introductory study : water indicators and statistical analysis of the hydrological data east of...
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INTRODUCTORY STUDY : WATER INDICATORS AND STATISTICAL ANALYSIS
OF THE HYDROLOGICAL DATA EAST OF
GUADIANA RIVER by
Nikolas Kotsovinos ,P. Angelidis , V. Hrissanthou , and A. Pechtelidis
Democritus University of Thrace -School of Engineering
Greece
PART II
SOME STATISTICAL ANALYSIS
AMARELEJA STATION
We choose one station (AMARELEJA) to test
rigorously the standardization procedure (probability
transformation) assuming normal, log–normal, and
gamma statistics for precipitation.
AMARELEJA station: Annual precipitation for the period 1932-2005.
Mean annual precipitation = 523 mm
Standard Deviation / mean = 0.30
0
100
200
300
400
500
600
700
800
900
1931
-219
33-4
1935
-619
37-8
1939
-40
1941
-219
43-4
1945
-619
47-8
1949
-50
1951
-219
53-4
1955
-619
57-8
1959
-60
1961
-219
63-4
1965
-619
67-8
1969
-70
1971
-219
73-4
1975
-619
77-8
1979
-80
1981
-219
83-4
1985
-619
87-8
1989
-90
1991
-219
93-4
1995
-619
97-8
1999
-00
2001
-220
03-4
HYDROLOGICAL YEAR
AN
NU
AL
PR
EC
IPIT
AT
ION
(m
m)
ANNUAL PRECIPITATION (mm) MEAN ANNUAL PRECIPITATION (mm)
AMARELEJA station: Mean monthly precipitation and standard deviation for the period 1932-2005
The standard deviation is very high
MEAN MONTHLY PRECIPITATIONAMARELEJA: 1932 -2005
65
53
60
52
38
22
3 3
24
6267
73
0
10
20
30
40
50
60
70
80
JA
N
FE
B
MA
R
AP
R
MA
Y
JO
YN
JO
YL
AU
G
SE
P
OC
T
NO
V
DE
C
PR
EC
IPIT
AT
ION
(m
m)
MEAN MONTHLY PRECIPITATION AVERAGE STANDARD DEVIATION
AMARELEJA station: The ratio of (Standard deviation / Mean monthly precipitation)
for the period 1932-2005.The ratio varies from 0.65 to 2.5.
MONTHLY PRECIPITATIONAMARELEJA: 1932 -2005
0.0
0.5
1.0
1.5
2.0
2.5
JA
N
FE
B
MA
R
AP
R
MA
Y
JO
YN
JO
YL
AU
G
SE
P
OC
T
NO
V
DE
C
ST
AN
DA
RD
DE
VIA
TIO
N /
ME
AN
.
Data and theoretical cumulative probability distributions formonthly precipitation at AMARELEJA station. Period 1932-2005.We tested three theoretical cumulative probability distributions:GAMMA LOG-NORMAL NORMALThe fit is not very good for the above three theoretical distributions
CUMULATIVE PROBABILITYFOR MONTHLY PRECIPITATION
AMARELEJA 1932 - 2005
0,00
0,10
0,20
0,30
0,40
0,50
0,60
0,70
0,80
0,90
1,00
0 50 100 150 200 250 300 350
MONTHLY PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
PART III
COMPUTING SPI
The SPI is computed by fitting a probability density function to the frequency distribution of precipitation summed over the time scale of interest. Each probability density function is then transformed into the standardized normal distribution.
Calculation:The monthly precipitation time series are modelled using different statistical distributions.
1. The first is the gamma distribution, whose probability density function is defined as
where
The calculations are quite complicated
2. Another possibility is the log–normal distribution. It has the advantage of simplicity since it is just a logarithmic transformation of the data, i.e. Y = ln(x) (for x > 0), with the assumption that the resulting transformed data are described by a Gaussian distribution.
3. Normal distribution. The central limit theorem suggests that, as we move to extended time periods in excess of 6 months, the resultant time averaging will tend to shift the observed probability distributions towards normal. Because the gamma distribution tends towards the normal as the shape parameter α tends to infinity, it would be computationally more efficient to standardize the data directly from a fitted normal distribution.
