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Stand Density Management
for Optimal GrowthMicky G. Allen II
12/07/2015
How does stand density affect growth?
Langsaeter’s Hypothesis (1941)Relationship valid for a given tree species, age and site
Source: Daniel, T.W., J.A. Helms and F.S. Baker. 1979. Principles of silviculture. Second Edition, Mc Graw-Hill.
Thinning and Growth: A full turn around
Patterns for periodic annual
total and merchantable volume
increment (Gross volume)
How is stand density defined?
How is merchantable volume
defined?
Source: Zeide, B. 2001. Thinning and growth: A full turnaround. J. For. 99(1):20-25.
Post-thinning relationship between
growth and density
Source: Nyland, R.D. 2002. Silviculture: Concepts and applications. Second Edition, McGraw-Hill. 682p.
Only a graphical representation
Relative density?
Total stem volume or
merchantable volume?
Research Purpose
Determine stand density that optimizes volume increment in loblolly pine
plantations
Use multiple operational studies
Thinning vs. No Thinning
Compare multiple definitions of stand density
Commonly used measures of stand density
Determine which is more correlated with volume increment
Compare multiple definitions of volume increment
Total vs. Merchantable
Gross vs. Net
Data: Non-intensively managed
plantations
Thinning study established 1980-1982 in 8-25 year old plantations
Designed to examine thinning across a range of ages
186 permanent plots established across the natural range of loblolly pine
Treatments:
Unthinned control
Light thin removing 30% of basal area
Heavy thin removing 50% of basal area
Management:
Site preparation
No genetic improvement
No fertilization or weed control
Data: Intensively managed plantations
Thinning study established 1996-2000 in plantations 3-8 years old
Stands thinned at a common point in stand development (45 ft, 13.7 m)
170 permanent plot locations across the natural range of loblolly pine
Treatments:
Unthinned control
Light thin removing 30% basal area
Heavy thin removing 50% basal area
Pruned, removing dead branches only
Management:
Site preparation
Genetically improved seedlings
Fertilization and weed control
Data: Region Wide 19 Thinning Study
Thinning and fertilization study established 2007-2012 in plantations 10-16
years old
8 permanent plot locations across natural range of loblolly pine
Treatments:
Thin to 1235 (500), 741 (300), 494 (200), and 247 (100) stems per hectare (acre)
No fertilization or fertilization with N and P
Four replications of each thin+fert combination at each location
Management:
Genetically improved seedlings
Site preparation
Data: Spacing Study
Trials designed to examine effects of different planting densities established
in 1983
Four locations established in Virginia and North Carolina
3 replications of study design at each location
Base design: Grid with four spacings (4, 6, 8, 12 ft) (1.2, 1.8, 2.4, 3.6 m)
Spacing varies in two-dimensions for a total of 16 plots
Ex: 4x4, 4x6, 4x8, 4x12, etc.
In this work only the initial planting density was considered
4x6 = 6x4 = 1815 stems/acre
9 initial planting densities
2722, 1815, 1361, 1210, 907, 680, 453, and 302 stems/acre
6727, 4484, 3363, 2989, 2242, 1681, 1494, 1121, and 747 stems/hectare
Definitions of Stand Density
Stems per Hectare (SPH)
Basal Area per Hectare (BPH, m2/ha)
Relative Spacing
Diameter: 𝑅𝑆𝐷 = 10,000/𝑆𝑃𝐻 𝑑𝑞, 𝑑𝑞 = quadratic mean diameter (m)
Height: 𝑅𝑆𝐻 = 10,000/𝑆𝑃𝐻 𝐻𝑑, 𝐻𝑑 = dominant height (m)
Stand Density Index*
Diameter: 𝑆𝐷𝐼𝐷 = 𝑆𝑃𝐻 25 𝑑𝑞−1.43089
Height: 𝑆𝐷𝐼𝐻 = 𝑆𝑃𝐻 30 𝐻𝑑−1.42324
Volume: 𝑆𝐷𝐼𝐷 = 𝑆𝑃𝐻 0.65 𝑣 −0.48895, 𝑣 = mean tree volume (ft3)
* Parameter estimates from: Burkhart, H.E. 2013. Comparison of maximum size–density relationships based on
alternate stand attributes for predicting tree numbers and stand growth. Forest Ecology and Management 289(0):404-
408.
