Modelling the roles of magnitude and frequency in the evolution of an upland catchment

T.J.Coulthard, M.J.Kirkby and M.G.Macklin
School of Geography,University of Leeds,Leeds, LS2 9JT


Despite recent advances in numerical modelling of our environment, the relative roles of random as opposed to more deterministic factors are frequently ignored. These include the non linear behaviour of a river basin in response to stochastic events such as extreme weather and the effects of climate change. Existing modelling approaches tend to ask how a landform develops, but not why. Whilst we have a good understanding of the processes involved in a catchments evolution, a gulf exists between our knowledge of the cause and consequent effect. To investigate this, a cellular automaton (CA) type model is being used, to simulate the evolution of a small catchment (>3km2) over a variety of climatic and cultural scenarios.

This innovative model is applied to the catchment of Cam Gill Beck, above Starbotton, North Yorkshire. This is divided into 1 million, 2m by 2m grid cells, to which a range of process laws are applied. These include approximate expressions for mass movement rates, soil creep, the influence of vegetation and hillslope hydrology, as well as fluvial erosion and deposition in ten grainsize fractions. This provides a good representation of valley floor geometry whilst retaining a fully dynamic interaction with the surrounding valley sides.

The model has allowed for the first time the relative roles of flood magnitude, frequency and vegetation change to be examined in great detail over time scales from seconds to 100 years. Furthermore, the model produces detailed stratigraphies, grainsize distributions, erosion, deposition, and examples of avulsion and channel change. Results show geomorphological change resulting from deforestation and especially from larger flood magnitudes, with a non linear sediment discharge.

1. Introduction

When studying the causes of change in geomorphic systems the two competing philosophies of catastrophism and uniformitarianism are frequently discussed. In fluvial geomorphology, this debate continues under the guise of the role of large, infrequent floods relative to the cumulative effects of more frequent floods of a lower magnitude in shaping channel form and valley floor morphology. Recent large flood events would appear to have a major impact upon the landscape, but how important are these events on the long term history and evolution of a catchment?

As deposits from the largest floods are usually well preserved, a seductive conclusion is that these extreme events are dominant. But, the equilibrium and dominant discharge concepts suggest that channel geometry is adjusted to relatively frequent floods of a moderate magnitude, as there are fairly strong empirical links between channel geometry and channel discharge statistics. Therefore, whilst workers have determined a strong link between the process and form, they have failed to make a good connection between the cause (big floods or many moderate events) and the effect (aggradation and valley floor transformation).

Similarly, modelling approaches have tended to concentrate on process-form relationships explaining how, but neglecting to ask why landforms have developed.

This is partly explained by the complexity of modelling the numerous processes interacting within a drainage basin over a wide range of spatial and temporal scales. For example, the previous inherited channel geometry, vegetation, lithology, sediment supply (Merrett & Macklin 1998), slope channel coupling and the flashiness and shape of the flood hydrograph (Knighton 1998) all have roles. Unlike many reach based modelling schemes (e.g Nicholas & Walling, 1997; Bates et al., 1997), a cellular automaton (CA) model developed at Leeds (Coulthard et al. 1996, 1998a, 1998b ) has succeeded in combining these factors within one high resolution framework. This is achieved by dividing the catchment into 2m2 grid cells within which process laws are applied in relation to each cells immediate neighbours. Examples from this model have shown the formation of berms, braids and terraces at a 1m2 resolution, as well as non-linear behaviour caused by interplay between these processes (Coulthard et al. 1996, 1997).

To establish firmer links between cause and effect, this model is applied to the catchment of Cam Gill Beck, a tributary of the River Wharfe, North Yorkshire, UK (Coulthard 1998b). Here, the impacts of one extreme flood event are compared to the effects of longer periods of moderate floods on bedload discharge and valley floor morphology.

2. Method

The model concept, which is summarised below, is simple, although its operation is complex and for a full description readers are referred to Coulthard et al. (1996, 1997, 1998a).

The catchment relief was digitised from 1:10 000 scale Ordnance Survey map contours. This data, supplemented by EDM surveyed detail for the valley floor was combined using the TOPOGRID command in ARC-INFO to create a 2 m2 resolution digital elevation model (DEM), of one million equal sized grid cells (Figure 1).

Figure 1. 3d projection of Cam Gill Beck DEM, viewed from south. Scale 1400 by 2800m. Area highlighted is detailed in figure 5.

