QUANTIFYING FOREST STAND DIVERSITY USING STAND STRUCTURE by Christina Lynn Staudhammer B.Sc, University of California (Davis), 1990 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF FORESTRY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA December 1999 © Christina Lynn Staudhammer, 1999 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver, Canada Date DE-6 (2/88) Abstract Stand structure is an important component of the overall description of biological diversity in a stand. The diversity of tree sizes within a stand affects tree growth and yield, and is highly correlated with the biodiversity of a stand. Four measures of stand diversity based on the stand distribution of basal area by height, diameter, and species were proposed, assuming a baseline of maximum diversity corresponding to a uniform distribution. First, the Extended Shannon Index (ESI), a measure based on Shannon's Index, was derived. Second, a measure based on fitting the univariate and bivariate distributions of diameter and height was investigated. The third measure, STRI, was derived as a modified R-squared, based on the fit of the distribution to that of a uniform distribution. The fourth measure, STVI, was derived based on comparing the variance of a distribution to that of a uniform distribution. The four measures were evaluated with simulated data and inventory data from the Malcolm Knapp Research Forest (MKRF). The ESI and STVI resulted in logical orderings of both the simulated data and the MKRF data. The advantages of the ESI include its known sampling distribution and accepted use in forestry; however, the ESI depends on allocating data to arbitrary classes. The STVI does not require classifying data; however, its sampling distribution is unknown. ii Since the measures derived require only normally collected inventory plot data, they are relatively inexpensive to use. However, a thorough investigation of the properties of each should be undertaken. Since evaluating structural diversity is becoming an increasingly important part of forest assessment, a quantitative measure is needed to measure diversity. These measures provide a starting point toward finding an inexpensive, practical measure of structural diversity which gives reasonable results. iii TABLE OF CONTENTS ABSTRACT ii TABLE OF CONTENTS iv LIST OF TABLES vi LIST OF FIGURES vii ACKNOWLEDGEMENT viii CHAPTER 1: INTRODUCTION 1 CHAPTER 2: COMMON DIVERSITY MEASURES 5 2.1 SHANNON'S INDEX 6 2.2 SIMPSON'S INDEX 7 2.3 T H E Q STATISTIC 8 2.4 RELATIONSHIPS A M O N G INDICES 9 2.5 USING INDICES TO MAKE INFERENCES ABOUT POPULATIONS 11 2.6 DISCUSSION OF COMMON DIVERSITY MEASURES IN U S E 12 CHAPTER 3: STRUCTURAL DIVERSITY MEASURES 16 3.1 MEASURES OF VERTICAL DIVERSITY 18 3.2 MEASURES OF HORIZONTAL DIVERSITY 20 3.3 OTHER STRUCTURAL DIVERSITY MEASUREMENTS 22 3.4 DISCUSSION OF MEASURES 24 CHAPTER 4: POSSIBLE MEASURES OF STAND DIVERSITY USING STAND STRUCTURE 26 4.1 DESIRABLE CHARACTERISTICS OF A MEASURES OF STAND DIVERSITY USING STAND STRUCTURE 27 4.2 POSSIBLE ALTERNATIVE MEASURES OF STAND DIVERSITY USING STAND STRUCTURE 29 4.2.1 Shannon's Index Extended to Diameter, Height, and Species 29 4.2.1.1 Post-hoc Method 30 4.2.1.2 Combination Method 31 4.2.1.3 Properties of the Extended Shannon's Index 32 4.2.2 Structural Index based on Curve Fitting 33 4.2.2.1 Univariate Curve Fitting 35 4.2.2.2 Bivariate Curve Fitting 36 4.2.2.3 Properties of a Structural Index Based on Curve Fitting 37 4.2.3 Structural Index Based on R2 (STRI) 38 4.2.3.1 STRI Univariate Method 39 4.2.3.2 STRI Bivariate Method 41 iv 4.2.3.3 Properties of STRI 42 4.2.4 Structural Index based on Variance Comparisons (STVI) 43 4.2.4.1 STVI Univariate Method 44 4.2.4.2 STVI Bivariate Method '. 49 4.2.4.3 Properties of STVI 52 CHAPTER 5: EVALUATION OF ALTERNATIVE MEASURES OF STRUCTURAL DIVERSITY 54 5.1 SIMULATED TREE DATA 54 5.1.1 Methods '. 54 5.1.1.1 Baseline Maximal Diversity 54 5.1.1.2 Simulated Stands of Varying Diversity 55 5.1.2 Results and Discussion 62 5.1.2.1 Shannon's Index Extended to Diameter, Height, and Species 62 5.1.2.2 Structural Index based on Curve Fitting 65 5.1.2.3 Structural Index based on R 2 (STRI) 71 5.1.2.4 Structural Index based on Variance Comparisons (STVI) 72 5.1.3 Overall Discussion of Proposed Indices Computed with Simulated Data 76 5.2 MALCOLM KNAPP RESEARCH FOREST DATA 76 5.2.1 Methods 77 5.2.2 Results and Discussion 80 5.2.2.1 Shannon's Index Extended to Diameter, Height, and Species 80 5.2.2.2 Structural Index based on Variance Comparisons (STVI) 84 5.2.3 Discussion of Extended Shannon Index and Univariate STVI Computed with MKRF Data 88 CHAPTER 6: CONCLUSION : 89 LITERATURE CITED 95 APPENDIX I 101 v List of Tables Table 2.1. Summary of performance and characteristics of selected diversity measures. 14 Table 5.1. Plot type abbreviations for simulated cases 60 Table 5.2. Extended Shannon indices computed using simulated stand data 64 Table 5.3. Resulting coefficients and RMSE for the univariate Weibull fitted to dbh using simulated stands 66 Table 5.4. Resulting coefficients and RMSE for the univariate Weibull fitted to height using simulated stands 67 Table 5.5. RMSE and resulting values of coefficient as for the bivariate Weibull fitted to simulated stands 69 Table 5.6. Resulting coefficients bi and bj for the bivariate Weibull fitted to simulated stands 69 Table 5.7. Resulting coefficients c/ and C2 for the bivariate Weibull fitted to simulated stands 70 Table 5.8. Univariate (STRId+h) and bivariate (STRIdh) STRI using simulated stands. 71 Table 5.9. Univariate (STVIa+h) and bivariate (STVIdh) STVI using simulated stands. 73 Table 5.10. Average sample variances for simulated cases compared to the variance of a theoretical uniform distribution 75 Table 5.11. Data for height-dbh equation development by species 78 Table 5.12. Descriptive statistics of selected MKRF plots 80 Table 5.13. Extended Shannon indices computed with MKRF data 82 Table 5.14. Univariate (STVI d b h, STVIh eight, and STVI d + h) and bivariate (STVId h) STVI using MKRF data 85 Table 5.15. Sample variances and average sample variances by species for chosen MKRF stands compared to the variance of a theoretical uniform distribution... 87 v i List of Figures Figure 4.1. STRI and three equal dbh ranges 44 Figure 4.2. Comparison of STVI values with m=l and varying values of pi and p2.47 Figure 4.3. Comparison of STVI values with pt=p2=2 and varying values of m 48 Figure 5.1. Possible combinations of dbh versus height 57 Figure 5.2. Graphical description of simulated dbh and height values 59 vii Acknowledgement I would like to extend thanks to the many people that helped me to successfully complete this thesis, especially my supervisor Dr. Valerie LeMay, and my thesis committee, Drs. Tony Kozak, Guillaume Therien, Rick Routledge (Department of Statistics, Simon Fraser University), and Peter Marshall. Each and everyone one of them contributed significantly to my education and my experience at U.B.C., and I will not forget their generosity. I would also like to thank Dr. LeMay for helping me to find funding for this project and securing the data necessary to complete this project. Last, I would like to thank my family and friends for their amazing support during this time in my life, and extend an especially warm thanks to William Kovacs, whose love of the forest was my first inspiration. Thanks for letting me grow. viii Chapter 1 Introduction Over the past two decades, the focus of the sustainable forestry debate in North America has been increasingly drawn toward the concept of diversity. The maintenance of forest diversity has been linked to robust bird and insect populations, and is seen as an indicator of the wellbeing of a biological system (Magurran 1988). Interest in forest diversity has prompted both the Canadian Council of Forest Ministers (Canadian Standards Association 1996) and the United States federal government (U.S. federal law 36 Code of Federal Regulations Sec. 217.27) to require forest managers to conserve biological diversity at natural levels. The term 'biodiversity' has gained wide acceptance in describing biological diversity, encompassing the total variety of a biological system, including species diversity of plants, animals, fungi and bacteria, as well as the genetic and individual variation that exists within those species (Dobson 1996). Biodiversity may be evaluated at many levels: over an entire ecosystem, in a forest stand, or within a species at the genetic level. At the stand level, biodiversity has been defined to include not only species diversity, but also the variation of size, age, and genetic composition of the stand (Lahde et al. 1999). Thus, the definition of stand diversity may include not only the composition of species, but also the size distribution of individuals of those species. 