Rainfall intensity duration frequency curve statistical analysis and modeling for Patna, Bihar

BOHR International Journal of Civil Engineering
and Environmental Science
2023, Vol. 1, No. 2, pp. 66–75
DOI: 10.54646/bijcees.2023.08
www.bohrpub.com
RESEARCH
Rainfall intensity duration frequency curve statistical analysis
and modeling for Patna, Bihar
Pappu Kumar1*, Madhusudan Narayan1 and Mani Bhushan2
1Department of Civil Engineering, Sandip University, Madhubani, India
2Department of Civil Engineering, RRSD College of Engineering, Begusarai, India
*Correspondence:
Pappu Kumar,
pappukumar.ce16@nitp.ac.in
Received: 21 July 2023; Accepted: 24 July 2023; Published: 30 August 2023
Using data from 41 years in Patna’ India’ the study’s goal is to analyze the trends of how often it rains on a
weekly, seasonal, and annual basis (1981−2020). First, utilizing the intensity-duration-frequency (IDF) curve and the
relationship by statistically analyzing rainfall’ the historical rainfall data set for Patna’ India’ during a 41 year period
(1981−2020), was evaluated for its quality. Changes in the hydrologic cycle as a result of increased greenhouse
gas emissions are expected to induce variations in the intensity, length, and frequency of precipitation events. One
strategy to lessen vulnerability is to quantify probable changes and adapt to them. Techniques such as log-normal,
normal, and Gumbel are used (EV-I). Distributions were created with durations of 1, 2, 3, 6, and 24 h and return
times of 2, 5, 10, 25, and 100 years. There were also mathematical correlations discovered between rainfall and
recurrence interval.
Findings: Based on findings, the Gumbel approach produced the highest intensity values, whereas the other
approaches produced values that were close to each other. The data indicates that 461.9 mm of rain fell during
the monsoon season’s 301st week. However, it was found that the 29th week had the greatest average rainfall,
92.6 mm. With 952.6 mm on average, the monsoon season saw the highest rainfall. Calculations revealed that the
yearly rainfall averaged 1171.1 mm. Using Weibull’s method, the study was subsequently expanded to examine
rainfall distribution at different recurrence intervals of 2, 5, 10, and 25 years. Rainfall and recurrence interval
mathematical correlations were also developed. Further regression analysis revealed that short wave irrigation,
wind direction, wind speed, pressure, relative humidity, and temperature all had a substantial influence on rainfall.
Originality and value: The results of the rainfall IDF curves can provide useful information to policymakers in
making appropriate decisions in managing and minimizing floods in the study area.
Keywords: IDF, hydrologic, vulnerability, rainfall distribution, Gumbel techniques, Weibull’s method
Introduction
Precipitation is a type of water that occurs when atmospheric
vapor is converted to water on hydrologic occasions. It
differs with existence. The information on total precipitation
and its appropriation design around the time of a spot
is critical for better harvest arranging, determining water
system and waste prerequisites of yields, planning and
development of hydrologic structures, and so on. References
(1) and (2) have proposed the utilization of daily, weekly,
monthly, occasional, and yearly precipitation disseminations
for crop arranging.
We also supported the use of yearly precipitation
conveyance for crop planning. Rajendra Nagar and
Kankarbagh remain low for the eighth day on Friday
PO, with the climate division issuing a warning for heavy
rains in the coming days. The situation is probably going
to deteriorate. Rajendra Nagar, one of the most severely
affected areas, is under four feet of stale water, adding to the
locals’ despair. However, the organization said it has gotten
66

10.54646/bijcees.2023.08 67
substantial siphons to flush out water, but circumstances
have not helped a lot.
There have also been reports of robberies in locked
houses in the Kadam-kuan police headquarters region. The
weighty downpours have guaranteed 73 lives up until now.
Individuals who were taking refuge at the rooftop are
presently leaving their homes.
A few groups have claimed that there was no game plan
from the public authority to provide drinking water and food.
Regardless, the organization has stated that authorities are
on the streets and roads assisting the affected individuals.
In the present, an endeavor has been made to assess the
precipitation dispersion example of Patna, Bihar.
The forecast of precipitation dissemination at various
repeat spans was done utilizing Weibull’s strategy (3). On
Monday, the state capital Patna remained among the most
noticeably terrible influenced by four- to six-foot profound
waterlogging in a few areas. The National Disaster Response
Force (NDRF) must protect Vice President Minister Sushil
Modi, who was abandoned at his Patna home.
