The role of top-down modelling for Prediction in Ungauged Basins (PUB)

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1 HYDROLOGICAL PROCESSES Hydrol. Process. 17, (2003) Published online in Wiley InterScience ( DOI: /hyp.5129 The role of top-down modelling for Prediction in Ungauged Basins (PUB) I. G. Littlewood, 1 * B. F. W. Croke, 2 A. J. Jakeman 3 and M. Sivapalan 4 1 Centre for Ecology and Hydrology, Wallingford OX10 8BB, UK 2 Integrated Catchment Assessment and Management Centre, Australian National University, Canberra, ACT 0200, Australia 3 Centre for Resource and Environmental Studies, Australian National University, Canberra, ACT 0200, Australia 4 Centre for Water Research, The University of Western Australia, Crawley, WA 6009, Australia *Correspondence to: I. G. Littlewood, Centre for Ecology and Hydrology, Wallingford OX10 8BB, UK. igl@ceh.ac.uk Introduction Solutions to hydrological and water resources problems are often required for locations where there are no suitable ground-based data available for making decisions rationally, or the data that do exist are sparse or of poor quality. So there is a continuing need for new and improved techniques for estimating hydrological variables. The situation has worsened in recent years as systematic ground-based monitoring of catchments, from which information can be transferred to areas without data, has tended to decrease in some regions. In response, the International Association of Hydrological Sciences (IAHS) initiated the Decade for Prediction in Ungauged Basins (PUB) (Hubert et al., 2002). This article presents and discusses some views on the merits of so-called top-down computer modelling in the context of PUB. Finer detail of the top-down approach is given elsewhere in the literature (e.g. Young, 2001; Sivapalan et al., in press). In this article, examples are given from personal experience of how a top-down rainfall-runoff modelling approach might contribute to both streamflow and water quality aspects of PUB, assisting with derivation of schemes for the transfer of hydrological information from gauged to ungauged basins (regionalization). PUB PUB recognizes that an improved capability for predicting hydrological responses in ungauged basins is essential for more informed management towards sustaining the world s water and land resources. Knowledge and quantitative prediction of the separate and combined effects of climate, land cover, and land and water management on streamflow, its pathways and associated contaminants is required to assess environmental impacts. In addition, prediction of the socio-economic outcomes of different policy options is required in order to choose which of those options to implement. Several major approaches will be taken within PUB towards making advances in the hydrological science that underpins practical methodologies for estimating variables in ungauged catchments, or in gauged catchments where data are sparse or of poor quality. A corollary of recognizing that such improvements are required is acknowledgement of the value of ground-based data from gauged Received 18 March 2003 Copyright 2003 John Wiley & Sons, Ltd Accepted 18 March 2003

2 I. G. LITTLEWOOD ET AL. catchments. It is essential that good-quality hydrometric data from gauged basins (raingauge and streamflow measurements, etc.) are available for analysis. Numerical (computer) models of gauged-catchment systems, and for transferring information (i.e. model parameters) from gauged catchments to ungauged catchments, form part of the science-based approach required for water resource system design and operation in all major regions of the world. The computer models require good-quality ground-based data for their calibration and subsequent application. The nature of the modelling capacity required in each case obviously depends on the problem being addressed, including the spatial and temporal scales for the prediction, the type and information content in the data available for constructing the predictive model, and the level of certainty (accuracy and precision) required to make the prediction useful. Therefore, a spectrum of techniques will be required for PUB if a range of problem types is to be addressed. The term top-down modelling implies that there is also bottom-up modelling. Indeed, history shows that, through the centuries, these two approaches have played important complementary roles in science. PUB will adopt this proven twintrack approach and encourage interaction and cross-over between the tracks. In practice, the spectrum of modelling approaches in hydrology stretches from purely statistical (or black box ) models at the extreme top-down end, to models at the extreme bottom-up end that include representations of fine-scale processes based on detailed experimental observations. Hybrid models, possessing features of both top-down and bottom-up approaches, are common. Simplistically, the spectrum can be considered in two halves; the top-down half and the bottom-up half (see Figure 1). Although the main topic here is the role of the hybrid approach in the top-down half of the modelling spectrum, reference will be made to