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Equations to describe crown allometry of Larix require local validation
Forest Ecology and Management 148 (2001) 109±116
Equations to describe crown allometry of Larix require local validation Daniel W. Gilmore* University of Minnesota, Department of Forest Resources, North Central Research and Outreach Center, 1861 Highway 169 East, Grand Rapids, MN 55744-3396, USA Received 21 January 2000; accepted 25 May 2000
Abstract Allometric equations were developed to describe crown attributes of three species of open-grown Larix in Minnesota and compared with similar data from Maine and published works from Austria. No differences were detected in allometric relationships to predict crown length, projected crown radius, and crown radius at different depths within the crown among tamarack, European larch, and hybrid EuropeanJapanese larch grown in Minnesota. Small (<1 m within the range of data), but signi®cant, differences in the prediction of crown length from total height were found between Minnesota and Maine. Allometric equations constructed from inventory data in Austria used a different model form and predicted larger crown attributes then equations generated with data from Minnesota. Results underscore the necessity for forest managers to validate allometric equations locally. With exotic species, however, it may be necessary to use published works until local data and resources for validation can be obtained. Applications include usage in the construction of stocking charts, stand density diagrams, and the prediction of DBH and volume from crown shape using remote sensing data. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Austria; Crown shape; Larch; Leaf area; Maine; Minnesota; Stem volume; Tamarack
1. Introduction The potential for fast-growing exotic larch to alleviate projected ®ber supply shortages in North America is well documented (Einsphar et al., 1984; Carter and Selin, 1987; Zaczek et al., 1994). Most site quality and growth and yield studies in North America for non-indigenous Larix, however, have been empirical (Aird and Stone, 1955; Parsonage, 1989; Gilmore et al., 1993, 1994; Gilmore and Briggs, 1996). Process-based and remote sensing models of tree growth
* Tel.: 218-327-4522; fax: 218-327-4126. E-mail address: [email protected] (D.W. Gilmore).
characterize the tree canopy primarily through estimates of leaf area. Crown shape has been shown to have a strong relationship to leaf area (Coyea and Margolis, 1992; Gilmore et al., 1996; Kene®c and Seymour, 1999) and for many process-based models (e.g. Mitchell, 1975) crown shape serves as a surrogate for leaf area. Allometric relationships are also useful in the construction of stocking guides (Seymour and Smith, 1987) and density management diagrams (Jack and Long, 1996). Once allometric equations are developed to describe attributes for a particular species, the tendency is for practitioners to apply them without local validation. One reason for the ubiquitous application of allometric equations is the cost associated with validation. Once in common usage, allometric
0378-1127/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 1 1 2 7 ( 0 0 ) 0 0 4 9 3 - X
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D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
equations are often assumed to be universally valid for a given species. In the special case of exotic species, local data often do not exist to validate allometric equations for local use. Nonetheless, the local validation of allometric equations is important because their principal application is to directly or indirectly estimate timber volume from which harvest levels are determined. The aim of this study is to develop allometric equations to predict crown attributes from readily available tree measurements (e.g. height, diameter, crown length) for three species of open-grown Larix, two of which are exotic to Minnesota. These results will be compared to European larch using data from Maine and published results from Austria.
Gilmore and Briggs, 1996). Plantations were located in 3 climatic zones (Briggs and Lemin, 1992), elevations ranged from 35 to 300 m asl, soils were representative of Spodosols, Inceptisols, and Entisols, of glacial or glacial-¯uvial origin, acidic, sand to silt loam in texture, and characterized by a frigid temperature regime. Latitude ranged from 448560 to 458160 N and longitude ranged from 688390 to 708350 E. Data used for comparison in this study were collected from the younger plantations that had not yet achieved crown closure. On average, trees in Maine were larger (average attributes (range in parentheses): DBH14.4 cm (7.3±24.5); crown length8.1 m (5.1± 12.4); total tree height8.7 m (5.5±13.7)) and site index (base age 20 years at BH) was higher (13.4± 19.6 m) than in Minnesota (Table 1).
2. Methods and materials
2.2. Data collection
2.1. Study locations
Total tree height (0.1 m), diameter at breast height (0.1 cm, measured at 1.37 m above ground line), diameter at base of the live crown (0.1 cm), crown length (0.1 m), and one-sided crown radius (0.1 m) were measured at the widest point of the crown, at the base of the live crown, midway between the widest point of the crown and the tree top, and midway between the widest point of the crown and base of the live crown were measured on single-stemmed tamarack, European larch, and hybrid larch in the fall of 1999 (Table 1). Total outside bark volume for individual trees was calculated from the equation:
2.1.1. Minnesota Single, 25 tree blocks of native tamarack (Larix laricina (Du Roi) K. Koch), European larch (Larix decidua Mill.), and hybrid larch (L. deciduaL. leptolepis(Sieb. and Zucc.)) were planted at a 2.7 m spacing in June of 1992 at the University of Minnesota, North Central Research and Outreach Center in Grand Rapids, MN (478140 N, 938310 W) at an elevation of 380 m asl. Prior to planting, the site was used for agriculture crops. Soil series is predominantly Rosy very ®ne sandy loam with inclusions of Cowhorn loamy very ®ne sand (Nyberg, 1987). Grass and herbaceous competition were mowed during the summer. Estimated site index (base age 20 years at BH) for European larch determined from 8-year tree height using the equation [SI203.081.537 (mean tree height at age 8) from Gilmore et al. (1993)] is 12.7 m. Data were collected prior to crown closure and the trees can best be described as open-grown. 2.1.2. Maine Crown length, height, and DBH measurements were collected from sample trees in all known European larch plantations planted in Maine prior to 1981 during 1989 and 1990 as part of a comprehensive growth and yield study (Gilmore et al., 1993, 1994;
