Vol. 5 No. 4 7 7 9 - - 7 8 7 A C T A S E I S M O L O G I C A SINICA Nov. , 1992

Time interval strike- slip fault* Shande Pan and J in Ma

Institute of Geology , State Seismological Bureau, Bei jing 100029 , China

of earthquake recurrence for

Abstract

On the basis of faul t ' s dynamic model of Knopoff et al. ( 1 9 7 3 ) , this paper has finally obtained a simple approximate formula to be able

to es t imate the recurrence t ime interval TR of ear thquake on strike-slip fault . Pre l iminary result holds that # and 6~--6f have not much

effect on Tn. Let a is the ratio of the coseismic displacement D, to the total displacement Dt in whole event course , i. e. , a~D,/DL, then

a = 1 / 3 m a y represent the s tandard theoretical state in which Tn is independent on # and 6~--6s. At this t ime , Tn is the ar i thmetic aver -

age of so/v and kd//~, where so is the l o n ~ t e r m preseismic accumulated sl ippage, t, is fault ' s ave rage displacement ra te , d is the f racture

length on the fault of seismic focal region and fl is shear wave velocity. In addit ion, k=vo/av , where v0 is the initial f racture velocity

of actual s t ructure at the coseismic instant .

Key w o r d s : earthquake recurrence, s tr ike slip fau l t , recurrence t ime interval .

Introduction

No mat ter whether the great ear thquakes had period or no t , but historical informat ion indicated

that the ear thquake had even recurred in some regions. Never theless , this recurrence of ear thquake has

not an exact period, and it is not sure that it recurred in the original place. In other words the actual

recurrence t ime interval of ear thquakes may deviate much from the mean in te rva l ; and even if they

occurred on the same fau l t , the loci often depart f rom each other. In addi t ion, the recurrence t ime in-

terval of var ian t regions could be d i f fe ren t , longer or shorter. For example , as for the same magni tude

of ear thquake as 7 or 8 , the shorter (As ear thquake of magni tude about 7 in W u q i a , Aksu , Xin j iang ,

China) is only 1 0 - - l l a , and the longer (As ear thquake of magni tude about 8 in Er ta i , Keketuohai ,

Xin j i ang , China) m a y reach 3000a. It is l ikely that they neither belong to fault with the same nature

nor to the same tectonic-physical factors. But ear thquake recurrence remains yet to be explained with

some consistent reason. That is , on one hand it is controlled by fault slip ve loc i ty , on the other hand

it is restrained by energy release fashion on fault . Thus , it is still possible to study it in simplest case.

The researches of seismogeology show that there are a good m a n y great intraplate earthquakes oc-

curred on strike-slip faults and even recurred along a part of this fault . According to Wal lace ' s fo rmu-

la (Ma and X u , 1989; Wa l l ace , 1 9 7 0 ) , the recurrence t ime interval TR as a result of the slip on faul t

m a y be estimated by

TR = Dt/v or Tn = D~/av (1)

where D, = D. q-D. and a=-D./Dt. Herein D~ and D~ are coseismic and non-coseismic displacement re-

spect ive ly , and v is the long- term average displacement rate on fault at ear thquake source. There fo re ,

we can get when the next ear thquake will occur so long as we know TR. Even if the ear thquake deviat-

* The Chinese version of this paper appeared in the Chinese edition of Acta Seismologwa SinWa, 14, 1 8 7 - - 1 9 4 , 1992.

780 ACTA SEISMOLOGICA SINICA Vol. 5

ed f rom the or ig inal place of the last even t or had not the exact r ecur rence per iod , ( 1 ) could be used

too.

Expe r imen t s indicate tha t D, is no t a cons tan t . Neve r the l e s s , if we take accoun t of the in t rap la te

tec tonic ea r thquakes wi th the same type o n l y , then we can assume tha t bo th v and D~ all are cons tan t .

F o r , as k n o w n as we know

D~ : Mo/#A ( 2 )

whe re # , A and M0 are shear m o d u l u s , f au l t a rea and seismic m o m e n t on fau l t a t e a r t h q u a k e source.

