MARSH BULLETIN
Marsh Bulletin 4(1)(2009) 60-74
Amaricf_Basra
[email protected] [email protected] [email protected]
Probability Analysis of Extreme Monthly Rainfall In Mosul City, North of Iraq A.S. Dawood Civil Eng. Dept., Eng. Coll., Basrah Univ., Basrah, Iraq Abstract This study is an attempt at evaluating the proper theoretical statistical distribution of extreme monthly rainfall in Mosul. That is why data for the period (1923-1958) have been made use of to have all relevant information about sets of the highest monthly rainfall then. The frequency analyses and all needed statistical tests were done using the final version 1.1 of HYFRAN Software that operate under windows environment. The following Five distributions are used: Normal, Pearson Type III, Lognormal, 3-parameter lognormal and Gumbal. We arrived at the estimation of the theoretical distribution by using maximum likelihood method. The adequacy test is carried out by applying the chi-square test. All five distributions have been found to be suitable for representing of extreme monthly rainfall in the area under investigation.
1-Introduction
needs to be analyzed in different ways
Rainfall analysis for any region is
depending on the problem under consideration.
essential in planning and design of irrigation
For example, analysis of consecutive days
and drainage systems and overall programme of
rainfall is more relevant for drainage design of
command area development. As the distribution
agricultural lands, whereas analysis of weekly
of rainfall varies over space and time, it is
rainfall data is more useful for planning
required to analyze the data covering long
cropping pattern as well as water management
periods and recorded at various locations to
practices than that of monthly, seasonal and
obtain reliable information. Further, these data
annual data. The analysis of rainfall data for
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A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
computation of expected rainfall of a given
those of the sample data. The first moment for
frequency is commonly done by utilizing
each observation is the sample mean, the
different probability distributions. Different
second is the variance, and the third is the
distribution
commonly used are
coefficient of skewness. R.A. Fisher developed
Gumbel, lognormal, Pearson type III, log
the method of maximum likelihood in 1922.
Pearson type III, and Gamma distribution. Since
Fisher’s method for determining the parameter
no single distribution can be described as the
of a probability distribution is finding the
best, it is required to compare the suitability of
parameter
different probability distributions.
likelihood of occurrence. The method of
functions
values
which
maximizes
the
With the possibility of including additional
maximum likelihood is considered the most
rainfall data using computer programs, a
theoretically correct for fitting probability
statistical
distributions.
analysis
is
possible for
many
locations. The output of a hydrologic system is
Frequency analyses of hydrologic data
treated as stochastic, space-independent, and
use probability distributions to relate the
time-independent when there is no correlation
magnitude of extreme events to their frequency
between
Random
of occurrence. The distribution functions most
variables are statistically described by a
often used when estimating hydrologic events
adjacent
observations.
probability distribution. The probability of an event is the chance that it will occur based on an observation of the random variable. The larger the sample size of random variables the better the estimate of the probability of the event. To determine the probability of events occurring, probability distribution functions are
were: Normal, 2 Parameter Log Normal, 3 Parameter Log Normal, Pearson Type III, Log Pearson Type III, and Gumbel Type I. Monthly precipitation events tend to follow the normal distribution, and distribution varies over a continuous range and is symmetric about the mean but allows negative values. However, hydrologic variables tend to be skewed and all
fit to the data to determine the appropriate
are non-negative. The log normal distribution
function to use for estimation of the events.
eliminates
There are two methods for fitting distributions
variables as the data are greater than zero,
to data: method of moments and method of
permits the skewness of the data, and does
maximum likelihood. Karl Pearson developed
require the data to be symmetric about the
the method of moments in 1902. Pearson’s
logarithm of the mean. The Pearson Type III
method of moments selects probability function
methods
parameters such that the moments are equal to
deviation, and the coefficient of skewness into
the
problem
transform
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the
of
non-negative
mean,
standard
A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74 the
three
parameters
distribution
of heavy rainfall, and are used in the design of
function. When the data are greatly skewed, the
hydraulic structures (Nguyen et al. :2002). Such
log transformation of the Pearson Type III is
studies, however, primarily focus on point
used to reduce the skewness. Next, the Gumbel
rainfall totals, rather than areal totals. When a
Type I distribution, also known as the Extreme
drainage system is designed, it must handle
Value Type I distribution, is a two-parameter
rainfall within a given area, not just rain from a
distribution.
single point.
