Post on 04-Feb-2023
Geo
phys
. J. R
. ast
r. SO
C. (1
982)
71,
175-
792
Larg
e-sc
ale c
hara
cter
istic
s of p
late
bou
ndar
y de
form
atio
ns re
late
d to
the
post
-sei
smic
read
justm
ent
of a
thin
ast
heno
sphe
re
F. K
. Leh
ner a
nd V
. C. L
i *D
ivis
ion
of E
ngin
eerin
g, B
rown
Uni
vers
ity,
Prov
iden
ce, R
hode
Isla
nd, U
SA
Rec
eive
d 19
82 A
pril
26; i
n or
igin
al fo
rm 1
981
Oct
ober
9
Sum
mar
y. T
he la
rge-
scal
e re
spon
se o
f an
elas
tic li
thos
pher
e, ri
ding
on
a ‘th
in’
visc
oela
stic
asth
enos
pher
e, t
o pe
riodi
cally
occ
urrin
g ru
ptur
es a
t a tr
ansf
orm
or
subd
uctio
n-ty
pe p
late
bou
ndar
y is
des
crib
ed a
ppro
xim
atel
y by
app
ropr
iate
lim
it cy
cle
solu
tions
for
a pl
ate/
foun
datio
n m
odel
intro
duce
d pr
evio
usly
by
Rice
. Th
e cy
clic
beh
avio
ur o
f th
ickn
ess-
aver
aged
disp
lace
men
ts, s
train
s an
d st
rain
rat
es,
thei
r de
cay
away
fro
m t
he p
late
bou
ndar
y, a
nd a
reso
lutio
n in
to
cose
ismic
and
pos
t-sei
smic
alte
ratio
ns a
re o
btai
ned
and
thei
r de
pend
ence
on
repe
at t
ime
and
a ch
arac
teris
tic r
elax
atio
n tim
e in
vesti
gate
d. A
com
paris
on
is m
ade
with
exi
sting
per
iodi
c so
lutio
ns f
or t
he s
urfa
ce d
efor
mat
ions
in
a N
ur-M
avko
ha
lf-sp
ace
mod
el.
This
sugg
ests
impo
rtant
ef
fect
s du
e to
vi
scos
ity s
tratif
icat
ion
on p
ost-s
eism
ic r
ebou
nd w
hen
earth
quak
e re
peat
tim
es
exce
ed re
leva
nt r
elax
atio
n tim
es b
y at
leas
t one
ord
er o
f mag
nitu
de.
1 In
trod
uctio
n
Mod
el s
tudi
es o
f po
st-s
eism
ic su
rface
def
orm
atio
ns a
t act
ive
plat
e bo
unda
ries
have
bec
ome
an i
mpo
rtant
met
hod
of in
vesti
gatin
g th
e su
bcru
stal
rheo
logy
of
the
Earth
. Beg
inni
ng w
ith
the
wor
k of
Nur
& M
avko
(19
74)
and
Smith
(197
4),
the
typi
cal e
arth
mod
el a
ssum
ed i
n su
ch th
eore
tical
ana
lyse
s is
that
of
a vi
scoe
lasti
c ha
lf-sp
ace w
ith a
n el
astic
sur
face
laye
r, th
e la
tter
repr
esen
ting
the
litho
sphe
re a
nd t
he f
orm
er a
n as
then
osph
eric
sub
stra
tum
cap
able
of
visc
ous
rela
xatio
n, u
sual
ly i
n th
e m
anne
r of
a li
near
Max
wel
l bod
y. In
the
two-
dim
ensi
onal
N
ur-M
avko
m
odel
an
edge
or
scre
w d
islo
catio
n is
intro
duce
d at
a c
erta
in d
epth
with
in th
e lit
hosp
here
to
repr
esen
t a s
udde
n un
iform
disp
lace
men
t al
ong
a di
p-sl
ip o
r str
ike-
slip
faul
t w
hich
ext
ends
inde
finite
ly a
long
stri
ke. T
he q
uasi
stat
ic su
rface
def
orm
atio
ns p
redi
cted
by
this
mod
el m
ay b
e co
mpa
red
with
geo
detic
obs
erva
tions
of
post-
seism
ic r
ebou
nd m
otio
ns
asso
ciat
ed w
ith l
arge
ear
thqu
akes
, th
us a
llow
ing
new
inf
eren
ces
of r
elax
atio
n tim
es a
nd
visc
ositi
es
for
the
asth
enos
pher
e.
This
earth
quak
e lo
adin
g pr
oble
m
has
since
be
en
mod
elle
d in
gre
ater
det
ail i
n st
udie
s whi
ch a
llow
for
fin
ite fa
ults
and
arb
itrar
y di
strib
utio
ns
of s
lippa
ge (
Bark
er 1
976;
Run
dle &
Jack
son
1977
a, b;
Run
dle
1978
; Mat
su’u
ra &
Tan
imot
o 19
80)
as w
ell
as f
or m
ore
com
plex
rhe
olog
ical
lay
erin
g (C
ohen
19
80,
1981
; Y
ang
&
Toks
oz 1
981)
. D
etai
led
com
paris
ons
betw
een
mod
el p
redi
ctio
ns a
nd g
eode
tic o
bser
vatio
ns
Now
at D
epar
tmen
t of C
ivil
Engi
neer
ing,
MIT
, Cam
brid
ge, M
assa
chus
etts
021
39, U
SA.
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
776
have
bee
n re
porte
d in
par
ticul
ar b
y Th
atch
er &
Run
dle
(197
9) a
nd T
hatc
her
et a
l. (1
980)
fo
r thr
ee m
ajor
Japa
nese
thru
st e
vent
s. H
ere
we
take
a s
peci
al i
nter
est
in t
he w
ork
of S
avag
e &
Pre
scot
t (1
978)
and
Spe
nce
&
Turc
otte
(19
79),
who
hav
e in
depe
nden
tly a
naly
sed
the
cycl
ic b
ehav
iour
dis
play
ed b
y a
Nur
-Mav
ko
eart
h m
odel
in r
espo
nse
to a
seq
uenc
e of
iden
tical
strik
e-sl
ip ev
ents
. One
oft
he
impo
rtan
t co
nclu
sion
s th
at m
ay b
e dr
awn
from
the
ir w
ork
is th
at t
he d
epth
of p
enet
ratio
n of
sig
nific
ant
post
-sei
smic
def
orm
atio
n in
to t
he p
late
int
erio
r de
pend
s no
t on
ly o
n th
e pr
esen
ce o
f a
visc
oela
stic
subs
trate
and
the
dept
h of
fau
lting
, but
also
stro
ngly
on
recu
rren
ce
time.
The
par
amet
er w
hich
gov
erns
the
latte
r dep
ende
nce
is es
sent
ially
a d
imen
sion
less
rat
io
of r
ecur
renc
e tim
e to
a c
hara
cter
istic
Max
wel
l rel
axat
ion
time
of th
e as
then
osph
ere
and
for
the
abov
e m
entio
ned
mod
els
this
num
ber
is ty
pica
lly o
f or
der
10. T
hus,
the
tim
e sp
ans
allo
wed
foi
visc
oela
stic
rela
xatio
n pr
oces
ses
by p
erio
dica
lly r
ecur
ring
earth
quak
es a
re s
uch
as
to w
arra
nt l
ittle
em
phas
is o
n rh
eolo
gica
l la
yerin
g be
yond
the
mod
ellin
g of
the
lay
er o
f lo
wes
t vi
scos
ity.
How
ever
, if,
as
has
ofte
n be
en p
ostu
late
d, t
his
low
visc
osity
zon
e is
conf
ined
to
a ‘t
hin
laye
r’,
then
pos
t-sei
smic
su
rfac
e de
form
atio
ns d
ue t
o vi
scoe
lasti
c re
laxa
tion
may
be
expe
cted
to
diff
er s
igni
fican
tly f
rom
def
orm
atio
ns p
redi
cted
by
half-
sp
ace
mod
els.
In o
ther
wor
ds,
the
ratio
of
recu
rren
ce t
ime
for
grea
t ea
rthq
uake
s ov
er
estim
ated
ast
heno
sphe
ric re
laxa
tion
time
appe
ars
to b
e sm
all e
noug
h to
exp
ect
a do
min
ant
influ
ence
of
the
zone
of l
owes
t visc
osity
in th
e Ea
rth’s
man
tle o
n po
st-s
eism
ic d
efor
mat
ions
. Su
ch d
efor
mat
ions
sho
uld
thus
be
indi
cativ
e of
the
exis
tenc
e of
a lo
w v
iscos
ity l
ayer
. In
the
follo
win
g w
e ex
plor
e th
is q
uest
ion
by d
evel
opin
g ap
prop
riat
e lim
it cy
cle
solu
tions
fo
r ‘in
finite
’ fa
ults
, usin
g a
simpl
e th
in l
ayer
mod
el w
hich
is
iden
tical
with
the
gen
eral
ized
El
sass
er p
late
mod
el in
trodu
ced
and
anal
ysed
pre
viou
sly b
y Ri
ce (
1980
) an
d Le
hner
, Li &
Ri
ce (
1981
), an
d by
a s
ubse
quen
t co
mpa
rison
with
the
hal
f-sp
ace
mod
el o
f Sa
vage
&
Pres
cott
and
Spen
ce &
Tur
cotte
. As m
ay b
e ex
pect
ed, p
ost-s
eism
ic d
efor
mat
ions
will
app
ear
mor
e cl
osel
y co
nfin
ed t
o th
e fa
ult
zone
in
the
thin
lay
er m
odel
. Mor
eove
r, th
ere
will
be
sign
ifica
nt q
uant
itativ
e di
ffer
ence
s in
the
res
pons
e of
the
tw
o m
odel
s. Th
e th
in l
ayer
ap
prox
imat
ion
prop
osed
her
e sh
ould
the
refo
re f
urni
sh a
rou
gh c
hara
cter
izat
ion,
in s
impl
e an
alyt
ical
term
s, o
f ast
heno
sphe
re st
ratif
icat
ion
effe
cts i
n po
st-s
eism
ic r
ebou
nd.
