859
Bulletin of the Seismological Society of America, 90, 4, pp. 859–869, 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,
 
c
, forAlaska, the western United States, and for the
 JUNEC
 earthquake catalog of Japan.
 M 
c
 was estimated based on its departurefrom thelinearfrequency-magnituderelationof the 250 closest earthquakes to grid nodes, spaced 10 km apart. In all catalogsstudied,
 M 
c
 was strongly heterogeneous. In offshore areas the
 M 
c
 was typically oneunit 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 thatdepend 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 com-mon value of 
 M 
c
. We believe that the data quality, as reflected by the
 M 
c
 level, shouldbe used to define the spatial extent of seismicity studies where
 M 
c
 plays a role. Themethod we use calculates the goodness of fit between a power law fit to the data andthe observed frequency-magnitude distribution as a function of a lower cutoff of themagnitude data.
 
c
 is defined as the magnitude at which 90% of the data can bemodeled by a power law fit.
 
c
 in the 1990s is approximately 1.2
0.4 in mostparts of California, 1.8
0.4 in most of Alaska (Aleutians and Panhandleexcluded),and at a higher level in the
 JUNEC
 catalog for Japan. Various sources, such as ex-plosions and earthquake families beneath volcanoes, can lead to distributions thatcannot be fit well by power laws. For the Hokkaido region we demonstrate howneglecting the spatial variability of 
 
c
 can lead to erroneous assumptions aboutdeviations from self-similarity of earthquake scaling.Introduction
The minimum magnitude of complete recording,
 M 
c
, isan important parameterformoststudiesrelatedtoseismicity.It is well known that
 M 
c
 changes with time in most catalogs,usually decreasing, because the number of seismographs in-creases and the methods of analysis improve. However, dif-ferences 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 asregions outside outer margins of the networks, are so radi-cally different in their reporting of earthquakes that theyshould not be used in quantitative studies together with thecatalogs for the central areas covered.In seismicity studies, it is frequently necessary to usethe maximum number of events available for high-qualityresults. This objective is undermined if one uses a singleoverall
 
c
 cutoff that is high, in order to guarantee com-pleteness. Here we show how a simple spatial mapping of the frequency-magnitude distribution (
FMD
) and applicationof a localized
 M 
c
 cut-off can assistsubstantiallyinseismicitystudies. 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 fre-quency-magnitude relationship, we routinely map the min-imum magnitude of completeness to define an area of uni-form 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 datawould contaminate the analysis. In seismicity studies wherestatistical considerations play a key role, it is important thatresults are not influenced by the choice of the data limits. If these limits are based on the catalog quality, then improvedstatistical robustness may be assured. For thisreasonwerou-tinely map the quality of the catalog for selecting the datafor our studiesofseismicquiescence;however,homogeneityin
 M 
c
 does not necessarily guarantee homogeneity in earth-quake reporting, since changes in magnitude reporting influ-ence the magnitude of homogeneous reporting (Habermann,1986; Habermann, 1991; Zuniga and Wyss, 1995; Zunigaand Wiemer, 1999).Our estimation of 
 
c
 is based on the assumption that,for a given, volume a simple power law can approximatethe
FMD
. The
 FMD
 (Ishimoto and Iida, 1939; Gutenberg and
 
860
 S. Wiemer and M. Wyss
Figure
 1.
 (a) Cumulative frequency-magnitudedistribution of events for the three catalogs investi-gated. (b) Number of events in each magnitude binfor these catalogs.
Richter, 1944) describes the relationship between the fre-quency of occurrence and magnitude of earthquakes:log
 
 a
 
 bM 
,
 (1)
10
where
 
 is the cumulative number of earthquakes havingmagnitudes larger than
 M 
, and
 a
 and
 b
 areconstants.Variousmethods have been suggested to measure
 b
 and its confi-dence 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 sourcelengthof about 10 m (Abercrombie and Brune, 1994) or approxi-mately magnitude 0 eventsize.Someauthorshavesuggestedchanges in scaling at the higher magnitude end (e.g.,Lomnitz-Adler and Lomnitz 1979; Utsu, 1999) or forsmaller events (Aki, 1987). However, neither of these sug-gested 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 magni-tudes. In Figure 1 we show the overall
 FMD
 in cumulative(Figure 1a) and noncumulative (Figure1b)formforthethreedata sets we investigate. We assume that the drop in thenumber of events below
 M 
c
 is caused by incomplete report-ing of events.Other studies that have addressed the completenessproblem have either used changes between the day andnighttime sensitivity of networks (Rydelek and Sacks, 1989,1992), comparison of amplitude-distance curves and thesignal-to-noise ratio (Sereno and Bratt, 1989; Harvey andHansen, 1994) or amplitude threshold studies (Gomberg,1991) to estimate
 M 
c
. Waveform-based methods that requireestimating the signal-to-noise ratio for numerous events atmany stations are time-consuming and cannot generally beperformed as part of a particular seismicity study. Using the
FMD
 to estimate completeness is probably the simplestmethod. Our study demonstrates that despite some obviousshortcomings, spatially mapping of 
 M 
c
 based on the
 FMD
 isa quick yet useful tool for seismicity analysis and should inour opinion be a routine part of seismicity related studies.
Method
The first step toward understanding the characteristicsof an earthquake catalog is to discover the starting time of the high-quality catalog most suitable for analysis. In addi-tion, we seek to identify changes of reporting quality as afunction of time. Issues connected with these problems arenot 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 weknow the starting date of the high-quality catalog, and thatthere are no changes of reporting (magnitude stretches andshifts) serious enough to corrupt the analysis we have inmind, so that we may proceed to map
 M 
c
.Our estimate of 
 