Volume Estimation
Total volume (ft3/acre)*
Unthinned: 𝑉𝑢 = 0.21949 + 0.00238𝐷2𝐻, D = diameter(in), H = height (ft)
Thinned: 𝑉𝑡 = 0.25663 + 0.00239𝐷2𝐻
Merchantable volume (ft3/acre)*
Unthinned: 𝑀𝑉𝑢 = 𝑉𝑢𝑒𝑥𝑝 −0.78579 ∗ 𝑑4.9206 𝐷4.55878
Thinned: 𝑀𝑉𝑡 = 𝑉𝑡𝑒𝑥𝑝 −1.04007 ∗ 𝑑5.25569 𝐷4.99639
Volumes converted to metric (m3/ha)
Pulpwood: Trees in the 13 cm DBH class and greater to 7.6 cm top(d)
Sawtimber: Trees in the 20 cm diameter class and greater to 15.2 cm top (d)
* Volume equations from: Tasissa, G., H.E. Burkhart, and R.L. Amateis. 1997. Volume and Taper Equations for Thinned
and Unthinned Loblolly Pine Trees in Cutover, Site-Prepared Plantations. Southern Journal of Applied Forestry
21(3):146-152.
Growth Definition
Net Productivity: Standing volume + Volume removed in thinning
Gross Productivity: Net + Volume of mortality
Productivity = Periodic annual increment (PAI)
𝑃𝐴𝐼 =𝑉𝑜𝑙𝑢𝑚𝑒2−𝑉𝑜𝑙𝑢𝑚𝑒1
𝑇𝑖𝑚𝑒2−𝑇𝑖𝑚𝑒1
Measurement periods ranged from 2 to 3 years
Increasing
Optimum
∆V
Stand Density
What kind of pattern?
1
0
aDensityaV )exp( 2 Densitya
Hypothesis Testing
)exp( 201 DensityaDensityaPAI
a
Test of hypothesis:
Ho : “Optimum pattern” between volume increment and stand density
HA : “Increasing pattern” between volume increment and stand density
or
Ho : all coefficients are significant
HA : all coefficients, except , are significant
Results: Definitions of Growth and
Density
Measures Based on diameter were more correlated with growth
BPH, RSD, SDID
Consistently explained the most variation in volume increment (20-30%)
Net and Gross growth had similar relationships in most cases
Results for Total Volume
Increment
Results: Is there a stand density that
optimizes total volume increment?
Dataset PAIDensity
SPH BPH RSD RSH SDID SDIH SDIV
NIMP Gross
Net 291
NIMP1t Gross 69.5 10.65
Net
IMP Gross 1476 5.77 560 909
Net 1498 5.70 554 896
IMP1t Gross 733 7.72 791 585 711
Net 731 7.76 766 590 693
RW19 Gross 1622 53.8 8.84 7.03 1297 718 1027
Net 1520 44.9 7.82 6.23 1065 587 837Spacing Study Gross 37.3 6.82 5.84 1282 765 1098
Net 27.7 5.87 4.66 1108 491 843
Hypothesis test
concluded an
optimal
relationship
Density values that
optimize total
volume increment
Results for Pulpwood
Volume Increment
Results: Is there a stand density that
optimizes pulpwood volume increment?
Dataset PAIDensity
SPH BPH RSD RSH SDID SDIH SDIV
NIMP Gross 3.31
Net 324 345
NIMP1t Gross 75.8
Net
IMP Gross 35.96 6.76 5.21 953 504 749
Net 35.42 6.70 5.18 949 502 746
IMP1t Gross 737 7.70 792 584 711
Net 736 7.75 768 590 695
RW19 Gross 1636 53.9 8.85 7.01 1296 714 1024
Net 1602 45.6 7.89 6.32 1090 605 859
Spacing Study Gross 2150 36.5 6.68 5.22 1084 570 803
Net 2440 28.4 5.93 4.28
Results for Sawtimber
Volume Increment
Results: Is there a stand density that
optimizes sawtimber volume increment?