F. K
. Leh
nera
nd V
. C. L
i
2 Th
e th
in la
yer
mod
el
We
re-d
eriv
e br
iefly
the
gov
erni
ng e
quat
ions
of t
he g
ener
aliz
ed E
lsass
er m
odel
intr
oduc
ed b
y Ri
ce (
1980
) an
d di
scus
sed
and
anal
ysed
in
deta
il by
Leh
ner
et a
l. (1
981)
. We
shal
l lim
it ou
rsel
ves
to th
e on
e-di
men
sion
al p
robl
ems
perta
inin
g to
a v
ery
long
rupt
ure
on a
stri
ke-s
lip
or t
hrus
t fa
ult,
in w
hich
the
rel
evan
t va
riabl
es w
ill b
e th
ickn
ess-
aver
aged
dis
plac
emen
ts a
nd
stres
ses
in t
he e
last
ic l
ithos
pher
e. F
ig.
1 sh
ows
the
sche
mat
ic g
eom
etrie
s as
sum
ed i
n m
odel
ling
a tra
nsfo
rm a
nd s
ubdu
ctio
n-ty
pe p
late
bou
ndar
y by
a l
ine
acro
ss w
hich
the
th
ickn
ess-
aver
aged
dis
plac
emen
t rH
u(
y, t)
= H
-’ I
u‘(y
, z, t
) dz
JO
suff
ers
an e
piso
dic
jum
p di
scon
tinui
ty in
reg
ular
rec
urre
nce
time
inte
rval
s, T
, dur
ing
whi
ch
ther
e m
ay b
e as
eism
ic s
lippa
ge a
long
dee
per
sect
ions
of
the
faul
t. N
otic
e in
par
ticul
ar t
he
sche
mat
ic p
ictu
re o
f a
subd
uctio
n-ty
pe b
ound
ary
at w
hich
the
stre
ss d
rop
and
cose
ismic
di
spla
cem
ent
of O
UI
mod
el a
re t
o be
gi
ven
the
inte
rpre
tatio
ns o
f th
ickn
ess-
aver
aged
qu
antit
ies
defin
ed a
t th
e lo
catio
n y
= 0
whi
ch c
orre
spon
ds p
erha
ps r
ough
ly t
o th
e tre
nch
axis.
Thu
s, th
e ac
tual
thru
st p
lane
lies
at y
< 0,
but
eve
nts
on it
are
rele
vant
to
the
mod
ellin
g of
pla
te s
tress
es a
nd d
efor
mat
ions
at
y >
0 on
ly i
n as
muc
h as
the
y in
fluen
ce b
ound
ary
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
Plat
e bo
unda
ry d
efor
mat
ions
77
7
eu
+
L H
/
Figu
re 1
. (a)
Str
ike-
slip,
and
(IJ)
unde
rthr
ust m
odes
of
plat
e bo
unda
ry a
s ide
aliz
ed in
thin
laye
r m
odel
.
cond
ition
s at
y =
0.
Also
sho
wn
in F
ig.
1 is
the
dire
ctio
n of
the
uni
form
pla
te m
otio
n at
sp
eed
V fa
r fr
om t
he p
late
bou
ndar
y, w
hich
driv
es t
he e
arth
quak
e cy
cle.
In
term
s of
th
ickn
ess-
aver
aged
stre
sses
and
shea
ring
tract
ions
, T
P,
actin
g on
the
low
er s
urfa
ce o
f th
e lit
hosp
heric
pla
te i
n th
e ne
gativ
e 0-
dire
ctio
n, th
e re
leva
nt e
xact
equ
ilibr
ium
equ
atio
ns in
the
plan
e of
the
plat
e ar
e
au,,/
ax,
= TP
/H.
(1)
Follo
win
g Ri
ce (
1980
), a
sim
plifi
ed c
oupl
ing
to a
Max
wel
lian
visc
oela
stic
asth
enos
pher
e is
now
ass
umed
thro
ugh
the
rela
tion
i,b/G
t
T,h/
q =
(2)
whe
re b
is
an e
ffec
tive
leng
th
for
shor
t-tim
e el
astic
cou
plin
g, w
hich
will
be
sele
cted
ap
prop
riate
ly f
urth
er b
elow
. Fo
r si
mpl
icity
we
shal
l as
sum
e he
re t
hat
the
shea
r m
odul
us
G at
tain
s a
unifo
rm v
alue
thro
ugho
ut th
e lit
hosp
here
and
ast
heno
sphe
re. F
or s
ubse
quen
t use
w
e de
fine
the
para
met
ers
a H
Gh/
q,
0 = bH
.
The
ratio
T =
p/a
det
erm
ines
the
rela
xatio
n tim
e in
an
Elsa
sser
-type
pla
te m
odel
that
invo
lves
a
Max
wel
lian
visc
oela
stic
foun
datio
n.
In t
he p
robl
ems
to b
e st
udie
d he
re,
as d
epic
ted
by F
ig.
1, th
ere
is on
ly o
ne n
on-z
ero
thic
knes
s-av
erag
ed d
ispl
acem
ent
com
pone
nt in
the
pla
ne o
f th
e pl
ate.
Thu
s, if
we
assu
me
a st
ate
of p
lane
stre
ss f
or th
e pl
ate,
the
stres
s-str
ain
rela
tions
for a
n is
otro
pic
elas
tic p
late
are
si
mpl
y
uxy
= G
au
/ay,
fo
r the
stri
ke-s
lip m
ode
(3a)
uyy =
[2/
(1 -v
)]G
au/a
y,
for t
he th
rust
mod
e.
(3b)
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
778
Subs
titut
ion
of (2
) an
d (3
) in
(1) t
hen
yiel
ds th
e di
ffer
entia
l equ
atio
n F.
K. L
ehne
r and
V. C
. Li
p2(a
+pa/
at)a
2u/a
y2 =
auja
t (4
)
whe
re p
2 3
1 f
or t
he s
trike
-slip
mod
e an
d p2
2/
(1 -
v)
for
the
thru
st m
ode.
It i
s see
n th
at
whe
n el
astic
pro
perti
es o
f th
e as
then
osph
ere
are
disr
egar
ded,
one
reco
vers
Elsa
sser
's (1
969)
di
ffus
ion
equa
tion
with
diff
usiv
ities
(Y fo
r th
e st
rike-
slip
mod
e an
d 2(
~/(1
- v) f
or th
e th
rust
m
ode,
an
equa
tion
empl
oyed
also
by
Bot
t &
Dea
n (1
973)
and
And
erso
n (1
975)
in st
udyi
ng
the
prop
agat
ion
of d
istu
rban
ces a
way
fro
m s
uch
plat
e bo
unda
ries.
Q
uite
obv
ious
ly t
here
are
a n
umbe
r of
sho
rtcom
ings
atta
ched
to o
ur m
odel
equ
atio
n (4
). It
is a
n eq
uatio
n in
a th
ickn
ess a
vera
ged
disp
lace
men
t an
d it
is ba
sed
on si
mpl
ifyin
g as
sum
p-
tions
, am
ong
othe
rs a
neg
lect
of
shea
r str
esse
s ux
y with
in t
he a
sthe
nosp
here
whi
ch, f
or th
e st
rike-
slip
mod
e, b
ecom
e im
port
ant
near
the
pla
te b
ound
ary.
We
emph
asiz
e, h
owev
er,.
that
we
sha
ll be
con
cern
ed w
ith q
uant
itativ
e as
pect
s of t
he p
late
def
orm
atio
n m
ostly
at d
ista
nces
of
the
ord
er o
f on
e lit
hosp
here
thi
ckne
ss f
rom
the
fau
lt, w
here
a p
late
the
ory
beco
mes
m
ore
appr
opria
te.
Also
, we
wish
to
take
adv
anta
ge h
ere
of th
e an
alyt
ical
sim
plic
ity o
f a o
ne-
dim
ensi
onal
m
odel
an
d se
arch
fo
r di
stin
ctiv
e qu
alita
tive
feat
ures
in
the
post
-sei
smic
de
form
atio
ns p
redi
cted
by
a th
in la
yer m
odel
, whi
ch la
ter m
ight
be
stud
ied
in g
reat
er d
etai
l.
