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 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 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° 30′ N latitude, 85° 15′ 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.
Figure 1. Location map of the study area.
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 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.
Table 1. Annual maximum rainfall depth, P (in mm) of different durations.
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:
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 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.
Table 2.
Overall, the regression model statistically substantially predicts the outcome variable, according to ANOVA (Table 3), which shows p = 0.000, which is less than 0.05. (i.e., it is a good fit for the data).
Table 3.
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(ShortWave 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.
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.
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):
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):
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):
Here, “w” is calculated as
In Eq. (3), “p” is the probability of occurrence in a specified return period “T” and its value calculated as
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):
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)].
Figure 2. Intensity-duration-frequency (IDF) curve by 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):
The frequency precipitation is calculated as
The intensity can be calculated by
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 () 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.
Table 4.
Table 5. Values of S and P for normal distribution.
Table 6. Rainfall intensity (I) computed from normal distribution.
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.
Table 7. Value of standard deviation (S*) and avg. precipitation (P*).
Table 8. Rainfall intensity (I) computed from log-normal distribution.
Figure 3. Intensity-duration-frequency (IDF) curve by log-normal 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:
The Gumbel distribution uses the following equation proposed by Chow:
where XT is the intensity in millimeters per hour, Xavg is the mean, S is the standard deviation, and KT is the frequency factor.
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 () 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.
Table 9. Values of standard deviation (S) and average precipitation (X).
Table 10. Rainfall intensity (I) computed from Gumbel distribution EV1.
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.
Figure 4. Intensity-duration-frequency (IDF) curve by Gumbel distribution.
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.
Figure 5. Graph between observed rainfall (in mm) in 24 h and their return period.
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.
Table 11. Chi-square goodness of fit test for various yearly rainfall patterns in years 1981–2019.
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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