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Singular Vectors (singular + vector)
Selected AbstractsThe characteristics of Hessian singular vectors using an advanced data assimilation schemeTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 642 2009A. R. Lawrence Abstract Initial condition uncertainty is a significant source of forecast error in numerical weather prediction. Singular vectors of the tangent linear propagator can identify directions in phase-space where initial errors are likely to make the largest contribution to forecast-error variance. The physical characteristics of these singular vectors depend on the choice of initial-time metric used to represent analysis-error covariances: the total-energy norm serves as a proxy to the analysis-error covariance matrix, whereas the Hessian of the cost function of a 4D-Var assimilation scheme represents a more sophisticated estimate of the analysis-error covariances, consistent with observation and background-error covariances used in the 4D-Var scheme. This study examines and compares the structure of singular vectors computed with the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecasting System using these two types of initial metrics. Unlike earlier studies that use background errors derived from lagged forecast differences (the NMC method), the background-error covariance matrix in the Hessian metric is based on statistics from an ensemble of 4D-Vars using perturbed observations, which produces tighter correlations of background-error statistics than in previous formulations. In light of these new background-error statistics, this article re-examines the properties of Hessian singular vectors (and their relationship to total-energy singular vectors) using cases from different periods between 2003 and 2005. Energy profiles and wavenumber spectra reveal that the total-energy singular vectors are similar to Hessian singular vectors that use all observation types in the operational 4D-Var assimilation. This is in contrast to the structure of Hessian singular vectors without observations. Increasing the observation density tends to reduce the spatial scale of the Hessian singular vectors. Copyright © 2009 Royal Meteorological Society [source] Singular vectors and excess growths in semi-implicit non-hydrostatic modelsTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 619 2006T. J. PAYNE Abstract In this note we show how the use of a semi-implicit time discretization in a linearized non-hydrostatic model can permit very large non-modal perturbation energy growth. This occurs in a single normal-length time step when a hydrostatically unbalanced increment at initial time is mapped to an approximately balanced increment. The energy growth is typically far larger than that of the leading meteorological singular vector (SV) over the course of a 12-hour optimization time interval so, unless action is taken, the leading SVs obtained with this discretization are meteorologically spurious. We show that the excess growths can be prevented by imposing a condition on the relation between the predictor (explicit) step and corrector (implicit) step of the semi-implicit method. We apply this condition to the Met Office's ,perturbation forecast' model to obtain SVs with admissible growth rates. Copyright © 2006 Royal Meteorological Society [source] The ECMWF operational implementation of four-dimensional variational assimilation.THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 564 2000II: Experimental results with improved physics Abstract A comprehensive set of physical parametrizations has been linearized for use in the European Centre for Medium-Range Weather Forecasts (ECMWF's) incremental four-dimensional variational (4D-Var) system described in Part I. The following processes are represented: vertical diffusion, subgrid-scale orographic effects, large-scale precipitation, deep moist convection and long-wave radiation. The tangent-linear approximation is examined for finite-size perturbations. Significant improvements are illustrated for surface wind and specific humidity with respect to a simplified vertical diffusion scheme. Singular vectors computed over 6 hours (compatible with the 4D-Var assimilation window) have lower amplification rates when the improved physical package is included, due to a more realistic description of dissipative processes, even though latent-heat release contributes to amplify the potential energy of perturbations in rainy areas. A direct consequence is a larger value of the observation term of the cost-function at the end of the minimization process when improved physics is included in 4D-Var. However, the larger departure of the analysis state from observations in the lower-resolution inner-loop is in better agreement with the behaviour of the full nonlinear model at high resolution. More precisely, the improved physics produces smaller discontinuities in the value of the cost-function when going from low to high resolution. In order to reduce the computational cost of the linear physics, a new configuration of the incremental 4D-Var system using two outer-loops is defined. In a first outer-loop, a minimization is performed at low resolution with simplified physics (50 iterations), while in the second loop a second minimization is performed with improved physics (20 iterations) after an update of the model trajectory at high resolution. In this configuration the extra cost of the physics is only 25%, and results from a 2-week assimilation period show positive impacts in terms of quality of the forecasts in the Tropics (reduced spin-down of precipitation, lower root-mean-square errors in wind scores). This 4D-Var configuration with improved physics and two outer-loops was implemented operationally at ECMWF in November 1997. [source] The structured total least-squares approach for non-linearly structured matricesNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 4 2002P. Lemmerling Abstract In this paper, an extension of the structured total least-squares (STLS) approach for non-linearly structured matrices is presented in the so-called ,Riemannian singular value decomposition' (RiSVD) framework. It is shown that this type of STLS problem can be solved by solving a set of Riemannian SVD equations. For small perturbations the problem can be reformulated into finding the smallest singular value and the corresponding right singular vector of this Riemannian SVD. A heuristic algorithm is proposed. Some examples of Vandermonde-type matrices are used to demonstrate the improved accuracy of the obtained parameter estimator when compared to other methods such as least squares (LS) or total least squares (TLS). Copyright © 2002 John Wiley & Sons, Ltd. [source] Can 4D-Var use dynamical information from targeted observations of a baroclinic structure?THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 651 2010E. A. Irvine Abstract Targeted observations are generally taken in regions of high baroclinicity, but often show little impact. One plausible explanation is that important dynamical information, such as upshear tilt, is not extracted from the targeted observations by the data assimilation scheme and used to correct initial condition error. This is investigated by generating pseudo targeted observations which contain a singular vector (SV) structure that is not present in the background field or routine observations, i.e. assuming that the background has an initial condition error with tilted growing structure. Experiments were performed for a single case-study with varying numbers of pseudo targeted observations. These were assimilated by the Met Office four-dimensional variational (4D-Var) data assimilation scheme, which uses a 6 h window for observations and background-error covariances calculated using the National Meteorological Centre (NMC) method. The forecasts were run using the operational Met Office Unified Model on a 24 km grid. The results presented clearly demonstrate that a 6 h window 4D-Var system is capable of extracting baroclinic information from a limited set of observations and using it to correct initial condition error. To capture the SV structure well (projection of 0.72 in total energy), 50 sondes over an area of 1×106 km2 were required. When the SV was represented by only eight sondes along an example targeting flight track covering a smaller area, the projection onto the SV structure was lower; the resulting forecast perturbations showed an SV structure with increased tilt and reduced initial energy. The total energy contained in the perturbations decreased as the SV structure was less well described by the set of observations (i.e. as fewer pseudo observations were assimilated). The assimilated perturbation had lower energy than the SV unless the pseudo observations were assimilated with the dropsonde observation errors halved from operational values. Copyright © 2010 Royal Meteorological Society [source] Singular vectors and excess growths in semi-implicit non-hydrostatic modelsTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 619 2006T. J. PAYNE Abstract In this note we show how the use of a semi-implicit time discretization in a linearized non-hydrostatic model can permit very large non-modal perturbation energy growth. This occurs in a single normal-length time step when a hydrostatically unbalanced increment at initial time is mapped to an approximately balanced increment. The energy growth is typically far larger than that of the leading meteorological singular vector (SV) over the course of a 12-hour optimization time interval so, unless action is taken, the leading SVs obtained with this discretization are meteorologically spurious. We show that the excess growths can be prevented by imposing a condition on the relation between the predictor (explicit) step and corrector (implicit) step of the semi-implicit method. We apply this condition to the Met Office's ,perturbation forecast' model to obtain SVs with admissible growth rates. Copyright © 2006 Royal Meteorological Society [source] Dynamics of Jovian atmospheres with applications of nonlinear singular vector methodINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 8 2007Zhiyue Zhang Abstract Nonlinear singular vectors (NSVs) of a Jovian atmosphere model are obtained numerically in this paper. NSVs are the initial perturbation, whose nonlinear evolution attains the maximal value of the cost function, which is constructed according to the physical problem of interest. The results demonstrate that the motions of Jupiter's atmosphere is relatively stable under some assumptions. Copyright © 2007 John Wiley & Sons, Ltd. [source] Multivariate chemometric approach to thermal solid-state FT-IR monitoring of pharmaceutical drug compoundJOURNAL OF PHARMACEUTICAL SCIENCES, Issue 8 2008Wei Jian Tan Abstract The study of thermal-related solid-state reaction monitored by spectroscopic method needs the use of advanced multivariate chemometric approach. It is because visual inspection of spectral data on particular functional groups or spectral bands is difficult to reveal the complete physical and chemical information. The spectral contributions from various species involved in the solid-state changes are generally highly overlapping and the spectral differences between reactant and product are usually quite minute. In this article, we demonstrate the use of multivariate chemometric approach to resolve the in situ thermal-dependent Fourier-transform infrared (FT-IR) mixture spectra of lisinopril dihydrate when it was heated from 24 to 170°C. The collected FT-IR mixture spectra were first subjected to singular value decomposition (SVD) to obtain the right singular vectors. The right singular vectors were rotated into a set of pure component spectral estimates based on entropy minimization and spectral dissimilarity objective functions. The resulting pure component spectral estimates were then further refined using alternating least squares (ALS). In current study, four pure component spectra, that is, lisinopril dihydrate, monohydrate, anhydrate, and diketopiperazine (DKP) were all resolved and the relative thermal-dependent contributions of each component were also obtained. These relative contributions revealed the critical temperature for each transformation and degradation. This novel approach provides better interpretation of the pathway of dehydration and intramolecular cyclization of lisinopril dihydrate in the solid state. In addition, it can be used to complement the information obtained from differential scanning calorimetry (DSC) and thermogravimetric analysis (TGA). © 2007 Wiley-Liss, Inc. and the American Pharmacists Association J Pharm Sci 97: 3379,3387, 2008 [source] The characteristics of Hessian singular vectors using an advanced data assimilation schemeTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 642 2009A. R. Lawrence Abstract Initial condition uncertainty is a significant source of forecast error in numerical weather prediction. Singular vectors of the tangent linear propagator can identify directions in phase-space where initial errors are likely to make the largest contribution to forecast-error variance. The physical characteristics of these singular vectors depend on the choice of initial-time metric used to represent analysis-error covariances: the total-energy norm serves as a proxy to the analysis-error covariance matrix, whereas the Hessian of the cost function of a 4D-Var assimilation scheme represents a more sophisticated estimate of the analysis-error covariances, consistent with observation and background-error covariances used in the 4D-Var scheme. This study examines and compares the structure of singular vectors computed with the European Centre for Medium-Range Weather Forecasts (ECMWF) Integrated Forecasting System using these two types of initial metrics. Unlike earlier studies that use background errors derived from lagged forecast differences (the NMC method), the background-error covariance matrix in the Hessian metric is based on statistics from an ensemble of 4D-Vars using perturbed observations, which produces tighter correlations of background-error statistics than in previous formulations. In light of these new background-error statistics, this article re-examines the properties of Hessian singular vectors (and their relationship to total-energy singular vectors) using cases from different periods between 2003 and 2005. Energy profiles and wavenumber spectra reveal that the total-energy singular vectors are similar to Hessian singular vectors that use all observation types in the operational 4D-Var assimilation. This is in contrast to the structure of Hessian singular vectors without observations. Increasing the observation density tends to reduce the spatial scale of the Hessian singular vectors. Copyright © 2009 Royal Meteorological Society [source] The value of observations.THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 628 2007II: The value of observations located in singular-vector-based target areas Abstract Data-assimilation experiments have been run in seven different configurations for two seasons to assess the value of observations taken in target regions identified either using singular vectors (SVs) or randomly, and located over the Pacific or the Atlantic Oceans. The value has been measured by the relative short-range forecast error reduction in downstream areas, specifically a North American region for observations taken in the Pacific Ocean, and a European region for observations taken in the Atlantic Ocean. Overall, results have indicated (1) that observations taken in SV-target areas are on average more valuable than observations taken in randomly selected areas, (2) that it is important that the daily set of singular vectors are used to compute the target areas, and (3) that the value of targeted observations depends on the region, the season and the baseline observing system. If the baseline observing system is data-void over the ocean, then the average value of observations taken in SV-target areas is very high. Considering for example winter 2004, SV-targeted observations over the Pacific (Atlantic) reduce the day-2 forecasts error of 500 hPa geopotential height forecasts in the verification region by 27.5% (19.1%), compared to 15.7% (14.9%) for observations taken in random areas. By contrast, if the baseline observing system is data-rich over the ocean, then the average value of observations taken in SV-target areas is rather small. Considering for example winter 2004, it has been estimated that adding SV-targeted observations over the Pacific (Atlantic) would reduce, on average, the day-2 forecasts error in the verification region by 4.0% (2.0%), compared to 0.5% (1.7%) for observations in random areas. These average results have been confirmed by single-case investigations, and by a careful examination of time series of forecast errors. These results indicate that more accurate assimilation systems that can exploit the potential value of localized observations are needed to increase the average return of investments in targeting field experiments. Copyright © 2007 Royal Meteorological Society [source] The value of observations.THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 628 2007III: Influence of weather regimes on targeting Abstract This paper assesses the value of targeted observations over the North Atlantic Ocean for different meteorological flow regimes. It shows that during tropical cyclone activity and particularly tropical cyclone transition to extratropical characteristics, removing observations in sensitive regions, indicated by singular vectors optimized on the 2-day forecast over Europe, degrades the skill of a given forecast more so than excluding observations in randomly selected regions. The maximum downstream degradation computed in terms of spatially and temporally averaged root-mean-square error of 500 hPa geopotential height is about 13%, a value which is 6 times larger than when removing observations in randomly selected areas. The forecast impact for these selected periods, resulting from degrading the observational coverage in sensitive areas, was similar to the impact found (elsewhere in other weather forecast systems) for the observational targeting campaigns carried out over recent years, and it was larger than the average impact obtained by considering a larger set of cases covering various seasons. Copyright © 2007 Royal Meteorological Society [source] Limited-area ensemble predictions at the Norwegian Meteorological InstituteTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 621 2006Inger-Lise Frogner Abstract This study aims at improving 0,3 day probabilistic forecasts of precipitation events in Norway. For this purpose a limited-area ensemble prediction system (LAMEPS) is tested. The horizontal resolution of LAMEPS is 28 km, and there are 31 levels in the vertical. The state variables provided as initial and lateral boundary conditions for the limited-area forecasts are perturbed using a dedicated version of the European Centre for Medium-Range Weather Forecasts (ECMWF) global ensemble prediction system, TEPS. These are constructed by combining initial and evolved singular vectors that at final time (48 h) are targeted to maximize the total energy in a domain containing northern Europe and adjacent sea areas. The resolution of TEPS is T255 with 40 levels. The test period includes 45 cases with 21 ensemble members in each case. We focus on 24 h accumulated precipitation rates with special emphasis on intense events. We also investigate a combination of TEPS and LAMEPS resulting in a system (NORLAMEPS) with 42 ensemble members. NORLAMEPS is compared with the 21-member LAMEPS and TEPS as well as the regular 51-member EPS run at ECMWF. The benefit of using targeted singular vectors is seen by comparing the 21-member TEPS with the 51-member operational EPS, as TEPS has considerably larger spread between ensemble members. For other measures, such as Brier Skill Score (BSS) and Relative Operating Characteristic (ROC) curves, the scores of the two systems are for most cases comparable, despite the difference in ensemble size. NORLAMEPS has the largest ensemble spread of all four ensemble systems studied in this paper, while EPS has the smallest spread. Nevertheless, EPS has higher BSS with NORLAMEPS approaching for the highest precipitation thresholds. For the area under the ROC curve, NORLAMEPS is comparable with or better than EPS for medium to large thresholds. Copyright © 2006 Royal Meteorological Society [source] Issues in targeted observingTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 613 2005(Invited paper for the Q. J. R. Meteorol. Abstract This paper summarizes successes and limitations of targeted observing field programmes starting from the Fronts and Atlantic Storm-Track Experiment in 1997 through recent programmes targeting winter storms and tropical cyclones. These field programmes have produced average reductions in short-range forecast errors of about 10 per cent over regional verification areas, and maximum forecast error reductions as large as 50 per cent in certain cases. The majority of targeting cases investigated so far involve sets of dropsondes and other observation data that provide partial coverage of target areas. The primary scientific challenges for targeting include the refinement of objective methods that can identify optimal times and locations for targeted observations, as well as identify the specific types of satellite and in situ measurements that are required for the improvement of numerical weather forecasts. The most advanced targeting procedures, at present, include: the ensemble transform Kalman Filter, Hessian singular vectors, and observation-space targeting using the adjoint of a variational data assimilation procedure. Targeted observing remains an active research topic in numerical weather prediction, with plans for continued refinement of objective targeting procedures, and field tests of new satellite and in situ observing systems. Copyright © 2005 Royal Meteorological Society [source] A singular vector perspective of 4D-Var: Filtering and interpolationTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 605 2005Christine Johnson Abstract Four-dimensional variational data assimilation (4D-Var) combines the information from a time sequence of observations with the model dynamics and a background state to produce an analysis. In this paper, a new mathematical insight into the behaviour of 4D-Var is gained from an extension of concepts that are used to assess the qualitative information content of observations in satellite retrievals. It is shown that the 4D-Var analysis increments can be written as a linear combination of the singular vectors of a matrix which is a function of both the observational and the forecast model systems. This formulation is used to consider the filtering and interpolating aspects of 4D-Var using idealized case-studies based on a simple model of baroclinic instability. The results of the 4D-Var case-studies exhibit the reconstruction of the state in unobserved regions as a consequence of the interpolation of observations through time. The results also exhibit the filtering of components with small spatial scales that correspond to noise, and the filtering of structures in unobserved regions. The singular vector perspective gives a very clear view of this