TVOL 0:037 0:000516 DBH2 H
(1)
where TVOL is total outside bark volume to the tree tip in m3, DBH the diameter at 1.37 m in cm and H the total tree height in meters (Gilmore et al., 1993). 2.3. Data analyses Simple linear regression models of the form: y b0 b1 x e
(2)
where, y is the dependent variable, x the independent variable, b0 and b1 are predicted coefficients, and e the error NID(0,s2), were used to predict crown length (CL) as a function of total height (H), CL as a function of diameter at breast height (DBH, measured at 1.37 m above groundline), projected crown radius (PCR) as a
D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
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Table 1 Average (range in parentheses) values for the dependent and independent variablesa used to construct predictive equations by species Species
DBH (cm)
DBLC (cm)
CL (m)
Tamarack (n23) 5.8 (4.1±8.4) 8.3 (5.1±11.3) 4.3 (3.0±5.3) European larch (n24) 9.9 (5.7±12.3) 13.8 (7.7±16.5) 6.0 (4.8±7.2) Hybrid larch (n23) 9.7 (4.2±12.6) 13.1 (6.4±17.0) 5.7 (3.6±6.8)
CR (m)
TVOL (m3)
H (m)
PCR (m)
4.7 (3.5±5.8) 6.3 (5.2±7.4) 6.1 (3.9±7.3)
0.99 (0.5±1.6) 0.88 (0.2±1.6) 0.093 (0.035±0.215) 1.32 (0.6±2.5) 1.07 (0.2±2.5) 0.334 (0.091±0.511) 1.38 (0.7±1.9) 1.07 (0.2±1.9) 0.331 (0.039±0.564)
a DBH, diameter at breast height (cm, measured at 1.37 m above ground line); DBLC, diameter at base of live crown (cm, measured immediately below lowest living branch); CL, crown length (m); H, tree height (m), PCR, projected crown radius (m); CR, crown radius at four different positions within the crown; TVOL, total outside-bark volume to the tree tip in m3.
function of CL, DBH as a function of CL, DBH as a function of PCR, and diameter at the base of the live crown (DBLC) as a function of DBH. Data collected from Maine were limited to crown length, height, and DBH and it was not possible to construct crown radius models specific to Maine. Data from Maine (Gilmore et al., 1993) were combined with the Minnesota data and regression with indicator variables was used to test for differences in parameter estimates to predict crown length from height using (Eq. (2)). Two model forms, a quadratic (Honer, 1971), and a log-transformed independent variable (Mitchell, 1975), models (3) and (4) respectively, were fit to predict crown radius (CR) at different portions of the crown as a function of distance from the tree tip and H: CR c1 H
h c2 H
c4 H and CR d1 ln
hH c3 H
h2 e
H
h d2
h2 =H (3)
1 e
(4)
where, CR is crown radius at (H h) distance from the tree tip to CR, H the tree height, h the distance from tree tip to measurement location of CR, and e the error NID(0,s2). Regression analyses with indicator variables (Neter et al., 1990) were used to test the hypotheses that coef®cients for ®tted models varied among species. A non-intercept regression model was used when intercept terms did not differ from zero (p>0.5). Normality of residuals was tested with a non-parametric one sample Kolmogorov±Smirnov Lilliefors test (SPSS Inc., 1999). Average bias was calculated as observed minus predicted values divided by number of observations. Percent bias is the average bias
expressed as a percent of the mean value for a given parameter. All statistical analyses were done with SYSTAT (SPSS Inc., 1999). 3. Results 3.1. Crown allometry No improvement (p0.5) in the performance of all ®tted model forms was detected through the inclusion of indicator variables to account for differences in parameters among species. Parameters for equations general to all three Larix species to predict CL, PCR, and DBLC are provided in Table 2. Equations constructed to predict CL and DBLC explained a high proportion of the variation (>90%) in the data. Equations to predict PCR were less precise. Intercept terms for the equations constructed to predict PCR from crown dimensions did not differ from zero (p>0.5), therefore no-intercept equations are presented (Table 2). The Lilliefors test suggested non-rejection of the hypothesis of normally distributed residuals, but only marginally so for the DBLC prediction model. Bias was low for all Larix species except for the reciprocal PCR and DBH prediction equations. A strong linear relationship between crown length and tree height for open-grown Larix is evident for both Minnesota and Maine (Fig. 1). The allometric relationship between tree height and crown length differs, however, between Minnesota and Maine. Differences (p<0.001) between b0 and b1 coef®cients were detected between Minnesota and Maine. Differences in tree size and site quality may contribute to the differences in the CL, H allometric relationship between Maine and Minnesota (Fig. 1).
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D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
Table 2 Parameters (standard errors in parentheses), model mean square error (MSE), coefficients of determination (R2), and bias among species for the models of the form: yb0b1xe, where, yaCL, PCR, or DBLC; xaH, DBH, PCR, or CL; b0 and b1 are predicted coefficients; and eNID(0,s2) b1
Model MSE
R2
Lilliefors p-Biasb value T
0.513 (0.082)
1.025 (0.014)
0.014
0.987
0.202
0.448 (0.121)
0.092 (0.014)
0.082
0.392
0.318
H
0.216 (0.006)
0.088
0.354
0.198
PCR
CL
0.231 (0.007)
0.089
0.342
0.104
DBH
CL
3.122 (0.717)
2.183 (0.132)
1.291
0.798
0.999
DBH
PCR
3.162 (0.829)
4.343 (0.645)
3.886
0.392
0.337
DBLC
DBH
1.290 (0.379)
1.230 (0.043)
0.805
0.923
0.054
Dependent variablea
Independent variablea
CL
H
PCR
DBH
PCR
b0
0.054 1.2% 0.251 25.3% 0.024 2.4% 0.012 1.2% 0.334 5.7% 1.629 28.1% 0.194 2.3%
E
H 0.054 0.9% 0.546 41.3% 0.038 2.8% 0.057 4.3% 0.029 0.02% 1.006 10.0% 0.301 2.2%
0.002 0.03% 0.416 30.11% 0.063 4.5% 0.060 4.3% 0.365 3.7% 0.579 5.9% 0.120 1.2%
a CL, crown length (m); H, tree height (m); PCR, maximum crown radius (m); DBH, diameter at breast height (cm, measured at 1.37 m above ground line); DBLC, diameter at base of live crown (cm, measured immediately below lowest living branch). b bias(S observed predicted)/n, T, tamarack; E, European larch; H, hybrid larch; % bias(bias/mean)100.