Dataset PAIDensity
SPH BPH RSD RSH SDID SDIH SDIV
NIMP Gross 877 58.48 8.96
Net 935 58.92 8.94
NIMP1t Gross 701 46.9 7.95
Net 727 48.7 8.12 7.21
IMP Gross 356
Net 350
IMP1t Gross 684 804 586 720
Net 689 804 601 725
RW19 Gross 1374 57.4 9.27 6.82 1438 679 1046
Net 1385 53.9 8.85 6.69 1346 662 993
Spacing Study Gross 877 38.9 7.05 5.53 773 462 634
Net 915 39.4 7.09 5.47 776 457 633
Conclusions
Results of hypothesis test were different depending on the dataset, measure
of growth, and measure of density
Approach was inconclusive at determining the relationship between growth
and density
Stand density measures based on diameter (BPH, RSD, and SDID) consistently
explained the most variation in volume increment (20% – 30%)
No consistent solution between these measures
Different Approach #1
Zeide (2004) argues that “optimal” density cannot be determined as in the previous manner
Relating volume increment as a function of density alone does not account for the effects of age and average tree size (quadratic mean diameter)
Age: as stands age any gaps in the canopy created by mortality become more difficult to fill by the residual stand
Volume increment can only be optimized when the canopy is full
Any gaps in the canopy reduce the total amount of light interception
Average tree size: Stands at similar densities can have largely different average diameters
Average diameter and density can determine diameter increment which is related to volume increment
Source: Zeide, B. 2004. Optimal stand density: A solution. Canadian Journal of Forest Research 34(4):846-854.
Formulating a model: Define the volume
of an average tree
Using a “local” volume equation
𝑣 = 𝛼𝐷𝑞𝛽
Using the “combined variable” equation
𝑣 = 𝛼𝐷𝑞2𝐻𝑑
Where
𝑣 = mean total tree volume (m3)
𝐷𝑞 = quadratic mean diameter (cm)
𝐻𝑑 = mean tree height (m)
α,β = parameters to be estimated
Formulating a model: Differentiate to obtain
individual tree volume increment equations
𝑑𝑣
𝑑𝑡= 𝛼𝛽𝐷𝑞
𝛽−1 𝑑𝐷𝑞
𝑑𝑡
𝑑𝑣
𝑑𝑡= 𝛼 2𝐷𝑞𝐻𝑑
𝑑𝐷𝑞
𝑑𝑡+ 𝐷𝑞
2 𝑑𝐻𝑑
𝑑𝑡
Where
𝑑𝑣
𝑑𝑡= individual volume increment (m3 / year)
𝑑𝐷𝑞
𝑑𝑡= diameter increment (cm / year)
𝑑𝐻𝑑
𝑑𝑡= height increment (m / year)
Formulating a model: Multiply volume
increment of average tree by Stems per
Hectare
𝑑𝑉
𝑑𝑡= 𝑆𝑃𝐻 ∗ 𝛼𝛽𝐷𝑞
𝛽−1 𝑑𝐷𝑞
𝑑𝑡
𝑑𝑉
𝑑𝑡= 𝑆𝑃𝐻 ∗ 𝛼 2𝐷𝑞𝐻𝑑
𝑑𝐷𝑞
𝑑𝑡+ 𝐷𝑞
2 𝑑𝐻𝑑
𝑑𝑡
Where
𝑑𝑉
𝑑𝑡= volume increment per hectare (m3 / hectare / year)
Formulating a model: Add modules to
account for the effects of age and density
𝑑𝑉
𝑑𝑡= 𝑆𝑃𝐻 ∗ 𝛼𝛽𝐷𝑞
𝛽−1 𝑑𝐷𝑞
𝑑𝑡∗ exp 𝛾 ∗ 𝑎𝑔𝑒 ∗ exp(
−𝐷𝑒𝑛𝑠
𝛿)
𝑑𝑉
𝑑𝑡= 𝑆𝑃𝐻 ∗ 𝛼 2𝐷𝑞𝐻𝑑
𝑑𝐷𝑞
𝑑𝑡+ 𝐷𝑞
2 𝑑𝐻𝑑
𝑑𝑡∗ exp 𝛾 ∗ 𝑎𝑔𝑒 ∗ exp(
− 𝐷𝑒𝑛𝑠
𝛿)
Where
𝛾,𝛿 = parameters to be estimated
Formulating a model: Formulate SPH as a
function of commonly used measures of
density
𝐵𝑃𝐻 = 𝐵𝐴𝑡𝑟𝑒𝑒 ∗ 𝑆𝑃𝐻
𝑆𝑃𝐻 =𝐵𝑃𝐻
𝐵𝐴𝑡𝑟𝑒𝑒=
𝐵𝑃𝐻
𝑑𝑞2∗𝑐= 𝑐 ∗ 𝐵𝑃𝐻 ∗ 𝑑𝑞
−2
𝑅𝑆𝐷 = 10,000/𝑆𝑃𝐻 𝑑𝑞
𝑆𝑃𝐻 = 𝑐 ∗ 𝑅𝑆𝐷−2 ∗ 𝑑𝑞−2
𝑆𝐷𝐼𝐷 = 𝑆𝑃𝐻 25 𝑑𝑞−1.43089
𝑆𝑃𝐻 = 𝑐 ∗ 𝑆𝐷𝐼𝐷 ∗ 𝑑𝑞−1.43089
Formulating a model: Replace SPH in
