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Timothy DelSole and Michael K. Tippett

norm. For instance, principal component analysis determines components that maximize variance, as defined by some norm. Singular vector decomposition depends on two norms: one for measuring “response” and another for constraining “initial condition.” Without a firm basis for choosing these norms, variance analysis could generate virtually any set of vectors by a suitable choice of norm. One approach to these problems is to define predictability precisely and then to choose norms to ensure

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Linus Magnusson, Martin Leutbecher, and Erland Källén

most of the probable range of developments in the atmosphere; another is to reflect the uncertainties in the analysis. Various perturbation methods focus on these different features. The European Centre for Medium-Range Weather Forecasts (ECMWF) uses the singular vector (SV) method to achieve maximum perturbation growth rate for a given optimization time ( Lorenz 1965 ; Palmer 1993 ). The National Centers of Environmental Prediction (NCEP) uses the Ensemble Transform (ET) technique ( Wei et al

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Hyun Mee Kim and Byoung-Joo Jung

1. Introduction Adaptive (or targeted) observation strategies have been applied to high-impact weather events to identify regions where additional observations have the potential to significantly improve weather forecasts. These regions may be considered “sensitive” in the sense that changes to the initial conditions in these regions are expected to have a larger effect on a particular measure of forecast skill than changes in other regions ( Kim et al. 2004 ). Singular vectors (SVs) are the

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Christopher L. Wolfe and Roger M. Samelson

development of disturbances is typically quantified using singular values and their associated singular vectors (SVs). Singular vectors are disturbances that produce the greatest linear growth in a specified inner product over a specified optimization time interval ( Lorenz 1965 ; Farrell 1989 ). There is considerable arbitrariness in the choice of an inner product and optimization interval, which can make their physical interpretation difficult. The asymptotic stability of trajectories on aperiodic

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Xiaobing Zhou, Youmin Tang, Yanjie Cheng, and Ziwang Deng

1. Introduction As an optimal perturbation method, singular vectors (SVs) have been widely applied in weather forecast and ENSO prediction, including the error-growth estimation and atmospheric predictability studies, the adaptive observation strategy, as well the construction of the initial fields for ensemble forecast (e.g., Lorenz 1965 ; Mureau et al. 1993 ; Penland and Sardeshmukh 1995 ; Xue et al. 1997a , b ; Moore and Kleeman 1996 , 2001 ; Moore et al. 2003 ; Fan et al. 2000

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Hyun Mee Kim and Byoung-Joo Jung

identify sensitive regions are adjoint sensitivity and singular vectors (SVs). The adjoint sensitivity is the gradient of some forecast measure with respect to the model control variables (e.g., Errico 1997 ) or to the observations ( Baker and Daley 2000 ), and has been used to detect sensitive regions for adaptive observations (e.g., Bergot 1999 ; Bergot et al. 1999 ; Pu and Kalnay 1999 ; Kim and Jung 2006 ; Wu et al. 2007 , 2009 ). Singular vectors are the fastest growing perturbations during

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Ed Hawkins and Rowan Sutton

observations (e.g., Palmer et al. 1998 ). Several techniques exist for generating such optimal perturbations, such as singular vectors (e.g., Buizza and Palmer 1995 ) or breeding vectors (e.g., Toth and Kalnay 1997 ). However, in NWP it is the rapidly growing weather modes that are important, but on longer (seasonal to decadal) time scales, it is necessary to remove any effects of the rapidly growing weather perturbations and instead focus on the fastest-growing climate perturbations. Several methods

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Olivier Rivière, Guillaume Lapeyre, and Olivier Talagrand

normal mode (NM) approach, that is, linearizing a model about a mean state and finding a solution asymptotically growing in time. Such a method presents the disadvantage that it fails to capture localized disturbances that can have a rapid growth over a limited period in time ( Farrell 1982 ). A second approach consists of identifying “optimal perturbations” [called singular vectors (SVs)] that maximize the growth rate over a given time interval ( Farrell 1982 ; Lacarra and Talagrand 1988 ; Farrell

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Xiaoli Li, Martin Charron, Lubos Spacek, and Guillem Candille

and different in-house analyses, and using LBCs provided by the National Centers for Environmental Prediction (NCEP) global ensemble prediction system (EPS) based on the breeding vector approach. Since 2003, SREF has been updated and uses three different models ( Du et al. 2003 ). Some limited-area EPSs based on global singular vector (SV) perturbations for ICs and LBCs obtained from the European Centre for Medium-Range Weather Forecasts (ECMWF) EPS have been developed ( Marsigli et al. 2001

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Hyun Mee Kim, Sung-Min Kim, and Byoung-Joo Jung

observation guidance). Various organizations have developed real-time adaptive observation guidance that suggests possible target regions for adaptive observations to improve tropical cyclone track forecasts ( Kim et al. 2008 ). The real-time guidance employed for T-PARC includes total energy singular vector (TESV; Peng and Reynolds 2006 ; Reynolds et al. 2010 ) and adjoint sensitivity ( Amerault and Doyle 2009 ; Reynolds et al. 2010 ) guidance, both used by the Naval Research Laboratory (NRL); TESV

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