Modification of FAO Penman–Monteith equation for minor components of energy

Evapotranspiration (ET) is a term used to describe water loss from the surface of the Earth to the atmosphere. It includes the process of evaporation of liquid water from water bodies and soil, etc. and transpiration from vegetation. The importance of ET has been revealed in a variety of scientific fields. As a controlling factor governing how the sun's energy will be partitioned into latent and sensible heat, ET is an important variable in atmospheric transport and the water cycle, and hence plays a key role in meteorology, hydrology, and agriculture. The amount of ET depends upon the availability of water at the surface, the amount of available energy (required to change the phase of the water, known as latent heat of evaporation, LE), the ability of the atmosphere to accommodate water vapor (i.e. vapor pressure deficit (VPD)), and a mechanism for water transport from the land surface to the atmosphere (mainly through turbulent transport). Net radiation, R_n, is the main source of energy, and wind-generated turbulence is generally the transport mechanism.

There are three primary causes for energy imbalance: measurement errors, the nonidentical elevations of these measurements, and the elimination of ‘insignificant’ components (i.e. MECs) contributing to the energy equation (Foken 2008). If the contributions of these MECs turn out to be significant in ET estimation, the FAO P-M equation should be modified accordingly. These terms that researchers generally presumed to be negligible may turn out to be significant under certain circumstances, such as storage terms (Oliphant et al. 2004; Jacobs et al. 2008; Moderow et al. 2009; Lindroth et al. 2010), or the advection term (Aubinet et al. 2010; Kochendorfer & Paw 2011), which can account for as much as 10% of the imbalance in the energy budget (Oncley et al. 2007). Even though the contribution of MECs can be significant under certain circumstances, there is no research including these terms in the FAO P-M equation. Therefore, the objectives of this paper are to: (a) investigate the significance of MECs for ET estimation by FAO P-M equation based on analyzing six years' records of in-situ micrometeorological measurements of four Iowan sites; and (b) explore the possibility of estimating MECs based on major energy components and available variables. The article will begin with an explanation of the data exploited in this study.

This work supports the Iowa Flood Center's (http://iowafloodcenter.org/) development of a flood warning system, where spatially resolved daily ET over Iowa is required by the model. Iowa is mostly cloudy: the city of Des Moines, IA, for example, reports a total number of only 105 sunny days, including the days with a cloud cover of at most 30% (Osborn 2015). The State is categorized as zone 2 according to AIA Climate Zones, a cold weather with less than 2,000 CDD (cooling degree days) and greater than 7,000 HDD (heating degree days) (RECS 2005); summers are hot and humid, with daytime temperatures sometimes near 32 °C. Average winters in the State have been known to drop well below freezing, even dropping below −28 °C. Analysis of land cover Cropland Data Layer (CDL) maps provided by National Agricultural Statistics Service (NASS 2015) shows that 80% of Iowa is covered by corn, soybeans, and prairie. Accordingly, we investigated four sites as being typical of Iowa and their specifications are reflected in Table 1. Sites #1, 2, and 3 were adjacent farms, and site #4 was 57 km away from them. The sites were of rectangular shape. Sites #1–3 were switched from corn to soybeans in an irregular fashion and shape each year. These sites are long-term sites maintained by the USDA National Laboratory for Agriculture and Environment (NLAE) with high-quality, well-maintained equipment (ARS 2015). We have recorded data for these agricultural fields quasi-continuously every 15 minutes since 2006, with minimal gaps due to field activities (i.e. planting), periodic calibrations, and sensor or data-logging problems. The annual precipitation in Iowa amounts to 889 mm, while the year 2012 was an exceptional drought year with 660 mm precipitation. Therefore, we selectively focused on the time series between the years 2007 and 2012. Due to data gaps, a total of 171,089 15-min data points (∼20% of the entire period), and a total of 4,911 daily data points (∼56% of the entire period) were left for analysis.

We exploited 13 variables measured by in-situ instruments to estimate the required energy fluxes and ET_o. The list of these variables along with the specification of instruments are reflected in Table 2. We computed turbulent fluxes of H, LE, and momentum from the micrometeorological sensors using an EC system operating at 20 Hz. The raw data were processed using software developed at the USDA ARS. R_n ­was measured by a pyranometer, and the temperature–humidity sensors contained thin-film platinum resistance thermometers thermally attached to a capacitive relative humidity (RH) sensor. Two soil heat flux plates (SHFPs) were buried at each site, one exactly below the vegetation row and the other between the rows. The depth of the SHFPs changed each year for different fields. We buried four soil thermocouples: two above each of the SHFPs were co-located and their elevation varied annually at 2, 4, and 6 cm below ground. Infrared gas analyzers (IRGAs) were open path sensors. All devices were wired to a Campbell Scientific CR3000 data logger, and powered with deep cycle marine batteries connected to solar panels. An in-depth procedure to estimate the energy components is covered in Varmaghani et al. (2016).

