Accuracy of Evapotranspiration Rates 

Determined by the Water-Budget 

Method, Gila River Flood Plain, 

Southeastern Arizona

GEOLOGICAL SURVEY PROFESSIONAL PAPER 655-L

Accuracy of Evapotranspiration Rates 

Determined by the Water-Budget 

Method, Gila River Flood Plain, 

Southeastern Arizona

By Ronald L. Hanson and David R. Dawdy

GILA RIVER PHREATOPHYTE PROJECT

GEOLOGICAL SURVEY PROFESSIONAL PAPER 655-L

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1976
UNITED STATES DEPARTMENT OF THE INTERIOR 

THOMAS S. KLEPPE, Secretary

GEOLOGICAL SURVEY 

V. E. McKelvey, Director

Library of Congress Cataloging in Publication Data

Hanson, Ronald Lee, 1934-

Accuracy of evapotranspiration rates determined by the water-budget method, Gila River flood plain, southeastern Arizona. (Gila

River phreatophyte project)

(Geological Survey Professional Paper 655-L)

Bibliography: p. 35

Supt. of Docs.: I 19.16:655-L

1. Evapotranspiration-Alaska-Gila River watershed-Measurement. I. Dawdy, David R., 1926- joint author. II. Title.

III. Series. IV. Series: United States Geological Survey Professional Paper 655-L. 

QE75.P9no. 655-L[QC915.7.U5] 557.3'08s[551.5'72'097917] 75-619341

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402

Stock Number 024-001-01819-1
CONTENTS

Page

Symbols ________________________________ IV

Metric conversion factors _____________________ V

Abstract _______________________________ LI

Introduction _____________________________ 1

Description of study area ___ ______________ ___ 2

The water-budget equation _ _______________ ___ 3

Evaluation of water-budget components and errors _ _ _ _ ___ 4

Streamflow ___________________________ 4

Channel storage _______ _______________ 7

Tributary inflow _________________________ 9

P-re

Evaluation of water-budget components and errors Continued

Precipitation ____ L10

Soil-moisture content 15

Basin-fill inflow _______ _ ____ _ _ 23

Downvalley ground-water flow __ __ _ 23

Computation ofET_and total error inET ____________________ 25

Computation of A ET and error in A ET _____________ ___ 27

Discussion of results _ ___ : _____ _ 33 

Development of equations describing unadjusted sampling

error, V _ ___________ - ___ - _ 34 

References cited ___ __ 35

ILLUSTRATIONS

FIGURE 1. 

2-17.

Page

Map showing study area and instrumentation location ____ _________ ___ _ ___ _ L3 

Graphs showing:

2. Average relation between instantaneous discharge and error in discharge at cross sections 1 and 9 6

3. Relation between average daily discharge and wetted cross-sectional area for cross sections 1, 9, 13, and 17 8

4. Relation between average daily discharge and average error in wetted cross-sectional area__ 9

5. Average relation between number of precipitation gages and missing-data error of average precipitation __ 13

6. Relation between average precipitation and total adjusted sampling error _ ___ __ __ 14

7. Average relation between number of access holes and missing-data error of moisture-content change 18

8. Average relation between number of access holes and total adjusted sampling error of moisture-content change 19

9. Estimates of average relation between sampling density of access holes and total adjusted sampling error of

moisture-content change _ ______________ - _ _ __ __ 20

10. Apparent specific yield values derived from relation between average change in water level and average

moisture-content change in capillary zones ______ _________ _____ _ 21

11. Average relation between number of ground- water wells and missing-data error in moisture-content change 23

12. Average downvalley ground-water slopes through cross section 9 in winter months _ 24

13. Relation between average downvalley ground-water slope through ends of each reach and width of flood plain at

downstream cross section _______ ________ ___________________ _____ - ___ _ 25

14. Sources of water contributing to ET for the 1964 water year, reach 1 __________________ 26

15. ET and corresponding errors in ET and outflow of Gila River at cross section 9 for 1964 water year 28

16. Water-budget ET values in reach 1 before and after clearing and average change in ET as a result of

phreatophyte clearing ________ _____ ____ ____ ___ ____ ___ __ _ _ ____ _______ 32

17. General relation for error in departure of average of m samples from average of n samples and error in departure

of average of m samples from estimated population mean ________ _ ___ _ __ 34

TABLES

TABLE 1. Evapotranspiration and water-budget components for 1964 water year, reach 1_____ L5

2. Evapotranspiration and water-budget components for the 21-day budget period 688-708 (August 18-September 7,1964)

in reach 1 and corresponding sampling and bias errors of each component _ -- 5

3. Daily discharges and errors atcross section 1 for budget period 688-708 ____________ -___ -- 5

4. Average daily discharge (q), corresponding wetted cross-sectional area of river channel (A), and error in the area (EA) for

cross sections 1 and 9 on budget period days 688 and 708 _________________ _ 8

5. Tributary runoff into reach 1 during budget period 688-708__________________________ 10

6. Area assigned to each precipitation gage in reach 1 and precipitation amounts observed at each gage during budget

period 688-708_______________________________________________________-___ 10

in
IV CONTENTS

Page

TABLE 7. Example of precipitation data array used to compute the average precipitation for each budget period (Pt), the departure 

of precipitation at a given gage from the average precipitation (R/t), the average departure for all budget periods 

(Rj), and the standard deviation of the average departure (sj) ___________ - 11

8. The average departure in precipitation of each gage in reach 1 (T?,-) and the standard deviation of the average departures

(sj) computed for three precipitation ranges with each range containing k budget periods of data __ 12

9. Example of array of mean departures of precipitation (Rmv ) and standard deviations of the mean departures (Smv) 12

10. Emv and Smv values for precipitation in upper range P>1.51 in. (38.4 cm) in reach 1 with gages arranged in order of

increasing Sj and in order of decreasing s/ _________ ______~___ 13

11. Average missing-data error (Sm ) of precipitation in each precipitation range from curves in figure 5 for m = 1 to 10 gages

in reach 1. Included are the unadjusted (V) and adjusted (Ep) sampling errors for m="10 gages 14

12. Soil moisture content measured in the soil and capillary zones of flood plain access holes in reach 1 for budget period days

688 and 708 __________ __________._________ ._________ __________ __________ _ 16

13. Average departure in moisture of each access hole (ff^) and the standard deviation of the average departure (s,) for the

soil and capillary ^ones of the flood plain and the soil, intermediate, and capillary zones of the terrace in r?ach 1 17

14. Missing-data errors (Sm ) from curve in figure 7 for the soil and capillary zones of the flood plain and for the soil,

intermediate, and capillary zones of the terrace, reach 1 _ ___ 19

15. Total adjusted sampling error in moisture change as defined with a complete set of moisture-content data (m =n) f T each

zone of the flood plain and terrace in reaches 1, 2, 2a, and 3_____________ 20

16. Average apparent specific yield (S'), number of budget periods (k) used to define S', total adjusted sampling erro~ in the 

measurement of average water-level change (E^), and the total adjusted sampling error of moisture change in the 

capillary zone (Ec& and ETC^), wnen the moisture change is derived from water-level change (equatior 34) __ 22

17. Ground-water level changes (AA) in flood plain wells of reach 1 used to compute AMc for budget period 688-708 __ 22

18. Average departure in ground-water level change (Rj) of each terrace well in reach 1 and the standard deviation (Sj) ofRj

computed from 20 budget periods of water-level data collected during the 1968 water year _ 23

19. Evapotranspiration (ED and total measurement error of ET (EET) for each budget period during water yearr 1963-

71 reaches 1, 2, 2a, and 3 ___________________________________________________ 29

20. ET values obtained defore and after clearing and their corresponding sampling, and total measurement errors for

selected 14-day budget periods during June and July, reach 1 ______________ _ __ 32

SYMBOLS

a constant defined by equation 23 

A area of reach, surface area assigned to a sample point, 

wetted cross-sectional area of Gila River channel, or 

subscript denoting "after clearing" 

6 constant defined by equation 24 or subscript denoting

a bias type of error

B subscript denoting "basin fill" or "before clearing" 

c constant defined by equation 25 

C subscript denoting "capillary zone of soil profile" 

AC change in Gila River channel storage 

d subscript denoting a given budget period day 

D total number of days in budget period 

e\ temporal variability of Gila River wetted crosssectional area 

^2 spatial variability of Gila River wetted cross-sectional

area 

63 error in Gila River wetted cross-sectional area due to

error in discharge used to compute area 

eq error in Gila River instantaneous (or average daily)

discharge 

Eq average standard error of mean daily discharge of the

Gila River for budget period

Ex bias error, adjusted sampling error, or total measurement error of water-budget component or parameter 

__ indicated by subscript x

ET, ET evapotranspiration, average evapotranspiration 

AET average change in evapotranspiration 

GB basin-fill inflow

GI, Gq_ downvalley ground-water inflow and outflow 

A/i, Ah ground-water level change, average ground-water 

level change

k

L 

m 

M,

t, AM

n 

n'

N

O

P 

P,P

q

Qi,Qo

QT

I/

" mv

downvalley ground-water slope

subscript denoting "inflow" or "intermediate zone of 

soil profile"

sample point precipitation gage, soil-moisture access hole, or ground-water well

total number of observations (budget perods)

length of Gila River channel

number of sample points «; n

soil-moisture content in zone z of hole j at the end of 

budget period t

change in soil-moisture content in zone z of hole; at the 

end of budget period t, and average change in 

soil-moisture content

total number of sample points in reach

largest even value =£ n

number of discharge measurements made during the 

budget period

subscript denoting "outflow"

total number of permutations

precipitation and average precipitation

instantaneous or average daily discharge

Gila River inflow and outflow during a budget period

tributary inflow

approximate correlation coefficient

departure of observation at sample point j from 

average of observations at n sample points for 

budget period t

average departure of k observations at sample point j

mean departure of observations at m sample points 

from observations at n sample points fir permutation v
CONTENTS

SET

SA

S

S' 

t

T

subscript denoting a sampling type of error

standard deviation of Rj

standard deviation of average ET

standard deviation of average change in ET

subscript denoting "soil zone of soil profile"

standard deviation ofRmv

average standard deviation of all Rmv computed fromp

permutations (missing-data error) 

apparent specific yield 

subscript denoting a given budget period 

transmissivity or subscript denoting "tributary"

TC, TI, TS

W 

z

v

subscripts denoting "capillary," "intermediate," and

"soil" zones of terrace soil profile 

subscript denoting a given permutation of the sample

points

width of saturated alluvium 

subscript denoting a given soil-moisture zone 

expectation _ _ 

residual error in ratio of S2m to S2zm 

estimate of population correlation coefficient between

sample points ji and 72 

unadjusted sampling error for a complete set of sam pie

points

METRIC-ENGLISH EQUIVALENTS

Metric unit English equivalent

Length

millimetre (mm) 

metre (m) 

kilometre (km)

= 0.03937 inch (in) 

= 3.28 feet (ft) 

= .62 mile (mi)

Area

square metre (m2 ) 

square kilometre (km2 ) 

hectare (ha)

= 10.76 square feet (ft2 ) 

= .386 square mile (mi2 ) 

= 2.47 acres

Volume

cubic centimetre (cm3 ) 

litre (1) 

cubic metre (m3 ) 

cubic metre 

cubic hectometre (hm3 

litre 

litre 

litre 

cubic metre

cubic metre

= 0.061 cubic inch (in3 ) 

= 61.03 cubic inches 

= 35.31 cubic feet (ft3 ) 

= .00081 acre-foot (acre-ft) 

= 810.7 acre-feet 

= 2.113 pints (pt) 

= 1.06 quarts (qt) 

= .26 gallon (gal) 

= .00026 million gallons (Mgal or 

10« gal) 

= 6.290 barrels (bbl) (1 bbl = 42 gal)

Weight

gram (g) 

gram 

tonne (t) 

tonne

= 0.035 ounce, avoirdupois (oz avdp) 

= .0022 pound, avoirdupois (Ib avdp) 

= 1.1 tons, short (2,000 Ib) 

= .98 ton, long (2,240 Ib)

Specific combinations

kilogram per square 

centimetre (kg/cm2 ) 

kilogram per square 

centimetre 

cubic metre per second 

(mVs)

= 0.96 atmosphere (atm) 

= .98 bar (0.9869 atm) 

= 35.3 cubic feet per second (ft3/s)

Metric unit

Specific

litre per second (1/s) 

cubic metre per second 

per square kilometre 

[(m*/8)/km*l

metre per day (m/d)

metre per kilometre 

(m/km) 

kilometre per hour 

(km/h) 

metre per second (m/s) 

metre squared per day 

(m»/d)

cubic metre per second 

<m»/s)

cubic metre per minute 

(m3/min) 

litre per second (1/s) 

litre per second per 

metre [(l/s)/m]

kilometre per hour 

(km/h) 

metre per second (m/s) 

gram per cubic 

centimetre (g/cm3 ) 

gram per square 

centimetre (g/cm5 ) 

gram per square 

centimetre

English equivalent

combinations Continued

= .0353 cubic foot per second

square mile [(ft3/s)/mi«]

conductivity) (ft/d) 

= 5.28 feet per mile (ft/mi)

= .9113 foot per second (ft/s) 

= 3.28 feet per second

= 10.764 feet squared per day (fta/d) 

(transmissivity)

= 22.826 million gallons per day 

(Mgal/d)

= 264.2 gallons per minute (gal/mir) 

= 15.85 gallons per minute

= 4.83 gallons per minute per foot 

[(gal/min)/ft]

 .62 mile per hour (mi/h)

= 2.237 miles per hour 

= 62.43 pounds per cubic foot (Ib/f s )

 .0142 pound per square inch (lb/ina)

Temperature

degree Celsius CO 

degrees Celsius 

(temperature)

= 1.8 degrees Fahrenheit ("F) 

= [(1.8X°C)+32] degrees Fahrenheit

GILA RIVER PHREATOPHYTE PROJECT

ACCURACY OF EVAPOTRANSPIRATION RATES

DETERMINED BY THE WATER-BUDGET METHOD,

GILA RIVER FLOOD PLAIN, SOUTHEASTERN ARIZONA

By RONALD L. HANSON and DAVID R. DAWDY

ABSTRACT

Evapotranspiration by phreatophytes (primarily saltcedar) was 

determined by the water-budget method for 5,500 acres (2,230 ha) of 

the Gila River flood plain in southeastern Arizona. The water budget 

consists of 12 components including surface and subsurface flow 

through the study area, precipitation on the area, and soil-moisture 

changes in the unsaturated soil profile.

Nine years (1963-71) of hydrologic data were collected on four 

reaches within the area. These data provided over 400 measurements 

of evapotranspiration for two- or three-week periods. Midway through 

the study the vegetation was removed from the flood plain. The 

evapotranspiration measurements are therefore defined for both 

natural vegetative cover and essentially bare-ground conditions.

This report shows how each component of the water budget was 

evaluated, demonstrates the significance of each component in 

relation to the total evapotranspiration, and describes the methods 

used to evaluate the relative accuracy of each component.

The two most significant components of the water budget are, 

generally, the Gila River inflow and outflow. One of the least 

significant is tributary inflow, which occurred only 4 percent of the 

time during the 9-year study. Soil-moisture change is highly 

significant during periods of low streamflow and is one of the more 

difficult components to measure. The ground-water flow components 

are the least variable in the water budget, fluctuating only in response 

to seasonal changes in the downvalley ground-water slope.

The total measurement error of each component consists primarily 

of a sampling error which is dependent on the number of observation 

points used to measure the component. This error is time variant, 

reflecting both the variability in repetitive measurements and the 

error due to missing data. Included in the total measurement error is 

a bias error which gives a constant overestimate or underestimate of 

the component. Only the ground-water flow components introduce 

a measurable bias error, but the direction of this error is unknown 

and its magnitude in relation to evapotranspiration is relatively 

insignificant.

The total measurement error in evapotranspiration is not related to 

the magnitude of evapotranspiration but rather to the total volume of 

water moving through the area. Thus, the minimum errors occur 

during the midsummer months of maximum evapotranspiration 

when streamflow is low and precipitation is negligible.

Evapotranspiration rates computed for reach 1 indicate that 

phreatophyte clearing reduced summer rates by nearly 45 percent. 

The average computed measurement errors in summer evapotranspiration rates, before and after clearing, are ±59 percent and ±113 

percent, respectively, and the average measurement error in the

change in summer evapotranspiration as a result of clearing is nea^y 

±200 percent. These large computed measurement errors are shown 

to overestimate substantially the true measurement variable in 

evapotranspiration. The computed errors do give, however, a good 

indication of the relative significance of each evapotranspiration 

value and provide a means of selecting those values which should be 

used in computing average evapotranspiration rates. Furthermore, 

the results of this error analysis show that reliable estimates of 

summer evapotranspiration can be determined and that a significant 

difference in summer evapotranspiration could be detected as a result 

of clearing phreatophytes from the flood plain.

INTRODUCTION

The determination of ET (evapotranspiration) by 

phreatophytes from a flood plain by the water-budget 

method requires that all significant movement of liquid 

water into and out of the area be measured.

Components of the water budget include surface and 

subsurface flow through the area, precipitation on the 

area, and soil-moisture content changes in the unsatrrated zone of the area. It has been the general opinion of 

most researchers that the measurement errors associated with these components are too large to provide 

reliable estimates of ET particularly when an estimate of water salvage as a result of phreatophyte 

removal is desired. To date (1976) few studies have been 

conducted which evaluated ET from a large area 

(Gatewood and others, 1950; Turner and Skibitzke, 

1952; Bowie and Kam, 1968), and little is known about 

the accuracy of these evaluations. In October 1962 a 

nine-year water-budget study began on 5,500 acres 

(2,230 ha) of the Gila River flood plain in southeastern 

Arizona. The primary objective of this study was to 

evaluate seasonal ET rates of phreatophytes from the 

area and water salvage following removal of the 

phreatophytes (Culler and others, 1970).

The study was designed and instrumented to measure 

independently each component of the water budget. Tt e

Li
L2 GILA RIVER PHREATOPHYTE PROJECT

purposes of this report are to show how each component 

of the water budget was derived, to describe the method 

used in estimating the accuracy of each component, and 

to demonstrate the significance of each component and 

its error in the resultingET values. The errors for m.ost 

of the water-budget components are expressed in terms 

of the standard error of their measurement. For some 

components, however, the measurement error can only 

be approximated. The assumptions and criteria used to 

arrive at these approximations were defined to provide a 

resultant error which, in most instances, should exceed 

the expected standard error in the measurement of the 

component. The derived error of each ET value is 

therefore considered to be only a relative indicator of 

measurement variability in ET.

Two types of error were investigated in this analysis the bias error and the sampling error. The bias 

error is a constant time-invariant error caused by 

consistent overestimates or underestimates of the true 

value of the component. When evaluating the average 

change in evapotranspiration ( &ET) such as occurs 

after clearing phreatophytes from the flood plain this 

error cancels and is thus iipt included in the determination of the accuracy of AET. However, when evaluating 

absolute values of ET, the bias error may, in some 

instances, be a significant part of the total error inET.

The sampling error reflects the variability in the 

measurement of a water-budget component due to 

insufficient sampling of the component. This error 

decreases with an increase in the number of observations at the sample point or with an increase in the 

number of sample points, and it increases with an 

increase in the magnitude of the component. The 

sampling error is time dependent for those components 

in which the number of sampling points and the 

magnitude changes during the study period.

The total measurement error of each ET value was 

obtained from the sum of squares of the bias and 

sampling errors defined for each component. Because of 

independence of the components no covariance term 

exists in the computation of the total measurement 

error. Furthermore, this total error term is considered 

to be only a relative indicator of the measurement 

variability in ET and not an estimate of the expected 

standard error in ET.

This study was conducted under the general supervision of R. C. Culler, project chief of the Gila River 

Phreatophyte Project. Transformation of basic field 

data into a form acceptable for analysis was performed 

by R. M. Myrick and F. P. Kipple. The authors are 

indebted to the San Carlos Apache Indian Tribe and the 

Bureau of Indian Affairs for the use of their lands and 

facilities, respectively, to make this study.

