Data assimilation in

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Marko Scholze

Strictly speaking, there are so far no DA activities in QUEST, but

• CCDAS (as part of core team activities)

• CPDAS and C4DAS (in the planning stage)

QuickTime™ and aTIFF (Uncompressed) decompressorare needed to see this picture.

A Carbon Cycle Data Assimilation System (CCDAS)

2FastOpt

31

Wolfgang Knorr and Marko Scholze

in collaboration withPeter Rayner1, Heinrich Widmann2, Thomas Kaminski3

& Ralf Giering3

Carbon Cycle Data Assimilation System (CCDAS) current form

BETHY+TM2only Photosynthesis,

Energy&Carbon Balance+Adjoint and Hessian code

Globalview CO2

+ Uncert.

Optimised Parameters + Uncert.

Diagnostics + Uncert.

Assimilation Step 2 (calibration) + Diagnostic Step

Background CO2 fluxes:ocean: Takahashi et al. (1999), LeQuere et al. (2000)emissions: Marland et al. (2001), Andres et al. (1996)

land use: Houghton et al. (1990)

veg. Index (AVHRR) + Uncert.

full BETHY

PhenologyHydrology

Assimilation Step1

Parameter Priors + Uncert.

CCDAS calibration step

• Terrestrial biosphere model BETHY (Knorr 97)delivers CO2 fluxes to atmosphere

• Uncertainty in process parameters from laboratory measurements

• Global atmospheric network provides additional constraint

€

J(r p ) =1

2r p −

r p 0[ ]

T Cp 0

-1 r p −

r p 0[ ] + 1

2

r M (

r p ) −

r D [ ]

T CD

-1 r

M (r p ) −

r D [ ]

covariance of uncertainty in measurements + modelcovariance of uncertainty

in priors for parameters

priors for parameters

observed concentrations

• Terrestrial biosphere model BETHY (Knorr 97)delivers CO2 fluxes to atmosphere

• Uncertainty in process parameters from laboratory measurements

• Global atmospheric network provides additional constraint

Gradient of J(p) providessearch directions for

minimisation.Second Derivative (Hessian)

of J(p) yields curvature of J,provides estimateduncertainty in popt

Figure taken from Tarantola '87

J(p)Minimisation and Parameter-Uncertainties

Space of p (model parameters)

€

Cp ≈ ∂ 2J(popt)∂pi, j

2

⎧ ⎨ ⎩

⎫ ⎬ ⎭

−1

Optimisation(BFGS+ adjoint gradient)

Posterior uncertainties on parameters

f ir st guess opt im ized pr ior unc. opt . unc. Vm ( Tr Ev) Vm ( EvCn) Vm ( C3G r) Vm ( Cr op)

µmol/m 2s µmol/m 2s % %Vm(TrEv) 60.0 43.2 20.0 10.5 0.28 0.02 -0.02 0.05Vm(EvCn) 29.0 32.6 20.0 16.2 0.02 0.65 -0.10 0.08Vm(C3Gr) 42.0 18.0 20.0 16.9 -0.02 -0.10 0.71 -0.31Vm(Crop) 117.0 45.4 20.0 17.8 0.05 0.08 -0.31 0.80

error covarianceexamples:

€

Cm ≈∂2J popt( )

∂pi, j2

⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

−1Use inverse Hessian of objective function to

approximate posterior uncertainties

Relative reduction of uncertainties

Observations resolve about 10-15 directions in parameter space

CCDAS diagnostic stepGlobal fluxes and their uncertainties

• Examples for diagnostics:• Long term mean fluxes to atmosphere

(gC/m2/year) and uncertainties• Regional means

Extension of concept1. More processes/components

• Have tested a version extended by an extremely simplified form of an ocean model:flux(x,t) = coefficient(i) * pattern(i,x,t)

• Optimising coefficients for biosphere patternswould allow the optimisation to compensate for errors (e.g. missing processes) in biosphere model (weak constraint 4DVar, see ,e.g., Zupanski (1993))

• But it is always preferable to include a process model, e.g for fire, marine biogeochemistry

• Can also extend to weak constraint formulation for state of biosphere model:include state as unknown with prior uncertainty estimated from model error

J(p) = ½ (p-p0)T Cp-1(p-p0)

+ ½ (cmod(p)- cobs) T Cc-1(cmod(p)- cobs)

+ ½ (fmod(p)- fobs)T Cf-1(fmod(p)- fobs)

+ ½ (Imod(p)- Iobs)T CI-1(Imod(p)- Iobs)

+ ½ (Rmod(p)- Robs)T CR-1(Rmod(p)- Robs)

+ etc ...

Extension of concept2. Adding more observations

Flux Data

•Can add further constraints on any quantity that can be extracted from the model (possibly after extensions/modifications of model)

•Covariance matrices are crucial: Determine relative weights!•Uses Gaussian assumption; can also use logarithm of quantity (lognormal distribution), ...

