Comparison of a physiological model and a statistical model for prediction of growth and yield in boreal forests

Ecological Modelling 161 (2003) 95 /116 www.elsevier.com/locate/ecolmodel

Comparison of a physiological model and a statistical model for prediction of growth and yield in boreal forests J. Matala a,*, J. Hynynen b, J. Miina c, R. Ojansuu b, H. Peltola a, R. Sieva¨nen b, H. Va¨isa¨nen a, S. Kelloma¨ki a a

Faculty of Forestry, University of Joensuu, P.O. Box 111, FIN-80101 Joensuu, Finland b Finnish Forest Research Institute, P.O. Box 18, FIN-01301 Vantaa, Finland c Finnish Forest Research Institute, P.O. Box 68, FIN-80101 Joensuu, Finland

Received 29 August 2001; received in revised form 5 August 2002; accepted 19 August 2002

Abstract The structural and functional properties of a physiological model (FinnFor) and a statistical model (Motti), developed independently, were analysed in order to assess whether the former would provide the same prediction capacity as the latter, which is based on a huge body of long-term inventory data. The predictions were compared in terms of (i) stand-level variables, (ii) analysis of volume growth graphs, and (iii) stand structure variables (diameter and height distributions). Both unmanaged and managed (thinned) stands of Scots pine (Pinus sylvestris ), Norway spruce (Picea abies ) and silver birch (Betula pendula ) growing on medium-fertility sites in central Finland were used for the comparison. In general, the outputs of the models agreed well in terms of relative growth rates regardless of tree species, with the implication that both predict competition within a stand and the effect of position on tree growth in a similar way. The statistical model was stable in its predictions, but not as sensitive to initial stand conditions and management as that based on physiological processes, but the two models agreed well in their dynamics and predictions. The processbased model may therefore be applied to practical management situations, in order to achieve more precise predictions under changing environmental conditions, as in the case of climate warming. On the other hand, some elements of process-model thinking could be incorporated into statistical models in order to make these responsive to changing conditions. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Physiological model; Statistical model; Growth; Yield; Boreal forests; Model comparison

1. Introduction * Corresponding author. Tel.:
/358132515320; fax:
/ 358132514444. E-mail address: [email protected] (J. Matala).

Process-based models of forest growth (physiological models, gap models, etc.) have been devel-

0304-3800/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 3 8 0 0 ( 0 2 ) 0 0 2 9 7 - 1

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oped until now (e.g. Ma¨kela¨ and Hari, 1986; Ryan et al., 1996; Kelloma¨ki and Va¨isa¨nen, 1997) independently of traditional statistical forest growth and yield models. These two approaches have the same aim, of providing tools for forest resource management and related research. Process-based models, which set out from processes such as photosynthesis, use weather and soil data as inputs and are mainly used to predict the productivity of sites (environmental conditions) and variation in productivity between years, for example (McMurtie et al., 1994; Mohren and Burkhart, 1994; Landsberg and Waring, 1997). Process-based models are also widely used in projections of the impacts of climate change on forests (Kelloma¨ki et al., 1997; Kirschbaum, 2000; Lindner, 2000) but they are typically of a complex structure, with many parameters (Mohren and Burkhart, 1994; Battaglia and Sands, 1998; Sands et al., 2000; Ma¨kela¨ et al., 2000a) and seldom provide a real option for practical management. Whenever locally focused predictions of forest growth and yield are needed, statistical models are preferable (Mohren and Burkhart, 1994) although these entail the assumption that the environment and the management practices undertaken remain the same as had prevailed in the past. The recent heightened awareness of the influence of changing environmental conditions on forest growth has led to increased interest in applying process-based models to forest management (Landsberg and Waring, 1997; Ma¨kela¨, 1997; Ma¨kela¨ et al., 2000a,b; Sands et al., 2000). The degree to which such models could become useful tools for this purpose nevertheless depends on how well they can answer the questions dealt with in forest management; i.e. process-based models should be able (i) to utilise inputs that can be readily obtained from current forest measures and (ii) to give outputs that are relevant to planning and management. Furthermore, special attention should be paid to validation, because processbased models typically consist of numerous interrelated components, making thorough validation difficult for technical reasons or due to the absence of appropriate data. The predictions given by process-based models have usually been validated against short-term

