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Frequency of intense rainfall for the territory of Poland
Journal of Hydrology 5 (1967) 158-162;
II;)
North-Holland Publishing Co ., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
FREQUENCY OF INTENSE RAINFALL FOR THE TERRITORY OF POLAND JULIAN LAMBOR Technical University, Warsaw , Poland
Abstract: On the basis of observations of rainstorms recorded at pluviographic stations the author calculated intensity-duration-frequency formula. For the computation daily data have also been used from stations, of which the observation period exceeds in some cases even 100 years. From records of stations situated in different geographical conditions and at different altitudes above sea level up to the inversion limit, the author finds a systematic variation in the coefficients of the formulas. The formula can be adjusted to different regions by introducing the mean annual precipitation.
In order to determine with a sufficient accuracy the recurrence interval of extreme values of rainfalls, it is necessary to analyse long periods of observation, which allow to carry out sufficiently reliable extrapolation for a period of time which is considerably longer than the period of observation. The longer the period of observation, the more reliable the results. It is particularly difficult to obtain long and continuous series of homogeneous observations. In hydro-meteorological research a 100 year period of observation should be considered as a long one allowing for good interpretation of the observed phenomena. Periods of 50 years should be considered as sufficient to determine the probability of occurrence for various design purposes, whereas periods below 20 to 30 years are too short to base on them certain conclusions, even for practical purposes. The intensity of a rainfall is a function of its period of duration and it is also a function of its frequency or of its probability of occurrence. For arriving at formulas representing these relationships in this study all the available observations carried out on the eleven most important stations, in Poland have been used . The observation periods often exceeded 100 years, and for some stations even exceeded 180 years. The observations comprise the following data: values detailed to I minute duration 1 hour and 24 hour values monthly sum values 158
2138 rainfalls 7836 rainfalls 3838 rainfalls
FREQUENCY OF INTENSE RAINFALL IN POLAND
159
The intensity of rainfall can be described as a function of the time of duration and the probability of occurrence:
I=f(t·p) with
I t
p
(1)
= intensity of rainfall in mm per hour = time of duration in hours = probability of occurrence in per cent, corresponding to the average number of occurrences in one hundred year.
There appears to be a great harmony in these intensity-duration-frequency formulas for different stations. This enables to use one general formula which can be applied for the whole territory of Poland up to 1500 m above sea level. The relation is expressed in the following form: a
+ b logp +d. (I + c)n
J=-
(2)
The procedure of calculation was based on finding for each station separately the values of the unknown regional constants a, b, c, nand d, corresponding in the best way to the observed values of I, t and p. After the unknown constants have been determined separately for each station, the remaining task is to find relations between these values in order to generalize the formula. The results of first calculations of the value of the exponent n for various stations show a relation between the elevation of the station and the value of the exponent n. Moreover, it has been found that the elevation of a station is characterised in the best way by the mean annual precipitation (calculated on the basis of years during which the observations have been carried out). The results of the above mentioned first calculations of the exponent n are given in the Table I where are also given mean annual rainfall H for each station (related to the years during which the observations have been carried out). The first calculation of values of the exponent n show a distinct increase of this exponent with the decreasing annual sum of rainfall. A smooth curve relating n to H is constructed. Further calculations have shown in an analogical manner that the coefficients a and b depend not only on the probability of occurrence p, but also on the mean annual rainfall H. A similar non-linear relation characterises also the coefficient c.
160
JULIAN LAMBOR TABLE
1
n obtained on the basis of calculation per station
H
mean annual precipitation inmm
No
Station
I
Kasprowy Wierch Hala Gasienicowa Zakopane Rabka Krakow Koszalin Skarzysko Kamienna Warszawa Poznan Blonie Wroclaw
2 3 4 5 6 7 8 9 10 11
*
1.676* 1.708* 1.137 0 .828 0.697 0.696 0.605 0.559 0.503 0.495 0.494** 0.576
0.474 0.498 0.600 0.665 0.651 0.685 0.667 0.692 0.697 0.698 0.720
Inversion.
** Different va lues for different per iods of observations. Final results of the intensity-duration-frequency analysis for the territory of Poland, are given by the following general formula : (38 - 1210gp) HO.2 8 1= - - -----:-------:--- -
(3)
(t + c)"
where: n=0.779-0.164 H I
c = - - (20.92 x H x 1000
pO.345 -
0.15
p -
2.0).
It has been found that factor d used in formula (2) is of significance only in the case of a very long time of duration, practically it can be completely neglected. Meanwhile factor c is of significance only in the case when the time of duration is short. The above formula is slightly different in comparison to those used in the USA and in the USSR. As an advantage of this formula must be considered its adaptation to particular regions, as a result of which it allows to calculate the required values in each physiographical region of Poland, with each region characterised by the mean annual rainfall H. In the USSR as well as in the USA only coefficients contained in the numerator are the variable quantities of the general formula. In addition
:J =:
P = ::J. t
(3B :-.12 [9y)f.jO.26
c
=
0,001(20,92 H'po,345_0,15p -2,0)
n • 0,779 -0,164H o
~
:=:: ~
1~~~ r-- r-- I-- r--: r=:::: t:::==::
"' 72//
; 4B1
~
f--
24/
-
10
5/
3 ........ f - -
2/
"'
Q)
::>
c
E
t--t--
0.2
o~
~
~
~
--
I~
~
I
C'\..'\: S
20
I
-
"' ~
Qj E
o
I
',,=
E
~
j
--
Q
i
I
~
~~
I
'" in metres-
i !
