Summary We built a simple tree growth model for Norway spruce (Picea abies (L.) Karst.) that describes the biomass and stem radial growth of one tree in a stand. Growth is controlled by an external height growth function that accounts for site quality. Crown recession is represented by an empirical func-tion that accounts for the limitafunc-tion to crown development caused by mechanical contacts with neighboring trees. The model describes biomass growth based on carbon budget (pho-tosynthesis, respiration and senescence) and carbon partition-ing between foliage, stem and root compartments. An internal regulation is introduced based on a functional balance between crown and root development. Stem annual growth is distributed along the stem by means of an empirical rule. Stem profile is the final output of the model and can be used to check the overall consistency of the model and as an aid in wood quality studies. The underlying assumptions of the model are de-scribed.
Keywords: carbon budget, process model, stem analysis.
Introduction
Most forest growth and yield models are empirical and are based on a statistical description of tree or stand growth (Houl-lier et al. 1991). Their validity is therefore limited to a particu-lar range of climatic, geographic and silvicultural conditions. Because these conditions vary from one region to another and are likely to change over time, it is necessary to update and calibrate these empirical models regularly. Such calibration usually requires generating large data sets from permanent plots and so there is a long delay before the models can be updated. It seems essential, therefore, to develop new, more biologically oriented models whose structure will reduce, or even avoid, the need for (and the cost of) periodic calibration procedures (Kimmins 1990, Sievänen 1993).
Process-based models, the result of a well-developed mod-eling approach, are often based on a detailed description of physiological processes, thus they are complex and mostly restricted to research or educational applications (Landsberg et al. 1991, Thornley 1991). We propose an alternative approach
intermediate between knowledge-based and empirically based modeling strategies. Our objective was to derive a forest man-agement-oriented model from general biological principles that can be fitted to empirical field data. Thus, our goal was not to describe tree growth processes in detail but to build a general model that approximates tree growth and can be extrapolated to varying growth conditions.
Most process-based growth models predict biomass growth (Reynolds and Thornley 1982, McMurtrie 1985, Mäkelä 1986) but do not provide a detailed description of the stem. Because radial growth is readily measured and because tree profile is of prime importance for timber quality (Väisänen et al. 1989), our model was designed to describe both radial and height growth and to predict the internal ring structure of the stem. One problem with this a priori choice is that the links between primary and secondary growth in the whole plant involve mechanisms and regulations that are not well understood. As stated by Sloboda and Pfreundt (1989), height growth is one of the unsolved problems in process-based growth modeling. To avoid this difficulty, Sievänen et al. (1987, 1988) and Sievänen and Burk (1991) used an allometric relationship between height and radial dimensions; Valentine (1985) and Mäkelä (1986) applied the pipe model theory; and Sloboda and Pfre-undt (1989) used an external height growth function.
We have built a single-tree model that is controlled by an empirical site-dependent height growth curve, which was fit-ted to field data, and by a mechanical constraint to crown development. The height growth function can be regarded as an expression of site quality and genetic control (cf. empirical forest growth models). The underlying assumptions of the model are explicitly stated and discussed in the context of potential applications and further refinements. We also com-pare the predictions of the model with field growth data.
Data Site
Data were obtained from an unthinned stand of pure, even-aged Norway spruce (Picea abies (L.) Karst.) located on a flat
Prediction of stem profile of
Picea abies
using a process-based tree
growth model
CHRISTINE DELEUZE
1and FRANÇOIS HOULLIER
21 Laboratoire de Biométrie, Génétique et Biologie des Populations, CNRS-URA 243, Université Claude Bernard Lyon I, 43 Boulevard du 11 novembre 1918, 69622 Villeurbanne Cedex, France
2 Unité Dynamique des Systèmes Forestiers, ENGREF, 14 rue Girardet, 54042 Nancy Cedex, France
Received April 12, 1994
area at Moncel-sur-Seille, near Nancy (northeastern France). The height of the dominant trees in the stand at 50 years was 26.9 m. The stand was 53 years old in 1991 when six trees (two dominant, two codominant and two suppressed trees; Houllier 1993) were selected, felled and measured.
