The deterraiination of earthquake mechanism,
using both longitudinal and transverse waves
V. I. K E Y L I S -
INTRODUCTION
T h e p r a c t i c a l a s p e c t of a m e t h o d of
determining earthquake mechanism
(the
" f a u l t p i a n e s o l u t i o n " ) is dea.lt w i t h h e r e .
This m e t h o d has m u c h in c o m m o n with t h e
w e l l - k n o w n m e t h o d s of B y e r l v ( e s p e c i a l l y ,
t a k i n g i n t o a c c o u n t t h e i m p r o v e m e n t s sugg e s t e d b y A . R . R i t s e m a ) a n d of J a p a n e s e
s e i s m o l o g i s t s ; t h e p r i n c i p a l d i f f e r e n c e lies
in t h e f a c t t h a t transverse waves are widely
used h e r e , w h i c h r e n d e r s t h e s o l u t i o n u n a m b i g u o u s (*).
The worked out theoretical
p r i n c i p l e s p e r m i t as w e l l t o u s e s i m p l y t h e
a m p l i t u d e r a t i o of d i f f e r e n t w a v e s , b u t in
p r a c t i c e i t is less r e l i a b l e .
I n § 1 t h e t h e o r e t i c a l p r o p e r t i e s of t h e
waves caused b y t h e dipole with m o m e n t
are described.
S u c h m o d e l of a seismic
s o u r c e is t h e m o s t o f t e n e n e o u n t e r e d .
The
m e t h o d is a p p l i c a b l e t o a n y o t h e r m o d e l s as well, b u t t h e i r t h e o r e t i c a l f e a t u r e s
are considered v e r y briefly.
W e d o n o t d e a l w i t h t h e e f f e c t of t h e
i n t e r f a c e s a n d t h e i n l i o m o g e n e i t y of t h e
m e d i u m o n t h e f o r n i a n d i n t e n s i t y of diss p l a c e m e n t s ; t h e m e t h o d s of e l i m i n a t i n g of
t h e s e f a c t o r s a r e d e s c r i b e d in d e t a i l in a n
other article.
I n § 2 t h e g e n e r a l p a t t e r n of t h e i n t e r p r e t a t i o n is g i v e n .
I n §§ 3-6 t h e s u c c e s s i v e s t e p s of t h e
i n t e r p r e t a t i o n of o b s e r v a t i o n s a r e p r e s e n t e d .
T h e Wolf s t e r e o g r a p h i c p r o j e c t i o n u s e d
f o r i n t e r p r e t a t i o n is d e s c r i b e d in t h e S u p plementi.
§ 1.
-
INITLAL
FORM ('LAS
E a r t h q u a k e foci a r e g e n e r a l l y e q u i v a l e n t
t o t h e d i p o l e w i t h m o m e n t (fig. 1).
The
axis x corresponds to t h e motion direction.
y = 0 - to the fault piane.
y
3>
Fig. 1 - T h e dipole w i t h m o m e n t (the s y m m e trical f a u l t ) .
I n this paragraph the main properties
of e l a s t i c w a v e s c a u s e d b y t h e d i p o l e w i t h
m o m e n t ( a n d s o m e o t h e r sources) i n a
h o m o g e n e o u s m e d i u m will b e c o n s i d e r e d .
1.
(*) Tlie experience in i n t e r p r e t a t i o n of a
g r e a t n u m b e r ( ^ 300) of sources (and even
t h e correlation of <S-pliases at different stations) a c q u i r e d since 1947, testifies t h a t t h e
s t u d y of t r a n s v e r s e waves is as reliable as
P P , P i t P , p P etc. (provided t h e s e i s m o g r a m s
a n d not t h e questionnaires are used).
BOEOK
The formulas
/ o r displacements
homogeneous
medium
at great
" r " from the
source.
in a
distance
A t g r e a t d i s t a n c e s r t h e d i s p ì a c e m e n t in
a l o n g i t u d i n a l w a v e is d i r e c t e d a l o n g t h e
r a y a n d is c o m p l e t e l y d e t e r m i n e d b y o n e
101)
(anv)
vaine
ua is
from
V. I. K E Y L I S - BOROK
c o m p o n e n t . W e aliali consider t h e
of t h e total d i s p l a c e m e n t v e c t o r ua\
regarded positive when it is directed
t h e source.
I n a t r a n s verse wave t h e displacement is
d e t e r m i n e d b y t w o components (the f l u i d
one can be found from t h e condition t h a t
t h e displacement vector is perpendicular to
t h e ray). W e shall consider two independ e n t c o m p o n e n t s , vj' in t b e piane of incidence ( S F wave) a n d uHh in t b e direction
p e r p e n d i c u l a r to t h e piane (SH wave) —
see fig. 2. u " is r e g a r d e d positive w h e n it
is directed clocfovise, assuming t h e focus to
be in t h e centre (or to t h e right if one looks
a t t h e observational p o b i t f r o m t b e focus),
is positive when its horizontal c o m p o n e n t
is directed frcm t h e focus. F o r instance,
fig. 2 shows itf > 0 a n d wf > 0 .
It would be convenient to introduce
two systems of coordinates (x, y, z) and
(x, y, z) (the beginning of b o t h systems
coinciding w i t h t h e focus) so t h a t t o obtain
m o r e compact, forni ulas for ua, u",
axes x a n d y relate to t h e direction of
dipole axes as it is sliown in fig. 1.
axis x is directed to t h e F a s t , y t o
The
the
The
the
Xorth, z upward.
A t g r e a t distances r t h e f o r m u l a s for
i n d e p e n d e n t c o m p o n e n t s a r e as follows
[V. I . Iveylis-Borok et al., 1957]:
4 7ioua ~
a r
K'
(t — ria)
[1]
T H E D E T E R M I N A T I C I OF E A R T H Q U A K E MECHANISM, ETC.
4 non"
~
-
ygH
K'
b3 r3
VJJ
b3Lr
4 Tiouf
^
o
K
(t — r/b)
' V ~
r/b)
[2]
2. The nodal lines and the distribution
signs of ua, u", u'h' in each point.
[3]
T h e nodal surfaces -wliere ali or some of
tlie displacements ua, u c o m e out zero
can b e easilv obtained f r o m f o r m u l a s
[l]-[3].
Ali t h e nodal surfaces go t h r o u g h t h e
focus. However, in p r a c t i c e t h e diplacem e n t s vanish only 011 t h a t p a r t of t h e
surfaces where t h e values of r are sufficiently g r e a t (i. e., w h e r e a p p r o x i m a t e
f o r m u l a s [l]-[3] are valid). F o r m u l a s [l]-[3]
give t h e following nodal lines for t h e dijiole
with m o m e n t :
Here,
o — d e n s i t y ; a, b — longitudinal a n d
t r a n s v e r s e wave velocities; K (t) intensitytime function.
V
=
ya-X — xfte
sin i
[4]
x cos' i •— zy x
sin i cos i
[5]
• anale of incidence
107
sin i
a) p i a n e y
< =
=
0, w h e r e ua
u"
of
=
0,
Fig. 3 - The nodal lines for SI 7 -wave 011 a Wolf stereographic projè'ction.
ax; /}x; y , — angles m a d e b y t h e axis x
andtlie axes x, y, z respectively (so t h a t
x = xax + yfix + zyx...
