Adaptive covariance inflation in the EnKF by Gaussian scale - - PowerPoint PPT Presentation

Adaptive covariance inflation in the EnKF by Gaussian scale mixtures

pdf

x

Patrick N. Raanes, Marc Bocquet, Alberto Carrassi

patrick.n.raanes@gmail.com

NERSC NERSC

EnKF Workshop, Bergen, May 2018

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Questions ∼answered in paper

Nonlinear models cause sampling error. Why and how? Can it be dissociated from non-Gaussianity? Does the inherent bias

• E[tr(¯

Pa)] < tr(Pa)

• cause

collapse ? divergence ? Other reasons for inflating in nonlinear contexts. Linear models attenuate sampling error. How? Is the covariance factor

1 N−1 optimal?

How does localization affect inflation ? How should inflation be defined as a parameter, rather than just a target statistic? How does the feedback of the EnKF-N compare to “unbiased” updates. What is the bias of the estimator ˆ βR ? Why is it better than ˆ βI or ˆ βML ?

Adaptive covariance inflation in the EnKF by Gaussian scale mixtures

pdf

x

Patrick N. Raanes, Marc Bocquet, Alberto Carrassi

patrick.n.raanes@gmail.com

NERSC NERSC

EnKF Workshop, Bergen, May 2018

Outline

Idealistic contexts (with sampling error)

Revisiting the EnKF assumptions = ⇒ Gaussian scale mixture (EnKF-N)

With model error

Survey inflation estimation

ETKF-adaptive EAKF-adaptive

EnKF-N hybrid Benchmarks

Outline

Idealistic contexts (with sampling error)

Revisiting the EnKF assumptions = ⇒ Gaussian scale mixture (EnKF-N)

With model error

Survey inflation estimation

ETKF-adaptive EAKF-adaptive

EnKF-N hybrid Benchmarks

Outline

Idealistic contexts (with sampling error)

Revisiting the EnKF assumptions = ⇒ Gaussian scale mixture (EnKF-N)

With model error

Survey inflation estimation

ETKF-adaptive EAKF-adaptive

EnKF-N hybrid Benchmarks

Outline

Idealistic contexts (with sampling error)

Revisiting the EnKF assumptions = ⇒ Gaussian scale mixture (EnKF-N)

With model error

Survey inflation estimation

ETKF-adaptive EAKF-adaptive

EnKF-N hybrid Benchmarks

Outline

Idealistic contexts (with sampling error)

Revisiting the EnKF assumptions = ⇒ Gaussian scale mixture (EnKF-N)

With model error

Survey inflation estimation

ETKF-adaptive EAKF-adaptive

EnKF-N hybrid Benchmarks

Outline

Idealistic contexts (with sampling error)

Revisiting the EnKF assumptions = ⇒ Gaussian scale mixture (EnKF-N)

With model error

Survey inflation estimation

ETKF-adaptive EAKF-adaptive

EnKF-N hybrid Benchmarks

Idealistic contexts (EnKF-N)

Assume M, H, Q, R are perfectly known, and p(x) and p(y|x) are always Gaussian.

EnKF

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

Revisiting EnKF assumptions

Denote yprior all prior information on the “true” state, x ∈ RM, and suppose that, with known mean (b) and cov (B), p(x|yprior) = N(x|b, B) . (1)

Computational costs induce:

≈ p(x|E) =

• N(x|b, B) p(b, B|E) db dB

= ⇒ “true” moments, b and B, are unknowns, to be estimated from E. Ensemble E =

• x1,

. . . xn, . . . xN

• also from (1) and iid.

EnKF prior

But p(x|E) =

• B
• RM

N(x|b, B) p(b, B|E) db dB (2) Recover standard EnKF by assuming N=∞ so that p(b, B|E) = δ(b − ¯ x)δ(B − ¯ B) , where ¯ x = 1 N

N

• n=1

xn , ¯ B = 1 N − 1

N

• n=1

(xn − ¯ x) (xn − ¯ x)T . (3) The EnKF-N does not make this approximation.

