nearby areas of Norway indicate that the seismic potential may be higher. ..... 1879, where - as for southern Sweden - various spatio-temporal inconsistencies.
Natural Hazards 7: 41-57, 1993. © 1993 Kluwer Academic Publishers.
41 Printed in the Netherlands.
Maximum Likelihood Estimation of Seismic Hazard for Sweden A N D R Z E J K I J K O a, E F T H I M I O S S K O R D A S b, R U T G E R W A H L S T R O M b, and P,~IVI MA_NTYNIEMI c alnstitute of Geophysics, Polish Academy of Sciences, Ksiecia Janusza 64, 01-452 Warsaw, Poland bSeismological Department, Uppsala University, Box 12019, S-750 12 Uppsala, Sweden Clnstitute of Seismology, University of Helsinki, Et. Hesperiankatu 4, SF-O0100 Helsinki, Finland (Received: 31 October 1991; in final form: 22 May 1992)
Abstract. The maximum magnitude, the activity rate, and the Gutenberg-Richter b parameter as earthquake hazard parameters, have been evaluated for Sweden. The maximum likelihood method permits the combination of historical and instrumental data, The catalog used consists of 1100 earthquakes in the time interval 1375-1989. The extreme part of the catalog contains only the strongest historical earthquakes, whereas the complete part is divided into several subcatalogs, each assumed complete above a specified threshold magnitude. The uncertainty in magnitude determination was taken into account. For southern Sweden, the calculations give b-values of 1.04 (0.05) for the whole area south of 60°N and 0.98 (0.06) for a subregion of enhanced seismicity in the Lake V~inern area. For the whole area north of 60° N, the b-value is 1.35 (0.06) and for the seismicity zone along the Gulf of Bothnia 1.26 (0.06). The number of annually expected earthquakes with magnitude equal to or larger than 2.4 [ML(UPP) or MM(UPP)] is 1.8 for the whole southern Sweden, 1.3 for the Lake V~nern region, 3.7 for northern Sweden, and 2.4 for the region along the Gulf of Bothnia. The maximum expected regional magnitude is calculated to 4.9 (0.5) for a time span of 615 years for southern Sweden and the Lake V~inern subregion, and 4.3 (0.5) for a time span of 331 years for northern Sweden and the Gulf of Bothnia subregion. However, several historical earthquakes with magnitude above 5 in nearby areas of Norway indicate that the seismic potential may be higher. Key words. Earthquake hazard (maximum magnitude, activity rate, b-value), Sweden, extreme and complete catalog parts, threshold magnitude, magnitude uncertainty.
1. I n t r o d u c t i o n A l t h o u g h a l o w - s e i s m i c i t y r e g i o n , q u a n t i t a t i v e k n o w l e d g e o f a n a t u r a l h a z a r d such as t h e seismic risk is i m p o r t a n t for S w e d e n f r o m t h e v i e w p o i n t o f t h e s a f e t y o f sensitive s t r u c t u r e s such as, e . g . , d a m s , n u c l e a r p o w e r p l a n t s , a n d d e p o s i t o r i e s for r a d i o a c t i v e w a s t e . T h e risk c a l c u l a t i o n m e t h o d o f K i j k o a n d Sellevoll (1990) c o m b i n e s s c a t t e r e d d a t a o f l a r g e h i s t o r i c a l e a r t h q u a k e s with m o r e c o n s i s t e n t lists o f m o r e r e c e n t e a r t h q u a k e s . This is a s u i t a b l e p r o c e d u r e f o r S w e d e n , w h e r e , for d i f f e r e n t p e r i o d s o f t i m e , s e v e r a l fairly h o m o g e n e o u s s u b c a t a l o g s c a n b e i d e n t i f i e d , e a c h w i t h its o w n t h r e s h o l d m a g n i t u d e . H o w e v e r , in t h e p r e s e n t s t u d y w e a p p l y t h e exact d i s t r i b u t i o n o f a p p a r e n t m a g n i t u d e s as t h e p r i n c i p a l m e t h o d a n d c o m p a r e t h e results with t h o s e using t h e a p p r o x i m a t e m e t h o d o f K i j k o a n d S e l l e v o l l (1990).
