l
"~I' T
.,
I
AVAILABLE SOIL MOISTURE AS
STOCHAS~IC
i\
PROCESS
by
Dale E. Cooper and David D. Nason
t
,\ rl
..
This research was done in cooperation with the Division of
Relations, Tennessee Valley Authority •
~gricultural
Institute of Statistics
Mimeo Series No. 270
December, 1960'
•
iv
TaBLE OF CONTENTS
Page
LIST OF TABLES • •
o
LIST OF FIGURES. " "
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CHAPl'ER
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INTRODUCTION "
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FORMULATION OF AVAILABLE SOIL MJISTURE AS A TIME
DEPENDENT STOCHASTIC PROQESS
"
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vi
" " " " • • • • • • • • • vii
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" " "
General " .. • • "~,, • • • "
Statement of Objectives'.. "
II.
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2 0 1 The Soil as a Storage S,rstemo " " " • " " • " •
2 2 M:>dels Expressing Available Soil M:>isture as
a Time Dependent Stochastic Process " • " " "
2.3 Available Soil M:>isture as a Markov Process. "
204 The Transition Matrix and Stationary State
Probabilities " • • • • • • " • • " " • • • •
2.5 Solution for Stationary Probabilities • " • • •
2.6 The Expected Value of Available Soil MOisture"
2.7 M:>isture Deficits " • " • • • • • • " 8 4> " " •
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1
1
3
• •
4
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4
0
III.
6
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• • • • 11
" • • • 15
• • • • 17
• • • • 19
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•
IV.
•
RESULTS ON THE FREQUENCY DIsrRIBUTION OF PRECIPITATION. • •
301 Possible Distribution Functions for Characterizing
Precipitation Frequencies • • " " " " " • • • " •
3,,2 Results Based on North Carolina Weather Station
Records • • • " • • • • " " • • • • • " " • • • •
303 Estimates of the Parameters of the Gamma
Distribution.... " • " •• " • • •• • • • • • •
3.4 The Distribution of Inputs into the S,ystem. • • • •
•
•
20
21
••
21
••
31
••
••
36
RESULTS ON THE DIsrRIBUTION OF AVAILAl3LE 50IL I"1OISTURE • ••
40
The General Shape of the Frequency Function of
Available Soil Moisture " " • • • • " " " • • "
Distribution Free Methods • • " " " • " • • • • •
Queueing Theory Results " • " • • • • " " " " • •
Distribution of Number of Drought Days Occurring
in N Days • • " • " • • • • • • • " " " " • • •
47
37,
• • • 40
• • • 45
• • • 47
• ••
v
TABLE OF CONTENTS (continued)
Page
V.
,I
APPLICATIONS • • • • • "
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• • • • • • • • • 50
5.1 The Crop Production Function. • • • " • • " " • • " • • 50
5.2 The Drought Index " " " • • • • • • " " •
51
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503 So il 1'10 isture Index • " " " " " " " • " • • • • " • "
5,,4 Decisions Concerning the Use of Supplemental
•
It
53
55
Irrigation. • "
5.5 Complementary Use of Long Term Weather Forecasts • • • • 58
5.6 Sequences of Drought Days • • • • • • • • • • • • • • • 59
0
VI.
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SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH. • • • • • • • •
Summary ••• " •• " •• "
Future Research " • • • " •
LIST OF REFERENCES • " "
APPENDIX .. " "
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71
vi
LISI' 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 • e e o . • • • • • • •
eo. • • • • •
75
0
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vii
LIST OF FIGURES
Page
i
2.1
Available Soil M::>isture as a Finite Queueing System • • • • •
3.1
Goldsboro -- April Observed Frequencies of Rainfall • • • ••
23
3.2
Goldsboro
Ms.y Observed Frequencies of Rainfall • • • • • •
23
3.3
Goldsboro
June Observed Frequencies of Rainfall. • •
·. "
24
3.4
Goldsboro
July Observed Frequencies of Rainfall. • • • • •
24
3.5
Goldsboro
August Observed Frequencies of Rainfall•• • • •
25
3.6
Goldsboro
September Observed Frequencies of Rainfall • • •
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
3.10
Lumberton -- August Observed Frequencies of Rainfall. " • ••
27
3.11
Kinston -- June Observed Frequencies of Rainfall. • • • • ••
28
3.12
Kinston -- August Observed Frequencies of Rainfall. • • • ••
28
3.13 Edenton -- June Observed Frequencies of Rainfall. • • • • ••
29
3.14
Edenton -- August Observed Frequencies of
Ra~nfall.
• • • ••
29
401
Available Soil Moisture Frequencies with
p
1 • • • • • ••
41
4.2
Available Soil l'40isture Frequencies with
p > 1 • • • • • ••
42
4.3
Available Soil Moisture Frequencies with
p
<
=1
• • • • • ••
43
• • • • • • • • • •
77
Appendix
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; however,
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 characterIt
ized by a drought indexo
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 condij
tionso
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 period.
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
conditionso
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
&1allshaw (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' (1958) applicable to areas in the
Tennessee ValleY9 and by van Bavel andVerlinden (1956) for areas in
North Carolinao
In both of these studies drought probabilites, based
on van Bavel's (1956) 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
102 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
0
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
CHA,PTER
II
FORMULATION OF AVAIL./U3LE SOIL MOISTURE AS A TIME DEPENDENT
STOCHASTIC PROCESS
201 The Soil as a Storage System
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 To L. Saaty (1959)· provides a resUme 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 rooisture 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
/
/
---
/
.
/
'0
M
•
r-I
•
(\J
(\J
o
6
202 M>dels 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
period, the precipitation which occurred during the time period, and
the moisture loss during the time period
(2.21)
where
Zt
=:
X
~ precipitation occurring in the tth time period
Lt
=:
t
available soil moisture at time t
. .m th e tth t·J.me perJ.o
. d ..
wat er 1oss occurrmg
M:ldel (2.21) defines a storage system with both input and output
as random variables.
This model is complicated by the fact thatLt is
difficult, if not impossible, to measure and is a function of numerous
variables.
Some of the factors which affect Lt 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
which 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 10) 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.
The following modification, based on
7
van Bavel's (1956) evapotranspiration method of estimating soil moisture
is proposed to allow for these possibilitieso
Let
At ... duration of the amount of precipitation Xt
•
R = maximum rate of moisture infiltration by the soil
It . . It
if
'" AtR if
It ~ AtR
Xt
> AtR
V ... potential evapotranspiration occurring during the tth t:iIne
t
period
o ""
maximum amount of plant available water which can be held by
the soil.
Then we can write
<
t < 0 + Vt
if
Vt
... 0
if
Zt + X >0 + Vt
=0
if
Zt +
Zt + X
t
(2.22)
Xi < Vt
All of the climatic variables (X , At and V ) involved in model
t
t
(2022) 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 + ~ - Vt
<
Zt + It < C ... Vt
if
Vt
... 0
if
Zt + It L 0 + Vt
... 0
if
Zt + It <. Vto
(2023)
8
In this model no recognition is given to runoff 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 Vt from available climatic data have been
proposed in recent years.. These methods are discussed by Van, Bavel (1956)
and Pelton et al o (1960)..
The method derived by Penman (1948) is general-
ly accepted as being more appropriate to the humid areas of the United
State s
Penman's formula as given by van Bavel (1956) is
0
H + 0027 Ea
/). + 0027
vt =
,
where
incremental change in vapor pressure
/).
=
H
= net
Ea
=a
heat adsorption at the surface
function of saturation deficit and wind velocity ..
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
O? 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
if
Zt + It ~. C + V
... 0
if
Zt +
=0
4(
C+ V
It < V
w1':en runoff is an important factor, and
if
V < Zt + Xt
.. C
if
Zt +- ~ >C + V
... 0
if
Zt + Xt
Zt+l .. Zt + It - V
<
C+ V
(2 0 25)
< V
vmen 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.
M:ldel (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.
M:ldel (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 approach..
M:ldels (2 .. 23) and (2 .. 24) are intermediate
between (2 .. 22) and (2 .. 25) in simplicity and departure from reality ..
