Abstract In this study, any-aged forestry (AAF) refers to forest management in which no explicit choice is made between even- and uneven-aged management, or between rotation forest management and continuous cover forestry. Optimal AAF is more profitable than optimal even- or uneven-aged management because AAF has fewer constraints. This study developed management instructions for optimal AAF. The instructions consist of four models, the first indicating the probability that an immediate cutting in the stand is the optimal decision. In case of cutting, the second model gives the probability that partial cutting (thinning) is optimal. If thinning is selected, the remaining two models indicate how many trees should be removed from different diameter classes. The models for optimal management were based on optimized cutting schedules of 2095 stands, located in different parts of Finland. The use of the model requires that discount rate is specified, and site fertility and temperature sum of the stand are known. The required growing stock characteristics are stand basal area, mean tree diameter and the basal area of pulpwood-sized trees (dbh 8–18 cm). High stand basal area and large mean tree size increase the probability that cutting is the optimal decision. High basal area of pulpwood-sized trees increases the probability that partial cutting is optimal. Thinning from above is the optimal type of cutting in most cases. The models were tested by comparing the model-driven stand management schedules with stand-level optimizations. Schedules based on the models resulted in equally good net present values as schedules based on optimizations. When the discount rate was 3 per cent or more, the models led to similar profitability as stand-level optimization. Introduction Silvicultural systems used in ‘high forests’ (forests of seedling origin) may be divided into even-aged and uneven-aged management, or rotation forest management (RFM) and continuous cover management (Schütz et al., 2012). Classifications that are more detailed have also been presented, for instance a division into clear-cutting, shelterwood and selection systems (Matthews, 1989). Continuous cover forest management (CCF) is a wider concept than uneven-aged management since stand structure does not need to be continuously uneven-aged in CCF. Uneven-aged management is close to the German Plenterwald concept while CCF resembles the German Dauerwald (Möller, 1922). The main feature of CCF is that forest cover is maintained continuously. The principles of CCF are to avoid clear-fellings other than small gaps (<0.25 ha), use natural regeneration, and harvest mainly financially mature, senescent and non-healthy trees (Möller, 1922; Schütz et al., 2012). Another approach to forest management is to allow combinations of silvicultural systems. This type of management has been referred to as any-aged forest management (Haight and Monserud, 1990) and freestyle silviculture (Boncina, 2011). The current study uses the term any-aged forestry (AAF) for management where all silvicultural options are available at any stand state. Management may include prolonged periods of uneven-type of silviculture but if the capacity of the stand to regenerate naturally decreases, regeneration methods of conventional even-aged silviculture, such as natural regeneration via seed trees or clear-felling and planting, can also be used. Selecting one silvicultural system (e.g. even-aged management or uneven-aged management) can be regarded as a constraint, which never increases profitability, as compared with optimal management without any constraints (Pukkala et al., 2014a). Logically, earlier studies indicate that CCF is often more profitable than uneven-aged management (Pukkala, 2015) because CCF is more flexible. When NPV was maximized so that all silvicultural systems were allowed, Haight and Monserud (1990) found that uneven-aged management was the most profitable management system for a range of stand types in the western United States, but understocked stands of some stand types should be clear-cut and planted with white pine (Pinus monticola). However, a single clear-cutting does not mean that RFM would be continuously optimal for these stands. Tahvonen (2011) showed that it is sometimes optimal to switch from uneven-aged to even-aged management, and Tahvonen (2009) concluded that the optimal management system of a stand may depend on its initial diameter distribution. Pukkala et al. (2014b) showed that the optimal management of even-aged plantations may consist of a long period of high thinnings (thinning from above) during which the stand is converted from even-aged to uneven-aged structure utilizing the gradual appearance of advance regeneration. CCF has usually been found to be more efficient than RFM when forest management objectives require the simultaneous delivery of several ecosystems services (Peura et al., 2017; Pukkala, 2017a). It may be concluded that AAF cannot be worse than CCF if AAF is implemented in an optimal way because none of the practices of CCF is ruled out in AAF. Pukkala (2014) found that when the net present value and carbon balance of forestry were simultaneously maximized, AAF was more efficient than CCF and RFM, and CCF was more efficient than RFM. AAF allows diversified management, which is an advantage for maintaining a large number of habitat types and ecosystem services (Knoke et al., 2008). Management diversification also decreases economic and biological risks (Reeves and Haight, 2000; Knoke and Wurm, 2006). Uncertainty of natural regeneration is one reason why it is not wise to stick to CCF although it may seem more efficient than RFM, especially in multifunctional forestry (Peura et al., 2017; Pukkala, 2017a). If regeneration and ingrowth are plentiful, it is usually optimal to continue removing large trees from the stand to release growth space for smaller trees and enhance regeneration (Pukkala et al., 2014b). However, if regeneration ceases for some reason, it may be optimal to conduct specific ‘regenerative’ cuttings, such as shelterwood cutting, seed tree cutting, or clear-felling and planting. Natural regeneration depends on weather conditions of several years in complicated ways (Manso et al., 2013). Weather affects flowering, seed maturation, dispersal and germination, as well as the dynamics of seed predators and pathogens. Since it is impossible to predict the weather conditions in the distant future, it is also impossible to tell for how long CCF management or repeated high thinning will be better than clear-felling and planting in a certain stand. Other reasons that may alter the ranking of