Introduction

Efficient irrigation water management strategies are vital for enhancing crop productivity with sufficient availability and minimum wastage of water. For optimizing irrigation water requirements, in situ measurements of reference evapotranspiration (ET0), crop evapotranspiration (ETc), and crop coefficient (Kc) are essential, especially for regions with limited water resources. ET0 is “the evapotranspiration from disease-free, well-fertilized crops, grown in large fields, under optimum soil water conditions, and achieving full production under the given climatic conditions” (Nandagiri and Kovoor 2006). Various studies have proved that FAO-56 Penman–Monteith (PM) method is the most reliable method for precise estimation of ET0 and evaluation of other empirical models (Lima et al. 2013; Pandey et al. 2014; Pereira et al. 2015), but FAO-56 PM requires all the variables that regulate energy exchange and corresponds latent heat flux. Due to this, various investigators felt the need to develop models to determine ET0 based on limited climatic parameters available for the particular agro-climate. Tyagi et al. (2019) utilized the DSSAT model to explore the trend of ET0 in eastern Uttar Pradesh having sub-humid climate and found a decreasing trend of ET0 during 1978–2003 and projected an increase during the 2040s for crops like wheat and rice. Laqui et al. (2019) developed ANN models for estimation of ET0 at Peruvian highlands using meteorological parameters as input parameters. Yirga et al. (2019) devised a model employing multiple linear regression which can be used to predict ET0 in the Megecha catchment. Mohsin and Lone (2020) developed regression models for the prediction of monthly ET0 using monthly weather data for three stations in the Kashmir Valley having temperate agro-climate. Kumar et al. (2020) investigated the impact of constrained meteorological data on evapotranspiration-based numerical modeling. However, the ET0 determined by the researchers using various techniques was not implemented in estimating ETc values for specific crops.

ETc is an important agro-meteorological parameter that assesses loss of moisture from the soil–plant system and is considered a critical component of water balance (Uniyal et al. 2019). Several investigators have used historical data for modeling ETc (Valipour et al. 2017). The field measured ETc-field provides an accurate assessment of crop water productivity (Nhamo et al. 2020), but its measurement involves the use of specific instruments and precise observation of various physical parameters of the soil water balance using Lysimeter. It is important to determine accurate ETc using the appropriate method for the study region. However, due to tediousness and instrumentation constraints, the measurement of ETc is not feasible (Minacapilli et al. 2009). Therefore, ETc is generally determined as “the product of Kc for the crop growth stage and the corresponding ET0”. The Kc value represents crop-specific water use, thus, correct values of Kc are important for the accurate estimation of irrigation requirements and can lead to adequate water savings. The Kc value for a crop varies throughout the entire crop period and is not only crop development stage-dependent but also the climatic conditions. The stage-wise Kc for various crops necessitates local calibration of Kc under given climatic conditions and crop canopy. In the absence of localized Kc values, Kc for different growth stages, as recommended by FAO, are widely utilized to determine ETc. However, field observed stage-specific Kc (Kc field) value for any agro-climate, i.e., the ratio of measured ETc and computed ET0, provides its accurate estimation.

Jamshidi et al. (2020) stated that there were inconsistencies in reported Kc values of various studies because of the unpredictability and complexity of climatic factors, irrigation management, crop physical and biological features. Mobe et al. (2020) estimated Kc using detailed observation of evapotranspiration, transpiration, soil attributes, weather, and tree physiological variables and concluded that the necessity for a method to derive precise Kc utilizing readily accessible information is essential for accurate water resources management. Allen et al. (1998) recommended that the Kc values should be obtained empirically for each crop based on lysimetric data and local meteorological parameters. However, only a few studies have been reported on ETc for field crops due to the complexity involved in the estimation technique and its necessity for soil parameters and daily meteorological data (Poddar et al. 2018). The Kc values attained through lysimeter-based experiments have not been enhanced for different crops under semi-arid climatic conditions in South Asian countries (Benli et al. 2006). Various researchers globally reported that for precise estimation of ETc, determining accurate Kc for local climatic conditions is an important task (Montazar et al. 2016). Numerous studies were performed for calibration of the Kc for various crops in several agro-climates (Poddar et al. 2020).

