12.1 The essentials of a catchment model

The derivation of relationships between the rainfall over a catchment area and the resulting flow in a river is a fundamental problem for the hydrologist. In most countries, there are usually plenty of rainfall records, but the more elaborate and expensive streamflow measurements, which are needed for the assessment of water resources or of damaging flood peaks, are often limited and are rarely available for a specific site under investigation. Modelling the way in which rainfall becomes river discharge has stimulated the imagination and ingenuity of hydrologists for over a century, but has been an important area of research in the last 40 years as digital computers have become more widely available. Catchment models are now routinely used in flood forecasting, the design of flood defences and urban drainage systems, water resources assessment, and predicting the response of ungauged catchments. The use of models in hydrology is widespread in the applications illustrated in the remaining chapters of this book.

As well as these practical applications, however, catchment models are also an important way of doing science in hydrology, since they provide a mechanism for formalising hydrological understanding. We can take the type of mathematical representations of hydrological processes that have been described in previous chapters, and implement them as computer programs that will produce predictions of the consequences of assuming that those representations are correct. Those predictions can then be compared with what we know about the response of a catchment to see whether they are satisfactory as hypotheses about how the hydrological system is functioning.

To facilitate comparisons it is usual to express values for rainfall and river discharge in similar terms. The amount of precipitation (rain, snow, etc.) falling on a catchment area is normally expressed in millimetres (mm) depth per unit area over a specified time period, but may be converted into a total volume of water, cubic metres (m3) falling on the catchment by multiplying by the catchment area. Alternatively, the river discharge (flow rate), measured in cubic metres per second (m3 s-1 or cumecs) for a comparable time period may be converted into total volume (m3), or expressed as an equivalent depth of water (in mm) by dividing by the catchment area. The discharge per unit area, often termed runoff, is then easily compared with rainfall depths over a defined time period. To make it easier to assess the water balance in a catchment, other variables such as storages and evapotranspiration flux can also be expressed as millimetres over the catchment area.

Estimating runoff or discharge from rainfall measurements is very much dependent on the timescale being considered. For short durations (hours) the complex interrelationship between rainfall and runoff is not easily defined, but as the time period lengthens, the connection becomes simpler until, on an annual basis, a straight-line correlation between the cumulative depths or volumes of rainfall and runoff may often be obtained. The purpose of an application will define what type of data time step is needed in developing relationships between rainfall and runoff. Relating a flood peak to a heavy storm will require continuous or short time-step records, but determining water yield from a catchment can be accomplished satisfactorily using relationships between totals of monthly or annual rainfall and runoff (noting, however, that accurate estimates of monthly or annual totals might also require continuous records, particularly in small or flashy catchments).

Naturally, the size of the area being considered also affects the relationship. For very small areas of a homogeneous nature - a stretch of impermeable motorway, say - the derivation of the relationship could be fairly simple; for very large drainage basins and for long time periods, differences in local rainfalls and runoff production are smoothed out giving relatively simple rainfall-runoff relationships. However, in general and for short time periods, great complexities occur when spasmodic rainfall is unevenly distributed over an area of varied topography, soil and geology characteristics with spatially heterogeneous antecedent soil water storage. For catchments with locally variable surface characteristics affected by a single severe storm - say catchments up to about 200-300 km2 in humid temperate climates - the direct relationship between specific rainfalls and the resulting discharge or runoff is extremely complicated and often quite non-linear (e.g. doubling the rainfall input will more than double the discharge output).

The non-linearity arises because, at intermediate scales of both area and time, other physical and hydrological factors, such as evaporation, infiltration, groundwater flow, and the processes of wetting and drying over a sequence of events, are very significant and thus any direct relationship between rainfall alone and runoff is not easily determined. In this chapter, models for estimating the runoff resulting from a rainstorm or continuous sequence of rainstorms will be discussed. These methods range from simple procedures derived by engineers for immediate practical use to complex computer models.

Any catchment model needs to address two fundamental characteristics of the relationship between rainfall and runoff. The first is the proportion of the volume of rainfall represented by the storm hydrograph (or that part of the hydrograph defined as stormflow over and above the baseflow that would have occurred if the storm had not happened, Fig. 12.1). We know that, given an accurate estimate of rainfall over the catchment, this proportion will be less than one, but may vary in non-linear ways with the rainfall pattern in time and space and the antecedent wetness of the catchment. The runoff generation process must therefore be treated non-linearly. The second characteristic is the distribution of the runoff generated in time to make up the shape of the storm hydrograph. In this case, hydraulic theory suggests that flow velocities should vary non-linearly with discharge rate, but many hydrograph estimation procedures make the assumption that the time distribution does not change very much with the nature of the storm, i.e. that the runoff routing process can be treated linearly.

Rainfall n

Discharge

Fig. 12.1 Definition of stormflow and baseflow.

Baseflow volume

Time

Fig. 12.1 Definition of stormflow and baseflow.

This is an assumption that seems to work quite well in practice, the reasons why will be discussed below.

There is now an enormous variety of catchment models available, ranging from the very simple to complex distributed models that make predictions in space as well as time. Two points are worth making before going on to consider the different types of models that might be used in practice. The first is that even the most complex model will still be only an approximate representation of the concepts of runoff generation described earlier in Section 1.6. The predictions will necessarily therefore be to some degree approximate or uncertain and it will be worthwhile considering carefully whether the assumptions of a particular modelling approach are appropriate for a particular application. The second is that the predictions of a model can only be as accurate as the input data used to drive the model. Poor input data will result in poor predictions. In some types of application it is possible to compensate for data deficiencies (see e.g. Section 12.4) but it is also worthwhile considering carefully the possible errors in the input data when using a model. A much more complete discussion of catchment models is provided by Beven (2001) while details of some specific models are given in the books edited by Singh (1995) and Singh and Frevert (2002a,b).

In practice, the choice of a method of modelling catchment response might depend on the nature of the application. There are very many applications, for example, that could not justify the expense of applying a complex distributed catchment model. This is nowadays not so much a matter of the expense of running the model, but rather the cost of the time necessary to assemble the data sets needed (if they exist at all). Thus simpler methods may often still be useful. The simplest methods of predicting runoff aim to predict only the hydrograph peak discharge. These derive from the so-called 'rational method, which is also the oldest form of catchment model, dating back to the work of Thomas Mulvaney in Ireland and Emil Kuiching in the USA in the nineteenth century. A modern implementation of the rational method, still used in the UK, is the ADAS345 method, so called as it is set out in Report 345 produced by the Agricultural Development and Advisory Service (ADAS) for the then UK Ministry of Agriculture, Fisheries and Food (MAFF, 1981), now Defra. The method is intended to predict the peak discharge from an area underlain by field drains and was based on an analysis of the response of small rural catchments. It is only recommended for use in small catchment areas less than 30 ha.

The ADAS345 model is the simple equation

where Q is the peak flow in L s-St is the soil type factor, which ranges between 0.1 for a very permeable soil to 1.3 for an impermeable soil; F is a factor which is a function of catchment average slope, maximum drainage length and average annual rainfall; and A is the area of the catchment being drained in hectares.

Guidance on the values of the above variables is given in the ADAS report, together with a nomograph that can be used to estimate the flow. The return period of the peak flow varies with the coefficient F. The ADAS345 model is based on empirical analysis rather than hydrological theory and, while the results would be expected to be rather uncertain in any particular application, effectively this provides a simple and low cost but accepted convention for the estimation of the runoff expected from a small area, perhaps a potential development site in a previously undeveloped rural area. In many applications, more sophisticated catchment models will be justified that attempt to predict both the amount of rainfall that becomes runoff and the timing of the hydrograph. However, it is worth noting that even the most complex hydrological models that are described later in this chapter depend, to some greater or lesser extent, on empirical representations of hydrological processes.

12.3 Estimating the proportion of rainfall equivalent to the stormflow hydrograph

At first sight this would appear to be a simple problem. A rainfall event occurs, the stream responds, discharge increases during the 'rising limb' to a peak and then falls during the 'falling limb' or 'recession' before the next event. This is the storm hydrograph shown in Fig. 12.1. By calculating the volume of rainfall input, and the volume represented by the stormflow, it should be easy to calculate that part of the rainfall represented in the stormflow hydrograph, otherwise known as the 'effective rainfall' .1 The proportion of the rainfall represented by the stormflow is also known as the event 'runoff coefficient'.

