122: Rainfall-runoff Modeling: Introduction

KEITH J BEVEN
Department of Environmental Science, and Lancaster Environment Centre, Lancaster University,
Lancaster, UK

This section provides an introduction to the theme of rainfall-runoff modeling in hydrology. It provides a summary
of the purposes of rainfall-runoff modeling; a classification of rainfall-runoff models; a brief account of process
descriptions in rainfall-runoff models; and a short history of rainfall-runoff modeling. It also discusses the
problem of model choice from the wide range of possibilities available, and the important question of model
calibration using different types of observational data before considering the prediction of the effects of future
change (when no observations are possible). Finally, a guide is given to the future of rainfall-runoff modeling.
Reference to the other sections in this article are given throughout, as well as to relevant available texts.

INTRODUCTION the local topography, soils, and land use alone. This “pre-
diction of ungauged catchments” problem is still essentially
The Many Purposes of Rainfall-runoff Modeling unsolved (see Chapter 133, Rainfall-runoff Modeling of
Ungauged Catchments, Volume 3). Thus, there is still a
No introduction to the range of rainfall-runoff modeling considerable body of modeling work that depends on local
techniques can be presented without considering the pur- calibration of models using measured rainfall inputs, and
poses for which the modeling is carried out. All modeling comparing observed and predicted discharges with a view
involves some form of extrapolation in either time or space, to improving the predictions. In the past it had been con-
but different modeling techniques are appropriate for differ- sidered sufficient to find some “optimal” model to make
ent hydrological problems. At the end of this article, there predictions. Today, the uncertainties inherent in the mea-
are reviews of modeling for flood forecasting (both in real sured inputs to the model, model definition, and model
time and for predicting flood frequencies), for flood inunda- calibration are more widely recognized, and techniques of
tion modeling, for integrated basin management, and for the assessing uncertainties in the predictions are being devel-
prediction of the effects of change. There are also sections oped (see Chapter 130, Fuzzy Sets in Rainfall/Runoff
that present different types of modeling methodologies in Modeling, Volume 3 and Chapter 131, Model Calibra-
more technical detail. tion and Uncertainty Estimation, Volume 3).
It has to be recognized straight away that, despite the Part of this uncertainty arises from the model definition
very real demand for good quantitative rainfall-runoff pre- itself. There are now very many different models in
dictions for these very practical problems, the degree to hydrology, representing variants on several different lines
which current models can satisfy that demand is limited. of development or families of models of different degrees
Much of hydrology measurement technique is constrained, of complexity (see Section “A classification of rainfall-
since so much of what is interesting in the hydrological runoff models”). They all remain, however, simplifications
response of a catchment to rainfall takes place beneath of what we perceive as the real complexity of hydrological
the ground surface. The development and application of processes. One of the major questions that arises in rainfall-
models is, therefore, also limited by the available mea- runoff modeling is how best to approximate the perceived
surement techniques. The result is that it remains very complexity of the real catchments. Ironically, perhaps,
difficult to predict the hydrological response of any arbi- making models more complex to take account of some
trary catchment area based on mapped information about of the perceived complexities does not necessarily lead to

Encyclopedia of Hydrological Sciences. Edited by M G Anderson.

