Transcript Edinburgh Forest Model

Bayesian calibration and comparison of
process-based forest models
Marcel van Oijen & Ron Smith
(CEH-Edinburgh)
Jonathan Rougier (Durham Univ.)
Contents
1. Bayesian calibration of a forest model
2. … what measurements to take?
3. Bayesian comparison of forest models
1. Bayesian calibration of a forest model
Process-based forest models
Environmental
scenarios
Initial
values
Nutr. C
H2O
H2O
NPP
Trees
Nutr.
Nutr. C
Soil
Parameters
Height
Atmosphere
C H2O
Nutr.
Subsoil (or run-off)
Model
H2O
Soil C
Using data
We need a method that:
1. Quantifies how uncertainty about inputs and model
structure causes output uncertainty
2. Efficiently uses data, on inputs & outputs, to reduce
uncertainties
BASic FORest model (BASFOR)
39 parameters
BASFOR
12 output variables
BASFOR: Inputs
Parameter Unit
BETA
(-)
CL0
(kg m-2)
CLITT0
(kg m-2)
CO20
(ppm)
CR0
(kg m-2)
CSOMF0
(kg m-2)
CSOMS0
(kg m-2)
CW0
(kg m-2)
FLITTSOMF (-)
FLMAX
(-)
FSOMFSOMS
(-)
FW
(-)
GAMMA
(-)
KCA
(m2)
KCAEXP
(m2)
KDL
(d-1)
KDLITT
(d-1)
KDR
(d-1)
KDSOMF
(d-1)
KDSOMS
(d-1)
KDW
(d-1)
KH
(m)
KHEXP
(-)
KLAIMAX
(m2 m-2 mm-1)
KNMIN
(kg m-2)
KNUPT
(kg m-2 d-1)
KTA
(degC-1)
KTB
(degC)
KTREE
(m2 m-2)
LUE0
(kg MJ-1)
NLCONMAX (kg kg-1)
NLCONMIN (kg kg-1)
NLITT0
(kg m-2)
NMIN0
(kg m-2)
NRCON
(kg kg-1)
NSOMF0
(kg m-2)
NSOMS0
(kg m-2)
NWCON
(kg kg-1)
SLA
(m2 kg-1)
Min
0.4
0.0001
0.15
320
0.0001
5
1
0.0001
0.4
0.25
0.01
0.52
0.4
3.65
0.333
0.0007
0.0007
0.000135
0.000028
0.0000028
0.00004
2.5
0.2
0.002
0.0005
0.0005
0.02
10
0.35
0.001
0.03
0.01
0.005
0.0001
0.02
0.2
0.05
0.0005
5
Max
0.6
0.01
0.6
380
0.01
10
3
0.01
0.8
0.35
0.1
0.62
0.6
14.6
0.5
0.0028
0.0028
0.00054
0.00011
0.000011
0.00016
10
0.33
0.008
0.002
0.002
0.04
30
0.65
0.003
0.05
0.03
0.02
0.002
0.04
0.4
0.2
0.002
15
BASFOR
12 output variables
BASFOR: Prior pdf for parameters
Prior parameter marginal probability distributions (uniform)
4000
4000
2000
0
4000
2000
0
CL0
0.005
0.01
4000
2000
0
0.5
FW
0.6
0.7
0
2000
0
CR0
0.005
0.01
0
4000
4000
2000
2000
0 GAMMA
0.4
0.6
0.8
0
0
CW0
0.005
0.01
4000
4000
2000
2000
0 BETA
0.4
0.6
0.8
4000
0
2
4
0
2000
CO20
350
0
300
400
4000
2000
KCA
4000
0
1
0
FLMAX
0.3
0.35
4000
2000
KCAEXP
0.5
0
0.25
2000
KDL
0
0.005
0.01
0
0
KDR
0.5
1
-3
x 10
4000
4000
4000
4000
4000
2000
2000
2000
2000
2000
0
2
KDW
4
6
0
3
KH
4
5
0
0.2
KHEXP
0.3
0.4
0
0
KLAIMAX
0.005
0.01
0
4000
2000
0
KNMIN
1
-5
2000
0
0.02
KTA
0.03
0.04
0
KNUPT
1
0
10
KTB
20
30
4000
4000
2000
2000
KTREE
0
0.4
0.6
0.8
0
1
LUE0
2
3
x 10
4000
4000
2000
2000
0 NLCONMIN
0.01
0.02
0.03
2
-3
x 10
4000
2000
0
-3
x 10
4000
2
NLCONMAX
0
0.04
0.05
0.06
-3
x 10
4000
4000
4000
4000
4000
2000
2000
2000
2000
2000
0
0.02
NRCON
0.03
0.04
0
0
NWCON
1
2
0
0
SLA
20
40
0
0
CLITT0
0.5
1
0
4000
2000
CSOMF0
6
8
10
0
1
CSOMS0
2
3
-3
x 10
4000
4000
2000
0
4000
2000
0
NLITT0
0.01
0.02
4000
2000
0
0.2
NSOMF0
0.3
0.4
0
4000
2000
0
NSOMS0
0.1
0.2
0
2000
0
NMIN0
1
2
-3
x 10
4000
4000
2000
0
4000
2000
0
KDLITT
2
4
-3
x 10
0
2000
KDSOMF
0
1
2
-4
x 10
0
0
KDSOMS
1
2
-5
x 10
4000
0
0.4
2000
FLITTSOMF
0.6
0.8
0
0
FSOMFSOMS
0.05
0.1
Example: Simulating growth of Norway spruce
Skogaby
BASFOR: Prior predictive uncertainty
20
Biomass
10
1.5
15
h
Cl
Cw
1
10
5
0.5
5
0
5000
10000
0
15000
3
0.5
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
5
0
0
5000
10000
-0.5
15000
0
0.1
0.04
5000
10000
0.05
10
0.02
0
5000
10000
0
15000
0.6
0
15000
NCl
0.06
0
5000
10000
150
Nmin
Miny
0.15
0.2
0.1
0.05
0
5000
10000
Time
15000
0
5
0
15000
0.2
0.4
0
0
10
0.15
0
0
15000
LAI
NPPy
Cr
Ntree
10000
1
1
Nsoil
5000
1.5
2
0
0
Csoil
0
Prior
uncertainty
for Skogaby
BASFOR: Predictive uncertainty
Height
0
5000
10000
Time
15000
100
50
0
Time
BASFOR: Predictive uncertainty
39 parameters
High input
uncertainty
12 output variables
BASFOR
High output
uncertainty
Calibration of
parameters
Data:
measurements of
output variables
Bayes’ Theorem
f = the model, e.g. BASFOR
P(|D) = P() P(D| ) / P(D)  P() P(D|f())
“Posterior
distribution of
parameters”
“Prior distribution
of parameters”
“Likelihood” of data,
given mismatch with
model output
Finding the posterior: MCMC
P(|D)  P() P(D|f())
MCMC: walk through parameter-space →set of visited points
approaches the posterior parameter distribution P(|D)
[e.g. using Metropolis-Hastings random walk]
Sample of 104 -105 parameter vectors from the
posterior distribution P(|D) for the parameters
MCMC: Metropolis-Hastings random walk
Metropolis (1953) algorithm
1.
Start anywhere in parameter-space: p1..39(i=0)
2.
Randomly choose p(i+1) = p(i) + δ
3.
IF: [ P(p(i+1)) P(D|f(p(i+1))) ] / [ P(p(i)) P(D|f(p(i))) ] > Random[0,1]
THEN: accept p(i+1)
ELSE: reject p(i+1)
i=i+1
4.
IF i < 104 GOTO 2
Sample of 104 -105 parameter vectors from the
posterior distribution P(|D) for the parameters
Forest data from Skogaby (Sweden)
Skogaby
Planted: 1966, (2300 trees ha-1)
Weather data: 1987-1995
Soil data: C, N, Mineralisation rate
Tree data: Biomass, NPP, Height,
[N], LAI
BASFOR: Prior predictive uncertainty
20
Biomass
10
1.5
15
h
5
0.5
5
0
5000
10000
0
15000
3
10000
0.5
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
5
0
5000
10000
-0.5
15000
0
0.04
10000
0.05
10
0.02
0
5000
10000
0
15000
0.6
0
15000
NCl
0.1
5000
Csoil
0
0.06
0
5000
10000
150
Nmin
Miny
0.15
0.2
0.1
0.05
0
5000
10000
Time
15000
0
5
0
15000
0.2
0.4
0
0
10
0.15
0
0
15000
LAI
NPPy
Cr
Ntree
5000
1
1
Nsoil
0
1.5
2
0
Data
Skogaby
Cl
Cw
1
10
0
Prior pred.
uncertainty
BASFOR: Predictive uncertainty
Height
0
5000
10000
Time
15000
100
50
0
Time
Data:
Göran Ågren
MCMC parameter trace plots: 10000 steps
-3
4
x 10
CL0
2
0
5000
0.6
FW
0.55
-5
0x 10 5000
14
12
10
8
6
Param.
