Training on tree allometric equations,
December 10-14th 2012, Lusaka, Zambia
Use and prediction
•
Dr. Laurent Saint-André,
Gael Sola, Dr. Matieu
Using a
biomass/volume
equation
Range of calibration
û
Confidence intervals for the predictions
û
Linear regression
Ä
∑
−
−
+
±
−
i
i
X
X
X
X
n
t
X
Y
2
2
2
/
1
)
(
)
(
1
ˆ
)
(
α
σ
Confidence interval for the mean
∑
−
−
+
+
±
−
i
i
X
X
X
X
n
n
t
X
Y
2
2
2
/
1
)
(
)
(
1
ˆ
)
(
α
σ
Confidence interval for an
individual prediction
2
/
1
−
α
Using a
biomass/volume
equation
Example: Isoberlinia doka
û
Red – model
Yellow – confidence interval for
an individual tree
Green – confidence interval for
the mean
Using a
biomass/volume
equation
General case for linear and non linear regression,
utilization of the delta-method (Serfling, 1980)
Ä
2
)
2
/
1
(
.
)
(
X
t
S
y
Y
±
−
α
with :
Confidence interval for the
mean
∂
∂
∑
∂
∂
=
β
β
βY
Y
S
y.
ˆ
.
' 2z
c
c
y
X
S
t
X
Y
ˆ
,
2
2
)
2
/
1
(
.
ˆ
.
ˆ
.
)
(
±
−
α
+
σ
σ
Confidence interval for an
individual prediction
The variance for
Y
∂
∂
β
Y
The matrix of the derivative of Y
toward the model parameters
'
∂
∂
β
Y
The transposed matrix of
∂
∂
β
Y
β
∑
ˆ
The covariance matrix of the parameters
2
ˆ
σ
The variance of errors of the given
compartment
c
c,
ˆ
σ
The variance of errors of the given
compartment regarding the whole
system of equation
z
X
ˆ
The weighting function
Confidence intervals for the predictions
Using a
biomass/volume
equation
0 50 100 150 200 250 300 350 0 0.1 0.2 0.3 0.4 0.5D2H (m3)
D
M
S
te
m
(
kg
)
M easured SimulatedAge 7.3 years
0 2 4 6 8 10 12 14 16 18 0 0.1 0.2 0.3 0.4 0.5D2H (m 3)
D
M
B
ar
k
(k
g
)
M easured SimulatedAge 7.3 years
-8 -6 -4 -2 0 2 4 6 8 0 0.005 0.01 0.015 D 2H ( m3) M easured SimulatedAge 1.12 years
-4 -2 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5D2H (m3)
D
M
D
ea
d
B
(
kg
)
M easured SimulatedAge 7.3 years
Example : Eucalyptus in Congo (Saint-André et al. 2005)
Using a
biomass/volume
equation
From the tree to the stand, Monte-Carlo
simulations
û
)
,
(
)
,
(
X
X
f
Y
=
β
+
ε
γ
Model for mean
Model for variance
h
d
age
e
age
5
2
4
3
2
1
(
β
β
β
β
)
β
µ
=
+
−
+
−
(
2
)
γ
2
)
γ
ε
=
N(0,
1
d
h
Y = f(input data, parameters, error term)
β
,
γ
: estimated by the fitting procedure. We got their
mean and their asymptotic standard deviation.
They are correlated (within a given compartment and
between compartments)
Diameter (d) and height (h). Both
contained errors. We assumed that
σ
=0.3 cm for diameter
σ
=3% of height if less than 15 m
Using a
biomass/volume
equation
Biomasse aerienne
10
15
20
25
30
35
01/10/2000 09/01/2001 19/04/2001 28/07/2001 05/11/2001 13/02/2002 24/05/2002 01/09/2002
Date
B
io
m
as
se
(
t/
h
a)
Y = a + b.X
+
Ν( 0 ,σ)
Only the error term vary
From the tree to the stand, Monte-Carlo
simulations
Using a
biomass/volume
equation
From the tree to the stand, Monte-Carlo
simulations
û
Error term & parameters of the mean vary
Biomasse aerienne
10
15
20
25
30
35
01/10/2000 09/01/2001 19/04/2001 28/07/2001 05/11/2001 13/02/2002 24/05/2002 01/09/2002Date
B
io
m
as
se
(
t/
h
a)
Y = a + b.X
+
Ν( 0 ,σ)
Ν( α
,σα)
Ν( β
,σβ)
Using a
biomass/volume
equation
From the tree to the stand, Monte-Carlo
simulations
û
Error term, parameters of the mean, and input data vary
Biomasse aerienne
10
15
20
25
30
35
01/10/2000 09/01/2001 19/04/2001 28/07/2001 05/11/2001 13/02/2002 24/05/2002 01/09/2002
Date
B
io
m
as
se
(
t/
h
a)
Y = a + b.
