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Bayesian calibration and uncertainty Bayesian calibration and uncertainty analysis of dynamic forest modelsanalysis of dynamic forest models
Bayesian calibration and uncertainty Bayesian calibration and uncertainty analysis of dynamic forest modelsanalysis of dynamic forest models
Marcel van Oijen
CEH-Edinburgh
Input to forest models and outputInput to forest models and outputInput to forest models and outputInput to forest models and output
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 input data
Input to forest models and outputInput to forest models and outputInput to forest models and outputInput to forest models and output
Soil
Trees
H2OC
Atmosphere
H2O
H2OC
Nutr.
Subsoil (or run-off)
H2OC
Nutr.
Nutr.
Nutr.
Model
Jmax
-100 0 100 200 300 400 500
Fre
quen
cy
0.00
0.04
0.08
0.12
0.16
Vmax
-50 0 50 100 150 200 250 300
0.00
0.05
0.10
0.15
0.20
0.25
umax,root
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.05
0.10
0.15
0.20
0.25
0.30
froot
-0.5 0.0 0.5 1.0 1.5
0.00
0.05
0.10
0.15
0.20
0.25
Initial Csoluble
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Cstarch
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Wtotal
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.0
0.1
0.2
0.3
0.4
0.5
Initial Nsoluble
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.00
0.05
0.10
0.15
0.20
Photosynthesis
Fre
qu
ency
Parameter value
Parameter value
Allocation
C-pools
N-pools
Jmax
-100 0 100 200 300 400 500
Fre
quen
cy
0.00
0.04
0.08
0.12
0.16
Vmax
-50 0 50 100 150 200 250 300
0.00
0.05
0.10
0.15
0.20
0.25
umax,root
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.05
0.10
0.15
0.20
0.25
0.30
froot
-0.5 0.0 0.5 1.0 1.5
0.00
0.05
0.10
0.15
0.20
0.25
Initial Csoluble
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Cstarch
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Wtotal
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.0
0.1
0.2
0.3
0.4
0.5
Initial Nsoluble
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.00
0.05
0.10
0.15
0.20
Photosynthesis
Fre
qu
ency
Parameter value
Parameter value
Allocation
C-pools
N-pools
[Levy et al, 2004]
Input to forest models and outputInput to forest models and outputInput to forest models and outputInput to forest models and output
Jmax
-100 0 100 200 300 400 500
Fre
quen
cy
0.00
0.04
0.08
0.12
0.16
Vmax
-50 0 50 100 150 200 250 300
0.00
0.05
0.10
0.15
0.20
0.25
umax,root
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.05
0.10
0.15
0.20
0.25
0.30
froot
-0.5 0.0 0.5 1.0 1.5
0.00
0.05
0.10
0.15
0.20
0.25
Initial Csoluble
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Cstarch
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Wtotal
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.0
0.1
0.2
0.3
0.4
0.5
Initial Nsoluble
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.00
0.05
0.10
0.15
0.20
Photosynthesis
Fre
qu
ency
Parameter value
Parameter value
Allocation
C-pools
N-pools
Jmax
-100 0 100 200 300 400 500
Fre
quen
cy
0.00
0.04
0.08
0.12
0.16
Vmax
-50 0 50 100 150 200 250 300
0.00
0.05
0.10
0.15
0.20
0.25
umax,root
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
0.00
0.05
0.10
0.15
0.20
0.25
0.30
froot
-0.5 0.0 0.5 1.0 1.5
0.00
0.05
0.10
0.15
0.20
0.25
Initial Csoluble
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Cstarch
-0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35
0.00
0.05
0.10
0.15
0.20
Initial Wtotal
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.0
0.1
0.2
0.3
0.4
0.5
Initial Nsoluble
Value
-0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06
0.00
0.05
0.10
0.15
0.20
Photosynthesis
Fre
qu
ency
Parameter value
Parameter value
Allocation
C-pools
N-pools
bgc
century
hybrid
bgc
0.0
0.1
0.2
0.3
0.4
century
Freq
uenc
y
0.0
0.1
0.2
0.3
0.4
hybrid
-40 -20 0 20 40 60 80
0.0
0.1
0.2
0.3
0.4
Ctotal / Ndepositedkg C (kg N)-1NdepUE (kg C kg-1 N)
[Levy et al, 2004]
Simpler models?Simpler models?Simpler models?Simpler models?
Goal:Robust models, predicting forest growth over 100
years, with low uncertainty
Effects that must be accounted for:N-deposition
CO2
TemperatureRain
RadiationSoil fertility
Management, e.g. thinning...
Are simple, robust models possible?
