Page History

...

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

jumpy

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

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.  

The ensemble spread is flow-dependent and varies for different parameters.  It usually increases with the forecast range, but there can be cases when the spread is larger at shorter forecast lead times than at longer.  This might happen when the first days are characterized by strong synoptic systems with complex structures but are followed by large-scale fair weather high-pressure systems.

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 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.     

Fig8.1.2.1: The diagram is a schematic plume showing the relation between the standard deviation of the ensemble members for the whole forecast range (shaded area), an individual forecast (green line), the ensemble control member (CTRL, blue line) and the ensemble mean (EM, red line).  The ensemble mean lies more or less in the middle of the ensemble spread whereas any individual ensemble member (green line), can lie anywhere within the spread.  The ensemble control, which does not constitute a part of the plume, can even on rare occasions (theoretically on average 4% of the time) be outside the standard deviation plume.

Clearly the error in a forecast increases through the forecast period and it It is useful to have an idea of the likely magnitude of the forecast error and how it varies with forecast lead-time.   The accuracy of the ensemble mean can be estimated by the spread of the ensemble; on .

On average, the larger the spread , the implies larger the expected error of the ensemble mean.  Assuming If one assumes a gaussian distribution of ensemble results then the ensemble mean should also give an indication of the variability.

   An analysis of the relationship between root mean square error of the ensemble mean against lead-time shows a strong similarity with to a measure of the spread of the ensemble members against lead-time.   Thus  Thus the greater the spread, the greater the likely error. 

On average, the spread increases with lead-time, but if less than normally seen at a given lead-time then the error is likely to be less than normally expected.

 The spread around the ensemble mean as a measure of theoretical accuracy applies only to the ensemble mean forecast error.  It does not apply to the median or the ensemble control (CTRL), even if they happen to lie mid-range within the ensemble.

...

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

...

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.  

Fig8.1.2.3: 500hPa ENS ensemble mean geopotential height (in red, 580dam isopleth crosses east Italy and north Greece) and spread of geopotential height among ensemble members (coloured according to the scale).  Forecast VT DT 12UTC 13 August 8 Aug 2017, T+120 DT VT 12UTC 8 Aug 13 August 2017.  The greatest spread shows the areas of greatest uncertainty.  The light green area indicates a spread of 4-5dam and this could be:

• where geopotential heights on many ENS ensemble members are lower than the ENS ensemble mean (with a few ENS ensemble members well above the ENS ensemble mean) suggesting the trough will probably be slower or broader.
• where geopotential heights on many ENS ensemble members are higher than the ENS ensemble mean (with a few ENS ensemble members well below the ENS ensemble mean) suggesting the trough will probably be faster or sharper.

...

 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. 

...

      Nstd is the Normalised Standard Deviation.
      Std is the Standard Deviation of the latest ENSensemble.
      Mstd is the Mean Standard Deviation of the spread of the 30 most recent 00UTC or 12UTC ENS ensemble runs.

The Normalised Standard Deviation highlights areas of unusually high or low spread, where the uncertainty is larger or smaller than over the last 30 days.  in a particular area, an Nstd value: 

...

Fig8.1.2.5(Right): Ensemble control PMSL (hPa) in blue and spread of the ensemble members (represented by their the standard deviation, purple shading).  Colour scale for spread in hPa shown above the chart.   

Fig8.1.2.5(Left): Ensemble mean PMSL (hPa) in blue with normalised standard deviation (coloured shading).  Normalised standard deviation (Nstd) is calculated by dividing the standard deviation (Std) (in right hand frame above) by a mean standard deviation (Mstd), which is a pre-computed mean of the standard deviations of the 30 most recent 00UTC (or 12UTC) ensemble forecasts for the given lead time (this is also a function of location).  Colour scale for normalised standard deviation in hPa shown above the chart - uncoloured indicates a similarity with previous ENS ensemble mean values.

The panel on the right in Fig8.1.2.5 gives an assessment of the reliability of the absolute values of the contoured ensemble mean forecast fields.  Relatively large/small absolute values of standard deviation tend to indicate relatively high/low uncertainty in forecasts.  No colouring or the paler purples imply high higher confidence, brighter purples/magentas imply low lower confidence. 

