The Norm Dependence of Singular Vectors

2004 ◽  
Vol 61 (23) ◽  
pp. 2943-2949 ◽  
Author(s):  
Zhiming Kuang
Keyword(s):  
Perturbation Theory ◽  
Singular Vector ◽  
Vector Analysis ◽  
Ensemble Forecasts ◽  
Singular Vectors ◽  
General Properties ◽  
Wide Range ◽  
Linearized System ◽  
Targeted Observations

Abstract For a linearized system such as ∂ψ/∂t = 𝗠ψ, singular vector analysis can be used to find patterns that give the largest or smallest ratios between the sizes of 𝗠ψ and ψ. 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 𝗠ψ and ψ, 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 norm-related ambiguity in problems, such as designing targeted observations or ensemble forecasts, as is often assigned to them.

On Ensemble Prediction Using Singular Vectors Started from Forecasts

2005 ◽  
Vol 133 (10) ◽  
pp. 3038-3046 ◽  
Author(s):  
Martin Leutbecher
Keyword(s):  
Singular Vector ◽  
Ensemble Prediction ◽  
Prediction System ◽  
Ensemble Forecasts ◽  
Ensemble Prediction System ◽  
Singular Vectors ◽  
The Tropics ◽  
The Impact ◽  
Very High ◽  
Tropical Cyclone Tracks

Abstract The impact on the ECMWF Ensemble Prediction System of using singular vectors computed from 12-h forecasts instead of analyses has been studied. Results are based on 34 cases in November–December 1999 and 28 cases in September 2003. The similarity between singular vectors started from a 12-h forecast and singular vectors started from an analysis is very high for the extratropical singular vectors in each of the 62 cases and for both hemispheres. Consistently, ensemble scores and spread measures show close to neutral impact on geopotential height in the extratropics. The sensitivity of the singular vectors to the choice of trajectory is larger in the Tropics than in the extratropics. However, the spread in tropical cyclone tracks is not significantly decreased if singular vectors computed from 12-h forecasts are used. The computation of singular vectors from forecasts could be used to disseminate the ensemble forecasts earlier or to allocate more resources to the nonlinear forecasts. Furthermore, it greatly facilitates the implementation of computationally more demanding configurations for the singular-vector-based initial perturbations.

scholarly journals Ensemble Forecasts and the Properties of Flow-Dependent Analysis-Error Covariance Singular Vectors

2003 ◽  
Vol 131 (8) ◽  
pp. 1741-1758 ◽  
Author(s):  
Thomas M. Hamill ◽  
Chris Snyder ◽  
Jeffrey S. Whitaker
Keyword(s):  
Total Energy ◽  
Linear Combination ◽  
Singular Vector ◽  
Energy Norm ◽  
Initial Time ◽  
Forecast Errors ◽  
Ensemble Forecasts ◽  
Singular Vectors ◽  
Error Covariance ◽  
Analysis Error

Abstract Approximations to flow-dependent analysis-error covariance singular vectors (AEC SVs) were calculated in a dry, T31 L15 primitive-equation global model. Sets of 400-member ensembles of analyses were generated by an ensemble-based data assimilation system. A sparse network of simulated rawinsonde observations were assimilated, and a perfect model was assumed. Ensembles of 48-h forecasts were also generated from these analyses. The structure of evolved singular vectors was determined by finding the linear combination of the forecast ensemble members that resulted in the largest forecast-error variance, here measured in a total-energy norm north of 20°N latitude. The same linear combination of analyses specifies the initial-time structure that should evolve to the forecast singular vector under assumptions of linearity of error growth. The structures of these AEC SVs are important because they represent the analysis-error structures associated with the largest forecast errors. If singular vectors using other initial norms have very different structures, this indicates that these structures may be statistically unlikely to occur. The European Centre for Medium-Range Weather Forecasts currently uses singular vectors using an initial total-energy norm [“total-energy singular vectors” or (TE SVs)] to generate perturbations to initialize their ensemble forecasts. Approximate TE SVs were also calculated by drawing an initial random ensemble with perturbations that were white in total energy and applying the same approach as for AEC SVs. Comparing AEC SVs and approximate TE SVs, the AEC SVs had maximum amplitude in midlatitudes near the tropopause, both at the initial and evolved times. The AEC SVs were synoptic in scale, deep, and did not appear to be geographically localized nor tilted dramatically upshear. This contrasts with TE SVs, which started off relatively smaller in scale, were tilted upshear, and had amplitudes typically largest in the lower to midtroposphere. The difference between AEC SVs and TE SVs suggests that operational ensemble forecasts based on TE SVs could be improved by changing the type of singular vector used to generate initial perturbations. This is particularly true for short-range ensemble forecasts, where the structure of the forecast ensemble is more closely tied to the analysis ensemble.

Singular vectors and the predictability of weather and climate

1994 ◽  
Vol 348 (1688) ◽  
pp. 459-475 ◽  
Keyword(s):  
Chaotic System ◽  
Singular Vector ◽  
Weather Prediction ◽  
Dynamical Evolution ◽  
Singular Values ◽  
Numerical Weather Prediction Model ◽  
Ocean Dynamics ◽  
Vector Analysis ◽  
Lanczos Algorithm ◽  
Singular Vectors

The local instability properties of a chaotic system are determined by the singular vectors and singular values of the dynamical evolution operator, linearized about a finite trajectory portion of the integral curves of the nonlinear equations. Knowledge of these quantities allows an assessment of the reliability of a finite-time forecast from a chaotic system. After a brief study of the Lorenz model, singular vector analysis is applied to study three predictability issues in atmosphere-ocean dynamics. The first concerns the predictability of weather forecasts of a few days, and singular vector calculations are made from a large-dimensional numerical weather prediction model using an iterative Lanczos algorithm. The second concerns the predictability of El Niño on seasonal to interannual timescales. Here singular vector calculations are made using a coupled ocean-atmosphere model of the tropical Pacific region. Finally we show results from a multi-decadal integration of a medium-resolution quasi-geostrophic model, and discuss the possible relevance of singular vector analysis for the problem of climate change.

scholarly journals Singular vector decomposition for sensitivity analyses of tropospheric chemical scenarios

2011 ◽  
Vol 11 (6) ◽  
pp. 16745-16799 ◽  
Author(s):  
N. Goris ◽  
H. Elbern
Keyword(s):  
Singular Vector ◽  
Weather Prediction ◽  
Emission Factors ◽  
The State ◽  
Vector Analysis ◽  
Sensitivity Analyses ◽  
Initial Time ◽  
State Estimate ◽  
Singular Vectors ◽  
Initial Values

Abstract. Observations of the chemical state of the atmosphere typically provide only sparse snapshots of the state of the system due to their insufficient temporal and spatial density. Therefore the measurement configurations need to be optimised to get a best possible state estimate. One possibility to optimise the state estimate is provided by observation targeting of sensitive system states, to identify measurement configurations of best value for forecast improvements. In recent years, numerical weather prediction adapted singular vector analysis with respect to initial values as a novel method to identify sensitive states. In the present work, this technique is transferred from meteorological to chemical forecast. Besides initial values, emissions are investigated as controlling variables. More precisely uncertainties in the amplitude of the diurnal profile of emissions are analysed, yielding emission factors as target variables. Singular vector analysis is extended to allow for projected target variables not only at final time but also at initial time. Further, special operators are introduced, which consider the combined influence of groups of chemical species. As a preparation for targeted observation calculations, the concept of adaptive observations is studied with a chemistry box model. For a set of six different scenarios, the VOC versus NOx limitation of the ozone formation is investigated. Results reveal, that the singular vectors are strongly dependent on start time and length of the simulation. As expected, singular vectors with initial values as target variables tend to be more sensitive to initial values, while emission factors as target variables are more sensitive to simulation length. Further, the particular importance of chemical compounds differs strongly between absolute and relative error growth.

Singular Vector Analysis for Atmospheric Chemical Transport Models

2006 ◽  
Vol 134 (9) ◽  
pp. 2443-2465 ◽  
Author(s):  
Wenyuan Liao ◽  
Adrian Sandu ◽  
Gregory R. Carmichael ◽  
Tianfeng Chai
Keyword(s):  
Singular Vector ◽  
Transport Model ◽  
Adjoint Operator ◽  
Chemical Transport ◽  
Vector Analysis ◽  
Optimal Placement ◽  
Time Interval ◽  
Transport Models ◽  
Singular Vectors ◽  
Chemical Transport Models

Abstract The singular vectors of a chemical transport model are the directions of maximum perturbation growth over a finite time interval. They have proved useful for the estimation of error growth, the initialization of ensemble forecasts, and the optimal placement of adaptive observations. The aim of this paper is to address computational aspects of singular vector analysis for atmospheric chemical transport models. The distinguishing feature of these models is the presence of stiff chemical interactions. A projection approach to preserve the symmetry of the tangent linear–adjoint operator for stiff systems is discussed, and extended to 3D chemical transport simulations. Numerical results are presented for a simulation of atmospheric pollution in East Asia in March 2001. The singular values and the structure of the singular vectors depend on the length of the simulation interval, the meteorological data, the location of the optimization region and the selection of optimization species, the choice of error norms, and the size of the optimization region.

scholarly journals Forest-Based and Semiparametric Methods for the Postprocessing of Rainfall Ensemble Forecasting

