Error analysis of evapotranspiration measurements

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Authors Hartman, Robert Kent.

Publisher The University of Arizona.

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ERROR ANALYSIS OF EVAPOTRANSPIRATION

MEASUREMENTS

by

Robert Kent Hartman

A Thesis Submitted to the Faculty of the

SCHOOL OF RENEWABLE NATURAL RESOURCES

In Partial Fulfillment of the RequirementsFor the Degree of

MASTER OF SCIENCEWITH A MAJOR IN WATERSHED MANAGEMENT

In the Graduate College

THE UNIVERSITY OF ARIZONA

1980

)2,DR. MARTIN M. F

DR. LOUIS H. , JR.Assistant Profes r of Watershed

Management

Date

STATEMENT BY AUTHOR

This thesis has been submitted in partial fulfillment of re-quirements for an advanced degree at The University of Arizona and isdeposited in the University Library to be made available to borrowersunder rules of the Library.

Brief quotations from this thesis are allowable without specialpermission, provided that accurate acknowledgment of source is made.Requests for permission for extended quotation from or reproduction ofthis manuscript in whole or in part may be granted by the head of themajor department or the Dean of the Graduate College when in his judg-ment the proposed use of the material is in the interests of scholar-ship. In all other instances, however, permission must be obtainedfrom the author.

SIGNED: 7...-;AJ;T: - )7

APPROVAL BY THESIS COMMITTEE

This thesis has been approved on the date shown below:

DR. L OYD W. GAY,/ esis DirectorProfessor of Watershed Management

)77a4L/Z 2, / 980Date

ACKNOWLEDGMENTS

This thesis is based upon research supported by the following

grants (Lloyd W. Gay, Principal Investigator): U.S. Department of the

Interior (Project C-6030) as authorized under the Water Resource Re-

search Act of 1964, as amended; by the Arizona Agricultural Experiment

Station, Hatch Project 04; and by the U.S. Geological Survey, Grant

No. 14-08-0001-G-617.

TABLE OF CONTENTS

LIST OF TABLES Page

LIST OF TABLES

LIST OF ILLUSTRATIONS

ABSTRACT

1. INTRODUCTION 1

11

Definition of ProblemScope and Objectives of Study

2. ENERGY BUDGET ANALYSIS 3

The Energy Budget 3Principal Energy Budget Components 4The Model 7

The Bowen Ratio Method 7Required Measurements 8Applying the Bowen Ratio Method 9

Basic Assumptions 9Sensor Positioning 10Time Averaging 12Other Problems 13

3. ENERGY BUDGET ERRORS 15

Sources of Errors 15Systematic Errors 16

Net Radiation 17Soil Heat Flux 17Bowen Ratio 18Sources Combined 19

Random Errors 20Net Radiation and Soil Heat Flux 22Bowen Ratio 23Combined Variance Sources 27

4. MINIMIZING ERRORS 28

Sampling Considerations 28

iv

TABLE OF CONTENTS--Continued

Page

Eliminating Biases in Gradient Measurements 30

5. APPLICATION TO FIELD MEASUREMENTS 37

Field Measurements 37The 1978 Study 40The 1979 Study 41

Error Analysis, 1978 42Error Analysis, 1979 48Future System Improvements 55

6. CONCLUSIONS 57

APPENDIX A: SYMBOLS AND DEFINITIONS 60

APPENDIX B: ENERGY BUDGET TABULATIONS AND GRAPHS 63

APPENDIX C: ERROR ANALYSIS DATA 89

REFERENCES 108

LIST OF TABLES

Table

1. Example of bias elimination in gradient measurements

Page

through the exchange method 32

2. Mastwise comparisons of average half hour energybudget flux densities for the 1978 field study . 43

3. Error analysis data for mast #1 on 27 May 1978 44

4. Error analysis data for mast #2 on 27 May 1978 45

5. Mastwise comparisons of average half hour energybudget flux densities for the 1979 field study . 50

6. Error analysis data for mast #1 on 15 July 1979 52

7. Error analysis data for mast #2 on 15 July 1979 53

vi

LIST OF ILLUSTRATIONS

Figure Page

1. Relationship between the wet-bulb bias and theuneliminated bias in the vapor pressuregradient 36

2. Mastwise comparison of 95% confidence intervalfor latent energy flux density on 27 May 1978 . 46

3. Mastwise comparison of 95% confidence intervalfor latent energy flux density on 15 July 1979 . . 54

vi i

ABSTRACT

Model, site, sampling and instrumentation requirements for the

Bowen ratio energy budget method were evaluated to provide guidelines

for proper application. An error analysis procedure was developed to

evaluate system performance and identify potential areas of improve-

ment.

The procedures were applied to two sets of field data to

identify the measurement errors. Bowen ratio energy budget measure-

ments were made over an extensive stand of mesquite (Prosopis pub-

escens) in the San Pedro River valley near Mammoth, Arizona in May,

1978 and over a kochea (Kochea scoparia) pasture adjacent to the Pecos

River near Roswell, New Mexico in July, 1979.

Mean daily evapotranspiration over the mesquite stand was

about 5 mm. The error analysis indicated mean half-hour 95 percent

confidence intervals for latent and sensible energy of 0.48 + 0.09 and

0.21 + 0.09 cal/cm2/min respectively, with a majority of the error

originating in the Bowen ratio measurement.

Diode circuitry and calibration procedures were improved for

the July 1979 study. The mean daily evapotranspiration over the

kochea pasture was about 3 mm. The 95 percent confidence intervals on

the mean half-hour estimates of latent and sensible energy were 0.27 +

0.01 and 0.23 + 0.01 cal/cm2 /min respectively. The increased

viii

ix

measurement precision was attributed to the improvements made in the

system.

CHAPTER 1

INTRODUCTION

Definition of Problem

In arid regions water represents a critical resource often

limiting economic growth and development. In Arizona the current an-

nual water use exceeds the total annual surface supply and thus ground

water storage is decreasing. Providing for future needs will require

intensive conservation practices and development of current and new

sources.

Phreatophytes transpire millions of acre feet of groundwater

from floodplains and watercourse areas in the western U.S.A. each year.

Previous research indicates that phreatophyte communities consume be-

tween four and eight feet of water annually depending on site and vege-

tation conditions (Horton and Campbell 1974).

Vegetation management in phreatophyte communities represents a

possible means of augmenting the water supply in the southwest. Fur-

ther research is needed to refine methods of evaluating evapotranspira-

tion over phreatophytes and other vegetation types to produce accurate

data from which sound vegetation management decisions can be made.

Scope and Objectives of Study

The primary purpose of this study is to identify application

requirements and improve measurement techniques for the Bowen ratio

1

2

method of evapotranspiration estimation. The techniques will be tested

through application to evapotranspiration measurements obtained over

well-watered vegetation during periods of high evaporative demand. The

specific objectives are:

1. To isolate the problems and requirements of accurate applica-

tion of the Bowen ratio method,

2. To develop recommendations for improving Bowen ratio measure-

ment techniques,

3. To develop an error analysis to determine the validity of

energy budget data and identify areas of potential improvement,

and

4. To illustrate the application of the analyses and demonstrate

the effects of system improvements by use of data from two

different field experiments.

CHAPTER 2

ENERGY BUDGET ANALYSIS

The purposes of this chapter are to: (1) explain the theoret-

ical basis of the energy budget and the Bowen ratio as a means of

solution; (2) define the principal energy budget components; and, (3)

summarize the conditions and requirements necessary for proper Bowen

ratio method application.

The Energy Budget

Before the energy budget can be strictly defined, the system

to which it applies must be identified. In general, a system is a

volume with prescribed boundaries, e.g., a box. Energy may be trans-

ferred into and out of the system through its boundaries and as the

system has volume and associated mass, energy may be stored. As energy

is always conserved, the energy budget for the system can be described

as a function of the energy inputs, outputs,andchanges in storage.

The system of interest for the energy budget analysis of evapotrans-

piration is generally a small portion of the earth's surface. The

horizontal boundaries of the system are restricted by the geographical

extent of the area for which measurements of energy transfer and stor-

age are representative. The lower boundary of terrestrial systems lies

at a depth in the soil medium where the vertical transfer of heat is

3

4

negligible. The upper boundary lies at the surface-atmosphere inter-

face or vegetation-atmosphere interface if the surface is vegetated.

Principal Energy Budget Components

The transfer of energy into and out of the system is accomp-

lished through the processes of radiation, convection, conduction, and

chemical transformation (Budyko 1956, Sellers 1965). Evaluation of

these processes will yield the principal components of the energy

budget.

