AVAILABLE SOIL MOISTURE AS
STOCHAS~IC
Ii
PROCESS
by
Dale E. Cooper and
Dav~d
Do Mason
This research was done in cooperation with the Division of
Agricultural Relations, Tennessee Valley Authority.
Institute of Statistics
Mimeo Series No. 270
December, 1960
iv
TABLE OF CONTENTS
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Page
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General ...... e~ .. " • •
Statement of Objectives" ..
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LIST OF TABLES
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LIST OF FIGURESo
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vi
. vii
CRAPI'ER
INTRODUCTION"
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FORMULATION OF AVAILABLE SOIL M>ISTURE AS A TIME
DEPENDENT STOCHASTIC PROGESS
e
e e e e
II
2.. 1
2.. 2
0
.,
II
The Soil as a Storage SWstem..
e o ..
M:>dels EJcpressing Available Soil M:>isture as
a Time Dependent Stochastic Process
Available Soil M:>isture as a Markov Process ..
The Transition Matrix. and Stationary State
Probabilities to .. .. .. .. .. .. .. e " " e " e .. ..
Solution for Stationary Probabilities "
The Expected Value of Available Soil MOisture ..
MOisture Deficits .. " e
to
" "
"
0
..
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0
0
2.3··
204
20 5
206
207
IIIo
0
0
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1
. .. ..
1
3
4
4
0
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0
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e
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11
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RESULTS ON THE FREQUENCY DIsrRIBUTION OF PRECIPITATION ..
15
.
17
19
..
20
. ..
21
3.. 1
Possible Distribution Functions for Characterizing
Precipitation Frequencies .. e o ..
302 Results Based on North Carolina Weather Station
Records " .. .. .. " " ." .. .. .. .. .. e .. " " .. " .. " .. ....
3.. 3 Estimates of the Parameters of the Gamma
Distribution
"
" .. "
"..
304 The Distribution of Inputs into the Slfstem.. .. " .. • ••
36
37
..' .
40
0
IV.
0
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RESULTS ON THE DIsrRIBUTION OF AVAILABLE SOIL I-DISl'URE
4.. 1
42
..
...
21
31
The General Shape of the Frequency Function of
Available Soil MOisture .. .. " " • " .. .. .. • • .. .. .... 40
Distribution Free Methods • " ..
e • •
•
•
•
..
45
4.3 Queueing TheoryRe sults • .. .. • .. .. 0
47
404 Distribution of Number of Drought Days Occurring
in N Days • .• " "
47
0
0
0
•
•
•
"
•
"
•
"
"
"
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v
TABLE OF CONTENTS (continued)
Page
V.
APPLICATIONS • • • • .. ..
5.1
5.2
5.3
,.4
,.,,.6
VI.
•
•
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.,
e
· ..
• • • • • • • • 50
e
..........
.. .. . ......
•
•
Q
•
•
0
. .. • 50
. .. .. 51
.. .. .. 53
• .
.. • ..
.. .. ..
•
",8,9
• • • •
61
• • • • • •••
61
SUMMARY AND SUGGESl'IONS FOR FurURE RESEARCH. •
o
.
•
..
•
.
•
•
•
Ii
_
.. .. . .. .. ..
LIST OF REFERENCES •
•
•
The Crop Production Function. • .. .. .. . .. .. .. .. •
The Drought Index . • • • . • .. • • • . • . .. .. .
So il 11:> isture Index .. .. .. .. .. .. .. . .. .. .. .. .. .. •
Decisions Concerning the Use of Supplemental
Irrigation. .. .. .. . . • • • . • • • .. • . . • •
Complementary Use of Long Term Weather Forecasts ..
Sequences of Drought Days .. .. • . • • • . • • • •
Summary ........ • • • • • •
Future Research • • • • • •
APPENDIX
•
• • • • • •
.. ..
•
•
•
0
•
64
68
..oo
••
oe
•••
71
vi
LIsr OF TABLES
Page
Coefficients of Skewness and Excess Kurtosis for
North Carolina Weather stations • • • • • • • •
e-
•
•
•
e-
•
32
Appendix
1.
Approximations (2.36) and (2.37) to the Transition
Probabilities Pk with Lower and Upper Bounds. • • • • • ••
74
Parameter Estimates for the Gamma Distribution based on
Daily Precipitation Records of North Carolina Weather
Stations • • • • • • • • • • • • • •
• • •
75
0
•
•
•
•
•
•
•
vii
LIST OF FIGURES
Page
Available Soil M:>isture as a Finite Queueing System. • • • •
5
Goldsboro
April Observed Frequencies of Rainfall •• • • •
23
Goldsboro
M9.y Observed Frequencies of Rainfall
.
23
Goldsboro
June Observed Frequencies of Rainfall. • • • • •
24
3.4
Goldsboro
July Observed Frequencies of Rainfall. • • • • •
24
3.. 5
Goldsboro
August Observed Frequencies of Rainfall. • • • •
3.6
Goldsboro
September Observed Frequencies of Rainfall • • •
25
25
3.. 7
Nashville
June Observed Frequencies of Rainfall. • • • • •
26
3.8
Nashville
August Observed Frequencies of Rainfall • • • • •
26
3.. 9
Lumberton -- June Observed Frequencies of Rainfall. • • • ••
27
3010
Lumberton -- August Observed Frequencie s of Rainfall. .. • ••
27
3.11
Kinston -- June Observed Frequencies of Rainfall. • • • • ••
28
3.12
Kinston -- August Observed Frequencies of Rainfall. • • • ••
28
Edenton -- June Observed Frequencies of Rainfall •• • • • • •
29
3.14
Edenton -- August Observed Frequencies of
R&~nfall.
• • • ••
29
401
Available Soil Moisture Frequencies with
p<l • •
..
• • • •
41
4.2
Available Soil Moisture Frequenc ie s with
P>
• • • • • • •
42
4.3
Available Soil Moisture Frequencies with
p
.
• • • • • •
43
.
77
-'-
=1
Appendix
1.
Stationary Probability of the Zero State
CHAPTER
I
INTRODUCTION
1.1 General
The agricultural industry is faced with two major sources of
uncertainty which give rise to large risks.
products and resources and 2) weather.
These are:
1) prices of
Considerable information is
available to aid the farm manager in view of uncertain .prices; hawever,
little has been done toward aiding him in making decisions whose outcomes depend on the weather.
Virtually all crop production planning
decisions are affected by the weather.
planning of an irrigation program.
An obvious example is the
The extent of such a program would
depend directly on the weather conditions during and previous to the
growing season.
Recent research has shown that the amount of fertilizer necessary
for economically optimum crop yields is in many cases a function of
soil moisture conditions throughout the growing season.
Parks and
Knetsch (1960) found that the economically optimum amount of nitrogen
fertilization for corn increased with decreased drought, as characterized by a drought index.
Similar results were reported by Havlicek (1959).
Other areas where farm operator decisions are affected by soil
moisture conditions are as follows:
1) The amount of capital reserves necessary for long run survival
2) The storage of livestock feed
3) Economically optimum crop stands
4) Weed control
2
These examples illustrate that any attempt to aid farm managers in
making rational decisions concerning production planning would in many
cases depend on a knowledge of probable weather or soil moisture conditionso
At the present time weather forecasts are not usually available
far enough in advance to provide a basis for production planning.
For
example, the farmer's decision concerning his fertilizer program is
usually made during the first few months of the growing season.
In
general, the management of a farm requires plans to be made in one time
period for a product which wiil be realized at a later time periods
Decisions could be made more nearly rational by a knowledge of the probabilities of future production yields.
For the majority of agricultural
products, these probabilities would depend on probable soil moisture
conditions o
In the arid regions of the world where irrigation is a common
practice and the limited precipitation occurs in a more or less definite
time of the year, the problem of predicting soil moisture conditions is
considerably simplified.
However, in humid and sub-humid regions,
particularly Eastern United States, where natural precipitation forms a
substantial source of soil water supply, the problem of predicting soil
moisture conditions is highly complicated by the erratic nature of both
the occurrence and amount of precipitation.
The need for a knowledge of probabilities of soil moisture conditions has been recognized by a number of workers, notably Knetsch and
Smallshaw (1958), Parks and Knetsch (1960), van Bavel and Verlinden
(1956), and Havlicek (1959).
Tables of drought probabilities have been
3
presented by Knetsch and Smallshaw'
(19.58) applicable to areas in the
Tennessee ValleY9 and by van Bavel and Verlinden
North Carolinao
on van BavelVs
(19.56) for areas in
In both of these studies drought probabilites, based
(19.56) evapotranspiration method of estimating soil
moisture conditions, were computed for each weather station within the
area for different values of moisture storage capacities as the percent
of occurrence in previous yearso
1 0 2 Statement of Objectives
The available moisture in the soil at any particular time represents an extremely complicated system dependent on numerous random
occurrences o No attempt is made in the present study to characterize
soil moisture to the degree of refinement necessar,y for plant behavior
studieso
Rather, an attempt is made to characterize soil moisture in
the overall situation as it affects crop yields o
The objectives are
to characterize available soil moisture as a time dependent stochastic
process and to study the probability distribution function of available
soil moistureo
4
CH1l,.PTER
II
FORMULATION OF AVAILABLE SOIL MOISTURE AS A TIME DEPENDENT
srOCHASTIC PROCESS
2.1 The Soil as a Storage §Ystem
The concept of available soil moisture as a stochastic process
is based on the analogy between· the soil as a storage system and the
storage systems ordinarily encountered in the theory of queues or
waiting lines o
Queueing theory has received considerable attention in
recent years and several mathematical and statistical journals devote
considerable space to problems arising from queueing situations.
A
recent book by T. L. Saaty (1959) provides a resnme of queueing theory
including areas of application.
A review article by Gani (1957) gives
a good account of the aspects of queueing theory applicable to the
present problem.
The analogy between soil rr.oisture and a queue appears to be farfetched; however, certain aspects of the two systems are similar..
The
arrival of a customer in a queue is analogous with the occurrence of
precipitation, the service time of the customer corresponds with the
amount of precipitation which enters the soil and is available for plant
use..
The queue busy-period is analogous to the period of adequate
moisture supply or non-drought, and the period of waiting for the next
customer corresponds to a period of drought.
The queue capacity is ana-
logous to the moisture storage capacity of the soil.
Figure 1 shows
available soil moisture as a finite queueing system with precipitation
occurring at times
t
= 2, 5, 9,
and 16.
5
s
Q)
of'
II)
~
tu>
s::
..-I
~
~
....,Q)
.r-!
s:::
~
ell
~
.~
....,
r.a
~
.r-!
....,
.r-!
0
~
r-i
.r-!
0
~
(I)
r-i
cd
r-i
,D
'@
~
•
r-i
•
N
[
~
o
N
•
o
•
r-i
exn~s1oW t~Os atq~t~~AV
o
6
2.2 Models Expressing Available Soil Moisture as a Time
Dependent Stochastic Process
Using the concept of the soil as a storage ,system it is possible
to express available soil moisture for a particular time period as a
simple function of the available soil moisture from the previous time
periodj) the precipitation which occurred during the time periodj) and
the moisture loss during the time period
where
Zt .. available soil moisture at time t
X
t
g
precipitation occUrring in the tth time period
1t .. water loss occurring in the t th time period.
Model (2.21) defines a storage system with both input and output
as random variables.
