A singular vector perspective of 4D-Var: Filtering and interpolation

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

Moist singular vectors and the predictability of some high impact European cyclones

The ECMWF full-physics and dry singular vector (SV) packages, using a dry energy norm and a 1-day optimization time, are applied to four high impact European cyclones of recent years that were almost universally badly forecast in the short range. It is shown that these full-physics SVs are much more relevant to severe cyclonic development than those based on dry dynamics plus boundary layer alone. The crucial extra ingredient is the representation of large-scale latent heat release. The severe winter storms all have a long, nearly straight region of high baroclinicity stretching across the Atlantic towards Europe, with a tongue of very high moisture content on its equatorward flank. In each case some of the final-time top SV structures pick out the region of the actual storm. The initial structures were generally located in the mid- to low troposphere. Forecasts based on initial conditions perturbed by moist SVs with opposite signs and various amplitudes show the range of possible 1-day outcomes for reasonable magnitudes of forecast error. In each case one of the perturbation structures gave a forecast very much closer to the actual storm than the control forecast. Deductions are made about the predictability of high-impact extratropical cyclone events. Implications are drawn for the short-range forecast problem and suggestions made for one practicable way to approach short-range ensemble forecasting. Copyright © 2005 Royal Meteorological Society.

The influence of physical processes on extratropical singular vectors

An investigation is made of the impact of a full linearized physical (moist) parameterization package on extratropical singular vectors (SVs) using the ECMWF integrated forecasting system (IFS). Comparison is made for one particular period with a dry physical package including only vertical diffusion and surface drag. The crucial extra ingredient in the full package is found to be the large-scale latent heat release. Consistent with basic theory, its inclusion results in a shift to smaller horizontal scales and enhanced growth for the SVs. Whereas, for the dry SVs, T42 resolution is sufficient, the moist SVs require T63 to resolve their structure and growth. A 24-h optimization time appears to be appropriate for the moist SVs because of the larger growth of moist SVs compared with dry SVs. Like dry SVs, moist SVs tend to occur in regions of high baroclinicity, but their location is also influenced by the availability of moisture. The most rapidly growing SVs appear to enhance or reduce large-scale rain in regions ahead of major cold fronts. The enhancement occurs in and ahead of a cyclonic perturbation and the reduction in and ahead of an anticyclonic perturbation. Most of the moist SVs for this situation are slightly modified versions of the dry SVs. However, some occur in new locations and have particularly confined structures. The most rapidly growing SV is shown to exhibit quite linear behavior in the nonlinear model as it grows from 0.5 to 12 hPa in 1 day. For 5 times this amplitude the structure is similar but the growth is about half as the perturbation damps a potential vorticity (PV) trough or produces a cutoff, depending on its sign.

Singular vector ensemble forecasting systems and the prediction of flow dependent uncertainty

The ECMWF ensemble weather forecasts are generated by perturbing the initial conditions of the forecast using a subset of the singular vectors of the linearised propagator. Previous results show that when creating probabilistic forecasts from this ensemble better forecasts are obtained if the mean of the spread and the variability of the spread are calibrated separately. We show results from a simple linear model that suggest that this may be a generic property for all singular vector based ensemble forecasting systems based on only a subset of the full set of singular vectors.

Estimating climatically relevant singular vectors for decadal predictions of the Atlantic Ocean

A key aspect in designing an ecient decadal prediction system is ensuring that the uncertainty in the ocean initial conditions is sampled optimally. Here, we consider one strategy to address this issue by investigating the growth of optimal perturbations in the HadCM3 global climate model (GCM). More specically, climatically relevant singular vectors (CSVs) - the small perturbations which grow most rapidly for a specic initial condition - are estimated for decadal timescales in the Atlantic Ocean. It is found that reliable CSVs can be estimated by running a large ensemble of integrations of the GCM. Amplication of the optimal perturbations occurs for more than 10 years, and possibly up to 40 years. The identi ed regions for growing perturbations are found to be in the far North Atlantic, and these perturbations cause amplication through an anomalous meridional overturning circulation response. Additionally, this type of analysis potentially informs the design of future ocean observing systems by identifying the sensitive regions where small uncertainties in the ocean state can grow maximally. Although these CSVs are expensive to compute, we identify ways in which the process could be made more ecient in the future.