42
ANDRZEJ KIJKO ET AL. magnitude
T/
uncertainty
T
TI
---i
~ i
2 Xma×
etc.>. :
Fig. i. An illustration of data which can be used to obtain basic seismic parameters by the applied procedure (after Kijko and Sellevoll, 1990). The applied approach permits the combination of data of the largest earthquakes in historical time, extreme part, with earthquake data assumed to be complete above a given threshold magnitude, complete part. It is possible to incorporate the largest known historical earthquake (Xm~) that occurred before the beginning of the catalog. The complete part can be subdivided into time periods with different threshold magnitudes (rnl, m2, • • .). 'Time gaps' (Tg) can be introduced when data are missing, uncertain or anomalous. The uncertainty in earthquake magnitude is also considered such that the upper and lower magnitude limits are specified (vertical error bars). The assumption is that such an interval contains the true magnitude.
A separation is made for earthquakes south and north, respectively, of 60°N latitude, and the calculations are also performed for two subregions of higher seismic activity, Lake V~inern in southwestern Sweden, and the coast of the Gulf of Bothnia. For each of the four regions, calculations of the b-value, activity rate, and maximum expected magnitude are performed. 2. Outline of Theory In the present procedure, the frequent character of earthquake catalogs to comprise mixed data is accounted for. In general, the extreme part of the catalog contains information on the strongest historical events and the complete part includes more recent observations, instrumental and/or macroseismic. The complete part can be divided into several subcatalogs, each complete above a given magnitude threshold. One or several periods of time can be omitted when seismic records are missing or uncertain. Such a period is here denoted 'time gap'. The errors in the observational data are taken into consideration by introducing the concepts of apparent magnitude and magnitude uncertainty interval. In the discussed model, the uncertainty of earthquake magnitude is specified by two values: x and g. x is the lower and g is the upper magnitude limit. Introducing an apparent magnitude value equal to x - 0.5(x_ + ~), the lower and the upper magnitude limits are equal to x = x - 6 and g = x + 6, where 6 is the measure of the magnitude uncertainty equal to 3 = 0.5 (g - x) (of. Kijko, 1988; Kijko and Sellevoll, 1990). Figure 1 shows an example of how the combined data can be arranged. The parameters to be determined are the maximum regional magnitude, m . . . .
LIKELIHOOD ESTIMATION OF SEISMIC HAZARD FOR SWEDEN
43
the activity rate, A, and the b parameter of the Gutenberg-Richter relation. The method is based on the principle of maximum likelihood, and the basic assumptions of the probability computations are the Poisson distribution of earthquakes with activity rate A and the doubly truncated Gutenberg-Richter distribution of earthquake magnitude x. We introduce the vector 0 = (/3, A), where/3 = b • ln(10). The density and cumulative, doubly truncated exponential distribution of earthquake magnitude can be expressed as follows (e.g. Page, 1968; Cosentino et al., 1977): f(x]m) =/3A(x)/(A1 - A2),
(1)
F(xlm) =
(2)
[A~ - A ( x ) J I ( A i - A2),
where m is the threshold magnitude, A1 = exp (-/3m), A2 = exp (-/3rnm~x), A (x) = exp(-/3x), and magnitude x belongs to the domain (m, mm~). The probability distribution function of the strongest earthquake during the time interval t, conditional on the earthquake existence, is given by
G(x [m0, t)
= exp{- A(m0)t[1 - F(x]mo)]}- exp [ - A(mo)t] 1 - e x p [ - A(m0)t]
(3)
(e.g. Benjamin and Cornell, 1970; Gan and Tung, 1983), where A(mo)= A [ 1 - F(mo Irnmin)] and mo is the 'threshold' magnitude for the extreme part of the catalog. The choice of mo is such that it is less than or equal to the lowest magnitude value in the extreme part of the catalog and at the same time mo t> rnmin, mm~nplays the role of the 'total' threshold magnitude and has a rather formal character. The only condition in the choice of its value is that m ~ cannot exceed the threshold magnitude of any part of the catalog, extreme as well as complete. If the uncertainty of earthquake magnitude is specified by the lower and upper magnitude limits, x, g, the density probability function of the apparent magnitude becomes the convolution of magnitude distribution (1) and the uniform distribution in the range ( - 8 , 8). The density probability function of the apparent magnitude for the discussed uncertainty model then becomes (Kijko and Sellevoll, 1992) f ( x t rn, 8) = C , ( x ] m , 8) /3A(x) A1 - A2'
(4)
where the correction function C~(x [m, 8) is given by {exp[/3(x - m)] - exp(-/38)}/2/38, cf,
for m - 8 ~
