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Modelling the spatial structure of complex stands by Point Processes
Marie Ange Ngo Bieng 1,2 C. Ginisty 1
F. Goreaud 2
1 Research Team « Écosystèmes Forestiers » (Nogent / Vernisson)
2 Laboratory of engineering for complex systems (Clermont - Ferrand), FRANCE
IUFRO SAULT, july 29- August 2, 2007:
Complex Stand Structures and Associated Dynamics: measurement indices and modeling approaches; Sault Ste. Marie, Ontario, Canada
Agricultural and Environmental Engineering Research
Overview
I. Modelling complex stands structure:
Why?
II. Characterising our oak-pine stands spatial structure
III. Modelling: reproducing the identified stands spatial structure
IV. Conclusions and perspectives
Modelling the spatial structure of complex stands
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I. Modelling Complex stands structure
I. Modelling complex stand structure
Why?
Complex stand
Individual Based Model (IBM) spatially explicit
Growth modelling
Modelling the spatial structure of complex stands
5
Initial state Simulation
proceed
IBM
Description and Location of each tree
Modelling Real stand structure Real stand Virtual stand
Modelling the spatial structure of complex stands
I. Modelling complex stand structure
Why?
Our aim: present a model of spatial structure of oak-pine mixed stands.
We are focusing on mixed stands of sessile oak (Quercus petraea) – Scots-pine (Pinus
sylvestris) of the Orleans forest (France).
Modelling the spatial structure of complex stands
I. Modelling complex stand structure
7
Our aim: modelling the spatial structure
real virtual
Real characteristics Reconstruction
2
1
2
1 Typology of spatial structure (part 2) Point Processes (part 3)
Modelling the spatial structure of complex stands
I. Modelling complex stand structure
II. Characterising our oak-pine stands spatial structure
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The stand is considered as a point pattern.
The stand S, a set of trees T n
Characterise by their locations (X n , Y n )
T
1(x
1, y
1) T
2(x
2y
2) .
. .
T
n(X
n, Y
n)
x y
Plot 20
Horizontal distribution of trees in space
Modelling the spatial structure of complex stands
II. Characterising stands spatial structure
The stand is considered as a spatial point pattern
A spatial Point Process is a stochastic model that governs the location of points in an area (Cressie, 1993).
An appropriate tool for examining spatial structure of trees in a forest stand
Modelling the spatial structure of complex stands
II. Characterising stands spatial structure
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Characterising the spatial point pattern by its second-order intensity, described by Ripley function (Ripley, 1977)
aggregation \random\ regularity attraction \ independence\ repulsion
random
regular aggregated
0 100
0 100
L(r) = 0 Κ( r)= π r 2
0 100
0 100
L(r) < 0
0 100
0 100
L(r) >0
Modelling the spatial structure of complex stands
II. Characterising stands spatial structure
Characterising the spatial point pattern by L(r) (Besag in Ripley, 1977)
L(r) = (K(r)/ π) 1/2 – r
Random Regular Aggregated
-3 -2 -1 0 1 2 3 4
10 20 30 40 50
L(r)
range r
Ripley curves for the 3 types of spatial distributions
Modelling the spatial structure of complex stands
II. Characterising stands spatial structure
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25 1ha mapped plots
Defining sub-populations:
Others
Canopy pines
Canopy and
Understorey oak
Modelling the spatial structure of complex stands
II. Characterising stands spatial structure
Plot 20
Modelling the spatial structure of complex stands
II. Characterising stands spatial structure
Type 3 :
- Random structure of oak - Slight aggregation of pine - Slight repulsion
-3 -2 -1 0 1 2 3
2 6 10 14 18 22 26 30 2 6 10 14 18 22 26 30 2 6 10 14 18 22 26 30
plot 7 plot 14 plot 17 plot 20 plot 21 plot 24
L O (r) real L P (r) real L OP real
Plot 20
0 25 50 75 100
-50 -25 0 25 50
canopy oak canopy pine
Typology of spatial structure
based on Ripley and Intertype computed 25 1 ha plots
Type 1 Type 2
Type 3 Type 4
15
III. Reproducing the identified stand spatial structure
The point processes used
Poisson process: random pattern
0 20 40 60 80 100
y
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
L(r) LIC- LIC+
Modelling the spatial structure of complex stands
III. Reproducing stands spatial structure
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The point processes used
Neyman-scott process: aggregated pattern
0 20 40 60 80 100
0 20 40 60 80 100
x
y
-2 0 2 4 6 8 10
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 L(r) LIC- LIC+
Modelling the spatial structure of complex stands
III. Reproducing stands spatial structure
The point processes used
Hard Core or simple inhibition process: regular or repulsive pattern
0 20 40 60 80 100
0 20 40 60 80 100
x
y
-10 -8 -6 -4 -2 0 2
2 4 6 8 10 12 14 16 18 20 22 24 26 28 30
L(r) LIC- LIC+
Modelling the spatial structure of complex stands
III. Reproducing stands spatial structure
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Our model of structure
rp d
5 Parameters:
- N1 Pines
1) Nb of clusters 2) Rp: radius
3) l: regularity distance
N2 oaks
4) d: intertype distance 5) Pa (probability when d
intertype is not respected)
1
III. Reproducing stands spatial structure
Modelling the spatial structure of complex stands
Our model of structure
L o (r) simulated L P (r) simulated L op simulated
III. Reproducing stands spatial structure
L o (r) real L p (r) real L op real
-2 -1 0 1 2 3
2 6 10 14 18 22 26 30 2 6 10 14 18 22 26 30 2 6 10 14 18 22 26 30
plot 7 plot 14 plot 17 plot 20 plot 21 plot 24
Modelling the spatial structure of complex stands
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Our model of structure: fitting procedure by the least square criterion between the real and the simulated curves
What are the parameters that minimize the difference between Lreal and Lsim?
∑
∑
∑ = = =
− +
− +
−
= 30
2
2 30
2
2 2
30
2
)) ( )
( (
)) ( )
( (
)) ( )
( (
r
th m sim
m r
th m sim
m th
m sim
m r
r LOP r
LOP r
LP r
LP r
LO r
LO SCE 1
∑
∑
∑ = σ σ
= σ σ
= σ − σ + − + −
= 30
2
2 30
2
2 30
2
2 ( ( ) ( )) 2 ( ( ) ( )) ( ( ) ( ))
r
th sim
r
th sim
r
th
sim r LO r LP r LP r LOP r LOP r
LO SCE
SCE = SCE 1 + SCE 2
III. Reproducing stands spatial structure
Euclidean distance between the real and simulated mean curves
Euclidean distance between the real and the simulated standard deviations
Modelling the spatial structure of complex stands
Our model of structure: fitted parameters
For Pines
Nb of clusters: 38 /ha Radius: 8m
Regularity distance: 10m
For Oak
Intertype distance: 4m Pa: 0.15
SCE min = 17.77
III. Reproducing stands spatial structure
Modelling the spatial structure of complex stands
0 25 50 75 100
0 25 50 75 100
canopy oak canopy pine
Simulated plot
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Our model of structure: comparison
III. Reproducing stands spatial structure
L simulated L real
-4 -2 0 2 4
2 8 14 20 26 6 12 18 24 30 4 10 16 22 28
Plot 20 Simulated plot
0 25 50 75 100
0 25 50 75 100
canopy oak canopy pine 0
25 50 75 100
-50 -25 0 25 50
canopy oak canopy pine