stand-level fluxes of water, carbon and nitrogen measured at intensively studied sites, for example (Van Grinsven et al., 1995; Ryan et al., 1996; Ma¨kela¨ et al., 2000a) or stand-level forestry data such as basal area (BA), volume or dominant height (Ma¨kela¨, 1988; Sieva¨nen, 1993; Landsberg and Waring, 1997; Valentine et al., 1997; Bartelink, 1998; Ma¨kela¨ et al., 2000a,b). Furthermore, the distribution of relative growth between trees of different size and position is among key questions involved in growth modelling when aimed at demonstrating the validity of the growth response to prevailing environmental conditions. In this respect, a method based on volume growth and survival graphs (VGSs) is relevant as it examines the performance of trees in different size classes (Sieva¨nen et al., 2000). VGSs reflect the properties of components such as photosynthesis, partitioning of growth and tree survival, and thus offer a means of evaluating the components of a model (growth or mortality) by comparing them with other models or measurements. In this context, the output of a well-validated statistical model based on a large body of inventory data may provide a solid base against which to compare the output of a process-based model. We set out here to compare a physiological model (FinnFor, Kelloma¨ki and Va¨isa¨nen, 1997) with a statistical model (Motti, Hynynen et al., 2002) in order to analyse whether the two approaches produce sufficiently similar results in terms of growth dynamics to provide a basis for further development of the Motti model to incorporate the effects of climate change. The comparison is based on (i) stand-level variables, (ii) analysis of volume growth graphs (VGs), and (iii) stand structure variables. The Motti model provides the baseline for the comparison, because its parameterisation utilises a huge body of inventory data on tree growth covering the whole of Finland in the interval 608/708N. latitude. The comparison applies to unmanaged and managed Scots pine (Pinus sylvestris L.), Norway spruce (Picea abies L.) and silver birch (Betula pendula Roth.) stands growing on medium-fertility sites in the southern boreal zone in central Finland.

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2. Outlines for the comparison 2.1. Outlines of the models In the physiological model (FinnFor) the dynamics of the forest ecosystem are linked directly to climate through photosynthesis, respiration and transpiration and indirectly through the hydrological and nitrogen cycles (Kelloma¨ki et al., 1993; Strandman et al., 1993; Va¨isa¨nen et al., 1994; Kelloma¨ki and Va¨isa¨nen, 1997). The hourly computations cover one entire year or several years, and thus represent both the active and the dormant seasons. The FinnFor model can provide predictions of forest growth in response to environmental conditions and management under current and changing climatic conditions. Parameterisation and validation of the model are based on findings reported earlier by other authors, data from long-term forest ecosystem and climate change experiments (Kelloma¨ki et al., 2000) measurements of short-term stand-level fluxes of water and carbon at intensively studied sites (Kramer et al., 2002) and stand-level forest inventory data. A comprehensive description of the FinnFor model is given in Kelloma¨ki and Va¨isa¨nen (1997). The statistical model (Motti), has been developed as a decision-support tool for use in standlevel analysis in forest management planning (Hynynen et al., 2002). The parameters were obtained from both extensive permanent sample plots and forest inventory data derived from series of permanent sample plots which form a subsample of the stands included in the 6th (1971 /1976) and 7th (1977 /1984) Finnish National Forest Inventories. Altogether, data from 3060 sample plots on mineral soils were employed, covering over 25 000 trees, of which 67% were Scots pines, 24% Norway spruces and 9% birches. Further complementary data on birch stands were obtained from repeated measurements of experimental stands. The Motti model uses a time step of 5 years. The main processes and variables typical of the FinnFor and Motti models for determining the dynamics of a tree stand are listed in Table 1. In simulations, similar input data on the number of

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sample trees, their diameter, height and age and the site characteristics of the stand could be used for both models, and the same mortality model and management schedules were employed in both simulations. 2.2. Description of environmental conditions in the stand 2.2.1. FinnFor The daily weather statistics needed for the simulations were provided using the weather simulator developed by Strandman et al. (1993), initialisation of which requires site-specific monthly and daily weather statistics. The weather parameters simulated consist of air temperature, cloudiness, solar radiation, precipitation, water vapour pressure, wind velocity and atmospheric CO2, with an hourly time resolution, except in the case of cloudiness and precipitation, which are calculated on a daily basis. For further details on the weather simulator, see Strandman et al. (1993). The radiation figures produced by the weather simulator constitute the main factor controlling the growth and yield of forests through photosynthesis. The radiation incidental on a tree crown is calculated as the sum of direct and diffuse radiation figures available for sample trees representing given cohorts. The computations of light are based on (i) the probability of diffuse and direct radiation passing through the crowns of other trees and reaching those of the given cohort, (ii) the length of the path through tree crowns in other cohorts, (iii) the thicknesses of the crown layers, and (iv) the shadow area produced on a given crown layer by another, as adopted from Oker-Blom (1985, 1986). Physiological processes (photosynthesis, transpiration) are linked to soil conditions through soil temperature (T), soil moisture and nitrogen, as described in Kelloma¨ki and Va¨isa¨nen (1997). The T and moisture gradients between the soil surface and lower soil layers drive the transfer of water and heat into the soil, using an hourly time step. The water on the soil surface represents direct precipitation (rain, snow), precipitation through the canopy and water from melting snow. Daily evaporation from the surface pool is calculated as