Radiation is an electromagnetic phenomenon by which heat or

energy can be transferred in the absence of a medium. The intensity

and spectral quality of the radiation varies as a function of the

emitting body's absolute temperature and emissivity (Charney 1945).

As radiation can be reflected or absorbed in the system and many

sources exist, the net exchange of radiation, Q*, is of interest in

the energy budget. Net radiation is the algebraic sum of short and

longwave radiation where flux into the system is considered positive

and flux out of the system negative. Net radiation can be written:

Q* = - Kt + - Lt (1)

where K and L represent short and longwave respectively and the arrows

indicate their direction.

Conduction is a process through which heat is transferred by

direct molecular contact. Conduction requires a medium, is directed

towards lower temperature, varies directly with the temperature gradi-

ent, and is influenced by the thermal characteristics of the medium.

5

In the energy budget, conduction is associated with the system's

energy storage and the eventual liberation of stored energy for trans-

fer out of the system. The soil represents the major storage medium

in most terrestrial systems. Changes in soil temperature when combined

with volumetric heat capacity yields the soil heat flux (G). When the

soil medium is liberating energy (cooling) the soil heat flux is con-

sidered positive (Vries 1963).

Convection is a process through which heat is transferred both

by conduction and by the displacement of molecules within a fluid.

Convection occurs in the direction of lower temperature. The intensity

of the convective flux is a function of the temperature difference and

the mechanism of displacement. Buoyant forces predominate in free con-

vection and external forces, e.g., wind, predominate in forced convec-

tion. The transfer of heat between the system and the atmosphere is

the sensible heat flux (H). Analytically, H is represented as follows:

H = p Cp Kh [d(Ta + rz/dz]

(2)

where p is the air density, C is the heat capacity of the air, Kb is

the transfer coefficient for sensible heat, and d(Ta

+ rz)/dz is the

change in temperature potential over the change in vertical separation

z, corrected for the dry adiabatic lapse rate, rz. Sensible heat flux

into the system is considered positive (Sellers 1965).

Chemical transformations within the system can account for an

appreciable portion of the energy budget. The transformation of water

between the liquid and vapor states is the most important of these and

6

requires a significant amount of energy. The latent heat of vaporiza-

tion for water, L, is the amount of energy required in the vaporization

or liberated in the condensation of a unit volume of water. The latent

heat of vaporization for water varies inversely with temperature as

given in the following equation.

L (cal/g) = 597.993 - 0.5475t (3)

where t is the temperature of the water in degrees Celsius (Beers

1945).

The flux of water vapor in and out of the system can be con-

sidered an energy flux equivalent to the latent heat required for its

vaporization. The transfer of water vapor in the atmosphere is

directed towards locations of lower water vapor concentration subject

to displacement mechanisms. The latent energy flux (LE) of vapor

evaporating out of or condensing in the system is calculated as

LE = (pLE/p) Ke (de/dz) (4)

where p is the air density, L is the latent heat of vaporization, c is

the ratio of the molecular weights of water to air, p is the atmos-

pheric pressure, Ke

is the transfer coefficient for water vapor, and

(de/dz), the vapor pressure gradient is the change in vapor pressure

over the change in the vertical separation z (Sellers 1965). Normal

surface evaporation represents an energy loss to the system and the

latent energy flux would be considered negative.

7

The magnitudes of additional energy budget components, such as

the chemical energy fixed in photosynthesis, are small in comparison

to the radiative, convective, storage, and latent components defined

previously. Since these small components are also difficult to mea-

sure, their exclusion is a common practice.

The Model

The energy budget is obtained by combining the major energy

flux densities.

Q* + G + LE + H = 0 (5)

The major energy flux densities are usually evaluated over a short

period of time, e.g., one hour, and presented as the mean for that

period.

The Bowen Ratio Method

Bowen (1926) first employed the ratio of convection to evapo-

ration to estimate evaporation from water surfaces. The method assumes

that the transfer coefficients of sensible heat and water vapor (Kh

and Ke) are equal. Bowen's original work applied specifically to water

surfaces, but the method is now being successfully used for evapotran-

spiration from terrestrial systems.

The Bowen ratio (r3 = H/LE) is formed by combining the sensible

and latent heat fluxes (Equations [2] and [4 1 ) based upon measurements

made over the vertical separation z.

a = H/LE = --aLE I dePC [d(Ta

+ rz (6)

8

Rearranging the energy budget Equation (5) and substituting a for HILEshows the significance of the Bowen ratio in terms of an energy budget

solution.

Q* + G + LE + H = 0

Solving for LE:

LE = -(Q* + G)/1 + a) (7)

Solving for H:

H = - (Q* + G)/f3(1 + (3)

(8)

Required Measurements

The evaluation of the energy budget using the Bowen ratio re-

quires the measurement of net radiation flux density, soil heat flux

density, the temperature gradient, and the vapor pressure gradient. Net

radiation and soil heat flux densities are measured with a net radi-

ometer and a soil heat flux disc, respectively. Typical flux density

units are watts per square meter or calories per square centimeter per

minute. The determination of the dimensionless Bowen ratio requires

that the temperature and vapor pressure gradients be measured in the

atmosphere near the evaporating surface. The gradients are commonly

determined through the use of wet-bulb psychrometers set near the sys-

tem's upper boundary. The calculation of saturated vapor pressure, es ,

is taken from Murray's (1967) approximation

(b*t)c+t

e = a * es

where a is 6.1078, b is 17.2694 and c is 237.3 for all temperatures, t,

above 0°C and e is the base of the natural logarithm. The vapor pres-

sure is then determined through evaluating the saturation vapor pres-

sure at the wet-bulb temperature, tw , and subtracting the vapor pressure

deficit as determined through the psychrometric formula

e = es - Ap (1 + 0.00115 * t) (t - tw w

) (5)

where A is apsychrometric constant, p is pressure, and t is the dry-

bulb temperature. In this discussion, units will be (°C/m) for tem-

perature gradients and (mb/m) for vapor pressure gradients.

Applying the Bowen Ratio Method

For proper application of the Bowen ratio method, a number of

site and instrumentation requirements must be met. Due to the assump-

tion made in the Bowen ratio, failure to meet the requirements will

consistently lead to erroneous nonrepresentative results.

Basic Assumptions. The primary assumption allowing a simpli-

fied assessment of the ratio of sensible heat to latent energy is that

the ratio of their transfer coefficients is unity. Under most condi-

tions the error in assuming Kh = Ke

is negligible when turbulent mixing

dominates the exchange, i.e., when Richardson numbers are in the range

of + 0.03 (Tanner 1963). The Richardson number (Ri) represents the

ratio of the rate at which mechanical energy for turbulent motion is

9

(9)

10

being dissipated (or produced) :Dy buoyant forces to the rate at which

mechanical energy is being produced by inertial forces (forced convec-

tion) (Sellers 1965). The Richardson number is

Ri =

-z2gAT h— z (11)

T(Au)2z1

where g is the acceleration due to gravity, AT is the potential tem-

perature gradient over the vertical separation z, T is the average

temperature of the layer, Au is the windspeed gradient over the verti-

cal separation z, and z1

and z2 are the heights at which the tempera-

ture and windspeeds are measured. It is most likely that the transfer

coefficients of sensible heat and latent energy will be equal when

winds are strong and friction contributes much more to turbulence than

does buoyancy (Ri is small) as the sensible heat and latent energy

fluxes will be transported via the same mechanism (Tanner 1960).

A recent study of transfer coefficients for sensible heat and

latent energy over alfalfa and soybeans indicated the Kh > Ke

during

advective conditions (Verma, Rosenberg,andBlad 1978). Their find-

ings were supported by Warhaft's (1976) theoretical analysis in which

he concludes the greatest departure of Kh/Ke from unity will occur when

temperature and humidity gradients are of opposite sign. Tanner (1960)

points out however that as long as f3 is not less than -0.5 the error in

LE is significantly less than the error in caused by assuming Kh = Ke

due to the form of Equation (7).

Sensor Positioning. Sensor positioning is also of importance

for proper measurement of temperature and vapor pressure gradients.

11

The lower sensor should be placed as low as possible to avoid the

effects of thermal stratification but must be high enough to avoid ir-

regular influences close to the vegetation (Tanner 1968, Webb 1965).

The upper sensor should be placed high enough to generate an instru-

ment separation over which the gradients can be assessed but below the

level where readings may be adversely affected by the influence of

adjacent surfaces.