Authorities say the state capital has not seen such
waterlogging since the 1975 floods. The Bihar government
has additionally requested two helicopters from the Air Force
for lifting and airdropping food parcels and drugs. The
Patna local organization has requested that all schools be
closed until Tuesday. The NDRF and State Disaster Response
Force (SDRF) are directing tasks in low-lying spaces of
the state capital.
The waterlogging has seriously influenced one of Patna’s
leading government emergency clinics, Nalanda Medical
College and Hospital (NMCH). A few trains and flights
have been dropped, rescheduled, or redirected due to
the circumstances. The investigation of precipitation
information is one of the main occasions in the
hydrological cycle.
It is an important part of the water cycle for collecting
the vast amount of water in the universe. The normal
precipitation in this nation is 1,200 mm per year. It varies
from 339 to 2,250 mm per year. Ordinarily, 80−85% of
the complete yearly precipitation in India is recorded from
June to September.
Precipitation is an interesting phenomenon that is
profoundly enhanced by space and time. So rainfall
investigation and daily rainfall calculation should be carried
out in order to work on the administration of water
asset application and the compelling usage of water.
This data is additionally utilized for some water in the
executive’s application, including the plan of major and
minor storm water, the board framework, sanitary sewer,
confinement lakes, course, span, dams, siphoning station,
and street, among others.
Predictions of precipitation are also an important and
controlling factor in the planning and activity methodologies
of any farming system in any random region. In this way,
accurate and unambiguous information about the pattern
of precipitation throughout time for a specific location has
ceased to be needed for proper and perfect planning of
the most important irrigation system and trimming design.
The precipitation that occurs during the storm season
provides a sizable amount of the country’s total annual
conjunctive water needs.
Precipitation circulation varies greatly from year to year.
Gulping flooding and hungrily dry times are the products
of our nation’s astoundingly far-reaching precipitation
conveyance sites. Data of outrageous precipitation trademark
is needed in hydrological plans of designs that control
spillover; such data is frequently communicated as a
connection between power length and frequency bend.
An intensity-term recurrence bend is a numerical function
that relates the precipitation force with the span and
frequency of the event, i.e., the return period (4). IDF
frequency bend for precipitation in Vietnam’s storm area;
they deduced a summarized IDF formula using precipitation
depth. Reference (5) developed a precise formula to assess the
precipitation force for the Riyadh region in Saudi Arabia, and
the results showed that the Gumbel method and other logical
approaches worked well together.
Based on an examination of rainfall data, inferred
precipitation profundity range, and frequency connection for
two Saudi Arabian locations, it was discovered that the results
obtained utilizing the Gumbel conveyance technique were
superior to the outcomes obtained utilizing appropriation,
for example, IPT III circulation (6). Reference (6) had
set up a precipitation IDF relationship for Basrah City,
Iran, utilizing the Gumbel technique; their outcome showed
the greatest forces happen over a short term with high
variety. Various specialists were directed to determine and
set up experimental precipitation assessment condition, and
IDF curves for various areas worldwide, particularly in
nonindustrial nations (7).
Battered by heavy precipitation for the past 48 h in 3
areas of Bihar, something like 29 individuals have kicked the
bucket in the state because of accidents brought about by the
storm, as indicated by the news agency ANI. Patna, the state
capital, remained among the most noticeably bad, influenced
by four- to six-foot-deep waterlogging in a few areas Monday.
The Bihar authorities say the state capital has not seen such
waterlogging since the 1975 floods.
The Bihar government has likewise requested two
helicopters from the Air Force for lifting and airdropping
food parcels and medications. When Bihar experiences
a waterlogging problem, the NDRF and SDRF lead
relief efforts in the state capital’s low-lying areas. The
waterlogging has seriously influenced one of Patna’s leading
government clinics, NMCH.
In this problem, the investigation of precipitation
information is one of the main occasions in the hydrological
cycle. It is an important part of the water cycle for collecting
the vast amount of water in the universe. The normal

68 Kumar et al.
precipitation in this nation is 1200 mm per year. It varies
from 339 to 2250 mm per year.
From June to September, India receives 80−85% of its total
annual precipitation. Precipitation is a special phenomenon
that is exceptionally expanded in both space and time. So
rainfall examination and calculation should be done to work
on the administration of water asset application and the
compelling usage of water.
This data is likewise utilized for some water in the
executive’s application, including the plan of major and
minor storm water, the board framework, sanitary sewer,
confinement lakes, courses, spans, dams, siphoning stations,
and streets among others. Predictions of precipitation are
also an important and controlling factor in the planning
and activity methodologies of any farming project in
any random region.
