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Bayesian calibration and comparison of process-based forest models . Marcel van Oijen & Ron Smith (CEH-Edinburgh) Jonathan Rougier (Durham Univ.). Contents. Bayesian calibration of a forest model … what measurements to take? Bayesian comparison of forest models. - PowerPoint PPT Presentation
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Bayesian calibration and comparison of Bayesian calibration and comparison of process-based forest models process-based forest models
Marcel van Oijen & Ron Smith (CEH-Edinburgh)
Jonathan Rougier (Durham Univ.)
ContentsContents
1. Bayesian calibration of a forest model
2. … what measurements to take?
3. Bayesian comparison of forest models
1. Bayesian calibration of a forest 1. Bayesian calibration of a forest modelmodel
Process-based forest modelsProcess-based forest models
Soil
Trees
H2OC
Atmosphere
H2O
H2OC
Nutr.
Subsoil (or run-off)
H2OC
Nutr.
Nutr.
Nutr.
Soil C
NPP
HeightEnvironmental scenarios
Initial values
Parameters
Model
Using dataUsing 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)BASic FORest model (BASFOR)
BASFOR 12 output variables39 parameters
BASFOR: InputsBASFOR: Inputs
BASFOR 12 output variables
Parameter Unit Min MaxBETA (-) 0.4 0.6CL0 (kg m-2) 0.0001 0.01CLITT0 (kg m-2) 0.15 0.6CO20 (ppm) 320 380CR0 (kg m-2) 0.0001 0.01CSOMF0 (kg m-2) 5 10CSOMS0 (kg m-2) 1 3CW0 (kg m-2) 0.0001 0.01FLITTSOMF (-) 0.4 0.8FLMAX (-) 0.25 0.35FSOMFSOMS(-) 0.01 0.1FW (-) 0.52 0.62GAMMA (-) 0.4 0.6KCA (m2) 3.65 14.6KCAEXP (m2) 0.333 0.5KDL (d-1) 0.0007 0.0028KDLITT (d-1) 0.0007 0.0028KDR (d-1) 0.000135 0.00054KDSOMF (d-1) 0.000028 0.00011KDSOMS (d-1) 0.0000028 0.000011KDW (d-1) 0.00004 0.00016KH (m) 2.5 10KHEXP (-) 0.2 0.33KLAIMAX (m2 m-2 mm-1) 0.002 0.008KNMIN (kg m-2) 0.0005 0.002KNUPT (kg m-2 d-1) 0.0005 0.002KTA (degC-1) 0.02 0.04KTB (degC) 10 30KTREE (m2 m-2) 0.35 0.65LUE0 (kg MJ-1) 0.001 0.003NLCONMAX (kg kg-1) 0.03 0.05NLCONMIN (kg kg-1) 0.01 0.03NLITT0 (kg m-2) 0.005 0.02NMIN0 (kg m-2) 0.0001 0.002NRCON (kg kg-1) 0.02 0.04NSOMF0 (kg m-2) 0.2 0.4NSOMS0 (kg m-2) 0.05 0.2NWCON (kg kg-1) 0.0005 0.002SLA (m2 kg-1) 5 15
BASFOR: Prior pdf for parametersBASFOR: Prior pdf for parameters
0 0.005 0.010
2000
4000
CL00 0.005 0.01
0
2000
4000
CR00 0.005 0.01
0
2000
4000
CW0
Prior parameter marginal probability distributions (uniform)
0.4 0.6 0.80
2000
4000
BETA300 350 4000
2000
4000
CO200.25 0.3 0.350
2000
4000
FLMAX
0.5 0.6 0.70
2000
4000
FW0.4 0.6 0.80
2000
4000
GAMMA0 2 4
0
2000
4000
KCA0 0.5 1
0
2000
4000
KCAEXP0 0.005 0.01
0
2000
4000
KDL0 0.5 1
x 10-3
0
2000
4000
KDR
2 4 6
x 10-5
0
2000
4000
KDW3 4 5
0
2000
4000
KH0.2 0.3 0.40
2000
4000
KHEXP0 0.005 0.01
0
2000
4000
KLAIMAX0 1 2
x 10-3
0
2000
4000
KNMIN0 1 2
x 10-3
0
2000
4000
KNUPT
0.02 0.03 0.040
2000
4000
KTA10 20 30
0
2000
4000
KTB0.4 0.6 0.80
2000
4000
KTREE1 2 3
x 10-3
0
2000
4000
LUE00.01 0.02 0.030
2000
4000
NLCONMIN0.04 0.05 0.060
2000
4000
NLCONMAX
0.02 0.03 0.040
2000
4000
NRCON0 1 2
x 10-3
0
2000
4000
NWCON0 20 40
0
2000
4000
SLA0 0.5 1
0
2000
4000
CLITT06 8 10
0
2000
4000
CSOMF01 2 3
0
2000
4000
CSOMS0
0 0.01 0.020
2000
4000
NLITT00.2 0.3 0.40
2000
4000
NSOMF00 0.1 0.2
0
2000
4000
NSOMS00 1 2
x 10-3
0
2000
4000
NMIN00.4 0.6 0.80
2000
4000
FLITTSOMF0 0.05 0.1
0
2000
4000
FSOMFSOMS
0 2 4
x 10-3
0
2000
4000
KDLITT0 1 2
x 10-4
0
2000
4000
KDSOMF0 1 2
x 10-5
0
2000
4000
KDSOMS
Example: Simulating growth of Norway Example: Simulating growth of Norway sprucespruce
Skogaby
BASFOR: Prior predictive uncertaintyBASFOR: Prior predictive uncertainty
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5N
PP
y
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass Prior uncertainty for Skogaby
BASFOR: Predictive uncertaintyBASFOR: Predictive uncertainty
BASFOR12 output variables
High output uncertainty
39 parameters
High input uncertainty
Data: measurements of output variables
Calibration of parameters
Bayes’ TheoremBayes’ Theorem
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
f = the model, e.g. BASFOR
Finding the posterior: MCMCFinding the posterior: MCMC
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
P(|D) P() P(D|f())
MCMC: Metropolis-Hastings random walkMCMC: Metropolis-Hastings random walk
Metropolis (1953) algorithm1. 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)Forest data from Skogaby (Sweden)
Planted: 1966, (2300 trees ha-1)Weather data: 1987-1995Soil data: C, N, Mineralisation rateTree data: Biomass, NPP, Height, [N], LAI
