Journal of Hydrology, 100 (1988) 341 r352 341 Elsevier Science Publishers B.V., Amsterdam --- Printed in The Netherlands

[4]

RAINFALL-RUNOFF MODEL ING - - PAST , PRESENT AND FUTURE

E. TODINI

University of Bologna

(Received January, 1988; revised and accepted February 4, 1988)

ABSTRACT

Todini, E., 1988. Rainfall runoff modeling -- Past, present and future. J. Hydrol., 100: 341-352.

A brief review of the historical development of mathematical methods used in rainfall runoff modeling is presented. A simple classification of the current available models based upon both a priori knowledge and problem requirements is proposed in order to assess the state of the art. Finally an analysis of emerging problems in hydrology is used to ascertain possible future develop- ments and trends.

INTRODUCTION

The analys is of the histor ical development of a branch of science should a lways be preceded by a discussion of the reasons which in the first place promoted, and constant ly motivated, the interest in that part icu lar line of research.

These mot ivat ions const i tute the basic key to understand past developments and future trends in a part icu lar branch of science.

In this paper an at tempt is made to provide a l ink between the problems that or ig inated and mot ivated the interest of hydrologists in the field of mathemat i - cal in terpretat ion of the ra in fa l l - runof f process, a small b ranch on the tree of hydrology. Histor ical developments and future trends of ra in fa l l - runof f model l ing are considered in the l ight of the evolut ion of problems and solutions at different points in time, within the f ramework of avai lable knowledge and computat iona l resources.

A HISTORICAL OVERVIEW OF METHODS

The origins of ra in fa l l - runof f model ing in the broad sense can be found in the second hal f of the 19th century, ar is ing in response to three types of engineer ing problems: urban sewer design, land rec lamat ion dra inage systems design and reservoi r spi l lway design.

In all these problems the design discharge was the major parameter of interest.

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According to Dooge (1957, 1973), during the last part of the 19th century and the earlier part of the 20th, most engineers used either empirical formulas, derived for particular cases and applied to other cases under the assumption that conditions were similar enough, or the "rational method" which may be seen as the first attempt to approach rationally the problem of predicting runoff from rainfall.

The method, which was derived for small or mountainous catchments was based upon the concept of concentration time; the maximum discharge, caused by a given rainfall intensity, happens when rainfall duration equals or is larger than the concentration time.

During the 1920s, when the need for a corresponding formula for larger catchments was perceived, many modifications were introduced in the rational method in order to cope with the nonuniform distribution, in space and time, of rainfall and catchment characteristics.

The modified rational method, based on the concept of isochrones, or lines of equal travel time, can be seen as the first basic rainfall runoff model based on a transfer function, whose shape and parameters were derived by means of topographic maps and the use of Mannings formula to evaluate the different travel times.

The types of problems to be solved were much the same as before, but hydrologists were trying to provide more realistic and accurate solutions, although still in terms of surface runoff.

Sherman (1932) introduced the concept of the unit hydrograph on the basis of the superposition principle.

Although not yet known at the time, the superposition principle implied many assumptions, i.e. the catchment behaves like a causative, linear time invariant system with respect to the rainfall/surface runoff transformation.

The unit hydrograph principle accelerated the interest of hydrologists who were now in a position to produce estimators, not only of the peak discharge, but also of the hydrographs caused by more complex storms.

The unit hydrograph method had however a number of problems: (1) the separation of surface runoff from base flow; (2) the effective rainfall determina- tion; and (3) the derivation of the unit hydrograph.

Solutions to all these problems involved an extremely high degree of subjec- tivity.

At the end of the thirties and during the forties a number of techniques were proposed in order to improve the objectivity of methods and results, and the techniques of statistical analysis were invoked.

A discussion on the different approaches and the relevant bibliography can be found in an interesting report by Dooge (1973).

