Reprinted frm the NEW ZEALAND JOURNAL OF GEOLOGY AND GEOPHYSICS, Vl. 9, N.3 A STATISTICAL SURVEY OF EARTHQUAKES IN THE MAIN SEISMIC REGION OF NEW ZEALAND PART 2-TIME SERIES ANALYSES D. VERE-JONES* and R. B. DAVIES Applied Mathematics Divisin, Department f Scientific and Industrial Research, Wellingtn (Receit1ed fr publicatin 24 March 1965) ABSTRACT Time series analyses are carried ut n earthquake data frm the main seismic regin f New Zealand fr the years 1942-61. Origin times nly are cnsidered, the energies and exact psitins f the shcks being largely ignred. The relevant statis tical thery fr the first and secnd rder prperties f the prcess is described, and simple prbability mdels fr earthquake ccurrence are put frward. On the basis f these results, the. data are examined fr peridic and gruping effects. N significant peridic effects are fund, either amng the shallw shcks (depths up t 100 km) r amng the deep shcks (depths 100 km r greater). Bth cmpnents shw strng evidence f gruping, and several alternative mdels t describe this effect are put frward and cmpared. l-introduction Time series methds have been used fr a lng time in the analysis f earthquake frequencies, but many f the papers make little use f the statistical techniques which are nw available. Our aim in this paper is t apply sme f these techniques, especially spectral analysis, t earthquake recrds frm New Zealand. Any statistical analysis relies ultimately n a prbability mdel f the prcess under cnsideratin, and in the curse f the paper we shall develp sme simple mdels fr earthquake ccurrence. Like the chemical prcesses emplyed by the Curies, these methds can be used t extract a small quantity f active infrmatin frm a large bdy f inert data. We hpe that in the present cntext they may prve useful t the seismlgist. In particular, it is pssible that the study f suitable stchastic mdels may ultimately thrw sme further light n earthquake mechanism. ;'The recrds frm a small, self-cntained seismic regin (f which the lli-ln seismic regin f New Zealand appears t furnish a reasnable cxlunple) reveal time series f extremely cmplex structure. Large fiuctua 'Jins in the numbers f shcks per year, cmplicated sequences f related shcks, dependence f statistical parameters n depth and perhaps magnitude, and fluctuatins f activity n a larger time-scale, all appear t be haracteristic features f such recrds. An earlier paper (Vere-Jnes, *Nw with Department f Statistics, Australian Natinal University, Canberra. N.z. J. Gel. GeaphyJ. 9: 251-84
J 252 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 Turnvsky, and Eiby, 1964) attempted t ascertain the slw fluctuatins in activity {in the frm f. linear trends) frm data cmpiled by the New Zealand Seismlgical Ob$ervatry ver the perid.1942-61. In this paper we shall use the same data as the basis f a time-series study. In general, we shall ignr: the, lcatins and magnihldes f the shcks, except in s far as they determmed the fbur classes f shcks described in the earlier paper: (a) The ttal shcks. (b) The shallw shcks (depths less than 100 km).* (c) The deep shcks (depths greater than r equal t 100 km). (d) The mdified list f shallw shcks, btained after the large swarms and aftershck sequences have been remved. We refer the reade'r t the earlier paper fr a mre detailed accunt f the data, and a discussin f its limitatins. It is restricted t the main seismic regin f New Zealand, and t shcks which have an instrumental magnitude f 4'5 r greater. After remval f linear trends, the data frm each class were assumed t represent data frm a statinary time series. There is, f curse, n reasn t suppse that all the nn-statinarity is cntained in the linear trends, but we culd regard ur prcedure as the first step in a series f apprximatins. Thus we ca:n use the data after remval f linear trends t frm a rugh estimate f the cvariance structure; knwing this, we can then estimate a mre cmplex trend, and s n. The techniques we shall use are standard methds f time-series analysis, mdified slightly t take int accunt the discrete nature f earthquake ccurrence. Sme f the ideas may be new in the present cntext, s that we have discussed them in mre detail than wuld usually be necessary. Sectin 2 cntains backgrund material, while sectins 2.1-2.2 are cncerned with particular mdels fr earthquake ccurrence. The technique f spectral analysis is described in sectin 2.3, and the fllwing three sectins cntain sme applicatins. In sectin 2.7 we cmment briefly n energies. Sectins 3'.1-3.4 cntain evaluatins f the results fr each f the fur classes f earthquakes in turn. 2.0-STOCHASTIC POINT PROCESSES The stchastic prcesses relevant t a descriptin f earthquake ccurrence are knwn as "stchastic pint prcesses". In these prcesses events ccur singly, r perhaps in small grups, at (t a first apprximatin) instants f time. They are t be distinguished frm the stchastic prcesses which develp cntinuusly in time. The prbability structure f a pint prcess is described by the distributins f the number f events in particular intervals r cllectins f intervals. Fr practical purpses, the first arid secnd rder prperties f these distributins (the means, variances, and cvariances), are especially imprtant. These quantities can, in general, be written dwn in terms f tw basic functins, the instantaneus rate met), *We have included the 100 km shcks with the deep shcks instead f the shallw shcks as in the earlier paper.
