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Chapter 1. Theory
Chapter 2. Computational
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1.1 Introduction
The ensemble prediction system (EPS) is a technique to predict the probability distribution of forecast states, given a probability distribution of random analysis error and model error. (At the time of writing the latter is not taken into account in the operational EPS.) More specifically, the operational EPS is a set of (50) integrations of a lower resolution (TL159) version of the operational model from initial conditions which are constructed by either adding or subtracting small dynamically active perturbations to the operational analysis for the day. In situations where the ensemble is tightly distributed around the operational integration, this forecast can be thought of as very likely to occur. More generally, the EPS gives a probability forecast of some given flow type, or some given category based on temperature, precipitation or other forecast variable.
Fig. 1.1 shows a schematic illustration of the phase-space evolution of the probability distribution function (PDF) of analysis error throughout the forecast range. A specific isopleth (e.g. the 1 standard deviation isopleth) is illustrated. It is assumed that at initial time the distribution is normal along each phase-space direction. At initial time (Fig. 1.1 (a)), the isopleth is shown as isotropic, i.e. bounding an
-sphere, where 
is the dimension of phase space 
for the ECMWF operational forecast model). In general,
this error will not be isotropic-analysis error is likely to be larger along
directions which are less well observed, and vice versa. However, it is
straightforward to define an inner product on phase space, with respect
to which the initial probability density function (PDF) is isotropic. This
inner product, defined from the analysis error covariance matrix, is fundamental
in the theory of singular vectors below. In the early part of the forecast, error growth is governed by linear dynamics. During this period an initially spherical isopleth of the PDF will evolve to bound an
-dimensional ellipsoidal volume (Fig. 1.1 (b)). The major axis of the
ellipsoid corresponds to a phase-space direction which defines the dominant
finite-time instability of that part of phase space (relative to the analysis
error covariance metric). The arrow shown in Fig.
1.1 (b) points along the major axis of the ellipsoid. It can be thought
of as evolving from the arrow shown in Fig. 1.1 (a). Note that the arrows in
Figs. 1.1 (a) and (b)
are not parallel to one another. This illustrates the non-modal nature of
linear perturbation growth. The arrows at initial and forecast time define the dominant singular vector at initial and final time (with respect to the analysis error covariance metric). At forecast time, the dominant singular vector defines the dominant eigenvector of the forecast error covariance matrix. See Section 1.2.1 for more details.
The growth of the (isopleth of the) PDF between Figs. 1.1 (b) and (c) describes a nonl inear evolution of the PDF. In Figs. 1.1 (c) the PDF has deformed from its ellipsoidal shape in Figs. 1.1 (b). The nonlinear deformation will cause the PDF to evolve away from a normal distribution. Put another way, in the nonlinear phase, the PDF in any give direction is partially determined by perturbations which, in the linear phase, were orthogonal to that direction. Finally, Fig. 1.1 (d) shows (schematically) the situation where the evolved PDF has effectively become indistinguishable from the system's attractor, so that all predictability has been lost.
The number of degrees of freedom of the operational ECMWF model is (very) much larger than the largest practicable ensemble size. This raises the question of whether any particular strategy is desirable in sampling the initial PDF. If initial errors can occur independently in all the phase-space directions, then a strategy of random-under-sampling could lead to an EPS whose reliability was poor, especially for cases of small ensemble spread. In particular, if the spread from a randomly under-sampled ensemble was found to be small on a particular occasion, this could either be because the flow was especially predictable, or because the ensemble perturbations poorly sample the unstable subspace in which the analysis error lay. From a credibility perspective, it is important to try to minimize the latter type of occurrence.
An alternative strategy is to base the perturbations on the singular vectors. Clearly, by focusing on the unstable subspace, the cases of small spread being associated with large forecast error should be minimized, at least in the linear and weakly nonlinear range. In addition to this, there are five related reasons why the initial perturbations for the ECMWF EPS are based on the dominant singular vectors.
Firstly, as shown by Rabier et al. (1996), the sensitivity of day-2 forecast error to perturbations in the initial state projects well into the space of dominant singular vectors. Rabier et al. have shown that cases of severe forecast failure can be dramatically improved if the analysis is modified using the sensitivity perturbations.
Secondly, provided the metric or inner product for the singular vectors is an accurate reflection of the analysis error covariance matrix then, as mentioned, the evolved singular vectors point along the largest eigenvectors of the forecast error covariance matrix. As such (see Ehrendorfer and Tribbia, 1997), perturbations constructed from the dominant singular vectors represent the most efficient means for predicting the forecast error covariance matrix, given a pre-specified number of allowable tangent model integrations.
Thirdly, singular vector perturbations may provide a relatively efficient means of sampling the forecast error PDF in the nonlinear range, particularly during transitions in weather regimes. For example, Mureau et al. (1993) have shown a case where the singular vector perturbations were successful in capturing a major transition to blocking, where random perturbations were inadequate. Gelaro et al. (1998) have documented further such cases. A more systematic study of the ability of ensembles to describe the probability of regime transitions in the weakly nonlinear forecast range has been made by Trevisan et al. (1998) using an intermediate-complexity model of the extratropical circulation. Relatively small ensembles initialised using firstly singular vectors, and secondly local Lyapunov vectos, were compared with a large Monte-Carlo ensemble based on random perturbations. It was found that small ensemble spread from the singular-vector ensemble was a reliable indicator of small ensemble spread from the Monte-Carlo ensemble. By contrast, small ensemble spread from the Lyapunov-vector ensemble was a much less reliable indicator of small spread from the Monte-Carlo ensemble.
Fourthly, in practice the initial PDF is only poorly known. Hence it is difficult to even define a truly random initial sampling.
Fiftly, from a purely pragmatic point of view, it would seem to be wasteful to integrate explicitly those perturbations that are likely to grow slowly, and thus resemble the control forecast. Such perturbations can be implicitly taken into account in constructing a forecast PDF, by increasing the weight given to the control forecast relative to the perturbed forecasts.
At present the EPS is entirely based on the notion that forecast uncertainty is dominated by error or uncertainty in the initial conditions. This is consistent with studies that show that, when two operational forecasts differ, it is usually differences in the analyses rather than differences in model formulation that are critical to explaining this difference. In order to include model error (or uncertainty) into the specification of the EPS, experiments are underway testing the impact of stochastic perturbations in the physical tendencies. The ratio of spread associated with initial error and model error can be tuned to the ratio of the influence of initial and model error obtained from studies of divergent operational forecasts.
Each EPS perturbation is a linear combination of the computed singular vectors. This is done so that a given perturbation covers as much of the Northern and Southern hemisphere as possible. The amplitude of the perturbation is then fitted to the statistics of analysis error. These processes are described in Section 1.2.
The EPS was first implemented operationally in 1992 (Palmer et al., 1993). A general description of the ECMWF EPS is described in Molteni et al. (1996). The singular vector computations are described in Buizza and Palmer (1995), and Barkmeijer et al. (1998).
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10.04.2002 |
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