Modeling individual tree mortality for Austrian forest species

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Abstract

Individual tree mortality models were developed for the six major forest species of Austria: Norway spruce (Picea abies), white fir (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 (D−1) to be highly significant 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 significant, except for oak. We likewise found basal-area-in-larger-trees (BAL) to be a highly significant predictor of mortality rate for all species except fir 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 D−1, 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 (fir, beech, spruce) than for shade intolerant species (larch, Scots pine, oak). Examination of graphs of observed vs. predicted mortality rates reveals that the species-specific mortality models are all well behaved, and match the observed mortality rates quite well. The D−1 transformation is flexible, 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.

Introduction

Recent statistics from the Austrian National Forest Inventory indicate that at least 40% of forest stands have a mixed-species composition (Schieler and Schadauer, 1993). Furthermore, only 45% of the inventory plots were sufficiently 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 reflect 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 efficiently 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 flexible 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 Hafley, 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 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 significant improvement in our ability to predict mortality using classical statistical methods. Recursive partitioning is best known by the acronym CART (Classification And Regression Trees; Breiman et al., 1984), although it is a similar classification tool to the SCREEN algorithm used by Hamilton and Wendt (1975)for efficiently identifying potential independent variables and relationships. CART employs heuristic methods with binary classification, 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 classified was very low. Working in a branch of artificial intelligence, Guan and Gertner, 1991a, Guan and Gertner, 1991bbuilt 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.

Section snippets

Methods

We rely on the remeasured permanent plots of the Austrian National Forest Inventory for mortality and survival data (Forstliche Bundesversuchsanstalt, 1981, Forstliche Bundesversuchsanstalt, 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

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 fit as well (Neter and Maynes, 1970). We take the classical approach to model the probability of mortality, the logistic equation:P=1+eb′X−1where b′X is a linear combination of parameters b and independent variables X, and e is the base of

Results

For all species, we found the hyperbolic D−1 to be highly significant 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 mortality rate for the larger diameters (Fig. 1). It was clearly superior to a simple D term.

Crown ratio CR was also consistently significant, except for oak (Table 2). We likewise found BAL to be a highly significant predictor of mortality rate for

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; Eckmüllner 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

Acknowledgements

This research was conducted when Monserud was Visiting Scientist and Universitätslektor at the Institut für Waldwachstumsforschung in Vienna, on a grant 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.

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