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The Maximum Likelihood Ensemble Filter as a non-differentiable minimization algorithm

  1. Milija Zupanski1,*,
  2. I. Michael Navon2,
  3. Dusanka Zupanski1

Article first published online: 24 JUN 2008

DOI: 10.1002/qj.251

Issue

Quarterly Journal of the Royal Meteorological Society

Quarterly Journal of the Royal Meteorological Society

Volume 134, Issue 633, pages 1039–1050, April 2008 Part B

Additional Information

How to Cite

Zupanski, M., Navon, I. M. and Zupanski, D. (2008), The Maximum Likelihood Ensemble Filter as a non-differentiable minimization algorithm. Quarterly Journal of the Royal Meteorological Society, 134: 1039–1050. doi: 10.1002/qj.251

Author Information

  1. 1

    Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, USA

  2. 2

    School of Computational Science and Department of Mathematics, Florida State University, Tallahassee, USA

*Cooperative Institute for Research in the Atmosphere, Colorado State University, 1375 Campus Delivery, Fort Collins, CO 80523-1375, USA.

Publication History

  1. Issue published online: 24 JUN 2008
  2. Article first published online: 24 JUN 2008
  3. Manuscript Accepted: 25 MAR 2008
  4. Manuscript Revised: 24 JAN 2008
  5. Manuscript Received: 25 SEP 2007

Funded by

  • National Science Foundation Collaboration in Mathematical Geosciences. Grant Numbers: ATM–0327651, ATM–0327818
  • NASA Precipitation Measurement Mission Program. Grant Number: NNX07AD75G

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Keywords:

  • unconstrained minimization;
  • ensemble data assimilation

Abstract

The Maximum Likelihood Ensemble Filter (MLEF) equations are derived without the differentiability requirement for the prediction model and for the observation operators. The derivation reveals that a new non-differentiable minimization method can be defined as a generalization of the gradient-based unconstrained methods, such as the preconditioned conjugate-gradient and quasi-Newton methods. In the new minimization algorithm the vector of first-order increments of the cost function is defined as a generalized gradient, while the symmetric matrix of second-order increments of the cost function is defined as a generalized Hessian matrix. In the case of differentiable observation operators, the minimization algorithm reduces to the standard gradient-based form.

The non-differentiable aspect of the MLEF algorithm is illustrated in an example with one-dimensional Burgers model and simulated observations. The MLEF algorithm has a robust performance, producing satisfactory results for tested non-differentiable observation operators. Copyright © 2008 Royal Meteorological Society

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