OREGON STATE UNIVERSITY
?I
!
APPLICATION OF GEOSTATISTICS TO
REGIONAL EVAPOTRANSPIRATION
AMEGEE Y. Kodjo
toRVAlL.IS 1985

---

APPLICATION OF GEOSTA.TISTICS TO
REG ION'AL E'lAPO'TRANSP I RAT I ON
by
Kodjo Y. Amegee
A THESIS
Submitted to
Ore<J5~JLState university
CON5EH. ArR!CAIN H MALGAChL'
: POUR i..'Ei'JSEiGNEMENT S\\j~):;;IT;'r.:~w
: C. A. M. E. S. -
OUAGADOUGOU
I' Arrivee " 22 NOV...1995_..... 1I
\\i _~nr_eg_istl~~ s~~~ n,# Q.:2 1) ~~··5~~l
in partial fulfillment of
the requirements of the
degree of
DOCTOR OF PHILOSOPHY
Completed April 19. 1985
Commencement June 1985
r1

---

AN ABSTRACT OF THE THESIS OF
Kodjo Y. Amegee for the degree of Doctor of Philosopp~ in
Civil Engineering
presented on
April 19, 1985.
Title:
Application of Geostatistics to Regional
•
Evapotranspiration
Abstract approved:
Richard H.
Cuenca
Regionalized variable analysis has been used to study
a number of meteorological and hydrological variables
including precipitation (Delhomme and Delfinier.
1973).
streamflow (Villeneuve et al .•
1979) and hydraulics of
groundwater aquifers (Gambolati and Volpi.
1979).
In the
research work reported in this thesis.
an attempt was
made to characterize the spatial variability of
evapotranspiration over the state of Oregon using methods
~f ~eostatistics.
):~~einiv~riograms wasreguired todescrii>e.·spatial variation
. ,"'-,. ' .
". ~
: ::.I!
..
.
,.;,
",.'
·'hr~'--~'ariable asafunctionof<dist~rice~ 'The
,"\\".',"
·-0~emiv~riogrammodels were derived from~eference
. < :
.'evapotranspiration rates computed from data at 175
weather stations during May through September of 1979.
The resulting semivariograms for all months but
September were anisotropic and indicated that the rate of

---

change in reference evapotranspiration was higher in the
east-west direction than in north-south direction.
The
semivariograms were fitted by spherical models and
applied to perform interpolation of evapotranspiration
using the geostatistical technique of kriging.
Kriging
I.
estimates of evapotranspiration were made at approximately
1,600 locations where no weather data existed, cor res-
ponding to corners of a 12.8 km by 12.8 km square grid
system laid over the state of Oregon.
Using computer
plotting routines, the estimates were transformed into
contour maps of evapotranspiration and of contour maps of
kriging variance.
The contour curves of reference
evapotranspiration agreed with the general distribution
of climatological and topographical features over the
state of Oregon.
A self-validation test performed on the
semivariograms revealed the existence of a certain bias
which could be removed by deriving individual
:~
semivariograms' for climatic subregions within the state.
The results of this research indicate that the
spatial variability of reference evapotranspiration over
large geographical areas can be described by semivario-
•
gram models and this information can be used to predict
evapo~,ansPiration rates by me~~s of kriging.
Such a
~..
~
proce~~Le could be effectively;applied to increase the
efficiency of water resources utilization in the arid
regions.

---

APPROVED:
•
Associate Professor of Civil Engineering in charge of
major
ring
Dean of
~c.l~L
Grad~choolCl
Date thesis is presented:
~A~p~r~i~l~=1~9~,~1~9~8~5
__
Typed by Valerie Hender for:
~K~o~d~.j~o~Y~.~A~m~e~g~e~e~ __

---

ACKNOWLEDGEMENTS
I would like to express my gratitude to the African-
American Institute.
the Oregon State Experiment Station •.
and people who have contributed to the completion of this
.: -: :
thesis.
Thanks to Dr. Marvin Durham~ foreign students'
,
,
advisor. wh~ made it possible for mywif~ and I· to attend'
.:'
Oregon Stats university.
My deepest gratitude to Dr. Richard H. Cuenca. for
his patient guidance and consistent encouragement during
the five years I spent under him.
He was available
always.
not
just as an advisor.
but" also as a friend.
I
know this friendship will continue.
Thanks to all the
other members of my Graduate Committee. Dr. P. Klingeman.
Dr. R.
Petersen. Dr. E. Schmisseur. and Dr.
P. Lombard.
My deep appreciation to Valerie Hender who spent many
weeken~s typing this thesis. and to Joanne Wenstrom for
her kindness and consistent support.
Thanks to my
friends and office mates.
the graduate students.
for
what we shared together in the "hut".
friendship and
solidarity.
•
I would like to thank Papa and Dada.
my parents.
for
the faith and hope they gave to me.
They always made me
feel
that my success was an answer from God to their
prayers.
My final gratitude to my wife.
Romance.
for her
patience when I was "burning the midnight oil"
in the
· 0
computer room and for her generous understanding.

---

I
I '
TABl.E OF CONTENTS
1.
INTRODUCTION -----------------------------------
1
1.1
Statement of the Problem-------~-------­
1
' . ' .
..
4
~. .
1.2
Alternative Methods o~Approach~-----~­
7 ....
1.3
Thesis Objectives anCi Scope -:---------~-
2.1
Advances ih Reference.
Evapotranspiration ---~-------:--~--~~-~- -10
2.1.1
Historical Development of Local
Evapotranspiration Estimates -----------
11
2.1. 2
Practical Methods of computing
Local Evapotranspiration ---------------
19
2.1. 3
Estimation of Regional
.
Evapotranspiration ---------------------
23
2.2
Theoretical Fundamentals of
Geostatistics --------------------------
35
2.2.1
The semivariogram -----------------7----
35
2.2.2
Stationarity Assumptions ---------------
39
2.2.3
Properties of the Semivariogram --------
44
2.2.4
Simple Kriging -------------------------
50
2.2.5
Universal Kriging ----------------------
61
3.
PROCEDURE AND DATA SELECTION -------------------
65
';iJ
3.1
Selection of Evapotranspiration
Data and Methods -----------------------
65
3.1.1
Available Weather Data -----------------
65
3.1.2
Choice of an Evapotranspiration
Methods --------------------------------
66'
3.1.3
Estimation of Missing Weather Data -----
69
3.2
Organization of Geostatistical
Computations ---------------------------
73
3,.2.1
Procedure for Computing ET
Semivariograms -------------------------
74
3.2.2
.•
Fitting a Sperical Model of
Semivariogram --------------------------
79

---

Page
3.2.3
Kriging for Mapping ET Contour
. Curves ---------------------------------
80
3.3
Modeling semivariograms with
,
83
Insufficient Data -----------~--~--~--~-
I' .'
I
3.4
Method for Testing Goodness of,
Est i ma t ion .- - - ~ - - - - -- - - _-:. - - _:.-:-~' '":" -: -":"",..;~ --:..
87
,
,
,
'
RESULTS', AND DISCUSSION ~_-....:
,,...'----,:;.-~.,;.-:....;:..:--:..---...:,i
89
"
.
!'.'.>
.
';
' . -
-'. '
4.'1"
'Loca 1 Estima tes, of' Reference>','
' , " .
, , Evapotranspi ra tion --.:.:.':..-"7-~.,:;.----~7"~;'":"-::...-
89
......
4.2
Evaluation of the Stationarity'"
Assumptions
91
-------~---------~----~::...----
4.3
Kriging Estimates of
Evapotranspiration ---------------------
101
4.4
Estimates of Kriging Variances ---------
102
~,
4.5
Self Validation of the
Semivariogram Model --------------------
103
4.6
SUbregional Model of semivariogram -----
105
4.7
Comparison of semivariogram Models
with Least Squares Models --------------
108
..-
4.8
Comparison of Simple Kriging with
Universal Kriging ----------------------
110
4.9
Contour Maps of Reference
Evapotranspiration ---------------------
113
4.10
Contour Maps of Kriging Variances ------
121
5.
CONCLUSIONS AND RECOMMENDATIONS
129
5.1
Conclusions ----------------------------
129
5.2
Recommendations for Future Research ----
130
6.
BIBLIOGRAPHY -----------------------------------
133.

---

APPENDIX A -------------------------------------
139
Program MAIN for Estimating the
Local Reference Evapotranspiration
Rates at Weather Stations --------------
140
Program VARIO for· Computing and
Plotting the Semfvario~rams ------------
144
Program KRIGX for
Interpolating
by Kriging' the Evapotranspiration.··
Rates Bet~een Weather Stations--~------ 149
APPENI:>IX B
154
Monthly Reference Evapotrans-
piration Rates for June 1979 at
Weather Stations -----------------------
155
Kriging Estimates of
Evapotranspiration Rates at
Unoccupied Grid Corners for
June 1979 ------------------------------
159
I~
Estimates of Kriging Variances
at Unoccupied Grid Corners for
June 1979 ------------------------------
162
Results of Self-validation Test
of State-wide Semivariogram Models
for June 1979 --------------------------
165
I.
APPENDIX C
169
Results of the "best" Sub-
regional Semivariogram Search
for June 1979 --------------------------
17~

---

LIST OF FIGURES
1
Reconstruction of homogeneous
evaporating surfaces by moving from
small to larger scales. -------------------
25
2
Regional (S) and local (s) scales of
evapotranspirating surfaces. --------------
27
3
Juxtaposition at regional scale of
homogeneous surfaces each developing
an internal- boundary layer. ---------------
29
4
Homogenization of latent heat flux at
regional scales 2 and 3 above elemen-
tary surfaces in 1, using measurements
made at the evaporating surface (soil),
surface boundary layer (SBL), and
planetary boundary layer (PBL). -----------
29
5
Ideal semivariogram curve. ----------------
45
6
Subdivision of the state of Oregon
into climatic subregions and wind
zones of influence. -----------------------
72
7
Grouping data into distance and angle
classes. ----------------------------------
75
8
Superimposition of a (12.8 x 12.8 km)
grid system over the state of Oregon.
82
9
Monthly variations of daily evapotrans-
pi ration throughout 1979 growing
season. -----------------------------------
90
10
Drifts in reference evapotranspiration
for June 1979. ----------------------------
93
•
11
Semivariograms of reference evapotrans-
piration for May 1979. --------------------
94
12
semivariograms of reference evapotrans-
piration for June 1979. -------------------
95
13
Semivariograms of reference evapotrans-
piration for July 1979. -------------------
96
14
Semivariograms of reference evapotrans-
piration fo~ August 1979. -----------------
97

---

15
Semivarioqram of reference evapotrans-
priation for September 1979. --------------
98
16
Contour map of reference evapotrans-
piration (mm/day) for May 1979. -------~-~-
115
17
Contour map of reference evapotrana-
pi ration (mm/day) for June 1979 .. ----~~.:..---.
116
18
Contour map of. reference evapot.rans-
'~"'.' '"
'~.,
:. '.
.:;.: ",-
piration (mm/day)- for July 1979:.,---'::'--:::~:':;:"'117·
"c'
.-.
" ~ ~. ; .,:
~
19
Contour map of reference evapotrans~·
"~;:"~">-,'
'.~
pi ration (mm/day) for August 1979~ ~ __ .:.S:,~'-~'
. :
118'
~..
. .."
,
,
. .
~ ", '.
.':-. . .
20
Contour map of reference evapotrans-
piration (mm/day) for September 1979.
119
I ::: : .
I"
I
21
Contour map of kriqinq variance
«mm/day)2) for May 1979. -----------------
122
".
22
Contour map of kriqinq variance
«mm/day)2) for June 1979. ----------------
123
23
Contour map of kriqinq variance
,.,.
{(mm/day)2) for July 1979. ----------------
124
24
Contour map of kriqinq variance
({mm/day)2) for Auqust 1979. --------------
125
25
Contour map of kriqinq variance
({mm/day)2) for September 1979.
126
•

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'i
LIST OF TABLES
Table
1
Secondary weather data st~tions. ----------
67
2
FAO modified Blaney-Criddle monthly
reference evapotranspiration (mm/day)'
:'-;.
for. the· climatic subreqions in 1979. ------.
. 92
3
Characteristics of. the s t a t e - w i d e : ·
100· ....
semivarioqrams.
-----~---------------------
"
4
Resuli~ of state-wid~ semivarioqram
:'validation tests.
----~---------~-~----~~~- 104
5
Characteristics of the reqional semi-
varioqram derived by jacknifinq. ----------
106
6
Linear reqression models. -----------------
109
-.
7
Self-validation for universal kriqinq
models for the state of Oreqon in 1979.
111
o

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Application of Geostatistics to
Regional Evapotranspiration
1.
INTRODUCTION
1.1
Statement of the Problem
Since the 1950's,
agricultural scientists and
engineers have been increasingly concerned with the
, -
quantification of crop water requirements,
especially in
arid and semi-arid irrigated areas of the world.
Crop
water requirements are defined by Doorenbos and Pruitt
(1977)
as
"the depth of water needed to meet the water
loss
through evapotranspiration
(ET
) of disease-free
crop
crops,
growing in large fields under non-restricting soil
conditions
including soil water and fertility and
achieving full
production potential under the given
growing environment".
The determination of crop water
requirements
is essential when an irrigation system must
be designed to supply water
for growing crops.
Crop water
requirements may be measured directly by
Lysimeter or estimated from a
suitable evapotranspiration
formula using meteorological data.
Evapotranspiration is
termed
reference evapotranspiration when it is estimated
for a
reference crop under a set of standard conditions.
Doorenbos and Pruitt
(1977) defined reference
evapotranspiration (ET)
as
"the rate of
r
evapotranspiration from an extensive surface of 8 to 15

---

2
centimeters tall green grass cover of uniform height,
actively growing, completely shading the ground, and not
short of water".
Earlier, Jensen et al.
(1971) proposed
that reference evapotranspiration be defined as "the
upper limit or maximum evapotranspiration that occurs
under given climatic conditions with a field having a
well-watered agricultural crop with an aerodynamically
rough surface, such !a~' a lfa lfa wi th 30 to 45 cm of top
growth".
Reference evapotranspiration may be considered
as a meteorological or climatic variable which varies in
response to atmospheric conditions.
When the
meteorological parameters used in an ET
estimation are
r
representative of a single location,
the estimated value
is termed the local estimate of reference
evapotranspiration.
Seguin (1977) used the words local
and regional evapotranspiration to apply for about 100
square kilometers and 10,000 square kilometers,
•
"~!
•
respectively.
These numbers are simple averages WhiCh
can be modified according to the evaporating surface
roughness and heterogeneity.
When irrigation systems are designed to supply water
required for growing crops in a given region,
the ET r
used is one locally estimated from a meteorological
station considered as representative of the project
site.
The fields to be irrigated may be hundreds or
thousands times larger than the supposedly representative

---

3
weather station site.
ET
estimates from weather station sites are
r
therefore extrapolated to larger fields.
This same
extrapolation is implicitly assumed when a water
resources balance has to be made over a region perhaps
hundreds of square kilometers in area and possibly
several kilometers away from a weather station from which
data are taken for the evapotranspiration estimate.
Such
an approximation incorporates errors which may cause
crops to be overirrigated or underirrigated and water
storage reservoirs to be overdesigned or underdesigned.
Worldwide problems of energy and water scarcity
and/or relatively high cost justify an effort to improve
:
the accuracy of regional ET
estimates and to quantify
r
the error associated with the application of a local
ET
estimate to a region a certain distance away from
r
the meteorological station.
A model of ET
spatial
r
variability will take into con~ideration the change in
ET
with respect to geographic coordinates and the
r
distance between project site and weather stations where
data are collected.
Such a model will provide estimates
based on the variation of ET
as a function of distance
r
between measurement points.
As a consequence.
the model
will provide a more accurate estimate of ET
at
r
locations where no climatic data are available compared
to a method which does not take into consideration the
=

---

4
t.
structure of ET
variability.
An accurate model of
r
ET
spatial variability could improve the efficiency of
r
irrigation scheduling by providing a means of determining
more precisely when to irrigate and how much water to
apply at locations where no weather data exist.
The
relationship determined between error and station density
could be used by engineers to determine required
hydrometric station density as a function of desired
accuracy.
MacMahon and Cronin (1980) provided curves
relating standard errors of hydrologic estimates to
gaging station density and studied the marginal cost and
benefit associated with increasing network density.
The
work proposed in this thesis would aid in such an
analysis.
1.2
Alternative Methods of Approach
This thesis is not the first attempt to model
:-~.
regional evapotranspiration.Recent progress in remote
sensing techniques has encouraged some scientists to
derive evapotranspiration formulas using the crop canopy
temperature.
The soil-plant canopy temperature sensed by
satellite or by other remote sensing instruments was used
in combination with ground based parameters measured from
traditional weather stations.
Although the resulting
formulas offered promising results,
their validity has
not been adequately verified with respect to soil

---

5
moisture and cloudy sky conditions to allow operational
estimates of regional ET
(Schmugge. 1978: Bernard et
r
al .• 1981).
The most direct procedure (Baier. 1979)
consists of applying local models to small homogeneous
units and to weight resulting values according to the
relative surface area in order to compute an overall
regional evapotranspiration.
To obtain reliable regional
ET estimates in such methods.
it is necessary to equip a
large number of data collection sites.
Recently. a theory termed the Theory of Regionalized
Variables was developed by Matheron (1965) in France.
This theory was partially based on a practical
statistical method designed by Krige (1960) to estimate
metal grade in gold mines.
This theory. which will be
extensively explained in the next chapter. was first used
by geologists and mining engineers to estimate metal
grade in ore deposits.
According to Journel and
Hujbregts (19781. a regionalized variable can be
characterized by the correlation between neighboring
measurements.
As was stated by Henly (1981),
the Theory of
Reqionalized Variables. or Geostatistics. can be applied
to a large variety of fields including geophysics.
meteorology. ecology, geography and oceanography.
This
method has also been applied to ground water level and
:
soil characteristics mapping. and even to investiqate air

---

6
pollution.
In all these applications.
the method uses
the correlation between neighboring measurements to
construct a model which characterizes the structure of
spatial variation of the parameter under study.
This
theory has been applied to estimate variables at
locations where few or no measurements were available.
The method has also provided a tool to quantify the error
of estimation.
It has aided in detecting to what degree
a variable is heterogeneous or anisotropic.
A
regionalized variable is isotropic if it exhibits the
same behavior in every direction.
The variable will be
anisotropic if it changes at a diff~rent rate from one
direction to another.
In water resources planning. it is beneficial to be
able to describe reference evapotranspiration
quantitatively over a large geographic area.
In
irrigation scheduling.
it is advantageous to be able to
predict crop water requirements at locations where no
weather stations are available and to quantify the error
associated with the predicted values.
In the design of
hydrometeorological networKs.
it is useful to be able to
relate the station density to errors associated with
estimations based on those densities.
During the preliminary studies of regional water
resources development.
it would be convenient and time-
saving to be able to rapidly access a map of

---

7
evapotranspiration contour curves on a weekly. monthly or
annual basis.
Such maps can be made relatively rapidly
if a model of spatial interpolation exists which can be
translated into computer language and routinely used to
perform interpolation based on the structure of
evapotranspiration spatial variability.
Although some of
the benefits mentioned above can be reached using
classical statistics methods based on random sampling.
Geostatisti~s offers the advantage for dealing with data
not necessarily randomly sampled.
A classical regression
between locally estimated periodic ET
and the
r
geographic coordinates of sampling locations can be used.
but this method is not necessarily accurate because the
least squares regression method does not exactly
reproduce the measured values previously included for the
model derivation (Neter and Wasserman. 1974).
On the
contrary.
the geostatistical estimation method termed
kriging. which will be explained in the next chapter.
is
a least squares method Which is also an exact estimation
method.
This property was described by Journel and
Hujbregts (1978) who noted that the kriged surface passed
through the experimental data points.
1.3
Thesis ObJectives and Scope
This thesis is an attempt to study how well the
Theory of Regionalized Variables can be applied to

---

8
evapotranspiration.
There are five objectives which
are:
1.
To characterize the spatial variability of
regional evapotranspiration.
This can be
done by relating the change in
evapotranspiration to geographic variables
using a geostatistical feature termed the
semivariogram.
This term is defined in the
next chapter.
2.
To apply regionalized variable analysis to
evapotranspiration at sampling locations and
compare the estimated variables to the
original data.
This operation constitutes a
test for the validity of the spatial
variability model obtained in the first
objective.
3.
To estimate reference evapotranspiration
where no weather data are available using
the model tested in the second objective.
4.
To use regionalized variable analysis to
quantify the error associated with an
estimation of evapotranspiration using such
a model.
S.
To repreBent.
by contour curves.
the spatial
variation of reference evapotranspiration
and the error of estimation over a region.

---

9
The scope of the objectives is limited to the State
of Oregon. where local reference evapotranspiration rates
will be estimated at 175 locations for 1979 during the
months of May through September.
For each month. ET r
data will be used with the geographic coordinates of
their locations to compute parameters which characterize
the spatial variability in terms of geostatistics.
Once
the spatial variability model is tested for its agreement
with the measured data.
the analysis will be carried on
through the fifth objective.
The final result will be
computer plots of reference evapotranspiration and maps
of the error of ET
estimation.
r
;

---

10
2.
LITERATURE REVIEW
This chapter focuses on the changing concepts of
evaporation through human history and recent advances in
regional evapotranspiration.
Later.
the theory of
regionalized variables will be reviewed along with the
specific methodology applied in this study.
2.1
Advances in Reference Evapotranspiration
Recent studies on reference evapotranspiration rates.
measured by highly accurate lysimeters and compared to
estimates based on formulas.
revealed that 95 percent of
variability in the rate of
reference evapotranspiration
can be correctly accounted for
by certain estimating
. formulas.
Doorenbos and Pruitt
(1977)
compared lysimeter-
measured evapotranspiration for grass at Davis.
California.
to evapotranspiration predicted from four
different climatic data-based estimating methods.
In
each case.
a very high correlation coefficient was found
(r ~ 0.97)
between measured and estimated
evapotranspiration.
This result.
among others.
indicates
how accurately evapotranspiration can be currently
estimated.
Such success has not been reached without
many philosophical debates.
laborious experiments. ·and an
accumulation of scientific knowledge since antiquity.

---

11
2.1.1
Historical Development of Local Evapotranspiration
Estimates
Brutsaert (1982) wrote an overview of the progress of
the evapotranspiration concept through history.
A great
deal of the material presented in this section was based
on his book, Evaporation into the Atmosphere.
Scientists
showed limited interest in evapotranspiration formulation
before Penman (1948)
pUblished the first scientifically
sound equation to estimate the evaporation rate of water
vapor from a free water surface into the atmosphere.
Penman relied on Dalton's
(1802)
formulation of water
vapor flow as the product of water vapor deficit and a
conductivity coefficient which was a function of wind
velocity.
Dalton's efforts were the first step in the
quantitative theory of evaporation.
Before Dalton
(1802),
most of the scientific and philosophic works on
the subject were intended to clarify Goncepts on the
causes and effects of the evaporation process.
Among the peoples of antiquity,
the Greeks
contributed the most to explain the relationship between
the sun,
the clouds and the rain.
The views of
Anaximander of Miletos
(565 B.C.) on the phenomenon of
evaporation were summarized by Aetius in Diels
(1934):
"Wind is a stream of air,
of the finest
in it and of
the moistest,
which are moved or dissolved by the sun."
..
According to Aetius in Diels
(1934),
Xenophanes of
Colophon (570-460 B.C.) said that:

---

12
11
• •
what happens in the sky is caused by the heat
of the sun;
for.
when the moisture is drawn up out of the
sea.
the sweat part. which is distinguished by its fine
texture.
forms a cloud.
and drips out as rain . . . 11
Later. Aristotle (384-332 B.C.) developed Herakleito's
concept of dual exhalation:
11 • • •
the sun not only draws up the moisture on the
earth's surface.
but also heats and so dries the earth
itself; and this must produce exhalations which are of.
the two kinds we have described,
namely vaporous and
smoky.
The exhalation containing the greater amount of
moisture is,
...•
the origin of rain:
the dry exhalation
is the origin and natural substance of winds . . . 11
Aristotle agreed that the moist eXhalation required solar
radiation.
but contrary to Anaximander.
he denied any
connection between evapotranspiration and the wind except
that both are separate eXhalations and were caused by the
sun.
Aristotle's opposition to the fact that wind may
also be a cause of evaporation constituted a setback that
influenced the understanding of the causes and effects of
evaporation through the era of the Roman Empire,
the
Middle Ages.
and up to the 17th century.
Descartes
(1637)
broke partially away from Aristotle's views in his
book The Meteors. where he said:
"
. although the winds are caused nearly only by
the vapors.
they are not composed only of vapors.
11
Later.
Perrault (1674).
based on experimental results he
obtained from the first evaporation experiments on
record,
broke completely away from the concept that wind
is not a cause of evaporation.
He wrote:
.-
IIAlthough Aristotle and all the other philosophers
give QClly one cause for the evaporation of water,
namely

---

13
heat,
I would be able to find two more,
one the cold,
its
contrary,
and the other the movement of the particles of
air."
Further experiments conducted by Halley (1691)
showed
that evaporation is only caused by heat and wind.
Le Roy (1751)
introduced the concept of
"degree of
saturation" of air in order to characterize the moisture
content of the air.
A major contribution to
thermodynamics was made by Dalton (1801,
1802) who
clearly expressed the laws of partial pressures known as
Dalton's Law;
that is the partial pressure of a gas in a
mixture of gases is independent of the presence of the
others.
From this point,
a
large horizon was opened to
the quantitative theories of evaporation .
•
Dalton's
(1802)
contribution to the quantification of
evaporation was expressed in the ~ollowing terms:
"The quantity of any liquid evaporated in the open
air
is directly as
the force of stream from such liquid
as
its temperature.
"
"Evaporation is greater'i·where there is a stream of
air
than where the air
is stagnant."
These concepts were formulated as:
E = fD(U)(e~ - ea)
(2.1)
where:
E = the evaporation rate
fD(U)
= a wind function
ij
= mean wind speed
Cs = saturation vapor pressure at the evaporating
surface
,r

---

14
'.
ea : vapor pressure in the air.
Due to the fact that the Dalton equation included
fD(O).
a wind function.
it was called an aerodynamic
equation.
Earlier in the eighteenth century.
it was well
known that wherever evaporation took place. it was
accompanied by a cooling process.
The discovery of the
latent heat of vaporization by Black. around 1760. made
it clear that evaporation required heat.
The discovery
by Pouillet
(1838) of the solar constant by means of a
pyrheliometer. an instrument used to measure direct solar
radiation. helped Daubree (1847) establish a numerical
relationship between the amount of water evaporated from
the earth's surface and the amount of solar energy
received at the outer boundary of the atmosphere.
Later.
Homen (1897) conducted experiments to give a detailed
analysis of the energy budget at the earth's surface.
Homen's works were partially based on the understanding
of radiation from the discovery by Stefan (1879) and
Boltzmann (1884) concerning radiant flux:
F
=·aT4
(2.2)
where:
F
radiant flux density
(W • m- 2 )
a = S tef an--·Bo 1tzmann cons tan t
T = absolu~e temperature. OK (degree Kelvin)
:
Later. Bowen (1926) and others contributed the energy
..,

---

15
budget approach to the estimation of
local evaporation.
Rn = E + H + G
(2.3)
where:
Rn = specific flux of net incoming radiation
E = the rate of evaporation
H = specific flux of sensible heat into the
atmosphere
G = specific flux of heat conducted into the
earth.
It can be noticed that Eq.2.1 and Eq.2.3 are two
different approaches to the evaluation of the evaporation
rate
(E).
However.
each of them contains a
term
difficult to either measure or estimate.
They are e s
•
in Eq.2.1 and H in Eq.2.3.
A major contribution to
!
science was made by Penman (1948).
who combined the two
equations
into a single formula by discovering the linK
between vapor pressure deficit.
e
-
e
• and the
s
a
sensible heat.
H.
He applied the la~of Fourier on heat
transfer.
FicK's Law on mass transfer.
and Newton's Law
on viscous shear stress.
He also used Reynold's
(1874)
analogy on the similarity of
the wind functions
involv~d
in heat transfer and mass transfer.
According to
Reynolds.
the transport mechanisms of heat and momentum
in turbulent flow might be similar.
using the laws
recalled above.
the components of
the energy budget
equation can be written as follows
(Thorn et al .•
1981):
••

---

16
( 2 . 4 )
where
fh(U)
= wind function for
sensible heat
transfer
Q = mean wind
speed for
the period considered
y
psychrometric constant
Ts = temperature at the evaporating surface
Ta = air temperature
From the Reynolds'
analogy (1874),
(2.5)
Replacing fD(u)
in Dalton's equation. Eq.2.1.
gives:
E
(2.6)
:
TlJ
'
sIng th e Tay 1 or
.
expansIon
.
serIes.
the term (e'
-
e
)
s
a
/ !
can be developed as follows:
e s
(2.7)
(ae')
\\3T
is the value of the first derivative of e(T)
at
d
the air temperature T .
It can be replaced by 6.
the
a
slope of the saturation vapor pressure curve at air
temperature.
T • which can be obtained in equation form
a
(Burman et al .•
19B3).
from the psychrometric chart or
from tables.
The term c is negligible since it
:
contains the second derivative of etT).
Therefore •
..

---

••
17
'.
equation (2.6)
can be written:
( 2 . 8 )
'rhis gives:
e.# (T )
-
e
E
a
a
(T
-
T
)
=
5
a
( 2 .9)
substitution of Eq.2.9 in Eq.2.4 gives:
e" (T )
E
a
(
-
t.,ofh(u)
(2. 10)
where:
,
e(T a ) :
saturation vapor pressure at Ta
measured at the surface where i:i is
..
measured .
substitution of this value of H in Eq.2.3 gives:
y
y
[e" (T )
-
e
]
R
-
G = E + -
o
E
-
-
0
fh(U)
a
a
(2.11)
n
!J.
Rearranging to solve for E:
y
E =
(R
-
G)
+
f, (u)
[e" (T )
-
e
1
n
a
a
y+6
Cl
Y+6.
(2.12)
Over a period of one qay the heat storage in the ground,
G,
is negligible.
Eq.2.12 then becomes:
(\\
E =
. R
[e'(T)
-
e
1
n
a
y+6
Cl
(2.13)
..
Eq.2.l3 was developed by Penman (1948).
It is often
.o

---

18
termed the combination equation because it contains an
energy balance component and an aerodynamic component.
This formula is based on theory but it contains a certain
degree of empiricism because of the empirical formulation
of the wind function f(a).
fen) was a straight line
equation fitted to a few observations of evaporation
losses versus wind velocity.
The Penman formula,
although originally developed to estimate evaporation
from a free water surface. was used to estimate
evapotranspiration of water from a soil and plant canopy
surface.
This was done by using in the Penman equation a
net radiation. R • representative of the net radiation
n
at the specific soil-canopy surface.
In 1965. Monteith developed a fully rigorous equation
which models evapotranspiration from vegetated areas in
terms of the meteorological conditions and plant
physiological factors:
f; • Rn
+ (J' C • (e" (T ) -
e
) / r
E =
P
a
a
a
L\\ +y. (1 + r
/ r
)
(2.14)
s
a
where:
p
density of air
Cp : specifi~ heat of air
aerodynamic resistance to the diffusion
of water from the evaporating surface to
the same reference level above where vapor
pressure is ea:
it is also termed aero-
dynamic resistance to water vapor transfer
;;
.

