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Western Mensurationists 2013 Leavenworth, WA, USA

Where’s Waldo? Stand structure indices and maximum density relationships.Ian Moss, PhD, RPF Tesera Systems Inc.Valerie LeMay, PhD, RPF University of British

Columbia

“It may be that Reineke’s index is not a good measure of density, but it is still the best we have. At least it does not lead to confusion between overstocked and understocked stands of the same species. However the relationship between the number of trees and diameter may not be as simple as Reineke believed.”

Boris Zeide 2005

Stand Structure - Definitions Let 𝑟𝐷𝑏ℎ𝑖𝑗 be the relative tree dbh equal to the tree dbh for a given tree,

j, minus the quadratic mean tree diameter, Dg, for the associated plot or stand, i.

𝑝𝑖𝑘𝐺 be the proportion of total basal area per hectare, G, greater

than or equal to a relative diameter diameter threshold, k, equal to 1,2,3 …, l, where l is equal to the maximum relative dbh within a given dataset or region rounded up to the nearest integer. The sum of the differences is taken over equal diameter intervals.

𝑝𝑖𝑘𝑁 be the proportion of total stems per hectare, N, greater than or equal to a diameter threshold k equal to 0,1, 2 …, l.

= tree dbh – Dg (relative dbh; not dbh/Dg)

= proportion of basal area perhectare relative dbh (cumulative distribution)

= proportion of trees per hectare relative dbh

(cumulative distribution)

Objectives • To evaluate relationships between Stand

Structure Indices (SSI) and Reineke’s (1933) SDI, i.e. Stand Structure = f (N, Dg, SSI )whereSSI = f(pG , pN | rDbh) ?

• … and Maximum Density ~ f(Dg, SSI)?

Stand Structure Indices

• Three Indices: – Gini Coefficient (e.g. Bonan 1988; GINI)– Lorenz Area and Lorenz Maximum

(LA or LM analogous to GINI)– Cumulative Distribution Index (CDI)

GINI

𝐺𝐼𝑁𝐼𝑖 = σ σ ห𝑥𝑗 − 𝑥𝑚ห𝑛𝑚=1𝑛𝑗=12𝑛ሺ𝑛− 1ሻ𝑥𝑏𝑎𝑟

Bonan, G. B. 1988. The size structure of theoretical plant populations: spatial patterns and neighborhood effects. Ecology 69:1721-1730.

For a given plot, i, the average absolute difference in size (x = dbh) amongst all pairs of individual trees, j & m, scaled in proportion to the mean (xbar = mean tree dbh).

Sen, A. 1973. On economic inequality. Clarendon, Oxford. Cited in Weiner, L. 1985. Size hierarchies in experimental populations of annual plants. Ecology 66(3):743-752.

Lorenz Area

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0.050.

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0.250.

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0.350.

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0.450.

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0.550.

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0.650.

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0.750.

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0.850.

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0.951

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rea

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Proportion of Trees Per Hectare

Line of absolute equality _-_

Line of perfect uniformity rev. J.

Maximally Bimodal -_-

𝐿𝐴𝑖 = ൫𝑝𝑖𝑗𝐷 − 𝑝𝑖,𝑗−1𝐷 ൯(𝑝𝑖𝑗𝑁− 𝑝𝑖,𝑗−1𝑁 )𝑛𝑗=1

Duduman, G. 2011. A forest management planning tool to create highly diverse uneven-aged stands. Forestry 84(3):301-314.

Lorenz Maximum 𝐿𝑀𝑖 = max ൫𝑝𝑖𝑗𝐺 − 𝑝𝑖𝑗𝑁൯

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0.250.

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0.350.

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0.450.

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0.550.

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0.650.

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0.750.

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0.850.

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0.951

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ortio

n of

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al A

rea

Per H

ecta

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Proportion of Trees Per Hectare

CDI 𝐶𝐷𝐼𝑖 = ൫𝑝𝑖𝑘𝐺 − 𝑝𝑖𝑘𝑁൯𝑙

𝑗=1

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-18

-10 -2 6 14 22 30 38 46 54 62 70 78 86 94 102

110

N1 N2 N3 N4 N5 G1 G2 G3 G4 G5

Numerical integration at standard rDbh intervals (1 cm)

Trees ranked (large to small)

Method of Evaluation: Part I• Plot level distributions: pG and pN vs. rDbh • Fuzzy C-Means classification:

5, 10, and 20 classes.• Find best index to explain differences amongst

classes.• Also classify plot-level Lorenz Curve’s

(pG at fixed pN intervals); compare results.• Classes ranked and renumbered according to

LA (low to high).

