## Context

Channel heads are important morphological features within drainage basins because they mark the locations of process transitions. Specifically, channel heads mark the boundary between the hillslope and channelled domains of drainage basins. From the perspective of drainage basin modelling, defining the locations of channel heads correctly is therefore an essential first step that enables the model to intelligently apply appropriate sediment transfer sub-models to the correct portions of the drainage basin. In addition, correctly defining the location of channel heads is fundamentally important in empirical morphometric analysis, as attributes such as the total stream channel length and thus drainage density and stream ordering are all sensitive to the correct delineation of the full extent of the drainage network. The locations of channel heads are not fixed, and channel heads have been known to migrate considerable distances up or downslope as a result of changes in climate or land use. For all these reasons, modelling the locations of channel heads within drainage basins is of considerable theoretical and practical significance.

Past research has shown that the locations of channel heads can often be predicted using an inverse relationship between the contributing drainage area above the channel head and the local slope at the channel head (Montgomery and Dietrich 1988; Prosser and Abernethy 1996). The reason that slope-area models of channel head locations within drainage basins often take this form can be explained theoretically, as described below.

## Theoretical Models of Channel Head Location

As discussed above, channel heads are key morphological 'hinge points' in drainage basins, marking the boundary between the hillslope (upbasin) and river channel (downbasin) domains. It is therefore essential to predict the location of channel heads in drainage basins accurately, as this affects estimates of key morphometric indices such as drainage density. In fact, there have been two main approaches to explaining the location of channel heads on hill slopes. Kirkby & Chorley (1967) and Smith & Bretherton (1972) argued that the channel head represents a change in the landscape from the dominance of hillslope transport processes to fluvial transport. In such a theory the channel head occurs at a spatial transition between diffusive, slope-dependent sediment transport processes and incisive, slope and discharge dependent fluvial transport processes (Kirkby & Chorley 1967). Mathematical modelling of a smooth surface which is subject to a sediment transport law as a function of slope and discharge under steady-state conditions revealed erosion on concave slopes, whereas convex slopes remained stable (Smith & Bretherton 1972). As a result such models predict that channel heads will occur at the point of inflection between convex and concave slope profiles.

An alternative approach was developed by Montgomery & Dietrich (1988) by building upon the work of Horton (1945), who proposed that erosional thresholds control the location of the channel head. This approach makes a lot of sense, not least because channel heads mark the locations of channel initiation, which is fundamentally an erosional process. Studies carried out by Dietrich et al. (1992) and Montgomery & Dietrich (1994a) have shown that the processes driving channel initiation and thus channel head location can be described mathematically through slope dependent contributing area erosion thresholds. This is because drainage (contributing) area and local slope are the key factors controlling the erosional processes (e.g. by overland flow, shallow landslides, and seepage erosion) that scour the hillslope surface to create channels. The thresholds are related to the contributing drainage area because drainage area is a reasonable proxy for flow discharge, whereas the slope controls the efficacy of landsliding and the magnitude of the shear stress exerted by overland flow.

Simple erosion threshold models for the three different channel incision processes (landsliding, overland flow, seepage erosion) have been provided by Dietrich et al. (1992), Montgomery & Dietrich (1994) and Prosser & Abernethy (1996), amongst others. For example Montgomery & Foufoula-Georgiou
(1993) define the critical drainage area per unit contour length, a_{cr}, required for **channel initiation by overland flow** as:

a |
(1) |

where C is a constant that varies in relation
to rainfall intensity and site specific physical field
characteristics and tanθ is the local slope at
the channel head. Channel initiation by overland flow is
mainly limited to regions of the landscape with moderate
gradients (Montgomery & Dietrich
1988). For channel heads located within steeper areas
of the landscape, channel initiation processes are more
likely to be dominated by shallow landsliding (Montgomery & Dietrich
1988). The critical area per unit contour length required
for **channel initiation by shallow landsliding** can be
determined thus:

a |
(2) |

where T is the soil transmissivity (the rate at which water infiltrates through the soil), q_{r} is the rainfall intensity, ρ_{s} and ρ_{w} are the density of soil and water respectively and φ is the internal friction angle of the soil (Montgomery & Foufoula-Georgiou 1993). Dietrich & Dunne (1993) describe the mechanics of seepage erosion, whereby channel head erosion is brought about through Coulomb failure or transportation of individual sediment grains through porous materials. The critical area per unit contour length required for **channel initation by seepage erosion** is given by:

a |
(3) |

The three simple threshold models presented above (Equations 1, 2 and
3) predict a systematic relationship between the
contributing drainage basin area and the local slope
at the channel head, the precise form of which depends
upon the dominant channel initiation process. This
is illustrated in Figure 1 (below) which shows that
at lower gradients, channel heads associated with
initiation by overland flow plot on a logarithmically
scaled graph as a * linear
inverse* area-slope
relationship (Montgomery & Dietrich 1994). At steeper gradients (roughly tanθ > 0.5)
channel heads associated with shallow landsliding
plot on a logarithmically scaled graph as a

__relationship between the critical support area and local slope (Montgomery & Foufoula-Georgiou 1993). Finally, channel heads associated with initiation by seepage erosion plot on a logarithmically scaled graph as a__

*non-linear inverse*__area-slope relationship (Montgomery & Dietrich 1994). By collecting data on the contributing area and local slope for channel heads, the geomorphologist can therefore interpret the type of geomorphological processes responsible for the formation of the observed channel heads by evaluating the shape of the plotted relationships between drainage area and slope.__

*positive***Figure 1**: Landscape process regimes at which
different sediment transport and channel initiation processes operate.
A = drainage basin area; S = local slope at the channel head. Note the
forms of the positive linear relationship for seepage erosion (marked 'saturation
threshold'),
the negative linear relationship for overland flow erosion,
and the negative non-linear relationship for channel heads formed by shallow
landsliding. *Source*:
Montgomery and Dietrich (1994)