1 Introduction

Of all the components of the hydrological cycle, evapotranspiration (ET) is the key component and its estimation is the prior task of researchers and practitioners working in the fields of land, crop, water, and atmosphere studies. The procedure for estimating ET rates of agricultural crops (crop ET, ETc.) involves two steps. As a first step, computation of reference evapotranspiration (ETo) is carried out using regularly recorded climatic data and ETo is multiplied by the crop coefficient (K c ) in the second step. The K c incorporates crop characteristics and averaged effects of evaporation from the soil (Doorenbos and Pruitt 1977). Accurate estimation of ETo is the basis for solving a wide array of problems such as crop water requirement computation, irrigation scheduling, water balance computation, evaluation of land use changes etc.

There exists a multitude of methods (e.g., Jensen et al. 1990; Jennifer and Sudheer 2001; George et al. 2002; Itenfisu et al. 2003) for measurement and estimation of ETo. Direct measurement of ETo using the lysimeter or the water balance approach is a costly and time consuming process (Adamala et al. 2014a,b). Therefore, based on the easily available location characteristics (elevation and latitude) and meteorological parameters many indirect methods have been developed for ETo estimation, viz. (i) temperature based (Thornthwaite 1948; Hargreaves and Samani 1985), (ii) radiation based (Priestley and Taylor 1972; Turc 1961), (iii) evaporation based (Christiansen 1968), and (iv) combination method (Penman 1948).

The reliability of ETo estimation by the above methods largely depends on the site characteristics, quality and quantity of input climatic data, chosen method, and assumptions related to its parameterization (Adamala et al. 2015). When the required set of climate data are available for a site, ETo is often calculated using the FAO-56 Penman-Monteith (FAO-56 PM) combination method. This method is recommended as a sole and standard method by the Food and Agriculture Organization (FAO) (Allen et al. 1998) for the estimation of ETo if all the required data are available.

ETo provides a measure of the integrated effect of climatic variables such as temperature (T avg ), humidity (RH avg ), wind speed (W s ), solar radiation (R s ). In humid and arid regions, ETo provides an upper limit for ETc and indicates the total available energy for ETc. Among the above climatic variables, only some of the input variables exert a greater influence on ETo, as compared to others. Thus, it is very important to understand the effect of a change in each climatic variable on estimated ETo before performing any analysis. Generally, the above objective can be achieved by a process called sensitivity analysis. Sensitivity analysis quantifies how changes in the independent variables (input) of the equation or model affect the dependent variable (output). The results of this analysis make it possible to determine the required accuracy for measuring climatic variables to be used in estimating ETo (Irmak et al. 2006).

In the past, a few studies (McCuen 1974; Saxton 1975; Coleman and DeCoursey 1976; Beven 1979; Piper 1989; McKenney and Rosenberg 1993; Ley et al. 1994; Rana and Katerji 1998) have been devoted to sensitivity analysis of evaporation or ETo combination models to different input and parametric data using single or multiple climatic stations. Hupet and Vanclooster (2001) quantified the effect of the sampling frequency of commonly measured climatic variables on ETo estimates by the FAO-56 PM equation in a moderately humid climate area in Belgium. The results showed that the R s and W s were the most sensitive to bias induced by the inadequate temporal sampling frequency. Goyal (2004) showed that ETo was less sensitive to increase in R s , followed by W s in comparison to T avg and increase in vapor pressure had a small negative effect on ETo. Gong et al. (2006) used non-dimensional relative sensitivity coefficients to predict responses of ETo to perturbations in four climatic variables: T avg , W s , RH avg , and sunshine duration. Results showed that RH avg was the most sensitive variable, followed by short-wave radiation, T avg and W s . Irmak et al. (2006) studied the sensitivity of the standardized ASCE PM ETo equation in different climates of the United States from semi-arid to humid conditions. The results indicated that the ETo was most sensitive to vapor pressure deficit (VPD) at all sites, while the sensitivity of ETo to the same variable showed significant variation from one site to another and at the same site within the year. Bormann (2010) compared 18 different potential evapotranspiration (PET) models with respect to their sensitivity to observed climate change. It was found that the PET models were sensitive to significant trends in climate data and all models showed different sensitivities. Kwon and Choi (2011) reported that an estimated ETo showed different sensitivity to variations of meteorological parameters in order of vapor pressure followed by W s and R s . Tabari and Hosseinzadeh Talaee (2014) reported that the sensitivity of ETo to wind speed and air temperature decreased and to sunshine hours increased from arid to the humid environment.