Extreme value distribution for singular measures

In this paper we perform an analytical and numerical study of Extreme Value distributions in discrete dynamical systems that have a singular measure. Using the block maxima approach described in Faranda et al. [2011] we show that, numerically, the Extreme Value distribution for these maps can be associated to the Generalised Extreme Value family where the parameters scale with the information dimension. The numerical analysis are performed on a few low dimensional maps. For the middle third Cantor set and the Sierpinskij triangle obtained using Iterated Function Systems, experimental parameters show a very good agreement with the theoretical values. For strange attractors like Lozi and H\`enon maps a slower convergence to the Generalised Extreme Value distribution is observed. Even in presence of large statistics the observed convergence is slower if compared with the maps which have an absolute continuous invariant measure. Nevertheless and within the uncertainty computed range, the results are in good agreement with the theoretical estimates.

Numerical computation of an analytic singular value decomposition of a matrix valued function

This paper extends the singular value decomposition to a path of matricesE(t). An analytic singular value decomposition of a path of matricesE(t) is an analytic path of factorizationsE(t)=X(t)S(t)Y(t) T whereX(t) andY(t) are orthogonal andS(t) is diagonal. To maintain differentiability the diagonal entries ofS(t) are allowed to be either positive or negative and to appear in any order. This paper investigates existence and uniqueness of analytic SVD's and develops an algorithm for computing them. We show that a real analytic pathE(t) always admits a real analytic SVD, a full-rank, smooth pathE(t) with distinct singular values admits a smooth SVD. We derive a differential equation for the left factor, develop Euler-like and extrapolated Euler-like numerical methods for approximating an analytic SVD and prove that the Euler-like method converges.

On the numerical integration of a class of singular perturbation problems

A three-point difference scheme recently proposed in Ref. 1 for the numerical solution of a class of linear, singularly perturbed, two-point boundary-value problems is investigated. The scheme is derived from a first-order approximation to the original problem with a small deviating argument. It is shown here that, in the limit, as the deviating argument tends to zero, the difference scheme converges to a one-sided approximation to the original singularly perturbed equation in conservation form. The limiting scheme is shown to be stable on any uniform grid. Therefore, no advantage arises from using the deviating argument, and the most accurate and efficient results are obtained with the deviation at its zero limit.

Algorithms for Robust Pole Assignment in Singular Systems

The solution of the pole assignment problem by feedback in singular systems is parameterized and conditions are given which guarantee the regularity and maximal degree of the closed loop pencil. A robustness measure is defined, and numerical procedures are described for selecting the free parameters in the feedback to give optimal robustness.

Properties of singular vectors using convective available potential energy as final time norm

We study the feasibility of using the singular vector technique to create initial condition perturbations for short-range ensemble prediction systems (SREPS) focussing on predictability of severe local storms and in particular deep convection. For this a new final time semi-norm based on the convective available potential energy (CAPE) is introduced. We compare singular vectors using the CAPE-norm with SVs using the more common total energy (TE) norm for a 2-week summer period in 2007, which includes a case of mesoscale extreme rainfall in the south west of Finland. The CAPE singular vectors perturb the CAPE field by increasing the specific humidity and temperature of the parcel and increase the lapse rate above the parcel in the lower troposphere consistent with physical considerations. The CAPE-SVs are situated in the lower troposphere. This in contrast to TE-SVs with short optimization times which predominantly remain in the high troposphere. By examining the time evolution of the CAPE singular values we observe that the convective event in the south west of Finland is clearly associated with high CAPE singular values.

Essential selfadjointness of singular magnetic Schrödinger operators on Riemannian manifolds

In this paper we extend the well-known Leinfelder–Simader theorem on the essential selfadjointness of singular Schrödinger operators to arbitrary complete Riemannian manifolds. This improves some earlier results of Shubin, Milatovic and others.