Data Assimilation

Meditation jargon

Particle Filter

...with Sequential Importance Resampling (SIR-PF)...

Data assimilation is basically the integration of near real-time observations in the forecasting process, i.e., observations, which represent the true state of interest, are combined with model outputs with the aim of improving the forcasted states. This technique is especially important and therefore widely applied in weather and hydrological forecasting.

DA methods in application include : EnKF, PF, 3/4D Var., etc. Here you can test the ensemble-based particle filtering algorithm proposed by Gordon et al. (1993). The models used are rather simple and only meant to showcase the algorithm's applicability.

Number of model integrations (t : 1 to T) :

Number of ensemble members (Np) :

Initial state (Xt=0) :

Measurement and/or system noise ((co-)variance) :

Select PDF (for ω and ν) & run (JS) :  

To bring you up to speed, here's what's happening behind the Run PF button :

The 'synthetic truth', Xt=1:T, is derived through f(Xt-1)+ω and then mapped through the observation model, Yt = h(Xt)+ν. Meanwhile, the initial state is perurbed to have Np uncertain prior particles (i=1:Np). These are similarly integrated through the process model,f(Xit-1), and consequently mapped through h(Xit). Mapped priors are then weighted against the 'truth' for subsequent resampling and derivation of the analysis/update. This update is then used to initialize the next timestep, t+1.

...Methods : models...

Process model : Xt = f(Xt-1) + ω = 0.5Xt-1 + 25Xt-1 / (1 +Xt-12) + 8cos(1.2(t-1)) + ω

Observation model : Yt = h(Xt) + ν = Xt2/20 + ν

...Methods : particle weighting and resampling...

Particle weight : wti ≈ 1/√(2πσ2)exp[ -(Y - Yi)2/2σ2 ]

Sequential Importance Resampling : prior Xi=M is selected as posterior such that ∑i=0M-1wt < ui ≤ ∑i=0Mwt

where X is the state, Y the mapped state, σ2 the variance, and ui is randomly drawn from the uniform distribution over [0,1]. ω and ν are, respectively, the process and observation model errors.