## M

Both sides of this equation represent deviations within distribution functions. The left-hand side is the deviation of a local storage deficit, Di, from the sub-catchment mean value, D, scaled by the parameter m of the exponential transmissivity function, while the right hand-side represents sum of a (negative) deviation of the local topographic index, ln(a/tan ¡3) from the mean sub-catchment value of the index, 1, and the deviation of the local value of To from the mean sub-catchment value. In the topographic index, a is the area drained per unit contour length at a point and tan 3 is the local slope gradient at that point, and

where A is the total sub-catchment area. In most applications it is not possible to know how the transmissivity varies in space and so both To and m are normally treated as

(tan

constants and the last bracketed term in (12.17) is assumed equal to zero (the local value is everywhere equal to the mean). The variation in local deficit is then only a function of the topographic index and the parameter m. In particular, for any value of D we can calculate where the local drainage deficit Dj is zero. This represents the saturated area. Saturation will be more likely when the area draining though a point is high (e.g. in convergent hollows) and where the slope gradient is relatively low (e.g. Fig. 12.11). The model will predict that the saturated area will expand and contract as the catchment wets and dries so that the mean deficit D gets smaller and larger.

The mean soil moisture deficit will increase with drainage of the saturated zone and evapotranspiration and decrease with rainfall (and snowmelt). Under the exponential transmissivity function assumption, it can be shown that drainage from the saturated zone, qy, is given by qb = q0 exp(-D/m) (12.19)

where the value of qo is a function of T0 and l, and m is the same constant as before.

The model has been well tested on several UK catchments over wet and dry periods and for different vegetation types with the best performance demonstrated in the humid and temperate climate and topographical conditions for which it was developed (e.g. Fig. 12.12). It compares very favourably in goodness of fit with other notable deterministic computer models which have a greater number of parameters, even using only measured and estimated parameters (Beven et al., 1984). However, the model assumptions will not be appropriate to all catchments, even if the parameters can be calibrated to give a good fit to a discharge record. Some relaxations of the assumptions

Fig. 12.12 Application of Topmodel to the River Eden at Kirkby Stephen, Cumbria, UK. Part of calibration period, with Nash-Sutcliffe efficiency = 0.896.

Fig. 12.12 Application of Topmodel to the River Eden at Kirkby Stephen, Cumbria, UK. Part of calibration period, with Nash-Sutcliffe efficiency = 0.896.

have been tested. By changing the saturation reference level from the soil surface, Pinol et al. (1997) show how the concepts can be applied to fast subsurface responses and Quinn et al. (1991) show how the model concepts can be applied to deeper water tables. Beven and Freer (2001a) have also shown how the steady-state assumption can be relaxed in a dynamic version of the model in a way that allows other definitions of similarity of response to be used in addition to topography while retaining computational efficiency. It is the computational efficiency of the model that has made it a very useful research tool in exploring uncertainties in model predictions (e.g. Beven, 1993; Beven and Freer, 2001b) and in using continuous simulation for flood frequency estimation (see Beven, 1987; Cameron et al., 2000).

The ability to check the spatial pattern of the model predictions has also proven to be interesting. Because of the simplifying assumptions used in the model, we would not expect the predictions to be correct everywhere and in going from the storage deficits used in the simplest form of the model to a measured water table depth, it is necessary to introduce an additional assumption and parameter. What has been found in a number of studies is that the broad pattern of modelled contributing areas is often well predicted (e.g. Beven and Kirkby, 1979; Seibert et al., 1997; Guntner et al., 1996), but that there are areas where saturation is not obviously related to the surface topography (Ambroise et al., 1996) and that local water table predictions can be improved if local transmissivities are calibrated (e.g. Lamb et al., 1998; Blazkova et al., 2002; Freer et al., 2004), although this does not necessarily result in improved discharge predictions. A review of Topmodel applications is provided by Beven (1997).

