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A Framework for Adaptive Inflation and Covariance Localization for Ensemble Filters Ahmed Attia 1 Emil Constantinescu 12 1 Mathematics and Computer Science Division (MCS), Argonne National Laboratory (ANL) 2 The University of Chicago, Chicago, IL

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SLIDE 1

A Framework for Adaptive Inflation and Covariance Localization for Ensemble Filters

Ahmed Attia 1 Emil Constantinescu 12

1Mathematics and Computer Science Division (MCS), Argonne National Laboratory (ANL) 2The University of Chicago, Chicago, IL

SIAM Conference on Mathematics of Planet Earth (MPE18) September 14, 2018

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SLIDE 2

Outline

Ensemble Filtering Ensemble Kalman Filter (EnKF) Inflation & Localization Optimal Experimental Design OED and the alphabetical criteria OED Inflation & Localization A-OED inflation A-OED localization Numerical Experiments Experimental setup Numerical Results

Adaptive Inflation and Localization [1/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 3

Outline

Ensemble Filtering Ensemble Kalman Filter (EnKF) Inflation & Localization Optimal Experimental Design OED and the alphabetical criteria OED Inflation & Localization A-OED inflation A-OED localization Numerical Experiments Experimental setup Numerical Results

Adaptive Inflation and Localization Ensemble Filtering [2/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 4

Ensemble Kalman Filter (EnKF)

Assimilation cycle over [tk−1, tk]; Forecast step

◮ Initialize: an analysis ensemble {xa k−1(e)}e=1,...,Nens at tk−1 ◮ Forecast: use the discretized model Mtk−1→tk to generate a forecast ensemble at tk:

xb

k(e) = Mtk−1→tk (xa k−1(e)) + ηk(e),

e = 1, . . . , Nens

◮ Forecast/Prior statistics:

xb

k =

1

Nens

Nens

  • e=1

xb

k(e)

Bk = 1

Nens − 1 Xb

k

  • Xb

k

T ;

Xb

k = [xb k(1) − xb k, . . . , xb k(Nens) − xb k] Model State Time

!"#$

Initial/Analysis Ensemble ~ &' ("#$ Model State

Forward Model !"#$%,→ "#

Time

"# "#(%

Forecast/Background Ensemble ~ *+ ,# Model State

Forward Model !"#$%,→ "#

Time

"# "#(%

Forecast/Background Ensemble ~ *+ ,#

Adaptive Inflation and Localization Ensemble Filtering [3/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 5

Ensemble Kalman Filter (EnKF)

Assimilation cycle over [tk−1, tk]; Analysis step

◮ Given an observation yk at time tk ◮ Analysis: sample the posterior (EnKF update)

Kk = BkHT

k

  • HkBkHT

k + Rk

−1

xa

k(e) = xb k(e) + Kk

  • [yk + ζk(e)] − Hk(xb

k(e)) ◮ The posterior (analysis) error covariance matrix:

Ak = (I − KkH) Bk ≡ B−1

k

+ HT

kR−1Hk

−1

Model State

Forward Model !"#$%,→ "#

Observation & Likelihood Time

"#(% "#

Observation: )#; Likelihood: * )#|,# Model State

Forward Model !"#$%,→ "#

Corrected Model State (Analysis) Time

"#(% "#

Observation & Likelihood Analysis Ensemble ~ *+ ,# Model State

Forward Model !"#$%,→ "#

Corrected Model State (Analysis) Time

"#(% "#

Observation & Likelihood Analysis Ensemble ~ *+ ,#

Adaptive Inflation and Localization Ensemble Filtering [4/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 6

Ensemble Kalman Filter (EnKF)

Sequential EnKF Issues

◮ Limited-size ensemble results in sampling errors, explained by:

  • variance underestimation
  • accumulation of long-range spurious correlations
  • filter divergence after a few assimilation cycles

◮ EnKF requires inflation & localization

Model State

Forward Model !"#$%,→ "#

Corrected Model State (Analysis) Time

"#(% "#

Observation & Likelihood

"#)%

Assimilation Cycle

Forward Model !"#,→ "#*%

Analysis Ensemble ~ ,- .# Sequentially Repeat the Assimilation Cycle Model State

Forward Model !"#$%,→ "#

Corrected Model State (Analysis) Time

"#(% "#

Observation & Likelihood

"#)%

Assimilation Cycle

Forward Model !"#,→ "#*%

Analysis Ensemble ~ ,- .# Sequentially Repeat the Assimilation Cycle

Adaptive Inflation and Localization Ensemble Filtering [5/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 7

