Content uploaded by Lelys Bravo de Guenni
Author content
All content in this area was uploaded by Lelys Bravo de Guenni on Jul 10, 2014
Content may be subject to copyright.
Scientific
Procedures
Applied to the
Planning,
Design
and
Management
of
Water
Resources Systems (Proceedings of the Hamburg Symposium, August 1983).
IAHSPubl.no. 147.
On the estimation of the mean and variance
of annual sediment
yield,
based on basin and
storm characteristics
JOSE
R,
CORDOVA, IGNACIO RODRIGUEZITURBE
&
LELYS BRAVO
DE
GUENNI
Graduate Programme in Hydrology and Water
Resources, Universidad Simon Bolivar,
AP no. 80.659, Caracas, Venezuela
ABSTRACT The first and second moments of the annual
sediment yield probability distribution are analytically
derived based on basin and storm characteristics. The
Modified Universal Soil Loss Equation (MUSLE) developed
by Williams, is used to estimate sediment yield as a
function of the direct runoff volume and the peak runoff
rate produced by individual storms. The peak runoff rate
is obtained convoluting a triangular IUH with an
effective rainfall of constant intensity throughout the
storm duration. Effective intensity is estimated
simplifying Eagleson's equations for runoff rate, and its
validity is corroborated by comparing its results with the
SCS method. The number of storms in a year is taken to be
Poisson distributed with the rainfall intensity and
duration assumed to be exponentially distributed and
independent random variables. The probability density
function of the effective storm intensity is derived and
the MUSLE equation is used as a system kernel to obtain
the mean and variance of the annual sediment yield.
De 1'estimations de la moyenne et de la variance de
1'apport annuel de sédiments basée sur les
caractéristiques du bassin versant et des averses
RESUME Le premier et le second moments de la distribu
tion de probabilité de l'apport annuel de sédiments sont
analytiquement déduits à partir des caractéristiques du
bassin versant et de l'averse. L'équation universelle
modifiée des pertes en sol (MUSLE) mise au point par
Williams est utilisée pour estimer la production de
sédiments comme une fonction du volume de ruissellement
direct et du taux de pointe de ruissellement produits par
les averses individuelles. Le taux de pointe du
ruissellement est calculé par convolution à partir de
1'hydrogramme unitaire instantané (IUH) supposé de forme
triangulaire avec une averse effective d'intensité
constante pendant toute la durée de la pluie. L'intensité
effective est estimée en simplifiant les équations
d'Eagleson pour le taux de ruissellement et sa validité
est vérifiée en comparant les résultats obtenus avec ceux
de la méthode de SCS. Le nombre d'averse dans une année
est supposé suivre une distribution de Poisson,
125
126 Jose R.Cordova et al.
l'intensité et la durée de la pluie étant considérées
comme des variables indépendantes aléatoires avec
distribution exponentielle. La fonction de densité de
probabilité de l'intensité effective de l'averse est
déterminée et l'équation MUSLE et utilisée comme un
système "kernel" pour obtenir la moyenne et la variance
de l'apport annuel de sédiments.
INTRODUCTION
Traditionally the average annual sediment yield of a basin has been
predicted estimating the average annual gross soil erosion with the
Universal Soil Loss Equation (USLE) developed by Wischmeier & Smith
(1978) and then multiplying the result by a delivery ratio. There
are five factors playing a role in the USLE equation:
R = rainfall and runoff erosivity index
K = soilerodibility factor
LS = topographic factor
C = croppingmanagement factor
P = conservation practice factor
Those factors have been extensively documented and with the
exception of R are not dependent on geographical location. The
dependence of R on geographical location makes the use of USLE
restricted to regions whose R has been mapped through actual
measurements. Not only that, but given the large variations in
rainfall and runoff conditions even in regions not very far apart,
makes the USLE equation difficult to apply except in well studied
areas.
The above comments are also valid for the delivery ratio
which multiplies the gross soil erosion to give the average annual
sediment yield (Foster,
1981).
In addition, the methodology gives
only an estimation of the mean value of the annual sediment yield,
giving no information about the variance of the process which is an
important measure of the variability one may expect around the mean
value.
Williams in 1972, developed a sediment yield model through a
modification of the USLE. The model developed by Williams retains
the basic USLE structure and the parameters representing soil
erodibility, slope, cropping management and erosion control practice.
The main change in the USLE structure was the replacement of the
rainfall erosivity factor with a runoff factor that is expressed in
terms of total volume of runoff and the peak outflow discharge of
the response hydrograph. This runoff factor incorporates both
erosion and sediment transport forces. The socalled Modified
Universal Soil Loss Equation (MUSLE) attempts the prediction of the
sediment yield in a stormevent basis and avoids the use of the
delivery ratio and the parameter R through the incorporation of the
main characteristics of the hydrograph.
The Williams sediment yield model is expressed as
y = 11.8 (Q
Qp)
0

