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So, for the option of 10% of climatology being 0, the absolute minimum possible forecast rank-mean is 5.5, while for 30% it will be 15.5 and for the totally dry climatology, where all 99 percentiles are zero, the rank-mean will be 50.5. The extent of which the rank-mean of the forecast increases depends on how many of the ensemble members will be non-zero and with which actual rank (determined by 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 of the forecast will always be 100 (and the expected forecast anomaly category 'Extreme high'), regardless of the actual ensemble member values (i.e. how much higher they are than 0). So, even if all forecast ensemble member river discharge values are very low, say from 0.12 to 0.23, the forecast rank-mean will still be 100 and the forecast anomaly category 'Extreme high'.
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| title | Non-zero climatology examples Examples with no 0-value section in the climatology with 5 ensemble forecast rank groups only |
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Few examples are shown here, when there is no zero 0-value in the climatology, so all ensemble forecast members can be ranked without any issue. For simplicity, 5 clusters groups are used in the forecast only. So, for 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 , again 10 members will be in the 2nd group with the rank of 45, and so on. Then, the The rank-mean of this simplified forecast distribution will be very close to 50 , as the mean os practically speaking the mean of the rank of the 5 groups with 40, 45, 50, 55 and 60, as the population of the 5 groups is almost exactly the same (other than the middle group with 11)(mean of 40-45-50-55-60 with almost the same population in each group) and the rank-std will be 7.0. This puts this forecast case into the 'Normal' expected anomaly category and the 'Low' uncertainty caegory. 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. | Number of ensemble members in each group | Common rank in each group | Rank-mean | Expected forecast anomaly category | Rank-std | Forecast uncertainty category |
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| 10 | 10 | 11 | 10 | 10 | 40 | 45 | 50 | 55 | 60 | 50.0 | Near normal (40-60) | 7.00 | Low uncertainty | | 10 | 10 | 11 | 10 | 10 | 30 | 40 | 50 | 60 | 70 | 50.0 | Near normal (40-60) | 14.00 | Medium uncertainty | | 10 | 10 | 11 | 10 | 10 | 10 | 30 | 50 | 70 | 90 | 50.0 | Near normal (40-60) | 28.00 | High uncertainty |
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| | 10 | 10 | 11 | 10 | 10 | 60 | 65 | 70 | 75 | 80 | 70.0 | Bit high (60-75) | 7.00 | Low uncertainty | | 10 | 10 | 11 | 10 | 10 | 50 | 60 | 70 | 80 | 90 | 70.0 | Bit high (60-75) | 14.00 | Medium uncertainty |
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| | 0 | 10 | 31 | 10 | 0 |
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| 50.0 | Near normal (40-60) | 3.13 | Low uncertainty | | 0 | 10 | 31 | 10 | 0 |
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| 50.0 | Near normal (40-60) | 6.26 | Low uncertainty | | 0 | 10 | 31 | 10 | 0 |
| 30 | 50 | 70 |
| 50.0 | Near normal (40-60) | 12.52 | Medium uncertainty |
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| | 2 | 10 | 27 | 10 | 2 | 1 | 45 | 50 | 55 | 100 | 50.03 | Near normal (40-60) | 14.21 | Medium uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 40 | 50 | 60 | 100 | 50.03 | Near normal (40-60) | 15.21 | Medium uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 30 | 50 | 70 | 100 | 50.0 | Near normal (40-60) | 18.64 | Medium uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 20 | 50 | 80 | 100 | 50.0 | Near normal (40-60) | 23.34 | High uncertainty | | 2 | 10 | 27 | 10 | 2 | 1 | 10 | 50 | 90 | 100 | 50.0 | Near normal (40-60) | 28.62 | High uncertainty |
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| title | Eamples with some percentage of 0-value section in the climatology |
