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## 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

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

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

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

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 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

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

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.

## 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.

## Abstract

The linear structures that produce the most in situ energy growth in the lower stratosphere for realistic wintertime flows are investigated using T21 and T42 calculations with the ECMWF 19-level forecast model. Significant growth is found for relatively large scale structures that grow by propagating from the outer edges of the vortex into the strong jet features of the lower-stratospheric flow. The growth is greater when the polar vortex is more asymmetric and contains localized jet structures. If the linear structures are properly phased, they can induce strong nonlinear interactions with the polar vortex, both for Northern Hemisphere and Southern Hemisphere flow conditions, even when the initial amplitudes are small. Large extensions from the main polar vortex that are peeled off during wave-breaking events give rise to a separate class of rapidly growing disturbances that may hasten the mixing of these vortex extensions.

## Abstract

The linear structures that produce the most in situ energy growth in the lower stratosphere for realistic wintertime flows are investigated using T21 and T42 calculations with the ECMWF 19-level forecast model. Significant growth is found for relatively large scale structures that grow by propagating from the outer edges of the vortex into the strong jet features of the lower-stratospheric flow. The growth is greater when the polar vortex is more asymmetric and contains localized jet structures. If the linear structures are properly phased, they can induce strong nonlinear interactions with the polar vortex, both for Northern Hemisphere and Southern Hemisphere flow conditions, even when the initial amplitudes are small. Large extensions from the main polar vortex that are peeled off during wave-breaking events give rise to a separate class of rapidly growing disturbances that may hasten the mixing of these vortex extensions.

## Abstract

The scale dependence of rapidly growing perturbations is investigated by studying the dominant singular vectors of T21 and T42 versions of the ECMWF model, which show the most linear energy growth in a 3-day period. A spectral filter is applied to the optimization process to determine which spatial scales are most effective in promoting energy growth. When the initial perturbation is confined to the top half of the total spherical harmonic wavenumber spectrum (high wavenumber end), the growth rates and final structures of the disturbances are changed very little from the case in which all wavenumbers are included. These results indicate that synoptic waves that become fully developed in a period of three days can arise from initial perturbations that are entirely contained at subsynoptic scales. Rapid growth is associated with initial perturbations that consist of smaller spatial scales concentrated near the effective steering level. The linear evolution of these initial perturbations in a highly complex basic flow leads to disturbances of synoptic scale that extend through most of the depth of the troposphere. Growth rates are approximately doubled when the model resolution is increased from T21 to T42, which is consistent with greater growth being associated with smaller spatial scales. When the initial perturbation is confined to the lower half of the total wavenumber spectrum, which describes the larger horizontal scales, the growth rates are significantly reduced and the initial and final structures are very different from the case in which all wavenumbers are included. These low wavenumber perturbations tend to be more barotropic in structure and in growth characteristics. As expected from their linear growth rates, when the low-wavenumber perturbations are inserted in the T63 forecast model, they grow more slowly and result in less forecast dispersion than the high wavenumber perturbations.

## Abstract

The scale dependence of rapidly growing perturbations is investigated by studying the dominant singular vectors of T21 and T42 versions of the ECMWF model, which show the most linear energy growth in a 3-day period. A spectral filter is applied to the optimization process to determine which spatial scales are most effective in promoting energy growth. When the initial perturbation is confined to the top half of the total spherical harmonic wavenumber spectrum (high wavenumber end), the growth rates and final structures of the disturbances are changed very little from the case in which all wavenumbers are included. These results indicate that synoptic waves that become fully developed in a period of three days can arise from initial perturbations that are entirely contained at subsynoptic scales. Rapid growth is associated with initial perturbations that consist of smaller spatial scales concentrated near the effective steering level. The linear evolution of these initial perturbations in a highly complex basic flow leads to disturbances of synoptic scale that extend through most of the depth of the troposphere. Growth rates are approximately doubled when the model resolution is increased from T21 to T42, which is consistent with greater growth being associated with smaller spatial scales. When the initial perturbation is confined to the lower half of the total wavenumber spectrum, which describes the larger horizontal scales, the growth rates are significantly reduced and the initial and final structures are very different from the case in which all wavenumbers are included. These low wavenumber perturbations tend to be more barotropic in structure and in growth characteristics. As expected from their linear growth rates, when the low-wavenumber perturbations are inserted in the T63 forecast model, they grow more slowly and result in less forecast dispersion than the high wavenumber perturbations.

## Abstract

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.

## Abstract

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.