7 N.3 VERE-JONES & DAVIES - EARTIlQUAKE ANALYSES 253 and cvariance.dsity C(u, t). The instantaneus rate at a given time t is efined as the limit f the mean rate f ccurrence f events in a small time mterval abut t, as the length f the interval tends t zer, m(l) = lim E[N(t - a) t +,8]/(a +,8) (1) a,/3... 0 (t1,.t2) her rresents the number f events (a randm variable) ccurnng ill the tllne mtervals (f 1, f 2 ). The prcess is assumed t be sufficiently regular fr the limit t exist and t be finite (and fr ur purpses bunded) in the whle f the time interval under cnsideratin. The cvariance density is defined as the limit f the cvariance f the numbers f events in tw small time intervals, divided by the lengths f the time intervals, as the tw lengths tend t zer, C(u, t) [N(t- a, t + {3), N(u -y, u + 8)] lim Cv ------------- a,/3,'y, 11-+0 (a+{3)(y+8) (2) Again it is assed that this limit exists and is finite fr all relevant pairs f values f u ana f.. The mean and variance f the numbers f events in any interval (t1' t2) are given by (3) I r t 2 C(u, t) du dt (4) J t1 and the cvariance f the numbers f events in tw disjint intervals (t1' t2) and (t8,/4) by (5) When the prcess is statinary-when its prbablity struture is unchanging with time-these equatins assume a partlrularly Simple frm. In this case m(t) must be a cnstant (say m) and C(u, t) mst reduce.t? a functin C(v) f the difference v = (t - u) alne. In particular (wntmg T = 11 - t 2 ), Equati()ns (3) and (4) take the frm (6) B[N(T)] = mt Var [.N(T)] = mt + 2 J: (T - u) C(u) du (7) Inset-l
254 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 2.1-THE POISSON MODEL The natural starting pint fr a discussin'f stchastic mdels fr earthquke ccurrenc is the Pissn' prcess. The fundamental prperty f the POIssn prcess IS that the numbers f events in any tw disjint intervals are inqependent randm variables. This prperty and the supplementary cnditins that the prbability shuld be asympttically prprtinal t the length f the interval, and that the prbability f mre than ne event in the small interval shuld be asympttically negligible ((dt) ), are sufficient t specify the whle prbability structure f the prcess. The prbability f exactly n events ccurring in the interval (t1' t 2 ) given by the frmula. P.. = A(t1' t2 ).. exp { - A(t1' t2)}in! (8) where A(t1,t2) = ft2 ;\(t)dt and ;\(t) is the instantaneus rate defined in 11 equatin (1). Frm equatins (3) and (4) we have E[N(t1' t2)} = Var [N(t1' t2)} = A(t!) t2) (9) If the prcess is statinary we can write ;\(t) = ;\ (a cnstant) and A(tv t 2 ) = (t2-11);\, Then E[N(t1' t2)} = Var [N(t1' t2)} = (t2 - t1);\ (10) Many prcesses apprximate quite well t' the Pissn prcess in spite f its restrictive definitin. One reasn fr this is that a prcess built up f a number f small, independent cmpnents will resemble a Pissn prcess mre and mre clsely as the number f. cmpnents increases, and the cntributin frm each decreases (Khinchin, 1955). It might be thught, therefre, that the prcess cnsidered here, representing the sum f cntributins frm many different areas and depths, wuld be f this type. We shall see that this is nt the case. A cnsiderable prprtin f the shcks in any interval appear t cme frm nly a small number f active surces. Sme reasns fr suppsing that the Pissn mdel is nt satisfactry were put frward in the earlier paper. This cnclusin is strengthened by a direct examinatin f the cvariance structure and the distributin f interval lengths. T determine the cvariance structure, an autcrrelatin analysis was carried ut n the numbers f shcks ccurring in successive 0'1 year perids. A separate analysis was carried ut fr the fur classes f shcks listed in sectin 1. In each case a linear trend was remved frm the uncrrected data and the autcvariances cmputed frm the frmula 1 N-s N- r YrYr+B is (11)
N.3 VERE-JONES & DAVIES - EARTltQUAKE ANALYSES 255 where N is the ttal number f intervals (here 200), the Yr are the transfrmed variables (after the trend and mean has been remved), and the "v" n the symbl dentes an estimated value. Values fr the first 10 autcvariances are given in Table 1a. Fully efficient tests fr dependence are hard t establish, but with 200 bservatins, the distributin f the cjf; shuld be apprximately that f the rdinary crrelatin cefficient, and n this assumptin, the first f the ratis cj/c can be used t test whether there is sme degree f crrelatin present. A ne-sided test is apprpriate, and the 5%, 2%, and 1% values fr each grup (scaled up by the factr C in each case) are listed in Table lb. In all cases the value f c 1 is significant at the 5% lel, and fr the shallw shacks and fr all shcks at the 1 % level. This evidence is f mt interest fr the deep shcks, fr which it has usually been assumed that there are few r n grups f related shcks. Hwever, as pinted ut in the earlier paper, the printed lists shw numerus grups f deep shcks, ccurring verlerids f a year r s, within very narrwly defined limits f latitude an depth. In this paper als, we find repeated evidence f dependence amng the deep shcks. The high value f c 1 in the furth rw f Table 1a pints in a similar manner t a residual dependence amng the shallw shcks, even after the bvius grups f swarms and after shcks have been remved. A mre sensitive indicatr f deviatins frm the Pissn mdel is the rati f the variance t the mean (the Pissn index f dispersin). This has the expect value-f a unity fr a Pissn prcess (Equatin 9), and fr a large number f bservatins its distributin is asympttically prp.rtinal t a X2 distributin with (N - 1) degrees f freedm. The values f this index are shwn in Table 1c. Since the 0'1 % pint is 1'33, fr all grups, the Pissn hypthesis is rejected at a very high level. The large fluctuatins are even mre striking if we take a