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,
M
c
, forAlaska, the western United States, and for the
JUNEC
earthquake catalog of Japan.
M
c
was estimated based on its departurefrom thelinearfrequencymagnituderelationof 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 common 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 frequencymagnitude distribution as a function of a lower cutoff of themagnitude data.
M
c
is deﬁned as the magnitude at which 90% of the data can bemodeled by a power law ﬁt.
M
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 explosions 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
M
c
can lead to erroneous assumptions aboutdeviations from selfsimilarity 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 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 asregions outside outer margins of the networks, are so radically 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 highqualityresults. This objective is undermined if one uses a singleoverall
M
c
cutoff that is high, in order to guarantee completeness. Here we show how a simple spatial mapping of the frequencymagnitude distribution (
FMD
) and applicationof a localized
M
c
cutoff 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 frequencymagnitude 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 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 thisreasonweroutinely map the quality of the catalog for selecting the datafor our studiesofseismicquiescence;however,homogeneityin
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; Zunigaand Wiemer, 1999).Our estimation of
M
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 frequencymagnitudedistribution of events for the three catalogs investigated. (b) Number of events in each magnitude binfor these catalogs.
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 havingmagnitudes larger than
M
, and
a
and
b
areconstants.Variousmethods 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 sourcelengthof about 10 m (Abercrombie and Brune, 1994) or approximately magnitude 0 eventsize.Someauthorshavesuggestedchanges in scaling at the higher magnitude end (e.g.,LomnitzAdler and Lomnitz 1979; Utsu, 1999) or forsmaller 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 (Figure1b)formforthethreedata sets we investigate. We assume that the drop in thenumber of events below
M
c
is caused by incomplete reporting 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 amplitudedistance curves and thesignaltonoise ratio (Sereno and Bratt, 1989; Harvey andHansen, 1994) or amplitude threshold studies (Gomberg,1991) to estimate
M
c
. Waveformbased methods that requireestimating the signaltonoise ratio for numerous events atmany stations are timeconsuming 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 highquality catalog most suitable for analysis. In addition, 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 highquality 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
M
c
is based on the assumption of aGutenbergRichter (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 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
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 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 magnitude 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 cutoff. 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
S

i i M
i
R
(
a
,
b
,
M
)
100
100
(2)
i
B
ii
where
B
i
and
S
i
are the observed and predicted cumulativenumber of events in each magnitude bin. We divide by thetotal number of observed events to normalize the distribution. 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, consequently, the goodness of ﬁt, measured in percent of the totalnumber of events, is poor. The goodnessofﬁt 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 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 former would be a drastic change of the completeness of recording 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 Katsumata, 1999). Grids with several thousand nodes spaced regularly at 1 to 20 km distances, depending on the size of thearea to be covered, the density of earthquakes, and the computer power available, are arbitrarily placed over the studyregion, and we construct the
FMD
at each node for the
N
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. Colorcoded 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 theobservedfrequencymagnitude 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 Earthquake 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
M
c
using asample size of
N
250 and a node spacing of 10 km. Thegrid we created excludes lowseismicity 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 Fairbanks, 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