Unfortunately it is not that simple because, except in some arid and semi-arid environments, where discharges are often zero between events, the storm discharges from individual events are not easily separated. The rising limb from one event always starts somewhere on the falling limb from the previous event. Thus, 'hydrograph separation' requires distinguishing between what will be called stormflow and what will be called baseflow (Fig. 12.1). The only really objective way to estimate the volume of discharge from any single event would be to try to estimate what would have happened if no new events had occurred (as suggested by the dotted line in Fig. 12.1; this was tried by Reed et al., 1975). But this will then include baseflow discharges that could have long timescales, much longer than what might be called 'stormflow'. Thus, in the past, many pragmatic methods of estimating the stormflow component in a hydrograph have been proposed (such as the solid separation line in Fig. 12.1) but it is important to remember that they are all rather arbitrary; in general there is no easily separated 'stormflow' component, particularly where much of the stormflow may be water displaced from pre-event storage by the event inputs so that the stormflow is not the same water as that which fell on the catchment during the storm (see discussion of runoff processes in Sections 1.2 and the hydrograph separations based on tracer concentrations in Section 11.3).

Having said that, this type of hydrograph separation has been rather important in the history of rainfall-runoff analysis because of the way it allowed estimates of input (the effective rainfall) and output (the stormflow) to be matched. Effectively, after baseflow separation has been carried out (by whatever method), the volume of effective rainfall (and therefore the storm runoff coefficient) required is known precisely. Mass balance is then assured in the analysis, even if not all the water is accounted for. Having matched the input and output volumes for many events for a catchment, the way in which the effective rainfall varies between storms, and the way in which the flow processes modify the time distribution of the hydrograph could be studied. It turned out that understanding how much of the rainfall became effective rainfall was rather difficult, but that, for many catchments, the time distribution stayed fairly constant (see the next section). In fact, we do not need to invoke any process interpretation for the generation of the 'stormflow'. We only need to insist that the stormflow is defined in a consistent way since the matching effective rainfall will be conditional on how stormflow is defined.

The simplest and perhaps most widely used method of determining the effective rainfall to match a given stormflow is the phi-index approach. Phi (the Greek letter $) is a coefficient that is used as a threshold on each time increment of rainfall in a storm. Any rainfall above the threshold is assumed to be effective rainfall (a modification is to allow some initial deficit to be satisfied before implementing the threshold but this introduces an additional variable parameter, since the initial loss might be different for wet and dry antecedent conditions). The value of $ is chosen such that the total volume of effective rainfall is equal to the total volume of stormflow. In this way, the greatest contribution to effective rainfall is given by time increments with high rainfall intensities (Fig. 12.2a). The value of $ is chosen on an event by event basis in analysis. The difficulty then remains to choose a value for $ in predicting the effective rainfall in predicting a new event.

12.4 Estimating the time distribution of runoff 12.4.1 The time-area method

The time-area method of obtaining runoff or discharge from rainfall can be considered as the first attempt to create a hydrological model that is distributed in space. The concept was (possibly) first introduced by Imbeaux (1892) in the Durance River in France, and later used by Ross, Zoch and Clark in the USA and Richards in the UK (see Beven, 2001). The method attempts to divide a catchment into areas based

1 234567 8 9 10 11 Time t

Fig. 12.2 Different ways of assigning a volume of effective rainfall based on total rainfall hyetograph: (a) phi index method; (b) proportional rainfall method. The volume of effective rainfall is the same in both cases.

on how long it will take runoff generated on that area to reach the catchment outlet. Once this areal discretisation of the catchment has been made, estimates of the runoff generated on each fractional part of the catchment can be delayed by the appropriate time delay and then added to calculate the storm hydrograph. The normal assumption is that the delays do not change over time. This implies that this method of routing runoff is linear, i.e. that if the estimate of the runoff generation is doubled, the result will be a simple doubling of the output hydrograph (see Sections 12.5 and 12.6 below).

Thus the storm hydrograph, Qt is the sum of flow-contributions from subdivisions of the catchment defined by time contours (called isochrones). The method is illustrated in Fig. 12.3a. The flow from each contributing area bounded by two isochrones (T — AT, T) is obtained from the product of the mean runoff generated on that area (i) from time T — AT to time T and the area (AA). Thus Q4, the flow at X at time t = 4 h is given by:

where NT is the number of time steps of length AT equivalent to the time of concentration Tc and ik is the effective rainfall generated on a fractional area Ak. Note that different sets of units can be used for this calculation. If runoff generation is expressed as a depth per unit time step and area as a fraction, then the predicted flows will also have units of depth per unit time (e.g. mm over the time step). If runoff generation is expressed as ms—1 (equivalent to mm h—1/(1000x3600)) and area in m2, then the predicted flows will have units of m3 s—1.

AT Time, T

Fig. 12.3 The time-area method: (a) division of storm rainfalls and catchment area into areas with equal travel time AT to the outlet. (b) time-area-concentration curve. (b) cumulative time-area curve.

The whole catchment is taken to be contributing to the flow after T equals Tc, which is called the time of concentration of the catchment. Hence, in deriving a flood peak for design purposes, a design storm with a critical sequence of runoff generation intensities can be used for the maximum intensities applied to the contributing areas of the catchment that have most rapid runoff.

To fix the isochrones considerable knowledge of the catchment is required, so that the times of overland flow and flow in the river channels may be determined. This is most easily achieved by deriving a representation of the time-area histogram from an analysis of storm data. The simple discrete form of the time-area concept can be generalised by making AT very small and considering increases in the contributing area to be continuous with increasing time. Thus, in Fig. 12.3c, the plot of catchment area against time is shown as a dashed line and this is known as the time-area curve. Its limits are the total area of the catchment and the time of concentration. For any value of T, the corresponding area A gives the maximum flow at the river outfall caused by a rainfall of duration T. The derivative of the time-area curve shown in Fig. 12.3b gives the rate of increase in contributing area with time, and is called the time-area-concentration curve, since the length of the time base is equal to the time of concentration of the catchment. It is a form of linear impulse-response function or transfer function for the catchment. A generalisation of the time-area transfer function that has been very widely used in hydrological analysis is the unit hydrograph.

A major step forward in hydrological analysis was the concept of the 'unitgraph' introduced by the American engineer Leroy K. Sherman (1932). Sherman (1942) later called this the unit hydrograph, which is the name that is normally used today in hydrology. The unit hydrograph was actually a rather neat idea since it was a way of generalising the time-area curve in a way that did not require the rather difficult task of estimating which parts of the catchment actually fell within each time increment. Sherman defined the unit hydrograph as the hydrograph of stormflow resulting from effective rainfall falling in a unit of time such as 1 h or 1 day and produced uniformly in space and time over the total catchment area. By averaging the runoff generation as a uniform effective rainfall over the catchment, and generalising the transfer function as the unit hydrograph, a useful predictive tool was provided.

In practice, a T-hour unit hydrograph is defined as resulting from a unit depth of effective rainfall falling in T h over the catchment. The magnitude chosen for T depends on the size of the catchment and the response time to major rainfall events. The standard depth of effective rainfall was taken by Sherman to be 1 in, but now 1 mm is more normally used. The definition of an effective rainfall-runoff relationship is shown in Fig. 12.4a, with 1 mm of uniform effective rainfall occurring over a time T producing the hydrograph labelled TUH. The units of the ordinates of the T-hour unit hydrograph are m3 s-1 per mm of rain. The volume of stormflow is given by the area under the hydrograph and is equivalent to the 1-mm depth of effective rainfall over the whole catchment area.

The unit hydrograph method makes several assumptions that give it simple properties that make it easy to apply.

(a) There is a direct proportional relationship between the effective rainfall and the stormflow. Thus in Fig. 12.4b, two units of effective rainfall falling in time T produce a stormflow hydrograph that has its ordinates twice the TUH ordinates, and similarly for any proportional value. For example, if 6.5 mm of effective rainfall fall on a catchment area in T h, then the hydrograph resulting from that effective rainfall is obtained by multiplying the ordinates of the TUH by 6.5.

(b) A second simple property, that of superposition, is demonstrated in Fig. 12.4c. If two successive amounts of effective rainfall, Rj and R2, each fall in Th, then the stormflow hydrograph produced is the sum of the component hydrographs due to R1 and R2 separately (the latter being lagged by T h on the former). This property extends to any number of effective rainfall blocks in succession. Once a TUH is available, it can be used to estimate design flood hydrographs from design storms.

Fig. 12.4 The unit hydrograph concept. (a) The T hour unit hydrograph (TUH) arising from 1 mm of effective rainfall in a time step of T h. (b) The linearity assumption: 2 mm of effective rainfall results in a time step of length T in a predicted hydrograph of 2 x the TUH. (c) The superposition principle: linear addition of the contributions of effective rainfall in successive time steps.

Fig. 12.4 The unit hydrograph concept. (a) The T hour unit hydrograph (TUH) arising from 1 mm of effective rainfall in a time step of T h. (b) The linearity assumption: 2 mm of effective rainfall results in a time step of length T in a predicted hydrograph of 2 x the TUH. (c) The superposition principle: linear addition of the contributions of effective rainfall in successive time steps.