56
K C P LS (1)
where y is sediment yield in tonnes; Q is volume of runoff in m
3
and
3 —
1
Qp is peak flow rate in m s
The product (Q Q
p
) is called the runoff factor. This factor
Estimation of mean and variance of sediment yield 127
needs to be obtained by any of many methods of hydrological models
each of which has its particular advantages and drawbacks.
The main advantage of the Williams model is that it uses
parameters for which there is a wealth of information, since most
of them are those which are not geographically dependent, and are
widely used through the well documented USLE (Wischmeier & Smith,
1978) .
Coupling the MUSLE equation with a hydrological model that uses
storm characteristics to give the volume of runoff and the peak flow
rate it is possible to obtain an estimate of the sediment yield for
any rainfall storm event.
Using a probability description of the storm characteristics this
approach allows the derivation of expressions for the mean and
variance of storm sediment yields. In addition, describing the
number of storms in a year in a probabilistic manner we may then
obtain the first and second moment of the annual sediment yield.
ESTIMATION
OF THE
VOLUME
OF
RUNOFF
Working with the Phillip's infiltration equation, Eagleson (1978)
derived an expression which relates the depth of storm surface
runoff with the intensity and duration of the rainfall
(i
r
,t
r
),
for
a given initial soil moisture content and soil parameters.
According to Eagleson (1978) :
R = i
e
t
e
=
<i,
A
0
)t
r
S*(t
r
/2)
*r S to
> t,
(2)
where
:
t„ =
•c» \ 1 y» O
2 (1  s
Q
)
5n K(l) f(l) *
i
(d,s
0
)
10.43 m
(3)
(4)
K(l)(l + s )  w
o
(5)
i
e
= effective rainfall intensity;
t = effective storm duration:
e
R = surface runoff;
n = porosity;
K(l) = saturated hydraulic conductivity;
f(l) = saturated soil matrix potential;
$
i
(d,s
Q
)
= infiltration diffusivity function;
w = capillary rise from the water table;
m = pore size distribution index;
d = diffusivity index;
c = pore connectivity index;
s
Q
= initial soil moisture content expressed as a percentage
of the soil moisture under saturation conditions.
Typical values of the parameters n, m, K(l), ¥(1),
$^(d,s
0
),
d and c
128 Jose R.Cordova et al.
are given by Eagleson (1978) as functions of soil texture.
In equation (2), S* is known as the sorptivity, A
Q
is the
gravitational infiltration rate and t
Q
is the time of initiation of
the surface runoff.
Our main interest in this study lies in the estimation of the
total annual sediment yield. This annual value depends mainly on
those storms that produce large values of runoff and high peak
discharges.
It is reasonable to assume that in most of these cases
s
0
will tend to 1 and S^ and t
0
can be neglected. This assumption
allows us to establish the following relations (from equations (2),
(3),
(4) and (5)):
t
Q
= 0
(6)
R

