Postseismic surface deformations are attributed to the inelastic flow of the subcrustal regions of the Earth following an earthquake. A multilayer representation of the Earth's rheological properties is used in conjunction with a finite element computational scheme to calculate time dependent displacements and strains subsequent to a strike slip earthquake. The deviatoric stress strain relations for the uppermost layer is assumed elastic. Lower layers are assumed to be, in order of increasing depth, a standard, linear, three element, viscoelastic solid; a linear viscoelastic fluid; and another elastic solid. Physically these layers correspond to the upper lithosphere, lower lithosphere, asthenosphere, and lower mantle, respectively. Elastic dilatational properties are assumed throughout. Appreciable postseismic displacements, possibly approaching meters, for large earthquakes, arise from viscoelastic relaxation following sudden coseismic slip. Furthermore, compared to the simpler case of an elastic lithosphere over a viscoelastic asthenosphere and the near fault postseismic shear, strain is increased, by a factor of two or more in some cases, by the presence of a viscoelastic lower lithosphere. Also, the duration of postseismic straining is increased by the relatively slow relaxation of this layer

Cohen, S. C.

English

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4(NASA-T4- 82089) POSTSEISNiC REBOUND DUE TO 	 1181- 19677
CREEP OF THE LOWER L11"HOS.PHERE AND
ASTHENOSPHERE (NASA) 1b p tic A02/fiF A u1
CSCL 04A	 U-nClas
G3/46 1882a
i	 TEchnicel Memorandum 82089
Postseismic Rebound due to
Creep of the Lower Lithosphere
and Asthenosphere
4
S. C. Cohen
:AS4 art ^^^^^
FEBRUARY 1981
National Aeronautics and
Space Administration
Goddard Space Flight Center
Greenbelt, Maryland 207 71
F
pF
peF
i
r	 TM 82089
Postseismic Rebound Due to Creep of the bower Lithosphere and Asthenosphere
Steven C. Cohen
Geodynamics Branch
Goddard Space Flight Center
Greenbelt, MD 20771
Abstract
Postseismic surface deformations are attributed to the inelastic flow of the
subcrustal regions of the earth follow ing an earthquake. A multilayer
representation of the earth's rheological properties is used in conjunction
with a finite element computational scheme to calculate time-dependent
displacements and stiains subsequent to a strike-slip earthquake. The
deviatori,c stress-strain relation for.the uppermost layer is assumed elastic.
Lower layers are assumed to be, in order of increasing depth, a standard
linear, three-element, viscoelastic solid; a linear, viscoelastic fluid; and
another elastic solid. Physically these layers correspond to the upper
lithosphere, lower . lithosphere, asthenosphere, and sower mantle, respectively.
Elastic dilatational properties are assumed throughout.' Appreciable post-
seismic displacements, possibly approaching meters, for large earthquakes,
arise from the viscoelastic relaxation following the sudden coseismic slip.
Furthermo•.e, compared to the simpler case of an elastic lithosphere over a
viscoelastic asthenosphere the near-fault postseismic shear strain is increased,
by a factor of two or more in some cases, by the presence of a viscoelastic
lower lithosphere Also the duration of postseismic straining is increased if
the viscosity'of the lower lithosphere is greater than that of the underlying
asthenosphere.
.
	
