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Hydrological Sciences–Journal–des Sciences Hydrologiques, 50(6) December 2005
1151
Uncertainty and sensitivity analysis of runoff and
sediment yield in a small agricultural watershed
with KINEROS2
MOHAMED M. HANTUSH & LATIF KALIN
United States Environmental Protection Agency, National Risk Management Research
Laboratory, 26 West Martin Luther King Dr., Cincinnati, Ohio 45268, USA
[email protected]
Abstract Using the Monte Carlo (MC) method, this paper derives arithmetic and
geometric means and associated variances of the net capillary drive parameter, G, that
appears in the Parlange infiltration model, as a function of soil texture and antecedent
soil moisture content. Approximate expressions for the arithmetic and geometric
statistics of G are also obtained, which compare favourably with MC generated ones.
This paper also applies the MC method to evaluate parameter sensitivity and predictive
uncertainty of the distributed runoff and erosion model KINEROS2 in a small experimental watershed. The MC simulations of flow and sediment related variables show that
those parameters which impart the greatest uncertainty to KINEROS2 model outputs are
not necessarily the most sensitive ones. Soil hydraulic conductivity and wetting front net
capillary drive, followed by initial effective relative saturation, dominated uncertainties
of flow and sediment discharge model outputs at the watershed outlet. Model predictive
uncertainty measured by the coefficient of variation decreased with rainfall intensity,
thus implying improved model reliability for larger rainfall events. The antecedent
relative saturation was the most sensitive parameter in all but the peak arrival times,
followed by the overland plane roughness coefficient. Among the sediment related
parameters, the median particle size and hydraulic erosion parameters dominated
sediment model output uncertainty and sensitivity. Effect of rain splash erosion
coefficient was negligible. Comparison of medians from MC simulations and
simulations by direct substitution of average parameters with observed flow rates and
sediment discharges indicates that KINEROS2 can be applied to ungauged watersheds
and still produce runoff and sediment yield predictions within order of magnitude of
accuracy.
Key words capillary drive; infiltration; KINEROS2; Monte Carlo simulation; runoff;
sediment transport; sensitivity; uncertainty; watershed model
Analyse d’incertitude et de sensibilité des simulations d’écoulement
et de transport solide de KINEROS2 dans un petit bassin versant
agricole
Résumé Grâce à la méthode de Monte Carlo (MC), cet article déduit des expressions
arithmétiques et géométriques, ainsi que les variances associées, du paramètre G de
l’écoulement capillaire net du modèle d’infiltration de Parlange, comme fonction de la
texture du sol et de l’humidité du sol antérieure. Des expressions approximatives des
statistiques arithmétiques et géométriques de G sont également obtenues, qui sont
comparables avec celles de la méthode MC. Cet article applique également la méthode
MC pour évaluer la sensibilité des paramètres et l’incertitude de prévision du modèle
distribué d’écoulement et d’érosion KINEROS2 pour un petit bassin versant
expérimental. Les simulations MC des variables d’écoulement et de transport solide
montrent que ces paramètres qui impliquent la plus grande incertitude sur les sorties du
modèle KINEROS2 ne sont pas nécessairement les plus sensibles. La conductivité
hydraulique du sol et l’écoulement capillaire net du front d’humectation, suivis par la
saturation relative effective initiale, dominent les incertitudes sur les simulations
d’écoulement et d’érosion à l’exutoire du bassin versant. L’incertitude de prévision
mesurée par le coefficient de variation décroît avec l’intensité de la pluie, ce qui
implique une meilleure confiance dans la modélisation pour les événements pluvieux
plus importants. La saturation relative antérieure est le paramètre le plus sensible, pour
tous les aspects sauf pour le temps de réponse, suivie par le coefficient plan de rugosité
de surface. Parmi les paramètres de transport solide, la taille médiane des particules et
les paramètres d’érosion hydrique dominent l’incertitude et la sensibilité par rapport aux
Open for discussion until 1 June 2006
Copyright  2005 IAHS Press
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Mohamed M. Hantush & Latif Kalin
simulations de transport solide. L’influence du coefficient d’érosion par effet splash est
négligeable. La comparaison des médianes des simulations MC et des simulations
obtenues pour des paramètres moyens avec des écoulements et des transports solides
observés indique que KINEROS2 peut être appliqué à des bassins versants non jaugés et
qu’il génère des simulations d’écoulement et de transport solide qui respectent un ordre
de grandeur de précision.
Mots clefs écoulement capillaire; infiltration; KINEROS2; simulation de Monte Carlo;
écoulement; transport solide; sensibilité; incertitude; modèle de bassin versant
INTRODUCTION
Sediment transport and fate studies arise in problems related to soil erosion, loss of
water storage capacity in reservoirs due to sediment filling, and water quality impairment due to nutrients and pollutants attached to sediments originating from natural and
anthropogenic sources. Distributed, process-based hydrological and transport models
are increasingly relied upon by resource managers and decision makers as planning
and management tools to mitigate storm related peak flows, reduce soil erosion, and
assess alternative abatement strategies to reduce sediment discharge to surface water
bodies. Although previous studies have questioned the true merits of complex, distributed models (Loague & Freeze, 1985; Loague, 1990; Wilcox et al., 1990; Refsgaard,
1994; Michaud & Sorooshian, 1994; Atkinson et al., 2003), some of these studies,
however, agreed that this class of models outperforms the simpler, lumped parameter
models in ungauged watersheds. It could also be argued that physically-based hydrological models have a wider range of applicability, because they simulate in some
detail the individual components of the entire surface flow, erosion, sediment transport, and deposition processes. Atkinson et al. (2003) compared eight models with
varying complexity and showed that improved process descriptions (i.e. added model
complexity) improved the accuracy of predictions to hourly flow at the outlet of
Mahurangi catchment in New Zealand. They indicated that complex fully-distributed
models outperformed lumped models under summer conditions, and that the latter are
adequate for wet winter conditions.
This paper tests predictive uncertainty and parameter sensitivity of the distributed
Kinematic Runoff and Erosion model, KINEROS2, in a small experimental watershed.
The KINEROS2 model (Smith et al., 1995b) is an extension of the original, eventbased model KINEROS which was developed by Woolhiser et al. (1990). Although
KINEROS2 was modified for a new soil moisture redistribution algorithm during
rainfall hiatus (Smith et al., 1993), the lack of a complete soil moisture balance
component renders the model unsuitable for continuous time simulations; particularly,
during periods of high evapotranspirative losses. Because it is event-based, one of the
main requirements of a successful KINEROS2 application is accurate inference of
antecedent soil moisture distribution; unless this quantity is made available, lack of a
complete soil moisture balance accounting will continue to limit the scope of the
model applications. To alleviate this problem, Goodrich et al. (1994) demonstrated the
use of remote sensing as a means to infer initial soil moisture content for event-based
model applications.