Typical model size: 50 - 311 parameters
Simple (semi-)empirical relationshipsSimple (semi-)empirical relationshipsSimple (semi-)empirical relationshipsSimple (semi-)empirical relationships
1. Lieth (1972, “Miami”-model): NPP = f(Temperature, Rain)
2. Monteith (1977): NPP = LUE * Intercepted light
3. Gifford (1980): NPP = NPP0 (1 + β Log([CO2]/[CO2]0) )
4. Gifford (1994): NPP = 0.5 GPP
5. Temperature ~ Light intensity
6. Roberts & Zimmermann (1999): LAImax Rain
7. Beer’s Law: Fractional light interception = (1-e-k LAI)
8. West, Brown, Enquist, Niklas (1997-2004): Height ~ Mass¼ ~ {fleaf, fstem, froot}
9. Brouwer (1983): Root-shoot ratio = f(N)
10. Goudriaan (1990): Turn-over rates, SOM, litter
BASic FORest model (BASFOR)BASic FORest model (BASFOR)BASic FORest model (BASFOR)BASic FORest model (BASFOR)
BASFOR 24 output variables39 parameters
BASFOR: InputsBASFOR: InputsBASFOR: InputsBASFOR: Inputs
BASFOR 24 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
Weather & soil: Skogaby (Sweden)
Forest data from Skogaby (Sweden)Forest data from Skogaby (Sweden)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: InputsBASFOR: InputsBASFOR: InputsBASFOR: Inputs
BASFOR 24 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
Weather & soil: Skogaby (Sweden)
BASFOR: InputsBASFOR: InputsBASFOR: InputsBASFOR: Inputs
BASFOR
Weather & soil: Skogaby (Sweden)
p1,min p1,max
P(p1)
p39,min p39,max
P(p39)
24 output variables
BASFOR: Prior predictive uncertaintyBASFOR: Prior predictive uncertaintyBASFOR: Prior predictive uncertaintyBASFOR: Prior predictive uncertainty
0 5000 10000 150000
0.2
0.4Tr
eeD
ens
0 5000 10000 15000-10
0
10
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 5000 10000 15000-0.5
0
0.5
Cl
0 5000 10000 15000-2
0
2
Cr
0 5000 10000 150000
0.5
1
Clit
t
0 5000 10000 150005
10
15
Cso
mf
0 5000 10000 150001
2
3
Cso
ms
0 5000 10000 15000-0.02
0
0.02
Nl
0 5000 10000 150000
0.01
0.02
Nlit
t
0 5000 10000 150000
0.2
0.4N
som
f
0 5000 10000 150000
0.1
0.2
Nso
ms
0 5000 10000 15000-0.2
0
0.2
Nm
in
0 5000 10000 150001000
1500
Rai
nCum
0 5000 10000 15000-1
0
1
NP
Py
0 5000 10000 150000
100
200
Nm
iner
alis
atio
nhay
0 5000 10000 150000
20
40
y(1)
0 5000 10000 150000
1
2
y(2)
0 5000 10000 15000-5
0
5
y(3)
0 5000 10000 15000-10
0
10
y(4)
0 5000 10000 150005
10
15
y(5)
0 5000 10000 15000-0.1
0
0.1
y(6)
Time (d)0 5000 10000 15000
0
0.5
1
y(7)
Time (d)0 5000 10000 15000
0.02
0.04
0.06
y(8)
Time (d)0 5000 10000 15000
0
1
2x 10
-3
y(9)
Time (d)
Wood C
Height
NPP
Skogaby, not calibrated (m ± σ)
BASFOR: Predictive uncertaintyBASFOR: Predictive uncertaintyBASFOR: Predictive uncertaintyBASFOR: Predictive uncertainty
BASFOR
24 output variables
High output uncertainty
39 parameters
High input uncertainty
Data: measurements of output variables
Calibration of parameters
CalibrationCalibrationCalibrationCalibration
f P(f(p))P(p)
DCalibration =“Find P(p|D)”
Bayesian calibration: P(p|D) = P(p) P(D|p) / P(D) P(p) L(f(p)|D)
“Posterior distribution”
“Prior distribution”
“Likelihood” given mismatch between
model output & data:
CalibrationCalibrationCalibrationCalibration
f P(f(p))P(p)
DBayesian calibration
P(f(p))P(p)
Data Skogaby (S)Data Skogaby (S)Data Skogaby (S)Data Skogaby (S)