---

19
rs
;
aerodynamic resistance to
the diffusion of
water through the evaporating surface;
i t
is also termed canopy resistance to water
vapor transfer.
Monteith (1965)
gave details on the derivation of
Eq.2.14.
Eq.2.14
is
the Monteith version of Penman
equation.
It is original
in the sense that i t gives a
more theoretical expression of the wind function than
that empirically determined in the Penman equation.
Monteith included canopy,
r
,
and aerodynamic,
r
,
s
a
resistances to water vapor
transfer
in order to eliminate
the empiricism of
the wind function.
However,
for
practical purposes Eq.2.14
is difficult to use because of
the difficulty in determining r
.
s
:
2.1.2
Practical Methods of Computing Local
~vapotranspiration
Since Penman (1948),
several equations have been·
developed
to model evapotranspiration.
Some are more
theoretical versions of Penman equation,
such as
Businger's
(1956),
Monteith'g
(1965)
and Van Bavel's
(1966).
They attempted
to
replace the empirical wind
function with more theoretical
relations.
These
scientists
included in the Penman equation theoretical
expressions for
the resistance of water diffusion through
and from the evaporating surface.
Other methods were
more practical than the Penman method.
The authors were
concerned with practical considerations and wanted to
.•

---

20
design simple methods which could yield the best estimate
from the data available.
These empirical methods are
widely used by engineers for
project design.
Erpenbeck (19B1)
identified seventeen empirical ET
methods.
Seven of
these methods used air temperature
only as
the primary weather parameter.
Some of them were
calibrated using secondary weather data.
Ten methods
used most of
the data available at a
relatively complete
weather station,
including solar radiation,
windspeed,
and saturation deficit of
the air.
Primary weather data refer to values available for
each day or month in each year.
Secondary weather
parameters are long-term values for
the period of record,
taken on a monthly basis.
The secondary weather
pa~ameters allow for adjustment based on the general
climatic conditions.
Such an adjustment has been termed
local calibration.
Doorenbos and Pruitt
(1977)
showed
consistently improved reference ET estimates by applying
local calibration or at
least an adjustment that
considers the. general climatic conditions.
Details on
different empirical methods were given by Erpenbeck
(1981).
Erpenbeck evaluated and compared the seventeen
ET methods at fourteen meteorological sites
in the state
of Washington.
The FAO-modified Blaney-Criddle method
(Doorenbos and Pruitt,
1977) was selected as the best
state-wide ET method.
The selection was based on the

---

21
weather data available throughout the state and on
statistical ranking using the coefficient of
determination for each estimating method compared to
lysimeter measurements.
Allen and BrocKway (1982)
compared four FAO-modified methods (i.e. FAO
Blaney-Criddle. FAO Penman. FAO Makkink. FAO Pan
methods). Jensen-Haise. SCS-modified Blaney-criddle.
standard Penman and Wright modified Penman methods.
The
comparison was done using daily weather data from the
USDA Snake River Water Conservation Laboratory at
Kimberly.
Idaho.
The FAO-modified Blaney-Criddle method
was selected as the best state-wide ET method for Idaho
based on accuracy,
responsiveness of the evaluation, and
the primary data requirement of air temperature only.
For the state of Oregon. no such comparative study of ET
methods has been done to date because no lysimeter is
available.
However.
the climate in western Oregon is
like the climate in western Washington, while the climate
in central and eastern Oregon is like the arid climate of
Idaho.
Of the seventeen methods compared by Erpenbeck
(1981), only the FAO Blaney-Criddle method will be
described.
From Doorenbos and Pruitt (1977):
f
(8.13 + 0.46 Ta)P
(2.15)
where:
f ~ Blaney-Criddle ~f'l factor.
•
.
Ta ~ air temperature (oC)

---

22
"
p = daily percentage of annual daytime hours
(percent)
(2.16)
h
=
an
(2.17)
where:
h an and hd are the annual and daily daytime
hours
(hr).
respectively
an = number of days per year.
Doorenbos and Pruitt (1977)
published a modified version
of the Blaney-Criddle method for a green grass reference
crop.
The following formula was derived:
ETr-BC = a + b.f
(2,18)
where:
a = -1.41 + 0.0043 RHmin -
n/N
(2.19)
RHmin is the minimum relative humidity (percent)
n/N is the ratio of actual to maximum possible
sunshine hours
b is a function of RH .
,
n/N,
and u .
U
is the
mln
2
2
daytime wind speed
(m/s) at 2 meters height.
Doorenbos
and Pruitt (1977) published calibration curves for
Blaney-Criddle ET.
Those curves are functions of
RH
,
n/N,
and u .
Burman et al.
(1983) gave a
min
2
regression equation for b as follows:
.•

---

23
+ a4 RHmin
nlN + as RHmin U2
(2.20)
Frevert et al.
(1962) determined the numerical values of
the regression coefficients
a
= 0.81914. a
= -0.0040322.
a
= 1.0705.
o
1
2
a
= 0.06546.
a
= 0.0059664.
as = -0.0005987
3
4
Allen and Brockway (1982) verified in Idaho the
recommendation made by Doorenbos and Pruitt
(1977)
to
make a
ten percent upward adjustment of the FAO
Blaney-criddle ET estimate for
every 1000 meters altitude
above sea level.
2.1.3
Estimation of Regional Evapotranspiration
There is a
large gap of available methods
to ma~e the
transformation between local and regional ET estimates.
In order
to reduce this gap,
several possibilities have
been examined.
These require either extension of
local
methods or development of new procedures.
A representation of weather condition on a
large
scale by a
single parameter implicitly involves smoothing
out the heterogeneities which exist at elementary
:
scales.
Such a
homogenization is assumed when local
•.

---

24
•
estimates are lumped together to represent an estimate at
a larger scale or when a representative value of the
large scale is measured directly.
Fig. 1 (Seguin. 1978)
illustrates the heterogeneities that must be smoothed out
in order to pass from one scale to a larger one.
The
reference evapotranspiration. ET • corresponds to
r
homogenization at scale D. since in the theory ET r
corresponds to measurement made from a lysimeter
surrounded by a mimimun of 100 meters diameter of
homogeneous cropping conditions identical to the
conditions which prevail on the lysimeter (Jensen.
1973).
The regional evapotranspiration, ETR, corresponds
to a homogenization at scale E.
Bouchet (1963) showed that variations of regional
actual evaporation, ETR, which is an experimental value,
lead to simultaneous variations of potential
evapotranspiration, ETP, which is the theoretical demand
of water due only to the climate.
The reason is that any
modification of regional ETR due to the reduction of
water supply at a regional scale results in an increase
in the remaining solar energy available for evaporating
water in terms of ETP at any neighboring surface where no
water restriction has been imposed.
Bouchet (1963)
proposed the following equation:
:
ETR + ETP = 2ETPO
..
or
ETP
ETPO a
ETPO
(2.21)

---

I.'.
25
".
...
Scale:
A . . .
Stomata
1 mm
B . . .
Leaf
1 to 10 cm
I
I
C
• . .
Plant
1 to la m
.,;.
x r l' r r T TT I T 1 IT Y
o ... Cropped
100 m to 1 km
parcel
E ...
Small region
la km
Figure 1.
Reconstitution of homogeneous evaporating
surfaces by moving from small to larger
scales.
'-;,

---

I
•
26
where ETP
is the potential evapotranspiration from
o
either a
local or regional surface when the supply of
water is not limited.
Eq.2.21 neglected the effect of
advection on ETP.
In fact,
this effect depends on the
size of the evaporating surface.
Seguin (1975) used the
definition sketch in Fig.
2 to illustrate the
relationship between ETP and ETR using their
corresponding evaporating surface sizes.
The following
I.· ,
relationship was established (Sequin, 1975):
ETP(s) - ETPO = [f(x/L,
L/Zo)].(ETPO - ETR(S»
(2.22)
where:
Zo = surface roughness parameter (m)
ETP(s)
= potential evapotranspiration from a surface
of size s, with extension x, which is not
short of water
ETPO = potential evapotranspiration where there is
no advection and no water shortage
ETR(S)
= actual regional evapotranspiration from
surface S, with extension L, where water
shortage has been imposed.
Seguin (1975) demonstrated the following:
f(x/L,
L/ZO)
=
f(x/L)
for
10 6 ~ L/ZO ~ 107
(2.23 )
Eq.2.22 becomes independent of the surface roughness
6
7
parameter,
ZO'
as LIZ o is between 10 and 10 .
seguin (1975)
assumed Zo = 1 cm, which is equivalent to
grass 7.5 centimeters tall as specified by Brutsaert
'.',f
, .

---

27
...•..... _
_
~ , -
"'T'
!
H
- ........~-
!
,
ETR(S)
h
I
_ _-=:-'
I -
_
,--
~
S
--J
:
-;(
_..
.
_.~.-
~_. -." x ...__.. -.i
I
:
I
~ .•. -
-.. -. L••..•. --_
_.- ~
Fig.
2.
Regional
(5)
and
local
(s)
scales of
evaporating surfaces.
u - windspeed
Rs
global solar radiation
Hand h are thicknesses of boundary layers for surfaces 5
at
the regional scale and s
at the local scale .
•.

---

I
•
28
I,·.
(1982).
For 2
= 1 cm, L lies between la and 100
0
kilometers.
Seguin (1978)
gave in Fig.
4 an illustration of
latent heat flux homogenization as one moved from local
to regional scales.
Fig.
3 shows the elementary boundary
layers produced individually by elementary homogeneous
surfaces.
If measurements of weather data are made very
close to the earth's surface at one specific location,
they will represent that particular
location.
The
computation of evapotranspiration from each of these
measurements will not represent the whole surface at
regional scale.
However,
if
the same measurements are
made at a
higher altitude outside the elementary boundary
layers.
the corresponding flux will better represent an
average of
the elementary fluxes and therefore be more
representative of
the regional evapotranspiration.
One
of
the applications of remote sensing is
to measure
representative weather conditions for
surfaces at the
regional scale.
In the united States.
research on regional
evapotranspiration has tended to estimate the variation
of
locally calibrated evapotranspiration as
it changes
from one location to another.
Jensen
(1974)
found
that
the peak monthly ET rate at Brawley.
California.
an arid
inland
location.
is 2.5
times
that at a coastal
location
at Lompoc.
California.
Earlier, Nixon et al.
(1963)

---

29
i'lll7, Tf;:-
I
I
I
•
I
I
Bare soi 1
I
Lake
I
I rri gated
Forest
lre 3.
Juxtaposition at
regional scale of homo-
geneous surfaces each developing an
internal boundary layer.
f f
,
,
i
11
~ •
' .
i
1
: j !
I
:; , ! I I !
.~
,
II
I
Bare soil
Lake
Irrigated
I
I
Forest
I
4.
Homogenization of latent heat
flux at
region21 scales
2 and 3 above elementary
surEace~ in 1, using measurements made
at evaporating 3urface
(soil),
surface
bounda ry
la/e r
(SBL),
and plane tar! boun-
dary
layer
(PBL).

---

30
found in a California coastal valley that ET 37
Kilometers (23 miles) inland was more than 1.5 times that
21 Kilometers (13 miles) nearer the ocean.
Trimmer
(1980) concluded that within 100 Kilometers radius of
weather station.
the ET data were representative of the
crop water use in the Nebraska High Plains. a relatively
flat area.
There is no clean-cut difference between the
local scale and regional scale on a practical basis.
However.
it has been customary to assume that at
distances greater than 16.5 Km (10 miles) away from a
weather station. adjustment may have to be made to ET
I
I·
estimates in order to take into consideration the
I •
variability due to distance.
surface heterogeneity and
micro-climatic modifications.
Seguin et al.
(19B2) stated that two main criteria
must be used to design techniques for estimating regional
ET.
Those estimates must be representative of the whole
surface and hence taKe into consideration the
heterogeneity of elementary surfaces which compose the
I-
regional scale.
The techniques also must be simple
enough to be used routinely for practical purposes using
traditional weather station networKS.
The local
estimation methods cannot be used for regional estimates
because the local estimates have been designed for a
I~
homogeneous surface and under very exclusive conditions
as specified in the ET
definition (Jensen et al .• 1971;
r

---

31
Doorenbos and Pruitt.
1977).
The three main techniques
used are summarized below.
The classical approach is to delimit small
homogeneous units for which local methods may apply.
compute the local estimate of ET for each elementary
homogenous surface then make a weighted average of these
elementary fluxes.
Seguin et al.
(1982)
noticed i t is
difficult to equip the required large number of sites.
If a catalog of soil and vegetation parameters can be
made available.
this method can be used in connection
with a hydrologic water balance model to derive the
regional evapotranspiration.
Brutsaert and Maw~sley (1976)
extended the Penman
j
aerodynamic method
to the planetary boundary layer.
For
surfaces
larger than 5 to 10 ~ilometers. the aerodynamic
boundary layer over the surface penetrates the
atmospheric boundary layer.
This requires measurement of
.~
weather data
(e.g.
wind velocity)
at 50 to 100 meters
height by airplane.balloons or poles.
Some successful
results have been obtained.
but only on a monthly basis.
Difficulties arise also from the inadequacy of radiosonde
networKS and errors in recording upper air data. Other
models were based on the measurement of weather data
in
the lower atmospheric boundary layer
(e.g.
traditional
weather stations)
at grid corners to compute surface
fluxes.
The most difficult problem was to define a

---

32
parameter that could represent the surface wetnass at
grid corners
(Brunet.
1982).
One approach described as the equilibrium evaporation
approach is to approximate regional ET by the global
radiative term of Penman's equation.
Rouse and Stewart
(1972)
estimated regional ET with a 10 percent precision.
compared to Penman's local ET estimate.
using this method
for hourly and daily values on moderately dry days.
Working with wheat.
Perrier et al.
(1980)
established
that equilibrium evapotranspiration accounted for
10
percent of observed ET variation.
navies and Alien
(1973)
proposed to extend the Priestley-Taylor
formulation by using a parameter a which depends upon
soil moisture conditions.
The Priestley-Taylor formula
for equilibrium ET is given as follows:
a'~ R
6+y
n
(2.24)
The terminology equilibrium ET refers to the assumption
that the evaporating surface is saturated and advective
effects are minimal.
This assumption allows one to
neglect
the aerodynamic component of
evapotranspiration.
In Eq.2.24.
a is an empirical
coefficient found
to be 1.26
(Erpenbeck.
1981) ~nd
O/(~+y) is as defined in Eq.2.13.
Another approach consists of using the Bouchet
(1963)

---

33
relationship. Eq.2.21.
Bouchet's formula appears as
valid as the equilibrium evaporation method (Seguin et
al .• 1982).
A difficulty in using this formula is that
it does not apply for ET
estimates in windy regions
r
sUbject to large scale advection.
Besides the
difficulties inherent to each individual approach and for
which improvements are possible. all these methods of
regional estimation provide only global values and their
precision with respect to the surface size has not been
defined.
Remote sensing can be used to collect. more rapidly
'I .
and on different scales. weather and surface parameters
needed for ET estimates.
Due to recent progress in
.'
technology (resolution.
frequency of passes. stability of
,
~
measuring systems) and associated techniques (data
-treatment.
image analysis. atmospheric corrections).
remote sensing now appears as a useful tool to improve
methods of local and regional evapotranspiration
estimation. especially by using thermal infrared or
1-
microwave imagery.
Using the basic energy balance
equation. Eq.2.3. Bartholic et al (1970) derived the rate
of evapotranspiration as follows:
R
-
G
n
E =
1 + YO(T
-
T )/[e~(T)
-
e'(T)}
(2.25)
a
c
s a s
c

---

34
I...
~' :
In Eq.
2.25 T
is canopy temperature measured by remote
c,
.
sensing and e
(T
)
IS
the saturated vapor pressure
s
c
obtained as a function of T .
All other terms are
c
defined as in the Penman equation Eq.2.l3.
Brown and
"
"
'
I'
Rosenberg (1973) ellmlnated e
(T ) -
e
(T ) from
s a s
c
Eq.2.25 using the aerodynamic resistance to water vapor
transfer,
ra'
and the specific heat capacity of air,
This method is formulated as follows:
(2.26)
where:
I:
p = the air density
Stone and Horton (1974)
found that Bartholic's method
yielded smaller estimates,
by 17 percent,
than the other
typical predictions of
local ET , whereas the
r
8rown-Rosenberg method produced estimates 22 percent
larger
than the traditional models
(i.e.,
Penman
method).
Besides these difficulties which are related to
local estimates.
the main problem concerns the
methodology of
incorporating remote sensing data.
Tc
,
and e
(T ),
and combining them with ground based
s
c
parameters in order to derive ET.
The use of
r
microwave techniques to remotely sense T
in case of
c
cloudy conditions requires more research in order
to
define
the exact significance of
those data relative to
surface soil moisture.

---

35
2.2
Theoretical Fundamentals of Geostatistics
The purpose of this section is to present the main
geostatistical tools.
termed the semivariogram.
estimation variance. and kriging. used by
geostatisticians to characterize spatial variability and
to estimate regionalized variables.
Elements of the
theory of regionalized variables will be covered.
In the
next chapter.
thls theory will be applied to identify the
spatial structure and variability of reference
evapotranspiration.
The theory will therefore be
explained using reference evapotranspiration as an
example variable.
2.2.1
The Semivariogram
Matheron (1962-1963) pUblished his treatise on the
theory of Regionalized Variables, after the empirical
wo~k done by Krige in South Africa to estimate ore
reserve in gold mines.
The development and applications
of this theory for mining industries led to the popular
name Geostatistics.
The application of Geostatistics to
the estimation of ore reserves in mining is its most well-
known use.
However,
it has been emphasized (Clark, 1979)
that this estimation technique can be used wherever a
continuous measure is made on a sample at a particular
location in space or time (i.e. where a sample value is
(
expected to be affected by its position and its

---

36
relationship with its neighbors).
Such a continuous
variable which is expected to take different values at
different sampling locations is termed a regionalized
variable.
All the realizations (z(x.),
i=l •... n) at n
1
locations in a given domain or geographical region
constitute a random variable.
According to this definition. the different values of
the reference evapotranspiration in a given region at the
same time or during the same period of time can be
considered as local realizations of a regionalized random
variable.
In addition,
the variable reference
evapotranspiration is continuous over space and its
realizations from one location to the next location a
certain distance away are random (i.e. not determined a
priori).
The random character of evapotranspiration is
due to the randomness of the variables it -was derived
from.
In fact,
temperature,
relative humidity. wind
velocity, and solar radiation are each a rando~ variable
from one location to another during the same period of
time.
Reference evapotranspiration, as well as any other
spatial variable such as soil permeability,
topographic
elevation,
population density,
tree diameter in a forest.
rainfall or temperature in a watershed. sediment mean
size distribution in an alluvial deposit.
is a
regionalized yariable.
Those variables have a spatial
characteristic defined by what ia called the

---

37
I '
semivariogram.
Most of
t.he following explanation on the
semivariogram definition and characteristics was taken
from a review of David
(1977)
and Journel and Hujbregts
(1978).
The material was greatly simplified in order to
present only the notions and properties which can be
applied to the spatial variability of evapotranspiration.
I
'
Let us assume a spatial variable.
z(x).
measured or
observed at N locations.
x.
(i=1.2 •... IN).
to be
1
represented by z(x ).
Z(x ) •...•
Z(~).
One way to
1
2
compare two values z(x)
and z(x + h).
separated by h
kilometers.
is to compute their difference
z(x)
-
z(x + h).
This differ~nce can be positive.
negative or zero.
If one is
just interested in the
absolute value of
the difference.
then the value to be
considered is
Iz(x)
-
z(x + h) I.
If
the main
interest is not
just to compare
two single observations
.but all
the n(h).
n(h) < N.
observations separated by the
distance h kilometers.
then it makes more sense to
compute the average of
the quantity [z(x)
-
z(x + h)J.
This average can be written as follows:
1
o(h)
D (h)
=
l:
[ z(x.)
-
z(x.
+ h)JI
.
1
1
(2.27)
n(h)
i=l
D(h)
is the estimate of the mathematical expectation of
.'
the difference
(z(x.)
-
z(x.
+ h)].
The plot of D(h)
1
1

---

"
38
I · '"
I
i :
for different values of h is termed the drift and its
departure fr'om zero indicates how much the region is
heterogeneous with respect to the property z(x).
If the
difference D(h) is zero for all values of h over the
region considered,
the regionalized random variable z{x)
is said to have first order ~tationarity (i.e. there is
no drift).
First order stationarity also indicates there
is no trend (Clark, 1979).
For variables which exhibit
first order stationarity,
E {z(x)
- z{x + h)} :: 0
(2.28)
I -
The drift, D{h}, can be positive,
negative or zero.
However,
in geostatistics one is interested in the
I •
absolute value of changes as functions of distance.
One
way to avoid dealing with the sign of D{h), without being
encumbered by the calculation of means of absolute
values,
is to compute the mathematical expectation of the
squared difference of zex.) - zex.
+ h) which is
1
1
defined as:
{2.29}
In its computational form, Eq. 2.29 is written as follows:
1
n(h)
2y(h)
=
2
[ [ ( z ( x . )
7. ( x.
+ h)]
n(h)
1.=1
1
1
(2.30)

---

39
where:
2y(h)
is
termed the variogram
n(h)
is
the number of pairs separated by the
distance h
y(h)
is termed the semivariogram and is written:
1
n(h)
2
y (h)
=
E
[z(x.)
-
z(x.
+ h)]
~
1
2n(h)
i=l
(2.31)
2.2.2
Stationarity Assumptions
It can be intuitively demonstrated that the
realizations of a
regionalized variable constitute a
random function as
they contain a certain degree of
randomness.
According to Matheron (1976a),
i t is not
..
possible to consider any experiment which would conclude
that a mining deposit
is not a
realization of a
random
function.
This same statement can be made of any
climatic variable which varies with location (e.g.
evapotranspiration).
Some assumptions have to be
introduced about this
type of
random function.
Since the
purpose of geostatistical analysis
is to characterize the
variability of a
parameter as a
function of geographic
.~
location,
one should
look for
a stationary random
function.
a
natural
property which remains
invariant
undsr spatial
translation.
The realization of
the
function should be allowed
to change from one location to
another.
but the mathematical expression of t h e '
.
.

---

40
underlying function should be stable.
This property is
called stationarity.
Going from the most restrictive to
the more general.
three possible assumptions on
stationarity can be made in geostatistics (David. 1977).
Weak Stationarity (Second Order stationarity):
This
assumption consists of two conditions seldom found in
natural phenomena:
The expected value of the regionalized variable z(x)
is the same over the entire field of interest.
This is
the first order stationarity written as follows:
E{z(x)} = m
or
E[z(x) - z(x + h)] = 0
(2.32)
where (x + h) is obtained from x through a vectorial
translation using the vector distance h.
For each pair of random variables {z(x).
z(x + h)},
the covariance exists and depends on the
separation h.
cov(x, x + h)
E{[z(X) - rn][z(x + h) - m]}
C(h)
(2.33)
The stationarity of the covariance implies the
stationarity of the variance or of the variogram.
Thus:
Var{z(x)}
E{[z(X) - rn]2}
C(O)
(2.34)
From Eq. 2.29.
it follows:
•.

---

41
y(h)
= ~ E{(z(X) - z(x + h)]2}
(2.35)
2
Adding and subtracting m and expanding, Eq.
2.35 becomes:
y(h)
= ~ E{(z(x) - m]2 - 2[z(x) - m][z(x + h) - m]
2
+
[z(x + h) -
m]2}
(2.36)
In case of first order stationarity,
E{z(X)}
= E{z(x + h)} = m
(2.28)
therefore:
y(h)
l{c(o) - 2C(h) + C(O)}
(2.37)
2
y(h)
= C(O) - C(h)
(2.38)
Relation (2.38)
indicates that,
under the hypothesis of
second order stationarity,
the covariance and the semi-
variogram are two equivalent tools for characterizing the
auto-correlations between two variables z(x)
and z(x + h)
separated by h.
A third tool termed the correlation,
p(h),
can be derived from Eq.
2.38 as follows:
p(h)
£iQl
1 -
y(h)
C(O)
C (0)
(2.39)
Second order stationarity supposes a priori existence of
variance, C(O),
and covariance, C(h),
independent of x.
However,
according to Journel and Huijbregts
(1978),
there are several natural phenomena and random functions
.,

---

42
which have an infinite capacity for dispersion,
i.e.,
which have neither a priori variance nor a covariance,
but for which a variogram can be defined.
The intrinsic
hypothesis can be made where the second order
stationarity is not rigorously applicable.
Intrinsic Hypothesis:
A random function is saia to
satisfy the intrinsic hypothesis when:
The mathematical expectation of z{x) exists and does
not depend on the location x,
E{z{x)} = m
(2.32)
For all vectors h,
the increment [z(x) - z{x + h)]
has a finite variance which does not depend on x,
Var{[z(x) - Z{x + h)]}
= E{[z(x) - z(x + h)]2 = 2y(h)
(2.40)
Eq.
2.38 and Eq. 2.40 indicate that second order
stationarity implies the intrinsic hypothesis, but the
converse is not true.
Quasi-Stationarity (Hypothesis of Universal Kriqinq):
This hypothesis is less restrictive than rhe two previous
ones.
It assumes the second moment,
Var{[z(x)
- 2(X + h)]}, has some stationarity within
a vicinity of restricted size and that the expectation
E{[Z(X)]},
which is no longer stationary, varies in a
regular manner in a vicinity bOO
If x and x + hare

---

43
taken in such a vicinity,
so that
Ihl < b o ' then
these relations follow:
E[z(x)]
= m(x)
E[z(x + h)]
= m(x + h)
(2.41)
D(h) = E[z(x) -
z(x + h)] ~ 0
(2.42)
D(h)
can be statisticallY derived by a linear regression
technique and used for
kriging,
which in the case where a
drift is present,
is termed universal kriging.
The
universal kriging is possible only if
the drift D(h)
is a
linear combination of simple known functions
(Journel and
Hujbregts,
1978) and
if
the semivariogram y(h)
of
the
original z(x),
defined by Eq.
2.31,
is a~cessible.
More
details on universal
kriging can be found
in David
(1977).
According to Journel and Hujbregts
(1978),
quasi-stationarity is a
compromise between data
availability and a strict stationarity because i t is
.~
always
possible by reducing the vicinity b o to produce
a zone so small that
the stationarity is verified.
However.
this
reduction would not always be possible
because of economical constraints related to station
density for data collection.
If no limitation is made on
data availability,
the stationarity of a continuous
random variable can always be verified as the vicinity is
adequately reduced.
Because of
this fact,
a
theoretical
..

---

44
test could never refute the hypothesis of stationarity of
a continuous random function (Journel and Hujbregts,
1978).
2.2.3
Properties of the Semivariogram
A semivariogram function represents all the possible
values of y(h) as h varies.
To compute a semivariogram
function,
one uses Eq. 2.31 for all possible h.
The
graphical representation of all the computed y(h) is
termed the experimental semivariogram.
This curve is
obtained by fitting a smooth curve to the cloud of points
representing the pairs (h. y(h)}.
In almost all
pUblications on geostatistics.
the term semivariogram is
indifferently used to refer to the computational value
y(h) for a given h and its graphical representation.
In order to avoid adding to the confusion.
this paper
will not use any specific new term to make a distinction
between those concepts.
Ideal Semivariogram Function:
Although an
experimental semivariogram can exhibit a large variety of
shapes, an ideal semivariogram plots as shown in Fig. 5.
An ideal semivariogram is expected to maintain the
following properties:
The semivariogram is always a positive quantity,
expected to go to zero as the distance h goes to zero.

---

45
..... 'f' .,
'..f •••.. '.,
•
I
....
. A
/'';'''''
. ,
.I
C(h) ,~
:
. /
,
I.
,
."
.
.
~a
a
3
h
Figure 5.
Ideal semivariogram curve.
Hence:
y(h)
y(-h) >
0
and
y(O)-
0
(2.43)
In practice. due to the difficulty to make h as small as
possible,
it may occur that:
y(o)
Co i= 0
(2.44)
Co is termed nugget effect.
It represents half the
squared error that one must account for when estimating
the variable z(x) within a distance less than the sample
distance.
The semivariogram.
in case of second order
stationarity. will approach asymptotically a maximum
value which is the sample variance. Var[z(x)].
as shown
(
,

---

46
in Fig. 5.
The maximum value is termed the sill value,
Co + Cl'
and is written:
(2.45)
When the nugget effect Co is zero, the sill value
becomes Cl.
Beyond a certain distance, a,
y(a) = Co + Cl
(2.46)
where E
is a negligible quantity.
a is termed the
a
range.
It represents the distance beyond which the
spatial structure of the variable no longer exists.
It
is also termed the range of influence in the sense that
it represents the zone within which the semivariogram can
be used to represent a spatial correlation and to
interpolate between data locations.
Beyond distance a.
the random variables z(x) and z(x + h) are no longer
correlated.
Isotropic-Models_~f Semivariogram:
When a
semivariogram function y(h) depends only on the
absolute value
Ihl.
the natural phenomenon it
describes is said to be isotropic and the corresponding
semivariogram is termed an isotropic semivariogram.
No
matter how well an experimental semivariogram fits the
points which represent the computed y(h),
the
,
experimental semivariogram cannot describe perfectly the
variability of a natural phenomenon in a region.

---

47
According to Ga:nbolati and Volpi
(1979)
"the true
semivariogram ('Mhich resides
in God's mind) will never be
known. 11
Howevel:.
experience has proven that almost all
the semivariograms observed
in Geostatistics fall
in the
following four
categories:
(1) Linear model:
y(h)
= Co + b'h
y(h)
= Co + Cl
h> a
(2.47)
where b
is the slope
(2) Spherical model or Matheron model:
3
J
y (h)
=
(
Co + Cl h- ~)
1
h
- "2 (--) )
0 < h < a
a
a
y (h)
.-
(2.48)
Co + Cl
for
h > a
This model
is
the most commonly chosen one
in
Geostatistics.
According to David
(1977),
i t has been
possible to estimate a
hundred ore deposits with only the
spherical and the exponential model and the
tendency is
now to use only the spherical model.
This model was
mentioned for
the first
time by Matheron.
(3)
Exponential model:
y (h)
C
+ C
[1
( h i
-
0
1
- exp -J a)J
o < h < a
Eor
h > a
(2.49)

---

48
In practice.
a
is such that yea)
= 95 percent of the
sill value at the range
(Vieira.
19B3).
(4) Gaussian model:
2
y(h)
= Co + C1[1 - exp (-~~ )]
(2.50)
where a
is the distance after which the semivariogram
becomes visually stable.
Anisotropic Models of semivariogram:
A property
is
said to be anisotropic when its variability is not the
same in every direction.
The structural function y(h)
which characterizes its spatial variability depends on
the direction of
the vector distance h.
When anisotropy
is suspected.
a
semivariogram function must be
experimentally determined for
each direction where the
characteristics
(i.e.
isotropic model.
nugget effect.
Co'
sill value.
Co + Cl'
and range.
a)
are
suspected of changing.
When anisotropy exists.
a single
isotropic semivariogram can no
longer be used to
characterize the spatial variability of
the property in
all directions.
Two
types of anisotropy have.been
identified
in Geostatistics:
Geometric anisotropy and
zonal anisotropy.
A semivariogram y(x,
y.
z)
in tridimensional ipace
has a geometric anisotropy when the anisotropy cap be
reduced to
isotropy by a
linear
transformation of
the
vector distance h as follows:

---

49
y(h)
y' <JX' 2
'2
'2'
=
+
Y
+
z
)
(2.51)
where:
x'
x
y'
=
[A]
Y
(2.52)
z'
z
[A]
represents the matrix of transformation of the
coordinates (x. y.
z)
into the new coordinates
(Xl.
y'.
z').
Details on the derivation of matrix [A]
will be found in David (1977) and Journel and Hujbregts
(1978).
Any anisotropy which cannot be reduced by a simple
linear transformation of coordinates is termed zonal
anisotropy.
It is identified by a significant difference
between the sills of the individual unidirectional
semivariograms.
A unique model of
zonal anisotropy can
be derived for
the whole domain as follows
(David 1977):
y (h)
Yiso(lhl)
+
Yzon(lhzl)
(2.53)
In the case of zonal anisotropy.
the total variability
y(h) can always be decomposed into an isotropic
component y.
( I hi)
and a zona 1 component
1S0
Y
( h)
For vector distance h within the region:
zon
z .
I h I i 6 the ,a bsol u t e va 1u e 0 f h
Ih \\' is the component of h in the direction of
Z
zonal anisotropy.

---

50
The direction of anisotropy is chosen on basis of
experience.
configuration of
the region.
and the specific
property under study.
As a general
rule.
an anisotropic
semivariogram y(h)
can always be decomposed into a
sum
of p isotropic semivariograms y. ( I h. I).
As
1
1
formulated by Journel and Hujbregts
(1978):
P
y (h)
= E
i=l
(2.54)
2.2.4
Simple Kriging
Kriging is a
local estimation technique which
provides
the best linear unbiased estimator
(abbreviated
to BLUE) of
the unknown characteristic studied
(David.
1977).
The name kriging appeared around 1960 to
designate that estimation technique which was created in
France by Matheron.
D.G.
Krige
(1951) was
probably the
first
to make use of spatial correlation and BLUE in the
field of mineral
resources evaluation.
Given a
property z(x)
observed at n locations
K. (i~l •...• n)
in a
region where the semivariogram
1
function
is available.
the best
linear unbiased estimate
of z(x)
can be made at a
location Xo where no
Observations
Z(X o) are available.
The
term "point
kriging"
refers
to estimation by the BLUE technique of
z(x)
at a single location x o '
"Block kriging"
refers
to the estimation by the BLUE technique of the mean value

---

51
of z(x) over a block or subregion.
A block is a surface
or volume smaller than the whole region or domain from
which data are collected.
The term "simple kriging" is
used to indicate kriging in case of second order
stationarity.
"Universal kriging" is application of the
BLUE technique which takes into account the existence of
adrift (i.e. non stationarity).
In many applications of kriging. such as
qeochemistry. hydrology contour mapping (Clark. 1979). or
local estimation of evapotranspiration. all that is
needed is point kriging on a regular grid system.
Given
n observations of z(x.) at locations x.(i=l, ...• n)
in
1
1
a region where a semivariogram function has been defined.
a kriging system of linear equations can be derived to
provide, at location xo. a kriging estimate z*(x )
o
expressed by David (1977) as follows:
n
[
W.
z (x. )
1
1
1=1
(2.55)
where:
Z(X ) {i=l •...• n} are observed data at location xi
i
z*(X ) is the kriging estimate at, location xO'
o
w. are weights that will be obtained from the
1
kriging system. Eq. 2.65.
Eq.
2.55 shows z*(X ) is a
o
linear estimate using z(x.) and w. (i=1. ... ,n).
The
1
1
weights w
are computed to agree with the following two
i
conditions:

---

52
(1) Non-bias Condition:
This condition implies the
kriging estimate z*(k.) at a data location is exactly
1
equal to the observation z*(x.) at that location.
This
1
condition is formulated as follows:
E[Z*(XO) - z(xO)] = 0
(2.56)
Replacing z*(xO) by its expression in Eg. 2.55 gives:
n
£{[
E
\\V
z(X
)] -
z(x
i
i
o)} = 0
i=l
(2.57)
Because E{
} is a linear operator, Eg. 2.57 becomes:
n
l:; { Z ( x . )}
[
W i
.- E { Z ( x o)}
-::: In
.1
i=l
(2.58)
Using the first order stationarity assumption stated by
Eq.
2.31,
E{z(xi> }
E {z (xO)}
=
m
(2.59)
Therefore:
n
I
\\01.
-:::
1
1
i=l
(2.60)
Eq.
2.60 is another expression of the non-bias condition.
(2) rhe Minimum Estimation Variance:
This condition
is written as follows~

---

53
Minimize
(2.61)
Replacing z*{xO)
by its value from Eq.
2.55 gives:
n
2
E{[ E
w.
z(x.)
-
Z(x
)]
}
O
(2.62)
i=l
1
1
Using the linear property of the operand E{
},
+ E{[Z{xO)]2}
-
2 E{[z*{xO)]
[z(xo)]}
(2.63)
Kriging System of Equations:
The kriging system of
linear equations is based on the non-bias and minimum
estimation variance conditions.
Developing the right
hand side of Eq.
2.63 term by term and using Eq.
2.33
leads to:
2
n
E{[z*(xo)l
} = E{[
2:
'd.
z(x.)]2}
1
1
i=l
n
n
l:
>=
w.
w.
Efz(x.)
z(x.)}
i=l
j=l
1
J
1
J
(2.64)
n
n
2
=
vI.
\\'1.
[C (x.,
x.)
+ m ]
L:
L:
1
J
1
J
i=l
j=l
n
L
w.
-
1
j=l
J
(2.60)
....

