Miscellanea 727 ANNALS OF GEOPHYSICS, VOL. 51, N. 4 August 2008 Key words seismicity model – information gain – Gutenberg-Richter relation – Kanto – Japan 1. Introduction A number of researchers (Utsu, 1977, 1982; Rhoades and Evison, 1979; Aki, 1981; Hama- da, 1983; Grandori, et al., 1988) have formulat- ed expressions of earthquake probabilities based on precursory anomalies detected by multidisciplinary observation. Their formulas assume that different precursory phenomena occur independently of each other. In such a case, the probability expected from detecting multiple precursory phenomena is given by the product of probability gains for respective ob- servations and the probability estimated from secular seismicity. Imoto (2006, 2007) extended their formulas to cases in which precursory anomalies are ob- served as continuous measurements. He further considered the effects originating from mutual correlations between two precursory anomalies, where he assumed two distributions for each precursory parameter: those associated with on- ly space-time volumes in the vicinity of target events (conditional density distributions) and those associated with space-time volumes ex- cluding target events within a short distance (background density distributions). Assuming normal distributions for the conditional and background densities from each discipline, he obtained analytical solutions for the general case in which mutual correlations exist among precursory anomalies in both distributions. In this paper we attempt to build statistic models for earthquake probability in Kanto, central Japan, based on three parameters: the a- Performance of a seismicity model based on three parameters for earthquakes (M ≥≥ 5.0) in Kanto, central Japan Masajiro Imoto National Research Institute for Earth Science and Disaster Prevention, Ibaraki-ken, Japan Abstract We constructed a model of earthquakes (M ≥ 5.0) in Kanto, central Japan, based on three parameters: the a and b values of the Gutenberg-Richter relation, and the ν- parameter of changes in mean event size. In our method, two empirical probability densities for each parameter, those associated with target events (conditional density distributions) and those not associated with them (background density distributions), are defined and assumed to have a normal distribution. Therefore, three parameters are transformed by appropriate relations so that new pa- rameters are normally distributed. The retrospective analysis in the learning period and the prospective test of testing period demonstrated that the proposed model performs better by about 0.1 units in terms of the informa- tion gain per event than the value summed up with those of the three parameters. The results are confirmed by a simulation with randomly selected model parameters. Mailing address: Dr. Masajiro Imoto, National Re- search Institute for Earth Science and Disaster Prevention, Tennodai 3-1, Tsukaba-shi, Ibaraki-ken, 305-0006, Japan; tel: +81-29-863-7594; fax: +81-29-863-7876; e-mail: imo- to@bosai.go.jp Miscellanea 9-03-2009 14:42 Pagina 727 728 M. Imoto and b- values of the Gutenberg-Richter relation and a parameter representing changes in mean event size. We evaluate model performance in terms of information gain per event (IGpe). We estimate IGpe for the parameters and combina- tions of the parameters, which are compared with values estimated from data in both the learning period and a testing period. 2. Method The hazard function is expressed as the ex- pectation of the number of earthquakes in a space-time volume dx (Daley and Vere-Jones, 2003). We consider unconditional and conditional probabilities of observing a parameter value of θ, which are represented by g(θ)dθ (the «back- ground» density) and f(θ)dθ (the «conditional» density) and are empirically determined with random samples in the whole study volume and samples conditioned on occurrences of earth- quakes. The hazard function at a space-time point, x, conditioned on a value of θ , is given by (2.1) where m0 is the number of target events and V0 is the space-time volume for study. Taking the Poisson model as the baseline, the information gain per event (IGpe : Daley and Vere-Jones, 2003; Imoto, 2004) for a large number of target events is given by (2.2) where the integral is performed within the whole space of θ defined, R. The above equation represents the fact that IGpe is equivalent to the Kullback-Leibler quantity of information expressing the distance between two probability distributions. Assum- ing that f(θ) and g(θ) are normal distributions of multi-variables, Imoto (2007) derived an ana- lytical form to estimate the IGpe value. Hereafter referring to Imoto (2007), the main results of previous works will be intro- duced for the sake of convenience in the later ( ) ( ) ( ) ,lnIGpe f g f d R θ θ θ θ= # ( | ) g ( ) f ( ) ,x x xh d V m d 0 0θ θ θ = application of the present study. For a single pa- rameter θ1, the IGpe value for θ1, IGpe(θ1), can be represented as follows: (2.3) where, µ1 and σ12 are the mean and variance of f(θ1) and those of g(θ1) have been fixed at 0. and 1. 2.1. Correlations in neither the background nor the conditional distributions Here, we consider n variables θ1, θ2, ...θn possessing joint density distributions f(θ1,θ2, ...θn) and g(θ1,θ2, ...θn) and their marginal distri- butions of θi are noted as fi(θi) and gi (θi) for the conditional and background distributions. If variables θ1, θ2, ...θn are mutually independent in both distributions and normally distributed, that is, (2.4) (2.5) the following equation can be obtained, where N(µi, σi2) refers to the normal distribution of the mean µI, and variance σi2 . (2.6) 2.2. Correlated conditional distribution We assume here that the correlation among the n variables θ1,θ2, ...θn can be observed only in the conditional density distribution f(θ1,θ2, ..θn) and that the covariance matrix C can be ex- pressed as (2.7) ( , , , ) 2 ( ) 1 2 1 ( ) ( ) C C det exp f 1 2 1 n n t gθ θ θ π θ µ θ µ = - - --& 0 ( , , ... ) lnIGpe n 2 1 1 2 2 2 1 2 n n n n 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 $ $$ $$$ $$$ θ θ θ σ σ σ σ σ σ µ µ µ = + + + + + + + + + - ( , , ... ) ( ) ( , )g g N 0 11 2 n i i i n i n 1 1 θ θ θ θ= = = = % % ( , , ... ) ( ) ( , ),f f N1 2 n i i i n i i i n 1 2 1 θ θ θ θ µ σ= = = = % % ( ) ,lnIGpe 2 1 1 2 1 1 1 2 1 2 1 2 θ σ σ µ = + + - Miscellanea 9-03-2009 14:42 Pagina 728 729 Performance of a seismicity model based on three parameters for earthquakes (M ≥ 5.0) in Kanto, central Japan and (2.8) where the superscript -1 refers to the inverse of a matrix, ρij the correlation coefficient between θi and θj. By introducing an appropriate transforma- tion of the coordinate system with an orthogo- nal matrix S, the covariance matrix can be ex- pressed in a diagonal matrix. At the same time, the vector µ is transformed into µ′ with the same matrix. Referring to the previous case, the IGpe val- ue is represented by (2.9) where trace denotes the sum of the diagonal el- ements and is an invariant parameter for a uni- tary transformation and λi2 (i=1,2,..n) eigen values. 2.3. Correlations in both distributions Now we consider a general case in which some correlations among parameters are ob- served in both distributions. It is assumed that the marginal distribution of each variable θi can be expressed in the form of a normal distribution and that correlations among variables are observed in the background distribution. Let the correlation coefficient between θi and θj be γij, thus the joint density distribution for the background distribution, g(θ1,θ2,...θn), is given as follows: (2.10) ( , , , ) ( )B B det exp g 2 1 2 1 n n t 1 2 1 gθ θ θ π θ θ = - -& 0 ( , , ... ) ( ) ( ) ' ln ln det IGpe n trace n C C 2 1 1 2 2 2 2 1 1 2 2 2 1 2 ' ' ' n n n n 1 2 2 2 2 1 2 2 2 2 1 2 2 2 2 2 $ $$ $$$ $$$ θ θ θ λ λ λ λ λ λ µ µ µ µ = + + + + + + + + + - = = + + - [ ]C 1 2 2 2 2 ij i j ij i j n j σ σ ρ σ σ ρ σ σ σ = J L K K K KK N P O O O OO where matrix B is the covariance matrix (equiv- alent to the correlation matrix in this case), rep- resented by (2.11) By operating successively appropriate transformations of the coordinate system with an orthogonal matrix and a diagonal matrix, the covariance matrix can be expressed in a unit matrix. The compatible transformations are ap- plied to covariance matrix C, and vector µ. Af- ter these transformations, the IGpe value can be estimated by eq. (2.9). Once we have estimated the means and vari- ances of the parameters together with the corre- lation matrices for both the conditional and background distributions, we can represent the hazard function of the combined model by sub- stituting eqs. (2.7) and (2.10) into eq. (2.1). This function estimates the hazard rate at any point of interest conditioned on the parameter values observed at the respective point. 