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Individual-tree diameter growth and mortality models for bottomland mixed-species hardwood stands in the lower Mississippi alluvial valley Dehai Zhaoa,*, Bruce Bordersb, Machelle Wilsona a b

Savannah River Ecology Laboratory, University of Georgia, Aiken, SC 29802 USA Warnell School of Forest Resources, University of Georgia, Athens, GA 30602 USA

Received 19 November 2003; received in revised form 4 February 2004; accepted 16 May 2004

Abstract Individual-tree diameter growth and mortality models were developed for the bottomland mixed-species hardwood stands in the Lower Mississippi Alluvial Valley (LMAV). Data came from 5-year remeasurements of continuous forest inventory plots. Six species groups were created according to diameter structure, tree growth, mortality, recruitment and light demand of species. A 5-year basal area increment model and logistic mortality model were calibrated for species groups. Potential predictor variables at tree-level and stand-level were selected based on the available data and their biological significance to tree growth and mortality. The resulting models possess desirable statistical properties and model behaviors, and can be used to update shortterm inventory. # 2004 Elsevier B.V. All rights reserved. Keywords: Individual-tree growth model; Distance-independent; Mixed-species; Bottomland hardwoods; Mortality

1. Introduction Recently, active management of mixed-species stands is becoming more prevalent and seems to be a worldwide trend. This change from pure, single species stand management to mixed-species stand management stresses the need for tools to support decision making in such stands. Specifically, reliable growth models for mixed-species stands are required in practice. Although mono-specific stands have been modeled extensively and rather successfully for decades, relatively few models or modeling strategies *

Corresponding author. Tel.: þ1 803 725 6181; fax: þ1 803 725 3309. E-mail address: [email protected] (D. Zhao).

have been developed and evaluated for more complex mixed-species stands (Burkhart and Tham, 1992). Mixed-species forests with a high diversity of tree species exhibit a huge range of life forms and stem sizes. In these forests, age is irrelevant as a modeling variable (Vanclay, 1995), and the DBH (diameter at breast height) distributions of species (species group, or stand) may be not unimodal, and thus may not be described by standard distribution functions. Therefore, some modeling approaches used successfully for mono-specific stands have limited utility in multispecific and heterogeneous forests (mixed forests). For example, whole stand models draw on stand-level parameters such as stocking (trees/ha), stand basal area, and standing volume to predict stand growth or yield. This well developed approach has been used

0378-1127/$ – see front matter # 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.foreco.2004.05.043

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only in stands with a limited number of species (Lynch and Moser, 1986; Murphy and Farrar, 1988). Matrix models have been applied widely in mixed-species forest modeling (Buongiorno and Michie, 1980; Solomon et al., 1986; Mendoza and Setyarso, 1986; Mengel and Roise, 1990; Osho, 1991; Buongiorno et al., 1995; Lin and Buongiorno, 1997). However, because of the reasonable diameter classes required and assumption of the uniform distribution of trees within classes, a matrix approach may underestimate or overestimate yields. Compared with matrix models, as well as other stand class models, individual-tree growth models ensure reliable predictions over all tree sizes, sites, and stand conditions, thus they can provide more detailed tree growth information. Individual-tree models simulate diameter (or basal area) increment and mortality for each individual-tree, then predict the growth and yield of stands, and thus avoid using a distribution function to describe the diameter distribution. Although stand-level simulations fit yield data better than tree level simulations, as a result of cumulated model errors from tree to stand-level, tree level approaches seem most appropriate for understanding stand growth as affected by competition between individuals of different species (Porte´ and Bartelink, 2002). Based on the requirement of the spatial location of trees, individual-tree growth models can be subgrouped into ‘distance-dependent tree models’ where the tree location is known and ‘distance-independent tree models’ where the tree location is unspecified (Porte´ and Bartelink, 2002). For most inventory data, spatial information from mapped tree locations is not available, thus distance-independent tree models have been used widely for growth and yield predictions (Wykoff et al., 1982; Belcher et al., 1982; Wykoff, 1990; Monserud and Sterba, 1999; Sterba et al., 2002; Yang et al., 2003). Bottomland hardwoods of the Lower Mississippi Alluvial Valley (LMAV) with a wide diversity of tree species are an important ecological resource providing many functions and values such as wildlife habitats and timber production. Several researchers including Putnam et al. (1960), Hodges (1997), Meadows and Stanturf (1997) discussed the development and management of these stands. However, there is no suitable growth and yield model for bottomland hardwoods in this area. The objectives

of this study were to develop distance-independent individual-tree diameter growth and mortality models for bottomland mixed-species hardwood stands in LMAV using continuous forest inventory (CFI) data. Based on the characteristics of diameter structure, growth, mortality, recruitment and light demand of species, tree species were grouped using ‘principal component and cluster’ analysis procedures. For each species group, we developed individual-tree basal area increment and mortality models that can be used in a stand simulator that uses a list of individual trees by species (or species group). Finally, the performance of the individualtree models for stand basal area prediction was examined.

2. Data The data used in this study are from continuous forest inventory plots located in bottomland mixed hardwood stands in LMAV. All plots used to estimate the parameters of the growth model belong to the Riverfront Hardwood Timber type (Putnam et al., 1960), in Arkansas, Louisiana and Mississippi. This type has been divided into two subtypes—say A and B. Type A is characterized by mixed soils in the ‘A’ horizon, or clay cap less than 36 in. (91.4 cm) deep. Sweet pecan, sycamore, elm and cottonwood are the predominate species associated with these soils. Type B will have a clay cap of 36 in. (91.4 cm) or more and is characterized by ash, hackberry, nuttall oak and overcup oak. The permanent plots were established in 1968, and were re-measured in 1968, 1973, 1978, 1983, 1988, 1993 and 1998 at 5year intervals. Each plot consists of three concentric circles with areas 0.08, 0.02, and 0.004 ha (1/5, 1/20 and 1/100 acre), respectively. The trees in the sawtimber size class (dbh 27.9 cm (11.0 in.)) were measured and numbered on the 0.08 ha (1/5 acre) plot, trees in the pole class (12.7 cm (5.0 in.) dbh 27.8 cm (10.9 in.)) were measured and numbered on the 0.02 ha (1/20 acre) plot, and finally trees in sapling class (2.5 cm (1.0 in.) dbh 12.6 cm (4.9 in.)) were measured and numbered on the 0.004 ha (1/100 acre) plot. Dbh measurements were carried to the nearest tenth inch. Plots that had been permanently cleared or partly cleared were re-established as new plots. There