---

54
0) }
E{[Z(X
)]2}
-
E{Z(X
O
O)
Z(X o +
2
= C(X ' X
O
o) + m
(2.65)
n
:: 2E{[
EW
z(x )
z(x )]}
i
i
O
i=l
n
z(X
)]}
, = 2
E
w. E{[Z(X.)
O
1
1
i=l
(2.66)
n
2
= 2
E
w. C (x. ,
x ) + 2m
1
1
O
i=l
Substituting Eq.
2.64
through Eq.
2.66
into Eq.
2.63
gives:
n
n
2
E{[z*(x
)
-
z(x
)]
}
=
Z
2
O
E
w. w.
O
[C(x' r
X . )
+ m ]
i=l j=l
1
J
1
J
n
2
+ Clx
) -I-
')
2
m
-
O '
x O
<.
E
w.
C (x. ,
XO)
- 2m
i==l
1
1
E{[Zk(X
)
- Z(X )]2}
O
=
O
C(X O' XO)
1~
(2.67)
n
n
n
+
"
L,
L:
w.
w. C (x. , x . ) - 2
E
w.
C (x. ,
XO)
i=l j=l
1
J
1
J
i=l
1
1
where:
C(xo. X )
o
is
the covariance of
z(x)
at Xo
C(x
)
o ' Xo
is zero if there is no nugget effect
C(xo ' x. ) is the covariance of z(x) at Xo and
1
the sampling point x.1

---

ss
C(X.,
x.)
is the covariance of z(x) at sampling
I
J
points x.
and x ..
I
J
C(Xi,
Xj)
= 0
f 01:
i
=
j
C(Xi.
Xj) :F 0
for
i "* j
(2.68)
Eq.
2.67 is to be minimized under conditions of Eq.
2.61.
The condition of minimum variance set in Eq.
2.62,
and
the condition of unbias expressed by Eq.
2.60 may be
combined in a single mathematical expression using the
Lagrangian parameter ~ as follows:
IF
= 0
(2.69)
where:
w.)
-
li
(2.70)
1
The last term in (
} in Eq.
2.70 is equal to zero.
The
factor 2 in front of ~ has besn chosen in order to
avoid non integer coefficients in the anticipated
expression of aF/aw..
Substituting Eq.
2.67 and
1
Eq.
2.70 into Eq.
2.69,
3F
n
() vI.
=
L
W.
w.
C(x.,
x.)}
1
j:::l
1
J
J.
J
a
n
{2
L
~~/~
W i
C(x i , x o)} + ;):'-:- [2\\[( ~
'd.)
-
I]}
1
i=l
1
i=l
1
(2.71)

---

56
Analyzing each term of Eg.
2.71 gives Egs.
2.72 through
2.75 as follow:
(2.72)
n
n
n
0
{ L
l:
w. w. C (x. , x . ) } = 2
l:
w. C (x. , x .)
ow.
1
J
1
J
j=l
1
1
J
1
i=l j=l
(2.73)
The factor 2 is mathematically justified by the fact that:
n
n
L
\\v.
=
L
w.
i=l
1
j=l
J
n
(2.74)
I:
w.
C(x "
x
)}
1
l
O
i=l
n
d
{2)..[(
LVI.)
-
l]}
=
2)"
(2.75)
'dVl i
i = 1
1
Adding up Eqs.
(2.72)
through (2.75), Eg.
2.69 becomes:
n
L
\\'1.
C (x .,
x.)
+ A =
i=l
J
1
J
n
(i=1,2, ... ,n)
2:
w.
=
1
i=l
1
(2.76)

---

57
The system of linear equations given by Eq.
2.76 is
termed kriging system.
It has n+l equations and n+l
unknowns.
In matrix form it is written:
[K] [W]
==
[R]
(2.77)
where:
C(Xl'
xl)
C(Xl'
x2)
C(Xl'
x n )
1
C(x2'
xl)
C(x2'
x2)
C(x2'
x n )
1
[K]
==
(2.78)
1
1
o
[K]
is termed the kriging matrix.
C(Xo.
xl)
C(Xo.
x2)
[W]
[R]
(2.79)
Solving Eq.
2.76 which is a system of n+l equations and
n+l unknowns.
w.
(i==1. . . . • n)
and f....
provides weights
1
w
needed to compute the kriging estimate z*(x )
i
o
given by Eq.
2.55.
The kriging estimation variance is
defined as follows:

---

58
(2.80)
Rearranging Eq.
2.76:
n
l:
w. C(x., x.)
i-l
]
1
J
(2.81)
and estimating the results
into Eq.
2.67 gives:
n
n
= C(x ' x
O
o) +
L:
w
C(x ,
x
)
-
L:
W.
A
i=l
i
i
O
i=l
1
n
-
2
L:
WiC(xi,x o)
i=l
n
n
= C(x o' x ) + [ w C(x , x
O
H.
i=l
i
i
o) - J..
1:
i=l
1
n
-
2
i=l
n
= C(x
,
) - ,\\
O '
x
)
-
1:
wiC(x
x
O
i
o
i=l
(2.82)
The kriging estimation variance is given by:

---

S9
n
C ( x 0'
x 0 )
-
L
Wi
C (x 0'
>: i )
- A
i-=l
(2.83)
The kriging system given by Eq. 2.76 assumes second order
stationarity (i.e. stationary of the covariance).
In case of the intrinsic hypothesis, only the
stationarity of the variance is assumed.
The kriging
system should then be modified in order to be written in
terms of semivariograms.
According to David (1977),
this
can be done using Eq.
2.38 which gives:
C(h)
C(O)
-
y(h)
(2.B4)
"
Eq.
2.84 can be written using a more general notation
which keeps track of the coordinates present in the
kriging system:
C (Xo.
xO)
y(XO.
xi)
(2.85)
C ( xi.
X j)
.=
C ( xo.
xo)
y(xi. Xj)
Replacing C(X o' x.) and C(x .• x.) in Eq. 2.66
1
1
J
through Eq.
2.76 gives the following formulas used in
case of kriging under the condition
of the intrinsic
hypothesis:

---

60
n
[
w.
y(x.,
x.)
+ 1-l
j=l
]
1
]
(i=1,2, . . . ,n)
11
L
w.
= 1
1
i=l
(2.86)
where ~ replaces ~ in Eq.
(2.71).
Therefore.
substituting Eq.
2.85 into Eq.
2.83.
it
follows that
(Rendu et al .•
1978):
n
C(X O' x ) -
1:
'w.
o
i=l
1
(2.87)
n
n
+
L: w.
y(x O' Xi) +
L:
w.
Y
i=l 1
i=l
1
which reduces to:
11
L:
(2.88)
i=l
where ~ ha~ replaced ~.
If h
and h
are vectors defined by the couples of
ij
Oi
points
(x .•
x.)
and
(x o' x.),
1
J
1
Y(Xi,
Xj)
y(hij)
Y (XO •
x 1. )
y(hOi)
Eq.
2.86 can then be developed as follows:

---

61
+ 0
= 1.0
(2.89)
solving the system of linear equation 2.89 will provide
the weights w.
needed in Eq.
2.55 to obtain:
1
n
2:
w.
z(x.)
i::.:l
1
1
(2.55)
~.
According to David
(1977),
the system given by Eq.
2.89
has proven to be true even when C(X o' Xo)
y(oo)
does not exist
(i.e.
if
the semivariogram does not have a
sill value).
2.2.5
Universal Kriginq
When a drift
is present which is not equal
to zero,
some geostatisticians recommend to
include i t in the
krlging matrix.
For simple drift represented by a second
degree polynominal,
David
(1977)
recommended the
following Kriging system in matrix form:
[K2][W]
[R2]
(2.90)

---

62
where:
1
1
2
y (h
)
y(h
)
y (h
)
1
X
Y
X
y 2
X y
nl
n2
nn
n
n
n
n
n n
..
1
1
1
0
0
0
0
0
0
[K ] =
2
Xl
X
X
0
0
0
0
0
0
2
n
Yl
Y2
Y
0
0
0
0
0
0
n
2
2
2
X
X
X
0
0
0
0
0
0
1
2
n
y2
y 2
y2
0
0
0
0
0
0
1
2
n
,>
Xl y 1
X Y
2 2
X Y
0
0
0
0
0
0
n n
(2.91)
and
[W]
and
[RZJ
are given by:
..

---

63
1
11 0
[W]
:::
fl 2
(2.92)
where:
n
1
X
:::
L:
X.
0
n
J
j:::l
n
1
Y
:::
-
E
Y.
0
n
J
j:::l
...,
n
-L.
1
:::
X
-
l.:
X~
0
n
J
j:::l
(2.93)
n
-2
1
2
:::
V
-
J:
Y.
o
n
J
j=l
n
1
:::
E
XOY
XjY
O
n
j
j:::l

---

64
where:
X. and Y. are the coordinates of x. with
)
)
)
respect to Xo(Xo ' Yo ).
The universal kriqinq may be done successfully only if
the main trend m(x)
is known a priori.
Otherwise.
there
is no certainty it will improve the estimates (Volpi and
Gambolati.
1978).
The choice between simple kriging and
universal kriginq is sometimes a matter of experience and
how much the main trend is known a priori.
The use of
kriging with drift is still a matter of research.
More
detail on universal kriging can be found in David
(1977).
-~

---

65
3.
PROCEDURE AND DATA SELECTION
In order to design a model suitable for the study of
evapotranspiration spatial variability,
the state of
Oregon was chosen as the geographic base.
This chapter
explains the procedure followed to select meteorological
data for ET calculations and to estimate missing data.
It also explains the computation of local
evapotranspiration at weather stations and states the
assumptions on which the computations of the
geostatistical characteristics were based.
3.1
Selection of Evapotranspiration Data and Method
From the literature review,
it is clear that weather
data such as solar radiation, windspeed, and air vapor
pressure are necessary for estimating evapotranspiration
using any theoretically-based method such as Penman's or
Monteith's.
However,
these data were available at only a
select number of weather stations in Oregon.
Therefore,
the only alternative left was to look for a convenient
empirical method.
The choice of such an empirical method
was made after the inventory of the available data.
The
weather data used to support this research were collected
ducing the 1979 growing season, from May through
September.

---

66
3.1.1
Available Weather Data
From the official publications of NOAA (National
Oceanic and Atmospheric Administration), weather data
have been provided regularly for most of the 284 weather
stations listed for
the state of Oregon in the monthly
Climatological Data bulletins
(NOAA,
1979).
Of these
stations,
175 provided the daily primary data of maximum
and minimum air temperatures,
T
and T ,
respectively.
x
m
In 1979,
eighteen stations provided daily windspeed
data.
At sixteen stations,
daily maximum and minimum
relative humidity were available,
while the solar
radiation direct measurements or estimates were available
at thirteen stations only.'
Some of these data were
provided by Oregon State Experiment Stations,
and others
by the Weather Natibnal Service,
mainly for aviation and
public
information purposes.
This study has been limited
to
the state of Oregon data base constituted by 175
o
stations which recorded
temperature data.
Twenty-two of
these stations are listed in Table 1.
These 22 stations
were equipped to record at
least one of the
three
secondary weather data described above.
3.1.2
Choice of an EvaQotranspiration Method
The reasons for
selecting the FAO Blaney-Criddle
method as
the ET method for
the states of Washington and
Idaho were presented in the previous chapter.
Methods

---

67
Table 1.
Secondary Weather Data Stations
Weather Stations
Rs
ID~
WRUN
ID*
RHmin
ID*
Astoria
Y
01
Y
01
Y
01
Bend
Y
02
N
N
Burns
Y
03
Y
03
Y
03
Coos Bay
Y
04
N
N
Corvallis
(Hyslop)
y
05
Y
05
Y
05
Eugene
Y
06
Y
06
Y
06
Hermiston
Y
07
Y
07
Y
07
Hood .Ri ver
N
Y
08
N
Illahe
(near Agnes)
Y
09
N
Klamath Falls
(near
Kingsley Field)
Y
10
Y
10
Y
10
La Grande
Y
11
N
N
Malheur Exp.
Sta.
(near Boise)
N
Y
12
Y
12
Medford
Y
13
Y
13
Y
13
Moro
N
Y
14
Y
14
North Willamette
N
Y
15
Y
15
Pendleton Exp.
Sta.
Y
16
Y
15
Y
16
Portland
Y
17
Y
17
Y
17
;1-
Redmond
N
Y
18
Y
18
Salern
N
Y
19
Y
19
Union Exp.
Sta.
N
Y
20
Y
20
Sexton Summit
N
Y
21
Y
21
Whitehorse Ranch
Y
22
N
N
Number of Stat10ns
13
18
16
Y and N:
data available
(Y)
and not available
(N)
Rs :
daily solar global radiation (langley/day)
:;,
1-lRUN:
daily wind run (miles/day)
_.
RHmin:
daily minimum relative humidity (percent)

---

68
more sophisticated than the FAO Blaney-Criddle method
could be used at specific sites where the data
requirements were met
in the state of Oregon.
However.
no option was
left other
than choosing,
for
the whole
state,
a
temperature-based method -- since only
temperature data were available throughout the state.
The fact
that researchers
in the adjacent states of
Washington and Idaho have adopted the FAO Blaney-Criddle
method as
the best state-wide ET method,
provided some
confidence in the choice of
that method.
The different
variables
required by the FAO Blaney-Criddle method were
given in Eqs.
(2.15)
to
(2.20).
A computer program
termed MAIN was written in FORTRAN to provide the FAO
Blaney-Criddle ET estimate for
all 175 locations,
given
the following
input data for
each location:
monthly
average maximum and minimum daily temperatures.
TMAX and
TMIN.
monthly minimum relative humidity.
RHMIN,
monthly
average solar
radia~ion. RS, monthly average wind run,
WRUN,
monthly average day to night wind ratio.
WRTIO,
height of anemometer,
HWIND,
longitud,e,
LONG,
latitude,
LAT,
and altitude.
ALT.
Program MATN is
listed
in Appendix A.
It provided
the following parameters
in tabular form:
clear sky
solar radiation,
RRAN,
and
the monthly percentage of
annual daytime hours,
PP.
as a
function of latitude and
month,
and a
local calibration table from Doorenbos and

---

69
Pruitt
(1977) used
to determine the factor
b.
The a
coefficient was calculated from Eq.
2.19.
A second
adjustment was made for
topography.
which allowed a
ten
percent increase of ET for
every 1000 meters in altitude
above sea-level
(Allen and Brockway.
1982).
Once this
final adjustment was made.
the local reference ET was
computed for all 175 locations.
along with the longitude
and latitude
(in Cartesian coordinates).
After
execution.
program MAIN provided an output file which
could then be used as
input to compute the geostatistical
characteristics of ET spatial variability.
3.1.3.
Estimation of Missing Weather Data
Missing Primary Data
(Temperature):
All
175 stations
which defined the weather data
base for
local ET
estimation were equipped
to provide at least
the daily
temperature.
However,
sometimes data were not recorded
during one or more days.
Missing data were due to
instrument breakdown or failure
to record measurements.
In such cases,
a day by day moving average was performed
to provide for
the missing data.
The number o~
observations used
to estimate each missing data depended
on the number of consecutive days when the observations
were not available.
Longer missing-data periods required
a greater number of observations
to estimate one datum.
When only one observation was missing.
i t was replaced by

---

70
the average of two observations, one before and one
after the missing observation.
When two consecutive
observations were missing.
the first was replaced by the
average of two observations before and one observation
after the set of missing data: the second missing datum
was replaced by the average of one observation before and
two observations after the set of missing data.
Missing Secondary Weather Data:
As indicated in the
previous section. only 22 stations were equipped to
measure data other than air temperature on a daily
basis.
This suggested the use of a temperature-based
method calibrated by secondary weather parameters which
in general are estimated.
In the case where the FAO
Blaney-Criddle method is used,
these secondary parameters
are estimated from available minimum relative humidity,
RH .
, solar radiation. R • and wind speed. U.
It
mln
s
has been found conve~ient to use the secondary data only
'for adjusting the ET estimates for the local climate.
Solar radiation and minimum relative humidity were not
expected to ~hange considerably over a short distance.
This was not true for windspeed.
Windspeed could change
dramatically over short distances, depending on
topographic and surface roughness variations.
The
decision of which secondary weather data should be used
at a specific primary weather station. where such a
secondary weather data was not available. was based on

---

71
1
empiricism.
consideration of terrain.
and advice from
Weather Service representatives.
Field trips were
required to visit some of the weather stations,
observe
their topographic and floral environments,
and discuss
meteorological data needs with weather data technicians
and specialists who have some experience with the local
climate.
Such an investigation helped determine
similaritie~ and dissimilarities between primary data
stations and their neighboring secondary data stations.
The main topographical features which affect climate
in Oregon are two predominantly north-so~th mountain
ranges,
the Coastal Range and the Cascades,
separated by
approximately 144 kilometers
(90 miles)
in the western
part of
the state.
and the high plateau and highlands
which are east of the Cascades.
West of the Cascades,
the climate is generally humid.
with annual precipitation
on the order of 1.000 ~illimeters (40 inches) in the
inter-mountain valley.
East of the Cascades.
semi-arid
to arid conditions prevail. with annual precipitation on
the order of 250 millimeters
(10 inches) or less.
Topographic features and local meteorological conditions
have led to the state of oregon being divided into the
five climatic regions.
as indicated in Fig.
6.
These are
labeled as A) coastal. B)
north inter-mountain valley.
C)
south inter-mountain valley. D)
north high plateau,
and
•
E) south high plateau.
Although the different climatic

---

.. •...
.
, .
. /
~
'
~'
. .'
/
"
....:
o wind, solar radiation and
air relative humidity mea-
~"\\"--"
'"
,.
"
surement sites.
o
"
/
"
.,-
•
" '((
'"
,
•
'"
lit
I
,
'" '"
I
.. Wind und uir relative humi-
,
"
\\
".
I
• "B·,
\\
I
'"
d; ty meiisurell:ent sites.
1
1
I
o
....
1
....
it'"
I
\\
I
\\
\\'
..
I
I
I
I
o
Wind measurement sites.
o
I
I
I
I
*
I
I
,
I
I
I
I
I
I
1
o
I
%}
Solar radiation measure-
I
I
--
ment sites.
_ _ _
I
---
-_, [
\\
': ~.
\\
\\
c
\\
\\
\\
\\
\\
\\
\\
o
\\
~
\\
o
\\
\\
'Figure 6.
Secondary weather stations, and subdi :Vision 'of:th~e::st~te'ofdregon into
climatic subregions
(solid lines)
and windspeed zonesof·irifluence
--.J
(dashed lines).
N
~;. '.~:

---

73
subregions are not separated in nature by straight lines.
such delineations were made as a first approximation for
allocating secondary weather data from secondary stations
to the primary stations.
Each climatic subregion was
subdivided into secondary weather data zones of
influence. using the technique of Thiessen polygons •. by
drawing bisectors between secondary weather stations.
All the primary data stations which fell in a given
Thiessen polygon received the same secondary weather data
corresponding to that polygon of influence.
Such a
sUbdivision is shown in Fig. 6 for the windspeed zones of
influence.
When no specific reasons based on experience
existed for not doing so.
the same criteria governed the
assignment of solar radiation. R ' and minimum relative
s
humidity. RH . • data to the primary weather stations.
mln
One station. Sexton Summit. has not been indicated on
Fig. 6 because its data were missing during May and
June.
Because of this inconsistency.
the wind and
relative humidity data for Sexton Summit were used only
for that station whenever they were available.
3.2
Organization of Geostatistical Computations
The geostatistical computations involved the
semivariograms. kriging estimates. and kriging estimation
variances.
The method used for constructing
aemivariograms was based on the spatial configuration of

---

74
the available data.
Journel and Hujbregts (1979) stated
that various cases could be distinguished according to
whether or not the data were aligned and whether or not
they were regularly spaced along these alignments.
In
the state of Oregon, weather stations were not aligned
nor regularly spaced.
3.2.1
Procedure for Computing ET Semivariograms
Since ET data were scattered throughout the state, it
was found convenient to group them into distance and
angle classes as is illustrated in Fig. 7.
According to
David (1977), such scattered data can be grouped into
classes using polar coordinates (h. a).
This technique
allowed the computation of experimental semivariograms
y*(a.
h) without losing too many data and with a
sufficient number of data pairs.
The star (*) adjacent
to y indicated that y*(a,
h) was a computed value.
different from the thioretical semivariogram.
Since the
ET data were not continuously observed throughout the
state, discrete distances,
hj' angles. ai'
tolerance distances, ~h. and tolerance angles, ~a,
were used to group them into distance and angle classes.
The intersection of the two regions (h
±. ~h) and
j
2
(a.
+ ~a) defined a class of data which could be
1
-
2
treated as located at (a .• h.).
Given a direction
1
J
a., all the semivariograms y*(a., h.) were
1
1
J

---

75
\\
\\
.'-i
'"
'Llh
2"
Xi
'\\~__-L
_
Figure 7.
Grouping data into distance
and angle classes.
computed and plotted as
the sernivariogram curve for
the
direction 8..
h.
represented the preselected
jth
1
J
lag distance.
A qomputer program named VARIO was written in FORTRAN
to compute the quantities y*(8 .•
h.).
For each
1
J
direction 8 .•
the program performed the computations
1
step by step.
h +
is obtained by adding ha to h ..
j
1
J
The quantities ha and ~h were termeG~basic lag
distance and distance tolerance.
After
these computa-
~8
tions were completed for
one direction.
8.
+ - .
1
-
2
tl8
the direction was changed
to 8.
1 + - - and the
1+
-
2
same computations of
the quantities y*(8.
l '
h.)
1+
J
were repeated.
For
the basic lag distance.
ha.
12.87
kilometers
(8.00 miles) was chosen,
based on the ranges
of regional ET scales previously discussed
in the
literature review.
The tolerance distance was 3.22 km

---

76
(2.00 miles).
so that samples separated by a distance
12.87 ± 3.22 kilometers belonged to the same
semivariogram y*{8.
12.87).
Moving radially outward
in the same angle class.
samples which were separated by
the distance 2(12.87) ± 3.22 kilometers.
belonged to the
same semivariogram defined as y*[8.
2{12.87»).
and so
forth.
The topographic configuration of
the state of Oregon.
where a more pronounced climatic change exists in the
east-west direction than in the north-south direction.
suggested that an anisotropy of ET spatial variability
could be expected.
To confirm or deny this assumption.
i t was necessary to plot at least two directional
semivariograms.
one for
the east-west direction and
another for the north-south direction.
The program VARIO
was written in a general form in order to provide the
plot of two other diagonal semivariograms.
i.e.
NE/SW and
NW/SE.
In the program. 6.
could take the values O.
1
U/4. u/2.
and -u/4. while the angle tolerance 66
was allowed to take any value.
For this project.
68
was chosen to be u/8 in order to allow enough margin
between angle classes and to allow most of the data to
fall
in one or another class.
Along with the plot of the
semivariogram.
the program VARIO could compute and plot
the drift. D(a .•
h.).
previously defined by
1
J
Eq.
2.27.
in each direction using the same lag distance

---

77
and distance tolerance.
The reason for plotting the
drift was that
it could be used to confirm or deny the
first order stationarity.
The shape of
ih~ semivariogram
could also be used to identify not only second order
stationarity but also first order stationarity.
Moreover,
the semivariogram plot could help detect the
vicinLty within or
the extent to which secondary order
stationarity was applicable.
Program VARIO is
listed in
Appendix A.
It used,
as
input,
local
reference ET data
and the Cartesian coordinates of
their
locations.
Its
o~tputs were mainly the semivariograms and drifts plots.
The next step was
to fit one of
the theoretical
models of semivariograms presented
in the
literature
review to each direction,
and evaluate its
range,
a,
and
sill value,
Co + Cl.
The comparison of these
characteristics could help identify the existence of any
anisotropy.
These characteristics had to be combined in
order to derive a unique semivariogram model of
anisotropy which could take into account not only the
variability due to the distances between weather
stations,
but also that due to their directional
relationships.
When anisotropy is present,
the
directional semivariograms can be combined into one
single anisotropic model.
In the case where the
directional semivariograms are spherical with no nugget
effect,
a unique anisotropic model can be derived from

---

78
Eg.
2.48 as follows
(David.
1977):
2
2(_5
v
2
1
"y (h)
(-2. __ )
2r -5
3
==
C { -;:- r X
+ (:- )
y
L
a
u
[I~ )
y
-
- "2
+ (a:~)
}
L
X
Y
X
.i
"
3
+ (C
- C ) {i l~J - i(41) }
x
y
<3 x
x
(3 . 1 )
where:
Cx and a x are- the sill value and range of
•
. · t
influence in East-West dIrectIon
Cy and ay are the sill value and range of
influence in North-South direction
X and Y are the Cartesian components of vector h
which links two locations xi and Xj.
.-
In Eg.
3.1. X corresponds to the direction of
anisotropy.
It is associated with the direction of
highest variability (i.e .•
highest s i l l value).
Eg.
3.1
is governed by the following boundary conditions:
if
Y = ay
if
2
2,°·5
if
(~~)
[
+ (~y) j
> 1
(3.2)

---

79
3.2.2
Fitting a Spherical Semivariogram Model
Geostatisticians,
such as M.
Armstrong
(Verly et al.,
1983),
advise against the systematic use of
the least
squares method
to fit a
theoretital
semivariogram model,
y(h},
to the computed semivariogram,
y*(h).
Such a
technique assumes the observations are random while the
purpose of
the Geostatistics is to account fcir
the
spatial correlation between observations.
Another
reason
.,
for
not systematically using th~ least squares method is
that
it generally leads to models different from the
simple theoretical ones described
in the
literature
review,
and which are suitable for
the kriging with the
underlying assumption that there is a
certain
;:
stationarity of the semivariogram.
The spherical model
has been the most commonly selected semivariogram model
in applications in hydrology.
The following steps are
used to fit
the spherical model to the computed
semivariogram,
y*(h),
based on the illustration shown
in Fig.
5.
1.
Fit a straight line to the first few points
represented by (y*(h).
h).
2.
The intercept of this straight line with the
y(h)
axis is the nugget effect CO'
3 .
Draw a
horizontal
line representing the sill
value,
Cl + CO' which is given by the sample
variance as follows:

---

80
r'l \\ 2
I
( 3 . 3 )
where:
M is the mean of all the N observations.
4.
The intersection of the line drawn in step 1 with
the line drawn in 6te~ 3 is assumed to represent
a distance 2a/3 from the origin of h axis and
this result is used to compute the range.
a.
If there is zd~al anisotropy. the highest sill value is
taken equal to the sample variance and any other sill
values are fitted visually.
The small errors associated
with the sill values and the range when such a visual fit
is made do not make significant difference in the kriging
estimates
(David.
1977).
3.2.3
Kriging for Mapping ET Contour Curves
The most convenient means of displaying the
distribution of evapotranspiration over a geographic
region (e.g.
the state of Oregon), was by using
computerized contour plotting routines,
such as COl~LOT,
available in some form in most mainframe computer
pacKages.
A computer program was written in FORTRAN to
draw the contour curves of ET.
It used a subprogram,
PLOTLIB.
of the COMPLOT package.
Such plotting programs
normally require that data be available at uniformly
spaced grid corners for efficient plotting of contours.

---

81
It became necessary to divide the state of Oregon into a
square grid system 12.87 kilometers (8.00 miles) on a
side. as it is shown in Fig. 8.
The elementary grid size
was selected to conform with the 12.87 kilometers (8.00
miles) previously chosen for the basic lag distance
ho' The grid size could be modified. depending on the
desired distance between contour curves.
Grid corners
located at points where no meteorological data existed
were termed unoccupied grid corners and were to be
provided with kriged estimates of evapotranspiration,
ETK, and a kriging variance. ERK.
A computer program named KRIGX was written in FORTRAN
to compute the kriging estimates.
This program is listed
in Appendix A.
It received as input the characteristics
of the directional semivariograms and a file of Er data
and the Cartesian coordinates of their locations.
Program KRIGX used the IMSL (International Mathematical
and Statistical Library) routine named LUDATF to solve
the system of linear equations given by Eq. 2.89 as the
kriging system.
To estimate ET at each unoccupied grid
corner.
the eight nearest weather stations were selected
and their corresponding semivariogram values used to
build a system of nine linear equations which conformed
to the kriging system given by Eq. 2.89.
It was decided to use eight weather stations because
whenever the data are regularly spaced and anisotropy is

---

82
c
o
tJ1
(l)
I-<
o
4-l
o
I-<
Q)
>
o
E
(])
.w
([J
>.
Ul
OJ
X
OJ
96
08
v9
8v
Z£
OJ
91
o
co
N I C e
o::t
a;;
N
IC
C O N \\ O O
co
,...
Lt")
o::t
N C O ' >
,...
N
\\0
C
l""')
N
N
-
0">
N
N
N
co
N
N
-
-
- -
-

---

83
expected,
a certain bias is avoided by selecting
multiples of four
in order to equally incorporate the
variability in the two orthogonal directions.
In ~he
analysis it was noticed that using eight points instead
of four
improved the kriging results,
while increasing
the number of points to 12 was not only costly in terms
of computer execution time,
but did not improve the
kriging results.
The limitation of the estimation
improvement as mo.re pointB are added is due to a
screening effect explained by Matheron (1965)
and by
David (1977).
The existence of zonal anisotropy is also
a reason for not using too many points.
A subprogram of KRIGX.
named GAMMA,
computed the
semivariogram ~(e., h.) for each couple (e.,
h.)
1
J
1
J
using the semivariogram model for anisotropy.
Program
KRIGX was written to output the kriging estimates, ETK;
and the kriging variance. ERK,
along with the Cartesian
coordinates of their
locations.
Program KRIGX was
written to perform simple kriging.
However,
it could be
slightly modified to perform universal kriging if a drift
existed that had to be taken into account.
3.3
Modeling semivariograms with Insufficient Data
It has been a crucial problem in Geostatistics to
decide whether all the data available on a
large and
heterogeneous region should be taken together to compute

---

84
the semivariograms,
or whether
i t is better to consider
separately each subregion and derive
its semivariogram
using only the few data available for
that sUbregion.
The advantage of restricting a
semivariogram to a small
homogeneous subregion was that the stationarity
assumptions were better sustained.
However,
a
semivariogram derived from too few data
points may not be
reliable.
For
this project,
an iterative method was designed
and a computer program written to search for
the optimum
semivariogram in each sUbregion.
The optimum
semivariogram was the one that produced kriging estimates
closest to the original data and kriging variances.
o~, closest to the the variances of the error.
Z(X
) -
z*(X
).
The trial-and-error
technique
i
o
developed
involved the following steps:
1.
Compute the sample variance.
By definition i t is
equal to the s i l l value,
Co + Cl.
It
is
reduced to Cl'
if the nugget effect,
Co'
is
zero.
Assuming a
zero nugget effect agrees with
the fact mentioned by Journel and Hujbregts
(1978)
that the existence of nugget effect is due
both to measurement errors and the discontinuity
in the property at very small distances.
Because
of
its continuous nature,
evapotranspiration
~.
cannot be discontinuous at very small dista~ces.

---

85
2.
Assume a range of
influence.
a.
3.
Assume the spherical model since the tendency has
been to use only this model for
kriging
estimation (David.
1977).
This model
is entirely
defined
by Cl and a.
given that Co is zero.
4.
At each step,
delete one of
the observed data,
z(x.),
and use
the assumed semivariogram to
1
estimate its value z*{x.)
by kriging.
Repeat
1
this operation for all data
locations.
A similaz
technique has been termed
jacknifing (Vieira,
1983) and used to validate the krigi~g and
semivariogram model.
S.
Test the reduced errors for
normal distribution.
The reduced error,
R{x.),
is defined as follows:
1
( 3.4)
where:
0k(X ) is the kriging standard diviation,
i
commonly termed kriging standard error,
and
computed from the kriging variance O~(Xi)'
This test is peiformed by te~ting if the mean and
variance of the reduced error are m
= 0 and
R
o~ = 1.
The mean of the reduced errors,
m ,
is given by:
R

---

86
n
1
E{R(X ) '1)
=
i:
R (x . \\
mn. .-
i
lJ
l '
i=J.
( 3 . 5)
The variance,
of the r~duced errors,
R(x.),
is given as follows:
1
2
a
-
Va~{R(x.)} =
F..
1
.
(3.6)
This method for testing the normality of the reduced
errors was suggested by Delhomme (1976)
and used by
Vieira
(1983).
It agreed also with Snedecor and Cochran
(1967).
In order to verify how efficient this trial and error
procedure could be.
ET data from each of the five
subregions shown in Fig.
6 were used separately to
generate subregional semivariogram curves.
For example,
it was meaningless to fit a semivariogram curve to the
semivariograms computed from the 41 ET data in the
Willamette Valley because the number of data pairs which
contributed to each semivariogram y{h.) computation
1
were too small to be used for modeling a semivariogram.
Journel and Huijbregts .(1978) suggested 30 pairs as an
order of magnitude although fewer number of pairs have
been used by other geostatisticians
(Verly et al.,
....
1983).
In the Willamette Valley,
as in the other.

