W35! 5-0 '86 #44 o s “5 MSU LIBRARIES m RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES will be charged if book is returned after the date stamped be1ow. / v SIMPLIFIED APPLICATION OF PENMAN'S EQUATION By Andres R. Fernandez A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Master of Science Department of Agricultural Engineering 1982 ABSTRACT SIMLPLIFIED APPLICATION OF PENMAN'S EQUATION By Andres R. Fernandez A sensitivity analysis was performed on the Penman equation; from this analysis, it was determined that wind velocity and the dew point temperature have relatively little affect on ET prediction by Penman's method which is more sensitive to changes in extraterres- trial radiation, maximum temperature, and percent of sunshine. Approximations were introduced to estimate wind velocity and percent of sunshine. In addition, minimum temperature of the night before was used instead of dew point temperature in the deter- mination of saturated vapor pressure. The simplified equation was evaluated in the summer of 1981, using evapotranspiration from corn, measured by the gravimetric method and evapotranspiration from potato, measured by the neutron scattering method. A very good correlation was obtained with the neutron scatter- ing method (R = 0.954) and the sum of the evapotranspiration obtained by both methods for the season was almost the same. The gravimetric method gave higher values of evapotranspiration and its correlation with the simplified equation was lower (R = 0.65). To my beloved father and in the memory of my mother. ii ACKNOWLEDGMENTS I would like to express my gratitude to my major professor, Dr. George Merva, for the support he gave me and his guidance, understanding, and friendship. I would like also to thank Dr. Fred Nurnberger and Dr. Ted Loundon for being on my committee. Finally, special thanks to James Jenkins who supplied me with some of the data used in this thesis. iii TABLE OF CONTENTS LIST OF TABLES . . . . . . . . . . . . LIST OF FIGURES . . . . . . . . . . Chapter I. INTRODUCTION . . . . . . . . . . II. LITERATURE REVIEW . . . . . . . . Plant Water Requirements . . . . . . Evaporation . . . . . . . . . Historical Developments . . . . . . General Facts . . . . . . . . Actual and Potential Evapotranspiration Crop Coefficient (Kc) . . . . . Methods for Determining Evapotranspiration Direct Measurements . . . . . . . Methods based on Climatological Data . Penman Equation . . . . . . . Modified Versions of the Penman Equation Effectiveness of Penman Equation . . . Limitations of the Model . . . . . III. PROCEDURE . . . . . . . . . . . IV. RESULTS AND DISCUSSION . Analysis of Sensitivity . . . . . . Simplifications . . . . . . . . Evaluation of the Simplified Penman Equation V. SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS . APPENDIX . . . . . . . . . REFERENCES . . . . . . . . . . . iv Page vi 47 53 53 62 74 83 85 9O Table LIST OF TABLES Conditions assumed for the examination of the sensi- tivity of Penman Equation to its different parameters . Simplified values of percent of sunshine based on monthly average values and standard deviations . . Crop coefficient for the different stages of the crops 0 I O O O O O O O I O O O O O I Table of ETM and ETP for corn for the different periOds O O O O O O I O O O O O O O 0 Table of ETM and ETP for potato for the different periods . . . . . . . . . . . . . . . Page 54 71 77 78 78 Figure 10. 11. 12. LIST OF FIGURES Sensitivity of the Penman Equation to extraterres- trial radiation; expressed in percent change . Sensitivity of the Penman Equation to maximum tem- perature, expressed in percent change . . . . Sensitivity to the Penman Equation to percent of sunshine, expressed in percent change . . . Sensitivity to the Penman Equation to Albedo, expressed in percent change . . . . . Sensitivity of the Penman Equation to minimum tem- perature, expressed in percent change . . . Sensitivity of the Penman Equation to dew point temperature, expressed hipercent change . . . Sensitivity of the Penman Equationtxrwind velocity, expressed in percent change . . . . . . . Regression of ET calculated by the Penman Equation and ET calculated by the Penman Equation with the wind velocity approximations . . . . . . . Regression of dew point temperature and minimum temperature for Lansing, Michigan . . . . . Regression of ET calculated by the Penman Equation and ET calculated by the Penman Equation with the minimum temperature substitution . . . . . . Regression of ET calculated by the Penman Equation and ET calculated with the wind approximations and the minimum temperature substitution . . . . . Regression of ET calculated with the Penman Equation and ET calculated with the percent of sunshine approximations . . . . . . . . . . . . vi Page 55 56 57 58 59 60 61 64 66 69 70 72 Figure 13. 14. 15. 16. 17. Page Regression of ET calculated with the Penman Equation and ET calculated with the wind and percent of sun- shine approximations and the minimum temperature substitution . . . . ., . . . . . . . . . 73 CrOp coefficient curve for corn . . . . . . . 75 Crop coefficient curve for potato . . . . . . . 76 Regression of ETM for corn and ETP for the different periods with a 45° line . . . . . . . . . . 8O Regression of ETM for potato and ETP, for the differ- ent periods with a 45° line . . . . . . . . . 82 Average kc value for initial crop development stage as related to level of ETo and frequency of irriga- tion and/or significant rain . . . . . . . . 89 vii CHAPTER I INTRODUCTION Irrigation is one of the most important management practices in agriculture. Its role can be easily explained by the fact that although irrigated areas represented only 13% of the global arable land, it accounts for 34% of the world's cr0p production (FAO, 1979). Through history, irrigation has enabled the establishment of flores- cent civilizations in otherwise useless land and the increase of productivity to sustain the increasing world pOpulation. In the U.S. there has been a steady increase in irrigated areas in the last 50 years, a fact which has played an important role in the agricultural revolution that has transformed agricultural pro-‘ duction, making economically feasible the utilization of the modern techniques used nowadays in agriculture (introduction of improved varieties of crops, use of pesticides, fertilizers, equipment for agricultural practice, etc.). The estimated irrigated area in the United States in 1979 was 24,746,0001uu3.,representing a growth of 136% in relation to the 1949 figures, which gives a 4.5% average rate of growth per year (Jensen, 1980). Michigan receives less rainfall than any other of the states east of the Mississippi River which is one of the reasons why crop yield increases had been lower than those of the other states (drought stress is a probability two years out of every five for each summer month) (Lucas and Vitosh, 1978). These facts help explain the large increase of irrigated area in Michigan, estimated at 23% for the last three years up to 161,1491EEL in 1980 (Vitosh et al., 1980). However, because of the high costs of the energy required in any method of irrigation, as well as the high investments needed for its implementation, the farmers must irrigate in the most efficient way in order to obtain benefits or even survive in today's business. In addition, the competition for limited fresh water supply between industrial, urban, and agricultural needs is already a matter of concern and will be a very hot issue in the future. Continuous deple- tion of the groundwater supply (by far the main source of irrigated water) is making pumping costs more and more expensive and putting more and more pressure on irrigation, the main consumer of water. In order to irrigate efficiently, it is imperative that the irrigators know the amount of water to apply to avoid an excess in application which increases the costs of operation and may damage the crop, or to avoid a deficiency of water which causes stress in the plant, thus a decrease in production. The amount of water to apply and the time to appy it is determined by the soil type (water holding characteristics), the crop and the water requirement of the crop. The water requirement depends mainly on evapotranspiration, leaching requirement, effective rainfall, soil moisture content, the ascension of groundwater and irrigation efficiencies. The relative importance of these parameters varies with the climate and some local conditions. For example in Michigan, under normal conditions, the important parameters are evapotranspiration, effective rainfall, and irrigation efficiencies. The role of evapotranspiration is very important in the determination of water requirement. As Shockley (1966) said: "Evapotranspiration is an important part of the over— all water requirement problem and accuracy in its determination is desirable. However, the relatively indeterminate nature of most of the other factors involved indicates that complex and time consuming procedures to achieve extreme precision seldom will be justified for farm irrigation planning for on-farm irrigation water management." One of the most used methods to calculate evapotranspiration is the Penman Equation. This is a theoretically sound method which yields relatively accurate evapotranspiration estimates (Doorenbos and Pruitt, 1977). However, the model requires some climatalogical data (percent of sunshine, wind velocity, and dew point temperature) which generally are not easily available for individual farmers. The objectives of this study are: 1. To analyze the sensitivity of the Penman equation to variation of its parameters. 2. To propose simplifications which will enable Penman's Equation to be used by individuals possessing hand-held programmable calculators and a minimum of climatologi- cal data gathering equipment. 