EnKF prior

But p(x|E) =

• B
• RM

N(x|b, B) p(b, B|E) db dB (2) Recover standard EnKF by assuming N=∞ so that p(b, B|E) = δ(b − ¯ x)δ(B − ¯ B) , where ¯ x = 1 N

N

• n=1

xn , ¯ B = 1 N − 1

N

• n=1

(xn − ¯ x) (xn − ¯ x)T . (3) The EnKF-N does not make this approximation.

EnKF prior

But p(x|E) =

• B
• RM

N(x|b, B) p(b, B|E) db dB (2) Recover standard EnKF by assuming N=∞ so that p(b, B|E) = δ(b − ¯ x)δ(B − ¯ B) , where ¯ x = 1 N

N

• n=1

xn , ¯ B = 1 N − 1

N

• n=1

(xn − ¯ x) (xn − ¯ x)T . (3) The EnKF-N does not make this approximation.

EnKF-N via scale mixture

Prior: p(x|E) =

• N(x|b, B) p(b, B|E) db dB

(4) (5) (6) (8)

EnKF-N via scale mixture

Prior: p(x|E) =

• N(x|b, B) p(b, B|E) db dB

(4) . . . . . . (5) . . . . . . (6) . . . ∝

• 1 +

1 N−1x − ¯

x2

εN ¯ B

−N/2

(7) (8)

EnKF-N via scale mixture

Prior: p(x|E) =

• N(x|b, B) p(b, B|E) db dB

(4) . . . ∝

• α>0

N

x − ¯

xεN ¯

B|0, α

p α|E dα

(5) . . . . . . (6) . . . ∝

• 1 +

1 N−1x − ¯

x2

εN ¯ B

−N/2

(7) (8)

EnKF-N via scale mixture

Prior: p(x|E) =

• N(x|b, B) p(b, B|E) db dB

(4) . . . ∝

• α>0

N

x − ¯

xεN ¯

B|0, α

p α|E dα

(5) . . . . . . (6) . . . ∝

• 1 +

1 N−1x − ¯

x2

εN ¯ B

−N/2

(7) Posterior: p(x|E, y) ∝ p(x|E) N

y|Hx, R

• (8)

EnKF-N via scale mixture

Prior: p(x|E) =

• N(x|b, B) p(b, B|E) db dB

(4) . . . ∝

• α>0

N

x − ¯

xεN ¯

B|0, α

p α|E dα

(5) . . . ∝ N

x|¯

x, α(x)¯ B

˜

p

α(x)|E

• (6)

. . . ∝

• 1 +

1 N−1x − ¯

x2

εN ¯ B

−N/2

(7) Posterior: p(x|E, y) ∝ p(x|E) N

y|Hx, R

• (8)

Mixing distributions – p(α| . . .)

1 2 3 4 5 6

pdf λ

Prior Posterior Likelihood

Prior: p(α|E) = χ−2(α|1, N−1) Likelihood: p(x⋆, y|α, E) ∝ exp

−1

2y−H¯

x2

αεNH¯ BHT+R

• =

⇒ Posterior: p(x⋆, α|y, E) ∝ exp

−1

2 D(α)

Summary – Perfect model scenario

Even with a perfect model, Gaussian forecasts, and a deterministic EnKF, “sampling error” arises for N < ∞ due to nonlinearity, and inflation is necessary. Not assuming ¯ B = B as in the EnKF leads to a Gaussian scale mixture. This leads to an adaptive inflation scheme, nullifying the need to tune the inflation factor, and yielding very strong benchmarks in idealistic settings. Excellent training for EnKF theory. Especially general-purpose inflation estimation.

Summary – Perfect model scenario

Even with a perfect model, Gaussian forecasts, and a deterministic EnKF, “sampling error” arises for N < ∞ due to nonlinearity, and inflation is necessary. Not assuming ¯ B = B as in the EnKF leads to a Gaussian scale mixture. This leads to an adaptive inflation scheme, nullifying the need to tune the inflation factor, and yielding very strong benchmarks in idealistic settings. Excellent training for EnKF theory. Especially general-purpose inflation estimation.

Summary – Perfect model scenario

Even with a perfect model, Gaussian forecasts, and a deterministic EnKF, “sampling error” arises for N < ∞ due to nonlinearity, and inflation is necessary. Not assuming ¯ B = B as in the EnKF leads to a Gaussian scale mixture. This leads to an adaptive inflation scheme, nullifying the need to tune the inflation factor, and yielding very strong benchmarks in idealistic settings. Excellent training for EnKF theory. Especially general-purpose inflation estimation.