Skogaby
BASFOR: Prior predictive uncertaintyBASFOR: Prior predictive uncertainty
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5N
PP
y
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass Prior pred. uncertainty
Data Skogaby
Data:Göran Ågren
MCMC parameter trace plots: 10000 stepsMCMC parameter trace plots: 10000 steps
0 5000 10000
2
4x 10
-3
CL0
0 5000 10000
246
x 10-3
CR0
0 5000 100002468
x 10-3Parameter trace plots
CW0
0 5000 10000
0.450.5
0.55 BETA
0 5000 10000
330340350360370 CO20
0 5000 100000.260.280.3
0.320.34 FLMAX
0 5000 10000
0.55
0.6 FW
0 5000 10000
0.450.5
0.55 GAMMA
0 5000 10000468
101214
KCA
0 5000 100000.350.4
0.45 KCAEXP
0 5000 100000.8
11.21.41.61.8
x 10-3
KDL
0 5000 10000
2
4
x 10-4
KDR
0 5000 10000
68
101214
x 10-5
KDW
0 5000 10000
4
6 KH
0 5000 10000
0.220.240.260.280.3
0.32 KHEXP
0 5000 1000034567
x 10-3
KLAIMAX
0 5000 100000.60.811.21.41.61.8
x 10-3
KNMIN
0 5000 100000.60.811.21.41.61.8
x 10-3
KNUPT
0 5000 10000
0.0250.03
0.035 KTA
0 5000 10000152025 KTB
0 5000 10000
0.40.50.6 KTREE
0 5000 100001.5
22.5
x 10-3
LUE0
0 5000 10000
0.0150.02
0.025 NLCONMIN
0 5000 10000
0.0350.04
0.045 NLCONMAX
0 5000 10000
0.0250.03
0.035 NRCON
0 5000 100000.60.811.21.41.61.8
x 10-3
NWCON
0 5000 1000068
101214 SLA
0 5000 100000.2
0.4CLITT0
0 5000 10000
6
8CSOMF0
0 5000 10000
1.52
2.5 CSOMS0
0 5000 100000.0060.0080.010.0120.0140.0160.018 NLITT0
0 5000 10000
0.250.3
0.35 NSOMF0
0 5000 100000.060.080.10.120.140.160.18 NSOMS0
0 5000 10000
0.51
1.5
x 10-3
Iteration
NMIN0
0 5000 10000
0.50.60.7
Iteration
FLITTSOMF
0 5000 100000.020.040.060.08
Iteration
FSOMFSOMS
0 5000 100001
1.52
2.5x 10
-3
Iteration
KDLITT
0 5000 10000
5
10x 10
-5
Iteration
KDSOMF
0 5000 10000
5
10x 10
-6
Iteration
KDSOMS
Steps in MCMC
Param. value
Posterior marginal distributions for Posterior marginal distributions for parametersparameters
0 2 4 6
x 10-3
0
1000
2000
CL00 0.005 0.01
0
1000
2000
CR00 0.005 0.01
0
2000
4000
CW0
Parameter probability distributions
0.4 0.60
1000
2000
BETA320 340 360 3800
1000
2000
CO200.25 0.3 0.350
1000
2000
FLMAX
0.5 0.6 0.70
1000
2000
FW0.4 0.60
2000
4000
GAMMA0 5 10 15
0
1000
2000
KCA0.3 0.4 0.50
1000
2000
KCAEXP0.5 1 1.5 2
x 10-3
0
5000
10000
KDL0 2 4 6
x 10-4
0
1000
2000
KDR
0 0.5 1 1.5
x 10-4
0
2000
4000
KDW2 4 6 8
0
1000
2000
KH0.2 0.3 0.40
1000
2000
KHEXP2 4 6 8
x 10-3
0
2000
4000
KLAIMAX0.5 1 1.5 2
x 10-3
0
1000
2000
KNMIN0.5 1 1.5 2
x 10-3
0
1000
2000
KNUPT
0.02 0.03 0.040
2000
4000
KTA10 20 30
0
2000
4000
KTB0 0.5 1
0
1000
2000
KTREE1 2 3
x 10-3
0
2000
4000
LUE00.01 0.02 0.030
2000
4000
NLCONMIN0.03 0.04 0.05 0.060
1000
2000
NLCONMAX
0.02 0.03 0.040
1000
2000
NRCON0.5 1 1.5 2
x 10-3
0
1000
2000
NWCON5 10 15
0
1000
2000
SLA0 0.5 1
0
1000
2000
CLITT04 6 8 10
0
1000
2000
CSOMF01 2 3
0
1000
2000
CSOMS0
0.005 0.01 0.015 0.020
1000
2000
NLITT00.2 0.3 0.40
1000
2000
NSOMF00 0.1 0.2
0
1000
2000
NSOMS00 1 2
x 10-3
0
1000
2000
NMIN00.4 0.6 0.80
1000
2000
FLITTSOMF0 0.05 0.1
0
1000
2000
FSOMFSOMS
0 1 2 3
x 10-3
0
1000
2000
KDLITT0 0.5 1 1.5
x 10-4
0
2000
4000
KDSOMF0 0.5 1 1.5
x 10-5
0
1000
2000
KDSOMS
Parameter correlationsParameter correlations
CL0
CR
0
CW
0
BET
A
CO
20
FLM
AX
FW GA
MM
A
KC
A
KC
AEX
P
KD
L
KD
R
KD
W
KH
KH
EXP
KLA
IMA
X
KN
MIN
KN
UPT
KTA
KTB
KTR
EE
LUE0
NLC
ON
MIN
NLC
ON
MA
X
NR
CO
N
NW
CO
N
SLA
CLI
TT0
CSO
MF0
CSO
MS0
NLI
TT0
NSO
MF0
NSO
MS0
CL0 1.00 0.60 -0.67 -0.58 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
CR0 0.60 1.00 -0.49 -0.54 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
CW0 -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
BETA -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
CO20 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
FLMAX -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
FW 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
GAMMA 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
KCA 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
KCAEXP 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
KDL 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 0.17 -0.61 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
KDR 0.59 0.96 -0.57 -0.61 0.09 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 -0.61 -0.69 -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
KDW 0.38 -0.19 -0.74 -0.66 -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 0.49 -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