value
KDW
6
4
2
10000 0
0.55
0.5
0.45
10000 0
6
4
0
5000
10000 0
KTA
0.035
0.03
0.025
-3
Parameter
trace plots
x 10
-3
x 10
CR0
CW0
8
6
4
2
10000 0
14
GAMMA
12
10
8
6
4
5000
10000 0
0.32
KH
0.3
0.28
0.26
0.24
0.22
KTB
10000 0
0.6
5000
Iteration
10000 0
5000
Iteration
10000 0
370
360
350
340
330
CO20
0.34
FLMAX
0.32
0.3
0.28
0.26
-4
10000 0x 10 5000
10000
10000 0x 10 5000
1.8
KCA
KCAEXP
KDL
KDR
1.6
0.45
4
1.4
1.2
0.4
1
2
0.35
0.8
-3
-3
-3
x
10
x
10
5000
10000 0
5000
10000 0
5000
10000 0x 10 5000
10000
1.8
7
KHEXP
KLAIMAX 1.8
KNMIN
KNUPT
1.6
1.6
6
1.4
1.4
1.2
1.2
5
1
1
4
0.8
0.8
3
0.6
0.6
-3
5000
10000 0x 10 5000
10000 0
5000
10000 0
5000
10000
0.025
KTREE
LUE0
NLCONMIN
NLCONMAX
0.045
2.5
0.02
0.04
2
0.015
0.035
1.5
5000
10000 0
5000
10000 0
5000
10000 0
5000
10000
5000
25
0.5
20
0.4
15
-3
0
5000
10000 0x 10 5000
10000 0
14
1.8
NRCON
NWCON
0.035
1.6
12
1.4
0.03
10
1.2
1
8
0.025
0.8
6
0.6
0
5000
10000 0
5000
10000 0
0.018
NLITT0
NSOMF0 0.18
0.35
0.016
0.16
0.014
0.14
0.3
0.012
0.12
0.01
0.1
0.25
0.008
0.08
0.006
0.06
-3
-5
-6
0x 10 5000
10000 0x 10 5000
10000 0x 10
10
10
2.5
KDLITT
KDSOMF
2
1.5
5
5
1
0
BETA
-3
5000
5000
0.55
0.5
0.45
10000 0
SLA
5000
CLITT0
0.4
5000
0.2
-3
10000 0x 10 5000
NSOMS0
5000
1.5
1
0.5
10000 0
KDSOMS
5000
NMIN0
5000
Iteration
CSOMF0
8
6
10000 0
0.7
0.6
0.5
10000 0
5000
2.5
2
1.5
10000 0
FLITTSOMF0.08
0.06
0.04
0.02
5000
10000 0
Iteration
CSOMS0
5000
10000
FSOMFSOMS
5000
Iteration
10000
Iteration
Steps in MCMC
10000
Posterior marginal distributions for parameters
Parameter probability distributions
2000
2000
4000
2000
2000
2000
1000
1000
2000
1000
1000
1000
0
0
CL0
2
4
2000
6
-3
x 10
1000
FW
0.6
0.7
2000
0
KDW
0.5
1
1.5
4000
1000
GAMMA
0
0.4
2000
-4
x 10
2000
KTA
0
0.02
2000
0.03
0
0
0.6
KH
4
KTB
30
1000
0
0.02
2000
NRCON
0.03
0
0.04
0.5
2000
1000
NWCON
1
1.5
2
-3
x 10
1000
0
0.005
2000
NLITT0
0.01 0.015
0
0.02
0.2
4000
1000
NSOMF0
0.3
0.4
1
KDLITT
2
3
-3
x 10
2000
0
0
5
15
KHEXP
0.3
0.4
0
KTREE
0.5
1
KDSOMF
0.5
1
1.5
-4
x 10
0
2000
2000
1000
1000
0
0.5
5
SLA
10
15
0
CO20
340
360
0
380 0.25
2000
2
KLAIMAX
6
8
4
-3
x 10
0
0.5
2000
KDL
1
1.5
2
-3
x 10
LUE0
2
3
-3
x 10
0
0.5
4000
KNMIN
1
1.5
2
-3
x 10
CLITT0
0.5
1
NLCONMIN
0
0.01
0.02
2000
0
0
4
CSOMF0
6
8
10
0
1000
1000
1000
1000
0
0
0
NMIN0
1
2
-3
x 10
0
KDSOMS
0.5
1
1.5
-5
x 10
KNUPT
1
1.5
2
-3
x 10
0.06
1000
2000
0.2
x 10
NLCONMAX
0
0.03 0.03 0.04 0.05
2000
2000
NSOMS0
0.1
6
-4
0
0.5
2000
2000
0
KDR
2
4
2000
2000
0
0
1000
1000
0
0.35
1000
2000
1
FLMAX
0.3
1000
1000
2000
1000
0
0
KCAEXP
0.4
4000
2000
2000
0
1000
1000
20
KCA
10
2000
2000
8
0
320
10000
0.6
5000
0
0.3
4000
0
4000
6
BETA
1000
0
0.2
2000
2
0
0.04
10
2000
1000
CW0
0
0.005
0.01
0.4
2000
0
0.01
0
2000
0.005
1000
4000
0
CR0
0
2000
0
0.5
4000
0
0
0
0.4
FLITTSOMF
0.6
0.8
0
1
0
CSOMS0
2
3
FSOMFSOMS
0.05
0.1
Parameter correlations
FW
GAMMA
KCAEXP
KDL
KDR
KDW
KH
KHEXP
KLAIMAX
KNMIN
LUE0
NLCONMIN
NLCONMAX
NRCON
NWCON
SLA
CLITT0
CSOMF0
CSOMS0
NLITT0
NSOMF0
NSOMS0
0.25
-0.16
0.51
0.46
0.26
0.12
0.64
0.59
0.38
-0.42
-0.07
0.71
-0.28
0.17 -0.64
-0.32 -0.58
0.23
0.55
0.52
0.12
0.50
-0.58
0.10
0.50
-0.66
-0.57
0.55
0.62
0.17
0.40
0.01
0.24
0.51
0.56
0.49
0.96
-0.19
-0.09
0.06
0.55
0.07
0.83
-0.60
-0.81
-0.21
-0.17
0.61
0.67
0.20
0.65
-0.54
-0.05
0.33
-0.29
0.05
0.46
0.61
-0.67
-0.49
1.00
0.91
0.24
0.45 -0.70
-0.82
-0.23
0.03 -0.74
-0.57
-0.74
0.77
-0.31 -0.98
0.76
-0.10
0.85
0.14
0.78
-0.61
-0.84
-0.91
0.51
-0.81
0.77
-0.30
-0.38
0.84
0.33 -0.88
-0.90
-0.58
-0.54
0.91
1.00
0.30
0.42 -0.78
-0.79
-0.46
-0.08 -0.79
-0.61
-0.66
0.81
0.04 -0.95
0.60
-0.32
0.94
0.17
0.61
-0.59
-0.98
-0.95
0.29 -0.94
0.84
0.01
-0.46
0.83
-0.01 -0.94
-0.96
0.25
0.17
0.24
0.30
1.00
0.05
-0.26
-0.41
-0.33
-0.28
0.11
0.09
-0.35
0.67
-0.02
-0.21
0.62
0.00
0.37
0.06
-0.22 -0.76
-0.33
-0.37
0.15
-0.19
0.57
-0.33
-0.34
-0.02
-0.28 -0.54
-0.36
-0.16
0.40
0.45
0.42
0.05
1.00
-0.69
-0.62
0.43
0.82
-0.56
0.25 -0.87
0.54
-0.05
-0.40
0.59
0.64
0.19 -0.81
0.74
-0.49
-0.31
-0.18
0.61
-0.33
0.06
-0.14
0.21
0.75
0.36
-0.35
-0.21
0.51
0.01 -0.70
-0.78
-0.26 -0.69
1.00
0.61
0.32
-0.18
0.56
0.05
0.86
-0.83
-0.28
0.77
-0.60
-0.16 -0.75
0.26 -0.55
0.76
0.68
0.58
-0.25
0.58
-0.63
-0.17
0.54
-0.77
-0.13
0.72
0.72
0.46
0.24 -0.82
-0.79
-0.41 -0.62
0.61
1.00
-0.05
-0.28
0.82
0.45
0.78
-0.82
0.19
0.75
-0.81
-0.06 -0.64
0.14 -0.72
0.63
0.80
0.73
-0.46
0.78
-0.65
0.49
0.06 -0.85
-0.31
0.87
0.67
0.26
0.51
-0.23
-0.46
-0.33
0.43
0.32
-0.05
1.00
0.84
-0.01
0.38
-0.10
-0.34
-0.49
0.39
0.07
0.72
-0.68
-0.69
0.35
0.30
0.49
0.51
0.47
0.37 -0.69
-0.49
0.86
0.05
0.54
0.45
0.62
0.12
0.56
0.03
-0.08
-0.28
0.82
-0.18
-0.28
0.84
1.00
-0.30
0.41
-0.48
0.00
-0.24
0.07
0.24
0.76
-0.36 -0.91
0.59
0.01
0.16
0.27
0.59
0.06
-0.48
-0.22
0.68
0.42
0.44
0.16
0.32
0.64
0.49 -0.74
-0.79
0.11 -0.56
0.56
0.82
-0.01
-0.30
1.00
0.64
0.56
-0.53