larger perid. v Fig. 1 shws the bserved variance (actually V (T)IT) fr the shallw shcks fr a perid f length T fr different values f T. The variances have been calculated frm the frmula (Murthy, 1961)., 1 JT - 7' T2 VeT) = N2(t, t + T)dt- _N2(0, T)(12) T - T 0 T2 where T is the ttal time f the sequence cnsidered. The lwer hrizntal line n the same figure represents the variance fr a Pissn prcess with the same rate (V(T) = AT). Final evidence f gruping cmes frm the distributin f the interval lengths between shcks. In Table 2 we have set ut the distributin f interval lengths between shcks fr tw series f 101 shcks, ne series fr a quiet perid (1954-56), and ne fr an active pid' (1946-49). In bth cases the numbers in each cell were cmpared Wlth the expected numbers frm an expnential distributin having th same mean. The X 2 value fr the quiet perid is. smewhat reaer than lts expect:d vale (8), bu nt significantly s. ThlS seems t mdlcate that fr a qulet pend the POlssn
IV VI ;. 0'> [ ' IOO ------,---------;;-:;-r---- 0,.6... 0 // 0// Pissn,(,) I- -7 Expnentil mdel, / / --------- 1st pwer law mdel I // 2nd pwer law mdel I / / z / /// N 50 '0' 1/ '/ ---L--125 Iwk -05yr -Iyr (- z l'" > --------------- fii -- 1501 --- 0 ii ------- ------- / 0 0 0 > Z ;;: 100 (/P-(f-------=-= =---.::--- - - -- -----------==... t1 50 - C) t'1 't:i I"'l en I I I I 0 0-5 1-0 1-5 2-0 FIG. 1--Plt. f C.(r) against perid l' fr different mdels (shallw shcks). Years <: 0 r '0
N.3 VERE-JONES & DAVIES - EARTHQUAKE ANALYSES 257 TABLE 1a-Autcvariances fr Numbers f Shcks in 0 1 Year Perids Shck Types C C, C. C, C. C. C. C, C. C Ct. All shcks 12 9 2 5 0 7 1 6 1 8 0 7 1 3 1.7 0 7 0 9 2 1 Shallw shcks 9 5 2 1-0 2 1 0 1 4-0 1-1 2 1 2 1 0 0 5 0 8 Deep shcks 3 5 0 4 0 1 0 3 0 3 0 4 0 1-0 1-0 1 0 0 0 0 Mdified shallw shcks 3 4 0 4 0 1 0 0 0 2 0 2 0 1 0 1 0 2 0 0-0 2 --------- TABLE 1b--Significance Levels fr Ct TABLE Ie-Mean and Pissn Index Shck Types 5% 2% 1% Mean Pissn Index All shcks 1 5 1 9 2 1 5 7 2 3 Shallw shcks 1 1 1 4 1 6 ; 2 3 0 Deep shcks 0 4 0 5 0 6 2 5 1 4 Mdified shallw shcks 0 4 0 5 0 6 2 3 1 5 TABLE 2-Distributin f Time Intervals between Shcks SERIES 1-(5.1.54-4.5.56) Intervals (days) 0 1 2 3 4 5 6-7 8-9 10--12 13+ Observed N. 12 12 7 9 3 6 10 8 13 20 Expected N. S l 11 1 9 8 S 6 7 6 6 7 11 1 S 7 9 5 20 9 Mean length = 8 0 days. x' (8 degrees f freedm) = 11. SERIES 11-(15.10.46-4.1.48) Intervals (days) 1 2 3 4 5 6-7 S-9 10-12 13+ Observed N. 24 13 10 S 12 5 11 5 :; 7 Expected N. 10 6 18 0 14 3 11 5 9 2 7 ; 10 5 6 7 5 8.6 1 Mean length = 4 5 days. X (8 degrees f freedm) = 23 0. The 5% and 1% vlues f x with 8 degrees f freedm are 15 5 and 20 1, respectively.
258 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 apprximatin may nt be unreasnable. Fr the active perid, hwever, the X 2 value is significant at the 1% level indicating that the Pissn apprximatin is then quite inadequate. 1n each the largest deviatin is at zerthere are mre clsely spaced shcks than there shuld be n the Pissn mdel. The psibi1i/ f describing earthquake ccurrenc by a Pissn press has been InvestIgated by a number f authrs, especially by V. N. GalSky, wh has studied the recrds f Central Asian earthquakes. These investigatins have chiefly cncerned the ccurrence f large shcks, fr which n appreciabte divergence frm the Pissn law is fund (Gaisky, 1961; Gaisky and Katk, 1960; Gaisky and Birman, 1962). These results d nt frm a real cntradictin t thse f the present paper since we are nt examining the distributin f large shcks, but the crrelatins amng shcks f small r mderate size. Indeed such results give an added plausibility t the mdels discussed in the next chapter, where it is suppsed' that grups f small shcks are initiated by majr events beying the Pissn Law. An earlier paper by Ljapunv and Fandjusina (1950) shwed that with Central Asian earthquakes, als, discrepancies frm the simple Pissn law may ccur when earthquakes f smaller size are cnsidered., 2.2-TRIGGER MODELS The results s far established pint t a significant degree f dependence amng earthquakes. There is a psitive crrelatin amng the numbers f earthquakes in successive time intervals, which prduces (In accrdance with Equatin 4) a high variance fr the numbers f shcks in an interval f fixed length. Such prperties place the earthquake prcess amng the general class f "cntagius prcesses" which have been widely studied in applicatin t eclgy and frestry (Matern, 1960), t csmlgy (Neyman and Sctt, 1958), and in ther cntexts. Fr earthquakes it is natural t suppse that the crrelatin effects arise frm the ccurrence f grups f statistically related shcks. The mdels that we shall cnsider in this sectin are all based n this idea. We shall nt, hwever, immediately identify these grups with earthquake swarms, after-shck sequences, and ther grups which are knwn t llave ccurred. In view f the practical difficulties f identifying all members f a given swarm r after-shck sequence we shall at first make n assumptins as t the nature f the grups, but attempt t ascertain indirectly, by estimating the larameters f certain mdels, the extent and character f the clustering e ect. The simplest mdel t allw fr gruping is the cmpund Pissn prcess, in which grups f events ccur tgether at the instants f a simple Pissn prcess. This mdel is nt satisfactry as it stands, since in practice the grup will spread ver a finite and cnceivably large segment f the time I axis. A mre suitable class ()If mdel can be built up in the fllwing way. Suppse that events initiating grups f shcks ("trigger events"-nt necessarily themselves shcks)* ccur at the instants f a *The case in which the trigger event is itself a shck can be brught within the scpe f the analysis f allwing A},,(t) t have a discrete cmpnent (atm) at t = O.