volume increment equations
𝑑𝑉
𝑑𝑡=∗ 𝛼 ∗ 𝐵𝑃𝐻 ∗ 𝐷𝑞
𝑐 𝑑𝐷𝑞
𝑑𝑡∗ exp 𝛾 ∗ 𝑎𝑔𝑒 ∗ exp(
− 𝐵𝑃𝐻
𝛿)
𝑑𝑉
𝑑𝑡=∗ 𝛼 ∗ 𝑅𝑆𝐷𝜃 ∗ 𝐷𝑞
𝑐 𝑑𝐷𝑞
𝑑𝑡∗ exp 𝛾 ∗ 𝑎𝑔𝑒 ∗ exp(
− 𝑅𝑆𝐷
𝛿)
𝑑𝑉
𝑑𝑡=∗ 𝛼 ∗ 𝑆𝐷𝐼𝐷 ∗ 𝐷𝑞
𝑐 𝑑𝐷𝑞
𝑑𝑡∗ exp 𝛾 ∗ 𝑎𝑔𝑒 ∗ exp(
− 𝑆𝐷𝐼𝐷
𝛿)
Results: Is there a stand density that
optimizes volume increment?
Both “local” and “combined variable” formulations with all measures of
density resulted in optimal relationships between density and volume
increment (for all measures compared here)
In all cases the “optimal” density was well outside the commonly held
theoretical maximum SDI for loblolly pine in the southern U.S. of 450, as
defined by Reineke (1933)
which equates to 450 10-inch trees per acre or 1112 25-centimeter trees per
hectare.
Using this approach it appears that maximum volume increment occurs near
the maximum observed densities
Different Approach #2
Use the RW19 data
Individual plots were thinned to a common number of stems per hectare
Treatments: 1235, 741, 494, and 247 residual SPH
Spacing Study data
Nine different planting densities
6727, 4484, 3363, 2989, 2242, 1681, 1494, 1121, and 747 SPH
Fit 2nd degree polynomial to each thinning treatment and planting density
Useful for determining if different relationships exist among treatments
𝑃𝐴𝐼 = 𝐵0 + 𝐵1 ∗ 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 + 𝐵2 ∗ 𝐷𝑒𝑛𝑠𝑖𝑡𝑦2
Results from the
RW19 Study
Dens.Thinning
Treatment
Total Gross PAI Pulpwood Gross PAI Sawtimber Gross PAI
R2 OptD OptP R2 OptD OptP R2 OptD OptP
BPH 247 SPH 0.6823 0.6766 0.5011
494 SPH 0.4929 39.60 27.89 0.4795 39.69 27.96 0.4008 27.67 26.74
741 SPH 0.3605 40.81 27.47 0.3421 41.24 27.69 0.3562 33.74 29.65
1235 SPH 0.0897 44.00 28.00 0.0850 44.06 28.23 0.1986 42.60 32.65
RSD 247 SPH 0.6818 0.6762 0.5010
494 SPH 0.4913 0.4777 0.4069 6.06 26.64
741 SPH 0.3604 7.90 29.05 0.3416 0.3599 6.62 29.45
1235 SPH 0.0930 7.48 27.90 0.0881 7.49 28.14 0.2012 7.37 32.47
SDID 247 SPH 0.6670 0.6616 0.4902
494 SPH 0.4806 0.4674 0.3993 549.23 26.65
741 SPH 0.3586 841.62 28.26 0.3403 854.60 28.62 0.3617 704.28 29.68
1235 SPH 0.0684 0.0660 0.1701 977.26 33.20
OptD = optimum density, OptP = PAI at which density is optimized
Results: Fitting 2nd degree polynomial to
thinning treatments
Results from
Spacing Study
Conclusions
Total Volume Increment
Maybe Langsaeter (1941) was partially right…
Volume seems to become near optimal over a wide range of densities although not
necessarily constant
A decrease in volume production at higher levels of density may not be
observable
Before a reduction in volume increment can occur stands self-thin
Adjustment to Langsaeters Hypothesis
Merchantable Volume Increment
Relationship between pulpwood volume increment and density is similar to
that of total volume increment
Merchantable volume increment can become optimal
Depends on:
Initial planting density
Thinning Intensity
Is this the final answer?
Questions?
