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Fig. 10.14 Principal axes
and diagonalization of the
moment tensor
for the representation of earthquake sources.
The moment tensor allows to build a simple
representation of arbitrarily oriented double
couples by a combination of double couples
that have force vectors oriented as the coordinate
axes. For example, for a double couple in the x 1 x 2
plane, we would have that M assumes the form:
M ii D 0, M 12 D M 21 D M 0 , M 13 D M 31 D M 23 D
M 32 D 0, where M 0 is the magnitude of the single
couples.
Using the moment tensor formalism and on
the basis of the superposition principle, we have
that for an arbitrary double couple the far-field so-
lutions ( 10.48 )and( 10.49 ) assume the following
general form:
entation. Therefore, the E-W directed coseismic
slip along the fault plane would have the same
direction as the main force couple. However, it is
not possible to infer a unique focal mechanism
for an earthquake, in terms of strike, dip, and
rake, starting from the observed radiation pattern.
For example, the pattern illustrated in Fig. 10.12
could have been generated either by an E-W
oriented right-lateral strike-slip fault or by an N-
S oriented left-lateral fault. This ambiguity in the
source mechanism is intrinsic in the moment ten-
sor representation and cannot be avoided. There
are always two complementary focal mechanisms
that are consistent with far-field seismic observa-
tions. The true fault plane is termed the primary
fault plane , while the other solution is called the
auxiliary fault plane . While the determination of
the correct primary fault plane is easy when direct
geological observation of the structures is avail-
able, in the case of deep faults (e.g., associated
with intra-slab deformation) it is necessary to
consider the distribution of aftershock locations .
In fact, all large earthquakes are followed by a
sequence of smaller earthquakes, the aftershocks,
which are distributed along the fault plane and are
associated with readjustments of the stress field
after the mainshock.
The symmetry of the moment tensor implies
that it can be diagonalized by solving an eigen-
value problem and applying the corresponding
similarity transformation. For a double couple
in the x 1 x 2 plane, the result of this operation is
illustrated in Fig. 10.14 and the components of
the diagonalized tensor will be: M ij D 0for i ¤ j ,
M jk .t r='/
r
1
2 ¡' 3
x i x j x k
r 3
u i .r;t/ D
(10.50)
ij
r 2 M jk .t r=“/
1
4¡“ 3
x k
r
x i x j
u i . r ;t/ D
r
(10.51)
Now we recall from the previous section that
double couples are systems of equivalent body
forces that produce the same radiation pattern of
real shear dislocations. Consequently, their orien-
tation is related to slip directions and fault plane
orientations associated with earthquakes. For ex-
ample, we expect that the pattern of displace-
ments for an E-W oriented right-lateral strike-
slip fault can be described through an equivalent
model that includes an E-W oriented couple of
forces and a conjugate couple having N-S ori-
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