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\documentclass[draft,grl]{agutexSI2018}

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\newcommand{\sscd}{$\Delta C$}
\newcommand{\qs}{$Q_s$}
\newcommand{\abssscd}{$\lvert\Delta C\rvert$}
\newcommand{\meanssc}{$\left<C\right>$}
\newcommand{\sscdws}{$\Delta C_{ws}$}
\newcommand{\2}{$^{-2}$}
\newcommand{\1}{$^{-1}$}
\newcommand{\3}{$^{-3}$}

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\begin{document}
\title{Supporting Information for ``Simple metrics predict salt-marsh sediment fluxes''}
\authors{D.J.\ Nowacki\affil{1}, N.K.\ Ganju\affil{2}}

\affiliation{1}{U.S.\ Geological Survey, Santa Cruz, CA 95060 USA}
\affiliation{2}{U.S.\ Geological Survey, Woods Hole, MA 02543 USA}


\begin{article}

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\noindent\textbf{Contents of this file}

\begin{enumerate}
\item Text S1
\item Figures S1 to S5
\item Table S1
\end{enumerate}

\noindent\textbf{Additional Supporting Information (Files uploaded separately)}
\begin{enumerate}
\item Table S2
\end{enumerate}

\noindent\textbf{Introduction}
The supporting material includes Text S1, which describes an analysis of estimated sediment-flux errors, as well as five figures and two tables.
\cref{fig:sscd_error} includes error analysis as described in Text S1.
\cref{fig:flux_bias} shows relationships between \sscd\ and \qs\ with different scaling of \qs\ on flood and ebb, as described in Text S1.
\cref{fig:calib} shows the turbidity--SSC calibration for Grand Bay.
\cref{fig:sscdiff_others,fig:sscdiff_mean_others} are similar to Figures 3 and 4 in the main text, but with additional data points from the other studies described in the text.
\cref{tab:SItab} presents similar information to that of Table 1 in the main text, but with separate fits for each site. 
Table S2 contains the turbidity and SSC values used for the Grand Bay turbidity--SSC calibration.

\noindent\textbf{Text S1.}

There are nine constituent terms that comprise the sediment flux, each with an associated error: ($a$) measured velocity, ($b$) slope and ($c$) intercept of regression between measured velocity and channel mean velocity, ($d$) measured pressure, ($e$) slope and ($f$) intercept between measured pressure reading and channel area, ($g$) measured turbidity, ($h$) slope and ($i$) intercept of regression between measured turbidity and SSC. 
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The errors for ($a$), ($d$), and ($g$) were obtained from manufacturer documentation, and the regression errors were computed using standard statistical techniques. 
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In this context, the sediment flux is defined as 
\[
Q_s = (ab + c) (d e + f) (g h + i), 
\]
and the total error in \qs\ is
\[
\delta Q_s = \sqrt{\left( \frac{\partial Q_s}{\partial a} \delta a \right)^2 + \cdots + \left( \frac{\partial Q_s}{\partial i} \delta i \right)^2}, 
\]
where $\delta a \dots \delta i$ are the errors associated with the constituent terms, following standard error-propagation techniques. 
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We are interested in whether the propagated error will meaningfully influence the predictive power of \sscd.
Using the Bayou Middle data as an example, the visual relationship between \sscd\ and \qs\ is maintained  when error bars derived from the above equations are included (\cref{fig:sscd_error}).
The fractional sign agreement between \sscd\ and \qs\  for Bayou Middle assuming no error is 0.77. 
Considering variability from the error, the sign agreement ranges from 0.67 to 0.79, suggesting that error does not dominate the relationship.

Error may also be introduced if conditions in the channel do not fully reflect the total flux into or out of the wetland complex.
For example, mass exchange between creeksheds via sheet flow over the wetland surface during periods of high water could reduce the representativeness of the channel measurements. 
\cref{tab:SItab} includes an assessment of the creekshed area delineation and the likelihood for mass exchange between creeksheds.
Of the 13 sites, five were very well constrained with little to no opportunity for cross-creekshed flow.
Creekshed boundaries in this category consisted of uploads, roads, and constructed levees.
Four sites were well constrained, with uplands, natural levees, and topographic highs serving as the creekshed boundaries. 
In this category, there was more opportunity for flow across creeksheds not captured by the channel measurements.
Four sites were relatively unconstrained, and creekshed boundaries typically consisted of topographic highs of the wetland platform, which could facilitate cross-creekshed exchange during high water levels. 
There is no apparent grouping or biasing based on this assessment of creekshed delineation in the \sscd--\qs\ relationships (Figure 3 and \cref{fig:sscdiff_others}).

