8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 1/17

Bulletinof the SeismologicalSocietyof America,Vol. 75, No. 6, pp. 1681-1697,December1985

E F F E C T S O F M A G N I T U D E U N C E R TA I N T IE S O N E S T IM AT I N GT H E PA R A M E T E R S IN T H E G U T E N B E R G - R IC H T E R

F R E Q U E N C Y- M A G N I T U D E L AW

BY STEFANO TINTI AND FRANCESC0 MULARGIA

ABSTRACT

T h e e v a l u a t i o n o f t h e p a r a m e t e r s in t h e G u t e n b e rg - R i c h t e r ( G R ) fr e q u e n c y -m a g n i t u d e l a w l o gN r = a - b m i s s h o w n t o b e s t r o n g l y a f f e c t e d b y m a g n i t u d eu n c e r t a in t i es . I f t h e m a g n i t u d e e r r o r s a r e a s s u m e d t o b e d is t ri b u t e d n o r m a l l yw i t h s t a n d a r d d e v i a t i o n o, t h e o b s e r v e d m a g n i t u d e t h a t w e c a l l t h e a p p a r e n tm a g n i tu d e , b e c o m e s a ra n d o m v a r i ab l e , a n d t h e f r e q u e n c y - a p p a r e n t m a g n i tu d el a w d i f fe r s f r o m t h e G R r e la t io n . W e s h o w t h a t t h e r e i s a r a n g e o f m a g n i t u d e sw i t h i n w h i c h t h i s l a w m a y b e a p p r o x i m a t e d a s l o gN a = a - b m + .y 2 l og (e ) . Here ,

~2 = ~2a2 /2 ' [~ =b / I o g e ) ] , a n d N a s t a n d s f o r t h e a p p a r e n t n u m b e r o f th ee a r th q u a k e s . A s b o t h th e t r u e a n d t h e a p p a r e n t m a g n i t u d e c u r v e s h a v e t h e s a m es l o p e o n a l o g a r it h m i c g r a p h , t h e e s t i m a t o r s u s u a l ly e m p l o y e d f o r ~ r e m a i n v a l id .H o w e v e r, t h e u s u a l e s t i m a t o r s f o r a a r e b i a s e d b y a q u a n t i ty w h i c h i s a q u a d r a t icf u n c ti o n o f t h e e r r o r s t a n d a r d d e v i a t io n a a n d o f ~ . T h i s i m p l i e s t h a t th e a p p a r e n tn u m b e r, N a , o f e a r t h q u a k e s e x c e e d i n g a g i v e n m a g n i tu d e t e n d s t o b e l a rg e r t h a nt h e r e a l n u m b e r N r a n d f o r r e a li s ti c v a l u e s o f a a n d b (o r/ ~'), N , i s e x p e c t e d t o b ee v e n a s l a r g e a s t w i c e N r . T h e i m p l i c a t i o n s o f t h e m a g n i t u d e v a r i a b i l i t y o n t h ese i smic r i sk ana lys i s a r e a l so cons ide red : i n pa r t i cu la r, t he eva lua t ions o f t hea t te n u a t i o n l a w p a r a m e t e r s a n d o f t h e e x c e e d a n c e p r o b a b i li ty o f t h e g r o u n dm o t i o n p e a k a c c e l e r a t i o n a r e s h o w n t o b e a f f e c t e d .

INTRODUCTION

T h e r e la t io n b e t w e e n t h e f r e q u e n c y a n d t h e m a g n i t u d e o f t h e e a r t h q u a k e s w a sf i r s t i n t r o d u c e d b y G u t e n b e rg a n d R i c h t e r 1 95 4) a n d i s g e n e r a l l y g i v e n t h e f o r m

log Nr = a bm 1)

w h e r e a a n d b a r e p a r a m e t e r s . I t m a y b e i n t e r p r e t e d e i t h e r a s b e i n g a c u m u l a t i v ere la t ionsh ip , i fN r i s t h e n u m b e r o f e a r t h q u a k e s i n a g i v e n p e r i o d l a rg e r t h a n m , o r

a s b e i n g a d e n s i t y l a w i f N r i s t h e n u m b e r o f e a r t h q u a k e s i n a c e r t a i n s m a l lm a g n i t u d e i n t e r v a l a r o u n d m . I t s v a l i d i t y i s s u p p o r t e d b o t h b y l a b o r a t o r y e x p e r i -m e n t s M o g i , 19 62 ; S c h o lz , 19 68 ) w h i c h d e m o n s t r a t e t h a t i t i s a p p r o p r i a t e e v e n fo rm i c r o f r a c t u r i n g i n r o c k s p e c i m e n s u n d e r c o n t r o l l e d lo a d i n g c o n d i t io n s a n d b yt h e o r e t ic a l m o d e l s w h i c h d e r i v e i t f r o m t h e c h a r a c t er i st i c s o f t h e e a r t h q u a k e s o u rc ereg ion see, e .g ., K an am or i and A nde rson , 1975). S ince i t was f i r s t p roposed ,e q u a t i o n 1 ) h a s b e e n a n d i s s ti ll c o m m o n l y u s e d i n v e r y m a n y a p p l i c at i o n s s u c h a si n t h e s t a t i s ti c a l a n a l y s i s o n r e c u r r e n c e t i m e o f e a r t h q u a k e s s e e E p s t e i n a n dL o m n i t z , 1 96 6), i n d e f i n i n g t h e s e i s m o t e c t o n i c c h a r a c t e r o f a r e g io n s e e K a r n i k ,1971; C ar t e r an d Berg , 1981), a nd in d e tec t ing s ign i f i can t chang es in the t im e

evo lu t ion o f se i sm ic i ty see Sm i th , 1981; M ula rg ia an d Tin t i , 1983).A l t h o u g h i t h a s b e e n s u c c e s s f u ll y a p p l i e d m a n y ti m e s , n e v e r t h e l e s s t h e r e r e m a i ni n s t a n c e s w h e r e e q u a t i o n 1 ) h a s b e e n f o u n d to b e n o t a d e q u a t e in d e s c r ib i n g t h eo b s e r v e d f e a t u r e s o f t h e r e c o r d e d s e i sm i c c a t a lo g s , e s p e c i al l y i n t h e l a rg e m a g n i t u d er a n g e , a n d a n u m b e r o f m o d i f ie d v e r s i o n s h a v e b e e n p r o p o s e d W y s s , 19 73 ; P u r c a r u ,

1681

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 2/17

682 STEF N O TIN TI ND FR NCESCO MUL RGI

1975 , Capu to , 1976 ; Guarn ie r i Bo t t ie t a l . 1980; Caputo , 1981; Makjani~ , 1982; Maina n d B u r t o n , 1 9 8 4 ) .