---

87
subregions,
most of the numbers of data pairs were fewer
than 15.
A computer program named VALID was written in
FORTRAN to compare the original ET
to the kriging
r
2
estimates ETK and to compute m
It could
R and OR'
also be used to test the goodness of kriginq estimation
as described in the next section,
3.4
Method for Testing Goodness of Estimation
The iterative technique described above could be used
to validate any semivariogram model.
including the case
in which the model was derived from a sufficient number
of data.
In the case of sufficient data.
what was
required was not to modify the semivariogram model by
iteration until
the optimum model was found.
but to test
whether or not
the estimation method was unbiased
(i.e.
did not result
in systematic errors).
According to
Delhomme
(1976),
the goodness of
the estimation method
could be verified
through two conditions:
1.
The mean of
the reduced errors.
m • must be
R
close to zero.
2.
The variance.
a~. of the reduced err~rs.
R(X
),
must be close to 1.
i
Verification of.
the
two conditions above meant
that the
estimation method was unbiased.
The test on the reduced
errors does not .reveal how close
the kriging estimates
.'
are to the observed evapotranspiration.
ET ,
Such
r

---

88
information could be obtained by testing the mean, mE'
of the deviations, ET
- ETK, for the null hypothesis:
r
HO:
mE = 0
against the alternative Ha:
mE # 0
It could be assumed that the data are statistically
paired, since ETK and ET
correspond to the same
-
r
location.
Program VALID includes the computation of ~
and o~, the variance of the deviations ET
- ETK.
r
The next chapter will give the results obtained when
following the procedures presented in this chapter.

---

89
4.
RESULTS AND DISCUSSION
This chapter presents results obtained using
procedures described
in the previous chapter with the
climatic data collected throughout the state of Oregon.
Occasionally,
intermediate results were needed to modify
previous assumptions and
to obtain more satisfactory
results.
4.1
Local Estimates of Reference Evapotranspiration
The local estimates of
referenc~ evapotranspiration
using the FAO modified Blaney-Criddle method 'were made
available for
175 locations.
The individual values of
monthly ET
given in mm/day at each location were
r
computed for May through September.
The results
for
the
month of June as 'an example are
listed
in Appendix B .
..
The weather data are shown for
the same month in Appendix
B.
In order
to
reveal
the existence of any trend of
ET
throughout
the growing season,
the monthly ET
r
r
were plotted in Fig.
9 for
five
locations,
each one
chosen inside one of
the five climatic sUbregions shown
in Fig.
6.
Those locations were Astoria
in sUbregion A,
Corvallis
in subregion B,
Medford in subregion C,
Hermiston in subregion D,
and Malheur Branch Station in
subregion E.
The plot confirmed the earlier results
'"
pUblished by CuenCd et al.
(1981)
which indicated the
parabolic shape of ET
variation throughout the
i;.
r

---

• ,c
•.
1
•
r
I;'
. f
\\ '
9.00
/.-.-._.~
8.00
/
.~
rvlalheur Branch
7.00
>,
IU
~:"
~.
/
-0
.O·~-
" .
-....
;;'.
.
.
~ 6.00
e/
it
c
--~............
, . - -
............
o
_ _ l:}
-.;:;
5.00
,.---
~
g-.-:-:"
.....................
"
'0. .......
IV
S-
Medford
.........{ t - _ _
.,.-
...-
0-
~
4.00
--...-
J:t-.--
IU
S-
Corvallis
~
+.J
r;>_ -
-
-
-
-
_ _ -I:]
_
o
' -EJ- - - - , - - -
0-
--~
~- -
-El
IU
3.00
::-
w
s--
Astoria
2.00
1.00
0.0
o
10
t 20
30
40
f 50
60
70
T80
90
100
f110
120
130
fi'40
May
June
Jul y
August
September
Days (May 1 = 1)
Figure 9.
Monthly variations of daily evapotranspiration throughout 1979
growing SE;ason.
'"
o

---

91
growing season.
Table 2 shows the monthly average of
daily reference evapotranspiration for weather stations
where all the climatic data needed were measured in each
of the 5 climatic sUbregions represented in Fig. 6.
These data plotted in Fig. 9 indicate that significant
variations in ET
could be expected at the same
r
location th~oughout the growing season.
For this reason,
it was decided to study the spatial variability of ET r
for each month separately.
Fig. 9 indicates a general
increase of reference evapotranspiration as one moves
from Astoria on the West Coast to Malheur Branch Station
in semi-arid Eastern Oregon.
4.2
Evaluation of the Stationarity Assumptions
It became clear from the literature review that first
order stationarity could be revealed by the plot of the
drift D(h) defined in Eq.2.27.
The plot of the drifts
are shown for June in Fig: 10.
They have a trend similar
to the ones shown by Vauclin et al.
(1982) for soil
surface temperature.
Fig. 10 is presented as an example
of the trend in the drift.
Such a trend also existed for
che other four months.
Adequacy of the assumption of
second order stationarity could also be made apparent if
the semivariogram function y(h) defined by Eq.2.35 was
plotted.
Figures 11 to 15 indicate plots of the
semivariograms for each month using data from all 175

---

. .
,
,. ~..
,,"
'I
I;,
. . \\'\\
Table 2.
FAO modified Blaney-Criddle monthly reference evapotrans-
piration (mm/day) for the climatic sUbregions in 1979.
Cl ima tic
SUbregions
Location
May
June
July
August
Sept.
A
Astoria
2.538
3.266
3.403
3.305
2.613
B
Corvallis Exp.
Sta.
3.910
4.878
5.784
4.461
3.866
C
Medford Exp.
Sta.
4.878
6.461
7.000
5.224
4.467
D
Hermiston Exp.
Sta.
4.973
6.244
7.241
5.776
4.289
E
Malheur Branch Sta.
5.796
8.387
8.638
6.708
4.009
I.D
N

---

,..
I'
.~
, "
' )
' J .
e,"
l
Ill'
•
<l
" ,
.
•
•
East-l~est
1.5
. / ......../
.----'.
1.
/
O.
......... / /" ---''''..1
0.5
•
- . - . . . . . . .
0
0
0
•
0
• • •
_
0 0
•
_
0
_
/
•
•
• • • •
o _ o . o o o o o
. , /
°
-
• •
0
_
0
_
0
°
>,
0.0
/ .
---------
~:=--... - - - - - - - -
•
rei
-0
.......- \\. .-........ I"
.........
E
,,/'
'\\.
',,,
E
-0.5
.
"
........ ,
.
+-'
"
.........
j\\
4-
'''-
'".
.
~
Cl
-1. O.
North-South
"""""'./\\
'-
. /
.........
.
I-
-1. 5
•
•
•
•
o
40
80
120
160
200
Distance (Miles)
Figure la.
Drifts in reference evapotranspiration for June 1979.
1..0
W

---

" .
4. .
'j'
..
f
, .
, ..
1.20
1.00
N .......
• " lC
0.88
I(
East-l'Jest (x)
re
..,.,
- ---
>,
-
~ 0.80
,.-
,.-
-
E
E
J
"
" , -
..
, /
... "
0.60
E
0.60
",,-
•
North-South ( .)
re
)'"
•
e. . _,; . . . . . . . . lA • • • • • • • • •
s....
...
0
..
CJ)
/
"
• •
•
/ l l
lAI
'S:: 0.40
/",
I
re
/
..
+
I
>
/
•
I
',-
..
I
E
,-
.
~ 0.20
/
. •
150
./ .'' •
/,' '"
120
/.
... .'. •
0.0
0
20
40
60
80
100
120
140
160
180
200
Di 5 tance (mil es)
Figure 11.
Semivariograms of reference evapotranspiration
for May 1979.
1.0
.l>o

---

..
'"
.,
_,
f
.~
2.40
10
2.20
It,
...
2.00
N
1.80
--.
1.82
East-West
(x)
~
>,
", 1t
" ,
'to
ru
" ,
'D
1.60
",
,
~
1-
E
'"
./
1.
o ·
•
E
~
1.40
'"
~
/
~
•
1.30
/
~
/
- - - - - - - North-South
(.)
,
E
"
.·t'
0
1.20
/
J'
ru
/
Il
....
.,
/
0'
/
"10
•
0
1.00
1 - ' ·
·rl
I
....
/1-
•
ru
>
0.80
...., ,
,,'
110
,. .
/
...
-
/
o.
o
•
:l!
/
U)
0.60
/ J' . '
,+
~
I
0.40
I
.'
/
." .
.. / ,,'
1.--- ..,.
0.20
170
1.1.0" •
.
,~
0.0
20
40
60
80
100
120
140
160 180 200
220
240
260 280
Distance
(miles)
Figure 12.
Semivariograms of reference evapotranspiration for
June 1979.
1.0
U1

---

"
.
.
t
',Cl
" l
\\ '
"
2.40
IC
2.39. East-West
(x)
l(
.-"
"
..-
, /
2.20
, /
.-
/
2.00
/
/
..
It
-
..
1.80
/
" It
-
I(
/
1. 70
( )
..-..
.. .. - - - - - North-South •
N
., /
•
~ 1.60
r1j
.
/
-
/
-0
•
•
I
~ 1.40
..
-
14,
•
I
-
---
---
/
E
1.20
..'JII'
ro
~
en
,I
0
-,
1.00
'r-
- ,
•
I
~
ro
'.
•I
>
~
/ -. ~
I
'E 0.80
- .
/
.
I
Q.J
I
Vl
I
.-
0.60
-
I
200
.I..
/
,a.
0.40
•
I .'
/ .' -
I ,-
0.20
/ .. .
/..
170
0.0
0
20
40
60
80
100
120
140 160
180 200
220
240
260
280
Distance (miles)
Fig4 r e 13.
Sernivariograrns of reference evapotranspiration for
July 1979.
I.D
0'1

---

.
.
.,
','.
..' \\;
n
•
&
"
11
..
K
1. 06
~ .. _----
East-West
(x)
...
" .
".
1L
....
/
/ .,./ .,. 11.
•
"
, .•...... ..•..
0.70
---- North-South
/ '
.' .. ,
.
(
)
..
/
"
, .
.'
•
"
" . "
11
•
11 /)1
"
. '
•I
/'
. '
,
/ " . .' "
•·
I
1&
'Y '".• .' ..
·;
180
/
.'
·
1L
/ . '
.'.
/
·
150
.,.
~. ...
•
o
20
40
60
80
100 120
140
160 180
200
220
Distance (miles)
Figure 14.
Sernivariograms of reference evapotranspiration
for August 1979.
~
---.l

---

. .
.
, ,
"_1'-
?, •.
:_l ". ",
~i·
"
'
\\'
0.70
•
I
•
0.60
~
• o ,
0.58
East-West
(x)
•
110
___.... __
... -.--
North-South (.)
.
N
, . " . ' )&
~ 0.50
'lI,.
, "
nJ
u
-,
, ,
..
J
,
E
E
0.40
1
•
,
•
' -
•
•
'li
,~
,
,
.-
'Il
, "
, "
•
E
1L
1I
.
re
"
l&
,
'to
~ 0.30
'lI
,
•
...
.
.f.
0
, "
•
I
,-
~
.-1.
I
re
"
I
>
...,
' ' -
0.20
,,'
•
I
•
I
E
1l • "
Y.
,
(l)
x "
•
I
VI
'"
,
.
"
• •
200
0.10
,
"" .II •
."
.
{'. "
•
0.0
L..4
I
0
20
40
60
80
100
120
140
160
180
200
220
Distance (miles)
Figur~ 15.
Semivariograrn of reference evapotranspiration for September 1979.
~
co

---

99
weather stations lumped together.
The procedure for
fitting these semivariogram models was described in the
previous chapter.
Other than September, each month
showed a zonal anisotropy, revealed by the difference
between the sill values of the north-south and the
east-west semivariograms.
Table 3 indicates these
differences as well as the ranges and characteristics of
the anisotropic models of semivariograms.
The plot of the drifts shown for June in Fig. 10
departed systematically from the zero axis.
This
indicated a lack of stationarity.
Obviously. kriging on
the basis of the derived semivariograms would not yield
unbiased estimates.
A certain error should be expected
on the average when each observati~n is intentionally
deleted and the kriging estimate performed for that
location using the neighboring observations.
This lack
of precision always occurs whenever a lack of first order
stationarity has beed ignored (Volpi and Gambolati.
1978).
However.
the most important criterion on which
the semivariogram models could be accepted was how well
the kriging estimates agreed with the observations.
More
details on this verification is given following
discussion of the kriging estimates.
The indicated semivariograms could be used to design
weather station networks for ET estimates.
This could be
done using the relationship which has been found,
through

---

.
••'11
•
.'
.!
..
....
t'
.. ~
Table 3.
Semivariogram characteristics
North - South
East - West
Month
Cy
ay(mi)
ay(km)
Cx
ax(mi)
ax(km)
May
0.60
120
192
0.88
150
240
June
1. 30
170
272
1. 82
170
272
July
1. 70
170
272
2.39
200
320
August:
0.70
150
240
1. 06
180
288
September
0.58
200
320
0.58
200
320
Cx and Cy = sill values «mm/day)2)
a x = range of influence in miles (mi) and in kilometers (km)
in the direction east-west.
ay
range of influence in miles
(mi) and in kilometers (km)
in the direction north-south.
/-'
o
o

---

101
the semivariogram.
between error and station density.
If
one is interest~d in del3igning a station network in order
to estimate ET with an average error of AET.
the
following can be done:
1)
Compute 1(h) = (6ET)2/2.
2)
Read the corresponding hEW value on east-west
semivariogram.
3)
Read the corresponding ~S value on the north-
south semivariogram.
4)
Oh the average.
the distances between adjacent
stations would be hEW and ~S'
Such a procedure assumed the stationarity of the drift
D(h) and of the semivariogram 1(h). as discussed in the
literature review.
When there is a lack of stationarity.
the station density is expected to vary from one
subregion to another.
(This can be verified if Table 3
is compared to Table 5 for the semivariogram
characteristics.
Table 5 corresponds to the subregional
semivariograms and is described in Section 4.6).
4.3
Kriging Estimates of Evapotranspiration
Kriging estimates of evapotranspiration were
performed at each unoccupied grid corner represented in
Figure B.
For each estimation.
the eight nearest weather
stations were selected.
The semivariogram characteristics
shown in Table 3 were used for kriging.
The geographical

---

102
coordinates of the weather stHtion locations and their
original evapotranspiration r(ltes were used to perform
the kriging estimates of the evapotranspiration and the
kriging variances at the unoccupied grid corners.
The
kriging estimates are listed in Appendix B for the month
of June, 1979.
It should be noticed that the kriging
estimate of ET is identical to the original ET wherever
the grid corner corresponds exactly to a weather
station.
The slight differences which might be noticed
when Figure 8, showing the coordinates of the locations,
is used to check this identity, came from errors due to
tranforming longitudes and latitudes into Cartesian
coordinates x and y.
It should also be noticed that all
the kriging estimates of ET lie between the lowest and
highest computed evapotranspiration rates at occupied
grid corners during the same month.
This is an
additional attribute of using kriging as an interpolation
technique.
4.4
Estimates of Kriginq Variances
Kriging techniques not only provide an estimation of
the regionalized variable,
but they also provide the
kriging variance for each location.
The kriging variance
was computed using Eq. 2.88 by the program KRIGX along
with the computation of kriging estimates, ETK, at each
grid corner of Fig. 8.
Whenever the grid corner

---

103
coincides with a weather station,
the kriging variance
O~(Xi) is identical to zero. On the contrary, O~(Xi)
is large when the grid corner is far away from the
weather stations.
These properties could be verified in
Appendix B, which lists the kriging variances for the
month of June, 1979.
The kriging variances near the borders of the state
of Oregon were high because they corresponded to
estimates which were made far from the weather stations.
This is consistent with the expectation that the error
should be high for any estimation made using observations
collected far away.
The square root of the kriging
variance is an estimator of the error made when using,
for a given location xo' the kriging estimator z*(xo)
instead of Z(X o), which could not be known unless
measuremen~s were made at that location.
4.5
Self-Validation of the Semivariogram Models
This test was described in Section 3.4 of the
previous chapter.
It is a means of evaluating how close
the kriging estimates,
based on the derived
semivariogram, are to the observations.
It is a test not
only for Z*(X ) but also for O~(Xo).
Table 4
o
gives the means of the reduced errors, m , and their
R
variances, a~, using the semivariogram models
represented by Figs. 11 through 15.
Table 4 is a summary

---

104
Table 4.
Results of statewide semivariogram validation
tests.
Month
. May
June
July
August
September
C
0.88
1. 82
2.39
1.06
0.58
x
C
0.60
1. 30
1. 70
0.70
0.58
Y
a
240
272
320
288
320
x
a
192
272
272
240
320
Y
m
-0.02
-0.01
-0.01
0.002
-0.01
R
2
0
1. 44
1. 09
1. 44
2.31
1. 80
R
'\\:
-0.01
-0.008
-0.009
0.00009
-0.006
2
0
0.22
0.28
0.45
0.27
0.13
E
ItEI
0.28
0.20
0.18
0.003
0.22
Cx and Cy are sill values in east-west and north-south
directions
«mm/day)2)
a x and ay are ranges of influence in east-west and
north-south directions
(10 3 meters)
mE and fiR are means of errors and reduced errors
(mm/day)
2
2
0E and oR are variances of errors
and reduced errors
ItEI
is the parameter of Student's t-test for ET r - ETK.

---

105
of the semivariogram models validation tests.
It shows
low values of ~' which are very close to zero, while
the values of a~, besides the month of June, are not
close enough to 1.0 to confirm a lack of bias.
On the
other hand,
the mean error, ~' of the deviation
ET
ETK are very small.
The results of Student's
r
t-test shows the mean ~ can be considered null at 5%
level of significance for each of the five months
analyzed during 1979.
The negative signs are indications
that kriging slightly underestimates ET
on the average.
r
4.6
SUbregional Models of Semivariogram
It was explained in the previous chapter how a
semiv3riogram model could be derived for the case where
only a few observations are available.
There were too
'few observations in each subregion,
taken separately,
to
compute a subregional semivariogram.
A computer program
named VALID was written in FORTRAN to compute the
parameters m
a~d a~ needed for the validation
R
tes t.
It also provided the mean, mE' and variance,
~~' of the differences, ET
- ETK.
The application
r
of the jacknifing technique to the ET data available for
June 1979 provided the results shown in Table 5 for the
five subregions.
Table 5 shows only the characteristics
,,,
of the subregional semivariograms retained as the
"best".
The term "best" semivariogram,
in this thesis.

---

...
11
of
_
I':
' ~'
j
Table 5.
Charactetistics of the subregional serni-
variograms derived by
jacknifing.
2
2
SUbregion
Cl
a
mR
oR
mE
ItEI
°E
A
0.138
52
-0.017
1. 005
-0.003
0.078
0.011
B
0.302
27
-0.072
1. 017
-0.04
0.266
0.078
C
0.137
12
-0.088
1.149
-0.04
0.175
0.096
D
0.65
73
-0.015
1. 008
-0.014
0.247
0.028
E
1. 57
150
-0.018
1.006
-0.026
0.380
0.042
Cl ~ Sill value «mm/day)2)
a = Range of
influence (miles)
fiR = Mean of the reduced ~rrors (mm/day)
2
.
OR = Varlance of the reduced errors «mrn/day)2)
mE = Mean of the deviation (mm/day)
?
0E = Variance of the deviation «mm/day)2)
ItEI
The parameter of Student's t-test for ET r - ETK
f-'
o
(j\\

---

107
is associated with the semivariogram for which mR and
a~ are the closest to zero and one. respectively.
The results of these tests also revealed the subregional
semivariograms were isotropic because the magnitudes of
their ranges of influence and sill values do not depend
on the direction.
Table 5 and the column for June in Table 4 reveal
that the variances, a~, for four out of the five
subregional semivariograms were closer to 1.0 than those
computed using the state-wide semivariogram.
This
indicated that. on the average, kriging with the
subregional semivariogram is less biased.
The Student's
t
parameters were at least three times lower for all
E
the subregional semivariograms than for the state-wide
semivariogram.
This is an indication that kriging
estimates using the subregional semivariograms are closer
to the original data values than kriging estimates using
the state-wide semivariogram.
Detailed results of the
subregional semivariogram validation tests which support
this conclusion are shown in Appendix C.
The state-wide
semivariogram for June results in the lowest variance of
reduced errors. a~. when compared to the other
stat~~wide semivariograms for~the other months.
Results
using the 6ubregional semivariograms for the other months
should therefore also be exp~cted to be better compared
with those obtained using the corresponding state-wide

---

108
semivariograms.
These results are partially explained by the
realization by Journel and Huijbregts (1978) that the
stationarity assumption is more and more verified as the
vicinity from where the data are taken to compute the
semivariograms and the drifts is reduced.
This attribute
was described in Section 2.2.2 of the literature review
as quasi-stationarity.
4.7
comparison of Semivariogram Models with Least Square
Models
The 175 computed reference evapotranspiration data
previously used for kriging have been regressed with
respect to the Cartesian coordinates, X, Y, and Z,
the
altitude of the weather stations.
A stepwise linear
regression analysis was performed starting with 9
parameters, X, Y,
Z, x2 , y2 , Z2, XY, XZ, and YZ.
Table 6 shows a summary of the model retained for each
2
month.
The highest coefficient of determination, R ,
was 0.76 and corresponded to June.
This indicates the
regression models cannot explain at least 24 percent of
the variabilities in ET.
The residuals means and
r
vari~nces are shown along with the regressions
coefficients in Table 6.
Comparing Tables 4 and 6 shows
h
.
2
f
k
.
.
t
at the varlances, 0E'
~r the
rlglng models are
lower than the ones found for the least squares models.

---

• 1';1
- "
~J
• i
--:"
",
.~'1'1
Table 6.
Linear regression models.
Regression
coefficient
~lay
June
July
August
September
b
3.062
3.484
3.582
3.819
2.667
1
b x
0.025
0.029
0.041
0.013
0.019
by
-0.0077
0
-0.0059
-0.0049
-0.0031
-5
5
-5
b xx
-2.89xlO-
0
-3.37xl0- 5
-3.02xl0
-3.85xl0
-5
b yy
0
-2.40xl0
0
0
0
-5
b
0
0
-3.15xl0
0
0
XY
-6
-6
-6
-6
b xz
-1. 88xl0
-1. 57xl0
-2.64xl0
-1. 51xl0
0
-
2
R
0.71
0.76
0.75
0.68
0.70
mE
0
0
0
0
0
2
0.24
0.37
0.59
0.32
0.16
°E
f-'
o
ID
;:,:
.' ....,

---

110
The mean error from the linear regression analysis was
always zero. which compared with the values of the mean
error close to zero in Table 4.
In addition.
kriging
models contain only two independent variables (X. Y).
while the least squares model for each month contains at
least 5 parameters.
For the reasons above. the kriging
model was preferred to the least squares model as a
better method for interpolating between the available
data.
Other traditional interpolation techniques. such as
moving average.
inverse distance. and inverse squared
distance methods. have not been compared to kriging in
this research work. since they are well known to be
biased (Henley. 1981; Journel and Huijbregts. 1978).
4.8
Comparision of Simple Kriging with Universal Kriging
Universal kriging. as explained in the literature
review.
is kriging which takes the non-zero drift into
account.
It has been noticed from Fig. 10 that the drift
is almost linear.
This allows to drop the second degree
. .
2
2
.
and InteractIon terms. X. Y • XY In Eq. 2.91.
The
results of the self-validation test. using such a kriging
system along with the semivariograms shown in Figs. 11
through 15. are presented in Table 7.
Comparison bf
these results to those obtained for simple kriging. shown
:.-
in Table 4.
indicates that simple kriging is a better

---

I I I
Table 7.
Self-validation of universal kriqinq models
for the state of Oregon in 1979.
Month
May
June
July
August
September
mR
-0.089
-0.095
-0.045
-0.060
-0.060
2
oR
1. 640
1. 304
1. 781
2.423
2.098
mE
-0.339
-0.050
-0.029
-0.025
-0.019
2
0.238
0.312
0.510
0.263
0.147
°E
ItEI
9.19
1.18
0.54
0.64
0.66
ffiR
Mean of the reduced errors
(mm/day)
o~
Variance of the reduced errors
«mm/day)2)
mE
Mean of the deviation ET r - ETK (mm/day)
o~
Variance of the deviation «mm/day)2)
It~1
The parameter of Student's t-test for ET r - ETK
.~.

---

112
model.
In fact.
the means of the reduced errors are
closer to zero for the simple kriqinq.
while their
variances are consistently closer to one.
The means and
variances of the deviations.
ET
- ETK,
are also
r
respectively closer to zero and one.
The Student's
t-tests mentioned in the previous chapter also favored
the simple kriqinq.
The Student's t, here tB'
is
calculated as follows:
(4.1)
where N is the sample size (N = 175).
Two main reasons could be applied to explain why
simple kriqing performed better than universal kriging.
while theory indicates it should be the opposite.
First.
the drift function.
D(h).
represents an average of the
difference.
z(x)
-
z(x + h).
over the state of Oregon
while h is maintained constant for each computed D*(h).
This averaging process
included different magnitudes of
z(x)
-
z(x + h).
However. when the eight nearest weather
stations are selected for
kriging.
most of these stations
could belong to a very homogeneous zone where the drift
is negligible.
Performing universal kriging using the
same drift function for
the whole state might remove too
much drift in some locations and too little drift in
other locations.
A second reason could be the use of a state-wide

---

113
semivariogram derived from data which are not detrended.
The question which arises is how to detrend a
semivariogram when the local drift is not known.
David
(1977) suggested an iterative technique which consisted
of applying the jacknifing technique described in Section
3.3 and using Eq. 2.90 for universal kriging.
This
technique is expected to be very expensive in computer
time.
Such a technique has been written into computer
programs such as BLUE PACK.
The cost of using such a
technique for plotting ET contour curves should be
evaluated in comparison with the gain in precision of the
estimates.
This cost and resulting precision should be
contrasted with that required for kriging with
subregional semivariograms.
The use of subregional
semivariograms for mapping over a region requires
additional research on how to define the semivariograms
that should be used for the transition zones between two
adjacent sUbregions.
4.9
Con~our Maps of Reference Evapotranspiration
The reference evapotranspiration rates shown in
Appendix B were estimated at unoccupied grid corners
using simple kriging on the basis of the Gemivariograms
shown in Figs. 11 through 15.
Chapter 3 discussed the
computer programs and routines used to generate the
iSO-ET
contour curves over the state of Oregon.
r
\\
\\

---

114
Figs. 16 through 20 show such maps.
As expected.
the
contour levels of ET
increase generally from the
r
relatively humid West Coast to semi-arid Eastern Oregon.
The general north-south orientation of the reference ET
curves agreed with the reference ET curves manually drawn
for California using lysimeter and pan evaporation data
(Cuenca et al., 1981).
Beside this general trend which
is consistent from May through September, there are
singularities which depend on the time of year and
topographical configurations.
South-east of Hood River
lies Mount-Hood where the reference evapotranspiration
rates are lower than expected from the general trend of
increasing ET
from West to East.
This same
r
singularity is noticed for the region which corresponds
to the Cascades.
It is also shown for the mountainous
region lying east and north of Baker.
The recorded daily
temperatures in that region were not very much different
from the adjacent zones where th0ET rates were higher.
The explanation for these low ET rates is due to the low
wind runs recorded at Union Experiment Station.
The
climatological data bUlletins showed relatively low wind
runs compared to nearby Pendleton throughout the growing
season.
This same explanation holds for Escadia mid-way
between Salem and Hood River.
The wind runs used for
that subregion were low and corresponded to the ones
~.
recorded at North Willamette Experiment Station.
Another

---

,r"
. .
..
"j'
.'
•
288
2L1 O
. I
'-v"
J 1
))/ noro
----
4a
,
,
~96
Vl
. r .
,
~
. , .
QJ
r -
'r-
~
-- 160
QJ
u
;::
ro
+.J
Vl
'r-
Cl
80
~~.
BUrns~nction
4.96
Fall 5
- .
0
o {
0
80
160
240
320
400
Distance (i'-'!iles)
f-'
f-'
V1
Figure 16.
Contour ma)J of reference evapotranspiration
(mm/day)
for May 1979.

---

·...
..
"
~'
"
I
~-+-
288tAstoria
~~'~------+-----
I
I
I
240
Newpor
In
QJ
.,....
:::E
160
QJ
U
C
ro
~
In
.,....
Cl
80
o o
80
160
240
320
400
Distance (Miles)
!-'
!-'
Figure 17.
Contour map of reference evaDotranspiration"i(m'rn~'diay)fo'r June 1979.
0'1

---

".
..
..
288
240
<J)
Cl;
......
'
~ 160
QJ
U
C
to
.;-'
<J)
'r-
o
80
0 L
0
80
160
240
320
400
Di s tan ce U1il es )
I-'
I-'
-J
Figure 18.
Contour map of reference evapotrans~iration (mm/day) for July 1979.

---

" .
..
"
\\'
'.
.
."
-}
24
---(f)Q)-r-
::E:
160
---Q)u
C
to
+.J
(f)
'r-
Cl
80
se,
0
s
-
. .
0 -0
80
160
240
320
400
Di s tan ce (Mi 1es )
I-'
I-'
ro
Figure 19.'
Contour map of reference evapotranspiration
(mm/day)
for August 1979.

---

.. l' .~
. .
240
i f 3.
osake r
'\\.;a 1e
f\\.
V1
(l)
3\\?1)
L
160
(l)
O
3 84
(4140
u
~
9:ugene
s::
~drnbnd
ro
-r.>
v;
.r-
Cl
80
.40
.96
Fall s
t ---,
-'"= / /
'\\
I
\\
l
I
J
t
o
I
I
I
I
o
80
160
240
320
400r
Distance (Miles)
f--'
f--'
Figure 20.
Contour map of refer~nce evaptranspiration (mm/day) for September 1979.
'-0

---

120
•
explanation could come from the high sensitivity of the
FAO-modified BlaneY-Criddle estimated rE!ference ET to
temperature (Allen and Brockway, 1982).
Since
temperature decreases with altitude,
in general,
this
change produces higher decreases in evapotranspiration
than the 1 percent per 100 meters upward correction
suggested by Doorenbos and Pruitt (1977).
The overall
change is therefore a decrease in reference
evapotranspiration with the altitude.
In the north of
the state,
the contour curves tend to depart from their
general north-south orientation because of the influence
of the Columbia Gorge, which is expected to increase the
relative humidity.
The contour maps are shown with few contour levels in
order to make the maps less crowded and thus easier to
read.
However.
the plotting routine was designed to
provide more contour levels if they are asked for.
The
changes in evapotranpiration rates from one month to
another are also revealed by the contour maps of ET.
The
maps, Figs. 16 through 20.
show that the reference
evapotranspiration rates for the same location increase
from May to July.
then decrease through September.
This
observation agrees with the reference evapotranspiration
tTends shown by Fig. 9.
For the reasons given above.
the contour maps appear
to be a good illustration of the general spatial
••

---

121
variability of evapotranspiration over the state of
Oregon.
It should be emphasi:~ed that the ET contour maps
shown represent the monthly means of the daily
evapotranspiration rates.
The use of these maps should
be restricted to the description of ET variability over
the state of Oregon on a monthly basis.
For project
design at a specific location.
i t is recommended to refer
to Appendix B where the kriging estimates of ET are given
at each 12.8 km by 12.8 km grid corner.
The values
indicated compare very well with the original ET data.
'while the maps are less precise because of the small
number of contour
levels and possibly the lack of
adjustment when computer plots of the contours were laid
over a separate drawing of the state of Oregon.
Beside
those manipulation errors,
the contour maps could be used
to demonstrate.
better than any other work done so far.
the spatial variability of the monthly average of the
daily evapotranspiration rate over the state of Oregon
during the months and year indicated.
4.10
Contour Maps of Kriging Variances
Kriging variances were computed by program KRIGX along
with the kriging estimates of ET.
The variances were
plotted by computer and are shown as contour maps of
.-
kriging variances
in Figs.
21 through 25.
These maps
show the distribution of estimation variances over the

---

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..
.'.
. .. "
' I
..
I
24
Vl
<lJ
-...... 160
:E
~
ClJ
U
C
tU
+-'
Vl
......
Cl
80
0 I-
0
80
160
240
320
400
Distance (Miles)
t-'
N
Figure 21.
Contour map oE kriging variance ((mm/day)2)
for May 1979.
N

---

.
~
,.
t.tll
>of
11
., ,
.
"
"
240
0 I
0
80
160
240
320
400
Di stance (~1il es)
I--'
2
Iv
Figure
22.
Contour map of kriging variance
( (mm/day)
)
for June 1979.
w

---

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'.
~.
, .
240
. (/)
cv
r -
.,....
~ 11~O
Q)
u
C
1'0.
+-'
(/)
.,....
a
80
o....'--------.-----
o
I
Distance (Miles)
I-'
Figure 23.
N
Contour map of kriging variance ((mm/day)2)
for JUly 1979.
,j::.

---

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. .
•
,J
. , " "
...:
.
240
Vl
llJ
-
'r"
-160
:E
-(l)u
C
It!
.....
Vl
.~
Cl
80
o
o
80
160
240
320
400
Distance (Miles)
~
N
U1
Figure
24.
Contour map of kriging variance
((mm/day)2)
for August 1979.

