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Bulletin of the Seismological Society of America, 90, 4, pp , August 2000 Minimum Magnitude of Completeness in Earthquake Catalogs: Examples from Alaska, the Western United States, and Japan by

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Bulletin of the Seismological Society of America, 90, 4, pp , August 2000 Minimum Magnitude of Completeness in Earthquake Catalogs: Examples from Alaska, the Western United States, and Japan by Stefan Wiemer and Max Wyss Abstract We mapped the minimum magnitude of complete reporting, M c, for Alaska, the western United States, and for the JUNEC earthquake catalog of Japan. M c was estimated based on its departure from the linear frequency-magnitude relation of the 250 closest earthquakes to grid nodes, spaced 10 km apart. In all catalogs studied, M c was strongly heterogeneous. In offshore areas the M c was typically one unit of magnitude higher than onshore. On land also, M c can vary by one order of magnitude over distance less than 50 km. We recommend that seismicity studies that depend on complete sets of small earthquakes should be limited to areas with similar M c, or the minimum magnitude for the analysis has to be raised to the highest common value of M c. We believe that the data quality, as reflected by the M c level, should be used to define the spatial extent of seismicity studies where M c plays a role. The method we use calculates the goodness of fit between a power law fit to the data and the observed frequency-magnitude distribution as a function of a lower cutoff of the magnitude data. M c is defined as the magnitude at which 90% of the data can be modeled by a power law fit. M c in the 1990s is approximately in most parts of California, in most of Alaska (Aleutians and Panhandle excluded), and at a higher level in the JUNEC catalog for Japan. Various sources, such as explosions and earthquake families beneath volcanoes, can lead to distributions that cannot be fit well by power laws. For the Hokkaido region we demonstrate how neglecting the spatial variability of M c can lead to erroneous assumptions about deviations from self-similarity of earthquake scaling. Introduction The minimum magnitude of complete recording, M c,is an important parameter for most studies related to seismicity. It is well known that M c changes with time in most catalogs, usually decreasing, because the number of seismographs increases and the methods of analysis improve. However, differences of M c as a function of space are generally ignored, although these, and the reasons for them, are just as obvious. For example, catalogs for offshore regions, as well as regions outside outer margins of the networks, are so radically different in their reporting of earthquakes that they should not be used in quantitative studies together with the catalogs for the central areas covered. In seismicity studies, it is frequently necessary to use the maximum number of events available for high-quality results. This objective is undermined if one uses a single overall M c cutoff that is high, in order to guarantee completeness. Here we show how a simple spatial mapping of the frequency-magnitude distribution (FMD) and application of a localized M c cut-off can assist substantially in seismicity studies. We demonstrate the benefits of spatial mapping of M c for a number of case studies at a variety of scales. For investigations of seismic quiescence and the frequency-magnitude relationship, we routinely map the minimum magnitude of completeness to define an area of uniform reporting for study (Wyss and Martyrosian, 1998, Wyss et al., 1999). Areas of inferior reporting (higher M c ), outside such a core area, are excluded because these data would contaminate the analysis. In seismicity studies where statistical considerations play a key role, it is important that results are not influenced by the choice of the data limits. If these limits are based on the catalog quality, then improved statistical robustness may be assured. For this reason we routinely map the quality of the catalog for selecting the data for our studies of seismic quiescence; however, homogeneity in M c does not necessarily guarantee homogeneity in earthquake reporting, since changes in magnitude reporting influence the magnitude of homogeneous reporting (Habermann, 1986; Habermann, 1991; Zuniga and Wyss, 1995; Zuniga and Wiemer, 1999). Our estimation of M c is based on the assumption that, for a given, volume a simple power law can approximate the FMD. The FMD (Ishimoto and Iida, 1939; Gutenberg and 859 860 S. Wiemer and M. Wyss Richter, 1944) describes the relationship between the frequency of occurrence and magnitude of earthquakes: log N a bm, (1) 10 where N is the cumulative number of earthquakes having magnitudes larger than M, and a and b are constants. Various methods have been suggested to measure b and its confidence limits (Aki, 1965; Utsu, 1965, 1992; Bender, 1983; Shi and Bolt, 1982; Frohlich and Davis, 1993). The FMD has been shown to be scale