1 Along with governmental and non-govemmental directives to conserve biological diversity comes the necessity to measure biological diversity. For example, U.S. federal law (36 C.F.R. Sec. 219.26) requires that inventories include "quantitative data making possible the evaluation of diversity". The evaluation of diversity, however, has long been a subject of heated debate. Many measures of diversity, in the form of indices, are in use in forestry and ecology; however, none are without critics (Magurran 1988). It has even been asserted that since there is no precise accepted definition of biological diversity, there can be no effective quantitative measure (Hurlbert 1971, Silbaugh and Betters 1995). Several traditional biodiversity indices, such as Simpson's index (Simpson 1949) and Shannon's index (Shannon and Weaver 1949) have gained wide acceptance in forestry as quantitative measures of species diversity (Swindel et al. 1984, Magurran 1988, Lewis et al. 1988, McMinn 1992, Silbaugh and Betters 1995). These indices rely on the distribution of individuals by species to arrive at a quantitative value of biological diversity. However, because of the variation that exists in tree size, traditional diversity indices are not entirely suited to the measurement of forest stand diversity (Lahde et al. 1999). Furthermore, constraints of time and money limit the forest manager to measuring only a subset of species represented in a single sample plot of forest. Surrogate measures for ecological composition are unavoidable, as the number of existing species in forested ecosystems is vast, and there are inherent difficulties in measuring them. To help remedy this, the use of indicator species has been proposed 2 (Noss 1990, Schneider 1997). Desirable criteria for species used as biodiversity indicators include (Noss 1990): • Wide distribution across regions • Potential to act as surrogate for other species » Inexpensive and easy to monitor • Pre-existing data available • Biology of species reasonably known • Sensitive to forest management disturbances • Sufficient sample size attainable • Population dynamics stable Since trees meet these criteria, a useful measure might be derived using data from forest inventories or other regular forest surveys (LeMay et al. 1997, Lahde et al. 1999). Stand structure has been defined in terms of the mixture of tree sizes and species, without explicit recognition of other flora and fauna (Buongiomo 1995). The diversity of stand structure indicated by the mixture of tree size and species has been shown to be an important element of total biological diversity (MacArthur and MacArthur 1961, Willson 1974, Franzreb 1978, Temple et al. 1979, Aber 1979, Ambuel and Temple 1983, Freemark 1986), with the highest diversity found in stands where there are multiple species and multiple tree sizes (Buongiomo et al. 1994). It has been argued that managing forests for biodiversity can be accomplished by managing for structural diversity (Onal 1997). Thus, a useful measure of the total diversity of a forest stand may be found by measuring the structural diversity of a stand. 3 Stand structure can be measured based on the distribution of tree ages and sizes (Lahde et al. 1999). Although age is difficult to measure quickly and cheaply, tree size may be readily measured by diameter at breast height (dbh) and height. An index of stand structure as a function of species composition, dbh, and height should be correlated with overall forest diversity, and would be a function of tree variables found in forest inventory samples. The objective of this study is to present possible measures of stand diversity that quantify the diversity in a stand based on both species and size diversity, using commonly measured tree attributes. Several measures are suggested and discussed. The measures were then evaluated using simulated stand data, and applied to the University of British Columbia's Malcolm Knapp Research Forest (MKRF), located in coastal British Columbia. The measures were evaluated using objective criteria and conclusions were drawn as to the performance of the suggested measures. 4 Chapter 2 Common Diversity Measures The most common diversity measures in use are species diversity measures. Species diversity measures can be applied at various spatial and temporal scales, incorporating landscape patterns, as well as stand level variation (LeMay et al. 1997). Biodiversity is often broken up into three spatial components. The a-component of the diversity of organisms inhabiting a variety of habitats is loosely defined as the average diversity within a single habitat (Routledge 1980). The (^ -diversity can be defined as the amount of turnover between habitat types (Routledge 1979). T-diversity describes the variation in a and P -diversity as one moves across regional landscapes and along major environmental gradients (Kimrnins 1997). This paper will focus on a-diversity, that is, diversity at the stand level. Species diversity can be broken into two elements. Species richness refers to the number of species. Species evenness, or species equitability, refers to how equally abundant the species are. From these definitions, species diversity can be classified into three categories (Magurran 1988): 1. Species richness indices are a measure of the number of species in a defined sampling unit. 2. Species abundance models describe the distribution of species as a function, or parameter of an associated function. 3. "Diversity indices" or "heterogeneity indices" are indices based on proportional abundances of species, combining richness and evenness into a single index. There are many measures of species diversity (see Magurran 1988). The diversity measures chosen for discussion in this thesis are Simpson's Index of Concentration, Shannon's Information Index (which is often incorrectly referred to as the Shannon-Weaver or Shannon-Wiener index (Magurran 1988)), and the Q statistic. Based on information theory, Simpson's Index and Shannon's Index are perhaps the most popular diversity measures. Both are nonparametric indices; that is, neither relies on any assumptions about the underlying species abundance distribution. The Q statistic is a diversity measure derived from an abundance model. 2.1 Shannon's Index Shannon's index is a measure of how uncertain we are that an individual picked at random from an infinitely large community will be of a certain species. The more uncertainty one has about the identity of an individual, the higher the diversity of the community. As well as assuming that the population is 'indefinitely large' and that individuals are selected at random, the calculation of this index requires that all species be represented in the sample. The failure to represent all species can be a source of considerable error (Magurran 1988). Shannon's index, H', is defined as follows (Shannon and Weaver 1949): H' = tpihPi [2-1] i=i where: p, = proportion of individuals in the t'th species; and S is the number of species in the population. 6 When the population proportions are unknown, Shannon's Index is calculated with the maximum likelihood estimator (MLE) of Pi = ntIN [2.2] where: «, = the number of individuals sampled of species /, and N - total number of individuals sampled. In practice, the terms pu nu and N are not necessarily computed with the number of individuals, as individual trees, plants, or animals can vary greatly in size. The proportion of a species has been measured in a variety of ways, including: number of individuals (Franzreb 1978, Swindel et al. 1991, Niese and Strong 1992, Condit et al. 1996), basal area (McMinn 1992, Harrington and Edwards 1995), stems per ha (McMinn 1992, Harrington and Edwards 1995), foliar cover (Swindel et al. 1984, Lewis et al. 1988, Qinghong 1994, Corona and Pignatti 1996), crown cover (Corona and Pignatti 1996), and biomass (Swindel et al. 1984, Swindel etal. 1991). Shannon's Index is often applied in the form of a relative index. This relative index is the ratio of absolute diversity to the maximum diversity possible. This index can be taken as a measure of evenness (Pielou 1977): E = H'IH'mx=H'l\xiS [2.3] 2.2 Simpson's Index Simpson's index of concentration, A, is the probability that two individuals selected at random from an infinite population are of the same species (Simpson 1949). Simpson's 7 index differs from Shannon's in that Simpson's index decreases as diversity increases. Thus, Simpson's Index is often expressed as l/A or exp(A). Simpson's index is defined as follows (Simpson 1949): where: /?, = proportion of individuals in the ith species. When a sample is chosen at random from a population, Simpson's index can be computed with the following unbiased estimator (Simpson 1949): where: n, is the number of individuals of the ith species; and N = total number of individuals sampled. As with Shannon's Index, the terms p„ n„ and N are not necessarily computed with the number of individuals. The proportion of a species has been measured in a variety of ways, including: number of individuals (Swindel et al. 1984, Condit et al. 1996), basal area (Harrington and Edwards 1995), stems per ha (Harrington and Edwards 1995), foliar cover (Swindel et al. 1984, Lewis et al. 1988), crown cover (Swindel et al. 1991), and biomass (Swindel et al. 1984, Swindel etal. 1991). 