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 past and current work. Very recently we developed the databe

IFTNC - STI Database is being use to identify the drivers of site quality and define site-type classes throughout the Inland Northwest Present work

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 Where each dot now represents a plot from a member of the coop

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

Log Density Log Diameter- (QMD) Lines approach a maximum line = self-thinning line Log Density Log Diameter- (QMD) self-thinning line

Limiting Density-Size Relationship and Reineke’s Max Stand Density Index 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 Since 1930 ‘s many methods have been proposed to fit the limiting relationship (manually locating the line, OLS, quantile regression) they give a ‘good’ average line curve but don’t describe the limithing beahviour. In this study we apply SFR, which has been shown to provide a true boundary to the data.

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 Lspecies limiting line through all sites in the inland NW. These are the 4 most popular and valueables species in the region. Parameter estiMATES AND CALCULATED max sdi. Some of these values for these species are somehow lower than those reported for these species (western larch)

Results: Species Limit Density-Size lines Graphically, scatterplots and fitted line , SFR line. You can see the huge amount if data.

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 After rock type, we investigated the effect of Climate variables on the limitng relationship.

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 5-clusters provided by proc varclust

Clusters of climate variables High correlations within a cluster, low between

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 5 climate variables to test in the limiting density – size model

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 2𝑖 ∙ 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) + b 2𝑖 ∙ 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)

Grand-fir: Parameter Estimates of the Multidimensional Limiting Density-Size Relationship (Response variable: Ln(Tres per acre)) Ln(TPA) = b0 + b1·Ln(QMD) + b 2𝑖 ∙ 𝑅𝑜𝑐𝑘𝑇𝑦𝑝𝑒 𝑖 + b3·CosAspect + b4·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)) Ln(TPA) = b0 + b1·Ln(QMD) + b 2𝑖 ∙ 𝑅𝑜𝑐𝑘𝑇𝑦𝑝𝑒 𝑖 + b3·Ln(ADI) + b4· Ln(Elevation) + b5· Ln(Prop. BA) + b6·Ln (Prop. BA) ·Ln(QMD) Rock type, elevation, ADI and prop BA

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)

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