Developing a Framework for Ensemble Data Assimilation in Geophysical Sciences

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Developing a Framework forEnsemble Data
Assimilation inGeophysical Sciences

• Adel Ahanin
• Ralph M. Parsons Laboratory

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Outline

• Review
• General
• Estimation in Geophysical Systems
• Challenges
• High-dimensionality
• Non-linearity
• Projection subspaces
• Ensemble Kalman Filter (EnKF)
• Reduced Rank Kalman Filter (RRKF)
• Hybrid Filter (COFFEE)

• Uncertainty in Null-space
• Formulation
• Results
• LTI
• Lorenz95
• Random Matrix Theory
• Concepts/Relevance
• Null-case
• Spiked Spectrum
• Preliminary Results
• Challenges/Future Work

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Review - General

• Estimation in Geophysical Systems
• Objective
• Forecasting (Past Obs.)
• Smoothing (Past and Future Obs.)
• Parameter Estimation
• Associated Uncertainties
• Given
• Physical constrains
• Governing Equations
• Boundary Condition
• Observations
• Direct Measurements
• Remotely Sensed Data
• Solution, Data Assimilation
• Variational Methods
• Sequential Methods

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Review - General cont.

• Sequential estimation of the State based on the
  past data
• Assumptions
• Gaussian Distributions wtrue(t)N(0, Qtrue(t)) ,
  v(t)N(0, R(t)) , w(t)N(0, Q(t))
• Reachability and Observability
• Stability
• Objective Minimize Xa(t) Xtrue(t)
• Optimal Solution in Linear Systems Kalman Filter

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Review - General cont.

• Nonlinearity
• Fluid dynamics equation
• Unstable linearization
• Non-Gaussian Distributions
• High-dimensionality
• State size O(105)
• Computational constraints
• Sparsity, Covariance Structure
• Sub-optimal Solutions
• Asymptotically optimal algorithms
• Approximate solutions Projection

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Review - Projection Subspaces - EnKF

• Projecting on a random subspace (Low rank vs.
  Full rank)
• Assumes Gaussian distribution
• Fully nonlinear propagation
• Asymptotic convergence
• Inefficiency (?)

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Review - Projection Subspaces - RRKF

• Projecting on a deterministic subspace
• Assumes Gaussian distribution and Linearity
• The best projection in linear dynamics
• Rank deficiency Considering proper process
  noise is essential
• Unstable/unbalanced linearization

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Review - Projection Subspaces - COFFEE

• Projecting on a combination of deterministic and
  random subspaces
• Assumes Gaussian distribution
• Assumes Linearity in the deterministic subspace
• Nonlinear propagation in the random subspace
• More robust than solely deterministic
• More efficient than pure random
• Additional cost for null space projection

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Uncertainty in Null-space
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Uncertainty in Null-space
Traditional Forecast Covariance Matrix After
Filling the Null-space
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Dispersion Model

• Governing Equation
• State space model

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Dispersion Model (cont)

• Properties
• Stable
• Completely Reachable (for case w/
  Q)
• Completely Observable
• KF is the optimal solution
• Proper process noise
• Performance Metric

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Dispersion Results
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18d-Lorenz95 System

• State space model

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18d-Lorenz95 System (cont)

• Properties
• Total energy is conserved
• Depending on F, the system is stable or chaotic
• For J18
• the critical value is F1.
• Dimension of the attractor less than 12
• All of the state is observed
• Proper process noise
• Performance Metric (RMSE, same as before)

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18-d Lorenz95 Results
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18-d Lorenz95 Results (cont)
RMSE of EnKF, RRKF and COFFEE 18d L95 model,
rank7, without process noise
With Qtrue
No Qtrue
RMSE of EnKF, RRKF and COFFEE 18d L95 model,
rank7, with process noise
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Conclusion and future work

• Main challenge in geophysical systems
• High-dimensionality Projection Subspace
• Projection Subspaces
• Deterministic (RRKF) Unstable in NL Dynamics
• Random (EnKF) Robust but Inefficient
• Hybrid (COFFEE) Stable and Robust
• Filter Divergence
• Null-spaces Development Proper process noise
• Future work
• Choice of Deterministic/Random ensembles in
  Hybrid
• Improving the process noise model
• Analysis of the ensemble covariance matrix as a
  Random Matrix