Modelling the risk of snow damage to forests under short-term snow loading

Forest Ecology and Management 116 (1999) 51±70

Modelling the risk of snow damage to forests under short-term snow loading Marja-Leena PaÈaÈtalo, Heli Peltola*, Seppo KellomaÈki University of Joensuu, Faculty of Forestry, P.O. Box 11, FIN-80101 Joensuu, Finland Received 8 December 1997; accepted 14 July 1998

Abstract Regression models are developed to assess the risk of snow damage to Scots pine (Pinus sylvestris L.), Norway spruce (Picea abies (L.) Karst) and birch (Betula spp.) stands based on simulated data, employing a mechanistic wind and snow damage model developed by Peltola et al., 1998a. The risk is predicted in terms of the critical windspeed needed to cause stem breakage and uprooting of trees at forest edges under short-term snow loading. Separate regression models are developed for each tree species using stem taper (breast height diameter of stem relative to tree height, d1:3 =h), stand density, snow loading and distance from the stand edge as variables, and a general model for stem breakage and uprooting is also proposed having tree species as an additional dummy variable. The overall risk of stem breakage and uprooting is shown to increase with snow loading and decrease with increasing stem taper and stand density for all three tree species, although Scots pines and Norway spruces are predicted to be much more susceptible to snow damage than birches, which, being lea¯ess, had much less crown area for snow attachment and wind loading. The greatest susceptibility to stem breakage and uprooting is seen at the stand edge, where the risk due to wind loading is much greater than inside the stand. Under these circumstances, slightly tapering Scots pines and Norway spruces are found to be the most vulnerable under a snow load of 60 kg mÿ2, suffering damage at windspeeds of <9 m sÿ1 at a constant height of 10 m above the ground, i.e. these windspeeds enhance the risk, whereas higher speeds can be expected to dislodge the snow from the crowns. Birches will only exceptionally be broken and uprooted at windspeeds of <9 m sÿ1 according to the models developed here. Since the general models give rise to somewhat greater residuals compared with the simulated data than do the single tree species models, it seems that the latter will give more reliable predictions of the risk of snow damage. The models could be useful when discussing the risk of snow damage in connection with alternative forms of stand management, especially in high risk areas, enabling high-risk trees to be removed during thinning. # 1999 Elsevier Science B.V. All rights reserved. Keywords: Critical snow load; Critical windspeed; Stem breakage; Uprooting; Mechanistic wind and snow damage model; Scots pine; Norway spruce; Birch; Stand management; Risk assessment; Regression models

1. Introduction

*Corresponding author. Tel.: +358-13-251-3639, fax: +358-13251-4444, e-mail:[email protected]

Damage caused by snow is a continuing source of economic loss in managed forests, especially in the boreal forest zone of Northern Europe (Suominen,

0378-1127/99/$ ± see front matter # 1999 Elsevier Science B.V. All rights reserved. PII: S0378-1127(98)00446-0

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1963; Matthews, 1989; Valinger and Lundqvist, 1992a; Valinger and Lundqvist, 1994). In some areas of Finland, for example, snow damage may occur almost every year, whereas in others the frequency can vary greatly. The limit between low and moderate risk is exceeded once every ®ve years on average in southern Finland, once every three years in northeastern Finland, and once every 8±17 years in other parts of the country (Solantie, 1994). Snow damage reduces the yield of recoverable timber and increases the costs of unscheduled thinnings, leading to disturbances in forest management planning. Furthermore, the presence of broken trees (most common type of snow damage) and uprooted trees (if the soil is unfrozen) in stands can lead to detrimental insect attacks on the remaining stems because of an increase in the amount of breeding material (Ravn, 1985; Schroeder and Eidmann, 1993; Valinger and Lundqvist, 1994). The total growing stock has increased over recent decades, making the structure of the forest ecosystem more prone to snow damage (Valinger and Lundqvist, 1994), i.e. stands in which management has been delayed or which have been left totally unmanaged are most vulnerable to damage. On the other hand, intensive thinnings and clear-fellings, the main method of timber harvesting, have led to a greater number of vulnerable stand edges (Neustein, 1965; Persson, 1972, 1975; Laiho, 1987; Lohmander and Helles, 1987). The risk may even be enhanced in the future at northern latitudes because of the potential decrease in soil freezing on account of a warmer, more humid climate, leading to a decline in anchorage so that uprooting could become a more common type of snow damage (Peltola et al., 1998b). It has been shown in numerous connections that the properties and position of a tree stand will affect the frequency and extent of snow damage. Certain tree and stand characteristics (e.g. tree species, taper, stand density) will place critical limits on snow loading beyond which damage will occur (Persson, 1972; Rottmann, 1985; Valinger et al., 1993; Solantie, 1994; NykaÈnen et al., 1997). The management of forests, e.g. thinnings, can increase the risk of snow damage for some years (Suominen, 1963; Abez and Prange, 1976; Johann, 1981; Valinger and Lundqvist, 1992a, b; Valinger et al., 1994). Local snow and wind conditions, extremes of climate and topography will determine how probable a

serious snow load is, the severity of the damage being mainly dependent on the amount of snow falling. Snowfalls of 20±40 cm or more at temperatures of around 08C appear to produce low-to-moderate risk conditions, whereas snowfalls of ca. 60 cm imply a very high risk of damage, as snow will then accumulate in the crowns of the trees (Solantie and Ahti, 1980; Petty and Worrell, 1981; Solantie, 1994). These snow loads can cause damage even under calm conditions, especially if they persist for some time, whereas windspeeds of <9 m sÿ1 are thought to intensify the risk of snow damage when combined with short-term snow loading. Above this windspeed, the snow would more probably be dislodged from the tree crowns (Solantie, 1994). On the other hand, strong winds can naturally cause heavily laden trees to break (Valinger and Lundqvist, 1992a). Mechanistic and empirical models (e.g. logistic models) have been developed in recent years for describing snow (and wind) damage to trees, especially coniferous species, but also birches spp. (Peltola et al., 1997, 1998a; Gardiner et al., 1998; Valinger et al., 1993; Valinger and Fridman, 1997). The logistic models developed by Valinger and Fridman (1997), for example, are based on real wind and snow damage data and serve to predict the probability of damage (by both, wind and snow) using various tree and stand characteristics. These models cannot be used to assess the silvicultural risks, e.g. how the timing of thinning or thinning intensity would affect the risk. The mechanistic model developed by Peltola et al. (1998a) correspondingly attempts to describe in full the mechanistic behaviour of trees under snow and wind loading in order to determine the snow load and windspeed required to cause damage. The model sets out from site and stand information and predicts the canopy-top mean windspeeds at which trees will break or be uprooted due to wind and/or snow loading at forest edges under a range of silvicultural conditions. To provide a measure of the total turning moment on the tree, the model calculates the wind force and gravitational force (caused by the weight of the stem, crown and attached snow). Correspondingly, the prediction of stem breakage relies on values for the modulus of rupture determined for different species of timber, and the resistance to uprooting is calculated using an estimate of the root-soil plate mass to derive a resistive moment. By virtue of its mechanistic

