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Extended Range - CDFs and EFIs

ecChart display.

To complement the EFI and SOT for two parameters in the extended range (introduced with IFS cycle 46r1 in June 2019) a facility to view extended-range Cumulative Distribution Functions (CDFs) for those parameters was also introduced (in February 2020).  Unlike ECMWF's pre-existing CDF-viewing tools, used for 24h periods at shorter ranges, which show absolute values, these Extended range extreme forecast index (EFI), shift of tails (SOT) and cumulative distribution functions (CDF) is available for 2m temperature and precipitation.  These CDFs depict anomalies (relative to the ER-M-Climate distribution).  They  This is unlike ECMWF the CDF used for 24h periods at shorter ranges which show absolute values.  The CDFs cover the following:

Parameters:

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Each CDF plot displays the latest forecast along with the Extended Range Model climate (ERMER-M-climate) corresponding to it, and also all previous extended range ensemble forecasts valid for the same 7-day period. The following percentiles are used to draw both ERMER-M-climate and ensemble forecast CDFs: 0 (minimum), 1, 2, 5, 10, 25, 50 (median), 75, 90, 95, 98, 99 and 100 (maximum). For the forecasts, which have far fewer realisations than the ERMER-M-Climateclimate, percentiles 0, 1 and 2 all take the same value, as do 98, 99 and 100.

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Fig8.2.6.2: The EFI and SOT for 2-metre weekly mean temperature anomalies, and CDF plots of temperature and precipitation anomalies for one site (see green pin). On the CDF widget the black curve represents the ERMER-M-climate and coloured curves represent the different extended-range forecasts valid for the same 7-day period.

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For both precipitation and temperature, zero on the x-axis (and the thicker vertical gridline) simply corresponds to ERMER-M-climate mean values (for the location, the time of year and the lead time displayed), because of course "anomaly" computations use these mean values as their reference points. This statement is true for all of the curves. However, for different lead times (i.e. the different coloured curves) the absolute value that is the mean will vary a little bit (due to model drift and under-sampling). In spite of such variations it is still reasonable, helpful and recommended to inter-compare the single black ERMER-M-climate curve with all the coloured curves (even if this is only strictly valid for the same lead time that it represents - i.e. the red curve).

Note, incidentally, that the ER-M-climate, as used here and on extended range meteograms, is now based on re-forecasts initialised from ERA5 data; this is higher quality output, and has greater compatibility with actual forecasts, than was the case previously when the re-forecasts were initialised from ERA-Interim data (i.e. before model cycle 46r1 was introduced in June 2019).

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For precipitation, the plot design is such that the x-axis always starts from the ERMER-M-climate minimum (black), for the given valid dates, for the lead time of the latest forecast (red). So the black curve should always begin from the graph's lower left corner (Fig. 8.2.6.3a). Note that over the vast majority of the world this will equate to zero precipitation in the 7-day period; it is of course impossible to get less than zero. As the ERMER-M-climate is a function of lead time the forecasts shown for different lead times (not red), which are referenced to (slightly) different ERMER-M-climates to calculate anomalies, may not have the same "zero" starting point. On Fig. 8.2.6.3a for example the end of the dashed purple curve, corresponding to the T+504-672h forecast, probably also corresponds to zero rain, but lies left of the plot area shown (and so is not visible). Equally, such a curve could end just to the right of the lower left point of the graph, but could still correspond to zero rain in absolute terms. It all depends on how the ERMER-M-climate for the given lead time compares with the ERMER-M-Climate climate shown (black curve). In some very wet locations the ERMER-M-climate CDF may have never been 0 (i.e. no completely dry weeks). In this case the x-axis would start from a value that in the ERMER-M-Climate climate (black) does not correspond to zero rain in absolute terms as shown on Fig. 8.2.6.3b.

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Fig8.2.6.3: CDF plots for 7-day total precipitation anomalies. The plot design caters for two scenarios: (a) - which is very common - used when the ERMER-M-Climate climate minimum (black) = 0mm in absolute terms (bottom left corner = 0mm), and (b) - which is very rare - used when the ERMER-M-Climate climate minimum is >0mm. On (a) the purple curve, for example, does not start from "0" (the point where x- and y-axis intersect) because of lead time dependence of the M-climate. A similar situation in which a curve "disappears" off the left of the plot could occur in case (b), but has not done so on this example.

For temperature, the x-axis starts from the overall minimum encountered within all the displayed CDFs (ERMER-M-climate and ensemble forecasts).

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Fig8.2.6.5: ecCharts of Extended Range EFI and SOT (quantile 10) for weekly T2m anomaly, from DT:00UTC 25 Jul 2019, for: Fig8.2.6.5A: VT:week ending 5 Aug 2019 (left),  Fig8.2.6.5B: VT:week ending 29 Aug 2019 (right).  Purple area on map shows where EFI is below -0.9.  SOT values (quantile 10) are shown in green boxes, the feint line is where SOT=1. Actual EFI and SOT values at the green pin site are shown in the lowest white box. The extended range meteogram with ERMER-M-climate (red), and the time-series (blue) of both EFI and SOT (quantile 10) values for Nizhniy Novgorod illustrate the expected evolution.

• Fig8.2.6.5A (week ending 5 Aug 2019) highlights an area where 2m temperatures are expected to be very much on the cold side of the ERMER-M-Climate climate distribution (as taken from re-forecasts).  The CDF diagram at Fig 8.2.6.6D would be similar to that for Nizhniy Novgorod.
• Fig8.2.6.5B (week ending 29 Aug 2019) shows that the distribution of possible mean 2m temperatures is overall close to the ERMER-M-Climate climate distribution, implying that the actual model forecast on this occasion is unable to add much to a (model-free) forecast that purely reflects climatological probabilities.  The CDF diagram at Fig 8.2.6.6C would be similar to that for Nizhniy Novgorod.

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• Dashed black isopleths show SOT values associated with the 10% of ENS results (quantile 10) showing the coldest temperatures.
• Solid black isopleths show SOT values associated with the 10% of ENS results (quantile 90) showing the warmest temperatures or most precipitation.

Extended Range CDFs.

Cumulative Distribution Function (CDF)s for ensemble temperature and rainfall forecasts may be constructed from ensemble extended range forecasts.   It is important to note that here it is anomalies from the "norm" that are considered rather than absolute temperature or rainfall values.  The anomalies for the extended range climate (ER-M-climate) (black line) are the frequencies of departures from the mean (here defined as the "norm") of the ER-M-climate for the date in question (i.e.the light green lines on the diagram indicate the value at 50% probability and marked as 0°C anomaly).   Some anomalies are positive, some in the tails of the plot are extremely positive; some are negative, some some in the tails of the plot extremely negative.   The CDF for the ensemble values is constructed from the anomaly of the temperature forecast by each ensemble member (red line) as a departure from the mean or "norm" of the ER-M-climate.

Extreme Forecast Index (EFI) and Shift of Tails (SOT) are derived in the same way as for the medium range products.

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Fig8.2.6.7B: The moderate positive EFI suggests the ensemble rainfall anomaly distribution is slightly above the ERMER-M-climate anomaly distribution.  The negative upper tail SOT (quantile 90) indicates there are very few if any ensemble members predicting extreme equivalent rainfall anomalies (above the 99th ER-M-climate percentile shown by the dashed green line).  This suggests uncertainty that larger than normal rainfall anomaly may be forecast (moderate EFI - but note 70% of ensemble members forecast less than about 1mm precipitation, equally 15% of ensemble members forecast more than about 2mm precipitation), but it is unlikely there will be an exceptional rainfall event (SOT -0.6).

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Reliability diagrams in Extended Range.

Reliability diagrams are available for extended range forecasts. This gives an assessment of the current model characteristics and allows some indication of the confidence one can have in the evolution shown within the extended ranges - unless there is good evidence to the contrary (e.g. a major change from previous forecasts in the evolution within the medium range).

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Fig8.2.6.9: Example of Cost/Loss diagrams (here at week1 and week5) illustrating two things in particular: that potential economic value for all users reduces as lead time increases, and that the range of users for whom the forecasts can have some intrinsic economic value reduces markedly as lead time increases.

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See Section 12B for description of the Reliability diagrams and ROC diagrams and interpretation of Cost/Loss diagrams.

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