All things considered, accurate and plain information
about the precipitation appropriation design throughout
time for a specific location is crucial for the right and
ideal planning of the necessary irrigation framework and
editing design. Precipitation that occurs during a storm
period contributes significantly to the nation’s overall
conjunctive water needs throughout the calendar year. There
is huge variety in the conveyance of precipitation from one
year to another.
The incredible limits of precipitation conveyance in our
country cause gushing floods and eagerly dry seasons.
Extreme precipitation data is required in hydrological plans
of designs that control storm overflow; such data is frequently
communicated as a link between force length and frequency
bend. An intensity span recurrence bend is a numerical
function that relates the precipitation force with the length
and frequency of events, i.e., the return period; it is an
intriguing factual strategy for assessing precipitation force
and advancing the IDF relationship utilizing outrageous
precipitation data.
The connection between precipitation information and
force and span for a bowl in Jordan; he guaranteed that
the outcome acquired from Gumbel’s strategy is comparable
with different techniques. The IDF frequency bent for
precipitation in the rainy area of Vietnam; they deduced
a condensed IDF formula using precipitation depth. The
following experimental formula was developed to evaluate
the precipitation force at the Riyadh location in Saudi Arabia:
he expressed a good match as an accomplishment between
Gumbel’s strategy and other insightful techniques.
Precipitation profundity length-frequency relationship
for two areas in Saudi Arabia through rainfall data
examination; discovered that the outcomes obtained
utilizing Gumbel appropriation strategy were superior to the
outcomes obtained utilizing dispersion, for example, IPT III
conveyance. Numerous technical articles using previous and
forthcoming rainfall forecast data to create IDF curves have
been published at the scientific level. For our study, we have
used numerous of these works as references. The papers are
listed in the section titled, “References.”
Materials and methods
The objective of the present study is to determine the IDF
curve and the statistical analysis of rainfall data for a record
of 41 years using log-normal, normal, and Gumbel (EV-I)
distribution methods.
Study area
Patna has been chosen as the study location. It is the capital
and largest city of the Indian state of Bihar. The daily rainfall
data for 31 years (from 1965 to 1995) were collected from
the meteorological observatory, located at the Agricultural
Research Institute, Patna (25◦ 300 N latitude, 85◦ 150 E
longitude, and 57.8◦m above mean sea level), for evaluation
of the rainfall distribution pattern.
According to the 2018 United Nations Population Report,
Patna has a population of approximately 2.35 (8). Its urban
agglomeration, the 18th biggest in India, spans 250 square
kilometers (97 square miles) and has a population of nearly
2.5 million. Mostly on the Ganges River’s southern bank is
where you’ll find the modern city of Patna.
Although earthquakes have not been common in recent
history, Patna is located in seismic zone IV of India,
demonstrating her vulnerability to severe tremors (9).
Additionally, Patna moves toward the storm and flood zone.
In Figure 1, the review area’s guidance is visible and starts at
this location in October and lasts until February.
The minimum temperature in Patna often fluctuates
between 12 and 30 degrees throughout the colder months
of the year and begins in March and ends in May. Due
to its location in the sub-equatorial rainforest, Patna has
FIGURE 1 | Location map of the study area.

10.54646/bijcees.2023.08 69
muggy, humid, late spring days. The base temperature is close
to 26 degrees, while the typical maximum temperature is
about 37 degrees.
The season runs from June to September. During the
monsoon, the city experiences hefty amounts of rain, which
can occasionally cause the city to flood. During these months,
the temperature and humidity remain relatively high. The
most precipitation ever recorded was 204.5 mm (8.05 inches)
in 1997 (9).
Data collection
The Patna Metrological Department in Bihar collected
rainfall data for the 40◦year period 1981−2020 in order to
create an intensity-duration-frequency (IDF) curve for the
research region. After that, a maximum yearly rainfall is
calculated using the data on annual rainfall depth that has
been gathered using Eq. (1). Table 1 displays the computed
yearly maximum rainfall depth for periods of 1, 2, 3,
6, 12, and 24 h.
Methodology and discussion
Precipitation information was broken down to insulate the
greatest precipitation profundity recorded in a day for a
year. A yearly most extreme precipitation series was derived
from precipitation profundity data. Eq. (1), a formula from
the Indian Meteorological Office, is used to calculate the
depth of precipitation across time periods of 60 min, 2, 3,
6, 12, and 24 h.