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, again for simplicity reasons, the climatological and |
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forecast values will only be in one of 2 categories, either 0-value or non 0-value. This way, the |
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main impact of the 0/non-0 value issue can be demonstrated |
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. In the tables below, the numbers and the related average ranks are given for the two groups of 0 and non-0 ensemble members, with the rank-mean, rank-std and expected anomaly and uncertainty categories determined from those cases. There are 4 tables, with 10% / 30% / 70% and 100% of 0 values in the climatology (i.e increasingly dry climate). For example, in the 7th row of the 1st table with 10% of 0 in the climatology, 11 ensemble members are 0-value and the remaining 40 are greater than 0. The average rank for the 0 members are 5.5 (as this is given by the method of handling the 0-value issue with equal representation, explained above), while the average rank for the non-zero members is given as 11. The related rank-mean is then 9.81, making this forecast into the 'Extreme low' expected category while the rank-std is 2.26, with low uncertainty category. These tables demonstrate the complex interaction between the dryness of the climatology and ensemble forecasts, reflected in the forecast rank-mean and rank-std values and the subsequent expected anomaly and uncertainty categories. They also demonstrate, how less likely it becomes to have negative anomalies as the climatology becomes drier and drier. | Expand |
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| title | Table with 10% of 0-value in climatology. Click here to expand... |
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| | Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Forecast anomaly category | Rank-std | Uncertainty category |
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| 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 |
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in a Below, there are examples with simplified rank distributions and specific cases of very dry rivers (0 values) that will demonstrate how the forecast anomaly category and uncertainty category generation work.
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:
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| 70 | (0 * 5.5 + 51 * 70)/51 = 70 |
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| 100 | (0 * 5.5 + 51 * 100)/51 = 100 | Extreme high (90<) | 0 | Low uncertainty |
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| | 11 | 40 | 5.5 | 11 | (11 * 5.5 + 40 * 11)/51 = 9.81 | Extreme low (<10) | 2.26 | Low uncertainty | | 11 | 40 | 5.5 | 20 | (11 * 5.5 + 40 * 20)/51 = 16.87 | Low (10-25) | 5.96 | Low uncertainty | | 11 | 40 | 5.5 | 50 | (11 * 5.5 + 40 * 50)/51 = 40.40 |
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| 5.5 | 70 | (11 * 5.5 + 40 * 70)/51 = 56.08 |
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| 100 | (11 * 5.5 + 40 * 100)/51 = 79.61 | High (75-90) | 38.86 | High uncertainty |
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| | 21 | 30 | 5.5 | 11 | (21 * 5.5 + 30 * 11)/51 = 8.73 | Extreme low (<10) | 2.70 | Low uncertainty | | 21 | 30 | 5.5 | 20 | (21 * 5.5 + 30 * 20)/51 = 14.02 | Low (10-25) | 7.13 | Low uncertainty | | 21 | 30 | 5.5 | 50 | (21 * 5.5 + 30 * 50)/51 = 31.67 | Bit low (25-40) | 21.90 | High uncertainty | | 21 | 30 | 5.5 | 70 | (21 * 5.5 + 30 * 70)/51 = 43.44 | Near normal (40-60) | 31.74 | High uncertainty | | 21 | 30 | 5.5 | 100 | (21 * 5.5 + 30 * 50)/51 = 61.08 | Bit high (60-75) | 46.50 | High uncertainty |
