ABSTRACT From Haskell&apos;s (1969) integral representations for the near-field displacements due to a propagating strike-slip and dip-slip dislocation, a solution is obtained for a dislocation &quot;line source&quot; by an analytic integration in the direction of the fault propagation. This reduces the numerical integration from a surface integral required for the usual evaluation of the near-field motion, to a one-dimensional integration over the fault width. Since the dislocation function modeled here is a Heaviside step function, these results may be extended to any arbitrary source time-function by convolving these displacements with the time derivative of the desired source function

David M Boore

John Boatwright

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Bulletin of the Seismological Society of America. Vo|. 65, No. 1, pp. 133-138. February 1975 
A SIMPLIFICATION IN THE CALCULATION OF MOTIONS NEAR A 
PROPAGATING DISLOCATION 
BY JOHN BOATWRIGHT AND DAVID M.  BOORE 
ABSTRACT 
From Haskell's (1969) integral representations for the near-field displacements 
due to a propagating strike-slip and dip-slip dislocation, a solution is obtained for a 
dislocation "line source" by an analytic integration in the direction of the fault 
propagation. This reduces the numerical integration from a surface integral 
required for the usual evaluation of the near-field motion, to a one-dimensional 
integration over the fault width. Since the dislocation function modeled here is a 
Heaviside step function, these results may be extended to any arbitrary source 
time-function by convolving these displacements with the time derivative of the 
desired source function. 
INTRODUCTION 
The ability to synthesize seismic motions produced by propagating faults of finite extent 
is of basic importance in the rapidly developing field of near-field seismology. One of the 
most common methods in current use is based on Haskell (1969). In this method a 
Green's function is integrated numerically over the fault surface. In principle, this method 
can simulate arbitrarily complex fault propagation, with the source time-function and 
rupture velocity given as spatial functions over a fault surface of any shape. In practice, 
however, most simulations (e.g., Anderson, 1974; Trifunac, 1974; Tsai and Patton, 1972; 
Battis and Turnbull, 1974) simplify the description of the fault considerably and treat a 
rectangular fault which ruptures instantaneously over the fault width and propagates at a 
uniform speed until the fault terminates. A more complex model of faulting can be 
obtained by summing the contributions from a number of these simple rectangular faults. 
It is possible to evaluate the motion from a simple rectangular fault model by first 
breaking the rectangle into a number of strips of small width. An analytic expression is 
given in this paper for the contribution due to the propagation along each strip. Thus the 
usual two-dimensional numerical integration is reduced to a simple sum over the contri- 
butions for each strip. Although the evaluation of the motion for a propagating strip was 
first solved by Knopoff and Gilbert (1959), an error was pointed out by Savage (1965) 
which casts doubt on the expressions contained in Knopoff and Gilbert (1959). Savage 
rederived the expressions and published wave forms based on the resulting equations 
(Savage, 1965), but did not publish the expressions themselves. Unfortunately, his 
derivations have since been lost (Savage, personal communication, 1973). With his 
encouragement we present here the solutions with the hope that other seismologists will 
find them useful in simulating near-field wave motions. 
THE MODEL 
We have assumed for a dislocation source function a propagating Heaviside step 
function. Thus the discontinuity across the fault surface (Auk) is given by 
Au k = D k H ( t -  ~ I / v ) [ H ( ¢ I ) - H ( ~ I  -L) ]  (1) 
133 
134 JOHN BOATWRIGHT AND DAVID M. BOORE 
where v is the rupture velocity and L is the fatdt length. D1, D 2  correspond to dislocation 
slip in the positive x l ,  x2 directions, respectively, on the positive x3 side of the fault. 
(See Figure 1 for the geometry.) 
The integration of Haskell's operator M~i,q across the fault "strip", [see Haskell, 1969, 
equations (2), (3.1-4.3)], was simplified through a transformation of variables from 41, 
the first component of the fault coordinates, to r j ,  the first component of the vector 