silvicultural systems include changes in financial markets, timber assortments and their prices, and silvicultural costs (Tahvonen, 2009, 2011). These uncertainties call for adaptive forest management (Lohmander, 2007; Pukkala and Kellomäki, 2012) in which management can be adapted to the prevailing conditions. Management instructions applicable to adaptive forestry should indicate the optimal management of certain stand, given the current stand state, timber assortments and their prices, etc. The instructions should produce information on whether the stand should be cut or would it be better to let it grow further. If cutting is the optimal decision, the instructions should indicate how many trees should be removed from the stand and from which diameter classes. One possible way to advise a forest landowner in the management of a specific stand is to optimize its management into the distant future and derive a recommendation from the optimization results. As explained above and shown in previous studies (Haight and Monserud, 1990; Tahvonen, 2011; Pukkala et al., 2014a), these optimizations should not be restricted to one silvicultural system, and they should allow changes from one system to another. Optimization may be conducted in several ways, for instance with or without considering the stochasticity of timber prices, regeneration, tree growth and survival. If deterministic optimization is used, the reliability of the recommendation may be inspected by sensitivity analyses. Optimality of cutting is related to the relative value increment of the stand (Davis and Johnson, 1987). A high relative value increment means that it is often profitable to let the stand grow on, whereas a low relative value increment is a sign that the stand is too dense or the trees are too large for maintaining a sufficient rate of value increment. Moreover, relative value increments of individual trees affect the optimal type of cutting. A tree is mature for cutting once its relative value increment falls below a guiding rate of interest, and the tree no longer has significant value increases in the future (Duerr et al., 1956; Davis and Johnson, 1987; Knoke, 2012). Removing financially mature trees from the stand often improves the relative value increment of the residual stand. The rate of value increment of a stand can be improved by conducting cuttings in such a way that the value of capital invested in timber production decreases considerably without an accompanying deterioration in the value increment of the stand. This effect is reached by removing those valuable trees from the stand whose increment has declined due to large tree size (Pukkala et al., 2016). Computational tools for analysing different options of AAF have been developed in earlier forest research (Haight and Monserud, 1990; Pukkala et al., 2014a; Vauhkonen and Packalen, 2017). However, the use of these tools may be too complicated for forestry practice, which often prefers straightforward, clear guidelines rather than computer tools which require simulation and optimization. This study developed instructions for any-aged management of Finnish forests when the aim is to maximize profitability of timber production. The instructions produce advice in optimal forest management without the need to use simulation and optimization. They consist of four models, the first of which indicates whether a certain stand is mature for cutting. If cutting is the optimal decision, the second model is consulted to see whether the cutting should be a thinning treatment or final felling. In case of thinning, the remaining two models indicate how many trees should be removed and from what different diameter classes. Materials and methods Choice of variables Variables that are used to derive instructions for optimal management should correlate with the financial maturity of the stand and they should be easily measurable in the forest. Preferably, they should be variables that are measured in routine forest inventories and stored in management databases. Examples of these variables are mean tree diameter, stand basal area, proportions of different tree species, site fertility and temperature sum of the region. Temperature sum is the sum of the mean temperature minus 5°C of those days of the growing season when the mean temperature is >5°C. The major value thresholds of trees growing in Finnish forests are illustrated in Figure 1, which shows the value of 1m3 if a tree is sold with stumpage prices and its stem is partitioned into timber assortments in an optimal way (Pukkala, 2017b). Trees approaching a sufficient size for the first pulpwood log (~8 cm in dbh) have a very high relative value increment in the coming years, as do trees close to the minimum size of one saw log (18 cm in conifers and 20 cm in birch). Since these tree sizes produce the highest return per invested capital in the coming years, it seems evident that they should be left to grow over the value threshold. After passing the major value thresholds a tree becomes gradually more mature for cutting. The exact time point of financial maturity depends on the rate of interest, affecting the opportunity cost of keeping the tree in the stand. In addition, the optimality of removing the tree depends on its growth rate, and the effect of removing the tree on the growth of other trees (Davis and Johnson, 1987; Pukkala et al., 2016). If artificial regeneration is obligatory after clear-felling, increasing regeneration costs postpone the optimal time of clear-felling and increase the likelihood that partial cutting is optimal. Figure 1 View largeDownload slide Value of 1 m3 in pine, spruce and birch stems (growing on mesic site in Central Finland) when the stems are crosscut in an optimal way. Value increases occur when the tree reaches the minimum size for a larger and more valuable timber assortment. Figure 1 View largeDownload slide Value of 1 m3 in pine, spruce and birch stems (growing on mesic site in Central Finland) when the stems are crosscut in an optimal way. Value increases occur when the tree reaches the minimum size for a larger and more valuable timber assortment. Figure 1 also gives insight about the type of cutting. If the stand has several trees smaller than the threshold of a major value increase, the optimal cutting is such a thinning from above, which removes trees that have passed the value threshold and leaves healthy and good-quality trees smaller than the threshold to continue growing. If all trees of the stand have passed all major value thresholds, and the relative value increment of the stand is smaller than the required rate of return, the optimal cutting may be