The Walk Through and Fro Solution to the Boundary Overlap Problem

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

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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

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

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

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.
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.
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 !
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.

Walk Through and Fro

Bounds on two sides

Use same formula

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.

INSIDE

Simpler formula

OUTSIDE OF STAND

r

FRO

THROUGH

T

S

bf

bt

Walk To and Fro

Bounds on two sides

Just measure bf

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

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 !
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.

×

1

×

×

×

0

1

2

Example Configurations

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4

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

×

1

×

×

×

0

1

2

Corner Configuration

Stand is 2 by 2.

Tree Plot has r = 2.

Plot area /stand area = π/4

EVALUATION - Corner
• For Tree at (1.8, 1.5) - red circle last slide
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.
OUTLINE
• Some current methods
• New Methods
• Any good?
• Theory
• Sources of Variance
• Remove the Center?
• Have Fun !
What’s a bow-tie?

Divide the tree centered plot into m bow-ties, each with angle π/m.

Let m approach infinity.

What’s a bow-tie?

Remember the complex boundary example?

What’s half a bow-tie? (Fro)

Keep triangle-shaped portion from the tree towards the sample point.

S

T

T

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

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

WalkThrough

Tree Concentric

INSIDE

INSIDE

OUTSIDE

OUTSIDE

W = 1.25

W = 1.0

W = 2

MORE VARIANCE

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.

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

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

Try it. No barbed wired fences to climb !

Thank you,