We recently had the pleasure of re-reading the Tribunes Libres of Ghislain de MARSILY (1994) and Jacques GANOULIS (1996), especially their discussions of a new typology for hydrological models and the analysis of uncertainty. It appears, however, that some confusion and alternative interpretations of hydrological modelling still persist. It is therefore important, notwithstanding our agreement with many of the authors' points, to re-examine some aspects of hydrological modelling in order to clarify certain ambiguities.
A distinction made by de MARSILY, between models conditioned by observable phenomena and the physically-based models employed when no phenomena have been observed, invites criticism in terms of the practices to which it leads. GANOULIS' argument, that physically-based models can provide a viable description of processes if differing spatial and temporal empirical coefficients are used, does not stand up to a detailed analysis of the effects of scale. In other words, the issues addressed by these authors arise from the impossibility of using purely physically-based modelling in practical applications due to the difficulty of taking into account and transcribing the characteristics and unique behaviour of each unit of landscape or sub-catchment. To this we can now respond that there are now other lines of thought concerning what are known as physically-based models.
Where distributed modelling is concerned, that all places have unique characteristics is a geographical aphorism. The fact remains that the limitations of modelling, expressed by de MARSILY (1994) as the three principles of uniqueness of place, action and time, can be better defined by performing more detailed analysis in the context of uniqueness. Uniqueness limitations partly explain the wide-ranging developments in modelling in respect of both the theory and tools specific to particular applications. One cannot help but notice that expectations of quantitative prediction in hydrology have increased in parallel with the availability and power of computers. This evolution, however, is essentially due to technological advances rather than real scientific progress. Why? Principally, because of the unique characteristics of catchments: in our view, catchments transcend all available theories concerned with hydrological modelling. Moreover, this does not change if better physical hypotheses are proposed, nor if predictions are made for the variables or "non-observable phenomena" discussed by de MARSILY.
In this paper, we address these questions and suggest a relevant approach to hydrological modelling for taking into account the unique character of catchments.
The uniqueness of place, action and time
If all places implicated in a specific analysis have unique characteristics (i.e., uniqueness of place), how can we attempt their generalisation in hydrology? In our opinion, two main options pertain: either by the use of flow models - including physically-based models founded on hydrological 'laws' which, in practical applications, are considered only as empirical generalisations (as in BEVEN, 1996a) - or by empirical regionalisation of variables of hydrological interest. These two approaches require the availability of useful measurements - in modelling, for parameter calibration and in regionalisation, for gathering a sample of values sufficient for the establishment of correlations. In the discussion that follows, we concentrate on modelling and on the problem of uniqueness as a set of parameters.
With few exceptions, hydrological measurements are carried out on point scales or on very small plots in comparison with the natural scale and spatio-temporal variability of hydrological processes. Consequently, measuring techniques well characterise the problem of uniqueness of place. In addition, because of the limited duration of data collection, we can reasonably conclude that data are also characteristic of the uniqueness of time and action. Only a continuous measurement of discharge or in-flow tracer concentrations furnishes a truly integrated measurement of the variability of responses to a process occurring upstream from the point of measurement, be it at the scale of the catchment, sub-catchment or even the slope in isolation. Nevertheless, neither point-scale nor integral measurements can give more than a simple indication of the heterogeneity of spatial and temporal responses without significant investigative effort in situ and its associated high cost. We therefore cannot, in a general way, know the entire variability characterising the uniqueness of a catchment, hence the element of surprise commonly associated with tracer measurements on small and large scales (e.g., FLURY et al., 1994; IORGULESCU, 1997; JOERIN, 2000).
It is true that we can make classifications of slopes through different indices based on topography, vegetation, soil or geologic substratum. Many modellers now possess Geographic Information Systems (GIS) and use them to this end. Such databases have already been used for catchment modelling (e.g., SCHULZ, 1996; FLÜGEL, 1995; HIGY and MUSY, 2000), but these give only a general indication of important hydrological variables. They do not directly specify the parameter values necessary for modelling. For example, it has been demonstrated that the flow regime in the saturated zone can be more influenced by the topography of the substratum than by that of the surface (McDONNELL et al., 1996; FREER et al., 1997); that soil hydraulic characteristics, obtained from physical measurements (a function of soil transfer) do not always yield good hydrological predictions (HIGY, 2000); and that it is possible to measure highly varied responses for a given type of vegetation. It follows that data obtained from GIS can be effectively utilised for hydrological modelling, but only with considerable resulting uncertainty. As yet the question of the unique character of hillslopes or catchments has not been resolved by the sole contribution of data sets, however pertinent and complete they might be.