The maximum amount of plant available water, C, is denoted as the
"base amount" in van Bavei's evapotranspiration method of estimating
soil moisture conditions on which models (2.22) through (2 .. 25) are
based.
The determination of C regulates the intensity of drought as
defined when Zt = 0..
van Bavel (1956) proposes that C be obtained as
the difference between field capacity and the wilting point, both
expressed on a volume basis"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;
.,
!.~
Zt represents readily available soil moisture..
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.
<
Q, where Q is
Although the results of this study are applicable
to either definition of C, the departures from reality of models (2 .. 22)
through (2 .. 25) become more serious when Zt < Qo
lIhen soil moisture is
below the wilting point, V is dependent upon Zt as well as weather cont
ditions. Given a mathematical expression relating V as a function of
t
Zt it is possible that the models could be modified to account for the
dependence of Vt on Zt"
11
203 Available Soil Moisture as a Markov Process
In order to keep the notation general, it will be convenient to
(2.22) through (2025) by the single model
denote models
..
Wt +l
= 1ft
if
1< Wt + Ut < r + 1
... r
if
Wt + Ut
>r
... 0
if
Wt + Ut
~
+ Ut - 1
+ 1
(2031)
1,
where
Wt
U
t
= Zt/M
for models
(2022) and (2.23)
= Zt/V
for models
(2024) and (2.25)
...
+ 1
for model
(2.22)
+ 1
for model
(2.23)
. . Xi/V
for model
(2024)
... Xt/V
for model
(2 025)
... e/M
for models
(2022) and (2.23)
... C/V
for models
(2.24) and (2,,25) ..
...
r
t - Vt )
(X
M
(Xt - Vt )
M
The quantity M is the maximum value of Vt' characteristic of a particular geographic area and time of year"
models
By introducing M and adding 1 in
(2.22) and (2 .. 23), the lower limit on Ut is zero for all of the
models ..
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
<
~:
0 <Wt
S2:
1 <"Wt < 2
1
..
..
..
•
r
i
_
1< Wt
~r'
where r' is the largest integer in r.
Let p.
Jk
be the transition probability of going from state
time t to state
Sk
at time t+L
S. at
J
The Markov property is satisfied if' 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, .... r'+l; !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(W + = 0
t l
I
Wt ... 0)
That this condition is
Clearly
P
jk
I:
0
for
j
>
Pr(k - 1 <Wt +1 ~ k)
k + 1
= Pr(k
since from (2031)
= Pr(k <
and since
(k
j
<
W
t
>k
<
< Wt
- 1
+ Ut - 1 <k)
W + U
t
t
<
k + 1)
U ::> 0, the upper limits on Wt which satisfy (2033) are
t
k + 1), but we are given that (j - 1 <W < j), hence, for
t
+ 1, P jk
= 00
In order to evaluate the Pjk' we need to have a knowledge of the
cumulative distribution function of UtO
If this distribution function is
Ok can be obtained explicitly in terms of F(U); io~.,
denoted by F(U), the P
= F(k
+ 1) - F(k)
PO(r ' +l)= 1 - F(r')
k
= 1,
since Pr(Wt
2, ° • r'
>
r)
= o.
POO • F(l)
(2.34)
14
The Pjk for
= 1,
j
2, 3, ••• r'
and
k
= 0,
1, 2, •• r'
known exactly until the distribution of 1f is known.
t
are not
In this case, from
the basic laws of probability
k
k
Pjk
=
-~
j
j
A
j
where G(Wt+ll
dG(Wt )
j
A
dG(Wt +1
I wt )
,
dG(W )
t
'W ' is the conditional cumulative distribution function
t
of W + given W and G(W ) is the marginal cumulative distribution
t
t l
t
function of 1ft " Since
k
k
~
dG(W +1
t
I
Wt )
F(k + 1 - 1ft ) - F(k - Wt ),
=
Pjk can be written
)
j
J~
dG(W )
t
!'bran (1954), in deriving the transition probabilities for the
amount of water in a dam, asserts that a suitable approximation to the
P Ok 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).
(2.36)
Another approximation can be obtained by assuming that Wt is uniformly
distributed on the interval (j - 1, j) so that (2.35) becomes
15
j
=:0
j
J:.
F(k + 1 - Wt ) - F(k - Wt ) dWt ,
(2 .. 37)
Bounds for Pjk can be obtained by setting Wt equal to j-land 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 f+l) different
values of Pjk and the notation can be simplified to
k = 0, 1, 2, ••• r' + 1.
POO
POI P02 P03
Po
PI
0
Po
P2
PI
P3
P2
Then the transition matrix can be written
P04 P05
P4
P3
•
P5
P4
•
"
"
POri
PO(r'+l)
•
"
•
Pr '
Prf +l
.. ..
r'-l
•
•
Prl-l 1 -
~ Pi
i=O
r f-2
0
0
Po
PI
P2
P3
•
"
•
"
Prl - 2 1 -
~ Pi
i=O
(2.41)
r'-3
0
0
0
Po
PI
P2
0
0
0
0
0
~ Pi
Pr '-3 1 - i=O
•
"
"
"
"
• Po
"
"
0
"
1 - Po
16
From (2041) 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 Markov chain is described by Feller
(1950) to be irreducible and aperiodic.
Let the stationary probabilities of state ~ be ~ at time t and
~ at time t+l" with
Then
.!:**
P* and P** the corresponding r'+2 column vectors.
= TV P*; i.~.,
~ = ~POO
+ P!P0
Pf* = ~Ol
+ ptPl +
~Po
If' = ~P02 + l!P2 + ~Pl + ~O
•
k+l
~ • ~pOk
+
•
~
i=l
J1Pk-i+l
rt+l
~
i=l
~+l= ~(l
rt
-
~.
~=O
r'-i+l
r'+l
POi) +
~
i=l
~(l
~
-
~
j=O
PJo).
If the system has been allowed to run until equilibrium is attained,
Pk = ~ •
Pk " the stationary probability" and
which becomes a set of r 1+2 independent equations if the last equation
is replaced by the restriction
r l +l
~
i ... O
Pi
=
10
2.5 Solution for Stationar! Probabilities
Several rrethods are available for solving (2.43) for the P •
k
]bran (1954) and Gani and Moran (1955) give a discussion of alternative
methods including ]bnte Carlo methods.
for programming on a computer:
1
= POO
+ G:tPO
•
•
Gk
k+l
= POk
+
~
GiPk_i+l
i=l
•
o
r1+l
lipa
=1
+
~
i=l
G••
3.
The following rrethod is proposed
18
The Gk are obtained from the Pjk by successive substitutions, starting
with
1 - Poa
G =--1
Po
Given the G , the P are easily obtained, since from the last equation
k
k
of (2.51)
1
r'+l
1+"~· G
i
i=1
Gk
--r..,.I+....1 :----'
1 ...
~
0
•
r'+l;
G
i
i=1
als~
k ... 1, 2, 3,
i f we let
1
then
1
and
Pal - 1
G
k
...
1
~
r
Ok
-
P 1
O(k-l)
, k ... 2, 3,
0
0
r'+l,
19
so that the stationary state probabilities are completely determined
~ l~
by either G or POk9 k
k
2, 3$ " " ri+lo
The discrete approximation to the continuous distribution of
Wt
= Zt
(constant) is given by
k
B(k) "" Pr(W <k)
t
&
~ Po ... Po
. a
1""
J.
A.n advantage to a solution in terms of the Gk rather than Pk is
that G is independent of r, and once the G are found for the largest
k
k
r p the value for Po with a smaller r is obtained from (2053) by dropping
the appropriate number of G in the summationo
i
206 The Expected Value of Available Soil M:>isture
The solution to the stationary probabilities.\} Pkp allows the
expected value of available soil moisture to be obtained in terms of
Wt~ !.o~o»
the discrete approximation to the distribution of
.
o
I'D
~
Po
~
k""l
(k"~)
i
G + P ( r;r
k
o
k G + Po G + ( r+r
ri l
k
2
r
&,
)
Gri +
)
..
l
D
Po (1
.. 1)
2
Pari
i
Po ~
k=2
kCL~
POk
GO
1
) ~ P ( ..l:.PO(k=l)
a POI
..