X
+
Ν( 0 ,σ)
Ν( α
,σα)
Ν( β ,σβ)
Ν( Ξ
,σξ)
Using a
biomass/volume
equation
For most compartments, standard errors were small with regard
to standing biomass (below 10%)
Total
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
180.00
200.00
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
Age (months)
B
io
m
a
s
s
(
t/
h
a
)
Biomass
Interval of confidence
G3A
G3B
G3C
G3D
G2
G1
At 100 months :
Total biomass = 128 ± 1.9 t/ha
Above-Ground = 104 ± 1.8 t/ha
Below-ground = 24 ± 1 t/ha
From the tree to the stand, Monte-Carlo
simulations
Using a
biomass/volume
equation
Except for the dead branches biomass (worst model)
At 100 months :
Dead B = 1.9 ± 0.3 t/ha
Dead Branches
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
0.00
20.00
40.00
60.00
80.00
100.00
120.00
140.00
160.00
Age (months)
B
io
m
a
s
s
(
t/
h
a
)
Biomass
Interval of confidence
G3A
G3B
G3C
G3D
G2
G1
From the tree to the stand, Monte-Carlo
simulations
Using a
biomass/volume
equation
Same principles of simulations can be applied also to ratios
(root-shoot, wood density, etc.)
From the tree to the stand, Monte-Carlo
simulations
Using a
biomass/volume
equation
From the tree to the stand, Monte-Carlo simulations
û
Biomass increments
Plot
Age (months)
Year
Start date
End date
Total biomass
increment (t/ha)
G3A
9 to 22
2001
23/03/2001
23/03/2002
19.5 1.6
G3A
22 to 34
2002
23/03/2002
02/04/2003
25.6 3.3
G3B
26 to 38
2001
08/01/2001
27/12/2001
12.0 0.9
G3B
38 to 49
2002
27/12/2001
28/11/2002
12.9 1.0
G3C
49 to 61
2001
12/01/2001
11/01/2002
9.6 1.9
G3C
61 to 74
2002
11/01/2002
31/01/2003
15.3 2.4
G3D
74 to 86
2001
13/01/2001
11/01/2002
14.7 2.9
G3D
86 to 99
2002
11/01/2002
31/01/2003
10.9 2.8
Standard errors where relatively large (from 7 to 25 % of the biomass increment)
For the eddy-correlation site,
∆
biomass2002 = 12.9 1.0 t/ha/year
Using a
biomass/volume
equation
Example Sicard et al. 2005 (accepted in Trees, Structure and Function).
Fertilization effect on Spruce and Douglas fir
Step 1 : analysis of the bar chart
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0
20
40
60
80
100
120
140
c
1.30(cm)
F
re
q
ue
n
cy
(%
)
f - plot
f - sampled trees
nf - plot
nf - sampled trees
1a) Norway Spruce
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0
20
40
60
80
100
120
140
c
1.30(cm)
F
re
q
ue
n
cy
(%
)
f - plot
f - sampled trees
nf - plot
nf - sampled trees
1a) Norway Spruce
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0
20
40
60
80
100
120
140
c
1.30(cm)
F
re
q
ue
n
cy
(%
)
f - plot
f - sampled trees
nf - plot
nf - sampled trees
1b) Douglas Fir (DF)
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
0
20
40
60
80
100
120
140
c
1.30(cm)
F
re
q
ue
n
cy
(%
)
f - plot
f - sampled trees
nf - plot
nf - sampled trees
1b) Douglas Fir (DF)
Behind the single estimation of biomass, the understanding of
the ecosystem functioning
Using a
biomass/volume
equation
Example Sicard et al. 2005 (accepted in Trees, Structure and Function).