2.3 The Q Statistic The Q statistic can be considered to be a species richness index, but it is a statistic derived from the species abundance distribution of a particular population. The Q statistic [2.4] 8 measures the inter-quartile slope of the cumulative species abundance curve, providing an indication of the diversity of the community without weighting toward the very abundant or very rare species (Kempton and Wedderburn 1978): 0.5n„, +yR2 '« +0.5nB1 Q = — [2.6] log(/?2//fl) where: nr is the total number of species with abundance r; Rl and R2 are the 25% and 75% quartiles, respectively; and nm and n«2 are the number of individuals in the classes where Rl and R2 fall, respectively. This approach takes into account the distribution of species abundances, but does not actually entail fitting a model. 2.4 Relationships Among Indices Hill's (1973) indices provide an elegant relationship between S, A, and H'. Hill's indices were derived from the observation that these indices can be arranged by their propensity to weight toward rare species or common species (i.e., toward species richness or evenness). Hill (1973) defined a diversity index as "reciprocal mean proportional abundance", with the ath order of diversity given by: A l = (5>r I"""" . for s o m e « > 0 [2-7] where: pi = proportion of individuals in the /th species. It has been proven that if an index of diversity, N, satisfies the following properties, it must be one of Hill's indices (Routledge 1979): 9 HI: N(l/S, 1/S,. . . , 1/5) = S . For a population with a given number of species, the measure of diversity will be at a maximum when all the species are present in equal proportions (or with maximum evenness). U2:N(pi, p2,...,ps)~~, = the proportion of profiles of type /. Ambuel and Temple (1983) designated sixteen profile types, using the presence/absence of vegetation in each possible combination of four height zones. Willson (1974) and Erdelen (1984) confirmed MacArthur and MacArthur's (1961) results, finding that FHD was highly correlated to bird species diversity. However, these relationships did not hold when only forested plots were included in the analyses. When Aber (1979) used MacArthur and MacArthur's (1961) layer definitions, FHD correlated strongly with stand age. On the other hand, there was not a significant relationship with stand age when FHD was calculated with four strata. Results of Ambuel and Temple's (1983) study revealed a strong relationship between FHDAT and the presence of long-distance migrants. f 3.2 Measures of Hor izonta l Diversity The first, measures of horizontal diversity emerged from forest planning models. Distributions of diameters are used as inputs to some natural resources mathematical programming models, and so can be used to juxtapose the evaluation of diversity with economic returns in a forest. Gove et al. (1991) measured diameter diversity using the dichotomous rarity measures presented by Patil and Taillie (1979, 1982), with nine two-inch diameter classes in place of species. In this approach, rarity is measured as: 1 —Vs 18+1 A,(g)= y , P > - \ [3.14] where: K.t = the proportion of individuals in diameter class i; and 20 s = the number of diameter classes. These rarity indices are synonymous with Hill's indices: when /?= -1, Afi (n) is the species count; when /?= 0, A^(^;)is Shannon's index; when /?= 1, A^(^)is Simpson's index (Gove etal. 1991). Later researchers used Shannon's Index to predict the effects of some management regimes on the diversity of tree sizes. Buongiomo et al. (1994) used the diameter distribution as a measurement of stand structure for several reasons. First, because it plays a key role in the structure of forest stands, it is a key element in the biological diversity of stands. Second, since diameter - height relationships are well defined and tree height is positively correlated with FHD, the diameter distribution is a good proxy for FHD. Third, a measure using diameters is more economically feasible than FHD. Buongiomo et al. (1994) measured diameter diversity in northern hardwoods in Wisconsin using Shannon's Index with p, = the proportion of trees in the ith two-inch diameter classes. In the French Jura Mountains, Buongiomo et al. (1995) used a similar technique with five-cm diameter classes. A "normalized absolute deviation" (NAD) index was used by Onal (1997) to measure divergence of a diameter distribution from an arbitrarily specified target distribution: [3.15] n 21 where: y, is the number of individuals in species i; Pi = yi I Zi yi; {Pi }is a specified distribution with /?, >0 and S, /?,; = 1 ; n is the number of classes; and £ = m i n ( # ) . This measure exhibits properties similar to that of Shannon's index when the target distribution is uniform (i.e., /?,= l/h). Like Shannon's index, the maximum value of NAD is reached when the proportion in each class is equal, NAD is scale neutral, and the value of NAD is independent of class ordering2 (Onal 1997). Onal (1997) used simulations to show that there is a strong significant quadratic relationship between NAD and Shannon's Index. 3.3 Other Structural Diversity Measurements Various other measures of structural diversity have been developed in the literature. However, none has gained widespread acceptance. The following methods are only a sampling of the various methods developed. Freemark and Merriam (1986) introduced an index of spatial variability. Their habitat heterogeneity index (HH) is a measure of plant species and forest structure diversity derived from Shannon's index (after Orloci 1970)3: HH=-±±XAn(Xij/Xi) • [3.16] /=i y=i 2 See section 2.4: HI, point 4 in H4, and point 5 in H4, respectively. 22 where: c = number of plots; r = number of classes; and Xij is! the proportion of individuals in the /th class of they'th plot. Equation [3.16] differs from Shannon's index in that the denominator contains the average value of the class, not the total value of the class. HH was computed separately for eight components: tree density, tree dbh class, canopy closure, foliar cover in five defined vertical bands, average canopy height, herb height, percent litter, and percent bare ground. Since components varied in units, magnitude, and number of classes, HH was scaled to facilitate comparability and combinability. To arrive at a total value of forest HH, the components were summed across forest type. Using this index, the authors concluded that the more hetrogeneous habitats in the agricultural landscape near Ottawa had more bird species (Freemark and Merriam 1986). Some researchers have included a measure of structure in Shannon's and Simpson's indices by modifying the way /?, is calculated. Cain and Shelton (1995) computed the p,'s with importance values: IV = relative density + relative height + relative frequency where: "relative" refers to each species' contribution to the total for all species. Then, /?, = proportion of TV in species /. The resulting values for both Shannon's and Simpson's indices were then used to compare the diversity of woody understory growth after disturbance in the southeastern United States. 3 Although this measure of diversity is presented as a forest-level measure (whereas the other measures presented are stand level measures), the measure could easily be modified to a stand level one by removing the second summation sign. 23 A variety of tree and stand variables have been used to arrive at an index of structural diversity. One such recent example (Lahde et al. 1999) uses seven indicator variables by species: stem distribution by diameter size, basal area of growing stock, volume of standing dead trees, volume of fallen dead trees, undergrowth density, occurrence of "special trees", and volume of charred wood. All variables were delineated into classes, e.g., trees were classed into three diameters groups and two basal area groups. Based on the values,of the indicator variables, stands were given diversity scores by species, which were then combined into an index for the entire stand. The index was found to differentiate between forest development types and site types fairly well (Lahde et al. 1999). 3.4 Discussion of Measures Reviews of bird communities and vegetation stand structure subsequent to MacArthur and MacArthur (1961) emphasized that the location of plots greatly affects the discriminant abilities of the foliage height diversity index. However, all but one of the studies cited was not directly comparable to MacArthur and MacArthur's original work. Willson (1974) and Erdelen (1984) assert that the mere existence of a tree layer in the canopy determines bird diversity, regardless of the foliage volume or distribution among layers; however, their results are not truly comparable as classes were not defined in exactly the same manner. Aber (1979) used four strata in the calculation of FHD, finding it uncorrelated with stand age. Again, the calculation of this FHD was quite different from MacArthur and MacArthur's since Aber's strata were of different sizes and at different heights, depending on 24 the sample site. This result is only comparable to MacArthur and MacArthur's results if stand age is seen as a proxy for bird diversity. A variety of techniques were used to assess vegetation cover and FHD, and results depended, at least in part, on the selection of class boundaries. Height and diameter are continuous variables, and as such, the division of data into classes will necessarily depend on the distribution of the data. Since many studies are incomparable, caution should be used in drawing conclusions (Erdelen 1984). Further, no study compared the use of vertical and horizontal stand structure indices, and thus, no comparison may be drawn. Many stand structure measures have been developed to characterize both vertical and horizontal stand structure. However, most measurements of stand structure are derived by modifying existing diversity measurements. Unfortunately, no one method has been used consistently, which makes comparisons and evaluations difficult. Furthermore, results depend on splitting continuous variables into classes. This class division could have a marked effect on results. Some researchers have also used variables such as volume of coarse woody debris or crown ratio, which are not always measured in common forest surveys, and thus, these measures would be difficult to use from an economic standpoint. A more practical measure of stand structure will be one that uses readily available data, does not depend on subjective division of continuous variables into classes, and directly correlates to overall stand biological diversity. 