12.8.4 The Système Hydrologique Européen model

The Système Hydrologique Européen (SHE) model is a physically based distributed model accounting directly for spatial variations in hydrological inputs and catchment responses. Originally developed jointly by hydrologists in the UK, France and Denmark

Rain and snow

Canopy interception model

Net precipitation Snowmelt model

Water table rise and fall

Rain and snow

Canopy interception model

Net precipitation Snowmelt model

One-dimensional unsaturated flow model for each grid element

Three-dimensional saturated flow groundwater model (rectangular grid)

* Exchange through seepage faces

Exchange across boundaries

Fig. 12.13 Schematic diagram of the Système Hydrologique Européen (SHE) model (with kind permission of the Danish Hydraulics Institute).

One-dimensional unsaturated flow model for each grid element

Three-dimensional saturated flow groundwater model (rectangular grid)

* Exchange through seepage faces

Exchange across boundaries

Fig. 12.13 Schematic diagram of the Système Hydrologique Européen (SHE) model (with kind permission of the Danish Hydraulics Institute).

starting in 1977, it has now been commercialised by the Danish Hydraulic Institute as MIKE-SHE and developed into a research tool as SHETRAN in the UK. In SHE, the catchment is discretised using a square grid. Past applications have used grid sizes ranging from 50 m or less in small catchments to 4 km in a large catchment in India (Jain et al., 1992). Finite difference methods are used to obtain the solutions of the non-linear flow equations representing overland and channel flow, unsaturated and saturated subsurface flow. The model structure is shown in Fig. 12.13 (Abbott et al., 1986; Refsgaard and Storm, 1995). In the unsaturated zone, the system is simplified with a one-dimensional vertical flow component used to link the two-dimensional surface flow and saturated groundwater components. The saturated zone can interact directly with the channel network along designated river segments.

The computational sequence is as follows. The precipitation rate is the data input to the interception model. A layered snowmelt model component may be applied next (different snow accumulation and melt models of different complexity are included). The evapotranspiration loss model operating from several vertical zones requires four meteorological inputs or a specified potential evapotranspiration rate. The overland-channel flow model requires boundary flow and initial flow depth conditions, topography of the overland flow plane and channels and particulars of any man-made discharge alterations. The unsaturated flow model is based on Darcy's law and controls infiltration and recharge and allows different soil layers and vegetation root patterns. It requires an initial water content profile for each grid square. The saturated flow model requires input data on boundary conditions and topography of the base aquiclude, initial water table levels or saturated thicknesses and any man-made interference to natural conditions (such as pumped wells or drainage tunnels) must be defined. It requires an initial depth of saturation for each grid square. In the first versions of the model the saturated zone was represented as a two-dimensional depth-averaged solution of the Darcy equation, but both MIKE-SHE and SHETRAN now have fully three-dimensional options, and MIKE-SHE also has a simple conceptual lumped store option. The different components can use different, automatically controlled, time steps. The operation is made flexible by a frame structure controlling the coupling of the individual component models so that they may be applied to a variety of catchment conditions.

The distributed nature of the model means that it can take account of the variability of rainfall and other meteorological variables over a catchment. An interface to a geographical information system (GIS) can also be used to facilitate the specification of the topography and soil and vegetation types. Model predictions can also be presented in a form compatible with GIS systems with on-screen animations of the distributed predictions.

There have been a number of interesting applications of the SHE model, including attempts to make predictions based only on physically based estimates of the vegetation, soil and groundwater parameters needed. If this were possible, then it would be important for two reasons. One is that with such a large number of parameters required for each grid square, and very many grid squares, calibration of the individual parameter values required by the model is not really possible (though limited calibration has been tried, e.g. Bathurst, 1986). A second is that, if the model can be shown to perform well using only estimated parameter values, then we might be able to have more belief in predictions of future conditions with, for example, parameters representing changed land use. Some of the earlier applications using SHE were greatly limited by computer time, which meant that the periods of simulations used were relatively short, so that results might have been unduly influenced by the specified pattern initial conditions. Later studies have shown some success in reproducing both discharges and internal state variables (Parkin et al., 1996; Bathurst et al., 2004; Thompson et al., 2004) but it is not yet clear that this type of model has real advantages over other rainfall-runoff models except where highly detailed local data are available (e.g. Ebel and Loague, 2006) and even then it is not clear that the Darcian descriptions of flow in the unsaturated and saturated zones on which these models are based are adequate to describe the complex flow pathways in reality (see Beven, 1989; and discussions in Abbott and Refsgaard, 1996). The model has been extended to include the transport of soluble contaminants, sediment transport and landslides.

