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This page describes the way the anomaly and uncertainty of the ensemble forecasts in the sub-seasonal and seasonal products are determined using the climatology as reference. This includes also how the dominant forecast anomaly category (the 'dominant' of the 7 predefined ones) and the uncertainty category (of the 3 predefined ones) of the ensemble forecasts are determined. This is a generic procedure, which is the same for both EFAS and GloFAS, as it is executed the same way for each river pixel, regardless of the resolution, and also the same for the sub-seasonal and seasonal products, as it works in the exact same way regardless of whether it is weekly mean values, as in the sub-seasonal, or monthly mean values, as in the seasonal.

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Climatological percentiles and forecast anomaly categories

Currently, From the sub-seasonal climate sample uses 660 reforecast values, while the seasonal uses 500 values. From the climate sample 99 climate percentiles are determined, which represent equally likely (1% chance) segments of the river discharge value range that occurred in the 20-year climatological sample (both sub-seasonal and seasonal is currently based on 20 years). Figure 1 shows an example idealised generic climate distribution, either based on weekly means or monthly means, with the percentiles represented along the y-axis. Only the deciles (every 10%), the quartiles (25%, 50% and 75%), of which the middle (50%) is also called median, and few of the extreme percentiles are indicated near the minimum and maximum of the climatological range indicated by black crosses. Each of these percentiles have an equivalent river discharge value along the x-axis. From one percentile to the next, the river discharge value range is divided into 100 equally likely bins (separated by the percentiles), some of which is indicated in Figure 1, such as bin1 of values below the 1st percentile, bin2 of values between the 1st and 2nd percentiles or bin 100 of river discharge values above the 99th percentiles, etc. 

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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 perfectly 'normal' condition. 

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Forecast anomaly category computation for the whole ensemble

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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 dominant forecast anomaly category (one of the 7 categories in Table 1) for the 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 normal', 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 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 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 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 24. Schematic of the forecast extremity ranking of the 51 ensemble members and the calculation of the dominant forecast anomaly category for the whole ensemble.

Forecast uncertainty category computation for the whole ensemble

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In addition to the dominant forecast anomaly computation for the whole ensemble, 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.

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For forecast uncertainty, three uncertainty categories are defined based on the rank-std value of the ensemble forecasts, using the easy to remember generic category split values of 10 and 20. (Table 2). These work well enough for all lead times and geographical areas.

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

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interpreting the methodology of the rank computation and the selection of the

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forecast 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 forecast anomaly category and uncertainty category computation 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.

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Example 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 dominant anomaly forecast anomaly category 'Extreme high'), regardless of the actual ensemble member values. So, even if all the forecast ensemble member river discharge is values are very low, say from 0.12 to 0.23, the forecast rank-mean will still be 100 and the dominant forecast anomaly category 'Extreme high'.

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