Pergamon Computers & Geosciences Vol. 22, No. 7, pp. 751-764, 1996

Copyright 0 1996 Elsevier Science Ltd

PII: SOO98-3004(96)00004-0 Printed in Great Britain. All rights reserved

009%3004/96 $15.00 + 0.00

DEFORMATION PRODUCED BY A RECTANGULAR DIPPING FAULT IN A VISCOELASTIC-GRAVITATIONAL

LAYERED EARTH MODEL. PART II: STRIKE-SLIP FAULT--STRGRv AND STRGRH FORTRAN PROGRAMS

TING-TO YU,‘t JOHN B. RUNDLE,’ and JO& FERNANDEZ2

‘Cooperative Institute for Research in Environment Sciences, University of Colorado, Boulder, CO 80309, U.S.A. and 21nstituto de a Astronomia y Gedesia (CSIC-UCM), Facultad de Ciencias Matematica,

Ciudad Universitaria, 28040-Madrid, Spain (e-mail: rundle@hopfield.colorado.edn)

(Received I7 January 1995; revised I5 December 1995)

Abstract-The theoretical and computational methods for the calculation of viscoelastic-gravitational displacements resulting from strike-slip faulting in a layered Earth model were described. We considered a medium composed of one elastic-gravitational layer over a viscoelastic-gravitational half-space. The FORTRAN programs STRGRV and STRGRH used to compute vertical and horizontal displacement, respectively, are described. Some examples of computed displacements with various parameters are presented. Copyright 0 1996 EIsevier Science Ltd.

Key Words: Viscoelastic-gravitational layered earth model, Strike-slip fault, Vertical and horizontal displacements.

INTRODUCTION

This is the second paper in a series dealing with the surface displacements due to an elastic layer overlying a viscoelastic-gravitational half-space as described in Yu, Rundle, and Fernandez (1996). Many models deal with a viscoelastic strike-slip fault, but none have calculated deformations for viscoelastic-gravitational media. Regions of high seismicity such as southern California require such models for computing the postseismic viscoelastic- gravitational displacement at various times after the main shock. Ma and Kusznir (1992, 1993, 1994a, b) computed the displacement, stress and strain changes for coseismic and fully relaxed stages at the surface and underground. However, they neither computed the time-dependent viscoelastic displacement at different times after the main shock, nor did they include the gravity effect in their models. To compute the stress changes induced by previous earthquakes at various times in the earthquake cycle, the time-dependent displacement fields are required. For using the high resolution data acquired by modern techniques such as global positioning systems (GPS), the effect of gravity has to be con- sidered. The basic approach and methodology in this paper are based upon Rundle (1981), the same as in the companion paper (Fernindez, Yu, and

TCurrent address: Institute of Earth Sciences, Academia Sinica, P.O. Box l-55, Nankang, Taipei, Taiwan (e-mnil: yutt@earth.sinica.edu.tw).

Rundle, 1996), and thus we describe only the differ- ences between the two models in the paper.

In the next sections we review briefly the defor- mation model, the construction of the elastic Green’s functions, the numerical formulation used, and the FORTRAN programs developed to compute vertical and horizontal deformation as a result of strike-slip faulting in an Earth model composed of an elastic-gravitational layer overlying a viscoelastic- gravitational half-space. Finally, illustrations of calculations are given for the vertical and horizontal displacements from a finite rectangular strike-slip fault. Because of the length of code, detailed tech- niques used to construct this code will not be dis- cussed here. The purpose of this paper is to provide an introduction and user documentation for STRGRV and STRGRH FORTRAN codes. The application and verification of this model will not be discussed. These topics are treated in Yu, Rundle, and Fernadez (1996).

DEFORMATION MODEL

We use the same technique to construct the sol-

ution for strike-slip faulting as paper I (Fernandez,

Yu, and Rundle, 1996) and readers should refer to equations (I)-( 11) in that paper. The major difference between the strike-slip fault and dip-slip fault is the source and kernel functions. The source functions [Q,,] for the six elementary dislocation sources have

751

752 T.-T. Yu, J. B. Rundle, and J. Fernandez

been derived by Ben-Menahem and Singh (1968). In the notation of Singh (1970) (jk) refers both to the direction of the force system and to the normal of plane in which it is applied.