The above-cited literature emphasizes the need for the determination of Kc values for different crops in various agro-climates. However, due to inherent inconsistencies in Kc values, a field study for local calibration of Kc values is needed. Hence, the present study is undertaken to estimate Kc field from FAO-56 modified Kc (Kc FAO-M) using regression modeling for seven crops grown in a humid sub-tropical agro-climate. The specific objectives are to:

  1. (a)

    Determine field Kc (Kc field) from actual ETc and ET0

  2. (b)

    Obtain Kc FAO-M for local agro-climate using the FAO-56 modification procedure

  3. (c)

    Develop new Kc (Kc-new) through regression modeling between modified and field Kc and

  4. (d)

    Compare and evaluate the Kc new based ETc with actual ETc.

Materials and methods

Study area

The field crop experiments were performed at Hamirpur (Himachal Pradesh, India). The experimental station is located at 31° 42′ 40.8′′ N latitude and 76° 31′ 33.3′′ E longitude, and the average elevation is 895 m above mean sea level. The geographical outline of the study location is shown in Fig. 1. The climate of the study is categorized as humid sub-tropical (Kumari et al. 2021). The meteorological data for the study period were obtained from an Automatic Weather Station (AWS) located at the National Institute of Technology Hamirpur, as shown in Fig. 2. The precipitation recorded using a digital rain gauge is presented in Fig. 3. Daily actual evaporation was observed from the ISI standard Pan (Modified class A) at 09:00 A.M. Indian Standard Time. The daily climatic data (relative humidity, maximum temperature, average temperature, minimum temperature, solar radiation, and wind speed) is given in Fig. 4. The soil is sandy loam in texture with sand, silt, and clay content of 54.98%, 23.83%, and 21.19%, respectively. The field crop experiments were executed according to the prevailing agricultural practices in the study area.

Fig. 1
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Geographical outline of the study location Hamirpur

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Fig. 2
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Automatic weather station (AWS) located at National Institute of Technology Hamirpur

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Fig. 3
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Precipitation for the period from January 1st, 2017 to December 31st, 2019

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Fig. 4
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Climatic parameters from January 1st, 2017 to December 31st, 2019

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Crop details

Seven crops, i.e., Wheat, Indian mustard, Potato, Maize, Sorghum, Guar, and Pea were grown in the experimental station and the lysimeters. The details of the crop duration, growth stages, and irrigation days for Maize, Pea, Wheat, Sorghum, Indian mustard, Guar, and Potato during the study period (2017–2019) are summarized in Table 1. The entire growth period of crops is divided into four stages: I initial (ground cover < 10%), II development (ground cover: 70–80%), III mid-season (full ground cover to time of the start of maturing), and IV late season (full maturity or harvest) (Xiang et al. 2020).

Table 1 Details of the field crops, duration, growth stages, and irrigation days
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The plant height is a variable that indirectly signifies the growth of the crop. The average height of crops grown was recorded on the observation days, along with root depth and leaf area. For this purpose, few crops were randomly selected (as a representation of the entire field crops) to measure height. The average height was recorded. Most of the crops achieve their maximum height in the mid-season stage. However, in the later stages, the crop gets slightly bent down, and the average height decreases. Figure 5 shows the variation of plant height for the crops considered.

Fig. 5
figure 5

Variation of plant height during crop period for Wheat, Indian mustard, Potato, Maize, Sorghum, Guar, and Pea

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Lysimeter set up

Two lysimeters for accurate and reliable measurement of ETc were installed in the centre of the experimental station. The lysimeters (drainage type) were 2 m deep with a surface area of 2.25 m2 as shown in Fig. 6. Soil-moisture measurement sensors (Watermark, Irrometer Inc. Riverside, CA) were embedded at 0.2, 0.4, 0.6, 0.8, 1.0, and 1.2 m depths to determine the soil moisture status throughout the crop season for all crops. At the bottom, a perforated barrier is provided to drain off the percolated water uniformly to the collecting arrangement. A tipping bucket arrangement is placed to collect water from the bottom of the lysimeter. The measurements involve the amount of precipitation/irrigation applied, the percolated water from the lysimeter, and the soil moisture status at different times.