 2005 John Wiley & Sons, Ltd.
2 RAINFALL-RUNOFF MODELING

more precise and less uncertain predictions. This is because In the twentieth century, therefore, more complex lumped
adding complexity in general, introduces more parameter models were developed to reflect the perceptions of dif-
values that are not easily measured directly or calibrated ferent hydrologists about the processes of hydrological
on the basis of indirect information. More complex models, responses to rainfall. One of the most important devel-
therefore, in general have more degrees of freedom in opments was that of the unitgraph (now called the unit
representing the catchment, and this might result in greater hydrograph) that attempted to predict the time distribution
uncertainty in prediction. Thus, there is a problem of trying of discharge as well as the peak. The unitgraph represented
to achieve the right balance between model complexity and the time distribution of one unit of effective or excess rain-
prediction uncertainty in situations that might have different fall. The effective rainfall was that part of the total rainfall
types of data available for model calibration or evaluation. inputs that contributed to the storm hydrograph for an event
It must be stressed that this is an ongoing research area (though it is known that in many catchments much of the
in hydrology. This is reflected in the sections on recent hydrograph is made up of water that was stored in the
modeling approaches, including fuzzy modeling techniques, catchment prior to an event, and which is displaced out
“top–down” approaches to modeling and the assessment of of storage during the event). There was still a runoff coeffi-
uncertainties in model calibration and model predictions. cient problem in predicting how much of the rainfall would
These new approaches have been developed out of a become effective rainfall for a given magnitude event and
recognition of some of the difficulties and limitations of antecedent conditions, on an event-by-event basis. The unit
modeling all the hydrological processes in any specific hydrograph is a linear approximation, which means that
catchment area; difficulties that become important when once a unit hydrograph has been determined for a catch-
trying to predict hydrological responses under more extreme ment area, different effective rainfalls will be predicted
(wet or dry) conditions than measured before, or trying as having the same time distribution, but with a simple
to predict the hydrological impacts of future change in linear scaling of the magnitudes. The linear approxima-
land management or climate inputs to inform policy and tion then produces the result that two units of effective
management decisions. A view on the future of rainfall- rainfall (rather than the total input) will produce twice as
runoff modeling is expressed below in “Predicting the much discharge.
effects of change” and a guide to further reading in “The The linear assumption of the unit hydrograph has been
future of rainfall-runoff modeling”. much criticized in the past, but has served rainfall-
runoff modelers quite well. Similar principles are still
A Classification of Rainfall-runoff Models used in transfer function models today (see Chapter 128,
Rainfall-runoff modeling: Transfer Function Models,
There are a variety of different ways of classifying rainfall- Volume 3). The greater problem is that of estimating how
runoff models (see e.g. Clarke, 1973; Wheater et al., 1993), much of the total input to a catchment should form the
but the most fundamental distinction that is usually made is effective rainfall. The effect of antecedent conditions on the
between lumped models and distributed models. Lumped effective rainfall is very much a nonlinear problem. Twice
models deal with a catchment as a single unit, attempting to as much rainfall with the same antecedent conditions will
relate precipitation inputs to discharge outputs without any generally produce more than twice the discharge, while the
consideration of the spatial patterns of the processes and same rainfall under dry antecedent conditions will produce
characteristics within the catchment. Distributed models much less runoff than under wet antecedent conditions. This
attempt to take account of the spatial patterns of hydro- nonlinearity problem was partly solved by the introduc-
logical response within a catchment area. tion of continuous simulation models, particularly as digital
Rainfall-runoff modeling started with lumped models computers started to become more widely available in the
back in the nineteenth century with the “rational method ” 1960s. One of the earliest lumped continuous simulation
that related discharge (usually peak discharge) directly models was the Stanford Watershed Model, developed by
to a measure of rainfall inputs, catchment area, and a Norman Crawford and Ray Linsley at Stanford University
runoff coefficient (see Beven, 2001b). The runoff coefficient in 1962 (for a fuller history see Section “A short his-
served the purpose of what would now be called a model tory of rainfall-runoff modeling” and Beven, 2001b). This
parameter that could be varied to reflect local conditions structure still survives, with many added components, in
in particular applications. The difficulty in applying the the HSPF (Hydrologic Simulation Package Fortran) pack-
rational method was therefore a difficulty of deciding on age now distributed by the US Environmental Protection
a value for the runoff coefficient. This was a particular Agency. It is an example of a lumped explicit soil moisture
difficulty in this prototype model because this parameter accounting (ESMA) model in that it has several storage
would vary not only from one catchment to another but elements to represent different hydrological processes and
also with the magnitude of an event and the state of the storages (interception, upper soil moisture, lower soil mois-
catchment prior to an event (the antecedent conditions). ture, groundwater etc.). The fluxes between these stores are
 RAINFALL-RUNOFF MODELING: INTRODUCTION 3

controlled by a variety of different parameters. Application or parameterizations or outputs, such that a given input
of the model, therefore, requires that these parameters be sequence will produce an uncertain prediction of the output
estimated or calibrated for each application area. The tem- variables. Unfortunately it can be very difficult to separate
poral changes in the antecedent conditions are taken into out the effects of the various sources of uncertainty that
account by continuously calculating the changes in storage contribute to the prediction process, particularly for this
in the model, but at the expense of introducing a consid- type of nonlinear system. There is, therefore, no agreement
erable number of parameter values (more than 30 in some about what approach should be used in assessing predictive
versions of the Stanford Watershed Model). uncertainty. Those methods that have a good theoretical
This type of model can be made more distributed by base often have restrictive assumptions that are difficult
applying it to different subcatchments and then routing to justify in real applications. Other methods are not so
the subcatchment outputs to the point of interest using a restrictive in their assumptions, but do not have such a
river routing component. This allows the spatial patterns of strong basis in theory. These issues are discussed further in
inputs to be taken into account, at least at the subcatchment (see Chapter 130, Fuzzy Sets in Rainfall/Runoff Model-
level. This might be important where there are strong spatial ing, Volume 3 and Chapter 131, Model Calibration and
patterns in inputs resulting from strong changes in rainfall Uncertainty Estimation, Volume 3).
with elevation, or the seasonal patterns of snowmelt as a
function of elevation, aspect, snow accumulation, and so Process Descriptions in Rainfall-runoff Models
on. The only problem with such a strategy is that each It is important to recognize the difference between the per-
subcatchment then requires its own set of parameter values ceived complexity of water flow pathways in a catchment
to be estimated or calibrated. and the relatively simplistic models that are used to predict
A similar problem applies to fully distributed models water fluxes and stream discharges. The hydrologist can
which attempt to make predictions of the response of all appreciate the complexity of the real processes in a qualita-
the elements in a spatial discretization of a catchment area. tive way (the perceptual model); it is much more difficult
The elements might be based on a square grid discretization to represent that complexity in a quantitative mathematical
(an approach that has become common with the widespread description (the formal or conceptual model). Thus, any
availability of raster digital elevation data) or more irregular rainfall-runoff model must be necessarily only an approx-
elements or control volumes. Each element can have its imate representation of the real processes. This is perhaps
own inputs and parameter values. The more complex easiest to appreciate for lumped catchment models, though
models also involve multiple layers in the vertical to these can still make useful predictions if the only variable of
predict fluxes in three dimensions. Thus, these models can real interest is the discharge hydrograph. The same consid-
have literally hundreds or thousands of parameter values eration also applies, however, to even the most distributed
that must be defined to run the model. Even so, these rainfall-runoff model, since they can only approximate the
models cannot reflect all the perceived complexity of the responses at the sub-control volume scale. Thus, all pro-
hydrological processes. In particular, they still require some cess descriptions in rainfall-runoff models rely on some
parameterization of the variability in soil water, runoff, parameterization of the sub-control volume complexities
and evapotranspiration fluxes at the sub-element scale. and heterogeneities.
Beven (1989) has therefore, suggested that even the most In all models, it is necessary to maintain mass continuity.
distributed hydrological models are still effectively lumped Water, as represented by the model, should not be gained
representations at the element scale. The representative or lost. Theoretically, this should also be the case for
elementary watershed approach to distributed modeling, continuity of energy and momentum as well, so that
described in (Chapter 13, Pattern, Process and Function: hydrological models could claim to preserve the second
Elements of a Unified Theory of Hydrology at the law of thermodynamics. Unfortunately, this is much more
Catchment Scale, Volume 1 and Chapter 127, Rainfall- difficult to ensure, with energy and momentum losses
runoff Modeling: Distributed Models, Volume 3), is an being accounted for implicitly by model parameters such
attempt to deal directly with this issue in a physically as resistance coefficients (but see Chapter 127, Rainfall-
rigorous way. runoff Modeling: Distributed Models, Volume 3). What
A final classification of rainfall-runoff models is into is often not realized, however, is that even for mass
models that are deterministic and models that are stochastic continuity, the water balance kept by the model may
in their predictions. A deterministic model will, with a given not be the same as that kept by a real catchment or
input sequence, produce a single prediction of all its output control volume. This is because there are components
variables. The majority of model applications in hydrology of the real mass balance that cannot easily be measured
are still run deterministically, even though the uncertainties accurately, especially as integral fluxes over an area. This
associated with such predictions are now widely appreci- is particularly true of precipitation inputs over a catchment
ated. A stochastic model allows for uncertainty in the inputs when rain falls from locally intensive convective cells or
4 RAINFALL-RUNOFF MODELING

