Adaptive Kernel Estimation and Continuous Probability Representation of Historical Earthquake Catalogs

Transcription

1 Bulletin of the Seismological Society of America, Vol. 92, No. 3, pp , April 2002 Adaptive Kernel Estimation and Continuous Probability Representation of Historical Earthquake Catalogs by Christian Stock and Euan G. C. Smith Abstract To develop spatially continuous seismicity models (earthquake probability distributions) from a given earthquake catalog, the method of kernel estimation has been suggested. Kernel estimations with a global (spatially invariant) bandwidth deal poorly with earthquake hypocenter distributions that have different spatially local features. For example, a typical earthquake catalog has several areas of high activity (clusters) and areas of low-background seismicity. An alternative approach is adaptive kernel estimation, which uses a bandwidth parameter that is spatially variable. Its performance compared to kernel estimations with spatially invariant bandwidths suggests that (discrete) earthquake distributions require different degrees of local smoothing to provide useful spatial seismicity models. Using adaptive kernel estimation, the (local) indices of temporal dispersion of any earthquake probability distribution can be estimated and used to model the spatial probability distribution of mainshocks. The application of these methods to New Zealand and Australian earthquake catalogs shows that the spatial features (earthquake clusters) in which the mainshocks occurred have been reasonably stable throughout the observation period. The observed regions of higher activity persisted throughout the observation period, and none of these regions decreased to background activity during that time. This suggests that these regions will continue to represent higher risk for the occurrence of moderate to large earthquakes within the next few years or decades. Furthermore, it has been observed that shallow earthquakes are mostly part of temporal sequences (e.g., aftershocks or swarms), whereas the earthquakes within the subduction zones in New Zealand showed only small temporal variation during the observation period. Introduction Probabilistic seismic hazard analysis is, in general, based on an appropriate model for the occurrence of earthquakes in space and time. Such models often use historical earthquake catalogs or other representations of discrete hypocenter distributions. Any seismicity catalog for some region represents a sample in time drawn from an (unknown) parent distribution of seismicity. Thus, any appropriate model for the parent distribution must pass the test that the actual earthquake catalog has a reasonable probability of being drawn from the modeled parent distribution. Our goal is to develop a model that is inferred from the data rather than to construct a model with the help of external parameters, such as the use of prior seismic zoning, which dictates the boundaries of seismic activity in the model. The advantage of such a model is its independence from subjective decisions. Furthermore, external parameters may be wrongly assumed to be controlling the observed seismicity. Also, commonly used seismic zoning has unnatural sharp borders of activity that are not to be expected in actual earthquake occurrence. There are several different ways to develop earthquake occurrence models. The first possibility is the use of parametric regression, which implies that all the parameters needed to develop an earthquake occurrence model are known. Since those parameters are currently not very well known, nonparametric estimation seems to be a better choice. The three most commonly used nonparametric estimations are orthogonal series estimations, kernel estimations, and spline estimations (e.g., Härdle, 1990). In orthogonal series estimation the model is estimated by estimating the coefficients of its Fourier expansion, and in spline estimation the local variation of the model is minimized by the introduction of a roughness penalty. Some work on spline estimation in connection with earthquake occurrence models has been done by Ogata et al. (1991). Also, there are other forms of nonparametric estimations that might be used for the estimation of occurrence models (e.g., Papazachos, 1999). Kernel estimation has been suggested and used in several studies to transform discrete earthquake distributions 904

