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 (a i ), the tree is sampled (I i =1). Easy and Unbiased - If a i is entirely in A.

6. Tree concentric – Direct Area What if π i (a i /A) ? IN OUT Fix a i OR use weight w = 1 / ( fraction of a i 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 Estimator Ducey, 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. Walkthrough Estimator Ducey, Gove and Valentine, 2004 OUT w = 2

10. 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 !

11. Walk Through and Fro OUTSIDE OF STAND INSIDE T bfbf btbt S THROUGH FRO r T = Tree Location S = Sample point b f, b t = distances to boundaries R = radius of trees plot.

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

13. Walk Through and Fro Complex intersection T r = 10 -10-7 03 6 10 IN WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS throughfro Green = inside of stand. Red = Outside of stand.

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

15. Walk To and Fro Bounds on two sides Just measure b f

16. Walk To and Fro Complex intersection T r = 10 -10-7 03 6 10 IN WEIGHT COMPUTATION USES SQUARES OF INTERSECTIONS throughfro

17. 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 !

18. 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.

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

20. EVALUATION For Tree at (1.5, 1.0) - red circle last slide EstimatorMean errMSE Uncorrected-0.154.256 Tree Concentric.000.359 Walkthrough.000.476 Walk Through and Fro.000.397 Walk To and Fro.000.647

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

22. EVALUATION - Corner For Tree at (1.8, 1.5) - red circle last slide EstimatorMean errMSE Uncorrected-.395.394 Treee Concentric.000.962 Walkthrough-.104.809 Walk Through and Fro.0001.099 Walk To and Fro.0005.066

23. 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.

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

25. Whats a bow-tie? Divide the tree centered plot into m bow-ties, each with angle π/m. Let m approach infinity.

26. Whats a bow-tie? Remember the complex boundary example?

27. Whats half a bow-tie? (Fro) Keep triangle-shaped portion from the tree towards the sample point. T T S

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

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

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

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

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

33. 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.

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

35. Try it. No barbed wired fences to climb ! Thank you,