Lava Flow Update Steps

[1] Calculate the reference eruption width $W_{e}$ based on the characteristic magmatic crust height $H_{mc(k)}$ for drainage basin $k$ using equation eq:eruption-width.

[2] Calculate minimum x-coordinate of the eruption domain $x_{e,min}$ using equation eq:x-eruption-min.

[3] Forecast the y-coordinate $y_{e,fc}$ of the eruption by interpolating the y-coordinate of the topography grid $(x_{topo}, y_{topo})$ at $x_{e,fc}$, which is calculated from eq:x-eruption-fc using the reference eruption width $W_{e}$ and the minimum x-coordinate of the eruption domain $x_{e,min}$.

[4] Calculate the characteristic volume per flow $V_{flow}^{char}$ from equation eq:char-flow-volume that uses the forecasted y-coordinate of the eruption $y_{e,fc}$ to determine subaerial or submarine conditions.

[5] Calculate the number of flows per model time step $N_{flow}$ from eq:num-flows using the characteristic volume per flow $V_{flow}^{char}$ and the total volume of volcanic material extruded from the drainage basin $V_{vol(k)}$ that was calculated during melt extraction.

[6] Calculate the volume of volcanic material per flow $V_{flow}$ from equation eq:flow-volume using the number of flows per model time step $N_{flow}$ and the total volume of volcanic material extruded from the drainage basin $V_{vol(k)}$.

[7] Define the initial y-coordinate of topography grid $y_{topo, init}$ by setting it equal to the current y-coordinate of the topography grid $y_{topo}$ (this will be used when the compaction section is added to the text).

[8] Initialize the integrated lava flow thickness $H_{flow,tot}$ to zero.

[9] Run Lava Flow Loop.

[10] Apply compaction correction to the y-coordinate of topography grid $y_{topo}$ using an approach similar to that used to account for sediment compaction during the sediment transport model but with the thickness of newly deposited lava flows $H_{flow,tot}$ replacing the sediment thickness. Pre-existing sedimentary and sticky markers are adjusted to account for compaction. The initial y-coordinate of the topography grid $y_{topo,init}$ is used to calculate the compaction correction.

Lava Flow Loop

For Each Lava Flow Do:

[1] Calculate the x-coordinate of the eruption site $x_{e}$ for the current flow as a normally distributed random variable using equations eq:random-eruption-location and eq:eruption-location-constraints, the minimum x-coordinate of the eruption domain $x_{e,min}$ and the eruption width $W_{e}$.

[2] Determine residual flow thickness $H_{res}$ from equation eq:residualflowthickness using the forecasted y-coordinate of the eruption $y_{e,fc}$ and the y-coordinate of sea level $y_{sl}$ to determine subaerial or submarine conditions.

[3] Decimate the topography grid $(x_{topo}, y_{topo})$ to obtain a grid with a lower resolution $(x_{topo}', y_{topo}')$ that will be used for lava flow calculations using a cellular automata approach.

[4] Calculate a the decimated grid spacing $\Delta x'$.

[5] Calculate the number of pulses per flow $N_{pulse}$ from equation eq:num_pulses using the flow volume $V_{flow}$, the decimated grid spacing $\Delta x'$, and the residual flow thickness $H_{res}$.

[6] Calculate pulse thickness, $H_{pulse}$, from equation eq:pulse_thickness using the flow volume $V_{flow}$, the number of pulses per flow $N_{pulse}$, and the decimated grid spacing $\Delta x'$.

[7] Initialize lava flow thickness $H_{flow}'$ to zero on the decimated grid.

[8] Run Lava Flow Pulse Loop.

[9] Calculate the updated thickness of the lava flow from multiple pulses on main grid $H_{flow}$ by interpolating $H_{flow}'$.

[10] Update the y-coordinate of topography $y_{topo}$ with flow thickness from equation eq:topo-flow-update using $H_{flow}$ after copying $y_{topo}$ to $y_{topo}^o$.

[11] Update total laval flow thickness integrating multiple flow events, $H_{flow,tot}$, using the following equation:

eq:total-laval-flow-thickness

\[ H_{flow,tot} = H_{flow,tot}^o + H_{flow}\]

where $H_{flow,tot}^o$ is the previous total lava flow thickness.

Lava Flow Pulse Loop

For Each Lava Pulse $p$ Do:

[1] Calculate the index of the eruption node $i_e$ on the decimated grid from eq:eruption-index using the x-coordinate of the eruption $x_{e}$ and the decimated grid spacing $\Delta x'$.

[2] Sort indices of the decimated grid $(x_{topo}', y_{topo}')$ based on distance from the eruption node $i_e$.

[3] Add the pulse thickness $H_{pulse}$ to the flow thickness grid $H_{flow}'$ at the eruption node $i_e$ using equation eq:initialize-pulse-thickness.

[4] Run Cellular Automata Lava Flow Update Steps.

Cellular Automata Lava Flow Update Steps

While Maximum Thickness Difference $>$ Tolerance Do for Each Node $i$:

• Update lava flow thickness $H_{flow}'$ using cellular automata approach

whereby the flow thickness for each cell is updated starting from node $i_e$ and radiating outward by allowing lava to flow from each cell with higher relative elevation to adjacent cells with the lower relative elevation.