l T I ., AVAILABLE SOIL MOISTURE AS i\ PROCESS by t Dale E. Cooper and David D. Nason ,\ rl .. This research was done in cooperation with the Division of Relations, Tennessee Valley Authority • Institute of Statistics Mimeo Series No. 270 December, 1960' •
89
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l"~I' T I.,
AVAILABLE SOIL MOISTURE AS i\
STOCHAS~IC PROCESS
by
t
Dale E. Cooper and David D. Nason
,\ rl
..This research was done in cooperation with the Division of
~gricultural Relations, Tennessee Valley Authority •
Institute of StatisticsMimeo Series No. 270December, 1960'
•
iv
TaBLE OF CONTENTS
Page
LIST OF TABLES • • o 8 0 • • G • • • • e & 0 • Q • • eo. • • • • • vi
LIST OF FIGURES. " " .. .. " " " " " • • • • • • • • • vii
General " .. • • "~,, • • • "Statement of Objectives'.. "
o 0 • \) 0
o e 0 0 0
" " " " " .. . .• 0 0 • • • • • •
13
20 1 The Soil as a Storage S,rstemo " " " • " " • " • • " ••20 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
Coefficients of Skewness and Excess Kurtosis forNorth Carolina Weather Stations e • • • • e • • .' . . . . . 32
Appendix
1. Approximations (2.36) and (2.37) to the TransitionProbabilities Pk with Lower and Upper Bounds. • • • • • •• 74
Parameter Estimates for the Gamma Distribution based onDaily Precipitation Records of North Carolina WeatherStations • e eo. • • • • • • • 0 • 0 • • eo. • • • • • 75
vii
LIST OF FIGURES
Page
23
24
24
25
25
26
26
• • •
• • •
• • •
• • •
• • •
· . "
• •
• •
• •July Observed Frequencies of Rainfall.
June Observed Frequencies of Rainfall.
June Observed Frequencies of Rainfall•• "
Ms.y Observed Frequencies of Rainfall ••••••
August Observed Frequencies of Rainfall••
August Observed Frequencies of Rainfall•••••
September Observed Frequencies of Rainfall
Available Soil M::>isture as a Finite Queueing System
Goldsboro
Goldsboro
Goldsboro
Goldsboro
Goldsboro
Nashville
Nashville
Goldsboro -- April Observed Frequencies of Rainfall • • • •• 23
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
2.1
i 3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3,,9
Appendix
Stationary Probability of the Zero State • • • • • • • • • • 77
It
CHAPTER I
INTRODUCTION
1.1 General
The agricultural industry is faced with two major sources of
uncertainty which give rise to large risks. These are: 1) prices of
products and resources and 2) weather. 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 out
comes depend on the weather. Virtually all crop production planning
decisions are affected by the weather. An obvious example is the
planning of an irrigation program. 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 character-
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
j
•
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 condi
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 pro
babilities 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 condi
tions 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
..
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 repre
sents 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
3
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 far
fetched; 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.
4
(\J
/
/--- .
//
o
'0M
5
•r-I•
(\J
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
~ precipitation occurring in the tth time period
t 1 .. th tth t· . d=: wa er oss occurrmg m e J.me perJ.o ..
where
Zt =: available soil moisture at time t
Xt
Lt
(2.21)
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 evapotrans-
piration, leaching, or as runoff. The following modification, based on
•
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 It ~ AtR
'" AtR if Xt > AtR
Vt ... potential evapotranspiration occurring during the tth t:iIne
period
o "" maximum amount of plant available water which can be held by
the soil.