3. Seismicity model based on three parameters As an application of the above formula, we attempt to build a seismicity model for moder- ate and larger earthquakes (target events M≥5.0) in Kanto, central Japan, based on three parameters: the a- and b- values in the Guten- berg-Richter relation (GR relation) and a pa- rameter representing temporal change in mean event size, ν- value (Imoto 2003). The latter pa- rameter is obtained from the difference between two mean event sizes, long-term and short-term mean. The long-term mean is defined as a sim- ple mean size of earthquakes (M≥2.0) within 20km of the point over a period of 960 days pri- or to the assessment. The short-term mean is calculated in the same way as the long-term one except that an exponentially decaying weight with a time constant of 400 days is used. These space and time windows are reasonably select- 1 1 1 B , ij ij j c c = J L K K K KK N P O O O OO Miscellanea 9-03-2009 14:42 Pagina 729 730 M. Imoto ed with careful consideration of characteristic features of seismic activity in Kanto, and opti- mization of respective parameters (Imoto, 2003). The ν-value could be considered a sort of b-value short-term variation since the mean event size is inversely proportional to the b-val- ue. In this study, we use the hypocenter param- eters for the period from 1980 to 2006 located by the Kanto-Tokai network operated by the National Research Institute for Earth Science and Disaster Prevention (NIED). Taking a bal- ance between the duration of the catalogue and stable estimation, a longer time window, 3650 days is selected for the a- and b- values. The same spatial window and cutoff magnitude as that of the ν- value are used so as to simplify conditions. The cut-off magnitude is selected at 2.0 since the magnitude-frequency relation of the earthquakes in the present study volume ex- hibits a linear relationship between size and the cumulative number of earthquakes down to a magnitude of 2.0 (Imoto and Yamamoto, 2006). We surveyed the a-, the b- and the ν- values within the study volume (200 x 200 x 90km3; Imoto and Yamamoto, 2006) from January 1990 to December 1999 for the region. In this specified space-time domain, clustering fea- tures of target earthquakes are not observed. The assessment was made at 2km-spaced points and at 10-day intervals. To ensure a reliable dis- tribution, we selected only those points of as- sessment with both 100 or more earthquakes for 3650 days and 20 and more for 960 days. We classified the selected points into two categories, points of conditional distribution and points of background. A point of condition- al distribution was defined as the grid point of assessment nearest to a target in space and im- mediately before it in time. This definition of a point belonging to the conditional distribution may be restrictive and other definitions could be explored. Although a study from this view- point would be important, it is beyond the scope of the present study. Only the present definition is considered in the study. The other grid points are classified as background. We thus obtained two distributions, the conditional distribution from 33 samples and the background distribu- tion from about 6.6x107, for each of the a-, b- and ν- values. Figures 1a, 1b and 1c illustrate the empirical background (solid line) and con- ditional (gray line) distributions for a- b- and ν- values. We have already proposed two types of seis- micity model, one based on the ν- value (Imo- to, 2003) and the other on the a- and b- values (Imoto, 2006). The hazard functions, hab(x| a,b) and hν(x| ν) are represented as (3.1) and (3.2) where B denotes the Beta function, and a1, b1, b2, c0, c1 and c2 are model parameters that have been optimized by the maximum likelihood method in the point process analysis. The hazard function of eq. (3.1) is not mo- notonic and takes its maximum at b=b1. Equa- tion (3.2) is similar to this. In transforming the b- and n- parameters into new ones with a nor- mal distribution, we make the hazard function be monotonically increasing with a transformed parameter. We assumed that the background distribu- tions of the a-, b- and ν- values are respectively represented by an exponential function (dashed line in fig. 1a), a normal function (dashed line in fig. 1b), and a Beta function (dashed line in fig. 1c), which are fitted by the maximum like- lihood method. Therefore, a- b- and ν- values are transformed into the a’, b’ and ν’ values as follows: (3.3) (3.4) where b0 and s denote the mean and the stan- dard deviation of the b-value in the background . s e db s e e dx 2 1 2 1 2 1 ( ) ( ) ' s b b b b s b b b x 2 2 2 2 0 2 2 0 2 2 π π π + = = 3 3 3 - - - - + - - - - # # # ( ) . exp a a a da e dx 2 1 2 1 2 2 1 0 0 ' a a x 2 2 2 π - - - - = = 3- - & 0# # ( | ) ( ( ) | , ) h x c x c c0 1 2ν νΒ=o ( | , ) 10h x a b a1 ( ) ab a b b b1 2 2 $= - - Miscellanea 9-03-2009 14:42 Pagina 730 731 Performance of a seismicity model based