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Modeling individual tree mortality for Austrian forest species
Forest Ecology and Management 113 (1999) 109±123
Modeling individual tree mortality for Austrian forest species Robert A. Monserud*, Hubert Sterba1 Rocky Mountain and Paci®c Northwest Research Stations, USDA Forest Service, 1221 SW Yamhill #200, Portland, OR 97205, USA Institut fuÈr Waldwachstumsforschung, UniversitaÈt fuÈr Bodenkultur Wien, Peter Jordan Strasse 82, A-1190, Vienna, Austria Received 2 February 1998; accepted 17 June 1998
Abstract Individual tree mortality models were developed for the six major forest species of Austria: Norway spruce (Picea abies), white ®r (Abies alba), European larch (Larix decidua), Scots pine (Pinus sylvestris), European beech (Fagus silvatica), and oak (Quercus spp.); a joint model for the remaining broadleaf species was also developed. Data came from 5-year remeasurements of the permanent plot network of the Austrian National Forest Inventory. Parameters of the logistic equation were estimated using maximum likelihood methods. For all species, we found the hyperbolic transformation of diameter (Dÿ1) to be highly signi®cant in predicting the high mortality rates for small diameter trees and decreasing mortality rates for larger diameters. For spruce, a quadratic transformation in D was needed to accurately model the increase in mortality observed for large, low-vigor trees with diameter >70 cm, which resulted in a U-shaped distribution. Crown ratio was also consistently signi®cant, except for oak. We likewise found basal-area-in-larger-trees (BAL) to be a highly signi®cant predictor of mortality rate for all species except ®r and oak. Predicted mortality rate increases as the basal area in larger trees increases and as crown ratio decreases. The resulting logistic mortality model had the same general form for all species, with the signs of all parameters conforming to expectations. In general, chi-square statistics indicate that the most important variable is Dÿ1, the second most important is crown ratio, and the third most important predictor is BAL. The relative importance of crown ratio appears to be greater for shade tolerant species (®r, beech, spruce) than for shade intolerant species (larch, Scots pine, oak). Examination of graphs of observed vs. predicted mortality rates reveals that the species-speci®c mortality models are all well behaved, and match the observed mortality rates quite well. The Dÿ1 transformation is ¯exible, as can be seen by comparing the rather different mortality rates of larch and Scots pine. Predicted and observed mortality rates with respect to crown ratio are quite close to the observed mortality rates for all but the smallest crown ratios (CR<20%), a class with very few observations. Finally, the logistic mortality models passed a validation test on independent data not used in parameter estimation. The key ingredient for obtaining a good mortality model is a data set that is both large and representative of the population under study, and the Austrian National Forest Inventory data satisfy both requirements. # 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Growth model; Mixed-species; Mortality; Stand simulation; Survival
1. Introduction *Corresponding author. Fax: +1-503-808-2020; e-mail: monserud/[email protected] 1 Fax: +43-1-47654-4242; e-mail: [email protected]
Recent statistics from the Austrian National Forest Inventory indicate that at least 40% of forest stands
0378-1127/99/$ ± see front matter # 1999 Published by Elsevier Science B.V. All rights reserved. PII: S0378-1127(98)00419-8
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have a mixed-species composition (Schieler and Schadauer, 1993). Furthermore, only 45% of the inventory plots were suf®ciently even-aged to allow usual site index determination (Monserud and Sterba, 1996). This lack of a stand age and site index on half of Austria's forests renders existing yield tables increasingly unreliable, and provided the impetus for the development of the individual-tree stand growth model PROGNAUS (Sterba et al., 1995; Monserud et al., 1997). To date, models have been developed for predicting basal area increment (Monserud and Sterba, 1996), height increment (Hasenauer and Monserud, 1997), crown ratio (Hasenauer and Monserud, 1996), natural regeneration (Schweiger and Sterba, 1997; Sterba et al., 1997), and harvesting (Monserud et al., 1997). In addition, the model has successfully passed a validation test (Sterba and Monserud, 1997), and has been the focus of an investigation on simultaneous equation systems (Hasenauer et al., 1998). The remaining key component is an accurate mortality model (Hamilton, 1990; Avila and Burkhart, 1992). Accordingly, our objective is to develop individual tree mortality models for the major forest species of Austria. We are looking for mortality models that have generality in two senses. First, we want to develop mortality models that are representative of all forest conditions in Austria, and for all major species. Second, we want to develop models that can be used not only in the PROGNAUS simulator, but in any stand simulator that uses a list of individual trees by species. Error propagation and budgeting analyses have shown that growth predictions are very sensitive to the underlying mortality model; furthermore, the contribution to total variability due to the mortality component increases as the projection period increases (Gertner, 1989). Guan and Gertner (1991a) considered this situation common in stand simulation modeling. Mortality remains one of the least understood components of growth and yield estimation (Hamilton, 1986). The key to a tree's survival is its genetic makeup and its environment (Spurr and Barnes, 1980). Growth modelers almost universally ignore a tree's genetic status (Monserud and Rehfeldt, 1990), as well as important environmental factors such as climatic extremes (e.g., wind, drought, killing frosts), insects, and diseases. Great detail is paid to environmental competition arising from neighboring trees,
however (Buchman et al., 1983), as well as measurable gross physical features of the tree and site. Perhaps mortality would appear less stochastic if relevant environmental variables were measured on permanent plots, and if the genetic status of the trees could be characterized. The literature on modeling mortality of forest trees is not small, but successes are rare. Realistically, mortality modelers mostly hope to capture the average rate of mortality, and relate it to a few reliable and measurable size or site characteristics. The key then is a large and representative sample of remeasured trees so that a rare event ± mortality ± can be observed frequently enough to predict it accurately. A representative sample must re¯ect both the full range in site variability as well as the diversity of management treatments in a given population (Hamilton, 1980). Because mortality data are most reliably and ef®ciently obtained from permanent plots, researchers are often forced to live with the limitations of the underlying permanent plot network. As a result, many studies rely on data from either unthinned plots (e.g., Zhang et al., 1997) or only lightly thinned plots (Dursky, 1997). Even if the data contain various treatments, the permanent plots often are clustered spatially and do not necessarily represent the average and dispersion of stands in the entire region of interest (e.g., Hasenauer, 1994). The most common methodology for modeling individual tree mortality is statistical. Generally, the parameters of a ¯exible non-linear function bounded by 0 and 1 are estimated using weighted nonlinear regression or a multivariate maximum likelihood procedure (Neter and Maynes, 1970). Although most cumulative distribution functions will work, the most popular is the logistic or logit (e.g., Monserud, 1976; Hamilton and Edwards, 1976; Buchman, 1979; Hamilton, 1986; Vanclay, 1995). Other applications have used the Weibull (Somers et al., 1980), the gamma (Kobe and Coates, 1997), the Richard's function (Buford and Ha¯ey, 1985), the exponential (Moser, 1972), and the normal or probit (Finney, 1971). Monserud (1976) found that both the probit and the logit produced similar results, even though the underlying functional forms are quite different. Vanclay's 1995 recommendation of the logistic for mortality models in tropical forests is surprisingly quite relevant to our Austrian temperate forest situation, for both cover a
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spectrum of different species mixtures and age structures, precluding the possibility of using either stand age or site index as predictor variables. Two additional procedures have been used to model individual tree mortality: recursive partitioning and neural networks. Unfortunately, neither has led to signi®cant improvement in our ability to predict mortality using classical statistical methods. Recursive partitioning is best known by the acronym CART (Classi®cation And Regression Trees; Breiman et al., 1984), although it is a similar classi®cation tool to the SCREEN algorithm used by Hamilton and Wendt (1975) for ef®ciently identifying potential independent variables and relationships. CART employs heuristic methods with binary classi®cation, with results presented as decision trees (Verbyla, 1987). Dobbertin and Biging (1998) compared CART to logistic regression for two species in northern California, and concluded that CART performed somewhat better, although the percentage of dead trees correctly classi®ed was very low. Working in a branch of arti®cial intelligence, Guan and Gertner (1991a, b) built a neural network mortality model that was as accurate as the corresponding logistic model with similar variables, and was better behaved because the same model form was not constrained to operate in all regions of the data space. Hasenauer and Merkl (1997) have recently compared a neural network to the logit for Austria, and found that both predict equally well, with a slight advantage to the neural network. 2. Methods We rely on the remeasured permanent plots of the Austrian National Forest Inventory for mortality and survival data (Forstliche Bundesversuchsanstalt, 1981, 1986). A systematic 3.89 km grid of permanent plots covering Austria was established in 1981±1985. Each year 20% of the grid locations are sampled (1100 clusters) in such a way that all of Austria is covered by the inventory each year. Each location was then remeasured from 1986±1990, exactly 5 years after establishment. The total inventory comprises 5500 clusters, consisting of 22 000 permanent plots. At each grid location, a cluster of four plots is located at the vertices of a square 200 m on a side. Using variable radius plot sampling (Bitterlich, 1948),
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all `in' trees larger than 10.4 cm in diameter are selected using a BAF of 4 m2 haÿ1. Trees are identi®ed by their distance and azimuth from plot center, their species, and size measurements. A hidden nail in the root collar at the ground line indicates that diameter at breast height is to be measured 1.3 m above it. All trees with diameters between 5 and 10.4 cm are recorded on a ®xed area plot located at plot center with radius 2.6 m, the limiting distance for sampling a 10.4 cm tree with BAF4. Trees smaller than 5 cm are not sampled. The survival status of the same individual trees is recorded, with mortality from natural causes distinguished from normal harvesting and thinning. Harvesting due to severe tree damage or death is coded as natural mortality. The hidden plot center is a buried metal rod located by a map and metal detectors. This hidden plot design is used to eliminate research plot bias (Bruce, 1977) and to ensure that the Forest Inventory is representative of both growing conditions and forest management throughout Austria. Repeated measurements from 1986±1990 were made on 43 615 trees. These 5-year observations of survival and mortality were converted to 5-year mortality rates using the standard estimator for unequal probability sampling (Cochran, 1963), which weights each observation by the number of trees per hectare it represents in the sample (Table 1). The Austrian Forest Inventory remeasured and evaluated the status of several thousand trees of the ®ve main species in Austria (Table 1): Norway spruce (Picea abies), white ®r (Abies alba), European larch (Larix decidua), Scots pine (Pinus sylvestris), and European beech (Fagus silvatica). We also considered Table 1 Total number of trees remeasured, 5-year mortality rate, and annual mortality rate, by species Species
Total no. of trees
Mortality rate 5-year (%)
1-year (%)
Spruce Fir Larch Scots pine Beech Oak Other broadleaf All species
26 699 1878 3015 4138 4484 784 2617 43 615
4.4 6.1 2.9 5.6 4.3 3.2 8.0 4.7
0.9 1.3 0.6 1.1 0.9 0.6 1.7 1.0
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building a model for oak (Quercus spp.), although the sample size is less than 800 trees. Finally, we combined the 2615 observations of the remaining broadleaf species into one additional tree class (consisting mostly of Carpinus, Fraxinus, Acer, Betula, and Alnus). Based on a sample of 43 615 trees, overall 5-year mortality rate is 4.7%, which corresponds to an annual rate of 1.0% (Table 1). The most common species, spruce, had a 5-year mortality rate of 4.4%. Conifer mortality rates (5-year) varied between a high of 6.1% for ®r and 2.9% for larch. The 5-year rate was 4.3% for beech, 3.2% for oak, and 8.0% for all other broadleaf species combined.
