# A Markov Chain Model for Simulating Wood Supply from Any-Aged Forest Management Based on National Forest Inventory (NFI) Data

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. The Markov Chain Model

#### 2.1.1. General Description

#### 2.1.2. Application of the EFDM for the Simulations of Any-Aged Forest Management

#### 2.2. Steps of the Analysis

- Preparing the initial forest inventory data and pairwise observations for modeling the transitions (Section 2.3);
- Parameterizing the EFDM model with respect to the initial state space, transition and activity probabilities, and output coefficients (Section 2.4);
- Running the EFDM with scenario assumptions specified in Section 2.5.

#### 2.3. Forest Inventory Data

#### 2.3.1. An Overview of Parameterizing the EFDM with the Input Data

- The initial state was determined based on the full NFI11 (or NFI10 in one validation analysis). The data processing involved the initial estimation (Section 2.3.2) and formatting the data for the EFDM (Section 2.4.1).
- Two alternative data sets were used to derive the pairwise observations to model the transitions due to natural processes. The data sets were permanent NFI10-11 and temporary NFI11 plots, which slightly differ in terms of estimating the pairwise observations, as detailed in Section 2.3.3 and Section 2.4.2 with respect to estimation and EFDM-specific formatting, respectively. The two plot types were used to derive two distinct sets of transition probabilities, which were switched as the input data when running the analyses, but the permanent and temporary plots were never merged when modeling the transitions. Notably, the transitions estimated based on both the permanent and temporary plot data included only growth and not potential reductions due to calamities or natural disturbances [32], for example. Including these effects is discussed in Section 4 and Section 5.
- A simulator was implemented to derive the pairwise observations from situations before and after thinning treatments. The thinnings took place in the beginning of each simulation period and the growth of the thinned forests in that period was simulated applying the transition probabilities of forests not managed. The forests affected by final fellings were forced to transit to the beginning of the even-aged rotation. The data processing involved the initial estimation (Section 2.3.3) and formatting the data for the EFDM (Section 2.4.2).
- The activity probabilities, which give the proportion of different types of forests to be managed, were defined in two steps with two assumptions. First, the initial allocation of the harvests to the different types of forests was assumed to follow either the proportion of harvests realized during the most recent five-year period (“business-as-usual allocation”, A
_{BAU}) or the proportion of forests that should be harvested strictly according to the instructions of forest management [41] (“schoolbook allocation”, A_{SB}). Second, the final values for the activity probabilities of both the allocations were obtained by iteratively adjusting the initial probabilities to produce the harvesting levels aimed at in the future large-area scenarios. The data processing related to both the steps is explained in detail in Section 2.4.3 and Section 2.5. - Output coefficients were prepared to translate the areas in different volume, age, and stem number classes to timber assortment and total volumes. The related steps are explained in Section 2.4.4.

#### 2.3.2. Measurements and Estimation Principles of the NFI Data

#### 2.3.3. Pairwise Observations

- Deriving the growth observations is explained separately for the two alternative data sets, namely (a) permanent and (b) temporary NFI plots:
- (a)
- Permanent NFI sample plots, which were matched between data sets with certainty and had no treatments between the two subsequent inventories, were identified from the data. The positive differences in the total volumes between NFI10 and NFI11 were recorded as the pairwise data.
- (b)
- Pairwise observations were derived from temporary NFI11 plots using the increment estimates. In the NFI, every seventh tree (continuously over plot clusters) is measured for diameter and height increment, from which a volume growth percent was estimated following Tomppo et al. [43]. Due to the low number of observations, the percent was estimated at the level of sampling regions and the percent computed for a sampling region was applied for all plots in the region. This percent was used to estimate the difference of volumes between the two subsequent inventories for the plots without treatments, i.e., the pairwise observations were obtained as the difference between the observed volume and that deducted by the estimated growth.