3. To evaluate the results of the application of these simplifications, relating them to field measurements of evapotranspiration. CHAPTER II LITERATURE REVIEW Plant Water Requirements There is no theory which completely explains the process of water passing through a plant, but water is known to be important in every stage of a plant. Water is essential in the photosynthesis process by which the plant produces its food. It is also indispensable in the respira- tion by which the plant uses this food for growing, reproducing or sustaining itself, depending on its stage of development. Water also acts as a solvent and medium of transportation of food within the plant. It gives internal support to the plant: "Under pressure within the plant cells, water furnishes support to the plant" (Merva, 1975). Finally, but not less important, water evaporating from the plant cells and soil absorbs much of the radiant energy the plant does not use for photosynthesis, thus keeping the plant temperature from being higher than those values convenient for its functions. The evaporation from plants and soil comprises more than 95%» of the water used by the plant (Kramer, 1969). Pallas et a1. (1962) conducted an experiment in a controlled environment growth room and found that transpiration cools the plants as generally acceptable. However, they reported that the amount that transpiration lowers the leaf temperature, and whether or not it does this below air tempera- ture depends on the species and the environment. They also suggested that their failure, as well as the failure of others who had worked on the topic, to recognize interactions between radiant energy, soil moisture tension, and air vapor pressure deficit, reflected existing discrepancies found in the literature as to what extent transpiration is important to the plant. Although only a small proportion of the water taken by the plants is used for photosynthesis and other biological processes in which water is needed, Pallas et al. in the same work noted a decrease in photosynthesis with an increase in moisture tension.£ In This they concurred with Gingrich and Russell (1957) who concluded that most plants show a decrease in production when moisture stress is more than one atmosphere. While evaporation may not be the most important water con- sumed by the plant because it constitutes such a great proportion of the total, it can be used to determine the water used by a crop. Evaporation Historical Developments Deacon, Priestley, and Swinbank (1958) in a review of litera- ture related to evaporation and the water balance, defined evapora- tion from natural surfaces such as open water, bare soil or plants as "a diffusive process by which water in the form of vapor is transferred from the underlying surface to the atmosphere." /.. ._\‘ ~..___ The first individual who understood the role of evaporation in the hydrologic cycle was Vetruvius Pollio (50 B.C.). He said that the sun heated up the water in rivers, springs, marshes, and seas, forming vapors which rise to form clouds. Leonardi da Vinci (1500) explained: "where there is life there is heat and where there is vital heat there is movement of vapor)’ Lavoisier regarded evapora- tion as the combination of fire and water while Benjamin Franklin said it was due to the solution of water in air (Penman, 1948a). Dalton (1834) reviewed by Deacon, Priestley, and Swinbank (1958) and Penman (1948a) was the first to study scientifically the evaporation process. His statements about the properties of vapors (mainly the partial pressure law and the dependence of vapor trans- fer on pressure differences) were the foundation for most of the scientific studies of evaporation up to the present time. He per- formed many experiments to explain the factors controlling evapora- tion and found that the Space above a water surface could have only a limited amount of water vapor whose maximum partial pressure was dependent on temperature. Dalton demonstrated that when the partial pressure was not at its maximum, there would be evaporation at a rate proportional to the partial pressure deficit and the wind speed. He deve10ped the formula: E = a (eS - ed) (1 + b u) (1) where: E = rate of evaporation e , e = saturation vapor pressures at the evaporating s d surface and dewpoint temperature respectively n = wind speed a, b = empirical constants During the following century, most of the work on evaporation was mainly based on the estimation of evaporation from small open water surfaces which was used as a standard for the calculation of evapora- tion from larger bodies of water. The majority of these studies were performed in arid areas of the United States and generally yielded models which followed the form of Dalton's Equation. Since the 19405 many scientists have studied evaporation with . different approaches, obtaining empirical formulas relating evapora- tion to temperature, or more theoretical models, using sophisticated parameters for calculating it. In the last 15 years the studies have focused mainly on modifying earlier formulas to adapt them to local conditions. More recently, computer models are being used to esti- / mate and predict evaporation. General Facts Natural evaporation can occur from Open water surfaces, from bare soil and from plants. Evaporation from Open water is known to be entirely dependent on weather conditions and may be influenced by the shape of the water body. However, evaporation from bare soil and plants can be limited by other factors. Penman (1948a) summarized the works of Eser, Wollny, and others who found that in addition to the effect of weather conditions, the amount of evaporation from bare soil is restricted by the moisture tension of the soil, and by the resistance of soil to transmit water at a moisture content less than saturation. The vaporation from plant surfaces (also called transpiration) is affected by biological-physical factors of the plants stomata aperture, .leaf area, root depth, stage of devel- ‘ 0pment, species, etc.), as well as by the factors which influence evaporation from bare soil. In their research on plant transpiration, Pallas et a1. (1962) found a strong relationship between transpiration and leaf area. White (1932) reported by Penman (1948a), studied the effects of the control of stomatal opening by daylight and other factors in alfalfa cropped under semi-arid conditions. The results showed the variations in the rate of transpiration during the day and a steady daily increase of transpiration rate until the alfalfa was cut. Especially notable was the cessation of effective transpiration after removing the evaporating surfaces and the gradual resumption when the surfaces grew again. Penman (1948a) concluded from the literature he reviewed that the crops with enough water available will have a transpiration rate similar to the evaporation rate from an open water surface. Also that they will exhibit a behavior similar to that of a highly con- ducting channel between a source (available water to the crop in 10 the soil) and a sink (atmosphere). Penman also cited the results of Schofield (1935) and coworkers who, investigating the thermo— dynamics of the soil-water-plant continuum, found that, measured in terms of the free energy of the soil water, the available moisture content resembled closely a soil moisture constant nearly independent of soil or crOp. Lawes and Gilbert (1871), Hendrick (1921) and Hendrick and Welsh (1938), also reported by Penman (1948a) found from experiments that when the available water is renewed adequately by irrigation or precipitation, transpiration is not limited by moisture content of the soil and the behavior of cropped soil in the summer is nearly the same as the behavior of bare soil, being entirely dependent on wea- ther conditions and independent of crop yield. McIlroy and Angus (1964), reviewed by Dilley and Shepherd (1972), working on pasture and potatoes at A5pendale, Victoria found that changes in plant especies composition caused grass evaporation to fall in spite of continued liberal irrigation. From that they concluded that even under potential conditions, some species can exert a physiological constraint on evaporation. Actual and Potential Evapotranspiration The most used definition of evapotranspiration (ET), describes. it as "the combined process by which water is transferred from the earth's surface to the atmosphere; this includes evaporation from soil and plant surfaces plus transpiration of water through plant ll tissues expressed as the latent heat transfer per unit area or its equivalent depth of water per unit area" (Jensen, 1980). Because of the difficulties involved in the estimation of evapotranspiration due to the fluctuations originating from physical characteristics and moisture conditions of the soil as well as the effects of physical and physiological plant features, a parameter was needed to estimate an evapotranspiration factor which did not take into account these fluctuations. Thornthwaite (1948) introduced the term‘ POTENTIAL EVAPOTRANSPIRATION, (PET) which has been universally adopted. The most widely accepted definition for potential evapotrans- piration was given by Penman (1956): "Potential evapotranspiration is the amount of water transpired in unit time by a short green crOp, completely covering the ground, of uniform height and never short of water." In recent years, there has been an increasing use of the term REFERENCE EVAPORTRANSPIRATION, (RET) to replace PET because of the vagueness involved in the interpretation of the later (Jensen, 1980). Jensen, Wright, and Pratt (1971) defined RET. as "the upper limit or maximum evapotranspiration that occurs under given climatic conditions with a field having a well-watered agri- cultural crop with an aerodynamically rough surface, such as alfalfa with 12 in. to 18 in. of top growth." However, the potential evapo- transpiration term is by far the most widely used. Both terms depend on meteorological conditions, mainly radiant energy available and the partial pressure difference between the evaporating surface and the surrounding atmosphere. 