Forest growth models

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function for site index, which is based on stand age and varying with latitudes. Elfving and Nyström (1996b) found site index trends while applying this correction function to the

independent data of Norway spruce in northern Sweden. Elfving (1994) also introduced stand age as independent variable in the Swedish site index prediction models (Hägglund and Lundmark, 1977). Similarly, a site index trend was found for Norway spruce in the Austrian NFI data and age was therefore included as an additional explanatory variable in the site index prediction models (Schadauer 1999). A significant interaction of the site index trend with the temperature sum in these models also indicated a regional variation in the site index trends. Schadauer (1999) suggests that site index trends are more likely caused by a real shift in growing conditions over time than by inappropriate time trends implicit in site index models. Albert and Schmidt (2010) also described strong trends in site indices after 1970 for Norway spruce and common beech in Germany.

The presently applied Norwegian site index models for Norway spruce and Scots pine (Tveite, 1977; Tveite and Braastad, 1981) are based on data from experimental permanent sample plots located in eastern and middle Norway, while western and northern regions are not represented. The data used for these models also inadequately represent poorer sites and data were completely lacking from higher altitudes. Significant deviations of dominant height developments based on these models have been reported for western Norway (Blingsmo, 1985; Øyen and Nes, 1997; Orlund, 2001) and northern Norway (Tveite, 1994).

2.2. Forest growth models

Modelling growth and yield has been an intrinsic part of forestry research for many years, but still remains an area of important and active research (e.g. Porte and Bartelink, 2002;

Vanclay, 1994). Growth models are useful tools for forest managers for various purposes such as inventory updating, evaluation of silvicultural alternatives, harvest scheduling, and management planning in general (Garciá, 1994; Amaro et al., 2003). Based on the

management objectives, access to the computational facilities and input data, growth models may operate either at stand level or at individual tree level.

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2.2.1. Stand-level growth models

Stand growth is commonly measured in terms of stand basal area growth, stand height growth or stand volume growth. Stand growth can be modelled as a function of stand variables such as site index, stand age, stand diameter (e.g. quadratic mean diameter), stand basal area, stand density index, and number of stems (e.g. Pienaar and Rheney, 1995; Huuskonen and Miina, 2007; Gizachew and Brunner, 2011). Stand growth models do not describe growth dynamics of individual trees and are usually applicable only for even-aged and homogenous stands.

Various stand-level growth models for Norway spruce and Scots pine have been developed as basis for decision-making tools for forest management planning in Norway. Most of these models are based on data from long-term experimental plots and on supplementary data from temporary sample plots administered by the Norwegian Forest and Landscape Institute (Andreassen et al., 2008). The models are either basal area growth models (Eide and Langsæter, 1941; Brantseg, 1969; Nilsen and Haveraaen, 1982; Andreassen et al., 2008), diameter growth models (Braathe, 1955; Braastad, 1974; Blingsmo, 1984; Andreassen and Øyen, 2002; Gobakken and Næsset, 2002), or volume growth models (Braastad, 1975;

Blingsmo, 1988). Stand-level mortality and recruitment models have also been developed using NFI data (Eid and Øyen, 2003; Lexerød and Eid, 2005). Many of the above-mentioned models have been used in Norwegian stand-level simulators for forest management planning such as BESTPROG (Blingsmo and Veidahl, 1994), AVVIRK2000 (Eid and Hobbelstad, 2000; Eid and Hobbelstad, 2005), and GAYA (Hoen and Eid, 1990).

2.2.2. Individual tree based growth models

As opposed to stand-level growth models, individual tree based growth models describe growth of individual trees in a stand. The growth of an individual tree, i.e., the subject tree (also called a focal tree or target tree), within a stand largely varies due to competition from other trees. Competition varies with competitor species, number, size, distance, and direction.

Individual tree based growth models are usually developed to describe growth dynamics for structurally complex and heterogeneous stands (Wykoff, 1990; Pretzsch et al., 2002; Uzoh and Oliver, 2006; Bollandsås and Næsset, 2009). In these models, the potential growth of individual trees is reduced by competition indices, which may be either spatially explicit (also called distance dependent) (Bella, 1971; Biging and Dobbertin, 1992; Ledermann and Stage,

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2001; Rivas et al., 2005) or spatially non-explicit (also called distance independent) (Wykoff, 1990; Uzoh and Oliver, 2006; Bollandsås and Næsset, 2009). Only spatially explicit

individual tree based growth models are sensitive to differences in the spatial arrangement of the trees.