Inventories

AtmosphericIsotope Ratios

Atmospheric Concentrations (could also be column integrated)

Earth-System Predictions

• to build an adequate Earth System Model that is computationally efficient QUEST’s Earth System Modelling Strategy

• to develop a tool that allows the assimilation of observations of various kinds that relate to the various Earth System components, such as climate variables, atmospheric tracers, vegetation, ice extent, etc. CPDAS & C4DAS

• Climate Prediction Data Assimilation System (CPDAS): Assimilate climate variables of the past 100 years to constrain predictions of the next 100 years, including error bars.

• Coupled Climate C-Cycle Data Assimilation System (C4DAS): Assimilate carbon cycle observations of the past 20 (flask network) and 100 years (ice core data), to constrain coupled climate-carbon cycle predictions of the next 100 years, including error bars.

• Step-wise approach, building on and enhancing existing activities such as CCDAS, C4MIP, QUEST-ESM (FAMOUS), GENIEfy, QUMP and possibly Paleo-QUMP.

• Using the adjoint (and Hessian, relying on automatic differentiation techniques) which allows – for the first time – to optimize parameters comprehensively in a climate or earth system model before making climate predictions.

• Scoping study to start next month (pot. users meeting).

CCDAS methodological aspects

• remarks: – CCDAS tests a given combination of observational data plus model formulation with

uncertain parameters– CCDAS delivers optimal parameters, diagnostics/prognostics, and their a posteriori

uncertainties– all derivative code (adjoint, Hessian, Jacobian) generated automatically from model code

by compiler tool TAF: quick updates of CCDAS after change of model formulation– derivative code is highly efficient– CCDAS posterior flux field consistent with trajectory of process model

rather than linear combination of prescribed flux patterns (as transport inversion)– CCDAS includes a prognostic mode (unlike transport inversion)

• some of the difficulties/problems:– Prognostic uncertainty (error bars) only reflect parameter uncertainty

What about uncertainty in model formulation, driving fields…?– Uncertainty propagation only for means and covariances (specific PDFs),

and only with a linearised model– Result depends on a priori information on parameters– Result depends on a single model– Two step assimilation procedure sub optimal– lots of other technical issues

(bounds on parameters, driving data, Eigenvalues of Hessian ...)

BETHY(Biosphere Energy-Transfer-Hydrology

Scheme)• GPP:

C3 photosynthesis – Farquhar et al. (1980)C4 photosynthesis – Collatz et al. (1992)stomata – Knorr (1997)

• Plant respiration:maintenance resp. = f(Nleaf, T) – Farquhar, Ryan (1991)growth resp. ~ NPP – Ryan (1991)

• Soil respiration:fast/slow pool resp., temperature (Q10 formulation) and

soil moisture dependant• Carbon balance:

average NPP = b average soil resp. (at each grid point)<1: source>1: sink

t=1h

t=1h

t=1day

lat, lon = 2 deg

Seasonal cycleBarrow Niwot Ridge

observed seasonal cycleoptimised modeled seasonal cycle

Parameters I• 3 PFT specific parameters (Jmax, Jmax/Vmax and b)• 18 global parameters• 57 parameters in all plus 1 initial value (offset)

Param Initial Predicted Prior unc. (%) Unc. Reduction (%)

fautleafc-costQ10 (slow)

(fast)

0.41.251.51.5

0.241.271.351.62

2.50.57075

3917278

(TrEv) (TrDec) (TmpDec) (EvCn) (DecCn) (C4Gr) (Crop)

1.01.01.01.01.01.01.0

1.440.352.480.920.731.563.36

25252525252525

7895629591901

Some values of global fluxes

1980-2000 (prior)

1980-2000

1980-1990

1990-2000

GPPGrowth resp.Maint. resp.NPP

135.723.544.0468.18

134.822.3572.740.55

134.322.3172.1340.63

135.322.3973.2840.46

Fast soil resp.Slow soil resp.NEP

53.8314.46-0.11

27.410.692.453

27.610.712.318

27.2110.672.587

Value Gt C/yr

Global Growth Rate

Calculated as:

observed growth rateoptimised modeled growth rate

Atmospheric CO2 growth rate

MLOSPOGLOB CCC 75.025.0 +=

Including the ocean • A 1 GtC/month pulse lasting for three months is used as

a basis function for the optimisation• Oceans are divided into the 11 TransCom-3 regions• That means: 11 regions * 12 months * 21 yr / 3 months =

924 additional parameters• Test case:

all 924 parameters have a prior of 0. (assuming that our background ocean flux is correct)

each pulse has an uncertainty of 0.1 GtC/month giving an annual uncertainty of ~2 GtC for the total ocean flux

Including the oceanSeasonality at MLOGlobal land flux

Observations

Low-res incl. ocean basis functions Low resolution model

High resolution standard model