3. Analysis Individual tree mortality is a discrete event. A datum can have only the value 0 (live) or 1 (dead). A dichotomous dependent variable calls for special consideration not only in parameter estimation (Hamilton, 1974), but in the interpretation of goodness of ®t as well (Neter and Maynes, 1970). We take the classical approach to model the probability of mortality, the logistic equation: ÿ1 0 (1) P 1 eb X where b0 X is a linear combination of parameters b and independent variables X, and e is the base of the natural logarithm. We used the CATMOD procedure (SAS Institute, 1987), which estimates the parameters of the logistic equation using maximum likelihood methods. Although weighted nonlinear regression can be used to estimate the parameters of a probability of mortality function, maximum likelihood methods are well behaved and more straightforward. Usual measures of goodness of ®t such as the coef®cient of determination or the correlation coef®cient are not appropriate for dichotomous variables (Neter and Maynes, 1970); it does not matter how close a prediction for a tree is to 0 or 1 as long as it classi®es the observation correctly (Monserud, 1976). The appropriate statistic is Chi-square (Neter and Maynes, 1970). The key question in modeling is the speci®cation of the set of independent variables after the model form
and error structure have been hypothesized. We begin by intentionally excluding the two most traditional variables in forest stand modeling: site index and age. Our reasons are simple; we want to develop a mortality model that can be used to accurately simulate the development of almost any forest stand growing in Austria, where the majority of those stands now fail to meet the accepted de®nition of even-aged (Monserud and Sterba, 1996). Tree size, of course, already can be understood as the integrated response of the tree to site quality and age, and thus implicitly contains such site and age effects. Site-speci®c variables include slope, aspect, elevation, soil type, vegetation type, soil depth, and several others listed by Forstliche Bundesversuchsanstalt (1981, 1986). At ®rst we considered them as possible predictors, but we soon realized that this was risky. Although we have an enormous sample of live trees, the number of dead trees is still relatively small for all species except spruce. Our fear was that it would be very easy to over-parameterize the resulting mortality model. We were thus left with a reduced set of variables describing tree size, competition, and growth. Diameter (such as diameter at breast height, 1.3 m) is an important and reliable measure of a tree's size. For woody plants, the larger the individual the greater its chances of competing for scarce resources. Thus, mortality rate should decrease as diameter increases. Nearly all mortality models include this variable (e.g., Monserud, 1976; Buchman et al., 1983; Vanclay, 1991; Avila and Burkhart, 1992; Hasenauer, 1994; McTague and Stans®eld, 1994; Dursky, 1997). For most species, the mortality rate for the smallest trees is quite high, and declines rapidly as individuals survive to larger diameter classes. A simple linear function of diameter is not suf®cient to capture this nonlinear effect. The TWIGS-model (Miner et al., 1988) uses a highly nonlinear transformation of diameter to gain more ¯exibility in the logit. We hypothesized that a hyperbolic Dÿ1 transformation of diameter should accurately track both the large mortality rates for small trees and decreasing mortality rates for larger diameters. Because the largest trees of a given species are often the oldest individuals, there is likely a size past which vigor declines (as indicated by increment), senescence
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becomes apparent, and the probability of mortality increases (Harcombe, 1987). A very large sample is required to detect this effect in a normal forest inventory. Lorimer and Frelich (1984) conclude that the death-rate curve for forest trees is probably U-shaped. The general model of Buchman et al. (1983) allows for this effect. We hypothesized that this U-shaped senescence effect could be modeled by including terms for D and D2 in the exponent of the logistic (D.A. Hamilton, personal communication). Because stand age is not de®ned for over half of the sample stands, we cannot simply break out the confounding in diameter between the rapid decrease in mortality rate for small trees and the increasing mortality rate (senescence effect) for large, older trees. Tree age is not available, so both hyperbolic and quadratic transformations of diameter are needed to model these different effects. A second important attribute for survival is the size of the tree's crown as an indicator of tree vigor. Generally, mortality rate should be lower for trees with larger crowns. Because total height and height to the base of the live crown were measured on all trees in the ®rst inventory, we calculated a dimensionless crown ratio as the ratio of crown length to total height. Interestingly, many of the mortality models using crown ratio as an independent variable do not include a diameter increment term (Avila and Burkhart, 1992; Hasenauer, 1994; Zhang et al., 1997), and vice versa (Monserud, 1976; Buchman et al., 1983; Dursky, 1997), indicating that probably only one indicator of tree vigor is needed.
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Competition can be expressed by a variety of stand density measures. Wykoff et al. (1982) and Wykoff (1990) introduced basal-area-in-larger-trees (BAL), a tree-speci®c measure of density that has proven quite useful in modeling stand dynamics. BAL is a simple count of the stand's basal area that is from trees with a diameter larger than the subject tree's diameter. BAL for the largest tree is 0.0, and for the smallest tree equals stand basal area minus that tree's basal area. BAL was also used in the Lake States version of TWIGS (Miner et al., 1988). We also considered crown competition factor (Krajicek et al., 1961) as a potential predictor of mortality rate. Combining all these effects, the hypothesized mortality model has the same general form for all species: ÿ1 2 (2) P 1 e b0 b1 =Db2 CRb3 BALb4 Db5 D where P is the probability of mortality (5-year), D is diameter (cm) at breast height (1.3 m), CR is crown ratio, BAL is basal area in larger trees (m2 haÿ1), and b0±b5 are species-speci®c parameters to be estimated. Furthermore, D, CR, and BAL are values at the beginning of the 5-year period. 4. Results For all species, we found the hyperbolic Dÿ1 to be highly signi®cant in predicting mortality rate (Table 2). Furthermore, it behaved properly in matching the rapid decline from high mortality for the smallest diameters to a more gradual decline in
Table 2 Estimated parameters b0±b5 for mortality model 2 Species
Intercept
Variable 1/D
CR
BAL
D
D2
0.0425 (0.0187)
ÿ0.00081 (0.000255)
Spruce
2.1283 (0.3877)
ÿ10.0745 (1.6467)
3.8251 (0.1695)
ÿ0.0186 (0.00207)
Fir Larch Scots pine Beech Oak Other broadleaf
2.0985 4.4070 4.1076 3.5734 4.4508 2.9223
ÿ10.9085 (1.9438) ÿ12.9395 (2.2204) ÿ18.9714 (1.3834) ÿ13.9542 (1.5835) ÿ12.0041 (4.3602) ÿ8.4877 (1.5631)
3.9311 2.2039 2.3267 3.1339
(0.5469) (0.6963) (0.4963) (0.395)
ÿ0.0326 (0.0081) ÿ0.0234 (0.0061) ÿ0.0161 (0.0047)
2.0609 (0.4069)
ÿ0.0228 (0.0048)
(0.3199) (0.5097) (0.301) (0.336) (0.4685) (0.3154)
D is diameter (cm) at breast height (1.3 m), CR is crown ratio, and BAL is basal-area-in-larger-trees (m2 haÿ1). Standard errors are given in parentheses.
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mortality rate for the larger diameters (Fig. 1). It was clearly superior to a simple D term. Crown ratio CR was also consistently signi®cant, except for oak (Table 2). We likewise found BAL to be a highly signi®cant predictor of mortality rate for all species except ®r and oak. The effect was as expected: predicted mortality rate increases as the basal area in larger trees increases and as crown ratio decreases. By comparison to BAL, crown competition factor (Krajicek et al., 1961) was uniformly insigni®cant as a predictor. All parameters listed in Table 2 are highly signi®cant (P<0.001). For all species except spruce, the coef®cients of D and D2 in the exponent were not signi®cant, indicating that a senescence effect could not be detected. As an overall measure of goodness of ®t, the likelihood ratio ranged from 165 for oak to 7965 for spruce (Table 3), all of which corresponded to error probabilities >0.9999. To judge the relative importance of the variables, standard errors are listed in Table 2 and Chi-square values (Wald, 1943) are provided in Table 3. All parameter estimates conform to expectations. The coef®cient of Dÿ1 is negative in all cases (Table 2), resulting in decreasing mortality rates with increasing diameter. For spruce, the quadratic transformation in D effectively increases the mortality rate for trees exceeding 60 cm, resulting in a U-shaped distribution (Fig. 2). The coef®cient of CR is positive in all cases, resulting in increasing mortality rates as crown ratio decreases; and the coef®cient of BAL is negative in all cases, indicating that mortality rate will be higher as basal area in larger trees increases and a tree's competitive status is less favorable. The Chi-
square statistics in Table 3 reveal that the most important variable is breast height diameter (except for ®r), the second most important is crown ratio CR, and the third most important predictor is BAL (except for larch). The relative importance of crown ratio appears to be greater for shade tolerant species (®r, beech) than for shade intolerant species (larch, Scots pine, oak), with spruce intermediate. Examination of Fig. 1 reveals that the species-speci®c mortality models are all well-behaved, and match the observed mortality rates reasonably well. The Dÿ1 transformation is ¯exible, as can be seen by comparing the rather different mortality rates of larch and Scots pine in Fig. 1. Predicted and observed mortality rates with respect to crown ratio are examined in Fig. 3. Generally, the predictions are quite close to the observed mortality rates for all but the smallest crown ratios (CR<20%). The underestimates in the smallest crown ratio class for spruce, Scots pine, and beech could be ameliorated by replacing CR by the inverse of CR in Eq. (2), but would result in an overestimate for larger crown ratios. Furthermore, the inverse CR does not perform well for ®r and larch. We further examined the model for spruce and found that predictions in the 0±10% CR class were rather close (32% observed vs. 28% predicted mortality rates), with most of the underestimates in the 10±20% class (34% observed vs. 21% predicted mortality rates). This lack of a consistent error indicates that the model is well-behaved with respect to crown ratio. An examination of predicted vs. observed mortality rates with respect to basal area in larger trees (BAL) turned up no problems with model behavior for any of
Table 3 Chi-square values for the parameter estimates in Table 2 (model 2) Species
Likelihood ratio
Chi-square statistic Variable
Spruce Fir Larch Scots pine Beech Oak Other broadleaf
7965 749 700 940 1391 165 1365
1/D
CR
BAL
D
D2
37 31 43 82 78 8 29
509 52 10 33 63 ± 26
80 ± 16 6 12 ± 22
5 ± ± ± ± ± ±
10 ± ± ± ± ± ±
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Fig. 1. Observed vs. predicted 5-year mortality rate vs. diameter at breast height (Dbh) for the six major forest species in Austria.