### 12.8.5 Other grid-based models

There are a number of other distributed-based models that do not aim to be as 'physically based' as the SHE model but which were developed to try to take account of the spatial patterns of inputs (including snow) and soil and vegetation characteristics in large catchments. A good example of this is the SLURP model developed by Kite and Kouwen (1992) in Canada where it has been used to model the Mackenzie and other large basins. SLURP is based on the concept of hydrological response units (HRUs) that have similar soil, vegetation and elevation characteristics, and are therefore expected to be similar in their hydrological response. In SLURP a single grid square may contain a number of different HRUs in an arbitrary pattern, with the areas of each being grouped together for calculation purposes. A similar approach, based on hexagonal units, has recently been applied across a range of scales in Russia (Semenova and Vinogradova, 2009). The HRUs can be defined automatically using overlays of GIS and remote sensing information. A somewhat similar approach has been taken with the LISFLOOD model of de Roo et al. (2000). This is now being used operationally at a 5-km grid scale at the European Union Joint Research Centre in Ispra, Italy, as the basis for a European Flood Warning System. The aim is to use ensemble numerical weather prediction (NWP) forecasts from the European Medium Range Weather Forecasting Centre to provide up to 10-day ahead forecasts of potential flood situations for all the major river basins in Europe. As noted in Section 12.8.2 above, a similar gridded approach in the grid-to-grid (G2G) model has been applied nationally at the Centre for Ecology and Hydrology, Wallingford.

Gridded models of this type are ideally suited to take advantage of the facilities of GIS, which can be used to store different layers of spatial information and to facilitate the presentation of distributed model outputs in graphical form. GIS systems have not traditionally been good at dealing with time-dependent spatial patterns needed in hydrological simulation (with some exceptions, such as the PC-Raster system that was used to create the LISFLOOD model). Some distributed models, such as SHE, have developed their own GIS interfaces. A general hydrological GIS toolkit has been developed in the Arc Hydro system (Maidment, 2002). Arc Hydro makes use of the proprietary GIS software ArcGIS, making use of spatial data layers for streams and hydrography, catchment drainage areas, channel sections, surface topography, soil and land use, and vegetation, together with time series information from rainfall and stream gauging sites. The system makes use of a Visual Basic interface to ArcGIS to allow flexibility in the programming of hydrological models.

### 12.8.6 Irregular element distributed models

Square grids are not the only way of discretising a distributed catchment model. In fact, the use of square or rectangular grids comes more from the ease of implementing approximate finite difference solutions to the subsurface flow equations than from any process considerations. Hillslope flow processes are not readily represented by a square grid in plan, even if that grid is rather fine, while sloping soil horizons and complex geological structures are not easily represented in the vertical by rectangular elements. Thus, it might be more realistic to have a more flexible discretisation.

Two main approaches have been followed. There are other solution techniques to the subsurface flow equations that are not based on a square or rectangular grid. Models based on the finite element solution method, that use triangular or quadrilateral elements to discretise the flow domain have also been available since the earliest days of distributed hydrological models. A recent finite element based model is the Integrated Hydrological Model (InHM) of VanderKwaak and Loague (2001). This has been used in a series of papers that have modelled the small R5 catchment at Coshocton, Ohio, gradually making the representation of the processes more complicated. The original perception of the hydrological response of this catchment was that it was dominated by surface runoff, but the final papers in the series have recognised that there may be an important subsurface component (Loague et al., 2005). The InHM model has been recently extended to include sediment and chemical transport.

The other approach starts off with a non-raster description of the topography. While most of the readily available digital maps of topography are in raster (grid) form, many of these have been derived from vector (contour) or point elevation data. In fact, some approximation is introduced in going from point and vector data to a raster grid. It is, however, possible to create spatial discretisations directly from point elevation data, e.g. using a triangular irregular grid (TIN) based on Delauny triangulation. There have been a number of models based on the TIN approach. One is still being developed at the Massachusetts Institute of Technology and now includes complex representation of vegetation processes as affected by interactions with the hillslope flow processes (see Ivanov et al., 2008).