As air moves from one surface to another the velocity, tempera-

ture and vapor pressure gradients change from those representing the

first to those representing the second. In the case of air moving from

a dry surface to a moist area, the air is colled and moistened by the

absorption of latent energy and the temperature and vapor pressure

gradients become modified (Webb 1965). The modification is not immed-

iate, rather the depth of the modified zone increases with increasing

distance from the boundary (Tanner 1968). The rate at which the modi-

fied zone develops depends upon the difference in surface conditions

and upon the wind speed and roughness of the second surface as turbu-

lence enhances mixing and thus the rate of modification.

For valid assessment of the temperature and vapor pressure

gradients the upper sensor must be located within the modified zone.

As the depth of the modified zone increases with fetch or distance from

the boundary, a height-fetch ratio is typically used to identify the

appropriate height of the upper sensor. A height-fetch ratio of 1:100

indicates that the distance from the zero plane of displacement (the

system's upper boundary) to the upper sensor is one one-hundredth of

12

of the horizontal distance from the instrument to the upwind boundary.

Due to the effects of surface differences, wind speed and surface

roughness, no universal height-fetch ratio can be applied. However,

Tanner (1963) suggests a minimum height-fetch ratio of 1:50 if the

discontinuity (change in surface conditions) is small and at least

1:100 or 1:200 if the discontinuity is large. For grass 15 to 20 cm

tall, Webb (1965) favors a height-fetch ratio of 1:200 with more or

less for smoother or rougher surfaces. In selecting an experimental

site, researchers should carefully consider the effects of surface

properties on the height-fetch ratio.

Time Averaging. Time averaging can be employed to reduce the

effects of transients occurring at the surface (e.g., vapor pressure

or temperature changes brought about by radiation variation, turbulence,

etc.). Short period evaluation of the energy budget during which

significant radiation variation occurs may be misleading due to the

time required for the temperature and vapor pressure gradients to ad-

just to a change in available energy ( 2* + G). Tanner (1963) suggests

minimum averaging periods of 10 to 30 minutes. Fluctuations in net

radiation, especially during days of intermittent cloud cover, often

require that mean flux densities be estimated for periods of at least

one half to one hour in length (Monteith 1973). Some success has been

reported for averaging periods of four to eight hours (Tanner 1963),

although errors may result from fluctuations in the Bowen ratio over

such a long period. Webb (1960) discusses the error introduced by a

fluctuating Bowen ratio when long period means of the temperature

13

and vapor pressure gradients are used and indicates how to correct for

it.

Other Problems. Problems in energy budget analysis often occur

when atmospheric conditions are unsteady. Times near sunrise and sun-

set occasionally yield unrealistic results due to rapid atmospheric

change. Conditions of patchy low clouds, which cause both unsteadiness

and horizontal non-uniformity may also cause problems if the instrumen-

tation is not of sufficient quality. For this reason Bowen ratio

energy budget analysis is ideally performed during periods of clear

weather and maximum evapotranspiration.

Canopy structure also affects application of vertical profile

measurement. If a canopy is sparse or broken or of such structure as

to permit horizontal advection of heat and water vapor, vertical pro-

file measurements will not represent the true flux. The main criteria

is not height but whether or not the structure permits airflow that

results in significant errors (Tanner 1963).

As the gradients of temperature and vapor pressure can be ex-

tremely small, instrumentation must be of high quality. Webb (1965)

suggests an accuracy of temperature gradient measurenents of 0.01 to

0.03°C will yield satisfactory results. The effects of sensor accur-

acy and precision will be discussed in Chapter 3.

Occasionally, the Bowen ratio method fails due to the formula-

tion of the Bowen ratio (Equation [61). When H and LE are equal in

magnitude and opposite in sign (.3 = -1), Equations (7) and (8) are un-

defined, In this case the available energy (Q* + G) must equal zero

14

as Q* + G + H + LE = 0 and therefore the condition occuis only during

periods of minimal flux such as during sunrise and sunset or at night.

Another undefined form arises when the vapor pressure gradient is zero.

Here, LE is zero and H is equal in magnitude and opposite in sign to

the available energy.

CHAPTER 3

ENERGY BUDGET ERRORS

Evaluation of the energy budget via the Bowen ratio method re-

quires the measurement of atmospheric variables. The measurement of

these variables is not without error and thus the evaluation of the

energy budget contains a related error. Error analysis of the Bowen

ratio method has been studied by Fritschen (1965b), Fuch and Tanner

(1970), Sinclair, Allen and Lemon (1975) and Revfeim and Jordan (1976)

among others. As the error analysis procedures depend upon the experi-

mental approach, the procedures found in the literature are often dif-

ficult to apply. This chapter will identify the types and sources of

errors in the measurement of evapotranspiration and develop an error

analysis procedure applicable to the experimental approach taken in

this study.

Sources of Errors

All measurements are subject to three types of errors; system-

atic errors, mistakes, and accidental errors. Systematic errors are

constant and affect all measurements alike. They are the result of

general imperfections and can usually be eliminated by applying the

proper corrections. Mistakes fail to follow any pattern and can only

15

16

be avoided through caution and experience. Accidental errors follow

the laws of chance and can usually be minimized.

The sources of errors (mistakes, systematic errors and acci-

dental errors) in an energy budget analysis include the errors in

individual sensors, in reading the sensors and in application. Errors

in application are mistakes and can be avoided by carefully analyzing

the site's suitability as discussed in Chapter 2. Systematic errors

arise from a constant offset or bias recorded by individual instru-

ments or the data acquisition system. Errors due to a fluctuating

Bowen ratio over an excessive sampling period (design problem) are

also systematic and can usually be corrected via the procedure in Webb

(1960). Accidental errors are random and occur in the instruments and

the data acquisition system. Systematic and random errors in the

energy budget will be examined in more detail.

Systematic Errors

Uncorrected systematic errors in the variables of a function

cause a systematic error in the resultant value. Letting Y be a func-

tion of n independent variables,

Y = f(x l , x2 , , xn) (12)

systematic errors in the x's, Ax, will cause a systematic error in Y,

AY, so that

Y + AY = f(X1 +Ax

1 , x

2 +A x

2 , , x

n +A x

n) . (13)

17

Taylor series expansion of the right hand member of (13) and subtrac-

tion of (12) yields the systematic error in Y

DY DyAY = Ax + Ax

2 +1 + x LAXn .Dx

1 ox2 D

n(14)

Thus, if the systematic error in the variables of the energy

budget can be defined, the resulting systematic error in latent energy

flux density and sensible heat flux density can be estimated.

Net Radiation

Systematic errors in the measurement of net radiation are dif-

ficult to determine as no true standard exists. Fritschen (1965b)

estimated the accuracy of the net radiometer alone to be + 0.02

cal/cm2 /min. Fuch and Tanner (1970) estimated net radiometer accuracy

based on a comparison of two independently calibrated sensors to be

+ 3 percent. From Equations (7), (8) and (14) the systematic error in

Q*, AQ*, would produce the following systematic errors in latent energy

and sensible heat flux densities:

aLEALEQ* = aQ* x AQ* -AQ* (14i3) (15)

AHQ*

aH= x AQ* -f3 x AQ*

(l+r3) (16)

Soil Heat Flux

Systematic errors in measurements with a soil heat flux disc

involve calibration, recording and design. Design errors are associ-

ated with dimensions and with failure of the heat capacity of the

AI3

(19)

At +(g)

At(g)

• Ae I(g)

• Ae (g)]

18

sensor to equal that of the soil (Fritschen and Gay 1979). Fritschen

(1965b) estimated the accuracy of the disc alone to be + 0.02 cal/cm2/

2.min. Since the normal range of G is about + 0.10 cal/cm /min this

estimate is much larger than the + 5 percent figure reported by Sin-

clair, Allen, and Lemon (1975). The effects of systematic error in G,

AG, on LE and H can be estimated from Equations (7), (8) and (14) as

3LE -AG ALE

G = x AG

• (1+6)(17)

and

AHG

BHx AG

-6 x AG (1+6)

(18)

Bowen Ratio

Systematic errors in the temperature and vapor pressure grad-

ients generate errors in the Bowen ratio From Equations (6) and (14)

the systematic error in 6 caused by systematic errors in the tempera-

ture and vapor pressure gradient measurements would be:

where t(g) and e (g) are the temperature and vapor pressure gradients

respectively. The effect of systematic error in 6, A6, on LE and H

can be estimated from Equations (7), (8) and (14) as

ALES =3LE LE

x A(3 - x(11-(3)

(20)

and

AH = x Af3 -LE x ABB DB (1143)

With the aid of periodic sensor exchange (Sargeant and Tanner

1967) it is possible to remove all the systematic error from the tem-

perature gradient measurement. However, even with exchange of sensors,

some systematic error will remain in the vapor pressure gradient mea-

surement when the systematic error in one or both wet-bulb temperature

sensors is large. The bias that remains in the vapor pressure grad-

ient measurement is caused by the difference in the slope of the vapor

pressure curve at the true versus measured temperature. The relation-

ship between bias remaining in the vapor pressure gradient and system-

atic errors in the wet-bulb temperature will be discussed in more de-

tail in Chapter 4.

