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Filtering with One-Step-Ahead Smoothing for Efficient Data Assimilation

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Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (Vol. IV)

Abstract

State-space systems arise in many applications in signal processing and data assimilation. In this context, the main goal is to estimate online the state of the system based on available observations, the so-called filtering problem. Standard filtering solutions are computed recursively as successive cycles of alternating time-update (forecast) and observation-update (analysis) steps. This path is however not the only recursive way to compute the filtering solution. In this context, new one-step-ahead smoothing (OSAS)-like filtering schemes have been introduced, reversing the order of the observation-update and the time-update steps. These involve two Bayesian-like update steps based on the same (present) observation: one for smoothing the previous state and one for analyzing the present one. These include new variants of Kalman filters (KF-OSAS), particle filters (PF-OSAS) and ensemble Kalman filters (EnKF-OSAS), depending on the size and the linear-Gaussian character of the underlying state-space system. While the standard KF and KF-OSAS provide the same (exact) estimator, the use of the same data twice in the estimation process generally leads to improved trade-off between estimation quality and computational burden for the PF-OSAS and EnKF-OSAS. This chapter offers a comprehensive presentation of the OSAS-like filtering algorithms, reviewing their derivations, detailing algorithmic and practical differences and similarities with their classical counterparts, and discussing their relevance for both small- and large-dimensional applications.

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Notes

  1. 1.

    Without loss of generality, throughout the chapter PF refers to the particular bootstrap algorithm of Gordon et al. (1993), which is commonly used in geophysics problems. Sequential Importance Sampling (SIS) or Sampling Importance Resampling (SIR) algorithms will designate the other importance sampling (IS)-based algorithms.

  2. 2.

    Without loss of generality, the term \(\mathbf{g}(\mathbf{m})\) in any modeling equation \(\mathbf{d} = \mathbf{g}(\mathbf{m}) + { noise}\), will be called operator, whereas the whole equation will be called model, i.e., equations in (1) refer to as dynamical and observational models; ditto for those in (25).

  3. 3.

    Notice that (13) is simply an “averaging” of \(p(\mathbf{y}_n | \mathbf{x}_{n})\) w.r.t. \(p(\mathbf{x}_n | \mathbf{x}_{n-1})\) to obtain \(p(\mathbf{y}_n | \mathbf{x}_{n-1})\), i.e., \(p(\mathbf{y}_n | \mathbf{x}_{n-1}) {\mathop {=}\limits ^{(13)}} \mathbb {E}_{p(\mathbf{x}_n |\mathbf{x}_{n-1})} [p(\mathbf{y}_n | \mathbf{x}_n)]\).

  4. 4.

    Recall that the OSAS-like filter  reverts the order of the time-update and observation-update steps (i.e., step (12) then (14)), involving the use of the new likelihood before the new transition density. Thus, for the sake of consistency, we revert the conventional order of the models in the new system.

  5. 5.

    Notice that another OSAS-like KF has been introduced in Ait-El-Fquih and Hoteit (2015), but is left out in this chapter. Its difference with the KF-OSAS lies in the fact that (i) it does not follow from the generic filter (12)–(15), but from splitting the state vector using the variational Bayesian approach, and (ii) it only computes an approximation of the analysis and smoothing pdfs.

  6. 6.

    Without abuse of language, we use iid throughout the chapter even though this is true only asymptotically (in M). Indeed, with finite M, the particles are identically distributed (id) from the associated density, but not independent.

  7. 7.

    The term \(p(\mathbf{y}_n |\mathbf{x}_{n-1}^{a,m})\) in (42) is not a probability density function, but the value of this function at point \(\mathbf{y}_n\) (i.e., the observed data).

  8. 8.

    Steps W and S commute in SIR-CID since the weights do not depend on the new particles \(\mathbf{z}_n^{m}\).

  9. 9.

