Evapotranspiration and Water Stress Coefficient for Deficit-Irrigated Maize

The crop coefficient method as described in Food and Agriculture Organization (FAO) Irrigation and Drainage Paper 56 (FAO-56) (Allen et al. 1998) is commonly used to estimate evapotranspiration of crops and other vegetated surfaces. In this method, the evapotranspiration of a nonstressed crop ($ETc$) is the product of the calculated evapotranspiration of a standardized reference crop ($ETref$) multiplied by a crop coefficient specific to a crop, the growth stage, and growing conditions. Dual crop coefficients are used to separate the contribution of wet soil evaporation from transpiration (plus minor diffusive evaporation through a dry soil surface) of a nonstressed cropwhere $Kcb$ = basal crop coefficient for the nonstressed or standard condition; and $Ke$ = wet soil evaporation coefficient. Under the nonstandard condition in which inadequate soil water restricts the ability of the crop to meet the evaporative potential of the atmosphere, a stress coefficient ($Ks$) is used to scale the depression of transpiration due to water deficitwhere $ETa$ = actual crop evapotranspiration. Plant transpiration is related to the ability of the plant to extract water from the soil into its roots and transport it to the leaves at a rate sufficient to meet the evaporative demand of the atmosphere at the stomates. When root water uptake is limited by large negative soil water potential due to soil water deficit (SWD), stomata close to preserve plant turgor. Stomata closure is represented in the Penman-Monteith combination equation by an increase in bulk surface resistance (Jensen and Allen 2016; Ortega-Farias et al. 2006).

Scientists have been studying the effects of inadequate soil water content on plant transpiration for more than $70\mathrm{years}$. Table 1 lists several plant water stress studies that have related soil water potential (SWP) or soil water content (SWC) to $ETa$ or $Ks$. The first article (Veihmeyer and Hendrickson 1955), supplemented by several discussion responses, reviews many of the studies prior to 1960. Many of the listed studies between 1960 and 1980 were conducted in lysimeters in which $ETa$ and changes in SWC could be precisely measured. Most concluded that there is a threshold SWP above which transpiration is not affected, although the proposed threshold varied among studies. Factors that were proposed to affect the threshold other than SWP included plant species, atmospheric evaporative demand, soil type, and root distribution. Most studies concluded that $ETa$ approached 0 as the SWP in the root zone approached about $\mathrm{1,500}\mathrm{kPa}$, commonly termed the permanent wilting point (PWP). Between the threshold and PWP, several studies observed that $ETa$ decreased linearly with the logarithm of SWP.

Although the plant’s ability to absorb water is directly related to the SWP near the roots, both SWP and soil hydraulic conductivity are related to SWC, which enables a relationship between $ETa$ and SWC. A relationship between $ETa$ and SWC is useful because SWC can be measured or predicted through water balance calculations. By scaling SWC between the SWP at about 30 kPa, referred to as field capacity to hold water following drainage (FC), and PWP, relationships between $Ks$ and relative soil water deficit (rSWD) have been developedwhere SWD = FC–SWC; and TAW = total available water = FC–PWP.

Most of the studies in Table 1 published after 1980 were designed to validate a $Ks:\mathrm{rSWD}$ relationship within the two-step (crop coefficient times $ETref$) approach [e.g., Eq. (2)] under water deficit conditions by comparing predicted $ETa$ with field water balance estimated $ETa$. Several $Ks:\mathrm{rSWD}$ relationships have been proposed. Jensen et al. (1970) proposed a logarithmic stress coefficient relationship for use in irrigation scheduling

This curvilinear relationship assumes the threshold is 0 such that $Ks$ approaches 1 as rSWD approaches 0.

Doorenbos and Kassam (1979) assumed a linear relationship between the threshold rSWD and PWPwhere $p$ = threshold rSWD; and $D_{rel}=(\mathrm{rSWD}-p)/(1-p)$, which increases linearly from 0 at the threshold to 1.0 at PWP. This relationship was later adopted by Allen et al. (1998) in FAO-56. In both Doorenbos and Kassam (1979) and FAO-56, proposed threshold values are near 0.5 for many cereal crops, slightly less than 0.5 for sensitive, shallow-rooted crops, and slightly greater than 0.5 for stress-tolerant deep-rooted crops. Both publications propose that $p$ is negatively related to $ETc$ and changes by 0.04 for each $1\mathrm{mm}{\mathrm{day}}^{-1}$ deviation of $ETc$ above or below $5\mathrm{mm}{\mathrm{day}}^{-1}$.

