RIDGE CHANNEL PATH INTERDEPENDENCE IN DRAINAGE BASINS

RIDGE/CHANNEL PATH INTERDEPENDENCE IN DRAINAGE BASINS BY CHRISTIAN WERNER University of California at Irvine, School of Social Sciences, Irvine, California, USA. Werner. C. I993: Ridge/channel path interdependence in drainage basins. Geogr. Ann. 75 A (3): 73-8]. ABSTRACT. It is readily apparent from both map and ﬁeld observation that the patterns of channels and ridges in ﬂu- vially eroded terrain are closely intertwined. The purpose of this paper is the formulation of some of the principles gov- erning their relationship. Speciﬁcally. the paper models the sequence and length distribution of ridge paths on the basis of the sequence and length distribution of channel paths within a given drainage basin. The modeling approach first codiﬁes the channel and ridge networks within drainage basins in graph theoretical terms; it then takes advantage of the fact that ridge lines tend to run quasi-parallel to neighboring channels thus permitting the estimation of the former from the latter. Data were ob- tained from two drainage basins of the (environmentally homogeneous) Cumberland Plateau of Kentucky and the (environmentally complex) San Gabriel Mountains of south- ern California. Comparison of the observed length values of ridge paths with their values as derived from channel path data de- monstrate the intimate relation between channels and ridges. thereby providing support for the model as a formal description of the ridge-channel interdependence. Key words: Channel networks: ridge networks: Network geometry: graph theory. Introduction and outline Whether observed on the map or in the ﬁeld, ﬂu- vially eroded terrains tend to display a close and systematic interplay of channels and ridges. It is the purpose of this paper to substantiate this im- pression through the measurement and compari- son of selected ridge and channel parameters fol- lowed by the formulation and testing of hypo- theses about their interdependence. In view of the conspicuous. wide spread. and seemingly repetitive and uniform interplay of channel and ridge patterns in maturely eroded areas the scarcity of research investigating their in- terrelation is puzzling. The first formal and rigor- ous conceptualization of the ridge/channel in- terplay seems to be the work by Warntz (1966, 1975) and Warntz and Woldenberg (1967) which was based on ideas published in the previous cen- Geografiska Annaler~ 75 A (1993) - 3 tury (Caley 1859; Maxwell 1870). But although its level of rigor and abstraction is impressive there has been little subsequent research building on the concepts of the Warntz/Woldenberg formalism be- cause their application leads to serious theoretical as well as practical difficulties (Mark 1979). An alternative approach. based on the concept of interlocking networks, has led to the formula- tion of several mathematical relations between channel and ridge nodes. links, paths and subnet- works (Werner l982, I988. 1992). However, they refer to network topology only, and their implica- tions for the spatial layout of ridge and channel patterns—if any—are uncertain. Finally, Ab- raham‘s (1980) study of the relation between chan- nel link lengths and the angles formed by adjacent divide lines successfully investigates underlying geometric dependencies but does so only at the local level of individual channel links and bifurca- tions; the associated issue of ridge/channel rela- tions at the level of entire networks is not adres- sed. In this study both spatial and graph theoretic in- terdependencies will be investigated; specifically, we will define and examine the sequential order and length distribution of channel and ridge paths in drainage basins where length is defined in both geometric and topologic terms: As a topologic par- ameter length is measured by the number of links of the path in question, and as a geometric para- meter it is measured in meters or some other unit of length in R2. In concrete operational terms the input information we use consists of the number. the sequence and the lengths of all channel paths of selected natural drainage basins, with particular identification of those paths that drain the areas along the basin boundary. From this information estimates of the number. sequence and lengths of the ridge paths in the basin are derived (Figure 2). The last sections of the paper compare estimated and observed data of the length distribution of ridge paths (Figures 5-7) and close with some 73