Then we can write
7
... 0
= 0
if Vt < Zt + Xt < 0 + Vt
if Zt + Xt >0 + Vt (2.22)
if Zt + Xi < Vt
All of the climatic variables (Xt
, At and Vt ) involved in model
(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 if Vt < Zt + It < C ... Vt
... 0 if Zt + It L 0 + Vt (2023)
... 0 if Zt + It <. Vto
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 alo (1960).. The method derived by Penman (1948) is general
ly accepted as being more appropriate to the humid areas of the United
States 0 Penman's formula as given by van Bavel (1956) is
v =t
where
H + 0027 Ea
/). + 0027,
/). = incremental change in vapor pressure
H = net heat adsorption at the surface
Ea = a 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 Vt as computed from the Penman
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 Vt in models (2.22) and
(2.23) by an average value, V, over some finite period of time and given
geographical area which gives
=0 C
... 0
if V < Zt + It 4( C + V
if Zt + It ~. C + V
if Zt + It < V
w1':en runoff is an important factor, and
Zt+l .. Zt + It - V if V < Zt + Xt < C + V
.. C if Zt +- ~ >C + V (2 0 25)
... 0 if Zt + Xt < V
vmen runoff except that in excess of the storage capacity can be ignored.
Thus, models (2.22) through (2.25) represent alternative formula
tions 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.. With this definition
the condition when Zt = 0 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 < Q, where Q is
the wilting point. 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, Vt is dependent upon Zt as well as weather con
ditions. Given a mathematical expression relating Vt as a function of
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
denote models (2.22) through (2025) by the single model
..Wt +l = 1ft + Ut - 1 if 1< Wt Ut < r + 1+
... r if Wt + Ut > r + 1 (2031)
... 0 if Wt + Ut ~ 1,
where
Wt = Zt/M for models (2022) and (2.23)
= Zt/V for models (2024) and (2.25)
Ut
(Xt - Vt )+ 1 for model (2.22)...
M
(Xt - Vt )+ 1 for model (2.23)... M
... Xi/V for model (2024)
... Xt/V for model (2 025)
r ... e/M for models (2022) and (2.23)
... C/V for models (2.24) and (2,,25) ..
The quantity M is the maximum value of Vt' characteristic of a partic
ular geographic area and time of year" By introducing Mand adding 1 in
models (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 < 1
S2: 1 <"Wt < 2
..
..
..
•r
i_
1< W ~r't
where r' is the largest integer in r.
Let p.k
be the transition probability of going from state S. atJ J
time t to state Sk at time t+L The Markov property is satisfied if' the
probability of being in state S. at time t is independent of the statesJ
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.. That this condition isJ
satisfied is evident from (2.31) since Wt
+l
is completely determined if
Wt and Ut are known.. The Pjk can be written
t:oo ... Pr(Wt +l = 0 I Wt ... 0)
Clearly Pjk I: 0 for j > k + 1 since from (2031)
Pr(k - 1 <Wt +1 ~ k) = Pr(k - 1 < Wt + Ut - 1 <k)
= Pr(k < Wt + Ut < k + 1)
and since Ut ::> 0, the upper limits on Wt which satisfy (2033) are
(k < Wt < k + 1), but we are given that (j - 1 <Wt < j), hence, for
j > k + 1, Pjk = 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
denoted by F(U), the POk can be obtained explicitly in terms of F(U); io~.,
=F(k + 1) - F(k) k =1, 2, ° • r' POO • F(l)(2.34)
PO(r ' +l)= 1 - F(r') since Pr(Wt > r) = o.