Measurements
For each tree, we measured total height (H), diameter at breast height (D), crown length (Lc) and the radius of the horizontal projection of the crown along eight directions (Rc). Height growth was described throughout the life of the tree, based on the scars located at the top of each growth unit. The angle, length and horizontal extension of one branch whorl per growth unit were measured to describe crown morphology. Eleven or 12 disks were then cut from each tree to obtain a description of annual increment along the stem.
Model structure
The model (Figure 1) is recurrent, that is, the state of the tree at date t determines the growth between t and t + 1 and, hence, the state of the tree at date t + 1.
Description of the tree
The tree is regarded as a simple functional system with three compartments: foliage, stem and roots (cf. McMurtrie 1985, Sievänen et al. 1987, 1988, Sievänen and Burk 1991,
McMur-trie 1991). Foliage and stem are described by both dry weight and geometrical dimensions.
Photosynthesis and carbon assimilation
The annual dry weight increment depends on annual leaf photosynthetic production (P), which is given by:
P=σcWf, (1)
where σc is specific photosynthetic activity and Wf denotes leaf biomass, which we assumed was correlated with current-year leaf area, Sf, by the equation: Wf = df Sf, where df is leaf weight per unit area. We used Sf because (i) it is directly related to the area that intercepts light, and (ii) leaves are approximately structured in successive shells corresponding to each year (Mitchell 1975). Miller (1986) suggests that the growth of evergreens is correlated more closely with the amount of foliage located in the external shell (current-year foliage) than with the total amount of foliage, because nutrients are translo-cated from older to younger leaves and older leaves have a lower photosynthetic efficiency than younger leaves.
We assumed that net assimilation (∆W), which is the differ-ence between photosynthesis and growth respiration (R), is proportional to growth (with a constant γ). Thus where ∆W = P−R and R = γ∆W:
∆W= P
1+γ . (2)
Carbon partitioning
Net assimilation (∆W) is partitioned between foliage, stem and roots by means of allocation coefficients: λf for foliage, λs for stem and λr for roots. Conservation of mass in this system implies that:
λf+λr+λs= 1. (3)
For each compartment there is a biomass balance: net growth = growth allocation − turnover − maintenance respira-tion, where turnover is the proportion (si) of living biomass that is lost by senescence, and maintenance respiration is the respi-ration necessary to maintain living biomass.
For a stand model, stem senescence (ss) signifies the death of some trees, but for this single tree model, the tree either dies or survives; therefore we did not take senescence of the stem into account and set ss = 0. For root and foliage, maintenance respiration is proportional (with a constant ri) to foliage biomass (Wf) and root biomass (Wr). However, for the stem, maintenance respiration is correlated more with stem area (Ss) than with stem biomass (Ws) (Kinerson 1975):
∆Wf=λf∆W−(rf+sf)Wf, (4)
∆Wr=λr∆W−(rr+sr)Wr, (5)
∆Ws=λs∆W−rsSs. (6)
Functional balance
We used the principle of functional balance (Reynolds and Thornley 1982, Mäkelä 1986) to link the photosynthetic activ-ity of foliage to the assimilation of nutrients by roots. We supposed implicitly that (i) carbon and nutrients are used in a constant ratio for dry matter growth (this should be true for a large range of environmental conditions), (ii) assimilation of elements is correlated with their utilization, and (iii) storage of carbon and nutrients is constant in the tree during the time of simulation (these reserves are affected by environmental per-turbations, and this first model could be improved by the introduction of a storage compartment that would play the role of a buffer). The principle of functional balance assumes that the quantity of assimilated carbon (σcWf) is proportional to the uptake of nutrients by roots (σnWr):
πnσcWf=σnWr, (7)
where σn is the specific assimilation coefficient and πn is a constant.
Resolution: external height growth function and crown shape function
At this stage, the model contains eight state variables (P, ∆W, Wf, Wr, Ws, λf, γr, γs) but only seven equations. Moreover, we
need a geometrical description of foliar and stem compart-ments to estimate Sf and Ss.