12 Crop Coefficient (Kc) To account for the difference between PET and the restric- tion on evapotranspiration imposed by crop species, crop growth stage, and crOp density, a dimensionless parameter, the crop coeffi- cient is used. The coefficient is the ratio of actual evapotranspiration (AET) to potential or reference ET for a crop: = AET c PET _ AET K c ' RET (2) or K The most popular method to measure AET is the use of a properly located sensitive weighing lysimeter with water table not affecting the data. PET or RET can be obtained by measuring the ET from the reference crop (grass or alfalfa) under ideal conditions or by using one of the well known methods for predicting PET. Many studies have been done to obtain coefficients for the most important agricultural crops in every climatic condition. Baier and Robertson (1968) took into account the moisture tension character- istics of the soils in computing Kc while Rijtema (1959) considered the apparent diffusion resistance of crops and its relation to degree of cover. Jensen, Wright, and others worked in the early 70s to develop cr0p coefficients and crop water requirement and irrigation scheduling at Kimberly, Idaho. They developed a crop coefficient which included the effects of wet soil surfaces in evapotranspiration (after irrigation or precipitation). l3 Jensen, Wright, and Pratt (1971) and Jensen, Rob, and Franzoy (1970) estimated: RC = KCOKa + KS (3) where: KC = crop coefficient KCO = the mean crop coefficient based on experimental data where soil moisture was not limited Ka = the relative coefficient related to available moisture KS = the increase in the coefficient where the soil surface is wetted by irrigation or rainfall The studies were performed using the Penman combination equa- tion for estimating PET, and soil sampling for estimating AET. In the following studies, conducted mainly by Wright in the late 703 and early 803, a revised procedure was used to obtain revised crop coeffi- cients for semi-arid conditions. A revised method for calculating reference evapotranspiration and weighing lysimeter data were used to obtain the revised coefficient for most of the irrigated crOps of the region. Wright (1981) developed a formula to adjust these revised coefficients for the effects of surface soil wetness and soil ten- sion properties: 14 = _ _ 12 KC ch + (l ch) [l (t/td) ] f(w) (4) where: RC = the adjusted crop coefficient ch = crop coefficient designed to represent dry soil surface conditions and called by Wright "basal ET crop coefficient." t = the number of days after major rain or irrigation td = the usual number of days for the soil surface to dry fw = the relative portion of the soil surface originally wetted Wright (1981) also reported a new mean crop coefficient for some crops at different stages of growth which gives seasonal esti- mates for typical crop development and local management practices where root zone soil moisture does not limit growth and for the rain- fall and irrigation patterns of the area. Doorenbos and Pruitt (1977), on a study published by the Food and Agricultural Organization (FAO), reported values of crap coefficients for most of the irrigated crops, where the physiological features of the crops are taken into account, as well as the pre- vailing relative humidity and wind speed of the locality. They gave two values for each crop, one fOr the mid-season stage and the other for the harvest/maturity stage. These numbers, along with another value obtained from an average KC for initial crop development stage 15 related to level of PET and frequency of irrigation and/or signifi- cant rain are used to develop a crop coefficient curve which gives daily value of KC during the crop season. Methods for DetermininggEvapotranspiration We will classify the methods for determining ET as: 1. Those which are based on direct measurements of water evaporated 2. Those which are based on the use of meteorological data Direct Measurements The most important are pan evaporation and water balance field measurement. Pan Evaporation This may be the most widespread method for estimating evapo- transpiration. Evaporation data from pans have been collected in many climatological stations for many years all over the world and have been widely used in empirical studies of the topic. The main objective has been to get means of applying this evaporation measure- ment to obtain indications of evaporation from large bodies of water and from bare and cropped soils. This is because both are influenced by the same climatic factors: solar radiation, wind velocity, tem— perature and humidity (Westesen and Hanson, 1981). Different types of pan evaporation. The most widely used is the U.S. Weather Bureau Class A pan which is well described by Jensen 16 (1980). Also important are the U.S. Bureau of Plant Industry Sunken plant (set into the ground), the USGS floating pan (placed on a raft for measuring lake evaporation), Colorado Sunken pan, Class B pan, wash tub evaporation pan, standard British tank and standard Aus- tralian evaporation tank. They differ only in shape, color, and/or size and are used to calculate PET using coefficients which have been estimated empirically. There have been many studies conducted to verify the rela- tionship between evaporation from pan or tank and evapotranspiration. Penman (1948a) in the Rothamsted experimental station in England, found a high correlation between evapotranspiration from short grass well watered and evaporation from open water surfaces. Pruitt (1966) reported a high correlation between pan evaporation data and evapo- transpiration from grass grown in a lysimeter. He also reported the success of similar comparisons in Nigeria, England, and Israel (Stanhill, 1958, 1961, 1962, 1963) using standard British tank, Zaire (Congo) (Brutsaert, 1965). Canada (Wilcox, 1963) using a Class A pan, a sunken pan, and a black Bellami plate anemometer, Denmark (Aslyng, 1965) using an Open 12 square meter tank, Australia (McIlroy and Angus, 1964) using the standard Australian evaporation tank. Sims and Jackson (1971) found that a NO. I wash tub could be used to acceptably measure evaporation. l7 Coefficients to be used with evaporation pans. TO calculate the potential evaporation rate from pan evaporation data, a pan coefficient is used with the relation: PET = K * E , RET = K * E (5) P P P P where: Ep = pan evaporation in mm/day Kp = dimensionless pan coefficient Jensen (1980) gives a table Of pan coefficients to be used with Class A and Class B pans with different values of relative humidity, pan exposure, wind velocity and distance Of homogeneous material to the windward side. A similar table is given by Doorenbos and Pruitt (1977). Mclllroy and Angus (Pruitt, 1966) found an average pan coef- ficient of 0.84 for Class A pans, 1.05 for standard Australian evapo— ration pans, and 1.6 for the 1.6 meter weighing evaporimeter using a grass-clover mixture as the reference crop. Pruitt also reported values Of 0.87 for shallow pans, 0.86 for Class A, and 1.13 for Bureau Of Plant Industry Sunken pans as found by Abon—Khaled at Davis, California, who used grass as the reference crop. Middleton et al. (1962) using a Class A pan evaporation at Washington State, found coefficients from about 0.8 for corn, grapes, and peaches in periods of near maximum vegetative cover with no cover crop to a maximum of 1.05 for Delicious Apples with a grass cover crop; 18 Middleton found also a 0.9 coefficient for sugar beets, soybeans, red beans, ladino cover, late potatoes, wheat, as well as alfalfa in humid western Washington. For green peas, early potatoes, rasp- berries, and peaches with an alfalfa cover crop, they found a coeffi- cient Of about 1.0. For arid Central Washington, they found a coeffi- cient Of 0.95 for alfalfa. Hargeaves and Christiansen (1966) reported by Westensen and Hanson (1981) found a seasonal pan coefficient of approximately 1.0 for alfalfa at full cover and 0.75 at 25% into the growing season; for small grain, they found 0.33 at 25% into the growing season and 0.90 at full cover (65% or more into the growing season). Disadvantages of the Pan Evaporation Method. There is some certainty in the effect of some local environmental factors as regards to pan evaporation. Pruitt (1966) showed a "very marked" effect of immediate upwind condition in evaporation pan readings, with a near linear decrease in evaporation as a function Of the logarithm of upwind fetch Of grass. He also recommended a standardi- zation of the local environment of evaporation pans and a considera- tion of the proximity Of any major difference in crop height or roughness. He also found a wide variation from the coefficient aver- age value during three strong wind, high advention days which could not be predicted by any single correction. From the same studies on evaporation pans, an effect of the pan size and shape on evaporation has been Observed. 19 Water Balance Field Measurement These methods are primarily used for calibrating and evaluat— ing other methods to measure evapotranspiration. Because they require actual measurements, the tendency has been to develop models to measure and predict ET from climatalogical data. The most important field measurement methods are as follows. Soil Moisture Budget. Despite the Objection to the validity of this method, it is widely used in the determination of water evapotranspired. It consists Of periodic measurements of root zone water content and an inventory of the water entering and leaving the soil-plant system (rainfall, irrigation, and drainage). The budget method considers evaportranspiration as the difference between water into the system minus water out. In an abstract about soil profile sampling, Davidson and Nielsen (1966) reported some of the objections usually used against this method. The most serious Objections were the errors introduced by the simultaneous redistribution of water during extended periods of drainage, the ascension of water from below the greatest sampling depth, and the assumption that no water moves out and/or into the sampling zone between sampling periods. The most widely used methods for determining the moisture content Of the soil are the gravimetric sampling technique and the neutron scattering method. Bowaman and King (1965) considered the neutron meter a more appropriate method because it averages over a larger volume of soil 20 and because the measurements are taken at the same location each time, which eliminates the distortions caused by nonuniform moisture distribution in the soil. They found a variation of 18% to 24% by volume in the gravimetric determination of moisture content of 500 cc soil samples taken within a 0.91 meter (3 ft) radius, in the A hori- zon, and a variation Of 7% to 31% by volume in the gravelly C horizon. In their study, Bowaman and King found the neutron scattering method "accurate" to 3.8 mm (0.15 in) of water for one week, or 17 mm (0.62 in) over a three-month period on a 1.3 mt (51 in) profile. The neutron scattering method has the disadvantages of an initial high investment in the equipment used and the time required for installation and calibration. Lysimeters. The use Of lysimeters (soil tanks in which crops are grown) has been traced back to 1688 in France (Kohnke, Dreibelkis, and Davidson, reported by Harrold, 1966), but insuse for determining ET began in the early 19005. After about 1930, attention was given to measurement of runoff and the effect of lysimeter cover different from that of the surrounding area. At the present, lysimeters are used for a broad range of purposes; Penman (1948b) worked with lysi- meters at Rothamsted to study the effects of weather and soil condi- tions on ET. Dilley and Shpeherd (1972) used lysimeters at Aspendale as a standard for evaluating pan evaporation and the McIllroy combina- tion formula. Jensen, Wright, and Pratt (1971) and Jensen, Robb, and Franzoy (1970) used weighing lysimeters to evaluate the Penman Equa- tion and the von Bavel Equation to develop a model for irrigation 21 scheduling; Wright and Jensen (1972) used the same lysimeters to determine peak water requirements of some crOps in Southern Idaho; von Bavel (1966) used precision weighable lysimeters to evaluate his combination equation; Pruitt (1966) reported the use of lysimeters at California and Washington to evaluate the application of pan evapora- tion to determine ET. Boonyatharokul and Walker (1979) used hydraulic weighing lysimeters at Colorado State University to study the effects Of depletion Of soil moisture on ET. To get meaningful results from lysimeters, they must meet certain physical requirements in location, construction, and Opera- tion. The most important requirements were reviewed by Harrold (1966) who reported that Makkind (1959) indicated that to yield real values, lysimeters have to represent what occurs in nature. Makkind suggested that discontinuities of vegetation in and out of the lysi- meter give erroneous results, and that small lysimeters with large unnatural borders allow extra radiation to reach the vegetation, causing higher values Of ET. Harrold also reviewed the conclusion Of Popov (1959) that the thermal regimen of the lysimeter must be simi- lar to that Of the soil its data are to represent. He also reported the Opinion of King, Tanner, and Suomi (1956) that errors introduced in measurements of ET are likely to be larger for small lysimeters than for larger ones. Harrold (1966) gives detailed information on physical and operational features of different kinds Of lysimeters. 