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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

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The Bow-Tie Theory of Sampling (alternate title)

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

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

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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.

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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

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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.

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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.

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Walkthrough Estimator Ducey, Gove and Valentine, 2004 OUT w = 2

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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 !

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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.

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Walk Through and Fro Bounds on two sides Use same formula

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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.

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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

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

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

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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 !

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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.

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

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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

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

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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

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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.

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

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

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

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

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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

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

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

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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.

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

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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.

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

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

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