PRFCED;NG PAGE E!l: ANK NOT FILMED
i
.4"0- max, e 
hThe time dependent deformation of the earth following an earthquake has
frequently been studied in terms of the rheological properties of the crust
and upper mantle. In particular, several studies have attributed surface
deformation to viscoelastic relaxation of the asthenosphere following the
sudden stress change associated with the coseismic slip. The most commonly
employed model has been that of an elastic layer (lithoophere) lying over a
viscoelastic, asthenosphere. The deviatoric stress-strain relation has been
taken to be that of a linear Maxwell. body (Nur and Marko, 1974; Rundle and
Jackson, 1977; Thatcher and Rundle, 1079) or a Maxwell body with strain rate
proportional to a non-unity power of stress (Melosh, 1976). Other models have
examined the role of fault slip at depth (Thatcher, 1974), flow of low
viscosity magma regions (Wahr and Wyss, 1980), and anelastic relaxation of
the lithosphere (Cohen, 1980) on postseismic rebound. Other factors that
might influence coseismic and/or postseismic deformation include, but are not
limited to, fault geometry, spatial variations in rigidity (Mahrer and Nur,
1979), and spatial variations in coseismic slip.
The conceptual understanding of the influence of the earth's rheological
properties on postseismic rebound has been influenced by studies of the creep
properties of rocks (see e.g., reviews by Weertman and Weertman (1975) and
Kirby (1977)). These studies suggest that the creep mechanism and the duration
and magnitude of the creep can be a sensitive function of temperature, stress,
and to a lesser extent pressure. There seems to be little doubt that flow of
the asthenosphere is an important process in a variety of geologic processes
`	 including tectonic plate motions and the accumulation of strain at plate
r	
boundaries as part of the earthquake cycle. Therefore it is not surprising
that asthenospheric flow has been suggested as a mechanism for postseismic
G	
deformation. There is also evidence to suggest that the conditions in the
lower lithosphere may also be appropriate for some form of creept. Temperatures
may well exceed 1000°C and stress may exceed 1 kilobar (Goetze and Evans, 1979);
1
conditions at which creep processes may be active.
With these thoughts in mind I have investigated, in a theoretical and
numerical model, the postseismic deformations to be expected from visco-
elastic relaxation of the lower lithosphere and asthenosphere following an
earthquake. The models I have considered are three- and four-layer
representations of the earth with a strike-slip fault in the uppermost layer.
As will be shown below, the surface deformations derived from this model are,
under appropriate conditions, appreciably larger than those computed by ignoring
the lower lithosphere viscoelasticity. On the other hand if the effective
rigidity of the lower lithosphere shows little time dependence, no effect of
this layer on postseismic rebound is expected. Similarly the surface deformations
are enhanced when the depth of slip extends to the vicinity of the flow region
and they are suppressed when the slip zone is shallow compared to these
viscoelastic regions.
The multilayer viscoelastic model is shown in Figure 1. A strike-slip
fault is embedded in the upper elastic layer (upper lithosphere with rigidity,
V ) from the surface to a depth, D. The elastic layer has thickness H 1 . Below
this layer, to a depth H2 , is a viscoelastic solid layer consisting of an
elastic element (rigidity, u 2a ) in series with a parallel combination of elastic
(rigidity, 
u2b) and viscous elements (viscosity, n 2 ). Below this lower
lithosphere layer is a Maxwell body consisting of a series combination of elastic
(rigidity, ti 3 ) and viscous elements(viscosity, n 3 ) extending to a depth H3 and
repesenting the asthenosphere. In some of our calculations we have included a
fourth layer to a depth H4 which consists of an elastic body and represents the
I
high viscosity region of the mantle'llying under the
	 fluid upper
asthenosphere. The presence or absence of this layer does not effect the
conclusions of this paper. The rheologi,cal elements are meant to describe the
deviatoric or shear properties of the earth, the dilatational or volumetric
properties are assumed elastic. A description of the creep properties of these
2
'	 a
i
_	 .. •	 ,^^ .,may:. _t^.^, _44 a	 ^^.:._..:,a.^^^v:'.,r .^ _ a
	 .-^ .-'"	
_.	 a►t^. .»,	 d.	 ^
linear elements can be found in stan,lard texts and are summarized in the
review article by Cohen (1979). In particular, for a suddenly Ppplied and
maintained constant s'.iear stress, the three-element solid has an initial
rigidity, P a , undergoes transient creep with a time constant T = n/P b$
P
a 11b
and has a long term rigidity -u-.t P	 Similarly the Maxwell substance hasa
an initial rigidityp,undergoes L;teady state creep and has no long term
rigidity. Under conditions of constant strain this body relaxed an applied
stress with a time constant T=n/p.
The results I will discuss in the following; 	 are derived using
a two-dimensional finite element scheme which employs a version of the computer program
developed by Melosh and Raefsky (1980a) which I havz modified to accomodate
the aforementioned rheological model for the lower lithosphere. The split
node techniques of the aforementioned authors (Mclosh and Raefsky, 1980b) has
been employed to specify the fault slip. The results are derived using an
explicit time integration algorithm. The parameters used in most of the
calculations are V, 
	 p2a "- 112b 	 11	 44	 5 . 101
` dyne/cm 2 1P n2 = 1.1021
poise, n 3 = 5 . 1019 poise, D = 20 km, fi t	40 km, H2 = 75 km, H3 = 400 krii
114 = 800 km, blip = im. Sample calculations using an approximate elastic
half-space, two	 theologically different viscoelastic half-spaces, and a
two-layer model of an elastic layer over a viscoelastic half space were
compared to analytic and previously published numerical solutions to check
for accuracy in the numerical procedure. As expected the uniform elastic and
viscoelastic half-space models showed only coseismic and not time-dependent
deformations, while the two-layer model showed the expected relaxation due
to astbenospheric flow.
Turning to numerical results, the postseismic displacements, AW(t) = W(t) - W(0),
where W is the surface displacement along the fault strike direction and time
t=0 is the time of the earthquake,is shown as a function of the distance from
the fault,	 in Figure 2. At the time chosen for the figure t = 5 x 10 9 sec
ry 159 years nearly all the postseismic relaxation due to both lower lithosphere
3
Land acthenosphero creep has been completed, at least within 500 km of' tilt,
fault. The figure reveals that the postseismic d
i
splacement may exceed
several tens-of -cent Imetero for major earthquakes involving several meters
of coseismio slip. There is also the suggostion that postseismic displacements
oxtend over a much broader di stance from the fault that do the cosoismic ones.
The point of maximum potitseinmic displacement (and v', oro post8eismic shoat,
strain, see below) is about "k'00 kill from the fault, although it must be
remembered that this two- d imonor,ona- mod(-1 assumes e ssentially infinitely
, I
long faults. The postseismic shear strain Ae 13 = 	 (engineoring :,train) I,.,
ax
shown at, o function of di stance from the fault In Figure 3, It can be shown
that the cosoismic strain drop x'10 (0)  is reduced from one-half its peak value
at ki distance X = D where it is acoumed that the slip Is uniform and the
fault ruptures the surface and extends, to the depth 1) (= 20 km in the pre y s'Ont
cans). By contrast the postseismic strain (an increase) initially decreases more
slowlv with distance from the fault ar)3 reaches one-half its peak valuo at
X ^ 60 kin, This broader region of significant pontseismic -,train is of courso
a roflc-etion of the deeper position of the source of the post soismic motion,
in thin eaoo the erooping, portions of the lithosphoro and astbonospherv.
While the. preceding figures have examined the cumulative surface deformation
versus divtonce from the fault, it I ,, also Interesting to consider the time
dependence of the deformation at a fixed location. Figure 4 shows the strain
versus time at a position close to the fault (averaged over an element of the
finite element grid at distances 1-10 km from the fault). Also shown for
comparison is a similar result for the case where lower lithsophore visco-
elastioity is ignored, i.e., computed with the elastic layer extending to
the top of the asthenosphere (at a depth of 75 km). It is clear that the
effect of the shallow zono of partial flow is to double the ultimate post-
seismic strain. Moreover, the results suggest that appreciable postsei6mic
4
dc,"ormaton can occur even if the asthenosphe.re is deep compared to the depth
of the cosei g mic slip zone provided the regiin of lithsopheric viccoolacticIty begins
at a sufficiently shallow depth. Figure 4 also reveals that the duration of
surface straining following an earthquake is longer in the multilayer model.
This is, of course, a direct consequence of the assumption of a higher
viscosity (and bonce longer response time) of the lower lithosphere layer
compared to that of the asthenosphere . The
magnitude of the postseismic deformation due to lithospheric relaxation can
be increased by reducing this layer's effective longer rigidity (i.e.,
&2, crease p.) and by bringing the layer close to the seismogenic zone.
(i{1 + D). A detailed analysis of the postseismic displacements, strains,
stresses and the time dependence and variation with depth, distance from
the fault, and model parameters will be published elsewhere (Cohen, 1981).
It woOd be desirable to compare the model predictions to geodetic data.
Unfortunately, since high precision surveys have been available only in recent
years and over limited networks (in ^p-ofTqphical location and size) and since major
strike-slip events are infrequent, albeit potentially important, there is
virtually no data for making such a comparison. Thatcher (1975) has reported
postseismic strains in the range 41.9 ± 9.1 to 72.1 i 15.5 listrain over a
period of 24 years in the immediate vicinity of the San Andreas fault
subsequent to the 1906 San Francisco earthquake. While it is not clear that
this model is applicable for such a shallow event (D = 10 km) we find that
with a slip of 3-6 meters and fit 
 