KH -0.42 -0.09 0.77 0.81 0.67 0.54 -0.83 -0.82 -0.34 0.00 -0.53 -0.20 -0.84 1.00 0.07 -0.78 0.85 0.08 0.80 -0.07 0.44 -0.93 -0.77 -0.73 0.30 -0.64 0.84 -0.25 -0.52 0.68 0.12 -0.92 -0.79
KHEXP -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 -0.26 0.14 0.00 -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
KLAIMAX 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 0.21 -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
KNMIN -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 -0.58
KNUPT 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
KTA -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 0.39 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
KTB -0.32 -0.81 0.14 0.17 0.06 -0.81 0.26 0.14 -0.69 -0.91 0.07 -0.69 0.49 -0.07 0.00 -0.21 -0.22 -0.83 0.39 1.00 -0.33 0.16 -0.25 -0.39 -0.46 -0.21 0.47 0.05 -0.52 -0.25 -0.22 -0.21 -0.38
KTREE -0.58 -0.21 0.78 0.61 -0.22 0.74 -0.55 -0.72 0.35 0.59 -0.81 -0.34 -0.73 0.44 -0.40 -0.70 0.58 0.28 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
LUE0 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 0.16 -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
NLCONMIN 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 0.92
NLCONMAX 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
NRCON 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 0.08 -0.46 0.66 -0.33 -0.16 -0.32 1.00 -0.22 -0.01 -0.46 0.34 0.44 0.31 -0.23 -0.21
NWCON 0.50 0.65 -0.81 -0.94 -0.19 -0.33 0.58 0.78 0.37 0.06 0.88 0.75 0.47 -0.64 -0.05 0.83 -0.42 0.48 -0.83 -0.21 -0.58 0.35 0.97 0.91 -0.22 1.00 -0.72 -0.03 0.23 -0.79 0.12 0.86 0.85
SLA -0.58 -0.54 0.77 0.84 0.57 0.06 -0.63 -0.65 -0.69 -0.48 -0.48 -0.57 -0.48 0.84 0.12 -0.82 0.60 -0.41 0.94 0.47 0.24 -0.72 -0.85 -0.87 -0.01 -0.72 1.00 -0.13 -0.75 0.51 -0.03 -0.93 -0.92
CLITT0 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 -0.38 0.07 0.05 -0.32 0.28 0.00 0.11 -0.46 -0.03 -0.13 1.00 -0.25 -0.15 -0.64 0.22 0.00
CSOMF0 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 0.33 -0.71 -0.52 0.15 0.56 0.41 0.46 0.34 0.23 -0.75 -0.25 1.00 -0.10 0.09 0.50 0.65
CSOMS0 -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 0.61 0.10 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 -0.68
NLITT0 -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 0.42 0.58 -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
NSOMF0 0.55 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 0.26 -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
NSOMS0 0.62 0.61 -0.90 -0.96 -0.36 -0.21 0.72 0.67 0.62 0.32 0.63 0.63 0.56 -0.79 0.00 0.96 -0.58 0.41 -0.99 -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 1.00
NMIN0 -0.16 -0.31 -0.47 -0.41 -0.64 -0.43 0.56 0.33 0.16 -0.09 -0.06 -0.30 0.66 -0.63 0.29 0.45 -0.72 -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
FLITTSOMF 0.48 0.60 -0.01 0.08 0.61 0.31 -0.43 0.03 -0.22 0.05 0.36 0.63 -0.39 0.40 0.15 -0.02 0.34 0.33 0.12 -0.39 -0.22 -0.62 0.01 -0.02 0.29 0.13 0.12 0.23 -0.28 -0.11 -0.40 -0.10 -0.10
FSOMFSOMS -0.66 -0.28 0.86 0.83 0.08 0.55 -0.89 -0.56 -0.33 0.08 -0.58 -0.27 -0.78 0.72 -0.04 -0.91 0.61 0.04 0.81 -0.03 0.69 -0.63 -0.69 -0.72 0.41 -0.62 0.65 0.07 -0.55 0.78 0.27 -0.70 -0.83
KDLITT 0.42 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 0.02 -0.83 -0.01 -0.63 0.80 0.84 0.88 -0.56 0.77 -0.80 0.34 0.37 -0.75 -0.16 0.92 0.87
KDSOMF 0.15 -0.43 -0.39 -0.31 -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 -0.27 0.55 -0.45 0.60 0.12 0.14 -0.51 0.04 -0.13 -0.14 0.29 -0.43 -0.25 0.20 0.29
KDSOMS -0.55 -0.18 0.83 0.81 0.13 0.80 -0.75 -0.92 0.12 0.47 -0.89 -0.35 -0.86 0.75 -0.12 -0.79 0.72 0.18 0.62 -0.32 0.89 -0.54 -0.76 -0.66 0.52 -0.77 0.51 -0.28 -0.03 0.98 0.39 -0.77 -0.65
39 parameters39
par
amet
ers
Bayesian calibration: overviewBayesian calibration: overview
0 0.005 0.010
2000
4000
CL00 0.005 0.01
0
2000
4000
CR00 0.005 0.01
0
2000
4000
CW0
Prior parameter marginal probability distributions (uniform)
0.4 0.6 0.80
2000
4000
BETA300 350 4000
2000
4000
CO200.25 0.3 0.350
2000
4000
FLMAX
0.5 0.6 0.70
2000
4000
FW0.4 0.6 0.80
2000
4000
GAMMA0 2 4
0
2000
4000
KCA0 0.5 1
0
2000
4000
KCAEXP0 0.005 0.01
0
2000
4000
KDL0 0.5 1
x 10-3
0
2000
4000
KDR
2 4 6
x 10-5
0
2000
4000
KDW3 4 5
0
2000
4000
KH0.2 0.3 0.40
2000
4000
KHEXP0 0.005 0.01
0
2000
4000
KLAIMAX0 1 2
x 10-3
0
2000
4000
KNMIN0 1 2
x 10-3
0
2000
4000
KNUPT
0.02 0.03 0.040
2000
4000
KTA10 20 30
0
2000