DESCRIPTION OF STUDY AREA

The study area includes a 15-mile (24 km) length of 

the Gila River flood plain above San Carlos Reservoir in 

southeastern Arizona (fig. 1). The flood plain averages 1 

mile (1.6 km) in width and has a gradual dovnvalley 

slope of about 1.5 ft per 1,000 ft. The water-bearing 

deposits in the flood plain consist of basin-fill deposits 

and alluvial deposits which overlie the basin fill. The 

basin-fill deposits are more than 1,000 ft (300 m) thick 

and consist of fine-grained material of low permeability. 

The alluvial deposits are as much as 60 ft (20 m) thick 

and consist of lenticular gravel, sand, and silt beds with 

a relatively high permeability. The alluvial deposits 

form the flood plain and lower terraces in the study area. 

The Gila River meanders across the alluvium in a 

channel averaging 110 ft (35 m) wide and 7 ft (2 m) deep. 

A detailed description of the geology of the stuCy area is 

given by Weist (1971).

The depth to ground water ranges from about 5 ft (IVz 

m) near the river to more than 20 ft (6 m) near the outer 

boundaries of the flood plain. Wells that penetrate 

through the flood-plain alluvium into the urderlying 

basin fill indicate that ground water in the basin fill 

flows vertically upward into the alluvium at a rate of 

about 0.3 ft (0.09 m) per year or 0.1 million gallons per 

day per acre (0.01 m3/s/ha). Down valley ground-water 

movement through the alluvium averages about 5.1 

acre-ft per day (0.0063 hm3/day) (Hanson, 1972). This 

downvalley flow is equivalent to 1.7 million gallons per 

day (180,000 m3/s).

Gila River inflow to the study area is derived from 

11,500 mi2 (29,800 km2) of drainage area. Most of the 

streamflow results from winter and late summer 

precipitation. The average discharge of the Gila River is 

250 ft3/s (7.1 m3/s) but can range from no flow for a few 

days in the summer to several thousand cubic feet per 

second during the winter and summer storm periods.

Tributary inflow is derived from 225 mi2 (583 km2) of 

drainage area bordering the study area. Annual runoff 

from these tributaries is small and generally occurs 

only for short periods during the summer as a result of 

thunderstorms.

Precipitation occurs primarily in December and 

January from large frontal storms and in July, August, 

and September from short-duration high-intensity 

convective storms. The average annual precipitation at 

San Carlos Reservoir is 14 in. (360 mm), but annual 

totals have ranged from 4.0 in. (102 mm) to 25.8 in. (655 

mm) during the period of record, 1882-1973.

Mean daily temperatures in the study area range 

from a minimum of 32°F (0°C) during the winter to a 

maximum of 100°F (38°C) in the summer. Pan evaporation at San Carlos Reservoir averages 97 in. (2,460 mm)
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L3

per year, and evaporation from the reservoir is 

estimated at 70 in. (1,780 mm) per year (F. P. Kipple, 

written commun., 1975).

When the study began (October 1962) nearly 70 

percent of the vegetation on the flood plain was 

saltcedar (Tamarix chinensis)!, with the remaining 

vegetation consisting of mesquite (Prosopis), willow 

(Salix), cottonwood (Populus), seepwillow (Baccharis), 

and seepweed (Suaeda). Vegetation was removed from 

the flood plain in segments of several acres at different 

periods during 1967-71. Virtually all the study area 

was cleared of vegetation by March 1971.

THE WATER-BUDGET EQUATION

Twelve components are significant in defining ET

'Also referred to as Tamarix pentandra and Tamarix gallica.

EXPLANATION

from the water budget of the Gila River flood plain. A n 

equation expressing these components is

ET=Qj-Q0+QT +AC+P+AMS +

where 

ET 

Qf 

Qo 

QT

AC =

= evapotranspiration from the area,

= surface inflow of the Gila River,

= surface outflow of the Gila River,

= surface inflow from tributaries borderirg

the area, 

change in channel storage in the Gila

River,

= average precipitation on the area, 

= average change in moisture content in the

unsaturated soil zone located immediately below the land surface,

114° 112° 110°

Boundary 36° of Gila River flood plain

Gaging station on the Gila River

2-inch access pipe for 

soil-moisture meter

4-inch observation well 

Precipitation gage

DETAIL OF TYPICAL CROSS SECTION

3 1

I I 

3 2

2

|

I I 

4

f

I

4

I I 

6

T

8

MILES

I I 

1010 KILOMETRES

FIGURE 1. Map showing study area and instrumentation location.
L4 GILA RIVER PHREATOPHYTE PROJECT

AM/ = average change in moisture content in the 

unsaturated intermediate zone located 

between the overlying soil zone and the 

underlying capillary zone,

average change in moisture content in the 

capillary zone located below the intermediate zone and within the zone of 

water-table fluctuation,

ground-water inflow vertically upward 

into the alluvium from the underlying 

basin fill,

ground-water inflow downvalley through 

the saturated alluvium,

ground-water outflow downvalley 

through the saturated alluvium,

average lateral ground-water movement 

through the capillary zone between the 

flood plain and the adjacent terrace area.

The basic data used to compute each component of the 

water budget were collected at the 13 cross sections 

shown in figure 1. Each cross section included three 

ground-water level observation wells equipped with 

recorders on each side of the Gila River, an access hole 

for measuring soil-moisture content adjacent to each 

well, and a nonrecording precipitation gage at both ends 

of each cross section (see inset showing detail of typical 

cross section in fig. 1). Recording precipitation gages 

were established at both ends of cross sections 1, 9, 17, 

and 23. Streamflow gaging stations were established at 

cross sections 1,9,13,17, and 23 to define the Gila River 

inflow and outflow through reaches 1, 2, 2a, and 3 as 

shown in figure 1. Tributary inflow was measured at 16 

continuous-recording gaging stations and 43 creststage gages (not shown in fig. 1) located along the 

perimeter of the study area. A more detailed discussion 

of the instrumentation is given by Culler and others 

(1970).

To solve for ET in equation 1 all basic field data were 

first transformed into terms described by the components in equation 1. Table 1 shows the water-budget 

components obtained from the basic field data in reach 1 

and corresponding ET values for the 22 budget periods 

included in the 1964 water year (Oct. 1, 1963, to Sept. 

30,1964). The budget period includes either a two-week 

or a three-week period, depending on when field 

measurements of soil moisture were obtained. The end 

date shown in column 1 of table 1 refers to the last day of 

each budget period. The project day shown in column 2 

refers to the ending day of the budget period referenced 

from the day the study began on October 1,1962. Table 2 

shows the water-budget components, the resulting ET, 

and their corresponding errors for the 21-day budget 

period 688 to 708 (Aug. 18 to Sept. 7, 1964). The

following sections of this report describe how each ET 

component in table 2 was derived and discusses the 

methods used in deriving the sampling and bias errors 

associated with each component.

EVALUATION OF WATER-BUDGET 

COMPONENTS AND ERRORS

STREAMFLOW

The Gila River inflow (Qj) and outflow (^Q) were 

obtained by summing the computed daily discharges 

over the budget period at the upstream and downstream 

ends of the reach, respectively. For reach 1, Q/ was 

obtained from the computed daily discharges at cross 

section 1, and QQ was obtained from the computed daily 

discharges at cross section 9 (fig. 1). The total volume of 

inflow during the 21-day period in table 2 is l,0f 1 acre-ft 

(1.296 hm3), and the total volume of outflow is 1,107 

acre-ft (1.365 hm3). Table 3 lists the daily discharges 

used for obtaining the volume of inflow Q/ through cross 

section 1 during budget period 688-708. The difference 

between Q7 and Qo is -56 acre-ft (-0.069 hm3), 

indicating a net inflow to the river through the reach.

The accuracy of the computed volume of water 

passing a gaging station during a budget period is 

dependent on the measurement error in discharge and 

the accuracy of the stage-discharge relation defined for 

the station. The channel of the Gila River is subject to 

considerable scour and fill; thus, good definition of the 

stage-discharge relation requires frequent discharge 

measurements. Burkham and Dawdy (1970, figs. 11 and 

12) developed curves of the relation between the 

standard error in a computed instantaneous discharge 

obtained from the rating curve and the frequency of 

discharge measurements for the stations at cross 

sections 1 and 9. The data used in their analysis was 

restricted to flows below a bankfull discharge of about 

4,000 ft3/s (100 m3/s). Their error curves were developed 

on the assumption that the standard error of any given 

measured discharge is 4 percent as indicated by Carter 

and Anderson (1963, fig. 1), and they show that the 

standard error in a computed instantaneous discharge 

obtained from the stage-discharge relation is greater for 

the summer months (July through October) than for the 

winter and spring months (November through June). 

Burkham and Bawdy's error curves for the summer 

months at cross sections 1 and 9 were averaged to define 

the error curves shown in the semilog plot of figure 2 for 

measurement frequencies of one measurement every 3 

days, every 5 days, and every 12 days. Because the 

curves in figure 2 were developed from only the summer 

data, their application to winter flows may give 

estimates of the standard error in a computed discharge 

which are too high. The frequency of discharge 

measurements for the Gila River during the study
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L5

TABLE 1. Evapotranspiration and water-budget components for 1964 water year, reach 1

[All values in acre-ft for indicated days. Flood-plain area is 1,723 acres (697 ha) ]

End 

date

1963 

Oct. 15-________

Oct. 29_________

Nov. 12 

Nov. 26_________

Dec. 10 __ ___

Dec. 24_________

1964

Jan. 21 _ _____

Feb. 4 .

Feb. 18- .

Mar. 3.. _ __ .

Mar. 17-_______.

Mar. 31_--._-__

Apr. 14 -______.

May 25 _ _ _ _

Jul. 6________.

Jul. 27 ______

Aug. 17_________

Sep. 7 ____ .

Sep. 28__ .

Project 

day

_______ 380

 _ 394

.--_ ___ 408

._- _ 422

_______ 436

_-..- _ 450

..______ 464

.__ ___ 478

.____-_ 506

.- _._ 534

.___ -562

 _ 582

._ __ __ 603

. ____ 645

.___-.. 666

.--. 687

________ 729

Number 

of days

14

14

14

14

14

14

14

14

14

14

14

14

14

20

21

21

21

21

21

21

21

ET

293

582

137

209

494

-604

97

9

196

112

269

410

0.79

424

1,056

1,903

513

2,799

0,

983

11,960

5,036

4,107

2,691

2,185

958

977

1,166

688

8

7,614

18,502

1,051

24,456

Qo

-960

-11,288

-5,022

-3,873

 2 269

QQQ

-890

-600

-144

-17,068

-21,821

AC

17

7

1 Q

 29

-5

7

5

0

4

7

0

00

QT

i

21

0

176

oeo

310

P

0

96

36

50

24

0

0

26

4

0

5

93

272

174

289

AA/c &

i -24

26 -23

0

-18

9

-20

6

-2

-17

16

23

17

15

6

-61

8

-73

Mj \MC

137

-128

30

-70

71

-17

1 A

-166

6

17

17

21

-51

27

127

162

145

36

-86

QO

GB

41

41

41

41

41

41

41

41

41

41

CQ

62

fio

62

62

62

CO

62

«/

87

87

85

83

83

82

QO

82

QO

00

84

85

QC

121

128

1 00

1 9.O

1 00

123

GO

-77

-77

 77

-77

-77

-76

-76

-76

76

-76

-76

76

-76

-107

-113

1 1 0.

-115

1 1Q

-112

-113

->«

46 -64

-5

-46

9

61

OQ

-191

46

5

1 Q

fifi

-42

77

174

-7

44

TABLE 2. Evapotranspiration and water-budget components for the 21-day budget period 688-708 (August 

18-September 7, 1964) in reach 1 and corresponding sampling and bias errors of each component

[All figures in acre-ft per 21 days]

ET Ql Qo AC QT AMc AM; G, 'O

Component value ___ 513 1,051 -1,107 23 252 174 8 __ 82 62 124 -112 -44 

Sampling error _.___ 334 107 121 12 252 16 20 __ 42 0 0 0 J38 

Bias error _______ 69 0 0 __ 00 0 62 23 21 0 

Total measurement error_______B£r=(3342 +692)'^ = ± 341 acre-ft per 21 days

TABLE 3. Daily discharges and errors in discharge at cross section 1 

for budget period 688-708

Date 

(1964)

18

19

20

21

?,?,

23

24

25

26

27

28

29

30

31

1

2

3

4

5

6

7

Qi

Eq 

EQl

i

(ft3/s)

August 

______ _______ 47.0

___ _ _____ ___ 29.0

_ __ __ _____ __ 32.0

___ _ ____ ____ 55.0

24.0

______ __ _. ___ 18.0

_. _ __ ______ 11.0

__ _ _ _______ 9.2

_ __._ . 185.0

_ ___ _ ______ 20.0

___________ _ ___ 11.0

 _ _ _ _______ 11.0

___ __ ______ 9.1

_ _ ___ ___ ____ 9.6

September 

_____ _ __ ____ 10.0

_ ______________ 9.6

__ _____________ 9.2

_ __ ________ __ 7.9

_______ ___ _ _ _ 7.2

8.2

7.0

Totals _ _ 530.0

e il

(dimensionless)

0.133 

.142 

.140 

.130 

.146 

.151 

.160 

.164 

.108 

.149 

.160 

.160 

.164 

.163

.162 

.163 

.164 

.166 

.168 

.166 

.169

= 530.0 x 1.9835 = 1,051 acre-ft/21 days 

= (581.6/21)*= = ±5.26 ft3/s = ±10.4 acre-ft/day 

= 21 ((5.26W4.2)* = ±53.9 fts/s/21 days = ±107 

days

(e q xg)2 

(ft3/s>2

39.1 

17.0 

20.2 

51.3 

12.2 

7.4 

3.1 

2.3 

395.6 

8.9 

3.1 

3.1 

2.2 

2.4

2.6 

2.4 

2.3 

1.7 

1.5 

1.8 

1.4

581.6 

acre-ft/21

1From figure 2 (or equation 2).

period averaged about one every 5 days; thus, the 5-dry 

curve in figure 2 was used in this analysis. The equation 

for this curve when q ^ 4,000 ft3/s (113 m3/s) is

eq =0.205 - 0.043 Iog10g, (2)

where eq is the error, expressed as a fraction of tl^ 

instantaneous discharge, q, in cubic feet per second. For 

discharges above 4,000 ft3/s (113 m3/s), the error is 

assumed to increase linearly by the relation

e9 =-1.75 + 0.50 (3)

Equation 3 assumes that the error in flows abo^e 

bankfull stage increases from 5 percent of the discharge 

at q =4,000 ft3/s (113 m3/s) to 25 percent of the discharge 

at g = 10,000 ft3/s (283 m3/s). Estimates of the error in 

discharge during overbank flooding in the Gila River as 

defined by equation 3 are believed to be high because 

most of the flow in the 4,000 to 10,000 ft3/s range is 

contained within the main channel where measurement errors are minimal. Equation 3 is not considered 

applicable to discharges above 10,000 ft3/s (283 m3/s). 

Burkham and Dawdy (1970) showed that the standard error, EQ, in the volume of flow passing a station 

during the budget period is
L6 GILA RIVER PHREATOPHYTE PROJECT 

INSTANTANEOUS DISCHARGE (q) . IN CUBIC METRES PER SECOND

0.4 0.6 0.8 1.0

j____i i i

60 80 100

i i i

200

100 1000 

INSTANTANEOUS DISCHARGE (q). IN CUBIC FEET PER SECOND

10,000

FIGURE 2. Average relation between instantaneous discharge and the error in discharge expressed as a fraction of the discharge for

summer (July through October) flows of the Gila River at cross sections 1 and 9.

(4)

where Eq is the average standard error in the mean 

daily discharge for the budget period expressed as a 

fraction of the discharge, D is the number of days in the 

budget period, and N is the number of discharge 

measurements made during the budget period. Eq is 

obtained from the relation

E.

D

 =[£( (5)

where qd is the average discharge for day d and eqd is the 

fractional error in qd as defined by equation 2 or 3. Only 

the discharge measurements provide an independent 

estimate of the daily discharge all other daily discharges are computed from the stage-discharge relation 

and are therefore dependent estimates. Equation 4 

considers this dependency by including the number of 

measurements, N =D/5, made during the budget period. 

The application of equation 2 or 3 to obtainEq assumes 

that the error for a daily discharge is the same as the 

error for an instantaneous discharge. This assumption 

is invalid only during wide fluctuations in discharge 

which occur for a few days during the winter snowmelt 

runoff and the late summer thunderstorm periods.

Included in table 3 are the computations for determining the standard error of the volume of inflow through 

cross section 1, Eqr during budget period 688-708. The 

fractional errors, eq , shown for each daily discharge 

were obtained from figure 2 (or equation 2). Applying 

equation 5 gives the average error for the 21 daily 

discharges ofEg =±10.4 acre-ft (±0.0129 hm3) perday. If

one discharge measurement is made every 5 days, then 

N=4.2 measurements during the 21-day budget period 

and£Q7 =±107 acre-ft (±0.132 hm3) per 21 days. Similar 

computations were made to obtain the outflow error 

through cross section 9 of EQ0=±l2l acre-ft (±0.149 

hm3) per 21 days. The values forEQj andEQ0 ar^ shown 

in table 2 as sampling errors for water-budget components Qi and QQ, respectively.

A comparison of the net change in discharge in reach 

1 of -56 acre-ft (-0.069 hm3) for budget period 688-708 

with the average sampling error in this change, 

computed as V (107)2+(121)2=±162 acre-ft (±0.200 

hm3), emphasizes the significance of the sampling error 

in the discharge components of the water budget.

No independent evaluation of the discharge error was 

made for the flow at the gaging stations at cross sections 

13, 17, and 23, because the flow characteristics and 

measurement conditions at these stations are similar to 

those at cross sections 1 and 9 and the freqMency of 

discharge measurements is the same. The error relation 

of figure 2 was therefore applied to the flow at cross 

sections 13, 17, and 23.

Of the 12 components included in the water budget, Q/ 

and Qo are generally the largest (table 1). A comparison 

of the ET values in table 1 with their corresponding 

discharge values suggests that a highET coincides with 

a high discharge. But, this is true only during periods of 

high runoff in the summer when thunderstorms provide 

sufficient moisture to satisfy the seasonally high potential ET. Thus, all ET values associated with a high 

runoff in table 1 reflect, in part, a large sampling error 

and are therefore not considered reliable estimates of 

the true ET.

The errors in the volume of discharge for those budget
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L7

periods containing days without streamflow were not 

evaluated and the ET rates computed for those periods 

were discarded.

Possible bias in the determinations of Q/ and QQ was 

investigated for the gaging stations at cross sections 1 

and 9. The difference between the measured discharge 

and the computed discharge obtained from the stagedischarge rating using the river stage observed on the 

day of the measurement was plotted against the 

computed discharge. The plots for both stations show a 

relatively uniform distribution of points about the line 

of zero difference, indicating no significant bias in the 

computed discharges. A further investigation of possible bias in streamflow was made by comparing the gains 

and losses in the computed flow in reach 1 for days 

midway between discharge measurements with the 

gains and losses on days when discharge measurements 

were made. The gains and losses obtained from the 

computed discharges appear to be equally distributed 

among the gains and losses from the measured 

discharges, thus supporting the assumption of no bias in 

the computed discharges at cross sections 1 and 9. A plot 

of these gains and losses with time of year for the period 

before the phreatophytes were removed indicates that 

the Gila River was a gaining stream during the winter 

months (October through February) and a losing stream 

during the summer months of high ET (March through 

June). No independent evaluation of streamflow bias 

was made for reaches 2a or 3.

CHANNEL STORAGE

The change in channel storage within a reach of the 

Gila River during a budget period is computed from the 

average change in wetted cross-sectional areas of the 

channel at the upstream and downstream ends of the 

reach. This change may be expressed as

AC = (6)

where AC is the change in channel storage in acre-feet, 

L is the length of the river channel within the reach in 

feet, and A/ and AQ are the wetted cross-sectional areas 

of the river channel in square feet at the upstream and 

downstream ends of the reach, respectively. Subscripts 

1 and 2 denote the first and last day of the budget period, respectively. A positive change in AC indicates a 

depletion in channel storage and a corresponding 

addition to ET. A negative change in AC indicates an 

increase in channel storage and a corresponding 

subtraction from ET.