[6]). T h e f o r m u l a s
for o t h e r sources m a y be derived f r o m [1],
[2], [3] if t o replace t h e f a c t o r " ? / " b y t h e
following symbols: a or b for simple force;
•X for t h e dipole w i t h o u t m o m e n t ; y
t h e superposition of t h e dipoles w i t h mom e n t a n d w i t h o u t m o m e n t (see fig. od);
y2la or y2jb — for t h e doublé dipole with
m o m e n t . . . a n d so on; for various sources
K is r e p r e s e n t e d in various units, which
can b e neglected in t h e present article.
b) p i a n e x
=
0, where ua
— 0,
H
e) p i a n e g = 0 or xc • y — yc • x =
= 0, w h e r e uff = 0.
H e r e xcì yc are t h e coordinates of a n y
point on t h e axis x. This piane is vertical
afor
n d includes t h e axis x.
d) cone gp = 0, orx(xja—zcx)
— zcy) = 0, w h e r e u'b' = 0.
+ y(ycz —
This one is a n elliptical cone; its opposite
ì ulings are t h e axes z, a n d x. T h e cone axis
108)
V. I. K E Y L I S - BOROK
lies in t h e p i a n e xz a n d biseets t h e angle
m a d e by x a n d z.
T h e projeetion of t h e eone on a horizontal p i a n e is a circle t h e d i a m e t e r of which
connects t h e epicentre witli t h e axis x.
T h e cone projeetion on a Wolf stereog r a p h i c p r o j e e t i o n is a n ovai passing t h r o u g h
t h e projeetion of t h e axis z (the centre) a n d
t h e p r o j e e t i o n of t h e axis x-, a f a m i l y of
such ovals for various inclinations of t h e
axis x is given in fig 3; t h e line of s y m m e t r y
in fig. 3 coincides w i t h u% = 0.
is also of g r e a t i m p o r t a n c e . Ali possible
combinations of signs (top t o b o t t o m : ua,
u
bi ub) a r e shown in fig. 4. Ali t h e signs
certainly can b e simultaneously
reversed.
T h e n o d a l lines are u n a m b i g u o u s l y related
to t h e m a i n p a r a m e t e r s of dislocation in a
focus: y = 0 d e t e r m i n e s t h e f a u l t piane;
axis x (a p o i n t on this piane) — t h e motion
direction.
Fig. 5 shows t h e nodal lines a n d t h e
correlation of signs for some o t h e r sources.
F o r details see [Keylis-Borok et al., 1957].
sed simultaneously.
T h e line « f = 0 can be o b t a i n e d on t h e
Wolf stereographic projeetion, w h e n t h e
axis x is k n o w n , b y r o t a t i n g t h e t r a c i n g
p a p e r with t h e axis x on it a b o u t t h e
centre of fig. 3 so t h a t t h e axis x should
coincide w i t h t h e line of s y m m e t r y ; t h e n
t h e ovai in fig. 3 passing t h r o u g h t h e axis
x will represent t h e line u'b' = Ó.
I n t e r p r e t a t i o n requires t h e knowledge of
t h e theoretical position of t h e nodal lines
on a Wolf stereographic projeetion. T h e y
are given in fig. 4. x — 0 a n d y = 0 are
t h e projeetions of two p e r p e n d i c u l a r planes.
u"
0 is a s t r a i g h t line passing t h r o u g h
t h e centre a n d t h e axis x (the pole of t h e
line x = 0). wf = 0 is one of t h e ovals
f r o m fig. 3.
The distribution of tlie signs of ua, Ubi Mb
in different regions between t h e nodal lines
3. Amplitude
ratios (*)
F r o m [l]-[6] follows t h a t
, u
le —a
Ut
x
=
r
=
xax + yfiLx + zyx
=
_
yax — xfix
. .
sin i
[7]
y H>a
le
=
X
[8]
xo-x + y P x + zy x
(xax + ypx) cos 2 i — zyx sin 2 i
ij _
H
ub
sin i cos i
gp __ (xax + yfix) cos 2 i — zyx sin 2 i
H
g
(yax — rfix) cos i
[9]
(*) This point may be omitted if as usually
we use only the displacement signs.
T H E D E T E R M I N A T I C I OF E A R T H Q U A K E MECHANISM, ETC.
a3
F o r m u l a s [l]-[3] give k =
b3
: however,
in practice it m a y be considerably less due
to t h e f a c t , t h a t t h e source is n o t a point,
a n d t h e m e d i u m is not ideally elastic.
Let
U
,
"
7 »
U
7
le —jr =h
u"b
"
7 p
: k • -p- = h
u\ o
U
(
.
CHyx
= 0
[7a]
Apax
+ BPpx
+ GPyx
= 0
[Sa]
A,ax
+ B,px
+
=
0
[9a]
C,YX
if t h e m e d i u m were homogeneous.
Then
applying t h e elastic w a v e t h e o r y for a
homogeneous m e d i u m we d e t e r m i n e t h e
source e q u i v a l e n t to t h e focus.
T h e i n t e r p r e t a t i o n includes t h e following
steps:
b
; f — —~
u"
b
T h e n [7]-[9] can be easily w r i t t e n down
as follows:
Ah ax + BHpx+
109
T h e f o r m u l a s for A, B, C are given in
table 1. These f o r m u l a s are also t r u e for
t h e sources (b), (e), (a), (/) (fig. 5).
1. T h e d e t e r m i n a t i o n of t h e initial
observations (ground displacement components).
2. T h e r e d u c t i o n of observations t o a
homogeneous m e d i u m .
a) t h e e l i m i n a t i n g of t h e effect of
interfaces a n d of t h e deflection of a r a y
f r o m a s t r a i g h t line.
b) t h e p l o t t i n g of t h e initial observations on a Wolf stereographic projection (see Supplement).
Table
1
i
Ai
Bi
GÌ
E
x — liH y cosec i/,
y + hH x cosec i;,
z
P
x (1 — > l P ctg ih)
y (1
f
x cos ih — yf
—hpctgih)
y cos ih + xf
x, y, z can be substitutec . as follows:
x = sin a • sin i/, ; y = cos a • sin i;,
;
— z sin ih tg ih
z = COS ih
INTER-
3. T h e d e t e r m i n a t i o n of t h e d y n a m i c
p a r a m e t e r s of a focus (fault p i a n e solution).
T h e initial d a t a for i n t e r p r e t a t i o n are
t h e signs of t h e arrivals of longitudinal
a n d t r a n s v e r s e seismic waves. T h e worlced
o u t t h e o r y p e r m i t s t o use a m p l i t u d e ratios
as well (without a n y a d d i t i o n a l c o m p u t a t ions, w i t h t h e help of special n o m o g r a p h —
figs. 13), h o w e v e r in f a c t it is o f t e n considerably less reliable.
T h e i n t e r p r e t a t i o n of observations first
implies t h e d e t e r m i n a t i o n of t h e initial d a t a
a) T h e d r a w i n g of nodal lines basing
u p o n t h e signs of displacements a n d t h e i r
correlation in each point, t h e a m p l i t u d e
ratio being t a k e n i n t o a c c o u n t w h e n e v e r
possible.
§2.
-
GENERAL
PRETATION
OF
PATTERN
OF
THE
OBSERVATIONS.
b) T h e e s t i m a t i o n of t h e a c c u r a c y
of i n t e r p r e t a t i o n .
N o w we shall describe ali t h e successive
steps.
110)
V. I. KEYLIS - BOROK
F i g . 5 - T h e n o d a l lines f o r v a r i o u s sources. Ali signs c a n be r e v e r s e d
simultaneously.
THE
DETERMINATICI
OF
E A R T H Q U A K E M E C H A N I S M , ETC.