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Table 1 Conceptual comparison of some model processes and important variables in the present study

Purpose Species Time step Productivity Tree interactions Growth Mortality Stand

FinnFor model

Motti model

Growth and succession in natural and managed forests, subject to climate change Scots pine, Norway spruce, birch spp. Multiple (1 year, 1 day, 1 h) Dependent on the temporal patterns of radiation, temperature, water and N availability and CO2 Through photosynthesis, which is affected by shading and water and nutrient uptake Photosynthesis, respiration, mass allocation

Growth and succession in natural and managed forests, under current climate in different regions in Finland Scots pine, Norway spruce, birch spp. 5 years SI, site variables: TS, altitude, lake and sea index, fertility classes Stand density and within stand competition RDF, BA, RDFL BA and height growth determined by tree crown ratio and variables of productivity and tree interactions Individual tree survival, self-thinning Dg, SI Trees in cohorts represented by sample trees, homogenously distributed

Individual tree survival, self-thinning (Dg, SI) Trees in cohorts represented by sample trees, homogenously distributed

the daily latent heat flow divided by the latent heat of evaporation. The downward flow of heat is the sum of heat conduction and convection. The soil moisture is given in terms of volumetric water content and water tension in individual layers. Decomposition of litter and humus, defined according to the algorithm developed by Chertov and Komarov (1995, 1997), makes the nitrogen bound in organic matter (litter and humus) in the soil available to the trees. Litter represents dead organic material from any mass compartment (foliage, branches, stem, course roots and fine roots), and its rate of decomposition is determined by its moisture, nitrogen and ash content and the soil temperature. The parameterisation of decomposition at different sites is presented by Chertov et al. (in press), although only the values for sites of the Myrtillus type are employed in the present calculations. 2.2.2. Motti The site description is based on site factors denoted by variables describing the geographical locations of stands and variation between stands on a local scale. The geographical variation is caused mainly by climatic differences and the local variation by differences in soil characteristics. For geographical variation, a temperature sum (TS) with a threshold value of
/5 8C (TS) is used together with altitude, lake index and sea index. The TS for each location is predicted by the

interpolation method of Ojansuu and Henttonen (1983). Local differences in site productivity are described in terms of fertility classes (Kuusela and Salminen 1969) based on the forest site types of Cajander (1949). The site index (SI) is defined as the average height of 50-year-old site trees (at breast height), of all trees thicker than the mean diameter weighted by tree BA, and can be predicted with a height development model in which height development is a function of site variables , height (H ) being expressed as a function of stand age (A ) at breast height: H a exp(bAg );

(1)

where a , b and g are parameters. Parameter a determines the asymptotic maximum height and is a function of the site variables, while parameters b and g determine the shape of the model and are independent of the site variables. A basic value for SI for all tree species is predicted as a function of the site variables. The model can be calibrated to a given single-storey, even-aged stand, if the stand age at breast height, diameter distribution and height of one or more site trees are known. The measure of competition among trees within a stand is obtained in terms of stand and tree-level measures, i.e. (i) stand density (relative density factor (RDF) or (BA)) and (ii) density of all trees larger than the object tree (RDFL). RDF gives the ratio between the total growing space of trees and

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the available growing space, where growing space is determined by the average minimum growing space achieved by self-thinning. The self-thinning line is denoted as a modification of the formula of Reineke (1933): b

N  b0 Dkg1 ;

(2)

where N is the number of stems per hectare, Dkg is mean diameter at stump height weighted by the BA of the stand, and b0 and b1 are species-specific parameters. Eq. (2) is based on two assumptions, (i) that all the growing space in stands undergoing self-thinning is used completely, and (ii) that the minimum growing space depends on the tree diameter. This average minimum growing space is dependent on the average tree diameter with a minimum growing space: b1

gai  b1 0 dki

;

(3)

where dki is the stump height diameter of a tree i predicted as a function of diameter at breast height. Stand density is the sum of the minimum growing spaces of all trees in the stand: RDF 

n X

wj gaj ;

(4)

i1

RDFLj 

respiration are calculated in terms of daytime respiration (Rday), maintenance respiration (Rm) and growth respiration (Rg). Maintenance respiration (Rm) is calculated over a year as the sum of the hourly values for each tree organ. The yearly values of Rm are subtracted from the net assimilation in foliage, which is gross photosynthesis minus the respiration related to the photosynthetic process (Rday). The rest of the assimilates are available for growth and the consequent growth respiration (Rg), which is related to the growth of each organ. The growth of the stem is used to calculate its height (h , m), diameter (at breast height) (d , cm) and volume (v , m3). The height increment in year t (Dh) is calculated on the basis of the allocation of mass to the stem (mass growth) (DM , kg) and the current mass of the stem (M , kg) Dh(t)c

