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 values below and above).  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 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 is useful to have an idea of the likely magnitude and how it varies with forecast lead-time.  The accuracy of the ensemble mean can be estimated by the spread of the ensemble; on average, the larger the spread, the larger the expected error of the ensemble mean.  Assuming 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 a measure of the spread of the ensemble members against lead-time.  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 Northern Hemisphere extratropics at various forecast-lead times. The relationship, on average, between ensemble root mean square error (full line) and ENS spread (dashed line), shows a strong correlation.   A low (or high) spread in the forecast implies low (or high) error on average (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.  

Fig8.1.2.3: 500hPa ENS 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 12UTC 13 August 2017 T+120 DT 12UTC 8 Aug 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 members are lower than the ENS mean (with a few ENS members well above the ENS mean) suggesting the trough will probably be slower or broader.
• where geopotential heights on many ENS members are higher than the ENS mean (with a few ENS members well below the ENS 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 (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.

The Normalised Spread is defined as:  Nstd = Std/Mstd

where, for a given forecast lead-time and location:

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

• near 1 implies the spread remains similar to previous spreads in that area, irrespective of whether the spread is large or small.
• >1 implies the spread is greater than recently. 
• <1 implies the spread is greater than recently. 

The normalised spread shows the increase or decrease in spread at a location.  It does not show the magnitude of the spread.  Therefore it highlights relatively low or relatively high uncertainty, but not the uncertainty itself.

ECMWF produces Mean and Spread charts and Normalised Standard Deviation charts for each ensemble run to aid understanding of the uncertainty of the forecast and whether the forecast is more or less uncertain in a given area at a given lead-time.

Fig8.1.2.5(Right): Ensemble control PMSL (hPa) in blue and spread of the ensemble members (represented by their 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 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 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 confidence, brighter purples/magentas imply low confidence. 

The panel on the left in Fig8.1.2.5, the normalised standard deviation, 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 a particular set of ensemble forecasts (right panel) seems to be large, but has of late also tended to be equally large at Day5 in the same area, then the left panel shading will denote a value that is close to 1 (uncoloured).  If the spread in a particular area at Day5 in the ensemble is greater/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 greater/less than 1 (purple/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.

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.

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 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 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.

Consider the charts for T+120 (Fig8.1.2.6) and T+72 (Fig8.1.2.7), both verifying at 00UTC 13 September 2017.

Over Scotland and northern England at T+120 (Fig8.1.2.6):

• 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 ENS. 

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 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. 

Elsewhere, comparing the charts for T+120 (Fig8.1.2.6) and for T+72 (Fig8.1.2.7):

• the the standard deviation of the ensemble surface pressure patterns is significantly less at T+72 than at T+120.
• the normalised standard deviation of the ensemble surface pressure pattern in mid-Atlantic:
  • at T+72 green area indicating less spread than recent forecasts in this area.
  • at T+120 white colouring indicating spread similar to recent forecasts in this area.
• the ensemble mean PMSL shows only small differences.

General points:

• Variability within the ensemble members (as measured by standard deviation) usually can be expected to increase as forecast lead-times increase.
• Large normalised standard deviation states only that the variability (or uncertainty) of the ensemble members is more than from recent ensemble forecasts at this forecast lead-time and location.  It does not necessarily imply greater uncertainty.  One would anyway expect greater variability in ensemble results in the vicinity of a forecasted deep depression.