow (%) | fF (%) | cc(Altitude Class 1/2) |

No harvest | 0 | 0 | 0 | 0.42/0.50 |

Harvest | 9 | 2 | 3 | 0.42/0.50 |

## 3. Results

#### 3.1. Simulations

^{−1}. It should be noted that the starting values of growing stock and stem numbers are true in the sense of being derived from the inventory data, also the initial harvest amounts are supported by the data through the actual activity probabilities. Thus the plausibility of the simulated outputs to some extent can be judged by the magnitude of their departure from the initial values as well as by their logical consistency.

**Figure 1.**Development of the Norway spruce forest on elevation class 1 (red line) and 2 (blue dash line) when no harvests are simulated. The calamity coefficients are 0.42 (elevation 1) and 0.5 (elevation 2).

#### 3.2. Comparisons with Historical Data and between Simulations

**Figure 2.**Development of the Norway spruce forest on elevation class 1 (red line) and 2 (blue dash line) when harvests are simulated. The calamity coefficients are 0.42 (elevation 1) and 0.5 (elevation 2). Probabilities for activities are: 9% (high thinning), 2% (low thinning), and 9% (final felling).

^{3}ha

^{−1}and total drain from 5.61 to 7.68 m

^{3}ha

^{−1}, which is higher than simulated. There is an annual drain of between 0.3 and 0.92 m

^{3}ha

^{−1}related to calamities. This could be compared to the simulated 2–3 m

^{3}ha

^{−1}of volume loss; the simulated figures are higher, however, it should be remembered that we used only a subset of Austrian forests and that not all volume lost in calamities is necessarily extracted. Also, differentiating between harvests due to calamities and ordinary harvests is very difficult, especially when the inventory takes place several years after the calamity. The higher drain from harvests and the lower drain from calamities partly even each other out.

**Figure 3.**Development of the Norway spruce forest on elevation class 1 using 1983 (green line) and 2001 (red line) initial states, respectively.

## 4. Discussion

**Figure 4.**The distributions of areas into the volume and stem number classes for elevation class 1 (Norway spruce) at the initial states “1983” (left) and “2001” (right) illustrated by fractions of the total area.

## 5. Conclusions

## Author Contributions

## Appendix

(Factor) | (Value) |
---|---|

Altitude: | (m) |

class 1 | <900 |

class 2 | 900–1200 |

class 3 | >1200 |

Dominant species: | |

class 1 | Norway Spruce |

class 2 | other conifers |

class 3 | broadleaved trees |

Stem number: | (lower bound, no. ha^{−1}) |

class 1 | 0 |

class 2 | 200 |

class 3 | 300 |

class 4 | 450 |

class 5 | 600 |

class 6 | 1000 |

class 7 | 1500 |

class 8 | 2250 |

class 9 | 3500 |

class 10 | 5000 |

Volume: | (lower bound, m^{3} ha^{−1}) |

class 1 | 0 |

class 2 | 63 |

class 3 | 92 |

class 4 | 134 |

class 5 | 188 |

class 6 | 259 |

class 7 | 347 |

class 8 | 452 |

class 9 | 575 |

class 10 | 710 |

class 11 | 852 |

class 12 | 994 |

class 13 | 1124 |

class 14 | 1232 |

Owner: | (owner group or ha) |

class 1 | <200 |

class 2 | 200–1000 |

class 3 | >1000 |

class 4 | Bundesforste AG ^{1} |

^{1}The federal forestry company.

## Supplementary Materials

## Conflicts of Interest

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**MDPI and ACS Style**

Sallnäs, O.; Berger, A.; Räty, M.; Trubins, R.
An Area-Based Matrix Model for Uneven-Aged Forests. *Forests* **2015**, *6*, 1500-1515.
https://doi.org/10.3390/f6051500

**AMA Style**

Sallnäs O, Berger A, Räty M, Trubins R.
An Area-Based Matrix Model for Uneven-Aged Forests. *Forests*. 2015; 6(5):1500-1515.
https://doi.org/10.3390/f6051500

**Chicago/Turabian Style**

Sallnäs, Ola, Ambros Berger, Minna Räty, and Renats Trubins.
2015. "An Area-Based Matrix Model for Uneven-Aged Forests" *Forests* 6, no. 5: 1500-1515.
https://doi.org/10.3390/f6051500