DM(t)d ; M(t)e

(6)

where c , d and e are species-specific parameters. Consequently, the current height is h(t)h 0


n X

Dh(t);

(7)

t1

where n is the number of cohorts and wj the number of stems per hectare in cohort j . In mixed stands, RDF is calculated separately for each tree species. The density of trees larger than the object tree in cohort j is calculated with the formula: nj X

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wi gai ;

(5)

i1

where nj is the number of cohorts larger than the object cohort j . RDFL is calculated only for all tree species together. 2.3. Growth of trees 2.3.1. FinnFor The rate of photosynthesis per unit area of foliage is calculated on an hourly basis by means of the biochemical model developed by Farquhar et al. (1980) and Von Caemmerer and Farquhar (1981), after which the losses of assimilates in

where h0 is the initial height (m) of the tree and n is the number of years simulated. Based on Marklund (1988), the mass of stem is given as a function of height and diameter:  M(d; h)exp b0
b1

 d
b2 h
b3 ln h ; d
l

(8)

where b0, b1, b2, b3 and l are parameters specific to each tree species. Using h from Eq. (8), the diameter can be solved from Eq. (8) as d(M; h)

lA(M; h) ; b1  A(M; h)

(9)

where l is a function of the parameters and variables A from Eq. (8) and   M A(M; h) ln : (10) exp(b0
b2 h
b3 ln h) The resulting values for height and diameter (at breast height) were introduced into the polynomial

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taper function (Eq. (11)) of Laasasenaho (1982) in order to obtain the radial growth along the stem with the consequent stem form: dl b1 x
b2 x2
b3 x3
b4 x5
b5 x8
b6 x13 d0:2h
b7 x21
b8 x34 ;

(11)

where d0.2h is the diameter at 20% of the height, dl is the diameter at a height of l m from the ground, x /1/1/l is the relative distance from the stem apex, and b1 /b8 are parameters specific to each tree species. The volumes of the stems are predicted by applying this polynomial model to the stem curve. 2.3.2. Motti The tree crown ratio (cr ), which is used as an independent variable in tree growth models, is predicted with a static model, being defined as the ratio of the length of the crown to tree height. The crown base is determined as the height of the lowest living branch over which the number of dead nodes is less than two. The crown ratio follows a non-linear equation cr1exp(fc (x)):

(12)

The formulation of the model restricts the predicted values to between 0 and 1. The effects of site, geographical location and stand and tree characteristics are included in the function fc(x), and the crown ratio follows a general age-dependent pattern, decreasing as the tree grows older. The stage of stand development is described by the mean height of site trees (Hs). The crown ratio within a stand at a given point in time varies according to the absolute tree size (d ) and competition between the trees (RDFL ), increasing with tree diameter, but decreasing with increasing stand density (RDF ), and also with increasing tree height in the model for broadleaved species. The effect of recent thinning is expressed by a categorical variable (TH ), and that of geographical location is incorporated into the models for conifers by means of the TS . Separate expressions for the fc model were developed for Scots pine, Norway spruce and silver birch. The equation for Scots pine, for example, is

fc (a1 a11 TH05 a12 TH510 )Hsa2 d a3 exp(a4 RDFL)TS a5 exp(a6 RDF )

(;13)

where TH0  5 is a categorical dummy variable referring to thinning treatment over the past 5 years and TH5  10 refers to thinnings that took place 5 /10 years earlier, Hs is the mean height of site trees, RDFL is the relative density factor summed for trees larger than the object tree, TS is the temperature sum, RDF is the relative density factor for the whole stand and a1 /a12 are parameters. The diameter growth of a tree is predicted with models for BA growth, the dependent variable in these calculations being the natural logarithm of BA growth during the next 5 years, on the assumption that the effect of growth factors on tree growth is multiplicative. The models were constructed as mixed linear models with a random plot effect. The fixed variables can be grouped as referring to site, phase of stand development, tree size, stand density, within-stand competition and stand treatment. The height growth of an individual tree is predicted with a model in which the reference growth is multiplied by a modifier function, i.e. the predicted height development of the dominant trees serves as a reference growth rate (Eq. (1)) to which the predicted height growth of individual trees can be related, as presented below for Scots pine and Norway spruce ih IHs [d=Ds]  (a1 IHsa2
a3 (cr=CRs)
a4 cr
a5 RDFL): (14) In Eq. (14) ih is the height increment of an individual tree during the next 5 years (m), IHs is the 5-year height increment of the site trees (increment of the 100 largest trees in the stand, i.e. the reference height growth), d is tree diameter, Ds is the mean diameter of the site trees, cr is the tree crown ratio, CRs is the mean crown ratio of the site trees, RDF is the relative density factor of the stand and RDFL is the relative density factor of trees larger than the object tree. Stem volume is predicted from the stem curve of Laasasenaho (1982), as in the FinnFor model, using Eq. (11). The stem curve is constructed using information on tree diameter and height.