2 Adaptive Kernel Estimation and Continuous Probability Representation of Historical Earthquake Catalogs 905 into spatially continuous probability distributions (e.g., Vere-Jones, 1992; Frankel 1995; Cao et al., 1996; Woo, 1996; Jackson and Kagan, 1999). In kernel estimation the hypocenters of earthquakes are redistributed in space, where the kernel function and its bandwidth dictate the shape and the amount of the redistribution of each hypocenter. Different kernels produce qualitatively similar results, whereas the choice of an appropriate bandwidth is critical for good results (e.g., Silverman, 1986). For the development of seismicity models, kernel estimation has been mostly used with global (spatially invariant) bandwidths. Due to the spatially clustered nature of earthquake occurrence it has to be expected that the optimal bandwidth is dependent on earthquake frequencies and locations. Hence, a local (spatially varying) bandwidth may give a better representation of the distribution of seismic activity in space. Most earthquake occurrence models developed in the past using kernel estimation suffer from ignoring the rather important effect of the bandwidth, and the resulting occurrence models reflect the particular choice of bandwidth. In this study, adaptive kernel estimation, which is one specific type of kernel estimation using a local bandwidth, has been developed for use with earthquake hypocenter distributions and it is qualitatively compared to kernel estimation with a global bandwidth. (We have developed a quantitative test and used it for comparing the performance of both kernel estimations in Stock and Smith [2002].) The bandwidth has been chosen to be independent of time and earthquake magnitude, which has the advantage of permitting the identification of non-poissonian time behavior and deviations from the Gutenberg Richter magnitudefrequency law in the earthquake data set. Once a spatially continuous data representation has been calculated for a given time interval, the data can be further analyzed for local activity fluctuations in time. For this purpose, local indices of dispersion have been calculated, which enables differentiation between earthquake clusters in time (e.g., aftershocks, swarms), Poisson time behavior and temporal earthquake repulsion (inhibition), which could be caused by stress release (i.e., the absence of earthquakes after a large earthquake). The indices of dispersion can be used to filter the original continuous earthquake frequency representation and produce a spatial representation of mainshock occurrence probabilities, where multiple events (e.g., swarms) are effectively reduced to single events. To demonstrate the performance of this method, it has been applied to New Zealand and Australian historical earthquake catalog. The method has been developed for the use in two dimensions to produce planar seismicity representations based on epicenters. In principle, it should also work for the production of three-dimensional representations using hypocenters, although the method might have to be altered to achieve a similar performance because specific kernels perform differently in different dimensions. Methods Adaptive Kernel Estimation Kernel estimation is commonly used for nonparametric density estimation, namely, to estimate the parent distribution of a sample without the use of a parametric model. It redistributes the sample data using a kernel function, which controls the shape of the redistribution of each data point, and a bandwidth, which controls how much of each data point is redistributed over space it controls the degree of smoothness (i.e., amount of high frequency) of the estimated continuous probability distribution. The two-dimensional kernel model (e.g., Silverman, 1986, p. 76) is given by N 2 i c i g(x) K (x x ), (1) Nc where K is the chosen kernel function, c is the bandwidth, N is the total number of earthquakes, x i is the location of each earthquake, and x is the location in the spatial seismicity representation. Different kernels have been suggested for the construction of representations of earthquake occurrences (e.g., Vere-Jones, 1992; Cao et al., 1996; Woo, 1996; Jackson and Kagan, 1999). In general, the choice of kernel should reflect the properties of spatial earthquake hypocenter distributions, but little research has been done in this field. For example, Woo (1996) suggested the use of studies of the fractal distributions of hypocenters (e.g., Robertson et al., 1995). The detection of fractal distributions is, however, problematic (e.g., Marzocchi et al., 1997). Another problem can be the choice of a rotationally invariant kernel, which assumes that the earthquake activity has no preferred direction. This is not the case for earthquakes located on welldefined long narrow fault systems, for example, plate boundaries. Much more critical than the choice of kernel function is the choice of an appropriate bandwidth (e.g., Silverman, 1986). Bandwidths that are too high lead to very smooth representations, that is, the sample data (earthquake catalog) is redistributed strongly over space. This can lead to blurring of local seismicity clusters (oversmoothing), that is, main features that are present in the sample data cannot be identified in the estimated parent distribution. In representations generated by bandwidths that are too low, very small seismicity features are preserved, even if these features are generated by single earthquakes, which would not be expected to be strongly represented in the parent distribution (undersmoothing). Earthquake occurrence is commonly characterized by areas consisting of local clusters of high activity and other regions consisting of more uniform activity. Such behavior is better modeled with different degrees of local smoothness, which a global bandwidth does not allow for. In the spatial frequency domain, a local bandwidth allows for sharp transitions between high- and low-activity levels in space (i.e., rapidly increasing or decreasing activity over