3 A
perio
dic
solu
tion
repr
esen
ting a
n ea
rthq
uake
cyc
le
We
seek
a s
olut
ion
to e
quat
ion
(4)
desc
ribin
g th
e di
spla
cem
ent
in a
pla
te w
hich
mov
es a
t a
unifo
rm s
peed
V fa
r fr
om th
e pl
ate
boun
dary
, but
at t
he b
ound
ary
exhi
bits
a q
uasi
-per
iodi
c m
otio
n w
hich
is
sepa
rabl
e in
to a
com
pone
nt o
f un
iform
mot
ion
at s
peed
V a
nd a
stri
ctly
pe
riodi
c di
spla
cem
ent,
the
perio
d T
fixin
g th
e re
curr
ence
tim
e of
sei
smic
eve
nts i
n an
infin
ite
sequ
ence
of
eart
hqua
ke c
ycle
s. Th
e pe
riodi
c th
ickn
ess-
aver
aged
dis
plac
emen
t at
the
plat
e bo
unda
ry w
ill i
nvol
ve c
ontr
ibut
ions
fro
m c
osei
smic
slip
page
as
wel
l as
inte
rsei
smic
fau
lt cr
eep
and - a
t an
unde
rthr
ust b
ound
ary - fr
om a
sthe
nosp
here
read
just
men
ts in
the
verti
cal
plan
e y
= 0
(cf. F
ig.
1).
The
gene
ral
appe
aran
ce o
f a
plot
ver
sus
time
of t
he t
hick
ness
av
erag
ed d
ispl
acem
ent
u+ =
u(O
+, t
) on
the
y =
O* s
ide
of th
e pl
ate
boun
dary
will
thus
be
of
the
kind
ind
icat
ed b
y th
e do
tted
lines
in
Fig.
2(a
) sh
owin
g co
seism
ic s
hifts
of m
agni
tude
Au
+ a
t t =
n T
(n =
0, f
1, f 2
, . . .
). T
o fix
idea
s, co
nsid
er f
irst
a tra
nsfo
rm f
ault
rupt
urin
g do
wn
to d
epth
D (cf.
Fig.
1) d
urin
g ea
ch s
eism
ic e
vent
and
ther
eby
cont
ribu
ting
an a
vera
ge
disp
lace
men
t (ta
ken
over
the
who
le l
ithos
pher
e) e
qual
to
Au+
= u
(O+,
n T'
) - u
(O',
n T-)
on
the
y=
O+
sid
e of
the
fau
lt. S
ubse
quen
tly,
cree
p on
dee
per
sect
ions
of
the
faul
t will
ac
coun
t fo
r a
furt
her,
inte
rsei
smic
ave
rage
dis
plac
emen
t, al
thou
gh th
e up
per s
ectio
ns o
f the
(a 1
(b)
(C)
Figu
re 2
. Sy
nthe
sis
of t
hick
ness
-ave
rage
d fa
ult
disp
lace
men
t, (a
) by
sup
erpo
sitio
n of
str
ictly
per
iodi
c di
spla
cem
ent,
(b)
of
'ele
men
tary
ea
rthq
uake
seq
uenc
e'
and
(c)
linea
rly
incr
easin
g di
spla
cem
ent
of
unifo
rm p
late
mot
ion.
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
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e bo
unda
ry d
efor
mat
ions
77
9
faul
t may
rem
ain
lock
ed b
etw
een
seism
ic e
vent
s. N
ot e
noug
h se
ems t
o be
kno
wn
abou
t the
se
inte
rsei
smic
his
torie
s an
d fo
r thi
s rea
son
we
shal
l dea
l her
e on
ly w
ith th
e ap
prox
imat
ion
of a
lin
ear i
nter
seis
mic
dis
plac
emen
t-tim
e re
latio
n as
rep
rese
nted
by
the
solid
line
s in
Fig.
2 (a
). Th
is in
clud
es t
he e
xtre
me
case
of
a ve
ry r
apid
pos
t-sei
smic
cre
ep a
djus
tmen
t al
ong
the
deep
er s
ectio
ns o
f th
e fa
ult,
in w
hich
Au+
= VT
and
the
plat
e re
spon
ds q
uasi
-sta
tistic
ally
as
for
a ru
ptur
e de
pth
D =
H. W
hen
D <
H, t
hen
on th
e as
sum
ptio
n of
neg
ligib
le i
nter
seis
mic
sli
p on
the
rupt
ure
surf
ace
one
has
u+ =
H-'
For g
reat
und
erth
rust
eve
nts,
on
the
othe
r han
d,
[u'(O
+, z,
n T
+) - u'
(O+,
z, n
T-)]
dz
= V
TD/H
.
u+=
VT
(6)
wou
ld
seem
a r
easo
nabl
e ap
prox
imat
ion
in v
iew
of
the
gene
rally
lar
ger
ratio
D/H
of
dow
n-di
p ru
ptur
e w
idth
to
litho
sphe
re t
hick
ness
(se
e, e
.g.
Dav
ies
& H
ouse
197
9; S
penc
e 19
77).
Fig.
2 i
llust
rate
s the
man
ner i
n w
hich
the
act
ual p
late
bou
ndar
y di
spla
cem
ent
is vi
ewed
as
a su
perp
ositi
on o
f st
rictly
per
iodi
c di
spla
cem
ent
ue
, for
min
g an
'ele
men
tary
ear
thqu
ake
sequ
ence
', an
d a
disp
lace
men
t ur
n whi
ch in
crea
ses a
t a u
nifo
rm ra
te V
, pre
cise
ly a
s for
mul
ated
pr
evio
usly
by
Sava
ge &
Pre
scot
t (1
978)
. Th
is d
ecom
posi
tion
mak
es c
lear
tha
t it
suff
ices
to
obta
in s
olut
ions
to
equa
tion
(4)
for
the
stric
tly p
erio
dic
boun
dary
dis
plac
emen
t u e
. Fo
r fin
ite T
thes
e va
nish
at
infin
ity, b
ut a
ddin
g ur
n eve
ryw
here
will
pro
duce
the
plat
e's r
espo
nse
to t
he b
ound
ary
cond
ition
fur
nish
ed b
y th
e so
lidly
dra
wn
line
in F
ig. 2
(a).
Inde
ed, w
hile
fu
lly d
eter
min
ing
stra
ins
and
stra
in
rate
s, t
he s
olut
ion
for
the
elem
enta
ry e
arth
quak
e se
quen
ce u
e w
ill y
ield
a d
ispl
acem
ent
mea
sure
d w
ith r
espe
ct t
o a
line
norm
al t
o th
e fa
ult
and
atta
ched
to
an o
bser
ver
'sitti
ng o
n th
e pl
ate'
at a
lar
ge d
ista
nce
from
a t
rans
form
bo
unda
ry. B
earin
g th
is in
min
d w
e no
w c
onsi
der
a so
lutio
n to
(4) o
f the
form
U(Y
, t)
= U
YY
, t>
+ u
"W +
U"Y
) (7
)
whe
re u
'(y)
is a
time-
inde
pend
ent
disp
lace
men
t th
at d
epen
ds o
n th
e ch
oice
of
refe
renc
e st
ate
and
~'(
0)
=
0. F
urth
erm
ore,
u"(t)
= Vt
- %
Au+
(8
)
and
u"
(y, t)
is
a so
lutio
n fo
r th
e el
emen
tary
ear
thqu
ake
sequ
ence
as
repr
esen
ted
by t
he
Four
ier
serie
s Au+
rn
1
7r n
=l
n ue
(O+,
t) =
~ - s
in(2
nnt/
T)
whi
ch is
the
perio
dic
exte
nsio
n of
the
func
tion
as s
how
n in
Fig
. 2(b
). A
sol
utio
n to
(4),
subj
ect
to (
9) m
ay n
ow b
e ob
tain
ed in
a m
anne
r sim
ilar
to t
hat
disc
usse
d by
Car
slaw
& J
aege
r (1
959,
sec
tion
2.6)
fo
r he
at c
ondu
ctio
n pr
oble
ms
with
tim
e-pe
riodi
c bo
unda
ry
cond
ition
s.
Acc
ordi
ngly
, we
co
nsid
er
first
a pa
rticu
lar s
olut
ion
of th
e fo
rm
u"
(y, t)
=u,(
y/d)
exp(
inw
t),
w=
2n/T
, ~
'=p
/"'~
. (1
0)
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
780
Subs
titut
ing
ths i
n (4
) we
get
F. K
. Leh
nera
nd V
. C. L
i
v”-A
Z,v,
= 0
,
AZ, =
in a/( 1/r +
in a).
Then
uE(y
, t) =
A,
exp
I- [R
e A, t
i Im
A,]
y/d
1 exp
(in a
t)
Reh
, --fy, =
f {
[l +
(1+
e~
)1/2
]/2(
1+e~
)}1/
2
Imh,
--fq
, =
-fe,
/i?-
(ite
~)[~
+(i
+e~
)~/~
]t~/
~ 0,
= T/
(2nn
r).
