Theoretical Basis for at-many-stations Hydraulic Geometry (AMHG)

Abstract:

At-many-stations hydraulic geometry (AMHG) is a recently discovered set of geomorphic relationships showing that empirical parameters of at-a-station hydraulic geometry (AHG: w=aQ^b, d=cQ^f, v=kQ^m) are functionally related along a river. This simple empirical conclusion seemingly refutes previous decades of research defining AHG as spatially independent and derived from cross sectional geometry. AMHG also enabled a recent methodology that successfully estimated river discharge solely from satellite imagery [Gleason and Smith, 2014]. Despite these important implications, AMHG has remained an empirical phenomenon without theoretical explanation. Here, we derive AMHG, showing that it arises when spatially independent AHG curves within a reach intersect at or near the same values of discharge and width, depth, or velocity. Thus, AMHG is shown to be a mathematical construct, and the strength (linearity) of AMHG is determined by the degree of this convergence: how tightly these rating curves cluster about their intersection point determines how linear the resultant AMHG equation will be. We thus suggest future interpretation of AMHG as an index of geometric variability (as rating curves will not cross if AHG exponents similar and rating curves are parallel) and hydraulic similarity. We also show how AMHG enables remotely sensed discharge estimation by defining the convergence point of rating curves in hydraulic space, thus generating a set of possible inverted discharge values that usually agrees with the range of true discharge when truncated by minimum and maximum imposed discharges and observed widths. Finally, we show that Gleason and Smith’s [2014] previously proposed remotely sensed proxy for AMHG is coincidental and should not be used, and we propose alternatives for remote sensing of discharge without in situ data. We conclude that AMHG discharge estimation remains robust, and that the hydrologic and hydraulic drivers of AMHG are fertile ground for further study.