Chapter 8
RAINFALL-RUNOFF MODELING
8.1
Introduction
A model can be defined according to Clarke (1972) as “ a simplified representation of a
complex system”. A mathematical model is thus defined as “a set of mathematical
expressions and logical statements combined in order to simulate the behavior of a given
system”. Rainfall-runoff models use mathematical expressions and logical statements to
simulate the conversion of rainfall into runoff. Rainfall-Runoff models can be applied to
a vast field of water resources problems. Some as the most common applications are

Simulation of natural discharge.

Operational forecasting.

Prediction of effects of future physical changes in a catchment.
Simulation of natural discharge means that the model is used to simulate runoff from
meteorological input data available in the catchment or in its neighborhood.
The
performance of the model is generally verified against a recorded runoff series. The
model can be used to extend runoff records by means of long records of meteorological
observations. It can also be used to tell artificial from natural variations in a catchment
where human influence or other changes in the hydrological regime are suspected.
Attempting, but rather difficult, application is the estimation of runoff in un-gauged
catchments. To do this with any certainty requires long experience with the model so that
its components can be related to the physiographic characteristics of the catchment.
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In operational forecasting, the model is first fitted and tested in order to verify its
capability of runoff simulation from meteorological data. Then meteorological forecasts
or recorded climatic series can be used to forecast discharge in rivers.
The most difficult point is to predict the effects of future physical changes in a catchment,
as it requires not only an accurate runoff model, but also that its components with
certainty can be related to the characteristics of the catchments, such as degree of
urbanization or percentage of clear cuttings in a forest. If so, the model can be used to
study the possible effect on the hydrological regime of a proposed activity in the
catchment.
8.2
Types of rainfall-runoff models
It is a known fact that hydrological processes which describe the hydrological cycle are
extremely complex, and difficult both to measure and understand in full detail. Rainfallrunoff models attempt to model the hydrological processes involved in the catchment by
representing the know processes by a series of simplified mathematical functions. Figure
8.1 presents the overview of the hydrological processes in the hydrological cycle.
Rainfall-runoff models fall into two broad categories namely, empirical “black box”
models and conceptual physically-based models.
8.2.1 Empirical “black box” model
Empirical” black box” models or sometimes referred to as system type of models simply
attempt to relate rainfall as input (It) to runoff as output (Qt), with little or no attempt to
simulate the individual hydrological processes involved. Central to the model is the
transformation operator by the catchment. Figure 8.2 presents a schematic representation
of a hydrologic “black box model”.
A “black box” model consists of an algebraic
equation (or equations) containing one or more parameters to be determined by data
analysis or other empirical means. The applicability of a black box model is restricted to
the range of data used in the determination of the parameter values.
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Examples of the black box models include the unit hydrograph method, the simple linear
reservoir model, linear difference equation models (Box and Jenkins, 1976) , the
Constrained Linear Systems Model (Natale and Todin, 1976) and the Linear Perturbation
Model (Nash and Basrsi, 1983).
Figure 8.1
Overview of the hydrological processes in the hydrological cycle
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Figure 8.2
Schematic representation of the “Black Box” or system type of model
8.2.2 Conceptual “physically based” model
Conceptual “physically-based models” , on the other hand , attempt to simulate to a
greater or lesser extent, the most important hydrological components of the catchment
response, e.g. interception, infiltration, groundwater flow, evapotranspiration, surface
water flow, etc. Conceptual models are simplified representations of the physical
processes, usually relying on mathematical descriptions (either in algebraic form or by
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ordinary differential equations), which simulate complex processes in the mean by
relying on a few key conceptual parameters.
Figure 8.3 presents a schematic
representation of the simplified hydrological processes in a catchment. The extensive use
of conceptual models in engineering hydrology reflects the inherent complexity of the
phenomena and the practical inability to account for deterministic components in all
instances.
The earliest examples of the conceptual physically-based process models were the
Stanford Watershed Model and Sacramento Model, (USA) and the Sugawara Tank
Model
(Japan). Other conceptual models are the NAM (Denmark), Pitman (South
Africa) and the Xinanjiang Model (China).
Conceptual models are more favorable to simulate hydrological processes than black box
models in that they have the potential for predicting the effects of, for example, landuse
changes or for application to ungauged catchments. Further more conceptual models are
able to account for volumetric losses from rainfall through a soil moisture budgeting and
also are intended to represent the non-linear effects of the catchment process better than
the “black box” counterparts.
8.2.3 Lumped Models
A lumped model refers to a model in which the parameters do not vary spatially within
the catchment. Therefore, catchment response is evaluated only at the outlet, without
explicit accounting of the response of individual subcatchments. Typical examples of
lumped parameter models are the unit hydrograph, HEC-1 and Tank model.
8.2.4 Distributed models
A distributed model refer to a model in which the parameters are allowed to vary
spatially within the catchment. This enables the calculation not only to consider the
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overall catchment's response but also of the response of individual subcatchments. An
example of the distributed model is the SHE model.
Evapotranspiration
Rainfall from atmosphere
Interception by vegetation
Evaporation
Surface detention
Direct runoff
Infiltration
Infiltration excess
Saturated runoff
Deep percolation
(Groundwater recharge)
Figure 8.3
Base-flow
Simplified hydrological process in a catchment
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The increased detail with which simulations can be made with a distributed model
renders it more computationally intensive than a lumped model.
This permits the
modeling of special features such as spatially varying rainfall and spatially varying
hydrologic abstractions. However, for the results of distributed modeling to remain
meaningful, the quality and quantity of available data must be commensurate with the
increased level of detail.
8.2.5 Deterministic models
A deterministic model is a model where two equal sets of inputs will always yield the
same output, if run through the model under identical conditions. The model has no
components controlled by chance. A vast majority of the models are deterministic.
8.2.6 Stochastic models
A stochastic model has some components of random character. Identical inputs may
result in unequal outputs, if run through the model under identical conditions. Virtually
no model if fully stochastic.
8.3
Model components and model construction
The basic catchment model components are