Fertilization effect on Spruce and Douglas fir
Behind the single estimation of biomass, the
understanding of the ecosystem functioning
û
Step 2 : analysis of the biomass equations
a: Leaves 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 c1.30(cm) B io m a s s (k g ) NS nf observations NS nf estimation NS f observations NS f estimation a: Leaves a: Leaves 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 (cm) B io m a s s (k g ) NS nf observations NS nf estimation NS f observations NS f estimation NS nf observations NS nf estimation NS f observations NS f estimation a: Leaves 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 c1.30(cm) B io m a s s (k g ) NS nf observations NS nf estimation NS f observations NS f estimation a: Leaves 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 c1.30(cm) B io m a s s (k g ) NS nf observations NS nf estimation NS f observations NS f estimation a: Leaves a: Leaves 0 5 10 15 20 25 30 35 40 45 50 0 20 40 60 80 (cm) B io m a s s (k g ) NS nf observations NS nf estimation NS f observations NS f estimation NS nf observations NS nf estimation NS f observations NS f estimation b: Stemwood 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 c1.30(cm) B io m a s s (k g ) DF nf observation DF nf estimation DF f observations DF f estimation b: Stemwood b: Stemwood 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 (cm) B io m a s s (k g ) DF nf observation DF nf estimation DF f observations DF f estimation DF nf observation DF nf estimation DF f observations DF f estimation b: Stemwood 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 c1.30(cm) B io m a s s (k g ) DF nf observation DF nf estimation DF f observations DF f estimation b: Stemwood 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 c1.30(cm) B io m a s s (k g ) DF nf observation DF nf estimation DF f observations DF f estimation b: Stemwood b: Stemwood 0 50 100 150 200 250 300 350 400 0 20 40 60 80 100 (cm) B io m a s s (k g ) DF nf observation DF nf estimation DF f observations DF f estimation DF nf observation DF nf estimation DF f observations DF f estimation
Douglas differences are only significant
for the TRUNK
Spruces differences are only significant
for the LEAVES
Using a
biomass/volume
equation
Behind the single estimation of biomass, the understanding of the ecosystem
functioning
û
Step 3 : decomposition of the fertilization effect on a per hectare basis
A
Control Fertilized Branches Leaves Stem bark Stem wood 24.1 26.0 81.3 9.0 81.3 9.0 17.2 22.2 17.0 18.3 59.9 6.9 72.4 8.2 15.0 19.0B
20.2 21.7 67.9 7.6 80.7 8.8 17.7 23.5C
Branches Leaves Stem bark Stem wood 7.5 14.0 83.5 12.4 101.2 12.4 7.5 14.0 9.2 17.3 97.5 14.6 123.3 9.6 18.1 15.2 14 27.3 121.5 18.9 151.3 13.6 26.4 18.8Control Fertilized Control Fertilized
Control Fertilized Control Fertilized Control Fertilized
Norway Spruce
Douglas fir
27.9
92
26
11
29.7
113
23
13
27.8
189
15
23
23
127
12
19
33
104
31
12
37
126
28
14
41
232
21
29
36
159
18
25
C
130=64.6cm
C
130=57.0cm
C
130=61.6cm
C
130=69.3 cm
C
130=70.4cm
A
Control Fertilized Branches Leaves Stem bark Stem wood 24.1 26.0 81.3 9.0 81.3 9.0 17.2 22.2 17.0 18.3 59.9 6.9 72.4 8.2 15.0 19.0B
20.2 21.7 67.9 7.6 80.7 8.8 17.7 23.5C
Branches Leaves Stem bark Stem wood 7.5 14.0 83.5 12.4 101.2 12.4 7.5 14.0 9.2 17.3 97.5 14.6 123.3 9.6 18.1 15.2 14 27.3 121.5 18.9 151.3 13.6 26.4 18.8Control Fertilized Control Fertilized
Control Fertilized Control Fertilized Control Fertilized