The IMD experimental decreasing recipe has been proven
to provide the best evaluation of short-term precipitation in
Chowdary:
pt = p24
r
t
24
(1)
P is the computed depth of the precipitation, P24 is the
yearly maximum precipitation lasting 24 h, and t can be
used to signify the time for which P is being calculated. To
calculate rainfall intensity, rainfall depth was divided by the
corresponding time periods. Previous research publications
have employed the IMD empirical formula (10).
Regression analysis
Regression analysis was applied to examine the strength of
relationship between Short Wave Irrigation, Wind Direction,
Wind Speed, Pressure, Relative Humidity, Temperature
(Predictor Variables) and Rainfall (Outcome Variable) by
using IBM-SPSS 25.
Table 2 the correlation between predictor variables and
outcome variables is 89.6 according to R-value. The adjusted
TABLE 1 | Annual maximum rainfall depth, P (in mm) of
different durations.
Years 1◦H 2◦H 3◦H 6◦H 12◦H 24◦H
1981 51.960 65.466 74.939 94.418 118.959 149.879
1982 17.276 21.767 24.917 31.393 39.553 49.834
1983 22.560 28.424 32.537 40.994 51.649 65.074
1984 33.900 42.711 48.892 61.600 77.611 97.784
1985 24.470 30.831 35.292 44.466 56.023 70.585
1986 41.989 52.903 60.558 76.299 96.130 121.116
1987 26.482 33.366 38.194 48.122 60.629 76.388
1988 25.314 31.894 36.509 45.999 57.955 73.018
1989 18.808 23.697 27.126 34.176 43.059 54.252
1990 16.3 20.611 23.594 29.727 37.453 47.188
1991 15.495 19.522 22.347 28.156 35.474 44.694
1992 13.994 17.631 20.183 25.429 32.038 40.365
1993 38.505 48.513 55.534 69.968 88.154 111.067
1994 33.168 41.789 47.836 60.270 75.935 95.672
1995 45.226 56.982 65.228 82.182 103.543 130.455
1996 19.286 24.299 27.816 35.046 44.155 55.632
1997 50.719 63.901 73.149 92.162 116.117 146.298
1998 32.298 40.692 46.581 58.689 73.943 93.163
1999 40.243 50.702 58.040 73.126 92.132 116.080
2000 31.407 39.571 45.297 57.071 71.904 90.594
2001 64.086 80.743 92.428 116.452 146.721 184.856
2002 27.552 34.714 39.737 50.066 63.079 79.475
2003 34.137 43.010 49.234 62.031 78.155 98.469
2004 21.781 27.442 31.414 39.579 49.866 62.827
2005 21.707 27.349 31.307 39.445 49.697 62.614
2006 38.285 48.236 55.217 69.569 87.651 110.433
2007 38.183 48.108 55.070 69.384 87.418 110.140
2008 18.229 22.967 26.291 33.125 41.734 52.582
2009 19.770 24.909 28.513 35.924 45.262 57.026
2010 16.103 20.288 23.224 29.260 36.866 46.448
2011 33.532 42.247 48.361 60.931 76.768 96.722
2012 20.472 25.794 29.526 37.201 46.870 59.052
2013 40.326 50.807 58.160 73.277 92.323 116.320
2014 32.682 41.176 47.135 59.387 74.823 94.271
2015 21.576 27.183 31.117 39.205 49.396 62.235
2016 27.020 34.044 38.970 49.099 61.861 77.940
2017 35.859 45.180 51.718 65.160 82.097 103.436
2018 22.538 28.397 32.506 40.955 51.600 65.012
2019 35.571 44.817 51.302 64.637 81.438 102.605
R2 is 0.802, indicating that short-wave irrigation, wind
direction, wind speed, pressure, relative humidity, and
temperature (an independent variable) explain 80.2% of the
variance in rainfall (a dependent variable); the remaining
19.8% is influenced by other factors. Durbin- Watson
is 1.839, which shows that there is no first-order linear
autocorrelation in the data.
Overall, the regression model statistically substantially
predicts the outcome variable, according to ANOVA

70 Kumar et al.
TABLE 2 |
Model summaryb
Model R R2 Adjusted
R2
Std. error of the
estimate
Durbin-Watson
1 0.896a 0.802 0.802 112.6256853 1.839
aPredictors: (Constant), Short Wave Irrigation, Wind Direction, Wind Speed, Pressure,
Relative Humidity, Temperature.
bDependent Variable: Rainfall.
TABLE 3 |
ANOVAa
Models Sum of squares Df Mean square F Sig.