Aj and AQ in equation 6 were not measured for each 

budget period but were derived indirectly from a

predetermined area-discharge relation. Figure 3 shoe's 

the data points defining the relation between measured 

discharge and corresponding wetted area using data 

from selected discharge measurements at cross section 9 

for water years 1963-69.

Similar relations were defined for cross sections 1,13, 

and 17. The curves defining the area-discharge relation 

for each cross section all lie within the scatter of the data 

defining the curve for any one cross section. Thus, only 

one relation approximating the average of all of tl v» 

curves has been used to obtain the A/ and AQ values in 

equation 6. This relation is expressed as

A =2.9^0.65, (7)

where q is the daily average discharge in cubic feet p?r 

second and A is the wetted cross-sectional area in 

square feet.

The change in channel storage for each budget period 

was determined by solving equation 7 for A/I} A/2 , AOp 

and Ao2 , using the inflow and outflow discharges 

computed for the beginning and ending days of the 

budget period and substituting these area values in 

equation 6. The discharge values and corresponding 

cross-sectional areas for budget period 688-708 are 

given in table 4. Substituting the A/ and AQ values in 

table 4 into equation 6, where L=36,800 ft (11,200 n) 

for reach 1, gives AC=23 acre-ft (0.028 hm3) per 21 days.

The accuracy of AC is dependent on the accuracy of 

the AI and AQ values in equation 6. The error in tl^ 

determination of A may be expressed as

Vfe (8)

where

EA = the total sampling error in the wetted

cross-sectional area of the river channel, 

e\ = the temporal variability of A at a given cro-^s

section as indicated by the scatter of data

points in the q versus A relation of figure

3,

62 = the variability of A between cross sections 

as indicated by the difference in the q 

versus A curves of figure 3,

63 = the variability in A due to the error in tl s 

discharge used in equation 7 to obtain A as 

indicated by the q versus eq relation of 

figure 2.

Equation 8 is applicable only if ei, 2, and 63 are 

independent estimates of error. This assumption is 

considered valid because no relation should be expected 

to exist between these three error terms.

The temporal variability in A at cross section 9 ranges 

from ±10 ft2 (±0.93 m2) when g = 10 ft3/s (0.28 m3/s) to 

±70 ft2 (±6.50 m2) when 9 =4,000 ft3/s (113 m3/s) (fig. 3)
L8 GILA RIVER PHREATOPHYTE PROJECT 

AVERAGE DAILY DISCHARGE (q). IN CUBIC METRES PER SECOND

1000

600 800 10CO

I I I

40 60 80 100

l ill

EXPLANATION

Data for cross section 9(1963-69 water years) 

Average relation for all cross sections 

Cross section 1 

Cross section 9 

Cross section 13 

Cross section 17

100

100 1000 4000

AVERAGE DAILY DISCHARGE (q), IN CUBIC FEET PER SECOND 

FIGURE 3. Relation between average daily discharge for the budget period and wetted cross-sectional area for cross sections 1,9,13, and 17.

TABLE 4. Average daily discharge (q), corresponding wetted crosssectional area of river channel (A), and error in the area (E^.) for 

cross sections 1 and 9 on budget period days 688 and 708

Budget Cross section

day

688 __ _ ____

708 _________

q

(ft3/s)

47.0

7.0

AI

(ft2 )

35.4

10.3

1

EA

(ft2 )

17.5

8.3

Cross section

q

(ft3/s)

52.0

5.1

A0

(ft2 )

37.8

8.4

9

EA

(ft2)

18.2

7.4

and can be expressed as ei=4.6g°-33 . The variability in 

A between cross sections ranges from± 2 ft2 (±0.2 m2) at 

q = W ft3/s (0.28 m3/s) to ±55 ft2 (±5.1 m2) at ?=4,000 

ft3/s (113 m3/s) (fig. 3) and can be expressed as 

e2=0.38<70- 60 . The error in A attributed to the error in 

the computed discharge can be derived from the q versus 

eq relation of figure 2. For this analysis the fractional 

error, eq , is assumed to be 0.13 for all discharges, and so

the error in q is 0.13q. When this discharge is included 

in equation 7, 63 can be expressed in term« of the 

expected values:

0.65x (9)

Applying a first order approximation of the Taylor 

series expansion to equation 9 and dropping all but the 

first terms in the series yields e3=0.17^°-65 . Inclusion of 

higher order terms in the Taylor series was found to 

have no effect on the expression for 63.

The above expressions for e\, 62, and 63, when 

substituted in equation 8, can be approximated by

EA=3.9q°-39 , (10)

where EA is in square feet and q is in cubic feet per 

second.

Figure 4 shows the relation of q toE^ both as defined
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L9

ERROR 'ERAGE INWETTED CROSS-SECTIONAL 

), (EIN AREA SQUARE FEETA

o S 

5 o^

AVERAGE DAILY DISCHARGE (q), IN CUBIC METRES PER SECONC 

0.2 0.40.60.81 2 4 6 8 10 20 406080100 200

1

-

^

EA ~3.9q*\^

-

»- o> NJ ^-» 

°bo> '-> kj* 

CROSS-SECTIONAL ERROR WETTEDIN ERAGE 

fE^;, SQUARE AREA INMETRES

10 100 1000 10,000

AVERAGE DAILY DISCHARGE (q). IN CUBIC FEET PER SECOND

FIGURE 4. Relation between the average daily discharge for a budget 

period and the error in wetted cross-sectional area of the Gila River 

channel.

by equation 8 and as defined by equation 10. Equation 

10 was used to obtain the errors in the wetted 

cross-sectional areas shown in table 4.

An estimate of the standard error in AC for any given 

budget period can be obtained from the square root of 

the sum of the variances of each At and AQ value in 

equation 6 or

+#V, V*.(iD 2x43,560

where the EA terms are in square feet and E^; is in 

acre-feet. Solving equation 11 for iheE^ values in table 

4 gives-EAC=± 12 acre-ft (± 0.015 hm3) for budget period 

688-708. This error is about 50 percent of the computed 

change in channel storage of +23 acre-ft (+0.028 hm3) 

but is only 2 percent of the total ET of 513 acre-ft (0.633 

hm3).

AC is generally not a significant component in the 

water-budget equation, and greater refinement in its 

computation was not considered justified. The determination of E&c in equation 11 assumes independence of 

the error terms EA{ andEA0 both with time and between 

cross sections.

The computed increase in channel storage (-AC in 

equation 1) is substantially underestimated during 

periods of high discharge when low-lying portions of the 

channel banks are overtopped and surface water goes 

into depression storage in the many small channels and 

low areas on the flood plain. ET values computed for 

these periods are commonly unrealistically high and 

actually indicate a large component of unmeasured 

water going into storage. This may partly explain the 

unrealistically high ET values for the budget periods 

ending on project days 394, 666, 687, and 729 (table

1) all periods of high discharge in the Gila River. The 

subsequent period of drainage from depression store <?e 

immediately following a high discharge frequently 

causes an underestimate of the decrease in channel 

storage (+AC in equation 1) resulting in computed ET 

values which are too low. Reliable field measurements 

of depression storage were not possible, particularly 

during high flow periods. However, the errors in 

discharge of the Gila River for these high flow periods 

are too large to give reliable estimates ofET and those 

ET values are generally disregarded.

TRIBUTARY INFLOW

Runoff from tributaries adjacent to the study area 

originates from 225 mi2 (583 km2) of drainage area. A 

total of 43 tributaries draining 95 percent of the area 

adjacent to reaches 1 and 2 were instrumented with 

either recording-stage gages or crest-stage gages. The 

stage data from the recording gages were used for 

estimating runoff volumes, and the crest-stage gages 

were used primarily to define periods of significant 

runoff. Tributary runoff into reach 3 was not measured, 

because collection of water-budget data on this reach 

was discontinued before instrumentation on the 

tributaries was fully established.

Normally the tributaries were not monitored during 

the winter season (November through April) because 

precipitation during this period is generally from 

frontal storm systems which may cover a large area but 

seldom produce significant flow volumes into the Gila 

River. Tributary inflow to the study area was observed 

during a few large winter storms, but these periods 

coincide with a high discharge in the Gila River and a 

corresponding large error in the water budget. The ET 

values for these periods have been discarded or are 

recognized as not reliable. The only significant tributary runoff observed in the project area during the 

9-year study occurred from May through October. Most 

of this runoff resulted from short, intense thunderstorms in July and August.

Tributary runoff occurred, on the average, less than 4 

percent of the time, or about 13 days out of the year, and 

runoff in any one tributary occurred, on the average, 

only 3 days per year. Tributary runoff occurred in about 

30 of the over 180 budget periods evaluated during the 

nine-year study, but only 15 of these periods had runoff 

volumes which were a significant part of the ET.

Estimates of the volume of tributary runoff into 

reaches 1,2, and 2a were obtained from stage-discharge 

relations and peak discharge-storm volume relations 

developed for each tributary by Burkham (1976). Tie 

runoff volumes during budget period 688-708 in each of 

the 20 tributaries bordering reach 1 (table 5) wore 

estimated from these relations.
L10 GILA RIVER PHREATOPHYTE PROJECT

TABLE 5. Tributary runoff into reach 1 during 

budget period 688-708

Tributary 

number

24

25

26

27

28

29

30

31

32

33

Subtotal

Total| .

Runoff 

(acre-ft)

0.4

3.0

5.0

 

_ 8.4

Tributary 

number

34

35

36

37

38

38.5

39

40

41

42

Runoff

(acre-ft)

7.0

14.0

2.0

5.0

50.0

42.6

2.5

1.4

120.0

243.9

. . 252.3

Burkham (1976) indicated that definition of the 

stage-discharge and peak discharge-storm volume relations are poor at best and estimated that the computed 

volume from a runoff event in any one tributary may be 

in error by 100 percent; however, periods when runoff 

did not occur were considered to be accurately defined. 

Because of the generally low volume of tributary inflow 

to the study reaches and their relatively infrequent 

occurrence, no evaluation was made of their standard 

error. All estimates of tributary inflow to the study area 

were thus assumed to be 100 percent in error. Accordingly, the tributary runoff of Qy=252 acre-ft (0.311 

hm3) for budget period 688-708 (table 2) was assumed to 

have an error of EQT=±252 acre-ft (±0.311 hm3).

PRECIPITATION

Accumulated precipitation during each budget period 

was obtained from wedge gages located at the ends of 

each cross section (see fig. 1). Visits to the gages were 

made at two- or three-week intervals which coincided 

(within two or three days) with the last day of the budget 

period and with the field measurements of soil moisture. 

In a few instances precipitation occurred during the 

two-day period required to visit all the gages in a reach, 

resulting in discrepancies in the total accumulated 

precipitation between gages for the budget period.

These occurrences were rare, and an attempt ws made 

to correct only the obvious discrepancies.

Each gage was assigned a portion of the total area in 

the reach using a method of proportioning whicl^ closely 

approximates the Thiessen method (1911). T^e total 

accumulated precipitation for the budget period was 

computed as an average weighted value from

(12)

.7 = 1

where 

P the average weighted precipitation for the

budget period, in inches, 

Pj = the accumulated precipitation at gage.; for

the budget period, in inches, 

AJ = the area assigned to gagej, in acre?, and 

n = total number of gages in the reach.

Table 6 shows the precipitation amounts observed at 

each gage in reach 1 for budget period 688-708 and the 

areas assigned to each gage. The average weighted 

precipitation of these 10 gages isP=1.21in. (30.7 mm), 

or 174 acre-ft (0.215 hm3) in volume for the budget 

period.

Occasionally the precipitation at a gage was not 

obtained. In such instances, the precipitation was 

estimated using observed data from nearby wedge 

gages or from the recording gages located at the ends of 

the reach. Thus, all budget periods contain a complete 

set of data.

The total measurement error in the computed 

average precipitation for a budget period can include 

both a bias error and a sampling error. A bias error 

commonly occurs when the gage is located too close to 

trees, buildings, or other obstructions which interfere 

with catchment in the gage. This type of error is not 

considered significant for the project area as a 11 of the 

wedge gages were located in areas of ample exposure. 

The measurement of precipitation by the gages may 

have been slightly low during the summer months 

owing to loss by evaporation from the gage; however, a 

thin film of oil was maintained in each gage to minimize 

evaporation, and this loss is not considered significant.

TABLE 6. Area assigned to each precipitation gage in reach 1 and precipitation amounts observed 

at each gage during budget period 688-708

Area (acres)

Precipitation (in.) _ _ _ _

0101

_ _ ___ . 154

_____ _ 1.50

0106

103 

1.58

0307

77 

1.60

0312

371 

1.25

Gage number 

0513 0518

55 

1.40

342 

1.00

0719

305 

1.10

0724

129 

1.10

0925

92 

1.15

0930

95 

1.15

Total area reach 1=1,723 acres

Average weighted precipitation reach 1, P=1.21 inches or 174 acre-ft.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA Lll

Thus, all bias errors involved in the measurement of 

precipitation are believed to be negligible.

The sampling error in the measurement of precipitation may be attributed to missing data and insufficient 

sampling points (gages) within the reach. Included in 

the sampling error is the human error resulting from 

misreading the gage and the instrument error resulting 

from leakage of a damaged gage or debris falling into 

the gage.

In this analysis the missing-data error was first 

evaluated. The rate of change in the missing-data error 

as the number of sample points increases was then 

evaluated to obtain an estimate of the sampling error 

for a complete set of sample points that is, with no 

missing data.

The missing-data error of the average precipitation 

per budget period decreases as the number of gages 

monitoring precipitation on the reach increases. Estimates of this error were obtained for reach 1 by 

evaluating the departure of the average precipitation 

computed using all gages in the reach Oi = 10 gages) 

from the average precipitation computed using m gages, 

where m <n. This analysis was made using a total of 114 

budget periods with average precipitation amounts of 

0.06 in. (1.5 mm) or more. The data were subdivided into 

three ranges 0.06 to 0.50 in. (1.5-12.7 mm), 0.51 to 

1.50 in. (13.0-38.1 mm), and greater than 1.50 in. (38.1 

mm) and sampling errors were evaluated for each of 

the three ranges. Of the 114 periods, 53 fell within the 

lower range defining an average precipitation of 0.31 in. 

(7.9 mm), 45 periods fell within the midrange defining 

an average precipitation of 0.85 in. (21.6 mm), and 16 

periods fell within the higher range defining an average 

precipitation of 2.06 in. (52.3 mm).

The following six steps outline the procedure used in 

evaluating the missing-data error, Sm. The procedure 

assumes that the observations of precipitation at each 

gage are independent and that Sm = 0 when m = 10 

gages. The computations described below refer to the 

precipitation data in a given range; these same 

computations were made for each range of data: 

1. Compute the average unweighted precipitation, Pt, 

for each budget period, t, from

10

 = 4r V Pi (13)

where Pjt is the accumulated precipitation at gage

j for budget period t. Table 7 gives a generalized

example of the array of Pjt data and resulting Pt

values involved in this step.

Compute the average departure of precipitation at

TABLE 7. Example of precipitation data array used to compute the 

average precipitation for each budget period (P^, the departure, of 

precipitation at a given gage from the average precipitation (Rjt), the 

average departure for all budget periods (Rj), and the standird 

deviation of the average departure (gj)

Gage

U>

1

#1,1

2 2>1

#2,1

Budget period ft)

 Rj Sj

2 - - - k

P.* P,,k

r> 2? i 1? c 

Klft K -i,k " ! 1

p p ,

#2,2 #2,/fe #2 S2

Pn,k

# #

'Pt

PI Pk

each gage from the average precipitation computed from all gages in the reach from

(14)

t=l

where

j =the average departure of precipitation at 

gage j, for k budget periods, and

(15)

The Rjt and Rj values defined by this step are 

included in table 7. The Rj values computed for 

each of the 10 gages in reach 1 are given in table 8. 

3. Compute the standard deviation of each Rj value 

from

k-l

(16)

where Sj is the standard deviation of the averrge 

departure in precipitation for gage j based on 

precipitation data from k budget periods. Table 7 

includes an example of the array of Sj values 

defined by this step, and table 8 lists the sy values 

for each of the 10 gages in reach 1. 

4. For each of m = 1, . . .,9 gages, arrange the gages into 

10 unique permutations (p = W) and for erch
L12 GILA RIVER PHREATOPHYTE PROJECT

TABLE 8. Average departure in precipitation of each gage in reach 1 (Rj) and the standard deviation of the average departure (sj) computed, 

for three precipitation ranges with each range containing k budget periods of data

[All values are in inches per 21 days]

j Gage No. ty &j

1 0101 -0.001 ±0.156 

2 0106 -.003 .180 

3 0307 -.002 .140 

4 0312 .005 .182 

5 0513 .002 .160 

6 0518 .006 .163 

7 0719 .002 .176 

8 0724 -.030 .127 

9 0925 .019 .161 

10 0930 .001 .155

permutation (u = l, . . ., 10) compute the mean 

departure in the average precipitation at m gages

for all m = l, . . ., 9 by the equation

i m 

~ \°"* ~

Rmv~ rn 2*i Rjv > (17) 

J = l

where R mv is the mean departure for the m gages 

in permutation v. A generalized example of the

array of Rmv values defined by this step is shown in 

table 9. 

5. For each of the u=l, . . ., 10 permutations, compute 

an approximation of the standard deviation of the 

mean departure of m gages for all m = l, . . ., 9 by 

the equation

Sm.«, = ^-(B l,i; +«2. V + " +Sm,v +2rs l,v s*,V

+ 2rS].,vS3,v + - ' +2rsltVsmtV (18) 

+ -+2rs(m _ 1)fl;s mtl;) % ,

where Sm>v is the standard deviation of the mean 

departure for m gages arranged in permutation v, 

sl,vl -sm,v are the standard deviations in the 

departures for 1 . . .m gages, respectively, in 

permutation v, and r is the correlation coefficient 

of precipitation measured at any two gages. 

Equation 18 assumes that the population mean 

and variance of the departures in precipitation are 

constant throughout the reach. Furthermore, the 

RJ values were computed in such a manner that

n 

2 fl/=0, (18a) 

J = l

R'j fy 3j $7

-0.007 ±0.321 0.046 ±0.511 

-.079 .261 -.109 .384 

.012 .288 .045 .439 

-.095 .252 .015 .344 

.030 .289 .033 .381 

-.018 .272 -.032 .567 

.002 .302 .001 .618 

.020 .337 .032 .636 

.049 .342 -.001 .556 

.086 .350 -.029 .438

TABLE 9. Example of array of mean departures of precipitation (R^y) 

and standard deviations of the mean departures (S nv)

Permutations (t>)

of u = l o=2 v=p 

gapes

^ m,i S>m,i "01,2 "m,a ' ' ' ^'m,f ^mj>

m = l 5^,5 &i,i RI# Si# Ki.p Sltp 

m=2 Rf,i S^,,. Rj.2 Sj,,2 «2]p Sfj)

m=n i Rn-'i,i Sn-'i,i Rn-i,a Sn -ll2 Rn -ivp Sn ii,p

and thus defining a "closed array" of departures. 

Chayes (1971, p. 40) showed that a closed array of 

values having a constant mean and variance 

defines a correlation coefficient which is negatively biased. This coefficient, r, can be approximated as a constant value from the relation

r=-r 1 > (19) n

where n is the total number of gages in tl^ reach. 

For reach 1, r -0.11 with re = 10 gages. Justification for using equation 19 to approximate the true 

correlation coefficient was investigated by 

evaluating Sm,v in equation 18 for selected 

combinations of gages using an estimate of the 

true correlation coefficient , /fyy 2 , computed from 

the relation

*u-2: (*j,«»tf)/(zX' ixo* .<2°>

*=i / *=i t=i

where pj^ is the correlation coefficient between 

gages 71 and 7 2 and Rjlt and-R,-^ are as defined by 

equation 15. A comparison of the Sm>v values 

calculated using pj^ with the SmjV values calculat-
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L13

ed using r= -0.11 showed close agreement, thus 

supporting the use of equation 19 to approximate 

the correlation coefficient.