in the range 10-20 km postseismic strains
of 7­36 pstrain are predicted with the time scale depending on the chosen
viscosities. Neither the observations nor the model predictions have been
adjusted for the possible effects of straining due to plate motion although
this effect may be significant (i.e., with e= 10- 6 /yr, Ac = 24 pstrain
after 24 years).
5
In summary, a multilayer *nodel o: earth has been used to examine post-
seismic deformation following a major utriko-slip earthquake.. The analysis
has suggested the viscoelastic relaxation of the earth's subsurface layers
includingthe lower litho ,.;phere and asthenosphere may be responsible for
significant deformations, reloading of the fault slip region, and broadening
of the initial coseicmic deformation region. These viscoelastir. effects are
of potentially greater significance for some major thrust earthquakes which
occur at greater depths than strike-slip events and hence in regions where
creel, processor, may be expected to have a higher degree of activation. I
hope to report on calculations of the postseismic deformations for the
thrust case at a later date.
ACKNOWLtDGMENT: I wish to thank Jay Meloob both for providing the computer
devel^pe b y M 2 
tie 
^rt^urfRaef 1ey,proAram61% w Bch f-Orme 	 10 asis or tie numerical computations and for many
helpful in.^ightful technical discussions.
. E
6
References
Cohen, 1S.C. , "Numerical W Laboratory simulation of Faiilt Motion and
Earthquake Occurrence," Rev. 922p-hy—S. §p—ace Thal. 1 17, 61-72, 1979.
Cohen, S.C., "Postseismic Viocoelaotic Sirface Deformation and Stres s
1, Theoretical Considerations, Displacement, and Strain Calculations,"
J. Geo2las. Res., 85, 3131-3150, 1980,
Cohen, S.C., "Time Dependent Deformation Following an Earthquake--A Multilayer
Model with Depth Dependent Rheological Properties," in press, 1981.
Goetze, C., and B. Evans, "Stress and Temperature in -the Bending Lithosphere
and Constrained by Experimental Rock Mechanics, 11 ,2^h s. J. R. Astr.
Soo., 59, 46$-478, 1079.
Kirby,	 "State of Stress in the Lithosphere; Inferences from the Flow
Laws of Olivine," Pure Appl.
,
Geophys., 115, 245-257, 1977.
Mahrer K.D., and A. Nur, "Static Strike-slip Faulting in a Horizontally Varying
Crust," Bull. Seismol. Soo. Am., 69, 975-1009, 1979.
Melosh, 114J. , "N. .online: ..r Stress Prwdagation in the Ear th's Upper Miantle,"
J. Geophys. Res. , 81, 5621-5621, 1976.
Melosh, H.J, and A. Raefsky, "The Dynamical Origin of Subduction Zone Topography,"
Geophys. J.R. Astr. Soc., 60, 333-35 1 , 198Oa.
Melosh, H.J. and A. Raefsky, "A Siriple and Efficient Method for Introducing
Faults into Finite Element Computations," in press, 1980b.
Nur, A., and 01 , Mauko, "Postneismic Viscoelastic Rebound," Science, 183,
204-206, lr^74.
Rundle, J.B., and D.D. Jackson, "A Three-Dimensional Viscoelastic Model of a
Strike-Slip Fault," Geophys. J. Roy. Astron. Soo., 49, 565-591, 1977.
Thatcher, W., "Strain Release Mechanism of the 1906 San Francisco Earthqtjake,"
Science, 1,84, 1283-1285, 1974,
Thatch, W., "Strain Accumulation and Release Mechanicsm of the 1906 San Francisco
Earthquake," J. Geophys. Res., 80, 4862-4872, 1975.
!F--
7
Thatcher, W., and J.H. Rundle, "A Model for the Earthquake Cycle in Under-
thrust zones," J. Geophys. Res., 84, 5 O-r- a56, 1979.
Wahr, J., and M. Wyss, "Interpretation of Post;ae lsmic Deformation with a
Viscoelastic Relaxation Model," J. Geo h s. Res., 86, 6471-6477, 1980.
Weertman, J., and J.R. Weertman, "High Temperature Creep of Rock and Mantle
Viscosity," Ann. Rev. Earth Plan. Sci., 3, 299-515, 1975.
8
_ .
	 -
3I
2
r.r
/I
I i	 ELASTIC SOLID
QUM-
STANDARD	 92b
VISCOELASTIC
	