-0.03
0.73
-0.39
0.07 -0.81
0.21
0.81
0.67
-0.25
0.88
-0.48
0.10
-0.02 -0.93
-0.25
0.70
0.63
0.59
0.96
-0.57
-0.61
-0.34
-0.10
0.70
0.72
0.09
0.75
-0.57
0.10
0.19
-0.42
-0.01
0.57
0.63
0.38
-0.19 -0.74
-0.66
0.49 -0.73
0.81
0.54
0.50 -0.60
0.47
-0.48
0.29
0.21 -0.81
-0.41
0.67
0.56
0.44 -0.93
-0.77
0.30 -0.64
0.84
-0.25 -0.52
0.12 -0.92
-0.79
-0.42
-0.09
0.77
0.09
0.17 -0.61
0.25
0.05
0.45
0.38
0.41
0.64
1.00
-0.06
-0.20
0.12
0.59
-0.07
0.75
-0.35 -0.87
0.86
0.78
-0.10
-0.48
0.56
-0.06
1.00
-0.84
0.12
0.70
-0.86
-0.49 -0.54
-0.82
-0.34
-0.20 -0.84
1.00
0.07 -0.78
0.85
0.08
0.80
-0.26
0.14
0.81
0.67
0.54
-0.83
0.00 -0.53
-0.07
0.06
-0.31
0.04
-0.02
-0.05
-0.28
0.19
-0.49
-0.24
-0.03
0.12
0.12
0.07
1.00
0.14
-0.43
0.71
0.55
-0.98
-0.95
-0.21
-0.40
0.77
0.75
0.39
0.07
0.73
0.59
0.70
-0.78
0.14
1.00
-0.67
-0.61
-0.69
-0.07
0.00
KTREE
FLMAX
-0.58
-0.49 -0.54
KTB
CO20
-0.67
1.00
KTA
BETA
0.60
0.60
KNUPT
CW0
1.00
KCA
CR0
CL0
CR0
CW0
BETA
CO20
FLMAX
FW
GAMMA
KCA
KCAEXP
KDL
KDR
KDW
KH
KHEXP
KLAIMAX
KNMIN
KNUPT
KTA
KTB
KTREE
LUE0
NLCONMIN
NLCONMAX
NRCON
NWCON
SLA
CLITT0
CSOMF0
CSOMS0
NLITT0
NSOMF0
NSOMS0
NMIN0
FLITTSOMF
FSOMFSOMS
KDLITT
KDSOMF
KDSOMS
CL0
39 parameters
39 parameters
-0.73
0.68
-0.40
-0.01
-0.12
0.15 -0.76
-0.05
0.12
0.72
-0.37
-0.05
-0.47
-0.02
0.00
0.21 -0.93
-0.21 -0.70
0.60
0.88
0.93
-0.38
0.83
-0.82
0.11
0.51
-0.83
-0.22
0.89
0.96
-0.58
-0.28
0.07
0.76
0.60
0.62
0.59
-0.60
-0.81
0.07
0.24
-0.39
-0.07 -0.86
0.85
-0.43 -0.67
1.00
0.38
0.53
-0.22
0.58
-0.86
-0.52
-0.59
0.66
-0.42
0.60
-0.63
-0.22
0.61
0.42 -0.73
0.17
0.83
-0.10
-0.32
0.00
0.64
-0.16
-0.06
0.72
0.76
0.17
0.75
-0.49
0.08
-0.26
0.21
0.38
1.00
-0.43 -0.83
0.28
-0.27
0.45
0.46
0.47
0.48
-0.41
-0.38
0.33
0.10
0.58
0.26
0.41
-0.64
-0.60
0.85
0.94
0.37
0.19 -0.75
-0.64
-0.68
-0.36 -0.61
-0.61
-0.54
0.80
0.14 -0.93
0.53
-0.43
1.00
0.40 -0.64
-0.92
-0.93
0.08 -0.83
0.94
0.07 -0.71
0.66
-0.05 -0.92
-0.99
0.07 -0.69
0.49
-0.07
0.00
-0.22 -0.83
0.39
1.00
-0.33
-0.25
-0.39
-0.46
-0.21
0.47
0.05 -0.52
-0.25
-0.22
-0.21
-0.38
-0.34 -0.73
0.44
0.58
0.40
-0.33
1.00
-0.26 -0.52
-0.51
0.66
-0.58
0.24
-0.32
0.15
0.91
0.60
-0.50
-0.48
0.16
-0.26
1.00
0.52
0.53
-0.33
0.35 -0.72
0.28
0.56
-0.45
-0.13
0.73
0.62
0.92
-0.32 -0.81
0.14
0.17
-0.58
0.78
0.61
-0.22
0.74
-0.55
-0.72
0.35
0.59
-0.81
0.23
-0.17 -0.61
-0.59
-0.76
-0.49
0.76
0.63
0.30
0.01
0.21
-0.10
0.81
-0.93
-0.01
0.60
-0.86
-0.27 -0.64
0.55
0.61
-0.84
-0.98
-0.33
-0.31
0.68
0.80
0.49
0.16
0.81
0.70
0.54
-0.77
-0.12
0.88
-0.52
0.45 -0.92
-0.25 -0.52
0.52
1.00
0.94
-0.16
0.97
-0.85
0.00
0.41 -0.77
0.10
0.95
0.52
0.67
-0.91
-0.95
-0.37
-0.18
0.58
0.73
0.51
0.27
0.67
0.72
0.50 -0.73
0.15
0.93
-0.59
0.46 -0.93
-0.39 -0.51
0.53
0.94
1.00
-0.32
0.91
-0.87
0.11
0.46 -0.67
0.05
0.92
0.96
0.12
0.20
0.51
0.29
0.15
0.61
-0.25
-0.46
0.47
0.59
-0.25
0.09 -0.60
0.30 -0.76
-0.38
0.66
0.47
-0.46
0.66
-0.33
-0.16
-0.32
1.00
-0.22
-0.01
-0.46
0.34
0.31
-0.23
-0.21
-0.33
0.58
0.78
0.37
0.06
0.88
0.75
0.83
-0.42
-0.21 -0.58
0.35
0.97
0.91
-0.22
-0.03
0.23 -0.79
0.12
0.86
0.85
0.06 -0.63
-0.65
-0.69
-0.48
0.12 -0.82
0.60
-0.41
0.94
0.24 -0.72
-0.85
-0.87
-0.03 -0.93
-0.92
-0.38
-0.21
0.06 -0.81
0.50
0.65
-0.81
-0.94
-0.19
-0.58
-0.54
0.77
0.84
0.57
0.26
0.14 -0.69
-0.91
-0.48 -0.57
0.47 -0.64
-0.48
0.84
-0.21
0.39
-0.40 -0.70
-0.05
0.10
-0.05
-0.30
0.01
-0.33
-0.14
-0.17
0.49
-0.49
-0.22
0.10
0.10
0.29
-0.25
0.72
0.11 -0.63
0.50
0.33
-0.38
-0.46
-0.34
0.21
0.54
0.06
0.86
0.68
-0.02
0.19
0.21 -0.52
-0.37
0.51
-0.22
-0.66
-0.29
0.84
0.83
-0.02
0.75
-0.77
-0.85
0.05
0.42 -0.93
-0.42 -0.81
0.68
-0.05 -0.83
-0.57
0.05
0.33
-0.01
-0.28
0.36
-0.13
-0.31
0.54
0.44
-0.25
-0.01
-0.41
0.12
-0.47
-0.22
0.46 -0.88
-0.94
-0.54
-0.35
0.72
0.87
0.45
0.16
0.70
0.57
0.67
-0.92
-0.02
0.89
-0.73
-0.58
0.55
0.28
0.48 -0.83
0.66
-0.25
0.91
-0.45 -0.77
-0.67
0.44 -0.79
0.51
-0.15
-0.10
1.00
0.39 -0.74
-0.05
-0.22
0.60
-0.13
0.10
0.05
0.31
0.12
-0.03 -0.64
0.09
0.39
1.00
-0.05
0.01
0.26 -0.92
-0.21
-0.50
0.73
0.95
0.92
-0.23
0.86
-0.93
0.22
0.50 -0.74
-0.05
1.00
0.92
0.41 -0.99
1.00
0.62
0.32
0.63
0.63
0.56
-0.79
0.00
0.96
0.33
0.16
-0.09
-0.06
-0.30
0.66
-0.63
0.29
0.45 -0.72
0.48
0.60
-0.01
0.31
-0.43
0.03
-0.22
0.05
0.36
0.63
-0.39
0.40
0.15
-0.66
-0.28
0.86
0.83
0.08
0.55
-0.89
-0.56
-0.33
0.28 -0.93
-0.89
-0.55
-0.51
0.73
0.87
0.25
-0.04
0.62
0.39
0.81
-0.91
0.26
0.90
-0.88
-0.08 -0.70
0.75
0.19
-0.03
-0.42
0.09
-0.46
0.75
-0.46
0.03
0.41
-0.49 -0.59
-0.75
-0.92
0.12
-0.35 -0.86
0.75
0.72
0.83
0.81
0.13
0.80
0.47 -0.89
-0.12 -0.79
-0.68
0.10
0.58
0.67
-0.18
0.65
0.42
0.56
-0.55
0.00
0.50
0.61
0.72
-0.31
0.22
-0.10
0.34
-0.43
-0.39
-0.15 -0.64
1.00
-0.46
0.46
-0.21
-0.43
-0.25
0.11
0.41
-0.36
0.15
1.00
-0.25
0.00
-0.41 -0.64
-0.04 -0.91
-0.13
0.23 -0.75
0.28
0.56
-0.96
0.72
0.51
0.15
-0.47
-0.27 -0.78
-0.13 -0.75
-0.32
-0.90
0.08 -0.58
1.00
0.05
0.61
0.42
-0.72
-0.52
-0.31
-0.02
1.00
-0.01 -0.72
0.07
0.62
0.61
0.47
0.44
0.33 -0.71
-0.16
0.08
0.08
0.16
-0.03
0.09
-0.38
-0.48
0.62
0.92
0.96
-0.21
0.85
-0.92
0.00
0.65
-0.68
0.01
0.92
-0.33
-0.40
0.25
-0.21
0.79
0.27
0.41 -0.66
0.16
-0.39
0.14
0.33
-0.23
0.06
0.42
0.45
0.34
0.33
0.12
-0.39
-0.22 -0.62
0.01
-0.02
0.13
0.12
0.23
-0.28
-0.11
-0.40
-0.10
-0.10
0.61
0.04
0.81
-0.03
0.65
0.07 -0.55
0.78
0.27 -0.70
-0.83
0.69
-0.63
-0.69
-0.72
-0.01 -0.63
0.80
0.84
0.88
-0.56
0.77
-0.80
0.34
-0.27
0.55
-0.45
0.60
0.12
0.14 -0.51
0.04
-0.13
0.62
-0.32
0.89
-0.54
-0.76
-0.77
0.51
0.02 -0.83
0.18
0.29
-0.66
0.41 -0.62
0.52
0.37 -0.75
-0.16
0.92
-0.14
0.29
-0.43
-0.25
0.20
0.87
0.29
-0.28
-0.03
0.98
0.39 -0.77
-0.65
Bayesian calibration: overview
4000
4000
4000
4000
20
2000
2000
2000
2000
2000
15
0.01
4000
2000
0
0.5
0
0
0.01
4000
4000
2000
2000
FW
0.6
CR0
0.005
0 GAMMA
0.4
0.6
0.7
0
0.8
CW0
0.005
0
0 BETA
0.4
0.6
0.01
0.8
4000
0
2
0
4
0
0
1
0.35
0
0.005
0
0.01
0
KDR
0.5
0
1
0
4000
2000
2
KDW
4
0
6
4000
2000
KH
4
3
5
4000
2000
KHEXP
0.3
0
0.2
0
0.4
0
0
0.01
0
-5
2000
0
0.02
2
KNUPT
1
0
-3
4000
KTA
0.03
0
0
10
0.04
KTB
20
4000
4000
2000
2000
KTREE
0
0.4
0.6
30
0
0.8
LUE0
2
1
4000
2000
2000
0 NLCONMIN
0.01
0.02
5000
10000
0
15000
1.5
0
NLCONMAX
0
0.04
0.05
0.5
0.03
0
5000
10000
-0.5
15000
0
0.06
0.1
0.04
5000
10000
0
0.04
NWCON
1
0
0
2
SLA
20
0
0
40
4000
2000
0
2000
CSOMF0
6
8
0
1
0
10
CSOMS0
2
1
0.05
0.02
0
0
3
-3
x 10
4000
4000
4000
4000
4000
4000
2000
2000
2000
2000
2000
2000
0
NLITT0
0.01
0
0.2
0.02
NSOMF0
0.3
0
0.4
NSOMS0
0.1
0.2
0
0
NMIN0
1
0
FLITTSOMF
0.6
0
0.4
2
0
0.8
FSOMFSOMS
0.05
0.1
0
-3
x 10
4000
4000
2000
2000
2000
0
0
KDLITT
2
0
4
KDSOMF
0
1
-3
0
2
KDSOMS
0
1
10000
15000
0
Parameter probability distributions
5000
10000
0
15000
2000
2000
20
1000
1000
1000
15
0
0.7
1000
GAMMA
0
0.4
2000
0
0.6
BETA
0
320
10000
0.6
1000
KCA
10
2000
1000
2000
5
15
0
380 0.25
2000
5000
0
0.3
4000
0
CO20
340
360
KCAEXP
0.4
0.5
FLMAX
0.3
KDL
1
1.5
2
-3
x 10
0
KDR
2
4
2000
0
5000
10000
-4
x 10
2000
2
6
2000
0
0.02
2000
KTA
0.03
KTB
20
30
1000
0
0.02
2000
NRCON
0.03
0
0.04
0.5
2000
1000
NWCON
1
1.5
2
-3
x 10
1000
0
0.005
2000
NLITT0
0.01 0.015
0
0.02
0.2
4000
1000
NSOMF0
0.3
0.4
0
3
-3
x 10
0
0
0
4
-3
x 10
2000
0
KTREE
0.5
1
0
2000
2000
1000
1000
0
2
4000
5
SLA
10
15
0
0
0.5
4000
2
-3
x 10
x 10
2000
2
NLCONMIN
0
3
0.01
0.02
-3 2000
x 10
CLITT0
0.5
1
0
4
CSOMF0
6
8
10
0
2000
2000
1000
1000
1000
1000
0
NSOMS0
0.1
0.2
0
0
NMIN0
1
2
-3
x 10
NPPy
Cr
x 10
0
1.5
-4
x 10
0
0
KDSOMS
0.5
1
0
5000
10000
1.5
-5
-0.5
15000
0
0
0.4
FLITTSOMF
0.6
0.8
0
1
CSOMS0
2
0.06
5000
10000
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
5
0
15000
0.1
0.04
0.06
3
0.05
0
0
FSOMFSOMS
0.05
0.1
10
0.02
0
5000
10000
0
15000
0.6
0
5000
10000
150
0.15
0.2
0.1
0.05
0
5000
10000
Time
15000
0
5
0
15000
0.2
0.4
0
x 10
0.5
0.15
1000
KDSOMF
0
0.5
1
0
10
0
2
-3
1000
2000
0
0
15000
1
1
KNUPT
1
1.5
NLCONMAX
0
0.03 0.03 0.04 0.05
2000
1000
0
10000
1000
LUE0
1
0
0.5
2000
2000
2000
2000
KDLITT
1
2
0.4
1000
0
0.04
10
2000
1000
0
0
4000
5000
-4
1000
KNMIN
1
1.5
0
1.5
NCl
1.5
1000
KLAIMAX
6
8
15000
1.5
5
0
15000
Nmin
1
KHEXP
0
8
0.2
0.3
2000
10000
0.5
3
6
Ntree
0
4000
KH
4
5000
1
10
0
0
Biomass
10
5
Nsoil
0
1000
KDW
0.5
0
50
BASFOR: Predictive uncertainty
Height
2
2000
15000
Time
0.35
1000
0
0.5
2000
10000
100
0
15000
LAI
2000
0
0.5
4000
FW
0.6
CW0
0
0.005
0.01
0.4
2000
0
0.01
0
2000
0.005
h
CR0
0
10000
Csoil
x 10
1000
0
4000
5000
Miny
6
5000
Cl
2000
2000
-3
0
Time
Cw
4000
1000
4
15000
Data
2000
CL0
2
10000
5
0
15000
0.1
Time
1000
0
10000
0.05
x 10
2000
0
5000
150
Bayesian
calibration
2000
5000
0.15
0.2
2
0
0.2
-5
x 10
5000
0.4
0
-4
x 10
0
0.6
Nsoil
0
4000
0
10
Csoil
4000
CLITT0
0.5
Miny
0
0.02
NRCON
0.03
NCl
2000
Nmin
4000
2000
15000
0.06
Ntree
4000
2000
10000
5
0
15000
-3
4000
2000
5000
0
0.15
x 10
4000
0
10
1
1
2
0
x 10
4000
3
0
15000
-3
x 10
2000
10000
2
2000
KNMIN
1
x 10
4000
5000
4000
2000
KLAIMAX
0.005
0
Cr
4000
2000
0.5
3
-3
x 10
4000
5
5
2000
KDL
1.5
1
10
4000
2000
KCAEXP
0.5
FLMAX
0.3
0
0.25
400
4000
2000
KCA
CO20
350
0
300
LAI
0
Biomass
10
NPPy
0
CL0
0.005
h
0
BASFOR: Predictive uncertainty
Height
Cl
4000
2000
Cw
Prior parameter marginal probability distributions (uniform)
4000
0
5000
10000
Time
15000
100
50
0
Time
Prior & posterior predictive uncertainty
20
Biomass
10
1.5
15
Cl
Cw
h
5
0.5
5
0
5000
10000
0
15000
3
10000
0.5
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
5
0
5000
10000
-0.5
15000
0
0.04
10000
0.05
10
0.02
0
5000
10000
0
15000
0.6
0
15000
NCl
0.1
5000
Csoil
0
0.06
0
5000
10000
150
Nmin
Miny
0.15
0.2
0.1
0.05
0
5000
10000
Time
15000
0
5
0
15000
0.2
0.4
0
0
10
0.15
0
0
15000
LAI
NPPy
Cr
Ntree
5000
1
1
Nsoil
0
1.5
2
0
Posterior
uncertainty
(using data
Skogaby)
1
10
0
Prior pred.
uncertainty
BASFOR: Predictive uncertainty
Height
0
5000
10000
Time
15000
100
50
0
Time
Partial corr. coefficients (PCC) parameters – outputs
39 parameters
Cw
Cl
Cr
Clitt
Csomf
Csoms
Nl
Nlitt
Nsomf
Nsoms
Nmin
RainCum
NPPy
Nimeralisationy
H
CA
LAI
Ctree
Csoil
Ntree
CL0
CR0
CW0
BETA
CO20
FLMAX
FW
GAMMA
KCA
KCAEXP
KDL
KDR
KDW
KH
KHEXP
KLAIMAX
KNMIN
KNUPT
KTA
KTB
KTREE
LUE0
NLCONMIN
NLCONMAX
NRCON
NWCON
SLA
CLITT0
CSOMF0
CSOMS0
NLITT0
NSOMF0
NSOMS0
NMIN0
FLITTSOMF
FSOMFSOMS
KDLITT
KDSOMF
KDSOMS
TreeDens
12 output variables
0.00
-0.04
0.05
-0.06
-0.18
-0.04
-0.02
0.05
-0.16
0.21
-0.21
-0.09
0.00
-0.08
-0.03
-0.06
0.00
0.02
-0.05
-0.11
-0.05
0.00
0.02
-0.04
-0.05
-0.02
0.01
-0.05
-0.01
0.00
0.00
0.10
0.00
0.01
0.01
-0.01
0.00
0.00
0.02
0.00
0.00
-0.04
0
0.01
-0.06
0.01
0.01
0.03
0.02
0.13
-0.01
0.00
-0.02
0.01
0.04
0.00
0.00
0.01
0.00
0.00
-0.04
-0.04
0.05
0.02
-0
0
0.00
0.05
0.08
0.02
-0.03
0.07
0.00
0.06
-0.02
-0.06
0.06
0.03
0.00
0.09
0.06
0.00
0.00
-0.01
0.06
0.05
0.05
-0
0.00
-0.01
-0.05
-0.03
-0.01
-0.05
0.02
0.05
0.02
-0.02
0.02
0.32
0.00
-0.06
0.02
-0.03
0.00
-0.01
-0.02
-0.05
0.00
0
0.00
0.09
0.66
-0.43
0.22
0.15
0.01
0.65
0.31
0.00
0.00
0.04
0.00
0.31
0.32
0.01
0.00
0.74
0.04
0.22
-0.17
0
0.00
0.94
0.39
-0.51
0.34
0.41
0.02
0.52
0.25
-0.08
-0.05
0.58
0.00
0.84
0.17
0.66
0.00
0.47
0.90
0.49
-0.02
0
0.00
-0.01
-0.18
-0.08
0.02
-0.18
-0.02
0.14
0.09
-0.06
0.03
0.75
0.00
-0.19
0.02
-0.01
0.00
-0.18
-0.05
-0.17
-0.02
0
-0.01
0.02
0.03
0.03
0.09
0.09
-0.02
0.05
0.07
-0.08
0.03
0.03
0.01
0.05
0.03
0.06
0.00
0.02
0.03
0.11
0.05
-0
0.00
0.00
0.00
-0.01
0.01
0.02
0.02
0.02
0.02
-0.06
0.10
0.02
0.00
-0.01
-0.02
0.05
0.00
-0.04
-0.01
0.02
0.00
0
0.01
0.19
-0.81
-0.55
0.20
0.33
0.11
-0.80
0.36
0.45
0.02
0.61
0.00
0.40
0.50
0.08
0.00
-0.88
-0.17
0.39
-0.67
0
0.00
0.33
0.08
-0.94
0.01
0.63
0.13
0.10
-0.02
0.86
0.00
0.45
0.00
0.41
0.45
0.09
0.00
0.17
-0.39
0.63
-0.91
0
0.00
-0.92
0.12
0.09
0.43
0.78
0.24
0.11
0.05
0.09
0.07
0.19
0.00
0.16
0.15
-0.63
0.00
0.03
-0.88
0.81
-0.19
0
0.00
-0.02
-0.09
-0.04
-0.08
-0.02
0.01
-0.10
-0.10
0.13
-0.02
-0.04
0.00
-0.07
-0.05
0.99
0.00
-0.02
-0.04
-0.05
-0.11
0
0.00
-0.02
-0.07
-0.05
0.01
-0.08
-0.06
-0.05
0.01
0.03
0.06
-0.03
0.00
-0.07
-0.03
0.97
0.00
0.00
-0.04
-0.09
-0.09
0
-0.01
0.13
0.80
-0.57
0.31
0.20
0.15
0.77
0.40
-0.08
0.11
-0.02
-0.01
0.42
0.40
0.08
0.00
0.85
0.07
0.31
-0.20
0
-0.01
-0.07
-0.07
0.03
0.06
0.01
0.04
-0.08
0.05
-0.04
0.01
0.09
0.00
-0.05
-0.06
0.02
0.00
-0.01
-0.06
0.03
0.02
-0
0.00
0.02
0.04
0.02
0.02
-0.04
-0.03
0.04
0.03
-0.02
-0.02
-0.13
0.00
0.01
0.05
0.06
0.00
-0.06
0.02
-0.03
0.02
-0
0.00
0.08
0.18
0.16
0.03
0.23
0.03
-0.15
-0.02
-0.08
0.06
-0.74
0.00
0.26
0.05
0.03
0.00
0.26
0.13
0.23
0.09
-0
0.00
0.03
0.21
0.13
0.06
0.23
-0.03
-0.08
-0.01
-0.03
-0.05
-0.77
0.00
0.22
0.01
0.03
0.00
0.18
0.08
0.23
0.10
-0
0.00
0.05
0.13
0.05
0.00
0.15
0.00
-0.09
-0.03
-0.01
0.04
-0.52
0.00
0.13
0.03
0.02
0.00
0.16
0.07
0.14
0.02
-0
0.00
0.09
0.24
0.15
0.04
0.20
-0.04
-0.11
-0.05
-0.03
0.00
-0.79
0.00
0.27
0.02
0.05
0.00
0.29
0.14
0.20
0.09
-0
0.00
0.01
-0.66
-0.38
-0.14
-0.54
-0.13
0.57
0.19
0.00
0.04
0.32
0.00
-0.49
0.20
-0.02
0.00
-0.73
-0.19
-0.56
-0.14
0
0.00
-0.75
-0.27
-0.22
-0.19
-0.53
-0.08
0.56
0.14
0.05
-0.03
-0.23
0.00
-0.63
0.29
-0.34
0.00
-0.32
-0.71
-0.56
-0.12
0
0.00
-0.91
-0.54
-0.89
-0.29
-0.89
-0.25
-0.69
-0.26
-0.14
0.03
-0.72
0.00
-0.93
-0.12
-0.59
0.00
-0.64
-0.92
-0.89
0.33
-0
0.01
-0.46
-0.14
-0.32
-0.15
-0.40
-0.13
-0.17
-0.01
-0.42
-0.09
-0.29
0.00
-0.51
-0.05
-0.20
0.00
-0.17
-0.46
-0.44
0.56
-0
0.00
-0.08
-0.89
0.77
-0.34
-0.04
0.02
-0.90
-0.52
-0.15
0.13
-0.68
0.00
-0.40
-0.54
0.02
0.00
0.95
0.04
-0.16
0.43
-0
0.01
0.01
-0.06
-0.02
-0.02
0.19
0.04
-0.09
-0.07
-0.01
0.10
-0.01
0.00
-0.02
-0.03
-0.04
0.00
0.03
0.00
0.19
-0.01
0
0.00
-0.14
-0.07
-0.04
0.01
0.95
0.80
-0.07
0.00
0.04
0.05
-0.11
0.00
-0.14
-0.13
-0.02
0.00
-0.06
-0.13
0.95
-0.08
0
0.00
-0.07
-0.06
-0.02
-0.03
-0.01
1.00
-0.03
-0.01
0.05
0.00
-0.01
0.00
-0.02
-0.02
-0.04
0.00
0.03
-0.07
0.91
-0.06
0
0.00
0.34
0.02
0.23
0.02
0.39
0.04
0.09
0.02
0.81
0.07
0.23
0.00
0.23
0.28
0.24
0.00
0.02
0.34
0.39
0.26
0
0.00
0.84
0.39
0.78
0.25
0.83
0.19
0.55
0.25
1.00
0.58
0.86
0.00
0.85
0.90
0.39
0.00
0.48
0.86
0.84
0.83
1
0.00
0.45
0.19
0.36
0.11
0.39
0.03
0.25
0.11
0.69
1.00
0.48
0.00
0.50
0.58
0.14
0.00
0.18
0.47
0.40
0.43
1
0.00
0.14
-0.01
0.04
0.08
0.10
0.01
0.00
0.11
-0.03
0.17
0.05
0.00
0.10
0.09
0.02
0.00
0.03
0.12
0.13
0.05
0
0.00
-0.74
-0.30
-0.65
-0.19
0.80
0.18
-0.39
-0.19
0.64
0.04
-0.74
0.00
-0.75
-0.81
-0.26
0.00
-0.35
-0.75
0.78
-0.71
0
0.00
-0.33
-0.07
-0.28
-0.05
-0.29
0.94
-0.15
-0.03
-0.60
0.90
-0.42
0.00
-0.40
-0.47
-0.08
0.00
-0.14
-0.35
0.04
-0.32
0
0.01
0.17
-0.02
0.16
-0.90
0.49
0.20
-0.01
-0.90
0.58
0.12
0.10
0.01
0.02
0.06
0.04
0.00
-0.02
0.18
-0.16
0.13
-0
0.01
0.88
0.48
0.83
0.31
-0.34
0.77
0.63
0.30
-0.89
0.66
0.90
0.00
0.90
0.93
0.49
0.00
0.56
0.89
-0.08
0.87
-0
0.00
0.44
0.16
0.40
0.08
0.40
-0.82
0.26
0.08
0.66
-0.94
0.53
0.00
0.50
0.57
0.06
0.00
0.22
0.47
0.26
0.47
-0
2. What kind of measurements would have
reduced uncertainty the most?
