1) Tensor of the seismic moment for pure shear. A point source is assumed.
Vector of displacement parallel to the fault plane. Unit vector in direction of
the displacement, unit vector of the fault normal. Isotropic medium. Rigidity.
2) Tensor in the coordinate system related to the fault. Note the two identical
components given by product of rigidity and displacement, = the surface
density of seismic moment.
3) Wave field due to the pure-shear source. Convolution to express principle of
linear superposition (reaction is dependent on the whole history of the action).
Summation to express arbitrary space orientation of the fault. [In case of a
finite fault we have also integration over the fault plane.] Green function is
action of a special orientation: Gip = i-th component of displacement due to a
single force applied along p-th coordinate. Gip,q = space derivative of the
Green function.
4) In our case only Gi1,3 and Gi3,1 are in effect. This means that the source is
equivalent to two force couples (the so-called double couple source, DC). The
source along axis 1, arm along axis 3, and vice versa.
5) Equation for the eigenvector and eigenvalues of the moment tensor.
6) The pure shear case has one zero eignevalue, and two eigenvalues of the same
size and opposite sign. This m is again surface density of scalar moment.
[Check Global CMT solutions, see similar values, although never exactly zero
value.]
7) Eigenvectors along axis 2 and at 45 degrees with respect axes 1 and 3 =
nothing but the well known B, P and T axes, P and T in the middle of the
negative and positive signs on the well-known beachballs.
8) The moment tensor is now diagonalized with the eigenvalues on the diagonal.
Note the sum of the diagonal terms = trace; zero trace means zero volume
component. Note the product of the diagonal terms = the determinant; zero
determinant = a zero eigenvalue = pure shear. The trace and determinant are
independent of the coordinate system; those are invariants of the moment
tensor.
9) Finally, a completely different representation of the same displacement when
the coordinate system of eigenvalues is employed: again two force couples are
in effect, but now those are the ‘degenarate couples’, each couple has forces
of the same direction (opposite polarity), with zero arm.
10) In practice (not shown here) we have a third, most common system of
coordinates, it is the geographic system; then the moment tensor is usually
‘full’ (no reason to have just a few non-zero components).
11) Remark: Complexity of Gip,q even in a homogeneous infinite medium: the far
and near-field effects.
Gip,q = delta function derivative for both P and S+ delta function for both P and S
+ a linear (trapezoid) pulse between P and S
M=step
U=M*Gip,q = delta function + step + a quadratic pulse between P and S
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