Nonlinear Coupled Data assimilation Peter Jan van Leeuwen

Nonlinear Coupled Data assimilation Peter Jan van Leeuwen Melanie Ades, Phil Browne, Javier Amezcua, Mengbin Zhu

Data assimilation: general formulation Bayes theorem: The solution is a pdf!

Nonlinear filtering: Particle filter Use ensemble with the weights.

What are these weights? • The weight is the normalised value of the pdf of the observations given model state. • For Gaussian distributed variables is is given by: • One can just calculate this value • That is all !!! • Or is it? More is needed for high-dimensional problems…

Fully nonlinear DA: Particle Filters e D n e g e t a r e

Particle Filters with resampling e D n e g e t a r e

Why doesn’t this work? Volume of a hypersphere with radius ry in an Ny-dimensional space Log 10 of Volume of hypersphere of radius 1. Number of observations

Reduce obs number via Localisation Different particles perform Differently over the domain. Particle 17 Particle 3 How do we glue different particles together? Particle 7 Interesting work by Poterjoy.

Another solution: proposal densities Use a different model and correct in the weights: The second model knows about future observations !

Examples of proposed models Use nudging: or use LETKF: or 4 DVar:

Note These are all ‘hybrids’ but this time without ad-hoc adjustments. There is solid maths behind all this!

Resulting weights The weights now contain contributions from 1. the observations via the likelihood p(y|x), 2. The use of a different model than the original model (proposal density). So:

Weight of a particle Weight of Particle i X X Position of particle i in state space

Optimal proposal density (1 timestep), implicit PF (window) This is a 4 DVar on each particle, with a perturbation added to it. But a special 4 DVar: - no initial errors, - include model errors (weak constraint) Weights proportional to:

Example simple Gaussian model Snyder et al 2008, 2011, 2015 Claim: Number of particles needed to avoid collapse grows exponentially with system size

Remedy With ai found from: With wtarget given by:

So, ensure that the weights are equal. . Likelihood weight Proposal weight Target weight

Equal-weight Particle filtering Define an implicit map as follows: with the mode of the optimal proposal density, e. g. a random draw from the density , with covariance of the optimal proposal density, the and chosen such that all particles have equal weight (using the expression for the weights).

Experiments, model error and observation errors Gaussian, H linear • Linear model of Snyder et al. 2008. • 1000 dimensional independent Gaussian linear model • 20 particles • Observations every time step

Implicit Equal-weights Particle Filter 1000 dimensional system, 20 particles…

Note on localisation This particle filter has localisation build into it because all updates are pre-multiplied by either the model error covariance or a covariance of the form In which Q is the model error covariance and R the observation error covariance. NO EXPLICIT LOCALISATION NEEDED !!

Example: Had. CM 3 climate model • Coupled ocean-atmosphere climate model used extensively in IPCC • 2. 3 million variables • No flux correction • Atmosphere: 3. 75 X 2. 5 staggered B, 19 levels • Ocean: 1. 25 X 1. 25 staggered B, 20 levels • Daily coupling • Atmosphere run first with 30 min time step, followed by ocean with 60 min time step

EMPIRE data-assimilation framework Fast coupling of any model to data assimilation codes via MPI, e. g. Had. CM 3 (2 million), Unified Model (300 million), etc.

The model is nonlinear… Pdf of meridional wind at a point in the mid North Atlantic.

Data-assimilation parameters • Identical twin experiment • 32 particles • Daily observations of Sea-Surface Temperature with uncertainty 0. 55 K • Model errors smaller than 0. 1 times deterministic model update • Correlation structure from snapshots of long model run.

Model error covariance Correlation atmospheric zonal flow and oceanic meridional flow

Results: Observed variable SST

Results: Ocean Temperature

Results: Meridional velocity

Results: Atmospheric Temperature

Rank Histograms SST (observed) Meridional wind high up in Atmosphere (unobserved)

Time evolution of particles Prior ensemble (yellow), posterior ensemble (blue), truth (red), for SST in two grid points

Estimated pdfs

Conclusions • Not only should particles be close to observations, we have to move them such in state space that their weights are equivalent. • We can use particle filters for climate models. • Nonlinear filtering moves problem of state covariances to covariances of model error • Efficient representation of pdf in highdimensional system is problematic.