Ensemble Kalman Filter (EnKF)

Inflation & Localization

◮ Covariance underestimation in EnKF is counteracted, by applying covariance inflation:

→ replace B, with an inflated version B

◮ Long-range spurious correlations are reduced by covariance localization (e.g., Schur-product)

→ replace B, with a decorrelated version B

Adaptive Inflation and Localization Ensemble Filtering [6/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 8

EnKF: Inflation

◮ Additive Inflation:

  • B := D + B;

s.t. D = diag (λ) , λ =

λ1, λ2, . . . , λNstate

T , 0 ≤ λl

i ≤ λi ≤ λu ◮ Multiplicative Inflation:

  • 1. Space-independent inflation:
  • Xb = √

λ xb(1) − xb , . . . , √ λ xb(Nens) − xb ; 0 < λl ≤ λ ≤ λu

  • B =

1 Nens − 1 Xb Ä XbäT = λ B

  • 2. Space-dependent inflation: Let D := diag (λ) ≡ Nstate

i=1

λieieT

i,

  • Xb = D

1 2 Xb ,

  • B =

1 Nens − 1 Xb Ä XbäT = D

1 2 BD 1 2 . ◮ The inflated Kalman gain

K, and analysis error covariance matrix A

  • K =

BHT H BHT + R−1 ;

  • A =

I − KH B ≡ B−1 + HTR−1H−1

Adaptive Inflation and Localization Ensemble Filtering [7/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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EnKF: Schur-Product Localization

State-space formulation; B−Localization

◮ space-independent covariance localization:

  • B := C ⊙ B;

s.t. C = [ρi,j]i,j=1,2,...,Nstate

◮ Entries of C are created using space-dependent localization functions †:

→ Gauss: ρi,j(L) = exp

  • −d(i, j)2

2L2

  • ;

i, j = 1, 2, . . . , Nstate , → 5th-order Gaspari-Cohn:

ρi,j(L) =

  

− 1

4

d(i,j)

L

5

+ 1

2

d(i,j)

L

4

+ 5

8

d(i,j)

L

3

− 5

3

d(i,j)

L

2

+ 1 , 0 ≤ d(i, j) ≤ L

1 12

d(i,j)

L

5

− 1

2

d(i,j)

L

4

+ 5

8

d(i,j)

L

3

+ 5

3

d(i,j)

L

2

− 5 d(i,j)

L

  • + 4 − 2

3

  • L

d(i,j)

  • ,

L ≤ d(i, j) ≤ 2L 0 . 2L ≤ d(i, j)

†

  • d(i, j): distance between ith and jth grid points
  • L: radius of influence, i.e. localization radius

Adaptive Inflation and Localization Ensemble Filtering [8/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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EnKF: Schur-Product Localization

Space-dependent formulation; B−Localization

◮ Space-dependent radii, i.e., L ≡ L(i, j): we need to define localization kernel C

→ Examples include †:

C :=

                                

Cr = [ρi,j(li)]i,j=1,2,...,Nstate Cc = (Cr)T = [ρi,j(lj)]i,j=1,2,...,Nstate

1 2 (Cr + Cc) = 1 2 ρi,j(li) + ρi,j(lj) i,j=1,2,...,Nstate

Cd = ρi,j(lmin (i,j))

i,j=1,2,...,Nstate

Cu = ρi,j(lmax (i,j))

i,j=1,2,...,Nstate 1 2 (Cd + Cu) = 1 2 ρi,j(lmin (i,j)) + ρi,j(lmax (i,j)) i,j=1,2,...,Nstate

CG = ρi,j

  • li lj
  • i,j=1,2,...,Nstate

◮ We focus here on the symmetric kernel:

C := 1 2 (Cr + Cc) = 1 2 [ρi,j(li) + ρi,j(lj)]i,j=1,2,...,Nstate †Ahmed Attia, and Emil Constantinescu. ”An Optimal Experimental Design Framework for Adaptive Inflation and Covariance Localization for Ensemble Filters.” arXiv preprint arXiv:1806.10655 (2018).

