Dept of MathematicsUniversity of Surrey

VAR and modelling the carbon cycle

Sylvain DelahaiesIan Roulstone

Dept of MathematicsUniversity of Surrey

NCEO Theme 7: Data Assimilation

GPP Croot

Cfoliage

Clitter

Ra

Af

Ar

Aw

Lf

Lr

Lw

Rh

D

Photosynthesis &plant respiration

Phenology &allocation

Senescence & disturbance

Microbial &soil processes

Climate drivers

GPP Croot

Cwood

Cfoliage

Clitter

CSOM/CWD

Ra

Af

Ar

Aw

Lf

Lr

Lw

Rh

D

Photosynthesis &plant respiration

Phenology &allocation

Senescence & disturbance

Microbial &soil processes

Climate drivers

DALEC evergreen

DALEC evergreen

Initial carbon pools: Cf, C

r, C

w, C

l, C

s

Parameters: p1, ...., p

11

Atmospheric Co2 concentration

4DVAR

4DVar data assimilation finds the trajectory that best combines a back-ground estimation of the control variable, the model and observations.

4D VAR

TT

dtLdtxfxtJL00

~

.0 ,0

(model) ,0

(adjoint) ,0

001

0

TTx

LxxB

x

L

xfxL

xfJx

L

b

T

Minimizing the cost function :

4DVAR

Conjugate gradient method

Preconditioning using the Hessian matrix

Minimization subject to box constraints

Dept of MathematicsUniversity of Surrey

Incremental 4D Var

Source Estimation

Testing VAR

• Relative error (TLM)

• Gradient test

4DVAR : linearized model and perfect observations

variable Relative error

Cf 0.59E-12

Cr 0.49E-05

Cl 0.24E-01

Cw 0.39E-05

Cs 0.33E-03

p1 0.18E-02

p2 0.68E-10

p3 0.45E-10

p4 0.77E-05

p5 0.12E-11

p6 0.25E-01

p7 0.24E-06

p9 0.39E-05

p10 0.32E-03

p11 0.98E-09

4DVAR : linearized, obs with small Gaussian error

variable Relative error

Cf 0.21E-03

Cr 0.89E+01

Cl 0.26E+05

Cw 0.39E+01

Cs 0.33E+03

p1 0.18E+05

p2 0.68E-03

p3 0.45E-03

p4 0.77E+01

p5 0.12E-03

p6 0.25E+05

p7 0.24E+00

p9 0.39E+01

p10 0.32E+03

p11 0.98E-03