This model is complicated by the fact that
~
is
difficult, if not impossible, to measure and is a function of numerous
variables.
Some of the factors which affect 1
t
are 1) moisture storage
capacity of the soil, 2) depth and extent of plant roots, 3) the wilting
range which depends on both soil and plant factors, 4) the tenacity with
mich moisture is held by the soil, ,) maximum rate of water infiltration
by the soil, 6) the intensity of precipitation, 7) slope of the terrain,
8) soil temperature, 9) relative humidity, and lO).wind speed.
The above factors serve to illustrate that model (2.21) must be
simplified if it is to be of any practical value.
In spite of the above
factors, water loss from the system can occur only through evapotranspiration, leaching, or as runoff.
Ajt',.,
,
The following modification, based on
7
van Bavel's
(1956) evapotranspiration method of estimating soil moisture
is proposed to allow for these possibilities"
~
Let
... duration of the amount of precipitation It
R .. maximum rate of moisture infiltration by the soil
It, . .
It
if
'" AtR if
It ~ AtR
It
> AtR
V .. potential evapotranspiration occurring during the t th tillJ.e
t
period
C ... maximwn amount of plant available water which can be held by
the soil"
Then we can write
(2022)
"" C
.,. 0
All of the climatic variables (It.? ~ and V t) involved in model
(2 0 22) can be measured or estimated from available climatic data"
The
variables Rand C are constant over time for a given soil and crop and
can be determined experimentally"
It is possible to further simplify the model to
Zt+l "" Zt + It ~ Vt
< Zt
+ It < C ... Vt
if
Vt
"" C
if
Zt + It
... 0
if
Zt + It <. Vt"
:c C + Vt
(2,,23)
8
In this model no recognition is given to rtllloff except that in excess
of the storage capacity.
van Bavel
(1956) asserts that the error
incurred by ignoring runoff is not very serious, particularly in Eastern
United States and areas where precipitation does not occur largely as
thunderstorms.
MOdel (2.23) approaches model (2022) if
Fr(Xt > AtR)
is smallo
A difficulty of both model (2.22) and (2023) lies in obtaining
estimates of Vto
Evapotranspiration is largely a function of incident
radiative energy which is associated with a number of climatic variables,
notably, temperature, cloudiness, windspeed, and relative humidity.
Several methods of estimating V from available climatic data have been
t
proposed in recent years.
and Pelton et.al.
These methods are di scussed by v.an. Bavel
(1956)
(1960). The method derived by Penman (1948) is general-
ly accepted as being more appropriate to the humid areas of the United
States.
Penman's formula as given by van Bavel
(1956) is
H + 0027 E
- - - - -a,,
t:. + 0027
where
H
= incremental change in va.por pressure
= net heat adsorption at the surface
Ea
=a
t:.
function of saturation deficit and wind velocityo
Since the climatic data needed for the solution of the Penman formula are
available only at United States Weather Bureau Class A Stations or their
eqUivalent, evapotranspiration rates for a particular location are
usually based on values obtained from the nearest station.
Knetsch and
9
Smallshaw (1958) present evidence that V as computed from the Penman.
t
formula does not vary appreciably for various areas within the Tennessee
Valley.
van Bavel (1956) points out that the variation in evapotranspiration
is small relative to the variation in precipitation and gives bounds for
Vt as
0 L Vt
>
0.35 inches per day for all t and any geographical area.
In view of this~ van Bavel proposes replacing V in models (2.22) and
t
(2.23) by an average value, V, over some finite period of time and given
geographical area which gives
if
V < Zt + It <: C + V
... C
if
Zt + It .~ C + V
"" 0
if
Zt + It < V
wren runoff is an important factor, and
if
V < Zt + It < C ... V
"" 0
if
Zt
of;
It >0 + V
... 0
if
Zt
of;
It < V
Zt+l ... Zt + Xt - V
(2025)
when runoff except that in excess of the storage capacity can be ignored.
Thus, models (2.22) through (2.25) represent alternative formulations of available soil moisture as a time dependent stochastic process.
MJdel (2.22), while the most complicated, is the most realistic in that
all three ways in which water is lost from the system are accounted for.
MJdel (2.25) expresses the change in the system as a function of only
10
one time variable, precipitation, and lends itself most readily to the
queueing theory approacho
10bdels (2.23) and (2024) are intermediate
between (2022) and (2025) in simplic ity and departure from reality 0
The maximum amount of plant available water, C, is denoted as the
"base amount" in van Bavel' s evapotranspiration method of estimating
soil moisture conditions on which models (2.22) through (2.25) are
~~
basedo
The determination of C regulates the intensity of drought as
defined when Zt
= 00
van Bavel (1956) proposes that C be obtained as
the difference between field capacity and the wilting point, both
expressed on a volume
basi~multiplied
by the depth of the root zone o
He defines agricultural drought as a condition in which there is
insufficient soil moisture available to a crop.
the condition when Zt
=0
With this definition
does not represent zero available soil
moisture but a condition of inadequate moisture for optimum plant
growth;
!. ~o,
Zt represents readily available soil moisture 0
When C
is defined as the total maximum plant available moisture, a drought
condition exists, as defined by van Bavel, when Zt
the wilting point o
<
Q, where Q is
Although the results of this study are applicable
to either definition of C, the departures from reality of models (2022)
through (2.25) become more serious when Zt~ Q..
When soil moisture is
below the wilting point, ' t is dependent upon Zt as well as weather conditions.
Given a mathematical expression relating 't as a function of
Zt it is possible that the models could be modified to account for the
dependence of 't on Zt.
11
2,,3 Available Soil Moisture as a Markov Process
In order to keep the notation general, it will be convenient to
denote models (2.22) through (2 .. 25) by the single model
if
1 < 1ft + Ut < r + 1
"" r
if
1ft + Ut
.. 0
if
if + U ~l,
Wt +l ... Wt + Ut - 1
t
>r
+ 1
(2,,31)
t
where
W't
U
t
... Zt /M
for models (2,,22) and (2.23)
. Zt/v
for models (2024) and (2025)
...
t - Vt )
(X
. (Xt
r
M
- Vt '
M
+ 1
for model (2.22)
+ 1
for model (2,,23)
. . Xi/V
for model (2.24)
... Xt/v
for model (2025)
... C/M
for models (2022) and (2023)
.. c/v
for models (2024) and (2"25),,
The quantity M is the maximum value of Vt ' characteristic of a particular geographic area and time of year.. By introducing Mand adding 1 in
An approximate solution to the problem of determining the probability
distribution function of Zt can be obtained by defining a finite number
of discrete soil moisture states which satisfy the properties of a
12
Markov chain.
The states defined in terms of the generalized variable
..
•
.
r' -1< Iit S',r i
where r i is the largest integer in r.
Let P.k be the transition probability of going from state
J
time t to state
~
at time t+l.
.3. at
J
The Markov property is satisfied i f the
probability of being in state S. at time t is independent of the states
J
at times t-2, t-3, t-4, • ••
for all
j "" 0, 1, 2, ... ri+lj !o~., the
probability of going from state Sj to state Sk is independent of the
manner in whic h the system arrived in state S..
J
satisfied is evident from (2.31) since
Wt and Ut are known.
t=oo
W + is completely determined i f
t l
The Pjk can be written
... Pr(Wt +l ... 0 I
lit ... 0)
That this condition is
13
POI
Pr(O < Wt +1 s.1
P02
Pr(l
< Wt +1 ~
2
I Wt
= 0)
I Wt
... 0)
"
•
PO(r'+l)
Pr(r l < Wt +1 < r
P
Pr(Wt +1 ... 0
P
Pr (0 < W +1 ~ 1
10
11
I Wt ... 0)
I 0 <: Wt~
t
I
1)
0 <:: Wt ~ 1)
"
"
... Pr(k - 1 < Wt +1 ~ k
P jk
Clearly
P
jk
... 0
for
j
>
k + 1
Ij
== Pr(k < 'lilt
U
t
;>
< 'lilt ~ j);
< Wt
>
0
0
r' ..
+ Ut - 1 <k)
+ Ut
<
k + 1)
0, the upper limits on Wt which satisfy (2 33) are
0
(k < 'lilt < k + 1), but we are given that (j - 1
j
j"k, ... 1,2,3,
since from (2.31)
Pr(k - 1 <Wt +1 ~ k) ... Pr(k - 1
and since
- 1
< 'tit <
j), hence, for
k + 1, P jk ... O.
In order to evaluate the Pjk' we need to have a knowledge of the
cumulative distribution function of Ut"
If this distribution function is
denoted by F(U)" the P
can be obtained explicitly in terms of F(U); i.~.,
Ok
• F(k +
1) -
PO(r'+l). 1 - F(r')
F(k)
k
=1, 2, ...
since Pr(Wt
>
r'
r) == O.
P
'" F(l)
OO
(2.34)
14
The P'k
for
J
= 1,
j
2, .3,
••• r'
and
k
= 0,
1, 2, •• r'
known exactly until the distribution of W is known.
t
are not
In this case, from
the basic laws of probability
k
kljA
j
dG(W )
dG(W +
I W)
t
t
t l
--------::-j-------,
j
where G(Wt+ll
J:.
dG(Wt '
Wt ' is the conditional cmnulative distribution function
of W + given W and G(W ' is the marginal cumulative distribution
t l
t
t
function of W •
t
Since
k
k
~
dG(Wt +1
I
•
Wt '
F(k + 1 - Wt ' - F(k - Wt ',
Pjk can be written
)
j
A
dG(Wt '
M:>ran (1954), in deriving the transition probabilities for the
amount of water in a dam, asserts that a suitable approximation to the
P'k is obtained by taking W as the midpoint of its bounds; i.e.,
t
J
-Pjk ~
F(k - j + 3/2) - F(k - j + 1/2).
Another approximation can be obtained by assuming that W is uniformly
t
distributed on the interval (j - 1, j) so that (2.35) becomes
15
j
-
j
h F(~
+ 1 - Wt ) -
F(~ - Wt ) dWt •
Bounds for Pjk can be obtained by setting Wt equal to j-l and j respectively; i.e., P' k lies between F(k-j+2) - F(k-j+l) and F(k-j+l) - F(k-j).
--
J
2.4 The Transition Matrix and Stationary State Probabilities
Let the r'+2 by r'+2 matrix of transition probabilities be denoted
by T
= (Pjk);
j, k
==
0, 1, 2, •• r'+l.
Notice that
Pjk
= P(j-l)(k-l)
for both (2.36) and (2.37); hence, there are only 2(r'+1) different
values of Pjk and the notation can be simplified to
k == 0, 1, 2, ••• r' + 1.
Then the transition matrix can be written
POO
POl
P02
P03 P04 P05
•
•
0
•
POri
PO(rl+l)
Po
Pl
P2
P3
•
•
•
•
Prl
Pr'+l
0
Po
Pl
P2
P4
P3
P5
P4
.. ..
rl-l
•
•
Prl-l 1 -
~ Pi
i==O
r'-2
0
0
Po
Pl
P2
P3
•
0
•
•
Prl - 2 1 -
]
i-O
Pi
(2.41)
0
0
0
Po
Pl
P2
0
0
0
0
0
r ' -3
Pr' - 3 1 - ~ Pi
i-O
•
0
·
•
0
• Po
•
•
0
0
1 - Po
16
From (2.41) it is seen that there is always a probability, PO
> 0,
that the system will move in a single transition from a given nonzero state into the next lowest state, and that any state can be
reached from the zero-state in a single transition.