Several applications of KINEROS have been reported in the literature, ranging from
applications to range lands (Osborn & Simanton, 1990), semiarid watersheds (Michaud
& Sorooshian, 1994), unpaved mountain roads (Ziegler et al., 2001), and agricultural
Copyright  2005 IAHS Press
Uncertainty and sensitivity analysis of runoff and sediment yield
1153
watersheds (Kalin et al., 2003, 2004a,b). Kalin et al. (2003) investigated the effect of
geomorphological resolution on flow and sediment transport by utilizing the KINEROS
model. Using the concept of unit linear sedimentographs, Kalin et al. (2004a,b)
combined KINEROS with linear optimization to identify sediment source areas in two
agricultural watersheds. Goodrich (1990) examined the relationship between model
spatial resolution and KINEROS model accuracy. He showed that, without calibration,
the model still produced accurate runoff simulations for small catchments (less than 0.1
km2). Woolhiser et al. (2000) applied the KINEROS2 model to estimate channel
infiltration from surface runoff at the Solitario Canyon watershed. Other applications of
KINEROS include Smith et al. (1995a, 1999), and Ziegler et al. (2000).
Since infiltration excess is the primary runoff generating mechanism in Hortonian
catchments, the importance of the infiltration process cannot be understated, especially
during moderate to low intensity storms. KINEROS2 describes the infiltration process
by solving the Parlange et al. (1982) equation, which is a generalization of the wellknown Green-Ampt and Smith-Parlange (Smith & Parlange, 1978) infiltration models.
The model utilizes the Smith et al. (1993) soil moisture redistribution algorithm during
rainfall hiatus. The infiltration process depends on three parameters: saturated
hydraulic conductivity, net capillary drive, and soil water deficit. While most of the
above cited applications of KINEROS2 focused on the effect of soil permeability and
antecedent soil moisture deficit, with the exception of Smith & Goodrich (2000), none
of these studies analysed thoroughly the effect of uncertainty in the net capillary drive
parameter on infiltration and runoff generation. Smith & Goodrich (2000) presented a
model based on Latin hypercube sampling that can simulate infiltration and runoff
taking heterogeneity in saturated hydraulic conductivity and the capillary drive
parameter into account.
Using the Monte Carlo method, this paper investigates the impact of soil, channel
and land cover parameters uncertainties on KINEROS2 performance and sensitivities
in a small, experimental watershed. The predominant runoff generation mechanism in
this watershed is infiltration excess (Kalin et al., 2003). Stochastic sensitivity
approaches have proven to give the best response in parameter uncertainty analysis
(Tiscareño-Lopez et al., 1993; Veihe & Quinton, 2000; Castillo et al., 2003). In this
study, model output spread in response to a parameter uncertainty is expressed in terms
of probability of exceedence curve and related coefficient of variation generated by the
Monte Carlo method. Therefore, model output variance is tested over the full range of
reported parameters values. The objectives of this study are: (a) to derive conditional
probability density functions and expressions for the first-two moments of the wetting
front capillary drive parameter as a function of antecedent soil moisture condition and
soil texture; (b) to explore the effect of parameter uncertainty on KINEROS2 output and
examine model parameter sensitivities as a guidance to model calibration in comparable
watersheds, and (c) to test model runoff and sediment discharge predictive uncertainty
under conditions of parameter uncertainty in an experimental, agricultural watershed.
DISTRIBUTED HYDROLOGICAL MODEL KINEROS2
The KINEROS2 model is a modified version of the KINEROS model, which is a
distributed, event-oriented, physically-based model describing the processes of surface
Copyright  2005 IAHS Press
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Mohamed M. Hantush & Latif Kalin
runoff and erosion from small agricultural and urban watersheds (Woolhiser et al.,
1990). The watershed is represented by cascades of overland flow planes, channels,
and ponds, in which flow and sediments are routed through a cascade of rectangular
plane elements and, ultimately, to the channels. The cascades allow rainfall, infiltration, runoff and erosion parameters to vary spatially. This model may be used to
determine the effects of various artificial features such as urban developments, small
detention reservoirs, or lined channels on flood hydrographs and sediment yield.
Background theory
Overland flow When the rainfall rate exceeds the infiltration capacity, Hortonian
overland flow begins. The KINEROS2 model assumes one-dimensional flow in each
plane and solves the kinematic wave approximation of the overland and channel flow
equations using implicit finite differences. The flow rate is related to the channel flow
cross-sectional area or overland flow depth through Chezy or Manning flow resistance
relationships. In the kinematic wave approximation, the channel or bed slope
approximates the friction slope.
Erosion and sediment transport The sediment transport equation is described by
the following mass balance equation (Woolhiser et al., 1990):
∂
∂
( AC ) + (QC ) − er ( x, t ) = q s ( x, t )
∂t
∂x
(1)
in which C is the volumetric sediment concentration [L3 L-3]; A is the channel crosssection area or, for overland flow, it is equal to the product of flow depth and plane
width normal to flow direction [L2]; Q is the is the channel or overland plane discharge
[L3 T-1]; er is sediment erosion/deposition rate [L2 T-1] given below; and qs is the rate
of lateral sediment inflow for channels [L3 T-1 L-1]. In KINEROS2, er is composed of
rainfall splash erosion rate gs and hydraulic erosion rate gh:
er = g s + g h
(2)
Rainfall splash erosion is given by (Woolhiser et al., 1990):
g s = c f exp(−c h h) i r
= 0
r>0
r<0
(3)
in which cf is a positive constant [T]; h is the flow depth [L]; ch is damping coefficient
for splash erosion [L-1]; i is the rainfall rate [L T-1]; and r is the excess rainfall (rainfall
rate minus interception and infiltration) [L T-1]. The exponential term represents the
reduction in detachment by raindrops caused by increasing depth of water. In channel
flow, this term is usually equal to zero: the accumulating water depth absorbs nearly all
the imparted energy by the raindrops. The hydraulic erosion represents the rate of
exchange of sediment between the flowing water and the soil over which it flows. Such
interplay between shear force of water on the loose soil or channel bed and the
tendency of the soil particles to settle under the force of gravity may be described by
this first-order rate expression:
Copyright  2005 IAHS Press
Uncertainty and sensitivity analysis of runoff and sediment yield
g h = c g (C * − C ) A
1155
(4)
in which C* is the volumetric concentration at equilibrium transport capacity [L3 L-3];
and cg is a transfer rate coefficient [T-1]. This relationship assumes that if C exceeds
equilibrium concentration, C*, deposition occurs and when C < C* erosion occurs. The
hydraulic erosion parameter cg represents erodibility when C* is greater than C, and is
usually very high for fine, noncohesive material, and very low for cohesive material.
Conversely, cg is a function of the particle fall velocity when deposition is occurring;
i.e. when C exceeds C* (Woolhiser et al., 1990). The particle fall velocity and cg
depend on the median particle diameter, d50 [L]. Several expressions for C* are
available in literature (see, e.g. Woolhiser et al., 1990). The KINEROS2 model
employs the Engelund & Hansen (1967) formula.
It is noted that C* decreases with median particle size, which means that deposition
of coarse-textured eroded soils is more likely than fine-textured soils. Equation (4)
predicts that fine textured soils are more susceptible to hydraulic erosion than coarser
particles because of greater transport capacity.