M.-L. PaÈaÈtalo et al. / Forest Ecology and Management 116 (1999) 51±70

approach, the model provides a tool for assessing the risk of snow- or wind-induced damage to trees by predicting the critical combination of snow load and windspeed at which trees will break or be uprooted under a range of silvicultural conditions, and even the probability of damage if the local extremes of wind and snow conditions are known. Simpler approaches would be needed, however, to assess the risk of snow damage with regard to practical forest management, i.e. to combine optimal stand growth and yield with the minimum risk of damage. The aim of this research was to formulate simple regression models to assess the risk of snow damage to pure Scots pine (Pinus sylvestris L.), Norway spruce (Picea abies (L.) Karst.) and birch (Betula spp.) stands in the presence of short-term snow loading on the basis of data simulated by the mechanistic wind and snow damage model developed by Peltola et al. (1998a). The risk of snow damage is predicted in terms of the critical windspeed needed to cause stem breakage and uprooting of individual trees under various snow loads based on tree and stand characteristics at newly formed stand edges. Regression models are developed separately for Scots pine, Norway spruce and birch, and general models are formulated that include the tree species as additional dummy variables.

These forces are manifested by the total turning moment applied at any point in height along the stem or in the root system. Trees are also assumed to de¯ect to a point of no return when acted upon by a wind of constant mean speed and direction. This makes it possible to calculate the maximum snow load that a tree can withstand, in interaction with windspeed, when the breaking strength of the stem and the rooting resistance are known. A tree is assumed to break if the breaking stress, acting on the stem, exceeds a certain critical value, namely the modulus of rupture (Sunley, 1968; Petty and Worrell, 1981; Petty and Swain, 1985). Similarly, it is assumed to be uprooted if the total turning moment exceeds the support provided by the root-soil plate anchorage (Coutts, 1986), assuming that the soil is unfrozen (Fig. 1). 2.1.2. Combined loading imposed on trees by snow and wind The mean wind load affecting the stem and crown systems under stand edge conditions is calculated at each height in the canopy, using a predicted logarithmic wind pro®le for the stand edge (Oliver and Mayhead, 1974; Grace, 1977; Smith et al., 1987; Peltola and KellomaÈki, 1993; Peltola et al., 1998a). At a height, z m, this will be F1 …z† ˆ 0:5  Cd    U…z†2  A…z†

2. Material and methods 2.1. The mechanistic model 2.1.1. Outlines of the model The data for the regression models were simulated with the help of mechanistic wind and snow damage model developed by Peltola et al. (1998a). This describes the mechanistic behaviour of trees at stand edges under wind and snow loading, setting out from site and stand information and predicting the mean windspeed at which the trees will break or be uprooted. The snow load attached to the tree crowns is assumed to interact with the wind loading to exert stresses on the crown components, stem and roots, possibly resulting in stem breakage, stem bending or uprooting. The forces acting upon a tree are thus divided into (1) the horizontal force exerted by the wind, and (2) the vertical force exerted by gravity, including the weight of the snow, stem and crown.

53

(1)

where F1(z) is the mean wind force, N, z the height along the stem, m, U(z) the mean windspeed, m sÿ1, A(z) the area of the tree projected against the wind, m2, Cd the drag coef®cient (dimensionless), and  the density of air, kg mÿ3 (Peltola and KellomaÈki, 1993; Peltola et al., 1997, 1998a). Once any substantial bending occurs in a tree, an additional force, that due to gravity, is present. The total mean gravitational force at height z arises from the green mass of the stem and crown segment, the snow load and the gravitational constant, as follows (Grace, 1977; Jones, 1983; Petty and Swain, 1985), F2 …z† ˆ M…z†  g

(2)

where F2(z) is the gravitational force, N, M(z) the green mass of the stem and crown along with the snow load, kg, and g the gravitational constant, m sÿ2. The snow load attached to the tree crown is estimated as the product of the area of the crown projected on the

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M.-L. PaÈaÈtalo et al. / Forest Ecology and Management 116 (1999) 51±70

Fig. 1. Outline of the mechanistic wind and snow damage model (see Peltola et al., 1998a).

ground surface, e.g. a snowfall of 60 cm accumulating in the crown corresponds to a snow load of 60 kg mÿ2 (Peltola et al., 1998a). Correspondingly, the weight of the snow is divided within the crown with respect to the horizontal crown area of each one-metre segment projected against the wind. The total turning moment at the base of the stem is obtained by multiplying the mean turning moment (caused by wind and gravitational forces) by the gust factor, as follows (see Peltola et al., 1998a), T…z† ˆ Gf  …F1 …z†  z ‡ F2 …z†  x…z††

(3)

where T(z) is the total turning moment, Nm, Gf the gust factor, x(z) the horizontal displacement of the stem from the upright position, m, and other symbols are as above. The gust factor (Gf) is obtained as a function of distance from the stand edge, stand density and tree height, and it relates the mean wind load over one hour to the extreme wind load in 3-s gusts by reference to wind tunnel observations (Gardiner and Stacey, 1996; Gardiner et al., 1997; Peltola et al., 1998a). The bending of the stem is assumed to be directly proportional to the mean force of the wind acting on the

M.-L. PaÈaÈtalo et al. / Forest Ecology and Management 116 (1999) 51±70

crown centre and inversely proportional to the stiffness of the stem (Pennala, 1980; see Peltola and KellomaÈki, 1993). 2.1.3. Stem resistance and root anchorage The prediction of stem breakage, the most prominent type of snow damage, relies on values for the modulus of rupture determined for different species of timber (Peltola et al., 1998a). Breakage occurs if the maximum strength of the stem, i.e. the modulus of rupture, is exceeded. The ability of a tree to resist stem breakage is expressed as follows (Jones, 1983) 3 †=32 Tcrit ˆ …  MOR  d1:3

(4)

where Tcrit is the critical turning moment for breakage at the base of the stem, Nm, d1.3 the stem diameter at breast height, m, and MOR the modulus of rupture, Pa. Moreover, stem breakage can occur at any height, and not necessarily at breast height. The ability of a tree to resist uprooting depends on the total turning moment in relation to tree anchorage which, in turn, depends on the depth, diameter and mass of the root-soil plate and the properties of the soil (Coutts, 1986; Peltola and KellomaÈki, 1993). The model calculates the resistance to uprooting using a resistive moment derived from a prediction of the rootsoil plate mass, because the contributions of factors other than weight are very complicated. The support provided by the roots is then modelled by the proportion of the total anchorage contributed by the root-soil plate weight, including other components in addition to this weight, i.e. windward roots, root-soil plate mass, the hinge effect and soil resistance (Peltola and KellomaÈki, 1993), RSsup ˆ …g  Mass  RSmean †=Arsw

(5)

where RSsup is the supporting moment of the total root-soil plate anchorage, Nm, Mass the fresh mass of the root-soil plate, kg, g the gravitational constant, m sÿ2, and RSmean the mean depth, m, of the root-soil plate, derived from its width and depth. Arsw is a parameter [%] expressing the weight of the root-soil plate as a proportion of total below-ground anchorage (dimensionless) (see Coutts, 1986). The properties of the mechanistic model and the validity of its output for podzol soil conditions (till formation) are discussed in detail by Peltola et al. (1998a) and Gardiner et al. (1998).

