Earthquake sources: Fault slip and the Okada model

To initiate a tsunami from an earthquake, it is necessary to generate a model of how the seafloor moves, which is generally specified in a dtopo file as described in Topography displacement files.

Fault slip

For historic earthquakes, it is generally possible to find many different models for the distribution of slip on one or more fault planes, see for example the pointers at Earthquake source models.

An earthquake subfault model is typically given in the form of a set of rectangular patches on the fault plane. Each patch has a set of parameters defining the relative slip of rock on one side of the planar patch to slip on the other side. The minimum set of parameters required is:

  • length and width of the fault plane (typically in m or km),
  • latitude and longitude of some point on the fault plane, typically either the centroid or the center of the top (shallowest edge),
  • depth of the specified point below the sea floor,
  • strike, the orientation of the top edge, measured in degrees clockwise from North. Between 0 and 360. The fault plane dips downward to the right when moving along the top edge in the strike direction.
  • dip, angle at which the plane dips downward from the top edge, a positive angle between 0 and 90 degrees.
  • rake, the angle in the fault plane in which the slip occurs, measured in degrees counterclockwise from the strike direction. Between -180 and 180.
  • slip > 0, the distance (typically in cm or m) the hanging block moves relative to the foot block, in the direction specified by the rake. The “hanging block” is the one above the dipping fault plane (or to the right if you move in the strike direction).

Note that for a strike-slip earthquake, rake is near 0 or 180. For a subduction earthquake, the rake is usually closer to 90 degrees.

Okada model

The slip on the fault plane(s) must be translated into seafloor deformation. This is often done using the “Okada model”, which is derived from a Green’s function sollution to the elastic half space problem. Uniform displacement of the solid over a finite rectangular patch specified using the parameters described above, when inserted in a homogeneous elastic half space a distance depth below the free surface, leads to a steady state solution in which the free surface is deformed. This deformation is used as the seafloor deformation. Of course this is only an approximation since the actual seafloor in rarely flat, and the actual earth is not a homogeneous isotropic elastic material as assumed in this model. However, it is often assumed to be a reasonable approximation for tsunami modeling, particularly since the fault slip parameters are generally not known very well even for historical earthquakes and so a more accurate modeling of the resulting seafloor deformation may not be justified.

In addition to the parameters above, the Okada model also requires an elastic parameter, the Poisson ratio, which is usually taken to be 0.25.

The GeoClaw routine $CLAW/python/pyclaw/geotools/ available starting in Version 4.6.3, is an improved version of $CLAW/python/pyclaw/geotools/ that allows specifying whether the latitude and longitude provided corresponds to the centroid, bottom center, or the top center of the fault plane (the original assumed top center). The specification of other parameters has also been modified, see the documentation in that file.

The Python module $CLAW/python/pyclaw/geotools/ (new in 4.6.3) provides tools to convert a file specifying a collection of subfaults into a dtopofile by applying the Okada model to each subfault and adding the results together (valid by linear superposition of the solutions to the linear elastic halfspace problems). These still need to be cleaned up and better documented, but an example of the usage can be found in the new application example $CLAW/apps/tsunami/chile2010b.

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