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?