115
y
Fig. 5 (continuation)
T h e nodal lines for
v a r i o u s sources. Ali
signs can be reversed
simultaneously.
/) t h e doublé dipole w i t h m o m e n t
e) t h e superposition of a) a n d b)
g) t w o dipoles w i t h m o m e n t .
§
3.
-
THE
DETERMINATION
OF
INITIAL
OBSERVATIONS.
1. Measurements
on
seismograms.
T h e signs a n d a m p l i t u d e s of t h e first
a r r i v a l a i n ali b o d y a n d d i f f r a c t e d ( h e a d )
s e i s m i c w a v e s ( t h e p a t h s of w h i c h a r e
k n o w n ) c a n s e r v e a s i n i t i a l d a t a (in a n y
combination).
L e t P , a n d Pv b e v e r t i c a l a n d h o r i z o n t a l
c o m p o n e n t s of t h e g r o u n d d i s p l a c e m e n t
(for t h e c a s e of a n i n c i d e n t l o n g i t u d i n a l
w a v e ) ; SH, 8v — h o r i z o n t a l c o m p o n e n t s o n
t h e g r o u n d d i s p l a c e m e n t in SH a n d SV
w a v e s (for t h e c a s e of a n i n c i d e n t t r a n s v e r s e
w a v e ) ; S , is t h e v e r t i c a l c o m p o n e n t of t h e
g r o u n d d i s p l a c e m e n t (for a n i n c i d e n t t r a n s v e r s e w a v e ) ; 8, a n d 8V a r e t h e c o m p o n e n t s
of t h e S F w a v e .
F o r t h e d i s t u r b a n c e of t h e e a r t h s u r f a c e
c a u s e d b y t h e a r r i v a i of a l o n g i t u d i n a l
w a v e ( P , SP, e t c . ) o n l y o n e c o m p o n e n t is
i n d e p e n d e n t - e i t . h e r v e r t i c a l or h o r i z o n t a l .
For the displacement caused by a transverse w a v e two i n d e p e n d e n t components
c a n b e m e a s u r e d : SH a n d Sv or S..
If t h e d a t a a r e n o t t o o n u m e r o u s , i t
happens to be useful to determine the
signs a n d t o m e a s u r e t h e a m p l i t u d e s f o r
ali t h e i n t e r p r e t e d p h a s e s o n ali a v a i l a b l e
r e c o r d s . N o t ali t h e m e a s u r e m e n t s c a n
give i n d e p e n d e n t initial d a t a ; b u t t h e
rest can be used for t h e control a n d
r e c o g n i t i o n of a n o m a l i e s in a z i m u t h s a n d
a n g l e s of e m e r g e n c e ( t h e s e a n o m a l i e s c a n
i n d i c a t e t h e n e c e s s i t y of a c c o u n t i n g f o r
t h e c o r r e s p o n d i n g i n h o m o g e n e i t y of t h e
medium).
V. I. K E Y L I S - BOROK
112)
2. The determination
of ground
displacements.
§ 4.
-
EEDUCTION
HOMOGENEOUS
Using t h e d a t e of t r a c e a m p l i t u d e s we
d e t e r m i n e t h e corresponding g r o u n d disp l a c e m e n t s applying t h e usuai m e t h o d s . I t
would b e sufbcient t o m e a s u r e t h e m in a n y
a s s u m e d u n i t s (not necessarily absolute).
1. General
OF
OBSERVATIONS
A s e m i - t a n g e n t t o t h e r a v in a h y p o c e n t r e will b e called a straightened
ray
/V
s o
\s
Sv--
v
V
/
•
I t should b e n o t e d now which of t h e
observations of g r o u n d displacements w i b
b e necessary for t h e d e t e r m i n a t i o n of t h e
signs of ua, w£, w f .
F o r t h e d e t e r m i n a t i o n of t h e signs of
ua it is sufbcient t o k n o w n only t h e sign
of Pz or Pv.
T b e d e t e r m i n a t i o n of t h e signs of
and
u £ r e q u b e t h e knowledge of t h e direction
of t h e t o t a l horizontal g r o u n d displacement
(it should b e resolved i n t o 8V a n d S H ) a n d
t h u s , — t h e values of t h e NS a n d EW
c o m p o n e n t s . W h e n only t h e signs of t h e s e
c o m p o n e n t s being k n o w n , t h e sign of one of
t h e c o m p o n e n t s SH or Sv can b e d e t e r m i n ed (fig. 6).
Finally, t h e sign of 8 t u n a m b i g u o u s l y
d e t e r m i n e s t h e sign of u'b', provided t h e
angle of incidence t o t h e e a r t h surface is
less t h a n criticai.
As a result of t h i s p a r t of i n t e r p r e t a t i o n
a t a b l e should b e m a d e for a b t h e f o u n d
signs a n d a m p l i t u d e s of P 2 , Pv, SH, Sv, Sz
in various waves.
A
statements.
N
Fig. 6 - The determination of the sig:
NS and E\Y are known.
TO
MEDIUM.
\
5
efiicentze
of SV or SH when only the signs of
(fig. 7). A w a v e t h a t would b e observed
f r o m t h e same focus if t h e m e d i u m were
homogeneous a n d i d e a b y elastic w i b b e
called a primary
wave.
Using t h e observed ground d i s p l a c e m e n t s
(they can be Pz, Pv, SH, 8V, Sz) we shall
t r y t o find such signs (and maghe 7-atios) of
Ua, u^, u» in the primary wave that would
he observed on the straightened rays at great
distance from the source.
The data sought
for are Constant along a s t r a i g h t e n e d r a y
beginning w i t h sufficiently g r e a t distance;
t h i s p e r m i t s t o ascribe t h e m to conventional
points of observations — t h e stereographic
p r o j e e t i o n of t h e s t r a i g h t e n e d rays.
T h e s a m e idea lies in t b e m e t h o d of
B y e r l y except t h a t h e talces a n o t h e r projeetion a n d does n o t consider t r a n s v e r s e
waves.
2. ZJse of
observations.
If t h e n a t u r e of t h e w a v e does n o t cliange
along its p a t i i t h e n ua is d e t e r m i n e d b y P,
or P„;
— b y SH, a n d w£ — b y Sv or S..
THE
DETERMINATICI
OF E A R T H Q U A K E M E C H A N I S M , E T C .
T h e o b s e r v a t i o n s of t h e w a v e s t h a t
change their n a t u r e along their paths m a y
c h a r a c t e r i z e o n l y ua o r up, a s t h e SH w a v e
c a n n o t c h a n g e i t s n a t u r e (*). ( F o r e x a m p l e
t h e o b s e r v a t i o n s of sP a n d pS m a y c h a r a c t e r i z e o n l y u' b a n d u a r e s p e c t i v e l y ) .
culations show t h a t t h e inhomogeneity a n d
r e f r a c t i o n a t t h e i n n e r b o u n d a r i e s in a
n u m b e r of cases h a v e n o e f f e c t o n t h e signs
a n d h a v e s u f f i c i e n t l y low e f f e c t o n t h e r a t i o s
of d i s p l a c e m e n t s [ K e y l i s - B o r o k e t al., 1957].
a) The signs of ua,
It is worih-while
any point,
where
and up may
be
to use the observations
one of the signs
of
in
ua,
determined.
113
u f f , and
u£.
T h e signs ( d i r e c t i o n ) of t h e a r r i v a l a c a n
change at t h e boundaries, which the r a y
traverses. P r a c t i c a l l y it h a p p e n s generally
a) n e a r s t a t i o n s
Fig. 7 - Conventional a n d real points of observation.
