COUPLING STOCHASTIC AND DETERMINISTIC HYDROLOGIC MODELS FOR DECISION-MAKING by William Carlisle Mills

COUPLING STOCHASTIC  AND DETERMINISTIC HYDROLOGIC MODELS FOR DECISION-MAKING by William Carlisle Mills
COUPLING STOCHASTIC AND DETERMINISTIC
HYDROLOGIC MODELS FOR DECISION-MAKING
by
William Carlisle Mills
A Dissertation Submitted to the Faculty of the
DEPARTMENT OF HYDROLOGY AND WATER RESOURCES
In Partial Fulfillment of the Requirements
For the Degree of
DOCTOR OF PHILOSOPHY
WITH A MAJOR IN HYDROLOGY
In the Graduate College
THE UNIVERSITY OF ARIZONA
1979
THE UNIVERSITY OF ARIZONA
GRADUATE COLLEGE
I hereby recommend that this dissertation prepared under my direction
William Carlisle Mills
by
entitled
Coupling Stochastic and Deterministic Hydrologic
Models for Decision Making
be accepted as fulfilling the dissertation requirement for the Degree
of
Doctor of Philosophy
N 7 ci
Xi/rid/ 6
Date
As members of the Final Examination Committee, we certify that we have
read this dissertation and agree that it may be presented for final
defense.
/1/0-. 6,
Date
/9 7 9
Date
kstro--.
'79
Date
Aron/ 1 ;7
Date
Date
Final approval and acceptance of this dissertation is contingent on the
candidate's adequate performance and defense thereof at the final oral
examination.
STATEMENT BY AUTHOR
This
requirements
is deposited
rowers under
dissertation has been submitted in partial fulfillment of
for an advanced degree at The University of Arizona and
in the University Library to be made available to borrules of the Library.
Brief quotations from this dissertation are allowable without
special permission, provided that accurate acknowledgment of source
is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by
the head of the major department or the Dean of the Graduate College
when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission
must be obtained from the author.
SIGNED:
ACKNOWLEDGMENTS
I wish to express my appreciation to a number of people who
have contributed to the completion of this dissertation. I am
especially grateful to
Dr.
Donald R. Davis, dissertation director, for
his patience and critical help throughout the writing of the dissertation and for putting up with the problems and inconvenience associated
with communicating at long distance between Athens, Georgia and Tucson,
Arizona. The patience and excellent work displayed by Mrs. Joy
Cornelius in typing preliminary and final drafts of the manuscript is
greatly appreciated. I wish to thank Mrs. Rebecca Slack for assistance
in drafting figures and mathematical symbols. The assistance provided
by Messrs. Tom Woody and Charles Burroughs in computer operations and
graph plotting is recognized and appreciated. Finally, I wish to
acknowledge support provided throughout the entire endeavor by my
employer, the USDA Science and Education Administration, Agricultural
Research.
TABLE OF CONTENTS
Page
LIST OF ILLUSTRATIONS
vi
LIST OF TABLES viii
ABSTRACT ix
1.
INTRODUCTION 1
2.
PRESENT METHODS FOR COUPLING STOCHASTIC AND
DETERMINISTIC MODELS 6
First Order Approach Analytic Method Solution by Enumeration Monte Carlo Simulation Limitations of Present Methods 3.
4.
PROBABILITY GENERATING FUNCTIONS AND PREVIOUS
APPLICATIONS 31
DERIVATION OF COUPLING TECHNIQUE USING PROBABILITY
GENERATING FUNCTIONS 37
Factoring Input Probability Generating Function
Deriving Output Probability Generating Function
Obtaining Output Probability Mass Function Simple Demonstration Example 5.
6
12
19
21
29
REASONABLENESS OF POISSON ASSUMPTION FOR APPLICATION
OF DERIVED TECHNIQUE Definition of Stochastic Rainfall Process Consideration of Basic Poisson Requirements Assumption by Others for Poisson Occurrence
of Rainfall Events Test of Poisson Assumption for Rainfall Event
Occurrence at Ahoskie, North Carolina iv
•
•
•
• •
38
41
44
60
68
68
71
80
84
V
TABLE OF CONTENTS—Continued
Page
6. DIRECT RUNOFF APPLICATION
Deterministic Transform Model
Stochastic Rainfall Model
Stochastic State Model
Demonstration Example
7. SEDIMENT YIELD APPLICATION
Deterministic Transform Model
Stochastic Rainfall Model
Stochastic State Model
Demonstration Example
88
98
100
102
114
121
142
143
153
155
159
160
162
165
Further Application
Limitations
Suggestions for Research
Conclusions
APPENDIX A: COMPUTER PROGRAMS FOR SIMPLE EXAMPLE
87
113
8. DISCUSSIONS AND CONCLUSIONS
APPENDIX B: SCS CURVE NUMBERS AND RELATIONS
171
APPENDIX C: AHOSKIE CREEK WATERSHED MAP AND DATA
176
APPENDIX D: PROGRAM FOR COMPUTING PROBABILITY MASS
FUNCTION OF CUMULATIVE DIRECT RUNOFF
180
APPENDIX E: PROCEDURE FOR OBTAINING MAXIMUM LIKELIHOOD
ESTIMATE OF WEIBULL PARAMETERS
185
APPENDIX F: PROGRAM FOR COMPUTING PROBABILITY MASS
FUNCTION CUMULATIVE SEDIMENT YIELD
190
LIST OF REFERENCES
196
LIST OF ILLUSTRATIONS
Figure
Page
1. Comparison of PDFs for Crop Season Runoff Obtained
by the Derived Recursive Technique and Exact
Analytic Method
2. Typical Sample Function of Rainfall Intensity
for a Point in Space
3. Histogram and Fitted Exponential PDF for Time
Between Rainfall Event Endings at Ahoskie, NC 4.
86
101
5. PDFs of Annual Direct Runoff Computed by Recursive
Technique for Ahoskie Creek Watershed
7. Relation to Event Duration of Log-Normal Parameter
Estimate 1^1 for Ahoskie Rainfall Depth
Probability Distribution
8.
Relation to Event Duration of Log-Normal Parameter
Estimate a for Ahoskie Rainfall Depth
Probability Distribution
69
• •
Histogram and Fitted Exponential PDF for Rainfall
Event Depths at Ahoskie, NC
6. Histogram and Fitted Weibull PDF for Rainfall
Event Durations at Ahoskie, NC
67
110
127
130
131
9. Histogram and Fitted Log-Normal PDF of Ahoskie
Rainfall Event Depths for Durations of
1 Hour or Less
132
10. Histogram and Fitted Log-Normal PDF of Ahoskie
Rainfall Event Depths for Durations of
1 to 3 Hours
133
11. Histogram and Fitted Log-Normal PDF of Ahoskie
Rainfall Event Depths for Durations of
3 to 6 Hours
134
vi
vii
LIST OF ILLUSTRATIONS--Continued
Figure
Page
12. Histogram and Fitted Log-Normal PDF of Ahoskie
Rainfall Event Depths for Durations of
6 to 12 Hours
135
13. Histogram and Fitted Log-Normal PDF of Ahoskie
Rainfall Event Depths for Durations
Greater Than 12 Hours
136
14. PDFs of Annual Sediment Yield Computed by
Recursive Technique for Ahoskie
Creek
Watershed
150
LIST OF TABLES
Table
Page
1. Relation of SCS Antecedent Moisture Condition
Class to Antecedent Rainfall
2.
3.
4.
5.
6.
103
Relation of SCS Curve Number to Antecedent
Moisture Condition Class
104
Input Data for Operation of Computer Program
for Direct Runoff Example
109
Relative Frequency of Rainfall Event Depth and
Duration at Ahoskie, NC
Chi-square Statistics and Exceedance Probabilities
Showing Goodness of Fit of Log-Normal
Distribution to Histograms of Ahoskie Rainfall
Depth for Five Duration Classes
Input Data for Operation of Computer Program for
Sediment Yield Example
v i ii
123
138
147
ABSTRACT
Many planning decisions related to the land phase of the
hydrologic cycle involve uncertainty due to stochasticity of rainfall
inputs and uncertainty in state and knowledge of hydrologic processes.
Consideration of this uncertainty in planning requires quantification
in the form of probability distributions. Needed probability distributions, for many cases, must be obtained by transforming distributions
of rainfall input and hydrologic state through deterministic models of
hydrologic processes.
Probability generating functions are used to derive a recursive
technique that provides the necessary probability transformation for
situations where the hydrologic output of interest is the cumulative
effect of a random number of stochastic inputs. The derived recursive
technique is observed to be quite accurate from a comparison of
probability distributions obtained independently by the recursive
technique and an exact analytic method for a simple problem that can
be solved with the analytic method.
The assumption of Poisson occurrence of rainfall events, which
is inherent in derivation of the recursive technique, is examined and
found reasonable for practical application.
ix
Application of the derived technique is demonstrated on two
important hydrology-related problems. It is first demonstrated for
computing probability distributions of annual direct runoff from a
watershed using the USDA Soil Conservation Service (SCS) direct runoff
model and stochastic models for rainfall event depth and watershed
state. The technique is also demonstrated for obtaining probability
distributions of annual sediment yield. For this demonstration, the
deterministic transform model consists of a parametric event-based
sediment yield model and the SCS models for direct runoff volume and
peak flow rate. The stochastic rainfall model consists of a marginal
Weibull distribution for rainfall event duration and a conditional
log-normal distribution for rainfall event depth given duration. The
stochastic state model is the same as employed for the direct runoff
application.
Probability distributions obtained with the recursive technique
for both the direct runoff and sediment yield demonstration examples
appear to be reasonable when compared to available data. It is
therefore concluded that the recursive technique, derived from
probability generating functions, is a feasible transform method that
can be useful for coupling stochastic models of rainfall input and
state to deterministic models of hydrologic processes to obtain
probability distributions of outputs where these outputs are cumulative
effects of random numbers of stochastic inputs.
CHAPTER
1
INTRODUCTION
In the field of hydrology we are concerned with the occurrence
and movement of water on the earth. The purpose of this concern is to
provide information and techniques for rational planning for beneficial
control and use of water and for planning to avoid or minimize damage
by water. This general twofold purpose related to planning for water
use and damage alleviation encompasses a broad spectrum of specific
problem areas. Several following examples chosen from throughout this
spectrum demonstrate the diversity of hydrologic problems encountered.
Planning for beneficial use of water quite often involves
design of surface reservoirs or groundwater recharge facilities for
storing water runoff from a watershed or drainage area. Such planning
might also involve modifying vegetation or treating the watershed
surface to increase water yield for storage and use.
Beneficial-use planning could, in some instances, be concerned
with soil moisture recharge from cumulative infiltration of water into
the soil prior to and during the growing season in
dryland and semi-
arid regions. Planning decisions related to this concern might involve
determining optimum
dryland farming or rangeland management practices.
1
2
Damage by water can result from flooding. The planning
decision related to flooding damage might be whether or not to locate
agricultural crops or urban development in a flood-prone area -- or
it might be whether to alleviate flooding by structural measures.
Erosion and sedimentation are water-related processes that also
cause damage (Guntermann, Lee, and Swanson 1975; Lee and Guntermann
1976; Pimentel et al. 1976). Decisions related to these processes may
be concerned with management practices on the land that affect soil
erosion rates, or they may be concerned with methods for handling
sediment after it has moved downstream.
Agricultural chemicals from upland areas transported by flowing
water either as a solute or adhered to sediment particles can be
damaging to reservoirs, lakes, and estuaries (Frere 1973). Decisions
concerning these potentially damaging pollutants may involve both
application rates of chemicals and practices to retard water transport
of chemicals (Train 1975; Pisano 1976).
Sometimes planning involves optimum tradeoff of benefits related
to beneficial water-use with those related to water-damage alleviation.
Such an example is the planning of a drainage system for a swamp that
contains an aquifer recharge area. In this case drainage would benefit
agricultural use of the land, but could detrimentally alter recharge to
underlying aquifers.
The examples presented here are common ones encountered in
planning for control and use of water or for alleviating damage from
water or water-related processes. Other similar examples could also be
given.
3
If we examine these examples we find that information needed
for planning decisions in each case is the cumulative effect over some
planning period of many random inputs to a hydrologic system. For the
first example this is cumulative water yield from a number of runoff
events during a season or other appropriate planning period. For the
second example it is cumulative infiltrated water from several rainfall
events. The third example requires cumulative damage from many
different size floods. The example on erosion and sedimentation
requires information on both cumulative erosion on-site and cumulative
sedimentation off-site caused by a number of rainfall and runoff events.
The next example requires information on cumulative chemical transport
to sites of concern due to several runoff events. The last example
requires information on cumulative recharge from a number of events.
The cumulative effect of many random inputs in each of these
examples is not a deterministic value, i.e., a value with probability
of occurrence of one, but is rather a probabilistic value. This
cumulative effect is a random variate. Thus, planning decisions based
upon consideration of risk, which is probable loss associated with
probability of that loss, require estimates of the probability measure
or distribution of the cumulative effect of concern.
Data for directly obtaining estimates of probability distributions for the examples given would be difficult to acquire in many
cases. Long-term records on cumulative water yield, infiltrated water,
flood damage, erosion and sedimentation, chemical transport, and
aquifer recharge are not available for many planning situations, and
4
even if available they may not be adequate since hydrologic systems
usually change during long time-periods. Moreover, to evaluate
alternate planning decisions, which is required for proper planning
of future hydrologic system changes, probability distributions of
cumulative hydrologic effects are needed for each change considered.
Historical data obtained under only one or even a mixture of hydrologic
conditions will not provide the necessary probability distributions for
evaluating alternate plans.
Deterministic models of hydrologic processes involved in
watershed runoff, soil moisture recharge, flood generation and
movement, erosion and sedimentation, chemical transport, and aquifer
recharge have been or presently are being constructed to provide output
information for given input and initial state. These models range from
simple algebraic equations to complicated computer programs. If models
of random or stochastic input and state are coupled to these
deterministic process models, then probability distributions for the
cumulative effects or process outputs of concern can be obtained
through transformation of hydrologic system input and state
probabilities.
This dissertation will consider the problem of coupling
stochastic input models and stochastic state models to deterministic
models of hydrologic systems to obtain probability distributions of
cumulative effects of inputs over pertinent planning periods. A
literature review will explore several methods presently used and
limitations of these methods will be discussed. Probability generating
5
functions will be reviewed for possible use in the stochasticdeterministic coupling problem. A technique using probability
generating functions will be derived for obtaining the probability
distribution of cumulative effect of stochastic inputs to deterministic
hydrologic systems with stochastic states. Both theoretical development
and computer implementation procedures will be given. The derived
technique will be demonstrated on a simple example for which an exact
analytic solution can be obtained for comparison of results.
Assumptions required for use of the technique and validity of these
assumptions in hydrologic application will be examined.
Application of the derived technique will be demonstrated on
two realistic hydrology-related problems similar to the planning
examples given above. In connection with these demonstration examples
various deterministic models of pertinent portions of the land phase of
the hydrologic cycle will be reviewed as a basis for selecting deterministic models appropriate for application of the derived technique.
Stochastic models of rainfall input and hydrologic state required for
the demonstration examples will be defined. Results obtained in the
application demonstrations will be compared to available data and
reasonableness of results considered.
CHAPTER
2
PRESENT METHODS FOR COUPLING STOCHASTIC
AND DETERMINISTIC MODELS
The present methods for coupling stochastic and deterministic
models for decision-making in hydrology, as well as in other fields,
fall into four general categories. These are
(1) first order approach,
(2) analytic method, (3) solution by enumeration, and (4) Monte Carlo
simulation.
First Order Approach
The first order uncertainty approach is described by Cornell
(1972). It essentially deals with means, variances, and covariances.
Thus, inherent in the method is the assumption that the behavior of a
random variable or stochastic process can be described adequately by
the first two moments of the probability distribution or stochastic
process. Additionally, first order theory requires that functional or
system relationships linking input, output, and other random variables
be simplified so that only the first order terms in a Taylor-series
expansion or a system perturbation analysis are retained. For example,
if Y is an analytic function of the random variable X, i.e., Y
= f(X),
it is assumed that
Y = f(P x )(X - P x )
6
d
T
)
(1)
7
df(X)
the derivative of f(X) with
x is the mean of X and dX is
ux
respect to X evaluated at
x' More generally, if Y is an analytic
function g of a column vector X of random variables it is assumed that
where p
(2)
bT(X - P x )
=
where p is a column vector of the means of the elements in X, and b T is
the transpose of a column vector b of partial derivatives such that
element i is
(agX).
If Y(t) is the output of a system that is
X.1
operating on stochastic input X(t), then it is assumed in first order
DX.1
theory that
Y(t) = L[px (t)] + ,c("1 ( * [X(t) - p x (t)]
(3)
where L[ilx (t)] denotes the operator of the system acting on the mean
dL * [X(t) p(t)] denotes convolution of
value function of X(t), and cyx-
the linearized system function and the zero-mean deviation process
X(t) - i(t) (Cornell 1972, p. 1247).
In the first order approach the mean p
2
Y
and variance
Y
function g of a vector X of random variables are obtained by the
of a
.
equations
p
and
y
= g( p )
—x
(4)
8
a
2
= b
T
E
b
(5)
where E is the covariance matrix of X and other symbols are as
—x
previously defined. If the elements in X are uncorrelated equation 5
reduces to
2
a2
Y
-ag(X)
aX.
i
• a2
(6)
where a 2 is the variance of the ith element of X. In the approximation
x.
given by equation 4, an additional term involving second derivaofp
Y'
tives and second moments is sometimes obtained by using three terms of
the Taylor-series expansion of g(X) instead of two. The additional
term is negligible, however, if the coefficients of variation of the X i
and the nonlinearity in the function are not large (Benjamin and
Cornell 1970, p. 184).
The mean value function p (t) and covariance function KY (s,t)
Y
of a stochastic process Y(t) that arises from operation of a linear
system with time-varying system function h(t,t) on an input stochastic
process X(t) are obtained as follows:
f px T
(
0
and
)
h ( t ,T )
(7 )
9
St
K (s,t) = f
Y
If
Y(t)
f K (u,v) h(s,u) h(t,v)
dv.
(8)
X(t)
the system
du
00 x
arises from a nonlinear operation on
function is modified as previously indicated to provide linearization
with respect to the random
conponent.
The modified system function is
then used in the above equations to obtain
(0 and K (s,t).
Y
Y
Another aspect of first order uncertainty theory that can be
quite useful is the "total expectation theorem," which Cornell
p.
1253)
expresses as
E[9(Y,X)]
where
g(Y,X)
=
E X {Ez[g(YIX)]}
E (.)
— Y
withrespecttoY,E
pp.
(9)
denotes a function of random variables Y and X,
indicates the same function of Y for given X,
E(-)
(1972,
g(YIX)
denotes expectation
( .) denotes expectation with respect to X, and
is total expectation. From this general relation, Cornell
1253-1254)
(1972,
derives expressions for the mean and variance of the sum
of a random number of random variables that have common mean and
variance. These expressions for the case where the random variables
are
uncorrelated
are given by the following equations:
Pz
and
= Pn
Pr
(10)
10
a
where p
r
and a
2
variables R. p
2 = 2 2 + p a 2
z U rn
n r
are the mean and variance, respectively, of the random
n
and a
2
are the mean and variance of the random number
of random variables R i , and
sum
(11)
and a
2
N
are the mean and variance of the
Z of the random number N of random variables R i .
The first order uncertainty approach was applied by Chamberlain
et al.
(1974) in connection with design of a water quality monitoring
system for streams. They related the means and standard deviations of
stream velocity, upstream dissolved oxygen (DO) deficit, and upstream
biochemical oxygen demand (BOO) to the mean and standard deviation of
downstream DO deficit by using first order theory and the deterministic
Streeter-Phelps equations. They assumed a normal distribution for
downstream DO deficit. With this assumed distribution and parameters
obtained by the first order approach, probability estimates for
violation of water quality standards were computed. These violation
probabilities were then used to identify potential sampling stations
and establish relative priorities for implementation of the stations.
Burges and Lettenmaier (1975) also used the first order approach
in connection with the Streeter-Phelps equations. In addition to
uncertainty in the upstream DO and BOO, they were also concerned with
uncertainty in downstream DO and BOO prediction due to uncertainty in
parameters such as the BOO decay constant, the
reaeration constant, and
the dissolved oxygen saturation concentration. Monte Carlo simulation
11
was used to check the accuracy of the first order approach. Results
obtained by the two methods coincided for small stream distances and
travel times but began to diverge for larger distances and times.
Fogel,
Hekman, and Duckstein (1976) employed the first order
approach to obtain estimates of the mean and variance of annual sediment
yield for two watersheds in Arizona. First, they calculated the mean
and variance of sediment yield per event from rainfall data by using a
relation expressing sediment yield as a function of rainfall depth and
duration. Then, with these calculated values and estimates for the mean
and variance of the number of sediment-producing events per year, they
computed annual sediment yield mean and variance using the first order
relations given by equations
10 and 11.
The obvious advantage of the first order approach is the ease
with which estimates of means and variances of output from certain
deterministic models linked to stochastic inputs can be obtained. If
mean and variance estimates are sufficient for a decision problem, the
first order approach should be considered. However, use of this
approach implies that either the form of the probability distribution of
interest is known and only the first two moments are needed to complete
its description or the cost function related to the decision problem of
concern is either linear or quadratic (Benjamin and Cornell
1970, pp.
151-153). Such is not the case with many hydrology related problems.
It is also pointed out by Cornell
(1972, p. 1259) that the first
order approach usually is inadequate for the treatment of problems that
have severe and important discontinuities. Such problems lack
12
continuous derivatives, which are required in first order theory. Many
hydrology related models contain discontinuities; therefore, problems
that require use of these models may also preclude use of the first
order approach.
Analytic Method
If a distribution function describing the full probability law
of a transformed random variable is needed, the analytic method can be
used in certain situations. The analytic method is discussed by
Benjamin and Cornell
(1970, pp. 100-134) under the general topic of
(1970, pp. 116-170) under
derived distributions and by Hogg and Craig
the topic of distributions of functions of random variables. Depending
upon properties of the deterministic transformation, the analytic
method may take any one of several slightly different forms.
If the transformation
g(.) relating the input random variable X
1
one to one and monotonically
with the output random variable Y is
increasing, 2 the cumulative probability distribution FY (y) of Y can be
derived from the cumulative probability distribution F(x) of X as
follows:
F Y (y) = PfY < y}
(
1. Meaning the transformation defines a one-to-one correspondence between points in the input and output spaces.
2. That is
g(X) does not decrease as X increases.
