Hydrology of Mountainous Areas (Proceedings of the Strbské Pleso Workshop, Czechoslovakia, June 1988). IAHS Publ. no. 190, 1990.

The modelling of mountain hydrology: the ultimate challenge

V. KLEMES President, International Association of Hydrological Sciences, National Hydrology Research Institute, 11 Innovation Blvd. Saskatoon, Saskatchewan, Canada

INTRODUCTION

There is little doubt that many will regard the title of this paper as an exaggeration and will tend to attribute its choice to the author's notorious predilection for irritating the fellow hydrologists. After all, why should the modelling of mountain hydrology be signed out from the great variety of hydrological modelling problems which all seem to be overwhelmingly difficult? If, as is often the case, one views hydrological modelling merely as the fitting of some more or less plausible mathematical con­structs to given sets of data so as to minimize the dif­ference between the modelled and recorded time series of runoff, then there probably is no reason to single out mountain hydrology as the most challenging object of hydrological modelling. This, however, is not what I mean by hydrological modelling. Rather, I regard it as a syn­thesis of observed empirical facts and their theoretical understanding, expressed in terms of general (as opposed to ad hoc) and internally consistent quantitative rela­tionships formulated in computationally feasible algo­rithms. Viewed in this way, the above title seems justi­fied since observations of the states of nature in moun­tainous terrain are the most difficult to make and the processes governing mountain hydrology cover the widest range thus posing the greatest demands. oh:\thèoretical ida'der standing.

Before coming to the problems of modelling, it may therefore be proper to say a few words about each of these two prerequisites of it in order to be able to put the modelling aspects into a proper perspective.

The States of Nature

Mountains do not give up their secrets easily. This does not apply only to the proverbial yeti and sasquash but also to such mundane things like precipitation, snowcover or streamflow. The problem often is not what, from the scientific point of view, should be measured and observed, but what can be observed given the available logistics. The most formidable problem is accessibility on a conti­nuous basis. In mountainous terrain, this is not assured even where roads and permanent settlements exist. Where

29

V. Klemes 30

they do not we simply do not have observations of the phenomena of interest but only what can best be termed their random glimpses and often only guesses. When we want to draw the precipitation profile across a mountai­nous territory, it still will look like the sketch in Fig.l, although more than a quarter of a century has pas­sed since the latter had been compiled, we still can draw it only with dashed lines over the mountain peaks.

Fig.l Orography and precipitation profile across western Canada along the Canada-U.S.A. border (after Bruce and Clark, 1966)

To measure the snow cover, we have to organize mountain­eering expeditions and hire Olympic ski champions as observes. To measure streamflow in a mountain creek, we first have to pollute it with some chemical and measure its concentration as the best proxy - and yet, if we take our job seriously enough, we can draw the peaks of hydrographs only as dashed lines. It seems that the only feasible way to measure flood flows in mountain streams may be acoustic or seismic methods, that is to record, from a safe distance, the thundering sound and ground vibrations accompanying the associated "sediment trans­port" - the genuine rock music of rolling stones which, by the way, is much more inspiring than its various man-made substitutes.

The problem of accessibility does not plague only measurement involving water but also geological mapping, the measurement of surface energy, wind speed and all

31 The modelling of mountain hydrology

other phenomena in the boundary layer. A great step for­ward has been the introduction of remote sensing tech­niques which have already borne the fruit in topographic surveying, land-use and soil-type mapping and, most sig­nificantly for mountain hydrology, in the mapping of snow cover and lately in the measurement of the snow water equivalent. However, the problem of accessibility will remain a permanent one since ground measurements will be required for the validation of remotely sensed data and for some phenomena the former will remain the only source of information.

The second problem is accuracy. In general, the accu­racy of measured hydrological variables is lower in mountainous terrain than it - is in flat terrain. The high slopes and strong wind affect the catch of precipitation gauges; the harsh conditions cause more frequent malfunc­tioning of instruments which leads to gaps in the records and causes inhomogeneity when instruments have to be changed or recalibrated; the accuracy of streamflow data suffers from the high instability of channel cross-sec­tions, frequent damage to the measuring weirs and stage recorders by coarse sediment and debris, or from the necessity to use dilution (tracer) methods which are inherently less accurate than conventional methods. An example illustrating the problem of accuracy is shown in Fig.2 adapted from Sevruk (1985). It represents only one aspect of the problem, namely the systematic error in precipitation measurement, but can serve as parts pro toto in demonstrating the inverse relationship between the accuracy of measurement and the "large-scale rough­ness" of the terrain.

