RELEVANCE OF CONSERVATION LAWS FOR AN ENSEMBLE KALMAN FILTER - PowerPoint PPT Presentation

RELEVANCE OF CONSERVATION LAWS FOR AN ENSEMBLE KALMAN FILTER Svetlana Dubinkina (CWI, Amsterdam)

LARGE-SCALE STRUCTURES Motivation is to predict behaviour of geophysical flows at large scales Jupiter Great Red Spot Gulf-stream rings

STATISTICAL EQUILIBRIUM THEORIES Statistical equilibrium theories aim at predicting coherent large- scale structures. In statistical equilibrium theories, such a coherent coarse-grained structure is the most probable macrostate and it is defined by conserved quantities associated with all possible microstates. How to find those microstates? Using evolution of the dynamics.

ILLUSTRATION OF STATISTICAL EQUILIBRIUM THEORY ON QUASI-GEOSTROPHIC FLOW ( x, y ) ∈ [0 , 2 π ) × [0 , 2 π ) q t = q x ψ y − q y ψ x ∆ ψ = q − h is the potential vorticity, is the stream function, ψ q is the orography, is the Laplace operator. ∆ h Conserved quantities are Z Energy E = − 1 / 2 ψ ( q − h ) dxdy Z Casimirs C f = f ( q ) dxdy

DEFINITIONS Microstate is q ( x, y ) Macrostate is defined by the probability density function ρ ( x, y, σ ) of having q ( x, y ) = σ The coarse-grained or macroscopic vorticity is defined as Z h q ( x, y ) i = σρ d σ The most probable macrostate is the maximiser of ρ ∗ ( x, y, σ ) Z S = − dxdyd σρ ln ρ subjected to satisfy conservations laws

DIFFERENT STATISTICAL EQUILIBRIUM THEORIES The QG model has an infinite number of conserved quantities: Z Energy E = − 1 / 2 ψ ( q − h ) dxdy Z Casimirs (with any smooth function) C f = f ( q ) dxdy When deriving a statistical equilibrium theory, one can only take into account some of these conserved quantities. Thus one needs to make a choice which of these is statistically relevant?

ENERGY STATISTICAL THEORY Assume that the only statistically relevant conserved quantity is energy Z E = − 1 / 2 ψ ( q − h ) dxdy Then the most probable macrostate is ρ ∗ = N − 1 exp( − λ E ) The coarse-grained vorticity and stream function are h q i = h h ψ i = 0

ENSTROPHY STATISTICAL THEORY Assume that the only statistically relevant conserved quantity is enstrophy (second order Casimir) Z q 2 dxdy Z = 1 / 2 Then the most probable macrostate is ρ ∗ = N − 1 exp( − α Z ) The coarse-grained vorticity and stream function are h ψ i = � ∆ − 1 h h q i = 0

ENERGY -ENSTROPHY STATISTICAL THEORY Assume that the only statistically relevant conserved quantity are energy and enstrophy Z Z q 2 dxdy E = − 1 / 2 ψ ( q − h ) dxdy Z = 1 / 2 Then the most probable macrostate is ρ ∗ = N − 1 exp( − β ( Z + µE )) The coarse-grained vorticity and stream function are h q i = µ h ψ i ( µ � ∆ ) h ψ i = h

ASSUMPTION OF ERGODICITY The most probable macrostate is the maximum of ρ ∗ ( x, y, σ ) Z S = − dxdyd σρ ln ρ subjected to satisfy conservations laws The coarse-grained or macroscopic vorticity is Z h q ( x, y ) i = σρ ∗ ( x, y, σ ) d σ Assumption of ergodicity is that Z t 0 + T 1 h q ( x, y ) i = lim q ( x, y, t ) dt T T →∞ t 0 subjected to conservative dynamical evolution of q

ARAKAWA DISCRETIZATIONS The QG model q t = q x ψ y − q y ψ x Arakawa discretizations are classical finite difference schemes based on the following equivalent formulation of the right hand side q x ψ y − q y ψ x ≡ ( q ψ y ) x − ( q ψ x ) y ≡ ( ψ q x ) y − ( ψ q y ) x Arakawa E discretisation (preserves energy E) Arakawa Z discretisation (preserves enstrophy Z) Arakawa EZ discretisation (preserves energy E and enstrophy Z)

COARSE-GRAINED FIELDS OBTAINED BY ARAKAWA DISCRETIZATIONS Arakawa EZ Arakawa E Arakawa Z h ψ i = � ∆ − 1 h ( µ � ∆ ) h ψ i = h h ψ i = 0 ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 2 4 6 0 2 4 6 0 2 4 6 0 h q i = h ψ i h q i = µ h ψ i h q i = 0 h ψ i 1 1 1 0.5 0.5 0.5 ⟨ q ⟩ ⟨ q ⟩ ⟨ q ⟩ 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩

HAMILTONIAN PARTICLE-MESH METHOD (EULERIAN-LAGRANGIAN METHOD) q t = q x ψ y − q y ψ x Y k + = X k Q ∆ ψ = q − h k q i q i q i,j q i,j ψ ψ i,j ψ i ψ i,j i ✓ x i − X k ◆ ✓ y j − Y k ◆ X q ( x i , y j , t ) = Q k φ φ r r k ∆ ψ = q − h dtX k = − ∂ d � ∂ y ψ ( x, y, t ) � ( x,y )=( X k ( t ) ,Y k ( t )) dtY k = + ∂ d � ∂ x ψ ( x, y, t ) � ( x,y )=( X k ( t ) ,Y k ( t ))