±
e
t
e
<i,
o r
(7)
Since t,
one gets
= (i
T
A
0
) (8)
The surface runoff generation during a typical storm situation is
shown in Fig.l where the shaded area represents the distribution in
time of the generated volume of surface runoff by the storm in
consideration. When s
Q
tends to 1 we will approximate the shaded
area of Fig.l by a rectangle of equal area and duration t
Thus,
i
e
will be taken as a constant throughout the duration of the storm
which had a constant rainfall intensity i .
From equations (7) and (8), the total volume of surface runoff for
a rainstorm of intensity i
r
and duration t
r
will be:
FIG.l Surface runoff generation during a typical storm
situation.
Estimation of mean and variance of sediment yield 129
Q = 10
11
A (i
r
 A
Q
) t
r
(9)
Q = 10" A i
e
t
e
where Q is the total volume of surface runoff in m
3
,
A is the area
of the basin in km
2
and i„, A_ and t_ are the storm intensity
—
i
• — i
(cm h ), the gravitational infiltration rate (cm h ) and storm
duration (h), respectively.
ESTIMATION
OF THE
PEAK FLOW RATE
With the assumption of a certain form of the IUH it is possible to
obtain an explicit relationship to link the storm characteristics
with the peak outflow discharge in a basin.
Henderson (1963) has shown that for an IUH of a triangular form
the following relationship holds:
2 t„i„A ,, to
e e

(1  ^T~) if t
0
< tv
t
h
zt
b
2t
>
X1 L
e S
L
b
(10)
if t
e
> t
b
where Q is the peak discharge produced by an effective rainfall of
constant intensity i
g
and duration t , A is the area of the basin
(i
e
A is the equilibrium discharge) and t
b
is the base time of the
IUH equal to the time of concentration t of the basin. Moreover,
Henderson (1963) also shows that the exact form of the IUH is not
very important for the estimation of Q
p
as long as its peak, q , is
correct.
Since q
p
x t
b
= 2, one may rewrite equation (10) as
t
e %
I i
e
t
e
A q
p
(1 ^) if t
e
< t
c
(11)
i
0
t
0
if t. > t.
e e
e ' c
The peak of the IUH, q , can be estimated with the methodology
proposed by Nash (1960) or using the geomorphoclimatic IUH developed
by RodriguezIturbe et al.
(1982).
We will use here the impulse response function of the Nash model:
h(t)
=
SW
<
t/K)N_1
e_t/K
<
12
>
where T(.) is the gamma function of N and K are parameters which
need to be estimated from rainfallrunoff event type of data by
any of several well known methodologies. The IUH peak is given by
the expression
e
Nl
= S_ (13)
4
P KF(N)
130 Jose R.Cordova et al.
STOCHASTIC DESCRIPTION
OF
STORM CHARACTERISTICS
The precipitation process is modelled as a sequence of rectangular
pulses of random height (storm intensity) and random length (storm
duration),
positioned in time by a Poisson process. Todorovic &
Yevjevich (1969) and Restrepo (1979) have successfully used this
model in many climatic conditions over the United States. The
Poisson distribution is expressed as:
(vt)
n
e~
vt
fn
t
<
n
> = —, n = 1, 2 ... (14)
where n is a number of storms in the interval (0,t) and v is a
positive constant which represents the average arrival rate of storms
in the time interval under consideration.
It has been observed that under many different climatic
conditions,
the intensity and duration of rainfall events can be
taken as independent random variables described by exponential
distributions (e.g. Grayman & Eagleson,
1969).
Thus
f
Ir
(i
r
)
= ae~
ai
r i
r
I 0 (15)
f
T
(t
r
) = ôe
_ôt
r t
r
> 0 (16)
where 1/a is the average storm intensity and 1/6 is the average storm
duration.
PROBABILITY DISTRIBUTION
OF THE
EFFECTIVE RAINFALL INTENSITY
Figure 2 shows the relationship between rainfall intensity and
effective rainfall intensity implied by equation (8). Under the
assumption that the rainfall intensity is exponentially distributed
and making use of equation (8), the cumulative distribution function
of the effective rainfall intensity can be written as:
F
I
e
(
i
e> =
F
I
r
(i
e
+ A
o> (17)
fie+A.
air
ae
x
di_ (18)
r
o
F
Ie
(i
e
)
= 1 
e

