# 7.1 Consideration of Probability

## Consideration of Probability

### Use of probabilities

Probabilities give no indication of the physical nature of the uncertainty.  A 25% probability of precipitation >5mm/24hr might be related to a showery regime or to the uncertainty of the arrival of a frontal rain band.  A 25% risk forecast for temperatures <0°C might be related to the possible early morning clearing of low cloud cover or the possible arrival of arctic air.

Probability forecasts cannot be linearly extrapolated into the future.  If two days ago the forecast probability of an event was 10% , yesterday it was 20%, and today it was 30%, there is no reason that the probability will further increase in the forecast tomorrow.  The probability could equally well remain at its current level or decrease (Fig7.1.1).

Fig7.1.1: A schematic illustration of what might be considered some "typical" event probability developments for a specific location over ten days.  The lines represent probability of the event:

• if using climatological probability in the absence of other information (horizontal green line).
• if at long lead-times the event is forecast by very few ENS members.  The event probability commonly (though not always) reduces to zero with time (descending orange dotted line).
• if day by day the event is forecast with increasing frequency by ENS members.  The event probability commonly (though not always) increases with time (turquoise line).  However, at any lead-time, such an increase may stop.  Event probabilities in subsequent forecasts may fall back to lower levels or to zero (dashed orange lines).

It is very important to recognise that an apparent trend in probabilities is unreliable (e.g. turquoise line in Fig7.1.1); a trend should not be extrapolated forwards.  In real scenarios probabilities may reduce, increase or remain the same, and indeed they may also go up and down as the event approaches.  In the turquoise line scenario, 6 days before the potential event there is ~40% chance that event will occur.  Equally there is ~60% chance that the event will not occur and equivalently a 60% chance that the probabilities in subsequent forecasts will decrease to zero (solid orange line).

In summary, forecasters should generally use the probability at a given time just as presented from the latest forecast.  They should not not try to change that probability based on any extrapolation procedures.  Sometimes some form of weighted average of the last two or three forecasts could be appropriate, particularly if the target day is more than a day or two away.

### Probabilities over time intervals

A increased probability of a forecast event is offset by uncertainty of when the event will actually occur.  The longer the time interval over which event probabilities are calculated, the higher will be their values on average.  The uncertainty in individual rain forecasts for days 5, 6 and 7 is always higher than for the whole three-day interval.  A statement of high probability of an event during a few days may convey a stronger message than a statement of lower probabilities for each day separately.  Thus a 70% probability of heavy rain (say >40 mm in 24hr) any time during Friday-Sunday is likely to have more impact than a 30% probability of heavy rain on Friday, Saturday and/or Sunday.

Probabilities cannot easily be combined: if the probability for an event in one time interval is 40% and for the next time interval 20%, there is normally no straightforward way to find out the probability over both time intervals together, except when the events are uncorrelated.  Depending on the correlation between the two time intervals, the combined probability that it will rain in either period might be anything between 40% and 60% and the probability that both periods will have rain can vary between 0% and 20% (see Fig7.1.2).  The only way to get a correct probability for combined time intervals is to count the proportion of members having rain in either or both of the time intervals in the original ENS data.

Note: ECMWF graphical products, including ecCharts, do not incorporate this "time windowing" approach to calculation of probabilities but may be considered in the future.  In the meantime tailored local processing of ECMWF output fields could be performed by specific users to achieve this goal.

Fig7.1.2:  If the events in the two adjacent time intervals are correlated, so that rain in the first interval is followed by rain in the second, the probability for rain at any time during the whole period is 40% (far left figure).  If they are anti-correlated (e.g. because of differing speeds of frontal passage), so that rain in the first period is followed by dry conditions in the second, and dry in the first period is followed by rain in the second, then the total probability is 60% (centre figure).  If the events in the two adjacent time intervals are non-correlated, the combined probability is (1 - (1 - 0.4) x (1 - 0.2)) =52% (far right figure).

### Probabilities over areas

Probabilities are normally calculated for individual locations.  Calculating probabilities with respect to several grid points within a certain geographical area normally increases the event probability.

A statement of high probability of an event over an area may convey a stronger message than a statement of lower probabilities over an individual location.  Thus a 70% probability of heavy rain (say >40 mm in 24hr) somewhere in Belgium is likely to have more impact than a 10% probability of heavy rain at Brussels Airport.

For hydrological applications there are two particularly important considerations related to areal probabilities:

• Heavy rainfall can have hydrological consequences far away from its immediate location.  There are practical advantages in calculating the probabilities of rain somewhere within specific groups of grid points that together define individual catchment areas.
• One may be particularly interested in the probabilities of rainfall volume integrated over a catchment.  In this case it would be helpful to sum the rainfall totals across the groups of grid points, and then compute probabilities for those totals.

Of course hydrology also involves further complexity, such as infiltration properties, and run-off lag times which will vary across each catchment.

### Probabilities of combined events

As with probabilities over longer time intervals or larger areas, probabilities for combined events such as “cloud cover <6/8 and temperatures >20°C” or blizzards (combination of heavy snowfall and strong winds) cannot be made from the separate probabilities but can be calculated from the ENS data.  ecCharts now includes the capacity to compute several types of combined probabilities (type "combined" in the Add layers Filter box on ecCharts).

### Modification of the probabilities

Event probabilities are calculated from the proportion of ENS members exceeding a certain threshold.  So, if 34% of ENS members forecast an event, then the probability for this event is considered to be 34%.  Since the number of ENS members is limited it is important not to assume that resulting probabilities exactly show the complete picture.  The probability of an event is not necessarily 0% because no member has forecast it. The probability of an event is not necessarily 100% because all members have forecast it.  Probabilities such as 1-2% and 98-99% could be assigned to situations when no or all members forecast an event, with intermediate probabilities adjusted slightly upwards or downwards accordingly.  However, this depends somewhat on the underlying mathematical and statistical assumptions, and the size of the ensemble.

### Calibration of probabilities

Forecast probabilities often show systematic deviations from the observed frequencies.  Low probabilities are often too low, high probabilities are often too high.  Calibration of probabilities or statistical post-processing (MOS) can improve the reliability of the probability forecasts.  This might affect the internal consistency between parameters.  If an over-prediction of rain is coupled to an over-prediction of cloud and perhaps under-prediction of temperature, ideally all the parameters would have to be calibrated jointly, in order to maintain a physical consistency.

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