Within this topographic representation the model assigns each grid cell properties of elevation, discharge, vegetation, water depth, and grainsize. For every model timestep, these values are altered in accordance only to their immediate neighbour and four groups of processes. The first set is a model of hill slope hydrology, using an adaptation of TOPMODEL (Beven & Kirkby 1979) with an exponential soil moisture store. The second part is a hydraulic routing scheme, utilising bed slope and calculating depth with an adaptation of Manning's formulae. Thirdly, fluvial erosion and deposition are calculated using the Einstein-Brown (1950) equation. This is applied to nine grainsizes ranging from 0.004 to 1.025 metres in whole phi classes incorporated with an eleven strata active layer system similar to that used by Parker (1990) and Hoey & Ferguson (1994). Bedrock and a cohesive vegetation mat are also represented and the stratigraphy is recorded for 2.1m below the channel bed. Finally, mass movement rates are calculated, incorporating a factor of safety which changes with the soil saturation. The model was run on the Silicon Graphics Cray Origin 2000 at the Manchester Computing Centre High Performance Computing (HPC) centre.

Periods of moderate flooding were simulated using a ten year hourly rainfall data set from Church Fenton North Yorkshire, running from January 1985 to December 1995. This was multiplied by a factor of 1, 1.5 and 2 to represent different climatic regimes whose hydrographs are shown in figure 2. The factor of 1.5 closely resembles data recorded from nearby Coverdale. These simulations were repeated for a further ten times, simulating 100 years of erosion. The extreme flood event was simulated using a ten year hourly rainfall data set from Church Fenton, which was edited to include two hours of extreme rainfall of 100 and 80 mm/hr each. These amounts are similar to those recorded by Evans (1996) of 192 mm in 2 hours in the severe flood of Wycoller Beck in the central Pennines. At the end of every simulation run, the elevations were saved and subtracted from the initial ones to determine the sediment discharge.

Figure 2. Hydrographs from Run 1, Run 1.5, Run 2 and Extreme.

Because of the catchment size, initial conditions could not be defined for every grid cell. Therefore, a spin up time is required to allow initial conditions to evolve. The catchment begins with equal distributions of grainsizes on every cell and develops its own channel network and armoured channel. The extreme flood simulation used initial conditions created by 1000 years simulation of the run with a rainfall magnitude of 1.

3. Results

Figure 3 shows the ten year bedload discharges from the three moderate flooding simulations. The initial peak is caused by the removal of fines from the initial conditions. These figures are compared to bedload yields measured from several other catchments in table 1 and figure 4.

Figure 3. Bedload discharge for simulations Run 1, Run 1.5 and Run 2.


Area (km2)

Bedload (m3 yr-1)

Bedload (m3 km2 yr-1)


Monachyle, Balquidder




Stott et al. (1986)
Kirton, Balquidder




Stott et al. (1986)
Beckthorn, Cumbria




Newson and Leeks (1985)
Coledale, Cumbria




Newson and Leeks (1985)
Lanthwaite, Cumbria




Newson and Leeks (1985)
Allt a'Mhuillin, Ben Nevis




Richards and McCaig (1985)
Allt a'Mhuillin, Ben Nevis




Richards and McCaig (1985)
Rough Sike, Moor House




Warburton and Evans (1998)
Trout Beck,

Moor House




Warburton and Evans (1998)
Cam Gill Beck (medium 1)




Cam Gill Beck (medium 1.5)




Cam Gill Beck (medium 2)





Table 1. Bedload volume calculated assuming 2.65 t m3 ( after Warburton and Evans 1998).

Figure 4. Bedload discharge against catchment area from table 1.

Figure 5 shows a plan view of a section of the valley floor highlighted in figure 1. Figure 5.1 is a field map of a large boulder berm deposit caused by a catastrophic flood in 1686 described in full by Coulthard et al.1998b. Figure 5.2 details the valley floor morphology, showing terrace / berm features in a similar location to those found in the field. These are highlighted in figure 5.3. The volumes of these features and those generated by the model are detailed in table 2.

Figure 5. Plan view of section highlighted in figure 1. (1) Map of flood deposits, (2) Shaded view of simulation results, (3) Shaded view with deposition highlighted.

Initial Conditions



Volume eroded



Volume deposited



Balance (E-D)



Volume Berm A



Volume Berm B



Table 2.(All units in m3.)

Table 3 describes the bedload discharges from the extreme simulation and compares them to those from the Wycoller beck flood (Evans 1996). For Cam Gill Beck, the figures in brackets are the sediment volumes immediately after the flood, the other figures after the rest of the ten year data set.


Wycoller Beck

Cam Gill Beck Simulation.

Volume eroded (m3)


5357 (4000)

Volume deposited (m3)


2285 (2200)

Sediment discharge (m3)


3071 (1800)

Catchment area




0.01 - 0.15

0.1 - 0.3

Table 3.

4. Discussion

After initial peaks, the results from the moderate flood simulations stabilise to produce a regular bedload discharge. This compares well with field measurements from other catchments (figure 4). There is considerable scatter in this chart, which highlights differences created by catchment process interactions. The figures from the three simulations fit well within these bounds and as the 1.5 run is assumed to represent current rainfall characteristics the model may be under estimating bedload yield. The scatter also shows that there is no simple linear relationship between catchment area and bedload discharge. Furthermore the fluctuations in the bedload discharge over the 100 years (figure 3 ) show that even within the same catchment and an identical rainfall sequence there is no simple relationship relating to the bedload discharge. These non-linear reactions are caused by the complexity of the interactions between process and form in a catchment and can have considerable implications for environmental modellers (Coulthard et al.1997).

