Indices of stand structural diversity: adding spatial diversity to a stand structural index

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1 Indices of stand structural diversity: adding spatial diversity to a stand structural index Valerie LeMay and Christina taudhammer Associate Professor, Department of Forest Resources Management, University of British Columbia, Main Mall, Vancouver, BC, Canada Valerie.LeMay@ubc.ca Assistant Professor, University of Florida, Gainesville, Florida, UA staudham@ifas.ufl.edu Extended Abstract tand structure is an important element of stand biodiversity (e.g., MacArthur & MacArthur, 96; Aber, 979; Freemark & Merriam, 986). High biodiversity is associated with stands where there are multiple tree species and sizes (Buongiorno et al., 994). Managing forests for biodiversity may be accomplished by managing for structural diversity (Önal, 997). Measures of stand structural diversity are also important for predicting future stand growth and development (Pretzsch, 997). everal indices of stand structure have been proposed based on tree attributes, particularly species and tree size; a few authors have suggested indices that combine a mixture of spatial diversity (arrangement) and tree attribute diversity into an overall structural index (Pommerening, 00). For this paper, the definition of most heterogeneous (most structurally diverse) stand is proposed as a uniform ribution over size and species, and a Poisson ribution of trees in clumps of varying sizes for the spatial ribution. Extensions of the tand Variance Index (TVI) developed by taudhammer and LeMay (00) for spatial diversity were examined. TVI was based on the variance of basal area per ha for diameter (; diameter outside bark at.3 m above ground) and height relative to the variance for a maximally diverse stand with a uniform ribution of and height. The developed diversity index, as shown for diameter (TVI i for a species i), was: U i, when i U U TVI = i i i max U U p, when where is the variance of for species i; and p are constants > 0 that define the shape of the curve; and m is a constant.0 that controls the value of the index when the ribution is maximally bimodal. ample variance was calculated by: n [ wi ( xi x) ] i= = n i= where x i is i or height i ; w i is the basal area per ha represented by the ith tree in the sample plot; x is the average of or height, weighted by basal area; and n is the number of trees in a sample plot. w i i U

2 The variance of a uniform ribution is given by: ( b a) U = where a and b define the range of the ribution, differing for versus height. The maximum possible variance of a ribution occurs when the ribution is maximally bimodal, when half the basal area is at a and half the basal area is at b. For this basal area ribution, the variance is: ( a + b) ( a + b) ( b a) max = a + b = 4 The values for, p, and m were chosen by placing three constraints on the index to yield certain index values under defined conditions. The index was constrained to equal 0.5 when: ) the variance is equal to that of a uniform ribution over half the maximum possible range ( 0. 5U ); and ) the variance is equal to that of a bimodal ribution, with half of the values uniformly ributed over the lower quartile, and the other half uniformly ributed over the upper quartile of the maximum possible range ( max ). The index was also constrained to equal 0. for the maximum variance (maximally bimodal stand). ince the variances used in defining these constraints are all functions of the variance of a uniform ribution,.4094, p , and m.8, for any defined size ranges from a to b (see Appendix of taudhammer & LeMay, 00 for details). To arrive at a measure of structural diversity for species i, TVIi and TVIht i were averaged, with a maximum value of (uniform for both diameter and for height, equally weighted). These were summed over all species in the plot (TVId+h), for a maximum value equal to the number of species. The separation by diameter and height ributions by species allowed for a more complete view of the stand structure, obtaining an overall index, indices by species, and indices for and for height by species, making the index potentially more relevant. The index is also somewhat size invariant, as the constraints are the same regardless of the choice of (a,b). To extend TVI to include spatial diversity, a separate component of the index for ance could be developed. The main reasons for developing and using this TVI rather than a well-known spatial index are: ) the standard for the index would be maximally diverse spacing, rather than Poisson spacing; ) each component (, height, ance) could be reported separately; and 3) the index could be easily combined with the separate and height indices to report one measure of structural diversity. All three structural measures would then indicate a value near zero for low diversity, and a value near one for high diversity, resulting in some consistency of interpretation. In extending the TVI for ance, one issue is how to define the most spatially diverse arrangement. Most spatial indices have been developed using a Poisson process as their basis. One possibility for the maximal spatial diversity index is a Poisson ribution of different sized clumps, resulting from microsite variability, inter-tree competition over different species and tree sizes, and/or urbance events. This basis is more subjective than the uniform ribution over a wide range of tree sizes used to develop the TVI index for and height. The number of clumps and ribution of clump sizes (number of trees and space occupied by the trees in the clump) would need to be specified. Another issue is what ance metric to use. As noted by Dale (999), the use of first nearest neighbours does not differentiate a clumped pattern from an even ribution of regularly sized clumps; moreover, information about the ribution is lost (Cressie, 993). Alternatives include using another neighbour that might better indicate primary neighbours, using all possible tree-to-tree ances, or using tessellated ances similar to Zenner and Hibbs (000). Another alternative, following the development of TVI for and for height, would be to calculate TVI for the x-direction and for the y-direction separately. The expected