Infiltration At the beginning of a storm and prior to ponding, the infiltration rate
is rain limited and equal to the rate of precipitation. If the rainfall intensity is greater
than the saturated hydraulic conductivity, then at the onset of runoff, the infiltration
rate approaches the infiltration capacity which is described by this equation (Parlange
et al., 1982):
f (t ) = K s +
αK s
exp{αF (t ) /[(G + h)(θ s − θ i )]} − 1
(5)
where f(t) is the infiltration capacity [L T-1]; F(t) is the cumulative depth of the water
infiltrated into the soil [L]; θs is the soil porosity [L3 L-3]; θi is the initial (antecedent)
soil moisture content; α is a parameter between 0 and 1, for sand α = 0 and for wellmixed loam α = 1; and Ks is the soil saturated hydraulic conductivity [L T-1]. The
value α = 0 reduces equation (5) to the familiar Green-Ampt infiltration method and
α = 1 simplifies equation (5) to the Smith-Parlange (1978) model. For most soils, α =
0.85 has been recommended (Parlange et al., 1982). The effective net capillary drive G
is the integrated capillary head across the wetting front (Smith et al., 1993):
ψi
G (ψ i ) = ò [ K (ψ ) / K s ] dψ
(6)
0
in which ψ is the soil water capillary head taken positive [L]; ψi is the initial
(antecedent) soil capillary head [L]. The parameter G [L] accounts for the effect of
capillary forces on moisture absorption during infiltration.
In general, G is a function of the antecedent soil moisture and the surface moisture
content above the wetting front. Equation (6) assumes that the soil surface is fully
saturated (i.e. ψ = 0). In KINEROS2, soil moisture redistribution during rainfall hiatus
is simulated using the Smith et al. (1993) equation, which equates the downward flux
movement at the wetting front with the reduction in water content in an assumed
rectangular wetted soil profile. During this period the net capillary drive parameter
becomes also a function of the shrinking surface soil moisture.
Copyright  2005 IAHS Press
1156
Mohamed M. Hantush & Latif Kalin
NET CAPILLARY DRIVE PARAMETER
Mathematical formula
If one substitutes the Brooks & Corey (1964) soil characteristic relationship for
unsaturated conductivity:
K (ψ ) = K s (ψ b / ψ )
2 +3λ
= Ks
ψ ≥ ψb
ψ < ψb
(7)
and performs the integration in equation (6) from 0 to ψb, with K(ψ) = Ks and from ψb
to ψi, with K(ψ) given by equation (7), one obtains:
é 1 − (ψ b / ψ i )1+3λ ù
G (ψ i ) = ψ b ê1 +
ú
1+ 3 λ
ë
û
(8)
in which ψb is the bubbling pressure [L]; and λ is the pore-size distribution index. The
specific case of ψi = ∞ leads to this simpler expression (Kineros-2 Manual,
http://www.tucson.ars.ag.gov/kineros; Ogden & Saghafian, 1997):
2 + 3λ
(9)
1 + 3λ
Relationship (9) obviously overestimates G; thus it over-predicts infiltration and
under-predicts runoff for initially wet soils. Relationship (8) provides estimates for the G
parameter for applications where direct measurements are not available. It can also be
expressed in terms of volumetric soil moisture using the Brooks & Corey (1964) equation:
G = ψb
ψb
= S 1/ λ
ψ
ψ > ψb
(10)
in which S is the effective relative saturation,
S=
θ − θr
θs − θr
(11)
where θ is volumetric soil water content [L3 L-3]; θs is total porosity [L3 L-3], i.e. θ
(ψ = 0); and θr is residual soil moisture [L3 L-3].
In terms of initial relative saturation, Si, equation (8) can be written as:
é 1 − S i 3+ (1 / λ ) ù
G ( S i ) = ψ b ê1 +
ú
1+ 3 λ û
ë
(12)
In the following section the Monte Carlo method is applied to equation (12) to
derive conditional probability distributions for the G parameter based on the US
Department of Agriculture (USDA) soil texture classification, and explicit approximate expressions for the first-two moments are also provided.
Conditional simulation
Rawls et al. (1982) compiled data for 1323 soils with about 5350 horizons from 32
states in the USA and derived statistics, among others, for the Brooks & Corey water
Copyright  2005 IAHS Press
Uncertainty and sensitivity analysis of runoff and sediment yield
1157
retention parameters, total porosity and saturated conductivities for the major USDA
soil textures classes. They provided the arithmetic and geometric means and the
corresponding standard deviations for λ and ψb according to the USDA soil texture
classes, and indicated that these two parameters are approximately lognormally
distributed. While the arithmetic and geometric statistics compiled by Rawls et al.
(1982) are consistent with a lognormally-distributed λ, those reported for ψb were not.
An obvious reason is that the following familiar one-to-one relationships:
E(ψ b ) = exp{E(ln ψ b ) + (1 / 2) var(ln ψ b )}
{
E(ln ψ b ) = ln E(ψ b ) / 1 + [CV (ψ b )]2
}
var(ψ b ) = [E(ψ b )]2 (e var (ln ψb ) − 1)
var(ln ψ b ) = ln{1 + [CV (ψ b )] 2 }
(13)
were not exactly satisfied by the reported arithmetic and geometric statistics of ψb,
which is a necessary condition for a lognormally distributed variate. E( ) is the
expectation operator; var is the variance; and CV is the coefficient of variation (ratio of
standard deviation to the mean). Through these transformations, one lognormal
probability density function should be inferred using either of the arithmetic or
geometric statistics and equation (13). Unfortunately, that was not the case here. As a
partial fix, two assumptions were invoked and the arithmetic and geometric statistics
of Rawls et al. (1982) for ψb were revised to better honour the transformations in
equation (13). The assumptions are: (a) ψb is indeed lognormally distributed (Rawls et
al., 1982; Smith & Goodrich, 2000); and (b) the data reported by Rawls et al. (1982)
were not sufficient to infer the actual or underlying probability density function. The
revised estimates of ψb are obtained as follows: (i) for each soil texture class, compute
the arithmetic mean and standard deviation from the reported (Rawls et al., 1982)
geometric mean and corresponding standard deviation using the top two expressions in
equation (13); (ii) compute the geometric mean and standard deviation from the
reported arithmetic statistics using the bottom two expressions in equation (13);
(iii) take the arithmetic average of the computed and reported arithmetic statistics; and
(iv) take the geometric average (i.e. square root of the product) of the computed and
reported geometric statistics. Figure 1(a) shows mean G generated by applying Monte
Carlo simulations to equation (12) by preserving either of the arithmetic or geometric
statistics of ψb reported in Rawls et al. (1982) and after the averaging procedure. The
revised ψb statistics had the effect of narrowing significantly the gap between mean G
values generated by preserving the arithmetic statistics of ψb and those generated by
preserving the geometric ones. The difference of step (iii) with step (iv) transformed
by the first of equation (13) was compared to the difference of step (iv) with step (iii)
transformed by the second of equation (13), and the former was smaller. Therefore, the
average of the geometric statistics (step (iv)) was selected to characterize the
lognormal distribution of ψb.