Prior predictive uncertainty & height-data
20
Biomass
10
1.5
15
Cl
Cw
h
5
0.5
5
0
5000
10000
0
15000
3
10000
0.5
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
0
5000
10000
15000
5
0
5000
10000
-0.5
15000
0
0.04
10000
0.05
10
0.02
0
5000
10000
0
15000
0.6
0
15000
NCl
0.1
5000
Csoil
0
0.06
0
5000
10000
150
Nmin
Miny
0.15
0.2
0.1
0.05
0
5000
10000
Time
15000
0
5
0
15000
0.2
0.4
0
0
10
0.15
0
0
15000
LAI
NPPy
Cr
Ntree
5000
1
1
Nsoil
0
1.5
2
0
Height data
Skogaby
1
10
0
Prior pred.
uncertainty
BASFOR: Predictive uncertainty
Height
0
5000
10000
Time
15000
100
50
0
Time
Prior & posterior uncertainty: use of height data
Biomass
10
1.5
15
h
Cl
Cw
1
10
5
0.5
5
0
5000
10000
0
15000
3
0.5
5000
10000
15000
0
5000
10000
-0.5
15000
0
0.04
10000
5
0.05
5000
10000
0
15000
0.6
0
5000
10000
Miny
Nmin
0.1
0.05
0
5000
10000
Time
15000
0
10000
15000
0
5000
10000
15000
0
5000
10000
15000
150
0.15
0.2
5000
5
0
15000
0.2
0.4
0
10
0.02
0
0
15000
NCl
0.1
5000
0
5000
10000
Time
15000
100
50
0
Posterior
uncertainty
(using
height data)
Height data
Skogaby
0
0.06
0
0
10
0.15
0
0
15000
LAI
NPPy
Cr
Ntree
10000
1
1
Nsoil
5000
1.5
2
0
0
Csoil
0
Prior pred.
uncertainty
BASFOR: Predictive uncertainty
Height
20
Time
Prior & posterior uncertainty: use of height data
Biomass
10
1.5
15
h
Cl
Cw
1
10
5
0.5
5
0
5000
10000
0
15000
3
0.5
5000
10000
15000
0
5000
10000
-0.5
15000
0
0.04
10000
5
0.05
5000
10000
0
15000
0.6
0
5000
10000
Miny
Nmin
0.1
0.05
0
5000
10000
Time
15000
0
10000
15000
0
5000
10000
15000
0
5000
10000
15000
150
0.15
0.2
5000
5
0
15000
0.2
0.4
0
10
0.02
0
0
15000
NCl
0.1
5000
0
5000
10000
Time
15000
100
50
0
Posterior
uncertainty
(using
height data)
Height data
(hypothet.)
0
0.06
0
0
10
0.15
0
0
15000
LAI
NPPy
Cr
Ntree
10000
1
1
Nsoil
5000
1.5
2
0
0
Csoil
0
Prior pred.
uncertainty
BASFOR: Predictive uncertainty
Height
20
Time
Prior & posterior uncertainty: use of height data
Biomass
10
1.5
15
h
Cl
Cw
1
10
5
0.5
5
0
5000
10000
0
15000
3
0.5
5000
10000
15000
0
5000
10000
-0.5
15000
0
0.04
10000
5
0.05
5000
10000
0
15000
0.6
0
5000
10000
Miny
Nmin
0.1
0.05
0
5000
10000
Time
15000
0
10000
15000
0
5000
10000
15000
0
5000
10000
15000
150
0.15
0.2
5000
5
0
15000
0.2
0.4
0
10
0.02
0
0
15000
NCl
0.1
5000
0
5000
10000
Time
15000
100
50
0
Posterior
uncertainty
(using
height data)
Posterior
uncertainty
(using
precision
height data)
0
0.06
0
0
10
0.15
0
0
15000
LAI
NPPy
Cr
Ntree
10000
1
1
Nsoil
5000
1.5
2
0
0
Csoil
0
Prior pred.
uncertainty
BASFOR: Predictive uncertainty
Height
20
Time
Summary of procedure
Prior P()
Model f
Data D ± σ
“Error function”
e.g. N(0, σ)
MCMC
Samples of 
(104 – 105)
Samples of f()
(104 – 105)
Posterior P(|D)
PCC
P(D|f())
Calibrated parameters,
with covariances
Sensitivity analysis
of model parameters
Uncertainty of model
output
3. Bayesian comparison of forest models
Uncertainty regarding model structure
Imperfect
understanding
Environmental
scenarios
Initial
values
Nutr. C
H2O
H2O
NPP
Trees
Nutr.
Nutr. C
Soil
Parameters
Height
Atmosphere
C H2O
Nutr.
Subsoil (or run-off)
Model
H2O
Soil C
Bayesian comparison of two models
Model 1
Model 2
Atmosphere
Atmosphere
Nutr.
H2 O
C
H2 O
Nutr.
Trees
Nutr.
Nutr.
C
H2 O
H2 O
H2 O
Bayes Theorem for model probab.:
Soil
C H2 O
C
Trees
Nutr.
Nutr.
Subsoil (or run-off)
P(M|D) = P(M) P(D|M) / P(D)
Nutr.
C
H2 O
Soil
C H2 O
Nutr.
Subsoil (or run-off)
The “Integrated likelihood” P(D|Mi) can be
approximated from the MCMC sample of
outputs for model Mi (*)
P(M1) = P(M2) = ½
P(M2|D) / P(M1|D) = P(D|M2) / P(D|M1)
The “Bayes Factor” P(D|M2) / P(D|M1)
quantifies how the data D change the
odds of M2 over M1
P( )
(*)
P( D | M ) 
 P( ) P( D |  )d 

(M )
 w P( D |  )  P( ) P( D |  ) P( D |  )


P( )
w
 P( ) P( D |  )

i
MCMC
MCMC
i
MCMC
MCMC
n
1
MCMC P ( D |  )
 harmonic mean of likelihoods in MCMC-sample (Kass & Raftery, 1995)
Bayes Factor for two big forest models
Parameter marginal probability distributions (truncated normal)
400
200
0
0
CL0
2
4
200
100
0
CR0
2
4
-3
0.7
200
100
CW0
0 BETA
0 0.005 0.01 0.4 0.6
200
100
0 CO20
0.8 300 350
200
100
0 FLMAX
400 0.25 0.3 0.35
-3
x 10
200
100
0 FW
0.5 0.6
6
400
200
0
x 10
200
100
GAMMA
0
0.4 0.6
0.8
200
100
0
0
KCA
2
4
200
100
0
0
KCAEXP
0.5
1
200
100
0
0
KDL
0.5
200
100
0
1
0
KDR
0.5
-3
200
100
0
2
KDW
4
6
200
100
0
KH
3
4
5
200
100
0
0.2
200
100
KHEXP
0
0.3 0.4 4
-5
200
100
0
0
0
KNUPT
1
2
-3
x 10
x 10
200
200
100
100
LUE0
NLCONMIN
NLCONMAX
0
0
2.5
3 0.01 0.015 0.02 0.04 0.05 0.06
200
100
KTA
0
0.03 0.04 10
200
100
KTB
0 KTREE
20
30 0.4 0.6
0.8
200
200
100
100
0 NRCON
0
0.02 0.03 0.04 0
200
100
NWCON 0 SLA
1
2
0
20
200
100
0 CLITT0
40 0
0.5
2
200
100
0
-3
x 10
200
100
0
x 10
KNMIN
1
2
-3
x 10
200
100
0
0.02
KLAIMAX
6
8
1
-3
x 10
Calculation of P(D|BASFOR)
MCMC 5000 steps
-3
x 10
200
100
0 CSOMF0
1
6 8
10
200
100
0
1
CSOMS0
2
3
-3
x 10
200
100
0
0
200
100
NLITT0
0 NSOMF0
0.01 0.02 0.2 0.3
0.4
200
100
0
0
400
200
NSOMS0
0
0.1 0.2 0
NMIN0
1
2
200
100
0FLITTSOMF
0.4 0.6
400
200
0 FSOMFSOMS
0.8 0
0.05 0.1
-3
x 10
400
200
0
0
KDLITT
2
4
200
100
0
200
100
KDSOMF
0 KDSOMS
0
0.5
1
0
1
-3
-4
x 10
Skogaby
2
-5
x 10
x 10
Parameter marginal probability distributions (truncated normal)
200
100
0 CB0T
0
1
2
200
100
0
CL0T
0
-3
0
CR0T
2
200
100
0
0.35 0.25
200
100
0 KDBT
0.5
1
200
100
0
0
5
200
100
0 CS0T
0
1