N.3 VERE-JONES & DAVIES-EARTIIQUAKE ANALYSES 259 si';dple Pissn trcess with cnstant rate J-t. Let '\(t) be a decay functin with the prperhes '\(t) 0 (t < 0) '\(t) > 0 (t >0) (13) f '\(t)dt = 1 and suppse that the cnditinal prbability that a shck will ccur in the small time interval (t + x, t +.'JC + dx), given that a trigger event ccurred at time t, is independent f t and equal t A'\(x). We shall assume als that A itself, the quantity characterising the size f the grups, is a randm variable with distributin F(a), finite mean a and variance v a= E(A) = f adf(a) (14) v = Var (A) = f (a - a)2 df(a) Finally we shall assume that separate trigger events generate independent sequences f shcks, and that the randm variables A, assciated with distinct trigger events, are als independent. It is nt difficult t shw that these assumptins define a statinary pint prcess, fr which the instantaneus rate is given by m=p.di (15) and cvariance density by C(u) =J-t(a 2 + v) f '\(t)'\(1 + u)dt (16) Hence (frm Equatins 6 and 7) the mean and variance f the numbers f events in an interval f length T are given by B[N(T)] pflt (17) _ C(r) = pat + 2.J-t(tP + v) rt (r - u)c(ti)du (18) where c(u) - J '\(t) '\(t + ti)dt. I J 0
260 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 Similarly the rth cvariance (the cvariance between the numbers f shcks in tw intervals rt apart, each f length T) is given by t er(t) = (a2 + v) (7 - u)[e(rt + u) + c(rt - u)]du (19a) Fr T large, (20) The right-hand side f this expressin is the variance f a cmpund Pissn prcess fr which grups f events ccur at rate ft, and the size f the grup has the distributin (21) This is als the distributin f the numbers f shcks in a grup initiated by a single trigger event. Thus Equatin (20) expresses the fact that when the interval becmes very large, we may ignre effects due t the shcks within a single grup spreading ut alng the time axis. It is imprtant in practice t knw hw quickly the asympttic frm (Equatin 20) is reached. The faster the functin A(t) appraches zer, the faster the asympttic frm is reached. The bserved variances (Fig. 1) suggest that the asympttic frm is reached nly slwly, and hence that the decay functin itself is nt a rapid ne.. Such mdels have a fair degree f bth intuitive appeal and f generality. Their secnd-rder prperties depend n the tw parameters p.a and p.(a2 + v) and n the decay functin A(t). Different chices f these quantities give rise. t particular mdels that can be tested fr fit against the bserved prcess. We shall cnsider in detail tw chices f A(t), ne with an expnential decay A(t) = pe-pf (p > 0) (22) and the ther with an inverse pwer-law decay AU) = pepi(e + t)p + 1 (e > 0) (23) 2.3-THE SPECTRUM Estimates fr the parameters f these mdels and the tests f gdness f fit will nt be based directly n the cvariances described in the previus sectin, but n their Furier transfrm, the spectrum,
N.3 VERE-JONES & DAVIES - EARTIiQUAKE ANALYSES 261 t (<0) = 00 C,,(T) cs nw (24) 'T -00 The bserved quantity crrespnding t the theretical spectrum is the peridgram I (w) T (25) (25a) where the C,,(T) = C-,,(T) are the bserved cvariances defined in Equatin (11), Xt is the number f shcks in the tth interval, N is the ttal number f intervals cnsidered, and xt is the average f the x t. Accrding t Equatins (24) and (25) the spectrum may be interpreted as an analysis f the ttal variance int cmpnents crrespnding t different frequencies. In the case f a cntinuus variable--say a vltage varying in time--the spectrum may be regarded as giving the average pwer transferred at different frequencies. In the case f a stchastic pint-prcess this interpretatin n lnger hlds, but the analgy may give sme idea f the meaning f the spectrum t thse nt familiar with the thery. In particular, a cyclic effect in the Xt with perid T (measured in units f the basic interval T) willgiverisetapeakini (w) atw = 27r/T. T Under wide cnditins, the rdinates Ij = 1(00;) (Wj = 27rj/N, j = 0, 1,..., N - 1) are apprximately uncrrelated. When the Xt are independent, the I j are als apprximately identically distributed, and in particular, the expected value f If is then apprximately equal t the cnstant value u = 2 vat (Xt) fr all values f j. Fr such a prcess the theretical spectrum is just a hrizntalline at height u2 These results are the basis f mst f the simple tests fr cyclic effects. In the case f a prcess with spectrum few), the rati 1/ = Ij/f f has prperties similar t thse f the peridgram arising frm independent Xt (Hannan, 1960). We shall use Ii' fr tests f gdness f fit f ur prpsed mdels. The peridgram is nt a cnsistent estimatr f the spectrum, and fr mst purpses it is necessary t smth it. Several methds have been suggested; we shall use Bartlett's frmula Inset-2 M-l l(w} 2 (1 -) c. cs (26) -M+ 1
------------------------"-----.-'--'''''-.--------.-.---. ----- i6i N.Z. JOURNAL OF GEOLOGY AND GEOPttYStCS VOL. 9 where M < N is chsen with regard t the degree f smthing required. This frm may be regarded as (apprximately) the result f averaging the values f I (w) in the vicinity f w, r alternatively (again apprximately) f dividing the ttal recrd f data in t N / M sectins f equal length, and averaging the resulting peridgrams. The use f spectral methds in the analysis f stchastic pint prcesses has been discussed recently by Bartlett (1%3). Bartlett uses the Furier transfrm f the cvariance density <I>(w) = J: C(t)e iwt dt +- m (27) as the basis f his discussin, where C(t) and m are defined as in sectin 2'0, and we may take C(O) =. f (w) is related t <I>(w) by T 2 00 4T sin 2(1w ) (28) where T -00 is the fixed interval length assciated with f (w). Fr the trigger mdel f the previus sectin, 1 foo 12 <I>(w) = f1j1 + p,(a + 2 v) A(t)eiwt dt I (29) I 0, The idea f a dubly stchastic prcess mtined in Bartlett's paper is als relevant. Such a prcess may be described as a Pissn prcess, whse parameter is itself a randm functin, say t(t).* The trigge,r mdels are f this type, where (t) is a "saw-tth" functin, whse peaks are lcated at the trigger events. The decay frm each peak is determined by the functin '\(t). Bartlett estimates <I> ( w) directly by the statistic T (30) where t. is the time f the sth event, and T is the ttal time cnsidered. Fr ur purpses f (w) has the advantage f the finite length, and is mre T simply estimated, but suffers frm "aliasing" (e.g., a peak at w in f (w) culd be due t peaks at any f x =,/T, x = (2n7T + W)/T, x = (2n7T - w)/r in <I>(x), althugh nrmally nly the first fur terms r s need be cnsidered). In practice the difference between Equatins (25) and (30) amunts t a difference in the chice f r. Perhaps the mst relevant factr here is the time-scale f the effects we wish t study. If we are interested in the *It is interesting t nte that the pectrum f HI) is just 4>«(0) - m (d. 27). T