snow accumulation is affected by wind redistribution. It is The subjective nature of model design was recognized
also true of evapotranspiration fluxes from heterogeneous early in the history of computer-based rainfall-runoff mod-
vegetation covers on hillslopes. This is not to suggest eling. It did not seem to be really scientific. There was
that models should not maintain mass continuity (although then a conscious attempt to make model definition more
some models include multipliers to adjust the input rainfalls objective by representing the processes by the physical
or potential evapotranspiration rates as parameters), only “laws” used to describe those processes. The seminal paper
that these physical laws must be viewed as theoretical describing such an approach was the “blueprint for a physi-
capacities of the system that might be difficult to verify cally based digitally simulated model” of Freeze and Harlan
by measurement (see discussion in Beven, 2002a). (1969). Very briefly, they showed how a model could be
In many conceptual models, the representation of pro- developed using the Richards’ equation (based on Darcy’s
cesses at the control volume scale is expressed in terms of law) for partially saturated subsurface fluxes and the St.
some function relating flux to storage in the control volume. Venant equations for depth-integrated surface fluxes. These
Most lumped catchment models are of this type, includ- equations could be coupled though appropriate boundary
ing the ESMA models. Many routing models, such as the conditions. The equations are both nonlinear partial dif-
unit hydrograph, can also be interpreted as a sequence of ferential with boundary conditions that vary in space and
storage elements in series or in parallel. These storage ele- time as hydraulic potentials and therefore infiltration rates,
ments can have storage-flux relationships that are either seepage faces, and other features of the system change
linear or nonlinear. With one or two exceptions, these rela- dynamically. There are no general analytical solutions to
tionships are not based on secure theoretical reasoning, but these equations; therefore, the model must be based on
on subjective choices that seem to give the right type of approximate numerical solution algorithms (originally finite
behavior. A linear store has often been used, for exam- difference methods but now more usually finite element
ple, to represent the recession curve from a catchment area. or finite volume techniques). This requires that the sys-
The equation of a linear storage gives a recession curve for tem be discretized into a large number of control volumes
to represent the dynamic distributed responses and the
which discharge declines exponentially with time. There is
way in which the fluxes will vary in response to pat-
no physical reason to suggest that the recession of the inte-
terns of hydraulic potentials on the hillslopes and in the
gral drainage fluxes of all sources of subsurface flow in the
stream network.
catchment or control volume should have the form of a lin-
Following the Freeze and Harlan blueprint, the model
ear store (except under some the very limiting assumptions:
description is objectively based on physical principles. It
see Chapter 126, Modeling Recession Curves and Low
does not, however, entirely eliminate the subjective ele-
Streamflows, Volume 3) but the form appears to work for
ment in model definition. This is because the characteris-
many cases. tics of all the control volumes representing the catchment
But not for all cases: other nonlinear storage elements must be defined in terms of appropriate parameter values
seem to be more appropriate for some catchments. Some before the model can be run. In principle, these param-
models have used first order or second order hyperbolic eters have physical meaning (they include, for example,
functions in time (see e.g. Tallaksen, 1995; Lamb and hydraulic conductivity and porosity of the soil, and the
Beven, 1997, for a recent discussion of the different forms roughness coefficient for channel flow). But we have no
that result from different assumptions about the subsur- techniques that will allow the values of those parameters to
face flow domain). The important point to take from these be measured at the control volume scale. Small-scale mea-
different parameterizations is that we do not have the inves- sures of hydraulic conductivity, for example, reveal that
tigative measurement techniques necessary to be secure there may be small-scale variability of orders of magni-
about what form these relationships should take, given that tude within a control volume, especially for soils containing
we know that the sub-control volume processes are subject macropore structures. This then implies that there will be
to considerable complexity, except by seeing which func- sub-control volume variability in the hydraulic gradients
tions might be appropriate in reproducing the discharges and velocities of flow in the soil. Because of the highly
at the catchment outlet (where we can take a measure- nonlinear relationships between unsaturated hydraulic con-
ment). It then follows that different types of function might ductivity and flow rate, this then implies that the small-
be appropriate for different catchments or control volumes scale physics (as represented here by Darcy’s law so that
within a catchment. This does not only apply to subsurface flux per unit area is assumed to be directly proportional
drainage but also to any other hydrological flux within a hydraulic gradient and local unsaturated hydraulic conduc-
control volume. Any subjective choice of function to rep- tivity) should not be used at the control volume scale. In
resent the fluxes at the control volume scale can usually a heterogeneous domain, the same form of relationship
only be calibrated and corroborated by comparison with cannot represent the nonlinear variability of local veloc-
measured discharges in this way. ities at the control volume scale except in the special
 RAINFALL-RUNOFF MODELING: INTRODUCTION 5

