ABSTRACT
In the latter part of the past century, geomorphic thought was first dominated by the closed-system, cycle-of-erosion model of W. M. Davis. About 25 years ago, the popularity of the Davisian approach was largely replaced by an open-system, dynamic-equilibrium model. An alternative model, that of allometric change, was first described explicitly for geomorphic purposes by W. B. Bull, although the technique had been used previously by many researchers.
Allometric analysis is the development of simple or multiple power-function equations that express the relative rates of change among the variables of a system. A principal geomorphic utility of the method is to show adjustment between two variables. For many geomorphic systems, however, and for fluvial systems in particular, an allometric relation unaffected by other variables rarely can be identified. Unless the effects of complicating variables are held constant and thereby eliminated from consideration, a bivariant allometric relation is likely to be in error. To avoid this difficulty and to provide for the development of reliable multivariant power functions, a simple modification to the allometric-change model is advocated for many studies of fluvial systems. This modification is the determination of a fixed or constant exponent for a bivariant power-function relation by holding the effects of other variables constant. Having established an exponent, that value is imposed on subsequent power-function equations, whether bivariant or multivariant. The coefficient must be evaluated accordingly. The technique seems especially applicable to fluvial systems owing to their complexity and lack of distinction between dependent and independent variables.
The determination of fixed exponents for width-discharge and gradient-discharge relations of alluvial stream channels serves to illustrate the use, advantages, and limitations of the invariant power-function technique. Among the advantages suggested by the examples are:
The method results in increased accuracy and sophistication of the adjustment between two variables for empirical studies.
When employing multiple regression (or a similar curve-fitting technique), a specified exponent for an independent variable avoids error that would otherwise be inherent in the computation owing to non-linear effects by other independent variables.
Conflict caused by defining separate regional relations between two variables is eliminated.
Preestablished exponents, based on numerous data, provide a measure of safety when relating and extrapolating very limited data.
34Invariant power functions provide a uniformity that permits the comparison of results within a study, or with other studies.
The method helps focus attention on geomorphic and hydrologic processes, whereas free bivariant analysis ignores process.
Limitations of the invariant power-function technique include its largely empirical approach and the assumption that a power function adequately describes the relation between two variables.