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2.4. Mortality

Mortality attributable to tree age is predicted using the following model:

The mortality model used in the comparison originated from the Motti model (Hynynen et al., 2002). Mortality is predicted with models for the survival of individual trees and stand-level selfthinning. At the beginning of the simulation step, the probability of survival is calculated for each tree by means of two sub-models: (i) the withinstand competition model and (ii) the life-span model. At the end of the simulation step, the simulated stocking level is checked to ensure that it falls below the self-thinning line, which determines the maximum allowable number of stems in a stand with a given mean diameter. The ratio of maximum stem number to simulated stem number is then calculated. If this ratio is less than one, the predicted stem number will be multiplied by the ratio to reduce the survival and increase the mortality rates until the simulated number of stems equals the maximum allowable number. The mortality of individual trees in the 5-year growth period is obtained with a model that combines the probability of mortality attributable to competition (pcomp ) with that attributable to age (pold ). It is assumed that the probabilities of mortality are independent and the total probability of death of a given tree: ptot 1(1pcomp)(1pold):

(15)

For the following model is used for mortality of Scots pine and Norway spruce caused by withinstand competition:   1 pcomp  ; 1
exp(a0
a1 d
a2 BA
a3 BAL) (16) where d is diameter at breast height (cm), BA is stand basal area (m2 ha1) and BAL is basal area of trees larger than the object tree (m2 ha 1). Mortality for deciduous tree species is predicted with the following sub-model:

pcomp 



101

pold 

f (age)  f (age
5) ; f (age)

(18)

where f is the age-dependent survival function. The numerator of Eq. (18) is the change in survival probability in the next 5 years and the denominator the survival probability at the beginning of the 5-year period. The survival function is f (age)

1   1
exp 10
10

age

 ;

(19)

0:82Amax

where Amax is the maximum age of a tree, and is dependent on the tree species. In order to ensure that stocking remains within reasonable limits during the simulation, the upper allowable level of stocking is controlled by the maximum density. This is done for 5-year intervals during the simulation by applying a modified form of the self-thinning model of Hynynen (1993) based on the relationship first introduced by Reineke (1933). The self-thinning model describes the relationship between stem number and mean diameter in an unthinned stand undergoing selfthinning. Separate mixed linear models were developed for different tree species. The following site-dependent model is employed for conifers: ˆ ln(N) a0
a1 ln(SI)
a2 ln(Dkg );

(20)

and for deciduous tree species: ˆ ln(N) a0
a1 ln(Dkg ):

(21)

In order to stabilize the performance of the model in the case of young stands, the mean diameter at stump height (weighted by stand BA at stump height, Dkg) was employed instead of the mean diameter at breast height.

1 1
exp(a0
a1 dk
a2 RDFL
a3 dk RDFL)

 :

(17)

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2.5. Tree stand description Both models treat the trees in a stand as cohorts represented by sample trees whose dimensions are the average values for the whole cohort under consideration. The age, diameter, height and number of trees in a cohort should be known from data on sample trees, and the structure of the crown (length, width and needle/foliage area density) also have to be calculated for the FinnFor model. Similarly, the crown ratio has to be in the initial data for the Motti model, or else it has to be calculated specifically (Eq. (12)). The stand database is updated at the end of each simulation period. Static models are employed to predict stem volume and to assess the technical quality of the stems. After all the tree-level variables have been updated, stand-level information is calculated by summing the tree information for the cohorts. 2.6. Validity of models The FinnFor model has previously been evaluated along with five other process-based models in order to determine their capacity for simulating CO2 and H2O fluxes monitored by means of the eddy covariance method (Kramer et al., 2002). The FinnFor model was found to be capable of simulating both fluxes quite accurately under boreal conditions. The Motti model was evaluated and calibrated during its development by reference to data from the temporary sample plots used in the Finnish National Forest Inventory (Hynynen et al., 2002). The performance of the models was then further analysed here based on simulations for few long-term thinning experiments run by the Finnish Forest Research Institute. Simulations were performed for three stands located in southern Finland: Scots pine in Kerima¨ki (61850?N, 29815?E), Norway spruce in Heinola (61808?N, 25857?E) and silver birch in Suomusja¨rvi (60826?N, 23844?E), extending over their whole measuring periods, with initial values representing the establishment of the plots. Thinning was simulated by using the BA in FinnFor and the stand density in Motti to define the thinning intensity. Unmanaged stands were available only for Scots pine and Norway spruce.