For a fault plane inclined at dip angle II/ to the horizontal

(jk) = (1, 2)cos i/j + (3, I)sin cl/ (1)

with each component given by

(jk) = (1,2): (D,), = PLyi (Ok)* = 2py (2)

(jk) = (3, 1): (D,)2 = 2y (Of), = - 2yi. (3)

Here, y = AU dZ/4rc, AU is the relative displace- ment along the crack, dC is an element of area on the crack, and 6 = l/(3 - 40). By summing the two source contributions, the displacement at the surface may be written

ll= s

Z k dk{[x:(O)P, + y:(O)B, + z:(O)C,]cos II/ 0

+[xt(O)P,+yt(O)B,+z~(O)C,lsin1(1. (4)

Substituting equation (4) from paper I for P,, B,,, , and C,, replacing e”“” and ie”“” by cos(m0) and -sin(mtJ), respectively, to obtain the real part, and splitting the displacement vector into its three components, we obtain

by Rundle (1981a, b), Fernandez and Rundle (1994a), Yu, Rundle, and Fernandez (1995), and Fernandez and Rundle (1994b) for the kernel’s XL and y!,,. We apply equations (92)-(96) of Rundle (1980) to compute kernel z!,,.

The kernel functions xf, y,!,, and zf, may posses poles (Rundle, 1981a; Fernandez and Rundle, 1994a) in the k-axis. They are located, if they ex- ist, close to the gravitational wave number k, given by equation (26) in Rundle (1981b). Rundle (1981a) solves this problem by integrating the kernel functions around the poles. We have found that poles contribute an insignificant part to the final results, while requiring many CPU-intensive calcu- lations. In fact, these integrals can be regarded as Cauchy principal-values integrals. Therefore, in our programs we achieve a considerably enhanced efficiency by setting the kernels equal to zero at the pole positions.

Accuracy of this model is about 1 x 10d3, and depends on the precision chosen for inte- grating Green’s function along both the strike and dip. The algorithms of STRGRV and STRGRH programs are similar to the codes dealing with dip- slip faulting; however, the kernel and source function computing schema is different between these codes.

u,=j;kdk{[f ’ (” ) Y,(O) akr J,(kr)+zl(O) (‘> r1

g J,(k ) cosfIcosti

}. (5)

z+,= -j;kdk{[f ' ('> Y,(O) kr J,(kr)+zt(O) z J,(k 1 (") rl sinecos$ +[y:(O)(~)J,(kr)+fz:(0)(~)~~(kr)]cos2H sin+ >. (6)

cc u:= - s {[ k dk f xl (O)J, (kr)sin 0 sin Ic, - xi(O)J,(kr) (7) 0

-

Equations (5)-(7) give the solution to the elastic- gravitational problem of a point nucleus of dilation in an elastic-gravitational layer over an elastic- gravitational half space.

The introduction of viscoelasticity into Equations (5)-(7) is carried out by techniques described by Rundle (1982).

NUMERICAL FORMULATION

We consider a gravitational medium composed of one elastic layer overlying a viscoelastic half-space, and refer to the layer properties as I,, pL, p,_, and the half-space properties as I,, pH, pH. The numerical formulation used in the program is described in detail

PROGRAMS

Using this formulation, we have developed two FORTRAN 77 programs to compute vertical and horizontal displacements, identified as STRGRV and STRGRH, respectively. We consider an Earth model composed of one elastic layer overlying a viscoelastic half-space. The fault plane is located entirely in the elastic layer. Both programs run on Sun Spare 10 Workstations and need about 95min to compute displacements for 50 data points.