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was no exact information on the history of harvesting and silvicultural operations in our available data, and measurement records of 1973 were not available. Therefore, the data used to develop models here were taken only from the 5-year measurement intervals where it is known that none of harvest cut, pulp cut or chemical treatments were employed during that interval and only from the last four growth periods (1978–1983, 1983–1988, 1988–1993, 1993–1998). The numbers of plots for the four growth periods are 157, 147, 120 and 122, respectively. To estimate the individual-tree diameter growth models and individual-tree mortality models, the data are pooled over the plots of these four periods (nonoverlapping growth intervals). The pooled 546 ‘plots’ can be treated as independent plots. For re-measured permanent plot data, serial correlation and spatial correlation may be expected. These correlations violate the assumption of independent error terms in most statistical methods, thus invalidate the t-test, the F-test, and the confidence intervals (Koak, 1997). Borders et al. (1987) found that for permanent plot data with more than three measurements, temporal correlation did not occur for nonoverlapping growth intervals, whereas obvious correlation occurred when all possible growth intervals were used. Huang (1997) concluded that for prediction purposes, whether the correlation was accounted for or not had little practical significance. In this study, the diameter growth and mortality models will be mainly used for prediction. In addition, the number of trees is large compared with the number of re-measurements on the same individual, and the models regress diameter (or basal area) increment on initial size rather than age. Therefore, the assumption of no serial correlation should be reasonable in our situation (Vanclay, 1991, 1995; Zhao, 2003). Since there are few mortality trees within each plot, the within-plot spatial correlation should be trivial. Furthermore, the data are pooled over many plots which would ‘dilute’ the within-plot correlation (Yang et al., 2003). Because the plots were widely spaced, the betweenplot correlation should also be small. Thus, we also assume no spatial correlation in the following models. In this study, there are 30 tree species. Some species have insufficient data for reliable parameter estimation. To provide unbiased prediction equations, six

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species groups were created using a cluster analysis procedure based on the variables describing the stand structure, tree growth, mortality and recruitment characteristics and species functional attributes, that is, maximum dbh (Dmax), minimum dbh average dbh (D), (Dmin), mean diameter growth increment for a 5-year maximum diameter growth incregrowth period (G), ment (Gmax), basal area proportion (PBA), mortality (PM), recruitment (PR), and shade tolerance class (TOL) from Burns and Honkala (1990) (Zhao, 2003). A detailed listing of the members of the species groups is presented in Table 1. species groups 1, 2 and 3 belong to soft hardwoods; species groups 4 and 5 belong to hard hardwoods; some small-sized species or less-abundant species combined with non-commercial species form species group 6. Individual-tree diameter growth and mortality models are developed separately for each species group. Only re-measured trees that were alive during both inventories of the growth periods were used to calibrate the individual-tree diameter growth models. Trees alive at the beginning of the growth period and whose mortality statuses (live or dead) taken at the end of the same growth period were used to fit individualtree mortality models. Included in these models, predictor variables at both the tree-level and stand-level were taken at the beginning of the same growth period. The summary of these variables by species group is given in Table 2.

3. Model development Based on the available data set and biological processes associated with tree growth and mortality, the following variables at the tree-level and stand-level are selected as potential predictor variables in the individual-tree basal area increment and mortality models developed later. 3.1. Diameter at breast height (D), diameter squared (D2), and the reciprocal of diameter (D1) Many variables such as age, top height and site quality used in plantation growth models have little relevance in mixed-species stands. Tree size (usually diameter) might be a better variable than tree age for growth and mortality models in these stands, and thus,

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Table 1 Species included in each species group for distance-independent Individual-tree growth models Species Code group

Common name

Scientific name

1 3 5 15 23

Cottonwood Willow Cypress River birch Cedar elm

Populus deltoides Salix nigra Taxodium distichum Betula nigra Ulmus crassifolia

10

Sugarberry or Hackberry

Celtis laevigata or occidentalis

4 6 9 12 20 30 31

Box elder Sweet gum Sycamore Sassafras American elm Silver maple Red maple

Acer negundo Liquidambar styraciflua Platinus occidentalis Sassafras albidum Ulmus americana Acer saccharinum Acer rubrum

32 40 41

Green ash Sweet pecan Bitter pecan

Fraxinus pennsylvanica Carya illinoenisis Carya aquatica

11 54 56 62

Honeylocust Nuttall oak Water oak Overcup oak

Gleditsia triacanthos Quercus nuttallii Quercus nigra Quercus lyrata

13 21 22 42 50

Coffee tree Red elm Winged elm Hickories Cherrybark oak

72 90 94 95 0

Walnut Persimmon Mulberry Osage orange Swamp privet, plum, Swamp dogwood, pawpaw, Sumac, Hawthorn, Button Willow, Water elm, etc.