invariant down to a source length of about 10 m (Abercrombie and Brune, 1994) or approximately magnitude 0 event size. Some authors have suggested changes in scaling at the higher magnitude end (e.g., Lomnitz-Adler and Lomnitz 1979; Utsu, 1999) or for smaller events (Aki, 1987). However, neither of these suggested changes in slope will be relevant for the estimate of M c because by far the dominant factor changing the slope of the FMD is incompleteness in reporting for smaller magnitudes. In Figure 1 we show the overall FMD in cumulative (Figure 1a) and noncumulative (Figure 1b) form for the three data sets we investigate. We assume that the drop in the number of events below M c is caused by incomplete reporting of events. Other studies that have addressed the completeness problem have either used changes between the day and nighttime sensitivity of networks (Rydelek and Sacks, 1989, 1992), comparison of amplitude-distance curves and the signal-to-noise ratio (Sereno and Bratt, 1989; Harvey and Hansen, 1994) or amplitude threshold studies (Gomberg, 1991) to estimate M c. Waveform-based methods that require estimating the signal-to-noise ratio for numerous events at many stations are time-consuming and cannot generally be performed as part of a particular seismicity study. Using the FMD to estimate completeness is probably the simplest method. Our study demonstrates that despite some obvious shortcomings, spatially mapping of M c based on the FMD is a quick yet useful tool for seismicity analysis and should in our opinion be a routine part of seismicity related studies. Method The first step toward understanding the characteristics of an earthquake catalog is to discover the starting time of the high-quality catalog most suitable for analysis. In addition, we seek to identify changes of reporting quality as a function of time. Issues connected with these problems are not the subjects of this article; they are dealt with elsewhere (Habermann, 1986; Habermann, 1991; Zuniga and Wiemer, 1999; Zuniga and Wyss, 1995). Here we assume that we know the starting date of the high-quality catalog, and that there are no changes of reporting (magnitude stretches and shifts) serious enough to corrupt the analysis we have in mind, so that we may proceed to map M c. Our estimate of M c is based on the assumption of a Figure 1. (a) Cumulative frequency-magnitude distribution of events for the three catalogs investigated. (b) Number of events in each magnitude bin for these catalogs. Gutenberg-Richter (GR) power law distribution of magnitudes (equation 1). To evaluate the goodness of fit, we compute the difference between the observed FMD and a synthetic distribution. For incomplete data sets, a simple power law cannot adequately explain the observed FMD, so the difference will be high. The following steps are taken to estimate M c : First we estimate the b- and a-value of the GR law as a function of minimum magnitude, based on the events with M M i.we use a maximum likelihood estimate to estimate the b- and a-values and their confidence limits (Aki, 1965; Shi and Bolt, 1982; Bender, 1983). Next, we compute a synthetic distribution of magnitudes with the same b-, a- and M i values, which represents a perfect fit to a power law. To estimate the goodness of the fit we compute the absolute difference, R, of the number of events in each magnitude bin between the observed and synthetic distribution Minimum Magnitude of Completeness in Earthquake Catalogs: Examples from Alaska, the Western United States, and Japan 861 Mmax i i M i Bi i B S R(a, b, M i) (2) where B i and S i are the observed and predicted cumulative number of events in each magnitude bin. We divide by the total number of observed events to normalize the distribution. Our approach is illustrated in Figure 2, which shows R as function of M i.ifm i is smaller then the correct M c, the synthetic distribution based on a simple power law (squares in Figure 2) cannot model the FMD adequately and, consequently, the goodness of fit, measured in percent of the total number of events, is poor. The goodness-of-fit value R increases with increasing M i and reaches a maximum value of R 96% at M c 1.8 in this example. At this M c, a simple power law with the assumed b-, a-, and M c value can explain 96% of the data variability. Beyond M i 1.8, R increases again gradually. In this study we map M c at the 90% level, that is, we define M c as the point at which a power law can model 90% or more of the FMD. For the example shown in Figure 2, we therefore define M c 1.5. Not all FMDs will reach the 90% mark. In some cases the FMDs are too curved or bimodal to be fitted satisfactory by a simple power law. This can be due to strong spatial or temporal inhomogeneities in the particular sample, or actual physical processes within the earth. An example of the former would be a drastic change of the completeness of recording during the investigated period; an example of the latter might be a volcanic region where distinct earthquake families and swarms are frequent. It is important