Modeling Reference Evapotranspiration with Wind Speed and Relative Humidity Data

-The objective of this research is to investigate whether evapotranspiration can be modelled using wind speed and relative humidity data only. A linear model based on wind speed and relative humidity has been formulated for estimation of pan coefficient (Kp). The equation is calibrated by using multivariate regression analysis. The coefficients of wind speed and relative humidity thus obtained were used for deciding the range of Monte Carlo simulations. Nash-Sutcliffe efficiency (NSE) is used as an objective function and the results of Monte Carlo simulations were used for sensitivity analysis. Sensitive model parameter shows differences in separation and form between the cumulative frequency distribution curves. Sensitivity analysis revealed that the simple linear model developed in this study is sensitive to relative humidity only; the wind velocity is an insensitive parameter. The impression after conducting this study is that the pan evaporation equation can be modelled using relative humidity and wind velocity data with the Nash-Sutcliff efficiency varying between 65 to 90 per cent which is acceptable. Keyword Evapotranspiration, Pan Evaporation, Monte-Carlo Simulations, Sensitivity Analysis


II. DATA USED
The meteorological data was collected form hydrological data user group (HDUG) Nashik, Maharashtra, India. The data comprises maximum and minimum temperature, relative humidity, wind velocity (measured at 2m above ground) and pan evaporation. The data were sorted year wise and calculations were done for aerodynamic resistance (Ra), bulk surface resistance (Rs), saturation vapour pressure deficit, net radiation (Rn), soil heat flux (G) etc. Using measured and calculated data the year wise daily reference evapotranspiration values based on FAO-56 penman's equation were calculated for each station. For missing data the average value of preceding and succeeding years for the particular metrological parameter has been adopted.
III. METHODOLOGY The Meteorological data was collected for the period of 2002 to 2008 for four Meteorological stations. However substantial amount of data (period of one month or more) was missing for the years 2000, 2001 and 2009. Hence, these years were excluded from the analysis. Multivariate analysis was performed for each year's data of every station. The results of multivariate analysis are tabulated in Table I. The intention behind carrying out the multivariate analysis was to find the coefficients of linear regression model with two variables, namely Relative humidity and wind velocity as given under, Here, RHmean corresponds to relative humidity (%), and U2 corresponds to the wind speed (m/s) measured at 2 m height. Once this value has been determined, reference evapotranspiration can be calculated as: The primary goal is to determine the set of coefficients i, such that model predicts the values of ET0 as accurately as possible compared to those produced by FAO-56 Penman-Monteith equation. The purpose was to investigate whether a linear model with only two variables (RHmean& U2) is adequate to fit the standard FAO-56 Penman-Monteith ET0 results. We also wanted to investigate the sensitivity of the parameters i by ranking the parameters according to Monte-Carlo realizations and examining the proximity of the yearly sample sets. However the results of multivariate analysis are not really encouraging except for some years, were NSE is above 60 per cent. But it served the purpose of finding parameter spaces for Monte Carlo simulations. Every model has a set of parameters that cannot be measured directly, but these parameters can be construed from the calibration process. The calibration process is generally a trial-and-error method that adjusts the parameter values in such a manner that the input-output behaviour of the model matches with the real world system which it represents. In evapotranspiration modelling the model generated values of evapotranspiration (ETp) are matched with those calculated using FAO-56 PM equation.
Manual calibration procedures are labour-intensive and time consuming. These difficulties with the manual calibration method have led to the evolution of automatic calibration procedures which utilize the speed and power of computers. Many studies using manual calibration have reported the difficulties in finding the optimum parameter estimate. Multiple local optimum parameter sets have been noticed while employing optimization algorithms irrespective of the modelling methodology. It still remains typically difficult, if not impossible to find a unique "Best" parameter set whose performance measure differs significantly from other parameter sets.

IV. MONTE CARLO SIMULATION TECHNIQUE
Repeated random sampling is the basis of all computational algorithms that are classified as Monte Carlo methods and are used to obtain numerical results. In this technique the distribution of unknown probabilistic entity is obtained by running over the simulations many times. Monte Carlo methods vary, but tend to follow a particular pattern:  Selecting imprecisely known model input parameters to be sampled.
 Assigning ranges and probability distributions for each of these parameters.
 Generating many sample sets (realizations) with random values of model parameters.
 Running the model for all realizations to estimate uncertainty in model outcomes.
The optimization of the parameters of a model requires the use of an objective function. Objective function is a reference numerical quantity enabling the calibration to be improved. The choice of objective function to be used for a given model is a subjective decision which influences the values of the parameters and the performance of the model. The objective functions for hydrologic simulations used in the present study are given below, The regression coefficient or the measure of the model efficiency discussed by Nash and Sutcliffe (1970), used in the present study to evaluate model performance is as given below,

VI. SENSITIVITY ANALYSIS
The primary objective of sensitivity analysis is to identify whether the perturbation of parameter significantly affect the model response i.e. the variable of interest. In case it is observed that impact of particular parameter is small, the relevant parameters may be replaced by constants or eliminated altogether. This strategy will not only help model construction but also model calibration on parameter estimation. In Pan evaporation modelling the main aim of carrying out sensitivity analysis is to investigate sensitivity of Kp to Relative humidity (RHmean) and wind velocity (U2).

VII. CONCLUSIONS
The multivariate analysis was done to find the optimum values of i, the coefficients of the evapotranspiration equation modeled using mean relative humidity and wind velocity. However the results of multivariate analysis did not yield acceptable results. Hence it can be concluded that multivariate analysis is not useful for optimization of models of pan evaporation studies. However it can used to find parameter spaces and hence the range of the dependent variables to be optimized using more advanced techniques. Monte Carlo simulation technique is used in this study to find the optimum parameter set for each year of observation. The Nash-Sutcliff efficiency was observed to vary between 65 to 90 percent which is acceptable. Other performance indicators (SSE, SLE & SAE) are in the acceptable range. The value of pan coefficient Kp on daily scale were observed to be greater than one as mentioned in the earlier literature. The averaging of the optimum parameter sets obtained for different years of analysis using MCS run worked well and the unique coefficients for the pan evaporation equation were obtained. Sensitivity analysis showed that Pan Evaporation equation is insensitive to the coefficient of wind velocity ( 2 ), However it is sensitive to the constant of the equation ( 0 ) and coefficient of relative humidity ( 1 ). The impression after conducting this study is that the pan evaporation equation can be modeled using relative humidity and wind velocity data based on the averaging of coefficients ( 1 ,  2 ) and a constant of equation ( 0 ) of yearly analysis.