We previously estimated all of the underlined variables in Equation (5) over the Iowan sites. Then, we adjusted the energy components based on an a posteriori analysis by means of constrained linear multiple regression (refer to Varmaghani et al. (2016) for details). An example of this correction is shown in Figure 1 for major fluxes. The figure illustrates the daily-averaged measurements of major fluxes vs. the adjusted ones over the Iowan sites for the years 2007–2012. The underlined variables in the X-axes denote in-situ measurements in Equation (5) while the Y-axes are the adjusted ones except that the Y-axes in subplots (a) and (b) represent the difference between the two fluxes. The subplots confirm that the latent and sensible heat fluxes are mainly underestimated while the most reliable measurement is usually attributed to R_n (Foken 2008). According to the subplots, as the absolute value of fluxes increases, the associated uncertainty in estimation tends to increase. The ratio of the adjusted flux over the measured flux, r, for these daily-averaged R_n, G, LE, and H were 0.98, 0.93, 1.20, and 1.34, respectively. This indicates that R_n, and G were rather overestimated.

As discussed earlier, when the fluxes are adjusted, Equation (4) will be closed while Equation (3) still suffers from lack of closure, and the term E_m should be estimated. Therefore, we calculated the term E_m over the four Iowan sites according to Equation (6) separately for each crop for the years 2007–2012. Figure 2(a) depicts the daily variation of E_m as a fraction of R_n for different crops throughout the year. It was expected that E_m should be commensurate with R_n. As can be seen, the contribution of MECs during the peak growing season is minimum (∼13%), confirmed by Meyers & Hollinger (2004), and maximum E_m occurs during winter (∼35%). There is no noticeable difference between the E_m trend of various crops. Figure 2(b) demonstrates the variation of E_m/R_n as a function of day of the year and hour of the day. The white colors indicate nonexistence of data; there are two opposite white bows (one at the mid-top (sunset) and one at the mid-bottom (sunrise) devoid of the term E_m/R_n since R_n approaches zero, and the atmosphere goes to the neutral phase. Except for the mentioned bows, there are no other trends observable in subplot (b).

In order to observe the influence of the term E_m in ET_o estimation, and subsequently in ET estimation, ET_o was computed from Equation (2) and from Equation (7). The computations were performed at 15-min temporal scale, and then summed up to daily. Figure 3 aptly illustrates the difference between these two values on a daily basis. It is clear from the subplots that the ET_o over corn and soybeans is subject to more correction than over prairie. Typically, corn shows higher correction (ET_om is 31% less than ET_o), and this percentage for soybeans and prairie is about 29 and 20%, sequentially. On average, the ET_om is 27% less than common ET_o, which indicates the importance of considering MECs in ET_o computation and consequently in ET estimations. Figure 3(d) depicts the ratio ET_om/ET_o vs. ET_o for different crops.

Adjustment of major/minor components of energy are pre-requisite to estimate the E_m term, implying that the knowledge of these components is required; however, with regard to large-scale ET estimation in practical applications, not only is the quantification of MECs not possible, but also estimation of the major components of energy is challenging (Li et al. 2009). The only available energy component in the P-M equation is R_n, and G is commonly calculated as a portion of R_n in an indirect way (Norman et al. 1995). Therefore, an estimation of the term E_m is highly desirable.

In this paper, four Iowan sites during the years 2007–2012 were investigated. The EC towers, installed in the middle of the farms (namely corn, soybeans, and prairie), provided 15-min records of micrometeorological variables. The major/minor components of energy had been previously adjusted due to lack of closure, and finally the term E_m was calculated. The annual variation of E_m (Figure 2) showed that the term E_m is minimum (∼13% of R_n) during the peak growing season, and is maximum (∼35% of R_n) during the winter for the three vegetation types. Figure 3 revealed that, even after flux adjustment, ignoring minor energy components, on average, can result in overestimation of daily reference ET by 27%. This indicates the importance of considering minor components of energy in reference ET computation and consequently in ET estimations. The regression among the potential variables (Figure 4) demonstrated that the minor energy components can be indirectly estimated as a function of net radiation, air temperature, and Monin–Obukhov length (Equation (9)) in a recursive solution. This was expected since the Monin–Obukhov length is an indicator of advection of energy (Varmaghani et al. 2016), and balance of energy between latent and sensible heat fluxes (Equation (8)). Since the three selected vegetation covers showed similar patterns (Figure 4), we hypothesize that similar patterns will be found in other vegetation types, and the term E_m as a function of net radiation, air temperature, and Monin–Obukhov length can be generalized. Since meso-scale estimation of friction (shear) velocity is challenging, it is suggested to pay particular attention to its mapping as future research. This issue led the authors to introduce Equation (10) for rough estimation of daily E_m as a function of net radiation and day of the year in the absence of friction velocity. The result in this study indicated that the FAO Penman–Monteith equation (namely, the reference ET equation) needs to be modified by considering the minor energy components in its energy term. Since the majority of ET models apply the simplified version of the energy balance equation (i.e. Equation (3)) to estimate ET as the residual of the energy equation (Li et al. 2009), the findings in this study strongly urge that ignoring the minor components of energy should be done with caution, and these energy balance ET models may require investigation of the importance of the proposed equations in their validation.

This work was accomplished under the grant support of Iowa Flood Center (http://iowafloodcenter.org/). We are grateful to the USDA National Laboratory for Agriculture and Environment (NLAE), Ames, IA for providing the micrometeorological data. Cropland Data Layer maps were provided by National Agricultural Statistics Service of USDA, Washington DC, USA. We also express our appreciation to the anonymous reviewers for their constructive remarks.