(and unrealistic) case of a soil with spatially homogeneous several attempts to predict the stream hydrographs using
characteristics. a model based on this concept and with more and more
Darcy’s law is still being used, however, in nearly all detailed information about the spatial patterns of local infil-
rainfall-runoff models based on the Freeze and Harlan tration characteristics in the catchment. The measurement
blueprint, because without detailed knowledge of the small- scale was, however, still smaller than the model control vol-
scale heterogeneity in soil structure and characteristics, it ume scale. None of these attempts was considered entirely
is difficult to develop an adequate alternative description. successful (see Loague and Kyriakidis, 1997) and the story
The continued use of Darcy’s law to represent control is now continuing with a new approach based on cou-
volume fluxes, therefore, depends on an implicit (and sub- pled surface and subsurface modeling (van der Kwaak and
jective) assumption that it is an adequate approximation to Loague, 2001). Similar problems have been encountered
the integral fluxes at the control volume scale. Physically, in trying to represent distributed field measurements of
this is much easier to justify for flows in the saturated soil water contents or water table levels in other studies
zone than in the unsaturated zone. Similar considerations (e.g. Lamb et al., 1998; Anderton et al., 2002). It should
apply to the representation of surface flows because of be noted, however, that such comparisons are also subject
the way in which velocities and depths can vary rapidly to scale problems since local water content measurements
across a hillslope (especially if covered with short veg- and even local water table measurements in shallow soils
etation) or in the variable cross-sections of the channel might not be directly equivalent to the variable of the
network. In all these cases, effective values of the model same name predicted by the model at the control vol-
parameters will be required at the control volume scale, ume scale.
but these effective values cannot easily be measured and There is one additional important consideration in the for-
their physical basis has been undermined by the effects of mulation of process representations of rainfall-runoff mod-
the sub-control volume nonlinearities. These laws are now els. In many field natural and artificial tracer experiments,
just one possible choice for a sub-control volume param- based, for example on the variations in concentrations of
eterization, awaiting the development of something more
the environmental isotopes of oxygen and hydrogen in the
realistic. For a modern interpretation of the Freeze and Har-
water molecule, it has been shown that much of the storm
lan blueprint, however, (see Chapter 127, Rainfall-runoff
hydrograph is made up of water that was stored in the
Modeling: Distributed Models, Volume 3).
catchment prior to an event, and not the rainfall that fell
In addition, the Freeze and Harlan blueprint is not
in that event. The stored water is being displaced into the
complete (Beven, 2002b). There are processes that were not
stream by the incoming water. A model that is predict-
included in the original description, including preferential
ing only discharge does not necessarily need to take this
flows in the soil and on the surface. Other processes
might also be important, such as surface crusting, the into account, but it might be an indication that the pro-
concentration of rainfalls reaching the soil surface as cess descriptions being used in a model are not appropriate
a result of the structure of the vegetation canopy, and (for example, if a model is based on an infiltration excess
the separation of different parts of the saturated flow runoff description but the tracer data indicate a high propor-
domain due to fracturing of the bedrock or small-scale tion of pre-event water in the hydrograph). It does become
undulations in the bedrock surface. There are many field important if predictions of water quality are required, since
studies that attest to the perception of such processes as the pre-event water and the event water might have quite
important in different catchments, but the development of different quality characteristics (though it has to be remem-
generally acceptable descriptions of such processes has bered that the event water might interact geochemically
proven to be very difficult. It is, for example, extremely with the vegetation and soil before reaching the stream even
difficult to obtain adequate descriptions of the bedrock if it does contribute to the storm hydrograph). In research
surface or soil structure even in research catchments. Some studies, it has proven very difficult to achieve adequate
models have added preferential flow components or depth simulations of both runoff and some quality characteris-
variations in surface flow within their sub-control volume tics, even for conservative tracers such as the oxygen and
parameterizations, but the effective parameter values will hydrogen isotopes. This is, at least in part, because the
be difficult to estimate. effective storage volumes for the prediction of fluxes might
The difficulty of applying this type of model, even with be quite different to the effective storage volumes for the
intensive field measurements to try and determine model prediction of concentrations, and neither might relate eas-
parameter values, is attested by the continuing story of ily to the storages that can actually be measured (locally)
modeling the R5 catchment at Chickasha, Ohio, by Keith in the soil profile. Thus, additional model parameteriza-
Loague and his coworkers. Their original model was based tions and additional effective parameter values, will be
on the perception that the stream hydrograph was the result required for the prediction of concentrations in addition
of a Hortonian infiltration excess runoff process. They made to fluxes.
6 RAINFALL-RUNOFF MODELING