The third major problem in this category is represen­tativeness . Even if observations could be made where necessary rather than where feasible and if their accu­racy could be brought under control, the problem of their areal representativeness would still remain a formidable obstacle to an accurate description of the state of nature. The areal representativeness of point observations is of course a general and pervasive problem in hydrology but it is most serious in mountainous ter­rain. Here the ubiquitous heterogeneity of the soil is combined with heterogeneities of the topography, ground cover, and with the extreme variability of most hydro-logical and meteorological components. The main hope for improvement resides in the perfection of remote sensing techniques and their ability to supply areal distribu­tions or at least areally integrated or averaged quan­tities, since a reliable reconstruction of the states of nature from point measurements is often virtually impossible in mountainous terrain.

An illustration of the combined effect of the major problems described above is the fact that in some mountain basins even such a basic hydrological state of

V. Klemes 32

Summer

Winter

Year

Fig. 2 Systematic errors (required corrections in %) of precipitation measurements in Switzerland (after Sevruk, 1985)

33 772e modelling of mountain hydrology

nature as the components of a long-term water balance cannot be reliably established. Thus, for example, for two out of nine analyzed basins in the Canadian Cordil-leran region, Halstead (1967) found the mean annual runoff to be about 1.7 times higher than the mean annual precipitation, without being able to tell whether this was due to error in the precipitation data, in drainage area, or due to glacier melt or other factors (e.g. springs fed from adjacent basins).

Theoretical understanding

The increased difficulty of theoretical understanding of the processes shaping mountain hydrology resides in their wide range as well as in the wide range of the physical conditions over which they operate. For example, in terms of vegetation, surface temperature, surface water and groundwater regime, a single mountain slope may be seen as telescoping conditions from subtropical to arctic, that is over thousands of kilometres and tens of degrees of latitude in flat region. Surface water velocities range from free conditions to supercritical and subcritical as the mountain slopes change from verti­cal walls to flat alluvial plains in the valeys. Response times to precipitation input may vary from minutes to centuries depending on whether it enters the basin on clear slopes with southern exposures or north slopes with glaciers. The orographic influence injects into areal precipitation distribution a variability surpassing the highest extremes observable in flat terrain and the same is true for the distribution of temperature. Sharp and temporally stable discontinuities in irradiation fre­quently develop as mountain ranges form vast cloud reservoirs on their one side while the other is under clear blue skies as one can frequently observe when, for instance, traveling over the St. Gotthard pass in the Alps or crossing the Coastal range in California.

It is in mountain hydrology where the claim that many hydrological problems require an interdisciplinary approach can perhaps best be demonstrated. Thus in mountain hydrology there meet the disciplines of hydro­logy, glaciology, climatology, boundary layer meteorolo­gy, geomorphology, geophysics and geology, and the interplay of the corresponding processes raises the complexity of the theoretical understanding of hydrologi­cal phenomena - which in many respects integrate the effects of the other processes - above the level encoun­tered in most other natural settings. This intrinsic theoretical complexity, combined with the aforementioned difficulties encountered in connection with gathering the empirical knowledge give a justification for the title of this paper and provide the necessary perspective for the hydrological modelling which aims at a simulation

V. Klemes 34

of the empirical facts based on a quantitative represen­tation of the underlying physical processes.

SOME IMPORTANT ASPECTS OF MODELLING IN MOUNTAIN HYDROLOGY

There is a large body of literature on hydrological modelling in mountainous areas whose systematic critical review is beyond the scope of one paper and would be more properly addressed in a monograph form. Here I would like to emphasize some aspects that have emerged as having a greater importance in the context of mountain hydrology than they may have in other contexts.

At a general level, an important problem is that of scale of the modelled prototype which to a large extent defines the very meaning of the designation "mountainous" basin. While in other contexts the scale usually has only the connotation of the areal size, in the context of mountain hydrology its important aspect is the height of the mountains since different processes will dominate the runoff say, in the Himalays and in the Jizera Moun­tains. The vertical aspect of scale has a direct rele­vance to the effective structuring of a hydrological model, its parametrization, the extent of lumping and, in particular, the type of lumping - while in low mountains, horizontal lumping by sub-basins may be effective, ver­tical lumping by elevation zones may be the only produc­tive way in high mountains. This immediately suggests that, for different climates, there may be transitional ranges of vertical scale where none of the lumping sche­mes may work well and, on the other hand, ranges with optimal performance potentials for different model struc­tures. Similar patterns may exist for the horizontal scale of lumping and, of course, the performance at various combinations of the vertical and horizontal scales will further depend on the scale at which the processes are lumped in the time dimension. In other words, in mountain hydrology the qualitative problems of scale discussed at an earlier occasion (Klemes, 1983) acquire one additional dimension.