I n t h e p r e s e n t s t u d y, h o w e v e r, w e s h a l l t a k e i n t o a c c o u n t o n l y t h e b a s i c f o r m o ft h e G u t e n b e rg - R i c h t e r G R ) l a w g i v en i n 1 ) w h i c h i n c lu d e s b o t h li m i t e d a n du n l i m i t e d m a g n i t u d e c a s e s, le a v i n g t h e g e n e r a l iz a t i o n o f t h e c o n c e p t s i l lu s t r a t e dh e r e to f u r t h e r i n v e s t i g a t io n s . N o t i c e t h a t , s t r ic t l y s p e a k in g , o n l y t h e u n b o u n d e dd i s tr i b u ti o n s h o u l d b e a t t r ib u t e d t o G u t e n b e r g a n d R i c h te r. T h e t r u n c a t e d e x p o -n e n t i a l l a w w h i c h w e w i l l s t u d y a s w e l l w a s p r o p o s e d a n d s t u d i e d o n l y m u c h l a t e rsee Corne l l and V anm ark e , 1969 ; Co sen t ino an d L uz io , 1976; s ee a l so Ber r i l l and

D a v i s , 1 98 0 , f o r a d e r i v a t i o n f r o m t h e m a x i m u m e n t r o p y p r i n c ip l e ) .D u e t o t h e a s s e s s e d p ra c t ic a l i m p o r t a n c e o f t h e l a w, m u c h a t t e n t i o n h a s b e e n

d e v o t e d t o t h e e s t i m a t i o n o f t h e p a r a m e t e r s a a n d b , b u t i n s p i te o f t h e g e n e r a li n t e r e s t f o r t h e r e l a t io n , t h e q u e s t i o n o f h o w m u c h t h e p a r a m e t e r e v a l u a t i o n i sb i a s e d a s a c o n s e q u e n c e o f m a g n i t u d e u n c e r t a i n t i e s h a s n e v e r b e e n g i v e n a s a ti s fa c -t o r y s o l u t i o n . I n t h i s p a p e r, w e s h a l l p r e s e n t a n o r i g i n a l a p p r o a c h t o t h e p r o b l e ma n d s e e t h a t t h e m a g n i t u d e v a r i a b i l i ty e n t a i l s re c o n s i d e r i n g t h e u s u a l e s t i m a t o r sf o r t h e a c t i v i t y p a r a m e t e r a . I n t h e n e x t s e c t i o n s , w e sh a l l f ir s t il l u s tr a t e o u r m o d e la n d p r o p o s e t h e e s t i m a t o r s a p p r o p r i a t e b o t h f o r b o u n d e d a n d u n b o u n d e d m a g n i t u d ed i s tr i b u ti o n s . T h e n w e s h a ll s h o w t h e v a l i d i t y o f o u r t h e o r y b y m e a n s o f n u m e r i c a ls i m u l a t i o n s a n d d i s c u s s s o m e i m p l i c a t i o n s o n t h e s e i s m i c r is k a n a l y si s .

M GNITUDE UNCERT INTIES

M a g n i t u d e , w h i c h w a s t h e f i r s t q u a n t i t a t iv e m e a s u r e r a t i n g t h e s t r e n g t h o f a ne a r t h q u a k e , i s s ti ll o n e o f t h e m o s t p o p u l a r e a r t h q u a k e d e s c r i p t o r s a n d i s r o u t i n e l yp u b l i s h e d i n e a r t h q u a k e b u l l e t i n s a n d c a t a l o g s . A s a n y o t h e r o b s e r v a b l e q u a n t i t y,m a g n i t u d e is a f f e c te d b y u n c e r t a i n t ie s t h a t m a y b e d i s ti n g u is h e d i n to t h r e e b r o a dc la s ses

1 . sys t ema t i c e r ro r s2 . d i sc re t i za t ion e r ro r s3 . r andom e r ro r s .

S y s t e m a t i c e r r o r s a r e d e r i v e d fr o m i n e f f ic i e n c ie s o f t h e m e a s u r i n g a p p a r a t u s a n df r o m u n c e r t a in t i e s i n th e v a l u e s o f t h e c o n s t a n t s t h a t a r e i n c o r p o r a t e d i n t h ef o r m u l a s o f m a g n i t u d e d e t e r m i n a t i o n t o a c c o u n t fo r r e g i o n a l a n d s t a t i o n l o c ale f f e c t s . G e n e r a l l y, t h e y a r e t r a n s p a r e n t t o t h e b u l l e t i n a n d c a t a l o g u s e r s , t o w h o mu n f o r t u n a t e l y q u i t e s c a rc e m e a n s a r e p r o v i d e d t o c o r re c t t h e m . I n t h e s e c o n d c la s s,w e i n c lu d e t h e u n c e r t a i n t i e s o r i g i n a t e d b y d i s c r e ti z in g t h e m a g n i t u d e , i.e ., b ys u b d i v i d i n g t h e m a g n i t u d e a x i s in t o i n t e r v a l s o f f i n it e l e n g th . T h e y a r e a l w a y s t ob e f a c e d in a n y a p p l i c a t io n s i n c e m a g n i t u d e d a t a a r e g i v e n t o o n e d e c i m a l d i g it a tm o s t a n d t h e r e f o r e a f i n e r r e s o l u t i o n i s c e r t a i n l y a r b i t r a r y. M o r e o v e r, i n t h e q u i t ei m p o r t a n t c a s e o f m a g n i t u d e s r e c o v e r e d fr o m i n t e n s i t y d a t a , o n l y m a g n i t u d e v a l u e sa f in i t e d i s t ance apa r t a r e ava i l ab l e , and magn i tude g roups a s w ide a s 0 .5 to 0 .7 a reusua l ly se l ec t ed .

F r o m r a n d o m e r r o rs , w e m e a n a ll e r r o r s t h a t c a n n o t b e c l a ss i fi e d i n t o th e p r e v i o u sc a t eg o r i es . F o r m u l a s o f m a g n i t u d e d e t e r m i n a t i o n s e e B A t h, 19 7 3) i n c l u d e d i r e c t lym e a s u r a b l e q u a n t i ti e s , s u c h a s th e m a x i m u m d i s p l a c e m e n t o f t h e r e c o rd e d t ra c ef r o m th e r e s t p o s i t io n , t h a t p o t e n t i a l l y m a y b e p e r t u r b e d b y r a n d o m l y a c t i n g f a c to r s .B e y o n d t hi s, t h e r e a r e a t l e a st tw o m o r e r e a s o n s w h y r a n d o m e r ro r s a r e w o r t hc o n s i d e ri n g . T h e m a g n i t u d e v a l u e s i s s u e d in o f f ic i a l p u b l i c a t i o n s a r e g e n e r a l ly m e a nv a l u e s o f a n e m p i r ic a l s a m p l e c o n s i s t in g o f a ll m a g n i t u d e d e t e r m i n a t i o n s p e r f o r m e do n t h e a v a i l a b l e r e c o r d e d s e i s m o g r a m s . S e c o n d , t h e m a g n i t u d e d a t a b a s e s o f t e n

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 3/17

EFFECTS OF MAGNIT UDE ON THE GUTE NBERG RICH TER LAW 683

happen to include magnitude values derived from other magnitude types or fromintens ity on the basis of empirical conversion laws. A magni tude value so dete rminedmust therefore be regarded as a value representat ive of a given distribut ion, and theassociated random error accounts well for the spread in the original magnitudedata.

In this paper, we shall essenti ally restrict our ana lysis to the correction of rand omerrors for purpose of clarity. The generalization of the model in order to deal withdiscretization errors is, however, ready and trivial. Our basic assumption aboutrandom errors is that deviations are normally distributed. This may sound arbitrary,but it is anyway quite reasonable and in agreement with most theoretical trea tmen tsof experimental errors, so that any differe nt choice of the error distribution shouldbe carefully justified. Our model, however, ma y be readily extende d to considerother forms of random noise, although we believe that the most important resultsof the present analysis would remain subs tantially unchanged.

BASIC MODEL

Magnitude m of an earthquake is postulated to be a random variable withprobability dens ity func tion pdf) given by

[ ( m ) = [3X e -~m m~ <-_ m <=m

[ ( m ) = 0 otherwise 2)

where the norma lizat ion const ant }, is defined as follows:

h = N t ( e - ~m l - e - ~ ~ ) - 1 . 3)

If we let m2 --* + ~, t he t run cat ed distr ibution in 2) becomes the u nlimited one. Inequation 2), ml and m 2 are the lower and upper cutoff magnitude, and N t is theexpected total number of earthquakes in the selected basic time period. Notice that,contrary to the conventional practice in probability theory, we prefer normalizing[ ( m ) such as f+ _: f ( m ) d m = N t , since this way definit ion 2) is equivalen t toexpression 1) commonly used by the seismologists. If equation 1) is taken as adensity law, it is easy to see tha t para meter s a an d b are related to )~ and fl as follows

a = log fl},)

b = fl log e).