1 Regression 754143768.949 6 125690628.158 9908.958 0.000b
Residual 185612946.902 14633 12684.545
Total 939756715.852 14639
aDependent Variable: Rainfall.
bPredictors: (Constant), Short Wave Irrigation, Wind Direction, Wind Speed, Pressure,
Relative Humidity, Temperature.
(Table 3), which shows p = 0.000, which is less than 0.05. (i.e.,
it is a good fit for the data).
Coefficients table shows the strength of the relationship,
i.e., the significance of the variable in the model and
magnitude with which it impacts the dependent variable.
Table No - reveals
• The Sig. value indicates that the significant difference
in rainfall caused by temperature is 0.028, which is less
than the allowed limit of 0.05.
• The significant change in rainfall caused by relative
humidity as a result of the Sig. value is 0.013, which is
less than the 0.05 limit.
• The difference in rainfall caused by pressure is
considered significant because the Sig. value of 0.030
is less than the 0.05 threshold.
• As a result of the Sig. value, the significant difference
between rainfall and wind speed is 0.014, which is less
than the permitted standard of 0.05.
• The significant variation in rainfall caused by wind
direction, as determined by the Sig. value, is 0.000,
which is below the permitted limit of 0.05.
• The Sig. value has caused a significant shift in rainfall
that is less than the permitted value of 0.05 or 0.000.
This is due to short-wave irrigation.
• Since VIF and tolerance are below the permissible
range, there is no evidence of multiple collinearities
among the variables, and as a result, the variance of beta
is not inflated in any way.
A multiple regression was run to predict rainfall from short
wave irrigation, wind direction, wind speed, pressure, relative
humidity, and temperature. These variables predicted rainfall
statistically significantly: F (6, 14633) = 9908.958, p < 0.05,
and adjusted R2 = 0.802. All six variables contributed
statistically significantly (p < 0.05) to the prediction. Hence,
linear regression established that there is a significant
impact of short wave irrigation, wind direction, wind speed,
pressure, relative humidity, and temperature on rainfall.
The regression equation:
Rainfall = 501.793 + (−0.892)Temparature
+ 0.528(Relative Humidity) + (−0.029)Pressure
+ 0.337(Wind Speed) + 0.140(Wind Direction)
+ 0.164(Short Wave Irrigation)
As previously discussed, force length recurrence bends are
used to track down plan precipitation power as a component
of the tempest term and return time of a specific period on
which the tempest water framework is based. Power span
recurrence bends are created for a series of tempest events
rather than a single tempest event. The quantity of the mean
and its takeoff from the mean may be used to describe the
power of any tempestuous event.
TABLE 4 |
Coefficients
Models Unstandardized Coefficients Standardized Coefficients t Sig. Collinearity Statistics
B Std. Error Beta Tolerance VIF
1 (Constant) 501.793 416.660 1.204 0.028
Temperature −0.892 0.358 −0.023 −2.493 0.013 0.152 1.577
Relative Humidity 0.528 0.058 0.049 9.102 0.000 0.465 1.151
Pressure −0.029 0.325 −0.001 −0.088 0.030 0.173 1.767
Wind speed 0.337 0.669 0.002 0.504 0.014 0.827 1.210
Wind direction 0.140 0.012 0.053 12.144 0.000 0.718 1.392
Short wave Irrigation 0.164 0.001 0.937 150.686 0.000 0.349 1.865
aDependent Variable: Rainfall

10.54646/bijcees.2023.08 71
TABLE 5 | Values of S and P for normal distribution.
Durations 1◦H 2◦H 3◦H 6◦H 12◦H 24◦H
P (in mm) 29.971 37.761 43.226 54.461 68.616 86.451
S (in mm) 11.451 14.427 16.515 20.808 26.217 33.031
FIGURE 2 | Intensity-duration-frequency (IDF) curve by normal
distribution.
The flight of the mean is interpreted as the product of the
standard deviation and the recurrence factor K. As a result,
“” is derived from Eq. (4). The return period is a function of
both the departure and the frequency factor K.
Chow (11) provides the frequency factor equation,
which may be used for a variety of hydrological
probability assessments.
Procedure for developing the IDF curves:
1. The precipitation data is separated into the series of
yearly most extreme precipitation for 1, 2, 3, 6, 12,
and 24 h. Precipitation power is determined for all the
precipitation profundities in millimeters per hour.