on three parameters for earthquakes (M ≥ 5.0) in Kanto, central Japan 0 0.2 0.4 0.6 0.8 1 2 2.5 3 3.5 4 Background Conditional a-value 0 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 b-value Background Conditional 0 0.2 0.4 0.6 0.8 1 0.4 6.05.0 ν-value Conditional Background Fig. 1a-c. Cumulative background and conditional distributions for the parameters. (a) Empirical background distributions for the a- value are plotted as a black solid curve, and conditional distributions as a gray step line. An exponential distribution is fitted to the background distribution (dashed curve). (b) Same as fig. 1a but for the b- value. A normal distribution is fitted to the background distribution (dashed curve). (c) Same as fig. 1a but for changes in mean event size. Beta function is fitted to the background distribution (dashed curve). a b c Miscellanea 9-03-2009 14:42 Pagina 731 732 M. Imoto distribution, and b+ and b- are related to the b- value by the relation: , (3.5) (3.6) where c3 and c4 are the maximum likelihood es- timates of the model parameters for the B func- tion fitted to the background distribution of the ν- value. The upper limit of the integral of the first term on the left side in eq. (3.6), ν- and the lower limit of the second term, ν’+ satisfy the re- lations as follow: (3.7) Figures 2a, 2b and 2c illustrate the back- ground distribution in black and the conditional distribution in gray for a’, b’ and ν’. Normal distributions fitted to the conditional and back- ground distributions are indicated with dashed lines in gray and black. These figures indicate that two distributions in each set are more or less appropriately approximated with normal distributions. The parameters of these normal | , | , | , , , c c c c c c 1 2 1 2 1 2 – –$ ν ν ν ν ν Β Β Β = = = + + ^ ^ ^ h h h ( | , ) ( | , ) , y c c dy y c c dy e dx 2 1 x 3 4 0 3 4 1 2 2 π Β Β+ = = 3 o o o - - - + # # # ,b b b b b b b b2 2 2 2 2 2 – –$- = - = -+ +] ] ]g g g distributions are summarized in table I. The standard deviations of these estimates for the conditional distributions are given below in parentheses since those of the background dis- tributions are much smaller. This could be jus- tified as follows. The sample size of the back- ground distributions is by far larger than that of the conditional distributions. The transforma- tions have been performed so the background distributions follow a normal distribution with mean 0 and variance 1. Only relative values of the conditional distribution to the background distribution are involved in calculating IGpe. Weak correlations among the three parameters are observed for the background distributions. Tables IIa and IIb summarize the correlation matrices in the background and conditional dis- tributions. In the latter table, the standard devi- ations of the correlation coefficients in the con- ditional distributions, which are estimated from Fisher transformation (Imoto, 2007), are given in parentheses. It can be concluded that at least the correlation between a′- and ν′- values is significant. This suggests that the formula of Aki and others is not applicable to the present case. 4. Results and discussion The last column in table I indicates IGpe for each parameter, where both distributions are as- Table I. Terms of normal distributions for each parameter and its IGpe value. The standard deviations for the estimates in the conditional distributions are given in parentheses. Background Conditional IGpe Av Std Av Std a′ 0 1 0.964 1.063 0.47 (0.180) (0.136) b′ 0 1 0.669 0.986 0.22 (0.172) (0.128) ν′ 0 1 0.283 0.649 0.18 (0.115) (0.088) Miscellanea 9-03-2009 14:42 Pagina 732 733 Performance of a seismicity model based on three parameters for earthquakes (M ≥ 5.0) in Kanto, central Japan 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 Background Conditional a´ -value 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 Background Conditional b´-value 0 0.2 0.4 0.6 0.8 1 -4 -3 -2 -1 0 1 2 3 4 Background Conditional ν´ -value Fig. 2a-c. Cumulative background and conditional distributions for the transformed parameters. (a). Empirical background distribution for the transformed a- value is plotted as a black solid curve, and the conditional distri- bution as a gray step line. Their fitted normal distributions are plotted as dashed lines in black and gray. (b) Same as fig. 1a but for the b- value. (c) Same as fig. 1a but for changes in mean event size. a b c Miscellanea 9-03-2009 14:42 Pagina 733 734 M. Imoto sumed to be normally distributed with the pa- rameters in the respective columns. If the three parameters are not