3.2. Crown shape prediction Parameters for the Honer (1971) and Mitchell (1975) model forms to predict crown radius are presented in Table 3. The Honer (1971) model, having a greater number of independent variables, explained a
greater percentage of variation in the data then the Mitchell (1975), but still did not perform exceptionally well with a coef®cient of determination of 0.44. The c4 parameter was marginally signi®cant (p0.6), but was retained to facilitate comparison of parameters generated from this study with those for other species. Lilliefors p-values suggest rejection of the hypothesis of normally distributed residuals for both models. Biases in prediction for each species were low for both model forms. 4. Discussion
Fig. 1. Comparison of allometric equations, including 95% confidence intervals, to predict open-grown crown length (CL) from tree height (H) from data collected in Minnesota (dashed lines, CL 0.5131.025H) and Maine (solid lines, CL0.574 0.862H).
Ek (1971) reported a linear relationship (R20.58) between crown length and tree height for open-grown red pine (Pinus resinosa Ait.) in the Lake States. Hasenauer (1999) reported a log±log relationship (R20.62) between the height to live crown and total height for open-grown European larch in Austria. Their lower coef®cients of determination are due in a large part from an increased variance in the data associated with tree size. Results from this study explain a greater proportion of the variation in the data because of the smaller tree sizes associated with
D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
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Table 3 Parameters (standard errors in parentheses), model mean square error (MSE), coefficients of determination (R2), and bias among species for the Honer (1971) and Mitchell (1975) model forms to predict crown shape Model form
Parameters
Model MSE
R2
Lilliefors p-value
Biasa T
E
H
Honer (1971)
c10.955 (0.186) c2 0.066 (0.031) c3 0.880 (0.265) c40.082 (0.043)
0.081
0.44
0.002
0.009 1.01%
0.011 1.02%
0.014 1.31%
Mitchell (1975)
d10.188 (0.070) d20.022 (0.039)
0.070
0.11
0.001
0.013 1.47%
0.019 1.77%
0.005 0.46%
a
bias(S observed predicted)/n; % bias(bias/mean)100, T, tamarack; E, European larch; H, hybrid larch.
younger stands. Both Ek (1971) and Hasenauer (1999) transformed their data during analyses to better satisfy assumptions of least-squares regression associated with increased levels of variance with increased tree size; Ek (1971) used a weighted least squares regression and Hasenauer (1999) used a log±log transformation. Models to predict crown radius were less precise then those developed in other studies. For example, Hasenauer (1999) predicted crown width from tree height and DBH for open-grown trees in Austria, including larch, using a logarithmic transformation of both the dependent and independent variable that explained better than 90% of the variation in his data. Seymour and Smith (1987) developed an equation to predict the crown radius of white pine (Pinus strobus L.)
using quadratic transformation of crown length and interactions with tree height, and explained more than 60% of the variation in their data. The model forms used by Hasenauer (1999) and Seymour and Smith (1987) were evaluated, but did not satisfy the assumptions of least-squares regression as well, or have as low of a bias as those presented (Table 2). A graphical comparison of the equation developed by Hasenauer (1999) and the equation developed from this study (Fig. 2) shows the Hasenauer (1999) equation to predict a greater crown radius with increasing differences in prediction between the two equations with increasing tree diameter. A greater prediction in crown radius was also observed in the Hasenauer (1999) study in comparing the allometric equations to predict crown radius from
Fig. 2. Comparison of allometric equations to predict projected crown radius (CR) from DBH from data in Minnesota (CR0.4480.092DBH) and published equations for Austria [ln (2CR) 0.33960.6823 ln (DBH)] (Hasenauer, 1999).
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D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
Fig. 3. Comparison of allometric equations to predict projected crown radius (CR) from tree height (H) from data in Minnesota (CR0.216 H) and published equations for Austria [ln (2CR) 0.35260.8680 ln (H)] (Hasenauer, 1999).
tree height (Fig. 3). The slopes of the two equations were similar. Differences in predictability of the models from Austria (Hasenauer, 1999) and those from this study may be partially due to uncorrected logarithmic bias (Baskerville, 1972) or an artifact of the different regression models ®t to the respective data from each study. Additional explanations include differences in tree size, genetics, site quality, and regional differences in tree growth and form. The model forms of Honer (1971) and Mitchell (1975) to describe crown shape have performed well
with other coniferous species; namely balsam ®r (Abies balsamea (L.) Mill.) (Honer, 1971; Gilmore and Seymour, 1997) and Douglas-®r (Peudotsuga menziesii (Mirb.) Franco) (Mitchell, 1975). Despite the low coef®cients of determination (Table 3) for these model forms with young Larix, their performance was reasonable as determined through low mean square errors and low bias for each species of Larix. A graphical comparison of the parabolic Honer (1971) and conic Mitchell (1975) model forms (Fig. 4) reveals a greater prediction in crown radius toward the
Fig. 4. Comparison of the quadratic Honer (1971) and parabolic Mitchell (1975) model forms for a tree with a height of 6 m and predicted radii measures at 0.25 m intervals from the tree top into the crown.