3 ributions in each direction may be more predictable. For example, given regular (square) spacing ribution, the frequencies will all be equal and spread widely over the ranges, but there will be discontinuities in ances, relating to regular gaps in x- and in y-directions. Conversely, for a Poisson ribution of a number of unequal sized clumps, unequal frequencies would be shown, with gaps in both directions. However, using two measures adds complication. Once decisions on what is maximally diverse, and which ance metric to use are made, the TVI for ance over all species would be:, when TVI = C p, when where is the target stand variance of ances for all species; is the variance for a most spatially diverse stand; C is the variance of a ribution of two maximally separated equal-sized clumps, with Poisson spacing within clumps. Then, using a similar concept as for tree size variables, values for, p, and m would be chosen by placing constraints on the index to yield certain index values under defined conditions. To maintain the constraint of 0 (low diversity) to (high diversity), and using similar additional constraints, TVI should equal:. Nearly 0 for very regular (e.g. square) spacing of trees, where the number of clumps is equal to the number of trees;. A bit larger than 0 for a Poisson ribution, where the number of clumps is less than or equal to the number of trees (e.g., 0.); 3. Nearly for a very highly diverse stand, possibly a Poisson ribution of different sized clumps ( maximally diverse ); 4. A value of 0.5 for a diverse stand with a Poisson ribution of different sized clumps, but half the diversity of maximal diversity; and 5. A value a bit larger than that of a Poisson ribution for two equal sized clumps, maximally dispersed in space (e.g., at diagonal corners in a square space) (e.g., 0.3). These additional constraints are quite arbitrary. Adding the constraints: 0. = TVI 0.5 = TVI = = Poisson 0.5 C p 0.3 = TVI = where Poisson is the variance of a Poisson ribution of trees; 0. is the variance of a ribution with ½ the diversity of a most spatially diverse stand; values for, p, and m could be calculated given expected values of the variances. ince expected values for the variances are not known for these ributions, regardless of which ance metric is selected, Monte Carlo simulations would be needed to obtain these variances as with other spatial indices (Dale, 999). This index could be calculated for any selected ance C C p 5 3

4 measurement, for x- and y-directions, separately, and pooled across or separated by species. If the TVI were calculated by species, then TVI components for diameter, height, and ance for each species i, (TVI i, TVIht i, and TVI i ) could be averaged (would be four values if x- and y-ance variances were separately calculated). This would result in a maximum value of for a most diverse stand, if each component was given equal weight in averaging. This type of stand would have a wide range of tree sizes (height and ), and a diverse ribution of trees in clumps for the species. Combining these indices over a number of species, the maximum value for the combined index, TVId+h+ would have a maximum value of, the number of species in the stand. We applied the spatial TVI index to a number simulated stands and to uneven-aged stand data (LeMay & taudhammer, 005). The extension of TVI to a spatial diversity measure would result in consistencies in the definitions of the three indices, and an overall index could be simply calculated. However, further thought is needed on what might be considered maximal spatial diversity to obtain meaningful and consistent index values. Acknowledgments The National cience and Engineering Council (NERC) of Canada provided funding. Reference Aber, J.D. (979). Foliage-height profiles and succession in northern hardwood forests. Ecology 60: 8-3. Buongiorno, J., Dahir,., Lu, H.C., and Lin, C.R. (994) Tree size diversity and economic returns in uneven-aged forest stands. Forest cience 40(): Cressie, N.A.C. (993) tatistics for spatial data: Revised edition. John Wiley & ons, Ltd., Toronto. Dale, M.R.T. (999) patial pattern analysis in plant ecology. Cambridge University Press, Cambridge. Freemark, K.E., and Merriam, H.G. (986) Importance of area and habitat heterogeneity to bird assemblages in temperate forest fragments. Biological Conservation 36: 5-4. MacArthur, R.H., and J.W. MacArthur. (96) On bird species diversity. Ecology 4: Önal, H. (997) Trade-off between structural diversity and economic objectives in forest management. American Journal of Agricultural Economics 79: Pommerening, A. (00) Approaches to quantifying forest structure. Forestry 75(3): Pretzsch, H. (997) Analysis of modeling of spatial stand structures: Methodological considerations based on mixed beech larch stands in Lower axony. Forest Ecology and Management 97: Qinghong, L. (994) A model for species diversity monitoring at the community level and its applications. Environmental Monitoring and Assessment 34: taudhammer, C.L, and LeMay, V.M. (00) Introduction and evaluation of possible indices of stand structural diversity. Canadian Journal of Forest Research 3(7): LeMay, V.M. and taudhammer, C.L, (005) Indices of stand structural diversity: mixing discrete, continuous, and spatial variables. Paper in draft form on website. 4

5 Zenner, E.K and Hibbs, D.E. (000). A new method for modeling the heterogeneity of forest structure. Forest Ecology and Management 9: