Sequential data assimilation in oceanography

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Sequential data assimilation in oceanography
Centre de Geostatistique
Centre de géostatistique

• TIES 2002, Genoa, 16th june

Bertino Laurent, Wackernagel Hans Centre de
géostatistique, ENSMP http//cg.ensmp.fr/bertino
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Outline

• Data assimilation (DA) problem
• Usual sequential DA methods
• Optimal Interpolation
• Kalman filter (KF)
• Further theoretical developments
• Normalisation for different supports
• Nonlinear estimation ( illustration)
• Case studies Odra lagoon hydrodynamics
• Assimilation of water level (easy)
• Assimilation of salinity (difficult)

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Issues in data assimilation

• Needed
• Monitoring stations.
• Dynamical model (partial diff. equations pde)
• Goals
• Combine both informations on line.
• Correct an imperfect model.
• Interpret data by nonlinear physical equations.

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Data assimilationas a hidden Markov chain model
Hidden state

• Model (pde) f
• Measurements Y

Observations

• Variational DA optimisation (meteo 1980)
• Sequential DA statistical estimation (oceano
  1990)

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Theoretical problems

• Nonlinear pde model and observations
• Linear models not really interesting.
• Is the gaussian approximation adequate?
• Different spatio-temporal supports
• Model defined grid and time step.
• Data sampling device and integration time.

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Practical problems

• Memory requirements
• Physical models require high resolution.
• State vector X of 106 parameters.
• Computing times
• Model propagations are CPU consuming.
• Operational constraint forecast must be
  delivered before the events go out of hand!

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Sequential DA

• Two-step recursive estimation
• 1. Propagation Xf (as forecast)
• 2. Correction Xa (as analysis)

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Optimal interpolation(Gandin 1965, Daley 1991)

• Stationary error, unbiased and gaussian
• X-Xf N (0,C)
• Multivariate covariance C to be specified
• Least squares estimation simple kriging.
• Primitive but cheap 1 model propagation!

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Kalman filter(Kalman 1960, Ghil 1981)
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2
3

• 3 random error fields
• 1 Initial error N(0,C0)
• 2 Model error N(0,?m)
• 3 Measurement error N(0,?o)
• Improvement error propagation by f
• nonstationary Xn(x) of covariance matrix Cfn

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Kalman filterlinear case

• Propagation

• Correction
• Kalman gain

• Update

F linear model H observation operator
The correction estimator is kriging with known
mean
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Kalman filtersnonlinear propagation

• Ensemble KF EnKF (Evensen 1994)
• Monte Carlo simulations drawing Xa,inL(XnY1n)
• avoids linearizing f
• strongly nonlinear dynamics
• 100 propagations

• Extended KF EKF (Jazwinski 1970)
• f linearised every step
• CPU consuming
• Sub-optimal schemes
• RRSQRT Reduced Rank Square Root (Heemink
  Verlaan 1995)
• approximation on dominant eigenvalues
• 50 model propagations

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Different supportsnormalisation

• The model propagation provides
• Output on grid cells Xn(vi)
• Block covariance matrix Cfn C(vi,vj)(i,j)
• Correction step
• Quasi punctual measurements Yn(x)
• C(x,vj) required
• How to increase the covariance resolution?

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Different supports

• Mercer decomposition theorem (eigenfunctions)

We have
We need
Estimation of ?(x) knowing all ?(vi)?
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Different supports(idea C. Lajaunie)

• Kriging
• Gaussian assumption.
• Specify a stationary structure C?(x-y)
• Good restitution of C(x,vi)
• Easy to implement with EKF, EnKF.

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Kalman filterlinear correction

• X and Y assumed gaussian, but f nonlinear
• Least squares estimation not optimal.
• Attraction towards the gaussian law.
• Non-physical Xan values (into f again at n1).