Gymnocladus dioica Ulmus fulva Ulmus alata Carya sp. Quercus falcata or Q. pagodifolia Juglans nigra Diospyros virginiana Morus rubra Maclura pomifera

1

2

3

4

5

6

has been widely used in modeling individual-tree growth and mortality (e.g. West, 1980; Belcher et al., 1982; Wykoff et al., 1982; Wykoff, 1990; Vanclay, 1995; Monserud and Sterba, 1999; Lessard et al., 2001; Yao et al., 2001; Yang et al., 2003). The shape of the DDS (periodic change in squared diameter) curve relative to dbh is unimodal with a maximum at a level of dbh depending on the species (Wykoff et al., 1982). That is, basal area increment increases to a maximum early in the life of a tree and then slowly decreases, approaching zero as the tree matures. Terms D and D2 included in growth model would represent this relationship. For woody plants, the larger the individual the greater its chance of competing for scarce resources. Thus, mortality rate should decrease as diameter increases. However, as diameter increases further, trees approach maturity with a very low growth rate, senescence becomes apparent, and the probability of mortality increases (Harcombe, 1987). Therefore, a U-shaped mortality trend is probably expected (Buchman et al., 1983; Lorimer and Frelich, 1984; Monserud and Sterba, 1999; Yang et al., 2003). This u-shaped mortality trend could be captured by diameter (D) and diameter squared (D2). For most species, the mortality rate for the smallest trees is quite high, and declines rapidly as individuals survive to larger diameter classes. A hyperbolic D1 transformation of diameter should capture this nonlinear effect (Monserud and Sterba, 1999). Therefore, individualtree diameter (D), diameter squared (D2), and the reciprocal of diameter (D1) are selected as predictor variables. ¯ ), and DBH and relative 3.2. Relative diameter (D/D ¯) diameter interaction term (D2/D In the PROGNOSIS model, relative diameter (D=D) that is ratio of tree DBH (D) to mean stand diameter and a DBH and relative size interaction term (D) were included. An increment prediction for (D2 =D) a small tree in one stand is more strongly affected by relative size than is the prediction for a large tree in a different stand, even though the two trees are at the same relative position in the DBH distribution within their respective stands (Wykoff, 1986). In this study, these two terms are also included as potential predictor variables.

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Table 2 Summary of the data for model development by species group and mortality status (number of trees (N); mean, minimum (min), maximum (max), and standard deviation (S.D.)) Species group

Variables

Live trees N

Dead trees Mean

S.D.

Min

Max

N

Mean

S.D.