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Fig. 2. Observed vs. predicted 5-year mortality rate vs. diameter at breast height (Dbh) for Norway spruce.
the species (Fig. 4). This generalization holds even for ®r and oak, whose mortality models do not include a BAL term. 4.1. Old growth spruce Because sample size for spruce was so large (n26 699), we were able to detect the classic Ushaped distribution expected by many (Buchman et al., 1983; Lorimer and Frelich, 1984; Harcombe, 1987; D.A. Hamilton, personal commun.). The observed spruce mortality rate declines in a regular fashion to below 2% for diameter classes up to 70 cm. The mortality rate then increases dramatically to 5.25% past 70 cm, with 21 dead trees out of 400 (Fig. 2). We examined these trees in detail. Fifteen out of the 21 dead trees are older than 140 years, the maximum age class coded in the inventory. The remaining six trees indicate a mortality rate of 1.5%, which is about the same rate as for the 60±70 cm diameter class. All 21 dead trees are from steep (median slope of 80%) Protection Forests that are at high elevations for spruce (1100±1600 m). Furthermore, nine of the dead trees were from stands described as being in the beginning stage of decline, with a general loss of vigor. From this, we conclude that for spruce larger than 70 cm, the
mortality process is fundamentally different than would be described in Eq. (2) if only the hyperbolic Dÿ1 term were used. Large old-growth spruce have a mortality rate nearly four times higher than would be expected from the decreasing rate normally associated with increasing size. The quadratic transformation in D increased the predicted mortality rate for these mature trees, resulting in the expected U-shaped distribution. These large, old trees lacking in vigor are reminiscent of the trees that exhibited lower survival with increasing diameter in the study of Buchman et al. (1983). We also observed a higher mortality rate in the largest diameter class for ®r and the other broadleaf species, but this high mortality rate resulted from the death of only two±four trees. In those cases, the rate for the largest diameter class is not signi®cantly different than the rate for the penultimate diameter class, and we could not support the quadratic diameter transformation in model 2. 4.2. A validation test Independent permanent plot data for validation were available from a thinning experiment in 22 mixed Norway spruce ± Scots pine stands from Litschau, in the Austrian part of the Bohemian Massif (Sterba and
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Fig. 3. Observed vs. predicted 5-year mortality rate vs. crown ratio for the six major forest species in Austria.
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Fig. 4. Observed vs. predicted 5-year mortality rate vs. basal area in lager trees for the six major forest species in Austria.
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Monserud, 1997). These data were not used for parameter estimation or model development, and were obtained from a separate sample with different objectives. Thus, they provide an independent set of observations for a validation test of the model. Management history included litter-raking until the 1950s, resulting in various degrees of soil degradation. Current stands were regenerated after clearcutting by planting Norway spruce. Volunteer spruce and Scots pine regeneration followed soon after, converting the plantations to mixed stands. Permanent research plots were established in 1977 and ®rst remeasured in 1982, a growth interval that included snow breakage in the winter of 1979±80. Dominant stand age in 1977 ranged from 10 to 110 years. Thinning was done in 1983. The next observation followed in 1987 and the third remeasurement was performed in 1992, thus providing three 5-year increment periods. Plot size varied from 400 to 2025 m2. While site factor variation between the plots was not large, there was considerable variation in dominant height (5± 30 m) and density (stem number between 450 and 28 000 trees haÿ1, basal area between 5 and 48 m2 haÿ1, and Crown Competition Factor (Krajicek et al., 1961) from 0 to 300). Thinning intensity in the plots varied from 6% to 420% of the 15-year basal area increment. The proportion of Scots pine varies between 0% and 73%. The plots are discussed in detail by Sterba and Monserud (1997). We compared our model predictions with observed mortality in the Litschau permanent research plots. Over the 1977±1992 observation period, mortality was somewhat higher in both Norway spruce and Scots pine than predicted by the model. The observed and predicted 5-year mortality rates for spruce were 6.1% vs. 5.6%; corresponding rates for Scots pine were 5.7% vs. 4.0% (Table 4).
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In Fig. 5 we compare observed with predicted number of dead trees by species for the 15-year remeasurement period. Predictions for spruce are close to the observed mortality, with no apparent trend with respect to crown ratio. Predictions for pine are underestimates, with greater mortality observed than predicted in the lower crown ratio classes. For both spruce and pine, mortality is underestimated in the smallest diameter classes. The situation between Scots pine and Norway spruce in the Litschau plots is interesting. First, some history is in order. Litter-raking in the Bohemian Massif ceased mid-century, which allowed for a return to the gradual buildup of organic matter and nutrients that facilitate normal soil building processes. Scots pine is well known as a species that can tolerate poor soil conditions, including both nutrient-poor sandy soils and water-logged bogs (Monserud et al., 1996). Formerly, clearcuts in the vicinity of Litschau were large, followed by plantations of Norway spruce. Because of the poor nutrient levels in the litter-raked soils, Scots pine established itself abundantly in these plantations, occasionally outnumbering the planted spruce. With the cessation of litter-raking and the increased levels of exogenous nitrogen deposition in recent decades (Kenk and Fischer, 1988; Skef®ngton and Wilson, 1988), soil conditions have improved and given a competitive advantage to spruce. Increment rates reported by Sterba and Monserud (1997) indicate that pine growth rates at Litschau have increased over expectation in the period 1977±92. The exact reason is unknown, of course, but contributing factors could be nitrogen deposition, improved soil nutrients, and climate warming. Sterba (1995) points out that overall site improvement will give a differential advantage to Norway spruce over Scots pine, for spruce has much greater potential growth, and is more tolerant of shade than the light-demanding Scots pine.
Table 4 Observed vs. predicted mortality rates (5-year (%) and 1-year (%)) for the Litschau permanent plots for the period 1977±92, by species Species
Spruce Scots pine
Mortality rate Observed No. dead
Observed 5-year (%)
Predicted 5-year (%)
Observed 1-year (%)
Predicted 1-year (%)
279 72
6.1 5.7
5.6 4.0
1.25 1.17
1.15 0.81
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Fig. 5. Observed vs. predicted mortality vs. diameter at breast height (Dbh), crown ratio, and basal area in larger trees for Norway spruce and Scots pine for the Litschau validation plots.
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Increased growth rates increase competition, with the net result that mortality rates should increase for Scots pine at a faster rate than for spruce (Sterba, 1995). This indeed happened at Litschau. In addition, the underestimation of mortality in the smallest diameter classes can be attributed to documented snow breakage in the winter of 1979±80. Snow breakage affects trees with the largest height/diameter ratios the most, usually the smallest diameter trees (PollanschuÈtz, 1974; Abetz, 1976). Based on the physics of tree form, Petty and Worrell (1981) con®rmed the large empirical literature in Central Europe documenting the relationship between large height/diameter ratios and the risk of snow breakage. The main canopy of Scots pines at Litschau was not dense enough to protect the small spruce from heavy snow. 5. Application Although our mortality model was developed as a submodel in the distance-independent stand simulator PROGNAUS (Sterba et al., 1995; Monserud et al., 1997), it should work equally well in a simulator with a different architecture (e.g., Sterba, 1983; EckmuÈllner and Fleck, 1989; Pretzsch, 1992). The key ingredient for obtaining a good mortality model is a large data set that is representative of the population under study. The Austrian National Forest Inventory data (Forstliche Bundesversuchsanstalt, 1981) satisfy this requirement. The one shortcoming of this sample is that it is restricted in time to only one decade, the 1980s. As additional remeasurements become available from future inventory cycles, the reliability of the time cross-section should be improved. The simulator architecture determines how mortality must be calculated, however. Because of the nature of the competition index calculation, mortality in a distance-dependent individual tree simulator must be a discrete event, with the tree either completely dead or alive. This is obviously a very stochastic process to simulate, especially when plot size is small. The stochastic rule compares the predicted mortality rate with a uniform random number on the interval 0±1; if the random number is less than the mortality rate, the tree is considered to have died (Weber et al., 1986). In contrast, mortality in a distance-independent simulator can be a continuous event, with the mortality rate
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Table 5 Percent correct classification, by species Species
Percent correct classification Live trees
Dead trees
Total
Spruce Fir Larch Scots pine Beech Oak Other broadleaf Total
88 90 89 92 91 92 81 89
44 41 46 42 38 25 42 42
87 89 88 90 89 91 79 86
continuously and smoothly reducing the number of trees/ha each sample tree represents. This advantage was ®rst seized by Stage (1973) in designing the Stand Prognosis Model. Thus, it is much easier to accurately predict mortality rates over large areas with distanceindependent models. Because one of the main applications of a stand growth simulator is to illustrate management alternatives for planning, this advantage is considerable. Weber et al. (1986) compared these stochastic and deterministic methods for determining mortality and found no practical differences in mean stand values. They concluded that there is no need for simulating multiple stochastic runs (with a distanceindependent model) that are later averaged when the main interest is in projected stand values. A second deterministic method can also be used to simulate mortality by using a threshold. If the threshold exceeds the estimated probability of mortality then the tree is considered dead. The best and most logical choice for a threshold is the average observed mortality rate for that species. Table 5 shows the result of this approach by using the overall mortality rate in Table 1 as the threshold for each species. We correctly classi®ed 89% of the live trees and 42% of the dead trees, for a total of 86% correct classi®cation (Table 5). This compares very closely to Monserud's (1976) results for northern hardwoods (viz., 88% live and 35% dead correctly classi®ed). Acknowledgements This research was conducted when Monserud was Visiting Scientist and UniversitaÈtslektor at the Institut fuÈr Waldwachstumsforschung in Vienna, on a grant
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from the Austrian Ministry of Agriculture and Forestry. Karl Schieler and Klemens Schadauer of the Federal Forest Research Center (Forstliche Bundesversuchsanstalt, Wien) generously made the Austrian Forest Inventory data available. We thank Dave Hamilton, Ralph Amateis, Karl Schieler, and the anonymous reviewers for helpful review comments. References Abetz, P., 1976. BeitraÈge zum Baumwachstum ± der h/d Wert. Forst und Holzwirt. 31(19), 389±393. Avila, O.B., Burkhart, H.E., 1992. Modeling survival of loblolly pine trees in thinned and unthinned plantations. Can. J. For. Res. 22, 1878±1882. Bitterlich, W., 1948. Die WinkelzaÈhlprobe. Allg. Forst-und Holzwirt. Zeitung Wien. 59, 4±5. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J., 1984. Classification and regression trees. Wadsworth, Belmont, CA. Bruce, D., 1977. Yield differences between research plots and managed forests. J. For. 75(1), 14±17. Buchman, R.G., 1979. Mortality functions. In: A generalized forest growth projection system applied to the Lakes States region. USDA For. Serv. N. Cent. Res. Stn. Gen. Tech. Rep. NC-49. pp. 47±55. Buchman, R.G., Pederson, S.P., Walters, N.R., 1983. A tree survival model with application to species of the Great Lakes region. Can J. For. Res. 13, 601±608. Buford, M.A., Hafley, W.L., 1985. Modeling the probability of individual tree mortality. For. Sci. 31, 331±341. Cochran, W.G., 1963. Sampling techniques, 2nd ed. Wiley, New York, 413 pp. Dobbertin, M., Biging, G.S., 1998. Using the non-parametric classifier CART to model forest tree mortality. For. Sci., in press. Dursky, J., 1997. Modellierung der Absterbeprozesse in Rein± und MischbestaÈnden aus Fichte und Buche Allg. Jagd- und Forstzeitung 168 (6±7) 131±134. EckmuÈllner, O., Fleck, W., 1989. Begleitdokumentation zum Wachstumssimulationsprogramm WASIM 1.0. Institut f. Forstliche Ertragslehre, Univ. f. Bodenkultur, Wien, 29 pp. Finney, D.J., 1971. Probit Analysis, 3rd ed. Cambridge University Press, New York, 333 pp. Forstliche Bundesversuchsanstalt, 1981. Instruktionen fuÈr die È sterreichischen Forstinventur 1981±1985. Feldarbeit der O Forstliche Bundesversuchsanstalt, Wien, 172 pp. Forstliche Bundesversuchsanstalt, 1986. Beiheft der DienstanweiÈ sterreichischen sung (Instruktionen fuÈr die Feldarbeit der O Forstinventur 1981±1985). Forstliche Bundesversuchsanstalt, Wien, 42 pp. Gertner, G.Z., 1989. The need to improve models for individual tree mortality. In: Proc. 7th Central Hardwood Conf. General Technical report NC-132, North Central Forest Experiment Station, USDA Forest Service, pp. 59±61.