D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
top of the tree for the Mitchell (1975) model with a convergence in the prediction of CR for both models further down the stem. Both models would predict a similar crown projection area. Average stem volume for tamarack was signi®cantly less then that of European and hybrid larch (p0.001, Table 1). The apparent contradiction of similar allometric relationships among these three species of Larix and different volume growth can be explained through variables not accounted for in this study. Branch angle and branch density both affect leaf area (Honda and Fisher, 1978). Leaf area, the timing of leaf emergence, and the timing of leaf senescence affect the carbon balance of individual trees. Exotic larches in Minnesota initiate their needle growth before and retain their needles longer than tamarack, which may account for a faster rate of above ground stem volume accumulation as determined from average height and diameter. 5. Conclusion Differences in the crown allometry of three species of open-grown Larix were not detected in Minnesota. Allometric equations describing the crown shape of larch from Minnesota differed from similar equations constructed for Maine and published equations for Austria. Forest managers, however, will likely continue to use allometric equations from non-local sources, particularly for exotic species, due to the absence of existing plantations or because of a lack of resources to develop localized equations. Immediate applicability includes the determination of desired planting densities and the subsequent construction of thinning schedules through the comparison of the crown radii of forest-grown trees to the predicted crown radii of open-grown trees. The presumed management objective would be to maintain as large of a tree crown as possible through the removal of adjacent trees. Results from different studies can be used to provide a range of minimum stocking densities. For example, Fig. 2 suggests two alternative stocking levels with higher levels of planting density suggested for Minnesota relative to Austria. Further enhancement of these allometric equations would be dependent on site quality, genetics, desired products, economic analyses, and land owner objectives.
115
Acknowledgements Research supported by the College of Natural Resources, Agricultural Experiment Station and North Central Research and Outreach Center, University of Minnesota, and the University of Minnesota/Institute of Paper Science and Technology Aspen & Larch Genetics Cooperative. In particular, I thank Dr. Andrew J. David, Director of the Aspen & Larch Genetics Cooperative for use of the Minnesota larch data. I also thank Egon Humenberger and Timothy C. O'Brien for assistance with data collection. Published as paper no. 004420027 of the Minnesota Agricultural Experiment Station. References Aird, P.L., Stone, E.L., 1955. Soil characteristics and the growth of European and Japanese larch in New York. J. For. 53, 425±429. Baskerville, G.L., 1972. Use of logarithmic regression in the estimation of plant biomass. Can. J. For. Res. 2, 49±53. Briggs, R.D., Lemin Jr., R.C., 1992. Delineation of climatic regions in Maine. Can. J. For. Res. 22, 801±811. Carter, K.K., Selin, L.O., 1987. Larch plantation management in the Northeast. North. J. Appl. For. 4, 18±20. Coyea, M.R., Margolis, H.A., 1992. Factors affecting the relationship between sapwood area and leaf area of balsam fir. Can. J. For. Res. 22, 1684±1693. Einsphar, D.W., Wyckoff, G.W., Fiscus, M.H., 1984. Larch Ð a fast-growing fiber source for the Lake States and Northeast. J. For. 82, 104±106. Ek, A.R., 1971. Size±age relationships for open grown red pine. For. Res. Notes No. 156, Univ. Wisconsin-Madison. Gilmore, D.W., Briggs, R.D., 1996. Empirical yield prediction equations for plantation-grown European larch in Maine. North. J. Appl. For. 13, 37±40. Gilmore, D.W., Seymour, R.S., 1997. Crown architecture of Abies balsamea from four canopy positions. Tree Physiol. 17, 71±80. Gilmore, D.W., Briggs, R.D., Seymour, R.S., 1993. Stem volume and site index equations for European larch in Maine. North. J. Appl. For. 10, 70±74. Gilmore, D.W., Briggs, R.D., Seymour, R.S., 1994. Identification of low productivity sites for European larch (Larix decidua Miller) in Maine USA. New For. 8, 289±297. Gilmore, D.W., Seymour, R.S., Maguire, D.A., 1996. Foliage± sapwood area relationships for Abies balsamea in central Maine, USA. Can. J. For. Res. 26, 2071±2079. Hasenauer, H., 1999. Dimensional relationships of open-grown trees in Austria. For. Ecol. Manage. 96, 197±206. Honda, H., Fisher, J.B., 1978. Tree branch angle: maximizing effective leaf area. Science 199, 888±889. Honer, T.G., 1971. Crown shape in open- and forest-grown balsam fir and black spruce. Can. J. For. Res. 1, 203±207.
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Jack, S.B., Long, J.N., 1996. Linkages between silviculture and ecology: an analysis of density management diagrams. For. Ecol. Manage. 86, 205±220. Kenefic, L.S., Seymour, R.S., 1999. Leaf area prediction models for Tsuga canadensis in Maine. Can. J. For. Res. 29, 1574±1582. Mitchell, K.J., 1975. Dynamics of simulated yield of Douglas-fir. For. Sci. Monog. 17, 1±39. Neter, J., Wasserman, W., Kutner, M.H., 1990. Applied Linear Statistical Models, 3rd Edition. Irwin, Homewood, IL. Nyberg, P.R., 1987. Soil Survey of Itasca County, Minnesota. USDA Soil Conservation Service and Forest Service,
and Minnesota Agriculture Experiment Station, St. Paul, MN. Parsonage, D.W., 1989. Soil±Site Relationships for Planted Japanese Larch (Larix leptolepis Sieb. and Zucc.) in Pennsylvania. M.Sc. thesis. Penn. State Univ., 155 pp. Seymour, R.S., Smith, D.M., 1987. A new stocking guide formulation applied to eastern white pine. For. Sci. 33, 469±484. SPSS Inc., 1999. SYSTAT 9.0. SPSS Inc., Chicago. Zaczek, J.J., Steiner, K.C., Shipman, R.D., 1994. Performance of Japanese and hybrid larch progenies in Pennsylvania. North. J. Appl. For. 11, 53±57.