The system could be solved either by introducing another constraint or another equation (e.g., using a hypothesis of growth optimization (McMurtrie 1985, 1991), or another physiological process, for example the pipe model assumption (Mäkelä 1986, Valentine 1985, 1987)), or by decreasing the number of state variables (e.g., setting the allocation coeffi-cients to constant values (McMurtrie 1985)).
A second possibility would be to link W and S by an allomet-ric relationship between biomass and area (the pipe model is such a relation (Shinozaki et al. 1964)) or between biomass and height (there are such empirical relations for stem but not for foliage: e.g., Ws = νD2H (Sievänen and Burk 1991)).
In principle, the pipe model theory solves both problems. However, we did not choose this solution because the range of conditions where it can be applied is not well defined. More-over the relationship between foliar mass and cross-sectional area at the base of the crown is altered by environmental variations (Granier 1981, Aussenac et al. 1981, Long and Smith 1984, Dean and Long 1986, Long and Smith 1988, Aussenac and Granier 1988), and may also be altered by thinning. We chose, therefore, to use an external description for height growth and crown development.
The height function is the Chapman-Richards equation:
H(t)=b1 parameter related to maximum height growth (b1b2 is the growth rate at beginning), and b3 is a form parameter that was set to 2.
We supposed that the shape of the crown does not change over the life of a tree and compared two simple equations: a cone (Equation 9) and a logarithmic function (Equation 10) proposed by Mitchell (1975) and Ottorini (1991):
B(h,t)=a(H(t)−h), (9)
In a closed stand, crown recession is determined by me-chanical and space-based tree-to-tree competition (Mitchell 1975, Ottorini 1991). Because we simulate only one tree, and do not have a detailed description of its neighbor trees in the stand, we used an empirical crown recession function (Fig-ure 2):
∆Hb=f(H)∆H
f(H)= c1
1+ c2
H
c3 ,
(11)
and c1 is the asymptotic value of the crown recession/height growth ratio, c2 is approximately the height where crown recession starts, and c3 is the shape parameter.
Equations 8, 9 or 10, and 11 allow the calculation of Wf for each year. The system can then be solved and ∆Ws determined. Stem growth is finally partitioned along the stem according to Pressler’s empirical rule (Figure 3), which is similar to the pipe model theory: ‘‘the area increment on any part of the stem is proportional to the foliage capacity in the upper part of the tree, and therefore is nearly equal in all parts of the stem, which are free from branches’’ (Assmann 1970).
Fitting the empirical external functions Height function
The Chapman-Richards equation was adjusted tree by tree with a nonlinear procedure (Table 1 and Figure 2) using a differential mode, i.e., annual height growth was adjusted instead of cumulated height.
Crown shape
The parameter a of Equation 9 was set to 0.197, which
corre-sponds to a constant angle of 11.17°, and Equation 10 was fitted, tree by tree, to crown data (Figure 4).
Use of Pressler’s rule as a method for estimating the base of the functional part of the crown
Our tree model requires information about the successive po-sitions of the base of the crown. The height to the base of the crown is not a usual measurement: in our data set we only knew its position in 1991. Moreover, it has not been established whether the theoretical base of the functional crown is directly related to the field measurements of the first living branch, the first living whorl or the first whorl free of any contacts with
Figure 2. Adjusted height growth curve (upper line) on height data (points) and theoretical crown recession (lower line) compared with adjusted crown base (points) using Pressler’s rule. Example for Tree a72.
Figure 3. Pressler’s rule: left is the cumulated (from top downward) foliar area (or mass), and right is the stem basal area increment.
Table 1. Estimated parameters for Equations 8 and 10. Abbreviations: N = number of experimental points, SSE = sum of squared errors, fixed denotes that this parameter was not estimated but fixed.1