The Rmv and Smv values for the upper precipitation range are shown in table 10 for two of the 

p=10 permutations used in this analysis. The 

gages are arranged in sequences of increasing sj 

(see table 8) for permutation v± and decreasing sj 

for permutation v%.

6. Compute the average standard deviation (£m) of the 

10 permutations of Rmv for each m using the 

relation

-[ >2 "mv

P

E (21)

where-Rmy andSmu are defined in equations 17 and 

18, respectively, subscript v is a given permutation 

of gages, and p=10 is thejtotal number of 

permutations. The departure, Rmv , is included in 

equation 21 to account for the average departure 

in the precipitation error; however, T[mv is of little 

significance as indicated in table 10 and could have 

been omitted.

The Sm values defined in equation 21 represent the 

average missing-data error associated with m gages. A 

plot of these values is shown in figure 5 for each of the 

three precipitation ranges with a solid curve drawn 

through the points to approximate the average rate of 

decrease in Sm as m increases. The broken curves 

bounding each solid curve approximate the maximum 

and minimum missing-data errors computed from the 

10 permutations.

Table 11 lists the I5m values obtained from the curves 

in figure 5 for each of the three precipitation ranges 

investigated. These 3^ values show the expected 

departure of an average precipitation value computed 

from data at m gages from the average value computed

TABLE 10. Rmv and Smv values for precipitation in the upper range P** 

.1.51 in. (38.4 mm) in reach 1 with gages arranged in order of 

increasing Sj and in order of decreasing sj

m

1 _

2 __ -

3 __ __

5 _._ _ _

6 _______

7 _ ____

8

9 -- _ .

Gages in sequence 

of increasing sj

Gage

No.

_ - 0312

 ___ 0513

__ 0106

_ __-_ 0930

__ _ 0307

 0101

_ .... 0925

_ - 0518

.. _ .. 0719

R mvi

(in.)

0.015 

.024 

-.020 

-.022 

-.009 

0 

0 

-.004 

-.004

S/nui 

(in.)

±0.344 

.166 

.187 

.130 

.114 

.149 

.124 

.078 

.071

Gages in sequence 

of decreasing s j

Gage

No.

0724 

0719 

0518 

0925 

0101 

0307 

0930 

0106 

0513

n

" mv* 

(in.)

0.032 

.016 

0 

0 

.009 

.015 

.009 

-.006 

-.002

Smu2 

(in.)

±0.636 

.312 

.290 

.224 

.114 

.087 

.080 

.041 

.038

0.3

0.2

S3 o- 1

I 

o 

? o

0.4

0.06</»<0.50 inches

-6

-4

- 2

0 j

0.3

0.2

0.1

0

0.6 

0.5

0.3

0.2

0.1

0

111111111

9 x O.SK/^1. 50 inches

Nv x"^

>s^ ^^

^^°^^o^"""-^

-~-~ ^""°^ o """""'-""-.^

~~ ~~~ "" "~°~~ ft ~^~

I I I I I I 1 I 1

- 1C

- 8

- 6

- 4

- 7

0

 s.

z

i?

7

0

H

E

0

DC

i i I \ i

.51 inches

123456789 

NUMBER OF PRECIPITATION GAGES (m)

10

FIGURE 5. Average relation between the number of precipitation 

gages and the missing-data error of average precipitation durf ng 

the budget period for three ranges of precipitation in reach 1.

from data at 10 gages. For example, referring to tal^v 

11, the departure of the average precipitation defin<>d 

with 5 gages from the average precipitation defined 

with 10 gages can be expected to have a mean standa rd 

deviation of 85 =±0.10 in. (±0.25 cm) when the average 

precipitation is in the middle range.

The development of the curves in figure 5 assuir^s 

that no error exists in the average precipitation 

computed from 10 gages. Each curve shows, however, a 

residual error (Sio) when m=10 gages. This residral 

error is attributed to two factors: (1) the use of a constant 

r to approximate the actual correlation coefficients and 

(2) differences in the standard deviations, s/, of the 

gages (table 8). These curves therefore overestimate the 

expected standard deviation of an average precipitation 

value computed from m ^W gages.

The primary objective of the analysis of precipitation 

error is not to determine the error attributed to missing 

data, as all budget periods contain a complete set of 

data, but rather to define the sampling error in 

precipitation computed from a complete set of data 

(m = 10 gages). This sampling error was estimated by 

evaluating the rate of change in Sm as m approaches 10
L14 GILA RIVER PHREATOPHYTE PROJECT

TABLE 11. Average missing-data errors ($$m)\of precipitation in each precipitation range from curves in figure 5 for m=l to 10 gages in reach 

1. Included are the unadjusted (V)lamf adjusted (Ep) sampling errors for m= 10 gages

Precipitation

range 

(in.)

0.06-0.50

0.51-1.50

>1.50

Average

precipitation 

(in.)

0.31

.85

2.06

Number of

periods 

(*)

53

45

16

_

Si

0.16

.31

.50

_

S2

0.12

.20

.31

S3

0.09

.17

.25

Average standard deviation o{R'mv 

(from curves in fi&. 5. in inches!

_

S4

0.08

.14

.21

_

S5

0.05

.10

.16

_

S6

0.05

.09

.13

_

S7

0.04

.07

.10

_

Ss

0.03

.05

.08

SB

0.02

.04

.06

_

Sio

0.02

.03

.04

Sampling error 

(in.)

Unadjusted Adjusted

V

±0.054

.092

.134

Ep

±0.056

.097

.141

(fig. 5). This sampling error, V, when expressed in terms 

of the gradient between 5m and S2m is of the general 

form

aV4 +6V2 +c=0,

where

n'/2

2

-$l . m 2m /'

_2 _

c=

_2_2 _6 _4_2

(22)

(23)

(24)

(25)

and

where n' is the largest even number s£n. The derivations of equation 22 and the coefficients a, b, and c are 

given in the section "Development of equations describing unadjusted sampling error, V."

The coefficients a, b, and c were determined for each of 

the three ranges of precipitation by substituting the 

appropriate Sm andS^m values of table 11 in equations 

23r 24, and 25. These coefficients were then used to solve 

equation 22 for V, which defines the sampling error of 

precipitation for ra = 10 gages. This value does not 

include the residual error, SIQ, (table 11) and is 

therefore referred to as the "unadjusted" sampling error 

as shown in table 11.

The total "adjusted" sampling error, Epm , in precipitation for any given number of gages includes both the 

missing-data error (Sm\) and the sampling error, V. This 

total error is computed from

(26)

where Sm is defined by equation 21 and V is defined by 

equation 22. The total adjusted sampling error for a

complete set of data (m = 10 gages) was obtained for each 

precipitation range by substituting the appropriate V 

and SIQ values of table 11 into equation 26. These Ep 

values are included in table 11 and are plotted against 

their corresponding average precipitation values in 

figure 6.

A curve closely approximating the relation between 

these sampling errors and their respective average 

precipitation values may be expressed as

(27)

where Ep is the sampling error in precipitation in 

inches for 10 gages and P is the average precipitation 

in inches.

Substituting the average precipitation for th ^ example period 688-708 of P = 1.21 in.(30.7 mm) (see table 6) 

into equation 27 gives a total adjusted sampling error of 

£p=±0.11 in. (±2.8 mm) or ±16 acre-ft (±0.020 hm3).

Independent evaluations of the precipitation errors 

for reaches 2, 2a, and 3 were not made, but the error 

relation of figure 6 is considered applicable to these 

reaches because the gage density of these reaches is 

similar to that for reach 1.

AVERAGE PRECIPITATION (PI, IN MILLIMETRES

4 6 8 10 20 40 60

-4 2

AVERAGE PRECIPITATION (PI. IN INCHES

FIGURE 6. Relation between average precipitation during the budget 

period and the total adjusted sampling error in th? average 

precipitation.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L15

A few conclusions can be drawn from this analysis 

regarding optimum gage density and variation in the 

areal distribution of precipitation on the study area. 

Referring to figure 5, the missing-data error curves 

show a relatively small rate of change in error as ra 

approaches 10 gages. This suggests that a large increase 

in gage density above 10 gages on reach 1 may not 

significantly reduce the error in computing average 

precipitation. In fact, precipitation per budget period in 

the two lower ranges can be defined from only six or 

seven gages with little loss in accuracy. Also, as noted 

previously, the average precipitation for a budget 

period is a weighted value reflecting the fraction of total 

area in the reach assigned to each gage. For purposes of 

simplicity in the error analysis, precipitation at each 

gage was given equal weight using equation 13. A 

comparison of average precipitation values weighted by 

area (equation 12) with the equally weighted values 

indicates no significant differences. Equation 27, which 

defines the error of an unweighted precipitation value, 

is therefore considered applicable to the weighted 

precipitation values in the water budget.

Finally, a correlation between precipitation on the 

left bank of the flood plain with precipitation on the 

right bank of the flood plain indicates that precipitation 

averages slightly greater (+0.02 in. or +0.5 mm per 

budget period) on the left bank. This is attributed to the 

orographic position of Mount Turnbull and the Santa 

Teresa Mountains which rise to an altitude of over 8,000 

ft (2,400 m) 8 miles (13 km) to the south (left bank) of the 

flood plain.

SOIL-MOISTURE CONTENT

The soil-moisture content was measured within two 

areas of each reach: (1) the flood plain which corresponds with the area for which ET is evaluated and (2) 

the adjacent terrace area which extends out from the 

flood plain to the contact of the saturated terrace 

alluvium with the basin fill. Measurements were made 

with a neutron probe at two- to three-week intervals 

thus defining the water-budget periods. The difference 

in the moisture content measured between the beginning and end of the period defines the change in 

moisture content for the budget period.

An access hole for measuring moisture content was 

located within about 15 ft (4.6 m) of each ground-water 

observation well in the study area. Each hole was 

classified as one of the following three types: (1) river 

hole located adjacent to the river, (2) flood-plain hole 

located between the river and the terrace, or (3) terrace 

hole located in the adjacent terrace. The river and 

flood-plain holes were used to obtain the change in 

moisture content in the unsaturated zone of the 

flood-plain alluvium and the terrace holes were used to

obtain the change in the unsaturated zone of the 

adjacent terrace alluvium. A detailed description of the 

installation of the access holes was given by MyricV in 

Culler and others (1970).

Neutron probes were used to obtain moisture-content 

readings in each hole at the Mj-ft (0.15 m) and 1-ft (C .30 

m) depths below the land surface and at 1-ft (0.30 m) 

intervals throughout the remaining depth of the hole 

which generally extended several feet below the 

ground-water level. The change in moisture content 

was determined for three zones of the profile: (1) the soil 

zone extending from the land surface to 2% ft (0.76 m) 

below land surface in the flood plain and to 5 ft (1.52 m) 

below land surface in the terrace, (2) the intermediate 

zone extending from the bottom of the soil zone to about 

3 ft (0.9 m) above the highest observed ground-water 

level, and (3) the capillary zone extending from the 

bottom of the intermediate zone to the bottom of the h ole 

in the flood plain and to about 3 ft (0.9 m) below the 

lowest observed ground-water level in the terrace. No 

intermediate zone was defined for the flood plair of 

reach 1 because of the relatively shallow ground-water 

level in the reach.

The change in moisture content in each of these three 

zones within the ET area of the flood_plain_corresponds 

to the water-budget components AMs, AM/, and AMc 

respectively in equation 1 and items 6, 7, and 8 

respectively in table 2. The change in moisture content 

in the capillary zone of the terrace corresponds to AAfyc 

in equation 1 and item 12 in table 2. Moisture changes 

in the soil and intermediate zones of the terrace (AMys 

and AM?1/ respectively) are not included in the 

water-budget equation when evaluating ET from the 

flood plain because the moisture in these two zones is 

considered to be removed solely by the overlying terrace 

vegetation which lies outside the boundaries of the ET 

area. Moisture in the capillary zone of the tenroe, 

however, is believed to be too deep (20 to 40 ft or 6 to 12 

m below land surface) to be readily extracted by the 

overlying terrace vegetation. Moisture changes in the 

terrace capillary zone are thus assumed to result from 

changes in ground-water levels in the adjacent floodplain alluvium. All significant movement of water out of 

the terrace capillary zone is assumed to be lateral and in 

the direction of the flood plain in response to an overall 

drop in ground-water levels with a general water-level 

gradient towards the Gila River. All significant 

movement of water into the terrace capillary zone is also 

assumed to be lateral but originating from the flood 

plain in response to an overall increase in ground-water 

levels with a general water-level gradient away from 

the river.

The average change in moisture content in a given 

zone of the reach for a budget period was computed from
L16 GILA RIVER PHREATOPHYTE PROJECT

where 

AM,

(28)

average weighted change in moisture 

content in zone z of the reach during 

budget period t,

Mzj (t_ l} - Mzjt , (29)

and Mzjt = measured moisture content in zone z of 

holej at the beginning (£-1) and end 

(t) of the budget period,

AJ = surface area assigned to holej, and 

n = total number of access holes in the

reach.

The surface area, AJ, assigned to each hole was 

determined using the same approximation of the 

Thiessen method that was applied in assigning areas to 

the precipitation gages. When moisture-content data 

were missing for an access hole, the change in moisture 

content for the hole was approximated using the 

average unweighted change computed from the measured access holes in the reach of the same type (river, 

flood plain, or terrace) as the unmeasured hole. Table 12 

gives the moisture-content data for the soil and 

capillary zones of each hole in the flood plain of reach 1 

measured on budget period days 688 and 708. A 

negative change in moisture content indicates an

increase of moisture in the profile (negative ET 

component) during the budget period, whereas a 

positive change indicates a loss of moisture in th^ profile 

(positive ET component). The AA?s and &M/c values 

shown in inches of moisture-content change at the 

bottom of table 12 were converted to acre-feet in table 2. 

Similar computations were made to obtain the 

AMyC = -44 acre-ft (-0.054 hm3) for the terrace 

capillary zone as shown in table 2.

The soil-moisture data in table 12 indicate that the 

amount of moisture change during a budget period is 

relatively small compared to the total moisture measured in the profile. For example, the total moisture 

content measured in the flood plain of reach 1 averages 

about 3 in. (8 cm) in the soil zone and about 28 in. (71 cm) 

in the capillary zone giving a total of 31 in. (79 cm) in the 

soil profile. The measured change in this moisture for 

budget period 688-708 includes 0.059 in. (0.150 cm) in 

the soil zone and 0.575 in. (1.460 cm) in the capillary 

zone giving a total change of 0.634 in. (1.610 cm), or only 

2 percent of the total moisture measured in the reach. 

Thus, a reliable estimate of this comparatively small 

moisture change requires that measurements of the 

total average moisture be highly accurate allowing for 

only a fraction of a percent error.

During periods of low streamflow_in the Gil a River, 

the moisture-storage components AMS , AM^, and AM^c 

are generally the most significant components of the

TABLE 12. Soil-moisture content measured in the soil and capillary zones 

offload plain access holes in reach 1 for budget period days 688 and 708

[All moisture content values in inches]

Hole 

type1

2

1

1

2

2-

2

1

1

2

2

1

1

1

2

2

1

1

2

1

1

2

Hole

No.

0102

0103

0104

0105

0106

0308

0309

0310

0311

0312

0514

0515

0516

0517

0720

0721

0722

0926

0927

0928

0930

Area 

(acres)

99.2

55.1

33.1

40.4

29.4

40.4

36.7

77.1

172.6

121.2

14.7

40.4

95.5

246.1

213.0

91.8

128.6

51.4

40.4

44.1

51.4

MS, 688

2.32

1.79

3.46

4.20

3~19

2.72

2.33

2.24

8.43

2~44

1.22

2.12

3.17

6~87

1.80

1.60

2.11

Soil zone

MS, 708

2.86

1.73

3.30

4.58

4.20

2.67

2.05

1.75

2.21

8.39

6.66

4.41

2.06

1.17

2.12

3.11

7~08

1.85

1.97

2.03

AMS

-0.54

.06

.16 -.38

2 -.05

.52

.67

.58

.03

.04 4.18

4.18

.38

.05

0

.06 4.18

-.21

-.05

-.37

.08

MC, 688

20.21

43.30

28.99

64.23

39~43

14.66

38.91

20.58

46.05

29~56

23.00

19.72

21.66

36761

22.64

34.26

26.72

Capillary zone

MC, 708

20.58

40.28

27.84

64.81

29.57

42.09

16.04

38.65

20.66

46.04

27.02

18.22

28.35

22.41

18.79

20.09

41.56

31.24

20.49

33.21

25.11

Mfc

-0.37

3.02

1.15 -.58

3 .13

-2.66

-1.38

.26 -.08

.01

5 1.13

5 1.13

1.21

.59

.93

1.57

5 1.13 -.63

2.15

1.05

1.61

Total area=1,723 acres AMo=0.059 in. AMC =0.575 in.

'Type of access hole l=river, 2=flood plain.

"Average measured moisture-content change in soil zone of type 2 holes.

3Average measured moisture-content change in capillary zone of type 2 holes.

4Average measured moisture-content change in soil zone of type 1 holes. 

5Average measured moisture-content change in capillary zone of type 1 holes.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L17

water budget. The heterogeneity of the alluvial deposits 

underlying the flood plain and terrace result in a wide 

variation in the measured moisture change between 

access holes as indicated by the wide range in the AMg 

and AMc values in table 12. A comprehensive evaluation of the measurement error of soil-moisture content 

was therefore undertaken to determine the reliability of 

these moisture-storage components.

The sampling error is the only significant error in the 

measurement of the change in moisture content. A bias 

error may exist in the measurement of the total 

moisture content of the soil profile because of calibration inaccuracies of the measuring equipment; but this 

error is one directional and nearly constant with time 

and thus essentially cancels when computing the 

change in moisture content as used in the water budget.

The sampling error in the measurement of the 

average change in moisture content in a reach may be 

attributed to the following factors:

1. Missing moisture-content data during flood periods 

when measurements could not be obtained at some 

access holes.

2. Insufficient sampling points (access holes) within the 

reach.

3. Improper placement of the neutron probe in the 

access hole or misreading the count of returning 

neutrons.

4. Variability in the count of returning neutrons.

The method for evaluating the total measurement 

error in moisture change is the same as was used in the 

precipitation analysis, that is, the missing-data error 

was first evaluated and then the rate of change in the 

missing-data error as the number of access holes 

increases was evaluated to obtain an estimate of the 

sampling error.

These errors were evaluated for each zone of the soil 

profile in reaches 1 and 2 of the flood plain and reaches 1, 

2,2a, and 3 of the terrace. These errors were then used to 

approximate the error in moisture change for reaches 2a 

and 3 of the flood plain, thus providing estimates of 

measurement error in moisture change for all of the 

areas included in the water-budget study.

The general form of the equations used in defining the

measurement errors in moisture change is identical to

the equations previously described for defining the

precipitation error. The steps used in applying these

equations are described below. All computations refer to

the moisture-content data in a given zone; the same

computations were made for each zone.

1. For each budget period containing a complete set of

moisture-content data, compute the change in

moisture content at each access hole in the reach,

bMjt, as defined by equation 29 and the average

weighted change in moisture content for the reach,

as defined by equation 28. (Note that the 

array of data illustrated in table 7 for precipitation 

also applies to jthe moisture-content data by 

replacing P and P in the table with AM and AM, 

respectively.)

2. ComputeRj for each hole in the reach using equation 

14 where Rj is now the average departure of 

moisture change in hole./, fork budget periods and

(30)

Moisture content data from a total of k =53 buc'get 

periods were used to obtain an.Rj value for each of 

the 21 access holes in the flood plain of reach 1, and 

data from k =66 budget periods were used to obtain 

anRj value for each of the nine terrace access holes 

in the reach. Table 13 lists theRj values computed 

for each hole and each zone in the flood plain and 

terrace of reach 1.