12)
13
F (y) = P{X < g-1(y)}
(13)
F(i)
= Fx [g -1 (y)]
y
(14)
Y
where
P
g -1 (.)
is probability measure and
is the inverse of
g(-).
If in addition to the above requirements the derivative of the
inverse transformation
function
f
Y
function f
g -1 (-)
is continuous, the probability density
(y) of Y can be obtained directly from the probability density
(x)
of
X.
This is demonstrated by the following sequence of
equations:
f (y)
Y dy [F y (y)]
•
fy (y) = -dcry- (F x
fy (y) = -c7
1
J
D
Q
-1
-1 (y)]}
[g
(Y)
f(x) dx]
-00
I f x [ g 1( y )] l
fy ( y )
d
-
-1(
dy '
Y )] 1)1
fY( Y ) = fx [ g
If the transformation
g(.)
is monotonically decreasing instead
of increasing, yet the other conditions specified above are satisfied,
the probability density function
f
Y
(y) can be obtained by the above
expression with the absolute value of
(
substituted for
4cay
L( '
that is,
14
f (y) = f [g -1 (y)]
Y
dx
dy
(20)
•
For situations where the deterministic transformation g(.) is
neither monotonically increasing nor decreasing or is not one to one,
the cumulative probability distribution of output Y can, in some cases,
be obtained analytically. The general approach used in such cases is
given by the following derived equations:
F()
Y
D
F y (Y) =
= P {Y < y }
sX takes on any value x i
I
l suc h that g(x) <y
F (y) = f f(x)
dx
x
Y
Y
where
R is that region where g(x) is less than or equal to y.
Y
If the output random variable Y is a function
input random variables X i ,
(1) of two or more
i = 1,2,...m, this general approach also
applies. The appropriate equation is
F (y) =
Y
ff
...f f x (x l , x 2 , •.. x m ) dx 1 dx 2dx m
(24)
Y
where
fx (x1,x2,•••xm) is the joint probability density function of
X i ,X 2 ,•.•X m and Ry is that region where (I) (x 1 ,x 2 ,...x m ) is less than or
equal to y.
15
If the transformation
(1, is one to one, monotonically increasing
or decreasing, and the derivative of the inverse transformation is continuous, the output probability density function can be obtained from the
joint input probability density function in a manner similar to the case
where these conditions are satisfied for single input random variables.
For multiple inputs it may be necessary to assume some dummy outputs,
since the number of input and output random variables must be the same.
The joint output probability density function derived is then integrated
over the dummy output region to obtain the marginal probability density
function of the desired output. The procedure is given by the following
equations:
-1
fy ( Y1' .Y2" — Ym ) = fx [1) 1 l'2'
-1
m' —4)M ( Y1 , Y2" — Ym )]1J I
(25)
and
f (y ) =
Y1 1
where
ff ...f
fy (yi ,y2 ...ym ) dy, dy 2 ...dy
(26)
Rd
f x is the joint input probability density function, f is the joint
output probability density function,
Y
is the marginal output
f
proba-
yl
bility density function of Y 1 , j is the inverse transformation that
maps
y i to x i , i = 1,2,...m, R d is the dummy output region, and IJ1 is
the absolute value of the
Jacobian of the inverse transformation.
For discrete random variables where output is obtained by a one
to one transformation of input, the output probability mass function can
be obtained from the input probability mass function by the equation
16
NY 1
= y i , Y 2 = y 2 , ...1 = P{X 1 =
gV(y 1 ), X 2 = g -2 1 (y 2 ),
(27)
where P is probability measure of the event specified and other symbols
are as previously defined.
The analytic method was applied by Eagleson (1972) in deriving
the cumulative probability distribution of peak streamflow from the joint
probability density of rainfall excess intensity and duration. In this
derivation he used the kinematic wave equations to express the transformation of rainfall excess to streamflow. The probability density
functions of rainfall excess intensity and duration were modeled as
exponentials. To get an analytic expression for the cumulative distribution F(q) of peak streamflow, Eagleson integrated the joint probability
density of rainfall excess intensity and duration over the region for
which peak streamflow is less than specified values of q. Expressions
relating peak streamflow with exceedence probabilities and return periods
were then developed from the peak streamflow cumulative probability
distribution function.
Important simplifying assumptions were made by Eagleson to
facilitate the analytic solution. The major assumptions were that storm
parameters are independent of each other and that parameters defining the
direct runoff-producing area are independent of the storm parameters.
Other assumptions eliminated the consideration of some uncertainty by
assuming average values. For example, the possibility of a random number
of events in a year was not considered; an average annual number was used
instead. A constant value was also assumed for the abstraction from
total rainfall to obtain rainfall excess.
17
More recently Eagleson (1978a, 1978b, 1978c, 1978d, 1978e, 1978f,
1978g) employed the analytic method in deriving probability distributions
of surface runoff, evapotranspiration, and groundwater runoff from distributions of rainstorm properties. He represented stochastic rainstorm
arrival by a Poisson distribution and rainstorm intensity and duration by
exponential distributions. The transform system was modeled by physically based dynamic and conservation equations which express infiltration,
exfiltration, transpiration, percolation to groundwater, and capillary
rise from the water table in terms of rainfall, potential evapotranspiration, soil and vegetal properties, and water table elevation. Many
simplifying assumptions were incorporated to make the derivations tractable. Probably the most important of these were the assumptions that
storm intensity and duration are statistically independent and that soil
moisture is constant at its space and time average.
Duckstein, Fogel, and Kisiel (1972) derived an expression for the
cumulative probability distribution of storm runoff volume through use of
the analytic method. They assumed the rainfall volume probability density
function to be exponential and used the Soil Conservation Service (SCS)
empirical rainfall-runoff relationship for transformation of rainfall
volume to runoff volume.
Klemes (1974) used the analytic approach to derive expressions
for the storage and output probability density functions of a linear
reservoir. These probability densities were expressed as infinite convolutions of probability densities obtained from the input density
functions by the change-of-variable procedure. He also showed how the
18
probability density for Markovian input could be expressed as an infinite
convolution of probability densities obtained from the density function
of the input random component by the change-of-variable procedure using
powers of the correlation coefficient to express variable transformation.
Woolhiser and Blinco (1972) derived an expression for the probability density of sediment yield from individual storms by using a slight
variation of the analytic method. A linear regression of storm sediment
yield on the product of total storm kinetic energy and maximum 30-minute
intensity (designated El) was used to express the transformation of rainfall inputs to sediment yield. The conditional probability density
function of sediment yield for given El was then assumed to be normal and
parameters of the density function were obtained from the regression.
The product of this conditional probability density function and the
density function for El provided an expression for the joint density of
sediment yield and EI. The marginal probability density of storm sediment yield was obtained by integrating the joint probability density
over the set of El values.
A solution identical to that obtained by Woolhiser and Blinco
(1972) could have been obtained with the standard analytic method that
was previously described. With this procedure, the joint density of
sediment yield and El would be expressed as
. I dl
, - lf" "
f
= 4 r, - l(„ " )
I
"
l x"1 uPY2 i ' 2 `.11""2)1
yul' .7 2 1
where f
x
(28)
is the input probability density function; f is the output
Y
probability density function; y l is storm sediment yield; y 2 is El;
19
g i (EI,E) = K(EI - m) +
E,
for E > K(M - El), g l = 0 otherwise, where E
is the normally distributed random variate of y l about the regression
K(EI - m); g 2 (EI,e) = El; g
of g 2 ; and J =
is the inverse of g l ; gi l is the inverse
1 K
0 1
Advantages of the analytic method are that an exact closed
form expression for the probability distribution of the transformed
variate can be obtained. Disadvantages are that the deterministic
transformation and usually the distributions of input random variables
must be highly simplified for the method to be tractable. Many of the
models presently being developed to more adequately represent hydrologic processes are too complicated for use with the analytic method.
Solution by Enumeration
For discrete input and output random variables it is theoretically possible to obtain the output probability mass function for a
system from the input probability mass function by enumeration of all
values of input that lead to a particular value of output. The probability of that output value is the sum of probabilities of the inputs
leading to it.
Benjamin and Cornell (1970, pp. 101-103) demonstrate the
enumeration approach with a simple example in high-speed groundtransportation systems.
In hydrology the enumeration method has been applied to the
solution of a number of problems. For example, Nnaji, Davis, and
Fogel (1974a) used the enumeration method along with a deterministic
20
watershed model to quantify uncertainty in the runoff hydrograph due
to uncertainty in location of storm center and rainfall depth at storm
center for a given raingage reading on a watershed. Also, Szidarovszky,
Duckstein, and Bogardi (1974) employed the enumeration approach to
obtain the probability of failure of a levee along the confluence of a
tributary and main river. In this work Szidarovszky et al. used a
nonuniform steady flow equation in discretized form to transform each
pair of probable input values of main river stage and flood stage in
the tributary to water level along the levee.
Laurenson (1974) expressed the enumeration method in matrix
form and demonstrated its application in transforming the flood frequency curve from an upstream point at Gravesend to a downstream point
at Pallamallawa on the Gwydir River in New South Wales. He also
demonstrated use of the method for obtaining the flood frequency curve
below the confluence of the Namoi and Peel Rivers when the correlated
flood frequency curves for each of these rivers upstream from the
confluence are known. In addition to these examples, he showed how
the enumeration method in matrix form can be used to compute the probability mass function of water storage in a reservoir where stochastic
annual inflows are serially uncorrelated.
These examples given above demonstrate that the enumeration
approach is useful in many situations. And as stated previously, it
is theoretically possible to apply the approach to any discrete system.
However, in practice enumeration of all possibilities may become
unwieldy for complex problems where several random variables are
21
involved, and especially where there is a random number of random
variables as in the examples given in the introductory chapter.
Monte Carlo Simulation
Monte Carlo simulation is the method presently used to obtain
the probability distribution of output from complicated deterministic
systems with stochastic inputs. The method essentially involves random
sampling of the stochastic inputs and transformation of the input
samples through a deterministic model of the system to obtain output
samples. The output samples form a histogram from which probability
estimates are made.
Generation of truly random input samples on a digital computer
is impossible, of course, since the computer is a deterministic device.
Nevertheless, algorithms have been devised for generating sequences of
pseudorandom numbers that pass certain statistical tests for randomness
(Hammersley and Handscomb 1964; Ralston and Wilf 1967; Lewis, Goodman,
and Miller 1969).
When using these algorithms, it is best to keep in mind advice
given by Wallis, Matalas, and Slack (1974, p. 213): "The generation
of such numbers is at best an art. . . . An analysis of a computer
system's arithmetic (both the hardware and the software) and the
built-in algorithms for generating pseudorandom numbers should be an
integral part of the studies for any important project where decisions
are dependent upon analyses utilizing such numbers."
The basic set of pseudorandom numbers generated by a digital
computer is uniformly distributed and must be transformed in most cases
22
to represent samples from input probability distributions that are
important in hydrology. If the inverse function of the cumulative
probability distribution for input can be expressed in closed form, the
desired input samples are simply obtained by the equation
S m = H (R m )
where S
(29)
-1
i s the
m is a sample from the desired input distribution F, F
inverse of F, and R
m
is a uniformly distributed pseudorandom number.
In cases where the inverse of the cumulative probability distribution
cannot be expressed in closed form, numerical techniques are required
for approximating the inverse function. Normally distributed pseudorandom numbers can be obtained from uniformly distributed pseudorandom
numbers by the Box-Muller transform (Box and Muller 1958) or by summing
a sequence of uniformly distributed pseudorandom numbers (Hamming 1962,
pp. 34 and 389; IBM 1970,
P. 77).
Hahn and Shapiro (1967, p. 242) used
known relationships among various distributions to derive algorithms
for transforming uniform and normal variates to variates with other
probability distributions. They summarized these algorithms for
several of the commonly known distributions in table form. This table
of algorithms is also presented in a slightly modified form by Hann and
Barfield (1973, p. 375). Wallis et al. (1974) give algorithms for
generating samples from the log normal, Gumbel, Weibull, gamma (Pearson
type 3), and Pareto (Pearson type 4) distributions by using either
uniform or normally distributed pseudorandom numbers as the basic
sample set.
23
The Monte Carlo method has been applied to numerous problems in
hydrology and water resources planning. Thomas and Fiering (1962)
introduced the use of Monte Carlo simulation to generate synthetic
streamflows as inputs to water resource planning models for evaluating
risks associated with alternate river basin plans.
Since then many
different methods have been explored for stochastic modeling of streamflows to provide for generating inputs to planning models. Matalas
(1975) gives an excellent summary of this work. Askew, Yeh, and Hall
(1971) demonstrate the use of Monte Carlo techniques to derive optimum
contract levels for multipurpose water resource systems subject to
constraints of maximum permissible probabilities of failure to meet
contract deliveries. They also point out that similar approaches can
be used to select optimum values for water resource system size, output,
and operating policy.
Stochastic streamflow models used for planning large river basin
systems are usually based upon streamflow records. However, streamflow
records are not generally available for small watersheds and may not be
adequate for some planning and evaluation purposes on large basins.
For these situations streamflows are sometimes generated through Monte
Carlo simulation of inputs such as rainfall and evapotranspiration with
subsequent transformation of these inputs to streamflows by a watershed
model. For example, DeCoursey and Seely (1969) generated synthetic
sequences of daily surface runoff from a watershed using a simple
deterministic watershed model with a probabilistic component. They
used Monte Carlo simulation to generate stochastic inputs of daily
24
rainfall and evapotranspiration. Values of the probabilistic component
of the watershed model were also generated by Monte Carlo. The method
was applied to a study for determining effect of land-use change on
runoff.
Ott and Linsley (1972) employed Monte Carlo techniques to
evaluate the uncertainty of the return period of extreme floods. They
generated 500 years of precipitation data for two different climatic
regions on the west and east coasts of the United States using a rainfall synthesis model developed by Franz (1970). The Stanford Watershed
Model (Crawford and Linsley 1966) was then used to convert the simulated
rainfall to streamflow. The Gumbel and Log-Pearson Type III methods of
frequency analysis were performed on each time series of simulated
streamflow. Comparison was made between the methods of moments and
maximum likelihood for estimating parameters of the Gumbel extreme value
distribution. Effects of applying the coefficient of skewness and
adjusting for record length with the Log-Pearson Type III analysis were
also studied. Use of the methods with streamflow records of various
lengths was evaluated.
Parmele (1972) simulated errors in daily potential evapotranspiration (PET) by Monte Carlo to determine effect of random error
in PET inputs on streamflows simulated by several different watershed
models.
Brajkovic, Jovanovic, and Dakkak (1971) used Monte Carlo
methods to generate flood waves at several control sections on the
Zapadna Morava River basin in Yugoslavia. They simulated 29 synthetic
25
series of daily rainfall with each series representing rain during the
high rainfall season (March-June). From each season one rainy spell
was selected that was expected to produce the highest peak discharge
for that season. Deterministic relations developed for each sub-basin
were used to transform the rainfall into streamflow hydrographs. A
frequency analysis of synthetic peak discharges was conducted and
results were compared with the frequency analysis of historical annual
floods.
Ahmed, van Bavel, and Hiler (1976) applied Monte Carlo methods
in generating a stochastic weather and precipitation pattern for a
soil-water-atmosphere-plant system model. This model was developed
for use as a tool for optimal irrigation decision-making. The simulated weather and precipitation pattern consisted of 4-hourly values of
rainfall, air temperature and humidity, wind speed, sky cover, and
radiation. Several different irrigation strategies were compared to
evaluate effectiveness.
Smith (1976a) employed Monte Carlo type methods to simulate
the effect of random areal variation in soil characteristics on
effective infiltration patterns. A physically based infiltration
model was used to describe the flow process. Parameters of this
model, such as sorptivity and saturated conductivity were treated
as random variables to represent areal heterogeneity. Monte Carlo
simulated values of the parameters were input to the infiltration
model and the response pattern was then computed.
26
Duckstein and Simpson (1975) generated synthetic sequences of
rainfall by Monte Carlo to obtain recharge input values to a karstic
aquifer in connection with studies to evaluate uncertainties associated
with karstic water resource systems.
Shih (1972) used Monte Carlo techniques involving random walk
structures for solving problems related to groundwater movement in a
flood area as affected by various flood stages. Inputs to the groundwater solution structure were uniformly distributed random numbers
provided by a random number generator.
Freeze (1975) made a stochastic analysis of one-dimensional
groundwater flow in nonuniform homogeneous media with the aid of
Monte Carlo methods. He assumed a joint probabilistic structure for
the three basic hydrogeologic parameters: hydraulic conductivity,
compressibility, and porosity. Aquifer compressibility and hydraulic
conductivity were assumed to be log normally distributed and aquifer
porosity was considered normally distributed. A multivariate normal
density function defined by the proper means, standard deviations,
and correlation coefficients was sampled by Monte Carlo techniques to
obtain values of the hydrogeologic parameters. The boundary value
problem was then solved using these parameter values. Repeated
samplings and boundary value solutions were carried out to obtain the
statistical distribution of hydraulic head through space and time.
Effects of stochastic hydrogeologic parameter distributions on predicted hydraulic heads were evaluated.
27
As previously discussed, Burges and Lettenmaier (1975)
employed Monte Carlo simulation to check the accuracy of first order
analysis for quantifying uncertainty in predicted stream quality due
to uncertainty in parameters of the Streeter-Phelps equations.
Brutsaert (1975) also applied Monte Carlo simulation to stream
quality modeling. He used the Streeter-Phelps equations as the
deterministic model and defined probability density functions for the
initial dissolved oxygen deficit, deoxygenation and reoxygenation
coefficients, velocity and temperature of the water, and the biochemicaloxygen-demand (BUD) load. The deterministic model output was dissolvedoxygen (DO) deficit. Monte Carlo sampling of the input probability
density functions and subsequent transformation to DO-deficit provided
for determining (1) the most probable time and distance downstream for
the critical DO-deficit to occur, (2) the probability of having a
critical DO-deficit greater than or less than a prespecified value, or
the probability for this value to fall between specified limits, and
(3) the probability of having a critical DO-deficit greater than or
less than a specified value after a certain period of time.
Esen and Rathbun (1976) used a Monte Carlo technique for
simulating a random walk process in a stochastic model developed for
estimating the distribution of dissolved oxygen deficit for points
downstream from a waste source. Process equations incorporated in
the water quality model were based upon conservation of mass. Both
the deoxygenation and reaeration coefficients were defined as normally
distributed random variables. The means and variances of these
28
coefficients were considered functions of travel time through a channel
reach. Biochemical-oxygen-demand (BOO) at the upstream end of a reach
was modeled as a normally distributed random variable. Provision was
made for dividing random variations of the deoxygenation coefficient
into variations at the upstream end of the reach and variations along
the reach. The model was applied to both a hypothetical example and
to actual data from the Sacramento River in California.
The Monte Carlo method has also been used in connection with
the simulation of sediment transport. For example, Murota and Hashino
(1969) developed a stochastic sediment model for the Arita River in
Japan that incorporated procedures for Monte Carlo simulation of daily
rainfall as input. The model included a deterministic relationship
between rainfall and runoff and a unit hydrograph procedure to distribute runoff amounts in time. Brown's sediment-discharge formula was
used to compute sediment-transport rates from stream-discharge rates.
Duckstein, Szidarovszky, and Yakowitz (1977) employed Monte
Carlo simulation to obtain probabilistic forecasts of sediment storage
required for a proposed reservoir site in Arizona. For this simulation,
they assumed Poisson occurrence of runoff-producing rainfall events and
used a bivariate gamma distribution to represent the joint probabilistic
structure of event rainfall amount and duration. The deterministic
transform was provided by a nonlinear relation expressing sediment yield
per event as a function of rainfall amount and duration. Simulated
sediment yield values were coupled to an economic loss function and
29
optimum designs for reservoir sediment storage were computed,
considering both natural uncertainty and uncertainty in rainfall
parameter estimates. Bayes risk, indicating loss due to uncertainty,
was evaluated for each design.
Wide use of the Monte Carlo method in hydrology and water
resource planning is demonstrated by the many examples given above.
Close examination of the method, however, reveals some disadvantages.
One disadvantage is that a large number of essentially identical
samples must be generated in order to define a histogram of system
output. This could be expensive in terms of computer time for problems
where many histograms are needed, such as decision problems where
parameters of a complicated deterministic model are treated as decision
variables to be optimized. Another disadvantage is that tails of output
probability distributions are usually not well defined by histograms
generated by Monte Carlo simulation, yet for many planning and design
problems in hydrology the distribution tails are extremely important.
A third disadvantage is that the Monte Carlo method introduces an
additional source of uncertainty into decision problems due to sampling
error.
Limitations of Present Methods
From this review of present methods for coupling stochastic and
deterministic models, it appears that each has some inherent limitation
or disadvantage. The first order approach is applicable only when mean
and variance estimates are sufficient for a decision problem and there
are no severe discontinuities involved in the hydrologic system of
30
concern. The analytic method can only be used with problems for which
the deterministic system transform and distributions of input random
variables are highly simplified. Solution by enumeration of all
possibilities can become unwieldy for complex problems. Monte Carlo
simulation requires generation of a large number of samples and
introduces an additional source of uncertainty into decision problems.
In consideration of these limitations, present decision
procedures usually involve selection of the method for which
limitations and disadvantages are least important for the particular
problem at hand. In many cases, however, limitations are still
significant. For example, when the Monte Carlo method is selected
because of the need to use a complicated deterministic model, the
sampling error problems and cost of generating a large number of
samples do not vanish. Therefore, it appears that the need exists
for development of an additional method for obtaining transformed
probability distributions for quantities important in hydrologyrelated decisions. The next chapter will explore a possible approach
for such a development.
CHAPTER 3
PROBABILITY GENERATING FUNCTIONS AND
PREVIOUS APPLICATIONS
In view of the limitations and disadvantages of methods for
coupling stochastic and deterministic models discussed in the previous
chapter, it was decided to explore the possibility of using probability generating functions in the solution of the coupling problem.
According to Feller (1968, p. 264): "In the theory of probability,
generating functions have been used since DeMoivre and Laplace, but the
power and the possibilities of the method are rarely fully utilized."