In this context, a reductionist argument can be still be made that if a better descriptive model of natural processes could be developed, one that included all the effects' heterogeneity, in some places a better representation of these processes would result. With this view, de MARSILY (1994) concludes his thinking on the sole manner of establishing the prediction of "non-observable" phenomena by emphasising three lines of development: geometrical representation of reality, the use of spatially distributed process representations and analyses of scenarios, and an insistence on bringing the focus of modelling down to a level that includes the underlying processes of modelled phenomena. To bolster this argument, de MARSILY uses examples of fluvial deposition modelling to characterise the particular structure of aquifers as well as the storage of radioactive wastes. GANOULIS also employs an argument in which physically-based modelling can succeed only if we have a theory for the aggregation of parameters at an appropriate scale. Such inquiry by means of exploring reasonable hypotheses is acceptable for furthering understanding of the flow processes, but let us recall that our understanding is in that case strictly hypothetical and non-site specific, even if based on observation. For in situ, practical applications, we believe this argument is impossible to justify (for a complete explanation, see BEVEN, 1995, 2000). Indeed, how many realistic models of a given domain are possible? How many parameter values lead to truly different results? How many physical measurements are necessary to carry out an appropriate aggregation or at least to reduce the number of simulations? Experiments carried out at Yucca Mountain (USA) and Sellafield (UK) on radioactive waste storage have underscored the difficulties of obtaining coherence between different models - or modellers - based on this approach. In such cases, at best we can obtain scenarios that are more or less realistic, but incorporating many remaining uncertainties nonetheless.
By nature, the uniqueness of place, action and time creates uncertainties of prediction, flow modelling or of direct regionalisation of the variables necessary for a particular application. We may observe that the problem of uniqueness will remain a limitation for applications even had we developed a 'perfect' theory of processes.
Uniqueness as a set of parameters
From one particular perspective, a model of flows in one catchment can be extended to another by reducing the unique character of the catchment to the uniqueness of the set of parameters , initial and boundary conditions characterising the basin. As de MARSILY (1994) shows, in cases where data are available, parameter calibration causes the model to be considered a 'black box,' even if physically-based. For his part, GANOULIS (1996) opposes the idea that physically-based models can exhibit black-box behaviour; this position, however, is not always defendable in practical applications (see BEVEN 1989, 1996a).
The main problem in characterising the uniqueness of catchments with parameter values is that when models are calibrated, most have sufficient degrees of freedom to produce simulations that are acceptable in comparison with available data. It is not usually difficult to find an acceptable set of parameters if acceptability is not too narrowly defined. This problem is also a result of errors in measurement and uncertainties in spatial and temporal interpolation of the data themselves.
Therefore, unquestionably, we must concede that we cannot represent catchment hydrological behaviour by a set of unique parameters. It is preferable, in such cases, to consider the 'equifinality' of the multiple parameter sets that produce acceptable simulations of a catchment's response (BEVEN 1993, 1996a, 2001). The foundation of this approach lies in the observation that as a general rule there is no one unique set of optimal parameters for model calibration but a family of solutions or surface regions of response, yielding results of similar quality. To avoid this problem, methods using multi-objective calibration have been developed for portions of a model's surface of response (GUPTA et al., 1998; YAPO et al., 1998; TARANTOLA, 1987); however, even if we can consequently reject more sets of parameters, a group of acceptable models will still remain.
At this point, the interpretation of calibrated parameter values becomes difficult and the 'validation' of models impossible, since parameter values depend essentially on the construction and mode of application of the model, on initial and boundary conditions, on the duration of acquisition of available data, and above all on the values of the other parameters. It follows that the extrapolation of parameter variables to other conditions will be highly uncertain, as is shown by PARKIN et al. (1996).
