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Ensemble mean and ensemble spread

Ensemble mean

The ensemble mean is the average of the forecast values of the ensemble members at a given forecast time (i.e. the sum of the values divided by the number of ensemble members).  The mean leans towards the values of a greater number of ensemble members and less weight is given to outliers.  It is mostly used with medium range forecasts where the mean tends towards the most probable value.

The ensemble mean is most suited to parameters like temperature and pressure because these usually have rather symmetric Gaussian distributions.

Ensemble median

The ensemble median is the middle value of the forecast values of the ensemble members at a given forecast time when sorted into a list (i.e. the same number of ensemble member values below and above the middle value).  The median lies at the centre of the range of the ensemble members and can be more descriptive of the data set than the mean.  It is mostly used with seasonal forecasts where the range of values can be quite large.

The ensemble median is is more suited to parameters like wind speeds and precipitation because these usually have skewed distributions.  

Ensemble spread

The ensemble spread is a measure of the difference between the members and is represented by the standard deviation (Std) with respect to the ensemble mean (EM).  Theoretically, on average, small spread indicates high forecast accuracy, large spread indicates low forecast accuracy.   The ensemble spread should reflect the diversity of all possible outcomes.  

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Two similar-looking forecast charts may display large differences in geopotential if they contain systems with strong gradients that are slightly out of phase.  Conversely, two synoptically rather different forecast charts will display small differences if the gradients are weak.  The spread refers to the uncertainty of the values of mean sea level pressure, geopotential height, wind or temperature, but not necessarily to the flow patterns.  Also “jumpy”, deterministic forecasts might indicate that very different weather developments are possible.   Such aspects are reflected in charts of ensemble spread.   

Relationship of ensemble mean and ensemble spread against forecast lead-time

The ensemble mean (or, on occasion, the ensemble median) forecast tends to average out the less predictable atmospheric scales.   As the forecast proceeds the variation between the results of ensemble members gradually increases.  The ensemble mean, of course, will lie within the envelope of ensemble members throughout the forecast. 

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Fig8.1.2.2: The graph shows the error, on average, in 850hPa temperature for the extra-tropical Northern Hemisphere at various forecast lead-times.  The relationship, on average, between ensemble root mean square error (full line) and ensemble spread (dashed line), shows a strong correlation.   A low (or high) spread in the forecast, on average, implies low (or high) error (though at the same time any individual ensemble mean forecast may by chance be good or bad).

Mean and spread charts

Special composite charts have been created to allow comparisons between the ensemble mean and the ensemble control (e.g. on ecCharts) (Fig8.1.2.3).  The coloured areas do not indicate the probability of the location of a feature, but merely indicate the magnitude of the uncertainty.   Users should refer to Postage Stamp charts (example chart), Spaghetti Plot charts, or Clustering (example chart) to assess probability of departure from the ensemble mean before making forecast decisions.  

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 Fig 8.1.2.4: An example of forecast mean sea level pressure (taken from part of an ECMWF mean and spread chart) highlighting the difference between the ensemble control (green) and the ensemble mean (black).  Absolute spread of ensemble members is shown by shading.  The ensemble mean is the average over all ensemble members.  It smooths the flow more in areas of large uncertainty (large spread).  If there is large spread, the ensemble mean can be a rather weak pattern and may not represent any of the possible states.  The ensemble mean should always be used together with the spread to capture this uncertainty.  Note in particular the small depressions forecast by the ensemble control near 35W (shown by arrows) and the additional uncertainty (darker purple) within the ensemble members nearby.  This suggests at least some of the ensemble members show something similar to the ensemble control although with timing and/or location differences. 

The normalised standard deviation

The ensemble spread tends to show a strong geographical dependence.  Geopotential and pressure generally show little spread at low latitudes.  Variability is greater at mid-latitudes and the spread is consequently rather higher.  This latitude dependence tends to obscure the features of a given situation and a normalised spread or standard deviation (Nstd) is more useful.  For this, the spread, measured by the standard deviation (Std) of ensemble member values at a given point and lead time, is normalised against the mean of the spread of the 30 most recent 00UTC ensemble members (Mstd) for 00UTC runs (or 12UTC ensemble members (Mstd) for 30 most recent 12UTC runs) for the same lead-times and geographical locations.

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So although the forecast for (say) longer lead-times in the ensemble (say days 8-10) will usually be of rather low confidence, there will be some occasions when one can be rather more confident than usual for this lead-time.  The normalised standard deviation will tend to show this by green shading.

An example of an analysis of ensemble mean and spread charts

Comparing the run-to-run changes in ensemble mean and spread charts and the normalised standard deviation charts can be informative and aid an assessment of confidence in the forecast.

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