The probability distributions of the G parameter conditioned on the antecedent soil
moisture are obtained by applying Monte Carlo simulations to equation (12). For each
initial effective relative saturation value, Si = 0, 0.1, …, 1, a conditional probability
density function is generated for the G parameter for each soil texture from
lognormally distributed ψb and λ. Table 1 lists the revised estimates of the geometric
means and exponent of the standard deviations of lnψb and those for λ (Rawls et al.,
1982), for 11 soil texture classes. A total of 40 000 statistically-independent random
Copyright  2005 IAHS Press
1158
100
Geometric mean of G
(cm)
200
Mean value of G (cm)
Geometric mean of G (cm)
Mohamed M. Hantush & Latif Kalin
Silt Loam
160
120
80
40
80
60
40
20
0
0
0.0
0.2
0.4
0.6
0.8
0
1.0
0.2
0.8
60
40
20
0
1
0
based on log(values)
based on averaging (geom.)
(b)
0.2
0.4
0.6
0.8
Si
clay
sand
silt loam
silty clay loam
based on artihmetic values
based on averaging (arth.)
0.6
80
Si
Si
(a)
0.4
100
MC (sand)
approx. (sand)
MC (clay)
approx. (clay)
(c)
Fig. 1 Monte Carlo computed G: (a) effect of averaging Rawls et al. (1982) ψb
arithmetic and geometric statistics on generated mean G; (b) geometric mean of G vs
effective relative saturation, Si, for four soils; and (c) comparison of MC simulated
geometric mean of G with the approximation in equation (16), for sand and clayey
soil.
Table 1 Estimates of geometric means, exp[µln(*)], and anti-log of the mean of ln of ψb and λ, exp[σln(*)].
All values in cm.
Sand
Loamy
sand
Sandy
loam
Loam
Silt
loam
Sandy Clay
loam
clay
loam
Silty
clay
loam
Sandy Silty
clay
clay
Clay
exp(µ ln ψ )
9.09
10.71
17.17
15.75
26.32
33.69
30.56
39.78
38.03
42.11
46.84
exp(σ ln ψ )
3.49
3.52
3.23
4.50
3.81
3.48
3.39
3.39
3.94
3.43
3.45
exp(µ ln λ )
0.59
0.47
0.32
0.22
0.21
0.25
0.19
0.15
0.17
0.13
0.13
exp(σ ln λ )
1.77
1.75
1.73
1.61
1.55
2.00
1.94
1.68
2.16
1.72
1.93
b
b
deviates for each of ψb and λ were generated and inserted in equation (12) to produce
40 000 values of G per Si value. This process is repeated for each of the 11 USDA soil
texture classes. The generated G values are lognormally distributed, and the generated
arithmetic and geometric first-two moments are consistent with equation (13).
Table 2 lists the fitted relationships for the arithmetic and geometric means and the
standard deviations of G as a function of Si over the soil texture classes. The fitted
relationships were of the form G ( S i ) = G (0) + [G (1) − G (0)] S i b , and were almost
exact with R2 > 0.99. The newly generated arithmetic and geometric statistics of G are
larger than those reported in the KINEROS2 manual (Woolhiser et al., 1990) for the
specific case of initially dry soil, Si = 0 (compare the last two columns with the second
and third evaluated at Si = 0). Figure 1(b) displays the fitted geometric mean of G (i.e.
exp{E[lnG]}vs Si for four soil textures. The dependence on Si increases from coarse- to
fine-textured soils and only when the soil is relatively wet, Si > 0.6. The arithmetic
mean of G displayed a similar behaviour, with a diminishing dependence on Si for
coarse-textured soils. The standard deviation of lnG, however, was almost invariant
with Si. It is clear that these relationships are relevant to fine textured soils.
Copyright  2005 IAHS Press
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Uncertainty and sensitivity analysis of runoff and sediment yield
Table 2 Fitted arithmetic and geometric statistics of parameter G (cm).
USDA soil
texture
Arithmetic mean
(cm)
Geometric mean
(cm)
SD
(cm)
exp(σlnG)
Sand
Loamy sand
Sandy loam
Loam
Silt loam
Sandy clay loam
27.3–7.3Si5.1
33.7–9.9Si5.6
52–17.5Si6.8
79.6–29.9Si8.53
105–39.8Si8.47
115.4–41.2Si7.85
12.37–3.24Si5.05
15.12–4.38Si5.53
25.84–8.61Si6.67
25.22–9.39Si 8.27
42.38–15.95Si8.26
52.33–18.51Si7.63
54.6–14.8Si5.16
68.3–20.3Si5.68
92.2–31.4Si6.98
239.8–90.5Si8.69
241.6–92.1Si8.65
230.4–83.4Si8.13
3.50
3.53
3.25
4.52
3.82
3.50
KINEROS manual
mean, Si = 0
Arithm.
Geom.
(cm)
(cm)
10.1
4.6
14.7
6.3
24.8
12.7
37.5
10.8
48.5
20.3
61.7
26.3
Clay loam
Silty clay loam
Sandy clay
Silty clay
Clay
105–39.9Si8.94
142.6–57.9Si11.04
161.6–62.8Si9.69
155.5–64.7Si11.97
173.1–71.1Si11.5
49.27–18.58Si8.62
66.91–26.97Si10.59
62.17–23.97Si9.33
72.12–29.85Si11.53
79.42–32.41Si11.06
201–77.1Si9.28
273–111.7Si11.54
393.2–154.1Si10.01
302.2–126.6Si12.46
340.8–141Si11.99
3.41
3.41
3.96
3.44
3.47
53.3
72.0
76.8
81.2
89.0
25.9
34.5
30.2
37.5
40.7
By exploring equation (12) more closely, one may be tempted to fix λ assuming
that it has negligible effect on the spread of G. Then by applying the expectation
operator to equation (12), one immediately obtains this approximate expression for
mean G:
µ G ≈ µ ψb
2 + 3λ µ ψb
(1+ 3 λ ) / λ
−
Si
1 + 3λ 1 + 3λ
(14)
where µ ψ b = E (ψ b ) , and λ = e E (ln λ ) is the geometric mean of λ. The variance of G can
also be approximated from equation (12):
2
2
σ G ≈ σ ψb
2
æ 2 + 3λ ö
2 + 3λ
(1+ 3 λ ) / λ
çç
÷÷ − 2 σ ψ b 2
Si
2
(1 + 3λ )
è 1 + 3λ ø
2
(15)
2
where σ G is the variance of G; and σ ψb is the variance of ψb. In arriving at equation
2 (1+ 3 λ ) / λ
(15), a term of order O[ S i
] was ignored. Similar approximations can be
obtained for the geometric mean and related standard deviation, by taking log of both
sides of equation (12), applying the expectation operator, then taking the antilog (i.e.
exponent for loge):
G ≈ ψb
ψb
2 + 3λ
(1+ 3 λ ) / λ
−
Si
1 + 3λ 1 + 3λ
(16)
in which G = e E [ln G ] is the geometric mean of G; and ψ b = e E [ln ψ b ] is the geometric
mean of ψb. Once again, taking the log of both sides of equation (12) and noting that λ
and Si are fixed, one immediately obtains:
σ ln G ≈ σ ln ψ b
(17)
where σ ln G and σ ln ψ b are the standard deviations of lnG and lnψb, respectively. Thus,
Copyright  2005 IAHS Press
1160
Mohamed M. Hantush & Latif Kalin
the standard deviation of lnG is approximately equal to that of lnψb and is invariant
with Si, a result that is also supported by the MC generated values (fifth column in
Table 2). Equations (14) and (16) provide excellent approximations to the arithmetic
and geometric means as comparisons with MC results show (e.g. Fig. 1(c)).