22 Methods Based on Climatological Data After years of checking and calibrations Of models used in the calculation of ET from climatological records, the use Of models has become more and more important. There are mathematical models for practically every climatic condition although "No single exist- ing method using meteorological data is universally adequate under all climatic regimes, especially for tropical areas and for high elevations, without some local or regional calibration" (Jensen, 1980). These methods yield potential or reference ET which are multiplied by the crop coefficient to estimate actual ET. Primary methods to predict PET based on climatological data are as follows. Empirical Equations Though almost all the equations used to calculate PET have some empirical approach, we define as empirical those which are based on empirical relations Of the parameters involved (usually two or three), yielding satisfactory results only in the conditions in which they are developed. Jensen (1966) gave three circumstances where the use of empirical equations is reasonable: 1. When there are not adequate meteorological and soil- crop data available for the use of completely rational equations 2. When there is no need for an accuracy beyond that supplied by the empirical equations 23 3. When the use of rational equations requires greater technical ability and experience in meterology, physics, and agronomy than that which the users Of ET data have or can obtain. He said that even though some rationally developed empirical methods of determinating ET using solar radiation approximate solu— tions based on the energy balance approach, qualified technicians have no justification using empirical methods when there are the meteorological parameters available for the use of rational equa- tions. This can be said also of the other empirical approaches. The first empirical equations were used to predict evapora- tion from open water surfaces, i.e., Dalton's formula and many others expressed in the same form (Deacon, Priestly, and Swinbank, 1958). Blaneyr-Criddle Method. In 1941-47, Morin and Blaney devel- Oped formulas for computing evaporation and consumption use from temperature, daytime hours, and humidity records (Blaney, 1955). Later, the formula was simplified by eliminating the humidity factor because humidity records were not readily available at many stations (Blaney and Criddle, 1962). The procedure was to develop coeffi- cients from the correlation of existing consumptive use data for different crops with monthly temperature, percentage of daytime hours, precipitation, frost-free (growing) period, or irrigation season. These coefficients are used to transpose the consumptive-use data from the area where they were developed to other areas for which only climatological data are available. The formula is: 24 _ KP (45.7: + 813) U ' 100 (6) where: U = monthly consumptive use, in mm K = empirical consumptive use crop coefficient for the growing period P = monthly percentage of daytime hours of the year t = mean monthly temperature, in °C The method has been widely used and it has been revised and varied many times. The crop coefficient and the percentage of day- time hours Of the year are easily found in literature on the topic. Lowry and Johnson Method. The Lowry and Johnson method (Lowry and Johnson, 1942; Israelsen and Hansen, 1962) was one of the first empirical models to be developed and was designed for comput- ing water requirements for irrigation projects Of the Bureau Of Reclamation. It applies to a region not to individual farms or individual crops. The formula assumes a linear relationship between the accumulation Of maximum daily temperatures above 32°F during the growing season (termed by them effective heat) and consumptive use. It was applied with good results to irrigation projects in Western U.S.A. where the data for its development were collected. The formula was: 25 U = 0.8 + 0.156F (7) where: U = valley consumptive use in acre-feet per acre F = effective heat in thousands of day degrees Thornthwaite Method. An empirical formula was deveIOped by Thornthwaite (1948), who found a close relationship between mean temperature and potential evapotranspiration when the variation in daylength is adjusted. Its computation only requires mean monthly values of temperature and the location of the station (latitude). The equation is Of the form: e - 1.6 (lOt/I)a (8) where: e = monthly PET in cm t = mean monthly temperature, in °C I = Xi for the year or growing season, the heat index i = (t/5)1°514, monthly index a = 0.00000067513-0.000077ll2 + 0.017921 + 0.49239 This yields unadjusted values Of PET which must be multiplied by a factor that varies with the month and with the latitude. This factor takes into account the variation in number Of days in a month (28-31 days) and the variation in the number of possible hours Of sunlight with the season and the latitude. 26 Thornthwaite gives a table with the factor for Northern and Southern Hemispheres at different latitudes. The formula was very popular in the humid Eastern U.S.A. Thornthwaite and Mather (1955) reported that annual average of PET had been computed for about 3,500 weather Bureau Stations in the United States, using the Thorn- thwaite Method. Empirical formulas based on solar radiation. The last group of empirical models to be considered are those in which solar radia- tion is the primary variable. Jensen (1966) gave simplified versions Of the models: LE = Ké (1)1 Rn LE = Ke $2 RS LE = Ke ¢3 Ra (9) where: Ke = a crop coefficient Rn’ RS, and Ra = net, solar, and extraterrestrial radiation, respectively ¢1, ¢2, $3 = net, solar, and extraterrestrial radiation coefficients, respectively. The crOp coefficient takes into account the period of leaf area development, minor differences between field crops at effective full crop canopy, and the stage of maturation of the crop. The product of the two other variables represents PET from agricultural 27 crops surrounded by sufficient buffer area (generally 30.5 m wide is enough) to avoid advection errors. Some of the most used empirical solar radiation equations were developed by Makkink (1957), Ture (1961), Jensen-Haise (1963), Stephens-Steward (1963), Grassi (1964), and Stephens (1965). These were reviewed by Jensen (1966) who pointed out the major advantages of empirical equations using solar radiation as being simplicity, facility Of calibration for an area, and reliability of estimates sufficient for most engineering or water management applications. Energy Balance Approach The thermal balance at the evaporating surface can be used to calculate evapotranspiration if there is a quantitative measure- ment of the other factors that contribute to this balance. This appraoch was first used by Schmidt (1915) and later by Angstrom (1920) (reported by Deacon, Priestley, and Swinbank, 1958, and Thornthwaite and Mather, 1955). In general, the heat budget at the evaporating surface can be expressed as: LE+H+G :6 II where: R = net radiative flux at evaporating surface E = rate of evaporation per unit area L = latent heat Of evaporation of water H = rate of transfer of sensible heat per unit area of the evaporating surface 28 C) II rate of heat storage per unit area below evaporating surface R and G can be measured thus: R - G = LE + H (11) Cummings (1925) assumed that the net radiation energy must be assigned to evaporation, neglecting the rate of transfer of sensi- ble heat. This assumption was rebutted by Bowen (1926) who showed that the heat losses by conduction and convection should be consid- ered in the energy balance. He regarded the process Of evaporation and diffusion of water vapor from a surface into the layer Of air above it as exactly the same as that Of the diffusion Of specific heat energy from the surface into the layer. Bowen developed the ratio, 8, Of the heat loss by diffusion to that by evaporation: B = 0.46 (T87 Ta ) P (12) es - ea 760 where: TS = the temperature Of the layer Of air in contact with the evaporating surface, °C eS = the vapor pressure Of the layer of air in contact with the evaporating surface, in mm of mercury Ta = the original temperature Of the air passing over the surface, in °C 29 ea = the vapor pressure of the air passing over the surface, in mm of mercury P = the atmospheric pressure, in mm of mercury From Equation (11), Cummings and Richardson (1927) derived an equa- tion for evaporation from lakes: _R—G LEN—(1+8) (13) They tested the formula experimentally and found that it supported Bowen's theory. Although developed for evaporation from water surfaces, the formula has been used for the determination Of potential evapotrans- piration by considering evaporation from a plant cover well watered and water surface as being proportional (Penman, 1949). Gerber and Decker (1961) stated as the main requirements for the use of the model, that the borders of the experimental area be large, that the measurements of the environmental factors used be taken a short distance above the top of the crop, and that the heat stored in the crOp mass be very small. Fritschen (1966) reported that the method had been tested on agricultural crops in humid and arid regions with valid results; how- ever, he warned that care be taken to assure proper instrumentation and compliance with the assumptions of the model. He emphasized the careful measurement of R under all conditions and 8 especially in arid conditions. 