Title Page
 Table of Contents
 List of Figures
 List of Tables
 Evapotranspiration and satellite...
 A system for automatic collection...
 Theoretical basis of the temperature...
 Verification of the temperature...
 Biographical sketch

Title: Evapotranspiration
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00089828/00001
 Material Information
Title: Evapotranspiration
Series Title: Evapotranspiration
Physical Description: Book
Language: English
Creator: Heimburg, Klaus
Publisher: Klaus Heimburg
Publication Date: 1982
 Record Information
Bibliographic ID: UF00089828
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000339630
oclc - 09617030

Table of Contents
    Title Page
        Page i
        Page ii
        Page iii
        Page iv
    Table of Contents
        Page v
        Page vi
    List of Figures
        Page vii
        Page viii
    List of Tables
        Page ix
        Page x
        Page xi
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        Page 5
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    Evapotranspiration and satellite data
        Page 12
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    A system for automatic collection of ET data
        Page 37
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    Theoretical basis of the temperature gradient response ET estimation methods
        Page 62
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    Verification of the temperature gradient response ET estimation methods
        Page 82
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    Biographical sketch
        Page 211
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Full Text








The work reported in this dissertation grew out of National Aero-

nautics and Space Administration (NASA) sponsored water resources re-

search. It was primarily supported by a grant from the Office of Water

Resources Technology in the U.S. Department of the Interior and Agri-

culture and Resources Inventory Surveys Through Aerospace Remote Sens-

ing (AgRISTARS) program funds administered through the U.S. Department

of Agriculture (USDA). Some support was also received in the form of an

assistantship from the Agronomy Department at the University of

Florida. All these sources of support.are gratefully acknowledged.

I would especially like to thank Dr. Wayne C. Huber and Dr. L.

Hartwell Allen, Jr. for signing the original research proposal as prin-

ciple investigators and seeing this project through to its completion.

Without their initial confidence in me and the day-to-day administra-

tive efforts of Dr. Allen none of the work would have been possible. I

would also like to thank them and the rest of my supervisory committee,

Dr. Howard T. Odum, Dr. Ralph W. Swain, and Dr. James P. Heaney for

improvements they made possible with their comments on the manuscript.

The research reported in this dissertation stretched over four

years and required the help and cooperation of many people. Sensors and

other equipment were borrowed from USDA, NASA, Center for Wetlands,

Fruit Crops Department, and Environmental Engineering Sciences Depart-

ment of the University of Florida. The Animal Science Department per-

mitted ET measurements in part of one of its pastures and the Agronomy

Department provided space for an instrument room. The people I owe spe-

cial thanks to are Bill Ocumpaugh and Fred McGraw for patiently working

around the measurement equipment and giving up some space; Johnny

Weldon for allowing me the use of the Agricultural Engineering Depart-

ment machine shop; Jim Hales for use of his tools and advice in fabri-

cating apparatus in the machine shop; Mark Lester for his fine machin-

ing; my brother Stephan Heimburg for conscientiously checking and ad-

justing the thermopile time constants; Mike Baker for designing and

helping build the scanning valve control electronics, and repairing the

dewpoint analyzer after a lightning strike; Wayne Wynn for help in

maintaining the measurement system; Dan Ekdahl at the Digital Design

Facility for .electronics repairs--especially a lightning-damaged compu-

ter-controlled voltmeter and similarly damaged computer interface

-.boards; and finally, Beth Chandler for expeditiously inking most of the

figures in this dissertation.

I also wish to thank Dr. Tom R. Sinclair for finally revealing why

Real Scientists don't do micrometeorology in a neat five-minute sermon-


I owe a debt of gratitude beyond words to three people who went

miles out of their way to help me. Gene Hannah was invaluable in the

original field installation and helped with problems throughout the

course of the project. I'm thankful to Ferris Johnson for his tireless

assistance in use of the computer system and trouble-shooting computer

hardware problems. Finally, I enthusiastically acknowledge the work of

Pattie Everett, who spared no effort and sacrificed evenings, weekends

and holidays in moving this manuscript through countless drafts toward


The post-defense party thrown in my honor made the frustrations

encountered in this work seem tolerable--even worthwhile. I have Pattie

Everett, Bill Campbell, Pierce Jones, and Lisa Lucille Biles to thank

for this totally awesome affair, not to mention that Wild and Crazy

Guy, Terry Spires, and the overwhelming Special Guest Appearance of

The Sublime Ms. Shavonne Rhodes. Get down, tiny dancers!



ACKNOWLEDGMENTS . . . . . . . . ... ....... ii

LIST OF FIGURES . . . . . . . . ... ....... vii

LIST OF TABLES . ... ... . . . . . . . . ix

ABSTRACT . . . . . . . . ... . . . . x

CHAPTER 1: INTRODUCTION . . . . . . . .... ... 1

Potential for Remote Evapotranspiration Estimates .. . 1
Scope of Research . . . . . . . . . '3
Research Approach . . . . . . . . . . 5
Experimental Site and Data Collection .......... 7
Organization of Dissertation . . . . . . ... 11


Overview . . . . . . . . . . . . 12
The Evapotranspiration Process . . . . . . .. 12
The Energy Balance Approach to ET Estimation . . . .. 17
The Energy Budget Equation . . .. ...... 17
Transport Similarity and Wind Models ........ 20
Latent and Sensible Heat Flux Expressions ... .. 23
Energy Budget ET Estimation Strategies . . ... 26
Remote ET Estimation Methods . . . . . . .... 29
Surface Temperature and Net Radiation . . . ... 29
Simulation Methods . . . . . . . ... .30
Steady-State Methods . . . . . . .... 33
Temperature Gradient Response Methods . . . ... 35


Overview . . . . . . . .. . ... . .37
Energy Budget/Profile Bowen Ratio Theory . . . ... 37
Sensor and Time Constant Considerations . . . ... 40
Data Collection Equipment . . . . . . .... 44
Data Collection Programs . . . . . . . ... 51
Operational Considerations. .... . . . . .58

ET ESTIMATION METHODS . . . . . . .... 62

Overview . . . . . . . ... ..... .62
Temperature Gradient Model . . . . . . .... 63
Strict Temperature Gradient Response Method . . ... 69
Average Temperature Gradient Response Method . . . .. 70
System Stationarity and Average Temperature Gradient
Response . . . . . . . . . . . 70
Use of Temperature Gradient/Net Radiation Correlation 73
Extension to Totally Remote ET Estimation Method . 77
Review of Assumptions . . . . . . . .... .79

ET ESTIMATION METHODS . . . . . . ... 82

Overview ............... ... . . . . 82
Validity of Assumptions . . . . . . . . 83
Radiation Temperature and Sensible Heat Transport . 83
Constancy of Parameters . . . . . . .... 86
Strict Temperature Gradient Response Method . . .. 95
Average Temperature Gradient Response Method . . . .. 97
Graphical Representation of the Average TGR Method .97
ET Estimates with the Average TGR Method ...... 103
Effects of Individual Parameter Variations . . .. 105
Generality of ATGR Latent/Sensible Partition .... .112
Tests of the ATGR Method .. ... ...... . ..114

CHAPTER 6: CONCLUSION . . . . . . . . ... ... 123

Summary of Results . . . . . . . . . . 123
The Average Temperature Gradient Response Method . 123
Method Limitations and Strengths . . . . .. 125
Recommendations for Future Research . . . . .... 128

REFERENCES . . . . . . . ... .... . . 131

APPENDIX A: LIST OF SYMBOLS . . . . . . . .. . 136


Program SET . . . . . . . . ... ...... 139
Program MEASR . . . . . . . . ... ... . 141
Program REPRT . . . . . . . . ... .... . 146
Program ANALZ . . . . . . . . ... .... . 149
Definition of Names . . . . . . . .... 152



BIOGRAPHICAL SKETCH . . . . . . . . ... .... . 211



Figure 1-1.

























Figure 5-2.











Figure 5-8.

Figure 5-9.

Location of the University of Florida Beef Re-
search Unit . . . . . . . . . . .

Field Apparatus and Sensor Locations . . . .

System Diagram of Generalized Evapotranspiring
Surface . . . . . . . . . . . .

Rough Calculation of Heat Storage in Pasture Canopy

Bowen Ratio Calculation from Measurements of Vapor
Pressure and Temperature . . . . . . .

Schematic of ET Measurement System . . . . .

Detail of Profile Measurement Mast Arm . . . .

Detail of Air Sampling Equipment . . . . .

Example of Intermediate Program Output . . . .

Example of Half-Hourly Data Report . . . .

Definition Sketch for Transport Properties

Components of Vapor Pressure Gradient . . . .

Total vs. Turbulent Temperature Gradients for a
Clear Day . . . . . . . . . . .

Total vs. Turbulent Temperature Gradients for a
Cloudy Day . . . . . . . . . . .

Heat Transport Coefficient Data . . . . ..

Moisture Availability Data . . . . . . .

Vapor Pressure Parameter Data . . . . . .

Soil Heat Flux Parameter Data . . . . . .

Clear Day Temperature Gradient/Net Radiation Corre-
lation . . . . . . . . . . .

Graphical Interpretation of Temperature Gradient/Net
Radiation Correlation . . . . . . . .

Cumulative ET Estimates . . . . . . .


* 10

. 14

. 19

. 41

. 45

* 48

. 50

* 53

* 56

* 64

* 67

. 87

. 89

. 91


. 94




Figure 5-10. Effect of Moisture Availability and Vapor Pressure
Parameters on Temperature Gradients . . . .

Figure 5-11. Effect of Heat Transport Coefficient and Soil Heat
Flux Parameter on Temperature Gradients . . .

Figure 5-12. Temperature Gradient Response of a Clear Day with
Constant Moisture Availability . . . . .

Figure 5-13. Temperature Gradient Response of a Partly Cloudy
Day . . . . . . . . . . . .

Figure 5-14. Generalized Clear Day H/E and E/R Patterns . .

Figure 5-15. Comparison of Measured and Estimated Bowen Ratios

Figure 5-16. Cumulative ET Estimates by the ATGR and Residual
Methods . . . . . . . . . . .

Figure 1.

Figure 2.

Figure 3.

Figure 4.

Figure 5.

Figure 6.

Figure 7.

Figure 8.

Figure 9.

Figure 10.

Figure 11.

Figure 12.

Appendix D: Supplementary Figures

Hypothetical Daytime Temperature Profile . . .

Simplified Temperature Profile . . . . . .

Simplified Temperature Profiles for a Clear Day . .

Simplified Temperature Profiles for an Overcast Day

Air Transport Coefficient for Average Conditions .

Soil Heat Flux Parameter for Average Conditions . .

Variation of the Daily Average Heat Transport
Coefficient with the Daily Average Windspeed . .

Data and ET Estimates for Oct. 17, 1981 . . . .

Data and ET Estimates for Oct. 18, 1981 . . . .

Data and ET Estimates for Oct. 21, 1981 . . . .

Data and ET Estimates for Oct. 22, 1981 . . . .

Data and ET Estimates for Oct. 23, 1981 . . . .










Table 1-1.

Table 3-1.

Table 3-2.

Table 3-3.

Table 4-1.

Table 5-1.

Table 5-2.

Table 5-3.


Spatial and Temporal Resolution in Satellites . 3

Data Acquisition System Identification . . .. 46

Sensor Identification . . . . . . . . 49

Variable Names and Units for Half-Hourly Reports . 57

Evapotranspiration Formulae for Average TGR Method . 74

Example Calculations with the Strict TGR Method . 96

Comparison of Average and Correlation Estimated
A and B . . . . . . . .... ... . 116

Quality of ET Estimates made with the ATGR Method .119

Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy



Klaus Heimburg

December 1982

Chairman: Wayne C. Huber
Major Department: Environmental Engineering Sciences

A generalized physical method is developed for making evapotran-

spiration (ET) estimates based on directly measured air temperature and

remotely sensed surface temperature and net radiation data. The method

is based on the correlation of surface-to-air temperature gradients and

varying net radiation loads; the slope and intercept of this correla-

tion are shown to be composite values of two groups of surface parame-

ters. Five equations are developed to calculate ET from these composite

values plus net radiation and some combination of two of the four sur-

face parameters (bulk air transport, moisture availability, saturation

deficit, and soil heat flux).

The method is validated using ET measurements made over a pasture

surface using the energy budget/profile Bowen ratio technique. An auto-

matic measurement system consisting of a computer-controlled data ac-

quisition system and air sampling arrangement, time-constant-matched

humidity, temperature, and radiation sensors, and four interacting

programs was developed to measure and calculate half-hour average sur-

face energy budgets and statistics. Data from 42 days in the spring and

fall of 1981 are reported.

It was found that the radiation surface temperature is in general

not the same as the effective heat transport surface temperature--it

may be necessary to correct remote surface temperature measurements

before using them with conventionally evaluated heat transport coeffi-

cients. Because parameters are assumed constant, instantaneous ET esti-

mates made with the developed method are at times systematically high

or low, but these errors tend to cancel in cumulative estimates.

The method is shown to be well-suited for use with 1- to 3-hour

time resolution satellite data. In effect it evaluates surface parame-

ters such as moisture availability, requires no interpolation for ET

.estimates between data sets, is adapted to the inevitable cloud-caused

loss of satellite surface temperature data, and reduces calculation of

cumulative ET to estimating total positive net radiation and duration

of positive net radiation in a particular estimation period. The meth-

od's ET estimates are shown to be as accurate as the state-of-the-art

simple residual method, which does not have these advantages.



Potential for Remote Evapotranspiration Estimates

The loss of water from the earth's surface by either evaporation

from soil and plant surfaces or transpiration by plants is called evapo-

transpiration (ET). Along with rainfall and runoff, it plays a very sig-

nificant role in determining the availability of water at the earth's

surface and the recharge to deep aquifers. Because water is critically

important to man's existence, ET estimation methods are important in

solving problems of water supply.

Water supply problems in relatively dry areas have long included

the estimation of crop water requirements, evaporation from reservoirs,

and evapotranspiration over aquifer recharge areas. As population has

grown, the demand for water has increased and interest in estimation

methods has become more widespread. Today, there is a growing need for

evapotranspiration estimates even in relatively wet areas, such as


Present methods of measuring and estimating ET are diverse, depend-

ing upon the specific purposes of the estimates and available data. On

the one hand are physically-based measurement techniques developed by

scientists. They provide accurate instantaneous ET rates for a specific

location, but require continuous measurements of such variables as air

temperature and vapor pressure, net radiation, and soil heat flux. Exam-

ples of these techniques are the eddy flux correlation, energy

budget/profile Bowen ratio and Penman methods (American Society of Agri-

cultural Engineers, 1966; Brutsaert, 1982).