Most individual tree based growth models describe radial growth at breast height. Radial growth of a tree is more affected by competition than height growth. Consequently, few individual tree height growth models have been developed (Hasenauer and Monserud, 1997;

Pretzsch et al., 2002; Fahlvik and Nyström, 2006; Nord-Larsen, 2006b; Uzoh and Oliver, 2006; Ritchie and Hamann, 2008; Vaughn et al., 2010). Except for a few (e.g. Hasenauer and Monserud, 1997; Pretzsch et al., 2002), all these models are based on limited data regarding quantity and representativeness. In recent years, data from NFIs supply repeatedly measured heights of individual trees. A weakness of such data, however, is that large measurement errors are involved. Alternatively, stem analysis data free from these errors could be used for modelling. However, stem analysis data suffer from missing descriptions of the competitive situation over time and seldom represent larger areas.

All existing individual tree based growth models in Norway are diameter growth models (Bollandsås, 2007; Bollandsås and Næsset, 2009) or basal area growth models (Andreassen and Tomter, 2003). Also mortality models for individual trees have been developed (Eid and Tuhus, 2001; Bollandsås, 2007). All these models are based on NFI data. Individual tree height growth models for Norway are lacking. The individual tree based forest simulator - T (Gobakken et al., 2008) developed for Norway comprises various modules (growth models, mortality models, recruitment models, height-diameter models, volume functions). The diameter growth models (Bollandsås, 2007) are driving the simulation processes. Dominant height growth models (Tveite, 1977; Tveite and Braastad, 1981) are used to predict height for individual trees in young stands, assuming all trees of a certain age and site index to attain the same height. For older stands, height-diameter models (Bollandsås, 2007) are used to predict heights. Since the presently applied height-diameter models (Bollandsås, 2007) lack sample plot-level random effects and dominant height as a covariate, the models are likely to be biased.

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Many individual tree based forest simulators have been developed for forest management planning in other European countries. Examples of such simulators are BWIN (Nagel, 1997) and SILVA (Pretzsch et al., 2002) for Germany, MOSES (Hasenauer et al., 2006) and PROGNAUS (Monserud and Sterba, 1996; Sterba and Monserud, 1996) for Austria,

HEUREKA (Lämås and Eriksson, 2003) for Sweden, and MOTTI (Hynynen et al., 2005) for Finland. SILVA and MOSES are spatially explicit and comprise many different models with different purposes (growth, crown expansion, mortality). PROGNAUS is a spatially non-explicit simulator and comprises various models (growth models, mortality, dynamic crown ratio, harvesting). HEUREKA and MOTTI are spatially non-explicit simulators. HEUREKA comprises production modules (growth models, volume functions, mortality models, and recruitment models), a treatment module (silvicultural and harvest operations), and an optimization module. In HEUREKA, simulations are also possible for different climate scenarios.

A number of challenges are still associated with all these simulators. In addition to growth, the simulators should be able to precisely simulate regeneration and recruitment, probabilities of survival or mortality, and wood quality (e.g. annual ring width, wood density, knot width), as well as growth and other processes under changing site and climate conditions. Climate sensitive forest models (e.g. Mäkelä et al., 2006; Albert and Schmidt, 2010; Schmidt, 2010;

Schmidt et al., 2011) may be implemented in the simulators. Process-based growth models accurately predicting wood quality (e.g. Mäkelä and Mäkinen, 2003; Kantola et al., 2007) may also be used as growth modules in the simulators. Another challenge related to forest simulators is the availability of regional data for region-specific models and predictions.

Getting appropriate individual tree information as required for different simulators may also be difficult. For such cases, missing input data may be generated with algorithms (e.g.

STRUGEN developed by Pretzsch (1997)).

Tree growth is determined by the availability of growth resources such as moisture, nutrients, light, and temperature. The radial growth of a tree is more affected by competition than height growth. The radial growth response to competition is nearly linear. Height growth of dominant trees, however, is often not affected by competition, although height growth of suppressed trees may be significantly reduced when competition increases above a certain level (Brunner and Nigh, 2000). Thus, height growth has an asymptotic response to

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competition. In order to model individual tree growth, expected height growth of individual trees may be obtained through the potential growth reduction approach. Potential height growth is either estimated simultaneously with a modifier for competition (Courbaud et al., 1993; Hasenauer and Monserud, 1997; Huang and Titus, 1999; Uzoh and Oliver, 2006;

Vaughn et al., 2010) or it is estimated separately (Biging and Dobbertin, 1992, 1995; Pretzsch et al., 2002).

Earlier individual tree based growth models assumed certain effects of distance (e.g. Pukkala 1989) and size ratio (e.g. Hegyi 1974) in competition indices, but recent models estimate distance effects and size ratio effects from the data (Miina and Pukkala 2000; Canham et al.

2004; Boyden et al. 2008; Bøhler et al., 2008; Richards et al., 2008; Peltoniemi and Makipaa 2011; Pommerening et al. 2011; Sabatia and Burkhart 2012). For models describing

individual tree height growth, the effects of distance and size ratio have rarely been estimated from the data.