3.3 Predicting the distribution parameters.
The parameters values fitted for each distribution are a function of\(d_{b}^{*}\) (Figure \(5\)). For the GAM, \(\alpha\) decreases and\(\beta\) increases with higher \(d_{b}^{*}\) (Figure \(5\)A) as the
distribution shape shifts from unimodal to monotonically decreasing for\(d_{b}^{*}>\sim 3\) . For LN, both \(\mu\) and \(\sigma\) increase
with \(d_{b}^{*}\) as the thickness of the tail increases to describe
longer \(t^{*}\) at higher \(d_{b}^{*}\) values (Figure \(5\)B). For FR,\(\beta\) increases and \(\mu\) decreases (Figure \(5\)C), as the
distribution flattens and its peak shifts towards longer \(t^{*}\) and,
lastly, for EXP, \(\rho\) decreases with \(d_{b}^{*}\). For all
distributions, the parameters attain constant values for high\(d_{b}^{*}\), reflecting the convergence to the RTD for a
quasi-infinite streambed. The regression formula for each of these
parameters (Table 3) capture a very large fraction of the variance in
the parameter values with dimensionless depth (\(R^{2}\geq 99.8\%\)with small Root Mean Square Error in all cases). These formulae are
applicable over the whole range \(d_{b}^{*}\) considered in this study
(from very shallow to quasi-infinite \(d_{b}^{*}\)), and hence provide a
useful tool for future modelling studies.