The cumulative probab ility function cpf) F of the variat e m is the n expressed as

F ( m ) = 0 m < ml

F ( m ) = h ( e - ~ - e -~ m ) m l ~ m ~ m2

F ( m ) = N ~ m 2 < m 4)

consistently with our normalization choice. We observe th at the cumulative GR lawis equivalent to expressing the cpf F, comple mented to N t , i.e., the func tion Nt - F,versus the magni tude in logarithmic scale. Indeed, we see tha t Art - F is byconstruction ju st N r , i.e., the real number of expected earthquakes exceeding m,

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 4/17

1684

a n d t h a t

STEF NO TINTI ND FR NCESCO MUL RGI

log Nt - F ) = log N,. = log[X e -~m e ~ r n 2 ]

wh ich su ff i c i en t ly f a r f rom the u pp e r end , m2 s impl i f i e s t o a li nea r r e l a t ion . O nc o m p a r i n g i t t o 1 ), w e f in d t h a t

a = log X

b = fl log e).

I n t h e f o l l o w i n g s e c t io n s , w e s h a ll d i s t i n g u i s h t h e r e a l n u m b e r o f e a r t h q u a k e s ,N r, f r o m t h e a p p a r e n t n u m b e r, N a , f o r r e a s o n s t h a t w i ll b e f u ll y e x p l a i n e d l a t e r o n .

A s a s e c o n d b a s i c a s s u m p t i o n , w e t a k e t h e m a g n i tu d e a s b ei n g p e r t u r b e d b y ar a n d o m n o i se , n, t h a t f o l lo w s a n o r m a l d i s t r i b u t i o n w i t h z er o m e a n a n d s t a n d a r d

dev ia t ion z , i.e ., w i th pd f g iven by

g~ n) - 1 e_ ~2/2~2). 5)

T h e o b s e r v e d a p p a r e n t m a g n i tu d e ,y , i s t h e t r u e m a g n i t u d e , m , d i s t o r t e d b y t h es t o c h a s t i c a l l y i n d e p e n d e n t n o i s e , n, a s f o l lo w s

y = m + n 6)

a n d r e s u l t s , t h e r e f o r e , t o b e a r a n d o m v a r i a b l e w i t h c p fH y ) e x p r e s s e d b y L l o y d ,1980)

H y ) = f~ m 2 [ [ m ) f _: -m g ~ n )d n ] dm . 7)

A f t e r s u b s t i t u t i n g d e f i n i t io n s 2 ) a n d 5 ) i n e q u a t i o n 7 ) a n d p e r f o r m i n g s o m ec u m b e r s o m e c a lc u l a ti o n s , t h e c p f H m a y b e r e w r i t te n a s fo l lo w s

XH y) = -~{u ml) - u m2 ) + e~2-~YIv ml) -v m2)l}. 8 )

H e r e ,

u x) = e -~{ 1 + ¢[O x)]} ,

¢ X) = ~ e -r2 dr

i s t h e e r r o r fu n c t i o n s e e A b r a m o w i t z a n d S t e g u n , 1 9 7 0 ), a n d

o x ) - y - x

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 5/17

E F F E C T S O F M A G N I T U D E O N T H E G U T E N B E R G - R I C H T E R L A W 1685

E v e n t u a l l y,

v x ) = ¢ [ ~ - O x ) ]

a n d

,~2 ~ 20 2- 9 )

2

i s a c o n s t a n t t h a t w i ll b e s h o w n t o p l a y a n i m p o r t a n t r o le in o u r a n a ly s i s . T h e p d fh m a y b e o b t a i n e d b y d i f fe r e n t i a t i n g e q u a t i o n 8 ) a n d r e s u l t s i n t h e f o l lo w i n ge x p r e s s i o n

h y ) = -~ {w ml) - w m 2 ) - e ~ - ~ Y [ z m l ) -z m2)]}-Z 10)

w h e r e

w x ) = 2 e -~ x g ~ y - x )

z x ) = f l y x ) + 2 g ~ ~ a 2 + x - y ) .

E x p r e s s i o n s 8 ) a n d 1 0 ), t o g e t h e r w i t h d e f i n i ti o n s 3 ) a n d 9 ), a r e t h e b a s i ce q u a t i o n s o f o u r t h e o r y a n d w i ll b e d i s c u s s e d i n d e t a i l i n t h e n e x t s e c t io n s .

H e r e , w e o n l y o b s e r v e t h a t t h e l e s s t h e r a n d o m n o i s e p e r t u r b s t h e r e a l m a g n i t u d e ,m , t h e m o r e t h e a p p a r e n t m a g n i t u d e , y, a p p e a r s t o b e d i s t r i b u t e d l i k e m . I t m a yi n d e e d b e e a s i l y v e ri f ie d t h a t

l im H y ) = F y )a-- )O

l i m h y ) = f y ) .~-.o 11)

A s a la s t r e s u l t o f t h i s s e c t i o n , it is w o r t h w h i l e t o s h o w t h a t t h e b a s i c e q u a t i o n s 8 )a n d 1 0 ) m a y b e s i m p l i fi e d i n t h e l i m i t in g c a s e o f a n u n b o u n d e d m a g n i t u d e

d i s t r ibu t ion a s fo l lows

tet~mll im H y ) =

m2-~ ~ 2

te~mll im h y ) - - - -

m~+~ 2

{u ml) + e ~ - ~ Y [ v m l ) - 1]}

{w m l ) - - e ~ - ~ Y [ z m l ) - ~]}.

12)

13)

A P P A R E N T M A G N I T U D E D I S T R I B U T I O N A N A L Y S IS

I n t h i s s e c ti o n , w e sh a l l s t u d y t h e b e h a v i o r o f t h e d i s t r ib u t i o n o f t h e a p p a r e n tm a g n i t u d e , y, a n d c o m p a r e t h i s w i t h t h a t o f t h e r e a l u n d i s t u r b e d m a g n i t u d e , m . W eh a v e c o m p u t e d o u r s o l u t i o n s f o r d i f f e r e n t v a lu e s o f fi a n d a . I n F i g u r e s l a a n d 2 a ,t h e c o m p l e m e n t e d c u m u l a t i v e p ro b a b i l i ty f u n c t i o n s c c p f) , d e f in e d a sN t - H y )

a n d a s N t - F y ) r e spec t ive ly, a r e d i sp l ayed in loga r i thmic sca l e fo r /~ = 1 .5 and

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 6/17

1686 S T E FA N O T I N T I A N D F R A N C E S C O M U L A R G I A

= 2.0. Figures lb and 2b show the related pdf s h y ) and f y ) in the same scale.In all cases, the domain of the real magnitude distribution ranges between valuesml = 3.5 and m2 = 7.5. Althou gh the values selected for the p arameter s do not referto any particular real situation, nonetheless, they are reasonable and realistic. Thebehavior of the GR pdf f y ) is rigorously log-linear inside the domain as expected,the value of ~ being relevant in determining the slope of the curves. The relatedccpf is log-linear in most of the domain, departure from log-linearity being evidentonly in the neighborhood of the upper end me. In addition to the GR distribution,