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| | 36 | 15 | 5.5 | 11 | (36 * 5.5 + 15 * 11)/51 = 7.11 | Extreme low (<10) | 2.50 | Low uncertainty | | 36 | 15 | 5.5 | 20 | (36 * 5.5 + 15 * 20)/51 = 9.76 | Extreme low (<10) | 6.60 | Low uncertainty | | 36 | 15 | 5.5 | 50 | (36 * 5.5 + 15 * 50)/51 = 18.58 | Low (10-25) | 20.27 | High uncertainty | | 36 | 15 | 5.5 | 70 | (36 * 5.5 + 15 * 70)/51 = 24.47 | Low (10-25) | 29.38 | High uncertainty | | 36 | 15 | 5.5 | 100 | (36 * 5.5 + 15 * 50)/51 = 33.29 | Bit low (25-40) | 43.05 | High uncertainty |
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| | 51 | 0 | 5.5 | NA (no member to rank) | (51 * 5.5 + 0)/51 = 5.5 | Extreme low (<10) | 0 | Low uncertainty |
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| title | Table with 30% of 0-value in climatology. Click here to expand... |
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| | Number of 0 members | Number of non-0 members | Average rank of 0 members | Average rank of non-0 members | Rank-mean | Forecast anomaly category | Rank-std | Uncertainty category |
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| 0 | 51 | NA (no member to rank) | 31 (the lowest possible rank for a non-zero member if 1-30 percentiles in the climatology are 0) | (0 * 15.5 + 51 * 31)/51 = 31 | Bit low (25-40) | 0 | Low uncertainty | | 0 | 51 | NA | 50 | (0 * 15.5 + 51 * 50)/51 = 50 | Near normal (40-60) | 0 | Low uncertainty | | 0 | 51 | NA | 70 | (0 * 15.5 + 51 * 70)/51 = 70 | Bit high (60-75) | 0 | Low uncertainty | | 0 | 51 | NA | 100 | (0 * 15.5 + 51 * 100)/51 = 100 | Extreme high (90<) | 0 | Low uncertainty |
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| | 11 | 40 | 15.5 | 31 | (11 * 15.5 + 40 * 31)/51 = 27.65 | Bit low (25-40) | 6.37 | Low uncertainty | | 11 | 40 | 15.5 | 50 | (11 * 15.5 + 40 * 50)/51 = 42.55 | Near normal (40-60) | 14.18 | Medium uncertainty | | 11 | 40 | 15.5 | 70 | (11 * 15.5 + 40 * 70)/51 = 58.24 | Near normal (40-60) | 22.41 | High uncertainty | | 11 | 40 | 15.5 | 100 | (11 * 15.5 + 40 * 100)/51 = 81.77 | High (75-90) | 34.75 | High uncertainty |
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| | 21 | 30 | 15.5 | 31 | (21 * 15.5 + 30 * 31)/51 = 24.61 | Low (10-25) | 7.62 | Low uncertainty | | 21 | 30 | 15.5 | 50 | (21 * 15.5 + 30 * 50)/51 = 35.79 | Bit low (25-40) | 16.97 | Medium uncertainty | | 21 | 30 | 15.5 | 70 | (21 * 15.5 + 30 * 70)/51 = 47.55 | Near normal (40-60) | 26.82 | High uncertainty | | 21 | 30 | 15.5 | 100 | (21 * 15.5 + 30 * 100)/51 = 65.20 | Bit high (60-75) | 41.58 | High uncertainty |
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| | 36 | 15 | 15.5 | 31 | (36 * 15.5 + 15 * 31)/51 = 20.05 | Low (10-25) | 7.06 | Low uncertainty | | 36 | 15 | 15.5 | 50 | (36 * 15.5 + 15 * 50)/51 = 25.64 | Bit low (25-40) | 15.71 | Medium uncertainty | | 36 | 15 | 15.5 | 70 | (36 * 15.5 + 15 * 70)/51 = 31.52 | Bit low (25-40) | 24.83 | High uncertainty | | 36 | 15 | 15.5 | 100 | (36 * 15.5 + 15 * 100)/51 = 40.35 | Near normal (40-60) | 38.50 | High uncertainty |
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| | 51 | 0 | 15.5 | NA (no member to rank) | (51 * 15.5 + 0)/51 = 15.5 | Low (10-25) | 0 | Low uncertainty |
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Example No-1: No zero section in climatology, 5 number/rank groups:
In these simplified examples, a fixed portion of the climatological and/or ensemble forecast distribution is 0, while the non-zero ensemble members have just one other rank for the very dry cases and 5 for the non-zero cases, with members having the same rank in each group. This way, the computation methodology can be demonstrated in a simple way that is easier to interpret.
in a Below, there are examples with simplified rank distributions and specific cases of very dry rivers (0 values) that will demonstrate how the forecast anomaly category and uncertainty category generation work.
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).
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Example No-2: Lowest 10% of the climatology is 0, 2 number/rank groups:
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