Theoretical basis for at-many-stations hydraulic geometry

At-many-stations hydraulic geometry (AMHG) is a recently discovered set of geomorphic relationships showing that the empirical parameters of at-a-station hydraulic geometry (AHG) are functionally related along a river. This empirical conclusion seemingly refutes previous decades of research defining AHG as spatially independent and site specific. Furthermore, AMHG was the centerpiece of an unprecedented recent methodology that successfully estimated river discharge solely from satellite imagery. Despite these important implications, AMHG has remained an empirical phenomenon without theoretical explanation. Here we provide the mathematical basis for AMHG, showing that it arises when independent AHG curves within a reach intersect near the same values of discharge and width, depth, or velocity. The strength of observed AMHG is determined by the degree of this convergence. Finally, we show that AMHG enables discharge estimation by defining a set of possible estimated discharges that often match true discharges and propose its future interpretation as a fluvial index.