One
of
parameter
the
62
is
the most
probable value of the distribution and the
2- Location of study area
second is a measure of dispersion. Extreme
Mosul is the second largest city in Iraq. It
value distributions have been widely used in
is located along the Tigris River in Ninawa. The
hydrology (Chow et al.,: 1988). Analyzing data
Tigris River runs through the center of Mosul,
for the largest or smallest observations from
bisecting the city into eastern and western
sets of data became the basis for using the Gumbel (Extreme Value) Type I distribution. In the early 1980s, a comparison of the various distribution functions available for analysis of rainfall frequencies showed that the Log Pearson Type III distribution became method of choice. As a result, in 1981 the U.S. Water Resources Council (now called Interagency Advisory Committee on Water Data) recomme-
halves.
This city is 396 km (250 miles)
northwest of Baghdad. The original city stands on the west bank of the Tigris River, opposite the ancient city of Nineveh on the east bank, but the metropolitan area has now grown to encompass substantial areas on both banks, with five bridges linking the two sides (see figure 1).
nded the Log Pearson Type III distribution be
Mosul city is located topographically in
used in an effort to promote consistency for
a depression surrounded by hilly lands from the
flood flow analysis. An independent study
east and west slopping toward city centre and
showed that this distribution was the most
Tigris River. This characteristic brought this
appropriate method of estimation of rainfall
city to deliver big quantities of surface runoff
data (Naghavi et al., 1991).
during rainfall, flowing through number of
Rainfall frequency analyses are often
main Wadies to Tigris River. Mosul located on
used to aid the design of hydraulic structures,
Latitude (360 20') North and Longitude (430 08')
such as the design of storm sewer network
East, and it has elevation (222 m) above sea
(Huff and Angel :1992), and Faires et al.
level.
:1997). These studies provide the information necessary for the development of a design storm, to represent the probability of occurrence
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A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
Fig.(1) map of study area.
Climate of study area
which starts in December and ends at the end of
The Mosul is of a temperate-warm
February.
Mosul may witness the fall some
climate. Mosul is called Um Al-Rabi'ain (The
frost. This indicates this city is really cold in
City of Two Springs), because autumn and
Winter.
spring are very much alike there.
Mosul
The city
The climate of the governorate of varies
according
to
topographical
enjoys the Mediterranean climate. That is, it is
differences. The temperature degrees are (3 °C
hot in Summer which starts in May and ends at
– 6 °C) in Winter and (30 °C – 40 °C) in
the end of September. The temperature varies a
Summer.
lot and it may approach 50 °C.
In Winter
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64
A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74 Mosul (Al Mawsil) shares the severe
rims. These gages consist of a funnel which
alternations of temperature experienced by
drains into a collector bottle inside a cylindrical
upper Mesopotamia . The summer heat is
over flow can. The precipitation data as shown
extreme, and in winter frost is not unknown .
in appendix (A) is presented as monthly totals
Nevertheless the climate is considered healthy
in millimeters. No rainfall which is indicated by
and agreeable; copious rains fall in general in
a zero (0.0). A "trace" is indicated by (T) and is
winter.
defined as rainfall of less than measurable Rainfalls due to the effect of winds
quantity. Data of rainfall is totaled in calendar
blowing from both the Mediterranean Sea Area
years and water years, the latter extending from
and the Red Sea Area. The Red Sea winds are
October of the pervious calendar year through
less frequent.