0 0.5 10
0.5
1Tr
eeD
ens
0.5 1 1.5
x 104
0
10
20
Cw
Model "basfor12"
0.5 1 1.5
x 104
0.5
1
1.5
Cl
0.5 1 1.5
x 104
0
2
4
Cr
0 0.5 10
0.5
1
Clit
t
0 0.5 10
0.5
1
Cso
mf
0 0.5 10
0.5
1
Cso
ms
0 0.5 10
0.5
1
Nl
0 0.5 10
0.5
1
Nlit
t
0 0.5 10
0.5
1
Nso
mf
0 0.5 10
0.5
1
Nso
ms
6000 8000 10000 120000
1
2x 10
-3
Nm
in
0 0.5 10
0.5
1
Rai
nCum
1.0584 1.0585 1.0586
x 104
0.5
1
1.5
NP
Py
1.0584 1.0585 1.0586
x 104
50
100
150
Nm
iner
alis
atio
nhay
7000 8000 9000 100000
10
20
y(1)
0 0.5 10
0.5
1
y(2)
1.0584 1.0585 1.0586
x 104
0
10
20
y(3)
0.5 1 1.5
x 104
0
10
20
y(4)
7000 8000 9000 100005
10
15
y(5)
0.5 1 1.5
x 104
0
0.05
0.1
y(6)
Time (d)7000 8000 9000 100000
0.5
1
y(7)
Time (d)0.5 1 1.5
x 104
0
0.02
0.04
y(8)
Time (d)0 0.5 1
0
0.5
1
y(9)
Time (d)
Wood C
HeightNPP
Calculating the posterior distributionCalculating the posterior distributionCalculating the posterior distributionCalculating the posterior distribution
Bayesian calibration: P(p|D) P(p) L(f(p)|D)
Calculating P(p|D) costs much time:
1. Sample parameter-space representatively
2. For each sampled set of parameter-values:a. Calculate P(p)b. Run the modelc. Calculate errors (model vs data), and their likelihood
Sampling problem: Markov Chain Monte Carlo (MCMC) methods
Computing problem: Computer power, Numerical software
Solutions
Markov Chain Monte Carlo (MCMC)Markov Chain Monte Carlo (MCMC)Markov Chain Monte Carlo (MCMC)Markov Chain Monte Carlo (MCMC)
Metropolis algorithm BASFOR (~ 30 lines of code)
MCMC: walk through parameter-space, such that the set of visited points approaches the posterior parameter distribution P(p|D)
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)) L(f(p(i+1))) ] /
[ P(p(i)) L(f(p(i))) ] > Random[0,1]
THEN: accept P(i+1) & i=i+1ELSE: reject P(i+1)
4. IF i < 104 GOTO 2
1. E.g. {SLA=5, k=0.4, ... <39 parameters> ...}
2. Use multivariate normal distribution for [δ1, ... ,δ39]
3. Run BASFOR.Assume normally distributed errors: L(output-dataj) ~ N(0,σj) with different σj for each datapoint
MCMC parameter trace plots: 10000 stepsMCMC parameter trace plots: 10000 stepsMCMC parameter trace plots: 10000 stepsMCMC parameter trace plots: 10000 steps
0 5000 10000
2
4x 10
-3
CL0
0 5000 10000
2
4
6x 10
-3
CR0
0 5000 10000
2468
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
0.32KHEXP
0 5000 1000034567
x 10-3
KLAIMAX
0 5000 100000.60.8
11.21.41.61.8
x 10-3
KNMIN
0 5000 100000.60.8
11.21.41.61.8
x 10-3
KNUPT
0 5000 10000
0.0250.03
0.035 KTA
0 5000 1000015
20
25 KTB
0 5000 10000
0.4
0.5
0.6 KTREE
0 5000 100001.5
2
2.5
x 10-3
LUE0
0 5000 10000
0.015
0.02
0.025NLCONMIN
0 5000 10000
0.0350.04
0.045 NLCONMAX
0 5000 10000
0.0250.03
0.035 NRCON
0 5000 100000.60.8
11.21.41.61.8
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 100000.0060.0080.01
0.0120.0140.0160.018 NLITT0
0 5000 10000
0.250.3
0.35 NSOMF0
0 5000 100000.060.080.1
0.120.140.160.18 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 10000
11.5
22.5
x 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
Marginal distributions of parametersMarginal distributions of parametersMarginal distributions of parametersMarginal distributions of parameters
0 2 4 6
x 10-3
0
1000
2000
CL0
0 0.005 0.010
1000
2000