(c) A third property of the TUH assumes that the effective rainfall-stormflow relationship does not change with time, i.e. that the same TUH always occurs whenever the unit of effective rainfall in T h is applied. Using this assumption of invariance, once a TUH has been derived for a catchment area, it could be used to represent the response of the catchment whenever required.

The assumptions of the unit hydrograph method must be borne in mind when applying it to natural catchments. In relating total rainfall to stormflow, the amount of effective rainfall will depend on the state of the catchment before the storm event. If the ground is saturated or the catchment is impervious, then a high proportion of the rain would be expected to be effective in producing stormflow. On unsatu-rated ground, however, the soil will have a certain capacity to take up rainfall before stormflow is generation. Only when any storage deficits have been made up and the rainfall becomes fully effective will extra rainfall in the same time period produce proportionally more runoff. The first assumption of proportionality of response to effective rainfall conflicts with the observed non-proportional behaviour of river flow. In a second period of effective rain, the response of a catchment will be dependent on the effects of the first input, although the second assumption makes the two component contributions independent (Fig. 12.4c). The third assumption of time invariance implies that, whatever the state of the catchment, a unit of effective rainfall in T h will always produce the same TUH. However, the response hydrograph of a catchment might be expected to vary according to the season: the same amount of effective rainfall will be longer in appearing as stormflow in the summer season when vegetation is at its maximum development and the hydraulic behaviour of the catchment will be 'rougher'. In those countries with no marked seasonal rainfall or temperature differences and constant catchment conditions throughout the year, then the unit hydrograph would be a much more consistent tool to use in deriving stormflow from effective rainfall.

Another weakness of the unit hydrograph method is the assumption that the effective rainfall is produced uniformly both in the time step T and over the area of the catchment. The areal distribution of rainfall within a storm is very rarely uniform, and, as discussed in Section 1.2, we expect stormflow contributing areas to expand and contract in space as the catchment wets and dries. For small or medium-sized catchments (say up to 100 km2), a significant rainfall event may extend over the whole area and, if the catchment is homogeneous in composition, a fairly even distribution of effective rainfall may be produced. More usually, storms causing large river discharges vary in intensity in space as well as in time, and the consequent response is often affected by storm movement over the catchment area. However, rainfall variations are damped by the integrating action of the catchment, so the assumption of uniformity of effective rainfall over a selected period T is less serious than might be supposed at first. The effect of variable rainfall intensities in time can be reduced by making T smaller. The effect of variable rainfall intensities in space can be reduced by developing TUH for different sub-catchments where the data are available to do so or developing TUHs for storm type of different origins showing different characteristic patterns.

Despite these conceptual limitations, the unit hydrograph method has the advantage of great simplicity. Once a unit hydrograph of specified duration T has been derived for a catchment area (and/or specific storm type), then for any sequence of effective rainfalls in periods of T, an estimate of the stormflow can be obtained by adopting the assumptions and applying the simple properties outlined above. The technique has been adopted and used worldwide over many years. It is therefore very useful for the hydrologist to understand the unit hydrograph concept and its simplifying assumptions, since it is still the basis for many analysis and estimation procedures in hydrology. Previous editions of Hydrology in Practice have dealt with aspects of unit hydrograph theory in some detail, including the derivation of the TUH from simple storms and multiple storms, changing the time step T by using the S-curve technique, and the instantaneous unit hydrograph (IUH) that is the (theoretical) response from an instantaneous unit input of effective rainfall (Nash, 1957). For the IUH, the summation of equation 12.3 is replaced by an integral and it is no longer necessary to assume that the effective rainfall is uniform over long time steps (though it is still necessary to assume that it is uniform in space). Discharge at any time t is then predicted as:

This is called a convolution integral, analogous to the discrete summation of (12.3), in which the time series of effective rainfalls R(t) is convolved with the transfer function that is the IUH, H(t), where time t is now treated as a continuous variable rather than a series of discrete time steps. This was not easy to evaluate for any arbitrary form of the IUH in the days before digital computers were widely available; hence the discrete TUH remained popular. Now, however, it is relatively easy to program a numerical solution to the convolution integral (12.4).

12.5.1 The unit hydrograph as the routing component of a catchment model

We have already seen how the unit hydrograph is essentially a linear model for routing runoff generation (or effective rainfall) to the catchment outlet. In the late 1950s, through the work of the Irish engineering hydrologists Jim Dooge and Eamonn Nash, it was realised that this allowed a link to be made to general linear storage models (Dooge and O'Kane, 2003, provide a detailed summary of this work). This allowed the unit hydrograph to be represented in terms of some simple functional forms with just a small number of parameters. Nash (1960), for example, proposed a model of the unit hydrograph that consisted of n linear stores in series, each of which has a mean residence time of K. The equation for a single storage element is then that outflow

where S is storage and K is a time constant. This has an impulse response or unit hydrograph of the form of an instantaneous rise followed by an exponential decline, or

H(t) = K exp(-T/K) and for n stores in series vi \ (T V-1 exp(-T/K)

This is the equation of a gamma distribution, with T(n) being the tabulated gamma function. It has the advantage that it can take on a variety of forms depending on the values of n and K (Fig. 12.5). In the general case, n need not be an integer value. Jim Dooge (1959) also showed how the unit hydrograph could be represented by a general class of linear models with combinations of fast and slow storages, how a constant time delay could also be incorporated into the model in cases where the rising limb did not start immediately, and how the same concepts could be used for routing flood waves in large rivers (see also Dooge and O'Kane (2003) and the discussion of the data-based mechanistic modelling concepts in Section 12.8.3 below).

Fig. 12.5 Unit hydrographs generated by the Nash cascade with mean residence time K = 2 time steps in each store and number of stores n = 1,2,4,6,10.

Time steps

Fig. 12.5 Unit hydrographs generated by the Nash cascade with mean residence time K = 2 time steps in each store and number of stores n = 1,2,4,6,10.

12.5.2 The geomorphological unit hydrograph and channel-width function

A more recent interesting development of the unit hydrograph concepts has been to try to relate the catchment response in the form of the IUH to the characteristics of the channel network. There have been two ways of doing this, one based on characterising the network in the form of statistics of stream links of different order, the other by representing the network as a distance histogram or 'channel network width function'.

The geomorphological instantaneous unit hydrograph (GIUH) depends heavily on the concept of stream order introduced by Robert Horton in his classic 1945 paper on the development of catchment geomorphology. The ordering system most commonly used is now that of Arthur Strahler (1956; Fig. 12.6). In Strahler's system, first-order streams are those segments of the channel network that originate in stream sources; second-order segments occur below the junction between two first-order segments; third-order segments below the junction of two second-order segments, and so on. Junctions with lower order streams do not change the order of a segment. Horton noticed that in a larger order network of channel segments the numbers, lengths and catchment areas of different order streams had ratios that were approximately constant over a range of orders, and proposed that these ratios could be considered as laws at the catchment scale (in fact, the very nature of a space-filling dendritic (treelike) network ensures that the ratios will be approximately constant; see the detailed discussion of the properties and development of channel networks in Rodriguez-Iturbe and Rinaldo, 1997).

Fig. 12.6 Strahler ordering system for channel links in the ^ = 5 River Hodder stream network. The River Hodder is a tributary of the River Ribble in northern England, draining the southern slopes of the Forest of Bowland. Not all network links are labelled for clarity. Stream order increases when two streams of equal lower order join.

Fig. 12.6 Strahler ordering system for channel links in the ^ = 5 River Hodder stream network. The River Hodder is a tributary of the River Ribble in northern England, draining the southern slopes of the Forest of Bowland. Not all network links are labelled for clarity. Stream order increases when two streams of equal lower order join.

The GIUH concept, as originally introduced by Ignacio Rodriguez-Iturbe and Juan Valdes (1979), makes use of these laws and a probabilistic argument that a drop of effective rainfall could be produced randomly anywhere in the catchment. It will then flow to the catchment outlet in the segments of the channel network. This can be viewed as a form of transition probability matrix between states in the system. In any time step there is a probability that a drop of effective rainfall on a hillslope (zero-order state) will reach a first or higher order stream; a probability that a drop in a first-order stream will reach a second or higher order stream; and so on for all the possible transitions until a drop reaches the catchment outlet which acts as a sink or trapping state. The model is completed by assuming that the distribution of waiting times in each channel state is an exponential distribution with a single parameter that depends on the order of the segment. This is equivalent to assuming that each state acts as a linear store as in the Nash model (Chutha and Dooge, 1990); more complex distributions could be used, but this will both increase the number of parameters that need to be defined and the difficulty of the mathematics (see Gupta et al., 1980). It can then be further assumed that the mean residence time in each channel state is related to the mean length of segments of that order and a constant routing velocity (see Section 14.4 for a discussion of the difference between routing velocity and actual flow velocity of the water).

