5In situ observations Transport through Mozambique Channel
6Data assimilationUncertainty points to use of probability density functions.P(u)0.00.51.0u (m/s)
7Data assimilation: general formulation Bayes theorem:Solution is pdf!NO INVERSION !!!
8How 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’s4DVar: smoother assumes Gaussian pdf for initial state and observations (no model errors)Representer method: as 4DVar but with Gaussian model errorsCombinations of these
9Prediction: 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.
10These are not the same for nonlinear problems !!! Filters maximize the marginal pdfSmoothers maximize the joint pdfThese are not the same for nonlinear problems !!!
11Example Nonlinear model 2 xn xn+1 = 0.5 xn + _________ + nn 1 + e (xn - 7)Initial pdfx0 ~ N(-0.1, 10)Model noisenn ~ N(0, 10)
12Example: marginal pdf’s Note: mode is at x= - 0.1Note: mode is at x=8.5x0xn
14And 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.
15Where do we want to go? Represent pdf by an ensemble of model states Fully nonlinearTime
16How do we get there? Particle filter? Use ensemblewiththe weights.
17What 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:
19Particle 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
20ProblemsProbability space in large-dimensional systems is ‘empty’: the curse of dimensionalityu(x1)u(x2)T(x3)
22Specifics of Bayes Theorem I We know from Bayes Theorem:Now use :in which we introduced the transition density
23Specifics of Bayes Theorem II This can be rewritten as:q is the proposal transition density, which might beconditioned on the new observations!This leads finally to:
24Specifics of Bayes Theorem III How do we use this? A particle representation ofGiving:Now we choose from the proposal transition densityfor each particle i.
25Particle filter with proposal density Stochastic modelProposed stochastic model:Leads to particle filter with weights
26Meaning 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
27Particle filter with proposal transition density
28Experiment: Lorentz 1963 model (3 variables x,y,z, highly nonlinear)x-valueMeasure onlyX-variabley-value
29Standard Particle filter with resampling 20 particles Typically 500 particles needed !X-valueTime
30Particle filter with proposal transition density 3 particles X-valueTime
31Particle filter with proposal transition density 3 particles Y-value(not observed)Time
32However: degeneracyFor 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:
33Equal weightsWrite down expression for each weight with q deterministic:Prior transition densityLikelihood2. When H is linear this is a quadratic function infor each particle.3. Determine the target weight:
34Almost Equal weights I143Target weight254. Determine corresponding model states, e.g. solving alpha in
35Almost equal weights II But proposal density cannot be deterministic:Add small random term to model equations from a pdf with broad wings e.g. GauchyCalculate the new weights, and resample if necessary
36Application: Lorenz 1995 N=40 F=8 dt = 0.005 T = 1000 dt Observe every other grid pointTypically 10,000 particles needed
37Ensemble mean after 500 time steps 20 particles Position
38Ensemble evolution at x=20 20 particles Time step
39Ensemble evolution at x=35 (unobserved) 20 particles
40Isn’t nudging enough?Only nudgedNudged and weighted
41Isn’t nudging enough? Unobserved variable Only nudged Nudged and weighted
42Conclusions The nonlinearity of our problem is growing Particle filters with proposal transition density:solve for fully nonlinear solutionvery flexible, much freedomapplication to large-scale problems straightforward
43FutureFully 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???