FOREST INVENTORY PREDICTIONS FROM INDIVIDUAL TREE CROWNS - Regression Modeling Within a Sampling Framework Jim Flewelling in association with ImageTree Corp. FIA SYMPOSIUM October, 2006

OUTLINE Make a Crown Map Make a Crown Map Sample the Crown Map Sample the Crown Map Model the Trees Model the Trees Model-Assisted Inference Model-Assisted Inference

Context Complete LiDAR & Digital photography Complete LiDAR & Digital photography 100% crown mapped. 100% crown mapped. Number of stands >> # of field plots. Number of stands >> # of field plots. Unbiased for population totals. Unbiased for population totals.

Crown Segmentation, Delineation & Attribution Identify individual crowns. Identify individual crowns. Locate center points. Locate center points. Delineate crown boundaries. Delineate crown boundaries. (non-overlapping) (non-overlapping) Attribute species. Attribute species. Attribute height. Attribute height.

Individual Tree Crown (ITC) Delineation Valley following Deep shade threshold Rule- based system 1995 Courtesy of Canadian Forest Service

Delineated Individual Tree Crowns At ~30 cm/pixel, 81% of the ITCs are the same as interpreted crowns Courtesy of Canadian Forest Service

Delineated and Classified ITCs Courtesy of Canadian Forest Service

Add in Stand Boundaries Individual stand on LiDAR image after tree polygon creation. A polygon now surrounds every visible tree crown. ©ImageTree Corp 2006

Completed Crown Map Census of individually delineated crowns. Census of individually delineated crowns. Location, Size, Shape Location, Size, Shape Center (centroid, or high-point) Center (centroid, or high-point) LiDAR Height LiDAR Height Species Assignment or Probability Species Assignment or Probability and more ? and more ? Stands Stands Boundaries Boundaries Auxiliary Data Auxiliary Data

OUTLINE Make a Crown Map Make a Crown Map Sample the Crown Map Sample the Crown Map Model the Trees Model the Trees Model-Assisted Inference Model-Assisted Inference

Two-Stage Sampling 1st Stage: Stands 1st Stage: Stands 2nd Stage: Plots on Crown Map 2nd Stage: Plots on Crown Map

Plots Random Plot Center Coordinates Random Plot Center Coordinates GPS to those locations GPS to those locations Establish fixed-area stem-mapped plot. Establish fixed-area stem-mapped plot. Co-locate plots to find true position. Co-locate plots to find true position. Accept or reject altered coordinates. Accept or reject altered coordinates. Make fixed-area circular crown plot. Make fixed-area circular crown plot.

Fixed-Area Crown Plot ©ImageTree Corp 2006 Green dots

Edge Bias Correction Tree-Concentric Method

Matched Trees and Crowns Errors in Segmentation Errors in Segmentation One delineated crown = 2 neighboring trees. One delineated crown = 2 neighboring trees. One real tree wrongly divided into 2 crowns. One real tree wrongly divided into 2 crowns. Trees entirely missed. Trees entirely missed. Ground vegetation seen as a tree. Ground vegetation seen as a tree. Understory trees don’t contribute. Understory trees don’t contribute. Technical improvements, but no absolute solution. Technical improvements, but no absolute solution.

Matched Trees and Crowns

TREE MATCHING SCHEMES Subjective Subjective potential for significant bias potential for significant bias Crown Captures ALL in tessellated area. Crown Captures ALL in tessellated area. Expand crown area. Expand crown area. Trees compete to be captured. Trees compete to be captured. Consider DBH, height, species … Consider DBH, height, species … Ground plot size > crown plot size. Ground plot size > crown plot size.

CROWN BASED SAMPLE FRAME REQUIREMENT Trees linked to segmented crowns. Trees linked to segmented crowns. Linkage must be independent of sampling. Linkage must be independent of sampling.  BUT Linkages need not be physically correct. Linkages need not be physically correct. Suppressed trees need not be linked if sampled another way. Suppressed trees need not be linked if sampled another way.