bottom-up modelling wherever this is useful to better illustrate features of the top-down approach. A model in one half of the modelling spectrum may sometimes be referred to as either just a top-down or a bottom-up model, whereas, it is a hybrid model to some extent. A problem of central importance in PUB is the estimation of streamflow, so the discussion will often refer to this variable. However, the principles and benefits of top-down modelling apply to wide ranges of hydrological variables and spatial scales. Top-down versus bottom-up Top-down numerical modelling in hydrology is concerned largely with establishing relationships (in computer models) that govern the behaviour of environmental systems such as catchment-scale rainfall-runoff, streamwater quality responses to environmental change, lake sediment accretion, snow-field dynamics, in-stream ecological assemblages, etc. An end-member top-down model (Figure 1) does not have any representation of the processes that act to convert the system inputs to system outputs, except, perhaps, very loosely through the choice of input and output variables. Such a model, e.g. a regression model, might be useful for some environmental management purposes but it cannot contribute to improved scientific understanding of within-system processes. A purist bottom-up modeller might say, therefore, that it lacks scientific interest. However, as will be shown later, by invoking simple conceptual representations of system-scale processes, hybrid top-down models can become more useful than purely statistical models. Furthermore, Top-down end-member Modelling spectrum Bottom-up end-member Hybrid top-down Hybrid bottom-up Figure 1. The modelling spectrum Copyright 2003 John Wiley & Sons, Ltd Hydrol. Process. 17, (2003)

3 some hybrid top-down models allow inferences concerning the nature of within-system processes, providing a focal point for dialogue with bottomup modellers who are typically more concerned with detailed experimental studies and scientific understanding of within-system processes. Dialogue between top-down and bottom-up modellers in this manner can lead to improvements in a top-down model such that it attains a degree of scientific credibility, and its utility to assist with environmental management is often greatly improved. An example will help. In order to model continuous streamflow in a gauged basin, a simply structured hybrid topdown rainfall-runoff model is often sought that characterizes the dynamic relationship between observed rainfall and streamflow. In order to promote model identifiability, such models tend to have not more than about six parameters (Jakeman and Hornberger, 1993) and are therefore referred to as parametrically parsimonious conceptual models (PPCMs). As will be discussed later, PPCMs can be useful in the development of schemes to estimate streamflow at ungauged sites. A bottom-up approach to modelling continuous streamflow would be to build a catchment model from interconnected sub-models of all the relevant within-catchment processes, paying close attention to the physics of how water moves through and out of the catchment. In principle, such models can be constructed without calibration because coefficients in the component sub-models will already be known from field-process or laboratory studies (but note that these coefficients may themselves be the result of modelling exercises). A bottom-up computer model formulated and constructed in this manner (if it were feasible) would comprise a detailed, spatially distributed, representation of sub-catchment processes and, when applied in simulation-mode to estimate streamflow at the catchment outlet, would require vast amounts of input data to describe the variable rainfall and evaporation fields, topography, vegetation, soils, etc. The costs would be prohibitively high. However, by spatially lumping the processes to catchment scale, the model structure can be simplified, rendering it more amenable to application with data sets more likely to be available (e.g. catchment areal rainfall estimated from available raingauge data). Although coefficients of the component submodels might be valid at the sub-system process scale, they are seldom so at catchment scale. The coefficients cannot be simply transferred to the spatially lumped model (though the developing field of scaling theory may offer a way forward). The structure of the revised (spatially lumped) model comprises representations of all the catchment-scale processes in the original model, with the coefficients effectively replaced by parameters that have to be calibrated against observations (by varying the parameters in search of minimal difference between observed and modelled streamflow). A bottom-up model of this type, because it attempts to represent all relevant processes, could have dozens of parameters, many of which might be related statistically to others due to parameter identifiability problems in such circumstances (not discussed further here). Such models are therefore referred to as parametrically generous conceptual models (PGCMs). In addition to problems of parameter identifiability in PGCMs, experience shows