Does the solution need to be gaussian?
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Kalman filterfor lognormal variables

• Nonlinear transform
• Difficulty nonlinear estimation of nonstationary
  variables.
• Use a Monte Carlo approximation (EnKF)

Valid for all transformations
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Lognormal model illustration Idealised case
1-D ecological model

• Spring bloom model (Evans Parslow, Eknes
  Evensen)

Nutrients
Herbivores
Phytoplankton
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Idealised 1-D ecological model

• Simulated data
• Equivalence between EnKF and RRSQRT

• Characteristics
• Sensitive to initial conditions
• Non-linear dynamics

Nutrients
Phytoplankton
Herbivores
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Lognormal assumption
Original histograms positive
N
P
H
logarithms
possible refinement with gaussian anamorphosis
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Results
Gaussian
Lognormal

• Gaussian assumption
• Truncated Hlt0
• Low H values overestimated
• False starts
• Lognormal assumption
• Positive values
• Errors dependent on values

N
P
H
RMS observed errors
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SIR filters(Doucet et al. 2001)the latest
estimation method

• Alternative Monte Carlo methods
• Sequential Importance Sampling Resampling
• Fully nonlinear estimation (? Kalman)
• Recent application in physical oceanography (van
  Leeuwen 2002)

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Application to the Odra lagoon

• Data assimilation
• fixed stations
• hydrodynamics TRIM3D model (Casulli 1994, Wolf)

Pl
D
Summer 1997 flood period
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The Odra lagoonsensitive to hydrodynamics and
transport
Water level
Salinity
Fluxes
Heat
Matter
Physical model
Ecological model
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Water levels TRIM3D model (Casulli 1994, Wolf)
Well known dynamics but ill-specified boundary
conditions

• Boundary conditions
• Water levels at the Baltic interface
• Wind fields
• Odra river discharge
• Almost homogeneous variations
• RRSQRT KF

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Water levelmeasurements

• 6 fixed stations flood period
• 1 boundary conditions
• 3 assimilated data
• 2 validation
• Measurements
• temporally rich
• spatially poor
• Are 3 stations enough?

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Model error at the Baltic sea interface

• Time series analysis
• cross-covariances shifts
• Assumption wave-like error propagation (Sénégas,
  1999)
• Invariant variograms
• shifted cross-covariances
• Spatial structure with large range
• ignored support effect

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Model validation without DATRIM3D boundary
conditions
Water levels at Odh1
meters
Time (days)
Arbitrary initial state
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DA ValidationRRSQRT KF
Water levels at Odh1
meters
Time (days)
Data not assimilated
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Salinitya more difficult example

• Slow transport
• More sensitive to initial conditions

2 stations only Influence range?
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DA of salinitysetup

• EnKF gt RRSQRT
• easier setup of initial error
• fewer parameters
• Unknown structures
• initial error
• error at boundaries
• support effect
• Ship campaign data analysis
• Ill-specified structures?

- Horizontal variogram
- Vertical variogram
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Results attractordecreasing variances
Sd(sx)
Cov(sodh2,sx)
First assimilation
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Results attractordecreasing variances
Cov(sodh2,sx)
Sd(sx)
After 1 day of assimilation
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Results attractordecreasing variances
Cov(sodh2,sx)
Sd(sx)
After 2 days of assimilation
The model is always right!
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DA of salinityinitial state
sx
Simulated sainity by TRIM3D
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Results
TRIM3D
EnKF
1 day assimilation
CPU time 25 real time
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Validation
TRIM3D
EnKF
A
A
B
B
B
A
5 days later
AB section ship campaign
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References

• Bertino, L. (2002). Assimilation de données pour
  la prédiction de paramètres hydrodynamiques et
  écologiques le cas de la lagune de lOder. PhD
  thesis, ENSMP
• Bertino, L. Evensen G. et Wackernagel, H.
  (2002). Combining geostatistics and Kalman
  filtering for data assimilation in an estuarine
  system. Inverse Problems (18), p. 1-23
• Evensen, G. (1994). Sequential data assimilation
  with a nonlinear quasi-geostrophic model using
  Monte Carlo methods to forecast statistics.
  Journal of Geophysical Research 99(C5), p.
  10143-10162
• Verlaan, M. Heemink, A.W. (1997). Tidal flow
  forecasting using reduced rank suqare root
  filters. Stochastic Hydrology and Hydraulics
  11(5), p. 349-363
• Wolf, T. Sénégas, J. Bertino, L. et
  Wackernagel, H. (2001). Application of data
  assimilation to three-dimensional hydrodynamics
  the case of the Odra lagoon. In Monestier, Allard
  et Froidevaux (Eds.), GeoENV III geostatistics
  for environmental applications, Amsterdam, p.
  157-168. Kluwer Academic

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PIONEER projectthe example of the Odra lagoon

• Physical data
• water levels, salinity

Biological data nutrients, chlorophyll
Fluxes
Heat
Matter
Physical model
Ecological model