Precipitation,

Hydrologic losses, and

Runoff.
Usually, precipitation is the modeling input, hydrologic losses are determined by the
catchment's properties and runoff is the modeling output.
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8.3.1 Precipitation
Precipitation, either in the form of rainfall or snowfall, is the process driving the
catchment model.
Surface runoff is a direct consequence of excess rainfall and/or
snowmelt. Rainfall can be described in terms intensity, duration, depth, frequency,
temporal distribution, spatial distribution and areal correction.
8.3.2 Hydrologic Abstractions
Hydrologic abstractions are the physical processes acting to reduce total precipitation into
effective precipitation. Eventually, effective precipitation goes on to constitute surface
runoff. There are many processes by which precipitation is abstracted by the catchment.
Among them, those of interest to engineering hydrology are the interception, infiltration,
surface storage, evaporation and evapotranspiration
The modeling objectives determine to a large extent which hydrologic losses are
important in a certain application. For event models, the emphasis is on infiltration. For
instance, the SCS runoff curve number method, which is widely used in event models,
takes explicit account of infiltration. All other hydrologic abstractions are lumped into an
initial abstraction parameter, defined as a fraction of the potential maximum retention.
8.3.3 Runoff
Two distinct modes of runoff are recognized for modeling purposes:
(1)
catchment runoff and
(2)
stream channel runoff.
Catchment runoff has three-dimensional features, but eventually this type of runoff
concentrates at the catchment outlet. After it leaves the catchment, runoff enters the
channel network, where it becomes stream channel flow. Unlike catchment runoff, the
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marked longitudinal orientation of stream channel flow generally justifies the assumption
of one-dimensionality.
8.4
Model construction and application
The construction of a catchment model begins with the selection of model components.
Once these are chosen, they are assembled as parts of the overall model, following a
logical sequence that resembles that of the natural processes. Rainfall and snowfall are
considered first, followed by hydrologic abstractions, subcatchment hydrograph
generation, reservoir and stream channel routing, and hydrograph combination at stream
network confluences.
The issue of model resolution must be addressed at the outset of model construction and
application. Resolution refers to the ability of the model to depict accurately certain
scales of problems. Resolution is related to catchment scale and modeling objective.
Modeling runoff from small catchments requires fine resolution, with typical time steps
on the order of minutes and correspondingly small subcatchments and short channel
reaches.
On the other hand, modeling runoff from midsize catchments requires an
average resolution to match the sub-catchment size and channel reach length.
The modeling objective can have an influence on the choice of model resolution. Event
models are short-term by definition and, therefore, are subject to relatively fast changes
in model variables. A fine resolution, usually with time steps ranging from several
minutes to a few hours, depending on catchment size, is usually required by event
models. Continuous-process models are designed to account for long-term processes,
with correspondingly lesser fluctuation in model variables. Therefore, a coarse model
resolution is possible in continuous-process models.
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8.5
Model calibration and verification
The essential ingredients of each model component are variables and parameters.
Variables are the physical quantities themselves, i.e., discharge, stage, flow area flow
depth, mean velocity, and so on. Parameters are the quantities that control the behavior
of the variables. Each model component may have one or more variables and parameters.
Model calibration is the process by which the values of model parameters are identified
for use in a particular application. It consists of the use of rainfall-runoff data and a
procedure to identify the model parameters that provide the best agreement between
simulated and recorded flows.
Parameter identification can be accomplished either
manually, by trial and error, or automatically, by using mathematical optimization
techniques.
Calibration implies the existence of stream flow data; for un-gauged catchments,
calibration is simply not possible. The overall importance of calibration varies with the
type of model. For instance, a deterministic model is generally regarded as highly
predictive; therefore, it should require little or no calibration. In practice, however,
deterministic models are usually not entirely deterministic, and therefore, a certain
amount of calibration is often necessary.
In conceptual modeling, calibration is extremely important, since the parameters bear no
direct relation to the physical processes. Therefore, calibration is required in order to
determine appropriate values of these parameters. Practical estimates of conceptual
model parameters, based on local experience, are sometimes used in lieu of calibration.