Boonyatharokul and Walker (1979) proposed a power curve stress coefficient relationshipin which $m$ and $n$ = coefficients dependent on the distribution of root water uptake, the saturated hydraulic conductivity, and $ETc$. Like Jensen et al. (1970), they assumed no threshold (i.e., $p=0$) such that $Ks$ approaches 1 as rSWD approaches 0.

The AquaCrop model (Raes et al. 2018) uses an exponential relationship to model crop stress effects, including water deficit, on $ETc$. The AquaCrop $Ks$ for $\mathrm{rSWD}>p$ iswhere $sf$ = shape factor. As previously, for $\mathrm{rSWD}<p$, $Ks=1(ETa=ETc)$. Similar to the FAO-56 relationship, the threshold is adjusted for $ETref$ values above or below $5\mathrm{mm}{\mathrm{day}}^{-1}$.

Past studies have shown that $Kcb$ is related to the crop interception of solar radiation, which is related to the fractional canopy ground cover ($fc$) (Allen and Pereira 2009; Pereira et al. 2020). Because water stress can reduce the rate of plant growth, prior cumulative soil water deficit may reduce the canopy ground cover and plant height (Trout and DeJonge 2017). Thus, current $ETa$ may be affected by both prior and current SWD.

The goal of this paper is to measure the impact of SWD on $ETa$ measured by field water balance for $6\mathrm{years}$ of maize deficit irrigation field trials. Our hypothesis is that both the impact of prior water deficit on plant growth and canopy ground cover and the impact of current water deficit on stomatal resistance must be considered and can be modeled.

Management of deficit irrigation requires estimation of $ETa$ so that target crop water use can be achieved and soil water maintained within a range that limits stress and enables production of acceptable yields. Although current plant stress levels can be monitored through plant measurements such as leaf water potential (van Bavel 1967; Kang et al. 2000; Rallo and Provenzano 2013; Ferreira et al. 2012), stomatal resistance (Ortega-Farias et al. 2006), or canopy temperature (Jackson et al. 1981; Colaizzi et al. 2003; Maes and Steppe 2012; Kullberg et al. 2017), $ETa$ must be estimated to manage irrigation and maintain the SWD within limits that reduce water use but do not cause excessive stress and yield loss. With accurate $Kcb$ and $Ks$ estimates, a daily water balance can be used to calculate $ETa$ and SWD and schedule deficit irrigation to achieve target water savings and crop yields.

In the FAO-56 approach to estimate $ETa$, a stress coefficient, $Ks$, is used to reduce daily $ETa$ based on rSWD. These data confirm that rSWD is a good predicter of the reduction in $ETa$ due to water stress. However, these data indicate that $ETa$ reductions occur at deficits less than the threshold value used in the FAO-56 linear $Ks$ relationship. These data indicate that for the sandy loam soils at the experimental site and the semiarid US western Central Plains climate, maize reduces $ETa$ at rSWD values above 0.25, and $Ks$ can be best modeled as curvilinear (concave downward) relationship for $\mathrm{rSWD}>0.25$. This relationship can be modeled by an exponential [Eq. (7)] or power [Eq. (6)] relationship. The logarithmic relationship of Jensen et al. (1970) [Eq. (4)] does not have sufficient flexibility to model these data. Although basing the logarithmic relationship on $D_{\mathrm{rel}}$ (e.g., with a threshold) rather than rSWD would improve flexibility, the very steep logarithmic relationship at high rSWD would not fit these data.

