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Figure 2 shows the process of determining the ranks for each ensemble member. In this example, the lowest member gets the rank of 54 (red r54 on the graph in Figure 2) by moving vertically until crossing the climatological distribution and then moving horizontally to the y-axis to determine the two bounding percentiles and thus the right percentile bin. In this case, the lowest ensemble member value is between the 53rd and 54th percentile, which results in bin-54. Then all ensemble members, similarly, get a bin number, the 2nd lowest values with bin-60 and so on until the largest ensemble member value getting bin-97, as the river discharge value is between the 96th and 97th percentiles.
Figure 2. Schematic of the forecast extremity ranking of the 51 ensemble members and the 7 anomaly categories in the context of the climatological distribution.
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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. |
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Expected anomaly category computation
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The ensemble forecasts have 51 members, which will be assigned an extremity rank each. Using these 51 ranks, the forecasts will be put in one of the 7 anomaly categories (as described in Table 1). This is done based on the arithmetic mean of the 51 ensemble member rank values (rank-mean) (see Figure 4). This rank-mean will also be a number between 1 and 100, but this time a real (not integer) number. If the anomaly is 50.5, that is exactly the normal (median) condition, i.e. no anomaly whatsoever. If the anomaly is below 50.5, then drier than normal conditions are forecast, if above 50.5, then wetter than normal. The lower/higher the anomaly value is below/above 50.5, the drier/wetter the conditions are predicted to be. The lowest/highest possible value is 1/100, if all ensemble members are 1/100 (the most extremely dry/wet). Then, based on this rank-mean, we define the expected forecast anomaly category (one of the 7 categories in Table 1) for the whole ensemble forecast, by placing the rank-mean into the right categories, as defined in Table 1 above. For example, all rank-mean values from 40.0 to 60.0, interpreted as 40.0<= <60.0, will be assigned to 'Near normalNormal', or category-4.
The ensemble forecast anomaly was not based on the most probable of the 7 anomaly categories, as that would make it prone to jumpiness. For example, in the super uncertain case of 6, 8, 7, 7, 7, 9, 7 members being in each of the 7 anomaly categories, the forecast category (the dominant expected one) would be the 'High' category (cat-6), as that has the most members (9). However, it is likely that nearby river pixels could easily be only slightly different with 7, 9, 7, 7, 7, 7, 7 members in each category, in which case the forecast anomaly forecast anomaly category would be the 'Low' category (cat-2), as now that has the most (again 9) members. It is worth mentioning that very uncertain cases are especially likely to happen at longer ranges. These two forecasts are only slightly different in terms of distribution, but the ensemble expected forecast anomaly categories would be almost the complete opposite of each other, making the signal look possibly very jumpy geographically. With the mean-rank definition, we avoid this and simply assign the 'Near normalNormal' category (cat-4) for both of these forecasts, as the mean of the ranks are certainly very close to each other and both will be quite near the median.
Figure 4. Schematic of the forecast extremity ranking of the 51 ensemble members and the calculation of the expected forecast anomaly category for the whole ensemble.
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