On the other hand, water use planners and water supply engineers

have developed methods which produce daily to monthly estimates for

larger areas. In locations where such records are kept, these methods

are based on climatologic data. They are generally founded on some phys-

ical correlation, but all involve empirical adjusting factors for vege-

tation type, air humidity, altitude and the like. Examples are the

Blaney-Criddle method, the radiation method, the Penman method, and the

pan evaporation methods (Doorenbos and Pruitt, 1977).

The weather stations which provide the base information for these

methods are widely scattered. On the average, each station in the United

States represents an area on the order of 100 mi square (Price, 1982).

Regional estimates of evapotranspiration are thus difficult to make and

of dubious accuracy. They are limited by insufficient data on highly

variable surface parameters such as soil moisture conditions and vegeta-

tion types.

By comparison to the weather station network, today's satellites

return remotely sensed information about the earth's surface with an

unprecedented level of detail. The surface area element or pixel sizes

and the time intervals between coverage of some of the satellites appro-

priate to regional scale studies are shown in Table 1-1. As a result of

the availability of this type of data and modern high-speed computers,

the potential exists to systematically monitor evapotranspiration on a

regional scale.

Development of this potential could benefit a variety of research.

areas. If remote-sensing methods are also developed to estimate rainfall

Table 1-1. Spatial and Temporal Resolution in Satellites

Satellite Orbit Pixel Time
Acronym Type Size Intervals

Landsat polar 80 x 80 m 18 da
HCMM polar .6 x .6 km 12 hr each 5 da
TIROS polar 1 x 1 km 12 hr.
GOES geostationary 8 x 4 km 30 min

on a regional basis and if streamflow is gaged, aquifer recharge over

wide areas can be estimated (Allen et al., 1980). The information on

surface energy fluxes gained by an ET estimation technique could also be

useful as boundary conditions for models of the atmosphere. It is also

possible that large-scale changes on the earth's surface such as defor-

estation and desertification could be monitored by observing longer-term

changes in ET patterns. Finally, the correlation of evapotranspiration

and yield in agronomic crops may lead to large-scale yield predictions

(Doorenbos and Pruitt, 1977; Chang, 1968).

The purpose of this research is to develop and test a generally

applicable method for estimating evapotranspiration based as much as

possible on remotely sensed data. Since it is ultimately intended for

use with satellite data from large diverse areas, criteria for this

method include that it be strictly physical, relatively easy to apply,

and compatible with the format and limitations of satellite data. The

research is also intended to identify factors critical to the accuracy

of the estimates which require more research, and factors which may im-

prove future satellite measurement for use in ET estimation.

Scope of the Research

Data returned from a satellite consist of the energy flux in a par-

ticular band of wavelengths coming from a particular surface area

element at a particular time. For environmental applications, the elec-

tromagnetic spectrum is usually resolved into visible and thermal bands.

With a clear sky and proper consideration of atmospheric transmission

properties, these measurements can be used to calculate the surface tem-

perature and the net radiation absorbed by the surface.

Net radiation and surface temperature estimates should lead to good

evapotranspiration estimates because they are very prominent variables

in the heat exchange processes that take place at the earth's surface.

Net radiation is the primary energy source used in changing water from

liquid to vapor at the surface, while surface temperature--because it is

a result of surface variables and energy exchange processes--is a com-

posite measurement of the effects of these variables.

However, it is a long step from measurements of net radiation and

surface temperature to an operational ET estimating system using satel-

lite data. The following questions illustrate the range of problems

faced in developing a method for such a system.

1. What is the best way to estimate net radiation from satellite
pixel information? How does one treat clouds or haze?

2. How is the radiation temperature of a complex surface like that
of vegetation interpreted? Does angle of view and height of
vegetation make a difference? How does one handle a canopy
underlain by a cool surface like a marsh or swamp? How does
one treat mountainous topography?

3. Is an interpolation technique required to compensate for the
temporal resolution of satellite data?

4. What level of detail is required in a practical ET estimation
method? How does one get the most acceptably accurate estimate
for the least effort in data collection and processing?

5. How are the effects of water availability, vegetation type,
cloudiness, and wind related? And how do they influence ET?
What is the minimum amount of data needed from ground-based

6. Can estimates made with area average data be of reasonable
accuracy when there are various vegetation types or net ra-
diation regimes in the same pixel?

7. Ultimately, what factor most limits the accuracy of a given
remote estimation scheme?

The work described in this dissertation addresses many of these

questions. The emphasis is on how to most efficiently account for all

the factors affecting evapotranspiration, and how to extract as much

information as possible about the surface and its environment from re-

mote data. Practical limitations such as the fact that satellite data

are available only at discrete time intervals and sometimes incomplete

because clouds prevent a surface temperature measurement are considered.

All ground-based measurements except air temperature were avoided; meth-

ods to eliminate this measurement are suggested, but their investigation

was considered beyond the scope of this research.

It is assumed that estimates of net radiation and surface tempera-

ture are, barring clouds, available at regular time intervals. The ques-

tion of complex radiation temperatures is side-stepped by considering a

relatively simple pasture grass surface. Although a parameter that in-

cludes the effect of wind on evapotranspiration is used, its functional

dependence on windspeed is not explored.

Research Approach

The overall approach to developing a remote evapotranspiration es-

timation scheme was to compare estimates made with trial methods to ac-

tual ET rates measured over a test surface. Accordingly, there were

three main areas of effort: the collection of a base of accurate ET

data, the theoretical development of an estimation method based on re-

motely sensed data, and the testing of that method against the actual ET


As suggested in the previous section, the enormous variety of ter-

rain and vegetation types present on the earth's surface introduce a

large number of complicating factors into ET estimation formulations. In

order to clearly assess the potential of a general method, as many of

these complicating factors as possible were avoided by choosing a rela-

tively homogeneous flat area of pasture as a test surface. The approach

was to develop a basic method which would work for simple surfaces; once

it is proven successful can be modified if necessary to deal with

more complex situations like mountainous terrain or swamp.

A micrometeorologic measurement technique was used to measure ac-

tual ET so that surface processes were left as undisturbed as possible.

The radiation surface temperature as well as net radiation was measured

for later use in method-testing. A great deal of effort went into devel-

oping a data collection system to assure the reliability of the ET mea-

surements. Special efforts were made to match the time constants of the

sensors involved and to reduce electrical signal noise. Control of the

measuring system, scheduling of the measurements, and calculations were

all performed by computer to minimize human error.

The fundamental assumption in method development was that satellite

data would be available in time intervals on the order of 1-3 hours.

After this assumption, the emphasis was on operational criteria--a prac-

tical method must have general applicability, computational simplicity,

and low data requirements. With these objectives in mind, an analytical

approach, rather than a simulation approach, was chosen. In order to

keep data requirements low yet take advantage of satellite data, the

level of detail was chosen to be somewhat intermediate between the

strictly physical ET measurement methods and the empirical estimation

methods. This required a set of assumptions, all of which are explicitly

identified in the derivation of the method.

The general objective of the method-testing was to validate the

general framework of the method and to assess the error contributions of

various parts of the method on instantaneous and cumulative ET esti-

mates. The assumptions made during the development of the method were

individually examined; in this way, the relative importance of ground-

gathered ancillary data such as air temperature, saturation water vapor

deficit, windspeed, and soil temperature could be judged. The testing

was done with ideally accurate on-site measurements of net radiation,

surface temperature, air temperature, and evapotranspiration.

Experimental Site and Data Collection

An area of pasture at the University of Florida's Beef Research

Unit was used as the research surface. The s-ite is loc-ated northeast of

Gainesville, Florida, as shown in Figure 1-1. It was chosen because it

is typical of northern Florida pasture areas, and was amenable to micro-

meteorologic measurement of a surface energy budget. The area was flat

with relatively uniform grass cover, and was large enough to ensure

well-developed temperature and vapor pressure profiles. The test surface

was a mixture of grasses: roughly 60-70% was bahiagrass (Paspalum

notatum), about 20-40% was smutgrass (Sporobolus poiretii), and 5-10%

was white clover (Trifolium repens).

Evapotranspiration was computed by an energy budget/profile Bowen

ratio method from measurements of net radiation, soil heat flux, and

gradients of temperature and water vapor pressure over the pasture sur-

face. A Hewlett-Packard 2100 computer and low-speed data acquisition

subsystem was used to automatically scan and measure the sensors,




0 1 2 3


0 UNIT -/

*... . "

*<~"-^ .. *.,^.'." .~. *
*** ^ .- *- **" *^
*- * ** r. r

University of
Florido a

***^ ~ ~ ...^ -- --:.-.
A. A'.
* .. ,. -. ... .... I ..,,,
7-.,.,. "" .-. '" ." 4 .. -"


,~.. ,... .. *,. A.. . J.

.-. CI.- I
,.-z. _____

Location of the University of Florida Beef Research Unit.

_ I ,

Figure 1-1.

convert the measurements to proper units, and compute averages. Average

energy budget components and temperature and vapor pressure gradients

were calculated and reported for half-hour periods.

The arrangement of sensors in the field is shown in Figure 1-2.

Aspirated thermopiles and air sampling ports were mounted on arms of a

2.5-m mast. The area within a 10-m radius of this mast was completely

unobstructed to meet the fetch requirements of the measurement method.

Radiometers were attached to the end of a guyed boom about 2 m over the

surface. The precision radiation thermometer was bolted to a camera tri-

pod atop an antenna tower 9.5 m above the grass surface; the windspeed

and direction sensors were mounted on the same tower at 7 m. A 14-m

tower served as lightning protection for the entire group of in-


Shielded buried signal cable connected the sensors- to the data ac-

quisition system which was housed in a building about 90 m away from the

sensors. For vapor pressure measurements, air samples from five separate

levels in the field were pumped continuously back to the building

through heated insulated tubing and mixing chambers to a dewpoint ana-

lyzer. Air samples were switched sequentially to this instrument by a

scanning valve controlled by the measuring computer program. The dew-

point was measured after a half-minute delay to allow time for the ana-

lyzer to settle on the dewpoint of the sample from the new level.

Altogether, ET data from 42 days were used in verifying the remote

ET estimation method developed. These data were collected in the spring

and fall of 1981.













Figure 1-2. Field Apparatus and Sensor Locations. This diagram is not
to scale.

Organization of Dissertation

Basic concepts underlying current understanding of the evapotran-

spiration process are reviewed in Chapter 2. These concepts are funda-

mental to both the evapotranspiration measurement and method develop-

ment portions of the study. Chapter 3 describes the computer-based evap-

otranspiration measurement system that was developed to collect a base

of accurate ET data. This chapter contains the theory of the meas-

urement technique, considerations made in designing the profile sensing

systems, brief descriptions of the programs that operate the system, and

an assessment of the strengths and weaknesses of the measurement sys-

tem. Two methods of calculating ET based on remotely sensed data are

derived in Chapter 4, one relatively rigorous with a minimum of added

assumptions, and a grosser less detailed one with extensive approxima-

tions. Both methods are based on a temperature gradient model which uses

net radiation and surface temperature data to determine surface parame-

ters. The performance of this model and these methods is compared to

actual ET measurements in Chapter 5. The method most suitable for use

with satellite data is tested component by component to clearly evaluate

its strengths and weaknesses. A summary of conclusions and suggestions

for further research are contained in Chapter 6.

Repeatedly used symbols are defined in Appendix A. (All symbols are

defined in the text where they are introduced.) Appendix B is a listing

of the programs developed for the automatic ET measurement system, along

with definitions of names for subroutines, functions, data arrays, and

indexes. Appendix C is a summary listing of the data collected, and sup-

plementary figures are presented in Appendix D.




The availability of satellite images of the earth's surface and the

resources to investigate their usefulness has resulted in a variety of

remote-sensing research projects. In recent years, there have been pro-

grams in which evapotranspiration estimation procedures were the objec-

tive, notably a joint effort among the National Aeronautics and Space

Administration (NASA), the Institute of Food and Agricultural Sciences

at the University of Florida, the Florida Water Management Districts

(Allen et al., 1980), and NASA's Heat Capacity Mapping Mission, or HCMM

(Goddard Space Flight Center, 1978).

Since the estimation techniques need to be applicable to many dif-

ferent surfaces, physical rather than empirical approaches are.required.

The physical methods that have been developed, including the one pre-

sented in this work, are all based on the energy budget concept of the

surface and on the similarity of transport among quantities in turbulent

flow. These ideas and various approaches to solving the energy budget

equation are developed in the first part of this chapter. Remote ET es-

timation techniques are reviewed in the second part, which concludes

with an introduction to the new method.

The Evapotranspiration Process

At the interface between a liquid and a gas, molecules are continu-

ally breaking and reforming the intermolecular bonds which hold them at

the surface as a liquid. The energy of the random molecular collisions

which cause the bonds to break is carried with the freed molecule; this

thermal (heat) energy is lost by either liquid or gas molecules near the

interface. Since this energy contributes only to the molecule's conver-

sion to the vapor state and not its temperature, it is called the latent

heat of vaporization. It is released to the molecules at the surface

should a free molecule collide with and be captured by molecules in the

liquid state.

When the concentration of vapor molecules is higher at the surface

than at some distance away from it, there is a net flow of molecules and

energy (in the form of latent heat) away from the surface. This process

is evaporation.

Evapotranspiration is the evaporation of water from soil or plant

surfaces together with transpiration by plants. In transpiration, water

evaporates from internal plant surfaces and diffuses into the air around

the plant through openings in the leaves stomataa). Like the process of

evaporation, evapotranspiration consists of three fundamental elements:

the absorption of thermal energy at a water-air interface, the change of

state of water from liquid to vapor, and the resulting net loss of vapor

molecules and their heat of vaporization from the surface due to a fa-

vorable vapor concentration gradient.

The heat energy consumed in the evapotranspiration process is lost

from the vegetation biomass. Therefore, all the energy fluxes to and

from the plant canopy and the factors influencing them play a part in

determining the evapotranspiration rate. Figure 2-1 is a simplified dia-

gram of the surface and its primary energy and water fluxes. It is pre-

sented in the diagramming language of Odum (1982), and embodies many of


Figure 2-1. System Diagram of Generalized Evapotranspiring Surface.
Symbols are from Odum (1982).


the concepts and simplifications conventionally applied in evapotranspi-

ration theory.

The heat energy stored in the plant canopy is represented by its

temperature (Ts). The bulk of this energy comes into the vegetation in

the form of direct or scattered solar short wavelength (0.3 to 3 um)

radiation (Qs); it also intercepts thermal or long wavelength (3 to

50 pm) radiation emitted by the atmosphere (Qa). A substantial fraction

of the shortwave radiation received by the surface is reflected (Qr), a

very small part is used to drive photosynthetic reactions in the plants,

and the remainder becomes heat energy absorbed and stored temporarily in

the biomass. Part of this energy is reradiated to the atmosphere (Q ).

The difference between the downwelling radiation (direct and atmo-

spheric) and the upwelling radiation (reflected and emitted) is referred

to as net radiation (R).