Fig. 6
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Detailed sectional view of Lysimeter set-up

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Computation of reference evapotranspiration

FAO-56 PM method is the most suitable indirect approach for accurate estimation of ET0 and evaluation of other empirical models (Pandey et al. 2016 ; Pereira et al. 2015; Poddar et al. 2018). Hence, during the present study, FAO-56 PM method was used to determine ET0 by using the following equation (Allen et al. 1998):

$$ET_{0} = \frac{{0.408\Delta \left( {R_{n} - G} \right) + \gamma \frac{{900}}{{T + 273}}u_{2} \left( {e_{s} - e_{a} } \right)}}{{\Delta + \gamma \left( {1 + 0.34u_{2} } \right)}}$$
(1)

Computation of field crop evapotranspiration

Field crop evapotranspiration (ETc-field) was determined by conducting water balance studies for the entire growth period of the crops. Since drainage type lysimeter was used, stage-wise ETc-field was determined. Precipitation (P), irrigation (Ir), and the quantity of water drained off from the bottom of the lysimeter (Dr) were carefully measured. The runoff component (RO) is assumed to be insignificant as the top level of the lysimeter was above ground level. The ETc-field was computed using the following water balance equation,

$$ET_{{C - field}} = P + I_{r} - D_{r} - RO \pm \Delta S.$$
(2)

The change in the soil moisture for the specific depth (dz) and the period was calculated as:

$$\left( {\Delta S_{z} } \right) = \left( {\theta _{{z,\,final}} - \theta _{{z,\,initial}} } \right) \times dz,$$
(3)

where ∆S = moisture storage change, θz, final, and θz, initial are final and initial water content in the soil profile in a discrete-time interval.

Modified crop coefficients

The standard crop coefficients (Kc-FAO) were modified using the modification equations given in FAO-56 (Allen et al. 1998). The procedure involves the computation of the impact of the time interval between wetting events, the magnitude of the wetting events, and the evaporative power of the atmosphere. The Kc ini values for the local agro-climate were computed using the following equation:

$$K_{{c~ini}} = K_{{c\;ini\left( {FAO} \right)}} + \frac{{\left( {I - 10} \right)}}{{\left( {40 - 10} \right)}}\left[ {K_{{cini\;\left( {heavywetting} \right)}} - K_{{cini\;\left( {lightwetting} \right)}} } \right].$$
(4)

The procedure for the modification of Kc mid and Kc end involves climatic variables and mean plant height ([mm] (0.1 mm < h < 10 mm) during the corresponding crop growth stage (h)). Kc mid and Kc end values were determined from the following equation:

$$K_{{c\,\,mid/end}} = K_{{c\,\,mid/end{\kern 1pt} (FAO)}} + \left[ {0.04\left( {u_{2} - 2} \right) - 0.004\left( {RH_{{\min }} - 45} \right)} \right]{\kern 1pt} \,\left( {\frac{h}{3}} \right)^{{0.3}} .$$
(5)

Field crop coefficients

The field crop coefficients (Kc-field) were estimated using the FAO-56 crop coefficient approach (Allen et al. 1998). According to this approach, Kc is defined as the ratio of crop evapotranspiration to the reference evapotranspiration. The following equation is used:

$$K_{{c - field}} = \frac{{ET_{{c - field}} }}{{ET_{0} }}.$$
(6)

Experimental and modeling methodology

The flowchart representing the methodology adopted in the present study is illustrated in Fig. 7. The modified Kc-FAO (Kc-FAO M) values are compared with the Kc-field values. Regression modeling is then applied to develop regression equations for predicting Kc-field from Kc-FAO M. The regression modeling has been implemented in Microsoft Excel. Kc values derived using the regression equations are used for computing new ETc. In the end, the comparison and evaluation between ETc values are performed.