It is also always
possible to' move from one to another of a given pair of states in a
finite number of steps
Such a M3.rkov chain is de scribed by Feller
0
(1950) to be irreducible and aperiodic.
Let the stationary probabilities of state ~ be
~ at time t+l, with
Then
~ = T'
~
=
time t and
P* and 1'** the corresponding r'+2 column vectors.
P*;
~Poo
1 at
!.~.,
+ F!Po
I!* = ~Ol + I1Pl +~Po
Pf = ~P02 +J!P 2 + ~Pl + ~o
•
•
k+l
~ = ~POk
+.
~
i=l
P!Pk-i+l
•
•
r'+l
~
i=l
r'-i+l
~(l - ~
~
j=O
Pj
).
If the system has been allowed to run until equilibrium is attained,
Pk = F!* = ·pk '
the stationary probability, and
which becomes a set of r'+2 independent equations if the last equation
is replaced by the restriction
rt+l
~
Pi
i ... O
=
10
205 Solution for Stationary Probabilities
Several rrethods are available for solving (2043) for the Pko
1-bran (1954) and Gani and Moran (1955) give a discussion of alternative
methods including I>:bnte Carlo methods.
for programming on a computer:
1
= POO
+ ~PO
o
o
k+l
Gk .. POk +
~
GiPk_i+l
J.=l
•
o
r'+l
I/Po = 1
+
~
. 1
J.=
G.
J.
0
The following rethod is proposed
18
The Gk are obtained from the Pjk by successive substitutions, starting
with
1 - Poa
Po
G ""---
I
o
Given the G , the Pk are easily obtained, since from the last equation
k
of (2051)
1
r'+l
1'+'~'
i=l
G
i
alsOj i f we let
1
k
+~.'
1
i ...l
G
i
then
1
POI
G
k
...
1
~
r
Ok
- 1
-
and
P 1
, k ... 2, 3,
O(k-l)
0
0
r'+l,
19
so that the stationary state probabilities are completely determined
by either G or POk9 k ~ l~ 2~ 3, • 0 rB+l o
k
The discrete approximation to the continuous distribution of
Wt
~
Zt (constant) is given by
k
B(k) "" Pr(W
t
< k)
&.
~
i~O
k
P .... P
1
0
~
P ~ G.
0 i~l 1
A.n advantage to a solution in terms of the G rather than P is
k
k
'that G is independent of r, and once the G are found for the largest
k
k
r, the value for Po with a smaller r is obtained from (2.53) by dropping
the appropriate number of G. in the summations
1
206 The Expected Value of Available Soil MOisture
The solution to the stationary probabilities~ PkJ) allows the
expected value of available soil moisture to be obtained in terms of
the discrete approximation to the distribution of
r
&
Wt~
io!.o ~
U
Po ~
k=2
... ( r+r U
2
k( L-i.l ... :. 1
) ... P ( ...L... 1)
POk
PO(k=l)
0 POl
)
P ( l
0 Po
... -1:..- ) _ Po
POrB
2
(
-.l- ...
POru
1)
20
and
E(Zt)
...
for models (2022) and (2023)
M E(W )
t
"" v E(Wt )
for models (2024) and (2025)0
It is also possible to approximate the higher moments ofZ from
t
E(~) :.
207 M:>isthre Deficits
Some of the recent research utilizing climatic variables in crop
production functions employ a drought index based on moisture deficitso
A moisture deficit occurs when available soil moisture is below some
critical point Q9 0
If the moisture deficit is denoted by Zt at time t,\1
Z9=QO_Z
t
t
... 0
if
Z
~Qi
if
Zt
>
t
(2071)
Q9
0
Then the probability that a moisture deficit occurs is Pr(W
t
where
q
g
< q),
Q9/M for models (2022) and (2023) and q"" Q9/V for models
(2 024) and (2025)0
The discrete approximation to this probability in
terms of the stationary state probabilities Pk is
q9
Pr(W
t
<
q)
.&
.~
k...O
where q 9 is q rounded to the nearest integero
P
:J
k
(2072)
The discrete approximation
to the expected value of moisture deficits is given by
(2073)
21
C HAP T E R I I I
RESULTS ON THE FREQUENCY DIBrRIBUTION OF PRECIPITATION
301 Possible Distribution Functions for Characterizing
Precipitation Frequencies
As indicated in the previous chapter.!> a knowledge of the frequency distribution of U is required in order to obtain the transition
t
probabilities 9 Pjk"
The variable U as defined for models (2,,22)>>
t
(2,,23) and (2 .. 24) is a function of at least two climatic variables ..
However» since It is involved in
all
of the models» a starting point in
studying the frequency distributions of U for all four cases would be
t
a knowledge of the frequency distribution of~..
approximations to Pk» k
= 09
The POk and the
l.!> 2» .. 0 rQ+l» can be obtained from the
frequency distribution of It for the case defined by model (2025)0
Nothing was said in the previ'ous chapter about the length of the
time intervalo
The choice of a time interval depends on two factors
which work in opposite directions..
It is desirable to choose a time
interval as small as possible in order to quantify available soil
moisture as nearly as possible as a dynamic system"
For example» soil
moisture probabilities based on monthly time periods would have little
value since a complete cycle from drought to storage capacity could
have occurred within a month..
On t.he other hand» it is desirable to
choose a long time period in order to justify» to some extent» the
independence assumption of the input variable U
t
0
The shortest time
period for which precipitation records are readily available is one
day..
Thus»any frequency curve fitting procedure must be based on daily
22
records or longer time periods!)
The shortest time period of one day
seems to be desirable since it is generally easier to derive a frequency
function for a long time period from a function for a short time pericd
than to derive a function for a short time period from fun~tions based
on a longer time period.
When the time peri©Jd is one day ~ the distribution function of
is discontinuous at zero$
!o!o,
~
'"
there exists a finite probability that
It ,.. 0..
However.!> the function may be aSSUIll:ld to be continuous for
It > 00
Then,!) if the cumulative distribution function of daily precipi=
tation is denoted by F1 (X):J it may be written
X
F1 (X) .. (1 - n) + n
f
f (x) dx:J
0+
where n is the probability that rain occurs during the time inteI""1Tal
and f(x) is the probability density function of the amount of rain o
In order to determine a distribution function which would suitably
characterize the frequency distribution of rainfall,!) twenty-five years
(1928-1952) of North Carolina rainfall records for the months April
through September were studiedo
The observed frequencies of daily rain=
fall are given in figures 3.. 1 through 3014 for some of the stations..
It
is evident from these frequencies and generally recognized in the litera=
ture (Chow,l) 1953) that the distribution of rainfall is positively skewed 9
the degree of skewness generally depends on t:m length of the time period..
When the time period is one day as in figures 3.. 1 through 30 14, the
distribution tends to be J shaped suggesting the exponential distribution
23
7060-
0.5
;:t.0
2.0
inches per day
Figure 3.1.
Goldsboro--!pril Observed Frequencies of Rainfall
70
60
50
2.0
inches per day
Figure 3.2.
Goldsboro--May Observed Frequencies of Rainfall
24
706050-
4030-
10l--l--L-,-JL--J---l-,-l---
0.5
Ii· r-{I
r--{"""""1-;
-i .. _r:-'L . L_l_
'-=l~...,t,...J·~===-·_-_
1.0
1.5
2.0
inches per day
Figure 3.3.
Goldsboro--JJl,tl6 Observed Frequencies of Rainfall"
rl
I
!
80-f
I
I
70-l
60J
I
50 -t
40J
I
0..5
inches per
Figure
3.4.
d~y
Goldsboro--July Observed Frequencies of Rainfall
90-'
80-
7060-
50-
3020-
10-
1.0
e,
105
2,,0
inches per day
Figure 3.5"
Goldsboro--~ugust Observed Frequencies of Rainfall
60-
20
10
0.5
inches per day
Figure 3.6.
Goldsboro--September Observed Frequencies of Rainfall
26
40302010-
0.5
inches per day
Figure 3.70
Nashville--June Observed Frequencies of Rainfall
•
6050-
4030-
10-
0.5
1.0
1.5
inches per day
Figure 3 8.
0
Nashville--A,ugust Observed Frequencies of Rainfall
27
50-
4030~
2010-
oS
1.0
inches per day
Figure 3.9.
Lumberton--June Observed Frequencies of Rainfall
70-
0050-
403020/
10-
inches per day
Figure 3.10 0
Lumberton--August Observed Frequencies of Rainfall
28
3020-
10-
005
1.0
1.5
2 00
inches per day
Figure 3011..
Kinston..;.-JuneObserved Frequencies of Rainfall
30
20
10
1.0
1.5
inches per day
Figure 3.12
0
Kinston-:"'.A,ugust Observed Frequencies of Rainfall
29
3020-
10-
0.5
2.0
inches per day
Figure 3 0130 Edenton--June Observed Frequencies of Rainfall
403020-
10-
0.5
Figure 3.140
1 0
1.5
inches per day
0
2.0
Edenton--August Observed Frequencies of Rainfall
30
The exponential distribution has been used by M:>ran (1955) for the dis=
tribution of inputs into a dam and many of the explicit results from
queueing theory make use of the exponential service time distribution
(Saaty, 1959).
However, due to geographic as well as seasonal varia-
tions in amounts of daily rainfall, a one parameter distribution
function, such as the exponential, would not appear to be sufficiently
flexible to have wide applicability.
The exponential distribution is a special case of the nore general
gamma or Pearson type III two parameter distribution
x'A.-l e-x/.t..,
which reduces to the exponential distribution for
bution is J shaped for
A. <1
considerable flexibility.
and bell shaped for
A. = 1.
This distri-
). > 1, which allows
The gamma distribution has been proposed by
several workers (M:>ran, 1955; Manning, 1950; Beard and Keith, 1955) in
fitting rainfall data.
Another distribution function which has been used extensively for
hydrologic data is the logarithmic normal (Chow, 1953 and 1951; }bIllwraith,
1953 and 1955; and Foster, 1924).
In
many
situations involving skewed fre-
quencies, it is possible to normalize the distribution by taking logs of
the observations.
If
x t = ln x,: then
The majority of the work done in fitting rainfall distribution
functions has been for the purpose of predicting floods (M:>ran, 1957;
31
Paulhus and Miller» 1957)~ . Recent advances along this line have been
made by Gumbel (1941~ 1945.9 1958) using the statistical theory of extreme
values,P and a similar approach has been used to predict drought (Gumbel,jl
1958).
However SJ drought as defined by van Bavel is not the extreme
drought as defined by Gumbel's theory of extreme value s approach but
rather a state of soil moisture conditions when the plant functions at
less than optimum because of moisture deficiency.
3.2 Results Based on North Carolina Weather Station Records
Following the Pearson system of curve fitting (Elderton s 1953)~ the
first four moments about the mean as well as the coefficients of skewness
and kurtosis were computed, using the 25 years of North Carolina weather
data for the stations shown in Table 3010
The following relationships are characteristic of the moments of
the gamma distribution mere foI.J! is the kth moment about the means
&Jt ...
..<}..
2
~
... ..<. }..
IJ.
"" 2..<3}..
3
1J.4
coefficient of skewness ""
==
3-<4}..(}.. + 2)
~l
...
IJ.j
-----:3~11":ll:2-
~2
iii
~4
~
1J.2
2
n
""2
coefficient of kurtosis ...