The real breakthrough came in the fifties when hydrologists became aware of system engineering approaches used for the analysis of complex dynamic systems. They finally realised that the unit hydrograph was the solution of a causative, linear time invariant system and that the use of mathematical techniques such as Z, Laplace or Fourier transforms could lead to the derivation of the response function from the analysis of input and output data.

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This was the period when "conceptual" models originated. The derivation of the unit hydrograph in discretized form (the unit graph) from sampled data (known as the inverse problem) still remained a big problem, due to the non- particularly linear behaviour of the system and the generally large errors in input and output data.

To overcome this problem, hydrologists found that shapes of the unit hydrograph could be provided on the basis of the solution of more or less simplified differential equations, such as for instance those describing the time behaviour of the storage in a reservoir or in a cascade of reservoirs (Nash, 1958, 1960). The unit hydrograph could then be expressed in terms of few parameters to be estimated from catchment characteristics or by means of statistical procedures: moments, cumulants, regression, maximum likelihood, etc.

A bloom of these model gave rise an unbelievable variety of solutions: a cascade of linear reservoirs, linear channels, linear channels and reservoirs, nonlinear reservoirs (Prasad, 1967), etc.

On the other hand, in deriving the unit graph shape from actual data, very few advances were made even using the transforms: a classical paper by Rao and Delleur (1971) shows the effect of noise in data, on the unit hydrograph derived by Fourier transforms. Only after the work of Wiener (1949) and Tikhonov (1963a, b) and the introduction of continuity and regularization constraints in the estimation phase [Eagleson et al. (1965), Natale and Todini (1977)] more realistic and reliable estimates of the unit hydrograph were obtained.

Studies for the representation of nonlinear systems where also carried out by means of Volterra integrals and polynominal projection (Amorocho and Orlob, 1961) or constrained piecewise linearization (Todini and Wallis, 1977).

Box and Jenkins (1970) provided hydrologists with an alternative method of expressing the unit graph in terms of parameters, i.e. the autoregressive exogenous variables form of the transfer function (ARX) or the autoregressive moving average (ARMA), and the analogies with existing "conceptual" models were pointed out (Spolia and Chander, 1974).

In subsequent developments these techniques, though satisfactory from the mathematical and philosophical point of view, lost more and more of their connection with the real world of hydrological problems, and became more or less mathematical games played by algebrists concerned only to prove the generality of their approach.

There is an obvious danger in models which fail to provide a reflection of reality: it is very easy to check a model on the basis of actual rainfall and runoff values, but it is extremely difficult to understand the quality of results e.g., in terms of continuity of mass, for a model which uses transformed input and output in the Box-Jenkins (1970) sense, in order to stationarize the time series (i.e. through successive applications of the difference operator).

In the sixties other approaches to rainfall-runoff modelling were considered. In search for a more physical interpretation of the process one could represent the behaviour of single components of the hydrolog cycle, at the catchment

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scale, by using a number of interconnected conceptual elements, each of which represented the response of a particular subsystem.

The need for such an approach arose from the following requirements: (1) to extend the use of the model to long continuous records avoiding the complexity of storm runoff and base flow separation; (2) to apply the model to complex watersheds with a large variety of soils, vegetation, slopes, etc.: and (3) to extend the model more or less without calibration to other similar catchments.

A number of these models appeared: Dawdy-O'Donnel 0965), Stanford model IV (Crawford and Linsley, 1966), Sacramento river (Burnash et al., 1973), U.S. Corps of Engineers (Rocwood and Nelson, 1966), Tank and ~o on (WMO, 1975), which represented differently the interconnected subsystems and were considered the top models of the sixties.

In theory, if the structural description was correct the parameters of the model, such as storativities (surface, saturated unsaturated zones), friction factors and threshold effects could be related to the actual physiographic charcteristics of the catchment.

Unfortunately, in many cases, the large number of parameters used in the models and the fact that these were calibrated on a "best fit" basis, lead to sets of unrealistic parameter values, generally incorporating errors in data and moreover errors in the basic description of the interrelations between simple process models.