2019 ◽  
Vol 34 (3) ◽  
pp. 617-634 ◽  
Author(s):  
Maxime Taillardat ◽  
Anne-Laure Fougères ◽  
Philippe Naveau ◽  
Olivier Mestre
Keyword(s):  
Heavy Rainfall ◽  
Hybrid Methods ◽  
Ensemble Prediction ◽  
Ensemble Forecasts ◽  
Ensemble Prediction System ◽  
Wide Range ◽  
Selection Step ◽  
Heavy Tailed ◽  
Ensemble Model Output Statistics ◽  
Model Output Statistics

Abstract To satisfy a wide range of end users, rainfall ensemble forecasts have to be skillful for both low precipitation and extreme events. We introduce local statistical postprocessing methods based on quantile regression forests and gradient forests with a semiparametric extension for heavy-tailed distributions. These hybrid methods make use of the forest-based outputs to fit a parametric distribution that is suitable to model jointly low, medium, and heavy rainfall intensities. Our goal is to improve ensemble quality and value for all rainfall intensities. The proposed methods are applied to daily 51-h forecasts of 6-h accumulated precipitation from 2012 to 2015 over France using the Météo-France ensemble prediction system called Prévision d’Ensemble ARPEGE (PEARP). They are verified with a cross-validation strategy and compete favorably with state-of-the-art methods like analog ensemble or ensemble model output statistics. Our methods do not assume any parametric links between the variables to calibrate and possible covariates. They do not require any variable selection step and can make use of more than 60 predictors available such as summary statistics on the raw ensemble, deterministic forecasts of other parameters of interest, or probabilities of convective rainfall. In addition to improvements in overall performance, hybrid forest-based procedures produced the largest skill improvements for forecasting heavy rainfall events.

AN EXPLICIT FORMULA FOR SOME SINGULAR VECTORS OF THE NEVEU–SCHWARZ ALGEBRA

1992 ◽  
Vol 07 (13) ◽  
pp. 3023-3033 ◽  
Author(s):  
LOUIS BENOIT ◽  
YVAN SAINT-AUBIN
Keyword(s):  
Explicit Expression ◽  
Explicit Formula ◽  
Singular Vector ◽  
Central Charge ◽  
Discrete Series ◽  
Virasoro Algebra ◽  
Verma Modules ◽  
Singular Vectors ◽  
Highest Weight ◽  
Highest Weight Representations

Similarly to the Virasoro algebra, the Neveu–Schwarz algebra has a discrete series of unitary irreducible highest weight representations. These are labeled by the values of [Formula: see text] (the central charge) and of the highest weight hpq = [(p (m + 2) − qm)2 − 4]/(8m (m + 2)) where m, p, q are some integers. The Verma modules constructed with these values (c, h) are not irreducible, however, as they contain two Verma submodules, each generated by a singular vector ψp,q (of weight hpq + pq/2) and ψm−p, m+2−q (of weight hpq + (m−p)(m+2−q)/2), respectively. We give an explicit expression for these singular vectors whenever one of its indices is 1.

scholarly journals ENSO Predictability of a Fully Coupled GCM Model Using Singular Vector Analysis

2006 ◽  
Vol 19 (14) ◽  
pp. 3361-3377 ◽  
Author(s):  
Youmin Tang ◽  
Richard Kleeman ◽  
Sonya Miller
Keyword(s):  
Singular Vector ◽  
Climate Model ◽  
Global Climate Model ◽  
Heat Content ◽  
Enso Cycle ◽  
Coupled Models ◽  
Error Norm ◽  
Singular Vectors ◽  
Enso Predictability ◽  
Fully Coupled

Abstract Using a recently developed method of computing climatically relevant singular vectors (SVs), the error growth properties of ENSO in a fully coupled global climate model are investigated. In particular, the authors examine in detail how singular vectors are influenced by the phase of ENSO cycle—the physical variable under consideration as well as the error norm deployed. Previous work using SVs for studying ENSO predictability has been limited to intermediate or hybrid coupled models. The results show that the singular vectors share many of the properties already seen in simpler models. Thus, for example, the singular vector spectrum is dominated by one fastest growing member, regardless of the phase of ENSO cycle and the variable of perturbation or the error norm; in addition the growth rates of the singular vectors are very sensitive to the phase of the ENSO cycle, the variable of perturbation, and the error norm. This particular CGCM also displays some differences from simpler models; thus subsurface temperature optimal patterns are strongly sensitive to the phase of ENSO cycle, and at times an east–west dipole in the eastern tropical Pacific basin is seen. This optimal pattern also appears for SST when the error norm is defined using Niño-4. Simpler models consistently display a single-sign equatorial signature in the subsurface corresponding perhaps to the Wyrtki buildup of heat content before a warm event. Some deficiencies in the CGCM and their possible influences on SV growth are also discussed.

Sign in / Sign up

Export Citation Format

Share Document