"" -
...
T6 + 3
e
e
e
Table 3.10 Coefficients of Skewness and Excess Kurtosis for North Carolina Weather Stations
Station M:>nth
.e
t
gl
g'1
g2
g2
1.8152
2.3108
2.6920
2.3809
406543
2.9040
105305
2.0482
2.7353
2..1270
404256
2.. 6591
3.,5137
6.2932
1102233
607868
29 .. 3796
10.. 6064
4.9424
8.0096
10.8702
8.,5030
32.4937
12.6498
2.,858
3.432
'" 0.704
30433
6.228
4.. 088
0.. 2661
0.,1780
0.,1799
0.2872
0.,1463
000800
- 1.039
~ 0.859
... 0.,755
~ 0.715
'" 0.664
'" 0.978
2.8346
2.4042
2.0433
2.4863
2.4754
2.7338
2.9473
204649
10 7186
2.2774
2.7383
2.,4777
13.0304
9.1138
4.4306
7.. 7802
11.. 2480
9.. 2086
12.0524
8.6702
6.2626
9.2725
9.,1914
11.2104
'" 10954
... 0.886
3.664
0.,3350
0.,2470
0.,4139
0.3240
0.,5257
0..1389
... 00618
0.702
'" 0.380
0.588
... 0.399
... 0 .. 663
1.8153
2.3186
3.2989
1.7366
1.5456
2.4900
10 7610
2.0724
3.3149
1.. 3729
1.1733
203055
4,,6518
6.4424
-16.. 4838
2.. 8273
2.0653
7.9736
4.9429
8.0638
16.3241
4.5236 '
3.5833
903001
0.582
3.244
'" 0.319
3.393
3.036
2.654
go
g2
gl
ASHEVILLE
~pril
~y
tlune
July
.£ugustSeptember
EDENTON
~ri1·'
~..
June
Jul ., ,
.' Y
c4ugustSeptember
EtIZA:BETHTOWN
April
~y
June
July
.I1,:ugust'
September
2~985
... 4.112
4.004
co
eo'
Ul
N
e
Table 3.,1.
(continued)
Station M:>nth
Fl(YETTEVI11E
April
~y
June
July
August
~eptember
e
e
gl
gi
g2
g2
2.,3544
3.047·9
1.,9261
1.6301
2.8002
3.. 2894
2.,1269
3.2568
1.. 7146
1.3919
207800
301239
6., 7856
15.. 9108
4.,4101
2.. 9061
1105934
1406384
8.3147
13.9345
5.,,647
3..9858
11.,7616
1602302
2..8374
2.7912
).0459
108223
304234
4.. 9121
11.6862
13.916)
4.. 9815
17.5803
8.3612
3601944
12.0762
1).5775
5.4939
17.. 8320
9.9328
33.4501
1"'215
2.. 4618
2.2676
2.5266
2.6954
3,,9137
3.4727
9.0909
7.7131
9.5756
1008981
22.9757
5.1626
9.8273
7.. 6994
9" 7744
12.0430
22.,8fD6
go
=
.e
g2
t
gl
3.059
3.,951
2.-309
2.,159
00336
30184
OOWSBORO
Apteil'
~y
June
Jury
August
~eptember
3~0086
1 0 9138
3.4479
2.5733
40722)
2.3£i)9
=
0.781
0.. 676
1.0~'
0.504
),,144
= $.488
KINSTON
~pril'
May
June
JulYu
A,ugust-
September
1.8552
2.5596
202656
2.5527
2.8335
3.9039
3.380
10474
= 0.,027
0.398
20290
"" 00229
0.4042
005063
00)105
006488
0.. 3386
001.355
0.l73
00191
0.545
0..527
= 0..178
= 0.. 240
=
=
\.J.)
\.J.)
e
Table 3.. 1..
e
(continued)
Station M::>nth
LUMBERTON
!Wril:
:May
June
July
Allgust~ptember
e
t
gl
t
g2
gl
gi
g2
g~
20 2528
1.8958
3.0294
1.. 7722
2.1244
4.2041
2.1701
1.6106
3.1807
1.,5289
1.. 9914
4,,2467
7.0646
3.8915
15,,1756
.3..50t4
5.9490
27,,0529
7,,6126
5.. 3910
13.7658
4.7110
6" 7696
2605116
1.. 097
2,,999
'" 2.818
2.409
1.. 641
'" 1.081
0.0511
0.2849
0.2898
0,,3381
0.. 2757
0..1189
'"
""
'"
'"
'"
'"
0,,969
0..526
0,,603
0.. 614
0,,835
00580
2.3831
2.2284
2,,6068
2.2012
2.8694
2,,6659
201135
2.. 0273
1.2684
109779
2,,8757
203914
6.7006
601654
20 4134
508684
1204046
8..5785
805187
7.4486
3.8727
702679
12.. 3501
1006605
3.. 637
,2,,567
2.919
20799
'" 0,,108
4..165
0,,3015
002591
0.3947
003478
002451
0.2220
'"
'"
'"
'"
'"
'"
00542
00806
0,,640
0,,452
0.. 688
0.596
go
-
N.(t,SHVILLE
;April
May
June
July
August
~eptember
~
35
The sample estimates of
given in Table 3010
that
~O
.. 0.
~O' ~i, ~2
- 3, denoted by go' glJ) and g2 are
The criterion for fitting the gamma distribution is
The sample estimates, go' deviate from zero in both direc""
tions although positive deviations are more prevalent.
Since there are
no available estimates of the variances of the ~, which is complicated
by correlation between gl and
g2~
it is difficult to make a decision as
to whether the deviations from zero could be due to random variation of
the samples.
By computing gi and g2' such that 3gi2 - 2g2
lIIl
0 and
3g1 2 - 2g .. OJ) it is possible to some extent to assess the deviations
2
of go from zero in terms of the coefficients of skewness and kurtosis
separately.
The values of gi am. g2 as shown in table 3.1 indicate
that the large values of
variation of the samples..
Igo I could reasonably be due
to random
This argument is empirical in that some
correlation exists between gl and g2"
However, when compared with
other distribution functions of the Pearson s,ystem, the gamma distribution generally gives a better fit.
The possibility that the log of rainfall follows the normal distribution was investigated for five of the North Carolina weather
stations.
For the normal distribution
~l
= ~2
- 3 .. oJ the corre-
sponding estimates are d. enoted by gl and g2 in Table 3.1. While log
"
x is considerably less skewed than the original observationsJ) the
skewness remains consistently greater than zero.
The estimates of
kurtosis become negative for log x with the exception of two of the
station-months, indicating that the frequency curve of log x is less
peaked than the normal curve"
36
Although these results are not conclusive j they indicate that the
frequency curve of daily amounts of precipitation can be fit reasonably
well with the gamma distribution.
The lack of fit maYj in part" be due
to a discontinuity in the right tail of some of the observed frequencies
as can be noted from figure 3. 6,
2:..~
•.9
large amounts of precipitat,ion
tend to occur more often than would be predicted from the ga:mm.a. dietribution..
These occurrences can probably be attributed to the influence
of tropical storms.
However, for the purpose of deriVing the transition
probabilities,\! the shape of the extreme right tail of the curve has
little effect since these heavy rains are generally in excess of the
storage capacity of the soil.
303 Estimates of the Parameters of the Gamma Distribution
With the assumption that f(x) in
(3011) follows the gamma distri-
bution, the problem arises of estimating the parameters .t.. and 'X. in (3.13) ..
The distribution can be written in terms of IJ. ... E(x) and 'X. by substituting
.t.. ...
t
in
(3.13) which gives
f(x) ,.,
1
r('X.)
x
Then the maximum likelihood estimate of
v~tions
'X....l
~
e
-x-'X.
IJ..
for a sample of n obser'"
is given by
and the maximum likelihood estimate of
~
is the solution of
37
where
r u (~)
is the first derivative of
rex)
and Ii1Y ...
This equation must be solved by iterative procedureso
~(if)
of
!J
~ .~ ln
Xi 0
Extensive tables
commonly known as the digamrna function, are given by Davis
(1933) e The estimates (~) can be obtained directly from In X - In·X
f'rom tables given by Chapman (1956) although these tables are not extensive enough to afford more than two digit accuraey in many caseso
A first estimate of ~ can be obtained f'rom X* "" r.2/s 2, where
is the sample variancee
A
I:
13
2
This is the method of moments estimate since
~/~ from (3.21)0 This estimate, though simpler, does not have
the desirable properties of the maximum likelihood estimate.
estimates of'
&J.
The
and A. are given in appendix Table 1 for the North
Carolina weather data.
An estimate of the probability that rain occurs, denoted by n
in (3.11) is obtained from n/N, where n is the number of days in which
rain occurred and N is the total number of days in which a record is
available.
The estimate n/N is given in appendix Table 1 for the North
Carolina weather data.
Since the probability that rain occurs is
likely to be greater for a particular day i f rain occurred on the
previous day than if no rain occurred on the previous day, a more
realistic estimate of nmight be obtained by a suitable weighting of
nl/(N-n) and (n-lJ.)/n, where ~ is the number of days in which rain
occurred,preceeded by a day in which no rain occurrede
3.4
The Distribution of Inputs into the
~stem
A knowledge of the frequency function of' It allows the study of
the stationary distribution of Zt following the procedures given in
38
Chapter II for the case defined by model (2.25).
In order to employ
models (2.22)~ (2.23), and (2.24), a lmowledge of the frequency fune""
tions of
It
Vt ,
=
When Ut ~
It -
(Xt --Vt '
M
Vt ,
+ 1
and
It,
respectively, are required.
as in model (2.23),
distribution of Yt is similar to that of
:I.e;
then, if It follows the
gamma distribution, this assumption would give
f(y) ...
-7
~eT:
1
e
-yu
/eT2
7y,
y
where
~
IIIl
E(Y) _"" n p. - V + M
The basis of such an assumption is the small variation in V relative
t
toX as shown by van Bavel and Verlinden (1956); however,\) the frequency
t
39
distribution of It sh:>uld be verified from data and (3041) is presented
only as a reasonable hypothesis.
For the case defined by zoodel (2.24) $ the frequency distribution
of
Xi
is required and curve fitting procedures require records of tte
duration of rainfa1l 9 At' as well as the amount of rainfallo The
frequen.cy function must be studied for a range of possible values of
R,p the maximum rate of zooisture infiltration by the soilo
One approach
would be to study the distribution of ~I~ so that the mean of
Xl
could be obtained from
E(Xi
I~?
0) •
E(~)
H(R) + R
E(~)
[1 ..
H(R~
,
where HeX') is the cumulative distribution function of~/At. Then
the frequency function for Xt,,:might be obtained by censoring the distribution of X •
t
40
CH 4P TE R IV
RESULTS ON THE DISTRIBUTION OF AVAILABLE SOIL MJISTURE
4.1 The General Shape of the Frequency Function of
Available Sb11Mbisture
With the assumption that dally amounts of precipitation follow
the gaJmlla distribution, it· is possible to obtain the stationary state
probabilities of available soil moisture following the procedures in
Chapter II for the case defined by model (2.2,).
Thus, we can study
the frequency distribution of available so11 moisture based on the
stationary state probabilities.
Figures 4.1, 4.2, and 4.3 are characteristic of the frequency
curves of Zt and show the. effect of different values of the parameters
on the shape of the curve; the parameter
1:'
=
Th~V
, is the argument
used in the tables of the incomplete gamma function (Pearson" 1947).
