Capturing intraoperative deformations: research experience at Brigham and Womens hospital
Large-scale characteristics of plate boundary deformations related to the post-seismic readjustment...
18
Geophys. J. R. astr. SOC. (1982) 71,175-792 Large-scale characteristics of plate boundary deformations related to the post-seismic readjustment of a thin asthenosphere F. K. Lehner and V. C. Li *Division of Engineering, Brown University, Providence, Rhode Island, USA Received 1982 April 26; in original form 1981 October 9 Summary. The large-scale response of an elastic lithosphere, riding on a ‘thin’ viscoelastic asthenosphere, to periodically occurring ruptures at a transform or subduction-type plate boundary is described approximately by appropriate limit cycle solutions for a plate/foundation model introduced previously by Rice. The cyclic behaviour of thickness-averaged displacements, strains and strain rates, their decay away from the plate boundary, and a resolution into coseismic and post-seismic alterations are obtained and their dependence on repeat time and a characteristic relaxation time investigated. A comparison is made with existing periodic solutions for the surface deformations in a Nur-Mavko half-space model. This suggests important effects due to viscosity stratification on post-seismic rebound when earthquake repeat times exceed relevant relaxation times by at least one order of magnitude. 1 Introduction Model studies of post-seismic surface deformations at active plate boundaries have become an important method of investigating the subcrustal rheology of the Earth. Beginning with the work of Nur & Mavko (1974) and Smith (1974), the typical earth model assumed in such theoretical analyses is that of a viscoelastic half-space with an elastic surface layer, the latter representing the lithosphere and the former an asthenospheric substratum capable of viscous relaxation, usually in the manner of a linear Maxwell body. In the two-dimensional Nur-Mavko model an edge or screw dislocation is introduced at a certain depth within the lithosphere to represent a sudden uniform displacement along a dip-slip or strike-slip fault which extends indefinitely along strike. The quasistatic surface deformations predicted by this model may be compared with geodetic observations of post-seismic rebound motions associated with large earthquakes, thus allowing new inferences of relaxation times and viscosities for the asthenosphere. This earthquake loading problem has since been modelled in greater detail in studies which allow for finite faults and arbitrary distributions of slippage (Barker 1976; Rundle & Jackson 1977a, b; Rundle 1978; Matsu’ura & Tanimoto 1980) as well as for more complex rheological layering (Cohen 1980, 1981; Yang & Toksoz 1981). Detailed comparisons between model predictions and geodetic observations Now at Department of Civil Engineering, MIT, Cambridge, Massachusetts 02139, USA. by guest on September 20, 2014 http://gji.oxfordjournals.org/ Downloaded from
Transcript of Large-scale characteristics of plate boundary deformations related to the post-seismic readjustment...
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
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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)
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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
Plat
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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Plat
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
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
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
by guest on September 20, 2014http://gji.oxfordjournals.org/Downloaded from
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
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