The following three points are emphasized: (a) the MC generated statistics of G in
Table 2 are based on Rawls et al. (1982) soil data and revised statistics for ψb;
(b) equations (14)–(17) provide means to revise the arithmetic and geometric statistics
of G whenever new ψb data becomes available; and (c) the results in Table 2 and
equations (16) and (17) are sufficient to define the conditional probability density
function for G, f G Si [G S i ] , since G is lognormally distributed.
MODEL APPLICATION
In this section, KINEROS2 model output response to uncertainties in Ks, G, θs, si
(antecedent soil saturation), np (overland plane manning roughness), nc (channel
Manning roughness), d50, cg, and cf, is investigated by application to a USDA
experimental watershed. Monte Carlo simulations were used to generate exceedence
probability or the complementary cumulative distribution function (ccdf) curves for all
parameters. Further, the condition number, CNyx = CVy/CVx, was chosen as a measure
of sensitivity of model output y to input parameter x, where CVx and CVy denote the
coefficient of variation of x and y, respectively. A ccdf curve associated with random
parameter x describes the probability of design variable y equal to or exceeding a given
value, while fixing all other parameters at their average values. In this manner
uncertainty of model outputs is tested over the full reported range of each parameter
and is measured by the corresponding CVy value. The design variables are peak flow
rate (m3 s-1), flow volume (m3), peak sediment discharge (kg s-1), sediment yield (kg),
time to peak flow (min), and time to peak sediment discharge (min).
Study area
The data used in this study come from a small USDA-operated, experimental
watershed named W-2, which is located near Treynor, Iowa, USA. The area of the
watershed is approximately 33.6 ha. Figure 2 depicts the location, topography, and
main soil texture distribution of the watershed. This watershed is one of the four
experimental watersheds established by USDA in 1964 to determine the effect of
various soil conservation practices on runoff and water-induced erosion. Runoff and
sediment load have been measured since then. There are two raingauges around the
watershed. A typical rainfall depth for a one-hour, 100-year return period storm is
approximately 9.5 cm; and 18 cm for a 24-h, 100-year return period. The W-2
watershed has a rolling topography defined by gently sloping ridges, steep side slopes
and alluvial valleys with incised channels that normally end at an active gully head,
typical of the deep loess soil (Kramer et al., 1990). Slopes usually range from 2 to 4%
on the ridges and valleys and 12 to 16% on the side slopes. An average slope of about
8.4% is estimated, using first-order soil survey maps. The major soils are well drained,
classified as fine-silty, mixed, mesics. The surface soils consist of silt loam (SL) and
Copyright  2005 IAHS Press
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Uncertainty and sensitivity analysis of runoff and sediment yield
IOWA
Fig. 2 Study watershed.
silty clay loam (SCL) textures (Fig. 2) that are very prone to erosion, requiring suitable
conservation practices to prevent soil loss (Chung et al., 1999). The cropped portions
of the watersheds cover the ridges, side-slopes and toe-slopes. The regional geology is
characterized by a thick layer of loess overlying glacial till that together overlay
bedrock. The loess thickness ranges from 3 m in the valleys to 27 m on the ridges.
Bromegrass was maintained on the major drainage ways of the alluvial valleys. Corn
has been grown continuously on W-2 since 1964. It is believed that nearly all runoff is
produced by the Horton mechanism, and that subsurface storm flow and saturation
overland flow are negligible.
Uncertainty analysis
Uncertainty of KINEROS2 output to the parameter set (Ks, G, θs, si, np, nc, d50, cg, cf)
and their sensitivities are investigated using Monte Carlo simulations for two rainfall
events which occurred on 26 August 1981 and 13 June 1983, as depicted in Fig. 3.
Table 3 lists model parameters as well as their reported or assumed probability
distributions for the SL and SCL soils in the W-2 watershed. The bottom part of
Table 3 applies to both SL and SCL soils. For those parameters whose distributions are
unknown, uniform distributions with typical range values were assumed, as reported in
the literature (Mullins et al., 1993; Knisel, 1980; Woolhiser et al., 1990). Some of the
sources of the statistical data for the parameters are shown below Table 3. Woolhiser et
al. (1990) provided relationships for cg and cf based on USLE parameters. However,
reported values in the literature show much higher variation (e.g. Smith et. al., 1999).
The lower and upper limits of cg and cf were decided based on reported model
applications in the literature (Smith et al., 1999; Ziegler et al., 2001).
In Table 3, si is the initial (antecedent) saturation, si = θi/θs, where θi is the
antecedent soil moisture content. The antecedent saturation si is related to the effective
relative saturation through this relationship: si = (1 – sr)Si + sr, in which sr is the
Copyright  2005 IAHS Press
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Mohamed M. Hantush & Latif Kalin
5/
00
00
00
13
:
12
:
2
31
/8
5/
25
/8
83
14
/
6/
6/
6/
12
/
13
/
83
83
6/
8/
8/
11
/
26
/8
1
25
/8
1
24
/8
23
/8
8/
5/
0
2
0
83
2
0
1
4
00
0
4
2
8/
2
2
8
6
27
/8
0
4
5/
12
5/30/82
6
10
:
i (cm/hr)
00
5:
4:
00
3:
2:
00
4
00
0
17
:0
0
i (cm/hr)
2
1
i (cm/hr)
6
8
10
11
:
16
4
8
2
6/13/83
21
:0
0
8
10
20
29
/8
8/26/81
6
i (cm/hr)
20
8
19
:0
0
10
Fig. 3 Selected rainfall events.
Table 3 Statistics of KINEROS2 model parameters; log(*,*) implies a lognormally distributed
parameter, with first value given in parentheses being the arithmetic mean and the second being the
standard deviation, and U(*,*) denotes a uniformly distributed parameter, with the arguments being the
lower and upper limits, respectively.
SCL
SL
Ks
(mm h-1)
ψb
(cm)
λ
θs
si
log(0.7,1.9)
log(4.5,12.3)
log(70,74)
log(64.4,143.9)
log(0.18,0.14)
log(0.23,0.13)
log(0.47,0.05)
log(0.50,0.08)
U(0.44,0.92)
U(0.27,0.97)
cg
(s-1)
cf
(s)
np
nc
d50
(mm)
U(0.01,0.3)
U(10,300)
U(0.03, 0.14)
U(0.03,0.07)
U(0.003,0.05)
Sources:
λ, θs, si: Rawls et al. (1982). ψb based on revised statistics.