Tree no. b1 b2 b3 N SSE a1 a2 a3 N SSE
a72 45.2431 0.03271 2 (fixed) 45 0.7638 0.828 (fixed) 1.1570 1.7884 17 0.7079
a88 39.5563 0.03666 2 (fixed) 45 1.6000 0.828 (fixed) 1.2690 1.8926 14 0.6757
a94 39.5255 0.03774 2 (fixed) 45 1.3167 0.828 (fixed) 2.0761 3.6428 10 0.2848
a124 44.3679 0.03542 2 (fixed) 45 1.6013 0.828 (fixed) 1.3231 2.0625 15 1.3647
a170 40.2753 0.03678 2 (fixed) 45 2.3114 0.828 (fixed) 2.3691 4.4032 12 1.2928
a201 44.6615 0.03345 2 (fixed) 45 2.2653 0.828 (fixed) 3.4941 7.6095 13 0.3601
1 We used the software ‘‘multilisa’’ developed in our laboratory by Jean Christophe Hervé.
neighbor trees (Colin and Houllier 1992). Therefore, to obtain information about the functional part of the crown and to determine the parameters for our crown recession function (Equation 11), we used Pressler’s rule.
Stem analysis provided an estimate of ∆as(h), the cross-sec-tional area growth of the stem at various heights h. According to Pressler’s rule, ∆as(h) is constant below the crown base and decreases above the crown base to ∆as(H) = 0. If we assume that the shape of the crown is conical, Pressler’s rule leads to a simple segmented model: the first segment (below the func-tional part of the crown) corresponds to a constant value, and the second segment (within the functional part of the crown) decreases linearly with crown height (Figure 5). We thus esti-mated the successive positions of the base of the functional crown by the following procedure. The stem analysis data were used to adjust the two linear segments for each date, the point where the two segments intersected being an estimate of the height of the base of the crown for each tree (see Figure 5).
Simulations
Parameters and initial conditions
Physiological parameters were obtained from the literature. Because Norway spruce is rarely studied in process models, some parameters were taken from other species (Table 2).The initial values of the model state variables were obtained from stem analysis data at age 20.
Simulation of internal stem structure
To simulate the model, a program was written in C++. Twelve simulations were done: two for each tree, that is one with a conical crown and one with the logarithmic function of crown shape. Crown and stem were assumed to be symmetric around the vertical axis and were geometrically defined by their hori-zontal radius at any height.
The program produces a schematic growth figure similar to a ‘‘herring-bone’’ (Figure 6). On the left side of Figure 6, each line represents the profile of the crown at one date; on the right
side of Figure 6, fine lines represent simulated stem profiles at each date, and thick lines represent the same but from observed stem analysis data at ages 25, 30, 40 and 50 years (simulation begins at age 20).
Radial growth at breast height versus date was drawn for each simulation, and compared with the observed stem analy-sis data (Figure 7).
Results and discussion
Qualitative validation: stem profile
For a conical crown, the simulated stem profile had a marked inflexion point at the top of the stem, a feature that does not appear in the data. This inflexion point was less evident when we used the logarithmic function for crown shape. Because the actual crown shape lies between the conical and logarithmic form, it is likely that the existence of this inflexion point in the simulation outputs was due to the Pressler’s rule which seems to be invalid in the top part of the stem. If Pressler’s rule, and Huber’s value (xylem cross-sectional area per foliar biomass above height h (Ewers and Zimmermann 1984)) are brought together, we find that the ratio of stem cross-sectional area increment/foliage biomass above is not constant and increases in the top part of the stem.
Quantitative validation
The comparison of simulated and actual diameter growth curves provided a first test of the model (see Figure 7). The intermediate parameters such as partitioning coefficients al-lowed an internal control of the model performance. Figure 8 shows a progressive stabilization of the coefficients to values (λf≈ 0.2, λr≈ 0.2, λs≈ 0.6), which were comparable to those obtained in other process models (Mäkelä 1986, McMurtrie 1991) and to proportions given by Cannell (1985, 1989). Improvement and perspectives
The model could be improved by better descriptions of foliage compartment (crown) and crown-to-stem relationships (Pressler’s rule) and by replacement of the analytic conic and logarithmic functions of crown profile by a true dynamic model that would relate branch growth to height growth and to distance to the tree top (Mitchell 1975). We could also replace the area of the crown, Sf, by the volume of the external shell of foliage, which includes the youngest leaves.