When the two fronts of winds
September of the calendar year concerned. In
meet over the Mosul, the city has heavy rain
general, rainfall observations have not been
that leads to the flooding of the streets of the
made
city. This flood took place in 1982.
normally there is no rainfall. Rainfall during the
History and description of precipitation data
months of May and October is usually such a
collection
very small percentage of annual rainfall. The
during the summer
months,
when
Rainfall is generally measured as an
metrological station is located on Latitude (360
accumulated depth over time. Measurements
19') and longitude (430 09'), and it has elevation
represent the amount caught by the gage
(222 m) above sea level.
opening and are valid only for the gage location. The amount collected may be affected
Extreme value theory (EVT)
by gage location and physical factors near the
Extreme value theory (EVT) is a
gage. Application over large areas requires a
separate branch of statistics that deals with
study of adjacent gages and determinations of a
extreme events. This theory is based on the
weighted
of
extremal types theorem, also called the three
precipitation was obtained from the report of
types theorem, stating that there are only three
hydrological survey of Iraq (summary of
types of distributions that are needed to model
monthly precipitation at stations in Iraq for the
the maximum or minimum of the collection of
period 1887-1958) . Rain gages installed by or
random
for the Directorate of Meteorology are British
distribution.
rainfall
amount.
The
data
Meteorological office standard non-automatic type with either eight or five inch diameter
observations
Broadly
from
speaking,
there
the
are
same
two
principal kinds of model for extreme values.
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A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
The oldest group of models are the block
Here two distribution functions H1 and H2
maxima models; these are models for the
are said to be of the same type if one can be
largest
derived from the other through a simple
observations
collected
from
large
samples of identically distributed observations. Let
(X1,
X2
…)
be
identically
distributed random variables with unknown underlying
distribution
location-scale transformation,
function.
H1 (x) = H 2 (Ax + B),
Very often, (2) is called the Gumbel type,
Then
common cumulative distribution function is F, F(x) = Pr {Xi ≤ x}
A>0
(3) the Fre'chet type and (4) the Weibull type. In (3) and (4), α >0
Also let Mn = max (x1, …, xn) denote the nth sample maximum of the process. Then Pr {Mn ≤ x} = F(x)
n
...(1)
The three types may be combined into a single Generalized Extreme Value (GEV) distribution:-
since it simply says that for any fixed (x) for
x - μ -1 ξ H(x) = exp - (1 + ξ ) + , …(5) ϕ
which F(x) 0 is a scale parameter and ξ is
Equation (1) is of no immediate interest,
a
Such that
corresponds to the Gumbel distribution, ξ > 0
M _ bn n Pr n ≤ x = F (a n x + b n ) → H (x ) an
The three type theorem originally stated
without detailed mathematical proof by Fisher
shape
parameter.
The
limit
ξ→0
renormalize: find an >0, bn
to the Fre'chet distribution with α = 1 ξ , ξ < 0 to the Weibull distribution with α = −1 ξ . Methodology
and Tippett (1928), and later derived rigorously
The program used in this research to do
by Gnedenko (1943), asserts that if a non
most of the statistical tests is commonly known
degenerate (H) exist (i.e., a distribution function
as HYFRAN; which is a name composed of the
which does not put all its mass at a single
first pairs of letters of the words: HYdrological
point), it must be one of three types:
FRequency
H(x) = exp (-e- x ), all x,
… (2)
developed at the National Institute of Scientific
… (3)
Research of the University of Québec, with
H(x) =
{
x 0 α
Analyses.
x 0
Sciences and Engineering Research Council … (4)
(NSERC).
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66
A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74 HYFRAN
includes
a
number
of
Where, k=rank and n= number of
powerful, flexible, user-friendly mathematical
observations, and for Weibul a=0, therefore
tools that can be used for the statistical analysis
F[x[k ]] = Pa = k n + 1
of extreme events. It can also perform more
The return period T (in years) is the
basic analysis of data time series. HYFRAN
reciprocal of Pa, or in mathematical notation T= 1/ Pa
can be used in any study that requires fitting of statistical distribution to an independent and
Normal, Pearson Type III, Lognormal,
identically distributed data series. Applications
3-parameter lognormal and Gumbal are the five
are found in various technical areas such as
distributions
engineering,
meteorology,
probability analyses of Extreme rainfall in this
As soon as the data entered
study. The (99%) confidence interval is also
into the spreadsheet, this software automatically
used here (figures from 4-8). Maximum
calculates the basic statistics as shown in table
likelihood method is
(1), and plots the non-exccedance probability
theoretical distribution parameters (table 2).
and histogram of observation data (see figures.
