This comprehensive text and reference work on numerical weather prediction, first published in 2002, covers not only methods for numerical modeling, but also the important related areas of data assimilation and predictability. It incorporates all aspects of environmental computer modeling including an historical overview of the subject, equations of motion and their approximations, a modern and clear description of numerical methods, and the determination of initial conditions using weather observations (an important science known as data assimilation). Finally, this book provides a clear discussion of the problems of predictability and chaos in dynamical systems and how they can be applied to atmospheric and oceanic systems. Professors and students in meteorology, atmospheric science, oceanography, hydrology and environmental science will find much to interest them in this book, which can also form the basis of one or more graduate-level courses.

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Atmospheric Modeling, Data Assimilation and Predictability

A theory for estimating the probability distribution of the state of a model given a set of observations exists. This nonlinear filtering theory unifies the data assimilation and ensemble generation problem that have been key foci of prediction and predictability research for numerical weather and ocean prediction applications. A new algorithm, referred to as an ensemble adjustment Kalman filter, and the more traditional implementation of the ensemble Kalman filter in which “perturbed observations” are used, are derived as Monte Carlo approximations to the nonlinear filter. Both ensemble Kalman filter methods produce assimilations with small ensemble mean errors while providing reasonable measures of uncertainty in the assimilated variables. The ensemble methods can assimilate observations with a nonlinear relation to model state variables and can also use observations to estimate the value of imprecisely known model parameters. These ensemble filter methods are shown to have significant advantag...

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An Ensemble Adjustment Kalman Filter for Data Assimilation

(2005). Atmospheric Modeling, Data Assimilation, and Predictability. Technometrics: Vol. 47, No. 4, pp. 521-521.

Atmospheric Modeling, Data Assimilation, and Predictability

A suboptimal Kalman filter called the ensemble transform Kalman filter (ET KF) is introduced. Like other Kalman filters, it provides a framework for assimilating observations and also for estimating the effect of observations on forecast error covariance. It differs from other ensemble Kalman filters in that it uses ensemble transformation and a normalization to rapidly obtain the prediction error covariance matrix associated with a particular deployment of observational resources. This rapidity enables it to quickly assess the ability of a large number of future feasible sequences of observational networks to reduce forecast error variance. The ET KF was used by the National Centers for Environmental Prediction in the Winter Storm Reconnaissance missions of 1999 and 2000 to determine where aircraft should deploy dropwindsondes in order to improve 24‐72-h forecasts over the continental United States. The ET KF may be applied to any well-constructed set of ensemble perturbations. The ET KF technique supercedes the ensemble transform (ET) targeting technique of Bishop and Toth. In the ET targeting formulation, the means by which observations reduced forecast error variance was not expressed mathematically. The mathematical representation of this process provided by the ET KF enables such things as the evaluation of the reduction in forecast error variance associated with individual flight tracks and assessments of the value of targeted observations that are distributed over significant time intervals. It also enables a serial targeting methodology whereby one can identify optimal observing sites given the location and error statistics of other observations. This allows the network designer to nonredundantly position targeted observations. Serial targeting can also be used to greatly reduce the computations required to identify optimal target sites. For these theoretical and practical reasons, the ET KF technique is more useful than the ET technique. The methodology is illustrated with observation system simulation experiments involving a barotropic numerical model of tropical cyclonelike vortices. These include preliminary empirical tests of ET KF predictions using ET KF, 3DVAR, and hybrid data assimilation schemes—the results of which look promising. To concisely describe the future feasible sequences of observations considered in adaptive sampling problems, an extension to Ide et al.’s unified notation for data assimilation is suggested.

Adaptive sampling with the ensemble transform Kalman filter. Part I: Theoretical aspects

Ensemble forecasting

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 opera...

Singular Vectors, Metrics, and Adaptive Observations.

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....

https://journals.ametsoc.org/doi/pdf/10.1175/1520-0469%281998%29055%3C1012%3ASAOFEA%3E2.0.CO%3B2

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

The dominant singular vectors of the tangent propagator of the European Centre for Medium-Range Weather Forecast (ECMWF) numerical weather-prediction model are an essential component of the ECMWF Ensemble Prediction System (EPS). These singular vectors describe the principal finite-time linear instabilities of the northern extratropical atmospheric circulation.

The impact of increasing the horizontal resolution of the tangent model from T21 to T42 on three different types of initial perturbation, which make use of these singular vectors, is considered. the increase of resolution allows the possibility of describing more accurately instabilities with an upscale cascade of energy from subsynoptic to synoptic scales. Two of the perturbations are referred to as the pseudo-inverse and sensitivity vectors. These are both diagnostic, and involve estimating from the short-range forecast error the component of initial error in the unstable subspace. the third type of perturbation is used to construct the set of initial states for the ensemble prediction scheme. Linear and nonlinear integrations are described using these different types of perturbation.

All the results point to the conclusion that the higher-resolution calculations lead to more accurate results. Specifically, it is shown that significant amounts of energy in the pseudo-inverse and sensitivity vectors reside in scales smaller than T21, and that the skill of the EPS is improved when generating the initial perturbations using T42 singular vectors. This is found to be especially true in a number of cases when the control forecast was particularly poor.

As a consequence, the predictability of synoptic-scale disturbances in the atmosphere on time-scales of a few days is likely to be determined by errors in the initial state on subsynoptic scales.

In addition to these conclusions, it is proved that the analysis of the relationship between the EPS and the pseudo-inverse perturbations can be used to define the initial amplitude of the EPS perturbations.

The impact of increased resolution on predictability studies with singular vectors

The effectiveness of 2 methods for targeting observations is examined using a T21 L3 QG modelin a perfect model context. Target gridpoints are chosen using the pseudo-inverse (the inversecomposed of the first three singular vectors only) and the quasi-inverse or backward integration(running the tangent equations with a negative time-step). The effectiveness of a target is measuredby setting the analysis error to zero in a region surrounding the target and noting theimpact on the forecast error in the verification region. In a post-time setting, when the targetsare based on forecast errors that are known exactly, both methods provide targets that aresignificantly better than targets chosen at random within a broad region upstream of theverification region. When uncertainty is added to the verifying analysis such that the forecasterror is known inexactly, the pseudo-inverse targets still perform very well, while the backwardintegration targets are degraded. This degradation due to forecast uncertainty is especiallysignificant when the targets are a function of height as well as horizontal position. When anensemble-forecast difference is used in place of the inexact forecast error, the backward integrationtargets may be improved considerably. However, this significant improvement depends onthe characteristics of the initial-time ensemble perturbation. Pseudo-inverse targets based onensemble forecast differences are comparable to pseudo-inverse targets based on exact forecasterrors. Targets based on the largest analysis error are also found to be considerably moreeffective than random targets. The collocation of the backward integration and pseudo-inversetargets appears to be a good indicator of target skill. DOI: 10.1034/j.1600-0870.2000.00071.x

R. Gelaro

Papers

Singular Vectors, Metrics, and Adaptive Observations.

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

The impact of increased resolution on predictability studies with singular vectors

Examination of targeting methods in a simplified setting

Examination of targeting methods in a simplified setting