Abstract

Ensemble-variational methods form the basis of the state-of-the-art for nonlinear, scalable data assimilation, yet current designs may not be cost-effective for reducing prediction error in online, short-range forecast systems. We propose a novel, outer-loop optimization of the ensemble-variational formalism for applications in which forecast error dynamics are weakly nonlinear, such as synoptic meteorology. In order to rigorously derive our method and demonstrate its novelty, we review ensemble smoothers that appear throughout the literature in a unified Bayesian maximum-a-posteriori narrative, updating and simplifying some results. After mathematically deriving our technique, we systematically develop and inter-compare all studied schemes in the open-source Julia package DataAssimilationBenchmarks.jl, with pseudo-code provided for these methods. This high-performance numerical framework, supporting our mathematical results, produces extensive benchmarks that demonstrate the significant performance advantages of our proposed technique. In particular, our single-iteration ensemble Kalman smoother is shown both to improve prediction / posterior accuracy and to simultaneously reduce the leading order cost of iterative, sequential smoothers in a variety of relevant test cases for operational short-range forecasts. This long work is thus intended to present our novel single-iteration ensemble Kalman smoother, and to provide a theoretical and computational framework for the study of sequential, ensemble-variational Kalman filters and smoothers generally.

Highlights

  • Ensemble-variational methods form the basis of the state-of-the-art for nonlinear, scalable data assimilation (DA) (Asch et al, 2016; Bannister, 2017)

  • The single-iteration ensemble Kalman smoother (SIEnKS) demonstrates significantly improved smoother accuracy over the Lin-iterative ensemble Kalman smoother (IEnKS) while remaining at a lower leading order cost. This suggests that the sequential multiple data assimilation (MDA) scheme of the SIEnKS is better equipped to handle highly nonlinear observation operators than the 4D-maxiumum a posteriori (MAP) formalism, which appears to suffer from a greater number of local minima

  • We provide a detailed review of the state-of-the-art for sequential, ensemble-variational Kalman filters and smoothers in perfect models within the Bayesian MAP formalism of the IEnKS

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Summary

Introduction

Ensemble-variational methods form the basis of the state-of-the-art for nonlinear, scalable data assimilation (DA) (Asch et al, 2016; Bannister, 2017). When 40 the linear-Gaussian approximation for the forecast error dynamics is adequate, nonlinearity in the DA cycle may instead by dominated by nonlinearity in the observation operator, nonlinearity in hyper-parameter optimization, and / or nonlinearity in temporally interpolating a re-analyzed, smoothed solution over the DAW In this setting, our novel formulation of iterative, ensemble-variational smoothing has substantial advantages in balancing the computational cost / prediction accuracy trade off for these estimators

45 1.2 Objectives and outline
Notations
Deriving the SIEnKS
The ETKF
The fixed-lag EnKS
The Gauss-Newton, fixed-lag IEnKS
Algorithm
Nonlinear observation operators
Adaptive inflation and the finite-size formalism
Multiple data assimilation
Asynchronous data assimilation
Algorithm cost analysis
Weakly nonlinear forecast error dynamics – linear observations
91 ETKS 7832 5654 3476 1298 110
91 EnKS-N SIEnKS-N Lin-IEnKS-N IEnKS-N
Weakly nonlinear forecast error dynamics – nonlinear observations
82 MLES 73 64 55 46 37 28 19 10 1
Strongly nonlinear forecast error dynamics – lag versus ∆t
Conclusion
52 ETKS 46
52 EnKS-N 46
Findings
1: Require
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