H t

H F I O -

HFIO

~=1.5

. . . . ~ . . ~ . ~ (7 ~ , 6

~ ( T = , 3

2 . 3 . 4 . 5 . 6 . 7 . 8 , Y( a )

N t ~ I O :

N t , l O

~ = 1 . 5

a=.6(T=.5

0 =.3

_ f I

J3 . 4 . 5 . 6 . 7 8 y

( b )

FIG. 1 . a ) Co mp lem ented cumu la t ive p robab i l i ty func t ion ve rsus ma gni tude comp uted fo r /3 = 1 .5 :o r ig ina l t run ca te d m2 = 7 .5) d i s t r ibu t ion dashed line ) ; appa ren t d i s t r ibu t ions fo r va r ious a -va lues so l idl ines ). b ) P robab i l i ty dens i ty func t ion ve rsus ma gni tude co mpu ted fo r /3 = 1 .5 : o r ig ina l t runc a ted rn2 =7 .5 ) d i s tr ibu t ion {dashed l ine ) ; appa ren t d i s t r ibu t ions fo r va r ious a -va lues so l id lines ).

each figure displays three more curves with different a values. As may be seen, theapparent magnitude distributio n progressively deviates from the purely expo nentialone as a grows larger. The most remarkable features are

1. The apparent magnitude, y, assumes values even outside the domain of m.2. The fro nt and rear tails of the d istribution grow along with ~.3. In a large portion of the domain of m, both the ccpf N t - H y ) and the pdf

h y ) resemble a straight line, paralleling to the GR line and progressivelydisplaced upwards as ~ increases.

The first two points are a foreseeable effect of the noise distortion and do not

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 7/17

E F F E C T S O F M A G N I T U D E O N T H E G U T E N B E R G - R I C H T E R L AW 1687

deserve further explanation here. The last one is, however, worth discussing sinceit becomes useful in practical applications. Let us come back to our basic solution(10), or in the case of unbounded distribution to (13), and see how they may beapproximated in the region of log-linearity. By this we mean, in this context, themagnitude interval where the error functions in (10) and (13) approach the limitingvalues +1. If we denote as u a value such that the error function absolute valuet ~b (x) t is suffic ient ly close to un ity as x is la rger t ha n g, then, remembering thearguments of ¢ in (10) and (13), we deduce the following set of inequalities, that

N t l O - ~

N t ~ l O - A

~=2.0

% ~ G = . 6

\_ N t - F

_ H t - H \ ~

3 . 4 , 5 . 6 . 7 . 8 ,{a]

~ = 2 . 0

N ~ ' \

G : . 6

0 : . 5

H t 0 =-

~ t - l O

2. 3 . 4 . 5 . (b} 6 . 7 . 8 . y

FIG. 2. a) The s ame as in Figure la but wit h/3 = 2.0. b) The s ame as in Figure lb, but with/3 = 2.0.

should be simult aneously satisfied:

I O m 2) l > ~ t

1 3: - O m ~ ) l > m

I Y - O ( m 2 ) ] > / x . (14)

The solution of (14) is easily found to be the interval

ml + ~ 2 + avr~ < y < m2 - tLa~/-2. (15)

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 8/17

1688 STEFANO TIN TI AND FRANCESCO MULARGIA

W i t h i n t h i s i n t e r v a l , e x p r e s s i o n s ( 8) a n d ( 1 0) m a y b e w e l l a p p r o x i m a t e d b y

H ( y ) h (e -~m~ - e ~2-~y)

h ( y ) ~ h f l e ~ : -~ y

16)

( i7 )

and a s m2 --~ + 0 the eq ua t io ns co r re spo nd in g to (12 ) and (13 ) be com e

l im H ( y ) ~- N t [ 1 - e ~-a(~-ml)] (18 )r r t~- i -ov

l ira h ( y ) N t f l e ~ 2- ~( y- ml) . (19)m2---. ¢~

P a s s i n g t o t h e l o g a r it h m s , i t is q u i t e e a s y t o f i n d t h e f o l l o w i n g e x p r e s s i o n f o r t h ec c p f

e~2-~y _ e-~-~log (Nt - H ) = log (Nt - F ) + loge_#y _ e_~ , ,~

wh ich su ff i c i en t ly f a r f rom th e up pe r en d m2 (o r equ iva l en t ly a s m2 --~ + oo) m ay b es i m p l y w r i t t e n a s

log (N t - H ) -- log (N t - F ) + -r21og(e). (20)

A n a l o g o u s l y, w e m a y a l s o w r i t e

log h ( y ) ~ log / (y) + ~ /21og(e) . (21)

A s m a y b e s ee n , w e h a v e r e a c h e d t h e r e m a r k a b l e c o n c l u si o n t h a t i n th e l o g - li n e a rr e g io n , t h a t f o r e q u a t i o n ( 15 ) i s c o n t a i n e d i n th e d o m a i n [ m l , m 2 ], t h e a p p a r e n tm a g n i t u d e c c p f a n d p d f a re p r o p o r t i o n a l t o t h e r e s p e c ti v e d i s tr i b u ti o n o f t h e r e alm a g n i t u d e , t h e m u l t ip l y i n g f a c t o r b e i n g e ~ > 1 . F r o m d e f i n i ti o n ( 9) , t h e v e r t i c a ld i s t a n c e b e t w e e n t h e s t r a i g h t p o r t i o n o f t h e c u r v e s i n t h e l o g a r i th m i c g r a p h si s a q u a d r a t i c f u n c t i o n o f t h e d e c a y p a r a m e t e r, fl, a n d o f t h e n o i s e s t a n d a r ddev ia t ion a .

PARAM ETER EVALUATIONT h e m a i n p r o b l e m t o s o l v e is to r e c o v e r th e p a r a m e t e r s o f t h e r e a l m a g n i t u d e

d i s t r i b u t i o n s t a r t i n g f r o m t h e a p p a r e n t m a g n i t u d e , t h e o n l y o n e a v a i l a b l e t o t h einves t iga to r s .

A s r e g a r d s t h e i n f e r e n c e o f f), o r e q u i v a l e n t l y o f b , w e h a v e a l r e a d y o b s e r v e d t h a tt h e p e r t u r b e d d i s t ri b u t i o n s , H a n d h , b o t h d e p e n d o n f~. T h i s m e a n s t h a t t h ee v a l u a t i o n o f fl m a y b e m a d e b y f i t ti n g t h e e x p e r i m e n t a l s a m p l e s w i t h t h e t h e o r e t i c a lc u r v es c o m p u t e d i n th e p r e v i o u s s e c t i o n s b y m e a n s o f t h e u s u a l t e c h n i q u e s o fl in e a r i z e d l e a s t - s q u a r e s o r m a x i m u m l ik e l ih o o d . S i n c e , h o w e v e r, th e a n a l y t i c a le x p r e s s i o n s o f t h e c u r v e s a r e q u i t e c o m p l i c a t e d , t h e r e s u l t in g e s t i m a t i o n p r o c e d u r e ,a l t h o u g h i n p r i n c ip l e c o r r e c t, w o u l d b e t o o s o p h i s t i c a t e d t o b e a p p l i e d r o u t in e l y.T h u s , w e h a v e tr i e d a n o t h e r s o l u t io n , b a s e d o n t h e c l o se r e s e m b l a n c e o f t h e r e a la n d a p p a r e n t m a g n i t u d e d i s t r i b u t i o n s i n t h e l o g - l i n e a r r e g i o n , w h e r e b o t h e x h i b i ta n e x p o n e n t i a l d e c a y g o v e r n e d b y t h e p a r a m e t e r ~ . A s a c o n s e q u e n c e , i f w e r e s t r i c t

