The daily rainfall data (1988-2018) were analysed to extrapolate maximum rainfall in Dharwad district of Karnataka state, India. Newer run-off model was formulated to study the different cropping patterns@ different significant levels. Absolute  rainfall data  were  divided into 23 sets viz. one- annual  and  four  seasons  (June to September) were  Standardized based on   different meteorological  weeks (23-39rd SMW). The model findings showed that, the annual maximum daily rainfall IQR ranged between (23.14-83.44) mm indicated  with large amount of fluctuation of rainfall because due to periodic changing of the weather parameters and global warming. Asper  the models diagnostic checking, the  runoff- probability  forecasted distribution model was found  to be  best fit  to optimize  the absolute daily rainfall. Our formulated  model will be very useful for the agricultural scientists for designing of  cropping  patterns  and also  it can be  served  as  navigation tool for the  meteorologists,  agricultural policy planners and researchers for  the  holistic development of soil conservation strategies and  irrigation purpose in the developing  countries. 

Maximum daily rainfall, Probability distributions, Kolmogorov-Smirnov test.

Rainfall is one of the most important natural resources for the growth of agricultural crops. The maximum rainfall is related to natural systems and unpredictable seasons. Paucity of mathematical/statistical computer-aided predictive modelling techniques makes the forecasting of rainfalls (uncertainties and likelihoods), the major biggest problems facing the world today. Rainfall and temperature variations are caused by environmental factors and are affected by the cyclone's periodic effects. One of the difficult worldwide challenges of the day is how to accurately identify rainfall and other meteorological conditions. The effectiveness of agricultural cultivation and irrigation will be determined by on time frame. The long-term study of rainfalls is necessary to carry out planning and cultivating crops. However, success and failure depends on various rainfall conditions. It is closely correlated with the weather forecast. Imbalanced weather conditions had a significant impact on human resources, farming, management and crop production. The weather is extremely complicated, making it difficult to choose the best choice for crop production. Due to the negative impacts of climatic conditions, the SARS-CoV-2 pandemic and other bacterial infections have spread throughout the world, discouraging farmers and agricultural specialists from focusing on managing agricultural resources, such as imports and exports to a global level (distribution of seeds, fertilisers and agricultural equipment's etc. Accordingly, the explanation of heavy rainfall encourages taking into account unexpected events that make projecting future output and economic losses.  For the management of crops and the adoption of modern agricultural policies for changing the cropping pattern to a greater extent, simulation data driven prediction support modelling is absolutely critical. The present analytical study will help the prediction of rainfall and management of water and natural resources. This navigated formulated model will be the greatest advantage for forecasting weather parameters, floods, cyclonic seasonal variations and drought prone areas at national level. This study attempt to determine to formulate various probability distribution models to optimize the maximum daily rainfall. 

Thrust area. Dharwad district was purposively selected for the study, it is located in the north-western part of Karnataka, co-ordinated @ 15°27′30″ North and East @ 75°00′30″. As experienced tropical wet climate, (the similar ≈741 meter above the sea level). The boundary was encompassed with a geographical area of 4263 sq. kms. South-west monsoon is the most common which can be received  maximum rain fall. An average annual rainfall was ≈ 864 mm (over five years).  Very less rainfall was recorded   in winter instead of summer season. Humid/dry weather in Feb to early June month, the highest quantity during precipitations with mean temperature (24.1°C). The hot climate in the summer (April–May) and nice throughout the year. The hot season begins from March to Jun with the highest temperature of (38oC) and lowest temperature (14oC) during the month of December (freezing month). After  SARS-CoV-2 pandemic attacked, all weather  parameters  have  been changed, periodically hitting  cyclones  and got  maximum rain fall  in the  year  2021 -2022 (regular rainfall > 1800mm). Under these conditions, the net crop sown area is very less in Dharwad district in 2020-2022; it will have a significant impact on both economic security and crop yields. The accurate likelihood estimation of rainfall is very essential for estimation of cause and effect relationship between the climatic conditions and cropping patterns at national level. In this practical manner, we have formulated newer run off -model for absolute estimation and its likelihoods. A preliminary stage, the sample data were used to demonstrate our projected models. The selection of sample has drawn from the total geographical areas it was determined by the following eqn (1)  

  (1)

Where, Xi = Random sample, i= 1, 2,…, n ;

CDF =F(x)= [number of observations ≤ x] .