---

'"
~ 'i.\\~
.'.
..
~
•
~. ~ I
:.I
.
i
240
Vl
Q..i
r -
'r-
:2:::
....... 160
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u
C
l\\J
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Vl
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80
01
' -
I
o
8~
1En
~~n
~An
AXn
Di stance Ulil es)
~
Figure =:5.
N
Contour map of kriging variance
2
((mm/day)
)
for September 1979.
O't

---

127
state.
The kriqinq variances. which are the estimates of
i
the squared errors.
increase from May to July and then
decrease through September.
The variance levels are
generally low where there is a combination of high
station density and low heterogeneity.
However. because
simple kriging (which assumes stationarity) was used. the
variances depended only on the station density.
SUbregions of high station density such as the Willamette
Valley are covered by low contour levels while those near
the state boundaries such "as in the north-east and
south-east where the station densities are low. exhibit
/
higher contour\\levels of kriginq variances.
The interest
of such maps of contour va~iances is to show how much the
estimation could be improved when additional weather
stations were added to the network.
This subject was
discussed in detail by Huqhes and Lettenmaier (1981).
Few contour levels are shown to make the maps easy to
read.
More detail is provided by Appendix B. where the
numerical values of kriqing variance are listed.
The shapes of the variance contour curves were
conserved from one month to the next while they changed
for the evapotranspiration rates through the same
period.
This consistency in the shape of the variance
contour curves is an indication that the relative changes
in kriging variances do not depend on the absolute values
of the observations. but rather on the station density

---

128
and the semivariogram model.
The months which have
higher sill values produced,
in general, higher contour
levels of kriging variances.

---

129
5.
CONCLUSIONS AND RECOMMENDATIONS
5.1
Conclusions
Weather data were collected from 175 stations over
the state of Oregon during the 1979 growing season.
These data were used to estimate the monthly reference
evapotranspiration rates, ET , by the FAO-modified
r
Blaney-criddle method.
This ET
estimation method was
r
selected among others because of its compatibility with
the weather data available throughout the state of Oregon.
The 175 ET
rates were used to compute
r
semivariograms to which spherical (or Matheron's)
semivariogram models were fitted.
For all months but
September,
the semivariograms were anisotropic and
revealed that the rate of change in ET
was higher in
r
the north-south than in the east-west directions.
These
semivariogram functions described the relationship
between the reference evapotranspiration variability and
the distance between the weather stations.
For each month,
the semivariogram models were used to
estimate the reference evapotranspiration rates,
by
kriging technique. at locations where no weather data
were available.
Such a technique was verified to provide
better estimates of ET
rates than the ordinary linear
r
regression models.
The comparison was made on the basis
of the number of parameters included in the model, and

---

130
the mean and variance of the residuals.
The use of jacknifing to test the model validation
revealed that simple kriging performed better than the
universal kriging.
The kriging estimates and the estimation of kriging
variances were made at approximately 1.600 locations over
the state of Oregon.
These estimates were translated
into contour maps of evapotranspiration Which agreed with
the general distribution of the climate over the state of
Oregon.
The contour curves of the kriging variance also
agreed with the change in the weather station density.
Subregions of low station density corresponded to high
contour levels of kriging variance and vice-versa.
5.2
Recommendation for Future Research
The results of this research could be improved by the
use of sUbregional semivariograms for estimating the
reference evapotc;Ospiration rates within each
sUbregion.
Such a method would incorporate a means of
modifying the subregional semivariograms for their use in
the transition zones between sUbregions.
The subregional
semivariograms derived by jacknifing for June constitute
the initial approach which must be improved.
This thesis assumed the constancy of wind velocity
inside each zone of influence.
A relationship could be
found between the few secondary weather data and the much

---

131
larger number of temperature data available.
Additional
research is required to express such a relationship.
Cokriging might be a way to approach this problem.
Cokriging is an unbiased estimation method of poorly
sampled variables, using the spatial correlation between
these variables and the ones abundantly sampled.
In any
case. an effort could be made to add automatic recording
weather stations which could be moved from one location
to another and even to remote sites.
Other traditional interpolation techniques such as
moving average,
inverse distance and inverse squared
distance interpolation methods could be used and compared
to the results of the semivariogram models.
These
methods have the weakness of assuming a priori a spatial
variability function.
However.
they have the advantage
of being simpler than the kriging technique.
Their use
to obtain estimates of secondary weather data at primary
weather stations may improve the local estimates of the
reference evapotranspiration at those stations.
Additional research is required on the relationship
between geographical configuration and the stationarity
of the drift. and the semivariogram before the kriging
variances could be used for designing station density.
The general approach has been explained assuming
stationarity of the drift and stationarity of the
semivariogram.
Any efficient design of weather station

---

132
density. using such an approach. should be based on a
long-term mean semivariogram curve and take into account
its calibration based on the local topographical
characteristics.
The ultimate goal of this research was efficient.
timely. computerized plotting of evapotranspiration
contours over large regions.
Such contours could be
drawn for different probabilities based on long
historical records of meterological variables for water
resources system and irrigation system design.
Contours
could also be drawn for data collected in the previous 24
to 48 hours for irrigation scheduling.
The application
of a method which accounts for the spatial structure of
evapotranspiration over a region could prove to be
beneficial for improving the productivity in irrigated
agriCUlture and water resources development.
i~
..

---

133
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f'~
., .

---

;
APPENDICES

---

139
I
APPENDIX A
l~IN:
Program for estimating the local reference
evapotranspiration rates at weather stations.
VARIO:
Program for computing and plotting the
semivariograms.
KRIGX:
Program for interpolating by kriging the
reference evapotranspiration rates between
weather stations .
.."

---

140
PROGRAM MAIN(INPUT,OUTPUT,TAPE1,TAPE3)
C
c
..
C
C
PROGRAM NAME: MAIN
C
WRITTEN BY
: KODJO AMEGEE, GRADUATE STUDENT
C
AT OREGON STATE UNIVERSITY
C
C
THIS PROGRAM COMPUTES THE FAO MODIFIED BLANEY
C
AND CRIDDLE REFERENCE EVAPOTRANSPlRATION ESTIMATES
C
ALONG WITH THE CONVERSION OF THEIR SITES LONGITUDES
C
AND LATITUDES INTO CARTESIAN COORDINATES.
C
C INPUT VARIABLES:
C
ID
: WEATHER STATION NAME
C
IYEAR, MONTH: YEAR AND MONTH OF DATA COLLECTION
C
TMAX,TMIN: MAXIMUM AND MINIMUM DAILY TEMPERATURES IN
C
DEGREES FARENHEIT
C
RHMIN
MINIMUM DAILY RELATIVE HUMIDITY IN PERCENT
C
RS
DAILY GLOBAL SOLAR RADIATION IN LANGLEY
C
WRUN
DAILY WIND RUN IN MILES/DAY
C
HWIND
ANEMOMETER HEIGHT IN METERS
C
WRTIO
RATIO OF DAY BY NIGHT WIND RUNS
c
ALT
ALTITUDE OF WEATHER SATION IN FEET
c
LAT
WEATHER STATION LATITUDE
c
LONG
II
"LONGITUDE
c
C OUTPUT VARIABLES:
C
X
CONVERSION OF LONG INTO CARTESIAN COORDINATE IN FEET
C
Y
II
LAT"
11
"
C
ET
EVAPOTRANSPIRATION ESTIMATE IN MM/DAY
C
ell
.
C
REAL LAT,LONG,NRATIO
oIMENS ION BB ( 6 ,6 , 6) , PP ( 11 , 12) , RRA N( 11 , 12)
DIMENSION ET(175),X(175),Y( 175)
READ(1,100) ZONE,CWIND,CSOLAR
100 FORMAT(2A10/F6.0,T16,F5.0,/)
WRITE(3,150)
150 FORMAT( r STI\\T
DATE
F
ET
X
Y
+
ALT t )
DO 22 I =1, 175
READ( 1,200) [D,IYEAR,MONTH,TMAX,TMIN,RHMIN,RS,WRL~,
+
ilWIND,WRTIO,ALT,LAT,LONG
200 FORMAT(A7,212,2F5.0,4F6.0,F5.0,2F6.0,F7.0)
C
PRINT*,TMAX,TMIN,LAT,MONTH
.
TAVG=(TMAX+TMIN)/2.
~

---

141
TAVG=(TAVG-32.)/1.8
RHIAG=RHMIN
U2MAVG=WRUN*WRTIO/(WRTIO+1.)*CWINO*1000./3600./12.*
+(2.0/ID~IND)**0.22
C TABLE OF MONTHLY PERCENT OF DAYLIGHT pp TO COMPUTE P AND THEN F
DATA PP
1 I
.267,.264,.261,.257,.252,.246,.239,.231,.220,.209,. 195,
2
.269, .268, .266, .264, .261 , .257, .253, .248, .243, .236, .228,
3
.269, .269, .269, .269, .269, .269, .268, .268, .268, .267, .266,
4
.269, .270, .272, .275, .278, .282, .286, .291 , .297, .303, .310,
5
.271 , .273, .276, .281 , .287, .294, .303, .312, .322, .334, .346,
6
.274, .280, .285, .291, .298, .307, .316, .328, .341, .355, .371,
7
.275, .281, .287, .293, .299, .305, .313, .321, .330, .341, .354,
8
.274, .278, .282, .287 , .291 , .295, .300, .304, .309, .315, .322,
9
.271,.277,.280,.281,.281,.281,.281,.281,.281,.281,.281,
A
.270, .269, .268, .267, .264, .261, .258, .254, .250, .245, .240,
B
.269, .267, .264, .260, .254, .247 , .240, .231 , .222, .211 , .200,
C
.268, .264, .262, .257, .250, .242, .232, .221, .209, .195, .180 I
LL=INT(LAT/5)*5
IF(LAT.GT.50.0) LL=50
L1=LL/5+1
L2=L1+1
IF(L2.GT.11) L2=11
FACP=(LAT-LL)/5.0
P=PP(L 1 ,MONTH)"+FACP*(PP(L2,MONTH)-PP(L 1 ,MONTH))
C
PRINT*,L1,MONTH,FACP,L2,P
F=P*(0.46*TAVG+8.13)
RS=RS*CSOLAR
C
THIS IS TO CALCULATE RA THEORETICAL SOLAR RADIATION
C
REACHING EARTH SURFACE IN ABSENCE OF ATMOSPHERE
C
rOR 12 MONTHS AND LATITUDE 0 TO 50 NORTH
DATA RRAN
1 I
15.8, 14. 1, 13.2, 12. 2, 11 .2, 10. 1 ,08.9 ,07 . 6 ,06 . 4 ,05. 1 ,03.8,
2
15."" 14.9, 14.3, 13.5, 12. 7 , 11.7, 10.7 ,09.6,08.5,07 . 3,06. 1 ,
3
15. 7 , 15.6 , 15. 3, 14 .9 , 14. 4, 13. 7 , 13.0, 12.2, 11 . 3, 10.3,09.3,
4
15.3,15.5,15.6,15.7,15.6,15.5,15.2,14.7,14.2,13.5,12. 7,
5
14.4,15.0,15.5,16.8,16.3,16.4,16.5,16.4,16.3,16.1,15.7,
6
13.9,14.6,15.2,15.8, 16.3, 16.7"7.0,17.~,17.3,17.3, 17.2,
7
14.1,14.7,15.3,15.8,16.3,16.6,16.7,16.0,16.7, 16.6, 16.4,
8
14.6,15.2,15.5,15.8,15.9,15.8,15.7,15.5, i5.1, 14.6, 14.0,
9
15.3,15.3,15.3,15.1,14.8,14.5,13.9,13.2,12.5,11.7, 10.9,
A
15.4,15.1,14.6,14.1,13.4,12.6,11.7,10.7,09.6,08.5,07.2,
B
15.1,14.4, 13.6,12.7,11.7,10.6,09.5,08.2,07.0,05.6,04.3,
C
14.8,13.9,13.0,11.9,10.3, 9.5,08.3,07.0,05.7,04.3,03.9 /
FACR=(LAT-LL)/5.0
RA=RRAN(L1,MONTH)+FACR*(RRAN(L2,MONTH)-RRAN(L1,MONTH))
NRATIO=2.0*(RS/RA-O.25)
IF(NRATIO.GT.1.0) NRATIO=0.999
IF(NRATIO.LT.O.O) NRATIO=O.O
C THIS SECTION INTERPOLATES ET1 USING A,B,AND F

---

142
:
DATA BB
1 I 0.84.0.80,0.74,0.64,0.52,0.38, 1.03,0.95,0.87,0.76,0.63,0.48,
2
1.22,1.10,1.01,0.88,0.74,0.57, 1.38,1.24,1.13,0.99,0.85,0.66,
3
1.54.1.37,1.25,1.09,0.94,0. 75, 1.68,1.58,1.36,'.18,1.04,0.84,
4
0.97,0.90,0.81,0.68,0.54,0.40, 1.19,1.08,0.96,0.84,0.66,0.50,
5
1.41,1.26,1. 11,0.97,0.77,0.60, 1.60,1.42,1.25,1.09,0.89,0.70,
6
1.79, 1.59,1.39,1.21,1.01,0.79, 1.98,1.74,1.52,1.31,1.11,0.89,
7
1.08,0.98.0.87,0.72,0.56,0.42, 1.33,1.18,1.03,0.87,0.69,0.52,
8
1.56,1.38,1. 19,1.02,0.82,0.62, 1.78,1.56,1.34,1.15,0.94,0.73,
9
2.00,1.74,1.50,1.28,1.05,0.83, 2.19,1.90,1.64,1.39,1.16,0.92,
A
1.18,1.06,0.92,0.74,0.58,0.43, 1.44,1.27,1.10,0.91,0.72,0.54,
B
1.70, 1.48, 1.27, 1.06, 0.85, 0.64, 1.94, 1.67, 1.44,1 .21 , 0.97 ,0.75,
C
2. 18, 1.86, 1.50, 1.34, 1.09, °.85, 2.39,2.03, 1.74, 1.46, 1.20, 0.95,
o
1.26, 1.11,0.96,0.76,0.60,0.44, 1.52,1.34,1.14,0.93,0.74,0.55,
E
1.79,1.56,1.32, 1.10,0.87,0.66,2.05,1.76,1.49,1.25,1.00,0.77,
F
2.30,1.96, 1.66,1.39,1.12,0.87, 2.54,2.14,1.82,1.52,1.24,0.98,
G
1.29,1.15,0.98,0.78,0.61,0.45, 1.58,1.34,1.17,0.96,0.75,0.56,
H
1.86, 1.61 , 1.36, 1. 13, 0.39, °.68, 2. 13, 1.83, 1.54, 1.28, 1.°3, 0.79,
I
2.39,2. °3, 1.71 , 1.43, 1. 15, 0.89, 2.63,2.22, 1.86, 1.56 , 1.27 , 1.00 I
XX=RHIAG
n=NRATIO
Z=U2MAVG
c
PR1NT*,XX,n,z
1l=INT(XX/20. )+1
12=11+1
1F(12.GT.6) 12=6
J1=INT(YY/O.2)+1
J2=Jl+1
1F(J2.GT.6) J2=6
K1=INT( V2)+ 1
K2=Kl+1
IF(K2.GT.6) K2=6
1F(Kl.GT.6) K1=6
Xl=(1l-1)*20
X2=( 12-1 )*20
Yl=(Jl-1)*0.2
Y2=(J2-1)*0.2
Zl=(Kl-1)*2
Z2=(K2-1)*2
fACX=O.O
FACY=O.O
FACZ=O.O
1F(Kl.NE.K2) PACZ=(Z-Zl)/(Z2-Z1)
Cl1
= BB(11,J1,K1)+FACZ*(BB(Il,Jl,K2)-BB(Il.Jl,Kl))
C12
= BB(11,J2,Kl)+FACZ*(BB(1l,J2,K2)-BB(Il,J2,Kl))
C21
= BB( 12,Jl ,Kl hFACZ*(BB(I2,Jl ,K2)-BB( I2,Jl ,Kl))
C22
= BB(I2,J2,Kl)+FACZ*(BB(I2,J2,K2)-BB(I2,J2,Kl))
C
PRINT*,PP(3,S),RRAN(10,8),8B(3,4,5)
C
PRINT*,Xl,Yl,Zl
IF(Jl.NE.J2) FACY=(YY-Yl)/(Y2-Yl)
.,

---

143
IF(I1.NE.I2) FACX=(XX-Xl)/(X2-Xl)
C
PRINT*,FACZ,FACY,FACX,Cl1,C12,C21,C22
Dl
= C1l +.FACY*(C12 - C11)
D2
= C2l + FACY*(C22 - C21)
BP = Dl+FACX*(D2-Dl)
AP=0.0043*XX-YY-l.41
C
PRINT*,U2MAVG,D1,D2,BP
ET(I)=AP+BP*F
C
CORRECTING FOR ALTITUDE
ET(I)=ET(I)*(1.+ALT*O.305/10000.)
C CONVERSION OF LATITUDE AND LONGITUDE INTO
C
X AND Y COORDINATES (WITH SALEM AS ORIGIN
C
SLONGO AND
SLATO)
IN MILES·
SLATO = 42.0
SLONGO = 124.50
LATO=INT(SLATO)
LONGO=INT(SLONGO)
NLAT=INT(LAT)
tfLONG=INT(LONG)
IF(NLAT-45) 101,102,103
101 FLAT= 68.093 + 0.061*(NLAT-40)/5.0
FLON= 53.063 - 4.068*(NLAT-40)/5.0
GOTO 104
102 FLAT= 69.054
FLON= 48.995
GOTO 104
103 FLAT= 69.054 + 0.06l*(NLAT-45)/5.0
FLON= 48.995 - 4.443*(NLAT-45)/5.0
104 X(I)=-FLON*«NLONG-LONGO)+(LONG-SLONGO+LONGO-NLONG)*100./60.)
Y( I) =FLAT*( (NLAT -LATO)+( LAT -SLATO+LATO-NLAT) * 100 ./60. )
WRITE(3,400) ID,IYEAR,MONTH,F,ET(I),X(I),Y(I),ALT
400 FORMAT(A6,2I2,5F10.3)
22 CONTINUE
STOP
END
.>

---

144
PROGRAM VARIO(INPUT,OUTPUT,TAPE2,TAPE3)
c
c ·
.
c
c
PROGRAM NAME: VARIO
c
WRITTEN BY
: KODJO AMEGEE, GRADUATE STUDENT
c
AT OREGON STATE UNIVERSITY.
c
c
SEMI-VARIOGRAM IN TWO DIMENSIONS
c IRREGULAR GRID.THERE HAY BE MISSING DATA.
c CALCULATION BY CLASS OF ANGLE AND DISTANCE.
cC •.••PARAMETERS
c VR(ND)
DATA ARRAY
C X(ND),Y(ND)
X AND Y COORDINATES OF POINTS
C NO
NUMBER OF POINTS
C KMAX
MAXIMUM NUMBER OF COMPUTATION LAGS
C PAS
LENGTH OF BASIC LAG
C
DP
WIDTH OF DISTANCE CLASS. IF DP=O.
-)
C
DP=PAS/2. IS TAKEN
.
C NDI
NUMBER OF DIRECTIONS
)
C ALP(NDI)
ANGLES DEFINING DIRECTIONS (WITH
C
RESPECT TO X AXIS IN DEGREES)
C DA
WIDTH OF ANGLE CLASS. IF DA=O. THEN
C
DA=45. DEGREES IS TAKEN
C NC(KMAX*NDI) NUMBER OF COUPLES/LAG/DIRECTION
C G(KMAX*NDI)
VARIOGRAM VALUES/LAG/DIRECTION
C D(KMAX*NDI)
AVERAGE DISTANCE/LAG/DIRECTION
C U
AVERAGE
/
C V
VARIANCE / OF DATA .GT.TEST
C N
NUMBER
C
C · ... OPTIONS
C IS.NE.l RESULTS ARE PRINTED
C
C · ... COMtl.iONS
C
C TEST
INFERIOR BOUNDARY OF EXISTING DATA
C
IF VR.LE.TEST MISS[NG OR ELIMINATED DATUM
C
C
C ................................................................................
C
·.
·.....,
· .

---

145
C DIMENSIONS AND PARAMETERS SPECIFICATIONS
REAL U,V,H
INTEGER N,ND,KMAX,KD,KR
Q1ARACTER REGION*30,STA*20,HEAD*4,TICK4*4,TICK5*5
CHARACTER*1 STAR,DOT,PLUS,BLANK
CHARACTER LlNE1(0:44}*45,LINE2(-10:10}*21
PAP~METER(KMAx=62,ND=175)
DIMENSION SCALE(0:11},D(KMAX},KMAX1(4},CAN(4},SAN(4},ALP(4}
DIMENSION G(KMAX},DR(KMAX},NC(KMAX},VR(ND},X(ND},Y(ND}
DATA (ALP(KD},KD=1,4) 10.,45.,90.,-45.1
BLANK=' ,
DOT=' . '
STAR='·'
PLUS='+'
TICK4='+ ••• '
TICK5=' +•••• '
C
PRINT*,'PAS=?,DP=?,DA=?,TEST=?,IS=?'
READ*,PAS,DP,DA,TEST,IS
PRINT*,'REGION NAME?'
READ(*,B}REGION
B FORMAT(A30}
PRINT*,'REGION LIMITS:XW,XE,YS,YN'
READ*,XW,XE,YS,YN
C
C
INITIALIZE
C
PI=3.14159265
IF(DA.LE.0}DA=45.
IF(D~.LE.0}DP=PAS/2.
C
DO 9 KD=1,4
ALPHA=PI*ALP(KU)/180. ~,.
CAN(KD)=COS(ALPHA)
SAN(KD}=SIN(ALPHA)
9 CONTINUE
THETA=PI*DA/180.
CDA=COS(THETA}
C READING IN THE ET DATA
NCOUNT=O
READ(2,99) HEAD
99 FOfU-1AT(A)
DO 5, 1=1 , 175
READ(2,100) STA,ET,XX,YY
100
FOR~AT(A20,3(Fl0.3))
IF(XX.LE.XE.AND.XX.GE.XW) THEN
IF(YY.LE.YN.AND.YY.GE.YS) THEN
NCOUNT=NCOUNT+1

---

146
':-,
VR(NcormT) =ET
l{ ( NCCUHT ) =XX
Y(UCOUNT) =YY
ELSE
ENDIF
ENDIF
5 CONTINUE
C
C INITIALIZATIONS CONTINUE
C
N=l
U=VR(l)
V=VR(l)*VR(l)
c
C COMPUTATIONS FOR EACH DIRECTION
C
KMAX1(1)=(XE-XW)/PAS+l
KMAX1(3)=(YN-YS)/PAS+1
DMAX=SQRT«XE-XW)*(XE-XW)+(YN-YS)*(YN-YS»
KMAX1(2)=KMAX1(4)=DMAX/PAS+l
DO 50,KD=1,4
DO 40, KK= 1,KMAX 1( KD )
NC(KK)=O
D(KK)=O.
OR(KK)=O.
G(KK)=O.
40
CONTINUE
DO 30,I=1,NCOUNT
11=1+1
IF(I1.GT.NCOUNT) GOTO 30
IF(VR(Il).LE.TEST) GOTO 30
IF(KO.EQ.l) THEN
N=N+l
U=U+VR(Il)
V=V+VR(Il)*VR(Il)
ELSE
ENOIF
DO 20,J=Il,NCOUNT
IF(VR(J).LE.TEST) GOTO 20
OX=X(I)-X(J)
OY='{( I)-Y(J)
H=SQRT(OX*OX+DY*OY)
IF(H.LT.l.E-03)GOTO 20
COSD=(DX*CAN(KD)+DY*SAN(KD»/H
COSD1=ABS(COSD)
IF(COSD1.GT.CDA) THEN
REST=MOD(H,PAS)
..:.:..
KR=H/PAS

---

147
IF(KR.LT.KMAX1(KD»
THEN
IF(REST.LE.3.*DPI2.AND.REST.GE.DPI2.)THEN
IK=KR+1
ELSE
GOTO 20
ENDIF
IF(COSD1.EQ.0.)THEN
VRR=VR(I)-VR(J)
ELSE
VRR=COSD*(VR(I)-VR(J»/COSD1
ENDIF
NC( IK) =NC( IK)+1
D(Il<:)=D(IK)+H
DR(IK)=DR(IK)+VRR
G(IK)=G(IK)+.5*VRR*VRR
ELSE
GOTO 20
ENDIF
ENDIF
20
CONTINUE
30
CONTINUE
C
C PRINT ALL THE ~~AX1 RESULTS FOR EACH DIRECTION
-t
C
·r
IF( KD. EQ. 1)THEN
")
V=(V-U*U/N)/N
U=U/N
\\
I
WRITE(3,2000) REGION,XW,XE,YS,YN
WRITE(3,2001) U,V,N
WRITE(3,2003 )
ELSE
ENDIF
IF(V.GT.0.5.AND.V.LE.1.0) SINCR=0.10
IF(V.GT.1.0.ANO.V.LE.1.5) SINCR=0.15
IF(V.GT.1.5.AND.V.LE.2.0) SINCR=0.20
IF(V.GT.2.0.ANO.V.LE.2.5) SINCR=0.25
00 35,.J=O,11
35
SCALE(J)=J*SINCR
WRITE(3,3001) KO, (SCALE(KK) ,KK=O,11)
WRITE(3,3011)(TICK4,J=1,11),(TICK5,L=1,4)
00 45,IK=1,KMAX1(KD)
IF(NC(IK).GT.1)THEN
D(IK)=D(IK)/NC(IK)
DR(IK)=OR(IK)/NC(IK)
G(IK)=G(IK)/NC(IK)
ELSE
ENDIF
:.., ...~"
IF(NC( IK) .GE.1.AND.G( rK) .LT .11. *SINCR)Tm~N

---

148
00 43,J::O,44
LINEl (.1) ::BLANK
LINE 1«()) ::ooT
43
CONTINUE
DO J~4, J =-10, 10
LINE2(,J) =BLANK
LINE2(-10)=LlNE2(0)=LIlffi2(10)=DOT
44
CONTINUE
HG=G( IK)iHlO. I (SINCR* 10. )+.5
IF(MG.GT.44) MG=44
LINE1 (MG}=STAR
IF{DR(IK}.EQ.O.)THEN
MD=O
ELSE
MD=DR(IK)*5.+.5*DR(IK)/ABS(DR(IK))
ENDIF
IF(MD.GT.l0)MD=10
IF(MD.LT.-10)MD=-10
LINE2(MD)=STAR
WRITE(3,3003)IK,NC(IK),D(IK),G(IK),IK,
1
LINE1,DR(IK),IK,LINE2
ELSE
WRITE(3,j002)IK,NC(IK),D(IK),G(IK),IK,DR(IK),IK
ENDIF
45
CONTINUE
50 CONTINUE
C
PRINT IF IS.NE.O
IF(IS.EQ.O) GO TO 6
JOOl FORMAT(II, 'DIRECTION',I2,24X,12(Fj.l,lX),11X,
1'-2.0 -1.0
0.0 +1.0 +2.0')
3011 FORMAT(j4X, '0 ',11A4, '+', 15X,4A5, '+')
c
2000 FORMAT(1H ,5X,AjO,2X,'IRREGULAR GRID 2 DIMENSIONS' ,3X,
l'ET .. SEMI-VARIOGRAM' ,jX,'AND DRIFT'/1H ,5X,'XW = ',F7.2,2X,
2' XE :: I, F7 . 2, 2X, I YS = I, F7 . 2, 2X, 'YN = ',F7. 2, 8X,
j'******************' ,jX, '*********')
2001 FORMAT(lH ,'AVE.!.AGE = 1,F7.4,6X,'VARIANCE = '
1,F7.4,' NUMBER OF DATA:: ',IS)
200j FORMAT(1H,' LAG
NC
AVD',6X,'1/2VAR',55X,'DRIFT
LAG')
JOOj FOP~AT(lH ,Ij,2X,I4,?X,F7.j,2X,F6.j,SX,Ij,1X,45Al,2X,F6.j,
ljX, I j, 1XI 21 A1)
j002 FOfu~AT(lH ,Ij,2X,I4,2X,F7.j,2X,F6.j,SX,Ij,lX, '.',
146X,F6.j,3X,IJ, lX,' .')
C
6 STOP
END
.."

---

149
PROGRAM KRIGX{INPUT,OUTPUT,TAPE5,TAPE6)
c
c
.
c
C
PROGRAM NAME: KRIGX
C
WRITTEN BY
: KODJO AMEGEE, GRADUATE STUDENT
C
AT OREGON STATE UNIVERSITY
C
C
THIS PROGRAM PERFORMS KRIGING ESTIMATES AND KRIGING
C
VARIANCES OF EVAPOTRANPIRATIONS AT GRID CORNERS
C
USING FEW OBSERVATIONS, SEMIVARIOGRAM CHARACTERIS-
C
TICS, AND GEOGRAPHICAL COORDINATES.
C
C INPUT VARIABLES:
C
X(ID)
EAST-WEST DISTANCE FROM A REFERENCE
C
Y(ID)
NORTH-SOUTH"
FROM THE SAME
"
C
C INPUT PARAMETERS:
C
NSAMP
NUMBER OF SAMPLES FOR EACH ESTIMATION
C
SILLX
SILL VALUE FOR EAST-WEST DIRECTION
C
SILLY
SILL VALUE FOR NORTH-SOUTH DIRECTION
C
RANGX
RANGE FOR EAST-WEST SEMIVARIOGRAM
C
RANGY
RANGE FOR NORTH-SOUTH SEMIVARIOGRAM
C
NDAT
TOTAL NUMBER OF OBSERVATIONS
C
NSYS
DIMENSION OF THE KRIGfNG LINEAR SYSTEM
C
C OUTPUT VARIABLES
C
ETK(IY, IX)
KRIGfNG ESTIMATE OF "ET" AT (IY,IX)
C
ERK(IY,IX): KRINGING VARIANCE AT GRID CORNER
C
IDENTIFIED BY THE ID. ~NMBERS (IY,IX)
C
C SUBROUTINES AND FUNCTIONS
C
GAM(
): COMPUTES SEMIVARIOGRAM BETWEEN POINTS
C
SOLVE(
): SOLVES FOR KRIGING WEIGHTS X USING THE
C
KRIGING SYSTEM OF LINEAR EQUATIONS
C
LUDATF,LUELMF: ARE COMPUTER ROUTINES FROM IMSL USED
C
TO PERfOHM HAfRL< TRANSfORMATIONS ON
C
THE KRIGING SYSTEM.
C
c
..
C
CHARACTER STA(17S)*10,HEAO*4
INTEGER NSAMP,NSYS,IER,NX,NY
~; ..
.
~

---

150
REAL SILLX,SILLY,RANGX,RANGY,XEW,YSN,XO,YO
PARA~ETER(NDAT=115,NSAMP=8)
PA~~METER(NSYS=NSAMP+l)
REAL A(NSYS,NSYS),B(NSYS),WEIG(NSYS)
REAL X(NDAT),Y(NDAT),ET(NDAT),DIS(NDAT)
REAL SX(NSAMP),SY(NSA~~),ETR{NSAMP)
REAL ETK(40,55),ERX(40,55),ALT{UDAT)
CO~~ON IVARIOI SILLX,SILLY,RANGX,RANGY
C
C READ THE DATA FROM VARIOGRAM AND FROM TAPE5
C
PRINT*,'SILLX=?,SILLY=?,P~NGX=?,RANGY=?'
READ*,SILLX,SILLY,RANGX,RANGY
C READ REGION LIMITS AND GRID DISTANCES
PRINT*,'XLW=?,XLE=?,YLS=?,YLN=?'
REAO*,XLW,XLE,YLS,YLN
PRINT*,'OX=?,DY=?'
READ*,DX,DY
NX=(XLE-XLW)/DX+l
NY=(YLN-YLS)/DY+1
C MAKE THE UPPER LEFT KRID CORNER THE ORIGINE OF X AND Y
READ(5,15)
15 FORMAT(A4)
DO 10,1= 1,NOAT
READ(5,20) STA(I),DUM,ET(I),X(I),Y(I),ALT(I)
Y(I)=Y(I)-YLS
XO )=X( I)-XLW
10 CONTINlJE
20 FO~1AT(A10,5Fl0.3)
C
C SEARCH FOR NSAMP LOCOTIONS IN KRINGING NEIGHBORHOOD
C
DO 50, IY =1, NY
YC= ( IY- 1P'DY
DO 50, IX= 1 ,NX
XC=(IX-1 )*DX
DO 21,I=1,NDAT
XS=xO )-XC
YS=Y(I )-YC
DIS(I)=XS*XS+YS*YS
21
CONTINUE
DO 22, 1= 1, NSAMP
10=1
DO 21,J::2,NDAT
IF(DIS(J).LT.DIS(ID»
ID=J
21
CONTINUE
SX(I)=X(ID)
..-
.
;
, .