Adaptive Inflation and Localization Ensemble Filtering [9/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 11

EnKF: Schur-Product Localization

Space-dependent formulation; R−Localization

◮ Localization in observation space (R−localization): ◮ HB is replaced with ”

HB = Cloc,1 ⊙ HB, where Cloc,1 = ρo|m

i,j

  • ; i = 1, 2, . . . Nobs ; j = 1, 2, . . . Nstate

◮ HBHT can be replaced with HBHT

  • = Cloc,2 ⊙ HBHT, where

Cloc,2 ≡ Co|o = ρo|o

i,j

  • ; i, j = 1, 2, . . . Nobs
  • ρo|m

i,j

is calculated between the ith observation grid point and the jth model grid point.

  • ρo|o

i,j is calculated between the ith and jth observation grid points. ◮ Assign radii to state grid points vs. observation grid points:

  • Let L ∈ RNobs to model grid points, and project to observations for Cloc,2 [hard/unknown]
  • Let L ∈ RNobs to observation grid points; [efficient; followed here]

Adaptive Inflation and Localization Ensemble Filtering [10/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 12

Inflation & Localization

Tuning the parameters

◮ Tuning the inflation parameter/factors λ

  • Bayesian approach for adaptive inflation exists, and still requires improvements
  • mostly for uncorrelated observation errors

◮ Tuning the localization radii of influence L

  • adaptive localization approaches are limited, especially in the vertical
  • mostly for uncorrelated observation errors
  • expert knowledge, especially with observation system, is required
  • theory is lacking

The parameters λ, L are generally tuned empirically!

Adaptive Inflation and Localization Ensemble Filtering [11/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 13

Outline

Ensemble Filtering Ensemble Kalman Filter (EnKF) Inflation & Localization Optimal Experimental Design OED and the alphabetical criteria OED Inflation & Localization A-OED inflation A-OED localization Numerical Experiments Experimental setup Numerical Results

Adaptive Inflation and Localization Optimal Experimental Design [12/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 14

Optimal Experimental Design (OED)

An OED problem seeks an optimal design w that solves

min

λ∈RNstate

ΨOED(w) + α Φ(w)

subject to

wl ≤ w ≤ wu

◮ ΨOED(w) is the specific design criterion ◮ For sensor placement, the design decides which sensors to activate ◮ The optimal design minimizes the uncertainty in the posterior state ◮ OED famous criteria:

  • 1. A-optimality: Trace of posterior covariance
  • 2. D-optimality: Determinant of the posterior covariance
  • 3. etc.

◮ Φ(λ) : RNs + → [0, ∞) is a regularization function (e.g.,ℓ1, ℓ0, etc.) ◮ α > 0 is a user-defined penalty parameter that controls the sparsity of the design Adaptive Inflation and Localization Optimal Experimental Design [13/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 15

Outline

Ensemble Filtering Ensemble Kalman Filter (EnKF) Inflation & Localization Optimal Experimental Design OED and the alphabetical criteria OED Inflation & Localization A-OED inflation A-OED localization Numerical Experiments Experimental setup Numerical Results

Adaptive Inflation and Localization OED Inflation & Localization [14/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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OED Approach for Adaptive Inflation

The A-optimal design (inflation parameter, λA−opt) minimizes:

min

λ∈RNstate

Tr A(λ) − α λ − 11

subject to

1 = λl

i ≤ λi ≤ λu i ,

i = 1, . . . , Nstate Remark: we choose the sign of the regularization term to be negative, unlike the traditional formulation

◮ Let H = H = I with uncorrelated observation noise, the design criterion becomes:

ΨInfl(λ) := Tr A =

Nstate

  • i=1
  • λ−1

i

σ−2

i

+ r−2

i

−1

◮ Decreasing λi reduces ΨInfl, i.e. the optimizer will always move toward λl Adaptive Inflation and Localization OED Inflation & Localization [15/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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OED Approach for Adaptive Inflation

Solving the A-OED problem, requires evaluating the objective, and the gradient:

◮ The design criterion:

ΨInfl(λ) := Tr A = Tr B − Tr

Ä

R + H BHT−1H B BHTä

◮ The gradient:

∇λΨInfl(λ) =

Nstate

  • i=1

λ−1

i

eieT

i (z1 − z2 − z3 + z4)

z1 = Bei z2 = HT R + H BHT−1 H Bz1 z3 = BHT R + H BHT−1 Hz1 z4 = HT R + H BHT−1 H Bz3 ei ∈ RNstate is the ith cardinality vector

Adaptive Inflation and Localization OED Inflation & Localization [16/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 18

OED Adaptive B−Localization (State-Space)

min

L∈RNstate

ΨB−Loc(L) + γ Φ(L) := Tr A(L) + γ L2

subject to

ll

i ≤ li ≤ lu i ,

i = 1, . . . , Nstate

◮ The design criterion:

ΨB−Loc(L) = Tr B − Tr

Ä

R + H BHT−1 H B BHTä

◮ The gradient:

∇LΨB−Loc =

Nstate

  • i=1

ei lB,i

  • I + HTR−1H

B−1 I + BHTR−1H−1 ei lB,i = lT

i ⊙

eT

iB

li =

∂ρi,1(li)

∂li , ∂ρi,2(li) ∂li , . . . , ∂ρi,Nstate(li) ∂li

T

ei ∈ RNstate is the ith cardinality vector

Adaptive Inflation and Localization OED Inflation & Localization [17/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 19

OED Adaptive: Observation-Space Localization

◮ So far, we assumed full state-space formulation, i.e. L ∈ RNstate

  • 1. the OED problem is solved to find LA−opt in the model state space
  • 2. LA−opt is projected, in the analysis step, into observation space to localize HB, and HBHT

◮ Pros:

  • reduces the cost of calculating the analysis

◮ Cons:

  • same cost for the optimization problem
  • projecting of LA−opt might be challenging or unknown

◮ Alternative: observation-space formulation:

→ formulate OED optimization problem in the observation space; i.e., L ∈ RNobs

Adaptive Inflation and Localization OED Inflation & Localization [18/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 20

OED Adaptive: Observation-Space Localization

◮ Assume L ∈ RNobs is attached to observation grid points ◮ HB is replaced with ”

HB = Cloc,1 ⊙ HB, with Cloc,1 = ρo|m

i,j (li)

; i = 1, 2, . . . Nobs ; j = 1, 2, . . . Nstate

◮ HBHT can be replaced with ’

HBHT = Cloc,2 ⊙ HBHT, with Co|o := 1 2

  • Co

r + Co c

  • = 1

2

  • ρo|o

i,j (li) + ρo|o i,j (lj) i,j=1,2,...,Nstate ◮ Localized posterior covariances: ◮ Localize HB:

  • A = B − ”

HB

T

R + HBHT−1” HB

◮ Localize both HB and HBHT:

  • A = B − ”

HB

T Ä

R + ’ HBHTä−1 ” HB

Adaptive Inflation and Localization OED Inflation & Localization [19/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 21

OED Adaptive R−Localization

Decorrelate HB

◮ The design criterion:

ΨR−Loc(L) = Tr (B) − Tr

Ä ”

HB ” HB

T

R + HBHT−1ä

◮ The gradient:

∇LΨR−Loc = −2

Nobs

  • i=1

ei lT

HB,i ψi

ψi = ” HB

T

R + HBHT−1 ei lHB,i = ls

i

T ⊙

eT

iHB

ls

i =

∂ρi,1(li)

∂li , ∂ρi,2(li) ∂li , . . . , ∂ρi,Nstate(li) ∂li

T

ei ∈ RNobs is the ith cardinality vector

Adaptive Inflation and Localization OED Inflation & Localization [20/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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OED Adaptive R−Localization

Decorrelate HB and HBHT

◮ The design criterion:

ΨR−Loc(L) = Tr (B) − Tr

  • ”

HB ” HB

TÄ

R + HBHT

  • ä−1

◮ The gradient:

∇LΨR−Loc =

Nobs

  • i=1

ei

  • ηo

i − 2 lT HB,i

  • ψo

i

ψo

i = ”