final felling if thinning does not rise the value increment of remaining trees to a sufficient level. Figure 1 suggests that the amount of pulpwood-sized and non-commercial small trees might be good predictors of both financial maturity and type of cutting. A high share of pulpwood-sized trees implies high relative value increment. If the stand has also sawlog-sized trees, it is evident that the most profitable type of cutting is thinning from above. The amount of trees smaller than pulpwood may also be a good indicator of the optimal type of cutting. For example, if a stand consists of log-sized trees, on one hand, and trees smaller than pulpwood size, the optimal treatment would most probably be to remove the larger trees and leave the small trees to continue growing. Based on this reasoning, the potential predictors of optimal stand management included site variables, mean tree size, basal areas of different tree species and the basal areas and numbers of pulpwood-sized (dbh 8–18 cm) and small (dbh < 8 cm) trees. All these variables except the amounts of 8–18 cm and <8-cm trees are always known and stored in management databases in Finland. The amounts of 8–18 cm and <8-cm trees can usually be predicted by the computer systems used in forest management, and they can be easily assessed in the field. Data generation The stands of five large forest holdings from different parts of Finland (2095 stands in total) were used as the dataset for developing instructions for optimal any-aged stand management (Table 1). The optimization method developed by Pukkala et al. (2014a) for any-aged management was used to optimize the next three cuttings in each stand. The optimization system does not need any choice between even- and uneven-aged management. If the tested cutting schedule reduces the stand basal area below the lowest allowed value (‘legal limit’, Anonym, 2013), the software checks whether advance regeneration is sufficient (fulfills another legal limit, Anonym, 2013). If this is not the case, artificial regeneration is simulated, consisting of cleaning the regeneration site, site preparation, planting or sowing, and a tending treatment of the young stand. In this study, the total cost of these treatments was 1300–2050 € ha−1, depending on site fertility. The legal limit for post-cutting stand basal area was taken from the regulations for uneven-aged management, and it ranged from 7 to 9 m2 ha−1, depending on site quality. Table 1 Description of the forests used as data source in the study. The total number of stands in the five forest holdings was 2095. The three next cuttings in each stand were optimized. Forest 1 Forest 2 Forest 3 Forest 4 Forest 5 Number of stands 593 275 415 255 557 Temperature sum, d.d. > 5°C 1300 1200 1100 950 800 Growing sites (%) Herb-rich or better 30.4 41.6 6.3 4.9 4.8 Mesic 45.7 38.2 42.8 76.9 52.6 Sub-xeric 14.8 16.4 33.2 18.1 36.5 Poorer than sub-xeric 9.1 3.8 17.8 0 6.0 Tree species (volume, m3 ha−1) Pine 43.0 48.6 58.6 35.3 102.1 Spruce 34.2 92.0 15.8 13.7 16.8 Broadleaf 38.8 47.4 24.3 11.1 14.5 Assortments (volume, m3 ha−1) Saw log 57.9 101.9 24.4 8.6 61.7 Small log 4.0 6.5 8.9 5.9 4.2 Pulpwood 54.2 79.9 66.3 45.7 67.6 Stage of stand development (%) Seedling and sapling stand 15.9 2.1 14.2 14.5 3.2 Seed and shelter tree stand 5.3 8.0 0 1.5 0.4 Pulpwood-sized forest 23.6 11.9 23.2 42.6 15.2 Sawlog sized thinning forest 33.4 41.5 56.5 30.1 30.6 Mature forest 21.8 36.5 6.0 11.4 50.5 Forest 1 Forest 2 Forest 3 Forest 4 Forest 5 Number of stands 593 275 415 255 557 Temperature sum, d.d. > 5°C 1300 1200 1100 950 800 Growing sites (%) Herb-rich or better 30.4 41.6 6.3 4.9 4.8 Mesic 45.7 38.2 42.8 76.9 52.6 Sub-xeric 14.8 16.4 33.2 18.1 36.5 Poorer than sub-xeric 9.1 3.8 17.8 0 6.0 Tree species (volume, m3 ha−1) Pine 43.0 48.6 58.6 35.3 102.1 Spruce 34.2 92.0 15.8 13.7 16.8 Broadleaf 38.8 47.4 24.3 11.1 14.5 Assortments (volume, m3 ha−1) Saw log 57.9 101.9 24.4 8.6 61.7 Small log 4.0 6.5 8.9 5.9 4.2 Pulpwood 54.2 79.9 66.3 45.7 67.6 Stage of stand development (%) Seedling and sapling stand 15.9 2.1 14.2 14.5 3.2 Seed and shelter tree stand 5.3 8.0 0 1.5 0.4 Pulpwood-sized forest 23.6 11.9 23.2 42.6 15.2 Sawlog sized thinning forest 33.4 41.5 56.5 30.1 30.6 Mature forest 21.8 36.5 6.0 11.4 50.5 Table 1 Description of the forests used as data source in the study. The total number of stands in the five forest holdings was 2095. The three next cuttings in each stand were optimized. Forest 1 Forest 2 Forest 3 Forest 4 Forest 5 Number of stands 593 275 415 255 557 Temperature sum, d.d. > 5°C 1300 1200 1100 950 800 Growing sites (%) Herb-rich or better 30.4 41.6 6.3 4.9 4.8 Mesic 45.7 38.2 42.8 76.9 52.6 Sub-xeric 14.8 16.4 33.2 18.1 36.5 Poorer than sub-xeric 9.1 3.8 17.8 0 6.0 Tree species (volume, m3 ha−1) Pine 43.0 48.6 58.6 35.3 102.1 Spruce 34.2 92.0 15.8 13.7 16.8 Broadleaf 38.8 47.4 24.3 11.1 14.5 Assortments (volume, m3 ha−1) Saw log 57.9 101.9 24.4 8.6 61.7 Small log 4.0 6.5 8.9 5.9 4.2 Pulpwood 54.2 79.9 66.3 45.7 67.6 Stage of stand development (%) Seedling and sapling stand 15.9 2.1 14.2 14.5 3.2 Seed and shelter tree stand 5.3 8.0 0 1.5 0.4 Pulpwood-sized forest 23.6 11.9 23.2 42.6 15.2 Sawlog sized thinning forest 33.4 41.5 56.5 30.1 30.6 Mature forest 21.8 36.5 6.0 11.4 50.5 Forest 1 Forest 2 Forest 3 Forest 4 Forest 5 Number of stands 593 275 415 255 557 Temperature sum, d.d. > 5°C 1300 1200 1100 950 800 Growing sites (%) Herb-rich or better 30.4 41.6 6.3 4.9 4.8 Mesic 45.7 38.2 42.8 76.9 52.6 Sub-xeric 14.8 16.4 33.2 18.1 36.5 Poorer than sub-xeric 9.1 3.8 17.8 0 6.0 Tree species (volume, m3 ha−1) Pine 43.0 48.6 58.6 35.3 102.1 Spruce 34.2 92.0 15.8 13.7 16.8 Broadleaf 38.8 47.4 24.3 11.1 14.5 Assortments (volume, m3 ha−1) Saw log 57.9 101.9 24.4 8.6 61.7 Small log 4.0 6.5 8.9 5.9 4.2 Pulpwood 54.2 79.9 66.3 45.7 67.6 Stage of stand development (%) Seedling and sapling stand 15.9 2.1 14.2 14.5 3.2 Seed and shelter tree stand 5.3 8.0 0 1.5 0.4 Pulpwood-sized forest 23.6 11.9 23.2 42.6 15.2 Sawlog sized thinning forest 33.4 41.5 56.5 30.1 30.6 Mature forest 21.8 36.5 6.0 11.4 50.5 Based on the post-cutting basal areas and post-cutting treatments, the optimized cuttings were divided into three categories: Thinning: remaining basal area was higher than the minimum allowed value (legal limit). Removal of upper canopy (equal to releasing advance regeneration): remaining basal area was below the legal limit but there was sufficient advance regeneration, making artificial regeneration unnecessary. Final felling: remaining basal area was below the legal limit and advance regeneration was insufficient, leading to obligatory artificial regeneration. When