Sources Combined

The total systematic error in latent energy and sensible heat

flux densities can be estimated by combining the errors in net radia-

tion, soil heat flux and the Bowen ratio.

aLE am[

amALE = x AQ* + aG x AG + at3 x AB

and

r B H 9HAH = x AQ* + x AG + x AB9Q*

The total systematic error can be evaluated only if the errors

19

(21)

(22)

(23)

in the variables Q*, G, and are known. If the systematic errors AQ*,

20

AG, and Af3 are unknown but are believed to fall within specific limits,

Equations (22) and (23) can be used to estimate the limits of the sys-

tematic error in LE and H through combining error limits and partial

derivatives in a maximal and minimal manner.

Random Errors

An analysis of accidental or random errors in the energy budget

provides a means of assessing the variability of the measurement sys-

tem. The analysis can be used to estimate the variability of previ-

ously collected data as well as to isolate problem areas for variance

reduction in subsequent measurements.

There have been a number of "probable error" analyses of Bowen

ratio measurements (Holbo 1973). Probable error is the interval which

will contain one half of the errors, and thus is synonomous with a

fifty percent confidence interval. The procedure based on Scarborough

(1966), is as follows.

Equation (14), which provides the systematic error in Y, AY,

holds for any kind of error. If the errors in the x's are random and

normally distributed, then the error in Y will be random and normally

distributed. If the probable error of the x's, denoted by ri , are

known then the probable error of Y, denoted by R, can be calculated.

2 2 2 3e

R = [Y2 3Y

r —2 ,

r3Y

+n2] (24)

nax

11 + ax2

r2 ax

This analysis of the probable error of the energy budget may be

incorrect, for the method requires that the probable errors of the

21

variables be known while in fact they can only be estimated. The vari-

anceterminEquation(24)isr..If r. is known, a normal or z dis-

1 1

tribution applies and Equation (24) is valid. If the variance term is

estimated, Student's t distribution applies. Use of the z distribution

in place of the t distribution may lead to a substantial underestimate

of the random error, and thus generate a narrower confidence interval

than is correct. This problem diminishes as the degrees of freedom

for the variance term becomes large, so that the t distribution ap-

proaches the z distribution.

Since the random errors in the energy budget variables are

estimated, rather than known, the proper procedure for confidence

interval estimation is to calculate the variance and apply the proper

t statistic. Once the variance of an energy budget component is esti-

mated, the probable error or any other confidence interval can be

established.

Proceeding as in Scarborough (1966), the variance in Y, V(Y),

where Y is a function of n independently measured variables is a func-

tion of the variance in xl , V(x1 ), in x2 , V(x 2 ), and so forth:

V(Y) = [— V(x ) +

ay ayV(x) + + V(xn) . (25)

ax1

1 a 2x2

Therefore the estimation of variance in latent energy flux density

requires the estimation of variance in Q*, G, and (3 and evaluation of

the partial derivatives of LE with respect to Q*, G and f3. The vari-

ance of sensible heat flux density is estimated similarly with the

partials of H with respect to Q*, G and (3.

22

Net Radiation and Soil Heat Flux

The sources of variance in the measurements of net radiation

and soil heat flux are the sensor's precision and the precision and

resolution of the data acquisition system. The most convenient way of

estimating the variances of Q* and G is through the use of the sensor's

calibration data. From the calibration data the variance in Q* and in

G would be:

V(Q*) = MSE (Xh (X 1 X) 1Xh

)

and

V(G) = MSE (Xh (X'X) - 1Xh )

where MSE is the mean squared error of the calibration regression, Xh

-1 iis the transposed mean response matrix, (X'X) s the inverse of the

data matrix premultiplied by its transpose, and Xh is the mean response

matrix. The matrix approach to regression analysis is discussed in

many statistics texts, e.g., Neter and Wasserman (1974). If the same

data acquisition system is used for calibration and field work, the

effects of the data acquisition system's precision and resolution will

be contained in the mean squared error of the regression.

When calibration data is not available the variance of Q* and

G can be estimated from the manufacturer's specifications. The pre-

cision of net radiometers and soil heat flux discs can be estimated

from the reported linearity of response, which will be taken here as

representing + 2 standard deviations. Further, the specified precision

(26)

(27)

23

of the data acquisition system is also taken here as representing + 2

standard deviations.

The resolution of the data acquisition system represents a

random error when many samples are taken; this can be manipulated to

represent + 2 standard deviations. Since the distribution of resolu-

tion errors is uniform and + 2 standard deviations covers 95% of the

maximum resolution error, the variance representing + 2 standard de-

viations would be 95% of the maximum resolution error. Thus the

sensor's linearity of response and the data acquisition system's pre-

cision and resolution, each representing 2 standard deviations, can be

combined to yield an estimate of the variance in the measurement of Q*

and G.

V(C) =

-

C( 17 ( 1r

+ sp

) + sr

) k2

(28) 2

where C is the measured component (Q* or G), k is the linear calibra-

tion coefficient, and 1 , s , and s are 2 standard deviations for ther p r

linearity of response, and the data acquisition system precision and

resolution respectively. Units are: 1r and s, percent; s r , units of

sensor output; and C, energy flux density units. The numerator of

Equation (28) is the combined random error representing 2 standard

deviations in the measurement of the component. Division by 2 yields

the standard deviation of C, the square of which is the variance.

Bowen Ratio

Variance in the Bowen ratio arises from the variance of the

temperature and vapor pressure gradient measurements. The effects of

24

variation in normal atmospheric pressure at a point are slight and

will be neglected. However, variations in pressure with change in

elevation should be taken into account as was done by Revfeim and

Jordan (1976).

The calculation of variance of the temperature and vapor pres-

sure gradient measurements is based upon individual sensor calibration

data. The sensors should be calibrated with the same system as used

in the field so the system variance is included in the mean squared

error for each calibration regression. The relationships that follow

apply specifically to the psychrometer exchange method. However, the

necessary adjustments should be minor for different measurement tech-

niques.

The variance of each individual dry-bulb temperature measure-

ment, t.,t., is based upon the mean squared error (MSE) from the previ-

ously defined calibration matrices:

V(ti ) = MSE (Xh (X'X)-1

Xh ). (29)

In the psychrometer exchange method, n is the number of exchange

periods in the averaging period, and the variance of the temperature

difference measurement, At, is

V( At )

E V(t. )up

v(t)i=1 i=l

dn(30)

n2

Calculation of the variance in the vapor pressure gradient

measurement is slightly more complex. The sources of variation are

25

the wet- and dry-bulb temperature measurements. Required calculations

are the variance of each wet- and dry-bulb temperature measurement and

the partial derivatives of vapor pressure with respect to each tempera-

ture, t. and tw..

1 1

V(twi ) = MSE (Xh (X'X) -1

)5h)

(31)

Be= -Ap - Ap(0.00115 * tw i ) (32)

[17.2694 *Be 17.2694 * 237.3

i i237.3 + tw

Bt .w1(237.34.t1.1.)2 * 6.1078 e.

1

(33)+ Ap - (Ap * 0.00115 * ti )

+ 2(Ap * 0.00115 * tw.)1

where A in Equations (32) and (33) is a psychrometric constant and p

is the atmospheric pressure. From Equations (29), (31), (32) and (33)

the variance of the vapor pressure measurement is

nBe

((V(t.) -B ) + (/(tw.) 712--)) up3. t Btwi=1

. .v(Ae) =

(34)

E ((v(tBe

+ (V(tw.)

i=1i

) )i

)) dnat 1i

The variance of the Bowen ratio measurement is determined as

for a ratio (Kendall and Stuart 1977).