Besides these radiant energy fluxes, the vegetation exchanges en-

ergy with its environment in several other ways. Thermal energy ex-

changed with the air by the process of molecular conduction and turbu-

lent diffusion is referred to as sensible heat flux (H); energy ex-

changed with the soil is soil heat flux (G). Energy used in the change

of state from water to water vapor is transported with water vapor and

is referred to as latent heat flux (E).

In this generalized view of the surface system, the plant canopy is

considered to have a uniform temperature representative of its heat con-

tent. There are complex energy exchange processes that occur within the

canopy because of differences in temperature. For example, radiation is

exchanged between plant surfaces, and sensible heat released from one

leaf may be reabsorbed and released from another as latent heat.

However, when the purpose is to make total evapotranspiration estimates,

these exchanges are ignored, and only the energy fluxes entering or

leaving its boundaries are considered.

Besides the radiant energy pathways (R) and heat energy stored in

the plant canopy, Fig. 2-1 shows the dependence of the surface energy

balance, and thus evapotranspiration, on factors in the environment of

the surface. Heat that is lost to (or gained from) the air as sensible

heat is not (is) available for evapotranspiration. This flux is depen-

dent on the air temperature (Ta) and the thermal transport properties of

the air, represented by the eddy thermal diffusivity (KH) in the figure.

When the air temperature is cooler than the surface temperature of the

canopy, sensible heat moves from the canopy into the air. When the can-

opy is cooler than the air, it absorbs heat energy from the air. There

is an analogous heat flux pathway to the soil, dependent on soil temper-

ature (T ) relative to the canopy temperature (T ), and the thermal con-

ductivity of the soil (X).

The right half of Fig. 2-1 shows the pathway of water through the

surface system. It originates in the soil and moves through plant tis-

sues into the leaves, where it evaporates. Depending on the vapor pres-

sure inside the leaves (ei), the vapor pressure in the surface air layer

outside the leaf (es), and the stomatal conductivity (C ), water vapor

then diffuses through stomata into the air around the leaves. From the

surface layer water vapor diffuses into the air, depending on the rela-

tive vapor pressures of the surface layer and air (es and ea) and the

eddy water vapor diffusivity (KW).

The Energy Balance Approach to ET Estimation

The Energy Budget Equation

The three elements of evapotranspiration (the absorption of water

from the soil or plant surfaces, the absorption of thermal energy from

the plant canopy, and the flux of water vapor through the air over the

surface) provide at least three fundamental approaches to evapotranspi-

ration measurement. These have been referred to as the water budget,

energy budget, and aerodynamic approach, respectively. All previously

developed remote ET estimation methods, the remote technique developed

in this study, and the ground truth measurement technique used in this

study are founded on the energy budget equation.

The energy balance of a vegetation and air layer can be written

R E -'H G P S = 0 2-1

where R is the net radiation flux absorbed (from p.15, R Q +Q -Q -Q )
E is the latent heat flux, a r
H is the sensible heat flux,
G is the soil 'heat flux,
P is the photosynthetic heat flux, and
S is the time rate of heat flux storage in the vegetation/air

Here energy "flux" is used to describe energy "flux density," i.e., the

energy flow per unit time through a unit area. All terms are in these


Because of inherent measurement difficulty and sensor limitations,

the energy budget components can only be measured to within about 10% of

their actual values (Sinclair et al., 1975). Since some of the smaller

components are actually indistinguishable from measurement error, they

need not be considered.


Usually the smallest component is photosynthetic heat flux. It can

be considered negligible because only 1 to 5% of the net radiation im-

pinging on vegetation is absorbed in this way (Allen et al., 1964).

It can be shown by a "worst case" calculation that the storage term

is also in the negligible range. Heat in the vegetation/air layer can be

stored as sensible heat in the air, latent heat in the air, sensible

heat in the biomass, and sensible heat in the litter surface layer. In a

strict sense, these are evaluated as follows:

a1 Ta(Z) 1 CaT ea(z)
S = c() dz + -- dz +
Sa t J at
1 aTb(z) Id T-(z)
Sb(z) dz + cd ()T z dz
b abt dz 9 at

where ca(z), cb(Z), and g (z) are volumetric heat capacities ( c) of
canopy air, plant biomass, and surface soil, respectively,
Ta(z), Tb(z) and Tg(z) are the temperatures of canopy air,
canopy biomass, and surface soil, respectively,
ea(z) is the vapor pressure of canopy air,
y is the psychrometric constant,
1 is the vegetation height,
d is the depth of the surface litter layer, and
z is the vertical space coordinate.

Using averages for the spatial variables, Eq. 2-2 can be rewritten:

AT (pCp ) ae
S = (pCp) h a + -- a h a +
Sa At y At
(Vc) b + (p) d
b At g At
where a, b, and g are subscripts referring to air, biomass, and soil
specific heats, densities, and temperatures, and
V is the mass of vegetation per unit area.

total sensible heat latent heat heat stored heat stored
heat = stored in + stored in + in + in top
storage canopy air canopy air vegetation soil layer

S= (P ) h ATa + (p)h a + (Vc) b + () d
pa pa t + (VC)b t +16gd Tf

S = .0012 x 0.24 x 20 cm x24 C + .0012 -g x 0.24 ca x 20 cm x 1.5 x 3mb +
cm cm

10,000 x 1.0 al 24 + 1.5 x 0.2 alx 1 cm x
ha gC 4 hr 3

S = .0006 al + .0004 cal + .01 al + .006 cal
cm min cm min cm min cm min

S .02 cal
cm min

Figure 2-2. Rough Calculation of Heat Storage in Pasture Canopy.

This expression is evaluated in Fig. 2-2 using values typical for

pasture grass. Heat storage in pasture biomass and the top litter layer

is approximately two orders of magnitude less than peak net radiation

loads; latent and sensible heat storage in the canopy air is about three

orders of magnitude less.

Since the values of the photosynthetic heat flux and the time rate

of canopy heat storage are negligibly small, the energy budget equation

may be written

R E H G = 0 2-4

The ET measurement method and the remote estimation method are based on

this simplified form of the equation. It is also the basis for all but

the empirical remote-sensing ET estimation methods. The following sub-

sections briefly review the fundamental analytical concepts and evalua-

tion techniques which are common to previously developed evapotranspira-

tion estimation methods based on the energy budget equation.

Transport Similarity and Wind Models

After the surface energy balance, the most important concept to ET

estimation techniques is that of transport similarity among momentum,

heat, and mass fluxes in the turbulent layer near the surface. This idea

is used in all forms of the energy budget approach to ET estimation,

both to evaluate transport properties and to avoid evaluating transport


The fundamental equations for the one-dimensional transport of mo-

mentum, heat, and water vapor are (Eagleson, 1970)

= pK 2au
T = pK t_. 2-5

H = pc KHDT 2-6
p H 3z

E = cLe K 2-7
p W a2

where T is momentum flux,
H is sensible heat flux,
E is latent heat flux,
KM, K, KW are the eddy diffusivities of momentum, heat, and water
u is the average horizontal windspeed,
T is temperature,
e is vapor pressure,
p is the air density,
c is the air specific heat at constant pressure,
Lp is the latent heat of evaporation,
P is the atmospheric pressure, and
e is the ratio of molecular weights of water and dry air.

The similarity hypothesis, which was developed in the last half of the

nineteenth and early twentieth centuries (reviewed by Brutsaert, 1982),

proposes that the eddy diffusivities of momentum, heat, and water vapor

are all the same:

KM = KH=K 2-8

It was not until Prandtl's (1932) development of the mixing length

concept that general analytical treatment of eddy diffusivity began.

According to mixing length theory, it is argued that

2 du
KM(z) = d 2-9

where a is the mixing length and
Ui is the average windspeed perpendicular to z (horizontal).

By postulating that the mixing length increased with distance from a

surface (k = KZ, where K is the von Karman constant), Prandtl went on to

derive an expression that accurately describes the variation of wind-

speed near a surface, the simple log wind profile. With parameters for

displacement height (D--with dense vegetation, that height above the

surface where the windspeed vanishes) and roughness height (z0--a

parameter included so that the windspeed is defined as zero when

z D = 0), the equation for the log profile can be written
ST z D + z2
u(z) p In 2-10

where TO is the shear stress at the surface. With this wind profile, the

eddy diffusivity can be evaluated between the surface and any level in

the air with average windspeed ua:

K u (z D + z
KM(z) z D + z0 2-11

With the assumption of transport similarity (Eq. 2-8), this expression

can be used to evaluate KH and KW. Similar treatments of eddy diffusiv-

ity can be found in many texts (e.g., Brutsaert, 1982).

With very precise experimental work it has been determined that the

turbulent transport of momentum, heat, and water vapor is strictly simi-

lar only under neutral stability conditions, e.g., Swinbank and Dyer

(1967). To describe eddy diffusivities under other conditions, diabetic

influence functions (tM' ,H' pW) have been developed. They are defined

such that

KH = KM and 2-12

KW KM 2-13

These are experimentally determined and expressed in terms of dimension-

less variables such as the Monin-Obukhov length or Richardson number

(Morgan et al., 1971; Businger, 1973).

A number of wind profiles and corresponding eddy diffusivity treat-

ments both with and without stability corrections have been developed.

(These are referred to as wind models.) None are used in this study, but

the fact that bulk air transport is theoretically and experimentally

adequately understood is important in supporting the remote-sensing

method developed. All remote-sensing methods involve a wind model of

some kind to help evaluate sensible and latent heat fluxes.

Latent and Sensible Heat Flux Expressions

In application, the flux between two specific points (z1 and z2)

that have a gradient between them must be evaluated. Since eddy diffu-

sivity in general varies with the distance from a surface (Eq. 2-11),

the latent and sensible heat transport equations (Eqs. 2-5, 2-6 and 2-7)

must be integrated along the direction of transport and between the

points of application (Monteith, 1973). Assuming that all parameters

except diffusivity are constant between the two levels and that the flux

in question is steady (or that flux storage in the layer between levels

is negligible),
PC (T1 T)
H = 1- 2 and 2-14
f z2 dz
z K

pLE (el e2
E p Jz2 dz 2-15

The integral in the denominator of these equations, when evaluated, rep-

resents the lumped transport properties between points z1 and z2 away

from the surface. From the preceding subsection, it is understood that

these integrals can be evaluated with various wind models for K (z).

The expressions for latent and sensible heat flux that are commonly

used are simplified versions of Eqs. 2-14 and 2-15. For sensible heat

flux from the surface to a reference level above the surface, the inte-

gral expression is abbreviated either as a bulk thermal conductivity or

as a bulk air resistance:
PCp(T Ta)
H = pCpK(T T ) p ,r 2-16
p s a r a

where T is the surface temperature,
Ta is the air temperature at a reference level above the surface,
K is the bulk thermal conductivity for the slab of air between
the surface and the reference level, and
r is the bulk resistance to heat transport of the slab of air
between the surface and the reference level.

In this study, the sensible heat flux expression is condensed even fur-

ther to

H = h(T Ta) 2-17

Where h is referred to as the bulk heat transport coefficient. Since the

fundamental definition of h is

h =-- 2-18
za dz

use of a wind model (to evaluate KH) is implied any time the bulk heat

transport coefficient or bulk air resistance is used (Monteith, 1973,

1975; Thom and Oliver, 1977).

Applying the similarity concept to a description of latent heat

flux is complicated because it is impossible to measure the vapor pres-

sure at the vegetation surface. The air inside the leaves is usually

assumed to be at the saturation vapor pressure corresponding to the sur-
face temperature [es = e (T)] A unitless parameter M, which varies
face temperature [e = e (T )]. A unitless parameter M, which varies

from 0 to 1, can be introduced to account for subsaturation of the sur-

face air:

M(e e,) = es ea 2-19

This formulation was suggested by Tanner and Pelton (1960) and applied

by Outcalt (1972), Pandolfo and Jacobs (1973), Nappo (1975), and Carlson

and Boland (1978), and in a slightly different form by Barton (1979).

The equation for latent heat flux can then be written in terms of the

heat transport coefficient and moisture availability parameter:
h *
E = h M(es ea) 2-20
y s a
where es is the saturation vapor pressure at the surface temperature,
e is the vapor pressure at the reference level a,
Ma is a unitless parameter interpreted as moisture availability,
h is the bulk heat transport coefficient, and
y is the psychrometric.constant (y = c P/Le).

The resistance formulation (Monteith, 1973) includes an additional re-

sistance term, rs, the bulk stomatal diffusion resistance (sometimes

referred to as the canopy resistance, r ) to account for the subsatura-

tion of air at the surface:

PCp (es e )
E = p (es a 2-21
Y(ra + r s)

Both of the transport coefficient resistance formulations are used

in the ET literature; analytic evaluation of either type of expression

is based on diffusivity integrals like those in Eqs. 2-14 and 2-15.

These formulations can be substituted for one another with the following

h -
r, and 2-22

M = 2-23
r +- r
a s
This study uses the conductivity formulation.

Energy Budget ET Estimation Strategies

There are two major ways in which the energy budget and gradient

equations can be solved. The physically more realistic method is based

on dynamic simulation of the heat transfer processes; the other method

is based on a cruder description of the surface and steady-state analy-

sis of the surface heat exchange processes.

Gradient expressions like those in Eqs. 2-5, 2-6 and 2-7 are used

in both approaches. The difference is that in simulations the expres-

sions are applied over arbitrarily short distances and time steps ac-

cording to the level of detail required in the application. When trans-

port is in one direction, as it is considered to be in most of the prob-

lems encountered in ET measurement or prediction, the medium through

which the flux is transported is thought of as consisting of layers per-

' pendicular to the direction of transport. Fluxes through each layer can

then be computed individually for each time step, allowing the treatment

of flux transients as well as the treatment of differing transport prop-

erties of the layers. In the steady-state approaches the gradient ex-

pressions are applied over the entire distance between measurements, and

transients are ignored.

Simulation models consist of an interdependent system of equations

which describe the exchange of latent, sensible, and soil heat flux with

the vegetation layer and the air or soil, and also the transport of la-

tent, sensible and soil heat between layers. This system of equations is

solved iteratively using solar and atmospheric radiation data as a forc-

ing function and quantities such as air temperature, vapor pressure, and

soil temperature as boundary conditions. Generally, unknown surface pa-

rameters are chosen such that simulated surface temperatures match

observed surface temperatures. The simulated ET flux is then assumed to

be the actual ET flux. Examples of evapotranspiration simulations are

Waggoner et al. (1969), Stewart and Lemon (1969), Sinclair et al.

(1971), Murphy and Knoerr (1970, 1972), Goudriaan and Waggoner (1972),

Lemon et al. (1973), and Sinclair et al. (1976). Dynamic.models of the

surface heat transfer processes are computationally orders of magnitude

more complex than the steady-state approaches, and were developed only

after the introduction of the electronic computer.

The earliest physical models of the surface energy exchange process

were based on steady-state analysis and the similarity of latent and

sensible heat transport. Three steady-state strategies for solving the

energy budget equation for evapotranspiration have been developed; they

are referred to as the simple residual, Penman, and Bowen ratio methods.

To more easily compare these methods, their equations have been written

in the same notation. Soil heat flux is included even though this compo-

nent is often assumed too small to be included for vegetated surfaces.

In the residual approach, the energy budget equation is solved for

latent heat flux, and a simple gradient expression is used to evaluate

sensible heat flux:

E = (R G) h(Ts Ta) .2-24

The transport coefficient for air conductivity is estimated from empiri-

cally derived wind functions or physical wind models as described previ-

ously. The biggest advantage of this method is that it requires no in-

formation on the surface moisture status. Its disadvantage is that it is

very sensitive to an accurate transport coefficient estimate. When the

sensible heat flux term is written in terms of a resistance, this method

is also called the resistance energy balance method (Rosenberg, 1974)

The Penman (1948) approach is very closely related to the residual

approach. In addition to the wind function, it includes an expression

that relates the temperature gradient to the vapor pressure gradient via

the linearized saturation vapor pressure curve,

es ea = s(T Ta) + 6ea. 2-25

where s is the slope of the saturation vapor pressure curve, and
6ea is the saturation deficit of the air.

This approach has since been generalized to include subsaturated sur-

faces (Barton, 1978), which allows ET to be expressed as a function of

net radiation, the moisture availability parameter (M), and the satura-

tion deficit (6ea):

E = Ms+ [s(R G) + h6ea] 2-26

.(See Chapter 4 for the full derivation.) Historically, Penman's method

was the first to combine the energy budget equation with a wind model to

evaluate ET. Although the residual approach also employs a wind model,

in common usage it is the Penman method that is referred to as the com-

bination method. The Penman method's main advantage is that it is not

explicitly dependent on measurement of a temperature gradient; its prin-

cipal disadvantage is that it requires information on moisture availa-

bility of the surface.

The Bowen ratio approach (Bowen, 1926) assumes that in the fully

turbulent layer over the surface, transport of heat and water vapor are

similar (i.e., KH = K). This allows eddy diffusivities to be avoided

altogether, and latent and sensible heat flux to be apportioned accord-

ing to the relative strength of the temperature and water vapor pressure


(R G)
E = 2-27
E y(T2 TI
(e2 e1)

where subscripts 1 and 2 refer to two levels in the fully turbulent air

layer. This approach is free of a wind model, but it requires very accu-

rate measurement of temperature and vapor pressure gradients. It is dis-

cussed in detail in Chapter 3.

Remote ET Estimation Methods

Surface Temperature and Net Radiation

Satellite-borne sensors can measure the amount of radiant energy

coming from a particular surface area element in a particular wavelength

interval. For environmental applications, the wavelength intervals mea-

sured are divided into the visible, thermal, and microwave regions of

the electromagnetic spectrum, yielding measurements of reflected solar,

emitted thermal, and microwave radiation. .So far, all ET estimation

methods designed for use with satellite data only employ the visible and

thermal wavelength ranges.

Net radiation is the largest component of the surface energy bud-

get, and surface temperature plays a role in determining all the energy

budget components. Usually, measurements of reflected solar and emitted

thermal radiation measurements are used to estimate net radiation and

surface temperature. Methods to estimate ET are then based on these net

radiation and surface temperature estimates.

With a clear sky and proper consideration of the atmosphere's

transmission properties, surface temperature can be determined directly

from emitted thermal radiation:

Qe = eT 2-28
e s

where e is the emissivity of the surface,
o is the Stefan-Boltzmann constant, and
T is the surface temperature.

Solving for Ts,
Ts = e 2-29
Y o
In principle, net radiation is calculated according to the equation

R = Qs + Q r Qr Qe 2-30

The upwelling components, reflected (Qr) and emitted (Q ) radiation, are

directly measurable by satellite given atmospheric transmission proper-

ties. The solar radiation incident at the surface (Qs) is known as a

function of date, time of day, location, and atmospheric absorption

(Tennessee Valley Authority, 1972). Atmospheric radiation (Qa) can be

similarly estimated.

Some of the ET estimation methods discussed in.-the following sec-

tions are designed for use with satellites that provide only thermal

data from the surface. These methods express the net shortwave radiation

as a function of estimated incident solar radiation (Rs) and albedo (a):

Qs Qr = (1 a)Qs 2-31
Simulation Methods

In 1978, NASA launched the Heat Capacity Mapping Mission (HCMM).

The polar orbit of the HCMM satellite was designed to collect maximum

and minimum temperatures of the earth's surface, and groups worldwide

were funded to study the maximum-minimum temperature data. Several

groups adapted or developed simulation methods to bridge the long time

intervals (12 hours) between data sets. Examples of models used are

Carlson and Boland (1978), Soer (1977), and Rosema et al. (1978).

The Carlson model is very general, having been developed for study

of urban and rural surfaces. It is based on the energy budget equation

and gradient transport equations for latent, sensible, and soil heat

flux. Soil thermal conductivity and heat capacitance are combined into a

thermal inertia parameter which is evaluated with an empirical relation-

ship to thermal conductivity. The model does not describe soil and plant

water transport. It introduces a moisture availability parameter as

shown in Eq. 2-19 to account for the subsaturation of the surface air.

Eddy diffusivities for latent and sensible heat are iteratively computed

using empirical stability corrections; there are, in fact, different

atmospheric models for daytime and nighttime.

Use of the Carlson model to determine daily heat budget components

is discussed in Carlson et al. (1980). Computed solar radiation is used

to force the model; measured windspeed, air temperature and.humidity,

and soil temperature are used as boundary conditions. By varying two

model parameters (thermal inertia and moisture availability) on succes-

sive model runs, sets of corresponding cumulative heat budget components

and 24-hour maximum and minimum temperatures are generated. Then a re-

gression equation expressing daily ET as a function of maximum and mini-

mum temperatures is developed. Given the ground-measured data for the

simulation and two extreme temperature maps from the HCMM satellite, a

map of daily ET is produced.

The Soer model (named TERGRA) is much the same as the Carlson

model, providing for stability conditions in the surface air layer and

requiring temperature, vapor pressure, and windspeed as boundary condi-

tions at a reference level. However, rather than a moisture availability

parameter, soil and plant water transport is modelled in detail. (The

TERGRA model was designed for grasslands, making this more detailed ap-

proach feasible.) It uses pseudo-empirical expressions for soil water

transport resistance and stomatal resistance, and requires a reference

soil moisture pressure as well as a soil temperature as a boundary con-


Use of the TERGRA model in obtaining cumulative ET estimates is

explained in Soer (1980). The procedure requires data on the boundary

conditions and radiation falling on the surface for the duration of the

simulation periods, and values of various parameters like soil hydraulic

conductivity and surface roughness. First, windspeed, roughness height,

air temperature, and remotely measured surface temperature are used to

compute the instantaneous ET rate for the time at which satellite data

are available. This is done with the simple residual method (see previ-

pus subsection), which requires no knowledge of surface moi-sture. Then

the TERGRA model is run with various soil moisture pressures to match

the ET rate at the time of the satellite overflight. The modelled cumu-

lative daily ET rates are then assigned to areas with matching instanta-

neous ET rates at the time of the overflight.

The Rosema et al. (1978) model (named TELL-US) is also constructed

around the surface energy budget, and similarly computes latent, sensi-

ble, and soil fluxes based on measured gradients and calculated trans-

port properties. It is more detailed in describing the surface; surface

slope and slope direction must be specified. Its parameters are soil

thermal inertia and surface relative humidity.

Given the daily course of boundary conditions and incident radia-

tion, the model is used to compute daily maximum and minimum tempera-

tures and cumulative daily evapotranspiration for various combinations

of thermal inertia and surface relative humidity. This procedure must be

repeated for each combination of surface roughness, slope, and slope

direction. Then satellite-measured maximum and minimum temperatures for

specific areas are matched to the modelled values to determine daily ET

for those areas.

Steady-State Methods

Most efforts to use remote-sensing data to estimate ET rates were

made with the simple residual method (Eq. 2-24). Remotely sensed data

were used to estimate net radiation, and sensible heat flux from the sur-

face was evaluated with a remotely-sensed surface temperature and a

ground-measured air temperature. Evapotranspiration was then calculated

as the net radiation less the estimated sensible and soil heat flux.

Studies that fall into this category are Allen et al. (1980), Seguin

(1980), Soer (1980), and Price (1982). Soer and Price extend their meth-

ods to cumulative daily ET estimates with the help of simulation models

described in the preceding subsection.

These methods differ primarily in how they treat the bulk heat

transport coefficient or transport resistance of the surface air layer.

Two methods of computing sensible heat flux were used in the Allen et

al. (1980) approach. For short vegetation (mostly pasture), a stability-

corrected thermal conductivity was computed using the log law wind model

and dimensionless empirical relationships developed by a group at the

University of California at Davis (Morgan et al., 1971). An empirical

resistance equation based on leaf length, windspeed, shelter factor, and

leaf area index (Monteith, 1965) was used for transport over areas

covered with trees. By using measured windspeed and air temperature, the

estimated tree resistances and a surface temperature map, it was

possible to construct a map of instantaneous evapotranspiration rates.

For regional estimates, the rates computed for subareas were weighted by

the total area with that particular ET rate and summed.

The Seguin (1980) approach to thermal conductivity in the surface

air layer was formulated in terms of a resistance. It used the simple

log law wind function with surface roughness to evaluate the resistance

to sensible heat flux; no stability corrections were made. Measured

windspeed, air and soil temperature, remotely measured surface tempera-

ture, and estimated albedo and soil conductivity were required to esti-

mate instantaneous ET rates. Regional ET rates were estimated by multi-

plying areas with different surface temperature and surface roughness

combinations by their individual ET rates.

Soer (1980) also used a resistance formulation of the sensible heat

flux. It included stability corrections based on the Monin-Obukhov

length and the Businger-Dyer semi-empirical mass and heat transport

equations. In other particulars it is practically identical to the

Seguin approach.

Price (1977, 1980) has developed the energy budget equation in

terms of time averages in an effort to determine surface thermal inertia

using remotely sensed maximum and minimum surface temperatures. He has

since (Price, 1982) used this approach in conjunction with the TELL-US

model to estimate daily ET rates. First a preliminary estimate is made

with a residual equation like Eq. 2-24, except that time average air and

surface temperatures and windspeed are used. The daily ET value obtained

is then corrected with a regression equation developed from a set of

corresponding Price method estimates and TELL-US simulation estimates.

A different approach to solving the residual equation was taken by

Menenti (1980). In his approach, the simple residual equation is simpli-

fied by Taylor series expansion around some central ET rate at a given

shortwave radiation level. All terms except those containing surface

temperature and albedo are eliminated, leaving the ET rate for a partic-

ular surface a function of the central ET rate, its surface temperature,

and its albedo. No means to make cumulative daily ET estimates were sug-


Temperature Gradient Response Methods

The two ET estimation methods developed in this study are steady-

state methods. They are based on the response of surface-to-air temper-

ature gradients to varying levels of net radiation. One of these meth-

ods, the average temperature gradient response method, is suitable for

use with satellite data.

The primary difference between this method and the simple residual

method is that it expresses the vapor pressure gradient in terms of the

temperature gradient, the slope of the saturation vapor pressure curve,

and saturation deficit--an innovation first made in Penman's (1948) pio-

neering work. This addition gives the method some protection against

"residual errors." For example, if the measured temperature gradient is

erroneously high, both the latent and sensible heat fluxes will be af-

fected; there will not just be an increase in sensible heat and an equal

decrease in latent heat flux. Also, the method allows ET to be expressed

as a function of net radiation and parameters only (without explicit

mention of surface and air temperature). This feature makes ET calcula-

ble when surface temperatures cannot be measured remotely but net

radiation can be estimated, as when there is cloud cover or in between

sets of satellite data.

A significant advantage of the estimation method developed is that

it, in effect, determines surface parameters like moisture conditions

almost completely from remote-sensing data. This is donewith an equa-

tion (hereafter referred to as the temperature gradient model) that re-

lates the surface-to-air temperature gradient to net radiation and pa-

rameters that describe the surface. By assuming that the parameters are

constant, two of them (e.g., moisture availability and saturation defi-

cit) can be determined from the correlation of the surface-to-air tem-

perature gradient and net radiation. Although surface temperatures are

required (implying clear skies) to determine parameters, they can be

used with cloudy condition net radiation estimates for cloudy condition

ET estimates.

The need for a.surface-to-air temperature net radiation correlation

calculation requires several daytime satellite data sets. Unlike the

HCMM methods, the remote ET estimation method developed in this study is

designed for use with satellite data that is available at least every 2

or 3 hours. At this time resolution, the average temperature gradient

response method can make reasonably accurate cumulative daily ET esti-

mates without the need for simulation. Because the parameters are con-

sidered constant, no interpolation scheme is needed to make cumulative

ET estimates; only an estimate of the cumulative daytime net radiation

is required.




The energy budget/profile Bowen ratio technique was used to make

the evapotranspiration measurements needed for a data base in this re-

search. It was selected because it is one of the methods that least dis-

turbs the surface being measured, and when correctly applied, permits

measurements with an error on the order of 10% (Sinclair et al., 1975).

The profile Bowen ratio method has been successfully applied to a vari-

ety of surfaces (Sinclair et al., 1975; Stewart and Thom, 1973; Black

and McNaughton, 1972, 1971).

The theoretical basis of this method is developed first, followed

by a discussion of considerations going into the choice and use of the

sensors and other apparatus. Next, the automatic data collection system

that is assembled to make and report energy budget measurements is de-

scribed. It consists of a computer-controlled scanner, voltmeter, gas

sampling arrangement, and a set of four interacting programs. The chap-

ter concludes with a discussion of practical considerations that are

important in maintaining a high level of accuracy in the measurements

and the limitations of the data collection system.

Energy Budget/Profile Bowen Ratio Theory

As described in Chapter 2, the energy balance of a vegetated sur-

face can be written:

R = E + H + G 3-1

where R is net radiation absorbed by the surface,
E is the evapotranspirative or latent heat flux,
H is sensible heat flux, and
G is the soil heat flux.

It has already been shown that the rate of heat storage in the vegeta-

tion layer and the rate of photosynthetic assimilation are negligible in

comparison to these terms.

The Bowen ratio is defined as the ratio of sensible heat flux to

latent heat flux:
6 3-2

In the energy budget/profile Bowen ratio measurement technique, net ra-

diation and soil heat flux are measured directly. Latent and sensible

heat fluxes are determined indirectly by first measuring the Bowen ra-

tio, and then computing the fluxes:

E (R G) and 3-3

H (R G) 3-4
S+ 1

The Bowen ratio can be calculated from air temperature and water

vapor pressure measurements at various heights over the surface, pro-

vided a number of experimental conditions are met. Over a uniform sur-

face with adequate fetch, latent and sensible heat fluxes may be consid-

ered to exist in the vertical direction only (no flux divergence). In

the turbulent boundary layer the fluxes at any instant can be described

as follows:
H = -pc K 3-5
p H 9z

S-EL e
E -L K 3-6
P W3Z '
where p is air density,
c is specific heat capacity at constant pressure,
E is the ratio of molecular weights of water and dry air,
L is the latent heat of evaporation of water,
P is the atmospheric pressure,

KH is the eddy thermal diffusivity,
K is the eddy vapor diffusivity,
T is the air temperature,
e is the vapor pressure, and
z is the vertical coordinate.