The adequacy test is achieved by using Chi-
2 and 3).
square test, these values are compared with
environment,
medical sciences.
selected for
applied to estimate
(table 3). According to this test, , all the
calculated using Weibul formula:
F[x[k ]] = [k − a ] [n − 2a + 1]
distributions are found to be appropriate for
0 < a ≤ 0.5
describing of extreme monthly rainfall in the study area.
Table (1) Basic statistics of data sets
Minimum Maximum Average Standard deviation Median Coefficient of variation (Cv) Skewness coefficient (Cs) Kurtosis coefficient (Ck)
the
tabulated values of (0.01) significant level
Empirical non-exccedance probability is
Basic statistics
representing
Value 63.5 182 114 31.1 111 0.273 0.388 2.14
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Number of data
36
67
A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
+
Observation curve
Fig.(2) Observation of dataset on probability paper
Fig.(3) Histogram of observation dataset
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A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
+
Observation curve Model Conf. Int. 99%
Fig.(4)Empirical, theoretical probabilities and confidence interval against extreme monthly rainfall (normal distribution)
+
Observation curve Model Conf. Int. 99%
Fig.(5) Empirical, theoretical probabilities and confidence interval against extreme monthly rainfall (Pearson type III distribution)
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68
69
A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
+
Observation curve Model Conf. Int. 99%
Fig.(6)Empirical, theoretical probabilities and confidence interval against extreme monthly rainfall (lognormal distribution)
+
Observation curve Model Conf. Int. 99%
Fig.(7)Empirical, theoretical probabilities and confidence interval against extreme monthly rainfall (3-parameters lognormal distribution)
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A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
+
Observation curve Model Conf. Int. 99%
Fig.(8) Empirical, theoretical probabilities and confidence interval against extreme monthly rainfall (Gumbal distribution)
Table (2) Fitting the models with the estimated parameters No.
1.
Distribution
Mathematical formula f ( x) =
Normal
2.
Pearson type III
3.
Lognormal
4.
3-parameters lognormal
5.
Gumbal
( x − µ )2 exp − 2σ 2 σ 2π 1
αλ f ( x) = (x − m)λ −1 e −α ( x−m) Γ(a ) f ( x) =
f ( x) =
[ln x − µ ]2 exp − 2σ 2 xσ 2π 1
[ln ( x − m) − µ ]2 1 exp − (x − m)σ 2σ 2
f ( x) =
x−u 1 x − u exp − exp − α α α
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Estimated parameters µ = 113 . 739 σ = 31 . 1066
α = 0.0675167 λ = 4.67117 m = 44.5535 µ = 4 . 69717
σ = 0 . 276707
µ = 4.71862 σ = 0.267044 mα= =−226 .29191 .5816 u = 98 .5719
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A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
Table (3) Adequacy test using Chi-square
Chi-Square test No.
Distribution
Tabulated value
Calculated value
(0.01) significant level 1.
Normal
13.277
4.00
2.
Pearson type III
11.344
4.89
3.
Lognormal
13.277
4.00
4.
3-parameters lognormal
11.344
4.00
5.
Gumbal
13.277
4.89
4- References
4- Conclusions The five distributions: Normal, Pearson
Chow, Ven Te., David R. Maidment, and Larry
Type III, Lognormal, 3-parameter lognormal
W. Mays. Applied Hydrology McGraw-
and Gumbal are selected to represent the
Hill, New York, 1988.
probability analyses of Extreme rainfall in city
Faiers,
G.E.,
B.D.