3 906 C. Stock and E. G. C. Smith short distances), whereas sharp transitions are filtered (smoothed) in regions where the data suggests a more uniform probability distribution (for example, in areas with very few individual earthquakes). Hence, a local bandwidth may give a better representation of the distribution of seismic activity in space. Since the aim of this study is to illustrate the difference between the performances of global and local bandwidths, the choice of kernel is not very important and the commonly used Gaussian function has been selected here: K(x) exp (x x i). (2) 2p 2 Adaptive kernel estimation is a three-step process (see Silverman, 1986, p. 101): N 2 1 (x x i ) 1 2pNc1 i 1 2c1 1. g (x) exp, 2 2 l 2. c 2(x), (3) g (x) 1 N (x x i ) 2 2pNc 1 i 1 c 2(x i) 2c1 c 2(x i) 3. g (x) exp, where c 1 is a global parameter and l is the global mean of earthquake activity per area during the observation period. In step 1 the pilot estimate g 1 (x) is calculated, which is used to determine the local bandwidth c 2 (x) in step 2. The probability distribution of earthquake occurrences g 2 (x) is estimated in step 3, by relating c 2 (x) to the location of each earthquake xi. When the region of interest has no seismic activity on its borders, locations with no activity should be excluded when calculating l, otherwise c 2 (x) is dependent on region size. A c 1 value of 1.0 (in units of x) worked very well for the examples presented in this article and seems to be a good initial choice. For other cases another value may give better results. Other ways of calculating c 2 (x) are possible and may improve the performance of this method. Index of Dispersion A typical, unmodified historical earthquake catalog includes short time activity features such as aftershocks and swarms, which are still present in the continuous representation g 2 (x). For earthquake hazard analysis, it is important to distinguish such sequences from regions of high activity that remain constant throughout the observed time period. To detect temporal activity fluctuations in g 2 (x), the kernel estimates g t (x) of individual years can be compared to the kernel estimate g T (x) of the whole time period T, and the variance can be calculated. The index of dispersion is defined by the variance over the mean: T T t 1 (g (x) g t(x)) 2 2 V T(x) T g (x). (4) In the examples presented in this article each g t (x) is estimated using the local parameter c 2 (x) obtained from the kernel estimate of g T (x) to avoid a spurious time dependency of c 2 (x), that is, c 2 (x) is the same for each individual year. We judge that there are not enough earthquake data in our data sets to estimate a reliable c 2 (x) for each year. For a Poisson process, the expected value of V T (x) is 1 (Johnson et al. 1992, p. 157). Thus, the measured index of dispersion gives the deviation from a Poisson process, with a value above 1 indicating temporal clustering. Values below 1 are expected in regions where earthquakes decreased the probability of subsequent earthquakes, which could be caused by stress relaxation. In statistical terms, such a condition is referred to as being repulsive in time. In this context, seismicity patterns related to decreased probabilities of subsequent earthquakes can be more adequately referred to as inhibitive in time. Regions of low numbers of events usually have a high bandwidth c 2 (x) (see equation 3) and thus are highly influenced by neighboring regions, which leads to a similar level of activity for every year throughout the observation period. In this case kernel estimation usually leads to low temporal variances (i.e., small fluctuations of earthquake activity) and thus to a low index of dispersion during the observation time. For longer observation times, more data become available, and it has to be expected that the temporal variance will increase. For these regions the small number of observed earthquakes is not sufficient to decide if the region behaves in a Poissonian or non-poissonian way because of the high uncertainty in the estimate of V T (x). Filtering Sequences The main interest of hazard studies is the probability of the occurrence of main events in the investigated region. With mainshock occurrence we mean the representation of multiple sequences (i.e., aftershocks, foreshocks, swarms) by one main event. Multiple sequences can distort this probability distribution because they produce transient local activity features. Thus it is an advantage if such sequences are taken out of the analysis. After the probability distribution of mainshocks is known, the hazard due to related earthquakes can be incorporated using an appropriate occurrence model for these sequences (e.g., Omori s law for aftershocks; see Reasenberg and Jones, 1989). Removal of individual earthquakes belonging to a sequence is often problematic (e.g., Davis and Frohlich, 1991; Savage and depolo, 1993) because it is often not clear if an earthquake belongs to a sequence or not. An alternative way to estimate probability distributions of mainshocks can be achieved by dividing the continuous representation g 2 (x)by the index of dispersion V T (x) 2 : T

