Paper 1: diameter growth... page 1

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1 Paper : diameter growth... page Modeling diameter growth in response to varying silvicultural treatments within mixedspecies stands located in coastal British Columbia Paper height growth. page 0 Modeling height growth in response to varying silvicultural treatments within mixedspecies stands located in coastal British Columbia 0 Leah C. Rathbun, Valerie LeMay, and Nick Smith Department of Forest Resources Management, University of British Columbia, 0 Main Mall, Vancouver, BC VT Z, Canada Corresponding Author: Valerie.LeMay@ubc.ca Nick Smith Forest Consulting, Kite Way, Nanaimo, BC V0T Z

2 Abstract In this study, diameter increment models for unthinned, thinned, and fertilized mixedspecies stands located on Vancouver Island were developed using a model based on metabolic processes. For this purpose, the Box and Lucas model was fitted for the three main species, Douglas-fir (Pseudotsuga menziesii var. menziesii (Mirb.) Franco), western hemlock (Tsuga heterophylla (Raf.) Sarg.), and western redcedar (Thuja plicata Donn) for the diameter growth series by tree, and then a parameter prediction approach was used to modify the parameters for size and competition variables. A time-since-treatment 0 variable was used to modify growth following fertilization. Reasonably, accurate results were obtained for the three species, with best results for hemlock. Keywords: Height growth, Box Lucas, thinning.

3 0 0 Introduction The successional processes of the forests located on Vancouver Island are defined by small-scale disturbances mainly brought on through windthrow (Gavin et al. 00). Gap dynamics define the forest structure of the area. Douglas-fir (Pseudotsuga menziesii var. menziesii (Mirb.) Franco) is the dominating, pioneer species with western hemlock (Tsuga heterophylla (Raf.) Sarg.) and western redcedar (Thuja plicata Donn) being the late-successional species aggregated within gaps (Getzin et al. 00). To mimic the natural disturbance regimes and maintain the structural complexity and biodiversity of these mixed-species stands, variable retention harvesting has become increasingly widespread on Vancouver Island. Variable retention harvesting is designed to retain individual or groups of trees over an area in order to maintain ecosystem structure and function, provide structural features for regenerating stands, and sustain connectivity within a landscape (Sullivan et al. 00). This type of harvesting system produces large differences in treatment applications and often thinning intensity, time of thinning, and the number of thins vary greatly from stand to stand. In addition to thinning, fertilization is a common silvicultural practice within this area. Stands of Douglas-fir in coastal British Columbia have shown a responsive to nitrogen fertilization (Weetman et al. ) and poorer sites in cutover areas of old-growth forests on Vancouver Island consisting of western hemlock and western redcedar have been identified as being nutrient poor in nitrogen and phosphorus (Weetman et al. ). One major need within forest management planning is the development of diameter growth models which can predict stand development under varying treatment methods (Palahi et al. 00).

4 Different model forms for diameter increment models exist and are generally categorized as either a growth potential independent model, where increment is expressed directly as a function of physical tree characteristics and site conditions, or a growth potential dependent model, where the potential growth is defined and then modified (Huang and Titus ). Regardless of the model form, the model and its components should be logically consistent and biologically realistic (Vanclay and Skovsgaard ). To model diameter increment and capture the variation due to different treatment applications found on Vancouver Island, a flexible individual-tree distance-independent diameter increment model is required. 0 0 The objective of the study was to develop a diameter increment model for variable retention harvested and fertilized mixed-species stands located on Vancouver Island. Our hypotheses were: ) a biological diameter increment model would outperform a common mathematical diameter increment model; ) plots which were fertilized would experience an initial increase in diameter increment growth during the first few years following application only; ) modifying the parameter estimates for inter-tree competition variables such as basal area of larger trees and stand basal are, would reflect the initial increase in diameter increment growth during the first few years following thinning; ) the increase in diameter increment growth due to the combination of thinning and fertilization can be represented additively; and ) consecutive treatments do not produce an additive effect in diameter increment growth.