I)
r+r i
I
I
Po
I
+ ( ) P ( - ... ) .. - ( .. 1)
2
a Po
Pari
2
Pari
20
and
E(Zt'
ME(Wt '
for models
(2022) and (2023)
"" v E(Wt '
for models
(2024) and (2025)0
""
It is also possible to approximate the higher moments of Zt from
(2062)
207 M::>istUre 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 Qi
0
If the llDisture deficit is denoted by Zt, at time to?
'"' 0
if
Z
t
if
Zt::> Q i
~Qi
(2 0 71)
0
Then the probability that a moisture deficit occurs is Pr(W
t
where
q ""
< q).9
QV/M for models (2022) and (2 0 23) and q = Qo/V for models
(2 024) and (2025)0 The discrete approximation to this probability in
tems of the stationary state probabilities Pk is
(2072)
where q v is q rounded to the nearest integero
The discrete approximation
to the expected value of moisture deficits is given by
ql
~
k=l
(k..,}) Pq v-k
0
(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 tm previous chapter,!) a lmowledge of the frequency distribution of U is required in order to obtain the transition
t
probabilities.9 PjkO
The variable U as defined for models (2022).9
t
(2023) and (2 0 24) is a function of at least two climatic variableso
However.9 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 ~ 0 The POk and the
approximations to Pk 9 k
= 0,
l,!) 2,!)
0
0
r V+l 9 can be obtained from the
frequency distribution of It for the case defined by model (2025)0
Nothing was said in the previous chapter about the length of the
time intervalo
The choice of a time interval depends on two factors
which work in opposite directionso
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 systemo
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 montho.
On the other hand,51 it is desirable to
choose a long time period in order to justify,51 to some extent$) the
independence assumption of the input variable U 0 The shortest time
t
period for which precipitation records are readily available is one
day 0
Thus 9 any frequency curve fitting procedure must be based on daily
22
records or longer time periods..
The shortest time period of one day
seems to be desirable since it Is generally easier to derive a frequency
function for a long time period from a function for a short time period
•
than to derive a function for a short time period from fumtions based
on a longer time period.
When the time period is one day ~ the distribution function of
is discontinuous at zero$
!Q~O.9
~
'"
there exists a finite probability that
It ... 00
However, the function may be assuned to be continuous for
It > 00
Then,\l i f the cumulative distribution function of daily precipi=
tation is denoted by F (1), it may be written
l
I
Fl(I) ...
(1 - n) + n
f
f(x) dx,
0+
where n is the probability that rain occurs during the time inter-val
and f(x) is the probability density function of the amount of rain..
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 studied..
The observed frequencies of daily
rain~
fall are given in figures 3.. 1 through 3.. 14 for some of the stations..
It
is evident from these frequencies and generally recognized in the litera=
ture (Chow, 1953) that the distribution of rainfall is positively skewed~
the degree of skewness generally depends on tm length of the time period o
When the time period is one day as in figures 3.. 1 through 3..14, the
distribution tends to be J shaped suggesting the exponential distribution
23
7060-
.
1.0
1.5
inches per day
Figure 3.1.
2.0
Goldsboro--.A,pril Observed Frequencies of Rainf'all
70
40
1.0
1.5
2.0
inches per day
Figure 3.2.
Goldsboro--May Observed Frequencies of Rainfa.ll
24
706050-
I--
40.30r-1
Ii
20=
i
10-
I--L--L...,.-I--L-~-- --L"b~~c=-C!-l--"::-"-i
, ---.-1:==1.0
1.5
2.0
_
inches per day
Figure 3..3.
Goldsboro--June Observed Frequencies of Rainfall ..
r-,
, I
I !
80-! I
7060
1
501
40J
30
I
I '
20
10
inches per day
Figure .3 .4.
Goldsboro--July Observed Frequencies of Rainfall
9080-
7060-
50-
403020-
10-
0.5
LO
1.5
inches per day
Figure
3.5..
Goldsboro--~ugust Observed Frequencies of Rainfall
60-
50
40
30
20
10
2 ..0
0.5
inches per day
Figure 3.6.
Goldsboro--September Observed Frequencies of Rainfall
26
.
4030~
20~
10-
inches per day
Figure 307..
Nashvi11e--June Observed Frequencies of Rainfall
60-
5040-
3020-
10-
0.5
Figure 308..
1.0
1 05
inches per day
2 ..0
Nashville--A,ugust Observed Frequencies of Rainfall
27
504030~
2010-
inches per day
Figure 3.9.
Lumberton--June Observed Frequencies of Rainfall
70-
0050403020I
10-
oS
inches per day
Figure 3.100
Lumberton--August Observed Frequencies of Rainfall
28
,30-
2010-
0.5
1.0
1 05
2 00
inches per day
Figure 3 0 11 0
Kinston"'-JuneObserved Frequencies of Rainfall
30
20
10
1.0
inches per day
Figure 3.12
0
Kinston--.A,ugust Observed Frequencies of Rainfall
29
3020-
10
1.0
2.0
inches per day
Figure 3.13. Edenton--June Observed Frequencies of Rainfall
403020-
10-
1.0
1.5
2.0
inches per day
Figure 3.14.
Edenton--August Observed Frequencies of Rainfall
30
The exponential distribution has been used by MJran (1955) for the dis=
tribution of inputs into a dam and m9.ny 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 mre general
gamma or Pearson type III two parameter distribution
1
f(x) = ~-- x
.t.}.. r(}..)
}..-l
e-x/.t.,
which reduces to the exponential distribution for
bution is J shaped for
}..
considerable flexibility.
<
1
and bell shaped for
}.. = 1.
}..
This distri-
> 1, which allows
The gamma dist-ribution 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; ~Illwraith.9
1953 and 1955; and Foster, 1924).
In many situations involvfug skewed fre-
quencies, it is possible to normalize the distribution by taking logs of
the observations.
If
x t ... In x,) then
The majority of the work done in fitting rainfall distribution
functions has been for the purpose of predicting floods (!'bran, 1957;
31
Paulhus and Mi11er,9 1957)~ . Recent advances along this line have been
made by Gumbel (1941,9 1945,9 1958) using the statistical theory of extreme
values~
and a similar approach has been used to predict drought (Gumbel~
1958)..
However;J drought as defined by van Bavel is not the extreme
drought as defined by Gumbel's theory of extreme values approach but
rather a state of soU 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 3 1953)3 the
first four moments about the mean as well as the coefficients of skewness
and kurtosis were computed" us::lng the 25 years of North Carolina weather
data for the stations shown in Table 3.. 1 ..
The follow::lng relationships are characteristic of the moment s .of
the gamma distribution mere
~
~
~
is the kth moment about the mean:
... .('4
2
.<. A.
1J.3
2.,(3A.
1J.4
= 3.,(4'4( A.
+ 2)
~
coefficient of skewness ... ~l .. -1J."""':l~MIf':::2- ...
2
n-'4-
b·
coefficient of kurtosis ...
~2
...
1J.4
~
1J.2
...
6
-r+ 3
..
e
e
e
Table 30L Coefficients of Skewness and Excess Kurtosis for North Carolina Weather Stations
Station M:>nth
ASHEVILLE
!\p:r.-il··
May
June
July
,tugust
~ptember
$
$
gl
gl
1
g2
g2
1081,2
203108
206920
203809
406,43
2.9040
1.,30,
200482
2073,3
201270
4042,6
206,91
3.5137
6.2932
11.2233
607868
29.3796
10.6064
409424
800096
1008702
805030
3204937
1206498
2.8,8
30432
= 00704
30433
00228
4.088
0.2661
001780
001799
002872
001463
000800
""
""
""
':'
""
2.8346
2.4042
200433
204863
2.47,4
207338
209473
2.. 4&t.9
107186
202774
207383
204777
13.0304
901138
4.4306
7.7802
1l.2480
9.2086
1200524
8.6702
602626
90272,
901914
1102104
= 109,4
= 00886
30664
0.33,0
0.. 2470
0.4139
003240
0.,2,7
0.1389
... 00618
- 00702
"" 0.380
... 00588
.. 00399
=00663
1081,3
203186
3.2989
1.7366
1.,4,6
2.4900
1. 7610
200724
303149
1.3729
101733
2030"
4.. 6,18
6.4424
-16.. 4838
2.8273
2.06,3
7.9736
409429
8.0638
16.3241
40,236 3.5833
903001
0.,82
3.244
"" 00319
30393
30036
206,4
go
gl
g2
1.039
00859
007"
0071,
00664
00978
EDENTON
APril",
~y
June
July -
~ugust
September
2~98,
." 4.112
40004
EtIZA,BETHTOWN
A,pril
~y
June
July
A.ugust·
September
\AJ
I\)
e
...
e
Table 3..1.
