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Ch12.3 Filter Performance with Regularly Spaced Sparse

Thursday, April 12, 12

Thursday, April 12, 12

Weakly Chaotic Regime

Thursday, April 12, 12

Strongly Chaotic Regime

Thursday, April 12, 12

Strongly Chaotic Regime

Thursday, April 12, 12

Fully Turbulent Regime

Thursday, April 12, 12

Fully Turbulent Regime

Thursday, April 12, 12

Fully Turbulent Regime

Thursday, April 12, 12

Super-Long Observation Times

Thursday, April 12, 12

Thursday, April 12, 12

Chapter 13: SPEKF for filtering turbulent signalswith model error

Consider a dynamical system that depends on a parameter �

u = F (u, t,�).

If � is not known, we could represent its uncertainty by modeling it

as a dynamic process

� = g(�).

The parameter � can be estimated as part of a filtering algorithm.

Common choices for g(�) are g = 0 and g = �W . The former

works best when � is constant, and the latter works well when �has small variations.

Even when F is linear in u, it is often nonlinear when � is

considered as a variable. As a result, one has to use nonlinear

filtering methods like EKF or EnKF.

The “mean stochastic model” (MSM) for a scalar (e.g. Fourier

coe�cient) has the form

u = (�� + i!)u + �W + f (t).

The parameters �, !, and f (t), and � can be inferred from

climatological statistics.

One can estimate the parameters in the MSM on the fly within the

filtering algorithm. SPEKF augments the MSM with SDEs for �,and F :

u = (��(t) + i!)u(t) + b(t) + f (t) + �W

b = (��b + i!b)(b(t)� b) + �bWb

� = �d�(�(t)� �) + ��W�

The parameters �b, d� , �b, �� are “model error” parameters that

are not necessarily directly tied to physical processes.

The foregoing system was the “combined” model. We can also

consider an “additive” model with �(t) = �, or a “multiplicative”

model with b = b.

The augmented equations are nonlinear

u = (��(t) + i!)u(t) + b(t) + f (t) + �W

b = (��b + i!b)(b(t)� b) + �bWb

� = �d�(�(t)� �) + ��W�

In a filtering framework that estimates the parameters � and b one

might use EKF, but EKF is notoriously inaccurate over long times.

It is possible to evolve the mean and covariance of the above

exactly (review chapter 13.1). We can therefore use the “NEKF”

(nonlinear EKF) instead of the EKF. In chapter 13.1, this strategy

was applied to filtering the scalar stochastic process with hidden

instabilities from chapter 8. The filter accurately infers the jumps

between stability and instability (last lecture).

We now consider filtering the spatially-extended turbulent model

from section 8.2. The true dynamics for each Fourier mode are

uk = (��k(t) + i!k)uk + �k wk + fk(t)

where �k(t) randomly switches between stable and unstable states

The random forcing is set to give energy spectrum k�3, with

Rossby wave dispersion relation !k = 8.91/k ; there are 52 Fourier

modes (105 grid points).

Filtering setup

I Sparse observations every 7 grid points (15 total)

I Obs time interval is 0.25

I Obs noise (white, ro = 0.3) is such that obs noise is larger

than climatological variance for k > 7

I Filtering using RFDKF: only the largest-scale mode in each

aliasing set is actively filtered, though the others are predicted

(using MSM, which is perfect).

I The observed/filtered modes are forecast using

I Perfect model

I MSM

I SPEKF-A, SPEKF-M, SPEKF-C

Robustness of the algorithm to di↵erent choices of the “model

error” parameters (damping rates and noise amplitudes for � and

b) are explored in chapter 13.2, but omitted here for brevity.

We now consider the QG model from chapter 11.

The true signal is generated by two-layer QG with 128⇥128 points

in each layer.

The observational grid is 6⇥ 6 points in the upper layer only.

The filters use the RFDKF strategy.

We compare MSM1 and MSM2 (see chapter 12), SPEKF, and a

local-least-squares EAKF using the perfect model (which is about

4 orders of magnitude more expensive).