that they tend to perform more poorly than PPCMs in simulation-mode, i.e. on periods of record not used for model calibration. The PPCM approach Collectively, we think we know what processes control how temporal patterns of rainfall become variations in streamflow at the catchment outlet, and can describe them qualitatively and make use of this knowledge in PPCMs. If we introduce simple, basin-scale, conceptualizations of what happens to rainfall after it falls on the catchment and travels to the basin outlet, the temporal pattern of continuous streamflow (even at daily and smaller time steps according to catchment type) can be modelled by PPCMs quite well in many cases. We can try different conceptual schemes for calculating the variable catchment storage status, and for routing notional effective rainfall through the catchment to become streamflow, and select a model structure that gives a best match between observed and modelled hydrographs when the model parameters are calibrated. Copyright 2003 John Wiley & Sons, Ltd Hydrol. Process. 17, (2003)

4 I. G. LITTLEWOOD ET AL. The better the model-fit, the more inclined we are to believe that the conceptualization (model structure) we have adopted captures the essence of how the catchment converts rainfall to runoff. However, there are still problems with such models that will need to be addressed within PUB. For ungauged (flow) basins, i.e. a major focus of PUB, the modelling of continuous streamflow from rainfall, etc. is a more difficult problem than for gauged basins, perhaps by an order of magnitude (see later). Some of the problems and issues that can arise in the calibration, selection and application of PPCMs to assist with deriving regionalization schemes are illustrated in the following methodological example. The unit hydrograph (UH) is iconic in hydrology and deserves further mention here because recent years have seen significant advances in the application of UH-based PPCMs for continuous simulation of streamflow. A UH-based PPCM converts inputs of rainfall and air temperature to streamflow. Apart from catchment size, no other data or information are required. The version of UH-based PPCM discussed here (Jakeman et al., 1990) has six parameters that, because of their conceptual meaning within the model structure, can be expected to be related to physical catchment descriptors (PCDs), which are referred to again later. The fine detail of how the model parameters are calibrated is beyond the scope of this article. In outline, however, a standard procedure for calibrating a preliminary best model using a software package (Littlewood et al., 1997) for this modelling methodology involves a trade-off between (a) accounting for a high proportion of the variance in observed streamflow (a high value for D c ) in calibration-mode and (b) obtaining good average precision on the UH parameters (a low value for A). Although the model calibration and selection procedure is fairly complex, iterative, and requires hydrological judgement by the analyst, modern computers and software assist greatly in rapid convergence towards a best provisional model of this type. It is essential that D c A trade-off models of this type are regarded as provisional. As a specific example, a daily time-step model, calibrated and selected as outlined above, accounts for about 86% of the variance in observed daily streamflow over its calibration period of 2970 days (model #1). But model #1 is not a good model. When flow duration curves (FDCs) for the observed and modelled (#1) flows are compared (Figure 2) it becomes clear that, in percentage terms (note the logarithmic 1000 observed model #1 model #2 100 Flow (m 3 s -1 ) % of daily mean flows exceeding Y Figure 2. Flow duration curves for models #1 and #2 Copyright 2003 John Wiley & Sons, Ltd Hydrol. Process. 17, (2003)

5 scale on the y-axis), low flows are increasingly overestimated by model #1. The best integer value ofpuretimedelayd, to give model #1, was 1 day. By introducing a non-integer pure time delay of 0.6 days, and selecting a best model as above but additionally on the basis of ensuring a good FDC over the whole range of modelled flows between Q 5 and Q 95, a better model than #1 can be obtained (Littlewood, 2002). The FDC for model #2 in Figure 2 agrees well with that for observed flows, and a time series plot (not shown) confirms that the temporal match between observed and modelled flow events has been preserved. The example of models #1 and #2 illustrates how FDCs can be effective for assisting with calibrating a model and checking that its parameters are accurate (see later). It is now widely accepted that rainfall-runoff PPCMs calibrated solely on the basis of a single objective function, e.g. D c, can lead to selection of a poor model. The D c A trade-off approach outlined above is an improvement, but it can still give very misleading results. The additional criterion of ensuring good agreement between FDCs over a wide range of flows gives better results, illustrating that a multiobjective approach to model calibration and selection is essential. Assessment of and convergence towards multi-objective