Min

Max

1 D D=D D2 =D BA RBA S CC

187 187 187 187 187 187 187

22.83 2.58 66.67 140.17 0.29 0.43 1.71

9.15 1.21 50.99 52.80 0.15 0.50 0.89

3.6 0.43 2.36 14.71 0.026 0 1

46.7 7.10 320.78 332.72 0.810 1 5

47 47 47 47 47 47 47

18.29 2.48 64.10 114.15 0.28 0.32 2.40

15.48 1.65 97.25 56.84 0.19 0.47 1.30

1.2 0.23 0.37 21.05 0.013 0 1

94.8 7.31 575.33 332.72 0.810 1 5

D D=D D2 =D BA RBA S CC

2623 2623 2623 2623 2623 2623 2623

11.68 1.45 19.98 113.77 0.51 0.21 2.68

5.39 0.79 17.30 39.98 0.24 0.41 1.03

1.0 0.09 0.09 12.14 0.008 0 1

29.9 5.77 118.92 332.72 1.0 1 5

280 280 280 280 280 280 280

10.04 1.42 19.33 112.84 0.46 0.20 3.20

6.23 0.98 20.71 39.58 0.24 0.40 1.10

1.0 0.13 0.13 12.20 0.007 0 1

28.8 4.72 96.69 332.72 1.0 1 5

D D=D D2 =D BA RBA S CC

2061 2061 2061 2061 2061 2061 2061

11.50 1.57 22.50 113.61 0.48 0.16 2.81

6.12 0.97 24.08 39.78 0.24 0.36 1.04

1.0 0.10 0.10 12.78 0.006 0 1

36.3 7.98 252.08 332.72 1.0 1 5

489 489 489 489 489 489 489

11.08 1.50 23.11 116.61 0.47 0.15 3.16

7.23 1.09 28.35 47.75 0.26 0.35 1.10

1.0 0.11 0.11 12.14 0.022 0 1

40.2 6.43 250.00 332.72 0.969 1 5

D D=D D2 =D BA RBA S CC

1444 1444 1444 1444 1444 1444 1444

14.06 1.80 30.08 117.25 0.40 0.33 2.32

6.81 0.99 28.74 40.74 0.21 0.47 1.11

1.0 0.12 0.12 22.51 0.008 0 1

40.2 7.79 279.51 332.72 0.972 1 5

157 157 157 157 157 157 157

10.72 1.52 25.38 119.28 0.36 0.33 3.15

8.71 1.18 34.75 38.69 0.20 0.47 1.21

1.0 0.14 0.16 42.17 0.009 0 1

38.4 5.58 175.53 196.00 0.880 1 5

D D=D D2 =D BA RBA S CC

227 227 227 227 227 227 227

13.49 1.88 30.77 107.54 0.22 0.58 2.56

6.96 1.17 30.14 42.67 0.16 0.49 1.14

1.2 0.22 0.41 25.21 0.014 0 1

35.7 5.83 169.51 332.72 0.689 1 5

34 34 34 34 34 34 34

8.46 1.46 26.48 109.62 0.20 0.35 3.44

9.37 1.62 49.16 38.65 0.14 0.49 1.16

1.1 0.20 0.27 54.49 0.017 0 1

32.3 6.33 204.45 176.99 0.525 1 5

D D=D D2 =D BA RBA S CC

542 542 542 542 542 542 542

4.58 0.74 5.56 106.21 0.21 0.23 3.74

4.21 0.61 9.44 44.92 0.19 0.42 0.65

1.0 0.12 0.13 12.78 0.005 0 1

20.0 3.86 68.78 256.34 0.835 1 5

225 225 225 225 225 225 225

3.90 0.67 4.71 106.71 0.20 0.24 3.83

3.84 0.61 9.23 45.11 0.21 0.43 0.57

1.0 0.10 0.10 12.20 0.004 0 1

19.2 3.52 67.58 256.34 0.835 1 5

2

3

4

5

6

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3.3. Competition measure Distance-dependent competition indices widely used in individual-tree models (Bella, 1971; Hegyi, 1974; Daniels, 1976; Waller, 1981; Mithen et al., 1984; Daniels et al., 1986; Weiner, 1982, 1984; Holmes and Reed, 1991; Radtke et al., 2003) require time-consuming and expensive information on locations of individual trees. As with most inventory data, spatial information from mapped tree locations is not available, meaning that we need distance-independent competition indices. Thus, the resultant models will be distance-independent. Stand basal area is a good measure of stand crowding, since it combines both tree size and density. Competition can be expressed by a variety of stand density measures. In mixed-species hardwoods, both one-sided competition and two-sided competition are likely to be present. In one-sided competition, larger trees are at a competitive advantage over smaller trees, but smaller neighbors do not affect the growth and survival of larger trees (Cannell et al., 1984). This type of competitive relationship may be expressed in aboveground competition for light. The basal area in larger trees (BAL) has been commonly used to capture one-sided competition in modeling tree growth (Wykoff et al., 1982; Wykoff, 1990; Lessard et al., 2001; Sterba et al., 2002) and mortality (Monserud and Sterba, 1999; Yang et al., 2003). In two-sided competition, all trees impose some competition on their neighbors, regardless of their sizes (Cannell et al., 1984). This type of competition is usually for below ground resources such as moisture and nutrients. On the other hand, mixed-species stands with a certain niche differentiation among different species may make more efficient use of light, nutrients and moisture. The litter from diverse species is usually beneficial to nutrient balance, and thus, may improve total site quality (Sterba et al., 2002). Therefore, tree species mixture should be included in the tree growth model. For this study, instead of BAL, we use stand basal area (BA) and the basal area proportion of species (RBA ¼ BAi/BA, where BAi is the basal area of the ith species group). 3.4. Site effects In the following models, we set a dummy variable (S) to identify the two habitat types of the Riverfront

Hardwood Timber: S ¼ 0 is for type A; and S ¼ 1 for type B. 3.5. Crown class (CC) Crown class, which is classified based on crown position in the canopy and crown condition, is recorded in five ordered categories: dominant (1), co-dominant (2), intermediate (3), suppressed (4) and no-top (5). Each tree with dbh 12.7 cm (5.0 in.) was assigned a value of crown class in the field survey. We set trees in the sapling class (dbh < 12.7 cm (5.0 in.)) at crown class 4. This variable is assumed to be an indicator of tree vigor. Many mortality models include terms describing individual-tree vigor such as growth increment from the preceding period (Monserud, 1976; Buchman, 1979; Belcher et al., 1982; Buchman et al., 1983; Hamilton, 1986, 1990; Wykoff, 1986; Zhang et al., 1997; Yang et al., 2003), crown related variables, e.g. crown width, crown ratio (the ratio of live crown length to total tree height) or crown class (Hamilton and Edwards, 1976; Wykoff et al., 1982; Monserud and Sterba, 1999). An interesting finding is that probably only one indicator of tree vigor is needed in these models (Monserud and Sterba, 1999). For this study, we include crown class as a predictor variable. Crown class is a discrete variable, so in the following models we set four dummy variables to identify these five crown classes: 1; if CC ¼ 2 1; if CC ¼ 3 C1 ¼ ; C2 ¼ 0; otherwise 0; otherwise 1; if CC ¼ 4 1; if CC ¼ 5 C3 ¼ ; C4 ¼ 0; otherwise 0; otherwise 3.5.1. Individual-tree diameter growth model The diameter growth of individual trees may be expressed as diameter increment or a corresponding basal area increment. Bella (1971) concluded that using basal area increment rather than diameter increment as the dependent variable provides higher values of R2. West (1980) and Shifley (1987), however, found no difference in the precision of diameter and basal area increment equations. Vanclay (1995) inferred that any differences in the fit may be due to the error structure and an implied functional relationship, rather than the superiority of one model over another.

D. Zhao et al. / Forest Ecology and Management 199 (2004) 307–322

Wykoff (1990) used the logarithm of periodic change in squared diameter as the dependent variable to develop basal area increment models for 11 conifer species in the Northern Rocky Mountains. His distance-independent models behaved well and had desirable statistical properties such as homogeneous residual variance, minimal multicollinearity, and linear parameters. In our preliminary analyses, we compared the 5-year diameter increment (DD ¼ D2 D1), 5-year diameter growth rate (D_rate ¼ (DD/D1)), 5year change in squared diameter (DDS ¼ D22 D21 ), and the natural logarithm of each as dependent variables. From the residual plots and fit statistics, we found ln(DDS) and ln(D rate) are better than the others. The periodic change in squared diameter is directly proportional to basal area increment and can be readily transformed to an estimate of diameter increment if the diameter at the start of the growth period is known (Stage, 1973). Hence, ln(DDS) is used as the dependent variable in this study. The general form of the linear basal area increment model for each species group is: 1 D lnðDDSÞ ¼ b0 þ b1 þ b2 D þ b3 D2 þ b4 D D D2 þ b5 þ b6 BA þ b7 RBA þ b8 S D 4 X þ b9i Ci

(1)

i¼1

where ln(DDS) is the natural logarithm of 5-year change in squared diameter, and the bs are the coefficients to be estimated; the values of the predictor variables are taken at the beginning of the 5-year periods. The ordinary least squares method (OLS) was used to estimate the unknown parameters. The best model for each species group was selected based on statistical properties and biological principles. Any relationship that violates accepted biological principles should be rejected, even if it results in efficient predictions for a particular data set (Hamilton, 1990). 3.5.2. Individual-tree mortality model Individual-tree mortality is a discrete event, either alive or dead. In this study, the dependent variable was coded as 0 for trees that were alive at both ends of a measurement interval, and 1 for trees that were alive at the beginning of a measurement interval but were dead

