carbon fusion 9-11 th may 2006 budgets and bias in data assimilation keith haines, essc&darc,...

Carbon Fusion 9-11th May 2006

Budgets and Bias in Data AssimilationKeith Haines, ESSC&DARC, Reading

Background: Marine Informatics• Assimilation algorithms in Ocean circulation models

Satellite and In Situ data setsPhysically based covariances + simple errors in big and Biased modelsBudget diagnostics based on assimilation

• Met Office FOAM, ECMWF Seasonal Forecasting collaborationsDARC-NCOF Fellow Dan Lea based in NCOF group at Met Office

• New project (Marine Quest) will look at assimilation constraints on Carbon within a coupled physics-biochemistry ocean model

• e-Science/Grid: Model and Satellite data viewed in Google Maps/Earthhttp://lovejoy.nerc-essc.ac.uk:8080/Godiva2


Ocean Box-Inverse solution Ganachaud and Wunsch (2000)

Transport in Sverdrups 1Sv = 106 m3 s-1

Budgets and OceanThermohaline Circulation

After Broeker

• Closed Budgets of .. Heat, Salt, Mass/Volume, Tracers..• Processes: Advection, Surface fluxes, Mixing, Data Assimilation


Ocean Box-Inverse Assimilation

• Key assumption is for Steady State system• Therefore can use asynoptic data (different

ocean sections observed at completely different times)

• Try to correct for known variability eg. Seasonal cycle (surface properties and wind induced transports)

• Deduce unknown box-exchanges (circulation and mixing rates) for closed system

• Often problem underconstrained => use some Occams razor or conditioning assumption (smallest consistent flows/mixing rates)


Transport in Sverdrups 1Sv = 106 m3 s-1


N. Atlantic Water Budgetby density class (11S-80N)

COADS surface fluxesCTD section at 11SSteady State (cf. Ocean Inverse) => Mixing

Transformation Flux (Sv)

Speer (1997)

27.72 28.11


Walin Budget diagnostics for HadCM3 climate model (100yr

average)

Transformation Flux (Sv)

Old and Haines 2006

27.72 28.11


Data Assimilation in a time-evolving model?

• Steady state box-inverse models estimate process rates or parametrisations like mixing from a 3D Variational problem

• Similar “Parameter Estimation” while matching time–evolving data often uses 4DVar Assimilation

• 4DVar very expensive computationally• The “budget within a box” concept is subsumed into

seeking a solution to the temporal model equations• Parameter tuning assumes process representations are

‘structurally’ correct

• Different approach: Assimilation corrects for model bias so evaluate assimilation as another process within Box Budgets

• A posteriori “Process Estimation”


Process Estimation v. Parameter Estimation

Parameter estimation4DVar. Cost function containing fit to observations, a-priori info.Tune: initial state, sources/sinks, model parameters (diffusion)…..


Data Assimilation in a time-evolving model?

• Steady state box-inverse models estimate process rates or parametrisations like mixing from a 3D Variational problem

• Similar “Parameter Estimation” while matching time–evolving data often uses 4DVar Assimilation

• 4DVar very expensive computationally• The “budget within a box” concept is subsumed into

seeking a solution to the temporal model equations• Parameter tuning assumes process representations are

‘structurally’ correct

• Different approach: Assimilation corrects for model bias so evaluate assimilation as another process within Box Budgets

• A posteriori “Process Estimation”


OCCAM Assimilation Experiment

• 1993-96• ECMWF

6hr winds• Monthly

XBT assim.

• 10-day-ly Altimeter assim.

• SST weakly relaxed to Reynolds

• SSS weakly relaxed to Levitus

Sea Level analysis 28th March 1996

1/4° x 36 levels Global Ocean Model

RUN1


Process Estimation: Local Heat Budget Wm-2

Local Trend = Convergence + Assimilation + Surface Flux (+ Mixing)

Assimilation Advection

Trend 1993-96 Surface Flux Mixing

•Bias•Patterns•Amplitudes•Space scales•Transients

(Haines; 2003)


Process Estimation: N Atlantic Box Budgets

-G/ = dV/dt -

G = (1) Surface Forcing, (2) Mixing, (3) Data Assimilation

G = Volume Transformation Rate (Sv) G = Volume Transformation Rate (Sv) (after Walin 1982)(after Walin 1982)

Thermodynamically Irreversible ProcessesThermodynamically Irreversible Processes

Fox and Haines (2003) JPO

16Sv

Run1


Process Estimation in the Ocean

• Locally assimilation corrects for wrong Advection: eg. Gulf stream overshoots, Eastern Pacific thermocline

• Basin average sense assimilation corrects for wrong forcing i.e. surface heat flux

• Characteristic of certain processes can help to attribute assimilation contributions to box-budgets, eg.– Advection is conservative between regions (no

sources or sinks)– Mixing also conservative AND always downgradient


Relevance to Carbon Budget Modelling and Assimilation?