55

2.2. Simulation of data for regression models The data were simulated along stand edge using distances of 0±5 tree heights from stand edge, i.e. `0' tree heights representing the stand edge (0±1 tree heights) and `5' tree heights further inside the stand conditions, respectively. In computations, it was also expected that there existed an in®nite upwind open area ahead of the forest, and a gap size of ten tree heights was used in computations, so that it would not reduce the wind loading at the windward edge. Snow loads of 20, 40, and 60 kg mÿ2 acting on the tree crowns were used in the computations, i.e. loads usually representing low-to-high risks of snow damage (Solantie, 1994). The heights selected for the Scots pines, Norway spruces and birches in the computations were 12, 16, 20 and 24 m, and the tapers (d1.3/h) 1 : 80, 1 : 100, and 1 : 120 in stands of varying density (20% for tree heights of 16 to 24 m and 25% for a tree height of 12 m). The crown shape was assumed to be accounted for by two triangles in the Scots pine and birch (in leaf) and one triangle in the Norway spruce, these triangles having a common base equal to twice the length of the longest branch in the crown (Hakkila, 1971; Peltola and KellomaÈki, 1993; Peltola et al., 1998a). The crown projection area of a birch without leaves is assumed to be ca. 20% of that of a birch in leaf, as deduced from the surface area distribution of a tree (see Halldin, 1985; Peltola et al., 1998a). Correspondingly, the stem area projected against the wind is compiled from the projections of stem segments one metre in length, based on the stem taper (Laasasenaho, 1982; Peltola and KellomaÈki, 1993). Furthermore, the crown area projected against the wind was assumed to be unstreamlined and not dependent on windspeed, because of snow loading. The crown mass is given as a proportion of the stem mass, and its vertical distribution is taken to be the same as that of the crown area projected against the wind (Peltola et al., 1997, 1998a). The crown weight-to-stem weight ratios used in the calculations were 30, 30 and 50 for the Scots pine, birch (in leaf) and Norway spruce, respectively. Since the crown weight of birch without leaves is ca. 68% of that in leaf (e.g. MaÈlkoÈnen, 1977), the corresponding crown weight without leaves is taken to be 20% of the stem weight (Halldin, 1985; Peltola et al., 1998a).

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Similarly, the crown weight was expected to be located on the uppermost 42% of the stem in the Scots pine, 50% in the birch and 75% in the Norway spruce (see NyyssoÈnen, 1954). The green mass of a stem segment is estimated from its volume and the mean green density of the stem (Laasasenaho, 1982; KaÈrkkaÈinen, 1985; Peltola and KellomaÈki, 1993). The rooting depth and root mass of Scots pine and Norway spruce are, in principle, based on the relationships presented by Hakkila (1971), although the rooting depth of the Norway spruce is further modi®ed by reference to Peltola et al. (1998a) and that of birch is expected to be equal to that of Scots pine, i.e. both are based on Finnish tree pulling data applying to podzol soils on till formations (see Peltola et al., 1998c). Furthermore, the root-soil plate width of trees of all species is expected to equal the crown width, again based on Finnish tree pulling data for podzol soils (Peltola et al., 1998c). The tree and stand characteristics used in the model computations are presented in Table 1. The trees were assumed to be liable to short-term snow loading, and ®gures equivalent to 85% of the modulus of rupture (MOR), derived from static tests for pure woods (see Fons and Pong, 1957; Lavers, 1969; Petty and Worrell, 1981; Petty and Swain, 1985), were used as the critical values for stem breakage (MOR), based on Finnish tree-pulling data, i.e. 39.1, 30.6 and 53.6 MPa for Scots pine, Norway spruce and birch, respectively (Peltola et al., 1998a, c). The modulus of elasticity used for calculating the stem displacement curve was assumed to be 7000, 6300 and 9900 MPa for the three species, respectively, values which are quite typical for green wood (Sunley, 1968; Lavers, 1969; Petty and Worrell, 1981). In addition, the supporting moment caused by the weight of the root-soil plate was assumed to be 30% (Arsw) of the total below-surface support in Scots pine and 20% in Norway spruce. The contribution of the root and soil weight was expected to be even more important for Scots pine than for Norway spruce due to its deeper rooting (see Laitakari, 1934; Kalela, 1949; Coutts, 1986; Peltola and KellomaÈki, 1993; Peltola et al., 1998a, c). The Arsw for Scots pine was also used for birch, in the light of Finnish treepulling results obtained on a till formation (Peltola et al., 1998c). Furthermore, values of 0.29 and 0.35 were used as the drag coef®cients (Cd) for Scots pine and

Norway spruce (Mayhead, 1973). The Cd value for Scots pine was also used for birch. The snow load attached to the tree crown was estimated as the product of the unstreamlined area of the crown projected on the ground surface, e.g. a snowfall of 60 cm accumulating in the crown corresponds to a snow load of 60 kg mÿ2 (Peltola et al., 1997, 1998a). Correspondingly, the weight of the snow within the crown was divided with respect to the horizontal crown area of each one-metre segment against the wind. Based on simulated data, an attempt was made to develop regression models for assessing the risk of snow damage to Scots pine, Norway spruce and birch stands in terms of the critical windspeeds needed to cause stem breakage and uprooting of single trees under short-term snow loading (dependent variables in the models). Although the aim was to formulate a separate regression model (using the stepwise procedure of SPSS for Windows 6.1) for each tree species, general models for all species were also aimed at. The aim was to use the following as independent variables in the single-species regression models, on the basis of simulated data using the mechanistic model: snow load (dummy variables), distance from forest edge (dummy variable), taper (relation of diameter at breast height to tree height), stand density (stems haÿ1), tree height, tree diameter and various modi®cations of tree 3 2 2 , d1:3 , h  d1:3 ). The intention was to use diameter (d1:3 tree species as additional independent dummy variables in the general models for stem breakage and uprooting. All the above were proposed for use as independent variables because of their potential applicability in practical forestry (easy to measure, etc.). 3. Results 3.1. Models for stem breakage and uprooting 3.1.1. Variables in the models The regression models were designed to predict the critical windspeeds for stem breakage and uprooting under conditions of short-term snow loading from data on stem taper, stand density, snow load and distance from stand edge regardless of tree species (Table 2). The location of the tree (stand edge, in the stand) and

