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An unavoidable consequence of modifying the initial conditions around the most likely estimate of the truth (i.e. the 4D-Var analyses) is that the perturbed analysis is on average slightly degraded.  The RMS distance from truth for a perturbed analysis is, in the ideal case, on average √2 times the RMS distance of the unperturbed analysis from the truth (see Fig5Fig51.1.3A).

 Fig5Fig51.1.3A: A schematic illustration of why the perturbed initial conditions will, on average, be further from the true state of the atmosphere than the control analysis.  For a specific grid point, the analysis state (a) is known, as well as the average error of an analysis state from the true state (a-t).  The true state (t) is therefore, on this diagram, the analysis offset by the average error and therefore can lie anywhere on the darker circle radius (a-t) centred on the analysis (a).  Any perturbed analysis state (p) can be very close to the truth, but is in a majority of the cases much further away; in the ideal case the average distance is √2 times the analysis error ((a-t)√2).

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Consequently, the proportion of the perturbed analyses that are better than the control (i.e. unperturbed analysis) for a specific location and for a specific parameter (e.g. 2m temperature or MSLP at Paris) is only 35% (see Fig5Fig51.1.4B).  Considering more than one grid point lowers the proportion even further.

Fig5Fig51.1.4B: A schematic illustration for a specific grid point of the special case where the perturbed analyses differ on average from the control analysis (radius of red circle) by as much as the control differs from the truth (TC). 

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With respect to the forecasts, in the short range only a small number of the perturbed forecasts are, on average, more skilful than the control forecast.  However, with increasing forecast range the average proportion of perturbed forecasts that are better than the control forecast increases, eventually approaching 50% asymptotically.

Fig5Fig51.1.5C: Schematic representation of the percentage of perturbed forecasts with lower root mean square error than the control forecast for regions of different sizes: Northern Hemisphere, Europe, a typical ““small”” Member State and a specific location.  With increasing forecast range, fewer and fewer perturbed members are worse than the control (from Palmer et al 2006).

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Skill levels vary with parameter, time of year and geographical location, so of course Fig 5.1.6 Fig51.D is not universally applicable. Rather it gives a general guide for predictive skill for the extra-tropical synoptic pattern.

Fig5Fig51.1.6D: Schematic image of the root mean square error of the ensemble members, ensemble mean and control forecast as a function of lead-time.  The asymptotic predictability limit is defined as the average difference between two randomly chosen atmospheric states.  In a perfect ensemble system the root mean square error of an average ensemble member is √2 times the error of the ensemble mean.

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