(continued)
Station M::>nth
FAYETTEVILLE
APril
~y
!Iune
July
.t\ugust
~eptember
e
gl
gi
g2
g2
2.,3.544
3.0479
1.,9261
106301
2..8002
3.. 2894
201269
3,,2568
107146
103919
207800
301239
607856
1509108
4.. 4101
209061
1105934
1406384
803147
13 .. 9345
505647
3.. 9858
11.. 7616
16.,2302
2..8374
1.9138
3..4479
2.5733
40722,3
2.. 7912
300459
108223
304234
2.3609
409121
1106862
1309163
4.,9815
1705803
803612
3601944
12.,0762
13.5775
5.. 4939
1768320
9.9328
33.4501
108552
2.. 5596
202656
2&5527
2.. 8335
3.. 90.39
105215
204618
2.,2676
2,,5266
2.,6954
309137
304727
9.. 0909
7.,7131
9.5756
1008981
2209757
5..1626
9.8273
7.. 6994
9.,7744
12.0430
22.,8tb6
go
=
.e
t
g2
gl
3.. 059
3.,951
20-309
2.,159
0.336
30184
GOLDSBORO
April
~y
June
July
August
~eptember
KINSTON
lPril-
May
June
July
A.ugust
September
3~0086
=
eo
=
=
0.. 781
0.. 676
1.,025
0.504
30144
$.488
30380
1.474
0.. 027
0,,398
20290
00229
004042
005063
003105
006488
003386
001355
0.,1.73
00191
0..545
0..527
= 0.,178
= 0.. 240
=
=
\..oJ
\..oJ
e
,
e
Table 3010
e
(continued)
Station 1Jbnth
.e
.e
gl
gi
g2
g~
202528
1.8958
3.0294
1.. 7722
2.1244
402041
2..1701
106106
301807
105289
109914
4.. 2467
7.0646
3.8915
1501756
.3",064
509490
2700529
706126
503910
13.. 7658
407110
607696
2605116
10097
20999
... 20818
20409
1 0 641
... 10081
0.. 0511
002849
0 2898
003381
002757
0..1189
.,.
'"
...
...
'"
""
0.. 969
00526
00603
00614
00835
00580
2.3831
202284
106068
2.. 2012
208694
2.. 6659
201135
2.. 0273
1.. 2684
109779
208757,
203914
6.7006
601654
211'4134
508684
1204046
8,,5785
8.. 5187
7..4486
3.. 8727
7.. 2679
12.. 3501
10,,6605
3.. 637
'2 .. 567
2.. 919
2.. 799
.,. 00108
40165
003015
002591
003947
003478
002451
002220
""
""
...
.,.
""
'"
0.. 542
0,,806
0.. 640
0.. 452
00688
00596
go
g2
gl
LUMBERTON
~ri1!
~y
June
July
August
~ptember
0
N~SHVILLE
APril
~y
June
July
August
§eptember
\;J
.I=""
35
2
.
The sample estimates of ~o' ~l' ~2 - 3, denoted by go' gl' and g2 are
given in Table 3..1..
that
~o
... 00
The criterion for fitting the gamma distribution is
The sample estimates, go' deviate from zero in both direc=
tions although positive deviations are more prevalento
Since there are
no available estimates of the variances of the go.ll 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 va.riation of
the sample So
3g1
2
- 2g
2 ..
By computing gi and g2' such that 3g:L2 - 2g2 -0 and
0, it is possible to some extent to assess the deviations
of go from zero in terms of the coefficients of skewness and kurtosis
separately.
The values of gi and
that the large values of
variation of the samples..
g~
as shown in table 3.1 indicate
Igo I could reasonably be due
to random
This argument is empirical in that some
correlation exists between gl and g20
However, when compared with
other distribution functions of the Pearson system, the gamma distribution generally gives a better fito
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 :: 0; the corre-
.e.e
sponding estimates are denoted by gl
and g2 in Table 3.10
WhOile log
x is considerably less skewed than the original observations,p 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 curveo
,36
Although these results are not conclusive, they indicate that the
frequency curve of daily amounts of precipitation can be fit reasonably
well with the gamma distribution..
The lack of fit may, 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" !"~"!J large amounts of precipitat,ion
tend to occur more often than would be predicted from the garoma. distri...
bution"
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) follo'tVs the gamma distribution, the problem arises of estimating the parameters .( and A in
0 .. 13).,
The distribution can be 'Written in terms of ~ ... E(x) and A by substituting
.( . . t
in (3 ..13) which gives
1
f(x) '""
rCx)
x
Then the maximum likelihood estimate of
A=l e-x-A
~
..
~
for a sample of n obser""
vations is given by
n
~. ~X.
~.~ i~ ~
n
--x
and the maximum likelihood estimate of A. is the solution of
37
where
r u (~)
is the first derivative of
r(x)
and InY ...
This equation must be solved by iterative procedures..
of
~(~f)
(1933)..
, commonly known as the digamma function,
~. ~ In
Xi"
Extensive tables
are given by Davis
The estimates ('X) can be obtained directly from In f - In I
from tables given by Chapman (1956) although these tables are not extensive enough to afford more than two digit accuracy in many cases..
A first estimate of X can be obtained from X* ""
is the sample variance..
xf!/6 2,
where
62
This is the method of moments estimate since
~ .. ~/~ from (3 .. 21).. This estimate, though simpler, does not have
the desirable properties of the maximum likelihood estimate.
The
estimates of I.l. 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 if 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
~/ (N-n) and (n.. ~ lin, where ~ is the number of days in which rain
occurred,preceeded by a day in which no rain occurred.
3.4 The Distribution of Inputs into the §>lstem
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 (202,).
In order to employ
models (2022), (2023).9 and (2.24).9 a knowledge of the frequency func..,
tiona of It
=
t,
Vt' X - V ' and X
t
t
When Ut '"
(Xt -'Vt '
M
+ 1
respectively.9 are required.
as in model (2.23).9
A curve fitting procedure for obtaining the frequency function of Y
t
is contingent upon the availability of daily records of V or an
t
acceptable method of estimating V from present climatic records.
t
Such a study is not attempted in "the present work due to the controversy
concerning methods of estilnatingV (see Pelton et al., 1960).
t
Certain limited information relevant to the distribution of Yt
can be obtained from the distribution of X •
t
The distribution function
of Y .9 unlike X .9 is completely continuous with
t
t
0 <Y
t
<
00
0 For
practical purposes, it may be reasonable to assume that the frequency
distribution of Yt is similar to that of It; then" if' Xt follows the
gannna distribution" this assumption would give
1
2
-yu fa
e ';( y!l
where
~
Y
= E(Y)
B
n ~ - V +M
The basis of such an assumption is the small variation in Vt relative
toX as shown by van Bavel and Verlinden (19,6); however, the frequency
t
39
distribution of Y soould be verified from data and (041) is presented
t
only as a reasonable hypothesise!
For the case defined by IOOdel (2.24), the frequency distribution
of Xl, is required and curve fitting procedures require records of the
duration of rainfall, At' as well as the amount of rainfall.
The
frequency function must be studied for a range of possible values of
RoP the maximum rate of IOOisture infiltration by the soilo
One approach
would be to study the distribution of ~/~ so that the mean of
Xl
could be obtained from
E(Xi
I~>
0) •
E(~)
H(R) + R
E(~)
[1 -
H(R~
,
where H(X t) is the cumulative distribution function of It/At.
the frequency function for
tribution of Xt •
Xt-:~ might
be
0 btained
Then
by censoring the dis-
40
CHJ\PTER
IV
RESULTS ON 'IRE DIBrRIBUTION OF AVAILilJ3LE SOIL MJISTUBE
4.. 1 The General Shape of the Frequency Function of
Available Soil M:>isture
With the assumption that daily amounts of precipitation follow
the gamma 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 02.5)..