Comparisons were made in terms of stem volume and the total production of stem wood (Fig. 1). The simulated trend in stem volume in Scots pine stands deviated from the observed trend to some degree regardless of the model and form of management. The difference between the models was related to the sensitivity of FinnFor to the initial values for tree populations and the allocation of growth between height and diameter; i.e. diameter growth relative to height growth is smaller in FinnFor than in Motti, with the consequence that mortality proceeds more slowly. The simulations for Norway spruce and silver birch represented good agreement with the measured values in the case of thinned stands, but those for unmanaged Norway spruce stands deviated from the measured values. The deviation was related to the mortality of trees regardless of the model, in the sense that FinnFor hardly attained the self-thinning line at all and predicted a low level of mortality compared with Motti, which seemed to overestimate mortality. 2.7. Calculations for comparing the predictions All the stands were located in central Finland (Kuopio 63800?N, 27848?E). The initial age of the Scots pine and silver birch stands growing on medium fertility sites in the southern boreal zone in Finland used for the comparison was 9 years and that for the Norway spruce stand 14 years. The simulations were run for 100 years without management (‘unmanaged stands’) or with a species-specific thinning schedule (‘managed stands’). The thinning schedules were modified for individual stands according to the recommendations of the Finnish Forestry Development Centre Tapio (Luonnonla¨heinen metsa¨nhoito, 1994). The models were compared first in terms of stand-level variables (stand density, BA, stem volume, total yield, dominant height and mean height and diameter weighted by BA), while secondly, VGs were used to analyse the growth dynamics of the trees (Ma¨kela¨ et al., 2000b; Sieva¨nen et al., 2000). In this analysis the volume-growth rate of an individual tree/cohort (DV ) in a stand was divided by the maximum

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Fig. 1. Comparison of living stem volume and total yield predictions of the FinnFor and Motti models with observed stand data.

volume growth rate (DV *) in the same stand, giving a ratio (DV /DV *) that could be plotted against a variable indicating the relative size of the

tree (d/dmax and h /hmax) in order to analyse the relationships between tree size and growth in stem volume for individual trees. Thirdly, stand struc-

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ture was used to analyse the diameter and height distributions at different phases of stand development. The ages of the pine and birch stands selected for the two latter comparisons were 39, 79 (only for VGs) and 109 years (only for distribution comparisons) and of the spruce stand 44, 84 (only for VGs) and 114 (only for distribution comparisons). These simulation ages were selected for two reasons: to be able to compare the model dynamics between a young and an old stand; and to give enough time for the model dynamics to affect the development of the stands after thinning.

3. Results 3.1. Comparison of stand characteristics The development of the Scots pine stands was predicted in a fairly similar manner by both models (Fig. 2), i.e. the number of trees in unmanaged and managed stands was virtually the same (Fig. 2A) and total stem wood production was only slightly higher in FinnFor than in Motti (7 and 8%, respectively) (Fig. 2D). The stem volume in unmanaged stands was the same in both models, but that in the managed stands was about 13% larger in FinnFor (Fig. 2C). There were also clear differences between the predictions in terms of height, diameter and BA (Fig. 2B, E, F and G), i.e. FinnFor predicted a greater dominant height, somewhat more slender stems and a smaller BA than did Motti and was also more sensitive to management, so that the dominant height at the end of the rotation was 0.5 m greater in the managed than in the unmanaged stand and the mean height 1.7 m greater. The trends in BA and stem volume were similar in unmanaged and managed silver birch stands regardless of the model (Fig. 3B and C). On the other hand, Motti predicted a smaller number of trees than FinnFor with both management schedules, but a larger total stem wood production (Fig. 3A and D). Furthermore, Motti predicted a greater dominant height, mean height and mean diameter for the birches than did FinnFor (Fig. 3E, F and G). Management did not affect the

trend in dominant height in the birches in either model, although it increased the mean diameter in both (Fig. 3E and G). The largest differences between FinnFor and Motti were found when simulations were performed for Norway spruce stands (Fig. 4), predicted tree growth being slower in the FinnFor model, as demonstrated by the total stem wood production, especially in the unmanaged stand (Fig. 4D). On the other hand, the results of the two simulations regarding thinned Norway spruce stands were quite close to each other, e.g. in stem volume and total stem wood production (Fig. 4C and D). Tree height was quite responsive to thinning in FinnFor, i.e. the dominant height was 3.8 m greater in the managed stand than in the unmanaged one (Fig. 4E). Although diameter was increased by thinning regardless of the model (Fig. 4G), the capability of the trees for reacting to the increased growing space available after thinning through height and diameter growth was larger in FinnFor, especially during the late part of the simulation period (Fig. 4E, F and G).