a(i
e
+A
o>
(19)
The probability density function of the effective rainfall
intensity fj (i
e
) is then obtained differentiating equation (19)
dF
T
(i )
f
T
(i
e
) = —— = isae e
(2
o)
e
di„
Estimation of mean and variance of sediment yield 131
STORM INTENSITY
FIG.2 Relationship between i
r
 i
e
and fj (i
r
)  fr (i
e
)
I
e
^e'
where
aA,
Computing the first and second moments of equation (20) the mean
E [I
e
] = B/a
var [I
e
] = g(2 e)/a
z
(21)
(22)
Note that the cdf of i
e
is a compound distribution, since it has a
spike at the origin equal to 1  3. This spike is calculated from
equation (19) as equal to:
P [i
e
= 0] = FT. (0) = 1
(23)
This value represents the fraction of the storms that do not
produce surface runoff.
MEAN
AND
VARIANCE
OF THE
ANNUAL SEDIMENT YIELD
Equations (9) and (11) may be substituted for Q and Q_ in
equation (1); one obtains in this manner an estimation of the
sediment yield for any given storm event
:
y =
0[W A
2
q
p
i
e
2
t
r
2
(1
0[W A
2
i
e
2
t
r
]
r
056
t
r
q
p
/4)]'
< t,
*r I *c
(24)
132 Jose R.Cordova et al.
where 0 = 11.8 K C P LS, W is a units coefficient equal to 27 800 and
2 ~ 1
the units of the different terms are as follows: A (km ), q
p
(h ),
i
e
(cm h

), tr (h), and y is in tonnes.
Under the framework of the Nash model specified by equation (12)
the parameters K and N remain constant throughout the different
storm events. This implies that q
p
is also a constant in
equation (24). The use of equation (10) assumed a triangular IUH
for which t
c
= 2/q
p
is also a constant and thus the only stochastic
variables in equation (24) are the effective intensity and duration
of a storm. The probability distribution of i
e
is given by
equation (20) and the distribution of t
r
is specified in
equation (16).
MEAN
OF
STORM SEDIMENT YIELD
The mean value of a storm sediment yield can be obtained as:
E[y] = [E
ie
[Et
r
[y/i
e
]]] (25)
where
E
tr
[yi
e
] =
ftr
6(WA
2
q
p
i
e
2
t
r
2
(1  t
r
q
p
/4))
0

56
6e
6t
r
0(WA
2
i
e
2
t
r
)'
ôt,
dt,
(26)
Integrating equation (26) yields:
E[yi
e
] = ew
0
'
56
6(i
e
A)
1
"
12
[q
p
0

56
I]
where
r(1.56,6t„)] (27)
IT
=
(1 
q
p
t
r
/4)
0.56
ôt,
dt,
(28)
r(a,x) = r<a) 
Y(a,x);
F
(a)
= gamma function;
y(a,x) = incomplete gamma function,
. 56
To integrate equation (28), the term (1  t q /4) will be
i p
approximated by a Taylor series expansion around the mean value of
,056 /, Z_/yi\056,
ri
,,,,.,. I %/ . s
/o»'*
1
!
(1  t
r
q
p
/4)
u
y
b
(1  t
r
q
p
/4)
u

bb
+ 0.14(t
r
 t
r
)(q
p
)AQ
c
0.0154 (t
r
 t
r
) q_ /2fi
l . i k
(29)
Introducing (29) in (28) and integrating it yields:
Estimation of mean and variance of sediment yield 133
I
I = C
1
6~
2

12
Y
(2.12,6t
c
) + C
2
<r
3

12
Y(3.12,ôt
c
)
:
3
where
+ Coô
l4

12
Y(4.12,6t„) (30)
C
±
= Q
0
'
56
+ 0.14 t
r
q
p
/Çl°
hh

0.0154
t
r
2
q
p
2
/2.Q
1
"
lk
C
9
= 0.14 q
/fi
0
'
1
"*
+
0.0153
q
n
2
/u
1,lk
(31)
C
3
= 0.0154 q
2
/2Q
1

1
'*,
where £2 = 1
t
r
q
p
/4
The next step is the integration of equation (27) over the pdf of
l
e.
E[y] = E [E [y/i ]] = GW
0