- To determine transitions due to management by thinning treatments, a plot-level simulator was implemented to assess the proportion of the total and assortment volume to be removed in future thinning treatments. The NFI11 plots were used as the input data. The simulator determined the thinning removal following two types of instructions:
- (a)
- A thinning from below corresponding to traditionally applied instructions for forest management [41] was simulated according to thinning curves downloaded from the repository of the SIMO software (Simosol Py, Riihimäki, Finland, [44]). The curves predict the thinning limit and remaining basal area as the function of tree species, site type and dominant height (estimated as mean stand height +1.75 m [45]). The lower curves of two available thinning intensity levels were used and the harvesting removal was determined as the difference between the thinning need and remaining basal area.
- (b)
- A thinning from above was simulated by predicting the remaining basal area using the plot-level forest attributes as predictors of Equation (2) of Pukkala et al. [46]. The thinning intensity is determined in terms of interest rate used as a predictor of the aforementioned equation. We used the curves corresponding to an interest rate of 3% and determined the harvesting removal as the difference between the initial and remaining basal area.

^{2}/ha and mean height >10 m and also the remaining basal area was forced to be at least 10 m

^{2}/ha to simulate treatments that are profitable and legal according to the instructions of forest management [41].

#### 2.4. Parameterization of the Markov Model

#### 2.4.1. State Space

- Known land-use restrictions: FAWS, FRAWS, FNAWS
- Forest ownership: private, public + other (cf. [42]).
- Site fertility: altogether, five categories corresponding to four taxation classes traditionally used in Finland + fifth class including all poorly productive forest land.
- Dominant species: pine, spruce, deciduous trees.

- In the input data for the simulations, every plot representing even-aged forest had a value of 0 as the stem number class and the values of age and volume ranged according Appendix A. Every plot representing uneven-aged forest had a value of 0 as the age class and the values of stem number and volume ranged according to Appendix A.
- In the input data for modeling the transition probabilities, the set of plots with pairwise observations was duplicated. Thus, every plot was included twice with either the age or stem number class set as 0, i.e., the data to model the transitions of both Age-Vol or N-Vol were always the same except for the variable combination.

#### 2.4.2. Transition Probabilities

#### 2.4.3. Activity Probabilities

_{BAU}) or schoolbook (A

_{SB}) allocation based on realized or proposed harvests, respectively, in the data classified according to the factor combinations. More specifically, A

_{BAU}was obtained as the proportion of areas with no management, thinning or final felling observed in the pairwise data from the permanent plots between NFI10 and NFI11. A

_{SB}was the proportion of areas, in which a need for thinning or final felling within the next five years according to the forestry guidelines [43] was recorded in the field inventory. In both cases, there was none or a very limited number of observations on the activities for a number of static and dynamic factor combinations, which would have resulted in unrealistic activity probabilities if computed as factor-specific based on the data. For this reason, both the A

_{BAU}and A

_{SB}were computed by pooling the activities for the Age-Vol or N-Vol classes over the static factors, i.e., the probability of a certain activity was the same for the given dynamic factors in all static factor combinations.

#### 2.4.4. Output Coefficients

^{3}/ha) of each volume class. The timber assortment drain was determined by computing the average proportion of log- and pulpwood relative to the total volume in each factor combination and multiplying it with the total volume. The coefficient values for thinnings were based on the simulations (Section 2.3.3) and determined according to the number of volume class to be applicable in simulations based on the both Age-Vol and N-Vol matrices. The coefficients for final fellings were based on all NFI11 plots and determined according to numbers of both age and volume classes.

#### 2.5. Harvesting Scenarios Used in the Simulations of Any-Aged Forest Management

^{3}/a (BAU) and target level of 80 mill m

^{3}/a in 2025 (NFP). However, as the total growing stock analyzed here represented approximately 95% of the total growing stock in Finland [48], we applied this percent also to the harvesting targets to obtain the final levels used in the analyses (61.75 mill m

^{3}/a and 76 mill m

^{3}/a in BAU and NFP, respectively). In NDV, the harvesting target was adjusted to a level, which did not decrease the volume of the growing stock. The amount was obtained by iterating the activity probabilities (Section 2.4.3) either in the beginning of the simulations or between the individual simulation steps.