CR0
0 0.005 0.010
2000
4000
CW0
Parameter probability distributions
0.4 0.60
1000
2000
BETA
320 340 360 3800
1000
2000
CO20
0.25 0.3 0.350
1000
2000
FLMAX
0.5 0.6 0.70
1000
2000
FW
0.4 0.60
2000
4000
GAMMA
0 5 10 150
1000
2000
KCA
0.3 0.4 0.50
1000
2000
KCAEXP
0.5 1 1.5 2
x 10-3
0
5000
10000
KDL
0 2 4 6
x 10-4
0
1000
2000
KDR
0 0.5 1 1.5
x 10-4
0
2000
4000
KDW
2 4 6 80
1000
2000
KH
0.2 0.3 0.40
1000
2000
KHEXP
2 4 6 8
x 10-3
0
2000
4000
KLAIMAX
0.5 1 1.5 2
x 10-3
0
1000
2000
KNMIN
0.5 1 1.5 2
x 10-3
0
1000
2000
KNUPT
0.02 0.03 0.040
2000
4000
KTA
10 20 300
2000
4000
KTB
0 0.5 10
1000
2000
KTREE
1 2 3
x 10-3
0
2000
4000
LUE0
0.01 0.02 0.030
2000
4000
NLCONMIN
0.03 0.04 0.05 0.060
1000
2000
NLCONMAX
0.02 0.03 0.040
1000
2000
NRCON
0.5 1 1.5 2
x 10-3
0
1000
2000
NWCON
5 10 150
1000
2000
SLA
0 0.5 10
1000
2000
CLITT0
4 6 8 100
1000
2000
CSOMF0
1 2 30
1000
2000
CSOMS0
0.005 0.01 0.015 0.020
1000
2000
NLITT0
0.2 0.3 0.40
1000
2000
NSOMF0
0 0.1 0.20
1000
2000
NSOMS0
0 1 2
x 10-3
0
1000
2000
NMIN0
0.4 0.6 0.80
1000
2000
FLITTSOMF
0 0.05 0.10
1000
2000
FSOMFSOMS
0 1 2 3
x 10-3
0
1000
2000
KDLITT
0 0.5 1 1.5
x 10-4
0
2000
4000
KDSOMF
0 0.5 1 1.5
x 10-5
0
1000
2000
KDSOMS
Parameter correlations (PCC)Parameter correlations (PCC)Parameter correlations (PCC)Parameter correlations (PCC)
CL
0
CR
0
CW
0
BE
TA
CO
20
FL
MA
X
FW
GA
MM
A
KC
A
KC
AE
XP
KD
L
KD
R
KD
W
KH
KH
EX
P
KL
AIM
AX
KN
MIN
KN
UP
T
KTA
KT
B
KT
RE
E
LU
E0
NL
CO
NM
IN
NL
CO
NM
AX
NR
CO
N
NW
CO
N
SL
A
CL
ITT
0
CS
OM
F0
CS
OM
S0
NL
ITT
0
NS
OM
F0
NS
OM
S0
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 parameters3
9 p
ara
me
ters
Posterior predictive uncertaintyPosterior predictive uncertaintyPosterior predictive uncertaintyPosterior predictive uncertainty
0 5000 10000 150000
0.2
0.4Tr
eeD
ens
0 5000 10000 150000
10
20
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 5000 10000 150000
1
2
Cl
0 5000 10000 150000
2
4
Cr
0 5000 10000 150000
0.5
1
Clit
t
0 5000 10000 150005
10
15
Cso
mf
0 5000 10000 150001
2
3
Cso
ms
0 5000 10000 150000
0.01
0.02
Nl
0 5000 10000 150000
0.01
0.02
Nlit
t
0 5000 10000 150000.2
0.3
0.4N
som
f
0 5000 10000 150000
0.1
0.2
Nso
ms
0 5000 10000 15000-2
0
2x 10
-3
Nm
in
0 5000 10000 150001000
1500
Rai
nCum
0 5000 10000 150000
1
2
NP
Py
0 5000 10000 150000
100
200
Nm
iner
alis
atio
nhay
0 5000 10000 150000
10
20
y(1)
0 5000 10000 150000
1
2
y(2)
0 5000 10000 150000
10
20
y(3)
0 5000 10000 150000
10
20
y(4)
0 5000 10000 150005
10
15
y(5)
0 5000 10000 150000
0.05
0.1
y(6)
Time (d)0 5000 10000 15000
0
0.5
1
y(7)
Time (d)0 5000 10000 15000
0
0.05
y(8)
Time (d)0 5000 10000 15000
0
0.5
1x 10
-3
y(9)
Time (d)
Wood C
HeightNPP
Skogaby, calibrated (m ± σ)
Posterior predictive uncertainty vs priorPosterior predictive uncertainty vs priorPosterior predictive uncertainty vs priorPosterior predictive uncertainty vs prior
0 5000 10000 150000
0.2
0.4Tr
eeD
ens
0 5000 10000 15000-20
0
20
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 5000 10000 15000-2
0
2
Cl
0 5000 10000 15000-5
0
5
Cr
0 5000 10000 150000
0.5
1
Clit
t
0 5000 10000 150005
10
15
Cso
mf
0 5000 10000 150001
2
3
Cso
ms
0 5000 10000 15000-0.02
0
0.02
Nl
0 5000 10000 150000
0.01
0.02
Nlit
t
0 5000 10000 150000
0.2
0.4