(B — conventional, A — •real) OB s t r a i g h t e n e d rays.
T h e r a t i o of t h e v a l u e s of t h e s e c o m p o n e n t s will b e u s e d o n l y in t h e c a s e w h e n
t h e y a r e d e t e r m i n e d f o r t h e w a v e s of t h e
s a m e p a t h r e c o r d e d a t o n e s t a t i o n (for
e x a m p l e , P a n d S, PP a n d SS, e t c . ) a n d
consequently correspond to sufficiently d o s e
c o n v e n t i o n a l p o i n t s ( t h e o r e t i c a l l y i t is p o s sible t o u s e t h e d i s p l a c e m e n t r a t i o i n diff e r e n t p o i n t s ; h o w e v e r , i t is m o r e c o m p l i c a t e d a n d , a s e x p e r i e n c e s h o w s , consider a b l y less r e l i a b l e ) .
3. Determination
of signs and
ratio in the primary
wave.
i n case of r e f l e c t i o n or in c a s e of s u c h w a v e s
t h a t change their nature along their paths.
I n a d d i t i o n , t h e c u r v a t u r e of t h e r a y c a u s e s
a c h a n g e of t h e sign of t h e $ F a r r i v a i i n
t r a n s v e r s e w a v e s e m e r g i n g downward
from
t h e f o c u s (fig. 8).
Table 2 can be utilized to determine t h e
signs of ua, up t a k i n g i n t o c o n s i d e r a t i o n
displacement
O n e s h o u l d e l i m i n a t e t h e e f f e c t of t h e
i n h o m o g e n e i t y of t h e m e d i u m so t h a t t o
determine t h e displacements in a p r i m a r y
wave.
I n practice one can eliminate t h e effect
of o n l y k n o w n b o u n d a r i e s , a n d t h e c u r v a t u r e of t h e r a y . H o w e v e r , t h e e x p e r i e n c e
in i n t e r p r e t a t i o n a n d t h e t h e o r e t i c a l cal(*) A n exception is t h e case of non-parallel
b o u n d a r i e s w h e n i t is necessary to resolve a
t r a n s v e r s e w a v e i n t o SH a n d SV for each
piane of incidence a n e w ; in t h i s case t h e surface
displacements SV a n d SH will be linear f u n c t i o n s
of u £ a n d u' b '.
S U due to t h e c u r v a t u r e of t h e r a y .
t h e signs of t h e o b s e r v e d d i s p l a c e m e n t s .
T h e t a b l e is c o m p i l e d t a k i n g i n t o a c c o u n t
t h e s u r f a c e of E a r t h , t h e c u r v a t u r e of t h e
r a y s a n d t h e t w o b o u n d a r i e s of t h e c r u s t
of p r i m a r y i m p o r t a n c e .
T h e e f f e c t of a n y o t h e r b o u n d a r y will b e
i n t r o d u c e d i n t o t h i s if w e t a k e i n t o a c c o u n t
t h e signs of t h e c o r r e s p o n d i n g coefficients
g i v e n i n t h e t h e o r y of p i a n e w a v e s .
V. I. K E Y L I S -
114)
Table
2
T h e occurrence of like a n d unlike signs
in t h e p r i m a r y w a v e displacements a n d t h e
e a r t h s u r f a c e displacements.
" + " indicates t h a t tlie signs are t h e same,
" — " t h a t t h e signs are different. F o r t r a n s verse w a v e s t h e sign corresponds to t h e horizontal Sv c o m p o n e n t of g r o u n d displacements.
I n a p r i m a r y w a v e u a or
can be d e t e r m i n e d
when t h e first l e t t e r in t h e w a v e i n d e x is P
or 8 respectively. u' b ' has t h e s a m e sign as Sh.
Wave
index
Sig n
+
+
P
P
pP
—
PP
—
pPP
PPP
PcP
pP*
Wave
index
Sign
Wave
index
Sign
PP*
sSS
+
sP
SSS
—
ScS
+
SP
—
8(*)
+
+
S
sS*
—
+
+
+
S(**)
—
pS
—
sS
—
PS
—
- -
SS
+
SS*
«Vj-'r„n
(*)
T h e r a y moves u p w a r d f r o m t h e
focus.
(**) T h e r a y moves d o w n w a r d f r o m
t h e focus.
B e s i d e s t a b l e 2 is c o m p i l e d o n t w o assumptions:
1) t h a t t h e E a r t h s u r f a c e d o e s n o t
c a u s e a c h a n g e i n t h e sign of t h e S V a r r i v a i .
I t is a l w a y s c o r r e c t o n l y if t h e a n g l e of
i u c i d e n c e t o t h e s u r f a c e is less t h a n t b e
criticai one; otherwise one should use a
n o m o g r a p h in [ M a b n o v s k a j a , p . 151];
2) t h e sign of PP, pP c h a n g e s ( a n d
sS, SS d o e s n o t ) a f t e r r e b e c t i o n a t t b e
s u r f a c e . I t is c o r r e c t p r a c t i c a b y f o r ali
e p i c e n t r a l d i s t a n c e s if a0 (P-wave
velocity
a t t b e s u r f a c e ) i s n o t m o r e t h a n 5.5-6 k m / s e c ,
w h i c h g e n e r a l l y o c c u r s . T h e case of g r e a t e r
a0 is c o n s i d e r e d b y J . H . H o d g s o n a n d B .
E . I n g r a m (Bull. Seism.
Soc. Amer.
46,
N . 3, 1956).
b ) Determination
of
amplitudes.
E o r t b e e b m i n a t i o n of t b e e f f e c t of t b e
b o u n d a r i e s t r a v e r s e d b y a r a y i t is n e c e s -
BOROK
sary to divide the ground displacements
b y piane w a v e reflection a n d refraction
c o e f b c i e n t s . T h e s e coefficients a r e g i v e n i n
[ K e y b s - B o r o k e t al., s u p p l . I V ] .
I t m u s t be borne in m i n d t h a t for t h e
a n g l e s of i n c i d e n c e g r e a t e r t h a n t h e c r i t i c a i
a n g l e u £ is t h e l e a s t r e b a b l e .
4. Plotting
of conventional
points
stereographic
projeetion.
on a Wolf
T h e c o o r d i n a t e s of a c o n v e n t i o n a l p o i n t :
t h e a z i m u t h a of t h e r a y ( f r o m e p i c e n t r e
t o t h e station) a n d t b e r a y i n c b n a t i o n ih
when emerging from the focus are determ i n e d b y u s u a i g e o m e t r i e m e t h o d s . If a
a n d in a r e n o t a c c u r a t e , t h e r e g i o n of p o s sible p o s i t i o n s of t h e c o n v e n t i o n a l p o i n t is
o u t l i n e d . T h e o b s e r v a t i o n s of l o n g i t u d i n a l
and transverse waves with identical rays
(for i n s t a n c e , P a n d S, PP a n d SS, pP a n d
sS a n d so on) a t o n e s t a t i o n m a y b e r e f e r r e d
(in first a p p r o x i m a t i o n ) t o o n e c o n v e n t i o n a l
p o i n t . S t r i c t l y s a y i n g , i t is n o t a b s o l u t e l y
accurate.
