Editorial Type:
Article
Article Type:
Research Article

Sensitivity Analysis of Forecast Errors and the Construction of Optimal Perturbations Using Singular Vectors

R. Gelaro European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, United Kingdom

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R. Buizza European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, United Kingdom

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T. N. Palmer European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, United Kingdom

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E. Klinker European Centre for Medium-Range Weather Forecasts, Shinfield Park, Reading, United Kingdom

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Print Publication:
01 Mar 1998
DOI:
https://doi.org/10.1175/1520-0469(1998)055<1012:SAOFEA>2.0.CO;2
Page(s):
1012–1037
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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.

* Current affiliation: Naval Research Laboratory, Monterey, California.

Corresponding author address: Dr. Ronald Gelaro, Naval Research Laboratory, 7 Grace Hopper Avenue, Monterey, CA 93943-5502.

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.

* Current affiliation: Naval Research Laboratory, Monterey, California.

Corresponding author address: Dr. Ronald Gelaro, Naval Research Laboratory, 7 Grace Hopper Avenue, Monterey, CA 93943-5502.

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