Guan, B.T., Gertner, G.Z., 1991a. Using a parallel distributed processing system to model individual tree mortality. For. Sci. 37(3) 871±885. Guan, B.T., Gertner, G.Z., 1991b. Modeling red pine tree survival with an artificial neural network. For. Sci. 37(5), 1429±1440. Hamilton, D.A., 1974. Event probabilities estimated by regression. USDA For. Serv. Intermt. Res. Stn. Res. Pap. INT152. 18 pp. Hamilton, D.A., 1980. Modeling mortality: a component of growth and yield modeling. In: Brown, K.M., Clarke, F.R. (Eds.), Proc. Forecasting Forest Stand Dynamics Workshop, School of Forestry, Lakehead Univ, Thunder Bay, Ontario, pp. 82±99. Hamilton, D.A., 1986. A logistic model of mortality in thinned and unthinned mixed conifer stands of northern Idaho. For. Sci. 32, 989±1000. Hamilton, D.A., 1990. Extending the range of applicability of an individual tree mortality model. Can. J. For. Res. 20, 1212± 1218. Hamilton, D.A., Wendt, D.L.R., 1975. SCREEN: a computer program to identify predictors of dichotomous dependent variables. USDA For. Serv. Gen. Tech. Rep. INT-22. 20 pp. Hamilton, D.A., Edwards, B.M., 1976. Modeling the probability of individual tree mortality. USDA For. Serv. Intermt. Res. Stn. Res. Pap. INT-185. 22 pp. Harcombe, P.A., 1987. Tree life tables. Bioscience 37(8), 557± 568. Hasenauer, H., 1994. Ein Einzelbaumwachstumssimulator fuÈr ungleichaltrige Fichten-Kiefern- und Buchen-FichtenmischbestaÈnde. Forstl. Schriftenreihe, Univ. FuÈr Bodenkultur, Wien. È sterr. Ges. f. WaldoÈkosystemforschung und experimentelle O Baumforschung an der Univ. f. Bodenkultur. Band 8. 152 pp. Hasenauer, H., Merkl, D., 1997. Forest tree mortality simulation in uneven-aged stands using connectionist networks. In: Bulsari, A.B., Kallio, S. (Eds.), Proc. Int. Conf. on Engineering Applications of Neural Networks, Stockholm, Sweden, 16±18 June 1997, pp. 341±348. Hasenauer, H., Monserud, R.A., 1996. A crown ratio model for Austrian forests. For. Ecol. Manage. 84, 49±60. Hasenauer, H., Monserud, R.A., 1997. Biased statistics and predictions for tree height increment models developed from smoothed `data'. Ecol. Modelling 98, 13±22. Hasenauer, H., Monserud, R.A., Gregoire, T.G., 1998. Using simultaneous regression techniques with individual tree growth models. For. Sci. 44(1), 87±95. Kenk, G., Fischer, H., 1988. Evidence from nitrogen fertilization in forests of Germany. Environ. Pollut. 54, 199±218. Kobe, R.K., Coates, K.D., 1997. Models of sapling mortality as a function of growth to characterize interspecific variation in shade tolerance of eight tree species of northwestern British Columbia. Can. J. For. Res. 27, 227±236. Krajicek, J.E., Brinkman, K.E., Gingrich, S.F., 1961. Crown competition ± a measure of density. For. Sci. 7, 35±42. Lorimer, C.G., Frelich, L.E., 1984. A simulation of equilibrium diameter distributions of sugar maple (Acer saccarum). Bull. Torrey Bot. Club 111, 193±199. McTague, J.P., Stansfield, W.F., 1994. Stand and tree dynamics of uneven-aged ponderosa pine. For. Sci. 40(2), 289±302.
R.A. Monserud, H. Sterba / Forest Ecology and Management 113 (1999) 109±123 Miner, C.L., Walters, N.R., Belli, M.L., 1988. A Guide to the TWIGS Program for the North Central United States. USDA For. Serv. Gen. Techn. Rep. NC-125, 105 pp. Monserud, R.A., 1976. Simulation of forest tree mortality. For. Sci. 22, 438±444. Monserud, R.A., Rehfeldt, G.E., 1990. Genetic and environmental components of variation of site index in inland Douglas-fir. For. Sci. 36(1), 1±9. Monserud, R.A., Onuchin, A.A., Tchebakova, N.M., 1996. Needle, crown, stem, and root phytomass of Pinus sylvestris stands in Russia. For. Ecol. Manage. 82, 59±67. Monserud, R.A., Sterba, H., 1996. A basal area increment model for even- and uneven-aged forest stands in Austria. For. Ecol. Manage. 80, 57±80. Monserud, R.A., Sterba, H., Hasenauer, H., 1997. The single-tree stand growth simulator PROGNAUS. In: Teck, R., Moeur, M., Adams, J. (Eds.), 1997. Proc. Forest Vegetation Simulator Conf., 3±7 February, Fort Collins, CO. Gen. Tech. Rep. INTGTR-373. Ogden, UT. USDA. For. Serv., Int. Res. Stn. pp. 50± 56. Moser, J.W., 1972. Dynamics of a uneven-aged forest stand. For. Sci. 18, 184±191. Neter, J., Maynes, S.E., 1970. On the appropriateness of the correlation coefficient with a 0, 1 dependent variable. J. Am. Stat. Assoc. 65(350), 501±509. Petty, J.A., Worrell, R., 1981. Stability of coniferous tree stems in relation to damage by snow. Forestry 54(2), 115±128. PollanschuÈtz, J., 1974. Erste ertragskundliche und wirtschaftliche Ergebnisse des Fichte-Pflanzweiteversuches `Hauersteig'. In: Egger, J. (Ed.), 100 Jahre Forstliche Bundesversuchsanstalt Wien. Eigenverlag der FBVA Wien, pp. 99±172. Pretzsch, H., 1992. Konzeption und Konstruktion von Wuchsmodellen fuÈr Rein und MischbestaÈnde. Forstl. Forschungsber. MuÈnchen, Nr. 115. 358 pp. SAS Institute, 1987. SAS/STAT Guide for personal computers, version 6 ed. Cary, NC. 1028 pp. Schieler, K., Schadauer, K., 1993. Zuwachs und Nutzung nach der È sterreichischen Forstinventur 1986/1990. O È sterr. Forstzeitung O 104(4), 22±23. Schweiger, J., Sterba, H., 1997. Natural regeneration establishment of Norway spruce (Picea abies L.) in Austria. For. Ecol. Manage. 97, 107±118. Skeffington, R.A., Wilson, E.J., 1988. Excess nitrogen deposition: issues for consideration. Environ. Poll. 54, 159±184.
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Somers, G.L., Oderwald, R.C., Harris, W.R., Langdon, O.G., 1980. Predicting mortality with a Weibull function. For. Sci. 26, 291±300. Spurr, S.H., Barnes, B.V., 1980. Forest ecology, 3rd ed. Wiley, New York, 687 pp. Stage, A.R., 1973. Prognosis model for stand development. USDA For. Serv. Res. Pap. INT-137, 32 pp. Sterba, H., 1983. Single stem models from inventory data with temporary plots. Mitt. d. Forstl. Bundesversuchsanstalt (Wien) 147, 87±102. Sterba, H., 1995. Forest decline and increasing increments ± a simulation study. Forestry 68(2), 153±163. Sterba, H., Hasenauer, H., Monserud, R.A., 1995. Prognaus-Ein È sterWaldwachstumssimulator fuÈr Rein-und MischbestaÈnde. O reichischen Forstzeitung 103(5), 19±20. Sterba, H., Monserud, R.A., 1997. Applicability of the forest stand growth simulator PROGNAUS for the Austrian part of the Bohemian Massif. Ecol. Modelling 98, 23±34. Sterba, H., Golser, M., Schweiger, J., 1997. Modelle fuÈr das Ankommen und das Wachstum der NaturverjuÈngung. Centralblatt f. Ges. Forstwesen. 111(1), 11±33. Vanclay, J.K., 1991. Mortality functions for north Queensland rain forests. J. Tropical For. Sci. 4(1), 15±36. Vanclay, J.K., 1995. Growth models for tropical forests: a synthesis of models and methods. For. Sci. 41(1), 7±42. Verbyla, D.L., 1987. Classification trees: a new discrimination tool. Can. J. For. Res. 17, 1150±1152. Wald, A., 1943. Tests of statistical hypotheses concerning general parameters when the number of observations is large. Trans. Am. Math. Soc. 54, 426±482. Weber, L.A., Ek, A.R., Droessler, T.D., 1986. Comparison of stochastic and deterministic mortality estimation in an individual tree based stand growth model. Can. J. For. Res. 16(5), 1139±1141. Wykoff, W.R., 1990. A basal area increment model for individual conifers in the northern Rocky Mountains. For. Sci., 36(4): 1077±1104. Wykoff, W.F., Crookston, N.L., Stage, A.R., 1982. User's guide to the Stand Prognosis Model. USDA For. Serv. Intermt. Res. Stn. Gen. Tech. Rep. INT-133. 112 pp. Zhang, S., Amateis, R.L., Burkhart, H.E., 1997. Constraining individual tree diameter increment and survival models for loblolly pine plantations. For. Sci. 43(3), 414±423.