The problem has yet another interesting implication. Often it is not possible to measure all parameter quantities necessary to a model's application; to find numerical values for the conditions of soil, say, or sub-soil, or 'similar' types of vegetation, a literature search is necessary. However, to obtain 'effective' values, it has usually been necessary to calibrate these values and, what is more, for different (unique) conditions of place and time - and, perhaps, for models other than the one being applied (though all may be 'physically-based'). This type of transfer of parameter values is often taken, for example, when modelling soil-vegetation-atmosphere transfers. The fact that the calibrations frequently result from optimisation further aggravates the problem. In such cases one single set of parameters is chosen among others that might be compatible with available data, but for the application of a different model! Moreover, the uncertainty resulting from the particular optimisation of published parameter values adds to the other sources of uncertainty. It is possible that this approach will be successful but subject to conditions related to the complex interactions among parameter values and to model structure, and as a function of initial and boundary conditions, and of the period of acquisition of data used for calibration (BEVEN, 1993, 1995, 2000).
Consequently, the use of predictions based on the outcome of a model combining different parameter values from diverse sources should be done with care; they may turn out to be good just as well as bad.
Uniqueness and uncertainty: Beyond the reductionist approach
This brief discussion illustrates the limitations of a reductionist approach where the hydrological behaviour of catchments is concerned. In particular, it is useful to distinguish between deductive modelling for the comprehension of hydrological processes in a well-defined area, and modelling for effective prediction of unique hydrological responses on the scale of an aquifer or catchment. A similar observation can be made for the 'non-observable' phenomena of DE MARSILY. With these, using currently available technology, it appears essentially impossible to construct a theory or model that depends only on physical measurements. It follows that calibration is inevitable and that predictions must invariably be associated with uncertainty.
Thus, if reductive description is impossible and representations at the scale of the catchment uncertain, how are we to approach modelling in the future? One way is to make predictions according to a conditioning structure that rejects invalidated models. A few viable schemas of such structures exist; the reader is directed to the examples given by de MARSILY (1994) as well as to the fuzzy-set logic examples suggested by GANOULIS (1996). We remain nonetheless convinced that such conditioning is also possible when we take into consideration the equifinality of models.
Consider for the moment that one or more models have been chosen for a practical application (even if for the prediction of non-observable phenomena). All models that predict the dependent variables are a priori potential models (we can nonetheless initially make a choice, taking more or less account of our scientific preferences). In all models, it will then be necessary to specify the values of a few parameters. These will be uncertain; however, by specifying a range of values - and, rarely, the associated covariances - it is possible to conduct simulations with multiple sets of parameters to evaluate predictive uncertainty.
Clearly, we could stop here (as do PARKIN et al., 1996), i.e., at the direct estimation of the distribution of predictions (reflecting all the spatio-temporal variations of a distributed and non-linear model) for use in a risk analysis. However, it is also pertinent to conduct an evaluation of such simulations in comparison with available data, be they observable variables or expert opinion. We can thus hope that such conditioning would reduce the range of predictions, and would allow the rejection of models that are not credible. Conditioning would thereby allow a more precise representation of the uniqueness of a particular catchment. Nonetheless, there would always be multiple models, all adequately compatible with the data (in the qualitative sense of "adequate"). Such conditioning is formalised in the GLUE methodology of BEVEN and BINLEY (1992), which employs the Monte Carlo method to explore the range of possible simulations.
In consequence, data availability assumes prime importance. This is why de MARSILY (1994) distinguishes between non-observable and observable phenomena. A question now arises: How do we arrange data for future conditions? This is obviously not possible, but, as GANOULIS indicates, there are always some opinions available that might be used in a conditioning process. To this end we can therefore use fuzzy or Bayesian techniques. We thus turn our attention to the efficacy of different types of data in the conditioning process and, particularly, in model rejection. To reiterate, what we propose in this context is a data evaluation completely different from the direct measurement of parameter values. In addition, we can also evaluate the effectiveness of data in constraining uncertainty in comparison with the data acquisition costs.
Such an approach would allow a different formulation of scientific hydrology in moving from smaller to larger hillslope or catchment scales. Despite our presentation of these ideas in a context of practical application, they are not limited to approaches for the practising engineer. Essentially, it is necessary to have a theory for describing and understanding the hydrological response at the scale of required predictions, one that also satisfies expectations of prior catchment responses. Such a theory could be in the form of a conceptual model - whether derived by empirical, physical or inductive means. Moreover, the approach could be based on an attempt to aggregate small-scale parameters into large-scale dependent values, or could also take into account a summary of expert opinion. All these approaches, with all the sets of necessary parameters, are candidates in the evaluative competition of conditioning or rejection. It is possible that not all the feasible possibilities will be evaluated, but this depends on making choices and therefore on an a priori rejection of some theories and models, choices which could be subjected to debate. It is at this point that we could evaluate which approaches are the most effective, and which data are the most appropriate for different applications.