a cutting was simulated, it was specified with a thinning intensity curve showing the proportions of trees removed from different diameter classes. The following logistic curve was used to express thinning intensity as a function of dbh (Jin et al., 2017): TI=11+exp(a1(a2−d)) (1) where TI is the thinning intensity (proportion of removed treed) at diameter, d, cm, and a1 and a2 are the parameters optimized for each cutting. If a2 is negative, small diameter classes are harvested more strongly than large ones, corresponding to thinning from below, and a1 > 0 corresponds to thinning from above. Parameter a2 gives the dbh at which thinning intensity is 0.5. During the optimization process, stand development was simulated in 5-year steps, using the individual-tree diameter increment, survival and height models of Pukkala et al. (2009, 2013) and the ingrowth models of Pukkala et al. (2013). The variables optimized for each cutting were time from start or previous cutting, and parameters a1 and a2 of the thinning intensity curve (three optimized variables per cutting). When a cutting was simulated, the removed trees were partitioned into timber assortments (Table 2) using the taper models of Laasasenaho (1982). A part of sawlog volume was moved to pulpwood due to quality reasons using the models of Mehtätalo (2002) and the results of Malinen et al. (2007). The income from cutting was calculated by subtracting harvesting costs from the roadside values (Table 2) of different assortments. Harvesting costs were calculated with the functions of Rummukainen et al. (1995). The cost of harvesting 1 m3 depended on harvested volume per hectare and mean volume of harvested trees, among other things. The type of cutting also affected harvesting costs so that final felling was cheaper than thinning if the mean size and total volume of harvested trees were the same. Table 2 Timber assortments and their stumpage and roadside prices used in the calculations. Assortment Minimum top diameter (cm) Minimum log length (m) Stumpage price (€ m−3) Roadside price (€ m−3) Pine saw log 15 4.3 50 57 Pine small log 13 3.4 22 32 Pine pulpwood log 8 2.0 15 30 Spruce saw log 16 4.3 50 57 Spruce small log 13 3.4 22 32 Spruce pulpwood log 9 2.0 15 30 Birch saw/veneer log 17 3.4 45 45 Birch pulpwood log 8 2.0 15 30 Aspen saw log 17 4.3 50 40 Aspen pulpwood log 8 2.0 10 20 Alder pulpwood log 8 2.0 10 20 Assortment Minimum top diameter (cm) Minimum log length (m) Stumpage price (€ m−3) Roadside price (€ m−3) Pine saw log 15 4.3 50 57 Pine small log 13 3.4 22 32 Pine pulpwood log 8 2.0 15 30 Spruce saw log 16 4.3 50 57 Spruce small log 13 3.4 22 32 Spruce pulpwood log 9 2.0 15 30 Birch saw/veneer log 17 3.4 45 45 Birch pulpwood log 8 2.0 15 30 Aspen saw log 17 4.3 50 40 Aspen pulpwood log 8 2.0 10 20 Alder pulpwood log 8 2.0 10 20 Table 2 Timber assortments and their stumpage and roadside prices used in the calculations. Assortment Minimum top diameter (cm) Minimum log length (m) Stumpage price (€ m−3) Roadside price (€ m−3) Pine saw log 15 4.3 50 57 Pine small log 13 3.4 22 32 Pine pulpwood log 8 2.0 15 30 Spruce saw log 16 4.3 50 57 Spruce small log 13 3.4 22 32 Spruce pulpwood log 9 2.0 15 30 Birch saw/veneer log 17 3.4 45 45 Birch pulpwood log 8 2.0 15 30 Aspen saw log 17 4.3 50 40 Aspen pulpwood log 8 2.0 10 20 Alder pulpwood log 8 2.0 10 20 Assortment Minimum top diameter (cm) Minimum log length (m) Stumpage price (€ m−3) Roadside price (€ m−3) Pine saw log 15 4.3 50 57 Pine small log 13 3.4 22 32 Pine pulpwood log 8 2.0 15 30 Spruce saw log 16 4.3 50 57 Spruce small log 13 3.4 22 32 Spruce pulpwood log 9 2.0 15 30 Birch saw/veneer log 17 3.4 45 45 Birch pulpwood log 8 2.0 15 30 Aspen saw log 17 4.3 50 40 Aspen pulpwood log 8 2.0 10 20 Alder pulpwood log 8 2.0 10 20 The net present value of the final growing stock (net present value of management actions performed after the third optimized cutting) was predicted with models as explained in detail in Pukkala (2015). These predictions assume that the stand would be managed in an economically optimal way also in the future. Since the value of final growing stock was discounted from distant future, its effect on the total NPV of the cutting schedule was low, especially with high discount rates. The stand variables at the beginning of each 5-year step were saved for modelling purposes, together with information on cuttings (no cutting, thinning, releasing advance regeneration, final felling). In case of thinning, the values of thinning intensity parameters (a1 and a2) were also saved. The management schedule of each stand was optimized with five different discount rates, 0.1, 1, 3, 5 and 7 per cent. This made it possible to use the interest rate as one predictor of optimal management. The optimized management schedules included 101 709 stand states. Cutting was simulated in 26 111 stand states leading to an average cutting interval of 19.5 years. Of all cuttings, 78 per cent were thinnings, 17 per cent were clear-fellings followed by planting or sowing, and 6 per cent were cuttings that corresponded to the removal of large trees above existing regeneration. Clear-fellings were common at low discount rates (Figure 2), most probably because the establishment of tree plantations was profitable at low discount rates. Discount rates of 3 per cent or more led to reduced use of clear-felling and artificial regeneration. Figure 2 View largeDownload slide Proportions of cutting types in the optimizations conducted with different discount rates (0.1%, 1%, 3%, 5% and 7%). Figure 2 View largeDownload slide Proportions of cutting types in the optimizations conducted with different discount rates (0.1%, 1%, 3%, 5% and 7%). The algorithm of Hooke and Jeeves (1961) was employed in optimizations. It is a heuristic search method, which has been used widely in stand management optimization. It finds good solutions but the solution found may sometimes be a local optimum. The direct search algorithm of Hooke and Jeeves starts from a set of initial values of optimized variables. The initial values were 15 years for all cutting intervals, 0.3 for all a1, and 20 cm for all a2 (these three variables were optimized separately for every cutting). The values were changed in alternating exploratory and pattern search modes using a certain step size. The step size was gradually reduced when the search proceeded, and the search was terminated when the steps size became smaller than a predefined stopping criterion. In this study, the initial step size was 0.1 times the range specified for the optimized variable (0–100 years for the time to the cutting, −1 to 8 for a1 and 10–40 cm for