HB

TÄ

R + HBHT

  • ä−1

ei ηo

i = lo B,i

Ä

R + HBHT

  • ä−1”

HB lo

B,i =

lo

i

T ⊙

eT

iHBHT

lo

i =

∂ρi,1(li)

∂li , ∂ρi,2(li) ∂li , . . . , ∂ρi,Nobs(li) ∂li

T

Adaptive Inflation and Localization OED Inflation & Localization [21/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 23

Outline

Ensemble Filtering Ensemble Kalman Filter (EnKF) Inflation & Localization Optimal Experimental Design OED and the alphabetical criteria OED Inflation & Localization A-OED inflation A-OED localization Numerical Experiments Experimental setup Numerical Results

Adaptive Inflation and Localization Numerical Experiments [22/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 24

Experimental Setup

◮ The model (Lorenz-96):

dxi dt = xi−1 (xi+1 − xi−2) − xi + F ; i = 1, 2, . . . , 40 ,

  • x ∈ R40 is the state vector, with x0 ≡ x40
  • F = 8

◮ Initial background ensemble & uncertainty:

  • reference IC: xTrue

= Mt=0→t=5(−2, . . . , 2)T

  • B0 = σ0I ∈ RNstate×Nstate, with σ0 = 0.08

xTrue

  • 2

◮ Observations:

  • σobs = 5% of the average magnitude of the observed reference trajectory
  • R = σobsI ∈ RNobs×Nobs
  • Synthetic observations are generated every 20 time steps, with

H(x) = Hx = (x1, x3, x5, . . . , x37, x39)T ∈ R20 .

◮ EnKF flavor used here: DEnKF with Gaspari-Cohn (GC) localization

All experiments are carried out using DATeS

  • http://people.cs.vt.edu/~attia/DATeS/
  • https://doi.org/10.5281/zenodo.1247464
  • Ahmed Attia and Adrian Sandu, DATeS: A Highly-Extensible Data Assimilation Testing Suite, Geosci. Model Dev. Discuss.,

https://doi.org/10.5194/gmd-2018-30, in review, 2018. Adaptive Inflation and Localization Numerical Experiments [23/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 25

Numerical Results: Performance Metrics

◮ RMSE:

RMSE =

Ã

1

Nstate

Nstate

  • i=1

(xi − xTrue

i

)2 ,

◮ KL-distance to uniform Rank histogram

1 2 3 4 5 0.0 0.1 0.2 0.3 0.4

Relative Frequency

β(0.45, 0.43) 2 4 6 8 10 0.00 0.05 0.10 0.15 0.20 β(2.00, 2.12) 5 10 15 0.00 0.02 0.04 0.06 0.08 0.10 β(1.11, 0.88) 5 10 15 20

Rank

0.00 0.02 0.04 0.06

Relative Frequency

β(1.27, 1.16) 10 20 30

Rank

0.00 0.02 0.04 0.06 0.08 β(0.68, 0.61) 10 20 30 40

Rank

0.00 0.01 0.02 0.03 0.04 β(2.18, 2.19)

→ The KL divergence between two Beta distributions Beta(α, β), and Beta(α′, β′):

DKL(Beta(α, β) | Beta(α′ β′)) = ln Γ(α + β) − ln(α β) − ln Γ(α′ + β′) + ln(α′ β′) + (α − α′) ψ(α) − ψ(α′) + (β − β′) ψ(β) − ψ(β′)

  • ψ(·) = Γ′(·)

Γ(·) is the digamma function, i.e. the logarithmic derivative of the gamma function

  • U(0, 1) ≡ Beta(α′ = 1, β′ = 1)
  • A small, e.g. closer to 0, KL distance to be an indication of a nearly-uniform rank histogram,i.e., indicates a well-dispersed ensemble.

Adaptive Inflation and Localization Numerical Experiments [24/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 26

Numerical Results: Benchmark

5 10 15 20 25 30 35 40 Ensemble Size 1.0 1.03 1.06 1.09 1.12 1.15 1.18 1.21 1.24 Inflation Factor

RMSE KL-Distance to U

100 130 160 190 220 250 280 Time (assimilation cycles) 10−1 4 × 10−2 6 × 10−2 2 × 10−1 log-RMSE Forecast OED-DEnKF (a) RMSE 10 20 Rank 0.00 0.01 0.02 0.03 0.04 0.05 Relative Frequency (b) Rank histogram

The minimum average RMSE over the interval [10, 30], for every choice of Nens, is indicated by red a triangle. Blue tripods indicate the minimum KL distance between the analysis rank histogram and a uniformly distributed rank

  • histogram. Space-independent

radius of influence L = 4 is used. Analysis RMSE and rank histogram of DEnKF with

L = 4, and λ = 1.05.