N.3 VERE-JONES & DAVIES - EARTHQUAKE ANALYSES 263 fine structure f the prcess-that is t say, in phenmena having a timescale f the same rder as the typical interval between events-then we shuld chse a value f 7 which is small with respect t this interval, and such a: chice crrespnds t the statistic (Equatin 30). On the ther hand, little infrmatin will be lst by taking a lnger value f T if we are interested in phenmena having a larger time-scale. The aliasing effect remains as a disadvantage f chsing intervals f equal length, but we shall nt enter int a discussin f ther pssibilities. Tw analyses will be described belw. Fr the first f these we have taken 7 = 0 1 year, and fr the secnd 7 = 1 day. The first analysis was made with the primary aim f investigating a pssible cyclic effect with a perid f tw t three years. It als yields infrmatin abut the cvariance structure. f the prces ver re.iatively lng perids. The secnd analysis yields cnstderably mre mfrmattn abut the fine structure f the prcess. In this case f «(0) is a reasnable apprximatin t <p(w). T. Te smthed peridgrams resulting frm the first analysis are shw m Ftgs. 2-5, and thse frm the secnd analysis in Figs. 6-9. Fr the first set N = 200 and M = 40 (in the ntatin f Equatin 26), and fr the secnd set N = 7,304 and M = 200. We have als marked in the values f the first few (unsmthed) periqgram rdinates; the value CO(7), which gives the mean height f the peridgram; theretical spectra frm several prpsed mdels; and apprximate significance levels against the hypthesis f independent x t with variance (12 = C( T) These levels refer t an verall maximum fr the peridgram, and nt t a peak at a particular value chsen in advance. The marked value f 20% is a cnservative estimate, btained by regarding the smthed peridgram as the average f N!M independent cmpnents. This appears t give a rather crude apprximatin, particularly when N!M is large, and a value f 5% (based n the frmulae in Hannan, 1960, sectin 3.3) might be mre apprpriate fr the ne-day spectra. In any case, the significance levels fr a prcess with such cmplex structure are likely t be very apprximate, and shuld be taken nly as a guide. Fr the interpretatin f theseperidgrams it is helpful t bear in mind the frms f the spectra fr the simple and cmpund Pissn prcesses mentined previusly. Fr the simple Pissn prcess with mean rate,\= p.a, f (w) =.p.th (31) 'T whereas fr the cmpund prcess t which ur trigger mdel apprximates fr lng intervals, f (w) =p.(a + a2 + v)-r (32) 'T Hence fr (0 crrespnding t perids lng cmpared with the lengths f grups f shcks (w 0) we might expect Equatin (32) t hld, and fr shrt perids «(O 71") Equatin (31) t hld. Is is apparent that this gives th general frm f the peridgrams-,..a regin f high values near the
264 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSiCS VOL. 9 rigin, subsequently.decreasing twards a value crrespnding t tl:le mean rte. We shall cnsier the peridgrs in mre detail in the ensuing sectins, wh.er they will be cmpared with the spectra f sme sme trigger mdels; It IS dear at nce, hwever, that n prnunced cyclic effects are present. In cnclusin it shuld be nted that the variance curve, the c<>variances, a!ld the spectrum are merely different ways f expressing the same infrmatin, the spectrum having the advantages f simpler interpretatin, and f transfrming simply under simple transfrmatins f the basic prcess. 2.4--TRIGGER MODEL WITH EXPONENTIAL DECAY If the decay functin has the expnential frm (Equatin 22), the cvariance density is als expnential, and frmulae (Equatin 18) and (Equatin 19) becme CO(T) -ppt + p.(az + V)T[l - (1 - e-pt)/pt} (33) C,,(T) - p.(a 2 + v)t e- npt (csh pt - l)/pt (34) The spectrum takes the frm cs w - f «1) = C(T) + 2C1 (T) ------- (35) T 1 - e-2pt - 2e-PT cs (1) e-pt 2p. sin2 (ia» sinh (pt) p.(at + a 2 + V)T - - (a2 + v) (35a) P csb(p1') - cs w In rder t estimate the parameters f the mdel we shall first suppse that the mean rate (!JIlT) is determined directly as the ttal number f shcks divided by the ttal time. The remaining parameters will be estimated by using the methd f least uares t fit a functin f the frm (Equatin 35) t the peridgram rdlllates I j subject t the cnditin These methds are a cmprmise between the demands f technique and cnvenience: The peridgram rdinates, althugh apprximately independent, are expnentially rather than nrmally distributed, while least-squares methds have ptimal prperties fr nrmal variates. A general thery f maximum-likelihd estimatin f. the spectrum has been develped by Whittle and thers (e.g., Whittle, 1952), but their frmulae are nt easy t apply. With as many as 100 rdinates it seems likely that any tw reasnable methds f estimatin will lead t similarestimates--althugh the leastsquares methd will tend t give mre weight t pints fr which few) is large, than wuld Whittle's methd.
N. 3 VERE-JONES & DAVIES - EARnIQUAKE ANALYSES 265 The least squares equatins lead, after sme reductin, t the frmulae and ----lgy A(y). where 'I = e-fj'f, A(y) = lg Y -(1 - 'I)' B(y) = 1/y- 2 lgy - 'I. N-l the dash dentes differentiatin, and PN('I) is the plynmial an 'In-I 1 with cefficients an = (1 - n/n)cn + (n/n)c N -... Since 'I is f the rder f 0'2 these equatins are readily slved by numerical methds, the plynmials n the L.H.S. being largely determined by their first tw r three terms. Althugh the spectrum deeends n bth a and 11, as well as n 'I' there are nly tw parameters in these equatins, 'I and the factrp.(a2 + 11)7. The parameters a and 11 cannt be separated Wlless sme particular functinal frm is assumed fr the distributin fwlcti F(a). This pint will be discussed in mre detail in a later section. We have shwn in Table 3 the three parameters piit'l.j-(tp + li)t and 'I = rfj'f, tgether with the half-life (descrlbed belw), and the rati R = (a + tp + 1I)/a. Thes are shwn fr the ttal, shallw, and deep shcks. The crrespnding spectra are shown by the dtted lines in Figs. 2-4 and 6-8. T assess the "gdness. f fit" f the mdel, we first transfrm the peridgram by frming Ii'..:... Ii/Ii where Ii is the fitted- spectrum at III = 27rJ/N. We then examine the fwlctin This functin fr the shallw shcks is shwn in Fig. 10. The maximum deviatin f this functin frm the line y = x will,. the hypthesis that theel is crrect, fllw the Klmgrv-Smirnv law (e.g., Hannail, 1960). On this basis 20% and 5% significance levels are shwn n the diagram. This test des nt indicate disagreement with the mdel. Hwever in n case is the agreement very gd near the rigin (Figs. 2-5), r with the ne-day spectra.. The behaviur f the spectrum near the