However, such practice is risky and can lead to gross errors. Calibration also plays a
major role in the determination of parameters of empirical models.
The calibration needs of time-invariant and time-variant processes and models are quite
different. To evaluate the predictive accuracy of a time-invariant model, it is customary
to divide the calibration process into two distinct stages:
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(1)
calibration and
(2)
verification.
For this purpose, two independent sets of rainfall-runoff data are assembled. The first set
is used in the calibration per se, whereas the second set is used in model verification, i.e.,
a measure of the accuracy of the calibration. Once the model has been calibrated and the
parameters verified it is ready to be used in the predictive stage of the modeling.
With time-variant processes and models, the calibration is quite involved. Since the
parameters vary in time (and with the model variables), a calibration and verification in
the linear sense is only possible within a narrow variable range. A practical alternative is
to select several variable ranges, e.g., low flow, average flow, and high flow, and to
perform a calibration and verification for each flow level. In this way, a set of model
parameters for each of several variable ranges can be identified. A typical example of
multilevel (i.e., multistage) calibration is that of stream channel routing.
8.6
Model efficiency criteria
The performance of a model must be judged on the extent to which it satisfies its
practical objectives (accuracy), on the extent to which the achieved level of accuracy
persists through different samples of data (consistency) and on the extent to which it can
sustain the achieved level of accuracy when subjected to diverse applications and tests
other than those used for calibrating the model (versatility). A considerable number of
numerical and graphical criteria are available to check the versatility of a model (WMO,
1975), while model consistency is checked by split-sampling, i.e. by breaking the record
into two distinct periods, in one of which the model is calibrated and in the other (the
verification period) it is tested.
Efficiency criteria, which express model accuracy, are generally linked with the objective
function used in calibration for optimizing its parameters. A commonly used objective
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function, even for the case of non-linear or conceptual models, is the sum of squares of
differences F between the observed and the estimated discharges, with the summation
taken over the whole of the calibration period, that is,
F

^
 ( y  y) 2
(8.1)
^
y being the model output estimate of the measured output y. The quantity F is an index
of residual error, which reflects the extent to which a model is successful in reproducing
the observed discharges. It is, therefore, an appropriate criterion for expressing model
accuracy. However, it is not a dimensionless quantity and, while it may be used to
compare various alternative-forecasting models on the same catchment, it is not suitable
for comparing the performance of a model on different catchments or with different
lengths of records.
Nash and Sutcliffe (1970) met this objection by defining the model 'efficiency' R2,
analogously to the 'coefficient of determination' in linear regression, as the proportion of
the initial variance accounted for by F. Defining the initial variance Fo as
F0


 ( y  y)
2
(8.2)
where

1 N
 yi is the mean of y in the calibration period and N is the number of data
N i 1
y 
points, the efficiency criterion may then be expressed as
R2

F0  F
F0
(8.3)
i.e. the proportionate reduction of the initial variance by means of the substantive inputoutput transformation model.
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In application to the calibration period, these quantities are all obtained with reference to
that period so that R2 is identical to the 'coefficient of determination' and varies between
zero and one. When applied in the verification period, however, the initial variance Fo is
still calculated as the sum of squares of deviations within that period from the mean of
the calibration period, as the R2 criterion expresses a comparison of the sum of squares of
model errors with the sum of squares of errors which would occur when, in the absence
of any model (i.e. the 'no-model' situation), the only forecast which could be made for the
verification period would be the mean value of the discharge in the calibration period.
Hence, R2 may take negative values in the verification period, when the model under test
produces output forecasts which are worse estimators than is the mean of the recorded
output over the calibration period.
For comparing the relative accuracies of different models (say models 1 and 2) using the
same data, the R2 criterion provides a convenient index of comparison of the
corresponding sums of squares of model residual errors. Similar criteria may also be
used to express the proportion of the initial variance unaccounted for by model (1), which
is subsequently accounted for by mode (2) or by the addition of a further component or
indeed by an updating procedure. Such criteria may be expressed in the form
r2