H o w e v e r , t h e d i s c r e p a n c y of
the conventional points usually m a y be
n e g l e c t e d ; b u t t h e p r o c e d u r e of t h e i n t e r p r e t a t i o n b a s e d o n signs will n o t c h a n g e
if f o r g r e a t e r a c c u r a c y w e d i v i d e t h e conventional points mentioned above.
a) N e a r
stations.
a c a n b e d i r e c t l y m e a s u r e d o n a m a p (if
there are no azimuthal anomabes pointing
o u t t o t b e p r e s e n c e of i n c b n e d irrterfaces).
ih is d e t e r m i n e d in a c c o r d a n c e w i t h t h e
s e i s m o g e o l o g i c a l c o n d i t i o n s of t h e r e g i o n .
F o r t h e c a s e of n o n - h o r i z o n t a l i n t e r f a c e s
a a n d ih m a y b e d e t e r m i n e d b y m a k i n g
c e r t a i n c o n s t r u c t i o n s o n a Wolf s t e r e o graphic projeetion.
These constructions
suggested b y E . N. Bessonova are described
i n [ K e y b s - B o r o k e t al.] a n d a r e s i g n i f i c a n t
when sharp incbned interfaces are observed
near t b e focus.
h) D i s t a n t
stations.
At distant earthquakes the inhomogen e i t i e s of t h e e a r t h c r u s t b a v e a less d i s t o r ting effect on t b e rays.
a can be d e t e r m i n e d on a Wolf stereog r a p h i c p r o j e e t i o n ; i t s h o u l d b e k e p t in
THE
DETERMINATICI
119
OF E A R T H Q U A K E MECHANISM, ETC.
m i n d t h a t t h e difference of a f r o m t h e
a z i m u t h f r o m t h e s t a t i o n to t h e epicentre
is n o t 180° d u e to non-parallel meridiana.
T h e plots f o r d e t e r m i n i n g ih a n d i0 (angle
of incidence t o t h e surface) for various
w a v e s a n d various h d r a w n on t h e basis
(for d i s t a n t stations) of t h e Jeffreys-Bullen
t r a v e l - t i m e c u r v e a r e given in figs. 1 6 a - f .
F o r t h e d e t e r m i n a t i o n of ih one can also
use t h e tables of " e x t e n t e d distances "
compiled b y J . H . H o d g s o n ; t h e s e distances a r e equal to ctgi A .
c) J o i n t i n t e r p r e t a t i o n of records w h e n
some r a y s a r e directed d o w n w a r d a n d some
— u p w a r d f r o m t h e focus.
T h e r a y s P , S, P P , PKP a n d so on are
directed d o w n w a r d a n d t h e r a y s pP, sS,
etc. — u p w a r d f r o m t h e focus. T h e r a y s
P , S, f r o m n e a r e a r t h q u a k e s a r e directed
u p w a r d , a n d t h e r a y s of h e a d waves —
d o w n w a r d f r o m t h e focus. H o w e v e r , ali
t h e c o n v e n t i o n a l p o i n t s (for ali waves)
should b e p l o t t e d on one semi-sphere (bett e r on t h e u p p e r one).
If t h e s t r a i g h t e n e d r a y does n o t t r a v e r s e
t h i s semi-sphere (the w a v e p r o p a g a t i n g
d o w n w a r d f r o m t h e focus), t h e conventional
p o i n t should b e t a k e n on t h e extension of
t h e r a y in t h e opposite direction (ik r e m a i n s
t h e same, t o a t h e v a l u e of 180° is added).
T h e signs a n d ratios of ua, w f ,
in
such a conventional p o i n t will b e t h e s a m e
as on t h e s t r a i g h t e n e d r a y for t h e dipole
w i t h m o m e n t a n d sources, c, 6, g (fig. 5),
etc.; for t h e sources a, f , etc. t h e signs
considered should b e reversed.
As a result of this part of
interpretation
ali the conventional points of observations are
plotted on a Wolf stereographic
projection,
if the sign of even one of the
displacement
components for these points being found-, determined signs of ua, u'b', ub are put near them
(top to bottom)-, the unlcnoivn signs are designated by a ivavy line <—
F o r t h e conventional points w h e r e ali
t h e a m p l i t u d e s of ua , u'b',
are k n o w n
it is necessary also to d e t e r m i n e t h e ratios
« > f , K K -
§ 5.
-
DETERMINATION
PARAMETERS
OF
OF A FOCUS
THE
DYNAMIC
(FAULT
PLANE
SOLUTION).
1. Drawing
of nodal
lines.
T h e n e x t a n d t h e principal p r o b l e m is to
d r a w one of t h e theoretical systems of nodal
lines (figs. 4, 5) on a Wolf stereographic
p r o j e c t i o n in accordance w i t h t h e observed
distribution of t h e signs of d i p l a c e m e n t s
(according t o § 1 t h e nodal lines d e t e r m i n e
t h e orientation of t h e f a u l t p i a n e a n d t h e
direction of motion).
A t t h a t it is necessary to t a k e into
a c c o u n t t h e " s t a n d a r d " (possible theoretically) correlation of signs in each p o i n t
(figs. 4, 5).
T h e w a y s of drawing t h e n o d a l lines
can b e explained best of ali b y e x a m p l e s
(here, as for a n y kind of i n t e r p r e t a t i o n , it
is difficult to m a k e t h e general rules
exhausting). I n figs. 9-12 f o u r examples are
given. T h e y are based on t h e experience
of practical i n t e r p r e t a t i o n , b u t greatly simplified.
F i g . 9. Two nodal lines a r e necessary
for dividing t h e signs of ua, however, t h e y
a r e n o t d r a w n u n a m b i g u o u s l y (fig. 9a a n d
fig. 96). I t can b e easily seen t h a t t a k i n g
i n t o a c c o u n t t h e signs of ub a n d u'b we can
d r a w t h e n o d a l lines only in one w a y , as
is shown in fig. 9b. T h e obtained combin a t i o n of signs is consistent w i t h t h a t of
t h e " s t a n d a r d " one (fig. 4).
Fig. 10. T w o nodal lines: P = 0 (i. e.
ua — 0) a n d y = 0 should be d r a w n to
divide t h e signs of ua; in t h e e x a m p l e
considered t h e y are d r a w n u n a m b i g u o u s l y .
I t is necessary to use t h e signs of ub a n d u'b
so t h a t t o find out which of t h e s e lines
corresponds to t h e f a u l t p i a n e y — 0; t h e
signs can b e dibided in two w a y s shown in
fig. 10« a n d 106. T h e comparison of b o t h
v a r i a n t s w i t h t h e " s t a n d a r d " one (fig. 4)
leads t o t h e conclusion t h a t such a combin a t i o n of signs as o b t a i n e d in fig. 10« c a n n o t
exist in reality.
Thus, t h e only possible v a r i a n t of interp r e t a t i o n is t h a t shown in fig. 96.
Fig. 11. T h e signs of ua
a n d ub, can
b e divided in t w o ways as if is shown in
T H E D E T E R M I N A T I C I OF E A R T H Q U A K E MECHANISM, ETC.
117
fig. I l a a n d 116. H o w e v e r , a f t e r comparison w i t h t h e " s t a n d a r d " distribution
t h e v a r i a n t of I l a should be neglected.
Fig. 12.
The interpretation with the
help of signs is a m b i g u o u s : t h e s y s t e m of
n o d a l lines can b e d r a w n in two w a y s
(fig. 12a a n d 126), b o t h v a r i a n t s a r e consistent w i t h fig. 4. H o w e v e r , it is possible
to establish which of t h e t w o indicated
v a r i a n t s corresponds to t h e reality if t h e
for in reducing t h e observations t o a homogeneous m e d i u m .