The definition of a generating function is given by Feller
(1968, p. 264) as follows: "Let a
a l' a 2' .. be a sequence of
real numbers. If A(S) = a + a S + a S 2 + ... converges in some
o
2
l
interval -S < S < S
then A(S) is called the generating function
o
o'
o'
of the sequence {a i }." In this definition, S is a dummy variable;
thus its value is arbitrary. If we assume 'SI < 1, a comparison with
the geometric series shows that A(S) converges if the sequence {a i } is
bounded. If the sequence { a.} represents a probability mass function
of a discrete random variable, then it is bounded and A(S) is a
probability generating function.
The method of probability generating functions is a special
case of the method of characteristic functions for defining the
probability law of a random variable. The method of characteristic
31
32
functions is discussed by Parzen (1960, pp. 394-399) and Feller (1971,
pp 498-526). The characteristic function of a random variable X is
defined as the expectation of the complex function
e iuX
with respect
to the probability law of X. It is thus a function of the real
variable u and can be shown to be the Fourier-Stieltjes transform of
the probability distribution of X. If the probability density function
of X exists, the characteristic function is the ordinary Fourier
transform of the probability density function of X.
To convert the characteristic function of a discrete random
variable into the generating function form defined previously, we must
make a change of variable so that
S = e iu .
With this change of variable the expectation E(e
(30)
iuX ) can be expressed
X
as E(S ). For a discrete random variable X with only positive values,
we have from the definition of expectation that
a x S X(31)
E(S A ) =
x=o
where a
x
is the probability of X for X = 0, 1, 2, ...
Now note that the right side of this equation is the form
previously given for a probability generating function.
33
The probability generating function method can be thought of as
a transform method whereby the probability mass function of a random
variable expressed as a number sequence is transformed to the generating
function domain where the probability law is expressed as a power series
in open or closed form. The incentive for making this transformation is
that in the generating function domain certain mathematical operations
can oftentimes be performed more easily than in the probability mass
function domain.
In addition to its use for defining the probability law of
discrete random variables, the generating function technique can be
applied to the solution of difference equations (Goldberg 1958, pp.
189-207). This application of the generating function technique draws
attention to the similarity of generating functions and Z transforms,
since Z transforms are also used for solving difference equations
(DeRusso, Roy, and Close 1965, pp. 175-176). A change of variable
S =
1
will convert a Z transform to the generating function form
previously described. Z transforms are derived from Laplace transforms
for functions of discrete variables (DeRusso, Roy, and Close 1965,
pp. 158-159). Thus, a link is established between generating functions
and Laplace transforms.
A review of the literature revealed that probability generating
functions have been sparsely used in the hydrology and water resources
field. A few examples of generating function applications are
described as follows.
34
Generating functions were used by Prabhu (1959) to solve the
set of difference equations derived by Moran (1954) to give the
probability mass function of storage in a finite reservoir.
Thayer and Krutchkoff (1967) derived probability generating
functions for the random variables defining organic pollution and
dissolved oxygen at points downstream from a pollution source such as
a municipal sewage treatment plant. They made assumptions that are
generally accepted regarding rates of change of pollution and dissolved
oxygen and then derived expressions for probabilities of discrete
changes in these state variables. Using the expressions obtained for
discrete-change probabilities, they derived a differential-difference
equation expressing the stochastic process of pollution and dissolvedoxygen state-change in a stream. By expanding this general equation to
a set of differential-difference equations representing all possible
initial states, multiplying each equation by appropriate powers of the
dummy variables S and R, and then adding all equations, they derived a
quasilinear partial differential equation with the probability generating
function for pollution and dissolved oxygen as dependent variable. They
then solved this differential equation for three different initial
conditions to obtain the desired generating functions. From these
generating functions for the three initial conditions, expressions for
probabilities of being in a given state of pollution and dissolved
oxygen at a given time were derived. Means and variances were also
obtained from first and second derivatives of the generating functions.
35
Kisiel, Duckstein, and Fogel (1971) used generating functions
to describe the probability law of the sum of ephemeral flow durations
from a random number of runoff events in the arid Southwest. They
computed the mean and variance of the flow duration sums from first
and second derivatives of the generating functions. This provided some
information germane to natural recharge of aquifers in arid lands, since
flow duration is related to recharge opportunity for aquifers underlying
the ephemeral streams. However, flow duration mean and variance is
only first order information and the complete probability distribution
was not provided.
Duckstein, Fogel, and Kisiel (1972) later used probability
generating functions to obtain the probability mass function of
cumulative point or areal rainfall from a random number of thunderstorms occurring during a summer season. They also suggested combining
the characteristic function of runoff, given that an event occurred,
with the generating function of the number of runoff-producing rainfall
events to obtain the generating function (characteristic function) of
seasonal water yield. However, they did not actually give the
generating function of seasonal water yield and did not derive the
corresponding probability distribution from the generating function.
Duckstein, Fogel, and Davis (1975), in developing a stochastic
model of precipitation for mountainous areas of the western United
States, derived the generating function for interarrival time of a
renewal process representing winter precipitation. They first
obtained the generating function for duration of a sequence of
36
precipitation events that occur within three days of each other. Then
they combined this generating function with the generating function for
the dry spell between two sequences to obtain the generating function
of interarrival time of the renewal process.
These examples demonstrating use of the generating function
technique, although few in number, indicate that further application
of generating functions to the hydrology field should be beneficial.
The next chapter explores the possibility of further application by
developing a general technique to be used in coupling stochastic
and deterministic hydrologic models to obtain transformed probability
distributions for decision-making in planning and evaluation.
CHAPTER
4
DERIVATION OF COUPLING TECHNIQUE USING
PROBABILITY GENERATING FUNCTIONS
As pointed out in the introductory chapter there are many
decision-making problems in hydrology in which we are concerned with
cumulative output from a hydrologic system over some period of time
in which there are a random number of random inputs to the system.
Inputs may be randomly distributed in both space and time and quite
often must be described by more than one random variable such as
depth and duration for rainfall input. For problems of this type we
are interested in the sum of a random number of random variables, each
random variable being the output from some random input. The probability distribution for such a sum is difficult to obtain except for
highly simplified conditions. This is because both convolution and
randomization with integration or summation is required. Transformation
to the generating function domain should help to simplify this type
problem since convolution in the probability mass function domain
becomes multiplication in the generating function domain. Using probability generating functions, we will proceed to derive a general
technique for system transformation problems in which a random number
of random variables is involved.
37
38
Factoring Input Probability Generating Function
To provide a basis for the derivation, we wish to first
demonstrate how the input probability generating function might be
factored. We will assume for the demonstration that the input can be
modeled by a Poisson process, which as the next chapter will show is a
reasonable assumption for many situations. The generating function for
the number of Poisson events occurring in a given time period is presented by Feller (1968, p. 268). We express this generating function as
A(S) = exp(-(3 + 13S)
(32)
where 13 is the expected number of events in the given time period.
Now suppose that the probability mass function of an input
random variable such as duration of rainfall given that a rain event
occurs, is the sequence {f i } with i = 1,2,3, ... and its generating
function is F(S). Then the cumulative duration of rainfall input would
be a compound Poisson random variable (Feller 1968, pp. 288-293). Its
generating function would be
B(S) = exp[-13 + (3F(S)]. If we consider the definition that F(S) =
(33)
f4Si, we obtain a
i=1
substitution for F(S) so we can write that
B(S) = exp[
+
cc
l' f i S i ]
i=1
(34)
39
03
And since 1 f 4 = 1, we can write that
i=1
B(S) = expH
f. +
i=1
1
f.S1).
i=1
(35)
1
Then rearranging, we have that
B(S) = exp[
i=1
+ r3f i S 1 d
(36)
or
-=
i=1
exP(-13f.1
f3f 1
(37)
Note that the first factor in this product is the generating function
for an ordinary Poisson random variable with expectation 13f 1 , the
second factor is the generating function for two times a Poisson random
variable with expectation (3f 1 , the third factor is the generating
function for three times a Poisson random variable with expectation 13f 3 ,
etc. Thus, the ith factor of this product can be interpreted as
generating function for i times the number of rainfall events of
duration i, and the expected value of the number of rainfall events of
duration i is (3f i .
Now if we assume that the generating function for the amount of
rainfall during an event of duration i is G i (S), we can write the
generating function for the total rainfall for the given time period
attributed to events of duration i as
40
D i (S) = exp[-af i + af i G i (S)].
(38)
This generating function can also be written as
D i (S) = exp[-af. +
af.
g..S1
ij
(39)
and finally by reasoning similar to the preceding, we have that
CO
D.(S) =
7 exp(-af.g.. + af i g ii Si).
j=1
1 1J
(40)
And we note that the jth factor in this product is the generating
function for j times the number of rains of duration i and amount j.
The expected value for the number of rains of duration i and amount j
is af i g
which is the expected number of rains during the given period
times the probability of rainfall duration i given that an event occurs
times the probability of rainfall amount j given that an event of
duration i occurs.
It can be seen from these developments that we can continue to
rearrange compound Poisson generating functions until we obtain the
generating functions for the number of rains of certain specified
characteristics. Of course, what we are actually doing is taking
advantage of the fact that the Poisson and compound Poisson distributions
are infinite convolutions of Poisson and compound Poisson distributions.
41
We make use of this property to classify rainfall events according to
their characteristics and obtain the generating functions for the number
of events in each of these classes.
Deriving_ Output Probability Generating Function
Using as a basis the ability to factor the input probability
generating function and thus classify inputs according to their characteristics, we now proceed to derive the generating functions for
cumulative output during a given time period for each input class and
then combine these to obtain the generating function for cumulative
output from the total of all input classes. First, we will consider
situations where we can assume that a hydrologic system is completely
deterministic. These are situations where all uncertainty is removed
by specifying the input, and where the output for the specified input
is some deterministic value. The total output from all events with the
specified characteristics is this deterministic value times the number
of events in that class. The number of events in the class is a random
variable and its generating function can be obtained by factoring as
demonstrated in the previous section.
Now suppose the generating function for the number of inputs
with certain specified characteristics i is given by
H i (S) = exp(-Is i + (3 i S)
(41)
42
and the output due to an input event with these specified characteristics
is the deterministic value m i . Then the generating function for the
total output due to all events with specified characteristics i would be
P i (S) = exp(-13 i +
(42)
The total output due to all classes of input would be the sum of the
outputs from each class, and the generating function for the total
output would be the product of the generating functions for the
individual classes. We will designate this generating function as
P(S). It can be expressed in terms of the generating functions for the
individual classes as
CO
P(S) = ii P.(S)
i=l 1
(43)
or as
CO
P(S) = 7 exp(-13. + f3.S m i).
1
1
i=l
(44)
In developing the generating function P(S) we assumed that the
output from an event with specific input characteristics was completely
deterministic. However, this may not always be the case. There may be
uncertainty in the output due to causes of variation other than input.
One of these causes of variation could be the initial state of the
system. Of course, the initial state could be specified as input,
43
thereby eliminating this source of variation. However, it is conceivable that it may be necessary or desirable in many cases to leave
one source of variation, which may be the initial state, unspecified
input, or some other factor. For this situation we will make use of
the fact that the compound Poisson distribution is' an infinite convolution of compound Poisson distributions.
If we assume as we did previously that the input to the
hydrologic system can be represented by a Poisson process and the
output given that an input event occurs is a random variable, then
the total output from all input events during a given period is a
compound Poisson random variable. We can also represent this total
output for the given time period as the sum of outputs from all classes
of input, where the output from each of these classes is a compound
Poisson random variable. The generating function for hydrologic system
output due to all events in a class with input characteristics i could
be expressed as
Ui(S) = exp[- i + t3 i Q i (S)]
where
ai
(45)
is the expected number of input events with characteristics i
and Q(S) is the generating function for the hydrologic system output
given that an event with characteristics i occurs. The generating
function for the total output from all input classes would be the
product of the generating functions for all classes. Thus, the
44
generating function for total system output for the situation where
there is additional uncertainty in the output due to initial state,
or from some other cause, could be expressed as
CO
U(S) = 7 U.(S)
i= .1 1
(46)
or as
CO
U(S) = 7
i=1
exp[-13. +
1
i( S)].
(47)
Obtaining Output Probability Mass Function
In order to implement the proposed technique for decision-making
concerned with cumulative output from a hydrologic system for some
planning period, we must determine the inverse transforms for the previously derived generating functions. These inverse transforms are the
probability mass functions needed for risk evaluation of alternate
decisions.
Theoretically, the inverse transform can be determined by two
principal methods. One involves expanding the generating function in a
power series, the coefficients of which are the desired probabilities.
The other makes use of the inversion integral. However, for most cases,
both of these methods essentially reduce to successive differentiation
to obtain the inverse transform and thus the desired probability mass
function.
45
More explicitly, to get the probability mass function fv k l for
k = 0, 1, 2, 3 ..., which has generating function V(S), we make use of
the relation
v k
where V
(°) (S)
V(k)(S)
k!
(48)
S=0
is V(S) and V(S) indicates the kth derivative of V(S)
with respect to S for k > 0.
We now consider the generating function for cumulative output
from a completely deterministic hydrologic system with random input for
some given planning period. We previously derived this generating
function as equation 44. It is repeated here as equation 49 for
convenience.
CO
m
P(S) = 7 exp(-13. + f3 1 S i)
i=1
where i is some combination of input characteristics,
(49)
is expected
number of input events with characteristics i, and m i is system output
for input with characteristics i. By rearranging the equation, we can
write that
.0
P(S) = expp
m
(-13. + f3 i S i)]
(5 0)
46
or
P(S) = exp[-
(3. +
i=1 1i=1
(51)
And with further rearrangement, we have that
P(S) = exp(-(3) • ex[t3.S m i)
i=1 1
(52)
where I3 is the expected number of input events during the given planning
period.
Now if we group all terms that contain S to the same jth power
and let ai be the sum of the coefficients of the terms containing S i ,
then we have that
P(S) = exp(-13)
exp[
a.S i i
(53)
j=0
where a. is defined by the equation
a
j =
i
all i
such that
output
is j
(54)
CO
Next we designate the function a.S as W(S) and the constant
j=0
exp(-) as C. We can then write that
47
P(S) = C • exp[W(S)].
(55)
To derive the probability mass function
differentiate
P(S)
which we designate
{p id
we must successively
with respect to S. The first derivative of
P (1)
is expressed as
p(1)= w(l) • C • exp(W)
where
W (1)
tuting
P
is the first derivative of
for
C • exp(W)
P(S),
W(S)
(56)
with respect to S. Substi-
we have that
p(1)
The second derivative of
P(S)
w(1) p .
( 5 7)
is then
p(2) w (1) p (l)
w (2)
p
.
(58)
Differentiating again we have the third derivative as
p (3)
w (1) p (2)
w (2) p (1)
w (2) p(1)
P
(59)
or as
p (3)
w (1) p (2)
2w (2) p (1)
w (3)
p
.
( 6 0)
▪
48
The fourth derivative is
p(4)
w (1) p (3) w (2) p (2)
• w (3)
p(1)
2w (2) p (2)
2w (3) p (1)
p
(61)
or
p (4)
w (1) p (3) 4. 314 (2) p (2)
314 (3) p (1)
w(4) p
.
(62)
Observing the expressions obtained for these derivatives, we note that
for k = 1, 2, 3, and 4
(k..1)! w(j) p(k - j)
p (k)
(j-1)!(k-j)!
i=1
(6 3)
where P (0) is defined as P.
If we multiply each term on the right side of equation 63 by
J
[11 we get
k
j
p(k) _
-
J=1 k
w(j) p (k-j)
• j! (k-j)!
or
(k) w (j) p (k-j)
(65)
p (k)
j=1
k
which is a series somewhat analogous to the binomial series obtained by
expanding (P+W) k . It differs from the binomial series in that the term
49
for j = 0 is missing and the terms j = 1, 2, 3, ..., k contain the
factor {1)
k
Also W
P
(k
-i) are jth and (k-j)th derivatives of
W and P, respectively, rather than W and P raised to the jth and
(k-j)th powers.
Now if we assume that equation 63 is true for k = n, where n is
any integer, we can obtain P (n+1) by differentiation as follows:
w
(11..1)! (j) p(n-j+1)
(j-1)! (n-j)!
j=1
p (n+1)
n
(n-j)
(n-1)! W (j+1) P
L
j=1
(66)
(j-1)! (n-j)!
Then letting i = j+1, we have that
( n ..1)1
17'1
p(n+1)
=
L
w (j) p (n-j+1)
kJ-I)!
n+1
i=2
(n-j)!
(i) (n-i+1)
(n...1)! w p
(67)
(i-2)! (n-i+1)!
Grouping terms containing derivatives of the same order gives
the equation
p (n+1)
(n-l)! + (n-1)1
0.1..j)! (j-2)! (n:j+1)
j=2 [ (j-1)!
w (1) p (n)
w (n+1) p (0) .
(n
W(i) P
-j+1)
(68)
50
Obtaining common denominators and adding fractions in the coefficients
results in the equation
p (n+1)
(j-2)! (n-j+1)! +(n-1)! (1-1)1 (n-j)!) w (j) p (n-j+1)
(j-1)! (n-j)! (j-2)! (n-j+1)!
j=2
w (1) p (n)
(n+1) p (0) .
(69)
Then rearranging we have that
p (n+1)
(n-1)! (j-2)! (n-j)! Un-j+1) + (j-l)]
(j-1)! (n-j)! (j-2)! (n-j+1)!
+W
(1 )
P
(n)
+ W
(n+1)
P
(0)
w (j) p (n-j+1)
(70)
.
With further rearranging we have that
p (n+1)
p (n-j+1)w(1)
! w(j)(n_j+.1)!
n n(j-1)!
p (n)
w (n+1) p (0)
(71)
and finally we have that
P
(n+l-j)
(n+1)n+1 (n+1-1)! w (j) p
•
(j-1)! (n+1-j)!
j=1
(72)
Thus we see that if equation 63 is true for k = n, it is true for
k = n+1. Since we have shown that it is true for k = 1, 2, 3, and 4,
we have proved by induction that it is true when k is any integer.
51
Now we are ready to obtain an expression for
p1<
k = 1, 2,
3, ... . We begin with the relation given by equation 48 and restate
it in terms of p k and P (k) (S) as follows:
p(k)(s)
Pk
(73)
S=0
k!
for k = 0, 1, 2, ... . We then substitute the expression given by
equation 63 for P
(k)
(S) and obtain the relation
(k
Pk
-
(k-1)! 14 (i) P - j )
31 k! Tj71)! (k-J)!
=
(74)
which reduces to
k
w(j) p (k-j)
(75)
Pk = j=1 k(j-1)!(k-j)! S=0
for k
1, 2, 3 ... .
Then from equation 73 we get the relation
p(k )(s )
= k! p
S=0
(76)
k
for k = 0, 1, 2, ... .
And from the definition of W(S) on page 46 we have that
k)
(77)
a.S J
S.0
j=0
S.0
52
which when evaluated becomes
= k! a
S=0
(7 8)
k
for k = 0, 1, 2, ... .
Substituting the expressions given by equations 76 and 78 into equation
75 provides the relation
k
(j! a i )[(k-j)! p (k _ j) ]
Pk = 3.Z
= 1
(79)
k (j-1)! (k-j)!
for k = 1, 2, 3, ... .
Then simplifying we get that
Pk =
1
„
-j r
=
j1
(80)
(k - j)
for k = 1, 2, 3, ... .
To obtain an expression for evaluating p o , we begin with
equation 73. It is restated for p o as
p(0 )(s)
Po
-
0!
(8 1)
S=0
Substituting the expression provided by equation 55 for
P(0)(),
and
evaluating 0! as 1 we have that
p o = C
(82)
exp[W(S)]
S =0
53
Then using the relation given by equation 78, we obtain the relation
p o = C • exp(a 0 )
(83)
p o = exp(-e, + a o )
(84)
which can be expressed as
since C was defined as exp(-e).
Equations 80 and 84 when taken together provide a recursive type
procedure for obtaining the probability mass function of cumulative
output from a completely deterministic hydrologic system for a planning
period in which there are a random number of random inputs to the system.
Parameters that must be evaluated in order to use these equations are e
and the coefficients a j for j = 0, 1, 2, ... . The parameter e is the
expected number of input events for the planning period and can usually
be estimated directly from data. To obtain the a j coefficients a
probability distribution for input characteristics, given that an input
event occurs, must be estimated. From this distribution, probabilities
are computed for discrete classes of input characteristics. These
probabilities are then multiplied by e to give expected number of
events f3. in each class i, for i = 1, 2, 3, ... . The a coefficients
are obtained by summing all Ws that give system output j. (See
definition of a j by equation 54.)
54
Now that we have derived a recursive type procedure for
obtaining the probability mass function of cumulative output for a
specified period from a completely deterministic hydrologic system with
random input, we wish to consider a similar approach for systems where
there is uncertainty in the output for a given input.
The generating function for cumulative system output for a
specified planning period with a random number of inputs when there is
uncertainty in the output for given input was derived previously as
equation 47. It is repeated here as equation 85:
CO
U(S) =r exp[-. +is,Q i( S)]
i-1
(85)
1
where i is some combination of input characteristics, 13 i is expected
number of input events with characteristics i and Q 1 (S) is the
generating function for output from an input event with characteristics
i. We can also write the equation as
U(S) = exp
i=11
t3-Q.(s)]}
1 1
(86)
or as
13.Q.(S)]
U(S) = exp[ .+
L=11i=1 "
(87)
55
And with further rearrangement it can be written as
U(S) = exp(-13) • exp[I 13 i Q i (S)]
(88)
where f3 is the expected number of input events.
From the definition of a generating function we have that
Q(S) =
q..S 3
j=0 •
(89)
where q ij is the probability of output j from input with characteristics
1.
We can now express the generating function U(S) as follows:
U(S) = exp(-13) • exp
E s. E
q..S i
i=1 1 j=0 1J
(90)
or
.
co
U(S) = exp(-) • exp
i=1 j=0
i
q
(91)
And since all terms of the double infinite series in equation 91 are
positive and the series converges we can express U(S) as
.
U(S) = exp(-I3) • exp
co
q..Si
ij
(92)
56
Then if we let
CO
b. =
J
1
i = 1
is.q..
1 IJ
(93)
for j = 0, 1, 2, ... ,
we can write that
U(S) = exp(-13) - exp
'.
1 b. S i
j=0 i
(9 4)
Designating 1 b.S i as X(S) and exp(-) as C we have that
j=0
J
U(S) = C • exp[X(S)].
