β-distribution, described by the general expression given by Harr (1977)
β
α
y a b y
1
f (y) =
(A7)
(b a)B(α + 1, β + 1) b a b a
(y a)2
α=
(1 y) (1 + y)
~
~
2
Sy
α+1
β=
(α + 2)
~
y
~
y a
y=
b a
where a and b are the minimum and maximum values of y, and B is a beta function. When
α and β have the same sign, the β-distribution is unimodal and bell-shaped. The coeffi-
cients of skewness and kurtosis of the β-distribution can be readily obtained. With this
distribution, the probability of y in a given range can be determined.
Each model that was used to apply the PEM in this study has the same main program
and a unique subroutine called EQN that contains the specific equation being solved. The
main program performs all input and output functions and applies the PEM. The input
variables and file structure are described in the comments contained in the main program,
as well as the definitions of all variables used. In subroutine EQN each independent
random variable has a number that gives its position in the array VAR. In the output file,
the mean, coefficient of variation, standard deviation, and mean plus and minus standard
deviation are given for each independent random variable by position in VAR. This list is
followed by the mean, variance, standard deviation, coefficient of variation, and estimates
of the maximum and minimum of the corresponding dependent variable. The EQN sub-
routines developed in this study are
ROSICTHK, used to obtain the change in ice thickness from the air temperatures in
the basin using the temperature index model,
ROSQICE, used to obtain the discharge loss from the stream that goes into storage
in a reach as ice,
ROSQST, used to obtain the discharge loss to a stream to satisfy a change in water
storage in a reach,
ROSQLCL, used to obtain the discharge from the subbasin to a stream reach,
ROSQYLD, used in cases with a negative local inflow from the subbasin to obtain
the flow loss per square meter of wetted stream channel area.
Listings of these subroutines and of the main program follow.
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