313

at the end of the measurement interval. The predicted probability of mortality should be in the range [0,1] with 0 being alive and 1 being dead. Logistic regression appears to be the best method for individual-tree mortality modeling and has been widely applied (e.g. Hamilton, 1986; Vanclay, 1995; Monserud and Sterba, 1999; Yao et al., 2001; Yang et al., 2003). Vanclay (1995) believes the logistic function fitted to individual-tree data may offer the best way to model mortality in tropical forests which cover a spectrum of different species mixtures and age structures, precluding the possibility of using either stand age or site index as predictor variables. For this study, we take the general form of the full mortality model using the logistic function for each species group: 1 pm ¼ 1 þ exp b0 þ b1 þ b2 D þ b3 D2 D D D2 þ b4 þ b5 þ b6 BA þ b7 RBA þ b8 S D D 1 4 X þ b9i Ci (2) i¼1

where pm is the 5-year probability of mortality of an individual-tree and the bs are the coefficients to be estimated. The values of predictor variables are taken at the beginning of the 5-year periods. The maximum likelihood method was used to estimate the unknown parameters. The best (reduced) model for each species group was selected based on the Hosmer–Lemeshow goodness-of-fit statistic and the ecological behaviors of the fitted models. For the Hosmer–Lemeshow goodness-of-fit test, in this study, the grouping method based on the percentiles of the predicted probabilities was used to create ten groups of roughly the same size (Hosmer and Lemeshow, 2000). The discrepancies between the observed and expected number of observations in these groups are summarized by the Pearson chi-square statistic, which is then compared to the critical value from a chisquare. A small corresponding P-value suggests that the fitted model is not an adequate model. If a function fits the data well, the P-value associated with that function should be larger than 0.05, indicating no significant difference between the fitted function and the data at the 95% confidence level. The best model with the largest P-value was selected from several potential functions.

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4. Model estimation and discussion 4.1. Individual-tree basal area increment model Parameter estimates for the individual-tree basal area increment model for each species group are given in Table 3. Variables whose associated parameter estimates were not statistically significantly different from zero are excluded from the model. The resultant models, therefore, are the reduced model from that shown in (1). From Table 3, we can see that for each species group tree basal area increment is a function of tree diameter as well as other variables. The positive coefficient for tree diameter (D), negative coefficients for the diameter squared (D2) and the reciprocal of the diameter (1/D) confirm the presumed relationship between tree growth and tree diameter, that is, for a given set of stand conditions the basal area increment increased as tree diameter increased to a certain level and then decreased thereafter. For species groups 1 and 4, the has a significant and relative diameter term (D=D) positive effect on the basal area increment prediction. has a The dbh-relative size interaction term (D2 =D) significant effect on tree basal area increment of species groups 4 and 6. The basal area increment is significantly and negatively related to stand total basal area for all species

groups with the exception of species group 6 (Table 3). This suggests, as expected, that basal area increments of trees of these species groups decrease as the competitive influences resulted from the increase of stand total basal area increase. Negative estimates of parameters b7 for species groups 1, 2, and 4 imply that intra-specific competition (competition between trees of the same species (or species group)) for these groups is also intensive, thus resulting in a decrease of basal area increment as the basal area proportion of these groups increases. In addition to intra-specific competition, inter-specific competition (competition between trees of different species (or species groups)) is also present in mixed stands. Variables such as stand total basal area and mixture type (e.g. the basal area percentage of species) explicitly included in distanceindependent individual-tree models can describe these two types of competition (Sterba et al., 2002). Crown classification of hardwood trees is based on two factors: crown position in the canopy (the amount of direct sunlight received by the crown) and crown condition (crown balance and crown size). Crown condition and amount of sunlight received are important factors affecting the growth of the tree. The negative estimates of parameter b9i suggest that the individual-tree basal area increment is significantly related to crown class of trees for all species groups (Table 3). As expected, dominant and co-dominant

Table 3 The estimated parameters and fit statistics of distance-independent individual-tree basal area increment models by species group Parameter

Species group 1

Intercept 1=D D D2 D=D D 2 =D BA RBA S C1 C2 C3 C4 Fit statistics

^ b 0 ^ b 1 ^ b 2 ^ b 3 ^ b 4 ^ b 5 ^ b 6 ^ b 7 ^ b 8 ^ b91 ^ b 92 ^ b 93 ^ b

2 2.9684 (0.3483)

0.1105 (0.0265) 0.0015 (0.0005) 0.1453 (0.0639) 0.0028 (0.0011) 0.8183 (0.3928) 0.3526 (0.1194) 0.3506 (0.1548)

94

R2 Root MSE

0.432 0.7084

Note: Standard errors are given in parentheses.

3

4

4.0027 2.4838 0.0808 0.0013

(0.1687) (0.2246) (0.0183) (0.0006)

2.6057 1.1034 0.1354 0.0020

0.0042 0.6340 0.0717 0.3304 0.6721 1.4219 1.4108

(0.0004) (0.0691) (0.0385) (0.0547) (0.0583) (0.0728) (0.1076)

0.0020 (0.0005)

0.591 0.7858

0.1474 0.4008 1.1281 1.1211 0.545 0.8645

(0.1670) (0.2267) (0.0163) (0.0005)

(0.0719) (0.0730) (0.0855) (0.1245)

5

2.7815 1.1346 0.1290 0.0011 0.1523 0.0114 0.0038 0.2416

(0.1935) (0.2991) (0.0181) (0.0006) (0.0850) (0.0041) (0.0005) (0.0999)

0.2419 0.6514 1.4103 1.4062

(0.0574) (0.0677) (0.0883) (0.1513)

0.631 0.7516

6

2.7136 2.6204 0.1545 0.0024

(0.4235) (0.7707) (0.0389) (0.0010)

0.6624 (0.2524) 0.4139 (0.0425) 0.0191 (0.0030) 0.0260 (0.0116)

0.0057 (0.0012) 0.2041 0.3175 0.5373 0.6524 1.2444 0.723 0.6978

(0.0971) (0.1467) (0.1515) (0.1954) (0.2876)

