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Filtering Partially Observed Multiscale Systems with Heterogeneous Multiscale Methods–Based Reduced Climate Models

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  • 1 Statistical and Applied Mathematical Sciences Institute, Research Triangle Park, North Carolina
  • | 2 Department of Mathematics, North Carolina State University, Raleigh, North Carolina
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Abstract

This paper presents a fast reduced filtering strategy for assimilating multiscale systems in the presence of observations of only the macroscopic (or large scale) variables. This reduced filtering strategy introduces model errors in estimating the prior forecast statistics through the (heterogeneous multiscale methods) HMM-based reduced climate model as an alternative to the standard expensive (direct numerical simulation) DNS-based fully resolved model. More importantly, this approach is not restricted to any analysis (or Bayesian updating) step from various ensemble-based filters. In a regime where there is a distinctive separation of scales, high filtering skill is obtained through applying the HMM alone with any desirable analysis step from ensemble Kalman filters. When separation of scales is not apparent as typically observed in geophysical turbulent systems, an additional procedure is proposed to reinitialize the microscopic variables to statistically reflect pseudo-observations that are constructed based on the unbiased estimates of the macroscopic variables. Specifically, these pseudo-observations are constructed offline from the conditional distributions of the microscopic forcing to the macroscopic dynamics given the macroscopic variables with the method-of-moments estimator. This HMM-based filter is comparable to the more expensive standard DNS-based filter on a stringent test bed, the two-layer Lorenz’96 model, in various regimes of scale gap, including the not so apparent one. This high filtering skill is robust in the presence of additional model errors through inconsistent pseudo-observations and even when macroscopic observations are spatially incomplete.

Current affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio.

Corresponding author address: Emily L. Kang, Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH 45221-0025. E-mail: kangel@ucmail.uc.edu

Abstract

This paper presents a fast reduced filtering strategy for assimilating multiscale systems in the presence of observations of only the macroscopic (or large scale) variables. This reduced filtering strategy introduces model errors in estimating the prior forecast statistics through the (heterogeneous multiscale methods) HMM-based reduced climate model as an alternative to the standard expensive (direct numerical simulation) DNS-based fully resolved model. More importantly, this approach is not restricted to any analysis (or Bayesian updating) step from various ensemble-based filters. In a regime where there is a distinctive separation of scales, high filtering skill is obtained through applying the HMM alone with any desirable analysis step from ensemble Kalman filters. When separation of scales is not apparent as typically observed in geophysical turbulent systems, an additional procedure is proposed to reinitialize the microscopic variables to statistically reflect pseudo-observations that are constructed based on the unbiased estimates of the macroscopic variables. Specifically, these pseudo-observations are constructed offline from the conditional distributions of the microscopic forcing to the macroscopic dynamics given the macroscopic variables with the method-of-moments estimator. This HMM-based filter is comparable to the more expensive standard DNS-based filter on a stringent test bed, the two-layer Lorenz’96 model, in various regimes of scale gap, including the not so apparent one. This high filtering skill is robust in the presence of additional model errors through inconsistent pseudo-observations and even when macroscopic observations are spatially incomplete.

Current affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio.

Corresponding author address: Emily L. Kang, Department of Mathematical Sciences, University of Cincinnati, 2815 Commons Way, Cincinnati, OH 45221-0025. E-mail: kangel@ucmail.uc.edu
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