Several authors have proposed that loss of turgor and stomatal closure is determined not only by the lack of soil water but also by high evaporative demand (Denmead and Shaw 1962; van Bavel 1967; Allen et al. 1998; Raes et al. 2018). This is based on the concept that high evaporative demand requires a high rate of water uptake and transport, which creates greater loss of energy between the water in the soil and the stomates. Because evaporative demand varies diurnally and from day to day, this data set based on water balance $ETa$ measurement was not able to establish a correlation between $Ks$ and evaporative demand. In this study, mean midseason daily ETo ranged between 5 and $6\mathrm{mm}{\mathrm{day}}^{-1}$ and occasionally exceeded $7.5\mathrm{mm}{\mathrm{day}}^{-1}$.

Validating a $Ks:\mathrm{rSWD}$ relationship is difficult. As shown by Eq. (2); to derive $Ks$; $ETa$, $ETr$, $Kcb$, and $Ke$ (or wet soil evaporation) must all be known simultaneously. A representative rSWD value is difficult to determine in nonhomogeneous soils and with unevenly distributed soil water. These data demonstrate the difficulty of deriving $Ks$ relationships based on field water balance calculated $ETa$. Without weighing lysimeters to precisely measure changes in SWC, $ETa$ can only be estimated with required accuracy over multiday time intervals. Because rSWD varies daily, especially in coarse-textured soils when $ETa$ is large, it is difficult to assign an appropriate time-averaged $ETa$ value to a current rSWD value. The result is an average $ETa$ and thus $Ks$ value being associated with a range of rSWD values.

One result of using multiday averaged $ETa$ is that, if there is a change in the slope of the $Ks$ relationship during the water balance measurement interval, $Ks$ is underestimated. Fig. 11 illustrates the effect of multiday averaged $ETa$ on derived $Ks$ if the true $Ks$ relationship is the linear two-segment FAO-56 relationship [Eq. (5)]. The data were generated based on daily $ETcb=6\mathrm{mm}{\mathrm{day}}^{-1}$ and $\mathrm{TAW}=100\mathrm{mm}$ with $ETab$ calculated daily based on $Ks$ calculated by Eq. (5) with $p=0.5$. The SWD was then calculated by water balance and the process repeated the following day. The daily $ETab$ values were then averaged over the multiday interval. Each series of horizontal data points represents the daily rSWD values and average $Ks=ETab/ETcb$ values for an initial rSWD value at the beginning of the multiday interval. These generated data are similar to the measured data in Fig. 8. Fig. 11(a) is based on an irrigation interval of 5 days, similar to that used in this study, during which the range of rSWD values is fairly narrow. Fig. 11(b) is for a 10-day irrigation interval during which the range of rSWD values is large.

The average $Ks$ values are both less than and greater than the true value during any interval when $\mathrm{rSWD}>p$. If the range of rSWD values over an interval are both less than and greater than threshold $p$ and the derived $Ks$ value is assigned to the average rSWD value over the interval, $Ks$ is less than the true value for the average rSWD value. This is illustrated by the curved dashed line in the figures. The effect of multiday averaging converts the abrupt slope change of the FAO-56 relationship to a gradual change. Although the effect of averaging on the $Ks$ relationship is fairly minor for a 5-day interval, Fig. 11(b) shows that the effect is substantial for a 10-day interval. The data set in this study was based on 4- to 7-day midseason irrigation intervals and most often, 4- and 5-day intervals, such that the range of rSWD within each interval was fairly small and the error resulting from $ETa$ calculated over intervals should be fairly small. This exercise shows that $ETa$ based on intervals that include a wide range of rSWD values should not be used to derive a $Ks$ relationship.

Although $Ks$ estimates from field water balance measurements are imprecise and may be biased, the advantage of a multitreatment and multiyear field study as presented here is the wide range of concurrent soil water deficits achieved over a range of growth stages and environmental conditions and consequent robustness of the results.

These data show that deficit irrigation–induced stress reduces maize canopy growth. Because prior stress may reduce $fc$ and $ETcb$ is related to $fc$, this effect must be included in $ETa$ calculations. Measurements or estimates of $fc$ can be made from aerial or satellite imagery (Johnson and Trout 2012) and $Kcb$ adjusted accordingly. Alternatively, vegetation indexes from satellite imagery have been directly related to $Kcb$ (Neale et al. 1989; Glenn et al. 2011; Mateos et al. 2013).