describing the relative position of the observation point to a point on the fault surface. 
The resulting displacements were artificially decoupled into P and S modes, i.e., 
ld i ~ lli p -~- Ui s 
where u~ is the ith component of displacement at the observation point. These 
"decoupled" wave motions are valid by themselves only for the study of first motions, 
since coupling terms (which represent a spectrum of velocities from ~ to fl), are included 
in uPi and u~v The sum of the P and S modes, however, correctly describes the total 
motion for all times. 
The integrated solutions (as functions of the variable of integration, 0 are listed for 
x 3 
P 
T 1' x 1 
l i 
x 2 
FIG. 1. The  fault of  width W i s  broken into a n u m b e r  of  strips of  width d~2, located at ~2. The  mot ion  on 
each strip starts  at x~ = 0 and  ends at x l  = L .  P ( x j ,  x 2 ,  xa)  is the  observat ion point. 
both strike-slip and dip-slip dislocations, (D1 and D2) and are divided in a natural way 
into three sets of terms; the far-field terms (l~k), the coupling terms (H~k), and the terms 
which give rise to finite deformations (F~k). 
These results have been thoroughly checked, in unconvolved form against Savage's 
(1965) results, and in a convolved form, with a numerical integration over the width, 
against some results supplied by John Anderson (written communication, 1974). The 
reader should be warned that some of the figures in Haskell's (1969) paper contain 
erroneous wave forms and cannot be relied upon to provide a check of simulations (e.g., 
see Boore and Zoback, 1974, Figure 6). 
DEFINITIONS 
r2 = x 2 -  42 (2a) 
r3 = x3 (2b) 
a z = r2 z d-r32 (2c) 
THE CALCULATION OF MOTIONS NEAR A PROPAGATING DISLOCATION 135 
c2(xl  - vt) + v(a2(c 2 - v 2) 3- c 2 ( x  1 - vt) 2) 1/2 
r .  = (c2_v2)  (3) 
(X12 3 - a 2 )  1/2 [(Xl--L)23-12] 1/2 L 
to,, - (4a) tLm = + - (4b) 
C C V 
sgn(m) = 3-1 for  rn = p 
- 1 for  m -- s. (5) 
In  equations (3) to (5) above,  m designates the wave mode,  either P or S, whose correspond-  
ing velocity is c, either ~ or ft. r,, is a function of  t ime and corresponds to the root  r 1 of  
the equation,  
t - x l  - r l  +(r12+a2)  1/2 
V C 
This equat ion gives the t ime required for  rupture  along the fault  f rom 41 -- 0 to ~ = 
x1 - r l ,  plus the time to travel to the observat ion point  as either a P- or a S-wave. Thus 
when the equat ion is inverted, the physical meaning of  r,. is that  at a given time t, x 1 - r . ,  
represents the distance along the rupture f ront  f rom which a radiated wave of  mode  m 
will arrive at the observat ion point  at the given time t. tom and tLm correspond to the t ime 
of  the arrival o f  the m wave f rom the ends of  the strip at  x l  -- 0 and xa = L, respectively. 
Hik(Y 1, Y2) = Hik(Y 1 )  - -  Hik(Y2) (6) 
F/'~(Yl, Y2) = F / '~(Yl) -F~(y2)  (7) 
where Yl and Y2 are d u m m y  variables, corresponding to ( in the listing of  the functions 
in equations (12) to (17). 
SOLUTION 
To evaluate the total  wave motion,  we simply sum the wave motions for  each mode,  
ui = uiP+u~ s (8) 
where the displacements due to each mode  are given by (for the ranges listed): 
Fo r  the range t < to., (or x l  < r,.), 
ui m = 0 (9a) 
Fo r  the range tom < t < tr..~ (or Xl - L  < r m < x l ) ,  
ui" - 4re Dk {I~k(rm)+sgn(m)Hik(Xl, rm)+ F]~(xl, r,~)} (9b) 
k = l  
For  the range fi.m < t (or r~ < x 1 - L ) ,  
uim - 4re Dk {sgn(m)Hik(Xl, x l  - L ) + F ~ k ( X l ,  x l  - L ) } .  (%) 
k = l  
136 JOHN BOATWRIGHT AND DAVID M. BOORE 
Using the dummy variable ~ and the ~ dependent definitions, 
r 2 = ( 2 + a 2  
ve  
O(c) - 
e -v~/r  
(1o) 
(11) 
The functions I/~(ff), Frk(0 and Hik(~ ) are given below. 
The I~(~) terms for 
Strike-slip 
P mode 
2(2r3 
1/',(0 = ~ o(~) 
2~rzr3 
1~(~) = ~ o(~) 
2~r~ 
/~1(~) ~--- ~ 0(~) 
Dip-s@ 
2~r2r3 ,~(~) = -~ -r~o(~)  
2r~r3 
I~2(~) - ~3r4 0(~) 
2r2r~ 
I~(O - ~ r ,  0(~) (12) 
S mode 
11,(¢) = ~ 1 -  O(fl) 
- 2 ( r z r  3 
I~1(~)-  fi3r4 O(fl) 
1~,(() = f13~ (1 2r~ -7)o@ 
--2(rzr 3 
112(¢) -  ~3r4 O(fl) 
r 3 // 2r22"x 
I~z(~) = p-r \ l l - T T ]  O(fi) 
r 2 / 2 r 2 \  
I~2(~) = ~ 2  ~1- 7 )  O(fl) (13) 
The F~(~) terms, 
Strike-slip 
P mode 
~r3 
F~I({ ) - 1--;fi 
~Za2 r 
_ r2r3  
1 
F~'I(0 = ~2--? \r2 
Dip-slip 
r2r  3 
F~2(~ ) = o~2r 3 
r3 F~2(() _ ~Z azr~r2 1 + ~2 ~ -  3 (14) 
THE CALCULATION OF MOTIONS NEAR A PROPAGATING DISLOCATION 137 