Summary – Perfect model scenario

Even with a perfect model, Gaussian forecasts, and a deterministic EnKF, “sampling error” arises for N < ∞ due to nonlinearity, and inflation is necessary. Not assuming ¯ B = B as in the EnKF leads to a Gaussian scale mixture. This leads to an adaptive inflation scheme, nullifying the need to tune the inflation factor, and yielding very strong benchmarks in idealistic settings. Excellent training for EnKF theory. Especially general-purpose inflation estimation.

Summary – Perfect model scenario

Even with a perfect model, Gaussian forecasts, and a deterministic EnKF, “sampling error” arises for N < ∞ due to nonlinearity, and inflation is necessary. Not assuming ¯ B = B as in the EnKF leads to a Gaussian scale mixture. This leads to an adaptive inflation scheme, nullifying the need to tune the inflation factor, and yielding very strong benchmarks in idealistic settings. Excellent training for EnKF theory. Especially general-purpose inflation estimation.

Summary – Perfect model scenario

Even with a perfect model, Gaussian forecasts, and a deterministic EnKF, “sampling error” arises for N < ∞ due to nonlinearity, and inflation is necessary. Not assuming ¯ B = B as in the EnKF leads to a Gaussian scale mixture. This leads to an adaptive inflation scheme, nullifying the need to tune the inflation factor, and yielding very strong benchmarks in idealistic settings. Excellent training for EnKF theory. Especially general-purpose inflation estimation.

With model error

Because all models are wrong.

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β, ) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

• ,

where ¯ C(β) = βH¯ BHT + R , (10)

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β, ) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

• ,

where ¯ C(β) = βH¯ BHT + R , (10)

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β, ) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

• ,

where ¯ C(β) = βH¯ BHT + R , (10)

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β, E) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

• ,

where ¯ C(β) = βH¯ BHT + R , (10)

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β,

• E) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

• ,

where ¯ C(β) = βH¯ BHT + R , (10)

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β,

• E) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

• ,

where ¯ C(β) = βH¯ BHT + R , (10)

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β,

• E) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

• ,

where ¯ C(β) = βH¯ BHT + R , (10)

Fundamentals

Suppose xn ∼ N

b, B/β , and N = ∞.

Then there’s no mixture, but simply p(x|β,

• E) = N

x

• ¯

x, β ¯ B

.

(9) Recall p(y|x) = N

y|Hx, R .

Then p(y|β) = N

y

• H¯

x, ¯ C(β)

= N ¯

δ

• 0, ¯

C(β)

,

where ¯ C(β) = βH¯ BHT + R , ¯ δ = y − H¯ x . (10)

ETKF adaptive inflation

Again, p(y|β) = N

¯

δ

• 0, ¯

C(β)

,

(11) where ¯ C(β) = βH¯ BHT + R ≈ ¯ δ¯ δT . (12) “yielding” (Wang and Bishop, 2003) ˆ βR = ¯ δ2

R/P − 1

¯ σ2 , where P = length(y) and ¯ σ2 = tr(H¯ BHTR−1)/P. Also considered: ˆ βI, ˆ βH¯

BHT, ˆ

β¯

C(1), ML, VB (EM).

ETKF adaptive inflation

Again, p(y|β) = N

¯

δ

• 0, ¯

C(β)

,

(11) where ¯ C(β) = βH¯ BHT + R ≈ ¯ δ¯ δT . (12) “yielding” (Wang and Bishop, 2003) ˆ βR = ¯ δ2

R/P − 1

¯ σ2 , where P = length(y) and ¯ σ2 = tr(H¯ BHTR−1)/P. Also considered: ˆ βI, ˆ βH¯

BHT, ˆ

β¯

C(1), ML, VB (EM).