5 .. Study Area Materials and Methods 0 The study area was located on Vancouver Island within the Coastal Western Hemlock (CWH) Biogeoclimatic Ecological Classification (BEC) zone of British Columbia. This BEC zone is divided into multiple subzones: a very dry maritime subzone in the east, a moist maritime subzone in the central area, and a very wet maritime and hypermaritime subzone in the west (Brown and Hebda 00). The temperatures in the CWH BEC zone range from. to 0. C, with a mean annual temperature of C (Meidinger and Pojar ). Study plots were located within latitudes ranging. to. N and longitudes ranging. to. W. This area is comprised of second growth uneven- and evenaged multi-species stands regenerated naturally and from plantings. The common species include: western hemlock, Douglas-fir, western redcedar, red alder (Alnus rubra Bong.), Sitka spruce (Pinus sitchensis), and yellow cedar (Chamaecyparis nootkatensis (D.Don) Spach)... Data 0 Permanent Sample Plot (PSP) data were provided by Island Timberland, Ltd. The database contained 0 plots, ranging in size from 0.00 to 0.0 ha. Measurements spanned the years to 00, with measurement intervals varying from to years for an average of. years. Densities at plot establishment ranged from. to 0 live trees per hectare, site index ranged from. to 0.0m, and basal area per hectare

6 ranged from 0.0 to.0 m /ha (Table ). At each measurement and for each tree within the plot boundaries, species, tree status (i.e., live or dead), and diameter outside bark at breast height (. m above ground, dbh) was recorded. Height (ht, m) was measured for a subset of trees, and remaining heights were estimated using heightdiameter functions. At the time of plot establishment, the average diameters were.,., and.cm (Table ); western hemlock, Douglas-fir and western redcedar, respectively, with corresponding maximum values of.0,., and.cm. The average heights for western hemlock, Douglas-fir and western redcedar corresponded to.,., and 0.m. 0 0 Plot treatments included untreated, fertilization, single thinning, multiple thinnings, and a combination of fertilization and thinning with plots receiving no treatment, plots received fertilization, plots received at least one thinning, and plots which received multiple treatments. The majority of fertilized plots received one application of nitrogen ranging in concentration from 0 to 00 kg/ha; a few plots additionally received ammonium phosphate. The majority of thinned plots,, were thinned only once, plots received two thinnings, and plots received three thinnings. Basal area cut varied from 0.0 to., for an average of 0.m /ha removing an average of,0 trees per hectare. Plots which received multiple treatments received a combination of thinning and fertilization treatments... Model Development Tree growth can be represented as the difference in anabolic gain, where molecules are constructed and catabolic loss, where molecules are broken down to their smaller

7 0 components for use as energy (Pienaar and Turnbull ). The base model form representing this metabolic relationship was presented by Box and Lucas (): [] k exp dbh exp dbh Dincq Where D inc is the diameter increment for the q-year period, k is an asymptotic constant, and are parameters to be estimated, and dbh was previously defined. Figure illustrates typical curves produced from the Box-Lucas function. These curves follow the basic sigmoidal pattern typical of biological growth, growth increment increases to a maximum, begins to decrease tapering off asymptotically toward zero. Unlike other diameter increment models which use a fixed period interval, this model allows diameter increment to be projected for any time interval. Anabolism is modeled by and catabolism by, with both parameters working in conjunction to describe a trees metabolism. Figure illustrates how changes in affects diameter increment. Increasing values, produce larger diameter increments for trees up to a specific diameter and then an inverse relationship is seen. Figure illustrates that an increase in produces a decrease in diameter increment no matter the size of the tree. An increase in the asymptotic constant maintains the basic shape of the curve while increasing the diameter increment. Note the maximum diameter increment rate is found for a diameter of (Huang and Titus ): dinc max ln( / ) 0 Where all variables are as previously defined. Variables which affect diameter increment are incorporated into the model as modifications to and. Huang and Titus () observed for white spruce (Picea glauca (Moench) Voss) a linear relationship between

8 and the following variables: basal area ha -, tree height, percent basal area by species, relative dbh, and site productivity index; and a linear relationship between and stems ha -. 0 When selecting variables to define and, it is important to understand the biological processes which affect diameter growth. As a result, predictor variables representing tree size and stage of development, site productivity, and inter-tree competition were considered. Tree size and stage of development were represented by dbh (cm), height (m) and the diameter increment of the previous growth period (dinc previous, cm). To represent site productivity, coastal Douglas-fir site index (SI, m) at a base age 0 and Growth Effective Age (GEA, years) were included to distinguish between differences in site quality found across plots. GEA is the age of a dominant tree with the same height and having the same site index as the tree of interest (Hann and Ritchie ). GEA was developed from site index equations for the dominant tree species of a stand. A number of stand level competition measures were considered. Curtis Relative Density, an index measure of density, was calculated as: CurtisRD G d q d q n i dbh n i 0 Where dbh i (cm) is the dbh for tree i, dq (cm) is the quadratic mean diameter for the plot, G (m /ha) is the stand basal area, and n is the number of all live trees across all species within a plot. Stand basal area has been commonly used to measure competition for