The Bowen ratio can then be written:

c PK
H = pH 3-7
E cLKw
W It
If temperature and vapor pressure measurements are made at the same

heights, the az terms may be cancelled. If the measurements are made at

the same instant, it can be assumed that the eddy diffusivity for water

vapor and heat are the same (KH = KW). This in effect states that turbu-

lent mixing is the dominant transport mechanism in the turbulent bound-

ary layer, and that bouyancy and stability effects cause no significant

differences in the transport of heat or.water vapor (Dyer, -1967;

Swinbank and Dyer, 1967; Webb, 1970; Dyer and Hicks, 1970; Garratt and

Hicks, 1973). Incorporating these conditions into the expression for the

Bowen ratio,

B j I. 3-8
L= e ae 3-8

Since the terms in brackets are physical "constants" (abbreviated as the

psychrometric constant, y), only aT/e needs evaluation. This can be

done with air temperature and vapor pressure measurements.

In this application of the energy budget/profile Bowen ratio con-

cept, air and dewpoint temperatures were measured at five heights--35,

60, 85, 135, and 225 cm over the surface. Vapor pressure was calculated

from the dewpoint temperature according to the Magnus-Tetens formula

(Tennessee Valley Authority, 1972). The ratio aT/;e was the slope of a

two-independent-variable linear regression (Kendall, 1968) calculated

using temperature data as the ordinate and vapor pressure as the ab-

scissa (see Fig. 3-1). In calculating the Bowen ratio, the specific heat

of the air, the atmospheric pressure, and the ratio of molecular weights

was considered constant; the latent heat of vaporization was a function

of the average air temperature.

Sensor and Time Constant Considerations

Although simple in principle, a great deal of care is required in

choosing sensors and collecting data for the calculation of the Bowen

ratio. Temperature and vapor pressure vary randomly from instant-to-in-

stant and level-to-level in the turbulent boundary layer, and the total

temperature and dewpoint differences across the air layer to be measured

are only 1 or 20C. In order to calculate the relative strengths of the

gradients, very precise measurements at several levels are required.

Sensors were chosen to eliminate, as much as possible, the error

introduced by sensor-to-sensor variability. This was avoided entirely in

the case of the vapor pressure profile; the same dewpoint analyzer was

used to measure the dewpoint at each level by use of a gas sampling ar-

rangement. In the case of the temperature profile, the effect of thermo-

couple-to-thermocouple differences was minimized by measuring tempera-

ture differences with thermopiles. Twenty-junction copper constantan

thermopiles, arranged with 10 junctions at each level, were used to mea-

sure temperature differences between levels. The temperature at the low-

est level was measured with a thermocouple using an Omega Engineering

MCJ-T electronic icepoint reference. Temperatures at the other levels

were obtained by adding the appropriate thermopile-measured temperature

differences to the one reference temperature measurement.

11.2 11.4 11.6 1.8 12.0








Figure 3-1.

19 20 21

-. e4


I I 11

e T

o T

,0 -",e
T 2
,-' T

- 3' 3
'4 BOWEN_ cT
I I- I

11.2 11.4 11.6 11.8 12.0
Bowen Ratio Calculation from Measurements of Vapor
Pressure and Temperature. Note that the scale used
to plot vapor pressure profile in upper graph is the
same as the vapor pressure scale in the lower graph.
Data are from October 20, 1981, 9:30 TST (see Fig. 3-6).


- A LEVEL NO. 5-

SA 4-






The apparatus used to collect temperature measurements and air sam-

ples was designed so that the sensors returned signals accurately repre-

sentative of the air layers being sensed. The thermopiles were nested

inside three aspirated radiation shields, with each shield wrapped in

highly reflective aluminum foil. Air samples were pumped continuously

from sampling ports near the thermopiles through about 100 m of 6-mm ID

polypropylene tubing and 11.3-L mixing chambers in the instrument room.

To prevent any danger of condensation, the air sampling system was

heated from sampling mast to dewpoint analyzer. The bundle of five tubes

from the mast was taped around a heater cable (3 W/ft) and packed inside

a 1.3-cm-thick foam rubber insulation tube. The mixing chambers and the

sampling valve were also heated.

The travel time of air samples from mast to instrument room was

approximately 1 min. Therefore, the dewpoint measurement corresponding

to a temperature measurement at a specific level was made 1 min later.

Also, the dewpoint temperature measurements were pressure-corrected be-

cause the arrangement of the air sampling system caused the pressure

rate at the dewpoint sensor to be %30 mb less than atmospheric pressure.

To ensure clean electrical signals, shielded signal cable with a

single common ground was used. In spite of these precautions, the Beef

Research Unit fence charger managed to induce significant voltage spikes

on the low level signals (e.g., the 0-200 microvolt thermopile signals).

This problem was solved with a filtering routine in the data collection


In addition to reducing the error sources from the sensors in every

practicable way, the temperature and vapor pressure signals were

physically smoothed. Smoothing was required because the measurement rate

was limited to one measurement every 2.5 min for the vapor pressure

profile measurements.

Vapor pressure in the Bowen ratio data collection system was com-

puted from a measurement of the dewpoint temperature. Since the same

dewpoint sensor was used for all five levels and a delay had to be

scheduled between measurements to allow the analyzer to settle on new

dewpoints, the response of the dewpoint analyzer was the factor limiting

the sampling rate. The analyzer, an EG&G Model 880 Dewpoint Hygrometer,

was tuned so that it could "lock on" to small dewpoint temperature

changes within about 15 sec. However, 30 sec per measurement were sched-

uled to allow the analyzer to stabilize on a given dewpoint under less

than ideal conditions. Since there were five levels to measure, the time

interval between measurements at the same level was 2.5 min.

The variability of temperature and vapor pressure in the turbulent

air layer is well documented; at any point in this layer, instantaneous

temperatures and dewpoints vary randomly (Desjardins et al., 1978). The

higher-frequency temperature and dewpoint fluctuations were smoothed in

order to get representative measurements with a sampling rate of one

measurement every 2.5 min.

In the case of an air-sampling system, this smoothing is conveni-

ently done by inserting a mixing chamber into the sample stream ahead of

the analyzer. An abrupt (or step) change in an air sample is translated

into a gradual, approximately exponential change by mixing in a chamber.

The exponential change is characterized by a time constant, which is

determined by the volume of the mixing chamber divided by the flow rate.

By harmonic analysis, it was determined that a time constant of 4 min.

would damp random signal variations occurring more often than every 2.5

min to 10% or less of their amplitude. In the case of the dewpoint sys-

tem, 11.5-L mixing chambers with a flow rate of 3 L/min were used.

To maintain the proper correlation between dewpoint and temperature

readings, it was necessary to introduce the same time constant into the

temperature-sensing system. The appropriate time constant was determined

experimentally by varying the air flow rate over the aspirated thermo-

piles and subjecting them to different temperature differences. It was

found that at a set air flow rate, measured time constants varied with

the temperature difference applied to the thermopiles. As a result, the

air flow rates were adjusted so that a 4-min time constant resulted for

temperature differences in the average operating range--temperature dif-

ferences in the range of 0.2 to 0.30C.

The 4-min time constant was also introduced into the surface tem-

perature and net radiation measurements. Sensor response was slowed dig-

itally by using weighted averages of the most recent 25 sensor readings.

Each time a complete temperature and vapor pressure profile was

measured (every 2.5 min), the correlation coefficient between tempera-

ture and dewpoint measurements was calculated. This provided a running

check on the quality of the measurements and the current similarity of

the profiles.

Data Collection Equipment

The overall schematic for the thermopile/air sampling system is

shown in Fig. 3-2. The major parts are the data acquisition system, the

air sampling mast, the mixing box, and the signal cables and tubing

which connect them.

A computer-controlled data acquisition system was used because of

the large number of measurements and extensive calculations that this


Flow Meters
Mixing Bottle




Figure 3-2.

Schematic of ET Measurement System. Details of profile measurement and air
sampling equipment are shown in Figs. 3-3 and 3-4.

technique requires. The central piece of equipment was a Hewlett-Packard

2100S Minicomputer with a disk resident Real Time Executive-2 operating

system. The system allowed editing and compilation of programs, swapping

programs between core and disk memory, scheduling programs for relative

or absolute start times, and "simultaneous" running of programs accord-

ing to priority. Input and output were by means of a HP-2126P terminal.

The peripheral equipment used in making the measurements and con-

trolling the gas sampling valve is listed in Table 3-1. The controlling

computer, disk drive, data acquisition equipment, and terminal were all

housed in an air-conditioned room.

Table 3-1 Data Acquisition System Identification (All components
are manufactured by Hewlett-Packard)

Component Model No. Serial No.

Minicomputer (32K Memory) HP-2100S 1420A05546
Scanner HP-2911A 737-00476
Scanner Controller HP-2911B 832-00412
Integrating Digital Voltmeter HP-2402A 1027A01060
Disk Drive HP-7901A 1321A-00255
Terminal HP-2621P 2102W03475

The field apparatus on the pasture site consisted of an air-sam-

pling mast, a radiation sensor boom, and a 9.5-m tower supporting a

precision radiation thermometer at its top, and windspeed and direc-

tion sensors at 7 m. Another taller tower was erected and equipped to

protect all instrumentation from lightning.

The 2-m radiation sensor boom was supported by an aluminum tripod

stand and guy wires about 1.8 m over the ground surface. Two Epply pyra-

nometers, oriented to measure incomin: and reflected radiation, and a

Swissteco net radiometer were mounted at its end. An aspirating pump and

dessicant container for the net radiometer were held in a weatherproof

box at the base of the tripod.

The air sampling mast consisted of a 2.5-m steel channel to which

five sensor arms (see Fig. 3-3) were attached at various levels. At one

end of each arm, teflon spacers centered two clusters of 10 thermocouple

junctions inside the smallest of three radiation shields. Individual

junctions were kept in thermal contact with a metal oxide conducting

paste. Air was drawn over the thermopiles, between radiation shields,

and through the length of the arm by a small fan at the opposite end.

Air samples were drawn from the air flowing through the arm. All wiring

(four 20-junction thermopiles and one thermocouple) and tubing (5 sample

lines) were contained inside the 3x3 cm channel down to its base, where

they ended in wire and tubing connectors. The mast and sensor arms as

well as the radiation, shields were wrapped in highly reflective aluminum


The sensors were connected to the scanner in the instrument room

with shielded signal and thermocouple wire. In the field, leads from the

sensors ran aboveground in wire harnesses to a junction box, where they

were connected to a signal cable via screw connectors. This cable ran

100 m underground to another junction box in the instrument room. From

this panel the signal lines were connected to one of two 50-pin connec-

tors, which plugged into a short piece of cable tied directly into the

scanner. The "quick-disconnect" plugs were included to rapidly isolate

the data acquisition system from possible lightning strikes in severe

weather; the junction boxes allowed signal problems to be quickly traced

to sensors, underground cabling, or the data acquisition system. The

sensors used are identified in Table 3-2.





Detail of Profile Measurement Mast Arm.

Figure 3-3.

Table 3-2 Sensor Identification

Measurement Sensor Make & Model No. Ser. No.

Net Radiation Swissteco Net Radiometer 6990
Incoming Shortwave Radiation Epply Pyranometer 8-48 12876
Reflected Shortwave Radiation Epply Pyranometer 8-48 10000
Surface Temperature Barnes IT-5 (Spring 1981) --
Barnes IT-3 (Fall 1981) 521
Dewpoint Temperature EG&G 880-C1 1409
Windspeed and Direction R.M. Young 6101 and 6301
Air Temperatures Custom-made Thermopiles
Reference Temperature Omega Engineering MCJ-T
Soil Heat Flux Micromet Heat Flow Disk 282

Air was pumped continuously from each sample port on the mast

through .100 m of heated insulated polypropylene tubing and the gas sam-

pling apparatus in the instrument room. In the "mixing box," air first

passed through flowmeters, then the mixing chambers, the scanning valve,

and the air pump. Samples from each level were drawn sequentially

through a sampling port, a separate sample flowmeter, and the dewpoint

analyzer. All equipment except the pump and analyzer were contained in-

side a heated, insulated plywood box (see Fig. 3-4) to prevent condensa-

tion problems.

The scanning valve was controlled from the data acquisition compu-

ter. The sampling port was turned from one air source port to the next

by an electric motor powered for a precise fraction of a second. This

was done by a relay control circuit that was designed to sense scanner

closure. Thus a program statement calling for a measurement of the scan-

ning valve control channel resulted in changing the position of the

valve. After each change, the valve position was checked to ensure that

the programs and valve were synchronized.

,Valve Control Electronics
- Sampling Valve

Analyzer Sample

Air to Analyzer
Air to Pump

5 Sample
Tubes from


-Foam Rubber

Figure 3-4. Detail of Air Sampling Equipment.


Data Collection Programs

A system of four programs was developed to collect, report, and

analyze the data required for the test surface energy budget. Program

MEASR makes the measurements and calculations, REPRT produces the half-

hourly summary reports, ANALZ does some analysis of data and performs

additional calculations, and SET schedules the other programs. Listings

of these programs appear in Appendix B; brief descriptions of their

functions and interactions follow.

Basically, all sensors are scanned in a computer program loop. De-

pending on the status of various indexes in this loop or the system

clock, control is passed to specific calculation and/or reporting rou-

tines. This fundamental loop is in program MEASR; it is repeated approx-

imately every 30 sec, the measuring rate determined by the dewpoint ana-


When a program calls for a measurement [i.e., CALL EXEC (1, 9,


measurement program statement is passed to the scanner controller, and

the program word indicating type of measurement, voltage range, and de-

lay time is passed to the voltmeter. After the scanner has closed on the

proper signal lines, the voltmeter has been set for the type of measure-

ment, and a programmed delay is complete, the voltmeter integrates the

signal for 1/60 second and passes the average back to the measuring pro-

gram. It resumes execution with the next program step.

During each execution of the measurement loop, one air temperature,

one dewpoint temperature, and all other sensors except soil thermocou-

ples are scanned. Immediately after the dewpoint measurement, the scan-

ning valve position is changed (Subroutine STEP) so that the dewpoint

instrument can begin to stabilize on a new dewpoint. A programmed delay

makes up the balance of the 30 sec required between measurements. At the

end of five scans (2.5 min), a complete temperature and dewpoint profile

is available to compute a Bowen ratio. A report on that profile is

printed at the option of the system operator (see Fig. 3-5).