This test was used to decide the samples   drawn   from hypothesized distribution. Finally, we have formulated probability distribution models to optimize the exact rainfall. The above methodology will also be used for studying the probability distribution pattern for other disciplines.  In case of model formulation, we used daily rainfall data besides with  different weather parameters  between   (1988-2022),  the data  was collected from the Department of Agricultural Meteorology, Government of  Karnataka  and University of Agricultural Science , Dharwad. The primary and secondary data were used for demonstration of the formulated model.

Model formulation 

                          (2)

Where,

Yijk = Observed rail fall for ith village jth district kth cluster  

αi = Effect of ith village from rain fall

βjk = Interaction effect of rail fall on jth districts kth cluster

tk = The rain fall seasons at kth clusters

εijk = Error associated with ith village jth districts kth cluster

In equation (1) we predicated the random process of and correlation of rainfall in selected villages. The average rainfall was estimated for building the interaction effects with noisy data sets in eqn (2) 

                                        (3)

ni = Number of years attributed to the average rainfall occurs in particular seasons

For  the  elimination of noisy data below  the  average  rainfall in the  particular districts,  we build  the  models  in following eqn (4)

                        (4)

Run- off  Predictive model (RPM). The  runoff- model is  versatile and  robust analytical  algorithmic  predictive model, that was confined  to estimates  the  absolute  rain fall  in the  selected  areas. Mainly we substituted infiltrations, evapotranspiration, and depression storage and drainage losses in particular agriculture crops. Evolving  of  runoff  model which is consisted  too- many factors of random variables ‘Xi’. The  random variables  ‘Xi’  is  normally distributed  (Gaussian distribution) with mean μ, and variance σ2.

The  cumulative estimation of  rain fall in  the Gaussian distribution   attributed  with  rv’s  is  estimated in the  eqn (5)               (5) The  most widely used non parametric  method  is  the  kernel density  estimates  .We describing  the  initial  losses of  random variables {X1, X2, …. Xn} with  kernel function k(.),  the probability density function (PDF)  can be estimated   in the  eqn (6) 

                                        (6)

where  ‘h’ is  smoothing  factor,   known as  the  bandwidth and K(.) is  the  Gaussian Kernel density   with distribution

                          (7) The bandwidth is the most important characteristics of a Kernel density estimates (KDE) with a strong association between shape and scale parameters of PDF  

            (8)

Where, the IQR is the Interquartile range of rainfall in the selected years @ time ‘t’ and ‘σ’ is the sample standard deviation. 

In case of rainfall-runoff modelling, the loss parameter is one of the most important parameter, which  can describe  the  total  amount of rainfall. Which was estimated by using   infiltration, evapotranspiration, interception, depression storage and transmission losses etc. The several parameters were considered for the formulation of   modelling eqn (9)

  (9) 

Where tα/2 = hypothetical table value (1.96)

N = Number of selected waste hr/soil parameters

β0, β1, β2 … βn is the  coefficients of  random variables of xi (Soil moisture,  water wastage from drainage, ground water layer, soil condition/types  of  soil, turbidity of  the  soil  etc).

These parametric distributions rely on the assumption that the population (which the observed data sample belongs to) has an underlying parental distribution. These results were observed from the kernel density function. A nonparametric approaches also been used to correlated prior and posterior information’s of rainfall.  However, we substituted both assumptions that have been demonstrated by underlying probability distribution. This method will be, relies heavily  depends on the  absolute   rainfall data and its empirical grouping of  individual  bins. This model based study has compared for both parametric and nonparametric distributions @   initial losses of rainfall and outlier effects on periodicity of cyclones. 