A re
al s
olut
ion,
van
ishin
g at
infi
nity
and
sat
isfy
ing
Au+
nn
u,
(O+,
t) =
~ sin
(2nn
t/T
)
is ob
tain
ed f
rom
(13
) for
iA,
= A
u”/n
n as
Au+
uE
( y, t
) = _
_ e
xp (-
Y, y
/d) s
in (2
n n t/
T - 77
, y/
d).
nn
Hen
ce, b
y su
perp
ositi
on,
Au+
- 1
n n
=ln
u
e(y,
t)=
~
1 - e
xp(-
yny/
d)si
n(2n
nr/T
-r),
y/d)
(13)
(14)
furn
ishe
s th
e so
lutio
n to
equ
atio
n (4
) fo
r th
e el
emen
tary
ear
thqu
ake
sequ
ence
(1)
. W
ritin
g th
is a
s a
Four
ier
serie
s, bu
t ob
serv
ing
that
the
sin
e se
ries
conv
erge
s to
war
ds a
pie
cew
ise
cont
inuo
us f
unct
ion
with
jum
p di
scon
tinui
ties
at t
= n
T, o
ne c
an w
rite
(15)
as th
e su
m o
f a
piec
ewise
co
ntin
uous
fun
ctio
n ue
(O+
, t) ex
p (-
y/d)
and
a c
ontin
uous
fun
ctio
n so
tha
t w
ithin
the
inte
rval
0 G
t Q
T
m
ue(v
, t) =
A,+
( 1 - i
) exp (-
y/d)
- A
u+
[a, c
os (2
nntl
T) +
6, s
in (2
rznt
/T)]
n
= 1
1 nn
1 nn
an =
-’ ~
XP
(.
- Yn ~
/d
)
sin (77
, ~
/d
)
(16)
6, =
- [
~X
P
(-~
/d) - ~
XP
&
Yn ~
/d
)
cos (7
7, .~
/d)l
.
Clea
rly, a
t la
rge
enou
gh n
such
that
0,
e 1,
7, -, I - 3/
26,
and
77, +
%On
. As
see
n fr
om (1
2),
the
limits
0, +
0 a
nd h
ence
the
limits
7, +
1 a
nd 77
, +
0 m
ay a
lso b
e in
terp
rete
d in
term
s of
a ve
ry l
arge
rel
axat
ion
time 7. T
he F
ourie
r co
effic
ient
s van
ish i
n th
is li
mit
and
this
mak
es
clea
r th
at i
t is
esse
ntia
lly t
he s
erie
s te
rm i
n (1
6) w
hich
fur
nish
es t
he c
ontr
ibut
ion
to th
e di
spla
cem
ent
due
to v
iscoe
lasti
c re
laxa
tion.
To
fin
d th
e co
seism
ic d
ispl
acem
ent
jum
p at
any
loca
tion
y 2 0
, we
eval
uate
(16
) at t
= 0
an
d t =
T a
nd th
us o
btai
n
Au
(y)=
ue(
-~?,
O
)-ue
(y, T
) = A
u* e
xp(-
y/d)
. (1
7)
This
exp
ress
ion
repr
esen
ts a
n in
stan
tane
ous
elas
tost
atic
resp
onse
to
a su
dden
slip
eve
nt o
n th
e fa
ult.
As s
uch
it m
ust
obvi
ousl
y be
inde
pend
ent o
f rel
axat
ion
prop
ertie
s of
the
asth
eno-
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e bo
unda
ry d
efor
mat
ions
78
1 sp
here
and
inde
ed m
ust
coin
cide
, as
it do
es, w
ith th
e ze
ro-ti
me
solu
tion
for a
sing
le is
olat
ed
slip
even
t as g
iven
pre
viou
sly
by L
ehne
r et a
l. Th
e to
tal
disp
lace
men
t as
def
ined
by
(7)
may
now
be
obta
ined
by
addi
ng t
o (1
6) t
he
expr
essio
ns fo
r u"(
t) an
d uo
(y).
For
the
for
mer
we
have
from
(5)
and
(8),
whi
le f
or u
o(y)
we
sele
ct
1 2 uo
((y)
= - A
u' [
l -ex
p(-y
/d)]
.
Acc
ordi
ngly
u(y,
t) =
Au+
[l-e
xp(-
y/d)
] +
Au+
C[.
. .],
0
9 1
9 T
T
whe
re t
he s
erie
s is
the
sam
e as
in (
16).
Her
e uo
((y)
has
been
sel
ecte
d so
as t
o m
ake
u(y,
t =
Term
by
term
diff
eren
tiatio
n of
(19)
now
yie
lds
the
follo
win
g ex
pres
sion
for
the
stra
in:
0')
= 0.
au
nu
+
y(y
t)=
-=-
' ay
d
[a;
cos(
2nnt
/T)
+ b;
sin (
2nnt
/T)]
a; =
da,/d
(y/d
),
b; =
db,
/d(y
/d).
(20)
The
cose
ism
ic e
last
osta
tic ju
mp
in s
train
is g
iven
by
MY
) = (v
, 0) - Y
(Y,T
) = - (A
u+/d
) exp
(-Y/
d)
and
is se
en to
diff
er f
rom
the
disp
lace
men
t jum
p (1
7) in
sig
n an
d by
a fa
ctor
l/d
.
by v
irtue
of r
elat
ions
(3) a
nd (2
1), i
s giv
en b
y B
oth
ue
and
y m
ay a
lso b
e ex
pres
sed
in t
erm
s of
a 's
tress
dro
p' A
a at
the
faul
t, w
hich
,
AU E
uap(
T, 0
') - ~
~p(0
,O')
= A
u'G
/d.
(22)
How
ever
, si
nce
uap
repr
esen
ts a
mea
n st
ress
whi
ch, e
spec
ially
for
the
cas
e D
< H
, is
not
easil
y re
late
d qu
antit
ativ
ely
to f
ault
stre
sses
, it
seem
s pr
efer
able
to
reta
in t
he d
ispl
acem
ent
jum
p A
u+ in
the
abov
e ex
pres
sion
s. Fi
nally
, by
ter
mw
ise
diff
eren
tiatio
n of
(20
) w
ith r
espe
ct t
o tim
e an
d an
app
ropr
iate
ex
tract
ion
of a
pie
cew
ise c
ontin
uous
par
t, w
e ob
tain
an
expr
essi
on f
or th
e st
rain
rat
e of
the
form
Au'
-
-_
_
[a,
cos(
2nnt
/T)
+ (3,
sin
(2nn
t/T
)]
rd
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782
The
cose
ism
ic ju
mp
in th
e st
rain
rate
is fo
und
to be
F. K
. Leh
nera
nd V
. C. Li
Ai.
(y) =
+(y
, O)-
j.(y
, T
) = 2 7
d
4 D
iscus
sion
and
com
paris
on o
f thi
n la
yer a
nd h
alf-
spac
e m
odel
s
In p
roce
edin
g no
w to
a d
iscu
ssio
n of
thes
e re
sults
we
first
em
phas
ize
agai
n th
at th
e un
ilate
ral
disp
lace
men
t jum
p A
u+ ap
pear
ing
in e
quat
ions
(16
), (1
7) a
nd (
19)-(
24)
is to
be
view
ed as
gi
ven
eith
er b
y re
latio
n (5
) or
(6)
in t
erm
s of
the
quan
titie
s V
, T an
d D
/H, w
hich
we
rega
rd
as k
now
n. B
ut it
is c
onve
nien
t to
def
er th
is s
ubst
itutio
n fo
r Au+
until
con
side
ratio
n is
give
n to
a pa
rticu
lar
case
. W
e sh
all
base
sub
sequ
ent
quan
titat
ive
inte
rpre
tatio
ns o
f ou
r re
sults
on
the
num
eric
al
valu
es q
= 2
.0 x
lOI9
Pas
(2.
0 x
10''
pois
e) f
or t
he v
isco
sity
of
the
asth
enos
pher
e an
d G
= 5
.5 x
10"
Pa
for
the
shea
r mod
ulus
of
the
crus
t an
d up
per
man
tle, s
o th
at v
/G =
10 y
r in
agr
eem
ent w
ith a
rece
nt e
stim
ate
of T
hatc
her
el a
l. (1
980)
bas
ed o
n ea
rthqu
ake
load
ing
data
. Fu
rther
mor
e, w
e sh
all
assu
me
H=
30km
for
the
thi
ckne
ss o
f th
e lit
hosp
here
and
h
= 1
50 km
for
the
low
vis
cosi
ty l
ayer
(as
then
osph
ere)
. Th
ese
estim
ates
yie
ld T =
fl/a
= 10
b/h
yr f
or t
he r
elax
atio
n tim
e of
the
thin
laye
r mod
el a
nd if
b is
giv
en th
e va
lue
(n/2
)'H,
as is
pro
pose
d fu
rthe
r bel
ow, o
ne h
as T
= 5 y
r fo
r the
thin
laye
r mod
el.
In F
ig. 3
a p
lot
of th
e no
rmal
ized
per
iodi
c so
lutio
n (1
6) v
ersu
s tim
e is
show
n an
d th
is is
be
st i
nter
pret
ed i
n te
rms o
f the
dis
plac
emen
ts m
easu
red
with
resp
ect t
o a
line,
per
pend
icul
ar
to a
tran
sfor
m f
ault
and
mov
ing
alon
g w
ith a
rem
ote
poin
t (y +
-)
on th
e pl
ate.