Thus, we can study
the frequency distribution of available soil 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
"C' ...
ThlJ.V
J
is the argument
used in the tables of the incomplete gamma function (Pearson, 1947).
The actual values plotted are G ... 'P~PO' k ... 1, 2, 3, .. .. • 2.5,
k
following the procedure of section 2040 The probabilities of the zero
state, PO' are plotted in the appendix: for a range of parameters..
Since
the G are independent of the storage capacity, they are plotted in
k
preference to the P ; the P are easily obtained from P ... Po G , ,or
k
k
k
k
directly from the P
since from (2.4.5)
Ok
..
It is evident from figures
4.. 1
apd
4.. 2
and
403
that the frequency dis...
tribution 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
.15
o
20
25
k
Figure
4.1.
.Available Soil M.oisture Frequencies with p < 1.
42
20
A .?, n=.4,
R
~·.2
15
10
o
10
20
k
Figure
4.20
,Available Soil Moisture Frequencies with p
> 10
43
0·.5
0.4
"t---
-J'
n=.3, '\"=.3
0.3
0.2
0.1
-1---------
5
15
10
25
20
k
Figure 403.
Available SJil Moisture Frequencies with P
= 1"
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 i f n~V ... 1.
In queueing theory the parameter n~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.
4.2,
In figure
p is greater than one and the curves are
skewed to the left with the probability concentrated at, or near.ll the
storage capacity.
In figure
constant at approximately
1t,
4.3,
p .. 1 and the curves tend to be
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; pa.rticularly when the storage capacity
is large.
The probability curves in ap};andix 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
state.
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
E(lt -
Vt } .. E(X
t} -
Ee l t
- Vt ) .. nlJ. - V
E(Xt
-
V)
E(Xt - V)
t) -
.. E(X
V
for model (2.22)
for model (2.23)
V
for roodel (2.24)
for model (2.25)
.. TtiJ. - V
SinceE(Xi' ~E(~), the consequence of ignoring rwloff tends to
overestimate 5, resulting in a negative bias in Pk , k "" 0, 1, 2,
•• rl+l, when k is zero or near zero and a positive bias when k is
near r.
When the present approach is applied to the st.udy of moisture
deficits, the expected value of the change in the system per unit of
time is given by
51 ..
and pi ..
:!...
TtIJ.
I:
E(V -
It' = V-
lip so that pi> 1 if P
<1.
TtIJ.
This implies that the
probability curves in appendix Figure 1 with P >1 are not applicable
to the study of the distribut.ion 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 variable U in the
t
!.~o,
no • frequency of Ut
< 1/2
~ • frequency of 1/2
< Ut <1
46
~ ==
frequency of kI 2
< Ut < 2"k + 1"2"
e
Then
(no + ~)
o
POO ..
N
(n 2k + n
N
2k+1 )
-,
k .. 1, 2,
e
e
r
k .. 1, 2,
0
..
r' ,
t
and
Po
..... no/N
.
Pk ...
(n
_ + n )
2k 1
2k
N
,
where N is the total number of days in which climatic records are
available ..
Recalling from section 2 e 2 that the generalized variable U is
t
defined as
for model (2.22)
for model (2423)
IV
t
U =Vt
for model (2024)
It
U "'t
V
for model (2425)
It is seen that the frequency classes for U will de]pElnd on either
t
M or V, ~•
.s... ,
of Vk < It
the frequency of k < U
t
< V(k
<k
+ 1 is equal to the frequency
+ 1) for model (2.25).
Such an approach may be impractical, especially if r is large, although it is feasible if the c1,imatic 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 po ssible sets of transition probabilities ..
4,!3 Queueing Theory Results
The general form of the equations representing availa.ble soil
IllOisture 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 lit' 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 pre sent problem
is given by Gani (1957)..
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 the N days one of the events
48
Zt .. 0 or Zt> 0 occurs.
The stationary probability that the event
Zt "" 0 is PO' and at equilibrium we can assume that the variable d
follows the binomial distribution wi. th
par~meters
N and PO'
!o!0,
..
However.ll since Zt is a time dependent stochastic variables> the act'u.al
probability that Zt ... 0 varies from day to day throughout the N dayso
If the variable d is divided into two parts, 15 and h,9 where
g '"' the number of initial drought days "" the number
of sequences,
h '"' the number of drought days which occur in a sequence
after an initial drought day has occurred;
then d "" 15 + h.
Given 15, h follows the negative binomial distribution
with parameters 15 and
Poe;
Pr(h/g)
!..~.,
B
g+h-l
15-1
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 o
If the events are
restricted to either initial drought days or drought days occurring in
a sequence.ll the variable h is the number of drought days occurring in
a sequence obtained in observing 15 initial drought days..
probability of occurrence is given by
and.
E(h/ g)
Poo
= i-poo
15
Then the
49
The moment generating function is given by
E( eht
Ig ) • (1 - Paa )g (1 - Paaet).g •
(4.44)
Equations (4e43) and (4.44) are of little use since g is not known
and is a random variable.
However, we can infer from (4.43) that the
average length of a sequence of drought days is given by
E(hlg'. 1) + 1...
.
PJa
1 - Paa
+1
and the higher moments can be obtained from (4.44), when g
= 1.
The results of this section are of practical value for obtaining
the expected value of drought variables in crop prediction equationse
Specific examples will be given in the following chapter.
,a
CHAPTER
V
!PPLICATIONS
,.1 The
Crop Production Function
Practical applications utilizing a knowledge of probable soil
moisture conditions depend on a knowledge of crop production functions
relating the yield of a crop (Y) as a function of input variables (I)
such as
where at least one of the Its is characteristic of soil moisture conditions.
The IVs can be broadly grouped into two classes, namely.9 l}
factors which depend on the environment, and 2) factors which depend on
techniques of production.
Let I
li
denote the class 1 variables and 1 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 a.s
follows:
Class 1
Class 2
Plant available phosphorous
in the soil· '.
Phosphate applied as
fertilizer
&il texture
Seedbed preparation
Moisture ponditions during
1st month of growing season
Date of planting
Weed population
Methods of weed control
&il moisture thrOughout
the growing season
Irrigation
0
51
It should be noted that more than one of the
~i
may be associated
with each X2j ' and vice versa.
The farm manager's problem is then to choose the X2j such that
•
Cost of I 2j
•
The resulting optimum value of I
2j
(5.12)
Price of Y •
, say I'2j' 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 value s cannot be
obtained until the growing season is complete and are of no use.
2j
this case, X
In
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.
A;D. index proposed
by Knetsch (1959) has the general form
the growing season is divided into m growth periods, based on th3 phases
of growth of the crop, and the number of drought days, n , occurring :in
i
th
the i
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 -'-ij may be zero ..
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
~
distribution with parameter PO' given by (4 .. 41), where Po is the
stationary probability that a drought day occurs"
Then tlie expected
value of the drought index can be approximated from
m
E(Dl )
&.
.~1
~=
m
-'-.
~
N.~ PO·~ +
+ N.~ PO·)
.~1 -'-i·~ Ni POi(l - PO·
.~
~
~=
m
+
~
-'-ij Ni Nj POi POj '
where POi is the stationary probability of a drought day appropriate
to the parameters of the i th growth period..
In order that the third
summation in (5 .. 22) 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 2Jl n , and n
3
4
are the number of days in which Zt
the months Ms.y, June, July, and August, respectively.
= a for
Further,
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 2.0
inches.
Then the estimates of V as given by van Bavel and Verlinden
53
(1956) are 0014, 0.17, 0.16, and 0.14 inches per day, respectively,
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
0.28
0 .. 34
0040
0 034
~
0.88
0 ..86
0.88
0.82
IJ.
il,V
't' ... IJ.
r ... 2.0/V
0~42
0.44
0.51
0.48
0.32
0.35
0.. 31
0.30
14.. 3
11.. B
12 05
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, 0026,1)
0012, and 0.20 for the months May through August, respectively.
