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Table 2: Uncertainty categories defined by the standard deviation of the ensemble member ranks.
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Simplified forecast examples to help interpreting the expected anomaly and uncertainty category information
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Examples to help interpreting the methodology of the rank computation and the selection of the forecast anomaly and the uncertainty categories
Below, there are examples In this section, examples are given 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.
In these examples, for simplicity reason, 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.
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 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.
in order to demonstrate the rank-mean and rank-std computation for the expected anomaly and uncertainty category calculations. Based on these examples, the users can have a feel on how the rank-mean and rank-std values change with the changing underlying distributions. In addition, the impact of the 0-value in the climatology and in the ensemble forecasts is also demonstrated for different severity of the 0-value problem, with the complexity of these dry cases.
The non-zero value examples highlight that shifting the same rank distribution 'up' or 'down' (i.e. for wetter or drier) does not change the standard deviation (and thus the uncertainty). Also, after 'narrowing' the rank distribution over the same mean value (i.e. making it cluster more), the mean does not change, but the uncertainty drops markedly. Moreover, in a similar manner, by adding extreme members (i.e. near 1 or 100 very extreme members), even if only very few members, the uncertainty will increased quite substantially.
For those forecasts, when some portion or all of the climatological percentiles are 0, it is a general rule of thumb that as the percentage of zero climate percentiles increase, it gets more and more difficult to end up with negative forecast anomalies. The lowest possible rank-mean values are going to happen for forecasts with all 0 values, in which case the forecast rank-mean is going to be determined by the size of the 0-value section of the climatology. For example, if the lowest 30% of the climatology is 0, then the rank-mean of the forecast of all 0-value will be about 15, which will put this forecast in the expected anomaly category of 'Low'. But, with 70% of climatology being 0, say further into the Sahara, there the driest possible ensemble forecast of only 0 values is only going to be rank-mean of 35, so with expected anomaly category of 'Bit low'. Drier than this anomaly is simply physically not possible for such a climatological place.
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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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).
Example No-1 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 substantiallyExample 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 forecast anomaly category 'Extreme high'), regardless of the actual ensemble member values. 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'.
Example No-1: No zero section in climatology, 5 number/rank groups:
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