Fig. 7
figure 7

Flowchart representing the methodology adopted in the study

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Statistical comparison

The comparative evaluation in the study is based on the error statistics i.e., square error (SE), coefficient of determination (R2) and bias error (BE) given by Eqs. (7), (8), and (9), respectively.

$$~Square~Error = \left( {x - ~x^{\prime}} \right)^{2}$$
(7)
$$R^{2} = ~\left[ {\frac{{n\sum x~X~x^{\prime} - \sum x\sum x^{\prime}}}{{\sqrt {\left[ {n\sum x^{2} - \left( {\sum x} \right)^{2} } \right]\left[ {n\sum x^{{'2}} - \left( {\sum x^{\prime}} \right)^{2} } \right]} }}} \right]^{2}$$
(8)
$$Bias~Error = \left( {x - ~x^{\prime}} \right),$$
(9)

where x = observed value; x’ = empirical/predicted value; and n = number of samples.

Result and discussion

Computed reference evapotranspiration

The ET0 was computed using the FAO-56 PM method (Eq. 1). The variation of ET0 values for the entire study period is shown in Fig. 8. During the respective crop periods, the maximum, minimum, and average ET0 (mm day−1) were 5.38, 1.23, and 2.49 for Wheat; 2.44, 1.23, and 1.83 for Indian mustard; 7.02, 0.55, and 3.39 for Potato; 6.54, 3.79, and 5.42 for Maize; 6.54, 2.84, and 4.85 for Sorghum; 5.49, 2.84, and 4.41 for Guar; and 5.49, 2.03, and 3.37 for Pea.

Fig. 8
figure 8

Variation of Reference evapotranspiration during the study period

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Computed field crop evapotranspiration

ETc-field measurements were conducted at specific intervals during each growth stage of the crop period. The cumulative ETc-field for Wheat, Indian mustard, Potato, Maize, Sorghum, Guar, and Pea are 353, 169, 182, 494, 416, 506, and 278 mm, respectively. The cumulative and stage-wise water balance components are summarized in Table 2 for Maize, Indian mustard, and Pea.

Table 2 Water balance components for Maize, Indian mustard, and Pea
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FAO modified K c

The FAO modified Kc (Kc FAO-M) values for respective crops at different growth stages along with the magnitude of parameters involved in the modification are shown in Table 3. Kc values during the crop development stage and late-season stage were calculated using the linear interpolation technique (Shankar 2007). Based on the values presented in Table 3, it is observed that the Kc ini (FAO-M) values show a significant increase when compared to the FAO-recommended Kc ini values. This emphasizes the importance of calibrating the FAO-recommended Kc ini values for this agro-climate. On the other hand, Kc mid/end values are found to be in close agreement with the FAO-recommended Kc values.

Table 3 Modified values of Kc for actual field conditions of the study agro-climate
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Field observed K c

Field observed values of Kc (Kc-field) were computed using Eq. (6). For maintaining brevity in the paper, a detailed calculation of the values is not given here. However, the stage-wise Kc-field values are described in Table 4. It should be noted that these values represent the average Kc-field in the duration considered for ETc-field estimation.

Table 4 Field observed crop coefficients obtained from field experiments
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Comparison of FAO modified and field observed K c

A comparison between Kc FAO-M and Kc-field is necessary to understand the accuracy of the modification procedure and reliability of the Kc FAO-M values in the considered agro-climate. In this study, the comparison is based on error statistics SE, R2, and BE as described earlier. Table 5 presents the values of error statistics for each crop. For all the crops considered, SE (0.0009–0.0225) and BE (− 0.09–0.15) values are small, and R2 (0.81–0.89) values are close to unity, indicating a satisfactory agreement between Kc-field and Kc FAO-M values for the study agro-climate. From BE values, it is observed that Kc FAO-M values overestimate Kc-field values in the case of Potato and Sorghum, whereas, for all other crops, it underestimates Kc-field values.

