A model is proposed for post seismic surface deformations attributing them to lithospheric and asthenospheric viscoelasticity. The model predicts that the deformations due to lithospheric viscoelasticity depend on the ratio of the effective shear modulus acting long after the lithospheric viscoelastic relaxation to that acting immediately following the earthquake. While such deformations are generally smaller than those associated with asthenospheric viscoelasticity, they occur on a shorter time scale and may be in opposite direction to both the motion occurring at the time of the earthquake and that occurring as the asthenospheric relaxation occurs

Cohen, S. C.

English

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Produced by the NASA Center for Aerospace Information (CASI) 
https://ntrs.nasa.gov/search.jsp?R=19790008309 2020-03-22T01:41:53+00:00Z
NOVEMBER 1978
National Aeronau.ics and
Space Administration
Goddard Space Flight Center
Greenoeit, Maryland 20771
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NASA
Technic;al Memorandum 79668
Postseismic Surface Deformations
Due to Lithospheric and
Asthenospheric Viscoelasticity
(NASA -TM - 79EFA)	 FOSTSEISMIC SURFACE
	 N79-16480
rEFOPMATICNS C[1F 10 LITHCSPEIERIC ANr
ASTfiF.NOSPHERIC VISCOELASTICITY (NASA)
	 13 )
HC A02/MF A01
	 CSCL 089
	 Jnclas
G3/46 13579Steven C. Cohen
R
R)STSIi1SMIC SURFACE DEFORMATIONS
DITE TO I.IT110SP11ERIC
AND ASTHFNOSPHI RI(' VISCOELASTICi,rY
Steven C. Cohen
Geodynarnics Branch
Goddard Spare Flight Center
Greenbelt, Maryland _'0771
ABSTRACT
This paper proposes a model for post-seismic surface deformations by attributing them to
lithospL-:ric and asthenospheric viscoelasticity. The model predicts that the deformations due to
lithospheric viscoelasticity depend oil 	 ratio of the effective shear modulus acting long after
the lithospheric viscoelastic relaxation to that acting immediately following the earthquake.
While such cleforr.iations are generally smaller th. , w Ihose associated with asthenospheric visco-
elasticity, they occur on a shorter time scale and may be ill direction to both the mo-
tion occurring at the tin g e of the earthquake and that occurring as the asthenospheric relaxation
occurs.
1 .	 ...
Ill
POSTSFISMIC SURFACE DFht)RMATIONS
DUE TO LITHOSPIfFRIC
A'ND ASTHFNOSPHERIC VISCOELASTICITY
There is considerable interest in the time dependent surface deformations following an
earthquake. The description of such deformations is important for understanding postseismic stress
relaxation and redistribution, the occurrence of aftershock., and in developing models of earth
structure and rheology. informations that occur on a scale which is long compared to the time for
elastic rebound in an earthquake are p,esumably due to anclastic properties of the earth. We intend
to model some of' these anclastic properties by using linear viscoclastic elements to describe the
earth's rheology. An important principle associated with the linear theory is hie correspondence
lrinciple which allows the time dependen, viscoclastic deformation to be deduced by applying
certain mathematical transformations to the solution for the associated clastic problem. The
application of the correspondence principle to a homogeneous half-space reveals that when the
vist.oelastic properties in shear and dilatation are identical there is no additional surface d_formation
beyond the purely elastic displacement. 'rhus for time dependent posts';smic displacement to
occur with the context of a linear theory either there must be discontinuities in the properties of
the half-space or the responses in shear and dilatation must he different. The latter view was
adopted by Singh and Rosemnan t 1974)
  who calculated surface deformations associated with a
displacement discontinuity in a half-spac-e which is elastic dilatational and Maxwell or Voigt
deviatoric. Conversely, Nur and Mavko 1 10741 considered a dislocation in an elastic layer lying
over a viscoclastic half-space. Physically the elastic layer is associated with the earth's litho-
sphere and the viscoclastic half-space with the underlying asthenosphere. They considered both a
thrust fault which was represented by an edge dislocation and a strike-slip fault which was repre-
sented by a screw dislocation. Referring to the strike-slip case they found that the time-dependent
displacement parallel to the direction of the initial fault motion is such as to increase the total
displacement. The relative magnitude of' the viscoclastic versus elastic displacement increases with
distance from the fault. Furthermore, the viscoelastic displacement increases as the depth of the
screw dislocation approaches the lithosphere-asthcnosphere boundary. The time constant associated
with the motion is several years, this being the time constant for the viscoelastic response of' the
asthenosphere.
In addition to the viscoclasticity of the asthenosphere, there is evidence to suggest that a
shorter ttnic scale viscoclasticity can he associated with the lithosphere. flock mechanics expen-
ments, for example, suggest viscoelastir response times of 10' - 10 5 records (Rohertson, 1964).
Episodic creep events along the San Andreas Iatllt have suntlm time constants (Crough and Burford,
1977) and might he associaied with lithospheric viscoclasticity as might sonic aftershocks.
With these thoughts in mind we propose the following model for postscismic deformations.
The lithosphere's instantaneous elasticity and subsequent partial flow is
rodeled by a standard viscoelastic solid with shear stress, 0,— strain,ep relation
(Flagges 1967)
\1 + 1't dt	 G= ^^lu + kIt dt) e	 1 I I
is the retard::tion time at con-The instantaneous shear modulus is 
	 The quantity rt = <lo
stint stress for the lithospheric layer.'rhe asthenosphere is modeled as a Maxwell viscoelastic
element with shear stress-strain relation
C
d 1	 de
I + h2 dt ° _ ^lz dt
y
liere the shear modulus is 
^z 
and the stress relaxation time for constant strain is ra = p, We
P2
shall have need of the following parameters a = ^ 9z and Q = 9t — 1. Nunnc rically a has a90 pt
%aluc near unity with the values 0.25 4.0 oeing used by Rybicki ( 1971); for 0 a value i = 3 is
typical (Dieterich, 1972). For the model of the fault wv choose the screw dislocation represen-
tation of' a strike-slip Fault used by Rybicki ( 1`)71 ) and Nur and Mavko (1974). We shall Iind it
convenient to assume ra > rt although this simplification is for physical clarity rathct than
mathrntatiral necessity.
(2)
. 1FF-
The starting point for the analysis is the equation for the displacements associated witi, a 	 •
dislocation, AU, in an el,+s 	 layer with shear modulus µ, overlaying a half-space with shear
modulus tae . With the coordinate system shown in Figure I , the only non-vanishing displacenicnt is
U 2 where ( Rybicki, 1971)
DUI n 
+tan' s 	 Z ( µi + JU A"tan-' 3n11
2 -1) ,11 + X ]	 (3)
To obtain tic viscoelastic solution we replace p, and N2 by the corresponding Laplace transformed
viscoelastic parameters µ, (s) ai++i N2 (s) where s is the transform variahle. We then replace Alt by
DU
s
since the solution is valid only for t > 0 and the Laplacc transform for this step function in time
1
is 
s	
Pik , resulting modified equation I gives the Laplace transform for the viscoelastic deforma-
tion. Taking the inverse transformation yields the desired time dependent displacements. To carry
out this procedure we use the standard results (Flugge. 1+167)
qo1+^ s
2 1+p,s
1	 (I2
µ2 ( S ) 
= - 
I + p2 S
Then
µ, (s) - µ2 (s) i ' I a, S2 + a 2 s + a3 n IG n(s) = µ i (S) + N2 (s)
	