(95)
To obtain the probability mass function fu k } for k = 0, 1, 2, ...
we must make use of the relation between the generating function U(S)
and the probability mass function fu k l. This relation was expressed in
general form by equation 48, and was used previously for systems with
completely deterministic transforms. The relation is given for {u k }
by the following equation:
U (k) (S)
k!
S=0
(96)
where U(S) is the kth derivative of U(S) with respect to S and
U (°) (S) is U(S).
To evaluate the right side of this equation we must successively
differentiate U(S) with respect to S. For the differentiation we can
57
make use of a relation similar to that given by equation 63, which was
previously derived for completely deterministic systems. The use of
this relation is justified since the expression for U(S) in equation 95
is of the same form as the expression for P(S) in equation 55. The
relation that provides for successive differentiation of U(S) is given
by the equation
- j)
u (k)(k-1)! X ( j ) u(k
(J-1)!
j.1
(97)
where U (k) is the kth derivative of U(S) for k = 1, 2, 3, ... and U (°)
is U(S).
If we substitute the expression for
(k)
given by equation 97
into equation 96 we get that
(k-1)!
x
(
u k = .3=1
L(k-j)!
•
for u k = 1 2 1 3f
which reduces to
."
k
uk =
for u
(98)
S =0
lk-')
r X
Li' 3 '
j .1 k (J - 1)!(k - j)! S=0
(99)
k = 1, 2, 3, ... .
Then from equation 96 we get the relation
= k!u
S=0
for k = 0, 1, 2, ... .
k
(100 )
58
And from the definition of X(S) on page 56 we have that
x m (s )
=
(k)
b
S=0
j=0
(101)
S=0
which reduces to
X (k) (S)
= k!b
S=0
k
(102)
for k = 0, 1, 2.....
Then substituting the expression given by equations 100 and 102
into equation 99 we obtain the relation
k (j!b.) Uk-j)!u (k _ j) ]
uk
=
j=1
k (j-1)! (k-j)!
(103)
for k = 1, 2, 3, ... , which simplifies to
u k =
1
1
jb
(104)
" j1
=
for k = 1, 2, 3, ... .
To obtain an expression for u o we use equation 96 for k = 0.
This gives us that
u0 = U( S)
(105)
S=0
.
59
Then substituting for U(S) the expression provided by equation 95 we
have that
u0 = C • exp[X(S)]
(106)
S=0
.
And with the relation given by equation 102 this becomes
u o = C • exp(b 0 ).
(107)
Then considering that C was defined as exp(-13) we can express u o as
u
0
(108)
= exp(- + b )
o
Equations 104 and 108 now provide a recursive type procedure
for obtaining the probability mass function of cumulative output for a
specified planning period from a hydrologic system where there is
uncertainty in the output for given input, and where there are a random
number of random inputs to the system. Note that these equations are
similar to equations 80 and 84, which were derived for systems with
completely deterministic transforms of input to output. The parameter
f3, which is the expected number of input events for the planning period,
is the same for both procedures. And as previously stated, 13 can
usually be estimated directly from data.
The coefficients b i for j = 0, 1, 2, ... are given by equation
93. To evaluate this equation we must have values for
where
60
i
1,2,3, ... . These are the expectations for numbers of input
events with characteristics i. Estimates for
are obtained by the
same method used with the completely deterministic system. That is,
probabilities for discrete classes of the input characteristics are
first extracted from a probability distribution of input characteristics, given that an input event occurs, and then multiplied by
(3 to obtain (3 i for i = 1,2,3, ... .
We also need values for q ii in order to evaluate equation 93.
This is the probability of output j from input with characteristics i.
These probabilities may be extracted from the probability distribution
of a random component incorporated in the otherwise deterministic
model or they may be obtained from a probability distribution for
initial state of the hydrologic system. If the output j from input
with specified characteristics i is essentially determined by initial
state, then we can get q ij by summing probabilities for all initial
states that give output j when input to the system has characteristics
i for i = 1,2,3, ... and j = 0,1,2, ... .
Simple Demonstration Example
We will now demonstrate hydrologic application of the derived
recursive type procedure and compare results with those obtained by
an exact analytic method. For this demonstration a relatively simple
deterministic transform model will have to be assumed in order to keep
the analytic solution tractable. The deterministic transform model
will not incorporate uncertainty in output for given input. However,
61
since the basic approach is similar for both a completely deterministic transform and a transform with uncertainty in output for a given
input, the results should provide a valid demonstration of how the
basic derived recursive procedure compares with an exact analytic
method.
The demonstration example will be concerned with cumulative
surface runoff from an agricultural watershed during a cropping season.
This is a realistic concern since some nutrients and pesticides used
for crop production are largely transported by surface runoff
(Frere
1976, Caro 1976). The cumulative surface runoff during a season
should give an indication of the quantity of nutrients and pesticides
that accumulates downstream from the watershed in a lake or estuary.
For the example, we will assume that the occurrence of rainfall
events that cause surface runoff can be modeled as a Poisson process
and that the amount of rain in such an event is exponentially
distributed. We will also assume that surface runoff volume is
directly proportional to rainfall volume for these events.
In deriving the analytic solution for the probability
distribution of cumulative surface runoff for a cropping season, we
first obtain the probability density function of surface runoff
volume for a runoff event given that an event occurs. This is
expressed by the equation
dx
f Y( Y ) = fx [g1(y)3
(109)
62
where f (y) is the probability density function of surface runoff
Y
volume y given that an event occurs, f x (.) is the probability density
function of event rainfall volume x, and
g -1
(y) is the inverse of g(x)
where g(x) is given by the linear relation y = Kx.
Substituting into equation 109 we have that
•
exp [
-
OE
[f] 1
.
(
110)
where a is the reciprocal of the expected rainfall volume per event
and K is the fraction of event rainfall that becomes surface runoff.
If we rearrange this equation we have that
f (y) = [
Y
ij •
exp[4y] .
Now we see that f (y) is the probability density function of
Y
an exponential random variable with expectation
• Thus the probability density function of the sum r of a given number n of runoff
events would be gamma. This density function is shown as follows:
n-1
Td •
f r (r)
for r > 0 and n > 0.
exp
r •
F(n)
[{]r1
a
(112)
63
To obtain the probability density function of cumulative
surface runoff volume
Z for a cropping season, we must randomize n
since the number of runoff events in a season is a random
variable.
This randomization is shown by the equation
n-1
expRild •
f z (z)
for
Esn • exp(-13)
n!
[-c-Lz]
= n=1
(113)
z > 0, where f3 is the expected number of runoff events per season
and the other variables are as previously defined.
Rearranging the above equation and substituting
m+1 for n, we
have that
f(z) = [i) • exp[-mz_]
(114)
•
m=0
(m+1)! into (m+l)m!, and substituting r(m+2) for (m+l)r(m+l)
Factoring
gives the equation
m+1
f z (z) = ( 21 • exp
z-d
•
m=0
Then multiplying the right side of equation
a z
K
2
a
(
-
z
1/2
) provides the equation
-
m!r m+2)
•
115 by the factor
(
115
)
64
la
f z ( z ) =
K f3
2 (ak] • ex p [it] z_]
P
•
y
L[aKl
17144,
2
m+½2
(116)
m!r(m+2) m=0
Rearranging we have that
(117)
And rearranging further we get that
—2m+1
1 1 exp[-Wza
K.
d
[ a
la
2
;3
1
L ml p(m+2)
m ="
0 -
.
2
(118)
Now note that the infinite sum in the above equation is the modified
Bessel function of order
1
with argument
[e‘7—z.
2 •
to express the probability density function of
(a
f(z) =
where I
1 (.)
11ï • expEwz_d
a
•
z
[
for
a1
]
f(z)
z > 0
as
(119
z
is the modified Bessel function of order
The density function
values of
2
z
This a ll
11 ows us
1.
can now be evaluated for selected
with given values for a,
K,
and
Modified Bessel func-
tion values can be obtained from tables published by the National
)
65
Bureau of Standards (Abramowitz and Stegun 1964, pp. 416-429) or with
a computer subroutine (IBM 1970, p. 366).
The probability distribution of Z is a mixed distribution.
The continuous part is the probability density function f z (z), which
is given by equation 119 for z > O. In addition, there is a finite
probability that Z = O. This is the probability that no runoff events
occur during the cropping season. The number of runoff events is
Poisson distributed; therefore, the probability that Z = 0 can be
evaluated by the equation
P r (Z = 0) = exp(-f3).
(120)
The probability distribution of cumulative surface runoff for
this example can be obtained with the recursive technique that was
derived through the use of generating functions by evaluating equations
80 and 84. The coefficients a i for j = 1,2,3, ... in equation 80 are
evaluated by discretizing the exponential probability density function
for event rainfall amount into classes and then obtaining the probability for each class through integration of the exponential density
function between class limits. The value for a i is the probability
obtained for class j multiplied by is. The value for a o in equation 84
is zero since there is no input class that gives zero surface runoff.
Operations for computing the probability distribution of
cumulative surface runoff for a cropping season were programmed in
Fortran for both the analytic method using equations 119 and 120 and
the recursive procedure using equations 80 and 84. Program listings
are included in Appendix A, along with specification of required inputs.
66
For the demonstration example, estimates of a, K, and 13 were
obtained from data collected on Ahoskie Creek watershed which is
located in the northeast section of North Carolina. These estimates
are as follows: a
0.806; K = 0.37; and = 3.0.
Using the computerized procedures and data from Ahoskie Creek,
discrete values of the probability density function (POE) of cumulative
runoff for a cropping season were obtained with the analytic method,
and a probability mass function, representing discrete runoff class
probabilities, was computed by the derived recursive technique. The
probability mass function computed with the recursive technique was
converted to a PDF in graphical form by computing average probability
density for each runoff class and plotting it at class midpoint. A POE
was also plotted from the values obtained with the analytic method.
Figure 1 shows a comparison of the PDF's of cumulative surface
runoff for a cropping season computed by both the analytic method and
the derived recursive technique. As one can observe, both methods
appear to give similar results.
This demonstration example with a simple deterministic transform
model provides a test of the derived recursive technique since results
obtained with the recursive technique are compared to those obtained
independently by an exact analytic procedure. The comparison of results,
which is quite favorable, gives confidence in accuracy of the recursive
technique and thus supports use of the technique with more complicated
hydrologic system models.
67
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CHAPTER 5
REASONABLENESS OF POISSON ASSUMPTION FOR APPLICATION
OF DERIVED TECHNIQUE
In the derivation of the recursive technique from generating
functions in the previous chapter, we assumed that stochastic occurrence
of input events could be modeled by a Poisson process. We now propose
to examine the reasonableness of this assumption for application of the
recursive technique to realistic hydrology-related problems. In this
examination, we will limit consideration of inputs to rainfall since
this is the input of concern for the hydrology-related problems
discussed in the introductory chapter. For problems in which other
types of input are considered it will be necessary to examine those
inputs in a similar manner to determine if the Poisson assumption is
appropriate.
Definition of Stochastic Rainfall Process
Before proceeding with the examination we wish to first define
the rainfall process in terms of a stochastic process. The most basic
descriptor of the rainfall process, without considering complex atmospheric dynamics, is the instantaneous rate of rainfall (rainfall
intensity) at points in space. Since instantaneous rainfall intensity
is a random function of time, it can be considered as a continuous
parameter stochastic process defined on the continuous time set. A
typical sample function of rainfall intensity defined on continuous
time for a point in space is shown in Figure 2.
68
69
TIME
Figure 2. Typical Sample Function of Rainfall
Intensity for a Point in Space.
70
From the rainfall intensity process we can define a rainfall
event for a given point in space as an occurrence in which the rainfall
intensity at that point is greater than zero for one or more rain
periods not separated by a period of zero intensity of duration equal
to or greater than some specified amount. For example, if the specified
period of zero intensity for event separation is six hours, the rainfall
occurrence shown in Figure 2 beginning at 7:00 A.M. and ending at
3:30 P.M. would be defined as an event; another event would begin at
10:30 P.M. since there is at least six hours of zero intensity prior
to resumption of rainfall. A rainfall event for an area in space, such
as a watershed or river basin, could be defined as an occurrence in
which rainfall intensity is greater than zero anywhere on the area and
rain does not cease everywhere on the area for a specified time period.
With either of these event definitions we can define a
stochastic process in terms of events. This stochastic process
definition would include a counting process for the number of events
occurring in a given time period and probability distributions for
event characteristics given that an event occurs. Event characteristics
might include such descriptors as total rainfall depth, duration of the
event, time distribution of rainfall, and spatial distribution of rain.
The counting process of rainfall event occurrence is of primary concern
in this chapter, however, and with this in mind we will proceed with
considering the basic requirements for Poisson modeling of the event
counting process.
71
Consideration of Basic Poisson Requirements
Requirements that the underlying physical mechanism generating
the occurrence of rainfall events must satisfy in order that the
stochastic counting process of event occurrence be homogeneous Poisson
are given by Benjamin and Cornell (1970, p. 240). These requirements
along with brief explanations l are presented as follows:
1. Stationarity. The probability of an incident in a short
interval of time t to t + h is approximately Ah, for
any t.
2. Nonmultiplicity. The probability of two or more events
in a short interval of time is negligible compared to
Ah (i.e., it is of smaller order than Ah).
3. Independence. The number of incidents in any interval
of time is independent of the number in any other
(nonoverlapping) interval of time.
Consideration of these requirements relative to their applicability to
rainfall event occurrence is given in the following paragraphs.
The stationarity requirement essentially means that the
expected rate of event occurrence must be constant throughout the
period of concern. However, for many cases in hydrology the period of
concern includes more than one season, and we know that because of
seasonal differences in weather patterns the expected rate of rainfall
event occurrence is not always constant throughout all seasons. The
probability of event occurrence is often greater in some seasons than
in others. Also in some climates and geographical locations, the
1. In the explanations of assumptions A is expected rate of
event occurrence.
72
expected rate of rainfall event occurrence varies within seasons. This
was observed by Lane and Osborn
(1972) and Smith and Schreiber (1973)
to be the case for the summer season in southeastern Arizona.
Observations such as these concerning
interseasonal and
intraseasonal variation indicate that the stationarity requirement
does not always hold for rainfall event occurrence. If we eliminate
this requirement from the Poisson requirements given by Benjamin and
Cornell
(1970, p. 240), we then have the requirements for a nonhomo-
geneous Poisson process (Parzen 1962, pp. 125-126). The assumption of
a
nonhomogeneous Poisson process for rainfall event occurrence,
however, does not pose a significant problem with the basic Poisson
assumption incorporated in the derivation of the recursive technique
from generating functions in Chapter
parameter
4. It only means that the
!3, which is the expected number of events in a specified time
period of concern, is expressed as
= f X(t) dt
(121)
instead of
= À • T
where
and
(122)
T is the time period of concern, t is time within this period,
x is expected rate of event occurrence. The maximum likelihood
estimator of
f x(t)dt is simply the average number of events observed
in time period T. This is also the maximum likelihood estimator of
X
T.
73
The nonmultiplicity requirement for the Poisson process is of
no significant concern with rainfall events as defined in this chapter.
These events occupy finite time periods and are mutually exclusive.
Thus, the probability of two or more events occurring in a short
interval of time is negligible compared to the probability of
occurrence of one event.
Rather than concern for multiple occurrence of rainfall events
in a short time interval, the concern, with the event definition in
this chapter, is that an event occurring in one time interval may
overlap into subsequent time intervals, thereby reducing the probability for independent event occurrence in those time intervals. This
calls into question the assumption of independence of event occurrence
required by the Poisson model. It is noted, however, that in practice
the effect of finite event duration on the independence assumption is
considered to be negligible if the ratio of average event duration to
average time between events is "small" (Eagleson 1978b). Gupta and
Duckstein (1975) suggest that this effect does not pose difficulties
if expected event duration is about 5 percent or less of expected time
between events. This would usually be the case for rainfall input
that is significant enough to produce output from many hydrologic
systems with which we are concerned.
Persistence in event occurrence is also of concern in
connection with assumptions of event independence. Persistence implies
lack of independence, and it has long been recognized that there is a
tendency for persistence in weather patterns. Meteorological conditions
74
conducive to the occurrence of rainfall often prevail in an area for
several days. This phenomenon is manifested in a greater frequency of
rain occurrence on days following rainy days than on days following
non-rainy days. Several investigators have modeled this daily
persistence phenomenon with a discrete two-state Markov chain, where
the probability of rain on a given day depends upon whether the previous
day was wet or dry (Gabrial and Neuman 1962, Caskey 1963, Weiss 1964,
Feyerherm and Bark 1965, Smith and Schreiber 1973).
The actual meteorological process, of course, occurs in
continuous time and Green (1964) considered the continuous time
sequence of dry and wet spells to be an alternating renewal process,
with exponential density functions for lengths of dry and wet spells.
It should be noted, however, that the alternating renewal process
where the two probability distributions are exponential is equivalent
to the continuous-time two-state Markov chain where the time-totransition random variables are independent exponential variates
(Crovelli 1971, p. 50).
When dealing with discrete time intervals, such as days, the
time-to-transition random variables, which are the lengths of wet- and
dry-day sequences, were considered by Hershfield (1970) to be geometric
random variables. This is consistent with Green's assumption (Green
1964) since the geometric distribution is the discrete analog of the
exponential distribution. Hershfield (1970) also pointed out that a
simple Markov chain model would result in geometric distributions for
75
lengths of wet- and dry-day sequences. This is in line with Crovelli's
recognition (Crovelli 1971, P. 50) for the continuous process.
Gabrial and Neuman (1962, p. 91), in presenting the Markov chain
model for representing daily rainfall occurrence at Tel Aviv, stated
that the model "is not suggested in terms of a physical explanation of
rainfall occurrence but merely as a statistical description of the
observations." Caskey (1963, p. 299) reemphasized this with the
statement that "the success of the Markov chain model in producing
theoretical probabilities that agree closely with observed probabilities should not be interpreted as meaning that these distributions
have been 'explained'."
Rather than base modeling of the persistence phenomenon upon
observations of actual rainfall occurrence and related statistics, it
would be more basic to visualize the expected rate of event occurrence
as changing when different meteorological conditions move into an area,
whether an event occurs or not. A concept somewhat analogous to this
was suggested by Le Cam (1960, p. 168) in which he considered a "hump"
in the expected rate of occurrence of rainfall showers as corresponding
to the passage of a storm and a "trough" in the expected rate as
corresponding to a period of clear weather. This approach recognizes
the underlying cause of persistence rather than just assuming that rain
on one day tends to cause rain on the following day.
With this approach, event occurrence can be assumed independent
within periods that have meteorological conditions conducive to
rainfall. We might also then assume that event occurrence is
76
appropriately represented as a Poisson process within these rain
conducive periods. The number of events in a fixed period of concern
given that conditions are conducive to rain for a
T,
constant portion of T,
would then be Poisson distributed, and the expected number of events in
T would be the Poisson parameter
We know, however, that the portion of the fixed period
T for
which rain conducive conditions are present is not a constant, but a
random variable. Thus, the Poisson parameter
13 is also a random
variable. To obtain the actual distribution for the number of events
in
T requires that we randomize This is expressed by the equation
o
f(nIT) =
where
f
n
û
- • g(3.11 )d(3
n!
n. e-'
(123
-
)
f(n1T) is the probability mass function for the number n of
events in
T, is is the Poisson parameter, and g(f311 ) is the probability
.
density function of
If the probability distribution of
(3 is only slightly dispersed
and is concentrated fairly close to some constant value, randomization
of may not be necessary for practical application. In this case the
Poisson model would be an adequate representation of the probability
distribution for the number of events in the period of concern T. If
randomization of
f3 is necessary, it has been observed that for certain
distributions of
(3 the resulting distribution for the number of events
in
T is negative binomial (Menzefricke 1978, Cunnane 1979).
77
Falls, Williford, and Carter
(1971) in searching for theoretical
distributions for making probability inferences in regard to thunderstorm activity at Cape Kennedy, Florida, concluded that the negative
binomial distribution could be used as a provisional model for number
of thunderstorm events per day. However, with the thunderstorm event
definition that they used, which specifies occurrence of a thunderstorm
event whenever thunder is heard following a period of at least
15
minutes of no thunder, the variance relative to the mean in number of
thunderstorm events per day might be expected to be high, since a
random number of such events would be likely to occur for each major
period of thunderstorm activity. For this situation then, a negative
binomial distribution would be more appropriate than a Poisson
distribution since variance relative to the mean is greater for a
negative binomial than for a Poisson. For rainfall events defined by
separation periods of no rain that are several times longer than
15
minutes, randomness in the number of events for each major period of
rainfall activity would likely not be as great. Therefore, variance
relative to the mean in the number of rainfall events defined by longer
separation periods would be expected to be less than variance relative
to the mean in the number of thunderstorm events per day with events
as defined by Falls et al.
(1971). This suggests that with events
defined by longer separation periods, a Poisson may be more appropriate
than a negative binomial for the distribution of number of rainfall
events in a given time period.
78
Todorovic and Woolhiser (1976) derived a rather complicated
general expression for the distribution of the number of rainfall events
in a given time period. They then considered two special cases of the
process of event occurrence. The first case led to the time-dependent
or nonhomogeneous Poisson process and the second case led to a timedependent generalization of the Polya-Eggenberger distribution, which
could be further reduced to the negative binomial distribution. They
stated that oftentimes it would suffice to choose the model from one of
these two types. However, they recognized that these special cases
represent extremes and that in general the appropriate model is something intermediate between Poisson and negative binomial.
It should be pointed out though that the first special case
considered by Todorovic and Woolhiser (1976), which led to a nonhomogeneous Poisson process and thus a Poisson distribution for the
number of rainfall events in a given time period, was based on the
assumption that the expected rate of rainfall event occurrence as a
function of time could be considered independent of the number of
events for which probabilities are desired. This does not seem to be
an unreasonable assumption for rainfall events that produce significant
hydrologic system output for problems such as those discussed in the
introductory chapter. For most hydrologic systems involved in such
problems there is significant output only when the rainfall input is
above some threshold. Rainfall events that produce significant output
would, in most cases, be sufficiently rare so that expected event
occurrence rate could reasonably be considered independent of number
79
of events for probability purposes, thus leading to the nonhomogeneous
Poisson process for such events.