56
6A
1
'
12
[q
0
'
56
I, +
6"
1
'
5E
T
(1.56,6t„)]
i t
r
e
P J
e
r
n
1.12 aie J•
ga I' e
e
di
e €
which results
in:
E[y]
=
GW
0
'
56
ÔA
1

12
[q
0,56
^
+ 6"
1
'
5 6
T
(
1.
56,
6t
c
)
]
Bo"
1
'
1 2
V
(2
.12)
(32)
VARIANCE
OF
STORM SEDIMENT YIELD
By definition the variance of the random variable y is equal to:
var[y] = E[y
2
]  E
2
[y] (33)
The second term of the righthand side of (33) can be computed
from (32).
Thus,
to compute the variance it is only necessary to
obtain an expression for the expected value of the square of a
random variable y.
From equation (24) we have that
y
I G
2
[WA
2
q
p
i
2
t
2
(1 
tj.qp/4)]
1,12
t
r
< t
c
(34)
j e
2
[WA
2
i
2
t
r
]
1

12
t
r
> t
c
Taking expectation over
t
conditional
on a
given
i , in
equation
(34) one
gets:
E
t
[y
2
/i
e
]
=
r
rt
e
2
W
1
'
12
6(A.i
)
2
'
24
[q
1
'
12
 V" (1  t q /4)
1
'
12
e
Ôt
r
d
t
r
+ i
t£
12
e"
ôtr
dt
r
(35)
134 Jose R.Cordova
et al.
To integrate this equation
we
will approximate
the
exponent
of
the term
(1 
t
r
q„/4)
from
1.12 to 1.
Performing
the
integration
of
(35),
it
yields:
E
t
[y
2
/i
e
]
= 9
2
W
112
<5(A.i
e
)
2

2
"
[q^'
12
ô""
3

2l
*Y(3.24,
6t
c
)

q
2

12
ô"" •
2l
*Y(4.24,ôt
c
)/4+ <T
2
"
12
T
(2 .12
,
6t
c
)
]
(36)
and integrating
(36)
over
i the
final result
is
E[y
2
] =G
2
W
1

12
ÔA
2

2t
a
_2
'
2l
*6r(3.24)
[q^
"
12
ô"
3
'
2l
*y(3
.
24,
6t„)

