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On the Orthogonality of Bred Vectors

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  • 1 Frontier Research System for Global Change, Yokohama, Kanagawa, Japan, and Proudman Oceanographic Laboratory, Liverpool, United Kingdom
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Abstract

The “breeding method” is a well-established and computationally inexpensive method for generating perturbations for ensemble integrations. One feature of bred vectors is that, although arbitrary initial perturbations generally converge toward the fastest growing directions, distinct bred vectors for global numerical weather prediction (NWP) models are generally quasi-orthogonal. However, an examination of the local structure of the vectors indicates that this conclusion may be somewhat misleading since the perturbations frequently have very similar shapes over local areas (consistent with the low local dimension of atmospheric physics). Therefore, there may be substantial redundancy when multiple independent breeding cycles are performed in parallel, and the vectors can be inefficient in spanning the range of locally growing perturbations. Experiments with the 3-variable Lorenz63 model indicate that orthogonalizing the bred vectors can result in significantly improved performance. A simple local orthogonalization technique has been applied to the spatially resolved 40-variable Lorenz96 model with encouraging results. It is suggested that a similar treatment to bred vectors might be helpful for global NWP or other applications.

Corresponding author address: J. D. Annan, Frontier Research System for Global Change, 3173-25 Showa-machi Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan. Email: jdannan@jamstec.go.jp

Abstract

The “breeding method” is a well-established and computationally inexpensive method for generating perturbations for ensemble integrations. One feature of bred vectors is that, although arbitrary initial perturbations generally converge toward the fastest growing directions, distinct bred vectors for global numerical weather prediction (NWP) models are generally quasi-orthogonal. However, an examination of the local structure of the vectors indicates that this conclusion may be somewhat misleading since the perturbations frequently have very similar shapes over local areas (consistent with the low local dimension of atmospheric physics). Therefore, there may be substantial redundancy when multiple independent breeding cycles are performed in parallel, and the vectors can be inefficient in spanning the range of locally growing perturbations. Experiments with the 3-variable Lorenz63 model indicate that orthogonalizing the bred vectors can result in significantly improved performance. A simple local orthogonalization technique has been applied to the spatially resolved 40-variable Lorenz96 model with encouraging results. It is suggested that a similar treatment to bred vectors might be helpful for global NWP or other applications.

Corresponding author address: J. D. Annan, Frontier Research System for Global Change, 3173-25 Showa-machi Kanazawa-ku, Yokohama, Kanagawa 236-0001, Japan. Email: jdannan@jamstec.go.jp

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