9 0 below ground resources (Fan et al. 00). In addition, tree-level competition measures such as the basal area of larger trees (BAL, m /ha), relative dbh (RDBH), and crown competition factor of larger trees (CCFL) were evaluated for the model. BAL was calculated as: BAL n i δ x BAtree i Where δ is an indicator variable ( if tree i has a dbh greater than the tree of interest, 0 otherwise), BAtree i (m /ha) is the basal area per hectare value for tree i, and n is the number of all live trees across all species within a plot. Relative dbh (RDBH) is a ratio of the diameter of the tree of interest to the average diameter within a plot and was calculated as: dbh RDBH i dbh Where dbh i is the dbh of tree i and dbh is the average dbh for the plot. CCFL (expressed as a percent) was calculated as: CCFL n i δ x MCA i MCA i π(cw ˆ ) i x,000,000 CW ˆ i 0.0 a 0. 0 dbhi b. Where: MCA i is the maximum crown area for tree i expressed as a percentage of a hectare that can be occupied by the maximum crown of tree i with dbh i in cm (Avery and Burkhart 00), Ĉ Wi (m) is the crown width of tree i, and the parameters a and b are

10 species-specific constants for imperial units found in Smith (). All other constants are used for unit conversion. 0 A parameter prediction approach was used to determine which variables best described and. A subset of the data was used to obtain estimates for and. The subset was defined from the original untreated data as trees which contained more than measurement periods. Each tree was fit individually to obtain parameter estimates for and across all species. Plots of and versus possible predictor variables were created to identify trends. Different combinations of the possible predictor variables in linear, log-linear, and nonlinear equations were created to estimate and. The resulting residual and actual versus predicted value plots were used to determine the appropriate model forms for and for each species individually. The model forms for and were inserted into the Box-Lucas model and refit using the resultant parameter estimates as starting values. 0.. Model Accuracy For comparison, a model developed for mixed-species stands located in Queensland (Vanclay ) was also fitted. This model is: [] ln( dinc 0.0) 0 dbh log(dbh) log(dbh) *SQ log(g) BAL Where SQ is a categorical variable for site quality defined within a species; for high quality sites, 0 for poor quality sites. Site quality was defined according to the BC Ministry of Forests () where: ) western hemlock: SQ=0 if site index <0., else SQ=; ) Douglas-fir: SQ=0 if site index <., else SQ=; 0

11 ) Western redcedar: SQ=0 if site index <., else SQ=. And all other variables were defined previously. 0 Fit statistics were calculated for Models [] and [] using all unthinned data by species, including ACI, Pseudo R, Mean Bias, and RMSE. AIC was calculated a: AIC logl( ˆ ) p Where logl(ˆ ) is the log likelihood function and p is the number of parameters in the model. Pseudo R values were calculated as: R =-RSS/TSS Where RSS is the residual sum of squares and TSS is the total corrected sum of squares. For each species, mean bias, and root mean-squared error (RMSE) values were calculated for the unthinned plots. Mean bias was calculated as: Mean Bias m j y j m yˆ j Where y j is the actual diameter increment for tree j, ŷ j is the predicted diameter increment for tree j,, and m is the number of trees. Root mean-squared error was calculated for each species as: RMSE m j y j yˆ m p j Where p is the number of predictor variables in the model and all other variables were previously defined.

12 .. Treatment Affects The base model developed from the unthinned data was modified to include the effects of fertilization. Additional categorical and continuous variables were added to the model as modifications to the,, and asymptote parameters. All three parameters were modified individually as well as each possible combination of parameters modified. For each modified model form, Akaike s Information Criteria (AIC) values were calculated using PROC NLMIXED of SAS, version... AIC values were used to determine the correct model modification. In addition the base model developed from the unthinned data was applied to the thinned plots without modification. 0 Results 0.. Model Development The best combination of predictor variables found for, the parameter representing anabolic growth include: the previous diameter increment, basal area per hectare, growth effective age, and stems per hectare. The best combination of predictor variables found for, the parameter representing the catabolic breakdown of cells for energy include: the previous diameter increment, basal area of larger trees, and the relative diameter. The final diameter increment model was found to be: Where Dincq k exp dbh exp dbh 0. exp( b b log(dinc prev ) bg b log(gea) bsph) exp( c c log(dinc prev ) c log( BAL) c log(rdbh))