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 9/17

EFFECTS OF MAGNITUDE ON THE GUTENBERG-RICHTER LAW 1689

the analysis to the log-linear region where approximations (16) to (19) hold,/~ maybe estimated exactly in the same way as with the pure GR distribution. This meansthat, if the discretization errors are negligible, the classic estimators deduced for acontinuous variate by Aki (1965) and by Page (1968) are still successful even in thepresent framework. Further, if for the available data it were more convenient toassume a discretized variate, then the estimators due to Utsu (1965) and Weichert(1980) for grouped data and studied by Bender (1983) as well as other well-testedtechniques (see Tinti and Mulargia, 1984) would be appropriate. In the following,consistent with the purpose expressed in the section on Magnitude Uncertaintiesof only dealing with random errors, we shall suppose we know the apparentmagnitude values with such an accuracy tha t y may well be taken as a continuousrandom variable and Aki and Page's formulas apply. We remember that, given anexperimental sample, if rh is the average of all magnitudes exceeding a lower cutoff,too, then Aki's estimator appropriate for an u nbound ed distribution is given by

1-~ = r h - t o o . 2 2 )

If the distribution admits the upper bo und M, then Page's relation holds

1 Me-~M-mo)-~ = rh -- mo -- (23)

1 -- e -~(M- m°)

In (22) and (23), ~ is the estima te of ~. Generally, after arranging the experimen tal

data into either a cumulative or a density graph, we may recognize whether anunlimited or a limited distribution is appropriate for the case in hand. Indeed, anapparent magnitude coming from a bounded true magnitude is expected to deviatefrom log-linearity at the largest values, and we observe that the introduction of arandom noise does not add a new, serious problem in this respect. There are,however, some new points just arising from the allowed variabil ity of the magnitudeworth discussing. The first regards the lower cutoff. We know that the truemagnitude distribution can not be satisfactorily ascertained for the smallest earth-quakes because of the finiteness of seismological networks and the incompletenessof the seismic records. At the lowest values, the distribution could be drastically

different from the exponential one, and for example, it could approach a normalcurve as proposed by Lomnitz (1974). This difficulty is not of great concern inestimation lacking random errors, since the analysis may be restricted to the regionof validity of the exponential law, the lower cutoff ml being easily found from theobservations. In our model, where for simplicity we have supposed ml is known andthere are no earthquakes with magnitude m < ml, we find tha t the apparentmagnitude shows a log-linear behavior from a certain value onward. In the moregeneral case, if the true magnitude pd f is assumed to be exponential for magnitudesm = ml and to reasonably continue for magnitudes m < ml, the log-linear regionis expected to still exist and begin at a lower magnitude. Therefore, if we take moto be used in inference formulas (22) and (23), a magnitude value inside the

observed log-linear region, we see th at it plays exactly the same role in our theoryas the lower cutoff ml in the conventional error-free models. Furthermore, anestima te of m~ could even be achieved by using equation (15), although we shall no tmake any attempts in the paper. A more delicate question arises about, the higher

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 10/17

1690 STEFANO TINTI AND FRANCESCO MULARGIA

cutoff. Page's formula (23) requires the knowledge of the upper bound M. As aconsequence, in the following applications, we shall suppose th at it is known, whichin some circumstances may appear as a rather severe requirement. However, if Mwere unknown, the magnitude uncerta inties do not allow us to make a satisfactoryindepend ent estimate of M. For example, the ma ximum observed earthquake, whichin the absence of errors has been shown to be a reasonable choice for M a t least forsufficien tly large samples, would now be cert ainly a bad estimate of M.

Therefore, M should be taken as an unknown parameter of the apparent magni-tude dis tribut ion like fl, and t he simple Page's fo rmula is no longer adequate. In theAppendix, we shall show an example of how to estimate joint ly fi and M.

To prove the validity of the above estimators, we have performed a number ofnumerical experiments where r andom samples, distributed like the variate y, havebeen gene rated all the same size, N = 10,000. The pa rame ters selected are the sameused for the analytical solution shown in the previous section. Figure 3, a and b,refers to the limited dist ribut ion experiment s (m2 -- 7.5) with different true valuesof ft. In each figure, the estimated fl versus mo is shown, the evaluations beingperformed for different a values by means of (23). Figure 4, a and b, refers to theunbound ed distribution case, and Aki's formula has been used. The graphs suggestsome observations

1. In any curve, /3 is underestimated for small too, but from a certain pointonwards, the estimates stabilize around the true value and become acceptable.

2. For a given fl, the stabilization region starts at increasing mo as a becomeslarger.

3. For a given a, the stab ilization region start s at increasing mo as/3 grows larger.4. For the limited distribution case, the stable plateau terminates at a point,

where/3 tends to be overestimated.All of the above features may be readily explained if we consider the extens ion of

the log-linear region given by expression (15) and if we think that, after all, toevaluate/3 is like estimating the slope of a straight line fitting the distributionsNt - H or h in logarithmic scale, given in Figures 1 and 2. As a conclusion of theanalysis concerning the evaluation of/3, we stress that in the plateau region, thedeparture of the curves from the true value is small and compatible with the errorin the es timates. To evaluate a~, we have used the expression given by Shi and Bolt(1982), which corrects a forma l inaccuracy in the work of Aki (1965), and we havefound that all are in the range o f 0.04 to 0.07.

We now devote our attention to the evaluation of the activity parameter, a, andwe shall see that it is particularly biased by the normal noise. One way of estimatinga is based on counting the observed earthquakes exceeding a given magnitude m.We distinguish between Nr, i.e., the real number and Na, i.e., the apparent numberof earthquakes greater than m and recall that only Na is observable. Since Na andNr may be written as Nt - H and Nt - F, the following equation becomes useful toestimate a, provided tha t it is applied in the log-linear region

6 = 6GR -- y21og(e) (24)

where 6oR, defined as

6GR = log N , b m (25)

is the commonly used estimator of a.

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 11/17

E F F EC T S O F M A G N I T U D E O N T H E G U T E N B E R G R I C H T E R L AW 1 6 9 1

2 . 0

?

1 . 5

1 . 0

0

0[]

O O 0 ~

a o . eD • •

B O O *

:

• •

e

= 1 . 5 ; m 2 = 7 . 5

0 0 = . 3

• a = . 5

• 6 = . 6

i I I I I I~ . ~ o ~ . o 6 0

a ) m o

2 . 5

2 . 0

1 . 5

1 . 0

[]El

0

e

O 0o

o

9 o •

D ~ e eo O ~0 0 0 0 0 0

0O O •

e

• O • *

• •• •

= 2 . 0 ; m 2 ~ 7 . 5

a ( 7 = . 3• 7 = . 5

. I 0 = . 6

I f I I I ]

3 . 4 . 0 5 . 0 6 . 0

b ) m 0

F I G . 3. E s t i m a t e s o f ~ p e r f o r m e d f o r d i f f e r e n t v a l u e s o f l o w e r m a g n i t u d e m 0 , i n c a s e o f t r u n c a t e de x p o n e n t i a l d i s t r i b u t i o n . E v a l u a t i o n s a r e m a d e f o r / 3 = 1 . 5 ( a ) a n d / 3 = 2 . 0 ( b ).

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 12/17

692 S T E F A N O T I N T I A N D FR A N C E S C O M U L A R G I A

2 0

1 5

1 0

. 5

2 . 5

D•

• •D •

D • *

- e

I

3 . 5 4 . 0

D D O D O 0 :. . . . ; ; l e m l t ; ; = = ; ~

, ~ = l . 5 ; r n 2 = m

[] 0+ = . 3• a = 5

. 0 = 6

I J I . . . . . I

5 . 0 6 . 0

la) mO

?