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Table 1 Tree and stand characteristics used in data simulation Tree height (m)

Crown d1.3, (cm)

Scots pine 12 10 12 12 12 15 16 13.3 16 16 16 20 20 16.7 20 20 20 25 24 20 24 24 24 30 Norway spruce 12 10 12 12 12 15 16 13.3 16 16 16 20 20 16.7 20 20 20 25 24 20 24 24 24 30 Birch 12 10 12 12 12 15 16 13.3 16 16 16 20 20 16.7 20 20 20 25 24 20 24 24 24 30

Root-soil plate

Mass roots (kg)

depth (m)

width (m)

area (m2)

depth (m)

width (m)

5.0 5.0 5.0 7.0 7.0 7.0 8.0 8.0 8.0 10.0 10.0 10.0

3.1 3.5 4.0 3.7 4.2 5.0 4.3 5.0 5.9 5.0 5.7 6.8

8.0 9.0 10.5 13.2 15.0 17.7 17.4 19.8 23.6 24.8 28.6 34.2

0.44 0.48 0.55 0.51 0.57 0.64 0.58 0.64 0.70 0.64 0.69 0.75

3.1 3.5 4.0 3.7 4.2 5.0 3.4 5.0 5.2 5.0 5.7 6.8

9.0 9.0 9.0 12.0 12.0 12.0 15.0 15.0 15.0 18.0 18.0 18.0

3.7 3.9 4.3 4.1 4.4 4.9 4.5 4.9 5.6 4.9 5.4 6.2

16.5 17.6 19.3 24.5 26.5 29.5 33.8 36.9 41.6 44.3 48.8 55.6

0.31 0.34 0.39 0.36 0.40 0.47 0.42 0.47 0.55 0.47 0.53 0.62

6.0 6.0 6.0 8.0 8.0 8.0 10.0 10.0 10.0 12.0 12.0 12.0

3.1 3.5 4.0 3.7 4.2 5.0 4.3 5.0 5.9 5.0 5.7 6.8

1.9 2.1 2.4 3.0 3.4 4.0 4.3 5.0 5.9 6.0 6.9 8.2

0.44 0.48 0.55 0.51 0.57 0.64 0.58 0.64 0.70 0.64 0.69 0.75

the snow load were statistically the most signi®cant variables in all the models, whereas tree height and diameter at breast height were not statistically signi®cant at the same time when a stepwise procedure was used. Taper was the only tree characteristic that was able to predict the windspeed needed to break or uproot a given tree, i.e. the only one included in the models. The general models for all tree

Stand density (stems haÿ1)

stem (kg)

crown (kg)

7 15 31 22 37 64 41 64 106 64 97 158

42 60 94 92 134 209 173 248 388 285 410 641

13 18 28 28 40 63 52 74 116 85 123 192

1600, 1280, 960 1600, 1280, 960 1600, 1280, 960 1036, 863, 690 1036, 863, 690 1036, 863, 690 828, 690, 552 828, 690, 552 828, 690, 552 662, 552, 442 662, 552, 442 662, 552, 442

3.7 3.9 4.3 4.1 4.2 4.9 4.5 4.9 5.5 4.9 5.4 6.2

9 21 43 30 51 90 57 90 151 90 137 225

38 55 86 87 126 197 167 239 374 277 399 623

19 27 43 44 63 29 83 120 187 138 199 312

1600, 1280, 960 1600, 1280, 960 1600, 1280, 960 1036, 863, 690 1036, 863, 690 1036, 863, 690 828, 690, 552 828, 690, 552 828, 690, 552 662, 552, 442 662, 552, 442 662, 552, 442

3.1 3.5 4.0 3.7 4.2 5.0 3.4 5.0 5.2 5.0 5.7 6.8

8 16 33 23 39 68 43 68 113 68 103 168

42 60 94 95 138 215 181 259 405 300 432 675

9 12 19 19 28 44 37 53 83 61 88 138

1600, 1280, 960 1600, 1280, 960 1600, 1280, 960 1036, 863, 690 1036, 863, 690 1036, 863, 690 828, 690, 552 828, 690, 552 828, 690, 552 662, 552, 442 662, 552, 442 662, 552, 442

species include species as a signi®cant dummy variable in addition to the other variables named above (see signi®cance of individual variables in Appendix A). 3.1.2. Residual analysis The predictions given by the regression models regarding stem breakage and uprooting in individual

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Table 2 Coefficients of independent variables, R2, standard errors, F-values and significance of regression models developed. Variable stand indicates the distance from stand edge (0ˆedge, 1ˆ5tree height). Snow40 and Snow60 are dummy variables for snow loads (20 kg/m2 as a default; both Snow40ˆ0 and Snow60ˆ0, 40 kg/m2; Snow40ˆ1 and Snow60ˆ0 and 60 kg/m2; Snow40ˆ0 and Snow60ˆ1). Taper, i.e. d1.3/h indicates the relationship between stem diameter at breast height and tree height, and Density stand density, stems haÿ1. In general models, tree species were used as dummy variables for tree species (Scots pine as a default; both Tresp1ˆ0 and Tresp2ˆ0, Norway spruce; Tresp1ˆ1 and Tresp2ˆ0 and birch; Tresp1ˆ0 and Tresp2ˆ1) Stem breakage models Scots pine Variables: Stand Snow40 Snow60 Taper Density Tresp1 Tresp2 Constant Statistics: R2 Standard error F-value Significance F