Equations to describe crown allometry of Larix require local validation Daniel W. Gilmore* University of Minnesota, Department of Forest Resources, North Central Research and Outreach Center, 1861 Highway 169 East, Grand Rapids, MN 55744-3396, USA Received 21 January 2000; accepted 25 May 2000
Abstract Allometric equations were developed to describe crown attributes of three species of open-grown Larix in Minnesota and compared with similar data from Maine and published works from Austria. No differences were detected in allometric relationships to predict crown length, projected crown radius, and crown radius at different depths within the crown among tamarack, European larch, and hybrid EuropeanJapanese larch grown in Minnesota. Small (<1 m within the range of data), but signi®cant, differences in the prediction of crown length from total height were found between Minnesota and Maine. Allometric equations constructed from inventory data in Austria used a different model form and predicted larger crown attributes then equations generated with data from Minnesota. Results underscore the necessity for forest managers to validate allometric equations locally. With exotic species, however, it may be necessary to use published works until local data and resources for validation can be obtained. Applications include usage in the construction of stocking charts, stand density diagrams, and the prediction of DBH and volume from crown shape using remote sensing data. # 2001 Elsevier Science B.V. All rights reserved. Keywords: Austria; Crown shape; Larch; Leaf area; Maine; Minnesota; Stem volume; Tamarack
1. Introduction The potential for fast-growing exotic larch to alleviate projected ®ber supply shortages in North America is well documented (Einsphar et al., 1984; Carter and Selin, 1987; Zaczek et al., 1994). Most site quality and growth and yield studies in North America for non-indigenous Larix, however, have been empirical (Aird and Stone, 1955; Parsonage, 1989; Gilmore et al., 1993, 1994; Gilmore and Briggs, 1996). Process-based and remote sensing models of tree growth
* Tel.: 218-327-4522; fax: 218-327-4126. E-mail address: [email protected] (D.W. Gilmore).
characterize the tree canopy primarily through estimates of leaf area. Crown shape has been shown to have a strong relationship to leaf area (Coyea and Margolis, 1992; Gilmore et al., 1996; Kene®c and Seymour, 1999) and for many process-based models (e.g. Mitchell, 1975) crown shape serves as a surrogate for leaf area. Allometric relationships are also useful in the construction of stocking guides (Seymour and Smith, 1987) and density management diagrams (Jack and Long, 1996). Once allometric equations are developed to describe attributes for a particular species, the tendency is for practitioners to apply them without local validation. One reason for the ubiquitous application of allometric equations is the cost associated with validation. Once in common usage, allometric
0378-1127/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 1 1 2 7 ( 0 0 ) 0 0 4 9 3 - X
110
D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
equations are often assumed to be universally valid for a given species. In the special case of exotic species, local data often do not exist to validate allometric equations for local use. Nonetheless, the local validation of allometric equations is important because their principal application is to directly or indirectly estimate timber volume from which harvest levels are determined. The aim of this study is to develop allometric equations to predict crown attributes from readily available tree measurements (e.g. height, diameter, crown length) for three species of open-grown Larix, two of which are exotic to Minnesota. These results will be compared to European larch using data from Maine and published results from Austria.
Gilmore and Briggs, 1996). Plantations were located in 3 climatic zones (Briggs and Lemin, 1992), elevations ranged from 35 to 300 m asl, soils were representative of Spodosols, Inceptisols, and Entisols, of glacial or glacial-¯uvial origin, acidic, sand to silt loam in texture, and characterized by a frigid temperature regime. Latitude ranged from 448560 to 458160 N and longitude ranged from 688390 to 708350 E. Data used for comparison in this study were collected from the younger plantations that had not yet achieved crown closure. On average, trees in Maine were larger (average attributes (range in parentheses): DBH14.4 cm (7.3±24.5); crown length8.1 m (5.1± 12.4); total tree height8.7 m (5.5±13.7)) and site index (base age 20 years at BH) was higher (13.4± 19.6 m) than in Minnesota (Table 1).
2. Methods and materials
2.2. Data collection
2.1. Study locations
Total tree height (0.1 m), diameter at breast height (0.1 cm, measured at 1.37 m above ground line), diameter at base of the live crown (0.1 cm), crown length (0.1 m), and one-sided crown radius (0.1 m) were measured at the widest point of the crown, at the base of the live crown, midway between the widest point of the crown and the tree top, and midway between the widest point of the crown and base of the live crown were measured on single-stemmed tamarack, European larch, and hybrid larch in the fall of 1999 (Table 1). Total outside bark volume for individual trees was calculated from the equation:
2.1.1. Minnesota Single, 25 tree blocks of native tamarack (Larix laricina (Du Roi) K. Koch), European larch (Larix decidua Mill.), and hybrid larch (L. deciduaL. leptolepis(Sieb. and Zucc.)) were planted at a 2.7 m spacing in June of 1992 at the University of Minnesota, North Central Research and Outreach Center in Grand Rapids, MN (478140 N, 938310 W) at an elevation of 380 m asl. Prior to planting, the site was used for agriculture crops. Soil series is predominantly Rosy very ®ne sandy loam with inclusions of Cowhorn loamy very ®ne sand (Nyberg, 1987). Grass and herbaceous competition were mowed during the summer. Estimated site index (base age 20 years at BH) for European larch determined from 8-year tree height using the equation [SI203.081.537 (mean tree height at age 8) from Gilmore et al. (1993)] is 12.7 m. Data were collected prior to crown closure and the trees can best be described as open-grown. 2.1.2. Maine Crown length, height, and DBH measurements were collected from sample trees in all known European larch plantations planted in Maine prior to 1981 during 1989 and 1990 as part of a comprehensive growth and yield study (Gilmore et al., 1993, 1994;
