The above described role of chaos is evident on all spatial and temporal scales in the atmosphere and ocean. So, ideally, all forecasts should be made from an ensemble of initial conditions. In fact in any nonlinear dynamical system this approach offers the best possible forecast with the maximum information content. Averaging the ensemble members provides in a statistical sense a forecast more reliable than any of the single forecasts, including that started from the control analysis (Leith, 1974). Additionally, from the spread of the ensemble we can assess the reliability of the predictions and, for a sufficiently large number of realizations, any forecast quantity can be expressed in terms of probabilities. These probabilities convey all the information available regarding future weather. Note that each individual model run is deterministic, i.e., uniquely defined by the initial conditions, but collectively the ensemble of forecasts from the set of slightly different analyses portrays the chaotic nature of the atmosphere. Since a priori any single ensemble member is no more or less likely than any other, forecasting can be viewed as deterministic only to the extent that the envelope of solutions is sufficiently small that the differences between forecasts are inconsequential to the user. Otherwise, the variety of solutions and the implied uncertainty reflect the more general stochastic nature of the forecast process, with the range and distribution of possible outcomes providing information on the relative likelihood of various scenarios.

At NCEP the ensemble approach has been applied operationally for the medium- and extended range (Tracton and Kalnay, 1993; Toth and Kalnay, 1993), using the Environmental Modeling Center's (EMC) Medium-Range Forecast Model (MRF), which has a global domain. This document contains information regarding the operational global ensemble forecasts. However, short-range ensemble forecasts are also created at NCEP on an experimental basis, using the ETA and Regional Spectral Models (Brooks et al., 1995; Hamill and Collucci, 1996). Planning is also underway to run the coupled ocean- atmosphere model of EMC in an ensemble mode (Toth and Kalnay, 1995).

The analysis error distribution, however, is far from being white noise (Kalnay and Toth, 1994): Consider the analysis/forecast cycle of the data assimilation system as running a nonlinear perturbation model. The error in the first guess (short-range forecast) is the perturbation which is periodically "rescaled" at each analysis time by blending observations with the guess. Since observations are generally sparse they can not eliminate all errors from the short-range forecast that is subsequently generated as the first guess for the next analysis. Obviously, any error that grew in the previous short-range forecast will have a larger chance of remaining (at least partially) in the latest analysis than errors that had decayed. These growing errors will then start amplifying quickly again in the next short-range forecast.

It follows that the analysis contains fast growing errors that are dynamically created by the repetitive use of the model to create the first guess fields. This is what we refer to as the "breeding cycle" or Breeding of Growing Modes (BGM). These fast growing errors are above and beyond the traditionally recognized random errors that result from errors in observations. Those errors generally do not grow rapidly since they are not organized dynamically. It turns out that the growing errors in the analysis are related to the local Lyapunov vectors of the atmosphere (which are mathematical phase space directions that can grow fastest in a sustainable manner). Indeed, these vectors are what is estimated by the breeding method (Toth and Kalnay, 1993, 1995).

At NCEP we use 7 independent breeding cycles to generate the 14 initial ensemble perturbations. The initiation of each breeding cycle begins with an analysis/forecast cycle which differs from the others only in the initially prescribed random distribution ("seed") of analysis errors. These initially random perturbations are added and subtracted from the control analysis, so that each breeding cycle generates a pair of perturbed analyses (14 in all). From this point on each breeding cycle evolves independently to produce its own set of perturbations. The perturbations are just the differences between the short-term forecast (24 hour) initiated from the last perturbed analysis and the "control" analysis, rescaled to the magnitude of the seed perturbation. Since these short-term forecasts are just the early part of the extended range ensemble predictions, generation of the perturbations is basically cost free with respect to the analysis system (unlike the singular vector approach of ECMWF). The cycling of the perturbations continues and within a few days the perturbations reach their maximum growth.

Note that once the initial perturbations are introduced, the perturbation patterns evolve freely in the breeding cycle except that their size is kept within a certain amplitude range. Also note the similarity in the manner errors grow in the analysis vs. breeding cycles. The only difference is that from the breeding cycle, the stochastic elements that are introduced into the analysis through the use of observations containing random noise are eliminated by the use of deterministic rescaling. The seven quasi-orthogonal bred vectors from the breeding cycles span a subspace of the atmospheric attractor that represents the highest sustainable growth in the modeled atmosphere, at the given perturbation amplitude.

The breeding method has one free parameter, which is perturbation amplitude. We use a perturbation amplitude which is on average on the order of 12% of the total climatological rms variance (~10 m at 500 hPa height). The sensitivity to the choice of this amplitude (for example as a function of season) is under investigation. Regarding the spatial distribution of estimated analysis errors, we use a geographical mask (Toth and Kalnay, 1995) to which perturbations are rescaled every day. As a result, in data void regions such as the ocean basins the perturbations are three times or so larger than over data rich continents.

Finally, keep in mind that there is no guarantee that the above methodology "finds" all the possible growing modes or, equivalently, the ensemble will reliably encompass all possible outcomes in every situation: we cannot run enough perturbed forecasts (with, for example, different initial perturbation sizes) to populate the whole forecast distribution all the time. Moreover, remember that the forecast model is not "perfect", and model error, as well as initial condition uncertainty, will contribute to the distribution of predictions within the ensemble (especially systematic errors which may drive all the solutions in the same - wrong - direction). Overall, however, verifications indicate that the ensemble system as now constructed does provide enhanced skill through ensemble averaging and usefully reliable probability estimates.

The above list is clearly not exhaustive nor possibly designed and presented in an optimum way; again, feedback from users is most welcome!

A final few words on ensemble derived probabilities. Many forecasters have come to know and use (and love?) probability statements for precipitation and temperature. These generally are the direct or somewhat modified POP's and POT's generated statistically by TDL via the MOS (or similar) approach. They basically describe the probability distributions given the parameters (or, more generally, the synoptic features) from a single (i.e., deterministic) prediction (NGM or MRF). The actual uncertainty consists of two components; that associated with the non-unique distribution of precipitation or temperature given a particular synoptic scenario AND that intrinsic to there being an array of alternative scenarios. MOS accounts now only for the first component while the probabilities derived from the direct model output (as described above) from ensembles include both. Of course, one could derive and combine precipitation and temperature probability distributions from each ensemble member, and TDL is actively pursuing this approach.

1. Documentation for WMO CBS ET-EPS report - November 2011

2. Ensemble implementation documentations

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