Thus,
the average number of drought days is given by
E(n )
l
:&
(31) (.31)
., 9.61
E(n )
2
E(n )
3
&.
(30)(026)
...
7.80
III
•
(31)(.12)
...
•
'"
(31)( .. 20)
..
3.72
E(n~) ...•
(31)( .12)
EB8
E(n4)
6.. 20
+ (31) ( .. 12~
==
17.11
and
5.3 So il 1'10 isture Index
The majority of the research utilizing an index to weather conditions in crop pr'oduction functions employs a drought index.
However,
,4
recent results from the North Carolina TVA. corn 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
llDisture for the growth periods instead of drought days in (5021) might
improve the index when excess soil misture is an important factor.
If
such an index is a linear function of the average soil moisture for the
growth periods,\) the expected value of the index can be obtained from
probability curves such as those in appendix Figure 1.
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).
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.
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
1.5
inches. 'From appendix Table 2
n :.. 0.30
1000
V
:.
0.17 (from van Bavel and Verlinden, 19,6)
1:
•
...
AV ...• 0.32
IJ.
8.8 •
55
Then from
(2.61)
1"0
:.
~
Po
k=2
k(..J- r Ok
~
Ok=l
) + (
~
~
) PO(l/POr .;. l/PO'!"'o)
-
and from appendix Figure 1 (i)
or-':
E(Wt )
~ .30~(1/064
- 1/.77)
+
3(1/.55 - 1/.64)
+
,(1/042 - 1/.48)
+
6(1/.37 - 1/042}
+
8(1/..32 - 1/.34)
+
8.4(1/..30 - 1/032)
+
+
4(1/.48 - 1/.,,)
7(1/.34 - 1/037)
+
1/077
=f (1/032 + l~i
'" 2.. 952
E(Zt)
~
V E(Wt
0.17 (2.952)
) :
= 0.. ,02 ..
'04 Decisions Concerning the Use of Supplemental Irrigation
When the farm manager wishes to plan his production with the possi=
bility of using supplemental irrigation" 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
Do
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 intel"Val around
the optimum value; for example, the confidence interval given by
PI' (a ~ D i ~ b)
> I - .(.
56
For the special case when D is the number of drought days occurring
~.~o,
in some critical growth period of the crop,
the silking period
for corn, the probability can be estimated from
mere N is the total number of days in the growth period and Po
~s
stationary probability that a drought day occurs, with a.s;: D'
b < No
As a hypothetical example, suppose D'
m
17 and a
= 0,
b
S;;
= 20.l'
the
and
we wish to find Pr(D < 20) for the time period June 6 to July 25 applicable to the Asheville, North Carolina weather station.
The parameter
est:iJnates from appendix Table 2 are
JUNE
JULY
n .&
0.47,
and V as estimated by van Eavel and Verlinden (1956) is 0015 and 0 .. 14,ll
respectively, for the two months.
The use of (5.41) is contingent upon
the assumption that a single value of Po is operative throughout the
t:iJne period and the validity of this assumption depends, to some extents
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 warrarrlied by the other assumptions
involved.
It should be clarified that the parameters were estimated by
months only as a matter of convenience and in reality distinct population boundaries do not exist from month to month, but rather a gradual
continuous
chang~
in the population occurs.
57
The combined estimates for the two months are obta:ined from
'Where the subscripts 1 and 2 denote the months June and July,
ively.
respect~
The combined estimates of A. and V cannot be obta:ined from the
information available in appendix Table 2; however, the simple averages
of the estimates should be satisfactory for the present example; i,,!. J
4 '&0.73
c
V :. 0.145
c
and
1:c
• 0.73 (0.145)
=
0.3014
= o. 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
(~O/\
The desired probability is given by
(0.26)k (0.73)5 0- k • 0.92
as evaluated from Romig's (1947) binomial tables.
ThUS, a farm manager
operating under these conditions would conclude that his chances are
quite
good of atta:ining the economic optimum without irrigation.
58
505 Complementary Use of LOng Term Weather Forecasts
It was pointed out in the introduction that farm manager 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 time o
In
the event of reasonably accurate long term weather forecasts,9 crop
production planning decisions will be able to be made with more confidenceo
The present approach of obtaining probable soil moisture condi-
tiona from a knowledge of the frequency function of the input variable
can be extended to make use of tb;l information available from long term
weather forecasts o
As an illustration, suppose the long term weather forecast for the
Kinston, North Carolina area indicates that the daily amounts c>f June
rainfall will be 20 percent below normal; then we can adjust the para""
meter IJ. to make use of this information in predicting the average avail",
,
able soil moisture or expected number of drought days.
In sectian
503 s
the average available soil moisture for· June in the Kinston area was
found to be approximately 0.50 inches with a storage capacity of 1.5
inches and IJ. & 0 0 53; the adjusted J.i..is given,by
and
,;V ,&
003970
Then the appropriate values in equation (2.61) are obtained from appendix
Figure 1 (i) with,; ...
0.4
which gives
59
E(Wt ' = 040 @(1/067 - 1/479) + )(1/0059 - 1/067) + 4(1/.53 - 1/.59)
+ 5(1/.49 - 1/.53} + 6(1/.45 - 1/.49} + 7(1/.43 - 1/.45)
+ 8(1/.41 - 1/.43} + 8.8(1/.40 - 1/041) + 1/.79
- ~ (1/.41
+
18
E(Zt) ~ (0017) (2.149)
= 2.149
= 0.366.
Similarly" adjustments can be made in the parameters
11:
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
11:.
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.
5.6 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,
i f the index of weather or soil moisture conditions in (5 ..11)i'8 a
function of average length of continuous drought the expected
given by (4045)
0
val~e
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 assmnption that amounts of rainfall follow the gamma.
distribution, POO can be obtained from table s of the incomplete
r
func,:'
tion (Pearson, 1947) as
Poo ... (1 ..
n) + n
I('t', 'A)o
From appendix. Table 2 the parameter estimates are
In this example, since 'A is approximately one, POO can be obtained
directly from
POD = (1 - n) +
I
fr e:~ , dx = (1 -
n) + n(l - e
-~) • 0.79,.
Then from (4.,45), if we assmne that at least one drought day occur~ the
average length of a sequence of drought days is given by
~ 0.795
E( h I g... 1) + 1 =
0.,205 + 1 ... 4 088 dayso
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 failure is approximately
f
t
and if pOO & 0.795 as in the previous example and m ... 20 say, then
\
t
19
Pr(h'2- mig'" 1) :: 1 .. 0.205 .. 0.205
.~ (Oo795)k ... (00 795,20 ~ 0 00100
k...l
61
C HAP T E R V I
SUMMARY .AND SUGGESTIONS FOR FurORE RESEARCH
6.1 Summary
The management of a farm generally requires plans to be made in
one time period for a produot whioh 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 time
of production planning,; thus crop production planning could be made
more nearly rational by a knowledge of probable soil moisture conditions ..
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 f:inite capacity
enables one to use the probability tmory 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) evapo-
transpiration method of estimating soil moisture conditions.
The
models have the general form
lit "'l .. Wt + Ut - 1,
where W is available soU 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' + 2 discrete soil moisture states, ranging from zero to
the storage capacity, which satisfy the properties of a Markov chaino
The states are defined as
So : 1ft
=0
~ : k - 1 <: Wt
<
k, k
= 1.ll2»
3J1
&
"
"
r'
Sri+l : r' <: Wt <: r
where r is such that availa.ble soil moisture is at storage capacity
when W
t
= rand
r' is the largest integer in r"
Then at equilibrium
the stationary state probabilities P are the solution of
k
P
= T'
P
where P is the r i +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
&
•
r' + 1.
r'+l
Since the rows of T sum to unity; the restriction that
~
k=O
is necessary to reduce Eo
= T'P to
The transition probabilities
P
k
=1
a set of r' + 2 independent equations"
POk' k
= 0,
1, 2, •• " r', can be evalu-
ated from the cumulative distribution function of Ut " The Pjk' j = l.ll
2, 3, " " " r i , k = 0, 1, 2, ~ " "r' are dependent on the unknown distribution function ofW "
t
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"
63
The solution of P '" T 'f is simplified by solving the equations in terms
of G
k
III
Pk/P
o since
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 Pjk can
be obtained from this frequency function when precipitation is the only
time dependent variable involved in Utl>
Following the Pearson system
of curve fitting, the gamma or Pearson type III distribution was found
to give the best fit when compared with the log normal distribution and
other curves in, the Pearson system.