The input data for both programs (see Table 1) are the characteristics of the Earth model (layer thickness H, Lame constants AMU and ALAM, and densities RHOL), fault characteristics (dislocation U, depth

Deformation produced by a rectangular dipping fault--II

Table 1. Input data format for program STRGRH, see text for explanation of symbols used

$REEDIN H=30.,HMIN=O.,W=30.OOOO,AL=l.OD2,THETA=!9O.,U=lOO.,

ALAM=3.,AMLJ=3.,ALAMl=3.,AMUl=3., $END

$YINIT YSTART=-100.O,DYOBS=16.,NNY=1,XSTART=100.l,DXOBS=l6.,NNX=l,INDIC=O,

$TIMVAR TSTART=O.,DELT=l .O,NTIME=5,NMTERM=6, INlTME=O, $END

$RELAX NREL=l , $END

$GRED RHOL=3.O,RHOH=3.O,RHOFCT=l.00, $END

5

1.

5.

10.

50. 100.

HMIN, fault plane width W, semi-length along the strike AL and dip angle THETA), and the X and Y coordinates of the points to be computed (Fig. 1). The NAMELIST function is used to input data according to the format shown in Table 1. ALAM 1, AMUI is the Lame parameter for the half-space, YSTART is the first value of Y coordinate perpen- dicular to strike, DYOBS is the increment of Y, NNY is the step length in Y coordinates, XSTART, DXOBS and NNX are the same parameters for X coordinates that are parallel to the strike of the fault. NSTEP is the number of integration intervals for Green’s function integration. If INDIC is greater than zero, the programs read individual points. In the TIMVAR NAMELIST, the necessary information is given for the times we wish to compute. TSTART is the initial time, increment is DELT, number of different times is NTIME, and number of times used to invert the Laplace transform is NMTERM. If INTIME is equal to zero, the program reads individ- ual times. If NREL is equal to 1, AMUI relaxes, if NREL is equal to 2 AMUI and ALAMl relax with

X .A.:.:..

Y /y %

,,:, jj~:.~~~i:,.~..

,,,, >::I:=;., : . . . . :.:::::::::::>>:.. Layer *,, (,,, i::::::::::::::::*: ,,,:. :_, ‘.:::j::::*~::::>? ..:.::::::::::::::::::~:::::.‘::,::.:~~~~~~~~~~.~ .:,., I’:::i:p I I//v

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . :x:::::::.:::+:::::;:;: . . . . .‘.’ .:.: :‘i,~:i:i:~~:::::::::::~,:~~ .::; ::::: : *....... __. ::=::::y::= ‘.‘i.,/ .:.:‘:.“..A..:.:.: .,.,.,.i,.,.i,.,., . . . . _. _. .i.:.:.. ‘iA .i_.____._..__. . . . . . . . . . . . . ‘->:.:.?r.z...,:.:.: .,.i,.....i,_,__.,.,._.,. 11 ‘.::::~:::::::::jg::::::::::x:~::.: /.. y:~::::::::~;:~:~~::.... ‘.::;:i:~:::.:.. .,_ llalf space

1 t Figure 1. Geometry and coordinate system for a rectangu- lar, finite, dipping fault in elastic-gravitational layer over viscoelastic-gravitational half-space. D is depth, W is fault plane width;21 is length along strike, ‘P is-dip angle, and

H is layer thickness (Rundle, 1982a).

constant Poisson ratio, and if NREL is equal to 3, AMUl and ALAMl relax with constant bulk modu- lus. In GRED NAMELIST, information about the densities is given, RHOL corresponds to the layer and RHOH to the half-space. The gravity effect can be suppressed by setting RHOFCT equal to a small value (for example, 10e3). The values that follow the density control are the time poles needed to compute the Laplace transform inversion.

The units used for the programs for the data inputs are km (lo3 m) for distances, g cmm3 (IO3 kg mm3) for densities and 10” dynecm-* (lo’* N mm*) for the shear modulus. The output of the programs is shown in Tables 2 and 3 for vertical and horizontal displace- ment, respectively, where the input data are written with respect to the computed displacements in each point.

The subprograms for STRGRV and STRGRH programs are summarized in Tables 4 and 5. A flow- chart of the major components of the code is shown in Figure 2. The ICSCCU, DCSQDU, MMBSJO, MMBSJI, DGECO and ZGECO subprograms from the IMSL library are used to perform the numerical integration and other tasks. Both programs calculate the displacements for the situation of a finite three- dimensional strike-slip fault only.