to identify these areas for studies of the FMD, because here a power law cannot be readily applied. Our method allows us to map the fit to a power law behavior at each node, based on the minimum value of R obtained. For mapping M c, we use the gridding technique applied in our studies of b-values and seismic quiescence (Wiemer and Wyss, 1997; Wiemer et al., 1998; Wiemer and Katsumata, 1999). Grids with several thousand nodes spaced regularly at 1 to 20 km distances, depending on the size of the area to be covered, the density of earthquakes, and the computer power available, are arbitrarily placed over the study region, and we construct the FMD at each node for the N nearest events and estimate M c using the approach outlined previously. At the same time, we compute a map of the goodness of fit to a power law by finding the minimum R from equation (2) at each node. The same type of spatial Figure 2. Explanation of the method by which we estimated the minimum magnitude of completeness, M c. The three frames at the top show synthetic fits to the observed catalog for three different minimum magnitude cutoffs. The bottom frame shows the goodness of fit R, the difference between the observed and a synthetic FMD (equation 2), as a function of lower magnitude cut-off. Numbers correspond to the examples in the top row. The M c selected is the magnitude at which 90% of the observed data are modeled by a straight line fit. 862 S. Wiemer and M. Wyss gridding can also be applied in cross sections (e.g., Wiemer and Benoit, 1996; Power et al., 1998). Data and Observations In the following section, we apply the spatial mapping of M c to three test cases: Alaska, Western United States, and Japan. Alaska The seismicity catalog compiled by the Alaska Earthquake Information Center (AEIC) for the period of January 1992 December 1998 contains a total of about 21,000 events for central and interior Alaska with a depth less than 60 km and M 0.5. An epicenter map also identifying the main faults is shown in Figure 3c. We mapped M c using a sample size of N 250 and a node spacing of 10 km. The grid we created excludes low-seismicity areas. The sampling radii are typically r 70 km, and none are larger than 200 km. M c varies from values near 1.4 in the interior, near Fairbanks, and in the south, between Anchorage and Valdez (blue/purple in Figure 3a), to values of M c 3 offshore and on Kodiak island (red in Figure 3a). In Figure 4a we show a comparison of the FMDs for three areas: The vicinity of Fairbanks (M c 1.4), the Mt. McKinley area (M c 2.1), and Kodiak Island (M c 3.3). A map of the goodness of fit to a GR distribution is shown in Figure 3b. Mapped is the parameter R; low R- values (R 90%), shown in hot colors indicate that only a poor fit to a GR distribution could be obtained. Several areas can be identified where the best fitting GR explains less than 90% of the observed distribution. The poorest fit to a GR Figure 3. (a) Map of central and southern Alaska. Color-coded is the minimum magnitude of completeness, M c, estimated from the nearest 250 earthquakes to nodes of a grid spaced 10 km apart. The typical sampling radii are r 75 km, and all r 200 km. (b) Map of the local goodness of fit of a straight line to the observed frequencymagnitude relation as measured by the parameter R in percent of the data modeled correctly. (c) Epicenters of earthquakes in Central and Southern Alaska for the period and depth 60 km. Major faults are marked by red lines. Minimum Magnitude of Completeness in Earthquake Catalogs: Examples from Alaska, the Western United States, and Japan 863 Southern California Off-shore (Figures 4c and 5a) the resolution is not as good (M c 2 0.2) as it is on land and within the network, where M c 1.7 in most parts, except for the Mojave Desert (Figure 5a). The best resolution (M c 0.8) is achieved in the less populated Landers (Figures 4c and 5a), Salton Sea, San Jacinto, and Malibu areas. Pt. Conception, west of Santa Barbara, seems to be poorly covered (M c 2.0). In the Los Angeles area a resolution of M c 1.5 is achieved, which is remarkable, given the industrial noise level. Northern California The poorest coverage with M c 2.7 is seen far off the coast off Cape Mendocino (Figures 4e and 5a). At Cape Mendocino itself, M c 2. Most of the seismicity in the San Andreas fault system (west of the Great Valley) is resolved at levels of M c 1.6. The best job (M c 0.8) is achieved south of the San Francisco Bay to Parkfield (Figures 4e and 5a) and in the Mammoth Lake area. The border region with Nevada is not well covered; M c rises above the 2.3 level in places. M c also increases to 2 at the boundary between the northern and southern California seismic networks, south of Parkfield. Figure 4. Frequency-magnitude distributions for selected areas. (a) Three areas in Alaska for which M c ranges from 1.4 (Fairbanks) to 2.1 (McKinley) to 3.3 (Kodiak). (b) At volcanoes like Mt. Spurr, a bimodal distribution is often observed. In this case it is difficult to estimate M c. (c) In Southern California, the