Nso
mf
0 5000 10000 150000
0.1
0.2
Nso
ms
0 5000 10000 15000-0.2
0
0.2
Nm
in
0 5000 10000 150001000
1500
Rai
nCum
0 5000 10000 15000-2
0
2
NP
Py
0 5000 10000 150000
100
200
Nm
iner
alis
atio
nhay
0 5000 10000 150000
20
40
y(1)
0 5000 10000 150000
1
2
y(2)
0 5000 10000 15000-20
0
20
y(3)
0 5000 10000 15000-20
0
20
y(4)
0 5000 10000 150005
10
15
y(5)
0 5000 10000 15000-0.1
0
0.1
y(6)
Time (d)0 5000 10000 15000
0
0.5
1
y(7)
Time (d)0 5000 10000 15000
0
0.05
y(8)
Time (d)0 5000 10000 15000
0
1
2x 10
-3
y(9)
Time (d)
Wood C
HeightNPP
Skogaby, calibrated (m ± σ)
Skogaby, not calibrated (m ± σ)
Partial correlations parameters – output Partial correlations parameters – output variablesvariables
Partial correlations parameters – output Partial correlations parameters – output variablesvariables
Tre
eD
en
s
Cw
Cl
Cr
Clitt
Cs
om
f
Cs
om
s
Nl
Nlitt
Ns
om
f
Ns
om
s
Nm
in
Ra
inC
um
NP
Py
Nim
era
lis
ati
on
y
H CA
LA
I
Ctr
ee
Cs
oil
Ntr
ee
Ns
oil
NlC
l
Rs
oil
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
24 output variables3
9 p
ara
me
ters
Wo
od
C
Wood C vs parameter-valuesWood C vs parameter-valuesWood C vs parameter-valuesWood C vs parameter-values
2 4
x 10-3
4
6
CL02 4 6
x 10-3
4
6
CR02 4 6 8
x 10-3
4
6
Variation of wood-C with parameters
CW00.45 0.5 0.55
4
6
BETA330340350360370
4
6
CO200.260.280.30.320.34
4
6
FLMAX
0.55 0.6
4
6
FW0.45 0.5 0.55
4
6
GAMMA4 6 8 10 12 14
4
6
KCA0.35 0.4 0.45
4
6
KCAEXP0.8 1 1.21.41.6
x 10-3
4
6
KDL2 4
x 10-4
4
6
KDR
6 8 10 12 14
x 10-5
4
6
KDW4 6
4
6
KH0.220.240.260.280.30.32
4
6
KHEXP3 4 5 6 7
x 10-3
4
6
KLAIMAX0.60.811.21.41.61.8
x 10-3
4
6
KNMIN0.60.811.21.41.61.8
x 10-3
4
6
KNUPT
0.0250.030.035
4
6
KTA15 20 25
4
6
KTB0.4 0.5 0.6
4
6
KTREE1.61.822.22.42.62.8
x 10-3
4
6
LUE00.015 0.02 0.025
4
6
NLCONMIN0.0350.040.045
4
6
NLCONMAX
0.0250.030.035
4
6
NRCON0.60.811.21.41.61.8
x 10-3
4
6
NWCON6 8 10 12 14
4
6
SLA0.2 0.4
4
6
CLITT06 8
4
6
CSOMF01.5 2 2.5
4
6
CSOMS0
0.0060.0080.010.0120.0140.0160.018
4
6
NLITT00.25 0.3 0.35
4
6
NSOMF00.060.080.10.120.140.160.18
4
6
NSOMS00.5 1 1.5
x 10-3
4
6
NMIN00.5 0.6 0.7
4
6
FLITTSOMF0.020.040.060.08
4
6
FSOMFSOMS
1 1.5 2 2.5
x 10-3
4
6
KDLITT5 10
x 10-5
4
6
KDSOMF5 10
x 10-6
4
6
KDSOMS
Partial correlations parameters – wood CPartial correlations parameters – wood CPartial correlations parameters – wood CPartial correlations parameters – wood C
1 2 3-1
0
1CL0
1 2 3-1
0
1CR0
1 2 3-1
0
1CW0
PCC of parameters with wood-C
1 2 3-1
0
1BETA
1 2 3-1
0
1CO20
1 2 3-1
0
1FLMAX
1 2 3-1
0
1FW
1 2 3-1
0
1GAMMA
1 2 3-1
0
1KCA
1 2 3-1
0
1KCAEXP
1 2 3-1
0
1KDL
1 2 3-1
0
1KDR
1 2 3-1
0
1KDW
1 2 3-1
0
1KH
1 2 3-1
0
1KHEXP
1 2 3-1
0
1KLAIMAX
1 2 3-1
0
1KNMIN
1 2 3-1
0
1KNUPT
1 2 3-1
0
1KTA
1 2 3-1
0
1KTB
1 2 3-1
0
1KTREE
1 2 3-1
0
1LUE0
1 2 3-1
0
1NLCONMIN
1 2 3-1
0
1NLCONMAX
1 2 3-1
0
1NRCON
1 2 3-1
0
1NWCON
1 2 3-1
0
1SLA
1 2 3-1
0
1CLITT0
1 2 3-1
0
1CSOMF0
1 2 3-1
0
1CSOMS0
1 2 3-1
0
1NLITT0
1 2 3-1
0
1NSOMF0
1 2 3-1
0
1NSOMS0
1 2 3-1
0
1NMIN0
1 2 3-1
0
1FLITTSOMF
1 2 3-1
0
1FSOMFSOMS
1 2 3-1
0
1KDLITT
1 2 3-1
0
1KDSOMF
1 2 3-1
0
1KDSOMS
p
x
Allocation to wood
Senescence stem+br.
SOM turnover
Max Nleaf
Nroot
Should we measure the “sensitive Should we measure the “sensitive parameters”?parameters”?