0.6786 (0.2123) 1.1905 (0.2250) 1.6302 (0.4881) 0.579 0.9184

D. Zhao et al. / Forest Ecology and Management 199 (2004) 307–322

trees have higher growth potential, while intermediate, suppressed and no-top trees have lower growth potential. Most of the trees of species group 1 were dominant, co-dominant and intermediate, the effects of crown classes 4 and 5 on the growth of this group cannot be detected statistically in this model. The negative parameter estimate of the dummy variable S for species groups 1 and 2 and positive estimate for species group 5 imply that sites of Timber type A are beneficial to the growth of soft hardwood species groups 1 and 2, while the sites of Timber type B are suitable to hard hardwood species group 5. This conforms to the actual situations. Cottonwood, elm and willow included in species group 1 are the predominate species associated with Timber type A,

while nuttall oak, overcup oak and water oak in species group 5 are characteristics of Timber type B. Slightly surprising is that species group 2, sugarberry and hackberry, has higher growth potential in Timber type A, although hackberry with ash, nuttall oak and overcup oak are the predominate species in Timber type B. Compared with species groups 1 and 5, we note, the site effect on the growth of species group 2 is small, although it is statistically significant. The values of R2 range from 0.432 to 0.723 and estimates of standard errors (natural logarithm of the 5-year change in squared diameter) range from 0.6978 to 0.918 in the final models. The residual plots of the final basal area increment models for each species group are given in Fig. 1. We

4.0

4.5

3 2 1

0

1

2

3

Species group 4

1 0 -2

2 1 -1 0

3

4

0

1

2

3

Species group 5

Species group 6

3

4

5

5

1 0 -1

1 0 -1

2

4

2

Fitted values

Standarized Residuals

Fitted values

Fitted values

4

2

Species group 3 Standarized Residuals

Fitted values

2

1

0

5.0

Fitted values

-2

0

-2

Standarized Residuals

1 0 -1

3.5

-3

Standarized Residuals

3.0

1

Standarized Residuals

Species group 2

-2

Standarized Residuals

Species group 1

2.5

315

0

1

2

3

Fitted values

Fig. 1. Plots of standardized residuals against the fitted values of distance-independent individual-tree basal area increment models for each species group.

316

D. Zhao et al. / Forest Ecology and Management 199 (2004) 307–322

Table 4 The estimated parameters and P-values of Hosmer–Lemeshow goodness-of-fit test of distance-independent individual-tree mortality models by species group Parameter Species group 1

Intercept 1=D D D2 D=D D2 =D BA RBA S C2 C3 C4 H–L’s statistic pHL

^ b 0 ^ b 1 ^ b 2 ^ b 3 ^ b 4 ^ b 5 ^ b 6 ^ b 7 ^ b 8 ^ b 92 ^ b 93 ^ b

2

3

4

3.9440 (0.2452) 2.4959 (0.1807) 1.3953 (0.4110) 1.9903 (0.3475) 0.2564 (0.0532) 0.0467 (0.0106) 0.0025 (0.0009) 0.7231 (0.2211) 0.5417 (0.0914)

3.1271 (1.3362) 1.2891 (0.4283) 2.1842 (0.6303)

94

10.804 0.2130

0.2578 (0.1503) 0.4565 (0.1830) 1.6005 (0.2077) 1.5253 (0.2980) 3.3449 0.9109

0.5488 (0.1454) 1.2167 (0.2070) 6.006 0.6465

5

1.9704 (0.4077) 4.2375 (0.6206) 4.4666 (1.2269) 0.2792 (0.0450) 0.0062 (0.0011) 0.4322 (0.1417) 0.0259 (0.0069) 0.0070 (0.0023)

6 1.1579 (0.1503) 0.6228 (0.2786)

0.8828 (0.462) 0.8330 (0.2595) 2.2551 (0.7035) 1.9311 (0.3725) 1.8108 (1.0231) 11.108 6.418 0.1956 0.6006

9.468 0.3043

Note: Standard errors are given in parentheses; H–L’s statistic is Hosmer–Lemeshow’s goodness-of-fit statistic; pHL is P-values of Hosmer–Lemeshow’s goodness-of-fit test.

can see that there are no observable patterns on these residual plots. Using the natural logarithm of the 5year basal area increment as the dependent variable may remove some heterogeneity in variance. 4.2. Individual-tree mortality model Parameter estimates for the tree mortality models are given in Table 4. All parameters listed in Table 4 are highly significant (P < 0.01) with the exception of the coefficients of S for group 3 (P ¼ 0.086) and for group 5 (P ¼ 0.056). Small-sized and less-abundant, or noncommercial species compose group 6, and are usually in the understorey. For this group, only the reciprocal of diameter is significant in predicting mortality rate (Table 4), thus implying a higher mortality rate for smaller diameter trees and that the mortality rate rapidly decreases as diameter increases. For species group 5, composed of five shade-intolerant species (honey-locust, nuttall oak and water oak) and one shade-intermediate species (overcup oak), tree mortality rate is most highly correlated to diameter. The odds ratio estimate for the reciprocal of diameter is 87.1, very high, which implies that tree mortality rate decreases with increasing tree size for term are also this group. Crown class and D2 =D

significant in predicting mortality rate for this group. Trees with a good crown and large amount of direct sunlight have a lower mortality rate, while suppressed and non-top trees have a higher mortality rate. For species groups 2, 5 and 6, the coefficients of D and D2 were not significant, indicating that a senescence effect could not be detected. For species groups 1 and 4, the coefficient of D is negative and the coefficient of D2 is positive, while for species group 3 the coefficients of 1/D and D are positive, capturing the U-shaped mortality trend. Crown class (CC) is consistently significant, except for species group 6. The coefficients of C3 and C4 are positive for species groups 1–5, indicating mortality rates of suppressed and no-top trees are higher than that of dominant and co-dominant trees. The differences due to effects of crown classes 1, 2 and 3 on tree mortality rate cannot be detected statistically, which implies that dominant, co-dominant and intermediate trees have the same lower mortality. The dummy variable S is significant only for species group 1. The negative coefficient of S suggested that trees of groups 1, 3 and 5 in Timber type A have a higher mortality rate than in Timber type B for a given set of conditions. The coefficient of BA is positive for species group 4, indicating that mortality rate will be higher as stand total basal area increases. The positive coefficient of RBA for species