---

151
SY(I)=Y(ID)
ETR( r)::ET(ID)
0IS([O)=9.0E20
22
CONTINUE
C COMPUTATION OF KRIGING SYSTEM
00 35, I= 1,NSAMP
A(I,NSYS'=A(NSYS,I)=1.0
B(I)=GAM(XC,YC,SX(I),SY(I»
DO 33,J=1,I
A(I,J)=GAM(SX(I),SY(I),SX(J),SY(J»
A(J,I)=A(I,J)
33
CONTINUE
35
CONTINUE
A(NSYS,NSYS)=O.
B(NSYS)=1.0
IF(IY.EQ.2.ANO.IX.EQ.3) THEN
c
PRINT*,A
c
PRINT*,B
c
PRINT* , , A(2,3) =' , A(2,3) , I A<3,2) =' , A(3,2)
ENOIF
CALL SOLVE(A,B,WEIG,IER)
IF(IER.NE.129) GOTO 31
ETK(IY,IX)=-9.99
ERK(IY,IX)=-8.88
GOTO 50
31
SS1 =0.
SS2=0.
DO 40,1= 1,NSAMP
SS1=SS1+WEIG(I)*ETR(I)
SS2=SS2+WEIG(I)*B(I)
40
CONTINUE
ETK ( IY , I X) =SS 1
ERK(IY,IX)=SS2+WEIG(NSYS)
50
CONTHIUE
C
PRINT*,'ETK=' ,ETK
C
PRINT*,'ERK=',ERK
C
C PRINT THE ET ESTIMATES AND KRIGING ERRORS
("'
v
K1=INT(XLW)
K2=INT(XLE)
K3=INT(DX)
WRITE(6,200)
200 FOru~ATU,' KRIGING EVAPOTRANSPIRATION ESTIMATES' ,f)
WRITE(6,350)(K,K=K1,K2,K3)
NTICY=INT(YLN+DY)
DO 60,IY=NY, 1,-1
."

---

152
NTICY=NTICY-INT(DY)
WRITE(6,400) NTICY,(ETK(IY,IX),IX=l,NX)
60 CONTIWJE
WRITE(6,300)
300 FORMAT(/,' KRIGING ESTIMATION ERRORS',/)
WRITE(6,350)(K,K=Kl,K2,K3)
NTICY=INT(YLN+DY)
DO 70,IY=NY,1,-1
NTICY=NTICY-IN1(DY)
WRITE(6,ijOO) NTICY,(ERK(IY,IX),IX=l,NX)
70 CONTINUE
350 FOrill~T(7X,24(I4,lX»
400 FORMAT(I4,3X,24(F5.2»
E~
C
C FUNCTION GAM TO COMPUTE SEMIVARINCES
FUNCTION GAM(XC,YC,XS,YS)
REAL SILLX,SILLY,RANGX,RANGY
COMMON /VARIO/ SILLX,SILLY,RANGX,RANGY
C
C CONSIDERING GEOMETRIC AND ZONAL ANISOTROPY
DSILL=SILLX-SILLY
SCX=ABS(XC-XS)
SCY=ABS(YC-YS)
IF(SCX.GE.RANGX) SCX=RANGX
IF(SCY.GE.RANGY) SCY=RANGY
SCD=SQRT(SCX**2+(SCY*RANGX/RANGY)**2)
IF(SCD.GE.RANGX) SCD=RANGX
G=1.5*SCD/RANGX-O.5*(SCD/RANGX)**3
DG=1.5*SCX/RANGX-O.5*(SCX/RANGX)**3
GAM=SILLY*G+DSILL*DG
SILL=MAX(SILLX,SILLY)
IF(GAM.GE.SILL) GAM=SILL
C
PRINT*,'SCX=',SCX,'SCY=' ,SCY,' SCD=',SCD,' G=',GAM
RETURN
END
C
C SUBROUTINE SOLVE TO SOLVE SYSTEM OF LINEAR EQUATIONS
C
SUBROUTINE SOLVE(A,B,X,IER)
INTEGER N,IER,IA,IDGT
PARAMETER(N=9)
REAL A(N,N),LU(N,N),D1,D2
REAL EQUIL(N),X(N),B(N)
REAL RES(N),IPVT(N),DX(N),WA
IDGT=3
IA=N
; ,

---

153
CALL LUDATF(A,LU,N,IA,IDGT,D1,D2,IPVT,EQUIL,WA,IER)
CALL LUELMF(LU,B,IPVT,N,IA,X)
C
PRIN'l*,X
HETUBN
END
·-..

---

154
APPENDIX B
Monthly reference evapotranspiration rates (mm/day) for
J).me 1979.
Kriging estimates of monthly reference evapotranspiration
rates (mm/day) for June 1979 at each (8x8 mi) grid corner
over the state of Oregon.
Estimates of kriging variances «mm/day)2) at the grid
corners mentioned above.
Results of self-validation test of state-wide
semivariogram models for June 1979.
;: -!.'

---

155
REFERENCE EVAPOTRA»SPlRATION FOR OREGON, JUNE 1979
STA
DATE
F
ET
X
Y
ALT
ASTORI79 6
5.182
3.266
45.701
286.625
8.000
BANDON79 6
4.925
2.894
21.936
79.384
20.000
BROKIN79 6
5.108
3.695
28.290
2.301
70.000
CAFBLC79 6
4.617
3.177
13.716
57.515
188.000
CLODAL79 6
5.078
3.175
45.729
222.124
80.000
COQUIL79 6
5.091
3.670
32.061
81.685
23.000
DORA2W79 6
5.399
4.009
42.185
80.535
90.000
ELKTON79 6
5.643
4.275
63.278
110.447
120.000
GOLBEC79 6
4.975
3.549
21.432
27.607
50.000
HONEPA79 6
4.984
3.565
37.123
133.457
115.000
ILLAHE79 6
5.687
4.353
40.291
43.711
348.000
LAURHT79 6
4.743
3.185
63.091
201.372
3740.000
NEWPOR79 6
4.854
2.974
39.017
181.810
154.000
NBENDF79 6
4.967
3.535
29.530
97.792
6.000
OTIS2N79 6
5.087
3.189
44.096
209.464
150.000
PORFOR79 6
5.095
3.677
22.957
51. 763
45.000
POWERS79 6
5·361
3.985
39.434
60.965
230.000
REDSP079 6
5.180
3.770
36.279
117.350
60.000
SEASID79 6
5.122
3.210
44.912
275.065
10.000
TIDWAT79 6
5.318
3.392
46.488
166.851
50.000
TLAMOK79 6
4.885
2.991
47.362
238.236
10.000
VALSET79 6
5.179
3.375
58. 110
196.769
1155.000
BONDAM79 6
5.773
4.876
141 .269
250.896
60.000
CASDIA79 6
5.272
4.725
117.050
165.700
860.000
COROSU79 6
5.495
4.878
81.354
181.810
225.000
CORWS079 6
5.478
4.912
68.902
173.755
592.000
COTGR079 6
5.244
5.043
89.433
123.103
650.000
CGRODA79 6
5.467
5.366
90.276
118.501
831.000
DOREDA79 6
5.392
5.265
94.495
123.103
820.000
EUGEUE79 6
5·382
5. 181
80.524
146.138
364.000
FERIDA79 6
5.472
5.302
76.373
146.138
386.000
FDRGR079 6
5.538
5.015
84.925
243.991
180.000
LACOMB79 6
5.494
5.352
105.428
180.659
520.000
LEABUR79 6
5.530
5.425
107.088
144.988
675.000
LOPODA79 6
5.503
5.395
-104.619
132.307
712.000
NOTI1N79 6
5·336
5.1]2
68.072
142.686
445.000
SCREFA79 6
5.234
4.290
108.749
197.920
1350.000
BEVET079 6
5.659
4.771
98.807
241.689
215.000
CHYROV79 6
5.384
4.034
77.575
235.935
650.000
CKLAUI79 6
5.405
3.480
74.565
283.171
92.000
; .!
DALLAS79 6
5.670
4.798
75.543
202.523
325.000
ESCADA79 6
5.516
4.142
123.304
225.576
410.000
FOSTDA79 6
5.484
4.914
107.919
166.851
550.000

---

156
3TA
DATE
F
ET
X
Y
ALT
HPTLDW79 6
5.521
4.690
131.470
238.236
7118.000
HILBOR79 6
5.660
4.764
90.641
242.840
160.000
t1HINVI79 6
5.533
4.221
80.842
223.275
148.000
ULAMET79 6
5.675
4.372
102.073
226.727
150.000
ORECIT79 6
5.432
4.613
109.422
231.331
4136.000
PTLDWS79 6
5.904
5.017
109.422
248.594
21.000
SHELEN79 6
5.743
4.850
98.807
267.009
100.000
SALEMW79 6
5.565
L~ .509
90.486
201. 372
195.000
SCODAH79 6
5.677
4.811
80.025
240.538
355.000
SCOMIL79 6
4.969
4.102
114.560
203·673
2315.000
SILVER79 6
5.658
4.640
101.256
207.162
408.000
TRODAL79 6
5.889
5.002
119.221
246.293
29.000
VNONIA79 6
5.210
4.319
80.842
267.009
625.000
ASHLAN79 6
5.649
6.410
108.872
14.954
1780.000
CAVEJC79 6
5.559
6.181
60.008
11.503
1280.000
DRAIN 79 6
5.730
6.252
76.777
115.049
292.000
CHTPAS79 6
5.784
6.453
78.011
29.908
925.000
HOARDA79 6
4.891
5.669
126.875
14.954
4567.000
IDLPAR79 6
5.337
5.810
94.495
94.340
1080.000
LEMLAK79 6
5.104
5.943
132.461
94.340
4077.000
LCREDA79 6
5.657
6.385
110.587
46.012
1580.000
HEDFOD79 6
5.723
6.461
101.157
20.705
1457.000
HEDFWS79 6
5.831
6.598
101.157
25.306
1312.000
PROSPC79 6
5.486
6.284
119.160
50.613
2482.000
ROSBUR79 6
5.721
6.272
75.090
82.836
465.000
RUCH
79 6 .
5.645
6.361
92.584
16. 104
1550.000
SSUMIT79 6
5.096
5.258
75.439
42.561
3836.000
BELKNA79 6
5.337
4.511
139.464
158. "{96
2152.000
DTRODA79 6
5.425
4.490
128.672
187.564
1220.000
CVTCAM79 6
4.699
4.110
151.068
227.878
3980.000
MARION79 6
5.273
4.477
143.615
179.509
2475.000
MKE~nI79 6
5.704
4.842
135.313
150.741
1478.000
OKRrDC79 6
5.530
5.522
120.650
120.802
1275.000
SANTPA79 6
4.650
3.988
147.766
166.851
4748.000
TRELYN79 6
5.412
4.615
135.553
215.218
1120.000
CHEMUL79 6
4.864
. 4.256
182.240
85. 137
4760.000
RONGROf9 6
4.903
5.8011
203.171
23.006
4888.000
SUMLAK79 6
5.465
7.613
208.315
65.567
4192.000
WIKIDA79 6
5.090
5.877
159. 1t60
116.200
4358.000
ANTEL079 6
5.380
5.493
195.084
201.372
2690.000
ALINT079 6
6.358
6.403
227.010
256.651
285.000
80RDMA79 6
6.303
6.611
251.508
264. '707
300.000
CONDON79 6
5.4~6
5.622
227.827
223.275
2830.000
DUt'UR 79 6
5.508
5.456
181.282
238.236
1330.000
HEPPNE79 6
5.803
7.442
259.673
226. "{27
32110.000
HERHIS79 6
5.988
6.244
271.922
263.556
624.000
.;. ~
HODRIV"{9 6
5.697
5.905
162.500
254.31i9
500.000
KENT
79 6
5.478
5.635
202.513
220.973
2720.000
.
;
~

---

157
STA
DATE
F
ET
X
y
AL'f
MIKAL079 6
5.986
6.141
219.661
2.39.387
1550.000
~HLTON19 6
6.197
7.591
314.385
272.763
970.000
MORO
"79 6
5.582
5.643
201.696
240.538
1870.000
PAKDAL79 6
5.409
5.745
159.234
241.689
1930.000
PDLTON79 6
5.'784
7.053
303. rr69
256.651
1487.000
POLTON'79 6
6.047
7.474
293.153
254.349
1492.000
PILOT179 6
5.813
. 7.146
294.787
240.538
1720.000
PINGR079 6
5.607
5.735
169.849
215.218
2220.000
THEDAL79 6
6.381
6.767
178.015
248.594
102.000
ADEL
79 6
5.613
6.909
253.750
12.653
4580.000
ALKALA79 6
5.381
6.694
248.606
66.717
4332.000
ADREMI79 6
5.667
7.261
323.188
37.960
4180.000
BARNST79
5.150
5.116
233.706
134.608
3970.000
BEND
79 6
5.231
5.190
175.160
142.686
3650.000
BROTHE19 6
4.931
4.903
214.301
124.253
4640.000
BURNsw19 6
5.477
6.819
292.765
109.297
4140.000
CHIL~79 6
4.780
5.503
153.450
40.260
4220.000
CRISVA79 6
4.962
5.991
210.082
85.137
4310.000
DAYVIL19 6
5.930
5.896
258.115
176.051
2230.000
OREWSE19 6
5.630
6.956
326.513
124.253
3516.000
FOSSIL79 6
5.367
5.469
226.194
201.162
2650.000
HAREFU79 6
4.904
5.124
266.609
37.960
5616.000
KLTFAL79 6
5.458
6.512
156.879
13.803
4098.000
KLTFAL19 6
5.285
6.292
158.594
11.503
4090.000
LAKVIE19 6
5.224
6.311
229.147
14.954
4178.000
MADRAS19 6
5.513
5.349
184.292
181.810
2230.000
HADRAS79 6
5.514
5.382
183.462
184. 111
2440.000
MALFUG19 6
5.527
6.489
303.133
88.588
4109.000
MALINS19 6
5.263
6. 31~8
180.883
.000
4621.000
MTOLUS19 6
5.316
5.210
181.801
178.358
2500.000
HITCHE19 6
5.543
5.461
232.440
177.201
2744.000
OCHOC079 6
4.880
4.146
219. 158
165.100
3915.000
OORANC19 6
5.482
6.422
219.266
88.588
4136.000
PAISLE19 6
5.431
6.519
221.174
l18.312
1.360.000
PAULIN79 6
5·364
6·353
242.402
147.289
3684.000
PLTNDA19 6
5.967
5.805
. 179.311
188.714
1410.000
PINEMT19 6
4.183
4.897
196.583
123.103
6240.000
P-RANC79 6
5.317
6.265
306.043
56.364
4195.000
REDMON19 6
5.390
5.305
180.141
156.495
3010.000
REDMDF79 6
5.404
5.330
183.462
156.495
3060.000
SISTER79 6
5.148
5.003
163.538
157.645
3180.000
SPRARI79 6
5.074
5.999
. 171. )\\53
31.058
4360.000
SQABUT79 6
5.385
6.161
260.704
102.394
4665.000
SUNTEX79 6
5.244
6.462
263.235
110.4111
4310.000
IJALFAL79 6
5.407
6.515
235.747
31.058
4580.000
WAGONT'l9 6
5.314
6.754
250.580
86.281
4726.000
~ ':
IHTHOR19 6
5.497
7.034
339.416
23.006
4200.000
AUSTIN19 6
4.893
4.691
315.454
118.358
4213.000
~\\. ~
.,' ..:-. "

---

158
~
STA
DATE
p,'
ET
X
y
ALT
".
,.
.
~
,
. . .
BAKSRF79 6
.' 5~486
5.396
349 )l90
195.618
3368.000
BAKEftK79 6
. , 5.561'"
5.510
351.981
191.016
3J144.000
COVE
796 .
, 5.485:,.,
5.328
344,598
227.878' 2920.000
ELGIN 79 6
5'.602:. :
5.446
338.882
246.293
2655.000
E&"fPHI79 6 '
5.166 ..•.
5.0i6· . 370.729·
237.085
3790.000
ENPRIS79 6
5·30'\\' .~ 5.131
376.445
255.500
3280.000
:HALFWA?9' 6.
·5.515' :,"'5.330
384.356
199.071
2670.000
HUNGT019 0' _,
:6~420·. " 8.985.
376.885
162.2}i8
2130.000
JOHNDA79 6
- S.61f:<··
5.520
293.01~ 1
168.002
3063.000 "
LAGR..'UJ796
5.731
.
5.576 ' 330.716
229.029
2755.000
LOUCRE79 6"
r-
099~'
:>.
':'"
4.914
235.569
187.564
3722.000
M...~NDA79 6
. 5.148-\\; " 5.005
340.359
184.111
3900.000
MINAN779 6
4.9H
4.641
354.391
254.349
3584.000
MOl~&\\jT79 6
5.730
5.511
269.791
194.468
1995.000
RICHLA79 6
6.009
5.916
381.866
191.016
2215.000
SENECA79 6
4.906
4.766'
292.210
148.440
4666.000
UKIAH 79 6
5.017
6.403
289.071
216.369
3355.000
UNIONX79 6
5.455
5.26
340.515
222.124
2765.000
UNITY 79 6
5.300
7.394
328.737
168.002 4031.000
WALALA79 6
5.590
7.207
326.322
276.265
2400.000
WALLOW79 6
5.393
5.204
357.663
246.293
2923.000
BEULAH79 6
5.853
8.246
337.481
132.307
3270.000
BURNJT79 6
5.829
8.353
359.193
54.064
3930.000
..
DANNER79 6
5.460
7.731
385.769
64.416
4225.000
IONSID79 6
5.531
7.797
341. 189
157.645
3915.000
JUTURA79 6
6.060
8.518
349.293
124.253
2830.000
MHEURB79 6
6.066
8.387
395.697
136.909
2240.000
MDRMIT79 6
5.544
7.939
358.336
28.757
4464.000
NYSSA 79 6
6.116
8.460
396.541
128.855
2175.000
ONTARI79 6
6.200
8.600
391.828
141.536
2145.000
OWYEDA79 6
5.899
8.130
383.885
113.899
2400.000
RIVSI079 6
5.735
8.046
337.481
106.996
3330.000
:1.'
ROCVIL79 6
5.462
7.619
390.635
94.340
3670.000
ROME2N79 6
5.778
8.142
369.480
59.815
3410.000
SHEVIL79 6
5.447
7.788
394.853
77.083
4620.000
VALE
79 6
6.178
8.585
383.885
136.909
2240.000
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",,~jG;NG
ESTII,IATES
Of
Ev;,PCn"",:;NSfC1R':'7IOI,
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OREGON,
JUNE
1979
\\)
8
1 l~
:'::4
::Si
4(1
48
5lo
(-)4
72
BO
Bb
.9 6
1 04
1 1 2
1 20
1 28
::'~;o
3.07 :3.07
3.11
3.13
3.16 3.22 3.::'7
J.J:2
:).37
3.45 3.69 3.97 4.19 4.38 4.53 4.63 4.87
20:'::
3.08
J.05
3.093.11
3.14
.3.1~) 3.25
3.34
3.44
3.55 3.83 4.14 4.37 4 ""
.~'-'
4.67
4.74 5.0~~
274
3.06 3.06 3.03
3.0b 3.1 I
J
17
3.~5 :3.':; 1 3.57
3.82 4.11
4. ~j8 4.61
4.75 4.89 4.94 5.04
266
3.04 3.04 3.04 3.0b 3.08
3.15 3.26 3.45 3.70 4.02 4.38 4.60 4.80 4.9i
4.97
4.99 5.04
'258
3.03 3.02
3.02 ~.02 3.02
3.11
3.23
3.4 Li 3.79 4. 12 4.59 4.76 4.81
4.94 5.00 5.01
4.97
250
:j . C'9
:'.05
2.98
2.97
2.99
3.03
3.15
"J .·13
3.81
4./3 4.77 4.89 4.88 4.91
5.01
5.01
4.93
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3. (;~(
3. Ob
3.03
3.02
2.99
2.~0 3.04
j . :Jtl
3.70 4.07
4.74 4.74 4.70 4.76 4.82 4.7tl
4.7J
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2 . 9 7
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3. U 1 :J. U 4
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3.86 4.20 4.44 4. '19 4.56 4.60 4.46 4.45
226
:~.06 3.0b 2.98 3.00 2.99 3. ID 3.16 3.31
3.48 3.89 4.:<1
4.33 4.39 4.40 4.36 4.22 -1.30
218
3.06 ~.92 2.93 2.96 3.07 3.1 1 3.~1
3.32
3.56 4.01
4.37
4.39 4.46 4.46 4.32 4.12 4.37
2 HJ
2.,132.83
2.932.96 3.Ul
1.1\\
3.:20
3
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3.50 4.21
4.58
4.58
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4.26 4.21
4.37
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2.96
2.9S
2.95
2.95
1.~9 3.G7
3. ~5 :j.24
3.38 4.37 4.73 4.60 4.S9 4.50 4.21
4.24 4.36
19~
3.102.95 2.94
2.95
2.95
3.03
3.~O j.45 3.82 4.44 4.72 4.76 4.78 4.75 4.53 4.43 4.44
86
J . 00
2. S18
2.96
2.95
2.94 3.01
3.29
3.b8
4.15 4.06 4.83 4.tl9 5.02 5.17 4.88 4.65 4.55
78
J.02
3.00
2.99
2.99
3.00 3.10 3.40 3.8,-) 4.40 4.92 4.93 5.00 5. 12 5.23 4.96 4.75 4.57
7lJ
3.06 3.06 3.05 3.03 3.0B 3.20 3.45
4.U2 4.60 5.00 5.0:l 5.04 5'.08 5.07 4.85 4.69 4.59
It;j:.!
3.06 3.05 3.073.\\13.\\9 3.36 3.68 -1.18
4.68 4.98 5.07 5.U8 5.12 5.0'/ 4.94 4.80 4.72
1:;4
3.02 3.05
3.09 :,.15 3.29
3.49 3.84 4.31
4.7'1
5.11
5.18 5.15 5.22 5.25 5.13 5.05 4.~O
, ~ II
2.90
3.08
3.22 3.30 3.42 3.62
3.92 4.36 4.89
5.21
5.19 5.20 5.27 5.40 5.34 5,21
5.09
133
~j.06 3.00 3.3U 3.37 3.~7 3.66 3.97 4.3~ 4.86 5.25 5.28 5.16 5.23 5.38 5.40 5.33 ",.22
lJG
3. 11
3. 17
3.20
3.33 3.55 373 4.03 4.41
4.83
5.35 5.43 5.14 5.23 5.39 5.45 5.42 5.43
122
:3. 16 3. 17
3.26 3.40 3.63 3.d0 4.00 4.29
4.76 5.55 5.79 5.33 5.34 5.43 5.51
5.53 5.55
114
.3.34 3. 18
3.27
3.3~ 3.64
3.82 3.95 4.11
4.44 5.56 6.03 5.69 5.60 5.53 5.61
5.64 5.70
., ,.,,~
. vO
.3.30
3. n
3.22 3.37 3.64 3.B2
3.9R 4.23 4;68
5.49 5.90 5.80 5.69 5.68 5.69 5.75 f.i. 8 1
92
3.263.18 3.13
3.32 3.63
:1.89
4.13
4.48
4.94
5.63 5.96 5.86 5.80 5.77 5.82 5.84 5.90
90
Cl.20
3. 1 ()
3.06 3.22 3.62 3.93
4.30
4.66 :].16 5.87 6. 13 5.98 5.92 s.nG 5.94 5.95 5.88
82
3.20 3.G6
2.95
3.06 3.64 3.93 4.35
4.86 !'S.26 5.94 6.17
6.08
5.96
6.05 6.05 6.04 5.99
74
3.22
3.08
3.01
3.18
3.60 3.96 4.42
4.89 5.30 5.78 6.05 6.04 6.01
6.09 6.08
6.08
6.07
66
3.37
3.21
3.13 3.33 3.68 4.00 4.41
4.19
5.1 G 5.51
5.BO 5.90 5.98 6.0S 6.15 6.146.D4
sa
3.55 3.35 :,,25 J.46 3.81
4.10 4.40 4.77
5.0b 5.33 5.64
5.04 5.99 6.14 6.20 6.:<4 6.U5
50
3.66 3.5\\
3.45
3.60 3.95 4.26 4.51
4.67
5.0b
5.23
5.57 5.87 6.04 6.24 6.35 6.24 6.00
42
3.64
3.57
3.54 3.69 3.97 4.33
4.74
5.04
5.24
5.'10 5.71
6.01
6.22 6.37 6.37
6. 19
5.95
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3.b3
3.57
3.51
3.66 4.02
4.48
4
9:1
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3.:;63.51
3.51
3.G6 4.04 4.51
5.07
:}.~)~
S. 'J"/
0.24
0.44 6.43
6.49
6.52
6.33
6.06 5.82
16
3.56 3.52
3.5U
3.01.3
4.05
4.101
5. lE
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G.18
G.3:3
6.40
b.3':! 6.39
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6.30 5.96 5.70
1 CJ
3.57
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3.51
3.64 3.99 4.57 5.19 S.b~ 6.19 Li.n 6.37
f).37
6.37
6.31
(;.19
5.96 5.65
-;:
3.61
3.56 3.56 :3. b5 3.99 '1.55 5. 13 5.\\"iG 5.07 0.13 6.33 6.34 6.33 6.21
G.18 5.99 5.98
,.....
V1
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---

, .
' "
.. ,
KRIGING
ESTIMATES
OF
t'jAPOTRAN:',PIR;;llON
fOR
Or<EGOr~. JU"f..
19'19
"
'.;
~ .'..
136
144
10:1
100
16U
1/6
184
192
:WO
208
:116
224
232
240
:148
~56
<~64
20d
5.23 5.38 5.65 6.LlO 6.09
b.37 6.26 6.3u 6.45 6.47
6.52 6.6<3
6.42 b. 3'/
G. :c;~) 6': 6"
6.54
2132
5.175.33 5.72 6.00 6.13 6.41
6.31
6.34 6.43 6.45 6.6::; 6.65 6.43 6.40 6.46 6.44.6.40
274
5.20
5.42 5.06 5.98 6. 1 7 6. 4 6
(i. 3 Gb. 4 :
6.37 6.40 6.57 6.58 6.44 6.46 6 . :; 1 6.41'\\ 6 .•i 2
266
5. 12 5.29 5.56 5.93 6.21
6.42 6.42 6.38
6.31
0.43 6.46 6.47 6.42 6.51
6.59 6.576.43
258
5.0~j 5.11
5.43 5.84 6.23 6.56 6.47 0.28 6.20 b.26 6.33 6.34 6.45 6.57 6.'646.676.62'
250
4.86 4.93 5.31
5.80 6.20 6.61
6.28
6.04 5.95 6.04 6.18 (j.2a 15.41
6.58 6.70 6. a 1 6.85 i
242
4.7\\
4.73 5.03 5.65 5.89 5.90 5.G6 569 5.64 5.8j 6.05 6.17 6.30 6.55 6.00 6.07 7.05
234
4.':;", 4.41
4.SG 5.14 5.55 5.61
5.48 S.52 5.59 5.75 5.87 5.92 6.10 6.43 6.83 7.16
7. 23
226
4.41
4.30
4.34
4.90 5.45 5.61
5.54
5.5\\
5.62 5.68 5.71
5.65 5.86 6.27
6.67
7.10 7,.16
21d
4.59 4.52 4.6c 5.10 5.b1
5.71
5.5:3 5.59
5.~8 5.61
5.59 5.55 5.74 6.05 6."41
6.73 6.'76
210
4.53 4.64 4.77 5.:2tJ 5.61
5.73 5.64 5.58
~.54 5.52 5.54 5.47 5.62 5.87 6.17
b.36.6;33
202
4.53 4.71
4.89 5.23 5.55, ~.75 5.65 5.53 5.47 5.40 5.40 5.43 5.52 5.78 5.97 6.06 5.95.
194
4.49 4.67
4.90 5.19 5.41"5.12 5.64 S.48 5.34 5.31
5.33 5.38 5.50 5.70 5.88 5.88 5.69'
186
4.49 4.48
4.69 5.01
5.27 5.54 5.38
5.30 5.23 5.19 5.20 5.30 5.45 5.65 5.76 5.77 5.65
178
4.49 4.36 4.46 4.79 5.10 5.22 5.23 5.18 5.10 5.00 5.06 5. 13 5.44 5.74 5.88 5.8S! 5.'18
170
4.424.17 4.24 4.65 5.02 5.23 5.25 5.17 5.06 4.9S 4.81
5.02 5:42 5.80 5.96 6.00 5.82
162
4.55 4.38 4.44 4.78 5.05 5.2j 5.j1
5.21
5.04 4.94 'l.85 5.07 5.47 5.91
6.07 5.99 5.81
154
4.79 4.65
4.69 4.92 5. 1 1 5. 23 5. 2~:) c;. 16 5.04 4.94 4.94 5.15 5.55 6.0'/ 6.236.115.94
146
5.00 5.00 5.03 5.16 5.2U 5.::'U 5.2U 5.12 5.02 4.84 4.95 5.11
5.42 6.01
6 .. 28 6.·12 5.86
133
5.24 5.22 5.26 5.3S 5.325.26 5.lEi S.lj5 4.99 4.92 4.9:3 4.98 5.13 5.68 6.09 6.03 5.93
13('
5.40 5.44 5.50 ':).5'1 5.46 5.28 5.14 S.U:; '-I . ~JO '1.9iJ 4.B9 5.02 5.21
5.65 6.04 6.04 6.06
12:2
5,S8
5.62 5.7'3 5.78 5.51
5.27
5.08
4.94
4.93
4.CJB
S.D3 5.18 5.36 5.72 6.06 6.29 6.24
: ',,,
5.70 5.77 5. B 1 5.17
5.455.13 'l.96 '1.95
4.96 5. I 1 5.26 5.40 5.59 5.90 6.19 6,39 6.43
106
5.83 5.79 5.73 5;56 5.27 4.8d 4.72 4.84 5.04 5. :!8 5.48 5.65 5.80 6.15 6.39 6.61
6.67
08
5.8':J 5.78 5.62 5.j7 5.03 4.69 4.57 4.85 5.16 5.55 5.73 5.90 6.07 6.34 6.56 6.71
6.72
90
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5.87 5.72 5.54 5 .<.~ 4.87 4.53 4.41
4.94 5.43 5.133 6.03 6.23 6.30 6.58 6.71
6.74 6.59
b '2
5.66 5.09 5.43 5.13 4.93 4.6~ 4.71
5
29 5."1
6.3'1 6.46 0.54 6.49 6.64 6.71
6.69 6.53
74
5.86 5.64 5.32 5.21
5.07
5.05
5.21
5.7~ 6.43
7.00
7.00 6.G3 6.70 6.68 6.71
6.60 6.47
6c
5.86 5.61
5.38 5.26 5.2j 5.40 5.57
6.14
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1.50
7.23 6.98 6.73 6.69 6.65 6.50 6.34
5d
:'.86 S.77
5.44 5.35 5.375.52 5.78 G.27 (i. f3
7. 10
7.01
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6. 35
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50
5.53
5.71
5.47 5.42 5.55 5.67 S.bS 6.2~ G.52 G.78 6.68 6.00 6.60 6.52 6.41
6.25 6.00
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5.69
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6.36
6.50 6.55 6.46 6.:30 6.00
26
5.64 5.94 6.06 [).14 G.l:3 6.09 602 5.94 :> . 8., 5.9b 6.10 6.27 6.44 6.56 6.59 6.50 6.30
18
~.37 6.06 6.31
6. j', G.26 6.20 6
11
G.D1
5.92
5.90 G.OiJ 6.22 6.38 6.59 0.74 6.75 6.49
10
5.95 6.14 6.30 0.30 6.J1
o.:~~ b.Ll
b
{j':J
G.03
6.07
(). 16 6.31
6.43 6.62 6.79 c.Bl
6.150
6.06
6.18
6.27
6.32 6.34 0.35 6.32 6,22
b.13 6. 17 6.24 6.38 6.49 6.62 6.74 6.74 6.7'L
I-'
m
o

---

;<.RIGING ESTlil·IAlES OF
t:VA;:'()TR"'I'~SI"lf<AllON
t'OR OREG()~I.
JlH'lE
1979
272
~80
2UB
~~5
304
312
320
j2U
336
344
352
360
368
376
384
392
400
~00
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2
(i.55
6.68 6.»2
7.32
7.42
7.51
7.45
7.12
0.61
6.17 5.81
5.63 5.52 5.4/ 5.45 5.44 5.46
4
6.40 6.64 6.94 7.31
/.41
7.54
7.34
Ci.94 6.41
5.92 5.54 5.38 5.32 5.33 5.34 5.34 5.38
b
",.26 [1.667.037.317.267.247.01
\\1.596.11
5.bl
5.205.125.155.205.245.105.11
::53
b.S']
6.817.18
7.377.05
b.90·G.b9 b.2Y 5.82 5.35 4.854.90 4.98 5.125.13 5.13 5.11
250
6.826.»4 7.16 7.26 6.99 E.GiJ
6.38
a.06 5.02 5.24 5.04 5.09 5.06 5.09 5.13 5.14 5.12
:'::-1 ~
6.~'7 7.027.047.096.836.506.195.815.535.365.225.185.07 5.07 5.12 5.115.14
2;) ~~
b.9;':
6.93 6.9LI 6.81
fj.58
(-).29 5.'09 5.n
~i.48 :'.:165.235.175.;105.105.125.145.17
2:'b
~.~2 6.76 ~.(j~J 6,.5~ (j.:I~ ~.}~ ~.08 ~.~~ ;~.~o 5.;9 5.~4 5.2,1 5.~? 5',19 ~.,17 5.1~ 5.21
218
Q.bl
6.46 6.39 b.2~ 6.0.
~.d~ ~.77 ~.~~
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~.19 5.2,
5.24
210
6.15 6.02 5.93 5.83 5.70 5.57 5.41
5.21
5.24 5.30 5.38 5.39 5.38 5,31
5.20 5.34 5.53
202
'0.7'2
5.57 5.495.475.20 5.U4 5.10 5.115.105.205.435.495.515.515.325.585'.81
194
5.435.225.135.175.044.084.77 5.014.975.155.495.745.905.955.856.046.18
180
5.42 5.13 5.01
5.05 4.95 4.71
4.63 5.27 5.20 5.29 5.83 6.20 6.46 6.63 6.60 6.646.69
178
5.595.295.215.195.054.955.416.\\06.126.226.576.86 7.20 7.457.387.317.22
170
".655.495.365.425.405.6:3 6.23
7.30
7.12
7,08
7.32 7.558.008.338.147.917.70
162
5.61
5.32 5.23 5.28 5.47 6.04
6.68 7.43 7.6$ 7.71
7.79 8.07 8.54 0.95 8.58 8.26 7.98
154
5.b5 5.33 4.98 5.04 5.48 6.15
6.81
7.46 7.86 8.08 8.27 8,39 8.65 8.80 8.71
8.48 8.22
146
'; 3"
5.65 5.36 5.09 5.17 5.58 6.03 6.58
7.30 /.98 8.25 8.39 8.47 8.60 8.69 8.68 8.58 8.35
5.80 5.63 5.49 5.56 5.83 6.3\\
6.74
7.33 8.04 B.34 8.48 8.57 8.59 8.6U 8.57 8.47 B.LI2
i30
6.01
5.92 5.90 5.90 6.12 6.41
6./5 7.20
7.99 8.37 8.51
8.50 8.49 8.47 0,43 8.42 8.39
1.~ 2
6.24 6.24 b.29 6.36 6.44
6.61
G.83 7.21
7.86 8.27 8.46 8.45 8.36 8.31
8.27 8.26 8.24
114
6.43 6.46 6.61
6.71
6.67
b.70
7.01
/.38
7.9U 8.21
8.32 8.30 8.20 8.13 8.09 B.06 8.08
i t"J6
6.58
6.59
6.68
6.75 6.74 6:85
7.06 7.39 7.90 6.14 8.23 8.18 8.06 7.98
7.90
7.B7
7.91
98
6.57 6.53 6.S7 6.60 6.66 d.78 7.01
7.30 7.B2
B.08 8.20 8.12 8.04 7.90 7.78 7.67
7.78
90
6.526.43 G.4<3 6.46 6.49 6.716.91/.297.728.00 B.13 8.10 8.017.897.757.687.74
52
6.50 6.43 6.4:3 6.44 6.45 6.70 7.02
7.34
7.68
7.89 8.16 8.11
8.02 7.90 7.76 7.74 7.78
'14
6.366.306.316.29'6.376.646.96 7.337.60 7.87 8.118.178.077.93 7.817.75,7.78
60
6.286.216.206.236.306.596.967.297.58
7.f37 8.16 H.20 8.127.977.787.747.77
55
G.07
6.09 6.14 6.20 6.24 6.56 6.90 7.29 7.46 7.85 8.07 8.32 8.19 8.02 7.65 7.76 7.75
5CJ
S.S3 6.016.116.246.336.647.017.317.487.748.038.248.16 8.037.907.797.76
'2
5,830.016.17 c.32 6.416.727.107.317.427.64
7.918.108.008.00 7.89 7.83 7.76
4
5.99 6.13 6.26 6.426.536.787.027.19
7.27
7.47
7.76 8.007.987.937.86 7.817.76
6
6.24
6.29 6.40 6.47
6.59 6.78
6.89 7.01
7,05 7.26
7.60
7.84
7.88 7.84 7.80 7.77
7.75
8
6.46 6.46 6.52 6.61
6.64 6.78
6.02
7.00 6.99 7.20 7.47
7.68 7.75 7.75 7.73 7.72 7.71
o
0.62 6.59 6.62 6.70 6.70 6.81
6.92
7.00 6.97
7.14
7.39 7.58 7.64 7.67
7.65
7.66 7.67
2
6.70 6.67
6.72 6.77
6.83 6.84 6.95
7.03 6.95 7.10 7.35 7.50 7.56 7.59
7.59 7.61
7.62
~
0'\\
t-'

---

'"
",.
'" "
ESTIMATES UF ",R I C; I "C, ·.ARIAhCE ,~OR OR [GOr, ,
JUNE
1979
U
[3
16
24
3'2
~U
.i; r)
~) (.)
l1 .:.~
72
ilCJ
Bd
96
104
l"i 2
120
128
,.,
29G
1.20
1.00
,ilCl
.59
,36
· 17
.01
,:!u
.L~
,18
.31
,JO
.46
.52
.60
6C:
•
:.J
28:':
1.16
.96
.to
L-:"r;
.~:.J
.34
1
1
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APPLICATION DE LA GEOSTATISTIQUE A L'EVAPOTRANSPIRATION REGIONALE
TRADUCTION FRANCAISE CONDENSEE
THESE DE DOCTORAT
Ph.D
Kodjo Y. AMEGEE

---

2
AVANT PROPOS
Cette
partie du
document constitue
la traduction condensée
en français de la thèse
"version anglaise".
Elle ne reprend pas
toutes les
formules du texte original.
Elle devra être lue avec
report aux illustrations contenues dans le texte anglais.