2. The mean and standard deviation were determined
for the given information. For instance, the mean
(average) utilizing Eq. (2) and the standard
deviation (SD) utilizing Eq. (3) for the yearly
greatest precipitation power series for 1◦h length are
determined. The same interaction is repeated every 2,
3, 6, 12, and 24 h.
TABLE 7 | Value of standard deviation (S*) and avg. precipitation (P ∗).
P* 3.331 3.562 3.698 3.929 4.160 4.391
S* 0.376 0.376 0.376 0.376 0.376 0.376
3. The value of consistent KT for a specific time period is
calculated using probability conveyance. The worth of
KT is different for every likelihood appropriation (12):
Pavg =
1
n
n
X
i=1
Pi (2)
S =
"
1
n
n
X
i=1
(Pi − Pavg)
#0.5
(3)
4. Next, rainfall intensity is determined using the
K, mean, and standard deviation values from Eq.
(2). A typical distribution and the most common
approach in statistics is called the normal (Gaussian)
distribution. Like all other approaches, this one also
calculates the rainfall intensities in order to determine
the rain intensities for a certain return time and every
storm length. The formula to calculate precipitation P
(in mm) using a given return period (T) and a given
duration (t) is shown below (13):
P = P + K∗
T
S (4)
Equations (5), (6), and (9) are used to get the frequency factor,
KT, which is equal to “Z” for both the log-normal and normal
distributions (7):
Z = w −
2.515517 + 0.802853w + 0.010328w2
1 + 1.432788w + 0.189269w2 + 0.001308w3
(5)
Here, “w” is calculated as
W = [1n(1n(1/P2
))]0.5
(6)
In Eq. (3), “p” is the probability of occurrence in a specified
return period “T” and its value calculated as
P = 1/T (7)
TABLE 6 | Rainfall intensity (I) computed from normal distribution.
Return period (T) Value of “Z” calculated by Eq. (5) Durations
1 h 2 h 3 h 6 h 12 h 24 h
2◦Years −1.0E–07 29.971 18.881 14.409 9.077 5.718 3.602
5◦Years 0.8414567 39.607 24.951 19.041 11.995 7.556 4.760
10◦Years 1.2817288 44.648 28.127 21.465 13.522 8.518 5.366
25◦Years 1.7510765 50.023 31.512 24.048 15.150 9.544 6.012
50◦Years 2.0541886 53.494 33.699 25.717 16.201 10.206 6.429
100◦Years 2.3267853 56.615 35.665 27.218 17.146 10.801 6.804

72 Kumar et al.
TABLE 8 | Rainfall intensity (I) computed from log-normal distribution.
Return periods (T) Value of “Z” calculated by Eq. (5) Durations
1 Hour 2 Hours 3 Hours 6 Hours 12 Hours 24 Hours
2◦Years −1.0E–07 27.978 17.625 13.451 8.473 5.338 3.363
5◦Years 0.8414567 38.389 24.184 18.455 11.626 7.324 4.614
10◦Years 1.2817288 45.299 28.537 21.777 13.719 8.642 5.444
25◦Years 1.7510765 54.040 34.043 25.980 16.366 10.310 6.495
50◦Years 2.0541886 60.563 38.152 29.116 18.342 11.555 7.279
100◦Years 2.3267853 67.099 42.269 32.258 20.321 12.801 8.064
For the case of p > 0.5, “p” in Eq. (3) is substituted by
(1 − p), and Z gives a negative value. Considering Eq. (1),
for a single time, “P” is the arithmetic average of the rainfall
records Moreover, “S” is the standard deviation, and the
multiplication of “S” and “KT” gives the output as departure
of a return period. Finally, to develop the IDF curve, the
rainfall intensity I (in millimeters per hour) with respect to
a specific return period “T” and storm duration “t” (in hours)
is calculated by using Eq. (5):
I =
PT
t
(8)
In our project, we use the previously mentioned as well as
the following procedures to find the expected intensities for
six different rainfall durations and six different return periods
using the normal distribution (14).
Now, on the basis of recorded rainfall data, the values of
standard deviation (SD) and average precipitation (P) are
calculated by Eqs. (2) and (3) and mentioned in Table 2. After
that, using the value of Z for six different return periods in Eq.
(4), the corresponding value of expected rainfall depth (PT)
is calculated and by using Eq. (8), corresponding value of
expected intensities for six different rainfall durations and six
different return periods is calculated, which are mentioned in
Table 3.
Using Table 6, the IDF curve is finally shown with rainfall
intensity on the y-axis and rainfall duration on the x-axis.
With the help of “Microsoft excel software,” which is
shown in [Figure 2;(15)].