qp'
12
Ô
1
*
21
* y(4.24 6t
c
)/4+
6"
2
'
12
T(2.12,
<5t
c
)]
(37)
MEAN
AND
VARIANCE
OF
THE
ANNUAL SEDIMENT YIELD
Equations
(32) and (37)
allow
us to
compute
the
mean
and
variance
of
the sediment yield,
y,
produced
by a
rainstorm.
The
annual sediment
yield,
Y, can be
expressed
as:
Y
= ^
=1 yj
(38)
where
yj is the
sediment yield
of
storm
j and N is the
number
of
storms
in a
year with distribution given
by
equation
(14).
The mean
and
variance
of the
annual sediment yield
are
then:
E[Y]
= vt E[y] (39)
var[Y]
= vt E[y
2
] (40)
where
t is one
year
and v is the
rate
of
occurrence
of
storms
in
units
of
[t]
_1
.
CASE STUDY
The methodology
has
been applied with good results
to
several river
basins
in
Venezuela.
We
will present here
as a
case study
the
application
to the
Santo Domingo River watershed. This river basin
is located
in the
Andes region
of
western Venezuela.
The
area under
consideration located upstream
of a
proposed
dam
covers
an
extension
of
428 km
with
a
mean annual precipitation
of 1400 mm.
In
a
previous study (Henao,
1983) the
main geomorphoclimatic
characteristics
of the
basin were determined.
A
summary
of the
climatic information that
is of
interest
in
this application
is the
following:
average number
of
storms
in a
year:
310
average storm intensity:
0.13 cm h
_1
average storm duration:
3.50 h
Henao (1983) also applied
the
geomorphoclimatic methodology
developed
by
Cordova
&
RodriguezIturbe (1983)
to
estimate
the
extreme flow probabilities.
In
this estimation,
it was
determined
that
the
soil conservation service curve number
to
describe
the
Estimation of mean and variance of sediment yield 135
infiltration in the basin which best fitted the analytical
distribution of extreme flows to the historical data was 88 (US
Bureau of Reclamation,
1965).
Since sediment yield in a basin is
basically determined by the large storm events whose infiltration
characteristics correspond to a SCS curve number of 88, we adopted
this value of curve number to estimate effective rainfall.
In order to corroborate the analytical results given by
equations (32), (37), (39) and (40), a simulation model was
implemented. The model was run for 100 years, in each year the
number of storms were obtained from a Poisson distribution and for
each storm event the values of intensity and duration were obtained
from exponential distribution. The historical parameters of the
Santo Domingo River basin were used in the simulation model.
After generating the series of storm events the effective
rainfall intensity was computed in two
ways:
through equation (8)
and by using the soil conservation service method (US Bureau of
Reclamation,
1965).
The objective here was to compare the results obtained with the
methodology developed in this paper with those obtained through a
methodology like the one of the SCS which computes the effective
rainfall characteristics without the assumptions involved in
equations (6), (7) and (8). After estimating i
e
with the SCS method
the peak of the IUH was estimated with the criteria of reproducing
the extreme value historical distribution shown in
Fig.3.
In other
Probability
FIG.3 Comparison between historical Gumbel and
simulated extreme flows.
136 Jose R.Cordova et al.
basins where good rainfallrunoff data are available we may
estimate q_ with any standard methodology. For a q_ = h
_1
the
simulated floods fitted quite well the historical data as it is
shown in
Fig.3.
Equations (9) and (10) were then used to compute
the storm runoff volume and the peak runoff rate which are input into
the MUSLE equation. The parameter Q = 11.8 K C P LS of the MUSLE
equation was estimated using the mean sediment yield value for the
Santo Domingo basin obtained by several authors using other
approaches and field measurements (G & Y Estudios y Proyectos,
1983).
Of course, 8 in a practical case would be estimated through the
standard methodology to calculate K, C, P, and LS.
The simulation model gave in the above manner estimates of the
mean and variance of the sediment yield per storm and at an annual
level of aggregation. The numerical values are shown in Table 1.
The second set of runs of the simulation model was carried out
estimating i
e
through equation (8). We used the same q
p
and 0 as
before but in the use of equation (8) the parameter A needs to be
specified. For the purpose of the comparison we are interested if a
previous simulation. This is indeed the case for A
0
= 0.26 cm h
_1
which produces simulation results very similar to the previous
ones.
The comparison is shown in Table 1 which also gives the numerical
values obtained through the analytical expressions. The analytical
results agree extremely well with those found through simulation.
The sensitivity of the mean and standard deviation of storm
and annual sediment yield to the value of A was also studied for