Modeling individual tree mortality for Austrian forest species Robert A. Monserud*, Hubert Sterba1 Rocky Mountain and Paci®c Northwest Research Stations, USDA Forest Service, 1221 SW Yamhill #200, Portland, OR 97205, USA Institut fuÈr Waldwachstumsforschung, UniversitaÈt fuÈr Bodenkultur Wien, Peter Jordan Strasse 82, A-1190, Vienna, Austria Received 2 February 1998; accepted 17 June 1998
Abstract Individual tree mortality models were developed for the six major forest species of Austria: Norway spruce (Picea abies), white ®r (Abies alba), European larch (Larix decidua), Scots pine (Pinus sylvestris), European beech (Fagus silvatica), and oak (Quercus spp.); a joint model for the remaining broadleaf species was also developed. Data came from 5-year remeasurements of the permanent plot network of the Austrian National Forest Inventory. Parameters of the logistic equation were estimated using maximum likelihood methods. For all species, we found the hyperbolic transformation of diameter (Dÿ1) to be highly signi®cant in predicting the high mortality rates for small diameter trees and decreasing mortality rates for larger diameters. For spruce, a quadratic transformation in D was needed to accurately model the increase in mortality observed for large, low-vigor trees with diameter >70 cm, which resulted in a U-shaped distribution. Crown ratio was also consistently signi®cant, except for oak. We likewise found basal-area-in-larger-trees (BAL) to be a highly signi®cant predictor of mortality rate for all species except ®r and oak. Predicted mortality rate increases as the basal area in larger trees increases and as crown ratio decreases. The resulting logistic mortality model had the same general form for all species, with the signs of all parameters conforming to expectations. In general, chi-square statistics indicate that the most important variable is Dÿ1, the second most important is crown ratio, and the third most important predictor is BAL. The relative importance of crown ratio appears to be greater for shade tolerant species (®r, beech, spruce) than for shade intolerant species (larch, Scots pine, oak). Examination of graphs of observed vs. predicted mortality rates reveals that the species-speci®c mortality models are all well behaved, and match the observed mortality rates quite well. The Dÿ1 transformation is ¯exible, as can be seen by comparing the rather different mortality rates of larch and Scots pine. Predicted and observed mortality rates with respect to crown ratio are quite close to the observed mortality rates for all but the smallest crown ratios (CR<20%), a class with very few observations. Finally, the logistic mortality models passed a validation test on independent data not used in parameter estimation. The key ingredient for obtaining a good mortality model is a data set that is both large and representative of the population under study, and the Austrian National Forest Inventory data satisfy both requirements. # 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Growth model; Mixed-species; Mortality; Stand simulation; Survival
1. Introduction *Corresponding author. Fax: +1-503-808-2020; e-mail: monserud/[email protected] 1 Fax: +43-1-47654-4242; e-mail: [email protected]
Recent statistics from the Austrian National Forest Inventory indicate that at least 40% of forest stands
0378-1127/99/$ ± see front matter # 1999 Published by Elsevier Science B.V. All rights reserved. PII: S0378-1127(98)00419-8
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have a mixed-species composition (Schieler and Schadauer, 1993). Furthermore, only 45% of the inventory plots were suf®ciently even-aged to allow usual site index determination (Monserud and Sterba, 1996). This lack of a stand age and site index on half of Austria's forests renders existing yield tables increasingly unreliable, and provided the impetus for the development of the individual-tree stand growth model PROGNAUS (Sterba et al., 1995; Monserud et al., 1997). To date, models have been developed for predicting basal area increment (Monserud and Sterba, 1996), height increment (Hasenauer and Monserud, 1997), crown ratio (Hasenauer and Monserud, 1996), natural regeneration (Schweiger and Sterba, 1997; Sterba et al., 1997), and harvesting (Monserud et al., 1997). In addition, the model has successfully passed a validation test (Sterba and Monserud, 1997), and has been the focus of an investigation on simultaneous equation systems (Hasenauer et al., 1998). The remaining key component is an accurate mortality model (Hamilton, 1990; Avila and Burkhart, 1992). Accordingly, our objective is to develop individual tree mortality models for the major forest species of Austria. We are looking for mortality models that have generality in two senses. First, we want to develop mortality models that are representative of all forest conditions in Austria, and for all major species. Second, we want to develop models that can be used not only in the PROGNAUS simulator, but in any stand simulator that uses a list of individual trees by species. Error propagation and budgeting analyses have shown that growth predictions are very sensitive to the underlying mortality model; furthermore, the contribution to total variability due to the mortality component increases as the projection period increases (Gertner, 1989). Guan and Gertner (1991a) considered this situation common in stand simulation modeling. Mortality remains one of the least understood components of growth and yield estimation (Hamilton, 1986). The key to a tree's survival is its genetic makeup and its environment (Spurr and Barnes, 1980). Growth modelers almost universally ignore a tree's genetic status (Monserud and Rehfeldt, 1990), as well as important environmental factors such as climatic extremes (e.g., wind, drought, killing frosts), insects, and diseases. Great detail is paid to environmental competition arising from neighboring trees,
however (Buchman et al., 1983), as well as measurable gross physical features of the tree and site. Perhaps mortality would appear less stochastic if relevant environmental variables were measured on permanent plots, and if the genetic status of the trees could be characterized. The literature on modeling mortality of forest trees is not small, but successes are rare. Realistically, mortality modelers mostly hope to capture the average rate of mortality, and relate it to a few reliable and measurable size or site characteristics. The key then is a large and representative sample of remeasured trees so that a rare event ± mortality ± can be observed frequently enough to predict it accurately. A representative sample must re¯ect both the full range in site variability as well as the diversity of management treatments in a given population (Hamilton, 1980). Because mortality data are most reliably and ef®ciently obtained from permanent plots, researchers are often forced to live with the limitations of the underlying permanent plot network. As a result, many studies rely on data from either unthinned plots (e.g., Zhang et al., 1997) or only lightly thinned plots (Dursky, 1997). Even if the data contain various treatments, the permanent plots often are clustered spatially and do not necessarily represent the average and dispersion of stands in the entire region of interest (e.g., Hasenauer, 1994). The most common methodology for modeling individual tree mortality is statistical. Generally, the parameters of a ¯exible non-linear function bounded by 0 and 1 are estimated using weighted nonlinear regression or a multivariate maximum likelihood procedure (Neter and Maynes, 1970). Although most cumulative distribution functions will work, the most popular is the logistic or logit (e.g., Monserud, 1976; Hamilton and Edwards, 1976; Buchman, 1979; Hamilton, 1986; Vanclay, 1995). Other applications have used the Weibull (Somers et al., 1980), the gamma (Kobe and Coates, 1997), the Richard's function (Buford and Ha¯ey, 1985), the exponential (Moser, 1972), and the normal or probit (Finney, 1971). Monserud (1976) found that both the probit and the logit produced similar results, even though the underlying functional forms are quite different. Vanclay's 1995 recommendation of the logistic for mortality models in tropical forests is surprisingly quite relevant to our Austrian temperate forest situation, for both cover a
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spectrum of different species mixtures and age structures, precluding the possibility of using either stand age or site index as predictor variables. Two additional procedures have been used to model individual tree mortality: recursive partitioning and neural networks. Unfortunately, neither has led to signi®cant improvement in our ability to predict mortality using classical statistical methods. Recursive partitioning is best known by the acronym CART (Classi®cation And Regression Trees; Breiman et al., 1984), although it is a similar classi®cation tool to the SCREEN algorithm used by Hamilton and Wendt (1975) for ef®ciently identifying potential independent variables and relationships. CART employs heuristic methods with binary classi®cation, with results presented as decision trees (Verbyla, 1987). Dobbertin and Biging (1998) compared CART to logistic regression for two species in northern California, and concluded that CART performed somewhat better, although the percentage of dead trees correctly classi®ed was very low. Working in a branch of arti®cial intelligence, Guan and Gertner (1991a, b) built a neural network mortality model that was as accurate as the corresponding logistic model with similar variables, and was better behaved because the same model form was not constrained to operate in all regions of the data space. Hasenauer and Merkl (1997) have recently compared a neural network to the logit for Austria, and found that both predict equally well, with a slight advantage to the neural network. 2. Methods We rely on the remeasured permanent plots of the Austrian National Forest Inventory for mortality and survival data (Forstliche Bundesversuchsanstalt, 1981, 1986). A systematic 3.89 km grid of permanent plots covering Austria was established in 1981±1985. Each year 20% of the grid locations are sampled (1100 clusters) in such a way that all of Austria is covered by the inventory each year. Each location was then remeasured from 1986±1990, exactly 5 years after establishment. The total inventory comprises 5500 clusters, consisting of 22 000 permanent plots. At each grid location, a cluster of four plots is located at the vertices of a square 200 m on a side. Using variable radius plot sampling (Bitterlich, 1948),
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all `in' trees larger than 10.4 cm in diameter are selected using a BAF of 4 m2 haÿ1. Trees are identi®ed by their distance and azimuth from plot center, their species, and size measurements. A hidden nail in the root collar at the ground line indicates that diameter at breast height is to be measured 1.3 m above it. All trees with diameters between 5 and 10.4 cm are recorded on a ®xed area plot located at plot center with radius 2.6 m, the limiting distance for sampling a 10.4 cm tree with BAF4. Trees smaller than 5 cm are not sampled. The survival status of the same individual trees is recorded, with mortality from natural causes distinguished from normal harvesting and thinning. Harvesting due to severe tree damage or death is coded as natural mortality. The hidden plot center is a buried metal rod located by a map and metal detectors. This hidden plot design is used to eliminate research plot bias (Bruce, 1977) and to ensure that the Forest Inventory is representative of both growing conditions and forest management throughout Austria. Repeated measurements from 1986±1990 were made on 43 615 trees. These 5-year observations of survival and mortality were converted to 5-year mortality rates using the standard estimator for unequal probability sampling (Cochran, 1963), which weights each observation by the number of trees per hectare it represents in the sample (Table 1). The Austrian Forest Inventory remeasured and evaluated the status of several thousand trees of the ®ve main species in Austria (Table 1): Norway spruce (Picea abies), white ®r (Abies alba), European larch (Larix decidua), Scots pine (Pinus sylvestris), and European beech (Fagus silvatica). We also considered Table 1 Total number of trees remeasured, 5-year mortality rate, and annual mortality rate, by species Species
Total no. of trees
Mortality rate 5-year (%)
1-year (%)
Spruce Fir Larch Scots pine Beech Oak Other broadleaf All species
26 699 1878 3015 4138 4484 784 2617 43 615
4.4 6.1 2.9 5.6 4.3 3.2 8.0 4.7
0.9 1.3 0.6 1.1 0.9 0.6 1.7 1.0
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building a model for oak (Quercus spp.), although the sample size is less than 800 trees. Finally, we combined the 2615 observations of the remaining broadleaf species into one additional tree class (consisting mostly of Carpinus, Fraxinus, Acer, Betula, and Alnus). Based on a sample of 43 615 trees, overall 5-year mortality rate is 4.7%, which corresponds to an annual rate of 1.0% (Table 1). The most common species, spruce, had a 5-year mortality rate of 4.4%. Conifer mortality rates (5-year) varied between a high of 6.1% for ®r and 2.9% for larch. The 5-year rate was 4.3% for beech, 3.2% for oak, and 8.0% for all other broadleaf species combined.