[V(x) E(x2)V(y) 2(E(x) * COV(x,y) (35)v(x/y) -

E(y2

) E(y4

) E(y3

)

26

Where V, E, and COV are the variance, the expectation, and the covari-

ance of the variable within the parenthesis. When the measurements of

the temperature gradient, x, and the vapor pressure gradient, y, are

independent the covariance is zero and the third term of Equation (35)

drops off.

When the vapor pressure is determined through wet-bulb psy-

chrometry, as in this study, the measurements of the temperature grad-

ient and vapor pressure gradient are dependent and the covariance

exists. The covariance of the ratio of the gradients of temperature

and vapor pressure measurements is extremely complex. However, if the

temperature and vapor pressure gradients are positively correlated

(e.g., a positive change in At causes a positive change in ne),

neglecting the covariance will tend to increase the variance estimate

of the Bowen ratio. Similarly, if the gradients are negatively corre-

lated, neglecting the covariance will tend to decrease the variance

estimate of the Bowen ratio. It is not difficult to visualize condi-

tions of both positive and negative correlation. During periods when

the Bowen ratio is stable the gradients are most probably positively

correlated. During periods when the Bowen ratio is undergoing sig-

nificant change the gradients are most probably negatively correlated.

Thus neglecting the covariance, as is done in the remainder of these

analyses, should cause a slight underestimation of the Bowen ratio

variance estimate in the early morning and evening hours and a slight

overestimation during the middle of the day.

27

The estimate of the Bowen ratio variance is then found by

evaluating Equation (35)

2

V() =CPE;) [V(At2)At

2V(Ae)]

Ae Ae4

where the symbols are already defined.

Combined Variance Sources

The variances of latent energy and sensible heat flux densi-

ties are now estimated by solution of Equation (25).

DLE 3LE , 3 LEIV(LE) = V(Q*) 2* + v(G) 5--a--- + v03) --.3-3 , and[

(37)

DH 3H

[

@HV(H) = v(Q*) + V(G) -,7-- + V(f3) -T3,- . (38)

@Q* dG

In the exchange method, n-is the number of exchange periods in each

averaging period, and the proper t statistic for n-1 degrees of free-

dom can be found in any basic statistics text. Calculation of the

confidence interval for the latent energy and sensible heat flux

densities over the sampling period would take the following form:

V(X)1 2C.I.

(1-a= + t

) — (1-a/2) n(39)

where a, R, and V(x) are the probability of a Type I error, the mean,

and the estimated variance of LE or H.

(36)

CHAPTER 4

MINIMIZING ERRORS

This chapter will: (1) discuss minimizing energy budget

errors through the selection of an appropriate sampling frequency; and

(2) discuss the effectiveness of sensor exchange in elimineting grad-

ient measurement biases.

Sampling Considerations

The objective of experimental design is to isolate and reduce

the effects of errors so as to produce results with an acceptable

level of confidence. The atmospheric variables measured in the Bowen

ratio method vary in time and space and considerations of instrument

time constants and sampling frequency can affect the degree to which

measured values represent the actual values.

The parameter relating the amount of time required for an

instrument to adjust to a new environment is called the time constant,

T. A discussion of the time constant may be found in Fritschen and

Gay (1979). One time constant is the time required for a 1-1/e or

63.2 percent adjustment to the environmental step change. The degree

to which the sensor output dampens short period components and lags

behind the input is a function of the sensor's time constant. The

functional relationship between a sensor's time constant and the

28

29

attenuation and lag of a fluctuating input signal are described in

Gill (1964). Since high frequency variation or "noise" in an environ-

mental parameter such as temperature is essentially random and evenly

distributed about a mean over a selectively short period, the mean

sensor output over the same period will be identical to the true mean

when corrected for its lag.

Assuming the mean sensor output adequately represents the in-

put mean when corrected for lag, the problem becomes a function of the

sampling frequency required to adequately assess sensor output. The

required sampling frequency for complete signal reconstruction has

been studied by Tanner (1963), Goodspeed (1968), Byrne (1970, 1972),

and Fuch (1972) among others. For reasons outlined in Gay (1974), the

procedure in Fuch (1972) appears to be the most valid. Fuch recommends

that the sampling interval At = carT, where a is an arbitrarily selected

attenuation factor depending on the periodicity of the signal and the

time constant. Fuch suggested that a be about 0.05 as a guideline for

a sampling interval that will permit complete signal reconstruction.

However, as Tanner (1963) points out, the required sampling intervals

may be increased where the mean is of primary interest. Tanner con-

cluded the hourly mean of temperature sampled at 2T intervals was just

as sound as the mean sampled at 0.8T intervals (T=30s).

The relationships between time constants, sampling frequency

and accuracy were investigated by Stiger, Lengkeek,andKooijman (1976)

in an effort to determine "worst case" errors of period means. Tests

were run under conditions of high variability (intermittent cloudiness)

30

so as to maximize the effect of errors. Comparisons were made between

10 minute means determined with sampling intervals of is and 17.5s.

Ten minute means differed by a mean maximum of 0.04 degrees Celsius

for a sensor with T = 35s. The difference should be far less under

less variable conditions. The error also tends to decrease as the

time period for the mean is increased. A maximum interval of 2T ap-

pears to be a reasonable guideline for sampling at a fairly accurate

level. This interval should be adjusted as necessary given a desired

level of accuracy, the length of the averaging period, and the degree

of atmospheric variation.

The optimum sampling interval in the Bowen ratio method is a

function of the time constants, averaging period length, and the magni-

tude and variability of the temperature and vapor pressure gradients.

When the temperature and vapor pressure gradients are large, the need

for extremely accurate gradient measurements is reduced. Since the

difference between the true and measured values increases as the

sampling frequency decreases, samples should be taken as often as is

practical and the sampling interval should not exceed 2T.

Eliminating Biases in Gradient Measurements

Bowen ratio energy budget analysis requires extremely accurate

measurement of the temperature and vapor pressure gradients, espe-

cially when the gradients are small. A number of schemes have been

suggested to eliminate or reduce the effects of a bias between the

sensors measuring the gradient. These include periodic leveling of

31

the sensors and periodic sensor exchange (Sargeant and Tanner 1967,

Black and McNaughton 1971, Rosenberg and Brown 1974).

In this study the psychrometers were periodically interchanged

during the time averaging period (Sargeant and Tanner 1967). This

technique can virtually eliminate small biases that exist between

psychrometers. The time averaging period is divided into n (n = even)

exchange periods. During the first portion of each exchange period

the sensor is exchanged and equilibrates, and the actual sampling

takes place in the second half. When the number of exchange periods

per time averaging period is even, both psychrometers will have been

in the upper and lower positions for the same number of exchange

periods. The net effect of this procedure is to proportion the bias

between sensors evenly into the upper and lower averages for the time

averaging period. When the lower average is subtracted from the upper

average a bias free temperature gradient and an essentially bias free

vapor pressure gradient are produced. The method cannot completely

eliminate vapor pressure biases if the wet-bulb sensor bias is large,

because the relation between temperature and vapor pressure is non-

linear.

The results of this procedure are illustrated in Table 1. For

simplicity in this example the upper and lower temperatures and vapor

pressures have been held constant throughout the averaging period, but

a bias of 0.5°C has been added to the dry-bulb and the wet-bulb of one

psychrometer. When the true values in Table lA are compared with the

biased values in Table 1B, the temperature error is removed completely

32

Table 1. Example of bias elimination in gradient measurements throughthe exchange method. -- Psychrometer X and Y are exchangedbetween positions 1 and 2 after each reading. (A) Resultswith no bias. (B) Results with a 0.50 °C bias in the wet-and dry-bulbs of psychrometer Y.