4000
KTB0.4 0.6 0.80
2000
4000
KTREE1 2 3
x 10-3
0
2000
4000
LUE00.01 0.02 0.030
2000
4000
NLCONMIN0.04 0.05 0.060
2000
4000
NLCONMAX
0.02 0.03 0.040
2000
4000
NRCON0 1 2
x 10-3
0
2000
4000
NWCON0 20 40
0
2000
4000
SLA0 0.5 1
0
2000
4000
CLITT06 8 10
0
2000
4000
CSOMF01 2 3
0
2000
4000
CSOMS0
0 0.01 0.020
2000
4000
NLITT00.2 0.3 0.40
2000
4000
NSOMF00 0.1 0.2
0
2000
4000
NSOMS00 1 2
x 10-3
0
2000
4000
NMIN00.4 0.6 0.80
2000
4000
FLITTSOMF0 0.05 0.1
0
2000
4000
FSOMFSOMS
0 2 4
x 10-3
0
2000
4000
KDLITT0 1 2
x 10-4
0
2000
4000
KDSOMF0 1 2
x 10-5
0
2000
4000
KDSOMS
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5
NP
Py
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntr
ee
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5
NP
Py
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntr
ee
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height BiomassHeight Biomass
0 2 4 6
x 10-3
0
1000
2000
CL00 0.005 0.01
0
1000
2000
CR00 0.005 0.01
0
2000
4000
CW0
Parameter probability distributions
0.4 0.60
1000
2000
BETA320 340 360 3800
1000
2000
CO200.25 0.3 0.350
1000
2000
FLMAX
0.5 0.6 0.70
1000
2000
FW0.4 0.60
2000
4000
GAMMA0 5 10 15
0
1000
2000
KCA0.3 0.4 0.50
1000
2000
KCAEXP0.5 1 1.5 2
x 10-3
0
5000
10000
KDL0 2 4 6
x 10-4
0
1000
2000
KDR
0 0.5 1 1.5
x 10-4
0
2000
4000
KDW2 4 6 8
0
1000
2000
KH0.2 0.3 0.40
1000
2000
KHEXP2 4 6 8
x 10-3
0
2000
4000
KLAIMAX0.5 1 1.5 2
x 10-3
0
1000
2000
KNMIN0.5 1 1.5 2
x 10-3
0
1000
2000
KNUPT
0.02 0.03 0.040
2000
4000
KTA10 20 30
0
2000
4000
KTB0 0.5 1
0
1000
2000
KTREE1 2 3
x 10-3
0
2000
4000
LUE00.01 0.02 0.030
2000
4000
NLCONMIN0.03 0.04 0.05 0.060
1000
2000
NLCONMAX
0.02 0.03 0.040
1000
2000
NRCON0.5 1 1.5 2
x 10-3
0
1000
2000
NWCON5 10 15
0
1000
2000
SLA0 0.5 1
0
1000
2000
CLITT04 6 8 10
0
1000
2000
CSOMF01 2 3
0
1000
2000
CSOMS0
0.005 0.01 0.015 0.020
1000
2000
NLITT00.2 0.3 0.40
1000
2000
NSOMF00 0.1 0.2
0
1000
2000
NSOMS00 1 2
x 10-3
0
1000
2000
NMIN00.4 0.6 0.80
1000
2000
FLITTSOMF0 0.05 0.1
0
1000
2000
FSOMFSOMS
0 1 2 3
x 10-3
0
1000
2000
KDLITT0 0.5 1 1.5
x 10-4
0
2000
4000
KDSOMF0 0.5 1 1.5
x 10-5
0
1000
2000
KDSOMS
DataBayesiancalibration
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5
NP
Py
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
inTime
0 5000 10000 150000
50
100
150
Min
y
Time
Height Biomass
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5
NP
Py
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
inTime
0 5000 10000 150000
50
100
150
Min
y
Time
Height BiomassHeight Biomass
Prior & posterior predictive uncertaintyPrior & posterior predictive uncertainty
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5N
PP
y
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass Prior pred. uncertainty
Posterior uncertainty (using data Skogaby)
Partial corr. coefficients (PCC) parameters – Partial corr. coefficients (PCC) parameters – outputsoutputs
Tree
Den
s
Cw
Cl
Cr
Clit
t
Cso
mf
Cso
ms
Nl
Nlit
t
Nso
mf
Nso
ms
Nm
in
Rai
nCum
NP
Py
Nim
eral
isat
iony
H CA
LAI
Ctre
e
Cso
il
Ntre
e
Nso
il
NlC
l
Rso
il
CL0 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 0.06 -0.05 -0.03
CR0 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 0.06 0.01 0.02
CW0 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 -0.02 -0.02 0.00
BETA 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 -0.05 -0.01 0.09
CO20 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 0.00 0.32 -0.05
FLMAX 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 0.17 -0.13 0.39
FW 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 0.01 0.55 0.63
GAMMA 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 0.00 0.77 -0.16
KCA -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 -0.05 0.02 0.02
KCAEXP 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 0.00 0.01 -0.01
KDL 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 0.67 0.23 0.52
KDR 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 0.91 0.14 0.42
KDW 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 0.18 0.05 0.84
KH 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 0.11 -0.05 -0.09
KHEXP 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 0.09 -0.03 -0.05
KLAIMAX -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 0.20 -0.43 0.52
KNMIN -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 -0.02 -0.10 -0.03
KNUPT 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 -0.02 0.07 0.01
