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Error analysis of evapotranspiration measurements
Item Type Thesis-Reproduction (electronic); text
Authors Hartman, Robert Kent.
Publisher The University of Arizona.
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Link to Item http://hdl.handle.net/10150/191699
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
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43
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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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50
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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