30 Mass Transfer Method There are two different approaches for the calculation Of evaporation with the mass transfer theory: Profile Method and Eddy Flux Method. The profile method is based on the measurement Of the verti- cal gradient Of water vapor concentration. Although there are differ- ent versions Of the method, they generally follow the approach reported by King (1966) who used the ratio of the general equations Of vertical water vapor and shear stress diffusion: E = PKw (3%) (14) T = pxm ( g2) (15) to get: E=—PUi-:1:2 :1 (16) m 2 1 where: E = the vertical flux density Of water vapor (ET) P = the density Of air Kw = the turbulent transfer coefficient -§3 = the vertical gradient of specific humidity 82 T = the vertical flux of horizontal momentum of shear stress K = the turbulent transfer coefficient for momentum 31 82 the vertical gradient Of wind speed c: ll * the friction velocity (U2 - U1) and q2 - ql) = the difference in wind speed and specific humidity between 22 and 21 above the ground Assuming Kw = Km and an adiabatic wind profile (neutral stability) he Obtained: (U2 - U1) U 2n (22/21) = K * (17) thus: -PK2(U-U)< -> 2 1‘12 q1 E = 2 (18) [En (22/21] The model given by Equation (18) is called the Thorthwaite-Haltzman aerodynamic equation which, besides the assumptions Of adiabatic profile and Kw = Km, is based on the assumption that the surface is flat and uniform and U* is constant with height. For diabatic conditions causing thermal stratification near the ground and change Of vapor and wind profiles, several equations have been developed to model the wind profiles. Monin and Obukhov (1954) proposed an equation on the form: 32 U* K 2 L' o where: a = constant with value from 0.6 to 10 Brooks (1963) suggested: * 1 U=—I-(—[£ni+2Y(—.>‘2] (20) and Swinbank (1964) developed the equation: U* =___ exp(z/L) U K 9‘“ [EXP(zoL) (21) where 3 U L' = ____*__ KgH/Cppe 6 = absolute potential temperature U* is determined from any of Equations (19) to (21) and substituted in Equation (16). Assuming Km = K.W in the equation, a value Of PET can be determined. There are contradictory reports Of the use of the method and it appears to be very difficult to reliably measure surface stress from a wind profile (Barry, 1966). The Eddy Blux Method is based on the mean and instantaneous fluctuations Of velocity, temperature, and fluid properties introduced 33 into the momentum, energy, and continuity equations. This yeilds a transfer equation containing a molecular diffusion term and a turbu- lent diffusion term. From Goddard and Pruitt (1966): dt H = Cp (Db-BE + Pw t ) (22) =_ 93 \\ LE L (De dz + Pw q ) (23) where: H = sensible heat flux LE = latent heat flux w = vertical velocity 2 = vertical distance t = air temperature q = absolute humidity D = molecular diffusivity Of heat D II molecular diffusivity of water vapor C = specific heat L = latent heat Of vaporization 1 = air density For atmospheric conditions the molecular terms can be neglected, then: H = Cp lw‘t“ (24) 34 LE = L Rw‘q‘ (251 The values thus Obtained are then used in the energy balance Equation (10). Combination Equations For the determination Of ET for the energy balance method and the mass transfer profile method, the surface temperature or the surface vapor pressure are required. Because Of the difficulty involved in the measurementscfifthese parameters, Penman (1948a, 1948b, 1949), used both methods to Obtain an equation which only requires parameters easily available in any "routine records of weather stations.‘ This is the combination equation. From Dalton's Law: E0 = (eS - ed) f (u) He represented by Ea’ the value Of E with ea instead Of eS so E8 = (ea - ed) f (u) and got the ratio E a 'E; = (ea - ed)/(eS - ed) (26) From Equation (13): E0 = (R - G)/(l + B) and B = r (TS - Ta)/(eS - ed) 35 the ratio ((R - C)/Eo) = l + v(TS - Ta)/(es - ea) and putting (eS - ea) = A (TS - Ta) then ((R - G)/Eo) = 1 + Y [(eS - ea)/(eS - ed)]/A (27) and finally, combining equations (26) and (27) and eliminating vapor pressure, his equation becomes: E0 = [MR - G) + YEal/(A + v) (28) where: E0 = rate of evaporation, mm/day R, ea, es, Ts’ Ta’ G are as given in Equations (10) and (12) A = the slope of the e:T curve at T = Ta Y = the psychrometric constant ed = saturation vapor pressure at deWpOint temperature f (u) = wind function Ferguson (1952), reported by Deacon, Priestley, and Swin- bank (1958), worked independently from Penman but following the same approach and Obtained a similar formula using implicitly the Bowen's ratio. He proposed a numerical integration Of the differential equa- tion representing the energy balance to determine TS and es. de Penman, on the other hand, substituted-ET for (eS - ea)/ (TS - Ta) and neglected heat storage. Although it has more 36 approximations, Penman's method gained popularity and presently is one of the most used models in the determination of ET. McIlroy developed another combination model to predict poten- tial evapotranspiration (Slatyer and McIlroy, 1961; McIlroy, 1968) which was reported by Dilley and Shepherd (1972). From the general form: ___S_.B__G. P. _ E-S+Y(L L)+L(D Do) (29) He neglected G (small in comparison to R for averages over a day or more). He also neglected Do for the case Of ET, when the liquid water supply to the evaporating surface is adequate for the demand, thus creating a near saturated condition in the air adjacent to the evaporating surface. So for PET: S R h = -+—D PET S+YL L (30) where: E = actual evaporation PET = potential ET -——-= a temperature dependent weighting function -% = an atmospheric conductance for the air layer from surface to reference height D = the wet-bulb temperature depression at reference level, °C 37 D0 = wet-bulb temperature depression at evaporating surface, °C -% = evaporation equivalent Of net heat flux into the ground, mm/day -% = evaporation equivalent of net radiation flux, mm/day y = ratio of specific heat of air to latent heat of vaporization of water van Bavel (1966) tried to eliminate the empiricism from the Penman Equation and developed a model to calculate the instantaneous evaporation: LE0 = A/y H + L Bvda (31) A/y + l where: Bv = a transfer coefficient for water vapor, in cal cm"2 min-lmbu1 d8 = the saturation vapor pressure deficit of air, mb E0 = potential evaporation rate, g cm- min- , mm hours - , or mm day- H = sum Of energy inputs at surface, exclusive of sensible heat transfer in air and LE (calcm_2min-l) and: 38 D 2 U BV= 15K 'un—zarnz ‘32) a 0 where: p = density of air, g cm- 8 = the water-air molecular weight ratio K = the Van Karman constant P = the ambient pressure Ua = the wind speed at height a, cm min- A8 = the elevation above the surface, cm Zo = the roughness parameter, cm Penman Equation Being aware Of the difficulty of measuring the parameters required for the solution of Equation (28), Penman (1948a, 1948b, 1949) tried to simplify the model. For the total amount of energy available for evaporation and heating Of the air, R, Penman used various empirical expressions which relate net long- and short-wave radiation to temperature, vapor pressure, cloudiness and the reflectance of the evaporation surface. Brunt (1932, 1939) had developed the expression: R = Rc (1 -y- u)-OT4[0.56 - 0.092(ed);§] (1 - 0.9g) (33) where: 10 OT ‘ m 10 39 short-wave radiation from sun and sky in the evaporation equivalent of mm/day radiation reflection coefficient (albedo) 0.05 for water surface fraction of Re used in photosynthesis theoretical black—body radiation with T as temperature in °K and O the Stefan-Boltzman constant saturation vapor pressure at dew point tempera- ture, in mm of mercury the fraction of sky covered by cloud (1 - 0.9m/10) takes into account the effect Of the sky cover on net radiation and [0.56 - 0.092 (ed) 15 ] the effect of vapor pres- sure On net radiation. From Brunt's expression, Penman neglected u (very small in realtion to the other parameters) and used the correlation: In 56 where: as a + b-— = constants the theoretically calculatable amount of radiation that would reach the earth in the absence Of the atmosphere 40 which had been used satisfactorily in Virginia, U.S.A., and Camberra, Australia, with different values for a and b. He found the values of a = 0.18 and b 0.55 as being the adequate values for Rothasmsted. So: W II n Ra (0.18 + 0.55 N) (34) He also considered the influence of cloud type on the con- trol of cloud cover on long-wave radiation and proposed to set m n 'IO = 1 --fi so that Equation (31) became: R = R3 (1 - v)(0.18 + 0.55 %) - 0T4 [0.56 - 0.092 1 (edfl (0.1 + 0.9 II-) (35) N where: R8 = the theoretically calculable amount of radiation that would reach the earth in the absence Of the atmosphere -§ = the ratio Of actual to possible hours Of sunshine For the determination of E3, Penman Obtained the wind function: f (u) = (1 + 0.0098 uz) (36) where U2 = the wind speed at a height of 2 meters, in miles per day (m.p.d.) 41 and developed the expression: E8 = 0.35 (l + 0.0098 uz) (ea - ed) (37) Equation (28) can then be solved with relatively readily available weather parameters, i.e., mean air temperature, mean dew point temperature, mean wind velocity, and duration of sunshine; and some information obtainable from standard souces (Ra’ a, A, O, v) to yield the amount of evaporation. The equation was tested at the Rothamsted experimental sta- tion (Penman, 1948b) on Open water surface, wet bare soil surface and turf with an adequate supply of water and the last two were found to be a fraction of the water surface evaporation. Also, Penman reported a gOOd agreement of the equation with other methods when the model was used with published data from four different places in America and EurOpe and also agreed closely with estimates Of evaporation from the British Isles. In order to use the equation to predict ET from turf with a plentiful water supply, Penman gave some seasonal values Of-gl for O southern England: Mid-Winter (November-February) 0.6 Spring and Autumn (March-April, September-October) 0.7 Mid-Summer (May-August) 0.8 Whole Year 0.75 42 In later studies, Penman (1963) indicated that the equation could be used tO estimate potential evapotranspiration directly using r = 0.25 (for grass) instead of 0.05 (for water). He suggested a modifi- cation Of the wind function: f(u)==(0.5 + 0.01u2) for water and f (u) = (1 + 0.01u2) for grass to account for the extra roughness of grass compared to that Of Open water