KTA 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 -0.08 -0.77 0.23
KTB 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 -0.08 -0.72 0.18
KTREE 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 -0.01 -0.60 0.13
LUE0 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 -0.07 -0.80 0.20
NLCONMIN 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 0.14 0.99 -0.48
NLCONMAX 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 0.13 0.96 -0.49
NRCON 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 -0.30 -0.71 -0.82
NWCON 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 -0.55 -0.19 -0.36
SLA 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 -0.42 -0.46 -0.54
CLITT0 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 0.01 -0.03 0.06
CSOMF0 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 0.08 -0.04 0.77
CSOMS0 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 0.06 0.01 0.16
NLITT0 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 0.88 0.12 0.21
NSOMF0 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 1.00 0.57 0.68
NSOMS0 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 1.00 0.17 0.25
NMIN0 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 0.14 0.04 0.11
FLITTSOMF 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 0.72 -0.43 -0.92
FSOMFSOMS 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 0.32 -0.19 -0.45
KDLITT 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 -0.13 0.01 0.05
KDSOMF 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 -0.87 0.65 0.94
KDSOMS 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 -0.47 0.23 0.38
12 output variables
39 p
aram
eter
s
2. What kind of measurements 2. What kind of measurements would have reduced uncertainty would have reduced uncertainty
the most? the most?
Prior predictive uncertainty & height-dataPrior predictive uncertainty & height-data
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5N
PP
y
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass Prior pred. uncertainty
Height data Skogaby
Prior & posterior uncertainty: use of height dataPrior & posterior uncertainty: use of height data
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5N
PP
y
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass Prior pred. uncertainty
Posterior uncertainty (using height data)
Height data Skogaby
Prior & posterior uncertainty: use of height dataPrior & posterior uncertainty: use of height data
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5N
PP
y
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntre
e
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass Prior pred. uncertainty
Posterior uncertainty (using height data)
Height data (hypothet.)
Prior & posterior uncertainty: use of height dataPrior & posterior uncertainty: use of height data
0 5000 10000 150000
5
10
15
20
h
0 5000 10000 150000
5
10
Cw
BASFOR: Predictive uncertainty
0 5000 10000 150000
0.5
1
1.5
Cl
0 5000 10000 150000
1
2
3
Cr
0 5000 10000 15000-0.5
0
0.5
1
1.5N
PP
y
0 5000 10000 150000
5
10
LAI
0 5000 10000 150000
0.05
0.1
0.15
Ntr
ee
0 5000 10000 150000
0.02
0.04
0.06
NC
l
0 5000 10000 150000
5
10
Cso
il
0 5000 10000 150000
0.2
0.4
0.6
Nso
il
Time0 5000 10000 15000
0
0.05
0.1
0.15
0.2
Nm
in
Time0 5000 10000 15000
0
50
100
150
Min
y
Time
Height Biomass Prior pred. uncertainty
Posterior uncertainty (using height data)Posterior uncertainty (using precision height data)
Summary of procedureSummary of procedureData D ± σModel fPrior P()
Calibrated parameters, with covariances
Uncertainty of model output
Sensitivity analysis of model parameters
“Error function” e.g. N(0, σ)
MCMC
Samples of (104 – 105)
Samples of f()(104 – 105)
P(D|f())Posterior P(|D) PCC
3. Bayesian comparison of forest 3. Bayesian comparison of forest modelsmodels
Uncertainty regarding model structureUncertainty regarding model structure
Soil
Trees
H2OC
Atmosphere
H2O
H2OC
Nutr.
Subsoil (or run-off)
H2OC
Nutr.
Nutr.
Nutr.
Soil C
NPP
HeightEnvironmental scenarios
Initial values
Parameters
Model
Imperfect understanding
Bayesian comparison of two modelsBayesian comparison of two models
Bayes Theorem for model probab.:P(M|D) = P(M) P(D|M) / P(D)
The “Integrated likelihood” P(D|Mi) can be approximated from the MCMC sample of
outputs for model Mi (*)
Soil
Trees
H2OC
Atmosphere
H2O
H2OC
Nutr.
Subsoil (or run-off)
H2OC
Nutr.
Nutr.
Nutr.
Model 1
Soil
Trees
H2OC
Atmosphere
H2O
H2OC
Nutr.
Subsoil (or run-off)
H2OC
Nutr.
Nutr.
Nutr.