To study the effects of environmental changes, we could improve the functional balance with the introduction of a storage compartment. This would in turn require another equa-tion to solve the system. Because the utilizaequa-tion of carbon and nutrients is linked to water use in the tree, the models could also be improved by the introduction of a water compartment. The procedure used to assess the parameters of the model could be improved. Because our objective was to build a soundly based model rather than to make precise and accurate predictions, we focused on the qualitative behavior of the model rather than on its statistical properties. To obtain a more rigorous statistical evaluation of this complex model, we need Figure 5. Adjustment of Pressler’s rule (example for Tree a170, ring
Table 2. Summary of all of the parameters of the model.
Name Meaning Unit Value Source
Wf Foliage biomass kgDW Variable
Sf Foliage area m2 Variable
df Specific foliage area density kgDW m−2 0.17 Ceulemans and Saugier 1991, Granier 1981
Wr Root biomass kgDW Variable
Ws Stem biomass kgDW Variable
Ss Stem external area m2 Variable
ρ Specific weight kgDW m−3 390 Assmann 1970, Pardé 1980, Cannell 1989
P Annual photosynthetic production kgDW Variable
∆W Total annual growth kgDW Variable
R Growth respiration kgDW Variable
γ Growth respiration coefficient unitless 0.23 Mäkelä 1986, Cannell 1989
λi Partitioning coefficient of compartment i unitless Variable
σc Specific photosynthetic activity unitless 2.7 Ceulemans and Saugier 1991,
Saugier and Garcia de Cortazar 1991
σn Specific root assimilation activity unitless 0.04 Mäkelä 1986
πn Constant of functional balance unitless 0.01 Mäkelä 1986
rf, rr Maintenance respiration coefficient of foliage and root unitless 0.1 Mäkelä 1986
rs Maintenance respiration coefficient of stem kg m−2 0.01 Mäkelä 1986
sf Senescence coefficient of foliage unitless 0.2 Mäkelä 1986, Cannell 1989
sr Senescence coefficient of roots unitless 0.5 Mäkelä 1986, Cannell 1989
H(t) Total height at t m Variable
h Height in the tree m Variable
∆sa(h) Wood area increment at height h m2 Variable
b1 Asymptote of height growth m see Table 1
b2 Height growth parameter: maximum height growth year−1 see Table 1
b3 Height growth parameter: shape of the curve unitless see Table 1
c1 Asymptotic value of the ratio ∆Hb:∆H unitless 0.78
c2 Height where crown recession starts m−1 10
c3 Form parameter unitless 6
a Constant for conical crown unitless 0.1974
a1 First parameter for logarithmic crown unitless see Table 1
a2 Second parameter for logarithmic crown m see Table 1
a3 Third parameter for logarithmic crown m see Table 1
to use a global fitting procedure (Sievänen et al. 1987). The sensitivity of the model also needs to be assessed (e.g., with Monte-Carlo simulations, see Mäkelä 1988).
Conclusion
The model provides a simple framework that can serve as a basis for other process models: it allows us to check the consistency of global and simple physiological hypotheses with dendrometrical data (stem analysis). The assumptions of the model are clearly defined, so that its range of validity can be stated and it is possible to modify the model for a particular aim.
The model requires only an external height growth function (which describes site and genetic control) and physiological parameters (which could be extrapolated to any similar stand). The model provides a qualitative description of tree growth conditional on a few parameters (site index and crown reces-sion due to competition). Moreover, the simulation of the inner ring structure of the stem gives indications about wood quality, because wood density is closely linked to ring width in Nor-way spruce (Nepveu et al. 1988). The inner ring structure could be introduced in wood quality simulation software (e.g., the ‘‘SIMQUA’’ software developed by Leban and Duchanois 1990).
The model could also be incorporated in a stand growth model where the external crown recession function could be replaced by a dynamic process describing the tree-to-tree crown contacts. Building such a tree-and-stand model would require an analysis of the qualitative behavior of the model at the tree and stand levels, and a quantitative comparison of simulated growth to observed data at both levels (Sievänen and Burk 1991).
Acknowledgments
The authors are grateful to J.C. Hervé, who assisted with the fitting procedure run on software ‘‘Multilisa’’ and to D. Rittié and M. Ravart for field measurements.
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