| Natural rivers exhibit power-functional variability in their width, depth, and velocity
with flow discharge (Leopold and Maddock, 1953). This relation named hydraulic
geometry has been empirically supported by many field studies across the world
(e.g., Leopold et al., 1964; Stall and Fok, 1968). The relationship appears either at a
fixed cross-section, showing temporal variability, or along a downstream direction
across an entire basin, showing spatial variability, the latter named downstream or
basin hydraulic geometry. Theoretical studies that attempt to explain the power-law
phenomenon (fractal), have assumed that the watershed is homogeneous hydrologically and
geologically.
Nevertheless, real watersheds are often subject to spatially heterogeneous conditions, due
to various reasons including partial area storm coverage (Sólyom and Tucker, 2004) and
transmission losses on bed and banks (Lane et al., 1997). In this setting, hydraulic geometry
relationships are likely to deviate from monotonic power-law relationship and to follow
rather more complex multi-fractal characteristics. In fact, deviation from single
power-law was reported for at-a-station relationship of midwest rivers in US (Dodov
and Foufoula-Georgiou, 2004). In the case of downstream variation, we identify
significant multi-fractal characteristics over the Colorado River basin where strong
heterogeneity in geological and hydrological settings presents. Conventional power-law
hydraulic geometry relationships cannot express the functional variability for these
cases.
Motivated by this fact, we generalize the hydraulic geometry functional formulation in
this study to express multi-fractal relationships. To do so, we couple the formulation of Paik
and Kumar (2004), which generalized at-a-station and downstream relationships, with the
formulation of Dodov and Foufoula-Georgiou (2004) which was proposed for multi-scaling
in at-a-station relationship. The proposed formulation is successfully evaluated with the case
of Colorado River basin. This study has potential to broaden our perspective on hydraulic
geometry.
Keywords: Hydraulic geometry; Multi-fractal; Heterogeneity; Colorado River
References
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