Model 2
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(M1) = P(M2) = ½
(*)
MCMCMCMC
MCMC
MCMCi
MCMCi
MDP
n
DPPP
DPDPP
P
w
DPwdDPPMDP
)|(1
)|()()(
)|()|()(
)()|()|()()|(
)(
harmonic mean of likelihoods in MCMC-sample (Kass & Raftery, 1995)
Bayes Factor for two big forest modelsBayes Factor for two big forest models
MCMC 5000 steps
MCMC 5000 steps
0 2 4
x 10-3
0200400
CL02 4 6
x 10-3
0100200
CR00 0.005 0.01
0200400
CW0
Parameter marginal probability distributions (truncated normal)
0.4 0.6 0.80
100200
BETA300 350 4000
100200
CO200.25 0.3 0.350
100200
FLMAX
0.5 0.6 0.70
100200
FW0.4 0.6 0.80
100200
GAMMA0 2 4
0100200
KCA0 0.5 1
0100200
KCAEXP0 0.5 1
x 10-3
0100200
KDL0 0.5 1
x 10-3
0100200
KDR
2 4 6
x 10-5
0100200
KDW3 4 5
0100200
KH0.2 0.3 0.40
100200
KHEXP4 6 8
x 10-3
0100200
KLAIMAX0 1 2
x 10-3
0100200
KNMIN0 1 2
x 10-3
0100200
KNUPT
0.02 0.03 0.040
100200
KTA10 20 30
0100200
KTB0.4 0.6 0.80
100200
KTREE2 2.5 3
x 10-3
0100200
LUE00.01 0.015 0.020
100200
NLCONMIN0.04 0.05 0.060
100200
NLCONMAX
0.02 0.03 0.040
100200
NRCON0 1 2
x 10-3
0100200
NWCON0 20 40
0100200
SLA0 0.5 1
0100200
CLITT06 8 10
0100200
CSOMF01 2 3
0100200
CSOMS0
0 0.01 0.020
100200
NLITT00.2 0.3 0.40
100200
NSOMF00 0.1 0.2
0100200
NSOMS00 1 2
x 10-3
0200400
NMIN00.4 0.6 0.80
100200
FLITTSOMF0 0.05 0.1
0200400
FSOMFSOMS
0 2 4
x 10-3
0200400
KDLITT0 0.5 1
x 10-4
0100200
KDSOMF0 1 2
x 10-5
0100200
KDSOMS
0 1 2
x 10-3
0100200
CB0T0 5
x 10-3
0100200
CL0T0 2 4
x 10-3
0100200
CR0T
Parameter marginal probability distributions (truncated normal)
0 1 2
x 10-3
0100200
CS0T0.4 0.6 0.80
100200
BETA300 350 4000
100200
CO20
0.25 0.3 0.350
100200
FB0.25 0.3 0.350
100200
FLMAX0.25 0.3 0.350
100200
FS0.4 0.6 0.80
100200
GAMMA0 2 4
0100200
KCA0.4 0.6 0.80
100200
KCAEXP
0.5 1 1.5
x 10-4
0100200
KDBT0 5
x 10-4
0100200
KDRT2 4 6
0100200
KH0.2 0.3 0.40
50100
KHEXP0 1 2
x 10-3
0100200
KNMINT0 1 2
x 10-3
0100200
KNUPTT
0.02 0.03 0.040
100200
KTA10 20 30
0100200
KTB0.4 0.6 0.80
100200
KEXTT4 6 8
0100200
LAIMAXT2 2.5 3
x 10-3
0100200
LUET0.01 0.015 0.020
100200
NCLMINT
0.04 0.05 0.060
50100
NCLMAXT0.02 0.03 0.040
100200
NCRT0 1 2
x 10-3
050
100
NCWT10 20 30
050
100
SLAT4 6 8
0100200
TRANCOT0 0.5 1
0100200
CLITT0
6 8 100
100200
CSOMF01 2 3
050
100
CSOMS00 0.01 0.02
0100200
NLITT00.2 0.3 0.40
100200
NSOMF00 0.1 0.2
0100200
NSOMS00 1 2
x 10-3
0100200
NMIN0
0.4 0.6 0.80
50100
FLITTSOMF0 0.05 0.1
0100200
FSOMFSOMS0 2 4
x 10-3
0100200
KDLITT0 1 2
x 10-4
0100200
KDSOMF0 0.5 1
x 10-5
0100200
KDSOMS
Calculation of P(D|BASFOR)
Skogaby
Rajec
Skogaby
Rajec
Calculation of P(D|BASFOR+)
Data Rajec: Emil Klimo
Bayes Factor for two big forest modelsBayes Factor for two big forest models
MCMC 5000 steps
MCMC 5000 steps
0 2 4
x 10-3
0200400
CL02 4 6
x 10-3
0100200
CR00 0.005 0.01
0200400
CW0
Parameter marginal probability distributions (truncated normal)
0.4 0.6 0.80
100200
BETA300 350 4000
100200
CO200.25 0.3 0.350
100200
FLMAX
0.5 0.6 0.70
100200
FW0.4 0.6 0.80
100200
GAMMA0 2 4
0100200
KCA0 0.5 1
0100200
KCAEXP0 0.5 1
x 10-3
0100200
KDL0 0.5 1
x 10-3
0100200
KDR
2 4 6
x 10-5
0100200
KDW3 4 5
0100200
KH0.2 0.3 0.40
100200
KHEXP4 6 8
x 10-3
0100200
KLAIMAX0 1 2
x 10-3
0100200
KNMIN0 1 2
x 10-3
0100200
KNUPT
0.02 0.03 0.040
100200
KTA10 20 30
0100200
KTB0.4 0.6 0.80
100200
KTREE2 2.5 3
x 10-3
0100200
LUE00.01 0.015 0.020
100200
NLCONMIN0.04 0.05 0.060
100200
NLCONMAX
0.02 0.03 0.040
100200
NRCON0 1 2
x 10-3
0100200
NWCON0 20 40
0100200
SLA0 0.5 1
0100200
CLITT06 8 10
0100200
CSOMF01 2 3
0100200
CSOMS0
0 0.01 0.020
100200
NLITT00.2 0.3 0.40
100200
NSOMF00 0.1 0.2
0100200
NSOMS00 1 2
x 10-3
0200400
