Introduction to Data Assimilation

Peter Jan van Leeuwen

IMAU

Basic estimation theory

T0 = T + e0

Tm = T + em

E{e0} = 0E{em} = 0E{e0

2} = s02

E{em2} = sm

2

E{e0em} = 0

Assume a linear best estimate: Tn = a T0 + b Tm

with Tn = T + en

Find a and b such that:

b = 1 - a

a = __________ sm2

s02 + sm

2

E{en} = 0

E{en2} minimal

Solution: Tn = _______ T0 + _______ Tmsm

2 s02

s02 + sm

2 s02 + sm

2

___ = ___ + ___1 1 1

sm2s0

2sn2

and

Note: sn smaller than s0 and sm !

Basic estimation theory

Best Linear Unbiased Estimate BLUE

Just least squares!!!

Can we generalize this?

• More dimensions

• Nonlinear estimates (why linear?)

• Observations that are not directly modeled

• Biases

P(u)

u (m/s)

1.00.5

The basics: probability density functions

The model pdfP[u(x1),u(x2),T(x3),..

u(x1)

u(x2) T(x3)

Observations

• In situ observations: e.g. sparse hydrographic observations, irregular in space and time

• Satellite observations: e.g. of the sea-surface

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

NO INVERSION !!!

Data assimilation: general formulation

Bayes’ Theorem

Conditional pdf:

Similarly:

Combine:

Even better:

Filters and smoothers

Time

Filter: solve 3D problem several times

Smoother: solve 4Dproblem once

Note: the model is (highly) nonlinear!

Model equation:

Pdf evolution: Kolmogorov’s equation(Fokker-Planck equation)

Pdf evolution in time

Only consider mean and covariance

At observation times:

-The mean of the product of 2 Gaussians is equal to linear combination of the 2 means: E{|d} = a E{} + b E{d|}

- Assume p(d|) and p() are Gaussian, and use Bayes

- But we have seen this before in the first example !

(Ensemble) Kalman Filter

with Kalman gain K = PHT (HPHT + R)-1

Kalman Filter notation: mnew = m

old + K (d - H mold)

Old solution: Tn = _______ T0 + _______ Tmsm

2 s02

s02 + sm

2 s02 + sm

2

But now for covariance matrices:

mnew = R (P+R)-1 m

old + P (P+R)-1d

(Ensemble) Kalman Filter II

The error covariance:tells us how model variables co-vary

PSSH SSH(x,y) = E{ (SSH(x) - E{SSH(x)}) (SSH(y) - E{SSH(y)}) }

PSSH SST(x,y) = E{ (SSH(x) - E{SSH(x)}) (SST(y) - E{SST(y)}) }

For example SSH at point x with SSH at point y:

Or SSH at point x and SST at point y:

Spatial correlation of

SSHand SST in the Indian

Ocean

x

x

Haugen and Evensen, 2002

Covariancesbetween

modelvariables

Haugen and Evensen, 2002

Summary on Kalman filters:

• Gaussian pdf’s for model and observations• Propagation of error covariance P If N operations for state vector evolution, then N2 operations for P evolution…

Problems:• Nonlinear dynamics, so non-Gaussian statistics• Evolution equation for P not closed• Size of P (> 1,000,000,000,000) ….

Propagation of pdf in time:ensemble or particle methods

Example of Ensemble Kalman Filter (EnKF)

MICOM model with1.3 million model variablesObservations:Altimetry, infra-red

Validated with hydrographicobservations

SST (-2K to +2K)

SSH (-10 cm to +10 cm)

RMS difference with XBT-data

?

Spurious covariances

Local updating: restrict update using only local covariances:

EnKF:

with Kalman gain

Schurproduct, or direct cut-off

Localization in EnKF-like methods

Ensemble Kalman Smoother (EnKS)

Basic idea: use covariances over time.

Efficient implementation: 1) run EnKF, store ensemble at observation times2) add influence of data back in time using covariances at different times

0

2

4

6

8

10

408 412 416 420 424 428 432 436 440 444 448 452 456

Probability densityfunction of layer thicknessof first layer at day 41during data-assimilation

No Kalman filterNo variational methods

Nonlinear filters

The particle filter(Sequential Importance Resampling SIR)

Ensemble

with

Particle filter

SIR-results for a quasi-geostrophic ocean model around South Africa with 512 members

Smoothers: formulation

Model error

Initial error

Observation error

Boundary errors etc. etc.

Smoothers: prior pdf

Smoothers: posterior pdf

Assume all errors are Gaussian:

model initial observation

Assume Gaussian pdf for model errors and observations:

in which

Find min J from variational derivative:J is costfunction or penalty function

model dynamics initial condition model-obs misfit

Smoothers in practice: Variational methods

Gradient descent methods

J

model variable

123 4 561’

Forward integrations

Backward integrations

Nonlinear two-point boundary value problemsolved by linearization and iteration

The Euler-Lagrange equations

4D-VAR strong constraintAssume model errors negligible:

In practice only a few linear and one or two nonlinear iterations are done….

No error estimate (Hessian too expensive and unwanted…)

Example 4D-VAR: GECCO

• 1952 through 2001 on a 1º global grid with 23 layers in the vertical, using the ECCO/MIT adjoint technology.

• Model started from Levitus and NCEP forcing and uses state of the art physics modules (GM, KPP).

• Control parameters: initial temperature and salinity fields

and the time varying surface forcing,

The Mean Ocean Circulation, global

Residual values can reveal inconsistencies in data sets (here geoid).

MOC at 25N

Bryden et al. (2005)

Error estimates

J

Local curvature fromsecond derivative of J,the HessianX

Other smoothers

Representers, PSAS, Ensemble Kalman smoother, ….

Simulated annealing (Metropolis Hastings), …

Relations between model variables

• Covariance gives linear correlations between variables

• Adjoint gives linear correlation between variables along a nonlinear model run (linear sensitivity)

• Pdf gives full nonlinear relation between variables (nonlinear sensitivity)

Parameter estimation

Bayes:

Looks simple, but we don’t observe model parameters….

We observe model fields, so:

in which Hhas to be found from model integrations

Example: ecosystem modeling

29 parametersof which15 were estimatedand 14 were keptFixed.

Estimated parametersfrom particle filter (SIR)

All other methods thatwere tried, including4D-VAR and EnKF failed.

Losa et al, 2001

Estimate size of model error

Brasseur et al, 2006

Why data assimilation?

• Forecasts• Process studies• Model improvements - model parameters - parameterizations• ‘Intelligent monitoring’

Conclusions

• Evolution of pdf with time is essential ingredient

• Filters: dominated by Kalman-like methods, but moving towards nonlinear methods (SIR etc.)

• Smoothers: dominated by 4D-VAR,

New ideas needed!