25 Chapter 4 Possible Measures of Stand Diversity Using Stand Structure An index of diversity is needed which will characterize the diversity of stand structure. Stand structure diversity has been defined to include the richness of tree species and sizes (Lande et al. 1999). Further, structural diversity increases as the number of species and number of sizes increases (Buongiomo et al. 1994). This situation may be described using a probability distribution where every value over a specified range is represented equally. That is, every possible value of dbh and height may be found in a stand with equal probability. The probability distribution of maximal diversity under these circumstances is described mathematically by the uniform distribution. Low diversity stands would be defined as those with few species and few tree sizes, and minimum diversity would occur where all trees in a stand are of single tree species and single size. This distribution would be a spike at a single point. In an effort to incorporate stand structure into a diversity measure, several possible indices were derived. For all of these indices, the standard for maximum diversity was a uniform distribution. However, this assumption may be challenged since there is no quantitative definition of high structural diversity. Although a highly diverse stand is well characterized with this distribution, proposed measures could be modified with relative ease for use with another distribution. 26 There are several characteristics which a stand diversity index should possess. The alternative measures will be evaluated with a goal of meeting all or most of these criteria. 4.1 Desirable Characteristics of Measures of Stand Diversity Using Stand Structure An index of forest stand structure should have the following characteristics: 1) The index should measure both richness and evenness of structural and species diversity. A diversity index of this type would take into account the number of individuals of different species and size, as well as the distribution of individuals over those species and sizes. 2) The index should use commonly measured tree attributes only. This would enable forest managers to use the index with inventory data or other common forest sample data without additional expense. 3) The index should not rely on combining data into diameter or height classes. Many of the indices presented in Chapter 3 require that data be segregated into arbitrary classes. This practice is undesirable, as a change in class definitions could cause a change in index. 4) The index should not depend on sample size or plot size. Sample plots from forest inventories may consist of one plot or a cluster of several plots. In order to make comparisons between populations, the index should be insensitive to cluster size and plot size. 5) Horizontal and vertical diversity should contribute equally to the index. It has been shown that both tree diameter and height contribute to diversity by creating a variety of habitats. Thus, given two populations with equal horizontal diversity, the population with greater vertical diversity would have a higher index. 27 6) Basal area should be used to measure proportions of trees in species and size distributions. Although some authors (for example, Lewis et al. 1988, McMinn 1992) have used stems per ha, LeMay et al. (1997) recommended basal area, as it better represented the proportion of resource use. 7) The index should take on values between zero and S. An index value of S should be associated with stands of high diversity, and an index value of zero should be associated with stands of low diversity. 8) The index should measure the deviation between a particular community and that of a maximally diverse community. a) Given a set range, the index should be tree size insensitive, or location invariant. For example, a population with small trees evenly distributed over a ten cm diameter range would have the same index value as a sample with large trees evenly distributed over a ten cm diameter range (assuming all other things being equal). This would ensure that larger individuals would not drive the index. b) The index should be related to the range of values in diameter or height for continuously occupied ranges. In other words, a population with a larger diameter and/or height range that occurs evenly over that range would yield a larger diversity index than a population with values occurring over a smaller range, or over only parts of the range (as in a bimodal distribution). This would ensure that stands that exhibit greater size diversity would have higher index values. c) The index associated with a bimodally distributed population with two separate and distinct canopy layers with ranges of Ranget and Range2, should be at least as great as that of a unimodal distribution with a continuous range = Range/ + Range2. For 28 instance, an old growth stand with regenerating trees in the understory would be considered at least as diverse as a stand with a wide single-canopy layer. If these characteristics are met, the index should be easy to use and interpret, and the maximum value of the index will result for uniform stands with the greatest range. Some of the criteria may not be met, and evaluations with data analysis should reveal whether these criteria are truly necessary. 4.2 Possible Alternative Measures of Stand Diversity Using Stand Structure Several alternative measures were derived based on existing measures, such as Shannon's Index, or based on techniques familiar to foresters and biologists, such as curve fitting and variance measurements. 4.2.1 Shannon's Index Extended to Diameter, Height, and Species Shannon's index (Shannon and Weaver 1949) has been computed in the forestry and ecology literature with a species distribution, a diameter distribution, or a height distribution. However, Shannon's index computed with diameter, height, and species was not found in the literature. The index could be used separately for each of these variables, producing H'cl, H'h, and H's, respectively, and then these indices could be combined to produce a stand index, H'd+h+s. Alternatively, data could be classed simultaneously into diameter, height and species classes, producing one index, H'ilhs. The former method will be referred to as the 'Post-hoc Method', and the latter will be referred to as the 'Combination Method'. 29 4.2.1.1 Post-hoc Method The procedure for computing the post-hoc Shannon's index for a sample plot is: 1) Tree data are combined into diameter classes using dbh. The proportion of trees in the ith diameter class, /?,, is the percentage of basal area in that diameter class. Then, Shannon's Index is computed with P l and equation [2.1] to yield H'(l. 2) Tree data are combined into height classes. The proportion of trees in the jth height class, Pj, is calculated as the percentage basal area in that height class. Then, Shannon's Index is computed with pj and equation [2.1] to yield H'h . 3) Tree data are combined by species. The proportion of trees in the kxh species, pk, is calculated as the percentage basal area in that species. Then, Shannon's Index is computed withpk and equation [2.1] to yield H's. 4) The final index is computed as the average of the diameter, height, and species indices, as it is a convenient way to keep the index on a similar scale of values as the original Shannon's Index: H'd+h¥, = average (H'd,H'h,H's)= H ' " + H > + H * [4.17] The minimum value of the Extended Shannon Index under the post-hoc method is zero. This minimum value results when only one species, one diameter class, and one height class are present in a sample. The maximum value of the Extended Shannon Index under the post-hoc method is reached when H'd, H'h , and H's are at their maximum values. The maximum value of the H's is ln(5) (Magurran 1988). This maximum is achieved when each species is represented equally within a plot (uniformly distributed). Similarly, if all diameter and 30 height classes are represented equally within a plot, the maximum values of H'd and H'h are \r\{ni) and ln(n2), respectively, where ni is the number of possible dbh classes and n2 is the number of possible height classes. The overall maximum of the Extended Shannon Index under these circumstances is the average of these maximum values: m a x ( „ - ^ ) = M ^ % l ± i ! ^ [ 4 . 