We computed the multi-temporal SPI values by modelling the precipitation data with three different statistical distributions.
•GAMMA•LOG-NORMAL•NORMAL
We concluded, that we can use for simplicity LOG-NORMAL or NORMAL distribution instead of GAMMA producing almost the same results
SPI OF 24 MONTHS
AMARELEJA STATION. Period 1932 -2005Data and three theoretical cumulative probability distributions
for the running sum of 24 months. The fit seems quite good for all the three theoretical distributions.
CUMULATIVE PROBABILITYFOR RUNNING SUM OF 24 MONTHS
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 200 400 600 800 1000 1200 1400 1600 1800 2000
RUNNING SUM OF 24 MONTHS PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION. Period 1932 -2005SPI 24 for GAMMA, LOG-NORMAL and NORMAL distribution.
The results are very close
SPI 24
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05GAMMA
SPI 24
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05LOG-NORMAL
SPI 24
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
NORMAL
AMARELEJA STATION. Period 1932 -2005Comparison of SPI 24 for GAMMA, LOG-NORMAL
and NORMAL distribution.The results are essentially coincided
SPI 24
-4
-3
-2
-1
0
1
2
3
41
93
21
93
31
93
41
93
51
93
61
93
71
93
81
93
91
94
01
94
11
94
21
94
31
94
41
94
51
94
61
94
71
94
81
94
91
95
01
95
11
95
21
95
31
95
41
95
51
95
61
95
71
95
81
95
91
96
01
96
11
96
21
96
31
96
41
96
51
96
61
96
71
96
81
96
91
97
01
97
11
97
21
97
31
97
41
97
51
97
61
97
71
97
81
97
91
98
01
98
11
98
21
98
31
98
41
98
51
98
61
98
71
98
81
98
91
99
01
99
11
99
21
99
31
99
41
99
51
99
61
99
71
99
81
99
92
00
02
00
12
00
22
00
32
00
42
00
5
GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION: Comparison of SPI 24 for GAMMA distribution:a) Period 1932-2005b) Period 1932-1968c) Period 1969-2005
Trend in SPI values indicates that the drought conditions has changed significantly during last decades
SPI 24
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.0019
3219
3319
3419
3519
3619
3719
3819
3919
4019
4119
4219
4319
4419
4519
4619
4719
4819
4919
5019
5119
5219
5319
5419
5519
5619
5719
5819
5919
6019
6119
6219
6319
6419
6519
6619
6719
6819
6919
7019
7119
7219
7319
7419
7519
7619
7719
7819
7919
8019
8119
8219
8319
8419
8519
8619
8719
8819
8919
9019
9119
9219
9319
9419
9519
9619
9719
9819
9920
0020
0120
0220
0320
0420
05
GAMMA 1932-2005 GAMMA 1932-1968 GAMMA 1969-2005 Linear (GAMMA 1932-2005)
SPI OF 12 MONTHS
AMARELEJA STATION. Period 1932 -2005Data and three theoretical cumulative probability distributions
for the running sum of 12 months. The fit seems quite good for all the three theoretical distributions
CUMULATIVE PROBABILITYFOR RUNNING SUM OF 12 MONTHS
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 100 200 300 400 500 600 700 800 900 1000
RUNNING SUM OF 12 MONTHS PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION. Period 1932 -2005SPI 12 for GAMMA, LOG-NORMAL and NORMAL distribution.