4 Adaptive Kernel Estimation and Continuous Probability Representation of Historical Earthquake Catalogs 907 g 2(x) g 3(x) 2. (5) V (x) In regions with an index of variation above 1, the index of dispersion lowers the activity in these regions. For an index of dispersion smaller than 1 the probability of earthquakes rises. However, as previously discussed, an index of dispersion below 1 indicates that earthquake occurrence is inhibited. Therefore, the mainshock activity should not be increased. Hence, equation (5) has been amended to T g 2(x) g 3(x) c 2. (6) (V (x) 1) Now the effective index of dispersion, (V T (x) 1) 2, cannot be less than 1 no matter how small V T (x) is. For indices of dispersion much larger than 1 the expression (V T (x) 1) 2 behaves asymptotically to V T (x) 2, and thus c should be 1. For an index of dispersion of 1, g 3 (x) 1/4 g 2 (x). As we stated before, an index of dispersion of 1 indicates Poissonian (cluster free) behavior and thus should not be changed. However, because shallow earthquakes generally produce aftershocks it has to be expected that locations with an index of dispersion of 1 are not free of temporal clusters, and the probability of mainshock occurrence has to be down weighted, as this approach does. Nevertheless, a connection to physical earthquake behavior (e.g., Musmeci and Vere- Jones [1986] attempted to relate spatial aftershocks features to the inverse-binomial distribution) would be helpful to justify g 3 (x) as a model for probability distribution of mainshock occurrence. In the following section, we show that the method produces very similar representations for independent time intervals for New Zealand and Australia. We have tried other variations of this method (for example amending V T (x) 2 to (V T (x) 2 1)), but the form of filtering in equation (6) gives the best results. T Table 1 Magnitude Completeness Year Magnitude New Zealand (crust) New Zealand (subduction) Australia Examples Data The earthquake catalogs of the New Zealand Seismological Observatory (e.g., Maunder, 1999) and the Australian Seismological Centre (e.g., McCue and Gregson, 1994) have been used for this work. To generate useful probability distributions for a chosen magnitude interval, the earthquake catalogs have to be complete for the considered magnitudes. For a chosen time interval, the lowest magnitude for which the data set is still complete (cutoff magnitude) can be identified by a change of the slope in the graph of the magnitude frequencies. Such graphs have been generated and evaluated (Stock, 2001). The resulting cutoff magnitudes and corresponding years of completeness are given in Table 1. The earthquake locations have been binned into and cells for New Zealand and Australian shallow earthquakes, respectively. For the two New Zealand subduction zones (Anderson and Webb, 1994) a depth profile has been generated by projecting the hypocenters of the earthquakes onto a vertical plane along the main seismic activity trend running from southwest to northeast. The data have been binned into cells that are 12 km along strike and 5 km in depth. Within each cell, the number of earthquakes above the cutoff magnitude has been summed and divided by the number of years to calculate an annual rate of activity. These data sets can be readily used for adaptive kernel estimation. The probability densities of earthquake activity have been plotted as the probability of one or more earthquakes in one year, assuming a Poisson process. This probability is given by P (x) 1 exp( g (x)). (7) 1 3 New Zealand: Shallow Figure 1 shows the different performance of kernel estimation with global bandwidth g 1 (x) compared to adaptive kernel estimation g 2 (x). Whereas a small global bandwidth (Fig. 1b) preserves high-activity clusters, low activity offshore (e.g., around latitude 46, longitude 171 ), which in Figure 1a appears to be regular in space, is clustered as well (undersmoothing). A greater bandwidth (Fig. 1c) generates a higher degree of smoothness in low-activity regions, but the high-activity clusters are now distributed over a wider space, leading to a decrease of activity in their centers and an increase in activity in their immediate neighborhood (oversmoothing). The adaptive kernel estimation separates high-activity clusters from low-background seismicity, and clusters of high activity can be readily identified. Another important improvement lies in the reduction of the uniform circularity of the activity features in the kernel estimation with global bandwidth, which is artificially imposed by the choice of a rotationally invariant kernel. Figure 2 shows the spatial index of dispersion V T (x) between two different time intervals. Temporal activity clustering (Fig. 2a) can be mostly matched to spatial clusters of high activity (Fig. 1d). Although the exact location of most of the individual temporal sequences is different between the two different time intervals (compare Fig. 2a and b), most temporal sequences of the later time interval are located near the temporal sequences of the earlier time interval. This sug-

