Earthquakes - Quantitative Plate Tectonics - page 290

Fig. 10.17 Stress drop
across a fault plane after an
earthquake. T s and T 0 s are
the local fields of shear
stress along S before and
after slip u

M ij D M 0 n i b u i

This quantity is clearly:

u j C n j b (10.56)

Z

As we expected, this expression gives a sym-
metric tensor. Furthermore, it results:

T s .x;y/ C T s .x;y/ dS (10.54)

1
S

T s D

S

Tr.M/ D M kk D 2M 0 n k b u k D 2M 0 .n b u / D 0

(10.57)

Now we note that T s S represents the total surface
force exerted across the fault plane. Therefore,
the total energy release during an earthquake will
be given by:

Therefore, the trace of the moment tensor is
always zero. The expression ( 10.56 )showsthat
the moment tensor can be separated into a scalar
strength, corresponding to the seismic moment
M 0 , and a kinematic component associated with
the tensor quantity m ij n i b

T s
M 0

E D T s S u D (10.55)

u i . The latter
can be rewritten to make explicit the relation
between the components of M and the focal
mechanism. In fact, the components of the versor
n depend from both the fault strike, ¥, and the dip,
•, while

u j C n j b

The stress drop associated with large earth-
quakes is generally estimated from the seismic
moment. In the case of shallow events, it varies
between 1 and 10 MPa. Often earthquakes along
plate boundaries have lower stress drops than
intraplate events (Shearer 2009 and references
therein). It is estimated that the average • T s is
3 MPa in the case of plate boundary earth-
quakes and 6 MPa for intraplate events. This
is possibly a consequence of a lower area of
intraplate faults with respect to the typical dimen-
sions of plate boundary faults.
The scalar seismic moment measures the
strength of an earthquake. Therefore, it is quite
intuitive that it must be related to the moment
tensor components M ij . In fact, it is possible to
show (e.g., Udías 1999 )thatif n is the versor
normal to a fault plane and

u depends from the rake œ. Therefore, it
is easy to prove that the components of M in a
local reference frame oriented as x N, y E, z
Down are given by:
8
<

b

M 11 D M 0 .sin • cosœ sin 2¥

C sin 2• sin œsin 2 ¥
M 12 DC M 0 .sin • cos œ cos2¥
C 0:5 sin 2• sin œ sin 2¥/
M 13 D M 0 .cos• cos œ cos¥
C cos 2• sin œ sin ¥ /
M 22 DC M 0 .sin • cos œ sin 2¥

(10.58)

:

sin 2• sin œcos 2 ¥
M 23 D M 0 .cos• cos œ sin ¥
cos2• sin œ cos¥/
M 33 DC M 0 sin 2• sin œ

b u is the seismic

slip versor, so that u D 0, then an analytic

expression for the moment tensor is:

n b