14
The Pjk for j =1, 2, 3, ••• r' and k = 0, 1, 2, •• r' are not
known exactly until the distribution of 1ft is known. In this case, from
the basic laws of probability
k j
k -~ j AdG(Wt ) dG(Wt +1 I wt )Pjk = j ,
j A dG(Wt )
where G(Wt+ll 'Wt ' is the conditional cumulative distribution function
of Wt +l given Wt
and G(Wt ) is the marginal cumulative distribution
function of 1ft " Since
k
k ~ dG(Wt +1 I Wt ) = F(k + 1 - 1ft ) - F(k - Wt ),
Pjk can be written
)
j J~ dG(Wt )
!'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 Wt as the midpoint of its bounds; i .. e.,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
=:0 j
j
J:. F(k + 1 - Wt ) - F(k - Wt ) dWt ,
15
(2 .. 37)
Bounds for Pjk can be obtained by setting Wt equal to j-land j respect
ively; 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. Then the transition matrix can be written
POO POI P02 P03 P04 P05 • • " " POri PO(r'+l)
Po PI P2 P3 P4 P5 • " • Pr ' Prf +l
r'-l
0 Po PI P2 P3 P4 .. .. • • Prl-l 1 - ~ Pii=O
r f-20 0 Po PI P2 P3 • " • " Prl - 2 1 - ~ Pi
i=O
(2.41)r'-3
0 0 0 Po PI P2 • " " Pr '-3 1 - ~ Pii=O
"
"0 0 0 0 0 0 " " " • Po 1 - Po
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 non-
zero 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
16
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 P* and P** the corresponding r'+2 column vectors.
Then .!:** = TV P*; i.~.,
~ = ~POO + P!P0
Pf* = ~Ol + ptPl + ~Po
If' = ~P02 + l!P2 + ~Pl + ~O
•
k+l
~ • ~pOk + ~ J1Pk-i+li=l
•rt+l
~i=l
r t r'+l
~+l= ~(l - ~. POi) + ~~=O i=l
r'-i+l
~(l - ~ PJo).~ j=O
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
~ Pi = 10i ...O
2.5 Solution for Stationar! Probabilities
Several rrethods are available for solving (2.43) for the Pk
•
]bran (1954) and Gani and Moran (1955) give a discussion of alternative
methods including ]bnte Carlo methods. The following rrethod is proposed
for programming on a computer:
1 = POO + G:tPO
•
•k+l
Gk = POk + ~ GiPk_i+li=l
•
o
r1+l
lipa = 1 + ~ G••i=l 3.
The Gk are obtained from the Pjk by successive substitutions, starting
with1 - Poa
G =---1 Po
Given the Gk
, the Pk
are easily obtained, since from the last equation
of (2.51)
18
1r'+l
1+"~· Gii=1
Gk--r..,.I+....1:----' k ... 1, 2, 3, 0 • r'+l;
1 ... ~ Gii=1
als~ if we let
then
1
1Pal - 1
and
G ...k
1 1~ - P , k ... 2, 3, 0 0 r'+l,r Ok O(k-l)
19
so that the stationary state probabilities are completely determined
by either Gk or POk9 k ~ l~ 2, 3$ " " ri+lo
The discrete approximation to the continuous distribution of
Wt = Zt (constant) is given by
k
B(k) "" Pr(Wt <k) & ~ Po ... Po. a J.1""
A.n advantage to a solution in terms of the Gk rather than Pk is
that Gk
is independent of r, and once the Gk are found for the largest
r p the value for Po with a smaller r is obtained from (2053) by dropping
the appropriate number of Gi in the summationo
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
the discrete approximation to the distribution of Wt~ !.o~o»
o..I'D
~ Po ~ (k"~) Gk + Po ( r;ri
) Gri +lk""l
k Gk
+ Po Gri
+l
( r+rD
) .. Po (1 .. 1)2 2 Pari
r i
&, Po ~ kCL~ GO 1 ) ~ P ( ..l:.- .. I)k=2 POk PO(k=l) a POI
r+r i I I Po I+ ( - ) P ( - ... - ) .. - ( - .. 1)
2 a Po Pari 2 Pari
and E(Zt' "" ME(Wt '
"" v E(Wt '
for models (2022) and (2023)
for models (2024) and (2025)0
20
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 ~Qit
if Zt::> Qi 0
(2 0 71)
Then the probability that a moisture deficit occurs is Pr(Wt < q).9
where q "" 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..,}) Pq v-k 0
k=l(2073)
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 fre-
quency distribution of Ut is required in order to obtain the transition
probabilities.9 PjkO The variable Ut as defined for models (2022).9
(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 Ut for all four cases would be
a knowledge of the frequency distribution of ~ 0 The POk and the
approximations to Pk9 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$) the
independence assumption of the input variable Ut 0 The shortest time
period for which precipitation records are readily available is one
day 0 Thus9 any frequency curve fitting procedure must be based on daily
21
•
22
records or longer time periods.. The shortest time period of one day
seems to be desirable since it Is generally easier to derive a frequency
function for a long time period from a function for a short time period
than to derive a function for a short time period from fumtions based
on a longer time period.