2"
The actual values plotted are Gk =·p~po' k = 1, 2, 3, •••
following the procedure of section 2 0 40 The probabilities of the zero
state,
Po'
are plotted in the appendix for a range of para.meterso
Since
the Gk are independent of the storage capacity, they are plotted in
preference to the P ; thePk are easily obtained from P = Po G , ,or
k
k
k
directly from the P
since from (2.45)
Ok
o
It is evident from figures 4.1 a;nd402 and 403 that the frequency distribution of Zt tends to be J shaped am that some of the curves are
approximately exponential.
However, some of the curves are skewed to
~
41
.20
20
25
k
Figure
..
4.1.
Available Soil Koisture Frequencies with p <
1.
42
20
15
10
5
n=.4, 'li=.'>
o
20
10
k
Figure
4.2" .t!.vailableSoil Moisture Frequencies
with p
> 1"
43
Q..5
0.4
~----_----/
n-.3, "1;=.3
0.3
0.2
-1_-------....
0.1
5
15
10
20
k
Figure
4.3.
Available s:>il Moisture Frequencies with P '"" 1.
25
44
the right and some are skewed to the left.
For model (2.25), the
expected value of the change in the system per time period is given
by
which is zero if' n~V ... 1.
In queueing theory the parameter
nw'v
is
known as the traffic intensity and generally denoted by p.
For figure
4.1
the curves are skewed to the right with the pro-
bability concentrated at, or near, the origin; p is less than one for
these curves.
In figure
4.2,
p is greater than one and the curves are
skewed to the left with the probability concentrated at, or near.ll the
storage capacity.
constant at
In figure
approximate~
4.3,
p "" 1 and the curves tend to be
n, and then increase at a constant rate.
These results are characteristic of queueing and storage systems
(Downton, 1956) and are not limited to the assumption that the amounts
of precipitation follow the gamma. distribution or the assumptions in...
volved in model (2.25).
Downton points out that when p
> 1,
the system
does not reach statistical equilibrium and, hence, the stationary state
probabilities become unrealistic~ particular~ when the storage capacity
is large.
The probability curves in appendix Figure 1 which approach
zero for large r are examples of the results obtained from a system with
p
> 1; these curves tend to underestimate the probability of the zero
sta.te.
Thus, a knowledge of the expected value of the change in t he,. system
per unit of time allows an inference to be made on the general shape of
the frequency curve of Zt.
These expectations are given by
45
t - Vt) = E(X.p
E(X
= rt~ -
E(Xt - Vt)
t-
E(X
- V
V
= E(Xt ) -
V)
= n~ -
E(Xt - V)
V
V
for model
(2.22)
for model
(2.23)
for model
(2.24)
for model
(2025)
SinceE(X.t) <E(~).9 the consequence of ignoring runoff tends to
overestimate 5, resulting in a negative bias in Pk.9 k
iii
0, 1, 2,
•• rt+l.ll when k is zero or near zero and a positive bias when k is
near r.
When the present approach is applied to the study of moisture
deficits, the expected value of the change in the system per unit of
time is given by
5'
and p'
= ..::L
= lip
n~
= E(V - It> ""
so that p'> 1 if P
V - n~
<1.
This implies that the
probability curves in appendix Figure 1 with P
> 1 are not applicable
to the study of the distribuliion of moisture deficits.
4.2
Distribution Free Methods
It should be pointed out that the transition probabilities can"
be estimated directly from climatic records and, hence, the stationary
probabilities of available soil moisture can be estimated without a
knowledge of the underlying distributions.
Let
~
th
k
class;
be the observed frequency of the input variableU in the
t
!.~0.9
no -frequency of U
t
< 1/2
nl "" frequency of 1/2
<
Ut
<
1
46
~
... frequency of kI 2
k
1
< Ut < 2" + ~
..
Then
(no + nl )
..
POO ...
N
(n 2k + n 2k+l )
N
-,
k ... 1, 2, •• r'
and
Po
...•
Pk
...•
no/N
(n
_ + n )
2k l
2k
N
,
k ... 1, 2,
0
• r' ,
where N is the total number of days in which climatic records are
available.
Recalling from section 2.2 that the generalized variable Ut is
defined as
U
t
(x' t
III
U ...
t
V)
M
(It - Vt)
M
+1
+1
for model (2.22)
for model (2 0 23)
Ii
t
"'r
for model (2 0 24)
It
U '"'V
t
for model (2.2,)
U
t
It is seen that the frequency classes for U will depend on either
t
M or V, !.i., the frequency of k < U
t
of Vk
< It < V(k
<k
+ 1 is equal to the frequency
+ 1) for model (2.2,).
Such an approach may be impractical, especially if r is large, although it is feasible i f the climatic data are available on punch cards
and a card sorter with a counting attachment is available.
Tables of
47
probabilities of soil moisture conditions based on the distribution free
approach are not feasible since there would be an extremely large number
of possible sets of transition probabilities.
4.3 Queueing Tmoq Results
The general form of the equations representing available soil
moisture as a stochastic process (2 .. 31) is the same form as many storage
and queueing systems with random input and unit output..
Such systems have
been studied extensively and proposals have been made for obtaining the
stationary service time distribution, which is equivalent to the distribution of
Wt~
based on a continuous time approach..
Explicit results obtained for the distribution of service time have
been based on infinite capacity and usually assumes that the inputs
follow an exponential distribution..
Hence, these results cannot be
applied directly to the present problem..
A good review article of the work relevant to the present problem
is given by Gani (19.57).
The continuous time approach requires the
assumption that the input variable, U , be independently distributed for
t
arbitrarily small time periods.
In the present case U is directly
t
dependent on precipitation which is not independently distributed and
the interdependency of the X increases as the time period becomes small..
t
The lack of independence of the U is not eliminated in the present
t
approach; however, it is less pronounced for discrete time intervals and
decreases as the time interval becomes larger.
4.4 Distribution of the Number of Drought Days Occurring in N Days
The number of days, d, in Which Zt
a dichotomous variable,
~"2..,
=0
during a total of N days is
for each of theN days one of the events
48
The stationary probability that the event
Zt
0 or Zt> 0 occurs..
Zt
0 is PO' and at equilibrium we can assume that the variable d
follows the binomial distribution with
par~meters N
and PO' !e!.. ,
However, since Zt is a time dependent stochastic: variable,!) the a©tual
probability that Zt
=0
varies from day to day throughout the N days ..
If the variable d is divided into two p:l.rts, g and h,9 where
g "" the number of initial drought days "" the number
of sequences,9
h ... the number of drought days which occur in a sequence
after an initial drought day has occurred;
then d ... g + h.
Given g, h follows the negative binomial distribution
with parameters g and
Poe; f.. ~. ,
Pr(hlg) ""
g+h-I
g-l
h
Poe (1 - Poo
)g
This follows since the negative binomial variable is defined as the
number of failures obtained in observing a fixed number of successes
where the probability of a success is constant..
If the events are
restricted to either initial drought days or drought days occurring in
a sequence,9 the variable h is the number of drought days occurring in
a sequence obtained in observing g initial drought days..
probability of occurrence is given by
and
g POO
E(h
I g) = I-poo
Then the
49
The moment generating function is given by
E( e ht
Equations
(4.43)
and
and is a random variable.
Ig ) = (1
(4.44)
-
Paa )g
(
1 -
Paae t).. g •
(4.44)
are of little use s;tnce g is not known
However, we can infer from
(4.43)
that the
average length of a sequence of drought da.ys is given by
E(h/g'. 1) + 1 =
PJa
Paa
+ 1
1 -
and the higher moments can be obtained from
(4.44) II
when g
= 1.
The results of this section are of practical value for obtaining
the expected value of drought variables in crop prediction equations.
Specific examples will be given in the following chapter.
,0
CHAPTER V
!PPLICATIONS
,.1 The Crop Production Function
Practical applications utilizing a knowledge of probable soil
wnisture conditions depend on a knowledge of crop production functions
relating the yield of a crop (Y) as a function of input variables (X)
such as
where at least one of the I's is characteristic of soil moisture condit ions.
The Xv s can be broadly grouped into two clas se s ~ namely.9 1)
factors which depend on the environment, and 2) factors which depend on
techniques of production.
Let Xli denote the class 1 variables and X
2j
denote the class 2 variables, where X2j is a technological practice which
alters the environmental factor characterized by at least one of the Xli
Possible examples of class 1 and corresponding class 2 variables are as
follows:
Class 1
Class 2
Plant available phosphorous
in the soil·
Phosphate applied as
fertilizer
Soil texture
seedbed preparation
M:>isture ponditions during
1st month of growing season
Date of planting
Weed population
Methods of weed control
Soil moisture throughout
the growing season
Irrigation
0
51
It should be noted that more than one of the Xli may be associated
with each X2j ' and vice versa.
The farm manager's problem is then to choose the X2j such that
Cost of X2j
•
Price of Y •
The resulting optimum value of X2j " say X' 2jJl is generally a function
of Xli for one or more values of .i.
When the Xli are characteristic of the soil, the use of soil
2j •
testing provides a means of evaluating X
However, when the Xli
are characteristic of weather conditions, the actual values cannot be
obtained until the growing season is complete and are of no use.
In
this case, Xb should be based on the value of Xli which is most likely
to occur on the average; this suggests the use of expected values or
5.2 The Drought Index
Recently, attempts have been made to characterize weather conditions
as they affect crop yield in a single index, D, say.
An index proposed
by Knetsch (1959) has the general form
m
Dl
=~
~).
..<.,
i=l
m
n i '"
~
~
i~
.,(iJ' n 4 n .;
... J
the growing season is divided into m growth periods, based on the phases
of growth of the crop, and the number of drought days, ni' occurring in
the i th growth period are given weights .(i with the equation extended
to include second order effects.
Some of the .,(i may be zero and any
or all of the .,(.. may be zero 0
1.J
The variable n
represents the number of drought days which occur
i
in a total of say N. days, and at equilibrium follows the binomial
1.
distribution with parameter PO' given by (4041), where Po is the
stationary probability that a drought day occurs.
Then the expec;ted
value of the drought index can be approximated from
m
~
m
.<. Ni PO· +
1.
.11.
1,...
~ ~1.1.
..
• 1
1.=
Ni PO·1.. (1 - PO·1. + N.1. PO·)
1.
m
+
~
o(ij Ni Nj POi POj '
where POi is the stationary probability of a drought day appropriate
to the parameters of the i th growth periodo
In order that the third
summation in <"022) be valid, the number of drought days must be
independent from period to period.
This condition may be unrealistic
for adjoining periods.
As a specific example, suppose an index of drought conditions is
given by
where np
n 2 , ny
and n4 are the number of days in which Zt = 0 for
the months May, June, July, and August, respectively.
Furtherjl
suppose we wish to obtain the expected value of D for use on a farm
l
in the Lumberton, North Carolina area with a storage capacity of 2 00
inches.
Then the estimates of V as given by van Bavel and Verlinden
,3
(19,6) are 0014, 0017, 0.16, and 0.14 inches per day, respect1vely~
for the four months May through August, and from appendix Table 2
the parameter estimates are obtained as follows a
"
Parameter
May
June
July
August
n
0028
0.34
0040
0034
X
0.88
0.86
0088
0.82
IJ.