The
figu
re
repr
esen
ts a
sin
gle
earth
quak
e cy
cle
and
its p
erio
dic
exte
nsio
n to
the
right
and
to
the
left
will
the
refo
re c
onst
itute
an
infin
ite e
arth
quak
e se
quen
ce.
The
stric
tly s
ymm
etric
per
iodi
c m
otio
n im
pose
d by
con
ditio
n (9
) at
y/d
= 0
giv
es w
ay t
o a
char
acte
ristic
dam
ping
and
ph
ase
lag
beha
viou
r w
ith g
row
ing
dist
ance
s fr
om t
he f
ault,
the
latte
r eff
ect b
eing
sole
ly d
ue
to a
sthe
nosp
here
rel
axat
ion.
Whe
n m
easu
red
in t
he m
ovin
g co
ordi
nate
sys
tem
of
Fig.
3,
disp
lace
men
ts w
ill t
hus
cont
inue
to
grow
'co
-dire
ctio
nally
', i.e
. in
the
dire
ctio
n of
the
0.4 7
7
U'/h
U+
0.2
-0.4
tn\
i
c 4 0
0 0.
25
0.5
0.75
1 TI
ME
t/T
Figu
re 3
. T
imep
erio
dic
thic
knes
s-av
erag
ed d
ispla
cem
ent
in e
lem
enta
ry e
arth
quak
e cy
cle
at v
ariou
s di
stan
ces from
the
faul
t. R
epea
t tim
e 15
0 yr
.
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
Plat
e bo
unda
ry d
efor
mat
ions
78
3
-0.4
ue/A
u+
-0.2
m B 2 z Q
0,2
@ 0.
0
a
0.4
- 0.4
u'/A
u+
- 0.2
M z; @
0.0
3 ti la 0<
2
0.4
0 1
2 3
4 5
DIS
TAN
CE
y/d
0 1
2 3
4 5
DIST
ANCE
j/
d
Figu
re 4
. T
hick
ness
-ave
rage
d dis
plac
emen
t in
elem
enta
ry ea
rthq
uake
cycl
e as a
func
tion
of d
ista
nce f
rom
th
e fa
ult,
at v
ario
us ti
mes
. Rep
eat t
imes
: (a)
150
yr; (
b) 4
0 yr
.
cose
ismic
dis
plac
emen
t, bu
t, af
ter
atta
inin
g a
max
imum
whi
ch d
epen
ds o
n th
e di
stanc
e fro
m th
e fa
ult,
will
sw
ing
back
. The
cos
eism
ic ju
mp
in (t
otal
) di
spla
cem
ent,
as g
iven
by
the
simpl
e ex
pone
ntia
l re
latio
n (1
7),
may
also
be
read
off
Fig
. 3
by t
akin
g th
e di
ffere
nce
betw
een
the
func
tion
valu
es a
t t =
0 a
nd t
= T
for
any
giv
en r
atio
y/d
. So
me
of t
hese
pr
oper
ties
appe
ar, h
owev
er, m
ore
clea
rly o
n a
plot
of t
he sa
me
disp
lace
men
t ver
sus d
istan
ce
from
the
fau
lt as
sho
wn
in F
ig.
4 fo
r re
peat
tim
es o
f 15
0 an
d 40
yr,
resp
ectiv
ely.
The
du
ratio
n of
a c
ycle
cle
arly
gov
erns
the
'pen
etra
tion
dept
h' o
f sig
nific
ant d
ispl
acem
ent.
Sinc
e th
e co
seism
ic d
ispla
cem
ent
jum
p ac
cord
ing
to e
quat
ion
(17)
is
unaf
fect
ed b
y re
peat
tim
e,
this
diff
eren
ce i
n pe
netra
tion
dept
h is
entir
ely
due
to t
he f
arth
er s
prea
d of
sig
nific
ant
rela
xatio
n in
the
asth
enos
pher
e du
ring
the
long
er c
ycle
. Thi
s also
poi
nts
to th
e di
ffic
ulty
of
infe
rring
mag
nitu
des
of p
ost-s
eism
ic m
otio
ns d
ue to
sub
crus
tal r
elax
atio
n pr
oces
ses
from
a
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
784
‘sin
gle-
even
t mod
el’ o
f fa
ultin
g at
a p
late
bou
ndar
y. S
uch
mod
els
will
tend
to
over
estim
ate
mot
ions
due
to
rela
xatio
n an
d in
deed
may
dis
play
a s
ensi
tivity
to
a sp
ectru
m o
f rel
axat
ion
times
, pe
rhap
s re
sem
blin
g st
ratif
icat
ion
effe
cts,
whi
ch w
ould
be
dras
tical
ly r
educ
ed i
n th
e ca
se o
f an
ear
thqu
ake
sequ
ence
to
a re
spon
se g
over
ned
sole
ly b
y la
yers
with
rela
xatio
n tim
es
shor
ter t
han
the
cycl
e le
ngth
. A f
urth
er im
port
ant
redu
ctio
n of
thi
s pe
netr
atio
n de
pth
will
of c
ours
e be
due
to th
e fin
ite le
ngth
of r
eal r
uptu
res.
How
ever
, as a
lread
y ap
pare
nt fr
om F
ig.
4, s
ome
of t
he m
ost
inte
rest
ing
obse
rvab
le d
efor
mat
iona
l fe
atur
es m
ay o
ccur
at l
ocat
ions
ar
ound
y =
2d,
i.e.
app
roxi
mat
ely
3 lit
hosp
here
thi
ckne
sses
fro
m a
(tra
nsfo
rm)
faul
t. Fo
r ru
ptur
e le
ngth
s of
sev
eral
hun
dred
kilo
met
res (
grea
t ea
rthqu
akes
) the
cro
ss-s
ectio
nal m
odels
di
scus
sed
here
will
ther
efor
e be
mea
ning
ful.
In F
ig. 5
the
tota
l thi
ckne
ss av
erag
ed d
ispl
acem
ent
acco
rdin
g to
(19)
has
bee
n pl
otte
d an
d m
ay b
e co
mpa
red
with
the
sur
face
dis
plac
emen
ts p
redi
cted
by
the
half-
spac
e mod
el (d
ashe
d lin
es)
for
the
sam
e tim
e ra
tio T
, TG
/q =
15 a
s w
ell
as f
or t
he li
mit
case
T, +
0 o
f pur
ely
elas
tic r
espo
nse
(dot
ted
lines
). W
e ha
ve o
mitt
ed t
he t
ime-
depe
nden
t co
ntri
butio
n fro
m th
e se
cond
te
rm i
n (1
9)
arisi
ng
from
fau
lt cr
eep
at d
epth
, as
sum
ing D=H, i.e
. a
rapid
post
-sei
smic
adj
ustm
ent.
Thes
e se
ts o
f cur
ves a
re la
belle
d by
val
ues o
f f/T
in s
teps
of
1/10
of T
and
evol
ve fr
om th
e ze
ro r
efer
ence
lin
e at
t =
O+,
i.e.
im
med
iate
ly a
fter
the
last
eve
nt, t
o a
final
sha
pe a
t t =
T,
i.e.
just
bef
ore
the
next
eve
nt d
urin
g w
hich
the
dis
plac
emen
t w
ill j
ump
to t
he v
alue
u/A
u+ =
1 ev
eryw
here
. Sur
face
dis
plac
emen
ts f
or t
he p
urel
y el
astic
hal
f-sp
ace
are
show
n on
ly f
or ti
mes
t =
0, 0
.1 T
, 0.2
T, a
nd f
or t
= T
whe
n th
e di
spla
cem
ent
beco
mes
inde
pend
ent
F. K
. Leh
nera
nd V
. C. Li
0.0
u/Au
+
02
W
0
a
0.6
0
0.8
10
0
I 2
3
4
5
DIS
TAN
CE
y/
d Fi
gure
5.
Tota
l th
ickn
ess-
aver
aged
disp
lace
men
t fo
r th
in la
yer
mod
el (
solid
line
s) a
nd
men
ts fo
r ha
lf-sp
ace
mod
el (d
ashe
d lin
es; d
otte
d lin
es fo
r el
astic
hal
f-spa
ce) a
s fun
ctio
ns
the
faul
t at
succ
essiv
e tim
es fr
om t
= 0
to t
= T
= 15
0 yr
(in
crem
ent 0
.1 T
).
surfa
ce di
splace
of
dista
nce f
mm
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
Plat
e bo
unda
ry d
efor
mat
ioiis
78
5
of t
he p
aram
eter
T,.
For
the
thin
lay
er m
odel
the
cos
eism
ic ju
mp
from
U(J
~, T
)/A
u' to
1 .O
is
of c
ours
e th
at g
iven
by
the
expo
nent
ial
in e
quat
ion
(17)
. Fo
r th
e ha
lf-sp
ace
mod
pl t
he
resu
lt of
Spe
nce
& T
urco
tte (
thei
r equ
atio
n 5.