The notion of scale is. also important for the meaning of the designations of model structures as "lumped" or "distributed", especially when the latter is applied to the areal dimension. The intrinsic relativity of such a differentiation follows from the fact that every element of a distributed model is in itself a lumped representation of many smaller-scale elements. On the other hand, a limped model may easily become an element of a distributed model of a larger basin. Awareness of this aspect of scale is especially important at inter­national meetings such as this where notions of scale and size are of necessity influenced by the national origins of the participants. This is illustrated in

35 The modelling of mountain hydrology

Fig. 3 where the basin of the South Saskatchewan River at the City of Saskatoon is drawn to the same scale as the territory of Czechoslovakia (the area of the former is 140 000 km2 and that of the latter 120 000 km 2). It may be noted that the relative prominence of the South Saskatchewan river basin in the Canadian context would be comparable to that of the Orava River basin (the shaded area on the map) in the Czechoslovakian context. To pursue these perceptional analogies one step further, it may be noted that a sub-basin slightly larger than that of the Orava River is identified as a basin of a creek in the context of the South Saskatchewan River. It is not unlikely that such seemingly superficial differences may influence the attitudes of hydrologists of different countries to the use of lumped and distributed models in similar settings (in absolute terms, the Orava River and the Willow Creek could be considered to have similar mountain basins).

Coming now to the specific of the modelling problem, the issues that deserve special attention in the mountain hydrology context will be listed under the following three categories; model inputs, model structure and model outputs.

Model Inputs

There are two important features that distinguish moun­tain hydrology models on the input side. The first is that the input is not identical with precipitation only as is the case in other hydrological models but includes energy inputs as an indispensable component. While in the classical "rainfall-runoff" type of model energy inputs are required only to determine the évapotranspiration output from the basin, in mountain hydrology models they are essential to determine the "active" portion of the total precipitation input, i.e. to separate the liquid part which immediately contributes to runoff from the solid part which remains temporarily inactive in storage. To achieve this, mountain hydrology models must incorpo­rate elements of snow-melt and glacier-runoff models.

The second feature which is less obvious to an out­sider but is perhaps the most important one is that, unlike in a standard hydrological model, the inputs typi­cally are not simply "entered" into it but must them­selves be first modelled. This necessity arises from the fact discussed earlier, i.e. that measurement of preci­pitation and energy inputs can often not be made where they are most needed but only where they are technically feasible - usually in the accessible mountain valleys. Thus what in the standard models is merely a "processing" of inputs (typically a simple or weighted areal averaging of point measurements) becomes input modelling in moun­tain hydrology. Its aim is to estimate, from the scarce

V. Klemes 36

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37 The modelling of mountain hydrology

and ineffectively located point measurements of precipi­tation and energy components, the areal and elevation distributions of (a) precipitation amounts, (b) precipi­tation form, and (c) energy (or at least temperature). Considering the complexities of mountain topography, the related variability of microclimate and the problem of accuracy of the available measurements (Fig.2), it is often the case that the "input modelling" represents the most important and elaborate part of a mountain hydrolo­gy model. A direct mapping of the distributions of the precipitation and energy inputs would represent a radical simplification of this aspect of mountain hydrology models and an improvement of their effectiveness. A good example of this potential is the model of Martinec (1975) , which uses the observed snow cover area as one of the major inputs (see below).

Model Structure

While it is customary to classify models, in terms of their structure, as either lumped or distributed, such a classification is not very helpful unless it is applied to a specific basin, i.e. unless related to scale as discussed earlier.

A more informative approach, especially in mountain hydrology modelling where the physical prototype is extremely complex and variable, may be to view models in appropriate position on a continuous "system spectrum" covering the range between purely black-box (correlation) models ignoring the physical mechanisms, to white-box models trying to account for them explicitely. This systems dimension will always be accompanied by two dimensions of the coarseness of the representation (in space and time) whose lower limits are one element, i.e. a totally lumped spatial representation, and one time interval, i.e. totally lumped temporal representation (long-term averages).

The nature of mountain basins generally requires whiter and finer-grid models than flat basins of compa­rable size to yield results of comparable accuracy. However, it would be wrong to press the whiteness and the spatial and temporal detail too far regardless of other consideration.

One of them is the purpose of the model. Models inten­ded, for example, for an assessment of the impact of land use change or acid precipitation on stream conditions for the support of various types of aquatic fauna will have to be whiter and have a finer resolution than models for an assessment of runoff volume for the purpose of hydro-power generation; a model for the simulation of a time series of daily flows may have a different opti­mal structure than a model for peak-flow forcasting of flash floods, etc.