methods and model diagnostics will be an important aspect of the development and application of PPCMs in PUB. PPCMs and regionalization Much research is currently under way to reduce, or further constrain, the uncertainty associated with regionalization schemes, providing a firm basis from which further progress can be made during the PUB Decade. Most commonly, existing schemes are either for high flows (for flood studies) or low flows (for drought and water-quality management studies). In contrast, the UH-based PPCM approach discussed in this article characterizes the daily mean flow regime over a wide range of flows, usually between about Q 5 and Q 95 reliably but also higher and lower flows quite well in many cases (Littlewood et al., 2002). PGCMs are not suitable for a statistical approach to transferring information from gauged to ungauged basins because of the difficulties of calibrating with precision more than about half a dozen parameters from the information in catchment-scale hydrometric and hydrometeorological data. Even models with six or fewer parameters (PPCMs) can lack good precision in some of their parameters, as shown below. Improving this aspect of PPCMs will be a major objective of PUB. Further discussion of models #1 and #2 helps to illustrate the problem. Figure 2 showed that model #2 is a better characterization of the Q 5 Q 95 flow regime than model #1, but the former model is also superior to the latter in terms of D c (88%). There was also a small increase in the average precision on the UH parameters, i.e. model #2 has a slightly lower value of A. The UH part of the model comprises two linear storages acting in parallel, leading to hydrograph separation to give dominant quick and slow response components of modelled streamflow (discussed further in the next section). For both models, indicative 95% confidence limits associated with the UH recession time constant for slow flow (about ±13%) are substantially greater than those associated with the time constant for quick flow (about ±4 5%). This is common for catchments dominated volumetrically by quick flow (the slow-flow index (SFI) for the catchment in question is about 0 37). For catchments dominated by slow flow (e.g. SFI > 0 8), the time constant for slow flow is often estimated with better precision than for quick flow. When establishing regionalization equations based on model results for catchments dominated volumetrically by quick flow, it is likely, therefore, that the equation linking the slow-flow time constant with PCDs will be statistically inferior to that for the quick-flow linkage to PCDs. In addition to the slow-flow UH decay time constant for model #1 being imprecise, it is also relatively inaccurate (25% higher than for model #2). Furthermore, in going from model #1 to model #2, the size of the conceptual catchment storage decreased by 43%, and a parameter that modulates the rate at which the catchment dries out in the absence of rainfall increased by 45%. Such large inaccuracies will inevitably influence the quality of regionalization equations linking model parameters and PCDs. Careful model calibration and selection for each catchment in the set used Copyright 2003 John Wiley & Sons, Ltd Hydrol. Process. 17, (2003)

6 I. G. LITTLEWOOD ET AL. to derive a regionalization scheme is, therefore, imperative. Hydrograph separation and water-quality hydrology The UH-based PPCM allows continuous separation of the modelled hydrograph to give separate quick- and slow-flow hydrographs, yielding a continuously variable SFI (CVSFI). Figure 3a and b shows CVSFI plotted against peak modelled (#2) flow and against peak observed flow respectively, for 80 peaks in a 2 year sub-period of the record used to calibrate the model (peaks were defined as having lower flows on the days either side and were selected from the observed series). As might be expected, there is an inverse relationship between flow and CVSFI; the higher the peak flow the lower the proportion of slow flow in the streamflow at that time, but with considerable scatter due largely to hysteresis (CVSFI is different for equal flows on the hydrograph rising and recession limbs of a runoff event). It is reasonable to assume that the quick- and slow-flow comprise waters that have travelled by surface (and near-surface) pathways and deeper pathways respectively. They can be expected, therefore, to have broadly different chemical signatures because of different residence times within different strata in the catchment (soil horizons, lithologies, etc.). The concentration of a streamflow constituent (e.g. calcium) at a given time might be expected, therefore, to be influenced (not totally determined) by the CVSFI at that time. Only for catchments not unduly impacted by anthropogenic effects, and for constituents largely unaffected by within-basin geochemical and biological sources and sinks, would it be reasonable to expect a relationship between CVSFI and concentration. In principle, tracer studies using natural isotopes (e.g. 18 O, 2 H) in carefully chosen basins offer the best opportunity to investigate catchment-scale mixing dynamics. Many such studies tend to