1 8 ] 4.2.1.2 Combination Method The procedure for computing the combination index for a sample plot representing one forest stand is to first combine the data simultaneously by species, diameter class, and height class. The proportion of basal area in the ith diameter, 7th height, and klh species class, />//*, is then calculated. Then, Shannon's Index is computed with and summed over all classes to yield H'dhs . As with the post-hoc method, the minimum value of the Extended Shannon Index under the combination method is zero. This value results when there is only one species in a plot, and this species occurs in only one diameter-height class combination. The maximum value of the Extended Shannon Index under the combination method will be reached when all diameter and height classes, as well as species are represented equally within a plot (uniformly distributed). This maximum value is: max(if ^ ) = ln(5 x n, x n2) = 3 x max(H'd+h+s) [4.19] 31 4.2.1.3 Properties of the Extended Shannon's Index The Extended Shannon Index, in both the post-hoc and combination versions, measures richness and evenness of stand structure, giving equal weight to horizontal and vertical diversity. The indices use basal area and require only commonly measured tree attributes in their computation. Shannon's Index also has been shown to be relatively insensitive to sample size (Magurran 1988), and attains its highest values when distributions are perfectly even (i.e., uniform). The Extended Shannon indices should be insensitive to tree size, as indices are weighted only on the proportion of basal area occurring in a particular class. Both forms of the Extended Shannon Index possess the desirable characteristics listed as items 1, 2, 4, 5 and 6 (Section 4.1). Item 8 cannot be considered without further using data. Item 7 is not satisfied because the maximum values of these indices are functions of the number of possible diameter and height classes. This dependence results in indices that have a lower bound of zero, but are not bounded by S. Item 3 is not met because the indices rely on pre-defined classes. This is undesirable, as this creates issues such as where to put class boundaries. A change in class boundaries or an increase in the number of classes would invariably change the value of the indices. Even if set classes were used, there would be pronounced differences simply due to geography. For instance, in Coastal British Columbia, trees are larger on average and have a wider range of sizes than in the interior of the province. More classes would be represented in Coastal areas, resulting in larger indices. On the other hand, more species may be represented in sample plots from Interior areas of the province, which would also result in larger index values. For this reason, care should be taken in comparing diversity indices (of any type) from different geographic areas. 32 To alleviate problems associated with classes, an integral form of the Extended Shannon Index could be presented. However, this option would rely on finding functional forms for species, diameter, and height distributions. While a functional form may be found, it may not have a closed form, or may not be flexible enough to fit the variety of distributions that may be found. This issue will be addressed in the next section. 4.2.2 Structural Index based on Curve Fitting When a continuous variable, such as height or dbh, is translated into a class variable, information about the distribution of that variable is invariably lost. In order to avoid using diameter and height classes, which may affect the index, an index based on a continuous distribution is desirable. Foresters and biologists have used continuous distributions to describe the frequency distributions of diameter and height both univariately and bivariately. These distributions have included the Beta, Johnson's SB, and Weibull, with varying degrees of success (Hafley and Schreuder 1977). In fitting such a distribution, the resulting functional form may be used in an integral form of Shannon's Index, or the Q statistic could be used to measure diversity based on the derived curve. However, if data are sparse, a derived curve may cover a range much greater than that of the data. The use of the Extended Shannon Index or the Q statistic in this type of situation would involve making an inference about the population using the fit of a distribution with a restricted range. This may be inappropriate, and could produce misleading results. Hence, this option was not further investigated. 33 Alternatively, in fitting a regression curve, the regression coefficients may indicate the centre, shape, or scale of a distribution. For instance, the two coefficients in the function describing a univariate normal distribution indicate the mean and variance of the distribution. A known distribution could be fit to the data and its regression coefficients could be analyzed as indicators of diversity. Of the candidate distributions, the Beta distribution offers the most flexibility in terms of skewness, kurtosis and shape of the curve (Hafley and Schreuder 1977). However, the Beta is a complex function, which involves the computation of Gamma function values and may be extremely difficult to fit. Johnson's SB and its bivariate form, Johnson's S B B have been shown to perform consistently better than the Weibull or Beta distributions in fitting diameter and height data (Schreuder and Hafley 1977). However, this probability density function does not have a closed-form cumulative density function (Schreuder and Hafley 1977), and thus, cannot be used without allocating data to classes or making an approximation with numerical integration. The Weibull distribution has been used extensively in forestry applications (Clutter et al. 1983). It has advantages in that it is fairly flexible in shape and much simpler than some other functions. Although the univariate and bivariate Weibull distributions can closely approximate a normal curve under certain conditions (Bailey and Dell 1973), approximations 34 to univariate and bivariate uniform distributions are only adequate over a portion of the range of the data. A desirable distribution for investigation would be one that is flexible in shape, but relatively easy to fit. Considering the advantages and disadvantages, the Weibull distribution (in both univariate and bivariate forms) was selected as the base curve for fitting. 4.2.2.1 Univariate Curve Fitting The cumulative univariate Weibull probability functions for diameter and height based on basal area are as follows: where: b/, b2, c/, and c2 are coefficients to be estimated; and dbh and height are the independent regression variables. Equations [4.20] and [4.21] would be fit for dbh and height separately by species using non-linear fitting methods. A good fit would be indicated by a small root mean square error (RMSE), computed as: V n — m where: y, is the dependent variable, the cumulative distribution of the sample (Fi(dbh), or F2( height)); [4.20] [4.21] [4.22] 35 y. = is the predicted value of the cumulative distribution under the assumed theoretical distribution; n = the number of observations; and m = the number of coefficients in the regression model. The estimated coefficients could be analyzed for patterns relating to diversity and a coefficient chosen as a representative diversity index. By species, the diversity index would be the average of the dbh and height diversity indices. The total diversity index for a plot would be computed as the sum of all species diversity indices. Thus, if two plots have the same structure, the plot with more species would have a higher diversity index. 4.2.2.2 Bivariate Curve Fitting The bivariate Weibull has been derived in many forms (see Lu and Bhattacharyya 1990); however, based on pilot investigations, the best performing form was given by Hafley and Schreuder (1976): G(dbh, height) = F, (dbh) x F 2 (height) x [l + a 3 x (l - F, (dbh))x (l - F 2 (height))] [4.23] where: Fi(dbh) and F2(height) are defined as above; and a3, along with bi, b2, Cj, and c2 from [4.20] and [4.21], are coefficients to be estimated. F(dbh, height) is computed as the cumulative basal area at each dbh - height combination represented in the plot. Equation [4.23] would be fit for dbh and height simultaneously, by 36 species, using non-linear fitting methods. Starting values for bj, b2, cj, and c2 could be obtained from the univariate regressions. The resulting estimated coefficients would then be analyzed for patterns relating to diversity. If possible, a coefficient would be chosen as a representative diversity index. As in the univariate case, the total diversity index for a plot would then be computed as the sum of all species diversities. Thus, all things being equal, if two plots have the same structure, one with more species would have a higher diversity index. 