A fast, single-iteration ensemble Kalman smoother for online, sequential data assimilation
Authors:
Colin Grudzien1, Marc Bocquet2
Special Thanks:
Eric Olson1 & Mihye Ahn1 for HPC logistics and support
- University of Nevada, Reno, Department of Mathematics and Statistics
- CEREA, A joint laboratory École des Ponts Paris Tech and EDF R&D, Université Paris-Est, Champs-sur-Marne, France.
The observation-analysis-forecast cycle
- Using Bayes' law, this can be written as \[ \begin{align} {\color{#d95f02} {p\left(\pmb{x}_{L} \vert \pmb{y}_{L:1}\right)} } &\propto {\color{#7570b3} { p\left(\pmb{y}_L \vert \pmb{x}_L\right) } } {\color{#1b9e77} { p\left(\pmb{x}_L\vert \pmb{y}_{1:L-1}\right) } } \end{align} \]
- When the models are linear, i.e., \[ \begin{align} \pmb{x}_k &= \mathbf{M}_k \pmb{x}_{k-1};\\ \pmb{y}_k &= \mathbf{H}_k \pmb{x}_k + \pmb{\epsilon}_k; \end{align} \]
- and the error densities are Gaussian, i.e., \[ \begin{align} p(\pmb{x}_0) &= N\left(\overline{\pmb{x}}_0, \mathbf{B}_0\right);\\ p\left(\pmb{y}_k \vert \pmb{x}_k \right) &= N\left(\mathbf{H}_k\pmb{x}_k, \mathbf{R}_k\right); \end{align} \]
- the posterior filtering density is Gaussian at all times, \[ {\color{#d95f02} {p\left(\pmb{x}_{L} \vert \pmb{y}_{L:1}\right)\equiv N\left(\overline{\pmb{x}}^\mathrm{filt}_k, \mathbf{B}_k^\mathrm{filt}\right)}} \] and can be computed analytically in a recursive formulation by the Kalman filter.
- The ensemble Kalman filter (EnKF) uses an empirical, ensemble-based forecast mean and covariance, along with the parametric update of the Kalman filter, to approximate the Bayesian posterior.
- Many different techniques have been developed to efficiently re-sample an ensemble from the posterior, using the above approximation.
Smoothing
- This recursively estimates the density of the current state, but a related question regards the past states.
- E.g., the observation at time \( t_3 \) can be used to update the conditional estimate for the model state at times \( t_0 \), \( t_1 \) and \( t_2 \).
- This produces a smoothing posterior estimate for the past states.
- The smoothing estimate conditions all past states within the data assimilation window (DAW), i.e., over the time series of all lagged states.
- A smoothing estimate may be produced in a variety of ways, exploiting different formulations of the Bayesian inference problem as a joint or marginal smoother.8
- The implementation also depends on whether the DAW is fixed in time, or if this window is shifted in time sequentially.
- For online, observation-analysis-forecast cycles, we will sequentially shift the DAW.
8. Cosme, E., Verron, J., Brasseur, P., Blum, J., & Auroux, D. (2012). Smoothing problems in a Bayesian framework and their linear Gaussian solutions. Monthly Weather Review, 140(2), 683-695.
Sequential Smoothing
- In the above, we see a schematic of how the DAW is sequentially shifted in time for a general fixed-lag smoother.
- Generally, we will consider a DAW of a lag length \( L \times \Delta t \), with a sequential shift of length \( S\times \Delta t \).
- After the conditional estimate is formed, we will shift the DAW by \( S \times \Delta t \);
- this will discard \( S \times \Delta t \) past states from the DAW, and \( S \times \Delta t \) new states and observations will enter the DAW.
- In the figure above, the \( S \times \Delta t \) new states and observations that enter the DAW are filled in black.
- The above fixed-lag formalism allows us to treat the smoothing problem like sequential filtering.
- This can come with large benefits to the precision of estimation by utilizing a full time series to form conditional estimates;
- however, it can also become extremely costly to produce conditional estimates this way when \( S \) is small and \( L \) is large.
The ETKS
- The ensemble transform Kalman smoother (ETKS) takes advantage of the simple form of the retrospective, right-transform analysis by including an additional, inner loop of the filtering cycle.
- In the above, time is the horizontal axis where right moves forward in time.
- At each time, we produce the standard filtering estimate by computing \( \boldsymbol{\Psi}_L \) from the cost function, and updating the forecast \[ \mathbf{E}^\mathrm{filt}_L = \mathbf{E}_L^\mathrm{fore} \boldsymbol{\Psi}_L. \]
- The information from incoming observations is passed backward in time using the right-transform to condition the ensemble at past times: \[ \begin{align} \mathbf{E}^\mathrm{post}_{k|L} = \mathbf{E}^\mathrm{post}_{k|L-1} \boldsymbol{\Psi}_L. \end{align} \]
The single-iteration ensemble transform Kalman smoother (SIETKS)
- Compared to the classic ETKS, this adds an outer-loop to the filtering cycle to produce the posterior, smoothing analysis.
- The information flows from the filtered state back in time from the retrospective analysis.
- This re-analyzed state becomes the initialization for the next cycle over the shifted DAW, carrying this information forward.
- The iterative cost function is only solved in the filtering estimate for the new observations entering the DAW.
- Combined with the retrospective analysis, this comes without the cost of iterating the model forecast over the DAW.
Numerical demonstrations – tuned inflation
Numerical demonstrations – adaptive inflation
Numerical demonstrations – tuned inflation
Numerical demonstrations – adaptive inflation
Numerical demonstrations – nonlinear observations (SDA)
Numerical demonstrations – nonlinear observations (MDA)
Numerical demonstrations – nonlinear observations (SDA)
Numerical demonstrations – nonlinear observations (MDA)
Numerical demonstrations – lag versus shift (SDA)
Numerical demonstrations – lag versus shift (MDA)
Numerical demonstrations – lag versus shift (SDA)
Numerical demonstrations – lag versus shift (SDA)