Therefore, only t h e a p p r o x i m a t e direction
of t h e axis x — t h e region of its p r o j e c t i o n
on a Wolf stereographic p r o j e c t i o n — can
b e d e t e r m i n e d , a n d it is f a r f r o m being
every case. H o w e v e r , it is n o t connected
w i t h additional c o m p u t a t i o n a n d o f t e n m a y
be useful.
E q u a t i o n s [7a]-[9a] § 2 can b e used for
projection of t h e axis x is d e t e r m i n e d using
t h e r a t i o of a m p l i t u d e s .
H e r e for t h e u p p e r r i g h t p o i n t we h a d :
hH = — 2.5; hp = 7.0; / = — 0.35; a n d
for t h e lower (second to t h e right) hH = 0.25;
h p = 0.5; / = + 0.5. I n this case t h e
v a r i a n t in fig. 12a should be neglected (see
t h e n e x t po nt).
I t would b e sufficient to d e t e r m i n e only
t h e region, w h e r e t h e p r o j e c t i o n of t h e
axis x lies.
d e t e r m i n i n g t h e axis x.
Each equation
d e t e r m i n e s t h e piane, containing this axis
(only t w o of t h e s e equations a r e independent). If ua , uff , up are k n o w n a t some
p o i n t t h e axis x can be f o u n d as t h e intersections of t w o such planes. T h e graphical w a y (on a Wolf stereographic projection)
of d e t e r m i n i n g t h e axis x is t h e m o s t conv e n i e n t one. T h e projections of t h e planes
m e n t i o n e d a b o v e on a Wolf stereographic
projection a r e arcs of a g r e a t circle (they
will b e called " «-arcs " f u r t h e r on); t h e
intersection of t h e m d e t e r m i n e s t h e projection of t h e axis x. I n practice t h e det e r m i n a t i o n of t h e axis x on a Wolf stereographic projection is p u r e l y graphical,
using t h e n o m o g r a p h s given in [Malinovsk a j a ] a n d in fig. 13 (*).
2. The determination
of the axis x (direction
of motion) using displacement
ratios.
T h e displacement ratios are considerably
g r e a t e r effect b y t h e inhomogeneities a n d
n o n ideallv elastic properties of t h e e a r t h
t h a n t h e signs; a t t h e s a m e t i m e t h e effect
of t h e s e f a c t o r s can only p a r t l y be allowed
(*) For a more complete set of nomographs
apply to the Institute of the E a r t h ' s Physics.
118)
V. I. K E Y L I S - B O R O K
O n t h e n o m o g r a p h s t h e r e a r e " x-arcs "
f o r v a r i o u s y a l u e s of h " , h p , /.
These
values are p l o t t e d near t h e " #-arcs ".
Diflerent n o m o g r a p h s corresponrl t o diff e r e n t v a l u e s of i h — t h e i n c l i n a t i o n of a
straightened ray.
ride w i t h t h e c e n t r e of t h e n o m o g r a p h ,
a n d t h e m a i ' k of t h e a z i m u t h of t h e conv e n t i o n à l p o i n t of o b s e r v a t i o n w i t h t h e
u p p e r e n d of t h e v e r t i c a l d i a m e t e r of t h e
n o m o g r a p h ( t h e n t h e c o n v e n t i o n a l p o i n t on
tlie t r a c i n g p a p e r will c o i n c i d e w i t h t h e
hp
Fig. 13o - T h e n o m o g r a p h s for d e t e r m i n i n g motion direction using
a m p l i t u d e ratio.
I n o r d e r t o flnd a n " z - a r c " i t is n e c e s s a r y : 1) t o c h o o s e a n o m o g r a p h w i t h t h e
v a l u e s of i — in (if t h e n o m o g r a p h w i t h
t h e r e q u i r e d v a l u e of ih is a h s e n t t h e w h o l e
c o n s t r u c t i o n is m a d e f o r t w o n e i g h b o u r i n g
v a l u e s of i/, a n d t h e n i n t e r p o l a t i o n is c a r r i e d o t ) ; 2) t o s u p e r i m p o s e t r a c i n g p a p e r
o n t h e n o m o g r a p h ; 3) t o m a k e t h e c e n t r e
of t h e p r o j e c t i o n o n t h e t r a c i n g p a p e r coin-
d o u b l e circle o n t h e n o m o g r a p h ) ; 4) t o c o p y
t h e " a;-arcs " c o r r e s p o n d i n g t o t h e g i v e n
v a l u e s of h H , h p , f o n t h e t r a c i n g p a p e r .
T h e i n t e r s e c t i o n of t h e " re-arcs " g i v e s
t h e p r o j e c t i o n of t h e a x i s x.
In practice
w e d o n o t d r a w t h e w h o l e a r c s a n d flnd
immediately
t h e i r p o i n t of i n t e r s e c t i o n .
T h e " « - a r c s " c o n s t r u c t e d f o r t h r e e displacement ratios a t one station intersect
THE
DETERMINATICI
OF
E A R T H Q U A K E MECHANISM, ETC.
in o n e p o i n t a s o n e of t h e s e v a l u e s is n o t
independent.
B e s i d e s , i t is n e c e s s a r y t o n o t e t h e foll o w i n g p r o p e r t i e s of " « - a r c s " : t h e " a ? - a r c s "
constructed for / and hH pass through
t h e p o i n t of o s s e r v a t i on a n d t h e p o l e of
119
t i o n c e n t r e , n o m a t t e r w h a t is t l i e a c t u a l
d i r e c t i o n of t h e a x i s x (since t h e poles of
ali t h e c o n v e n t i o n a l p o i n t s g a t h e r a r o u n d
t h e centre).
T h e v a l u e of le s h o u l d b e f o u n d e m p i r i cally b y : 1) s e l e c t i n g s u c h e a r t h q u a k e s f o r
Fig. 13 6 - T h e n o m o g r a p h s for d e t e r m i n i n g motion direction using
a m p l i t u d e ratio.
t h i s p o i n t r e s p e c t i v e l y ; a n d t h e " a;-arcs "
c o n s t r u c t e d f o r Tip h a v e t h e a z i m u t h p e r p e n d i c u l a r t o t h e a z i m u t h of t h e s t a t i o n .
These properties should be t a k e n into
account w h e n trying to deterraine t h e axis
of x u s i n g " a - a r c g " f o r t h e a m p l i t u d e r a t i o s
of t h e s a m e c o m p o n e n t s a t v a r i o u s s t a t i o n s .
F o r i n s t a n c e , ih b e i n g g r e a t , t h e n ali t h e
" aj-arcs " f o r hH c o m e closer t o t h e p r o j e c -
w h i c h t h e a x i s x is d e t e r m i n e d o n l y f r o m
signs a n d 2) d e t e r m i n i n g le so t h a t t h e
d i s p l a c e m e n t r a t i o s h o u l d g i v e a close r e s u l t .
T h e s e s a m e v a l u e s of h m a y b e u s e d f o r
s t u d y i n g t h e focii of a g i v e n r e g i o n .
I t p r o e e e d s f r o m e x p e r i e n c e t h a t le s h o u l d
b e t a k e n less t h a n (a/6) 3 . F o r t h e T a n g o
e a r t h q u a k e 1927, 1 < le < 3 ; it w a s e s t a b l i s h e d in t h e p r o c e s s of i n t e r p r e t a t i o n of
120)
V. I. K E Y L I S -
a v a l u a b l e s e t of s e i s m o g r a m s c o l l e c t e d b y
Dr. H o d g s o n , E . A. (Dominion O b s e r v a t o r y ,
Canada), and kindly sent to me by Dr.
