1 / 51

Data Assimilation

Data Assimilation. Tristan Quaife, Philip Lewis. What is Data Assimilation?. A working definition: The set techniques the combine data with some underlying process model to provide optimal estimates of the true state and/or parameters of that model. What is Data Assimilation?.

mya
Download Presentation

Data Assimilation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Data Assimilation Tristan Quaife, Philip Lewis

  2. What is Data Assimilation? • A working definition: The set techniques the combine data with some underlying process model to provide optimal estimates of the true state and/or parameters of that model.

  3. What is Data Assimilation? • It is not just model inversion. • But could be seen as a process constraint on inversion (e.g. a temporal constraint)

  4. e.g. Use EO data to constrain estimates of terrestrial C fluxes • Terrestrial EO data: no direct constraint on C fluxes • Combine with model

  5. Data Assimilation is Bayesian • Bayes’ theorem:

  6. What does DA aim to do? • Use all available information about • The underlying model • The observations • The observation operator • Including estimates of uncertainty and the current state of the system • To provide a best estimate of the true state of the system with quantified uncertainty

  7. Kalman Filter DA: MODIS LAI product Data assimilation into DALEC ecological model

  8. Lower-level product DA • Ensure consistency between model and observations • Assimilate low-level products (surface reflectance) • Uncertainty better quantified • Need to build observation operator relating model state (e.g. LAI) to reflectance • Example of Oregon (MODIS DA) • Quaife et al. 2008, RSE

  9. Modelled vs. observed reflectance Red NIR Note BRF shape in red: can’t simulate with 1-D canopy (GORT here)

  10. NEP results No assimilation Assimilating MODIS (red/NIR) DALEC model calibrated from flux measurements at tower site 1

  11. Integrated flux predictions

  12. Flux Tower 4.5 km 15 65 gC/m2/year Mean NEP for 2000-2002 Spatial average = 50.9 Std. dev. = 9.7 (gC/m2/year)

  13. NEP – Site2 (intermediate) parameters, with/without DA Model running at Site 2, Oregon Site 1 model EO-calibrated at site 2NEP observations from Site 2 Shows ability to spatialise

  14. Data assimilation • Low-level DA can be effective • ‘easier’ data uncertainties • Can be applied to multiple observation types • Requires Observation operator(s) • RT models • Requires other uncertainties • Ecosystem Model • Driver (climate) • Observation operator

  15. Specific issues in land EOLDAS • No spatial transfer of information • Require full spatial coverage • Atmosphere dealt with by an instantaneous retrieval (i.e. no transport model) • All state vector members influence observations • We are not interested in other variables!

  16. Nominal classification of DA • Sequential • Smoothers • Variational

  17. Sequential methods • Kalman Filter • Variants - EKF • Ensemble Kalman Filter • Variants – Unscented EnKF • Particle filters • Lots of different types • true MCMC technique

  18. The Kalman filter • Propagation step: • x = Mx- • P = MP-MT + Q • Analysis step: • x* = x + K( y – Hx ) • K = PHT( HPHT+R )-1 Model Covariancematrix Stochastic forcing Kalman gain Observation covariance matrix Observation operator State vector Observation vector

  19. The Kalman Filter • Linear process model • Linear observation operator • Assumes normally distributed errors

  20. The Extended Kalman filter • Propagation step: • x = m(x-) • P = M'P-M'T + Q • Analysis step: • x* = x + K( y – h(x) ) • K = PH'T( H'PH'T+R )-1 Jacobian matrix Jacobian matrix

  21. The Extended Kalman Filter • Linear process model • Linear observation operator • Assumes normally distributed errors • Problem with divergence

  22. The Ensemble Kalman filter • Propagation step: • X = m(X-) + Q • no explicit error propagation • Analysis step: • X* = X + K( D – HX ) • K = PHT( HPHT+R )-1 State vector ensemble Perturbed observations

  23. Xa = h(X) X The Ensemble Kalman Filter • P estimated from X • Non linear observations using augmentation:

  24. The Ensemble Kalman Filter No assimilation Assimilating MODIS surface reflectance bands 1 and 2

  25. The Ensemble Kalman Filter • Avoids use of Jacobian matrices • Assumes normally distributed errors • Some degree of relaxation of this assumption • Augmentation assumes local linearisation

  26. Particle Filters • Propagation step: • X = m(X-) + Q • Analysis step: • e.g. Metropolis-Hastings

  27. Particle Filters

  28. Particle Filters No available observations

  29. Particle Filters • Fully Bayesian • No underlying assumptions about distributions • Theoretically the most appealing choice of sequential technique, but… • Our analysis show little difference with EnKF • Potentially requires larger ensemble • But comparing 1:1 is faster than EnKF

  30. Sequential techniques • General considerations: • Designed for real time systems • Only consider historical observations • Only assimilates observations in single time step • Can lead to artificial high frequency components

  31. Smoothers • Extension of sequential techniques • All observations effect every time step • Analogous to weighting on observations • [ smoothing-convolution / regularisation ] • Difficult to apply in rapid change areas

  32. Smoothers - regularisation x = (HTR-1H + λ2BTB)-1HTR-1y B is the required constraint. It imposes: Constraint matrix Lagrange multiplier Bf = 0 and the scalar λ is a weighting on the constraint.

  33. Regularisation

  34. Regularisation Quaife and Lewis (2010) Temporal constraints on linear BRDF model parameters. IEEE TGRS, in press.

  35. Regularisation • Lots of literature on the selection of λ • Cross validation etc • Permits insight into the form of Q

  36. Variational techniques • Expressed as a cost function • Uses numerical minimisation • Gradient descent requires differential • Traditionally used for initial conditions • But parameters may also be adjusted

  37. Background Observations 3DVAR • J(x) = ( x-x- ) P-1 ( x-x- )T + • ( y-h(x) ) R-1 ( y-h(x) )T

  38. 3DVAR • No temporal propagation of state vector • OK for zero order approximations • Unable to deal with phenology

  39. J(x) = ( x-x- ) P-1 ( x-x- )T + ( y-h(xi) ) R-1 ( y-h(xi) )T Σi Time varying state vector 4DVAR

  40. Variational techniques • Parameters constant over time window • Non smooth transitions • Assumes normal error distribution • Size of time window? • For zero-order case 3DVAR = 4DVAR • 4DVAR for use with phenology model • Absence of Q - propagation of P?

  41. Building an EOLDAS • Lewis et al. (RSE submitted) • Sentinel-2

  42. EOLDAS

  43. Assimilation Assume model Uncertainty known

  44. EOLDAS • Base level noise

  45. Cross validation

  46. Cross validation

  47. EOLDAS • Cross validation

  48. Double noise

  49. EOLDAS • Double noise

  50. Conclusions - technique • DA is optimal way to combine observations and model • Range of options available for DA • Sequential • Smoothers • Variational • Require understanding of relative uncertainties of model and observations • Require way of linking observations and model state • Observation operator (e.g. RT)

More Related