Mathematics of Light interception in Australian Woodlands

1. Relationships between forest
1
structure measurements
Peter Scarth
John Armston
Tim Danaher
Rob Hasset
John Carter
Statewide Landcover and Trees Study

2. Outline
C a n o p y S tru c tu re
F ie ld S it e s &
S a m p lin g
C lu m p in g L e a f A n g le
C lu m p in g M o d e l LA I ↔ BA M odel
FPC ↔ CCP M odel
Conclusions

3. Primary canopy structural measurements
Stand structure → 2 components
Structural complexity
Structural attributes
Structural attributes
Projected Foliage Cover
Projected Canopy Cover
Basal Area
Foliage clumping - Ω
Leaf Area Index
•LAI
•LAIe → Leaf Angle & Ω

4. Measuring Canopies - Where
139° 142° 145° 148° 151°
154°
18° 18°
X
X
X
X #
Charters Towers
#
Mount Isa
21° 21°
X
X X
X
X Rockhampton
#
#
Longreach
24° 24°
X
X X
X
X X
XX
X
X
Charleville X
X
X XX
X X
X
#
27°
X
X
X 27°
X
X #
X
X
Brisbane
154°
139° 142° 145° 148° 151°

5. Measuring Canopies - Why
Open Woodland Woodland Forest Closed Forest
40
Tall
30
Height
20
10
Low
Very Low
0
0 0.2 0.4 0.6 0.8 1
FPC

6. Measuring Canopies - Sampling Methods
Measurements
Cover
FPC
Basal Area
Clumping
3 × 100m + Trac
Distance
Branch
Crown
Green
Dead
1 1 1
2 1 1
3 1 1
4 1 1
5 1 1
6 1 1
7 1 1
8 1 1
9 1 1
10 1 1

7. Introduction to CCP ↔ FPC Model
Crowns ∈ Forest
P( )=ccp
− λπ r 2
ccp = 1 − e
Leaves ∈ Crowns
P( )=fpc
− GΩL
fpc = 1 − e
Leaves ∈ Forest
P( )=P( )×P( )
fpc
ccp = Ψ
1 − ( 1 − fpc ) Ω

8. Clumping in Forests
CCP ↔ FPC relationship
depends on “clumping”
Clumping (Ω,Ψ) is
deviation from “Random”
Modelled using Neyman
ccpclumped
= eb (ccp −1)
ccp

9. Fitting the Clumping Ratio Model
Invert FPC↔CCP and fit – looks OK 
Fitted Clumping Parameters
1
gold01_
02
gold02_ 03
suns01_
02
Ω log ( 1 − fpc )
= = a ccp eb (ccp −1) chin04_ 01
gold01_
chin03_
03 01
Ψ
0.8
 fpc  suns01_
01
gold02_ 01 02
chin04_
01
adav02_
log 1 −
ccp 
chin04_
04
chin02_
02
  chin02_
03
0.6
€€
€€
€
€
1
adav01_
03
adav02_
01
chin01_ 01
chin01_
chin02_ 03
01
02
aram02_
0.4 adav01_ 02
chin04_
1n 02
1r1a 0.4
gunp01_
02
2n 02
2a adav03_ 0.38
sr chat01_
2r adav03_
0301
adav01_ 03
quil02_
01 0.36
0.2 quil01_
01 01
b
aram02_
suns01_
04 0.34
quil01_
quil02_04
01
quil02_
02
cade01_ 03
01 03
quil01_
gunp01_ 0.32
adav01_02
cade01_
04
quil01_ 04
quil02_
02
adav03_03
chat01_
aram02_
aram02_ 02
chat01_
02
01
04 0.3 a
gunp01_01
0.955 0.96 0.965 0.97 0.975 0.98 0.985
0
0.2 0.4 0.6 0.8 1
CCP

10. Fitted CCP ↔ FPC Model
Apply clumping model to FPC↔CCP model
No analytical inversion – use root finding 
1 gold02_ 02
gold01_ 02
suns01_ 03
chin03_ 01
gold01_ 01
chin04_ 03
adav02_ 02
chin04_ 01
suns01_ 01
gold02_ 01
Projected Crown Cover
0.8
chin04_ 04
chin02_ 02
chin02_ 03
0.6 adav01_ 03
fpc
ccp =
aram02_ 03
adav02_ 01
chin01_ 01
chin02_
chin01_ 02
01
a ccp eb (ccp −1)
1 − ( 1 − fpc )
chin04_ 02
adav01_ 102 n
1 a gunp01_ 02
1 r
adav03_ 02
0.4
2 n
chat01_ 01
adav03_ 03
2 a
sr 2 r
quil02_ 03
adav01_ 01 aram02_ 01
Basal Area
50
quil01_ 01
suns01_ 04
quil01_ 04
quil02_ 01
0.2
quil02_ 02
gunp01_ 03
quil01_ 03
cade01_ 01
adav01_
cade01_
quil02_
quil01_ 04 02
02
04
30
chat01_ 03
aram02_ 04
adav03_ 01
chat01_ 02
10
gunp01_ 01
aram02_ 02
0
0 0.2 0.4 0.6 0.8 1
Projected Foliage Cover