        (10)

Where,

f (x) = The function f(x) describes the absolute rainfall yield with (x) random process 

n = Number of weather parameters responsible for RF

k = Number of success of geographical areas selected for likelihood estimation

ai = Regression intercept (point of infelcxion)

βij = Regression coefficient with respect to different parameter under concern

(At ith traits with jth areas )

εij = Residual error with ith traits  with jth areas.

(α = k + 1) ; (β = n – k + 1)

Example

α = k + 1 = (7 + 1) = 8, β = 10 – 7 + 1 = 4

 The random behaviour  of  proportion of  runoff  rate  was  modelled  by β-distribution. It provides  powerful tool quantify  the likelihood estimation  tasks. The probability density function is 

                                    (11)

Beta distribution is the conjugate of Binomial distribution 

BD                          (12)  

                        (13)

One day maximum daily rainfall corresponding date for the period of 31 years (1988-2018) is presented in the Table 1. The maximum (83.44 mm) and minimum (23.14 mm) annual one day maximum rainfall was recorded during the year 2000 (11th June) and 2016 (28th June) respectively. This indicates that the most fluctuations were observed during the decade 2000-11. The average for these 31 years rainfall was found to be 41.23 mm. It was also observed that 14 years (45.16%) received one day maximum daily rainfall above the average, however, no general trend in rainfall occurrence was observed during the study period. Fig. 1 shows the variation in one day maximum daily rainfall. The distribution of one day maximum rainfall received during different months in a year is presented in Fig. 2. From the Fig. 2, it can be seen that June received the highest amount of one day maximum rainfall (23%) followed by July (16%), September (16%) and October (16%) (Singh et al., 2012). 

The data was classified into 23 sets as mentioned in the Table 2 viz., 1 annual, 1 seasonal, 4 seasonal months and 17 seasonal SMW’s to study the distribution pattern of rainfall at different levels.

Fig. 1. Geographical distribution of   rainfall in Karnataka State.

Fig. 2. Absolute  mean  earth surface temperature  in Dharwad  District.

Table 1: One day maximum daily rainfall for the period of 1988 to 2018.

    3.91 10.85 meters below ground level in 2015 to 3.91 meters below ground level in 2019.

Table 2: Parameters of the best fit probability distributions for maximum daily rainfall.

Fig. 3. Year wise annual maximum daily rainfall (in mm).

Fig. 4. Distribution of one day maximum annual rainfall in a year.

Fig.  5. Extrapolation of minimum rainfall  by exponential smoothening  by Gaussian.