RESULTS AND CONCLUSION

We computed the horizontal displacements caused by a 90” strike-slip fault with gravity effect in Figure 3. To show the spatial effect of both com- ponents, the profile is made at X = 200 km, which is 100 km off the fault tip. These effects cannot be seen in the simplified two-dimensional models with infi- nitely long faults. The difference produced by includ- ing the gravity effect is shown in Figure 4. It is determined that gravity affects the displacement mostly near the extension of the strike (y/H < IO),

754 T.-T. Yu, J. B. Rundle, and J. Fernindez

Table 2. Example of output from program STRGRV

$reedin h= 30.oooooooO0000, hmin= O., w= 3O.ooooooooooo0, al= 100.000000000000, theta= 90.000OOOOO0000, u= 100.000000000000, alam=

3 .ooOOOoOOOOOOO, alaml= 3.OOOOooooooooo, amu= 3.0000000000000, amul= 3 .oooooOOOOOOOo

$end $timvar tstart= O., delt= 1.OOOOOOOOOOOOO, ntime= 5, nmterm= 6, intime= 0 $end $yinit ystart= -100.0OOOOOO00000, dyobs= 16.000000000000, nny= 1 , xstart= 100.100000000000, dxobs= 16.00000OOOOOO0, nnx= 1, indic= 0 $end $gred rhol= 3.0000000000000, rhoh= 3.0000000000000, rhofct= 1 .OOOOOOOOOOOOO $end

FINITE RECTANGULAR FAULT

AMUl RELAXES

TIME POLES ARE--

0.50000000 1 .ooooooo 5 .ooooooo 10.000000 50.000000 100.00000

x= 100.10000 Y= -100.00000

T= 0. COSEISMIC UZ = -0.97979438 T= 1 .OOOOOOO POSTSEISMIC UZ = -0.20025738E-02 T= 2.0000000 POSTSEISMIC UZ = -0.4 1796455E-02 T= 3.0000000 POSTSEISMIC UZ = -0.5398 1609E-02 T= 4.0000000 POSTSEISMIC UZ = -0.606 16236E-02

Deformation produced by a rectangular dipping fault-11

Table 3. Example of output from program STRGRH

LAYER THICKNESS: H= 30.00 KM MINIMUN VERTICAL DEPTH: HMIN = 10.00 KM FAULT LENGTH DOWN DIP: W = 20.00 KM FAULT SEMILENGTH: AL = 120.00 KM DIP ANGLE: THETA = 90.00 DEGREES DISLOCATION: u = 100.00

LAME CONSTANTS. LAYER: AMU = 3.000 DYNE*CM**-2

ALAM = 3.000 HALFSPACE:

AMUl = 3.300 ALAMl = 3.300

FINITE RECTANGULAR FAULT

DENSITIES. LAYER: RHOL = 3.300 G*CM**-3

HALFSPACE: RHOH = 3.500

DX: ELASTIC PART= -.38062085 ADDITIONAL PART=O. 16665 127E-02 DY: ELASTIC PART= -.38900130 ADDITIONAL PART=O.l0493622E-02

I’= 1 .OOOOOOO POSTSEISMIC UX=-.30855389 UY=-.28844400 T= 2.0000000 POSTSEISMIC UX=-.67880228 UY=-.65239834 I-= 3.0000000 POSTSEISMIC UX=- 1.0306748 UY=-.99862922 T= 4.0000000 POSTSEISMIC UX=- 1.3294405 UY=- 1.29 16952 T= 5.0000000 POSTSEISMIC UX=-1.5699502 UY=-1.5267 120 l-= 6.0000000 POSTSEISMIC UX=-1.7580790 UY=-1.7096948 I?= 7.0000000 POSTSEISMIC UX=-1.9024715 UY=-1.8493215 T= 8.0000000 POSTSEISMIC UX=-2.0115567 UY=-1.9540097 T= 9.0000000 POSTSEISMIC UX=-2.0926566 UY=-2.03 10556

756 T.-T. Yu, J. B. Rundle, and J. Ferngndez

Table 4. Major subroutines used and their tasks in program STRGRH

PROGRAM STRGRV: Reads data and computes viscoelastic-gravitational vertical displacement from a finite dimensional strike-slip fault ( Rundle, 198lb, 1982; Yu and Rundle, 1994).