3), w
hen
wri
tten
in o
ur n
otat
ion,
is
a(5)
In t
he e
last
ic li
mit
of T
, -+ 0
this
bec
omes
exp (-
t) s
inh .$
, 0
-G t Q
T.
(36)
2A
u't
'ITT
arct
an (
y/H
),
0 Q
t Q
T.
- ___
The
surf
ace
stra
ins
pred
icte
d by
the
half-
spac
e mod
el a
re o
btai
ned
by d
iffer
entia
tion
of (
25)
and
in th
e el
astic
lim
it Ts
+ 0
one
has
dire
ctly
from
(26)
the
sim
ple
resu
lt
au
2 Au'
t 1
3.Y
'ITH
T 1 +
(y/H
P
Yb, t)
= - (
Y, t
) = ~
and
henc
e th
e ju
mp
in s
train
for
the
half-
spac
e mod
el
This
resu
lt is
now
use
d fo
r det
erm
inin
g th
e ef
fect
ive
elas
tic th
ickn
ess 6
whi
ch e
nter
s int
o th
e th
in l
ayer
mod
el t
hrou
gh e
quat
ion
(2),
but
has
been
lef
t un
spec
ified
so
far.
We
fix 6
by
requ
iring
tha
t th
e ju
mps
(21)
and
(28
) pre
dict
ed b
y th
e tw
o m
odel
s mat
ch a
t the
fau
lt, th
at
is at
y =
0.
Sinc
e, u
nder
the
ass
umpt
ion D
=H
, the
jum
p in
dis
plac
emen
t ha
s th
e sa
me
mag
nitu
de A
u+ fo
r bot
h m
odel
s, it
is se
en th
at th
is m
atch
requ
ires
d =
p"' =
('IT
/Z)H
or
b
= (n
/2)'
H.
(29)
It wi
ll be
not
iced
tha
t fo
r th
e pr
esen
t di
sloc
atio
n pr
oble
m t
he a
ppro
pria
te el
astic
thic
knes
s b i
s fou
r tim
es la
rger
than
it is
for t
he a
nalo
gous
cra
ck p
robl
em (L
ehne
r et a
l. 19
81 ).
Here
and
subs
eque
ntly
it s
houl
d be
kep
t in
min
d th
at a
ll di
spla
cem
ents
disc
usse
d fo
r the
ha
lf-sp
ace m
odel
are
sur
face
dis
plac
emen
ts.
In p
lotti
ng th
ese
in F
ig. 5
the
argu
men
t ji
/H h
ad
to b
e re
plac
ed b
y (n
/2)y
/d so
that
the
com
paris
on w
ith th
e th
in la
yer
mod
el p
erta
ins
to th
e str
ike-
slip m
ode,
whi
le th
e so
lutio
n (1
9) re
mai
ns o
f cou
rse
valid
inde
pend
ently
for
the
thru
st
mod
e, if
d is
take
n eq
ual t
o 1/
(1-
v)nH
/2. T
he c
osei
smic
jum
p in
sur
face
dis
plac
emen
t fo
r the
hal
f-spa
ce m
odel
app
ears
in F
ig. 5
and
is g
iven
by
Au(
y)=
Au*
{1-(2
/7r)
arct
an [
(n/2
)j?/
d]}.
(3
0)
A co
mpa
rison
with
the
jum
p (1
7) f
or t
he t
hin
laye
r m
odel
sho
ws
that
the
lat
ter
pred
icts
so
mew
hat
too
larg
e po
st-s
eism
ic d
ispl
acem
ents
at
grea
ter
dist
ance
s fr
om t
he f
ault.
With
in
abou
t tw
o lit
hosp
here
thi
ckne
sses
fro
m t
he f
ault,
how
ever
, th
e di
scre
panc
y in
tot
al p
ost-
sei
smic
disp
lace
men
t at
t =
T re
mai
ns s
mal
l for
the
tw
o m
odel
s. In
con
tras
t her
ewith
, pos
t- sei
smic
disp
lace
men
ts e
arlie
r in
the
cycl
e di
ffer
ver
y m
arke
dly
as m
ay b
e se
en f
rom
the
att
itude
of
the
plot
s fo
r t/T
= 0
.1 a
nd 0
.2, f
or e
xam
ple.
Thi
s per
mits
us
to c
oncl
ude
that
the
28
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
786
F. K
. Leh
ner a
nd V
. C. L
i
0.6
yd/A
u+ 0.2
0.0
i
--4
-02
I- 0
1 2
3
4 5
DIST
ANCE
v/
d -,
Fip
re 6
. Th
ickn
ess-
aver
aged
str
ain
for
thin
lay
er m
odel
as
a fu
ncti
on o
f di
stan
ce f
rom
the
fau
lt, at
va
riou
s tim
es. R
epea
t ti
me
150
yr.
Das
hed
line
repr
esen
ts to
tal
elas
tic s
urfa
ce s
trai
n ac
cum
ulat
ed during
sam
e ea
rthq
uake
cycl
e ac
cord
ing
to h
alf-
spac
e m
odel
.
thin
lay
er m
odel
exh
ibits
a g
enui
ne s
tratif
icat
ion
effe
ct w
hich
man
ifest
s its
elf
durin
g the
ea
rlier
par
t of
an
eart
hqua
ke c
ycle
in
an a
mpl
ifica
tion
and
conc
entr
atio
n of
pos
t-seis
mic
disp
lace
men
ts n
ear
the
faul
t. Th
e am
ount
of v
iscoe
lasti
c re
laxa
tion
may
be
infe
rred
for e
ach
mod
el u
pon
com
parin
g, a
t t/
T =
0.1
or
0.2,
the
rele
vant
cur
ves w
ith th
e do
tted
lines
for t
he
pure
ly e
last
ic re
spon
se.
In F
ig. 6
the
dim
ensi
onle
ss s
train
, acc
ordi
ng t
o eq
uatio
n (2
0),
is pl
otte
d ve
rsus
dist
ance
fr
om t
he f
ault
for
vario
us t
imes
t/T
. Cor
resp
ondi
ng t
o ou
r ch
oice
of
refe
renc
e sta
te, th
is st
rain
is
zero
at
t = O
+ ju
st a
fter
the
even
t. Th
e va
lue
of y
at
t = T
is th
eref
ore
iden
tical
in m
agni
tude
with
the
stra
in ju
mp
as g
iven
by
(21)
. Thi
s m
ay b
e co
mpa
red
with
the
dashe
d lin
e, r
epre
sent
ing
the
tota
l ac
cum
ulat
ed s
urfa
ce s
train
for
the
hal
f-sp
ace
mod
el a
t t=
T ac
cord
ing
to (
27)
with
y n
orm
aliz
ed b
y d
= (n
/2)H
, as a
ppro
pria
te fo
r the
stri
ke-s
lip mo
de.
Clos
e to
the
faul
t, th
at is
at y
< H
the
tot
al s
train
acc
umul
atio
n di
ffer
s on
ly sl
ight
ly fo
r the
two
mod
els,
the
disc
repa
ncy
read
ing
at 6
per
cen
t at
y =
H. B
eyon
d th
is d
istan
ce th
e post
. se
ismic
stra
ins
pred
icte
d by
the
thi
n la
yer
mod
el a
re l
ikel
y to
be
affe
cted
by
the
large
r re
lativ
e di
scre
panc
y in
cos
eisr
nic
disp
lace
men
t be
twee
n th
e tw
o m
odel
s an
d ar
e the
refore
le
ss s
uite
d fo
r qua
ntita
tive
com
paris
ons.
Ther
e ar
e, h
owev
er, q
ualit
ativ
e fe
atur
es o
f inte
rest,
amon
g w
hich
we
notic
e sig
n re
vers
al o
f po
st-s
eism
ic s
train
s whi
ch, a
s is
also
appa
rent
from
Fig.
5, i
s du
e to
ast
heno
sphe
re r
elax
atio
n. In
the
refe
renc
e sy
stem
sel
ecte
d he
re, t
he st
rain
will
atta
in m
easu
rabl
e ne
gativ
e va
lues
that
per
sist o
ver l
arge
fra
ctio
ns o
f an
earth
quak
e cycl
e at
gre
ater
dis
tanc
es f
rom
the
fau
lt. A
gain
, as
with
dis
plac
emen
t, th
e pe
netra
tion
depth
of
sign
ifica
nt p
ost-s
eism
ic s
train
s will
incr
ease
stro
ngly
with
cyc
le le
ngth
. O
f pa
rticu
lar
inte
rest
is
the
beha
viou
r of
the
stra
in r
ate
give
n by
(23
) wh
en p
lottad
ag
ains
t di
stan
ce f
rom
the
pla
te b
ound
ary
as in
Fig
. 7 f
or tw
o di
ffer
ent r
epea
t tim
es. M
ost
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Plat
e bo
unda
ry d
efor
mat
ions
78
7
0.06
0.04
+d/Au+
0.02
F 2 0.00
rn -0
.02
8 -0.04
-0.0
% 0
1 2
3 4
5 D
ISTA
NC
E y
/d
0.06
OB4
%d/
Au+
0.02
-0,02
-0-04
-0.06
c 1 1
0 1
2 3
4 5
DIST
ANCE
?/
a Fi
gure
7. T
hick
ness
-ave
rage
d str
ain
rate
for
thi
n la
yer
mod
el a
s a
func
tion
of d
istan
ce fr
om th
e fa
ult,
at
vario
us ti
mes
. Rep
eat t
ime:
(a)
150
yr; (
b) 4
0 yr
.
