Geoscience Reference
In-Depth Information
(d)
can be used to calculate the probability of exceeding the
T
r
year flood each year during 5
subsequent years; that probability is [(
T
r
−
1)
/
T
r
]
5
;
(e)
the probability of not exceeding the
T
r
year event during the first3yofa5yperiod, and of
exceeding that event in each of the remaining2yis(1
/
T
r
)
2
(1
−
1
/
T
r
)
3
.
13.12
An annual flood record for a certain river is given below.
Maximum flow rate
Maximum flow rate
(m
3
s
−
1
)
(m
3
s
−
1
)
Year
Year
1991
269
1998
331
1992
374
1999
309
1993
207
2000
427
1994
241
2001
204
1995
393
2002
402
1996
289
2003
229
1997
535
These data represent a sample from a population with some unknown probability distribution. Do
not assume that the data obey some a priori distribution for parts (a), (b), (c) and (d). (a) Estimate
the median flood from this sample. (b) Estimate the mean flood from this sample. (c) Estimate
the 7 y flood from this sample. (d) Estimate, from this sample, the probability that next year the
maximum flow rate will lie between 331 and 393 m
3
s
−
1
. (e) Assume now that these data can
be fitted by the exponential distribution. The density function is
f
(
x
)
=
λ
e
−
λ
x
for
x
≥
0, and
f
(
x
)
=
0 otherwise. Estimate the parameter
λ
of this function from the available record by means
of the method of moments.
13.13
What is the probability that a single observation will exceed the mean
μ
when the probability
distribution function is the first asymptote (13.52)?
13.14
At a river gaging station, which has been operated for a very long time, it has been found that the
probability distribution of the annual maximal flows can be described by
F
(
Q
)
=
Q
/
(
A
+
Q
),
in which
Q
is the magnitude of these annual events and
A
is a constant. Derive the probability
distribution for decadal peak discharges (i.e. the maximal flows experienced in non-overlapping
periods lasting 10 successive years) from the distribution of the annual peaks. Give the result in
terms of
A
and
Q
.
It has been observed that the annual peak discharges
Q
(in m
3
13.15
s
−
1
) for a given river can be described
by Fuller's formula, as follows
Q
=
294 (1
+
0
.
3ln
T
r
), where
T
r
is the recurrence interval (in
years) of the peak discharge of magnitude
Q
(a) Derive the probability distribution function
F
=
F
(
Q
) from Fuller's formula. (b) What is the probability that 700 m
3
s
−
1
will be exceeded
every single year of a given 4 y period? (c) What is the probability that the 700 m
3
s
−
1
flood will
be exceeded only once, namely at the end of this 4 y period? In other words, what is the probability
that this flood will not be exceeded during the first 3 y and then will be exceeded during the last
year?
.