2 . 0

1 . 5

1 . 0

+I+0m+ o - - - -

° ~ e W I :~ rl r+l n r l D

[]

D

0 ~ ~

0

~,- /~ = ~).0 ; 1112= ooQ

• ~ u O + - . 3

+ • O ~ . 5

+ * a . ~ . 6

l t 1 I I3 . 5 4 . 0 5 . 0 6 . 0

(b) moF IG . 4 . E s t i m a t e s o f fl p e r f o r m e d f o r d i f f e r e n t v a l u e s o f lo w e r m a g n i t u d e m o , i n c a s e o f u n l i m i t e d

e x p o n e n t i a l d i s tr i b u t i o n . E v a l u a t i o n s a r e m a d e f o r / ~ = 1 .5 a ) a n d / ~ = 2 . 0 b ).

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 13/17

EFFECTS OF MAGNITUDE ON THE GUTEN BERG RICHTER LAW 693

H e r e , u s e h a s b e e n m a d e o f e q u a t i o n s 1 ) a n d 2 0). E q u a t i o n 2 4) s h o w s t h a t t h ec o r r e c t e s t i m a t o r ~ is a l w a y s s m a l l e r t h a n t h e G R e s t i m a t o r aGR a n d t h a t t h e b i a s~ 21 og e ) i s a q u a d r a t i c f u n c t i o n o f fl a n d ~ . I n o t h e r t e r m s , w e c a n s a y t h a t a s a ne f f ec t o f m a g n i t u d e u n c e r t a i n t i e s , t h e o b s e r v e d n u m b e r o f e a rt h q u a k e s , N a , e x c e e d -i n g a g i v e n m a g n i t u d e , m , a p p e a r s , a t l e a s t i n t h e l o g - l i n e a r r e g i o n , al w a y s l a rg e rt h a n t h e r e a l n u m b e r , N r.

IMPLICATIONS ON SEISMIC RISK ANALYSIS

T h e m a i n g o al o f a s ei sm i c r i sk a n a l y s is m a y b e s c h e m a t i c a ll y su m m a r i z e d a st h a t o f e v a l u a t in g t h e e x c e e d a n c e p ro b a b i li t y e p) o f a gi v en g r o u n d m o t i o n p a r a m -e te r e .g ., acce le ra t ion) du r ing a spec i f ied in te rv a l o f t ime . To th i s purpose , thee a r t h q u a k e o c c u r r e n c e m o d e l, t h e s p ac e d i s t r ib u t i o n o f th e s o u rc e s, a n d t h e a t t e n -u a t i o n l a w a r e n e e d e d . A ll t h e i n p u t g i v e n m a y b e d e t e r m i n i s t ic , b u t r e c e n t l ya c c e p t a n c e h a s b e e n g a i n e d b y t h e s t o c h a s t i c a p p r o a c h , w h e r e v a r i a b il i t y i n t h ei n p u t p a r a m e t e r s i s t a k e n i n t o a c c o u n t . I n p a r t i c u l a r, t h e v a r i a b i l i t y i n t h e a t t e n -u a t i o n l a w, i n v o l v in g a c c e l e r a t i o n a n d m a g n i t u d e , a n d i n t h e r e l a t i o n b e t w e e n f a u l tr u p t u r e l e n g t h a n d m a g n i t u d e h a v e b e e n i n v e s t i g a t e d s e e M e r z a n d C o r n e l l, 1 97 3;We i c h e r t a n d M i l n e , 1 9 7 9 ; B e n d e r, 1 9 8 4 a a n d b ) . H o w e v e r, t h e v a r i a b i l i t y i n t h ef r e q u e n c y - m a g n i t u d e r e l a ti o n h a s n o t y e t b e e n i n t r o d u c e d i n t h e s e is m i c ri s ke s t i m a t e s , a n d w e s h a l l s h o w, b y m e a n s o f a s i m p l e e x a m p l e , t h a t t h e e v a l u a t i o n o ft h e c o e f f i c ie n t s o f t h e a t t e n u a t i o n l a w m a y b e a f f e c te d . L e t u s a t t a c k f o r s a k e o fc l a ri t y t h e e l e m e n t a r y p r o b l e m o f a P o i s s o n i a n p o i n t s o u r ce w i t h e x p o n e n t i a le a r t h q u a k e d i s tr i b u t i o n . L e t u s f u r t h e r s u pp o s e , fo l lo w i n g M e r z a n d C o r n e l l 1 9 73 ),t h a t t h e n a t u r a l l o g a r i t h m o f t h e p e a k a c c e l e r a t io n , x , d u e t o a m a g n i tU d e , m ,e a r t h q u a k e , a t a s i t e p l a c e d a d i s ta n c e , R , f r o m t h e s o u r c e , b e g i v e n b y

In x = g R) + km + E 26)

w h e r e e i s a n o r m a l v a r i a t e w i t h s t a n d a r d d e v i a t i o n z ~. T h e d i s t a n c e f u n c t i o n , g R ) ,t h e c o e f f i c i e n t, k , a n d z , a r e t o b e e s t i m a t e d f r o m t h e e x p e r i m e n t a l d a ta . W e o b s e r v et h a t t h e d a t a s e t u s e d f or s t u d y i n g t h e a t t e n u a t i o n l aw m a y d i f fe r i n p r a c ti c e f r o mt h a t o n e u s e d i n d e r i v i n g t h e c o e f f i c ie n t s o f t h e m a g n i t u d e d i s t r i b u t i o n o f t h ee a r t h q u a k e s , s i n c e th e b a si c c r i t e r i u m t o ju d g e t h e h o m o g e n e i t y o f t h e d a t a f o r t h ea t t e n u a t i o n a n d f r e q u e n c y - m a g n i t u d e l aw s a r e d i f f e re n t .

F o r e x a m p l e , t o e v a l u a t e t h e s e i sm i c p o t e n t i a l o f a r e g i o n i.e ., t h e e a r t h q u a k eo c c u r r e n c e r a t e , N r, a n d t h e d e c a y p a r a m e t e r, ~ ) , g e n e r a l l y o n l y t h e r e s t r i c t e ds e q u e n c e o f t h e m a i n s h o c k s g e n e r a t e d in t h a t r e g io n i s t a k e n i n t o a c c o u n t . H o w e v e r ,t o e s t i m a t e t h e p a r a m e t e r s i n t h e a t t e n u a t i o n l a w [ i.e ., g R ) , k , a n d ~ ,) , th ea f t e r s h o c k s a r e a l so c o n s i d e r e d a n d , d u e to t h e p a u c i t y o f t h e s t r o n g - m o t i o nr e c o r d i n g s , t h e d a t a s e t m a y c o n s i s t o f e a r t h q u a k e s o c c u r r i n g in r e g i o n s p o s si b lyd i f f e r e n t i n p o t e n t i a l , b u t s i m i l a r i n a t t e n u a t i o n s e e C a m p b e l l , 1 98 1). F o r t h e s er e a s o n s w e s h a l l a s s u m e t h a t t h e t w o d a t a s e t s a r e d i st i n c t , a n d w e s h a l l r e f e r t ot h e m a s t h e a t t e n u a t i o n a n d t h e m a g n i t u d e d i s tr i b u t io n d a t a s e t s in t h e f o l lo w i n gd i s cu s s io n . I n o r d e r to p e r f o r m t h e e s t i m a t e s i m p l i e d b y e q u a t i o n 2 6), w e a r e f o r c e dt o u s e t h e a p p a r e n t m a g n i t u d e , y, d e fi n e d i n e q u a t i o n 6 ), fo r w e d o n o t k n o w t h et r u e m a g n i t u d e , m . E x p r e s s i o n 2 6) m a y b e p r o v e n t o c h a n g e a s f o ll o w s i n t e r m so f y

In x = g R ) k y ~ k 5

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 14/17

6 9 4 S T E FA N O T I N T I A N D F R A N C E S C O M U L A R G I A

w h e r e 5 i s a re s i d u a l v a r i a t e w i t h p d f p ( 5 ) g i v e n b y

p 5 ) = P m - y [ y ) = f m ; ~ l ) g y - m ; 0 .~ ) /h y ;~1, 0.1).

