1. 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 ,

2. Outline of Presentation
Introduction
Background: Cooperative Site Type Initiative (IFTNC - STI)
Maximum Stand Density-Size Relationships in the Inland Northwest

3. 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

4. IFTNC – Research Regions

5. 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

6. 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

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

8. 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

9. The Settings: Inland Northwest
Where each dot now represents a plot from a member of the coop

10. 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

11. 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

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

13. 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)

14. 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.

15. 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

16. Fitting the Self-thinning line using Stochastic Frontier Regression
Fitting and data analysis performed using SAS 9.2 (proc qlim)

17. 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)

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

20. Results: Limiting Species Density- Size Relationship by Rock Type
Are the self-thinning lines (and the corresponding SDI Max) affected by soil parent material ?

21. Results: Limiting Density-Size by Rock Type

22. 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

23. 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

24. 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.

25. Results: Bootstraps 95% Confidence Intervals for Max SDI

26. Results: Bootstraps 95% Confidence Intervals for Max SDI

27. Results: Bootstraps 95% Confidence Intervals for Max SDI

28. Results: Bootstraps 95% Confidence Intervals for Max SDI

29. 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.

30. Effect of Climate Variables on the Limiting Density-Size relationship

31. 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

32. Results: Clusters of climate variables
5-clusters provided by proc varclust

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

34. 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

35. 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)

36. 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 )
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

37. 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

38. 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)

39. 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)

40. 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

41. 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)

42. 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 Addressing climate change in the forest vegetation simulator to assess impacts on landscape forest dynamics.

43. Douglas-fir: Regional Geospatial Map

44. 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