_{BAU}) or proposed (A

_{SB}) treatments (Section 2.4.3), i.e., thinnings from below were applied due to the seldom historical use of other types of thinnings in FAWS. In areas marked as FRAWS, thinnings of young stands were carried out as thinnings from below similar to FAWS, but final fellings were replaced by thinning from above. This combination allowed managing FRAWS for wood production, but always retained a minimum forest cover as specified in the legislation. In areas marked as FNAWS, no management actions were allowed, i.e., the probabilities of all other activities were set to 0.

## 3. Results

#### 3.1. Effects of the Parameterization of the EFDM on the Projections

#### 3.2. Results of the Scenario Analyses

^{3}/a, and an end volume of 2451 mill m

^{3}(+9.6% from the start of the simulation), on average, based on the current land-use restrictions and all simulations.

^{3}/a and the variation shown in Figure 6 and Figure 7 (right column). According to the goal, the growing stock did not decline. Land use did not markedly differ from that of the BAU scenario (compare the upper row of Figure 6 and Figure 7). Figure 6 and Figure 7 also illustrate the differences of applying A

_{BAU}vs. A

_{SB}as activity probabilities when implementing the harvests. The main difference was that in the allocation based on A

_{SB}, more harvests were focused on older and heavily stocked stands. As a result, final fellings yielded more drain and the harvesting targets could be achieved by managing less area.

## 4. Discussion

#### 4.1. Parameterization of the Markov Model for the Simulations of Any-Aged Forest Management

#### 4.2. On the Use of the EFDM Approach for Projecting Future Wood Supply

_{BAU}or A

_{SB}in Figure 6 and Figure 7). The probabilities used in our analyses were factor- combination-specific ratios of areas of each activity to the total area in the initial state and determined in the beginning of the simulations. Although these probabilities were adjusted to yield the desired harvesting level during the simulations (Section 2.4.3), the applied iteration procedure did not change the relative proportions of the activities. Especially with demanding harvest goals, the forest areas in classes with high initial harvesting probabilities reduced considerably faster than the in- growth to these classes. Although the harvesting goals were fulfilled by increasing other classes to be harvested, a considerably different harvest allocation might have been obtained by considering where the harvests are most feasible with respect to the forest structure distribution in each future simulation step. As can be seen by comparing the proportions of timber assortments harvested using A

_{BAU}or A

_{SB}(Figure 6 and Figure 7), the harvesting objectives were more efficiently achieved using A

_{SB}with more final fellings. Even higher proportion of final fellings (or thinnings from above) could have been proposed to reach demanding harvest targets, but the present iteration did not increase their proportion very much due to the lower proportion in the beginning. It is overall intuitive that future management practices and their proportions should be determined according to the predicted state of the forest. Although no such functionality currently exists in the EFDM, considerable future opportunities are seen related to such implementations (see Section 4.3).

#### 4.3. Recommendations for Further Research

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

Volume Classes ^{1} | Age Classes ^{1} |
---|---|

1. (∞, 0.0000) | 1. (∞, 0) |

2. (0.0000, 10.5884) | 2. (0, 5) |

3. (10.5884, 29.8618) | 3. (5, 10) |

4. (29.8618, 51.3846) | 4. (10, 15) |

5. (51.3846, 73.6844) | 5. (15, 20) |

6. (73.6844, 96.5440) | 6. (20, 25) |

7. (96.5440, 122.8744) | 7. (25, 30) |

8. (122.8744, 153.2864) | 8. (30, 35) |

9. (153.2864, 191.3348) | 9. (35, 40) |

10. (191.3348, 248.0352) | 10. (40, 45) |

11. (248.0352, 303.3532) | 11. (45, 50) |

12. (303.3532, ∞) | 12. (50, 55) |

13. (55, 60) | |

Stem number classes ^{1} | 14. (60, 65) |

1. (∞, 0.0000) | 15. (65, 70) |

2. (0.0000, 69.8725) | 16. (70, 75) |

3. (69.8725, 330.6332) | 17. (75, 80) |

4. (330.6332, 509.0106) | 18. (80, 85) |

5. (509.0106, 676.9061) | 19. (85, 90) |

6. (676.9061, 875.1651) | 20. (90, 95) |

7. (875.1651, 1108.6395) | 21. (95, 100) |

8. (1108.6395, 1384.8279) | 22. (100, 105) |

9. (1384.8279, 1754.0609) | 23. (105, 110) |

10. (1754.0609, 2398.5651) | 24. (110, 115) |

11. (2398.5651, 2975.8412) | 25. (115, 120) |

12. (2975.8412, ∞) | 26. (120, ∞) |

^{1}Units in m

^{3}/ha for volume, 1/ha for stem number, and years for age.