P2a
H	
SOLID4QQ
21	 712
MAXWELL VISCOELASTIC FLUID
µ3	
773
ELASTIC SOLID
{4
µ4
Figure 1. Rheological properties and parameters of
multilayer postsehnic deformation model.
9
y
r
Ezw2w
a.
U
W
wv)
Oa.
0
z
0
U_
w
WO0
50
48
46
44
42
40
38
36
34
32
30
28
26
24
22
20
18
16
14
12
10
8
6
4
2
0
AWu W(t) -W(0)
t-5x 101 SEC sw 159 yrs
Qµ20 Km
H, 40 Km
H 2 = 75 Km772 = 5 x 10 20 POISE
H 3 = 400 Kmi7 3 = 5 x 1019 POISE
H4 =800Km
ALL A's = 5 x 10" DYNE/CM2
FAULT SLIP= 1 METER
W(0)
AW
* i
0	 100 200 300 400 500
DISTANCE FROM FAULT (Km)
Figure 2. Computed surface displacements parallel to fault strike
versus distance front fault. For postseismic displacements, AW,
curve shows total displacements after 5 . 10 9 seconds — 159 yrs.
10
3
z
<t 10 -8
U)
U_
2U
W`
U)
On
0 10-7
co
W
_U
W
W
O
L)
1 0-1
A613 = E13(t) _ E13(0)
t = 5 x 101SEC -- 159 yrs
E13(0) D=20  KmALL µ's= 5 x 1011
 DYNE/CM2
H 1 = 40 Km
H 2 = 75 Km ?12=5 x 1020 POISE
H 3 = 400 Km ?73 = 5 x 10 11 POISE -
H 4 = 800 Km
FAULT SLIP 1 METER
AE13 AE 13
13(0)
0)
SIGN REVERSAL
+ to -
10-8
0	 100	 200	 300	 400	 500
DISTANCE FROM FAULT (Km)
Figure 3. Computed strain in horizontal surface plane versus distances from
fault. For postseismic displacements, Ae 13, curve snows total strain after
	
