Geoscience Reference
In-Depth Information
p
=
y
cos
δ
+
d
sin
δ
,
q
=
y
sin
δ
−
d
cos
δ
,
y
=
η
cos
δ
+
q
sin
δ
,
(2.27)
d
=
η
sin
δ
−
q
cos
δ
,
R
2
=
2
+
2
+
q
2
,
ξ
η
X
2
=
2
+
q
2
.
ξ
enter into expressions (2.25) and (2.26) in
the form of a combination, which for practical calculations is conveniently ex-
pressed via the respective velocities of longitudinal and transverse seismic waves,
c
p
and
c
s
,
The Lame constants
λ
and
µ
c
s
c
p
−
µ
=
.
λ
+
µ
c
s
Usually, this ratio varies within the range from 0.3 to 0.5. More precise informa-
tion for a concrete region can be obtained, for instance, from the Reference Earth
Model (http://mahi.ucsd.edu/Gabi/rem.html).
Under special conditions some terms in formulas (2.22)-(2.26) become singular.
To avoid all singularities, the following rules should be obeyed:
ξη
/
qR
)=0 in equations (2.22)-(2.24)
(i) When
q
= 0, set arctan (
ξ
= 0, set
I
5
= 0 in equation (2.25)
(iii) When
R
+
(ii) When
η
= 0, set all the terms which contain
R
+
η
in their denominators
to zero in equations (2.22)-(2.26), and replace ln(
R
+
η
) by
−
ln(
R
−
η
) in
equations (2.25) and (2.26).
To assist the development of a computer program based on expressions (2.21)-
(2.27), several numerical results, permitting to check it, are listed in Table 2.1.
A medium is assumed to be
λ
=
µ
in the all cases, and the results are presented
in units of
U
i
.
Table 2.1 Chec
klist for numerical calculations. Adapted from [Okada (1985)]
u
x
u
y
u
z
= 70
◦
;
L
= 3;
W
= 2
Case 1:
x
= 2;
y
= 3;
d
= 4;
δ
Strike
−
8
.
689
E
−
3
−
4
.
298
E
−
3
−
2
.
747
E
−
3
Dip
−
4
.
682
E
−
3
−
3
.
527
E
−
2
−
3
.
564
E
−
2
Tensile
−
2
.
660
E
−
4
+1
.
056
E
−
2
+3
.
214
E
−
3
= 90
◦
;
L
= 3;
W
= 2
Case 2:
x
= 0;
y
= 0;
d
= 4;
δ
Strike
0
+5
.
253
E
−
3
0
Dip
0
0
0
Tensile
+1
.
223
E
−
2
0
−
1
.
606
E
−
2
90
◦
;
L
= 3;
W
= 2
Case 3:
x
= 0;
y
= 0;
d
= 4;
δ
=
−
Strike
0
−
1
.
303
E
−
3
0
Dip
0
0
0
Tensile
+3
.
507
E
−
3
0
−
7
.
740
E
−
3