a2). Search was terminated when the step size was less than 0.01 times the initial step. Since three cuttings were optimized, the total number of optimized variables was 9 (time, a1 and a2 for three cuttings). Modelling Data from optimized management schedules were used to fit a set of models, which show the optimal management action for any given stand state. The first model indicates whether an immediate cutting of the stand would be the optimal decision. The second model tells whether the cutting should be a thinning treatment. In case of thinning, two additional models give the parameters of the thinning intensity curve equation (1), indicating the optimal type and intensity of thinning. No model was developed for choosing between final felling and removal of upper canopy. This is because such a model would not be of great help in forestry practice as the need for artificial regeneration after removing the large trees from the stand can be seen in the field. The following logistic model was fitted for the probability of cutting (probability that cutting is the optimal decision) and for the probability that thinning is the optimal type of cutting: p=11+exp[−f(x)] (2) where x is a vector of site and growing stock variables. Then, the thinning events were used to fit models for the two parameters of the thinning intensity curve equation (1). The variables that were used to predict the optimal management action for a stand included mean tree diameter, stand basal area, basal areas of different tree species, basal area of pulpwood-sized trees (dbh 8–18 cm), basal area and number of small trees (dbh < 8 cm), site variables (temperature sum and forest site type), and discount rate. Many of these variables and their transformations were statistically significant predictors, due to the high number of observations. However, parsimonious models were pursued, seeking low number of predictors with high statistical significance. The absolute t value (parameter estimate divided by the standard deviation of parameter estimate) had to be at least 10 for all predictors. The effect of variables left out from the models on R2 and RMSE was small, although these left-out variables were often statistically significant. Validity testing Management prescriptions obtained from the models developed in this study were tested by simulating and visualizing cuttings, as advised by the models, in a few example stands. Another part of validity test consisted of comparing the net present values of prescriptions based on the models with the net present values of optimized management schedules. The cuttings of each stand of the study material (2095 stands) were simulated by using the models to decide when and how the stands are cut. Cutting was simulated if the cutting probability was higher than 0.5, and the cutting was thinning if the thinning probability was higher than 0.5. The next three cuttings in each stand were simulated, similarly as in optimization. Since the prescription depends on the rate of interest, all simulations were repeated with five different discount rates (0.1, 1, 3, 5 and 7 per cent). Results Probability of cutting Function f(x) was as follows in the logistic model equation (2) for the probability that cutting the stand now is the optimal decision: f(x)Cut=16.032−1.098G+2.806lnD×lnG−0.573D×G+0.000088G×TS+0.00454G×R+0.000486TS×lnR−3.878lnTS−0.944FS−0.814VT where D is the basal-area-weighted mean diameter of trees (cm), G is the stand basal area (m2 ha−1), TS is the temperature sum (degree days >5°C), R is the discount rate (per cent), FS is the indicator variable for fertile growing sites (mesic or better) and VT is the indicator variable for sub-xeric site. High stand basal area and large mean tree size increased the probability of cutting (Figure 3). The probability of cutting increased rapidly at mean diameters between 10 and 25 cm, and basal areas between 10 and 25 m2 ha−1, staying high above these ranges. Improving site quality decreased and higher discount rate increased the probability of cutting (Figure 4), which means that stands should be cut earlier with a higher rate of interest and when site quality is low. Increasing temperature sum decreased cutting probability implying that southern stands are to be cut at higher basal area or mean tree size than northern stands. The model includes also some interactions, indicating for example that the effect of temperature sum decreases with increasing stand basal area and discount rate. Figure 3 View largeDownload slide Probability that cutting is optimal decision as a function of stand basal area and (basal-area-weighted) mean diameter of trees when discount rate is 3%, site is mesic (MT) and temperature sum is 1200 d.d. Figure 3 View largeDownload slide Probability that cutting is optimal decision as a function of stand basal area and (basal-area-weighted) mean diameter of trees when discount rate is 3%, site is mesic (MT) and temperature sum is 1200 d.d. Figure 4 View largeDownload slide Effect of site fertility and discount rate on the probability that cutting is the optimal decision when temperature sum is 1200 d.d. In the left-hand-side diagram, the stand basal area is 28 m2 ha−1, and in the right-hand-side diagram, the mean tree diameter is 25 cm. OMT refers to herb-rich (fertile) site and CT refers to xeric (poor) site. Figure 4 View largeDownload slide Effect of site fertility and discount rate on the probability that cutting is the optimal decision when temperature sum is 1200 d.d. In the left-hand-side diagram, the stand basal area is 28 m2 ha−1, and in the right-hand-side diagram, the mean tree diameter is 25 cm. OMT refers to herb-rich (fertile) site and CT refers to xeric (poor) site. When a predicted probability of 0.5 was used to prescribe cutting, the logistic model gave the same prescription as the optimizations in 77 per cent of the cases. The model predicted a no-cutting decision for 71 125 of those 75 571 stand states, which had a no-cutting prescription in optimizations. The area under the ROC curve (AUC, area under receiver operating characteristic curve) of the model was 0.789, which indicates good performance. The Nagelkerke R2 statistic was 0.255. Probability of thinning Function f(x) was as follows in the logistic model equation (2) for the probability that thinning is the optimal type of cutting: f(x)Thin=−12.208+10.103lnG−4.004G+0.00731D×R+0.856Gpulp−0.785FS where Gpulp is the basal area of pulpwood-sized trees (dbh 8–18 cm) in m2 ha−1. The logistic model predicted the cutting type correctly in 81.3 per cent of the cases. The AUC statistic was 0.813, which indicates good performance. The Nagelkerke R2 