5 .10 9
 seconds 159 yrs. Engineering strain Ae,3
	
aX is plotted.
11
441	 ...	 ..un. .......^.1 ... •i 	 i:" I,	 ^TSIGl^6RrRA<.._s	 ..	 ..	 .
10-5 1^ r
^
r	 A613(t)	 E 13(t) C- 13 (0)X^1-10 Km	 D=20Km
FAULT SLIP = 1 METER
H 3 = 400 K m 'q 3 = 5 x 10 19 POISE
H 4 = 800 Km
ALL A's = 5 x 10 11 DYNE/CM'
H 1 = 40 Km
H 2 = 75 Km 772 = 5 x 1020 POISE
—
	 {
H, = 75 Km
NO "2" LAYER ^---
M
W
Z
Q
X
Cf)	 10 - 6U
2i	 `	 r
C10
O
a-
	
10 YRS	 25 YRS 50 YRS 100 YRS 200 YRS
	
10-7	 11	 1	 t	 1	 1	 11	 1	 1	 1 11	 1	 11	 11	 I	 I	 1
10 8
	 0'9	 1010
TIME (SECONDS)
Figure 4, Computed postscismic strain versus time following earthquake. Dashed line shows corresponding result
for elastic lithosphere-viscoclastic astlienospllere model with no creep in lower lithosphere,
12
`9


Postseismic rebound due to creep of the lower lithosphere and asthenosphere

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