To compensate for the approximately one-minute air sample travel

time from field to mixing chamber, temperature and dewpoint measurements

are offset by two levels. For example, the dewpoint at level 1 is mea-

sured in the same sensor scan as the temperature at level 3. This ac-

counts for extra statements at the beginning of the program which ensure

proper initialization, and for extra branching after sensor scans which

deal with the offset completion times of the temperature and dewpoint


To guarantee that the dewpoint analyzer is receiving the air sample

from the level called for in the program, a mark voltage channel is mea-

sured and checked in each scan of the sensors. In one particular posi-

tion of the scanning valve, 12 volts are expected on this channel. If

the voltage measured is low or 12 volts are measured when not expected,

the data for the profile being collected are discarded and a message to

the operator is printed. The program makes one attempt to reposition the

valve and restart data collection. If this fails, another message is

printed and the programs are terminated.

When temperature and dewpoint measurements at all five levels are

complete, the data are passed to subroutine RATIO, which calculates a

linear temperature versus dewpoint regression relationship. Its slope is

multiplied by the appropriate constants (Eq. 3-8) to give the Bowen

9: 16:27
B =1.723

B =1.813

B -1.612

B -1.590

PROF< 10
P1 =1.904

B =1.679

PROF# 12
B =1.618


RfD. T.







2. 18



11. z79


.4 8







V. P.

V. P,

9 .5

I 1 .

21 .4

12 .

21 .8
. 9.9
10 .1


1 .6



21 .0
1 .
9. 9


21 .5



11 .5

11 .7


9. 9





20. 1




19 .1


8. 8
1 1.





R = ,9y/

R =. .99b

R = .988

R .999

R =1 .000

Figure 3-5.

Example of Intermediate Program Output

This report is printed if switch #3

on face of HP 2100 computer is on. Data are from the 15 min preceding half-
hour report shown in Fig. 3-6.

ratio. The ratio and corresponding correlation coefficient are returned

to the calling program.

Function FILT was added to MEASR after it was discovered that the

shielding system did not prevent the Beef Research Unit electric fence

charging system from inducing noticeable spikes on the signal lines.

These 10-50 microvolt spikes were shorter than the voltmeter measurement

cycle, and thus lent themselves to being filtered digitally. FILT takes

10 measurements, looks for three in a row that are the same within a

tolerance, and compares the rest of the measurements to one of them. Any

measurement varying more than a specified tolerance is dropped, and the

average of the "good" measurements is passed back to MEASR. If more than

half of the measurements are noisy (out of tolerance), a warning is

printed to notify the operator.

Subroutine TMTCH is included to match the time constants of the net

radiation and precision radiation thermometer to that of the temperature

and dewpoint measurements. This matching is done by using the weighted

average of the 25 most recent (collected in the last 12.5 min) measure-

ments to calculate a matched measurement. The weights assigned to older

measurements decrease exponentially with a time constant of 4 min. The

same weighting scheme is used for the net radiation and surface tempera-

ture because their sensor response time constants are 8 and 2 seconds,

respectively. At a sampling rate of one measurement every 30 sec, their

responses are, in effect, instantaneous.

Program REPRT produces a half-hourly data summary report. It calcu-

lates half-hourly average profiles of the heat budget components, wind-

speed and direction, Bowen ratio, and profiles of soil and air tempera-

ture, air vapor pressure and relative humidity. Most of this program is

concerned with formating and printing the summary report. An example

report is shown in Fig. 3-6, and Table 3-3 lists the variable names


Program ANALZ makes ancillary calculations and produces the last

five lines of the half-hourly report. It has a search routine which com-

putes the displacement height of the temperature and vapor pressure pro-

files. With an assumed value of the roughness parameter (z0) and trial

values of the displacement height (D), it computes the correlation of

temperature or vapor pressure and height over the surface with

z D + z0
T, e = B ln + A 3-9

The assumed roughness height, the displacement height producing the best

correlation, and other profile parameters are printed out.

ANALZ also computes a variety of other quantities which may be of

use in data analysis or operation of the system. Among these are atmo-

spheric and stomatal resistances, albedo, optical air mass and atmo-

spheric transmission coefficient, zenith and hour angle of the sun, and

the equation of time.

The fourth program, SET, is the executive program. It is used to

properly start the acquisition of data and determine whether and when

the other programs should be run. In a "cold" start, SET positions the

scanning valve, initializes counters and statistics, and schedules MEASR

to start so that profile collection is completed at specified times. On

occasions other than a "cold" start, it determines whether the other

programs should be run, depending on flags in MEASR or operator input

via switches on the face of the HP-2100. Its most valuable function i.s

to schedule MEASR to begin at an absolute clock time at the beginning of


TULSD-AY, OCTOBER 2G, 1981 (JIJL.I N DAY 293) TIME 09:31:27 TST

.45 LY/M .02 LY/M .28 LY/M- .I- LY/M- 3.06 M/S
*16 LY/M .02 LY/M .28 LY/11 1 LY/ 11 33.06 M/S

I l '

(LY/H) (CM) (-C) (CM) (M1B)

.451 (5.0) 225 19.2 (4,2) 225 11.2 (3.0)
.802 (14.1) 135 19.6 (4.1) 135 11. 3 (2.9)
.177 (3.7) 85 20.0 (4. 1) 85 11.5 (2.8)
.109 (16.) 60 20.4 (4.0) 60 I 1 .G (3.1
.61G (5.5) 35 21 .1 (3.9) :35 11 .9 (3.1)
162 MM/H1R 0 23.4/ (5.5) .- .95+ BR =1.760,


22 .-0



-I .000
-1. 000
-1 .000


RSQ.>.95 B.R. -~VG.R.

12 OF 12 1.753 .999

(CM) (.) (CM) (-C)

225 50.2 0 16.8
135 49.8 -2 11.6
85 4,9.2 -S 18.2
60 48.6 -10 19.2
35 47.6 -25 20.1
+OR-.11-** -50 21.7




iN I R S, R TM(S/M)

A[EDO S110
.22 1.09

.238,4.350 .311,4353 .348,4,350 .397,4.358 ,436,4.365

0 II



-41 :2

.2579 9.48 9.25

Figure 3-6. Example of Half-Hourly Data Report.. .Variable names and units are listed in Table 3-3.


Table 3-3. Variable Names and Units for Half-Hour Reports

RSQ.>.95 Number of temperature and vapor pressure profiles with
correlation coefficient better than .95
B.R. Bowen Ratio
AVG.R. Correlation coefficient of half-hour average profiles
NET Net radiation (ly/min)
ISW Incoming shortwave radiation (ly/min)
RSW Reflected shortwave radiation (ly/min)
ALW Atmospheric longwave radiation (ly/min)
ELW Emitted longwave radiation (ly/min)
ET Evapotranspiration rate (mm/h)
.95+BR Average and standard deviation of individual profile Bowen
ratios with greater than .95 correlation coefficient
ZO Roughness height (cm)
TO Temperature at ZO by temperature profile extrapolation (oC)
DH Displacement height for heat (cm)
U*H Friction velocity as determined by fit of profile (m/min)
RCH Correlation coefficient for temperature profile
EO Vapor pressure at ZO by vapor pressure profile extrapo-
lation (mb)
DE Displacement height for vapor pressure (cm)
U*E Friction velocity as determined by fit of vapor pressure
profile (m/min)
RCE Correlation coefficient for vapor pressure'profile
RAIR Air diffusion resistance (s/m)
RSTM Stomatal diffusion resistance (s/m)
ABDO Albedo (fraction)
SWIO Shortwave insolation without atmosphere (ly/min)
OAM Optical air mass (atmospheric diameters)
ATC Atmospheric transmission coefficient
[absorption coefficient = -ln(ATC)]
ZNGL Zenith angle of sun (degrees)
HRNGL Hour angle of sun (degrees)
EOT Equation of time (h)
E.S.T. Eastern standard time
T.S.T. True solar time
DAY Day of year

each half hour. This prevents the data reports from processing out of

synchronization with the system clock.

SET also enables REPRT and ANALZ to be swapped between core and

disk so as not to interfere with the measurement schedule. At the end of

a typical half hour (1.5 min past the clock hour or half hour, when mea-

surement of the twelfth profile has just been completed) MEASR calls for

SET to run immediately and ends. SET schedules MEASR to start again at

an absolute clock time, 2 min into the new half hour, or roughly 30 sec-

onds after the last measurement made. It then loads and runs REPRT and

ANALZ. When it is time for MEASR to start, whichever program is in core

is moved back to disk, and MEASR is loaded. MEASR makes its first scan,

and during the usual delay between scans, REPRT and/or ANALZ are re-

loaded and run to completion. MEASR is then swapped back to core to be

continued at the end of the programmed delay;

The programs can be halted from the computer terminal or with

switches on the face of the computer. When switches 1 and 2 are on,

MEASR ends with the next profile and REPRT computes averages for all the

data collected in that half hour. When only switch 2 is on, MEASR ends

at the next normal half-hour reporting time.

Operational Considerations

The thermopile/air sampling system required a great deal of care in

setting up and maintaining the instrumentation involved. It also re-

quired an awareness of the theoretical and practical limitations of the

measurement method. Proper calibration and tuning of the dewpoint analy-

zer were most critical for good measurements. Sensor cleaning and output

calibration procedures are well documented in the EG&G Model 880 Dew-

point Hygrometer Manual (1977). However, to achieve optimum response

times, it was necessary to tune the instrument slightly differently than

Manual specifications. It was made more sensitive by setting the THK

potentiometer so that voltage on the test points was 200-260 mV, and

made faster by setting the GAIN potentiometer so that the test voltage

was 150-210 mV. The new settings sacrificed dewpoint analyzer response

time in large step changes in order to improve response time in the

smaller step changes normally encountered in the profiles. To ensure

that the dewpoint analyzer actually had time to settle on readings be-

fore being read by the voltmeter, its output was spot-monitored on a

millivolt recorder.

The most difficult problem was the individual and cross-correlation

of the dewpoint analyzer, the thermocouple/thermopile air temperature

sensors, and the precision radiation thermometer. The dewpoint analyzer

S-output was calibrated according to the EG&G manual. Temperatures at the

bottom and top of the scale were simulated by substituting precision

resistances for the mirror-temperature sensing thermistor; the analyzer

output at these simulated temperatures was matched to the factory stan-

dard instrument output. The radiation thermometer was calibrated by mea-

suring its output for known surface temperatures produced with a stirred

constant-temperature bath. A regression equation for the temperature vs.

output correlation was calculated and used in the programs.

It was not possible to cross-calibrate these temperature sensors

until the system was run in a light drizzle on Nov. 5, 1981. This situa-

tion resulted in the same temperatures at all measured levels, near-zero

net radiation, and air that was near saturation, so the dewpoint, air,

and surface temperatures were approximately the same. The temperature

differences between sensors were used to correct the rest of the data.

(It should be noted that this correction did not affect the Bowen ratio

calculation, since it used only relative changes. The correction did

affect surface temperatures, which were not used in computing the energy


The radiation sensors were calibrated against a recently purchased

(and calibrated) Epply Pyranometer.

On the whole, the thermopile/air sampling system developed worked

very well and produced excellent data. However, there were some situa-

tions in which it could not function well. The system was protected from

almost all of these situations because calculation of a complete energy

budget was made conditional on temperature and dewpoint profile similar-

ity. Latent and sensible heat fluxes were not calculated when the pro-

file correlation coefficient was less than 0.95.

Profiles were regularly dissimilar for a few time-periods in the

early.morning and late afternoon, while temperature and dewpoint pro-

files were reversing in direction. This dissimilarity occurred because

changes in the temperature profile generally preceded changes in the

dewpoint profile.

Sensible heat generated at the surface of the outermost radiation

shields was usually carried away by the air flowing over them. At very

low windspeeds, however, the warm air produced at the outer surface of

the lowest radiation shields could become entrained in the aspiration

air of mast arms above. This problem showed in profile correlation coef-

ficients but was usually not so bad that energy budgets could not be

calculated. Under clear skies this effect was not as marked, presumably

because the radiation shields could more effectively radiate heat away.

When the system was run at night, some condensation took place in

the air sample lines because the air sampling mast was not heated. Water

accumulated in the tubing in proportion to the length of the tubing sec-

tion in the mast. As a result, the fifth level produced obviously high

dewpoint temperatures until the tubing had dried. The temperature dew-

point correlation made it obvious at what time all condensation had been

evaporated from the sample lines.

The situation most hazardous to data quality occurred on very

sunny, dry days. At these times, the air temperature of the instrument

room (21-240C) was quite a bit higher than the dewpoint temperature of

the outside air. At some point the analyzer would no longer be able to

cool its sensor mirror low enough to get dew formation. Since air sam-

ples from different levels have different temperatures, the coolest mir-

ror temperatures possible varied also. A false dewpoint profile, which

correlated very well with air temperatures, would be measured and thus

passed through the correlation coefficient screen. Evidence for this

condition was the brightly-lit cooling circuit lamp on the dewpoint ana-

lyzer. With experience this condition could be anticipated, and its ef-

fects minimized by unplugging the heater cables to the sampling lines

and mixing box.

In spite of precautions taken, susceptibility to lightning damage

was the system's greatest weakness. The system was damaged twice by

lightning. In both cases, instrumentation and computer equipment was

damaged by current surges in the AC power system, in spite of power-

surge arrestors. The only solution was the most fundamental--unplugging

all sensor cabling and all AC power cords.




The key problem in developing a remote ET estimation method is de-

scribing the vegetation and air layer at the surface so as much informa-

tion as possible about its energy budget can be gained from the surface

temperature and net radiation. In addition, there is the question of how

much ancillary data is necessary for acceptable levels of accuracy in

the estimates. Previous approaches to these problems were outlined in

Chapter 2.

The methods developed in this chapter are based on the response of

surface-to-air temperatures to varying net radiation loads. First, a

functional relationship that describes the dependence of the surface-to-

air temperature gradient on net radiation and other factors is derived.

This temperature gradient response (TGR) model is used with surface tem-

perature, air temperature and net radiation data to evaluate surface

parameters, which are then used in an adapted version of the combination

equation to estimate evapotranspiration. Two methods of making estimates

are developed. The first is physically strict, with a minimum of assump-

tions; the second is more approximate with correspondingly fewer data


For the sake of simplicity, the derivations that follow are in

terms of surface temperature (Ts) and net radiation (R) rather than the

direct remote measurements, reflected (Qr) and emitted (Q ) radiation.