Uprooting models

Norway spruce

Birch

General

Scots pine

Norway spruce

Birch

General

Coefficients: 4.6483 ÿ1.5208 ÿ2.5626 ÿ0.1876 0.0012 ± ± 27.2231

5.2248 ÿ0.7381 ÿ1.3783 ÿ0.1933 0.0007 ± ± 28.4306

12.3574 ÿ3.2849 ÿ5.6478 ÿ0.4629 0.0042 1.2026 17.2856 67.2546

7.4102 ÿ1.8479 ÿ3.1962 0.2814 0.0020 ± ± 34.8200

4.2462 ÿ1.4473 ÿ2.4012 ÿ0.1469 0.0026 ± ± 21.1988

4.9187 ÿ0.7394 ÿ1.3364 ÿ0.1417 0.0034 ± ± 20.6213

9.7183 ÿ2.7485 ÿ4.4961 ÿ0.3012 0.0064 ± ± 43.9274

6.2944 1.6451 ÿ2.7445 ÿ0.1967 0.0041 1.5474 12.1097 24.0414

0.94 0.99 708.07 0.0000

0.94 1.01 720.38 0.0000

0.95 2.39 772.52 0.0000

0.90 3.30 840.75 0.0000

0.94 0.86 691.66 0.0000

0.96 0.77 934.96 0.0000

0.94 1.94 638.84 0.0000

0.91 2.29 924.76 0.0000

tree species were fairly well in line with the data simulated by the mechanistic model (Fig. 2, and Tables 3 and 4), i.e. the residuals were overall quite small ones. For windspeeds of <9 m sÿ1, however, i.e. in circumstances when damage is most likely to occur, the models for Scots pine and Norway spruce underestimated the windspeeds needed for damage in some cases (by as much as 1.8 m sÿ1) and, thus, overestimated the risk (i.e. when critical wind speeds of 4.0± 8.8 m sÿ1 were predicted by the mechanistic model) (Tables 3 and 4). According to models, especially for Scots pine, an overestimation of windspeed occurred almost as often as underestimation, however, residuals for overestimations seem to be smaller than for underestimations. The regression models failed, to some extent, to classify the risk correctly, especially in cases when windspeeds ranged from 7.4 to 9.1 m sÿ1 according to the mechanistic model. These were mainly overestimations of windspeed. Incorrect classi®cations occurred both at the edge and within the stand. The regression models overestimated the windspeed and underestimated the risk, especially to very slender Scots pines and Norway spruces within the stand

and that associated with all snow loads when the predicted windspeeds were close to 9 m sÿ1. Also, at the edge, similar incorrect classi®cations occurred for trees with taper 1 : 100 (and for some 1 : 80). Within incorrect classi®cations, there were only a few underestimations of windspeeds (and overestimation of risk). In addition, the models both, overestimated and underestimated the windspeed (by even more than 2 m sÿ1 for Scots pine and Norway spruce) when it was predicted to be 10±20 m sÿ1, especially inside the stand. Damage is unlikely to occur under these circumstances, however. The individual tree species models for Norway spruce classi®ed the damage cases more correctly in relation to the mechanistic model than did those for Scots pine. For birch, the regression models predicted few damage cases at windspeeds of <9 m sÿ1. In these cases, the classi®cation was incorrect with regard to the predictions given by the mechanistic model and the risk was overestimated. All in all, models of birches both, overestimated and underestimated windspeed more than models for Scots pine and Norway spruce (although there were less incorrect classi®cations for birches).

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59

Fig. 2. Comparison of critical windspeeds simulated by the mechanistic model and those predicted by the regression models developed for (A) stem breakage and (B) uprooting of Scots pines.

The general models for all tree species predicted the risk more correctly for stem breakage of Scots pine and for stem breakage and uprooting of birch relative to the data simulated by the mechanistic model than did the individual tree species models, where it came to the number of damage cases, except that they classi®ed more Norway spruce cases incorrectly. In addition, these cases generally involved underestimation of windspeed, whereas in the cases classi®ed

incorrectly by the models for individual tree species the windspeed was overestimated. The general models, nevertheless, overestimated the risk more than did the individual species models in terms of residuals. Thus, the windspeed differences were much greater in the general models than in the individual models and in some cases no additional wind was needed to cause damage. In this way, the models for individual tree species gave more realistic predictions of stem break-

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Table 3 Summary results for residual analysis including all cases, cases with no damage and damage for stem breakage models Single tree species models

All cases Windspeed difference range (m sÿ1) a Residuals within ÿ2±‡2 Residuals within ÿ1±‡1 No damage Windspeed difference range (m sÿ1) a Residuals within ÿ1±‡1 Damage cases Mechanistic model Regression model Both Windspeed difference range (m sÿ1)b Misclassifications Underestimation of windspeed Windspeed difference range m sÿ1 Residuals within ÿ2±‡2 Residuals within ÿ1±‡1 Overestimation of windspeed Windspeed difference range (m sÿ1) Residuals within ÿ2±‡2 Residuals within ÿ1±‡1

General models for all species

S. pine

N. spruce

Birch

S. pine

N. spruce

Birch

216 0±2.8 96% 67% 124 0±2.8 69% 92 91 81 80 0.02±1.8 12 1 0.1 100% 100% 11 0.2±1.6 100% 55%

216 0.01±2.5 94% 64% 147 0.01±2.5 69% 69 69 55 55 0.01±2.1 14 0 Ð Ð Ð 14 0.1±2.1 92% 64%

216 0.01±6.6 57% 32% 211 0.01±6.6 58% 5 0 5 0 3.6±4.2 5 5 3.6±4.2 0% 0% 0 Ð Ð Ð

216 0.02±4.9 63% 31% 122 0.02±3.5 28% 94 91 93 90 0.04±4.9 4 3 0.3±0.6 100% 100% 1 1.0 100% 100%

216 0.01±5.6 63% 35% 137 0.01±4.2 42% 79 69 73 63 0.2±5.6 16 10 0.5±2.2 80% 40% 6 0.09±1.3 100% 83%

216 0±15.6 30% 15% 216 0.01±15.6 15% 0 0 0 0 Ð 0 0 Ð Ð Ð 0 Ð Ð Ð

a

Includes both, under- and overestimation of windspeed, mainly overestimation. Includes both, under- and overestimation of windspeed, mainly overestimation with models for individual tree species of Scots pine and Norway spruce and underestimation for birches and general models. b

age and uprooting for all the tree species, by comparison with the mechanistic model, than did the general models. 3.2. Examples of predictions given by the models for individual tree species 3.2.1. Stem breakage According to the case examples, based on the models for individual tree species, the trees most likely to suffer stem breakage were slightly tapering Scots pines and Norway spruces growing under stand edge conditions, whereas damage was fairly unlikely within the stand (®ve tree heights from the edge) (Fig. 3). At the stand edge, Scots pines and Norway spruces with tapers of 1 : 120 and of varying heights were most liable to stem breakage under shortterm snow loading of 60 kg mÿ2, independent of the stand density used in the simulations (variation 20±25%).