NMIN00.4 0.6 0.80
100200
FLITTSOMF0 0.05 0.1
0200400
FSOMFSOMS
0 2 4
x 10-3
0200400
KDLITT0 0.5 1
x 10-4
0100200
KDSOMF0 1 2
x 10-5
0100200
KDSOMS
0 1 2
x 10-3
0100200
CB0T0 5
x 10-3
0100200
CL0T0 2 4
x 10-3
0100200
CR0T
Parameter marginal probability distributions (truncated normal)
0 1 2
x 10-3
0100200
CS0T0.4 0.6 0.80
100200
BETA300 350 4000
100200
CO20
0.25 0.3 0.350
100200
FB0.25 0.3 0.350
100200
FLMAX0.25 0.3 0.350
100200
FS0.4 0.6 0.80
100200
GAMMA0 2 4
0100200
KCA0.4 0.6 0.80
100200
KCAEXP
0.5 1 1.5
x 10-4
0100200
KDBT0 5
x 10-4
0100200
KDRT2 4 6
0100200
KH0.2 0.3 0.40
50100
KHEXP0 1 2
x 10-3
0100200
KNMINT0 1 2
x 10-3
0100200
KNUPTT
0.02 0.03 0.040
100200
KTA10 20 30
0100200
KTB0.4 0.6 0.80
100200
KEXTT4 6 8
0100200
LAIMAXT2 2.5 3
x 10-3
0100200
LUET0.01 0.015 0.020
100200
NCLMINT
0.04 0.05 0.060
50100
NCLMAXT0.02 0.03 0.040
100200
NCRT0 1 2
x 10-3
050
100
NCWT10 20 30
050
100
SLAT4 6 8
0100200
TRANCOT0 0.5 1
0100200
CLITT0
6 8 100
100200
CSOMF01 2 3
050
100
CSOMS00 0.01 0.02
0100200
NLITT00.2 0.3 0.40
100200
NSOMF00 0.1 0.2
0100200
NSOMS00 1 2
x 10-3
0100200
NMIN0
0.4 0.6 0.80
50100
FLITTSOMF0 0.05 0.1
0100200
FSOMFSOMS0 2 4
x 10-3
0100200
KDLITT0 1 2
x 10-4
0100200
KDSOMF0 0.5 1
x 10-5
0100200
KDSOMS
Calculation of P(D|BASFOR)
Calculation of P(D|BASFOR+)
Data Rajec: Emil Klimo
P(D|M1) = 7.2e-016
P(D|M2) = 5.8e-15
Bayes Factor = 7.8, so BASFOR+ supported by
the data
0 1 2 3 4
x 104
0
20
40
h
0 1 2 3 4
x 104
0
10
20
Cw
Model "BASFORC6e": Expectation +- s.d. and MAP-output
0 1 2 3 4
x 104
0
0.5
1
1.5
Cl
0 1 2 3 4
x 104
0
1
2
3C
r
0 1 2 3 4
x 104
0
0.5
1
1.5
NP
Py
0 1 2 3 4
x 104
0
10
20
30
LAI
0 1 2 3 4
x 104
0
0.05
0.1
Ntree
0 1 2 3 4
x 104
0
0.02
0.04
0.06
NC
l
0 1 2 3 4
x 104
0
10
20
30
Cso
il
0 1 2 3 4
x 104
0.2
0.4
0.6
0.8
Nso
il
Time0 1 2 3 4
x 104
-0.01
0
0.01
0.02
Nm
in
Time0 1 2 3 4
x 104
0
50
100
150
Min
y
Time
Summary of procedureSummary of procedure
Data DPrior P(1)
Updated parameters
MCMC
Samples of 1
(104 – 105)
Posterior P(1|D)
Model 1
MCMC
Prior P(2)
Model 2
Samples of 2
(104 – 105)
Posterior P(2|D)Updated parameters
P(D|M1) P(D|M2)
Bayes factorUpdated model odds
ConclusionsConclusionsBayesian 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
AppendicesAppendices
Bayesian calibration of big modelsBayesian 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())
Sampling problem: Markov Chain Monte Carlo (MCMC) methods
Computing problem: increased processor speed
Solutions
Bayes Factor for two big forest modelsBayes Factor for two big forest models
0 5
x 10-3
020004000
CB0T0 0.005 0.01
020004000
CL0T0 0.005 0.01
020004000
CR0T
Prior parameter marginal probability distributions (uniform)
0 5
x 10-3
020004000
CS0T0.4 0.6 0.80
20004000
BETA300 350 4000
20004000
CO20
0.25 0.3 0.350
20004000
FB0.25 0.3 0.350
20004000
FLMAX0.25 0.3 0.350
20004000
FS0.4 0.6 0.80
20004000
GAMMA0 2 4
020004000
KCA0 0.5 1
020004000
KCAEXP
0.5 1 1.5
x 10-4
020004000
KDBT0 0.5 1
x 10-3
020004000
KDRT2 4 6
020004000
KH0.2 0.3 0.40
20004000
KHEXP0 1 2
x 10-3
020004000
KNMINT0 1 2
x 10-3
020004000
KNUPTT
0.02 0.03 0.040
20004000
KTA10 20 30
020004000
KTB0.4 0.6 0.80
20004000
KEXTT4 6 8
020004000
LAIMAXT1 2 3
x 10-3
020004000
LUET0.01 0.02 0.030
20004000
NCLMINT
0.04 0.05 0.060
20004000
NCLMAXT0.02 0.03 0.040
20004000
NCRT0 1 2
x 10-3
020004000
NCWT0 20 40
020004000
SLAT4 6 8
020004000
TRANCOT0 0.5 1
020004000
CLITT0
6 8 100
20004000
CSOMF01 2 3
020004000
CSOMS00 0.01 0.02
020004000
NLITT00.2 0.3 0.40
20004000
NSOMF00 0.1 0.2
020004000
NSOMS00 1 2
x 10-3
020004000
NMIN0
0.4 0.6 0.80
20004000
FLITTSOMF0 0.05 0.1
020004000
FSOMFSOMS0 2 4
x 10-3