A Short History of Rainfall-runoff Modeling function to describe the infiltration capacity of the soil, with
two parameters to be identified for each soil type, though
The nineteenth century origins of rainfall-runoff model in
he viewed this function as very much a representation of
the rational method were noted in the Section “A classifi-
the surface controls on infiltration, rather than a control
cation of rainfall-runoff models”. The rational method, in
resulting from flow through the bulk soil as it is represented
its very simple equation,
using the Richards’ equation in most process based models
today. The combination of Horton’s and Sherman’s ideas
Qp = CAP (1)
then provided one of the first models to predict the full
where Qp is peak discharge, P is the volume of input storm hydrograph and has been very widely used. It has
precipitation over the catchment area A in a defined time also been widely modified with different types of runoff
period, and C is a runoff coefficient parameter, already has generation function, such as the US Soil Conservation
all the elements of a rainfall-runoff model and, indeed, the Service method that has its origins in the analysis of runoff
method is still in use today. The problem in its application, data from plots and small catchment areas and does not
and the reason why we need more complex methods, lies explicitly assume an infiltration excess mechanism (see the
in the parameter C. C cannot be treated as a constant for a summary of this and other methods in Beven, 2001b).
catchment area, but will vary from event to event. It must We now know, of course, that runoff generation is more
incorporate the effects of the space and time variability in complicated than that (see the Section below on “Process
antecedent conditions, precipitation inputs and surface and descriptions in rainfall-runoff models”), that surface runoff
subsurface runoff generation and the effects of routing of can be generated by saturation excess as well as infiltration
that runoff to the catchment outlet or where predictions excess mechanisms and that in many environments much
are required. of the storm hydrograph is made up of water that was
There was obvious scope, therefore, in later develop- stored in the catchment prior to an event and which is
ments of rainfall-runoff models to separate out the processes then displaced out of subsurface storage by the input
of runoff generation from runoff routing. This was first rainfalls. Thus, particularly as digital computers allowed
done in the 1920s by C. N. Ross in Australia when he more complex calculations to be made, there were new
used the idea of zones of different travel time to the outlet models developed that tried to take more explicit account
in a catchment to allow for different runoff coefficients in of the various storages in a catchment and their responses
different zones. The runoff generation in each zone could to rainfalls. These ESMA models, referred to in the Section
then be delayed by the appropriate travel time and summed “A classification of rainfall-runoff models” have been very
to produce a total discharge. A similar approach was used widely used and many different variants can be found. In
by Zoch and Clark in the United States, and by Richards the 1960s and 1970s there were probably nearly as many
(1944) in the United Kingdom in one of the very first books ESMA models as there were hydrological modelers. All of
on rainfall-runoff modeling. The approach is dependent on them had parameters of the functions controlling the fluxes
the assumption that the routing times do not change for between the storage elements that had to be calibrated for a
different events. It is, therefore, equivalent to a linear trans- particular catchment, although they varied in the number
fer function for the predicted runoff generation. Sherman of parameters to be calibrated. They all worked, more
(1932) took this idea and proposed that the transfer function or less, because given a historical period of rainfall and
could be represented as a time distribution for routing runoff discharge data the parameters could be varied until a good
from a catchment area without any direct link to the areas or atleast acceptable fit was obtained between observed and
involved. He called the transfer function the unitgraph, now predicted discharges.
more commonly referred to as the unit hydrograph. The This calibration was either carried out manually, with
unit hydrograph approach to runoff routing is still widely a visual evaluation of how well the model was fitting
used, and can be treated within a general linear transfer the data (this was how the Stanford Watershed Model
function methodology (see Chapter 128, Rainfall-runoff was calibrated) or by an automatic optimization method,
modeling: Transfer Function Models, Volume 3). using a quantitative measure of model performance, such
There remained the (nonlinear) question of how much of as the sum of the squared errors between observed and
the input rainfalls would be available as effective rainfall to predicted discharges during the calibration period. The
create the output hydrograph from a catchment. At the same manual method had the advantage of using “hydrological
time as Sherman proposed the unitgraph, Robert Horton reasoning” and the experience of the modeler to improve the
(1933) came up with a model of runoff generation that model performance (though this is difficult in practice once
has since been very widely applied. He proposed that the more than 4 or 5 parameters are being varied); the automatic
storm hydrograph, in excess of a baseflow component, was optimization method has the advantage of objectivity (at
predominantly made up of surface overland flow in excess least once the quantitative criterion of performance has been
of the local infiltration capacity of the soil. He provided a chosen). But, using either technique it was often difficult
 RAINFALL-RUNOFF MODELING: INTRODUCTION 7