offshore data (M c 2) cannot be resolved to the same low-magnitude levels as on land, where M c 1 for many locations, such as near Landers. (d) An example of explosions, which, mixed into the earthquake catalog, can lead to unnatural FMDs. (e) In Northern California, some of the lowest M c 0.4 are observed in the San Francisco Bay area, and the highest values of M c 2.5 are observed far off Cape Mendocino. (f) An example of an FMD, which may be a hybrid of two populations because of a possible change of reporting rate as a function of time. law (R 80%) can be observed near the volcanoes Mt. Spurr and Mt. Redoubt. In this area, the observed FMD (Figure 4b) is bimodal; two different populations of events are contained in the sample. Western United States We used the seismicity catalog compiled by the Council of the National Seismic Systems (CNSS) for the period of January 1995 May 1999 and events with a depths less than 30 km. Figure 5c shows the epicenter map of the area investigated. The spatial distribution of M c is plotted in Figure 5a. The goodness-of-fit map (Figure 5b) indicates numerous small regions where the fit to GR is less than 90% satisfactory. Japan Our analysis is based on the Japan University Network (JUNEC) catalog for the period of January 1986 December Japan is covered by regional seismograph networks, which are run separately by universities and government research laboratories. JUNEC combined these data in a single catalog. The epicenter map for earthquakes with depths shallower than 35 km is shown in Figure 6c, the spatial distribution of M c is imaged in Figure 6a. The M c in the subducting slab beneath central Japan is shown in Figure 7. In this catalog the seismicity located farther than about 100 km offshore is resolved only above the M c 3.2 level (Figure 6a). Within approximately 100 km from the coast, the completeness magnitude varies between 2.5 and 2.8. The areas in central Japan, near Tokyo and near Kyoto show the lowest M c of about 1.3, whereas in northern Japan, M c 2.4 in this catalog. The cross-sectional analysis in the Kanto region, central Japan (Figure 7) reveals the lowest M c level (M c 2.) onshore. With increasing distance from land, M c increases gradually to values greater than 3. Deep earthquakes in the subducting plate show a M c 3 at 200 km depth, and M c 3.5 at 400-km depth. This analysis does not represent the full extent of information available on seismicity in Japan. The catalogs of individual organizations such as Bosai and Tohoku University, for example, resolve the areas covered by their networks to significantly lower magnitudes than the JUNEC catalog does. However, since the JUNEC catalog is readily available, and because it covers all of Japan, the variations of M c in it are of interest. 864 S. Wiemer and M. Wyss Figure 5. (a) Map of the western United States. Color-coded is the minimum magnitude of completeness, M c estimated from the nearest 250 earthquakes to nodes of a grid spaced 10 km apart. The typical sampling radii are r 50 km, and all r 150 km. (b) Map of the local goodness of fit of a straight line to the observed frequencymagnitude relation as measured by the parameter R in percent of the data modeled correctly. (c) Epicenters of earthquakes in the western United States for the period and depth 30 km. Red lines mark major faults in California. Discussion and Conclusions Spatially Mapping M c The magnitude of completeness varies spatially throughout all seismic networks. By assessing the goodness of fit to a power law, we can reliably and quickly map out the spatial variability of M c, based purely on catalog data. A solid knowledge of M c is important for many seismicity and probabilistic earthquake hazard studies, and we propose that spatially mapping M c should be performed routinely as part of seismicity studies. (The software used in this study can be freely downloaded as part of the ZMAP seismicity analysis package; Knowing the spatial distribution of M c is important for regional studies that use bulk b-values. By excluding high M c areas, the magnitude threshold for analysis can be lowered and the amount of data available for analysis increased. For studies that address the spatial variability of the FMD, knowledge of the spatial distribution of M c is imperative. From the M c maps (Figures 3, 5, and 6) we see that in most of Alaska the seismicity is resolved to M c , in most of California the completeness level is at M c , and in the JUNEC catalog of Japan it is not better than in Alaska. In the Aleutian islands of Alaska, the seis- Minimum Magnitude of Completeness in Earthquake Catalogs: Examples from Alaska, the Western United States, and Japan 865 Figure 6. (a) Map of Japan. Color-coded is the minimum magnitude of completeness, M c estimated from the nearest 250 earthquakes to nodes of a grid spaced 10 km apart. The typical sampling radii are r 62 km, and all r 150 km. (b) Map of the local goodness of fit of a straight line to the observed frequency-magni

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