SUBROUTINE PRONY: Computes the expansion coefficients for the inverse Laplace transform using the prony series technique (Rundle, 1981b, 1982).

SUBROUTINE STRVSLP: Evaluates the exact displacements for an uniform halfspace with Poisson ratio of 0.25 ( Uses’Okada’s equation, 1985).

SUBROUTINE STRVTHFLTT: Integrates the Green function for a strike-slip point source over a rectangular fault surface ( Yu and Rundle. 1994).

SUBROUTINE STRVCALKRN: Computes the integration kernel differences for xl lR, x12i. xllR is the real part of x11; xl2i is the imaginary part of xl2 ( Rundle, 1978; Yu and Rundle. 1994).

SUBROUTINE EMAT: Computes [El matrix given by equation (36) of (Rundle, 198 1 b).

SUBROUTINE AMAT: Computes layer matrix given by equation (32) of (Rundle, 198lb).

SUBROUTINE STRVFM: Evaluates [Fm] matrices for the strike-slip point source (Rundle, 1981b. 1982; Yu and Rundle, 1994).

SUBROUTINE ZGOMPC: Calculates the [Z,(z)] matrix for k less than the gravitational wave number k, ( Rundle, 198 1 b).

SUBROUTINE ZOINVC: Computes the [Z,,(z)-‘] matrix for k less than the gravitational wave number k, ( Rundle. 198 1 b. 1982)

SUBROUTINE ZCOMPR: Calculates the [Z,(z)] matrix for k greater than the gravitational wave number k, ( Rundle. 198lb).

SUBROUTINE ZOINVR: Computes the [Z,(z)-‘] matrix for k greater than the gravitational wave number k, ( Rundle, 198 1 b. 1982)

SUBROUTINE STRVGFINT: Performs the integration of the Green’s function (Rundle, 1981 b. 1982; Yu and Rundle, 1994)

SUBROUTINE BJCALC: Calculates the Bessel functions needed by STRVAKNINT subroutine ( Rundle, 1978)

SUBROUTINE STRVAKNINT: Integrates the kernel function to obtain the Green’s function (Rundle. 1981b. 1982: Yu and Rundle. 1994).

Deformation produced by a rectangular dipping fault-11 751

Table 5. Major subroutines used and their tasks in program STRGRV

PROGRAM STRGRH: Reads data and computes viscoelastic-gravitational horizontal displacement from a finite dimensional strike-slip fault ( Rundle, 198 lb, 1982; Yu and Rundle, 1994).

SUBROUTINE PRONY: Computes the expansion coefficients for the inverse Laplace transform using the prony series technique (Rundle, 1981b, 1982).

SUBROUTINE STRSLP: Evaluates the exact horizontal displacements for an uniform halfspace with Poisson ratio of 0.25 (Uses Okada’s equation, 1985).

SUBROUTINE STRTHFLTT: Integrates the Green function for a strike-slip point source over a rectangular fault surface ( Yu and Rundle, 1994).

SUBROUTINE STRCALKRN: Computes the integration kernel differences for y 11 R, y 12i, zll and 212. yl1R is the real part of yll; y12i is the imaginary part of y12 ( Rundle, 1978; Yu and Rundle, 1994).

SUBROUTINE EMAT: Computes [E] matrix given by equation (36) of (Rundle, 198 1 b).

SUBROUTINE AMAT: Computes layer matrix given by equation (32) of (Rundle, 1981b).

SUBROUTINE STRFM: Evaluates [F,] matrices for the strike-slip point source (Rundle, 1981b, 1982; Yu and Rundle, 1994).

SUBROUTINE ZCOMPC: Calculates the [Z,,(z)] matrix for k less than the gravitational wave number kr ( Rundle, 198 1 b).

SUBROUTINE ZOINVC: Computes the [Z,(z)-]] matrix for k less than the gravitational wave number k, ( Rundle, 198 1 b. 1982)

SUBROUTINE ZCOMPR: Calculates the [Z,(z)] matrix for k greater than the gravitational wave number k, ( Rundle, 198 1 b).