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
788
F. K
. Leh
ner a
nd V
. C. Li
0.08
0.06
T-r
d/A
u+
0.04
-0,0
2
-0,0
4
-0,0
6 0
0.25
0.
5 0,
75
1
TIME t/T
08
8
0.06
jrd/
Au+
0.04
E 0.
02
vl
0'00
8 -0
82
0 0.
25
0.5
0.75
1
TIME
t/T
Figu
re 8
. Tim
e-pe
riod
ic th
ickn
ess-
aver
aged
stra
in r
ate
for
thin
lay
er m
odel
for
one
ear
thqu
ake C
ycle,
at va
rious
dist
ance
s fro
m th
e fa
ult.
Rep
eat t
ime:
(a)
150
yr; @
) 40
yr.
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
Plat
e bo
unda
ry d
efor
mat
ions
78
9
cons
picu
ous
is th
e ap
pear
ance
of
the
cose
ismic
jum
p, g
iven
by
equa
tion
(24)
, whi
ch a
ttain
s its
max
imum
Au+
/(Td
) at y
= 0
, rev
erse
s its
sig
n at
y =
d a
nd r
each
es a
noth
er p
rono
unce
d ex
trem
um
at
y =
2d
of
mag
nitu
de
exp
(- 2
) Au+
/(2d
T). T
hese
lo
catio
ns
and
jum
p m
agni
tude
s ar
e in
depe
nden
t of
rep
eat
time
and
inde
ed w
ould
be
of t
he s
ame
for
a sin
gle
even
t. Th
e st
rain
rat
es t
hem
selv
es a
nd t
heir
tem
pora
l dec
line
will
, how
ever
, dep
end
stro
ngly
on
repe
at ti
me
as m
ay b
e se
en f
rom
the
two
plot
s.
A p
erha
ps s
urpr
isin
g fe
atur
e is
the
jum
p ex
trem
um a
t y
= 2d
, tha
t is
at c
onsi
dera
ble
dist
ance
fro
m t
he f
ault.
For
a g
reat
ear
thqu
ake
with
Au+
= 4
m, s
ay, a
nd o
ur c
hoic
e of
5 y
r fo
r th
e re
laxa
tion
time
T,
the
jum
p in
stra
in r
ate
at y
= 2
d w
ill b
e yr
-',
i.e. j
ust
mea
sura
ble,
at
leas
t fo
r th
e lo
nger
rep
eat
time
of 1
50 y
r, w
hen
pres
eism
ic s
train
rat
es a
re
negl
igib
ly s
mal
l. An
atte
mpt
to
dete
ct a
fea
ture
suc
h as
the
pron
ounc
ed e
xtre
mum
in A
T at
y=
2d
by
mon
itorin
g st
rain
rat
es a
long
a s
urve
y lin
e pe
rpen
dicu
lar
to t
he f
ault
wou
ld
ther
efor
e se
em q
uite
just
ified
. In
Fig
. 8
norm
aliz
ed s
train
rat
es a
re p
lotte
d ag
ains
t tim
e fo
r va
rious
dis
tanc
es f
rom
the
faul
t. Cl
early
, a
few
mea
sure
men
ts d
urin
g th
e fir
st y
ears
fol
low
ing
the
even
t w
ith a
rep
eat
time
of 1
50 y
r w
ill s
uffic
e. E
xtra
pola
ting
back
to
t = 0'
and
dete
rmin
ing
the
loca
tion
ym
, sa
y w
here
the
(ne
gativ
e) s
train
rat
e ju
mp
of g
reat
est
mag
nitu
de I
AT I,,,
occu
rred
, one
has
th
e es
timat
es
and
from
I AT
Im =
exp
(- 2
) Au+
/(? d7
):
r -- 2
Au+
/(15
ym 1 A
T lm).
(32)
Nex
t we
show
that
the
half-
spac
e mod
el p
redi
cts a
qua
litat
ivel
y sim
ilar b
ehav
iour
for
sur
face
str
ain
rate
s, b
ut t
hat
ther
e ar
e si
gnifi
cant
qua
ntita
tive
diff
eren
ces
betw
een
thes
e an
d th
e av
erag
ed s
train
rat
es p
redi
cted
by
the
thin
lay
er m
odel
. Th
e re
leva
nt s
train
rat
e is
obta
ined
up
on d
iffer
entia
tion
of (2
5) w
ith re
spec
t to
y an
d t,
givi
ng
and
from
this
we
read
ily d
eriv
e th
e ju
mp
quan
tity
Corre
spon
ding
exp
ress
ions
for
i. a
nd A
T co
uld
be d
educ
ed f
rom
the
res
ults
of
Spen
ce &
Tu
rcot
te u
nder
less
res
trict
ive
assu
mpt
ions
rega
rdin
g in
ters
eism
ic f
ault
slip
page
, but
we
shal
l he
re lim
it th
e di
scus
sion
to a
com
paris
on o
f (3
3) an
d (3
4) w
ith r
esul
ts f
or t
he t
hin
laye
r m
odel.
A
plot
of
stra
in r
ate
vers
us d
ista
nce,
per
tain
ing
to s
trike
-slip
faul
ting,
is s
how
n in
Fig
. 9
and
may
be
com
pare
d w
ith t
hat
in F
ig. 7
. The
max
imum
jum
p A
q(0)
whi
ch o
ccur
s at
the
faul
t is e
qual
to 2
Au+
G/(
37iH
~)
and
from
this
one
has
the
estim
ate
q/G
= ?
Au+
/[3n
HA
j.(O
)].
(35)
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
790
F. K
. Leh
ner a
nd V
. C. L
i 0.
100
0.07
5
jq d
/'GA
u"
0.05
0
0.00
0
-0 0
25
DIST
ANCE
y,'id
Fi
gure
9.
Surf
ace
stra
in r
ate
for
half
-spa
ce m
odel
as
a fu
ncti
on o
f di
stan
ce f
rom
the
fau
lt, a
t vari
ous
times
. R
epea
t tim
e 15
0 yr
.
It is
of c
ours
e cl
ear
that
thi
s sim
ple
resu
lt w
ill lo
se i
ts v
alid
ity,
whe
neve
r in
ters
eism
ic fa
ult
cree
p be
com
es im
port
ant.
Ther
e is
agai
n an
out
er e
xtre
mum
in A
?, lo
cate
d at
ym
= nH
= I
d, i
.e. t
he sa
me
distan
ce
as w
as f
ound
for
the
thi
n la
yer m
odel
, and
its m
agni
tude
I A?
1, =
Au*G
/( 1
.5
~~
~~
).
W
e also
re
tain
re
latio
n (3
1) a
s a
mea
ns
for
dete
rmin
ing
H
from
y,
and
the
rela
xatio
n tim
e ap
prop
riate
for
the
half-
spac
e mod
el m
ay b
e es
timat
ed f
rom
q/G
1 A
u'/(l
511,
I Ay
Im ).
(36)
It fo
llow
s th
at f
or t
he s
ame
valu
es o
f A
uf, q
/G a
nd H
in
both
mod
els,
the
jum
ps in
strai
n ra
te a
re re
late
d by
1 A?
Im t
hin
laye
r =
O.8
(h/H
) I A'?
1 m h
alf-
spac
e.
(37)
Con
clus
ions
A sim
ple
plat
e/fo
unda
tion
mod
el o
f an
ela
stic
lith
osph
ere
ridin
g on
a 't
hin'
asth
enos
pheri
c su
bstra
tum
sug
gests
that
visc
osity
stra
tific
atio
n in
the
upp
er m
antle
may
alte
r po
st-sei
smic
defo
rmat
ions
due
to
visc
oela
stic
rela
xatio
n ef
fect
s sig
nific
antly
in c
ompa
rison
with
defor
ma-
tions
pre
viou
sly
pred
icte
d fo
r a
Nur
-Mav
ko
half-
spac
e m
odel
. In
the
thi
n lay
er m
odel
rela
xatio
n ef
fect
s du
ring
an e
arth
quak
e cy
cle
are
conf
ined
to
an e
ven
narro
wer
zon
e abo
ut th
e pl
ate
boun
dary
, its
'pe
netr
atio
n de
pth'
dep
endi
ng s
trong
ly o
n th
e ra
tio o
f rec
urren
ce
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Plat
e bo
unda
ry d
efor
mat
ions
79
1 tim
e to
rel
axat
ion
time
and
beco
min
g m
inim
al in
the
elas
tic li
mit
whi
ch is
virt
ually
atta
ined
wh
en t
his
ratio
is
less
tha
n or
equ
al t
o on
e. A
ccor
ding
to
this
ana
lysi
s, si
gnifi
cant
pos
t- se
ismic
visc
oela
stic
rela
xatio
n ef
fect
s fo
r re
curr
ence
tim
es o
f th
e or
der
of 1
00 yr
are
in
dica
tive
of v
iscos
ities
of
the
orde
r of
lO
I9 P
as (
10”
P) o
r le
ss f
or l
ithos
pher
e/as
then
o-
sphe
re t
hick
ness
ratio
s of
the
ord
er o
f 11
5. Th
e ea
rly p
ost-s
eism
ic d
istri
butio
n of
stra
in r
ates
alo
ng a
lin
e pe
rpen
dicu
lar
to a
ver
y lo
ng t
rans
form
fau
lt in
bot
h th
in l
ayer
and
hal
f-sp
ace
mod
els e
xhib
its a
pro
noun
ced
extr
emum
at
abou
t th
ree
litho
sphe
re th
ickn
esse
s aw
ay f
rom
th
e fa
ult,
whi
ch m
ay b
e ob
serv
able
afte
r a
larg
e ev
ent
and
wou
ld t
hen
perm
it in
depe
nden
t es
timat
es
of
asth
enos
pher
e re
laxa
tion
time
and
litho
sphe
re
thic
knes
s. Fo
rmul
ae
for
obta
inin
g su
ch e
stim
ates
are
giv
en h
ere
for
a th
in la
yer
as w
ell
as f
or a
hal
f-sp
ace
mod
el.