It was found that the resulting unit hydrograph expressions could be well represented by approximations for the peak, qp, and time to peak, tp as:

0.55

where v is the routing velocity, LQ is the length of the stream of highest order Rl is the ratio of mean stream lengths in different orders, RB is the ratio of stream numbers in different orders, and RA is the ratio of mean stream catchment areas in different orders.

Renzo Rosso (1984) also shows how these functions can be used to define a full unit hydrograph in the form of a gamma distribution (similar to that in the Nash cascade of (12.6) and Fig. 12.5) so that

where the two parameters, a and k are given by a = 3.29

0.78

RBRL

0.48

0.07

LQv~

The GIUH provides a unit hydrograph that, with some simple assumptions, is related directly to the form of the channel network as generalised by Horton's 'laws' of stream numbers, lengths and areas. It has been used quite widely and, with some adjustments of the velocity parameter by calibration, can produce good predictions given a good estimate of the effective rainfall.

It is not, however, necessary to simplify the channel network in this generalised way. In developing the GIUH Rodriguez-Iturbe and Valdez (1979) did so for two reasons. The first was to simplify the mathematical treatment in moving from one stream order to another; the second was to continue Horton's attempt to provide a general synthesis of catchment hydrology and geomorphology. In doing so, however, some

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 Distance from outlet (km)

Fig. 12.7 Network width function for the River Hodder network shown in Fig. 12.6.

information about the detailed form of the channel network is lost, in fact unnecessarily, since the actual channel network (as represented by the blue lines on a map) is generally one of the very easiest pieces of information to obtain for any catchment. By making the same constant routing velocity assumption, travel times in the network can then be determined directly from mapping the pattern of distances to the outlet along the channels. This is similar to the time delay histogram of Section 12.4.1 above, but now only considering the channel network. At any given distance, there may be one or more channel segments (see Fig. 12.6) and the histogram of numbers of channel segments against distance is known as the channel network width function (Fig. 12.7). This has been used directly as a linear flow routing algorithm (e.g. Kirkby, 1976; Beven, 1979) and with the addition of a dispersive component for each network link as a simple routing algorithm (e.g. Franchini and O'Connell, 1996), including at continental scale (Naden et al., 1999). It has been incorporated into several catchment rainfall-runoff models, including Topmodel (see Section 12.8.4). Other ways of deriving routing models from network geomorphology have also been considered (see, e.g. Shamseldin and Nash, 1998; Cudenec et al., 2004, Nourani et al., 2009). Routing of flood waves along the main channel of a large catchment is considered later in Chapter 14.

The unit hydrograph has proven to be a very useful technique in hydrological analysis and prediction. It has been widely used since its introduction by Sherman in the 1930s. As noted earlier in this chapter, it actually requires two components: one to determine how much rainfall becomes effective rainfall, and one (the unit hydrograph itself) to distribute that effective rainfall in time to predict the resulting hydrograph. In the 1930s, the combination of Sherman's unitgraph with the Horton infiltration excess concept of stormflow generation to predict the effective rainfall, meant that a practical rainfall-runoff model at the catchment scale was available. In fact it was so useful that later work often concentrated on improvements to ways of deriving the unit hydrograph, or relating the unit hydrograph to catchment characteristics, or improving the prediction of effective rainfalls, than on querying the rather strong assumptions on which it is based. The method is still used extensively in the UK as part of the FEH flood hydrograph method (see Section 13.5).

There is no doubt that, in some catchments, these assumptions will not be valid. The study of Minshall (1960), for example, is well known for showing that, in a small catchment, the derived unit hydrograph might change with the magnitude of the peak flow (see Beven, 2001). If this is the case, then the assumptions of linearity, superposition and invariance are not tenable. Such an analysis will be dependent on the rules that have been used to separate stormflow from baseflow. The derived unit hydrograph will vary, depending on what technique of hydrograph separation has been used. Many studies have suggested that it is difficult to carry out such a separation objectively, even if once a set of rules is chosen they are applied consistently. A general framework for relating both the unit hydrograph response and the actual travel times of water in a catchment in a way that properly reflects the time variability of the responses has been provided by Botter et al. (2010).