Keim,
and
K.K.
of Mosul. For the purpose of estimating the
Hirschboeck, (1999), a synoptic evaluation
theoretical
the
of frequencies and intensities of extreme
Maximum Likelihood Method was applied to
three- and 24-hour rainfall in Louisiana.
the data in this research. estimate. As a result of
Professional Geographer 46(2):156-163.
to the comparison between the values that
Fisher, R.A. and Tippett, L.H.C. (1928),
distribution
parameters,
obtained from the adequacy test (Chi-square
Limiting
values) with their counter-part tabulated values
distributions of the largest or smallest
of (0.01) significant
member of a sample. Proc. Camb.
level,
All the five
distributions were found to be suitable for representing of extreme monthly rainfall in Mosul city.
forms
of
the
frequency
Phil.
Soc. 24, 180{190. Gnedenko, B.V. (1943), Surla distribution limite du terme maximum d'une serie aleatoire. Ann. Math. 44, 423{453. HYFRAN Hydrological frequency analysis, primarily developed by Bernard Bobe´e.A
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72
A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74 demo
version
is
available
via
www.wrpllc.com/books/hyfran.html. Huff, F.A., and J.R. Angel, (1992), Rainfall frequency
atlas
of
the
Midwest.
Champaign, Illinois: Illinois Water Survey. Naghavi, Babak, Vijay P. Singh, Fang Xin Yu.
Nguyen, V.T.V., T.D. Nguyen, and F. Ashkar, (2002), Regional frequency analysis of extreme
rainfalls.
Summary of monthly precipitation at stations in Iraq