s	 f (s-s^(s-s 2 ) ] s	 (4)f	
J	 LL
where
r. - I	 a-1 -p 	 a
a , - a+ r	 a2
-71-(a+ 	 a3 = TaTr (a + I
la+ I +Q)	 a
s, _ - i (a + 1 1
	
s2	 a+ +	 ra
T lic simple forms for s, and s2 result from the assumption that ra ;ii> rr , Taking the inverse Laplace
transformation by the method of residues we Lind
3
1^7 _
I 
+ (n , I f' L
l.ln -I (s' )+ F 2
n-1 (, 2 1]	 IS)
where
I n-1 - dn-I 
rI A S- _1011 est sl I
' 	 (1411-1 L	 SI	 s
11-1	 do-1	 S	 n
F 2	 - -- B s^ - I )n est ^-
dsn-1 I
	
s
and
01-1	 a_ a-1 -Q
A = a 1 =n+1 B
=s1 = a+T+^/
We are most interested in determining the surface displacements at three times. At nine t = 0 we
recover the instantaneous elastic displacements. at a later time 7 a ^> t );- T 1 . the deformation asso-
ciated with viscoelasticity of the lithosphere has occurred and at yet a later time t )> 7  the
deformation associated with viscoelasticity of' the asthenosphere has also taken place. In these
three limits equation 5 becomes particularly easy to evaluate and
U	
,AU2 
+ tan"' - Z A ll tan-' -	
2X1 ()	 1
	