4.2.2.3 Properties of a Structural Index Based on Curve Fitting Both the univariate and bivariate methods possess the desirable characteristics listed as items 1, 2, 3, 5, and 6 (Section 4.1). An index derived from the Weibull distribution would measure richness and evenness using basal area and commonly measured tree attributes. The index would not rely on combining data into classes, and would account for vertical and horizontal diversity equally. However, items 4, 7, and 8 cannot be considered without further evaluation using data. Both the univariate and bivariate methods presented above rely on the fit of the data to a known distribution. To assure that no inappropriate inferences are made, this method should be used only if samples yield similar fit statistics, such as small residual error and bias. A more significant problem may arise in that no coefficient may be adequate as a measure of diversity. 37 4.2.3 Structural Index Based on R 2 (STRI) The coefficient of determination, or R , is the measure of how much of the variation in a regression line (or curve) is explained by the independent variable(s). Thus, a function representative of maximum diversity, such as the uniform, could be put into the form of a cumulative distribution function (cdf), and compared to a cumulative empirical sample distribution. If the cumulative empirical sample is regressed on the function of the theoretical cdf, the resulting R 2 will give a measure of the deviation of the sample from maximum diversity. A high R 2 (close to one) would indicate that the empirical sample came from a distribution that was similar to that of a high diversity population; a low R 2 (close to zero) would indicate that the sample came from a population with a distribution that deviated substantially from that of a high diversity population. R is traditionally computed as follows: R 2 _ 1 residual sum of squares _ 1 2X1 (y i~& ) 2 [424] total corrected sum of squares (y -y)2 where: y,- is the dependent variable, the cumulative distribution of the sample (F(dbh), F(height), or F(dbh, height)); y,. = the predicted value of the cumulative basal area distribution under the assumed theoretical distribution; y = the mean value of the cumulative basal area distribution of the sample; and n - the number of observations. If the theoretical distribution is taken to be a uniform distribution, then the regression form would necessarily go through the origin. This changes the formula for R to: 38 total uncorrected sum of squares residual sum of squares = 1- SL,(yj-y,-)2 i ; = , ( ^ , ) 2 [4.25] Although, [4.25] is algebraically correct, it no longer has the same interpretation as [4.24] (Kozak and Kozak 1995). However, this R 2 could be used as a measure of diversity, as it describes the amount of dispersion between some theoretical distribution of maximal though it may fall below zero in cases where the shape of the empirical distribution is much steeper than that of the thoretical. Since this is a variation on the R value, this measure will be known as STRI. 4.2.3.1 STRI Univariate Method For the univariate case, dbh and height are regressed separately by species. The cumulative empirical basal area distributions of dbh and height for species k are labelled as: F, (dbh)), and F2 (height)k. If it is assumed that a uniform distribution represents the highest possible diversity in a community, then the theoretical predicted values of the cumulative distribution functions of dbh and height for any species are represented by: diversity and a sample. R 2 gives a numerical value between zero and one in most cases, a4 < dbh < bA [4.26] F2 (height) = height — a , a5 < height < b, [4.27] where: a4 and as are the minimum values of dbh and height, respectively; and b4 and bs are the maximum values of dbh and height, respectively. Then, S T R L j b h and STRIheight for species k are: 39 S T R I - i a f a ^ ^ - ^ ^ J R 4 ? 8 1 S T R I d b h t - l [4.28] Z , - = i ( * i ( ^ , - ) i t ) i ( F 2 ( ^ g f a j ) t ~ ^2(heightj)f S T R I H E I G H T T -1 v"2 /j? ,u • , V [ 4 - 2 9 ] L^Wiiheight^) where: dbhj corresponds to the ith value of dbh; heightj corresponds to they'th value of height; ni is the number of observations with measured dbh; and n2 is the number of observations with measured height. To arrive at a measure of structural diversity for species k, STRLjbh^ and STRIheight^ are averaged to produce STRI ( d + h ) t : S T R I j k h "t" STRI h- i ( > h t STRI ( d + h )^ = A ^ - 2 ^ [4.30] An overall measure of diversity for a sample plot will beSTRI d + h , the sum of the values of all STRI ( d + h ) j over a plot: S T R I d + h = l t i S T R I ( d + h ) t [4-31] The maximum value of the index is S, the number of species. This maximum would occur if both of the cumulative distributions for dbh and height match that of the uniform for each species. The minimum value of this index is negative, and would occur if dbh or height were a single value at either the minimum or maximum value of the possible range. 40 4.2.3.2 STRI Bivariate Method For the bivariate case, dbh and height are considered simultaneously by species. The cumulative empirical sample distribution of dbh and height for species k is labelled as: G(dbh, height)k. If the assumption is made that a bivariate uniform distribution represents the highest possible diversity in a community, then the theoretical predicted cumulative basal area distribution function of dbh and height for species k is: height - a7 G(dbh, height) k dbh - a6 b 6 ~ a 6 x a6 < dbh < b6 [4.32] a7 < height < b7 b7 -a7 where: and 0,7 are the minimum possible values of dbh and height, respectively; and bf, and by are the maximum possible values of dbh and height, respectively. Then, STRIdh for species k is: Q T D T 1 (G^dbhJ • j )k - G(dbht, height j )f O I K I J U — 1 7 r - 4.33 llUllMdbh^height;),)2 1 where: (dbhi, height/) corresponds to the ith value of dbh andy'th value of height. An overall measure of diversity for a sample plot is STRIdh, the sum of the values of all STRLjh^ over a plot: S T R I d h = X L S T R I d h t [4.34] The maximum value of STRI^ is S. The maximum would occur if the distribution of the sample was a bivariate uniform for each species in the sample. The minimum value of this index is negative, and would occur in cases where the surface of the empirical joint distribution is much steeper than that of the bivariate uniform distribution. 41 4.2.3.3 Properties of STRI Both the univariate and bivariate STRI possess the characteristics 1, 2, 3, 5 and 6 listed in Section 4.1. An index derived from the R 2 would measure richness and evenness using basal area and commonly measured tree attributes. The index would not rely on combining data into classes, and would account for vertical and horizontal diversity equally. Item 4 will not be met, as R is dependent on sample size. If n is equal to the number of coefficients in the regression equation, then R 2=l; as n gets large (greater than 30), R 2 decreases, regardless of the fit of the data (Neter et al. 1996). Items 7 and 8 will only partially be met. While the index would be correlated to the range of values in dbh and/or height and would adequately describe bimodal distributions, the index will be sensitive to tree size. For example, consider a forest type where the maximum range of dbh values is 0 to 100 cm, and where there are three stands which have three univariate uniform dbh distributions, covering three equal portions of this 100 cm dbh range: 0.0 to 33.3 cm, 33.3 to 66.7 cm, and 66.7 to 100 cm. If the three distributions have the same proportions of basal area equally spaced over their respective ranges, the denominators of each of the three STRI's derived from the distributions will be equal. The numerator of the univariate STRI is dependent on the area between the empirical and theoretical cdf's; it is the squared distance between the two curves. Graphically, this is represented by the shaded areas in Figure 4.1. 42 Figure 4.1. STRI and three equal dbh ranges. 0 10 2 0 3 0 4 0 5 0 6 0 7 0 80 90 100 EUi(on) The grey region and the blue region (corresponding to the small stand and the large stand) have equal areas. However, it can be proven mathematically that the yellow region (corresponding to the mid-range stand) has a smaller area than that of the large and small stands. Thus, the STRI associated with a uniform distribution centred in the middle of the theoretical range will be higher than that of uniform distributions centred either at the low end or high end of the range. 4.2.4 Structural Index based on Variance Comparisons (STVI) The distribution of diameters and/or heights in a structurally diverse forest may be described in terms of variances. The uniform distribution yields a very high variance, though not the highest possible variance for any distribution over a given range. The diversity of a stand may be evaluated on the measured stand variance relative to that of a uniform distribution. 