Dept of Mathematics University of Surrey VAR and modelling the carbon cycle Sylvain Delahaies Ian Roulstone Dept of Mathematics University of Surrey NCEO.

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