SUBROUTINE ZOINVR: Computes the [Z,(z)-‘] matrix for k greater than the gravitational wave number k, ( Rundle, 198 1 b. 1982)

SUBROUTINE STRGFINT: Performs the integration of the Green’s function (Rundle.198 1 b 1982; Yu and Rundle, 1994)

SUBROUTINE B JCALC: Calculates the Bessel functions needed by STRAKNINT subroutine ( Rundle, 1978)

SUBROUTINE STRAKNINT: Integrates the kernel function to obtain the Green’s function (Rundle. 198 1 b. 1982; Yu and Rundle, 1994).

758 T.-T. Yu, J. B. Rundle, and J. FemPndez

r 1 4

t STFtHZL (Halfspace)

STRTHFLTT ht. along Dip

PRONY

II

DGECO -

I

DGESC 4

STFiFM

, STRCALKRN

* (Cal. Kernel)

I -IL

AMAT (Layer Matrix)

cz ZCOMPR [Zn(z)] Matrix

I EMAT 1 IFMatrir\ I L

l-l (EKLIJ Matrix) I

b End

Figure 2. Flowchart of major subroutines for program STRGRH. There are no pole calculations for this code.

Deformation produced by a rectangular dipping fault-11 159

(X=200 km, Dip= 90”) 6 ““I”“I”“I”“I”“I””

-. f \ / \

4- Gravitational I \ \

I \ . . . . . . . \

I : --.._ \ .

\ -4- \ /

\ \ I --- 47. -

. / _./

-6 I I I I I I I I I I I I I I I I I I I I I I Iti I I I I I

-15 -10 -5 0 5 10 15 y/H (II= Thickness of Elastic Layer)

. . . . . . . 17.

-2 II.1 I III, I III, I ,,,I I,.a.r;_74?=, -15 -10 -5 0 5 10 15

Y/H

Figure 3. Surface displacements due to vertical strike-slip fault. Solid curved line is coseismic displacement and dashed curve lines represent the postseismic displacements. Horizontal dashed line is boundary of elastic layer and viscoelastic half-space is 30 km. Vertical short line represents fault line, which is vertical.

Thick horizontal line is ground surface.

760 T.-T. Yu, J. B. Rundle, and J. FernPndez

X=200 km 6 ““I”“I”“I”“I~“‘J””

4- Gravity effect / >I. .

/ . / .

. I __----__ -. _.

z- :

/ .’ _... . . ._

Z I:.’

. . . ..__ __ I..._

4.’ .=...._..,__

,” 1’ !Z

5

0 t 0 ‘Ij, I tr

S ii _*---._._

..-_._____ :i

-2- -=___ __._..

. ..__ .“I

-_---.__-__.-_ ,._._.__1: _._,._;I&_. _.-_____-.__-_-___-_______-__._- 30 . . . /

. / . .._ ___. IT, _

-4- . / .-

--- 47. -

6 ““I”“I”“I”“I”“I””

/-\ / -. / \ / \

4- , \ _’ \ / \

/ \ / \

Z / \

/ _..._. _I__--.. \ . . rl /

_,m- .._. ,,_- : . . . . . . *, \

L 2- / ,’ ‘.

.*. ‘. \ / ‘. \ . .

\ , .I. ‘. \

sh / : ‘. \

/ ,’ . . \ , ,_ .

a , : .L_ - / : _. .

;-- . . . ,.m* .\ u

_.I’ -.._ __.’ .-._. 4

0. 0 S

?? 9

-_-~-._.__-.____-____~~.__-_~.__-.___.-.__-~-.__~~~.__-_~_-.-~~~-- 30 - .---. IT.

Figure 4. Difference of postseismic displacements, with and without gravity effect, due to vertical strike-slip fault profile at 100 km off fault tip.

Deformation produced by a rectangular dipping fault-11 761

(X=200 km, Dip= 30”) 10 m m r ’ 1 ’ 7 8 1 I’ I ’ 8 I’ ” n II z - ’ II ’ @ 8

Gravitational

--- 4T.