surface. Modified Versions Of the Penman Equation The Penman Equation is one Of the most popular methods for calculating ET and it has been modified by many authors to adapt it to specific conditions, or to try to improve its accuracy. Wright and Jensen (1972) developed a new wind function: f (u) = (0.75 + 0.0185 (u (38) 2) for alfalfa, to account for the roughness Of the surface of alfalfa in relation to that of a grass surface. Tanner and Penton (1960) Obtained another wind function using the Businger (1956) neutral profile approach. Doorenbos and Pruitt (1977) defined a wind function to be used in any climate to avoid the need for local calibration in the wind function: f (u) = 0.27 (l + 0.01 u2) (39) where: u2 is given in Km/day at 2 mt height 43 They recommended an adjustment factor to take into account the dif- ferences in weather conditions during day-night which affect the level of ET. Effectiveness of Penman Equation The Penman Equation has been tested more than any of the other equations used for predicting ET and most of the reports show the Penman model to be the most effective, in terms of its accuracy and adequacy. Even van Bavel (1966) who reported a poor agreement between the Penman Equation and measured evapotranspiration on a 24-hour basis on alfalfa, found an excellent agreement for 24-hours when all empiricism was excluded form the model. Tanner and Pelton (1960) concluded that the Penman Equation was suitable for estimating PET for periods as short as one day. However, they found the estimates to be too low for vegetation rougher than short grass, unless the wind function was modified. This was due to the fact that the wind and the vapor pressure are affected by surface roughness, so the wind function has to be modified to combine it with the vapor pressure deficit measured over the rough surface. Jensen, Wright, and Pratt (1971) found that over long time periods, Penman's version gave the same results as van Bavel's version and both yielded satisfactory results on their irrigation scheduling computer program. Wright and Jensen (1972) in a study concerning peak water requirements determined from lysimeter measurements and a slightly modified Penman Equation, found an acceptable agreement 44 between both methods in Southern Idaho. Wright and Jensen (1978) used several years of data from two precision weighing lysimeters and their modified version of Penman Equation to develOp a crop coeffi- cient to be used in the USDA-ARS Computerized Irrigation Scheduling Program. They concluded that the expected errors in computing PET were well within acceptable limits and the expected errors in esti- mating daily actual ET using the crop coefficients developed, were about the same as those in estimating PET. The Irrigation Water Requirements Technical Committee of the American Society of Civil Engineers (Jensen, 1974), published the results of a study of the sixteen common methods Of predicting ET. It was concluded that the combination methods were the most accurate with local calibration of the wind function and vapor pressure deficit terms, but even without calibration, they performed better than the other methods. The methods were ranked considering the accuracy of seasonal estimates in percent of the measured ET for the season and the root mean square of the monthly differences. The Penman method ranked second:finrinland—semi-arid to arid regime after the Jensen-Raise and van Bavel—Businger methods which were tied for first place. Kruse et al. (1977) made a study to check ET estimates from the USDA Irrigation Scheduling Program with lysimeters measurements Of ET from irrigated corn and alfalfa, in Colorado. A modified version Of the Penman Equation was used, along with the crOp coeffi- cients used in the USDA-ARS Scheduling Program (Jensen et al., 1970); 45 they concluded that the estimates of PET given by the modified Penman Equation were in good agreement with the values measured by the lysimeters. Doorenbos and Pruitt (1977) on the study by FAO reviewed earlier, worked on four of the best known methods to predict ET (Blaney-Criddle, Radiation Method, Penman Method and pan evaporation method). They stated that the Penman Model Offered the best results, with possible error of 10% in the summer and a maximum of 20% in the winter (low evaporative conditions). Shih et a1. (1981) evaluated five different methods of esti- mating potential ET as they compared with the basin-wide water budget method used in the Everglades, Florida. The methods evaluated were the Penman Method, the pan evaporation method, Thornthwaite Method, Blaney-Griddle Method, and the modified Blaney-Griddle Method. They showed that the average annual difference between the Penman and water budget method was insignificant. They also calculated an absolute deviation between the water budget and the other methods evaluated which was used as a criterion for testing the applicability Of the methods, the smaller absolute deviation indicating a better prediction. The Penman Equation gave the best prediction and the Thornthwaite Method gave the worst. Limitations of the Model The Penman Equation has been criticized by its empiricism and the assumptions upon which it was developed. van Bavel (1966) expressed his concern in the empiricism in the radiation part of the 46 formula and in the wind function, and in the use of average values for temperature, humidity, and wind speed (he believed instantaneous values were required). Deacon, Priestley, and Swinbank (1958) warned that for relatively short periods, the heat storage (neglected by Penman) may have to be considered and that the approximation Penman used to eliminate the surface temperature and vapor pressure through substi- tution of: e e — e (a). f” 5—1—5— 3 a could seriously underestimate evaporation, mainly in the case of a strongly evaporating vegetation cover. They also concluded that the formula to estimate incident solar energy in terms of cloud cover does not discriminate as to times of cloud occurrence. Tanner and Pelton (1960) remarked that all the work in the Penman Equation was related to short grass and that because most of the agricultural crOps were taller and aerodynamically rougher than grass, data were needed for rougher surfaces. However, as Penman (1956) pointed out, "any meteorological physicist will find easy to criticize the formulae for their sweeping simplifications of complex meteorological phenomena. Soil physicists, however, might support a claim that crude as the method is, it can give estimates of changes in soil water content as accurate as any simple field method at our disposal." CHAPTER III PROCEDURE A sensitivity analysis Of the Penman Equation was performed. The most used techniques for performing sensitivity analysis are the log derivative method and the probable error analysis, both methods based on partial derivatives. Another technique consists in varying the values Of a variable to compare the changes in the vari- ables and the amount of change in the equation introduced by the changes in the variables. This technique is very useful to analyze equations whose derivatives are difficult to evaluate (Fritschen and Gay, 1979). Such is the case with the Penman Equation and this is the technique followed in this research. The approach was the following: A computer program for the Penman Equation was developed. The equation used was basically the one developed by Penman, but modified to acceptlhe inputs in SI units, as reported by Schwab (1981): A(Rn - G) + y Ea A + y PET = where: 47 PET The net radiation used: where: 48 potential evaportranspiration, cal/cm2 day slope Of the saturated vapor vs. temperature curve (mb/°C) psychrometric constant, mb/°C net radiation, cal/cm2 day soil heat flux toward the surface, generally assumed to be 0, cal/cm2 day 15.36 (1 + 0.0062 u2) (ea - ed) saturation vapor pressure at mean air tempera- ture, mb saturation vapor pressure at mean dewpoint temperature, mb wind speed at 2 net, Km/day equation proposed by Penman (Equation [35]) is 2 4 *2 R3 (1 - r)(0.18 + 0.55 N) 0 Ta (0.56 - 0.092ed) n (0.1 + 0.9 N) extraterrestrial radiation, cal/cm2 day albedo ratio Of actual to possible hours of sunshine in percent Stefan-Boltzman constant, 11.7lx10'.8 cal/cm2 day mean daily temperature, °K 49 The saturation vapor pressures are calculated using the expression by Bosen (1960): eS = 33.8639 [(0.00738t + 0.8072)2 - 0.000019 I1.8t + 48| + 0.0013116] (40) where the vapor pressure is in milibars and t the mean daily tempera- ture or the mean dew point temperature is in degrees Celsius. The delta function (slope of the vapor pressure-temperature curve) is obtained from the first derivative of Equation (40). A = 2.00 (0.00738T + 0.8072)7 - 0.0016 (41) For the psychrometric constant, the expression proposed by Brunt (1952) was used: = 0.386P L (42) where: P = average barometric pressure, mb L = latent heat of vaporization, cal/g P is assumed to be constant for a given location and can be calculated using a straight line approximation of the U.S. standard atmosphere (Jensen, 1980): P = 1013 - 0.1055E (43) where: E = sea level elevantion, m. 50 L is calculated by the following expression by Brunt, 1952: 315595 - 0.51T (44) where: T is temperature in °C. The following step was to select average values for the para- meters, normal in the humid area of the North Central United States. Three situations were chosen: low, average, and high values for the parameters, which would give low, average, and high PET values, respectively. The parameters in the equation were then varied independently from minus 20% of the given values to plus 20% Of the given values. Ra was held constant at 937 cal/cm2 day for all three situations because it does not have much lower or higher values in the cases of interest; albedo was held at 0.22 inasmuch as this is the value assumed apprOpriate for grass. The variable analyzed were maximum temperature, minimum tem- perature, wind speed, percent of sunshine, albedo, extraterrestrial radiation, and mean dew point temperature. The parameters considered for modifications were those whose values are most difficult to Obtain without having access to a first- Order weather station: wind speed, percent of sunshine and mean dew point temperature. Once the decision had been made as to how to modify the use of a particular parameter, the accuracy of prediction was tested 51 using the values Obtained from the Local Climatological Data sheets (LCDs) for the Lansing Airport. PET as predicted by a modified parameter was correlated to PET as predicted by the original data and the same was done with the combination of the modifications. Values for June, July, and