Maximum Density-Size Relationships for
Douglas-fir, Grand-fir, Ponderosa pine and
Western larch in the Inland Northwest
Roberto Volfovicz-Leon1, Mark Kimsey,
Terry Shaw, and Mark Coleman
Intermountain Forest - Tree Nutrition Cooperative (IFTNC)
1 robertov@uidaho.edu , 208-885-8017
Outline of Presentation
• Introduction
• Background: Cooperative Site Type
Initiative (IFTNC - STI)
• Maximum Stand Density-Size Relationships
in the Inland Northwest
Background:
IFTNC Site Type Initiative (STI)
• Identify site factors driving carrying capacity
and optimal productivity
• Develop models to estimate site quality based
on factors that control max density (max SDI)
and optimal productivity
• Create regional, geospatial tools that
predict site quality
IFTNC – Research Regions
IFTNC – Site Type Initiative Drivers of Forest Site Quality
• Principal factors:
– Light
• Aspect, latitude, cloudiness, slope
– Moisture
• Precipitation, soil available water, aspect
– Temperature
• Soil/air temperature, elevation, slope/aspect, latitude
– Nutrients
• Parent material elemental composition, rock weathering,
organic matter, water availability
• Site quality is an expression of a complex
interaction among these factors
IFTNC – Site Type Initiative:
Past and current work
• Developed a database of IFTNC member
forest stand cruise and permanent plot
growth data
• Database was merged with geospatial
representation of physiography, climate,
soils and geology
IFTNC - STI
• Database is being use to identify the
drivers of site quality and define site-type
classes throughout the Inland Northwest
Objective of Present Study
• Identify the effects of Soil parent material (Rock
type), Topography and Climate variables on
Reineke’s Maximum Stand Density Index (SDI max)
in the Inland NW for:
– Douglas-fir
– Grand-fir
– Ponderosa pine
– Western larch
The Settings: Inland Northwest
IFTNC- STI Dataset
• + 150,000 plot data
• ~ 4,000,000 individual tree data
• 28 species
• ~ 100 variables: stand and tree level
variables, climate, topography, soil parent
material characteristics
Background: Limiting Density-Size
Relationship
Reineke (1933) observed a limiting linear
relationship (on the log-log scale) between
number of trees N and quadratic mean
diameter Dq in even-aged stands of full
density
ln N  ln    ln D q
Log Density
Lines approach a maximum line
= self-thinning line
Log Diameter- (QMD)
Limiting Density-Size Relationship and
Reineke’s Max Stand Density Index
ln N  ln    ln D q
SDI = e α +β Ln(Dq)
Max SDI is the number of trees per unit area
with a specified diameter (Dq = 10 in)
Fitting the Self-thinning line: SFR
• Fitting Method: Stochastic Frontier
Regression SFR (Comeau et al. 2010,
Weiskittel et al. 2009, Bi 2001)
• Econometrics fitting technique used to
study production efficiency, cost and profit
frontiers
Fitting the Limiting Density-Size line: SFR
SFR Model:
• Ln(TPA) = α + β*Ln(QMD) + v - u
• v = two-sided random error
• u = non-negative random error
• Maximum likelihood techniques are
used to estimate the frontier
Fitting the Self-thinning line using Stochastic
Frontier Regression
Fitting and data analysis performed using SAS
9.2 (proc qlim)
Results: Species Limit Density-Size lines
and Max SDI
Results: Species Limit Density-Size lines
Results: Limiting Species Density- Size
Relationship by Rock Type
• Are the self-thinning lines (and the
corresponding SDI Max) affected by soil
parent material ?
Results: Limiting Density-Size by Rock
Type
Comparing Max SDI: Bootstrap 95% Confidence
Intervals
• Stochastic frontier models introduce
skewed error terms
• Assumption of normality of errors is not
valid and traditional statistical tests
cannot be applied
• Bootstrapping provides approximate
Confidence Limits for estimation of Max SDI
Comparing Max SDI: Bootstrap 95% Confidence
Intervals
• Nonparametric bootstrap percentile
confidence intervals
• 1,000 bootstrap replications with
replacement
• SFR Intercept, Slope and corresponding
Max SDI were obtained from each sample
Comparing Max SDI: Bootstrap 95% Confidence
Intervals
• The bootstrap distribution of each
regression coefficient was compiled
• 2.5th and 97.5th percentiles of the
empirical distribution formed the limits for
the 95% bootstrap percentile confidence
interval.