statistic was 0.321. The most significant predictor was the basal area of pulpwood-sized trees, which increased the probability that thinning is the optimal type of cutting. The model suggests that whenever a stand with plenty of pulpwood-sized trees is cut, the cutting should be a thinning treatment. Increasing stand basal area increased and increasing mean tree diameter decreased the probability that thinning is the optimal cutting type (Figure 5). Also increasing discount rate and lower site productivity led to choosing thinning. The result can be interpreted so that investors with a high guiding rate of interest should use silviculture based on partial cuttings, especially on poor sites. Figure 5 View largeDownload slide Effect of site fertility and discount rate on the probability that the optimal cutting type is thinning. In the left-hand-side diagram, the stand basal area is 28 m2 ha−1, and in the right-hand-side diagram, the mean tree diameter is 25 cm. OMT refers to herb-rich (fertile) site and VT refers to sub-xeric (rather poor) site. Figure 5 View largeDownload slide Effect of site fertility and discount rate on the probability that the optimal cutting type is thinning. In the left-hand-side diagram, the stand basal area is 28 m2 ha−1, and in the right-hand-side diagram, the mean tree diameter is 25 cm. OMT refers to herb-rich (fertile) site and VT refers to sub-xeric (rather poor) site. Thinning type and intensity In case of thinning, the thinning intensity and type were determined by parameters a1 and a2 of the thinning intensity curve equation (1). The models for these parameters were: a1=11.324−0.440G−4.360lnD+0.597G×D+0.000832TS×lnR+0.000170G×TS−0.000117D×TS+0.175lnGpulp−0.0126D×R a2=(0.980+0.964D−0.000619D×G−0.0630lnD×lnG)2+0.4232 The RMSE of the model for a1 was 2.242 and the adjusted R2 was 0.216. The latter model was fitted for the square root of a2. The RSME of the fitted model was 0.423, and the R2 statistic was 0.537. The models imply that the thinning type is almost always thinning from above, because a1 is often larger than zero (Figure 6). High amount of pulpwood-sized trees increased the value of a1. At high values for a1, the type of thinning would resemble dimension cutting removing almost all trees larger than a2, and leaving almost all trees smaller than a2. Figure 6 View largeDownload slide Effect of mean diameter (left) and stand basal area (right) on the parameters of the thinning intensity curve equation (1). If parameter a1 is larger than zero, the type of cutting is thinning form above. Parameter a2 gives the diameter at which thinning intensity is 50%. In the left-hand-side diagrams, the stand basal area is 28 m2 ha−1, and in the right-hand-side diagrams, the mean tree diameter is 25 cm. Figure 6 View largeDownload slide Effect of mean diameter (left) and stand basal area (right) on the parameters of the thinning intensity curve equation (1). If parameter a1 is larger than zero, the type of cutting is thinning form above. Parameter a2 gives the diameter at which thinning intensity is 50%. In the left-hand-side diagrams, the stand basal area is 28 m2 ha−1, and in the right-hand-side diagrams, the mean tree diameter is 25 cm. Parameter a2 gives the diameter at which thinning intensity is 50 per cent. Figure 6 shows that a2 follows the mean diameter (weighted by basal area) of the stand but the thinning (from above) is to be extended to smaller diameters when the basal area of the stand increases. Increasing mean diameter also leads to thinning where a2 is clearly smaller than mean diameter, which usually means that stands of large trees should be thinned more heavily than stands of small trees. Simulations The models were visualized in different forest structures (Figures 7 and 8). For the three diameter distributions that represent even-aged stands (Figure 7, left panel), the models proposed cutting with increasing probability when the mean size of trees increased. The probability that the optimal cutting type is thinning was smaller when the trees of the stand were larger. For the most mature stand, the recommendation was clear-felling because the probability that thinning is the optimal type of cutting was less than 0.5 (Figure 7, bottom left). Figure 7 View largeDownload slide Effect of diameter distribution on the probability that cutting is optimal decision (Cut xxx) and the probability that thinning is the optimal type of cutting (Thin xxx) when discount rate is 3% and temperature sum is 1200 d.d. The removal of trees from different diameter classes is based on the models for parameters a1 and a2 of the thinning intensity curve equation (1). G xx/yy is the stand basal area before (xx) and after (yy) thinning (m2 ha−1). Figure 7 View largeDownload slide Effect of diameter distribution on the probability that cutting is optimal decision (Cut xxx) and the probability that thinning is the optimal type of cutting (Thin xxx) when discount rate is 3% and temperature sum is 1200 d.d. The removal of trees from different diameter classes is based on the models for parameters a1 and a2 of the thinning intensity curve equation (1). G xx/yy is the stand basal area before (xx) and after (yy) thinning (m2 ha−1). Figure 8 View largeDownload slide Effect of discount rate (DR) on the probability that cutting is optimal decision on fertile site when temperature sum is 1200 d.d. The removal of trees from different diameter classes is based on the models for parameters a1 and a2 of the thinning intensity curve equation (1). ‘Remaining G’ is the basal area (m2 ha−1) of the post-cutting stand. Figure 8 View largeDownload slide Effect of discount rate (DR) on the probability that cutting is optimal decision on fertile site when temperature sum is 1200 d.d. The removal of trees from different diameter classes is based on the models for parameters a1 and a2 of the thinning intensity curve equation (1). ‘Remaining G’ is the basal area (m2 ha−1) of the post-cutting stand. The left panel of Figure 7 shows that the thinning removed a larger part of the trees when the mean tree size increased. If the diameters of removed and remaining trees are compared with the diagrams of Figure 1, it can be concluded that in the thinning of the youngest stand, most of the removed trees were large enough for one saw log (Figure 7, top left) whereas in the more mature stand most removed trees were large enough for two saw logs. Pulpwood-sized trees were not removed in thinning. The optimal cutting type of the uneven-aged stand (Figure 7, top right) was thinning, with very high certainty. The thinning was close to dimension cutting, removing all trees larger than 20 cm (20 cm