It appears from the foregoing discussions that the number of
rainfall events in a given time period of concern might be appropriately
modeled as either a Poisson or negative binomial random variable,
depending upon the specific data and event definition. If data on
rainfall event occurrence is such that a negative binomial model seems
to be more appropriate, the recursive technique derived in this
dissertation can be extended to apply to that situation, since the
negative binomial can be treated as a compound distribution, with a
Poisson as the compounded distribution and a gamma as the compounding
distribution (Menzefricke 1978, Cunnane 1979). This would essentially
involve randomization of f3, which is expected number of events in the
time period of concern.
To sum up considerations in this section, we note that
rainfall event occurrence may not strictly meet all requirements for
Poisson modeling of the distribution of the number of events in a
given time period; however, it appears from these limited observations
that deviations from the Poisson requirements may not be serious for
many applications in hydrology. For further examination of the
appropriateness of the Poisson model we will consider situations
where others have investigated the Poisson assumption for rainfall
event occurrence.
80
Assumption by Others for Poisson Occurrence
of Rainfall Events
The Poisson assumption for rainfall event occurrence has been
used and tested by several investigators. Examples are presented as
follows.
Todorovic and Yevjevich (1969) tested a theoretical precipitation model that was based upon the assumption of Poisson event
occurrence. For the test they used daily precipitation data from
Durango, Colorado, Fort Collins, Colorado, and Austin, Texas. Hourly
data from Ames, Iowa, were also employed. In the treatment of the
data, storms were defined by two different criteria: (1) each rainy
day or rainy hour was considered to be an individual storm event,
whether or not it was preceded or followed by a rainy day or rainy
hour, and (2) a storm was defined as an uninterrupted sequence of
rainy days or rainy hours. Years were divided into 28 intervals of
approximately 13 days each and the number of storm endings (representing
storm events) in each interval was determined. The average number of
storm events per unit of time, called the "density of storms in time,"
was computed for each 13-day interval of the year. Variance in the
"density of storms" was also computed for these intervals.
For the Poisson model of storm occurrence to be appropriate,
the mean and variance of the number of storms in each interval should
be equal; i.e., the ratio of the variance to the mean should be one.
However, because of sampling variations, this would not be exactly
true. The ratio of sample variance to sample mean would be expected
81
to fluctuate randomly about one. It was found by Todorovic and
Yevjevich (1969) that the variance-to-mean ratio fluctuated about a
value above one for storms defined by the first criterion and below one
for storms defined by the second criterion. This was attributed to loss
of information about the true number of storms when precipitation is
accumulated into daily and hourly values. With daily and hourly values,
the first definition for storms tends to overestimate the number of
storms and the second definition tends to underestimate the number of
storms. The first definition results in a greater variance relative to
the mean by breaking some of the true longer storms into several
shorter ones, while the second storm definition results in less variance
relative to the mean by combining some of the true shorter storms into
longer ones. Thus, the variance relative to the mean in the number of
true storms would be expected to be somewhere between that obtained
with these two definitions using daily and hourly data.
From these observations, Todorovic and Yevjevich (1969, p. 56)
concluded that "the number of storms in a time interval is Poisson
distributed if storms are properly defined."
In a study of convective storm rainfall on the Atterbury
Experimental Watershed near Tucson, Arizona, Fogel and Duckstein
(1969) defined an event as the occurrence of at least one storm center
over the 20-square-mile watershed and observed (Fogel and Duckstein
1969, p. 1233) that "there appears to be a trend to indicate that a
Poisson variate can adequately describe the distribution for the number
of events per year."
82
Using warm season thunderstorm rainfall data from a dense
network in the Chicago metropolitan area, Fogel,
Duckstein, and Kisiel
(1971) further tested the Poisson hypothesis. In the test they
investigated several definitions designating the occurrence of an
event. These definitions (Fogel et al.
1. Network mean greater than
1971, p. 311) are as follows:
0.05 inch.
2. Network mean greater than 0.50 inch and at least one
gage more than 1.0 inch.
3.
Similar to
(1) except 0.75 inch used.
4.
Similar to
(2) except 0.75 inch and 1.25 inches used.
5. Any gage greater than 0.50 inch and the difference
between any two more than 0.50 inch.
Employing these definitions with the data, they obtained both an
observed frequency histogram and an estimate of the Poisson parameter
for each storm definition. Then applying the
Kolmogoroff-Smirnov test
for comparing observed and theoretical distributions, Fogel et al.
(1971, p. 311) "found that the Poisson distribution could not be
rejected for any of the cases at significant levels of at least
10
percent."
Duckstein, Fogel, and Kisiel (1972) later used definition 2
(Fogel et al.
1971, p. 311) for designating event occurrence with
summer type rainfall data from New Orleans, Louisiana, and found that
the Poisson assumption for number of events per season also holds for
that area.
In an analysis concerned with the stochastic occurrence of
extreme droughts, Gupta and
Duckstein (1975, p. 225) assumed that
83
"the termination epochs of rainfall events follow a homogeneous
Poisson process." With this assumption as a basis, they derived
a probability density function for extreme droughts. The derived
density function was validated for summer rainfall data from Austin,
Texas, and Chicago, Illinois, thus implying validation of the underlying Poisson assumption.
Duckstein, Fogel, and Davis (1975) in developing a winter
precipitation model for mountainous areas of the southwestern United
States compared a Poisson model for rainfall event occurrence with a
more complicated model that employed a mixed distribution for describing
event occurrence. For both models an event was defined as a sequence of
consecutive days on which a measurable amount of rain was recorded.
Daily precipitation data for the winter season (defined as November 1
to March 31) at San Antonio, Texas, were used in the comparison. All
hypotheses were tested at the 0.95 level with a Kolmogoroff-Smirnov
test and the hypothesis of a Poisson probability mass function could
not be rejected. It was then concluded that the more complicated model
for event occurrence did not seem to possess substantially better
features than the simpler Poisson model.
With this support for the Poisson assumption of rainfall event
occurrence provided by findings of others, it seems reasonable to
assume that for many problem situations in hydrology the assumption of
a Poisson counting process for rainfall event occurrence as input to the
hydrologic system is adequate. However, before we demonstrate application of the recursive technique derived from generating functions, we
84
wish to specifically test the Poisson assumption for occurrence of
rainfall events at Ahoskie, North Carolina, since data from this
location will later be used in demonstrating application of the
recursive technique.
Test of Poisson Assumption for Rainfall Event
Occurrence at Ahoskie, North Carolina
•
The test for Poisson occurrence of rainfall events at Ahoskie,
North Carolina, actually consisted of testing for exponential distribution of time between event endings. This is an appropriate test only
for the homogeneous Poisson event counting process, but for the humid
Ahoskie area where the expected rate of event occurrence is not highly
variable with season it should be adequate for practical application
purposes.
For this test, rainfall events were defined by the criterion
set forth in the first section of this chapter; i.e., events are rain
periods separated by a specified minimum period of no rain. Several
specified periods of no rain for event definition were employed (1 hour,
2 hours, 3 hours, 4 hours, and 6 hours), and the entire Ahoskie
rainfall intensity record from May 28, 1964, to September 14, 1973, was
used with each definition. Histograms of time between event endings
were obtained for each event definition and goodness of fit of the
exponential distribution to each histogram was tested using the chisquare test.
From these goodness-of-fit tests, it was found that the
exponential distribution is quite adequate as a probability model for
85
time between event endings for the Ahoskie area with events defined as
rain periods separated by 6 or more hours of no rain. The chi-square
statistic computed for these events was 13.87 with 16 degrees of
freedom. The probability of getting a chi square statistic greater
than this through sampling error is 0.6084. Figure 3 shows the
histogram and fitted exponential probability density function of time
between event endings with events defined by a 6-hour separation period
of no rain.
From the results of these tests for exponential distribution of
time between event endings, it appears that the assumption of Poisson
occurrence of rainfall events is valid for the general vicinity of
Ahoskie, North Carolina, if events are defined as rain periods separated
by 6 or more hours of no rain. With this assurance then, we will
proceed in the next two chapters to demonstrate application of the
derived recursive technique for obtaining probability distributions of
direct runoff and sediment yield from the Ahoskie Creek watershed which
is located near Ahoskie, North Carolina.
86
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CHAPTER 6
DIRECT RUNOFF APPLICATION
Direct runoff from a watershed appears in the stream during a
storm or shortly thereafter and consists of surface runoff (or overland
flow) and in some cases shallow subsurface runoff (or interflow). It
is the immediate response of rainfall on a watershed and is therefore
subject to random variation. When planning measures for non-point
source pollution control, the components of direct runoff are important
indices of the hazard for transport of certain chemicals used in agricultural production. Thus, probability distributions quantifying the
random variation of direct runoff can be quite useful in evaluating
environmental risks associated with agricultural chemical applications
on watershed lands. In addition to this use related to agricultural
pollution risks, probability distributions of direct runoff are also
needed for optimum design of methods for capture and storage of runoff
water in surface or groundwater reservoirs. In this chapter we will
demonstrate how the recursive technique can be applied for transforming
probability distributions of rainfall and hydrologic system state through
a deterministic transform model to obtain probability distributions of
annual direct runoff from a watershed.
87
88
To provide a basis for selecting the deterministic transform
model for the direct runoff demonstration, we will first review
rainfall-to-runoff deterministic transform models that might possibly
be used with the derived recursive technique. We will then select
one of the deterministic transform models for the demonstration and
define the required concomitant stochastic rainfall and state models.
Deterministic Transform Model
Numerous models have been proposed for transforming watershed
rainfall input to direct runoff. They range from simple runoff
coefficients that express runoff as a fraction of rainfall to rather
complex computational procedures based upon explicit expressions of the
physics of saturated and unsaturated soil water movement. Examples of
models throughout this range of complexity are described and discussed
briefly in the following paragraphs.
Constant runoff coefficients, which can be multiplied by
rainfall to give runoff, for various types of drainage surfaces are
given by Linsley and Franzini (1972, p. 46). Runoff coefficients
such as these have been commonly used in the past for design of storm
drains and small water-control projects. Linsley and Franzini,
however, express the opinion that the constant coefficient approach is
best suited for urban drainage problems where the amount of impervious
area is large. They recommend that this approach be avoided for rural
or other areas where the impervious portion is only a small percentage
of the total drainage area.
89
A modification of the constant runoff coefficient approach was
used by Grayman and Eagleson (1969) in a study conducted with the
objective of evaluating streamflow record length needed for modeling
catchment dynamics. In this study they expressed the runoff coefficient
as a function of soil, vegetation, and total antecedent precipitation
for the previous 30 days. This function is given by the equation
C
1 - exp(-B 4 • AP30)
(124)
where C is the runoff coefficient, B 4 is a soil-vegetation type
parameter, and AP30 is antecedent precipitation for the previous
30 days.
Another approach that has been commonly used in the past to
estimate runoff volume, especially from large areas, involves the use
of infiltration indices (Linsley, Kohler, and Paulhus 1975, p. 273).
The q) index, which is probably the simplest of these, is defined as the
rainfall rate above which rainfall volume equals runoff volume. The W
index, which is defined as the average infiltration rate for the time
that rainfall intensity exceeds infiltration capacity, is essentially
equal to the (1) index with the average rate of retention by interception
and depression storage subtracted.
The Soil Conservation Service (SCS) of the U. S. Department of
Agriculture uses runoff curve numbers to calculate direct runoff from
storm rainfall (USDA Soil Conservation Service 1972). These curve
numbers, which vary from 0 to 100, reflect available watershed storage
90
and thus potential runoff. The
SCS equations relating storm rainfall,
watershed storage, and direct runoff are as follows:
(P - I a ) 2
(P - I a ) + S
(125)
a = 0.2 S
(126)
I
S = 1000
CN
where
10
(127)
Q is direct runoff in inches; P is storm rainfall in inches; I a
is initial abstraction consisting mainly of interception, infiltration,
and surface storage;
S is a watershed storage parameter; and CN is the
runoff curve number. Runoff curve numbers for average moisture conditions are defined for various combinations of hydrologic soil group
and watershed land cover. Hydrologic soil group is determined by
runoff potential of the soil; and watershed cover is specified by land
use, treatment, and hydrologic condition. Tables giving runoff curve
numbers for several combinations of hydrologic soil group and watershed
cover are presented in Appendix B. In addition to hydrologic soil
group and watershed cover, the
SCS runoff curve number is also related
to soil moisture. Relation of curve number to three soil moisture
classes defined in terms of antecedent
5-day rainfall is included in
abbreviated form in a subsequent section on watershed state, and in
complete form in Appendix B.
In recent years, a number of digital computer models that
simulate the various watershed hydrologic processes have been constructed. These models include procedures for obtaining components of
91
direct runoff for given rainfall inputs. The following is
a brief
review of several of the computer oriented watershed models
and their
runoff components.
One of the earliest of the computer oriented watershed models
was the Stanford Model (Linsley and Crawford 1960, Crawford
and Linsley
1966). This model is based on water-balance accounting. For this
accounting, rainfall on a watershed segment is first reduced by an
interception loss until interception storage is filled. The remaining
rainfall is then partitioned between surface runoff, interflow, and net
water input to a lower zone. Partitioning is a function of the ratio
of soil moisture to nominal water-holding capacity of the different
soil zones. Water in the lower soil zone is largely depleted by base
streamflow and deep groundwater storage. Water in interception storage
and the upper soil zone is assumed to evaporate at the potential rate.
Evapotranspiration of lower zone soil moisture is determined by the
ratio of soil moisture to soil water-holding capacity.
Since its introduction, the Stanford Watershed Model has been
modified by a number of different users so that several versions are
available (James 1965; Ligon, Law, and Higgins 1969; Claborn and Moore
1970; Shanholtz and Lillard 1971; Ricca 1974). Some of these versions
are in different computer languages. Nevertheless, the basic structure
is essentially the same for all versions.
The USDAHL-74 watershed model, developed by Holtan et al. (1975),
has some similarity to the Stanford Model in that it is also a moisture
accounting system. The spatial structure, however, is different. The
92
USDAHL-74 model incorporates the concept of dividing a watershed into
hydrologic response zones that typify the elevation sequence of upland,
hillside, and bottom land. Actual delineation of the hydrologic zones
is accomplished by grouping land capability classes that are common to
each zone. Rainfall input and water routed across a hydrologic zone
are separated into infiltrated water and surface runoff by an equation
developed by Holtan (1961) for expressing infiltration capacity. The
general form of the Holtan equation is
f = a-S n + f c
(1 28 )
where f is infiltration rate, S is available storage in surface soil
layer, f c is steady-state infiltration capacity, a is an index of cover
which can vary with season, and n is an index sometimes set to 1.4 but
defined as the ratio of plant available water to gravitational water.
In the USDAHL-74 watershed model, infiltrated water for a zone,
computed by the Holtan equation, is apportioned to interflow, downward
seepage, and evapotranspiration. Interflow and surface runoff from
zones of higher elevation are routed to lower zones and ultimately to
the stream channel.
Other watershed models have been developed that use the Holtan
infiltration equation. Three of these are the Purdue model developed
by Huggins and Monke (1970), the Arizona model developed by Nnaji,
Davis, and Fogel (1974b), and a model developed at VPI by Li, Shanholtz,
Contractor, and Carr (1977). The Purdue model is structured so that a
watershed is viewed conceptually as a grid of square elemental areas.
93
The processes of interception, infiltration, depression storage and
surface flow are characterized in the model. Interflow is neglected.
Infiltration, which determines surface flow volume, is represented by
a slight modification of the Holtan equation. This modified equation
has the form
'S c - F
(129)
Pt
where f is infiltration rate, f c is steady-state infiltration capacity,
F is cumulative infiltration, P t is total porosity of the soil above
the impeding stratum, S c is storage capacity of the soil above the
impeding stratum (total porosity minus antecedent soil moisture), and
A and m are constants.
The Arizona model (Nnaji, Davis, and Fogel 1974b) is structured
somewhat similar to the Purdue model except the watershed is divided
into elemental areas that are polygons of various sizes and shapes
rather than squares. Since the Purdue model was intended for use on
small areas, square elements are adequate for defining areas of uniform
physical characteristics. The Arizona model, however, is oriented to
larger areas, thus elemental areas are defined as polygons in order to
have a small number of elements and uniform physical characteristics
within an element. Components of the runoff phenomenon included in the
Arizona model are interception, depression storage, and infiltration.
The Purdue modified version of the Holtan equation is used to simulate
infiltration. Interflow is not included in the Arizona model since the
model is essentially oriented to semi-arid environments.
94
The watershed model developed at VPI by Li et al. (1977)
provides for subdividing the drainage area into hydrologic response
units. The response units are defined so that soils and land use
within a unit are fairly homogeneous. A soil moisture model that
incorporates the Holtan infiltration equation is used to simulate
rainfall excess on each response unit. The rainfall excess is routed
as surface water to the watershed outlet. Interflow is not considered
in the model.
A watershed model that incorporates a grid system for input of
spatially distributed watershed and rainfall information is also being
developed at Athens, Georgia, by the Southeast Watershed Research
Program of USDA's Science and Education Administration (Mills et al.
1976). This model contains a retention function proposed by Snyder
(1971) for partitioning rainfall into portions that are effective and
not effective in runoff production. Runoff from watershed grid elements
is routed by convolution with the impulse response of one or more
linear reservoirs. The reservoir time constant is related to Manning's
equation and a feedback mechanism is incorporated so that the time
constant varies with watershed storage. This provides for piecewise
linearization of the nonlinear watershed response.
Examples of watershed models that incorporate infiltration
components based upon the partial differential equations for saturated
and unsaturated water flow in soil are those developed by Dawdy,
Lichty, and Bergmann (1972); Freeze (1972a, 1972b); Engman and
Rogowski (1974); and Rovey, Woolhiser, and Smith (1977).
95
The model developed by
Dawdy et al. (1972) deals with three
components of the hydrologic cycle
-- antecedent moisture, infiltration,
and surface runoff. The antecedent moisture accounting component
simulates moisture redistribution in the soil and evapotranspiration
from the soil; it specifies initial conditions for the infiltration
component. The infiltration component determines amount of water
available for surface runoff and is based upon the Green and
approach to infiltration simulation (Green and
Ampt
Ampt 1911). With this
approach it is assumed that soil water moves as a unit with a sharp
wetting front, thus providing a simplification to the unsaturated flow
equation and making it amenable to analytic solution.
The Freeze runoff simulation model (Freeze
1972a, 1972b)
couples three-dimensional, transient, saturated-unsaturated subsurface
flow with one-dimensional, gradually varied, unsteady channel flow.
The entire subsurface regime is treated as a unified whole so that
flow in the saturated zone is integrated with flow in the unsaturated
zone. Water input to the channel can be from three possible sources:
(1) rainfall in the channel, (2) overland flow, and (3) subsurface
inflow. The subsurface flow component uses the line successive overrelaxation technique to solve the Jacob-Richards diffusion equation.
In the channel flow component the full shallow water equations are
solved numerically with the single step
Lax-Wendroff explicit
technique.
The watershed model constructed by
Engman and Rogowski (1974)
is based upon the partial contributing area concept. The area
96
contributing to runoff is considered to be a subwatershed that expands
in time and space in accord with soil infiltration capacity and storm
duration and intensity. Philip's approximate solution to the
unsaturated flow equation for infiltration (Philip 1969) is used to
calculate infiltration capacity. This equation is of the form
+ A
f -
(130)
2t
where f is infiltration rate; t is time; S is sorptivity reflecting
soil matric suction and conductivity; and A is a parameter that is
taken to be soil conductivity at the wetting front. With this
equation, rainfall excess is computed and then routed over the length
of contributing area, taking detention storage and reinfiltration
into account. The result is a surface runoff hydrograph from the
contributing area.
The Rovey-Woolhiser-Smith watershed model (Rovey et al. 1977)
incorporates a parametric infiltration component with a kinematic
surface routing component. The parametric infiltration component was
constructed from numerical experiments based upon the Richard's
equation for unsaturated soil water flow (Smith and Woolhiser 1971;
Smith 1972). The form of the parametric infiltration equation
obtained directly from the numerical experiments is
f = foo + A(t - t o ) -a(131)
97
where
f is infiltration rate; f. is the steady-state infiltration
rate;
t is time; t o is the vertical asymptote of the infiltration
decay function; and A and a are parameters unique to a soil, initial
moisture, and rainfall rate.
To obtain a normalized infiltration equation Smith
(1972)
modified the above equation by using the following dimensionless
variables:
i* = j
dimensionless
rainfall
132)
f- ,(
f* =
t*
where
dimensionless infiltration
7r-,
-_ t
(133)
(134)
dimensionless time
f, f., i, and t are as previously defined and T o is designated
as a normalizing time. The normalizing time is defined by the equation
r
o
0
,
AS-- ds = f • T o'
(135)
Rovey-Woolhiser-Smith
The normalized infiltration equation used in the
watershed model has the form
f * = 1 + (1 - a) (t * - t e ) a .
-
(136)
98
In application of the derived recursive technique for obtaining
a direct runoff probability distribution, any of the hydrologic models
discussed above could serve as the deterministic transform of rainfall
to direct runoff. For this demonstration, however, we have chosen to
use the SCS method. It is recognized that there has been criticism
of the SCS approach (Smith 1976b) and that attempts have been made to
modify it (Reich 1973; Williams 1976; Hawkins 1978). However, the
SCS method is presently being used extensively to estimate runoff from
ungaged watersheds, and is about the only widely documented procedure
for computing runoff volumes from a general soil and land cover
classification. Other methods either require calibration with streamflow data, or other input information that is not readily available.
Thus, for demonstration of the recursive technique with input data
from an actual watershed, it was judged appropriate to use the SCS
rainfall-runoff model in order to avoid complications related to
obtaining input information required for the other watershed models.
Stochastic Rainfall Model
Rainfall event characteristics that need to be stochastically
modeled for application of the recursive technique are specified by
the deterministic transform model chosen. For deterministic models
incorporating infiltration components that express infiltration as a
function of time or accumulated moisture, stochastic modeling of
rainfall event depth, duration, and time distribution is required.
In addition, spatially distributed watershed models require modeling
99
of rainfall spatial characteristics. For the SCS direct runoff model,
however, which we have chosen for the recursive technique demonstration,
only rainfall event depth is needed in the stochastic input model.