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If mass were preferentially delivered to the wetland via the channel on floods but drained via other pathways (potentially across creekshed boundaries) on ebb, or vice versa, the channel would not be representative of conditions in the creekshed.
To simulate this condition, we constructed two scenarios: in the first, we scaled flood fluxes by 80\%, and in the second we scaled ebb fluxes by 60\%. 
Considering the 14 d averaging interval, the scaled relationships between \sscd\ and \qs\ have poorer correlations than that of the unscaled data (\cref{fig:flux_bias}).
The two-sided $p$-value for all relationships, however, is small, suggesting that we can reject the null hypothesis that the slope between \sscd\ and \qs\ is zero, and that a meaningful relationship between these quantities exists even when scaled in this manner.
\end{article}
\clearpage

\begin{figure}[p]
\centering
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	\caption{\sscd\ versus \qs\ for Bayou Middle at the diurnal interval with estimated errors shown. }
	\label{fig:sscd_error}
\end{figure}

\begin{figure}[p]
\centering
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	\caption{\sscd\ versus \qs\ at the 14 d averaging interval for (a) no scaling, (b) a scenario where floods are scaled by 80\%, (c) a scenario where ebbs are scaled by 60\%. Symbols are the same as in Figure 3.} 
	\label{fig:flux_bias}
\end{figure}

\begin{figure}[p]
\centering
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	\caption{Turbidity--SSC calibration for Grand Bay. }
	\label{fig:calib}
\end{figure}

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\begin{figure}[p]
\centering
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	\caption{\sscd\ versus \qs\ for full deployments. ``Other studies'' are the four additional investigations from the literature described in the main text.} 
	\label{fig:sscdiff_others}
\end{figure}


\begin{figure}[p]
\centering
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	\caption{Absolute value of \sscd\ versus \meanssc\ for full deployments.
	Circled symbols indicate negative \sscd. ``Other studies'' are the four additional investigations from the literature described in the main text.} 
	\label{fig:sscdiff_mean_others}
\end{figure}




\setcounter{figure}{1}

\begin{table}[p]

\centering
\caption{Statistics for the nine sites considered in this study, and the four other studies. Creekshed area in km$^2$, slope ($m$) and standard error and intercept ($b$) for 24.8 h least-squares regression between \sscd\ and \qs, Pearson's $r$, average \meanssc, average \sscd, and an assessment of the creekshed boundaries. BH: Bayou Heron, BM: Bayou Middle, BW: Blackwater, FB: Fishing Bay, RC: Reedy Creek, DC: Dinner Creek, OG: Ogunquit, SB1: Seal Beach 1, SB2: Seal Beach 2, S77: \textit{Settlemyre et al.} 1977, W81: \textit{Ward} 1981,  D84: \textit{Dankers et al.} 1984, S99: \textit{Suk et al.} 1999.}
\begin{center}

\begin{tabular}{l ScSSSSl }
{Site} & {Area} & {$m$} & {$b$} & {$r$} & {$\left<C\right>$} & {$\Delta C$} & {Creekshed}\\ \midrule
BH & 4.51 & $0.105 \pm 0.013$ & -0.167 & 0.65 & 17.91 & 2.14 & Relatively unconstrained\\
BM & 1.52 & $0.134 \pm 0.011$ & -0.131 & 0.70 & 21.84 & -0.47 & Relatively unconstrained\\
BW & 70.00 & $0.042 \pm 0.001$ & 0.068 & 0.99 & 63.53 & -17.14 & Very well constrained\\
DC & 4.16 & $0.175 \pm 0.009$ & 0.090 & 0.82 & 16.33 & 1.83 & Relatively unconstrained\\
FB & 1.74 & $0.314 \pm 0.014$ & 1.416 & 0.94 & 38.51 & 5.06 & Well constrained\\
OG & 1.37 & $0.102 \pm 0.014$ & 0.109 & 0.69 & 3.67 & 0.55 & Very well constrained\\
RC & 0.52 & $0.270 \pm 0.018$ & 0.215 & 0.83 & 9.50 & -0.77 & Well constrained\\
SB1 & 0.46 & $0.057 \pm 0.141$ & 0.130 & 0.05 & 9.24 & 0.05 & Very well constrained\\
SB2 & 0.63 & $0.230 \pm 0.033$ & 0.577 & 0.67 & 18.09 & -1.67 & Very well constrained\\
S77 & 1.08 & $0.310\pm0.016$ & 0.329 & 0.98 & 34.19 & -3.27 & Well constrained\\
W81 & 5.00 & $0.093\pm0.010$ & 0.118 & 0.95 & 36.99 & -8.02 & Well constrained\\
D84 & 0.18 & $0.019\pm0.003$ & -0.262 & 0.84 & 257.02 & 109.71 & Very well constrained\\
S99 & 0.20 & $0.086\pm0.033$ & 0.018 & 0.71 & 13.21 & 2.30 & Relatively unconstrained\\






\end{tabular}
\end{center}
\label{tab:SItab}
\end{table}

\setcounter{figure}{2}

\begin{table}

\caption{Turbidity and SSC values used for the Grand Bay turbidity--SSC calibration.}
\end{table}
\label{tab:sscsamples}

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\end{document}