Hodgson, J . H.
A n y e x c e s s in t h e a c c e p t e d v a l u e of 1; c a n
b e e a s i l y n o t i c e d f r o m t h e d i s p l a c e m e n t of
BOROK
s y s t e m of t h e i n i t i a l d a t a d e t e r m i n e s t h e
n o d a l lines in s u c h a w a y t h a t t h e y c a n
b e t r a n s f e r r e d in c e r t a i n l i m i t s w h i c h indicate t h e possible errors.
I t should be borne in m i n d t h a t these
l i m i t s e s s e n t i a l l y d e p e n d o n t h e e r r o r s of
Fig. 13 c - T h e n o m o g r a p h s for d e t e r m i n i n g motion direction using
a m p l i t u d e ratio.
t h e axis x to t h e corresponding
tional points.
§ G. -
ESTIMATION
OF
conven-
ACCURACY
T h e a c c u r a c y of t h e i n t e r p r e t a t i o n of
e a c h e a r t h q u a k e is e s t i m a t e d d i r e c t l y in
t h e p r o c e s s of d r a w i n g t h e n o d a l lines: e a c h
t h e c o o r d i n a t e s of t h e c o n v e n t i o n a l p o i n t s
t h e m s e l v e s (i. e. of t h e r a y d i r e c t i o n s i n t h e
hypocentre).
T h e m a i n s o u r c e of e r r o r s (especially f o r
near eartliquakes recorded by high frequenc y i n s t r u m e n t s ) is t h e u n k n o w n s t r u c t u r e
of m e d i u m , f o r e m o s t , t h e i n t e r f a c e s n e a r
t h e focus.
THE
DETERMINATICI
121
O F E A R T H Q U A K E M E C H A N I S M , ETC.
T l i e c o m p l e t e e s t i m a t i o n of t h e a c c u r a c y
requires t h a t the interpretation should be
c o n d u c t e d f o r t h e e x t r e m e possible p o s i t i o n s
of t h e c o n v e n t i o n a l p o i n t s .
F o r t h e i n t e r p r e t a t i o n o n a l a r g e scale i t
is n e c e s s a r y t o i n v e s t i g a t e t l i r o u g h l y t h e
REFERENCES
V. I. KEYLIS-BOROK, Proc.. TJ. G. G. I.
Rome.
V.
Congr..,
I . K E Y L I S - B O R O K , E . N . BESSONOVA, 0 .
D . GOTZADZE, S. D . JLOGAN, T . T . K U K C H .
Fig. 13d - T h e n o m o g r a p h s for d e t e r m i n i n g motion direction u r i n g
a m p l i t u d e ratio.
p o s s i b l e e r r o r s of a t l e a s t o n e or h y p o c e n t r e s
of e a c h g r o u p .
T h e a u t h o r is g r e a t l y o b l i g e d t o P r o f . P .
Caloi f o r h i s v a l u a b l e r e m a r k s a n d h e l p i n
p u b h c a t i o n of t h e p r e s e n t a r t i c l e .
Institute
Ac.
of the Earth's
Sci. USSE,
Physics,
Moscow.
TIKOVA, I . V .
KIRILOVA, L . N .
SKAYA, A . A . SORSKY,
skogo Instituta
il. 40, 1957.
Trudi
Akademii
MAI.INOV-
Geofizitclie-
Nauk
SSSR,
L . N. MALINOVSKAYA., Trudi
Instituta
Akademii
Nauk
1954.
Gdufisitclieskogo
SSSR,
n. 22,
A . R . RITSEMA.,
Geoph.,
n. 1, 1955.
Ind.
J.
Met.
v.
6,
122)
SUPPLEMENT.
ON A W O L F
V. I. K E Y L I S PRINCIPAL
CONSTRUCTIOXS
STEREOGRAPHIC
PROJECTION.
K g . 1 4 a r e p r e s e n t s a Wolf s t e r e o g r a p h i e
p r o j e c t i o n w h i c h is a s t e r e o g r a p h i e p r o j e c t i o n of a s e m i - s p h e r e on a p i a n e .
The
BOROK
t i o n of t h e s p h e r e b y p l a n e s p e r p e n d i c u l a r
t o N S ) . T h e s e lines s h o u l d n o t t a k e n f o r
g e o g r a p h i c a l m e r i d i a n s a n d p a r a l l e l s since
t h e c e n t r e of t h e s p h e r e c o i n c i d e s w i t h t h e
hypocentre.
T h e angle
is c o u n t e d off a l o n g t h e
5
Fig. 1 4 a - Wolf stereographie p r o j e c t i o n .
s c h e m e of t h e s e m i - s p h e r e b e i n g p r o j e c t e d
is s h o w n i n fìg. 14/;.
T h e c e n t r e of t h e
p r o j e c t i o n w i t h f o u r p o i n t s a r o u n d it corr e s p o n d s t o t h e v e r t i c a l a x i s 00. T h e circ u m f e r e n c e b o u n d i n g t h e p r o j e c t i o n corr e s p o n d s t o t h e circle u p o n w h i c h t h e
s e m i - s p h e r e is s u p p o r t e d t o s t a n d .
L e t t h e e n d s of t h e d i a m e t e r N S (fìg. 146)
b e t h e p o l e s of t h e s p h e r e ; d r a w t w o s y s t e m s
o.f lines o n t h e s p h e r e : m e r i d i a n s (sections of
the sphere b y planes forming different angles w i t h t h e a x i s 00) a n d p a r a l l e l s (see-
horizontal diameter f r o m the centre to a
given meridian. The angle y for each parallel is c o u n t e d off a l o n g a n y o n e of "the
m e r i d i a n s (each u n i t is e q u a l t o 2° i n
fìg. 14«).
T h e s t r a i g h t line a n d t h e p i a n e " p a s s i n g
t l i r o u g h t h e c e n t r e of t h e s p h e r e a r e r e p r e sented on t h e Wolf stereographie p r o j e c t i o n
respectively b y a point a n d b y such an are
which a f t e r being r o t a t e d coincides w i t h
o n e of t h e m e r i d i a n s .
T h e p o l e of t h e
p i a n e m e a n s t h e p r o j e c t i o n of i t s n o r m a l .
THE
DETERMINATICI
OF
E A R T H Q U A K E M E C H A N I S M , ETC.
T h e p o l e of t h e l i n e is t h e p r o j e c t i o n of
i t s n o r m a l l y i n g in o n e v e r t i c a l p i a n e w i t h
t i h s Une.
Ali c o n s t r u c t i o n s o n a W o l f p r o j e c t i o n
can b e m o s t conveniently fulfllled on a
t r a c i n g p a p e r being superimposed on t h e
123
a of t h e h o r i z o n t a l p r o j e c t i o n ) is g i v e n .
F i n d t h e p r o j e c t i o n of t h i s line (i. e. t h e
p o i n t h a v i n g t h e c o o r d i n a t e s a , ih).
O n t h e o u t e r circle ( t h e b o u n d of t h e
s t e r e o g r a p h i c p r o j e c t i o n ) c o u n t off t h e a n g l e
a clockwise a n d m a l t e a m a r k .
Pig. 14 6 - Meridians a n d parallels t r a c e d on a Wolf stereographic p r o j e c t i o n .
stereographic projection.