13 k is an asymptotic constant defined for a species:. for western hemlock,. for Douglas-fir and 0. for western redcedar and the parameter estimates for b to b and c to c are listed in Table. 0.. Model Accuracy The adjust R values indicate that model fits the data better than model (Table ). The bias estimates are all positive regardless of species for model and are all negative for model (Table ). The RMSE values are smaller for model for all species. Figure illustrates the average annual diameter increment both actual and predicted by dbh. On average, western hemlock predicts diameter increment well across all dbh sizes. The average diameter increment for Douglas-fir is greatly under-predicted for trees greater than 0cm in dbh. Western redcedar on average predicts well for trees less than 0cm in dbh, the model tends to under-predict for trees greater than 0cm in dbh... Treatment Affects The best model form for fertilized plots was found from modifying both the asymptote and the value. The model form which returned the smallest AIC value for the fertilized plots (Table ) is of the form: Where Dincq k F exp dbh exp dbh k F k a * F time F 0 0. exp( b b log(dinc prev ) bg b log(gea) bsph)

14 exp(c c log( dinc prev ) c log( BAL) c log( RDBH) c F time F is a categorical variable defined as for fertilized plots and 0 for unfertilized plots, time F is the time since fertilization, a is a parameter to be estimated and all other parameters were as described above. Figure illustrates the change in average diameter increment through time. The modifier was developed to decrease as time since fertilization increases. F ) 0 Discussion The Box and Lucas model used to model diameter increment for Douglas-fir, hemlock, and western redcedar has the advantage of being based on the metabolic processes governing tree growth. Unlike other models, using a process-based model allows interpretation of the models, as well as can result in accurate predictions. In this study, results indicated that the use of the Box and Lucas model as the base function, modified for distance independent measures of competition resulted in accurate predictions, particularly for hemlock, in untreated stands. 0 Several studies have shown a significant growth response to fertilization, including an increase in growth as a result of phosphorus and nitrogen fertilization for western redcedar (Devine and Harrington 00). The impacts of fertilizer on growth is a direct result of nutritional improvement and also an indirect effect due to changes in stand structure through time (McWilliams 0). In this study, the model was modified for effects of fertilization through adding a years-since-fertilization variable. Generally, the

15 resulting models performed as expected, with a change in growth following fertilization to a maximum. In terms of implementing these models in a growth and yield model, it was expected that impacts of partial harvest could be addressed through the same model as the untreated growth model, since the main impact of thinning is reduction in competition which would be reflected in the post-thin competition measures. However, since there is a lag in the growth response following thinning, the number of years since treatment was also important. 0 For thinned and fertilized stands, Brix () noted that nitrogen uptake after fertilization was similar in unthinned and heavily thinned plots over the first -year period, with the uptake taking place in the first year in unthinned plots and later in thinned plots for Douglas-fir. For Douglas-fir and western hemlock stands in coastal BC, Omule and Britton () showed the extra growth accumulated on fewer stems (thinned) and thus produced a greater amount of valuable wood. To estimate the impacts of thinning and fertilization on diameter increment, a two-step additive approach was tested, in that diameter increment was modified due to fertilization, and then again because of thinning. However, further research of this approach is needed. 0

16 Conclusion The Box and Lucas model was used to model diameter increment for three BC Coastal species and a parameter prediction approach was used to modify the processes for changes in competition. The impacts of treatments were modelled by including a yearssince treatment variable. Reasonably, accurate results were obtained for the three species, with best results for hemlock. Further testing of these models is warrented before they can be implemented in a growth model. 0 Acknowledgements We would like to thank Ken Epps and Island Timberland for providing the data for this project, and the Forest Science Program for providing research funding to Island Timberland. We would also like to thank Dr. Peter Marshall, Dr. Lori Daniels, and Dr. David Hann.

17 0 0 References Avery, T.E. and Burkhart, H.E. 00. Forest Measurements, th Edition. McGraw-Hill Companies Inc. New York, N.Y. BC Ministry of Forests,. Strathcona timber supply area analysis report. Box, G.E.P. and Lucas, H.L.. Design of experiments in non-linear situations. Biometrika (/): -0. Brix, H.. Fertilization and thinning effect on a Douglas-fir ecosystem at Shawnigan Lake: A synthesis of project results. For.Can. and B.C. Min. For., Victoria, BC FRDA Rep. No. Brown, K.J. and Hebda, R.J. 00. Origin, development, and dynamics of coastal temperate conifer rainforests of southern Vancouver Island, Canada. Can. J. For. Res. : -. Fan, Z., Kabrick, J.M., and Shifley, S.R. 00. Classification and regression tree based survival analysis in oak-dominated forests of Missouri s Ozark highlands. Can. J. For. Res. : 0-. Gavin, D.G., Brubaker, L., and Lertzman, K.P. 00. Holocene fire history of a coastal temperate rain forest based on soil charcoal radiocarbon dates. Ecol. () -0. Getzin, S., Dean, C., He, F., Trofymow, J.A., Wiegand, K. and Wiegand, T. 00. Spatial patterns and competition of tree species in a Douglas-fir chronosequence on Vancouver Island. Ecography : -. Hamilton, D.A., Jr.. A logistic model of mortality in thinned and unthinned mixed conifer stands of northern Idaho. For. Sci. (): -000.