August from 1978 to 1981 were used. Three months were selected because they are the critical months for irri- gation practices in the area. To evaluate the effectiveness of the simplifications intro— duced in the Penman Equation, the equation was used in the summer of 1981 to calculate daily potential evapotranspiration for June, July, and August, and after adjusting potential evapotranspiration with the crOp coefficient, the actual evapotranspiration values were correlated to the evapotranspiration values Obtained from a water balance, with the soil moisture determined by two methods: gravi- metric and neutron scattering technique. The measurements were made at the Crop Science Research Center, Michigan State University, in a sandy loam soil with a bulk density of 1.50 g/cm3 in the first 30 cm, 1.69 g/cm3 from 30 to 61 cm (there is a clay layer in the profile) and 1.58 g/cm3 from 61 to 91 cm. The gravimetric measurements were made by the author in irri- gated and nonirrigated corn. The neutron measurements were made by James Jenkens, Crop and Soil Science M.S. candidate in irrigated potato, as part of an experiment, to evaluate the effectiveness of the method for his thesis. The potato and the irrigated corn were irrigated by sprinkler and the precipitation was measured by a pluviometer located in the 52 research center. The crop coefficient for each day was determined from a crop coefficient curve for the season for each crop as was proposed by Doorenbos and Pruitt (1977). CHAPTER IV RESULTS AND DISCUSSION Analysis Of Sensitivity Table 1 presents the values used to perform the sensitivity analysis, for low, average, and high evapotranspiration conditions; Figures 1 through 7 persent the results of the analysis. The graphs illustrate how ET changes when the specific parameter of each graph is varied, with the other parameters held constant; the three situa- tions (low, average, and high ET) are presented in each figure with the values in percent. As it can be seen from the graphs, the extraterrestrial radia- tion is the most influential parameter in the Penman Equation. It has almost the same influence at each of the three levels, affecting slightly more at the low values (Figure l). The equation sensitivity to maximum temperature is about half that of extraterrestrial radia- tion and has about the same influence at average and high ET values and slightly less at low values (Figure 2). The percent of sun- shine has almost the same influence as temperature (a little less) at average and high levels, but its influence decreases more at low level, (Figure 3). The albedo is as influential as percent of sun- shine, but its effects are more constant at the different values and it varies inversely (negative slope) (Figure 4). 53 54 TABLE 1. Conditions assumed for the examination of the sensitivity of Penman Equation to its different parameters Condition Parameters l 2 3 Low ET Average ET High ET Albedo 0.22 0.22 0.22 Ext. Rad. (Ra) ca1/cm2 day 937.0 937.0 937.0 T °C 21.7 29.4 32.8 max T °C 6.7 15.6 23.3 min ° 6. 16. 22.2 Tdew point 7 7 Percent Of sunshine 67.0 85.0 94.0 Wind @ 6.09m (Km/day) 120.0 160.0 210.0 1Data for Lansing, July 5, 1972. 2Data for Lansing, July 17, 1972. 3 Data for Lansing, July 20, 1972. 55 O . C? 0 CD- c N E: m D ‘3 C3. II. C) c: Extraterrestrial Radiation . v . a ‘20-00 ‘10000 .000 10000 20000 O 9 O “H I D 0 c5 Key: a = low level cud b - average level I c a high level Figure 1. Sensitivity of the Penman Equation to extrater- restrial radiation; expressed in percent change. 56 20.00 ET 10.00 m .00 Tma 1 I .00 10.00 20.00 r W -20.00 -10.0 “10.00 Key: high level - average level low level (3‘93 1 D II II '20 I00 Figure 2. Sensitivity of the Penman Equation to maximum temperature, expressed in percent change. 57 D 0 C3. N 94 LL} 0 D C CD. a "' b O ‘2 Percent of sunshine l j I j -20-00 -10.00 .00 10.00 20.00 0 D O D "d I D :2 Key: a = average and DJ high levels fir b = low level Figure 3. Sensitivity of the Penman Equation to percent of sunshine, expressed in percent change. 20.00 10.00 I ”'20 000 Figure 4. “30000 cpoo ‘10000 ~00 SJ I 58 ET Albedo 1000 30.00 b8 C Key: a = high level b = average level c 8 low level Sensitivity of the Penman Equation to Albedo, expressed in percent change. 59 20.00 00 10- l Figure 5. ' Juun__, .00 10.00 20.00 Kay: a = high level b = average level c = low level C3 0 C O 3‘ Sensitivity of the Penman Equation to minimum temperature, expressed in percent change. 60 20.00 ET 10.00 a b Tdew " 000 000 100 600 00 1 I high level average level low level Key: 3 '10000 D II II -20.00 Figure 6. Sensitivity of the Penman Equation to dew point temperature, expressed in percent change. 61 20.00 ET 10.00 Wind Velocity a __— - . -10.00 .00 110.00 2000 .00 c? ”10000 a Key: a = high, average, and low levels “20000 Figure 7. Sensitivity of the Penman Equation to wind velocity, expressed in percent change. 62 Minimum temperature has a little less effect than albedo and it varies more in the three levels than the rest of the parameters, having less effect at low level of ET and more effect at high level (Figure 5). The dew point temperature effect in the Penman Equation is very little as it is shown in Figure 6, and it differs from the other parameter by the fact that its slope is negative at low and average values and positive at high levels of ET. The wind is the less influential value in the Penman Equa— tion and in relation to most of the terms it could be considered insignificant. It has about the same effect at the three levels of ET. Simplifications The influence of wind in the calculation of ET is minimal, expecially in the humid North Central Region. Wind has two differ- ent effects on the rate of transpiration. First, wind over a crOp canopy creates shear; the turbulence produced by the shear induces a vertical transport of vapor from the canopy to the atmospheric bound- ary layer. Most probably, any wind, regardless of its speed, will cause enough vertical transfer of vapor to prevent the microclimate within the canOpy from becoming saturated with water vapor. 0n the other hand, the wind has a more direct effect on evapotranspiration in the boundary layer thickness of the air around an individual leaf. The transfer of water vapor within the boundary layer is by molecular diffusion and the thinner the layer, the shorter the period of molecular diffusion. The determination of the exact 63 thickness over a point on any specific leaf would be very difficult because it requires such data as orientation and magnitude of the wind velocity vector with regard to the leaf. As the leaf orienta- tion is broadly more or less random and with the presence of turbu- lence in the canOpy, the wind velocity vector will also be random, the effects of the boundary layer thickness must average out. These two facts explain the little effect wind has in the determination Of ET. From Figure 7 it is evident that only gross values of wind speed are necesary for the determination of ET by the Penman Equation. A wind speed change of -20% of low value to +20% of high value alters ET by only about 2%, a value well within the range of error accept- able in irrigation measurements. It is suggested that the irrigator select three levels of wind speed based on data from the closest weather station (low, average, and high) and that the irrigator use for the period in question a low, average, or high value, based on his individual judgment. For the Lansing, Michigan, areas values of 10 km/day as a low value, 35 Km/day as an average value, and 85 Km/ day as a high value (wind speed at 2 m), appear to be usable values. These values were used for the calculation of ET for the summer months of June, July, and August for the period 1978 to 1981 inclusive, and correlated to ET calculated with the exact values of wind speed for the same period. The wind speed data used to corre- late the approximations were taken at 6.096 m (20 ft): so, the expression reported by Schwab (1981) was used: 64 l PENMRN ET WITH WIND SPEED SUBSTITUTION VS PENMHN ET 0.50 J 1 0.40 1 EVHfiQiRRNSPIRRTION. (CH) R I 0.9997 0.20 1 MODIFIED PENHRN 0.10 cPL°° 6 O 01 70160 10 0120 0130 0140 0150 PENHHN EVHPOTRRNSPIRRTION. (CH) Figure 8. Regression of ET calculated by the Penman Equation and ET calculated by the Penman Equation with the wind velocity approximations. 65 = log 2 u2 u1 a log h (45) where: 112 = wind speed at 2 m u = measured wind speed h = height at which the measurement was made Both ET were calculated using weather data from the Lansing Airport and the correlation coefficient Obtained was .9997 with a slope of 0.996 for the regression curve (see Figure 8). That shows that little gain was obtained with the use of wind velocities more accurate than the suggested approximations. It was also shown how little influence the dew point tempera- ture has in the Penman Equation (variation from -20% of low value to +20% of high value in dew point temperature causes only a varia- tion of t 2.3% in calculated ET); so, the determination Of dew point temperature does not require a great deal of accuracy for its use in the Penman Equation. Some authors (Gentilli, 1955; Pachop, Morton, and Cornia, 1973) have found a close relationship between dew point and minimum temperature, mainly in the humid climate. Figure 9 is a regression of dew point temperature against minimum temperature for the summer months of June, July, and August from 1978 to 1981 inclusive. The correlation coefficient was 0.90 and the slope of the regression line was 0.94. Because of the difficulty involved in the determination of dew point temperature and the other humidity variables and the 66 30.00 CDRRELFITIDN DF 1 TMIN V8 TDEN 25.00 D O “b duo 6100 .00 Ca .00 2'0 .00 2‘5 .00 3‘0 .00 . DEG . CELS . —O—- OO— m 2 Figure 9. Regression of dew point temperature and minimum tem- perature for Lansing, Michigan, daily values. 