ETKF adaptive inflation

Again, p(y|β) = N

¯

δ

• 0, ¯

C(β)

,

(11) where ¯ C(β) = βH¯ BHT + R ≈ ¯ δ¯ δT . (12) “yielding” (Wang and Bishop, 2003) ˆ βR = ¯ δ2

R/P − 1

¯ σ2 , where P = length(y) and ¯ σ2 = tr(H¯ BHTR−1)/P. Also considered: ˆ βI, ˆ βH¯

BHT, ˆ

β¯

C(1), ML, VB (EM).

ETKF adaptive inflation

Again, p(y|β) = N

¯

δ

• 0, ¯

C(β)

,

(11) where ¯ C(β) = βH¯ BHT + R ≈ ¯ δ¯ δT . (12) “yielding” (Wang and Bishop, 2003) ˆ βR = ¯ δ2

R/P − 1

¯ σ2 , where P = length(y) and ¯ σ2 = tr(H¯ BHTR−1)/P. Also considered: ˆ βI, ˆ βH¯

BHT, ˆ

β¯

C(1), ML, VB (EM).

ETKF adaptive inflation

Again, p(y|β) = N

¯

δ

• 0, ¯

C(β)

,

(11) where ¯ C(β) = βH¯ BHT + R ≈ ¯ δ¯ δT . (12) “yielding” (Wang and Bishop, 2003) ˆ βR = ¯ δ2

R/P − 1

¯ σ2 , where P = length(y) and ¯ σ2 = tr(H¯ BHTR−1)/P. Also considered: ˆ βI, ˆ βH¯

BHT, ˆ

β¯

C(1), ML, VB (EM).

ETKF adaptive inflation

Again, p(y|β) = N

¯

δ

• 0, ¯

C(β)

,

(11) where ¯ C(β) = βH¯ BHT + R ≈ ¯ δ¯ δT . (12) “yielding” (Wang and Bishop, 2003) ˆ βR = ¯ δ2

R/P − 1

¯ σ2 , where P = length(y) and ¯ σ2 = tr(H¯ BHTR−1)/P. Also considered: ˆ βI, ˆ βH¯

BHT, ˆ

β¯

C(1), ML, VB (EM).

ETKF adaptive inflation

Again, p(y|β) = N

¯

δ

• 0, ¯

C(β)

,

(11) where ¯ C(β) = βH¯ BHT + R ≈ ¯ δ¯ δT . (12) “yielding” (Wang and Bishop, 2003) ˆ βR = ¯ δ2

R/P − 1

¯ σ2 , where P = length(y) and ¯ σ2 = tr(H¯ BHTR−1)/P. Also considered: ˆ βI, ˆ βH¯

BHT, ˆ

β¯

C(1), ML, VB (EM).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

Renouncing Gaussianity

Assume H¯ BHT ∝ R. The likelihood p(y|β) = N

¯

δ

• 0, ¯

C(β)

becomes

: p(y|β) ∝ χ+2 ¯ δ2

R/P

• (1 + ¯

σ2β), P

• .

(13) Surprise !!!: argmax p(y|β) = ˆ βR , A further approximation is fitted: p(y|β) ≈ χ+2(ˆ βR|β, ˆ ν) . (14) Likelihood (14) fits mode of (13). Fitting curvature = ⇒ ˆ ν = ⇒ same variance as in Miyoshi (2011) !!! Likelihood (14) conjugate to p(β) = χ−2(β|βf, νf) , yielding νa = νf + ˆ ν , (15) βa = (νfβf + ˆ ν ˆ βR)/νa , (16) again, as in Miyoshi (2011).

EAKF adaptive inflation

Anderson (2007) assigns Gaussian prior: p(β) = N(β|βf, V f) , (17) and fits the posterior by a “Gaussian”: p(β|yi) ≈ N(β|ˆ βMAP, V a) , (18) where ˆ βMAP and V a are fitted using the exact posterior (“easy” by virtue of serial update). Gharamti (2017) improves via χ−2 and χ+2 (Gamma).

EAKF adaptive inflation

Anderson (2007) assigns Gaussian prior: p(β) = N(β|βf, V f) , (17) and fits the posterior by a “Gaussian”: p(β|yi) ≈ N(β|ˆ βMAP, V a) , (18) where ˆ βMAP and V a are fitted using the exact posterior (“easy” by virtue of serial update). Gharamti (2017) improves via χ−2 and χ+2 (Gamma).