V. Klemes 38

Another consideration is the available data and their quality. If the structural refinement of the model ex­ceeds the "carrying capacity" of the data the effective­ness of the model will decrease rather than increase. This is an extremely important consideration in view of the various deficiencies of inputs in mountainous terrain discussed earlier.

Yet another factor which carries much more significance in mountain hydrology models than elsewhere is the type of parametrization employed. There are as yet few general rules and much depends on the specific physical condi­tions of the basin being modelled. However, it seems to be borne out by the experience available so far that robust features integrating the effects of several causal factors are more effective as model parameters than a detailed simulation of the integrating process within the model because of the danger of the cumulative effect of even minor systematic errors that may be present in the measurements.

A related problem of the structure is the dichotomy between "conceptual" and "direct" parametrization, i.e. between the use of optimized parameters derived by cali­bration of the model on measured outputs, and parameters computed directly from the measured physical characteris­tics of the prototype. The advantage of the former appro­ach is that the various systematic errors in the data can be filtered out but this is achieved at a cost of the model's reduced applicability and, indeed, credibility since the physical interpretation of the parameters is compromised; the effects of errors in the data and in model structure can compensate each other so that the model may give plausible results for the wrong reasons and may fail under conditions different from those pre­vailing in the calibration data set.

Some of these aspects of mountain hydrology modelling can be profitably studied in the recently released WMO report on Intercomparison of Models of Snowmelt Runoff (WMO, 1986) where several models with different structu­res were applied to several mountainous basins ranging in area from 8.4 km2 to 2170 km2 and in elevation diffe­rence from a few hundred to 3000 m.

An illustrative example from the WMO (1986) study is presented in Figs. 4a and 4b. They show simulated stream-flows for two different basins as obtained by two models with different structures. One of them is the above mentioned model of Martinec which has a rather coarse physically based empirical structure and uses only six parameters, all of them externally derived (i.e. none obtained through calibration). It has been selected here because it has the smallest number of parameters of all the models featured in the study and the only one not requiring calibration. The other selected model, develo­ped by Morin (WMO, 1986, III.6), represents the other

39 The modelling of mountain hydrology

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Fig. 4a Observed and computed daily streamflow hydrographs in the Durance river (France) for four consecutive years (from top to bottom, 1975-6 to 1978-9). Left: Morin model; right: Martinec model (adapted from WMO,1986)

V. Klemes 40

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Fig. 4b Observed and computed daily streamflow hydrographs in the Dischma basin (Switzerland) for four consecutive years (from top to bottom, 1975-6 to 1978-9). Left: Morin model; right: Martinec model (adapted from WMO, 1986)

41 The modelling of mountain hydrology

end of the spectrum and has the highest number of para­meters; their number is 31 (plus four basin-related parameters for each sub-basin) and 2 8 of them are derived by calibration. Despite this large difference between the two models, their results are comparable, both on the relatively large Durance basin and the small Dischma basin, the Morin model showing an apparent exaggeration of the basin responsiveness to inputs, especially during dry periods, while the Martinec model showing a slight systematic underestimation of the low flows. It would be wrong to "rate" the performances of the models but the results demonstrate that appropriate parametrization and direct inputs of the right kind (in this case the snow cover area in Martinec's model) may significantly in­crease the parsimony of a model without necessarily sacrificing its performance accuracy.

The problem of the "shade of grey" of models is often blurred, especially in the so called conceptual models where the often unknown internal processes and their interactions are replaced by postulated ones which are fitted into structures designed more to satisfy our preconceived ideas than to describe the actual physical prototype. While on the surface such models may appear rather white, in reality they may be very dark grey since many of their "physical" elements with calibrate parame­ters may be just regression variables in disguise, their real function being to supply the number of the degrees of freedom necessary for the success of the model fitting exercise.

One general comment on the structure of mountain hydrology models may be of interest. Most mountain hydro­logy models incorporate, in one form or another, an assumption that the snowpack, the glacier subsystem, the groundwater subsystem, or sections of the whole basins behave like linear systems, i.e. that during input-free periods their water releases follow exponential recession curves which can be superimposed to yield the total runoff. While this may be so in certain situations, there is no reason why it should be a general rule. Actually, there are indications that strong nonlinearities and threshold effects may operate under many types of conditions typi­cal to mountainous watersheds, for example the "ridge effect" of a sudden saturation of some soil strata follo­wed by a sudden surge of groundwater runoff, release of liquid water from firn, temperature-controlled glacier melt and infiltration of meltwater into the soil and, especially in high mountains in warm climates such as the Himalayas, the rather frequent occurrences of sudden transfers of large volumes of snow and ice into low elevation zones by avalanches with the consequent acce­leration of their melting rates. It is well known that the same recession curves may result from many different combinations of linear and nonlinear types of storage

V. Klemes 42

release and that relying on them for system identifica­tion is unreliable. It is therefore important to study the actual behaviour of the various components and their interactions because without this knowledge and its incorporation into models it is unlikely that the present­ly poor model transferability will be significantly improved.