Plot Configuration 0.12 ac. Analysis Plot

Completed Sample Stands Weights Stands Weights Crowns Crowns Size, Color, height, species guess, etc. Size, Color, height, species guess, etc. Weights (from edge effects) Weights (from edge effects) Associated Trees (Sp., DBH, Height) Associated Trees (Sp., DBH, Height) Unassociated Trees Unassociated Trees

OUTLINE Make a Crown Map Make a Crown Map Sample the Crown Map Sample the Crown Map Model the Trees Model the Trees Model-Assisted Inference Model-Assisted Inference

Model Trees from Crowns Trees = f(Stand, Crown, Window) Trees = f(Stand, Crown, Window) Pr{Crown has no trees} = f(….) Pr{Crown has no trees} = f(….) Pr{Crown has one tree} = f(…) Pr{Crown has one tree} = f(…) Pr{1st Tree = Pine} = f(….) Pr{1st Tree = Pine} = f(….) DBH(1st tree|Species) = f(…) + e DBH(1st tree|Species) = f(…) + e Ht(1st tree|Species) = f(Lidar ht,..) + e Ht(1st tree|Species) = f(Lidar ht,..) + e Predictors for unassociated trees Predictors for unassociated trees

Model Predictions Expected results - crown or stand level. Expected results - crown or stand level. DBH Distribution too narrow DBH Distribution too narrow (R-square < 1.00) (R-square < 1.00) Variance added through simulation or “tripling”. Variance added through simulation or “tripling”.

OUTLINE Make a Crown Map Make a Crown Map Sample the Crown Map Sample the Crown Map Model the Trees Model the Trees Model-Assisted Inference Model-Assisted Inference

Sample Design Development Set of Sample Stands Development Set of Sample Stands Used for fitting Equations. Used for fitting Equations. Calibration Set of Sample Stands Calibration Set of Sample Stands Random Selection, With Replacement Random Selection, With Replacement Current: Probability proportional to Area. Current: Probability proportional to Area.

Use of Calibration Set Ratio Model Ratio Model Crown level or Plot level. Crown level or Plot level. BA = k  (predicted BA) + e BA = k  (predicted BA) + e Asymptotically unbiased for key attributes: Asymptotically unbiased for key attributes: BA, TPA, BA times Lorey height, by species. BA, TPA, BA times Lorey height, by species. Variance for population mean (design- based). Variance for population mean (design- based). MSE of Stand-level estimates. MSE of Stand-level estimates.

Combine Development & Calibration Samples Fix Estimated Population Totals. Fix Estimated Population Totals. Refit the Models, with constrained totals. Refit the Models, with constrained totals. Improved MSE’s, but difficult to estimate. Improved MSE’s, but difficult to estimate. Alternatives with Single Data Set. Alternatives with Single Data Set. Model-assisted approach (Sarndol) Model-assisted approach (Sarndol) Generalized Regression Generalized Regression

Generalized Regression Pred. Total =  (pred y) + Ratio Est Error Pred. Total =  (pred y) + Ratio Est Error Little (2004): “Design consistency - estimator converges to the population quantity … as the sample size increases, in a manner that maintains the features of the sample design.” Little (2004): “Design consistency - estimator converges to the population quantity … as the sample size increases, in a manner that maintains the features of the sample design.” Still need to allocate errors back to the model. Still need to allocate errors back to the model.

Summary - Statistics Trees and Segmented Crowns are not 1:1 Trees and Segmented Crowns are not 1:1 Data can be collected that allows for design-based inference of totals. Data can be collected that allows for design-based inference of totals. Totals are unbiased. Totals are unbiased. MSE’s at stand level from plot-level results. MSE’s at stand level from plot-level results. Edge-bias avoidance. Edge-bias avoidance. Estimator properties greatly changed. Estimator properties greatly changed.

Summary - Application Attractive technology. Attractive technology. Best for which forest types (?) Best for which forest types (?) Irregular spatial tree distributions. Irregular spatial tree distributions. Some multi-species situations. Some multi-species situations. Areas of difficult access. Areas of difficult access. Large Areas, Fast Results (future) Large Areas, Fast Results (future)

Acknowledgments Some slides were provided by Francois Gougeon and are courtesy of Natural Resources Canada, Canadian Forest Service. Some slides were provided by Francois Gougeon and are courtesy of Natural Resources Canada, Canadian Forest Service. Other slides were provided by ImageTree Corporation. Other slides were provided by ImageTree Corporation.

Resources 2005 Silviscan Silviscan ISPRS Laser-Scanner for Forest and - ser_forest/ 2004 ISPRS Laser-Scanner for Forest and - ser_forest/ ser_forest/ ser_forest/ ImageTree Corp. ImageTree Corp. Pacific Forestry Center Pacific Forestry Center Precision Forestry Coop (U.W.) Precision Forestry Coop (U.W.)

References Little, R To model or not to model? competing modes of Inference for finite population sampling. J Am. Stat. Assoc. 99: Little, R To model or not to model? competing modes of Inference for finite population sampling. J Am. Stat. Assoc. 99: Sarndal, C, Swensson and Wretman Model assisted survey sampling. Springer. Sarndal, C, Swensson and Wretman Model assisted survey sampling. Springer.