### 12.9 Uncertainty in rainfall-runoff modelling

The models described above are typical examples of those used in practice. It is clear that even the most complex distributed models can only approximate the complex response of a real catchment and the parameter values can only approximate the complex characteristics of a real catchment. In addition, in any practical application to a real catchment, we will only have approximate estimates of the rainfall input and evapotranspiration output boundary conditions for any model that is applied. It should therefore be expected that the predictions of any chosen model will be necessarily approximate. This implies that the predictions will be associated with some error. Traditionally, this issue was addressed by practitioners only by trying to minimise the error in model calibration, i.e. to find the best or optimal model possible in some sense. This was achieved either by a visual comparison of plots of observed and predicted variables, or by a maximising a performance measure, such as the Nash-Sutcliffe efficiency measure (12.10).

Recently, however, the uncertainties of model predictions have started to be addressed more explicitly. This is because there is now more computer power available to enable the multiple model runs that are needed for the prediction uncertainties to be estimated. The simplest form of uncertainty analysis makes assumptions about the nature of the sources of uncertainty (typically rainfall inputs and model parameters) and then carries out a forward uncertainty analysis using either analytical (for linear models) or Monte Carlo simulation techniques to propagate the assumed uncertainty through the model. The outcomes from such an analysis will be stochastic, but will clearly depend totally on the assumptions made about the nature of the uncertainties and their interactions.

A more interesting case arises when there are some observations available with which to try and constrain the uncertainties. In this case the performance of different runs of one or more models can be compared and those that give better performance can be given more weight (or greater likelihood) in assessing a distribution of predicted variables. Studies of this type have shown that there is not generally a clear optimal model, and that models with different structures will often result in similar performance. This is, at least in part, a result of the fact that we normally do not know too much about the uncertainties in the input data provided to a model. This is often a limiting factor in model performance and the first stage in any modelling study should be to check the hydrological consistency of the data set.

When model predictions are being compared to observed discharges, a similar issue arises. We should not expect the performance of any model to be better than the accuracy of the observations with which it is being compared. Remember that it is rare that discharge is measured directly. More often water level is measured and converted to discharge by means of a rating curve. This introduces uncertainty in the discharge measurements, particularly when the rating curve has to be extrapolated to flood discharges well above the measured values on which the rating curve is based. Thus, where possible, it is also worth checking the rating curve before carrying out any model runs to assess the range of discharges for which the observations could be considered reliable.

Given the potential for error in both model inputs and discharge observations, there is a possibility of over-fitting a model in calibration, particularly when only an optimal model is sought. This means that the fit of the model is partly a result of the errors in the calibration data. When such a model is used for prediction (where the input errors can be expected to be different), over-fitting may result in poorer predictions. Over-fitting is more likely with a complex model with many parameters to be calibrated (more degrees of freedom). One way of checking for over-fitting is to make a split record test. This is where only part of the data available is used in calibrating the model. The rest is retained for use in evaluating the predictions. If model performance deteriorates badly in the evaluation period, then the model may have been over-fitted. Split record tests should be used whenever possible. They can be used to assess model predictions both with and without an estimation of prediction uncertainties in the calibration period.

There are many different methods for uncertainty estimation that have been used in hydrological modelling. More detail on the techniques available can be found in Beven (2009).

### 12.10 Real-time forecasting models

Uncertainty of predictions is also an important issue in real-time forecasting because decisions about flood warnings or water resource management might have important economic and human consequences. In this respect, forecasts need to be both accurate and timely in the sense of providing adequate 'lead time' for any decision to be implemented. Accuracy and adequate lead time are conflicting requirements. One hour ahead predictions of flood flows in a river might well be accurate, but would not provide a sufficient lead time of potential over-topping of flood defences to be useful for flood warning purposes. There would just not be time for those who would be most affected to respond to any warning. A much less accurate 6- or 12-hour ahead forecast might be much more useful in preventing loss of life and allowing those who might be affected to try to protect their property. A 24-hour ahead forecast, but still less accurate forecast, might be useful in deciding about where to commit resources in deploying temporary or demountable flood defences.