-10 I9 I I I a I I I I I I I I I I II I I,, I I,, ,

-15 -10 -5 0 5 10 15 y/H (H= Thickness of Elastic Layer)

-_

5-

, / , “Is. _

- ,’ --- 47.

4 -101 I I I I I I I I I I I I I I I 111,,,,,,,,,,, I

-15 -10 -5 0 5 10 15 Y/H

Figure 5. Same as Figure 3, except dip angle is changed from vertical to 30”.

762 T.-T. Yu, J. B. Rundle, and J. Femkdez

(X=200 km, Dip= 45”) 10 T m ‘I 1 ’ s ” 1 c “I 1 ,‘I m 1 “I ” r ’ m s

Gravitational

. . . . . . . IT. _

--- 47.

-10 I L I I I I I I I I I I P I I I I I I I I I I I I I I I I _ -15 -10 -5 0 5 10 15

y/H (II= Thickness of Elastic Layer)

g ;:

c 3

_ . . . . . . , , _ .*

. . 17. -

--- 47.

-10. 0. I I I I I. I I I I * I I * I I I I. a I I I I Ia t -15 -10 -5 0 5 10 15

Y/H

Figure 6. Same as Figure 3, except dip angle is changed from vertical to 45”.

Deformation produced by a rectangular dipping fault-11 763

(coseismic at X= 100 km, Dip= 90”)

-15 -10 -5 0 5 10 15 y/H (I-I= Thickness of Elastic Layer)

Figure 7. Coseismic and postseismic surface vertical displacements due to vertical strike-slip fault with profile located at edge of fault.

764 T.-T. Yu, J. B. Rundle, and J. Fernandez

and increases as time elapses. In this profile, 10/3 times H away from the fault tip, the gravity effect is

as much as 60% in both components. Figure 5 shows

the surface displacements as a result of a 30" dipping

fault, with the patterns of the displacement field

changed dramatically compared to the situation of a

vertical fault (Fig. 3). The most distinct feature in the dipping fault solution is the long-wavelength postseis- mic displacement in both components, which cannot be modeled by the purely elastic models. The pattern of the postseismic displacement field is no longer similar to the coseismic field as in the vertical fault solution. The dip angle is changed from 30” to 45” in Figure 6 and the displacement pattern is similar to the previous plot in both coseismic and postseismic stages. The vertical coseismic and postseismic displacements due to a vertical strike-slip fault are shown in Figure 7. Here, the major component of the viscoelastic displacements for the strike-slip fault is parallel and perpendicular to the strike of the fault. It is about SO-100 times larger than the magnitude of the vertical displacement (depending on the dip angle). In vertical displacement, the magnitude of postseismic displacement is about only 1% of coseis- mic, unlike the U., and U,L, where the postseismic displacement is equal to or larger than than coseis- mic. The dip angle, depth of fault and rupture length of the fault will determine the magnitude. The pattern of displacements is sensitive to the depth of the fault and distance to the fault, for both coseismic and postseismic states. Therefore, the location of the profile and depth of the fault will alter the magnitude and pattern of the displacement field significantly. Source codes for the programs are available by anonymous FTP from the server IAMG.ORG. Pro- grams can also be purchased on diskette (See Com- puters & Geosciences v. 20, no. 6 for instructions). These codes have been tested on a Sun Spare work- station, an IBM-R6000 and PC platforms, and per- form normally on these systems. However, the NAMELIST input format is different in SUN and IBM environments, please refer to the included READE.ME file for detailed instructions.

Acknowledgments-The research of J. B. Rundle and T.-T. Yu was supported by NASA grant NAG52353 to the University of Colorado. The research of J. Fernlndez was suouorted with funds from oroiect EVSV-CT92-0173 of the European Union in the frame of the Environment Program, and by the Consejeria de Education y Cultura of the Comunidad de Madrid, which gives funds for his stay in

CIRES, University of Colorado at Boulder. We thank two anonymous reviewers who gave us valuable comments for improving this manuscript.

REFERENCES

Ben-Menahem, A., and Singh, S. J., 1968, Multipolar elastic fields in a layered half-space: Bull. Seismol. Sot. Amer., v. 58, no. 5, p. 1519-1572.

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