The results are very close
SPI 12
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
GAMMA
SPI 12
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
LOG-NORMAL
SPI 12
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05NORMAL
AMARELEJA STATION. Period 1932 -2005Comparison of SPI 12 for GAMMA, LOG-NORMAL and
NORMAL distribution.The results are essentially coincided
SPI 12
-4
-3
-2
-1
0
1
2
3
41
93
21
93
31
93
41
93
51
93
61
93
71
93
81
93
91
94
01
94
11
94
21
94
31
94
41
94
51
94
61
94
71
94
81
94
91
95
01
95
11
95
21
95
31
95
41
95
51
95
61
95
71
95
81
95
91
96
01
96
11
96
21
96
31
96
41
96
51
96
61
96
71
96
81
96
91
97
01
97
11
97
21
97
31
97
41
97
51
97
61
97
71
97
81
97
91
98
01
98
11
98
21
98
31
98
41
98
51
98
61
98
71
98
81
98
91
99
01
99
11
99
21
99
31
99
41
99
51
99
61
99
71
99
81
99
92
00
02
00
12
00
22
00
32
00
42
00
5
GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION:Comparison of SPI 12 for GAMMAdistribution:a) Period 1932-2005b) Period 1932-1968c) Period 1969-2005
Trend in SPI values indicates that the drought conditions has changed significantly during last years
SPI 12
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
GAMMA 1932-2005 GAMMA 1932-1968 GAMMA 1969-2005 Linear (GAMMA 1932-2005)
SPI OF 6 MONTHS
AMARELEJA STATION. Period 1932 -2005Data and three theoretical cumulative probability distributions
for the running sum of 6 months.The fit seems quite good for all the three theoretical distributions
CUMULATIVE PROBABILITYFOR RUNNING SUM OF 6 MONTHS
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 100 200 300 400 500 600 700
RUNNING SUM OF 6 MONTHS PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION. Period 1932 -2005SPI 6 for GAMMA, LOG-NORMAL and NORMAL distribution.
The results are to close, especially for GAMMA and LOG-NORMAL distributions
SPI 6
-5
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
GAMMA
SPI 6
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
LOG-NORMAL
SPI 6
-4
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
19
69
19
70
19
71
19
72
19
73
19
74
19
75
19
76
19
77
19
78
19
79
19
80
19
81
19
82
19
83
19
84
19
85
19
86
19
87
19
88
19
89
19
90
19
91
19
92
19
93
19
94
19
95
19
96
19
97
19
98
19
99
20
00
20
01
20
02
20
03
20
04
20
05
NORMAL
AMARELEJA STATION. Period 1932 -2005Comparison of SPI 6 for GAMMA, LOG-NORMAL
and NORMAL distribution.The results are essentially coincided,
especially for GAMMA and LOG-NORMAL distributions
SPI 6
-6
-5
-4
-3
-2
-1
0
1
2
3
41
93
21
93
31
93
41
93
51
93
61
93
71
93
81
93
91
94
01
94
11
94
21
94
31
94
41
94
51
94
61
94
71
94
81
94
91
95
01
95
11
95
21
95
31
95
41
95
51
95
61
95
71
95
81
95
91
96
01
96
11
96
21
96
31
96
41
96
51
96
61
96
71
96
81
96
91
97
01
97
11
97
21
97
31
97
41
97
51
97
61
97
71
97
81
97
91
98
01
98
11
98
21
98
31
98
41
98
51
98
61
98
71
98
81
98
91
99
01
99
11
99
21
99
31
99
41
99
51
99
61
99
71
99
81
99
92
00
02
00
12
00
22
00
32
00
42
00
5
GAMMA LOG-NORMAL NORMAL
SPI OF 3 MONTHS
AMARELEJA STATION. Period 1932 -2005Empirical and three theoretical cumulative probability distributions
for the running sum of 3 months. The fit is not very good for Log-normal distribution.
CUMULATIVE PROBABILITYFOR RUNNING SUM OF 3 MONTHS
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 100 200 300 400 500 600 700
RUNNING SUM OF 3 MONTHS PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
AMARELESA STATION. Period 1932 -2005SPI 3 for GAMMA, LOG-NORMAL and NORMAL distribution.
The results for GAMMA and LOG-NORMAL are very close
SPI 3
-5
-4
-3
-2
-1
0
1
2
3
4
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
GAMMA
SPI 3
-4
-3
-2
-1
0
1
2
3
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
LOG-NORMAL
SPI 3
-4.00
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
1932
1933
1934
1935
1936
1937
1938
1939
1940
1941
1942
1943
1944
1945
1946
1947
1948
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
1963
1964
1965
1966
1967
1968
1969
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
NORMAL
CONCLUSIONS
We concluded, that we can use for simplicity LOG-NORMAL or NORMAL distribution instead of GAMMA producing almost the same results.
