Topologic record analysis

Within the last three decades, there has been a steady progress in understanding the diverse physical mechanisms o f runoff production from hillslopes. The long-known empirical relationships between the hydraulic geometry o f channels in a network and Horton order, lend strong support to the idea that basic connections do exist between network morphology and runoff production. Despite all this recent progress, there still remain unresolved questions pertaining to the fundamental connections between hydrologie regime (rainfall-runoff) and river network adjustment. The inability of the random model to predict the network structure in the elevation space have led Gupta and Mesa (1988) to propose that future research in quantitative network geomorphology would have to go beyond planimetric properties, if the fundamental connections between river basin hydrology and channel network geometry are to be understood.

Kirkby (1976) proposed a stochastic function known as the network width function, in his attempt to test whether actual networks conformed to Shreve’s (1966) random network model. This function, N(x), 0<x<L, denotes the number of links (channels) at a flow distance x from the basin outlet, where L is the main channel length. The

network width function was further expanded on by Troutman and Karlinger (1985) in a highly mathematical treatise. It is considered as an important measure because o f its connection with the Geomorphological Instantaneous Unit Hydrograph GIUH o f a river basin, and secondly, because o f the influence o f stream channel pattern on peak streamflow. That index which is viewed as a far superior tool to the GIUH for assessing the hydrologie behaviour of the fluvial network, essentially describes the spatial distribution o f the network, thereby allowing more information about the network to be retained (Mesa and Mifflin, 1986). It analyses the network as a family tree, describing bifurcation and other properties in terms of number of units from the outflow, or in other words in terms of distance from the outflow. The importance of the network width function is in allowing a spatial description o f the network to be employed explicitly in the synthesis o f catchment response (Wharton, 1994). The problem of quantifying the temporal variability o f the width function remains open (Gupta and Mesa, 1988).

The function has the advantages that it is easily derived from maps (Naden, 1992). She employed the network width function in developing a methodology for implementing a semi-distributed unit hydrograph approach to rainfall-runoff modeling, by separating catchment response into a hillslope response function and a network response function. More recently, Naden et al. (1999) used mapped blue line data in the form o f the network width function for a river routing model, testing it for the River Amazon basin. Furthermore, the overall shape of the network width function is not very sensitive to map scale or data quality. Under the assumptions of uniform drainage density and constant flow velocity, the network width function is equivalent to a time-area diagram.

Rodriguez-Iturbe et al. (1982) suggested that a promising avenue in studying hydrologie response of a basin may lie in the description of the network through the density of link occurrences at different altitudes. Gupta et al. (1986) introduced a different stochastic function, named the link concentration function, N(h), 0<h<H, defined as the altitudinal projection of a network, claiming it to be the missing link in the connections among climate, geomorphology and hydrologie response. The

link concentration function is quite simply the number of links at an elevation h above the outlet, where H is the total network height. They postulated that the whole network must embody a deep sense o f regularity in formal relations between its parts. This deeper regularity is contained in two dimensions in Horton laws and in many other relationships derived subsequently for the two-dimensional characterisation o f the channel network. This measure was hailed as being hydrologically significant because of the connection that it makes between network height and runoff and sediment production in a river basin: “A well defined three- dimensional structural regularity of the network probably holds the key not only for how the basin routes the effective rainfall to make up the runoff hydrograph but also for some still undiscovered unifying principles regarding the transformation of rainfall to runoff at the basin scale” (Rodriguez-Iturbe, 1993, p. 66).

The objective o f the thesis is to establish a pattern o f change in the network width function and the link concentration function for some reference network representative of the channel evolution process and then use this as a guide against changes in the particular study area of N. England. This novel approach will provide us with a robust reference frame of the particular mechanics of network growth of an evolving experimental fluvial network which in turn can be used as a guideline for the changes observed in natural catchments. No worker up to this present day has attempted to use both of these morphometric indices in tandem, as diagnostic means of drainage network evolution through time. It is therefore clear that a combination o f the network width function with the link concentration function constitutes an innovative approach which is very pertinent to the project. This will be explored in depth for the networks of upland Britain.

It is believed that this is a sound approach, since it provides a quantitative analysis of the macroscopic stability and o f fluctuations in network structure in a large number o f upland basins. The synthesis o f these two random functions should provide a tool to examine the topology of the network structure. In essence, the development of this model, would facilitate the understanding of dynamic equilibrium of channel networks through time. Stated in another way, collecting large amounts of empirical

data from a number of different catchments would address how networks adjust to natural evolution and a combination o f anthropogenic and climatic changes. It should however be emphasised that measured changes in the network structure would provide us only with an indication of the general direction and trend of change as depicted on maps. This methodology allows us to identify and quantify recent network changes in this particular environment, in a more precise manner and with a better degree o f sophistication than previously achieved.

CHAPTERS

MODEL ASSESSMENT ON AN EXPERIMENTAL