Before rotating
t h e t r a c i n g p a p e r a b o u t t h e c e n t r e of t h e
W o l f s t e r e o g r a p h i c p r o j e c t i o n i t is n e c e s s a r y t o m a r k o n i t t h e p o s i t i o n of t h e c e n t r e
a n d t h e e n d of t h e v e r t i c a l d i a m e t e r of
the stereographic projection. W h e n making
constructions we shall consider t h a t t h e
d i r e c t i o n u p w a r d (z axis) is p r o j e c t e d i n t h e
c e n t r e of t h e s t e r e o g r a p h i c p r o j e c t i o n , t h e
a x i s N b e i n g d i r e c t e d t o t h e N o r t h (ÌJ) t h e
a x i s E t o t h e E a s t (x).
Consider now t h e principal constructions
on a Wolf p r o j e c t i o n .
T h e p l a n e s a n d t h e s t i ' a i g h t lines cons i d e r e d b e l o w p a s s t h r o u g h t h e c e n t r e of
t h e sphere.
1. T h e d i r e c t i o n of a s t r a i g h t line ( t h e
i n c l i n a t i o n ih t o t h e v e r t i c a l , t h e a z i m u t h
B y rotating the tracing paper m a k e this
m a r k c o i n c i d e w i t h o n e of t h e d i a m e t e r s
(it m a k e s n o d i f f e r e n c e e i t h e r h o r i z o n t a l
or vertical).
C o u n t off t h e a n g l e ih a l o n g t h e d i a m e t e r
f r o m t h e c e n t r e of t h e s t e r e o g r a p h i c p r o j e c t i o n t o t h e d i r e c t i o n of t h e m a r k .
N o w t h e p o i n t b e i n g f o u n d h a s t h e coord m a t e s a, ih2. T h e a z i m u t h A of t h e d i p p i n g a n d t h e
d i p e of a p i a n e a r e g i v e n . F i n d t h e p r o j e c t i o n of t h e p i a n e .
M a k e a m a r k for t h e a z i m u t h A on t h e
b o u n d of t h e s t e r e o g r a p h i c p r o j e c t i o n . R o tating the tracing paper m a k e this m a r k
coincide with t h e horizontal diameter.
C o u n t off t h e d i p a l o n g t h e s a m e h o r i z o n t a l
diameter from t h e stereographic projection
124)
V. I. K E Y L I S -
REPRESENTATIOII
flF SEISMIC
BOROK
FAULTS
UN
' . t MKMK.'J
OFOISLDCATIOMS IN T H E ORlGlNS OF EARTHQLWKES
•f 'ne fa-i; f.lan€
Ammattì
-4/iinu'h
Ptunge
55
- v
Uipote w i t h m o m e n t
|
of motion direction
30
~ focus
S
a e p t h («in)
Heavy àn&-that
Uss
definite fault
ikle of fautt,
urkuh mmrei
upva/icL.
planes
N
IO
- Azimuth of the horizontai projection of
motion direction
} Simple force
• Plungf of motion direction
(<vuj aiymm*foti>a£ feudty
Focus depth (nm)
s
N
- .Azimuth of the horiiontal projection
of motion direction
- Plu/lge of motion direction
500
| ,,nIy , h e
dirreti(m
. ..,,.•„,„
u
!' d e t e r m i n e d
focus dirli! hm\
Fig. 15 - T h e r e p r e s e n t a t i o n of t h e f o u n d f a u l t piane solutions on m a p s .
T h e h e a v y line corresponds to t h a t side of f a u l t which m o v e s u p w a r d .
THE
DETERMINATICI
129
OF E A R T H Q U A K E M E C H A N I S M , E T C .
b o u n d w h i c h is o p p o s i t e t o t h e m a r k of
the azimuth.
N o w t h e meridian passing trhough the
f o u n d p o i n t is t h e p i a n e b e i n g s o u g h t f o r
( t h e c o o r d i n a t e s of t h i s p o i n t , a s is e v i d e n t ,
a r e ISO + A , 90 — e ) .
3. T w o p o i n t s o n a. W o l f s t e r e o g r a p h i c
p r o j e c t i o n a r e g i v e n (i. e. p r o j e c t i o n s of
t h e s t r a i g h t lines p a s s i n g t h r o u g h t h e c e n t e r
of t h e s p h e r e ) . F i n d t h e p r o j e c t i o n of t h e
p i a n e c o n t a i n i n g b o t h lines.
R o t a t e the tracing paper with the two
p o i n t s p l o t t e d o n i t a b o u t t h e c e n t r e of
t h e p r o j e c t i o n u n t i l t h e y f a l l o n o n e of
t h e m e r i d i a n s ; d r a w this m e r i d i a n on t h e
t r a c i n g p a p e r , a n d it will b e t h e p r o j e c t i o n
being sought for.
4.
Find
until
with
The
its
the
one
p r o j e c t i o n of a p i a n e is g i v e n .
pole. R o t a t e t h e t r a c i n g p a p e r
p r o j e c t i o n of t h e p i a n e coincides
of t h e m e r i d i a n s . T h e n c o u n t off
200
400
600
<900
•//j
0
70 20 30 40 Off Off 70 00 00700
h = t h e d e p t h of t h e source is shown n e a r
eacli curve.
Figs. 16 a-f - T h e values of ih a n d iu for
various waves.
T h e s y m b o l s of waves are d r a w n in eacli
figure.
A,KM
o m
w
soo eoo 4 HM
0 200 W
000 SOO d, hai
i-miM
b-SM/w
Pig. 16 c
J °
TL=/S0/FM
h^lOOkM
0
70
74
k-emkH
Fig. 166
h-HSl"
V. I. KEYLIS - BOROK
PcP
THE
DETERMINATICI
OF E A R T H Q U A K E MECHANISM, ETC.
90° along t h e horizontal d i a m e t e r f r o m t h e
p i a n e to t h e direction of t h e centre. Now
t h e f o u n d p o i n t is t h e pole.
5. T h e pole of a p i a n e is given. F i n d t h e
p r o j e c t i o n of t h e p i a n e (the e q u a t o r of
t h e given pole).
127
s t r a i g h t lines — are given. Measure t h e
angle m a d e b y t h e m .
Rotating the tracing paper make both
points fall on one of t h e meridians; now
t h e angle s o u g h t for is c o u n t e d off along
this m e r i d i a n .
ti=670kM
0
20
40
fO
#0
700/1'
?/=7ff7tM
M a k e t h e pole fall on t h e horizontale
d i a m e t e r a n d count off 90° along it to t h e
direction of t h e centre. D r a w t h e m e r i d i a n
t h r o u g h t h e f o u n d p o i n t ; t h e n this m e r i d i a n
is t h e e q u a t o r bring sought for.
6. Two
points
—
the
projections
of
7. T h e projections of a p i a n e ancl a
s t r a i g h t line are given. F i n d t h e angle
made by them.
M a k e t h e p r o j e c t i o n of t h e p i a n e coincide
w i t h one of t h e m e r i d i a n s ; now t h e angle
sought for is c o u n t e d off along t h e parallel
passing t h r o u g h t h e given p o i n t .
128
V. J. KEYLIS - BOROK
ABSTRACT
The interpretational
aspect of determining
fault pinne solution for earthquake sources with
the use of both longtitudinal and transverse
waves of various types is described.
The
first arrivai direction of P, SV, SE and
the ratios of their amplitudes can be employed.
The use of the arrivai directions P, SV and
SH {and especially their combination at each
point) sharply lessens the quantity of observations required and makes results
unambiquous.
The properties of various sources (the
dipole loith moment in most detail) are considered.
Wolf stereographie projection discribed in
Supplement
is used for the
interpretation.