is the lower limit of the 21-cm diameter class) and maintaining all smaller trees. The treatment was similar in the other uneven-sized stand having two peaks in the diameter distribution (Figure 7, middle right). In the stand consisting of two very distinct canopy layers, cutting was optimal with 0.585 probability, and thinning was the optimal cutting type with a probability of 0.513. The thinning removed almost all trees from the upper canopy layer (Figure 7, bottom right). The no-thinning treatment of this stand would be the removal of upper canopy (to release advance regeneration), with practically the same outcome as in optimal thinning. The effect of discount rate is visualized in Figure 8. With very low discount rate it was optimal to let the stand grow without cutting. However, if the stand was thinned, the optimal type of thinning was from below (Figure 8, top left). The stand was financially mature for cutting at ~2 per cent discount rate. The optimal type of cutting was thinning from above. Figure 8 shows that the remaining post-cutting stand basal area decreased with increasing discount rate, but the decrease was not fast if the discount rate was 0.1 per cent or higher. Validity test The models presented above were applied to all 2095 stands, which were used to produce the datasets for modelling optimal any-aged management. The development of each stand was simulated in 5-year steps. In the beginning of each step, the models were used to decide whether the stand was cut and whether the cutting was thinning. In case of thinning, the thinning intensity curve equation (1) was derived using the models developed for parameters a1 and a2. Simulation was continued until three cuttings had been carried out. The results of each simulation were used to calculate the NPV of the schedule, exactly in the same was as in the optimizations. The NPVs of the model-based simulations correlated closely with the NPVs of optimized management schedules (Figure 9). With a 1 per cent discount rate, optimized schedules usually had higher NPVs than schedules in which the cutting prescriptions were based on the models developed in this study (Figure 9, top left). However, the performance of the models improved with increasing discount rate so that at higher rates the models often led to even more profitable management schedules than optimization. This was possible because the used optimization method (Hooke and Jeeves) did not find the global optimum with certainty. Sometimes it found a local optimum, producing a slightly lower NPV than the global optimum. Figure 9 View largeDownload slide Correlation between the net present values of optimized management schedules (x axis) and management schedules based on the models of this study (y axis) with four different discount rates in the southernmost forest (Forest 1 in Table 1). Figure 9 View largeDownload slide Correlation between the net present values of optimized management schedules (x axis) and management schedules based on the models of this study (y axis) with four different discount rates in the southernmost forest (Forest 1 in Table 1). A summary of the comparisons of the NPVs of model-based schedules in relation to the optimized ones (Figure 10) also shows that the models developed in this study were comparable to optimization when the discount rate was 3 per cent or higher. At a 1 per cent discount rate, model-based schedules decreased NPV by ~10 per cent, compared with optimized schedules. Figure 10 View largeDownload slide Relative net present values (NPV) of the five forests when the management of individual stands is based on the models of this study, in relation to NPVs based on stand-level optimizations. ‘Average’ is the average performance (relative NPV) of the models in the five forest holdings. Figure 10 View largeDownload slide Relative net present values (NPV) of the five forests when the management of individual stands is based on the models of this study, in relation to NPVs based on stand-level optimizations. ‘Average’ is the average performance (relative NPV) of the models in the five forest holdings. Discussion The four models presented in this study can be used to find the optimal management for any stand in Finland, without the need to decide or name the silvicultural system. Since the instructions are for any-aged management, it can be expected that the suggested management is at least as profitable as the recommended or optimal even-aged or continuous cover management. The use of the models requires that two site variables (forest site type and temperature sum) and the diameter distribution of the stand are known (more exactly: mean diameter, stand basal area, and basal area of pulpwood-sized trees). The validity tests showed that if the discount rate is 3 per cent or more, the instructions will lead to equally profitable management as stand-specific optimizations. Most probably, the averaging effect of models had a stronger impact on the results at low discount rates. Use of the models leads to frequent use of thinning from above where financially mature trees are removed and smaller trees with lower value but higher relative value increment are left to continue growing. This recommendation agrees with several studies conducted in Finland (Valsta, 1992; Vettenranta and Miina, 1999; Hyytiäinen et al., 2005; Pukkala et al., 2014a, 2014b, 2015, 2016) and other countries (Haight et al., 1985; Roise, 1986; Haight and Monserud, 1990; Solberg and Haight, 1991; Jin et al., 2017). The instructions developed in this study do not rule out final fellings. Final felling is optimal if all trees of the stand are financially mature and discount rate is low. However, the most important reason for choosing final felling instead of thinning is not a large mean size of trees. The mean reason is lack of small trees, which means that it is impossible to conduct such a thinning that the post-cutting stand would consist of trees having high relative value increment. The main reason why final felling was selected more often when low discount rate was used in optimization (Figure 2) is the fact that artificial regeneration after clear-felling is profitable at low discount rate but unprofitable at higher rates (Hyytiäinen and Tahvonen, 2002). The net present value of all costs and cuttings obtainable after final felling is negative if the discount rate is high, and the stand establishment operations in the beginning of the rotation are obligatory (Hyytiäinen and Tahvonen, 2002). Negative bare land value with high discount rate is a straightforward consequence of discounting. The result in Figure 2 should not be interpreted so that even-aged