Distance-Variable Estimators for Sampling and Change Measurement
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Distance-Variable Estimators for Sampling and Change Measurement
Western Mensurationists June 2006
Hugh Carter MSc (Candidate), RFT
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Kim Iles PhD.
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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
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Change Over Time
Distance-Variable Method0 1 2 3 4 5 6 7 8 98 7 6 5 4 3 2 1 0 1
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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!
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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
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Measurement
BA
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Measurement
BA
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Measurement
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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
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Measurement
BA
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Measurement
BA
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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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Mortality
Total
Mortality
Total
Mortality
Survivor
On-Growth
Total
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Measurement
BA
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Measurement
BA
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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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Measurement
BA
/ha
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On-growthSurvivor Mortality
Total
On-growthSurvivor Mortality
Total
Mortality Tree
Survivor Tree
On-Growth Tree
Total
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Measurement
BA
/ha
Total
Mortality Tree Survivor Tree
On-Growth Tree
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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
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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
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Measurement
Tre
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olu
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Tree 1
Tree 2
Tree 3
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Measurement
Tre
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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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