D. Zhao et al. / Forest Ecology and Management 199 (2004) 307–322

317

Table 5 Chi-square values for the parameter estimates in Table 5 (Model 4) Species group

Chi-square statistic Variable 1/D

1 2 3 4 5 6

D 23.3

11.5 32.8

19.5 38.5

D2 6.6

D=D

D2 =D

BA

10.7 35.1

RBA

S

5.5

9.1 6.2 2.9

31.6

9.3

13.3 5.0

group 1 indicates that mortality rate will increase as the basal area proportion of species group 1 increases. This may imply that intra-specific competition is more likely to cause tree mortality for species group 1. The calculated Hosmer–Lemeshow’s goodness-offit statistic is from 3.345 to 11.108 and the corresponding P values are from 0.9109 to 0.1956 (Table 4). The P values of the mortality models for each species group are larger than 0.05, which suggests that the mortality functions for all species groups fit the data well. To judge the relative importance of the variables, standard errors are listed in Table 4 and chi-square values are provided in Table 5. The chi-square statistics reveal that breast height diameter and crown class CC are important variables.

5. Model prediction Individual-tree diameter growth and individual-tree mortality models constitute an individual-tree growth model which can be used to predict stand growth. Usually, model evaluation may be made for the individual-tree diameter growth model and individual-tree mortality model, separately. Based on the results of model evaluations, some improvements may be made to each component of the tree growth model. Then, the performance of the whole individual-tree growth model for stand growth prediction should be examined. Stand basal area is the sum of individual basal areas of surviving trees, and thus should reflect the interrelated performance of both tree mortality and diameter growth prediction equations. In this study, therefore, we directly test the accuracy of the growth model based on 5-year stand basal area predictions.

C2

9.0 14.1

6.7

C3

C4

12.0 59.4 14.2 10.3 10.3

26.2 34.5 26.9 3.1

A total of 148 ‘plots’ were randomly selected (about 27%) as test data from the four measurement periods. To examine the performance of individual growth models in detail over stand densities, the test data sets were separated into 11 BA categories of 4.6, 9.2, . . . , 50.5 m2/ha (20, 40,. . ., 220 ft2/acre) according to initial stand basal area. Stand growth was predicted by simulating the growth and death of individual-trees within a stand. That is, individual-tree diameter growth and mortality models were applied to project the tree list of each plot based on characteristics of individual-trees and stands. Stand basal area was obtained by summarizing the updated tree list. At the projection step, probabilistic mortality and deterministic mortality methods can be used to assign mortality to individual-trees (for details see Belcher et al., 1982). A comparison of the predicted stand basal area of these two approaches indicated no significant difference in mean stand basal area, which confirms the finding of Belcher et al. (1982) and Weber et al. (1986) (Zhao, 2003). The following discussion is based only on results using the deterministic mortality method. The stand basal area obtained from the projection does not include the basal area of the ingrowth. For comparison purposes, the basal area of the ingrowth is estimated to get the predicted stand total basal area. In this study, we cannot find a suitable ingrowth function separately by species group. The pooled ingrowth of all species was expressed as a function of the stand basal area: I ¼ 360:257 23:545BA

(3)

where I is the ingrowth defined as the expected number of live trees of all species per hectare that

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Table 6 Actual and predicted average stand yields (m2/ha) after 5 years for stands of different densities

4.3 9.7 12.9 18.5 23.0 27.2 31.9 37.0 39.7 46.5 50.5 23.7

Total basal area

Actual

Predicted

Difference (%)

Actual

Predicted

Difference (%)

6.1 11.9 13.6 19.4 22.9 27.1 32.5 34.7 39.4 47.8 53.3 24.1

4.2 11.0 14.0 19.1 23.5 28.0 32.5 37.0 40.4 43.3 50.1 24.3

30.6 7.5 3.0 1.2 2.5 3.5 0.0 6.5 2.6 9.3 6.0 1.0

7.7 13.7 15.5 21.2 24.9 28.2 34.4 35.5 41.1 48.4 53.7 25.7

4.7 11.4 14.4 19.4 23.8 28.2 32.7 37.1 40.5 43.4 50.1 24.6

39.4 17.1 7.2 8.2 4.3 0.3 5.1 4.5 1.4 10.4 6.8 4.2

TREES/ACRE

300

BA = 20 ft^2/acre

BA = 40 ft^2/acre

0

100

200

expected basal area of the ingrowth is expressed as 0.00138 I (m2/ha). The initial average stand basal area, actual and predicted average stand basal area excluding

0

TREES/ACRE

grow into the smallest size class during 5-year growth period; BA, the stand basal area at the beginning of the growth period. The quadratic mean diameter of ingrowth trees is about 4.2 cm, thus, the

300

4.6 (n ¼ 6) 9.2 (n ¼ 9) 13.8 (n ¼ 17) 18.4 (n ¼ 23) 22.9 (n ¼ 30) 27.5 (n ¼ 27) 32.1 (n ¼ 15) 36.7 (n ¼ 11) 41.3 (n ¼ 5) 45.9 (n ¼ 2) 50.5 (n ¼ 3) All (n ¼ 148)

Basal area (excluding ingrowth)

200

Initial

100

Basal area (m2/ha)

2

4

6

8

10

12

14

16

18

20

22

24

26

2

4

6

8

10

12

14

16

18

20

22

24

26

150

150

BA = 60 ft^2/acre

BA = 80 ft^2/acre

0

50

TREES/ACRE

250

DIAMETER(in.)