200
100
0
2
200
100
0 BETA
0.4 0.6
100
50
0
6 0.2
0.8
KHEXP
0.3
0.4
200
100
0
200
100
0
0
0
-4
x 10
KTB
20
30
CO20
350 400
200
100
0
4 0.4
KCA
2
KCAEXP
0.6 0.8
KNMINT
1
2
200
100
0
0
KNUPTT
1
2
-3
x 10
200
100
KTA
0
0.03 0.04 10
200
100
0
300
0.8
x 10
200
100
GAMMA
0
0.35 0.4 0.6
KH
4
2
-3
x 10
200
100
FLMAX
FS
0
0.3 0.35 0.25 0.3
KDRT
-4
4
-3
x 10
200
100
0 FB
0.25 0.3
200
100
0
0.02
200
100
0
-3
x 10
1.5
5
0.8
200
100
0
LAIMAXT
4
6
8
200
100
0
2
LUET
2.5
3
Data Rajec: Emil Klimo
-3
x 10
200
100
0 KEXTT
0.4 0.6
Rajec
x 10
200
100
0 NCLMINT
0.01 0.015 0.02
-3
100
200
50
100
0 NCLMAXT
0
0.04 0.05 0.06 0.02
NCRT
0.03 0.04
100
50
0
0
100
50
0
2 10
NCWT
1
x 10
200
200
100
100
SLAT
TRANCOT
0
0 CLITT0
20
30
4
6
8
0
0.5
1
-3
x 10
200
100
0 CSOMF0
6
8
10
100
50
0
1
CSOMS0
2
3
200
100
0
0
200
200
200
100
100
100
NLITT0
0 NSOMF0
0 NSOMS0
0
0.01 0.02 0.2 0.3
0.4 0
0.1 0.2 0
NMIN0
1
2
-3
x 10
100
200
50
100
0 FLITTSOMF
0
0.4 0.6 0.8 0
200
100
FSOMFSOMS 0 KDLITT
0.05 0.1 0
2
4
-3
x 10
200
100
0 KDSOMF
0
1
2
-4
x 10
200
100
0
0
KDSOMS
0.5
1
-5
x 10
Calculation of P(D|BASFOR+)
MCMC 5000 steps
Bayes Factor for two big forest models
Parameter marginal probability distributions (truncated normal)
400
200
0
0
CL0
2
4
200
100
0
CR0
2
4
-3
0.7
200
100
CW0
0 BETA
0 0.005 0.01 0.4 0.6
200
100
0 CO20
0.8 300 350
200
100
0 FLMAX
400 0.25 0.3 0.35
-3
x 10
200
100
0 FW
0.5 0.6
6
400
200
0
x 10
200
100
GAMMA
0
0.4 0.6
0.8
200
100
0
0
KCA
2
4
200
100
0
0
KCAEXP
0.5
1
200
100
0
0
KDL
0.5
200
100
0
1
0
KDR
0.5
-3
200
100
0
2
KDW
4
6
200
100
0
KH
3
4
5
200
100
0
0.2
200
100
KHEXP
0
0.3 0.4 4
-5
200
100
0
0
KNUPT
1
2
-3
x 10
x 10
200
200
100
100
LUE0
NLCONMIN
NLCONMAX
0
0
2.5
3 0.01 0.015 0.02 0.04 0.05 0.06
200
100
KTA
0
0.03 0.04 10
200
100
KTB
0 KTREE
20
30 0.4 0.6
0.8
200
200
100
100
0 NRCON
0
0.02 0.03 0.04 0
200
100
NWCON 0 SLA
1
2
0
20
200
100
0 CLITT0
40 0
0.5
2
0
-3
x 10
200
100
0
200
100
0
P(D|M1) =
7.2e-016
Calculation of P(D|BASFOR)
MCMC 5000 steps
x 10
KNMIN
1
2
-3
x 10
200
100
0
0.02
KLAIMAX
6
8
1
-3
x 10
-3
x 10
200
100
0 CSOMF0
1
6 8
10
200
100
0
1
CSOMS0
2
3
Model "BASFORC6e": Expectation +- s.d. and MAP-output
40
20
1.5
-3
0.4
0
NSOMS0
0.1 0.2
400
200
0
0
200
100
NMIN0
0FLITTSOMF
1
2 0.4 0.6
400
200
0 FSOMFSOMS
0.8 0
0.05 0.1
20
0
0
1
2
3
-3
-4
x 10
2
3
2
2
1
1
20
0.5
x 10
0
0
1
2
3
0
4
0
1
2
3
4
200
100
0 KDBT
0.5
1
200
100
0
CR0T
2
0
-4
4
5
-3
200
100
0
2
200
100
0
300
CO20
350 400
100
50
0
6 0.2
0.8
KHEXP
0.3
0.4
200
100
0
200
100
0
0
0
200
100
0
4 0.4
KCA
2
KNMINT
1
2
200
100
0
KCAEXP
0.6 0.8
0
1
2
3
200
100
0
4
LAIMAXT
6
8
20
0.02
0
4
0
1
2
3
200
100
0
2
100
0
KNUPTT
1
2
-3
x 10
200
100
LUET
0 NCLMINT
2.5
3 0.01 0.015 0.02
2
3
Time
4
4
x 10
1
2
3
-0.01
0
1
2
Time
3
4
4
4
0.01
0
4
x 10
0.6
1
0
4
150
0
3
x 10
0.02
0.4
2
10
0
4
0.8
0.2
-3
0.8
0
0.04
4
x 10
200
100
KTB
0 KEXTT
20
30 0.4 0.6
0.05
1
4
30
x 10
x 10
200
100
KTA
0
0.03 0.04 10
0.8
-3
-4
x 10
4
x 10
0.06
Csoil
200
100
0 BETA
0.4 0.6
x 10
200
100
GAMMA
0
0.35 0.4 0.6
KH
4
2
0
4
NCl
200
100
0 CS0T
0
1
x 10
200
100
FLMAX
FS
0
0.3 0.35 0.25 0.3
KDRT
3
10
0
4
Miny
200
100
0
0.35 0.25
200
100
0
0.02
0
x 10
200
100
0 FB
0.25 0.3
1.5
200
100
0
-3
x 10
2
x 10
Nmin
-3
5
Nsoil
CL0T
0
Ntree
Parameter marginal probability distributions (truncated normal)
200
100
0
1
4
30
x 10
2
0
x 10
1.5
-5
x 10
0.1
200
100
0 CB0T
0
1
0
4
4
NPPy
KDLITT
2
4
1
x 10
3
200
100
KDSOMF
0 KDSOMS
0
0.5
1
0
1
0
x 10
Cr
0
0
4
4
-3
200
100
0
10
0.5
x 10
400
200
0
1
Cl
200
100
0
LAI
0
200
100
NLITT0
0 NSOMF0
0.01 0.02 0.2 0.3
h
200
100
0
Cw
x 10
Bayes Factor = 7.8, so
BASFOR+ supported by
the data
50
0
0
4
x 10
1
2
3
Time
4
4
x 10
Data Rajec: Emil Klimo
-3
100
200
50
100
0 NCLMAXT
0
0.04 0.05 0.06 0.02
NCRT
0.03 0.04
100
50
0
0
100
50
0
2 10
NCWT
1
x 10
200
200
100
100
SLAT
TRANCOT
0
0 CLITT0
20
30
4
6
8
0
0.5
1
-3
x 10
200
100
0 CSOMF0
6
8
10
100
50
0
1
CSOMS0
2
3
200
100
0
0
200
200
200
100
100
100
NLITT0
0 NSOMF0
0 NSOMS0
0
0.01 0.02 0.2 0.3
0.4 0
0.1 0.2 0
NMIN0
1
2
-3
x 10
100
200
50
100
0 FLITTSOMF
0
0.4 0.6 0.8 0
200
100
FSOMFSOMS 0 KDLITT
0.05 0.1 0
2
4
-3
x 10
200
100
0 KDSOMF
0
1
2
-4
x 10
200
100
0
0
KDSOMS
0.5
1
-5
x 10
Calculation of P(D|BASFOR+)
MCMC 5000 steps
P(D|M2) =
5.8e-15
Summary of procedure
Model 1
Model 2
Prior P(1)
Data D
Prior P(2)
MCMC
MCMC
Samples of 1
(104 – 105)
Samples of 2
(104 – 105)
P(D|M1)
P(D|M2)
Posterior P(1|D)
Bayes factor
Posterior P(2|D)
Updated parameters
Updated model odds
Updated parameters
Conclusions
Bayesian calibration using MCMC:
•
Improves model predictive capacity, by updating parameters
•
Quantifies uncertainty in parameters and output
Forest model calibration:
•
Benefits from high-precision tree height measurement
Bayesian model comparison:
•
Same probabilistic approach as Bayesian calibration
•
Bayes Factor shows how new data change the odds of models
•
Aid in model development
Appendices
Bayesian calibration of big models
P(|D)  P() P(D|f())
Calculating P(|D) costs much time:
1. Sample parameter-space representatively
2. For each sampled set of parameter-values:
a. Calculate P()
b. Run the model to calculate likelihood P(D|f())
Solutions
Sampling problem: Markov Chain Monte Carlo (MCMC) methods
Computing problem: increased processor speed
Bayes Factor for two big forest models