Most studies have focused on the sensitivity of different evaporation or ETo combination models for a single region/climatic station. However, Irmak et al. (2006) carried out a sensitivity analysis for ASCE PM ETo model for different regions (semi-arid, Mediterranean, coastal humid, inland humid and semi-humid, and island). The sensitivity of the FAO-56 PM combination-based equation to climatic variables for different regions (semi-arid, humid, and sub-humid) in India has not yet been studied. Thus, the objectives of this study are:

  1. 1.

    To perform sensitivity analysis of the FAO-56 PM method to climate variables in the following regions of India: semi-arid (Kovilpatti and Parbhani), humid (Mohanpur), and sub-humid (Ludhiana and Ranichauri).

  2. 2.

    To derive sensitivity coefficients for each of the climatic variables and quantify daily changes in ETo per unit of change in each climatic variable.

  3. 3.

    To evaluate the seasonal trend of change in ETo.

2 Materials and Methods

2.1 Study Area and Climate Data

For this study, five climatic stations (Fig. 1) in different agro-ecological regions of India were considered. Study regions include semi-arid (Kovilpatti and Parbhani), humid (Mohanpur), and sub-humid (Ludhiana and Ranichauri). The daily climatic data of 5-year (2001–05) period for the selected stations were collected from a project called All India Coordinated Project on Agro-meteorology (AICPAM), Central Research Institute for Dryland Agriculture (CRIDA), India. Table 1 presents information related to latitude, longitude, elevation, and meteorological characteristics of the chosen sites. The study area covers a wide range of variation in altitude (10 m at Mohanpur to 1600 m at Ranichauri above the mean sea level) and mean annual rainfall (680 mm at Ludhiana to 1500 mm at Mohanpur). The characteristics of long-term average monthly climatic variables (Table 2) shows T avg ranges from 14.66 °C at Ranichauri to 28.67 °C at Kovilpatti. The maximum RH avg of 79.52 % is observed at the humid site of Mohanpur and minimum RH avg of 55.21 % is observed at Parbhani (semi-arid). Stronger winds with W s of 6.72 km h−1 are observed at Kovilpatti (semi-arid) and weak winds of 1.53 km h−1 are observed at Mohanpur (humid). The R s ranges from 16.31 MJ m−2 day−1 at Ranichauri to 25.37 MJ m−2 day−1 at Mohanpur.

Fig. 1
figure 1

The locations of selected five climatic stations as study sites in India

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Table 1 Location specifications and climate characteristics during the period 2001–05 of the selected study sites
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Table 2 Characteristics of long-term average monthly climatic variables for each study site
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2.2 Evapotranspiration Computation Method

There are many indirect methods to estimate ETo, but it is difficult to select the best ETo estimation method for the available data and climatic conditions. To overcome this problem, a decision support system (DSS) for ETo estimation, i.e., DSS_ET (Bandyopadhyaya et al. 2012) was developed, which supports 22 ETo estimation methods. It can serve as a research tool with its user-friendly features like options for calculation of various intermediate parameters, generalized data input format with copy–paste option from spreadsheet applications, visualize/check input data and results, features to estimate missing data, and user-friendly graphical user interface (GUI) that enhances its applicability. In the present study, the FAO-56 PM method was considered for daily ETo estimation and was computed using ‘DSS_ET’. The equation for the estimation of daily ETo can be written as (Allen et al. 1998):

$$ E{T}_o=\frac{0.408\varDelta \left({R}_n-G\right)+\gamma \frac{900}{T_{avg}+273}{W}_s\left({e}_s-{e}_a\right)}{\varDelta +\gamma \left(1+0.34{W}_s\right)} $$
(1)

where ETo = reference evapotranspiration (mm day−1); R n  = net radiation at the crop surface (MJ m−2 day−1); G = soil heat flux density (MJ m−2 day−1); T avg  = average daily air temperature at 2 m height (°C); W s  = wind speed at 2 m height (m s−1); e s  = saturation vapor pressure (kPa); e a  = actual vapor pressure (kPa); Δ = slope of saturation vapor pressure versus air temperature curve (kPa °C−1); γ = psychometric constant (kPa °C−1).