S mode Strike-slip Dip-slip 
~3r 3 r2r 3 
F~,(() - fl2aZr3 F~z(( ) - fl2ra 
( F~i(~ ) = -tiEr3 F~2(( ) = ~ 3 -  
J (r2r2 / (2) 
Finally, the Hik terms are given below. (The notation should not be confused with the 
Heaviside step notation of equation 13 
Strike-s@ 
H11(0 = r3 ~ r 3 t r2 1 "4 vr 3 - \ - ~  - 4 
3( t -x i /v )2( (  ~4 2 ~ a  4 r -4 r2 +1)} 
H21(~) = r2r3{vl-~ra(3a2--5)+2(t~a~/v)~3tr 2 -- 3 ( 5 _ ~  2) 
3 
r 5 (t-xl/v) 2} 
r 2 [ ] [ ' 3 a  2 \ 1 / a 23 2(t-Xl/V)( 3 
H31(() 
+va ---42r (t--X1/V) ~3 t(5 -- 3(2"~7 ) -] (t--x1/V)2r3 ( 1 -  3r~r 2// (16a) 
Dip-slip 
H12(() = H21(() 
f r2(3 (5 3(2"~_ (3 +2( t_x i / v ) (13r2  ~ 
H/2(~) = r3)v2~X~rZ \ r2 ) vZa2r3 vr ~ \ r2 ] 
a 6 (t--XllV)2 ~r 15-- ~ + r4 )-I a a -r 
~ r~(3 (5 -3(2"~- ~3 b 2 ( t - ~  (1-3r~'~ 
r21v2aar~ t r 2 )  v2a2r3 vr Iv) r 2 )  H32(0 
+ ' 3 (t_Xl/V)2 15-- q" 
a 6 -~- + 74~ a *  r 
× (~z-3)}.  (16b) 
138 JOHN BOATWRIGHT AND DAVID M. BOORE 
The computation of a seismogram for a strip at ~2, at particular time (t) and observa- 
tion point (x 10 x2, x3), begins with the evaluation of of G and rp from equation (3). These 
values are then used as arguments in equations (9)i where the individual terms are defined 
in equations (12), (14), and (16) (for the P mode) and in equations (13), (15), and (16) 
(for the S mode). The contributions for each mode are then summed according to equa- 
tion (8). The result has dimensions of  (units of dislocation)/(units of fault width). After 
the time series for each strip is evaluated, the total response is obtained by summing the 
contributions of each strip. This summation corresponds to a numerical integration and 
as such can be done with varying complexity. The final step is a convolution with the 
time derivative of the desired source time-function. 
The step-function response contains step discontinuities at the arrival of discrete 
phases from the beginning and end of the fault, and in some cases it may be desirable to 
evaluate the step-function response for each strip at unequal time steps such that the 
discontinuities fall on the discrete time points. In this way the discontinuities can be 
represented accurately with relatively large time spacing. Since each strip will contain 
discontinuities at different times, it may be most convenient to smooth, by convolution, 
the time series corresponding to each strip and interpolate to a common time base before 
summing over all the strips. This degree of sophistication is usually not necessary unless 
several numerical derivatives of the motion are taken (as in constructing an accelero- 
gram), in which case the high-frequency components are accentuated. When working 
with displacement simulations, the number of strips required to adequately represent the 
motion produced by the finite width of  the fault can be very small (1 to 3). 
ACKNOWLEDGMENT 
This research was supported by the Earth Sciences Section, National Science Foundation, NSF Grant 
GA 36135XI. 
REFERENCES 
Anderson, J. (1974). A dislocation model for the Parkfield earthquake, Bull. Seism. Soc. Am. 64, 671-686. 
Battis, J. C. and L. S. Tumbull, Jr. (1974). Analysis of two California earthquakes using a moving 
dislocation source, EOS (abstract), 55, 352. 
Boore, D. and M. D. Zoback (1974). Near-field motions from kinematic models of propagating faults, 
Bull. Seism. Soc. Am. 64, 321-342. 
Haskell, N. A. (1969). Elastic displacements in the near-field of a propagating fault, Bull. Seism. Soc. 
Am. 59, 865-908. 
Knopoff, L. and F. Gilbert (1959). Radiation from a strike-slip fault, Bull. Seism. Soc. Am. 49, 163-178. 
Savage, J. C. (1965). The effect of rupture velocity on seismic first motions, Bull. Seism. Soc. Am. 55, 
263-275. 
Trifunac, M. D. (1974), A three-dimensional dislocation model for the San Fernando, California, 
earthquake of February 9, 1971, Bull. Seism. Soc. Am. 64, 149-172. 
Tsai, Y. B. and H. J. Patton (1972). Near field small earthquakes--dislocation motion, Semi-Annual 
Report 1, Texas Instr. Inc. (prep. for Air Force Off. Sci. Res., Contract F44620-72-C-0073). 
DEPARTMENT OF GEOPHYSICS 
STANFORD UNIVERSITY 
STANFORD, CALIFORNIA 94305 
Manuscript received June 3, 1974 


A SIMPLIFICATION IN THE CALCULATION OF MOTIONS NEAR A PROPAGATING DISLOCATION

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A SIMPLIFICATION IN THE CALCULATION OF MOTIONS NEAR A PROPAGATING DISLOCATION

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