Parameter marginal probability distributions (truncated normal)
Prior parameter marginal probability distributions (uniform)
4000
2000
0
0
CL0
0.005 0.01
4000
2000
0
0
CR0
0.005 0.01
4000
2000
0
0
4000
2000
CW0
0 BETA
0.005 0.01 0.4 0.6
400
200
0
4000
4000
2000
2000
0 CO20
0 FLMAX
0.8 300 350 400 0.25 0.3 0.35
0
CL0
2
4
200
100
0
CR0
2
4
-3
4000
2000
0 FW
0.5 0.6
4000
2000
0 GAMMA
0.7 0.4 0.6
4000
2000
0 KCA
0.8
0
2
4
4000
2000
0
4000
4000
2000
2000
KCAEXP
0 KDL
0 KDR
0 0.5
1
0 0.005 0.01 0 0.5
1
-3
x 10
4000
2000
0
2
KDW
4
6
4000
2000
0
3
4000
2000
0
5 0.2
KH
4
KHEXP
0.3 0.4
4000
4000
2000
2000
0 KLAIMAX
0
0 0.005 0.01 0
KNMIN
1
2
-5
4000
2000
KTA
0
0.02 0.03
0
KNUPT
1
2
-3
x 10
BASFOR
39 params
4000
2000
0
-3
x 10
4000
2000
0
0.04 10
4000
2000
0 KTREE
30 0.4 0.6
KTB
20
0.8
4000
2000
0
1
x 10
Calibration
MCMC 10000 steps
NWCON
1
2
4000
2000
0
4000
2000
0 CLITT0
40
0 0.5
SLA
0
20
0.8
200
100
0
KCA
0
2
4
200
100
0
0
KCAEXP
0.5
1
0
x 10
4000
4000
2000
2000
NLITT0
NSOMF0
0
0
0.01 0.02 0.2 0.3 0.4
0
4000
2000
NSOMS0
0
0.1 0.2 0
200
100
0
2
KDW
4
6
200
100
0
200
100
0
5 0.2
KH
3
4
KHEXP
0.3 0.4
200
100
0
4
200
100
KLAIMAX 0
6
8
0
-5
0
KDLITT
2
-3
4000
2000
0 KDSOMS
2
0
1
-4
x 10
200
100
0
0.02
CB0T
0
5
4000
2000
0
0
CL0T
0.005 0.01
200
100
0
0.03
200
100
KTA
0
0.04 10
KTB
20
30
200
100
0 KTREE
0.4 0.6
0.8
200
100
0
2
LUE0
2.5
3
NWCON
1
2
200
100
0 SLA
0
20
0
x 10
200
200
100
100
NLITT0
NSOMF0
0
0
0.01 0.02 0.2 0.3 0.4 0
0
0
-3
0
4000
4000
2000
2000
0 FLMAX
0 FS
0.35 0.25 0.3 0.35 0.25 0.3
4000
2000
0 KDBT
0.5
1
4000
2000
0 KDRT
1.5
0
0.5
-4
(Penman eq.,
corrections)
KTA
0.03
1
4000
2000
0
2
4000
2000
0 GAMMA
0.35 0.4 0.6
4000
2000
KH
0
4
6
0.2
0.8
KHEXP
0.3 0.4
4000
2000
0
0
0
KTB
20
KCA
2
4
KNMINT
1
2
4000
2000
0
200
100
0 CLITT0
40 0
0.5
400
200
NSOMS0
0
0.1 0.2 0
-3
0.8
4000
2000
0
LAIMAXT
4
6
8
NCWT
1
2
4000
2000
0
CSOMF0
6 8 10
1
CSOMS0
2
3
4000
2000
0
0
4000
2000
0
1
4000
4000
2000
2000
NLITT0
0 NSOMF0
0
0.01 0.02 0.2 0.3 0.4
0
-3
x 10
-3
x 10
NSOMS0
0.1 0.2
4000
2000
0 KDSOMS
2
0
1
-4
x 10
0.06
200
100
0 CSOMF0
1
6 8
10
200
100
0
1
CSOMS0
2
3
NMIN0
1
2
200
100
0FLITTSOMF
0.4 0.6
400
200
0 FSOMFSOMS
0.8 0
0.05 0.1
4000
2000
0
1
2
0
Calibration
MCMC 10000 steps
-3
5
2
100
50
0
6 0.2
0.8
KHEXP
0.3
0.4
200
100
0
200
100
0
0
0
200
100
0
4 0.4
KCAEXP
0.6 0.8
KCA
2
KNMINT
1
2
200
100
0
0
KNUPTT
1
2
-3
x 10
200
100
KTA
0
0.03 0.04 10
CO20
350 400
x 10
200
100
GAMMA
0
0.35 0.4 0.6
KH
4
200
100
0
0.8 300
-3
x 10
200
100
0
200
100
0 BETA
2 0.4 0.6
-4
x 10
200
100
0
0.02
200
100
0 CS0T
4
0
1
CR0T
0
2
200
100
FLMAX
FS
0
0.3 0.35 0.25 0.3
KDRT
-4
4000
2000
0 NCLMINT
3 0.01 0.02 0.03
LUET
2
1.5
5
200
100
0
-3
200
100
0
-3
x 10
200
100
KTB
0 KEXTT
20
30 0.4 0.6
0.8
200
100
0
4
LAIMAXT
6
8
200
100
0
2
x 10
200
100
LUET
NCLMINT
0
2.5
3 0.01 0.015 0.02
-3
100
200
50
100
0 NCLMAXT
0
0.04 0.05 0.06 0.02
NCRT
0.03 0.04
100
50
0
0
100
50
0
2 10
NCWT
1
x 10
200
200
100
100
SLAT
0 TRANCOT
0 CLITT0
20
30
4
6
8
0
0.5
1
-3
x 10
0
NMIN0
1
2
-3
4000
2000
0 KDSOMF
4
0
1
200
200
100
100
0 NLCONMIN
0NLCONMAX
0.01 0.015 0.02 0.04 0.05
x 10
x 10
200
100
0 KDBT
0.5
1
x 10
4000
4000
4000
2000
2000
2000
0 FLITTSOMF
0 FSOMFSOMS
0 KDLITT
0.4 0.6 0.8
0 0.05 0.1
0
2
0
KNUPTT
1
2
-3
4000
2000
0
CL0T
0
4000
4000
2000
2000
0 TRANCOT
0 CLITT0
40
4
6
8
0
0.5
SLAT
0
20
2
200
100
0
200
100
0
0.35 0.25
x 10
4000
2000
0
200
100
0 CB0T
0
1
200
100
0 FB
0.25 0.3
-3
4000
2000
0
2
-3
x 10
-5
x 10
KCAEXP
0.5
1
x 10
4000
4000
4000
2000
2000
2000
NCRT
0 NCLMAXT
0
0
0.04 0.05 0.06 0.02 0.03 0.04 0
-4
x 10
0
x 10
4000
2000
0 KEXTT
30 0.4 0.6
200
100
KDSOMF
0 KDSOMS
0
0.5
1
0
1
x 10
-3
x 10
4000
2000
0
0.04 10
200
100
0
-3
4000
2000
0
-3
x 10
4000
2000
0
0.02
KDLITT
2
4
x 10
4000
2000
0 FB
0.25 0.3
KNUPT
1
Parameter marginal probability distributions (truncated normal)
4000
2000
CO20
0
0.8 300 350 400
-3
x 10
BASFOR +
41 params
4000
2000
0 BETA
5
0.4 0.6
CS0T
0
x 10
-3
Skogaby
4000
2000
0
1
-3
x 10
200
100
0
x 10
400
200
0
2
CR0T
0.005 0.01
KDR
0.5
-3
4000
4000
2000
2000
NMIN0
0 FLITTSOMF
0 FSOMFSOMS
1
2 0.4 0.6 0.8 0 0.05 0.1
x 10
4000
2000
0
0
-3
x 10
200
200
100
100
0 NRCON
0
0.02 0.03 0.04 0
Prior parameter marginal probability distributions (uniform)
4000
2000
0
200
100
0
-3
4000
4000
2000
2000
0 CSOMF0
0 CSOMS0
1
6 8 10
1
2
3
-5
x 10
1
KNMIN
1
2
-3
x 10
-3
4000
2000
0 KDSOMF
4
0
1
KDL
0
0.5
x 10
x 10
4000
2000
0
200
100
0
x 10
-3
4000
2000
0
200
100
0 FLMAX
400 0.25 0.3 0.35
x 10
200
100
GAMMA
0
0.7 0.4 0.6
-3
x 10
4000
4000
2000
2000
0 NRCON
0
0.02 0.03 0.04 0
200
100
0 CO20
0.8 300 350
-3
4000
4000
2000
2000
0 NLCONMIN
0 NLCONMAX
3 0.01 0.02 0.03 0.04 0.05 0.06
LUE0
2
6
200
100
CW0
0 BETA
0 0.005 0.01 0.4 0.6
-3
x 10
200
100
0 FW
0.5 0.6
400
200
0
200
100
0 CSOMF0
6
8
10
100
50
0
1
CSOMS0
2
3
200
100
0
0
200
200
200
100
100
100
NLITT0
0 NSOMF0
0 NSOMS0
0
0.01 0.02 0.2 0.3
0.4 0
0.1 0.2 0
NMIN0
1
2
-3
x 10
2
-5
x 10
100
200
50
100
0 FLITTSOMF
0
0.4 0.6 0.8 0
200
100
FSOMFSOMS 0 KDLITT
0.05 0.1 0
2
200
100
0 KDSOMF
4
0
1
-3
x 10
2
-4
x 10
200
100
0
0
KDSOMS
0.5
1
-5
x 10
Bayesian methods
Bayes’ Theorem
MCMC
Forest models
Crop models
Probability theory
Bayes, T. (1763)
Metropolis, N. (1953)
Green, E.J. / MacFarlane, D.W. /
Valentine, H.T. , Strawderman,
W.E. (1996, 1998, 1999, 2000)
Jansen, M. (1997)
Jaynes, E.T. (2003)