XV
't' " " IJ.
r ... 200/V
0.42
0.44
00,1
0.48
0.32
0.3,
0031
0.30
14.3
11.8
12.,
14.3
Then from the probability curves of appendix Figure 1, the stationary
probabilities of the zero state are found to be approximately 0.21~ 0.26»
0.12, and 0 020 for the months May through August, respectively.
ThusS)
the average number of drought days is given by
E(n )
l
:&
(1)(.31)
E(n )
2
E(n )
3
:&
(0)(.26)
E(n4)
0
(31)(.12)
...•
(31)( .20)
I:
E(n~) ...0
lit
..
.
..
9.61
7.80
3.72
6.20
(1)(.12) ES8 + (31) (.12~
...
17.11
and
,.3 So il No 1sture Index
The majority of the research utilizing an index to weather conditions in crop production functions employs a drought index.
However,
,4
recent results frol)l the North Carolina TVA. com fertility project
indicate
that~
particularly on poorly drained soils, excess moisture
conditions have a significant effect on crop yields.
These results
suggest the need for an index which would be indicative of both drought
and excess moisture conditions.
The use of the mean available soil
m:;isture for the growth periods instead of drought days in (5c21) :might
improve the index when excess soil moisture is an important factor.
If
such an index is a linear function of the average soil moisture for the
growth periods,\l the expected value of the index can be obtained from
probability curves such as those in appendix Figure 10
If the index:
involves quadratic or higher order terms)/ the expected value of these
terms can be obtained from the higher moments given by (2.62) 0
Evaluation of the expected value of functions involving the product of
average soil moisture from two adjoining periods is complicated by the
lack of independence from period to period o
As a numerical example, suppose we wish to obtain the average
available soil moisture for the month of June, using the parameter
estimates from the Kinston, North Carolina weather station, assuming
a storage capacity of
n
1.,
inches •. From appendix Table 2
& 0.. 30
A & LOO
V & 0017 (from van Bavel and Verlinden, 19,6)
~ ~
A V :. 0.32
1"
808.
:.
lJ.
55
Then from (2.61)
and from appendiX Figure 1 (i)
-r.
E{Wt ) ~ .30 ~(1/064 - 1/.77) + 3{1/.55 - 1/.64) + 4(1/.48 - 1/.55)
+ 5(1/042 - 1/.48) + 6(1/.37 - 1/042) + 7(1/.34 - 1/037)
.. 8(1/.32 - 1/~34) + 8.4(1/,,30 - 1/032) .. 1/077
=
}
(1/032 +
l~i
". 2.952
E(Zt) ~ V E(Wt ) ~ 0.17 (2.952)
= 0.502.
504 Decisions Concerning the Use of Supplemental Irrigatio?
When the farm manager wishes to plan his production with the possi=
bility of using supplemental irrigation.ll the optimum value of drought
or soil moisture index is obtained from
where R is the ratio of the cost of reducing drought to the price of
the crop.
Then i f
D8
is the resulting optimum va,lue of D, the decision
concerning the feasibility of an irrigation program could be based on
Pr(a -< D < b) where a and b represent a suitably chosen interval around
the optimum value; for example, the confidence interval given by
Pr (a ~ D i ~ b) > 1 - -<.
56
For the special case when D is the number of drought days occurring
in some critical growth period of the cropJl
~.~o,
the silking period
for corn» the probability can be estimated from
Pr(a
~D
<b)
where N is the total number of days in the growth period and PO~s the
stationary probability that a drought day occurs, with a.s;:Dv=:;;: b < No
As a hypothetical example, suppose D'
= 17
and a
= 0,
b
we wish to find Pr(D <20} for the time period June 6 to July
cable to the Asheville, North Carolina weather station.
= 20,
and
25 appli=
The parameter
estimates from appendix Table 2 are
JUNE
JULY
and V as estimated by van Bavel and Verlinden (1956) is 0.,15 and O.. 14p
respectivelYJl for the two months.
The use of (5 .. 41) is contingent upon
the assumption that a single value of Po is operative throughout the
time period and the validity of this assumption depends, to some ext.ent.,\l
on how closely the parameter estimates agree for the two months..
In
the present example, the assumption appears to be reasonable, particularly
to the degree of apprOXimation warranted by the other assumptions
involved 0
~
It should be clarified that the parameters were estimated by
months only as a matter of comrenience and in reality distinct population boundaries do not exist from month to month, but rather a gradual
continuous change in the population occurs.
57
The combined estimates for the two months are obtained from
mere the subscripts 1 and 2 denote the months June and July, respectively
0
The combined estimates of A and V cannot be obtained from the
information available in appendix Table 2j however, the simple averages
of the estimates should_be satisfactory for the present example; i ..!
AC
"
0073
Vc
&:
0.145
.. ,
and
= 0.41.
Po is found to be apprOXimately 0.26 from appendix Figure 1, assuming
a storage capacity of 2.0 inches.
20
Pr(D <: 20) :.
~
k=O
as evaluated from Romig's
(~OJ\
The desired probability is given by
(0.26)k (0.73)'D-k = 0.92
(1947) binomial tables.. Thus, a fam manager
operating under these conditions would conclude that his chances are
quite
good of attaining the economic optimum without irrigation..
,8
,., Complementary Use of Long Term
~ather
Forecasts
It was po:1nted out in the introduction that farm nanager decisions
which are affected by uncertain weather conditions cannot usually be
based on weather forecasts since,' at the present time, accurate weather
forecasts are not generally available for long periods of timeo
In
the event of reasonably accurate long term weather forecasts,!) crop
production planning decisions will be able to be made with more confi=
dence.
The present approach of obtaining probable soil moisture condi-
tions from a lmowledge of the frequency function of the input variable
can be extended to make use of the ;information available from long term
weather forecasts o
.As an illustration, suppose the long term weather forecast for the
Kinstonj) North Carolina area indicates that the daily amounts Qf June
rainfall will be 20 percent below normal; then we can adjust tM para-
.
meter IJ. to make use of this information in predicting the average avail,
able soil moisture or expected number of drought days.
In section
'03,
the average available soil moisture for 'June in the Kinston area was
found to be approximately 0.,0 inches with a storage capacity of 1.,
inches and IJ. & 0
0
,3;
the adjUsted &1.is given,by
1J.1I& IJ. - .21J. :.
0.5343 - 0..1068 .. 0.. 4275,
and
Then the appropriate values in equation (2.61) are obtained from appendix
Figure 1 (i) with
1: ..
0.4 which gives
59
E(Wt
)
= .40 2(1/.67 - 1/.79)
~
)(1/0.59 - 1/.67)
~
,(1/.49 - 1/.,3)
+
6(1/.45 - 1/.49)
+
8(1/.41 - 1/.43)
+
8.8(1/.40 - 1/.41)
- 21 (1/.41
E(Z+)
...
+
\l
+ 1~
+
+
4(1/.53 - 1/.,9)
7(1/.43 - 1/.45)
+
1/.79
= 2.149
(0.17) (2.149) = 0.366•
Similarly, adjustments can be made in the parameters n and V i f
additional information is available on these parameters from weather
forecasts.
Forecasts to the effect that precipitation will occur less
frequently or more frequently will affect the parameter n.
Forecasts
predicting deviations of temperature from normal will affect evapotranspiration although adjustments in V are contingent upon the relationship
between air temperature and potential evapotranspiration which is complicated because air temperatUre lags behind radiative energy, the
determining factor in estimating potential evapotranspiration.
,.6
Sequences of Drought Days
The results in section
4~4
on the distribution of sequences of
drought days can be applied to problems of obtaining the expectation
and probabilities of drought days occurring in a sequence.
For example,
if the index of weather or soil moisture conditions in ('.11)1's a
function of average length of continuous drought the expected
given by
(4.4,).
val~.ie
is
To obtain this expected value for the parameter esti-
mates given by the Edenton, North Carolina weather station for the month
of June, we need the transition probability
60
Then under the assumption that amounts of rainfall follow the gamma
distribution, POO can be obtained from tables of the incomplete
tion (Pearson,
r funq:-
1947) as
Poo ..
(1 -
n) + n I(~,
A).
From appendix Table 2 the parameter estimates are
In this example, since l is approximately one, POO can be obtained
directly from
POO =
Then from
(1 - n)
(4.45),
+
fr
x
V
-r e:~ \ dx = (1
V
- n)
+ n(l - e -jj:')
.. 0.795.
if we assume that at least one drought day occurs, the
average length of a sequenoe of drought days is given by
E(h
Ig
=
1)
+
1& g:~8§
+
1
=
4.88 days.
Another possible area of application would be estimating the
probability of crop failure.
If m is the maximum number 'of drought
days occurring in a sequence that a particular crop can survive, then
the probability of crop £aill.lI'e is approximately
m-l
Pr(h~m',1 g
and i f pOO .&
Pr(h~m
=""1)'; 1 - (1 - pOO)
~ p~o
k=0
0.795 as in the previous example and m - 20 say, then
19
"" (0.795) k .. (0.795)
. 20
I g - 1) & 1 - 0.205 - 0.205 ,~
k-l
.&
0.010.
.....
----------- ~~~~~~-_
61
CHAPTER
VI
SUMMARY AND SUGGEsrIONS FOR FUTURE RESEARCH
6.1 Summary
The management of a farm generally requires plans to be made in
one time period for a product which will be realized at a later time
period..
Production yields of crops depend on the soil moisture condi-
tions during the growing season which are generally unknown at the tilne
of production planning; thus crop production planning could be made
more nearly rational by a knowledge of probable soil moisture conditionso
The objective of the present study is to characterize available soil
moisture as a stochastic process and to study the probability distribution function of available soil moisture as it affects crop yield ..
The concept of the soil as a storage system with a finite capacity
enables one to use the probability
t~ory
of storage systems and waiting
lines in studying the probabilities of soil moisture conditions..
Four
alternative models are proposed relating available soil moisture as a
time dependent stochastic process based on van Bavel's (1956) evapotranspiration xrethod of estimating soil moisture conditions.
The
models have the general form
Wt +l • \'It + Ut - 1,
where 1f is available soil moisture at time t multiplied by a constant
t
and U is the ratio of the amount of water which enters the soil and is
t
available for plant use to the evapotranspiration loss per unit of time ..
An approximate solution of the probability distribution of 'W
t
is obtained
62
by defining r
B
+ 2 discrete soll.moisture states, ranging from zero to
the storage capacity, which satisfy the properties of a Markov Chain4
The states are defined as
So : 1f = 0
t
<
~ : k - 1
<;: 1ft
Srl+l : r'
< Wt < r
k, k = l.l12}) 3.11 ••
0
r'
where r is such that available soil moisture is at storage capacity
when 1f = rand r
t
V
is the largest integer in r.
Then at equilibrium
the stationary state probabilities P are the solution of
k
P = TV P
where P is the r!+2 by 1 vector of the stationary state probabilities
and T .. (Pjk) is the r'+2 by r'+2 matrix of transition probabilities»
j, k
= 0,
1, 2, ••
0
e
•
r
i
+ 1.
r'+l
~
Since the rows of T sum to unity, the restriction that
k=O
is necessary to reduce
~
POk ' k =0, 1, 2, • • • r
ated from the cumulative distribution function of Ute
= 0,
1, 2'0 •• r'
tribution function of 1f •
t
=1
=. T'P to a set of r' + 2 independent equations..