Exchange Period

1

2 3 4 5 6

(A)

Level 2 Psychrom. Y X Y X Y X

T (Deg. C) 20.00 20.00 20.00 20.00 20.00 20.00

TW (Deg. C) 15.00 15.00 15.00 15.00 15.00 15.00

E (MB) 14.03 14.03 14.03 14.03 14.03 14.03

Level 1 Psychrom. X Y X Y X Y

T (Deg. C) 20.50 20.50 20.50 20.50 20.50 20.50

TW (Deg. C) 15.50 15.50 15.50 15.50 15.50 15.50

E (MB) 14.59 14.59 14.59 14.59 14.59 14.59

L.1 - L.2 Delta T

0.50 0.50 0.50 0.50 0.50 0.50

Delta TW

0.50 0.50 0.50 0.50 0.50 0.50

Delta E

0.56 0.56 0.56 0.56 0.56 0.56

Ave. Delta T (Degr. C) = 0.500

Ave. Delta TW (Degr. C) = 0.500

Ave. Delta E (MB) = 0.560

(B)

Level 2 Psychrom. Y X Y X Y X

T (Deg. C) 20.50 20.00 20.50 20.00 20.50 20.00

TW (Deg. C) 15.50 15.00 15.50 15.00 15.50 15.00

E (MB) 14.59 14.03 14.59 14.03 14.59 14.03

Level 1 Psychrom. X Y X Y X Y

T (Deg. C) 20.50 21.00 20.50 21.00 20.50 21.00

TW (Deg. C) 15.50 16.00 15.50 16.00 15.50 16.00

E (MB) 14.59 15.16 14.59 15.16 14.59 14.59

33

Table 1. -- Continued

Exchange Period

1

2 3 4 5 6

L.1 - L.2 Delta T

0.00

Delta TW

0.00

Delta E

0.00

Ave. Delta T (Degr. C) =Ave. Delta TW (Degr. C) =Ave. Delta E (MB) =

1.00 0.00

1.00

0.00

1.00

1.00 0.00

1.00

0.00

1.00

1.13 0.00

1.13

0.00

1.13

0.500

0.500

0.565

34

from the gradient (0.500°C vs. 0.500°C) and the error in the vapor

pressure gradient is negligible (0.560 mb vs. 0.565 mb).

The relationship between remaining vapor pressure gradient

bias and the dry- and wet-bulb biases can be estimated through the use

of Equations (13) and (14).

;e1@e

2e3 e4 4_

ae53 e

6 1Bias (Ae) = At (w Ft7--

1a atw2 " 9tw

3 Dtw

4 ' 3tw

5 9tw

6 '-

9e12

e3

@e4

Be5 4. /6+ At (

3t5 at6

where Atw is the wet-bulb bias, At is the dry-bulb bias, and the

numbered subscripts in the partial derivatives are the exchange period

numbers.

It is important to consider polarity when evaluating Equation

(40); At and Atw are considered positive and are assigned to the sensor

that reads the higher of the two. The partial derivatives of vapor

pressure with respect to wet- and dry-bulb temperature are evaluated

for the temperature measurement that contains the positive bias. Equa-

tion (40), as stated, is for the condition when both biased sensors

(wet- and dry-bulb) are in the lower position for the first exchange

period. If the biased sensor is in the upper position for the first

exchange period, the polarities of the partial derivatives for the

sensor should be reversed. In the event that the biased wet-bulb

sensor was in the upper position and the biased dry-bulb sensor was

in the lower position for the first exchange period, Equation (40)

would read as follows.

(40)

35

ae1Be23

Be4

De5

Be6Bias (Ae) = Atw (-

Btw1 Btw

2 atw

3 3tw

4+ )/6

Be Be Be Be.4 Be De6,)/6+ At (

Bt: 5Tat: Dt61

(41)

The effects of various wet- and dry-bulb temperature measure-

ment biases on the remaining bias in the vapor pressure gradient were

evaluated with a computer simulation over a wide range of temperatures,

vapor pressures, and gradients. The residual bias in the vapor pres-

sure gradient was highly dependent on the wet-bulb bias, and signifi-

cantly less dependent on the dry-bulb bias.

The remaining vapor pressure gradient bias can be conveniently

estimated if one first neglects the effect of the dry-bulb bias. Re-

gression analysis indicated the relationship between the remaining

vapor pressure gradient bias and the wet-bulb bias is essentially

linear, with a y-intercept of zero and a slope of 0.01 to 0.02 mb/°C

over the range of conditions tested. Figure 1 shows the relationship

for the temperature and humidity conditions specified in Table 1.

From the analysis of remaining vapor pressure gradient bias,

it is apparent that the exchange method is quite effective in reducing

vapor pressure gradient biases, even in the presence of a substantial

error in one of the wet-bulb sensors. The error generated in the

vapor pressure gradient will be less than 0.01 mb for a wet-bulb bias

of as much as 0.6°C.

36

4

H(1.

04H

4-)

3-PG)

0

Cqw) IN3I0V219 3WISS3ld tICIdVA NI bObin -worm

CHAPTER 5

APPLICATION TO FIELD MEASUREMENTS

Field studies of evapotranspiration were undertaken in May of

1978 and July of 1979. The data provided a basis for evaluating

errors in the energy budget. The analysis of the data collected in

1978 led to changes in sensor calibration and system operation thereby

substantially improving the precision of the data collected in 1979.

Field Measurements

The 1978 and 1979 studies used similar instrumentation and

experimental designs.

Two masts were used, supporting exchange mechanisms (Gay and

Fritschen 1979) with germanium diode wet-bulb psychrometers (Gay 1972,

Black and MacNaughton 1971) and a miniature net radiometer (Fritschen

1965a). Soil heat flux was measured with one soil heat flux disc

buried about one centimeter into the soil at a representative location.

The psychrometer sensors (germanium diodes) were lab cali-

brated using a constant temperature bath, an Accurex Autodata 9, and

a Tektronics 4051. A platinum resistance thermometer was used as the

standard. Calibration coefficients were determined through polynomial

regression. The calibration coefficients for the net radiometers and

soil heat flux disc used in the studies were those provided by the

manufacturer.

37

38

Data collection has been described by Gay (1979). The system

includes an Accurex Autodata 9 with its programmable, microprocessor

controlled, digital voltmeter capable of monitoring 40 channels. Fea-

tures include interval and continuous scanning, an integrating digital

voltmeter, a real-time clock, a printer, and RS-232C output. The sys-

tem has scales of 100 mV, 1 V, and 10 V and has an accuracy of + 0.012

percent at full scale on high resolution.

The Autodata 9 was linked to a Tektronics 4051 graphics calcu-

lator and a Texas Instruments 810 printer so the data could be reduced,

presented and stored as it was collected. Real-time analysis provides

for identification of problems and evaluation of system performance.

This technique maximizes the utility of the time spent in the field.

The development of the averaging period over which the com-

ponent flux densities were evaluated required considerations of sensor

time constants, exchange method requirements, data acquisition system

capabilities, and atmospheric variation. The time constant for the

psychrometer sensors ( 1N2326 germanium diodes) is about one minute

(Sargeant 1965). The data acquisition system's capability of develop-

ing integrated averages from continuous samples eliminates the effects

of any phase differences between the wet- and dry-bulb temperature

measurements caused by slightly different wet-bulb time constants

within the ceramic wick. The data acquisition system can scan at a

maximum rate of approximately 1.8 channels per second. The three

minute integrated averages were based on 19 samples for the 1978 study

and 14 samples for the 1979 study. Sensor design improvements in the

39

1979 study required additional measurements (i.e., channels) and thus

the number of samples per three minute period were reduced. Sampling

intervals of 0.16T in 1978 and 0.21T in 1979 with continuous scanning

were well within the guidelines discussed in Chapter 4.

Due to equilibration requirements and programming limitations

the length of each exchange period was set at six minutes. The first

three minute integrated average determined during sensor exchange and

equilibration was discarded and the second three minute integrated

average was used for analysis. The second three minute integrated av-

erage was considered a point measurement centered in the second half

of the exchange period. Six exchange periods were then combined to

form an averaging period of one half hour. The data from the final

exchange period became the endpoint of that averaging period; this was

then considered the first exchange period (starting point) of the fol-

lowing averaging period. The use of half hour averaging periods was

consistent with the application requirements discussed earlier. A

half hour averaging period was as follows:

0757 - sensors in equilibrium, begin scanning.

0800 - end of exchange period #1; psychrometers exchange and

begin to equilibrate; data printed and transferred to

4051; 4051 data reduction and magnetic tape storage

(ave. per. 0730-0800); line printer output (ave. per.

0730-0800) on T.I. 810.

0803 - sensors in equilibrium, begin scanning.

0806 - end of exchange period #2; psychrometers exchange and

begin to equilibrate; data printed and transferred to4051.

0809 - sensors in equilibrium, begin scanning.

40

0812 - end of exchange period #3; psychrometers exchange andbegin to equilibrate; data printed and transferred to4051.

0815 - sensors in equilibrium, begin scanning.

0818 - end of exchange period #4; psychrometers exchange andbegin to equilibrate; data printed and transferred to4051.

0821 - sensors in equilibrium, begin scanning.

0824 - end of exchange period #5; psychrometers exchange andbegin to equilibrate; data printed and transferred to4051.