Compute Sj for each hole in the reach using equation 

16 where S is now the standard deviation of the

TABLE 13. Average departure in moisture change of each access hole 

(Ri) and the standard deviation of the average departure (sj)/br the 

soil and capillary zones of the flood plain and the soil, intermediate, 

and capillary zones of the terrace in reach 1

[All values are in inches]

A. Flood plain (area=1,723 acres) 

£=53 budget periods

Hole 

No.

Soil zone Capillary zor e

0102

0103

0104

0105

0106

0308

0309

0310

0311

0312

0514

0515

0516

0517

0720

0721

0722

0926

0927

0928

0930

-0.089 

.013

-.003 

.098 

.034

-.008

-.001

-.045 

.034 

.170 

.017 

.043

-.068

-.055

-.026 

.061 

.098 

.006 

.008 

.047 

.000

0.495

.184

.213

.369

.300

.148

.198

.272

.570

.918

.255

.491

.581

.441

.605

.353

.565

.620

.199

.301

.420

-0.125

-.001 

.052 

.095

-.071 

.018 

.034 

.004 

.051 

.044 

.021

-.038 

.064

-.190

-.070

-.038 

.143

-.029 

.003 

.009 

.012

0.711

.522

.439

1.014

.586

.471

.422

.602

1.730

.982

.432

.451

.926

2.761

1.373

.693

1.409

.550

.374

.390

.527

B. Terrace (area = l,855 acres) 

k =66 budget periods

Hole 

No.

Soil zone Intermediate zone Capillary zone

0101

0307

0312

0513

0518

0719

0723

0724

0925

0.000

.001

-.002 

.003 

.006 

.002

-.001 

.001

-.004

0.038

.060

.045

.043

.064

.044

.032

.128

.034

0.001

.002

.000

.000

.001

.002

-.001

-.003

-.001

0.029

.064

.039

.055

.046

.033

.042

.201

.030

0.005

-.003 

.007 

.000 

.012 

.005 

.005

-.022

-.005

0.112

.160

.166

.208

.247

.070

.097

.323

.110
L18 GILA RIVER PHREATOPHYTE PROJECT

average departure in moisture change in hole j 

based on k budget periods/Table 13 includes the sj 

values computed for each By in the flood plain and 

terrace of reach 1.

4. Arrange the n access holes in the reach into five 

unique permutations (p=5), jmd for each permutation (u = l, . . ., 5) compute Rmu for alljn = l, . . ., 

(n-l) holes using equation 17 where Rmu is now 

the mean departure in moisture change form holes 

in permutation v.

5. For each of the u = l, . . ., 5 permutations, compute 

Smv for all m = l, . . . , (n 1) holes using equation 

18 where Smv is now the standard deviation of Rmu 

for m holes arranged in permutation v. As in the 

precipitation analysis, the mean and variance of 

the departures in moisture content were assumed 

to be constant within the reach. Thus, the correlation coefficient, r, included in equation 18 was 

approximated by equation 19. Equation 19 gives 

r=-0.05 for the 21 flood plain access holes and 

r= 0.12 for the 9 terrace access holes in reach 1.

To test the reliability of r as an approximation of 

the actual correlation coefficient, Pj I j 2 (see equation 20), the Smv values for selected combinations 

of access holes in the flood plain of reach 1 were 

computed using r=-0.05 and compared with the 

Smv values derived using the actual correlation 

coefficients, pjj2 , computed from equation 20. The 

Smv values derived by these two methods showed 

relatively close agreement, both for the soil zone 

and the capillary zone. Particularly close agreement exists between the Smv values as m 

approaches the total number of access holes in the 

reach.

No attempt was made to evaluate pj lj 2 for each 

combination of the nine access holes in the terrace 

of reach 1; however, r=-0.12 is considered a 

reasonable approximation of the actual correlation coefficients. _

6. Compute the Sm of the p permutations of Rmv for all

m = l, . . , (n-l) access holes using equation 21

where Sm is now the average standard deviation of

the p permutations for m access holes. Figure 7

shows a plot of the .Sm values for the soil and

capillary zones of the flood plain (fig. 7^4) and for

the soil, intermediate, and capillary zones of the

terrace (fig. IB}. The curves drawn through the

data points are estimates of the average error in

the computed change in moisture content where

data (access holes) are missing.

Table 14 lists the curve values of Sm for the flood plain

and the terrace of reach 1. As with the precipitation

error curves (fig. 5), a residual error also exists in the

moisture-content error curves when m=n holes. This

residual error is attributed to differences in thes- values 

of the access holes (see table 13) and the use of a constant 

r to approximate each pjj2 . These curves are thus 

assumed to overestimate the expected standard deviation in the average moisture change.

The curves in figure 1A show that the error in the 

computed average change in moisture content for the 

flood plain of reach 1 does not decrease substantially 

beyond about 12 holes. Thus, moisture change could 

have been obtained for this area from about one-half the 

access holes actually used without a significant increase 

in error. _

The rate of change in Sm as m increases (fig. 7) 

provides an estimate of the sampling error in moisture 

change when the change is computed from a complete 

set of data (m =n holes). To estimate this_samplir <* error, 

equations 22-25 were applied using theSm values taken 

from the curves in figure 7 and listed in table 14. The

1.2

1.0

0.8

- 0.6

0.4

0.2

3.0

2.5

2.0

1.5

1.0

0.5

A. FLOOD PLAIN

h-ao

- 2.5

-- 2.0

1.5

1.0

 0.5

2 4 6 8 10 12 14 16 18 20

-7

- 6

-5

-4

-3

- 2

- 1

234567 

NUMBER OF ACCESS HOLES (m)

FIGURE 7. Average relation between number of access holes in reach 

1 and the missing-data error of moisture-content change p^r budget 

period for (A) the soil and capillary zones of the flood plain and (B) 

the soil, intermediate, and capillary zones of the terrace.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L19

TABLE 14. Missing-data errors (Sm)from curves in figure 7 for the soil 

and capillary zones of the flood plain and for the soil, intermediate, 

and capillary zones of the terrace, reach 1

[Sm computed fromp=5 permutations]

A. Flood plain Sm, in inches

Number of 

holes

j

2 _ ...

3 _ _

4 _ -. .

5 __ ...

6 - ....

7 __ ...

8 __ ___

9 _______

10 _______

11 _ _ _

Soil Capillary 

zone zone

+033

OA

97

.24

99

9ft

.17

.16

.14

iq

1 9

±1.22 

.84 

.65 

.54 

.47 

.42 

.37 

.33 

.31 

.28 

.25

Number of 

holes

12 - 

1 Q

14 

15 

17 

1 Q

19

20 

21 - -

Soil 

zone

- +0.11

.10

09

.08

.07

.07

.05

.04

.04

Capillary 

zone

±0.23 

.22 

.20 

.18 

.17 

.16 

.15 

.14 

.13 

.13

B. Terrace Sm, in inches

Number of 

holes

1 _____ .

2 ___ ____

3

5

6

7 _______

8 _ __ __

9 _______

Soil Intermediate 

zone zone

.- __-_-_ - +1.13

. __ - ___ ______ .55

- _- __-_. - .32

 __ _ .29

_ . ___ - -_ _ .23

--_. ___ .... .20

_ __ . ..._ .17

... __ __________ .14

.-. _. __________ .12

±2.39 

1.13 

.74 

.58 

.46 

.38 

.31 

.26 

.21

Capillary 

zone

±2.84 

1.73 

1.20 

.90 

.71 

.58 

.48 

.41 

.36

TOTAL SAMPLING MOISTURE-CONTENT ), ERROR INCHANGE (E hcj - 

m5 PERIOD ACRE-FEET PER BUDGET IN »aa o I

» f ocj. 

JOPSH'M ro CO-> -*» .§ £ -.S-B8 S* _o o

\ A. FLOOD PLAIN 

\xCapitlary zone ~

- - _ _ _Soil zone " ' ___________ ~

i i i l i i 1 i i i

) 2 4 6 8 10 12 14 16 18 20 

NUMBER OF ACCESS HOLES (m)

i 1 1 

B. TERRACE

N^apiM -"-y zone

\ Intermediate ""^.zone

^^oil zone

1 1 1 l

-0.35 

-0.30 

-0.25 

-0.20 

-0.15 

-0.10

- 0.15 

- 0.10 (J6

- 0.05 g

z

0 0

SAMPLING ERROR MOISTURE-CONTEN1 LIN IN CUBICHECTOMETRES PER BUDGET PEF

) 2 4 6 8 H 

UMBER OF ACCESS HOLES (m) t3. Average relation between the number of access hotes in 

1 and the total adjusted sampling error of moisture-cortent 

e per budget period for (A) the soil and capillary zones of the 

>lain and (B) the soil, intermediate, and capillary zones of the

solution of these equations give the following V errors 

for the flood plain and terrace of reach 1:

Flood plain soil zone (n=21 holes): Vg =±0.118 in. 

(±0.300 cm)

Flood plain capillary zone (n=2l holes): V^ = ±0.244 

in. (±0.620 cm)

Terrace capillary zone (n=9 holes): V^c = ±0.776 in. 

(±1.97 cm)

These V values represent the unadjusted sampling 

error in moisture change when m=n access holes.

As with precipitation, the total adjusted sampling 

error in moisture change (EZm) for any given number of 

access holes includes both Sm and V. Thus, the Sm 

values in table 14 and the corresponding V values 

determined above were substituted in equation 26 to 

define the total error curves shown in figure 8 for each 

zone of the flood plain and terrace in reach 1.

As indicated in table 12, soil moisture at 17 of the 21 

access holes in the flood plain was measured during 

budget period 688-708. Entering m = 17 in figure SA 

gives the total adjusted sampling errors for the flood 

plain of£?s 17 =±20 acre-ft (±0.025 hm3) in the soil zone 

and JEc 17 =±42 acre-ft (±0.052 hm3) in the capillary 

zone. Similarly, soil moisture at eight of the nine access 

holes in the terrace was measured during budget period 

688-708 giving, from figure SA, £TC8 =±138 acre-ft 

(±0.170 hm3) in the terrace capillary zone. Table 2 

shows each of these sampling errors under their 

respective zone.

The procedure described above for determining the

total adjusted sampling error in moisture chang? of 

each zone was also applied to the soil-moisture content 

data for the flood plain and terrace of reach 2 and the 

terraces of reaches 2a and 3. Table 15 lists these errors 

in moisture change computed from a complete se+ of 

data (m=n).

As indicated previously, moisture change in the soil 

and intermediate zones of the terrace are not included in 

the water budget (equation 1) when evaluatingET from 

the flood plain. However, the errors for these two terrace 

zones were independently evaluated and are included in 

table 15.

The estimated error in moisture change is frequently 

as large or larger than the measured change in moisture 

content for the budget period. These errors are 

relatively small, however, when compared with the 

total volume of moisture measured in the reach. For 

example, the total volume of moisture in the soil and 

capillary zones of the flood plain of reach 1 averages 

about 31 in. (79 cm) or 4,450 acre-ft (5.49 hm3) (p. 16). 

The total error in moisture change for the flood plain 

of reach 1, assuming no missing data, is (Eg2 +.Ec 2) ^ = 

(182 +392)^=±43 acre-ft (±0.053 hm3) from table 15. 

Because this error is derived from two measurements of 

moisture volume one at the beginning and one at the 

end of the budget period the total error for one 

measurement is V432/2=±30 acre-ft (±0.037 hm3) or 

only 0.7 percent of the total volume.

No independent evaluation of the sampling errors in 

moisture change (Ez) was made for the flood plain of
L20 GILA RIVER PHREATOPHYTE PROJECT

TABLE 15. Total adjusted sampling error in moisture change as defined with a complete set of moisture-content data (m=n) for etch zone

of the flood plain and terrace in reaches 1, 2, 2a, and 3

Flood Plain

Reach

1

2 _ . _____ .

2a . .......

3 _ _ .._ 

Area

(acres) 

A

... .. .. 1,723

. .. .. ___ 2,307

. _______ 1,374

1 '440

Number 

of holes 

n

21 

23 

13

17

Density 

(acres/hole) 

A/n

82 

100 

106 

85

Error 

(acre-ft/budget period) 

ES El Ec

18 

18 '24 

'20

13 Z 18 

Z 15

39 

33 Z46 

338

Area

(acres) 

A

1,855 

1,363 

992 

602

Number 

of holes 

n

9 

8 

5 

6

Terrace

 Density 

(acres/hole) 

A/n

206 

170 

198 

100

Error 

(acre-ft/budgrt period)

ETS ETl ETC

34 

42

47 

23

82 

54 

49 

23

132 

88 

97 

56

'Estimated from equation 31. 2Estimated from equation 32. 'Estimated from equation 33.

reaches 2a and 3; rather, an approximation of the errors 

for these two areas was obtained from the previously 

derived errors for the flood plain of reaches 1 and 2 and 

the terrace of reach 3. These previously derived errors 

are considered to be applicable to the flood plain of 

reaches 2a and 3 because the areas have similar 

sampling densities (table 15).

To estimate the total sampling error in moisture 

change for the flood plain of reaches 2a and 3, the 

previously derived Ezm \ values in figure 8 were first 

expressed in terms of sampling density by converting 

the ra associated with each error value to the ratio Aim. 

The relation between Ezm and Aim for each zone was 

then plotted on a semilog scale as shown in figure 9. 

Finally, a straight line approximating an average 

relation for each zone was drawn to estimate the 

average errors in the measured moisture-content 

change for the flood plain of reaches 2a and 3. Equations 

for these average relations are

E>s = -51+37 Iog10 (Aim) 

£>r=-43+30 log10 '(A/m) 

Ec=-112+781og10 (A/m),

(31)

(32)

(33)

where ES:, EI? and EC are the errors in measured 

moisture change in acre-feet for the soil, intermediate, 

and capillary zones, respectively, A is the flood-plain 

area of the reach in acres, and m is the number of access 

holes. Equations 31-33 were solved using m=n to 

obtain the total sampling errors shown in table 15 for 

the flood plain of reaches 2a and 3.

Moisture-content measurements could not always be 

obtained at every access hole in a reach during a field 

visit; however, ground-water levels were generally 

recorded at most of the wells adjacent to the access 

holes. When moisture-content data from more than half 

of the access holes in a reach were missing for the budget 

period, a more reliable estimate of the change in 

moisture content in the capillary zone could generally 

be obtained from the water-level data. The relation used 

to obtain the capillary moisture-content change from 

the average water-level change is

(34)

where AMp (or Mfyc;) is the average moisture-content

SAMPLING DENSITY fA/m ). IN SQUARE HECTOMETRES PER HOLE

20 -

120

52 100 -

200 300 400

I I I I I I I I 

Intermediate zone -""

t =-43+30 log, Q l"/m ) 

I I I I 1 I I

80 100 200 400 600 800 1000 

SAMPLING DENSITY tA/m ). IN ACRES PER HOLE

EXPLANATION

Reach 1 Flood plain

 Reach 2 Flood plain

 Reach 3 Terrace

 Average for reaches 2a 

and 3 Flood plain

FIGURE 9. Estimates of average relation between sampling density 

of access holes and total adjusted sampling error of moisturecontent change per budget period for the soil, intermediate, and 

capillary zones of reaches 1, 2, 2a, and 3.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L21

change in the capillary zone of the flood plain (or 

terrace) in acre-feet, AA is the average change in the 

ground-water levels in the flood plain (or terrace) of the 

reach, in feet (positive for a rise and negative for a drop 

in water level), S' is the apparent specific yield of the 

aquifer in the zone of water-level change (dimensionless), and A is the area of the flood plain (or terrace) in 

acres. An average value of S' was determined for both 

the flood plain and terrace areas of each reach by

relating the average water-level change of the area, M, 

to the corresponding measured average moisturecontent change in the capillary zone, ^M0(or AM^c1 ), 

using budget periods containing a complete set of wrter 

level and moisture content data. Figure 10 shows p?ots 

of this relation for the flood plain and terrace of reach 1. 

The slope of the line drawn to average the data points in 

each relation defines the S' values used in equation 34. 

The M versus AJtfcl relation for some of the areas

-300

AVERAGE WATER-LEVEL CHANGE (&h), IN MILLIMETRES 

-200 -100 0 100 200 300

HI

H

O

o 

til

DC

D

CO

O

0.2

0.1

-0.1

-0.2

0.3

0.2 -

A. FLOOD PLAIN 

Specific yield=0.20

±0.04 ft 

or ±69 acre-ft 

o

k-40 

20 

0

 20

 40

 -60

I I I

B. TERRACE 

Specif icyield=0.17

Standard error=±0.08 ft 

or ±148 acre-ft

0.6 0.8 1.0 1

AVERAGE WATER-LEVEL CHANGE (Ah). IN FEET

FIGURE 10. Apparent specific yield values derived from relation between average change in water levels and average moisture-content

change in capillary zones for (A) flood plain and (B) terrace of reach 1.
L22 GILA RIVER PHREATOPHYTE PROJECT

indicate a slight difference inS' between recharge and 

drainage of the aquifer; however, the variability of the 

data used to define the relation and the normally small 

changes in A/& did not justify defining this difference in 

S'. Table 16 lists the S' values and the number of budget 

periods, k, used to define S' for the flood plain and 

terrace areas of each reach.

The dotted lines paralleling the average line in each 

plot of figure 10 bound two-thirds of the data points, 

thus approximating the standard error of the measured 

moisture-content change in the capillary zone. These 

standard error values include, not only the sampling 

(measurement) error in moisture change for any given 

A/i, but also the actual variability in moisture change 

due to temporal variations in ET. Thus, the error 

values in the figure cannot be compared directly with 

the sampling errors, Ec and ETC, given for reach 1 in 

table 15.

An illustration of the application of water-level data 

to compute the average moisture change in the capillary 

zone of the flood plain of reach 1 for budget period 

688-708 is given in table 17. The average weighted 

change in ground-water levels for the 21 wells in the 

flood plain is AE= -0.386 ft (-11.8 cm). Applying 

equation 34 where S'=0.20 (table 16) and A = 1,723 

acres (697 hm2 ) gives AMc = 133 acre-feet (0.164 hm3). 

This value differs considerably from the A5?c =82 

acre-feet (0.101 hm3) computed from the capillary 

moisture-content data in 17 of the 21 flood plain access 

holes (see table 2). But this discrepancy can be expected 

as indicated by the large standard error of the data 

points (±69 acre-feet or ±0.085 hm3) in the A/& versus 

AMc I relation of figure 1QA.

A brief examination of the error in estimating AM^c 

from the average change in water levels was made using 

water-level data collected at the 10 terrace wells in 

reach 1. The method of analysis was identical to that 

used for evaluating the sampling errors in precipitation 

and moisture change and therefore will not be described 

in detail here. Water-level data collected during 20 

budget periods for the 1968 water year were used in this 

analysis because the data provide a wide range in. 

water-level changes. Table 18 gives the average

TABLE 16. Average apparent specific yield (S'), number of budget periods (k) used to define S', total adjusted sampling error in the measurement of average water-level change (EAh), and the total adjusted sampling error of moisture change in the capillary zone (^C^h anc* 

-) when the moisture change is derived from water-level chpnge (equation 34)

departure in water-level change (Rj) derived from 

equation 14 and the standard deviation of these 

departures (sy) derived from equation 16 for eacl of the 

10 wells. The missing-data error values, Sm, derived 

from equation 21 are plotted in figure 11 and, ar in the 

previous evaluations, indicate a residual error of Sm 

=±0.035 ft (±^07 cm) when m = 10 wells. The application of these Sm\ values in equations 22-25 give an 

unadjusted sampling error in water-level change of 

V =±0.15 ft (±4.6 cm). Combining the residual ei~or and 

the unadjusted sampling error as in equation 2~ gives 

an adjusted sampling error in A^ of .£^=±0.16 ft (±4.9 

cm) for m = 10 wells. The use of water-level data to 

estimate moisture-content change in the water budget 

was seldom required, and the error in A^ for th^ other 

flood plain and terrace areas was not evaluated. The 

error E^=±0.16 ft (±4.9 cm) was therefore applied 

throughout the study area when using AA to estimate 

AMc and AMrc . Using equation 34 to express this error 

in terms of acre-feet of moisture change for the capillary 

zone of the terrace in reach 1 gives E^=± 50 acre-feet 

(±0.062 hm3) where S' in equation 34 is 0.17 (trble 16) 

and A is 1,855 acres (751 ha). Table 16 lists these E^ 

values for each reach in the study area.