The transition probabilities
2, 3, •• e r', k
P
k
i,
The
can be evaluPjk' j
= 1.11
are dependent on the unknown dis-
Two approximations are proposed for obtaining
these probabilities from the distribution function of U and evidence is
t
presented in appendix Table 1 to support the use of the approximations.
6,3
The solution of P
of G
k
II:
Pk/P
III
o since
T '! is simplified by solving the equations in terms
G is independent of the storage capac ityo
k
The frequency function of daily precipitation was studied for 25
years (1928-1952) of North Carolina weather station records since the
input variable U is a function of precipitation in all four of the
t
available soil moisture models and the transition probabilities p Ok can
J
be obtained from this frequency function when precipitation is the only
time dependent variable involved in UtI>
Following the Pearson system
of curve fitting, the gamma or Pearson type
In
distribution was found
to give the best fit when compared with the log normal distribution and
other curves in. the Pearson system.
Assuming that
da~
amounts of precipitation follow the gamma
distribution, frequency curves representative of the probability distribution function of available soil moisture were studied for selected
sets of parameters and the stationary probabilities of the zero state
are presented in appendix Figure 1 for a range of parameters based on
parameter estimates from the North Carolina weather stationsl>
The
results are in agreement with results from other storage and queueing
systems; !.I>.!o, when the expected value of Ut is less than unity the
probability tends to be concentrated at the origin and when t he expected
value of U is greater than unity the probability tends to be concent
trated at the storage capacity.
Some of the applications utiliZing probable soil moisture conditions are as follows:
64
1)
Obtaining the expected value of a drought index based on
a function of the number of drought days occurring in each
of m growth periods of a crop.
2)
Evaluating the expected value of available soil moist'are
for a particular growth period, and of a soil moisture index
based on an additive function of the average available soil
moisture for each of m growth periods.
3)
Obtaining approximate probabilities of a given number of
drought days occurring in a particular growth period as an
aid to determining the need for supplemental irrigation.
4)
Determining the average number of drought days which occur
in a sequence and the probability of exceeding a critical
number of continuous drought days,
.!_!.,
the probability of
crop failure.
The approach of obtaining probabilities of soil moisture conditions from
a lmowledge of the frequency function of the input variable can be modified to make use of long term weather forecasts by adjusting the· parameters in the distribution function which are affected by the forecast.
6.2 Future Research
It is hoped that the result s in Chapter II 't<1ill have wide applicability for obtaining probabilities of soil moisture conditions in areas
where natural precipitation is a primary source of available soil moisture o
The results in Chapter IlIon the distribution function of daily amounts
of precipitation are based on North Carolina weather records and hence
the hypothesis that the frequency of daily amounts of precipitation can
65
be obtained from the gamma distribution needs to be investigated for
other geographical areas in which no previous information is available.
The use of models (2.22) through (2.24) needs to be investigated to
determine i f the accuracy gained in using the mOr'e complicated models
outweighs the simplicity of model (2.25).
In the event of adequate
records of the climatic variables involved in models
(2022)~
(2023)9
and (2.24), or a satisfactory rrethod of estimating them from existing
climatic data, the frequency function of the input variable U should
t
be investigated for these models.
Practical applications utilizing probability curves such as those
in appendix Figure 1 are contingent upon the following conditions which
give rise to areas of future research:
1)
The input variable Ut approximately follows the gamrra distri..
bution with a finite probability that Ut "" 0 0
When this
condition is not satisfied, the transition probabilities can
be estimated directly from climatic data following the methods
proposed in section 4.2, or i f the observed frequencies indicate
another distribution function should be used,the stationary
state probabilities can be obtained based on the result:i.ng
transition probabilities.
2)
The approximation (2036) for the transition probabilities does
not result in serious error.
The close agreement between the
two approximations to the transition probabilities and the
narrow bounds as shown in appendix Table 1 indicate that this
condition is not too serious, particularly to the degree of
approximation warranted by the other conditions.
It is po ssible
66
that the transitfonprobabilities could be made more exact
by substituting 'the stationary state probability P. for the
J
denominator of (2.35) and deriving an empirical function for
F(k+l-Wt ) - F(k...Wt ) g*(Wt ) dWt
Pj
where P.J and g*(w:
}are obtained from an initial solution based
.t ,
on the approximation (2.36)0
3)
The range of parameters includes the estimated parameters.
This condition is trivial since the probability curves can be
extended following the same procedures presented in this stuQy.
4)
The discrete states of available soil moisture adequately
describe the dynamic systemo
The error involved in studying
available soil moisture as discrete states can in theory be
made as small as we like by defining n states within each of
the r V+2 states; then as n becomes large the discrete approximation to the stationary probabilities approaches the continuous distribution of available soil moisture.
The obvious
disadvantage of this approach is that n(r D+2) equations must
be solved for the stationary probabilities.
Also, as n
approaches infinity the transition probabilities approach
zero and even for relatively large values of n the transition
probabilities, with the exception of POO and PO' may be zero
to six or more decimal places.
67
5)
The lack of independence among the Ut can be ignored without
serious deviation from reality.
This condition represents a
primary weakness of the present approach.
In a continuous
time approach with arbitrarily short time intervals, it is
evident that the inputs into the system do not rep:re sent an
independent random variables and hence the so called exact
results based on continuous time are not directly applicable.
The same problem arises in the theory of dams as well as other
storage and queueing systems.
The theoretical workers in
these fields have assumed independent inputs as a matter of
course and have little to offer for the many practical problems
in which the independence assumption is unrealistic.
(1957)
Kendall
suggests that, in lieu of procedures for coping with
lack of independence of the input variable, solutions obtained
assuming independence should be regarded as approximations.
Since the problem of interdependence of the input variable will
arise in any approach to obtaining probable soil moisture conditions,
our present state of knowledge does not allow an exact solution to the
problem.
However, it is hoped that the present work will provide a
nucleus for future studies which will give rise to a more nearly
rational basis for making crop production planning decisions, than the
present guessing game usually based on the farm manager's intuition
incorporated with his experience.
68
LIST OF REFERENCES
Beard, L. R. and ,Keith, H• .A. 1955. Discussion of tIThe Log Probability
Law and its Engineering ,Applications". Proc. Amer. Soc. Civ. Eng.
sep 665, 81: 22-29.
Chapman, D. G. 1956. Estimating the parameters of a truncated gamma
distribution. Ann. Math. Stat. 27: 498-506.
Chow, V. T. 1953. Frequency analysis of hydrologic data. with special
application to rainfall intensities. Bulletin No. 414, Illinois
Engineering Experiment Station, Urbana, Illinois.
Chow, V. T.
tions.
19~4..
The log probability law and its engineering applicaProc. A,mer. Soc. Civ. Eng. 30: sep 536.
Davis, H. T. 1933. Tables of the higher mathematical functions, Vol.
I. The Principia Press, Inc., Bloomington, Indiana.
Elderton, W. P. 1953.
Washington, D.C.
Frequency Curves and Correlation.
Downton, F. 1957. ,A note on Moran I s theo ry of dams.
Oxford (2) 8: 282-286.
Haven Press,
~art.
J. Math.
Fel1er,W. 1950. ~ Introduction to Probability Theory and Its Applications. Vol. 1. John Wiley and Sons, Inc., New York.
Foster, H. A. 1924. Theoretical frequency curves.
Civ. Eng. 87: 142-173.
Trans. Amer. Soc.
Gumbel, E. J. 1958. The statistical theory of floods and droughts.
J. Inst. Water Eng. 12: 157-173.
Havlicek, J. Jr. 1960. Choice of optimum rates of nitrogen fertilization for corn on Norfolk-like soil in the coastal plain of North
Carolina. Unpub1ish€)d Ph. D. thesis, North Carolina State College,
Raleigh. (University Microfilms, Ann Arbor).
69
LIST OF REFERENCES (continued)
Kendall, D. G. 1957. Some problems in the theory of dams.
Statist. Soc., B, 19: 207-233.
J. R.
Knetsch, J. 1. 1959. Moisture uncertainties and fertility response
studies. J. Farm Econ. 41: 70-76.
Knetsch, J. 1. and Smallshaw, J. 1958. The occurrence of drought in
the Tennessee Valley. Tennessee Valley Authority Report T .58-2AE,
Knoxville, Tennessee.
Manning, H. 1. 19.50. Confide~ce limits of expected monthly rainfall.
J. Agri. Sci. 40: 169-176.
McIllwraith, J. F. 19.5.5. Discussion of "The Log Probability Law and
its Engineering Applications ll • Proc. ,Amer. Soc. Civ. Eng. 81:
sep 66.5.
Moran, P. ,A. P. 1954. .A probability theory of dams and storage systems.
Aust. J. ,App. Sci. 5: 116-124.
Moran, P. A,. P. 1955. .A probability theory of dams and storage systems:
modifications of release rules. Aust. J. J\pp. Sci. 6: 117-130.
Moran, P. A.. P. 1956. .A probability theory of a dam with a continuous
release. Quart. J. Math. Oxford (2), 7: 130-137.
Moran, P. A. P. 1957. The statistical treatment of flood flows.
.A,mer. Geophys. Union 38: 519-523.
Trans •
Parks, W. L. and Knetsch, J. 1. 1960. Utilizing drought days in evaluating irrigation and fertility studies. Soil Sci. Soc. Amer.
Froc. 24: 289-293.
Paulhus, J. 1. and Miller, J. F. 1957. Flood frequencies derived from
rainfall data. Proc . .A,mer. Soc. Civ. Eng. 83: sep 1451.
Pearson, K. (Ed.) 1946 reissue. Tables of the Incomplete Gamma Function.
Cambridge University Press.
Pelton, W. 1., King, K. M., and Tanner, C. B. 1960. An evaluation of
the Thornthwaite and mean, temperature methods for determining
potential evapotranspiration. Agron. J. 52: 387-395.
Penman, H. L. 1948. Natural evaporation from open water, bare soil
and grass. Froc. Roy. Soc. A., 193: 120-145.
Romig, H. G. 1947. 50 - 100 Binomial Tables.
New York.
John Wiley and Sons, Inc.,
70
LIST OF REFERENCES (continued)
Saaty, T. L. 1959. Mathematical Methods of Operations Research.
McGraw-Hill Book Co., Inc., New York.
van Bavel, C. H. M. 1956. Estimating soil moisture conditions and
time for irrigation with the evapotranspiration method. U. S. D. A.
ARB 41-11.
van 13avel, C. H. M. and Verlinden, F. J. 1956. ,Agricultural drought
in North Carolina. Technical Bulletin No. 122, North Carolina
,Agricultural Elcperiment Station, Raleigh.
71
APPENDIX
With the assumption that daily amounts of precipitation follow
the gamma distribution, the transition probabilities, POk' and the
approximations given by (2 •.36) to Pk' k .. 0, 1, 2,
..
..
r'!J can be
obtained from (4.11) using tables of the incomplete r -function (Pearson
1947) to evaluate the integral
!
x
,,>.-1
.-x/-< =
I(",X, >'-1),
where I('t'X, X-l) is the value of the integral obtained from the tables
with u .. 't'Xand p .. X-I in Pearson's notation.
POO
=
POk
lllI
n
Po
.r:
(l-n) .. n I('t'/2, A.-I)
Pk
.It.
n I [(k +
Thus,
(l-n) .... n I('t', )..-1)
i: ~k"lh, X-~
A-l), k = 1, 2, 3,.. . r l ,
n
I(~,
Tt
I Uk -
and
~)'t', X-~
-
~)'t', A-~
, k = 0, 1,
0
..