Also, in application, temperature differences are used to evaluate tem-

perature gradients. For that reason, differences are used in the equa-

tions developed and are referred to interchangeably as differences and


Temperature Gradient Model

The simplified energy balance of a vegetated surface was developed

in Chapter 2:

R = E + H + G 4-1

The purpose of this section is to express the components of the surface

energy budget as much as possible in terms of net radiation and surface

temperature, so that a useful relationship between the two can be de-


Because of heat storage in the surface air layer, surface tempera-

tures lag net radiation. This lag is complicated by the fact that the

passage of clouds usually makes the net radiation absorbed by the system

vary randomly. For this reason, a method containing time as a variable

has been avoided. This was done by modeling the response of surface-to-

air temperature gradients to changes in net radiation.

In describing heat flux using a surface temperature (i.e., between

the surface and some plane above the surface), at least two layers with

different transport properties must be considered (see Fig. 4-1). The

first is the surface layer, in which molecular diffusion is the primary

transport mechanism. It is the thin layer of air immediately next to the

plant surfaces, represented by the layer between zs and z0 in the dia-

gram. The second is the fully turbulent layer between z0 and za, where

turbulent eddies are the primary transport mechanism. Following the de-

velopment shown in Chapter 2, the heat flux between the surface (at


Za ---- T- -- --


Air KH h Turbulent
(3 Layer

S,.. Surface-.
'Zo.- - ;~-t : |Layer I-"
S "'. -* y I'-
ZT P, lant gpDQ;,e tcS 1I;


Figure 4-1. Definition Sketch for Transport Properties. The
surface layer which is dominated by laminar air
flow (molecular thermal diffusivity, kH) is repre-
sented by the layer between zs and za. The heat
transport coefficient is used to represent the
combined transport properties of both layers.

temperature Ts) and some level in the air above the surface (at Ta) is

described by
(pCp )a (Ts Ta)
H= ,p 4-2
f zO dz Za dz
zk H z H

where p is air density,
c is the specific heat of air at constant pressure,
kP is the molecular thermal diffusivity of air, and
KH is the eddy thermal diffusivity of air.

(The first term in the denominator is equivalent to the resistance of

the laminar surface air layer, and the second term in the denominator is

equivalent to the resistance of the surface turbulent boundary layer.)

Treating latent heat flux analogously,

(pCp)a (es ea)
E = 4-3
fZO lz fza dz)
z W k z
s 0 -

where e is the water vapor pressure,
y is the'psychrometric constant (- = c P/Le),
kw is the molecular water vapor diffusivity, and
K is the eddy water vapor diffusivity.

It has been shown that for a wide range of stability conditions normally

found (Dyer, 1967; Swinbank and Dyer, 1967; Webb, 1970; Dyer and Hicks,

1970; Garratt and Hicks, 1973):

Z a dz z a dz 44
z W z H
z0 0
For simplicity, it is also assumed that
zI0 dz z dz
z W z H
s s
This assumption is unvalidated, but shared by the majority of theoreti-

cal treatments. Literature values for the molecular water vapor and

thermal diffusivities are in fact at least approximately equal [e.g.,

Eagleson (1970) quotes values of 0.1 and 0.13 cm2/sec, respectively].

With the above assumption, transport of latent and sensible heat

can be considered similar from the surface to a reference level in the

air. The simple expressions developed in Chapter 2 (Eqs. 2-17 and 2-20)

can then be used to describe these fluxes:

H = h(Ts Ta) and 4-6

h *
E = M(e e ) 4-7
y s a

The moisture availability parameter (M) is included to account for the

subsaturation of the surface air layer. However, use of Eq. 4-7 as a

hard equality will force M to include minor differences due to inequali-

ty of molecular diffusivities of latent and sensible heat (Jarvis et

al., 1971), any differences due to stability effects, and any differ-

ences due to dissimilar sources and sinks of latent and sensible heat

within the vegetation system.

The dependence of the vapor pressure gradient on the surface-to-air

temperature gradient is shown in Fig. 4-2. It shows that the vapor pres-
sure difference (e ea) is in part due to the greater temperature of
the surface relative to the air, and in part due to the saturation defi-

cit of the air. Considering the saturation vapor pressure curve linear

in the neighborhood of the surface and air temperatures,

e ea = s(Ts T ) + ea 4-8

where s is the slope of the saturation vapor pressure curve between
Ts and Ta, and
6ea is the saturation deficit of the air.

Substituting this expression into the latent heat flux equation (Eq.

4-7) gives the latent heat flux in terms of the temperature gradient:

E = M [s(T T ) + 6e] 4-9
Y s a a





Figure 4-2.


T e
O -
o0 20 30 40

es'-ea a+b
es*-eas(Ts-Ta) +ea

Components of Vapor Pressure Gradient. Component (a)
can be calculated from the surface-to-air temperature
difference (Ts T,) and the slope (s) of the saturation
water vapor pressure curve [e*(T)]. Component (b) is
the saturation deficit (6ea) of the air.


The advantage of this substitution is that whatever error occurs in the

temperature gradient measurement affects both latent and sensible heat

fluxes, not just the sensible heat flux, as in the simple residual ap-

proach. Substituting Eqs. 4-6 and 4-9 into Eq. 4-1,

R = G + h(T T ) + hM [s(Ts Ta) + 6ea] 4-10

Rearranging terms and dividing by constants,

T Ta = h(Ms + ) [y(R G) hMea] 4-11

Equation 4-11 explicitly states the dependence of the surface-to-

air temperature gradient (difference) on other variables--net radiation

(R), soil heat flux (G), saturation deficit (6ea), bulk air conductivity

(h), and moisture availability (M). The equation is generally applica-

ble; the psychrometric "constant" can be adjusted to various atmospheric

pressures (altitudes) and the slope of the saturation vapor pressure

curve can be adjusted for various temperature ranges. The resistance

formulation of this equation has been used by Jackson et al. (1980).

In the strictest sense Eq. 4-11 is true only instantaneously. How-

ever, with the assumption of some degree of system stationarity, various

approaches to remote ET estimates can be made. Two are developed here.

The first makes a minimum of additional assumptions and uses ground-mea-

sured air temperature, saturation deficit, and soil heat flux measure-

ments. The second assumes that only surface and air temperatures change

in response to changes in radiation, and uses only remote measurements.

Both use the response of temperature gradients to varying net radiation

loads to evaluate surface parameters; these are then used in calculating


Strict Temperature Gradient Response Method

From Eq. 4-11, one recognizes that

H = h(Ts Ta) = y [y(R G) hM6ea] 4-12

Substituting into the energy budget equation (Eq. 4-1),

E=R-G Ms+y
E = R G Ms + [y(R G) -hM6e a] 4-13

E = Ms [s(R G) + hea] 4-14

This is a version of the combination equation for evapotranspiration

(Penman, 1948). When the surface is saturated (M = 1), this equation

reduces to the potential evapotranspiration equation of Tanner and

Fuchs (1968).

The slope of the saturation vapor pressure curve (s) is a known

function of temperature, and the psychrometric constant is a known func-

tion of atmospheric pressure. Net radiation (R), soil heat flux (G), and

saturation deficit (6ea) are measurable. Only the parameters for bulk

heat transport (h) and moisture availability (M) are unknown. These can

be evaluated with Eq. 4-11, assuming that they can be considered con-

stant for some period of time.

Supposing that measurements of Ts, Ta, 6ea, G, and R are available

for two different radiation loads,

y(R G)1 hM6e
(Ts Ta h(Ms + ) and 4-15

y(R G)2 hMse
(T T2 h(Ms ) a2 4-16
.s a 2 h(Ms+y)

The numerical subscripts identify the two sets of data corresponding to

the two differing radiation regimes. If the measurements are made close

enough in time so that moisture availability and the bulk thermal

conductivity can be considered constant, this pair of equations can be

solved for the two unknowns M and h,

RieI RiAe2
h = R -Ae1- e2 4-17
6elAT2 6e2AT1

y=(R2AT1 + RAT2)
M = 4-18
RjAe2 + R2Ae1

where R1 = (R G). 4-19

ATi = (Ts Ta)i and 4-20

Aei =(es e )i = SAT + 6e 4-21
1 s a i i a .

Equations 4-17 and 4-18 can be substituted into Eq. 4-14 to yield:

R2?AT RIAT2 R2Ael R1Ae2
R = R6ej 2 Reel] s(R G) + 6ea 6e lT2 6e2AT1 4-22
Rfse2 R^6e1 a 6e1ATT RetA2]]

Now any evapotranspiration rate between the two measured radiation re-

gimes can be calculated by substituting intermediate values of R, G, and

6ea. As a practical consideration, the measured radiation regimes must

be near enough in time such that the constant M and h assumption is

valid, but far enough apart in radiation load so that reasonably accu-

rate calculation of the parameters is possible.

Average Temperature Gradient Response Method

System Stationarity and Average Temperature Gradient Response

The most restrictive problem faced in estimating ET from above the

atmosphere is that very often clouds make surface temperatures unobserv-

able. As shown in Eqs. 4-17 and 4-18, surface temperatures are necessary

to estimate surface moisture availability and bulk air conductivity.

This fact requires that remote ET estimates be made in two stages. The

first stage consists of using clear sky net radiation and surface

temperature data to evaluate the surface parameters. The second stage

consists of using those parameters and measured or estimated net radia-

tion data alone to make ET estimates. It is assumed that through a com-

bination of cloud-reflected radiation and cloud-top temperatures, a net

radiation estimate for the surface is still possible.

The success of this scheme is limited by the period over which the

parameters can or must be considered constant. They do vary; moisture

availability changes as dew evaporates and the bulk air conductivity

changes with the windspeed. But for the strict temperature response

method to work, the parameters must be considered constant for the time

interval between clear sky data sets, which is limited by the time reso-

lution of the data collection system and cloud cover.

This unavoidable necessity motivates viewing the surface as a sys-

tem with approximately stationary parameters for the duration of a lon-

ger measurement period (i.e., assuming that the surface-to-air tempera-

ture gradients in Eq. 4-11 are a function of net radiation and con-

stants). Information on the average parameters can then be extracted

from the correlation of clear sky surface-to-air temperature gradients

and net radiation, and used to estimate ET for clear or cloudy skies.

A soil heat flux parameter is required in order to fully parameter-

ize the temperature gradient model. Soil heat flux is the smallest of

the energy budget components, usually accounting for less than 10% of

net radiation under a vegetated surface. It lags net radiation in time,

but because its magnitude is in the range of error expected in the esti-

mates, it can be safely and conveniently treated as a constant fraction

of net radiation:

= g 4-23

The quantity used in calculations is (R G), which for convenience can

be written

R G = (1 g)R = fR 4-24

Typical daytime values of "g" in the literature range from approximately

0.0 to 0.2, so for a vegetated surface "f" will have values between 1.0

and 0.8.

Substituting Eq. 4-24 into Eq. 4-11,
T T hMs (YfR hMea) 4-25

Formalizing the approximation of system stationarity,

h, M, f, s, 6ea i f(t) 4-26

Eq. 4-25 reduces to the form

Ts T = AR B 4-27

where A and B are constants:

A f 4-28
h(Ms + y) and

B 4-29
(Ms + ) ,

and the parameter values are averages for the time period over which the

surface system is considered stationary. The constants A and B can be

evaluated by correlating the temperature differences (T, Ta) and net

radiation (R) data. Simple linear regression equations can be used:

A = and 4-30
E (Rt)2 nR2

B = AT AR 4-31

where the summations are done with clear sky data only,
AT = (Ts Ta), and
t is a subscript denoting the time of the measurement.

As a practical matter, the time period for which A and B are calcu-

lated (and the parameters are considered constant) needs to be at least

a day. This period must be extended if enough clear sky data are not

available for a reasonable estimate of A and B. It should also be noted

that for normal daytime conditions, A and B must be positive for physi-

cally real parameters. This implies that the intercept of the surface-

to-air temperature gradient/net radiation correlation must always be

negative (zero at most), and the slope must always be positive (see Eq.


Use of the Average Temperature Gradient/Net Radiation Correlation

Incorporating the definition of the soil heat flux parameter (Eq.

2-24) into the equation for evapotranspiration (Eq. 4-14),
E =Ms (sfR + h6ea) 4-32

The slope of the saturation vapor pressure curve (s) is a known function

of temperature, the psychrometric constant (y) is relatively constant at

a given altitude, and net radiation (R) can be estimated directly from

satellite data. There are four unknown parameters: moisture availability

(M), the soil heat flux parameter (f), the bulk transport coefficient

(h), and the saturation deficit (6ea).

The equations for A and B, Eqs. 4-28 and 29, are in these four un-

knowns. Since all four are required to estimate ET, two must be approxi-

mated from average conditions or a rough daily measurement. Equations

4-28 and 29 can then be solved for the other two and substituted into

Eq. 4-32. Table 4-1 shows the ET formulae derived for the possible com-

binations of known and unknown parameters and the correlation constants

A and B.

Evapotranspiration Formulae for Average TGR Method. Conditions listed are necessary to
physically real solutions and prevent numerical problems. It is understood that A must

positive and unless otherwise noted, B must be
appear at right.

greater than or equal to zero.

always be

Equation numbers

Known Unknown Substitutions Special Evapotranspiration Equation
Parameters Parameters Constraints Formulae Number

--- M,f,h,6e A=- a 0 M < 1 E M (sfR + h6e) 4-32
a h(Ms +y) B Ms + .8 f < 1 Ms + a

Ah5e 6e
h,6e M,f M= yBf a 0 < B < a E = Bh sAR + 4-33
aa afa Bea Bs ea Bs s- Bs

a a a
M f(6ea Bs) 6e
f6 Mh h = 0 < B < a E = Ba (AsR + 6e, Bs) 4-34

h,f M,6ea M = (f hA) Bsf A < f E = (f hA)R + hB 4-35
a hAs ea (f hA) h

Mf h,= yf a = B(Ms + none E Mf sR +B 4-36
a A(Ms + y) a M Ms + y MA

_Ah(Ms + ) B(Ms + _)Ms
M,h f,6e, f = Ah(Ms e M none E = h MsAR + B] 4-37

M,6ea h,f --- -- --- no solution

Table 4-1.

All of the equations in Table 4-1 have the same basic form because

A, B and the parameters are considered constant for the measurement per-


E = CR + D 4-38

With A and B determined from clear sky data (requires surface tempera-

ture, air temperature, and net radiation), and two independently esti-

mated parameters, all that is needed in any of the formulae is a net

radiation estimate. If net radiation can be estimated for partly cloudy

or cloudy skies, ET rates for these conditions can also be calculated.

A significant advantage of considering the parameters approximately con-

stant arises in computing cumulative evapotranspiration. Using the gen-

eralized version of the equations in Table 4-1 (Eq. 4-38), the cumula-

tive ET rate for some period (p) can be calculated as follows:

Ep = Edt =. (CR + D)dt

= Cfp Rdt + DJp dt

Ep = CR + Dtp 4-39

where E is the cumulative ET over the estimating period,
Rp is the cumulative positive net radiation over the period,
tp is the duration of positive net radiation during the
P estimating period, and
C and D are constants calculated from A, B, and estimates of two
parameters as shown in Eqs. 4-33 through 37.