The total error of moisture_ change in the capillary 

zone when derived from the A/& vs. AM relation includes 

not only the sampling error in AA as defined above, but 

also the sampling error in AM as defined previously for 

the capillary zones of the flood plain and terrace areas of

TABLE 17. Ground-water level changes (Ah) in flood plair. wells of 

reach 1 used to compute AMc/or budget period 688-708

Hole

No.

0102 . ....

0103 __ ....

0104

0106

0309 __ ....

0310 . _

0311

0312 ........

Area 

(acres)

____ _ 99.2

. . _ 55.1

33.1

29.4

___ ___ 40.4

. - - 36.7

__ 77.1

172.6

. . 121.2

Average weighted change AA = 

AJ^,= -(-0.386)x0.20x 1,723 =

M 

(ft)

+0.02 

-.57 

-.46 

-.21 

-.06 

-.33 

-.35 

-.31 

-.18 -.06

Hole 

No.

0514 _ _ ....

0515 . . .

0516 . _ _

0517 ........

0720 .. . .

0721 ........

0722 ...

0926 ........

0927 - _ .

0928

0930 ._. __ _

-0.386 ft. 

= 133 acre-ft.

Area

(acres)

. - 14.7

_- ... 40.4

_ _ - 95.5

.. _ 246.1

213.0

 ... .. 128.6

__ . 51.4

. . _ 40.4

 - 44.1

. .. . 51.4

M

(ft)

-0.60 

-.81 

-.76 

-.55 

-.36 

-.47 

-.57 

-.18 

-.55 

-.53 

-.48

Flood Plain Terrace

S' S'

(acre-ft) (acre-ft) (acre-ft) (acre-ft)

1 -_ _

2 ___

2a._ ________

3 ___ __ . __ .

. __ _ . 54

 _ __ ____ 31

14

7

0.20

.26

.18

.20

+55

96

40

46

+67

101

64

60

68

46

31

15

0.17

.15

.17

.09

+50

33

27

9

+ 141

94

101

57
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L23

TABLE 18. Average departure in ground-water level change (Rj) of 

each terrace well in reach 1 and the standard deviation (sj) of Rj 

computed from 20 budget periods of water-level data collected during 

the 1968 water year.

Well 

No.

0101 _______ ___ __ _

0106 ______ _ ________

0307 _________________

0312 _ _______ __ ___

0513 _________________

0518 _________ _ _____

0719 __________ _ ____

0724 ____________ _ __

0925 ________ _____ _

0930 _________________

(ft) 5

____ ___ __ _ ___-0.03

_________________ .02

______ _ ___ _ __ -.02

________ __ _____ -.04

_____________ __ .02

.05

 _______._______ -.01

_________________ .02

_____________ _ _ .02

______ ___ ______ -.03

,2,

+0.33

.33

.43

.41

.18

.19

.28

.36

.34

49

each reach. This total error may be expressed as

(35)

where EM is given in table 16 and EC is the total 

adjusted sampling error in the capillary zone when 

m=n (table 15). For the terrace of reach 1 this total 

error is ETC^=(502 + 1322 )^=±141 acre-feet 

(±0.173 hm3). The EC& and ETC^ error values were 

computed from equation 35 for the capillary zone of each 

reach using the EM, values in table 16 and the EC and 

ETC values in table 15. A list of these total error values 

is included in table 16.

BASIN-FILL INFLOW

Basin-fill inflow (Gfi) to the study area is derived from 

deposits of low permeability which underlie the 

alluvium of the flood plain. This component moves 

vertically upward into the alluvium, and estimates of 

its rate of flow range from 0.07 to 1.3 ft (0.02 to 0.40 m)

0-4

LU

O 0.3

o.i

co I ^ o

h-12 £5

8"

hlO VQ.

- 8

6 -

- 4

- 2

(3 (3

2468 

NUMBER OF WELLS (m)

10 0 i5

FIGURE 11. Average relation between number of ground-water wells 

and missing-data error in moisture-content change per budget 

period for capillary zone of terrace in reach 1.

per year (Hanson, 1972, p. F27). If all basin-fill inf ow 

surfaced as evapotranspiration, these values wculd 

represent anET rate between 15 and 265 acre-ft (0.018 

and 0.326 hm3) per 21 days for reach 1. The constraints 

and limitations associated with the methods used to 

derive these Gg values make these estimates questionable.

A subsequent analysis of the moisture movement in 

the capillary zone of the deep terrace wells of reach 1 

did, however, provide a better indication of basin-fill 

inflow to the study area. In the analysis, only moisture 

data from the winter months, when ET is minimal and 

cross-valley ground-water slopes are negligible, was 

evaluated. The results of this study indicate that the 

basin-fill inflow is about 0.3 ft (0.09 m) per year per unit 

area of flood plain or GB=62 acre-ft (0.076 hm3) per 21 

days for reach 1. This value is believed to be a irore 

reasonable estimate of the true rate of basin-fill inflow.

The basin-fill inflow was assumed to remain constant 

throughout the year, unaffected by seasonal variations 

in barometric pressure, temperature, or ground-water 

levels. Because GQ is considered time invariant, its 

sampling error is zero as indicated under item 9 of 

table 2.

No evaluation of the bias in the estimate of basin-fill 

inflow was possible. Thus, it is assumed that the 

estimate is 100 percent in error as indicated by the lias 

error value ofEoB =±62 acre-ft (±0.076 hm3) under item 

9 in table 2. This bias may be significant when 

evaluating ET for a given budget period, but the bias 

cancels when computing the change in ET f~om 

before-clearing and after-clearing ET data.

DOWNVALLEY GROUND-WATER FLOW

Ground-water movement downvalley through the 

upstream and downstream ends of each reach was 

calculated from

G=iTWD, (36)

where 

G

i =

T =

W =

D =

downvalley ground-water flow through the 

alluvium in acre-feet per budget period,

average downvalley gradient of the groundwater level during the budget period 

through the upstream or downstream end of 

the reach,

transmissivity of the alluvium in acre-feet 

per day per foot,

width of saturated alluvium at the upstream 

or downstream end of the reach, in feet, and

number of days in the budget period.

The transmissivity, T, of the alluvium was assumed 

to be constant throughout the study area and was
L24 GILA RIVER PHREATOPHYTE PROJECT

determined by Hanson (1972, p. F27) to be 28,000 ft3 per 

day per foot or 0.644 acre-ft per day per foot (2,600 m3 

per day per metre). The width of saturated alluvium, W, 

was determined by measuring the distance between the 

points of contact of the alluvium with the basin fill at the 

water table on each side of the flood plain. The 

downvalley slope, i, was computed from the average 

ground-water levels for the budget period measured at 

the river wells and flood-plain wells on and adjacent to 

the cross sections at the ends of the reach. For example, 

the slope through the upstream end of reach 1 (cross 

section 1 in figure 1) was computed from the average 

water levels measured in the river wells and flood-plain 

wells at cross sections 1 and 3. Similarly, the slope 

through the downstream end of the reach (cross section 

9) was computed from the average water levels in the 

wells at adjacent cross sections 7 and 11. The 

calculations used to obtain G/ and GQ in table 2 are

G7 = 0.00158x0.644x5,800x21 = 124 acre-ft/21

days (0.153 hm3 721 days) 

GQ = 0.00148x0.644x5,600x21 = 112 acre-ft/21

days (0.138 hm3/21 days).

The sampling error associated with the G/ and GQ 

components is dependent only on the sampling error of i 

in equation 36 because i is the only factor in the 

equation which is measured for each budget period. The 

factors T and W do not have a sampling error, because 

they are considered constant with time (actually T and 

W may vary slightly with large changes in water level) 

and T is assumed to be constant throughout all reaches.

Seasonal variations in downvalley slope through 

most cross sections are generally less than 5 percent 

during periods of minimum ground-water level fluctuations. Most of this variability reflects changes in the 

ground-water level caused by precipitation, changes in

the river stage and seasonal variations in ET. As a 

result, no detailed evaluation of the sampling error in 

slope was possible. An approximation of this er^or was 

obtained, however, by examining the variabilitj^ in the 

measured downvalley slope during the winter months 

when ET is negligible and the ground-wate** level 

remains relatively stable. Figure 12 shows the average 

downvalley slopes through cross section 9 for the winter 

months (November through February) of water years 

1964,1965,1967,1969, and 1970. Variability about the 

general trends in these slopes suggests that the error in 

i for any given budget period is probably less than 

±0.005xlO~3 or ±0.3 percent of the slope. An error of 

±0.3 percent gives an average error in the downvalley 

ground-water movement of ±0.4 acre-ft (±0.0005 hm3) 

per 21 days for the ground-water inflow and outflow 

components in table 2. Because this sampling error is so 

small, it is considered zero as shown under the G/ and 

Go components in table 2.

Any bias error in the ground-water components, G/ 

and Go , is attributed solely to W and T in equation 36. 

The bias error in G/ and GQ resulting from an inaccurate 

determination of W was estimated to be ±200 ft (±60 m) 

or 4 percent of W. This estimate is probably high and is 

believed to be closer to ± 100 ft (±30 m). Considering that 

the downvalley ground-water inflow to reach 1 is 

G7 '=124 acre-ft (0.153 hm3) for budget period 688-708, 

the error in W of 4 percent gives a bias error in G/ of ±5 

acre-ft (±0.006 hm3).

The bias in Gl and Go attributed to using a constant 

average T for all reaches was estimated by assuming 

that the spatial variability in downvalley slopes, not 

explained by differences in flood-plain width, was a 

direct measure of the variability in T. The average 

downvalley slopes between the cross sections at the ends 

of each reach are plotted in figure 13 against the width

GC 

111 I- 

<

is

Z x

D -

OS

GC

e> 

>

LJJ

1.66

1.64 -

1.62 

1.60 -

1.58

FIGURE 12. Average down valley ground-water slopes through cross section 9 for winter months of the 1964,1965,1967,

1969, and 1970 water years.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L25

FLOOD-PLAIN WIDTH AT DOWNSTREAM CROSS SECTION (W), 

IN METRES

2.0

1.5

1.0

0.5

800

I

1000

I

1200 

I

1400 

I

1600 

I

1800 

_I

Standard error=±0.25x10"3\

21-23

Note: Numbers (15-17) 

are cross sections 

used to compute 

slope

2000 3000 4000 5000 6000

FLOOD-PLAIN WIDTH AT DOWNSTREAM CROSS SECTION (W). 

IN FEET

FIGURE 13. Relation between average downvalley ground-water 

slope through the upstream and downstream ends of each reach in 

the study area and width of the flood plain at the downstream cross 

section used to compute the slope.

of the downstream cross section used in the slope 

computation. The plot indicates that the spatial 

variability in slope, after the effect of differences in 

cross-section width is removed, is ±0.25xlO~3 or about 

18 percent of the mean slope of 1.42 xlO~3 . Assuming 

that this variability reflects the spatial variation in T, 

the bias error in the ground-water components attributed to using a constant T is 18 percent of the value of 

the component. For the ground-water inflow component 

G7 = 124 acre-ft (0.153 hm3 ), this bias error is 

±0.18x 124=±22 acre-ft (±0.027 hm3). The total bias in 

Gj due to the bias errors in W and T is then EG{ = 

V 52 +222=23 acre-ft (±0.028 hm3). Similar computations show that the total bias inGo is^G0 =±21 acre-ft 

(±0.026 hm3) for budget period 688-708. These bias 

errors are shown under the Gj and GQ components of 

table 2.

Because the bias error is computed as a fraction of G, 

the error will approach zero as G approaches zero. It was 

assumed in this analysis, however, that a bias error 

always exists because of the uncertainty in the 

measurement of a zero slope. Thus, a bias error of ±0.8 

acre-ft (±0.001 hm3) per day, which corresponds to the 

average measurement error in slope (fig. 13), was 

arbitrarily set as the minimum total bias error.

COMPUTATION OYET AND TOTAL ERROR INET

The total ET for a budget period is obtained by

algebraically summing the 12 components of the wrter 

budget as expressed in equation 1. For the exairrile 

budget period in table 2, this summation gives.ET=513 

acre-ft (0.633 hm3) per 21 days.

The components of the water budget in reach 1 for 

each budget period of the 1964 water year (table 1) have 

been grouped into four principal sources of water (fig. 

14) to illustrate the relative significance of each source. 

The algebraic summation of the bar graph values for 

any given budget period in the figure gives ET in acr-vft 

per 14 days. The graph of surface water sources does not 

indicate the amount of discharge in the Gila River and 

its tributaries but rather the loss (or gain) of f ow 

through the reach during the budget period. mhe 

primary components in the surface water sources are 

the Gila River inflow (Q/) and outflow (Qo). The channel 

storage (AC) and the tributary inflow (Qj) components 

are generally only a small part of the total surfecewater source (see also table 1). The graph indicates that 

surface water is the most significant source contributing toET during the winter and late summer month * of 

the 1964 water year. _ _

_The soil-moisture components (AMg, AMc, and 

AM^c) are the most significant sources of water in the 

water budget during May and June, when the contribution from surface water is minimal and ET rates are 

approaching a maximum. The precipitation (P) and 

ground-water sources (GB , G/, and Go ) are relatively 

insignificant during most of the year, with the 

ground-water components contributing a nearly constant 50 acre-ft (0.062 hm3) per 14 days to ET 

throughout the year.

This report has shown that nine of the components 

contain significant sampling errors and three of the 

components contain significant bias errors. Even 

though some of these components are interrelated, the 

measurement of each component is based on an 

independent observation. Thus, the estimate of the total 

error in.ET is treated as an expected value of the error 

variance of each term. The total sampling error in ET 

may therefore be obtained from

=E+E+E ( Q,Q0 QT

where EET is tne total sampling error in ET and the 

error terms on the right side of the equation are as 

defined previously. For budget period 688-708, this 

error is £#^=±344 acre-ft (±0.412 hm3).

The total bias error in ET may be obtained from

(38)
WATER-BUDGET COMPONENTS, IN ACRE-FEET PER 14 DAYS

888888 8 8

0

0

H

 z.

O

o

m

O

c_

 z.

Tl

m

00

TJ

7

c_

C z.

m

c_

C

r

c

Q

00

mT3

Z]

Zl

Z

Zl

1

 1

Zl

I]

1

 1

1 1

__ 1

=j

Zl

Z]

zi

p

Zl

Z]

Zl

Z]

1

Q

o

D

a

i

<D

J)

 + 

^ 

1

O

O

 

1 1 1

C

ZD

6 6

01

i I

or

 + 

6 6

N3 -»

TJ 8" 

 5

ff

Zl J

1

zz

3-

Zl

I I

3 8 8 c

-

i

C

I i

i i i

6 6 6 c

CO k> -»

.888888

 i i i i i i 

| 

1 

1

1 

Zl 

Zl 

1

1 III 

1 1 1 1 1 1 1 

3 pooopoop

WATER-BUDGET COMPONENTS, IN CUBIC HECTOMETRES PER 14 DAYS
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L27

whereE^Tb ig the expected bias error inET and the error 

terms on the right side of the equation are as defined 

previously. As in equation 37, equation 38 also assumes 

that the bias errors of each component are independent 

and unknown as to direction. For budget period 

688-708, EETb =±G3 acre-ft (±0.085 hm3).

The total measurement error inET attributed to both 

the bias and the sampling errors is

EET = (E ETs+E ET^ (39)

For budget period 688-708, this total error is EE? 

=V (334)2+(69)2 =±341 acre-ft (±0.420 hm3 ), which is 

66 percent of the computed ET.

Figure 15A shows the ET values computed for each 

budget period in reach 1 during the 1964 water year and 

the errors associated with eachET value. The brackets 

bounding these values define the total measurement 

error (E^T} in ET. Included within these brackets are 

bars indicating the error in ET attributed to the 

streamflow components Q/ and QQ and the error 

attributed to the soil-moisture change components 

AMS , AMc, and AM^c The hydrograph of the Gila River 

at cross section 9 in figure 15B shows that the 

magnitude of the streamflow errors is directly related to 

the discharge, with the largest errors occurring during 

periods of highest discharge.

The seasonal trend inET is indicated in figure 15A by 

the average potential ET curve. This curve was determined by using the average daily temperatures and 

the number of daylight hours for the study area (Blaney 

and Griddle, 1962) and by assuming that sufficient 

moisture is always available to satisfy the demand for 

vaporization; therefore, the curve approximates the 

upper limit of ET throughout the year. Actual ET may 

exceed this potential curve, however, because the curve 

is only an estimate of the potential rate and does not 

account for all the factors controlling ET. Most of the 

water-budget ET values in figure 15A which exceed this 

curve contain measurement errors that fall well below 

the curve, suggesting that the measurement errors are, 

in most instances, at least as large as the expected 

standard error in ET. Also, those ET values with the 

lowest measurement error follow, in general, the trend 

defined by the potential curve. Some ET values in the 

water budget are negative, but their measurement 

errors are generally large and extend into the positive 

ET range. In a few instances the computed measurement errors do not explain large negative ET values (as 

in January in fig. 15A) or unrealistically high ET 

values. These outliers generally occur during periods of 

high streamflow and are assumed to reflect large 

unmeasured changes in the stage-discharge relations 

which are not fully accounted for in the streamflow error

analysis. They may also reflect unknown quantities of 

surface water moving into or out of depression storage 

as described on page 9.

One of the most important points realized f~om 

figures 14 and 15 is that the total measurement errc** in 

ET is dependent on the volume of water moving through 

the reach and not the magnitude of ET. This is 

emphasized in figure 15A by the nearly constant error 

in moisti change for each ET value reflecting not the 

large varit oion in moisture change shown in figure 14 

but rather the ir l al volume of soil moisture measured in 

the reach whic i fluctuates relatively little with time.

None of the 12 water-budget components has both a 

sampling error and a bias error. This circumstance is 

unique to this study area and should not be expected to 

occur in other areas having the same type of components, particularly if the components are of different 

hydrologic significance. For example, in areas wl ^re 

ground-water movement is comparatively large, the 

sampling error may also be large and contribute 

significantly to the total measurement error in ET. A 

bias error in any one of the water-budget components 

may also become significant if the frequency of data 

collection or the sampling density do not adequately 

describe the temporal and spatial changes in the 

component.

During the 9-year study, a total of 416 ET values were 

computed from reaches 1, 2, 2a, and 3. Table 19 lists 

these ET values for each budget period and each reach 

and gives the sampling error (EET^ and total error 

(EET) for each ET value. About 60 percent of the ET 

values have a measurement error which exceeds theET 

value. However, as noted previously, the assumptions 

and criteria used in this analysis give measurement 

errors which, in most instances, would be expected to 

exceed the standard error of estimate.

COMPUTATION OF AfT AND ERROR IN

One of the principle objectives of the Gila R: ver 

Phreatophyte Project is to determine the salvage of 

water as defined by the change in evapotranspiration 

following removal of phreatophytes from the flood plain. 

The average change in evapotranspiration derived from 

ET data obtained before and after clearing for the 

June-July period is presented in this section to 

illustrate both the magnitude and the measurement 

variability of this ET change.

The average change (AET) is

-ET, (40)

where ETg and ET^ are the average evapotranspiration rates for given periods of time before and after
L28 GILA RIVER PHREATOPHYTE PROJECT

REACH 1 

1964 WATER YEAR

w 

CC 

O 

CC

CC w 

UJ >.