The approximation given by (2 •.37) as
is difficult to evaluate for the gamma distribution since an explicit
expression for F(k+l-Wt ) - F(k-Wt ) cannot be obtained except for
An evaluation can be obtained by series expansion which gives
.. . ),
>.. •
1.
r v..
72
where
&.
o
vq
..
"q+l
~ q ...
r (q...l) ~q'(A+q)
&q ...
1, 2" 3, •
Then
F(k~Wt)
F(k+l-W ) t
&0 (k+l·"'W
ft
-
t
) 'A 1 - &1 (k+l=W ) • &2 (k+l-Wt ,2
t
~ (k+l-Wt )3.
... &
2
0
t ) 'A 1 - ~ (k-Wt )
= &0 (k-W
(k=W )2 _ B... (k=W )3
t
-)
t
•
0
•
which can be integrated from j-l to j with respect" to W which
t
results in
•
where
k ... 1, 2, 3,
&1
q
...
•
_&q.:.-- '
o
r
•
o
o
•
•
•
)
t,
q .. 1, 2, 3,
(,,-.q+l)
~ converges quite rapidly for small values of k since
usually less than one.
For k in the neighborhood of
VA/1Jo
is
25 and larger,
and V\I~. near unity, the necessary number of terms in ~ for
4 digit
accuracy becomes prohibitive.
In order to compare the two approximations both were computed
for" ... 1, -r'"
y),v/"" ..
0.2, n ... 0.2 and 0.4.
The results are pre-
sented in appendix Table 1 along with the upper and lower bounds
given by
•
•
73
Po (upper)
F(l)
POO
Po (lower)
reO)
l=n
Pl (upper)
F(l)
Pl (lower)
F(2) - F(l)
Pk (upper)
F(k)
Pk (lower)
~
=
=
F(O}
... POO - (l-n)
POl
B
F(k-l) = PO(k-l)
F(k+l) - F(k) ...
POk~
k ...
2~
3$ •• rUe
The difference in the two approximations is seen to be triVial to
three significant digits which indicates that the simpler approximation
(2.36) proposed by Moran (1954) is adequate fer practical purposes.
Appendix Figure 1 shows the
state~
~
st~tionary
probability of the zero
for the range of parametersg l ... 0.7, 0.8~ 0.9 9 1.0, 1.2;
... 0.1, 0.2, 0.3, 0.4, 0.5, 0.61 and n
= o.a~
0.3,
004~
0.50
The
parameters were selected on the basis of the estimates from the North
Carolina weather stations as given in appendix Table 2. The computations
for appendix Figure 1 were made on an IBM 650 using the solution
procedure of section 2.5.
e
e
~pendix
e
Table 1
Approximations (2.36) and (2.37) to the Transition Probabilities (Pk) with Lower and Upper Bounds
,"", == 1,
1:' ..
~
0.2, n .... 0.2
k
(2.36)
(2.37)
lower
upper
0
1
.819033
.032810
.026861
.021992
.018010
.014790
.012070
..009882
.008091
.,006624
.005423
.0044Lo
.003635
.,002976
.002437
..001995
..001633
.001337
..001095
~000896
..000734
.000601
.000492
.000403
.000330
.818734
.032.858
.026902
.02202'6
.018033
.014764
.012088
.009,897
.008103
.006634
.005431
..004447
.003641
.002981
,,002441
0001998
.001636
.001339
.,800000
.029682
.024301
.019896
.016290
.013337
..010919
..008940
.007319
.005993
.004906
.o040i7
.003289
.002693
.002205
.001805
.001478
..001210
.000991
.999311
.000664
' ..000544
,,000445
..000364
.000298
.836253
.046253
.029782
.024301
.019896
.016290
.013337
.010919
.008940
.007319
.005993
,,004906
.004017
.003289
.002693
.002205
.002805
.001478
.001210
0000991
..000811
.,000664
.000544
.000445
.000364
2:
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
.oq1097
.000898
..00OD5
.000602
,,000493
"oooLa.>
,,000330
... 1,
1:'
==
0.2,
11. "",
0 .. 4
(2.36)
(2.37)
lower
upper
.638065
.065617
..053723
.043985
.036011
.029572
.024139
.019763
.016181
.013248
.010846
.008880
.007271
.005953
.004874
.00,3990
..003267
.002675
..002190
.001793
0001468
.001202
,,000984
.000806
.,000660
.637462
.065717
.053804
.0440$1
.,036066
'.029528
.024176
.019794
.016206
.013268
.010863
.008894
.007282
.005962
.004881
.003996
.003272
.002679
.002193
.001796
,,001470
,,001204
.000986
0000801
...000660
.600000
.059364
.048603
.039793
.032579
.026674
.021839
.017880
.014639
.011985
.009813
.008034
'.006578
00 05385
.,004409
.003610
.002956
.002420
.001981
.001622
..00132~
,,001087
..000890
,,000729
,,000597
..672506
.072506
.059346
.048601
.039793
.032579
.026674
, .021839
.017880
.014639
.011985
.009813
,,008034
.006578
..005385
.004409
",003610
0002956
.002420
..001981
..001622
0001328
..001087
.000890
.000729
.,,)
p-.
75
Appendix Table 2.. Parameter Estimates for the Gamma Distribution Based
on Precipitation Records of North Carolina Weather
Stations
Station Month
i
n/N
f/s 2
!SHVILIE
April
May
Jun~
July
August
September
0.3386
0.3870
0.42.13
0.4748
0.3922.
0.3000
0.2901
0.2375
0.2971
0.30520.3215
0.3202
0.7172
0.6230
0.5943
0.6270
0.3556
0.4511
0,.729
0.768
0.766
0.699
0.670
1.094
0.2733
0.2709
0.2786
0.3380
0.3135
0.2506
0.4022
0.4029
0.5492
0.6557
0.5661
0.6395
0.7815
0.8508
0.8429
0.6920
0.8448
0.5681
0.993
1.048
0.9940.872
0.916
0.24110
0.2658
0.3013
0.3987
0.3354
0.2573
0.4490
0.3974
005073
0.5137
0.5670
0.5782
1.0083
0.7455
0.6668
0.8306
0.9782
0,.6771
0.2h40
0.2516
0.2800
0.3354
0.2993
0.2493
0.5513
0.4607
0 4989
0.5758
0.6649
0.5794
0.8225
0.7596
0.9346
0.9630
Q.6484
0.5125
EIBNTON
~ri1
May
June
July
August
September
ELIZABETHTarlN
J\pri1
May
June
July
,August
September
F.fcrETTEVILLE
April
May
June
July
,August
september
0
0.80~··
76
.APpendix Table 2- ..
(continued)
X
Y./2
,s
0 0 3080
0 0 3006
0 ..3586
0.4038
0 3690
0.3053
0 0 ).+094
0 0 4089
0 4583
0 0 6106
0 ..4732
0.4579
0 0 5922
0 0 6:)03
0 0 7668
0 4930
0.5231
0 0 4362
0 0 2320
0.2309.
0 ..2960
0 0 3380
0.3032
002480
0.453;20 4966
0.5343
0.7020
005585
0 ..6268
0 0 9997
0 ..8147
0.,8528
0 ..9316
0.6796
0 ..5382
1.228
0.896
1.003
1.132
0 ..931
0.917
0 ... 2573
. 002825
0 ..3360
0.4038
0.3380
0 ..2840
0 ..4583
0 .. 4161
0 ..4432
0.5127
0.4822
0~:4775
0 ..7102.
0.7545
0.,,6576
0.8193
0.7315
0 ..4212
0.874
0.876
0.858
0.878
0.815
0..711
0 ..2973
0.2864
0.3173
0 0 3780
0 ..3354
0.2626
0.4047
0 0 4570
0.4562
0 0 4969
0.,5728
0 ..6017
0.6788
0.6988
0 8263
0.7630
0 ..5657
0.5519
0 ..861
0.813
0 0 835
0.918
0 ..729
0.742
Station Month
n/N
GOLreBORO
J\pril
May
June
July
August
September
0
0
0
KINSTON
aPril
May
June
July
August
September
0
LUMBERrON
aPril
May
June
July
August
September
NASHVILLE
April
f:1ay
June
July
August
September
0
e
e
e
-'
.9
X'" 0.7
(a)
~...
~BO.8
(b)
0.2
n • 0.2
.8
• 6
-6
..
.7
l!!II
..---1+
.,. 4
.
'
•3
.2
.1·
0' ,
9
,
,
5
!
,
,
•
1
10
,
,
t
,
,
15
!
,
!
,
'
20
,
!
,
,
,
25
Ii!
;
,
5
,
,
g
B
'
,
!
j
10
r~
&:>pendix Figure 1. Stationary Probability of Zarc state
!
'
15
,
'
,
F
i
j
ii.'
20
.-..J
--J
e
e
_
Cc)>., ... 0.9
'l;t • 0.2
>.,'" 1.0
(d)
n ... 0.2
•
~.6
.;;-.5
,
\
"
.5
Po
I
o
\
.......
'f' • ' , , ,
5
I
!
,
JrI
,
,
,
J
!
1~
,
!
F
-.5
-,;....4
"-
\
,;-.3
---------
""
----
'"
'\.
- ~ \\~~
"
-.,4
I
,\.
'
!
20
!
!
r
"
.2$
I " ! , , ! , , , !
$
10
8
!
'jI
'j
,1$
!
I
,
,
I
J
!
J
J
,2Q
r ---+
--J
APpendix Figure 1 (continued)
Q)
e
e
(e)
e
A. '" 1.2
(r)
n= 0 0 2
... ~6
o~ \\~
-1\\\\
'\.
).. "" 0.7
n ... 0 0 3
~
,,=.6
1;=05
----
'h-
"C.
..3
I
\
""
"C.
~.
03
\\
--..!.
~
4
•
·:bl'
• ....1...-i:-'....1......1..'....1...-'-:':;:-;;1....1..1-L.'-LI~,;-;'_I
o .-+-....1..
f
5
10
15
1
,
20
25
5
10
15
20
r-~
-..J
Appendix Figure 1 (continued)
l.
'0
e
e
e
e
e
e
... 1.0
1.t ,. 0.3
(i)~
'.
(j)
~
.. 1.2
n ...
0~3
··6
"-..-.
~·.,6
.5
Po
:-
o
5
10
15
20
25
r.:;
10
1.5
20
r~
ex>
.APpendix Figure 1 (c ontinued)
I-'
•
e
.9-
(k)
z.
0.7
1t ""
0.4
).
e
(1)
~
.. 0.8
n: ..
0.4
.,8-
",=.6
~.6
.2:-
1
.1
o
10
20
r~
OJ
-i\ppendix Figure 1 (continued)
I\)
•
e
.9
(m)
~.
'l}. I :
e
0.9
(n)
0.4
A'" 1.0
11t ..
0.4
";""06
Po
.3
.
..2.
.1-
";·01
o
5
Jr'
15
20
25
.5
10
15
20
r ->
co
Appendix Figure 1
((~onti.nued.)
Iv>
l
•
e
..9
(0)
A
=<
11 ...
--
1 ..2
(p)
0 .. 4
A
a
.n ...
o.. 7
oS
..7
Po
.1
a
5
10
15
20
25
5
10
1~
20
r~
en
~ppendix
FiglJre 1 (continued)
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