~
!
>---
i
f '....
!
I
Hin metres
~~ ~
~~ lONOc;o",U'lV
"";":~cicicSo
~. VaLu.e .c: in minutes
~ 800 1000 "-1500
20
50
100
2,6
3.6
4,1
3,4
4,5
5,1
7,0
5,8 8,6
I !
I
~,
20
~ ~ 1--
~
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~~~
r-....~~ ~ to
Fig.!.
~ m
5 min.
4 7 10
I
I
<;
t:;l
0,2
0,1
N
--
cs ec
~~o
CCIl'l'q"
00·0
40 70 100 200
~
Z
~t"" Z
4~ hr
I
II
I
z
0,5
10 15 20 30
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II
I
r-;
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p
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o
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\ I--_~
r-,
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- r--
20 -, 10 \ l -
0,1
t
~ ~~ t-- t-- tr---:::: r--:::: t - - t-t-- l-- r-r-t - - r-t-- t-- ~ r-- r--
1-l30 ,
5
0
C>
168~
144.-:1,
0
o
en '"
U
+
~
.........
,.....
,.....
0\
162
JULIAN LAMBOR
the values of the above coefficients are determined as regional constants depending on the geographical position. The present formula is more general and all the constants depend on the local conditions. In order to facilitate the use of the formula (3) a nomogram has been worked out, presented as Figure 1.
II;)
North-Holland Publishing Co ., Amsterdam
Not to be reproduced by photoprint or microfilm without written permission from the publisher
FREQUENCY OF INTENSE RAINFALL FOR THE TERRITORY OF POLAND JULIAN LAMBOR Technical University, Warsaw , Poland
Abstract: On the basis of observations of rainstorms recorded at pluviographic stations the author calculated intensity-duration-frequency formula. For the computation daily data have also been used from stations, of which the observation period exceeds in some cases even 100 years. From records of stations situated in different geographical conditions and at different altitudes above sea level up to the inversion limit, the author finds a systematic variation in the coefficients of the formulas. The formula can be adjusted to different regions by introducing the mean annual precipitation.
In order to determine with a sufficient accuracy the recurrence interval of extreme values of rainfalls, it is necessary to analyse long periods of observation, which allow to carry out sufficiently reliable extrapolation for a period of time which is considerably longer than the period of observation. The longer the period of observation, the more reliable the results. It is particularly difficult to obtain long and continuous series of homogeneous observations. In hydro-meteorological research a 100 year period of observation should be considered as a long one allowing for good interpretation of the observed phenomena. Periods of 50 years should be considered as sufficient to determine the probability of occurrence for various design purposes, whereas periods below 20 to 30 years are too short to base on them certain conclusions, even for practical purposes. The intensity of a rainfall is a function of its period of duration and it is also a function of its frequency or of its probability of occurrence. For arriving at formulas representing these relationships in this study all the available observations carried out on the eleven most important stations, in Poland have been used . The observation periods often exceeded 100 years, and for some stations even exceeded 180 years. The observations comprise the following data: values detailed to I minute duration 1 hour and 24 hour values monthly sum values 158
2138 rainfalls 7836 rainfalls 3838 rainfalls
FREQUENCY OF INTENSE RAINFALL IN POLAND
159
The intensity of rainfall can be described as a function of the time of duration and the probability of occurrence:
I=f(t·p) with
I t
p
(1)
= intensity of rainfall in mm per hour = time of duration in hours = probability of occurrence in per cent, corresponding to the average number of occurrences in one hundred year.
There appears to be a great harmony in these intensity-duration-frequency formulas for different stations. This enables to use one general formula which can be applied for the whole territory of Poland up to 1500 m above sea level. The relation is expressed in the following form: a
+ b logp +d. (I + c)n
J=-
(2)
The procedure of calculation was based on finding for each station separately the values of the unknown regional constants a, b, c, nand d, corresponding in the best way to the observed values of I, t and p. After the unknown constants have been determined separately for each station, the remaining task is to find relations between these values in order to generalize the formula. The results of first calculations of the value of the exponent n for various stations show a relation between the elevation of the station and the value of the exponent n. Moreover, it has been found that the elevation of a station is characterised in the best way by the mean annual precipitation (calculated on the basis of years during which the observations have been carried out). The results of the above mentioned first calculations of the exponent n are given in the Table I where are also given mean annual rainfall H for each station (related to the years during which the observations have been carried out). The first calculation of values of the exponent n show a distinct increase of this exponent with the decreasing annual sum of rainfall. A smooth curve relating n to H is constructed. Further calculations have shown in an analogical manner that the coefficients a and b depend not only on the probability of occurrence p, but also on the mean annual rainfall H. A similar non-linear relation characterises also the coefficient c.