18 0 0 Hamilton, D.A., Jr. and Edwards, B.M.. Modeling the probability of individual tree mortality. USDA For. Serv. Intermt. Res. Stn. Res. Pap. INT-. pp. Hann, D.W. and Ritchie, M.W.. Height growth rate of Douglas fir: a comparison of model forms. For. Sci. (): -. He, F. and Barclay, H.J Long-term response of understory plant species to thinning and fertilization in a Douglas-fir plantation on southern Vancouver Island, British Columbia. Can. J. For. Res. 0: -. Huang, S. and Titus, S. J.,. An individual tree diameter increment model for white spruce in Alberta. Can. J. For. Res. :-. McWilliams, E. R. G. and G. Therien.. Fertilization and thinning effects on a Douglas-fir ecosystem at Shawnigan Lake: -year growth response. For.Can. and B.C. Min. For., Victoria, BC FRDA Rep. No.. Meidinger, D. and Pojar, J., eds.. Ecosystems of British Columbia. The Research Branch, BC Ministry of Forests, Victoria, BC. pp.-. Omule, S.A.Y. and Britton, G.M.. Basal area response nine years after fertilization and thinning western hemlock. For.Can. and B.C. Min. For., Victoria, BC FRDA Rep. No.. Palahi, M., Pukkala, T., Miina, J., and Montero, G. Individual-tree growth and mortality models for Scots pine (Pinus sulvestris L.) in north-east Spain. 00. Ann. For. Sci. 0: -0. Pienaar, L.V. and Turnbull, K.J.. The Chapman-Richards generalization of Von Bertalanffy s growth model for basal area growth and yield in even-aged stands. For. Sci. : -.

19 0 Smith, J.H.G.,. Studies of crown development are improving Canadian forest management. In: Proc. VI World Forestry Congress, Madrid. Vol., pp. 0. Sullivan, T.P., Sullivan, D.S., and Lindgren, P.M.F 00. Influence of variable retention harvests on forest ecosystems. I. Diversity of stand structure. Journal of Applied Ecology : -. Vanclay. J.K.. Aggregating tree species to develop diameter increment equations for tropical rainforests. For. Ecol. And Manage. :-. Vanclay, J.K. and Skovsgaard, J.P.. Ecological Modelling. ():-. Weetman, G.F., Prescott, C.E., Kohlberger, F.L., and Fournier, R.M. Ten-year growth response of coastal Douglas-fir on Vancouver Island to N and S fertilization in an optimum nutrition trial. Can. J. For. Res. : -. Weetman, G.F., Fournier, R.M., Barker, J., and Schnorbus-Panozzo, E. Foliar analysis and response of fertilized chlorotic western hemlock and western redcedar reproduction on salal-dominated cutovers on Vancouver Island. Can. J. For. Res. : -.

20 Table. Plot summary statistics at plot establishment and the time of treatment. Treatment Plot Establishment Time of Treatment Type Variable* Mean Min Max Mean Min Max Stems ha All / Untreated Fertilized Thinned Multiple Treatments G Site Index Curtis' RD Stems ha G Site Index Curtis' RD Stems ha G Site Index Curtis' RD Stems ha G Site Index Curtis' RD * G is basal area per hectare, Curtis RD is Curtis Relative Density, and SI is Site Index 0

21 Table. Tree summary statistics at plot establishment by species. Species Variable Mean Min Max dbh. 0.. htfit Douglasfir bal rdbh. 0.. geage dbh. 0.. htfit Western bal Redcedar rdbh geage dbh htfit Western bal Hemlock rdbh geage. 0.0.

