Fault basis

Slip and slip-rate are defined with respect to a local fault basis. In this document the conventions for the fault basis are introduced. The direction of movement is defined in terms of the hanging wall and the foot wall:

“The foot wall (hanging wall) is defined as the block below (above) the fault plane. (…) the hanging wall moves up with respect to the foot wall and the fault is known as reverse. (…) the opposite happens and the fault is said to be normal.” [J. Pujol, Elastic Wave Propagation and Generation in Seismology]

The sign of the fault normal is chosen such that

\[n \cdot n_{\text{ref}} > 0.\]

We define that the fault normal points from the foot wall to the hanging wall. In this way the reference normal \(n_{\text{ref}}\) selects the foot and the hanging wall.

The first component of the slip or slip-rate vector is defined w.r.t. to the normal direction of the fault. Due to the no-opening condition the first component is zero.

The third component of the slip or slip-rate vector is defined w.r.t. to the strike direction. The latter is defined such that a hypothetical observer standing on the fault looking in strike direction sees the hanging wall on his right. Thus, the strike direction is

\[s := u \times n,\]

where \(u\) is the direction of “up”, given in the configuration file. E.g. using the enu convention, up would be the vector \(u=(0, 0, 1)\).

The second component of the slip or slip-rate vector is defined w.r.t. to the dip direction, which we define to point “down”. That is, the dip direction is

\[d := s \times n\]

Left-lateral, right-lateral, normal, reverse

The slip vector is given by \(u=[u_n] n + [u_d] d + [u_s] s\), where the square bracket operator for a scalar field \(q\) is defined as

\[[q] := q^- - q^+ = \lim_{\epsilon \rightarrow 0} q(x-\epsilon n) - q(x+\epsilon n)\]

Recall that the normal points from the foot wall to the hanging wall. Thus, if \([u_d] > 0\) we have a reverse fault. Conversely, if \([u_d] < 0\) we have a normal fault.

For strike slip fault, i.e. \([u_s] \neq 0\), we have to distinguish two cases:

“In a left-lateral (right-lateral) fault, an observer on one of the walls will see the other wall moving to the left (right).” [J. Pujol, Elastic Wave Propagation and Generation in Seismology]

If \([u_s] > 0\) then we have a right-lateral fault and if \([u_s] < 0\) then we have a left-lateral fault.

Special-case: Flat fault

Don’t do that.