Singular vectors, metrics, and adaptive observations

Singular vectors of the linearized equations of motion have been used to study the instability properties of the atmosphere-ocean system and its related predictability. A third use of these singular vectors is proposed here: as part of a strategy to target adaptive observations to 'sensitive' parts of the atmosphere. Such observations could be made using unmanned aircraft, though calculations in this paper are motivated by the upstream component of the Fronts and Atlantic Storm-Track Experiment. Oceanic applications are also discussed. In defining this strategy, it is shown that there is, in principle, no freedom in the choice of inner product or metric for the singular vector calculation. However, the correct metric is dependent on the purpose for making the targeted observations (to study precursor developments or to improve forecast initial conditions). It is argued that for predictability studies, where both the dynamical instability properties of the system and the specification of the operational observing network and associated data assimilation system are important, the appropriate metric will differ from that appropriate to a pure geophysical fluid dynamics (GFD) problem. based on two different sets of calculations. it is argued that for predictability studies (but not for GFD studies), a first-order approximation to the appropriate metric can be based on perturbation energy. The role of observations in data assimilation procedures (constraining large scales more than small scales) is fundamental in understanding reasons for the requirement for different metrics for the two classes of problems. An index-based tensor approach is used to make explicit the role of the metric. The strategy for using singular vectors to target adaptive observations is discussed in the context of other possible approaches, specifically, based on breeding vectors, potential vorticity diagnosis, and sensitivity vectors. The basic premises underlying the use of breeding and singular vectors are discussed. A comparison of the growth rates of breeding and singular vectors is made using a T21 quasigeostrophic model. Singular vectors and subjective potential vorticity (PV) diagnosis are compared for a particular case study. The areas of sensitivity indicated by the two methods only partially agree. Reasons for disagreement hinge around the fact that subjective PV diagnosis emphasizes Lagrangian advection, whereas singular vector analysis emphasizes wave propagation. For the latter, areas of sensitivity may be associated with regions of weak PV gradient, for example, mid to lower troposphere. Amplification of singular vectors propagating from regions of weak PV gradient to regions of strong PV gradient is discussed in terms of pseudomomentum conservation. Evidence is shown that analysis error may be as large in the lower midtroposphere as in the upper troposphere.