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**Figure 1.**Transitions from the initial (Class#, t0) to the subsequent class (Class#, t1) in volume classes derived from permanent (

**left**) and temporary plots (

**middle**) and stem number classes derived from permanent plots (

**right**). The thick horizontal lines depict the median, the bottom and top of the boxes the interquartile range between the 25th and 75th percentiles, the whiskers the lowest data within 1.5 times the interquartile range, and dots the data not included in categories above.

**Figure 2.**A comparison of Age-Vol (vertical bars) and N-Vol (dashed lines) matrices derived from the permanent plots in the simulations of the natural process. The subplots illustrate the development of the volume class indicated in the plot along the number of simulation steps of 5 years given in the x-axis (step 0 indicates the initial state). Note that the scale of y-axis of the volume class 12 differs from the others.

**Figure 3.**A comparison of Age-Vol (vertical bars) and N-Vol (dashed lines) matrices derived from the temporary plots in the simulations of the natural process. The subplots illustrate the development of the volume class indicated in the plot along the number of simulation steps of five years given in the x-axis (step 0 indicates the initial state). Note that the scale of y-axis of the volume class 12 differs from the others.

**Figure 4.**A comparison of the volume (

**left**) and stem number (

**right**) of the simulated harvests.

**Top row**: Values before and after thinning from below, black and green dots corresponding to simulated values and pairwise observations of the permanent plots, respectively.

**Middle row**: The correspondence of the simulated removal with the difference computed from the pairwise observations.

**Bottom row**: Values before and after simulated thinning from above.

**Figure 5.**The development of the volume class distribution in one simulation step. The black solid lines indicate the class distribution observed in the NFI10 and NFI11 data. The red and green broken lines represent predictions based on forecasting from NFI10 data or backcasting from the NFI11 data, respectively, with the transition probabilities derived from the permanent (

**left**) and temporary (

**right**) plots for pairwise observations.

**Figure 6.**Land use, growing stock, and timber assortment drain given by bars corresponding to the eight simulation steps, when the activity probabilities A

_{BAU}(bars) or A

_{SB}(broken lines) were iterated between the individual simulation steps to obtain the harvesting goals BAU (left column), NFP (middle column), and NDV (right column). From top to bottom, the colors within the bars represent the proportion of (

**top row**) FAWS managed for final felling, FAWS thinned, FAWS not treated, FRAWS thinned, FRAWS not treated, and FNAWS; (

**middle row**) standing volume in FAWS, FRAWS, and FNAWS; (

**bottom row**) pulpwood from thinnings, sawnwood from thinnings, pulpwood from final fellings, and sawnwood from final fellings, respectively.

**Figure 7.**Land use, growing stock and timber assortment drain, when the activity probabilities were set in the beginning of the simulation. Refer to the caption of Figure 6 for the interpretation of the image.

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Vauhkonen, J.; Packalen, T.
A Markov Chain Model for Simulating Wood Supply from Any-Aged Forest Management Based on National Forest Inventory (NFI) Data. *Forests* **2017**, *8*, 307.
https://doi.org/10.3390/f8090307

**AMA Style**

Vauhkonen J, Packalen T.
A Markov Chain Model for Simulating Wood Supply from Any-Aged Forest Management Based on National Forest Inventory (NFI) Data. *Forests*. 2017; 8(9):307.
https://doi.org/10.3390/f8090307

**Chicago/Turabian Style**

Vauhkonen, Jari, and Tuula Packalen.
2017. "A Markov Chain Model for Simulating Wood Supply from Any-Aged Forest Management Based on National Forest Inventory (NFI) Data" *Forests* 8, no. 9: 307.
https://doi.org/10.3390/f8090307