Abstract

Targeted dropsonde data have been assimilated using the operational ECMWF four-dimensional variational data assimilation (4DVAR) system for 10 cases of the North Pacific Experiment (NORPEX) campaign, and their impact on analyses and corresponding forecasts has been investigated. The 10 fastest-growing “analysis” singular vectors (SVs) have been used to define a subspace of the phase space where initial conditions are expected to be modified by the assimilation of targeted observing. A linear combination of this vector basis is the pseudoinverse, that is, the smallest perturbation with the largest impact on the forecast error. The dropsonde-induced analysis difference has been decomposed into three initial perturbations, two belonging to the subspace spanned by the leading 10 SVs and one to its complement. Differences and similarities of the three analysis components have been examined, and their impact on the forecast error compared with the impact of the pseudoinverse.

Results show that, on average, the dropsonde-induced analysis difference component in the subspace spanned by the leading 10 SVs and the dropsonde-induced analysis difference component along the pseudoinverse directions are very small (6% and 15%, respectively, in terms of total energy norm). In the only case where dropsonde data were exactly released in the area identified by the SVs, the different components of the dropsonde-induced analysis difference and the pseudoinverse had consistent impacts on the forecast error. It is concluded that the poor agreement between the dropsonde location and the SV maxima is the main reason for the relatively small impact of the NORPEX targeting observations on the forecast error.

Abstract

Singular vectors with maximum energy at final time inside a verification area are used to identify the target area where extra observations should be taken, at an initial time, to reduce the forecast error inside the verification area itself. This technique is applied to five cases of cyclone development in the Atlantic Ocean, with cyclones reaching the British Isles at the final time. Three verification areas centered around this region are considered.

First, the sensitivity of the target area to the choice of the forecast trajectory along which the singular vectors are evolved, to the choice of the verification area where singular vector energy is maximized, and to the number of singular vectors used to define the target area is investigated. Results show little sensitivity to the choice of the verification area, but high sensitivity to the choice of the trajectory. Regarding the number of singular vectors used, results based on the first 4 or the first 10 singular vectors are shown to be very similar.

Second, the potential forecast error reduction that could be achieved by taking extra observations inside the target area is estimated by contrasting the error of a forecast started from the unperturbed analysis with the error of a forecast started by subtracting so-called pseudo-inverse perturbations (estimated using the leading singular vectors) to the unperturbed analysis. Results indicate that root-mean-square errors in the verification region could be reduced by up to 13% by adding targeted observations.

Overall, results suggest that linear models can be used to define the target area where adaptive observations should be taken.

Abstract

The impact of ensemble size on the performance of the European Centre for Medium-Range Weather Forecasts ensemble prediction system (EPS) is analyzed. The skill of ensembles generated using 2, 4, 8, 16, and 32 perturbed ensemble members are compared for a period of 45 days—from 1 October to 15 November 1996. For each ensemble configuration, the skill is compared with the potential skill, measured by randomly choosing one of the 32 ensemble members as verification (idealized ensemble). Results are based on the analyses of the prediction of the 500-hPa geopotential height field. Various measures of performance are applied: skill of the ensemble mean, spread–skill relationship, skill of most accurate ensemble member, Brier score, ranked probability score, relative operating characteristic, and the outlier statistic.

The relation between ensemble spread and control error is studied using L ², L ⁸, and L ^∞ norms to measure distances between ensemble members and the control forecast or the verification. It is argued that the supremum norm is a more suitable measure of distance, given the strategy for constructing ensemble perturbations from rapidly growing singular vectors. Results indicate that, for the supremum norm, any increase of ensemble size within the range considered in this paper is strongly beneficial. With the smaller ensemble sizes, ensemble spread does not provide a reliable bound on control error in many cases. By contrast, with 32 members, spread provides a bound on control error in nearly all cases. It could be anticipated that further improvement could be achieved with higher ensemble size still. On the other hand, spread–skill relationship was not consistently improved with higher ensemble size using the L ² norm.

The overall conclusion is that the extent to which an increase of ensemble size (particularly from 8 to 16, and 16 to 32 members) improves EPS performance, is strongly dependent on the measure used to assess performance. In addition to the spread–skill relationship, the measures most sensitive to ensemble size are shown to be the skill of the best ensemble member (particularly when evaluated on a point-wise basis) and the outlier statistic.

Abstract

The performance of two ensemble prediction systems, that of the European Centre for Medium-Range Weather Forecasts (EC-EPS) and that of the Australian Bureau of Meteorology (BM-EPS) are compared over the Southern Hemisphere annulus (20°–60°S) and over the Australian region. Ten-day ensemble forecasts for 152 daily cases (from 2 April to 31 August 2001) of 500-hPa geopotential height are examined.

A comprehensive set of verification measures documents the different spread and skill characteristics of the BM-EPS and EC-EPS. Overall, EC-EPS deterministic (i.e., unperturbed control) products and the probabilistic ensemble-based products are more skillful than the corresponding BM-EPS products. The utility of the BM-EPS for the Australian region is indicated.

Abstract

The local phase-space instability Of the atmospheric global circulation is Characterized by its (nonmodal) singular vectors. The formalism of singular vector analysis is described. The relations between singular vectors, normal modes, adjoint modes, Lyapunov vectors, perturbations produced by the so-called breeding method, and wave pseudomomentum are outlined. Techniques to estimate the dominant part of the singular spectrum using large-dimensional primitive equation models are discussed. These include the use of forward and adjoint tangent propagators with a Lanczos iterative algorithm. Results are described, based first on statistics of routine calculations made between December 1992 and August 1993, and second on three specific case studies.