EAKF adaptive inflation

Anderson (2007) assigns Gaussian prior: p(β) = N(β|βf, V f) , (17) and fits the posterior by a “Gaussian”: p(β|yi) ≈ N(β|ˆ βMAP, V a) , (18) where ˆ βMAP and V a are fitted using the exact posterior (“easy” by virtue of serial update). Gharamti (2017) improves via χ−2 and χ+2 (Gamma).

EAKF adaptive inflation

Anderson (2007) assigns Gaussian prior: p(β) = N(β|βf, V f) , (17) and fits the posterior by a “Gaussian”: p(β|yi) ≈ N(β|ˆ βMAP, V a) , (18) where ˆ βMAP and V a are fitted using the exact posterior (“easy” by virtue of serial update). Gharamti (2017) improves via χ−2 and χ+2 (Gamma).

EAKF adaptive inflation

Anderson (2007) assigns Gaussian prior: p(β) = N(β|βf, V f) , (17) and fits the posterior by a “Gaussian”: p(β|yi) ≈ N(β|ˆ βMAP, V a) , (18) where ˆ βMAP and V a are fitted using the exact posterior (“easy” by virtue of serial update). Gharamti (2017) improves via χ−2 and χ+2 (Gamma).

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

EnKF-N hybrid

Use two inflation factors: α and β, dedicated to sampling and model error, respectively. For β, pick simplest (and ∼best) scheme: ˆ βR. Algorithm: Find β (via ˆ βR) Find α given β (via EnKF-N) Potential improvements:

Determining (α, β) jointly (simultaneously). Rather than fitting the likelihood parameters, fit posterior parameters (similarly to EAKF). Matching moments via quadrature Non-parametric (grid- or MC- based) De-biasing ˆ βR

Testing “improvements” did not yield significant gains.

Two-layer Lorenz-96

Evolution

dxi dt = ψ+

i (x)

+ F − hc b

10

• j=1

zj+10(i−1) , i = 1, . . . , 36, dzj dt = c bψ−

j (bz) + 0 + hc

bx1+(j−1)/

/10 ,

j = 1, . . . , 360, where ψi is the single-layer Lorenz-96 dynamics.

−4 −2 2 4 6 8 10 1 4 8 12 16 20 24 28 32 36

1 40 80 120 160 200 240 280 320 360

Example snapshots RMSE = 1 T

T

• t=1

1

M ¯ xt − xt2

2 .

N = 20, no localization.

Two-layer Lorenz-96

Evolution

dxi dt = ψ+

i (x)

+ F − hc b

10

• j=1

zj+10(i−1) , i = 1, . . . , 36, dzj dt = c bψ−

j (bz) + 0 + hc

bx1+(j−1)/

/10 ,

j = 1, . . . , 360, where ψi is the single-layer Lorenz-96 dynamics.

−4 −2 2 4 6 8 10 1 4 8 12 16 20 24 28 32 36

1 40 80 120 160 200 240 280 320 360

Example snapshots RMSE = 1 T

T

• t=1

1

M ¯ xt − xt2

2 .

N = 20, no localization.

Illustration of time series

0.5 1.0 1.5

ETKF tuned

0.5 1.0 1.5

EAKF adaptive

0.5 1.0 1.5

ETKF adaptive

2500 2600 2700 2800 2900 3000

DA cycle (k)

0.5 1.0 1.5

EnKF-N hybrid Inflation RMS Error RMS Spread

Benchmarks

5 10 15 20 25 30

Forcing (F) both for truth and DA

0.0 0.2 0.4 0.6 0.8 1.0

RMSE ETKF tuned

Benchmarks

5 10 15 20 25 30

Forcing (F) both for truth and DA

0.0 0.2 0.4 0.6 0.8 1.0

RMSE ETKF tuned ETKF excessive

Benchmarks

5 10 15 20 25 30

Forcing (F) both for truth and DA

0.0 0.2 0.4 0.6 0.8 1.0

RMSE ETKF tuned ETKF excessive EAKF adaptive

Benchmarks

5 10 15 20 25 30

Forcing (F) both for truth and DA

0.0 0.2 0.4 0.6 0.8 1.0

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive

Benchmarks

5 10 15 20 25 30

Forcing (F) both for truth and DA

0.0 0.2 0.4 0.6 0.8 1.0

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid

Benchmarks

5 10 15 20 25 30

Forcing (F) both for truth and DA

0.0 0.2 0.4 0.6 0.8 1.0

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid

Summary

Paper highlights: Cataloguing of reasons to inflate. Inflation-centric re-derivation of the dual EnKF-N. Formal survey of adaptive inflation methods. A simple hybrid of EnKF-N and ˆ βR, which is shown to systematically (but moderately) improve filter accuracy (no re-tuning!).

