Modeling Landform Evolution through time: the cellular automata approach

Most approaches to landscape evolution modeling either solve directly for the effects of water discharge by solving hydrodynamic equations, or use contributing area as a proxy for discharge (e.g., Ahnert, 1987; Willgoose et al., 1991; Howard 1994). An alternative approach is the rule-based Cellular Automata (CA) Algorithm that is simple to implement but still maintain a close analogy between model structure and physical system being modeled. The model presented here is a CA model and is implemented using Java technology. We chose Java applet because it is designed to run on different computer hardware and operating systems and we want the model to have the widest possible accessibility. The CA algorithm simulates first order processes associated with fluvial erosion by randomly dropping rainfall events, termed precipitons, onto an initial topographic grid and by using simple local rules and a few parameters to control the subsequent movement and behavior of the precipitons.

The CA algorithm works as follows: (1) A precipiton is dropped randomly onto a cell of the topographic grid (e.g., in cell 1 in Figure 3); (2) Diffusion, which simulates mass wasting and soil creep, is carried out by downhill transport between the target cell and its four nearest neighbors (e.g., cells 3, 5, 7 and 9 in Figure 3). The amount of diffusion is proportional to local slope and the material is eroded from a higher cell and deposited in a lower cell (e.g., from cells 5, 7 and 9 to 1 and from cell 1 to 3 in Figure 3); (3) The precipiton continues to move to the lowest of the 8 neighboring cells (e.g., from cell 1 to cell 2 in Figure 3). Erosion occurs if the parameterized carrying capacity of the storm is not exceeded. The magnitude of erosion is proportional to the amount of rainfall, the local slope, and the erodibility of the material at current cell (bedrock and sediment have different erodibility). (4) If the carrying capacity is exceeded, deposition occurs. (5) A precipiton keeps moving from current cell to the lowest of the 8 surrounding cells, until the sediment carrying capacity of the precipiton is exceeded, the precipiton reaches the edge of the grid, or the precipiton lands in a pit. At that time, the current iteration stops and a new iteration starts, i.e., the program loops back to step (1).

Figure 3
Schematic Diagram showing how CA model works. A precipiton falling on cell 1 will cause local diffusion at its 4 direct neighboring cells (#3, #5, #7 and #9) and move to the lowest of its 8 neighboring cells (#2). Material will be eroded from cell #1and deposited in cell #2. The precipiton will continue to move to the lowest neighboring cell and erode and deposit material along the way until it reaches the edge of the grid, land in a pit or its carrying capacity is exceeded. Deposition of carried material occurs when carrying capacity is exceeded.

The rules described above are in a sense analogous to the natural processes. For example, precipitons move to lower elevations, simulating water running downhill; the amount of erosion is proportional to the local slope and to the erodibility of the rock, simulating speedier erosion of steeper slope and less erosion of hard rocks or vegetation protected areas. The advantage of this approach is that it is simple and yet can still produce realistic first-order geomorphologic features and provide insights into landform evolution processes. This approach is similar to studying heat transport or mechanics in physics by dealing with large-scale laws rather than worrying about the motions of individual particles (Chase, 1992). The global pattern of landform occurs after the same local rules are applied to many precipitons (i.e., hundreds thousands to millions of iterations). The purpose of the model is not to capture every details of the physical process, but to gain some insights into the interactions and coupling between fluvial, climatic, and tectonic factors and their overall effects on landform evolution.

WILSIM Java Applet Interface and Parameters

WILSIM Java applet is composed of two components. The animation display window and the parameter selection control. The animation display window visually shows the landform evolution. The parameter selection control allows you to specify different options that will simulate different scenarios of landform evolution.

This is the default main display window that dynamically shows the animated landform evolution over time. The slide bars are used to control the viewing angle. The vertical bar controls elevation angle and the horizontal bar controls the azimuth angle. The Run button is used to start and suspend the animation. The simulation will start from the initial condition. The Reset button is used to reset the values and continue the animation. You can suspend the simulation, adjust the parameters and then continue the simulation. The simulation will continue from where it paused.

This tab displays snapshots of landforms different time intervals (at every 25% of the total iterations). This allows the user to compare still images of the landform at different stages of development.