Ks: US EPA/600/R-93/046, 1993, Mullins et al. (1993).
np: Beasley & Huggins (1981). For smooth disked ground (0.03–0.07), for cultivated row crop with
chisel plowed (0.10–0.14).
nc: CREAMS manual (Knisel, 1980) gives the range 0.04–0.06 for grass higher than flow depth with
poor condition. For good condition the range is 0.08–0.10. The range used in this study: 0.03–0.07.
d50: KINEROS manual (Woolhiser et al., 1990).
cg and cf: based on papers by Smith et al. (1999) and Ziegler et al. (2001).
residual saturation, sr = θr/θs. Rawls et al. (1982) provide the means and standard
deviations of θr and θs as a function of soil texture classes. Random values of Si are
generated from U(0,1) (note, by definition 0 ≤ Si ≤ 1), and together with the statistical
data provided by Rawls et al. (1982) for θr and θs, are used to generate si random
values for the SL and SCL soils using si = (1 – sr)Si + sr. All parameters in Table 3
were sampled from their respective probability distributions prior to Monte Carlo
simulations (MCS). The G parameter was generated using the statistics in Table 2 for
SL and SCL for an initially dry soil. The generated G values with Si = 0 are then
updated by KINEROS2 internally for each sampled random deviates of Si. Once model
parameters values were generated, KINEROS2 computations proceeded by spatially
averaging soil parameters in each overland flow plane element, with the weights being
the fractions of the element occupied by the two soil textures, SL and SCL.
The ccdf curves for the rainfall events on 26 August 1981 and 13 June 1983, and
coefficients of variation of model outputs in response to parameter uncertainty are
shown in Figs 4 and 5. The average rainfall intensity of the relatively smaller event (26
August 1981) is 1.9 cm h-1, and for the relatively larger event (13 June 1983) it is
4.8 cm h-1. In each MC simulation, out of 1000 runs, the parameter of interest is
Copyright  2005 IAHS Press
1163
Uncertainty and sensitivity analysis of runoff and sediment yield
1.0
1.0
(CV=0.73)
(CV=0.76)
G
Ks
(CV=0.78)
(CV=0.82)
Si
(CV=0.51)
Si
(CV=0.60)
nc
(CV=0.07)
nc
(CV=0.01)
np
(CV=0.26)
np
(CV=0.07)
0.8
exceedance probability
exceedance probability
0.8
G
Ks
0.6
0.4
0.6
0.4
0.2
0.2
(a)
(b)
0.0
0.0
0.0
0.5
1.0
1.5
2.0
2.5
0
3.0
2
4
6
peak flow (m3/s)
1.0
10
12
G
Ks
(CV=0.11)
(CV=0.17)
Si
(CV=0.03)
nc
(CV=0.03)
np
(CV=0.04)
G
Ks
exceedance probability
0.8
0.6
0.4
0.2
Si
(CV=1.01)
nc
np
(CV=0.09)
(CV=0.60)
cf
(CV=0.00)
0.4
0.2
0.0
(CV=1.06)
(CV=0.99)
d50 (CV=0.41)
cg (CV=0.09)
0.6
(d)
(c)
0.0
40
50
60
70
80
90
100
0
50
time to peak flow (min)
100
150
200
250
300
peak sediment discharge (kg/s)
1.0
1.0
0.6
G
Ks
(CV=0.08)
(CV=0.17)
Si
(CV=0.03)
nc
(CV=0.03)
np
(CV=0.04)
d50
(CV=0.01)
cg
(CV=0.00)
cf
(CV=0.00)
0.8
exceedance probability
0.8
exceedance probability
14
1.0
0.8
exceedance probability
8
total flow (m3)
0.4
0.2
0.6
0.0
(CV=1.18)
(CV=1.07)
Si
(CV=1.17)
nc
(CV=0.00)
np
(CV=0.42)
d50
(CV=0.45)
cg
(CV=0.08)
cf
(CV=0.00)
0.4
0.2
(e)
G
Ks
(f)
0.0
40
50
60
70
80
time to peak sediment discharge (min)
90
100
0
2000
4000
6000
8000
10000
sediment yield (kg)
Fig. 4 Monte Carlo generated probability of exceedence curves for the smaller event
(26 August 1981). Each plot corresponds to a particular random parameter while all
others fixed at their arithmetic or geometric averages.
randomly sampled from its particular distribution, while all other parameters are fixed
at their respective arithmetic or geometric averages. A wider transition from 1 to 0 (or
spread) of the ccdf curve implies greater uncertainty of the output to uncertainty of the
parameter in question, which is consistent with the larger coefficient of variation, CV,
shown next to each parameter in the legends. Conversely, a sudden drop in the
probability from 1 to 0 implies essentially a deterministic model output and a
negligible coefficient of variation. The model produced uncertainties of peak flow,
flow volume, peak sediment discharge and sediment yield with respect to Ks, G and the
Copyright  2005 IAHS Press
Mohamed M. Hantush & Latif Kalin
1.0
1.0
0.8
0.8
0.6
0.4
G
Ks
(CV=0.26)
(CV=0.32)
Si
(CV=0.09)
nc
(CV=0.02)
np
(CV=0.08)
exceedance probability
exceedance probability
1164
0.6
Si
(CV=0.26)
nc
(CV=0.00)
np
(CV=0.02)
0.4
(a)
(b)
0.0
0.0
0
2
4
6
0
8
10
20
30
40
total flow (m3)
peak flow (m3/s)
1.0
1.0
G
Ks
(CV=0.03)
(CV=0.04)
Si
(CV=0.01)
nc
(CV=0.01)
np
(CV=0.02)
0.8
exceedance probability
0.8
exceedance probability
(CV=0.41)
(CV=0.46)
0.2
0.2
0.6
0.4
0.2
0.6
G
Ks
(CV=0.48)
(CV=0.49)
Si
(CV=0.33)
nc
(CV=0.05)
np
(CV=0.24)
d50
(CV=0.18)
cg
(CV=0.22)
cf
(CV=0.00)
0.4
0.2
(c)
(d)
0.0
0.0
80
85
90
95
100
0
200
time to peak flow (min)
400
600
800
1000
1200
peak sediment discharge (kg/s)
1.0
1.0
0.6
G
Ks
(CV=0.03)
(CV=0.04)
Si
(CV=0.01)
nc
(CV=0.01)
np
(CV=0.02)
d50
(CV=0.01)
cg
(CV=0.00)
cf
(CV=0.00)
0.8
exceedance probability
0.8
exceedance probability
G
Ks
0.4
0.2
0.6
G
Ks
(CV=0.66)
(CV=0.63)
Si
(CV=0.53)
nc
(CV=0.01)
np
(CV=0.23)
d50
(CV=0.21)
cg
(CV=0.19)
cf
(CV=0.00)
0.4
0.2
(e)
(f)
0.0
0.0
80
85
90
95
time to peak sediment discharge (min)
100
0
10000
20000
30000
40000
50000
sediment yield (kg)
Fig. 5 Monte Carlo generated ccdf (probability of exceedence) curves for the larger
event (13 June 1983). For time to peak computations, a cut-off value of 0.001 m3 s-1
was used, below which flow is assumed to be 0 m3 s-1.
antecedent saturation, Si, are greater than those for all other parameters, as shown by
their larger spreads and CVs (Fig. 4(a), (b), (d) and (f)). The parameter np produced
relatively smaller spreads in the ccdf curves and smaller CVs for these variables,
whereas the effect of nc appears to be marginal. The smaller CVs associated with nc
compared to np may be attributed to the much smaller range from which the former
was sampled compared to that for the latter (Table 3).