for
period
Republic
Maps
Baghdad, Iraq, 62 p.
I-D-F-
Curves
(Summary
Science and
Technology 45(2):75-81.
“LADOTD 24-Hour Rainfall Frequency and
Water
Report)”. Louisiana State University, Baton Rouge, LA., August 1991.
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of
Iraq,
(1887-1958),
(1964),
Developing
Board,
73
A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74 Appendix (A) Monthly meteorological data of Mosul station for period (1923-1958)
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A.S. Dawood / Marsh Bulletin 4(1)(2009)60-74
ﺍﻟﺘﺤﻠﻴل ﺍﻻﺤﺘﻤﺎﻟﻲ ﻟﻘﻴﻡ ﺍﻷﻤﻁﺎﺭ ﺍﻟﺸﻬﺭﻴﺔ ﺍﻟﻌﻅﻤﻰ ﻓﻲ ﻤﺩﻴﻨﺔ ﺍﻟﻤﻭﺼل ﺸﻤﺎل ﺍﻟﻌﺭﺍﻕ ﻋﻤﺎﺭ ﺴﻠﻤﺎﻥ ﺩﺍﻭﺩ ﻗﺴﻡ ﺍﻟﻬﻨﺩﺴﺔ ﺍﻟﻤﺩﻨﻴﺔ ,ﻜﻠﻴﺔ ﺍﻟﻬﻨﺩﺴﺔ ,ﺠﺎﻤﻌﺔ ﺍﻟﺒﺼﺭﺓ ,ﺍﻟﺒﺼﺭﺓ ,ﺍﻟﻌﺭﺍﻕ ﺍﻟﻤﻠﺨﺹ ﺒﺎﻹﻤﻜﺎﻥ ﺍﻋﺘﺒﺎﺭ ﻫﺫﻩ ﺍﻟﺩﺭﺍﺴﺔ ﻤﺤﺎﻭﻟﺔ ﻟﺘﻘﻴﻴﻡ ﺍﻟﺘﻭﺯﻴﻊ ﺍﻻﺤﺘﻤﺎﻟﻲ ﺍﻟﻨﻅﺭﻱ ﺍﻟﺨﺎﺹ ﺒﻘﻴﻡ ﺍﻷﻤﻁﺎﺭ ﺍﻟﺸﻬﺭﻴﺔ ﺍﻟﻌﻅﻤﻰ ﻓﻲ ﻤﺩﻴﻨﺔ ﺍﻟﻤﻭﺼل .ﻭﻟﻬﺫﺍ، ﻓﻘﺩ ﺘﻡ ﺍﺴﺘﺨﺩﺍﻡ ﺍﻟﻘﻴﻡ ﺍﻟﻤﺴﺠﻠﺔ ﻟﻸﻤﻁﺎﺭ ﻁﻴﻠﺔ ﺍﻟﻔﺘﺭﺓ ) .(1958-1923ﻭ ﻟﺘﺤﻠﻴل ﺍﻟﺒﻴﺎﻨﺎﺕ ﺍﻟﻤﺘﻭﻓﺭﺓ ﻟﺩﻴﻨﺎ ﻗﻤﻨﺎ ﺒﺈﺠﺭﺍﺀ ﺍﻟﺘﺤﻠﻴﻼﺕ ﺍﻟﺘﺭﺩﺩﻴـﺔ ﻭﺠﻤﻴـﻊ ﺍﻻﺨﺘﺒﺎﺭﺍﺕ ﺍﻹﺤﺼﺎﺌﻴﺔ ﻭﺫﻟﻙ ﺒﺎﺴﺘﻌﻤﺎل ﺍﻟﻨﺴﺨﺔ 1,1ﻟﺒﺭﻨﺎﻤﺞ ﻫﺎﻴﻔﺭﺍﻥ ) (HYFRANﺍﻟﺫﻱ ﻴﻌﻤل ﺘﺤﺕ ﻨﻅﺎﻡ ﻭﻴﻨـﺩﻭﺯ) :(WINDOWSﻭﺘـﻡ ﺍﺨﺘﺒﺎﺭ ﺨﻤﺱ ﺘﻭﺯﻴﻌﺎﺕ ﻫﻨﺎ ﺃﻻ ﻭﻫﻲ ,ﺍﻟﻁﺒﻴﻌﻲ ,ﻭﺒﻴﺭﺴﻭﻥ ﺍﻟﻨﻭﻉ ﺍﻟﺜﺎﻟﺙ ,ﺍﻟﻠﻭﻏﺎﺭﺘﻴﻤﻲ ﺍﻟﻁﺒﻴﻌﻲ ,ﺍﻟﻠﻭﻏﺎﺭﺘﻴﻤﻲ ﺍﻟﻁﺒﻴﻌﻲ ﺫﻱ ﺍﻟﺜﻼﺙ ﻤﻌﺎﻟﻡ ,ﻭﻜﺎﻤﺒل. ﻭﻗﺩ ﺘﻡ ﺍﻟﺘﻭﺼل ﺇﻟﻰ ﺘﺨﻤﻴﻥ ﺍﻟﺘﻭﺯﻴﻊ ﺍﻟﻨﻅﺭﻱ ﺒﺎﺴﺘﻌﻤﺎل ﻁﺭﻴﻘﺔ ﺍﻻﺤﺘﻤﺎل ﺍﻷﻜﺒﺭ ﺃﻤﺎ ﺍﺨﺘﺒﺎﺭ ﺍﻟﻤﻭﺍﺌﻤﺔ ﻓﻜﺎﻥ ﻋﻥ ﻁﺭﻴﻕ ﺍﺴﺘﺨﺩﺍﻡ ﻤﺭﺒﻊ ﻜﺎﻱ .ﻭﺜﺒﺘـﺕ ﺍﻟﻨﺘﺎﺌﺞ ﺃﻥ ﺠﻤﻴﻊ ﺍﻟﺘﻭﺯﻴﻌﺎﺕ ﻫﻲ ﺍﻟﻤﺜﻠﻰ ﻟﻘﻴﻡ ﺍﻷﻤﻁﺎﺭ ﺍﻟﺸﻬﺭﻴﺔ ﺍﻟﻌﻅﻤﻰ ﻟﻤﻨﻁﻘﺔ ﺍﻟﺩﺭﺍﺴﺔ.
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