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

qDN lnlnln

Log D

en

sity

Log Diameter- (QMD)

Lines approach a maximum line = self-thinning line

self-thinning line

qDN lnlnln

SDI = e α +β Ln(Dq)

Limiting Density-Size Relationship and Reineke’s Max Stand Density Index

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 variables5 Clusters

R-squared with 1-R**2 Cluster Variable Own Next Ratio

Cluster Closest

Cluster 1 d100 0.97 0.67 0.08 dd5_f 0.98 0.60 0.05 mat_f 0.97 0.80 0.13 mmax_f 0.90 0.62 0.27 mtwm_f 0.94 0.64 0.16 Cluster 2 fday 0.88 0.18 0.15 ffp 0.95 0.28 0.08 gsdd5 0.92 0.69 0.25 mmindd0 0.72 0.73 1.03 sday 0.93 0.41 0.12 Cluster 3 dd0 0.95 0.78 0.25 mmin 0.90 0.53 0.22 mtcm 0.96 0.56 0.08 Cluster 4 adi_sqroot 0.93 0.44 0.12 gsp_inches 0.90 0.34 0.15 map_inches 0.87 0.33 0.19 sdi 0.83 0.54 0.38 Cluster 5 pratio 0.26 0.08 0.80 smrpb 0.80 0.30 0.28 smrsprpb 0.97 0.26 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 log-likelihood ratio test

Results: Multidimensional Limiting Density-Size Relationship

• Douglas-fir final model:Ln(TPA) = b0 + b1·Ln(QMD) + b·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))

Parameter

Estimate Standard Approx

Error Pr > |t|

Intercept

12.957 0.409 <.0001 Rock Type CaMetased -0.063 0.018 0.0007 Rock Type Extrusive 0.021 0.016 0.1765 Rock Type Glacial -0.051 0.017 0.0021 Rock Type Intrusive -0.132 0.016 <.0001 Rock Type Metasedimentary -0.081 0.016 <.0001 Rock Type Sedimentary

Baseline .

Ln(QMD)

-1.716 0.209 <.0001 Cos_Aspect

0.074 0.005 <.0001

Ln(ADI)

-0.398 0.012 <.0001 Ln(Elevation)

-0.524 0.050 <.0001

Ln(Prop.BA)

-1.080 0.039 <.0001 Ln(QMD)*Ln(Elev)

0.074 0.026 0.0043

Ln(QMD)*Ln(Prop BA) 0.281 0.021 <.0001

Ln(TPA) = b0 + b1·Ln(QMD) + b·Ln(ADI) + b5·Ln (Elevation) + b5· Ln(Prop. BA) + b6·Ln (Elevation)·Ln(QMD) + b7·Ln (Prop. BA) ·Ln(QMD)

Grand-fir: Parameter Estimates of the Multidimensional Limiting Density-Size Relationship

(Response variable: Ln(Tres per acre))

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

Ln(TPA) = b0 + b1·Ln(QMD) + b·Ln(ADI) + b5· Ln(Prop. BA) + b6·Ln (Prop. BA) ·Ln(QMD)

Ponderosa pine: Parameter Estimates of the Multidimensional Limiting Density-Size Relationship

(Response variable: Ln(Tres per acre))

Parameter

Estimate Standard Approx

Error Pr > |t|

Intercept

7.579 0.202 <.0001 Rock Type CaMetased -0.216 0.037 <.0001 Rock Type Extrusive -0.011 0.019 0.5595 Rock Type Glacial -0.186 0.024 <.0001 Rock Type Intrusive -0.276 0.022 <.0001 Rock Type Metasedimentary -0.202 0.023 <.0001 Rock Type Sedimentary

Baseline

Ln(QMD)

-1.169 0.013 <.0001 Ln(ADI)

0.052 0.018 0.0047

Ln(Elevation)

0.081 0.023 0.0004 Ln(Prop.BA)

-1.056 0.061 <.0001

Ln(QMD)*Ln(Prop BA) 0.322 0.032 <.0001

Ln(TPA) = b0 + b1·Ln(QMD) + b·Ln(ADI) + b4· Ln(Elevation) + b5· Ln(Prop. BA) + b6·Ln (Prop. BA) ·Ln(QMD)

Western larch: Parameter Estimates of the Multidimensional Limiting Density-Size Relationship

(Response variable: Ln(Tres per acre))

Parameter Estimate Standard Approx

Error Pr > |t|

Intercept 9.974 0.520 <.0001 Ln(QMD) -0.961 0.031 <.0001 Cos_Aspect 0.068 0.014 <.0001 Ln(ADI) -0.421 0.041 <.0001 Ln(Elevation) -0.189 0.061 0.0018 Ln(Prop.BA) -1.477 0.088 <.0001 Ln_QMD*LN(Prop.BA) 0.509 0.051 <.0001

Ln(TPA) = b0 + b1·Ln(QMD) + b·Cos_Aspect + b·Ln(ADI) + b4· Ln(Elevation) + b5·Ln (Prop. BA) + b6·Ln (Prop. BA) ·Ln(QMD)

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