At this stage in our reflections, we can reasonably ask if hydrological science as we have described it conforms to the relativist epistemology described by FEYERABEND (1975) as opposed to the much more methodical structures based on demarcation criteria, such as the falsification of POPPER (1998). The answer is unapologetically affirmative. The central idea of this paper is that uncertainty, equifinality and relativism follow from the uniqueness of place, action and time, and this is so until we have sufficient data to allow the rejection of some possibilities. Falsification may be possible in certain cases, while for the rest we can evaluate only indices of similitude or of possibility of acceptability. However, prior to any rejection stage, it is useful to specify the rules of competition between models and model building schemes. This way, other researchers will have a chance to repeat this 'game of sets,' as it were,... to evaluate data, improve the rules, carry out experiments to test hypotheses and eventually, we hope, to reduce the uncertainty of predictions. If the proposed approach is relativistic, nonetheless it is a scientific relativism.
Modelling as a Mapping of THE landscape space in A modelspace
It is possible to adopt an entirely different view concerning the uncertainty perceived as the product of the uniqueness of place, action and time. If we have multiple representations of a landscape (e.g., catchment, sub-catchment, hillslope, hydrological response unit, plot) all compatible with the available data, we have in sum a sort of uncertain or fuzzy mapping (or projection) of the landscape in the modelling space, in which the axes are defined by the free parameters (and perhaps also by the types of structure of the retained models). Each parcel of the landscape can be represented by multiple sets of parameters (or modelling structures) that are compatible with available data. In the same way that prediction uncertainties are evaluated through a GLUE-type methodology, we can attribute a measurement of plausibility to each set of parameters. These measurements work similarly to an empirical mapping between landscape and modelling space - a complex mapping, as the space of a set of parameters representative of a landscape unit is not necessarily trivial or bounded in a given region of the model space (FRANKS and BEVEN, 1997; BEVEN and FRANKS, 1999).
From this perspective, the possibilities for further development are of some interest, the principal goal of hydrological modelling being to evaluate hydrological functioning and water transfers in the landscape. Still, if we have a high degree of confidence in a model - such as when it is based directly on experiment - knowledge of hydrological functioning in model space can in principal be perfectly known, with computer calculating capacity as its sole limitation. In principal this is also true for stochastic modelling, once we allow that "perfection" here refers to the stability of the statistics employed. Thus, the uncertainty is not in the modelling itself but in the fuzzy mapping of the landscape into the model space. Cognisant of this inversion of the modelling process, we can seek techniques or data in order to constrain the mapping possibilities in a scientific manner. We can imagine, for example, examining the functioning of the model in space to determine which hypotheses can be subsequently tested in the field in order to reduce populations of parameter sets or of possible model structures (BEVEN, 2001). Such a schema thus leads us to join our relativistic view to an approach based on a criterion of falsifiability.
The exploration of equifinality in hydrological modelling will require still more new concepts. In this Tribune Libre, we have exposed possible lines of thought related to modelling and to the conditioning of parameters within the limits imposed by available data. We hope that this Tribune Libre will at least give the reader cause for reflection on the very nature of scientific activity and the role of models as predictive and explanatory schemas of physical phenomena.
Keith Beven extends thanks to Ion Iorgulescu, who called his attention to the Tribune Libre of Ghislain de Marsily; l'Institut d'aménagement des terres et des eaux de l'EPFL for its support during his stay in Lausanne; Bruno Ambroise for his perceptive criticism; and his French teacher, Angela Bolton, for her encouragement, friendship and perseverance with a difficult pupil. André Musy and Christophe Higy thank Keith Beven for the suggestions and comments he offered on each of his visits to Lausanne.
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Keith J. Beven, Environment
Lancaster, Institute of Environmental
and Natural Sciences, Lancaster University, Lancaster LA1 4YQ, UNITED KINGDOM
André Musy, Soil and Water Management Institute,Swiss
Federal Institute of Technology, CH-1015 Lausanne, SWITZERLAND
Christophe Higy, Soil and Water Management Institute, Swiss Federal Institute of Technology, CH-1015 Lausanne, SWITZERLAND