The Data

1/10th hectare plots 174 Combined variable and fixed radius plots 247Fdi leading species 233 Pl leading species 184Other (SX, VV) 4

Table. Study area summary statistics for quadratic mean diameter (Dg),

Lorey’s mean height (H), number of stems per ha (N), basal area per ha (G)

and volume per ha (V) (n=421 plots).

Dg

(cm)

H

(m)

N

( stems ha-

1)

G

(m2ha-1)

V

(m3ha-1)

Minimum 0.8 2.2 90 0.1 0

Median 11.4 16.1 2699 31.6 115.2

Maximum 32.6 37.6 39642 94.4 726.6

Results I: Indices

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CDI

CDI

Lorenz Area

CDI

GINI

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CDI

Lore

nz A

rea

Lorenz Area

Lore

nz A

rea

GINI

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GIN

I

0 0.2 0.4 0.6 0.8 1GI

NI

CDI

LA

GINI

CDI LA GINI

Results: Indices Interpretations

Coefficient of Variation

Bendel, R.B., Higgins, S.S., Teberg, J.E., and Pyke, D.A. 1989. Comparison of skewness coefficient, coefficient of variation, and GINI coefficient as inequality measures within populations. Oecologia 78:394-400.

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100

0 20 40 60 80 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1

CDI LA GINI

Results I: Classification - RDBH

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Stand Structure Class (RDBH) Title Title

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Stand Structure Class (RDBH) Title Title

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1 2 3 4 5 1 2 3 4 5 6 7 8 9 10 1 3 5 7 9 11 13 15 17 19

CDI

5

LA

GINI

10

20

All Species Stand Structure Class (Lorenz Curve)

Results I: Lorenz Classification

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CDI

5

LA

GINI

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All Species Stand Structure Class (Lorenz Curve)

Method of Evaluation: Part II• Use Frontier regression (frontier in R) sfa routine

to evaluate maximum density as a function of Dg, structural index (SSI), and Dg*SSI with, and without species differentiation.

Coelli, T. and Henningsen, A. 2012. Stochastic Frontier Analysis. R package ‘frontier’. Version 0.997-14. http://frontier.r-forge.r-project.org/ [accessed May 27 2013] .

Maximum Density = f(CDI)

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0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

LN(N

)

LN(Dg)

0 1 2 3 4 0 1 2 3 4+LN(CDI)

LN(N) = 10.9920 -1.0829 LN(Dg) + 0.5424*LN(CDI) - 0.1835 LN(Dg) LN(CDI) +

Structure Leading Species

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FD PL

CDI

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FD PLLo

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FD PL

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Leading Species

Maximum Density = f(Species)

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0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2

LN(N

)

LN(Dg)

FD PL FD PL

FD: LN(N) = 12.7406 -1.6646 LN(Dg) + PL: LN(N) = 11.4554 -1.2881 LN(Dg) +

Conclusion I1. The Cumulative Distribution Index (CDI)

• Fits within the concept of SDI (N, Dg) as residual information describing the degree of difference amongst diameter distributions.

• Is a reliable index of differences in stand structure.• Provides better differentiation of plots with respect

to leading species (vs. LA and GINI).• Probably includes better accounting for differences

related to the shapes of distributions (i.e., skewness) that is not included in LA, and may come at a cost of being less precise with respect to CV.

Conclusion II

Maximum Density:• Differences related to species can/may be

explained by structural differences.• Differences in species (can) explain differences

in maximum density ; differences in stand structure may (Pl) or may not (Fd) have a demonstrable significant effect.

• The maximum density differences are most pronounced with respect to changes in stand structure in the region of low CDI, i.e. stands with trees distributed within a narrow range of diameters.

Question

When compared with PL, FD has higher maximum density due to:

… greater complexity in stand conditions?… more shade tolerance?… a combination of the above?… nature of dataset?