11. Measuring Clumping (Ω)
L a rge C a n o p y G a p C o n tin u o u s C a n o p y L a y e r G a p in te r s p e r s e d w ith s m a ll c a n o p ie s
1800
1500
Trac
Downwelling PAR (mMol/s/m2)
1200
Measures
canopy “gaps” 900
Compares 600
distribution to
300
“Random”
Difference is 0
33 35 37 39 41 43 45 47 49
clumping Ω Distance along transect (m)

12. Fitting Trac Omega (Ω) ↔ Clumping Model
1
Omega ↔ CCP
Trac Clumping
modelled clumping
0.95
0.9 There is a fit but…
Omega is a function
0.85
of Zenith!
0.88 0.9 0.92 0.94 0.96
Modelled Clumping
Correct for  1.02
Clumping Correction
zenith
Zenith ↔ residuals 1.00
•Try 1st to 4th order fits
•3rd is best (⇓ error σ2) 0.98
•Agrees with published
work (Chen)  0 12 24 36
Zenith Angle
48 60

13. Improved Omega (Ω) ↔ Clumping Model
Same Relationship but improved confidence 
But limited number & variety of sites 
1
0.95
Trac Clumping
0.9
0.85
0.86 0.88 0.9 0.92 0.94 0.96
Modelled Clumping

14. Clumping ↔ CCP Models
Can recover within canopy clumping (Ψ)
Possible clumping > 1 implies “Regularity” (overdispersion)
Clumping Factors
1
0.95
Global (Ω)
0.9
Clumping
0.85
Crown (Ψ)
0.8
Ψ eb (1−ccp)
0.75
Ψ = Ω = Ωtrac
Ω a ccp
0.7
0 0.2 0.4 0.6 0.8 1
CCP

15. LAI ↔ Basal Area
Theoretical models → more leaf ≈ more BA
Log(LAI) = a + b log(BA)
•As reported in literature
LAI = a BAb
•b ⇒ 1 4
∴LAI = a BA
•Significant 3
Effective LAI (LΩ)
outliers 
2
1
log(1 −fpc)
LΩ=−
0 G →0.5
0 10 20 30 40
Basal Area

16. Leaf Angle Distribution ↔ Basal Area
Estimate G from Trac & Field Data
Pushing data to its limits 
Poor fit but sensible result
0.8
Leaf Angle Distribution
50o
0.6
70o
0.4
57o
 fpc 
log 1 −
 ccp 

G =− 0.2
LΨ
0 10 20 30 40
Basal Area

17. Leaf Angle Corrected LAI ↔ Basal Area
Slightly different estimator
Improved confidence 
High BA site has “flatter” leaves
log(1 − fpc)
4 LΩ = −
a + b BA
Effective LAI (LΩ)
3
2
1
0
0 10 20 30 40
Basal Area

18. FPC ↔ Basal Area Nonlinear Fit
Final model has simple form 
But still has wide confidence intervals on Statewide data 
1
BA
−
0.8
fpc = 1 − e a + b BA
0.6
FPC
0.4
CCP
100%
0.2
60%
20%
0 10 20 30 40 50
Basal Area

19. FPC ↔ LAI Final Relationship
We have estimated G and Ω
LAI ≈ 1.7 as FPC ⇒ 0 ∴ Mature trees have leaves 
fpc = 1 − e − GΩL
8
Leaf Area Index
Steeper Leaf Angles ≈ 70o
6
Within
Canopy
FPC≈ 50%
4
Fitted Leaf Angles ≈ 70o to 50o
2
Shallower Leaf Angles ≈ 50o
0.2 0.4 0.6 0.8
FPC

20. Conclusions & Future Work
Canopy parameters are interrelated
Probably different for regrowth – Sample?
No single “best” structure measure
Leaf scale parameters important
LAI & Leaf Angle
Clumping
Thanks to:
Rob Hasset
Site selection & photography
Sel Counter-
Field campaign
Slatters
Additional field sites