This tab allows user select various parameters that control the simulation. There are 4 subtabs: Initial Conditions, Erodibility, Climate, and Tectonics.

Under this subtab, you can change the following parameters related to initial model conditions:

Grid size:the number of columns (x) and rows (y) of the grid. The default is 60x100. You can change each by clicking on the radio buttons on the right of each option and selecting a value using the slide bar.

End time: the maximum number of iterations you want this simulation to run. The default is 100,000. You can adjust this value by clicking on the radio button on the right of this option and selecting a value using the slide bar.

Topography: you can adjust the slope of the initial topographic grid. The default is 0.01 slope. You can adjust this value by clicking on the radio button on the right of this option and selecting a value using the slide bar. (Note: a very small amount of random roughness is added in the initial topography to simulate natural surface and to avoid precipiton running downhill along straight lines)

Erodibility: the erodibility of the bedrock. This parameter controls how easy the bedrock will be eroded. There are 3 options:

(1)uniform:0.05, each cell has the same erodibility of 0.05. This is the default value but it can be changed by selecting its radio button and using the slide bar.

(2)break at x: meaning the erodibility changes at a column (x) specified by user with the slide bar. The actual erodibility on either side of the break line is specified by clicking on the left and right button and selecting a value using the slide bar.

(3)break at y: similar to break at x except that the change will occur at a row (y) specified by user.

This tab is used to control the amount of rainfall over time. There are three options:

(1)Constant rainfall: meaning the amount of rainfall will be constant over time. The default value is 0.10. You can also adjust this value by clicking on the button and select a value using the slide bar.

(2)Increasing rainfall: meaning the rainfall will increase linearly over the time period of your simulation. You can specify the low end and the high end of the linear function by clicking on the Low and High radio button and selecting a value using the slide bar.

(3)Decreasing rainfall:meaning the rainfall will decrease linearly over the time period of your simulation. The high and low end of the linear function can be changed similarly.

This tab allows you to change the tectonic uplift rate. The uplift is applied to the topographic grid after each iteration. There are three options:

(1)Fixed at 0 (no uplift): meaning the uplift rate will be 0 (i.e., no uplift). This is the default.

(2)Break at x: meaning the uplift rate changes at a column (x) specified by user with the slide bar. The actual uplift rate on one side of the break line is specified by clicking on the "Left" or "Right" radio buttons and selecting a value using the slide bar. The other side is fixed at 0 (no uplift).

(3)Break at y: similar to break at x, except that the change will occur at a row (y) specified by user.

This tab displays profiles (or cross-sections) of the landform at different time intervals. There are 3 subtabs: Average Profile, Column Profile, and Row Profile

This subtab displays average profiles along y (column) direction (i.e., the elevation at each row is the average of all the cells at that row). The top panel displays the surface elevation (total height in blue) and bedrock elevation (bedrock in red). The bottom panel displays the sediment depth (the difference between surface elevation and bedrock elevation). The profiles are displayed at every 10% of the total iterations.

This subtab displays the profile of one individual column selected by the user. Other features of the profile are similar to the average profile.

This subtab displays the profile of one individual row selected by the user. Other features of the profile are similar to the average profile.

This tab displays the hypsometric curve of the whole model topographic grid at every 10% of the total iterations. The hypsometric curve displayed here is slightly different from the traditional hypsometric curve in that it is defined for the whole model tropographic grid, not a watershed.

Continue to read about a Tutorial with step by step instruction for simulating different scenarios.

References Cited

Ahnert, F., 1987, Approaches to dynamic equilibrium in theoretical simulations of slope development, Earth surface processes and landform, v. 12, p. 3-15.

Chase, C. G., 1992, Fluvial landsculpting and the fractal dimension of topography, Geomorphology, v. 5, p. 39-57.

Howard, A. D., 1994, A detachment-limited model of drainage basin evolution, Water Resources Research, v. 30, p.2261-2285.

Willgoose, g., Bras, R.L., and Rodriguez-Iturbe, I., 1991, A coupled channel network growth and hillslope evolution model, 1. Theory, Water Resources Research, v. 27, p.1671-1684.

 
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