to decide whether one model or parameter set was really spatial context and compared with spatial observations.
better than another. Because of the simplifications made, however, the spatial
At the end of the 1960s, an alternative approach was predictions are expected to be much more approximate than
proposed with a view to avoiding some of these problems for the fully distributed models. The best known examples
by making the representation of the processes as “physically of this type of model are the Probability Distributed Model
based” as possible as in the Freeze and Harlan (1969) (PDM) in which a distribution function of storage capacities
blueprint for a physically based distributed description of is based on a purely statistical distribution with parameters
surface and subsurface flow processes in time and space. to be determined by calibration; and TOPMODEL in
Although, with even the best computers available in the which a distribution function of an index of hydrological
early 1970s, only very approximate (and indeed inaccurate) similarity is based on an analysis of catchment topography.
solutions of these equations could be achieved there were In the latter case, the similarity assumptions mean that
many perceived advantages of this family of models and predictions only need to be made for representative values
they have continued to be developed and applied to the of the topographic index, but knowing the pattern of the
present day. The best known current example of such a topographic index allows the predictions to be mapped back
model is the Système Hydrologique Européen (SHE) which into space (see Beven, 2001b, for a full description of this
is now available in a number of different versions. The and other distribution function models).
advantages of such models included the potential to apply
the models using only measured (rather than calibrated) The Problem of Model Choice
parameters, the ability to take account of spatial variability
of inputs and catchment characteristics in their correct Given this wide variety of different rainfall-runoff mod-
spatial context; the ability to make spatially distributed els available, there is a real problem of model choice
predictions of water fluxes that could then be used as the for the practitioner in any practical application. Beven
basis for other predictions of sediment and contaminant (2001b) summarizes a number of criteria on which to base
transport. All of these advantages appeared to make the model choice.
model applications more realistic and more objective. • Is a particular model readily available, or could it be
Unfortunately, for a number of reasons, it has proven made available if the investment of time and money
difficult to take full advantage of these attractive features appeared to be worthwhile?
of the Freeze and Harlan distributed model blueprint. It • Does the model predict all the variables required for a
is now more widely appreciated that the equations of particular aims of a project?
these models, when applied at the model grid scale, are • Are the assumptions of the model likely to be limiting
still only very approximate representations of the actual given what is known about the nature of the hydrolog-
processes. Thus, it is still generally necessary to resort to ical responses at a site?
some form of calibration to apply these models in real • Can all the inputs required by the model (flow domain,
catchments. There are very few applications reported in boundary and initial conditions, model parameters)
the literature where results based only on measured and be provided within the time and cost constraints of
directly estimated parameter values are reported and these the project?
results have generally not been that accurate for predictions
of either discharge or internal state variables such as water More correctly these are criteria for model rejection, and
table levels or soil moisture storage. For internal state it is all too common that all the available models could
variables, there is, in fact, a problem in making such a be rejected on the basis of one or more of these criteria.
comparison, since the local measurements are themselves This is clearly not very helpful in practical applications, so
generally at a much smaller scale to the element scale used some compromise may have to be reached. It is important,
in the distributed model. however, that the user be aware of what compromises are
These difficulties, however, do not take away the demand being made so that the associated uncertainties can, as far
for predictions that are distributed in space, so there has as possible, be evaluated.
also been a family of models that have attempted to take
account of the spatial heterogeneity in catchment areas, but
The Problem of Model Calibration
in a parsimonious way (that is, with only a small number of
parameters). This type of model represents the variability Nearly, all model applications require some form of model
in responses by some statistical distribution function of calibration to identify the model parameter values appropri-
characteristics within the area of the catchment rather than ate to a site. This is because we do not have the techniques
making spatially distributed predictions directly. In some that would allow the direct measurement or independent
cases, these models have been constructed so that the estimation of parameter values at the scale required by the
predictions can still be mapped back into their correct model (i.e. the control volume scale for a distributed model
8 RAINFALL-RUNOFF MODELING

or the catchment scale for a lumped model). There is no firm theoretical basis for model evaluation as is the case
general agreement, however, about how best to approach where strong assumptions can be made about the nature of
the model calibration problem. This is because the cali- the errors in statistical inference.
bration problem is associated with many different sources What is important in modern model calibration, however,
of error that cannot be easily separated. There is the error is that some attempt should be made to evaluate the uncer-
associated with the model structure, in that even the best tainty in the model predictions. It is no longer sufficient
model structures available are only an approximate rep- to find an optimal model and make a single deterministic
resentation of the actual processes. There are the errors prediction of the rainfall-runoff response. There is then too
associated with the definition of the subsurface flow domain much chance of being wrong in prediction. The problem of
and its boundary conditions. There are the errors associ- model calibration and uncertainty estimation is discussed
ated with the specification of the initial storages within further in (Chapter 131, Model Calibration and Uncer-
the flow domain. There are the errors associated with the tainty Estimation, Volume 3).
measurements of the input precipitation (and other forc-
ing variables) to the model, and their spatial and temporal Predicting the Effects of Change
variability. There are the errors associated with measure-
One of the most interesting problems in rainfall-runoff
ments of the discharges (and other internal variables) with modeling is predicting the impacts of future change
which the predictions of the model will be compared. Nor- (see Chapter 132, Rainfall-runoff Modeling for Assess-
mally, there is no way of estimating any of these different ing Impacts of Climate and Land Use Change, Vol-
sources of error independently. We can evaluate how well ume 3). This might be change in land management strate-
a particular model predicts the observed discharges, given gies or change in climate as inferred from the global
the specified inputs, but we do not know what is the real predictions reported by the International Panel on Climate
source of the modeling error. Change (IPCC, 2001; see www.ipcc.ch). In both cases, it
In many rainfall-runoff modeling situations, this then may not only be changes in runoff and subsurface storage
makes it quite difficult to use the theories of parameter that are of interest, but also other variables that depend on
inference that have been well developed in statistics. These runoff such as sediment production, nutrient export, plant
theories require that the structure of the different sources water stress, and so on. In addition, it may be changes in
of error be defined, so that the probability or likelihood the extreme hydrological responses (floods and droughts)
function of predicting a particular observation, given the that may be of most interest to water managers. The inter-
model, can be assessed. The form of the likelihood function esting thing about such prediction problems is that there
depends on the assumptions that are made about the nature are no data available from the future for recalibration of
of the errors. It is possible to lump all sources of error into the hydrological model.
a single “modeling” error, assume a particular structure for Prediction of the effects of change for a particular
that total error, and use the statistical approach. This is an catchment is, therefore, essentially an extrapolation prob-
objective approach in that the validity of the assumptions lem; extrapolation from what can be learned about the
about the modeling error can be tested in comparison with catchment responses today into some future scenario. This
the actual modeling results to see if those assumptions are can be difficult because there will certainly be additional
justified, at least approximately. uncertainties involved, but it may be very difficult to esti-
Some progress has been made in using this type of mate the magnitude of those uncertainties. For example,
approach, particularly in real time forecasting of river flows the various global model outputs reported by the IPCC
when predictions are only required for a few time steps usually provide estimates of the change in monthly rain-
into the future (e.g. Young, 2002; Krzysztofowicz and falls to be expected later in the twenty-first century as a
Kelly, 2000). However, for the case of longer time rainfall- result of changes in atmospheric composition. Since these
runoff simulations, using nonlinear distributed models, models take so long to run even on the best of today’s
it is far more difficult to justify simple structures for supercomputers, such predictions are not associated with
the errors in space and time, especially in the face of any direct estimates of uncertainty (though different global
potential errors in both the inputs and available model models from different modeling groups currently produce
structures. Thus, some alternative approaches have been quite widely differing predictions). Thus, it is impossible to
developed, generally based on Monte Carlo simulation associate any probability estimate with these scenarios. In
in which very large numbers of model runs are made addition, to model the impact of change on the hydrological
using random choices of inputs and/or parameter sets and extremes, it will be necessary to interpret these predicted
compared with the available observations. Those models mean monthly changes into changes in the distributions of
that perform well in this evaluation are then kept for use in rainfalls (and other climate variables) when it is not clear
prediction. However, this approach also cannot separate out if the relationships seen between atmospheric and ocean
the different sources of error and does not have the same circulation patterns and rainfall extremes in a region today
 RAINFALL-RUNOFF MODELING: INTRODUCTION 9