---

3
•
APPLICATION
DE
LA
GEOSTATISTIQUE
A
L'EVAPOTRANSPIRATION
REGIONALE
1. INTRODUCTION
1.1. POSITION DU PROBLEME
Depuis les années 50,
chercheurs et ingénieurs agronomes se
penchent
plus
sérieusement
sur
l'évaluation des
besoins
en eau
des cultures,
notamment dans
les régions arides et semi-arides.
Les besoins en eau des cultures ont été définis par Doorenbos et
Pruitt (1977) comme" la hauteur d'eau nécessaire pour compenser
les
pertes
d'eau
par
évapotranspiration
(EJcrop)
d ',une culture
sans
maladie,
en
pleine
croissance
sur
un
soil
ne
souffrant
d'aucun
déficit
y
compris
celui
en
eau
et
en
fertilité,
et
jouissant des conditions de plein rendement".
La détermination
des besoins en eau des cultures est essentielle lorsqu'un système
d'irrigation doit être conçu pour fournir de l'eau en vue de la
production agricole.
L'évapotranspiration
de
référence
(ETr)
est
une
évapotranspiration
relative
à
une culture di te de
référence et
soumise à des conditions standardisées définies par Doorenbos et
Prui t t
(1977).
Elle
peut
être
considérée
comme
une
variable
météorologique
ou
climatique
dépendante
des
conditions
atmosphériques.
Lorsque
les
paramètres
météorologiques
utilisés
dans l'évaluation de l'ETr sont représentatifs d'un lieu donné,
la valeur
estimée de
l'ETr
est
désignée
par
évapotranspiration
locale de référence.
D'après Seguin (1977),
l'évapotranspiration
locale s'applique à
une localité dont
la taille ne dépasse pas
100
km 2 ,
alors
que
l'évapotranspiration régionale
réfere
à
une
surface allant jusqu'à 10.000 km 2 .
Lors de la conception des systèmes d'irrigation destinés à
fournir de l'eau aux cultures, la superficie à irriguer peut être
plus de 100 à 1000 fois supérieure à celle de la petite station
météorologique sur laquelle les paramètres climatiques auront été
mesurés.
Cette
extrapolation
revient
à
u t i l i s e r
\\

---

4
l'évapotranspiration
locale
à
la
place
de
l' évapotranspiration
régionale. Une telle pratique se traduit en réalité par une sur-
irrigation
ou
par
une
sous-irrigation.
Il
en
résul te
une
utilissation
irrationnelle
de
l'eau,
des
pertes
en
rendement
agricole
et
une
perte
d'énergie
lors
du
pompage
de
l'eau
d'irrigation.
Ces conséquences,
néfastes sur le plan économique,
rendent nécessaire un effort en vue de modéliser la variabilité
spatiale
de
l ' ETr en
fonction
des
coordonnées
géographiques et
des
distances
entre
les
stations
de
climatologie.
Ce
modèle
permettrai t
d'estimer
l ' ETr
à
des
points
où
aucune
station
météorologique
n'est
disponible,
avec
une
précision
supérieure
par rapport à une interpolation qui ne tiendrait pas çompte de la
structure
des
observations.
Ce
modèle
permettrait
également de
calculer
l'erreur
dl estimation
en
fonction
de
la
densité
des
stations. La relation entre erreur d'estimation et la densité des
stations
peut
être
utilisée
pour
concevoir
un
schéma
d'échantillonnage
en
fonction
de
la
précision
que
l'on
désire
attreindre.
1.2. Différentes Méthodes d'Approche
Cette thèse ne constitue pas la première tentative en vue de
modéliser
l'évapotranspiration
régionale.
Les
progrès
en
télédétection
ont
permis
à
certains
chercheurs
d'estimer
l'évapotranspiration régionale en combinant des mesures faites à
partir de satellite d'une part et de celles effectuées au sol sur
des
stations
météorologiques
traditonnelles
d'autre
part
(Schmugge,
1978; Berard et al.,
1981). La méthode la plus directe
(Baier,
1979) consistait à faire une moyenne pondérée à partir de
mesures
effectuées
sur de
petites
surfaces
homogènes
de
taille
variable.
Cette
dernière
méthode
nécessite
malheurèusement
un
nombre important d'observations et par conséquent un équipement
coûteux.
Ma theron
(1965)
avai t
développé
en
France
une
théorie
appelée
théorie
des
Variables
Régionalisées.
Cette
théorie qui

---

5
1
est
basée sur une méthode statistique
conçue
par Krige
(1960)
pour estimer la teneur en métal des gisements d'or en Afrique du
Sud
fera
l ' obj et
d'ample
explication dans
le chapitre suivant.
D'après
Journel
et
Hujbregts
(1978)
une
variable
régionalisée
peut être définie à partir de la corrélation qui existe entre des
observations
voisines
les
unes
des
autres.
La
théorie
des
variables
régionalisées
s'applique
à
un
large
spectre
de
variables telles que les variables météorologiques,
écologiques,
géographiques,
océanographiques
(Henley,
1981).
Il Y a
lieu de
croire,
que
cette
technique
doit
pouvoir
s'appliquer
en
irrigation,
pour
estimer
avec
précision
la
valeur
de
l'évapotranspiration à
des endroits où aucune observation n'est
disponible et pour évaluer la précision d'une telle estimation.
Elle
peut
permettre
de
cartographier
la
variation
de
ces
paramètres.
Autre avantage de la théorie est qu'elle n'exige pas
nécessairement
que l'échantillonnage soit randomisé.
1.3. Objectifs et Limitations de cette Thèse
,
.~
Ce travail
de
recherche
vise à étudier comment
la
théorie
des
Variables
Régionalisées
pourraient
s'appliquer
à
la
variabilité spatiale de l'évapotraspiration . Cinq objectifs sont
visés à savoir:
1.
Définir
à
l'aide
du
Sémivariogramme
le
modèle
suivant
lequel l'évapotranspiration régionale varie en fonction des
coordonnées géographiques
2.
Tester
la
validité
du
modèle
en
comparant
les
données
observées à leurs estimations
3.
Estimer les valeurs de l'évapotranspiration de référence
aux
locali tés
où
les
observations
n'auront
pas
été
effectuées
4. Quantifier les erreurs associées aux estimations
5. Représenter par des isolignes,
la variabilité spatiale de
l'évapotranspiration ainsi que les erreurs d'estimation sur
toute la région étudiée.

---

6
La
région
étudiée
est
l'Etat
d'Oregon
aux
USA
où
les
observations
journalières existent à
175
stations
au cours
des
mois
de
Mai
à
Septembre
de
l'année
1979.
Pour
chaque
mois
la
moyenne
sera
calculée
en
même
temps
que
les
coordonnées
géographiques associées à la station.
2. REVUE
BIBLIOGRAPHIQUE
Ce
chapi tre
sera
consacré
à
1 1 évol ution
du
concepte
d'évaporation et d'évapotranspiration à
travers
l'histoire,
des
explications
sur
la
Théorie
des
Variables
Régionalisées
en
insistant sur ses aspects spécifiques adaptés à cette étude.
2.1.
Evolution du Concept d'Evapotranspiration de Référence
De récentes études ont permis de comprarer les résultats de
mesure
d' évapotranspiration
de
référence
par
lysimétrie
à
ceux
obtenus
par
de
méthodes
dites
empiriques
basées
sur
l'utilisation de relevés météorologiques.
Ces études ont montré
d'après Doorenbos et Pruitt (1977),
que ces
méthodes empiriques
permettent
d'atteindre
une
précision
d'environ
95
pourcent.
Un
tel
succès
n'a
peu
être
atteint
qu'après
de
multiples
débats
phisolophiques sur le concept d'évaporation, de laborieux travaux
de
recherche
et
une
énorme
accumulation
de
connaissances
scientifiques depuis l'Antiquité.
Nous
renvoyons
à
Brutsaert
(1982)
qui
a
donné
un
aperçu
sur
les
progrès
du
concept
de
l'évapotranspiration à travers l'histoire.
2.1.1. Historique du Concept de l'Etat.
Parmi
les
gens
de
l ' Antiqui té,
les
Grecs
sont
le
plus
contribué à expliquer les relations qui existent entre le soleil,
les nuages et la pluie. D'après Anaximander de Miletos (565 B.C.)
dont les points de vue ont été rappelés sur la question par Diels
(1934):

---

7
"Le Vent est un courant d'air dont la partie la plus sfine
et la plus humide est extraite e~-dissoute par le soleil."
Plutard
Perrault
(1674)
montrera
que,
contrairement
à
la
croyance des Grecs,
la chaleur (soleil) n'est pas la seule cause
de l'évaporation mais aussi le vent.
Il faut attendre l'avènement
de
DaI ton
(1801,
1802)
pour
que
ces
concepts
de
l'évaporation
soient
défini tivement
clarifiés
et
que
le
phénomène
soit
quantifié en terme d'équation de Dalton (Eq.
2.1).
Dans
la
formule
de
DaI ton,
le
rôle
de
la
température
bien
que
sous-entendu
n'est
pas
explicite.
Il
faudra
attendre
la
contribution
de
Stefan
(1879) ,
Boltzmann
(1884) ,
des
lois
de
Fourier sur le transfert de chaleur,
de Ficks sur le, transfrt de
masse,
de
Newton
sur
le
cisaillement
en
milieu
visqueux,
l'analogie
de
Reynold
sur
la
similitude
des
fonctions
aérodynamiques
impliquées
dans
les
phénomènes
de
transfert
de
chaleur
et
de
transfert
de
masse
pour
que,
Penman
(1948)
établisse
la
formule
d' évaporation
qui
porte
son
nom
et
qui
consti tue
le
point
de
départ
d'une
quantificatiion
précise
de
l'évapotranspirapion. Cette équation
s'écrit (Eq.
2.13
).
2.1.2.
Méthodes
Pratiques
de
Calcul
de
l'evapotranspiration
Locale
Bien
que
cette
formule
ait
été
établie
sur
des
bases
théoriques,
elle
demeure
quelque
peu
empirique
à
cause
de
la
fonction
aérodynamique
f( u)
dont
l'évaluation
ne
peut
se
faire
que sur la base d'hypothèses.
Due à cette difficulté,
nombreuses
sont
l e s
formules
é t a b l i e s
sur
l ' é v a l a t i o n
de
l'évapotranspiration
à
partir
d' hypothèses.
Parmi
ces
méthodes
figurent
la
formule
de
Blaney
et
Criddle
modifiée
par
FAO
(Eq.2.15
à
Eq.
2.20)
qui
s'est
avérée
précise
dans
l'état
d'Oregon. Cet Etat constitue le site expérimental pour ce travail
de recherche .
.'

---

8
2.1.3. Estimation de l'Evapotranspiration Régionale
Il
existe
très
peu
de
méthodes
permettant
de
passer
de
l'évapotranspiration locale à
l'évapotranspiration régionale.
La
représentation d'un phénomène météorologique
,
comme la vitesse
du
vent,
sur
une
grande
surface
par
un
paramètre,
implique
un
lissage grossier des valeurs observées sur de petites surfaces.
Ainsi
l ' évapotranspiration
(ETR)
calculée
à
partir
de
ces
paramètres
résultants,
représente
elle-même
une
estimation
moyenne. Une telle homogénéisation à grande échelle conduit à une
estimation
supposée
représenter
la
région.
Les
paramètres
mété 0 r 0 log i que s u t i l i s é s
dan s
l ' é val ,u a t ion
d e
l'évapotranspiration
sont
mesurés
à
des
altitudes
faibles
(1
à
2m
)
du
sol.
Des
mesures
à
une
telle
al ti tude
sont
peu
représentatives de la région (la grande échelle).
Pour percevoir
le paramètre représentatif,
i l
faudrat effectuer les mesures au
niveau
de
la
couche
limite
correspondante
à
la
taille
de
la
région
en
question.
Les
figures
1/
3
et
4
illustrent
quali tativement
les
erreurs
imputables
à
une
méconnaissance de
cette réalité.
Il
existe des
difficultés,
non
seulement sur le
plan
de
la
mesure
par
télédétection
de
ces
paramètres
météorologiques
régionaux,
mais
également
sur
le
plan
méthodologique
en
ce
qui
concerne
la
combinaison
de
ces
paramètres
avec ceux mesurés
au
niveau du
sol,
pour en déduire
l'estimation de l'ETR.
2.2.
Les Bases Théoriques de la Géostatistique
Le but de ce chapitre est de faire la lumière sur certains
outils
utilisés
par
les
géostatisticiens
pour
caractériser
la
variabilité
spatiale d'une variable régionalisée.
Il
s'agit de:
semivariogramme, variance d'estimation,
krigeage.

---

9
2.2.1.
Le Semivariogramme
Matheron
(1962,
1963)
publia son traité sur la Théorie des
Variables Régionalisées à la suite de travaux empiriques réalisés
par
Krige
en
vue
d'estimer
des
réserves
en
minerais
dans
des
mones d'or en afrique du Sud.
Le terme Géostatistique réfère à
l'exploitation et à la mise en application de cette théorie. Bien
que la Géostatistique ait
été conçue pour
son utilisation dans
l'estimation
des
réserves
minières,
Clarck
(1979)
soutient
que
cette
science
peut
être
utilisée
à
profit
sur
toute
variable
dépendante
de
l'espace
ou
du
temps.
Ainsi
toute
variable
qui
dépend de sa position géographique est une variable r~gionalisée.
A cause de la continuité spatiale des réalisations
[z(xi),
i
=
l,
2, .. ,n]
de l'évapotranpiration de référence (ETr)
en
une
région donnée
au
cours
d'un même
instant
ou d'une même
période,
cette
variable
ETr
peut
à
priori
être
qualifiée
devariable
régionalisée
par
définition.
D'autre
part
cette
variable
est
une
variable
aléatoire
à
cause
du
caractère
aléatoire
des
paramètres
climatiques
qui
rentrent
dans
sa
'-
formulation
à
savoir,
l'intensité
du
rayonnement
solaire,
la
vitesse
du
vent,
la
température,
l'humidité
relative,
etc.
Une
telle variable,
di te
régionalisée,
possède en
une
région et en
une
période
données
une
structure dont
la
variabilité
spatiale
est
caratérisée
par
le
semi variogramme.
La
maj eure
partie
des
explications
données
ici
sur
le
semivariogramme
proviennent de
David (1977) et de Journel et Hujbregts (1978). Dans ce document,
la théorie a édté simplifiée por n'enretenir que ses aspets qui
s'appliquent à la variabilité spatiale de l'évapotranspiration.
Soi t
une
variable
régionalisée
(Spatiale
ou
géographique)
mesurée à N endroits,
xi (i = 1,2, ... ,N) et dont les réalisation
en
ces
localités
sont
z(xl),
z(x2)'.'.'z(xn ).
Une
manière
de
comparer deux valeurs z(x) et z(x + h) séparées par la distance h
kilomètres est de calculer leur différence [z(x)
-
z(x + h)].
si
au
lieu
de
deux
valeurs
z(x)
et
z(x
+
h)
i l
existe
n(h)
réalisations de cette différence avec n(h)
comme fonction de h,

---

10
alors
i l
est
logique
de
calculer
plutôt
la
moyenne
des
différences
[z(x)
-
z(x + h)]
et ceci par la formule (Eq.
2.27).
Dans
cette
équation,
D( h)
est
l'estimation
de
l'espérance
mathématique
de
la
différence
[z(xi)
z(xi+h)].
La
représentation
graphique
de
D( h)
en
fonction
de
h
s'appelle
déri ve
et
sa
déviation
par
rapport
à
zéro
indique
combien
la
région est hétérogène en ce qui concernee la propriété z(x).
Si
quelque soit h,
D(h) demeure nulle de façon significative,
dansz
toute la région,
on dit que cette variable régionalisée z(x)
a
une stationarité de premier ordre ( c'est-à-dire qu'il n'y a pas
de
dérive).
Cette
propr iété
est
indiquée
par
la
formule
statistique (Eq. 2.32).
E [z(x) - z(x + h)]
=
a
La dérive,
si elle existe,
peut être négative ou positive,
mais
en géostatistique
il
suffi t
de
savoir
que
la
dérive
existe ou
non. Ceci revient à se limiter au calcul soit de
E
(x)
-
z(x + h)]
OU
à celui de la formule
(Eq.
2.29).
Cette dernière formulation
permet
d' évi ter
de
s'encombrer
avec
le
signe
"valeur absolue".
Sur le plan pratique cette formule est équivalente à
la formule
(Eq.2.30).
On en déduit
la fonction semivariogramme r(h)
donnée
par l'équation (Eq. 2.31)
où
:
r(h) est la fonction sémivariogramme
n(h) est le nombre de paires séparées par la distance h.
2.2.2. Hypothèses de Stationarité
En
allant
de
la
plus
contraignante
à
la
plus
générale,
trois
hypothèses
servent
de
bases
à
l'application
de
la
Géostatistique.
La technique de Géostatistique à utiliser en vue
d'estimer
les
réalisations
de
variables
régionalisées
dépende
des conditions suivantes:
1) Stationarité de premier ordre:
Cette stationarité suppose est praduite par une vérification de
.......,

---

Il
la relation:
E [(x) - z(x +h)]
=
a
Cette
stationari té
de
premier
ordre
qui
est
très
rarement
observée dans l'étude des phénomènes ou propriétés
naturels.
2) Stationarité de deuxième ordre:
Cette
condition
indique
que
la
covariance
existe
et
qu'elle
dépend de la distance h en moyenne et non pas de la localité xi.
Elle se traduit par (Eq.
2.33):
Cov(x, x + h) = E [(z(x) - m)*(z(x + h) - m»)
La stationarité de la covariance implique celle de la variance et
par
voie
de
conséquence,
la
stationarité
de
la
fonction
semivariogramme.
Il en résulte que (Eq.
2.34):
Var[z(x)]
= E [(z(x) - m)2] = C(O)
La
fonction
semivariogramme
peut
donc
être
calculée
par
la
différence entre la variance et la covariance (Eq. 2.38).
r(h) = C(O) - C(h)
La stationarité de deuxième ordre suppose donc que C(o) et C(h)
existent et sont indépendantes de x à l'intérieur de la région.
3) Hypothèse intrinsèque:
On
dit
qu'une
variable
aléatoire
satisfait
l'hypothèse
intsrinsèque lorsque
E
[z(x)]
= m
m étant une valeur finie et que (Eq. 2.40):
Var[z(x) - z(x+h)]
=
E [(z(x) - z(x+h»2]
=
2 (h)
existe et ne dépend pas de x.
Cette
hypothèse
est
vérifiée
dans
le
cas
de
stationari té
de
second ordre mais la réciproque n'est pas vérifiée.
3) Quasi-stationarité (Hypothèse du Krigeage Universel):
Cette hypothèse est moins contraignante que les deuxs premières.
Elle suppose que
Var [z (x)
-
z (x
+ h)]
est
stationnaire dans
les sous-régions de la
zone d'étude et que
E[z(x)]
qui n'est
plus
une
constante
varie
de
façon
régulière
dans
chacune
des
sous-régions.
Il en résulte que pour
h 5 b o
(bo définissant le
voisinage de la sous-région) on a:
E [z(x)]
= m(x)
Il
.
1
i,

---

12
E [z(x + hl]
=
m(x+h)
D(h) = E [z(x) - z(x + hl]
Dans
ce
cas,
la
dérive
D(h)
est
non
nulle.
Si
elle
peut
être
obtenue par une regression en fonction de h alors elle peut être
utilisée
dans
le
krigeage.
Ce
krigeage
est
appelé
krigeage
universel.
Le
Krigeage
universel
est
toujours
possible
si
aucune
contrainte financière ne s'impose car il est toujours possible de
réduire suffisamment le domaine de définition de b o de manière à
respecter les conditions de la stationarité de deuxième ordre.
2.2.3. Propriétés du Semivariogramme
La fonction semivariogramme représente toutes les valeurs de
. r( h)
en
fonction
de
h.
La
représentation
graphique
de
cette
fonction par des
données
numériques est appelée semivariogramme
expérimental.
Dans
presque
toutes
les
publications,
le
terme
semivariogramme est utilisé aussi bien pour désigner la fonction
r( h)
que
pour
identifier
la
représentation
graphique
de
cette
fonction.
Bien
que
cela
que
cela
puisse
mener
une
certaine
confusion,
nous
maintiendrons
dans
ce
texte
cette
absence
de
distinction a
fin
de
ne
pas
entraîner d' autres
confusions avec
des termes analogues utilisés en Statistique classique.
La Fonction Semivariogramme Idéale
Cette
fonction
est
représentée
par
Fig. 5.
Le
semivariogramme
est
toujours
une
fonction
posi tive.
D'Où
( Eq. 2 • 43 ) :
r(h)
= r(-h) ~ a
et
(0)
= a
Dans la pratique,
il peut arriver que (Eq.
2.44):
r( 0) = Co =
a
Dans le cas de stationarité de deuxième ordre,
le semivariogramme
sera asymptotique à la droite horizontale d'équation (Eq. 2.45):
r( h)
=
Co + Cl
Co + Cl est appelé valeur plafond u semivariogramme.
A partir d'une certaine valeur (h
a),
la différence (Eq. 2.46)

---

13
r(h) -
(Co + Cl) = Ea
devient négligeable. La distance,
a,
s'appelle portée. Au delà de
cette valeur de h,
la·correlation entre z(x)
et
z(x + h)
cesse
d'exister de façon significative.
Modèles Isotropiques de Semivariogramme
Tous les
semivariogrammes observés correspondent à
l'un ou
l'autre des quatre modèles suivants
(1) Modèle Linéaire (Eq. 2.47)
r(h) = Co
b*h
0 ~ h ~ a
r(h) = Co + Cl
h > a
oû b est la pente de la fonction
(2) Modèle Sphérique ou Modèle de Matheron (Eq. 2.48)
r(h)
Co + Cl[1,5(h/a) - 0,5(h/a)3]
0 ~ h' ~ a
r(h) = Co + Cl
h > a
D'après David
(1977),
ce modèle est le plus observé et la
tendance
actuelle
consiste
à
utiliser
uniquement
le
modèle
Sphérique.
(3) Modèle Exponentiel (Eq. 2.49)
r(h) = Co + Cl[l - exp(-3(h/a»]
r(h) = Co + Cl
h
> a
(4) Modèle gaussien (Eq. 2.50)
r(h) = Co + Cl[l - exp(-3(h/a)2)]
oû
a
est la valeur de h à partir de laquelle le semivariogramme
devient visiblement stable.
Modèles Anisotropigues de Sémivariogramme
Une
propriété
est
di te
anisotropique
lorsqu'elle
ne
se
reprodui t
par
identiquement
dans
toutes
les
directions.
Si
la
variable
regionalisée
qui
fait
l'objet
de
l'étude est
supposée
anisotropique il
est recommandé d'en étudier le semivariogramme
dans
chacune
des
directions
oû
des
caractéristiqùes
(modèle
isotropique,
Co,
Co+Cl,
a
pourraient
varier.
En
cas
d'anisotropie
un
seul
semivariogramme
isotropique
ne
peut
plus
être utilisé,
il faudra combiner ces différents semivariogrammes
pour en déduire un modèle anisotropique de semivariogramme avant
de l'utiliser pour le krigeage.

---

14
Il
existe
deux
types
de
l'anisotropie
l'anisotropie
géométrique et l'anisotropie zonale. Un semivariogramme
(x,y,z)
dans un espace à 3 dimensions possède une anisotropie géométrique
lorsqu'il peut être transformé en un modèle isotropique par une
transformation linéaire du vecteur distance h.
Ce qui revient à
transformer le semivariogramme d'une direction à
l'autre par un
simple changement de coordonnées
suivant les
formules
(Eq.
2.51
et Eq. 2.52).
Toute anisotropie qui ne peut pas être reduite à l'isotropie
par cette transformation linéaire est une anisotropie zonale. Un
modèle d'anisotropie
zonale
peut être décomposé en
la
somme de
deux composantes dont l'une est isotropique et l'autre zonale
comme l'indique le formule (Eq.
2.53):
r(h)
riso( Ihl) +
r zon ( Ihzl)
où :
est la valeur absolue de h
est
la
valeur
absolue
de
la
composante de
h
dans
la
direction d'anisotropie zonale.
Avec
une
transformation
appropriée
des
coordonnées
un
semivariogramme
anisotropique
peut
toujours
être
redui t
à
une
somme
de
semivariogrammes
isotropiques
(Journel
et
Hujbregts,
1978) par la formule (Eq.
2.54).
2.2.4. Krigeage Simple
Le Krigeage est une méthode d'estimation locale qui produit
le
meilleur
estimateur
linéaire
non
biaisé
appelé
en
abbrégé
BLUE
(the
Best
Lineaire
Unbiaised
Estimator)
exprimé
par David
(1977) selon la formule (Eq.
2.55) où :
z(xi)
i
= 1, ..... ,n
sont les valeurs observées aux
localités Xi
z*(xo ) est l'estimation par krigeage au point xo
wi
(i = l, ... ,n)
sont les poids obtenus par résolution de
l'équation 2.65.
Le
calcul
des
poids
wi
doi t
répondre
aux
deux
conditions

---

15
suivantes :
(1) Condition d'Absence de Biais
Cette
condi tion
exige
que
l'estimateur
de
krigeage
z * (xi)
reproduire au point d'observation exactement la valeur observée.
Cela revient à
E [z * (xo ) - z ( X o )] = 0
Ce qui entraîne les équations (Eq.
2.57 à Eq.
2.60).
(2) Condition de la Variance Minimum d'Estimation
cette condition s'écrit
Minimizer
E ([z*(xo ) - z(xo )]2)
Le Système d'Equations de Krigeage
L'application des deux conditions décrites ci-~essus permet
d'aboutir au système d'équations linéaires de krigeage (Eq.2.76),
dans lesquelles:
wi
et
Wj
sont
les
poids
identiques
à
ceux
exprimés
dans
l'équation de krigeage
est le paramètre de Lagrange
C(xi '
Xj) est la covariance de z(x) aux points
xi
et
Xj
Sous forme matricielle l'équation (2.76) devient:
[ K ]
[ W ]
[ R ]
où
[ K ],
[ W ]
et
[
R ] sont donnés par les formules
(Eq. 2.78
et
Eq.
2.79).
K
est
appelée
matrice
de
krigeage.
Quant
à
la
variance d'estimation elle est donnée par la formule
(Eq.2.88).
Quant au système d'équations de krigeage il est donné
par la formule (Eq.2.88) où:
hij et hoi sont des vecteurs définis par les couples (xi,
Xj) et
(xo ' xi)·
La. résolution de ce
systhème donne
les
poids wi
qui
irytroduits
ensui te
dans
l ' équa tion
de
krigeage
permet
d'obtenir
l'estimation z * (xo ).
2.2.5. Krigeage Universel
Lorsqu'une
dérive
non
nulle
e x i s t e ,
certains

---

16
géostatisticiens
recommandent
de
l'incorporer
au
système
de
krigeage. Celé suppose que la dérive puisse être mise en équation
par
une
regression
linéaire.
Il
st agit
alors
de
krigeage
universel.
Journel
(1977)
fournit
un exemple de la manière dont
le système d'équations de krigeage est modifié lorsque la dérive
peut
être
représentée
par
un
polynôme
du
second
dégré.
Il
est
donné sous forme matricielle par la formule (Eq.2.90):
[ K2 ]
[ W ]
[ R2 ]
où l K2 ],
[ W ] et [ R2 ] sont donnés par les formules (Eq.2.9l
à Eq. 2.93) .
Le
krigeage
universel
permettent
d'améliorer
la
précision
des
calculs par rapport au krigeage simple.
Il faut pou~ celà que la
dérive
soit
une
fonction
simple
pouvant
être
mise
en
équation
(Gambolati,
1978). Le succès du krigeage universel tient beaucoup
de
l'expérience.
Cette
question
continue
de
faire
l'objet
de
recherche (David,
1977).
3. PROCEDURE ET COLLECTE DES DONNEES
L'Etat
d'Oregon
(U.S.A.)
a
été
choisi
comme
site
géographique
expérimental
pour
tester
le
model
conçu
pour
l'étudier
de
la
variabilité
spatiale
de
l'évapotranspiration
(ET).
Ce
chapitre
explique
la
procédure suivie
pour
rassembler
les
données
météologiques
en
vue
du
calcul
de
ET
et
pour
l'estimation
des
données
manquantes.
Le
chapitre
explique
également le mode de calcul des ET
locaux et pose les hypothèses
de base de l'analyse géostatistique.
3 . 1 .
Collecte
des
Données
et
Choix
de
la
Méthode
d'Evapotranspiration
La
revue
bibliographique
a
permis
de
comprendre
que
l'estimation de
l'évapotranspiration par les méthodes de Penman
ou
de
Montei th
qui
ont
une
base
théorique
nécesssi te
que
certaines données
telles que
le rayonnement solaire,
la vitesse

---

17
du
vent,
et
la
tension
de
vapeur
d'eau
dans
l'air
soient
disponibles.
Malheureusement
ces
données
n'étaient
disponibles
qu'à
un
nombsre
réduit
de
stations climatologiques
dans
l'Etat
d'Oregon.
Il
en
résulte
la
contrainte
de
choisir
une
méthode
empirique pouvant utiliser les données disponibles.
3.1.1. Données Météorologiques Disponibles
Ces
données
sont
consignées
dans
les
publications
officielles
sous
formes
de
bulletins de
NOAA
(National
Oceanic
and Atmospheric Administration,
NOAA 1979).
Des 284 stations de
NOAA
dans
l'Etat
d'Oregon,
175
produisent
les
,températures
minimum
et
maximum,
Tx
et
Tm,
journalières.
Quant
aux
autres
données
météorologiques
utilisées,
le
Tableau
l
fai t u n e
récapitulation de leurs nombres disponibles.
3.1.2. Choix d'une Méthode d'Evapotranspiration
La méthode de FAO Blaney-Criddle est une méthode qui utilise
fondamentalement
les
températures.
D'autres
part
des
études
compa~atives de précision entre les méthodes de Penman et de FAO
Blaney-Criddle dans
les Etats de Washington au Nord d'Oregon et
d'Idaho à
l'Est ont montré que la méthode de FAO Blaney-Criddle
est plus précise que celle de FAO Penman.
Pour ces deux raisons
la méthode de FAO Blaney-Criddle a été choisie.
La méthode a été
tradui te
en
langage
informatique
FORTRAN.
Le
programme
permet
également de traduire les longitudes et latitudes des stations en
coordonnées
cartésiennes.
Elle
tient
compte
également
de
l'al ti tude
de
chaque
localité
afin de
produire
des
corrections
de ET par rapport au niveau de la mer (Allen et Brockway,
1982).
Les détails de calcul
sont consignés dans
le
programme MAIN en
Annexe A.

