Use and prediction

(1)

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Use and prediction

Laurent Saint-André, Gael Sola, Matieu Henry, Nicolas Picard

To cite this version:

Laurent Saint-André, Gael Sola, Matieu Henry, Nicolas Picard. Use and prediction: Using a biomass

- volume equation. Training Workshop on Tree Allometric Equations, May 2014, Colombo, Sri Lanka.

pp.19 slides. �hal-02792950�

(2)

Training workshop on tree allometric equations,

Forest Department Sri Lanka, May 20 – 24th 2014

Use and prediction

• Dr. Laurent Saint-André, Gael

Sola, Dr. Matieu Henry, Dr.

Nicolas Picard

(3)

Using a

biomass/volume

equation

Range of calibration

û

Confidence intervals for the predictions

û

Linear regression

Ä

Confidence interval for the mean

Confidence interval for an

individual prediction

quantile at the level 1-

α

/2 for the student low at n-2 degrees of freedom

∑

−

+

±

₋

i

X

n

t

X

Y

₂

2

2 /

1 )

(

)

(

1 ˆ

)

(

_α

σ

∑

−

+

±

₋

i

X

n

t

X

Y

₂

2

2 /

1 )

(

)

(

1 ˆ

)

(

_α

σ

2 /

1 −

α

t

(4)

Using a

biomass/volume

equation

Example: Isoberlinia doka

û

Red – model

Yellow – confidence interval for

an individual tree

Green – confidence interval for

the mean

(5)

Using a

biomass/volume

equation

General case for linear and non linear regression,

utilization of the delta-method (Serfling, 1980)

Ä

with :

Confidence interval for the

mean

Confidence interval for an

individual prediction

The variance for

The matrix of the derivative of Y

toward the model parameters

The transposed matrix of

The covariance matrix of the parameters

The variance of errors of the given

compartment

The variance of errors of the given

compartment regarding the whole

system of equation

The weighting function

Confidence intervals for the predictions

û

2 )

2 /

1 (

.

)

(

X

t

S

_y

Y

±

₋

_α









∂

∑









∂

=

β

Y

S

_y

.

ˆ

.

' 2

z

c

y

X

S

t

X

Y

ˆ

,

2

2 )

2 /

1 (

.

ˆ

.

ˆ

.

)

(

±

₋

_α

+

σ

Y









∂

β

Y

'









∂

β

Y









∂

β

Y

β

∑

ˆ

2 ˆ

σ

c

c,

ˆ

σ

z

X

ˆ

(6)

Using a

biomass/volume

equation

0 50 100 150 200 250 300 350 0 0.1 0.2 0.3 0.4 0.5

D2H (m3)

D

M

S

te

m

(

kg

)

M easured Simulated

Age 7.3 years

0 2 4 6 8 10 12 14 16 18 0 0.1 0.2 0.3 0.4 0.5

D2H (m 3)

D

M

B

ar

k

(k

g

)

M easured Simulated

Age 7.3 years

-8 -6 -4 -2 0 2 4 6 8 0 0.005 0.01 0.015 D 2H ( m3) M easured Simulated

Age 1.12 years

-4 -2 0 2 4 6 8 10 12 14 0 0.1 0.2 0.3 0.4 0.5

D2H (m3)

D

M

D

ea

d

B

(

kg

)

M easured Simulated

Age 7.3 years

Example : Eucalyptus in Congo (Saint-André et al. 2005)

(7)

Using a

biomass/volume

equation

From the tree to the stand, Monte-Carlo

simulations

û

Model for mean

Model for variance

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

σ

=1 m if above 15 m

)

,

(

)

,

(

X

f

Y

=

β

+

ε

γ

h

d

age

e

age

₅

2

4

3

2

1 (

β

)

β

µ

=

+

−

+

−

(

2 )

γ

2 )

γ

ε

=

N(0,

1 d

h

(8)

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

(9)

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 01/09/2002

Date

B

io

m

as

se

(

t/

h

a)

Y = a + b.X

+

Ν( 0 ,σ)

Ν( α

,σα)

Ν( β

,σβ)

(10)

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)

(11)

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

(

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

(12)

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

(

t/

h

a

)

Biomass

Interval of confidence

G3A

G3B

G3C

G3D

G2

G1

From the tree to the stand, Monte-Carlo

simulations

(13)

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

(14)

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)

(15)

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

(16)

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

(17)

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 19.0

B

20.2 21.7 67.9 7.6 80.7 8.8 17.7 23.5

C

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 18.8 Control 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

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 19.0

B

20.2 21.7 67.9 7.6 80.7 8.8 17.7 23.5

C

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 18.8 Control Fertilized Control Fertilized