22 Table. Parameter estimates for unthinned model. Parameter Estimate Variable Western Douglasfir Western Hemlock redcedar b b b b b c c c c

23 Table. Fit statistics for models and by species. Western Hemlock Douglas-fir Western redcedar Fit Statistic Model Model Model Model Model Model AIC Adusted R Mean Bias RMSE

24 Table. AIC values for base model with fertilization modifications. Variable Modified AIC Value -0-0 and - k -0 k and - k and - k,, and -

25 Diameter Increment (cm) Diameter Increment vs. Diameter Theta=0.0 Theta=0. Theta= Theta=0.0 Theta=0.0 Theta=0.0 Theta=0.0 Theta=0.0 Theta=0.0 Theta=.0 Theta=0. Theta= Diameter (cm) Figure. Box-Lucas model for varying and values, all forms indicate an increase in diameter increment to some diameter followed by a decrease which levels off toward zero.

26 Diameter Increment (cm) Diameter (cm) Figure. Box-Lucas model for varying values, increasing values indicate an increase in diameter increment for smaller diameter trees and a decrease in diameter increment for larger diameter trees.

27 Diameter Increment (cm) Diameter (cm) Figure. Box-Lucas model for varying values, increasing values indicate an increase in diameter increment no matter the size of tree.

28 Figure. Actual and predicted average annual diameter increment by cm dbh class.

29 species= size= species= size= annualdincmean annualdincmean time species= size= annualdincmean time species= size= annualdincmean time species= size= annualdincmean time species= size= annualdincmean time time Figure. Predicted (red) and actual (black) average annual diameter increment over time for fertilized plots. The time of fertilization is given as 0 (size is <.0 cm; size is.0 cm).

30 Modeling height growth in response to varying silvicultural treatments within mixed- species stands located in coastal British Columbia Leah C. Rathbun, Valerie LeMay, and Nick Smith 0 0 Department of Forest Resources Management, University of British Columbia, 0 Main Mall, Vancouver, BC VT Z, Canada Corresponding Author: Valerie.LeMay@ubc.ca Nick Smith Forest Consulting, Kite Way, Nanaimo, BC V0T Z 0

31 0. Introduction Two model forms for height increment models exist and are generally categorized as either a growth potential independent model, where increment is expressed directly as a function of physical tree characteristics and site conditions, or a growth potential dependent model, where the potential growth is defined and then modified (Huang and Titus ). Regardless of the model form, the model and its components should be logically consistent and biologically realistic (Vanclay and Skovsgaard ). Models have been developed such that they describe relationships purely mathematically or are formed from biological functions which describe chemical relationships. To model height increment and capture the variation due to different treatment applications found on Vancouver Island, a flexible individual-tree distance-independent height increment model is required. The objective of the study was to develop a height increment model for variable retention harvested and fertilized mixed-species stands located on Vancouver Island. Two models, a common mathematical model and a biological model, were developed and analyzed for unthinned data. Future work will be done to investigate the effects of thinning and fertilization. 0.. Study Area The study area was located on Vancouver Island within the Coastal Western Hemlock (CWH) Biogeoclimatic Ecological Classification (BEC) zone of British Columbia. This BEC zone is divided into multiple subzones: a very dry maritime subzone in the east, a moist maritime subzone in the central area, and a very wet maritime and hypermaritime subzone in the west (Brown and Hebda 00). The temperatures in the CWH BEC zone

32 range from. to 0. C, with a mean annual temperature of C (Meidinger and Pojar ). Study plots were located within latitudes ranging. to. N and longitudes ranging. to. W. This area is comprised of second growth uneven- and evenaged multi-species stands regenerated naturally and from plantings. The common species include: western hemlock, Douglas-fir, western redcedar, red alder (Alnus rubra Bong.), Sitka spruce (Pinus sitchensis), and yellow cedar (Chamaecyparis nootkatensis (D.Don) Spach). 0 0 Permanent Sample Plot (PSP) data were provided by Island Timberland, Ltd. The database contained untreated plots, ranging in size from 0.00 to 0.0 ha. Measurements spanned the years 0 to, with measurement intervals varying from to years for an average of. years. Densities at plot establishment ranged from. to 0 live trees per hectare, site index ranged from. to.m, and basal area per hectare ranged from 0.0 to. m /ha (Table ). At each measurement and for each tree within the plot boundaries, species, tree status (i.e., live or dead), and diameter outside bark at breast height (. m above ground, dbh) was recorded. Height (ht, m) was measured for a subset of trees, and remaining heights were estimated using heightdiameter functions. Only trees with actual height measurements were used in the analysis. At the time of plot establishment, the average diameters were 0.,.0, and.cm (Table ); western hemlock, Douglas-fir and western redcedar, respectively, with corresponding maximum values of.,., and.cm. The average actual heights for western hemlock, Douglas-fir and western redcedar corresponded to.,.0, and.m.