the case of the Santo Domingo River basin. Figure 4(a) and (b) show
the results obtained analytically and through simulation. Some
aspects of these figures are worth noting.
At the storm level the standard deviation of the yield is
considerably larger than the mean and even for large values of A
Q
(little production of runoff and sediment) the standard deviation is
much larger than the mean. This points out the difficulty in making
estimates of storm sediment yield and how little this can mean in
TABLE 1 Results of simulation models and analytical expressions of
mean and standard deviation of storm and annual sediment yields
Description Results of the simulation Analytic
model: results
Using SCS Using equation (8)
NC = 88 A„ = 0.26 cm IT
1
Average storm sediment
yield (t) 798.6 789.1 809.2
Standard deviation of storm
sediment yield (t) 4603.3 4657.4 4802.0
Average annual sediment
yield (t x 10
3
) 248.53 244.62 250.84
Standard deviation of annual
sediment yield (t x 10
3
) 93.32 83.17 85.74
Estimation of mean and variance of sediment yield 137
(a)
O Simulated
(b) io i
<
f
co
n
z.
<r
T
<
Q
tu
CO
i
IT
o
h
CO
.20
PARAMETER
.30
A„
(cm/h)
FJG.4 Comparison between simulated and analytical mean
and standard deviation of annual sediment
yield.
138 Jose R.Cordova et al.
practical cases if no mention of the standard deviation is provided.
At the annual level the mean is larger than the standard
deviation with a sharp decrease with an increase of A . As expected
the annual sediment yield has a much smaller statistical dispersion
around its mean value than the storm yield; nevertheless the mean
value is now affected much more by the value of A
Q
than what it was
in the cases of the storm event analysis.
CONCLUSIONS
AND
RECOMMENDATIONS
The methodology developed in this paper has two implicit assumptions
which are the validity of the MUSLE equation and the goodness of
Henderson's approach for estimating peak runoff rate. Given that
these two equations are reasonable representations of the real
world, the methodology presented here allows the estimation of the
mean and the variance of storm and annual sediment yields, in a quick
and reliable manner.
The results obtained in the case study corroborate the validity
of the derived equations and all the assumptions implicit in the
problem formulation.
ACKNOWLEDGEMENTS This research was funded by the Venezuelan
National Research Council
(CONICIT),
under the grant SI,1212.
J.R.Cordova and I.RodriguezIturbe also thank Fundacion Instituto
Internacional de Estudios Avanzados
(IIDEA),
for research support.
REFERENCES
Cordova, J.R. & RodriguezIturbe, I. (1983) Geomorphoclimatic
estimation of extreme flow probabilities. J. Hydrol. 65, 159173.
Eagleson, P.S. (1978) Climate soil and vegetation. 5. A derived
distribution of storm surface runoff. Wat. Resour. Res. 14 (5),
741748.
Foster,
G.R. (1981) Modeling the erosion process. In: Hydrologie
Modeling of Small Watersheds (ed. by C.T.Haan, H.P.Johnson &
D.L.Brakensiek),
297379. An American Society of Agricultural
Engineers monograph.
Grayman, W.M. & Eagleson, P.S. (1969) Streamflow record length for
modeling catchment dynamics. Report no. 114, Hydrodyn. Lab.,
Dept of Civil Engineering, Massachusetts Institute of
Technology, Cambridge, USA.
G. & Y.Estudios y Proyectos (1975) Complejo hidroelectrico General
Jose Antonio
Paez.
Sedimentos en el embalse. CADAFE. Empresa
nacional de energfa eléetrica. Caracas, Venezuela.
Henao,
A. (1983) Estudio geomorfoclimâtico de la cuenca alta del r£o
Santo Domingo. Tesis de Master. Graduate program in Hydrology
and Water Resources, Universidad Simon Bolivar, Caracas,
Venezuela.
Henderson, F.M. (1963) Some properties of the unit hydrograph. J.
Geophys. Res. 68 (16), 46854793.
Nash,
J.E. (1960) A unit hydrograph study with particular reference
Estimation of mean and variance of sediment yield 139
to British catchments. Proc. Instn Civ. Engrs 17,
249282.
Restrepo,
P.
(1979) Some géomorphologie inferences from hydrology.
S.M.Thesis, Massachusetts Institute
of
Technology, Cambridge,
USA.
RodriguezIturbe,
I.,
Gonzalez Sanabria,
M. &
Caamano,
G.
(1982)
On
the
climatic dependence
of the IUH: a
rainfallrunoff analysis
of
the
Nash model
and the
geomorphoclimatic theory.
Wat.
Resour.
Res.
18
(4), 887903.
Todorovic,
P. &
Yevjevich,
V.
(1969) Stochastic process
of
precipitation. Hydrol. Pap. 5, Colorado State University, Fort
Collins, USA.
US Bureau
of
Reclamation (1965) Design
of
Small Dams.
US
Government
Printing Office, Washington,
DC, USA.
Williams,
J.R.
(1972) Sediment yield prediction with universal soil
loss equation using runoff energy factor.
In:
Proceedings
of
Sediment Yield Workshop, USDA, Sedimentation Laboratory, Oxford,
Mississippi.
Wischmeier,
W.H. &
Smith,
D.D.
(1978) Predicting Rainfall Erosion
Losses  a Guide to Conservation Planning. USDA Agricultural
Handbook
no. 537.