plantation forestry is commonly optimal at low discount rates. The result only means that among the stands used in this study, it was frequently optimal to conduct one final felling followed by artificial regeneration. The optimal management of the plantation established after clear-felling would in most cases consist of repeated high thinnings, especially if advance regeneration is plentiful (Pukkala et al. 2014b). The optimal management of plantations would therefore resemble CCF. The reason for the frequent choice of final felling is the past management of Finnish forests, consisting of low thinnings and cleaning the stand from advance regeneration to make the thinning easier for the machinery. The basal area or number of small commercial trees (dbh < 8 cm) was not selected in any of the models as a predictor. However, an increasing amount of small trees affects the optimal choice between clear-felling and releasing advance regeneration in favour of the latter. A model for this choice was fitted in the course of this study but this model was not reported since the choice can be easily made in the field without the need for a model. The instructions developed in this study advice in optimal stand management when the aim is to maximize the net present value of timber production. Non-wood management objectives often lead to less frequent use of final felling (Miina et al. 2010; Jin et al. 2017) and sometimes also higher growing densities, especially when carbon sequestration is one management objective (Pohjola and Valsta, 2007; Niinimäki et al., 2013). These effects can be achieved by using a lower threshold probability for thinnings (leading to decreased use of final felling) and higher threshold probability for cutting (leading to higher stand densities). Use of lower discount rate also leads to higher average density of forests. Modern forest planning systems often consist of two steps (Kangas et al., 2008; Jin et al., 2016). The first phase simulates several alternative management schedules for the stands and the second phase employs combinatorial optimization to find such a combination of the simulated schedules that best fulfils the forest-level management objectives and constraints. The instructions developed in this study can be used when simulating management alternatives for combinatorial optimization. Alternatives can be produced by varying the discount rate, or the threshold probabilities for cutting and thinning. The degree of explained variance was not very high for the models presented in this study. For the model for cutting probability, the main reason is the fact that different time points of cuttings may be practically equally good in terms of for NPV. This is the case when the relative value increment of a stand is close to the discount rate, which is a very common situation at certain stage of stand development. In addition, the Hooke and Jeeves optimization algorithm sometimes finds local optima, in which the cutting time may differ from the global optimum. The model for parameter a1 of the thinning intensity curve also had a low degree of explained variance. However, all values of a1 larger than 0.5 lead to fairly similar thinning (thinning from above) and it does not matter much whether the predicted value of a1 is 1 or 4. Optimization produced much variation in a1 whereas the model predicts rather constant values. However, the effects of these differences on the NPV of the management schedule are small. Because of these reasons, it is more relevant to evaluate the models via the net present values of the management schedules which are based on the models. This comparison indicated a very good performance of the models, especially when the discount rate was not very low. The use of the models has an averaging effect in the sense that model-based management varies less than optimization-based management. If more diverse management is desirable, for instance, for risk management or biodiversity reasons (Knoke et al., 2008), it can be achieved for instance by varying the threshold probabilities for cutting and thinning. This would slightly reduce the expected net present value but, on the other hand, it would reduce some risks and improve the biological quality of the forest. For practical use, the models presented in this study can be converted into simple computer tools or even diagrams. Figure 11 is an example of such a diagram. It shows the probability that cutting is optimal decision as a function of mean tree diameter and stand basal area. The diagram shows that a stand consisting of small trees can be grown in many different densities but high densities must be avoided when the mean tree size gets bigger. This is because of the high opportunity cost of large trees; they must be grown at low stand density to keep the relative value increment of the stand higher than the guiding rate of interest. Figure 11 View largeDownload slide Probability that cutting is the optimal decision as a function of stand basal area and mean diameter on mesic site when discount rate is 3% and temperature sum is 1200 d.d. The probability of cutting is 0.3–0.4 in the lower light gray area, 0.4–0.6 in the dark grey area, and 0.6–0.8 in the upper light grey area. Figure 11 View largeDownload slide Probability that cutting is the optimal decision as a function of stand basal area and mean diameter on mesic site when discount rate is 3% and temperature sum is 1200 d.d. The probability of cutting is 0.3–0.4 in the lower light gray area, 0.4–0.6 in the dark grey area, and 0.6–0.8 in the upper light grey area. Although the instructions developed in this study were not based on stochastic optimizations the instructions may be regarded adaptive in the sense that only one management decision is made at a time. The next decision depends on the true development of the stand (advance regeneration, growth of lower canopy layer, etc.) and also on the developments in financial markets affecting discount rate. This is an advantage compared with management in which a sequence of cuttings is optimized and a certain long-term management schedule is then implemented. For adaptive management, the instructions could be further improved by including timber prices as model predictors. Conflict of interest statement None declared. References Anonym . 2013 Laki Metsälain Muuttamisesta. (The Law on the Change of the Forest Act). 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This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)
Forestry: An International Journal Of Forest Research – Oxford University Press
Published: Dec 1, 2018
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