The ability to make accurate forecasts with adequate lead time depends very heavily on the scale of the catchment. In large catchments such as the Rhone, Rhine or Danube, it will take several days for a flood wave generated in the mountain headwaters to move downstream. Even for the much smaller catchments in the UK, at catchment scales of 1000 km2 and above, given rain-gauge and radar rainfall estimates in real time, can usually provide warnings to downstream towns with a lead time of 6-12 h or more. At smaller scales, with much shorter times of concentration in a catchment, then this is not possible as was seen in the Boscastle event on 16 August 2004 (see Section 9.1.2). On the basis of numerical weather prediction (NWP) warnings of severe rainfalls were issued by the UK Met Office, but they could not be precise about either the location or the rainfall amounts (one grid square in the regional NWP model is much larger than the catchment area draining to Boscastle). In the event, up to about 200 mm of rain fell upstream of the village, but the flood wave occurred during the day and while many people had to be rescued by helicopter, no-one was killed. Some 58 properties were affected by the flood and some £2m of damage was caused, including cars that were swept out to sea. Fifty years before, in the night of the 15 August 1952, in a flood in the village of Lynmouth, further up the Devon coast, a flood wave caused by up to 300 mm of rain in the catchments of the East and West Lyn rivers, swept through the village at night, killed 34 people, and deposited 200 000 tons of debris. In these types of flash flood events on small catchments, it will not be possible to provide warnings with adequate lead times until the accuracy of short-term NWP rainfall predictions improves significantly. Despite the continuing improvements in NWP, this still seems to be some way off for this type of event even though post-event local predictions using an atmospheric model with a 1-km grid have shown some success in reproducing the nature of the Boscastle event (e.g. Golding et al., 2005).

Forecasting during droughts is not quite so time critical. Things happen much more slowly during drought periods, and water resource management decisions can consider the 'worst-case' scenario of no rainfall into the future. It is still, however, the case that any model predictions might start to deviate from what is actually being observed in real time and one of the features of the forecasting problem is to try to take account of whether a model is currently under- or over-predicting in improving the future forecasts. This is called adaptive forecasting or data assimilation. It is used routinely in weather-forecasting models but, until recently, has not been used widely in hydrological forecasting.

There are two basic approaches to adaptive forecasting for real-time applications. In both, the starting point is a model that has been calibrated and shown to work well for some historical datasets. The first approach is to update the state variables in a model (e.g. values of storage) that has already been calibrated on historical datasets, to try to match the current observations on the basis that the updated model will provide better predictions into the future. For linear models and near-linear models, a useful robust methodology for state updating is the Kalman filter (KF). The KF can make use of one or more observables in updating the model. At each update step or innovation it uses the latest set of residuals between observed and predicted variables (or innovations) to update the states of the model (e.g. storages) recursively (i.e. the estimates at the last time step provide the starting point for the updating at this time step). The amount of updating depends on the magnitude of the residual and an estimate of the covariance of the state estimates. Thus, not only are the predictions improved by the updating, but the covariance matrix can be used to give estimates of the uncertainty in the predictions. It is also worth noting that the model parameter values can also be included in the updating, so that both storages and parameters can be updated. However, there will only be a limited amount of information in the innovations, so that, if it is attempted to update too many variables, then the estimates may fluctuate rapidly without great benefit in improving the forecasts (this is equivalent to the over-fitting problem in model calibration). The very simplest form of updating in this way is to use only the gain parameter in the updating algorithm (see Young, 2002).

For the non-linear models that are more usual in hydrological modelling, the extended Kalman filter (EKF) is more appropriate. The EKF uses a form of linearisation at each time step in doing the state and parameter updating. This means that the EKF can be subject to stability problems for highly non-linear models or strongly interacting states in the model. To avoid the approximation arising due to the linearisation, a new method called the ensemble Kalman filter (EnKF) has been introduced recently. This uses an estimate of the covariance of the states and parameters at a time step to sample multiple realisations of the model. These are then used to provide an ensemble of predictions at the required lead time, including all effects of non-linearities in the model. As a new set of observables is received, a form of the KF filter is used to update the covariance matrix. Then a new ensemble of models is sampled and used to project the predictions forward to the required lead time and so on. Further details on the various forms of Kalman filter and other data assimilation methods may be found in Beven (2009).