Benchmark EnKF Results

Adaptive Inflation and Localization Numerical Experiments [25/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 27

Numerical Results: OED Adaptive Space-Time Inflation I

100 130 160 190 220 250 280 Time (assimilation cycles) 10−1 3 × 10−2 4 × 10−2 6 × 10−2 log-RMSE Optimal DEnKF OED-DEnKF (a) RMSE; α = 0.14 10 20 Rank 0.00 0.01 0.02 0.03 0.04 0.05 Relative Frequency (b) Rank histogram; α = 0.14 100 130 160 190 220 250 280 Time (assimilation cycles) 10−1 3 × 10−2 4 × 10−2 6 × 10−2 2 × 10−1 log-RMSE Optimal DEnKF OED-DEnKF (c) RMSE; α = 0.04 10 20 Rank 0.00 0.01 0.02 0.03 0.04 0.05 Relative Frequency (d) Rank histogram; α = 0.04

The localization radius is fixed to L = 4. The optimization penalty parameter α is indicated under each panel.

Adaptive Inflation and Localization Numerical Experiments [26/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 28

Numerical Results: OED Adaptive Space-Time Inflation II

100 140 180 220 260 300 Time (assimilation cycles) 1.012 1.014 1.016 1.018 1.020 Inflation factors λi

(a) α = 0.14

100 140 180 220 260 300 Time (assimilation cycles) 1.06 1.07 1.08 1.09 1.10 Inflation factors λi

(b) α = 0.04

Box plots expressing the range of values of the inflation coefficients at each time instant, over the testing timespan [10, 30].

Adaptive Inflation and Localization Numerical Experiments [27/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 29

Numerical Results; A-OED Inflation Regularization I

Choosing α

6.4 6.6 6.8 7.0 7.2 7.4 λ1 250 260 270 280 290 300 Time

α = 0.1400, RMSE = 0.0547 α = 0.1200, RMSE = 0.0548 α = 0.2200, RMSE = 0.0552 α = 0.0900, RMSE = 0.0555 α = 0.1300, RMSE = 0.0558

0.06 0.08 0.10 0.12 0.14 0.16 0.18

L-curve plots are are plotted for 25 equidistant values of the penalty parameter, at every assimilation time instant over the testing timespan [0.03, 0.24]. The values of the penalty parameter α that resulted in the 5 smallest average RMSEs, over all experiments carried out with different penalties, are highlighted on the plot and indicated in the legend along with the corresponding average RMSE.

Adaptive Inflation and Localization Numerical Experiments [28/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 30

Numerical Results; A-OED Inflation Regularization II

Choosing α 6.50 6.75 7.00 7.25 7.50 λ1 0.05 0.10 0.15 0.20 ΨInfl(λ)

α = 0.1400, RMSE = 0.0547 α = 0.1200, RMSE = 0.0548 α = 0.2200, RMSE = 0.0552 α = 0.0900, RMSE = 0.0555 α = 0.1300, RMSE = 0.0558

(a) Cycle 100 6.50 6.75 7.00 7.25 7.50 λ1 0.05 0.10 0.15 0.20 ΨInfl(λ)

α = 0.1400, RMSE = 0.0547 α = 0.1200, RMSE = 0.0548 α = 0.2200, RMSE = 0.0552 α = 0.0900, RMSE = 0.0555 α = 0.1300, RMSE = 0.0558

(b) Cycle 150

L-curve plots are are plotted for 25 equidistant values of the penalty parameter at assimilation cycles 100 and 150, respectively.

