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At-many-stations hydraulic geometry (AMHG), while useful for estimating river discharge from satellite data, remains empirical and has yet to be reconciled with the at-a-station hydraulic geometry (AHG) from which it was originally derived. Here we present evidence, using United States Geological Survey field measurements of channel hydraulics for 155 rivers, that AMHG can be hydraulically and geomorphically reconciled with AHG. Our results indicate that AMHG is rightly understood as an expression of a river-wide model of hydraulics driven by changes in slope imposed upon AHG physics. The explanatory power of AHG and this river-wide model combine to determine whether AMHG exists: if both AHG and the river-wide model adequately describe hydraulics, then we show that AMHG is a necessary mathematical consequence of these two phenomena. We also orient these findings in the context of river discharge estimation and other applications.

Plain Language Summary Hydraulic geometry (HG) is an empirical phenomenon that predicts river width, depth, and velocity given river discharge and is fundamental to our ability to predict floods, river habitats, and water availability for human and ecosystem use. Recently, a new form of HG was discovered (at-many-stations HG, or AMHG), and it has been successfully deployed in a range of applications and proven to exist in a wide variety of rivers. However, the novel phenomenon of AMHG remains empirical with puzzling and seemingly contradictory links to other, traditional forms of HG, which are well understood and have been studied since the 1950s. AMHG is also not manifested in all rivers, adding to confusion as to its origin. Here, we show for the first time that we can reconcile AMHG with all other traditional variants of HG and gain a complete understanding of how and when AMHG occurs in rivers. This puts the most puzzling aspects of AMHG-why is it observed in some rivers but not others, and what causes the phenomenon-to rest. We have fundamentally changed the conception of AMHG and suggest this work as a basis for all future AMHG research.