So long as # and A do not change too m u c h as the e a r t h q u a k e of the same magn i tude r e c u r r e d , we can

consider t ha t D, does not va ry . In add i t ion , v is also considered cons tan t in a re la t ive ly not too long

t ime in te rva l . T h u s , to ca lcu la te Ts fo rma l ly t r a n s f o r m s to ca lcula te a and D~ (or Dr). This paper in-

tends to get r e l e v a n t i n f o r m a t i o n , and also obta in the possible r ecur rence t ime in te rva l of ea r thquakes

f r o m above me thod .

Theoretical analysis of earthquake recurrence m o d e l

In the l ight of the way of Knopof f et al. ( 1 9 7 3 ) , we coun t the slip fau l t as a separa ted par t ic le

sys tem. Let part ic le mass be M , the dis tance be tween part icles be Az, and there a re the l inear spr ings

wi th coeff ic ient of e last ic i ty of spr ing cons tan t k a m o n g each pair of ad jacent part icles.

Let the lef test (i . e. the f i r s t ) part ic le l ink wi th rigid wal l by the spr ing. W h o l e sys tem of in f in i te

part icles s tands on a rigid body (F igu re 1 ) . A t the b e g i n n i n g , it stops on this body because of stat ic

f r ic t ion. It moves w h e n the part icle sus ta ins a force of in tens i ty to exceed f r ac tu re s t r eng th at this

place. Hence the part ic le m o v e m e n t m a y be wr i t t en by

ML}-f- k (2U. - - U~_~ - - U.+~) = T. - - D. , U, > 0 ( 3 )

where T~ is n o w a d a y s " tec ton ic f o r c e " , D~"dynamic f r i c t i o n " , U~ the d i sp lacement of i - t h part ic le re la-

t ive to its ini t ia l posi t ion.

T~ Tz T~ T, T~+l

Figure 1 Separated particle system standing on the plane and acted by friction due to springs

and tectonic stresses.

Take Az---~0. let M = p • :~c and k=/z /Xx .

T h u s , we get

And let f ( x ) = lim (T , - -D , ) /Y . x , and z = l imn • Ax.

pu~(x,t) -- ~u~(z , t ) = y ( z ) ( 4 )

No. 4 Pan,S. D. et al. :TIME INTERVAL OF EARTHQUAKE RECURRENCE 78]

Generally, f ( x ) is called stress drop along the fault , but it is not seismic stress drop. In fact , it repre-

sents long-term effect of fault action. In short, shear slip on fault is realized the limit case of slip of

separated spring parts, and also it is represented by non-homogeneous wave equation which has char-

acteristic velocity fl= (#/p)~/z and slippage u (x, t) on fault.

The crack tip of fault in fracture process would sustain resistance by the applied body. Let F, be

~ t + v

the force. And let g ( ~ ( t ) ) = - - l i m 1 F~dr , so g(~( t ) ) is no other than the fracture strength of ~-o ._\t J '

this system at the crack tip. According to Knopoff et al. (1973) , it should have the fracture condition

(5 )

[u , ] q- ~ [ u , ] = 0 (6)

Where [ ] stands for jump value (for instance, [ u , ] = u + - u ; - , ~(t) is the distance from particle posi-

tion on crack to the origin, and ~ fracture velocity of particle.

Experimental studies under different surround pressure (Ma and Xu, 1989) show that the influ-

ence of pressure and rock type on displacement distribution is not large, and (4) explains that the ini-

tial displacement distribution of long-term effect on fauW s activity is decided by stress drop function.

Therefore, the earthquake recurrence on straight fault can be simulated by non-coseismic initial static

displacement distribution u. (x) undergoes a whole earthquake process and goes back to a same displace-

ment distribution uo (D.d-x) after a translation, where Dr is the displacement between the former and

the latter earthquakes. Hence, the earthquake process satisfies the partial differential equation

p u ~ ( z , t ) - - #u**(z, t ) = f ( z ) (4 )

and the initial condition

u [ , = o = u°(x),u[,:% = uo(D, q- x) ;gu (7)

where v is slip rate ( cm/s ) in model. The non-coseismic initial displacement distribution uo (z) to char-

acterize long-term effect is decided by f ( x ) , and as we know the fracture is initial at x = 0 ( t = 0) and

Dr (t----~TR). Therefore,

~ u a ( x ) ,u az ~ ÷ f ( z ) = 0

uo(O) = O, #-~Uo(X) = ~ ( 0 )

