Nonlinear Coupled Data assimilation

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Nonlinear Coupled Data assimilation

• Peter Jan van Leeuwen
• Melanie Ades, Phil Browne, Javier Amezcua,
  Mengbin Zhu

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Data assimilation general formulation
Bayes theorem
The solution is a pdf!
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Nonlinear filtering Particle filter
Use ensemble
with
the weights.
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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

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Fully nonlinear DA Particle Filters
Degenerate
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Particle Filters with resampling
Degenerate
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Why doesnt this work?
Volume of a hypersphere with radius ry in an
Ny-dimensional space
Log10 of Volume of hypersphere of radius 1.
Number of observations
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Reduce obs number via Localisation
Different particles perform Differently over the
domain. How do we glue different particles
together? Interesting work by Poterjoy.
Particle 17
Particle 3
Particle 7
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Another solution proposal densities
Use a different model and correct in the weights
The second model knows about future observations !
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Examples of proposed models
Use nudging or use LETKF or 4DVar
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Note
These are all hybrids but this time without
ad-hoc adjustments. There is solid maths behind
all this!
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Resulting weights

• The weights now contain contributions from
• the observations via the likelihood p(yx),
• The use of a different model than the original
• model (proposal density).
• So

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Weight of a particle
Weight of Particle i
X
X
Position of particle i in state space
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Optimal proposal density (1 timestep), implicit
PF (window)
This is a 4DVar on each particle, with a
perturbation added to it. But a special 4DVar
- no initial errors, - include model errors
(weak constraint) Weights proportional to
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Example simple Gaussian modelSnyder et al 2008,
2011, 2015
Claim Number of particles needed to avoid
collapse grows exponentially with system size
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Remedy
With ai found from
With wtarget given by
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So, ensure that the weights are equal..
Target weight
Likelihood weight Proposal weight
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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 the
covariance of the optimal proposal
density, and chosen such that all
particles have equal weight (using the
expression for the weights).
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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

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Implicit Equal-weights Particle Filter1000
dimensional system, 20 particles
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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 !!

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ExampleHadCM3 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

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EMPIRE data-assimilation framework
Fast coupling of any model to data assimilation
codes via MPI, e.g. HadCM3 (2 million), Unified
Model (300 million), etc.
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The model is nonlinear
Pdf of meridional wind at a point in the mid
North Atlantic.
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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.

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Model error covariance
Correlation atmospheric zonal flow and oceanic
meridional flow
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Results Observed variable SST
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Results Ocean Temperature
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Results Meridional velocity
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Results Atmospheric Temperature
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Rank Histograms
SST (observed)
Meridional wind high up in Atmosphere
(unobserved)
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Time evolution of particles
Prior ensemble (yellow), posterior ensemble
(blue), truth (red), for SST in two grid points
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Estimated pdfs
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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
  high-dimensional system is problematic.