2.3 Sensitivity Analysis and Sensitivity Coefficients

There is no single, universally accepted procedure for analyzing the sensitivity and computing sensitivity coefficients. A common approach for analyzing sensitivity is to explore the effects of change in input variables, one at a time, on an output variable. The sensitivity coefficients for each climatic variable (i.e., T max , T min , R s , RH avg , and W s ) were derived from the ratio of change in ETo to the unit of change (either increase or decrease) in each climatic variable on a daily basis. It is represented as follows (Smajstrla et al. 1987; Irmak et al. 2006):

$$ {C}_s=\frac{C{H}_{E{T}_o}}{C{H}_{CV}} $$
(2)

where C s  = sensitivity coefficient, \( C{H}_{E{T}_o} \) = change in ETo with respect to change in climate variable, and CH CV = change (increase or decrease) in climate variable.

The absolute sensitivity analysis approach was used in this study. It was assumed that if any error encountered during data measurement, the quantity of error would not be any relative proportion of actual measurement. This error would be any absolute value which depends on the device used for measurement and/or operator.

2.4 Analysis of Sensitivity of ETo with respect to Climate Variables

The sensitivity of the FAO-56 PM ETo equation for each study site was quantified with respect to each climatic variable using the procedure reported by Irmak et al. (2006). Figure 2 shows the process flowchart of sensitivity analysis. It was assumed that maximum error could be encountered in data measurement is up to 5 units, either positive or negative. Therefore, sensitivity analysis was performed by increasing and decreasing the climate variables such as T max (°C), T min (°C), R s (MJ m−2 day−1), and RH avg (%) by one unit of increment and decrement up to five units (except for W s ) while keeping the other variables and parameters constant. However, wind speed W s (km h−1) was only increased with an increment of one km h−1 up to five km h−1 because of the lower value of W s . Here it is worth to mention that in the preliminary analysis of this study, it was found that T max and T min are not equally affecting ETo in all the study sites. Therefore, T max and T min were considered for analysis rather than average temperature (T avg ). In natural condition, these climatic variables are inter-related to each other and errors in one variable may change the other variable. But if any error is encountered at the time of measuring of any variable, it does not affect the measurement of other parameters. This is because measurement data are not inter-related to each other. Therefore, for sensitivity analysis one parameter was changed at a time with other parameters remained constant during the analysis period. The most commonly used unit was considered for the unit of each climatic variable (T max and T min (°C), R s (MJ m−2 day−1), W s (km h−1), and RH avg (%)).

Fig. 2
figure 2

Process flowchart of sensitivity analysis

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The daily average values for each climatic variable were obtained after taking an average value for a period of 5 years (i.e., 2001–05) climate data and the base ETo values for different stations were estimated using this data without any change (Fig. 2). Further, each climate variable was either increased and decreased or only increased (in the case of W s ) individually from one unit up to five unit with one unit interval, and a new set of ETo values was estimated followed by the change (either increase or decrease) in ETo on daily basis. Thus, for each study site, base ETo, new ETo and change in ETo values were computed.

Quantitative and qualitative effect of climate variable on ETo was analyzed by computing daily C s and the slope of the linear regression line between change in ETo with respect to change in each climate variable. For each increase or decrease in climate variable, C s was computed. After that, average C s was obtained on a daily basis. Similarly, the average change in ETo was also obtained. For example, to determine daily average C s for W s at Ludhiana, the change in ETo was determined as the difference between the computed base ETo and new ETo values for each day considering one to five unit increase in W s . After that, the corresponding change in ETo was divided by one, two, three, four, and five separately for each day. This value indicates C s for one, two, three, four, and five unit increase in climate variables, respectively. This procedure was also repeated for the condition when the climate variables were decreased by one unit to up to five units (not applicable for W s ). After getting C s with respect to each increase W s (also include a decrease in climate variable for other cases), the average of these values indicates daily average C s of W s for that particular day.

3 Results and Discussion

The ETo values were estimated by the FAO-56 PM method for three regions using mean daily climate data related to T max , T min , R s , RH avg , and W s . Figure 3 shows estimated mean daily ETo values for all selected sites. ETo was maximum at Parbhani (1935 mm) and minimum at Ranichauri (1087 mm) among the chosen sites. In the mid-year ETo values were higher as compared to start and end of the year for all the sites. Among the five sites, least variation in ETo was found at Mohanpur. Further, maximum ETo during the monsoon period (June-October) was observed at Kovilpatti. For FAO-56 PM estimated daily ETo values, the mean and standard deviation were found to be 5.30, 3.94, 4.98, 4.85 and 2.98 mm, and 1.37, 1.84, 0.68, 1.49 and 1.13 mm for Kovilpatti, Ludhiana, Mohanpur, Parbhani, and Ranichauri, respectively.