For stochastic modeling of rainfall depth, the exponential
model has been used by a number of researchers. Eagleson (1972), in
deriving peak streamflow for catchments in the New England area,
assumed exponential distributions for rainfall event depth given certain
duration ranges. Smith and Schreiber (1974), in an analysis of
thunderstorm rainfall in southeastern Arizona, found that cumulative
histograms of daily point rainfall above certain thresholds could be
approximated by cumulative exponential distributions. Todorovic and
Woolhiser (1976), in deriving the distribution function of the sum of
n-day precipitation, assumed an exponential distribution for daily
rainfall based upon observations at Austin, Texas. Moreover, Fogel and
Duckstein (1969) found that a geometric distribution, which is the
discrete version of an exponential, seemed to be appropriate for
rainfall depth of convective storms in Arizona.
The exponential distribution is a relatively simple model
with only one parameter that is easily estimated from rainfall event
depth data. This can be seen from the exponential probability density
function which is expressed by the equation
f(d)
=
a
exp(-ad)
(137)
where d is rainfall event depth, f(d) is probability density, and a is
the reciprocal of the expected value of rainfall event depth.
100
Adequacy of the exponential probability model for rainfall
event depth in the vicinity of
with data from a rain gage on
Ahoskie, North Carolina, was tested
Ahoskie Creek watershed, which was chosen
for the direct runoff demonstration example. The chi-square test for
goodness of fit was used with approximately ten years of rainfall event
data. A computed chi-square statistic of
14.3 with 11 degrees of
freedom would not allow rejection of the exponential distribution
since there is a probability of about
by sampling error. Figure
0.217 of exceeding that value
4 shows a histogram of the Ahoskie rainfall
event depth data with an exponential probability density function (POE)
fitted to it. For the fitting, the exponential parameter was estimated
by the maximum likelihood estimator. This estimator for the exponential
is the reciprocal of the sample average.
Stochastic State Model
The stochastic state model required for application of the
derived recursive technique is also specified by the deterministic
transform model chosen. For digital computer models that simulate the
individual watershed hydrologic processes, the state model must reflect
probability of various soil moisture conditions and possibly interception capacity of the plant canopy. Both evapotranspiration and soil
moisture redistribution would be involved in determining these state
probabilities.
For the
SCS direct runoff method, state is indicated by
antecedent moisture condition (AMC) class and corresponding curve
number. The AMC class is determined by
5-day antecedent rainfall;
1 01
i
.. LO
4-)
z
CL)
>
LU
0
=
.,—
C)
CO
LO
C\ I
CI
C•J
LC)
r-
A3N3110321J 3ALLV132:1 1.002J3d
C)
1--
LO
102
this relation is given in Table 1. The curve number corresponding to
an AMC class is obtained from the relation between curve number and AMC
class. A condensed version of this relation is given in Table 2. To
use Table 2, the curve number for AMC Class II must be obtained from
watershed soils and land-cover information. With the curve number for
AMC Class II determined, corresponding curve numbers for AMC Classes I
and III are then read from the same line in Table 2 that contains the
AMC Class II curve number.
For the stochastic state model to be used in this demonstration
with the SCS method, an estimated probability mass function for the AMC
classes for the Ahoskie Creek watershed was obtained from rainfall data.
This was accomplished by computing relative frequency of the three
antecedent rainfall classes at rainfall event beginnings. The estimated
probability mass function is given in Table 3 (p. 109, following
section) along with other data required for the direct runoff example.
Demonstration Example
For the direct runoff demonstration example, both hydrologic
and watershed physical data from the Ahoskie Creek watershed were used.
This watershed, near Ahoskie, North Carolina, was instrumented for
measuring rainfall and streamflow during the period July 1964 through
September 1973. A map of Ahoskie Creek watershed and pertinent
descriptive data are given in Appendix C. More detailed information on
Ahoskie Creek watershed is available in a watershed report published by
the Southeast Watershed Laboratory (1977) of the Agricultural Research
Service, USDA.
103
Table 1. Relation of SCS Antecedent Moisture Condition
Class to Antecedent Rainfall
(USDA Soil Conservation Service 1972, p. 4.12)
Antecedent Moisture Condition Class
Five-Day Total Antecedent Rainfall (inches)
Dormant Season
Growing Season
< 0.5
< 1.4
II
0.5 - 1.1
1.4 - 2.1
III
> 1.1
> 2.1
104
Table 2. Relation of SCS Curve Number to Antecedent
Moisture Condition Class a
Antecedent Moisture Condition Class
100
87
78
70
63
57
51
45
40
35
31
22
15
9
4
0
II
III
100
95
90
85
80
75
70
65
60
55
50
40
30
20
10
0
100
98
96
94
91
88
85
82
78
74
70
60
50
37
22
0
a. This is a condensed version of Table 10.1
of SCS National Engineering Handbook, Section 4 (USDA
Soil Conservation Service 1972, p. 10.7). The
complete table is given in Appendix B.
105
In this demonstration example, the objective is to obtain a
probability distribution of annual direct runoff from the upper 24square-mile drainage of Ahoskie Creek watershed. Such a distribution
might be coupled with an economic loss function to evaluate downstream
agricultural pollution risk associated with alternate watershed plans
and activities. For computation of the annual direct runoff probability distribution with the derived recursive technique, equations 108
and 104 from Chapter 4 are employed. Equation 108 gives the probability
u
u
0
for zero annual direct runoff and equation 104 gives probabilities
k
for annual direct runoff in classes k, for k = 1,2,3,... . For this
example, 3,000 annual direct runoff classes with widths of 4 acre-feet
are specified in addition to the zero class. Therefore, equation 104
provides probability values for annual direct runoff up to 12,000
acre-feet, which is assumed to be the point at which the economic loss
function levels off. The probability for annual direct runoff greater
than 12,000 acre-feet is computed as the difference between one and
the cumulative probability for annual direct runoff up to 12,000
acre-feet.
To evaluate equations 108 and 104, values of the coefficients
b. for j = 0,1,2,...,3000 are required. These are obtained from
equation 93. In equation 93, the
coefficients are expected number
of rainfall events with depth in class i, for i = 1,2,3,... . These
t3. coefficients are computed by multiplying the total expected number
of events per year by the probability of a rainfall event having depth
in class i. The probability for rainfall event depth in each class i
106
is obtained by integrating the exponential density function for rainfall
depth (the stochastic rainfall model) between class limits.
For this
demonstration example, rainfall depth class widths are 0.10 inch and the
number of rainfall depth classes is 100. Thus, probabilities are
obtained for rainfall event depths up to 10 inches. The probability of
getting a depth greater than 10 inches is only 2.9 x 10 -6 for the
Ahoskie area, and is, therefore, disregarded as insignificant.
The quantities q
ij
in equation 93 are the probabilities for
event direct runoff in class j given event rainfall depth in class i for
i = 1,2,3,...,100 and j = 0,1,2,...,3000. These probabilities are
obtained by summing probabilities for all antecedent moisture condition
(AMC) classes that give event direct runoff in class j for rainfall
event depth in class i. The Soil Conservation Service (SCS) direct
runoff model (deterministic transform model) provides event direct
runoff values for given rainfall event depths and runoff curve numbers
associated with the AMC classes. Probabilities for AMC classes and
associated SCS curve numbers are provided by the stochastic state model.
To provide for computer operation of the derived recursive
technique for use with the demonstration example, equations 108 and 104
were incorporated in a Fortran computer program. Equation 93 was also
included to give the required coefficients. The SCS direct runoff model
and associated stochastic rainfall and state models were embodied in
program subroutines that furnish the quantities necessary for evaluating
equation 93. A listing of the computer program and required input
specifications are presented in Appendix D.
107
In the computer program given in Appendix D, subroutine INPUT
contains the stochastic rainfall model and provides for discretizing
rainfall depth into classes and for computing the probability for each
class. These rainfall class probabilities are multiplied by expected
number of events in the main program to obtain the
coefficients in
equation 93. The q ij quantities in equation 93 are obtained in the
computer program by accumulating probabilities for event direct runoff
in output class j for event rainfall depth in class i. These probabilities are obtained from the state probabilities incorporated in
subroutine STATE. Subroutine DETERM contains the SCS direct runoff
model and gives event direct runoff for a given rainfall depth and
system state, thereby providing for assignment of the state probability
to an output class j. In the main program, the coefficients b i computed
with equation 93 are used in equations 108 and 104 to compute the
probability mass function of annual direct runoff for discrete direct
runoff classes up to the direct runoff value where the economic loss
function of concern levels off. The probability of annual direct runoff
beyond this point is then computed as the difference between one and the
sum of class probabilities up to that point.
Input data needed for operation of the computer program
incorporating the derived recursive technique consists of rainfall
statistics, SCS curve numbers, probabilities of watershed AMC classes,
and annual direct runoff value at which the economic loss function
levels off. In addition to these data, the number and width of classes
for incrementing rainfall depth, the number of AMC classes, and the
108
number of output classes are required. Table 3 provides a summary of
program input values used in the demonstration example.
With the above required data as input, the Fortran program
incorporating the recursive technique was run on an IBM 370/158
computer. Approximately 2.29 seconds were required for compilation and
2 minutes and 23.81 seconds for execution. Total machine cost of the
run was $9.02.
From the computer run a probability mass function of annual
direct runoff was obtained. A probability density function was then
constructed from this probability mass function by plotting the average
density for each class of runoff at class midpoint and connecting
points. The resultant probability density function is shown in Figure 5.
To demonstrate effect on the annual direct runoff probability
density function (PDF) of a change in watershed land use, the SCS curve
number for AMC class II was changed from 80 to 87. Corresponding AMC
class I and III curve numbers were then changed from 63 to 73 and 91 to
95, respectively. These curve number changes reflect clearing of all
woodland on the watershed and converting it to cropland. The PDF
resulting from the change in input data is shown in Figure 5, along with
the original PDF. Computation of PDF's for possible watershed changes
such as this could provide for evaluating hydrology related risks of
alternate watershed plans before they are put into effect, thus
providing for optimum decision-making in the face of future meteorologic
or other uncertainties.
109
Table 3. Input Data for Operation of Computer Program
for Direct Runoff Example.
Estimate of expected number of rainfall events
per year
Estimate of exponential parameter for rainfall
event deptha
Number of rainfall event depth classes
Width of rainfall event depth classes
67.9
1.58
100
0.10 inch
AMC Class I curve number
63
AMC Class II curve number
80
AMC Class III curve number
91
Probability estimate of AMC Class I
0.79
Probability estimate of AMC Class II
0.12
Probability estimate of AMC Class III
0.09
Number of AMC classes
3
Annual direct runoff value at which economic
loss function levels off
Number of annual direct runoff classes
12,000 acre-feet
3,001
a. Reciprocal of average rainfall depth per event in inches.
110
(i)
LL
cL
LC)
AlISN30 AlI1I8V9OU
111
To check the probability density functions of
annual direct
runoff obtained in this example, we must have data on direct
runoff from
the watershed. These data for Ahoskie Creek are not readily available
even for existing watershed land use, since measured streamflow includes
base flow as well as direct runoff. Nevertheless, a rough check
can be
made by examining storm runoff hydrographs and estimating direct
runoff.
Nineteen Ahoskie Creek storm hydrographs were examined for 1967, which
was a low runoff year, and 16 for 1969, which was a high runoff year.
From these examinations it was estimated that annual direct runoff for
the eight years of record (1965-1972) ranged from 2,688 acre-feet to
7,808 acre-feet. The average annual direct runoff was estimated at
4,275 acre-feet. This estimated average value compares reasonably well
with the expected value of 3,879 acre-feet computed from the probability
distribution of annual direct runoff obtained with the recursive
technique for 75 percent woodland on Ahoskie Creek watershed. The
estimated range for annual direct runoff appears to be slightly high on
the low end when compared to the computed POE for 75 percent woodland
given in Figure 5. This might lead one to suspect that the SCS model
for direct runoff, which was used as the deterministic transform model,
gives slightly lower than actual values. However, in a statistical test,
the hypothesis that the annual direct runoff value of 2,688 acre-feet is
the minimum of a sample of eight from the computed POE could not be
rejected at the 0.05 level. Therefore, it was concluded that even though
the SCS model may contain some error, the probability distributions
computed in this example seem adequate for demonstration purposes.
112
In summary, this direct runoff application of the derived
recursive technique demonstrates the usefulness of this method for
obtaining probability distributions of hydrologic system output
resulting from a random number of stochastic inputs. It also illustrates that the recursive technique can handle uncertainty in hydrologic
system state as well as uncertainty in input, since stochastic
antecedent moisture condition classes and associated curve numbers are
included in the example. This is a definite advance from any previous
use of generating functions with problems of this type. Furthermore,
since the derived recursive technique does handle uncertainty in both
state and input, it can be used with more complicated watershed models
such as those described in the first section of this chapter.
To further illustrate usefulness and diversity of the recursive
technique, we will demonstrate application of the technique for
obtaining probability distributions of watershed sediment yield. This
is dealt with in the next chapter.
CHAPTER
7
SEDIMENT YIELD APPLICATION
Sediment, which is produced by erosion of watershed soils, is
sometimes called the greatest water pollutant (Robinson
1971). It
clogs channels and fills reservoirs and harbors, thereby diminishing
beneficial water transport and storage capacity and increasing the
cost of harbor maintenance (American Society of Civil Engineers
p.
1975,
2). It settles on fertile alluvial land, causing damage to crops
and reducing soil productivity (McDowell and
Grissinger 1976). In
addition to being a pollutant itself, sediment is also a carrier for
chemical pollutants such as plant nutrients and pesticides used in
agricultural production (Caro
1976, Frere 1976, Mulkey and Falco 1977).
When evaluating environmental effects of sediment producing activities
on watershed lands, probability distributions of sediment yield are
needed since erosion and sediment yield can vary widely due to random
variation of rainfall amounts and intensities, which greatly influence
sediment production and transport. If average values of sediment yield
rather than probability distributions are used in planning and design,
nonoptimal decisions may result (Smith, Davis, and Fogel 1977;
Duckstein, Szidarovszky, and Yakowitz 1977).
113
114
In this chapter we will demonstrate application of the
previously derived recursive technique for obtaining probability
distributions of watershed sediment yield by transforming probability
distributions of sediment producing rainfall characteristics and
watershed state through a deterministic transform model. First, as a
basis for selecting the deterministic transform model we will briefly
review several methods and models for computing sediment production and
transport. This will give an indication of the requirements for
deterministic sediment models appropriate for use with the recursive
technique. After selection of a deterministic model for sediment
production and transport, the stochastic models required for rainfall
input and hydrologic state will be discussed and the stochastic models
to be used for the demonstration example will be specified.
Deterministic Transform Model
Several methods for estimating long term and average annual
sediment yield from drainage basins have been available for a number of
years
(Holeman 1972, Strand 1972, Livesey 1972). Such methods, however,
do not provide the deterministic transform needed for deriving
probability distributions of sediment yield, since they provide only
one sediment yield estimate for the total period of concern. For
derivation of sediment yield probability distributions, models are
required that represent the transform of rainfall and water flow energy
to sediment output from drainage basins on an individual event basis.
Most models of this type have only recently been developed, or in many
cases are still in the development stage.
115
The event-based sediment yield models that have been developed,
or are presently under development, range from empirical parametric
approaches to models using theoretical equations describing interactions
of physical processes. An example of the empirical parametric approach
is the sediment yield model developed by Williams (1972) through
modification of the Universal Soil Loss Equation (USLE). The USLE was
developed by Wischmeier and Smith (1965) and has been widely used for
predicting average annual soil loss from agricultural fields and
construction sites.
In developing the event-based sediment yield model from the
USLE, Williams (1972) retained the basic USLE structure and parameters
that represent soil erodibility, slope, cropping management, and erosion
control practice. The only modification to these parameters is the
provision for computing areally weighted average values so that
parameter values are representative of an entire heterogeneous watershed.
The major modification by Williams of the USLE was the replacement of
the USLE rainfall erosivity factor with a runoff factor that includes
both total volume of runoff and peak flow rate. The Williams sediment
yield model is expressed in mathematical form by the equation
S = 95(Q • q p )
0.56 • K • LS • C - P (138)
where S is sediment yield in tons, Q is volume of runoff in acre-feet,
q is peak flow rate in cubic feet per second, K is the soil erodibility
116
factor, LS is the slope length and gradient factor,
management factor, and
P is the erosion-control-practice factor.
Values for the parameters
be obtained from
Smith
C is the cropping
K, LS, C, and P in equation 138 can
USDA-ARS Agriculture Handbook No. 282 (Wischmeier and
1965) and subsequent supplementary material (Wischmeier, Johnson
and Cross
1971; Wischmeier 1974; Barnett 1976; Williams and Berndt
1976).
The runoff factor in the Williams sediment model can be computed
with any hydrology model that provides runoff volume and peak runoff
rate for given rainfall events. Williams and Berndt
runoff-volume model based upon the
(1977) used a
SCS curve number technique and a
soil-moisture-index accounting procedure. They obtained peak runoff
rate by convolving rainfall excess with an instantaneous unit
hydrograph.
Onstad and Foster (1975) also developed an event-based sediment
yield model that incorporates
USLE parameters and structure. In the
Onstad-Foster model, soil detachment and transport are calculated
separately for each slope segment of a watershed. Detachment capacity
for a slope segment is represented by the equation
W. (KCPS).
Ej -
where
width;
185.58
1 '5)
• (Xl.' 5 - X j-1
(13 9)
E j is detachment capacity on segment j in pounds per foot of
X j. is distance from upper end of slope to lower end of segment j
in feet; W. is the detachment energy term, which is a function of both
rainfall and runoff; and
K, C, P, and S are the USLE parameters for
117
soil erodibility, cropping management, erosion control practice, and
slopegradient.Thedetachmentenergyterm W j is given by the equation
W j. := 0.5 R
st
+ 15 Q. q 1/3
.
J PJ
(140)
whereR st isthestormrainfallfactorinEwitsoftheusLE,.is
Qj
the storm runoff volume in inches over segment j, and q pj is the storm
peak runoff rate for segment j in inches per hour.
The transport phase of the Onstad-Foster model is expressed by
the equation
WTSCPx 1.5
C185.58
where T
c
(141)
is transport capacity in pounds per foot of width at point X
on a slope; K is an average value of soil erodibility weighted on the
basis of contribution of each segment to the sediment load; and S, C,
and P values are assumed to be identical to those used for calculating
detachment. In the Onstad-Foster model, actual sediment transport and
deposition is calculated for each slope segment by comparing total
soil detachment and transport capacity. If transport capacity exceeds
the detached sediment load for a segment plus any input from upslope,
sediment yield for the segment is the sum of the detached load and the
upslope input. If the transport capacity is less than the total soil
available for transport, sediment yield is equal to the transport
capacity and the remaining soil is deposited on the segment. Calculations
118
are carried out in this manner in the downslope direction for all
segments so as to obtain total sediment yield from the watershed.
An event-based parametric sediment model that is somewhat
different from the Williams or Onstad-Foster models in that it does
not incorporate the USLE parameters is one that was constructed by a
team of U. S. Department of Agriculture researchers at Athens and
Watkinsville, Georgia (Bruce et al. 1975). This model, which was
developed around structural concepts defined previously by Snyder
(1972), represents processes governing water, sediment, and chemical
pesticide runoff from small watersheds. In this model, water runoff
rate is calculated by convolving computed effective rain with a
variable response function. The sediment component of the model is
built on the concepts of rill and interrill erosion developed mainly
by Foster and Meyer (1972) and Meyer, Foster, and Romkens (1972).
Interrill erosion is expressed as a function of rainfall intensity and
soil susceptibility to erosion. Rill erosion is expressed as a function
of water runoff and rate of change of water runoff. Pesticide concentrations in the runoff are expressed as functions of amount of runoff,
sediment concentrations derived from rill and interrill erosion, and
pesticide concentration in the respective runoff-erosion zones.
This water-sediment-chemical model has undergone limited
testing on a small watershed in the Southern Piedmont with excellent
results. In order to use the model for simulation, however, several
of the parameters must be calibrated with watershed runoff data.
119
Another sediment model that might be classed as parametric is
one developed by Negev
(1967). This model was designed to be used in
conjunction with the Stanford Watershed Model (Crawford and Linsley
1966) for simulating suspended sediment load. In this model the amount
of soil splash is expressed as a function of hourly precipitation, and
the transport of splash residue is a function of overland flow rate and
residue in storage on the ground surface. Gully erosion, which is
divided into two portions according to soil particle size, is also
related to overland flow. For simulation, the Negev procedure involves
a number of coefficients that must be obtained by calibration, although
a few coefficients can be assigned values based on physical information.
A number of sediment models that incorporate theoretical
equations describing interactions of physical processes are at various
stages of development. These include models proposed by
Hjelmfelt,
Piest, and Saxton (1975); Smith (1976c); Shirley and Lane (1978); and
Alonso,
DeCoursey, Prasad, and Bowie (1978). Essentially these models
are based upon the partial differential equations, which were presented
by Bennett
(1974), for expressing the conservation of mass and momentum
for water flow and the conservation of mass for sediment. In these
models, the kinematic approximation is used to describe water flow,
and various relations are employed to describe the effects of rainfall
and water flow on sediment detachment and deposition.
For the sediment yield demonstration of the previously derived
recursive technique, any of the sediment models described in this
section could be used as the deterministic transform if the appropriate
120
stochastic rainfall and state models and other required information
are available. However, since many of these sediment models either
require calibration with input-output data or have not been developed
to the operational stage, the choice for this demonstration was narrowed
to the Williams sediment model. For the Williams model, the parameters
have either been evaluated, or information is readily available for
evaluating them, since most parameters are the same as the widely used
and documented
USLE.
The hydrology model chosen for this demonstration to provide
runoff volume and peak runoff rate needed for computation of the
runoff factor in the Williams sediment model consists of the Soil
Conservation Service
(SCS) direct runoff model and an SCS technique
for estimating peak runoff rate when runoff volume and duration are
known (USDA Soil Conservation Service
1972, pp. 16.6-16.8). The SCS
direct runoff model is described in Chapter
it will not be described here. The
6 and Appendix B; thus
SCS peak-flow model is expressed
by the following equation:
484
A
Q
qp - 0.5D + 0.6 T c
where
(142)
q is peak runoff rate in cubic feet per second, A is area of
the watershed in square miles,
Q is volume of runoff expressed in
inches over the watershed, D is duration of runoff-producing rainfall
in hours, and
T c is time of concentration for the watershed, also
expressed in hours. When using the
SCS peak flow model, A and T c are
121
provided as watershed input data,
Q is obtained from the SCS direct
runoff model, and D is computed by multiplying rainfall duration by the
ratio of rainfall volume after initial abstraction to total rainfall
volume.