The simulation of an extreme flood event produced depositional features that closely resemble those found in the field generated by a catastrophic event in 1686 (Coulthard et al. 1998b). Some differences exist in the size of the features (Figure 5), namely the upstream deposit A is larger in the simulation than berm B. This is because the resolution of the contour data used for the DEM fails to pick up the valley floor features in great detail. Consequently the valley floor is wider at A and narrower at B than in the field, allowing more material to be held at A and not transported down to B. Furthermore, as less material is trapped at B, more is transported down to C. However, the total volume of the deposits (350 m3 field versus 331 m3 simulated) and their location does closely match those observed in the field. The comparison to the Wycoller Beck flood (Evans 1996; Table 3) shows Cam Gill Beck to have a greater bedload discharge for the given area. However Cam Gill Beck is steeper than Wycoller Beck, with fewer low gradient reaches allowing deposition. It would therefore be expected to flush more sediment out from the basin.

These simulations give us new insights into the roles of extreme floods and periods of lower magnitude flood events in valley floor development. The bedload yield of the extreme flood is 3000 m3 compared to 3277, 1443 and 996 m3 per 10 years for the more moderate floods. Therefore, in the terms of work done removing sediment from the basin, the extreme flood is equivalent to 10 years at 150% greater rainfall magnitude, 20 years at current levels and 30 years at 75%. If extrapolated over 100 years, the more moderate storm regimes cause 350, 500 and 1000% more bedload discharge than the extreme event. It could be argued that the ten year data set used is unrepresentative and may contain extreme floods itself. Indeed, the sequence runs from 1985 to 1995, capturing a full spectrum of storms including Hurricane Charley, August 1986, the wettest day in the UK's record (Institute of Hydrology, 1988). However, the hydrographs (figure 2) show the peak discharges from runs 1 , 1.5 and 2 ranging from 2.5, 5 and 8.5 m3 s-1 respectively, compared to 82 m3 s-1for the extreme flood. Thus, a ten year storm sequence giving a maximum flood of 2.5 m3 s-1 (run 1) erodes more material in 30 years than one massive flood. Furthermore, the storms producing extreme events especially those resulting from thunderstorms are often very localised. For example, Evans (1996) estimates that the Wycoller Beck flood was caused by a small cell of rain approximately 500m wide. The frontal weather systems associated with periods of sustained rainfall operate over regional scales of hundreds of square kilometres (Longfield & Macklin, 1998). Therefore considering this and results from the model, long term periods of more moderate storm events will have a far greater impact on basin sediment yield and lowering.

Despite indications that an extreme event are not as effective over long periods, the model shows they have a large impact upon the valley floor morphology leaving deposits A B and C (figure 5.3) standing up to 2m above the main channel. However, other simulations have shown that for up to two years after the extreme flood, large sections of deposit B were removed by smaller floods of up to 6m3s-1 (Coulthard et al. 1998b). But, this feature was preserved if these smaller floods were removed allowing a channel to incise next to the deposit and vegetation re-growth to stabilise it.

Notwithstanding these encouraging results, it must be remembered that this is only one catchment. As shown in figure 4 there is considerable variance in bedload yield between catchments of different sizes. Similarly, basins will be affected in different ways by extreme events. Larger catchments have the capacity to absorb or buffer extreme rainfall events and coupled with the small 'footprint' of large storms will show less effect than smaller basins. The authors feel that this modelling approach captures many of the complex interactions within a catchment and shows great potential for application to larger areas. Despite being site specific in its operation, the models design is generic, allowing it to be applied to any small catchment which can be digitised. To study the effects of varying rainfall intensities further the model needs to be applied to a larger catchment area which includes more extensive floodplains and depositional zones. Cellular Automaton models lend themselves to applications of parallel programming. The next step is to parallel code the model, running sections or sub catchments on separate processors simultaneously. In theory (despite a diminishing returns effect) the only limitation on the size of area to be studied is the number of processors available. Likewise, The spatial resolution could be increased to study a smaller area in greater detail over longer time scales.

5. Conclusions

Investigations of causality using a computer simulation of Cam Gill Beck, show that a single extreme flood event causes a large sediment discharge and create large depositional features changing the valley floor morphology. However, long periods of moderate flood events produce a significantly larger sediment discharge than this extreme event and therefore can have a larger, cumulative effect on basin evolution.


We would like to thank Stephen Merrett for his advice, comments and surveying assistance, Dr Joanna Schmidt of Leeds University Computing Services for help with the HPC facilities, and the Natural Environment Research Council for the provision of the research studentship no. GT4/95/147/F to TJC.


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