Under a snow load of 60 kg mÿ2, the Scots pines (individual trees) with a taper of 1 : 120 and a heights of 12, 16, 20 or 24 m required windspeeds of 3.7, 3.1, 3.0 and 2.8 m sÿ1 for stem breakage at the stand edge, indicating a high risk of damage (medium-dense stands with densities of 1280, 863, 690 and 552 stems haÿ1, see Table 1). Similarly, Scots pines with a taper of 1 : 100 and heights of 12, 16, 20 or 24 m broke under a snow load of 60 kg mÿ2 at the stand edge with windspeeds of 7.4, 6.9, 6.7 and 6.6 m sÿ1, although similar trees with a taper of 1 : 80 required windspeeds of 11.2, 10.7, 10.5 and 10.3 m sÿ1 (Fig. 3). Thus, these Scots pines were unlikely to suffer from stem breakage because the snow would be more likely be dislodged from the tree crowns by such winds (unless ®rmly attached). Snow loads of 20±40 kg mÿ2 were also found to be enough to cause stem breakage in slightly tapering trees, i.e. they would have suffered damage at wind-

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61

Table 4 Summary results for residual analysis including all cases, cases with no damage and damage for uprooting models Single tree species models

All cases Windspeed difference range m sÿ1 Residuals within ÿ2±‡2 Residuals within ÿ1±‡1 No damage Windspeed difference range (m sÿ1) a Residuals within ÿ1±‡1 Damage cases Mechanistic model Regression model Both models Windspeed difference range (m sÿ1) b Misclassification cases Underestimation of windspeed Windspeed difference range (m sÿ1) Residuals within ÿ2±‡2 Residuals within ÿ1±‡1 Overestimation of windspeed Windspeed difference range m sÿ1 Residuals within ÿ2±‡2 Residuals within ÿ1±‡1

General models for all species

S. pine

N. spruce

birch

S. pine

N. spruce

Birch

216 0.01±4.2 98% 76% 110 0.01±4.2 79% 106 105 94 93 0.01±1.7 13 1 0,4 100% 100% 12 0.3±1.7 100% 42%

216 0±1.9 100% 80% 150 0±1.9 81% 66 64 62 60 0.03±1.7 6 2 0.3±0.6 100% 100% 4 0.2±0.5 100% 100%

216 0±10.3 72% 43% 205 0±10.3 45% 11 0 11 0 1.9±3.4 11 11 1.9±3.4 9% 0% 0 ± ± ±

216 0.01±4.3 80% 43% 108 0.01±3.4 41% 108 105 97 94 0.01±4.3 14 3 0.3±1.0 100% 100% 11 0.07±2.7 46% 36%

216 0.01±4.1 82% 56% 144 0.01±2.7 66% 72 64 71 63 0.09±4.1 9 8 0.2±1.9 100% 88% 1 1.0 100% 100%

216 0.01±11.9 47% 21% 216 0.01±11.9 21% 0 0 0 0 ± 0 0 ± ± ± 0 ± ± ±

a

Includes both, under- and overestimation of windspeed, mainly overestimation. Includes both, under- and overestimation of windspeed, mainly overestimation with models for individual tree species of Scots pine and Norway spruce and under estimation for birches and general models. b

speeds of <9 m sÿ1 at the stand edge. It was the magnitude of the snow load that had the greatest signi®cance as a single factor affecting the risk of stem breakage in Scots pines, i.e. if snow load increased from 20 to 40 or 60 kg mÿ2, the windspeeds needed to cause damage decreased to 24 and 40% of the previous value for trees of height 12 m and a d1.3 of 10 cm. The difference in absolute values for the critical windspeed at the different stand densities used in the computations were small, and all trees with tapers of 1 : 120, independent of density, would have been damaged (Fig. 4). A fairly high percentage difference in windspeed was recorded for Scots pines of height 12 m with a 1 : 120 taper at the stand edge, however. The windspeed needed to break a tree increased by 11% in a high density stand and decreased by 10.8% in a low density stand as compared with medium density (1280 stems haÿ1). The risk of stem breakage also increased along with tree height (for constant taper), because less wind is needed in the case of taller trees to

cause damage at a certain constant height above the ground (e.g. at height of 10 m above the ground as used here for comparison). Correspondingly, slightly tapering Norway spruces (1 : 120) of height 12, 16, 20 and 24 m in mediumdense stands under a snow load of 60 kg mÿ2 needed very moderate windspeeds of 4.8, 4.4, 4.4 and 4.2 m sÿ1 to cause stem breakage at the stand edge. These trees would even have broken under a load of 20 or 40 kg mÿ2 at windspeeds ranging from 4.9 to 6.1 m sÿ1. Similar trees with tapers of 1 : 100 would break under a snow load of 60 kg mÿ2 with windpeeds of 8.6, 8.3, 8.2 and 8.1 m sÿ1. Trees with taper 1 : 80, would not break with any of these snow loads (windspeeds ranging from 11.9 to 14.1 m sÿ1 at the edge). Norway spruce required slightly greater windspeeds for similar tree heights and tapers than did Scots pine. On the other hand, the snow load was of greater signi®cance for Scots pine than for Norway spruce, which is obviously at least partly due to their different crown characteristics. Some birches suffered minor

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Fig. 3. Effect of stem tapering on stem breakage of Scots pines (A) at the stand edge and (B) within a stand, and correspondingly for Norway spruce (C and D) and birch (E, F), under a snow load of 60 kg mÿ2 (in medium-dense stands, see Table 1).

stem breakage from snow loading at windspeeds <9 m sÿ1 at the stand edge (Fig. 3). Furthermore, very slender Scots pines would have suffered stem breakage under windspeeds of 9 m sÿ1 within the stand (®ve tree heights from the edge).

3.2.2. Uprooting A slight taper also reduces the windspeed required to uproot individual trees when the soil is unfrozen. In addition, the windspeeds needed to uproot trees were somewhat lower than those needed to break a tree

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63

Fig. 4. Effect of stand density on stem breakage of Scots pines of height 12 m with a stem taper of 1 : 120 under a snow load of 60 kg mÿ2 (A) at the stand edge and (B) within the stand.

under snow loading, especially in the case of the taller trees. Scots pines with a taper of 1 : 120 and heights of 12, 16, 20 and 24 m, for example (at stand densities of 1280, 863, 690 and 552 stems haÿ1), required very low windspeeds of 4.5, 3.4, 3.0 and 2.6 m sÿ1, respectively, for uprooting under short-term snow loading of 60 kg mÿ2 (Fig. 5). Likewise, Scots pines with a taper of 1 : 100 and heights of 12, 16, 20 and 24 m were uprooted under a snow load of 60 kg mÿ2 at windspeeds of 7.4, 6.4, 5.9 and 5.5 m sÿ1 (stand density of 1280, 863, 690, and 552 stems haÿ1), whereas trees with a taper of 1 : 80 and having similar heights and densities required windspeeds of 10.4, 9.3, 8.8, and 8.5 m sÿ1 for uprooting under a snow load of 60 kg mÿ2, i.e. there was not a high risk of their being uprooted. Even snow loads of 20 to 40 kg mÿ2 were found to be enough to cause uprooting of slightly tapering trees at the stand edge, but when the snow load increased from 20 to 40 and 60 kg mÿ2, the windspeed needed to uproot the trees decreased by 20 and 35%, respectively (for Scots pines of height 12 m and diameter at breast height 10 cm at the forest edge). The effect of stand density on the absolute values for critical windspeed was not very great at the densities compared here, but it was somewhat more important for uprooting than for stem breakage (Fig. 6). The relative difference in windspeed for a Scots pine of height 12 m with a taper of 1 : 120, for example, was considerable: an increase of 18% in the