H e r e , t h e s y m b o lP A I B ) d e n o t e s t h e p r o b a b i l i t y o f o c c u r re n c e o f e v e n t A g i v e nB , a s u s u a l i n t h e p r o b a b i l i t y t h e o r y, a n d t h e d i s t r i b u t i o n s f , g , a n d h a r e g i v e n inexp res s ion s (2 ) , (5 ) , and (10 ) r e spec t ive ly. In t he ab ove fo rmu la , we have ex p l i c i t lyi n d i c a t e d t h e d e p e n d e n c e o n /~ 1 a n d 0.1 t h a t a r e t h e d e c a y p a r a m e t e r a n d t h e n o i s es t a n d a r d d e v i a t io n o f t h e d i s t ri b u t io n o f t h e e a r t h q u a k e s i n th e a t t e n u a t i o n d a t aset .

T h e f ir s t t w o c e n t r a l m o m e n t s o f t h e d i s t ri b u t io n o f 5 m a y b e w e l l - a p p ro x i m a t e din the log - l ine ar reg ion a s fo l lows: E( 5) = - fll0 .12 and 0 .2= o . 1 2 .I f w e i n t r o d u c e t h en e w v a r i a t e p d e f i n e d a s

p = e + k5 + k~ l0 1 2

t h a t h a s t h e r e l e v a n t p r o p e r t i e s o f h a v i n g m e a n z e r o a n d v a r i a n c e 0 .2 g i v e n b y

0.p2 = 0.2 4- k20.z , (2 7)

i t f o ll o w s t h a t t h e a t t e n u a t i o n l a w m a y b e g iv e n th e e x p r e s s i o n

In x = ~ R ) + k y + p (28)

w h e r e

~, R ) = g R ) - ki l l0.1 e . (29)

A n d , o n c o m p a r i n g e q u a t i o n ( 28 ) w i t h e q u a t i o n ( 2 6) , w e e a s il y s e e t h a t t h e f o l l o w i n gr e s u l t s a re e x p e c t e d f r o m a r e g r e s si o n a n a l y s i s o f t h e e x p e r i m e n t a l d a t a

1 . T h e e s t i m a t e o f k is n o t b i a s ed ;2 . T h e e s t i m a t e o f t h e d i s t a n c e f u n c t i o ng R ) i s b i a s ed b y the qu an t i t y -k~10.12,

d e p e n d i n g o n t h e a t t e n u a t i o n a s w e l l a s o n t h e a p p a r e n t m a g n i t u d e d i s tr i b u ti o ncha rac t e r i s t i c s .

3 . T he s t an da rd e r ro r r e su l t i ng f ro m the r eg res s ion i s 0.p wh ich i s j o in t ly d ue tot h e v a r i a b i l i t y i n t h e a t t e n u a t i o n a n d i n t h e m a g n i t u d e , a s w a s a l r e a d y s e e nb y J o y n e r a n d B o o r e ( 1 98 1 ), a l t h o u g h o n a d i ff e r e n t th e o r e t i c a l a p p r o a c h .

L e t u s n o w t u r n t o t h e p r o b l e m o f e v a l u a t i n g t h e e p o f t h e p e a k a c c e l e r a t io n , x .F r o m e q u a t i o n ( 26 ), w e s e e t h a t t h e r e d u c e d v a r i a t ez x ) = [ln x - g R ) ] / k i s t hes u m o f t h e t r u e m a g n i tu d e , m a n d o f t h e n o r m a l v a r i a te1 / k I n e w i t h s t a n d a r dd e v i a t i o n 0.~ = 0.~/k.H e n c e , w e m a y w r i t e

P x > X o ) = P z > t o o ) ~- e ~ 2 ~ } / 2 ) P m > t o o ) 30)

w h e r e m o = Z X o )a n d , a c c o r d i n g t o e q u a t i o n ( 4) ,P m > to o) = 1 - F m o ) / N ri s thee p o f t h e e a r t h q u a k e s w h i c h m u s t b e d e r i v e d f r o m t h e m a g n i t u d e d i s t r ib u t i o n d a t as e t. O f co u r s e , t h e a p p r o x i m a t i o n i n ( 3 0) h o l d s i n t h e l o g - l i n e a r r e g i o n g i v e n b ye q u a t i o n ( 15 ) ( s ee a ls o B e n d e r, 1 9 8 4 a) . N o w w e o b s e r v e t h a t

1 . From equat ion (27) , 0 .z2 = ape - 0.12.

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 15/17

E F F E C T S O F M A G N I T U D E O N T H E G U T E N B E R G R I C H T E R L A W 6 9 5

2. Fr om eq ua t ion (29), mo = rh0- ~ 1 o - 1 2 w he re rh0 = [ lnX o - ~ ( R ) ] / k .3 . F ro m eq ua t ion s (9) and (16),P ( m > t oo ) = e ( ~ / 2 ) P ( y > t oo ).H e r e , P ( y > to o) = 1 - H ( m o ) / N ri s t h e e p o f t h e a p p a r e n t m a g n i t u d e . E q u a t i o n

(3 0) m a y t h e r e f o r e b e r e w r i t t e n s o l e ly in t e r m s o f q u a n t i t i e s d i r e c t l y c o m p u t a b l ef r o m e x p e r i m e n t a l d a t a

P ( x > X o) ~- e(~2/2)(~*~-~12-~2)P(y > tho - f l la~ 2).

I f w e n o w c o n s i d e r t h a t , f o r a n e x p o n e n t i a l d i s t ri b u t io n P ( w ) , w i t h d e c a y p a r a m e t e rfl, t he fo l lowing p rop er ty ho lds

P W l -4-w 2 ) - - e-~P wl),

a t l e a s t s u f f ic i e n t l y f a r f ro m t h e u p p e r e x t r e m e . W e m a y c o n c l u d e

P ( x > X o) a P ( y > th o ) (31)

w h e r e t h e c o r r e c t i o n f a ct o r, a , h a s t h e e x p r e s s i o n

c~= exp{ f l2ap2 /~[fl (o-2+O1 2)- 2/~10 12]} (32)

W e n o t i c e t h a t t o n e g l e c t b o t h t h e a t t e n u a t i o n a s w e ll as t h e m a g n i t u d e v a r i ab i li tyi s e q u i v a l e n t t o a r b i t r a r i l y c h o o s e a c o r r e c t i o n f a c t o r e q u a l t o o n e . O n t h e o t h e r

h a n d , i f t h e t w o d a t a s e t s h a p p e n t o e x h i b i t t h e s a m e d i s t r i b u t io n p a r a m e t e r s (i.e .,fl = /~1 an d a = a l ) , t he f ac to r r edu ces to a = e (~p /2 ). T he above d i scuss ion show st h a t t h e i m p l i c a t io n s o f t h e t r u e m a g n i t u d e u n c e r t a i n t i e s o n t h e s e i sm i c r i sk a n a l y s i sa r e r e l e v a n t , e v e n t h o u g h f o r a f u l l u n d e r s t a n d i n g o f t h e c o n s e q u e n c e s , w e s h o u l dt a k e i n t o a c c o u n t m o d e l s m o r e re a l i st i c t h a n t h e s i m p l e p o i n t s o u r c e a n a l y z e d h e re .