3.2. Comparison of volume growth rate in relation to tree size The VGs graphs (Figs. 5 /7) were in general slightly convex in shape, indicating dependence of volume growth on the size of the tree. The shapes, slopes and locations of the graphs for the unmanaged Scots pine stand (Fig. 5), were almost the same in both models, but some differences existed, i.e. the height ranking in FinnFor corresponds to the ranking of the rate of volume growth, whereas in Motti the diameter ranking was more in line with the latter. On the other hand, Motti produced a narrower diameter and height distribution than did FinnFor. The VGs for the unmanaged silver birch stands were similar with both models (Fig. 6), but growth rates in the managed stand varied between trees less in FinnFor than in Motti, as seen in the higher location of the FinnFor curve for year 79. In the case of the Norway spruce stands, the differences

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Fig. 2. Characteristics of unmanaged and managed Scots pine stands.

between the models lay in the fact that the lines in FinnFor were longer, indicating wider diameter and height distributions (Fig. 7), and the variation

in growth rate between trees in the stand was also greater in FinnFor, as indicated by the lower position of the VGs lines (Fig. 7).

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Fig. 3. Characteristics of unmanaged and managed silver birch stands.

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Fig. 4. Characteristics of unmanaged and managed Norway spruce stands.

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Fig. 5. Volume growth of sample trees (cohorts) related to maximum volume growth of a tree within a stand as a function of relative tree diameter and height in Scots pine stands.

3.3. Comparison of diameter and height distributions In the Motti model, diameter was almost normally distributed over the rotation in the unmanaged Scots pine stand, whereas in FinnFor the diameter distribution had two peaks in the course of simulation, especially in the unmanaged stand (Fig. 8). The diameter distributions in the managed stands were more similar regardless of the model, but again two peaks were found in the FinnFor simulations (Fig. 9). A diameter or height distribution with two peaks is uncommon in Scots pine stands (Finnish Forest Research Institute, unpublished stand inventory). On the other hand,

the height distribution was wider in the FinnFor model than in Motti throughout the simulations regardless of management (Figs. 8 and 9). Both the diameter and height distributions in the silver birch stand were similar in shape regardless of management and or the model used (Fig. 10, results for the managed stand not shown here), although there were more trees in the smaller diameter class in the unmanaged birch stand in FinnFor than in Motti (Fig. 10). Motti also predicted taller trees, whereas FinnFor predicted a narrower height distribution at the end of the rotation (Fig. 10). Norway spruce showed much wider distributions for diameter and height than the other species (Fig. 11), which is to be

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Fig. 6. Volume growth of sample trees (cohorts) related to maximum volume growth of a tree within a stand as a function of relative tree diameter and height in silver birch stands.

expected in view of the high shade tolerance of this species. Furthermore, FinnFor predicted wider diameter and height distributions in unmanaged spruce stands than did Motti. Management influenced the distributions by increasing the number of trees in the largest diameter and height classes regardless of the model (Fig. 11). 3.4. Evaluation of the performance of the models The predictions given by the models agreed well in terms of stem volume and total stem wood production in the case of the pine and birch stands, the largest differences being found regarding tree size and total stem wood production for Norway

spruce and in the size distributions for pine. In the birch stand, the predictions differed mainly in terms of tree size. Motti gave larger values for the height and diameter of birches than did FinnFor, which in turn simulated taller but more slender pines than Motti (Figs. 2 and 3). These differences are repeated in the stocking and total growth of tree stands (Figs. 2 and 3). In this sense, spruce clearly deviated from pine and birch, being predicted to be thicker in FinnFor than in Motti (Fig. 4). On the other hand, the diameter distribution showed that FinnFor simulated both larger and smaller trees than Motti did (Fig. 11). In particular, the number of trees in the smaller diameter classes was

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Fig. 7. Volume growth of sample trees (cohorts) related to maximum volume growth of a tree within a stand as a function of relative tree diameter and height in Norway spruce stands.

greater in FinnFor. This difference leads to larger stem volumes and total production in Motti (Figs. 4 and 11). Thinning increased the diameter of all three tree species regardless of the model (Figs. 2/4), and tree height (dominant height) also increased in the case of pine and spruce in FinnFor (Figs. 2 and 4) but not in Motti. This difference in the thinning response pointed to structural differences between the models, i.e. the height growth of the dominant tree in a stand is fixed according to site conditions in Motti (Eq. (14)), whereas in FinnFor it increases whenever the growth space increases. This pattern also holds well for birch, but to a lesser extent than for the conifers. This difference between the tree

species in FinnFor is partly caused by the different initial stand size distributions used, as the shape of a birch stand was evidently normally distributed, a change in growth conditions was unlikely to have any effect on dominant height because the latter would be determined by only a few large trees. The models agreed well in terms of relative growth rates (Figs. 5 /7), especially in unmanaged stands. This implies that competition within a stand and the effect of tree position on tree growth, with the consequent growth dynamics, are quite similar in both models even though the absolute values for tree volumes or dimensions differ to some degree. These differences are propagated in the performance of the mortality