33 .. Model Development Model Forms Two models were analyzed for height increment growth: a common mathematical model and a biological model. The mathematical model form was: [] H incq exp(f (X)) Where f(x) is a linear function of the predictor variables, X. Biological tree growth can be represented as the difference in anabolic gain, where molecules are constructed and catabolic loss, where molecules are broken down to their smaller components for use as energy (Pienaar and Turnbull ). The base model form representing this metabolic relationship was presented by Box and Lucas (): [] k exp height exp height Hincq Where H inc is the diameter increment for the q-year period, k is an asymptotic constant, and are parameters to be estimated, and height was previously defined. Figure illustrates typical curves produced from the Box-Lucas function. These curves follow the basic sigmoidal pattern typical of biological growth, growth increment increases to a maximum, begins to decrease tapering off asymptotically toward zero. Unlike other height increment models which use a fixed period interval, this model allows height increment to be projected for any time interval. Anabolism is modeled by and catabolism by, with both parameters working in conjunction to describe a trees metabolism. Increasing values, produce larger height increments for trees up to a specific height and then an inverse relationship is seen. An increase in produces a decrease in height increment no matter the size of the tree. An increase in the asymptotic

34 constant maintains the basic shape of the curve while increasing the height increment. Note the maximum height increment rate is found for a height of (Huang and Titus ): Hinc max ln( / ) 0 Where all variables were as defined above. Variables which affect height increment are incorporated into the model as modifications to and. Huang and Titus () observed for white spruce (Picea glauca (Moench) Voss) a linear relationship between and the following variables: basal area ha - and site productivity index; and a linear relationship between and previous diameter increment and percent basal area ha - for a by species Variable Selection and Model Comparison When selecting variables to define and, it is important to understand the biological processes which affect height growth. As a result, predictor variables representing tree size and stage of development, site productivity, and inter-tree competition were considered. Tree size and stage of development were represented by dbh (cm), height (m) and the height increment of the previous growth period (Hinc previous, cm). To represent site productivity, coastal Douglas-fir site index (SI, m) at a base age 0 and Growth Effective Age (GEA, years) were included to distinguish between differences in site quality found across plots. GEA is the age of a dominant tree with the same height and having the same site index as the tree of interest (Hann and Ritchie ). GEA was developed from site index equations for the dominant tree species of a stand. A number

35 of stand level competition measures were considered. Curtis Relative Density, an index measure of density, was calculated as: CurtisRD G d q d q n i dbh n i 0 Where dbh i (cm) is the dbh for tree i, dq (cm) is the quadratic mean diameter for the plot, G (m /ha) is the stand basal area, and n is the number of all live trees across all species within a plot. Stand basal area has been commonly used to measure competition for below ground resources (Fan et al. 00). In addition, tree-level competition measures such as the basal area of larger trees (BAL, m /ha), relative dbh (RDBH), and crown competition factor of larger trees (CCFL) were evaluated for the model. BAL was calculated as: BAL n i δ x BAtree i Where δ is an indicator variable ( if tree i has a dbh greater than the tree of interest, 0 otherwise), BAtree i (m /ha) is the basal area per hectare value for tree i, and n is the number of all live trees across all species within a plot. Relative dbh (RDBH) is a ratio of the diameter of the tree of interest to the average diameter within a plot and was calculated as: 0 dbh RDBH i dbh Where dbh i is the dbh of tree i and dbh is the average dbh for the plot. CCFL (expressed as a percent) was calculated as:

36 CCFL n i δ x MCA i MCA i π(cw ˆ ) i x,000,000 CW ˆ i 0.0 a 0. 0 dbhi b. Where: MCA i is the maximum crown area for tree i expressed as a percentage of a hectare that can be occupied by the maximum crown of tree i with dbh i in cm (Avery and Burkhart 00), Ĉ Wi (m) is the crown width of tree i, and the parameters a and b are species-specific constants for imperial units found in Smith (). All other constants are used for unit conversion. 0 0 Akaike s Information Criteria (AIC) values were calculated using PROC NLMIXED of SAS, version... AIC values were used to determine best combination of predictor variables for model. AIC was calculated as: AIC logl( ˆ ) p Where logl(ˆ ) is the log likelihood function and p is the number of parameters in the model. A parameter prediction approach was used to determine which variables best described and for model. A subset of the data was used to obtain estimates for and. The subset was defined from the original untreated data as trees which contained more than measurement periods. Each tree was fit individually to obtain parameter estimates for and across all species. Plots of and versus possible predictor variables were created to identify trends. Different combinations of the possible predictor variables in linear, log-linear, and nonlinear equations were created to