266 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 rigin has a reciprcal relatin with the behaviur f the cvariance functin fr large values f t; a sharp peak at the rigin crrespnds t a slw rate f decay f the cvariance density and vice versa. The expnential spectrum in Figs. 2-5 is t flat near the rigin, crrespnding t the fact that the bserved wvariances fall ff much mre slwly than expnentially. Indeed, the values f C" calculated n the basis f Equatin (34) are negligable fr n greater than abut seven, while -several f the bserved values, fr example C 10, are quite large. It is hard t assess the significance f these later values, but it seems likely that they culd apprach the 50/0 level, if the expnential mdel was taken as the null hypthesis. Similarly, it can be seen frm Fig. 1 that the expnential mdel underestimates V (T) fr large values f T. The disagreement with the ne-day spectra is f the ppsite type. Fr the shrt perid, the expnential functin is t fiat. This effect is partly due t the fact that we have nrmalised ur mdel.s that CO(T) = CO(T), when T = 0'1 year. Hwever, these results nly enhance the general cnclusin that the expnential mdel cannt match the bserved prcess ver an extended range. Fr shrt perids the expnential functin rises at the rigin t slwly; fr lnger perids itdecays t fast. T summarise, it appears that the expnential mdel des nt affrd a gd descriptin f the data, althugh the disagreement is nt sufficiently prnunced fr the mdel t be rejected by the Klmgrv-Smirnv test applied t the 0'1 year spectra. These results have been btained n the basis f the data fr the shallw shcks, but they are supprted als fr the data fr the deep shcks. Indeed, the peak at the rigin in Fig. 3 is even sharper than that in Fig. 2, indicating that fr large T the cvariance density decays even mre slwly fr the deep shcks than it des fr the shallw shcks. This remark pints t an imprtant qualitative difference between the deep and shallw shcks, which is cnfirmed by the results set ut in Table 3. A "half life" f a grup f shcks may be defined as the time after which, n the average, half the shcks f the grup have ccurred. With the expnential mdel this equals (lge 2) / p. The higher value f the half-life fr the deep shcks suggests that fr these the time scale is lnger than fr the shallw shcks. TABLE 3-Parameters fr Trigger Mdel with Expnential Decay IJ4T p.(a' + V)T Y = r=pi R i-life (days) ----------- ----- All shcks 5'7 12 8 0'14 3'2 12') Shallw shcks 3'2 10'5 0'11 4'3 11'5 Deep shcks 2'5 1'9 0 22 1'8 16'9 -----
N.3 VERE-JONES & DAVIES - EARTHQUAKE ANALYSES 267. 2.5-WESTPORT EARTHQUAKE SEQUENCE-1962 S far we have nt islated any particular earthquake sequence fr examinatin, but analysed the list as a whle. In this sectin we cnsider a particular earthquake sequence that ccurred near Westprt, New Zealand, in 1962, in rder t btain sme justificatin for ur next chice f decay functin. Our data is frm Adams and Le Frt (1963). The cumuliative graph,f the number f shcks f magnitude greater than r equal t three, against time, is shwn in Fig. 11, tgether with a graph f the functin N=A[l - ()P ] with A = 100, p = 0'25, and c = 0'25 days. This suggests a decay functin f the frm.\(t) = pcp/(c + 1)1 +p (23) might give abetter fit than the expnential functin. It shuld be nted that we have included here earthquakes f a lwer magnitude than were were used in the verall study, and that we have been cncerned with a relatively shrt perid. The functin (Equatin 23) decays extremely slwly; even at the end f the perid shwn, the cumulative graph has reached little mre than 75% f its asympttic value. The use f such a functin in relatin t after-shck sequences is by n means new,and in particular it has been the subject f an extensive study by Utsu (1961) f after-shck sequences in Japan. Whereas Utsu was largely cncerned with the analysis f individual sequences, in the sectin belw we shall examine hw far the same hypthesis is capable f explaining verall features f the data, as reflected in the peridgrams f Figs. 2-9.. As a final remark, it is wrth pinting ut that in Fig. 11 there is a tendency fr the shcks t ccur in grups within the aftershck sequence itself. 2'6-TRIGGER MODEL WITH INVERSE POWER-LAW DECAY where k is numerical cnstant, apprximately equal t 0'9 when p = 0'25. The cvariance's 'T ) 1-p -(Hp) Cn(T),..., p.(a 2 + v)cp (. -c- n (37)
268 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 f.r nand () large. There des nt apear tg be a simple fgrmula fr the spectrum, but the ngn-aliased fgrm <I>(x) = p.a + p.(a2 + v) exp (- ip7r/2) r(1 -,p) (ex)p,..., pix + p.(a! + v)p2/(ex)2 fgr ex large. 00 (ex)i +p-- (38) (j _ p) j! The main feature.f <I>(x) and hence f (x) is the very sharp peak at the T.rigin-see Fig. 3, This cannt be reprduced n a smthed perid.gram f.r tw reasns, Firstly, frm EquatiGn (25a), the effect.f centring the.bservatins at the.bserved mean Xt is tg reduce the value.f the peridgram at the rigin tg zer, Remval.f the linear trend will have a similar effect. Secndly, the smthing itself will tend t.. remgve any sharp peaks. The unsmthed I; shuld ngt be greatly affected by this, and S. we have sh.wn IclIO in Figs. 2-5. These values have the cmpensating disadvantage that. their standard deviatin is f the same.rder as their expected value, SG that little significance can be placed.n their psiti.ns as individuals. Estimating the parameters.f the mdel is again difficult. The previgus article suggests taking p = 0'25. In this case the variance can be expressed in terms.f elliptic integrals, VeT) = CO(T) =!JIlT + p.(a 2 + V)T (1 - I(q) ] (39) 4 where I(q) = - 2( 2a )1 [2P(ep,1T/4) - B(ep, 7r/4) } 3 1 a 4 4 (1 - a)a 4 + and a =.COS ep = {e/(e + t)}i, and E (ep, 1T/4), P(ep,7I'/4) are elliptic integrals.f the first and secgnd kinds, respectively. If we then equatep.at X C = C C1 = c1 we find fgr the shallgw shcks, p.(a 2 + v)t = 31'1, e = 2'3 days, and f.r the deep shcks, p.(# + V)T = 6'2, e = 3'4 days. The c.rresp.nding "half-lives" are 35 days and 51 days respectively, althugh fgr such pwerlaw mdels the "half-life" gives a pogr indicati.n.f the time range, and the dependence remains appreciable fr several years.