TVOL 0:037 0:000516 DBH2 H
(1)
where TVOL is total outside bark volume to the tree tip in m3, DBH the diameter at 1.37 m in cm and H the total tree height in meters (Gilmore et al., 1993). 2.3. Data analyses Simple linear regression models of the form: y b0 b1 x e
(2)
where, y is the dependent variable, x the independent variable, b0 and b1 are predicted coefficients, and e the error NID(0,s2), were used to predict crown length (CL) as a function of total height (H), CL as a function of diameter at breast height (DBH, measured at 1.37 m above groundline), projected crown radius (PCR) as a
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Table 1 Average (range in parentheses) values for the dependent and independent variablesa used to construct predictive equations by species Species
DBH (cm)
DBLC (cm)
CL (m)
Tamarack (n23) 5.8 (4.1±8.4) 8.3 (5.1±11.3) 4.3 (3.0±5.3) European larch (n24) 9.9 (5.7±12.3) 13.8 (7.7±16.5) 6.0 (4.8±7.2) Hybrid larch (n23) 9.7 (4.2±12.6) 13.1 (6.4±17.0) 5.7 (3.6±6.8)
CR (m)
TVOL (m3)
H (m)
PCR (m)
4.7 (3.5±5.8) 6.3 (5.2±7.4) 6.1 (3.9±7.3)
0.99 (0.5±1.6) 0.88 (0.2±1.6) 0.093 (0.035±0.215) 1.32 (0.6±2.5) 1.07 (0.2±2.5) 0.334 (0.091±0.511) 1.38 (0.7±1.9) 1.07 (0.2±1.9) 0.331 (0.039±0.564)
a DBH, diameter at breast height (cm, measured at 1.37 m above ground line); DBLC, diameter at base of live crown (cm, measured immediately below lowest living branch); CL, crown length (m); H, tree height (m), PCR, projected crown radius (m); CR, crown radius at four different positions within the crown; TVOL, total outside-bark volume to the tree tip in m3.
function of CL, DBH as a function of CL, DBH as a function of PCR, and diameter at the base of the live crown (DBLC) as a function of DBH. Data collected from Maine were limited to crown length, height, and DBH and it was not possible to construct crown radius models specific to Maine. Data from Maine (Gilmore et al., 1993) were combined with the Minnesota data and regression with indicator variables was used to test for differences in parameter estimates to predict crown length from height using (Eq. (2)). Two model forms, a quadratic (Honer, 1971), and a log-transformed independent variable (Mitchell, 1975), models (3) and (4) respectively, were fit to predict crown radius (CR) at different portions of the crown as a function of distance from the tree tip and H: CR c1 H
h c2 H
c4 H and CR d1 ln
hH c3 H
h2 e
H
h d2
h2 =H (3)
1 e
(4)
where, CR is crown radius at (H h) distance from the tree tip to CR, H the tree height, h the distance from tree tip to measurement location of CR, and e the error NID(0,s2). Regression analyses with indicator variables (Neter et al., 1990) were used to test the hypotheses that coef®cients for ®tted models varied among species. A non-intercept regression model was used when intercept terms did not differ from zero (p>0.5). Normality of residuals was tested with a non-parametric one sample Kolmogorov±Smirnov Lilliefors test (SPSS Inc., 1999). Average bias was calculated as observed minus predicted values divided by number of observations. Percent bias is the average bias
expressed as a percent of the mean value for a given parameter. All statistical analyses were done with SYSTAT (SPSS Inc., 1999). 3. Results 3.1. Crown allometry No improvement (p0.5) in the performance of all ®tted model forms was detected through the inclusion of indicator variables to account for differences in parameters among species. Parameters for equations general to all three Larix species to predict CL, PCR, and DBLC are provided in Table 2. Equations constructed to predict CL and DBLC explained a high proportion of the variation (>90%) in the data. Equations to predict PCR were less precise. Intercept terms for the equations constructed to predict PCR from crown dimensions did not differ from zero (p>0.5), therefore no-intercept equations are presented (Table 2). The Lilliefors test suggested non-rejection of the hypothesis of normally distributed residuals, but only marginally so for the DBLC prediction model. Bias was low for all Larix species except for the reciprocal PCR and DBH prediction equations. A strong linear relationship between crown length and tree height for open-grown Larix is evident for both Minnesota and Maine (Fig. 1). The allometric relationship between tree height and crown length differs, however, between Minnesota and Maine. Differences (p<0.001) between b0 and b1 coef®cients were detected between Minnesota and Maine. Differences in tree size and site quality may contribute to the differences in the CL, H allometric relationship between Maine and Minnesota (Fig. 1).
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Table 2 Parameters (standard errors in parentheses), model mean square error (MSE), coefficients of determination (R2), and bias among species for the models of the form: yb0b1xe, where, yaCL, PCR, or DBLC; xaH, DBH, PCR, or CL; b0 and b1 are predicted coefficients; and eNID(0,s2) b1
Model MSE
R2
Lilliefors p-Biasb value T
0.513 (0.082)
1.025 (0.014)
0.014
0.987
0.202
0.448 (0.121)
0.092 (0.014)
0.082
0.392
0.318
H
0.216 (0.006)
0.088
0.354
0.198
PCR
CL
0.231 (0.007)
0.089
0.342
0.104
DBH
CL
3.122 (0.717)
2.183 (0.132)
1.291
0.798
0.999
DBH
PCR
3.162 (0.829)
4.343 (0.645)
3.886
0.392
0.337
DBLC
DBH
1.290 (0.379)
1.230 (0.043)
0.805
0.923
0.054
Dependent variablea
Independent variablea
CL
H
PCR
DBH
PCR
b0
0.054 1.2% 0.251 25.3% 0.024 2.4% 0.012 1.2% 0.334 5.7% 1.629 28.1% 0.194 2.3%
E
H 0.054 0.9% 0.546 41.3% 0.038 2.8% 0.057 4.3% 0.029 0.02% 1.006 10.0% 0.301 2.2%
0.002 0.03% 0.416 30.11% 0.063 4.5% 0.060 4.3% 0.365 3.7% 0.579 5.9% 0.120 1.2%
a CL, crown length (m); H, tree height (m); PCR, maximum crown radius (m); DBH, diameter at breast height (cm, measured at 1.37 m above ground line); DBLC, diameter at base of live crown (cm, measured immediately below lowest living branch). b bias(S observed predicted)/n, T, tamarack; E, European larch; H, hybrid larch; % bias(bias/mean)100.