When the time period is one day ~ the distribution function of ~'"
is discontinuous at zero$ !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 if the cumulative distribution function of daily precipi=
tation is denoted by Fl
(1), it may be written
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
70-23
60-
2.01.0 1.5
inches per day
..
Figure 3.1. 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
where P. and g*(litlaI'e obtained from an initial solution basedJ .
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 approxi-
mation to the stationary probabilities approaches the con-
tinuous 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 ProbabilityLaw and its Engineering ..Applications". Proc. Amer. Soc. Civ. Eng.sep 665, 81: 22-29.
Chapman, D. G. 1956. Estimating the parameters of a truncated gammadistribution. Ann. Math. Stat. 27: 498-506.
Chow, V. T. 1953. Frequency analysis of hydrologic data with specialapplication to rainfall intensities. Bulletin No. 414, IllinoisEngineering 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. Frequency Curves and Correlation. Haven Press,Washington, D.C.
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. Trans. Amer. Soc.Civ. Eng. 87: 142-173.
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 byMonte Carlo methods. 1l,.ust. J. App. Sci. 6: 267-273.
Gumbel, E.J. 1941. The return periods of flood flows. Ann. Math.Stat. 12: 163-190.
Gumbel, E. J. 194.5. Floods estimated by probability methods. Eng.News Rec. 134: .a33-337.
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 NorthCarolina. 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. J. R.Statist. Soc., B, 19: 207-233.
Knetsch, J. L. 1959. Moisture uncertainties and fertility responsestudies. J. Farm Econ. 41: 70-76.
Knetsch, J. L. and Smallshaw, J. 1958. The occurrence of drought inthe 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 andits 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 continuousrelease. Quart. J. M~th. Oxford (2), 7: 130-137.
Moran, P. A. P. 1957. The statistical treatment of flood flows. Trans.~er. Geophys. Union 38: 519-523.
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 ofthe Thornthwaite and mean· temperature methods for determiningpotential evapotranspiration. Agron. J. 52: 387-395.
Penman, H. L. 1948. Natural evaporation from open water, bare soiland grass. Proc. Roy. Soc. A., 193: 120-145.
Romig, H. G. 1947. 50 -100 Binomial Tables. John 'Wiley and Sons, Inc.,New York.
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 andtime for irrigation with the evapotranspiration method. U. S. D. A.ARS 41-11.
van Bavel, C. H. M. and Verlinden, F. J. 1956. ,Agricultural droughtin North Carolina. Technical Bulletin No. 122, North CarolinaAiricultural 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 Vf) can be
obtained from (4.11) using tables of the incomplete f-function (Pearson
1947) to evaluate the integral
xj 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. Thus,
and
POO" (l-n) ... n I('t', A-I)
POk" n i: ~k+lh, A-~ n I(k't', A-I), k .. 1, 2, 3, 0 0 TV,
Po &: (l-n) ... n I('t'/2, A-I)
Pk .It. 11: I [(k .. ~)'t', A-~ - 11: I Bk - ~)'t', A.-~ , k .. 0, 1,1) 0 • r v •
The approximation given by (2 •.37) as
is difficult to evaluate for the gamma distribution since an explicit
expression for F(k+l-W't) - F(k-Wt ) cannot be obtained except for A. .. 1.
An evaluation can be obtained by serie s expansion which gives