3 u , (D, ) = b, # j o ( x ) = ~ ( 0 ) ( ~ ( D , ) = ~ ( 0 ) )

.limu, (x) finite

(8 )

Here a , (0 ) is the static friction stress at x z 0, and uo(x) is estimated from the initial point of fracture

and so we take u o ( 0 ) = 0 (because it is the origin at x = 0 . But b#:0 because there is accumulation at

Dr after an earthquake process. In addition, there is a boundary condition #~gu/3x=o" s ( O ~ t ~ T R ) for

782 A C T A S E I S M O L O G I C A S1NICA Vol. 5

strike-slip faul t f rac tures , and ~r s can be taken as a constant approximately . According to two former

expressions of ( 8 ) , u~(x) can be reduced like this expression

aa,(x) u . ( x ) - - (1 - - e x p ( - - x / a ) ) , O ~ x ~ D . (9 )

#

where (r~(x) is , n ame ly , the static friction distribution along fault .

H o w e v e r , the curve of whole ear thquake recurrence process f rom exper iment may be shown by

Figure 2 ( M a and Xu 1989) . F rom ( ~ ) + 0<,<r~. , : r~= as/ /z + and ( d u ) _ 0<,<r~. ,=%= ' a ~ " c)x "

~ , and according to (7 ) we can take ( ) - '=o.rR=at ' , and so f rom (7 ) we get /Z x=O, 1)

o~ + t= o ( ~ - ) = a v - ~ ( a s / ~ + - ~ ( ~ ) / z - ) ( 1 0 )

z=0

where ~ and #+ are preseismic and postseismic shear modulus. And owing to ( 5 ) , we also have

~ ( ¢ ( t ) ) = ~ ( p + - p-)a~, + ,~s(1 - p+ ~ / ~ + ) - ,~ , (¢ ) ( l - p+ ~ /~ , ) (11)

If we let U ( x , t ) = u ( z , t ) - - u . ( x ) , then U ( x , t ) comes to be only relat ive to displacement distri-

bution as ear thquake occurs. That is , we can let D, ( z , t) = U ( z , t ) . Thus , the displacement distribu-

tion D, ( x , t ) which is only relat ive to ear thquake occurrence satisfies the fol lowing wave equation

p u . ( z , t ) - / ~ U . ( z , t ) = 0 (12)

and the initial condit ion

I : T R

Of course, there is still the corresponding boundary condition while the strike-slip f racture of

faul t occurs. In addi t ion, the initial distribution (9 ) should have the conditions of fol lowing form

o" ~(D.) - - a s ( 0 ) , b - - aa"(0--) (1 - - e x p ( - - Dr~a)) ( 1 4 ) a ~

And so, the whole process of the ear thquake recurrence on a single straight fault may be decom-

posed to two problems: one is decided by the non-coseismic displacement u ° ( x ) , i . e . the curve ( 9 )

under the condition ( 1 4 ) , and another is the displacement distribution D, ( z , t) expressed by (] 2) and

( 1 3 ) as ear thquake occurs. In fac t , since u° ( 0 ) = 0 and uo (D,) = aa, (0 ) ( 1 - - exp ( - - D~/a ) ) there is

the fol lowing relation

u ° ( D ~ ) = u,;(0) q- a a ~ ( 0 ) (1 - - e x p ( - - D . / a ) ) ( 1 5 ) #

In other words , u . ( 0 ) and u.(Dr) do not equal usually.

Above inferences are got under the condition of # = const. But /z m a y change at ear thquake

sources along the fault . H o w e v e r , in order to get Tn, it appears that its variat ion is not the most im-

portant. The re fo re , we do not consider it in this paper.

No. 4 Pan,S. D. et a/. :TIME INTERVAL OF EARTHQUAKE RECURRENCE 783

Coseismic displacement distribution Ds ( x , t )

Now, the coseismic displacement distribution is only determined by the wave equation (12) satis-

fying the initial condition (13) . Thus,

l ~ ~ p J 1 r~ . . . .