The main problem with the unit hydrograph approach, however, has always been in calculating the effective rainfall for an event, both in analysis and prediction. As noted above, in analysis this problem is minimised by setting the volume of effective rainfall to the volume of stormflow after hydrograph separation to impose a mass balance. In prediction, it is somewhat more problematic. We have the measured rainfalls, we know that the effective rainfall should depend on both the antecedent state of the catchment and the pattern of rainfalls, but we have not had very good ways of predicting just how much of the rainfall becomes effective rainfall in any particular event.

In fact, it would be better to avoid hydrograph separation and effective rainfall calculations completely by trying to predict the whole hydrograph. Young and Beven (1994), e.g. have shown how a general discrete time linear transfer function model could be used to represent both the storm and baseflow components of the hydrograph, in combination with a non-linear function on the input that would provide an estimate of the total effective rainfall for both components. This provided a complete catchment model with a structure inferred directly from the data, what Peter Young calls a data-based mechanistic model (see Section 12.8.3). This approach has since been developed further for both rainfall-runoff simulation and flood forecasting (Young, 2001, 2002, 2003, 2009; Romanowicz et al., 2008). Unit hydrographs in the form of linear transfer functions also provide the routing components for many different conceptual catchment rainfall-runoff models, as will be seen in the descriptions of some of the available models below. Routing of flood waves along the main channel of a large catchment is considered later in Chapter 14.

Unit hydrograph models of catchment response are examples of conceptual rainfallrunoff models. Although relatively simple, they reflect the two most important aspects of the hydrograph: a loss function to account for the fact that not all the rainfall in an event becomes streamflow, and a time distribution to account for the delays between runoff generation and the hydrograph measured at a catchment outlet. Rainfall-runoff modelling has become one of the most important tools available to the hydrologist. Many different models have been developed in the last 50 years, once digital computers became available to researchers and, later, practitioners. There are now so many, in fact, that a number of texts have appeared to provide detailed information about modelling techniques and particular models (see Beven, 2001; Abbott and Resfsgaard, 1996; Singh and Frevert, 2002a,b; Wagener et al., 2004). Here we will take an overview of the types of models used in practice and give some examples of their use.

The best way to classify hydrological models is by purpose. There are distributed models for predicting the spatial pattern of flow processes, there are lumped models for predicting the hydrograph response without worrying too much about what is happening in space, and there are real-time forecasting models that are used for flood forecasting or drought management within a data assimilation framework with the aim of improving the predictions as a flood or drought event unfolds. Models are also sometimes classified as to whether they are deterministic or stochastic. A deterministic model run has only one possible outcome; a stochastic model takes account of uncertainties in the representation of the system and might have a distribution of outcomes. Today this distinction has become somewhat blurred, however, since deterministic models can be run many times with randomly chosen inputs or parameter values in Monte Carlo simulation to produce distributions of outcomes (see Section 12.7 below).

The first widely recognised catchment model on a digital computer was the Stanford watershed model, developed by Norman Crawford during his Ph.D. with Ray Linsley at Stanford University in the early 1960s. This was a lumped, deterministic model that was later developed commercially and applied worldwide by their company called Hydrocomp. Versions of the model are still being used and can be freely downloaded as part of the US Environmental Protection Agency (EPA) software suite as HSPF (hydrologic simulation package fortran).2 Recent versions include components for non-point pollutant sources, including transport and modification in river channels. It has been extensively used in the EPA's Chesapeak Bay programme.

The Stanford watershed model set the framework for many of the rainfall-runoff models that followed: it consisted of a set of storage elements linked in series and in parallel, to represent different parts of the catchment system (surface runoff, the upper soil zone store, lower soil zone store, a ground water store, channel routing, etc. The use of digital computers meant that the equations describing the operation of each store did not have to be simple and did not have to be linear. The computations could be easily programmed in a high-level language such as Fortran. The Stanford watershed model had some 25 parameter values (34 if a snowmelt component was included). Some could be estimated on the basis of catchment characteristics, others had to be calibrated by fitting observed discharges.

There are many models of this type. They vary in the numbers of stores and the equations used for each store. They all, however, have a similar problem in applications. Many of the equations used will contain parameters that must be specified before a run of the model can be made. But the parameter values are not generally easily determined on the basis of catchment characteristics. Therefore, it is often the case that the model user tries to calibrate the parameters by adjusting the values until a good fit is obtained to an observed discharge time series. In the early days of using such models, this was done by trial and error, with the results assessed by eye. More recently, automatic optimisation methods have been developed, with goodness of fit being assessed by one or more numerical performance measures, of which perhaps the most widely used in hydrology is the Nash and Sutcliffe (1970) model efficiency measure, NSE, defined by:

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