Modification of a method o f Galanopoulos f o r determining earthquake risk R . M A A Z ( * ) R i c e v u t o il 20 M a g g i o 1969 SUMMARY. — A m e t h o d f o r the determination of e a r t h q u a k e risk basing on the e n e r g y - f r e q u e n c y - r e l a t i o n proposed b y Galanopoulos is m o d i f i e d . RIASSUNTO. — N e l l a nota viene modificato il m e t o d o per la determi- nazione del rischio legato ad un t e r r e m o t o , m e t o d o già proposto precedente- m e n t e da A . G. Galanopoulos e basato sulla relazione che intercorre t r a energia e f r e q u e n z a . In the relation lg N = a — bM [1] between earthquake magnitude M and annual number N of earth- quakes of the magnitude class with centre M the coefficient a appears to depend mainly on the energy release in the considered region V with area G, whereas b depends on the mean depth of the focus in /'. After reducing [1] to a general standard area G* the accordingly trans- formed relation [1] informs us about the relative frequency of such earthquakes. Supposing all earthquakes have equal depth, then a measure of the earthquake risk would be found by this. To com- pensate the actual variation of the mean focal depth and the varia- bility of b connected with it Galanopoulos (') proceeds as follows. May lg N' = a0 — b0M [2] ( * ) I n s t i t u t P h y s i k der Erde, Bereich 1 ( G e o d y n a m i k und P h y s i k des E r d i n n e r n ) . Jena ( D D R ) . 112 K . MA AZ substitutes [1] where b0 is a fixed reasonable value, possibly 0.8. The formulas [1] and [2] are connected by postulating that the magnitude M„ corresponds to a certain value N0 in both cases M = M0 * N'(M0) = N(M0) = N0 . [3] Therefore from [1] and [2] follows a« = a b° + ( l — M lg N0 . [4] In the next step Galanopoulos (') gets to the frequency N'* related to the standard area by thinking the earthquake events being distributed homogenously in F and F* cz F, too. That means N'* _ G* N' ~ G whence lg N'* = a'* — b0M , a'* = a0 + lg — . [5] (r Now we consider two regions 71, and F2, the intersection of which need not be empty, with a, ^ at, but. having equal parameters a'*, b a'*, = a'*2 , b, = b2 # b0 . [6] For such regions it holds naturally = Gt N2 G2 [6] helps with [5], [4], [7], and [1] to [ 7 ] 0 = Therefore b0\AToi N 02 N o t _ N 0 2 G, ~ G2 ~7,0 is a quantity which must be chosen uniquely for all regions r and N0 cannot be independent of F. With this condition we get «'* = ~ (a — lg UoG) + lg n0G* as modified measure of earthquake risk in the sense of Galanopoulos. M O D I F I C A T I O N OF A M E T H O D OF G A L A N O P O U L O S 1 1 3 The last term is an arbitrary number independent of a, It and G, suitably defined to zero: >io0 ( , (r \ The values of b0 and G* must be chosen definitly and according to reality, possibly in conformity with Galanopoulos as />„ = 0.8 , G* = 10' km- . The arbitrariness in choosing b„, G* and n0G* can be compre- hended as a foible of the method. Another difficulty is that the in- vestigated region /' cannot be defined without free choice. Moreover, there remains the open question mentioned also by Galanopoulos to which degree the basing relation [1] fits to reality. Furthermore, we consider the difference between a'* and the analogous quantity after Galanopoulos who generally postulates No = n0G = 1 , [10] instead of [8] where N0 is defined in [3] and who, therefore, gets b G* a*=}°a + [11] The difference b„ G 1°' b) * G* to [9] depends on the chosen values b0 and G* and of course on b and G. I n Tab. I in the paper of Galanopoulos, this difference does not exceed the amount of 0.11 in the case of South California where a = 5.25, b - 0.86, G = 29.61 G*, and a* = 3.41, a'* = 3.52. Finally, we can see at a consequence of [10] that this postulate is unsuitable. Let 1\ and I\ be two regions with equal relative quan- tities a* and b, b different from b0: a*i =