correlated in the two distri- butions, the resultant IGpe value, which is the case of Aki (1981) and others, is given by the summation of each IGpe value in the second to the last column. In the actual case, some corre- lation is observed in both distributions, as indi- cated in table IIa (background distribution) and table IIb (conditional distribution). Following the procedure given in the previous sections, we can calculate an IGpe value expected from a model combining three parameters. The first row of table III lists the IGpe values thus ob- tained. In order to confirm the validity of IGpe es- timates, we estimate the probability gain of each target event directly from the hazard func- tion given in eq. (2.1) for each parameter. We can obtain the hazard function of the combined model by substituting eqs. (2.7) and (2.10) into eq. (2.1) with the correlation matrices. Here- after, the model with the parameters fixed is re- ferred to as the combined model. Using this hazard function, the retrospective analysis has been performed for data from 1990 to 1999 (learning period) and the forward test done for that from 2000 to 2006 (testing period). The re- sults obtained are listed in the second and third rows in Table III. Comparing each IGpe value in the second row to the corresponding value in the first row, we can conclude that the parame- ter estimation and the model construction are consistent. Results obtained from data inde- pendent of model construction (testing period) indicate some gaps between expectation and observation. A gain of 0.51 units and a loss of 0.09 units in IGpe value from expectation are observed for a′- and b′- values. These gaps re- sult in gains of 0.48 units for the simple sum- Table IIa,b. Correlation matrix (a) Observed in background distribution (b) Observed in conditional distribution. The standard deviations are given in parentheses. a′ b′ ν′ a′ 1.000 0.049 0.108 b′ 0.049 1.000 0.079 ν′ 0.108 0.079 1.000 a′ b′ ν′ a′ 1.000 -0.215 -0.402 (0.225) (0.201) b′ -0.215 1.000 -0.022 (0.225) (0.236) ν′ -0.402 -0.022 1.000 (0.201) (0.236) Table III. Summary of the IGpe values. Those expected from the formula are listed in the first row. Those cal- culated as average values of probability gains for each target event are listed in the second row for the learning period and third row for the testing period. Those obtained by simulation are listed in the last row, where the standard deviations of these estimates are given in parentheses. The sum in the third to last column indicates the simple summation of the three IGpe values. The IGpe value of the combined model is given in the second to last column. The difference between these two values gained by an effect of correlations is given in the last column. Events a′ b′ ν′ Sum_up Combined Diff. Estimation 0.47 0.22 0.18 0.88 0.98 0.10 Learning Period 33 0.43 0.22 0.17 0.82 0.98 0.16 Testing Period 37 0.98 0.13 0.25 1.36 1.41 0.05 Simulation 0.49 0.25 0.21 0.95 1.16 0.21 (0.17) (0.12) (0.10) (0.23) (0.30) (0.20) Miscellanea 9-03-2009 14:42 Pagina 734 735 Performance of a seismicity model based on three parameters for earthquakes (M ≥ 5.0) in Kanto, central Japan mation and 0.43 units for our proposed com- bined model. Even in this case, it is notable that the combined model performs better than sim- ple summation. To confirm the above results, we estimated each IGpe value and its standard deviation us- ing 1000 sets of randomly generated samples. In generating a set of samples, we only consid- er the variations of parameter values in the con- ditional distributions, which are given in tables II and IIb. For each set of samples, we calculate values corresponding to each IGpe in table III, in just the same way as that of the first row. The means and standard deviations are estimated from the 1000 acquired sets of IGpe values and listed. The values in the first row are more or less similar to the respective values in the last row. A relatively large gap is observed in the combined case, the second to the last column, but it is still in the range of the standard devia- tion. The difference in the IGpe value in the last column suggests that the estimate by the simu- lation is no smaller than its standard deviation, and means that the IGpe value produced by the combined model is probably larger than that es- timated from the formula by Aki and others. It should be noted again that the application of the latter formula is no longer appropriate in the present case with a significant correlation in the conditional distribution between two parame- ters. Accordingly, it can be concluded that the combined model proposed in the present paper is more appropriate than that formulated by Aki and others, from the viewpoints of its perform- ance and assumption of independency. 