3.2. Crown shape prediction Parameters for the Honer (1971) and Mitchell (1975) model forms to predict crown radius are presented in Table 3. The Honer (1971) model, having a greater number of independent variables, explained a
greater percentage of variation in the data then the Mitchell (1975), but still did not perform exceptionally well with a coef®cient of determination of 0.44. The c4 parameter was marginally signi®cant (p0.6), but was retained to facilitate comparison of parameters generated from this study with those for other species. Lilliefors p-values suggest rejection of the hypothesis of normally distributed residuals for both models. Biases in prediction for each species were low for both model forms. 4. Discussion
Fig. 1. Comparison of allometric equations, including 95% confidence intervals, to predict open-grown crown length (CL) from tree height (H) from data collected in Minnesota (dashed lines, CL 0.5131.025H) and Maine (solid lines, CL0.574 0.862H).
Ek (1971) reported a linear relationship (R20.58) between crown length and tree height for open-grown red pine (Pinus resinosa Ait.) in the Lake States. Hasenauer (1999) reported a log±log relationship (R20.62) between the height to live crown and total height for open-grown European larch in Austria. Their lower coef®cients of determination are due in a large part from an increased variance in the data associated with tree size. Results from this study explain a greater proportion of the variation in the data because of the smaller tree sizes associated with
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Table 3 Parameters (standard errors in parentheses), model mean square error (MSE), coefficients of determination (R2), and bias among species for the Honer (1971) and Mitchell (1975) model forms to predict crown shape Model form
Parameters
Model MSE
R2
Lilliefors p-value
Biasa T
E
H
Honer (1971)
c10.955 (0.186) c2 0.066 (0.031) c3 0.880 (0.265) c40.082 (0.043)
0.081
0.44
0.002
0.009 1.01%
0.011 1.02%
0.014 1.31%
Mitchell (1975)
d10.188 (0.070) d20.022 (0.039)
0.070
0.11
0.001
0.013 1.47%
0.019 1.77%
0.005 0.46%
a
bias(S observed predicted)/n; % bias(bias/mean)100, T, tamarack; E, European larch; H, hybrid larch.
younger stands. Both Ek (1971) and Hasenauer (1999) transformed their data during analyses to better satisfy assumptions of least-squares regression associated with increased levels of variance with increased tree size; Ek (1971) used a weighted least squares regression and Hasenauer (1999) used a log±log transformation. Models to predict crown radius were less precise then those developed in other studies. For example, Hasenauer (1999) predicted crown width from tree height and DBH for open-grown trees in Austria, including larch, using a logarithmic transformation of both the dependent and independent variable that explained better than 90% of the variation in his data. Seymour and Smith (1987) developed an equation to predict the crown radius of white pine (Pinus strobus L.)
using quadratic transformation of crown length and interactions with tree height, and explained more than 60% of the variation in their data. The model forms used by Hasenauer (1999) and Seymour and Smith (1987) were evaluated, but did not satisfy the assumptions of least-squares regression as well, or have as low of a bias as those presented (Table 2). A graphical comparison of the equation developed by Hasenauer (1999) and the equation developed from this study (Fig. 2) shows the Hasenauer (1999) equation to predict a greater crown radius with increasing differences in prediction between the two equations with increasing tree diameter. A greater prediction in crown radius was also observed in the Hasenauer (1999) study in comparing the allometric equations to predict crown radius from
Fig. 2. Comparison of allometric equations to predict projected crown radius (CR) from DBH from data in Minnesota (CR0.4480.092DBH) and published equations for Austria [ln (2CR) 0.33960.6823 ln (DBH)] (Hasenauer, 1999).
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D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
Fig. 3. Comparison of allometric equations to predict projected crown radius (CR) from tree height (H) from data in Minnesota (CR0.216 H) and published equations for Austria [ln (2CR) 0.35260.8680 ln (H)] (Hasenauer, 1999).
tree height (Fig. 3). The slopes of the two equations were similar. Differences in predictability of the models from Austria (Hasenauer, 1999) and those from this study may be partially due to uncorrected logarithmic bias (Baskerville, 1972) or an artifact of the different regression models ®t to the respective data from each study. Additional explanations include differences in tree size, genetics, site quality, and regional differences in tree growth and form. The model forms of Honer (1971) and Mitchell (1975) to describe crown shape have performed well
with other coniferous species; namely balsam ®r (Abies balsamea (L.) Mill.) (Honer, 1971; Gilmore and Seymour, 1997) and Douglas-®r (Peudotsuga menziesii (Mirb.) Franco) (Mitchell, 1975). Despite the low coef®cients of determination (Table 3) for these model forms with young Larix, their performance was reasonable as determined through low mean square errors and low bias for each species of Larix. A graphical comparison of the parabolic Honer (1971) and conic Mitchell (1975) model forms (Fig. 4) reveals a greater prediction in crown radius toward the
Fig. 4. Comparison of the quadratic Honer (1971) and parabolic Mitchell (1975) model forms for a tree with a height of 6 m and predicted radii measures at 0.25 m intervals from the tree top into the crown.
