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1. The Walk Through and Fro Solution to the Boundary Overlap Problem Jim Flewelling flewelling@seattlebiometrics.com Jacob L. Strunk jacob.strunk@oregonstate.edu Presented at: Western Mensurationists Missoula, Montana June 21, 2010

2. The Bow-Tie Theory of Sampling (alternate title)

3. OUTLINE • Some Current Methods • New Methods • Any good? • Theory • Sources of Variance • Remove the Center? • Have Fun !

4. PROGRESS • Some current methods • Horovitz-Thompson Estimator • Tree Concentric • Mirage • WalkThrough • New Methods • Any good? • Theory • Sources of Variance • Remove the Center? • Have Fun !

5. Horovitz-Thompson Estimator(before edge effects) General For tree centered plots - If a sample point falls within this area (ai), the tree is sampled (Ii =1). Easy and Unbiased - If ai is entirely in A.

6. Tree concentric – Direct Area • What if πi  (ai/A) ? IN OUT Fix ai OR use weight w = 1 / ( fraction of ai within the stand) Unbiased, Lowest Variance

7. Mirage estimator • Each sample location near a boundary is reflected across the boundary. (fence climbing may be required). • Effectively double weight part of each tree centered plot.

8. Walkthrough EstimatorDucey, Gove and Valentine, 2004 • Walk from plot center to sample tree. • Continue walking an equal distance. • Double count the tree if now outside of the stand. (w = 2) • Method is easy to apply. • Unbiased for some boundaries; at least half of each plot must be within the stand.

9. OUTLINE • Some current methods • New Methods • Walk Through and Fro • Walk To and Fro • Any good? • Theory • Sources of Variance • Remove the Center? • Have Fun !

10. Walk Through and Fro INSIDE OUTSIDE OF STAND r FRO THROUGH T S bf bt T = Tree Location S = Sample point bf, bt = distances to boundaries R = radius of tree’s plot.

11. Walk Through and Fro Bounds on two sides Use same formula

12. Walk Through and FroComplex intersection r = 10 r = 10 T fro through IN IN IN 6 0 3 10 -10 -7 WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS Green = inside of stand. Red = Outside of stand.

13. Walk To and Fro( less walking. S to T, then fro to plot or stand bound ) INSIDE Simpler formula OUTSIDE OF STAND r FRO THROUGH T S bf bt

14. Walk To and Fro Bounds on two sides Just measure bf

15. Walk To and FroComplex intersection r = 10 r = 10 T fro through IN IN IN 6 0 3 10 -10 -7 WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS

16. OUTLINE • Some current methods • New Methods • Any good? • Evaluation Method • Sample Configurations • Overlap a straight boundary • Overlap a corner boundary • Conclusions • Theory • Sources of Variance • Remove the Center? • Have Fun !

17. Any Good?EVALUATION METHOD • Very small stand • One sample point per stand • One tree per stand – circular tree plot • Estimate ∑ (tree plot areas) / [Stand Area] • Calculate error. • Integrate over all possible sample points. • Report mean error and MSE.

18. × 1 × × × 0 1 2 Example Configurations Stand is 2 by 2. Tree Plot has r = 2. Plot area /stand area = π/4

19. EVALUATION • For Tree at (1.5, 1.0) - red circle last slide

20. × 1 × × × 0 1 2 Corner Configuration Stand is 2 by 2. Tree Plot has r = 2. Plot area /stand area = π/4

21. EVALUATION - Corner • For Tree at (1.8, 1.5) - red circle last slide

22. Evaluation - Conclusions • Tree Concentric always the best! • WalkThrough can be biased. • Walk Through and Fro is unbiased. Has MSE slightly better than Walkthrough. • Walk To and Fro is unbiased. - can have high MSE.

23. OUTLINE • Some current methods • New Methods • Any good? • Theory • Sources of Variance • Remove the Center? • Have Fun !

24. What’s a bow-tie? Divide the tree centered plot into m bow-ties, each with angle π/m. Let m approach infinity.

25. What’s a bow-tie? Remember the complex boundary example?

26. What’s half a bow-tie? (Fro) Keep triangle-shaped portion from the tree towards the sample point. S T T

27. Theory - Math Bow-ties Sample Trees, with fk referring to sampled bow-tie. Fraction of area in stand for sampled bow tie. w = 1 / f Weights

28. OUTLINE • Some current methods • New Methods • Any good? • Theory • Sources of Variance • Remove the Center? • Have Fun !

29. Sources of Variance WalkThrough Tree Concentric INSIDE INSIDE OUTSIDE OUTSIDE W = 1.25 W = 1.0 W = 2 MORE VARIANCE

30. Sources of Variance WalkThrough Walk Through and Fro INSIDE INSIDE OUTSIDE OUTSIDE W = constant W = 1.0 W = 2 S For any particular through and fro line, w is invariant to sample location.

31. OUTLINE • Some current methods • New Methods • Any good? • Theory • Sources of Variance • Remove the Center? • Have Fun !

32. Remove the Center • Through and fro and to-and-fro may be difficult if the sample point is near coincident with the tree. • Directional line can not be established. • Either method can be modified by excluding a small central area from the bow-tie. w formulas modied slightly. Tree concentric formula used for small center circle. Will improve MSE of To and Fro extimator.

33. OUTLINE • Some current methods • New Methods • Any good? • Theory • Sources of Variance • Remove the Center? • Have Fun !

34. Have Fun! Try it. No barbed wired fences to climb ! Thank you,