The second approach, called error correction, is to try to model the residual between the current observations and a calibrated model to try to improve the predictions into the future. This second approach requires that there be some structure in the residuals that allows projection into the future. If the residuals were purely random, then there would be no additional predictability, but this is not often the case with hydrological models, which will often under-predict or over-predict for successive time steps (the residuals can exhibit bias and autocorrelation). This might, of course, be so only because the inputs to the model are under- or over-estimated, but this still requires some adaptive capability to improve the forecasts. The advantage of this approach is that while the model might be highly non-linear, it might be possible to model the residuals with a linear model. Techniques such as the Kalman Filter might then be used to update the model of the residuals recursively as new observables are received, although the nature of the residuals of rainfall-runoff models does not normally conform to simple statistically assumptions such that it may be difficult to specify an adequate model.

There is one important feature of real-time forecasting that makes it different from other applications of hydrological simulation models. In forecasting, it is not so very important that we represent the processes controlling the response in any explicit way. The only requirement is to produce a forecast at the required lead time with good accuracy and minimum uncertainty. Hence, the usefulness of adaptive forecasting where, for example, the adaptation may mean that the model does not necessarily maintain mass balance. Indeed, we would not want to maintain mass balance in the model if, for example, an underestimate of the true rainfall given the available rain-gauges, would lead to an under-prediction in forecasting. An interesting example of this is provided by the study of Romanowicz et al. (2008). In developing a flood forecasting

Fig. 12.14 Real-time adaptive flood forecasting in the River Eden at Sheepmount Carlisle. The forecasts of water level are made with a 6-h lead time. Left, shows the calibration event, right, predictions for the January 2005 flood event. Note large difference in the maximum stage for the two events. Shaded area represents 95 per cent prediction limits for the forecasts.

Fig. 12.14 Real-time adaptive flood forecasting in the River Eden at Sheepmount Carlisle. The forecasts of water level are made with a 6-h lead time. Left, shows the calibration event, right, predictions for the January 2005 flood event. Note large difference in the maximum stage for the two events. Shaded area represents 95 per cent prediction limits for the forecasts.

system for the River Severn in the UK, they included rainfall-runoff models based on the DBM methodology of Peter Young described above. The inputs to the forecasting model were measured rainfalls, but the outputs were water levels not discharge. There was therefore no attempt to retain mass balance, even in the calibrated model. Their argument was that the water level is the actual observable and can be measured to good accuracy, whereas conversion to discharge requires the use of a rating curve that introduces error, especially when extrapolation is required to high flood levels. In addition, it is very often the water level that is the required predicted variable in flood warning since it is level that controls when a flood defence will be overtopped or houses will be flooded. The DBM model is based on a non-linear transformation of the rainfall input and a linear transfer function. The adaptive forecasting model in this case uses a combination of a Kalman filter on the states of the transfer function and a final gain updating step to improve the forecasts. Fig. 12.14 shows an application of this methodology to the January 2005 flood in Carlisle. By cascading rainfall to water level and level-to-level components in the main stream river, up to 6-h ahead forecasts were possible in this case (Fig. 12.14) which is adequate for issuing flood warnings to the public. A recent detailed discussion of techniques for adaptive forecasting in hydrology is provided by Young (2009).

Notes

1 The accepted use of the term effective rainfall is perhaps unfortunate since, as discussed in Sections 1.2 and 11.3, the discharge from a catchment measured in an event hydrograph may not be all rainfall from that event, but may be, at least in part, stored water from past inputs that is displaced by that storm rainfall. Thus the effective rainfall in an event may not actually be the rain water falling in that event. This may not be important if we only need to estimate the volume of runoff or the peak discharge in the channel. It might be important if trying to predict water quality since the stored water might have a quite different chemistry from the rainwater (see also Section 11.3).

2 See http://www.epa.gov/ceampubl/swater/hspf/index.htm

3 See http://www.es.lancs.ac.uk/cres/captain/

4 See http://www.es.lancs.ac.uk/hfdg/freeware/hfdg_freeware_tfm.htm

5 A demonstration freeware of the basic version of Topmodel can be found at http://www.es. lancs.ac.uk/hfdg/freeware/hfdg_freeware_top.htm

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