1. Data-Assimilation Research Centre
Nonlinear Data Assimilation using an extremely efficient Particle Filter
Peter Jan van Leeuwen
Data-Assimilation Research Centre
University of Reading

2. The Agulhas System

3. In-situ observations

4. In-situ observations

5. In situ observations Transport through Mozambique Channel

6. Data assimilation
Uncertainty points to use of probability density functions.
P(u)
0.0
0.5
1.0
u (m/s)

7. Data assimilation: general formulation
Bayes theorem:
Solution is pdf!
NO INVERSION !!!

8. How is this used today?
Present-day data-assimilation systems are based on linearizations and search for one optimal state:
(Ensemble) Kalman filter: assumes Gaussian pdf’s
4DVar: smoother assumes Gaussian pdf for initial state and observations (no model errors)
Representer method: as 4DVar but with Gaussian model errors
Combinations of these

9. Prediction: smoothers vs. filters
The smoother solves for the mode of the conditional joint pdf p( x0:T | d0:T) (modal trajectory).
The filter solves for the mode of the conditional marginal pdf p( xT | d0:T).
For linear dynamics these give the same prediction.

10. These are not the same for nonlinear problems !!!
Filters maximize the marginal pdf
Smoothers maximize the joint pdf
These are not the same for nonlinear problems !!!

11. Example Nonlinear model 2 xn xn+1 = 0.5 xn + _________ + nn
1 + e (xn - 7)
Initial pdf
x0 ~ N(-0.1, 10)
Model noise
nn ~ N(0, 10)

12. Example: marginal pdf’s
Note: mode is at x= - 0.1
Note: mode is at x=8.5 x0 xn

13. Example: joint pdf
Mode joint pdf x0 xn
Modes marginal pdf’s

14. And what about the linearizations?
Kalman-like filters solve for the wrong state: gives rise to bias.
Variational methods use gradient methods, which can end up in local minima.
4DVar assumes perfect model: gives rise to bias.

15. Where do we want to go? Represent pdf by an ensemble of model states
Fully nonlinear
Time

16. How do we get there? Particle filter?
Use ensemble
with
the weights.

17. What are these weights?
The weight w_i is the pdf of the observations given the model state i.
For M independent Gaussian distributed observation errors:

18. Standard Particle filter

19. Particle Filter degeneracy: resampling
With each new set of observations the old weights are multiplied with the new weights.
Very soon only one particle has all the weight…
Solution:
Resampling: duplicate high-weight particles are abandon low-weight particles

20. Problems
Probability space in large-dimensional systems is ‘empty’: the curse of dimensionality
u(x1)
u(x2)
T(x3)

21. Standard Particle filter
Not very efficient !

22. Specifics of Bayes Theorem I
We know from Bayes Theorem:
Now use :
in which we introduced the transition density

23. Specifics of Bayes Theorem II
This can be rewritten as:
q is the proposal transition density, which might be
conditioned on the new observations!
This leads finally to:

24. Specifics of Bayes Theorem III
How do we use this? A particle representation of
Giving:
Now we choose from the proposal transition density
for each particle i.

25. Particle filter with proposal density
Stochastic model
Proposed stochastic model:
Leads to particle filter with weights

26. Meaning of the transition densities
= the probability of this specific value for the random model error.
For Gaussian model errors we find:
A similar expression is found for the proposal transition

27. Particle filter with proposal transition density

28. Experiment: Lorentz 1963 model
(3 variables x,y,z, highly nonlinear)
x-value
Measure only
X-variable
y-value

29. Standard Particle filter with resampling 20 particles
Typically 500 particles needed !
X-value
Time

30. Particle filter with proposal transition density 3 particles
X-value
Time

31. Particle filter with proposal transition density 3 particles
Y-value
(not observed)
Time

32. However: degeneracy
For large-scale problems with lots of observations this method is still degenerate:
Only a few particles get high weights; the other weights are negligibly small.
However, we can enforce almost equal weight for all particles:

33. Equal weights
Write down expression for each weight with q deterministic:
Prior transition density
Likelihood
2. When H is linear this is a quadratic function in
for each particle.
3. Determine the target weight:

34. Almost Equal weights I 1 4 3
Target weight 2 5
4. Determine corresponding model states, e.g. solving alpha in

35. Almost equal weights II
But proposal density cannot be deterministic:
Add small random term to model equations from a pdf with broad wings e.g. Gauchy
Calculate the new weights, and resample if necessary

36. Application: Lorenz 1995 N=40 F=8 dt = 0.005 T = 1000 dt
Observe every other grid point
Typically 10,000 particles needed

37. Ensemble mean after 500 time steps 20 particles
Position

38. Ensemble evolution at x=20 20 particles
Time step

39. Ensemble evolution at x=35 (unobserved) 20 particles

40. Isn’t nudging enough?
Only nudged
Nudged and weighted

41. Isn’t nudging enough? Unobserved variable Only nudged
Nudged and weighted

42. Conclusions The nonlinearity of our problem is growing
Particle filters with proposal transition density:
solve for fully nonlinear solution
very flexible, much freedom
application to large-scale problems straightforward

43. Future
Fully nonlinear filtering (smoothing) forces us to concentrate on the transition densities, so on the errors in the model equations.
What is the sensitivity to our choice of the proposal?
What can we learn from studying the statistics of the ‘nudging’ terms?
How do we use the pdf???