We had analyzed a long data series of monthly precipitation for the period 1932 – 2005. Are the above conclusion valid for sorter data series?
In order to answer this question, we made the same analysis:a) For the half data series (1932-1968)b) For the 10% of the data series, e.g. 1932 – 1939.
LETS TAKE THE HALF DATA SETS 1932 - 1968
AMARELEJA STATION. Period 1932 -1968Data and three theoretical cumulative probability distributions
for the running sum of 24 months. The fit seems quite good for all the three theoretical distributions
CUMULATIVE PROBABILITYFOR RUNNING SUM OF 24 MONTHS
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
600 800 1000 1200 1400 1600 1800
RUNNING SUM OF 24 MONTHS PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION. Period 1932 -1968Comparison of SPI 24 for GAMMA, LOG-NORMAL
and NORMAL distribution.The SPI values are essentially coincided
SPI 24
-3
-2
-1
0
1
2
3
419
3219
3319
3419
3519
3619
3719
3819
3919
4019
4119
4219
4319
4419
4519
4619
4719
4819
4919
5019
5119
5219
5319
5419
5519
5619
5719
5819
5919
6019
6119
6219
6319
6419
6519
6619
6719
68
GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION. Period 1932 -1968Data and three theoretical cumulative probability distributions
for the running sum of 12 months. The fit is good enough for all the three theoretical distributions
CUMULATIVE PROBABILITYFOR RUNNING SUM OF 12 MONTHS
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 100 200 300 400 500 600 700 800 900 1000
RUNNING SUM OF 12 MONTHS PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION. Period 1932 -1968Comparison of SPI 12 for GAMMA, LOG-NORMAL
and NORMAL distribution.The SPI values are essentially coincided
SPI 12
-3
-2
-1
0
1
2
3
19
32
19
33
19
34
19
35
19
36
19
37
19
38
19
39
19
40
19
41
19
42
19
43
19
44
19
45
19
46
19
47
19
48
19
49
19
50
19
51
19
52
19
53
19
54
19
55
19
56
19
57
19
58
19
59
19
60
19
61
19
62
19
63
19
64
19
65
19
66
19
67
19
68
GAMMA LOG-NORMAL NORMAL
LETS TAKE ONLY 10% OF THE DATA SETS
1932 - 1939
AMARELEJA STATION. Period 1932 -1939Data and three theoretical cumulative probability distributions
for the running sum of 12 months.The fit is not very good for all the three theoretical distributions
CUMULATIVE PROBABILITYFOR RUNNING SUM OF 12 MONTHS
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 100 200 300 400 500 600 700 800 900 1000
RUNNING SUM OF 12 MONTHS PRECIPITATION (mm)
CU
MU
LA
TIV
E P
RO
BA
BIL
ITY
.
DATA GAMMA LOG-NORMAL NORMAL
AMARELEJA STATION. Period 1932 -1939SPI 12 for GAMMA, LOG-NORMAL and NORMAL distribution.
The SPI values are essentially coincided
SPI 12
-3
-2
-1
0
1
2
3
1932
1933
1934
1935
1936
1937
1938
1939GAMMA
SPI 12
-3
-2
-1
0
1
2
3
1932
1933
1934
1935
1936
1937
1938
1939
LOG-NORMAL
`
SPI 12
-3
-2
-1
0
1
2
3
1932
1933
1934
1935
1936
1937
1938
1939
NORMAL
CONCLUSIONS
We can use for simplicity LOG-NORMAL or NORMAL distributions instead of GAMMA, producing almost the same results, especially with GAMMA and LOG-NORMAL.
This conclusion was verified not only for very long data series, but also for sorter ones.
The verification was made for SPI values of 24, 12, 6 and 3 months.?????/////