    It follows from (54) that \(\mathbf{y}_n - \mathbf{y}_{n}^{f,m} = \mathbf{y}_n - \tilde{\mathbf{v}}_n^m - \tilde{\mathbf{h}}_{n-1} (\mathbf{x}_{n-1}^{a,m})\), suggesting that the Kalman correction in (53) is based on observations perturbed with \(-\tilde{\mathbf{v}}_n^{m} \sim \mathcal{N}(\mathbf{0}, \tilde{\mathbf{R}}_n)\). Similarly to the classical EnKF, perturbing the observations for each ensemble member enables matching with the error statistics in the KF-OSAS smoothing step when the state-space system is linear-Gaussian.

  10. 10.

    Note that members \({\boldsymbol{\upsilon }}_{n}^m\) correspond to (28) and their covariance, \(\mathbf{P}_{{\boldsymbol{\upsilon }}_n}\), is an ensemble-based approximation of the form (29) of the innovation matrix \(\mathbf{V}_n\). One could also perturb the observations by \(\mathbf{v}_n^{m}\) instead of \(\tilde{\mathbf{v}}_n^{m}\) (i.e., \(\mathbf{y}_{n}^{m}\) become \(\mathbf{y}_{n}^{m} = \mathbf{y}_n - \mathbf{v}_n^{m}\)); in this case the members \({\boldsymbol{\upsilon }}_{n}^m\) become \({\boldsymbol{\upsilon }}_{n}^m = \mathbf{y}_{n}^m - \mathbf{H}_n \mathbf{x}_{n}^{f,m}\) (which correspond to the form (33) of the innovation) and their covariance, \(\mathbf{P}_{{\boldsymbol{\upsilon }}_n}\), corresponds to the form (34) of \(\mathbf{V}_n\).

  11. 11.

    This assumption was forgotten in Ait-El-Fquih et al. (2016), Raboudi et al. (2018).

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  1. King Abdullah University of Science and Technology (KAUST), PSE, Thuwal, Saudi Arabia

    Boujemaa Ait-El-Fquih & Ibrahim Hoteit

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  1. Boujemaa Ait-El-Fquih
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  1. Climate and Energy Systems Engineering, Ewha Womans University, Seoul, Korea (Republic of)

    Seon Ki Park

  2. Marine Meteorology Division, Naval Research Laboratory, Monterey, CA, USA

    Liang Xu

Appendix

Appendix

Property 1

(Hierarchical sampling Ait-El-Fquih et al. 2016) Assuming that one can sample from \(p(\mathbf{x}_1)\) and \(p(\mathbf{x}_{2}|\mathbf{x}_{1})\), then a sample, \(\mathbf{x}_2^{*}\), from \(p(\mathbf{x}_2)\) can be drawn as follows:

  1. 1.

    \(\mathbf{x}_1^{*} \sim p(\mathbf{x}_1)\);

  2. 2.

    \(\mathbf{x}_2^{*} \sim p(\mathbf{x}_2|\mathbf{x}_1^{*})\).

Property 2

(Conditional sampling Ait-El-Fquih et al. 2016) Consider a Gaussian pdf, \(p(\mathbf{x} , \mathbf{y})\), with \(\mathbf{P}_{xy}\) and \(\mathbf{P}_{y}\) denoting the cross-covariance of \(\mathbf{x}\) and \(\mathbf{y}\) and the covariance of \(\mathbf{y}\), respectively. Then a sample, \(\mathbf{x}^{*}\), from \(p(\mathbf{x}|\mathbf{y})\), can be drawn as follows:

  1. 1.

    \((\tilde{\mathbf{x}} , \tilde{\mathbf{y}}) \sim p(\mathbf{x} , \mathbf{y})\);

  2. 2.

    \(\mathbf{x}^{*} = \tilde{\mathbf{x}} + \mathbf{P}_{xy} \mathbf{P}_y^{-1} [\mathbf{y} - \tilde{\mathbf{y}}]\).

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Ait-El-Fquih, B., Hoteit, I. (2022). Filtering with One-Step-Ahead Smoothing for Efficient Data Assimilation. In: Park, S.K., Xu, L. (eds) Data Assimilation for Atmospheric, Oceanic and Hydrologic Applications (Vol. IV). Springer, Cham. https://doi.org/10.1007/978-3-030-77722-7_3

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