The complete deterministic transform model chosen for
demonstrating application of the derived recursive technique for
obtaining probability distributions of sediment yield is thus composed
of three components. These are the Williams event-based sediment yield
model, the
SCS direct runoff model, and the SCS technique for computing
peak runoff rate.
Stochastic Rainfall Model
As was the case with the direct runoff application, rainfall
event characteristics that must be stochastically modeled for obtaining
a distribution of sediment yield with the recursive technique depend
upon the deterministic sediment model chosen. The
Onstad-Foster model
(Onstad and Foster 1975) and the water-sediment-chemical model developed
by Bruce et al.
(1975) require rainfall depth, duration, and time
distribution pattern. This is also true to some extent for the Negev
model (Negev
1967) since hourly rainfall amounts are required. Sediment
models based upon the partial differential equations expressing conservation of mass and momentum of water, and conservation of mass of
sediment also require time distribution of rainfall intensity. Moreover,
if the sediment model is spatially distributed, spatial distribution
patterns of rainfall intensity are required in addition to time
distribution.
122
With the deterministic model selected for the sediment yield
demonstration, which consists of the Williams sediment model (Williams
1972) coupled with the SCS models for obtaining direct runoff volume and
peak flow rate (USDA Soil Conservation Service 1972), stochastic
modeling of rainfall event depth and duration is needed. Since rainfall
event depth and duration are generally considered to be correlated
(Eagleson 1970, p. 186), a bivariate distribution is required for the
stochastic input model. Crovelli (1971) proposed a bivariate gamma for
modeling the joint distribution of rainfall depth and duration; however,
it was found that the Crovelli model is not appropriate for the Ahoskie
rainfall data which are used in this demonstration. The joint
occurrence pattern for rainfall event depth and duration at Ahoskie is
depicted by the bivariate frequency distribution of Ahoskie rainfall
depth and duration data presented as Table 4.
In order to obtain a stochastic model appropriate for the
Ahoskie data it was deemed best to express the bivariate model of
rainfall event depth and duration as the product of a marginal density
function for rainfall duration and a conditional density function for
rainfall depth given duration.
In searching for a model to represent the marginal distribution
of rainfall event duration, an exponential, a Weibull, and a gamma
distribution were each fitted to the Ahoskie rainfall event duration
data. From this fitting it was found that the exponential distribution
did not fit the Ahoskie duration data, and the gamma and Weibull
distributions both fitted the Ahoskie data about equally as well. The
Weibull distribution is very similar to the gamma distribution; however,
123
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the integral of the Weibull density function can be expressed in
closed form and the integral of the gamma density function cannot.
Probability evaluations needed for the recursive technique are easier
when the integral of the density function is available in closed form.
Thus, in this demonstration, the Weibull was chosen as the model to
be used for the marginal distribution of rainfall event duration.
The Weibull distribution has been widely employed as a probability model in life testing of components and systems where the
probability of failure varies with time (Hahn and Shapiro 1967, p.
108). Therefore, in view of this previous application it does not
appear unreasonable to use the Weibull distribution to model the life
or duration of rainfall events where the probability of termination
also usually varies with time.
The commonly used forms of the cumulative Weibull distribution
and probability density functions are
F(x) = 1 - exp[I Y]
(143)
and
(144)
f(x)[`(-)Y
exP E
where x is a value of the random variable of interest, F(x) is the
cumulative distribution, f(x) is probability density, x is the scale
parameter, and y is the shape parameter.
125
In order to simplify derivation of the maximum likelihood
estimating equations, Cohen (1965) wrote the density function in a
slightly different form. By designating g as e he was able to express
the Weibull density as
f(x) = (i) x ï-1 exp [-
(145)
The Weibull cumulative distribution is then expressed as
F(x) = 1 - exp [-
xi
(146)
The maximum likelihood estimating equations derived by Cohen
(1965) for parameters e and y are
n
i=1
i
,
e ln x.
11
n
/;
x4
=1 '
1 _ 1 n1 ln
x.1
„
n
i=1
Y
(147)
and
n x.
e = /
1
i=1
(148)
where x. for i = 1,2,...,n are samples and e and y are the parameter
estimates.
126
To obtain an estimate of the Weibull shape parameter y for the
Ahoskie rainfall event duration data, equation 147 was solved by an
iterative procedure. To initiate the iterative procedure, a first
approximation for y was obtained with a method developed by Cohen (1965)
which relates y to the sample coefficient of variation. After getting
the iterative solution of equation 147, an estimate of the parameter e
for the Ahoskie data was then computed using equation 148. Descriptions
of the iterative procedure and computer program incorporating the
procedure are presented in Appendix E, along with the method for
obtaining the first approximation for y.
A histogram of the Ahoskie rainfall event duration data with the
Weibull probability density function fitted by using the maximum likelihood parameter estimates obtained with the above procedure is shown in
Figure 6. Goodness of fit of the Weibull density function to the
Ahoskie data was tested with the chi-square test. A chi-square statistic
of 30.79 with 21 degrees of freedom was computed. With this value of
the chi-square statistic the hypothesis that the Ahoskie rainfall event
duration data is Weibull distributed was not rejected at the 0.05 level.
To obtain the conditional distribution of rainfall event depth
for given duration, the Ahoskie rainfall events were divided into five
duration classes: (1) one hour or less, (2) 1 to 3 hours, (3) 3 to 6
hours, (4) 6 to 12 hours, and (5) greater than 12 hours. Histograms of
rainfall event depth were constructed for each of these duration classes.
Parameters of the gamma, Weibull, and log-normal distributions were also
estimated from rainfall event depth data from each duration class. The
127
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A3N11032:1J 3ALLV131 .0130213d
128
chi-square goodness-of-fit test was then performed for each distribution
and duration class. From these goodness-of-fit tests, it was found that
the log-normal distribution fitted the
the gamma or
Weibull
Ahoskie
data better than either
distributions. Thus, the log-normal was chosen as
the model to be used for the conditional distribution of rainfall event
depth for given duration.
The conditional log-normal distribution in probability density
form is expressed by the equation:
f(y1x) =
vzex13)11
a(x)y
2
[ln
y
-
(x)]
2.a1 (x)
where y is rainfall event depth,
x
is event duration,
f(y1x)
is the
conditional probability density of depth given duration, and the
parameters
u(x)
and
a(x),
which are functions of event duration
x,
are
respectively the expected value and standard deviation of the natural
logarithm of rainfall depth.
The parameters
estimated from the
p
and a for the log-normal distribution were
Ahoskie
rainfall event data for each duration class
by the equations:
/
_ i=1
ln
yi
(150)
and
a-
(151)
129
where n is the number of samples in a duration class, y i for i =
1,2,...,n is a rainfall depth sample, and p and a are estimates of
; and a.
To determine relations of the log-normal parameters to event
duration, parameter estimates obtained by the above equations were
plotted against average duration of events in the corresponding duration
classes. Figure 7 shows this plotting for p vs. duration and Figure 8
shows the a vs. duration plotting. A correlation coefficient of 0.938,
indicating high linear correlation, was computed for ; and duration
for the first four duration classes. A linear relationship of p to
event duration for durations equal to or less than 12 hours was then
fitted by least squares. For rainfall event durations greater than
12 hours, the estimate of p obtained with equation 150 was assumed to
apply. The linear relationship for event durations equal to or less
than 12 hours and a horizontal line representing the estimate of p
for durations greater than 12 hours are shown in Figure 7.
The plot of a vs. event duration, which is Figure 8, indicates
very little relation between a and duration. Therefore, this parameter
is assumed constant for all rainfall event durations at the average
value of 0.876. This is indicated in Figure 8 by a horizontal line.
Figures 9 through 13 show histograms of Ahoskie rainfall event
depth for the five duration classes with log-normal probability density
functions fitted. For the fittings, estimates of the parameter p were
obtained from the relation of ; to duration in Figure 7. The estimate
for a was taken as the average value shown in Figure 8.
130
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Goodness of fit of the log-normal distribution with parameters
estimated by the above described procedure was tested with the chisquare test. The chi-square statistics computed and corresponding
exceedance probabilities are given in Table 5.
It is noted from Table 5 that the probability of exceeding the
chi-square statistic due to sampling error is greater than 0.02 for all
duration classes with the exception of the 1 to 3 hour class. The
histogram for this class, shown in Figure 10, appears to be somewhat
unusual in that it has both a high peak and a thick tail. Such a
histogram is difficult to fit and it appears from observing Figure 10
that the log-normal POE shown is about the best fit that one could
expect. Thus, we do not reject the hypothesis that the log-normal
distribution with parameters estimated by the relations given in Figures
7 and 8 is an adequate model for the conditional distribution of Ahoskie
rainfall event depth given duration.
Using the estimated relation of the log-normal parameter m to
rainfall duration x given in Figure 7, we can express the conditional
probability density function of rainfall depth given duration as
1 exp
a
31/TT-
2
[ln y - (2.79 + 0.0243x)] J',
y > 0, 0 < x < 72
(152)
f(y1x) =
a
11
ex[-
2a
yiTT7
(ln y - 4.53)1,
y > 0, x > 72.
138
Table 5. Chi-square Statistics and Exceedance Probabilities
Showing Goodness of Fit of Log-normal Distribution to
Histograms of Ahoskie Rainfall Depth
for Five Duration Classes
Duration
Class
(Hours)
Chi-square
Statistic
Degrees of
Freedom
Probability of Exceeding
Chi-square Statistic
through Sampling Error
0 - 1
11.48
6
0.0747
1 - 3
20.02
4
0.0005
3 - 6
11.30
4
0.0235
6 - 12
9.70
6
0.1379
Over 12
12.31
7
0.0911
139
Then combining the marginal probability density function (PDF)
for rainfall duration with the conditional PDF for rainfall depth, we
have the bivariate joint PDF for rainfall depth and duration as
f(Y,x) = f(y1x) • f(x)
(153)
or
y-1
y x
xp
eae
1
[
Y)
2 [ln y -(2.79 + 0.0243x)] 2 ,
2a
y > 0, 0 < x < 72
f(Y,x) =
(154)
y
exp
x:
1 (ln y - 4.53)1,
2
ea ,4T717-2a
y > 0, x > 72.
This combination of the Weibull and log-normal distributions
given in PDF form by equation 154 is the input probability model of
rainfall event depth and duration used with the recursive technique in
the sediment yield demonstration.
For computer implementation, which is required for the sediment
yield demonstration, the Weibull-log-normal model must be discretized
to obtain depth-duration cell probabilities. This is accomplished by
discretizing the relation
140
2.79 + 0.0243x, 0 < x < 72
u
(155)
[
4.53, x > 72
so that for a given duration class i, for i
1 2 3 ,•• • ,
x. + x .
1+1
, 0 < x < 72
[2.79 + 0.0241
2
•=
1
(156)
4.53, x > 72
where p i is the estimate of p for duration class i, x i is the lower
bound for duration class i, and x i+1 is the lower bound for duration
class i + 1. The cell probability P for depth-duration cell i,j is
expressed as
P (x i < X < x 1 +1 , y i < Y < Y i+1 )
(157)
Y4 + 1 x i+ ,
f(y,x) dx dy
f
fJ
x•
y .
J
1
or
P (x i < X <
yi < Y < Y i+ 1)
(158)
y. +1
= f Jf
y.x•
J 1
f(y1x) f(x) dx dy
depth class j,
where y i , for j = 1,2,3,..., is the lower bound for
and other symbols are
y i4. 1 is the lower bound for depth class j + 1,
a constant for x i < x < xi+1,
as previously defined. Now since p i is
141
for x i
f(y1x) does not vary with x
bility
for depth-duration cell
P
(x i<
i,j can be expressed
X <
the cell proba-
< x < x i+1 . Then
as
yi < Y <
x.1+1
Yj+1
f(x)dX] dy
= I f(yix)[
(159)
-
x.
Yj
or
P
X < x i+1 , y i<
(x i<
Y
< y i4.1 )
(160)
x. +1Yi 4.1
= f lf(x) dx • p ' f(ylx) dy
x.
Yj
or
P
(x i<X
< x i+1 , y i <Y < Yj + -1)
(161)
X Yi+ -1
xY
exp
exp
• Y.4.1
fJ
f(ylp i ) dY
Y•
where
1
- a
yvTT
1
exp[1: --2- (ln
2a
y
- pi )
(162)
142
And with numerical integration by Simpson's rule, the cell probability
is finally expressed as
y. < y
1+1 ' j
P (x i < X< x.
Y
Xi
exp -
Yj+1
X
exp
Y
i+1
I
• (Yi + 1 - Yj)
S
1 .g.t
7ET
Yi2
4
1^ 1
'Pi ) + 6 f,[
1
+ -6 - "F(Yj+11Pi )
( 1 63)
Stochastic State Model
The stochastic state model required for obtaining a sediment
yield probability distribution with the derived recursive technique
depends upon the deterministic transform model chosen, just as the
stochastic input model does. The deterministic transform model chosen
to be used with the sediment yield demonstration, which consists of the
Williams event-based sediment yield model coupled with the SOS direct
runoff and peak flow models, only requires antecedent moisture
condition (AMC) class and corresponding curve number for state. Thus
the same stochastic model for state is used with the sediment yield
example as was employed in the direct runoff example of the previous
chapter.
This stochastic state model essentially consists of a probability mass function representing probabilities of the AMC classes
and corresponding SCS curve numbers. Table 6 in the next section
143
gives the probability mass function and curve numbers for Ahoskie Creek
watershed along with other data required for the sediment yield example.
A more detailed description of the state model is given in Chapter 6
and will, therefore, not be repeated here.
Demonstration Example
Rainfall and watershed physical data obtained from Ahoskie
Creek watershed and used in the direct runoff example were also
employed for the sediment yield demonstration example. The rainfall
data are presented in histogram form in the section on the stochastic
rainfall model, and the watershed physical data, together with a map
of Ahoskie Creek watershed, are given in Appendix C.
The objective for the sediment yield demonstration example is
to obtain a probability distribution of annual sediment yield from the
upper 24-square-mile drainage of Ahoskie Creek watershed. Such a
distribution would be useful in evaluating cropping and tillage
practices on the upper watershed in terms of effect on cost of sediment
removal from the downstream channel, which must be kept open since it
carries flood water past the town of Ahoskie.
For computing the probability distribution of annual sediment
yield in this example, equations 108 and 104 from Chapter 4 are employed.
Two thousand annual sediment yield classes with widths of 15 tons each
are specified in addition to the zero class. Thus, equation 104
provides sediment probability values up to 30,000 tons. This is assumed
to be the point at which the economic loss function for annual sediment
yield from Ahoskie Creek watershed levels off. The probability for
144
annual sediment yield greater than 30,000 tons is computed as the
difference between one and the cumulative probability for annual
sediment yield up to 30,000 tons.
To evaluate equations 108 and 104, coefficients b i for
j = 0,1,2,...,2000 are computed by using equation 93. The coefficients
6 i in equation 93 are the expected number of rainfall events with depth
and duration in cell i, for i = 1,2,3,... . These 6 i coefficients are
derived by multiplying the total expected number of events per year by
the probability of a rainfall event having depth and duration in cell i.
The probability for rainfall event depth and duration in each depthduration cell is computed with equation 163, which is given in this
chapter in the section on the stochastic rainfall model.
For this sediment-yield demonstration example, rainfall event
depth is discretized into 200 classes of 0.05-inch widths. Event
duration is discretized into 50 classes of 2-hour widths. Thus, 10,000
depth-duration cells are specified. The probability of getting an event
depth or duration greater than that included in these 10,000 cells is
only 1.3 x 10 -3 . This is disregarded as insignificant.
The quantities q ii in equation 93 are the probabilities for
event sediment yield in class j given event rainfall depth and duration
in cell i for i = 1,2,3,...,10000 and j = 0,1,2,...,2000. These probabilities are obtained by summing probabilities for all antecedent
moisture condition (AMC) classes that give event sediment yield in
class j for rainfall event depth and duration in cell i. The deterministic model, which consists of the SCS direct runoff and peak flow-rate
145
models coupled with the Williams sediment yield model, provides event
sediment yield values for given event rainfall depth and duration and
SCS runoff curve number associated with given AMC class. Probabilities
for AMC classes and associated curve numbers are furnished by the
stochastic state model as in the direct runoff example.
To provide for computer operation of the above described
procedures, the computer program used for the direct runoff example
was modified for the sediment yield example by incorporating appropriate
subroutines for the chosen stochastic and deterministic models. The
subroutine INPUT used for the direct runoff application, which embodies
an exponential model for rainfall depth, was replaced with a subroutine
INPUT that incorporates the bivariate model consisting of a Weibull
distribution for rainfall event duration and a log-normal distribution
for rainfall event depth given duration. The subroutine DETERM for
direct runoff was replaced by a subroutine DETERM embodying the
runoff
Williams sediment model combined with the SCS models for direct
and
volume and peak runoff rate. A listing of the computer program
required input specifications are given in Appendix F.
INPUT
In the computer program given in Appendix F, subroutine
provides for discretizing rainfall depth and duration into depthduration cells and for computing cell probabilities. These cell
events in the main
probabilities are multiplied by expected number of
program to obtain the
coefficients in equation 93. The q ii
quantities in equation 93 are obtained by accumulating probabilities
rainfall event depth
for event sediment yield in output class j for
146
and duration in cell i. These probabilities are obtained from the
state probabilities incorporated in subroutine STATE as with the direct
runoff application. Subroutine DETERM, which contains the sediment
yield deterministic model, furnishes event sediment yield for given
rainfall depth and duration and hydrologic state. This provides for
assignment of state probabilities to the appropriate output class j.
As was the case with the direct runoff example, the coefficients
b i for j = 0,1,2,..., computed with equation 93, are used in equations
108 and 104 to compute a probability mass function of annual sediment
yield up to the sediment yield value where the economic loss function of
concern levels off. The probability of getting annual sediment yield
beyond this point is then computed as the difference between one and the
sum of sediment yield class probabilities up to that point.
Computer input data needed for the sediment yield example
consists of rainfall statistics, estimated probabilities of watershed
moisture condition classes, and watershed physical data -- including
drainage area, time of concentration, SCS curve numbers, and USLE
parameters. In addition, the numbers and dimensions of classes for
incrementing rainfall event depth and duration and the number of AMC
classes are also specified, along with the number of output classes and
the sediment yield point at which the economic loss function levels off.
Table 6 gives a summary of these data.
The Fortran computer program described above and given in
Appendix F was run on an IBM 370/158 computer with the data in Table 6
as input. Approximately 3.27 seconds were required for compilation and
147
Table 6. Input Data for Operation of Computer Program
for Sediment Yield Example
Estimate of expected number of rainfall events
per year
67.9
Maximum likelihood estimate of Weibull parameter e
for rainfall event durationa
18.5
Maximum likelihood estimate of Weibull parameter y
for rainfall event durationa
0.770
Estimate of log-normal parameter a for rainfall
event deptha
0.876
First coefficient in linear regression equation
relating rainfall depth log-normal parameter
estimate 1] to durationa
2.79
Second coefficient in linear regression equation
relating rainfall depth log-normal parameter
estimate fsl to durationa
0.0243
Estimate of rainfall event depth log-normal
parameter p for durations greater than
12 hoursa
4.53
Number of rainfall event duration classes
Number of rainfall event depth classes
50
200
Width of rainfall event duration classes
2 hours
Width of rainfall event depth classes
0.05 inch
AMC class I curve number
63
AMC class II curve number
80
AMC class III curve number
91
148
Table 6, Continued
Probability estimate of AMC class I
0.79
Probability estimate of AMC class II
0.12
Probability estimate of AMC class III
0.09
Number of watershed states
3
Watershed drainage area
24.0 sq. mi.
Watershed time of concentration
10.0 hours
Coefficient of Williams runoff factor
95
Exponent of Williams runoff factor
0.56
USLE soil erodibility factor
0.17
USLE slope length and gradient factor
0.5
USLE cropping management factor
0.20
USLE erosion-control-practice factor
0.6
Sediment yield point at which economic loss
function levels off
Number of annual sediment yield classes
30,000 tons
2,001
a Parameter estimates for bivariate rainfall model based upon
rainfall duration in ten-minute units and rainfall depth in hundredths
of an inch.
149
3 minutes and 20.57 seconds for execution. The total machine cost for
a run was $11.10.
A probability mass function (PMF) of annual sediment yield from
Ahoskie Creek watershed was obtained as output from the computer run.
A probability density function (PDF) was then constructed from the PMF
in the same manner as with the direct runoff example. This PDF,
resulting from conventional tillage of Ahoskie Creek watershed cropland,
is shown in Figure 14. Also shown in Figure 14 is a PDF of annual
sediment yield obtained by changing the USLE cropping management factor
and SCS curve numbers to indicate a change from conventional tillage of
the watershed cropland to minimum tillage of the cropland. The USLE
cropping management factor was changed from 0.20 to 0.17 and curve
numbers for watershed moisture condition classes I, II, and III were
changed from 63, 80, and 91 to 59, 77, and 89, respectively.
This example demonstrates how the computer program incorporating
the derived recursive technique can be used to evaluate the effects of
alternate tillage practices on watershed cropland in terms of the
probability distribution of annual sediment yield. The evaluation might
also be made in terms of risk associated with cost of removing the annual
sediment yield from a downstream channel or reservoir by coupling an
appropriate economic loss function to the computed probability
distributions. Such evaluations can provide for optimum planning in the
face of future uncertainties in rainfall and watershed factors that
affect sediment production and transport.
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Checking the probability distributions of annual sediment yield
computed in this example directly is difficult since no sediment data
have been collected on Ahoskie Creek watershed. However, if we look at
sediment yield data obtained from reservoir surveys (Dendy and Champion
1978) in the area consisting of the Chowan, Roanoke, Tar, Neuse, and
Cape Fear River Basins in which Ahoskie Creek watershed is located, we
can get a rough estimate of Ahoskie Creek sediment yield and its
variability. From these surveys, average annual sediment yield per
square mile of net drainage area is reported for one period each for
six reservoirs in the area and two periods each for two reservoirs. If
we make ergodic assumptions concerning the sediment yield process in
time and space, we can consider these annual sediment yield values as
10 samples somewhat representative of annual sediment yield from Ahoskie
Creek watershed.
For the 10 reservoir samples, annual sediment yield values
ranged from a low of 70 tons per square mile of drainage area to a high
of 2,103 tons per square mile. The average value was 568.2 tons per
square mile.