37 estimate and. The resulting residual and actual versus predicted value plots were used to determine the appropriate model forms for and for each species individually. The model forms for and were inserted into the Box-Lucas model, model, and refit using the resultant parameter estimates as starting values. Pseudo R fit statistics were calculated for Models [] and [] using all data by species. Pseudo R values were calculated as: R =-RSS/TSS Where RSS is the residual sum of squares and TSS is the total corrected sum of squares Results and Conclusions For all three species, the best combination of predictor variables found for model included height, dbh, GEA, G, and BAL. While adding CurtisRD for Douglas-fir returned a smaller AIC value, the difference in values was negligible. The same was true for western redcedar and RDBH. The final mathematical model form was: 0. Hincq exp( b0height b / dbh bgea bg bbal) The best combination of predictor variables found for, the parameter representing anabolic growth include: dbh, G, GEA, and stems ha -. The best combination of predictor variables found for, the parameter representing the catabolic breakdown of cells for energy include: dbh, BAL, RDBH, SI, and height. The final height increment model was found to be: Where Hincq k exp height exp height

38 0. exp( b b log( dbh) bg b log( GEA) bsph) exp( c c log( dbh) c log( BAL) c log( RDBH ) csi c log( height )) k is an asymptotic constant defined for a species: 0. for western hemlock,. for Douglas-fir and. for western redcedar. The parameter estimates for all other parameters for both model forms are listed in Table. The pseudo-r values for all three species are larger for model (Table ). The pseudo-r values indicate that for all species the biological model, model is a better fit for the unthinned data. More work will be done to investigate the effects of thinning and fertilization using the biological model. In addition, a growth potential model will be investigated. 0

39 Acknowledgements We would like to thank Ken Epps and Island Timberland for providing the data for this project. We would like to thank Nick Smith for his insightful input and questions throughout the entire process. We would also like to thank Dr. Peter Marshall, Dr. Lori Daniels, and Dr. David Hann.

40 References Brown, K.J. and Hebda, R.J. 00. Origin, development, and dynamics of coastal temperate conifer rainforests of southern Vancouver Island, Canada. Can. J. For. Res. : -. Huang, S. and Titus, S. J.,. An individual tree diameter increment model for white spruce in Alberta. Can. J. For. Res. :-. Meidinger, D. and Pojar, J., eds.. Ecosystems of British Columbia. The Research Branch, BC Ministry of Forests, Victoria, BC. pp.-. Vanclay, J.K. and Skovsgaard, J.P.. Evaluating forest growth models. Ecological Modelling. ():-. 0

41 Table. Plot summary statistics at plot establishment (m= plots). Variable * G (m /ha) Establishment Mean Minimum Maximum Stems 0 ha - CurtisRD SI (m)... * G is basal area per hectare, Curtis RD is Curtis Relative Density, and SI is Site Index

42 Table. Tree summary statistics at plot establishment by species. Douglas Western Variable * Western hemlock -fir redcedar dbh (cm) Mean Minimum 0... Maximum... Height (m) Mean... Minimum... Maximum... BAL (m /ha) Mean.0..0 Minimum Maximum..0. GEA (years) Mean... Minimum Maximum *dbh is diameter at breast height, BAL is basal area of larger trees, and GEA is growth effective age

43 Table. AIC values for possible models using the log-linear model form. Western Douglas Western Model Variables* hemlock -fir redcedar Height Height and dbh Height, dbh and SI Height, dbh, SI, and G Height, dbh, GEA, and G Height, dbh, GEA, G 0., and BAL Height, dbh, GEA, G 0., and CCFL Height, dbh, GEA, G 0., BAL, and RDBH Height, dbh, GEA, G 0., BAL, and CurtisRD * dbh is diameter at breast height, SI is Site Index, G is basal area per hectare, BAL is basal area of larger, CCFL is crown competition factor of larger, RDBH is relative diameter at breast height, and Curtis RD is Curtis Relative Density

44 Table. Parameter estimates for Box-Lucas and log-linear models. Parameter Western Hemlock Box-Lucas Model Western Hemlock Western Hemlock Log-Linear Model Douglasfir Douglasfir Western Redcedar b b b b b c c c c c c. -0..

45 Table 0. Psuedo R values for models and by species. Species Model Model Douglas-fir Western Hemlock Western Redcedar 0. 0.