Stress may not only reduce canopy growth but may also affect canopy structure and light interception through loss of turgor (leaf wilt, curl, and droop). Loss of turgor varies with the combined effect of rSWD, air temperature, and evaporative demand and may reduce $fc$ daily with the greatest effect during hot, dry afternoons. The impacts of water stress on $fc$ should be separated between effects of prior stress on growth and canopy size and effects of current stress on leaf turgor. The approach used in this study was to measure maize $fc$ prior to daily peak evapotranspiration demand and loss of turgor with the intent of documenting growth impacts of prior stress. This $fc$ was used to estimate $Kcb$. Current stress effects, which may include loss of turgor effects on $fc$ as well as increased stomatal resistance, were attributed to the estimate of $Ks$. Further research is needed to understand the effect of water stress and evaporative demand on diurnal $fc$ fluctuation, and how diurnal $fc$ reduction is related to $ETa$.

The impacts of prior and current water stress were combined to predict $Kcb$ and $Ks$ and thus $ETab$. The model estimated $ETab$ over multiple days with sufficient accuracy to manage deficit irrigation. Precise accuracy is not required because of feedback in the processes. If $ETab$ is estimated greater than the true value, water uptake and the increase in SWD will also be overestimated, which will reduce predicted $Ks$ and $ETab$ the following days. If $ETab$ is estimated less than the true value, the increase in SWD will also be underestimated, which will increase future predicted $ETab$. Occasional measurements of SWD are adequate to correct $Ks$ and $ETab$ projections.

Accurate predictions of $ETa$ are also less critical under deficit irrigation conditions when in-season precipitation is small compared to seasonal $ETc$. Irrigation efficiency with deficit irrigation is usually high such that most applied water is used for evapotranspiration (Trout et al. 2020). If precipitation is a small portion of $ETc$ (less than one-third), high SWD associated with deficit irrigation usually results in most precipitation being consumed as evapotranspiration. Thus, the water balance is dominated by the inputs of irrigation and precipitation with runoff and deep percolation being minor and often negligible losses. With deficit irrigation and low rainfall, accurate measurement of irrigation and precipitation and estimates of TAW and initial SWD are sufficient for accurate water balance estimates of $ETa$.

Prediction of $ETab$ under stress conditions is most valuable for irrigation management when combined with a water production function (WPF) such as the WPF developed for this data set by Trout and DeJonge (2017) and Comas et al. (2019). A WPF describes the relationship between crop yield and $ETa$. A farm manager can use an evapotranspiration-based WPF to select an economically viable level of yield and water savings (English 1990; Trout et al. 2020). Prediction of $ETa$ is then used to schedule irrigations to achieve the target yield and water savings appropriate to water and other input costs and availability.

Prediction of ET with deficit irrigation depends both on the impacts of prior stress on plant growth and current stress on stomatal resistance. Prior stress effects on $ETc$ can be predicted through the reduction in canopy ground cover. Current stress effects on $ETa$ can be predicted by the effect of the soil water deficit on the stress coefficient. These effects can be incorporated into the FAO-56 dual crop coefficient approach via the basal crop coefficient and stress coefficient. For maize, a linear function of canopy cover predicts unstressed $Kcb$ well. An exponential function of rSWD predicted the stress coefficient better than the linear relationship often used. These relationships allow prediction of the basal $ETa$ well enough for deficit irrigation management purposes.

All data used during the study are available in an online repository on the USDA National Agricultural Library Ag Data Commons: USDA-Agricultural Research Service Colorado Maize Water Productivity Dataset 2008–2011 (https://doi.org/10.15482/USDA.ADC/1254006) and USDA-Agricultural Research Service Colorado Maize Water Productivity Dataset 2012–2013 (https://doi.org/10.15482/USDA.ADC/1439968).

This study was based on a very large amount of field data. The authors thank the many USDA-Agricultural Research Service scientists, technicians, and students that assisted in the data collection and summarization. The work was funded by the USDA-Agricultural Research Service.

The use of trade, firm, or corporation names in this publication is for the information and convenience of the reader. Such use does not constitute an official endorsement or approval by the U.S. Dept. of Agri. or the Agricultural Research Service of a product or service to the exclusion of others that may be suitable.