Table 5 Error statistics between FAO modified and field observed crop coefficients of Maize, Pea, Wheat, Sorghum, Indian mustard, Guar, and Potato
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Regression modeling

The values of error statistics, as shown in Table 5, are acceptable and indicates the suitability of Kc FAO-M values for the agro-climate under consideration. However, a certain degree of refinement in Kc FAO-M values for the agro-climate considered will improve their reliability and minimize the errors associated. In the present study, this is achieved by performing regression modeling and developing regression equations between Kc-field (as dependent variable), and Kc FAO-M (as an independent variable). The developed regression equations are listed in Table 6. The scatter plots of the comparison between Kc field and Kc FAO-M values for all the crops are given in Fig. 9.

Table 6 Developed regression equations for Maize, Pea, Wheat, Sorghum, Indian mustard, Guar, and Potato
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Fig. 9
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Scatter plot of the comparison between observed and modified Kc values for all the crops

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New crop coefficients (Kc-new) were predicted from the developed regression equations which exhibited a strong correlation with the Kc-field values as indicated from high values of R2 (0.94–0.97). For each crop, the regression equations developed in the present study are useful in estimating the actual Kc-field from Kc FAO-M. The developed equations are applicable for the study area as well as regions with similar agro-climate.

Crop evapotranspiration based on K c-new

Regression modeling derived Kc-new values were multiplied with the corresponding ET0 to obtain new crop evapotranspiration (ETc-new) values. Table 7 shows the cumulative ETc-new values for the crops considered in the present study. To assess the performance of Kc-new values, a comparison was carried out between ETc-new and ETc-field values. This comparison was based on stage-wise ETc values. The error statistics R2 and BE are computed and summarized in Table 7. It is observed, that for each crop considered, BE values are small (10–24 mm), and R2 values are high (0.90–0.93), indicating a strong agreement between ETc-field and ETc-new values. This observation suggests that the Kc-new values obtained from regression modeling are reliable in computing the ETc.

Table 7 Crop evapotranspiration values for Maize, Pea, Wheat, Sorghum, Indian mustard, Guar, and Potato
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Summary and conclusions

Crop coefficients (Kc) of seven crops in a humid sub-tropical agro-climate were calibrated using the FAO-56 modification procedure. Field observed Kc (Kc-field) values were obtained by computing the ratio between field crop evapotranspiration (ETc-field) and reference evapotranspiration (ET0). The FAO modified Kc values (Kc FAO-M) were found to provide acceptable estimates of Kc values when compared with the Kc-field. The error statistics i.e., SE (0.0009–0.0225) and BE (− 0.09–0.15) values were small, and R2 (0.81–0.89) values are close to unity, indicating a satisfactory agreement between Kc-field and Kc FAO-M values for the crops considered. The Kc FAO-M values were further refined to improve their reliability in the considered agro-climate by performing regression modeling and developing regression equations between Kc-field (dependent variable) and Kc FAO-M (independent variable). Regression modeling derived new field crop coefficients (Kc-new) exhibit a strong correlation with the Kc-field values as indicated from high values of R2. The performance of the Kc-new values is assessed by comparing new crop evapotranspiration (ETc-new) with ETc-field. Based on the error statistics, it is observed, that for each crop considered, BE values are small (10–24 mm), and R2 values are high (0.90–0.93), indicating a strong agreement between ETc-field and ETc-new values.

Following conclusions are drawn based on the results obtained in the study:

  • Developed regression equations can be efficiently used for estimating Kc-field values from the Kc FAO-M for the study agro-climate.

  • The comparative analysis between ETc-new and ETc-field suggests the efficacy of regression modeling in predicting crop coefficients for estimating ETc.

  • The regression modeling approach can be applied to other crops in different agro-climates to validate and generalize the findings of the study.