67 little influence of dew point temperature in the equation, along with the closeness of fit of the variables, this author believes the substitution of minimum temperature for dew point temperature is a valid and sound approach toward the estimation of one of the most difficult to obtain weather data mong the parameters used in the Penman Equation. The cause of the high correlation between minimum and dew point temperatures, anywhere there is dew formation at night and the immediate area is not affected by air flowing over large bodies of water is related to the lack of moisture entering an air mass which must travel long distance over land and the large quantity of latent energy surrendered by condensing dew. Because the latent heat Of vaporization is so large (590 cal/g), a great deal of energy is released into the atmosphere in the process of dew formation. This energy counters the energy lost by radiation, hence the minimum tem- perature at night remains reasonably near the dew point temperature. During the daytime, solar radiation warms the air, but the saturation vapor pressure is determined by the quantity of water vapor present. Since there is little additional humidity available to air moving over a land mass, there will be little change in the saturation vapor presure and therefore the dew point temperature is in general constant during the day. This will not be true where there is not dew formation or when there is change in the air mass. If the latter is the case, a recent minimum temperature which occurred under the influence of the 68 air mass present when most of the ET occurred should be used. For example, if a cold front enters an area early in the morning, the minimum temperature from the following evening should be used since the saturated vapor pressure of the air accompanying the cold front air mass would differ significantly from that of the air present before the cold air arrived. Figure 10 represents the results of the correlation between Et calculated using minimum temperature to determine e and ET cal— d culated using dew point temperature to calculate e The correla- d' tion coefficient was 0.997 and the slope of the regression line was 0.945, indicating how little accuracy was lost with the substitution. Figure 11 is a graph of the correlation of ET calculated using the wind simplification along with the minimum temperature-dew point temperature substitution vs ET calculated with these parameters unchanged. The correlation coefficient was 0.997 and the slope of the line was 0.942, which is an excellent agreement. The most difficult parameter to obtain a simplification for in the Penman Model is the percent of sunshine. Figure 3 indicates the importance of accurate assessment of this parameter to reliable ET estimates. Unfortunately, dependable data are gathered only at well-equipped weather stations and normally are not broadcast or published, except in LCDs. It is possible to estimate, in a rough manner, the degree of sunshine on any given day, however. As a first approximation, we can adOpt the approach used for wind speed and proceed from this point. Long term records for Lansing, Michigan, 69 0-80 J PENMRN ET NITH ' MINIMUM TEMPERRTURE SUBSTITUTIDN VS PENMRN ET 0.60 1 0-40 1 0.30 1 0.20 J R - 0.997 0.10 l MODIFIED PENMHN EVHPOTRHNSPIRHTION. (CM) cP-°° 0110 0120 0130 0140 0150 0150 PENMHN EVRPDTRRNSPIRRTIDN. (CM) Figure 10. Regression of ET calculated by the Penman Equation and ET calculated by the Penman Equation with the minimum temperature substitution. 70 0.80 PENMRN ET WITH :- WIND RND TMIN SUBSTITUTIDN VS PENMFIN ET (CM) 0.60 0-40 I 0.30 l EVHPOTRRNSPIRRTION. 0.20 l R - 0.997 ED PENMHN 0:10 MODIFI cp-OO E: :3 EM 10 0120 0130 0140 0150 10150 PENMRN EVRPOTRRNSPIRHTION. (CM) Figure 11. Regression of ET calculated by the Penman Equation and ET calculated with the wind approximations and the minimum temperature substitution. 71 indicate the average percent of sunshine for June as 75% with July and August slightly less at 67 and 58%. Analysis of the four years of Lansing data from 1978 to 1981 yielded standard deviations that are probably reasonable inasmush as the means obtained for the period agreed reasonably well with the long—term means. Based on the above analysis, it was decided to use values as shown in Table 2 to calculate ET. Figure 12 gives the results of the correlation of ET TABLE 2. Simplified values of percent of sunshine based on monthly average values and standard deviations. Month June July August Low values (m - s.d.) 41% 36% 35% Average (m) 67% 64% 63% High values (m + s.d.) 92% 92% 92% Note: m = mean s.d. = standard deviation using the percent of sunshine substitution as given in Table 2 and ET using the actual value. The agreement is reasonable with a correlation coefficient of 0.949 and a slope of 0.810. This approach was adOpted as a reasonable first approximation and a final set of data were obtained using all simplifications proposed above. The results are presented in Figure 13 and yielded a correlation of 0.945 and a slope of 0.763. 0-80 J 0.50 l l 0.40 0:30 0.20 l MODIFgfig PENMHN EVHPOTRHNSPIRHTION. (CM) 72 PENMHN ET WITH I! PERCENT SUNSHINE u SUBSTITUTION VS 8 a: PENMHN ET HI .00 cPL°° Figure 12. 0.10 0120 0.30 0.40 0150 0100 PENMHN EVHPOTRHNSPIRHTION. (CM) Regression of ET calculated with the Penman Equation and ET calculated with the percent of sunshine approxi- mations. 73 3 PENMRN ET WITH - g WIND. TMIN RND SUNS- . ~a SUBSTITUTION vs ,1 PENMHN ET I 0.40 1 RN EVR%Q%FHNSPIRRTION. l 0.20 l MODIFIED PENM 0.10 cp.00 .00 0150 0110 0120 0130 0140 0150 PENMHN EVRPOTRHNSPIRHTION. (CM) Figure 13. Regression of ET calculated with the Penman Equation and ET calculated with the wind and percent of sun- shine approximations and the minimum temperature substitution. 74 The extraterrestrial solar radiation required for the Penman approach presents no problem if a programmable calculator or hand held calculator is used. In the current analysis, R3 was calculated for Lansing, Michigan, using the expression: R8 = 660 - 342 cos (0.0172d - 2.78) (46) where: d = day of the year measured from January 1 Appropriate values can be determined for any latitude. Evaluation of the Simplified Penman Equation Along with the water balance made in the corn and potato fields, estimations of some of the parameters used in the Penman Equation were made. The percent of sunshine and the wind velocity were estimated daily. The former was classified as high, average, adn low, depending on the cloud conditions. The wind was also termed as high, average, and low. After giving to those classifications the values assigned in the simplifications (Tables 1 and 2) and using minimum temperature to estimate e, the daily potential evapotrans— piration was calculated and adjusted to evapotranspiration with the crop coefficient. The crop coefficient curves for corn and potato for the season was developed following the FAO method (see Figures 14 and 15 and Appendix A). The emergence day for corn was June 1 and for potato was May 25; the average irrigation or precipitation period was a 75 1.20 1.00 S CORN CROP ”S COEFFICIENT CURVE 43.00 51.00 0500 105.00 187.00 DHYS INTO THE SERSON ,p .00 £4 .00 .00 Figure 14. Crop coefficient curve for corn. 76 .20 .00 1 0.80 J 1 0-60 POTHTO CROP COEFFICIENT CURVE POTRTO CROP COEF. 0.40 0.20 5‘3 .00 111.00 1‘33 .00 2‘3 .00 4b .00 5‘7 .00 ORYS INTO THE SERSON 41.00 6 0 Figure 15. Crop coefficient curve for potato. 77 little less than 4 days for both crOps in the initial stage and the level of PET was 4 mm/day. The crop coefficient curves were drawn from the crop coefficient values at different stages given in Table 3. TABLE 3. CrOp coefficient for the different stages of the crOps. Corn Potato Kc Duration Kc Duration (daYS) (days) Initial stage 0.75 20 0.75 25 Crop development Variable 35 Variable 30 Mid-season 1.1 40 1.1 45 Late stage 0.55 30 0.70 30 The values came from Figure 6 and Table 21 of Doorenbos and Pruit (1977). The daily values of ET for each crop were classified by periods, each period corresponding to the interval between field measurement of moisture content in each plot. The period, as well as the evapotranspiration measured by the water balance (ETM) and by the modified Penman Equation (ETP) are given in Tables 4 and 5. For the water balance method, the following model was used (Bowman and King, 1965): 78 TABLE 4. Table of ETM and ETP for corn for the different periods Period ETM (cm) ETP (cm) June 26 - July 1 2.78 2.04 July 2 - July 9 3.57 3.20 July 10 - July 16 4.65 ' 3.11 July 17 - July 23 2.96 2.90 July 24 - July 30 3.25 2.80 July 31 - August 6 3.37 2.85 TOTAL 20.58 16.90 TABLE 5. Table of ETM and ETP for potato for the different periods ETM (cm) ETP (cm) June 1 - July 6 11.25 11.25 July 7 - July 12 3.87 3.40 July 13 - July 19 3.30 3.16 July 20 - July 26 2.82 2.85 July 27 - August 2 4.20 2.83 August 3 - August 9 2.68 2.74 August 10 - August 16 1.79 2.65 August 17 - August 22 2.65 2.38 August 23 - August 30 1.02 2.89 TOTAL 33.58 24.15 AM =P +c -G +AR -ET (47) av av av av av av where: AMav = net change in soil moisture in an arbitrary depth 0.76 M for both crops in this case) and for the specified period averaged over the plot P = precipitation during the period av Cav = capillary rise from below the 0.76 m during the period Gav = percolation of water to below the 0.76 m during the period ARav = net runoff in the plot during the period ETav = evapotranspiration during the period In this case Cav’ Gav’ and Rav were assumed to be zero during the period. The values of ETM presented in Table 4 are the average of irrigated and nonirrigated plots. For each plot, two sample loca- tions were chosen for the gravimetric moisture content measurements and the moisture difference for each period calculated. The correlation of the values given in Table 4 is presented in Figure 16. The correlation coefficient was 0.65, the intercept was 0.5 cm and the slope 0.51. Obviously, the gravimetric method gave higher values than the modified Penman Equation and the correla~ tion was low in relation to other values found in some of the pre- vious research. This was probably due to the fact that no data were 80 PENMHN VS MEHSURED ET 1 PIRHTION 4:00 1 3.00 1 MERSUREO EVHPOTRRNS 2.00 1,00 R - 0.65 1100 2100 3100 4100 5100 0.00 PENMRN EVHPOTRHNSPIRRTION Figure 16. Regression of ETM for corn and ETP for the different periods with a 45° line to show what l-l correspon- dence would be. 81 taken about the uniformity of application in the irrigated plot. Other factors may have been soil variability, faulty soil moisture monitoring equipment and/or inaccurate measurement of irrigation (the author did not have control of the irrigation; instead, he was sup- plied with the information about the amounts applied). In the case of the potatoes, three sets of measurement were made using a Troxler Neutron Moisture Meter, Model 3222. The moisture content in each set was obtained and the three values averaged. Figure 17 represents a correlation of these values with the modified Penman Equation results. A very good correlation coefficient was found (0.954) with an intercept of -0.11 cm and a SIOpe of 1.012 for the nine periods. The last period (August 23-30) was the only one when the values did not agree substantially. This may be explained by the fact that 1.27 cm was applied on the 24th and 1.27 cm on the 26th and after the irrigation there was a precipitation of 0.79 cm on the 26th, 0.84 cm on the 27th, l.37