The Norm Dependence of Singular Vectors

For a linearized system such as ]c/]t 5 Mc, singular vector analysis can be used to find patterns that give the largest or smallest ratios between the sizes of Mc and c. 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 Mc and c, as noted in various applications. This causes complications because the choices of norm are generally nonunique. Based on perturbation theory, a derivation of how singular vectors change with norms typically used in the atmosphere–ocean literature is provided, and it is shown that the norm dependences observed in previous studies can be understood as general properties of singular vectors. This will hopefully clarify the interpretation of these observed norm dependencies, and provide guidance to new studies on how singular vectors would vary for different norms. It is further argued, based on these results, that there may not be as much normrelated ambiguity in problems, such as designing targeted observations or ensemble forecasts, as is often assigned to them.