Weak-constraint 4D-Var: when the model is allowed to be wrong

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Where it sits. When the model is wrong: estimating the state and the model error together. The full page is a self-contained document with operator boxes (inputs, outputs, shapes and units), figures and code sketches, which opens on its own and links back here.

TL;DR: Same cost, plus a price for every model error¶

• The transition gets noise: $\vect{x}_k = f_k(\vect{x}_{k-1}) + \gvect{\eta}_k$ with $\gvect{\eta}_k \sim \mathcal{N}(\vect{0}, \gmat{\Sigma}_\eta)$, where $f_k$ is the one-step model map, $\vect{x}_k = \vect{x}(t_k) \in \mathbb{R}^{D_x}$ and $\gmat{\Sigma}_\eta \in \mathbb{R}^{D_x \times D_x}$. The cost gains $\frac{1}{2} \sum_k \Vert \gvect{\eta}_k \Vert^2_{\gmat{\Sigma}_\eta^{-1}}$, and strong constraint is the limit $\gmat{\Sigma}_\eta \to \mat{0}$.

• The controls grow from one state to a whole trajectory, so $D_x$ values become $D_x (T+1)$. The analysis can now fit observations which the model alone could not reach.

• The gradient is still one adjoint sweep in the forcing formulation, whereas in the state formulation it becomes $T$ independent pieces, parallel in time.

• $\gmat{\Sigma}_\eta$ is the hard part. If it is too small, the model error leaks into the initial state; if it is too large, the analysis chases observation noise. A basis for the model error turns $\gmat{\Sigma}_\eta$ into a modelling choice.

• On Lorenz-96 with a biased model, strong constraint leaves a window error of 1.66. Weak constraint cuts it to 0.25, and a one-number bias estimate cuts it to 0.12 while recovering the bias itself (0.99, true 1.0).

On the page¶

1. Intuition

2. Where it comes from

3. The mathematics

4. The gradient

5. Formulation 1 · Model errors as controls (forcing)

6. Formulation 2 · States as controls (time-parallel)

7. Formulation 3 · Reduced model error

8. The three side by side

9. Application: the Gulf Stream window

10. Notes on vardax