Sim
ilar
rela
tions
may
be
expe
cted
to
exis
t al
so f
or m
ore
com
plex
thr
ee-d
imen
sion
al
prob
lem
s.
Ack
now
ledg
men
ts
This
rese
arch
was
sup
port
ed b
y th
e N
atio
nal
Scie
nce
Foun
datio
n G
eoph
ysic
s Pr
ogra
m a
nd
the
US
Geo
logi
cal S
urve
y Ea
rthq
uake
Haz
ards
Red
uctio
n Pr
ogra
m. W
e th
ank
Prof
esso
r J. R
. Ri
ce f
or h
elpf
ul d
iscus
sions
and
wish
to
ackn
owle
dge
valu
able
com
men
ts m
ade
by t
he
revi
ewer
s. N
atio
nal
Scie
nce
Foun
datio
n G
eoph
ysic
s Pr
ogra
m G
rant
EA
R78
- 129
48 0
1 an
d US
Dep
artm
ent o
f Int
erio
r, U
S G
eolo
gica
l Sur
vey
Con
tract
14-
08-0
001-
1979
3.
Refe
renc
es
And
erso
n, D
. L.,
1975
. Acc
eler
ated
pla
te te
cton
ics,
Sci
ence
, 18
7, 1
077-
1079
. Ba
rker
, T. G
., 19
76. Q
uasi
-sta
tic m
otio
ns n
ear
the
San
And
reas
faul
t zon
e, G
eoph
ys. J
. R. a
str.
Soc.
, 45,
Bot
t, M
. H. P
. & D
ean,
D. S
., 19
73. S
tres
s diff
usio
n fr
om p
late
bou
ndar
ies,
Nat
ure,
243
, 339
-341
. Ca
rslaw
, H. S
. & J
aege
r, J.
C.,
1959
. Con
duct
ion
ofH
eat
in S
olid
s. 2
nd e
dn, O
xfor
d U
nive
rsity
Pre
ss.
Cohe
n, S
. C.,
1980
. Pos
tsei
smic
vis
coel
astic
sur
face
def
orm
atio
n an
d st
ress
, 1, T
heor
etic
al co
nsid
erat
ions
, di
spla
cem
ent a
nd s
train
cal
cula
tions
, J. g
eoph
ys. R
es.,
85, 3
131-
3150
. Co
hen,
S.
C.,
1981
. Po
stse
ism
ic r
ebou
nd d
ue t
o cr
eep
of t
he l
ower
lith
osph
ere
and
asth
enos
pher
e,
Geo
phys
. Res
. Let
t., 8
,493
-496
. D
avies
, J.
N. &
Hou
se,
L.,
1979
. A
leut
ian
subd
uctio
n zo
ne s
eism
icity
, vol
cano
-tren
ch
sepa
ratio
n, a
nd
thei
r rel
atio
n to
gre
at th
rust
-typ
e ea
rthq
uake
s, J.
geo
phys
. Rex
, 84,
4583
-459
1.
Elsa
sser
, W.
M.,
1969
. Con
vect
ion
and
stre
ss p
ropa
gatio
n in
the
upp
er m
antle
, in
The
App
licaf
ion
of
Mod
ern
Phys
ics
to t
he E
arth
and
Pla
neta
ry I
nter
iors
, pp
. 22
3-24
6,
ed.
Run
corn
, S.
K.,
Wile
y (I
nter
scie
nce)
, N
ew Y
ork.
Le
hner
, F.
K.,
Li, V
. C. &
Ric
e, J
. R.,
1981
. Stre
ss d
iffus
ion
alon
g ru
ptur
ing
plat
e bo
unda
ries,
J. g
eoph
ys.
Res.,
86,
615
6-61
69.
Mat
su’u
ra, M
. &,
Tani
mot
o, T
., 19
80. Q
uasi
-sta
tic d
efor
mat
ions
due
to
an in
clin
ed, r
ecta
ngul
ar f
ault
in a
vi
scoe
last
ic h
alf-
spac
e, J
. Phy
s. Ea
rth,
28,
103
-118
. Nu
r. A
. & M
avko
, G.,
1974
. Pos
t-sei
smic
vis
coel
astic
rebo
und,
Sci
ence
, 183
, 204
-206
. Ri
ce, J
. R.,
1980
. The
mec
hani
cs o
f ea
rthq
uake
rupt
ure,
in P
hysi
cs o
fthe
Eur
th ’s
Infe
rior
, pp.
555
-649
, ed
s Dzi
ewon
ski,
A. &
Bos
chi,
E., I
talia
n Ph
ysic
al S
ocie
ty/N
orth
Hol
land
, Am
ster
dam
. R
undl
e, J
. B.,
1978
. Vis
coel
astic
cru
stal
def
orm
atio
n by
fin
ite, q
uasi
-sta
tic so
urce
s, J
. geo
phys
. Res
., 83
, 59
37-5
945.
R
undl
e, J
. B.
& J
acks
on,
D.
D.,
1977
a. A
thr
ee-d
imen
sion
al v
isco
elas
tic m
odel
of
a st
rike
slip
fau
lt,
Geo
phys
. J. R
. ast
r. So
c.,
49,5
75-5
91.
Run
dle,
J. B
. & J
acks
on, D
. D.,
1977
b. A
kin
emat
ic v
isco
elec
tric
mod
el o
f th
e Sa
n Fr
anci
sco
eart
hqua
ke
of 1
906.
Geo
phys
. J. R
. ast
r. So
c., 5
0,44
1-45
8.
Sava
ge, J
. C. &
Pre
scot
t, W
. H
., 19
78. A
sthe
nosp
here
rea
djus
tmen
t and
the
eart
hqua
ke c
ycle
, J. g
eoph
ys.
Res.
, 83,
336
9-33
76.
Smith
, A. T
., 19
74. T
ime-
depe
nden
t st
rain
acc
umul
atio
n an
d re
leas
e at
isl
and
arcs
: im
plic
atio
ns f
or t
he
1946
Nan
kaid
o ea
rthq
uake
, PhD
the
sis,
Mas
sach
uset
ts I
nstit
ute
of T
echn
olog
y, C
ambr
idge
, 295
pp.
689-
705.
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
792
F. K
. Leh
nera
nd V
. C Li
Spen
ce, D
. A. &
Tur
cotte
, D. L
., 19
79. V
isco
elas
tic r
elax
atio
n of
cyc
lic d
ispl
acem
ents
on
the
San
And
reas
Fa
ult,P
roc.
R. S
OC
. A, 3
65,1
21-1
44.
Spen
ce,
W.,
1977
. Th
e A
leut
ian
arc:
tec
toni
c bl
ocks
, ep
isod
ic s
ubdu
ctio
n, s
train
dif
fusi
on, a
nd m
agm
a ge
nera
tion,
J. g
eoph
ys. R
es.,
82, 2
13-2
30.
That
cher
, W.
& R
undl
e, J
. B.,
1979
. A m
odel
for
the
ear
thqu
ake
cycl
e in
und
erth
rust
zon
es,J
. geo
phys
. R
es.,
84,5
540-
5556
. Th
atch
er,
W.,
Mat
suda
, T.
, K
ato,
T.
& R
undl
e, J
. B.
, 19
80.
Lith
osph
eric
loa
ding
by
the
1896
Rik
u-u
eart
hqua
ke,
Nor
ther
n Ja
pan:
im
plic
atio
ns f
or
plat
e fl
exur
e an
d as
then
osph
eric
rhe
olog
y, J
. ge
ophy
s. R
es.,
85,6
429-
6435
. Y
ang,
M.
& T
okso
z, M
. H.
, 19
81.
Tim
e-de
pend
ent
defo
rmat
ion
and
stre
ss r
elax
atio
n af
ter
strik
e-sli
p ev
ents
, J. g
eoph
ys. R
ex, 8
6,28
89-2
901.
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from