Sediment Transport Update Steps

[1] Copy the topography grid $(x_{topo,i}, y_{topo,i})$ to $(x_{topo,i}, y_{topo,i}^o)$ to define the initial topography grid.

[2] Calculate the initial thickness of sediments and volcanics $H_{sed,i}^o$ on the topography grid by searching for the shallowest y-coordinates $y_{sed,shallow,i}$ and deepest y-coordinates $y_{sed,deep,i}$ of sediment and volcanic markers at each x-coordinate $x_{topo,i}$ of the topography grid. A low-pass filter is applied to remove high-frequency noise:

eq:low-pass-filter

\[ y_i^{LP} = \frac{ \displaystyle \sum_{k=0}^{2N_{smooth}} \left(y_{(i - N_{smooth} + k)} \right) }{2N_{smooth}+1}\]

where $y_i^{LP}$ is the $i$-th y-coordinate of either the top or bottom sediment grid grid with a low-pass filter applied and $N_{smooth}$ is the number of grid points that extend to the left and right from the current grid point $i$ that will be included in the running average. Initial sediment thickness is calculated as $H_{sed,i}^o = y_{sed,deep,i}^{LP} - y_{sed,shallow,i}^{LP}$ .

[3] Classify the shape of the topography at each x-coordinate $x_{topo,i}$ of the topography grid. The shape of the topography is classified as either a sloping segment, a flat segment, local minimum or local maximum.

[4] Define drainage divides $x_{divides}$ on the topography grid at each node classified as a local minimum, local maximum or a flat segment.

[5] Calculate downstream distances $D_{stream,i}$ on the topography grid as the distances between drainage divides located to the left and right of each topography grid node:

eq:downstream-distances

\[ D_{stream,i} = x_{divides,right} - x_{divides,left}\]

[6] Calculate water depth $W_i$ on the topography grid $(x_{topo,i}, y_{topo,i}')$ using the following equation:

eq:water-depth

\[ W_i = y_{topo,i}^o - y_{sl} \text{.}\]

where $y_{sl}$ is the y-coordinate of the seal level.

[7] Run Sediment Transport Time Loop.

[8] Set y-coordinate of topography grid $y_{topo,i}$ equal to $y_{topo,new,i}$.

Sediment Transport Time Loop

For each sediment transport time step do:

[1] Calculate transport diffusivity $\kappa_{s,i}$ using equation eq:sediment-transport-diffusivity with water depth $W_i$ and downstream distance $D_{stream,i}$.

[2] Calculate the y-coordinate of topography grid after sediment transport $y_{topo,i}^{trans}$ by solving equation eq:sediment-transport with $\kappa_{s,i}$ and $y_{topo,i}^o$ as the initial condition.

[3] Calculate the shallowest y-coordinates $y_{sticky,shallow,i}$ and deepest y-coordinates $y_{sticky,deep,i}$ of sticky markers at each x-coordinate $x_{topo,i}$ of the topography grid, and apply Eq. to remove high-frequency noise.

[4] Calculate the thickness of the sticky layer $H_{sticky,i}$ on the topography grid using $H_{sticky,i} = y_{sticky,shallow,i} - y_{sticky,deep,i}$.

[5] Calculate the newly deposited sediment thickness at maximum compaction state $\Delta H_{sed,i}^o$ using equation eq:depo-thickness-max-compaction with $y_{topo,i}^o$ and $y_{topo,i}^{trans}$.

[6] De-compact newly deposited compacted sediment thickness $\Delta H_{sed,i}^o$ to obtain the newly deposited sediment thickness $\Delta H_{sed,i}^f$ by applying the conservation of mass and the porosity-depth relationship from equation eq:sediment-porosity.

[7] Calculate the compacted initial sediment thickness $H_{sed,i}^f$ at a submud-depth $\Delta H_{sed,i}^f$. To accomplish this, the sedimentary basin is discretized into vertical 1D meshes, and each 1D mesh is divided into 20 cells. Average properties for equation eq:sediment-porosity including $\phi_o$, $\lambda_{comp}$ and $y_{submud}^{max}$ are calculated for each cell using all markers contained within a given cell. For each 1D mesh cells are compacted using the conservation of mass and the porosity-depth relationship from equation eq:sediment-porosity starting with the top cell and moving downward. Cell compaction only occurs if the new submud depth of the bottom of the cell exceeds the average maximum submud depth encountered by all markers within the cell. $H_{sed,i}^f$ is calculated by summing the compacted thickness of all cells associated with 1D meshes at each topography grid node.

[8] Calculate compaction displacement for cell of 1D compaction meshes by summing thickness changes for all cells below a given node. Compaction displacement for each marker $\Delta y_{comp,m}$ is calculated by mapping compaction displacement from 1D meshes to markers assuming a linear change in displacement within the cell.

[9] Calculate the top of the y-coordinates of uncompacted sedimentary basin $Y_{topo,precomp,i}$ at topography grid nodes by searching for the shallowest sedimentary markers at each x-coordinate $x_{topo,i}$ and applying eq:low-pass-filter to remove high-frequency noise.

[10] Update y-coordinates of sediment markers $y_m$ due to compaction displacement after copying $y_m$ to $y_m^o$ using the following equation: \begin{equation} ym = ym^o + \Delta y_{comp,m} \quad \text{if composition of marker $m$ is sediment} \end{equation}

[11] Calculate the top of the y-coordinates of compacted sedimentary basin $y_{topo,postcomp,i}$ at topography grid nodes by searching for the shallowest sedimentary markers at each x-coordinate $x_{topo,i}$ and applying Eq. to remove high-frequency noise.

[12] Calculate the compaction displacement at the sediment surface $\Delta y_{comp,i}$ at each node of the topography grid using the following equation: \begin{equation} \Delta y{comp,i} = y{topo,postcomp,i} - y_{topo,precomp,i} \end{equation}

[13] Adjust sticky air and water markers assuming a linear change in the sticky layer with zero displacement at the top of the model domain and a maximum displacement at the top of the sediments equal to $\Delta y_{comp,i}$.

[14] Calculate the new y-coordinate of the topography grid $y_{topo,new,i}$ taking into account compaction using equation eq:sediment-compaction-correction with $H_{sed,i}^o$, $\Delta H_{sed,i}^o$, $H_{sed,i}^f$ and $\Delta H_{sed,i}^f$.

[15] Set $y_{topo,i}^{o}$ equal to $y_{topo,new,i}$.

[16] Calculate downstream distances $D_{stream,i}$ using updated y-coordinates of topography grid $y_{topo,new,i}$.

[17] Calculate water depth $W_i$ on the new topography grid using updated y-coordinates of topography grid $y_{topo,new,i}$.