• Budget-box representation of terrestrial ecosystem• Conserved quantities: Carbon, Nitrogen/Nitrates?......• Understand cycling rates in model control (seasonal etc..

dependencies)• Assimilation will try to constrain Amounts of conserved

properties in each box. Unlikely to observe Transformation process rates?

• Success of assimilation may depend on;– Frequency of assimilation– Rate at which model transformation processes act– Any feedback between Amounts of property and transformation rates– Generation of unwanted transient processes as model adjusts to new

data


Shelf Seas: Carbon+Biochemistry Modelling

Hetero-trophs

Bacteria

Meso-Micro-

Particulates

Dissolved

Phytoplankton

Consumers

Pico-fDiatomsFlagell

-atesNO3

PO4

NH4

Si

CO2

Nutrients

Dino-f

Meio-benthos

AnaerobicBacteria

AerobicBacteria

DepositFeeders

SuspensionFeeders

Detritus

NutrIents

OxygenatedLayer

Reduced Layer

RedoxDiscontinuity

Layer

AtmosphereO2 CO2 DMS

3D

IrradiationWind Stress

Heat Flux

0D

Cloud Cover

Riv

ers

and

boundari

es

1D

Forcing Ecosystem

Physics

GOTMPOLCOMS

UKMO

ERSEM - key features

Carbon based process model

Functional group approach

Resolves microbial loop and POM/DOM dynamics

Complex suite of nutrients

Includes benthic system

Explicit decoupled cycling of C, N, P, Si and Chl.

Adaptable: DMS, CO2/pH, phytobenthos, HABs.


Bias and Data Assimilation• Assimilation often correcting for Process Biases

• In OCCAM model: – Locally assimilation corrects for wrong Advection: eg. mesoscale

eddies in the wrong location or biased advection eg. Gulf stream overshoots

– Basin average sense assimilation corrects for wrong forcing i.e. surface heat flux

• Characteristics of certain processes can help to attribute assimilation contributions to box-budgets, eg.– Advection is conservative between regions (no sources or sinks)– Mixing also conservative AND always downgradient

• May try to Account for bias when assimilating data as it should alter the error weighting between model and observations


Accounting for Bias in Data Assimilation

• Dee (2006) Review in QJRMS• Variational formulation easiest to understand (derivable from Bayesian

analysis; Drecourt et al; 2006)

2J(x,b,c) = (y-b-x)TR-1(y-b-x) +(x-xf+c)TB-1(x-xf+c) +

(b-bf)TO-1(b-bf) +(c-cf)TP-1(c-cf)

y =observation R =observation error covariance x =model state B =model background error covarianceb =observation bias O =observation bias error covariancec =model forecast bias P =model forecast bias error covarianceSuperscript f are forecast valuesObservation operators have been omitted


Accounting for Bias in Data Assimilation

• Solution (Analysed variables a)xa = (xf-cf) + K {(y-bf) – (xf-cf)} K = (B+P) [B+P+O+R]-1

ba = bf + F {(y-bf) – (xf-cf)} F = O [B+P+O+R]-1

ca = cf + G {(y-bf) – (xf-cf)} G = P [B+P+O+R]-1

or xa = (xf-ca) + K1{(y-ba) – (xf-ca)} K1 = B [B+R]-1

y =observation R =observation error covariance x =model state B =model background error covarianceb =observation bias O =observation bias error covariancec =model forecast bias P =model forecast bias error covariance

Usual problems are: (i) Knowing the Covariance errors(ii) Sequential 3DVar requires bias models for bf(t+1)= Mb[ba(t)]; cf(t+1)= Mc[ca(t)];


Comments on Bias Modelling

• Known Biases {bf (t); cf(t) known a priori eg. previous runs}– xa = (xf-cf) + K {(y-bf) – (xf-cf)} K = (B+P)[B+P+O+R]-1

– bf (t) = 0; cf(t) = 0 is particular case

– (B+P) total model err cov.; (O+R) total obs. err.