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Fig. 8. Diameter and height distributions in the unmanaged Scots pine stand with age.

model, for example, in that Motti shows the trees to grow more in diameter than does FinnFor, with the consequence that the number of trees decreases at a higher rate in Motti than in FinnFor. This resulted in a further increase in mortality, with a somewhat larger total stem wood production in Motti, even though stocking attains virtually the same level in both models. Stem volumes were almost at the same level in these simulations, but the number of trees was smaller in Motti.

4. Discussion This study was an attempt to compare a physiological and a statistical model for predictions of forest growth in order identify differences

and similarities in their performance, the mechanisms behind these and possible differences in their predictive capacity. The comparison covered three levels: (i) stand-level variables, (ii) analysis of VGs (Sieva¨nen et al., 2000), and (iii) stand structure (diameter and height distributions). Stand-level analysis was widely used in the comparison relative to analyses based on VGSs and stand structure, both of which indicate the performance of individual trees in the stand (Korol et al., 1996; Lindner et al., 1997; Ma¨kela¨ et al., 2000a,b; Sieva¨nen et al., 2000). VGSs have an advantage for evaluation purposes in that structure, management and species-specific characteristics have only a minor influence on them (Sieva¨nen et al., 2000). The basic work of validating the models has been done separately elsewhere (Kelloma¨ki and

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Fig. 9. Diameter and height distributions in the managed Scots pine stand with age.

Va¨isa¨nen, 1997; Hynynen et al., 2002; Kramer et al., 2002), but is further elaborated here. Although the use of comparison as a single tool for validating models has been questioned (Rastetter, 1996), we found, like Bolliger et al. (2000) that comparisons of models that have different theoretical approaches are valuable in helping us to understand their behaviour. As stressed by Soares et al. (1995), however, thorough evaluation requires different approaches and criteria. In our case, since FinnFor has previously been evaluated using flux data (Kramer et al., 2002) and Motti with a large body of inventory data (Hynynen et al., 2002), our interest in developing these models side by side in future led us to place emphasis on comparisons between them as such and in terms of their dynamics.

The FinnFor and Motti models agreed well in the relative growth rates for all three tree species, and were also close to each other as regards volume growth rates in pine and birch stands. Motti, being a statistical model, was found to be quite stable in its predictions and not as sensitive to initial stand conditions and management as the FinnFor model was. These characteristics and differences between the models in predicting the growth of different tree species might lead to differences in success and in the predictions obtained if they were used in their current form as practical forest management tools. On the other hand, the predictions of a statistical model alone may fail in the future if environment and forest management practices change, because the parameterisation of statistical models is based on past

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Fig. 10. Diameter and height distributions in the unmanaged silver birch stand with age.

data. The fact that the predictions obtained with these models agreed well with each other and with the observed stand data under current conditions implies that it might be possible to incorporate certain elements of process thinking into statistical models to make more precise predictions for practical forestry under the changing climatic conditions of the future.

Acknowledgements This study was carried out under the Finnish Centre of Excellence Programme (2000 /2005) at the Centre of Excellence for Forest Ecology and Management (Project no. 64308), co-ordinated by Prof Seppo Kelloma¨ki, Faculty of Forestry, Uni-

versity of Joensuu. Support provided by the Academy of Finland, the National Technology Agency (Tekes) and the University of Joensuu is gratefully acknowledged as the work took the form of co-operation between the Faculty of Forestry, University of Joensuu, and the Finnish Forest Research Institute (Vantaa and Joensuu Research Centres) within the framework of the Project ‘Dynamics and Modelling of the Functioning and Structure of a Forest Ecosystem with Implications for the Sustainability of Forest Production and Climate Change Impacts’ (Project no. 47087), funded by the Academy of Finland. It is also related to the EC Project ‘Strategies for response to climatic change in the management of European Forests’ (SilviStrat, contract no. EVK2-CT-2000-00073). We also thank Ms Sanna

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Fig. 11. Diameter and height distributions in the unmanaged Norway spruce stand with age.

Leinonen for drawing the figures, Dr Joann von Weissenberg and Mr Malcolm Hicks for checking the language of different versions of this article and two anonymous reviewers for their constructive criticism and comments on a previous version.

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