Should we measure the “sensitive Should we measure the “sensitive parameters”?parameters”?
Yes, because the sensitive parameters:• are obviously important for prediction
No, because model parameters:• are model-specific• are correlated with each other, which we do not measure• cannot really be measured at all
So, it may be better to measure output variables, because they:• are what we are interested in• are better defined, in models and measurements• help determine parameter correlations if used in Bayesian
calibration
The value of NPP-dataThe value of NPP-dataThe value of NPP-dataThe value of NPP-data
0 5000 10000 150000
0.2
0.4Tr
eeD
ens
0 5000 10000 15000-10
0
10
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 5000 10000 15000-0.5
0
0.5
Cl
0 5000 10000 15000-5
0
5
Cr
0 5000 10000 150000
0.5
1
Clit
t
0 5000 10000 150005
10
15
Cso
mf
0 5000 10000 150001
2
3
Cso
ms
0 5000 10000 15000-0.02
0
0.02
Nl
0 5000 10000 150000
0.01
0.02
Nlit
t
0 5000 10000 150000
0.2
0.4
Nso
mf
0 5000 10000 150000
0.1
0.2
Nso
ms
0 5000 10000 15000-0.2
0
0.2
Nm
in
0 5000 10000 150001000
1500
Rai
nCum
0 5000 10000 15000-2
0
2
NP
Py
0 5000 10000 150000
100
200
Nm
iner
alis
atio
nhay
0 5000 10000 150000
20
40
y(1)
0 5000 10000 150000
1
2
y(2)
0 5000 10000 15000-5
0
5
y(3)
0 5000 10000 15000-20
0
20
y(4)
0 5000 10000 150005
10
15
y(5)
0 5000 10000 15000-0.1
0
0.1
y(6)
Time (d)0 5000 10000 15000
0
0.5
1
y(7)
Time (d)0 5000 10000 15000
0.02
0.04
0.06
y(8)
Time (d)0 5000 10000 15000
0
1
2x 10
-3
y(9)
Time (d)
Wood C
HeightNPP
Skogaby, calibrated on NPP-data only (m ± σ)
Skogaby, not calibrated (m ± σ)
Data of height growth: poor qualityData of height growth: poor qualityData of height growth: poor qualityData of height growth: poor quality
0 5000 10000 150000
0.2
0.4
Tre
eDen
s
0 5000 10000 15000-10
0
10
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 5000 10000 15000-0.5
0
0.5
Cl
0 5000 10000 15000-2
0
2
Cr
0 5000 10000 150000
0.5
1
Clit
t
0 5000 10000 150005
10
15
Cso
mf0 5000 10000 15000
1
2
3
Cso
ms
0 5000 10000 15000-0.02
0
0.02
Nl
0 5000 10000 150000
0.01
0.02
Nlit
t
0 5000 10000 150000
0.2
0.4N
som
f
0 5000 10000 150000
0.1
0.2
Nso
ms
0 5000 10000 15000-0.2
0
0.2
Nm
in
0 5000 10000 150001000
1500
Rai
nCum
0 5000 10000 15000-1
0
1
NP
Py
0 5000 10000 150000
100
200
Nm
iner
alis
atio
nhay
0 5000 10000 150000
20
40
y(1)
0 5000 10000 150000
1
2
y(2)
0 5000 10000 15000-5
0
5
y(3)
0 5000 10000 15000-10
0
10
y(4)
0 5000 10000 150005
10
15
y(5)
0 5000 10000 15000-0.1
0
0.1
y(6)
Time (d)0 5000 10000 15000
0
0.5
1
y(7)
Time (d)0 5000 10000 15000
0.02
0.04
0.06
y(8)
Time (d)0 5000 10000 15000
0
1
2x 10
-3
y(9)
Time (d)
Wood C
HeightNPP
Skogaby, calibrated on poor height-data only (m ± σ)
Skogaby, not calibrated (m ± σ)
Data of height growth: high qualityData of height growth: high qualityData of height growth: high qualityData of height growth: high quality
0 5000 10000 150000
0.2
0.4
Tre
eDen
s
0 5000 10000 15000-10
0
10
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 5000 10000 15000-0.5
0
0.5
Cl
0 5000 10000 15000-2
0
2
Cr
0 5000 10000 15000-1
0
1
Clit
t
0 5000 10000 150000
10
20
Cso
mf0 5000 10000 15000
1
2
3
Cso
ms
0 5000 10000 15000-0.02
0
0.02
Nl
0 5000 10000 15000-0.02
0
0.02
Nlit
t
0 5000 10000 150000
0.2
0.4N
som
f
0 5000 10000 150000
0.1
0.2
Nso
ms
0 5000 10000 15000-0.5
0
0.5
Nm
in
0 5000 10000 150001000
1500
Rai
nCum