R22  R12
1  R12
(8.4)
or simply
r2

F1  F2
F1
(8.5)
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8.7
Model-Output "Residual" Errors
Model errors may be due to the following:
(1)
an incorrect or an inadequate representation in the model structure of the
actual physical hydrological processes involved and of their interaction.
(2)
data errors, i.e. errors in either the input or the output data, or in both
categories.
(3)
failure to optimise the model parameters properly or adequately,
particularly if the surface of the objective function used to calibrate the
model is very complex having many local optima.
Data errors may be due to the following:
(1)
sampling errors in the recorded precipitation and evaporation data.
(2)
non-representativeness of lumped (areal weighted) precipitation series or
of the pan-evaporation data and
(3)
errors in the observed flows (which may be either random in nature or
systematic as when based on an inadequate rating curve).
8.9
Sensitivity analysis
Uncertainties in catchment-modeling practice have led to increase reliance on sensitivity
analysis, the process by which a model is tested to establish a measure of the relative
change in results caused by a corresponding change in model parameters. This type of
analysis is a necessary complement to the modeling exercise, especially since it provides
information on the level of certainty (or uncertainty) to be placed on the results of the
modeling.
The issue of model sensitivity to parameter variations is particularly important in the case
of deterministic models having some conceptual components. Because of the conceptual
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components, calibrations are strictly valid only within narrow variable ranges; therefore,
errors in parameter estimation need to be ascertained in a qualitative way.
Sensitivity is usually analyzed by isolating the effect of a certain parameter. If a model is
highly sensitive to a given parameter, small changes in the value of this parameter may
cause correspondingly large changes in the model output. It is, therefore necessary to
concentrate the modeling effort into obtaining good estimates of this parameter. On the
other hand, insensitive parameters can be relegated to a secondary role.
In catchment modeling, the choice of parameters for sensitivity analysis is largely a
function of problem scale. For instance, in small catchments, the model's output is highly
sensitive to the abstraction parameter(s), e.g., the runoff coefficient in the rational
method. Therefore, it is imperative that the runoff coefficient be estimated in the best
possible way
In midsize catchment modeling, the model's sensitivity usually hinges on the temporal
rainfall distribution, infiltration parameters, and unit hydrograph shape. The selection of
rainfall distribution is crucial from the design standpoint. Catchment models are usually
very sensitive to infiltration parameters, which need to be evaluated carefully, with
particular attention to the physical processes.
For instance, a high-intensity, short-
duration storm may result in a high flow peak, due primarily to the high rainfall intensity.
However, a low-intensity, long-duration storm may also result in a high flow peak, this
time due to the long rainfall duration, which causes the hydrologic abstractions to be
reduced to a minimum.
In large-catchment modeling, the model's sensitivity focuses on the spatial distribution of
the storm, although the temporal distribution and infiltration parameters continue to play
a significant role. In any case, a careful evaluation of model sensitivity is needed for
increased confidence in the modeling results.
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Sensitivity analyses provide an effective means of coping with the inherent complexities
of catchment modeling, including the associated parameter uncertainties. In this sense,
distributed models, while being widely regarded as deterministic, can often show a
distinct probabilistic flavor.
References
Clarke R. T., 1972: Areview of some mathematical models used in hydrology, with
observation on their calibration and use. Journal of hydrology, No. 19.
Box, G.E.P. and Jenkins, G.M., 1976: Time series analysis : Forecasting and acontrol,
Holden-Day Inc., Oakland, CA.
Natale, I., and Todini, E., 1976: A stable estimator for linear models II: Real world
hydrologic applications. Water Resour. Res./ 12:672-676.
Nash, J.E. and Barsi, B.I., 1983. A hybrid model for model for flow forecasting on large
catchments. J. Hydrol., 65: 125-137.
World Meteorological Organization (WMO), 1975: Intercomparison of conceptual
models used in operational hydrological forecasting. WMO, Geneval, Report No. 429.
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