13
Log - Normal distribution
80
70
60
2 Years
5 Years
10 Years
20 100 Years
10
Rainfall
intensity
(inmm/
hr.)
FIGURE 3 | Intensity-duration-frequency (IDF) curve by log-normal
distribution.
Log-normal distribution
By means of the log-normal distribution with the interference
of logarithm variables, the frequency of precipitation can
be calculated, which is like the normal distribution.
Calculations for average precipitation and standard
deviations are done through logarithmically transformed
data (16):
P∗
= log(Pi) (9)
P
∗
=
1
n
X
i=1
nP∗
(10)
S∗
=
1
n
i
X
i=1
n(P∗
− P
∗
)2
(11)
The frequency precipitation is calculated as
PT∗
= P
∗
+ KT∗
S∗
(12)
The intensity can be calculated by
I = PT/t (13)
where PT is the antilogarithm of PT and KT is the
frequency factor with the same value as “Z” in the normal
distribution. In our project, the earlier discussed as well as
the following procedures are utilized to find the expected
intensities for six different rainfall durations and six different
return periods by log-normal distribution (17). Now, on
the basis of recorded rainfall data, the first values of
P∗ for different durations are calculated using Eq. (9)
and Table 1 and mentioned in Table 4. After that, the
values of standard deviation (S∗) and average precipitation
TABLE 9 | Values of standard deviation (S) and average
precipitation (X).
X 29.971 18.881 14.409 9.077 5.718 3.602
S 11.451 7.214 5.505 3.468 2.185 1.376

10.54646/bijcees.2023.08 73
TABLE 10 | Rainfall intensity (I) computed from Gumbel distribution EV1.
Return periods (T) Value of “KT” calculated by Eq. (14) Durations
1 Hour 2 Hours 3 Hours 6 Hours 12 Hours 24 Hours
2◦Years −0.164 28.090 17.696 13.504 8.507 5.359 3.376
5◦Years 0.719 38.210 24.071 18.369 11.572 7.290 4.592
10◦Years 1.305 44.910 28.291 21.590 13.601 8.568 5.398
25◦Years 2.044 53.375 33.624 25.660 16.165 10.183 6.415
50◦Years 2.592 59.656 37.581 28.679 18.067 11.381 7.170
100◦Years 3.137 65.889 41.508 31.676 19.955 12.571 7.919
TABLE 11 | Chi-square goodness of fit test for various yearly rainfall patterns in years 1981−2019.
Probability of
occurrences P (%)
Return
period (T)
Observed rainfall
depth (in mm) for
24◦H duration
Expected rainfall depth (in mm)
for 24◦H duration calculated by
using probability distribution
Chi-square test values for different
probability distribution
Normal Log-normal Gumbel Normal Log-normal Gumbel
50 2 76.7 86.5 80.7 81.0 1.093 0.195 0.227
20 5 110.8 114.2 110.7 110.2 0.105 0.000 0.003
10 10 136.5 128.8 130.7 129.5 0.465 0.263 0.377
4 25 170.6 144.3 155.9 154.0 4.787 1.385 1.792
2 50 196.3 154.3 174.7 172.1 11.443 2.678 3.417
1 100 222.1 163.3 193.5 190.1 21.149 4.206 5.394
Total 39.042 8.727 11.209
(P
∗
) are calculated by Eqs. (10, 11), respectively, and
mentioned in Table 5. After that, by using the value of Z
for six different return periods in Eq. (12), corresponding
values of expected rainfall depth (PT∗) are calculated,
and again by using Eq. (13), corresponding values of
expected intensities for six different rainfall durations and six
different return periods are calculated, which are mentioned
in Table 6.
Finally, using Table 8, the IDF curve is displayed with
rainfall intensity on the y-axis and rainfall duration on the
x-axis, with the help of “Microsoft Excel software,” which is
shown in Figure 3.
Gumbel Distribution
7
0
2
4
5
3
0
10 Years
25 Years
1
0
1 2 3 6 12 24
Duration (in hrs.)
Rainfallintensity(in
mm/hr.)
FIGURE 4 | Intensity-duration-frequency (IDF) curve by Gumbel
distribution.
Gumbel distribution (EV1)
After the name of the developer, Gumbel, the functionality is
termed, and it is also called “type 1 distribution of maxima.”
Utilizing the Gumbel distribution, the IDF curves are studied
and assessed as fitting maxima for attaining appropriateness.