Fig. 3
figure 3

Daily average FAO 56 PM estimated ETo for each study site

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3.1 Response between Change in ETo to Change in Each Climatic Variable

A likely change in ETo is expected to change in climatic variables (T max , T min , R s , RH avg , and W s ); however, it is important to analyze which variable has a significant effect on ETo estimation under different regions. The amount of change in ETo (mm day−1) with respect to a unit change in each climate variable is presented in Fig. 4 for all selected sites. Five separate lines corresponding to each climatic variable (T max , T min , R s , RH avg , and W s ) are shown in each figure (Fig. 4). The magnitude of the effect of a change in each climate variable on the change in ETo showed considerable variations among variables and sites (Fig. 4).

Fig. 4
figure 4

Increase or decrease in ETo (mm day−1) with respect to increase or decrease in climate variables for each study site

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The regression coefficients (slope and intercept of the regression line) between the changes in ETo relative to the unit change in climatic variables for each study site are given in Table 3. The slope of the regression lines (Table 3) represents the average slope for the entire year, but it does not provide information on the seasonal changes in slope. In general, the response of ETo to all climatic variables (T max , R s , RH avg , and W s ) for all sites was linear with r 2 values = 0.99 (r 2 ≥ 0.97 for T min ). Irmak et al. (2006) also reported the least effect of T min on ETo. Based on the slope, the change in ETo was most sensitive (maximum slope) to R s for Kovilpatti, Mohanpur, and Ranichauri and to W s for Ludhiana and Parbhani.

Table 3 Regression coefficients between change in ET o (mm day−1) with respect to change in climate variables
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3.2 Daily Variation in Sensitivity Coefficients

The daily variation in sensitivity coefficients (C s ) can provide important information on how ETo responds to each climate variable throughout the year in different sites. Daily values of C s were computed for each variable in chosen sites (Fig. 5). These C s values represent the sensitivity of ETo to errors in measurement of a specific climatic variable based on the assumption that other variables were accurately measured and climatic conditions were constant during the analysis period. Monthly and annual average of these coefficients for each variable are given in Table 4. All C s values showed a large degree of daily fluctuations at all sites. The sensitivity of ETo to the same climate variable showed variation within the site.

Fig. 5
figure 5

Daily changes in sensitivity coefficients

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Table 4 Monthly and annual average sensitivity coefficient with respect to change in each climate variable
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The sensitivity of evapotranspiration to W s decreased from semi-arid to the humid climate. In the semi-arid region, the wind flow most probably replaces the moist air very rapidly with dry air especially in summer months, and causes an increase in ETo compared to other regions. In humid and sub-humid regions, due to the high RH avg and the presence of clouds, the ETo demand is low. Under these conditions, the wind replaces the saturated air and removes heat energy. As a result, the effect of W s on ETo in humid and sub-humid regions was less compared to semi-arid conditions, where small variations in W s may result in larger variations in the ETo rate (Estevez et al. 2009; Tabari and Hosseinzadeh Talaee 2014).

The C s values for W s were higher and lower during the summer and winter months, respectively in Parbhani, Ludhiana, and Kovilpatti. At Mohanpur and Ranichauri, the C s values for W s were very less for both the winter and summer months (except February, March and April). For Parbhani, C s varied from 0.05 in August to 0.34 in April with an annual average value of 0.19. At Ranichauri, the C s was almost zero in September with an annual average value of 0.05, and at Mohanpur it was zero in the months of July and September with an annual average value of 0.05.