D,(x,t) = | 1 [ '+~ ' r (16) I ~ j . _ ~ , L a V - - ~ (o±/,~ + - o . . ( ¢ ) / u - ) ] d ¢ , , ~ fit

i .e . D~ (x , t ) is only determined by av on the fault of earthquake source and fracture velocity as earth-

quakes occur and a , ( ~ ) / g --~rff/~ +. However, /1+ and /1- could not be gained easily in the practical

fracture process, and the difference between ~ ( ¢ ) / / i and e f t # + may mainly be that between a , (~)

and ~r s only. This enables us tduse mean /~ to replace #+ and /z-. Thus, D.~(x,t) is determined by /~ at

the position of earthquake source and ~, ( ~ ) - a~ which is called effective stress.

(2)

(3)

I--~.---.I D(cm) D(cm)

(b)

t----D.--.4 D. ~

•

t (s) a t(s)

/ I / ' , / V ',

t(s) t(s)

Figure 2 Stress-displacement curve ( 1 ) , coseismic displacement-time curve (2 ) and stress

time curve ( 3 ) in whole process of earthquake recurrence activity ( ( a ) theoretical

model, (b) experiment).

The fracture velocity ~ which is lower than the shear wave velocity fl exists and is once observed.

Hence, on the basis of distributions of ~(q) and a , (~) with ~ D~(x, t) may be obtained by calcula-

tion of (16) . Since ~ (~rs (~) - - ~ i ) / # > 0, the coseismic displacement distribution is

784 ACTA SEISMOLOGICA S1NICA Vol. 5

-- , ~ (~ t - - x) + ~ e, f'~' + ' ('~,G) -- °V)//~]d~, O < ~ x < fit D , ( x , t ) = ! ] r ~+~, (17)

[ @ j . - . , ) / . ] de, • >

The second formula is apparently consistent with the experimental results (Figure 2b). By way of sim-

plification the first one is in agreement with the experimental result to some extent in the following cal-

culating example.

It is necessary to render D ~ ( x , t ) be worked out. For example, take p and a~--as are constants,

and let ~ be a parabolic distribution with ~ (Figure 3). It is similar to fault ' s healing on resistance af-

ter an earthquake occurs. At this t ime, calculating and rewriting ( 1 7 ) , we have

D~(x,t) d

{4,( 7 + ~s) _ _ + 4 . . . . . . [I -

~- • ( ( 1 -~- 4 1 ) 2 Jr- ( 1 -}- ~ 1 ) ( 1 - - >q) -~- ( l - - ) , 1 ) 2 ] }

(where 4, = fitt, and 0 ~ ;q < l ) X

& . 7 { ~ _ x as,+ 4" o'~ ~ o'~.~_o~O d [ 1 - 3 " 1 -dx [l

3-. T ( ( 1 + &)2 + (1 + &)( l - &) + (l - &)~)]}

[ ( w h e r e & = z / B t a n d 0 ~ 4 ~ < 1)

(18)

It should be pointed out that the former and the latter in the two above formulas have similar

forms. It suggests that they all are consistent with the experiment result, and predicts that there is

some distinction between them while we regard them as the approximate formula of displacement on

fault at earthquake source region and use them to practical earthquake.

&cm/s)

,,2 + - - , , 2 4 ¢(ma)

Figure 3 The variation of fracture velocity ~ with fracture position ,~ is a parabolic curve.

Values of D~ , Dt and TR

In a not too long geological term, v can not change too large. The former and the latter in formu-

las (18) represent the displacement distribution of ruptured and disrupted parts of fault after the earth-

No. 4 Pan ,S . D. et al. :TIME INTERVAL OF EARTHQUAKE RECURRENCE 785

quake occurs. They have a similar f o r m , and as both 2~ and )~2 approach to 1 and both x / d and f l t / d

approach to 1 / 2 , D ~ ( x , t ) / d all approach to the most large displacement at source region

D~ : ( a v - ~ - 2 a~ - - c~ s . "~o) d 3- " 27 (19)

Thus,

TR = D f f a v = ( J - F 2 a .¢-- a s • ~o ) d ( 2 0 ) 27

This is a calculat ing result on model. Hence , here ~0, fl and v all are in c m / s ; D~ and d are in

c m , and TR in s.