N.3 VERE-JONES & DAVIES - EARTHQUAKE ANALYSES 269 These values f c are cnsiderably larger than the value fr the Westp()ft sequence, and fr this reasn a secnd pwer-lllw mdel fr the shallw shcks was taken, with c = 0'37 days. Fr this mdel we have set pfr = x and (in rder t secure a gd fit with the ne-day spectrum ) /L(d? + v)t = 18'3. The "half-life" crrespnding t his value f c is 5.2 days.. The spectra fr the first mdels are shwn by the dashed lines in Figs. 3 and 6 (shallw shcks), and Figs. 4 and 7 (deep shcks). The secnd pwer-law mdel fr the shallw shcks is shwn by the half-dashed line in Figs. 3a.nd 6. The crrespnding graphs f V('T)/'T are shwn fr the shallw shcks in Fig. 1. Taking the shallw shcks first, we see that bth the pwer-law spectra fit the 0'1 year peridgrams reasnably well. As cmpared with the expnential mdel, there is an imprvement f fit near the rigin, but the verall shape f the graph is nt greatly altered. The Klmgrv-Smirnv test again indicates n significant disagreement with the mdel. The results shwn in Fig. 1 and the ne-day peridgram are mre interesting. Frm Fig. 1 it can be seen that the pwer law with the larger value f c gives a better picture f the grwth f the cvariance fr mderate. values f 'T. On the ther hand, the behaviur f the prcess fr small values f 'T appears t be better described by the pwer law with the smaller value f c. This is shwn up particularly well by the ne"day spectra (Fig. 6). Again it appears that n smgle mdel will give a satisfactry descriptin f the prcess ver the- whle range f values f 'T. The ne-day spectra shw further discrepancies with the bserved peridgram near W = 7/'. This effect shws up mst strngly in the scaled peridgram I/(w) ::::: J(w)/!(IJl), which is shwn in Fig. 12. The peak at 7T indicates a marked deviatin frm the behaviur t be expected frm any f the simple trigger mdels. It is significant at abut the 21% level.* The interpretatin f this peak will be taken up in Sectin 3. Figs. 4 and 7 shw that fr the deep shcks the pwer-law mdel gives a reasnable fit bth with the 0'1 year and the ne-day peridgrams. A further mdel (with a "blck decay" functin) will be described in sectin 3. The deep shcks shw n sign f the peaks near 7T n the ne-day peridgram. 2.7-ENERGY SPECTRA It might be thughtpsible t use a peridgram calculated frm the energy released by earthquakes in successive perids. In fact it is unsatisfac *Care has t be taken in assessing the significance f this peak, since the peridgram rdinates at 0, 7T are (tughly speaking) based n distributins with half the usual number f degrees f fr«<lm. A further difficulty is that the theretical spectrum f(",) has nt been scaled s that it enclses the same area as the bserved peridgram (i.e., the variances have nt been exactly matched). Hwever, fr any simple mdel, the spectrum shuld apprach the value pa as '" appraches 7T, and. rescaling (by altering the value fp.(ti' + V)T) will mdify the spectrum near", = 0, nt near '" = 7T. Thus the significance f the peak at '" = 7T will nt he greatly affected by such changes.
270 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 try t d this directly because f the very large fluctuatins in the energy released. The peridgram cnveys very little infrmatin since it is almst entirely determined by the few earthquakes f large magnitude; thus n gruping effects are apparent in such a study. The lgarithm f the energy in each interval (plus a cnstant t keep the lgarithm finite when n earthquakes are recrded) is better behaved, and its peridgram shws evidence f sme cvariance structure. Other pssibilities (fractinal pwers f the energy released) might als be cnsidered, but their. physical interpretatin is nt clear. Sme recent studies (Keilis-BrOk and Malinvskaya, 1964) have suggested that this might prve a fruitful field fr further investigatin. We shall nt be able t cnsider it further in the present paper. EVALUATION OF THE RESULTS 3.1-SHALLOW SHOCKS N significant peridic effects are apparent, at least within the ranges cvered by the 0'1 year and ne-day spectra (say three mnths t three years and 3-15 days), The gruping effect is certainly significant. This is shwn bth bya direct examinatin f the cvariances and by the spectra. Thus the simple Pissn mdel cannt be accepted as an adequate descriptin f earthquake ccurrence amng the shallw shcks. Nne f the prpsed mdels seems t give a fully satisfactry accunt f the gruping effect. There are disagreements bth fr very lng perids and fr very shrt perids. The cntinued grwth f the variance fr large values f T tells heavily aga:inst the expnential mdel, and suggests that the dependence amng shcks may last fr many mnths. The first pwerlaw mdel (c = 2'6) gives a reasnable fit fr mderate values f T, but even this mdel appears t underestimate the variance when.,. is very large. This fact pints t the pssibility that sme f the basic assumptins under lying the mdel may be at fault. The fluctuatins in activity. frm year t year may be t large t be represented in terms f any simple statinary mdel, even after the remval f linear trends. Alternatively we may be faced with a prcess whse variance diverges. The validity f any secnd. rder analysis wuld have t be examined carefully in this case. The disagreements fr shrt perids are als very interesting. A feature f the cvariances e" fr the ne-day analysis is that they drp away quite slwly frm a very high initial value C. Fr this reasn the expnential and first pwer.law mdels, which give a reasnable fit fr the 0'1 year spectra, are here quite inapprpriate. A much better fit is given by the secnd pwerlaw, which has a smaller time scale (the parameter c being reduced by a factr f 6-7). On. the ther hand this mdel agrees less well with the behaviur f the prcess fr larger values f T, particularly as regards the variance curve shwn in Fig. 1. A secnd remarkable feature f the ne-day analysis is the peak in the peridgram at <t) = 7r. This peak may be partly a genuine peridic effect, but it seems rather t mark a failure f the peridgram t decay t the mean rate as quickly as the theretical spectra. This is mst likely an indica
N.3 VERE-JONES & DAVIES - EARTIIQUAKE ANALYSES 271 tin f a secndary gruping effect within the main prcess. Such an eff wuld' have a characteristic time scale f the rder f ne t five days, as cmpared with characteristic time scale f ne t six mnths fr the main effect The recrds f after-shck sequences such as the Westprt sequence lend sme direct supprt t this hypthesis. All f these ideas need further investigatin. As was pinted ut in a previus sectin, the quantities p" a, and v cannt be separated by the present analysis, withut makmg sme further hyptheses abut the distributin f the number f shcks in a grup. Hwever, It wuld seem unwise t try and give a particular frm t this distributin, because f the extremely skew frm suggested by a direct examinatin f the grups (see Table 7 in the earlier paper). In fact the appeaqnce f this distributin raises again the unpleasant pssibility f infinite variance. Despite these difficulties it may' be f sme interest t cmpare different estimates f the rati R = p,(a + a2 + v)/p,a which can be interpreted as the rati f the mean-square grup-size t the mean grup-size. Fr the first pwer-law mdel R = 11, and fr the secnd R = 6. An estimate can als be btained directly frm the spectrum by using the first few peridgram rdinates t estimate p, (a + a2 + v) (Equatin 32). This methd suggests R = 8. These values are larger than wuld be btained frm the distributin f grup sizes given in the earlier paper, r frm the expnential mdel.. 3.2-DEEP SHOCKS Again there is n evidence f significant peridic effects. Bth the 0'1 year and ne-day peridgrams shw marked differences frm the crrespnding peridgrams fr the shallw shcks. The 0'1 year spectrum has a high narrw peak at the rigin. This may be partly due t secndrder r higher trends which have nt been remved, but the cvariances alne leave little dubt that a significant gruping effect is present. Thus the Pissn mdel is rejected fr this class als. The high, narrw peak suggests a small dependence extending ver very lng perids. The. expnential and pwer-law mdels suggest that the- time scale fr the deep shcks may be anything frm ne and a half t three times lnger than fr the shallw shcks. As an alternative methd f btaining a rugh estimate f this time scale, we have fitted t the data a trigger mdel having a' decay functin f "blck" type, In this case A(t) lit (0 < t < T) therwise. <I> (x) pil + 4p,(a2 + v) sin2 (jxt)lx2ta
272 N.Z. JOURNAL OF GEOLOGY AND GEOPHYSICS VOL. 9 The parameters were estimated frm the height f the first tw peri.dgram rdinates, and the pint at which the spectrum drps dwn t Its lwer value ptj. The results are T - 4 years p.(a2 + V)T 12 R 6. The value f T may seem rather lng cmpared with thse given by the ther mdels, but it shuld perhaps be cmpared with the "75% life" rather than the "half-life", the frmer having the values 34 days and 2'4 years fr the expnential and pwer-law mdels, respectively. : The ne-day peridgram shws little evidence f the fine structure.which is present with the shallw earthquakes. The tw peaks crrespndmg t perids near three and fur days apprach significance, and may be due t a phenmenn similar t that which causes the peak at a perid f tw days fr the shallw shcks. Such an interpretatin wuld cnfrm t the idea f a dilated time scale at greater depths. 3.3-TTAL SHOCKS We will nt prpse a mdel here because f the different frms Of clustering, which apparently predminate at different depths. It is f interest t cmpare the sum f the shallw- and deep-shck peridgrams with that f the ttal shcks. If the shallw and deep shcks are independent the sum f their peridgrams shuld be equal t that f the tcal shcks. N significant difference shws up, althugh there is sme difference near the rigin. This may pint twards a crrelatin with a cnsiderable time delay, but a crss-crrelatin analysis wuld be needed befre any mre definite statement culd be made. 3.4-MDIFIED LIST OF SHALLOW SHOCKS As pinted ut earlier and in the previus paper, the Pissn mdel is still unsatisfactry here. This indicates that sme gruping is still present. The ne-day peridgram shws that the clse gruping has been largely remved, -althugh there are still sme signs f a peak near 7r crrespnding t a peri"d f tw days. Mre extensive and varied types f "purging" might prvide anther methd leading t infrmatin abut the gruping effect. 4-DISCUSSION OF METHOD AND CONCLUSIONS (a) Te. thery f stchastic pint prcesses prvides a suitable setting fr a statistical study f earthquake ccurrence. Useful infrmatin can be O?tl1ed. frm te cvariances, peridgrams, variance-time curves, and the distnbutins f mterval lengths between shcks. Fr a preliminary survey
N.3 VERE-JONES & DAVIES - EAR11IQUAKE ANALYSES 273 perhaps the mst useful tl is a peridgram analysis based n the numbers f shcks ccurring in successive intervals f equal length. The chice f interval-length shuld be related t the time scale f the effect being studied. (b) The methds have the limitatins f utilising nly the time crdinates f the shcks, and nly first- and secnd-rder mments. Infrmatin is lst by neglecting the ther crdinates and the higher-rder mments. In additin, the techniques used require that the first-, secnd, and third-rder. prperties f the prcess are finite, but the results s far btained suggest that the pssibility f divergent mments must be brne in mind. (c) The "cntagius prcesses" prvide a suitable classf mdels which can be applied t earthquake ccurrence. In particular, earthquake ccurrence canbe described in a general way by trigger mdels, where the (cnditinal) prbability f a shck ccurring at a time t after a trigger event is prprtinal t a decay functin.\.(t). The pwer-law frm fr.\.(t) gives a better general fit than the expnential frm, but the tests used did nt shw up any significant difference between them. Nne f the mdels discussed can be cnsidered fully satisfactry, as they fail t predict the cntinued grwth f the variance fr lng intervals, and the details f the shrt-term structure, (d) The New Zealand data shw n evidence f any genuine peridic effect with a perid f mderate length (say ne mnth t tw years). On the ther hand, each class f data shws strng evidence f clustering, and fr each class the Pissn mdel must be rejected in favur f sme mre cmplex prcess. The time-perid cvered by the analysis is barely lng enugh t give an accurate idea f the gruping effect, especially with the deep shcks, and certainly n effects invlving times cmparable with r greater than 20 years culd have been detected. (e) There are imprtant qualitative differences between the gruping effects fr the deep ana shallw shcks. Estimates f the quantity R (the rati f the mean-square grup-size t the mean grup-size) vary frm 4'3 t 11 fr the shallw shcks, and frm 1'8 t 6 fr the deep shcks, Mrever the, time scale f the gruping effect appears t be lnger fr the deep shcks than fr the shallw shcks, being f the rder ne t six mnths fr the shallw shcks and up t fur years fr the deep shcks. A secndary gruping effect with a perid f tw days appears in the shallw shcks, but is nt present in the deep shcks, althugh these shw sme signs f secndry effects with perids f three t fur days. ACKNOWLEDGMENTS We are again grateful t the members f the Applied Mathematics Divisin staff wh have helped with the calculatins, and t Dr F. F. Evisn and the staff f the Sesmlgical Observatry, Wellingtn, f their cmments nd cntinued interest in this wrk.. We shyld hk t thank especially, Mr G. E. Dlckmsn fr handling the prgrammlog asciated With the spectral analyses, and Dr H. Thmpsn fr his help 10 he preparation f tl;1e fil drafts f the manuscript. We are indebted t Prfessr Galsky fr cmmuoicahng his wrk n the statistics f Central Asian earthquakes.