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Probability forecasts cannot be linearly extrapolated into the future.  If an event was assigned a 10% probability in the forecast two days ago, 20% in yesterday’s forecast and 30% in today’s, there is no reason that it will necessarily further increase in tomorrow’s forecast; it 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:

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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).  

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Increased certainty in forecasting an occurrence is gained by sacrificing knowledge of exactly when the event will occur; the .  The longer the time interval over which event probabilities are calculated, the higher will higher will be their values on average.  The uncertainty Uncertainty in individualrain forecasts for days 5, 6 and 7 is always higher than for the whole three-day interval.  A statement of

Statements of probability can convey stronger or weaker messages.  It depends on the requirements of the user but the impact of:  

• high probability over a few days (e.g. “70% risk of precipitation >40 mm/24hr any time during Friday - Sunday”)

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• might appear stronger.
• lower probabilities for each day separately (e.g. "30% risk of precipitation >40mm/24hr on Friday, Saturday and/or Sunday") might appear less strong.

Probabilities cannot easily be combined: if .  If the probability for of 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 the combined probability that it will of rain in either period might be anything between 40% and 60% and .  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 bothof the time intervals in the original ensemble data.

Note: current graphical ECMWF products, including ecCharts, do not incorporate this "time windowing" approach to calculation of probabilities.  This may be something that ECMWF considers in future.  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).

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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 over an area

Statements of probability can convey stronger or weaker messages.  It depends on the requirements of the user but the impact of:  

• high probability over a few days (e.g. “70% risk of precipitation >40 mm/24hr somewhere in Belgium”)

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• might appear stronger.
• lower probabilities for

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• each day separately (e.g. "10% risk of precipitation >40mm/24hr at Brussels Airport") might appear less strong.

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.  Calculate the probabilities of rain somewhere within specific groups of grid points that together define individual catchment areas.
• For rainfall volume integrated over a catchment.  Sum the rainfall totals across all the groups groups of grid points that together define individual catchment areas, and then .  Then compute probabilities for those totals.

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

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Event probabilities are calculated from the proportion of ensemble members exceeding a certain threshold (e. g. if  If 34% of ensemble members forecast 2mm/12hr or more, then the probability for this event is considered to be 34%).  The As the number of ensemble members is limited, so .  So the probability of an event is not necessarily: 

• 0%

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• because no member has forecast it

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• .  
• 100% because all members have forecast

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• it.

Depending  Depending on the underlying mathematical-statistical assumptions and the size of the ensemble, 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.  

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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, then ideally all the parameters should be calibrated jointly, in order to maintain a physical consistency.

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