Statistical derivation of downstream hydraulic geometry equations

Hydraulic geometry (HG) describes how width, depth and velocity vary with changing water flow in natural rivers. These relationships or ratings are modelled by simple power laws and apply at a river cross-section or at-a-station (AHG), along a river at bankfull flow (DHG) and between cross-sections (AMHG). Numerous theoretical attempts have been made to derive the relationships, but none are fully accepted. We presented a completely new statistical approach and show that coefficients and exponents in power law models of DHG can be determined using statistical randomization. For an entire river as opposed to a reach, we derive DHG equations and our results show the linkage between AHG, DHG and AMHG and that ordering of coefficient and exponent values with coefficients increasing and exponents decreasing in value along a river is a prerequisite for DHG and AMHG to occur. We introduce a cross-section shape factor as a component of HG.