3. Analysis Individual tree mortality is a discrete event. A datum can have only the value 0 (live) or 1 (dead). A dichotomous dependent variable calls for special consideration not only in parameter estimation (Hamilton, 1974), but in the interpretation of goodness of ®t as well (Neter and Maynes, 1970). We take the classical approach to model the probability of mortality, the logistic equation: ÿ1 0 (1) P 1 eb X where b0 X is a linear combination of parameters b and independent variables X, and e is the base of the natural logarithm. We used the CATMOD procedure (SAS Institute, 1987), which estimates the parameters of the logistic equation using maximum likelihood methods. Although weighted nonlinear regression can be used to estimate the parameters of a probability of mortality function, maximum likelihood methods are well behaved and more straightforward. Usual measures of goodness of ®t such as the coef®cient of determination or the correlation coef®cient are not appropriate for dichotomous variables (Neter and Maynes, 1970); it does not matter how close a prediction for a tree is to 0 or 1 as long as it classi®es the observation correctly (Monserud, 1976). The appropriate statistic is Chi-square (Neter and Maynes, 1970). The key question in modeling is the speci®cation of the set of independent variables after the model form
and error structure have been hypothesized. We begin by intentionally excluding the two most traditional variables in forest stand modeling: site index and age. Our reasons are simple; we want to develop a mortality model that can be used to accurately simulate the development of almost any forest stand growing in Austria, where the majority of those stands now fail to meet the accepted de®nition of even-aged (Monserud and Sterba, 1996). Tree size, of course, already can be understood as the integrated response of the tree to site quality and age, and thus implicitly contains such site and age effects. Site-speci®c variables include slope, aspect, elevation, soil type, vegetation type, soil depth, and several others listed by Forstliche Bundesversuchsanstalt (1981, 1986). At ®rst we considered them as possible predictors, but we soon realized that this was risky. Although we have an enormous sample of live trees, the number of dead trees is still relatively small for all species except spruce. Our fear was that it would be very easy to over-parameterize the resulting mortality model. We were thus left with a reduced set of variables describing tree size, competition, and growth. Diameter (such as diameter at breast height, 1.3 m) is an important and reliable measure of a tree's size. For woody plants, the larger the individual the greater its chances of competing for scarce resources. Thus, mortality rate should decrease as diameter increases. Nearly all mortality models include this variable (e.g., Monserud, 1976; Buchman et al., 1983; Vanclay, 1991; Avila and Burkhart, 1992; Hasenauer, 1994; McTague and Stans®eld, 1994; Dursky, 1997). For most species, the mortality rate for the smallest trees is quite high, and declines rapidly as individuals survive to larger diameter classes. A simple linear function of diameter is not suf®cient to capture this nonlinear effect. The TWIGS-model (Miner et al., 1988) uses a highly nonlinear transformation of diameter to gain more ¯exibility in the logit. We hypothesized that a hyperbolic Dÿ1 transformation of diameter should accurately track both the large mortality rates for small trees and decreasing mortality rates for larger diameters. Because the largest trees of a given species are often the oldest individuals, there is likely a size past which vigor declines (as indicated by increment), senescence
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becomes apparent, and the probability of mortality increases (Harcombe, 1987). A very large sample is required to detect this effect in a normal forest inventory. Lorimer and Frelich (1984) conclude that the death-rate curve for forest trees is probably U-shaped. The general model of Buchman et al. (1983) allows for this effect. We hypothesized that this U-shaped senescence effect could be modeled by including terms for D and D2 in the exponent of the logistic (D.A. Hamilton, personal communication). Because stand age is not de®ned for over half of the sample stands, we cannot simply break out the confounding in diameter between the rapid decrease in mortality rate for small trees and the increasing mortality rate (senescence effect) for large, older trees. Tree age is not available, so both hyperbolic and quadratic transformations of diameter are needed to model these different effects. A second important attribute for survival is the size of the tree's crown as an indicator of tree vigor. Generally, mortality rate should be lower for trees with larger crowns. Because total height and height to the base of the live crown were measured on all trees in the ®rst inventory, we calculated a dimensionless crown ratio as the ratio of crown length to total height. Interestingly, many of the mortality models using crown ratio as an independent variable do not include a diameter increment term (Avila and Burkhart, 1992; Hasenauer, 1994; Zhang et al., 1997), and vice versa (Monserud, 1976; Buchman et al., 1983; Dursky, 1997), indicating that probably only one indicator of tree vigor is needed.
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Competition can be expressed by a variety of stand density measures. Wykoff et al. (1982) and Wykoff (1990) introduced basal-area-in-larger-trees (BAL), a tree-speci®c measure of density that has proven quite useful in modeling stand dynamics. BAL is a simple count of the stand's basal area that is from trees with a diameter larger than the subject tree's diameter. BAL for the largest tree is 0.0, and for the smallest tree equals stand basal area minus that tree's basal area. BAL was also used in the Lake States version of TWIGS (Miner et al., 1988). We also considered crown competition factor (Krajicek et al., 1961) as a potential predictor of mortality rate. Combining all these effects, the hypothesized mortality model has the same general form for all species: ÿ1 2 (2) P 1 e b0 b1 =Db2 CRb3 BALb4 Db5 D where P is the probability of mortality (5-year), D is diameter (cm) at breast height (1.3 m), CR is crown ratio, BAL is basal area in larger trees (m2 haÿ1), and b0±b5 are species-speci®c parameters to be estimated. Furthermore, D, CR, and BAL are values at the beginning of the 5-year period. 4. Results For all species, we found the hyperbolic Dÿ1 to be highly signi®cant in predicting mortality rate (Table 2). Furthermore, it behaved properly in matching the rapid decline from high mortality for the smallest diameters to a more gradual decline in
Table 2 Estimated parameters b0±b5 for mortality model 2 Species
Intercept
Variable 1/D
CR
BAL
D
D2
0.0425 (0.0187)
ÿ0.00081 (0.000255)
Spruce
2.1283 (0.3877)
ÿ10.0745 (1.6467)
3.8251 (0.1695)
ÿ0.0186 (0.00207)
Fir Larch Scots pine Beech Oak Other broadleaf
2.0985 4.4070 4.1076 3.5734 4.4508 2.9223
ÿ10.9085 (1.9438) ÿ12.9395 (2.2204) ÿ18.9714 (1.3834) ÿ13.9542 (1.5835) ÿ12.0041 (4.3602) ÿ8.4877 (1.5631)
3.9311 2.2039 2.3267 3.1339
(0.5469) (0.6963) (0.4963) (0.395)
ÿ0.0326 (0.0081) ÿ0.0234 (0.0061) ÿ0.0161 (0.0047)
2.0609 (0.4069)
ÿ0.0228 (0.0048)
(0.3199) (0.5097) (0.301) (0.336) (0.4685) (0.3154)
D is diameter (cm) at breast height (1.3 m), CR is crown ratio, and BAL is basal-area-in-larger-trees (m2 haÿ1). Standard errors are given in parentheses.
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mortality rate for the larger diameters (Fig. 1). It was clearly superior to a simple D term. Crown ratio CR was also consistently signi®cant, except for oak (Table 2). We likewise found BAL to be a highly signi®cant predictor of mortality rate for all species except ®r and oak. The effect was as expected: predicted mortality rate increases as the basal area in larger trees increases and as crown ratio decreases. By comparison to BAL, crown competition factor (Krajicek et al., 1961) was uniformly insigni®cant as a predictor. All parameters listed in Table 2 are highly signi®cant (P<0.001). For all species except spruce, the coef®cients of D and D2 in the exponent were not signi®cant, indicating that a senescence effect could not be detected. As an overall measure of goodness of ®t, the likelihood ratio ranged from 165 for oak to 7965 for spruce (Table 3), all of which corresponded to error probabilities >0.9999. To judge the relative importance of the variables, standard errors are listed in Table 2 and Chi-square values (Wald, 1943) are provided in Table 3. All parameter estimates conform to expectations. The coef®cient of Dÿ1 is negative in all cases (Table 2), resulting in decreasing mortality rates with increasing diameter. For spruce, the quadratic transformation in D effectively increases the mortality rate for trees exceeding 60 cm, resulting in a U-shaped distribution (Fig. 2). The coef®cient of CR is positive in all cases, resulting in increasing mortality rates as crown ratio decreases; and the coef®cient of BAL is negative in all cases, indicating that mortality rate will be higher as basal area in larger trees increases and a tree's competitive status is less favorable. The Chi-
square statistics in Table 3 reveal that the most important variable is breast height diameter (except for ®r), the second most important is crown ratio CR, and the third most important predictor is BAL (except for larch). The relative importance of crown ratio appears to be greater for shade tolerant species (®r, beech) than for shade intolerant species (larch, Scots pine, oak), with spruce intermediate. Examination of Fig. 1 reveals that the species-speci®c mortality models are all well-behaved, and match the observed mortality rates reasonably well. The Dÿ1 transformation is ¯exible, as can be seen by comparing the rather different mortality rates of larch and Scots pine in Fig. 1. Predicted and observed mortality rates with respect to crown ratio are examined in Fig. 3. Generally, the predictions are quite close to the observed mortality rates for all but the smallest crown ratios (CR<20%). The underestimates in the smallest crown ratio class for spruce, Scots pine, and beech could be ameliorated by replacing CR by the inverse of CR in Eq. (2), but would result in an overestimate for larger crown ratios. Furthermore, the inverse CR does not perform well for ®r and larch. We further examined the model for spruce and found that predictions in the 0±10% CR class were rather close (32% observed vs. 28% predicted mortality rates), with most of the underestimates in the 10±20% class (34% observed vs. 21% predicted mortality rates). This lack of a consistent error indicates that the model is well-behaved with respect to crown ratio. An examination of predicted vs. observed mortality rates with respect to basal area in larger trees (BAL) turned up no problems with model behavior for any of
Table 3 Chi-square values for the parameter estimates in Table 2 (model 2) Species
Likelihood ratio
Chi-square statistic Variable
Spruce Fir Larch Scots pine Beech Oak Other broadleaf
7965 749 700 940 1391 165 1365
1/D
CR
BAL
D
D2
37 31 43 82 78 8 29
509 52 10 33 63 ± 26
80 ± 16 6 12 ± 22
5 ± ± ± ± ± ±
10 ± ± ± ± ± ±
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Fig. 1. Observed vs. predicted 5-year mortality rate vs. diameter at breast height (Dbh) for the six major forest species in Austria.