A C K N O W L E D G M E N T S

T h e a u t h o r s w i s h to t h a n k P r o f e s s o r E . B o s c h i f o r s ti m u l a t i n g d i s c u s si o n s a n d D r. G . L e n z i f o r t h ea p p r e c i a t e d c o n t r i b u t i o n t o t h e n u m e r i c a l c o m p u t a t i o n s i n t h e A p p e n d i x . We a r e a l s o g r a t e f u l t o M .B a c c h e t t i f o r d ra w i n g t h e f i gu r e s a n d M . C . J a n n u z z i f o r ty p i n g t h e m a n u s c i p t .

R E F E R E N C E S

Abramowi tz , M. and I . A . S tegun 61970) .H a n d b o o k o f M a t h e m a t i c a l F u n c t i o n s ,9th ed . , Dover, NewYork .

Aki , K . 1965). M axim um l ike l ihood es t ima te o f b in the fo rm ula logN = a - b m and i t s conf idencel imits , Bul l . Ea r thqu ake Res . Ins t . , Tokyo U niv. 43 ,237-238.

B~th, M. 1973) .In t roduc t ion to Se i smology,Birkh~user Ver lag , Base l , Swi tze r l and .Bender, B . 1983). M axim um l ike l ihood es t ima t ion o f b va lues fo r magni tude g roup ed da ta ,Bul l . Se i sm.

Soc . Am. 7 3 , 8 3 1 - 8 5 1 .Bender, B . 1984a) . Incorp ora t ing acce le ra t ion va r iab i l i ty in to se i smic hazard ana lys i s ,Bull . Seism. Soc.

A m . 7 4 , 1451-1462.Bender, B . 1984b) . Se i sm ic hazard es t im a t ion us ing a f in i t e - fau l t rup tu re mode l ,Bul l . Se i sm. Soc . Am .

74 , 1899-1923 .Ber r i l l , J . B . and R . O . Dav is 1980). M axi mu m en t rop y and the ma gni tude d i s t r ibu t ion ,Bul l . Se i sm.

Soc . Am .70 , 1823-1831 .Cam pbe l l , K . W . 1981). Nea r- source a t t enu a t ion o f peak hor izon ta l acce le ra t ion ,Bul l . Se i sm. Soc . Am.

71 , 2039-2070 .Caputo , M. 1976). M ode l and obse rved se ismic i ty represen ted in a two d imens iona l space ,Ann. Geofis.

29 , 277-288 .

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 16/17

6 9 6 S T E F N O T I N T I N D F R N C E SC O M U L R G I

Caputo, M. (1981). A note on a random stress model for seismicity statistics and ear thquakes prediction,Geophys. Res. Letters8, 485-488.

Carter, J. A. and E. Berg (1981). Relative stress variations as determined by b-values from earthquakesin Circum-Pacific subduction zones, Tectonophysics23, 283-297.

Cornell, C. A. and E. H. Vanmarcke (1969). The major influences on seismic risk, Proc. 4th Wo rld Conf .E a r t h q u a k e E n g . S a n t i a g oChile, 1, 69-83.

Cosentino, P. and D. Luzio (1976). A generalization of the frequency-magnitude relat ion in the hypothesisof a maximum regional magnitude, Ann. Geofis .29, 3-8.

Epstein, B. and C. Lomnitz (1966). A model for the occurrence of large earthquakes, N a t u r e 211,954-956.

Guarnieri Botti, L., V. Pasquale, and M. Anghinolfi (1980/81). A new general frequency-magnituderelationship, PA G E O P H 119, 196-206.

Gutenberg, B. and C. F. Richter (1954). Seismic i ty o f the Ear th2nd ed., Princeton University Press,Princeton, New Jersey.

James, F. and M. Roos (1975). MINUIT, C o m p u t e r P h y s . C o m m .10, 343-367.Joyner, W. B. and D. M. Boore (1981). Peak horizontal acceleration and velocity from strong motion

records including records from the 1979, Imperial Valley, California earthquake, Bull. Seism. Soc.A m . 71, 2011-2038.

Kanamori, H. and D. L. Anderson (1975). Theoretical basis of some empirical relations in seismology,Bull . Seism. Soc. Am.65, 1073-1095.

Karnik, V. (1971). Seismic i ty o f the European AreaReidl Publishing Co., Dordrecht, The Netherlands.Lloyd, E. (1980). Handbood o /Appl icab le Mathemat icsVol. II: Probability Wiley Publishers, Chichester,

England.Lomnitz, C. (1974). Global Tectonics and Earthquake RiskElsev ier Scientific Publishing Co., Amsterdam,

The Netherlands.Main, I. G. and P. W. Burton (1984). Information theory and the earthquake frequency-magnitude

distribution, Bull . Seism. Soc. Am.74, 1409-1426.Makjani6, B. (1982). On the generalized exponential distribution of earthquake intensity and magnitude,

Bull . Seism. Soc. Am.69, 1209-1214.Merz, H. A. and C. A. Cornel l (1973). Seismic risk analysis based on a quadrat ic magnitude-frequency

law, Bull . Seism. Soc. Am.63, 1999-2006.Mogi, K. (1962). Study of elastic shocks caused by the fracture of heterogeneous materials and its relat ion

to earthquake phenomena, Bull . Ear th quak e Res. Ins t . Toky o Univ.40, 125-173.Mulargia, F. and S. Tinti (1983). Relative stress evolution of the Strait s of Messina area in the period

1950-1980 as determined from seismicity, I l Nuo vo Cimento Le t t .38, 405-409.Page, R. (1968). Aftershocks and microaftershocks, Bul l . Se i sm. Soc . Am.58, 323-335.Purcaru, G. (1975). A new magnitude frequency relation for earthquakes and a classification of relat ion

types, Geophys. J.42, 61-79.Scholz, C. H. (1968). The frequency-magnitude relation of microfracturing in rock and its relation to

earthquakes, Bul l. Se i sm. Soc . Am .58, 399-415.Shi, Y. and B. A. Bolt (1982). The standard error of the magnitude-frequency b-value, Bull. Seism. Soc.

A m . 72, 1677-1687.Smith, W. D. (1981). The b-value as an earthquake precursor, N a t u r e 289, 136-139.Tinti, S. and F. Mulargia (1984). An efficient iterative procedure for evaluating the parameters in the

Gutenberg-Richter relationship, Boll. Geofis. Teor. Appl.101-102, 29-34.Utsu, T. (1965). A method for determining the value of b in a formula log n = a - bM showing the

magnitude-frequency relation for earthquakes, Geophys. Bul l . Hokkaido Univ.13, 99-103.Weichert, D. H. (1980). Estimation of the earthquake recurrence parameters for unequal observation

periods for different magnitudes, Bull . Seism. Soc. Am .70, 1337-1346.Weichert , D. H. and W. G. Milne (1979). On Canadian methodologies of probabilistic seismic risk

estimation, Bul l . Se ism. Soc . Am .69, 1549-1566.Wyss, M. (1973). Towards a physical understanding of the earthquake frequency d istribution, Geophys.

J. R. Astr. Soc.31,341-359.

DIPARTIMENTO DI FISIC

SETTORE DI GEOFISICAUNIVERSITA DI BOLOGNAVIALE BERTI PICHAT 840127 BOLOGNA, ITALY

Manuscript received 16 November 1984

8/13/2019 BSSA_1985

http://slidepdf.com/reader/full/bssa1985 17/17