Lecture 10

Our main interest in Musgrave's paper (see Readings) is the physical erosion model, section 4.

This class will also include some concluding remarks on fBm in terrain from Lecture 9 and may include some ideas on weathering in a paper by Dorsey in SIGGRAPH about 1999.

Erosion on Heightmaps

The amount of water that passes from one vertex v to the neighboring vertex u is

\begin{align} \Delta w = min (w_t^v, (w_t^v + a_t^v) - (w_t^u + a_t^u) \end{align}

with w = water, subscripts give time step, superscripts give location, a gives altitude.
So either all the water is passed from v to u, or just the difference is passed from v to u. Which ever is less.

If the change in water is less than or equal to 0, then sediment is deposited at v which increases the altitude at v.

\begin{array} {l} a_{t+1}{v} = a_t^v + K_d s_t^v \\ s_{t+1}{v} = (1-K_d)s_t^v \end{array}

with K-sub-d as the deposition capacity (how much of the sediment disolved is deposited) and s as the amount of sediment in the water at time t and location v.

If the change in water is greater than 0, then

\begin{array} {l} w_{t+1}^v = w_t^v - \Delta w \\ w_{t+1}^u = w_t^u + \Delta w \\ c_s = K_c \delta w \end{array}

with c sub s as the sediment capacity for that much water moving. This means that at most $c_s$ sediment will be carried away by this particular water motion. That new sediment will be added to the existing sediment.

If the moving water picks up some sediment from v, then the altitude at v needs to be reduced by the amount of new sediment because some material was eroded (and carried away in the water). Also, we need to decide how much of the newly eroded sediment is suspended in the water at v, suspended in the water at u and is deposited or removed from the terrain a v.

There are two possibilities. If $s_t^v \geq c_s$, that is, if the amount of sediment already in the water at $v$ exceeds the amount of sediment picked up through the motion of the water, then we pass the maximum amount of sediment to $u$ (because we can), increase the altitude at $v$ (because some left over sediment movement is be deposited at v) and marginally increase the amount of sediment in the water at $v$ (because we have to conserve sediment?).

\begin{array} {l} s_{t+1}^u = s_t^u + c_s \\ a_{t+1}^v = a_t^v + K_d (s_t^v - c_s) \\ s_{t+1}^v = (1 - K_d)(s_t^v - c_s) \end{array}

But, if $s_t^v < c_s$, then there's less sediment in the water than can be transported to the neighboring point. So we take all the sediment from from the water at v, pass it to u, add some more sediment from erosion at v and decrease the altitude at v accordingly, to wit:

\begin{align} \begin{array}{l} s_{t+1}^u = s_t^u + s_t^v + K_s(c_s - s_t^v) \\ a_{t+1}^v = a_t^v - K_s(c_s - s_t^v) \\ s_{t+1}^v = 0 \end{arary} \end{align}

in which $K_s$ is a measure of the soil softness.

Care must be taken to divy up the water and sediment to all neighbors proportional to their elevation below v.

This is a fine example of a global process simulated through local interactions.

although rain in nature doesn't fall uniformly, especially in the mountains where orographics play a dominant role, they allowed a fixed amount of rain to fall at regular intervals. It turns out that rain patterns in the Nile river follow a $1/f$ noise distribution. Would something like that work too?

Thermal weathering on heightmaps.

This is process by which "knocks material loose, which material then falls downto pile up atthe bottom of an incline." This process simply reduces steep angles to the angle of repose in hte terrain

\begin{align} a_{t+1}^u = \left\{ \begin{array}{l} a_t^v - a_t^u > T: a_t^v + c_t(a_t^v -a_t^u - T) \\ a_t^v - a_t^u \leq T: a_t^u \end{array} \right. \end{align}
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