Assuming that daily 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' stations.
The
results are in agreement with results from other storage and queueing
systems;
.. ,
!.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 moisture
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 will have wide applicability for obtaining probabilities of soil moisture conditions in areas
where natural precipitation is a primary source of available soil moisture"
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
6,
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 more complicated models
outweighs the simplicity of model (2.2')0
In the event of adequate
records of the climatic variables involved in models (2022)~ (2023)9
and (2 0 24), or a satisfactory method 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 U approximately follows the gamma distrit
bution with a finite probability that U
t
= O.
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 stationar,y
state probabilities can be obtained based on the resulting
transition probabilities.
2)
The approximation (2 0 36) 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 possible
66
that the transitionpro-babilities could be made more exact
by substituting the st.ationary state probability P. for the
J
denominator of
(2.35) and deriving an empirical function for
say g*(Wt)~,!~!~,
(
Pjk
:.
F(k+l-W ) - F(k=W ) g*(W ) dW
t
t
t
t
.l'j...l;;;..-----~P j - - - - - - - - ,
where P.J and g*(litlaI'e
obtained from an initial solution based
.
on the approximation (2.36)~
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 study.
4)
The discrete states of available soil moisture adequately
describe the dynamic system.
The error involved in studying
available soil moiSture as discrete states can in theoI7 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 8+2) equations must
be solved for the stationary probabilities.
Uso, 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 realityo
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 represent 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.
Kendall
(1957) 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 managerls intuition
incorporate~
with his experience.
68
LIST OF REFERENCES
Beard" L. R. and ,Keith, H. a. 1955. Discussion of "The 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. 1954. The log probability law and its engineering applications. Proc. Amer. 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.
Haven Press,
Downton, F. 1957. ..A note on Moran's theory of dams. Quart. J. Math.
Oxford (2) 8: 282-286.
Feller,W. 1950. .An Introduction to Probability Theory and Its Applications. Vol. I. John Wiley and Sons, Inc., New York.
Foster, H. A. 1924. Theoretical frequency curves.
Civ. Eng. 87: 142-173.
Trans. Amer. Soc.
Gani, J. 1957. Problems in the probability theory of storage systems.
J. R. Statist. Soc., B, 19: 182-206.
Gani, J. and Moran, P. A. P. 1955. The solution of dam equations by
Monte Carlo methods. 1l,.ust. J. App. Sci. 6: 267-273.
Gumbel, E.J. 1941. The return periods of flood flows.
Stat. 12: 163-190.
Ann. Math.
Gumbel, E. J. 194.5. Floods estimated by probability methods.
News Rec. 134: .a33-337.
Eng.
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. Unpublish€ld 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. L. 1959. Moisture uncertainties and fertility response
studies. J. Farm Econ. 41: 70-76.
'.
Knetsch, J. L. and Smallshaw, J. 1958. The occurrence of drought in
the Tennessee Valley. Tennessee Valley Authority Report T 58-2.AE,
Knoxville, Tennessee.
Manning, H. L. 1950. Confide~ce limits of expected monthly rainfall •
.J. Agri. Sci. 40: 169...176.
McIllwraith, J. F. 1955. Discussion of liThe Log Probability Law and
its Engineering .Applications lI • Proe. !mer. Soc. Civ. Eng. 81:
sep 665.
Moran, P. i\. 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. A,ust. J. APP. Sci. 6: 117-130.
Moran, P. .A. P. 1956. .A probability theory of a dam with a continuous
release. Quart. J. M~th. Oxford (2), 7: 130-137.
Moran, P. A. P. 1957. The statistical treatment of flood flows.
~er. Geophys. Union 38: 519-523.
Trans.
Parks, W. L. and Knetsch, J. L. 1960. Utilizing drought days in evaluating irrigation and fertility studies. Soil Sci. Soc. Amer.
Proc. 24: 289-293.
Paulhus, J. L. and Miller, J. F. 1957. Flood frequencies derived from
~ainfall data. Proc. ~er. Soc. Civ. Eng. 83: sep 1451.
Pearson, K. (Ed.) 1946 reissue. Tables of the Incomplete Gamma Function.
Cambridge University Press.
Pelton, W. L., 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. Proc. Roy. Soc. A., 193: 120-145.
Romig, H. G. 1947.
New York.
50 -100 Binomial Tables.
John 'Wiley and Sons, Inc.,
70
1IST OFREFERENOES (continued)
Saaty, T. 1. 1959. Mathematica.l Methods of Opera.tions Research.
McGraw-Hill Book 00., Inc., New York.
van Bavel, C. H. M. 1956. Estimating 80il moisture conditions and
time for irrigation with the evapotranspiration method. U. S. D. A.
ARS 41-11.
van Bavel, C. H. M. and Verlinden, F. J. 1956. ,Agricultural drought
in North Carolina. Technical Bulletin No. 122, North Carolina
Airicultural Experiment 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
V f)
can be
obtained from (4.11) using tables of the incomplete f-function (Pearson
1947) to evaluate the integral
j
x
x),-l e -x/"-
~
I (",X, ),-1),
where I('t'X, A-I) is the value of the integral obtained from the tables
with u ,. 't'Xand p .. A.-I in Pearson's notation.
POO"
Thus,
(l-n) ... n I('t', A-I)
POk" n
i: ~k+lh, A-~
n I(k't', A-I), k .. 1, 2, 3,
0
TV,
0
and
Po
&:
(l-n) ... n I('t'/2, A-I)
Pk
.It. 11:
I [(k ..
~)'t', A-~
-
11:
I Bk -
The approximation given by (2 •.37)
~)'t', A.-~
, k .. 0, 1,1)
0
•
as
is difficult to evaluate for the gamma distribution since an explicit
expression for F(k+l-W't) - F(k-W ) cannot be obtained except for A. .. 1.
t
An evaluation can be obtained by serie s expansion which gives
•
),
r
v•
72
where
6.
o
V).,.
,.
~
}..
vq
··1
>..q+l
3,
q ... 1, 2,
:J
q ... r(q...l) IJ.q(}..+q)
r(}.....l)>>
0
6.
Then
F(k+l-W ) - F(k=W ) == &O(k+l.,;Wt »).,. 1 t
t
- &3 (k+l-'Vlt ) 3
+&2 (k=Wt )
2
0
0
~(k.l~Wt) + &2(k+l-Wt ,2
-
&0 (k-'Wt)A.
... &3 (k=wt ,3
0
0
1 -
8
1 (k-Wt )
•
which can be integrated from j';"l to j with respect· to W which
t
results in
Po
.. 1 - 11 ... 11L:J..
PI
.. 11(~ - L:J..)
Pk
.s.
11(~+1 - 2~" ~-l)' k ==
2,3,4,
o
•
o
o
...
..
where
k .... 1, 2, 3,
&1
q
...
0
,
6.
q
)
r' ,
q ..
1, 2, 3,
0
(}..+q+l)
~ converges quite rapidly for small values of k since
usually less than one.
and
vVlJ.
For k in the neighborhood of
VV',J.
is
25 and larger,
near unity, the necessary number of terms in ~ for
4
digit
accuracy becomes prohibitive.
In order to compare the towo approximations both were computed
for}.. ...
1,
~
- yXVJIJ.
= 0.2,
11 == 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)
=
F(O)
=<
l-n
PI (upper)
.. F(l) = F(O)
•
PI (lower)
=
., POI
Pk (upper)
.. F(k)
Pk (lower)
=
F(2) - F(l)
=
F(k=l)
pOO = (l-n)
= PO(k=l)
F(k+l) ~ F(k) = POk 9 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 stationary probability of the zero
state, for the range of parameterSt l .. 0.7, 0.8~ 0 093 1.0, 1.2;
1:' ..