020004000
KDLITT0 1 2
x 10-4
020004000
KDSOMF0 1 2
x 10-5
020004000
KDSOMS
0 0.005 0.010
20004000
CL00 0.005 0.01
020004000
CR00 0.005 0.01
020004000
CW0
Prior parameter marginal probability distributions (uniform)
0.4 0.6 0.80
20004000
BETA300 350 4000
20004000
CO200.25 0.3 0.350
20004000
FLMAX
0.5 0.6 0.70
20004000
FW0.4 0.6 0.80
20004000
GAMMA0 2 4
020004000
KCA0 0.5 1
020004000
KCAEXP0 0.005 0.01
020004000
KDL0 0.5 1
x 10-3
020004000
KDR
2 4 6
x 10-5
020004000
KDW3 4 5
020004000
KH0.2 0.3 0.40
20004000
KHEXP0 0.005 0.01
020004000
KLAIMAX0 1 2
x 10-3
020004000
KNMIN0 1 2
x 10-3
020004000
KNUPT
0.02 0.03 0.040
20004000
KTA10 20 30
020004000
KTB0.4 0.6 0.80
20004000
KTREE1 2 3
x 10-3
020004000
LUE00.01 0.02 0.030
20004000
NLCONMIN0.04 0.05 0.060
20004000
NLCONMAX
0.02 0.03 0.040
20004000
NRCON0 1 2
x 10-3
020004000
NWCON0 20 40
020004000
SLA0 0.5 1
020004000
CLITT06 8 10
020004000
CSOMF01 2 3
020004000
CSOMS0
0 0.01 0.020
20004000
NLITT00.2 0.3 0.40
20004000
NSOMF00 0.1 0.2
020004000
NSOMS00 1 2
x 10-3
020004000
NMIN00.4 0.6 0.80
20004000
FLITTSOMF0 0.05 0.1
020004000
FSOMFSOMS
0 2 4
x 10-3
020004000
KDLITT0 1 2
x 10-4
020004000
KDSOMF0 1 2
x 10-5
020004000
KDSOMS
BASFOR39 params
BASFOR +41 params
(Penman eq., corrections)
SkogabySkogaby
MCMC 10000 steps
MCMC 10000 steps
0 2 4
x 10-3
0200400
CL02 4 6
x 10-3
0100200
CR00 0.005 0.01
0200400
CW0
Parameter marginal probability distributions (truncated normal)
0.4 0.6 0.80
100200
BETA300 350 4000
100200
CO200.25 0.3 0.350
100200
FLMAX
0.5 0.6 0.70
100200
FW0.4 0.6 0.80
100200
GAMMA0 2 4
0100200
KCA0 0.5 1
0100200
KCAEXP0 0.5 1
x 10-3
0100200
KDL0 0.5 1
x 10-3
0100200
KDR
2 4 6
x 10-5
0100200
KDW3 4 5
0100200
KH0.2 0.3 0.40
100200
KHEXP4 6 8
x 10-3
0100200
KLAIMAX0 1 2
x 10-3
0100200
KNMIN0 1 2
x 10-3
0100200
KNUPT
0.02 0.03 0.040
100200
KTA10 20 30
0100200
KTB0.4 0.6 0.80
100200
KTREE2 2.5 3
x 10-3
0100200
LUE00.01 0.015 0.020
100200
NLCONMIN0.04 0.05 0.060
100200
NLCONMAX
0.02 0.03 0.040
100200
NRCON0 1 2
x 10-3
0100200
NWCON0 20 40
0100200
SLA0 0.5 1
0100200
CLITT06 8 10
0100200
CSOMF01 2 3
0100200
CSOMS0
0 0.01 0.020
100200
NLITT00.2 0.3 0.40
100200
NSOMF00 0.1 0.2
0100200
NSOMS00 1 2
x 10-3
0200400
NMIN00.4 0.6 0.80
100200
FLITTSOMF0 0.05 0.1
0200400
FSOMFSOMS
0 2 4
x 10-3
0200400
KDLITT0 0.5 1
x 10-4
0100200
KDSOMF0 1 2
x 10-5
0100200
KDSOMS
0 1 2
x 10-3
0100200
CB0T0 5
x 10-3
0100200
CL0T0 2 4
x 10-3
0100200
CR0T
Parameter marginal probability distributions (truncated normal)
0 1 2
x 10-3
0100200
CS0T0.4 0.6 0.80
100200
BETA300 350 4000
100200
CO20
0.25 0.3 0.350
100200
FB0.25 0.3 0.350
100200
FLMAX0.25 0.3 0.350
100200
FS0.4 0.6 0.80
100200
GAMMA0 2 4
0100200
KCA0.4 0.6 0.80
100200
KCAEXP
0.5 1 1.5
x 10-4
0100200
KDBT0 5
x 10-4
0100200
KDRT2 4 6
0100200
KH0.2 0.3 0.40
50100
KHEXP0 1 2
x 10-3
0100200
KNMINT0 1 2
x 10-3
0100200
KNUPTT
0.02 0.03 0.040
100200
KTA10 20 30
0100200
KTB0.4 0.6 0.80
100200
KEXTT4 6 8
0100200
LAIMAXT2 2.5 3
x 10-3
0100200
LUET0.01 0.015 0.020
100200
NCLMINT
0.04 0.05 0.060
50100
NCLMAXT0.02 0.03 0.040
100200
NCRT0 1 2
x 10-3
050
100
NCWT10 20 30
050
100
SLAT4 6 8
0100200
TRANCOT0 0.5 1
0100200
CLITT0
6 8 100
100200
CSOMF01 2 3
050
100
CSOMS00 0.01 0.02
0100200
NLITT00.2 0.3 0.40
100200
NSOMF00 0.1 0.2
0100200
NSOMS00 1 2
x 10-3
0100200
NMIN0
0.4 0.6 0.80
50100
FLITTSOMF0 0.05 0.1
0100200
FSOMFSOMS0 2 4
x 10-3
0100200
KDLITT0 1 2
x 10-4
0100200
KDSOMF0 0.5 1
x 10-5
0100200
KDSOMS
Calibration
Calibration
Bayesian methodsBayesian methods
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)
Bayes’ Theorem
MCMC
Forest models
Crop models
Probability theory
