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For example, when the X% of the climatological distribution is 0, then the average rank of the 0-value ensemble members will always be X/2+0.5 with the even rank representation for the 0-value case explained above (e.g. for 10% being 0 in the climate, the average rank of the 0-value forecast ensemble members will be 5.5).

Example No-1 : No zero section in climatology, 5 number/rank groups:shows few examples when there is no zero value in the climatology, so all ensemble forecast members can be ranked without any issue. Here 5 clusters are used for simplicity. The even distribution is represented first, for which it is shown that by shifting the same rank distribution up or down does not change the standard deviation (and uncertainty). This is true for any variety of rank distributions. Also, after 'narrowing' the rank distribution, the mean does not change, but the uncertainty drops markedly. Moreover, in a similar manner, by adding extreme members (i.e. 1 or 100 or near that), even if only with very few members (2 in this example below), the uncertainty can be increased quite substantially. 

Example No-2/3/4/5 then demonstrate the complexity of the dry cases, when some portion or all of the climatological percentiles are 0. It is generally noticeable that as the percentage of the zero climate percentiles increase, it gets more and more difficult to end up with negative anomalies. The absolute lowest possible rank-mean values are when all ensemble forecast members are 0 and their average rank is determined by the 0-value section of the climatology. So, for the option of 10% of climatology being 0, the minimum possible rank is 5.5, while for say 30% it will be 15.5 and for the option of all 99 percentiles being zero, the rank-mean will be 50.5. The extent of which the rank-mean increases depends on how many of the ensemble members will be non-zero and with which actual rank (determined in the non-zero section of the climatology). For example, one of the most extreme cases is when all 99 climatological percentiles are 0 and none of the ensemble forecast members are 0. For this super unlikely to occur event the rank-mean will always be 100 (and the dominant anomaly category 'Extreme high'), regardless of the actual ensemble member values. So, even if all the forecast ensemble member river discharge is very low, say from 0.12 to 0.23, the forecast rank-mean will still be 100 and the dominant anomaly category 'Extreme high'.

Example No-1: No zero section in climatology, 5 number/rank groups:

Example No-2: Lowest 10% of the climatology is 0, 2 number/rank groups:

Example No-2: Lowest 10% of the climatology is 0, 2 number/rank groups:

Example No-3: Lowest 30% of the climatology is 0, 2 number/rank groups:

Example No-3: Lowest 30% of the climatology is 0, 2 number/rank groups:

Example No-4: Lowest 70% of the climatology is 0, 2 number/rank groups:

Example No-4: Lowest 70% of the climatology is 0, 2 number/rank groups:

Example No-5: All percentiles of the climatology (1-99) are 0, 2 number/rank groups:

• If all ensemble members are 0, then the ranks will spread evenly from 1 to 10 and the rank-mean will be around 5.5, so in the Extreme low category.

then the ensemble forecast anomaly (defined by the rank-mean) simply can not fall into the same extreme dry category, and the lowest possible is the 'Low' category with 10-25%. Similarly, if the lowest 25% is zero in the climatology, then the lowest possible anomaly signal is 'Bit low', so the category of 25-40%. Then, if 40% is zero, then there can not be anything lower than 'Near normal' anomaly for the ensemble forecast. All this makes sense, as actually it does not mean anything for those dry places to have below, say, 40th percentile, in case all of those lowest 40 percentiles are 0, as we can not go below zero. Or in the most extreme case, when even the 90th percentile is zero in the climatology, then the forecast can either be 'Near normal' or Extreme high. For this last case, if enough real time forecast ensemble member is above zero, then the rank-mean will exceed 90 and the dominant All this means, for these mixed-dry or super dry areas the number and distribution of the positively anomalous ensemble members will determine whether the anomaly will stay as 'Near normal' or will increase into one of the high categories. If enough members will be above the non-zero climate percentiles and thus high enough fraction of the 51-member ensemble forecast will get high enough ranks, then the distribution of the 51 ensemble member ranks will show a pronounced enough shift from the neutral/normal situation and the rank-mean will be high enough to fall into one of the high anomaly categories.

Forecast uncertainty category computation for the ensemble forecast

In addition to the forecast anomaly computation, as one of 7 anomaly categories, the forecast uncertainty is also represented in the sub-seasonal and seasonal products, namely on the new river network and basin summary products. The forecast uncertainty is defined by the standard deviation (std) of the ensemble member ranks (rank-std). If the members cluster well, and the spread of the ranks is low, then the forecast uncertainty will be low and conversely the confidence will be high.

The standard deviation of the even distribution with values ranging from 1 to 100 is (100-1)/sqrt(12) = 28.86, while the most extreme std value is when half of the members are with rank 1 and the other half with rank 99, in which case the std = 49.5. Obviously, the lowest std value is 0, when all ranks are the same. For forecast uncertainty, three uncertainty categories are defined, based on the rank-std value of the ensemble forecasts. Table 2 shows the categories, as defined by the std values of <10, 10<= <20 and 20<=. 

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Example No-5: All percentiles of the climatology (1-99) are 0, 2 number/rank groups: