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There is a consequence of the 0-value problem treatment over dry or very dry areas (described above), as some or all of the low anomaly signals will not be impossible to occur. Please consider some of these idealised examples , with as a help to demonstrate the situation with the 0-value problem. In these, for simplicity reason, a fixed portion of the ensemble members are being 0 and the rest having a simple average rank, in order to simplify . This way the rank-mean and dominant anomaly category computation :

...

is simpler and easier to interpret.

In the below examples, is X% of the climatology is 0, then the average rank of the

...

ensemble members which are also 0 will always be

...

X+0.5 with the evenly distributed rank representation (e.g.  for 10%, it is 1-10 evenly spread, so the average of those members will be 5.5).

Example No-1: Lowest 10% of the climatology is 0:

Number of 0 membersNumber of non-0 membersAverage rank of 0 membersAverage rank of non-0 membersRank-meanDominant anomaly category
051NA (no member to rank)11 (the lowest possible rank for a non-zero member if 1-10 percentiles in the climatology are 0)(0 * 5.5 + 51 * 11)/51 = 11Low (10-
25%
25)
051NA20(0 * 5.5 + 51 * 20)/51 = 20Low (10-
25%
25)
051NA50(0 * 5.5 + 51 * 50)/51 = 50Near normal (40-
60%
60)
051NA70(0 * 5.5 + 51 * 70)/51 = 70Bit high (60-
75%
75)
051NA100(0 * 5.5 + 51 * 100)/51 = 100Extreme high (
90%<
90<)






11405.511(11 * 5.5 + 40 * 11)/51 = 9.81Extreme low (
<10%
<10)
11405.520(11 * 5.5 + 40 * 20)/51 = 16.87Low (10-
25%
25)
11405.550(11 * 5.5 + 40 * 50)/51 = 40.40Near normal (40-
60%
60)
11405.570(11 * 5.5 + 40 * 70)/51 = 56.08Near normal (40-
60%
60)
11405.5100(11 * 5.5 + 40 * 100)/51 = 79.61High (75-
90%
90)






21305.511

(21 * 5.5 + 30 * 11)/51 = 8.73

Extreme low (
<10%
<10)
21305.520(21 * 5.5 + 30 * 20)/51 = 14.02Low (10-
25%
25)
21305.550(21 * 5.5 + 30 * 50)/51 = 31.67Bit low (25-
40%
40)
21305.570(21 * 5.5 + 30 * 70)/51 = 43.44Near normal (40-
60%
60)
21305.5100(21 * 5.5 + 30 * 50)/51 = 61.08Bit high (60-
75%
75)






36155.511

(36 * 5.5 + 15 * 11)/51 = 7.11

Extreme low (
<10%
<10)
36155.520(36 * 5.5 + 15 * 20)/51 = 9.76Extreme low (
<10%
<10)
36155.550(36 * 5.5 + 15 * 50)/51 = 18.58Low (10-
25%
25)
36155.570(36 * 5.5 + 15 * 70)/51 = 24.47Low (10-
25%
25)
36155.5100(36 * 5.5 + 15 * 50)/51 = 33.29Bit low (25-
40%
40)






5105.5NA (no member to rank)

(51 * 5.5 + 0)/51 = 5.5

Extreme low (
<10%
<10)

Example No-2

...

: Lowest 30% of the climatology is 0:

Number of 0 membersNumber of non-0 membersAverage rank of 0 membersAverage rank of non-0 membersRank-meanDominant anomaly category
051NA (no member to rank)31 (the lowest possible rank for a non-zero member if 1-30 percentiles in the climatology are 0)(0 * 15.5 + 51 * 31)/51 = 31Bit low (25-
40%
40)
051NA50(0 * 15.5 + 51 * 50)/51 = 50Near normal (40-
60%
60)
051NA70(0 * 15.5 + 51 * 70)/51 = 70Bit high (60-
75%
75)
051NA100(0 * 15.5 + 51 * 100)/51 = 100Extreme high (
90%<
90<)






1140
5
15.5
11
31(11 * 15.5 + 40 * 31)/51 = 27.65Bit low (25-
40%
40)
1140
5
15.550(11 * 15.5 + 40 * 50)/51 = 42.55Near normal (40-
60%
60)
1140
5
15.570(11 * 15.5 + 40 * 70)/51 = 58.24Near normal (40-
60%
60)
1140
5
15.5100(11 * 15.5 + 40 * 100)/51 = 81.77High (75-
90%
90)






2130
5
15.5
11
31

(21 * 15.5 + 30 * 31)/51 = 24.61

Low (10-
25%
25)
2130
5
15.550(21 * 15.5 + 30 * 50)/51 = 35.79Bit low (25-
40%
40)
2130
5
15.570(21 * 15.5 + 30 * 70)/51 = 47.55Near normal (40-
60%
60)
2130
5
15.5100

(21 * 15.5 + 30 * 100)/51 = 65.20

Bit high (60-
75%
75)






3615
5
15.5
11
31

(36 * 15.5 + 15 *

11

31)/51 = 20.05

Low (10-
25%
25)
3615
5
15.550(36 * 15.5 + 15 * 50)/51 = 25.64Bit low (25-
40%
40)
3615
5
15.570(36 * 15.5 + 15 * 70)/51 = 31.52Bit low (25-
40%
40)
3615
5
15.5100(36 * 15.5 + 15 * 100)/51 = 40.35Near normal (40-
60%
60)






510
5
15.5NA (no member to rank)

(51 * 15.5 + 0)/51 = 15.5

Low (10-
25%
25)


Example No-3

...

: Lowest 70% of the climatology is 0:

Number of 0 membersNumber of
Number of 0 membersNumber of
non-0 membersAverage rank of 0 membersAverage rank of non-0 membersRank-meanDominant anomaly category
051NA (no member to rank)71 (the lowest possible rank for a non-zero
member
members if 1-
30
70 percentiles in the climatology are 0)(0 *
15
35.5 + 51 *
31
71)/51 =
31
71Bit
low
high (
25
60-
40%
75)
051NA
50
100(0 *
15
35.5 + 51 *
50
100)/51 =
50
100
Near normal (40-60%
Extreme high (90<)
0






11
51
40
NA
35.5
70
71(
0
11 *
15
35.5 +
51
40 *
70
71)/51 =
70
63.34Bit high (60-
75%
75)
051
NA100(0 * 15.5 + 51 * 100)/51 = 100Extreme high (90%<)
1140
5
35.5
11
100(11 *
15
35.5 + 40 *
31
100)/51 =
27
86.
65
08
Bit low
High (
25
75-
40%
90)
11






21
40
30
5
35.5
50
11

(

11

21 *

15

35.5 +

40

30 *

50

71)/51 =

42

56.

55Near normal (40-60%)11405.570(11 * 15.5 + 40 * 70)/51 = 58.24Near normal (40-60%)11405.5100(11 * 15.5 + 40 * 100)/51 = 81.77High (75-90%)21305.511

(21 * 15.5 + 30 * 31)/51 = 24.61

Low (10-25%)21305.550(21 * 15.5 + 30 * 50)/51 = 35.79Bit low (25-40%)21305.570(21 * 15.5 + 30 * 70)/51 = 47.55Near normal (40-60%)21305.5100

(21 * 15.5 + 30 * 100)/51 = 65.20

Bit high (60-75%)36155.511

(36 * 15.5 + 15 * 11)/51 = 20.05

Low (10-25%)36155.550(36 * 15.5 + 15 * 50)/51 = 25.64

38

Near normal (40-60)
213035.5100

(21 * 35.5 + 30 * 100)/51 = 73.44

Bit high (60-75
Bit low (25-40%
)






3615
5
35.5
70
11

(36 *

15

35.5 + 15 *

70

71)/51 =

31

45.

52

94

Bit low
Near normal (
25
40-
40%
60)
3615
5
35.5100(36 *
15
35.5 + 15 * 100)/51 =
40
54.
35
47Near normal (40-
60%
60)






510
5
35.5NA (no member to rank)

(51 *

15

35.5 + 0)/51 =

15

35.5

Low
Bit low (
10
25-
25%
40)




  • If all ensemble members are 0, then the ranks will spread evenly from 1 to 10 and the rank-mean will be around 5.5, so in the Extreme low category.

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