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. This is often done by starting with a description of an earthquake fault, broken up into a collection of subfaults, with various parameters defined on each subfault. A seismic modeling code would take these parameters and compute the elastic waves generated in the earth as a result. However, for tsunami modeling all we need to know is the motion of the seafloor, which is one boundary of the seismic domain. Moreover the high-frequency ground motions during the earthquake have little impact on the resulting tsunami. For these reasons it is often sufficient to use the “Okada model” described below, which gives the final deformation of the sea floor due to specified slip on each subfault.

The Jupyter notebook $CLAW/apps/notebooks/geoclaw/Okada.ipynb illustrates how the Okada model works and how to generate the seafloor deformation needed in GeoClaw using this model.

The Python module $CLAW/geoclaw/src/python/geoclaw/dtopotools.py 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). See dtopotools module for moving topography for more documentation and illustrations.

Fault slip on rectangular subfaults

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.

For kinematic (time-dependent) rupture, it is also necessary to specify the rupture_time and rise_time of each subfault, as discussed below.

A fault can be specified in GeoClaw as an instance of the dtopotools.Fault class, instatiated e.g. by:

from clawpack.geoclaw import dtopotools
fault = dtopotools.Fault()

Then set fault.subfaults to a list of subfaults as instances of the class dtopotools.SubFault. Each subfault has attributes corresponding to the parameters listed above. In addition, coordinate_specification should be set for each subfault to one of:

• “bottom center”: (longitude,latitude) and depth at bottom center

• “top center”: (longitude,latitude) and depth at top center

• “centroid”: (longitude,latitude) and depth at centroid of plane

• “noaa sift”: (longitude,latitude) at bottom center, depth at top, This mixed convention is used by the NOAA SIFT database and “unit sources”, see: http://nctr.pmel.noaa.gov/propagation-database.html.

• “top upstrike corner”: (longitude,latitude) and depth at corner of fault that is both updip and upstrike.

For example, a simple single-subfault model of the 2010 Chile event can be specified by:

subfault = dtopotools.SubFault()
subfault.length = 450.e3             # meters
subfault.width = 100.e3              # meters
subfault.depth = 35.e3               # meters
subfault.strike = 16.                # degrees
subfault.slip = 15.                  # degrees
subfault.rake = 104.                 # degrees
subfault.dip = 14.                   # degrees
subfault.longitude = -72.668         # degrees
subfault.latitude = -35.826          # degrees
subfault.coordinate_specification = "top center"

fault = dtopotools.Fault()
fault.subfaults = [subfault]

Starting in Version 5.5.0, it is also possible to specify a set of triangular subfault patches rather than rectangles. Doing so requires a different set of parameters, as described below in Fault slip on triangular subfaults.

Once the subfaults have been specified, the function fault.create_dtopography can be used to create a dtopotools.DTopography object, and then written out as a dtopofile for use in GeoClaw, e.g.:

x,y = fault.create_dtopo_xy(dx=1/60., buffer_size=2.0)
fault.create_dtopography(x,y,times=[1.])
dtopo = fault.dtopo
fault.rupture_type = 'static'
dtopo.write('chile_dtopo.tt3', dtopo_type=3)

This will create a file chile_dtopo.tt3 that can be used as a dtopofile in GeoClaw. It will cover a region buffer_size = 2.0 degrees larger on each side than the surface projection of the rectangular fault, with a resolution of one arcminute (dx = 1/60. degree).

In addition to dtopotools.Fault, the dtopotools has several other derived classes that simplify setting up a fault from a specified set of subfaults:

• CSVFault: reads in subfaults from a csv file with header,

• SiftFault: sets up a fault based on the NOAA SIFT unit sources, see http://nctr.pmel.noaa.gov/propagation-database.html,

• UCSBFault: reads in subfaults in UCSB format,

• SegmentedPlaneFault: Take a single fault plane and subdivides it into recangles, to allow specifying different subfault parameters on each.

See dtopotools module for moving topography for more details, and the Jupyter notebook $CLAW/apps/notebooks/geoclaw/dtopotools_examples.ipynb for more examples.

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 solution to the elastic half space problem, following [Okada85]. 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.

Kinematic rupture

It is also possible to set a rupture_time and a rise_time for each subfault in order to model a time-dependent rupture process. This is called a “kinematic rupture” since the these values are specified.

To specify a kinematic rupture, create a dtopotools.Fault object fault with fault.rupture_type = ‘kinematic’. (For backward compatibility, you can also specify this as ‘dynamic’. However, the term “dynamic rupture” often refers to modeling the rupture process itself.)

A kinematic rupture is not modeled by via modeling the seismic waves that would be generated by the specified subfault motions. There are seismic codes that do this, based on the same set of fault parameters, but this is not supported directly in GeoClaw. If desired, output from such a code could be converted by the user into a dtopo file for use in GeoClaw.

Once a dtopotools.Fault object has been created with the desired subfaults, a dtopotools.DTopography object can be computed using the dtopotools.Fault.create_dtopography function in GeoClaw (and written out as a dtopo file using its write function.) The moving dtopo generated in this manner is the sum of the Okada solutions generated by each subfault, sampled at a set of specified times t. For subfaults with subfault.rupture_time > t, no displacement is included, while if subfault.rupture_time + subfault.rise_time <= t the entire deformation due to this subfault is included, with linear interpolation between these at intermediate times.

Warning

Starting in Version 5.5.0, the subfault parameter rise_time now refers to the total rise time of a subfault, while rise_time_starting is the rise time up to the break in the piecewise quadratic function defining the rise. By default rise_time_ending is set equal to rise_time_starting. (In earlier versions, rise_time read in from csv files, for example, was erroneously interpreted as rise_time_starting is now.) See the module function rise_fraction in dtopotools module for moving topography for more details.

Fault slip on triangular subfaults

Starting in Version 5.5.0, it is also possible to specify a set of triangular subfault patches rather than rectangles.

Specifying a subfault as a triangular patch rather than as a rectangle can be done by setting subfault.coordinate_specification = ‘triangular’ and specifying subfault.corners as a list of three (x,y,depth) tuples, along with the slip and rake. In this case you do not set the attributes length, width, depth, strike, or dip, since the corners of the triangle are sufficient to determine this geometry. Internally a strike direction is calculated by intersecting the plane defined by the triangle with the ground surface, and choosing the direction so that the plane of the triangle dips at an angle between 0 and 90 degrees relative to the strike direction. The specified rake is again interpreted as degrees counterclockwise from this strike direction.

For an example see [need to add a notebook].