• Persistent Biases {bf(t+1)= ba(t); cf(t+1)= ca(t) }– xa = (xf-cf) + K {(y-bf) – (xf-cf)} K = (B+P)[B+P+O+R]-1

– ba = bf + F {(y-bf) – (xf-cf)} F = O[B+P+O+R]-1

– ca = cf + G {(y-bf) – (xf-cf)} G = P[B+P+O+R]-1

– If O,P i.e. F,G are small => may hope to converge to ~ constant b,c

– Simplifications also arise if P=αB; O=βR => all Innovations proportional

• Attribution of Bias: When are O,P sufficiently different to allow identification of misfits {(y-bf) – (xf-cf)} ?

• Should always check misfits are consistent with B+P+O+R


Example: Bias Modelling applied toAltimeter Data Assimilation

Bias Error Covariance O on Mean Sea Level

Mean Sea Level



Mean Sea Level Bias ba Corrected Mean Sea Level


CONCLUSIONS

• Biased model parameterisations can be tuned through 4DVar but only as far as structural errors and computational resources allow

• Alternatively build assimilation increments into box-budgets and seek to understand bias as process. Retains physically intuitive interpretation of Bias and Assimilation increments

• Having identified bias it should be accounted for during assimilation as it impacts on error weighting of model and data. Will need a bias model eg. understand its persistence, spatial structure, diurnal/seasonal cycling.


Altimeter Assimilation

Displacement h => Gross Isopycnal geometry

+ Currents (geostrophy)

•Volume and T/S properties preserved on isopycnals

• Adiabatic (Thermodynamically Reversible)

T Profile Assimilation

T(z) => Isothermal Water Volumes •T/S properties preserved (since salinity is not observed)

•Volumes and T/S preserved below deepest observation

S(T) Assimilation

S(T) => Isopycnal Water Properties

•Large scale, slow variations associated with ventilation and climatic change

Conservation properties of assimilation


Box Budgets and Assimilation

Nutrient recyclingfast

Nutrient recyclingfast

Transformation(slow)


Thermohaline Schematic

BroekerBroeker

Schmitz (1996)


WOCE Atlantic Section A16

SS NNNote: Water mass origins AIW, NADW, ABW

Long-lived Lagrangian properties of water used to trace spreading pathways. “Core method” Wust (1935)

Currents,Circulationrates andMixing ratesnotdetermined from Coremethod


Dissolved Inorganic Carbon


WOCE ComparisonN-S Pacific Temperature section P14

TP+ERS1 data 1993

Simulation

XBT Assimilation

XBT and Altimeter

Run available onLive Access Serverwww.nerc-essc.ac.uk/

godiva

WOCE Cruise

How to quantify the role of assimilation in maintaining thermocline?


Relevant Ideas

• Can we use assimilation methods to perform budgets?

• Focus on conservative properties of system (total carbon?) and processes converting between reservoirs

• Tune assimilation impact on processes rather than on model parameters


Based on Web Services

HadOCC


MARQuest proposal

• Assimilation of physical ocean data (temperature profiles, satellite data..) => constrain surface temperature and mixed layer depth to observations

• Study different ecosystem models embedded into physical model with data assimilation. Compare carbon cycling processes!

• Must develop treatment for ecosystem variables for when physical ocean data are assimilated. Careful attention to ecosystem and carbon budgets.

• Work with Hadley centre/Met Office FOAM assimilation system.


Assimilationresults

Ship ValidationWOCE Cruise

Marine Assimilation in Global Ocean Models

500m

0m

55 N15 S

•Extensive experience developing new assimilation algorithms eg. most recently for ARGO data •Assimilation of hydrography => vertical T gradients• Assimilation of altimetry => horizontal T gradients and currents• Algorithms used operationally at Met Office, ECMWF, France,US• Assimilation control of surface T and mixed layer depth will also constrain Ecosystems


MarQuest: Assimilation impact on Ecosystems • Assimilation controls and corrects seasonal thermocline T and MLD• Biological production will be strongly influenced by assimilation

HadOCC thermocline and chlorophyll conc.No Data Assimilation

FOAM thermoclineWith Data Assimilation

High resolution FOAM

All data from www.nerc-essc.ac.uk/godiva


Ideas

• Get Icarus ERSEM pictures of carbon cycle• Get Oschlies results figures• More reference figure on inverse modelling• Contact new MIT woman about land surface

assim

carbon fusion 9-11 th may 2006 budgets and bias in data assimilation keith haines, essc&darc,...

Documents

model bias

process rates

box concept

assimilation constraints

process representations

satellite data

timeevolving model

ocean inverse