Indices of stand structural diversity:
adding spatial diversity to a stand structural index
Valerie LeMay1 and Christina Staudhammer2
1
Associate Professor, Department of Forest Resources Management,
University of British Columbia, 2045-2424 Main Mall, Vancouver, BC, Canada
Valerie.LeMay@ubc.ca
2
Assistant Professor, University of Florida, Gainesville, Florida, USA
staudham@ifas.ufl.edu
Extended Abstract
Stand structure is an important element of stand biodiversity (e.g., MacArthur & MacArthur,
1961; Aber, 1979; Freemark & Merriam, 1986). High biodiversity is associated with stands
where there are multiple tree species and sizes (Buongiorno et al., 1994). Managing forests
for biodiversity may be accomplished by managing for structural diversity (Önal, 1997).
Measures of stand structural diversity are also important for predicting future stand growth
and development (Pretzsch, 1997). Several 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, 2002). For this paper, the definition of most
heterogeneous (most structurally diverse) stand is proposed as a uniform distribution over
size and species, and a Poisson distribution of trees in clumps of varying sizes for the spatial
distribution. Extensions of the Stand Variance Index (STVI) developed by Staudhammer and
LeMay (2001) for spatial diversity were examined.
STVI was based on the variance of basal area per ha for diameter (dbh; diameter outside
bark at 1.3 m above ground) and height relative to the variance for a maximally diverse
stand with a uniform distribution of dbh and height. The developed diversity index, as shown
for diameter (STVIdbhi for a species i), was:
p1
2
⎧ ⎛ S2
⎞
dbh U − S dbh i
2
2
⎟ , when S dbh
⎪1 − ⎜
≤ S dbh
2
i
U
⎜
⎟
⎪⎪ ⎝
S dbhU
⎠
STVI dbh i = ⎨
p2
⎞
⎪ ⎛
S 2dbh i − S 2dbh U
⎜
⎟
, when S 2dbh i ⟩ S 2dbh U
⎪1 − ⎜
2
2
⎟
×
−
m
S
S
⎪⎩ ⎝
dbh max
dbh U ⎠
2
where S dbhi is the variance of dbh for species i; p1 and p2 are constants > 0 that define the
shape of the curve; and m is a constant ≥ 1.0 that controls the value of the index when the
distribution is maximally bimodal. Sample variance was calculated by:
∑ [w × ( x
n
S =
2
i =1
i
− x)2
i
]
n
∑w
i =1
i
where xi is dbhi or heighti; wi is the basal area per ha represented by the ith tree in the
sample plot; x is the average of dbh or height, weighted by basal area; and n is the number
of trees in a sample plot.
1
The variance of a uniform distribution is given by:
(b − a )2
SU2 =
12
where a and b define the range of the distribution, differing for dbh versus height. The
maximum possible variance of a distribution occurs when the distribution is maximally
bimodal, when half the basal area is at a and half the basal area is at b. For this basal area
distribution, the variance is:
2
⎡ 1 ⎛ (a + b )
(a + b ) ⎞ 2 ⎤ = (b − a )2
⎞ 1 ⎛
2
= ⎢ ×⎜
− a ⎟ + × ⎜b −
S max
⎟ ⎥
2 ⎠ ⎥⎦
4
⎠ 2 ⎝
⎢⎣ 2 ⎝ 2
The values for p1, p2, 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: 1) the variance is equal to that of a uniform distribution over half the maximum
possible range ( S 2 0.5U ); and 2) the variance is equal to that of a bimodal distribution, with
half of the values uniformly distributed over the lower quartile, and the other half uniformly
distributed over the upper quartile of the maximum possible range ( S 2 max ). The index was
also constrained to equal 0.1 for the maximum variance (maximally bimodal stand). Since
the variances used in defining these constraints are all functions of the variance of a uniform
distribution, p1≈2.4094, p2≈0.5993, and m≈1.1281, for any defined size ranges from a to b
(see Appendix of Staudhammer & LeMay, 2001 for details). To arrive at a measure of
structural diversity for species i, STVIdbhi and STVIhti were averaged, with a maximum value
of 1 (uniform for both diameter and for height, equally weighted). These were summed over
all species in the plot (STVId+h), for a maximum value equal to the number of species.