AREA PRESERVATION 2 π 2 π time 2 π 2 π We initialise K particles on Area associated with each Q k a uniform grid with vorticity is preserved over time under the divergent-free flow Q k , k = 1 , . . . , K

LEVEL SETS OF VORTICITY 2 π We initialise K particles on a uniform grid with vorticity Q k , k = 1 , . . . , K Denote vorticity levels as σ l 2 π σ l = Q k , l = 1 , . . . , L, where L ≤ K Meaning that we can have Q k = Q k 0 But we can’t have σ l = σ l 0 An example could be Q 1 = 1 , Q 2 = 1 , Q 3 = − 1 , Q 4 = 1 , Q 5 = − 1 ( K = 5) σ 1 = 1 , σ 2 = − 1 ( L = 2)

PRIOR DISTRIBUTION Let’s denote as the number of particles with vorticity level K l σ l Then the area associated with is σ l Π l = K l ∆ a 2 (2 π ) 2 This area is also preserved as it trivially follows from area- preservation of area associated with each Q k X Note that Π l = 1 l We take to be the prior distribution on vorticity Π l

CONSERVATION PROPERTIES OF HPM Π l = K l ∆ a 2 Area preservation of vorticity level sets (2 π ) 2 Z Energy conservation E = − 1 / 2 ψ ( q − h ) dxdy Z Conservation of circulation (first order Casimir) C = qdxdy

STATISTICAL THEORY BASED ON PRIOR The most probable macrostate is the maximum of ρ ∗ ( x, y, σ ) Z dxdyd σρ ln ρ S = − Π subjected to satisfy conservations laws of energy and circulation Then the most probable macrostate is ρ ∗ = N − 1 exp[( � β h ψ i + α ) σ ] Π The coarse-grained vorticity and stream function are X h q i = σ l ρ ∗ ( x, y, σ l ) ∆ h ψ i = h q i � h l

COARSE-GRAINED FIELDS OBTAINED BY THE HMP METHOD Fix energy and circulation. But change the prior. Choose the prior as normal distribution. Then the coarse- Π l grained fields are ⟨ ψ ⟩ 1 6 0.5 4 ⟨ q ⟩ 0 2 -0.5 -1 0 -1 0 1 0 2 4 6 ⟨ ψ ⟩

COARSE-GRAINED FIELDS OBTAINED BY THE HMP METHOD We use the same values for energy and circulation. Choose the prior as gamma distribution. Then the coarse- Π l grained fields are ⟨ ψ ⟩ 1 6 0.5 4 ⟨ q ⟩ 0 2 -0.5 -1 0 -1 0 1 0 2 4 6 ⟨ ψ ⟩

RELATION TO DATA ASSIMILATION By changing the prior we can obtain different coarse-grained fields by the HPM. We ask the following question: can data assimilation correct model error? Meaning that if we take the HPM model for the truth but an Arakawa model for the background, can data assimilation correct the model error? And moreover, are the conservation properties of an Arakawa model relevant for good estimations?

EXPERIMENTAL SETUP We derived observations from the HPM model with gamma prior distribution, since none of the Arakawa models are able to obtain such a non-linear behaviour no matter the initial conditions. The observations were assimilated into Arakawa EZ model (that preserves energy and enstrophy) Arakawa E (that preserves energy) Arakawa Z (that preserves enstrophy) As a data assimilation method we choose an Ensemble Kalman Filter with perturbed observations.

IS DATA ASSIMILATION CONSERVATIVE? In general, data assimilation is non-conservative with a few exceptions: particle fileters conservation of linear properties by an EnKF without localisation specially derived data assimilation methods that ensure conservation laws (e.g. “The Maintenance of Conservative Physical Laws within Data Assimilation Systems” by Jacobs and Ngodock, 2003; “Conservation of mass and preservation of positivity with ensemble-type kalman filter algorithms” by Janji ć et al, 2014)

COARSE-GRAINED FIELDS OBTAINED BY ARAKAWA DISCRETIZATIONS Arakawa EZ Arakawa E Arakawa Z 1 1 1 0.5 0.5 0.5 ⟨ q ⟩ ⟨ q ⟩ ⟨ q ⟩ 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 2 4 6 0 2 4 6 0 2 4 6

ASSIMILATION OF STREAM FUNCTION Arakawa EZ Arakawa E Arakawa Z 1 1 1 0.5 0.5 0.5 ⟨ q ⟩ ⟨ q ⟩ ⟨ q ⟩ 0 0 0 -0.5 -0.5 -0.5 -1 -1 -1 -1 0 1 -1 0 1 -1 0 1 ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ ⟨ ψ ⟩ 6 6 6 5 5 5 4 4 4 3 3 3 2 2 2 1 1 1 0 0 0 0 2 4 6 0 2 4 6 0 2 4 6

INSTANT ERRORS OBTAINED BY THE ENKF ψ q 1 10 EZ EZ E E 0.8 Z Z 8 0.6 RMSE RMSE 6 0.4 4 0.2 2 0 0 0 5000 10000 0 5000 10000 time time

LOCALIZATION AND INFLATION Arakawa EZ Arakawa E Arakawa Z (a) (b) (c) 1 1 1 0 0.5 0.5 -1 -2 0 0 -3 -4 -0.5 -0.5 -5 -1 -6 -1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 -1 -0.5 0 0.5 1 scheme mean std skewness HPM -0.32 0.30 0.34 EZ -0.37 0.29 0.48 E -5.27 1.70 1.99 Z -0.51 0.25 0.45