Utilization of the maximum rainfall values and extreme
data with ease is done by the Gumbel method. When
using the “likely to normal” function approach to estimate
precipitation frequency, a different occurrence factor K is
used, which is supplied by:
kt =
√
6
π
0.5772 + 1n 1n
T
T − 1
!!
(14)
200,0
180,0
160,0
140,0
120,0
100,0
80,0
60,0
1,0 10,0 100,0
8
9
,
0
5
+
)
x
(
n
l
3
5
1
,
7
3
=
y
9
0
6
9
,
0
=
²
R
Observed
rainfall
(in
mm)
y = 37, 153ln(x) + 5
5
5
50,
0,
0
0,
0,
0,
0,
0
0,
0,
0,
0,
0
0
0,
0
0,
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 98
98
98
98
8
8
98
98
98
8
98
98
98
98
8
8
98
8
8
8
8
8
98
8
98
8
98
8
8
8
98
8
98
8
8
98
98
8
9
9
98
8
8
98
9
9
98
8
9
9
9
9
9
9
9
9
9
9
9
9
9
9
9
R ² = 0,
0,
0,
0,
0,
0
0,
0
0,
,
0,
0
0
0,
,
0
0
0
0,
0
0
0
0
0
0
0
0
0
0
0 96
96
96
96
6
6
6
96
6
96
96
96
96
96
96
6
96
96
96
96
96
6
96
6
96
96
96
96
96
96
96
6
9
96
6
96
6
9
9
96
96
6
9
96
6
9
9
9
9
9
9
9
9
9 09
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
FIGURE 5 | Graph between observed rainfall (in mm) in 24 h and their
return period.

74 Kumar et al.
The Gumbel distribution uses the following equation
proposed by Chow:
XT = Xavg + K∗
TS (15)
where XT is the intensity in millimeters per hour, Xavg is the
mean, S is the standard deviation, and KT is the frequency
factor.
X =
1
m
m
X
i
xi (16)
In the present study, the earlier discussed as well as the
following procedures are utilized to find the probable
rainfall intensities for six dissimilar rainfall durations and six
different return periods by Gumbel distribution.
Firstly, on the basis of recorded rainfall data series, rainfall
intensity (X) data series for different durations are calculated
from Table 1 by simply dividing the value of rainfall depth
by their duration, as mentioned in Table 9. After that, the
values of the standard deviation (S) and average precipitation
(X) are calculated by Eq. (16), and mentioned in Table 9.
Further, by using Eq. (14), the frequency factor for different
return periods is calculated, and finally, corresponding values
of expected rainfall intensity are calculated by using Eq. (15)
for six different rainfall durations and six different return
periods, which are mentioned in Table 10.
Finally, the IDF curve is designed with rainfall period
on the x-axis and rainfall intensity on the y-axis by using
Table 10 with the help of “Microsoft Excel software,” which
is shown in Figure 4.
Goodness of fit
The chi-square test is typically used to see how closely the
values anticipated by the theoretical distribution fitted to
the data and the values actually observed during the return
period, T, match up.
The chi-square values with the lowest values
provided the best match.
Now, before carrying out a chi-square test, difference in
observed rainfall depth (in millimeters) between 39 years of
24 h duration and their return period is plotted on a log scale,
which is shown in Figure 5, and its variation is analyzed.
The aforementioned chi-square test of goodness of fit was
conducted for various distributions of the maximum annual
rainfall in the years 1981−2019, and its value for various
probability distributions was computed using Eq. (18) and
mentioned in Table 11.
Results
The relationship between rainfall intensity and time
durations, also known as the return period, can be generated
using the normal distribution, log-normal distribution, and
Gumbel distribution (EV1). In this paper, we calculated
the intensity, and the result shows that with the increase in
rainfall, the intensity of the return periods also increases.
This is shown in Tables 3, 4, 10. The intensity was
calculated with the help of return periods with respect to
probability distributions.
Conclusions
The observed rainfall data were used to formulate the
probability distribution function, and it represents the
suitable probability distribution. The rainfall pattern depends
upon the observed rainfall data. It was discovered that rainfall
patterns vary by location.
Data on rainfall were compared statistically at 1, 2, 4, 10,
20, and 50 percent probability using the chi-square test for
goodness of fit. It demonstrates that when compared to the
normal distribution and the Gumbel distribution technique,
the log-normal distribution has the lowest value. Prediction
using the log-normal distribution approach was therefore
determined to be the best model for the Patna city region.
Conflict of interest
During the study, there were no financial or commercial ties
that could be interpreted as potential conflicts of interest.
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