Model Outputs

It is quite a general feature of mountain hydrology mo­dels that the site at which the streamflow is to be simu­lated or forecast is itself outside the mountain environ­ment proper, i.e. in a well established part of the river channel in the valley, which may be rather far from the mountain slopes where the runoff is generated. Since most models are fine-tuned by calibration on the measured streamflow at such a site their conceptual soundness (whiteness) can be corrupted while their performance is being "improved". The point is that the streamflow at the output site contains components not explicitly represented by the model, such as hidden sinks and sources, effects of unaccounted for internal mechanisms (e.g. embedded evaporation - condensation cycles, unknown heterogenei­ties in the soil or glacier matrix, etc.) and last but not least, systematic and random errors in streamflow measurements. As noted earlier, the calibration process forces the compensation of these factors by adjustments of parameters which, especially in automatic calibration producers, are often not done on the basis of their physical soundness but rather on the basis of output sensitivity to the various changes. In this manner, the soundest components of a model can be corrupted while components of dubious value may remain undetected. This all points to the fact that a systematic improvement of models relying on output-based calibration is difficult which is especially true for mountain hydrology models where there are more unknowns and where the physical setting is more complex that in other environments.

CONCLUSIONS

Because of the higher complexity and variety of the processes to be modelled and the greater difficulty of their observation and measurement than is the case in other natural settings, mountain hydrology modelling highlights some important latent problems of contempo­rary hydrology and points the way to their solution more clearly than it would otherwise be apparent.

Thus it calls for a restructuring of hydrological education along the line of the geophysical disciplines treating the processes that shape the hydrologie cycle. This seems to be an obvious precondition for an informed

43 The modelling of mountain hydrology

interpretation of the observed geophysical phenomena that are to be modelled but so far it does not seem to have been recognized. Hydrologists are still being trained mostly as engineering technologists rather than as earth scientists. It has been adopted as one of the priorities of the IAHS to initiate, in collaboration with UNESCO and its International Hydrological Program, a change in this direction.

Another aspect of hydrology that mountain hydrology modelling makes painfully obvious is the importance of areal mapping of hydrological and other geophysical variables and the inadequacy of the traditional point measurements which are the legacy of the century old technology. It is embarrassing that in an era when man can measure physical conditions on the outer planets and, indeed, in distant galaxies, he often has no direct observations of conditions on this earth, conditions which are much more vital to his welfare. It is doubly embar­rassing since much of the needed technology is now avai­lable and it often is only a matter of redistribution of resources and of an updating of the scientific and tech­nology background of operational personnel in order that it can be employed. An important initiative in this direction has just been started jointly by the World Climate Research Program of WMO and the International Council of Scientific Unions. It is called the Global Energy and Water Cycle Experiment (GEWEX) and one of its main goals is to improve, with the aid of advanced space-based technology, the observations of areal distribution of phenomena important for the modelling of land-surface processes, the most important being identified as the hydrological processes. Since mountain areas belong to the most important source areas of surface runoff, moun­tain hydrology can greatly benefit from the GEWEX project and the IAHS will welcome ideas along these lines from the hydrological community.

REFERENCES

Bruce, J.P. and Clark, R.H. (1966) Introduction to hydro-meteorology. Pergamon, Oxford

Halstead, E.C. (1967) Cordilleran hydrogeological region. In: I.C.Brown (editor) Groundwater in Canada, Geologi­cal Survey of Canada.. Ottawa, pp. 159-172

Klemes, V. (1983) Conceptualization and scale in hydrolo­gy. Journal of Hydrology, Vol.65, pp.1-23

Martinec, J. (1975) Snowmelt-runoff model for streamflow forecasts. Nordic Hydrology, 6(3), pp. 145-154

Sevruk, B. (1985) Systematischer Niederschlagsmessfehler in der Schweiz. In: B.Sevruk (editor) , Per Nieder-schlag in der Schweiz, Geogr.V. Bern, pp. 65-75

WMO (1986) Intercomparison of snowmelt runoff. Operatio -nal hydrology Report No.23, WMO, No.646, Geneva