0.1, 0.2, 0.3" 0 04, OS, 0.6, and
11
=<0.2, 0.)" 0.4, 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
~ppendix
e
Table 1
APproxima.tions (2.36) and (2.37) to the Transition Probabilities (Pk) with Lower and Upper Bounds
·~·"'1,
'(;' == 0.2,
1t
= 0.2
~
k
(2.36)
(2.37)
lower
upper
0
1
23
4
5
6
7
8
9
10
.819033
.032810
.026861
.021992
.018010
..014790
.,012070
..009882
.008091
.,006624
.005423
.0044llo
0003635
.,002976
.002437
..001995
.001633
•.001337
..001095
.818734
.032B58
.026902
.02202'6
.018033
.014764
.012088
.009,897
.008103
.006634
.005431
.004447
.003641
.002981
.002441
.001998
.001636
.001339
.001097
.000898
..000735
..000602
..000493
,,000403
.000330
,,800000
.029682
.024301
.019896
.016290
.013337
.010919
.008940
..007319
.005993
.004906
.004017
.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
.000991
,,000811
..000664
<11'000544
.000445
.000364
11
12
13
14
15
16
17
18
19
20
21
22
23
24
~000896
..000734
.000601
..000492
.000403
.000330
... 1, '(;' = 0.2, n ~ 0.4
(2.36)
(2.37)
lower
upper
.638065
.065617
..053723
.043985
.036011
.029572
.024139
.019763
.016181
.013248
.010846
.008880
.007271
.005953
.004874
.003990
.003267
.002675
..002190
.001793
.001468
.001202
..000984
.000806
0>000660
.637462
.065717
.053804
.044051
,,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
.005385
",004409
.003610
.002956
.002420
.001981
.001622
..001328
.001087
,,000890
..000729
..000597
,,672506
,,072506
.059346
.048601
.039793
.032579
.026674
.•021839
.017880
.014639
.011985
.009813
..008034
.006578
,,005385
.004409
..003610
•.002956
.002420
.001981
..001622
,,001328
..001087
,,000890
•.000729
·~·l
+::-
75
APpendix Table Z. Parameter Estimates for the Gamma Distribution Based
on Precipitation Records of North Carolina Weather
Stations
7'
n/N
X
fls 2
0.3386
0.3870
0.4213
0.4748
0.3922.
0.3000
0.2901
0.2375
0.2971
0.305Z
0.3215
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
0.273}
0.2709
0.2786
0.3380
0.3135
0.2506
0.4022'
0.4029
0.5492
0.6557
0..5661
006395
0.7815
0.8508
0.8429
0.6920
0.8448
0.5681
0.993
1.048
0.9940.872
0 916
o.80W
0.2440
0.2658
0.3013
0.3987
0.3354
0.2573
0.4490
0.3974
0.5073
0.5137
0.5670
0.5782
1.0083
0.7455
0.6668
0.8306
0.9782
0 6771
0.2440
0.2516
0.2800
0.3354
0 2993
00249J
0.5513
0.4607
0.4989
0.5758
0.6649
0.5794
0.8225
0 07596
0 9346
0.9630
0.6484
0.5125
Station Month
.e.sHVILIE
.A,pri1
May
June
July
August
September
0
EDENTON
APril
May
June
July
August
September
ELIZABETHTCWN
April
May
June
July
,August
September
0
F-UETTEVILLE
.April
May
June
July
.august
September
0
0
0
76
.APpendix Table 2. ..
•
Station Month
(continued)
n/N
I
i/s 2
GOLDSBCRO
Jl.pril
May
Jlllle
July
,August
September
0 ..3080
0 ...3006
0.,,3586
0.4038
0 ..3690
0 ..3053
0 ..4094
0 ..4089
0 ..4583
0 ..6106
0 ..4732
0 ..4579
005922
0.,6003
0 ..7668
0 ..4930
0.5231
0 ..4362
0 02320
0.2309·
0 ..2960
0 ..3380
0 ..3032
0..2.480
0 04532
004966
005343
0 ..7020
005585
0 ..6268
0 ..9997
0.8147
00.8528
0.,9316
006796
0.5382
1.228
0.896
1 0003
1 0132
0 ..931
0 ..917
0 ..4583
0 ..4161
0.4432
005127
0.4822
0 ..7102.
007545
0",6576
0 0 819.3
6..874
KINsrON
J\pril
May
Jlllle
July
August
September
LUMBERl'ON
aPril
May
June
July
August
September
0 ..2573
. 0.,,2825
0.,3360
0.4038
0~3380
0.2840
0~h775
0 ..2973
0.2864
0 ..317.3
0 ..3780
0 ..3354
0 ..2626
0 ..4047
0 ..4570
0 ..4562
0.4969
0.5728
0 ..6017
0.876
0.,4212
0 ..858
0.878
0.815
0.711
0.6788
0 ..6988
0",826.3
0 0 76.30
0.,,5657
0.5519
0.861
0.813
0 ..835
0.918
0 ..729
0.742
0.7315
NASHVILLE
-
April
I:1ay
June
July
August
September
e
....
~
e
•
'--
e
-'
X 0.7
(a)
(b)
0.2
~
X-OoB
n ... 0.2
... 6
-·6
..
""
. .·--4
•
"" 4
...
0' '
il
,
,
5
,
t
,
!
!
10
,
!
!
,
,
15
!
,
!
,
,
20
J
!
,
t
'
25
It'
it'
5
t
!
,
,
,
!
i
10
r~
J
I
15
!
!
i
f
iIi
J_ '
20
.....J
.APpendix Figure 1.
S,tationary Probability of Zaro state
-J
..,
e
,.
e
0.9
(0»
1J.
Cd)
0.2
-_ ... -
_u
-.,4
-
-
=1
_~.6
---
, ,\, "
-I
.51
-
- - -- -
,;-.5
.~ \\~ ~
>.. .. 1.0
0.2
11 ...
..
Po
e
,;--.5
\\~~.
-,;-.4
\\~
,;-•.3
.4•.3
o
5
1(")
1~
20
25
.5
1'::'
15
20
r ---+
--J
~pendix
Figure 1 (continued)
en
,
e
.9
(e)
---
.~ ,,\~
e
A. '" 1.2
Tl '" 0.2
(f)
-1\\\\
'\.
~
I
\
"""
0.3
,,=.6
1;=05
'1:'-
..3
A.'" 0.7
n ...
=.6
e
"
.".=~
~
\\
~
.".- 4
---.!.
•
•.1
l
o ,."
'.."
. . . .15" , •••
" . " 25 I " , 5•• " .10" . . "
, ,7'
""
5
10
20
15
20
r-'ii>"
-.l
Appendix Figure 1 (continued)
'0
e
"
e
'-
,
e
e
{.
e
"
.9
(i)
,~
.1;t
l!
... 1.0
"It
(j)
0.3
e
.~
A" 1.2
0.3
n,-
.. 6
'C- •.6
.5
-s
Po
.4~1
o
5
}o
15
20
25
5
10
15
20
r~
(Xl
.APpendix Figure 1 (c ontinued)
~
.
e
,e
(k) '4 =- 0.7
'Jt =: 0.4
.9-
..
•
(1)
e
}",. 0.8
IT
Q
0.4
.,8-
-;=.6
-;-.6
.2-
1
.1
o
10
15
20
25
r;
10
15
20
r~
co
./tppendix Figure 1 (continued)
i\)
e
WI
.9
,
e
(m)
>.. ... 0.9
1):
==
(n)
0.4
•
.".
>.. ... 1.0
1:[
""
0.4
'C=06
Po
.4,
.3
..2
.1'
-';·01
o
t)
r
11"\
1,5
20
25
5
10
15
20
-->
(»
APpendix Figure 1 (Gontinned.)
w
...
e
..9
(0)
...
e
A == 1,,2
11 ...
(p)
0 .. 4
•
~
laO,,?
= 0.,,5
. IT
,,8
..7
Po
o
5
r
10
15
20
2.5
5
10
15
20
----?>
ex>
~ppendi.x
Figure 1 (continued)
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