densest stand and a decrease of 18% in the sparsest one as compared with the medium density (1280 stems haÿ1). Norway spruces seemed to suffer just as much as Scots pines from uprooting under stand edge conditions when faced with the critical combination of snow loading and windspeed (Fig. 5). Under a snow load of 60 kg mÿ2, slightly tapering (1 : 120) Norway spruces with heights of 12, 16, 20 and 24 m in medium-dense stands needed windspeeds of 6.6, 5.2, 4.7 and 4.2 m sÿ1 to be uprooted. Trees with a taper of 1 : 100 were also quite likely to be uprooted. Those of height 16, 20 or 24 m in medium-dense stands at the forest edge would have uprooted by windspeed of only 8.0, 7.5 and 7.0 msÿ1. None of trees with a taper of 1 : 80 would have been uprooted with any snow load (windspeeds ranging from 9.5 to 19.6 m sÿ1). Thus, Norway spruce required slightly greater windspeeds than Scots pine for similar tree heights and tapers. Very slender Norway spruces (and also some with taper 1 : 100) would also have been uprooted under snow loads of 20 and 40 kg mÿ2 at windspeeds from 4.4 to 8.8 m sÿ1. Birch seemed to run a greater risk of being uprooted under a snow load of 60 kg mÿ2, while the soil was unfrozen, than of suffering stem breakage (Figs. 3 and 5), although trees of 12, 16, 20 and 24 m height with a stem taper of 1 : 120 at the forest edge (stand densities of 1280, 863, 690 and 552 stems haÿ1) would still have needed windspeeds of 11.5, 8.7, 7.6 and

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Fig. 5. Effect of stem tapering on uprooting of Scots pines (A) at the stand edge and (B) within a stand, and correspondingly for Norway spruce (C and D) and birch (E, F) under snow load of 60 kg mÿ2 (in medium-dense stands, see Table 1).

6.8 m sÿ1 for damage to ensue. Thus, the risk of being uprooted still appeared to be small. On the whole, the simulation examples obtained from the regression models for individual species showed that uprooting

was most likely to occur at the stand edge in all three species, and that only some Scots pines and Norway spruces might suffer damage within the stand as well.

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65

Fig. 6. Effect of stand density on uprooting of Scots pines of height 12 m with stem taper of 1 : 120, under a snow load of 60 kg mÿ2 (A) at the stand edge and (B) within the stand.

4. Discussion and conclusions This research was aimed at formulating simple regression models based on simulated data to assess the risk of snow damage to Scots pine, Norway spruce and birch stands in terms of critical windspeeds needed to cause this damage. The mechanistic model used for the data simulation was assumed to be capable of properly describing the mechanism of damage (Peltola et al., 1998a). The resulting regression models showed that the risk of stem breakage and uprooting was explained by stem taper (d1.3/h), the short-term snow load and stand density in the case of all three tree species. The trees most likely to suffer stem breakage and uprooting were Scots pines and Norway spruces growing at the stand edge, where the risk was much greater than inside the stand due to the greater wind loading. In regard to these results, it is expected that a distance of `0' tree heights from stand edge used in computations would represent the stand edge conditions (for 0±1 tree heights) and a distance of `5' tree heights from the edge further inside the stand conditions, respectively. This assumption is done, because the wind loading will decrease most rapidly within 1±2 tree heights from the edge, i.e. as a function of distance from stand edge, but also depending on stand density and tree height (see Gardiner et al., 1998). Scots pines and Norway spruces varying in height from 12 to 24 m, especially those with only a slight taper of 1 : 120, were predicted to be the most vulner-

able under these circumstances (regardless of the stand density used in the simulations), particularly under a snow load of 60 kg mÿ2. They would have suffered stem breakage and uprooting at windspeeds of <9 m sÿ1 (at a constant height of 10 m above the ground). These ®ndings are in line with those of Petty and Worrell (1981), who suggested that slightly tapering trees (at least 1 : 120) run a risk of being damaged by a snow load of 60 kg mÿ2. Scots pine and Norway spruce with taper of 1 : 80 did not have a high risk of snow damage, because the snow would probably be dislodged from the tree crowns at the windspeeds needed to in¯ict the damage (unless ®rmly attached). Conifers with slightly tapering stems have also been found elsewhere to be the most susceptible to stem breakage and uprooting (Kramer, 1975; Abez and Prange, 1976; Cremer et al., 1982; Rottmann, 1985; Blackburn and Petty, 1988; Galinski, 1989; Macurrach, 1991; Peltola and KellomaÈki, 1993; Valinger et al., 1993). Trees with a taper of less than 1 : 100 and with highly elevated crowns have been mentioned as being particularly susceptible (Johann, 1981; Petty and Worrell, 1981; Petty and Swain, 1985; Rottmann, 1985). Thus, a pronounced taper would appear to increase resistance to breakage by the combined effects of snow and wind. The increase in the risk of snow damage with tree height to be seen in these predictions is in line with the earlier ®ndings by Suominen (1963), Persson (1972) and Valinger and Lundqvist (1992b). On the other hand, Rottmann (1985) claims that all slender trees of

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height 10±20 m are equally liable to snow damage. Suominen (1963) also suggested that Scots pine and Norway spruce stands, especially if of medium density, are equally likely to suffer from snow damage. On the other hand, Solantie (1994) suggested equal susceptibility for medium-sized Scots pine, Norway spruce and birches, i.e. about 15±20 m tall. Trees accustomed to growing in dense stands in high-risk areas seem to be especially liable to snow damage, because the trees in such stands usually taper relatively little and have asymmetric crowns (Persson, 1972; Petty and Swain, 1985; Rottmann, 1985; Valinger et al., 1994). In contrast, trees growing in stands with wider spacing, and/or intensively thinned stands, usually develop highly tapering stems and symmetric crowns, making them less vulnerable to damage (e.g. Petty and Worrell, 1981). Scots pines and Norway spruces were predicted here to be much more susceptible to snow damage than birches, which, being lea¯ess, had much less crown area for snow attachment and wind loading. Scots pine and Norway spruce have usually been found previously to be more likely to be damaged by snow than birch (Mikola, 1938; Suominen, 1963; Samuelson, 1970; Rottmann, 1985; Valinger and Lundqvist, 1992b). On the other hand, as suggested by Suominen (1963) and Gill (1974), birches could be seriously damaged, too, especially if in leaf. Scots pine and Norway spruce were not seen to differ much in their propensity for snow damage if of the same taper and height, although the former, particularly if the crown is asymmetric and short and the centre of gravity of the snow loading is concentrated outside the longitudinal axis of stem, would be more likely to being damaged by snow loading than the latter (Kangas, 1959; PerttilaÈ, 1987). The Norway spruce, with its greater crown depth, should be better able to resist damage, especially stem breakage, because the centre of gravity of the snow load is lower (see Rottmann, 1985). Under certain circumstances, however, the Norway spruce has been found to be even more likely to suffer damage than the Scots pine, e.g. in dense stands (Suominen, 1963), because a Norway spruce with a crown wider than that of a Scots pine can accumulate a heavier snow load. By contrast, the Norway spruce has also been found to be more resistant at high altitudes and latitudes than the Scots pine, as it usually develops a long, narrow crown,