---

lB
3.1.3. Estimation des Données Météorologiques Manquantes.
Les données manquantes
sont de premier ordre
(température)
et
de
second
ordre
(humidité
relative,
rayonnement
solaire,
vitesse du vent).
Relevés Météorologiques de Premier Ordre
175
stations
étaient
toutes
équipées
pour
fournir
au moins
les
températures
journalières.
Mais
i l
arrivait
parfois
qu'une
station
soit
défectueuse
pour
un,
deux
ou
quelques
jours.
Ces
données
manquantes
étaient
estimées
par
une
moyenne
glissante.
Plus
la série de données manquantes
est
longue
plus
nombreuses
étaient
le
nombre
d'observations
utilisées.
Si
Ip
série
est
consti tuée
par
une
seule
donnée
manquante,
i l
suffisait
de
la
remplacer
par
les
deux
observations
adj acentes
(celle
de
la
veille
et
celle
du
lendemain).
Par
contre
si
la
série
est
constituée de deux données manquantes,
trois observations étaient
utilisées pour chacune des deux obdservations.
Celle qui devait
être estimée était remplacée par la moyenne de deux observations
adj acentes
à
la
manquante
et
une
seule
adj acente
à
l'autre
observation manquante.
Relevés Météorologiques de Deuxième ordre
Comme
l'indique
le
Tableau
1,
seulement
22
stations
étaient
équipées
pour
fournir
des
observations
météorologiques
journalières de deuxième ordre.
Vue la rareté de ces données (22
sur
175
données
de
températures),
la
méthode
de
FAO
Blaney-
Criddle
pouvait
être
efficacement
utilisé
avec
nécessité
de
corriger (étalonner) les estimations par l'emploi de ces quelques
données
de deuxième ordre.
La décision quant
à
savoir
laquelle
des
22
observations
devrait
être
utiliser
pour
corriger
l'estimation
de
ET
à
chacune
des
175
stations
dépendait
des
conseils
de
spécialistes,
des
particularités
topographiques
et
microclimatiques de
la région et des conseils recueillis auprès
des spécialistes en météorologie et travaillant sur le site.
Le
facteur
topographique
était
le
plus
important
et
dépendait
en
majeure partie de deux chaînes montagneuses.
La Chaîne Côtière et

---

19
les Cascades séparées par 144 kilomètres et divisant ainsi l'Etat
d'Oregon en 3 régions
du Nord au Sud à savoir la Chaîne Côtière
sur
l'Océan
Pacifique,
le
Plateau
Central
et
les
Collines
du
l'Est.
La Chaîne Côtière et le Plateau Central sont humides avec
une précipitation annuelle de 1.000 mm/an dans la vallée. La zone
Est des Cascades
est
semi-aride
à
aride
avec
une
précipitation
annuelle inférieure ou égale à 250 mm/an.
Toutes
ces
conditions
ont
permis
de
subdiviser
l'Etat
d' Orégon
dont
la
superficie
est
de
280.000
km 2
environ,
en
5
sous-régions dénommées par les lettres A, B, C, D, E. Chacune
de
ces
sous-régions
est
ensui te
morcellée
en
zones
d'influence
en
utilisant la technique des polygones de Thiessen.
Ces zones sont
concentrées au-tour des stations de deuxième ordre.
La figure 6
indique
une
carte
de
l'Etat
d'Oregon
avec
les
subdivisions
en
sous-régions et en zones d'influence.
ces zones d'influence sont
relatives aux observations de vitesse de vent.
3.2. Organisation des Calculs Géostatistiques
Les
calculs
g é o s t a t i s t i q u e s
comprennent
les
semivariogrammes,
les estimations
par krigeage et
les variances
de
ces
estimations.
La
méthode
de
calcul
des
semivariogrammes
dépend de la configuration géographique des données disponibles.
D'après
Journel
et
Huj bregts
(1978),
ces
méthodes
diffèrent
suivant que les données sont alignées où non,
sont régulièrement
espacées ou non.
3.2.1. Méthode de Calcul des Sémivariogrammes
Les stations d'observations
météorologiques
et
les
données
d'évapotranspiration
ET
qui
en
résultent
étant
dispersées
à
travers
l'Etat
d'Oregon,
il
était
nécessaire de
les grouper en
classes
définies
par
des
coordonnées
polaires
( e,
h) .
Cette
technique permet
de calculer
les
semivariogrammes
expérimentaux
sans perdre trop de données. Dans
r* l'étoile (*) indique que

---

20
r*(s, h) est une valeur calculée différente des semivariogrammes
théoriques
r( S, h).
Etant
donné
le
caractère
discret
des
distances
hj
et des
angles
Si,
i l
était nécessaire d'utiliser
les
tolérances
de
distance
oh
et
d'angle
oS
pour
grouper
les
données de ET en classes de distance et d'angle.
L'intersection
d'une classe de distance (hj + oh/2) et d'un; classe d'angle
(Si
+
oS/2)
définie une classe appartenant au point ou localité
(Si
h j ).
Etant
donnée
une
direction
Si,
tous
les
semivariogrammes
r* (Si
h j ) étaient calculés et représentés
graphiquement comme le semivariogramme de la direction Si' Notons
que hj représente l'intervalle de distance d'ordre j.
Sur
cette
base,
un
programme
informatique
nommé
VARIO
fut
écrite en FORTRAN pour grouper les données
de
ET en classes et
calculer,
pour chacune des 4 directions supposées d'anisotropie,
un
semivariogragamme
en
évoluant
par
intervalle
(décalage)
de
12.07km
(8mi)
avec
une
tolérance de
3.22
km.
Pour passer d'une
direction
à
la
suivante,
l 'intervale
est
de
n/4
avec
une
tolérance
de
n/8.
Les
directions
d'anisotropie
sont
les
suivantes:
Nord-Sud,
Est-Ouest,
Sud-Est-Nord-Ouest,
Sud-Ouest-
Nord-Est.
Le Programme VARIO était prévu pour calculer et représenter
graphiquement les semivariogrammes,
pour calculer et représenter
graphiquement
les
dérives.
L'Annexe
A
donne
entièrement
le
programme
VARIO.
Une
fois
le
semivariogramme
dessiné
par
ordinateur,
i l
fallait
lui
adapter
un
semivariogramme
idéal
choisi
par
affinité
parmi
les
4
semivariogrammes
dont
les
équations
sont données
dans
le Chapitre
2.
Ceci
étant
fait,
i l
fallai t
déterminer
pour
chaque
semivariogramme
la
portée
d'influence,
a,
ainsi
que
la
valeur
plafond
[Co
+
Cl].
La
comparaison de ces
caractéristiques
pour
les
4
semivariogrammes
permet d'identifier la présence d'anisotropie.
David
(1977)
donne
l'équation
du
modèle
anisotropique
(Eq.3.1
et
Eq.
3.2)
pour
le
cas
où
les
semivariogrammes
directionnels
sont
du
type
sphérique
avec
absence
d'ordonnée
à
l'origine.

---

21
3.2.2. Adaptation du Modèle de Semivariogramme Sphérique
Les
géostatisticiens
tels
que
M.
Armstrong
(Verly et
al.,
1983)
mettent
en
garde
contre
l'utilisation
systématique
des
méthodes classiques de statistique (regression linéaire,
méthode
des moindres carrées) pour adapter une courbe
r(h) théorique â
une courbe
r*(h) observée,
car une telle pratique peut éloigner
des conditions de stationarité condition nécessaire au krigeage.
Le modèle sphérique est celui communément choisi pour application
de la géostatistique à des phénomènes hydrologiques.
Les étapes
à
suivre
pour
adapter
un
modèle
sphérique, r( h)
â
un
semivariogramme expérimental
r*(h) sont les suivantes:
1) Faire coïncider une ligne droite aux premiers points
(proches des petites valeurs de h)
représentés par [r*(h)
, h].
La pente de cette droite est Cl
2)
L' intersection
de
cette
ligne
avec
l'axe
des
(h)
est
l'ordonnée à l'origine ou l'effet pepite Co.
3)
Tracer
une
horizontale
passant
par
le
point
d'ordonnée
[Cl + Co] donnée par la formule (Eq. 3.3) où M est la moyenne des
N observations.
4) L'intersection de cette horizontale avec la droite tracée
â
l'étape 1 se trouve â
la distance 2a/3 de l'axe des r(h).
On
en déduit la valeur de la porte d'influence,
a.
Au cas
où
la
courbe
r( h)
n'admet
pas
d'asymptote,
il y
a
une
anisotropie
zonale.
Il
est
alors
recommandée
de
faire
correspondre
la
plus
grande
des
valeurs
plafond,
[Cl
+
Co],
variance de toutes les observations de
l'échantillon et d'apter
visuellement les autres valeurs plafond.
3.2.3. Krigeage et Cartographique des Isolignes
d'Evapotranspiration
La représentation par isoligne de la variabilité spatiale
de
l'évapotranspiration
sur
l'Etat
d'Orégon
nécessitait

---

22
l'utilisation de logiciels tel que COMPLOT et PLOTLIB disponibles
dans
plusieurs
librairies
de
systèmes
informatiques.
L'utilisation
de
ces
logiciels
nécessite
que
les
données
d'ET
soient
disponibles
sur
un
réseau
à
maille
quadrilatère.
Un
programme
informatique
nommé
KRIGX
a
été
mis
au
point
pour
estimer
par
krigeage
les
ET
à
tous
les
points
d'un
réseau
à
maille carrée de côté 12.87 km (8 miles).
L'estimationn de l'ET a
en
un
point
du
réseau
nécessitait
l'utilisation
des
8
observations les plus proches.
La
figure
8
indique ce
réseau maillé
tel
qu'il est généré
par
l'ordinateur.
Il
en
résulte
que
le
krigeage
d'un
point
nécessitait la résolution d'un système de krigeage d~ 9 équations
à
9 inconnues.
Huit inconnues sont les poids et la 9 ème est le
paramètre de Lagrange.
L'annexe
A
présente
l'ensemble
du
programme
KRIGX.
Ce
programme
utilise
le
logiciel
LUDATF
de
l ' IMSL
(International
Mathematical
and
Statistical
Library).
Ce
logiciel
permet
la
résolution des systèmes d'équations linéaires.
Il comprend aussi
un
sous
programme
GAMMA
qui
calcule
les
valeurs
des
semivariogrammes
r( ei'
h j ) sur la base du modèle anisotropique
observé dans le cadre de ce travail de recherche.
3.3.
Construction
de
Semivariogramme
avec
des
Données
Insuffisantes.
C'est un problème crucial pour le géostatisticien,
face au
problème de krigeage sur une vaste étendue hétérogène, de décider
s ' i l
doit
utiliser
toutes
ses
données
a
fin
de
définir
un
semivariogramme
anisotropique
ou
isotropique
pour
toute
la
région,
ou s ' i l est préférable de subdiviser la région en sous-
régions
plus
homogènes
et
de
calculer
pour
chacune
d'elle
un
semivariogramme.
Cette dernière
alternative entraîne
souvent
le
problème
de
n'avoir
disponible
qu'un
nombre
trop
réduit
d'observations.
Jounel et Huijbregts (1978)
suggèreent 30 paires
de
données
comme
un
ordre
de
grandeur
pour
le
calcul
d'un

---

23
semivariogramme
bien
que
des
quantités
inférieures
aient
été
utilisées
par
d'autres
géostatisticiens
(Verly
et
al.,1983).
Dans
cette
étude
et
pour
certaines
sous-régions,
il
arrivait
qu'un
nombre
de
paires
inférieur
à
15
soit
disponible
pour
le
calcul de semivariogramme.
Face à cette difficulté un programme
nommé
VALID
a
été
écrite
pour
rechercher
parmi
les
semivariogrammes
utilisables
pour
la
sous-région
celle
qui
permettrait la meilleure estimation,
ou en d'autres termes,
des
estimations
les
moins
biaisées.
Un
tel
semivariogramme
était
désigné par semivariogramme optimum et testé par la technique de
la technique de Jacknifing.
Cette technique consiste à suprimer
chaque fois une observation et à l'estimer à l'aide-des autres,
puis à comparer cette estimation avec l'observation.
Les écarts
représentés par
[z(xo ) - z*(xo )] sont utilisés pour calculer la
moyenne mR et
la variance
oR2 des erreurs.
Le semivariogramme
optimum est celui dont mr et
OR2 sont les plus proches de zéro
et
un.
Le texte
implique
l'utilisation des
équations
(Eq.3.4
à
Eq.3.6).
Ce test de normalité des erreurs réduites a été suggeré par
Delhomme
(1976),
utilisé
par
Viera
(1983)
et
en
accord
avec
Snedecor
et
Cochran
(1967).
Le
programme
VALID
permet
entre
autres de faire ce test.
3.4. Méthode pour Tester la Justesse de l'Estimation.
La
méthode
itérative
décri te
ci-dessus
peut
être
utilisée
pour tester la validité d'un semivariogramme même si celui obtenu
par un nombre insuffisant d'observations.
En ce qui concerne la
vérification
de
l'absence
de
biais
ou
"test
de
justesse
de
l'estimation" la procédure est la suivante
1) Vérifier que la moyenne,
mR,
des erreurs réduites,
est
proche de zéro de façon significative.
2) Vérifier que leur variance,
OR 2 est proche de l'unité
de façon significative
3) En plus de ces deux conditions,
il faudra faire le test

---

24
de Student
Ho
mE = 0
contre l'alternative Ha
o
où
ETr - ETK
évapotranspiration observée
ETK
Evapotranspiration estimée par krigeage au même
endroit.
Le programme VALID permet également de calculer mE et OE 2 .
4. RESULTATS ET INTERPRETATION.
Ce chapitre présente
les
résultats
obtenus
à partir de la
méthodologie
adoptée
dans
le chapitre
précédent
et
des
données
climatiques rassemblées à travers l'Etat d'Oregon.
4.1. Estimation de l'Evapotranspiration Locale de Référence.
En
u t i l i s a n t
la
méthode
FAO
Blaney-Criddle,
l'évapotranspiration
de
référence,
ETr,
a
été
calculée
en
175
localités.
Ces valeurs ont été calculées pour chacun des mois de
Mai à Septembre 1979 et sont exprimées en mm/jour les résultats
des
calculs
pour
le
mois
de
Juin
ainsi
que
les
données
météorologiques
pour
ce
mois
sont
présentées
dans
l'annexe
B.
Pour
chacune
des
5
sous-régions
délimitées
dans
le
chapitre
précédent,
la variation saisonnière de l'évapotranspiration a été
représentée
graphiquement
pour
une
station choisie
sur
la
base
que tous les relevés climatiques nécessaires aux calculs de ETr y
sont disponibles.
La
Fig. 9
montre
ces
courbes
pour
Astoria,
Corvallis,
Medford,
Hermiston
et
Malheur-Branch,
correspondant
respectivement aux sous-régions A,
B,
C,
D et E .
Le Tableau 2
indique les moyennes de l'ETr pour ces 5 stations. A,
B, C, D, et
E.
L'allure de ces courbes indique l'ETr varie fortement au cours
de la campagne agricole qui va de Mai à septembre.
Par conséquent

---

25
nous avons décidé d'étudier mensuellement la variabilité spatiale
de
l' ETr.
Ces courbes montrent également que
l' ETr croit de
la
zone la plus humide
(Astoria sur la côte)
vers
la zone la plus
sèche (Malheur-Branch dans la zone semi-aride à l'Est d'Oregon).
4.2. Evaluation des Hypothèses de Stationarité.
La
Fig. 10 représente
la dérive
spatiale de
l'ETr au cours
du
mois
de
Juin.
Ce
qui
met
en
évidence
l'absence
de
la
stationari té
de
premier
ordre.
Ces
types
de
dérive
ont
été
observés pour les autres mois.
En ce qui concerne la stationarité
de deuxième ordre, celle-ci est mise en évidence par ,l'allure des
semi variogrammes
qui
toutes
tendent
vers
une
asymptote
(valeur
plafond,
Co
+
Cl).
Les
Figures
10
à
15
présentent
les
semi variogrammes
pour
chacun
des
5
mois.
A
part
le
mois
de
Septembre, tous les semivariogrammes sont anisotropiques dans les
directions Nord-Sud et Est-Ouest.
Du fait
de l'inexistence la stationarité de premier ordre,
le krigeage simple des ETr aux postes d'observations ont produit
des
estimations
biaisées,
ayant
pour
conséquence
une
erreur
systématique
dans
ces
estimations.
Ces
erreurs
apparaissent
toujours
lorsque la stationarité de premier ordre a été ignorée
(Volpi et Gambolati,
1978).
Cependant,
le plus important est de
voir si
les valeurs krigées aux stations climatiques coïncident
suffisamment avec les observations.
4.3. Estimation par Krigeage des Valeurs de l'Evapotranspiration.
La
Fig.
8
représente
le
réseau
carré
déployé
sur
l'Etat
d'Oregon.
Ce
réseau
comprend
approximativement
1600
points
d'intersection de
lignes Nord-Sud et Est-Ouest.
Certains de ces
points
coïncident,
dans
les
limites
de
tolérance
supposée
au
chapitre
précédent,
aux
stations
d'observations.
Aux
autres
points
non
occupés
par
les
stations,
les
valeurs
de
ETr
sont
estimées
par
krigeage
à
partir
des
caractéristiques
de

---

26
sémivariogrammes
indiquées
par
le
Tableau
3.
Pour
estimer
une
valeur de ETr,
les 8 observations les plus proches sont utilisées
en associ~tion avec le semivariogramme obtenu à partir des 175
observations.
Les valeurs estimées pour le mois de Juin 1979 sont
consignées en annexe B avec leurs coordonnées cartésiennes.
4.4. Estimation des Variances de Krigeage.
La
technique
de
krigeage
permet
de
calculer
aussi
la
variance
de
chaque
estimation.
Cette
variance
est
appelée
variance
de
krigeage,
est
donnée
par
l ' Eq.
2.88.
A
chaque
intersection
non
occupée
du
réseau,
le
programme
K~IGX calcule
cette
variance.
Elle
prend
la
valeur
zéro
au
niveau
des
intersections
occupées
par
les
observations.
Au
voisinage
des
limites de l'Etat d'Oregon,
où des observations sont inexistantes
de
l'autre
côté
de
la
frontière,
les
variances
de
krigeage
prennent des valeurs nettement élevées.
Les Figures 21 à 25 sont
les
représentations
graphiques,
obtenues
automatiquement
par
ordinateur,
de
la
variabilité
spatiale
de
la
variance
de
krigeage.
La
constance
des
courbes
d'iso-variance,
de
krigeage
d'un mois à
l'autre,
constitue la preuve
(Hughs et Lettenmaier,
1981) que la variance de krigeage dépend de la densité du réseau
d'observations et non pas de valeur numérique des observations.
Elle
dépend
cependant
du
semi variogramme
et
spécialement de
sa
valeur plafond
[Co"+ Cl].
Ainsi,
plus
[Co + Cl]
est élevé plus
importante est la variance de krigeage,
toute autre facteur étant
constant.
La
racine
carrée
de
la
variance
de
krigeage
est
un
estimateur
de
l'erreur
commise,
lorsqu'en
la
localité
de
l'estimation
de
krigeage,
z*(xo ) est utilisé à
la
place
de
l'observation z(xo ) qui n'est pas connue à moins que mesurée.
4.5. Auto-Validation des Modèles de Semivariogramme
Le
test
d'auto-validation
du
semivariogramme,
permet
de

---

27
mettre en évidence la précision avec laquelle les estimations par
krigeage sont en accord avec le semivariogramme obtenu à partir
des observations.
Les étapes de ce test sont consignées dans le
paragraphe 3.4.
Le Tableau 4 donne les valeurs des paramètres mr , mE' OR2 ,
OE 2 ,
tE
du
test,
ainsi
que
les
caractéristiques
des
semivariogrammes.
Les
valeurs de mE
y
sont
faibles
(proches de
zéro)
alors
que
les
valeurs
de OR2
à
part
le
mois
de
Juin,
sont
assez
proches
de
1.0.
Ce
qui
confirme
le
fait
que
les
estimations ne sont pas biaisées. D'autre part,
les valeurs de mE
(ETr - ETK) sont très petites et montrent que les valeurs ETr et
ETK (valeurs estimées)
sont pratiquement égales,
à
5% de niveau
de signification.
4.6. Modèles de Semivariogramme Sous-Régional
Sur
la
base
des
explications
du
Paragraphe
3.3.,
des
semivariogrammes
ont
été
obtenus
à
partir
de
quelques
observations disponibles dans chacune des 5 sous-régions A à E.
Le tableau
5 donne
les caractéristiques de ces semivariogrammes
ainsi
que
les
paramètres
correspondant
aux
tests
d'auto-
validation
pour
le
mois
de
Juin
1979.
La
comparaison
des
résul tats
figurant
aux
Tableaux
5
et
4,
montre
que
les
semivariogrammes
sous-régionaux
conduisent
à
de
meilleures
estimations
(moins
biaisées)
que
celles obtenues
à
partir d'un
semivariogramme régional (tout l'Etat d'Oregon). Cette conclusion
provient du
fait
que
les
tE et
les
oR 2
des
semi variogrammes
sous-régionaux,
sont respectivement plus petites et plus proches
de l'unité que ceux du semivariogramme régional.
Ces constations
sont en accord avec l'affirmation de Journel et Huijbregts (1978)
que l'hypothèse de stationarité est d'autant plus vérifiée que la
zone de définition de la fonction semivariogramme est réduite.
Il
s'agit
en
fait
de
la
quasi-stationarité
expliquée
dans
le
Paragraphe 2.2.2.

---

28
4.7. Comparaison des Modèles de Semivariogramme et des Modèles de
Moindre Carré.
Les
175
observations
d'évapotranspiration
de
référence
précédamment
utilisées
pour
le
krigeage
ont
fait
l'objet
d'une
regression linéaire en fonction des coodonnées cartésinnes X,
Y,
Z des lieux d'observations. Cette régression qui a été effectuée
par étape a débuté avec 9 variables à savoir X,
Y,
Z, X2, Y2,
Z2,
XV,
XZ et YZ.
Le Tableau 6 montre les différents cooefficients de
régression,
ainsi que les coefficients de détermination relatifs
à
chaque
mois.
Le
meilleur
coefficient
de
détermination
correspond
au
mois
de
Juin
soit
0,76.
Ceci
indique
dans
le
meilleur des
cas,
que
la
régression
linéaire ne
peut expliquer
que 76% des variabilités de ETr.
Le Tableau 6 indique aussi les
moyennes et variances des résidus. Ces valeurs comparées à celles
du
Tableau
4
indiquent
que
le
krigeage
dans
ce
cas,
est
préférable
à
la
régression
linéaire
et
cela
d'autant
que
le
krigeage
ne
fait
intervenir de deux variables
X et
Y alors
la
regression linéaire en utilise 9.
D'autres
méthodes
d'interpolation
telles
que
la
"moyenne
glissante",
l'" inverse des distances",
l'''inverse des carres des
distances",
n'ont pas fait l'objet d'une telle étude comparative
puisqu'elles
sont
à
priori
biaisées
(Henley,
1981;
Journel
et
Hujbregts, 1978).
4.8. Comparaison des Krigeages Simple et Universel
Le krigeage universel comme il a été expliqué dans la Revue
Bibliographique,
est
celui
qui
incorpore
la
forme
de
la dérive
aux
calculs,
alors
que
le
krigeage
simple
suppose
nulle
ou
négligeable la dérive.
La Figure 10 montre que
les dérives sont
linéaires,
ce qui permet d'éliminer X2,
Y2 et XY de l'Eq.
2.91.
L'estimation
par
krigeage
universel
a
été
effectuée
et
les
résultats du test d'auto-validation de ce krigeage sont consignés
-

---

29
dans le Tableau 7.
La comparaison des Tableaux 4 et 7 montre que
le krigeage simple dans le cadre de cette recherche constitue un
modèle meilleur que celui du krigeage universel.
En effet,
pour
le
krigeage
simple
les
moyennes
et
les
variances
des
écarts
réduits
sont respectivement
plus
proches
de
zéro et de 1.0.
Le
test
"t"
de
Student
permet
également
de
conf irmer
cette
conclusion.
Le
paramètre
du
test
est
calculé
de
la
manière
suivant la formule
:
tE = mE IN/crE
où
N est la taille de l'échantillon (N = 175).
Deux raisons principales permettent d'expliquer pourquoi le
krigeage simple est ici meilleur que le krigeage uniyersel alors
que
la
théorie
soutient
le
contraire.
La
première
est
que
la
fonction D(h) représente une situation moyenne pour l'ensemble de
l'Etat
alors
que
la
D* (h) ,
utilisée
pendant
le
krigeage
en
un
point,
est basé sur 8 observations seulement par rapport à
175.
Ainsi,
ces 8 observations peuvent appartenir,
avec le point x o à
kriger,
à
une
zone
très
homogène
et
où
la
dérive
en
réalité
serait
nulle.
Ainsi
le
krigeage
universel
pourrait
amener
au
retrait automatique d'une dérive qui en fait n1existait pas dans
ce cas.
La
deuxième
raison
est
que
la
technique
du
krigeage
universel
à
tant
que
méthode
d'estimation
n'est
pas
encore
suffisamment au
point.
La
technique suivant
laquelle,
la dérive
doit être retranchée des
observations n'est pas encore parfaite
et continue de faire l'objet d'études et de controverses.
4.9. Cartes d'Iso-Evapotranspiration de Référence
Les
cartes
d' Iso-évapotranspiration
sont
obtenues
à
partir
des
valeurs
estimées
ETK
de
krigeage
et
leur
exploitation
par
ordinateur
pour
générer
les
courbes.
Ces
courbes
sont
ensui te
calquées
sur
un
fond
représentant
la
carte
(les
limites)
de
l'Etat d'Oregon.
Les
Figures
16
à
20
représentent
ces
cartes
d' iso-

---

30
évapotranspiration. Comme on devait s'y attendre,
l'intensité des
isolignes croit de la zone humide de la côte Ouest vers la zone
semi-aride de l'Est Oregon.
L'allure Nord-Sud des isolignes,
est
en
accord
avec
un
travail
de
recherche
analogue
effectué
par
Cuenca et al.
(1981)
pour l'Etat de Californie,
au Sud d'Oregon
et où les isolignes d'évapotranspiration étaient dessinées à
la
main.
Peu d'isolignes sont représentées sur les cartes,
dans le
souci de les rendre bien lisibles.
En réalité le logiciel utilisé
est conçu pour donner plus de détails selon qu'il est nécessaire.
La
tendance
saisonnière
de
l' ETr
observée
dans
la
Figure
9
se
retrouve
lorsque
les
cartes
d'iso-évapotranspiration
sont
comparées d'un mois à l'autre. Les valeurs lues sur les isolignes
correspondent bien aux observations des stations de météorologie.
Pour toutes ces raisons,
nous pensons que ces cartes offrent une
bonne
i l l u s t r a t i o n
de
la
v a r i a b i l i t é
spatiale
de
l'évapotranspiration de référence à travers l'Etat d'Oregon.
(
4.10 Cartes d'Isovariance de Krigeage
Les
variances
de
krigeage
sont
calculées
par
le programme
KRIGX
après
chaque
estimation
de
ETK.
Ces
variances
sont
cartographiées sous forme d'isolignes grâce à des logicielles et
sont représentées par les Figures 21 à 25.
Ces figures indiquent
que
les
variances
de
krigeage
sont
faibles
là
où
la
densité
d'observations est forte,
comme
c'est le cas dans la vallée de
Willamette,
alors qu'elles sont faibles sur les frontières Nord-
Est de l'Etat où les observations sont dispersées.
L'intérêt que
présente
les
cartes
d' isovariance
de
krigeage,
est
d'indiquer
comment
l'erreur
d'estimation
peut
être
diminueé
au
fur
et
à
mesure que
les
nouvelles
stations
sont
ajoutées
au
réseau
des
stations d'observations.
L'examen
des
Figures
21
à
25,
montre
que
l'allure
des
isolines demeure
relativement
stable
d'un mois
à
l'autre.
Ceci
confirme l'assertion selon laquelle les écarts relatifs entre les
variances de krigeage varient non pas en fonction de la taille

---

31
des variances mais plutôt en fonction de la densité du réseau des
observations.
5. CONCLUSION ET RECOMMANDATIONS.
5.1. Conclusion
Des données climatiques ont été observées sur 175 stations
météorologiques à
travers
l'Etat d'Oregon.
Ces observations ont
été
utilisées
pour
estimer
les
taux
d' évapotranspiration
mensuelles,
exprimés
en
moyenne
journalière.
Ces
taux
ont
été
utilisés
pour
déterminer
des
modèles
de
semivariogramme
qui
apparurent anisotropiques pour la plupart,
tout en indiquant une
plus
grande
variabilité
des
taux
d' évapotrë\\nspiration
dans
la
direction Nord-Sud que dans la direction Est-Ouest.
1
Ces
semi var iogrammes,
qui
consti tuent
des
modèles
de
/
variabilité
des
taux
d'évapotranspiration
en
fonction
des
distances,
ont
permis
d'estimer
par
krigeage
les
taux
d'évapotranspiration
à
1600
localités
où
les
observations
étaient inexistantes.
Ces estimations ont permis de dresser les
cartes d'iso-évapotranspiration et
des
cartes
d'iso-variance de
krigeage.
5.2. Recommandations pour des Recherches Futures
Les
résultats
de
ce
travail
de
recherche
peuvent
être
améliorés
si
des
semivariogrammes
sous-régionaux
peuvent
être
modifiés
de
manière
à
pouvoir être
utilisés
dans
les
zones de
transition
entre
les
sous-régions.
Le
semivariogramme
sous-
régional obtenu par Jacknifing pour le mois de Juin constitue une
approche de solution qui doit être améliorée.
La précision des résultats d'estimation des taux
d'évapotranspiration
calculés,
pouvaient
en
principe
être
améliorée,
si un plus grand nombre de relevés météorologiques de
second
ordre,
étaient
disponibles
ou
pouvaient
être
estimés
à

---

32
partir du nombre élevé des données de température.
Le cokrigeage
à
partir de données de premier et de second ordres enrégistrées
par des stations météorologiques mobiles constitue un domaine à
exploiter.
D' autres
méthodes
d'interpolation
basées
sur
la
"moyenne
glissante",
l'''inverse des distances",
l'''inverse des carrés des
distances" ,
bien
qu'elles
supposent
à
priori
un
modèle
de
variabili té
spatiale
ont
par
contre
l'avantage
d'être
simples.
Ces méthodes peuvent être testées dans
le cadre de cette étude
pour estimer des données de second ordre.
D' autres
investigations
sont
nécessaires
sur
les
relations
qui existent entre,
d'une part la configuration géo~raphique et
la
stationarité
de
la
dérive,
et
d'autre
part
le
modèle
de
semivariogramme, avant que la technique de krigeage soit utilisée
1
,
de
façon
systématique pour concevoir l'installation des
réseaux
d'observation.
1
1
Le but ultime de ce travail de recherche,
est de permettre
j
une
cartographie
automatique
par
ordinateur
des
courbes
d' iso-
1
,
évapotranspiration
sur
de
vastes
régions.
Ces
courbes
peuvent
être obtenues
pour différents
niveaux
de
probabilité
basée
sur
des
données
historiques,
en
vue
de
concevoir
des
systèmes
d'exploitation
de
ressources
hydrauliques
et
d'irrigation.
La
méthode
mise
au
point
peut
également
permettre
de
tracer
ces
isolignes
pour
des
données
relatives
aux
dernières
24
ou
48
heures
en
vue
de
programmer
les
doses
d'eau
à
appliquer
en
irrigation.
L'application de cette méthode peut ainsi
permettre
une
amélioration
de
la
productivité
en
agriculture
irriguée et
une meilleure exploitation des ressources hydrauliques.

---