Cell Averaged Bed Elevations and Bed Change

The average bed elevation and bed change over the wet and dry portions of the cell are

1)	$\begin{array}{l}\displaystyle z_{b}=\frac{1}{A}\sum _{i}^{M}a_{i}z_{bi}\end{array}$

2)	$\begin{array}{l}\displaystyle \Delta z_{b}=\frac{1}{A}\sum _{i}^{M}a_{i}\Delta z_{bi}\end{array}$

in which

$\begin{array}{l}i\end{array}$ : subscript indicating sediment subregion

$\begin{array}{l}A\end{array}$ : total cell horizontal area [L²]

$\begin{array}{l}ϕ_i^W = 1- ϕ_i^D\end{array}$ : wetted fraction of subarea i (between 0 and 1) [-]

$\begin{array}{l}ϕ_i^D\end{array}$ = dry fraction of subarea of subarea i (between 0 and 1) [-]

$\begin{array}{l}a_{i}\end{array}$ = area corresponding to subarea i [L²]

The average wet bed change $\begin{array}{l}\Delta z_b^W\end{array}$ is used for the bed change correction to sediment concentrations. $\begin{array}{l}\Delta z_b^W\end{array}$ is also used to compute an average bed change rate for the bed slope term. The average dry bed change $\begin{array}{l}\Delta z_b^D\end{array}$ is not used in any calculations except the average bed change rate which is a model output.

The cell averaged bed elevation and bed change are given by

3)	$\begin{array}{l}\displaystyle z_b^W = \frac{1}{A^W}\sum _i a_i^W z_{bi}^W\end{array}$

4)	$\begin{array}{l}\displaystyle z_b^D = \frac{1}{A^D}\sum_i a_i^D z_{bi}^D\end{array}$

5)	$\begin{array}{l}\displaystyle \Delta z_b^W = \frac{1}{A^W} \sum_i a_i^W \Delta z_{bi}^W\end{array}$

6)	$\begin{array}{l}\displaystyle \Delta z_b^D = \frac{1}{A^D} \sum_i a_i^D \Delta z_{bi}^D\end{array}$

in which

$\begin{array}{l}i\end{array}$ : subscript indicating sediment subregion

$\begin{array}{l}A=\sum _{i}^{M}a_{i}\end{array}$ = cell area [L²]

$\begin{aligned} a_{i}^{W}&=\phi _{i}^{W}a_{i} \\ a_{i}^{D}&=\phi _{i}^{D}a_{i} \end{aligned}$

$\begin{array}{l}a_{i}\end{array}$ : subregion i area [L²]

$\begin{array}{l}{A_{}}^{W}\end{array}$ : wetted area determined from high-resolution elevate

$\begin{array}{l}A^D = A−A^W\end{array}$ : dry area of a cell [L²]

$\begin{array}{l}ϕ_i^W = 1 - \phi_i^D\end{array}$ : wetted fraction of subarea i (between 0 and 1) [-]

$\begin{array}{l}ϕ_i^D\end{array}$ : dry fraction of subarea of subarea i (between 0 and 1) [-]