adapted to cope with snow loading (Mikola, 1938; Rottmann, 1985). The present simulations assumed symmetric crown forms for all three species, however, and the crown depth was taken as being a constant proportion of tree height. On the other hand, Norway spruce, with its lower stem strength, i.e. lower modulus of rupture (Lavers, 1969), would require a smaller turning moment for stem breakage than pine, given an equivalent height and taper. Similarly, a smaller turning moment is needed to uproot a Norway spruce, because of its shallower root system, which is also usually con®ned to the weaker soil layers near the surface (see Laitakari, 1934; Kalela, 1949; Hakkila, 1972; MaÈlkoÈnen, 1977; Coutts, 1986). On the whole, the risk of being uprooted would seem to be greater than that of stem breakage in sudden snowfalls in late autumn, when the soil is still unfrozen, because the resistance to stem 3 , whereas breakage increases as a function of d1:3 resistance to uprooting increases more or less as a 2 (i.e. in terms of root-soil plate weight, function of d1:3 see Peltola et al., 1997). If thinning takes place, the relative risk of snowinduced (and wind-induced) damage increases immediately afterwards, due to increased wind loading in the sparser stand (Cremer et al., 1982; Laiho, 1987; Valinger and Lundqvist, 1992a; Valinger and Lundqvist, 1992b, b; Gardiner et al., 1997). This can also be seen in the models developed here, i.e. a much lower windspeed in percentage terms was needed to cause damage in the sparser stands. The risk of snow damage can be minimised, however, by identifying and avoiding the high-risk management of stands that would leave the trees liable to snow damage. Heavy thinnings should be avoided, for example, especially in high risk areas, and slightly tapering high-risk trees should be removed in the course of thinning, i.e. less intensive thinning in areas where large deposits of snow in the crowns of the trees can be expected (Persson, 1972; Valinger et al., 1993; Valinger and Lundqvist, 1994; Valinger et al., 1994). Regression models like those developed here may, in time, be able to offer simple tools for assessing the risk of snow-induced damage to trees. They can be used to predict the critical combination of snow load and windspeed at which trees may be broken or uprooted, and to estimate the probability of damage in a range of silvicultural situations provided that the

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67

Appendix A Variable statistics of regression models based on t-tests (variable names, see Table 2.) Model Stem breakage models

Scots pine:

Norway spruce:

Birch:

General:

Uprooting models

Scots pine:

Norway spruce:

Birch:

General:

Variable

SE

Coefficient

t

Sig p

Stand Snow40 Snow60 Taper Density Stand Snow40 Snow60 Taper Density Stand Snow40 Snow60 Taper Density Stand Snow40 Snow60 Taper Density Tresp 1 Tresp 2

0.134651 0.164913 0.164913 0.004120 0.000211 0.138097 0.169134 0.169134 0.004226 0,000216 0.325859 0.399094 0.399094 0.009971 0.000510 0.259521 0.317847 0.317847 0.007944 0.000406 0.317847 0.347847

0.563701 ÿ0.173875 ÿ0.292996 ÿ0.743449 0.092063 0.612795 ÿ0.081616 ÿ0.152413 ÿ0.740869 0.055689 0.594240 ÿ0.148929 ÿ0.256059 ÿ0.727428 0.129072 0.353474 ÿ0.083106 ÿ0.143746 ÿ0.438415 0.062129 0.054085 0.777385

34.521 ÿ9.222 ÿ15.539 ÿ45.529 5.638 37.834 ÿ4.364 ÿ8.149 ÿ45.742 59.152 37.923 ÿ8.231 ÿ14.152 ÿ46.422 8.237 28.553 ÿ5.814 ÿ10.056 ÿ35.415 5.019 3.784 37.762

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0007 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0002 0.0000

Stand Snow40 Snow60 Taper Density Stand Snow40 Snow60 Taper Density Stand Snow40 Snow60 Taper Density Stand Snow40 Snow60

0.117532 0.143947 0.143947 0.003596 0.000184 0.105020 0.128623 0.128623 0.003213 0.000164 0.263505 0.322726 0.322726 0.008063 0.000412 0.179613 0.219981 0.219981

0.596496 ÿ0.191690 ÿ0.318024 ÿ0.674552 0.234918 0.670119 ÿ0.094968 ÿ0.171659 ÿ0.630716 0.296954 0.632113 ÿ0.168550 ÿ0.275717 ÿ0.640313 0.266428 0.415505 ÿ0.102384 ÿ0.170812

36.128 ÿ10.055 ÿ16.681 ÿ40.855 14.228 46.836 ÿ5.748 ÿ10.390 ÿ44.082 20.755 36.881 ÿ8.517 ÿ13.932 ÿ37.359 15.545 35.044 ÿ7.478 ÿ12.476

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

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Appendix A (Continued ) Model

Variable

SE

Coefficient

Taper Density Tresp1 Tresp2

0.005498 0.000281 0.219981 0.219981

ÿ0.424157 0.174727 0.096303 0.753673

probability distribution of long-term extremes in wind and snow conditions at the site are known. In other words, they could be used to evaluate the risks likely to arise out of given forest management measures. Such models could also be linked to those predicting available growth and yields, which would make it possible to assess the risk of snow damage to individual stands at different stages in tree or stand development, based on critical windspeeds and snow loads and the longterm probability distribution of extreme wind and snow conditions at the site. This would also make it possible to evaluate the in¯uence of wind and snow loading on trees and stands, growing under different exposure conditions, brought about by forest management (e.g. by thinning), and to evaluate the risk while optimising tree or stand growth and yield. Acknowledgements This study was funded by the EC under the research project "Silvicultural strategies for predicting damage to forests from wind, ®re and snow: integrating tree, site and stand properties with geographical information systems and regional environmental models to evaluate options for forest management" (AIR3CT94-2392). The work was carried out at the Faculty of Forestry, University of Joensuu, co-ordinated by Prof. Seppo KellomaÈki. Support from the Faculty of Forestry, University of Joensuu, and the Department of Ecological and Environmental Sciences of the University of Helsinki, in Lahti, is acknowledged. The authors would like to thank Prof. Taneli KolstroÈm and Dr. Erik Valinger for their valuable comments and constructive criticism. Mrs. Erja Koponen and Mrs. Riitta Honkanen are acknowledged for drawing the ®gures for this paper and Mr. Malcolm Hicks for revising the English of the manuscript.

t ÿ35.774 14.737 7.034 37.672

Sig p 0.0000 0.0000 0.0000 0.0000

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