0 50

TREES/ACRE

DIAMETER(in.)

2

4

6

8 10

14

18

22

26

30

34

2

4

6

8 10

14

18

22

26

30

34

26

30

34

DIAMETER(in.)

100

BA = 120 ft^2/acre

0

50

TREES/ACRE

50 100

BA = 100 ft^2/acre

0

TREES/ACRE

150

DIAMETER(in.)

2

4

6

8 10

14

18

22

DIAMETER(in.)

26

30

34

2

4

6

8 10

14

18

22

DIAMETER(in.)

Fig. 2. Actual (–- * -–), and predicted average diameter distributions through individual-tree growth model (– & –) after 5 years for stands at densities of 4.6, 9.2, 13.8, 18.4, 22.9, and 27.5 m2/ha (20, 40, 60, 80, 100 and 120 ft2/acre), respectively.

D. Zhao et al. / Forest Ecology and Management 199 (2004) 307–322

120 80 0

50

BA = 160 ft^2/acre

40

TREES/ACRE

150

BA = 140 ft^2/acre

100

4.5% to 10.4%. For the BA categories of 4.6 and 9.2 m2/ha, the individual-tree growth model extremely under-predicted the total stand basal area. In these stands with lower basal area (less than 9.2 m2/ha), the soft hardwood species, that is the first three groups of species, are predominant, more than 75% in terms of basal area. The basal area increments of trees of these species groups may be larger than the expected from Model (1) because competition among trees is not strong enough to retard their growth. In these stands, there may be density-independent mortality, but tree mortality resulted from competition may not be obvious, thus the actual died number of trees is smaller than the predicted from Model (2). In addition, the ingrowth of these stands was also underestimated by Model (3). The comparison of actual and predicted diameter distributions obtained from the individual-tree models was shown in Figs. 2 and 3. We can see that the predicted distributions for larger diameter classes

0

TREES/ACRE

ingrowth, and actual and predicted average stand total basal area were calculated for each BA category. The results were given in Table 6. Between the actual and predicted average stand basal area excluding ingrowth, the difference for the BA category of 4.6 m2/ha was 30.6%; for other categories, the differences ranged from 6.5 to 9.3%; the overall difference, for all categories, was 1.0% (Table 6). On average, the individual-tree growth models slightly over-predicted stand basal area. From Table 6, we can see that the overall difference between the actual and predicted stand total basal area for all categories was 4.2%, which indicated on average stand total basal area was under-predicted. This is due to the fact that under-estimation of the ingrowth part of stand basal area obtained from Model (3) overcame the over-predicted stand basal area obtained from the projection. For the BA categories of more than 9.2 m2/ha, the differences between the actual and predicted average stand total basal area ranged from

319

2

4

6

8 10

14

18

22

26

30

34

2

4

6

8 10

14

18

22

26

30

34

26

30

34

DIAMETER(in.)

2

4

6

8 10

14

18

26

30

300

TREES/ACRE

22

BA = 200 ft^2/acre

0 100

60

BA = 180 ft^2/acre

0 20

TREES/ACRE

100

DIAMETER(in.)

34

2

4

6

8 10

14

18

22

DIAMETER(in.)

60 20

40

BA = 220 ft^2/acre

0

TREES/ACRE

80

DIAMETER(in.)

2

4

6

8 10

14

18

22

26

30

34

DIAMETER(in.)

Fig. 3. Actual (–- * -–), and predicted average diameter distributions through individual-tree growth model (–- & -–) after 5 years for stands at densities of 32.1, 36.7, 41.3, 45.9, and 50.5 m2/ha (140, 160, 180, 200 and 220 ft2/acre), respectively.

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agree closely with actual diameter distributions, whereas there are big differences for smaller diameter classes, such as 5.1 and 10.2 cm (2 and 4 in.) classes. In contrast to the matrix model approach, which requires a fixed number of diameter classes, individual-tree growth models ensure predictions over all tree sizes. Thus, individual-tree growth models can provide more reliable diameter distributions with a larger number of classes. In summary, the individual-tree growth models predict stand basal area in the short-term very well except for stands with lower densities (less than 9.2 m2/ha). Combined with the ingrowth model, the developed individual-tree growth models seem to give valid predictions of stand total basal area for stands with initial stand basal areas more than 9.2 m2/ha.

appropriate use for the developed individual-tree growth models would be short-term inventory updating. In original data, the exact time that harvest cut, pulp cut or chemical treatments were employed is unknown for these plots, therefore, the data presented here were taken only from the 5-year intervals where it is known that none of the above treatments were employed during that interval. If the information on thinning time, thinning intensity, etc. were available, the models could be modified. The performance of these models has demonstrated that the distance-independent individual-tree modeling approach can be used for bottomland mixed-species hardwoods in LMVA. The next step is to further test these models with new data, and to compare with other modeling approaches such as a multi-species density-dependent matrix growth modeling.

6. Conclusion Using Principal Component and Cluster analysis procedures based on the indices related to the characteristics of diameter structure, growth, mortality, recruitment and light demand of species, six species groups were created for the bottomland mixed-species hardwood stands in LMAV. Distance-independent individual-tree basal area increment and mortality models, which are two primary components of individual-tree growth models, were developed and fitted for each species group. Similar to many individual-tree diameter increment and mortality models, multiple predictor variables were used in this study. Potential predictor variables were selected based on their ecological importance to tree growth and mortality rather than only on fitting statistics, which are related to tree size, relative size, crown class, site effect, stand density, proportion of species and competition. The individualtree basal area increment models were fitted using ordinary least squares and the parameters of individual-tree mortality models were estimated using the maximum likelihood method. The fitted models possess desirable statistical properties and are biologically reasonable. The individual-tree growth models used in conjunction with stand ingrowth model behave well in prediction of stand basal area. Because some variables such as tree crown class cannot be predicted, the most

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