This lack of a one-to-one relationship between model and reality gave rise to a research effort by a number of hydraulic and hydrological institutions, the Danish Hydraulic Institute (Denmark), the Institute of Hydrology (U.K.) and SOGREAH (France), to produce a model which would integrate the partial differential equations expressing the continuity of mass and momentum and linking the subprocess models by matching the relevant boundary conditions. The product of such an effort, the "SHE" (Syst~me Hydrologique Europ~en) is now available as a basic laboratory tool, allowing for the simulation of the internal as well as external effects of catchment behaviour (Abbot et al., 1986a, b).

Development of the SHE was, indeed, an ambitious undertoken. Previous physically based models were relevant to small urban catchments (Wooding, 1965-1966) or small mountain catchments (Freeze and Witherspoon, 196~ 1968), and generally reproduced only the rainfall-surface runoff process.

On the other hand the basic differential equations for all the component processes were available (Richards for the infiltration in the unsaturated zone, Darcy for the groundwater flow and De Saint Venant for the overland and channel flow) and the advent of large powerful computers made the project feasible.

It was hoped that the effort would be justified by the fact that the model could be calibrated using more physically based parameters, thus allowing for easy transfer to ungauged catchments, and that it would allow a reliable "internal description" of the catchment as an answer to newly growing em vironmental problems.

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The seventies saw a growing awareness and concern for soil erosion and degradation problems, pollutants diffusion, and in general of the environmen- tal impact of anthropization and land use changes.

Another type of rainfall-runoff model was developed in the late seventies and in the eighties: the real-time forecasting model as an answer for the need of warnings in flood prone areas, and as a tool for reservoirs or hydraulic structures management.

Generally based on recent updating and recalibrating techniques such as Kalman filters (Kalman, 1960; Kalman and Bucy, 1961; Todini, 1978; Todini and Wallis, 1978; Kitanidis and Bras, 1980a, b; O'Connell, 1980; Wood, 1980; Wood and O'Connell, 1985), real-time forecasting models must be reliable, updatable and constitute part of an entire forecasting system which must include automatic real-time data acquisition, data validation, forecasting procedures and dissemination of forecasts (Nemec, 1986; Bacchi et al. 1986).

Although many rainfall-runoff conceptual models have been set in a real- time forecasting mode (recently an intercomparison of models was carried out in Vancouver on behalf of WMO (1988), there is not yet a clear understanding on the advantages (or disadvantages) of recalibrating in real time all model parameters.

AN ASSESSMENT OF PRESENT

In order to find a path among the plethora of different rainfall-runoff mathematical models available today it seems necessary to introduce an additional classification of models (which unfortunately will also increase the plethora of different classifications).

A mathematical model in broad sense, is a combination of two basic components. The first one expresses all the a priori knowledge that one has on the phenomenon to be represented and can be referred to as the physical component.

The second, the stochastic component, expresses in statistical terms what cannot be explained by the degree of a priori knowledge already introduced in the physical component (see also Clarke, 1973).

The a priori knowledge can be introduced in the models in many different ways, ranging from the total a priori ignorance, thus reducing the model to a pure stochastic process model where not even the cause and effect postulate is advocated, to the full description of system dynamics based upon the partial differential equations describing the balance of mass and momentum.

The stochastic component is then added to explain what is not explained by the physical component, and therefore becomes conditional upon it.

The level of a priori knowledge on the system under study is here taken as the basis for model classification both in terms of model structure and in terms of parameters.

On these grounds it is reasonable to assume four classes of model structures with increasing level of a priori knowledge: (1) purely stochastic; (2) lumped

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integral; (3) distr ibuted integral; and (4) distr ibuted differential. Models 2, 3, and 4 are causal models. Two classes of parameters can be considered as a funct ion of methods used for their estimation: (a) stochastic; and (b) physical.