Q <

Z Q

< *

Z Z

O E 

 ill K- Q2000

1500 -

1000 -

500 -ir

-

i 

1 1

- T<

 + 

 +( 

^-Cc

!' i

-*-

EXPLANATION

atal + error in £7 

error in streamflow componc 

srror in soil-moisture compo 

mputed £7 for budget peric

;i :

i, i I-L^^ 1

i

;

1

i :

j

" Ijnts 

nen 

id

 -* 

i

1

Q/ a 

ts AA/

^^r

1

nd QQ 

1S .AMC .M

Potential l

f****1^ ! 

1 

1

E7_ -^-*

i

1 1

i

1 1

^

1

^-

- -500 - L

-1000

2.5

- 2.0

- 1.5

- 1.0

0.5

--0.5

-1.0

O

HI

I 

O

to

D 

O

z

w 

CC W

o >

CC < 

CC Q

Q *

OCT NOV DEC JAN FEB MAR APR MAY JUNE JULY AUG SEPT 

1963 1964

B

FIGURE 15. Graphs of (A) ET and corresponding errors in ET per budget period and (B) outflow of Gila River at cross

section 9 for the 1964 water year, reach 1.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L29

TABLE 19. Evapotranspiration (ET) and total measurement error of evapotranspiration (EgT) for each budget period during water years

1963-71 reaches 1, 2, 2a, and 3

[All values are in acre-feet per budget period]

Reach 1

1963 water year

Budget 

period

03-19-63 .

04-02-63 __

04-16-63 

04-30-63 ..

05-14-63 _

05-28-63 .

06-11-63

06-25-63 

07-09-63 -

07-23-63 -

08-06-63 --

08-20-63 

09-17 63 -

Number 

of days

.... 14

_ . .. _ 14

. . _ __ _ 14

__________ 14

 14

__ .. _ _ 14

__ __ .. _ 14

 . -- 14

 _ 14

 - 14

__-_____ 14

ET

521 

30 

129 

228 

-171 

312 

389 

220 

306 

254 

446 

851 

4122 

2556

EET

±478 

249 

248 

215 

194 

193 

178 

158 

159 

165 

598 

490 

1997 

1336

1966 water year

Budget 

period

10-11-65 __

11-01-65 _

11-22-65 ..

01-24-66 ..

02-14-66 ._

03-07-66 

03-28-66 .

04-18-66 _.

05-09-66 -

05-30-66 __

06-20-66 __

07-11 66 ._

08-01-66 __

08-22-66 ..

09-12-66 .

Number 

of days

-- _ ___ _ 21

-_ -_-_-_ 21

 21

 63

- _ ... _ 21

 ______ 21

... _ ______ 21

 _ _ 21

 21

 . 21

 __, 21

 21

 21

_ 21

. __ 21

ET

436 

302 

265

-228 

1702

999 

683 

671 

738 

405 

911 

564

EET

±206 

192 

221

1467 

1464

1044 

456 

228 

188 

428 

501 

633

1969 water year

Budget 

period

10-14-68 _

10-28-68 .

11-11-68 .

11-25-68 ..

12-09-68

12-23-68 __

01-06-69

01-20-69

02-03-69

02-17-69

03-03-69

03-17 69

03-31-69

04-14-69

04-28-69

05-12-69

05-26-69

06-09-69

06-23-69

07-07-69

07-21-69

08-04-69

08-18-69

09-01-69

09-15-69

09-29-69

Number 

of days

,_-.-____- 14

._ . 14

. _ __ ___ 14

. _ 14

._ 14

. __ _ __ 14

. __ _ 14

14

- 14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

14

ET

-42 

161 

97 

83 -31 

244 

-235 

317 

78 

199 

105 

99 

0 

56 

96 

98 

200 

138 

192 

228 

183 

89 

154 

-17 

-233 

183

EET

175 

225 

384 

426 

327 

501 

535 

663 

573 

350 

251 

225 

234 

213 

212 

191 

184 

176 

176 

261 

183 

223 

193 

702 

279

1964 water year

Budget 

Period

10-01-63 

10-15-63 

11 12-63 -

12-10-63 -

12-24-63 -

01 21 64 -

03-31 64 -

05-25-64 __

06-15-64 __

07 06-64 __

07-27-64 _

08-17-64 __

09-07-64 __

09-28-64

Number 

of days

_ _ _ 14

 - _. 14

 __._ 14

 . __ 14

- _- 14

 _ _- 14

 ._-__ 14

. _ _ _ 20

- ____ 21

 . .- 21

- _- 21

- .- 21 1

. _-_- 21 1

 21

 21 5

ET

728 

293 

582 

137 

209 

525 

494 

230 

604 

-40 

97 

9 

195 

196 

112 

269 

410 

372 

424 

056 

903 

513 

.799

EET

±727 

210 

1106 

496 

419 

364 

315 

311 

438 

445 

296 

200 

192 

197 

195 

200 

184 

172 

161 

819 

1427 

341 

1732

1967 water year

Budget 

period

10-24-66 -

11 14-66 

12-05-66

12-19-66 -

01-16-67 

02-27 67 -

04-10-67 __

05-01-67 ._

06-12-67 __

07-03-67 _

07 24-67

08-14-67

09-04-67 __

09-25-67

Number 

of days

 .__ 21

- __ 21

 . 21

- ._ 21

 .._ 28

2i

91

91

 . _ 21

 ._. 21

 21

 . . 21

 . __. 21

21

21

--__._ ___ 21

______ 21

ET

366 

109 

350 

263 

110 

44 

144 

105 

273 

199 

75 

221 

327 

200 

413

738

EET

±1096 

264 

258 

325 

215 

286 

363 

206 

206 

190 

181 

169 

162 

166 

1048

810

1970 water year

Budget 

period

10-13-69

10-27 69 -

11-10-69 __

11 24-69 __

12-08-69 -

12-22-69 -

01-05-70 

01-19-70 -

02-02-70 _

02-16-70

03-09-70 __

03-23-70 -

04-06-70 _.

04-20-70 __

05-04-70 -

05-18-70 _-

06-01-70 __

06-15-70 _

06-29-70 _.

07-13-70 

07-27-70 ._

08-10-70 __

08-24-70 __

09-07-70

09-21-70 __

Number 

of days

- --- 14

. __-.- 14

-_ _ __. 14

- ____._. 14

- ____.__ 14

- _ _ _ -_ 14

 -_ _ 21

 _ 14

. ._ 14

. __.__ 14

---- ___ __ 14

- . ____ 14

 . __ 14

_ . . 14

. _ _ __ ___ 14

. ___ _ ._ 14

.__ . 14

ET

208 

207 

-67 

112 

86 

-87 

106 

117 

98 

40 

127 

81 

135 

18 

19 

72 

23 

136 

269 

154 

209 

153 

183 

262 

193

EET

±169 

221 

221 

292 

346 

360 

227 

247 

215 

208 

435 

242 

215 

207 

204 

186 

177 

170 

166 

164 

176 

282 

235 

168 

196

1965 water year

Budget 

period

10-19-64 -

11 09-64

12-21-64 __

02-01 65 __

02-22-65 __

04-05-65

05-17-65 _.

06-28-65 

07 19-65

09-20-65 ...

Number 

of days

. _ - 21

 ..__.. 21

 ... 21

.. __ ...... 21

-_.- -_ 21

. ..___.- 21

21

............ 21

..._ _ ..... 21

..... .... 21

-__ . _ 21

91

21

21

21

91

21

ET

229 

132 

132 

113 

42 

-193 

465 

247 

33 

378 

82 

407 

448 

469 

1979 

265 

1224

EET

±409 

177 

262 

275 

408 

776 

757 

655 

545 

470 

335 

187 

159 

164 

1294 

600 

1247

1968 water year

Budget 

period

10-16-67 __

12-18-67 __

04-01 68

04-22-68 -

05-13-68 __

07-08-68 ...

07 22-68 ..

08-05-68 ...

08-19-68 ..

09-02-68 -

Number 

of days

 .. 21

.- ..___ 21

21

. _ .. .... 21

21

21

91

 _____ 21

- ..-___ 21

21

_-. - 21

_ . __.-. 21

 - 21

... _ _____ 14

 __ .... 14

_ _ _____ 14

-. . _ _ 14

._ __ __ 14

______ 14

ET

446 

159 

212 

397

1767

1228

-999

752 

403 

351 

25 

254 

-55 

55 

348 

90

EET

±805 

306 

302 

464

2181

3589

1649

882 

314 

211 

173 

426 

998 

632 

317 

178

1971 water year

Budget 

period

11 02-70

11-16-70

11 30-70

12-28-70 ...

01 11-71 ...

01 25-71 ...

02-08-71

02-22-71 ...

03-08-71 ...

03-22-71 ...

04-19-71 ...

05-17-71 ...

05-31-71 ...

06-14-71 ...

06-28-71

07 12-71 ...

07 26-71 -

08-23-71 ..

09-20-71 ..

Number 

of days

-.-.._.- 14

14

14

14

__ 14

 - 14

. 14

 - 14

_ 14

_ __ 14

.. .--. 14

___ ___ _ 14

__ __ 14

_ . ..... 14

__ ___ _ 14

___ . ... 14

____ .._ 14

- _-.. 14

__ ___ 14

... _ _ 14

. ... ___ 14

.- _ 14

. ___ ___ 14

._____ 14

... ___ 14

ET

568 

113 

97 

41 

14 

55 

5 

126 

-40 

91 

356

_ g

-88 

27 

74 

-20 

41 

82 

93 

106 

203 

192 

-728

518 

-183

EET

 ±750 

306 

177 

187 

179 

184 

184 

328 

376 

351 

326 

274 

212 

186 

184 

180 

173 

169 

165 

163 

165 

177 

462

626 

463
L30 GILA RIVER PHREATOPHYTE PROJECT

TABLE 19. Evapotranspiration (ET) and total measurement error of evapotranspiration (EET) for each budget period during water years

1963-71 reaches 1, 2, 2a, and 3 -Continued

Reach 2

Budget 

period

07-09-63 _

08-06-63 -

08-20-63 -

09-17-63

Budget 

period

10-04-65 .

10-25-65 -

11-15-65 __

12-06-65 _

02-07-66 .

02-28-66

03-21-66 -_

04-11 66 .

05-02-66

05-23-66

06-13-66 ..

07-04-66 _

07-25-66

08-15-66

09-05-66

09-26-66 .

Budget 

period

11 04-68

11 18-68 -

12-16-68 -

12-30-68

01 27 69

02-24-69 _

03-10-69 .

03-24-69 .

04-07 69 ._

05-05-69 __

06-02-69 -

06-30-69 _

07-14-69 __

07 28-69 __

08-11-69 -_

08-25-69 ..

09-22-69 --

1963 water year

Number 

of days ET

-- - . 14 427

 _ ... 14 573

 .._ 14 -195

- _ _ . 14 -455

 - - ._ 14 -755

1966 water year

Number 

of days ET

----- . 21 763

- __ 21 562

__ _ 21 250

 ___ 21 253

_ ___ . 60

__ _. - 21 -838

 ___ 21

 21

. ___ . 21

__ _ .... .. 21

_ . . 21

 _- ___ 21

_ ..... 21

21 604

. . 21 460

« . 21 -98

1969 water year

Number 

of days ET

14

__ 14

. . 14

. - - .. 14

. _ .. _. 14

. . . - - 14

.. - . 14

. . ._ 14

-_ ___. 14

_ .. .. 14

 . _ - __ 14

. .. __ . 14 289

 ... 14 527

__ . . _ 14 360

_ _. . .. 14 510

__ . .. 14 1026

.. - 14 721

._ . .. ... 14 576

 _ _ _ 14

EET

+ 154

566

468

1275

EET

+346

290

221

738

1117

EET

+ 139

119

125

183

125

155

Budget 

period

10-22-63 -

11 19-63 -

12-31 63 -

01 28-64 -

02-25-64 .

03-10-64

03-24-64

04-07 64

04-27-64

05-18-64 .

06-08-64

06-29-64 .

07-20-64 .

08-10-64 ..

08-31-64 ..

09-21-64 .

Budget 

period

03-27 67 ..

04-17-67 .

05-08-67

05-29-67 ..

06-19-67 .

07-10-67

07-31-67 ..

08-21-67 ..

09-11-67 .

Budget 

period

1 A_ori CQ

11 17 69

1 O fk 1 CQ

19 1 PL flQ

03-16-70

03-30-70 ._

07 06-70 -

08 03 70

08-17 70

09 14 70

09-28-70

1964 water year

Number 

of days

- _ .__ _- 14

_- _. 14

-_-._ - 14

14

14

20

-,____ _ 21

 --- . 21

-_._ - 21

__ . 21

 21

21

..... _ .... 21

1967 water year

Number 

of days

91

91

 28

 . 14

21

- . ..... - 21

21

21

21

. ... 21

___ ....... 21

21

1970 water year

Number 

of days

_ _ .. __ - 14

..._ __ .... 14

 - 21

14

14

. ___ .. __ 14

__ _ ....... 14

.. .... 14

ET

-293

111

535

437

_cq

-246

Q7

27

167

92

104

302

103

246

143

387

728

789

1058

1068

537

439

ET

795

Q.9Q

64

132

48

54

225

17

285

271

720

739

974

531

464

ET

194

419

194

344

474

220

1

28

-62

179

306

165

270

257

272

314

520

583

587

479

558

505

275

EET

+749

967

457

OCA

219

453

222

191

185

159

168

153

13&

141

647

1257

763

791

EET

314

210

355

214

184

163

146

135

131

170

991

664

EET

122

183

352

243

228

198

179

174

436

202

179

179

162

145

133

124

122

125

136

280

154

156

Budget 

period

10-12-64 ..

11-23-64

12-14-64

01-04-65 .

01-25-65 ..

02-15-65

03-08-65 ..

03-29-65

04-19-65 .

05-10-65

05-31-65 ._

06-21-65

07-12-65 _

08-02-65 .

09-13-65

Budget 

period

10-02-67 ..

10-23-67 ..

11-13-67

12-04-67 __

12-25-67

01-15-68 .

02-05-68 _

02-26-68

03-18-68 .

04-08-68

04-29-68

05-20-68

06-10-68

07-01-68

07-15-68

07-29-68

08-12-68 .

08-26-68 ..

09-09-68

09-23-68

Budget 

period

10-12-70

10-26-70 -

11-09-70

12-21 70 -

01-04-71 _.

01-18-71 ..

02-01-71

02-15-71 .

03-01-71

03-15-71 .

03-29-71 -

04-12-71

05-10-71 ..

06-07 71 -

07-19-71 ..

08-02-71 .

08-16-71 ..

08-30-71

09-13-71 ..

09-27-71

1965 water year

Number 

of days ET

- . _ . 21 -409

... _ ... 21 363

--_ - 21 8,5

21 145

21 197

21 283

21 293

21 32"'

 . 21 -391

21 -65

21 793

 __. -- 21 61<5

 21 563

___ . 21 837

 - - 21 1195

_ ._ _ . 21 84',

. _ 21 1W

1968 water year

Number 

of days ET

. ... .. 21 -212

21 219

. __ .. .. 21 40^

__ ..... _ 21 318

... __ 21

. ... ___ .. 21

21

. . .. _ 21

. _ ....___ 21

21

..._ 21

21

21

_ .. 21

14

14

. _ 14

14

__ . 14

14

1971 water year

Number 

of days ET

_ __ 14

. _ . 14 187

__ . 14 T

. . _ . - 14 177

... . ___ 14 122

. .. __ 14 110

._ - _- 14 177

. . ... 14 -91

. - 14 31

. . 14 -134

_ . - - 14 260

_____ ___ - 14 53

._ . _ 14 8

..._.. .. 14 170

_ . _ _ 14 268

----- 14 125

__ _ __ 14 127

_ ... __ - 14 205

14 188

 14 245

-_ _. 14 340

14

-_ __ 14

_..__ _- 14

14

EET

+ 1495

177

202

269

227

712

728

680

564

491

368

171

128

129

946

821

1184

EET

+ 1070

421

291

323

EET

+ 143

147

140

146

148

199

353

351

318

270

226

161

146

143

143

135

125

122

122

154

429
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L31

TABLE 19. Evapotranspiration (ET) and total measurement error of evapotranspiration (Egr) for each budget period during water years

1963-71 reaches 1, 2, 2a, and 3 Continued

Reach 2a

1966 water year

Budget 

period

10-04-65 -

10-25-65 -___.

02-28-66 .....

03-21-66 _ .

04-11-66 _ __

05-02-66 .....

05-23-66 _____

06-13-66 ....

07 04-66 .....

07 25-66 .___.

08-15-66 __ .

09-05-66 __ .

Number 

of days

_ ___- 21

.. ___ .. 21

_ . . 21

_.__ _ .. 21

_ - 63

... _ . . 21

_ ... _ 21

_ ... - 21

21

21

___ _ _ 21

21

21

21

___....- 21

21

ET

656 

663 

203

EET

±178 

154 

223

1969 water year

Budget 

period

10-07-68 ._

10-21-68 __

11-04-68 _ .

11-18-68 .....

12-02-68 ....

12-16-68

12-30-68 _____

01-13-69 _ .

01-27-69 . .

02-10-69 _ ..

02-24-69

04-07-69 .....

05-19-69 . .

06-16-69 .....

A/3 Qf\ £Q

f\fj OQ_CQ

AQ_Afi_.«Q

f\Q 99 AQ

Number 

of days

.___- 14

14

-__.-- 14

14

. .._ 14

14

..... 14

14

14

14

_. _ _ 14

14

__. .. _ 14

14

_ .. 14

-_.- _ 14

14

 _ 14

ET

379 

232 

27 

106 

230 -31 

42 

-325 

-111 

-706 

47 

55 

186 

102 

141 

223 

45 

68 

218 

146 

187 

302 

301 

350

EET

±139 

144 

170 

255 

416 

357 

418 

487 

567 

683 

452 

251 

207 

193 

193 

169 

169 

142 

130 

128 

126 

165 

132 

174

1967 water year

Budget 

period

11 07 66 

19 19 fifi

01 30-67 _

02-20-67 _ _

03-13-67 _-.

04-17-67 ___

05-08-67 _-.

05-29-67 ...

06-19-67 ...

07-10-67 .

09-11-67 ___

Number 

of days

.. .______ 21

.. ______ 21

9ft

_____ _ ... 21

._ ___ .... 21

. _ _______ 21

._ ___ 21

..________. 21

 __ . 21

_ __ _ 21

 __ ... _ 21

._ _ ___ 21

ET

-621 

228 

180 

198 

92 

-125 

-78 

102 

28 

195 

86 

388 

343 

323

EET

±349 

205 

312 

209 

255 

361 

219 

212 

177 

167 

149 

137 

139 

183

1970 water year

Budget 

period

10-06-69 .. _

10-20-69 __..

11-03-69

11-17-69 ..

12-01-69 __..

12-15-69

12-29-69 ..

01-12-70

01-26-70

02-09-70

02-23-70

03-16-70

03-30-70

04-13-70

04-27-70

05-11-70

05-25-70 ....

06-08-70 _

W nfi. 7n

08-03-70

08-17-70 ,

AQ_Qi 7n

AO_9Q 7A

Number 

of days

.... 14

. . 14

14

14

14

14

14

14

14

14

21

14

14

14

14

14

14

14

ET

34 

249 

45 

269 

114 

262 

35 -9 

-36 

131 

132 

327 

125 

202 

160 

161 

66 

111 

153 

229 

180 

151 

169 

232 

262 

40

EET

±153 

128 

198 

186 

321 

356 

248 

228 

199 

178 

171 

425 

198 

177 

174 

162 

145 

134 

125 

122 

120 

135 

290 

149 

130 

159

1968 water year

Budget 

period

11-13-67 -

12-25-67 -

01-15-68 .

02-05-68 ..

02-26-68 ..

03-18-68 ..

04-08-68 __

04-29-68 ..

05-20-68

06-10-68 ..

07-01-68

07-15-68

07-29-68 ..