From the formulated model the analysis observed that majority of the data sets Log-normal distribution was found to be the best fit probability distribution. Similar results were obtained by Manikandan et al. (2011). Further, majority of the study periods were observed to be scale dominated which showed large variation in the distribution of rainfall. The best fit probability distributions for maximum daily rainfall on different sets of data were identified based on the criteria of Kolmogorov-Smirnov test. A similar study was conducted by Nassif et al. (2021). The distributions such as Normal, Log-normal, Gamma (2P), GEV and Weibull (2P) were found to be the best fit for all the study periods. During the month of August, 25th, 28th, 35th and 38th SMW’s, Weibull (3P) distribution fitted well along with the distributions mentioned above. For annual, seasonal, June, August and September study periods, Log-normal was found to be the best fit probability distribution. For July month, GEV distribution fitted well with the lowest test statistic value and the highest p-value. For the 24th, 25th, 29th, 31th, 33th and 34th SMW’s, Log-normal was found as the best fit probability distribution, for 28th, 30th, 36th and 39th SMW’s, Gamma (2P) was found as the best fit probability distribution, for 27th, 32nd, 35th and 37th SMW’s, Weibull (2P) was found as the best fit probability distribution, for 23rd and 26th SMW’s, GEV was found as the best fit probability distribution and for 38th SMW, Weibull (3P) was found as the best fit probability distribution with the lowest test statistic value and the highest p-value. In this formulated  model  we  demonstrated  real data  sets  at possible  iteration to optimize  by means  of  combining  state  variables. We  start with above  formulated predictive models, which  are  supposed  to have  been developed  in  design phase  of  RF estimation. As  an  example,  eqn (10) parameter  of  the  model  we  can optimize by means of  historical  data  sets  as  part  of  the optimizing  techniques, as  model  construction  we substituted  different state  variables  like  too many weather  parameters, RF geographical  area, runoff  surface  flow, wind  blow, earth surface  temperature  , wind  pressure and  ocean current etc. Due  to outlier  state variable model (heavy rainfall due  to  occurrence  of  periodic  cyclone, flood, drastic  changes  of  climate for heavy RF, biotic  intervention etc), the model  was  rebooted and  lead  with  large  inherent uncertainty (estimated  errors has  deeply penetrated  to reproduce  imbalance  forecasted figures).The  above  critical circumstances, the  model  can able  to allow  the  substitution  parameters to render  the  absolute forecasted  figures (set of  forecasted  quality)   for  usual  interpretations. For a regression model eqn () without unobserved components, such as predicted model, the error statistics are the same as forecast error statistics. A similar   forecasted  model  was fitted  by (Erich J. Plate 2015), Asper  his  model predicted values, the     errors  are  varied  by means of  substitution of  large  number  of  parameters and also every iteration can be precluded  unusual  values .However, our  predicted  model will be very easy and  earnestly  produced  accurate  absolute values  of  minimum rain fall  even we substituted larger  group  of  parameters  with highest credibility and accuracy.The  variance (σ2)  of the  model was  considered  with weighted  parameters  for  standardization of  data  sets  because  due  to  floods , cyclonic  hit and  climatic  weather  parameter  variation. All observations are normally distributed with mean μ and common variance  i.e.  We presumed that; all observed values have followed    the Gaussianity with weighted values (Weighted logistic regression)   . Our predictive run-off errors were produced the best likelihood estimation, it was reduced   up to  95% (R2=0.93, Negenkal R square value was 108.8). The  present forecasted  compartment -model  can able  to yield  good  approximate   values, even  if  the  data series  will follow  the  heterology   distribution. Model iteration has restored at the 100pecent protection @ the  time  of  reproducibility. In the  relative  comparison of  RF (Dharwad  district ), the maximum   distribution of  RF was seen in  different  Progressive months between  June (maximum) followed  by  July (23%) , September (16%) and October (16%) respectively. The results  of the predictive  rainfall model identified  the best  fit  which has  extrapolated various patterns for  unobserved components different with  larger  number of traits (using run off model). The data showed that,  the annual maximum daily rainfall will  be received @  any specified  range  between 23.14 mm (minimum) to 83.44 mm (maximum) indicated at large account of  fluctuations were seen  in the  month of Jun and  July and  also we  observed that, maximum rainfall can received  with mean temperature  (32-33oC)   with moderate  precipitation.  Log-normal distribution was the best fit probability distribution for majority of the study periods. This model will help us  to explore the newer  ideas about the prediction of annual maximum daily rainfall to design with small and medium hydraulic pressure. Indirectly we will quantify the soil and water conservation structures, irrigation, drainage works, vegetative waterways and field diversions in the farmers’ field. 

Future line of work can be the current model, though focused on Dharwad district, can be extended to other agro-climatic zones in India to generalize the model's applicability. Coupling the predictive models with real-time meteorological data feeds and satellite observations can improve forecasting capabilities. Linking the rainfall prediction model with crop simulation software (e.g., DSSAT, APSIM) can help in optimizing sowing dates and irrigation schedules.