As mentioned earl ier apart from the purely stochastic models one can always add to the causal models a stochastic component to explain as much as possible the uncerta inty content of model residuals. In synthesis, the a priori knowledge based classif ication of models can be represented by Table 1.

Purely stochastic models

The purely stochastic model is a model that can be written as:

y = g(x ,~,~)

where one does not even imply a cause and effect relat ionship between the output variables y and the input variables x; the parameters ~ are estimated from input and output time series via statist ical techniques, general ly by minimizing a funct ional in terms of model residuals e.

The level of information introduced in this model is minimal and the results are valid on average. Since it is always possible to postulate at least cont inuity of mass, this type of model is general ly avoided for ra infa l l - runof f modelling, or used only when dealing with time increments larger than the actual system dynamics (for instance monthly time increments).

TABLE1

Model classification

a priori

knowledge

S.

P.

M 0 D E L S

S. L . I . D. I . D, D.

S, = stochastic; L.I. = lumped integral; D.I. = distributed integral; D,D. = distributed differen- tial; P. = physical.

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Lumped integral models

The lumped integral model is a model where the system dynamics are rep- resented in integral form (i.e., the impulse response, the unit hydrograph, etc.) and relates to a catchment or a subcatchment as a whole, by considering its overall behaviour.

A great variety of present rainfall-runoff mathematical models can be accommodated in this class, which includes most hydrological and conceptual models formulated in terms of impulse response function (linear channel, linear reservoir, cascade of reservoirs, etc.).

In this class, one can also include those transfer function models, such as the Constrained Linear Systems (CLS), where the a priori knowledge on the system in introduced in the model by means of constraints expressing for instance continuity of mass or regularity conditions (Natale and Todini, 1977).

Parameters are generally estimated using statistical techniques since, due to the complex internal relationships, they can hardly be interpreted on the basis of physical catchment characteristics.

Distributed integral models

The distributed integral models attempt to overcome this problem of physically meaningful parameters. This class of models which include most of the more complex '~conceptual" models (Stanford, Sacramento, and similar) is based upon the idea of representing all the phenomena at a subcatchment scale using either empirical formulas or the impulse response of the subsystem (infiltration, surface runoff, etc.) in integral form and combining all the components by matching their "boundary conditions".

Unfortunately, boundary conditions have to be assumed a priori in order to derive a physically meaningful response function, which in turn means that one cannot really match the boundary conditions when dealing with submodels represented by their integral impulse response function (apart, obviously, from continuity of mass when the time interval is sufficiently large to hide system dynamics).

One should also consider that these models, cannot provide a really distrib- uted output, unless one subdivides the catchment in extremely small size subunits which is generally impracticable due to the large number of parameters (16-23) used by this type of models for each subunit (WMO, 1975, 1983).

The distributed integral models should therefore be more fruitfully considered as an extension to the lumped integral models in order to provide a better interpretation of the overall behaviour of a large spatially variable catchment but not really adequate for quantitative computation of internal fluxes.

Anyway these models are widely used at present and their parameters are estimated either on physical grounds, within limitations described above, or on the basis of error minimization.

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Distributed differential models

Finally the distributed differential type, of which very few examples are known (e.g. SHE), represents the catchment behaviour in terms of all the differential equations discretized in time and space, expressing mass and momentum balance for each subsystem and linking together the subsystems by matching at each step in time their mutual boundary conditions.

The difficulty in the integration and the high computer requirements, together with the need for large amounts of data has reduced, for the time being, the applicability of this type of model, although interesting perspectives can be foreseen in future for its ability in actually describing in a distributed manner the internal phenomena at the subcatchment scale. This may allow for, e.g., the study of internal transport and diffusion phenomena as well as the analysis of the effect of averaging and lumping hydrological quantities or model parameters at a subcatchment scale in order to obtain more realistic lumped integral or distributed integral models.