The ETo is primarily affected by an increase in temperature due to higher capacity of air to hold water vapor, which transfers energy to the crop and exerts as such a controlling influence on the rate of ETo. The slopes of the regression line for T max were greater in semi-arid region (Kovilpatti: 0.0963) and Parbhani: 0.0818) as compared to humid (Mohanpur: 0.0786) and sub-humid region sites (Ludhiana: 0.0706), and Ranichauri: 0.0594) (Table 3). At all sites, C s values for T max were higher during the summer months as compared to the winter months. The C s values for T max varied from 0.05 (November) to 0.13 (June) with an annual average of 0.10 at Kovilpatti. Further, the C s values for T max are the largest coefficients among all sites (Table 4). Low C s values (annual average) of 0.07, 0.08, 0.08, and 0.06 were observed in Ludhiana, Mohanpur, Parbhani, and Ranichauri, respectively. Tabari and Hosseinzadeh Talaee (2014) also observed that ETo was more sensitive to air temperature in semi-arid climate as compared to humid climates. The C s values for the least sensitive variable, T min showed less variation during all months for all the selected sites. The effect of T min on change in ETo was low (less slope) at Ranichauri (slope = 0.0158) followed by Ludhiana (slope = 0.0223), Parbhani (slope = 0.0226), Kovilpatti (slope = 0.0245), and Mohanpur (slope = 0.0468) (Table 3). FAO-56 PM gave a positive slope for both T max and T min . The positive slope indicates an increase of ETo value with an increment in climate variable.

R s is the largest energy source and has the capability to change large quantities of liquid water into water vapor. Decreased cloudiness and increased R s would increase ETo and vice versa. Under humid region, ETo was more affected by R s in winter months than in summer months compared to other sites (Table 4). Under semi-arid and sub-humid regions, the opposite was observed, i.e., a greater sensitivity of ETo to a unit change in R s during the summer months compared to winter months. Similar studies were also reported by Smajstrla et al. (1987) who observed a greater sensitivity of the Penman (1948) model for a unit change in R s during the summer months compared to the winter months in Florida. Also, Irmak et al. (2006) observed the dominance of R s during the summer months in several semi-arid regions. Among the chosen sites, average annual C s values for R s was maximum (0.21) at humid region, e.g., Mohanpur, and minimum (0.10) at Ranichauri (Table 4). R s only affects the net radiation estimations and not an aerodynamic term of the ETo. R s has a seasonal pattern based on the angle of incidence and the distance between Earth and Sun; this variable did not show any clear relationship between its magnitude and the C s values in this study. Similar is the case for temperature, even though C s values decreased when R s was lower during winter months (Estevez et al. 2009).

The difference between the water vapor pressure at the evapotranspiring surface and the surrounding air (VPD) is the determining factor in the vapor removal. VPD of the air measures its dryness and e s increases exponentially with increasing T avg (Eq. 1). If all other factors remain unchanged, warming should cause drier air and hence increase ETo. The change in ETo with respect to change in RH avg also showed the greater slope at Kovilpatti (slope = −0.027), whereas it was less at Mohanpur (slope = 0.0058) (Table 3). Here, the most interesting point observed was that, all sites except Mohanpur showed negative slope for RH avg (Table 3). The negative slope indicates decrease of ETo value with an increment in a climate variable and vice-versa. Coefficients exhibited constant value (−0.01) for all months at Ranichauri. The C s values at Ludhiana ranged from −0.03 to 0.00. In contrast to all other sites, the coefficients were largest during the summer months at Kovilpatti with an average annual C s value of −0.02. At Mohanpur, C s values for RH avg varied from 0.0 to 0.01. Monthly average C s values were zero during the period of March to July.

4 Conclusions

The FAO-56 PM method is recommended as the standard method for estimating ETo, if all the required climatic data are available. In this study, the sensitivity of FAO-56 PM method was evaluated to change in climatic variables. The 5 years daily climatic data (T max , T min , R s , RH avg , and W s ) were used as an input for the estimation of ETo by FAO-56 PM method and analyzing sensitivity at Kovilpatti, Ludhiana, Mohanpur, Parbhani, and Ranichauri sites. These stations are in semi-arid (Kovilpatti and Parbhani), humid (Mohanpur), and sub-humid (Ludhiana and Ranichauri) regions of India. The response of ETo to changes in all climatic variables was linear with r 2 ≥ 0.97 for all the sites. The ETo was most sensitive to R s at Kovilpatti, Mohanpur, and Ranichauri, and to W s at Parbhani and Ludhiana. Thereafter, T max was the most sensitive variables for most sites. ETo was less sensitive to RH avg followed by T min at all sites. Results showed that the emphasis should be given to precise measurements of R s , W s and T max . The sensitivity of ETo to climate variables showed significant variations among the sites. The ETo was found to be differently sensitive to the climatic variables under different sites. Daily C s values were derived for each climatic variable and results showed that the considerable fluctuations over the seasons and the amplitude of the same coefficients showed considerable variations among the sites. This study reveals the need of accurate measurement of required climatic variables to estimate the FAO-56 PM ETo under different agro-ecological regions in India.