Summary

Paper highlights: Cataloguing of reasons to inflate. Inflation-centric re-derivation of the dual EnKF-N. Formal survey of adaptive inflation methods. A simple hybrid of EnKF-N and ˆ βR, which is shown to systematically (but moderately) improve filter accuracy (no re-tuning!).

Summary

Paper highlights: Cataloguing of reasons to inflate. Inflation-centric re-derivation of the dual EnKF-N. Formal survey of adaptive inflation methods. A simple hybrid of EnKF-N and ˆ βR, which is shown to systematically (but moderately) improve filter accuracy (no re-tuning!).

Summary

Paper highlights: Cataloguing of reasons to inflate. Inflation-centric re-derivation of the dual EnKF-N. Formal survey of adaptive inflation methods. A simple hybrid of EnKF-N and ˆ βR, which is shown to systematically (but moderately) improve filter accuracy (no re-tuning!).

Summary

Paper highlights: Cataloguing of reasons to inflate. Inflation-centric re-derivation of the dual EnKF-N. Formal survey of adaptive inflation methods. A simple hybrid of EnKF-N and ˆ βR, which is shown to systematically (but moderately) improve filter accuracy (no re-tuning!).

Summary

Paper highlights: Cataloguing of reasons to inflate. Inflation-centric re-derivation of the dual EnKF-N. Formal survey of adaptive inflation methods. A simple hybrid of EnKF-N and ˆ βR, which is shown to systematically (but moderately) improve filter accuracy (no re-tuning!).

References 1

https://github.com/nansencenter/DAPPER 2011: Marc Bocquet. Ensemble Kalman filtering without the intrinsic need for inflation. Nonlinear Processes in Geophysics. 2012: Marc Bocquet and Pavel Sakov. Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems. Nonlinear Processes in Geophysics 2015: Marc Bocquet, Patrick N. Raanes, and Alexis Hannart. Expanding the validity of the ensemble Kalman filter without the intrinsic need for inflation. Nonlinear Processes in Geophysics 2018: Patrick N. Raanes, Marc Bocquet, and Alberto Carrassi. Adaptive covariance inflation in the ensemble Kalman filter by Gaus- sian scale mixtures. QJRMS (minor rev), arxiv.org/abs/1801.08474

References 2

Jeffrey L. Anderson. An adaptive covariance inflation error correction algorithm for ensemble filters. Tellus A, 59(2):210–224, 2007.

• M. E. Gharamti. Enhanced adaptive inflation algorithm for ensemble filters. Monthly

Weather Review, In review(0):0–0, 2017. Takemasa Miyoshi. The Gaussian approach to adaptive covariance inflation and its implementation with the local ensemble transform Kalman filter. Monthly Weather Review, 139(5):1519–1535, 2011. Xuguang Wang and Craig H. Bishop. A comparison of breeding and ensemble transform Kalman filter ensemble forecast schemes. Journal of the Atmospheric Sciences, 60(9):1140–1158, 2003.

UKF reinventing localization

Appendix

Parametric distributions – Table

Parametric distributions – Properties

EnKF-N mixing distribution

Instead, we assign the Jeffreys (hyper)prior: p(b, B) ∝ p(B) ∝ |B|−(M+1)/2 , (19) and recall the likelihood: p(E|b, B) ∝

N

• n=1

N(xn|b, B) , (20) yielding p(b, B|E) = N(b|¯ x, B/N)

• p(b|B,E)

W−1(B|¯ B, N−1)

• p(B|E)

, (21) where W−1 is the inverse-Wishart distribution (c.f. Table 2).