Parameter classes

In order to complete this classification one should include an indication of the method used to estimate parameters.

In fact, in addition to the a priori knowledge in the assumptions for model structure specification, one has to indicate the a priori knowledge of parameters.

To clarify the effects of this assumption on parameters let us consider the consequences of parameter estimation.

Given the physical model structure M and a set of parameters, one can write:

y = f (M,~,x ) + g(~)

where y represents the observed output quantity (e.g. discharge) in terms of model structure M, parameters value u and causal input x (e.g. rainfall): f (M, ~, x) represents the physical model component; and g (D represents the stochastic model component.

In order to preserve the physicality of parameters a one has to "assume" or "adjust" parameters, within the range of physically meaningful values following basic physical justifications. As a consequence, one will obtain a residual, conditional upon model M and parameters ~ and will proceed to analyze the stochastic component with a hypothesis on its structure g (D.

Conversely if one '~calibrates" the model parameters ~ by means of a statisti- cal technique (least squares, regression, meximum likelihood, etc.) based upon residuals, regardless of the physicality of the model, this is equivalent to assuming a stochastic model:

y = g'(M, ~, x, e)

of structure M, thus loosing a great deal of the physical meaning.

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In other words, if one knows a priori that the system is well represented by model M and parameters ~, one should capitalize on that, thus eliminating a great deal of uncertainty inherent in any statistical analysis.

To clarify this classification on parameter estimates, the parabolic-type response function model, which is generally used for propagating flow overland and in channels can be considered a stochastic-parameter lumped integral model if its parameters (diffusivity and convectivity coefficients) are determined on the basis of residuals minimizations, or a physically-based parameter lumped integral model if the parameters are derived on the basis of their physical interpretation.

TOWARDS FUTURE DEVELOPMENTS

With an increasing awareness of the environmental impact of human activities there is a need for increasing monitoring and managing of critical situations such as floods, droughts or accidental pollutions, all of which will require the development of specific real time forecasting models.

In particular the effect on the environment of phenomena such as acid rains, soil degradation, diffusion of fertilizers and pollution in general can be studied only by dynamic, physically based models which provide the means for adding the chemical and biological conservation equations.

The availability of computerized maps as well as the increase in satellite- based remote sensing, allowing for the preparation of large data banks which will include many physical characteristics and the advent of more powerful computers, allowing for the solution of the differential equations on extremely large discretization grids, will more and more attract the interest to the distri- buted differential type models.

On the other hand the need for more or less simplified lumped models for water balance and water resources management studies, will address the interest of research on problems connected to the effects of lumping: i.e. integrating (and thus averaging) phenomena or parameters in space on the line already followed by Zhao (1977), Moore and Clarke (1981), Beven (1975) in order to produce simplified models at a subcatchment scale.

In particular, real-time forecasting models could incorporate these simplified models combined with stochastic models of their residuals in order to allow for updating.

CONCLUSIONS

Following the history and development of rainfall-runoff mathematical modeling one can perceive the interaction between the requirements for hy- drological models on one hand and the solutions proposed by hydrologists on the basis of their present knowledge and available technology on the other hand.

Nowadays problems and trends will constitute the basis for future develop- ments, and three types of models are foreseen for use in the nearby future: (1)

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differential distributed models; (2) distributed or lumped integral models derived from integrating the differential distributed models; and (3) distributed or lumped integral models plus stochastic component for real-time updating.

As the differential models will become an essential tool for providing answers to more complex problems involving internal flows, quantitative analysis and the effect of lumping on the overall outflow, the lumped models will synthesize and reflect this knowledge by reducing the number of parameters to the essential few. The new information systems based upon satellite and radar technologies will provide the means for acquisition of distributed data and the size and speed of computers will allow for the solution of large distributed differential systems.

On the other hand micro- and minicomputers will be used tor ~very day rainfall runoff modeling or for the management of the real-time forecasting systems and models at very low costs.

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