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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 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 dominant 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 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 normal' category (cat-4) for both these forecasts, as the mean of the ranks are certainly very close to each other and both will be quite near the median.
Figure 2. Schematic of the forecast extremity ranking of the 51 ensemble members and the 7 anomaly categories in the context calculation of the climatological distribution.dominant anomaly category.
Forecast uncertainty category computation for the ensemble forecast
In addition to the dominant forecast anomaly computation, as one of 7 predefined 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 ensemble member ranks 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 of ranks, with values ranging from 1 to 100, which is the perfect match to the climatologically expected spread, is going to be (100-1)/sqrt(12) = 28.86. On the other hand, the most extreme rank-std value is when half of the members are with rank 1 and the other half with rank 99 (perfectly split ensemble), in which case the rank-std is 49.5, while the lowest rank-std value is 0, when all ranks are the exact same, so there is no variability in the ranks at all.
For forecast uncertainty, three uncertainty categories are defined based on the rank-std value of the ensemble forecasts (Table 2).
| Uncertainty categories | Name | Rank STD |
|---|---|---|
| Cat-1 | Low uncertainty | 0-10 |
| Cat-2 | Medium uncertainty | 10-20 |
| Cat-3 | High uncertainty | 20< |
Table 2: Uncertainty categories defined by the standard deviation of the ensemble member ranks.
Examples to help interpret the methodology of the rank computation and the selection of the dominant anomaly and the uncertainty categories
Below, there are examples with simplified rank distributions and specific cases of very dry rivers (0 values) that will demonstrate how the dominant anomaly category and uncertainty category computation work.
In these examplesThere is a consequence of the 0-value problem over dry or very dry areas (described above), as some or all of the low anomaly signals will not be impossible to occur. Please consider these idealised examples as a help to demonstrate the situation with the 0-value problem. In these, for simplicity reason, a fixed portion of the climatological and/or ensemble forecast distribution is 0, while the ensemble members are being 0 and the rest having a simple average ranknon-zero ensemble members have just one or few more groups that have the same rank in each group. This way, the rank-mean and dominant anomaly category computation is simpler and computation methodology can be demonstrated in a simple way that is easier to interpret.
In the below examples, is For example, when the X% of the climatology climatological distribution is 0, then the average rank of the 0-value ensemble members which are also 0 will always be X/2+0.5 with the evenly distributed even rank representation for the 0-value case explained above (e.g. for 10% , it is 1-10 evenly spread, so being 0 in the climate, the average of those rank of the 0-value forecast ensemble members will be 5.5).
Example No-1: Lowest 10% of the climatology is 0:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Dominant anomaly category | Rank-std | Uncertainty category |
|---|---|---|---|---|---|---|---|
| 0 | 51 | NA (no member to rank) | 11 (the lowest possible rank for a non-zero member if 1-10 percentiles in the climatology are 0) | (0 * 5.5 + 51 * 11)/51 = 11 | Low (10-25) | 0 | Low uncertainty |
| 0 | 51 | NA | 20 | (0 * 5.5 + 51 * 20)/51 = 20 | Low (10-25) | 0 | Low uncertainty |
| 0 | 51 | NA | 50 | (0 * 5.5 + 51 * 50)/51 = 50 | Near normal (40-60) | 0 | Low uncertainty |
| 0 | 51 | NA | 70 | (0 * 5.5 + 51 * 70)/51 = 70 | Bit high (60-75) | 0 | Low uncertainty |
| 0 | 51 | NA | 100 | (0 * 5.5 + 51 * 100)/51 = 100 | Extreme high (90<) | 0 | Low uncertainty |
| 11 | 40 | 5.5 | 11 | (11 * 5.5 + 40 * 11)/51 = 9.81 | Extreme low (<10) | ||
| 11 | 40 | 5.5 | 20 | (11 * 5.5 + 40 * 20)/51 = 16.87 | Low (10-25) | ||
| 11 | 40 | 5.5 | 50 | (11 * 5.5 + 40 * 50)/51 = 40.40 | Near normal (40-60) | ||
| 11 | 40 | 5.5 | 70 | (11 * 5.5 + 40 * 70)/51 = 56.08 | Near normal (40-60) | ||
| 11 | 40 | 5.5 | 100 | (11 * 5.5 + 40 * 100)/51 = 79.61 | High (75-90) | ||
| 21 | 30 | 5.5 | 11 | (21 * 5.5 + 30 * 11)/51 = 8.73 | Extreme low (<10) | ||
| 21 | 30 | 5.5 | 20 | (21 * 5.5 + 30 * 20)/51 = 14.02 | Low (10-25) | ||
| 21 | 30 | 5.5 | 50 | (21 * 5.5 + 30 * 50)/51 = 31.67 | Bit low (25-40) | ||
| 21 | 30 | 5.5 | 70 | (21 * 5.5 + 30 * 70)/51 = 43.44 | Near normal (40-60) | ||
| 21 | 30 | 5.5 | 100 | (21 * 5.5 + 30 * 50)/51 = 61.08 | Bit high (60-75) | ||
| 36 | 15 | 5.5 | 11 | (36 * 5.5 + 15 * 11)/51 = 7.11 | Extreme low (<10) | ||
| 36 | 15 | 5.5 | 20 | (36 * 5.5 + 15 * 20)/51 = 9.76 | Extreme low (<10) | ||
| 36 | 15 | 5.5 | 50 | (36 * 5.5 + 15 * 50)/51 = 18.58 | Low (10-25) | ||
| 36 | 15 | 5.5 | 70 | (36 * 5.5 + 15 * 70)/51 = 24.47 | Low (10-25) | ||
| 36 | 15 | 5.5 | 100 | (36 * 5.5 + 15 * 50)/51 = 33.29 | Bit low (25-40) | ||
| 51 | 0 | 5.5 | NA (no member to rank) | (51 * 5.5 + 0)/51 = 5.5 | Extreme low (<10) |
Example No-2: Lowest 30% of the climatology is 0:
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Example No-4: All percentiles of the climatology (1-99) are 0:
| Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Dominant anomaly category |
|---|---|---|---|---|---|
| 0 | 51 | NA (no member to rank) | 100 (the lowest possible rank for a non-zero member if 1-99 percentiles in the climatology are 0) | (0 * 50.5 + 51 * 100)/51 = 100 | Extreme high (90<) |
| 11 | 40 | 50.5 | 100 | (11 * 50.5 + 40 * 100)/51 = 89.32 | High (75-90) |
| 21 | 30 | 35.5 | 100 | (21 * 50.5 + 30 * 100)/51 = 79.61 | High (75-90) |
| 36 | 15 | 35.5 | 100 | (36 * 50.5 + 15 * 100)/51 = 65.05 | Bit high (60-75) |
| 51 | 0 | 35.5 | NA (no member to rank) | (51 * 50.5 + 0)/51 = 50.5 | Near normal (40-60) |
- 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.
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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.
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