But the t ime and space scales of exper imenta l model and actual tectonics are ve ry d i f fe ren t , espe-

c ia l ly , the fault slip veloci ty is in order of m m / a and the coseismic displacement veloci ty in order of

c m / s , and so they could not be calculated on unified units. H o w e v e r , the real long- term geological

tectonic ac t iv i ty should be added to the short term coseismic act ivi ty. If both ~0 and fl are in k m / s , d

in kin , v directly in m m / a and D, in r am, then we must introduce an amplif icat ion factor K to the first

terms in the brackets on the right sides of ( 1 9 ) and ( 2 0 ) . The va lue K character izing the dif ference

between t ime and space scales m a y have a certain explanat ions. In fac t , the initial veloci ty v0 of actual

tectonics as ear thquake occurs is higher than av , but far lower than shear wave veloci ty ft. In the

practical process of ear thquake it is an indefini te quant i ty . Rea l ly , the dif ference of several former

terms in the large brackets of the first and second items of fo rmula ( 1 8 ) demonstrates that it is neces-

sary to introduce v0. Hence , let v o = a K v and R = ~0/r0. N o w , we call R the veloci ty ratio concisely ,

and K is the parameter of initial veloci ty.

The exper imenta l in terference of a has never been over 0 . 5 at present, a in exper iment increases

as the n u m b e r of times of s t ick-s l ip increases . The va lue a of the first few s t ick-s l ip events is on-

ly 0. 1 - - 0 . 15 , and then increases to 0. 2 5 - - 0 . 35 (Ma and X u , 1989) . The fol lowing study shows

that a = 1 / 3 has a special theoretical meaning. There fo re , it may be considered that the va lue of a

much approaches 1 / 3 .

The va lue of a f rom field informat ion is between 0. 3 and 0 . 9 (Sykes and Qui t tmeyer , 1981) .

Est imation of displacement veloci ty f rom earthquakes of magni tude greater than 6 and f rom distortion

of water systems on Xianshuihe fault in southwest area of China , the value of a are 0. 4 - - 0 . 5. It

means that the field va lue of a is higher than the exper imental .

In exper iment , the slip of fault is f rom the stable slip veloci ty v~ direct ly to av (See the second

fo rmula of ( 7 ) ) . But f rom above discussion, the slip of field faul t is f rom stable slip veloci ty z,~ di-

rect ly to a K v . Hence , the dif ference of the va lue a in the exper iment and field informat ion actual ly is

the va lue K. In other words in the light of above informat ion a K is between 0. 3 and 0. 9 ( for in-

s tance , a K is between 0 . 4 and 0. 5 on Xianshuihe f a u l t ) , and we can calculate that K is not less than

1 a / s at least and is not more than 9 a / s at most as a is between 0. 1 and 0 . 3 5 .

Thus D~ and TR of actual tectonic act ivi ty become

K d 2 as - - as ) ( 2 1 ) D~ -= a v ( 1 Jr- ~ - R • , 2 f l

and

- - K d 2 a~ as) - - ( 2 2 ) TR = D~/av = (1 ÷ T R • t, 2/~

786 ACTA SEISMOLOGICA SINICA Vol. 5

where K is in the unit a / s characterizing t ransformat ion of uni t , but according to its meaning it is rela-

t ive to the initial veloci ty as ear thquake occurs. In a word , it, a~ and a s in the formulas (21 ) and

( 2 2 ) may be taken in M P a , d in kin , fl in k m / s and K in a / s , and R is dimensionless. Numer ica l cal-

culation shows that R is a sensitive value. The re fo re , the precise est imation may strictly come from the

starting of calculat ing the values of D. an D,.

The non-coseismic displacement Do in the process of ear thquake recurrence m a y be taken as the

va lue of D~(x , t ) at z = f l t ~ - . d as a representa t ive , and Do(x , t ) may be obtained by it which is added

the initial displacement distribution of the former strong ear thquake after it calm down to an displace-

ment distribution of the latter strong ear thquake just b~. fore it occurs. According to ( 9 ) , the initial dis-

placement distribution of the form aa~ ( 1 - - e x p ( - - x / a ) ) / # in km, after we take the t ransformat ion of

i k m = 1 0 5 mm into account , may be taken as ala~ • 1 0 s / ~ , which is the initial displacement of the

position prone to rupture. The fault displacement of long- term earthquake generat ion process may be

simplified as t, Tn ~ - 6 , where 6 is the slippage while the original stable slip veloci ty r~ accelerates to the

initial veloci ty v0 as ear thquake occurs. The displacement just before ear thquake happens may be taken