D.W. Gilmore / Forest Ecology and Management 148 (2001) 109±116
top of the tree for the Mitchell (1975) model with a convergence in the prediction of CR for both models further down the stem. Both models would predict a similar crown projection area. Average stem volume for tamarack was signi®cantly less then that of European and hybrid larch (p0.001, Table 1). The apparent contradiction of similar allometric relationships among these three species of Larix and different volume growth can be explained through variables not accounted for in this study. Branch angle and branch density both affect leaf area (Honda and Fisher, 1978). Leaf area, the timing of leaf emergence, and the timing of leaf senescence affect the carbon balance of individual trees. Exotic larches in Minnesota initiate their needle growth before and retain their needles longer than tamarack, which may account for a faster rate of above ground stem volume accumulation as determined from average height and diameter. 5. Conclusion Differences in the crown allometry of three species of open-grown Larix were not detected in Minnesota. Allometric equations describing the crown shape of larch from Minnesota differed from similar equations constructed for Maine and published equations for Austria. Forest managers, however, will likely continue to use allometric equations from non-local sources, particularly for exotic species, due to the absence of existing plantations or because of a lack of resources to develop localized equations. Immediate applicability includes the determination of desired planting densities and the subsequent construction of thinning schedules through the comparison of the crown radii of forest-grown trees to the predicted crown radii of open-grown trees. The presumed management objective would be to maintain as large of a tree crown as possible through the removal of adjacent trees. Results from different studies can be used to provide a range of minimum stocking densities. For example, Fig. 2 suggests two alternative stocking levels with higher levels of planting density suggested for Minnesota relative to Austria. Further enhancement of these allometric equations would be dependent on site quality, genetics, desired products, economic analyses, and land owner objectives.
115
Acknowledgements Research supported by the College of Natural Resources, Agricultural Experiment Station and North Central Research and Outreach Center, University of Minnesota, and the University of Minnesota/Institute of Paper Science and Technology Aspen & Larch Genetics Cooperative. In particular, I thank Dr. Andrew J. David, Director of the Aspen & Larch Genetics Cooperative for use of the Minnesota larch data. I also thank Egon Humenberger and Timothy C. O'Brien for assistance with data collection. Published as paper no. 004420027 of the Minnesota Agricultural Experiment Station. References Aird, P.L., Stone, E.L., 1955. Soil characteristics and the growth of European and Japanese larch in New York. J. For. 53, 425±429. Baskerville, G.L., 1972. Use of logarithmic regression in the estimation of plant biomass. Can. J. For. Res. 2, 49±53. Briggs, R.D., Lemin Jr., R.C., 1992. Delineation of climatic regions in Maine. Can. J. For. Res. 22, 801±811. Carter, K.K., Selin, L.O., 1987. Larch plantation management in the Northeast. North. J. Appl. For. 4, 18±20. Coyea, M.R., Margolis, H.A., 1992. Factors affecting the relationship between sapwood area and leaf area of balsam fir. Can. J. For. Res. 22, 1684±1693. Einsphar, D.W., Wyckoff, G.W., Fiscus, M.H., 1984. Larch Ð a fast-growing fiber source for the Lake States and Northeast. J. For. 82, 104±106. Ek, A.R., 1971. Size±age relationships for open grown red pine. For. Res. Notes No. 156, Univ. Wisconsin-Madison. Gilmore, D.W., Briggs, R.D., 1996. Empirical yield prediction equations for plantation-grown European larch in Maine. North. J. Appl. For. 13, 37±40. Gilmore, D.W., Seymour, R.S., 1997. Crown architecture of Abies balsamea from four canopy positions. Tree Physiol. 17, 71±80. Gilmore, D.W., Briggs, R.D., Seymour, R.S., 1993. Stem volume and site index equations for European larch in Maine. North. J. Appl. For. 10, 70±74. Gilmore, D.W., Briggs, R.D., Seymour, R.S., 1994. Identification of low productivity sites for European larch (Larix decidua Miller) in Maine USA. New For. 8, 289±297. Gilmore, D.W., Seymour, R.S., Maguire, D.A., 1996. Foliage± sapwood area relationships for Abies balsamea in central Maine, USA. Can. J. For. Res. 26, 2071±2079. Hasenauer, H., 1999. Dimensional relationships of open-grown trees in Austria. For. Ecol. Manage. 96, 197±206. Honda, H., Fisher, J.B., 1978. Tree branch angle: maximizing effective leaf area. Science 199, 888±889. Honer, T.G., 1971. Crown shape in open- and forest-grown balsam fir and black spruce. Can. J. For. Res. 1, 203±207.
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Jack, S.B., Long, J.N., 1996. Linkages between silviculture and ecology: an analysis of density management diagrams. For. Ecol. Manage. 86, 205±220. Kenefic, L.S., Seymour, R.S., 1999. Leaf area prediction models for Tsuga canadensis in Maine. Can. J. For. Res. 29, 1574±1582. Mitchell, K.J., 1975. Dynamics of simulated yield of Douglas-fir. For. Sci. Monog. 17, 1±39. Neter, J., Wasserman, W., Kutner, M.H., 1990. Applied Linear Statistical Models, 3rd Edition. Irwin, Homewood, IL. Nyberg, P.R., 1987. Soil Survey of Itasca County, Minnesota. USDA Soil Conservation Service and Forest Service,
and Minnesota Agriculture Experiment Station, St. Paul, MN. Parsonage, D.W., 1989. Soil±Site Relationships for Planted Japanese Larch (Larix leptolepis Sieb. and Zucc.) in Pennsylvania. M.Sc. thesis. Penn. State Univ., 155 pp. Seymour, R.S., Smith, D.M., 1987. A new stocking guide formulation applied to eastern white pine. For. Sci. 33, 469±484. SPSS Inc., 1999. SYSTAT 9.0. SPSS Inc., Chicago. Zaczek, J.J., Steiner, K.C., Shipman, R.D., 1994. Performance of Japanese and hybrid larch progenies in Pennsylvania. North. J. Appl. For. 11, 53±57.