EnKF-N mixing distribution

Instead, we assign the Jeffreys (hyper)prior: p(b, B) ∝ p(B) ∝ |B|−(M+1)/2 , (19) and recall the likelihood: p(E|b, B) ∝

N

• n=1

N(xn|b, B) , (20) yielding p(b, B|E) = N(b|¯ x, B/N)

• p(b|B,E)

W−1(B|¯ B, N−1)

• p(B|E)

, (21) where W−1 is the inverse-Wishart distribution (c.f. Table 2).

EnKF-N mixing distribution

Instead, we assign the Jeffreys (hyper)prior: p(b, B) ∝ p(B) ∝ |B|−(M+1)/2 , (19) and recall the likelihood: p(E|b, B) ∝

N

• n=1

N(xn|b, B) , (20) yielding p(b, B|E) = N(b|¯ x, B/N)

• p(b|B,E)

W−1(B|¯ B, N−1)

• p(B|E)

, (21) where W−1 is the inverse-Wishart distribution (c.f. Table 2).

Benchmarks

100 101

Speed-scale ratio (c) both for truth and DA

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44

RMSE ETKF tuned

Benchmarks

100 101

Speed-scale ratio (c) both for truth and DA

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44

RMSE ETKF tuned ETKF excessive

Benchmarks

100 101

Speed-scale ratio (c) both for truth and DA

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44

RMSE ETKF tuned ETKF excessive EAKF adaptive

Benchmarks

100 101

Speed-scale ratio (c) both for truth and DA

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive

Benchmarks

100 101

Speed-scale ratio (c) both for truth and DA

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid

Benchmarks

100 101

Speed-scale ratio (c) both for truth and DA

0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid

Benchmarks (single-layer)

7.0 7.5 8.0 8.5 9.0

Forcing (F) DA assumes F = 8

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

RMSE ETKF tuned

Benchmarks (single-layer)

7.0 7.5 8.0 8.5 9.0

Forcing (F) DA assumes F = 8

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

RMSE ETKF tuned ETKF excessive

Benchmarks (single-layer)

7.0 7.5 8.0 8.5 9.0

Forcing (F) DA assumes F = 8

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

RMSE ETKF tuned ETKF excessive EAKF adaptive

Benchmarks (single-layer)

7.0 7.5 8.0 8.5 9.0

Forcing (F) DA assumes F = 8

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive

Benchmarks (single-layer)

7.0 7.5 8.0 8.5 9.0

Forcing (F) DA assumes F = 8

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid

Benchmarks (single-layer)

7.0 7.5 8.0 8.5 9.0

Forcing (F) DA assumes F = 8

0.35 0.40 0.45 0.50 0.55 0.60 0.65 0.70

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid

Benchmarks (Lorenz-63) – N = 3

10-4 10-3 10-2 10-1 100 101

Magnitude (Qii/∆) of noise on truth

0.4 0.5 0.6 0.7 0.8 0.9

RMSE ETKF tuned

Benchmarks (Lorenz-63) – N = 3

10-4 10-3 10-2 10-1 100 101

Magnitude (Qii/∆) of noise on truth

0.4 0.5 0.6 0.7 0.8 0.9

RMSE ETKF tuned ETKF excessive

Benchmarks (Lorenz-63) – N = 3

10-4 10-3 10-2 10-1 100 101

Magnitude (Qii/∆) of noise on truth

0.4 0.5 0.6 0.7 0.8 0.9

RMSE ETKF tuned ETKF excessive EAKF adaptive

Benchmarks (Lorenz-63) – N = 3

10-4 10-3 10-2 10-1 100 101

Magnitude (Qii/∆) of noise on truth

0.4 0.5 0.6 0.7 0.8 0.9

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive

Benchmarks (Lorenz-63) – N = 3

10-4 10-3 10-2 10-1 100 101

Magnitude (Qii/∆) of noise on truth

0.4 0.5 0.6 0.7 0.8 0.9

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid

Benchmarks (Lorenz-63) – N = 3

10-4 10-3 10-2 10-1 100 101

Magnitude (Qii/∆) of noise on truth

0.4 0.5 0.6 0.7 0.8 0.9

RMSE ETKF tuned ETKF excessive EAKF adaptive ETKF adaptive EnKF-N hybrid