will be the same under a changed climate. Many different whether this will result in a real improvement in model
scenarios could be envisaged for how these relationships accuracy and use because the problems inherent in the cur-
might change, but again it will be impossible to associate an rent generation of distributed environmental models do not
uncertainty measure with a potential scenario. Similar con- necessarily easily go away with improvements in space and
siderations hold for potential changes in land management time resolutions of the component models.
that might have more to do, for example, with future agri- We can distinguish two different (albeit overlapping)
cultural subsidy policies than with the impacts of climate types of models here in terms of constraints. In the first, the
change. accuracy of solutions is still constrained by computational
Thus, we know that predictions of future change will resources; in the second model accuracy is primarily a result
be uncertain but cannot easily quantify that uncertainty. of lack of knowledge of appropriate process representations
About the only strategy that can be taken in this situation and boundary conditions. In the first type of models, real
is to assess the uncertainty in the predictions of a model advances may still be possible as computational constraints
under current conditions and then run different future sce- are relaxed. In atmospheric modeling, for example, there
narios with that model taking account of similar input and is still scope for improvements of the representation of
parameter uncertainty assumptions. Any relative possibility local convection and rainfall forecasts, in the representation
estimate for different scenarios will be necessarily sub- of sub-grid spatial variability of energy fluxes, and in
jective, but such a procedure will give the most realistic the representation of topography by finer grid scales.
assessment of the potential future hydrological responses. Ultimately, however, this type of model will be constrained
The implication, however, is that it will be very important by the need to know finer and finer detail of boundary
to continue monitoring catchments into the future so that as conditions and parameter values, as is already the case in
the responses change they can be used to revise and update the second type of model. An example of this second type
the scenario predictions. is classical distributed hydrological models.
New developments in environmental modeling will allow
The Future of Rainfall-runoff Modeling a new approach to be taken to this problem based on scale-
New computer technologies seem likely to change the way dependent model objects, databases, and spatial objects in
that rainfall-runoff models are constructed and used as com- practical applications to specific places. One of the most
ponents of large-scale integrated modeling frameworks for exciting benefits of the possibilities provided by the GRID
environmental management. In particular, the possibility of in environmental modeling is the potential to implement
using large-scale (the GRID; see www.gridcomputing. models available from different institutions as a process
com; www.gridforum.org; www.globus.org) computer of learning about specific places. It will be possible, in
networking to link together distributed database and com- fact, to have models of all places of interest. However,
putational engines means that it will be possible to as argued by Beven (2000, 2001a, 2002b, 2004), as a
couple together models of many more different envi- result of scale, nonlinearity, and incommensurability issues,
ronmental systems across disciplinary boundaries and the representation of place will be inherently uncertain so
across national administrative boundaries. This is, in fact, that this learning process should be implemented within an
already possible and is already happening on a lim- uncertainty estimation framework.
ited basis as demonstrated, for example, in the regional Sites of interest for a particular prediction can be
water resources models under construction in Denmark implemented as active objects, seeking information across
or the national models for environmental management the GRID to achieve a specified purpose, and using the
being used in the Netherlands. In Japan, the very large- power of parallel computing resources to estimate the
scale “Earth Simulator” is aiming to implement global uncertainty associated with the predictions as constrained
scale models (see Earth Simulator Center, 2003; see by site-specific observations, including those accessed over
www.es.jamstec.go.jp/esc/eng/ES/index.html). the GRID in real time. Initially model results, based perhaps
There is then, however, a real question raised about how on only GIS databases and limited local information,
this type of interdisciplinary model might be best imple- may be relatively uncertain but experience in monitoring
mented. In the past, comprehensive modeling systems have and auditing of predictions will gradually improve the
been constructed as large complex computer programmes. representation of sites and boundary conditions. It is this
These programmes were intended to be general, but have learning process that will be critical in the development
proven to be expensive to develop, difficult to maintain and of a new generation of environmental models that are
difficult to apply because of their data demands and needs geared toward the management of specific places, rather
for parameter identification. With GRID computing tech- than general process representations.
nology, it will be possible to continue in the same vein, but That is not to say that models of places will not require
with more coupled processes and finer spatial and tempo- process representations, but there is a real research question
ral resolutions for the predictions. It is not clear, however, about how detailed a process representation is necessary to
10 RAINFALL-RUNOFF MODELING

