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For the forecast rank computation in the 0-value singularity case, a special solution was developed. All the 0 ensemble member values (all below 0.1 m3/s) get an evenly-representing rank assigned from any of the percentiles that have 0 values (i.e. below 0.1 m3/s) in the model climatology. In practice, this will mean, the 'rank-undefined' section of the ensemble forecast is going to be spread evenly across the 'rank-undefined' section of the climatology during the rank computation.

Figure 3 demonstrates the process on an idealised example, where the lowest 77 percentiles are 0 in the climatology and 23 out of 51 ensemble members are also 0 (see Figure 3a). The 23 ensemble members with 0 value then are spread across the 0-value range of the climatology from 1 to 77 (see Figure 3b). This way the ranks of the 23 members will be assigned from 1 to 77 with equal as possible spacing in between (see Figure 3c). Finally, the remaining non-zero ensemble members also get their ranks in the usual way, as described above in Figure 2. Finally, the schematic of ranks of all 51 members are provided in Figure 3d.

Figure 3. Schematic of the forecast extremity ranking calculation for areas with 0 river discharge values.

In the extreme case of all climate percentiles being 0, which happen over river pixels of the driest places of the world, such as the Sahara, the ensemble forecast member ranks can either be 100 for any non-zero value, regardless of the magnitude of the river discharge, or the evenly spread ranks from 1 to 100, as a representation of the totally 0 climatology. In the absolute most extreme case of all 99 climate percentiles being 0 and all 51 members being 0 in the forecast, the ranks of the forecast will be from 1 to 100 in equal representation. This means, this forecast will be a perfect representation of the climatological distribution, or with another word a forecast showing climatologically expected probabilities for all anomaly categories from Extreme low to Extreme high.

Few examples are shown here, when there is no 0-value in the climatology, so all ensemble forecast members can be ranked without any issue. For simplicity, 5 groups are used in the forecast only. The table below shows the numbers and the related average ranks for the 5 groups, with the rank-mean, rank-std and expected anomaly and uncertainty categories determined from those cases. For example, in the first row, 10 ensemble members are in the first group, which will all have the rank of 40. Then 10 members will be in the 2nd group with the rank of 45, and so on. The rank-mean of this simplified forecast distribution will be very close to 50 (mean of 40-45-50-55-60 with almost the same population in each group) and the rank-std will be about 7. This puts this forecast case into the 'Normal' expected anomaly category (rank-mean between 40 and 60) and the 'Low' uncertainty category (rank-set below 10).

The even distribution is represented first below, 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.