0 5000 10000 15000-1
0
1
NP
Py
0 5000 10000 150000
100
200
Nm
iner
alis
atio
nhay
0 5000 10000 150000
20
40
y(1)
0 5000 10000 150000
1
2
y(2)
0 5000 10000 15000-5
0
5
y(3)
0 5000 10000 15000-10
0
10
y(4)
0 5000 10000 150005
10
15
y(5)
0 5000 10000 15000-0.1
0
0.1
y(6)
Time (d)0 5000 10000 15000
0
0.5
1
y(7)
Time (d)0 5000 10000 15000
0.02
0.04
0.06
y(8)
Time (d)0 5000 10000 15000
0
1
2x 10
-3
y(9)
Time (d)
Wood C
HeightNPP
Skogaby, calibrated on poor height-data only (m ± σ)
Skogaby, not calibrated (m ± σ)
Skogaby, calibrated on good height-data only (m ± σ)
Model application to forest growth in Rajec Model application to forest growth in Rajec (Czechia)(Czechia)
Model application to forest growth in Rajec Model application to forest growth in Rajec (Czechia)(Czechia)
Rajec (CZ):Planted: 1903, (6000 trees ha-1)Tree data: Wood-C, Height
Skogaby
Rajec
Rajec (CZ): Uncalibrated and calibrated on Rajec (CZ): Uncalibrated and calibrated on Skogaby (S)Skogaby (S)
Rajec (CZ): Uncalibrated and calibrated on Rajec (CZ): Uncalibrated and calibrated on Skogaby (S)Skogaby (S)
0 2 4
x 104
0
0.5
1
Tre
eDen
s
0 2 4
x 104
0
10
20
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 2 4
x 104
0
0.2
0.4
Cl
0 2 4
x 104
0
2
4
Cr
0 2 4
x 104
0
0.5
1
Clit
t
0 2 4
x 104
0
10
20
Cso
mf0 2 4
x 104
0
2
4
Cso
ms
0 2 4
x 104
0
0.005
0.01
Nl
0 2 4
x 104
0
0.01
0.02
Nlit
t
0 2 4
x 104
0.2
0.3
0.4N
som
f
0 2 4
x 104
0
0.1
0.2
Nso
ms
0 2 4
x 104
-0.2
0
0.2
Nm
in
0 2 4
x 104
500
1000
Rai
nCum
0 2 4
x 104
0
0.5
1
NP
Py
0 2 4
x 104
0
100
200
Nm
iner
alis
atio
nhay
0 2 4
x 104
0
20
40
y(1)
0 2 4
x 104
0
1
2
y(2)
0 2 4
x 104
0
2
4
y(3)
0 2 4
x 104
0
10
20
y(4)
0 2 4
x 104
0
20
40
y(5)
0 2 4
x 104
0
0.05
0.1
y(6)
Time (d)0 2 4
x 104
0
0.5
1
y(7)
Time (d)0 2 4
x 104
0
0.05
y(8)
Time (d)0 2 4
x 104
0
1
2x 10
-3
y(9)
Time (d)
Wood C
HeightNPP
Rajec, Skogaby-calibrated (m ± σ)
Rajec, not calibrated (m ± σ)
Rajec (CZ): Uncalibrated and calibrated on Rajec (CZ): Uncalibrated and calibrated on Skogaby (S)Skogaby (S)
Rajec (CZ): Uncalibrated and calibrated on Rajec (CZ): Uncalibrated and calibrated on Skogaby (S)Skogaby (S)
0 2 4
x 104
0
0.5
1
Tre
eDen
s
0 2 4
x 104
0
10
20
Cw
Model "basfor12"
0 2 4
x 104
0
0.2
0.4
Cl
0 2 4
x 104
0
2
4
Cr
0 2 4
x 104
0
0.5
1
Clit
t
0 2 4
x 104
0
10
20
Cso
mf0 2 4
x 104
0
2
4
Cso
ms
0 2 4
x 104
0
0.005
0.01
Nl
0 2 4
x 104
0
0.01
0.02
Nlit
t
0 2 4
x 104
0.2
0.3
0.4N
som
f
0 2 4
x 104
0
0.1
0.2
Nso
ms
0 2 4
x 104
-0.2
0
0.2
Nm
in
0 2 4
x 104
500
1000
Rai
nCum
0 2 4
x 104
0
0.5
1
NP
Py
0 2 4
x 104
0
100
200
Nm
iner
alis
atio
nhay
0 2 4
x 104
0
20
40
y(1)
0 2 4
x 104
0
1
2
y(2)
0 2 4
x 104
0
2
4
y(3)
0 2 4
x 104
0
10
20
y(4)
0 2 4
x 104
0
20
40
y(5)
0 2 4
x 104
0
0.05
0.1
y(6)
Time0 2 4
x 104
0
0.5
1
y(7)
Time0 2 4
x 104
0
0.05
y(8)
Time0 2 4
x 104
0
1
2x 10
-3
y(9)
Time
Wood C
HeightNPP
Rajec, Skogaby-calibrated (m ± σ)
Rajec, not calibrated (m ± σ)
Rajec (CZ): further calibration on Rajec-dataRajec (CZ): further calibration on Rajec-dataRajec (CZ): further calibration on Rajec-dataRajec (CZ): further calibration on Rajec-data
0 2 4
x 104
0
0.5
1
Tre
eDen
s
0 2 4
x 104
0
10
20
Cw
Model "basfor12": Calibration and Uncertainty Analysis
0 2 4
x 104
0
0.2
0.4
Cl
0 2 4
x 104
0
2
4
Cr
0 2 4
x 104
0
0.5
1
Clit
t
0 2 4
x 104
0
10
20
Cso
mf0 2 4
x 104
0
2
4
Cso
ms
0 2 4
x 104
0
0.005
0.01
Nl
0 2 4
x 104
0
0.01
0.02
Nlit
t
0 2 4
x 104
0.2
0.3
0.4N
som
f
0 2 4
x 104
0
0.1
0.2
Nso
ms
0 2 4
x 104
-0.2
0
0.2
Nm
in
0 2 4
x 104
500
1000
Rai
nCum
0 2 4
x 104
0
0.5
1
NP
Py
0 2 4
x 104
0
100
200
Nm
iner
alis
atio
nhay
0 2 4
x 104
0
20
40
y(1)
0 2 4
x 104
0
1
2
y(2)
0 2 4
x 104
0
2
4
y(3)
0 2 4
x 104
0
10
20
y(4)
0 2 4
x 104
0
20
40
y(5)
0 2 4
x 104
0
0.05
0.1
y(6)
Time (d)0 2 4
x 104
0
0.5
1
y(7)
Time (d)0 2 4
x 104
0
0.05
y(8)
Time (d)0 2 4
x 104
0
1
2x 10
-3
y(9)
Time (d)
Wood C
HeightNPP
Rajec, Skogaby-calibrated (m ± σ)
Rajec, not calibrated (m ± σ)
Rajec, Skogaby- and Rajec-calibrated (m ± σ)
Summary of procedureSummary of procedureSummary of procedureSummary of procedure
Data D ± σModel fPrior P(p)
Calibrated parameters, with covariances
Uncertainty analysis of model output
Sensitivity analysis of model parameters
“Error function” e.g. N(0, σ)
MCMC
Samples of p(104 – 105)
Samples of f(p)(104 – 105)
Posterior P(p|D) P(f(p)|D)PCC
Model selectionModel selectionModel selectionModel selection
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
Imperfect output data
Imperfect input data
Model selectionModel selectionModel selectionModel selection
Bayesian model selection: P(M|D) P(M) L(fM(pM)|D)
Bayesian calibration: P(p|D) P(p) L(f(p)|D)
0 2000 4000 6000 8000 10000 12000 140000
2
4
6
8
10
12
14
Cw
Model "basfor12"
Time0 2000 4000 6000 8000 10000 12000 14000
0
2
4
6
8
10
12
14
W
Model "expolinear"
Time
BASFOR(39 parameters)
Expolinear(4 parameters)
Max(log(L)) = -5.7 Max(log(L)) = -6.9“By-products”
of MCMCMean(log(L)) = -6.4 Mean(log(L)) = -8.7
Conclusions (1)Conclusions (1)Conclusions (1)Conclusions (1)
Reducing parameter uncertainty:
• Reduces predictive uncertainty• Reveals magnitude of errors in model structure• Benefits little from parameter measurement:
i. model parameter what you measureii. parameter covariances are more important than
variances• Requires calibration on measured outputs (eddy fluxes, C-
inventories, height-measurement, ...)
Calibration:
• Requires precise data• “Central” output variables are more useful than
“peripheral” (NPP/gas exchange > height)
Conclusions (2)Conclusions (2)Conclusions (2)Conclusions (2)
MCMC-calibration
• Works on all models• Conceptually simple, grounded in probability theory• Algorithmically simple (Metropolis)• Not fast (104 - 105 model runs)• Produces:
1. Sample from parameter pdf (means, variances and covariances), with likelihoods
2. Corresponding sample of model outputs (UA)3. Partial correlation analysis of outputs vs parameters (SA)
Model selection
• Can use the same probabilistic approach as calibration• Can use mean model log-likelihoods produced by MCMC
AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgements
• Göran Ågren (S) & Emil Klimo (CZ)
• Peter Levy, Renate Wendler, Peter Millard (UK)
• Ron Smith (UK)
Appendix 1: Calculation times per MCMC-Appendix 1: Calculation times per MCMC-stepstep
Appendix 1: Calculation times per MCMC-Appendix 1: Calculation times per MCMC-stepstep
0 1 2 3 4 5 60
500
1000
1500
2000
2500
3000
3500
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 100000
5
10
15
20
25
MCMC: to doMCMC: to doMCMC: to doMCMC: to do
1. Burn-in
2. Multiple chains
3. Mixing criteria (from characteristics of individual chains and from comparison of multiple chains)
4. Better (dynamic? f(prior?)) choice of step-length for generating candidate next points in p-space
5. Other speeding-up tricks?