Results: Bootstraps 95% Confidence
Intervals for Max SDI
Results: Bootstraps 95% Confidence
Intervals for Max SDI
Results: Bootstraps 95% Confidence
Intervals for Max SDI
Results: Bootstraps 95% Confidence
Intervals for Max SDI
Effect of Climate Variables on the Limiting
Density-Size relationship
• Climate variables from the US Forest
Service Moscow-ID Laboratory
• Thirty-year averaged monthly values for
maximum, minimum and mean daily
temperatures and monthly precipitation
• Derived climatic variables
Effect of Climate Variables on the Limiting
Density-Size relationship
Climate Variable Reduction for Modeling using
Clustering
• In high dimensional data sets, identifying irrelevant
inputs is usually more difficult than identifying
redundant (highly correlated) inputs
• Our strategy was to first reduce redundancy and
then tackle irrelevancy in a lower dimensional
space
• Variable clustering (proc varclus SAS 9.2) was used
to reduce the number of redundant climate
variables to use as input in the self-thinning model
• Cluster representatives for each group were
selected using 1 – R2 ratio
Results: Clusters of climate variables
Clusters of climate variables
5 Clusters
Cluster Variable
Cluster
1
d100
dd5_f
mat_f
mmax_f
mtwm_f
Cluster
2
fday
ffp
gsdd5
mmindd0
sday
Cluster
3
dd0
mmin
mtcm
Cluster
4
adi_sqroot
gsp_inches
map_inches
sdi
Cluster
5
pratio
smrpb
smrsprpb
R-squared with
Own
Next
Cluster Closest
1-R**2
Ratio
0.97
0.98
0.97
0.90
0.94
0.67
0.60
0.80
0.62
0.64
0.08
0.05
0.13
0.27
0.16
0.88
0.95
0.92
0.72
0.93
0.18
0.28
0.69
0.73
0.41
0.15
0.08
0.25
1.03
0.12
0.95
0.90
0.96
0.78
0.53
0.56
0.25
0.22
0.08
0.93
0.90
0.87
0.83
0.44
0.34
0.33
0.54
0.12
0.15
0.19
0.38
0.26
0.80
0.97
0.08
0.30
0.26
0.80
0.28
0.05
Effects of Climate Variables on the Density-Size
relationship
• We select one representative from each cluster,
reducing the number of climate variable to include
in the self-thinning model from 20 to 5:
• Annual degree-days >5 °C (based on monthly mean
temperatures: dd5
• Length of the frost-free period: ffp
• Mean temperature in the coldest month: mtcm
• Annual Dryness Index: ADI (temperature/precipitation)
• Summer/Spring precipitation balance
(jul+aug)/(apr+may): smrsprpb
Effect of Topographic variables on the
Density-Size Relationship
•
•
•
•
Elevation (ft)
Slope
Aspect
The joint effect of Slope and Aspect was
modeled using the cosine and sine
transformation (Stage ,1976)
Testing the Significance of Climate, Topographic,
Soil Parental Material and Stand Variables on the
Density-Size Relationship
• Self-thinning relationship as a multidimensional surface
(Weiskittel et al . 2009)
• The selected climatic, topographic and stand factors
(Skewness of DBH^1.5 distribution, proportion of
basal area in the primary species, PBA) were tested
for relevance in the limiting function
• Significance of final covariates was tested using loglikelihood ratio test
Results: Multidimensional Limiting
Density-Size Relationship
• Douglas-fir final model:
Ln(TPA) = b0 + b1·Ln(QMD) + b2𝑖 ∙ 𝑅𝑜𝑐𝑘𝑇𝑦𝑝𝑒 𝑖 + b3·CosAspect + b4·Ln(ADI)
+ b5·Ln (Elevation) + b5· Ln(Prop. BA)
+ b6·Ln (Elevation)·Ln(QMD)
+ b7·Ln (Prop. BA) ·Ln(QMD)
where
Rock typei : represents a set of 6 indicator variables taking
values 0 and 1 for rock types (baseline sedimentary)
Prop.BA: proportion of basal area in the primary species
All other variables defined as before
Douglas-Fir: Parameter Estimates of the Multidimensional
Limiting Density-Size Relationship
(Response variable: Ln(Tres per acre))
Ln(TPA) = b0 + b1·Ln(QMD) + b2𝑖 ∙ 𝑅𝑜𝑐𝑘𝑇𝑦𝑝𝑒 𝑖 + b3·CosAspect + b4·Ln(ADI)
+ b5·Ln (Elevation) + b5· Ln(Prop. BA)
+ b6·Ln (Elevation)·Ln(QMD)
+ b7·Ln (Prop. BA) ·Ln(QMD)
Parameter
Intercept
Rock Type
CaMetased
Rock Type
Extrusive
Rock Type
Glacial
Rock Type
Intrusive
Rock Type
Metasedimentary
Rock Type
Sedimentary
Ln(QMD)
Cos_Aspect
Ln(ADI)
Ln(Elevation)
Ln(Prop.BA)
Ln(QMD)*Ln(Elev)
Ln(QMD)*Ln(Prop BA)
Estimate
12.957
-0.063
0.021
-0.051
-0.132
-0.081
-1.716
0.074
-0.398
-0.524
-1.080
0.074
0.281
Standard Approx
Error
Pr > |t|
0.409
<.0001
0.018
0.0007
0.016
0.1765
0.017
0.0021
0.016
<.0001
0.016
<.0001
Baseline
.