0827 - sensors in equilibrium, begin scanning.

0830 - end of exchange period #6; psychrometers exchange andbegin to equilibrate; data printed and transferred to4051; 4051 data reduction and magnetic tape storage(ave. per. 0800-0830); line printer output (ave. per.0800-0830) on T.I. 810.

The 1978 Study

Energy budget measurements were made from 27 May 1978 through

31 May 1978 above a mesquite thicket on the San Pedro River north of

Tucson, Arizona.

The elevation of the San Pedro River site, near Mammoth, Ari-

zona, is approximately 2300 feet above sea level. The San Pedro River

valley near Mammoth is oriented southeast to northwest. The San Pedro

River has a mean annual flow of 33,760 acre feet at Winkleman, Arizona

(U.S.G.S. 1975) approximately ten miles downstream from Mammoth. The

river was dry adjacent to the site during the study. The depth of the

water table during the study was about twenty feet.

The bottomland of the San Pedro River valley supports a healthy

stand of mesquite (Prosopis pubescens). The stand is closed over an

41

area about one half mile wide by three miles long with occasional small

openings. The height of the stand averaged approximately thirty-five

feet in the area surrounding the instrumentation. The understory at

the site consisted of dry grasses and forbs.

The masts were erected through the mesquite canopy approxi-

mately 100 feet apart in orientation with the axis of the river valley.

Fetch requirements were easily met as the masts were centrally located

within the extensive stand.

Energy budget tabulations and graphs for the 1978 study are

found in Appendix B.

The 1979 Study

Energy budget measurements were made on 15 July and 16 July

1979 in the floodplain of the Pecos River near Roswell, New Mexico.

The elevation of the site was approximately 3450 feet above sea level.

The Pecos River has a mean annual flow of 135,500 acre feet at Acme,

New Mexico (U.S.G.S. 1978), about twenty miles upstream from the site.

The river was flowing during the study. The depth of the water table

during the study was about six feet. The vegetation at the site con-

sisted primarily of kochea (Kochea scoparia) with assorted forbs and

grasses.

The weather during the study was moderately warm and inter-

mittently cloudy. A strong southerly wind prevailed throughout the

study.

General instrumentation for the 1979 study was the same as for

1978, except that the masts were located about thirty feet apart

42

perpendicular to the prevailing wind. Fetch requirements were easily

met as the kochea pasture extended a considerable distance in all

directions from the instrumentation.

Energy budget tabulations and graphs for the 1979 study are

found in Appendix B.

Error Analysis, 1978

The reliability of the energy budget analysis can be assessed

through comparison of results from the two masts. Table 2 shows the

comparison of average half hour flux densities and percent deviations

from the mean of the two masts for each day of the study. Over the

entire period, the average deviation from the mean of the two masts

for net radiation, sensible heat and latent energy were 0.98%, 34.34%,

and 15.63% respectively. It is evident that the net radiation measure-

ments compare much more favorably than either sensible heat or latent

energy measurements. It is difficult to judge whether the differences

between the two masts were real or a function of the measurement sys-

tem since the mesquite canopy was slightly irregular and very few

energy budget studies have incorporated two masts.

In an effort to evaluate measurement precision and identify

areas of potential improvement, the random error analysis procedure

discussed in Chapter 3 was performed on this data. The t statistic

used in Equation (39) for the half hour periods (n-1 = 5 degrees of

freedom) was 2.571. The results of the error analysis for 27 May 1978

are presented in Tables 3 and 4, and plotted in Figure 2. Similar

04

4-100

•00)Z P4

•

(1)

caE-1

43

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en en OD H riM L.c)

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rn •:r h •rr o crN IN ("1 Cr .1, m

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0 0 0 0 0 0I I I 1 t t

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44

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error analysis tabulations for 28 May through 31 May 1978 are found in

Appendix C.

The results tabulated in Tables 5 and 6 and Appendix C show

that the 95 percent confidence interval for mean latent energy flux

density was 0.48 + 0.09 cal/cm2 /min. over the entire period. The 95

percent confidence interval for mean sensible heat flux density was

0.21 + 0.09 cal/cm2 /min.

Inferences concerning latent energy measurement differences

between the two masts can be made through data provided from the ran-

dom error analysis. Accepted statistical procedures were used to test

equality of individual half hour and daily means of latent energy

estimates from the two masts. Fifty percent of the tests on individual

half hour estimates showed the estimates were significantly different

at the 95 percent level. The tests of daily means indicated the esti-

mates from mast #1 and mast #2 were significantly different at the 80

percent confidence level. The mean probability of a Type I error

(conclude LE1 LE2 when LE1 = LE2) was therefore approximately 20

percent for the five days. From this there is reason to believe the

actual evapotranspiration rates measured by the two masts were indeed

different.

Regression analysis of the two sets of net radiation measure-

ments showed

Q2 = 0.007 + 0.982 Ql (42)

48

where Ql and Q2 are the net radiation measurements from mast #1 and

mast #2 respectively, in cal/cm2 /man. Based on the excellent correla-

tion between the net radiation measurements, the differences in latent

energy estimates were generated primarily by differences in the Bowen

ratio measurements.

Throughout the study the Bowen ratio measurement contributed

over 99 percent of the variance in LE and H, so significant improve-

ments (i.e., greater confidence in concluding LE1 p LE2) would result

from a reduction in the Bowen ratio measurement variance. A reduction

in the individual variances of each wet- and dry-bulb temperature

measurement would result from improved calibration procedures. This

would yield significantly smaller mean squared errors and contribute

to improved measurement precision. An increased number of samples

would increase the degrees of freedom and reduce the value of the t

statistic, thus also increasing precision. However, the relatively

long time constant of the ceramic wick psychrometers limits the number

of exchanges to 6 (n-1 = 5 degrees of freedom) per half hour period.

Finally, a more rapid sensor exchange should improve the results by

allowing more time for sensor equilibration.

Error Analysis, 1979

The diode circuitry and calibration procedures were changed

and the psychrometer exchange time reduced for the 1979 measurements.

The adverse effects of current fluctuation from the diode power supply

(Black and McNaughton 1971) were eliminated by making an additional

measurement of the voltage drop across a precision resistor in each

49

diode circuit. The value of current flow thus determined was then

used with the measurement of voltage drop across the diode to calcu-

late the diode resistance. Thus the diodes were calibrated as a func-

tion of resistance against the temperature standard. In addition, the

number of data points taken between 0 and 50°C was increased from 15

in 1978 to 46. A fourth degree polynomial (third degree in 1978) re-

lated diode output (ohms) and temperature ( °C). These changes wereeffective in reducing the average regression MSE for the eight diodes

from 0.0018°C in 1978 to 0.0003°C in 1979.

Psychrometer exchange times were reduced from 56 to 11 seconds

by replacing the 4 rpm exchange mechanism motors with 20 rpm motors.

Neglecting equilibration that takes place during the exchange, the

faster motors increased the adjustment at the end of the three minute

exchange and equilibrium period from 87 to 93.5 percent.

Table 5 shows the comparison of average half hour flux densi-

ties and percent deviation from the mean of the two masts for each day

of the study. Over the entire period, the average deviation from the

mean of the two masts for net radiation, sensible heat and latent

energy were 0.03%, 2.64% and 2.21% respectively. Since the kochea

pasture was a relatively smooth, homogenous surface, the results from

the two masts were expected to be the same. The excellent correlation

in net radiation totals (+ 0.03%) is somewhat misleading as the means

from the two days tended to compensate each other.

The effectiveness of the system improvements on random errors

were evaluated with the random error analysis. The results of the

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51

error analysis are presented in Tables 6 and 7 for 15 July 1979. The

mastwise comparison of the generated 95 percent confidence intervals

for latent energy flux density are presented for 15 July 1979 in

Figure 3. Tables 6 and 7 show a significant decrease in both the con-

fidence interval widths for LE and H and the percent contribution of

the Bowen ratio measurement to the intervals. Similar error analysis

tabulations for 16 July 1979 are included in Appendix C. For the en-

tire two day period, the 95 percent confidence intervals for mean

latent energy and sensible heat flux densities were 0.27 + 0.01 and

0.21 + 0.01 cal/cm2 /min respectively.

The system changes were effective in significantly reducing

random errors. A comparison of Figures 2 and 3 shows the dramatic in-

crease in precision. Mean error limits (95 percent) for LE and H were

reduced from + 18.8% to + 3.7% and + 42.9% to + 4.8 percent respec-

tively. The error analysis showed a significant decrease in the con-

tribution of the Bowen ratio measurement to the total random error,

dropping from about 99 percent in 1978 down to 27 percent in 1979.

