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This section describes the way the anomaly and uncertainty of the ensemble forecast forecasts are determined, using the climatology as reference. And generally, how the probability dominant one of the 7 anomaly categories and the three uncertainty categories 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 on the weekly (in the exact same way regardless of whether it is weekly mean values, as in the sub-seasonal) , or monthly (seasonal) mean discharge values the same way again.values, as in the seasonal.

The characterisation of the forecast signal in both the sub-seasonal and seasonal is based on the ensemble member's extremity in the context of the model climatological distribution. 

Climatological percentiles and forecast anomaly categories

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Based on the percentiles and the related 100 bins, there are seven categories defined, which will be used as anomaly categories defined (Table 1). These are also indicated in Figure 1 by shading. The two most extreme categories are the bottom and top 10% of the climatological distribution (<10% as red and 90%< as blue). Then the moderately low and high river discharge categories from 10-25% (orange) and 75-90% (middle-dark blue). The smallest negative and positive anomalies are defined by 25-40% and 60-75% and displayed by yellow and light blue colours in Figure 1. Finally, the normal condition category is defined as 40-60%, so the middle 1/5th of the distribution, coloured grey in Figure 1.

Table 1: Definition and description of the 7 anomaly categories.The possible value ranges in the 'Ranks column' are inclusive at the start and exclusive at the end, so for example for Cat-1 the possible ranks are 1, 2, 3, ... and 10. Depending on the products, sometimes the middle three categories (Cat3, Cat-4 and Cat-5) are combined into one extended 'Near normal' category.

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The forecast has 51 ensemble members, again for both EFAS/GloFAS and both sub-seasonal or seasonal, regardless. The members are all checked for climatological extremity and placed in one of the 100 climate bins. This will be the anomaly or extremity level of the ensemble members, which can be called hereafter rank, as one of the values from 1 to 100. For example, 1 will mean the forecast value is below the 1st climate percentile (i.e. extremely anomalously low), then 2 will mean the value is between the 1st and 2nd climate percentiles (i.e. slightly less extremely low), etc., and finally 100 will mean the forecast value is above the 99th climate percentile (i.e. extremely high as higher than 99% of all the considered reforecasts (representing the model climate conditions for this time of year, location and lead time).

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The probability of the 7 anomaly categories is calculated by the count of ensemble members in each category and then dividing by 51, the total number of members. In the example of Figure 2, there is no member in the 3 low flow anomaly categories, while the 'Near normal' category has 2, resulting in 3.9% probability, the bit 'Bit high' category 13, with 27.5%, the high 'High' category 17, as 33.3%, and finally the extreme 'Extreme high' category has 18 ensemble members, with 35.3% probability. The inset table in Figure 2 show shows the numbers and the probability, but also shows the size (in terms of probabilities) of the 7 categories. This highlights, e.g., that the normal flow category's 3.9% probability is much lower than the climatologically expected probability of 20%, however, the 3 highe high flow categories have much higher probability probabilities than the climatological reference probability, especially the extreme high category, where the forecast probability (35.3%) is more than double the corresponding climatological probability (15%).

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The forecast extremity rank computation can be done for any value above 0 m3/s. However, it becomes undefined when the values drop to 0, as there is no way to differentiate the rank for the same value. The simulations are less reliable when we approach 0, so everything below 0.1 m3/s will be considered as 0 for the sub-seasonal and seasonal products. This problem can also happen for non-zero values, but normally the simulation should not produce a lot of identical non-zero values, unless there is some specific process, like reservoir operation rule, etc., which might generate such signal. There is no indication that the non-zero constant value is an issue at all, but it is clear that the 0 values value is actually a major problem, as large parts of the world has dry enough areas often combined with small enough catchments to have near zero or totally 0 river discharge values.

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Figure 3 demonstrates the process on an example, where the lowest 77 percentiles are 0 in the climatology and 23 out of 51 ensemble members are also 0 (see Figure ea3a). The 23 0 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 by as equal spacing 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. The schematic with the ranks of all 51 members are provided in Figure 3d.

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The ensemble forecasts have 51 members, which will be assigned an extremity rank each. Using these 51 ranks the forecast needs to get assigned forecasts will be put in one of the 7 anomaly categories. This is done with the arithmetic based on the arithmetic mean of the 51 ensemble member rank values (rank-mean). 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 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 or 100 (the most extremely dry/wet).  ThenThen, based on this rank-mean, we define the 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 as 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 would easily change from this distribution to an only very slightly different one 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. Very 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 (although not checked) and both quite near the median.

There 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 be impossible to occur. If only the lowest 10% of the climatological distribution is 0, 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 anythiong 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. For all 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 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 of them will be high enough to fall into one of the high anomaly categories.

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The standard deviation of the even distribution with values ranging from 1 to 100 is (100-1)/sqrt(12) = 28.86, while the most extreme std value is when half of the members are with rank 1 and the other half with rank 99, in which case the std = 49.5. Obviously, the lowest std value is 0, when all ranks are the same. For forecast uncertainty, three uncertainty categories are defined, based on the rank-std value of the ensemble forecasts. Table 2 shows the categories, as defined by the std values of <10, 10<= <20 and 20<=. 

Table 2: Uncertainty categories defined by the standard deviation of the ensemble member ranks.