The ccdf curves for the peak sediment discharge and sediment yield show np and
d50 imparting moderate uncertainty, and nc, cg and cf having negligible CVs. Parameters
Copyright  2005 IAHS Press
1165
Uncertainty and sensitivity analysis of runoff and sediment yield
Ks and G dominated uncertainties of the arrival time of peak flow and peak sediment
discharge rates, and produced much higher CVs than np, nc and Si (Fig. 4(c) and (e)),
whereas d50, cg and cf contributed negligible uncertainty to the arrival times (Fig. 4(e)).
In general, arrival times showed less uncertainty than all other design variables.
The coefficients of variation of the simulated output variables are reduced when
simulating the relatively larger event (13 June 1983), except for the hydraulic erosion
parameter, cg, which produced relatively greater CVs for peak sediment discharge and
sediment yield (Fig. 5). However, the dominance of Ks, G and Si on the spread of ccdf
curves for peak flow, flow volume, peak sediment discharge and sediment yield was
not affected by the size of the storm (Fig. 5(a), (b), (d) and (f)). The increased
coefficient of variation of peak sediment discharge and sediment yield with respect to
cg may be attributed to greater transport capacity C* as a result of increased flow
velocities for the larger storm. It is not surprising that the effect of random cf remained
negligible for this larger event. Relative to the smaller event, CVs for the time to peak
flow and time to peak sediment discharge were too small for these variables to even be
considered random.
To explore more closely the relationship between model output uncertainties and
rainfall intensity, 10 storm events having 120 min duration and constant rainfall
intensity, varying from 0.15 to 15 cm h-1 were considered. For each rainfall event,
1000 MC simulations were conducted by generating statistically independent random
parameter sets from the probability density functions in Table 3. Figure 6 shows the
coefficient of variation, CV, of the design variables as a function of rainfall intensity in
a log-log scale. The CV of peak flow rate, total flow volume, peak sediment discharge
and sediment yield decreased by order of magnitude as rainfall intensity i (cm h-1)
increased by one order of magnitude. This means that the influence of parameters
10
1
10
Time to peak flow
Total flow
CV
Peak flow
1
1
0.1
0.1
0.1
1
10
10
0.1
0.1
1
10
1
Sediment
yield
10
1
10
100
1
Peak sediment
discharge
CV
0.1
0.1
Time to peak
sediment discharge
1
0.01
0.1
0.1
1
10
0.1
0.1
1
10
0.1
1
10
i (cm/hr)
Fig. 6 Effect of rainfall intensity on the coefficient of variation of various model
outputs. Horizontal axis represents rainfall intensity, whereas vertical axis is the
coefficient of variation (CV).
Copyright  2005 IAHS Press
1166
Mohamed M. Hantush & Latif Kalin
uncertainty changes with rainfall intensity, and reliability of model predictions
diminishes with smaller rainfall events. Soil characteristics dominate runoff during
smaller rainfall events, whereas for larger events flow and sediment transport approach
equilibrium relatively faster and become storm-driven. Similarly, the CVs of peak flow
and peak sediment discharge rate arrival times showed an order of magnitude
reduction with i.
Sensitivity analysis
So far this paper has addressed model response to parameters’ uncertainty, rather than
model output sensitivity to the parameters. The former provide a measure for model
output spread to reported ranges of input parameters, whereas the latter measures how
a relative perturbation of the parameter is propagated into the relative perturbation of
the prediction, a measure of which is the condition number, CN.
Figure 7(a)–(d) shows a break-up of parameter sensitivities measured by CN for
flow and sediment model outputs. Interestingly, for both events, Si was the most
sensitive parameter except for the peak flow and peak sediment discharge arrival
times, followed by np for all but total flow volume. Parameters G and Ks were the
second and third most sensitive parameters, respectively, only for total flow volume,
and both were less sensitive than np in all other flow and sediment model outputs. The
high sensitivity of KINEROS2 to Si for low- as well as high-intensity storms is worth
8/26/81
2.0
(b)
3
CN
1.5
CN
8/26/81
4
(a)
1.0
2
1
0.5
0.0
0
Peak flow
Ks
G
Peak time
Si
Total flow
nc
Peak sed. disch.
np
Ks
G
Peak sed. time
Si
nc
np
Sed. yield
d50
cg
cf
2.0
6/13/83
6/13/83
(c)
0.8
(d)
1.5
CN
CN
0.6
1.0
0.4
0.5
0.2
0.0
0.0
Peak flow
Ks
G
Peak time
Si
Total flow
nc
np
Peak sed. disch.
Ks
G
Peak sed. time
Si
nc
np
Sed. yield
d50
cg
cf
Fig. 7 Conditional numbers as parameter sensitivity measures for peak flow, peak
flow arrival time, total flow volume, peak sediment discharge, peak sediment discharge arrival time, and sediment yield for the smaller event on 26 August 1981 (a, b)
and the larger event on 13 June 1983 (c, d).
Copyright  2005 IAHS Press
Uncertainty and sensitivity analysis of runoff and sediment yield
1167
noting as previous studies limited the sensitivity with respect to the initial saturation of
peak runoff and runoff volumes to low-intensity storms (Osborn & Simanton, 1990;
Michaud & Sorooshian, 1994; Castillo et al., 2003).
For both events, the parameter d50 was more sensitive than G and Ks for peak
sediment discharge and sediment yield. Also, these sediment outputs were more
sensitive to cg than G and Ks for the relatively larger event (13 June 1983), but less for
the smaller event. Time to peak flow and time to peak sediment discharge showed
greater sensitivity to nc followed by np and Si. It’s worth noting that the maximum
overland distance used in the model computations is about 183 m, whereas the
maximum travelled distance in the channels is approximately 800 m. On average, the
model may have computed greater resistance to flows in the channels than in the
planes, thus resulting in a greater sensitivity to nc than np.
Peak flow and total flow showed the least sensitivity for nc. All sediment-related
output variables showed almost no sensitivity to cf. Peak sediment discharge and
sediment yield were most sensitive to Si and np, followed by d50. The sensitivity of the
peak sediment discharge to d50 and cg is almost reversed for the two events; they are
more sensitive to d50 than cg for the smaller event (Fig. 7(b)).
Simulations with random initial soil moisture content
Figure 8 shows results derived from 1000 Monte Carlo simulations for each of the
three events in Fig. 3, along with observed flow (m3 s-1) and sediment discharge rates
(kg s-1) at the W-2 outlet. The antecedent saturation, si, was assumed to be uniformly
distributed, i.e. U(0.78,0.88) for SCL and U(0.66,0.76) for SL, for the event of 30 May
1982. Since no planting took place prior to this event, the lower limit was chosen to be
at field capacity and the upper limit fixed at the field capacity + 0.1. For the event on
13 June 1983, si ∼ U(0.44,0.54) was selected for SCL and (0.27,0.37) for SL. Noting
that corn was planted on 17 May, the lower limit was chosen to be at wilting point and
the upper limit to be at the wilting point + 0.1. The event of 26 August 1981 occurred
during the growing season, thus, the selected distribution for si was U(0.44,0.64) for
SCL and U(0.27,0.47) for SL, with the upper limit chosen relatively higher to reflect
the wet conditions produced by two storms that occurred 1–3 days earlier (Fig. 3).