)~2--~0 and x / d = 1 in the second formula of (] 8) . F ina l ly , considering v o = a K v , we have

d-=D° a'-2-~/t " -d-a1 . 105 _~_ --~(vTn -q- 6) 4- a v ( l - - ~-R-4 a,~ --/~ as) Kfl ( 2 3 )

But according to ( 2 1 ) , we have

D, a v ( l + 2 a, - - a f ) K ( 2 4 ) T = y R - /z 2fl

The re fo re , if we let So a~ al • 1 0 ~ ~ vTn-~-6 - - = - - • - - - - - , then d /~ d R

D~ D~ + D~ so 3 a~ K d - - ~ =-d +av(Y-R" --~r) (25) 7

In above fo rmulas , a~ and d is in km, and Do, D,, D, and So all in mm. Thus , sold is in m m / k m . Fi-

na l l y , on defini t ion a = D , / D t and to solve for K , we have

K : 2flso ( 2 6 )

[ ( 1 - - 3a ) -? 2 ( 3 + a ) R • a~ - - aS]vd /1 where So represents a slippage of long- term accumulat ion before an ear thquake occurs. As a = 1 / 3

K : 3 . fls___L . # ( 2 7 ) 2 Rvd (r~ - - a s

T h e r e f o r e , there is a theoretical meaning as a = I / 3 . At this t ime ( the strict condition is I 1 - - 3 a ] <<

2 ( 1 / 3 q - a ) R • a ' - - a s ) K is reversely proport ional to R. /z

Theoretical formula

Take a = 1 / 3 as the theoretical standard case. T h e n , f rom ( 2 7 ) we get

R = 3 . flSo z (28) 2 K d v a, - - a s

Substi tuting it into ( 2 2 ) , we obtain

No. 4 Pan,S . D. et al. :TIME INTERVAL OF EARTHQUAKE RECURRENCE 787

Tn = l__(s0 ~_d) 2 v @ (29)

That is, at this time TB is only dependent on so/v and Kd/•, and independent on (a~--as)//~. In spite

of a may deviate from 1 / 3 , numerical calculation indicates that (29) still holds approximately. It

means that Tn can be taken from (29) as a first approximation. If we take K = 5 a / s , then Tn is the

mean value of so/v and 5d/fi , namely. For example, let f l = 3 .5 km/s and for a 8 magnitude earth-

quake we probably have T n = 100 a and d : 50 k m, then so/v= 128.6 a. And, if fl does not change

and for a 8 magnitude earthquake let TR=3000 a and d = 150 kin, then we also get s 0 / v = 5 7 8 5 . 7 a.

Conclus ion and discussion

This paper, f inally, gets an approximate formula (29) of recurrence time interval TR estimating

strong earthquake of strike-slip type on the basis of Knopoff et al. (1973) . Preliminary result is that

it may be taken a = 1/3. At that t ime, TR is not only independent on shear modulus # but also inde-

pendent on effective stress a~- -a s. Strictly speaking, Tn should be dependent on ( a , - - a s)//z. In fact ,

the influence of # and a~ - - a s can be obtained by adopting other formulas in this paper.

a and K are two important parameters. The exact effect relative to other physical factors remains

to study further.

This paper is a part of contracted item of State Seismological Bureau Tectonic Physical Study

of Earthquake Recurrence Period and Characteristic Magnitude. Authors acknowledge for support of

Scientific-Technical Supervise and Survey Department, SSB.

References

Knopoff,L. , Mouton .J .O. and Burridge,R. , 1973. The dynamics of an one-dimensional fault in the presence of friction. C-eophys.

J . G . astr. Soc . , a s , 169--184.

Ma , J . and X u , X . Q. , 1989. Experimental study on earthquake recurrence time interval and the ratio of coseismic and nonseismic dis

placement. Earthquake, No. 1, 10-- 18 (in Chinese ).

Sykes,L. R. and Quittmeyer,R. C. , 1981. Repeat times of great earthquakes along simple plate boundaries. In: Ear/hquake Prediction,

D . W . Simpsons and P . G . Richards (eds) , 2 1 7 - - 2 4 7 .

Wallace,R. E. , 1970. Earthquake recurrence intervals on the San Andreas fault, California. Bull. GevL Soc. Am. , 81, 2875- -

2890.