5 908 C. Stock and E. G. C. Smith Figure 1. (a) Probability inferred from the cumulative number of crustal earthquakes (above 40 km, from 1962 to 1997) in New Zealand in 0.1 squares. (b) and (c) Corresponding probability distribution generated by kernel estimation with global bandwidths 1.0 and 2.5, respectively. (d) For comparison, the probability distribution generated by the adaptive kernel estimation is shown. gests that the area generating these temporal clusters is bigger than the spatial extent of any temporal cluster. Some spatial clusters of medium activity in Figure 1d (e.g., around latitude 39, longitude 176 ) indicate little or no temporal clustering in Figure 2a. They probably represent centres of persistent medium-rate activity. Figure 3 shows the seismicity probability distributions g 3 (x), with sequences filtered out, during two different time intervals. The two distributions reveal very similar spatial seismicity features (Fig. 3a and b), although they are independent, representing the activity of different time intervals and magnitudes. There are a few differences in detail, which could be explained by the shorter observation time for Figure 3b compared with Figure 3a. Whereas the uncertainty increases for shorter observation times, mainshock activity is expected to undergo real fluctuations over a time interval of decades (e.g., latitude 40, longitude 174 ). Nevertheless, the similarity of Figure 3a and b suggests that these temporal fluctuations are only of secondary order. Some differences may be due to the use of better instrumentation in the later years, which leads to more accurate hypocenters in Figure 3b. The similarity of the two figures suggests that the main

6 Adaptive Kernel Estimation and Continuous Probability Representation of Historical Earthquake Catalogs 909 Figure 2. The index of dispersion of crustal earthquakes in New Zealand (a) from 1962 to 1990 above magnitude 4.0 and (b) from 1991 to 1999 above magnitude 3.0. seismicity features in New Zealand have been stable over the last 40 yr. New Zealand: Subduction Zone Figure 4 shows a depth profile of the seismicity probability distribution g 2 (x), the spatial index of dispersion V t (x), and the seismicity probability distribution with earthquake sequences filtered out g 3 (x). In contrast to the shallow activity, temporal clustering was nearly absent for the deep earthquakes in the subducted slabs during the observation period (compare Fig. 4a and b), and the index of dispersion of the deep earthquakes looks very uniform. Hence there is no need to filter temporal sequences out of the representation of earthquake occurrence probability (Fig. 4a and c look alike). One explanation for this result would be that deep earthquakes do not produce many temporally clustered sequences, such as aftershocks and swarms, as it has been observed for large deep earthquakes in the past. The low temporal variation further suggests that the spatial earthquake occurrence features have been very stable during the observation period. Australia Figure 5 shows the seismicity probability distribution g 3 (x), with earthquake sequences filtered out, during different time intervals. The two probability distributions (Fig. 5a and b) reveal similar seismicity features, although they do not match as well as the corresponding figures for New Zealand. This is not surprising because the activity in Australia is much lower and a higher uncertainty in g 3 (x) istobe expected due to the lower sample size. The region of highest activity occurred at the location of the Tennant Creek earthquake sequence (latitude 20, longitude 134 ) on 22 January 1988, which had one event above magnitude 6.5 and two other events above magnitude 6.0 (Gaull et al., 1990). Before these events, the area around this earthquake has been reasonably inactive (Fig. 5c). There is little doubt that this earthquake started a spatial cluster of high activity, which persists today. It is possible that other clusters of persistent high activity, which have produced large earthquakes in historical times, have been started by one or a few large events. On the other hand, no spatial clusters of persistent high activity in New Zealand and Australia decreased to background activity during the observation time. Discussion In probabilistic seismic hazard analysis, the main interest lies in the occurrence of damaging earthquakes. The presented analysis includes earthquakes below magnitude five, which (in most cases) do not cause great damage. The Gutenberg Richter law predicts power-law behavior between small and large earthquakes and justifies the extrapolation from small to large earthquakes. In contrast, the characteristic earthquake model (e.g., Wesnousky, 1994; Stirling et al., 1996) predicts a proportionally larger number of large earthquakes in regions dominated by long faults. It is possible that some spatial high-activity clusters only produce medium and small size earthquakes but no large earthquakes. Examples of such clusters can be seen in the Taupo Volcanic Zone in New Zealand (latitude 39, longitude 176 ), where no earthquakes above magnitude 7.0 have been observed during historical times (e.g., Smith and Webb, 1986; Reyners, 1989). The Australian cluster at latitude 34 and longitude 122 is mining induced (Gaull et al.,