To compare these values with the probability distribution of
annual sediment yield computed for Ahoskie Creek watershed, we multiply
by the Ahoskie Creek drainage area, which is 24 square miles. The
sample range for annual sediment yield is then 1,680 tons to 50,472
tons, and the sample average is 13,637 tons. This sample average
compares favorably with the expected value of 12,456 tons computed
from the probability distribution of annual sediment yield obtained
152
with the recursive technique for conventional tillage on
Ahoskie
Creek
watershed. The sample range for annual sediment yield also appears to
be reasonable when compared with the computed
tillage in Figure
PDF
for conventional
14.
To summarize the sediment yield demonstration, it is noted that
the derived recursive technique, when used with the Williams model
coupled with the
SCS
models for direct runoff volume and peak runoff
rate, provides a direct transform of both rainfall event and state
probability distributions to sediment yield distributions for a given
period of concern. This is an advance from previous work in that a
direct transform such as this has not previously been provided for
obtaining sediment yield distributions from a random number of
stochastic rainfall inputs under stochastic state conditions.
CHAPTER 8
DISCUSSIONS AND CONCLUSIONS
This dissertation has dealt with the general problem of
coupling stochastic and deterministic hydrologic models to provide
probability distributions required for considering uncertainty in
decision-making related to activities that depend or impact upon
the land phase of the hydrologic cycle. The specific problem
investigated was that of obtaining probability distributions of
cumulative hydrologic system output for given periods of concern in
which there are random numbers of stochastic inputs. With the use
of probability generating functions, a recursive technique was
derived that provides for obtaining the desired output probability
distributions through deterministic transformation of hydrologic
input and state probabilities.
The derived recursive technique was incorporated in a computer
program and tested against an exact, independently derived analytic
procedure. For this test, a simple hydrologic problem that could be
solved by the analytic procedure was chosen, and probability
distributions were independently obtained by both the recursive
technique and the analytic procedure. These distributions were
compared and results indicate quite accurate probability computations
153
154
for the recursive technique since the distribution computed by the
recursive technique appears to be almost identical to the distribution
obtained with the exact procedure.
To determine the extent of possible hydrologic application of
the recursive technique, the assumption of Poisson occurrence of
rainfall input events, which is inherent in derivation of the technique,
was examined for reasonableness. From this examination it was observed
that the Poisson model for rainfall event occurrence appears to be
appropriate for many situations. More specifically, a test with
rainfall data from Ahoskie, North Carolina, indicated Poisson
occurrence of rainfall events when these events are defined as periods
of continuous or intermittent rainfall without a break of at least six
hours. It was therefore concluded that the assumption of Poisson
occurrence of rainfall events, especially when events are defined as
above, is quite appropriate for application in the humid area for
which the Ahoskie data are representative.
With results of the above tests insuring confidence in the
derived recursive technique, it was then applied to two important
hydrology-related problems. First the technique was used to obtain
probability distributions of annual direct runoff from a watershed
land
near Ahoskie, North Carolina, considering both present watershed
use and possible modification of land use by converting all woodland
technique
to cropland. The second application of the recursive
involved computation of probability distributions of annual sediment
tillage of
yield from the watershed both for current conventional
155
watershed cropland and for possible conversion to minimum tillage of
cropland. For these applications, uncertainties in both rainfall
input
and hydrologic system state were considered.
The probability distributions of annual direct runoff and
sediment yield computed with the recursive technique for present
watershed land use and tillage practice were tested for reasonableness
using available data. Direct runoff data were obtained by estimation
from measured total streamflow hydrographs. Sediment yield was
estimated from reservoir surveys in the river basin containing the
Ahoskie watershed and adjoining river basins. Results of these tests
indicate that the computed distributions of annual direct runoff and
sediment yield for the Ahoskie watershed are reasonably accurate.
These applications of the recursive technique, concerning
direct runoff and sediment yield, serve to demonstrate usefulness of
the technique for obtaining probability distributions needed for
planning decisions. To extend this usefulness beyond the two examples
demonstrated, possible further application of the technique will be
explored in the next section.
Further
Application
Further application of the recursive technique derived from
generating functions can be made in a number of hydrology-related
problem areas. For instance, all of the examples given in the introductory chapter are suitable for use of the technique. For these
examples, probability distributions are needed for water yield, soil
moisture recharge, flood damage, erosion and sedimentation, chemical
156
pollutant accumulation, and aquifer recharge for given time periods
of concern. Application can conceivably be made for any problem for
which the decision quantity of interest is the cumulative output or
effect of a random number of stochastic inputs to a hydrologic system.
For some situations this may require looking at the decision problem
in a different, but perhaps more realistic light. For example, with a
flood damage application, the real quantity of interest for decisionmaking should be cumulative damage over some planning period in the
future, rather than just damage from one large rare event.
If we consider cumulative flood damage over some time period of
concern as a decision quantity for future planning, then the derived
deterministic
recursive technique is applicable. For this application, a
transform model is required that gives flood damage for a single rainfall
event. Single-event flood damage for some cases can be computed within
flood damage to
the structure of the deterministic model by relating
computed peak
streamflow rate. For other cases it may be desirable to
relate flood damage to the total runoff
hydrograph. If so, this flood-
damage-hydrograph relation can be incorporated within the deterministic
model structure along with the
hydrograph computation. Additional
recursive technique
requirements for a flood damage application of the
and sediment yield
would be similar to those for the demonstrated water
stochastic state model to
applications. These requirements are a
immediately prior to flood
provide probabilities of watershed conditions
occurrence probabilievents, and a stochastic rainfall model to provide
ties of flood-producing rainfall characteristics.
157
The flood damage problem as stated here could conceivably be
treated with the computerized recursive technique using the same
bivariate rainfall model for event depth and duration and the Soil
Conservation Service (SCS) models for direct runoff and peak flow that
were employed in the sediment yield application. To obtain flood
damage for each event, a relation between flood damage and peak flow
would have to be included in the deterministic model containing the
SCS runoff and peak flow models.
The derived recursive technique, when used in conjunction
with a deterministic model of a hydrologic system along with required
stochastic input and state models, provides a means for evaluating
proposed changes in the hydrologic system in terms of output
probability distributions. The direct runoff and sediment yield
applications demonstrate this capability for changes in land use and
cropland tillage methods. Other planning applications such as the
examples concerning soil moisture recharge, flood damage, chemical
pollutant accumulation, and aquifer recharge, which are given in the
introductory chapter, could also make use of the capability. In this
manner, risk reflecting probability for loss or damage associated
with a proposed change in a hydrologic system can be evaluated before
the change is implemented.
Use of the recursive technique to couple deterministic and
stochastic hydrologic models and thereby compute probability
distributions of system output also provides a means for model
evaluation. Such evaluations can be accomplished by incorporating
158
the deterministic and/or stochastic models to be compared and evaluated
into the computer program containing the recursive technique. Probability distributions obtained with the different models can then be
compared with each other and with data. If an economic loss function
is available, risk associated with the different models can be computed
and evaluated.
Since the recursive technique was derived for hydrologic
systems having uncertainty in the output for given input due to
uncertain state or other causes, it can be used to evaluate the effect
of uncertainty in knowledge of the hydrologic model parameters. This
could include parameters of both deterministic and stochastic models.
Uncertainty in knowledge of model parameters is usually due to
limitations in data available for making parameter estimates. Thus,
the value of additional data for parameter estimation could be
obtained by comparing hydrologic system output probability distributions and risks computed with the recursive technique using
distributions of parameter estimates for available data and for
possible additional data. This application may require coupling of
the derived recursive technique with Bayesian decision theory. In
such coupling the Bayesian method would provide posterior parameter
distributions for additional data from prior parameter distributions
obtained with available data.
159
Limitations
Limitations in application of the recursive technique
essentially consist of two restrictions inherent in the derivation:
(1) The recursive technique as derived is limited to situations
where the hydrologic system output of interest is the cumulative
effect of a random number of stochastic inputs.
(2) Limitations in
the derivation also require that the counting process for input event
occurrence be such that it is appropriately modeled as a Poisson
process.
Although these limitations may appear restrictive at first
sight, it is possible and quite easy in many cases to redefine the
problem of concern or the definition of input events so that the
restrictions do not pose a serious problem in application of the
derived recursive technique.
At this point we would like to emphasize that application
of the recursive technique is not restricted by complexity of the
deterministic input-output system, dimension of the stochastic input,
or dimension of the stochastic system state. The only limitations
caused by complexities of this type are related to availability of
the necessary deterministic and stochastic system models in operational
form.
Although not a limitation to application of the recursive
technique, there is one other item that should be pointed out here.
When using the recursive technique, the cumulative output of interest
should be
discretized rather finely so that classes are relatively
160
small. This is needed to hold down uncertainty inherent with large
output classes. Obtaining small classes does not seem to be a problem,
however, since the recursive technique is quite computer efficient,
thereby allowing fine discretation of the cumulative output.
Suggestions for Research
Suggestions for research related to further development and
application of the derived recursive technique for computing hydrologic
system output distributions are largely concerned with three major
areas of deterministic and stochastic hydrologic model development.
These are discussed in the following paragraphs.
Additional work is needed in the area of developing operational
event-based deterministic models representing various aspects of the
land phase of the hydrologic cycle. These event-based models are
needed as input-output transforms for incorporation into the computer
program containing the recursive technique. Much work has already
been done in this area of model development, especially with regard to
water movement; however, operational models for which required input
data are readily available seem to be rare. Modeling of sediment
production and transport and chemical movement on an event basis has
only recently begun to be actively undertaken. Thus, only crude
operational models for sediment and chemical movement are available
for use. Ultimately, spatially distributed models representing
event-based response in terms of water, sediment, and chemicals will
be needed for use with the derived recursive technique, as well as
161
with other methods of stochastic transformation, for
nonpoint source
pollution considerations. Such models are required to evaluate
environmental effects of spatial modifications on hydrologic systems.
Research is needed for quantifying the uncertainties of
hydrologic system state and the uncertainties of prediction with
deterministic models. Uncertainty of prediction may be present in
models developed by regression using input-output data. For example,
the Williams sediment yield model has a deterministic structure that
was obtained by regression of sediment yield on runoff volume and
peak runoff rate. The model predicts average values of event sediment
yield for given inputs, yet actual sediment yield values vary around
these average values. Thus, there is uncertainty in prediction of
actual event sediment yield with the Williams model. Quantifications
of state and prediction uncertainty should be in the form of stochastic
models that can be incorporated into the computer program containing
the derived recursive technique. Uncertainty quantification is
required for a variety of hydrologic systems concerned with water,
sediment, and associated chemical output. Examples presented in the
introductory chapter concerning water yield, soil moisture recharge,
flood damage, erosion and sedimentation, chemical pollutant
accumulation, and aquifer recharge give an indication of the types
of systems for which uncertainty quantification is needed.
In order to use the recursive technique with more detailed
deterministic models of hydrologic systems, further research will be
necessary to develop stochastic models of rainfall inputs required
162
for operation of the deterministic models. This requires extending
the bivariate probability model for rainfall event depth and duration
developed in this dissertation to include time and spatial characteristics of rainfall inputs.
Conclusions
The following conclusions were drawn from this study:
(1) Probability generating functions were used to derive a
recursive technique that provides the probability mass function of
discretized cumulative output for a given time period from a hydrologic
system with stochastic states and a random number of stochastic inputs.
The stochastic states and inputs can be multidimensional and the
hydrologic system model may contain uncertainties of prediction if they
are quantified in a stochastic manner.
(2) The recursive technique, derived through the use of
generating functions, can be incorporated into a computer program that
contains an event-based deterministic transform model, a stochastic
state model, and a stochastic input model. This computer program can
be so structured that different deterministic and stochastic models
can be readily incorporated for use with different hydrologic systems
and decision problems.
(3) Accuracy of the recursive technique in computing probability distributions of cumulative system output for given time periods
of concern compares favorably with accuracy obtained with an exact
analytic method.
163
(4) The stochastic counting process for number of rainfall
events in the humid area near Ahoskie, North Carolina, can be
appropriately modeled as Poisson when events are defined as periods
of continuous or intermittent rainfall without a break of at least
six hours. This observation provides a basis for application of the
recursive technique in the Ahoskie area.
(5) The computerized recursive technique can be combined with
the Soil Conservation Service (SCS) model for direct runoff volume, a
probability mass function for antecedent moisture condition (AMC) class,
and an exponential rainfall event depth model to compute probability
distributions of annual direct runoff volume for a watershed with
specified soil, vegetation, and land management conditions. Probability
distributions of direct runoff computed in this manner for a watershed
near Ahoskie, North Carolina, appear to be reasonable when compared
with watershed runoff data.
(6) The joint probabilistic occurrence of rainfall event depth
and duration for the Ahoskie, North Carolina, area can be adequately
modeled with a bivariate distribution that consists of a Weibull
marginal distribution for rainfall duration and a log-normal conditional
distribution for rainfall depth given duration. For the log-normal
conditional distribution, the estimate for parameter p (which is the
average of the logarithms of rainfall event depths) is linearly related
to rainfall event duration for durations equal to or less than 12
hours.
164
(7) The recursive technique can be used to obtain probability
distributions of annual sediment yield from a watershed with specified
soil, vegetation, and land management conditions. For this application,
a deterministic sediment model consisting of a parametric event-based
sediment yield model and the SCS models for direct runoff volume and
peak flow can be coupled by the computerized recursive technique to a
probability mass function for AMC class and a Weibull-log-normal
bivariate probability model for rainfall event depth and duration.
Probability distributions of annual sediment yield computed with this
method for the Ahoskie watershed appear to be reasonable when compared
with reservoir sediment survey data from the general area.
APPENDIX A
COMPUTER PROGRAMS FOR SIMPLE EXAMPLE
Fortran computer programs incorporating the derived recursive
technique and analytic method used with the simple demonstration
example in Chapter 4 are presented on the following pages.
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1.17
APPENDIX
B
SCS CURVE NUMBERS AND RELATIONS
The Soil Conservation Service (SCS) runoff curve numbers for
antecedent moisture
cover
condition (AMC) Class II are related to soil-
complexes which are determined by land use, treatment or practice,
hydrologic
condition, and hydrologic soil group. Table B-1 gives this
relation for agricultural and forested areas and Table B-2 shows the
relation for urban and suburban areas.
The AMC Class I and III curve numbers are determined by
relations with the AMC Class II curve numbers. These relations are
given
in Table B-3 where corresponding AMC Class I, II, and III curve
numbers
are shown on the same horizontal line.
171
172
Table B-1. SCS Runoff Curve Numbers for AMC Class II for Hydrologic
Soil Cover Complexes on Agricultural and Forested Areas
(USDA Soil Conservation Service 1972, p. 9.2)
Cover
Land Use
Treatment
or Practice
Fallow
Straight row
Row crops
Straight row
Small grain
Close-seeded
legumes or
rotation
meadow
Hydrologic
Condition
Hydrologic Soil Group
ABCD
77
86
91
94
Poor
Good
72
67
81
78
88
84
91
89
Contoured
Poor
Good
70
65
79
75
84
82
88
86
Contoured and
terraced
Poor
Good
66
62
74
71
80
78
82
81
Straight row
Poor
Good
65
63
76
75
84
83
88
87
Contoured
Poor
Good
63
61
74
73
82
81
85
84
Contoured and
terraced
Poor
Good
61
59
72
70
79
78
82
81
Straight row
Poor
Good
66
58
77
72
85
81
89
85
Contoured
Poor
Good
64
55
75
69
83
78
85
83
Contoured and
terraced
Poor
Good
63
51
73
67
80
76
83
80
Poor
Fair
Good
68
49
39
79
69
61
86
79
74
89
84
80
Poor
Fair
Good
47
25
6
67
59
35
81
75
70
88
83
79
Pasture or
range
Contoured
173
Table B-1, Continued
Cover
Land Use
Treatment
or Practice
Hydrologic
Condition
Hydrologic Soil Group
ABCD
Meadow
Good
30
58
71
78
Woods
Poor
Fair
Good
45
36
25
66
60
55
77
73
70
83
79
77
Farmsteads
59
74
82
86
Roads (dirt)
72
82
87
89
74
84
90
92
(hard surface)
174
Table B-2. SCS Runoff Curve Numbers for AMC Class II
for Urban and Suburban Areas
(USDA Soil Conservation Service 1975,
P.
2-5)
Hydrologic Soil Group
Land Use and Hydrologic Condition
ABCD
Open Spaces, lawns, parks, golf courses,
cemeteries, etc.
Good condition: grass cover on 75% or
more of the area
Fair condition: grass cover on 50% to
75% of the area
39
61
74
80
49
69
79
84
Commercial and business areas (85% impervious)
89
92
94
95
Industrial districts (75% impervious)
81
88
91
93
77
61
57
54
51
85
75
72
70
68
90
83
81
80
79
92
87
86
85
84
98
98
98
98
98
76
72
98
85
82
98
89
87
98
91
89
Residential:
Average lot sizeAverage
1/8 acre or less
1/4 acre
1/3 acre
1/2 acre
1 acre
%
impervious
65
38
30
25
20
Paved parking lots, roofs, driveways, etc.
Streets and roads:
Paved with curbs and storm sewers
Gravel
Dirt
175
Table
B-3. Corresponding SCS Runoff Curve Numbers
for AMC Classes I, II, and III
(USDA Soil
Conservation Service
AMC CTass
1972,
p.
10.7)
AMC Class
AMC Class
I
II
III
I
II
III
I
II
III
100
97
94
91
89
100
99
98
97
96
100
100
99
99
99
57
55
54
53
52
75
74
73
72
71
88
88
87
86
86
31
30
29
28
27
50
49
48
47
46
70
69
68
67
66
87
85
83
81
80
95
94
93
92
91
98
98
98
97
97
51
50
48
47
46
70
69
68
67
66
85
84
84
83
82
26
25
25
24
23
45
44
43
42
41
65
64
63
62
61
78
76
75
73
72
90
89
88
87
86
96
96
95
95
94
45
44
43
42
41
65
64
63
62
61
82
81
80
79
78
22
21
21
20
19
40
39
38
37
36
60
59
58
57
56
70
68
67
66
64
85
84
83
82
81
94
93
93
92
92
40
39
38
37
36
60
59
58
57
56
78
77
76
75
75
18
18
17
16
16
15
35
34
33
32
31
30
55
54
53
52
51
50
63
62
60
59
58
80
79
78
77
76
91
91
90
89
89
35
34
33
32
31
55
54
53
52
51
74
73
72
71
70
12
9
6
4
2
0
25
20
15
10
5
0
43
37
30
22
13
0
APPENDIX C
AHOSKIE CREEK WATERSHED MAP AND DATA
A map of the 24-square-mile Ahoskie Creek watershed and
downstream channel is presented on the following page as Figure C-1.
Physical watershed data concerning soils, land-slope length and
gradient, and land use are given in tables C-1, C-2, and C-3.
176
177
178
Table C-1. Soil Name, Hydrologic Group, and Percentage of
Total Area for Ahoskie Creek Watershed Soils
Soil
Name
SCS Hydrologic
Group
Coxville
D
D
Lenoir
Craven
48
24
10
Chastain
Marlboro
Duplin
Norfolk
Percentage
of Total Area
4
D
4
3
3
D
Dunbar
1
Caroline
Faceville
2
1
100
179
Table C-2. Average Land-Slope Length and
Gradient for Ahoskie Creek Watershed
Slope length
Slope gradient
500 feet
2 %G.
Table C-3. Land Use and Percentage of
Total Area for Ahoskie Creek Watershed
Percentage
of Total Area
Land Use
75
Woodland
Row Crops
Pasture
Roads and Homesites
22
2
1
100
APPENDIX D
PROGRAM FOR COMPUTING PROBABILITY MASS FUNCTION
OF CUMULATIVE DIRECT RUNOFF
The Fortran computer program, incorporating the derived
recursive technique that was used with the direct runoff application,
is presented on the following pages.
180
181
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E
APPENDIX
PROCEDURE FOR OBTAINING MAXIMUM LIKELIHOOD
ESTIMATE OF WEIBULL PARAMETERS
The maximum likelihood estimate of the
y
Weibull
shape parameter
was obtained for rainfall event duration by iterative solution of
Equation
147.
Froberg (1965,
For this solution, a general iterative method given by
p.
33)
was employed. This iterative method requires
that the shape parameter estimate y be expressed as a function of y.
To obtain the required expression, Equation
Equation
E-1
Y -
n
for
i
.,
xY1
x.1
was rearranged to give
as follows:
..
where
147
i = 1,2,3,...,n
ln
n
1
x.1
ln
(E-1)
n x.1 Z XI:1
are rainfall event duration samples and
is the number of samples.
In performing the iterative method with Equation
E-1,
a value
„
of y is used along with values for x i and
n
to evaluate the right side
,,
of the equation. This evaluation gives a new estimate y which is then
used in the next iteration to again evaluate the right side of Equation
E-1.
The iterative process continues in this manner for a specified
„
number of iterations or until successive parameter estimates y differ
by an amount less than some desired value.
185
186
To begin this iterative procedure, an initial approximation for
A
y
is needed. This initial approximation can be obtained from a
relation, derived by Cohen (1965, pp. 584-585), that expresses the
Weibull coefficient of variation CV as a function of the shape parameter
y. Equation E-2 gives this relation.
CV -
// r[(2/y) + 1] - r 2 [(1/y) + 1]
r[(1/y) + 1]
(E- 2)
To provide for readily obtaining an approximation for y, the
relation of CV to y is given in graphical form in Figure E-1. When
using the graph of Figure E-1, one must first obtain the sample
coefficient of variation. The graph is then entered with this value
and a corresponding approximation for y is determined.
For implementing the iterative procedure represented by
Equation E-1, a Fortran computer program is given on the following
page. In addition to iteratively solving for the maximum likelihood
estimate of the Weibull shape parameter y, the program also provides
for computing the maximum likelihood estimate of the Weibull parameter
e.
187
CV
I(
Figure E-1. Relation of Weibull coefficient of variation
to shape parameter y (Cohen 1965, pp. 584-585)
CV
188
88
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APPENDIX F
PROGRAM FOR COMPUTING PROBABILITY MASS FUNCTION
OF CUMULATIVE SEDIMENT YIELD
The Fortran computer program,
incorporating the derived
recursive technique that was used with the sediment yield application,
is presented on the following pages.
190
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