08-12-68

08-26-68

09-09-68 ._

09-23-68 __

Number 

of days

- ____ ... 21

. _____ .. 21

. ____ .... 21

... ... ... ... 21

.. _ .. _ .. 21

. ___ ..... 21

___.____-- 21

.. 21

 . . _ 21

. ___ .. __ 21

... ... 21

___ .._..._ 21

. 21

_ _ ........ 21

-.-.-._. 14

__ __ 14

..... _ _ _ 14

14

.___.. 14

__ ---. . 14

ET

314

188

151

328

EET

±290 

325

177 

173

1971 water year

Budget 

period

10-12-70 -

10-26-70 -

11-09-70

11-23-70 ...

12-07-70 

12-21-70 __.

01 04-71 

01 18-71 - -

02-01-71 

02-15-71 ---

03-01-71 . .

03-15-71

03-29-71 

04-12-71 ..

04-26-71 ...

05-10-71 . .

05-24-71 

06-07-71 

06-21-71 ...

07-05-71 ...

07-19-71

08-02-71

08-16-71 ...

08-30-71 . .

09-13-71 __

09-27-71 -

Number 

of days

_____ 14

. . 14

... _ . . 14

.. _ 14

______ _ .. 14

14

... . ... 14

-.-.- 14

. . _ . . 14

--.--. 14

_ _ __ 14

. _ . ... 14

_ ___. __ 14

. . . _ . 14

. _ 14

... . 14

_-._-- . 14

. _____ 14

_ . ___ .. 14

. . . 14

. . ... 14

_____-_ . 14

.__ .. . 14

. . 14

_ __ _ .. 14

_ _ _ . 14

ET

55 

73 

32 

138 

-11 

216 

237 

123 

10 

-7 

95 

121 

77 

112 

92 

109 

77 

155

219

EET

±141 

145 

147 

199 

351 

344 

321 

272 

228 

159 

145 

143 

137 

128 

125 

121 

120 

131

365

Reach 3

Budget 

period

10-22-63 -

12-17-63 -

12-31-63 __

01 28-64 __

02-11-64 __

03-10-64 __

03-24-64 ._

04-07-64 __

05-11-64 .

06-01-64 .

06-22-64 ..

07-13-64 __.

08-03-64 __

08-24-64 ..

09-14-64 __.

1964 water year

Number 

of days

__ __. .. . 21

. ...____ __ 14

-.--_.___ 14

. .. __ ____ 14

..-..... 14

- . ---.__ 14

_ .-..__ 14

-__. .. . 14

.__ .. .. 14

_ _ ..... 14

... . . 14

__ ___ . . 14

...--.___ 14

_ _ _ 34

- ___-.. 21

.. . 21

__ ___ .. 21

21

21

21

ET

25

36 -27

47

-34

70

246

433

541

580

EET

+446

361

209

168

170

165

190

95

94

90

Budget 

period

10-26-64 __

12-07 64 -

01 18-65 ._

02-08-65 __

03-01-65 __

04-12-65 __

05-03-65 _.

05-24-65 ..

06-14-65

07-05-65

07-26-65 ._

08-16-65 __

09-06-65 __

09-27-65 __

1965 water year

Number 

of days

._-_..._____ 21

... . ___ 21

--- 21

..- ... 21

... . . 21

_ .... __ __ 21

___ .. 21

___ . _ 21

21

21

21

____..._____ 21

21

_ . ._ _ 21

__________ 21

ET

46

1 Aft

400

374

661

702

EET

246

390

191

100

83
L32 GILA RIVER PHREATOPHYTE PROJECT

clearing, respectively.

Table 20 gives ET values determined from reach 1 

during the season of high potential ET in June and July 

for the before- and after-clearing period of the study. 

Included in the table are the sampling error (EETS) and 

total measurement error (EET) for each ET value. The 

total bias error is nearly constant at £#^=±46 acre-ft 

(±0.057 hm3) per 14 days and has been omitted from the 

table. The ET values for some budget periods are 

obviously unreasonable and generally correspond with 

a large measurement error. Thus, those values with a 

total measurement error of Eg7i >±550 acre-ft (±0.678 

hm3) are not included in the subsequent computations. 

The criteria used in selecting this error limitation was 

arbitrarily established such that no data were accepted 

in which the error exceeded the average June and July 

potential ET of approximately 550 acre-ft (0.678 hm3) 

per 14 days. A plot of the ET values in table 20 with 

measurement errors less than ±550 acre-ft (±0.678 

hm3) is shown in figure 16.

_These ET data define average 14-day rates of 

ETs=320 acre-ft (0.395 hm3) before clearing and 

ETA = 18l acre-ft (0.223 hm3) after clearing. The 

average change (reduction) mET as a result of clearing 

is AET=320-181 = 139 acre-ft (0.171 hm3) per 14 days 

for the June-July period in reach 1.

The standard deviation of the 12 before-clearing ET 

values in table 20 is SETB =±^ acre-ft (±0.097 hm3) 

or ± 25 percent of ETfi. The standard deviation of the 19 

after-clearing ET values is S£;7iA =±77 acre-ft 

(±0.095 hm3) or ±43 percent of ETA . The near-equal

TABLE 20. ET values obtained before and after clearing and their 

corresponding sampling, and total measurement errors fo" selected 

14-day budget periods during June and July, reach 1

[All values in acre-it per 14 days]

Before clearing

Day1

6-11-63.

6-25-63_

7 9-63_

7-2S-63.

8-06-633 

6-15-642 

7- 6-642 

7-27-64 2 

6- 7-65z 

6-2S-652 

7-19-652 

6-20-662 

7-11-662 

8- 1-66Z

ETB

______ 389

_______ 270

____- 306

_______ 254

 ___ 446

_ _ . 248

___ _ 283

?_____ 704

______ 271 

 ___ 299

___-._ 313

______ 447

_ ___ 492

______ 270

EETS

±172 

151 

153 

159 

596 

153 

141 

671 

161 

138 

142 

192 

160 

352

EETB

±178 

158 

159 

165 

598 

163 

152 

673 

171 

149 

153 

200 

170 

357

After clearing

Day1

6-12-67* .

7- S-672

6-24-6S2 

7 8-68. _.

7-22-68...

8- 5-68__.

6- 9-69__

6-2J-69 _

7- 7-69 .

7-2 1-69 __

8- 4-69__

6-15-70. _

6-29-70 .

7 13-70__

7-27-70. _

6-28-71__.

7 26-71___

ETA

-___ 218

__ 133

..___ 275

_ _ 269

___.. 351

_ __ 25

_..._ 254

___.. 138

_____ 192

_____ 228

_ . _ 183

___.. 89

_____ 136

_____ 269

__.._ 154

__ _ 209

_.___ 93

..___ 106

__.._ 203

___.. 192

EETS

±140 

143 

857 

261 

205 

167 

424

170 

170 

25S

163 

160

15? 

170 

159 

157

I?'1

EETA

±151 

154 

859 

267 

211 

173 

426 

184 

176 

176 

261 

183 

170 

166 

164 

176 

165 

163 

165 

177

fe = 12 periods 

ET.. =320

D

B

£==; =±182

fe = 19 periods

ET . =181 A

=±205

 Day refers to last day of 14-day budget period.

2ET and error values originally computed for 21-day budget period but adjusted to 14-day 

budget period. 

3ET and error values excluded from computations because £gy>±550 acre-ft.

values ofsETB and SETA indicate that the variaHlity in 

the computed values of ET is a function of the total 

volume of water passing through the reach which does

BEFORE CLEARING AFTER CLEARING

OVJU 

CO

> 400-

Q

£ 300-

Q.

1-

UJ

UJ 

U.

UJ 200 -

CC

U

z

K 100-

Uj

0 -

*

 .

'

9

*~1*

9

*

_

 

ETR =320 acre-feet

1

_ \ J

 A£7=1 39 acre-feet

1 H

- _ r

ETA =181 acre-feet

^~

9 ^

_ <

9

* ^

*

»

*

«

____ Q g

CO

Q

-0.5 ^

CC

UJ 

Q.

 0.4 co

UJ

CCIUJ

 03 ^

0 

U

UJ

-0.2 x 

y

CD

D

-0.1 ^

iu

 n

JUNE JULY JUNE JULY

FIGURE 16. Water-budget ET values in reach 1 for the before-and after-clearing periods in June and July and the average change in 

ET as a result of phreatophyte clearing. Values are plotted on the day corresponding to the middle of the budget period.
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L33

not change appreciably with time and not the magnitude of ET.

_ The average measurement errors associated with 

and ETA may be computed from

k

f=l

where EET is the average total measurement error of 

ET for k budget periods and EETt is the total measurement error inET for budget period t. Applying equation 

41 using the total measurement errors EETB and EETA 

in table 20 gives EETB =±1&9 acre-ft (±0.233 hm3) and

A =±205 acre-ft (±0.253 hm3). _ 

Assuming independence between ETB and ETA > the

standard deviation of AET is S&ET= Vs2£TB +S^ETA =~ 

V792+772=±11Q acre-ft (±0.136 hm3) or ±79 percent of 

AET. Much of this variability is real, reflecting year to 

year variations in the potential ET rate and in the 

moisture available for evaporation and transpiration. 

The remaining variation results from measurement 

errors in the ET components. __

The average measurement error for AET includes 

only the sampling errors (EETS ) computed from equation 37; the bias errors (EETb ) are omitted because they 

are one directional and essentially constant for all 

estimates of ET and thus cancel in the error computation. The average sampling errors (EETS ) of ±182 acre-ft 

(±0.224 hm3) before clearing and ±199 acre-ft (±0.245h 

m3) after clearing were obtained by substituting the 

corresponding EETS error values of table 20 in equation 

41. The average measurement error in AET is then

= V 1822 + 1992 =±270acre-ft (±0.333 hm3), which 

is nearly ± 200 percent of AET and exceeds the standard 

deviation of AET by 2V6 times.

The fact that these measurement errors in ET and 

AET are significantly greater than their standard 

deviations indicates that the assumptions and criteria 

used to obtain the measurement errors produce an 

overestimate of the true measurement variability in 

ET. These total measurement errors must therefore be 

considered only an indicator of the relative significance 

of each ET value.

No evaluation was made of the winter ET rates in this 

report; however, Hanson, Kipple, and Culler (1972, fig. 

4) showed that the winter rates average substantially 

lower than the summer rates before clearing and that no 

significant change in the winter rates can be detected 

after clearing. The measurement errors inET are also 

generally higher during the winter than during the 

spring and early summer months as indicated in figure 

15A. Estimates ofET for the winter months of typically 

low rates are therefore less reliable than the summer 

estimates.

DISCUSSION OF RESULTS

Of the 12 components of the water budget, the G: la 

River inflow (Q/) and outflow (Qo) are generally the 

most significant, reaching maximum rates during the 

winter and spring snowmelt period and during the late 

summer thunderstorm period. Tributary inflow (Py) 

occured only 4 percent of the time during the nine-year 

study period, and even though some of these events did 

produce large volumes of inflow, this component is 

considered to be one of the least significant during the 

study period. Of the more important components in the 

water budget, moisture-content changes AMg, AM/, 

AMc, and AM^c are the most difficult to measure. 

Moisture change, particularly in the capillary zone, 

generally becomes significant during periods of low 

streamflow. Except for basin-fill inflow, which was 

assumed constant, the ground-water inflow (G/) rnd 

outflow (Go) are the least variable components durng 

the year, fluctuating only in response to seasonal 

changes in the down valley ground-water slope. The 

only component of any consequence in the water budget 

that was not measured in this study is depression 

storage that water which fills side channels ?nd 

depressions in the flood plain during overbank flooding. 

The relatively infrequent occurrence of depression 

storage and the difficulties in measuring this component did not justify including it in the water-budget 

analysis.

The total measurement error for most of the 

water-budget components consists primarily of a ss mpling error which is dependent on the number of 

observation points used to evaluate the component. The 

sampling error is time variant reflecting both the 

variability in repetitive measurements and the error 

due to missing data. Included in the total measurement 

error is a bias error which reflects a consistent 

overestimate or underestimate of the water-budget 

component. Only the basin-fill inflow and the groundwater inflow and outflow components introduce a 

measurable bias in the computation of ET. Because of 

the uncertainty in the estimate of the basin-fill inflow, 

the bias of this component is assumed to equal the tctal 

basin-fill inflow. The bias in the ground-water inflow 

and outflow components reflect possible errors in the 

determination of the average transmissivity for the 

study area and inaccurate measurements of the width of 

saturated alluvium at the inflow and outflow cross 

sections of each reach.

The magnitude of the measurement error of ET is 

directly related to the total volume of water moving 

through the reach and not the magnitude ofET. Thus, 

ET computed from a budget period of high streamflow 

has a correspondingly large measurement error. Fortunately, high streamflow is generally limited to the
L34 GILA RIVER PHREATOPHYTE PROJECT

winter months, whenl?T is minimal and a few weeks in 

late summer when runoff from thunderstorms occurs. 

During the midsummer months of maximum ET, the 

measurement errors become minimal because of low 

streamflow and negligible tributary inflow and precipitation.

The measurement errors of ET for the summer 

periods investigated in this report (table 20) are ±59 

percent of the computed average before-clearing value 

of ET and ±113 percent of the computed average 

after-clearing ET rate. The measurement error of the 

average change in ET as a result of clearing is nearly 

±200 percent of the computed change for these summer 

periods. The measurement errors of ET and change in 

ET for the winter periods are generally even greater 

than for the summer periods.

The large measurement errors computed in this study 

would make it appear that theET rates derived from the 

water budget do not provide reliable estimates of the 

true ET rates. Most of these computed errors can be 

assumed, however, to exceed the actual measurement 

errors of ET and AET because the criteria used to 

estimate the error of each component give values that 

would be expected to exceed their standard error. This is 

substantiated by a comparison of the EET values with 

the significantly lower sgf values in table 20. Because 

SB? includes both the true measurement errors in the 

data and real variations reflecting year to year 

differences in moisture available for evapotranspiration, it is apparent that the computed measurement 

errors are too high. These data show, in fact, that 

reliable estimates of ET for the summer periods can be 

obtained and that a significant difference inl?T could be 

detected as a result of clearing the phreatophytes from 

the flood plain.

Even though most of the computed measurement 

errors for ET probably exceed the actual measurement 

errors, they do provide a good indication of the relative 

significance of each ET value. These measurement 

errors were used as the basis for selecting the most 

reliable ET estimate in evaluating the average before 

clearing and after clearing ET rates from all reaches in 

the study. A discussion of the application of these 

measurement errors to the evaluation of the average ET 

rates will be included in a subsequent paper in this 

series.

Studies have been carried out to evaluate the 

variability in the ET data due to differences in moisture 

available for vaporization and differences in the 

potential to remove the available water. In addition, the 

differences inl?T between reaches due to differences in 

vegetative cover has been evaluated. The results of 

these studies will also be included in a subsequent 

report.

DEVELOPMENT OF EQUATIONS DESCRIBING 

UNADJUSTED SAMPLING ERROR, V

An estimate of the average value of a hydrologic 

variable for a given area such_as precipitation approaches the population mean (P) as the number of 

sample points used for the estimation in creases. 

Measures of most hydrologic variables are sample 

realizations of the time series in which they occur and 

therefore are frequently autocorrelated. Thus, the rate 

at which any estimate of a given variable approaches its 

mean value is unknown. _

Ifn sample points are used to estimate the meanPn , 

the standard deviation of the departure between Pn and 

Pm (where m<n) decreases as m approacher n in a 

manner indicated by curve Sm of figure 17. In this report 

Sm is referred to as the average missing-data error. A 

sampling error, V, exists at m=n, and a relation 

between sample size and total error which includes both 

Sm and V can be described by curveEpm in figure 17. An 

approximation to the shape of curve Epm can be found by 

estimating V such that

(26)

where E2pm is the variance of the departure between Pm 

and an estimate of the population mean, and is 

inversely proportional to the sample size m.

The procedure for determining l§m for all values of m 

has been described previously in this report. The 

purpose of this section is to describe the development of 

the equations used in estimating the sampling error, V.

An approximation of V may be obtained from an 

evaluation of the rate of change in Sm as m approaches 

n. If the variance S2m is defined from m sample-* and the

NUMBER OF SAMPLES (m) »-

FIGURE 17. General relation for error in departures of average of m 

samples from average of n samples (curve Sm), and error in 

departure of average of m samples from estimate of population 

mean (curve Ep ).
ACCURACY OF EVAPOTRANSPIRATION RATES DETERMINED BY WATER-BUDGET METHOD, ARIZONA L35

variance S22m is defined from 2m samples, then the ratio

S2m +V2

 ^ g ig e(lual to 2 with some residual error, gm 2m . It

IJ 0 ~\~v 2m 

can be assumed that the average rate of change in the

square of this residual error will approach zero as m 

increases or expressed in equation form,

d V M 

where

i

a v

sf+v 2

s+v

+v

(42)

(43)

(44)

(45)

(46)

and

7?' is the largest even value less than or equal to n. 

Solving the partial differential equation

S2 +V2

-fLf2 _ <L/o ___ t 

rtVM,2-flVr 2

(47)

By replacing each -^r £ 2 term in equation 42 with a 

general expression for equation 47 gives the previously 

shown quadratic equation

where

n 72 =S(

777=1 

77 72

=^(

777=1

(22)

(23)

(24)

7? 72

and

_ t

(25)

777=1

The positive root of V in equation 22 may then be 

obtained from

2a

(48)

Solving for V in equation 48 using the coefficients a, 6, 

and c obtained from equations 23-25 gives the le^st 

squares best-fit curve for V.

REFERENCES CITED

Blaney, H. F., and Criddle, W. D., 1962, Determining consumptive use

and irrigation water requirements: U.S. Dept. Agriculture Tech.

Bull. 1275, 59 p. 

Bowie, J. E., and Kam, William, 1968, Use of water by riparian

vegetation, Cottonwood Wash, Arizona: U.S. Geol. Survey

Water-Supply Paper 1858, 62 p. 

Burkham, D. E., 1976, Flow from small watersheds adjacent to tHe

study reach of the Gila River Phreatophyte Project, Arizona: U.S.

Geol. Survey Prof. Paper 655-1, 19 p. 

Burkham, D. E., and Dawdy, D. R., 1970, Error analysis of streamflow

data for an alluvial stream: U.S. Geol. Survey Prof. Paper 655-C,

13 p. 

Carter, R. W., and Anderson, I. E., 1963, Accuracy of current meter

measurements: Am. Soc. Civil Engineers Proc., Hydraulics D: v.

Jour., v. 89. no. HY4, p. 105-115. 

Chayes, Felix, 1971, Ratio correlation A manual for students of

petrology and geochemistry: Chicago, Univ. Chicago Press, 99 p. 

Culler, R. C., and others, 1970, Objectives, methods, and

environment Gila River Phreatophyte Project, Graham County, Arizona: U.S. Geol. Survey Prof. Paper 655-A, 25 p. 

Gatewood, J. S., Robinson, T. W., Colby, B. R., Hem, J. D., and

Halpenny, L. C., 1950, Use of bottom-land vegetation in lower

Safford Valley, Arizona: U.S. Geol. Survey Water-Supply Paper

1103, 210 p. 

Hanson, R. L., 1972, Subsurface hydraulics in the area of the Gila

River Phreatophyte Project, Graham County, Arizona: U.S. Geol.

Survey Prof. Paper 655-F, 27 p. 

Hanson, R. L., Kipple, F. P., and Culler, R. C., 1972, Changing the

consumptive use on the Gila River flood plain, southeaster

Arizona, in Age of changing priorities for land and water: Am.

Soc. Civil Engineers, Irrigation and Drainage Div. Special'y

Conf, Proc., p. 309-330. 

Thiessen, A. H., 1911, Precipitation for large areas: Monthly Weath°r

Rev., v. 39, p. 1082-1084. 

Turner, S. F., and Skibitzke, H. E., 1952, Use of water by

phreatophytes in 2000 foot channel between Granite Reef and

Gillespie Dams, Maricopa Co., Arizona: Am. Geophys. Union

Trans., v. 33, no. 1, p. 66-72. 

Weist, W. G., Jr., 1971, Geology and ground-water system in the Gila

River Phreatophyte Project area, Graham County, Arizona: U.S.

Geol. Survey Prof. Paper 655-D, 22 p.

«GPO 690-978-1976