Abstract
Abstract. The basis and challenge of strongly coupled data assimilation (CDA) is the accurate representation of cross-domain covariances between various coupled subsystems with disparate spatio-temporal scales, where often one or more subsystems are unobserved. In this study, we explore strong CDA using ensemble Kalman filtering methods applied to a conceptual multiscale chaotic model consisting of three coupled Lorenz attractors. We introduce the use of the local attractor dimension (i.e. the Kaplan–Yorke dimension, dimKY) to prescribe the rank of the background covariance matrix which we construct using a variable number of weighted covariant Lyapunov vectors (CLVs). Specifically, we consider the ability to track the nonlinear trajectory of each of the subsystems with different variants of sparse observations, relying only on the cross-domain covariance to determine an accurate analysis for tracking the trajectory of the unobserved subdomain. We find that spanning the global unstable and neutral subspaces is not sufficient at times where the nonlinear dynamics and intermittent linear error growth along a stable direction combine. At such times a subset of the local stable subspace is also needed to be represented in the ensemble. In this regard the local dimKY provides an accurate estimate of the required rank. Additionally, we show that spanning the full space does not improve performance significantly relative to spanning only the subspace determined by the local dimension. Where weak coupling between subsystems leads to covariance collapse in one or more of the unobserved subsystems, we apply a novel modified Kalman gain where the background covariances are scaled by their Frobenius norm. This modified gain increases the magnitude of the innovations and the effective dimension of the unobserved domains relative to the strength of the coupling and timescale separation. We conclude with a discussion on the implications for higher-dimensional systems.
Highlights
As the world of climate modelling has moved towards coupled Earth system models which combine ocean, atmosphere, sea-ice, and biogeochemical effects, it is essential to understand how the respective domains of disparate spatiotemporal scales covary and influence each other
This study presents an initial understanding of the transient dynamics associated with the Kalman filter forecast error covariance matrix for nonlinear multiscale coupled systems
We have explored the varying rank of the error covariance matrix related to the transient growth in the stable modes of the system, and in particular the applicability of this varying rank to different configurations of strong coupled data assimilation (CDA)
Summary
As the world of climate modelling has moved towards coupled Earth system models which combine ocean, atmosphere, sea-ice, and biogeochemical effects, it is essential to understand how the respective domains of disparate spatiotemporal scales covary and influence each other. In the context of state estimation, strongly coupled data assimilation (CDA) in multiscale systems allows the quantification of cross-domain dynamics. Strong CDA uses the cross-covariances amongst all components to influence the analysis increment, meaning that unobserved subsystems are directly adjusted in the analysis step regardless of observation set and coupling strengths (Laloyaux et al, 2016; O’Kane et al, 2019). Ensemble DA methods rely on properly sampling the variance of the model, implying that very large ensemble sizes are needed for high-dimensional systems. In practical implementation this is often not possible due to computational costs and limitations. It is necessary to investigate and develop new methods of accurately representing error growth in multiscale systems
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