43 4.2.4.1 STVI Univariate Method The variance of an empirical dbh or height distribution can be given by: S2 = ^ ' - l L ' — [4.35] S,-=iw« where: xt = dbhi or heightf, x is the mean of dbh or height; Wj is the weight of the ith cell, defined as proportion of basal area of the plot represented by the ith dbh or height point; and n is the number of dbh or height points. This estimate of the variance differs from that of the usual estimator used in forestry, which is weighted by trees per hectare. This weighting is consistent with the other proposed measures. The variances of univariate uniform distributions of dbh and height weighted by basal area are given by: ^ = ^ f ^ [4.36] where: a8 and ag are the minimum values of dbh and height, respectively; and bs and b9 are the maximum values of dbh and height, respectively. The maximum possible variance of a distribution occurs when the distribution is maximally bimodal. For the univariate case with dbh alone, this occurs when half of the basal area is at as and half the basal area is at bs. In this situation, the variance estimator weighted by basal area is given by: 44 'dbhm — X 2 +—x 2 [4.38] Similar results can be obtained for height by substituting coefficients a9 and b9 for as and respectively. Assuming that a uniform distribution represents a maximally diverse population, the difference between the variance of the dbh distribution from an extremely diverse population (Sdbhaiv) a n d t n a t ° f t n e uniform distribution (Sdhhu ) will be close to zero. However, the difference between the variance of a low diversity dbh distribution (Sdhh] ) and that of a uniform is not as predictable. If the distribution consists of trees that are all of the same dbh sdbhkm w i l 1 b e c lose to zero and Sdhh]m -Sdbhu will be large and negative (at most, -Sdhhv ). If the distribution is maximally bimodal, then Sdhh^ will be very large and Sdhhi -Sdhhv will be large and positive (at most, Sdhhm^ - Sdhhu ). In order to use these results to yield an index that measures the difference between the variance of a sample plot and that of the maximally diverse case, both instances must be considered. The following formula can be used to define a diversity index, STVLjbh for a species k: STVI d b h t = < 1-f ^2 _ n Jdbh,, °dbk 2 y 'k V 'dbh. ,when5 2 2 [4.39] where: Sdhh - is the variance of dbh for species k; 45 pi and p2 are constants > 0; and m is a constant > 1.0 . The constants pi and p2, define the shape of the curve relating the value of the index to the sample variance: when pt (or p2)< 1, the curve is concave upward; when pt (or pi) = 1, the curve is segmented linear; when pt (or p2) > 1, the curve is concave downward (Figure 4.2). If pi =p2 > 1, then a smooth, continuous function results. Figure 4.2. Comparison of STVI values with m= \ and varying values of pi and p2. 1.000 "j 7 1 - ^ 1 1 The coefficient m controls the value of the index when the distribution is maximally bimodal (i.e., equation [4.38], for dbh). If m= I, then the index will be zero for a maximally bimodal distribution; as m gets larger, the index value increases for the maximally bimodal case (Figure 4.3). 46 Figure 4.3. Comparison of STVI values with p1-p2=2 and varying values of ra. 1.000 0.000 -m=l •m=l . l •m=1.2 • uniform • maximum Variance The values for pi, p2, and ra may be chosen by placing three constraints on the index to yield certain index values under defined conditions. First, when a stand is uniform over half the maximum possible range, the value of the index was constrained to equal 0.5. Second, when a stand is distributed bimodally, such that half of its values are uniformly distributed over the lower quartile of the maximum possible range and half of its values are uniformly distributed over the upper quartile of the maximum possible range, the index was constrained to equal 0.5. Third, when a stand is distributed such that its variance is at the maximum value, the index was constrained to equal 0.1. Algebraically, these constraints are: 0.5 = STVI d b h i =1-( K1 - ?2 °dbhu °dbhil}u 'dbh,. [4.40] 0.5 = STVI d b h t = 1-f s2 -s2 ^dbtl^ ^dbhB m X S d b h m M ~Sdbhu \P2 [4.41] 47 where: Sdhhn is the variance associated with the half-range uniform distribution described in the first constraint; and Sdhh is the variance associated with the bimodal distribution described in the second constraint. After algebraic manipulations, [4.40], [4.41], and [4.42] simplify to: ln(l/2) „ A n n A " - l ^ 3 2 - 4 0 9 4 [ 4 ; 4 3 ] „2= 1 ^ = 0.5993 [4.44] ln(3/8) and m — — 3 1 + e x p f ln(0.9)xln(0.75)-(ln(2))2 ^ V ln(9/5) 1.1281 [4.45] With these choices of/?/, p2, and m, plots with variances close to that of the uniform would have index values close to one. Plots that are of low diversity and unimodal (e.g., the variance is close to zero) would have index values close to zero. Plots that are of low diversity and bimodal (e.g., variance is close to the maximally bimodal case) would have index values close to 0.1. 48 Similarly, a diversity index, STVIheight, based on height could be developed. To arrive at a measure of structural diversity for species k, S T V L j b h ^ and S T V I h e i g h ^ are then averaged to produce S T V I ( d + h ) t : , STVI d b h +STVI h e i e h t S T V I ( d + h ) t = [4.46] An overall measure of diversity for a sample plot is labelled as S T V I d + h , the sum of the values of all S T V I ( d + h ) t over a plot: S T V I d + h = 2:L S T V I ( d + h ) , [4-47] The maximum value of STVI^+h is S. The maximum occurs when the univariate distributions of dbh and height are uniform for each species in the plot. The minimum value is zero, and results in cases where each of dbh and height only exist in a single value for each species represented in the plot. 4.2.4.2 STVI Bivariate Method The variance of an empirical bivariate distribution is usually described with the generalized variance (Johnson and Wichern 1998), which is equal to the determinant of the variance-covariance matrix: ' dbh, height = det 'dbh cov(dbh, height) cov(dbh, height) ' height [4.48] where: Sdhh and Sleight, are given by equation [4.36]; and cov(dbh, height) = • S"=i lwi x (dbfy - dbh) x (height, - height) Ti=\wi [4.49] 49 where: wt is the weight of the ith cell, defined as proportion of basal area of the plot represented by the ith (dbh, height) pair; and n is the number of (dbh, height) pairs. The variance of the bivariate uniform distribution of dbh and height is given by the determinant of its variance-covariance matrix: det 'dbh,. cov(dbh, height) v cov(dbh, height) v heightu [4.50] where: Sdhh{j and S%eightu are given in [4.36] and [4.37]; and cov(dbh, height) v - 0. Thus, [4.50] simplifies to: (b8-a^)2 det 12 0 0 (b9 -a9)2 12 (h - Q 8 > 2 > < ( f e 9 ~ a 9 ) 2 144 [4.51] As in the univariate case, the maximum possible variance occurs when the population is maximally bimodal. This occurs when one quarter of the basal area is, at (as, b$), one quarter is at (as, bg), one quarter is at (a% b8), and one quarter is at (ag, b9). In this situation, the variance of dbh is as in [4.37] and the variance of height can be computed similarly. The covariance of dbh and height is identical to that of the uniform: cov(dbh, height)^ = 0 [4.52] The maximum variance of a joint distribution of dbh and height is given by the determinant of its variance-covariance matrix: 50 SL, = det Q>%~a*Y 4 0 0 (b9-a9)7 _ (fr 8 -a 8 ) 2 x(fr 9 -a 9 Y 16 [4.53] The formula given as [4.39] for STVIdbh can now be modified to use the generalized variance to arrive at an index of diversity for a species k using a bivariate distribution: STVI d , = \ 1-f ° U ^k s2 - s2 , when S; < 5 0 \P2 [4.54] , when S2k > S2, where: S% is the generalized variance for species k; pi and p2 are constants > 0; and m is a constant > 1.0 . As in the univariate case, the powers, pi and p2, define the shape of the curve, and the coefficient m controls the value of the index when the distribution has the maximum variance. Imposing the same constraints on the index as in the univariate case, the same values of pi, p2, and m would be used in the bivariate case. Using these values of pi, p2, and m, sample plots with variances close to that of the uniform would have index values close to one. Sample plots that are of low diversity would have index values close to zero if they were unimodal or 0.1 if they are bimodal. An overall measure of diversity for a sample plot is labelled as STVLjh, the sum of the values of all STVI d h ; over a plot: S T V I d h = l f = 1 S T V I d h A [4.55] 51 The maximum value of STVLjt, is S. The maximum occurs when the joint distribution of dbh and height is uniform for each species in the sample. The minimum value is zero, and results when there are data at only one dbh - height combination for each species represented in the sample. 