The panel on the left in Fig8.1.2.5, shows the normalised standard deviation , which aims to put the standard deviation measure into the context of the general ensemble behaviour within the chart area over the last 30 days.   It tells whether the most recent ensemble is showing greater or less spread (and hence uncertainty) than recent ensemble results.  If the spread at Day5 of

For a particular set of ensemble forecasts (right panel) in a certain area at day5, if the spread:

• is similar to the spread that had recently been seen there at day5, then the shading of the normalised standard deviation (left panel) has a value near 1 (uncoloured).
• seems to be large, but lately has

...

• also been similarly large at day5, then the shading of the normalised standard deviation (left panel) has a value close to 1 (uncoloured):
• is greater than the spread that had recently been seen there at day5, then the shading of the normalised standard deviation (left panel) indicates a value rather greater than 1 (purple shading).
• is less than the spread that had recently been seen there at

...

• day5, then the shading of the normalised standard deviation (left panel) indicates a value rather

...

• lower than 1 (

...

• green shading).

  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.

...

Fig8.1.2.6: Mean and Spread charts DT 00UTC 8 September 2017, for T+120 , VT 00UTC 13 September 2017.

Fig8.1.2.6 Right: Ensemble control PMSL (hPa) in blue and spread of the ensemble members (represented by their the standard deviation, purple shading).  Colour scale for spread in hPa shown above the chart.
Fig8.1.2.6 Left: Ensemble mean PMSL (hPa) in blue with normalised standard deviation (coloured shading, see Fig8.1.2.5).  Normalised standard deviation is a function of lead time and of geographical location.  Colour scale for normalised standard deviation in hPa shown above the chart.

...

Fig8.1.2.7: Mean and Spread charts DT 00UTC 10 September 2017, for T+72 , VT 00UTC 13 September 2017.

Fig8.1.2.7 Right: Ensemble control PMSL (hPa) in blue and spread of the ensemble members (represented by their the Standard Deviation, purple shading).  Colour scale for spread in hPa shown above the chart.  
Fig8.1.2.7 Left: Ensemble mean PMSL (hPa) in blue with Normalised Standard Deviation (coloured shading, see Fig8.1.2.5).  Normalised Standard Deviation is a function of lead time and of geographical location.  Colour scale for Normalised Standard Deviation in hPa shown above the chart.

...

• the standard deviation of the surface pressure pattern among the ensemble members is moderate (4hPa - 7hPa).  This implies some variation (and hence uncertainty) among ensemble members regarding MSLP values in this area, or in the location of any low pressure centres.  Some ensemble members may have developed a deeper low pressure centre or sharp pressure trough in the area while others may not have; this can be resolved by inspection of the corresponding postage stamp charts.  The large standard deviation is unsurprising as one would expect variability at longer lead-times. 
• the normalised standard deviation is relatively high (1·2 - 1·8).  This gives an indication of the variability among ensemble members regarding MSLP in this area compared to the variability expected at this forecast lead-time in this area.  Here there is more variability (or uncertainty) than from recent ensemble forecasts, probably due to the uncertainty in the depth and movement (or even existence) of low pressure centres developed (or not) by ensemble members.
• the ensemble mean PMSL shows a broad pressure trough over northern Britain.  This probably relates to the large normalised spread; it is likely that some ensemble members also have this feature.  Ensemble control shows development of a fairly deep depression (~987hPa) but the ensemble control should only be considered as one member of the ENSensemble. 

Over Scotland and northern England at T+72 (Fig8.1.2.7):

• the standard deviation of the surface pressure pattern among the ensemble members is moderate (4hPa - 7hPa) but of less spacial extent than seen at T+120 (Fig8.1.2.6).  This implies less widespread variation (and hence uncertainty) among ensemble members in this area regarding MSLP values or location of any low pressure centres, although the detail of any low pressure centre or trough and/or its location is imprecise.
• the normalised standard deviation is much greater (2·5 - 5·0) than seen at T+120 (Fig8.1.2.6).  This implies variability among ensemble members is significantly higher in this area regarding MSLP compared to the variability expected at this forecast lead-time.  Here this This is probably due to the depth and movement of possibly deeper low pressure centre(s) developed by ensemble members.
• the ensemble mean PMSL shows a sharp pressure trough (sharper than at T+120 (Fig8.1.2.6)) over northern Britain.  The large standard deviation suggests some ensemble members develop a low pressure centre or sharp pressure trough in the area.  However some ensemble members may not develop any low pressure at all.  This can be resolved by reference to the corresponding postage stamp charts.  The ensemble control shows development of a rather deeper and more vigorous depression (~983hPa) (deeper than at T+120 (Fig8.1.2.6)).  However, the ensemble control should only be considered as one member of the ensemble. 

...