Since the parameters are approximated as constant, no interpolation is

necessary for cumulative ET estimates. Use of an interpolation scheme

may be required to more accurately determine R depending on the time

interval between satellite data sets.

Use of particular versions of the ET equation listed in Table 4-1

is discussed in the remainder of this section. It is difficult to pre-

dict particular applications or weaknesses of these equations; they

depend both on the accuracy with which A and B can be determined, and

which of the parameters are most conveniently supplied from other mea-


In general, the most difficult parameter to estimate independently

is moisture availability. Equations 4-33, 4-34, and 4-35 assume that it

and one of the other parameters are unknown. It can be anticipated that

the first two of these equations will have difficulties with low satura-

tion deficits; Eq. 4-34 predicts infinite ET as the saturation deficit

nears zero. This is a result of having to express the parameters in

terms of the saturation deficit. From the substitutions, it is clear

that 6ea must remain larger than Bs in order for physically real (posi-

tive) parameter values to result. Equation 4-33 seems to be the most

sensitive to low saturation deficits since it is meaningless even when

6ea is equal to Bs; Eq. 4-34 reduces to E = fR with this condition (no

sensible heat is generated; all available net radiation is consumed in


Equations 4-33 and 4-34 are also very sensitive to the value of B

obtained from the temperature gradient net radiation correlation. This

presents another potential problem in their use because the clear sky

data collected in a given estimation period may not be able to estimate

B at an accuracy level commensurate with its influence on the ET esti-

mate. Once the behavior of B is better known, however, these equations

may become useful in areas where the least is known about the surface.

Saturation deficit is probably the parameter with the least spatial var-

iability, and since it is directly measurable it can be estimated for

large areas. Equation 4-33 might be useful over a large grassland area

or large area of short crops since the transport coefficient of this

type of area may also be approximated. For large forested areas, where

it can be assumed that f=l, Eq. 3-34 might work well. In both equations,

the only other estimate required is the slope of the saturation vapor

pressure curve, which is a known function of temperature.

Equation 4-35 assumes that moisture availability and saturation

deficit are unknown. The evapotranspiration equation generated has the

advantage that neither the psychrometric constant nor the slope of the

saturation vapor pressure curve needs to be evaluated. This makes the

equation easier to use, and since no division is involved, it is compu-

tationally safe. The additional parameters required are the bulk trans-

fer coefficient and the soil heat flux parameter. The former may be cal-

culable from measurements of windspeed and estimates of surface rough-

ness. This approach has a long history in the literature (see Chapter

.2), but its adaptability for use with a surface temperature has not been

demonstrated. The soil heat flux parameter only varies %20%, and might

possibly be estimated from near-infrared remotely sensed data, which

gives a good idea of vegetative cover. Because of its potential, simpli-

city, and easy graphical interpretation, Eq. 4-35 is used in the method

verification part of this study.

The last two equations, Eq. 4-36 and 4-37, assume that moisture

availability and either the soil heat flux parameter or the bulk tran-

sport coefficient are known. Since M is a normalized parameter, it can

be assumed to have various values in a constrained range. It is conceiv-

able that these equations might then be of some use.

Extension to Totally Remote ET Estimation Method

The final step in making a method depend only on remotely sensed

data is to generate air temperature measurements from surface

temperature measurements. Air temperatures are determined by the mixing

of air from higher in the atmosphere with near-surface air that has been

warmed (or cooled) by the surface. Therefore, one would expect air tem-

peratures, particularly those near the surface, to be primarily a func-

tion of the surface temperature. There is precedent for this approach.

In an effort to meet the boundary condition requirements of numerical

weather models, mesoscale climate modelers have begun developing "tem-

perature response functions" (Idso, 1982).

There are practical reasons for this approach as well. A network of

ground stations to supply this measurement would be very expensive, and

calibrating the ground-based sensors against a satellite sensor is in-

herently difficult. In addition to requiring knowledge of the atmo-

sphere's transmission properties, temperature measurements made by a

contact sensor (e.g., a thermocouple or thermistor) must be matched to

measurements made by a thermal radiation sensor. Since temperature dif-

ferences are required, a system based completely on remotely measured

surface temperatures would not be as sensitive to such absolute error


There are several ways in which air temperature measurements might

be generated. The simplest way is to choose a vegetation surface like

forest, whose surface temperature is closer to air temperature because

of the large fraction of surface elements visible to a satellite in

shade. The surface temperature of some reference pixel could then be

used as the air temperature for all pixels. In effect, the reference

pixel would never see a temperature gradient; its evapotranspiration

would exactly equal its available net energy (fR) since there could be

no sensible heat flux.

Another approach would be to simulate air temperature with a linear

combination of particular pixel temperatures. This involves empirically

choosing a set of reference pixels and determining coefficients for an

equation of the form:

Ta= a0 + alT+ a2 T + . 4-40

Both the length of time these coefficients are accurate and the extent

of the pixels for which they can be used to generate air temperature

measurements are important considerations in this approach. It should be

noted that similar considerations are involved in generating air temper-

ature measurements for all pixels from a limited number of ground sta-


The approaches suggested above are not developed any further in

this study.

Review of Assumptions

At the outset, the surface was conceived as a radiation-absorbing

(vegetation) layer, in contact with an air layer above and a soil layer

below. Energy fluxes inside the absorbing layer were considered irrele-

vant, and energy fluxes to and from the surface were considered one-di-

mensional and normal to the surface. It was assumed that the sensible

heat flux could be calculated from the surface-to-air temperature dif-

ference (i.e., that the radiation surface temperature is representative

of the effective heat transfer surface temperature). It was also assumed

that the vapor pressure gradient could be represented in terms of this

temperature difference and the saturation deficit of the air by way of

the linearized saturation vapor pressure curve. The time derivative term

and the photosynthetic heat flux term were considered negligible in the

energy budget equation.

These assumptions are typical in the evapotranspiration literature.

However, they amount to assuming a uniform surface with adequate fetch

and close to steady-state heat transfer. Realistically, the system is

always transient and there is some energy storage in both the surface

and air layer. Remotely sensed surface information in general does not

come from flat homogeneous surfaces.

Both the strict and totally remote temperature gradient response

methods are dependent on some degree of system stationarity. The former

assumes that only moisture availability and bulk air conductivity are

constant between data sets. It requires surface and air temperature, net

radiation, soil heat flux, and saturation deficit measurements.

To lessen the data requirements, these assumptions are extended and

a new parameter is introduced in the totally remote method. Moisture

availability, bulk air conductivity, and air saturation deficit are con-

sidered constant for a day or more, and soil heat flux is treated as a

constant fraction of net radiation. The surface is assumed to be a sta-

tionary system with four unknown parameters (M, h, 6ea, and f), and one

known parameter (s), the slope of the saturation vapor pressure curve.

To make up for data that cannot be collected, it is assumed that two of

the unknown parameters can be estimated from some prior knowledge of the

surface; the other two can be determined from the temperature gradient

response to various radiation loads. Once the parameters are determined,

evapotranspiration for any net radiation load is calculable.

Several elements of these methods were considered beyond the scope

of this study, but assumed possible. It is taken for granted that rea-

sonably accurate remote measurements of surface temperature and net ra-

diation can be made, further, that estimates of surface net radiation


are possible even with cloudy conditions. The pivotal assumptions are

that the surface temperature can be used as the effective heat transfer

temperature, and that air temperature measurements can be generated from

surface temperatures.




The central idea of the temperature gradient response (TGR) ap-

proach is to use the changes in the surface-to-air temperature differ-

ence relative to corresponding changes in net radiation in lieu of sur-

face parameters to estimate ET. A temperature gradient model is used to

determine the surface parameters from the temperature gradient re-

sponse; these are then used in calculating ET rates.

Use of the temperature gradient model, which expresses the func-

tional relationship between temperature gradients, net radiation, and

surface parameters,'involves a number of assumptions and approxima-

tions. Since the accuracy of the eventual ET estimate is limited by

this model and ancillary approximations, verification of the TGR meth-

ods is carried out in stages. Initially, the assumptions required are

individually examined. Then the implication for both the strict and

daily average TGR methods is demonstrated. For the most part, this is

done with data from a clear fall day, October 17, 1981.

The daily average TGR method is most suited for use with satellite

data. It is tested with practically all data collected. First, measured

temperature gradient/net radiation correlations are compared to those

predicted with independently measured constant parameters. Then the ET

estimates made with the correlations are compared to measured ET rates.

Validity of Assumptions

Radiation Temperature and Sensible Heat Transport

A simple expression involving a temperature difference and a bulk

heat transport coefficient (Eq. 4-6) was proposed to describe sensible

heat flux. It considers the sensible heat transport to and from the veg-

etation layer to be the same as that across a relatively thick layer of

air above the surface. This layer is at the average grass surface tem-

perature on one side, and at the average temperature of the air at a

specific reference level (a) above the surface on the other side. It is

assumed that the heat capacitance of air is very small and therefore

that the heat transport across the layer is close to steady. All the

properties of the air relevant to heat transport through this layer are

accounted for in the heat transport coefficient (h).

The most critical assumption in this formulation is that.the aver-

age radiation temperature of surfaces visible to an overhead sensor

(referred to as the surface temperature) and the effective heat trans-

fer temperature are the same, or related in some predictable way. In a

vegetation canopy, many small areas at different levels in the canopy

contribute toward a radiation surface temperature measured from above.

These represent only a small fraction of the total plant surfaces. All

surfaces, including those not seen by a radiation sensor, contribute

sensible heat to the air. These contributions depend on how well venti-

lated the canopy is at various levels.

Since radiation and sensible heat transfer are completely differ-

ent and a vegetation surface is so complex, the radiation surface tem-

perature and the effective heat transfer surface temperature may be dif-

ferent. The only way to compare these two temperatures is to infer

their relationship from observations of the temperature profiles over

the grass surface. Sensible heat flux was calculated using five temper-

ature measurements from the turbulent layer via the profile Bowen ratio

technique. One would therefore expect the heat flux to be proportional

to the temperature difference between the lowest and highest air tem-

perature measurements. The heat transport coefficient of this fully

turbulent layer can be calculated by solving Eq. 4-6 for the heat

transport coefficient

ht T T 5-
0 a
where TO is the lowest temperature measurement (35 cm) and Ta is the

highest (235 cm). If heat transport through the total air layer (between

the surface and highest air temperature measurement) is steady, the same

heat flux passes through it as the turbulent layer. Then the heat trans-

port coefficient of the total air layer can be calculated:
h = 5-2
s a
If the radiation surface temperature is the same as the effective heat

transfer surface temperature, one would expect the ratio of h/ht to re-

main constant over the course of a day. This ratio,

h T0 Ta
ht T Ta

can be most easily examined by plotting T Ta vs. TO Ta (see Fig.


The points in this figure would lie on a straight line intersect-

ing the origin if the relationship between the turbulent temperature

gradients (or effective heat transfer gradient) and the total surface-

to-air temperature gradients was constant. However, relative to a given







Figure 5-1.

To-Ta (0C)


Total vs. Turbulent Temperature Gradients for a Clear
Day. The total surface-to-air temperature difference
was calculated by subtracting the temperature measured
at 235 cm from the surface temperature. The turbulent
temperature difference was calculated by subtracting the
235-cm temperature from the 35-cm temperature. Data ar.e
from October 17, 1982; numbers indicate true solar time
at end of half-hour averaging period. Individual temper-
ature profiles for this day are plotted in Fig. 3 of
Appendix D.

effective heat transfer gradient, surface-to-air gradients are larger

in the afternoon than they are in the morning. This is because in the

afternoon, radiating surfaces lower in the canopy have also become

warm. Apparently, these surfaces make a relatively greater contribution

to the radiation surface temperature than they do to the sensible heat

flux via the effective heat transfer surface temperature. Although it

is not as extreme, this pattern is also observed on cloudy days, under

a diffuse radiation regime (Fig. 5-2).

One could reasonably expect radiation geometry to play a role in

creating differences between the radiation temperature and the effec-

tive heat transfer temperature. At high sun angles, when direct sun-

light is coming from angles close to the viewing angle of the radiation

sensor, less shaded area is visible to the sensor. The radiation tem-

peratures should peak relative to effective heat transfer temperatures

when the angle of incidence of direct sunlight coincides with the angle

of view of the sensor. The slight upward curvature in the total/turbu-

lent gradient correlation (Fig. 5-1) seems to confirm this effect, but

it is small in comparison to the morning/afternoon radiation tempera-

ture hysteresis.

The apparent difference in the effective heat transfer surface

temperature and radiation surface temperature must be viewed as a poten-

tial problem which may require modification of the equation used to com-

pute surface temperature. A time-varying factor may be required, espe-

cially in cases where the radiation geometry is further complicated by

surface slopes, as in mountainous areas.

Constancy of Parameters

The heat transport coefficient (h) and moisture availability (M)

are considered parameters in the strict TGR method, and they are

.4 .6
To-Ta (0C)

Figure 5-2.

Total vs. Turbulent Temperature Gradients for a Cloudy
Day. Temperature differences were calculated as in
Fig. 5-1; data are from October 30, 1981. Note that
compared to Fig. 5-1 the temperature scale is expanded
by a factor of five. Temperature profiles for this day
are plotted in Fig. 4 of Appendix D.





1.0 -





required to remain constant between sets of data. In the average TGR

method, the slope of the saturation water vapor pressure curve (s),

the saturation deficit (6ea), and the fraction of net radiation conduc-

ted into the soil (f) are also required to be approximately constant

for periods of a day or more. Though some of these variables are known

functions of measurable variables (e.g., s is a known function of tem-

perature), they must be considered parameters. This section shows how

these parameters vary over the course of a day.

The sensible heat flux is plotted as a function of the surface-

to-air temperature difference in Fig. 5-3a. The average heat transport

coefficient is represented by the slope of a line passing through the

origin and the center of gravity of the plotted points. The bulk air

conductivity for any half-hour period is computed as in Eq. 5-2 and has

been plotted in Fig. .5-3b.

It was shown in Figs. 5-1 and 5-2 that radiation surface tempera-

tures in the morning appeared cool relative to the effective heat trans-

fer surface temperature. Barring other factors, the resulting lower tem-

perature gradients would lead to higher calculated thermal conductivi-

ties for morning time periods. This does not show in Fig. 5-3b, how-

ever. The only apparent effect seems to be lowered conductivities

around noon resulting from apparently higher surface temperatures while

relatively less shaded areas are visible to the sensor.

It is difficult to say anything conclusive about the heat trans-

port coefficient in the early morning or late afternoon. Temperature

gradients are in the process of changing direction, making the calcula-

tion of h somewhat unreliable.


> 4-

0 2 4 6 8 10

.06 Ts-Ta (oC)

.02 h=.034 LY/MC


8 10 12 14 16
TIME (TST OCT. 17, 1981)

Figure 5-3. Heat Transport Coefficient Data.

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