0.209
<.0001
0.005
<.0001
0.012
<.0001
0.050
<.0001
0.039
<.0001
0.026
0.0043
0.021
<.0001
Grand-fir: Parameter Estimates of the Multidimensional Limiting
Density-Size Relationship
(Response variable: Ln(Tres per acre))
Ln(TPA) = b0 + b1·Ln(QMD) + b2𝑖 ∙ 𝑅𝑜𝑐𝑘𝑇𝑦𝑝𝑒 𝑖 + b3·CosAspect + b4·Ln(ADI)
+ b5· Ln(Prop. BA) + b6·Ln (Prop. BA) ·Ln(QMD)
Parameter
Estimate Standard Approx
Error
Pr > |t|
Intercept
9.132
0.046 <.0001
Rock Type
CaMetased
-0.144
0.035 <.0001
Rock Type
Extrusive
-0.190
0.032 <.0001
Rock Type
Glacial
-0.134
0.035 0.0002
Rock Type
Intrusive
-0.207
0.034 <.0001
Rock Type
Metasedimentary
-0.172
0.033 <.0001
Rock Type
Sedimentary
Baseline
Ln(QMD)
-1.139
0.016 <.0001
Cos_Aspect
0.079
0.008 <.0001
Ln(ADI)
-0.280
0.017 <.0001
Ln(Prop.BA)
-0.828
0.055 <.0001
Ln(QMD)*Ln(Prop BA)
0.259
0.030 <.0001
Ponderosa pine: Parameter Estimates of the Multidimensional Limiting
Density-Size Relationship
(Response variable: Ln(Tres per acre))
Ln(TPA) = b0 + b1·Ln(QMD) + b2𝑖 ∙ 𝑅𝑜𝑐𝑘𝑇𝑦𝑝𝑒 𝑖 + b3·Ln(ADI) + b4· Ln(Elevation)
+ b5· Ln(Prop. BA) + b6·Ln (Prop. BA) ·Ln(QMD)
Parameter
Intercept
Rock Type
CaMetased
Rock Type
Extrusive
Rock Type
Glacial
Rock Type
Intrusive
Rock Type
Metasedimentary
Rock Type
Sedimentary
Ln(QMD)
Ln(ADI)
Ln(Elevation)
Ln(Prop.BA)
Ln(QMD)*Ln(Prop BA)
Estimate
7.579
-0.216
-0.011
-0.186
-0.276
-0.202
-1.169
0.052
0.081
-1.056
0.322
Standard Approx
Error
Pr > |t|
0.202
<.0001
0.037
<.0001
0.019
0.5595
0.024
<.0001
0.022
<.0001
0.023
<.0001
Baseline
0.013
<.0001
0.018
0.0047
0.023
0.0004
0.061
<.0001
0.032
<.0001
Western larch: Parameter Estimates of the Multidimensional Limiting
Density-Size Relationship
(Response variable: Ln(Tres per acre))
Ln(TPA) = b0 + b1·Ln(QMD) + b2·Cos_Aspect + b3·Ln(ADI) + b4· Ln(Elevation)
+ b5·Ln (Prop. BA) + b6·Ln (Prop. BA) ·Ln(QMD)
Parameter
Intercept
Ln(QMD)
Cos_Aspect
Ln(ADI)
Ln(Elevation)
Ln(Prop.BA)
Ln_QMD*LN(Prop.BA)
Estimate Standard Approx
Error
Pr > |t|
9.974
0.520
<.0001
-0.961
0.031
<.0001
0.068
0.014
<.0001
-0.421
0.041
<.0001
-0.189
0.061
0.0018
-1.477
0.088
<.0001
0.509
0.051
<.0001
Predicting Max SDI: Geospatial Maps
Datum: NAD 1983
Climate ~ 600 meters
Topographic ~ 20 meters
Parent Material: Feature Class geology polygons at scales of
1:100,000 or 1:250,000 scale were rasterized. Raster pixel size
was set to equal topographic layer resolution.
• Geospatial predictions are bounded by geology limits and are
restricted to geographical zones where the species is estimated
to have >25% viability per Crookston, NL, GE Rehfeldt, GE
Dixon, and AR Weiskittel 2010- Addressing climate change in the
forest vegetation simulator to assess impacts on landscape
forest dynamics.
•
•
•
•
Douglas-fir: Regional Geospatial Map
Next Steps
• Developing models to estimate site quality
and productivity based on these identified
factors
• Develop regional geospatial tools that
predict site quality for Grand-fir,
Ponderosa pine, and Western larch