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Fig. 2. Observed vs. predicted 5-year mortality rate vs. diameter at breast height (Dbh) for Norway spruce.
the species (Fig. 4). This generalization holds even for ®r and oak, whose mortality models do not include a BAL term. 4.1. Old growth spruce Because sample size for spruce was so large (n26 699), we were able to detect the classic Ushaped distribution expected by many (Buchman et al., 1983; Lorimer and Frelich, 1984; Harcombe, 1987; D.A. Hamilton, personal commun.). The observed spruce mortality rate declines in a regular fashion to below 2% for diameter classes up to 70 cm. The mortality rate then increases dramatically to 5.25% past 70 cm, with 21 dead trees out of 400 (Fig. 2). We examined these trees in detail. Fifteen out of the 21 dead trees are older than 140 years, the maximum age class coded in the inventory. The remaining six trees indicate a mortality rate of 1.5%, which is about the same rate as for the 60±70 cm diameter class. All 21 dead trees are from steep (median slope of 80%) Protection Forests that are at high elevations for spruce (1100±1600 m). Furthermore, nine of the dead trees were from stands described as being in the beginning stage of decline, with a general loss of vigor. From this, we conclude that for spruce larger than 70 cm, the
mortality process is fundamentally different than would be described in Eq. (2) if only the hyperbolic Dÿ1 term were used. Large old-growth spruce have a mortality rate nearly four times higher than would be expected from the decreasing rate normally associated with increasing size. The quadratic transformation in D increased the predicted mortality rate for these mature trees, resulting in the expected U-shaped distribution. These large, old trees lacking in vigor are reminiscent of the trees that exhibited lower survival with increasing diameter in the study of Buchman et al. (1983). We also observed a higher mortality rate in the largest diameter class for ®r and the other broadleaf species, but this high mortality rate resulted from the death of only two±four trees. In those cases, the rate for the largest diameter class is not signi®cantly different than the rate for the penultimate diameter class, and we could not support the quadratic diameter transformation in model 2. 4.2. A validation test Independent permanent plot data for validation were available from a thinning experiment in 22 mixed Norway spruce ± Scots pine stands from Litschau, in the Austrian part of the Bohemian Massif (Sterba and
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Fig. 3. Observed vs. predicted 5-year mortality rate vs. crown ratio for the six major forest species in Austria.
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Fig. 4. Observed vs. predicted 5-year mortality rate vs. basal area in lager trees for the six major forest species in Austria.
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Monserud, 1997). These data were not used for parameter estimation or model development, and were obtained from a separate sample with different objectives. Thus, they provide an independent set of observations for a validation test of the model. Management history included litter-raking until the 1950s, resulting in various degrees of soil degradation. Current stands were regenerated after clearcutting by planting Norway spruce. Volunteer spruce and Scots pine regeneration followed soon after, converting the plantations to mixed stands. Permanent research plots were established in 1977 and ®rst remeasured in 1982, a growth interval that included snow breakage in the winter of 1979±80. Dominant stand age in 1977 ranged from 10 to 110 years. Thinning was done in 1983. The next observation followed in 1987 and the third remeasurement was performed in 1992, thus providing three 5-year increment periods. Plot size varied from 400 to 2025 m2. While site factor variation between the plots was not large, there was considerable variation in dominant height (5± 30 m) and density (stem number between 450 and 28 000 trees haÿ1, basal area between 5 and 48 m2 haÿ1, and Crown Competition Factor (Krajicek et al., 1961) from 0 to 300). Thinning intensity in the plots varied from 6% to 420% of the 15-year basal area increment. The proportion of Scots pine varies between 0% and 73%. The plots are discussed in detail by Sterba and Monserud (1997). We compared our model predictions with observed mortality in the Litschau permanent research plots. Over the 1977±1992 observation period, mortality was somewhat higher in both Norway spruce and Scots pine than predicted by the model. The observed and predicted 5-year mortality rates for spruce were 6.1% vs. 5.6%; corresponding rates for Scots pine were 5.7% vs. 4.0% (Table 4).
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In Fig. 5 we compare observed with predicted number of dead trees by species for the 15-year remeasurement period. Predictions for spruce are close to the observed mortality, with no apparent trend with respect to crown ratio. Predictions for pine are underestimates, with greater mortality observed than predicted in the lower crown ratio classes. For both spruce and pine, mortality is underestimated in the smallest diameter classes. The situation between Scots pine and Norway spruce in the Litschau plots is interesting. First, some history is in order. Litter-raking in the Bohemian Massif ceased mid-century, which allowed for a return to the gradual buildup of organic matter and nutrients that facilitate normal soil building processes. Scots pine is well known as a species that can tolerate poor soil conditions, including both nutrient-poor sandy soils and water-logged bogs (Monserud et al., 1996). Formerly, clearcuts in the vicinity of Litschau were large, followed by plantations of Norway spruce. Because of the poor nutrient levels in the litter-raked soils, Scots pine established itself abundantly in these plantations, occasionally outnumbering the planted spruce. With the cessation of litter-raking and the increased levels of exogenous nitrogen deposition in recent decades (Kenk and Fischer, 1988; Skef®ngton and Wilson, 1988), soil conditions have improved and given a competitive advantage to spruce. Increment rates reported by Sterba and Monserud (1997) indicate that pine growth rates at Litschau have increased over expectation in the period 1977±92. The exact reason is unknown, of course, but contributing factors could be nitrogen deposition, improved soil nutrients, and climate warming. Sterba (1995) points out that overall site improvement will give a differential advantage to Norway spruce over Scots pine, for spruce has much greater potential growth, and is more tolerant of shade than the light-demanding Scots pine.
Table 4 Observed vs. predicted mortality rates (5-year (%) and 1-year (%)) for the Litschau permanent plots for the period 1977±92, by species Species
Spruce Scots pine
Mortality rate Observed No. dead
Observed 5-year (%)
Predicted 5-year (%)
Observed 1-year (%)
Predicted 1-year (%)
279 72
6.1 5.7
5.6 4.0
1.25 1.17
1.15 0.81
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Fig. 5. Observed vs. predicted mortality vs. diameter at breast height (Dbh), crown ratio, and basal area in larger trees for Norway spruce and Scots pine for the Litschau validation plots.
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Increased growth rates increase competition, with the net result that mortality rates should increase for Scots pine at a faster rate than for spruce (Sterba, 1995). This indeed happened at Litschau. In addition, the underestimation of mortality in the smallest diameter classes can be attributed to documented snow breakage in the winter of 1979±80. Snow breakage affects trees with the largest height/diameter ratios the most, usually the smallest diameter trees (PollanschuÈtz, 1974; Abetz, 1976). Based on the physics of tree form, Petty and Worrell (1981) con®rmed the large empirical literature in Central Europe documenting the relationship between large height/diameter ratios and the risk of snow breakage. The main canopy of Scots pines at Litschau was not dense enough to protect the small spruce from heavy snow. 5. Application Although our mortality model was developed as a submodel in the distance-independent stand simulator PROGNAUS (Sterba et al., 1995; Monserud et al., 1997), it should work equally well in a simulator with a different architecture (e.g., Sterba, 1983; EckmuÈllner and Fleck, 1989; Pretzsch, 1992). The key ingredient for obtaining a good mortality model is a large data set that is representative of the population under study. The Austrian National Forest Inventory data (Forstliche Bundesversuchsanstalt, 1981) satisfy this requirement. The one shortcoming of this sample is that it is restricted in time to only one decade, the 1980s. As additional remeasurements become available from future inventory cycles, the reliability of the time cross-section should be improved. The simulator architecture determines how mortality must be calculated, however. Because of the nature of the competition index calculation, mortality in a distance-dependent individual tree simulator must be a discrete event, with the tree either completely dead or alive. This is obviously a very stochastic process to simulate, especially when plot size is small. The stochastic rule compares the predicted mortality rate with a uniform random number on the interval 0±1; if the random number is less than the mortality rate, the tree is considered to have died (Weber et al., 1986). In contrast, mortality in a distance-independent simulator can be a continuous event, with the mortality rate
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Table 5 Percent correct classification, by species Species
Percent correct classification Live trees
Dead trees
Total
Spruce Fir Larch Scots pine Beech Oak Other broadleaf Total
88 90 89 92 91 92 81 89
44 41 46 42 38 25 42 42
87 89 88 90 89 91 79 86
continuously and smoothly reducing the number of trees/ha each sample tree represents. This advantage was ®rst seized by Stage (1973) in designing the Stand Prognosis Model. Thus, it is much easier to accurately predict mortality rates over large areas with distanceindependent models. Because one of the main applications of a stand growth simulator is to illustrate management alternatives for planning, this advantage is considerable. Weber et al. (1986) compared these stochastic and deterministic methods for determining mortality and found no practical differences in mean stand values. They concluded that there is no need for simulating multiple stochastic runs (with a distanceindependent model) that are later averaged when the main interest is in projected stand values. A second deterministic method can also be used to simulate mortality by using a threshold. If the threshold exceeds the estimated probability of mortality then the tree is considered dead. The best and most logical choice for a threshold is the average observed mortality rate for that species. Table 5 shows the result of this approach by using the overall mortality rate in Table 1 as the threshold for each species. We correctly classi®ed 89% of the live trees and 42% of the dead trees, for a total of 86% correct classi®cation (Table 5). This compares very closely to Monserud's (1976) results for northern hardwoods (viz., 88% live and 35% dead correctly classi®ed). Acknowledgements This research was conducted when Monserud was Visiting Scientist and UniversitaÈtslektor at the Institut fuÈr Waldwachstumsforschung in Vienna, on a grant
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from the Austrian Ministry of Agriculture and Forestry. Karl Schieler and Klemens Schadauer of the Federal Forest Research Center (Forstliche Bundesversuchsanstalt, Wien) generously made the Austrian Forest Inventory data available. We thank Dave Hamilton, Ralph Amateis, Karl Schieler, and the anonymous reviewers for helpful review comments. References Abetz, P., 1976. BeitraÈge zum Baumwachstum ± der h/d Wert. Forst und Holzwirt. 31(19), 389±393. Avila, O.B., Burkhart, H.E., 1992. Modeling survival of loblolly pine trees in thinned and unthinned plantations. Can. J. For. Res. 22, 1878±1882. Bitterlich, W., 1948. Die WinkelzaÈhlprobe. Allg. Forst-und Holzwirt. Zeitung Wien. 59, 4±5. Breiman, L., Friedman, J.H., Olshen, R.A., Stone, C.J., 1984. Classification and regression trees. Wadsworth, Belmont, CA. Bruce, D., 1977. Yield differences between research plots and managed forests. J. For. 75(1), 14±17. Buchman, R.G., 1979. Mortality functions. In: A generalized forest growth projection system applied to the Lakes States region. USDA For. Serv. N. Cent. Res. Stn. Gen. Tech. Rep. NC-49. pp. 47±55. Buchman, R.G., Pederson, S.P., Walters, N.R., 1983. A tree survival model with application to species of the Great Lakes region. Can J. For. Res. 13, 601±608. Buford, M.A., Hafley, W.L., 1985. Modeling the probability of individual tree mortality. For. Sci. 31, 331±341. Cochran, W.G., 1963. Sampling techniques, 2nd ed. Wiley, New York, 413 pp. Dobbertin, M., Biging, G.S., 1998. Using the non-parametric classifier CART to model forest tree mortality. For. Sci., in press. Dursky, J., 1997. Modellierung der Absterbeprozesse in Rein± und MischbestaÈnden aus Fichte und Buche Allg. Jagd- und Forstzeitung 168 (6±7) 131±134. EckmuÈllner, O., Fleck, W., 1989. Begleitdokumentation zum Wachstumssimulationsprogramm WASIM 1.0. Institut f. Forstliche Ertragslehre, Univ. f. Bodenkultur, Wien, 29 pp. Finney, D.J., 1971. Probit Analysis, 3rd ed. Cambridge University Press, New York, 333 pp. Forstliche Bundesversuchsanstalt, 1981. Instruktionen fuÈr die È sterreichischen Forstinventur 1981±1985. Feldarbeit der O Forstliche Bundesversuchsanstalt, Wien, 172 pp. Forstliche Bundesversuchsanstalt, 1986. Beiheft der DienstanweiÈ sterreichischen sung (Instruktionen fuÈr die Feldarbeit der O Forstinventur 1981±1985). Forstliche Bundesversuchsanstalt, Wien, 42 pp. Gertner, G.Z., 1989. The need to improve models for individual tree mortality. In: Proc. 7th Central Hardwood Conf. General Technical report NC-132, North Central Forest Experiment Station, USDA Forest Service, pp. 59±61.