4.2.4.3 Properties of STVI The STVI in both the univariate and bivariate forms possesses the characteristics presented as items 1, 2, 3, 5, 6, and 7 (Section 4.1). Both forms of the STVI would measure richness and evenness using basal area and commonly measured tree attributes. The indices would not rely on combining data into classes, and would account for vertical and horizontal diversity equally. In order to satisfy criteria 4, the indices should not depend on sample size; however, this criteria would need to be evaluated using data. Since the indices are based on the variances of the dbh and height distributions, weighted by basal area, they should be insensitive to nominal tree size with respect to the distribution of basal area . However, since basal area is used as a weight, larger trees are given more weight. The indices should also be j correlated to range, as the variance of distributions with wider ranges tends to be larger. Furthermore, providing that pi,p2, and m are well chosen, the index value for a bimodal distribution with two distinct ranges should be the same as that of a unimodal distribution with a continuous range equal to the sum of the bimodal ranges. Thus, the desirable characteristics listed as items 8a, 8b, and 8c (Section 4.1) should be met; however, further evaluation is necessary. 52 The STVI may be the most flexible of the proposed indices presented, as the shape of the curve and its response to bimodal distributions may be modified to fit the users' expectations. Furthermore, this index does not require fitting a distribution to the data, making its application more simple. However, a disadvantage of these indices are their lack of theoretical background. Existing indices, such as Shannon's Index, have been well studied and have well-defined sampling distributions. The sampling distributions of these proposed indices will not be known unless further research is conducted. 53 Chapter 5 Evaluation of Alternative Measures of Structural Diversity Evaluations of the alternative measures with respect to item 8 (Section 4.1) were performed using simulated stands and using data collected during the 1995 Malcolm Knapp Research Forest (MKRF) inventory. Methods for generating the simulated data and collecting the MKRF data are presented, followed by results in using the proposed indices. 5.1 Simulated Tree Data Proposed measures were evaluated using simulated stands of varying structural diversity. Simulated stands are an invaluable tool in testing indices in that the shape of the distribution of dbh and height can be controlled, allowing for the evaluation of well-defined cases, which may be difficult to find in natural data sets. 5.1.1 Methods A baseline of maximally diverse structure was first chosen, then simulated stands were compared to this baseline. 5.1.1.1 Baseline Maximal Diversity High diversity stands were defined as those stands that have a large number of tree sizes and species present, and maximum diversity would occur when the maximum number of tree sizes and species are present. Mathematically, this is described with a uniform distribution: fW = , * • ,min(X): II 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 0.25 0.2 0.15 S 0-' 0.05 0 CQ Species: Douglas-fir 3$ - — D B H class Species: Douglas-fir 0.3 0.25 0.2 0.15 0.1 c I 0-05 a. 0 CQ 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Western Hemlock S °- 5 " CL g 0.4 -< 13 0.3 -2 0.2 -a o.i -0 -• T i - R A! 5£ •pi 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Red Cedar 0.45 -X 0.4 -H 0-35 -| 0.3 -< 0.25 -| 0.2 -? 0.15 -I a i 1 | 0.05 -0 -i D B H class Species: Red Cedar 0.5 £ 0.45 h 0.4 « 0.35 « 0.25 I 0.2 S 0.15 § 0.1 0.05 0 ?€' C 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Dougias-fir Z 0.6 -| 0.5-< 0.4 -1 0.3 -CQ I 0 2 ' y o.i -0 -Species: Douglas-fir 0.7 X 0.6 « ° ' 5 < 0.4 I 0.3 CQ I o.i 0 2.5 7.5 12.5 17.5 : 2.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Western Hemlock 0.35 -? 0.3 -« 0-25 -< 0.2 -1 0.15 -CQ s 01 -I 0.05 -0 -1 it Am Species: Western Hemlock 0.4 X 0.35 8. 0.3 a 0.25 •5 0.2 1 0.15 I °-' cS 0.05 0 i t - - - r " 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 103 DBH Distribution Plot: Low-Lrg Height Distribution Species: All = 0.5 &. g °-4 < •3 0.3 = 0.2 £ 0.1 Q. (I s . — ^ r - T - n D B H class Species: All 0.5 = 0.4 J 035 | 0.3 •3 0-25 | 0.2 I 0 1 5 S o.i °" 0.05 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Red Cedar 0.5 X 0.45 S 0.4 « 0.35 o °-25 | 0.2 ~ 0.15 3 0.1 £ 0.05 0 : = : h = : «v r r n — - • Species: Red Cedar 0.35 X 0.3 « 0 2 5 I 0.2 I 0.15 CQ S 0.1 S3 0.05 a. 0 '4% it -22.5 27.5 32.5 Height class JL 0.6 -0.5 • | 0.4 -< 8 0 3 " • 0.2 -a o.i -0 -Species: Douglas-fir Species: Douglas-fir = 0.3 £ 0.25 < 0.2 1 0.15 CQ S °'' S 0.05 0 13 l i 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class <* 1-2 " X 15 1 -c K 0-8 " < •3 0.6 -CQ 0.4 -g 0.2 -°" 0 -Species: Western Hemlock 1.2 1 ' 0.8 0.6 0.4 0.2 0 Species: Western Hemlock 2.5 7.5 12.5 17.5 : !.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 104 V Plot: Low-Med DBH Distribution Height Distribution 0.3 • * 0.25 f -3 °- 2 • < •3 0.15 -I 0.1-S 0.05 • Species: All o m-[,.i">.. •» 0.45 cd 0.4 1 0.35 It 0.3 < 0.25 1 0.2 aa s 0.15 | 0.1 £ 0.05 0 Species: All II 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Red Cedar S 0.25 c | 0.2 < •3 0.15 £ o.i § 0.05 0 M h Species: Red Cedar 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 0.7 -= 0.6 4 - • I 0-5 -< 0.4 -1 0.3 -co S ( U " I o.i -°~ 0 -Species: Douglas-fir Species: Douglas-fir 0.35 = 0.3 « O-25 K < 0.2 1 0.15 CQ y o.os a. 0 ts --2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Western Hemlock 0.7 = 0.6 | « < 0.4 1 0.3 CQ S 0.2 u S °-' a. 0 lip A' kj> oS> N * N * Nfe*> s * Species: Western Hemlock 0.7 K 0.6 It °'5 < 0.4 1 0.3 CO 1 ° ' 2 S3 0.1 a. 0 *>* > 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 105 Plot: Low-Sml DBH Distribution Species: All 0.6 0.5 | 0.1 0 1 D B H class Height Distribution Species: All 0.45 0 0.4 | 0.35 s °-3 < 0.25 1 0.2 CQ g 0.15 I 0.1 £ 0.05 0 _ * ft y 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5'52.5 57.5 62.5 67.5 Height class 0.6 X 0.5 CH 0.4 0.3 CQ 0.2 s 0.1 0 Species: Red Cedar p 4> DBH class t 0.5 8. S 0.4 | 0.3 2 0.2 c a. 0 Species: Red Cedar if,1 t " S( 7.5 12.5 Height class Species: Red Alder •a 0.6 4- -I 3 1 1 ? 0.4 a 0.2 • ct 0 J DBH class 0.7 * 0.6 0 0-5 < 0.4 1 0.3 CQ 1 °-2 | 0.! 0 Species: Red Alder A' S 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class a L 2 " X 6 1 " c B 0.8 -1 M " ™ 0.4 -c S 0.2 -Species: Paper Birch * 4> DBH class 8. |3 0.8 | 0.6 ™ 0.4 I 0.2 CL 0 Species: Paper Birch 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Heighi class 106 X 0.6 s. 0.5 < 0.4 1 0.3 m 0.2 (3 0.1 0 Spccics: Douglas-fir AY S T L 4» ^ DBH class Species: Douglas-fir 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Western Hemlock = 0.6-8. g 0-5 -< 0.4 • 1 0.3 • cc S 0.2 -| a i i o -DBH class Species: Western Hemlock 0.9 X O.X S. 0.7 g 0.6 < 0.5 1 0.4 2 0.3 § 0.2 £ 0.1 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class s. I 0.8 -a 0.6 -0.4 i S 0.2 -0 -CQ Species: Vine Maple D B H class 8 M ll,.J i0-6 ™ 0.4 S 0.2 a. 0 Species: Vine Maple 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 107 DBH Distribution Plot: Med-Norm Height Distribution Species: All 0.35 E 0.3 a 0.25 < 0.2 1 0.15 ca S 0.1 1 0.05 CL 0 Species: All t 0.25 4- -8. g 0.2 -< 1 0''5 " 2 o.i 4- -g 0.05 -a. 0 -2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Red Cedar 0.6 -8 05 +• -g 0.4 -< 1 "'3 " 2 0.2 -| 0.1 4- - - -* scJ> DBH class 0 Species: Red Cedar 0.4 I 0.35 | 0.3 4- -_=Ji £ 0.25 4- • 1 0.2 • I 0.15 -I 0.1 f -1 0.05 • 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Douglas-fir g 0.2 4--< •a 0.15 -3 ? 0.1 • S 0.05 -0 -4> stf DBH class Species: Douglas-fir 0.45 X 0.4 4-K. 0.35 g 0.3 < 0.25 1 0.2 4-0.15 g 0.1 £ 0.05 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Western Hemlock a 0.4 -? 0.35 4- -i l ° ' 3 " | 0.25 f -•a 0.2 -1 0.15 -1 0.1 -| 0.05 -0 -DBH class Species: Western Hemlock 0.3 * 0.25 g. S 0.2 < t 0.1 c I 0.05 a. 0 --ft 3 -M l tt e IKS i f vm 8 Wits m 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 108 DBH Distribution Plot: Low-Norm Height Distribution Species: Al l 0.45 • = 0.4 f -8. 0.35 J g 0 .3-< 0.25 -1 0.2-2 0.15 -5 0.1 -£ 0.05 -0 -4> 4> D B H class g 0.4 | 0.3 t 0.2 c 1 0.1 a. 0 Species: A l l 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Amabalis Fir 0.7 -0.6 -' 0.5 -0.4 -0.3 -0.2 -0.1 -oJ ft DBH class 0.45 £ 0.4 8. 0.35 2 0.3 < 0.25 1 0.2 2 0.15 c g 0.1 I 0.05 0 Species: Amabalis Fir • -2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Douglas-ftr Z 0.6 8 S 0 5 < 0.4 1 0.3 CQ S 0.2 I 0.1 CL 0 [5*1 ' l' Species: Douglas-ftr 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Western Hemlock 0.6 0.4 0.2 0 D B H class Species: Western Hemlock 0.8 3 0.7 8 0.6 | °'5 1 0.4 co 0.3 I 0.2 | 0.1 0 -.8 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class 109 Plot: Med-Bim DBH Distribution Height Distribution Species: A l l « 0 2 5 " x 8. 0.2-< 0.15 -1 0 .1 -| 0.05 -CL 0 -I'M Species: Al l t 0.25 8. g 0.2 < I 0.15 ? 0.1 B § 0.05 a. 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class Species: Red Cedar t 0.3 j 8. < 0.2 J a 0.15 -CO I 0.05 -0 J . DBH class Species: Red Cedar 0.45 X 0.4 8. 0.35 g 0.3 < 0.25 1 0.2 * 0.15 I 0.1 I 0.05 0 I l l p 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Height class n 0.3 -X S 0.25 -c g 0.2 4 < 0.15 -0.1 -S 0.05 -o -a Species: Douglas-fir DBH class Species: Douglas-fir t 0.5 8. g 0.4 | 0.3 " 0.2 I 0.1 a. 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Heighi class 0.6 -in-g 0.4 4- -< •3 0.3 -a m 0.2 -m Species: Western Hemlock **- V & DBH class Species: Western Hemlock 0.6 ! "-5 CL g 0.4 < | 0.3 * 0.2 C g 0.1 CU 0 2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5 52.5 57.5 62.5 67.5 Heighi class no ~~