## 2 Materials and Methods
### Reach-Scale Metabolism Estimates  
Reach-scale metabolism rates were estimated based on changes in dissolved oxygen (O\({}_{2}\)) through time with a two-station method [46; 47]. The two-station method was appropriate due to expected low reeration coefficients (\(K\)) and a short distance to source water which can confound dissolved gas flux estimates if experimental systems are not treated discretely [48; 49]. We used 2 HOBO(r) (U26-001, Onsetr Computer Corporation, Bourne, ME, USA) loggers placed in the middle of each ditch at upstream and downstream ends to measure dissolved O\({}_{2}\) and temperature at 15 min intervals during each 24 h nutrient loading event. A Bayesian inverse modeling procedure was used to estimate GPP, ER and \(K\) from dissolved O\({}_{2}\) data using the following equation [50]: \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{2}{c}{**2 h HRT**} & \multicolumn{2}{c}{**4 h HRT**} & \multicolumn{2}{c}{**6 h HRT**} \\ \hline
**Stream Fluxes (g)** & **Veg** & **UnVeg** & **Veg** & **UnVeg** & **Veg** & **UnVeg** \\ \hline NO\({}_{3}\)-N load & 15.5 \(\pm\) 0.6 & 15.5 \(\pm\) 0.6 & 15.5 \(\pm\) 0.6 & 15.5 \(\pm\) 0.6 & 15.5 \(\pm\) 0.6 & 15.5 \(\pm\) 0.6 \\ NO\({}_{3}\)-N net uptake & 8.4 \(\pm\) 0.9 & 4.2 \(\pm\) 0.6 & 4.8 \(\pm\) 0.4 & 2.2 \(\pm\) 0.3 & 3.9 \(\pm\) 0.4 & 1.9 \(\pm\) 0.4 \\ Denitrification & 1.3 \(\pm\) 0.3 & 0.7 \(\pm\) 0.0 & 2.0 \(\pm\) 0.8 & 1.0 \(\pm\) 0.1 & 1.1 \(\pm\) 0.2 & 0.8 \(\pm\) 0.2 \\ \% N removal & 63\% & 32\% & 44\% & 21\% & 32\% & 17\% \\ PO\({}_{4}\)-P load & 9.9 \(\pm\) 0.4 & 9.9 \(\pm\) 0.4 & 9.9 \(\pm\) 0.4 & 9.9 \(\pm\) 0.4 & 9.9 \(\pm\) 0.4 \\ PO\({}_{4}\)-P net uptake & 1.6 \(\pm\) 0.3 & 1.4 \(\pm\) 0.2 & 1.4 \(\pm\) 0.2 & 0.9 \(\pm\) 0.3 & 1.3 \(\pm\) 0.2 & 0.8 \(\pm\) 0.2 \\ \% P removal & 16\% & 14\% & 14\% & 10\% & 13\% & 8\% \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Estimated total 24 h loads, net nutrient uptake for nitrate and soluble reactive phosphorus, and denitrification by vegetation and HRT treatment. \[O_{2\text{down}(t+\tau)}=\frac{O_{2\text{up}(t)}\ +\ \left(\ \frac{\text{GPP}}{z}\times\sum_{t}^{t+\tau} \frac{\text{PPFD}}{\text{PPFD}_{\text{total}}}\right)\ +\ \frac{\text{ER}}{z}\tau\ +\ K\tau\frac{(O_{2\text{, satup}(t)}-O_{2\text{up}(t)}+O_{2\text{, satdown}(t+\tau)})}{2} \tag{2}\]  
where O\({}_{2\text{up}(0)}\) is the upstream O\({}_{2}\) concentration (g O\({}_{2}\)/m\({}^{3}\)) and O\({}_{2\text{down}(t+\tau)}\) is the downstream O\({}_{2}\) concentration of that same parcel of water following travel time, \(\tau\). GPP and ER are expressed in g O\({}_{2}\) m\({}^{-2}\) d\({}^{-1}\) and represent positive and negative rates of O\({}_{2}\) production and consumption, respectively. \(z\) is mean depth (m) calculated as stream volume (m\({}^{3}\)) divided by surface area (m\({}^{2}\)), O\({}_{2\text{,satup}}\) and O\({}_{2\text{,satdown}}\) are the saturation concentrations upstream and downstream (g O\({}_{2}\) m\({}^{-3}\)), \(K\) (d\({}^{-1}\)) is the gas exchange rate, and PPFD is the sum of the photosynthetic photon flux density (\(\mu\)mol m\({}^{-2}\) s\({}^{-1}\)) accumulated in the time interval from \(t\) to (\(t+\tau\)) divided by the daily total of PPFD (PPFD\({}_{\text{total}}\)) for any given parcel of water. We used the function metrop in the _mcmc_ package [51] in R to sample the posterior probability distribution P (\(\mu\) l D). Each chain was run for 20,000 iterations following a 1000 iteration burn-in, with no chain thinning, and probability distributions adjusted to achieve an acceptance rate of \(\sim\)20%. We used minimally informative prior probability distributions for GPP, ER, and \(K\) (GPP \(\sim\) N [\(\mu\) = 5, SD = 10]; ER \(\sim\) N \(\mu\) = -5, SD = 10]; \(K\) (\(\mu\) = 5, SD = 5]).