Guan, B.T., Gertner, G.Z., 1991a. Using a parallel distributed processing system to model individual tree mortality. For. Sci. 37(3) 871±885. Guan, B.T., Gertner, G.Z., 1991b. Modeling red pine tree survival with an artificial neural network. For. Sci. 37(5), 1429±1440. Hamilton, D.A., 1974. Event probabilities estimated by regression. USDA For. Serv. Intermt. Res. Stn. Res. Pap. INT152. 18 pp. Hamilton, D.A., 1980. Modeling mortality: a component of growth and yield modeling. In: Brown, K.M., Clarke, F.R. (Eds.), Proc. Forecasting Forest Stand Dynamics Workshop, School of Forestry, Lakehead Univ, Thunder Bay, Ontario, pp. 82±99. Hamilton, D.A., 1986. A logistic model of mortality in thinned and unthinned mixed conifer stands of northern Idaho. For. Sci. 32, 989±1000. Hamilton, D.A., 1990. Extending the range of applicability of an individual tree mortality model. Can. J. For. Res. 20, 1212± 1218. Hamilton, D.A., Wendt, D.L.R., 1975. SCREEN: a computer program to identify predictors of dichotomous dependent variables. USDA For. Serv. Gen. Tech. Rep. INT-22. 20 pp. Hamilton, D.A., Edwards, B.M., 1976. Modeling the probability of individual tree mortality. USDA For. Serv. Intermt. Res. Stn. Res. Pap. INT-185. 22 pp. Harcombe, P.A., 1987. Tree life tables. Bioscience 37(8), 557± 568. Hasenauer, H., 1994. Ein Einzelbaumwachstumssimulator fuÈr ungleichaltrige Fichten-Kiefern- und Buchen-FichtenmischbestaÈnde. Forstl. Schriftenreihe, Univ. FuÈr Bodenkultur, Wien. È sterr. Ges. f. WaldoÈkosystemforschung und experimentelle O Baumforschung an der Univ. f. Bodenkultur. Band 8. 152 pp. Hasenauer, H., Merkl, D., 1997. Forest tree mortality simulation in uneven-aged stands using connectionist networks. In: Bulsari, A.B., Kallio, S. (Eds.), Proc. Int. Conf. on Engineering Applications of Neural Networks, Stockholm, Sweden, 16±18 June 1997, pp. 341±348. Hasenauer, H., Monserud, R.A., 1996. A crown ratio model for Austrian forests. For. Ecol. Manage. 84, 49±60. Hasenauer, H., Monserud, R.A., 1997. Biased statistics and predictions for tree height increment models developed from smoothed `data'. Ecol. Modelling 98, 13±22. Hasenauer, H., Monserud, R.A., Gregoire, T.G., 1998. Using simultaneous regression techniques with individual tree growth models. For. Sci. 44(1), 87±95. Kenk, G., Fischer, H., 1988. Evidence from nitrogen fertilization in forests of Germany. Environ. Pollut. 54, 199±218. Kobe, R.K., Coates, K.D., 1997. Models of sapling mortality as a function of growth to characterize interspecific variation in shade tolerance of eight tree species of northwestern British Columbia. Can. J. For. Res. 27, 227±236. Krajicek, J.E., Brinkman, K.E., Gingrich, S.F., 1961. Crown competition ± a measure of density. For. Sci. 7, 35±42. Lorimer, C.G., Frelich, L.E., 1984. A simulation of equilibrium diameter distributions of sugar maple (Acer saccarum). Bull. Torrey Bot. Club 111, 193±199. McTague, J.P., Stansfield, W.F., 1994. Stand and tree dynamics of uneven-aged ponderosa pine. For. Sci. 40(2), 289±302.
R.A. Monserud, H. Sterba / Forest Ecology and Management 113 (1999) 109±123 Miner, C.L., Walters, N.R., Belli, M.L., 1988. A Guide to the TWIGS Program for the North Central United States. USDA For. Serv. Gen. Techn. Rep. NC-125, 105 pp. Monserud, R.A., 1976. Simulation of forest tree mortality. For. Sci. 22, 438±444. Monserud, R.A., Rehfeldt, G.E., 1990. Genetic and environmental components of variation of site index in inland Douglas-fir. For. Sci. 36(1), 1±9. Monserud, R.A., Onuchin, A.A., Tchebakova, N.M., 1996. Needle, crown, stem, and root phytomass of Pinus sylvestris stands in Russia. For. Ecol. Manage. 82, 59±67. Monserud, R.A., Sterba, H., 1996. A basal area increment model for even- and uneven-aged forest stands in Austria. For. Ecol. Manage. 80, 57±80. Monserud, R.A., Sterba, H., Hasenauer, H., 1997. The single-tree stand growth simulator PROGNAUS. In: Teck, R., Moeur, M., Adams, J. (Eds.), 1997. Proc. Forest Vegetation Simulator Conf., 3±7 February, Fort Collins, CO. Gen. Tech. Rep. INTGTR-373. Ogden, UT. USDA. For. Serv., Int. Res. Stn. pp. 50± 56. Moser, J.W., 1972. Dynamics of a uneven-aged forest stand. For. Sci. 18, 184±191. Neter, J., Maynes, S.E., 1970. On the appropriateness of the correlation coefficient with a 0, 1 dependent variable. J. Am. Stat. Assoc. 65(350), 501±509. Petty, J.A., Worrell, R., 1981. Stability of coniferous tree stems in relation to damage by snow. Forestry 54(2), 115±128. PollanschuÈtz, J., 1974. Erste ertragskundliche und wirtschaftliche Ergebnisse des Fichte-Pflanzweiteversuches `Hauersteig'. In: Egger, J. (Ed.), 100 Jahre Forstliche Bundesversuchsanstalt Wien. Eigenverlag der FBVA Wien, pp. 99±172. Pretzsch, H., 1992. Konzeption und Konstruktion von Wuchsmodellen fuÈr Rein und MischbestaÈnde. Forstl. Forschungsber. MuÈnchen, Nr. 115. 358 pp. SAS Institute, 1987. SAS/STAT Guide for personal computers, version 6 ed. Cary, NC. 1028 pp. Schieler, K., Schadauer, K., 1993. Zuwachs und Nutzung nach der È sterreichischen Forstinventur 1986/1990. O È sterr. Forstzeitung O 104(4), 22±23. Schweiger, J., Sterba, H., 1997. Natural regeneration establishment of Norway spruce (Picea abies L.) in Austria. For. Ecol. Manage. 97, 107±118. Skeffington, R.A., Wilson, E.J., 1988. Excess nitrogen deposition: issues for consideration. Environ. Poll. 54, 159±184.
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Somers, G.L., Oderwald, R.C., Harris, W.R., Langdon, O.G., 1980. Predicting mortality with a Weibull function. For. Sci. 26, 291±300. Spurr, S.H., Barnes, B.V., 1980. Forest ecology, 3rd ed. Wiley, New York, 687 pp. Stage, A.R., 1973. Prognosis model for stand development. USDA For. Serv. Res. Pap. INT-137, 32 pp. Sterba, H., 1983. Single stem models from inventory data with temporary plots. Mitt. d. Forstl. Bundesversuchsanstalt (Wien) 147, 87±102. Sterba, H., 1995. Forest decline and increasing increments ± a simulation study. Forestry 68(2), 153±163. Sterba, H., Hasenauer, H., Monserud, R.A., 1995. Prognaus-Ein È sterWaldwachstumssimulator fuÈr Rein-und MischbestaÈnde. O reichischen Forstzeitung 103(5), 19±20. Sterba, H., Monserud, R.A., 1997. Applicability of the forest stand growth simulator PROGNAUS for the Austrian part of the Bohemian Massif. Ecol. Modelling 98, 23±34. Sterba, H., Golser, M., Schweiger, J., 1997. Modelle fuÈr das Ankommen und das Wachstum der NaturverjuÈngung. Centralblatt f. Ges. Forstwesen. 111(1), 11±33. Vanclay, J.K., 1991. Mortality functions for north Queensland rain forests. J. Tropical For. Sci. 4(1), 15±36. Vanclay, J.K., 1995. Growth models for tropical forests: a synthesis of models and methods. For. Sci. 41(1), 7±42. Verbyla, D.L., 1987. Classification trees: a new discrimination tool. Can. J. For. Res. 17, 1150±1152. Wald, A., 1943. Tests of statistical hypotheses concerning general parameters when the number of observations is large. Trans. Am. Math. Soc. 54, 426±482. Weber, L.A., Ek, A.R., Droessler, T.D., 1986. Comparison of stochastic and deterministic mortality estimation in an individual tree based stand growth model. Can. J. For. Res. 16(5), 1139±1141. Wykoff, W.R., 1990. A basal area increment model for individual conifers in the northern Rocky Mountains. For. Sci., 36(4): 1077±1104. Wykoff, W.F., Crookston, N.L., Stage, A.R., 1982. User's guide to the Stand Prognosis Model. USDA For. Serv. Intermt. Res. Stn. Gen. Tech. Rep. INT-133. 112 pp. Zhang, S., Amateis, R.L., Burkhart, H.E., 1997. Constraining individual tree diameter increment and survival models for loblolly pine plantations. For. Sci. 43(3), 414±423.