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The forecast has 51 ensemble members, which all are sorted into one of check where they fall in the 100 bins along of the climate distribution. This bin number will be the anomaly or extremity of each ensemble member, with value, called hereafter a rank, of the ensemble members as one of the values from 1 to 100.   This means, we rank the verifying WB, or the forecast values (every member) in the 99-value percentile model climatology. So, for example +3 would mean, the forecast is between the 52nd and 53rd percentiles, while -21 would mean the forecast is between the 29th and 30th percentiles. The lowest value is -50 (forecast is below the 1st percentile), while the largest is +50 (forecast is larger than the 99th percentile).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).

This is possible to do for values above 0 mean river discharge, however, the rank computation becomes undefined when the values drop to 0. This is actually a major problem, as large parts of the world has dry enough areas combined with small enough catchments to have near zero or totally 0 river discharge values. For this singularity case we have a special treatment. All river discharge values below 0.1 m3/s are treated effectively as 0. All the 0 ensemble member values then get a randomly assigned rank from any of the percentiles that have 0 (i.e. below 0.1 m3/s) values in the model climatology. This effectively means, the 'rank-undefined' section of the ensemble forecast is going to be spread across the 'rank-undefined' section of the climatology during the rank computation.

Example-1:

Consider for example the following hypothetical example in Table 1. For simplicity, we represent the ensemble forecast by 21 members only (instead of 51). We also indicate below which values are considered as 0 (i.e. below 0.1 m3/s). In the 1st part of Table 1, you see the climatological percentile values, whether they are 0 or not, the possible rank values that can be assigned and the range for each of these rank values. Also, you see in the 2nd part in Table 1 the 21 ensemble member mean river discharge values, whether they are 0 or not, and then the assigned rank.

In this example, out of the 99 climate percentiles 58 are 0 (or considered 0), so this section is the undefined section where the rank computation is not directly possible. From the forecast, from the 21 represented ensemble members, 6 are (considered) 0. These 6 members will get ranks that come from the 1-58(/59) section of the 0 climatology, equally spanning it. For this example, the ranks would follow as 0, 58/5, 2*58/5, 3*58/5, 4*58/5 and finally 5*58/5, so 0, 11.6, 23.2, 34.8, 46.4 and 58. However, we can not assign fractional ranks, so we always choose the nearest integer number as rank. In this case, the ranks then would be for these 6 ensemble members: 0, 12, 23, 35, 46 and 58.

For the remaining ensemble members only few more are defined here, and the 61-96 percentiles and the related ranks are not listed here due to the limited space.

Table 1: Hypothetic example-1 of climate distribution and the related forecast ensemble distribution represented with only 21 members.

Example-2:

In this other super extreme example (Table 2) there is no measurable river discharge in the climatology at all. this, for example can happen somewhere in the middle of the desert in the Sahara, where basically all of the daily river discharge values in the longterm reanalysis are effectively 0, and thus all the 99 percentiles of the model climatology are 0.

In the forecast, however, there are few members which have noticeable discharge, so effectively are very extreme in the climatological context, even though the absolute values are not that large necessarily. So, out of the 21 members 17 are 0 (or considered 0), and those 17 members will get ranks assigned from the 0-section of the climatology, which is in this case all the way from 0 to 99. Thus, the ranks will be as follows: 0, 99/16 (so 6 as rounded value), 2*99/16 (12), 3*99/16 (19), ..., 15*99/16 (93) and 16*99/16 (99).

Moreover, all the forecast ensemble members, which are bigger than 0.1 m3/s, will get the rank of 99 for this case with the whole climatology being below 0.1 m3/s.

Table 2: Hypothetic example-2 of climate distribution and the related forecast ensemble distribution represented with only 21 members.

With this special treatment of the 0 singularity values, there is no need to mask areas on the new sub-seasonal and seasonal products. On the old version of the GloFAS seasonal forecasts, areas and time ranges, where the values were too small (effectively 0), were masked and the forecast values simply were not shown as 'undefined'.

For the new revised products, this special treatment will mean all of these 0 or near 0 cases will simply fall into an evenly distributed, very uncertain, on average near normal condition. Just as, following on from the 2nd example above, with no discharge in the climatology and no discharge in the forecast either, we will end up with a unified distribution of ranks from 0 to 99, evenly spanning the whole climatological range. Which means, if we average these ranks, we will get exactly the middle, the climate median, so no anomaly whatsoever.

This also means, for dry or very dry areas, there will never be a dry forecast anomaly, as that does not have any meaning (we can not go below 0), and therefore only neutral or positive anomalies are possible. The magnitude or severity of those positive anomalies then will be determined by the number and distribution of ensemble members being above the non-zero climate percentiles. If enough members will be above the non-zero climate percentiles and thus high enough fraction of the forecast will get high enough ranks, then on average the distribution of the 51 ensemble member ranks will show a pronounced enough shift from the neutral situation (i.e. when everything is 0 and the ranks are evenly distributed and the forecast end up looking totally normal). 

• For the real time forecasts, instead we will have a 51-value ensemble of ranks (-50 to 50). So, for defining the dominant category, we have options:
  • Most populated: We either choose the most populated of the 7 categories. But this will be more problematic for very uncertain cases, when little shifts in the distribution could potentially mean large shift in the categories. For example assuming a distribution of 8/7/8/7/7/7/7 and 6/7/7/7/8/9/7. These two are very much possible for the longer ranges. It is a question anyway, which one to choose in the first example, Cat1 or Cat5, they are equally likely. But then, in the 2nd case we have Cat6 as winner. But the two cases are otherwise very similar.
  • ENS-mean: Alternatively, we can rather use the ensemble mean, and rank the ensemble mean value and define the severity category that way. for the above example, this would not mean a big difference, as the ensemble mean is expected to be quite similar for both forecast distributions, so the two categories will also be either the same, or maybe only one apart. I think this is what we need to do!

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