Accepted statistical procedures were used to test equality of

individual half hour and daily means of latent energy estimates from

the two masts. Sixty percent of the tests on individual half hour

estimates showed the estimates were significantly different at the 95

percent confidence level. The tests of daily means indicated the esti-

mates from mast #1 and mast #2 were significantly different at the 50

percent level. The mean probability of a Type I error (conclude LE1

—LB2 when LE1 = LE2) was therefore approximately 50 percent. While the

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latent energy estimates from the two masts were extremely close, the

reduction in variance allowed for much finer differentiation of period

and daily means. Since the Bowen ratio estimates from the two masts

were extremely close, a significant portion of the differences in

latent energy estimates were attributed to differences in net radia-

tion measurements. Regression of the two sets of net radiation mea-

surements showed

Q2 = 0.023 + 0.962 Ql (43)

where Ql and Q2 are the net radiation measurements from masts #1 and

#2 respectively, in cal/cm2 /min. The failure of Ql to equal Q2 caused

unwanted differences between LE1 and LE2 as a function of the Bowen

ratio. From Equation (15) it can be seen that the absolute value of

the partial derivative of LE with respect to Q* decreases as (3 in-

creases. Therefore, the effect of Ql Q2 will be most pronounced

when the Bowen ratio is small (i.e., over an evaporating surface).

The failure of Ql to equal Q2 was a function of differences in

surface conditions, leveling problems, and possibly calibration differ-

ences. As the temperature and vapor pressure gradient measurements

represent areal averages, the use of mean net radiation ((Q1+Q2)/2)

would represent a logical solution to the problem.

Future System Improvements

The largest variance component remaining in the 1979 data was

associated with the measurement of net radiation. Reducing the

variance of the net radiation measurements will require improved

56

calibration procedures. Net radiation in the principal component of

available energy (Q* + G), from which LE and H are partitioned, so the

use of a point estimate of net radiation is somewhat risky. Not only

are the calibration procedures poorly defined, but surface factors may

unduly affect a single instrument. The use of a number of scattered

instruments to estimate mean net radiation will enhance the accuracy

of the energy budget measurements.

Further improvements in Bowen ratio measurement precision may

be realized through replacing the germanium diodes with metal resis-

tance elements (platinum or nickel-iron). The temperature sensitivity

of a germanium diode is slightly dependent on current (Hinshaw and

Fritschen 1970), while metal resistance elements follow Ohm's Law

(V=IR) and will thus reduce the need for a highly precise power supply.

Preliminary tests have shown the nickel-iron resistance elements yield

regression mean squared errors four to eight times smaller than the

germanium diodes under the same conditions. In addition, the shorter

time constant of the metal resistance elements may allow for a greater

number of exchange periods per half hour period. An increased number

of samples would increase the degrees of freedom and reduce the value

of the t statistic, thus also increasing precision.

CHAPTER 6

CONCLUSIONS

This study has identified instrumentation and site require-

ments for valid application of the Bowen ratio method.

The Bowen ratio method is a valid means of estimating the

surface energy budget provided specific conditions of site and instru-

mentation are met. The necessary assumption of Kh = Ke

is most likely

valid under non-advective conditions with predominantly forced convec-

tion (small Ri). Measurements should be taken over nearly homogenous

surfaces, for these conditions will minimize horizontal divergence of

heat and water vapor. The lower psychrometer must be placed high

enough to avoid the effects of surface irregularities. The upper

psychrometer must be within the modified zone as determined through a

conservative height-fetch ratio, yet the vertical distance between the

psychrometers should be great enough to allow for measureable gradi-

ents.

Instrumentation must be of high quality, especially for the

measurement of the temperature and vapor pressure gradients. Psychrom-

eter sensors should be calibrated with a great deal of care and local

control. Small biases should be corrected if possible; they can cause

large errors if the gradients are small. The exchange method provides

57

58

an effective means of eliminating biases and becomes essential when

the gradients are small, and when the atmosphere is dry so that the

wet-bulb depression is large.

The minimum sampling frequency should be determined from the

instrument time constants and the expected environmental fluctuations.

The sampling interval should be less than 2T; since the error in-

creases with decreasing sampling frequency it is recommended that the

samples be taken as often as is practical. The energy budget is best

solved on the basis of period means of one half to one hour.

The random error analysis provided a variety of information

from which instrumentation and measurement conclusions were made. In

1978 the 95 percent confidence interval for mean latent energy and

sensible heat flux densities were 0.48 + 0.09 and 0.21 + 0.09 cal/

2 .cm /man respectively. Mean daily evapotranspiration rates at the two

masts were found to be significantly different at the 80 percent con-

fidence level. The analysis identified the Bowen ratio measurement as

the primary contributor of variance in the estimates of H and LE. Im-

provements in psychrometer sensor calibration and circuitry for the

1979 study led to significant improvements in Bowen ratio measurement

precision. In 1979 the 95 percent confidence interval for mean latent

energy and sensible heat flux densities were 0.27 + 0.01 and 0.21 +

0.01 cal/cm2 /man respectively. Tests on latent energy estimates from

the two masts indicated there was insufficient data to conclude the

daily means differed significantly.

59

The random error analysis procedure demonstrated the need for

highly precise calibration of psychrometers. The sensors should be

calibrated in the same circuit and with the same data acquisition sys-

tem as used in the field. The measurement of diode resistance, as

opposed to diode voltage drop, can minimize the effects of current

fluctuation from the power supply. The power supply fluctuations can

be eliminated completely through appropriate circuitry and the use of

metal resistance elements in the psychrometers.

Due to the effectiveness of the exchange method in eliminating

biases in the temperature and vapor pressure gradients, the most sig-

nificant increase in energy budget accuracy would come from an increase

in the measurement accuracy of net radiation. Improved net radiometer

calibration procedures and the use of the mean net radiation from a

number of scattered instruments will inhance the accuracy of energy

budget measurements.

The field studies used here to illustrate the error analyses

certainly confirm the necessity for replication of the Bowen ratio

apparatus. All too often, past studies have based the entire analysis

upon a single pair of psychrometers on a single mast. This question-

able practice should now be discarded in future work.

APPENDIX A

SYMBOLS AND DEFINITIONS

Symbol Definition

A psychrometric constant, ° C-1

Celsius; energy budget component

COV covariance

specific heat of the air, 0.24 cal/gm C

expectation; evaporation mass flux density, g/cm2/min

G soil heat flux density, cal/cm2 /min

H sensible heat flux density, cal/cm2 /min

electrical current, amperes

incoming shortwave radiation flux density, cal/cm2 /min

Kt reflected shortwave radiation flux density, cal/cm2 /min

Ke

transfer coefficient for water vapor, cm2/sec

K. transfer coefficient for sensible heat, cm2/sec

latent heat of vaporization

L4,incoming longwave radiation flux density, cal/cm2/min

Lt outgoing longwave radiation flux density, cal/cm2/min

LE latent energy flux density, cal/cm2 /min

Ql net radiation, mast #1

Q2 net radiation, mast #2

Q* net radiation flux density, cal/cm2/min

60

61

Symbol Definition

probable error; resistance, ohms

Ri Richardson number

temperature, °C

Ta

air temperature, ° C

Td temperature difference, °C

V variance; voltage

X sensor calibration data matrix

Xh sensor mean response matrix

Y dependent variable

vapor pressure, mb;

base of the system of natural logarithms

e (g)vapor pressure gradient, mb/m

function

acceleration due to gravity, 980 cm/sec2

linear calibration coefficient

1r linearity of response, percent

number of samples; number of exchange periods per samplingperiod

atmospheric pressure, mb

probable error of independent variable

seconds

data acquisition system precision, percent

Sr

data acquisition system resolution

time; dry bulb temperature, ° C

t temperature gradient, °C/°C/mn

62

tw wet bulb temperature, °C

wind speed

independent variable

average

vertical distance or height

a attenuation factor; probability of a Type I error

Bowen ratio

emissivity; ratio of molecular weights of water to air, 0.622

adiabatic lapse rate, 1 °C per 100 meters

A difference between values; systematic error

partial derivitive

7 period

density of air, 1.02 x 10-3

gm/cm3

time constant, seconds

APPENDIX B

ENERGY BUDGET TABULATIONS AND GRAPHS

63

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APPENDIX C

ERROR ANALYSIS DATA

89

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