Adaptive Inflation and Localization Numerical Experiments [29/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 31

Numerical Results; A-OED Inflation Regularization III

Choosing α 0.00 0.05 0.10 0.15 0.20 0.25 α 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Average RMSE Rank histogram DKL to U

Average RMSE and KL-divergence from a uniform rank histogram resulted for 22 equidistant values of the penalty parameter in the interval [0.03, 0.24]. Values of the penalty parameter α that led to filter or optimizer divergence are indicated by red x marks.

Adaptive Inflation and Localization Numerical Experiments [30/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 32

Numerical Results: OED Adaptive Space-Time Localization I

100 130 160 190 220 250 280 Time (assimilation cycles) 10−1 3 × 10−2 4 × 10−2 6 × 10−2 log-RMSE Optimal DEnKF OED-DEnKF (a) RMSE; γ = 0 10 20 Rank 0.00 0.02 0.04 0.06 Relative Frequency (b) Rank histogram; γ = 0 100 130 160 190 220 250 280 Time (assimilation cycles) 10−1 3 × 10−2 4 × 10−2 6 × 10−2 log-RMSE Optimal DEnKF OED-DEnKF (c) RMSE; γ = 0.001 10 20 Rank 0.00 0.02 0.04 0.06 Relative Frequency (d) Rank histogram; γ = 0.001

The inflation factor is fixed to λ = 1.05. The optimization penalty parameter γ is shown under each panel.

Adaptive Inflation and Localization Numerical Experiments [31/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 33

Numerical Results: OED Adaptive Space-Time Localization II

100 130 160 190 220 250 280 Time (assimilation cycles) 10−1 4 × 10−2 6 × 10−2 log-RMSE Optimal DEnKF OED-DEnKF (a) RMSE 10 20 Rank 0.00 0.01 0.02 0.03 0.04 0.05 Relative Frequency (b) Rank histogram

Results for λ = 1.05, and γ = 0.04.

100 140 180 220 260 300 Time (assimilation cycles) 10 20 30 State variables 5 9 13 17

(a) γ = 0.0

100 140 180 220 260 300 Time (assimilation cycles) 10 20 30 State variables 5 9 13 17

(b) γ = 0.001

100 140 180 220 260 300 Time (assimilation cycles) 10 20 30 State variables 1

(c) γ = 0.04

Localization radii at each time points, over the testing timespan [10, 30]. The optimization penalty parameter γ is shown under each panel.

Adaptive Inflation and Localization Numerical Experiments [32/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 34

Numerical Results: OED Adaptive Space-Time Localization III

100 150 200 250 300 Time (Assimilation cycles) 10−1 3 × 10−2 4 × 10−2 6 × 10−2 log-RMSE localize B localize HB localize (HB, HBHT)

A-OED optimal localization radii L found by solving the OED localization problems in model state-space, and

  • bservation space respectively. No regularization is applied, i.e., γ = 0

Adaptive Inflation and Localization Numerical Experiments [33/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 35

Numerical Results: OED Adaptive Space-Time Localization IV

Rank histogram for A-OED localization solved in model state-space, and observation space respectively. Space-time optimal localization radii over the testing timespan.

Adaptive Inflation and Localization Numerical Experiments [34/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 36

Numerical Results; A-OED Localization Regularization I

Choosing γ

10 20 30 40 50 60 L2 250 260 270 280 290 300 Time

γ = 0.0000, RMSE = 0.0569 γ = 0.0010, RMSE = 0.0613 γ = 0.0020, RMSE = 0.0630 γ = 0.0030, RMSE = 0.0685 γ = 0.0040, RMSE = 0.0743

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55

L-curve plots are shown for values of the penalty parameter γ = 0, 0.001, . . . , 0.34.

Adaptive Inflation and Localization Numerical Experiments [35/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.

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SLIDE 37

Concluding Remarks

◮ Introduced an OED approach for adaptive inflation and localization ◮ Either A-OED inflation or localization is carried out each cycle ◮ Can create a weighted objective to account for both inflation and localization ◮ Regularization is a must for adaptive inflation ◮ Regularization may not be needed, in general, for adaptive localization ◮ Definiteness of the localization Kernel D ◮ Regularization norm ◮ Other OED criteria; e.g., D-optimality ◮ Adaptive Bayesian A-OED!

Thank You

Adaptive Inflation and Localization Numerical Experiments [36/36] September 14, 2018: SIAM-MPE 18; Ahmed Attia.