2(t =0)= n 1	 `
	
I	 4n' I1 2 -1) 2 + X, 2
U 2(Ta Y t s 7 1 ) = DU ^- I + tan ' 1 IBn tan ' — _?X' ^ —
((	 I	 4n=112 - U2
 + Xr I
(6)
U 2 (t ^ ra) = 7U _ a + tan-' j - tan-'
2X1 U
4n2 III _ 1) 2 + Xi 2
Notice that the deformations associated with the lithosphere viscoelasticity depend critically on the
parameter B which in turn is strongly influenced by the quantity 1 +Q. This latter qu:nrtity is the
ratio of the effective shear modulus for t *, rr to that for t = 0. As this ratio increases so do the
deformations.
4
1As an illustration of the deformations predicted by these equations we consider the case a = 1,
Q = 3, O = 11/2, The results are presented in Figure 2. The displacements that occur on the time scale
of the lithospheric viscoelasticity act to reduce the initial elastic displacements while those occurring
on the time scale of the asthenospheric viscoelasticity increase the displacements. At first it might
scan unlikely that file lithospheric viscoelasticity acts to reduce the initial elastic displacements. In
fact, in an carlicr paper concerned with the role of viscoelasticity in fault dynamics we argued that
viscoelasticity could be responsible for forward post-seismic creep and aftershocks along a fault
(C'olren, 1978). "file present behavior can be explained by a simple one dimensional argument using
the standard linear solid shown in Figure 3. Prior to the earthquake shear stress is slowly loaded on
the system with the result that just prior to the earthquake the shear force is
I i = Naµl ' L
Pa+Pb
where L is the initial stretch of the element. hurmg the earthquake there is a sudden drop in the
shear stress and stretch. Only the spring with elastic constant µ a can respond oil
	 time scale of
all
	 so immediately after the earthquake the shear force is
F r = Pa 1 + LL 
1i - µ a OX
where OX is the displacement occurring in the earthquake. In the fault motion model there is
sliding against friction during the earthquake and after the earthquake friction prevents further
motion until the stress partially recovers to an amount exceeding the I'rictional resistance. In the
present model for the motion off the fault there is no friction and the initial dislocation is relaxed
by the sliding so that L = AX. Thus the elastic shear stress is negative immediately after the
earthquake and viscoclastic flow acts to reduce the initial elastic displacement. We are, therefore,
ignoring any additional forward displacement induced by post-seismic creep along the fault.
The increased displacement occurring when the asthcnosphere adjusts to the earthquake is the
more usual viscoclastic rebound discussed by Nur and Mavko (1974) and is due to asthenosphcric
flow and associated drag on the lithosphere.
5
Althouph we have not yet investigated whether present geodetic or seismic data permits a
definitive test for the validity of the concepts presented here, there arc a number of potentially
significant implications of the theory if it is correct. One of the most important is that surveys
taken during the days to months following an earthquake may he misleading if interpreted in terim
of the elastic rebound occurring at the time of the earthquake. Furthermore faAurc to achieve
closure ill network survey Wright be attributed to the continued motion occurring as the network
is traversed. Oil 	 positive side, repeated simultaneous determinations of the location of survey
sites in the weeks following a mayor earthquake might produce mush information about the earth's
nccology and tectonic proccsse,,.
'I'll ,! present theory has suggested that viscoclasticity in both the asthcnosplier; and litho-
sphere may contribute to post-seismic surface detOrmations, A detailed quantitative model
will re(lidre additional consideration of the three dimensional nature of the problem with the
possibility that the dilational and deviaturic responses may he different. The model can he
developed by applying the procedure used in this paper to the elastic deformation equations of
kuudle and Jackson (1077). The labor involved in such calcuations is more arduous than that
involved in those we have reported here; we are in the process of working out the details.
6
Rl+'FRF.NCl:S
Cohan, S.0 , "The Viscoelastic Stiffness Model of Seismicity,"J. Geophys. /?., in press 1978
Crough, S.T., and R.U. Burl'ord, "Empirica! Law for Fault- C reep Events," Tectonopht:ylrs,
42, 753-759, 1977
Diete•.ich, J.H., "'Time Dependent Friction as a Possible Mechanism for Aftershocks," J. Geophys.
hrs., 77, 3771 -3791, 1972
Flogge, W., Vlsroelastiriti . Blaisdell. Waltham. Mass., 1967
Nur, A. and (I. Mavko, "Postseisnuc ti iscoelastic Rebound," Srien(e, 183, 204-200, 1974
Robertson, F.C., "Viscoelasticity of Rocks" in .State of Stress in the 1.*arth's Crust, W.R. Judd (Ed.),
Elsevier. New York, 1904
Rundle, J. B., and 1). 1). Jackson, "A Three-Dinik • nsional Viscoelastic Model of a Strike-Slip
Fault," Geophys J. R. Astr. Soc., 49, 575-591, 1977
Rybicki. K.. "The Elastic Residual Ficid of a Very Long Strike-Slip Fault in th ►- Presence of a
Discontinuity." thilt..Seismol.Soc. Amer.. 61. 79-92, 1971
Singh, S.J., and M. Rosenman, "Quasi-Static Deformat :n of a Viscoelastic lialf'-Space by a Dis-
placcmrn; Dislo.•ation." Ph ,rs. Earth Planet. hiterinrs, 8, 87-102, 1974
7
FIGURE CAPTIONS
figure I. Model for Lithospheric and Asthenosplieric Viscoclasticity. A Strike-Slip Fault is
Modeled as a Screw Dislocation in the I.ithosplicric Layer.
Figure 2. Computed Surface Deformations as a Function of Distance from the fault for Various
Times Following an Farthyuake.
Figure 3. (theological Element for Standard Viscoelastic Solid.
x 	 x,
Figure I. Model for Lithosphcric and Astlienosphcric Viscoelasticity. A Strike-Slip Fault
is Modeled as a Screw Dislocation in the Lithosnccric Layer.
8
mN
X I=
r
C^
C^	 Ci	 O	 O	 C)AV
n I
Figure 2. ('umh111Cd Surface Wformatious as a I unction ut Distance from the Fault
for Various 'runes i-ollowittg an F:artltquake.
T
c^
Pb
N,
Figure 3. khrological Hemvnt for Standard Viscoclastic Solid
10


Postseismic surface deformations due to lithospheric and asthenospheric viscoelasticity

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