The Norm Dependence of Singular Vectors.

Kuang, Z.

J. Atmos. Sci., accepted.

Abstract

For a linearized system such as t   =   È  È , singular vector analysis can be used to find patterns that give the largest or smallest ratios between the sizes of MÈ and È. Such analyses have applications to a wide range of atmosphere-ocean problems. The resulting singular vectors, however, depend on the norm used to measure the sizes of MÈ and È, as noted in various applications. This causes complications because choices of norm are generally non-unique. Based on perturbation theory, I provide a general derivation of how singular vectors change with norm, and show that the norm dependences observed in previous studies can be understood as general properties of singular vectors. This is hoped to clarify the interpretations on these observed norm dependencies, and provide guidance to new studies on how singular vectors would vary for different norms. I further argue, based on these results, that there may not be as much norm-related ambiguity in problems such as designing targeted observations or ensemble forecasts as is often assigned to them.
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Zhiming Kuang <kzm@atmos.washington.edu>