filtering and interpolating by the 4D-Var algorithm and shows that the appropriate specification of the a priori statistics is vital to extract the largest possible amount of useful information from the observations. Copyright © 2005 Royal Meteorological Society [source] Forcing singular vectors and other sensitive model structuresTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 592 2003J. Barkmeijer Abstract Model tendency perturbations can, like analysis perturbations, be an effective way to influence forecasts. In this paper, optimal model tendency perturbations, or forcing singular vectors, are computed with diabatic linear and adjoint T42L40 versions of the European Centre for Medium-Range Weather Forecasts' forecast model. During the forecast time, the spatial pattern of the tendency perturbation does not vary and the response at optimization time (48 hours) is measured in terms of total energy. Their properties are compared with those of initial singular vectors, and differences, such as larger horizontal scale and location, are discussed. Sensitivity calculations are also performed, whereby a cost function measuring the 2-day forecast error is minimized by only allowing tendency perturbations. For a given number of minimization steps, this approach yields larger cost-function reductions than the sensitivity calculation using only analysis perturbations. Nonlinear forecasts using only one type of perturbation confirm an improved performance in the case of tendency perturbations. For a summer experiment a substantial reduction of the systematic error is shown in the case of forcing sensitivity. Copyright © 2003 Royal Meteorological Society. [source] High-resolution limited-area ensemble predictions based on low-resolution targeted singular vectorsTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 582 2002Inger-Lise Frogner Abstract The operational limited-area model, HIRLAM, at the Norwegian Meteorological Institute is used at 0.25° latitude/longitude resolution for ensemble weather prediction over Northern Europe and adjacent parts of the North Atlantic Ocean; this system is called LAMEPS. Initial and lateral boundary perturbations are taken from coarse-resolution European Centre for Medium-Range Weather Forecasts global ensemble members based on targeted singular vectors (TEPS). Five winter and five summer cases in 1997 consisting of 20 ensemble members plus one control forecast are integrated. Two sets of ensembles are generated, one for which both initial and lateral boundary conditions are perturbed, and another with only the initial fields perturbed. The LAMEPS results are compared to those of TEPS using the following measures: r.m.s. ensemble spread of 500 hPa geopotential height; r.m.s. ensemble spread of mean-sea-level pressure; Brier Skill Scores (BSS); Relative Operating Characteristic (ROC) curves; and cost/loss analyses. For forecasts longer than 12 hours, all measures show that perturbing the boundary fields is crucial for the performance of LAMEPS. For the winter cases TEPS has slightly larger ensemble spread than LAMEPS, but this is reversed for the summer cases. Results from BSS, ROC and cost/loss analyses show that LAMEPS performed considerably better than TEPS for precipitation, a result that is promising for forecasting extreme precipitation amounts. We believe this result to be linked to the high predictability of mesoscale flows controlled by complex topography. For two-metre temperature, however, TEPS frequently performed better than LAMEPS. Copyright © 2002 Royal Meteorological Society [source] Interpretations of the total energy and rotational energy norms applied to determination of singular vectorsTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 566 2000Ronald M. Errico Abstract The interpretation of the commonly-used energy norm is examined in the context of a simple vertically-discrete model. The norm is shown to include expressions for kinetic and available potential energy in addition to an expression for a portion of unavailable potential energy. Another norm is then introduced that only includes the rotational-mode contribution to these. The characterization of the two norms in terms of corresponding covariance functions is shown to be quite different, with that for the latter norm looking more like prior error statistics used in synoptic-scale data assimilation. The leading singular vectors are determined for both norms. Those computed for the new norm have slower associated growth. Their corresponding structures are similar at the initial time, however, with some notable differences, but after 24 hours their shapes are almost identical. The new norm has advantages over the old norm for some applications; e.g. for effectively filtering ageostrophic, convectively-driven singular vectors and for being more consistent with a spatially and dynamically correlated error norm. [source] Understanding African easterly waves: a moist singular vector approachATMOSPHERIC SCIENCE LETTERS, Issue 3 2009Rosalind J. Cornforth Abstract Moist singular vectors (MSV) have been applied successfully to predicting mid-latitude storms growing in association with latent heat of condensation. Tropical cyclone sensitivity has also been assessed. Extending this approach to more general tropical weather systems here, MSVs are evaluated for understanding and predicting African easterly waves, given the importance of moist processes in their development. First results, without initial moisture perturbations, suggest MSVs may be used advantageously. Perturbations bear similar structural and energy profiles to previous idealised non-linear studies and observations. Strong sensitivities prevail in the metrics and trajectories chosen, and benefits of initial moisture perturbations should be appraised. Copyright © 2009 Royal Meteorological Society [source] | |||||||||||||