Results define three dominant geographical areas of instability in the Northern Hemisphere: the two regions of storm track cyclogenesis, and the North African subtropical jet Singular vectors can amplify as much as tenfold over 36 hours, and in winter there are typically at least 35 independent singular vectors, which quadruple in amplitude over this timescale. Qualitatively, the distribution of singular vectors can be associated with a simple diagnostic of baroclinic instability from the basic-state flow. However, this relationship is not quantitatively reliable, as, for example, the chosen diagnostic takes no account of the horizontal or time-varying structure of the basic-state flow.

Three basic types of singular vector are identified The most important and most frequent is located in mid latitudes. At initial time, the singular vector is localized in the horizontal, with most amplitude in the lower troposphere. Energy growth can be interpreted qualitatively in terms of wave pseudomomentum propagation into the jet, resulting in peak amplitudes in the upper troposphere at optimization time. During evolution the dominant horizontal wavenumber of the singular vector decreases. Singular vector growth is therefore fundamentally nonmodal. Singular vectors 1ocalized first in the tropical upper troposphere. and second with equivalent barotropic structure in the high-latitude troposhpere, are also identified.

Abstract

The forecast skill of the European Centre for Medium-Range Weather Forecasts Ensemble Prediction System (EPS) in predicting precipitation probabilities is discussed. Four seasons are analyzed in detail using signal detection theory and reliability diagrams to define objective measure of predictive skill.

First, the EPS performance during summer 1997 is discussed. Attention is focused on Europe and two European local regions, one centered around the Alps and the other around Ireland. Results indicate that for Europe the EPS can give skillful prediction of low precipitation amounts [i.e., lower than 2 mm (12 h)⁻¹] up to forecast day 6, and of high precipitation amounts [i.e., between 2 and 10 mm (12 h)⁻¹] up to day 4. Lower levels of skill are achieved for smaller local areas.

Then, the EPS performance during summer 1996 (i.e., prior to the enhancement introduced on 10 December 1996 from 33 to 51 members and to resolution increase from T63 L19 to T_L159 L31) and summer 1997 are compared. Results show that the EPS has been remarkably more skillful during summer 1997 than summer 1996, with the gain in predictability up to 3 days for the highest [5 and 10 mm (12 h)⁻¹] amounts of precipitation.

Finally, the EPS performance during wintertime is analyzed. Two issues are investigated: the seasonal variability of the forecast skill of the new EPS, and the impact of the system upgrade on the wintertime performance. The comparison of the performance of the new EPS system during winter 1996/97 and during summer 1997 indicates that the EPS is more skillful during winter than during summer, with differences in predictive skill around 3 days for precipitation amounts larger than 2 mm (12 h)⁻¹. The comparison of the EPS performance before and after the system upgrade on 10 December 1996 during winter confirms the summer conclusion that the upgraded system is more skillful than the old one.

Abstract

A methodology for evaluating ensemble forecasts, taking into account observational uncertainties for catchment-based precipitation averages, is introduced. Probability distributions for mean catchment precipitation are derived with the Generalized Likelihood Uncertainty Estimation (GLUE) method. The observation uncertainty includes errors in the measurements, uncertainty as a result of the inhomogeneities in the rain gauge network, and representativeness errors introduced by the interpolation methods. The closeness of the forecast probability distribution to the observed fields is measured using the Brier skill score, rank histograms, relative entropy, and the ratio between the ensemble spread and the error of the ensemble-median forecast (spread–error ratio). Four different methods have been used to interpolate observations on the catchment regions. Results from a 43-day period (20 July–31 August 2002) show little sensitivity to the interpolation method used. The rank histograms and the relative entropy better show the effect of introducing observation uncertainty, although this effect on the Brier skill score and the spread–error ratio is not very large. The case study indicates that overall observation uncertainty should be taken into account when evaluating forecast skill.

Abstract

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.

Abstract

The sensitivity of forecast errors to initial conditions is used to examine the optimality of perturbations constructed from the singular vectors of the tangent propagator of the European Centre for Medium-Range Weather Forecasts model. Sensitivity and pseudo-inverse perturbations based on the 48-h forecast error are computed as explicit linear combinations of singular vectors optimizing total energy over the Northern Hemisphere. It is assumed that these perturbations are close to the optimal perturbation that can be constructed from a linear combination of these singular vectors. Optimality is measured primarily in terms of the medium-range forecast improvement obtained by adding the perturbations a posteriori to the initial conditions. Several issues are addressed in the context of these experiments, including the ability of singular vectors to describe forecast error growth beyond the optimization interval, the number of singular vectors required, and the implications of nonmodal error growth. Supporting evidence for the use of singular vectors based on a total energy metric for studying atmospheric predictability is also presented.

In general, less than 30 singular vectors capture a large fraction of the variance of the Northern Hemisphere sensitivity pattern obtained from a T63 adjoint model integration, especially in cases of low forecast skill. The sensitivity patterns for these cases tend to be highly localized with structures determined by the dominant singular vectors. Forecast experiments with these perturbations show significant improvements in skill in the medium range, indicating that singular vectors optimized for a short-range forecast continue to provide a useful description of error growth well beyond this time. The results suggest that ensemble perturbations based on 10–30 singular vectors should provide a reasonable description of the medium-range forecast uncertainty, although the inclusion of additional singular vectors is likely to be beneficial.

Nonmodality is a key consideration in the construction of optimal perturbations. There is virtually no projection between the contemporaneous unstable subspaces at the end of one forecast trajectory portion and the beginning of a second, consecutive portion. Sensitivity and ensemble perturbations constructed using the evolved singular vectors from a previous (day−2) forecast are suboptimal for the current (day+0) forecast initial conditions. It is argued that these results have implications for a range of issues in atmospheric predictability including ensemble weather prediction, data assimilation, and the development of adaptive observing techniques.