focus on the variable proportions of old and new water in streamflow, where old refers to water in the catchment before the causative rainfall and new water is the rainfall (or snowfall). Although there has been considerable progress recently in the design and interpretation of natural isotope and other conservative tracer field studies, there is still disagreement and uncertainty surrounding the old versus new water issue in the context of streamflow-generation processes. Our understanding of this centrally important issue in scientific hydrology is incomplete. Coupled rainfall-runoff PPCM research and tracer studies within PUB, e.g. to elucidate the physical meaningfulness of CVSFI, could help to advance our knowledge in this area, leading to better estimation of streamflow and water-quality dynamics. Concluding remarks Some of the strengths and weaknesses of the top-down and bottom-up modelling approaches have been discussed, with particular reference to top-down modelling of continuous streamflow in gauged and ungauged basins. Issues involved in model calibration and selection for that purpose, in terms of the accuracy and precision of model parameters, have been illustrated and discussed CVSFI CVSFI (a) modelled flow (m 3 s -1 ) (b) observed flow (m 3 s -1 ) Figure 3. CVSFI against flow: (a) modelled; (b) observed Copyright 2003 John Wiley & Sons, Ltd Hydrol. Process. 17, (2003)

7 The IAHS PUB Decade provides a focus for bringing together many areas of hydrology towards reducing the uncertainty in estimates of hydrological variables in ungauged basins. The technical problems may be difficult to solve, but, with a concerted effort, good progress can be expected over the PUB Decade. In order to develop practical guidelines concerning which techniques to use in which circumstances, it will be necessary to compare models and schemes using agreed criteria, otherwise only favourite models (rather than a best one for a given purpose) will continue to be applied. A key facet of the PUB programme should be the assessment and comparison of models in terms of the uncertainty (accuracy and precision) in their predictions of hydrological variables in gauged and ungauged basins. Unless models and schemes for estimating streamflow (and other hydrological variables) in ungauged basins are tested on gauged basins, their worth cannot be assessed objectively. Groundbased hydrometric data are essential to progress in hydrological science, and no amount of PUB will change that. Indeed, when the uncertainties in outputs from rainfall-runoff models based on conventional ground observations are compared with outputs from models involving remotely sensed input variables, it is anticipated that the true value of good-quality ground-based data will become even more apparent during the PUB Decade. It almost goes without saying that the best estimate of streamflow at a given site will always be one based on flow measurements (not forgetting that there are uncertainties there too, of course). Clearly, we cannot measure at every point of interest. Whether estimates of hydrological variables are based on measurements at the site in question or remote sensing or regionalization schemes, PUB should help to deliver guidelines about what measurement network densities and data quality are required in order to meet a range of estimation objectives (including the desired level of uncertainty). PUB should also provide a new stimulus for advancing, and integrating, the science of hydrology by focusing a wide range of expertise on a key set of problems. REFERENCES Hubert P, Schertzer D, Takeuchi K, Koide S (eds) PUB Communications. IAHS Decade for Prediction in Ungauged Basins, Brasilia, November. URL: / iahs/index.html. Jakeman AJ, Hornberger GM How much complexity is warranted in a rainfall-runoff model? Water Resources Research 29(8): Jakeman AJ, Littlewood IG, Whitehead PG Computation of the instantaneous unit hydrograph and identifiable component flows with application to two small upland catchments. Journal of Hydrology 117: Littlewood IG Improved unit hydrograph characterization of the daily flow regime (including low flows) of the River Teifi, Wales: towards better rainfall streamflow models for regionalisation. Hydrology and Earth Systems Science 6(5): Littlewood IG, Down K, Parker JR, Post DA The PC version of IHACRES for catchment-scale rainfall streamflow modelling: user guide. Institute of Hydrology Software report. Littlewood IG, Jakeman AJ, Croke BFW, Kokkonen TS, Post DA Unit hydrograph characterisation of flow regimes leading to streamflow estimation in ungauged catchments (regionalisation). In PUB Communications. IAHS Decade for Prediction in Ungauged Basins, Brasilia, November, Hubert P, Schertzer D, Takeuchi T, Koide S (eds). URL: ensamp.fr/ iahs/index.html. Sivapalan M, Zhang L, Vertessy R, Blöschl G. In press. Downward approach to hydrologic prediction. Hydrological Processes. Young PC Data-based mechanistic modeling and validation of rainfall-flow processes. In Model Validation: Perspectives in Hydrological Science, Anderson MG, Bates PD (eds). Wiley: Copyright 2003 John Wiley & Sons, Ltd Hydrol. Process. 17, (2003)