7 910 C. Stock and E. G. C. Smith data, including geological data (e.g., faults), into the seismicity models. For the procedures introduced in this article to work well, a minimum number of earthquakes are needed. Unfortunately, the number of historically observed large earthquakes (e.g., above magnitude 6) is often too low to produce good results. The more data that are available, the better the detection and resolution of activity structures will be and the better the sequence filtering will work. For example, Australia has a relatively higher apparent rate of earthquakes occurring in high-activity regions inferred from magnitude 4 and above (Fig. 5a) compared to the activity inferred from magnitude 3 and above (Fig. 5b). This might be due to a lower temporal variance in the first representation, which probably results from the low number of counted aftershocks above magnitude 4. This would lead in turn to a higher uncertainty in the index of dispersion. There are more aftershocks counted above magnitude 3, which thus leads to greater filtering and lower estimated probability densities in the high-activity clusters. We suggest that one should generate earthquake probability distributions with the lowest possible cutoff magnitude to identify spatial and temporal earthquake distribution features. The additional consideration of probability distributions with a higher cutoff magnitude may help to identify important differences from the seismicity features of smaller earthquakes (e.g., the identification of high-activity clusters that have not produced large earthquakes in historical times). The generated representations of earthquake occurrence probabilities should not be used for the prediction of earthquakes for a time period longer than the observation period because activity fluctuations on timescales that exceed the observation time cannot be predicted with this approach. Conclusions Figure 3. The probability distribution of crustal earthquakes in New Zealand with sequences filtered out (a) from 1962 to 1990 above magnitude 4.0 and (b) from 1991 to 1997 above magnitude ) and has not produced any large earthquakes since inception. Kagan and Jackson (1999) found that the recurrence times of doublets of large earthquakes (above magnitude 7.0) show power-law behavior and conclude that large earthquakes cluster in time and space, similar to smaller-size earthquakes. Combined with our result, that no spatial clusters of persistent high activity in New Zealand and Australia decreased to background activity during the observation time, such clusters appear to be potential sources of future moderate to large earthquakes for at least a few decades. Currently, work is in progress to incorporate longer-term Adaptive kernel estimation can better resolve local seismic features in distributions of earthquake activity than kernel estimations with global bandwidths because of its ability to adapt the degree of smoothing to the data. The introduced local index of dispersion and procedure to filter temporal earthquake sequences has revealed the following results in New Zealand and Australian earthquake distributions: 1. Whereas shallow earthquakes are mostly part of temporal earthquake sequences (such as mainshocks with aftershocks), the subducting slab in New Zealand has been nearly free from temporal activity variations for the last 40 yr, suggesting that deep earthquakes do not produce many multiple events, as observed in the past. 2. The probability distribution of main events has been stable in New Zealand during the last 40 yr. Apart from the Tennant Creek earthquake, which introduced a new spatial high-activity cluster, the probability distribution of main events in Australia has been reasonably stable over the last 20 yr.

8 Adaptive Kernel Estimation and Continuous Probability Representation of Historical Earthquake Catalogs 911 Figure 4. The cross section of New Zealand shows (a) the probability distribution generated by adaptive kernel estimation, (b) the index of dispersion, and (c) the probability distribution with sequences filtered out for earthquakes down to 350 km. (d) The strike of the cross section. 