The separation by diameter and height distributions by species allowed for a more complete
view of the stand structure, obtaining an overall index, indices by species, and indices for
dbh 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 STVI to include spatial diversity, a separate component of the index for distance
could be developed. The main reasons for developing and using this STVIdist rather than a
well-known spatial index are: 1) the standard for the index would be maximally diverse
spacing, rather than Poisson spacing; 2) each component (dbh, height, distance) could be
reported separately; and 3) the index could be easily combined with the separate dbh 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 STVI for distance, 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 distribution of
different sized clumps, resulting from microsite variability, inter-tree competition over
different species and tree sizes, and/or disturbance events. This basis is more subjective
than the uniform distribution over a wide range of tree sizes used to develop the STVI index
for dbh and height. The number of clumps and distribution of clump sizes (number of trees
and space occupied by the trees in the clump) would need to be specified. Another issue is
what distance metric to use. As noted by Dale (1999), the use of first nearest neighbours
does not differentiate a clumped pattern from an even distribution of regularly sized clumps;
moreover, information about the distribution is lost (Cressie, 1993). Alternatives include
using another neighbour that might better indicate primary neighbours, using all possible
tree-to-tree distances, or using tessellated distances similar to Zenner and Hibbs (2000).
Another alternative, following the development of STVI for dbh and for height, would be to
calculate STVI for the x-direction and for the y-direction separately. The expected
2
distributions in each direction may be more predictable. For example, given regular (square)
spacing distribution, the frequencies will all be equal and spread widely over the ranges, but
there will be discontinuities in distances, relating to regular gaps in x- and in y-directions.
Conversely, for a Poisson distribution 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 distance metric to use are made,
the STVI for distance over all species would be:
where S 2 dist
p1
2
⎧ ⎛ S2
⎞
dist MD − S dist
2
2
⎟ , when S dist
⎪1 − ⎜
≤ S dist
2
MD
⎟
⎜
⎪⎪ ⎝
S dist MD
⎠
STVI dist = ⎨
p2
2
2
⎞
⎪ ⎛
− S dist
S dist
2
MD
⎟ , when S dist
⟩ S 2dist MD
⎪1 − ⎜⎜
2
2
⎟
⎪⎩ ⎝ m × S dist 2−C − S dist MD ⎠
is the target stand variance of distances for all species; S 2 dist MD is the variance
for a most spatially diverse stand; S 2 dist 2 − C is the variance of a distribution of two maximally
separated equal-sized clumps, with Poisson spacing within clumps. Then, using a similar
concept as for tree size variables, values for p1, p2, 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 1 (high diversity), and using similar additional
constraints, STVIdist should equal:
1. Nearly 0 for very regular (e.g. square) spacing of trees, where the number of clumps is
equal to the number of trees;
2. A bit larger than 0 for a Poisson distribution, where the number of clumps is less than or
equal to the number of trees (e.g., 0.2);
3. Nearly 1 for a very highly diverse stand, possibly a Poisson distribution of different sized
clumps (“maximally diverse”);
4. A value of 0.5 for a diverse stand with a Poisson distribution of different sized clumps,
but half the diversity of maximal diversity; and
5. A value a bit larger than that of a Poisson distribution 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:
⎛ S 2dist MD − S 2Poisson ⎞
⎟
0.2 = STVI dist = 1 − ⎜
2
⎟
⎜
S
dist MD
⎠
⎝
p1
⎛ S 2dist 0.5 MD − S 2dist MD
0.5 = STVI dist = 1 − ⎜
⎜ m × S 2dist − S 2dist
2−C
MD
⎝
⎞
⎟
⎟
⎠
⎛ S 2dist 2−C − S 2dist MD
0.3 = STVI dist = 1 − ⎜
⎜ m × S 2dist − S 2dist
2 −C
MD
⎝
⎞
⎟
⎟
⎠
p2
p2
where S 2 Poisson is the variance of a Poisson distribution of trees; S 2 dist0.5 MD is the variance of a
distribution with ½ the diversity of a most spatially diverse stand; values for p1, p2, and m
could be calculated given expected values of the variances. Since expected values for the
variances are not known for these distributions, regardless of which distance metric is
selected, Monte Carlo simulations would be needed to obtain these variances as with other
spatial indices (Dale, 1999). This index could be calculated for any selected distance
3
measurement, for x- and y-directions, separately, and pooled across or separated by
species.
If the STVI were calculated by species, then STVI components for diameter, height, and
distance for each species i, (STVIdbhi, STVIhti, and STVIdisti) could be averaged (would be
four values if x- and y-distance variances were separately calculated). This would result in a
maximum value of 1 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 dbh), and a
diverse distribution of trees in clumps for the species. Combining these indices over a
number of species, the maximum value for the combined index, STVId+h+dist would have a
maximum value of S, the number of species in the stand. We applied the spatial STVI index
to a number simulated stands and to uneven-aged stand data (LeMay & Staudhammer,
2005). The extension of STVI 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 Science and Engineering Council (NSERC) of Canada provided funding.
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