c
 is based on the assumption of aGutenberg-Richter (GR) power law distribution of magni-tudes (equation 1). To evaluate the goodness of fit, we com-pute the difference between the observed
 FMD
 and a syn-thetic distribution. For incomplete data sets, a simple powerlaw cannot adequately explain the observed
 FMD
, so thedifference will be high.The following steps are taken to estimate
 M 
c
: First weestimate the
 b
- and
 a
-value of the GR law as a function of minimum magnitude, based on the events with
 M 
 M 
i
. Weuse a maximum likelihood estimate to estimate the
 b
- and
a
-values and their confidence limits (Aki, 1965; Shi andBolt, 1982; Bender, 1983). Next, we compute a syntheticdistribution of magnitudes with the same
 b
-,
 a
- and
 M 
i
 val-ues, which represents a perfect fit to a power law. To esti-mate the goodness of the fit we compute the absolute dif-ference,
 R
, of the number of events in each magnitude binbetween the observed and synthetic distribution
 
 Minimum Magnitude of Completeness in Earthquake Catalogs: Examples from Alaska, the Western United States, and Japan
 861Figure
 2.
 Explanation of the method by which we estimated the minimum magni-tude of completeness,
 M 
c
. The three frames at thetop showsyntheticfitsto theobservedcatalog for three different minimum magnitude cutoffs. The bottom frame shows thegoodness of fit
 R
, the difference between the observed and a synthetic
 FMD
 (equation2), as a function of lower magnitude cut-off. Numbers correspond to the examples inthe top row. The
 M 
c
 selected is the magnitude at which 90% of the observed data aremodeled by a straight line fit.
 M 
max
|
 B
 
 
 |
 i i M 
i
 R
(
a
,
 b
,
 
)
 100
 
 100
 (2)
i
 B
 ii
where
 B
i
 and
 
i
 are the observed and predicted cumulativenumber of events in each magnitude bin. We divide by thetotal number of observed events to normalize the distribu-tion. Our approach is illustrated in Figure 2, which shows
 R
as function of 
 M 
i
. If 
 M 
i
 is smaller then the ‘correct’
 M 
c
, thesynthetic distribution based on a simple power law (squaresin Figure 2) cannot model the
 FMD
 adequately and, conse-quently, the goodness of fit, measured in percent of the totalnumber of events, is poor. The goodness-of-fit value
 R
 in-creases 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 simplepower law with the assumed
 b
-,
 a
-, and
 M 
c
 value can explain96% of the data variability. Beyond
 M 
i
1.8,
 R
 increasesagain 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 canmodel 90% or more of the
 FMD
. For the example shown inFigure 2, we therefore define
 M 
c
1.5.Not all
 FMD
s will reach the 90% mark. In some casesthe
 FMD
s are too curved or bimodal to be fitted satisfactoryby a simple power law. This can be due to strong spatial ortemporal inhomogeneities in the particular sample, or actualphysical processes within the earth. An example of the for-mer would be a drastic change of the completeness of re-cording during the investigated period; an example of thelatter might be a volcanic region where distinct earthquakefamilies and swarms are frequent. It is important to identifythese areas for studies of the
 FMD
, because here a powerlaw cannot be readily applied. Our method allows us to mapthe fit to a power law behavior at each node, based on theminimum value of 
 R
 obtained.For mapping
 M 
c
, we use the gridding technique appliedin our studies of 
 b
-values and seismic quiescence (Wiemerand Wyss, 1997; Wiemer
 et al.,
 1998; Wiemer and Katsu-mata, 1999). Grids with several thousand nodes spaced reg-ularly at 1 to 20 km distances, depending on the size of thearea to be covered, the density of earthquakes, and the com-puter power available, are arbitrarily placed over the studyregion, and we construct the
 FMD
 at each node for the
 
nearest events and estimate
 M 
c
 using the approach outlinedpreviously. At the same time, we compute a map of thegoodness of fit to a power law by finding the minimum
 R
from equation (2) at each node. The same type of spatial
 
862
 S. Wiemer and M. Wyss
Figure
 3.
 (a) Map of central and southern Alaska. Color-coded is the minimummagnitude of completeness,
 M 
c
, estimated from the nearest 250 earthquakes to nodesof 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 theobservedfrequency-magnitude relation as measured by the parameter
 R
 in percent of the data modeledcorrectly. (c) Epicenters of earthquakes in Central and Southern Alaska for the period1992–1999 and depth
60 km. Major faults are marked by red lines.
gridding can also be applied in cross sections (e.g., Wiemerand Benoit, 1996; Power
 et al.,
 1998).
Data and Observations
In the following section, we apply the spatial mappingof 
 M 
c
 to three test cases: Alaska, Western United States,andJapan.AlaskaThe seismicity catalog compiled by the Alaska Earth-quake Information Center (
AEIC
) for the period of January1992–December 1998 contains a total of about 21,000events for central and interior Alaska with a depth less than60 km and
 M 
0.5. An epicenter map also identifying themain faults is shown in Figure 3c. We mapped
 
c
 using asample size of 
 N 
250 and a node spacing of 10 km. Thegrid we created excludes low-seismicity areas.Thesamplingradii are typically
 r 
70 km, and none are larger than 200km.
 M 
c
 varies from values near 1.4 in the interior, near Fair-banks, and in the south, between Anchorage and Valdez(blue/purple in Figure 3a), to values of 
 M 
c
3 offshore andon Kodiak island (red in Figure 3a). In Figure 4a we showa comparison of the
 FMD
s 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 isshown in Figure 3b. Mapped is the parameter
 R
; low
 R
-values (
 R
90%), shown in hot colors indicate that only apoor fit to a GR distribution could be obtained. Severalareascan be identified where the best fitting GR explains less than90% of the observed distribution. The poorest fit to a GR
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