The plane roughness, np, is generated differently for the events. For the two events
on 30 May 1982 and 13 June 1983, random np values were generated from a uniform
distribution U(0.03,0.07) since the crop was not fully grown yet, and for the event
26 August 1981 higher values were generated from U(0.10,0.14) due to the availability
of crop. All other parameters were generated from their distribution in Table 3.
Parameter G was generated for SL and SCL from Table 2 with Si = 0, and was updated
internally in KINEROS2 for the randomly generated Si values.
The median (50th percentile), 25th, and 75th percentiles of the flow and sediment
discharge rates are computed from corresponding MC simulations (Fig. 8). In all plots,
the observed values of flow and sediment discharge fall within the uncertainty (50%
probability) band.
It is a common practice to insert average values for model input parameters
whenever data are not sufficient to permit a meaningful model calibration. Therefore,
flow and sediment discharge rates were simulated based on ensemble average
Copyright  2005 IAHS Press
1168
Mohamed M. Hantush & Latif Kalin
60
75%
5/30/82
50%
0.8
flow (m3/s)
sediment discharge (kg/s)
1.0
25%
avg-par
0.6
obs.
0.4
(a)
0.2
75%
5/30/82
50
50%
25%
40
avg-par
30
obs.
20
(b)
10
0
0.0
30
70
110
150
30
190
70
time (min)
sediment discharge (kg/s)
75%
flow (m3/s)
50%
25%
4
avg-par
obs.
2
(c)
6/13/83
190
75%
800
50%
25%
600
avg-par
obs.
400
200
0
(d)
0
50
70
90
110
130
150
50
70
8/26/81
(e)
sediment discharge (kg/s)
1.2
1.0
75%
50%
25%
avg-par
obs.
0.8
0.6
0.4
90
110
130
150
time (min)
time (min)
flow (m3/s)
150
time (min)
1000
6/13/83
6
110
0.2
0.0
(f)
8/26/81
40
75%
50%
25%
avg-par
obs.
30
20
10
0
40
60
80
100
time (min)
120
140
40
60
80
100
time (min)
120
140
Fig. 8 Comparison with observed values of Monte Carlo generated 75%, medians
(50%), 25% percentiles, and simulated model flow rates and sediment discharges
based on average model parameters (avg-par). In the Monte Carlo simulations all
parameters were generated according to the probability distributions in Table 3.
parameter values. Model response to average input parameters is shown by the thick,
solid line (avg-par). Regardless of the event size, comparison indicates that the median
and predictions based on average input parameters are within order of magnitude of the
observed values. This is interesting as these results were obtained without calibration,
except for making an informed guess of the range of antecedent moisture content. With
the exception of the simulated sediment discharge for the event on 13 June 1983,
Copyright  2005 IAHS Press
Uncertainty and sensitivity analysis of runoff and sediment yield
1169
model simulations based on average input parameters compared closely to, although in
most part slightly overestimated, the predicted median flows and sediment discharges.
Both simulated flow and sediment discharge rates overestimated observed values for
the larger event (Fig. 8(c) and (d)). However, the uncertainty band of both quantities
narrows down with the storm size. The simulated flows based on average parameters
compared well to the median and observed values for the events on 30 May 1982 and
26 August 1981; similarly for the simulated sediment discharge during the event on
26 August 1981. This clearly underscores the advantage physically-based hydrological
models have over empirically-oriented ones. The latter cannot be used in ungauged
watersheds, because their parameters usually do not have physical meanings and often
require calibration.
CONCLUSIONS
Using the Monte Carlo method (MC), probability density functions of the net capillary
drive parameter, G, conditioned on antecedent soil moisture were derived from a
national statistical soil database (Rawls et al., 1982) and revised arithmetic and geometric statistics of the bubbling pressure parameter ψb. The MC-generated probability
density functions for G were lognormally distributed. Curve-fitted relationships were
obtained for the arithmetic and geometric means, the standard deviation, and the
exponent of the standard deviation of lnG, as a function of antecedent relative effective
soil saturation and soil texture class. Approximate expressions for the arithmetic and
geometric statistics were also provided which compared very well to the fitted
relationships. These approximations are useful for generating lognormally distributed
G, conditioned on initial soil moisture, whenever new data of ψb and the pore-size
distribution index, λ, are made available. Further, they could be used in rainfall–runoff
models should the Smith-Parlange (1978) and Parlange et al. (1982) infiltration models
be selected.
KINEROS2 flow and sediment related outputs response to uncertainty in key soil
and erosion parameters was examined using the MC method. The computed
complementary cumulative distribution functions and coefficients of variation of
model outputs showed that the saturated hydraulic conductivity, Ks, the net capillary
drive parameter, G, and the antecedent relative effective soil saturation, Si, dominated
uncertainty of flow and sediment model outputs, whereas plane roughness coefficient,
np, the median particle diameter, d50, and the hydraulic erosion coefficient, cg, had a
moderate influence, particularly on peak sediment discharge and sediment yield. The
rain splash erosion coefficient, cf, and channel roughness coefficient, nc, had no
influence on sediment output uncertainty. The arrival times of peak flows and sediment
discharge showed much smaller coefficient of variation, and for the larger event, were
even negligible.
Monte Carlo simulations demonstrated that the coefficients of variation of flow
and sediment model outputs decreased with rainfall intensity, leading to the conclusion
that the KINEROS2 model performs more reliably under intense rainfall events than
for smaller events.
Parameter sensitivity was estimated by the condition number. In this watershed,
parameters that imparted greater uncertainty to the model output were not necessarily
Copyright  2005 IAHS Press
1170
Mohamed M. Hantush & Latif Kalin
the most sensitive ones. The latter are important in the model calibration phase.
Antecedent soil moisture content (Si) was the most sensitive parameter, for both
events, for all model outputs except for the arrival times of peak flow and peak
sediment discharge. The peak flow and peak sediment discharge arrival times were
most sensitive to channel and overland plane roughness coefficients. The order of
sensitivities of parameters other than Si varied with the design variable and event size.
In general, sediment outputs were not sensitive to the rain splash erosion parameter in
this watershed.
Monte Carlo simulated median streamflow and sediment discharge at the
watershed outlet were close to those based on average parameters in all but one event.
At worst, the simulated medians of flow rate and the sediment discharge and those
based on average input parameters were within order of magnitude of the observed
values. This was achieved without KINEROS2 calibration. Given the high sensitivity
of the model to antecedent soil moisture, methods for measuring or estimating this
parameter (e.g. remote sensing, or through soil moisture mass balance) are needed for
any successful application of the model. This study provided guidance for KINEROS2
model calibration in comparable watersheds, and produced revised statistics for the net
capillary drive parameter which can be useful in model applications that make use of
Parlange-type infiltration relationship.
Acknowledgements The US Environmental Protection Agency through its Office of
Research and Development funded the research described here through in-house
efforts and in part by an appointment to the Postgraduate Research Program at the
National Risk Management Research Laboratory administered by the Oak Ridge
Institute for Science and Education through an interagency agreement between the US
Department of Energy and the US Environmental Protection Agency. The authors
thank the two anonymous reviewers whose comments and critique improved
substantially the content and quality of the paper.
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Received 28 June 2004; accepted 21 July 2005
Copyright  2005 IAHS Press