## Scientific Issues

If you have the solution to either of the problems below please email me.

### More accurate description of tsunami origin

At present slowmo takes a single geographical point as initial condition for the tsunami. In reality a tsunami is often created along a rift in the ocean floor. It would therefore make more sense if slowmo could take a list of positions as input. This would only involve a very small change to the slowmo source code.

The scientific problem is to figure out how an earthquake at a given position is translated into a list of points that can be used as initial conditions for the calculation. I don't know anything about how to do that but I would guess that you would need at least a map describing the rift zones and knowledge about the earthquake strength. Maybe this article can be of help:

Surface deformation due to shear and tensile faults in a half-space, Y. Okada, 1985, Bulletin of the Seismological Society of America, Vol. 75.

### Travel time calculation

Presently the calculation nodes are situated at the center of the bathymetry grid boxes, see Figure 1. The travel time is simply calculated as half the distance between two points divided by the wave phase speed in the box (sqrt(gravity*depth)) containing one point plus half the distance divided by the wave phase speed in the box containing the other point. The distance is calculated using a great circle formula.

Figure 1: Current grid layout. The boxes outlined by black lines represent the bathymetry while the red circles represent the calculation nodes. A given bathymetry box is characterized by a single depth. The blue line shows the shortest path from the upper left node to the lower right node when following a great circle. The green line shows the shortest path using a plane approximation.

The travel time calculation could be changed to be more accurate. I have three suggestions for doing this. All three suggestions are a matter of finding analytical solutions. In order of ascending complexity they are:

1. Write the phase speed as a function of where it is on the green (or blue) line in Figure 1 using linear interpolation between the two points. Integrate 1 divided by the phase speed from point 1 to point 2. The result is an analytical solution for the travel time between the points.
2. As above but for the green line only using a bilinear interpolation between all four surrounding grid points.
3. As suggestion 2 but along the great circle curve.

For this exercise you would need to know a bit about bilinear interpolation. The last suggestion requires that the position on the great circle (blue curve in Figure 1 is parameterized in terms of distance from the first position. When the position is known it is possible to use the bilinear interpolation to calculate the depth and thus the phase speed. 1 divided by the phase speed can then be integrated along the curve. More information on great circles can be found here.

UPDATE: The first solution suggestion has now been implemented in slowmo. It will be included in slowmo-1.1.3.