Distance-Variable Estimators for Sampling and Change Measurement

Western Mensurationists June 2006

hugh.carter@jsthrower.com

Hugh Carter MSc (Candidate), RFT

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Kim Iles PhD.

Outline

2. Bias (or lack of)

3. Shapes

5. Compatibility

9. Summary

4. Change over time

6. Simple example

8. Future Work

1. Background

7. Edge

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Background

• Need a solution for applying VRP for measuring change over time.

• Problems encountered include:

- High variability due to on-growth. - Extending concepts to variables other than volume and BA. - Providing a solution that is easily applied and understood.

• A reminder of why we might want to use Variable Radius Plots (VRP) for measuring change:

- Efficiency (cost and time). - Remeasurement of existing plots. - Increase precision?

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• Attempts have been made to solve these problems, however none have covered them all.

• Distance-Variable estimators reduce variability, extend to any variable for any object of interest, and provide an easy to apply method.

Background Continued

• Distance-Variable estimators are an extension of the “Iles method” to any variable of interest on any sampled object of interest.

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Bias

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Horvitz-Thompson Estimator

Potential random sample points

Object of interest

Inclusion circle

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Bias Continued

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Expectation of

Potential random sample points

Object of interest

Inclusion circle

Distance-Variable Estimator

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Shapes

Why Use a Cone?

3x Value

0x Value

• Average at all potential sample points will give estimate

• Easy to use and visualize - height at point is 3x value - height at base is 0x value

• Can get a simple “Value Gradient”

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Shapes Continued

How do they work?

111 m2/s2/kg

0 m2/s2/kg

Average of all sample points is 37 m2/s2/kg

• Units no longer an issue

• Average at sample points give estimate

• Sample point is ¼ of distance from edge

• Estimate = ¼ * 111m2/s2/kg = 27.37m2/s2/kg

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Change Over Time

Traditional Subtraction Method0 1 2 3 4 5 6 7 8 98 7 6 5 4 3 2 1 0 1

Change Over Time

Distance-Variable Method0 1 2 3 4 5 6 7 8 98 7 6 5 4 3 2 1 0 1

Compatibility0 1 2 3 4 5 6 7 8 98 7 6 5 4 3 2 1 0 1

12 VVV

Both methods are compatible, however the traditional subtraction method is more variable!

0 1 2 3 4Tree 1 0 0 10 10 10Tree 2 10 10 10 10 10Tree 3 10 10 10 0 0Total 20 20 30 20 20

Measurement Time

Basal Area Example

Traditional Method (BAF 10m2/ha)

Distance-Variable Method (BAF 10m2/ha)

0 1 2 3 4Tree 1 0 0 3 8 12Tree 2 8 12 17 22 25Tree 3 12 13 14 0 0Total 20 25 34 30 37

Measurement Time

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On-growth

Total

On-growth

Total

Mortality

Survivor

On-Growth

0

5

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30

35

40

0 1 2 3 4

Measurement

BA

/ha Total

0

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0 1 2 3 4

Measurement

BA

/ha

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40

0 1 2 3 4

Measurement

BA

/ha

0 1 2 3 4Tree 1 0 0 10 10 10Tree 2 10 10 10 10 10Tree 3 10 10 10 0 0Total 20 20 30 20 20

Measurement Time

Basal Area Example

Traditional Method (BAF 10m2/ha)

Distance-Variable Method (BAF 10m2/ha)

0 1 2 3 4Tree 1 0 0 3 8 12Tree 2 8 12 17 22 25Tree 3 12 13 14 0 0Total 20 25 34 30 37

Measurement Time

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Survivor

Total

Survivor

Total

Mortality

Survivor

On-Growth

Total

0

5

10

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30

35

40

0 1 2 3 4

Measurement

BA

/ha

0

5

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40

0 1 2 3 4

Measurement

BA

/ha

0 1 2 3 4Tree 1 0 0 10 10 10Tree 2 10 10 10 10 10Tree 3 10 10 10 0 0Total 20 20 30 20 20

Measurement Time

Basal Area Example

Traditional Method (BAF 10m2/ha)

Distance-Variable Method (BAF 10m2/ha)

0 1 2 3 4Tree 1 0 0 3 8 12Tree 2 8 12 17 22 25Tree 3 12 13 14 0 0Total 20 25 34 30 37

Measurement Time

0 1 2 3 4 5 6 7 8 98 7 6 5 4 3 2 1 0 1

Mortality

Total

Mortality

Total

Mortality

Survivor

On-Growth

Total

0

5

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40

0 1 2 3 4

Measurement

BA

/ha

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0 1 2 3 4

Measurement

BA

/ha

0 1 2 3 4Tree 1 0 0 10 10 10Tree 2 10 10 10 10 10Tree 3 10 10 10 0 0Total 20 20 30 20 20

Measurement Time

Basal Area Example

Traditional Method (BAF 10m2/ha)

Distance-Variable Method (BAF 10m2/ha)

0 1 2 3 4Tree 1 0 0 3 8 12Tree 2 8 12 17 22 25Tree 3 12 13 14 0 0Total 20 25 34 30 37

Measurement Time

0

5

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30

35

40

0 1 2 3 4

Measurement

BA

/ha

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On-growthSurvivor Mortality

Total

On-growthSurvivor Mortality

Total

Mortality Tree

Survivor Tree

On-Growth Tree

Total

0

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40

0 1 2 3 4

Measurement

BA

/ha

Total

Mortality Tree Survivor Tree

On-Growth Tree

Edge

• Existing techniques for correcting edge remain applicable.

- Walk-through

- Toss-back

- Mirage

• Unbiased if inclusion areas are symmetrical through the tree.

• If extra sample points are needed the DV estimator is used instead of the traditional estimator.

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Future Work

• Variance control through different shaped estimators.

• Density surface mapping.

• Efficiency/Precision gains?

• Non-stationary object sampling.

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Summary

• Unbiased

• EXTENDS TO ANY VARIABLE FOR ANY OBJECT!!

• Easy to apply and understand

• Compatible

• Smoothes change/growth curves

• Works with existing edge techniques

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Distance-Variable Method

Acknowledgements

Kim Iles & Associates

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Volume Example

0 1 2 3 4Tree 1 0 0 2 2.5 3Tree 2 1.1 1.3 1.7 2.1 2.2Tree 3 0.9 1.1 1.2 0 0

Measurement Time

0 1 2 3 4Tree 1 0 0 0.1 0.4 0.9Tree 2 2.2 2.5 2.9 3.4 3.6Tree 3 1 1.3 1.4 0 0

Measurement Time

0

0.5

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1.5

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3.5

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Measurement

Tre

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olu

me

Tree 1

Tree 2

Tree 3

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3.5

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Measurement

Tre

e V

olu

me

Tree 1

Tree 2

Tree 3

Traditional Method

Distance-Variable Method

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Summary

2. Bias (or lack of)

3. Shapes

5. Compatibility

9. Summary

4. Change over time

6. Simple example

8. Future Work

1. Background

7. Edge

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