160
JULIAN LAMBOR TABLE
1
n obtained on the basis of calculation per station
H
mean annual precipitation inmm
No
Station
I
Kasprowy Wierch Hala Gasienicowa Zakopane Rabka Krakow Koszalin Skarzysko Kamienna Warszawa Poznan Blonie Wroclaw
2 3 4 5 6 7 8 9 10 11
*
1.676* 1.708* 1.137 0 .828 0.697 0.696 0.605 0.559 0.503 0.495 0.494** 0.576
0.474 0.498 0.600 0.665 0.651 0.685 0.667 0.692 0.697 0.698 0.720
Inversion.
** Different va lues for different per iods of observations. Final results of the intensity-duration-frequency analysis for the territory of Poland, are given by the following general formula : (38 - 1210gp) HO.2 8 1= - - -----:-------:--- -
(3)
(t + c)"
where: n=0.779-0.164 H I
c = - - (20.92 x H x 1000
pO.345 -
0.15
p -
2.0).
It has been found that factor d used in formula (2) is of significance only in the case of a very long time of duration, practically it can be completely neglected. Meanwhile factor c is of significance only in the case when the time of duration is short. The above formula is slightly different in comparison to those used in the USA and in the USSR. As an advantage of this formula must be considered its adaptation to particular regions, as a result of which it allows to calculate the required values in each physiographical region of Poland, with each region characterised by the mean annual rainfall H. In the USSR as well as in the USA only coefficients contained in the numerator are the variable quantities of the general formula. In addition
:J =:
P = ::J. t
(3B :-.12 [9y)f.jO.26
c
=
0,001(20,92 H'po,345_0,15p -2,0)
n • 0,779 -0,164H o
~
:=:: ~
1~~~ r-- r-- I-- r--: r=:::: t:::==::
"' 72//
; 4B1
~
f--
24/
-
10
5/
3 ........ f - -
2/
"'
Q)
::>
c
E
t--t--
0.2
o~
~
~
~
--
I~
~
I
C'\..'\: S
20
I
-
"' ~
Qj E
o
I
',,=
E
~
j
--
Q
i
I
~
~~
I
'" in metres-
i !
~
!
>---
i
f '....
!
I
Hin metres
~~ ~
~~ lONOc;o",U'lV
"";":~cicicSo
~. VaLu.e .c: in minutes
~ 800 1000 "-1500
20
50
100
2,6
3.6
4,1
3,4
4,5
5,1
7,0
5,8 8,6
I !
I
~,
20
~ ~ 1--
~
I--
~~~
r-....~~ ~ to
Fig.!.
~ m
5 min.
4 7 10
I
I
<;
t:;l
0,2
0,1
N
--
cs ec
~~o
CCIl'l'q"
00·0
40 70 100 200
~
Z
~t"" Z
4~ hr
I
II
I
z
0,5
10 15 20 30
I
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I
r-;
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~
I
~,
, ,
I
.
,,
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----'
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~
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1"'-"\ ~" [I''\. <,
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-:~'\. f'I' ' -',
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50
1
~
tOlI"lq'"
~ 1'1''''\ ~
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100
~
___ -::::--
e:--
-
t---
I
~ 0..
t--
t--
t--
_'
L--k[::::::=r:::-l::::=::[::::::~ 50DO
I--
r-,"-, 1'..""'- I''\.-,
0,5
....-
----,
Il"'!(',lOCO ~
:g DO
p
I
o
0
0 to
-- r-t--- r-J.--- L----J.----' r-- t-V ' 1..--L----L----- J.----- J-----t:: 1--r-- t-- ~ ~ v t=+: t--r-- l-.J--l--Ii , C----t::-r------ V L-------"J--- V r-tt-- t---. l-- V t:=t--r
\ I--_~
r-,
..--L-:::= ~ E:::::== L---::: k k f:::::: ~ b:::: r> p p p ~ t;::::~ ~ V ;:: ~ f::::::: ....- L--:::: ....- '....»> t.:::: r> ~ r..---L--- J.----t::::-~ ....- r:::::: ~ r:- r::= . ..-- v t---. t-- t::::: ~ L.--::: :....- 8 ~ 1---r-- t-- ~~k t:::::-~~ l..------- J.----- J.----~ t::- l-....L-::::: -=--- , ~ vt-- r-J..------t-- ~ : I : ! __ L---L-- J.----r-- t-- r.-. ~~t:::::::: ~~ ..-- J.----t---.
- r--
20 -, 10 \ l -
0,1
t
~ ~~ t-- t-- tr---:::: r--:::: t - - t-t-- l-- r-r-t - - r-t-- t-- ~ r-- r--
1-l30 ,
5
0
C>
168~
144.-:1,
0
o
en '"
U
+
~
.........
,.....
,.....
0\
162
JULIAN LAMBOR
the values of the above coefficients are determined as regional constants depending on the geographical position. The present formula is more general and all the constants depend on the local conditions. In order to facilitate the use of the formula (3) a nomogram has been worked out, presented as Figure 1.