5. Conclusions This paper introduces a way to combine multi-disciplinary observations into one hazard function and demonstrates its superior perform- ance to that expected from the well known for- mula of Aki and others. The combined model was confirmed using actual data taken from both the learning period and the subsequent testing period. The quantitative matching be- tween the IGpe values predicted by our formu- la and those observed for the learning period is reasonable and demonstrates only the logical consistency in our derivations. However, this is not the case for the data taken from the testing period. The combined model performs only 0.05 units better than Aki’s formula, but by 0.43 units better than that of the learning period. This gap is primarily because the IGpe value of the a′ parameter for the testing period is 0.51 units greater than that observed for the learning period. This corresponds to an increase of about 1.6 times the probability gain. This change could be interpreted by an increase of this fac- tor in the number of sample earthquakes since such increases raise the hazard rate by the same factor through eq. (3.1) with the baseline (the Poisson rate) resuming the same value as be- fore. The average number of earthquakes condi- tioned on a target event in the testing period ex- ceeds that observed in the learning period by a factor of 1.5. This evidence is consistent with the fact that the seismicity rate of target events in the testing period, 37 events in 7 years, is 1.5 times larger than that in the learning period, 33 events in 10 years. In summary, we attempted to build up a seismicity model based on three parameters, the a- and b- values of the Gutenberg-Richter mag- nitude frequency relation and a parameter of changes in mean event size. Applying the for- mula derived by Imoto (2007) to the data ob- served by the Kanto-Tokai network (NIED), we estimated that the IGpe value of the model be- comes about 0.1 units greater than that of the simple summation formulated by Aki and oth- ers. The model performs consistently with this estimation both in the learning period and the testing period. Acknowlegments This manuscript was greatly improved by the comments of two anonymous reviewers and editor Massimo Cocco. REFERENCES AKI, K. (1981): A probabilistic synthesis of precursory phe- nomena, in Earthquake Prediction, edited by D.W. SIMPSON and P.G. RICHARDS, 566-574, Agu. DALEY, D. J. and D. VERE-JONES (2003): An introduction to the Miscellanea 9-03-2009 14:42 Pagina 735 736 M. Imoto theory of point processes, vol. 1, Elementary theory and methods, Second edition, (Springer, New York), pp. 469. GRANDORI, G., E. GUAGENTI and F. PEROTTI (1988): Alarm systems based on a pair of short-term earthquake pre- cursors, Bull. Seism. Soc. Am, 78, 1538-1549. HAMADA, K. (1983): A probability model for earthquake prediction, Earthquake Prediction Res., 2, 227-234. IMOTO, M. (2003): A testable model of earthquake probabil- ity based on changes in mean event size, J. Geophys. Res., 108, ESE 7.1-12 No. B2, 2082, doi:10.1029/ 2002JB001774. IMOTO, M. (2004): Probability gains expected for renewal process models, Earth Planets Space, 56, 563-571. IMOTO, M. (2006): Statistical models based on the Guten- berg-Richter a and b values for estimating probabilities of moderate earthquakes in Kanto, Japan, in Proceed- ings of The 4th International Workshop on Statistical Seismology, January 9-13, 2006, ISM Report on Re- search and Education, ISM, Tokyo, Japan, 23, 116-119. IMOTO, M. (2006): Earthquake probability based on multi- disciplinary observations with correlations, Earth Planets Space, 57, 1447-1454. IMOTO, M. (2007): Information gain of a model based on multidisciplinary observations with correlations, J. Geo- phys. Res., 112, B05306, doi: 10.1029/ 2006JB004662. IMOTO, M. and N. YAMAMOTO (2006): Verification test of the mean event size model for moderate earthquakes in the Kanto region, central Japan, Tectonophysics, 417, 131-140. RHOADES, D. and F. EVISON (1979): Long-range earthquake forecasting based on a single predictor, Geophys. J. R.astr. Soc., 59, 43-56. UTSU, T. (1977): Probalities in earthquake prediction, Zisin II, 30, 179-185, (in Japanese). UTSU, T. (1982): Probabilities in earthquake prediction (the second paper), Bull. Earthq. Res. Inst., 57, 499-524, (in Japanese). (received July 6, 2007; accepted July 2, 2008) Miscellanea 9-03-2009 14:42 Pagina 736