3. No spatial high-activity clusters have been observed to decrease to background activity during the observation time. We infer that there is a higher risk of the occurrence of medium to large earthquakes associated with these clusters during the short to medium term (a few years to decades). The model comparisons in this study are qualitative, as there exists no objective comparison. Any form of comparison has to be suited to the purpose for which the kernel estimation is used (e.g., estimation of the parent distribution, earthquake forecasting, etc.). An approach to quantitatively compare different kernel models (i.e., kernel types and bandwidth choices) for the estimation of parent distributions has been developed in a separate study (Stock and Smith, 2002). Acknowledgments We would like to thank David Vere-Jones for support on the adaptive kernel method and other helpful comments related to this work. Further, we would like to thank Martha Savage and John Taber for reviews and constructive remarks. This work was supported by the Earthquake Commission of New Zealand Project 97/259. References Anderson, H., and T. Webb (1994). New Zealand seismicity: patterns revealed by the upgraded National Seismograph Network, New Zealand J. Geol. Geophys. 37, Cao, T., M. D. Petersen, and M. S. Reichle (1996). Seismic hazard estimate from background seismicity in southern California, Bull. Seism. Soc. Am. 86, no. 5, Davis, S. D., and C. Frohlich (1991). Single-link cluster analysis, synthetic

9 912 C. Stock and E. G. C. Smith Figure 5. The probability distribution of crustal earthquakes in Australia (above 50 km) above magnitude 4.0 in 0.5 squares with sequences filtered out (a) from 1978 to 1996 above magnitude 4.0, (b) from 1991 and 1996 above magnitude 3.0, and (c) from 1978 to earthquake catalogues, and aftershock identification, Geophys. J. Int. 104, no. 2, Frankel, A. (1995). Mapping seismic hazard in the central and eastern United States, Seism. Res. Lett. 66, Gaull, B. A., M. O. Michael-Leiba, and J. M. W. Rynn (1990). Probabilistic earthquake risk maps of Australia, Aust. J. Earth Sci. 37, Härdle, W. (1990). Applied Nonparametric Regression, Cambridge, Cambridge University Press. Jackson, D. D., and Y. Y. Kagan (1999). Testable earthquake forecasts for 1999, Seism. Res. Lett. 70, no. 4, Johnson, N. L., S. Kotz, and A. W. Kemp (1992). Univariate Discrete Distributions, Second Ed., in Probability and Mathematical Statistics, Wiley series in probability and mathematical statistics, Wiley, New York. Kagan, Y. Y., and D. D. Jackson (1999). Worldwide doublets of large shallow earthquakes, Bull. Seism. Soc. Am. 89, no. 5, Marzocchi, W., F. Mulargia, and G. Gonzato (1997). Detecting lowdimensional chaos in geophysical time series, J. Geophys. Res. 102, no. B2, Maunder, D. E. (Editor) (1999). New Zealand seismological report 1997, Seismological Observatory Bulletin E-180, Institute of Geological & Nuclear Sciences Science Report 99/20, Lower Hutt. McCue, K. and P. Gregson (Editor) (1994). Australian Seismological Report, 1991, Australian Geological Survey Organisation Record 1994/ 10, Australian Geological Survey, Canberra. Musmeci, F., and D. Vere-Jones (1986). A variable-grid algorithm for smoothing clustered data, Biometrics 42, Ogata, Y. M. Imoto, and K. Katsura (1991). 3D spatial variation of b-values of magnitude frequency distribution beneath the Kanto District, Japan, Geophys. J. Int. 104, no. 1, Papazachos, P. (1999). An alternative method for a reliable estimation of seismicity with an application in Greece and the surrounding area, Bull. Seism. Soc. Am. 89, no. 1, Reasenberg, P. A., and L. M. Jones (1989). Earthquake hazard after a mainshock in california, Science 243, Reyners, M. (1989). New Zealand seismicity : an interpretation, New Zealand J. Geol. Geophys. 32, Robertson, M. C., C. G. Sammis, M. Sahimi, and A. J. Martin (1995). Fractal analysis of three-dimensional spatial distributions of earthquakes with a perlocation interpretation, J. Geophys. Res. 100, no. B1, Savage, M. K., and D. M. depolo (1993). Foreshock probabilities in the western Great-Basin Eastern Sierra Nevada, Bull. Seism. Soc. Am. 83, no. 6, Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, Monographs on Statistics and Applied Probability, Chapman and Hall, London. Smith, E. G. C., and T. H. Webb (1986). The seismicity and related deformation of the central volcanic region, North Island, New Zealand, in Late Cenozoic Volcanism in New Zealand, Ian E. M. Smith (Editor), R. Soc. New Zealand Bull. 23, Stirling, M. W., S. G. Wesnousky, and K. Shimazaki (1996). Fault trace complexity, cumulative slip, and the shape of the magnitudefrequency distribution for strike-slip faults: a global survey, Geophys. J. Int. 124, no. 3, Stock, C. (2001). A consistent geological-seismological model for earthquake occurrence in New Zealand, Ph.D. Thesis, Victoria University of Wellington, Wellington, New Zealand. Stock, C. and E. G. C. Smith (2002). Comparison of seismicity models generated by different kernel estimations, Bull. Seism. Soc. Am. 92, no. 3, Vere-Jones, D. (1992). Statistical methods for the description and display of earthquake catalogs, in Statistics in the Environmental and Earth Sciences, A. T. Walden and P. Guttorp (Editors), Arnold Publishers, London, Wesnousky, S. G. (1994). The Gutenberg-Richter or characteristic earthquake distribution, which is it? Bull. Seism. Soc. Am. 84, no. 6, Woo, G. (1996). Kernel estimation methods for seismic hazard area source modeling, Bull. Seism. Soc. Am. 86, no. 2, School of Earth Sciences Victoria University of Wellington PO Box 600 Wellington, New Zealand Manuscript received 30 August 2000.