be useful in predicting the dominant modes of response of ment/water-framework/index-en.html; www.defra.

a system, given the uncertainties inherent in representing gov.uk/environment/water/wfd) are increasing de-
the processes in places that are all unique. This appropriate mands for predictions of this type about the responses of
complexity issue has become obscured by the desire to build specific locations to change in a way that integrates hydro-
more and more scientific understanding into models, includ- logical and ecological considerations in management. The
ing physical, chemical, and biological components. This system would need to be powerful enough to be used for
desire is perfectly understandable, it is a way of demonstrat- assessing uncertainties in model predictions and the conse-
ing that we do understand the science of the environment, quent risks of potential outcomes. It should also be able to
but it results in models that have lots of parameter values be used off-line for “what-if” management purposes or deci-
that cannot be easily measured or estimated in applications sion support including developing strategies for risk-based
to real places. There is always a certain underlying principle sustainable management in the context of climate and other
in science, that as we add more understanding and eliminate changes. This will include the evaluation of management
empiricisms, then the application of scientific principles of subsystems including for licensing of airborne emissions
should become simpler and more robust. This does not seem and effluents to water courses; strategies for remediation of
to have been the experience in the practical application of contaminated land, rivers and estuaries, and so on.
environmental models. An essential element of this strategy will be the need, as
Events such as the river flooding in the UK in 2000, 2002 far as possible, to “future proof” the model and database
and 2004 and the consequences of the 2001 fires in the US, systems used; avoiding, for example, a strict raster based
have demonstrated the need for a new generation of systems approach or a commitment to one particular modeling
for environmental forecasting. The subtle (and sometimes framework. The key will need to be flexibility. Raster
not so subtle) coupling between atmospheric forcing, catch- databases will continue to be driven by remote sensing
ment response, river runoff, and transport processes requires imaging inputs to the modeling process, and, in some
the dynamical coupling of many components to capture cases, by convenient numerical solution schemes for partial
these subtleties. Components would be a representation differential equations. However, it is often inappropriate to
of the regional atmosphere and the terrestrial surface and force an environmental problem into a raster straightjacket.
subsurface hydrology that would interact through differ- Treating places as flexible active objects might be one way
ent boundary conditions. Built on the fluxes within those around this future proofing problem. Defining the spatial
models air and water pollutant transport models and bio- domain of a prediction problem would allow that place,
geochemical models could be implemented locally within as an active object, to search on the GRID for appropriate
the regional-scale domain. Each component would be able methods and data for resolving that problem, and also for
to assimilate data transmitted from field sites and assess appropriate methods and data for providing the boundary
the uncertainty in the predictions. The components would conditions for the problem (which might then involve other
share 4-D/5-D visualization tools with appropriate inter- modeling or data extrapolation techniques).
active user interfaces. Users will be able to access the There are some interesting implications of such an
current data, visualize predictions for particular locations approach. One is that the variety of modeling methods
and play what-if scenario games over different timescales. available across the GRID to solve a prediction problem
The structure of the system would be such as to facili- might be able to be compared more readily, leading to better
tate and even stimulate improvements to the representation understanding of issues of appropriate model complexity
of different components and the constraint of predictive for different modeling problems. This will especially be
uncertainty by field data collection and data assimilation. the case if, as part of the learning process, simulations
The potential capabilities of the GRID underlie all these are saved to be compared with later observations of
components, though much could already be achieved using the real outcome. This use of “post-audit” analysis has
the Web technology of today. Examples of steps toward been rarely used in environmental modeling, but has been
this type of integrated system (albeit essentially raster instructive in the field of groundwater modeling (Konikow
based) include the US Inter-Agency Object Modeling Sys- and Bredehoeft, 1992; Anderson and Woessner, 1992) and
tem (OMS), (see Chapter 129, Rainfall-runoff Modeling is routine in atmospheric modeling in the evaluation of
for Integrated Basin Management, Volume 3). forecast skill (although the evaluation of global climate
Such an integrated system should operate both in real model predictions still requires an element of compromise
time, assimilating data and boundary conditions from larger at the regional level).
scale models, and displaying the “current state of the envi- To be useful, however, the process of model application
ronment”, as well as providing the potential to update will require the definition of a self-coding system attached
model predictions into the future under different scenar- to places to record and retrieve the methods that have been
ios. Initiatives such as the European Union Water Frame- applied to (or by) that place in the past so that they can
work Directive (see www.europa.eu.int/comm/environ be easily reviewed and evaluated by the user. There is then
 RAINFALL-RUNOFF MODELING: INTRODUCTION 11

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